Download Defects and Doping in Cu2O - Francesco Biccari

Defects and Doping in Cu2O
Scuola di Dottorato in Scienze Astronomiche,
Chimiche, Fisiche e Matematiche “Vito Volterra”
Dottorato di Ricerca in Fisica – XXII Ciclo
Candidate
Francesco Biccari
ID number 688774
Thesis Advisors
Prof. Mario Capizzi
Dr. Alberto Mittiga
A thesis submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy in Physics
December 2009
Thesis defended on 16 February 2010
in front of a Board of Examiners composed by:
Prof.
Prof.
Prof.
Prof.
Paolo Calvani (chairman)
Luca Biferale
Walter Kob
Paolo Mariani
Francesco Biccari. Defects and Doping in Cu2 O.
Ph.D. thesis. Sapienza – University of Rome
© 2010
version: 31 July 2010
website: http://biccari.altervista.org
email: [email protected]
Le mieux est l’ennemi du bien
(Voltaire, 1772)
Contents
1 Introduction
1
2 Cu2 O: a review
2.1 Crystalline structure . . . . . . . . . . . . . . . . . . . . . .
2.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Electronic band structure and effective masses . . . . . . . .
2.4 Hole mobility . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Optical Properties . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Excitons . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Cu–O phase diagram . . . . . . . . . . . . . . . . . . . . . .
2.7 Equilibrium concentration of intrinsic point defects . . . . .
2.7.1 Defect concentration from mass action law . . . . .
2.7.2 Deviation from stoichiometry: experimental results .
2.7.3 Electrical properties at high temperatures . . . . . .
2.7.4 Formation enthalpies of intrinsic defects . . . . . . .
2.7.5 Formation entropy of the copper vacancy . . . . . .
2.7.6 Theoretical predictions . . . . . . . . . . . . . . . . .
2.7.7 Self-compensation mechanism . . . . . . . . . . . . .
2.8 Self-diffusion of the lattice ions and intrinsic point defects .
2.8.1 Experimental techniques . . . . . . . . . . . . . . . .
2.8.2 Experimental results . . . . . . . . . . . . . . . . . .
2.9 Growth of Cu2 O . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1 Oxidation process . . . . . . . . . . . . . . . . . . .
2.10 Electrical properties at low and intermediate temperatures .
2.10.1 Conductivity . . . . . . . . . . . . . . . . . . . . . .
2.10.2 Persistent PhotoConductivity . . . . . . . . . . . . .
2.11 Other defect characterization techniques . . . . . . . . . . .
2.11.1 Photoluminescence . . . . . . . . . . . . . . . . . . .
2.11.2 DLTS . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.3 Magnetic Properties and EPR . . . . . . . . . . . .
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3 Sample preparation and experimental setup
3.1 Undoped Cu2 O preparation . . . . . . . . . . . . . . . . . . .
3.1.1 Starting material: copper . . . . . . . . . . . . . . . .
3.1.2 Oxidation Setup . . . . . . . . . . . . . . . . . . . . .
3.1.3 Copper Oxidation . . . . . . . . . . . . . . . . . . . .
3.1.4 Structural properties of Cu2 O obtained from oxidation
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v
3.2
3.3
3.1.5 Equilibrium of point defects and oxidation
3.1.6 Substrate characterization . . . . . . . . .
Device fabrication . . . . . . . . . . . . . . . . .
3.2.1 Devices with ohmic contacts . . . . . . . .
3.2.2 Junction based devices . . . . . . . . . . .
Experimental setup . . . . . . . . . . . . . . . . .
procedure
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4 Electronic properties and metastability effects
4.1 Acceptor and donor concentrations . . . . . . . . .
4.1.1 Mobility . . . . . . . . . . . . . . . . . . . .
4.1.2 Defect concentration . . . . . . . . . . . . .
4.2 Persistent Photo-Conductivity . . . . . . . . . . . .
4.2.1 Interpretation of PPC . . . . . . . . . . . .
4.3 Defect response . . . . . . . . . . . . . . . . . . . .
4.3.1 Admittance Spectroscopy . . . . . . . . . .
4.3.2 Capacitance transients . . . . . . . . . . . .
4.4 Failure of the model of Lany and Zunger applied to
4.4.1 The model of Lany and Zunger . . . . . . .
4.4.2 The model of Lany and Zunger for Cu2 O .
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Cu2 O
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5 A model based on donor–acceptor complexes
5.1 The idea . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Shift of energy levels upon pairing . . . . . .
5.1.2 Dynamics of pair association and dissociation
5.2 The model based on the complexes (VCu – VO ) . . .
5.3 Analysis of carriers vs temperature experiments . . .
5.4 Analysis of PPC . . . . . . . . . . . . . . . . . . . .
5.5 Analysis of the capacitance transients . . . . . . . .
5.5.1 The first capacitance transient: τ1 . . . . . .
5.5.2 The second capacitance transient: τ2 . . . . .
5.6 CV profile and aging . . . . . . . . . . . . . . . . . .
5.7 Limitations of the D–A complex model . . . . . . .
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6 Doping
6.1 Review: doping of Cu2 O . . . . . . . . . . . . . . . . .
6.1.1 Chlorine . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Nitrogen . . . . . . . . . . . . . . . . . . . . . .
6.1.3 Silicon . . . . . . . . . . . . . . . . . . . . . . .
6.1.4 Transition metals . . . . . . . . . . . . . . . . .
6.1.5 How difficult is the doping of Cu2 O? . . . . . .
6.1.6 Passivation . . . . . . . . . . . . . . . . . . . .
6.2 Preparation of doped substrates of Cu2 O . . . . . . .
6.3 Experimental results . . . . . . . . . . . . . . . . . . .
6.3.1 Morphology of the doped sample . . . . . . . .
6.3.2 Electrical properties and defect concentrations
6.4 Experiments on chlorine doped samples . . . . . . . .
6.4.1 Carrier concentration vs temperature . . . . . .
6.4.2 Persistent Photo–Conductivity Decay . . . . .
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vi
6.5
6.4.3 Discussion of conductivity data . . . . . . . . . . . . . . . . . 142
6.4.4 Solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7 Excitonic effects
7.1 Excitons and free carriers . . . . . . . . . . . . . . . . .
7.1.1 Exciton concentration under equilibrium . . . . .
7.1.2 Exciton concentration out of equilibrium . . . . .
7.1.3 Excitons as an alternative recombination channel
7.2 The exciton role in the solar cell operation . . . . . . . .
7.3 Excitonic effects on solar cells based on Cu2 O . . . . . .
7.3.1 Literature data . . . . . . . . . . . . . . . . . . .
7.3.2 Excitons concentration . . . . . . . . . . . . . . .
7.3.3 Exciton diffusion length . . . . . . . . . . . . . .
7.3.4 Exciton role in Cu2 O solar cells . . . . . . . . . .
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8 Conclusions and perspectives
153
A Irreducible representations of the O h point group
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B Conductivity of a compensated semiconductor
B.1 Completely ionized donors, ND+ = ND . . . . . .
B.1.1 High temperature case . . . . . . . . . . .
B.1.2 Intermediate temperature case . . . . . .
B.1.3 Low temperature case . . . . . . . . . . .
B.2 Degeneracy factors . . . . . . . . . . . . . . . . .
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C Correction for the series resistance
C.1 Admittance correction . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1.1 Error propagation for the Rs correction . . . . . . . . . . . .
C.2 Voltage drop on the series resistance . . . . . . . . . . . . . . . . . .
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D Complex concentration
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D.1 Thermodynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . 167
D.2 Concentration of complexes in the deep depletion region . . . . . . . 168
E Exciton equilibrium considering excited states
171
Bibliography
175
vii
Chapter 1
Introduction
The Copper(I) Oxide (Cu2 O, Cuprous Oxide) was the first substance known to
behave as a semiconductor, together with selenium. Rectifier diodes based on
this material were used industrially as early as 1924 and most of the theory of
semiconductors was developed using the data on Cu2 O based devices.
Historically the first real solar cell with Cu2 O was fabricated at the end of
the 1920 [264, 417]. However at that time, and until the first space explorations,
the energy production from the sun by the photovoltaic effect was just a curiosity.
Moreover electronic devices based on Cu2 O, like rectifiers [263] had a relatively
short life. In the early fifties doping of silicon and germanium was discovered, these
elements became the standard and the attention on Cu2 O declined.
One of the interesting aspect of Cu2 O is the easiness of its preparation process.
For example monocrystals can be obtained by a simple oxidation of copper in furnace
at high temperature while thin films can be grown by electro-deposition or sputtering.
Moreover the Cu2 O is composed of two very abundant and non-toxic elements. It
shows also several interesting characteristics for photovoltaic applications: a good
absorption coefficient for the light above the gap, a good mobility for the majority
carriers and a diffusion length of the minority carriers of several micrometers. For
these reasons, even if its direct energy gap of about 2 eV at room temperature is quite
high to match the solar spectrum and even if the maximum conversion efficiency
obtained up to now is only 2% [543], Cu2 O has always been considered a material
suitable for the realization of low cost solar cells [867].
As a matter of fact, during the oil crisis of the seventies a great research effort
was devoted to the study of alternative energy supplies and Cu2 O was one of the
most studied material for a low-cost photovoltaic cell [689].
Presently the interest in Cu2 O as photovoltaic material has increased again
because of the good results obtained from materials which have analogous problems,
both under the physical and technological aspects. These materials are CuInSe2
(CIS), CuGaSe2 (CGS) and the intermediate alloys Cu(In, Ga)Se2 (CIGS), which
now can give a photovoltaic conversion efficiency up to 20%. The presence of an
element like indium, rare and expensive, in these compounds shed a negative light
on the future of these semiconductors.
This renewed interest in Cu2 O as a photovoltaic material is supported also by
other two new ideas in the photovoltaic field: polycrystalline multijunction solar
cells and Intermediate Band solar cells. A multijunction cell is a photovoltaic device
1
2
1. Introduction
in which two or more semiconductors with different band gap are used to absorb a
larger region of the solar spectrum.
The interest in the fabrication of multijunction solar cells started in the early
1960s, but this idea was put in practice only when advanced epitaxial techniques
(MBE and MOVPE) were developed. Today, the most common multijunctions are
single monolithic cell in which the different junctions are grown one on the top of
the other, in order to obtain a two–terminal device.
However the epitaxial growth is not necessary to fabricate a useful multijunction
solar cell. For example, in addition to the epitaxial solar cells made using III–V
semiconductors, tandem and triple junctions made from amorphous semiconductors
are well developed and their production capacity is quickly increasing.
The fabrication of polycrystalline multijunction solar cells, based on CIGS
alloys, is instead still under consideration and only a few studies, both from a
theoretical [139] and experimental [905] point of view, were performed. For this kind
of new family of multijunctions Cu2 O could be useful, thanks to the high value of
its band gap, as the material for the top cell.
In the Intermediate Band (IB) solar cells the purpose is the same of the multijunction cells: to absorb a larger part of the solar spectrum with respect to that
absorbed by a normal solar cell. However in this case the strategy is to highly
dope the material with impurities which introduce deep levels in the band gap. If
the impurity concentration is sufficiently high, the defect levels form a band. In
this way the absorption of photons with energy lower than the band gap can be
obtained in such devices by a two steps mechanism. For example an electron is first
excited from the valence band to the intermediate band and then is promoted to
the conduction band by an absorption of another photon. According to theoretical
calculations [489], see Fig. 1.1, the maximum conversion efficiency which can be
obtained under maximally concentrated solar light is 63.2%. The optimal material
to obtain this maximum has a gap of about 1.9 eV which is very similar to the gap
of Cu2 O.
Obviously before using Cu2 O for these advanced applications, it is necessary to
reach a reasonably good conversion efficiency from cells with conventional structure.
As we said above, the maximum conversion efficiency obtained up to now for
a single cell based on Cu2 O is about 2% [543] and it is very low compared to the
theoretical limit of 22% under Air Mass 1.5 where we have considered a value of
Eg = 2 eV (Jsc = 14 mA/cm2 , J0 = 10−30 A/cm2 , n = 1, Voc = 1.65 V, FF = 92%).1
Unfortunately this theoretical value is almost impossible to achieve because is based
on the radiative recombination only.
The other two phenomena which can increase the J0 and therefore give a smaller
Voc are the presence of recombination centers in the space charge region and the
injection of the minority carriers. Assuming a reasonable value for the concentration
of the recombination centers, 1013 cm−3 , we obtain a maximum efficiency of about
21.9% (Jsc = 14 mA/cm2 , J0 = J0scr ≈ 10−17 A/cm2 , n = 2, Voc = 1.8 V, FF = 87%).
Finally let us consider the injection term. Considering an highly doped n-type
region, a estimated diffusion coefficient for the electrons Dn = 1.3 cm2 /s, a diffusion
length Ln = 10 µm and a concentration of holes at the equilibrium p = 1013 cm−3 ,
1
In the literature there are similar estimations of the maximum efficiency but under Air
Mass 1 [635, 638, 848].
3
70
65
1.71
1.93
Intermediate Band solar cell
2.14
2.36
1.48
2.56
Maximum efficiency
(%)
60
55
2.76
1.40
1.44
1.50
1.57
1.63
2.96
1.71
3.17
Tandem solar cell
1.79
50
(series connected)
1.87
1.95
45
40
Single gap solar cell
35
T
T
30
sun
cell
= 6000 K
= 300 K
25
0.5
0.6
0.7
0.8
0.9
1.0
Lowest energy gap
1.1
1.2
1.3
(eV)
Figure 1.1. Theoretical limiting efficiencies, under maximally concentrated solar light,
for an intermediate band solar cell, a two-terminal ideal tandem cell (tandem is a
multijunction with only two stacked cells) and for a normal cell based on a single gap.
The efficiency is plotted against the lowest band gap value of the cell. The numbers
reported near the curves correspond to the optimal band gap of the host material for
the IB cell and to the sum of the lowest and optimum highest band gaps for a tandem
cell. After [489].
the maximum efficiency turns out to be about 17% (Jsc = 14 mA/cm2 , J0 = J0inj ≈
5 × 10−26 A/cm2 , n = 1, Voc = 1.4 V, FF = 89%).
We conclude that the conversion efficiency of Cu2 O solar cell is limited by the
injection of the minority carriers which gives a realistic limit for the efficiency of
about 17%.
Cu2 O attracted much attention as a dilute magnetic semiconductor (DMS). This
class of materials is very important because of the potential use of both the charge and
spin of electrons for spintronic devices, but a Curie temperature greater than room
temperature is required. Observations of room temperature ferromagnetism in the Al
and Co co-doped Cu2 O film [26, 402] and in the Mn doped Cu2 O films [367, 480, 885]
have been reported with Curie temperatures far greater than room temperature.
Another recent technological application is the fabrication of low-cost non-volatile
memories, based on the effect of the switching resistivity [115, 116].
Cu2 O is important not only for its applications but also because is a test material
for several important fundamental physical phenomena. Cu2 O was the first substance,
together with CdS, where the excitons were observed [431]. Nowadays it still remains
a textbook example and fundamental innovative studies are performed in search
of the Bose–Einstein condensation of excitons [340, 904]. Its role in the study of
polaritons [381] and phonoritons [295] is very important also.
***
Despite the interest of these applications and the many studies, the current
experimental results do not form a coherent picture and therefore a satisfactory
4
1. Introduction
model of the material is not available.
Cu2 O shows spontaneously a p-type conductivity and the comparison between the
conductivity and Capacitance–Voltage measurements shows that it is a compensated
material i.e., both intrinsic acceptors and, in smaller number, donors are present.
The compensation ratio NA /ND , i.e. the acceptor concentration over the donor
concentration, in most cases is just slightly larger than 1 and in any case less than
10 [64, 84, 677].
The nature of the intrinsic point acceptors and donors is not completely clear.
Both high temperature experiments [679] and ab–initio calculations [686] suggest that
the p-type conductivity is due to the presence of copper vacancies which introduce an
acceptor level at about 0.3 eV above Ev [686]. Instead the nature of the compensating
donor is still controversial (simple candidates could be oxygen vacancies [679]) and
also their identification with intrinsic defects is not assessed.
Cu2 O shows also a series of metastability and instability effects, first of all the
persistent photoconductivity (PPC) [64,737,811]: after some minutes of illumination
with white light the material shifts to an high conductivity state which is stable in
the dark at room temperature. The state of equilibrium can be restored, for example,
heating the material to 400 K for several tens of minutes. These metastabilities
show up also in the Cu2 O junctions which show a dependence of their characteristics
(capacitance, series resistance (Rs ), open circuit voltage (Voc ) and short circuit
current (Jsc )) on the history of the sample [64, 542]. This phenomenon appears also
in other materials but in Cu2 O there is not a definitive theory of its origin.
Another important aspect regards the difficulties in the doping process. Currently
the world record efficiency for a solar cell based on Cu2 O was obtained in our
laboratory using non-doped substrates.
A p-type doping would give a cell with a lower Rs and a better fill factor (FF).
Moreover the lowering of the Fermi level could give a higher Voc .
In the literature there are many attempts to dope Cu2 O p-type, but only nitrogen
and chlorine gave clear results in the decrease of resistivity. The former has been
tested only on thin films and, due to the strength of the bond of the molecule N2 , it
is difficult to be inserted in bulk samples.
The p-type doping has never been optimized and a material which has the same
time both a high enough conductivity and a good minority carrier diffusion length
was never obtained.
An n-type doping would allow us instead to realize homojunctions, avoiding
the formation of the interface defects and thus reducing then the recombination of
carriers.
Unfortunately, the n-type doping seems to be impossible and this is probably
attributable to the mechanism of self-compensation. The formation energy of the
copper vacancies becomes negative when the Fermi level is above half the gap,
meaning that it is very easy to create copper vacancies. This is the accepted
explanation for the impossibility to dope Cu2 O n-type.
Up to now also the technological processes for the heterojunction formation
have not given good results. This failure is probably connected to the formation of
intrinsic defects in the interface region but these phenomena have not been elucidated
yet [849]. Moreover it has to be noted that the solar cells prepared with Cu2 O thin
films have always shown worse performance with respect to the solar cells made
with a bulk of Cu2 O.
5
The experiments also show that the impurities have an important role, not only
in the determination of the sample conductivity, but also in the crystal growth and
carrier lifetime. Most of the samples used contain a concentration of impurities
greater than 1 ppm which corresponds to a concentration of 1016 cm−3 and this is
greater than the typical intrinsic defect concentration found in Cu2 O. This is indeed
another source of scattering in the experimental data.
***
The present thesis is devoted to clarify some fundamental aspects of Cu2 O which
can be important both for device production and for a better understanding of
physical phenomena observed in Cu2 O. Our long run goal is to make high–efficiency,
stable and low–cost solar cells based on Cu2 O.
The next chapter (2) is a brief review of the most common properties of Cu2 O,
where a particular attention is devoted to emphasize the open questions and problems.
In the chapter 3 we discuss the preparation of the Cu2 O substrates, the fabrication
of the devices and the experimental setup used to characterize this material.
The chapter 4 is devoted to the investigation of the metastable effects of Cu2 O and
their relationship with the material defects. We show the results of several types of
electrical measurements: conductivity vs temperature, persistent photo–conductivity
decays, capacitance–voltage and, for the first time, capacitance transients.
These experiments allow to characterize the defects, measuring their concentrations and estimating some other physical parameters, like carrier capture cross
sections and ionization energies. However they show some inconsistencies with
electronic models used in the past or in similar semiconductors like chalcopyrites.
In chapter 5 we propose a new model, based on the association/dissociation of
point defects of Cu2 O, which can give a better explanation of the experiments.
In chapter 6 we discuss the doping of Cu2 O. After a brief review of the literature,
we present the results of our doping tests made with six different dopants: chlorine,
silicon, chromium, iron, silver, and sodium. The most effective dopant, the chlorine,
is then analyzed in depth.
Cu2 O presents very strong excitonic effects, indeed in this material a non negligible fraction of free carriers are actually bounded in excitons. In the chapter 7 we
discuss the possible role of the excitons in the electrical properties of Cu2 O.
Finally the conclusions and the future perspective of the work on Cu2 O are
discussed in chapter 8.
Chapter 2
Cu2O: a review
This chapter is a review of the most relevant properties of Cu2 O. We will focus our
attention mainly on the identification and characterization of the intrinsic point
defects.
2.1
Crystalline structure
Cu2 O crystallizes in the so called cuprite structure shown in Fig. 2.1, a simple cubic
Bravais lattice with the symmetry of the 224th space group (Oh4 , Pn3̄m). The only
other compounds with the same structure are Ag2 O, Pb2 O, Cd(CN)2 and Zn(CN)2 .
The cyanides have reversed roles of the ions (an antistructure). [4, 175, 727]
Inside the unit cell the oxygen ions are located on a bcc sub-lattice, while copper
ions on a fcc sub-lattice. Choosing the origin of the coordinate system on an oxygen,
the copper ions are in the positions (1/4, 1/4, 1/4), (1/4, 3/4, 3/4), (3/4, 1/4, 3/4)
and (3/4, 3/4, 1/4), in unit of lattice constant a. In other words the fcc sub-lattice
is translated by a(1/4, 1/4, 1/4) respect to the bcc sub-lattice.
The copper ions are on the vertices of a tetrahedron centered on the oxygen sites
and are twofold coordinated with the oxygen ions (D3d site symmetry), whereas the
oxygen ions are fourfold coordinated with copper ones (Td site symmetry). This
Figure 2.1. Unit cell of cuprous oxide lattice (cuprite structure): a) origin on a oxygen
site; b) origin on a copper site. Dashed lines represent the bonds whereas solid lines are
only a guide for the eyes. This structure contains two Cu2 O formula per unit cell.
7
8
2. Cu2 O: a review
Table 2.1. X–ray powder diffraction pattern for Cu2 O at atmospheric pressure and at
room temperature. The starred lines correspond to the most visible peaks. JCPDS International Centre for Diffraction Data. PDF, Powder Diffraction File, #05–0667.
(1996).
2θ
Int
h
k
l
29.579*
36.450*
42.334*
52.501*
61.400*
69.635
73.597*
77.400*
92.480*
103.882
107.688
124.402
139.541
9
100
37
1
27
1
17
4
2
4
3
3
3
1
1
2
2
2
3
3
2
4
3
4
4
5
1
1
0
1
2
1
1
2
0
3
2
2
1
0
1
0
1
0
0
1
2
0
1
0
2
1
structure is centrosymmetric, i.e. has the inversion symmetry. If we choose a
copper ion as the origin of the unit cell the transformed unit cell under the inversion
symmetry is identical to the original. Instead, if we choose an oxygen atom as the
origin, the inversion has to be followed by a translation of a(1/2, 1/2, 1/2).
The measurement of the lattice constant gives a value of a = 4.27 Å, the Cu–
O bond length is dCu−O = 1.85 Å, the O-O is dO−O = 3.68 Å and the Cu-Cu is
dCu−Cu = 3.02 Å [891]. The concentration of copper and oxygen ions are, with simple
calculation, [O] = 2.52 × 1022 cm−3 and [Cu] = 5.05 × 1022 cm−3 . The density of
the material is ρ = 6.10 g/cm3 , the molar mass is M = 143.092 g/mol and the molar
volume is Vm = 23.46 cm3 /mol.
Lattice constant and crystal structure have been usually measured by X–ray
powder diffraction (XRD) using Cu Kα radiation in θ/2θ mode [361]. The XRD
experimental lines for Cu2 O and their relative intensities are reported in Tab. 2.1.
Cu2 O transforms from the cuprite to an hexagonal structure at a pressure of
10 GPa (a = 4.18 Å at p ' 10 GPa). This hexagonal form changes in the pressure
range from 13 to 18 GPa into another hexagonal form, with CdCl2 –type structure.
Up to 24 GPa, the highest pressure generated, no decomposition of Cu2 O into Cu
and CuO was observed. For relevant data of phase transitions, structures, lattices
parameters and volume changes, see [891].
It is important to note that the given values are almost independent of the
temperature, because the Cu2 O has a very small expansion coefficient (the lattice
constant changes less than 0.5 % from 0 to 600 K). However a very interesting
behaviour of the Cu2 O is its negative thermal expansion below 300 K. See [540] and
references cited therein.
Finally we report some fundamental thermal properties [825]: the thermal
conductivity of Cu2 O is k = 4.5 W/(K m), the specific heat capacity is Cp =
70 J/(K mol) and therefore the thermal diffusivity is α = kVm /Cp = 0.015 cm2 /s.
2.2 Phonons
9
Table 2.2. Symmetries, energies and descriptions of the zone center phonons in cuprite.
The descriptions refer to a globe centered on an oxygen, large enough to put sea level
through the center of the four nearest copper atoms, with the equator lying in the (001)
plane. After [615, 672].
symmetry
energy [meV]
description
3 −
Γ15
0
3 acoustic modes.
3 −
Γ25
10.8
3 optical modes. Rigid rotation of the Cu4 O tetrahedron,
bending Cu-O-Cu bonds
2 −
Γ12
13.6
2 optical modes (silent). All four Cu move toward the equator,
then toward poles; Cu moves along lines of constant latitude,
twisting the tetrahedron
3 −
Γ15
18.8(TO)
19.1(LO)
3 optical modes (infrared active). All Cu move north, then
south, along lines of constant longitude: O move in opposition
along [001] to keep the center of mass fixed
1 −
Γ2
43.4
1 optical mode (silent).
Cu tetrahedron
3 +
Γ25
63.9
3 optical modes (Raman active). Relative motion of the two
simple cubic O lattices
78.5(TO)
82.1(LO)
3 optical modes (infrared active). While northern Cu move
toward the central O, the southern Cu move away; all O
move in opposition along [001] to keep the center of mass
fixed
3 −
Γ15
2.2
Breathing mode of the
Phonons
Six atoms in the unit cell means 6 · 3 = 18 phonon modes. We expect fifteen optical
phonon branches and three acoustic branches. These 18 phonon modes, in the
3 −
3 −
3 +
1 − 1
zone–center, have the symmetries 2 Γ−
12 ⊕ 3 Γ15 ⊕ Γ25 ⊕ Γ25 ⊕ Γ2 .
−
Of the three sets of triply degenerate Γ15 phonons, one set corresponds to the
acoustic phonons while the other two are the infrared–active optical modes. The
Γ+
25 mode is Raman-active. Optical phonons which are neither infrared-active nor
−
−
Raman-active are said to be silent. The Γ−
2 , Γ12 and Γ25 modes in Cu2 O are silent
modes. Since the crystal has the inversion symmetry, the infrared-allowed modes
are Raman forbidden and vice-versa.
In Tab. 2.2 a summary of Cu2 O phonons is reported; the vibrational mode
description is shown in Fig. 2.2. For a review of the techniques used to identify
the phonons, usually photoluminescence phonon replica and infrared spectroscopy,
see [57,366,672]. In [57] a study of phonon dispersion relations by neutron scattering
is also reported.
Considerable attention has been paid to the phonon spectrum in Cu2 O. After
numerous infrared and Raman studies the phonon spectrum of cuprous oxide is now
well understood. One of the main problem was that the observed number of bands
1
The notation used in this thesis for the group representations is a mix, commonly used in
literature, between the new Koster and the old BSW notations. Further details can be found in
App. A.
10
2. Cu2 O: a review
Figure 2.2. Pictorial representation of the vibrational modes of Cu2 O listed in the Tab. 2.2.
For a comparison of the notations of the Oh point group irreducible representations see
App. A.
is much larger than that deduced from the group theory. This fact is now explained
interpreting the spectra in terms of bi-phonons or multi-phonons processes in light
absorption. An example are the absorption band at 0.14 eV and 0.10 eV [94, 398].
2.3
Electronic band structure and effective masses
The Cu+ ion electronic structure ends 3d10 4s0 , with the 4s orbitals only slightly
higher in energy than the 3d levels. The Cu 3d levels form the valence band of Cu2 O
and the empty Cu 4s levels form the conduction band [184]. These means also that
the bands have the same parity. It can be shown that Cu2 O is a semiconductor with
a direct gap at the center of the Brillouin zone (Γ point).
The spatial wave function part of the highest valence band is triple degenerate
+
+
with symmetry Γ+
25 and is six degenerate if electron spin is included Γ25 ⊗ Γ6 . The
lowest of the conduction band is single and transforms with Γ+
1 ; if the electron spin
is included the total wave function belongs to Γ+
.
Spin
orbit
coupling lifts the
6
+
degeneracy of the six degenerate valence bands, separating two Γ7 states from four
Γ+
8 states with an energy split of ∆SO = 133.8 meV independent of temperature [860].
(Formally the direct product of the representations of the spatial and spin wave
+
+
+
functions are reducible: Γ+
25 ⊗ Γ6 = Γ7 ⊕ Γ8 ). Experiments on exciton absorption
+
showed that Γ7 is higher in energy. Another conduction band higher in energy, with
−
a spatial part belonging to Γ−
2 and with a total wave function to Γ8 , is formed by
Cu 4p orbitals. See Fig. 2.3.
+
The measured energy gap is ∆E(Γ+
7 − Γ6 ) = Eg = 2.1720 eV at 4.2 K, obtained
as the limit of the yellow exciton series [591, 860]. As usual it decreases with
temperature and its dependence on the temperature [359] is showed in Fig. 2.4.
The expression used to fit the data, based on the Toyozawa model, is [359]:
Eg (T ) = Eg (0) + S }ω − S }ω coth(}ω/2kT )
(2.1)
2.3 Electronic band structure and effective masses
11
Figure 2.3. The band structure of Cu2 O near the center of the Brillouin zone (Γ point).
The bands are labeled by the irreducible representation to which they belong. The
corresponding effective mass names are also reported. On the right the transition giving
rise to the four exciton series (yellow, green, blue and indigo) are shown.
2.18
2.17
2.16
(eV)
2.15
2.14
E
g
2.13
2.12
Cu O Floating Zone
2
2.11
Cu O Natural
2
Fit
2.10
2.09
0
50
100
150
T
200
250
300
(K)
Figure 2.4. Eg as a function of temperature for two different Cu2 O samples, after [359].
The data are obtained by luminescence experiments, observing the threshold energy
of lower edge of the XO − Γ−
12 phonon replica, where XO indicates the orthoexciton
emission line. The solid lines is a fit of the experimental data by the expression (2.1).
12
2. Cu2 O: a review
Table 2.3. Effective masses of carriers in Cu2 O, expressed in unit of the free electron mass
m0 . The experimental data are taken from [323] while theoretical predictions from [251].
band
me
mLH
h
mHH
h
mSPH
h
Γ+
6
Γ+
7
Γ+
7
Γ+
7
theory
ΓX ΓM ΓR
0.92
0.36
2.83
0.21
0.92
0.36
0.91
0.25
0.92
0.36
0.72
0.27
experiment
0.99
0.58
where Eg (0) = 2.173 eV is the energy gap value at T = 0 K, S = 1.89 is a constant
related to the material and }ω = 13.6 meV is the phonon energy of the emitted
phonon, Γ−
12 in this case.
At intermediate and high temperature the variation of the gap with temperature
can be described by the empirical formula proposed by Varshni
Eg (T ) = Eg (0) −
αT 2
T +β
(2.2)
where α and β are two empirical constants which take into account the effect of
thermal expansion and electron-phonon interaction. In the case of Cu2 O [780,
887] from the experimental fit of exciton luminescence the constants are α =
4.8 × 10−4 eV/K and β = 275 K.2
+
Other inter-band transition energies at 4.2 K are: ∆E(Γ+
8 − Γ6 ) = 2.304 eV
+
obtained as the limit of the green exciton series [281], ∆E(Γ7 − Γ+
8 ) = 2.624 eV
−
obtained as the limit of the blue exciton series [152], ∆E(Γ+
−
Γ
8
8 ) = 2.755 eV
obtained as the limit of the indigo exciton series [152].
The effective hole masses are labeled with LH (Light Hole) for the Γ+
7 and HH
(Heavy Hole) and SPH (spin-orbit SPlit-off Hole) for the Γ+
(an
explanation
of
8
the acronyms can be found in [578]). The effective masses of the free carriers are
measured by experiments of cyclotronic resonance [323] and the results are reported
in Tab. 2.3 together with the best available theoretical predictions [578].3
For symmetry reasons me and mLH
have to be isotropic whereas mHH
and
h
h
SPH
mh
have to be anisotropic. These symmetry considerations were confirmed by
calculations [578] and experiments [619].
Actually the effective masses are polaron effective masses. Indeed, being the
Cu2 O partly ionic, the free carriers are surrounded in their motion by a cloud of
2
In the article [887] the author claims that the Cu2 O is a non-polar crystal. This has been
proven to be false [625].
3
In [323] (see also [252]) the authors observed another peak with an effective mass of about
0.69 m0 which appears in some samples or under sub band gap illumination, in place of the 0.58 m0
peak. It was wrongly attributed to the carriers in the highest valence band.
Goltzené and Schwab [251] showed that the peak at 0.58 m0 , observed in high purity Cu2 O, shifts
to higher masses with the increase of the iron impurity concentration and therefore they concluded
that 0.58 m0 is the real polaron mass of the carriers in the highest valence band while the hole peak
with a higher mass, observed in some samples, is due to the defects which probably alter the band
structure of Cu2 O.
2.3 Electronic band structure and effective masses
13
phonons which increases their mass. Bare effective masses, strictly related to the
curvature of the bands are few per cent points smaller [323].
Using the experimental values of me and mLH
h we obtain the values of the effective
density of states, Nc and Nv , of the lowest conduction band and the higher valence
band of Cu2 O respectively:
2πme kT
Nc (T ) = 2
h2
3/2
2πmLH
h kT
Nv (T ) = 2
h2
= 4.76 × 1015 T 3/2 cm−3
!3/2
= 2.13 × 1015 T 3/2 cm−3
(2.3a)
(2.3b)
where the temperature T is expressed in kelvin. At room temperature we have
Nc (300 K) = 2.47 × 1019 cm−3
19
−3
Nv (300 K) = 1.11 × 10 cm
(2.4a)
(2.4b)
Finally the values of n2i :
n2i (T ) = 1.014 × 1031 T 3 e−Eg (T )/(kT ) cm−6
n2i (300 K)
3
−6
= 2.12 × 10 cm
(2.5a)
(2.5b)
where we have used Eg (300 K) = 2.09 eV in the last expression.
From the theoretical point of view, several ab-initio calculation of the Cu2 O
electronic structure have been performed, by several different techniques. We have to
remember that the self-consistent techniques based on Local Density Approximation
(LDA) or Generalized Gradient Approximation (GGA) underestimate the energy of
the excited states (i.e. the conduction bands) whereas those based on the HartreeFock approximation overestimate it. However the shape of the conduction bands are
qualitatively correct and therefore they can give an idea of the real conduction bands
simply shifting them in the energy. The only calculations with correct estimation
of the band gap are semi-empirical rather than self-consistent. For a brief review
see [91, 332, 641]. In Fig. 2.5 and 2.6 self-consistent [426] and semi-empirical [715]
bands calculation are shown.
The correctness of band calculations was investigated by several kind of spectroscopic experiments (x-ray photo-emission (XPS), x-ray absorption (XAS), x-ray
emission (XES), ultraviolet photo-emission (UPS), resonant photo-emission (RPES),
angle resolved photo-electron (ARPES) Auger electron (AES) and bremsstrahlung
isochromat spectroscopy (BIS, also called inverse photo-emission); low-energy electron-diffraction (LEED); predictions on absorption coefficient). See [332, 641] for a
review.
A good agreement between the calculated bands and the experiments has been
found and the ionic-covalent character of the bonds was experimentally seen.
In contrast with some other transition metal oxides, the electronic correlation is
expected to be relatively small, thanks to the closed shell (d10 configuration, in a
pure ionic model Cu+ ,O2– ). Indeed the valence band calculations are not affected
by many body effects (one-electron band theory is enough) and this was confirmed
by the experiments.
Finally, as we have seen, Cu2 O shows an unusual linear coordination O−Cu−O
of copper atoms. This coordination, and therefore the cuprite structure, can not be
14
2. Cu2 O: a review
Figure 2.5. Electronic bands calculated
by a self-consistent LDA approach with
localized Gaussian basis and Slater exchange, after [426]. a) Conduction
bands. b) Valence bands. Notice the
typical underestimation of the band gap
due to the Local Density Approximation.
Figure 2.6.
Electronic
bands calculated by
parameterized
semiempirical tight-binding
method, after [715].
2.4 Hole mobility
15
quantity
χ
Φ
I
Eg
EF
Figure 2.7. Schematic band diagram of
a semiconductor near the surface. χ
is the electron affinity, Φ is the work
function, I is the photo-threshold energy, Eg is the band gap and µ =
χ + Eg − Φ is the electron chemical potential. Even if µ is improperly called
Fermi energy (EF ), according to the
tradition we will use this wrong term
along this thesis.
value (eV)
3.20
≈ 4.90
5.25
2.09
≈ 0.35
Table 2.4. Surface potential values for
Cu2 O at 300 K. I and Φ are calculated
from the experimentally values of χ,
Eg and EF . Remember that χ does
not depend on temperature whereas Φ,
I, Eg , and consequently EF , do. EF
and Φ depend on doping concentration
also.
explained by simple ionic forces which, instead, predict an unstable structure. The
cited experiments found that the hybridization between orbitals O2p–Cu4sp exists,
confirming a long time hypothesis for cuprite stability.
For the sake of completeness, in Tab. 2.4 we report the characteristic energy
values of Cu2 O near the surface (see Fig. 2.7 for an explanation). Electron affinity
χ is taken from [638]. According to these values, near the surface an electron can be
excited up to the fourth conduction band before escaping from the material.
2.4
Hole mobility
The mobility of the free carriers is one of the most important parameter to describe
the electrical properties of a material. In Cu2 O no experiments on the electron
mobility were performed.4 On the other hand several measurements of the hole
mobility exist, however these values show some scattering because the mobility
depends also on the morphology of the crystal. The most recent systematic studies
in literature, Shimada and Masumi [760], Tazenkov and Gruzdev [817], Zouaghi et
al. [967] are performed on large grain Cu2 O, grown by similar techniques, giving the
possibilities to compare the results.
In Fig. 2.8 the mobility data of Tazenkov and Gruzdev [817] and Zouaghi et
al. [967] obtained on polycrystals, are reported together with the data of Shimada
and Masumi [760] obtained on monocrystals of Cu2 O. In the first two works the
mobility is obtained by the regular Hall technique whereas in the work of Shimada
and Masumi the conductivity below 77 K is so low that the Hall technique is applied
to the holes obtained by the photo-excitation of the material.
4
Actually a measurement of the electron mobility at liquid helium temperatures was performed
in [323] by transverse magnetoconductivity. They found that the electron mobility is slightly higher
than that of holes: respectively 9 × 104 cm2 /(V s) against 8 × 104 cm2 /(V s) in their samples.
16
2. Cu2 O: a review
Figure 2.8. Hole mobility in Cu2 O. After [454, 760, 817]. The curves are two theoretical
predictions: a) considers both high and low energy LO phonons while b) considers only
low energy LO phonons in the carrier–phonon scattering mechanism.
At very low temperature, below 30 K, the hole mobility is constant with temperature. This means that the main scattering process is due to the neutral impurities.
Before going ahead to analyze the rest of the curve we want to estimate the
concentration of neutral defects from the mobility. The mobility is given by
eτ
µ=
(2.6)
m
with e the electron charge, m the effective mass of the carrier and τ the relaxation
time which can be seen as the time between two scattering. Considering a carrier
with velocity v in a material with a density of N neutral defects with cross section
σ, τ is given by
1
τ=
(2.7)
σvN
From the scattering theory it can be shown that the cross section of a neutral
impurity for a free carrier is [749]
σ=
20adef
k
(2.8)
where adef is the Bohr radius of the neutral defect inside the host crystal and k is the
wave number of the electron. Putting all together and using the relation p/m = }k
we have for the mobility at very low temperature:
e
µneutral defects =
(2.9)
20adef N }
2.5 Optical Properties
17
This expression is independent of temperature. Knowing the mobility, we can also
find an estimate for the defect concentration:
N=
1
20adef }µ
(2.10)
where } is expressed in eV s (} = 6.582 × 10−16 eV s). Using the mobility value
measured by Shimada at low temperature where the mobility is constant, µ '
1.3 × 105 cm2 /V s and a value for adef ' 1 nm, gives N ' 5 × 1015 cm−3 . This value
of defect concentration is quite reasonable as we will see in the following.
Between 10 K and 30 K there is not a dominating scattering mechanism. The
mobility is determined by the scattering of carriers with neutral impurities, longitudinal optical phonons and maybe acoustical phonons. Above 30 K the main process
is the scattering from longitudinal optical phonons. The curve a and b are two
numerical fits performed by Shimada and Masumi, considering high (82.1 meV) and
low (19.1 meV) frequency LO phonons in the first case and just the low (19 meV)
frequency phonons in the second case.
The disagreement between theoretical predictions and experimental data above
200 K has been tentatively explained by several processes but none of them is satisfactory. The first idea was to consider the scattering from ionized impurities [89,760]
but the mobility versus the temperature would have the opposite trend, increasing
instead of decreasing, and moreover an improbably high concentration of impurities,
greater than 1018 cm−3 , is needed. Another hypothesis, suggested also for other
materials (AgBr, TlBr), to explain the rapidly decreasing mobility with temperature
is based on the model of Toyozawa and Sumi [760, 832]. In this model the decrease
of mobility with temperature is explained by the auto-trapping of the holes in
metastable states which originate only at temperature above 200 K. However, as we
said above, a clear explanation of the trend of mobility as a function of temperature
does not exist.
Another important aspect which has to be considered is the effect of the surface
since the Hall mobility is generally measured using a planar setup with all four
contacts on the same face. The influence of the surface conductivity on the Hall
measurements was studied by Dobrovol’skii and Gritsenko [172]. In addition to
the planar setup, they used also a transverse setup measuring the Hall current
generated by the Lorentz force. This new approach allows the direct measure of the
bulk mobility, avoiding the surface effects. They concluded that the Hall mobility
measured in the regular way equals the true bulk mobility when the the resistivity
is lower than 104 W cm. For higher values of resistivity, the difference increases up to
one order of magnitude.
Unfortunately, data of mobility above 500 K do not exist. This information would
be important to get the correct values of defect formation entropies and enthalpies
from conductivity measurements at high temperatures.
2.5
Optical Properties
The study of the optical properties of Cu2 O was very fruitful. For instance, today
Cu2 O still remains the best textbook example for excitons. Moreover a great effort is
devoted to the Bose–Einstein condensation of excitons and to the polariton properties
18
2. Cu2 O: a review
10
-1
cm )
8
(x10
5
6
4
Cu O
2
2
absorption coefficient
0
2
3
E
4
5
(eV)
Figure 2.9. Absorption coefficient of Cu2 O at room temperature in linear scale obtained
combining data from [56, 281, 361, 582].
and applications, in which Cu2 O is a perfect candidate for experiments (the polariton
effects are so strong that the light inside Cu2 O moves with a velocity equal or smaller
than that of sound!).
The absorption coefficient at room temperature is reported in Fig. 2.9. As we will
see in the next section, lowering the temperature, and looking near the absorption
edge a series of peaks emerge, revealing the excitons.
We have seen that Cu2 O has a direct forbidden band gap. A consequence of
the forbidden dipole transition between the two higher valence bands and the lower
conduction band can be directly observed in Fig. 2.9. In fact Cu2 O starts to absorb
strongly the light only above about 2.4 eV, which is the dipole allowed transition
between the higher valence band and the second lower conduction band.
For a such semiconductor, neglecting the excitonic effects, the absorption
coefficient α just above the absorption edge depends on the photon energy }ω
as [185, 281, 493, 928]
(
α(ω) ∝
(}ω)−1 (}ω − Eg )3/2
0
for }ω > Eg
for }ω < Eg
(2.11)
Compared to direct but allowed absorption edges, which gives an absorption coefficient proportional to (}ω)−1 (}ω − Eg )1/2 , a forbidden edge increases more slowly
with energy in the vicinity of the band gap. As a result, forbidden absorption edges
are difficult to identify except when there are strong exciton effects, as in the case of
Cu2 O.
The absorption coefficient in the presence of hole-electron interaction, in the form
of excitons, was studied in 1957 by Elliott [185]. A typical absorption spectrum for
a direct band gap semiconductor, both allowed and forbidden, near the band gap is
composed by several contributions. At energies below the band gap there are narrow
peaks due to the direct creation of excitons, just below the band gap these peaks
2.5 Optical Properties
19
are so near among each other that a continuum appears. The last effect below the
band gap due to the excitons is the presence of two or more continuum absorptions
for each excitons given by the exciton creation with the simultaneous creation or
absorption of a phonon. Finally above the band gap there is a continuum absorption
increased respect with the non-interaction case by the presence of excitons.
The absorption coefficient of the cuprous oxide near the absorption edge, see
Fig. 2.11, shows the resonant peaks of exciton creation only at low temperature. At
high temperature remain the other contributions. The direct forbidden absorption
trough the gap and the absorption by an exciton (the orthoexciton) with emission
or absorption of a phonon (a Γ−
12 phonon) [56, 185]. Grun et al. [281] claims that
the continuum absorption due to the merging of the exciton peaks at large main
quantum number n is visible at high temperature.
If }ω < 2.4 eV we can write [185, 281, 493, 928]
α(ω) = Cabs (}ω − (Eg − Ebx − Ephonon ))1/2 +
+ Cem (}ω − (Eg − Ebx + Ephonon ))1/2 +
(2.12)
+ Ccont (}ω)(}ω − (Eg − Ryx )) + Cdf (}ω)−1 (}ω − Eg )3/2
where Cabs , Cem , Ccont , Cdf are four factors, weekly dependent on ω, Ephonon is
the energy of the phonon involved in the absorption, Eg is the energy gap, Ryx is
the excitonic Rydberg and finally Ebx is the binding energy of the orthoexciton.5
Notice also that Cabs and Cem are not independent because they can be written as
Cabs = C(nphonon + 1) and Cem = Cnphonon where nphonon is the mean number of
phonons of energy Ephonon at temperature T (Bose–Einstein statistics).
Finally we report the optical parameters of Cu2 O at room temperature [153]:
index of refraction n ≈ 2.7 at low frequencies and n ≈ 2.6 at high frequencies, relative
static permittivity εr (0) ≈ 7.4, high frequency relative permittivity εr (∞) ≈ 6.5.
From these values the effective ion charge of copper ions e∗ = 0.33 is obtained.
2.5.1
Excitons
Excitons [431] play a fundamental role in the history of Cu2 O and Cu2 O plays a
fundamental role in the history of excitons.
The first observations of excitons in Cu2 O were made in the early years of
1950 [270, 307, 308]. Since then an impressive amount of work was dedicated to the
study of this subject and related phenomena. In particular, from 1980 [340], a great
effort was dedicated to the possibility of Bose-Einstein condensation of excitons
in Cu2 O, a phenomenon predicted long time before (see [416] and references cited
therein). The feasibility of this experiment arise from the fact that excitons are
bosons and in Cu2 O they have a long life, a relatively large radius and an high
binding energy, giving the practical possibility to reach the condition a3x nx 1,
where ax is the exciton radius and nx their concentration, at which the excitons
should begin to condensate. However no clear results about this phenomenon has
been observed yet [615].
Excitons are usually classified in two categories, Frenkel and Mott–Wannier
excitons, which correspond to the limit cases of the ratio between the excitonic
5
To avoid a cumbersome notation we omitted the step functions because it is obvious in which
interval each term can give a contribution.
20
2. Cu2 O: a review
radius aexc and the lattice constant a. Frenkel excitons have a radius of the order of
the lattice constant or smaller, whereas Mott–Wannier excitons have a much larger
radius than the lattice constant.
The Mott–Wannier exciton can be easily understood with a hydrogen-like model.
Since the ratio ax /a is large, several atoms are inside the exciton orbit, giving
the possibility to treat the underlying lattice of atoms as a medium of relative
permittivity εr in which the electrons and the holes exist as free particles with
effective masses me and mh . According to this model, we report useful simple
quantities of excitons:
mx = me + mh
1
1
+
me mh
mr,x 1
Ryx =
Ry
mr,H ε2r
mr,H
aB
ax = n2 εr
mr,x
Ryx
Ebx = 2
n
exciton mass
(2.13)
exciton reduced mass
(2.14)
excitonic Rydberg
(2.15)
exciton radius
(2.16)
exciton binding energy
(2.17)
−1
mr,x =
where Ry ≈ 13.6 eV is the Rydberg, mr,H is the reduced mass of the hydrogen atom
(very close to the electron mass), aB ≈ 0.53 Å is the Bohr radius and n an integer
number describing the various energetic levels.
The peaks in the absorption spectrum of a direct band semiconductor like Cu2 O
are due to the resonant absorption of photons, giving rise to excitons. The energies
at which these lines are observed are given by:
E(n) = ∆E −
Ryx
= ∆E − Ebx
n2
(2.18)
where ∆E is the energy gap between the bands at which electron and hole of the
exciton belong to.
Measuring the absorption spectra of light at low temperature, see Fig. 2.10, four
hydrogenic excitonic series were found: yellow, green, blue, indigo (or blue-violet) in
order of increasing energy. The yellow series is generated by the electron-hole pairs
+
+
+
+
across Γ+
7 and Γ6 , the green series across Γ8 and Γ6 , the blue series across Γ7 and
−
Γ8 and finally the indigo series is generated by the electron–hole pairs across the
−
Γ+
8 and Γ8 . See Fig. 2.3.
The excitons of Cu2 O are Mott-Wannier excitons. In Fig. 2.11 the yellow exciton
series of Cu2 O is shown. In this plot seven peaks are visible but some author [525]
identifies up to 12 peaks!
Both in the yellow and in the green series, the n = 1 lines do not follow the
previous relation. Moreover by luminescence experiments it was discovered that
these lines (1s excitons) are splitted by the exchange interaction into a singlet and a
triplet, called paraexciton (total angular momentum J = 0, symmetry Γ+
2 for the
+
+
yellow and Γ+
+
Γ
for
the
green)
and
orthoexciton
(J
=
1,
Γ
for
both
yellow
12
15
25
6
and green) respectively, the first lying 12 meV lower than the second.
6
Notice that singlet and triplet does not mean spin-singlet and spin-triplet. Indeed if so, the
2.5 Optical Properties
21
7
10
B
6
10
I
5
10
n
(cm
-1
)
n
4
10
Y
G
= 2
= 2
3
10
2
10
T = 4.2 K
T = 300 K
1
10
0
10
2.0
2.2
2.4
2.6
photon energy
2.8
3.0
(eV)
Figure 2.10. Absorption coefficient of Cu2 O at 4.2 K (black line) showing all the four
excitonic series: Y (Yellow), G (Green), B (Blue), I (Indigo). After [323]. For comparison
the absorption coefficient at 300 K, the same of Fig. 2.9, is also reported (red line). Notice
that the two curves can not be directly compared because the shift of the energy gap
with temperature has to be considered.
30
phonon emission
(square root behaviour)
25
T=77 K
-1
(cm )
20
X
15
O
10
X
P
(invisible)
5
phonon absorption
(square root behaviour)
0
2.01
2.02
2.03
E
photon
2.04
(eV)
Figure 2.11. Left: Absorption coefficient divided by the index of refraction n ' 2.7 at the
absorption edge at 6 K [883]. Excitons peaks are visible. Right: Absorption coefficient
near the absorption edge at 77 K [616]. XO and XP are the resonant absorption peaks
for the orthoexciton and paraexciton respectively. As expected XP peak does not
appear. The square root behaviour of the absorption coefficient is explained in the text.
This dependence is due to the emission and absorption of the Γ−
12 phonon with the
simultaneous absorption of a photon and the creation of an orthoexciton.
22
2. Cu2 O: a review
The binding energies of the four series of Cu2 O are described, according to the
Eq. (2.18), by the following relations [10, 152, 280, 323, 582], derived principally by
light absorption experiments at 4 K:7
yellow:



Ebx (1, para) = 151.3 meV
E (1, ortho) = 139.3 meV
bx


E (n) = 97.43/n2 meV
bx
(
green:
Ebx (1) = 174.2 meV
Ebx (n) = 149.7/n2 meV
if n = 1, paraexciton
if n = 1, orthoexciton
if n > 1
if n = 1
if n > 1
(2.19)
(2.20)
blue:
Ebx (n) = 55.8/n2 meV
(2.21)
indigo:
Ebx (n) = 62.6/n2 meV
(2.22)
As said above, both the orthoexciton and paraexciton energies of the yellow series
and the green series do not follow the law used for the other peaks and moreover
these excitons give a very small signal in absorption spectra. The first observation
is explained by the fact that the 1s yellow and green exciton has a much smaller
radius than the other excitons (about 7 Å for the yellow one, comparable with lattice
constant, 4 Å), and therefore plane waves are not a good approximation for electron
and hole wave functions. In fact large values of k are required to localize the exciton
and describe correctly its wave function. The effective mass approximation, valid
only for small values of k is not accurate enough and therefore central cell corrections
are needed [414]. This can explain also the difference in the yellow exciton mass
obtained by the sum of the electron and hole masses measured by cyclotron resonance,
1.57m0 , and the one measured directly from Raman experiments, about 3.0m0 for
paraexciton and 2.7m0 for orthoexciton.8
The second observation, the weakness of the absorption line, was explained in
the fundamental work by Elliott [184, 225, 281]. For an exciton level the total parity
is determined by the product of the parities of the valence band, the conduction
band and the exciton envelope wave function.9 Since the lowest conduction band
and the highest valence band (which give rise to the yellow series) have the same
even parity, the total parity is determined by the parity of the exciton wave function.
For an excitonic s–state, the parity is even, and the transition to the ground
order of the orthoexciton and paraexciton levels would be inverted. Instead the paraexciton is a
pure total spin S = 1 while the orthoexciton wave function has both an S = 1 term and a S = 0
term. It is this last term that is subject to the exchange interaction and rise the orthoexciton
energy level 12 meV above the paraexciton level. See [377, 615]. Moreover notice that the yellow
paraexciton binding energy of 151 meV is comparable with the spin-orbit splitting ∆SO = 130 meV
of the valence bands, which gives rise to a mixing of the yellow and green n = 1 excitons.
7
At 4 K the energy gaps are in order of increasing energy, ∆E(yellow) = Eg = 2.1725 eV,
∆E(green) = 2.3042 eV, ∆E(blue) = 2.6243 eV, ∆E(indigo) = 2.755 eV.
8
However Dasbach [149] pointed out that exciton mass measured by Raman experiments includes
some additional terms due to the dependence of hole–electron exchange interaction on the wave
vector of the center of mass motion.
9
Notice that the parity of the electronic bands is a good quantum number only at two points of
the Brillouin zone, the Γ point and the R point.
2.6 Cu–O phase diagram
23
state is dipole forbidden, but quadrupole allowed. For a p–state, the parity is
odd, and the transition to the ground state is dipole allowed.10 Considering the
exchange interaction, the orthoexciton level is dipole forbidden while paraexciton is
forbidden at all perturbation levels because ∆J = 1 is required in a single photon
process. Therefore the peak observed in light absorption spectrum is due to the
orthoexciton in the electric quadrupole transition. To observe paraexcitons there
are two possibilities: an experiment involving the concurrent absorption of two
photons [860] or the application of a mechanical stress to remove some selection
rules at Γ point. For the same reasons discussed above, also the n = 1 line of the
green series is weak and does not follow the relation valid for n > 1.
Since the paraexciton and orthoexciton binding energies are quite high and
radiative recombination is forbidden to all orders for the former and at first order for
the latter, excitons exist also at room temperature and it is possible to observe them
for example by photoluminescence experiments. At low temperatures, orthoexciton
lifetimes (tens of nanoseconds at very low temperature) is limited by the conversion
in paraexciton while the paraexciton lifetime is very variable, from tens of ns up to
14 ms, because it is limited by the concentration of impurities and defects (mostly
VCu ) [377].
2.6
Cu–O phase diagram
Cu2 O (cuprite, κ) and CuO (tenorite, also called melaconite, τ ) are the only two
chemical compounds formed by oxygen and copper, which can be found in nature.11
The phase diagram of the binary system Cu–O in terms of temperature T and
oxygen partial pressure pO2 is reported in Fig. 2.12. First of all we have to note
that Cu2 O is unstable in air at room temperature. However the kinetics of the
transformation in CuO is so slow at room temperature that Cu2 O can be considered
stable for practically all applications [821, 933]. In Fig 2.13 is reported the phase
diagram in terms of temperature T and atomic percent oxygen 100 · x(O). A detailed
description near x = 0.3̄ (Cu2 O) will be given in the next section.
For completeness the standard formation values (p = 1 atm) at T = 298.15 K
of Cu2 O and related substances are reported in Tab. 2.5. Notice that usually, the
formation enthalpies calculated by a first principles approach (not reported here)
differ substantially from the experimental values. A detailed description of the
thermodynamic potentials of the copper-oxygen system can be found in [739].
2.7
Equilibrium concentration of intrinsic point defects
Most of the properties of cuprous oxide of interest are determined by the small
deviation from stoichiometry that can occur in this compound. The extent of
non-stoichiometry and the electrical conductivity as a function of temperature and
oxygen pressure are particularly important to identify the dominant defects.
10
This is the reason of labelling the exciton lines as 1s, 2p, 3p, 4p, . . . Because the only visible
lines are formed by odd–parity envelope wave function, except for the first line since the n = 1 is
not degenerate in angular momentum.
11
Actually two other non-stable compounds exist: Cu3 O2 and Cu3 O4 (paramelaconite). They
are two compounds in which copper has both the oxidation numbers. The first one is the most
common and can be found as a defect in the surface of Cu2 O. See [530, 674, 822].
24
2. Cu2 O: a review
6
10
5
10
1354K
1551K
401Torr
4
10
5
9.5x10 Torr
3
10
p(O ) in air, 159Torr
L2
2
2
2
p(O )
(Torr)
10
1
1502K
1395K
1293K
CuO
10
0.274Torr
0
10
-1
10
Cu
-2
( 19
y=0
O
2-y
90
e
Xu
)
1621K
10
0.501Torr
1496K
0.09Torr
-3
10
-4
10
-5
10
0
y=
-6
10
(1
9
9
4
o
P
ra
1338K
t)
L1
0.00205Torr
Cu
-7
10
900
1000
1100
1200
T
1300
1400
1500
1600
(K)
Figure 2.12. Phase diagram pO2 -T of Cu2 O. After [129, 296, 739, 912].
1650
(21.5%,1621K)
-4
1600
0
-2
log
10
( p(O ) [atm] )
2
L2+O
L2
2
L1+L2
1550
1551K
1502K
1496K
L1
Cu O + L2
L1+Cu O
2
2
1400
2
CuO + L2
31.0%
CuO+O
9.57%
1450
T
1358.02K
1350
1354K
1338K
39%
1.7%
Cu+Cu O
Cu
1250
2
2
0
10
20
100 x(O)
30
-
Cu O+CuO
CuO
1300
Cu O
(K)
1500
2
40
50
Atomic Percent Oxygen
Figure 2.13. T -x diagram for Cu2 O. After [129, 296].
60
2.7 Equilibrium concentration of intrinsic point defects
25
Table 2.5. Thermodynamic standard potential for Cu2 O and related substances [470].
2.7.1
Formula
∆Hf◦
kJ/mol
∆G◦f
kJ/mol
S◦
J/(K mol)
Cu(s)
Cu(s)
O2 (g)
O2 (g)
1
Cu(s) + 2 O2 (g)
2 Cu(s) + 12 O2 (g)
0
0
−157.3
−168.6
0
0
−129.7
−146.0
33.2
205.2
42.6
93.1
CuO(s)
Cu2 O(s)
Defect concentration from mass action law
Standard structural inorganic chemistry explains in broad terms that an oxide will
naturally be cation–deficient if its cation is oxidizable, i.e., can have a more positive
oxidation number. Indeed, the existence of an oxide compound with higher oxidation
number of the metal cation indicates the propensity of the oxide to exist in a more
cation-deficient composition (Cu(II)O in our case).
Before going ahead we introduce the notation used in this thesis to represent
the point defects. The main symbol identifies the nature of the point defect, where
“V ” stands for vacancy, the subscript indicates where the atom or vacancy is placed,
where “i” means interstitial, and the superscript indicates the electric charge in
unit of positron charge respect to the ideal crystal, where 0 means “neutral”. For
0 means a vacancy (V ) of copper (Cu subscript) without any
example, in Cu2 O, VCu
0
charge (zero superscript); OCu
means an oxygen atom (O) in the place of the regular
position of a copper atom (Cu) without any charge; VO+ means a vacancy (V ) of
oxygen (O subscript) with charge +1 (+ superscript).12
The possible intrinsic point defects in Cu2 O are vacancies VCu and VO , interstitials
Cui and Oi , Frenkel defects (VCu − Cui ) and (VO − Oi ), Schottky defect (2VCu − VO ),
antisites CuO and OCu . Frenkel defects and Schottky defects are stoichiometric
defects, in other words their presence does not change the stoichiometry.
Usually the parameter used in literature to describe the non-stoichiometry is y,
defined as the deficiency deviation of copper from the perfect stoichiometry 2, giving
rise to the chemical formula Cu2-y O. Mathematically:
y =2−
nCu
[Cu]
=2−
,
nO
[O]
x=
1
,
3−y
−∞ ≤ y ≤ 2
(2.23)
where nCu and nO are the number of moles of copper and oxygen respectively and x
is the fraction of atomic oxygen. Square brackets indicates here and in the rest of the
document a concentration in terms of number of particles for unit of volume, usually
expressed in cm−3 . Writing [Cu] and [O] as a function of defect concentrations and
considering only small defect concentrations respect with the concentration of lattice
12
This notation is slightly different from the famous Kröger-Vink notation where × , • and 0
are used respectively in the place of 0 , + and − . Using the notation of this thesis, as well as the
Kröger-Vink notation, gives rise to quasi-chemical formulas. It means that, in principle, mass action
law can not be applied. Another very common notation, which permits to apply the mass action
law, is the Schottky notation. However, due to the simple form of our chemical formulas, applying
mass action law to our notation gives the same result obtained in the Schottky notation.
26
2. Cu2 O: a review
sites, we can write
y≈
h
1
−
0
[VCu
] + [VCu
] − [Cui0 ] − [Cui+ ]+
[Cu2 O]
+ 2 [Oi0 ] + [Oi− ] + [Oi2− ] − [VO0 ] − [VO+ ] − [VO2+ ]
i
(2.24)
where [Cu2 O] = [O] = 12 [Cu] is the ideal concentration of Cu2 O entities.
The neutrality condition is13
−
[VCu
] + [Oi− ] + 2[Oi2− ] + n = [Cui+ ] + [VO+ ] + 2[VO2+ ] + p
(2.25)
Considering the chemical reaction of formation of defects and using the mass
action law, we can write the following relations, where aO2 is the activity of the
oxygen:
1/4
1
4 O2
1 0
2 OO
0
+ V Cu
0
[V Cu
]/[Cu2 O] = kV 0 aO
1
4 O2
1 0
2 OO
−
+ V Cu
+ h+
−
[V Cu
]p/[Cu2 O] 2 = kV − aO
(2.26b)
[Cui0 ]/[Cu2 O] = kCu 0 aO
(2.26c)
(2.26d)
1 0
2 OO
0
+ CuCu
1
4 O2
+ Cui0
(2.26a)
2
Cu
1/4
2
Cu
−1/4
2
i
OO0
1
2 O2
+ V O0
[V O0 ]/[Cu2 O] = kV 0 aO
OO0
1
2 O2
+ V O+ + e −
[V O+ ]n/[Cu2 O] 2 = kV + aO
−1/2
O
2
−1/2
1/2
1
2 O2
Oi0
[Oi0 ]/[Cu2 O] = kO 0 aO
1
2 O2
Oi− + h +
[Oi− ]p/[Cu2 O] 2 = kO − aO
+h
0
(2.26f)
2
i
1
Oi2− + 2 h +
2 O2
OO0
Oi2− + V O2+
−
2 V Cu
+ V O2+
0
−
+
e
(2.26e)
2
O
1/2
(2.26g)
2
i
1/2
[Oi2− ]p2 /[Cu2 O] 3 = kO 2− aO
(2.26h)
[Oi2− ][V O2+ ]/[Cu2 O] 2 = kFrenkel
(2.26i)
i
− 2
[V Cu
] [V O2+ ]/[Cu2 O] 3
np = n2i
2
= kSchottky
(2.26j)
(2.26k)
Remember that the activity of a gas, in the case of an ideal gas, is numerically equal
to the pressure expressed in atmospheres. In the present case of copper-oxygen
system, in the stability region of Cu2 O the oxygen can be treated as an ideal gas.
Studying the dependence of y on partial oxygen pressure at equilibrium it
is possible to extract some information on the nature of defects while from the
temperature dependence their formation energies can be deduced.
The equilibrium constant of a reaction is related to the standard Gibbs free
energy of reaction according to
K(T ) = e−∆G°(T )/(kT ) = e∆S°/k e−∆H°/(kT )
(2.27)
Once the dependence of K(T ) on temperature has been obtained by experiments, it
is possible to extract the standard entropy and the enthalpy of reaction.
We point out also that the formation enthalpy and entropy of a charged defect
can be seen as the sum of two contribution. Indeed every chemical reaction, having
13
We have used p for the free positive hole concentration and n for the free negative electron
concentration in the place of [h + ] and [e – ] respectively.
2.7 Equilibrium concentration of intrinsic point defects
27
a charged defect as a product, can be decomposed in the chemical reaction for
the neutral defect plus the reaction of ionization of that defect. For instance, the
formation of a negatively charged copper vacancy:
1
4 O2
can be decomposed in
(1
4 O2
0
V Cu
1 0
2 OO
−
+ V Cu
+ h+
(2.28)
0
1 0
2 OO + V Cu
−
V Cu
+ h+
(2.29)
The equilibrium constants are respectively

0

[V Cu
]

∆Sf /k −∆Hf /(kT )

k
e
=
0 ≡ e

V

1/4
 Cu
[Cu2 O]aO
2

−

[VCu
]p

∆SI /k −∆HI /(kT )


k
≡
k
≡
e
e
=
0
−
I
 V /V
0
Nv
e−(EA −Ev )/(kT )
g[Cu2 O]
(2.30)
where in the last step of the second equation we have used the relation p =
Nv e−(EF −Ev )/(kT ) and the occupation probability of the copper vacancy in term of
the Fermi level. Finally g is the ratio of the degeneracy of the neutral over charged
copper vacancy and EA ≡ EVCu (0/−) is the acceptor energy level introduced by the
copper vacancy in the gap.
The enthalpy of ionization of a donor defect is defined as the energy required
to bring an electron from the defect to the conduction band, ∆HI ≡ Ec − E(+/0),
while the enthalpy of ionization of an acceptor defect is the energy required to bring
an hole from the defect to the valence band, ∆HI ≡ E(0/−) − Ev , and therefore, in
our case, is equal to ∆HI ≡ EVCu (0/−) − Ev .
The product of the two equilibrium constants (2.30) is equal to the equilibrium
constant of the total reaction (2.28), giving
Cu
[V Cu ][Cu2 O]
Cu
(
=
0
∆Hf (V −
Cu ) = ∆Hf (V Cu ) + EV Cu (0/−) − Ev
0
∆Sf (V −
Cu ) = ∆Sf (V Cu ) + ∆SI
(2.31)
Now if the formation enthalpies of both charge states of a defect are measured, it is
possible to extract the ionization enthalpy ∆HI = EVCu (0, −) − Ev , that is the value
of the electronic level introduced in the gap by that defect.
It is important to note that the formation enthalpies of the defects, both neutral
and charged, do not depend on the Fermi level. Instead, especially in theoretical
works, the formation energy, which depends on the Fermi level, is commonly used.
For a defect A, the formation energy ∆Ef is simply defined as the energy in the
exponential of the defect concentration expression at equilibrium
[A] = cost · e−∆Ef /(kT ) .
(2.32)
From this definition it follows that for a neutral defect the formation energy is
equal to the formation enthalpy, whereas for a singly charged acceptor defect
∆Ef = ∆Hf −(EF −Ev ) for a doubly charged acceptor defect ∆Ef = ∆Hf −2(EF −Ev )
and so on. For a charged donor EF − Ev is replaced by Ec − EF .
28
2. Cu2 O: a review
1600
(L1+L2)
L2
L2
1502 K
1500
1496 K
1400
Cu O+L2
2
L1+Cu O
2
O
2-y
1354 K
1338 K
(K)
1300
Cu
T
1200
1100
Cu O+CuO
Cu+Cu O
2
2
1000
900
33.20
33.25
33.30
100
x(O)
33.35
-
33.40
33.45
33.50
Atomic Percent Oxygen
Figure 2.14. Magnification of the phase diagram reported in Fig. 2.13 near the perfet
stoichiometry of Cu2 O (x = 1/3).The red points are experimental data from Porat and
Riess [678]. The dashed lines are postulated.
As an example, consider a doubly charged oxygen interstitial:
Oi0
Oi2− + 2 h + ,
(2.33)
[Oi2− ] = p−2 [Oi0 ] = Nv−2 e+2(EF −Ev ) e∆Sf /k e−∆Hf /(kT )
(2.34)
[Oi2− ] = cost · e−[∆Hf −2(EF −Ev )]/(kT ) = cost · e−∆Ef /(kT ) .
(2.35)
◦
◦
and finally
◦
2.7.2
Deviation from stoichiometry: experimental results
A few papers were devoted to the study of deviation from stoichiometry in the
Cu2 O. Early experiments based on chemical analysis of quenched samples [869] or on
microbalance [556,630] were not very accurate. More recent measurements performed
by coulometric titration [678, 835] and thermo–gravimetric analysis (TGA) [911]
has been used to measure y as a function of oxygen pressure and temperature,
determining the region of existence of Cu2-y O pure phase in the phase diagram
reported in Fig. 2.13 and then the nature of the dominant point defects.
In Fig. 2.14 the magnification of the region near the perfect stoichiometric Cu2 O
is shown. It extends from y ≈ −3 × 10−5 to y ≈ 5 × 10−5 at 873 K and from
y ≈ −1.8 × 10−3 to y ≈ 4.5 × 10−3 at 1245 K. The red points are the experimental
data obtained from Porat and Riess [678]. These data have been extrapolated using
an activated law to the invariant points to complete the diagram. The dashed lines
are postulated. It is evident that Cu2 O can also be formed with oxygen deficiency.
In Fig. 2.15 the maximum and minimum deviation from stoichiometry at a given
temperature is translated into the maximum concentration of copper and oxygen
vacancies considered as the sole present defects.
The experimental curves of y(pO2 , T = cost) have an S shape (see Fig. 2.16)
and this means that there are at least two types of defects with an exponential
2.7 Equilibrium concentration of intrinsic point defects
29
20
10
V
Cu
V
Concentration
-3
(cm )
O
19
10
18
10
0.80
0.85
0.90
0.95
1000/T
1.00
1.05
1.10
1.15
-1
(K )
Figure 2.15. Filled circles: the maximum positive deviation from stoichiometry is translated
into the maximum concentration of copper vacancies considered as the sole present defect.
Filled triangles: the maximum negative deviation from stoichiometry is translated into
the maximum concentration of oxygen vacancies considered as the sole present defect.
The deviation from stoichiometry is taken from [678].
dependence on pO2 , one with a positive exponent (1/n) and the other one with a
negative exponent (−1/m):
1/n
−1/m
y = A(T )pO − B(T )pO
2
2
where n, m, A and B are constants at fixed temperature.
0 (n = 4)
The fits show that at high oxygen pressure the dominant defects are VCu
regardless of temperature. Also a predominance of interstitial oxygen with a single
negative charge can give n = 4, but this would give a conductivity proportional to
1/4
pO while the experimental behaviour shows an exponent between 1/6 and 1/8.
2
On the other hand for the negative exponent, that is at low oxygen pressure, the
results are less clear. Porat and Riess [678] find an m which increases as T decreases:
from 3.3 at 1245 K to 5 at 873 K and they conclude that at low temperature Cui0
prevail (m = 4) whereas at high temperatures VO0 prevail (m = 2).
Xue and Dieckmann [911] find instead an a value for m near 3 and say that it
is quite possible to obtain an equally good fit even using m = 2, concluding that
the neutral oxygen vacancies are the predominant defects at low oxygen pressure,
regardless of the temperature.
Notice that positively charged oxygen vacancies VO+ can not be the dominant
defects since their predominance would give an n-type conductivity which was never
observed.
The Dieckmann group claims [15] that their interpretation is the correct one since
if copper interstitials would be the majority defects one would expect these defects
to contribute to copper diffusion in Cu2 O. Indeed the data concerning the copper
diffusion at low pO2 [671] do not indicate any contribution of cation interstitials
but in our opinion this is not a proof because, even if it is very unlikely, the cation
30
2. Cu2 O: a review
0.004
Ap(O )
T=1245 K
+1/4
2
Stability limits
y
0.002
0.000
-0.002
-Bp(O )
-1/2
2
-4
10
-3
10
-2
-1
10
10
p(O )
2
0
10
1
10
2
10
(Torr)
Figure 2.16. The y parameter as a function of oxygen pressure in equilibrium
at 1245 K [678].
Figure 2.17. Deviation from stoichiometry in Cu2 O at equilibrium as a function
of mean grain size [15].
interstitial could be the dominant defect if their diffusion coefficient is low enough.
If the Xue et al. interpretation were correct, the deviation from stoichiometry
would be described by the following equation:
1/4
−1/2
y = kV 0 aO − kV 0 aO
Cu
2
O
2
(2.36)
However it has to be noted that a lot of conflicting data, concerning the values of
the equilibrium constant, can be found in literature. A partial answer to this problem
has been suggested by Aggarwal et al. [15, 169], finding that the variation from
stoichiometry y is greatly influenced by the dimension of the grains. In fine-grained
samples the value of y is greater than in the large grained samples at the same
condition of temperature and oxygen pressure, see Fig. 2.17. This means that along
the grain boundaries an higher defect concentration exist. It was also found that
increasing the temperature the effect of the grain boundaries decreases. The effect
of grain boundaries becomes negligible when the average dimension of the grains is
about 1 mm2 or greater, as in the case of Xue et al. which used large grain samples
for their experiments [911].
In diffusion studies [169, 670] has been observed that the oxygen partial pressure
dependence of radioactive oxygen diffusion along boundaries and in the bulk is the
same. The same observation was made in other materials on the dominant defects.
These observations suggest that the defects present in the grain boundaries and
in the bulk are the same. Fitting the data of Fig. 2.17, it can be shown that the
formation enthalpies of neutral copper vacancies is lower in the grain boundaries,
about 57 kJ/mol, than in the bulk, about 74 kJ/mol.
2.7.3
Electrical properties at high temperatures
We have seen in the previous section that neutral defects are dominant in Cu2 O
at high temperature. If we want to collect information about the charged defects,
we have to consider physical phenomena influenced only by these defects, like conductivity (σ) and Seebeck coefficient (Q, also called thermopower or thermoelectric
power (TEP)), and study their behaviour as a function of partial oxygen pressure
2.7 Equilibrium concentration of intrinsic point defects
31
and temperature at equilibrium.14
Measurements of conductivity at high temperature were performed many times
in the history of Cu2 O [76, 162, 181, 514, 611, 656, 679, 788, 830]. The most recent and
extensive set of measurements was performed by Porat and Riess in [679].
In general, conductivity is given by the sum of three terms, ionic conductivity,
electron conductivity and hole conductivity
σ = σp + σn + σi
(2.37)
however the ionic conductivity σi can be neglected because usually is two order of
magnitude lower than electrical conductivity [163]. Using np = n2i and introducing
the electron µe and hole µh mobilities, we can rewrite the Eq. (2.37) as
σ = eµh p + eµe n = eµh p + eµe n2i /p
(2.38)
Since n2i has a low value we can neglect the electron contribution because the observed
conductivity is always p-type. As the dominant point defects are the neutral copper
vacancies, the dominant charged point defects at high oxygen pressure are the
single negatively charged copper vacancies. We can see from Eq. (2.26b), that,
−
using the neutrality condition p = [VCu
], the equilibrium concentration of charged
1/8
copper vacancies at equilibrium is proportional to aO . Therefore the conductivity,
2
considering only the copper vacancies, is given by
1/2
1/8
1/8
σ(T, pO2 ) = eµh kV − [Cu2 O]aO = A(T )pO
2
Cu
(2.39)
2
which is in fairly good agreement with the experimental results.
However accurate measurements of conductivity [679] have shown a more complex
behaviour of this quantity as a function of oxygen pressure. This is shown in Fig. 2.18.
The numbers reported in the figure correspond to
1/k =
∂ log σ
∂ log p(O2 )
(2.40)
which is always positive, meaning a p-type conductivity, except at high temperature
and low oxygen pressure where it approaches zero, suggesting a contribution of
electrons to conductivity. In the high pO2 range, 1/k varies from ∼ 1/6 at high
temperature to ∼ 1/9 at low temperature, while in the low pO2 range, 1/k varies
from almost zero at high temperature to ∼ 1/11 at low temperature.
Three interpretation were given to describe these results:
1. Porat and Riess [679] used a model in which the dominant charged defects are
negative copper vacancies and doubly negatively charged oxygen interstitials.
1/8
The former dominates at temperature below 950 K (σ ∝ pO ) and the latter
1/6
2
at temperatures above 1150 K (σ ∝ pO ). At intermediate temperatures both
2
defects exist in similar concentrations.
14
Conductivity at thermodynamic equilibrium is very different from the conductivity at low
temperature where the dynamics of defects is frozen. In the former case the number of defects
change with temperature and pressure, whereas in the latter case is constant. Therefore also the
activation energies of conductivity have different meanings.
32
2. Cu2 O: a review
Figure 2.18. Conductivity as a function of oxygen partial pressure for several temperatures.
∂ log σ
The numbers correspond to 1/k = ∂ log
p(O ) . After [679]
2
2. The second model, by Peterson and Wiley [671], is similar to that of Porat
and Riess but assumes, along with negatively charged copper vacancies, the
presence of singly charged oxygen interstitials.
3. Finally the third model, by Maluenda et al. [514] and Ochin [611], explains
the conductivity data by the formation of associates of neutral and charged
0 − V − ).
copper vacancies, (VCu
Cu
In my opinion, this last interpretation is the least acceptable because the stability
of the associated defects should be given by a binding energy much higher than the
thermal energy, a condition difficult to be satisfied especially when one of the two
0 − V − ) does not fit very
defects is neutral. Moreover the model with associates (VCu
Cu
well the experimental data.
The model of Peterson and Wiley was applied only to a limited range of conductivity measurements and maybe it cannot explain the whole range of 1/k values
observed by Porat and Riess (assuming that the measurements at very low oxygen
pressure are correct). Moreover, according to Peterson and Wiley, the formation
enthalpies of charged and neutral copper vacancy is the same within the experimental
error. This implies that the energy level EVCu (0/−) of the copper vacancy is equal
2.7 Equilibrium concentration of intrinsic point defects
33
3
10
2
10
p(O ) in air, 159Torr
2
L2
1
10
h
CuO
0
10
O
V
2
p(O )
(Torr)
-1
10
+
Cu
2-
Cu
+
-2
10
V
-
~O
Cu
10
-4
V
10
-
h
i
+
0
y=
Cu
-5
V
9
Cu
-6
-7
V
0
9
4
o
P
ra
O
t)
V
-
2-
i
0
O
2V ~O
0
Cu
10
10
(1
h >e
2-
-3
10
O
2-y
i
0
V
- h
L1
i
Cu
+
Cu
O
-8
10
900
1000
1100
1200
T
1300
1400
1500
1600
(K)
Figure 2.19. Phase diagram of cuprous oxide showing the different intrinsic point defects
predominating in the various stability regions according to the model of Xue et al. [15,
169, 911] for neutral defects and according to the model of Porat and Riess [679] for
charged defects.
to the energy of the top of the valence band Ev , which is quite strange and brings
to a different analysis of the dominant defects in Cu2 O.
On the other hand, even if the hypothesis of Porat and Riess is consistent with
the most extensive set of conductivity data, their interpretation gives strange values
for the formation energies of defects (see next section).
Besides conductivity, another experimental technique used to probe the nature
and the properties of charged defects is the measurement of Seebeck coefficient Q
as a function of temperature and oxygen pressure. The only systematic measure of
Q was performed by Porat and Riess [679] (see references cited therein for further
papers on Seebeck coefficient). The expression of Seebeck coefficient as a function of
oxygen pressure is rather complicated and we do not report it here. We mention only
the fact that the interpretation of Porat and Riess is consistent also with Seebeck
coefficient data.
A summary of this section and the previous one is condensed in Fig. 2.19. Over
the phase diagram of Cu2 O, the main defects, both neutral (dominant) and charged,
are reported in the region where they are most important, following the interpretation
of Xue and Dieckmann [15, 169, 911] for the neutral defects and the interpretation
of Porat and Riess [679] for the charged ones. However, as we have seen before, no
consistent model exists in literature to describe the defects of Cu2 O in equilibrium
at high temperature.
34
2. Cu2 O: a review
Table 2.6. Experimental enthalpies and entropies of formation of neutral defects according
to the model of Xue et al. [15, 169, 911] and of charged defects according to the model of
Porat and Riess [679]. The entropy for the neutral copper vacancy is taken from [678].
Defect
∆Hf (kJ/mol)
∆Hf (eV/defect)
∆Sf (J/(K mol))
0
VCu
74
0.77
−10
VO0
294
3.06
−
VCu
174
1.81
0.9
0.009
Oi2−
300
3.12
1.4
0.014
0
(VO0 − 2VCu
)
440
4.58
2.7.4
∆Sf (eV/defect)
−0.10
Formation enthalpies of intrinsic defects
In Tab. 2.6 the enthalpies and entropies of formation of the main defects in Cu2 O
are reported. These quantities rely on the model chosen to analyze y and σ as a
function of temperature. While for neutral defects the situation is almost clear,
for charged defects, as we have already seen in the previous section, several model
exist. We think that the most correct interpretation for the dominant neutral defects
is that one of Xue and Dieckmann [15, 169, 911] while for the charged defects the
correct interpretation is that one of Porat and Riess [679].
However it has to be noted that a simple analysis, which considers the copper
vacancy as the dominant charged defect at high temperature and high oxygen
pressures, gives completely different results for the formation enthalpy of the charged
copper vacancy. Indeed consider the activation energy of the conductivity of Cu2 O
at high temperature. If the conductivity is given only by negatively charged copper
vacancies the conductivity can be written as (see Eq. (2.39))
1/8
1/2
σ(T, pO2 ) = eµh aO [Cu2 O]kV − =
2
=
1/8
eµh aO [Cu2 O]
2
Cu
e
0 )+∆S )/(2k)
(∆Sf (VCu
I
0
e−(∆Hf (VCu )+EA )/(2kT )
(2.41)
where EA = EVCu (0/−) − Ev . Therefore, if the hole mobility µh is considered
constant with temperature [553, pp. 259 and 260], the activation energy of the
conductivity is
1
1
−
0
Eσ = ∆Hf (VCu
) = (∆Hf (VCu
) + EA ).
(2.42)
2
2
The experiments give a value of Eσ ≈ 0.65 eV and from Tab. 2.6 we see that
0 ) ≈ 0.77 eV. If this simple model is correct, the energy level introduced
∆Hf (VCu
by the copper vacancy lies about at 0.53 eV from the top of the valence band and
−
∆Hf (VCu
) ≈ 1.3 eV.
On the other hand, remembering the Eq. (2.31), we can calculate EA directly
from the values reported in Tab. 2.6. For the copper vacancy we obtain that the
electronic level should be more than 1 eV above the top of the valence band. We will
see in the following that this high value of the EVCu (0/−) − Ev is in disagreement
with the value obtained from low temperature conductivity measurements, which is
about 0.3 eV.
2.7 Equilibrium concentration of intrinsic point defects
2.7.5
35
Formation entropy of the copper vacancy
Using the values from Tab. 2.6, we can find also the entropy of the copper vacancy
at high temperature (1000 ◦C) as
1
1
0
S(VCu
) = ∆Sf − S(OO0 ) + S(O2 )
2
4
(2.43)
Using ∆Sf ≈ −10 J/(mol K) from the previous table and S(OO0 ) ≈ 13 S(Cu2 O) ≈
0 ) ≈
66 J/(mol K) and S(O2 ) ≈ 251 J/(mol K) from literature, we obtain S(VCu
20 J/(mol K). This value is obviously positive because a presence of a defect always
increase the disorder of the crystal.
Using the simple model in which the atoms in the crystal interact by harmonic
forces [630], it can be readily shown that the formation entropy is given by:
∆Sf = k
X
log(ωi /ωi0 )
(2.44)
i
where the sum is over all neighbors of the vacancy (usually nearest or second nearest
neighbors) and ω is the angular frequency of the harmonic oscillator between the
considered site and the ith neighbourhood. The prime refers to the defect crystal
site. The entropy is positive because of lowering the vibration frequencies of the
neighbouring copper atoms (it can be shown that oxygen vibrational frequencies are
almost unchanged by the presence of a copper vacancy).
2.7.6
Theoretical predictions
Few works [606,686,784,907] were devoted to the study of the Cu2 O defects from first
principle calculations. Wright and Nelson [907] predict the existence of two types of
copper vacancies. These two vacancies differ for their position in the lattice: the
regular vacancy is a simple vacant copper site, with the rest of the crystal unchanged;
instead the split vacancy is a configuration in which another copper atom relaxes
from its position to partly fill a simple copper vacancy, see Fig. 2.20. The authors
predict for the split vacancy an enthalpy of formation 0.1 eV/particle lower than
that of the simple copper vacancy. This is qualitatively due to the formation of four
bonds between the displaced copper and the nearby oxygen atoms, at the cost of
distorting lattice, whereas the simple vacancy retains the oxygen dangling bonds.
Nolan and Elliot [907] found similar results of the first paper but predicted
that the formation energy of the simple copper vacancy is 60 meV lower than that
of the split vacancy. Therefore the former has to be the stable copper vacancy
configuration.
Another and more refined work [686] devoted to the study of defects of Cu2 O,
by Raebiger, Lany and Zunger, reports that copper vacancies are the dominant
defects in Cu2 O and that their maximum stable concentration is about 1020 at
high temperature. The copper vacancies show an acceptor like behaviour with an
electronic level at 0.28 eV from the valence band and a formation enthalpy in the
neutral state of 0.7 eV in accordance with experiments.
In these works the formation energy ∆ED,q of a defect D in the charge state q,
is defined and calculated as
∆ED,q (EF , µ) = (ED,q − EH ) +
X
α
nα (µelem
+ ∆µα ) + q(EV + ∆EF )
α
36
2. Cu2 O: a review
Figure 2.20. a) Regular Cu2 O lattice; b) simple copper vacancy; c) split vacancy.
where EH is the energy of the pure host, ED,q is the energy of the host with one
defect D in the charge state q, µα = µelem
+ ∆µα is the chemical potential of a
α
reservoir of atom α, nα is +1 or −1 if an atom is added or removed from the
reservoir, respectively, and finally µelec = EV + ∆EF is the chemical potential of
an electron reservoir. Along this equation, the chemical equilibrium is imposed by
2µCu +µO = µCu2 O . The calculated energies of formation ∆E and the transition levels
ε(q/q 0 ), the values of µelec at which ∆ED,q and ∆ED,q0 intersect, are then used in
self-consistent thermodynamic simulations to calculate the equilibrium concentration
of defects using cD,q (µelec , µ, T ) = N e−∆ED,q /(kT ) . It is straightforward to note that
ε(q/q 0 ) is, by definition, equal to the energy level E(q/q 0 ).15
The defect formation energies versus Fermi level are reported in Fig. 2.21. If
in general a defect can exist in two or more charge states, for a given Fermi level
only the configuration with the lowest formation enthalpy is reported. We have to
note that the split vacancy (1.0 eV) is higher in energy respect to the simple vacancy
(0.7 eV). This is in agreement with Nolan and in contrast with the calculation of
Wright where the split vacancy level is 0.1 eV below the simple vacancy level. The
oxygen vacancy is predicted to be always neutral and the copper interstitial is
predicted to have an amphoteric behaviour, i.e. it can behave like an acceptor or a
donor. Finally the oxygen interstitials are predicted to be an important defect in
Cu2 O but the charge state in the region of predominant neutral copper vacancies is
neutral, which is not consistent with the analysis of Porat and Riess.
Similar results for the formation energies of the intrinsic defects in Cu2 O were
recently obtained by Soon et al. [784]. In that paper the formation energies of
antisites are reported also, see Fig. 2.22.
The defect concentration as a function of temperature are reported in Fig. 2.23.
Even if the simulation has been performed both near the Cu2 O/Cu boundary and
Cu2 O/CuO region of the stability diagram, y, i.e. the deviation from stoichiometry,
is always positive, in contrast to the measurements discussed above.
In the same figure the value of hole concentration p is also reported, both at the
equilibrium and at room temperature supposing an instantaneous quenching of the
material from the equilibrium temperature to the room temperature.
A partial agreement between the theoretical predictions and the experiments
exists as long as we consider the equilibrium properties of Cu2 O at high temperature.
15
We remember that the relation between the formation energy of a defect and its formation
enthalpy is ∆Ef = ∆Hf + q|EF − Eband |. In order to obtain graphically the formation enthalpies of
charged defect it is sufficient to extrapolate the segment toward the EF = Ev for the acceptors and
toward EF = Ec for the donors.
2.7 Equilibrium concentration of intrinsic point defects
37
Figure 2.21. Calculated formation enthalpies of defects in Cu2 O as a function of Fermi
level. After [686]. The labels c and d refer to different interstitial positions inside the
unit cell. Each defect can exist in two or more charge states indicated by the + and −
signs.
Figure 2.22. Calculated formation enthalpies of defects in Cu2 O as a function of Fermi
level [784]. The labels c and d refer to different interstitial positions inside the unit cell.
38
2. Cu2 O: a review
Figure 2.23. Concentration of defects at equilibrium and dependence of the Fermi level
versus temperature. After [686]. The value of hole concentration p is also reported, both
at the equilibrium and at room temperature supposing an instantaneous quench of the
material from the equilibrium temperature to the room temperature.
Indeed the value the value of p can be calculated from the conductivity using a
reasonable value for the hole mobility µp . Assuming µp ≈ 3 cm2 /(V s) at T = 1200 K
the concentration of hole carriers of about 1019 cm−3 is obtained from the measured
conductivity values [679]. This value is in good agreement with the theoretical
predictions.
Using the Eq. (2.24) we can obtain also the non-stoichiometry parameter y
theoretical prediction and we can compare this value with the experimental value [678].
At 1200 K we have
y th = 1.7 × 10−3
y exp = 2.8 × 10−3 .
(2.45)
The agreement is fairly good however, considering that the predominant contribution to the deviation from stoichiometry is due to the copper vacancies, this
agreement concerns the copper vacancy only whereas for the other defects no reliable
experimental values are available.
The other theoretical prediction regarding the concentration in samples subjected
to an instantaneous quenching, are not easily verified, both for technical reasons and
for the possible partial re-equilibration due to the quite high diffusion coefficient of
the copper vacancies (see Sec. 2.8).
Moreover it must be noted that all the intrinsic defects that could show a donor
behaviour have an high formation energy. We will see in the following that at low
temperature the copper vacancies are compensated by donors, which are quite often
identified with oxygen vacancies. These calculations suggest instead that these
donors are of extrinsic nature.
2.8 Self-diffusion of the lattice ions and intrinsic point defects
2.7.7
39
Self-compensation mechanism
The n-type conductivity has never been observed in undoped Cu2 O and every
attempt of n-doping on Cu2 O has given negative results, obtaining no doping or, at
best, a p-type doping.16
This can be explained observing the dependence of the defect formation energies
versus Fermi level in Fig. 2.21. The formation energy of charged defects varies
linearly with the Fermi level and can reach a zero value by an appropriate value of
the Fermi level. For example the energy required to create a charged copper vacancy
0 )+E
becomes very small when the Fermi level approaches ∆Hf (VCu
VCu (0, −1) ' 1 eV
and, for higher Fermi level, the formation energy becomes zero or negative. This
means that the crystal prefers, because of the entropy and the enthalpy gain, to
compensate the increasing Fermi level, producing defects with the appropriate charge
state.
This mechanism, called self-compensation [857], is at the base of the doping
problem in many wide gap semiconductor.
2.8
Self-diffusion of the lattice ions and intrinsic point
defects
The diffusion properties of the lattice atoms and of the point defects are very important because they can give several information about the growth and degradation of
materials and about the nature of the defects.
The self-diffusion takes place without a difference in chemical potential, resulting
only in a random-walk of an ideally labeled particle. The self-diffusion coefficient D
is defined by the first Fick’s law in the limit of zero gradient concentration.
In non-stoichiometric oxides the self-diffusion of lattice atoms proceed usually by
the vacancy mechanism, consisting of successive jumps of atoms or ions from lattice
sites to neighbouring vacancies. Obviously the direction of diffusion of the vacancy
in this process is opposite to that of the atom and consequently the vacancy type
of diffusion can ben regarded as either the movement of atoms or the equivalent
movement of vacancies. However the self-diffusion coefficient of vacancies, DVA , is
not equal to that of atoms, DA , being related by
DA = DVA
[VA ]
[A]
(2.46)
This relation simply states that the self-diffusion coefficient of a lattice atom A is
equal to the diffusion coefficient of the corresponding vacancy VA times the probability
for an atom A that a given neighbouring site is vacant. The presence of the mole
fraction of vacancies introduces a dependence of the self-diffusion coefficient of atoms
on the oxygen partial pressure with which the oxide is in equilibrium. Therefore, the
oxygen partial pressure dependence enables to identify the defects responsible for
the migration of the studied species whereas the temperature dependence provides
16
Actually there are some papers [241, 769, 770] where the authors claim that cuprous oxide,
prepared by electrodeposition under certain conditions, shows an n-type conductivity. However they
used, for technical reasons, a technique which does not give a direct measure of the conductivity
type. Therefore further studies are necessary to conclude that those samples are really n-type.
40
2. Cu2 O: a review
information about the enthalpies and entropies of formation and migration of these
defects. On the other hand, if the vacancy concentration is small, so that the
vacancies do not interact among each other, their self-diffusion depends only on the
temperature, being only a pure random walk.
The diffusion can proceed also by the so called interstitial mechanism. The simple
interstitial mechanism consists in successive jumps of an atom from one interstitial
position to another one and it can occurs only for interstitial defects. A movement
of the lattice atoms as in the case of the vacancy mechanism, can be realized by
the so called interstitialcy mechanism: an atom of the lattice is displaced by its
interstitial neighbour to an interstitial position and after the jump the displacing
atom occupies the lattice site.
2.8.1
Experimental techniques
The best technique to measure a self-diffusion coefficient (D) of a lattice atoms or
ions is by the radioactive tracer diffusion (D∗ ) because the different mass number
acts like a label for the radioactive atoms.
In general D∗ = f D, where f is a correlation factors, called Haven coefficient,
which depends only on the type of sub-lattice where the diffusion takes place and on
the type of diffusion mechanism. If an independent measurement of D is available
together with the tracer diffusion data D∗ , it is possible to extract the value of
f . For example in Cu2 O, the self-diffusion of copper atoms takes place on a face
centered cubic sub-lattice by a vacancy diffusion mechanism and the corresponding
correlation factor can be calculated as f = 0.78146 [553, Sec. 2.2]. This value was
confirmed by the experiments of Peterson and Wiley [671].
If the self-diffusion coefficient of one atom is known, the self-diffusion coefficient
of the corresponding vacancy can be obtained by Eq. (2.46), provided that the
molar fraction of vacancies is determined by another independent measurement.
Self-diffusion of copper and oxygen in Cu2 O by radioactive tracers 64Cu and 18O
were performed by Moore [547–549], Perinet [668–670] and Peterson [671].
Another general technique to extract the self-diffusion coefficient of a lattice
atoms or ions is by measuring the chemical diffusion which occurs in the presence of
a concentration gradient or in general in a chemical potential gradient. In this case
the diffusion can be enhanced by the presence of an electric field generated by the
non-uniform distribution of charged defects.
The total flux of defects is the algebraic sum of the fluxes of the cationic (Jc )
and anionic (Ja ) defects:
J = Jc + Ja .
(2.47)
The chemical diffusion D̃ is defined relating the total flux with the the total
concentration gradient of the diffusing components, c, through the Fick’s first law:
J = D̃
∂c
.
∂x
(2.48)
The chemical diffusion characterizes the re-equilibration processes and the chemical diffusion coefficient is a mean diffusion coefficient of the two kinds of defects.
Let us consider a sample of Cu2 O in equilibrium with the external pO2 at the
temperature T . In this condition there will be both cationic and anionic defects.
If the external oxygen partial pressure is rapidly changed to a different value, the
2.8 Self-diffusion of the lattice ions and intrinsic point defects
41
sample will reach the new equilibrium after a certain time in which the defects will
diffuse inside the oxide. Obviously if the diffusion coefficient of the two species are
different enough the re-equilibration process will have two characteristic times, the
first one due to the fast defect and the second due to the slow defect.
From these consideration it is easy to understand that D̃ will be in general a
complicated function of the defect concentrations. Therefore it is useful only if a
relation with the self-diffusion coefficients can be found and only if a diffusing species
is dominant. In this case it can be shown [305, 553] that
D̃ = (1 + α)D
(2.49)
where D is the self-diffusion coefficient of the dominant defect species and α is the
ionization degree of the this defect.
The transient towards the equilibrium can be experimentally followed by thermogravimetric measurements or by conductivity of the sample. In Cu2 O the chemical
diffusion was measured by conductivity at high temperature [98, 513, 788] and by
microthermogravimetric reequilibration [282]. However the first method can be
applied only if the charged defect fraction (α) does not vary in the range of temperature studied. This is not the case of Cu2 O, and indeed the measurements based on
conductivity are inconsistent and should not be considered.
2.8.2
Experimental results
After this brief introduction we present the most reliable literature data on the
diffusion coefficients.
The self-diffusion coefficient of the copper ions, show a dependence on the oxygen
1/4
partial pressure as pO confirming the fact that the copper ions migrate through the
2
copper vacancies. Therefore the self-diffusion coefficient of copper ions and copper
vacancies are related by:
DCu = DVCu
[VCu ]
1/4
∝ pO
2
[Cu]
(2.50)
One of the best determination of the diffusion coefficient of the copper vacancies
is given by Grzesik, Migdalska and Mrowec [282]:
DVCu = 3 × 10−3 e−0.55 eV/(kT ) cm2 s−1
(2.51)
In their analysis they assume that the diffusion coefficient is the same for neutral
and charged vacancies. This assumption was also made by other authors and
it is reasonable considering that during the diffusion process the copper vacancy
continuously and quickly changes its charge state.
From these experimental data we can obtain the migration entropy of the copper
vacancy [630]. From a microscopic point of view the diffusion of an ion can be
written, according to Zener, as:
D = αd2 κν e∆Sm /k e−∆Hm /(kT )
(2.52)
where α is a geometrical coefficient, d is the jump distance, κ is the transmission
coefficient, i.e. the probability that an ion will jump to another position and will
42
2. Cu2 O: a review
not return to its original position, ∆Hm is the enthalpy of migration, ∆Sm is the
migration entropy and ν is the taken as the vibration frequency of a cation next to
a vacancy.
An estimation of ν, using a simple model where an ion of mass M moves in a
parabolic potential well, is:
s
2
∆Hm
ν=
(2.53)
πa0
M
where M is the copper atomic mass and a0 is the lattice constant.
Using α ≈ 1, κ ≈ 1 and the value of ∆Hm obtained from the experiment, we
obtain ν = 1.91 × 1012 s−1 and ∆Sm = 4.5 J/mol (4.7 × 10−5 eV/defect).
The self-diffusion coefficient of the oxygen ions was measured by a radioactive
tracer experiment by Perinet et al. [668]:
−1.55 eV/(kT )
DO = 2.11 × 10−4 p0.4
cm2 s−1
O2 [Torr] e
(2.54)
This dependence on the oxygen partial pressure as p0.4
O2 suggests that the diffusion
of the oxygen ions proceeds by an interstitial mechanism, in particular by singly
negatively charged oxygen interstitials. This does not implies that the main defect
on the oxygen lattice is the oxygen interstitial but only that the migration proceeds
by these defects. Other types of defects can be present, even at higher concentration
respect with the oxygen interstitial, but they must have a much smaller diffusion
coefficient.
Finally notice that the diffusion coefficient of the oxygen interstitials can not
be obtained by the previous experimental data because the concentration of these
defects are not known.
In Fig. 2.25 we present a comparison of the diffusion coefficients as a function
of temperature for the most important experiments in the literature. The relation
for the oxygen self-diffusion is scaled to an oxygen partial pressure of 10 Torr for
comparison.
2.9
Growth of Cu2 O
Synthetic crystals of Cu2 O have been prepared in several ways, mainly by oxidation
of copper in furnace [183,571,829], by electrodeposition [770] and by sputtering [674].
Natural crystals are available also. Further techniques, as hydrothermal growth [424],
floating zone [361], growth by melt (with and without a crucible) [90,423,482,738,845]
were sometimes used, not only to oxidize copper, but also to recrystallize and purify
the sample obtained from oxidation in furnace producing larger crystal. See also [738]
and references cited therein. In this work we will deal only with oxidation in furnace.
2.9.1
Oxidation process
The simplest way to obtain Cu2 O consists in oxidizing a copper sheet in an atmosphere containing oxygen. Obviously it necessary to choose the temperature and
the oxygen partial pressure of the oxidation in the region of stability of Cu2 O (see
Fig. 2.12). Experimentally it was found that copper, like almost all metals, oxidizes
in the presence of oxygen gas in the form of a uniform film of oxide which, except
for very thin films, if the temperature and oxygen partial pressure are constant,
2.9 Growth of Cu2 O
43
-5
4x10
1974_Mrowec
1999_Haugsrud
-5
2008_Grzesik
2
(cm /s)
3x10
-5
D
VCu
2x10
D
VCu
is indipendent of
oxygen partial pressure
-5
10
0.70
0.75
0.80
1000/T
0.85
-1
(K )
Figure 2.24. Experimental values of the diffusion coefficient of the copper vacancies in
Cu2 O. The data of Haugsrud and Mrowec are obtained by the analysis of oxidation
kinetics as explained in Sec. 2.9.1
1958_Moore
1980_Perinet
-9
p
O2
=10 Torr
2
(cm /s)
10
-10
D
O
10
D
O
depends on the
oxygen partial pressure
-11
10
0.65
as
p
0.4
O2
0.70
0.75
0.80
0.85
T
1000/
0.90
0.95
1.00
-1
(K )
Figure 2.25. Experimental values of the self-diffusion coefficient of oxygen in Cu2 O.
44
2. Cu2 O: a review
increases in thickness proportionally with the square root of the time (parabolic rate
constant), suggesting an underlying diffusion process.
Two regimes are usually observed. The first one is when copper is oxidized
at pressure below the CuO dissociation pressure (the line between the cupric and
cuprous oxide in the pressure-temperature phase diagram); in this case a single layer
of Cu2 O is formed over the copper. The second one is observed at oxygen pressure
above the dissociation pressure. In this case the oxidation is called double layer
oxidation because in the first moments a layer of CuO is formed over copper, but
subsequently a layer of Cu2 O is formed between copper and cupric oxide. These two
layers increase both with time. In this work we will deal with only with single layer
oxidation, that is the oxidation of copper in the stability region of Cu2 O.
The rate of growth of Cu2 O increases rapidly with increasing temperature, and
also depends on the oxygen pressure. At high oxygen pressure we can summarize
the experimental evidences in this way:
L=
1/4
q
kp = kp0 pO e−Ep /(kT )
2kp t
2
(2.55)
where L is the the oxide thickness with L = 0 at t = 0 and kp is called the parabolic
1/4
rate constant where p stands for parabolic. The pO dependence is explained by the
2
fact that the oxidation of copper proceeds by neutral copper vacancies.
The most recent data of Zhu et al. [950] give kp0 = 3.19 × 10−4 cm2 s−1 Torr−1/4
and Ep = 1.02 eV. Another set of data, by Matsumura et al. [526], deduced from
low temperature experiments (200 °C < T < 350 °C), gives instead a prefactor
kp0 = 2.14 × 10−4 cm2 s−1 Torr−1/4 and an activation energy Ep = 0.97 eV. Older
experiments, e.g. [548, 828], give instead much greater activation energies (1.5 eV
and 1.7 eV respectively) but this could be due to the presence of CuO at the surface
(see Fig. 2.26).
On the other hand at low oxygen pressure Haugsrud and Kofstad [303] found
1/7
that the parabolic rate constant has a dependence of pO on the oxygen partial
1/4
2
pressure which becomes the usual pO at high pressure. They concluded that at low
2
oxygen pressure the oxidation takes place, no more by neutral copper vacancies but
by charged oxygen vacancies.
The presented values mean that, for example, at 1040 ◦C and with a partial
oxygen pressure of 159 Torr (air), the time needed for a complete oxidation of a
copper foil of thickness 100 µm is t = L2 /2kp ≈ 17 min, where L ≈ 165 µm is the
final thickness of Cu2 O given by the initial thickness of copper multiplied by the
ratio Vm (Cu2 O)/(2Vm (Cu)) ≈ 1.65. This oxidation time is valid if the oxidation
proceeds only from one of the two faces of the copper foil. It is more common that
the oxidation takes place from both surfaces at the same time. In this case the
oxidation time has to be divided by 4 which means, in our example, an oxidation
time of about 4 min.
The theory of oxidation of metals was proposed by Wagner [869] on the assumption that, if the conduction of the resulting oxide is mainly electronic, the growth of
the oxide is controlled by lattice diffusion of ions, metal ions toward oxide/oxygen
interface or oxygen ions toward metal/oxide interface. The Wagner model assumes
the chemical and thermodynamic equilibrium at the interfaces.
In the case of Cu2 O, the oxidation takes place by copper ions diffusion toward
the oxide/oxygen interface. This was experimentally demonstrated by a typical
2.9 Growth of Cu2 O
45
-6
10
p(O
-7
10
-2
2
-8
) = 10
k
k
k
k
10
p
-9
10
p
p
-10
k
p
2
(cm /s)
10
p
atm
(Zhu 2004)
(Tomlinson 1977)
(Moore 1951)
(Matsumura 1996)
-11
10
-12
10
-13
10
-14
10
-15
10
-16
10
-17
10
0.0005
0.0010
0.0015
T
1/
0.0020
0.0025
0.0030
-1
(K )
Figure 2.26. Experimental values of the parabolic rate constant kp for an oxygen pressure
of 10−2 atm.
technique in which a thick foil of copper is oxidized horizontally with one or more
platinum wires lying on its surface. After the oxidation process, if the wires will
be found inside the oxide, the oxidation process has taken place by the diffusion of
metal ions toward the interface, whereas if the wires will be found on the surface
of the oxide, the oxidation process has taken place by oxygen diffusion toward the
metal. In Cu2 O the first case was found (see [718, 858] and [553, Sec. 2.4]).
The oxidation process was then explained as follows:
• the oxidation takes place at the Cu2 O/O2 interface between the oxygen in the
air and the copper ions, migrated from the copper through the Cu2 O to the
Cu2 O/O2 interface;
• the copper atoms migrate because there is an opposite diffusion of copper
vacancies from the surface to the metal. This diffusion is due to a gradient in
the concentration of copper vacancies between the Cu/Cu2 O interface (lower
concentration) and the Cu2 O/O2 interface (higher concentration). Equilibrium
concentration of copper vacancies depends on the oxygen pressure in contact
with oxide and temperature but it does not depend on oxide thickness. The
0
reaction occurring at the surface is 4 Cu + O2
2 Cu2 O + 4 VCu
;
• if a part of vacancies are ionized, the electrical neutrality is maintained by a
current of holes from the oxygen surface to the metal balancing the current of
negatively charged copper vacancies in the opposite direction;
Now we want to show a simple version of Wagner theory. If dL is the increase in
the oxide thickness after a time dt, the number of copper atoms in this infinitesimal
oxide layer can be written as
[Cu]SdL = Φ(Cu)Sdt
(2.56)
46
2. Cu2 O: a review
where Φ(Cu) is the copper flux toward the surface S and [Cu] is the copper atomic
density in the oxide; Therefore the growth rate of the oxide is given by
dL
Φ(Cu)
=
.
dt
[Cu]
(2.57)
The copper flux Φ(Cu) is equal to the flux of vacancies in the opposite direction
Φ(Cu) = −Φ(VCu ). The flux of copper vacancies is the sum of two terms, the first
is for neutral vacancies and is simply given by the gradient in the concentration of
neutral copper vacancies
0 ]
∂[VCu
.
(2.58)
Cu
∂x
The second term is given by the diffusion of single negatively charged copper
vacancies. They move under the influence of the gradient concentration, enhanced
by the electrical field. This flux is balanced by an equal and opposite flux of positive
holes. It is readily shown that
0
Φ(VCu
) = −DV 0
−
∂[VCu
]
.
(2.59)
Cu
∂x
0 ] + [V − ] and [V − ] =
Summing these two terms and using the fact that [VCu ] = [VCu
Cu
Cu
α[VCu ] where α is the degree of ionization of copper vacancies, assuming DV − =
Cu
DV 0 ≡ DVCu we obtain
−
Φ(VCu
) = −2DV −
Cu
∂[VCu ]
(2.60)
∂x
If we are in the steady state condition, the concentration gradient of defects will not
change in time and moreover, assuming a small concentration of defects and a local
equilibrium at the interfaces, the flux has to be divergenceless, giving rise to a linear
concentration gradient. This reads:
Φ(VCu ) = −(1 + α)DVCu
−
−
[VCu
](x, t) = [VCu
]metal/oxide +
−
∆[VCu
]
x
L
(2.61)
and
∂[VCu ]
∆[VCu ]
=
∂x
L
Using the previous equation we finally obtain
dL
kp
= ,
dt
L
kp = (1 + α)DVCu
(2.62)
∆[VCu ]
[Cu]
(2.63)
which have the mathematical form of the experimental results.17
The correctness of this theory for the oxidation of copper was demonstrated
by Wagner himself [869], then in 1946 by Bardeen, Brattain and Schokley [47] and
subsequently by several authors.
It is important to say that at lower temperatures and in the presence of small
grains the diffusion of inner copper toward the Cu2 O/oxygen interface takes place
partly through the grain boundaries, lowering the activation energy of the kp .
For further aspects of copper oxidation and a partial review of the literature
data see Zhu [950].
It is interesting to note that by definition Φ(VCu ) = −D̃ ∂[V∂xCu ] , where D̃ is the chemical diffusion
coefficient of the reaction, and equating this to our expression for Φ(VCu ) we find D̃ = (1 + α)DVCu
which is commonly found in literature.
17
2.10 Electrical properties at low and intermediate temperatures
47
Diffusion coefficients from the oxidation kinetic
We mention here another way to measure the self-diffusion coefficient of defects
in the oxides, and in particular in Cu2 O, which is related to the kinetic of the
oxidation process. Usually, as in Cu2 O, the crystal lattice disorder is limited to one
sub-lattice, therefore the oxidation proceeds thanks to the migration of the ions of
that sub-lattice. For this reason only this self-diffusion coefficient can be measured
in this way.
From the experimental measure of kp , if a relation between kp and the selfdiffusion coefficient can be found, is possible to extract the self-diffusion coefficient.
In general,
kp = D̃Nd
(2.64)
where Nd is the molar fraction of defect in the “active” sub-lattice at the phase
boundary where they are formed.
A first method, proposed by Rosenburg, is rather complicated but permit to
extract separately Nd and D̃. In Cu2 O it was applied by Haugsrud [305] and its
data are reported in Fig. 2.25.
Other simpler methods are based on a direct relationship between kp and D.
Among them the Fueki-Wagner, the Mrowec-Stokłosa and the most recent KrögerKofstad methods. The last one is the simplest but most restrictive in the experimental
condition [553].
Starting from the Wagner theory and assuming a linear concentration gradient
of defects in the oxide, Kröger and Kofstad showed that
kp = (1 + α)DCu = (1 + α)DVCu
[VCu ]
[Cu]
(2.65)
where α is the degree of ionization of defects. The assumption of a linear gradient
of defects means a chemical diffusion coefficient independent of concentration. This
requirement is fulfilled when the concentration of defects is small.
In Cu2 O the oxidation kinetics was used by several authors to determine the selfdiffusion of copper. The oxidation process was followed by thermogravimetry [555]
and by direct observation of the oxide thickness [343, 556, 828]. The data of [556] are
reported in Fig. 2.25.
2.10
Electrical properties at low and intermediate temperatures
The study of the electrical properties of Cu2 O on samples at low and intermediate
temperatures is another important tool to characterize the defects of this material.
For low and intermediate temperatures we mean a range of values where the time for
reaching the thermodynamic equilibrium is much longer than the time of measurement
and therefore the defect concentrations can be considered fixed.
We have anticipated before that the Cu2 O is a spontaneously p-type semiconductor, indeed the Hall effect and the hot probe experiments always show such
behavior.
In this section first we will discuss briefly the most important behaviors of the
dark conductivity and the Persistent Photo–Conductivity. Then we will list the
48
2. Cu2 O: a review
interpretations and models proposed in the literature to explain these experimental
results.
2.10.1
Conductivity
First of all we want to present the overall dependence of dark conductivity on the
temperature. This was measured on monocrystal by O’Keeffe [628]. In Fig. 2.27 the
measurements of O’Keeffe are reported. Even if a great dependence of the activation
energies on the preparation condition exists, we can distinguish three temperature
ranges, described by three different activation energies. At high temperature we have
the usual activation energy of 0.6 eV, at low temperature we have a value ranging
from 0.2 eV to 0.5 eV, while at intermediate temperature the activation energy is the
highest and it has a value of about 0.9 eV.
O’Keeffe explained these three activation energies in a simple way. The activation
energy at high temperature is considered, as usual, as half the formation enthalpy of
charged copper vacancies, given by the sum of the formation enthalpy of neutral
copper vacancies plus their ionization energy.
At low temperature, the formation of defects are frozen and the only remaining
activation energy is half of the ionization energy. This is probably wrong since the
material is compensated and therefore the activation energy is simply equal to the
acceptor ionization energy.
At intermediate temperature, the effective pressure felt by the bulk of cuprous
oxide is different from the external pressure because the stable compounds at those
temperatures is the cupric oxide. The effective pressure is therefore a function of the
temperature and follows the dissociation pressure of CuO. According to O’Keeffe,
this pressure variation increases the activation energy of the conductivity.
Anyway it is surprising that the sample can reach the equilibrium during the
measurement. Another possible explanation takes into account the formation and
dissociation of defect complexes.
A great amount of work was done on the conductivity of Cu2 O at low temperature.
In fact electric measurements are relatively simple, can give information on the
concentration of defects and on the energy levels introduced by them and moreover
is an important aspect to take into account to develop a device.
However this kind of measurements depend strongly on the preparation conditions
of the sample because, we remember, they are performed out of the thermodynamic
equilibrium.
Another problem is the presence of the so called Persistent PhotoConductivity
(PPC), also called Photo Memory (PM) effect. If we illuminate a sample of Cu2 O
for few minutes, after the illumination it remains in a highly conductive state (about
an order of magnitude), which persists for several days at room temperature. To
return quickly to the electronic equilibrated state the sample has to be heated at
high temperature (e.g. 100 ◦C) for some minutes. We will discuss in detail the PPC
in the next section. The measures of this section are intended to be performed on
equilibrated samples.
2.10 Electrical properties at low and intermediate temperatures
49
2
10
1
10
E
Cu2O single crystal
= 0.6 eV
Equilibrated at
0
a:
10
b:
E
-1
10
p(O ) = 1.4 Torr
p(O ) < 10 Torr
2
-5
2
= 0.9 eV
-1
cm )
-2
10
-3
(
-1
10
-4
10
-5
a
10
-6
10
0.2 eV <
E
< 0.5 eV
-7
10
b
-8
10
0.5
1.0
1.5
2.0
1000/
2.5
T
3.0
3.5
-1
(K )
4.0
Figure 2.27. Conductivity of a
Cu2 O crystal as a function of
temperature [628]. The measurements were performed
equilibrating the crystal at
1000 ◦C with an oxygen partial pressure of 1.4 Torr and
< 10−5 Torr. Three activated
region are clearly visible.
Compensation
One of the milestone in the analysis of the conductivity is the work of Brattain
in the 1951 [84]. Analyzing the conductivity between 100 K and 400 K and the
Capacitance-Voltage (CV) experiments on Schottky junctions based on a substrate
of Cu2 O, he found that the cuprous oxide is a compensated material, i.e. both
acceptors and donors are present.
Now we will see how this result was achieved. Measuring the hole concentration
as a function of temperature usually one obtains an exponential trend at low
temperature, with an activation energy of about 0.3 eV, and a saturation value, at
temperature of about 400 K, of the order of 1014 − 1015 cm−3 . See Fig. 2.28.
Brattain tried to analyze this curve first with a model which assumes only the
presence of an acceptor level and then by a model which assumes the presence of an
acceptor level and one or more deep donor levels. In the former case the exponential
low temperature regime is described by (see Eq. B.9)
s
p=
Nv NA −EA /(2kT )
e
gA
and the saturation value corresponds to the total number of acceptors NA . In the
latter case, where both acceptors and donors are present, the low temperature region
is described by (see Eq. B.10)
Nv
p=
gA
NA
−1
ND
e−EA /(kT )
50
2. Cu2 O: a review
Figure 2.28. Hole carrier concentration as a function of reciprocal absolute temperature
for two different samples of Cu2 O. The plot is for the original paper of Brattain [84].
2.10 Electrical properties at low and intermediate temperatures
51
Figure 2.29. Simple band scheme for Cu2 O
after the work of Brattain. Notice the
presence of both acceptor levels and donor
levels, with the formers at about 0.3 eV
from valence band and the latters enough
deep so that the donors are all ionized.
and the saturation value is equal to NA − ND .
Using the first model to fit the exponential part of the data gives a concentration
of acceptors of the order of 1019 cm−3 . This value should be equal to the saturation
value, but it is not, in fact the saturation value is four order of magnitude lower.
Instead, the second model, which assumes a compensated material, can describe
better the experimental data. The exponential fit gives a compensation ratio NA /ND
of the order of unity and using also the saturation value, we obtain a similar
concentration, about 1014 − 1015 cm−3 , for both acceptors and donors.
According to the previous discussion, Cu2 O is a compensated material, with an
acceptor level situated about 0.3 eV above the valence band and a deep donor level
with a concentration slightly lower than that of the acceptors. See Fig. 2.29 .
However we have to note that the saturation value is not clearly observed, neither
in the following works on conductivity by other authors. Usually, after a beginning
of saturation the conductivity rises rapidly and this was observed by Brattain itself.
This can be due to the instability of the defects at those temperatures.
For this reason, to check better the hypothesis of the compensation, Brattain performed several Capacitance-Voltage measurements on Cu/Cu2 O Schottky junction.
It is known that, from the slope of C −2 versus the bias voltage at high reverse bias,
one can deduce the net concentration of charged defects, in our case NA − ND . The
measurements of Brattain give a value compatible with the compensation hypothesis.
This confirm the conclusion obtained from conductivity analysis that Cu2 O is a
compensated material.
Unfortunately also the C-V measurements are not straightforward to be analyzed,
because, as we will see in the following, several metastability effects are present.
However we are reasonably sure that a concentration of copper vacancies of the order
of 1019 cm−3 is impossible because this high concentration of defects would give a
strong optical absorption signal which has never been observed. Therefore today
the idea of Brattain that Cu2 O is a compensated material is commonly accepted.
Actually this is only a starting point for more complicated model to describe
all the properties of Cu2 O. In fact, apart from general characteristics, a complete
model still lacks.
52
2. Cu2 O: a review
Figure 2.30. Conductivity measured at 286 K (right) and activation energy of the conductivity (left) as a function of the oxygen partial pressure used for the Cu2 O re-equilibration.
Obviously all experimental points have been obtained quenching the samples in the same
way.
Dark conductivity
A lot of measurements of dark conductivity were performed on Cu2 O but a great
scattering of the data is observed in literature. This was due to the influence of the
preparation conditions on the properties of Cu2 O and is one of the problem for the
correct interpretation of conductivity.
An attempt to understand the relation between the conductivity of samples
of Cu2 O and their growth, annealing and cooling conditions was performed by
Bloem [70]. He made also an analysis of the possible point defects present at low
temperature, suggesting the identification of the acceptors with the copper vacancies
and the donors with the oxygen vacancies.
From a very simple point of view, one can imagine that the higher the oxygen
pressure during the oxidation, the higher the concentration of copper vacancies at the
growth temperature and then at room temperature. According to this interpretation
a monotonic relation between the conductivity at room temperature and the partial
oxygen pressure at which the Cu2 O sample was equilibrated, is expected. At most,
assuming a re-equilibration of the sample even at lower temperature, a constant
value of conductivity versus the oxygen pressure is expected.
Bloem instead performed an experiment of this kind re-equilibrating a set of
samples at 960 ◦C at different oxygen pressure and then quenching them in water
in less than 5 seconds. In the left part of Fig. 2.30 it can be seen that the curve
of conductivity vs the oxygen partial pressure has a non-monotonic behaviour. He
observed, not only a non-monotonic variation of the absolute value of conductivity,
but also a non-monotonic variation of its activation energy, ranging between 0.22 eV
to 0.47 eV, as can be seen in the right part of Fig. 2.30.
Bloem observed also a strong dependence of conductivity on the cooling rate
and suggested that, during the cooling of the samples, the point defects cannot be
reduced quickly because the re-equilibration with the outside atmosphere is frozen
and therefore the material reacts to this supersaturation condition associating the
defects. The dependence on the cooling rate is therefore explained but not the
dependence on the oxygen pressure.
The idea of defect complexes, formed by two or three point defects, or clusters of
2.10 Electrical properties at low and intermediate temperatures
53
defects is very common and it is proven to be thermodynamic favourable in several
materials [553, Sec. 1.4]. The idea of complexes association and dissociation was
used, as we will show in the next section, also by Kužel et al. [448, 661] to explain
other properties of Cu2 O, especially PPC.
Another detailed study of Cu2 O was done by a group of researcher of the
University of Strasbourg.
After a first experiment in [956], more detailed and controlled experiments were
reported in [809, 954] and summarized in [810].
The measurements were performed in high vacuum in the dark using a guard ring
technique on monocrystals of Cu2 O obtained oxidizing a foil of copper at 1050 ◦C
with a partial oxygen pressure of 100 Torr for 48 h. The samples were then cooled to
room temperature in different ways. Quenched samples have lower bulk resistivity
(106 < ρ < 108 W cm) and lower activation energies (0.4 < E < 0.6 eV) than slowly
cooled specimens (108 < ρ < 1011 W cm and 0.6 < E < 0.8 eV). For quenched
crystals the oxygen partial pressure has no effect. On the other hand for the slowly
cooled samples the bulk resistivity and the activation energy seem to increase with
decreasing pressure.
After this first analysis they made systematic measurements of resistivity varying
the annealing temperature only. The samples, previously prepared by oxidation
and quenching, were annealed at different temperatures, from 350 ◦C to 850 ◦C, for
about 16 hours in vacuum and they were cooled every time in the same way, with
a cooling rate of 80 K/min. The resistivity after the annealing was higher for the
higher temperatures. In Fig. 2.31 we report the figure of the original paper. The
resistivity varies from 104 W cm to 1013 W cm with activation energies ranging from
0.1 eV to 1 eV respectively. Notice that an activation energy of 1 eV corresponds to
the half of the band gap, meaning that the material is perfectly compensated or
nearly free of defects.
Concentration of point defects after the cooling process
It is generally supposed that the properties of non-intentionally doped Cu2 O are
controlled by the concentration of intrinsic defects. This hypothesis is not completely
justified since the contamination level of the starting copper used in most experiments,
and also in our own, is of the order of 10 ppm. This issue will be discussed in details
in the chapter 6.
Supposing anyway a negligible role of the impurities, let us consider what level
of intrinsic defects can be expected after the oxidation or annealing process.
Using the diffusion coefficient of the copper vacancies given in (2.51), it is easy to
see that at 450 ◦C the copper vacancies can diffuse up to 70 µm in two minutes only!
At that temperature the equilibrium concentration value of the copper vacancies
is lower than that calculated at the CuO dissociation pressure and from the data
shown in Fig. 2.15 we obtain [VCu ] ≈ 5 × 1016 cm−3 .
Therefore we can say that if the cooling is not too fast, in any sample at room
temperature we will find a copper vacancy concentration of this order of magnitude.
This is not true in quenched samples and indeed the conductivity measurements
on samples quenched from high temperature to room temperature, found a higher
conductivity respect with non-quenched sample, but not high as expected. This
suggests that the copper vacancies, even if they are subjected to a very rapid
54
2. Cu2 O: a review
quenching, can aggregate forming di-vacancies or small cluster of CuO [631].
Unfortunately we do not know the diffusion coefficient of the other intrinsic
defects. For this reason, if the diffusion coefficient of one or more of the other defects
is much smaller than that of the copper vacancies, it is possible that during the
oxidation process only a partial re-equilibration of the defects occurs.
For example, let us suppose that during one of the oxidation step a defect with
donor behaviour is formed, even at a relatively low concentration, e.g. 1014 cm−3 . If
the subsequent steps are not able to equilibrate these defects, at room temperature
their concentration will be about the same of the high temperature whereas the
concentration of copper vacancies will be smaller than that at high temperature,
because they can equilibrate quite fast. This could explain the compensation of the
material and the difficulties to obtain a Cu2 O with a high conductivity.
A proof that the intrinsic donors can be the dominant compensating defects
comes from the experiments of Tapiero et alia. They started with a quenched single
crystal oxidized in air at high temperature which was then annealed at different
increasing temperature, between 350 ◦C and 850 ◦C in vacuum (p < 10−5 Torr).
After 16 hours of annealing at each chosen temperature, the sample was cooled down
slowly (80 ◦C/h) to room temperature. After each annealing, they measured the
conductivity of the sample as a function of temperature, obtaining the data reported
in Fig. 2.31.
We can explain the general trend of their data considering that the original sample
has a certain amount of copper vacancies and a similar but smaller concentration of
donors (let say oxygen vacancies) because it is oxidized at high oxygen pressure and
then quenched.
After each annealing of the Tapiero’s samples, we can imagine that the copper
vacancies concentration are almost always the same because the sample is cooled
to room temperature very slowly and therefore, as we have seen before, the copper
vacancies have time to reach the equilibrium.
The data show that the annealings at 400 ◦C and 500 ◦C give a lower resistivity.
This can be explained considering that even if the partial oxygen pressure is very
low, the system is in the stability region of CuO. Therefore the initial concentration
of oxygen vacancies will be reduced by the annealing at these low temperatures.
However when the temperature of the annealing is raised, the system goes in
the stability region of Cu2 O and at the highest annealing temperatures, it falls in
the stability region of copper. The annealings in these regions increase greatly the
concentration of oxygen vacancies, which then are not eliminated during the cooling
of the sample because their diffusion coefficient is much smaller than that of the
copper vacancies.
Therefore the higher is the temperature of the annealing, the higher is the
resistivity of the sample, because the material is much more compensated.
Variation of activation energy
Tapiero et al. also studied the dependence of activation energy of conductivity as a
function of the resistivity taken at a conventional temperature.
Plotting the activation energy as a function of the resistivity of the sample, taken
conventionally at 20 ◦C, they obtained the Fig. 2.32. It is clear that a linear relation
between the activation energy and the logarithm of the annealing temperature, and
2.10 Electrical properties at low and intermediate temperatures
55
Figure 2.31. Conductivity as a
function of the temperature
for a sample annealed in vacuum at several different temperatures [810].
Figure 2.32. Activation energies of the curve reported in
Fig. 2.31 as a function of the
resistivity of the sample at
20 ◦C [810]. The linear relation is a typical example of the
Meyer-Neldel rule.
56
2. Cu2 O: a review
hence the resistivity of the samples, exists. This characteristic of cuprous oxide could
be a manifestation of the so called Meyer-Neldel Rule [534]. This law, observed in
several systems, says that the activation energy (E) and the pre-exponential factor
(ρ0 ) of an Arrhenius law is related. The pre-exponential factor is empirically given
by:
ρ0 = ρ00 eE/EMN
(2.66)
The energy EMN is a characteristic energy of the process and corresponds to a
characteristic temperature TMN = EMN /k where all the curves intersect. Today no
clear explanation of this experimental law still exists.
However it is important to note that this variation of the activation energy is
not observed on samples with a resistivity of 104 W cm or less.
An explanation of these experiments was given by the same authors some years
later [604]. They proposed an electronic model with a band of acceptor levels in
the gap instead of the usual single level. For example, such distribution could be
due to the spreading of a single electronic level if the surroundings is not the same
for the all defects. According to this model, the subsequent annealings in vacuum
would change the compensation ratio of the material, increasing easily the Fermi
level inside this band. This change is attributed to an internal rearrangement of
the defects in the crystal. However to fit the experimental data they are forced to
introduce a dependence of the hole mobility on the conductivity activation energy.
2.10.2
Persistent PhotoConductivity
The cuprous oxide exhibits a very interesting phenomenon which was called Photo
Memory (PM) effect until the 1970’s while now it is called Persistent PhotoConductivity (PPC). It consists in an increase of the conductivity which persist for several
days at room temperature, even if the sample is kept in the dark. This metastable
state is generated by illuminating the sample for some minutes. The PPC discharge
can be easily speed up by the increase of the temperature, for example at 120 ◦C
the decay time of PPC is few seconds.
This phenomenon has been observed in similar p–type semiconductors, like CIS,
CGS and CIGS (chalcopyrites) [187] however in this case the PPC discharge is much
faster than in the Cu2 O, for example at room temperature the decay time constant
for CIGS is about 1 hour while for Cu2 O is nearly infinite. The explanation of this
phenomenon in chalcopyrites is based on a complicated model [456,457]. In summary,
during the illumination, the minority carriers are captured by deep defects (actually
complexes) which in turn go in a new relaxed state. After the light is switched off
the electron re-emission or hole capture from these defects is very slow because they
are prohibited by an high energy barrier formed after the defect relaxation.
The PPC in Cu2 O was first observed in 1955 by Klier et al. [427]. Experiments
on monocrystalline samples proved that PPC is not related with the grain boundaries [448] and a study of the influence of the pre-illumination on the work function
of Cu2 O proved that the PPC is a bulk effect [449]. Finally Zouaghi [967] showed
that the PPC results from an increase of the carriers and not from an increase of
the mobility, which remains almost unchanged.
Currently, two interpretation exists. The older interpretation was proposed by
Kužel et al. [427, 445, 448, 661]. It explains the PPC by a mechanism of association
and dissociation of intrinsic point defects which occurs respectively during the
2.10 Electrical properties at low and intermediate temperatures
57
heating and pre-illumination of the sample. More recently the PPC was explained
by an electronic mechanism in which the electrons are trapped in deep centers and
then re-emitted slowly with time [811, 953].
The model of the complexes
The first idea was the formation of neutral complexes between two neutral copper
vacancies [427, 445, 661]. The authors supposes that the acceptors in the cuprous
oxide are both single copper vacancies and complexes formed by two copper vacancies.
At the equilibrium in the dark a large number of copper vacancies are coupled to
form these complexes. Illuminating the sample the complexes dissociate and each
complexes gives rise to two copper vacancies, therefore increasing the number of
acceptors to, at most, twice the original number. Heating the sample in the dark, the
copper vacancies can readily migrate in the material forming again the complexes
and reducing the number of acceptors. This can be described by the following
relation:
heat
0
0
V Cu
+ V Cu
(V Cu −V Cu ) 0
(2.67)
illumination
A more detailed analysis [448] of the pre-exponential factors of conductivity
under equilibrated and pre-illuminated condition for various samples, suggests the
presence of the following additional reactions to explain the PPC:
+
V−
Cu + V O
0
+
V Cu
+ (V −
Cu −V O )
heat
illumination
heat
illumination
+
(V −
Cu −V O )
(2.68)
0
+
(V Cu
−(V −
Cu −V O ))
(2.69)
In the first case the acceptors are the copper vacancies and the donors are the oxygen
vacancies. Their association makes disappear one donor and one acceptor, leaving
unaltered the difference NA − ND . The last reaction is in analogy with the F centers
in alkali halides. Another possibility is the formation of acceptors composed by more
than only two copper vacancies.
All these three types of complexes were proposed also by Bloem [70] to explain
the variation of the electrical properties with oxygen pressure.
The electronic models
Several purely electronic model were proposed to explain the PPC and the other
properties of Cu2 O.
A simpler approach to explain the PPC was introduced by Schick and Trivich in
1972 [737]. They assumes a trapping mechanism as the explanation of the PPC. In
this case the illumination induces a transfer of electrons into the compensating donor
states, reducing the compensation ratio and therefore increasing the conductivity.
This increase in conductivity persists also after the end of the illumination (PPC)
and if the donor level is deep enough, the trapped electrons will be emitted after
very long time unless the sample is heated.
To characterized the levels involved in this model, they performed a typical
experiments called Thermally Stimulated Conductivity (TSC). In this method the
sample is illuminated at low temperatures and then, in the dark, the temperature
is increased at constant rate with time, while the conductivity is measured. The
58
2. Cu2 O: a review
Figure 2.33. A typical TSC experiments [737], where the excess conductivity is plotted
against the temperature. The indicated numbers are the temperature rate at which
the sample is heated. Two peaks, one at 25 ◦C and the other one at 100 ◦C are clearly
visible.
difference between the conductivity of a pre-illuminated sample as a function of
temperature and the normal dark conductivity as a function of temperature is the
TSC, σ ∗ . In general some peaks emerge from this measurements. When these peaks
are due to the thermal emission of the electron by the traps, the temperature of
the TSC maximum gives information regarding the trap depth (the position of the
energy level in the gap), the area under the TSC curve gives information about the
density of traps and in some cases the detailed shape of the TSC curve may give
information about the capture cross section of the trap.
In the case of Cu2 O, two peaks appears at temperatures of about 25 ◦C and
100 ◦C as showed in Fig. 2.33. Their model consists of two trap donors, whose
energies are extrapolated by the peaks of the TSC, and one acceptor. The two
donor levels are identified with the two charge states of the oxygen vacancy and the
acceptor level is identified, as usual, with the copper vacancy. Analyzing the TSC as
a function of the heating rate, they found that the donor levels are located 1.03 eV
and 1.34 eV below the conduction band.
Schick and Trivich measured also the intensity of the excess conductivity as a
function of the photon energy, see Fig. 2.34. They interpreted the energy at which
the excess conductivity is zero as the the minimum energy to fill the trap which
corresponds to the energy difference between the trap level and the top of the valence
band. Unfortunately the extrapolated values, 1.1 eV and 1.3 eV, are inconsistent
2.10 Electrical properties at low and intermediate temperatures
59
Figure 2.34. Excess conductivity as a function of the photon energy [737]. The left scale
applies to the left curve and is in units of 10−8 W−1 . The right scale applies to the right
curve and is in units of 10−9 W−1 . The curves correspond to the two peaks observed in
TSC.
with those found in TSC experiments. In fact according to this model, the sum
of the energies obtained from TSC peaks analysis with the corresponding energies
found by the excess conductivity as a function of the photon energy, should be equal
to the band gap. Instead the sum of these two terms is 2.44 eV for one trap level
and 2.33 eV for the other level.
Another important feature to study is the decay of the PPC as a function of
temperature. Pollack and Trivich [677] found that the decay curve of the PPC can
be described by a sum of two exponentials, which is in agreement with the fact that
there are two electron traps (the two peaks of the TSC). In Fig. 2.35 the two time
constants from 400 K to 500 K are reported. They show an activated behaviour, in
agreement with a PPC decay given by emission of electrons from the traps. Studying
the activation energies, Pollack and Trivich conclude that the deeper trap is charged
(σn ∝ T −3 ) and located at 1.33 eV from the conduction band while the second trap
is neutral (σn ≈ 8 × 10−16 cm2 ) and is located at 1.04 eV from the conduction band.
In the same years, Tapiero et al. [811] made also TSC de-excitation experiments,
observing four peaks: two connected with the deep traps, at 0 ◦C and 75 ◦C (slightly
lower than those found by Schick and Trivich), and two connected with the shallow
traps, at −110 ◦C and −80 ◦C. Moreover it was found that the deepest traps cannot
be filled by light at temperature below 100 ◦C. This was explained by a strong
decrease of their electron capture cross sections with decreasing temperature.
Finally the Strasbourg group, collecting all information acquired in these years
proposed a final electronic model [811, 953] to explain PPC and the other features of
Cu2 O. They proposed a model with four donor levels, two shallow and two deep,
and one band of acceptors extending from 0.2 eV from valence band to the middle of
the band gap, see Fig. 2.36. In this model is present also a band of recombination
centers, extending from 0.45 eV to 0.65 eV from valence band. It was introduced
to explain the PPC spectrum and both the value and the time constant of the
photo-conductivity saturation. The authors do not discuss the nature of these levels.
However we have to mention the fact that all the models based on the trap
60
2. Cu2 O: a review
Figure 2.35. The PPC decay constants as a function of temperature. The points are
obtained by the fit of the decay curve of PPC as a function of time at constant
temperature. The fit was performed by a sum of two exponentials. According to Pollack
and Trivich [677], the two different time constants correspond to the electron de-trapping
process of the two electronic traps observed in TSC experiments (Fig. 2.33).
2.11 Other defect characterization techniques
61
Figure 2.36. Electronic model proposed by the Strasbourg group.
Four traps are present, two deep
and two shallow. The acceptor
levels are distributed in a band
into which lies another smaller
band of recombination centers.
mechanism have a problem. In fact in order to act as a trap the hole capture rate,
Rcp = νth σp p, has to be much smaller than the electron emission rate, Ren = en . It
can be readily seen that, using the PPC decay time constants found above, the
capture cross section for holes has to be σp 10−23 cm2 , a value very uncommon
compared to typical cross section of charged defects (10−15 cm2 ).
2.11
Other defect characterization techniques
Several other techniques have been used to characterize the intrinsic defects of Cu2 O.
We will discuss briefly some of them.
2.11.1
Photoluminescence
The photoluminescence is a technique much used to characterize the defects of a
semiconductor but in Cu2 O it was mainly used to study the exciton spectra. Anyway,
apart from those of excitons, three peaks emerged in the emission spectra at 720 nm,
820 nm and 910 nm or, in energy units, respectively 1.72 eV, 1.51 eV and 1.36 eV, see
Fig. 2.37. The bands are asymmetric, especially at low temperature and have a full
width at half height of 0.10 − 0.15 eV.
The origin of these bands is not clear. The band at 910 nm is always present and
clearly visible and therefore it was attributed to the copper vacancy level. Notice
that near the band at 910 nm another weaker band appears at 1010 nm.
Much more difficult is the situation for the other two bands. Bloem [70, 71]
observed them in samples annealed under low oxygen partial pressure and therefore
attributed them to the luminescence from the two charge states of the oxygen
vacancy (VO+ and VO++ ). This interpretation was reinforced by the fact that the in
all his samples the ratio between the intensities of these two peaks seems to be the
same.
However more recent experiments have shown that the relative intensities of
these bands are not correlated as claimed by Bloem. A different dependence
62
2. Cu2 O: a review
Figure 2.37. Photoluminescence emission spectrum on two samples of Cu2 O. After [71].
on the illumination intensity was observed [827], the 820 nm peak dominates at
low illumination intensity whereas the 720 nm peak dominates at high intensity.
Moreover the ratio between the intensities of these two bands varies with the sample
preparation [359].
Excitation spectra of photoluminescence [242, 359] were interpreted with the
above relations between the bands and the intrinsic defects but the authors claim
that the luminescence of the 720 nm and 910 nm bands is not due to the direct
recombination of carriers into the defect levels but instead to the recombination of
relaxed excitons.
The photo-luminescence decay of the discussed peaks was studied in few papers
and up to now, because of the unknown underlying recombination processes, no
clear explanation of the observed time constants exist [301, 818, 819].
2.11.2
DLTS
A good way to obtain information about the defect concentration and energy levels
is the Deep Level Transient Spectroscopy (DLTS).
A first attempt to use the DLTS on Cu2 O was done by Papadimitriou [650],
especially to characterize the acceptor band proposed by Noguet et al. [604]. Actually
they used another technique together with DLTS, the Space Charge Limited Current
(SCLC) on Schottky junctions. The results of these two techniques are in agreement
and the measured concentration of traps is of the order of 1011 − 1012 cm−3 . Unfortunately this value is very low compared to the expected value of 1014 − 1015 cm−3 .
For this reason he performed more accurate measurements [649], on Cu/Cu2 O
Schottky junctions made with Cu2 O single crystals. The DLTS measurement were
performed on a rather limited temperature range (200 − 300 K). A single hole trap
signal was observed. Using the standard DLTS analysis, Papadimitriou found for
this hole trap an energy level at 0.475 eV above the top of the valence band and an
hole capture cross section of 9.52 × 10−14 cm2 . Papadimitriou attributes these levels
to an acceptor like center which seems to be justified by the high value of the cross
2.11 Other defect characterization techniques
63
section typical of an attractive center. He does not identify these hole traps with the
acceptors which dope the material since the trap density estimated by the DLTS
is very low and is about 1011 cm−3 as found in his previous work. Maybe exists a
relation between this measurement and the band of recombination centers proposed
by the Strasbourg group.
Moreover the DLTS transients are non-exponential and therefore Papadimitriou
performed also a different data analysis based on a Gaussian distribution of the trap
energy levels. In this way both the energy level and the capture cross section are
reduced (Et = 0.41 eV, σp = 1.93 × 1015 cm2 ).
Recent measurements of DLTS were performed by Paul et al. [663, 664] on
heterojunctions ZnO/i-ZnO/Cu2 O made by reactive magnetron sputtering. In these
thin film devices the hole concentration is significantly higher (about 1017 cm−3 )
than that of the oxidized samples (about 1013 cm−3 ). The DLTS measurement were
performed on a larger temperature range (100 − 350 K) than the measurements of
Papadimitriou. In this way two hole traps were identified which are called A and
B. Trap B has an emission activation energy of 0.45 eV and can be identified as
the same trap observed by Papadimitriou. Trap A has a lower activation energy of
0.25 eV. Both the traps have concentration above 1016 cm−3 which are similar to
that of the free holes. Analyzing the dependence of their intensities as a function
of the partial oxygen pressure during Cu2 O growth and the temperature of the
substrate, the authors propose that trap B is due to the single copper vacancy while
A is due to copper di-vacancy.
2.11.3
Magnetic Properties and EPR
Pure Cu2 O should be a diamagnetic material because neither Cu+1 , which has full
d–shell, nor O2− are magnetic ions [784]. In experiments, Cu2 O is found to show
diamagnetic (negative susceptibility) or paramagnetic (small positive susceptibility)
properties, with a room temperature mass susceptibility of −2.5 × 10−9 m3 /kg <
χmass < 3.8 × 10−9 m3 /kg.18 However O’Keeffe [631] showed that the positive value,
which was obtained in one paper only [667], is most likely due to the presence of
CuO in the Cu2 O.
Moreover O’Keeffe found that the mass susceptibility as a function of temperature
is nearly constant for a stoichiometric sample of Cu2 O while for a cation deficient
sample it follows the law
A
χm = χm,0 +
(2.70)
T
where χm,0 is the mass susceptibility of the stoichiometric Cu2 O, A is a positive
constant and T is the absolute temperature in kelvin. For these reasons the most
reliable measurement of χm is that of the perfectly stoichiometric Cu2 O which is
χm = χm,0 = −1.95 × 10−9 m3 /kg [631].
Assuming that the paramagnetic component is due to the presence of the cation
vacancies, it can be shown that the experimental coefficient A would give a copper
vacancy concentration about 30 times lower than that expected value from the
quenched samples from high temperature.
18
Notice that in order to convert the values of mass magnetic susceptibility from the emu-cgs
unit system (cm3 /g or emu/(g Oe)) to the International System of Units (m3 /kg), they have to be
multiplied by 4π10−3 .
64
2. Cu2 O: a review
Further details can be obtained by the analysis of the Electron Paramagnetic
Resonance (EPR) experiments. Even if this technique is perfect to analyze the
properties of the defects of a semiconductor, especially vacancies, few papers was
devoted to study of EPR in Cu2 O [237, 250, 627, 814].
O’Keeffe [627] did not find any EPR signal and explained this result with a
pairing of the copper vacancies, which, in paired form, are not EPR active. On
the other hand, Goltzené [250] detected EPR signals whose hyperfine structure
was incompatible with any simple intrinsic defect of Cu2 O. The signals seems to
originate from a I = 7/2 nuclear spin and they can be attributed to cobalt impurities
present in his samples at a concentration of about 1017 cm−3 .
Chapter 3
Sample preparation and
experimental setup
In this chapter we will show the preparation method of our undoped cuprous oxide
samples. Then we will describe the device preparation based on Cu2 O and finally
we will present the experimental setup used to characterize these devices.
3.1
Undoped Cu2 O preparation
The simplest method to obtain cuprous oxide in laboratory is by the oxidation of
copper in furnace. Now we will describe the used copper, then the experimental
apparatus by which copper was oxidized and finally the oxidation procedure.
3.1.1
Starting material: copper
In order to prepare our Cu2 O substrates by oxidation in furnace we have used
and tested several types of commercial copper. They are identified by the name
alpha1, alpha2, alpha250 (bought from Alpha–Aesar) and OFE (a gift of KME
Italia). Their thickness and the impurity analysis provided by manufacturers are
reported in Tab. 3.1. However, all the samples analyzed in this thesis, except those
used to check the impurity content influence on the cuprous oxide properties, are
made with the 99.999% pure copper alpha1 or alpha2.
The copper foil is cut in rectangular sheets of about 5 cm × 3 cm. Two little holes
along the long side are made in order to hang the sheet to a platinum cage, thus
greatly reducing the contamination or reaction with the tube furnace material [968].
Just before the oxidation, the copper sheet are cleaned in acetone to remove
non-polar substances and then etched in nitric acid 4 M for about ten seconds. After
each step the copper is rinsed in deionized water and dried with nitrogen.
It has to be noted that the copper purity influences several properties of the
resulting Cu2 O. While it has been demonstrated the influence on oxidation kinetics [950] and inner porous layer (voids layer) formation [948], the effect of the
impurities on the electrical properties [84, 635, 842] is still unclear, even if it is quite
reasonable that they play a fundamental role. Indeed 1 ppm of impurities corresponds
to a concentration of about 1016 cm−3 which is much greater than the typical hole
concentration, about 1014 cm−3 found in Cu2 O. The effect on the grain size has
65
66
3. Sample preparation and experimental setup
Figure 3.1. Schematic experimental setup for the oxidation of copper.
been observed by us and a brief discussion is reported in the section dedicated to
the oxidation procedure. Instead, the purity does not play an important role in the
microstructural defects [738].
3.1.2
Oxidation Setup
The oxidation setup is composed by a furnace with horizontal alumina tube and
a computer controlled gas system to regulate the N2 /O2 gas mixture composition
fluxed inside the furnace. In Fig. 3.1 we show a scheme of our setup.
The furnace is equipped with two alumina shields at its extrema which make
the flux homogeneous and reduce the heat dispersion. The total pressure inside the
tube is 1 atm because the furnace outlet is open.
During the oxidation two parameters have to be controlled: oxygen partial
pressure and temperature. Mixing the fluxes coming from two cylinders, one
containing a mixture 80%/20% of nitrogen and oxygen and the other one containing
only nitrogen, we can regulate the oxygen partial pressure. However the partial
pressure near the surface of Cu2 O could be different from the nominal pressure
because of the copper and oxygen reaction. Indeed a gradient of oxygen always
exists near the growing oxide because some oxygen is adsorbed by the oxidation
process.
During our typical oxidation process, the total flux is 5 L/min which gives a
velocity of the gas inside the tube of about 52.5 cm/min. The oxidation phase is 90
min long with an oxygen flux of 18 sccm corresponding to 8.0 × 10−4 mol/min.
However, since our tube has an radius of 5.5 cm, one can ask what fraction of
the flowing oxygen can be actually absorbed by the sample. The effective oxygen is
the one flowing within a radial distance from the sample for which the diffusion time
is lower than the time needed for the flux to cover the sample length. Our samples
are 5 cm long and therefore this last time is about 5 seconds.
Since the diffusion coefficient of O2 in N2 at 910 ◦C is of the order of 1 cm2 /s, in
5 seconds we can collect the oxygen flowing within a distance of about 2 cm from
the sample surface.
3.1 Undoped Cu2 O preparation
67
Table 3.1. Copper impurity content. The reported values are the detectable limits unless
they are highlighted, indicating a detected element.
impurity
alpha1 100 µm
ppm
alpha2 100 µm
ppm
alpha250 250 µm
ppm
OFE 150 µm
ppm
Ag
Al
Ca
Cl
Fe
Ni
O
P
Pb
S
Se
Te
Ti
—
—
—
—
—
—
3
0.5
0.7
5
0.2
0.5
—
0.01
0.007
0.005
0.002
0.005
0.005
—
0.002
0.003
0.11
0.02
0.05
0.02
3
1
1
—
1
1
—
200
1
—
—
10
5
10
—
—
—
2
2
—
—
1
8
1
1
—
As
Au
B
Ba
Be
Bi
Br
Cd
Ce
Co
Cr
Cs
F
Ga
Ge
Hf
Hg
I
In
Ir
K
La
Li
Mg
Mn
Mo
Na
Nb
Nd
Os
Pd
Pt
Rb
Re
Rh
Ru
Sb
Sc
Si
Sn
Sr
Ta
Th
Tl
U
V
W
Y
Zn
Zr
—
—
—
—
—
1
—
1
—
—
—
—
—
—
—
—
0.1
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
1
—
0.002
0.5
0.0005
0.0005
0.001
0.002
0.005
0.05
0.001
0.0005
0.002
0.005
0.002
0.005
0.02
0.001
0.01
0.002
0.05
0.001
0.005
0.01
0.001
0.001
0.001
0.002
0.001
0.0005
0.005
0.001
0.002
0.001
0.001
0.001
0.001
0.005
0.005
0.0002
0.005
0.01
0.0001
5
0.0001
0.001
0.0002
0.0002
0.002
0.0002
0.05
0.0005
30
3
1
1
—
1
—
1
—
2
1
20
—
1
1
—
100
—
2
20
50
—
1
1
1
1
1
—
—
3
10
10
20
—
5
5
3
—
1
1
2
—
—
10
—
1
10
—
10
10
1
—
—
—
—
1
—
1
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
0.5
—
—
—
—
—
—
—
—
—
—
—
1
—
—
1
—
—
—
—
—
—
—
—
1
—
68
3. Sample preparation and experimental setup
The corresponding area is about 1/5 of the tube section and therefore the useful
oxygen flux is 1.6 × 10−4 mol/min.
A copper sheet 5 cm × 3 cm and 100 µm thick contains 6.4 × 10−3 mol of pure
copper and needs 3.2 × 10−3 mol of atomic oxygen (1.6 × 10−3 mol of molecular
oxygen) to reach the complete oxidation.
Since the time for complete oxidation, is about 31 min at 910 ◦C (considering
kp = 1.85 × 10−8 cm2 /s at 910 ◦C, p(O2 ) = 2.7 Torr and the oxidation proceeding
from both copper foil faces), the mean flux of required oxygen is 5.2 × 10−5 mol/min
which is much less than the supplied oxygen.
In the case of the temperature, the problem is much more easy. Because of the
fluxing cold gas, the temperature measured outside the tube, where the thermocouple
is placed, is higher than the real temperature in the center of the tube. In order to
calibrate the temperature we have used an N type thermocouple placed near the
sample. After the calibration the thermocouple was removed from the tube to avoid
contamination.
We conclude that the oxygen partial pressure and the nominal temperature in
the furnace can be considered the real parameters in which oxidation occurs. For
the oxygen partial pressure the error is about 5% while for the temperature is about
1%.
3.1.3
Copper Oxidation
Several oxidation procedure were developed in the past to oxidize copper in order
to obtain Cu2 O crystals with particular characteristics. The simplest method is an
oxidation in air of a copper foil at high temperature, for example at 1000 ◦C which
gives a polycrystalline material.
One of the most important study is that of Toth et al. [829] where the preparation
of Cu2 O single crystals, by copper oxidation in furnace, is discussed. Toth process
starts with an initial oxidation step about at 1040 ◦C which produces a polycrystalline
material with grain size comparable to the copper sheet thickness.
Large grains or even monocrystals can be obtained by a long annealing under
atmospheric oxygen partial pressure (grain growth method) at higher temperature
during which few large grains grow at the expenses of the other ones.
Toth found that thicker samples need longer times and lower temperature to
achieve the grain growth: samples with thickness lower than 400 µm were annealed
for about 10 hours at 1120 ◦C while a sample with a thickness of 760 µm needed an
annealing 144 hours long at 1090 ◦C.
Toth says also that the annealing temperature has to be in the interval 1070 –
1130 ◦C and that the samples with a thickness greater than 900 µm did not show the
grain growth (probably they need an exceedingly long annealing time).
Finally, in the Toth procedure, the sample is quenched in deionized water from
the annealing temperature to room temperature. In Fig. 3.5 a picture of a sample
prepared by us according to the Toth method is shown.
It is interesting to briefly discuss the the effect of the impurities on the grain size.
Zhu et al. [950] observed that the consequences of the impurities diminishes with
increasing the temperature and above 900 ◦C the difference in the grain size between
a 99.99% and 99.9999% pure copper vanishes. According to our experiments this
conclusion has not a general validity. Using copper of different purity grade and
3.1 Undoped Cu2 O preparation
1200
69
grain growth
1000
copper
oxidation
T
(°C)
800
600
annealing
400
quenching
200
Toth-Trivich method
0
0
2
4
6
8
10
t
12
14
16
18
20
(h)
Figure 3.2. The Cu2 O sample preparation according to Toth and Trivich method.
performing some doping tests introducing the impurities in copper of high purity,
we have demonstrated that the grain growth of Cu2 O is influenced by the presence
of some impurities. As we will see in Sec. 6.3, all the tested impurities, evaporated
on the surface of copper before the oxidation, show the obvious effect of reducing
the grain growth.
We also found that the purest copper we used, alpha2, does not give large grains.
A different surface with respect to the other copper sheets can give rise to a different
nucleation or alternatively a different average dimension of the copper grains in the
alpha2 with respect to the others (all our copper are polycrystalline) can influence
the final dimension of the Cu2 O grains. Finally we can imagine, even if unlikely,
that one or more particular impurities, uniformly distributed in the copper bulk,
can enhance the grain growth of Cu2 O. Looking at the Tab. 3.1, the candidate
impurities which could help the grain growth are reasonably silver or sulfur.
Unfortunately the Toth process produces highly resistive materials. The resistivity
of the sample prepared by the Toth method and cooled from high temperature
(1000 ◦C) to room temperature, both slowly or by quenching, is of the order of
104 –105 W cm [650, 849]. In order to obtain higher conductivity suitable for device
production, Trivich [650, 653, 848, 849] showed that a subsequent annealing of the
substrate in air at 500 ◦C for several hours and a quench in water, gives a resistivity
of the order of 102 Wcm.
In Fig. 3.2 the Toth method of oxidation and the Trivich step to increase the
conductivity are graphically presented.1
On the other hand Tapiero et al. [808] tried to reduce the resistivity of their
1
The annealing at 500 ◦C can be performed soon after the grain growth step or later. For example
Papadimitriou, after the grain growth, cooled his samples, grinded and polished half of samples
to remove the inner porous layer of voids and then he re-annealed them at 500 ◦C in air (private
communications).
70
3. Sample preparation and experimental setup
6
5
10
(
cm)
10
4
10
0
20
40
60
80
100
annealing time
120
140
160
180
200
(min)
Figure 3.3. Resistivity as a function of annealing time at 500 ◦C in air. The experimental
points are obtained using different parts of a sample of Cu2 O oxidized according the
Toth method. The resistivity at t = 0 is the resistivity of the original sample.
samples in this way but they obtained a decrease at most of a factor of 7 instead of
100. We also tried to reproduce the effect of this annealing on our Cu2 O samples.
They were prepared by the Toth method with the alpha250 copper and usually
show a resistivity of 105 W cm. From one of these samples, slowly cooled to room
temperature in nitrogen without performing the Trivich step, several square pieces
with a side of 1 cm were cut.
These Cu2 O squares were then annealed in air at 500 ◦C one at a time, varying
the annealing time. After each annealing was completed the sample was quenched in
deionized water and the resistivity was measured by the Van Der Pauw method. In
Fig. 3.3 the resistivity of the samples as a function of the annealing time is reported.
We can see that the reduction in the resistivity does not exceed the factor of 7, as
already reported by Tapiero.
The reason of the great decrease in resistivity obtained by the groups of Trivich
and Papadimitriou is still unclear, but it is probably correlated with a lower purity of
the starting copper or with a contamination of the furnace. A further confirmation
of the relevance of the impurities present in the starting copper or introduced during
the Cu2 O preparation is given by the fact that similar low values of resistivity were
obtained by other oxidation processes [263, 538, 539, 839].
We see therefore that the procedure of Trivich is not easily reproducible and
moreover the thick layer of CuO formed on the Cu2 O surface during this step needs
to be removed by grinding and polishing, not just etching the surface. For this
reason in our laboratory we have developed another oxidation procedure which is
more reproducible but gives a material with grain size of the order of 1 mm and
a typical resistivity of 2000 W cm. All our samples are made by 99.999% copper,
100 µm thick.
Our method proceeds as follows. The temperature is raised with a rate of
10 ◦C/min up to 910 ◦C in nitrogen flux. Then a 90 min long oxidation in a partial
3.1 Undoped Cu2 O preparation
71
1200
grain growth
100
1000
copper
annealing
10
600
(Torr)
oxidation
1
p
T
(°C)
800
400
quenching
0.1
Cu O75
2
= 1250
200
cm
2
= 101 cm /(Vs)
0.01
0
0
2
4
6
8
t
10
12
14
16
18
(h)
Figure 3.4. Temperature and partial oxygen pressure versus time for our method of
preparation of Cu2 O sample.
oxygen pressure of 2.7 Torr is done. Notice that soon after the oxidation step, the
grain size is of the order of the copper sheet thickness, like in the Toth oxidation
method. The oxidation is followed by two annealings: the first one at 1125 ◦C for
60 min and the second one at 780 ◦C for 600 min, both under a 0.27 Torr oxygen
partial pressure. The first annealing is needed to increase the grain dimensions up
to 1 mm while the second to get a partial equilibration of the sample to a lower
defect density to increase the mobility. Finally, after switching to the nitrogen flux
only, the sample is cooled down to 450 ◦C and then quenched to room temperature
dipping it into deionized water.2
Without the quenching the resistivity of the sample is of the order of 2000 W cm
whereas with quenching the resistivity is about 1000 W cm [543].3 In Fig. 3.4 a
graphical representation of our Cu2 O preparation method is reported and in Fig. 3.6
a picture of a sample prepared according to our method is shown. From now on all
samples are assumed to be oxidized with our method.
CuO and surface etching
It is very common that CuO is formed on the surface of the Cu2 O during the cooling
process of the samples oxidized in furnace. Indeed during this operation it is almost
impossible to keep the conditions inside the existence region of Cu2 O. One of the
obvious reactions is the surface oxidation of cuprous oxide (Cu2 O) to cupric oxide
(CuO):
Cu2 O + 12 O2
2 CuO
2
The oxygen impurity content in nitrogen is about 5 ppm which corresponds about to
3.8 × 10−3 Torr (5 mbar) oxygen partial pressure
3
In this cited paper the reported resistivity without quenching the sample is about 22 000 W cm
but it is a typo.
72
3. Sample preparation and experimental setup
Figure 3.5. Etched Cu2 O sample obtained
oxidizing copper with the Toth procedure.
The large grains are clearly visible. The
sample width is 5 cm.
Figure 3.6. Etched Cu2 O sample obtained by our oxidation method.
The grains are of the order of 12 mm. The sample width is about
2.5 cm.
If this layer is thick enough, it can be easily detected by an X-rays diffraction (XRD)
measurement or simply observing a dark layer on top of the material.
It has been reported [526] that the CuO is practically not formed if the oxidation
temperature is well below 350 ◦C. For example at 250 ◦C, even after 11 hours in a
partial oxygen pressure of 1 atm, no variation of the electrical and optical properties
of the sample indicate the presence of CuO. On the other hand, above 350 ◦C a
layer of cupric oxide is formed.
Several chemical etching techniques have been used in the past to eliminates the
CuO from the surface or in general to clean the surface of Cu2 O. Usually among
acids the preferred is the nitric acid because it has also an oxidant power which
helps the etched surface to keep its stoichiometry. Another very common technique
is based on bromine-methanol [635]. Bromine is in fact a strong oxidazing agent. It
can remove the residual copper on the Cu2 O surface, produced, for example, by a
non-perfect nitric acid etching.
However nitric acid does not attack, or attacks very slowly, the cupric oxide. For
this reason nitric acid is used only to etch Cu2 O surface almost free of CuO. In
order to eliminate a thick layer of cupric oxide we have to use the mechanically way,
grinding and polishing the Cu2 O surface.
If the CuO is not very thick we can use the chemical way, using HCl. Notice that
in general the acids hardly attack the cupric oxide. A thin layer of 1 or 2 µm is easily
removed because the acids penetrates under the layer trough the grain boundaries
or trough the region not covered by CuO. For this reason, thicker layer needs to be
removed mechanically otherwise a very rough surface is obtained.
However the etching with HCl leaves an irregular surface. In any case, after one
of these treatments, the sample has to be etched in nitric acid.
In our oxidation procedure the CuO layer is very thin because the cooling is
performed under a nitrogen atmosphere, with a very low oxygen partial pressure,
and indeed is not observed from XRD experiments (see Fig. 3.7). However in order
3.1 Undoped Cu2 O preparation
73
to clean the surface and remove an eventual thin layer of CuO, we perform, just
before the device preparation, a wet etching procedure divided in two steps. First
the sample is etched for 10 s in a 7 M HNO3 solution saturated with NaCl and then
rinsed in deionized water. Then the sample is etched for 10 s in HNO3 (65%) and
rinsed in deionized water. Finally the etched sample is put in a beaker in deionized
water and subjected to ultrasound for 5 minutes. The etching rate of our technique
is about 10 µm for each complete treatment.
The NaCl salt is needed to have chlorine ions in the solution which help the cupric
oxide layer removal, in fact a similar result can be obtained using diluted HCl directly.
Unfortunately this first etching leaves a surface rich in copper. The stoichiometry is
re-established after the second etching, thanks to the oxidant properties of HNO3 .
In the literature several empirical etching procedures are discussed. For example
Trivich [837] claims that HNO3 8 M is the best nitric acid concentration to have a
good surface for devices. Another very good etchant for surface treatment for solar
cells are proved to be the cyanides [841].
Trivich [838, 848] found also that surface treatment with K2 S after the etching
procedure helps to obtain ohmic contact with gold and silver. However, if gold is
used, a good ohmic contact is obtained in any case.
Finally it is interesting to cite the etching made with 1 part of HNO3 and 9 part
of H3 PO4 which selectively removes Cu2 O with an etching rate of 10 µm/min.
3.1.4
Structural properties of Cu2 O obtained from oxidation
Several micro-structural defects have been reported in literature both in single crystal
and in polycrystals.
The first problem is the formation, in the middle of the sample of a plane of
voids (also called inner porous layer). As we have seen before, during the oxidation,
the copper ions migrate from copper toward the oxygen/Cu2 O interface. At the
end of the oxidation the two growing layers of Cu2 O met in the center of the initial
copper foil. In that plane small voids, up to tens of micrometers are present [948].
The simplest explanation is that the two oxide/metal interfaces are not perfectly
flat, maybe because they reproduce the initial morphology of the copper surfaces.
Other defects are black spots of various dimensions, given by copper vacancies
agglomeration producing CuO or by transformation of Cu2 O into CuO thanks to
the trapped oxygen in the crystal voids [738, 948].
All these defects are particularly evident for sample oxidized at high oxygen
pressure, for example in air. However it has been shown that most of these defects
are eliminated by annealing the sample at high temperature under a low oxygen
pressure [738].
Another problem arise from the fact that Cu2 O obtained in the furnace, starting
from a copper foil, can grow only in the direction perpendicular to the copper surface.
For this reason, the expansion due to the transformation from copper to cuprous
oxide can not be accommodated in the direction parallel to the copper foil and a
great compressive stress is introduced in the sample. If the oxidation rate is not
uniform along the sample surface, this stress can introduce a sample bending.
74
3.1.5
3. Sample preparation and experimental setup
Equilibrium of point defects and oxidation procedure
As we have already discussed it is impossible to exclude the hypothesis that conductivity of non-intentionally doped Cu2 O are controlled by the concentration of
impurities.
Supposing anyway a negligible role of the impurities, let us consider what level
of intrinsic defects can be expected after our typical oxidation process.
Using the diffusion coefficient of the copper vacancies given in (2.51), it is easy
to see that at 800 ◦C in 11 hours these defects can diffuse up to 5 mm. Therefore
since the oxygen partial pressure is such that the system is well inside the Cu2 O
stability region, the copper vacancies will be able to diffuse to the surface and to
reach their concentration equilibrium value which from Fig. 2.17 can be estimated
as 2.5 × 1018 cm−3 .
It is interesting to note that even at 450 ◦C the copper vacancies can diffuse up
to 70 µm, the minimum distance to to reach the surface of our 150 µm thick samples,
in two minutes only! At that temperature the equilibrium concentration value of the
copper vacancies has to be calculated at the CuO dissociation pressure and from
the data shown in Fig. 2.15 we obtain [VCu ] ≈ 5 × 1016 cm−3 .
Unfortunately we do not know the diffusion coefficient of the other intrinsic
defects. For this reason, as we have discussed in Sec. 2.10.1, the concentration of
the other intrinsic defects cannon be estimated.
3.1.6
Substrate characterization
Some substrates were characterized by XRD. In Fig. 3.7 a typical XRD pattern of
our samples is shown. The analysis show that the samples are made of Cu2 O and
that the CuO is not detectable. Notice that XRD experiments are performed on
polycrystalline material, not powder.
Looking at the ratio between the expected intensity for a Cu2 O powder and the
intensity of our sample, we reasonably conclude that the crystalline planes parallel
to the surface are mainly (110).
Some samples were analyzed also at the Scanning Electron Microscopy (SEM)
equipped with a microanalysis. The first observation was that our samples have very
few and very small central voids, as shown in Fig. 3.8. This is most likely related
to the purity of copper alpha1 and alpha2 [948]. Another result was that, in the
detectability limit of 1000 ppm no trace of impurities were found.
Finally the resistivity and the mobility are measured in the Van der Pauw
configuration for all samples under ambient light illumination at room temperature.
3.2
Device fabrication
In this section we will discuss the fabrication of the devices used in this thesis. We
made two types of devices: devices with ohmic contacts, one of which is semitransparent, in order to analyze the conductivity of the bulk in different condition of
illumination, and junction based devices, in order to make AC analysis to probe
the electronic characteristics of defects. All these devices have all an active area of
0.5 cm2 and are fabricated by evaporation.
3.2 Device fabrication
75
1200
Cu O (220)
2
1000
Cu O (111)
2
(a.u.)
800
intensity
600
Cu O (110)
2
400
200
Cu O (311)
Cu O (211)
2
2
0
20
30
40
50
60
70
80
2
Figure 3.7. XRD pattern for a typical Cu2 O sample obtained by our oxidation methods.
No CuO peaks are visible.
Figure 3.8. Scanning electron microscopy image of the transverse section of a Cu2 O sample
oxidized with our method. The interface where the oxidation process ends is clearly
visible. Notice also that the central voids are few and of the order of 5 µm.
76
3. Sample preparation and experimental setup
Figure 3.9. Section representation of a photo-conductors device.
The devices are labeled in the following way. Devices with ohmic contacts: S
+ the number of the substrate (e.g. S81). Devices with Schottky junction: Sc +
material of the junction + number of the substrate (e.g. ScAl90). Devices with
heterojunction: C + substrate number (e.g. C85) (C stands for cell).
3.2.1
Devices with ohmic contacts
The simplest device used is a Cu2 O square, cut from the original Cu2 O sheet, covered
by ohmic contacts on both faces. It is formally called photo-conductor because one
of the ohmic contact is semitransparent. The back contact is made by 1200 Å of
evaporated gold. The front contact is made evaporating a 70 Å thick semitransparent
front gold contact covered by an anti–reflection layer of ZnS (320 Å) and by a gold
grid with a shadow factor of about 11%. (See Fig. 3.9)
3.2.2
Junction based devices
Our junction based devices can be divided in two categories: Schottky junctions and
heterojunctions.
Schottky junction
Solar cells based on Schottky junctions are made in the same way of the photoconductors, but substituting the gold of the front contact with a 70 Å thick semitransparent copper or aluminium, evaporated keeping the substrates at 80 ◦C.
We have fabricated also opaque devices, where the front contact is made only by
a 1200 Å thick layer of copper, aluminium or zinc, evaporated on substrate at room
temperature.
Heterojunctions
Solar cells based on heterojunctions were also used in this thesis to understand the
underlying physics of cuprous oxide. They were prepared for a previous work in our
laboratory, where the world record of 2% efficiency for a cell based on Cu2 O was
obtained [543].
Two materials were used to prepare the heterojunctions, In2 O3 (ITO) and ZnO,
two Transparent Conductive Oxide (TCO). Both were deposited by Ion Beam
Sputtering (IBS) and their resistivity is about 1 × 10−3 W cm and 4 × 10−3 W cm
respectively.
The cells based on ZnO have the structure Au (120 nm) / Cu2 O (150 µm) /
ZnO (50 nm) / In2 O3 (300 nm) / Cu grid (120 nm) / MgF2 (80 nm) while the cells
based on In2 O3 have the structure Au (120 nm) / Cu2 O (150 µm) / In2 O3 (300 nm)
3.3 Experimental setup
77
Figure 3.10. Section representation of a solar cell based on a Cu2 O/ZnO heterojunction.
/ Cu grid (120 nm) / MgF2 (80 nm). The MgF2 was used as antireflection layer. In
Fig. 3.10 the section of a solar cell based on Cu2 O/ZnO heterojunction is shown.
3.3
Experimental setup
In this section we will briefly show the experimental apparatus used to electrically
characterize our devices, both in DC and AC mode.
A first characterization of the substrate is done measuring the resistivity at room
temperature and the hole mobility by Hall effect in the Van der Pauw configuration.
To perform this measurement we have used a Bio-Rad HL5900 Hall profiler.
For all the other electrical measurements the sample is placed in a cryostat in
order to perform measurement as a function of temperature avoiding the oxidation
of the surfaces. The cryostat is equipped with a pumping system composed by a
turbo-molecular pump Balzers TPH240 connected in series with a rotative pump.
The highest vacuum we can reach is 10−6 Torr (about 10−4 Pa).
The cryostat is a Janis VPF100, working with liquid nitrogen, which can operate
between 77 to 475 K. The maximum temperature reached in our measurements is
430 K, because some plastic screws, used to hold the sample, start to melt at that
temperature.
The temperature is set and read by a Lakeshore 330 Temperature Controller
which drives an heater in contact with the sample holder. The temperature is
measured by two silicon diodes, one on the heater and one near the sample.
The sample holder acts as electrical contact for the back contact of the devices
and it has an L shape so that the sample can receive light from one of the cryostat
quartz windows. The lamp used is a 200 W halogen lamp with a photon flux of about
5 × 1017 cm−2 s−1 and a wavelength 350 nm < λ < 750 nm). In Fig. 3.11 and 3.12 a
scheme of the cryostat and a picture of the sample holder are shown.
In order to have a good thermal contact between the sample and the sample
holder, we have used a silver paint (TAAB S270). We have recently discovered that
at high temperature and after a long time the silver diffuse through the gold of the
back contact [329, 342, 375]. When the silver reaches the cuprous oxide, it transforms
the ohmic contact Au/Cu2 O in a rectifying junction Ag/Cu2 O. This process can
affect the correct interpretation of the measurements. For this reason in the future
we will use a different material for the back contact, not subjected to the silver
diffusion, or we will simply substitute the silver paint with a gold paint.
The DC measurements are performed by a Source-Measure Unit Keithley 236
whereas the AC measurements are performed by an Impedance Analyzers HP 4195A.
The AC measures of the last part of this thesis (doping) are performed with a new
instrument, recently acquired by our laboratory, the Impedance Analyzer Agilent
4594A.
78
3. Sample preparation and experimental setup
Figure 3.11. Scheme of cryostat and connections.
Figure 3.12. Cryostat internal showing the sample holder.
3.3 Experimental setup
79
The scheme of Fig. 3.14 represents the connections used in DC and AC modes.
Finally in Fig. 3.13 a picture of the entire experimental setup is shown.
80
3. Sample preparation and experimental setup
Figure 3.13. Overview of the experimental setup.
Figure 3.14. DC (left) and AC (right) measurement connections.
Chapter 4
Electronic properties and
metastability effects
The study of the intrinsic defects is fundamental to understand the problems and
the solutions for a technological use of Cu2 O. This chapter is mainly focused on its
metastability effects. We will prove that they are inconsistent with the accepted
model of Cu2 O where the PPC is explained by a simple electron emission from the
deep donors.
4.1
Acceptor and donor concentrations
In this section we will give an estimation of the acceptor and donor concentration for
a representative sample of Cu2 O, using two experiments: conductivity as a function
of temperature and capacitance–voltage profiling.
4.1.1
Mobility
Our Hall effect setup operates at room temperature only, therefore we cannot use it
to measure directly the carrier concentration as a function of temperature.
As an alternative we measure the conductivity σ of the sample as a function
of temperature and obtain the carrier concentration p, neglecting the electron
contribution which is extremely low, from the relation σ(T ) = eµp (T )p(T ), where e
is the module of the electron charge and µp is the hole mobility.
For the mobility we use the literature data [760] presented in Sec. 2.4 and rescaled
to the measured room temperature value of each sample.
We have fitted the literature data with an exponential function in the temperature
range used in our experiments (from 100 to 420 K) obtaining µp (T ) = 5754 ·
10−0.00643T where the temperature T is expressed in kelvin and the mobility µp in
cm2 /(V s), see Fig. 4.1. Notice that the mobility value at room temperature obtained
from the previous expression is 68 cm2 /(V s).
Then, for every sample, we have measured the value of mobility at room temperature and we have rescaled the pre-factor of the previous expression so that the
curve passes trough this measured point. For example, the sample 81 has a mobility
value of 100 cm2 /(V s) which gives, translating the fit of the literature data
µp (T ) = 8511 · 10−0.00643T .
81
(4.1)
82
4. Electronic properties and metastability effects
5
10
Literature data
Exponential fit
4
(cm
2
-1
-1
V s )
10
3
10
2
10
1
10
100
200
T
300
400
500
(K)
Figure 4.1. Literature mobility data obtained by Hall effect measurement as a function of
temperature for a Cu2 O sample [760]. The solid line is the fit of these data in the range
100–420 K.
The expression of the mobility as a function of temperature obtained in this way
has been used to extract the carrier concentration from the conductivity measurements.
We point out that the mobility value measured at room temperature is not
influenced by the surface conductivity. Indeed, according to Dobrovol’skii and
Gritsenko, the standard planar measurement gives the correct bulk mobility value
only if the resistivity is smaller than 104 W cm (see Sec. 2.4) and our samples have a
resistivity of about 2000 W cm.
Finally we want to remember that, as already mentioned in Sec. 2.4, the mobility
remains equal to the dark value for samples in the preilluminated state even if it
slightly increases during a strong illumination of the sample.
4.1.2
Defect concentration
We have performed two experiments to know the defect concentration of our sample:
conductivity versus temperature and Capacitance–Voltage (CV) profiling.
The experiments were performed in two conditions of the sample, equilibrated
and preilluminated. A sample which shows the PPC effect needs several hours at
high temperature to reach the equilibrium. We have left our sample at 400 K in
the dark and in vacuum until its conductivity reaches a minimum; we call this
state the equilibrated. Another condition is the preilluminated state: the sample is
illuminated for several minutes with white light at room temperature in vacuum until
the conductivity reaches a maximum. After these procedures, the sample is kept in
the dark and the temperature is lowered to 77 K to maintain the metastability state
unaltered.
In our experiment we have measured the current I vs temperature T with an
4.1 Acceptor and donor concentrations
83
15
10
14
10
13
10
-3
(cm )
12
10
11
p
10
Sample S81
10
10
equilibrated
preilluminated at 295 K
9
10
10
fit
8
2
3
4
T
1000/
5
(K
-1
6
)
Figure 4.2. Hole concentration as a function of temperature for the sample S81. The
experimental data were fitted with the Eq. (4.3). The results of the fit are reported in
Tab 4.1.
applied constant voltage of V = 0.1 V. The conductivity was then obtained by
σ=I
l
SV
where S is the active surface of the photoconductor.
The carrier concentration is then obtained by the following relation
p(T ) =
σ(T )
eµp (T )
(4.2)
where the minority carriers are neglected, e is the module of the electron charge, µp
is the mobility and σ the conductivity.
The result of this experiment on a representative sample of Cu2 O, sample S81,
is reported in Fig. 4.2. For the mobility we have used the empirical expression (4.1)
found in the previous section.
It can be seen that the preilluminated curve has a concentration of carriers about
7 times greater than that of the equilibrated one.
The results of this experiment has been analyzed by a simple model with one
single level acceptor and one or more deep donors.
For a p–type semiconductor with a concentration of NA acceptors compensated
by ND+ single ionized donors per unit volume, experimental data can be fitted with
the following expression (see App. B):
1
p(T ) = −
2
Nv
eEA /kT
+
ND+
1
+
2
s
Nv
eEA /kT
+
ND+
2
+
4Nv NA
eEA /kT
(4.3)
84
4. Electronic properties and metastability effects
Table 4.1. Acceptor concentration NA , ionized donor concentration ND+ and acceptor
energy level EA obtained from the fit of p(T ) using the Eq. (4.3). The compensation
ratio NA /ND+ is also reported.
state
NA (cm−3 )
ND (cm−3 )
EA (eV)
NA /ND+
equilibrated
preilluminated
5.72 × 1014
10.30 × 1014
4.91 × 1014
4.94 × 1014
0.325
0.322
1.16
2.07
where Nv = 2.13 × 1015 T 3/2 cm−3 is the valence band effective density of states, EA
is the acceptor energy level, k is the Boltzmann constant and where the degeneracy
of the acceptor is assumed to be 1. This expression holds if the concentration of
ionized donors ND+ remains constant during the measurement and if the acceptors
are described by a single electronic level. The former assumption is most likely
correct because, as we have seen in the Chapter 2, the donor level is very deep and
therefore we can consider ND+ = ND in the equilibrated case.
However, since a variation of the activation energy of p(T ) among different
samples is often observed, the latter assumption was criticized by some authors who
suggested that the acceptors are distributed in a band [811].
If this hypothesis were true, a different low temperature activation energy of
the p(T ) curves in the equilibrated and in the preilluminated conditions should be
observed as a result of a change in the compensation ratio as reported in [811]. On
the other hand this change was not observed in [737].
In our sample S81 only a very small variation, of the order of 0.01 eV, is observed.
Using the same model of [811] it can be shown that the acceptor band in this sample
is quite narrow, less than 0.025 eV. However analyzing a larger set of samples we
found that a few of them show indeed a larger difference of the activation energy
between the preilluminated and equilibrated states. It seems therefore that the
acceptor band width can vary from sample to sample. The modelling of the behaviour
of a sample with a large acceptor band is much more complicated and for these
reason in this chapter we will focus on a sample with a narrow band of acceptors.
As already shown in Sec. 2.10.2 and as it will be seen in the next section, the
PPC decay time is much greater than the measurement time, therefore it is possible
to assume, for preilluminated samples too, a constant concentration of ionized donor
during data acquisition, up to 350 K. The Eq. (4.3) can thus be used to fit both
the equilibrated and preilluminated curves of Fig. 4.2. The results of the fits are
reported in Tab. 4.1.
Notice that the saturation of the hole concentration expected at high temperature
is not very clear. This introduce a great error in the determination of defect
concentration, about 10%.1
To check the correctness of these numbers we have also performed several Capacitance–Voltage (C–V) measurements to obtain the concentration of the net fixed
charge, Nfix . This charge is obtained using the Schottky relation:
Nfix (x) = −
C3
e εA2
dC
dV
−1
(4.4)
1
It is interesting to note that the samples prepared by the Toth method or, in general, equilibrated
at high oxygen pressure, do not show any saturation of the p(T ) curve. For this reason in these
samples NA and ND can be extracted only by the aid of the CV measurements.
4.1 Acceptor and donor concentrations
85
4
T = 300 K
3
preilluminated
14
-3
cm )
Solar cell C80F
(10
equilibrated in the dark
A
-
N N
D+
2
1
0
0
1
2
3
x
4
(
5
6
7
m)
Figure 4.3. Net fixed charge Nfix as a function of the distance from the interface for
the solar cell C80F. The constant value of Nfix far from the interface is an estimate
of NA − ND+ and is consistent with the value of obtained from the conductivity vs
temperature fit.
where C(x) is the capacitance of the junction, e is the module of electron charge, ε
is the dielectric constant of Cu2 O, V is the applied voltage bias, A = 0.5 cm2 is the
device active area and x = εA/C is the distance from the Cu2 O/metal interface.
The experimental results, obtained on the cell C80F with a probe signal of 19 kHz
and an amplitude of 20 mV, are reported in Fig. 4.3.
The value of Nfix for large x is an estimate of the quantity NA− − ND+ [64]. Since
the acceptor level is about at 0.3 eV from the top of the valence band and has a fast
response, we can assume that under reverse bias NA− = NA . Instead the occupancy
of the donor level remains practically unaltered, because its response time is very
slow as we will see in Sec. 4.3. For these reason we can obtain NA − ND+ from the
C–V measurements.
Even if the results are not very clear we can state that they are compatible with
the data of Tab. 4.1.
According to the electronic model of PPC, the charged donor concentration should
have increased between the equilibrated and the preilluminated case, because with
light the donors become neutral and the material remains in a smaller compensated
state until they become again positive by electron emission or hole capture. A very
important behaviour of our results is that this is not the case. On the contrary
we observe an increase of the acceptors and an almost constant number of charged
donors. A similar effect is observed in chalcopyrites [457].
Finally we observe that the ionization energy of the acceptor agree with the
literature data.
86
4. Electronic properties and metastability effects
0.0014
light off
S81
0.0012
T = 393 K
0.0008
(
-1
-1
cm )
0.0010
0.0006
0.0004
0.0002
light on
0.0000
0
25
t
50
75
100
(min)
Figure 4.4. Typical experiment of PPC decay. Initially the sample is in equilibrium in the
dark at the given temperature (T = 393 K in this figure). At a certain point the shutter
of the lamp is opened and the sample is illuminated. After the conductivity saturates
the shutter is closed and the PPC discharge is monitored as a function of time. The
decay curve is well described by a single exponential.
4.2
Persistent Photo-Conductivity
In this section we show the measurements of the PPC decay time constants which
can give important information on the properties of donors.
The experiments were performed at three different temperature, 353 K, 373 K and
393 K. At lower temperature the time become too long to be practically monitored.
First of all we set the sample temperature in the cryostat to the desired temperature. Then the sample is illuminated until the photo-conductivity has reached
a maximum. After this stationary state is reached, the light is rapidly switched
off and the conductivity is monitored until it reaches its minimum, that is at the
equilibrium. This process is illustrated in Fig. 4.4 where the PPC at 393 K has been
measured as a function of time.2
In Fig. 4.5 the PPC curve for the sample S81 are reported for three different
temperatures in logarithmic scale.
After the illumination is switched off, the curve is well described by a single
exponential. In that figure the fits are also shown and the results are reported in
Tab. 4.2.
Representing the decay time constant in an Arrhenius plot it is evident that
they have an activated behaviour. This plot is reported in Fig. 4.6 together with
the literature data of Pollack and Trivich [677]. Our experimental points are fitted
2
The illumination of the sample of Cu2 O has been made by an halogen lamp which has a wide
frequency spectrum. This means that most of the light is absorbed in the first few microns of the
material giving a inhomogeneous illumination. An homogeneous illumination could be done by a
laser with energy slightly absorbed, that is near the Cu2 O gap (red light).
4.2 Persistent Photo-Conductivity
87
10
T = 353 K
-
0
) /
0
1
(
T = 373 K
0.1
T = 393 K
S81
Experimental data
Exponential fit
0.01
0
100
200
t
300
400
(min)
Figure 4.5. PPC decay as a function of time for three temperatures of the sample. All
three curves are weel described by a single exponential. The PPC is expressed as a
variation respect with the dark value of conductivity.
Table 4.2. PPC decay times at three different temperatures for the sample S81. The time
constants have an activated behaviour described by a prefactor τ0 and an activation
energy EPPC .
sample
S81
τPPC (353 K)
(s)
τPPC (373 K)
(s)
τPPC (393 K)
(s)
EPPC
(eV)
τ0
(s)
784
5132
35110
1.13
2.22 × 10−12
88
4. Electronic properties and metastability effects
5
10
Pollack e Trivich
Sample
4
10
PPC
PPC
short
1
2
3
,
,
,
,
4
,
3
PPC
(s)
10
2
10
1
10
Our sample
S81
0
10
Exponential fit
1.9
2.0
2.1
2.2
2.3
2.4
1000/T
2.5
-1
(K
2.6
2.7
2.8
2.9
)
Figure 4.6. PPC decay time constants in an Arrhenius plot. For comparison the data of
Pollack and Trivich are also reported [677].
with the expression
τ = τ0 eEPPC /(kT )
(4.5)
and the results for prefactor and activation energy are reported in Tab. 4.2.
Pollack and Trivich fit the decay curve with a sum of two exponentials. This is
related to the their TSC experiment which show two peaks as we have reported in
Sec. 2.10.2. In our TSC experiments the second peak is almost undetectable and
we have decided to use a single exponential for this work. The appearance of a
second peak in TSC, and therefore the use of two exponentials to describe the PPC
decay experiments, is related to the preparation conditions of the sample, indeed
also a Pollack sample, the fourth, does not show a second peak (this sample was
equilibrated at high oxygen pressure and temperature).
4.2.1
Interpretation of PPC
As we have discussed in Sec. 2.10.2, the most common accepted interpretation of
PPC is based on a purely electronic model, in which one or more deep donors trap
the electrons. These electron are then emitted in conduction band very slowly, giving
the typical PPC decay.
Now we want to emphasize some weak points of this interpretation.
During the illumination, the donors are filled with electrons and becomes neutral.
After the light is switched off, the processes which can empty the donor traps are
the electron emission and the hole capture. These processes are described by the
rate equation
df
= −en f − pcp f
(4.6)
dt
where f is the occupation probability of the trap, en is the electron emission coefficient,
p is the hole concentration and cp is the hole capture coefficient.
4.2 Persistent Photo-Conductivity
89
4.0
T = 373 K,
g = 1, E = 0.32 eV, N = 6x10 cm
N = 5x10 cm , f (0) = 1, e = 2x10
14
3.5
A
A
)
D
n
,
-1
s
quasi equilibrium
2.5
simple exponential
-3
cm
14
-4
2.0
1.5
p
(10
-3
D
3.0
-3
A
14
1.0
0.5
0.0
0
10
20
t
30
3
(10
40
50
s)
Figure 4.7. PPC simulation with the parameters reported in figure. The quasi equilibrium
expression is very similar to a simple exponential of the form p = peq + ∆p e−t/τ .
Now, if the donor has a very long re-equilibration time compared to the typical
time of the interaction between bands and the acceptors, we can consider a quasiequilibrium condition for all points of the decay process of PPC. In this way we can
calculate numerically p(t).
However since the decay curve of p(t) is well described by an exponential, we have
to reject the possibility of hole capture which would give a strong non–exponential
transient for p(t). The PPC decay process is therefore described by the rate equation
df
= −en f
dt
(4.7)
Notice that neglecting cp , the solution for fD is a simple exponential but this
is not always true for p. Indeed, using the equation 4.7 and the quasi-equilibrium
assumption, the p(t) is obviously given by the Eq. (4.3), where ND+ has been replaced
by
(4.8)
ND+ (t) = ND 1 − fD (0)e−t/τ
where fD (0) is the initial electron occupancy of donors and τ = 1/en .
However a simple simulation (see Fig. 4.7) of p(t) with typical values of our
materials shows that this curve departs slightly from a single exponential.
The single exponential used by us to fit the experimental data is
p(t) = peq + (p(0) − peq ) e−t/τ
(4.9)
where peq is the value of p at the equilibrium when the PPC is disappeared, p(0) is
the value of p at the initial moment, when the light is switched off and the material
has reached the quasi-equilibrium, and τ = 1/en .
90
4. Electronic properties and metastability effects
This expression is obtained considering that NA− goes from the quasi equilibrium
value at t = 0 to the equilibrium value reached at t = ∞ with an exponential, with
the same time constant of the donor discharge.
From the previous expression we can obtain also the ratio of neutralized donors
at t = 0 but this number has a big error. However we have seen that it is compatible
with 1, that means that all donors have been neutralized by light.
As en can be written as
en = cn vth Nc e−(Ec −ED )/(kT )
(4.10)
the activation energy EPPC = 1.09 eV and the prefactor τ0 = 7.1 × 10−8 s of τ ,
recalculated to take into account the T 3/2 dependence of Nc , gives respectively3
σn ≈ 10−15 cm−3
and
ED ≈ 0.91 eV
(4.11)
This value of en has to be much greater than the hole capture and therefore we
can obtain a limit for σp from the condition pcp en . The most strict condition is
obtained at 353 K and, choosing a mean value for hole concentration of p ≈ 1014 cm−3 ,
it gives
σp 10−25 cm2
(4.12)
and consequently
ep = vth σp Nv e−(ED −Ev )/(kT ) 10−12 s−1
(4.13)
These limits for σp and ep are very uncommon. However they could be explained
by a strong lattice relaxation in the neighbour of a donor which can occur when the
donor becomes from positive to neutral. Indeed in this case the hole capture cross
section has an activated behaviour due to a energy barrier:
σp = σ0 e−Eσ /(kT )
(4.14)
Choosing a reasonable value for the prefactor, for example σ0 = 10−18 cm2 , a
relaxation which gives rise to a energy barrier of 0.5 eV can explain the extremely
low values for σp and ep . A pictorial view of a similar model is given in Fig. 4.8,
where a configuration coordinate diagram is represented.
Roughly speaking this diagram is a representation of the total energy (lattice and
electronic) of the donor and its surrounding as a function of one of the generalized
coordinate Q which describe the surrounding. In the simplest case we can imagine
the most symmetrical configuration where the nearest neighbors of a donor are
equidistant from it and where the coordinate Q is simply the distance between one
nearest neighbor and the donor. Assuming an elastic force between the particles, we
obtain a parabola for the total energy of the system in the configuration coordinate
diagram.
In Fig. 4.8 the lowest parabola represents an empty donor and its surroundings
in the equilibrium state. If we excite an electron and an hole across the gap, we
obtain the same parabola translated of 2 eV, the energy gap. If the donor becomes
3
Notice that the cross section is an effective cross section which includes also the entropy term:
σeff = Xσ where X is a factor which can span from 1 to 100 [740].
4.2 Persistent Photo-Conductivity
91
4
+
Total Energy
(eV)
D
+ e + h
cp =
p
vth exp(-E /(kT))
ep =
p
vth Nv exp(-(ED+E )/(kT))
3
2
ep
cp
1
+
D
E
0
D
+ h
opt
E
0.5 eV
ED = 0.9 eV
0
Q1
Q0
Generalized Coordinate
Figure 4.8. Sketch of a possible configuration coordinate diagram which can explain
the extremely low values of σp and ep obtained from the PPC decays analysis. The
relaxation of the donor in the neutral state introduces an energy barrier called Eσ for
the hole capture.
neutral, the configuration of minimum energy can be reached for different positions
of the neighbors, represented in figure by the value Q1 .
Since the Born–Oppheneimer approximation is valid in this system, the optical
transition will occur vertically in the diagram, that is the lattice can not relax in
the time of transition. This brings to a difference in the optical energy required to
neutralize the donor respect with the thermal energy, being the former greater than
the latter, Eopt > ED .4
After the optical transition, the system is in an highly excited vibrational state
and, if the radiative disexcitation is relatively slow compared to the typical phonon
frequencies, the system reaches the minimum at Q1 . Once this minimum has been
reached the system can not recombine radiatively because there is no state to
conserve the crystal momentum.
In order to go to the positively charged state the donor has to emit the electron or
to capture an hole. The capture process can take place only if the system overcomes
the energy barrier denoted with Eσ . This process can give the very uncommon
values of σp . In the proposed diagram the thermal emission of the electron remains
instead unchanged, with the usual Eg − ED activation energy.
4
It has to be noted that the optical process shown in figure to neutralize the donor is not the
main mechanism involved in our experiment with white light. In our experiments the neutralization
of the donor comes from the capture of the electrons from the conduction band.
92
4.3
4. Electronic properties and metastability effects
Defect response
In order to characterize the intrinsic defects of Cu2 O we can use capacitance measurements on junctions. A very powerful capacitance transient spectroscopy is the
DLTS. It has been already used for Cu2 O, but due to the slow response of the compensating donors, it gave informations only for acceptors, as we have seen in 2.11.2.
A simpler technique which can probe such slow response is the measurement of a
few capacitance transients taken at different temperatures, a technique widely used
before the DLTS era [740].
Before this analysis we will show the results of two experiments which demonstrate
that acceptors have a fast response time while the donors have a response so slow
that they are unable to follow the DC bias variation.
4.3.1
Admittance Spectroscopy
Generally a first characterization of the defect response can be obtained from the
measurement of the admittance on a large range of temperatures and frequencies [483, 871].
It can be seen that a defect can follow the AC signal, i.e. it can change its charge
state according to the variation of the band bending, if the AC signal frequency
f is lower than a characteristic frequency f0 which depends on temperature and
other intrinsic properties of the material. This periodic variation of the charge in
the defect can contribute to the total differential capacitance of the device. Notice
that the charge variation can occur only for the levels which lie in the lower half
part of the band gap (see page 96).
In principle, for a typical metal/Cu2 O device, reverse biased with V = 0.4 V, we
expect three steps in the capacitance variation: one due to the donors, one due to
the acceptors and finally one due to the carrier freeze-out. This last step occurs
when ρε ω −1 (ρε is the semiconductor dielectric relaxation time). After this
last step the capacitance should approach the geometrical capacitance of the device
which in our case is about
εA
C=
≈ 22 pF
(4.15)
d
where d = 150 µm is the thickness of the device, A = 0.5 cm2 is its active area and ε
is the permittivity of Cu2 O.
The results obtained on a typical Cu2 O junction are shown in Fig. 4.9. The C–f
curves show a single step with the high frequency values approaching the geometrical
capacitance of the device. The capacitance “freeze-out” effect masks the capacitance
decrease due to the acceptor response time. No other capacitance step due to the
donors is visible but this is expected since the donors are very deep. Moreover this
observation leads to conclude that there is not any donor level with a fast response
time in the lower half part of the band gap.
Even if the donors are not able to follow the AC signal, it is important to
determine at what temperature they become able to follow the DC bias variation.
To check this point we have used a technique similar to Thermally Stimulated
Capacitance (TSCAP) [740], see Fig. 4.10 .
In the first run the sample is equilibrated at −1.4 V at 400 K and then it is cooled
to 100 K. Then the bias is changed to −0.4 V and the temperature is cycled up to
4.3 Defect response
93
0.7
ScCu90A
Cu/Cu2O Schottky junction
0.6
C
( nF )
0.5
360 K
0.4
0.3
0.2
200 K
0.1
0.0
2
3
10
4
10
10
f
5
6
10
10
( Hz )
Figure 4.9. Admittance spectroscopy performed varying the frequency at fixed temperature
for several temperatures. Capacitance versus frequency at different temperatures (from
200 K to 360 K with 20 K steps).
0.7
ScCu90A
0.6
Cu/Cu O
2
Schottky junction
C
( nF )
0.5
0.4
0.3
0.2
equil. -0.4 V, meas. -1.4 V
equil. -0.4 V, meas. -1.4 V
0.1
equil. -1.4 V, meas. -0.4 V
0.0
160
200
240
280
T
320
360
400
( K )
Figure 4.10. Capacitance–Temperature cycles. The green curve and the black dots refer
to the sample equilibrated at −0.4 V and measured at −1.4 V while the red curve refers
to the sample equilibrated at −1.4 V and measured at −0.4 V.
94
4. Electronic properties and metastability effects
300 K and back to 100 K. If the donors were able to equilibrate to the new DC bias
an hysteresis should appear. The absence of any hysteresis means that, up to 300 K
and on the minutes time scale, the hole capture rate from donors is negligible. Then
we have repeated the experiment equilibrating the sample at −0.4 V and changing
the bias to −1.4 V. In this case also no hysteresis is seen meaning that the hole
emission rate by donors is negligible also. It must be noted that if the maximum
cycle temperature is raised to 420 K the hysteresis appears (black dots in Fig. 4.10).
We conclude that there are no donors in the lower half of the gap other than
those usually responsible for the metastability effects of Cu2 O. We conclude also that
during C −V measurements at 300 K the charge in the donors can be considered fixed
and therefore the standard C − V analysis (based on the calculation of d(1/C 2 )/dV )
is useful and gives a charge profile corresponding to Nfix (x) = NA (x) − ND+ (x).
4.3.2
Capacitance transients
Until now we showed experiments which confirm that the donors occupancy is
unchanged up to room temperature. However in order to have an estimation of their
capture cross section we have measured the capacitance transient after a sudden
change of reverse bias on the equilibrated sample at different temperatures.
The experimental technique
This technique is well know and it was extensively used in the past, before the DLTS
era, in order to measure the cross sections and energy levels of the defects of a
semiconductor [740].
A typical result of this experimental technique is reported in Fig 4.11. The
capacitance is monitored as a function of time and at certain points the bias voltage
is suddenly changed. Until the point A the junction is at equilibrium, at the instant
tA the junction is reverse biased and a rapidly decrease of capacitance is observed
(B). After the junction has reached the new steady–state (C) with the new applied
voltage, the bias is re-established at the previous original value and consequently
the capacitance show a rapid increase (D). After a certain amount of time the
capacitance reaches the original value (E=A).
The capacitance behaviour shown in Fig. 4.11 is that expected in a typical
material with a deep level and can be easily explained in terms of emission and
capture processes.
We know that in the depletion layer approximation the junction capacitance can
be written as
s
r
eεr ε0
Nscr
C=A
(4.16)
2
Vbi − V
where A is the area of the device, e is the module of the electron charge, εr is relative
permittivity of the material, ε0 is the static permittivity, Nscr is the net fixed charge
in the space charge region, considered spatially constant, Vbi is the built-in potential
of the junction and V is the applied bias.
In the simple electronic model of Cu2 O, Nscr = NA− − ND+ . NA− can be
approximated with the total concentration of acceptors NA if we perform all the
4.3 Defect response
95
D
2.0
1.5
V
(a.u.)
C
V =V
V <V
A
B
1.0
D
A
A
A
E
C
0.5
0.0
B
-0.5
0
100
200
t
300
400
(a.u.)
Figure 4.11. A typical result of a capacitance transient experiment. The temperature
is fixed. In the point A the sample is in the steady state with a certain applied bias.
Suddenly the reverse bias is increased and the capacitance goes in the state B in a time
of the order of carrier sweep out. After a certain amount of time the sample reach the
new steady state. The reverse process, when the reverse bias is set to the original value,
is described by the points from C to E.
experiment under reverse bias. Then we can write
s
C=A
eεr ε0
2(Vbi − V )
s
s
ND (1 − fD (t))
ND (1 − fD (t))
= C0 1 −
1−
NA
NA
(4.17)
where fD is the occupancy function for donors.
If the donor concentration is much smaller than the acceptor one, we can take
the first order expansion in Taylor series of the square root.
C ≈ C0 1 −
ND (1 − fD (t))
2NA
(4.18)
Now we have to find an expression for fD (t). The rate equation which governs
the occupancy of the donor level can be analytically solved only in the simple case
of constant n and constant p. This is reasonably true in the quasi–neutral charge
regions. In this case we have
fD (t) = fD (∞) + (fD (0) − fD (∞)e−t/τ )
(4.19)
where fD (0) is the initial condition,
fD = τ (ep + ncn )
(4.20)
τ = (en + ncn + ep + pcp )−1 .
(4.21)
and
96
4. Electronic properties and metastability effects
In Fig. 4.12 the band scheme for our simple model is sketched for the four point,
A, B, C, D, of the Fig. 4.11. In these figures and in the following analysis of our
experiments the Fermi level is supposed to be constant even when the device is
reverse biased. This assumption is valid in a region extending from the bulk to the
point where the energy difference between the Fermi level and the top of the valence
band is equal to the barrier height of the device: EF − Ev ≤ ΦB . For ideal Schottky
junction where ΦB = ΦM − χ with ΦM the work function of the metal and with χ
the electron affinity of the semiconductor.
In the remaining part of the space charge region, near the interface, the Fermi
level bends keeping EF − EV ≈ ΦB .
This picture is valid not only for Schottky junctions but it can be applied also
to the other category of our devices, the heterojunctions, because their junctions are
made with a degenerate semiconductor (usually ZnO). Therefore, for our purposes,
a degenerate semiconductor can be considered like a metal.
From the measurement of the open circuit voltage on our Schottky junctions and
heterojunctions solar cells, Vbi ≈ 0.6 V, and considering that the quantity EF − EV
is about 0.3 eV in the bulk we obtain a value of ΦB ≈ 1 eV ≈ Eg /2 [542]. This means
that if an electronic level in the Cu2 O bulk is situated over the half of the band gap,
in a device its occupancy cannot be changed by reverse bias application. Moreover
this means also that no inversion layer exists at the interface and therefore that the
electron concentration is always negligible in these devices.
Before tA the device is in the steady–state under the applied voltage V (point
A) and with a donor occupancy f (A) ≈ 0. Straight after the application of the
reverse bias at time tA the material is in the state represented in B where only the
free carriers have been depleted resulting in a greater depletion region and therefore
in a smaller capacitance.5 Now the donors have to be filled with electrons by hole
emission or electron capture process, however this second process is neglected because
n ≈ 0 in our case. This transient can be described with the following quantities:
fD (0) = f (A),
fD (∞) = f (C) = ND ep τ,
τ = (en + ep )−1 .
(4.22)
After the device reaches the equilibrium (point C) the reverse bias is removed
and external condition returns identical to the initial ones (point D). Now the donors,
in order to reach the steady state, which is identical to the original one, have to
become positives or by hole capture or by electron emission. The return to the
steady–state is described by
fD (0) = f (C),
fD (∞) = f (D) = f (A) = ND ep τ,
τ = (en + pcp + ep )−1 . (4.23)
Substituting these expression of fD (t) in the Eq. (4.18) we obtain always a
single exponential both in the decreasing and increasing of capacitance. Fitting
the experimental results we can obtain ND (1 − fD (0))/NA and the capture cross
sections.
Results and analysis with a simple electronic model
We have performed capacitance–transient experiments on several devices, some based
on Schottky junction (ScCu90A, ScAl90A) and some on heterojunctions (C80F,
5
The depletion process takes place in few hundreds of picoseconds: t = W/vp , W ≈ 1 µm,
vp = Eµp , E ≈ 104 V/cm, µp ≈ 100 cm2 /(V s).
4.3 Defect response
97
(B)
(A) = (E)
E
c
C
n
E
C
E
D
E
E
E
F
D
E
A
E
E
F
V
A
E
V
x
e
x
W
A
A
x
D
A
B
B
D
D
Cu O
ZnO
ZnO
W
p
2
Cu O
2
(D)
(C)
E
E
C
C
e
E
n
E
D
D
E
E
F
A
E
E
F
E
A
V
E
V
W
c
C
x
ZnO
Cu O
2
p
C
x
D
D
ZnO
D
x
C
W
D
D
Cu O
2
Figure 4.12. Simple representation of the band scheme corresponding to the four points of
Fig. 4.11.
98
4. Electronic properties and metastability effects
C85C). We used a frequency of 19 kHz and an amplitude of 20 mV for the probe
signal superimposed to a reverse bias given by



−0.4 V
if t < tA
V = −1.4 V if tA < t < tC


−0.4 V if t > t
C
We could use V = 0 V for t < tA and t > tC but we chose a reverse bias even in these
cases in order to reduce the interface states contribution. In fact interface states are
most likely present at the metal/Cu2 O junction and their occupancy variations can
influence the capacitance measurements. However, if we keep our devices always
under reverse bias, the region near the interface remains in deep depletion condition
and therefore no variation of their occupancy occurs during the capacitance transient
measurements. A typical result for the Schottky junction ScCu90A is reported in
Fig. 4.13
If we analyze the absolute values of the capacitance, we observe that the fast
capacitance variations (between the points A and B and the points C and D) are
consistent with the new applied bias considering only the free carriers sweep out.
Instead the slow variations are due to change of occupation or density of the deep
defects.
All the transients are fairly well described by a single exponential and their time
constants are reported in Tab. 4.3 and represented in an Arrhenius plot in Fig. 4.14
(τ1 refers to the increasing capacitance transient and τ2 to the decreasing one). The
activation energy and prefactor for τ1 and τ2 are also reported
τ = τ0 e−Ea /(kT )
(4.24)
In the same figure we report also the PPC decay time constant which has values
very similar to those of τ2 suggesting that they are due to the same phenomenon.
Finally we have monitored the capacitance transient measurement of a preilluminated
junction. In particular we have illuminated the sample C85C under reverse bias
V = −1.4 V for about 10 minutes and then we have measured the capacitance as
a function of time soon after the light was switched off and at the same time the
voltage was set to V = −1.4 V. We called this time constant τ2op .
From the experimental data we can conclude that:
• τ1 () is always less than τ2 (N);
• τ2 (N), τPPC (•) and τ2op (F) are very similar and with similar activation
energies.
From the latter observation we can reasonably conclude that, as it is expected,
the τ2 is governed by the same mechanism of the PPC discharge. However the data
show some scattering which we explain with the ageing of the device. It is known
the the devices based on Cu2 O undergo to an aging process which is accelerated
by long annealing at high temperature. In particular they show a decrease of their
capacitance and a increase of both τ1 and τ2 . This phenomenon can be the cause
of the scattering of the data and it seems to be confirmed by the experiments
on the cell C85C which has been measured before and after a long annealing at
4.3 Defect response
99
1.8
2
(nF/cm )
1.6
C/A
1.4
ScCu90A
1.2
Cu/Cu O Schottky junction
2
T = 353 K
Experimental data
1.0
Exponential fit
0
200
400
t
600
800
1000
1200
(min)
2.0
C/A
2
(nF/cm )
1.8
1.6
ScCu90A
1.4
Cu/Cu O Schottky junction
2
T = 373 K
1.2
Experimental data
Exponential fit
1.0
200
300
t
400
500
600
(min)
2.4
2.2
C/A
2
(nF/cm )
2.0
1.8
1.6
ScCu90A
Cu/Cu O Schottky junction
2
T = 393 K
1.4
1.2
Experimental data
Exponential fit
1.0
50
60
70
t
80
90
100
110
(min)
Figure 4.13. Experimental results of a capacitance transient experiment performed on a
Schottky device based on Cu2 O. The transients have been measured at three different
temperatures.
100
4. Electronic properties and metastability effects
Table 4.3. Time constants of capacitance transient experiments.
device
Ea (eV)
τ0 (s)
ScCu90
τ1
τ2
T = 393 K
177
695
T = 373 K
1020
4760
T = 353 K
5050
52383
1.00
1.29
2.8 × 10−11
1.7 × 10−14
C80F
τ1
τ2
T = 393 K
148
775
T = 373 K
788
4423
T = 353 K
5407
30745
1.06
1.10
3.53 × 10−12
6.06 × 10−12
ScAl90
τ1
τ2
T = 399 K
29
227
T = 378 K
268
2041
T = 356 K
2765
20132
1.31
1.29
7.5 × 10−14
1.1 × 10−14
C85C new
τ1
τ2
T = 399 K
41
296
T = 378 K
303
2412
T = 356 K
3126
25512
1.23
1.27
1.06 × 10−14
2.85 × 10−14
C85C aged
τ1
τ2
T = 399 K
132
434
C85C preill
τ2
T = 399 K
488
T = 378 K
5303
T = 356 K
48330
1.30
1.74 × 10−14
5
10
,
2
op
2
and
PPC
4
(s)
10
3
1
10
1
2
1
2
1
2
10
2
1
2
1
2
op
2
ScCu90
C80F
ScAl90
C85C
C85C aged
C85C preilluminated
PPC
2.5
2.6
2.7
2.8
T
1000/
2.9
3.0
-1
(K )
Figure 4.14. Arrhenius plot for the time constants obtained from the capacitance transient
experiments.
4.4 Failure of the model of Lany and Zunger applied to Cu2 O
101
high temperature. We have to note that this ageing phenomena do not proceed
indefinitely and therefore after the maximum ageing the capacitance and the time
constants do not vary anymore.
Two aspects of the capacitance transients are inconsistent with the simple
electronic model. The first one is the magnitude of the transient from the point B
to the point C. Indeed in the deep depletion region the occupancy of the donor level
is given by
ep
.
(4.25)
fD =
en + ep
Using the values ED = 0.91 eV, σp 10−25 cm2 and σn ≈ 10−15 cm2 which we
derived to explain the PPC, we obtain
fD (353 K) 3.5 × 10−8
fD (373 K) 5.1 × 10−9
fD (393 K) 8.0 × 10−10
(4.26)
This means that the steady state occupancy of the donor would be always nearly
zero and therefore the capacitance increase from point B to point C would not be
detectable.
Another inconsistent aspect of the capacitance transients with the simple electronic model is that τ1 is always less than τ2 .
As we have discussed before in both cases
τ = (en + ep + ncn + pcp )−1
(4.27)
Therefore since during the increase of the capacitance (from B to C with time constant
τ1 ) the hole concentration is smaller than during the decrease of the capacitance
(from D to E with time constant τ2 ), it is obvious that τ1 τ2 must hold.
Another fundamental aspect of the capacitance transient is the dependence of
their time constants on the applied reverse bias. It is well known [740] that, if one of
the transient is due to an emission mechanism, this mechanism can be enhanced by
a greater electric field which can lower the barrier for the emission. We will discuss
this phenomenon after the presentation of our new model for Cu2 O.
4.4
Failure of the model of Lany and Zunger applied to
Cu2 O
In this section we want to apply the accepted model [457] for the metastability
effects showed by chalcopyrites to Cu2 O. Indeed these materials are very similar to
Cu2 O. They are wide gap semiconductor and are spontaneously p-type. They show
the persistent photo–conductivity.
Mainly two facts have to be explained: the increasing of the acceptor concentration after the illumination and the τ1 shorter than τ2 in the capacitance transients.
We will demonstrate that even this model can not explain the experimental
results.
4.4.1
The model of Lany and Zunger
In the chalcopyrites, like CIS, CGS and intermediate alloys CIGS, the metastability
effects, both induced by the illumination and by the reverse bias in the devices, are
102
4. Electronic properties and metastability effects
explained by the model of Lany and Zunger [457]. Here we will roughly present this
model in the case of CGS but it is completely analogous for the other compounds.
In the CGS the p–type conductivity is given by the presence of the copper vacancies, VCu , while the defects responsible for the metastability effects are complexes
formed by a selenium vacancy and a copper vacancy, (VSe − VCu ).
Theoretical predictions, see Fig. 4.15, show that the copper vacancy introduces
a simple acceptor level near the valence band. On the other hand they also show
that the selenium vacancy VSe is a double donors with a negative correlation energy,
+
U < 0 (there is only the transition 2 + /0 which means that the VSe
state is unstable
0
and quickly captures an electron, shifting to the VSe state). Moreover VSe introduces
other two deep acceptor levels and therefore it is an amphoteric defect.
These calculations predict also the presence of the complex (VSe − VCu ). The
binding energy is such that at room temperature all the selenium vacancy, which
are in concentration smaller than the copper vacancies, can be considered bound to
a copper vacancy.
The energy levels introduced by the (VSe − VCu ) complex are clearly related to
those of the VSe because the charge states of these two defects differ only by one
negative charge due to VCu . The transition energies are however shifted toward the
conduction band, an effect which could be partly due to the electrostatic interaction.
Since the transition energy between the shallow donor level (green line) and
the shallow acceptor level (red dashed line) is located at an energy greater than
the typical Fermi level in this material, at equilibrium the complex is in the donor
configuration (VSe − VCu )+ .
The negative correlation energy of this defect is due to the fact that when it is
filled with an electron (by illumination or reverse bias) it goes in the (VSe − VCu )0
configuration which, however, is highly unstable. A great relaxation of the distance
of the surrounding gallium ions occurs and the donor level, indicated with the
letter a shifts in the valence band after the relaxation. For this reason the complex
spontaneously emits an hole and goes in the acceptor configuration (VSe − VCu )− .
See Fig. 4.16.
This latter transition produces an increment of the acceptor concentration like
observed in the persistent photo–conductivity experiments. Moreover this state is
metastable because, in order to come back to the original donor configuration, it has
to capture simultaneously two holes. This capture process is further slowed down by
the presence of the barrier energy denoted with E2 in Fig. 4.16.
In addition to the light, the donor/acceptor transition can be triggered raising
the Fermi level by the reverse bias application. The transition occurs when EF
becomes greater than the level indicated with ε(+/−).
4.4.2
The model of Lany and Zunger for Cu2 O
Now we want build a model similar to that of CGS for Cu2 O where we replace the
complex (VSe − VCu ) with a complex composed by a copper vacancy and an oxygen
vacancy, (VO − VCu ).
An hypothetical configuration coordinate is represented in Fig. 4.17. For EF <
ε(+/−) the ground state of the complex is the donor configuration (1) while for
EF > ε(+/−) the stable state is the complex in the acceptor configuration (3).
4.4 Failure of the model of Lany and Zunger applied to Cu2 O
103
Figure 4.15. Defect formation energies
in CGS as a function of Fermi level.
After [457]. The solid circles mark the
equilibrium transition energies ε(q, q 0 ).
In the lower part of the plot the binding
energy of the defect complex (VSe −
VCu ) is reported.
Figure 4.16. a) Schematic energy level
diagram for the (VSe − VCu ) vacancy
complex in CGS. b) Configuration coordinate diagram for CGS. The red
square indicates the acceptor configuration.
104
4. Electronic properties and metastability effects
Figure 4.17. Possible configuration coordinate diagram for Cu2 O according to the similar
model of Lany and Zunger for chalcopyrites. On the right the corresponding band
diagram is shown.
The (1) and (3) are linked by a neutral configuration (2) which is highly unstable.
The energy level in the band gap corresponding to the transition + → 0 is called
ET and represents the minimal thermal energy required to fill the complex with one
electron.
Following the model of Lany and Zunger, we assume that the neutral state has
a negative correlation energy (U < 0) which gives rise to the level ET + U in the
valence band. As the neutral level is resonant in the valence band the complex
has an high probability for the emission of another hole (e0p ) which can bring the
complex in the acceptor state (VO − VCu )− (3).
Under illumination the complex captures an electron, becomes neutral and goes
spontaneously in the acceptor configuration emitting an hole. Instead in the devices
if the reverse bias is sufficiently high that EF > ε(+/−) the complex which is in the
state (VO − VCu )+ emit two holes going in the acceptor state (VO − VCu )− .
Reasonable values for the parameters are ε(+/−) = 0.75 eV, U = −2 eV and
therefore [749] ET = ε(+/−) − U/2 = 1.75 eV and ET + U = ε(+/−) + U/2 =
−0.25 eV.
We point out that this model, with a correct choice of the parameters, can
explain the increase in the acceptor concentration observed in the Cu2 O sample
after the illumination.
Now we want to investigate if it can reproduce also the observed behaviour of
the capacitance transients, that is τ1 < τ2 .
4.4 Failure of the model of Lany and Zunger applied to Cu2 O
105
Capacitance transients
We call the probability to find the the amphoteric complex (VO − VCu ) in the neutral,
positive and negative states respectively f 0 , f + and f − . They are related by
f0 + f+ + f− = 1
(4.28)
The rate equations for the f − and f + functions are
 −
df

−
0 0
0

= −f − (e−

n + pcp ) + f (ep + ncn )
dt
(4.29)
+


 df = +f 0 (e0 + pc0 ) − f + (e+ + nc+ )
n
p
p
n
dt
which, using the relation between the occupancy probabilities, can be written as
 −
df

−
0
0
+ 0
0
0
0

= −f − (e−

n + pcp + ncn + ep ) − f (ep + ncn ) + (ep + ncn )
dt
+


 df = −f − (e0 + pc0 ) − f + (e0 + pc0 + e+ + nc+ ) + (e0 + pc0 )
n
p
n
p
p
n
n
p
(4.30)
dt
This system of differential equations can be re-written with the matrix form
Y 0 = AY + Q
where
!
Y =
f−
,
f+
and where
(4.31)
!
A=
−a −b
,
−c −d
!
b
c
Q=
(4.32)
−
0
0
a = e−
n + pcp + ncn + ep
b = nc0n + e0p
(4.33)
c = e0n + pc0p
+
d = e0n + pc0p + e+
p + ncn
Notice that the previous coefficient are not all independent because they are
related by the principle of detailed balance.
The solution of the system are obtained diagonalizing the matrix:

λβ + d
b(d − c)
λα + d

−


f
=
−
k1 eλα t −
k2 eλβ t

ad − bc
c
c

c(a − b)

+

+ k1 eλα t + k2 eλβ t
f =
(4.34)
ad − bc
where λα and λβ are the eigenvalues of the matrix A:
λα,β
a+d
=−
1∓
2
s
ad − bc
1−4
(a + d)2
!
(4.35)
and k1 and k2 are two integration constants determined by the initial conditions.
As it is easily seen the evolution of the donor occupancy are described by two
exponential with time constant τα = 1/λα and τβ = 1/λβ . The greater between the
106
4. Electronic properties and metastability effects
two eigenvalues, λβ , it is extremely large with reasonable assumption and therefore
gives a very short τβ . Indeed, as the acceptor level is resonant in the valence band,
0
e−
n can be neglected while ep is the most important process of the neutral defect.
From these considerations derives that
a, b c, d
(4.36)
λβ λα ⇒ τβ τα .
(4.37)
and therefore
The large difference is compatible with the fact that in the capacitance transient
experiments only one time constant is observed. However it has to be necessary that
the integration constant kβ is very small or almost zero otherwise no capacitance
transient would be observed in the experiments.
Now we want to deduce two approximate expression for the capacitance transient
characteristic times, τ1 and τ2 , according to this model.
During the first capacitance transient, when the capacitance increasing is described by the time constant τ1 the concentration of the carriers, both hole and
electrons, can be neglected and we can write
a ≈ e0p ,
b ≈ e0p ,
c ≈ e0n ,
d ≈ e0n + e+
p
and therefore
λα ≈ −e+
p
τ1 = τα ≈
(4.38)
1
e+
p
(4.39)
For the second capacitance transient, n can be considered again zero but p can
not be neglected anymore:
a ≈ e0p + pc−
p,
b ≈ e0p ,
c ≈ e0n + pc0p ,
0
d ≈ e0n + e+
p + pcp
(4.40)
and therefore
pc−
p
+
(e0 + pc0p + e+
λα ≈ −
p ) + ep
e0p n
"
#
τ2 = τα ≈
1
pc−
p
e0p
+
(e0n + pc0p + e+
p ) + ep
(4.41)
Thus, even in this case, τ1 > τ2 which is incompatible with the experiments.
From numerical simulation over a wide range of variability of the parameters, it
seems that the inequality τ1 > τ2 is a general rule valid for all ideal Schottky junctions
where the minority carriers can be neglected and in which the deep levels can only
emit or capture carriers. This is intuitive because in the second transient, when the
majority concentration rapidly increases because the reverse bias is removed, the
number of processes which help the junction to reach the equilibrium is increased.
We conclude that no electronic model of Cu2 O, based exclusively on deep levels
which can only emit or capture the carriers, can explain the behaviour of the
capacitance in the capacitance transient experiments.
In the next chapter we will present a new model which instead is compatible
with all experiments shown up to now.
Chapter 5
A model based on
donor–acceptor complexes
We have seen in the previous chapter that the models of Cu2 O, based on one or
more electronic levels which can only emit or capture carriers, can not explain the
experimental results.
In this chapter we will introduce a new model, based on association and dissociation of the defect complex (VCu − VO ), which describe the observed opto-electrical
properties of Cu2 O.
5.1
The idea
In semiconductors, the association of two or more point defects to form defect
complexes is very common [646]. In the case of an ionized donor and an ionized
acceptor, the lowering of the total energy through the formation of a bound complex
is particularly obvious: the two defects attract each other through electrostatic
forces. Another simple process is the aggregation of neutral vacancies where the
internal surface of the crystal is reduced in order to lower the total energy.
In Cu2 O associates of point defects have been hypothesized in various papers. In
chapter 2 we have already encountered the (VCu − VCu ) complex, proposed by Kužel
to explain the PPC and by O’Keeffe to explain the missing EPR signal. (VCu − VCu )
together with (VCu − VO ) complex was also proposed by Bloem to explain the
observed anomalies in the conductivity at low temperature.
5.1.1
Shift of energy levels upon pairing
In order to understand the effect of the electrostatic interaction on the electronic
levels of the isolated defects, now we will describe the Iron–Boron complexes in the
p–type silicon.
The boron is a substitutional impurity for silicon and in the isolated configuration
introduces an acceptor level B 0/ – at 0.045 eV from the valence band. On the other
hand the iron is an interstitial impurity which acts as a double donor and in the
isolated configuration introduces two deep donor levels, Fe +/0 at 0.38 eV from the
valence band and Fe ++/+ resonant in the valence band. See Fig. 5.1.
107
108
5. A model based on donor–acceptor complexes
Figure 5.1. Band diagram of p–type silicon with boron and iron impurities. After [357].
When an iron and a boron interact by the electrostatic force their ionization
energies, in other words their electronic levels in the band gap, are modified. If the
defects are far enough we can think that the electrons are localized on the defects.
In this case the acceptor level shifts downward by the quantity
∆EB =
qFe
4πε0 εr r2
(5.1)
where r is the distance between the donor and the acceptor and qFe is the iron charge
(0e, +1e or +2e). The same applies to the iron levels which shift upward by the
quantity
qB
∆EFe =
.
(5.2)
4πε0 εr r2
When the distance decreases the electrons can not be considered anymore on
one or the other defect and we can think them as an effective donor–acceptor
complex W ≡ (Fei −BSi ) with four charge states: ++, +, 0, −. The corresponding
energy levels are difficult to be calculated. First of all, when two defects are nearest
neighbours, the energy shift of their levels can be greatly influenced by the lattice
relaxation around them and moreover they cannot be considered point charges
anymore, therefore complex calculations are needed to obtain reliable results. Then
we have to point out that even the neutral defects can interact electrostatically at
low distance. This interaction is similar to a Van der Waals force for defects.
However the levels of the iron–boron complex can be estimated by a simple
approach. Let us consider the complex without its valence electrons W ++ =
(Fe ++ −B 0 ), with iron and boron at their minimum distance, r = 0.235 nm. If we
assume that the energy levels of the iron and boron influence each other only by the
electrostatic force, the lowest complex level, W ++/+ , corresponds to the boron level
shifted downward by the presence of the doubly positive iron, as shown in Fig. 5.1.
5.1 The idea
109
Figure 5.2. Schematic representation of the barrier heights which play a fundamental role
in the association and dissociation dynamics of charged point defect complexes.
Therefore if we rise the Fermi level over the level W ++/+ the complex becomes W +
and if we assume that the electron can be assigned to only one of the two components,
the boron, the other two energy levels of the complex, W +/0 and W 0/− , are simply
given by the iron energy levels shifted upward by the presence of the singly negatively
0/−
charged boron. We can therefore imagine that W ++/+ ≡ (Fei++ −BSi )++/+ ,
++/+
+/0
− +/0
− 0/−
W +/0 ≡ (Fei
−BSi
)
and W 0/− ≡ (Fei −BSi
)
as reported in Fig. 5.1.
5.1.2
Dynamics of pair association and dissociation
The dynamics of association and dissociation of the charged point defects can be
analyzed considering a model in which the electrostatic potential is convoluted with
the periodically varying potential of the host crystal, as represented in Fig. 5.2.
This model is characterized by three fundamental energies, Eb , Ediss and Em .
Eb is the binding energy of the complex, that is the difference between the total
energy of the two defects at infinite distance and the total energy of the same defects
when they are bound. Ediss is the dissociation energy of the bound complex, defined
as the energy required to overcome the barrier between the next nearest neighbour
site. Finally Em is the migration energy of the most mobile isolated ion, that is the
height of the energy barrier that an isolated defect has to overcome to jump from a
lattice site to another nearest neighbour one.
The dynamics of association and dissociation is obviously determined by the
charge of the defects and therefore it can be influenced by the light, the bias voltage
in junction based devices and the temperature. However this dynamics is frozen if the
temperature is not enough high to permit the migration of the defects (kT Em ).
Instead, if the temperature is such that the thermodynamic equilibrium can
be reached in a reasonable time, the association and dissociation of isolated point
defects to form complexes can be described by the following chemical equation
Fe+ + B−
(Fe−B) 0
(5.3)
whose mass action law reads
[(Fe−B) 0 ]
keq
Z Eb /(kT )
=
=
e
+
−
[Si]
[Si]
[Fe ][B ]
(5.4)
110
5. A model based on donor–acceptor complexes
where Z is the number of possible configurations of complex with the same energy
and [Si] is the concentration of the silicon lattice sites.
While the energy Eb is responsible for the equilibrium concentrations the energies
Em and Ediss are related with the dynamics to reach this equilibrium.
The dynamics of associations and dissociations can be described with a random
walk of the mobile point defects in which the electrostatic interaction is considered.
However this very interesting approach is very complex. A simpler approach, assumes
a continuum distribution of the mobile species, described by a density ρ(~x, t). This
theory, initially developed by Ham [286] to describe the precipitation of the solute
in a supersaturated solution, was then used to describe the kinetics of association of
point defects in semiconductors by several authors [702].
According to this theory the concentration of the mobile species reaches the
equilibrium value as an exponential function with a time constant
τass =
1
4πDN Rc
(5.5)
where D is the diffusion coefficient of the mobile species, N is the volume concentration of the fixed species and Rc is the distance from the fixed ion within the mobile
ion is considered captured.
According to Reiss [702], for Rc we can use simply the distance at which the
thermal energy equals the coulombian attraction
Rc =
qA qD
4πε0 εr kT
(5.6)
where qA and qD are the charges of the acceptor and the donor respectively.
Substituting the previous equation in the Eq. (5.5), we obtain
τass =
ε0 εr kT
ε0 εr kT
=
eEm /(kT ) .
DN qA qD
D0 N qA qD
(5.7)
where in the second expression we have used
D = D0 e−Em /(kT ) .
(5.8)
Therefore, the measurement of τass as a function of temperature can give useful
information about the migration energy barrier Em and about the concentration of
the fixed species.
On the other hand, if the concentrations of the isolated defects at a given
temperature are smaller than the equilibrium value, the complexes have to dissociate.
Also the dissociation dynamics follows an exponential law with a time constant
τdiss = τ0 e−Ediss /(kT ) .
(5.9)
Notice that the dissociation kinetics can be greatly influenced, for example, by
the illumination or by carrier injection. In p–type silicon, for the iron–boron
pairs [357, 422] an activation energy much smaller than Em was observed when
minority carriers were injected. An explanation of this fact is that, since the complex
is usually a deep defect which can act as a recombination center, the energy released
by the electron–hole recombination on the complex is converted in vibrational energy
which can help the mobile species to dissociate from the complex.
5.2 The model based on the complexes (VCu – VO )
5.2
111
The model based on the complexes (VCu – VO )
We want now to develop a model based on a mechanism analogous to that involved
in the dynamics of iron–boron complexes in silicon. Therefore in this model we must
include the pairing between a double donor and an acceptor. Even if the nature of
the donors in Cu2 O is not clear, we assume that the acceptor is the copper vacancy
while the double donor is the oxygen vacancy.
In this hypothesis the mobile species is the copper vacancy while the oxygen
vacancy is thought to be fixed. In fact we know that the diffusion coefficient of the
copper vacancies at high temperature is (see Sec. 2.8)
DVCu = 3.0 × 10−3 e−0.55 eV/(kT ) cm2 s−1 .
(5.10)
Instead the diffusion coefficient of the oxygen vacancies is unknown: it can not
be derived from the self-diffusion coefficient of the oxygen because this diffusion
process proceeds by an interstitial mechanism and not by a vacancy mechanism.1
On the other hand from the deviation of stoichiometry at low oxygen pressure we
have seen (Sec. 2.8) that the oxygen vacancies are the dominant defect. Haugsrud and
Kofstad [303] observed that even at low oxygen pressure the oxidation mechanism
is governed by the diffusion of copper vacancies. From these two facts it can be
deduced that, at high temperature, the diffusion coefficient of the oxygen vacancies
is much smaller than that of the copper vacancies.
Unfortunately at low temperature the defect diffusion coefficients in Cu2 O are
quite difficult to be measured and therefore they are not known. However, since it
seems quite likely that the activation energy of the copper vacancy diffusion is smaller
than that of the oxygen vacancy, we can conclude that even at low temperature
DVCu DVO .
The copper vacancy and the oxygen vacancy can form a complex (VCu − VO ),
called W , which can exist in four possible charge state: ++, +, 0, −. Assuming that
the interaction is mainly electrostatic, only two charge configurations are stable, W +
and W 0 , because in the other cases one of the two defects is neutral.
The binding energy of these complex can be estimated if the distance between
the two charge distributions of the vacancies, RDA , is known.
If we assume the charge of the copper vacancy located on a nearest neighbour
oxygen site, and the charge of the oxygen vacancy located in the middle of the three
nearest neighbour copper atoms, we can state that RDA ≈ 0.48 nm (see Fig. 5.3).
This value of RDA gives a binding energy of 0.42 eV for the complex W 0 and a
binding energy of 0.84 eV for the complex W + . Finally using the Eq. (5.4) we can
obtain the fraction of paired complex at a given temperature, f . At 393 K, if we
−
assume an acceptor concentration [VCu
] ≈ 1015 cm−3 we have
−
Z[VCu
] E + /(kT )
[W + ]
=
e b
≈ 4200
++
[Cu]
[VO ]
(5.11)
−
Z[VCu
] E 0 /(kT )
[W 0 ]
e b
≈ 2 × 10−2
=
+
[Cu]
[VO ]
(5.12)
f (W + ) =
and
f (W 0 ) =
1
However this result is an indication of the fact that the diffusion coefficient of the interstitial
oxygen is greater than that of the oxygen vacancies.
112
5. A model based on donor–acceptor complexes
Figure 5.3. Cu2 O lattice: a) ideal; b) with a copper vacancy; c) with a complex formed
by an oxygen vacancy and a copper vacancy. In b) is represented the distance between
two oxygen and in c) the distance between the charge centers of the copper vacancy and
oxygen vacancy.
where we have used [Cu] = 5 × 1022 cm−3 for the copper site density in Cu2 O and
Z = 4 for the number of possible configurations of the copper vacancy around an
oxygen vacancy.
This calculation shows that in the temperature range of our experiments, the
complex W + is stable while the complex W 0 tends to dissociate.
In our model the isolated copper vacancy is a single acceptor which introduces an
0/−
electronic level VCu at about 0.3 eV from the valence band while the isolated oxygen
vacancy is a double donor, with a positive correlation energy U , which introduces
++/+
+/0
two electronic levels, VO
and VO (see Fig. 5.4).
0/−
The complex W introduces three levels in the band gap: W ++/+ ≡ (VCu −VO++ ),
++/+
+/0
−
−
W +/0 ≡ (VCu
− VO
) and W 0/− ≡ (VCu
− VO ), whose energies are related to
those of the isolated defects by the binding energies found above. At the minimum
distance RDA , the oxygen vacancy levels shift upward by 0.42 eV while the copper
vacancy level shifts downward by 0.84 eV which are represented in Fig. 5.4.
In order to obtain the best agreement with the experiments, the levels of the
isolated oxygen vacancy are chosen in order to have the level W +/0 well above the
Fermi level, which is usually located at about 0.3 eV from the top of the valence
band. In this way the stable electronic configuration of the complex is the W + .
++/+
Moreover the fact that the level VO
is below the Fermi level whereas the level
+/0
VO is above the Fermi level is not accidental (see Fig. 5.4). Indeed this means that
the stable charge state of the isolated oxygen vacancy is VO+ . If the stable charge
state were the double positive one, the complexes would never dissociate. On the
other hand if the oxygen vacancy were always neutral, no complex would exist.
According to this model when a negative copper vacancy during its random walk
meets a single positive oxygen vacancy, they form a complex W 0 . We have seen
before that this complex is not stable, unless it captures an hole, becoming a W + .
Therefore the equilibrium concentrations of the species will be given by a balance
between the capture process of an hole and the thermal dissociation of the neutral
complex.
Now we will analyze the consequences of this model and we will show that it
can explain both the capacitance transient measurements and the persistent photoconductivity. We emphasize the fact the the PPC decay and the second transient of
the capacitance transients can be explained by the same physical mechanism.
5.3 Analysis of carriers vs temperature experiments
113
Figure 5.4. Representation of the band scheme of Cu2 O in the our model based on
(VCu − VO ) complexes.
5.3
Analysis of carriers vs temperature experiments
In this section we will show how to interpret the measurements of carrier concentration
as a function of temperature seen in Sec. 4.1.
According to our model the dark measurement of p(T ) has to be interpreted in
the usual way, with an acceptor at about 0.3 eV from the valence band and a deep
donor. However, while the acceptor is still identified with the copper vacancy, the
donor levels in the new model are due to the complex W + instead of the oxygen
vacancy.
Notice that, as described in the previous section, during the entire measurement
of p(T ) the charge state of the complex is unaltered and the complexes are all stable.
Now we can state the correspondence between the values obtained in Tab. 4.1 for
the equilibrated curve and the quantities of our model. The acceptor concentration
NA corresponds to the copper vacancy concentration [VCu ]. This number does not
include the vacancies which are bound in the complexes because these last ones do
not act as acceptors.
Instead, the donor concentration ND corresponds to the total number of positive
complexes which is equal to the total number of the oxygen vacancies [VO ]tot because
we have an oxygen vacancy for each complex.
It is interesting to note that the total number of copper vacancies, both bound
and unbound, is given by [VCu ]tot = [VCu ] + [VO ]tot . Therefore, in order to have
always a p-type Cu2 O, [VCu ]tot has to be always greater than 2[VO ]tot .
Using the data of the curve measured on the equilibrated sample in Tab. 4.1, we
114
5. A model based on donor–acceptor complexes
+
VO
VCu
VCu
VCu
VCu
VO+
Figure 5.5. Pictorial representation of the defects before and after the illumination.
obtain for the sample S81 the following numbers:
[VO ]tot = 4.92 × 1014 cm−3
[VCu ]tot = 1.06 × 1015 cm−3
(5.13)
The interpretation of the preilluminated curve of p(T ) will be given in the next
section.
5.4
Analysis of PPC
Consider a sample of Cu2 O in equilibrium at a given temperature T . As the
concentration of copper vacancies is greater than that of the oxygen vacancies, we
can assume that each oxygen vacancy is coupled with one copper vacancy and a
certain number of copper vacancies remain free. This situation is illustrated in the
first panel of Fig. 5.5.
Now, if the sample is illuminated, the complexes will become neutral, or even
negative, mainly by the capture of the photo-generated electrons. If the hole capture
and the electron emission are not too high, the great part of the complex will
dissociate, because, as we have seen in the previous section, the neutral complex is
unstable at the temperatures of our experiments, and even more unstable will be,
obviously, the negative complex.
However the isolated double charged oxygen vacancy tends to emit an hole
++/+
becoming a VO+ because, as we have seen before, the level VO
is resonant
in the valence band. In this way, after the complex dissociation and after the
light is switched off, the final concentration of donors is equal to that before the
illumination, because the for each W + only one VO+ is formed. On the other hand the
concentration of the copper vacancies has increased and this increase is responsible
for the conductivity increase, the PPC (see Fig. 5.5).
Now it is straightforward to understand that the PPC discharge will be related
to the rate of the association of the copper with oxygen vacancies.
The kinetics of re-association, using the theory of Ham (see Sec. 5.1.2), is given
by
d[VO+ ]
d[VCu ]
−
=
= −4πDVCu Rc [VO+ ][VCu
]
(5.14)
dt
dt
5.4 Analysis of PPC
115
−
where [VCu
] is the concentration of isolated and charged copper vacancies, DVCu
is their diffusion coefficient, Rc is the capture radius of the oxygen vacancy for a
copper vacancy and [VO+ ] is the concentration of isolated and singly charged oxygen
vacancies.
Since in Cu2 O the concentration of the copper vacancies is greater than the
concentration of the oxygen vacancies the boundary conditions are:
and
[VO+ ](t = ∞) = 0
(5.15)
[VO+ ](t = 0) = [VO+ ]0 = fdiss [VO ]tot
(5.16)
where fdiss is the ratio of dissociated fraction of complexes and [VO ]tot is the total
number of oxygen vacancies, both isolated and bound in the complexes.
Finally, the last equation needed to solve the system is the neutrality condition
−
p + [VO+ ] + [W + ] = [VCu
]
(5.17)
−
p + [VO ]tot = [VCu
]
(5.18)
which can be re-written as
using the relation [VO ]tot = [W + ] + [VO+ ].
−
It is interesting to note that if p [VO ]tot , [VCu
] is nearly a constant (≈ [VO ]tot )
and the solution of the Eq. (5.14) is a simple exponential. However this condition
can be applied only at the end of the PPC decay transient because at the beginning
p is usually of the same order of magnitude of the oxygen vacancies.
In order to have the neutrality condition in a useful form, it has to be expressed
as a function of [VO+ ]. This can be easily done considering that
−
[VCu
]=
1+
[VCu ]
(E
e A −EF )/(kT )
(5.19)
where [VCu ] is the concentration of isolated copper vacancies, regardless of their
charge state, and that
[VCu ] = [VCu ]tot − [W + ] = [VCu ]tot − [VO ]tot + [VO+ ].
(5.20)
Moreover using the relation p = Nv e−EF /(kT ) and the two previous equations we
can write the neutrality condition (5.18) as
p2 + (aNv + [VO ]tot ) p + aNv (2[VO ]tot − [VCu ]tot − [VO+ ]) = 0
(5.21)
where we have called a = e−EA /(kT ) .
Solving this second degree equation, we obtain p = p([VO+ ]) which can be
−
substituted in Eq. (5.18) to express [VCu
] in terms of the [VO+ ] only.
Finally the differential equation (5.14) can be solved numerically in terms of the
five constant parameters [VCu ]tot , [VO ]tot , EA , DVCu and fdiss .
Before going ahead we can now show how to read the data of the preilluminated
curve in Tab. 4.1. According to our model, the donor concentration ND is the same
both in equilibrated and in preilluminated state because for each dissociated complex
W + , a VO+ is formed. Instead, the acceptor concentration NA in the preilluminated
116
5. A model based on donor–acceptor complexes
14
10
p-p
0
(cm
-3
)
T = 353 K
13
10
T = 373 K
T = 393 K
12
10
0
200
t
400
600
800
(min)
Figure 5.6. Numerical fit of the PPC decay curves for the sample S81. The fit has been
performed solving numerically the system composed of Eqs. (5.14), (5.18), (5.21) and
with the initial condition (5.16).
state is equal to the free acceptor concentration in the equilibrated state plus the
concentration of dissociated complexes, because for each W + , a VCu is formed:
NApreill = NAeq + fdiss NDeq
(5.22)
with 0 ≤ fdiss ≤ 1.
Looking at the data of Tab. 4.1, we can see that indeed the donor concentration is
unchanged from equilibrated to preilluminated curve while the acceptor concentration
shows a clear increase.
In Fig. 5.6 the numerical fit of the PPC decay curves of sample S81 are reported.
We have fixed three of the five parameters with the values obtained from the analysis
of the carrier concentration vs temperature. In particular
[VO ]tot = 4.92 × 1014 cm−3
[VCu ]tot = 1.06 × 1015 cm−3
EA = 0.32 eV.
(5.23)
We can see that the fit is quite good. In Tab. 5.1 we report the parameters of
the fit for all three temperatures. For comparison we report also the experimental
diffusion coefficient extrapolated from high temperature to the temperatures of our
experiments.
A comment about the strong difference between the diffusion coefficient obtained
from the numerical fit and its experimental value extrapolated from high temperature
is necessary.
In Fig. 5.7 the data of the copper diffusion coefficient at high temperature (see
Sec. 2.8), its extrapolation at low temperatures and the values obtained from our fit
are reported.
As we have already seen, the experimental data at high temperatures follow the
law
DVCu = 3.0 × 10−3 e−0.55/(kT ) cm2 /s,
(5.24)
5.4 Analysis of PPC
117
-4
10
D
-5
10
VCu
is indipendent of
oxygen partial pressure
-6
10
-7
D
VCu
2
(cm /s)
10
-8
10
-9
10
2008_Grzesik
-10
10
-11
2008_Grzesik
10
extrapolation
-12
10
-13
PPC decay fit
10
with our model
-14
10
-15
10
1.0
1.5
2.0
T
1000/
2.5
3.0
3.5
-1
(K )
Figure 5.7. Copper vacancy diffusion coefficient. Red squares: direct measure at high
temperature by Grezsik et al. (see Sec. 2.8). Red line: extrapolation at low temperature
of the Grezsik’s data. Black squares: values obtained by the fit of the PPC decay by our
model of complexes.
whereas the data of our fit have an activated behaviour given by
DVCu = 6.2 × 102 e−1.20/(kT ) cm2 /s.
(5.25)
Three reasons can explain this big discrepancy:
• our model is wrong;
• our model is correct but the acceptor species is not the copper vacancy but
another defect with a lower diffusion coefficient
• our model is correct but the values of the VCu diffusion coefficient at high
temperature cannot be simply extrapolated at low temperatures because other
processes become important as, for example, a trapping mechanism.
Most probably this third option has to be rejected because an extremely high
concentration of trapping centers would be needed.
Table 5.1. Parameters obtained by the fit of the PPC curves with our model of complexes.
DVHT
is the copper vacancy diffusion coefficient measured at high temperature and
Cu
extrapolated at the temperature of our experiments in order to compare it to the
diffusion coefficient obtained from the fit, DVfitCu .
T (K)
353
373
393
τPPC (s)
35110
5132
784
Rc (Å)
fdiss
DVfitCu (cm2 /s)
DVHT
(cm2 /s)
Cu
67
63
60
0.54
0.54
0.54
5.10 × 10−15
4.15 × 10−14
2.80 × 10−13
4.21 × 10−11
1.11 × 10−11
2.65 × 10−10
118
5. A model based on donor–acceptor complexes
Another strange result is the value of fdiss . This parameter was intentionally left
free in the fitting of the PPC decay curves. From the p(T ) measurements performed
soon after the device preparation we obtained fdiss ≈ 0.9. The PPC decay transients
were done many days after the p(T ) measurements and the value obtained from the
fit is fdiss ≈ 0.5. This shows that in the samples another still slower degradation
mechanism is operating.
5.5
Analysis of the capacitance transients
The capacitance transients are explained with a mechanism very similar to that used
for the PPC, where the illumination is replaced by the reverse bias. An important
difference is that, while with the illumination the complex can become even negative,
with the bias only the complex can not.
Consider the levels in Fig. 5.4. The level W +/0 of the complex lies below Eg /2
and therefore its charge state can be altered by the application of a reverse bias in a
junction.
If the reverse bias is high enough, the complex, which is in the state W + under
equilibrium, emits an hole and becomes neutral. As we have already seen, this
configuration is unstable and the complex dissociation reaction is shifted to the
right by an amount given by the balance of the electron re-emission process and the
dissociation process.
The copper vacancies, during their random walk in the material, can re-form
the neutral complexes with the single charged oxygen vacancies. However these
complexes dissociate soon after their formation because they cannot capture an hole
and stabilize in the W + condition.
The characteristic time of the capacitance transient after the application of a
reverse bias is then simply an electronic process which, however, has to be stabilized
by the complex dissociation.
The time constant of the capacitance decrease after the removal of the reverse
bias is instead explained by a completely different process. As in the PPC, the
−
decrease of the capacitance is limited by the association rate of the defects VCu
and
+
VO .
Indeed, when the reverse bias is removed, the hole capture by the W 0 is very
likely, since the hole concentration is restored to its equilibrium value and therefore
the complex can easily be stabilized into the state W + .
Using the values of σp estimated above it is easy to see that the capture of an
hole can be considered instantaneous respect with the typical time τ2 .
We want to stress that this explanation of τ1 and τ2 permit to have τ1 < τ2
which it is impossible in a purely electronic model.
5.5.1
The first capacitance transient: τ1
The kinetics of the first transient can be described by the system

d[W + ]


= [W 0 ]en − [W + ]ep

dt
0


 d[W ] = −[W 0 ]e + [W + ]e − k [W 0 ] + k [V − ][V + ]
n
p
ass Cu
diss
O
dt
(5.26)
5.5 Analysis of the capacitance transients
119
where kdiss and kass are respectively the rate of dissociation and association of the
complexes.
The first equation can be rewritten in the form
d[W + ]
= ([VO ]tot − [VO+ ] − [W + ])en − [W + ]ep
dt
= −[W + ](en + ep ) + ([VO ]tot − [VO+ ])en
(5.27)
where we have used the fact that
[VO ]tot = [VO+ ] + [W + ] + [W 0 ].
(5.28)
If [VO+ ] [VO ]tot , the term [VO ]tot −[VO+ ] ≈ [VO ]tot and the [W + ] reaches the new
equilibrium under reverse bias with an exponential behaviour with a time constant
τ1 =
1
ep + en
(5.29)
In order to understand if [VO ]tot − [VO+ ] can be considered constant we need an
estimate of the fraction of the dissociated complexes after the application of the
reverse bias. This number can be obtained by the analysis of the capacitance voltage
measurements.
In Fig. 5.8 the C-V profiling of the net fixed charge in a Cu2 O/ZnO heterojunction
is reported. The three curves correspond to three different condition of the sample,
equilibrated in the dark, equilibrated in the dark under a reverse bias of −1 V and
preilluminated at room temperature.
The observed charge excess respect with the equilibrated curve has to be attributed to the light and the reverse bias. In particular the reverse bias can change
the complex occupation only in a region extending from the interface to a distance
at which the Fermi level crosses the complex level W +/0 .
A rough estimation of the increase of the net fixed charged gives about 1 × 1013
– 2 × 1013 cm−3 which corresponds to about 2 – 4% (fdiss ≈ 0.03) of the total oxygen
vacancies [VO ]tot .
From the knowledge of fdiss we can obtain the ratio en /ep (see App. D) which
is of the order of 103 . This result is quite surprising because the kinetics of the
complexes is such that the first capacitance transient is determined by the en instead
of ep :
1
1
τ1 ≈
≈
(5.30)
ep + en
en
This means that the data of τ1 in Tab. 4.3 has to be analyzed according to this
new conclusion. The activation energies of τ1 T 2 , 0.95 eV < E < 1.26 eV, are referred
to the conduction band and the cross sections, 10−16 cm2 < σ < 10−11 cm2 , are the
electron capture cross sections.
On the other hand, as it can be easily seen from Fig. 5.8 the dissociation coefficient
fdiss is much higher in the case of the preilluminated device. The light is much more
efficient to dissociate the complexes and this is probably due to the fact that the
light can bring the complex in the state W − which is very unstable.
120
5. A model based on donor–acceptor complexes
4
T
= 300 K
preilluminated
3
equilibrated in the dark
at
14
-3
cm )
Solar cell C80F,
V
= 0 V
(10
equilibrated in the dark
at
V
= -1 V
A
-
N N
D+
2
1
0
0
1
2
3
x
4
(
5
6
7
6
7
m)
1.0
Solar cell C80F,
0.8
equilibrated in the dark at
cm )
equilibrated in the dark at
-3
14
= 300 K
V
V
= 0 V
= -1 V
0.6
0.4
A
-
N N
D+
(10
T
0.2
0.0
0
1
2
3
x
4
(
5
m)
Figure 5.8. Capacitance voltage profiling of the net fixed charge in a Cu2 O/ZnO heterojunction. The three curves correspond to three different condition of the sample, equilibrated
in the dark, equilibrated in the dark under a reverse bias of −1 V, preilluminated at
room temperature. The lower figure is a magnification of the upper figure.
5.5 Analysis of the capacitance transients
16
121
C80F
V
T = 393 K
14
a
= 3.2 V
Poole-Frenkel
-1
s )
12
8
1/
1
(10
-3
10
6
V
a
4
= 0.8 V
phonon-assisted tunneling
2
0
0
1000
2000
3000
E
4000
5000
6000
7000
(V/cm)
Figure 5.9. Time constant, τ1 , of the capacitance transient shows by a Cu2 O based junction
when a reverse bias is applied. τ1 is plot against the electrical field of the space charge
layer.
τ1 dependence on the electric field
One would expect that τ1 is the same among various devices because it is directly
related to the emission coefficient of the electrons. However we have seen in Tab. 4.3
and in Fig. 4.14 that a great variability of the τ1 data exists.
It is known that the the electric field can influence the emission rate and this is
a common problem in determining the correct cross section of the defects [740]. To
check if this dependence is important in our case, we have measured τ1 as a function
of the applied reverse bias at fixed temperature, T = 393 K. The device C80F was
used for this experiment.
The electric field was considered linear in the space charge region and it was
estimated by
V + Vbi
E=
(5.31)
Wscr
where V is the applied reverse bias (positive), Vbi ≈ 0.6 V is the built-in potential of
the junction and Wscr , the width of the space charge region, is given by
Wscr =
ε0 εr
C
(5.32)
with C calculated by the mean of the capacitance values at the beginning and at
the end of the capacitance transient described by τ1 .
The experimental results are reported in Fig. 5.9 and they show a clear dependence
of the emission rate on the strength of the electric field. This can be a proof that
the τ1 is governed by an emission process.
The enhancement of the emission probability with electric field strength is usually
explained by two mechanisms [236]: the Poole–Frenkel effect and the Phonon Assisted
Tunnelling.
122
5. A model based on donor–acceptor complexes
Figure 5.10. Graphical representation of the
emission process from a deep trap in a
semiconductor. If the trap after the emission is neutral, the Poole–Frenkel effect is
usually neglected.
These effects are graphically represented in Fig. 5.10. In the Poole–Frenkel effect
the emission probability is enhanced because the barrier height is lowered by the
electric field. Instead in the phonon-assisted tunnelling the fundamental parameter
is the width of the barrier which is decreased by the electric field.
In the Poole-Frenkel effect, the ratio between the emission under the presence of
an electric field, e(E), and the same quantity under no electric field, e(0), is
√
e(E)
∝ eA(T ) E
e(0)
where
s
A(T ) = 2
Ze3 1
ε0 εr kT
(5.33)
(5.34)
while for the phonon assisted tunnelling the ratio is described by
where
s
Ec2 =
e(E)
2
2
∝ eE /Ec
e(0)
(5.35)
3m∗ }
,
e2 t2
(5.36)
t2 =
}
± t1 .
2kT
The fit of our experimental data with these two theories, see Fig. 5.9, shows
that the Poole–Frenkel must be rejected. The fit with phonon-assisted tunnelling is
instead in a fairly good agreement with the data and its extrapolation to zero field
gives a value of τ1 ≈ 200 s. This value is however much smaller than the value of
τ2 = 775 s obtained at the same temperature and on the same device. Therefore we
conclude that, even considering the effect of the electric field in the space charge
region, the condition τ1 < τ2 is still valid and therefore the electronic models have
to be rejected because inconsistent with it.
It has to be noted that different values of activation energies of τ1 are observed
in the capacitance transients experiments: lower values of τ1 correspond to larger
activation energy.
5.6 CV profile and aging
123
This correlation cannot be explained by the dependence of the phonon assisted
tunnelling on the temperature. Indeed this mechanism would give the opposite
trend: at lower temperatures the electric field becomes much more important and
τ1 becomes larger, giving a greater activation energy. The fresh samples should
therefore show lower τ1 and larger activation energies: the former effect is indeed
observed while the latter it is not.
5.5.2
The second capacitance transient: τ2
As we have already seen, the second capacitance transients, which occurs when the
−
reverse bias is removed, is limited by the re-association of the defects VO+ and VCu
.
0
After the formation of the complex W , this defect capture an hole because its level
W +/0 is above the Fermi level and goes in the state W + , decreasing the capacitance
of the junction.
We have seen that the dissociated complexes at the end of the first transient
are just a few percent of the total. Therefore we can state that the copper vacancy
concentration during the re-association can be considered approximatively constant
which is equal to the value at the equilibrium.
Considering that all our experiments of capacitance transients are performed
always under a reverse bias (−1.4 V for τ1 and −0.4 V for τ2 ) we can state that all
the copper vacancies can be considered ionized
−
] = [VCu ].
[VCu
(5.37)
Therefore, looking at the Eq. (5.14), the second capacitance transient should be
described by a single exponential with a time constant
τ2 =
1
4πDVCu Rc [VCu ]
(5.38)
Using the values (see Tab. 5.1) of DVCu obtained from the fit of the PPC decay ,
the calculated values of Rc and finally the measured concentration of isolated copper
vacancy at equilibrium (≈ 5 × 1014 cm−3 ), we can estimate the expected values for
τ2 :
τ2th (353 K) = 40 857 s,
τ2th (373 K) = 5340 s,
τ2th (393 K) = 831 s
(5.39)
These values are in good agreement with the experiments (see Tab. 4.3) and
gives the following activated behaviour for τ2 :
τ2th = 1014 e1.16/(kT ) s
(5.40)
Notice that if the copper vacancy concentration can not be considered constant,
the simulation of the second transient is very complicated because the neutrality
condition can not be applied.
5.6
CV profile and aging
Since the diffusion coefficient of the copper vacancies is quite high, even at low
temperature, we can suppose that almost all the non bound copper vacancies can
drift in the space charge region toward the interface.
124
5. A model based on donor–acceptor complexes
12
ScAl90B
Al/Cu O Schottky junction
T
8
6
N
fix
(10
13
cm
-3
)
10
2
= 300 K
4
after 24 min at 373 K
after 63 min at 373 K
2
after 146 min at 373 K
after 433 min at 373 K
0
0
1
2
3
x
4
(
5
6
7
m)
Figure 5.11. Net fixed charge Nfix plotted against the distance from the junction interface.
The four curves correspond to different stages of ageing of the sample. The circled points
correspond to V = 0 V, while the squared points are those used for the estimate of the
width of the highly compensated region.
If the migration would proceed indefinitely the remaining vacancies will be, at
minimum, those needed to maintain the p-type character of Cu2 O. For this reason
after an highly compensated region is created.
We have to mention that this phenomenon was already observed by Weichman
and Kužel [889], who attributed it to a grain boundary effect. However we have
observed this increase of the highly compensated region also in monocrystals and
therefore we are reasonably sure that the hypothesis involving the grain boundaries
has to be rejected. On the other hand, Rosenstock et al. [719] already used the idea
of copper vacancy migration to explain the strange behaviour of the current-voltage
measurements on Cu2 O.
We want to analyze the spatial distribution of charged defects as a function of
the aging. In Fig. 5.11 the fixed charge Nfix has been obtained by the Schottky
relation
C3
dC −1
Nfix (x) = −
.
(5.41)
e εA2 dV
The four curves correspond to different stages of aging, obtained heating the sample
at 373 K for several minutes. All the C–V are then measured at room temperature.
An highly compensate region is clearly visible near the interface and it increases
with the aging. This result can be interpreted in the framework of our model.
We can describe the kinetics of this phenomenon using a simple model in which
the back diffusion of copper vacancies from the interface to the material is neglected.
Obviously this rough model will be valid only in the first part of the aging, because
the diffusion will be more important at the end of the aging process.
In the highly compensated region the charged acceptor concentration can be
5.7 Limitations of the D–A complex model
125
−
considered constant, in particular equal to the donor concentration ([VCu
] ≈ [W + ]),
and therefore the flux of copper vacancies from the bulk to the interface is given by
−
φ = [VCu
]µVCu E = [W + ]
eDVCu Vbi
kT WHCR
(5.42)
where the electric field E was assumed to be linear and given by Vbi /WHCR where
Vbi is the built-in potential and WHCR is the width of the highly compensated region.
The expansion rate of the WHCR is
[W + ]
dWHCR
φ
eDVCu Vbi
=
=
dt
[VCu ] − [W + ]
kT WHCR [VCu ] − [W + ]
(5.43)
and solving
2
WHCR
(t)
=
2
WHCR
(t
[W + ]
2eDVCu Vbi
t
kT
[VCu ] − [W + ]
!
= 0) +
(5.44)
If we assume that the value of WHCR is given by the flex of the experimental
curves, we can plot WHCR as a function of the ageing time and fit these point with
the simple model described above.
In Fig. 5.12 the experimental data and their fit are reported. We obtain a slope
of B = 2.97 × 10−12 cm2 /s, which is then used to estimate the diffusion coefficient
of the copper vacancy
DVCu = 2.97 × 10−12 cm2 /s ·
kT
2eVbi
[VCu ]
− 1 ≈ 1.2 × 10−13 cm2 /s
[W + ]
(5.45)
where we used the value T = 373 K, Vbi = 0.6 V, [W + ] = 1014 cm−3 , [VCu ] =
2.5 × 1014 cm−3 .
This value of diffusion coefficient is similar to that obtained by the fit of the
PPC at 373 K (see Tab. 5.1).
Notice that this analysis is very rough. After a first parabolic behaviour, the
increase in WHCR slows down.
This experiment can partly explain the instability of the Cu2 O solar cells observed
in the past [542]: all the parameters which describe a solar cells, Jsc , Voc , ecc. . . ,
vary with time. In particular the Jsc tends to increase. This latter observation can
be explained by the increase of the HCR: indeed the larger is the HCR, the larger
is width of the SCR, that is the region where a non-negligible electric field exists.
Therefore the collection of the carriers becomes easier because aided by the presence
of the electric field over a wider region.
5.7
Limitations of the D–A complex model
In this paragraph we want to point out several limitations of the D–A complex
model. The first clear limitation is due to the fact that we have used a single
discrete acceptor level. Even if in many samples (including the sample 81) the
activation energy of the p(T ) does not show any variation between the equilibrated
and preilluminated state, in other samples a difference can be observed.
126
5. A model based on donor–acceptor complexes
12
10
6
HCR
W
2
(
2
m )
8
ScAl90B
4
Al/Cu O Schottky junction
2
Experimental data
2
Linear fit
0
0
5000
10000
15000
t
20000
25000
30000
(s)
Figure 5.12. Squared width of the highly compensated region as a function of time.
Moreover also the value of the activation energy of p(T ) shows a significant
scattering among different samples (0.32 eV < EA < 0.42 eV).
These two experimental results suggest that the acceptors are distributed in a
band whose width can be neglected in some samples but it is significant in others.
From the mathematical point of view a refinement of the model which includes
an acceptor band of finite width would not be a big problem. On the other hand
the shape of this band would be completely arbitrary and the experimental data do
not contain enough information to fix the model parameters.
Chapter 6
Doping
It is known that the impurities in semiconductors play a fundamental role. Even a
very small concentration of impurities can alter significantly the properties of the
semiconductor and therefore a systematic study of the most common impurities and
of their effects as a function of the concentration is necessary.
From the point of view of the photovoltaic conversion, a doped material is an
important goal to achieve for several reasons. A p–type doped material with higher,
controlled, conductivity allows to build solar cells with smaller Rs and therefore with
a better Fill Factor and a better Jsc . Moreover, lowering the Fermi level, an increase
in the Voc can be obtained. With regard to the scientific aspect, if p–type doping
gives a shallow acceptor below the intrinsic levels, more studies could be possible,
especially the characterization of the intrinsic levels because a lot of experimental
techniques are limited by the carrier freeze–out which masks the response of the
intrinsic defects. A negligible Rs could be also very useful for practical reasons in the
characterization of junctions, since an high value of Rs spoils the true capacitance
value.
On the other hand, the availability of a n doping processes would make possible
to fabricate homojunctions avoiding the formation of interface defects and therefore
limiting the non–radiative recombination. This would give a reduction of the dark–
current (J0 ) of the cell and consequently a further increase of Voc .
Anyway n–type doping seems to be prohibited, indeed all the attempts to dope
Cu2 O, also with impurities that should behave as n dopants, bring at most to the
p–type doping. A possible explanation, often invoked to explain similar problems in
other semiconductors, could be the self-compensation mechanism: the more donor
impurities are inserted, the more the energy of formation of acceptors is lowered; so
donors are always less than acceptors (see Sec. 2.7.7).
A study of the effect of the impurities in Cu2 O is very important, not only for
the doping, but also because the typical defect density found in Cu2 O is of the order
of 1015 cm−3 while the impurities concentration from the starting copper was always
larger than 5 × 1016 cm−3 (1 ppm)!
Indeed a large scattering of the data exists in literature, especially for the
electrical properties. Therefore it would be very useful a systematic study of the
impurities, from the point of view of the morphology, electrical properties and
minority carrier diffusion length. In this way a better optimization of the devices
will be possible.
127
128
6. Doping
In this chapter, after a brief review of the literature on the doping attempts on
Cu2 O, we will show the results of our experiments. We have tried eight different
impurities: chromium (Cr), iron (Fe), silver (Ag), silicon (Si), sodium (Na), sulphur
(S), phosphorus (P) and chlorine (Cl).
The choice of these elements was dictated, in addition to their immediate
availability in our laboratory, by different reasons. Chromium and iron were tried
to understand if the low resistivity shown by some Trivich’s samples, obtained by
rolling very pure copper rods between two stainless steel cylinders, is due to the
introduction of contaminants from the steel.
He observed [841, p. 35] that while the rolled copper gave samples with ρ ≈
100 − 200 W cm, the untreated material gave a Cu2 O with a resistivity of the order
104 W cm.
Silver and sulfur were tried because they are impurities commonly found in most
of the commercial copper and therefore this experiment is useful also to investigate
the Ag and S role in the grain growth process. Moreover silver is the element with
the largest solubility in Cu2 O [856] and therefore this trial is the first step toward
the heavy doping needed to form an intermediate band in Cu2 O.
Sodium was tested because it showed a very positive effect in chalcopyrites,
increasing the conductivity and the grain size [695].
Silicon was tested because it produces p-type doping in thin films. Phosphorus
was tested because it is in the fifth group like the nitrogen, which is a good dopant
for Cu2 O thin films.
Finally the chlorine doping was investigated in details because it is the only one
which has produced a clear decrease of the resistivity in bulk samples. Unfortunately
the solar cell performances degrade as well.
6.1
Review: doping of Cu2 O
So far in the literature doping of Cu2 O has been attempted using several elements.
Among the impurities investigated up to now, only three of them have given a clear
conductivity increase: silicon, nitrogen and chlorine.
Doping technology is anyway still very primitive for all these impurities since their
concentration was never accurately controlled. Moreover, except for the nitrogen
case, even the physical mechanism leading to p-doping is not understood.
Let us now review the results obtained for each element investigated up to now
by experiments or numerical simulations.
6.1.1
Chlorine
Its preferred oxidation state is −1 and, if it substitute to the oxygen, it should behave
as a donor. Anyway the experiments show that it behaves as an acceptor [558, 635].
Two possible explanations were proposed: the first one is that it prefers to sit in
an interstitial position acting as an acceptor, and the second one is that it substitutes
the oxygen, acting as a donor, which rises the Fermi level and consequently lowers
the formation enthalpy of copper vacancies giving a less compensated material as
net effect.
The first experiments were done by the Olsen’s group [635] flowing the mixture
Ar/O2 (unknown ratio) in a furnace containing copper in the center and MgCl2 at
6.1 Review: doping of Cu2 O
129
the input extreme heated at temperatures between 100 and 400 ◦C. Olsen found
that the higher the temperature of the MgCl2 , the lower the resistivity of Cu2 O. In
these experiments a minimum resistivity of about 66 Wcm were obtained but no data
about the chlorine content were reported. Olsen reported only a uniform distribution
of chlorine in the sample and the presence of magnesium precipitate at the grain
boundaries.
In a more recent experiment [558] a flux of Ar was used as a carrier gas to
transport “vapors” of HCl in the oxidation furnace (at atmospheric pressure). In
this case the minimum resistivity obtained was 250 Wcm and again no data on the
chlorine content were reported.
6.1.2
Nitrogen
Its preferred oxidation state is −3 and therefore, if it can substitute an oxygen, it
should behave as an acceptor. This prediction was experimentally confirmed by
growing thin films by RF reactive sputtering in an O2 /N2 atmosphere and using a
glass substrate maintained at temperatures of 400 – 500 ◦C [350].
Undoped films show a resistivity of the order of 100 W cm with an activation
energy of 0.25 eV and a mobility of 50 cm2 /(V s). Different N2 /O2 flux ratio, up to
10%, were used giving a minimum resistivity of about 15 W cm, an activation energy
of 0.14 eV and a mobility of 20 cm2 /(V s).
6.1.3
Silicon
A recent work has shown that silicon behave as an acceptor in Cu2 O thin films [351].
The films were grown by RF reactive sputtering in an O2 /Ar atmosphere using as
substrate glass maintained at 400 ◦C. The silicon was introduced laying a small piece
of wafer on the target.
The lowest resistivity obtained was 12 Wcm and this sample showed a Hall
mobility of about 10 cm2 /(V s) and an activation energy of 0.19 eV. The hole density
of 3 × 1016 cm−3 is quite low compared to the Si concentration of about 0.5%
measured by electron-probe microanalysis in this sample. From these data the
authors calculated a very low doping efficiency, less than 10−3 .
From the IR spectrum the authors deduce that the Si should be coordinated
by 4 oxygen atoms instead of 2 like the copper. Therefore Si is probably not a
substitutional impurity but an element that produces SiO2 inclusions.
On the other hand many years ago, there was an attempt to dope Cu2 O bulk by
silicon [574]. The authors started from a bulk of Cu2 O which was then melt (in a
tube furnace of quartz and and a quartz crucible!) with a small amount of silicon in
the crucible.
The samples doped with 0.5% of silicon, shower a resistivity (2.6 × 104 W cm)
one order of magnitude greater than that of the undoped sample, a semiconducting
p-type conductivity and a mobility about one half with respect to the undoped
samples.
6.1.4
Transition metals
Many other attempts have been made to dope Cu2 O using transition metals. Except
for silver, thallium and copper all these elements should behave as donors, both
130
6. Doping
in substitutional and interstitial position, because they usually have an oxidation
number greater than 1. On the other hand some experiments (on cadmium) and
also some numerical simulations (on nickel) suggest that in some cases they could
behave as acceptors.
Cadmium
Cu2 O was doped with Cd in three ways. In the first one [849] a thin film of Cd (0.04
weight % and 0.06 weight %) was evaporated on the copper sheet (OFE) and then
the sample was oxidized at 1000 ◦C in air. The samples obtained from the oxidation
process have shown a minimum resistivity of 150 Wcm for a Cd concentration of
0.06% and gave solar cells of good enough quality.
In the second way, a liquid alloy CdO/Cu2 O was prepared (avoiding the use of
crucibles which can produce contamination) and pouring it into a copper crucible.
The material obtained from the liquid alloy contained an higher Cd content (0.9%)
and a lower resistivity (9 Wcm) but gave instead solar cells with Voc = 0 V [849].
Papadimitriou et al. [647, 648, 651] oxidized a copper/cadmium alloy (0.05 weight
%) by the grain growth method. They obtained monocrystalline samples of Cu2 O
with a resistivity of the order of 103 W cm which becomes about 50 W cm after an
annealing in air at 500 ◦C but they did not show any solar cell fabricated with these
doped substrates.
Finally Tapiero et al. [808] stated that the cadmium doping of Cu2 O does not
give a resistivity of the samples lower than that obtainable by the simple heat
treatments. However they do not report the cadmium concentration used in their
experiments.
Nickel
Nickel is an interesting impurity because a first principle calculation [520] predicted
that it behaves as an acceptor. A check of these predictions was attempted by
Kikuchi [419].
The Cu2 O was prepared by Pulsed Laser Deposition (PLD) on SiO2 substrates,
using CuO:Ni targets with different nickel concentration. The sample thickness was
about 0.1 µm.
Carrier density does no vary with the Nickel content (about 1015 cm−3 ) but the
mobility shows a dependence on the Ni concentration and deposition temperature.
The most important conclusion is that the Ni impurities do not affect the carrier
concentration but only the mobility. This means that Ni impurities act as neutral
scattering centers for carriers, because the mobility is independent of temperature for
doped films. The lowest resistivity (about 100 Wcm) was obtained for the undoped
film.
The measurement of carrier concentration versus temperature [420] show that
the activation energies of nickel doped and undoped samples are the same (0.22 –
0.25 eV) within the experimental error, confirming the previous hypothesis.
Other transition metals
Among them, Beryllium (Be) [628] was tried oxidizing at high temperature a copper
foil containing 0.022 weight % of Beryllium. The doped substrates showed an increase
6.1 Review: doping of Cu2 O
131
of the resistivity, suggesting that Beryllium can act as a donor or greatly reduces
the mobility.
Thallium (0.033 weight %) [849] and Silver (0.04 weight %) [849] were tried
evaporating a thin layer of dopant on OFE copper sheet but they did not give any
evidence of resistivity reduction. The same happened with the Iron [808]
The Cu2 O doping with cobalt [402] and manganese [885] are quite important
because of the magnetic properties of these two elements.
Cobalt preferred oxidation state is +2 and, if it substitutes to the copper, it
should behave as a donor. Its solubility was investigated by Tsur and Riess [855]
by sintering Cu2 O/CoO powder mixtures and they found a solubility limit of about
0.1%. The authors do not present any experimental measurements of conductivity
but they say the sample was still p-type (private communication).
Manganese [885] has a solubility in Cu2 O of about 0.5% and XRD suggested
substitutional doping. The resistivity of the doped samples was lowered by a factor
of 2 with respect to the undoped ones.
Aluminium and indium have a preferred oxidation state +3 and they should
behave as a double donors, if they substitute a copper.
Some experiments on their introduction into Cu2 O were performed by Trivich [849]
evaporating a thin film on a copper sheet (OFE) and then oxidizing the sample at
1000 ◦C in air.
The aluminium doping did not produce any noticeable conductivity variation.
Indium instead produced a slight conductivity increase: the minimum resistivity
reported was 2300 Wcm for a concentration of In of 0.022% and the solar cells
produced with this material were of good enough quality.
Ab-initio calculations [907] show that indeed aluminium and indium behave as
a double donors but they easily bind with two copper vacancies forming a neutral
complexes with an high binding energy. This could explain the negative or weak
doping results.
As a proof of the complex formation, the authors mention the fact that In
implantation decreases the Cu2 O reactivity probably because of a lower diffusion
coefficient of copper vacancies.
6.1.5
How difficult is the doping of Cu2 O?
According to the hydrogenoid model, the ionization level for a shallow impurity
in Cu2 O is about 160 meV. However this energy gives a radius of the orbit, 6 Å,
comparable with the lattice constant of the Cu2 O, 4 Å. This means that most
probably the impurities in Cu2 O cannot be treated by simple shallow impurities but
several corrections have to be taken into account.
Indeed, as we already know, the acceptor level in undoped Cu2 O has a ionization
energy of about 0.3 eV and similar values of the activation energy of the conductivity
are reported in the literature. See Tab. 6.1.
If we suppose that all these materials are compensated, the hydrogenoid limit
is reached only in doped and passivated thin films. The lowest value obtained for
the bulk is instead EA = 0.21 eV. It must be noted that the compensation ratio in
the thin films is not known, leaving open the possibility that the activation energy
decrease is due to a simple passivation of the donors.
132
6. Doping
Table 6.1. Summary of literature data regarding the measures of conductivity vs temperature in doped substrates.
dopant
work
substrates
Eσ
(eV)
Eσ (undoped)
(eV)
Be
Cd
Cd
Cl
N
N
[628]
[647]
[651]
this work
[350]
[353, 623]
bulk
bulk
bulk
bulk
film
film
0.65
0.21 – 0.24
0.21 – 0.35
0.3
0.14
0.11 – 0.14
0.3
0.37
—
0.35
0.25
—
Si
[351]
film
0.19
0.25
notes
passivation with H
and cyanide
However an important observation has to be done. Even if it is impossible to
obtain a shallow acceptor in Cu2 O, this could not represent a big problem for the
solar cells.
For example let us consider a Jsc ≈ 10 mA/cm2 , a Cu2 O thickness of d ≈ 150 µm
and a maximum allowed voltage drop in the bulk of 10 mV (compared to the open
circuit voltage of about 600 mV).
For a such cell the resistivity has to be lower than about 50 W cm, that means a
value of p ≈ 1015 cm−3 .
If we assume that the donor concentration does not vary with respect to the
undoped substrates (ND ≈ 1014 cm−3 ) we have that a concentration of acceptors of
about 1015 –1016 cm−3 with an ionization energy of 0.3 eV is high enough to satisfy
our requirements. It is important to note that these concentrations are well below
the solubility limits of the impurities in Cu2 O [855, 856].
Therefore the problem is only to find a suitable impurity for Cu2 O which introduces a relatively shallow acceptor level and does not introduce a new recombination
channel for the minority carriers.
6.1.6
Passivation
A few attempts have been performed to control the Cu2 O conductivity by passivation
of defects using hydrogen and cyanide.
Tabuchi et al. [803] exposed Cu2 O thin films to hydrogen at temperatures from
room temperature to 300 ◦C. The atomic hydrogen was formed by an hot tungsten
wire (from 1050 to 1450 ◦C) as catalyst, starting from molecular hydrogen.
The Cu2 O films were obtained oxidizing thin films of copper deposited by RF
sputtering at room temperature on quartz substrates in Ar atmosphere. The films
were oxidized at 300 ◦C in pure oxygen atmosphere at 1 atm. Two thickness were
used: 0.1 µm and 0.4 µm.
At wire temperature higher than 1300 ◦C Cu2 O is reduced to Cu. Instead at
1050 ◦C no trace of copper is observed.
Fixing the wire temperature at 1050 ◦C they varied the exposure time, substrate
temperature and H2 pressure. However the thicker films (0.4 µm) were not affected
by hydrogen, probably because they are too tick for an efficient hydrogen diffusion.
6.2 Preparation of doped substrates of Cu2 O
133
For the 0.1 µm thick films instead a lower carrier concentration and higher mobility
can be obtained. Using an exposure time of 30 minutes, pH2 = 1 mTorr and Tsub =
300 ◦C they succeeded to reduce the carrier concentration from 1.7 × 1016 cm−3 to
2 × 1015 cm-3 and to increase the mobility from 5.3 to 28.9 cm2 /(V s).
Therefore it seems that hydrogen can effectively passivate the acceptors. The
authors explained this mobility increase by the lower effect of ionic impurity scattering
due to the lower carrier concentration and then by the fact that the dangling bonds
at the grain boundaries are passivated by the hydrogen reducing the barrier height.
On the other hand at least three works [351, 353, 486] show that a treatment of
N-doped Cu2 O films in an hydrogen plasma leads to a conductivity increase.
Moreover the group of the university of Tsukuba has shown that a better
passivation can be obtained by a treatment in a solution containing KCN [353].
They report that the nitrogen doped sample has EA = 0.14 eV and NA /ND = 1.36,
the hydrogen passivated N-doped sample has a EA = 0.11 eV and NA /ND = 2.11 and
finally cyanide passivated N-doped sample has a EA = 0.12 eV and NA /ND = 5.04.
6.2
Preparation of doped substrates of Cu2 O
We have tried eight different dopants: chlorine (Cl), silicon (Si), chromium (Cr),
iron (Fe), silver (Ag), sodium (Na), sulfur (S) and phosphorus (P).
A test with 20 ppm of element respect with the copper was performed for all
dopants. Chromium, silver and chlorine was tried also at higher concentration.
All dopants have been evaporated on both surfaces of the starting alpha1 or
alpha2 copper and then they have been covered by a 2000 Å thick layer of evaporated
copper in order to avoid the evaporation of the dopant during the oxidation. For the
chlorine we have evaporated CuCl, for silicon the SiO, for the sodium the NaOH,
for sulphur the CuS and for phosphorus the P2 O5 , while the others elements were
evaporated by their elemental form (Cr, Fe, Ag).
The oxidation process for all this substrates is the same described for undoped
substrates in chapter 3.
The chlorine doping was performed, besides the CuCl evaporation, also with the
classical method with MgCl2 [635]. Chlorine is obtained by the thermal decomposition of dried MgCl2 put in a chamber along the gas mixture flow.1
During the oxidation step at 930 ◦C the chamber temperature is raised up to
230 ◦C and the Cl2 vapour is brought inside the furnace by the gas flux.
The analysis of these substrates have been made using the same devices used for
intrinsic samples (see chapter 3). We plan to perform for each doped sample the
following experiments: p(T ) both in the equilibrated and preillumated conditions,
PPC decay transients and CV charge profiling in Schottky devices. These last part
of the work is still in progress.
6.3
Experimental results
Now we will present the results of the experiments performed on the doped substrates.
1
MgCl2 it is commonly in hydrated form MgCl2 · 6 H2 O. We have heated this salt before its
usage in order to remove the water, otherwise it could influence the oxygen partial pressure in the
furnace.
134
6. Doping
Figure 6.1. SEM images of the surface of the doped Cu2 O samples S145 (left) and S147
(right), doped respectively with 20 ppm of iron and 100 ppm of chromium.
6.3.1
Morphology of the doped sample
All the samples with evaporated dopants have shown a reduction on the grain size
with respect to the undoped samples obtained by the same oxidation procedure.
In the undoped substrates the grain size is of the order of 1 mm or more and it is
obtained with the one hour long step at 1150 ◦C. The doped substrates have instead
a grain size of about 100 µm, very similar to that of the undoped sample before the
grain growth step.
The only exception is the sample S99 in which the chlorine was fluxed during the
grain growth step at 1150 ◦C unlike the sample S98 for which the flux was started
during the oxidation step at 930 ◦C.
In Fig. 6.1 two SEM images of the samples S145 and S147, doped with iron
(20 ppm Fe) and chromium (100 ppm Cr) are reported.
This is quite obviously due to the presence of the impurities which can segregate
at the grain boundaries, blocking the grain boundaries diffusion and therefore the
grain growth.
Another consequence, observed in Cu2 O by Zhu et al. [948] and discussed in
Sec. 3.1.4, is that the voids formation is assisted by the presence of impurities,
which probably reduce the plasticity of the copper and of the growing cuprous oxide.
Moreover Zhu observed also that after the oxidation the impurities tend to segregate
at the metal/Cu2 O interface.
In Fig. 6.2 two SEM images of the section of doped samples are reported. In
the upper figure we report the sample S147, doped with 100 ppm of chromium. In
this case the number of voids has increased so much that the two Cu2 O scales are
almost separated by a central highly defective Cu2 O layer, most probably containing
a larger Cr concentration.
On the other panel, we can see that in the sample S149, doped with 20 ppm of
silicon, the voids are not concentrated in the middle of the sample but they are
widely dispersed along the whole thickness.
In Tab. 6.2 a summary of the morphology results are reported.
6.3 Experimental results
135
Figure 6.2. SEM images of the cross section of the doped Cu2 O samples S147 (left) and
S149 (right), doped respectively with 100 ppm of chromium and 20 ppm of silicon.
136
6.3.2
6. Doping
Electrical properties and defect concentrations of doped samples
All our trials, except for the chlorine, do not show any decrease of the resistivity.
The sample S147 shows an increase of the resistivity which however can be explained
by the fact that the effective thickness of the sample is roughly half of the other
samples (see Fig. 6.2). If we scale the resistivity for the real thickness we obtain a
resistivity similar to the other samples.
The samples S156 and S158, respectively doped with sulfur and phosphorus,
show very bad electrical properties. We suspect that these results are not very
reliable because, for example, to furnace contamination. We will repeat these doping
attempts in the future.
With all the doped substrates we fabricated samples with ohmic contacts (photoconductors) and solar cells based on the Schottky junction Cu2 O/Cu. The first
type of device was used to measure the carrier concentration vs temperature and
the PPC decays, while the second to measure the C–V , obtaining an estimate of
the difference NA − ND+ , and the Quantum Efficiency (QE), obtaining an estimate
of the minority carrier diffusion length, Ln .
We can obtain Ln analyzing the long wavelength data of the external QE, between
500 nm and 650 nm, with the simple expression [107, 635]
T (λ)
EQE(λ) =
1+
α(λ)kT W
2Vbi
1 − e−α(λ)W
1
1 + α(λ)Ln
(6.1)
where T (λ) is the transmittance of the bilayer ZnS/Cu, T is the absolute temperature
and α(λ) is the absorption coefficient of the cuprous oxide [389].
This expression holds for a Schottky junction with an infinite thickness and a
constant value of T (λ). The retro-diffusion of the majority carriers is taken into
account, assuming a constant value of the electric field in the space charge region,
given by E = 2Vbi /W . The values of W were obtained by C–V measurements.
In Tab. 6.2 the parameters obtained by these experiments are reported.
As already mentioned, the data of S156 and S158 are not reliable and further
tests will be done. All the samples, except those doped with chlorine, are very similar
to the intrinsic samples. They show similar activation energies of the conductivity
and a similar value of the compensation ratio. Moreover the PPC effect has the
same intensity among the various samples and also the time constants of the PPC
decays are very similar to those found in undoped substrates.
On the other hand the samples doped with chlorine, both doped with MgCl2
and CuCl, show a clearly decrease of the resistivity, a very weak PPC effect and
much longer PPC decay times. We will discuss deeply the effect of chlorine in the
next section.
The results obtained on the substrates doped with Ag, Fe and Cr can be
understood considering the numerical simulation reported in the work of Wrigth [907].
He found that aluminium and indium in Cu2 O easily form neutral complexes with
two copper vacancies, becoming electrically inactive. These complexes have a very
large binding energies of the order of 3 eV.
If these results can be extended to all the transition metals it follows that the
control of these contaminants is not a critical issue in the Cu2 O preparation.
intr.
98
99
121
122
145
146
147
149
150
151
156
158
sample
—
Cl (MgCl2 )
Cl (MgCl2 )
Cl (CuCl)
Cl (CuCl)
Fe
Cr
Cr
Si (SiO)
Na (NaOH)
Ag
S (CuS)
P (P2 O5 )
impurity
—
—
—
20
20
20
20
100
20
20
20
20
20
level
(ppm)
1000
100
1000
100
100
50
100
100
150
100
100
100
100
grains
(µm)
90
90
80
92
93
92
89
78
88
85
83
76
77
µp
2
(cm /(V s))
3000
255
175
500
341
3000
2000
8000
2900
2500
3900
5000
8300
ρ(300 K)
(W cm)
0.33
0.33
0.31
—
—
0.33
0.34
—
0.32
—
0.40
-
EA
(eV)
1.2
6.72
4.88
—
—
1.15
1.10
—
1.06
—
3.48
-
NA /ND+
yes
no
no
no
no
yes
yes
—
yes
yes
yes
-
PPC
3.5
1
1
2.1
3.1
3.5
3.5
—
3
1.3
3.5
3
-
Ln
(µm)
large and dispersed voids
two large layers of voids
dispersed voids
wavy surface
probably non uniform
probably non uniform
notes
Table 6.2. List of doped Cu2 O samples. The mean grain diameter, the hole mobility µp , the resistivity under ambient light illumination ρ and the
electron diffusion length Ln are reported. The activation energy EA and the compensation ratio NA /ND+ obtained by the fit of the exponential
part of the p(T ) curve are also shown.
6.3 Experimental results
137
138
6. Doping
10
10
10
14
13
12
p
-3
(cm )
10
15
10
10
11
10
S94 equilibrated
S94 preilluminated
10
9
S97 equilibrated
S97 preilluminated
10
10
8
S98 equilibrated
S98 preilluminated
7
2.0
3.0
4.0
T
1000/
5.0
(K
6.0
-1
)
Figure 6.3. Hole carrier concentration as a function of temperature for three substrates
S94, undoped and quenched, S97, undoped and non–quenched and S98, doped and
non–quenched. Each measurement is performed in two initial conditions of the sample,
equilibrated in the dark or preilluminated for several minutes with white light.
6.4
Experiments on chlorine doped samples
Resistivity measurements at room temperature, performed using Van der Pauw
configuration, show a decrease of the Cl doped samples resistivity of about one
order of magnitude with respect to the best undoped ones. For example the doped
substrate S98 has a resistivity of 200 W cm under ambient light illumination while
the undoped S94 has a resistivity of 2000 W cm under the same conditions. Hall
effect measurements show the same p–type character and the same mobility, about
90 cm2 /(V s), for both doped and undoped substrates. Therefore the increase in
conductivity has to be attributed to a greater concentration of holes.
6.4.1
Carrier concentration vs temperature
As in undoped sample we have measured p(T ) in the equilibrated and preilluminated
states. The former was reached by annealing the sample at 400 K for some hours in
the dark until the conductivity reached its minimum, while the latter was obtained
illuminating the sample with white light (about 5 × 1017 photons cm−2 s−1 with
350 nm < λ < 750 nm) for about 15 minutes.
The comparison of the hole concentration p as a function of temperature T
of three typical doped and undoped samples are reported in Fig. 6.3, for both
equilibrated and preilluminated conditions.
It is possible to note two distinct effects of chlorine doping. The first is the
increment of hole density in the chlorine doped sample equilibrated in the dark, that
is greater by about one order of magnitude with respect to the intrinsic samples in
the same condition. The second effect is that the introduction of chlorine strongly
reduces PPC: the curve of preilluminated samples is almost superimposed to the one
6.4 Experiments on chlorine doped samples
139
Table 6.3. Acceptor concentration NA , ionized donor concentration ND+ and acceptor energy
level EA obtained from the fit of p(T ) using the Eq. (6.2) on the equilibrated samples.
The value of free carrier concentration p at room temperature and the compensation
ratio NA /ND+ are also reported.
sample
quench.
doping
p(300 K)
(cm−3 )
NA
(cm−3 )
ND+
(cm−3 )
EA
(eV)
NA /ND+
S94
S97
S98
yes
no
no
no
no
MgCl2
3.0 × 1012
3.3 × 1012
8.6 × 1013
1.93 × 1014
0.71 × 1014
8.74 × 1014
1.03 × 1014
0.23 × 1014
1.29 × 1014
0.385
0.402
0.330
1.87
3.06
6.72
obtained in equilibrated condition, while for undoped cases it is translated upward
of about a factor of 7 with respect to the dark curve.
For a p-type semiconductor with a concentration of NA acceptors compensated
by ND+ ionized donors per unit volume, experimental data can be fitted with the
following expression (see App. B):
1
p(T ) = −
2
Nv
eEA /kT
+ ND+
1
+
2
s
Nv
eEA /kT
+ ND+
2
+
4Nv NA
eEA /kT
(6.2)
where Nv = 2.13 × 1015 T 3/2 cm−3 is the valence band effective density of states, EA
is the acceptor energy level, k is the Boltzmann constant and where the degeneracy
of the acceptor is assumed to be 1. This expression holds if the concentration of
ionized donors ND+ remains constant during the measurement and if the acceptors
are described by a single electronic level. The small variation of the activation energy
of p(T ) observed in our samples has been neglected. See the discussion at page 84.
As PPC decay time is much greater than the measurement time (see Fig. 6.5
below), it is possible to assume, for preilluminated samples too, a constant concentration of ionized donor during data acquisition up to room temperature. The
Eq. (6.2) can thus be used to fit both the equilibrated and preilluminated curves of
Fig. 6.3. The use of this equation even for the doped samples will be discussed in
Sec. 6.4.3.
The results of the fits performed on the equilibrated samples are reported in
Tab. 6.3. It can be seen that the chlorine doping has the following consequences:
the activation energy is reduced, both the donor and acceptor concentrations are
increased as well as the compensation ratio NA /ND+ . The decrease of the activation
energy of p(T ), observed in our samples, is probably related to the different resistivity
of the materials as already reported in the past [604].
To check the correctness of these numbers we have also performed several Capacitance–Voltage (C–V) measurements to obtain the concentration of the net fixed
charge, Nfix . This charge is obtained using the Schottky relation:
C3
Nfix (x) =
eε
dC
dV
−1
(6.3)
where C is the capacitance of the junction, e is the module of electron charge, ε is
the dielectric constant of Cu2 O, about 7ε0 , V is the applied voltage bias and x is
the distance from the Cu2 O/metal interface.
140
6. Doping
10
6
4
N
fix
(10
14
-3
cm )
8
2
0
0.0
0.5
x
1.0
(
1.5
m)
Figure 6.4. Net fixed charge Nfix profile of the doped substrate S98 obtained by C–V
measurement on a Al/Cu2 O Schottky junction. The constant value of Nfix far from the
interface is an estimate of NA − ND+ and is consistent with the value of obtained from
the conductivity vs temperature fit.
The value of Nfix for large x is an estimate of the quantity NA− − ND+ [64]. Since
the acceptor level is about at 0.3 eV from the top of the valence band and has a fast
response we can assume that under reverse bias NA− = NA . Therefore we can obtain
NA − ND+ from the C–V measurements and we find that it is in agreement with the
value obtained from the fit of carrier concentration vs temperature. For example in
Fig. 6.4 the charge profile for a Schottky junction based on the doped S98 substrates
is shown. The value of NA − ND+ is about 7 × 1014 cm−3 which is in good agreement
with the values reported in Tab. 6.3.
6.4.2
Persistent Photo–Conductivity Decay
Now we report the measurements of the decay time constants of the PPC in the Cl
doped substrates, comparing them to those of the undoped ones.
The preillumination of the sample was done as described above, illuminating the
photoconductors until its photoconductivity reached its maximum value. The light
was then switched off and the decay transient of PPC was monitored at the same
temperatures of illumination.
All transients show an exponential behavior both for doped and undoped samples.
Experimental data for the S94, undoped, and S98, doped, are reported in Fig. 6.5
together with the exponential fit. The results of the fit are summarized in Tab. 6.4.
We note that PPC decay time constants in doped samples are about one order of
magnitude greater than in the intrinsic case, with a similar activation energy.
In agreement with the conductivity measurements as discussed in the previous
section, the results show that the persistent photoconductivity effect is greatly
reduced in Cl–doped samples. At the same time the photoconductivity is much
smaller: while in intrinsic substrates conductivity increment under illumination is
6.4 Experiments on chlorine doped samples
141
10
S94
T = 353 K
= 36814 s
1
)/
0
PPC
-
0
T = 373 K
= 4226 s
(
PPC
T = 393 K
= 523 s
PPC
0.1
0
100
200
t
300
(min)
0.2
S98
0.1
T = 373 K
0.09
= 50490 s
PPC
0.07
0.06
0.05
(
-
0
) /
0
0.08
T = 393 K
0.04
= 9220 s
PPC
0.03
0.02
0
200
400
t
600
800
(min)
Figure 6.5. PPC decay for the undoped S94 and the doped S98 samples. The black circles
are the experimental data while the red lines are the exponential fits.
142
6. Doping
about 700% of the dark value, this increment is only 10% in doped samples.
6.4.3
Discussion of conductivity data
Chlorine can enter the lattice in two different ways: at an interstitial site (Cli ),
introducing an acceptor level and in a substitutional position for oxygen (ClO ),
resulting in a donor level. The latter configuration can also be seen as a way for
the chlorine to reduce the concentration of the simple intrinsic defect VO (a double
donor).
The neutrality condition reads:
−
+
p + [D+ ] + [ClO
] = [VCu
] + [Cli− ].
(6.4)
where D+ represents the intrinsic donor concentration (VO or the complex VO − VCu ).
This charge neutrality condition is in general not compatible with the simple
expression seen before (Eq. 6.2). However the fit of the p(T ) data of the doped
samples with the simple expression is still good. We therefore hypothesize that it
should be possible to divide the defects in two groups according to the difference
between their energy levels and Fermi level (Ei − EF ). The former group contains
levels with Ei − EF kT , so that their charge does not depend on temperature and
it is described by the ND+ value while the latter contains levels with Ei − EF kT ,
whose charge changes with temperature and it is described by NA .
Since both NA and ND+ increase with chlorine doping, the simplest explanation
of our data is that chlorine ions act both as interstitial acceptors, with an energy
level very close to the VCu one, and also as substitutional donors. Therefore we
suggest to identify NA with [VCu ] + [Cli ] and ND+ with [D+ ] + [Cl+
O ].
Another effect is that the PPC is greatly reduced. We are therefore induced to
believe that chlorine passivates the intrinsic donors responsible for PPC.
Moreover the PPC effect connected to the residual VO concentration shows a
time decay about one order of magnitude greater than that observed in the undoped
samples. According to our model based on defect complexes [541], the time constants
−
should be roughly inversely proportional to the VCu
concentration. To estimate the
−
residual [VCu ] we can invoke the self–compensation mechanism [856, 857]: while the
neutral defect formation energy is independent of Fermi level, the formation energy
of charged intrinsic acceptors increases linearly with decreasing EF and therefore
−
−
[VCu
]dop = [VCu
]undop
pundop
.
pdop
(6.5)
Table 6.4. Persistent Photo–Conductivity decay time constants, τPPC , at different temperatures. The values are obtained by an exponential fit of the experimental data. EPPC
and τ0 are the slope and the prefactor of the activated behaviour of τ (T ).
Sample
S94
S97
S98
τPPC (393 K)
(s)
τPPC (373 K)
(s)
τPPC (353 K)
(s)
EPPC
(eV)
τ0
(s)
523
4226
36814
53613
1.17
6.9 × 10−13
9220
50490
1.07
1.56 × 10−10
6.4 Experiments on chlorine doped samples
0.6
143
Schottky junctions
Cu/Cu2O
Undoped
0.5
External QY
Doped
0.4
0.3
Exciton
0.2
generation only
0.1
0.0
300
400
500
600
700
(nm)
Figure 6.6. External quantum yield for Cu/Cu2 O Schottky junction solar cells based on
S75 undoped substrate (squares) and on S98 chlorine doped substrate (triangles). The
resistivity of the substrates are ρ(S75) = 1250 W cm, ρ(S98) = 255 W cm. In the figure is
also shown the band gap value of the Cu2 O at room temperature: below that energy
value (above that wavelength) the light absorption can not produce free carriers directly,
but only excitons.
Since the hole concentration in doped samples is more than one order of magnitude
larger than that in the undoped ones, the τPPC increase is consistent with this
hypothesis.
These arguments indicate that chlorine strongly reduces all the intrinsic defect
concentration. Therefore the data from Tab. 6.3 can be interpreted such that in
the doped substrate S98, the chlorine has produced roughly 8 × 1014 acceptors and
1 × 1014 donors per unit volume.
6.4.4
Solar cells
Having improved the conductivity of Cu2 O samples by chlorine doping, it is natural to
check if these substrates can produce better solar cells with reduced series resistance
and increased fill factor. We made several devices based on Schottky junctions, as
described in Sec. 6.2. Unfortunately their efficiency did not improve as expected
since the short circuit current was reduced, see Fig. 6.6.
To understand the reason of this decrease, we performed Quantum Efficiency
(QE) measurements on both doped and undoped cells (see Fig. 6.6).
The measurements show a reduced performance at long wavelengths of devices
made with doped substrates. The simplest explanation for these results is a shorter
lifetime of photogenerated electrons, which could simply be imputed to the action of
chlorine impurity as recombination centers as already suggested by Olsen [635].
The electron diffusion lengths were determined to be Ln = 3.5 µm for the undoped
device and Ln = 1 µm for the doped device.
√
The simple electron lifetime τn can be calculated using the relation Ln = Dn τn ,
where Dn is the electron diffusion coefficient which is given by Dn = µn kT . The
144
6. Doping
mobility µn at room temperature has been estimated rescaling the hole mobility by
the ratio of the effective masses, mh /me ≈ 0.58, obtaining a value of Dn ≈ 1.3 cm2 /s.
Using this number the electron lifetimes were about 100 ns for the undoped substrate
and 8 ns for the doped one.
In the next chapter we will analyze the possible role of the excitons in Cu2 O
and in particular we will analyze if the decrease of the QE observed in the Cl–doped
substrates can be imputed to a greater exciton contribution in doped samples. We
will show that this idea, even if reasonable, will be rejected, and therefore the
smaller diffusion length of the minority carriers has to be attributed to the chlorine
impurities which introduce a new recombination channel for minority carriers.
6.5
Conclusions
The morphology of doped samples is greatly influenced by the presence of some
impurities on the copper surface. The grains remain of the order of 100 µm while the
concentration and the dimensions of the voids increase as predicted in the literature.
A 20 ppm of concentration of the dopants Fe, Cr, Na, Si, Ag, S, P in the starting
copper does not lead to a decrease of resistivity in Cu2 O, except for chlorine. The
mobility is not influenced by these impurities up to high concentration. This confirms
that the main scattering mechanism is with phonons. While 20 ppm corresponds to
a concentration of 1018 cm−3 the concentration of electrically active defects in doped
samples are always of the order of 1015 cm−3 .
On the other hand, the chlorine doping of Cu2 O shows a decrease in the resistivity
of about one order of magnitude with respect to the best undoped.
The typical metastability effect observed in Cu2 O, the Persistent Photo–Conductivity, commonly attributed to the presence of oxygen vacancies, is greatly reduced
in the doped substrates. This suggests that the chlorine can passivate the intrinsic
donors, most likely acting as a substitutional ion for oxygen.
Free carrier vs temperature measurements show that both acceptor and donor
concentrations increase. They suggest therefore that chlorine acts as an interstitial acceptor, with an energy level close to that of the copper vacancy, and as a
substitutional donor also.
However, the solar cells made with Cl doped substrates show a lower efficiency.
From the analysis of their QE a reduction of the electron lifetime is observed. This
is likely due to recombination centers introduced by chlorine doping.
A similar effect is observed on the Na doped substrates. However this is not a
problem because Na is not a dopant and it is not present in the commercial copper.
Chapter 7
Excitonic effects
As we have already seen in chapter 2 the excitonic effects in Cu2 O are very strong.
In this chapter we analyze the possible role of the excitons in the solar cell operation.
We try also to understand if excitons are connected to the observed decrease of the
QE of the solar cells based on Cl doped Cu2 O substrates.
However, even if the exciton contribution to the cuprous oxide were negligible,
the study of the excitons recombination could give important information about the
minority carrier lifetime and the concentration of defects in Cu2 O.
Indeed, the minority carrier lifetime was measured by a time resolved photoluminescence, performed at energies slightly lower than the band gap, in CIS [618] and
CdTe [533]. Further there have been some attempts to correlate the exciton lifetime
in Cu2 O with the concentration of intrinsic defects [644].
7.1
Excitons and free carriers
The thermodynamic properties of a system composed by free electrons and holes with
the formation and dissociation of excitons was studied long time ago by Lipnik [475],
and then by other authors: Nolle [609], Barrau [51], Combescot [131] and Wolfe [902].
At the same time the role of the excitons as recombination channel for carriers in
semiconductors was studied in several papers [266, 609] and Hangleiter [293] found
that the excitons can influence the Auger recombination channel.
In this first section we derive the equations governing the free carrier concentrations and exciton concentration.
7.1.1
Exciton concentration under equilibrium
Let us consider a semiconductor with a concentration of carriers below the Mott
transition (screening radius of carriers smaller than the intercarrier spacing) and let
us assume only one type of exciton with only one stable energy level, its ground state.
We neglect the possibility of other electron interactions, e.g. electron–hole droplets
or electron–hole plasma. In this way we can treat the exciton and electron–hole
system as a perfect gas mixture (Combescot [131, 404]).
Now we will obtain the ratio between the exciton concentration and the minority
carrier concentration at the thermodynamic equilibrium.
145
146
7. Excitonic effects
The chemical reaction between electrons and holes is
e− + h+
x
(7.1)
The chemical potential µi of the i-th particle species is
ni
,
ni n0i T 3/2
(7.2)
µi = −Ei + kT log
n0i T 3/2
where the chemical potential in the reference state, Ei , is defined as the energy
required to bring the particle of the i-th species from the system to the infinity at
rest:
Ee = Evac − Ec ,
Eh = −(Evac − Ev ),
Ex = Ebx + Ee + Eh = Ebx − Eg (7.3)
while the densities of states are given by
mi 3/2 3/2
mi k 3/2 3/2
15
T
cm−3
(7.4)
T
≈
2.4
×
10
g
i
2π}2
m0
where mi are the particle mass of the i species, m0 is the electron mass and gi is a
degeneracy factor. gi contains the spin degeneracy (2) and the number of minimum
(maximum) M of the conduction (valence) band: gi = 2M .
At the thermodynamic equilibrium, the following equation must hold
n0i T 3/2 = gi
µe + µ h = µx
(7.5)
ne nh ≡ np = n∗ nx
(7.6)
which gives the mass action law
where n, p and nexc are the concentrations of electrons, holes and excitons respectively,
and n∗ is a term which depends only on temperature such that the higher the value
of n∗ , the lower is the number of excitons nexc .
The expression of n∗ is:
n0e n0h 3/2 −Ebx /kT
n∗ =
T e
(7.7)
n0x
Actually the value of n∗ will vary with n because the screening of the Coulomb
interaction leads to changes in the exciton binding energy [258]. Using the theory of
screening of ionized impurities we can write
rc 2
(7.8)
rs
where rc is the critical radius at which the Mott transition occurs and rs is the
screening radius given by the Debye formula:
unscreened
Ebx ≈ Ebx
1−
"
rs =
4πe2
εkT
!
#1/2
(n + p)
(7.9)
However the typical doping concentrations used in solar cells generally allow the use
of the unscreened value of exciton binding energy.
In Tab. 7.1 the values of n∗ for Si, CdTe and Cu2 O, together with the constants
needed for its calculation, are reported. With respect to the silicon, which has
n∗ = 8 × 1017 cm−3 , the CdTe has a smaller value of n∗ , and therefore a greater
concentration of excitons (for equal values of p and n) thanks to the smaller values
of Nc . On the other hand Cu2 O shows an even smaller n∗ due to the larger exciton
binding energy.
7.1 Excitons and free carriers
147
Table 7.1. Comparison among different silicon, cadmium telluride and cuprous oxide. For
silicon the values of me and mh , taken from [257], correspond to the mean value of the
effective masses.
Si
CdTe
Cu2 O
7.1.2
Ebx
(meV)
gx
mx
(m0 )
ge
me
(m0 )
gh
mh
(m0 )
Nc
(cm−3 )
Nv
(cm−3 )
n∗
(cm−3 )
20.6
24.8
151
48
8
4
≈ 0.88
0.66
3
12
2
2
1.09
0.11
0.98
4
4
2
1.15
0.55
0.58
2.86 × 1019
5.6 × 1017
2.5 × 1019
3.1 × 1019
6.2 × 1018
1.1 × 1019
8 × 1017
3.4 × 1016
3 × 1015
Exciton concentration out of equilibrium
We are obviously interested to investigate the role of excitons under photo-excitation,
because this is the condition in which a solar cell operates.
If the semiconductor is not in thermodynamic equilibrium, excitons can be treated
in equilibrium with the bands only if their recombination is very slow compared to
the formation and dissociation characteristic times. In this case we can apply the
Eq. (7.6) using the values of p and n under illumination.
However this condition can not be taken for granted. It can be easily shown that,
under illumination with light of energy greater than band gap, at the steady state
the exciton concentration will be lower or, at most, equal to that at equilibrium.
Indeed, for illumination with photons of energy greater than Eg , we can neglect
the generation term in the exciton rate equation. In a p-type semiconductor and in
the low injection approximation (p ≈ pequil ) we have therefore

dn
n


= Gn + nx αdiss −
− bnp

dt
τn

dn
n
x
x


= +bnp −
− nx αdiss
dt
τx
(7.10a)
(7.10b)
where Gn is the generation rate of free electron–hole pairs, αdiss is a coefficient
related to the dissociation probability of an exciton, b is the binding coefficient of an
electron and an hole (non-geminate formation) and τx is the exciton lifetime, due to
both radiative and non-radiative recombination.
From the principle of detailed balance, equating the rate of exciton formation
and dissociation at equilibrium (bnp = nx αdiss ) and using Eq. (7.6), we obtain
αdiss = bn∗
that holds in general.
At the steady state,
dn
dt
= 0,
dnx
dt
(7.11)
= 0, and, by simple calculations, we obtain

n
n∗
1



 n = p + bpτ
x
x

G
bpτ
n
x τn


 nx =
∗
1 + b(n τx + pτn )
(7.12a)
(7.12b)
Now it is obvious to note that, if the excitons can go at the equilibrium with the
steady state number of electrons and holes (τx → ∞), n/nx becomes smaller. This
proves that, if the excitons are negligible at the equilibrium, they will be negligible
also in non-equilibrium conditions.
148
7. Excitonic effects
The above result simply derives from the fact that a finite lifetime of excitons
means having a further recombination channel and therefore a number of excitons
at the steady state lower than the equilibrium one.
Another important parameter which determines the exciton concentration out of
equilibrium is the binding coefficient b. Unfortunately its value has been estimated
theoretically for silicon only in two papers. The first one, by Nolle [609], gives
b = 3 × 10−7 cm3 s−1 at 300 K. Another calculation was later performed by Barrau
et al. [51] and the value of b extrapolated at 300 K is 7 × 10−7 cm3 s−1 . The same
group performed some experiments [50] which seem to confirm this prediction.
7.1.3
Excitons as an alternative recombination channel
The solution (7.12) of the system (7.10) can be interpreted in term of an effective electron lifetime τneff which includes both non–radiative recombination and
recombination via excitonic states:
Gn =
n
.
τneff
(7.13)
By this definition and using the relations (7.12), we can easily obtain the following
expression for τneff
τneff = τn
1 + τx αdiss
1/b + τx n∗
= τn
1 + τx αdiss + τn bp
1/b + τx n∗ + τn p
(7.14)
where obviously the effective electron lifetime must be always less than the electron
lifetime without the exciton recombination channel: τneff < τn . Notice that τneff
increases with τx and τn , whereas it decreases as p increases.
7.2
The exciton role in the solar cell operation
Despite all the studies cited before, the interaction among electrons was always
neglected in the analysis of electronic devices until 1993, when Kane and Swanson [404] studied the possible influence of the excitons in the apparent band gap
narrowing in a p-n junction. Their calculations suggest that excitons can be present
in concentrations approaching those of minority carriers at doping levels of interest
in solar cells. They also state that their calculations are in fairly good agreement
with the experiments of Hangleiter [292] which imply that in silicon, with a doping
concentration of 1017 cm−3 , the exciton concentration at room temperature is about
2 × 1016 cm−3 (see Tab. 7.1 ).
The role played by the excitons in solar cells was first analyzed by Green’s group
in 1995 [138, 258]. They introduced the so called three particle model (electron, hole
and exciton) for describing the solar cell operation.
Besides the minority carrier diffusion, they introduced also the exciton diffusion,
assuming that they are dissociated by the strong electric field as soon as they reach
the space charge region.
They considered both the photogenerated excitons (to calculate the spectral
response) and those generated by minority carrier injection in forward bias (to
calculate the J0 value and therefore the I–V characteristics).
7.3 Excitonic effects on solar cells based on Cu2 O
149
Their theory predict a smaller value of the J0 and an enhancement of the quantum
yield, if the exciton diffusion length were greater than that of the minority carriers.
However these predictions were never tested experimentally.
A more refined model was proposed by the work of Zhang et al. [938], which
assumed that the exciton concentration at the edge of the space charge region is
equal to the equilibrium value of the bulk, instead of zero as supposed by the Green’s
group. With this new more reasonable hypothesis they predicted an opposite trend
on J0 which increases with increasing exciton concentration. Instead, the effect on
the quantum yield remained almost unchanged.
The model of Zhang et al. was applied to CdTe [405] also and it showed that a
large increase of J0 and therefore a worse behaviour of the cells is expected.
7.3
Excitonic effects on solar cells based on Cu2 O
Since the exciton binding energy ∆Eb in Cu2 O is quite large (139 meV for the threefold degenerate orthoexciton state and 151 meV for the paraexciton, see Sec. 2.5.1)
it is reasonable to assume that the excitons can play a fundamental role in transport and recombination mechanisms in Cu2 O. Therefore we have to consider the
hypothesis that the charge transport is also due to diffusion of excitons which are
dissociated by electric field when they reach the edge of depletion region.
It must be noted that at room temperature the excitonic gap of the material is
1.95 eV (635 nm) for the orthoexciton and therefore for energies lower than the real
band gap (2.09 eV corresponding to 593 nm), the absorbed photons produce excitons
and not free carriers.
7.3.1
Literature data
In Cu2 O the effect of excitons was investigated long before the works cited in the
previous section.
In 1987 [852] Tselepis measured, for the first time, the QE of back wall Schottky
solar cells illuminating at low temperature (from 2 to 170 K). Their experiments were
performed with photon energies below the gap, which directly generates excitons
instead of free carriers. A similar experiments, performed in order to obtain the
exciton diffusion coefficient, was done by Fortin [214].
Several years later similar experiments were conducted by a Japanese group [550,
551, 569], but also this time at low temperatures and only on a natural sample.
Unfortunately, even if in these measurements a clear exciton signal is visible in
the photocurrent response (see Fig. 7.1 ), it is impossible to draw any conclusion
about the effect of the excitons at room temperature.
Notice that in order to give an non-negligible contribution to the charge collection
mechanism, the excitons must be a significant fraction with respect to minority
carriers and they must have a relatively large diffusion length. Moreover it is
necessary, once arrived in the space charge region, that they dissociate in a free
electron and a free hole, so that the charge collection can take place.
The exciton dissociation in the space charge region is driven by the electric
field. However the required field strength to dissociate the excitons is too high to be
reached with the typical defect concentration of the Cu2 O. Klingshirn [428, 429, 564]
suggests that, at least in the Schottky junctions, the excitons can dissociate because,
150
7. Excitonic effects
Figure 7.1. Photocurrent spectra of Cu2 O recorded in the back-cell geometry for three
different Cu2 O thickness. Curves indicated by 1, 2 and 3 correspond to temperatures
T = 6 K, T = 77 K, T = 170 K. The arrows indicate the position of the n = 2 exciton
absorption line. The dashed part of the lines correspond to the ionization continuum
above the band gap. After [852].
when they reach the metal/semiconductor interface, the electrons are captured by
the metal.
A simpler explanation, valid for any kind of junction, could be that in the space
charge region the free carrier concentrations are low and therefore the equilibrium
described by Eq. 7.6 is strongly shifted to the left. However this hypothesis relies on
a fast dissociation kinetics and therefore on the values of b and αdiss which are not
known in Cu2 O.
7.3.2
Excitons concentration
In order to understand if the exciton can have a non-negligible role in Cu2 O, first
of all we calculate the maximum value of the ratio nx /n in Cu2 O in equilibrium at
room temperature (300 K).
Using me = 0.99m0 , mh = 0.58m0 , mx = 3.0m0 and the degeneracy factors
ge = 2, gh = 2 and gx = ge gh = 4 given by the spin degeneracy of the bands and finally
unscreened = 151 meV, we
considering the binding energy of the yellow paraexciton Ebx
obtain
n∗ ≈ 2.01 × 1014 T 3/2 e−0.151/(kT ) ≈ 3 × 1015 cm−3 .
(7.15)
Now we need to choose a limiting maximum value for p. We have seen that
undoped samples have a concentration of holes under illumination (≈ 1014 cm−3 ) one
order of magnitude greater than that in the dark (≈ 1013 cm−3 ). Instead, the samples
doped with chlorine have a greater concentration of holes in the dark (≈ 1014 cm−3 )
but the increment due to the illumination is very small.
For this reason we consider p ≈ 1014 cm−3 as the limiting maximum concentration
of holes in our Cu2 O samples, both doped and undoped, obtaining
nx
p
= ∗ ≈ 0.033.
(7.16)
n
n
Therefore in our samples the excitons are only few percent of the total number
of electrons. Some authors [258] have proposed that the calculation of n∗ should
7.3 Excitonic effects on solar cells based on Cu2 O
151
3
10
Trauernicht
Wolfe
2
(cm /s)
(paraexcitons)
(both paraexcitons
2
10
O'Hara
and orthoexcitons?)
D
(orthoexcitons)
1
10
1
10
T
100
(K)
Figure 7.2. Literature data on the exciton diffusion coefficient. The reported data are
from Trauernicht and Wolfe [833], O’Hara, Gullingsrud and Wolfe [616] and from Wolfe
and Jang [904]. The extrapolation of the diffusion coefficient at 300 K is indicated by
the red point which was obtained by a third degree polynomial fit of the Trauernicht’s
data. The circled point was not used in this fit.
consider not only the ground state of the exciton but also all the other excited state.
This approach is criticized in App. E.
7.3.3
Exciton diffusion length
To date there are no clear measurements of the diffusion coefficient and the lifetime
of the excitons in cuprous oxide at room temperature.
The measurement of the diffusion coefficient at low temperature has been done
illuminating a definite portion of the Cu2 O surface and monitoring how the luminescence spatial distribution broadens with time.
A summary of known literature data is reported in Fig. 7.2. Extrapolating these
data to room temperature, gives a value of Dx (300 K) ≈ 10 cm2 /s at 300 K.
The room temperature exciton lifetime shows a huge variability. Indeed the
radiative lifetime of the orthoexciton and the paraexciton are so long, 14 µs and
about 10 ms respectively [406, 407, 562, 563, 615], that the real lifetimes are limited
by other processes. The nature processes is not completely clear, but the great
variability [377, 563] observed in samples of different origins points to a key role of
the impurities and/or native defects. For example, natural samples of Cu2 O are the
only one which exhibit very long exciton lifetimes [563] even if they contain a large
amount of metallic impurities.
The variability range in the exciton lifetime is from about 1 ns to about 1 µs
depending on the sample quality. Using these values we obtain the following limits
152
7. Excitonic effects
for the exciton diffusion length ad room temperature:
1 µm . Lx . 30 µm.
7.3.4
(7.17)
Exciton role in Cu2 O solar cells
The simplest possible contribution of the excitons in a solar cells is in the charge
collection. If the exciton concentration is non negligible with respect to the minority
carrier concentration and if the exciton diffusion length is greater or similar than
that one of the electrons, a part of the charge collection is due to the exciton diffusion
toward the space charge region.
The maximum exciton diffusion length estimated previously (30 µm) is larger
than that measured in Cu2 O devices, but in our materials with p ≤ 1014 cm−3 , the
excitons are only a few percent of the minority carriers. We conclude that at room
temperature the collected charge due to the exciton diffusion is at best only a few
percent of the total one.
On the other hand, if the diffusion length of the excitons is much lower than
that of the minority carriers, the excitons act only as an alternative recombination
channel for the electrons. This effect could be more important in Cl doped samples
because the larger density of holes leads to a greater probability of exciton formation.
Now, using the Eq. (7.14) for τneff , we can show that this mechanism can not explain
the reduction in the effective electron lifetime observed in Cl-doped samples.
First of all we have to consider that τn is always greater than τneff . Considering
the effective lifetime, 100 ns, of minority carriers obtained for an undoped substrates
at room temperature, we can write
τn > 100 ns.
(7.18)
In order to use the Eq. (7.14), we have to know the value of b, however, as we have
already said, this value in Cu2 O is not known. We assume the value of the binding
coefficient b in silicon, b ≈ 3 × 10−7 cm3 /s, as a reference at room temperature. We
remember also that the variability range of the exciton lifetime is 1 ns . τx . 1 µs.
Now considering that in doped samples p is at most 1014 cm−3 , even in the worst
case of τx = 1 ns and b 3 × 10−7 cm3 s−1 (τx bn∗ 1), we would obtain
and therefore
1
τn < τneff < τn
4
(7.19)
τneff > 25 ns
(7.20)
which is much greater than the effective minority carrier lifetime, 8 ns, observed in
Cl-doped Cu2 O substrates.
It seems therefore very unlikely that the exciton formation can be the dominant
recombination channel for electrons, so that the decrease of the device QE caused
by doping is very likely related to the action of chlorine as a recombination center
as supposed in the previous chapter. Obviously a measurement of exciton lifetime
in our samples would be necessary to confirm this conclusion.
Chapter 8
Conclusions and perspectives
In this work we have tried to answer to a few basic questions on Cu2 O. These
questions regard the defect concentration and nature, how the concentration of
these defects can be controlled by the optimization of the Cu2 O synthesis process
(oxidation in furnace in our case) and the influence of these defects in the device
performances.
Measurements of electrical conductivity as a function of temperature and the
measurement of charge profiles by capacitance–voltage measurements have allowed
an initial characterization of the material.
These experiments confirmed that the material contains at least one deep donor
and one acceptor at about 0.3 eV from the valence band. The concentration of these
defects is of the order 1015 cm−3 .
Since the experiments evidenced various metastable phenomena in Cu2 O, these
measurements were conducted in two different condition of the samples: equilibrated
and preilluminated.
According to the simplest electronic model of Cu2 O, the donor concentration
should decrease from equilibrated to the preilluminated condition while the acceptor
concentration should remain constant. We have instead observed the opposite trend.
Since the estimate of defect concentrations is affected by a large error, we have
used other experiments to confirm the fact that the simple electronic model of Cu2 O
is wrong.
First of all we have analyzed the Persistent Photo-Conductivity (PPC) decay
time constants. The simplest interpretation of these experiments is the electronic
one: the effect of PPC is due to the fact that the donor level is very deep and has
a very low probability for the electron emission and a even lower probability for
hole capture. We have demonstrated that such a low probability needs an extremely
low capture cross section for holes: σp < 10−25 cm2 , much smaller than typical
ordinary deep defects. However this improbable result could be explained by a large
relaxation of the lattice near the defect.
Another fundamental experiment was the measurement of the capacitance transients in response to an instantaneous change of the applied bias voltage. These
experiments were performed here for the first time on devices in Cu2 O, both on
ZnO/Cu2 O/Au heterojunction solar cells and on Schottky junctions.
The application of a reverse bias gives rise to a transient in the capacitance that
grows reaching a constant value with a time constant τ1 . Instead, the subsequent
153
154
8. Conclusions and perspectives
reduction of the bias generates a decreasing capacitance transient, which it is also
well described by a single exponential with time constant τ2 . For all analyzed devices
τ1 is less than τ2 . This result has a fundamental importance because any electronic
model predicts the opposite behaviour.
Similar value of τ2 were obtained illuminating the junction and monitoring the
capacitance decrease. Moreover the PPC decay time, τPPC , is always very similar to
the τ2 in the same sample. This result shows that the phenomena of metastability
induced by light and from reverse bias in the devices were due to the same physical
mechanism.
We have therefore proposed a new model for the metastability phenomena in
Cu2 O based on the formation and dissociation of complexes between mobile acceptors
(probably copper vacancies) and fixed donors (probably oxygen vacancies).
The metastability effects shown by Cu2 O can be interpreted by assuming that
illumination of the samples, or the application of reverse bias to junctions, generate
the dissociation of pairs, leading to an increased density of acceptors which manifests
itself as an increased electrical conductivity in substrates and an increased capacitance
in the junctions.
With a suitable choice of the energy levels and of the electrostatic binding energy
of the pair, the concentration of ionized donors does not depend on the degree of
dissociation of complex but remains constant, as observed in the experiments.
According to this model the PPC decay and the capacitance transient decay are
limited by the acceptor diffusion. These two phenomena are therefore governed by the
same physical mechanism, in agreement with what is suggested by the measurement
of their time constants. The diffusion coefficient of the mobile acceptor can be
therefore experimentally obtained by the fit of the PPC decays.
The model also predicts the formation of a highly compensated region near the
junctions due to the drift of the mobile acceptors. The expansion of this region was
experimentally detected and it was found to be in fairly good agreement with the
diffusion coefficient of the acceptors given by the PPC decay fit.
A consequence of this phenomenon is that at a fixed reverse bias, the electric field
in the depletion region decreases with the aging of the devices and this is responsible
for the changes of τ1 observed in several samples.
The expansion of this highly compensated region can partly explain the solar cell
instability already observed in the literature. Especially it can explain the increase
of the short circuit current with time.
In the second part of this work we have analyzed the effect of the impurities in
Cu2 O.
In particular, one of the tested impurities, chlorine, proved to be an effective
dopant. A decrease in the resistivity of about one order of magnitude is observed
between the best undoped and Cl doped samples and the typical metastability
effect observed in Cu2 O, the PPC, commonly attributed to the presence of oxygen
vacancies, is greatly reduced in the doped substrates. This suggests that the chlorine
can passivate the intrinsic donors, most likely acting as a substitutional ion for
oxygen.
Moreover free carrier vs temperature measurements show that both acceptor
and donor concentrations increase. This suggests therefore that chlorine acts as an
155
interstitial acceptor, with an energy level close to that of the copper vacancy, and as
a substitutional donor also.
Unfortunately, the solar cells made with doped substrates show a lower efficiency
and from the analysis of their QE a reduction of the electron lifetime is observed.
This is likely due to recombination centers introduced by chlorine doping.
We have also considered the possible contribution of excitons both to transport
and recombination mechanisms, showing that their effects are most likely negligible.
***
In this thesis we tried to investigate several fundamental aspects of Cu2 O but a
lot of work still remains.
The first open question to be answered is: are the Cu2 O electrical properties
really controlled by intrinsic defects or are they controlled by impurities? This
dilemma is quite similar to that still affecting the ZnO community [528].
It is therefore necessary to understand the effects of the most common extrinsic
impurities in Cu2 O, otherwise all characterization experiments or doping attempts
could give different results due to an uncontrolled purity of the starting copper.
The first attempts reported in this thesis have not shown a large effect of the
studied impurities. However many others must be checked.
This study is also fundamental to decide if a certain impurity can be tolerated,
must be eliminated or can be used as a useful dopant. To speed up this investigation
it would be desirable to develop a technique for the measurement of the minority
carrier diffusion length independent of the cell fabrication.
If it turns out that the intrinsic defects are the main responsible for the electrical
properties a study of their nature is mandatory.
A systematic and deeper study of these defects could make a better use of the
photoluminescence (which was used mainly for excitons and only marginally for
the defects) and of Electro-Paramagnetic Resonance (a very powerful technique
which was applied to Cu2 O only two times). Another possibility is the Positron
Annihilation Spectroscopy, very useful to study the vacancies in a semiconductor
and never used for Cu2 O.
This improved understanding is necessary to solve some inconsistencies of the
D-A pair model proposed in this thesis as for example the fact that the predicted
diffusion coefficient of the mobile species is much lower than that expected for copper
vacancies.
From the technological point of view we need a better control of junction formation
and characterization, together with a study of the ohmic contacts formation and
stability.
If a good p-type doping, which decreases the resistivity and leaves unaltered the
diffusion length of the electrons, will reveal to be impossible, the attention should
shift to the thin film technology where the resistivity of the absorber is much less
important.
Appendix A
Irreducible representations of
the O h point group
Table A.1. Table of correspondence between the Koster (K) and the BouckaertSmoluchowski-Wigner (BSW) notations for the irreducible representations of the Oh
point group. A mixed notation is also presented because commonly used in Cu2 O papers.
Finally the notation usually found in old Cu2 O papers and in general in molecular
physics is reported.
Koster
BSW
Mixed
Molecular
Γ+
1
+
Γ2
Γ+
3
Γ+
4
Γ+
5
Γ−
1
Γ−
2
Γ−
3
Γ−
4
Γ−
5
Γ1
Γ2
Γ12
Γ015
Γ025
Γ01
Γ02
Γ012
Γ15
Γ25
Γ+
1
+
Γ2
Γ+
12
Γ+
15
Γ+
25
Γ−
1
Γ−
2
Γ−
12
Γ−
15
Γ−
25
A1u
A2u
Eu
T1g , F1g
T2g , F2g
A1g
A2g
Eg
T1u , F1u
T2u , F2u
157
Appendix B
Conductivity of a compensated
semiconductor
Now we will derive the relation between the free hole concentration and the temperature in a non-degenerate p–type semiconductor with one acceptor level and one
donor level [789].
Considering a semiconductor with a concentration of NA acceptors and ND
donors such that NA > ND , we can write the following system:

p + ND+ = NA− + n






p = Nv e−β(EF −EV )




2


 n = ni /p
N
D

ND+ =



1 + (gD 0 /gD+ )eβ(EF −ED )





NA



 NA− = 1 + (g /g )e−β(EF −EA )
0
A−
A
(B.1a)
(B.1b)
(B.1c)
(B.1d)
(B.1e)
where g is the degeneracy of the indicated defect charge state and β = 1/(kT ).
In a p–type semiconductor, n p, n ND+ and n NA− , and therefore the
electron concentration can be neglected and the previous system can be written as

p + ND+ = NA−






p = Nv e−β(EF −EV )




N
N
+
D
=
D

1 + (gD 0 /gD+ )eβ(EF −ED )





NA



 NA− =
−β(EF −EA )
1 + (gA0 /gA− )e
(B.2a)
(B.2b)
(B.2c)
(B.2d)
Choosing Ev = 0 as the zero for the energies, defining
gD 0 −βED
gA− −βEA
1 −βEA
A=
e
=
e
D=
e
= gD e−βED
gA 0
gA
gD+
and substituting the last three equations in the neutrality condition we obtain:
p3 + (NV A + NV D + ND ) p2 + NV A (NV D + ND − NA ) p − NV2 NA AD = 0 (B.3)
Even if it is possible to solve analytically this equation, we will analyze it only
in the limit of completely ionized donors.
159
160
B. Conductivity of a compensated semiconductor
17
10
N
16
10
15
A
= 10
-3
15
<
0 cm
10
-3
cm ,
N
D
E
A
15
< 10
= 0.3 eV
-3
cm ,
N
+
D
=
N
D
14
10
HT
13
MT
10
12
N
10
-3
D
= 0 cm
p
(cm
-3
)
11
10
10
10
9
10
LT
8
10
7
10
6
10
5
10
4
10
N
15
D
~ 10
N
-3
14
D
= 5 x 10
-3
cm
cm
3
10
0.000
0.002
0.004
T
1/
0.006
0.008
0.010
(K)
Figure B.1. Simulation of carrier concentration p vs the temperature T for a semiconductor
with one acceptor and one donor. The parameters are indicated in the figure labels.
B.1
Completely ionized donors, ND+ = ND
Assuming that the energy level ED = ED (0/+) lies high enough in the energy
band that ED − EV kT , we can consider the donors all ionized, ND+ = ND .
Substituting D → 0 in the Eq. B.3 we obtain
p2 + (NV A + ND ) p − NV A (NA − ND ) = 0
(B.4)
This is a simple equation, which gives:
1
p = (NV A + ND ) −1 +
2
s
4NV A(NA − ND )
1+
(NV A + ND )2
!
(B.5)
or in another form
p=
NV A(NA − ND )
NV A + ND
2
r
1+
1+
4NV A(NA −ND )
(NV A+ND )2
(B.6)
The previous expressions of p can be used to fit the experimental data of p
versus temperature. The parameters of the fit are NA , ND , EA and gA . In Fig. B.1
a simulation of the previous equation is shown. The parameters used for the
simulation are similar to those found in Cu2 O. Three region are labeled with
HT, high temperature, MT, medium temperature and LT, low temperature. The
explanation is given below.
161
B.1 Completely ionized donors, ND+ = ND
B.1.1
High temperature case
If the temperature is such that all acceptors are practically ionized we can approximate the Eq. B.5 as

p=
then if
Nv
gA
1
2

v
u
NV
u
(N
−
N
)
4
A
D 
NV

gA
+ ND −1 + u
2 
t1 + gA
NV
gA
(B.7)
+ ND
NA − ND , which is very often satisfied in several situations, we have
p = NA − ND
(B.8)
Formally this is valid if EA /k T ED /k.
B.1.2
Intermediate temperature case
If the temperature is such that the number of carriers is ND p NA then we
can approximate the Eq. B.5 as
s
p=
Nv NA −βEA /2
e
gA
(B.9)
Notice that this condition is not always possible because assumes that ND NA . If
ND ≈ NA this region disappears.
Formally the condition is
ND NA
which reads
k log
B.1.3
E
D
Nv
4ND gA
4NA
1
Nv A
T k log
E
A
Nv
4NA gA
Low temperature case
At sufficiently low temperatures, the product Nv A in the Eq.
√ B.5 becomes so small
that Nv A + ND ≈ ND and the
square
root,
with
the
form
1 + x, can be expanded
√
by Taylor series near x = 0, 1 + x ' 1 + 21 x.
We can readily obtain
p=
NV (NA − ND )
NV (NA − ND ) 1 −βEA
A=
e
ND
ND
gA
if the condition
kT EA log
NV
ND gA
with
or
kT EA log
are fullfilled.
4NV (NA − ND )
ND2 gA
4
NA < ND < NA
5
!
with
4
ND < NA
5
(B.10)
162
B.2
B. Conductivity of a compensated semiconductor
Degeneracy factors
The degeneracy factors play a fundamental role in the analysis of carrier concentration.
In the previous discussion we have used defects with only two charge states, neutral
and charged, and with only one energy level for each charge state. In this simple
view the degeneracy factors are gD = 2 for substitutional donor impurities, gA = 2
for substitutional acceptor impurities which can become gA = 4 if the highest valence
band is composed of two degenerate valence bands, and finally gV = 1/2 for vacancy,
both with acceptor and donor behaviour.
If the acceptor or donor can have a bound electron, not only in the ground state
considered by us, but also in several other excited levels the computation of g is
much more complicated and have a general expression only in the hydrogen defect
model.
Moreover the degeneracy of the levels of the defect can be removed by the
presence of several perturbation in the crystal, in this case is much more difficult to
calculate the value of g [863].
Throughout this thesis we assume g = 1. This choice is adopted by all authors
which have studied the electrical properties of Cu2 O.
Appendix C
Correction for the series
resistance
A Schottky diode can be thought as a junction capacitance C, a junction conductance
G = 1/R and a series resistance Rs , as shown in Fig. C.1. The series resistance is
due to the bulk resistivity and to the contact resistances.
To understand if a diode has a not negligible series resistance, one can measures
the Cp as a function of AC signal frequency. If the capacitance increases by reducing
the frequency, then a not negligible Rs exists.
C.1
Admittance correction
To remove the effect of the series resistance and obtaining the true impedance of the
junction we have to know Rs by an independent measurement.
Impedance meters usually assumes the device to be represented by the parallel
equivalent circuit in the right side of Fig. C.1.
Equating the impedance of the two circuit we obtain the junction parameters (C,
G, Rs ) as a function of the measured ones (Cp , Gp ) by the impedance meter [254,740].
From Zp = Z + Rs (holds in general), let
Ap = G2p + ω 2 Cp2
and
A = G2 + ω 2 C 2
(C.1)
Figure C.1. Left: schematization of a junction device under alternate signals. Right: the
most common equivalent circuit used by impedance meters.
163
164
C. Correction for the series resistance
the system becomes

G
Gp



 A = A − Rs
p

C
C
p


=

A
(C.2)
Ap
From the ratio of the first and the second equation
C
(Gp − Rs Ap )
Cp
G=
(C.3)
Finally substituting the previous relation in the second equation of the system
C=
Cp
1 − 2Gp R + Ap Rs2
G=
Gp − RAp
1 − 2Gp R + Ap Rs2
(C.4)
These are the inverse relations:
Cp =
C.1.1
C
2
(1 + Rs G) + (ωRs C)2
Gp =
G(1 + Rs G) + Rs (ωC)2
(1 + Rs G)2 + (ωRs C)2
(C.5)
Error propagation for the Rs correction
A = Cp2 + ω 2 G2p
C=
(C.6)
Cp
1 − 2Gp R + Ap R2
(C.7)
C
(Gp − RAp )
Cp
(C.8)
G=
If we know ∆Cp , ∆Gp , ∆R, we can calculate the error on C and G by the simple
propagation of uncorrelated errors.
v
u
u
∆C = t
∂C
∂Cp
∂C
∂Cp
∂C
∂Gp
v
u
u
∆G = t
∆Cp2
!2
!2
∂C
∂Rs
!2
∆Rs2
!2
∆G2p
+
C
− 2ω 2 R2 C 2
Cp
=
!2
C4
4R2 (Gp R − 1)2 2
Cp
∆G2p =
2
∂G
∂Cp
∆Cp2
+
∂C
∂Gp
∂C
∂Rs
2
∆Rs2
∆Cp2
!
(C.9)
(C.10)
∆G2p
(C.11)
C4
4 2 (AR − Gp )2 ∆R2
Cp
!
=
!2
∆Cp2
+
∂G
∂Gp
!2
∆G2p
+
∂G
∂Rs
2
(C.12)
∆Rs2
(C.13)
C.2 Voltage drop on the series resistance
C.2
165
Voltage drop on the series resistance
Both the IV characteristic and the CV measurements rely on the knoledge of the
effective voltage drop on the diode. However, if the series resistance is not negligible,
this voltage does not correspond to the external applied voltage because part of the
voltage drops across the series resistance.
For example it is known that the IV characteristic of an ideal diode with a shunt
resistance and a series resistance is given by
I = I0 eqV −Rs I/(nkT ) − 1 +
V − Rs I
Rsh
(C.14)
which is obviously an implicit and transcendental equation in V and I. It can be
shown (by implicit function derivation) that for V → ∞, I/V → 1/Rs .
Starting from bare diode parameters we obtain the measured IV parameters by
this chain of transformations, due to shunt resistance and then to series resistance:
(VD , ID )bare −→ (VD , I = ID + VD /Rsh ) −→ (V = VD + Rs I, I)
(C.15)
and viceversa
(V, I) −→ (VD = V − Rs I, I) −→ (VD , ID = I − VD /Rsh )
(C.16)
The same situation occurs for a capacitance–voltage measurement. After the
correction of the capacitance shown above, we have to know the effective potential
drop on the junction, which is obtained by the simple relation
Vjunction = V − Rs I.
(C.17)
Appendix D
Complex concentration
In this appendix we will find an equation to describe the concentration of the
complexes, both under thermodynamic equilibrium and in the deep depletion region
of a device, as a function of the known quantities from experiments.
D.1
Thermodynamic equilibrium
If we know [VCu ]tot , [VO ]tot , the energy level EW (+/0) of the complex W , and
its binding energy Eb , we can obtain the concentration of the complexes W at
equilibrium.
From the occupation probability of the complex W , considering that [W + ] =
[W ] − [W 0 ] we can write
[W 0 ]
= e−(EW (+/0)−EF )/(kT ) ≡ k2
[W + ]
(D.1)
[VCu ]tot = [VCu ] + [W 0 ] + [W + ]
(D.2)
[VO ]tot = [VO+ ] + [W 0 ] + [W + ]
(D.3)
Remembering that
and using the Eq. (5.12) we can write
[W 0 ]
Z Eb /(kT )
=
e
≡ k1
+
[Cu]
[VO ][VCu ]
(D.4)
−
where we have used the approximation [VCu
] ≈ [VCu ].
The system composed by the previous four equations we obtain
[VO+ ]2
k2
[VO ]tot k2
+ [VCu ]tot − [VO ]tot +
[VO+ ] −
=0
k1 (1 + k2 )
k1 (1 + k2 )
(D.5)
which gives the following solution
[VO+ ]
1
k2
=−
[VCu ]tot − [VO ]tot +
+
2
k1 (1 + k2 )
1
+
2
s
[VCu ]tot − [VO ]tot +
167
k2
k1 (1 + k2 )
2
+
4k2 [VO ]tot
k1 (1 + k2 )
(D.6)
168
D. Complex concentration
1.0
E
E
E
E
(+/0) -
F
W
F
W
F
W
F
= 0.12 eV
= 0.20 eV
= 0.25 eV
= 0.30 eV
0.6
0.4
f
= [
W ] / [V
O
]tot
0.8
E
E
(+/0) - E
(+/0) - E
(+/0) -
W
0.2
V ] = 5 x 10
[V ] = 1 x 10
Eb = 0.42 eV
[
14
O
tot
Cu
tot
-3
cm
15
-3
cm
0.0
100
200
300
400
T
500
600
700
800
(K)
Figure D.1. Fraction of bound oxygen vacancies [VO ]bound /[VO ]tot (which corresponds to
the fraction [W ]/[VO ]tot ) at the thermodynamic equilibrium as a function of temperature.
The parameters used for these simulation are reported in the labels.
The previous expression gives the concentration of isolated oxygen vacancies at
the thermodynamic equilibrium.
Therefore the fraction of associated oxygen vacancies
f =1−
[VO+ ]
[W ]
=
[VO ]tot
[VO ]tot
(D.7)
can be easily calculated if we consider EF approximately constant. Some simulations
are reported in Fig. D.1.
For these simulations we have used a value of Eb = 0.42 eV while for the
total copper and oxygen vacancies we have chosen [VCu ]tot = 1 × 1015 cm−3 and
[VO ]tot = 5 × 1014 cm−3 .
In our model we suppose, as suggested by the experiments, that in the temperature
range of our experiments all the oxygen vacancies are coupled with the copper
vacancies, which means [W + ] = [VO ]tot .
According to the simulations reported in Fig. D.1 this situation is realized
when EW (+/0) − EF > 0.25 eV and therefore, considering EF = 0.3 eV, when
EW (+/0) ≥ 0.55 eV. For example in this case at 400 K we have [W 0 ][W + ] < 0.0007.
D.2
Concentration of complexes in the deep depletion
region
In the deep depletion region the occupancy of an electronic level is given by
f=
ep
.
ep + en
(D.8)
D.2 Concentration of complexes in the deep depletion region
169
The concentration of neutral complex will then given by
[W 0 ] = f [W ] =
ep
ep
[W ] = [W + ] ≡ k2inv [W + ]
ep + en
en
(D.9)
which is completely analogous to the Eq. (D.1).
Since the expressions (D.2), (D.3) and (D.4) remain unaltered, the solution of
the system is identical to the Eq. (D.6) where k2 is substituted by k2inv .
From this last expression we can obtain, if the ratio [VO+ ]/[VO ]tot = fdiss is known,
we can obtain the ratio ep /en :
1
k2inv
=
1 − fdiss
en
=
.
2
ep
k1 fdiss [VO ]tot + k1 fdiss ([VCu ]tot − [VO ]tot )
(D.10)
Appendix E
Exciton equilibrium considering
excited states
According to some authors [258], the calculation of n∗ should consider not only the
ground state of the exciton but also all the other excited state.
In this case several reaction can occur:


e+h






e+h


x1,para
x1,ortho
e+h




e + h



···
x2
x3
each of which has an equilibrium constant n∗i .
The total exciton concentration, regardless of the electronic state, is
xtot = x1,para + x1,ortho + x2 + x3 + · · ·
= np
=
1
n∗1,para
+
np
n∗tot
1
n∗1,ortho
1
1
+ ∗ + ∗ + ···
n2 n3
!
(E.1)
The general equilibrium constant can be written as
n∗ (n, l) =
n0e n0h
T 3/2 e−Eb (n,l)/kT
(2l + 1)n0x
(E.2)
If we consider the binding energies degenerate in the angular momentum l (Eb (n, l) =
Eb (n)), we can write
n−1
X
n∗ (n) =
l=0
1
∗
n (n, l)
!−1
=
n0e n0h 3/2 −Eb (n)/(kT )
T e
n2 n0x
(E.3)
The equilibrium constant n∗tot for Cu2 O at T = 300 K is therefore given by
n∗tot
=
1
n∗1,para
+
1
n∗1,ortho
N
X
1
+
i2 eEb (i)/(kT )
5.256 × 1014 T 3/2 i=2
171
!−1
cm−3
(E.4)
172
E. Exciton equilibrium considering excited states
16
10
15
10
1/n* = 1/n*1 + 1/n*2 + ... +1/n*N
p = 10
14
14
10
-3
cm ,
T = 300 K
-3
(cm )
13
10
12
n*
10
11
10
10
10
9
10
0
200
400
600
N
800
1000
Figure E.1. n∗tot as a function of the number of excited exciton states considered.
The summation is limited by the Mott transition of the excitons which occurs
roughly when the exciton diameter is greater than the mean exciton-exciton distance.
The value of i at which the Mott transition occurs is denoted with N .
From this definition N is such that
3
rexc
nexc = 1
Remembering that
rexc = n2 aexc = n2 εr
(E.5)
mr,H
aB
mr,exc
(E.6)
where εr is the relative static permittivity, mr,H is the reduced mass of the hydrogen
atom, mr,exc is the reduced mass of the exciton and aB is the Bohr radius.
Using the previous equations, we can obtain N from the numerical solution of
this equation
N
X
6
N =A
2 B/m2
!−1
m e
(E.7)
n0e n0h 3/2 1
T
>0
n0x
npaexc
(E.8)
m=1
where, considering n and p as constants,
A=
and
Ryexc
> 0.
(E.9)
kT
Under thermodynamic equilibrium there are few electrons and few excitons and
therefore N is quite large.
On the other hand under illumination, considering n = p = 1013 cm−3 the
value of N is 35. With this value we obtain that at room temperature the exciton
B=
173
concentration is half of the electron concentration. However it has to be noted that
the binding energy for i = 35 is very small, much smaller than the thermal energy
and the radius of an exciton of i = 35 is almost 1 µm.
From a physical point of view such an highly excited state has not much in
common with an exciton in the ground state: a complete treatment should include
a separate equation for each excited state which however is not practical.
A more reasonable choice could be that of considering an N such that
Eexc =
Ryexc
= kT
N2
(E.10)
at T = 300 K we obtain N = 3 and therefore a value of n∗tot ≈ 1.8 × 1015 cm−3 .
Considering, as we have done before, p = 1014 cm−3 as the maximum value for the
hole concentration we have
nx /n = p/n∗ ≈ 0.055.
(E.11)
Therefore with this choice we obtain again that the excitons are only a small fraction
of the minority carriers.
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