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DEFECT-INDUCED ELECTRICAL/OPTICAL PROPERTIES OF SrTiO3-X (001)
BY PHOTO-ASSISTED TUNNELING SPECTROSCOPY
Asa Frye
A Dissertation in Materials Science and Engineering
Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
1999
Dissertation Supervisor
Graduate
Group
Chairperson
ABSTRACT
DEFECT-INDUCED ELECTRICAL/OPTICAL PROPERTIES OF SrTiO3-X (001)
BY PHOTO-ASSISTED TUNNELING SPECTROSCOPY
Asa Frye
Dawn A. Bonnell
The (001) surface of monocrystalline strontium titanate is used in a variety of commercial
applications as an active substrate or electrode and therefore represents an important
technological material. Its functionality is enabled by defect states energetically located in the
forbidden gap which are introduced upon removal of oxygen from the lattice. Studies by
conventional surface analysis techniques as well as first principles calculations have not yet
resulted in agreement regarding the microscopic origin of the acceptor type surface states. The
combined techniques of optical spectroscopy and scanning tunneling spectroscopy, however,
afford a unique opportunity to probe the local origins of deep level surface states by direct
modification of the surface charge density through optical excitation.
The technique of photo-assisted tunneling spectroscopy (PATS) using continuous
illumination of mono-energetic light was applied to study the optical responsivity of a series of
samples with increasing degrees of reduction. The surface structures were characterized by
conventional scanning tunneling microscopy (STM), photo-assisted tunneling microscopy
(PATM), and low energy electron diffraction (LEED). A theoretical model was developed to
iv
generate
tunneling
spectra
and
facilitate
interpretation
of
the
experimental results.
This work reports the first STM images and STS spectra obtained on undoped and
transparent single crystalline SrTiO3-x. The surface structure and optical responsivity was found
to strongly depend on the processing conditions, where the latter increased with increasing
degree of reduction. The results were explained in terms of variations of the local surface
potential induced by local charge transfer mechanisms, where evidence of both increasing and
decreasing surface charge was observed depending on the incident photon energy.
It has been determined that oxygen vacancy association is necessary to introduce a deep
level gap state centered at 1.77 eV below the conduction band edge and that this state is localized
on surface terrace sites. This work represents the first successful demonstration of spectroscopic
PATS, combined with theoretical modeling, as a strong metrological tool to study the local
electrical/optical
properties
of
wide
band
v
gap
semiconducting
oxide
materials.
To my mother,
the love that put me on the right track;
and to my wife,
the love that keeps me from derailing.
ii
ACKNOWLEDGMENTS
I’d like to thank GOD for giving me the gift of imagination and the courage to use it wisely.
I’d also like to thank my thesis advisor, Prof. Dawn Bonnell, who provided me the opportunity to
work on a challenging and unique research problem. Special thanks and appreciation is owed to
my
thesis
committee
members
—
Prof.
Peter
Davies,
Prof. Takeshi Egami, Prof. Jack Fischer, and Prof. Roger French — all of whom have offered
valuable insight and guidance towards the success of my studies and accomplishments, as well as
my development as a scientist.
I am indebted to my wife Solita Moran-Frye and my son Atiba Rivera who have endured
six years of sacrifice while remaining both supportive and encouraging.
Deep appreciation is extended to my office mates, Kelly Brown, Bryan Huey, Sergei
Kalinin, James Kiely, Marilyn Nowakowski, Jack Smith and Paul Thibado who have shared and
contributed in special ways to make my experience at Penn both enjoyable and rewarding.
Additional deep appreciation is extended to: Dr. Fred Allen for his friendship and
genuine interest in my success as a graduate student; Prof. John DiNardo, Prof. B. Graham, Prof.
C.
Graham,
Prof.
John
Vohs,
Prof.
Alan
T.
“Charlie”
Johnson,
Dr. Xiaomei Li, Dr. Xue-Feng Lin, and Dr. Dave Carroll who have all generously shared
equipment and/or time and expertise that helped to broaden my technical skills;
Prof. L. A. Girifalco for insightful scientific discussions and helping me to see the value in
“sticking to my guns”; and Prof. David Luzzi for recognizing both my strengths and weaknesses
and, most importantly, telling me about them.
iii
And last, but not least, a thousand thanks are extended to Irene Clements, Pat Overend,
Donna Samuel, Donna Hampton, and Cora Ingrum, all of whom have always been there to offer
meaningful
words
of
support,
encouragement
iv
and
understanding.
Table of Contents
Abstract ............................................................................................................................. iv
List of Tables..................................................................................................................... ix
List of Figures .................................................................................................................... x
Chapter 1
Introduction and Background .................................................................... 1
1.1: Motivation for study..................................................................................................... 1
1.1.1 SrTiO3: a critical technological material.............................................................. 1
1.1.2 Photo-assisted tunneling microscopy and spectroscopy........................................ 2
1.2: Background on SrTiO3 ................................................................................................. 3
1.2.1 Bulk structure and properties................................................................................. 3
1.2.2 Surface structure and properties.......................................................................... 16
1.3: Photo-assisted tunneling spectroscopy....................................................................... 25
1.3.1 Introduction to PATS............................................................................................ 25
1.3.2 Three basic photon absorption mechanisms ........................................................ 27
1.3.3 Limitations of PATS ............................................................................................. 30
1.4: Thesis objectives ........................................................................................................ 35
References ......................................................................................................................... 36
Chapter 2 Experimental .............................................................................................. 42
2.1: Photo-assisted tunneling spectroscopy....................................................................... 42
2.1.1 Experimental arrangement................................................................................... 42
2.1.2 Experimental method............................................................................................ 47
2.1.3 Experimental noise............................................................................................... 49
vi
2.2: Sample preparation and characterization ................................................................... 51
2.2.1 Sample processing history.................................................................................... 51
2.2.2 Methods of characterization ................................................................................ 56
References ......................................................................................................................... 57
Chapter 3
Tunneling Spectroscopy ............................................................................ 58
3.1: Introduction ................................................................................................................ 58
3.1.1 Quantum mechanical tunneling and the WKB approximation............................. 58
3.1.2 The purpose of modeling tunneling spectra ......................................................... 61
3.2: The tunneling model .................................................................................................. 63
3.2.1 One-dimensional quantum transmission.............................................................. 63
3.2.2 Effects of specular transmission........................................................................... 68
3.2.3 The potential distribution functions ..................................................................... 73
3.2.4 The potential barrier functions ............................................................................ 79
3.2.5 Determination of the defect-induced current ....................................................... 90
3.3: Sample of calculation................................................................................................. 93
3.3.1 Simulated vs experimental spectra....................................................................... 93
3.3.2 Parametric study of tunneling model ................................................................... 97
3.3.3 Discussion .......................................................................................................... 103
References ....................................................................................................................... 106
Chapter 4 Characterization of The Bulk ................................................................. 108
4.1: Bulk properties of reduced SrTiO3 ........................................................................... 108
4.1.1 Hall/resistivity measurements ............................................................................ 108
4.1.2 Optical measurements ........................................................................................ 112
vii
4.1.3 Discussion .......................................................................................................... 120
4.1.4 Conclusions ........................................................................................................ 123
References ....................................................................................................................... 125
Chapter 5 Characterization of Vicinal SrTiO3 (001) .............................................. 126
5.1: Structure and chemistry of reduced SrTiO3 (001).................................................... 126
5.1.1 LEED/Auger observations.................................................................................. 126
5.2: Morphological structure by STM............................................................................. 133
5.2.1 Surface morphology of V–930............................................................................ 133
5.2.2 Surface morphology of V–1100.......................................................................... 138
5.2.3 Surface morphologies of the H series ................................................................ 144
5.2.4 Surface morphology of V–930Nb ....................................................................... 148
5.2.5 Summary of observed morphologies .................................................................. 151
5.3: Surface electronic properties by STS and PATS ..................................................... 153
5.3.1 Terrace and step edge electronic properties by STS.......................................... 153
5.3.2 Terrace optical responsivity by PATS................................................................ 158
5.3.3 Summary of observed optical responsivity......................................................... 180
References ....................................................................................................................... 181
Chapter 6
Discussion and Conclusions .................................................................... 182
6.1: Discussion of results ................................................................................................ 182
6.1.1 Photo-assisted tunneling microscopy and spectroscopy.................................... 182
6.1.2 Surface structures and morphologies................................................................. 183
6.1.3 Defect-induced electronic properties ................................................................. 186
6.1.4 Conclusions ........................................................................................................ 191
viii
References ....................................................................................................................... 193
Chapter 7 Summary of Dissertation......................................................................... 194
Appendix A: Franck-Condon principle and the spectroscopic resolution .............. 196
Appendix B: Semiconductor defect statistics ............................................................. 201
Appendix C: Mathematica code for modeled tunneling spectra .............................. 208
ix
List of Figures
Figure 1.1
Coordinated octahedra structure as adopted by strontium titanate. ............... 5
Figure 1.2
Top: Calculated bulk electronic band structure of SrTiO3............................. 6
Figure 1.3
Ordering of oxygen vacancy point defects in nonstoichiometric cubic
perovskite ...................................................................................................................... 12
Figure 1.4
Sphere packing model showing the ideal (001) termination of a ABO3
perovskite surface.......................................................................................................... 16
Figure 1.5
TiO2 termination of (001) SrTiO3 showing titanium adatoms at a terrace site
and at a step edge. ......................................................................................................... 23
Figure 1.6
Surface photovoltage effect upon illuminating a n-type depletion
semiconductor with energies equal to or greater than the band gap energy, Eg............ 27
Figure 1.7
Other photoabsorption mechanisms............................................................. 29
Figure 1.8
The laser induced thermovoltage versus irradiance..................................... 31
Figure 1.9
Electric field intensity between tip and sample versus irradiance ............... 33
Figure 2.1
Experimental arrangement for photo-assisted tunneling spectroscopy. ...... 43
Figure 2.2
Spectral response of experimental optics..................................................... 45
Figure 2.3
Quantification of current variance ............................................................... 50
Figure 2.4
Laue back-reflection photograph showing 〈001〉 orientation....................... 52
Figure 2.5
AFM image showing the stepped surface of SrTiO3 ................................... 53
Figure 2.6
STM images showing a stepped surface...................................................... 53
Figure 2.7
A 500 nm × 500 nm AFM image showing a stepped surface...................... 55
Figure 2.8
AFM images of heavily reduced SrTiO3 (001)............................................ 55
x
Figure 3.1
A particle wave of energy E propagating within a piecewise constant
potential or within a continuous potential function....................................................... 60
Figure 3.2
Schematic representation of the energy band structure of a metal with
respect to a semiconductor in a non-equilibrium (Va ≠ 0) configuration...................... 69
Figure 3.3
Equivalent circuit for a metal-vacuum-semiconductor tunnel junction at
forward bias................................................................................................................... 73
Figure 3.4
Calculated potential across the sample, Vs, as a function of the total applied
bias, Va. ......................................................................................................................... 77
Figure 3.5
An equilibrium (Va = 0) configuration for a metal-vacuum-semiconductor
tunnel junction separated by a gap of width s. .............................................................. 78
Figure 3.6
Calculated spatial and voltage dependent vacuum potential barrier............ 81
Figure 3.7
Calculated surface potential (i.e., band bending) as a function of the voltage
component across the sample........................................................................................ 87
Figure 3.8
Comparison of equilibrium band bending for monovalent and divalent
donors in a semiconductor with band gap energy Eg = 3.2 eV. .................................... 89
Figure 3.9
Calculated tunneling spectrum using the parameters in Table 3.1. ............. 94
Figure 3.10
Comparison of calculated and experimental spectra. ................................ 95
Figure 3.11a,b
a) Increasing carrier density: 1, 5, and 10 × 1019 cm-3. b) Increasing
surface potential: 0.25, 0.30, and 0.35 eV..................................................................... 98
Figure 3.11c,d
c) Increasing static dielectric constant: 100, 210, and 300.
d) Increasing effective mass: 5, 12, and 50. .................................................................. 99
Figure 3.11e,f
e) Increasing tunneling gap: 8, 9, and 10 Å. f) Increasing electron
affinity: 2.6, 3.0, and 3.4 eV. ...................................................................................... 100
xi
Figure 3.12
The predicted effect of increasing surface charge density in steps of
∼1.40×10-7 coulombs per cm2. .................................................................................... 102
Figure 4.1
Resistivity and carrier density of undoped single crystal SrTiO3 .............. 110
Figure 4.2
The dispersion curves for the optical constants (n and k) of SrTiO3......... 114
Figure 4.3
The dispersion curves for the absorption coefficient of SrTiO3. ............... 117
Figure 4.4
Dielectric function of SrTiO3 below anomalous dispersion. ..................... 119
Figure 5.1
Chart recorder traces showing AES spectra of vacuum reduced SrTiO3
(001) surface............................................................................................................... 127
Figure 5.2
Two distinct LEED patterns from SrTiO3-x (001) vicinal surfaces............ 128
Figure 5.3
AES spectra of sample STO–8 heat treated successively at: 500, 1000,
1100, and 1200 °C for 5 minutes each. ....................................................................... 130
Figure 5.4
The
2 × 2R45o superstructure corresponding to the LEED pattern of
Figure 5.2b. ................................................................................................................. 132
Figure 5.5
Multiple unit cell high step edges observed on V–930.............................. 134
Figure 5.6
Comparison of dark and illuminated surfaces, with incident photon energy
of 3.6 eV...................................................................................................................... 134
Figure 5.7
Comparison of dark and illuminated surfaces, with incident photon energy
of 1.9 eV...................................................................................................................... 135
Figure 5.8
Comparison of dark versus illuminated sections of a single surface. ........ 136
Figure 5.9
Step with apparent holes along the edge and at the kink. .......................... 137
Figure 5.10
Surface hole formed by local chemical attack. ........................................ 138
Figure 5.11
Surface morphology of V–1100 showing wandering step edges. ........... 139
Figure 5.12
Apparent cluster-free surface with concave and convex step edges........ 140
xii
Figure 5.13
Series of convex step edges separated by ∼20 Å...................................... 141
Figure 5.14
Terrace cluster structure of heavily reduced SrTiO3 (001)...................... 142
Figure 5.15
Comparison of dark and illuminated surfaces, with incident photon
energy of 2.95 eV. ....................................................................................................... 143
Figure 5.16
Terrace and step edge morphology of H–700.......................................... 145
Figure 5.17
Terrace and step edge morphology of H–1000........................................ 146
Figure 5.18
Local terrace cluster structure.................................................................. 147
Figure 5.19
Comparison of dark and illuminated surfaces, with incident photon
energy of 3.8 eV. ......................................................................................................... 147
Figure 5.20
Terrace and step edge morphology of V–930Nb..................................... 148
Figure 5.21
Terrace and step edge morphology of V–930Nb..................................... 150
Figure 5.22
Local terrace cluster structure of V–930Nb............................................. 150
Figure 5.23
Linear and semi-log plots comparing the terrace electronic properties
in the H series.............................................................................................................. 154
Figure 5.24a
Step edge on V–930Nb where local electronic structure is observed to
vary as shown by the tunneling spectra in Figure 5.24b. ............................................ 155
Figure 5.24b,c Terrace versus step edge electronic behavior.. ................................... 156
Figure 5.25
Dark versus light spectra for H–700 illuminated with 3.8 eV light......... 160
Figure 5.26
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.040 eV; ∆β = 3.5 × 10-4 C/V; ∆χ = 0.80 eV. .................................................. 162
Figure 5.27
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.016 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. .................................................. 163
xiii
Figure 5.28
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.023 eV; ∆β = - 3.3 × 10-4 C/V; ∆χ = - 0.65 eV. ........................................... 164
Figure 5.29
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.035 eV; ∆β = 0 C/V; ∆χ = 0.40 eV. ............................................................... 165
Figure 5.30
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.145 eV; ∆β = 1.0 × 10-4 C/V; ∆χ = 0.70 eV. .................................................. 166
Figure 5.31
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.040 eV; ∆β = - 6.0 × 10-4 C/V; ∆χ = 0.20 eV................................................. 167
Figure 5.32
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.150 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV................................................. 168
Figure 5.33
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.170 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV................................................. 169
Figure 5.34
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.054 eV; ∆β = 0 C/V; ∆χ = 0.30 eV. ............................................................... 170
Figure 5.35
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.090 eV; ∆β = 0 C/V; ∆χ = 0.10 eV. ............................................................. 171
Figure 5.36
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.140 eV; ∆β = 3.0 × 10-4 C/V; ∆χ = - 0.70 eV. ............................................. 172
Figure 5.37
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.018 eV; ∆β = 0 C/V; ∆χ = 0.26 eV. ............................................................... 173
Figure 5.38
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.043 eV; ∆β = - 1.5 × 10-4 C/V; ∆χ = - 0.50 eV. ........................................... 174
xiv
Figure 5.39
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.032 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV. .................................................. 175
Figure 5.40
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.014 eV; ∆β = 9.6 × 10-4 C/V; ∆χ = 0.40 eV. .................................................. 176
Figure 5.41
The photo-induced change in surface charge determined by modeling
the observed changes in the tunneling spectra. ........................................................... 177
Figure A.1
Illustration of molecular energy as a function of internuclear distance.... 197
Figure A.2 Scheme to depict defect thermal ionization energies; scheme to depict the
same defect spectroscopic energies............................................................................. 198
Figure B.1 a) A defect-related state occupied by two electrons of opposite spin; b) the
same defect-related state with one electron removed to the conduction band. ........... 203
xv
List of Tables
Table 1.1: The irreps of high symmetry points and directions in a simple cubic lattice.... 7
Table 1.2: The ten irreps and corresponding term symbols for point group Oh................. 7
Table 1.3: Schottky and “Schottky-like” defect reactions in cubic SrTiO3 ..................... 10
Table 1.4: Theoretical and observed defect-induced ionization energies in SrTiO3-x...... 24
Table 3.1: Parameters used to calculate the tunneling spectrum in Figure 3.9. ............... 93
Table 4.1: Thermal history and Hall/resistivity measurements...................................... 109
Table 4.2: Optical transition energies deduced from Figure 4.3a. ................................. 118
ix
Chapter 1: Introduction and Background
Recent unresolved issues regarding the structure and properties of oxygen deficient SrTiO3 (001)
are presented in this chapter. Following a general description of the current understanding of
bulk and surface structures, the method of photon-assisted tunneling spectroscopy is described
and the objectives of the present study are proposed.
1.1 MOTIVATION FOR STUDY
1.1.1 SrTiO3: a critical technological material
Transition metal oxides (TMO) are an important class of technological materials that have
received a great deal of attention in recent years. The variability in oxidation state of the
transition-metal cation largely accounts for the observation of various stable bulk and surface
structures as well as versatility in physical properties. Considerable progress has occurred over
the past several decades in the development of experimental tools and techniques aimed towards
elucidating the fundamental nature of a variety of processes that occur on surfaces [1]. The
application of many of these methods to the study of TMOs is well-represented in the literature
[2]. Despite these gains, the details of the microscopic mechanisms of processes on oxide
surfaces remain largely undetermined.
Particular knowledge of the effect of local surface
geometric and electronic structure, for example, in determining the efficiency of surface
reactions is of vital use to several industries.
Strontium titanate is an excellent example of a model TMO that has found widespread
application in various technologies. It has been identified as a good substrate candidate for use
in photoelectrochromic devices where a charge transfer (redox) reaction is facilitated by a defect
state energetically located in the band gap of the oxide [3]. Similar surface defect-related
10
properties have made SrTiO3 the focus of research in the fields of photocatalysis and solar
energy conversion [4,5]. Advancements in other technologies where SrTiO3 has been identified
as a critical or potentially critical material, such as gas sensors [6], superconducting thin film
growth [7], and memory storage devices [8], depend on increased understanding of corrosion
mechanisms, high temperature reconstructions, defect interactions, etc., at the surfaces and grain
boundaries. Much of the photo- and chemical-reactivity of the (001) surface have been linked to
extrinsic states in the forbidden energy gap. The pursuit to determine the microscopic origin of
these states or the microscopic mechanisms of these reactions, however, has given rise to
controversies [2].
1.1.2 Photo-assisted Tunneling Microscopy and Spectroscopy
Recent developments in scanning probe techniques have permitted the surface science
community to witness the marriage of scanning tunneling microscopy and optical spectroscopy.
In principle, the high energy resolution of optical transition processes [9] combined with the high
spatial resolution of the STM presents a unique opportunity for the characterization of adsorption
modes and photocatalytic (or other charge-transfer) reactions. The former was demonstrated
recently in a study of water adsorption on RuS2 and TiO2 electrodes [10]. Increasing humidity
was observed to substantially increase photocurrent efficiencies. This was explained in terms of
a molecular adsorbate-induced channel for hole annihilation at the surface of the electrode via
recombination with tunneling electrons. It was thus suggested that the enhanced photocurrent
can be used to distinguish between molecular and dissociative adsorption. Other charge-transfer
mechanisms have been observed by photo-assisted tunneling spectroscopy (PATS) as will be
discussed further in section 1.3.
11
There are three primary ways (depending on the energy of the incident light) in which
photoabsorption may be detected in tunneling spectra using the method described in Chapter 2.
Comparison of dark and illuminated spectra can give (1) a surface photovoltage, (2) a direct
photocurrent, or (3) modified band bending. The last effect suggests the potential to correlate
local trapped surface charge with associated surface defect structure.
1.2
BACKGROUND ON SrTiO3
The physical properties of strontium titanate have been studied extensively for well over thirty
years.
These research efforts have accumulated a profusion of facts leading to deeper
understanding of some phenomena and greater bewilderment of others (compare, for example,
[11] with [12] and [13] with [14]). This section reviews the current level of knowledge of bulk
and surface structure and properties of monocrystalline SrTiO3.
1.2.1 Bulk structure and properties
The cations Sr and Ti, in terms of the ideal ionic model, assume their group oxidation states. It is
thus clear from the chemical formula SrTiO3 that all ions have closed-shell electronic
configurations. The attractive component of the lattice energy is dominated by electrostatic
interactions and may be estimated using the formal charges Z by an equation of the form [15]
E coh
1
=
2Nc
Z iZ je 2
∑i ∑
rij
j ≠i
[1.1]
where the indices i and j vary independently over all ions in the crystal, r is the separation
between two ions i and j, and Nc is the number of formula units (or unit cells) in the crystal. This
energy (also known as the Madelung potential) is the most significant contribution to the
cohesion of the solid. A recent calculation of the cohesive energy gives a value of approximately
12
149 eV per formula unit [16]. This energy may be compared to the lattice energies for other
oxides such as TiO2 (126 eV), Al2O3 (165 eV), ZrO2 (116 eV) or SrO (33.4 eV) [17].
At room temperature SrTiO3 adopts the ideal cubic perovskite structure which may be
described as a close packing of Sr+2 and O-2 ions with Ti+4 occupying one quarter of the
octahedral interstices. Alternatively, one may consider the structure as a network of polyhedra,
as illustrated in Figure 1.1, from which its simple cubic symmetry (crystallographic space group
Pm3m) is readily apparent. The basic structural unit is the Ti+4-O6-2 octahedron and the crystal
consists of corner shared octahedra with Sr+2 occupying the icosahedral interstices. Each oxygen
is coordinated to two Ti ions (linearly) and to four Sr ions, where the Ti-O bond length is smaller
than the Sr-O bond length. There are eight other oxygen ions surrounding each oxygen with an
O-O bond length equivalent to the Sr-O bond length. The unit cell edge is 3.905Å [18] so that
the bond lengths are approximately 1.95Å and 2.76Å for the Ti-O bond and Sr-O (O-O) bond,
respectively.
Early determinations of the electronic structure of strontium titanate [19–21] utilized the
LCAO (linear combination of atomic orbitals) or tight binding approach of Slater and Koster
[22] or molecular-orbital (MO) methods based on a local density approximation for electron
correlation. The limitations of both methods are well-known — the former overestimates the
band gap energy while it is underestimated by the latter. These calculations, however, afford
significant insight into the electrical and optical properties of SrTiO3 and several predictions
based on these results are consistent with experimental observations [23–25].
13
Figure 1.1
Coordinated octahedra structure as adopted by strontium titanate. The small black
spheres are Ti ions; the large sphere is the Sr ion.
14
Figure 1.2
Top: Calculated bulk electronic band structure of SrTiO3. The vertical energy scale
is measured in Rydbergs (≈ 13.6 eV) [ref. 20]. Bottom: The Broullouin zone for a simple cubic
lattice, where the symmetry points and directions are labeled in terms of the BouckaertSmoluchowski-Wigner symbols [ref. 26].
15
Table 1.1: The irreps of high symmetry points and directions in a simple cubic lattice.
Symbol
Symmetry
Γ (zone center)
Oh
∆ ([001])
C4v
Λ ([111])
C3v
Σ ([110])
C2v
M,X
D4h
R
Oh
T
C4v
Z,S
C2v
Table 1.2: The ten irreps and corresponding term symbols for point group Oh.
BSW symbol
Term symbol
Γ15
Γ25
′
Γ25
Γ1
Γ1′
Γ2
Γ2′
Γ12
Γ12′
Γ15′
A1
A1
A2
A2
Eg
Eu
T1g T1u T2g T2u
The band structure calculation of L. F. Mattheiss [20] and the Broullouin zone
corresponding to a direct lattice with cubic symmetry are shown in Figure 1.2 where the
representations of the points and directions of high symmetry are given by the conventional
BSW symbols [27] as outlined in Table 1.1. Of particular interest to the present study is the
behavior of the electronic structure along the ∆ direction from the center (Γ) to the edge (Χ) of
the zone. The ten irreducible representations (or irreps) for the cubic point group Oh and their
corresponding term symbols are listed in Table 1.2. The terms highlighted in bold represent the
symmetry classes to which the oxygen s and p Bloch sums are adapted in order to interact with
the metal d orbitals. This interaction gives the “molecular orbitals” which overlap and form the
energy bands of the solid [28]. For [001] propagation, the five Ti d states reduce to the following
16
classes of the ∆ group: ∆1, ∆2, ∆2', and ∆5. The s and p Bloch sums propagating along [001] may
be classified into all but the ∆2' class. Consequently, the t2g state with ∆2' symmetry does not
“mix” with the oxygen states giving a flat lowest energy conduction band. All other d states
(two t2g states with ∆5 symmetry and two eg states with ∆1 and ∆2 symmetry, respectively) mix
with the appropriate symmetry adapted oxygen s and p states to form the higher energy
conduction bands. The lower band states consist of non-bonding (Γ15 and Γ25) as well as
bonding (Γ15) oxygen Bloch sums which predominately constitute the valence band.
Several experimental investigations have produced results supporting as well as disputing
the
qualitative
features
of
the
band
structure
calculations
of
Mattheiss
and
others [19]. For example, Cardona [24] studied the electronic structure using reflectivity spectra
which contained features consistent with the calculated splitting of the valence bands at zone
center. Perkins and Winter [29] later demonstrated the correlation between Cardona’s results
and the calculated joint density of states based on theoretical band structure. The early transport
measurements of Frederikse and co-workers [23] reported a mobility effective mass of
approximately 16me (where me is the free electron mass) in close agreement with that predicted
by
the
band
structure
calculations
of
Kahn
and
Leyendecker [19]. Although some groups attributed the fundamental absorption edge to a direct
transition at zone center (Γ15→Γ25) in agreement with Mattheiss’ results, it has been
demonstrated [30] that this excitation is in fact indirect (Γ15→Χ3) and assisted by an optical
phonon mode.
It is well established that the energy of this transition (the optical band gap) lies near 3.21
eV at room temperature. It should be noted, however, that a recent approximate calculation
based on electron and hole formation energies suggests a thermal band gap energy of 4.35 eV
17
[16]. Moreover, Reihl et. al. [31] found it necessary to assume a gap energy of 4.5 eV in order to
achieve the best fit between theoretical density of states (DOS) and experimental DOS obtained
from photoemission studies. It is not unusual that the band gap energy is an adjustable parameter
for theoretical band structure calculations.
The Frank-Condon principle explains why the
thermal excitation energy for a particular ionization process is expected to differ from the optical
excitation energy for the same process. Based on this principle, however, and in contrast to what
is suggested in the discussion above, the optical excitation energy is expected to exceed the
thermal ionization energy.
(Appendix A contains a more thorough discussion on optical
excitation processes which are accompanied by large vibrational coupling.)
Nominally pure SrTiO3 is an electronic insulator at room temperature. Incorporation of
point defects into the lattice can generate free charge carriers or charged ionic species. In the
latter case, the defects may be associated or unassociated. In semiconducting SrTiO3 at room
temperature the charge carriers are predominately electrons introduced by donor impurity doping
or heating in a reducing atmosphere.
The latter treatment introduces an approximately
equivalent density of oxygen vacancies which are known to exhibit a significantly large lattice
mobility,
particularly
at
elevated
temperatures [32]. SrTiO3 is thus considered a mixed electronic-ionic conductor.
Oxygen vacancies may also form as a mode for incorporation of acceptor-type impurities
or intrinsic acceptor defects. Acceptor-type impurities, such as Al or Fe, were once believed to
be present in oxides at levels typically no less than 10–100 ppma (parts per million atomic) [33].
In a given sample of SrTiO3 this suggests an accidental impurity density of the order 1017–1018
cm-3.
Intrinsic acceptor defects — i.e., strontium vacancies — are formed during high
temperature processes such as sintering or annealing. Note that the dominant ionic disorder in
18
this close-packed lattice has been shown to be Schottky and “Schottky-like” [16]. Table 1.3 lists
the reactions that have the lowest defect formation energies. Highly charged defects such as VTi−4
are energetically unfavorable in ionic structures. It has thus been suggested that a deficiency in
Sr should be observed for samples processed at sufficiently high temperatures. The stability of
strontium vacancies in the structure is further supported by the observation of a finite solubility
(up to 1000 ppma) of excess TiO2 in SrTiO3 and an associated modification of the equilibrium
conductivity as expected from the following incorporation scheme [14]:
TiO 2 → TiTi + 2OO + VSr−2 + VO+2 .
[1.2]
In the above reaction, the ionic defects are assumed unassociated. It should be noted that an
earlier study [13] suggested excess TiO2 compensation by neutral (i.e., associated) vacancy pairs
(V
−2
Sr
+2
,VO
) since the observed conductivity behavior remained unaffected.
Excess SrO, resulting
from reaction I in Table 1.3, may be easily consumed by the formation of Ruddlesden-Popper
phases [34], and thus also leave the equilibrium conductivity behavior unaffected.
Table 1.3: Schottky and “Schottky-like” defect reactions in cubic SrTiO3
Formation energy (eV)*
Reaction
−2
Sr
+2
O
I SrTiO3 → V + V + SrO
1.53
II SrTiO3 → VTi−4 + 2VO+2 + TiO 2
2.48
III SrTiO3 → VSr−2 + VTi−4 + 3VO+2 + SrTiO3
* see reference 16
19
1.61
The structural accommodation of oxygen vacancies in perovskite oxides may be
illustrated as shown schematically in Figure 1.3. The far left image shows stoichiometric AMO3.
Removal of one oxygen from the lattice reduces two M+4 to M+3 and replaces two octahedra with
two square-pyramids, as shown by the center image. The limiting structure consisting of all M+4
in the lattice being reduced to M+3 with all octahedra being replaced with square-pyramids is
shown by the far right image. The perovskite structure cannot support further removal of oxygen
from the lattice. Note, however, that this limiting phase has a reduced lattice point symmetry
(i.e., C4v) which suggests the lifting of energy band degeneracies. Evidence of this type of
oxygen vacancy ordering in perovskite oxides has been observed in heavily reduced CaMnO3
[35,36]. Grossly oxygen deficient strontium titanate, in the limit corresponding to the formula
SrTiO2.5, will contain an oxygen vacancy density of the order 1021 cm-3. It should be noted,
however, that under extreme reducing conditions ( PO 2 ≈ 10-13 Torr) the oxygen vacancy density
has been reported to only be in the range 2.0–7.6 × 1019 cm-3 [13]. This corresponds to one
oxygen vacancy per 512 unit cells, or chemical formula SrTiO2.998.
20
Figure 1.3
Ordering of oxygen vacancy point defects in nonstoichiometric cubic perovskite
[ref. 36].
The introduction of band gap states, due to removal of oxygen from the lattice, has been
investigated theoretically using a tight-binding model including consideration for the effects of
lattice relaxation [37]. The results for an isolated vacancy, obtained from a second-neighbor
approximation, suggest that a doubly occupied (i.e., charge neutral) state with Eg symmetry lies
0.116 eV below the conduction band edge. The singly ionized state was found to have an energy
and symmetry that depended on the strength of the attractive potential assumed for the defect
site. Less attractive potentials give 0.27 eV, where the state has Eg symmetry. More attractive
potentials give 0.41 eV, where the state has Eu symmetry. These energies may be compared to
those determined from resistivity and Hall data which were found to be in the range 0.07–0.14
eV for carrier densities in the range 3×1018–8×1016 cm-3, respectively [38]. The discrepancy
between these values may be due to the neglect of mutual screening effects from neighboring
oxygen vacancies (see below). Relaxation of the titanium ions towards the vacancy increases all
defect binding energies, while the latter decrease when the titaniums are relaxed away from the
21
vacancy site. It is concluded that the oxygen vacancy may be characterized as an ionic species
— i.e., the localized gap states result only from the potential due to the charge of the defect.
Similar conclusions have been drawn from recent ab initio supercell band structure
calculations [39]. The incorporation of two or more oxygen vacancies in a supercell containing
eight unit cells gives rise to a variety of unique density of state functions depending on the
distribution of defects in the supercell. The defect densities considered (one to three defects per
supercell) are already of the order 1021 cm-3. Metallic behavior is thus predicted in every case.
The significant finding is that vacancy clustering is necessary in order to form well-defined band
gap trap states.
In the most heavily reduced case, corresponding to three homogeneously
distributed vacancies per unit cell (i.e., SrTiO2.625), conduction band states were found to extend
to 1.5 eV below the Fermi level. When all three defects reside on the same octahedra, a more
narrow band of states result at 0.5 eV below the conduction band edge.
In light of the previous discussions, one might expect singly-ionized oxygen vacancies to
predominate at room temperature for heavily reduced samples, while doubly-ionized oxygen
vacancies are expected to predominate for moderately reduced samples. Furthermore, for lightly
reduced samples, it has been proposed that transport occurs via the conduction band at room
temperature whereas at sufficiently low temperatures (< 50 K) transport occurs via an oxygen
vacancy induced defect band [38]. As the oxygen vacancy density increases, it is assumed that
the width of this band also increases, while the ionization energies decrease. In heavily reduced
SrTiO3 (as in Nb-doped crystals) it is often assumed that the defect state may be described by the
hydrogenic model according to which the binding energy of an electron to a singly-ionized
defect is given by [28]
22
Eb =
RH (m* m e )
κ 2st
,
[1.3]
where RH is the Rydberg constant (13.6 eV), m * is the density of states effective mass, and κ st is
the static dielectric constant of the undoped/stoichiometric crystal. Clearly, the large dielectric
constant of SrTiO3 (~ 300) ensures that all defect states will be ionized down to very low
temperatures, as observed by Yamada and Miller [18]. Assuming an electron effective mass of
12, [1.3] gives Eb ≈ 1.8 meV. Therefore, in heavily reduced and Nb-doped SrTiO3, transport
occurs exclusively via the conduction band. On the other hand, if a significant degree of defect
interaction occurs in the oxygen deficient case, a localized defect band is expected to trap
electrons at room temperature such that the conduction band carrier density will exactly equal the
density of oxygen vacancies in the lattice.
Much of the uncertainty regarding the nature of the oxygen vacancy defect and its
occupancy may be due to the variety of observed optical activity which is strongly correlated
with the degree of sample reduction (see Table 1.4, page 24). Optical transmission studies have
established a window of transparency for pure stoichiometric SrTiO3 in the energy range 0.25–
3.1 eV [40,41]. The lower energy is determined by the onset of optical phonon excitation while
the upper energy is determined by the onset of charge transitions between the valence and
conduction bands. Additional absorption bands are observed in reduced SrTiO3 and BaTiO3
single crystals. In heavily reduced SrTiO3, free carrier absorption gives rise to a 0.9 eV peak.
There is also a 2.9 eV peak that is observed to saturate with increased reduction [42,43], and a
2.14 eV peak introduced in strongly reduced SrTiO3 which is speculated to be due to divacancies
(i.e., two associated singly-ionized oxygen vacancies) [44]. Both reduced and Nb-doped samples
have been reported to exhibit absorption at 2.4 eV which was assumed to be due to a charge
23
transition from the lowest conduction band to a higher conduction band of mixed Ti-4p and O-3p
nature [43,45].
For moderately reduced samples, absorption at 1.77 eV is observed for
nominally pure samples but not for Nb-doped samples [43,44]. It is believed that this is due to
excitation of a single charge from a singly-ionized oxygen vacancy, or vacancy-related defect, to
the conduction band. Both the 1.77 eV and 2.4 eV energies were also reported to explain
transient absorption (i.e., photochromic) behavior in reduced SrTiO3 in terms of charge
transitions from band gap states to the conduction band [46]. A similar effect was observed in
reduced BaTiO3 where the energies were reported to be 1.8 eV and 2.6 eV [47].
It can be concluded that the exact nature of the oxygen vacancy defect, in particular the
mechanism of optical absorption in the bulk, remains largely unresolved. A similar uncertainty
holds
for
the
nature
of
the
oxygen
vacancy
and
associated
defects
on
the
(001)-terminated surface. Indeed, there is doubt as to whether the assumption of surface band
gap states is necessary to explain the often observed photoelectrochemical activity of reduced
SrTiO3 since the same behavior can be explained in terms of a bulk response [48].
The
following section discusses the presently understood properties of the (001)-terminated surface
of strontium titanate. The format begins with a description of the structure of the ideal bulktruncated lattice. Next, the mechanisms of relaxation and restructuring are discussed for both the
stoichiometric and non-stoichiometric surfaces. Finally, the effects of the above geometrical
considerations on the electronic structure at the surface are described and discussed in light of
recent experimental observations.
1.2.2 Surface structure and properties
Truncation of the bulk lattice along 〈001〉 can result in one of two possible terminations: one with
stoichiometric composition SrO or one with stoichiometric composition TiO2. This is illustrated
24
in Figure 1.4 where the large black spheres represent Sr ions, the small black spheres represent
Ti ions and the white spheres are the oxygen ions. The oxygen coordination of Sr decreases
from the bulk value of 12 to a surface value of 8. Similarly, the oxygen coordination of Ti
decreases from the bulk value of 6 to a surface value of 5. In order to understand the relative
stability of these two possible terminations, one must consider the manner in which charge is
redistributed at the surface and how this gives rise to shifts in atomic positions.
Figure 1.4
Sphere packing model showing the ideal (001) termination of a ABO3 perovskite
surface. The large black spheres are A ions and the small black spheres are B ions. This model
also
shows
the
geometry
of
a
surface
oxygen
vacancy
on
the
BO2
plane [ref. 2].
The mechanism of charge redistribution depends on the surface type.
Based on
electrostatic criteria, P. W. Tasker [49] established a classification of surface types. A type I
surface requires that the charge in each plane parallel to the surface is distributed such that these
25
planes are charge neutral. If these planes are not charge neutral but are stacked to give a repeat
unit (or bi-layer) with a net zero dipole moment, this is called a type II surface. A type III
surface is a polar surface — i.e., each layer parallel to the surface has a finite charge σ and the
stacking supports a net dipole between each layer throughout the crystal. Type III surfaces are
associated with a high surface energy and the instability of such surfaces has recently been
explained in terms of a simple electrostatic effect as discussed below. The (001) surfaces of
SrTiO3 are generally considered type I (assuming formal charges on the ions). As pointed out by
Noguera and co-workers, however, the real charges on the ions are not likely to be the formal
charges so that perovskite (001) surfaces are likely to have type III properties [50]. They
proposed to describe the (001) termination as “weakly polar” to distinguish it from standard
polar surfaces such as ZnO terminated at the basal planes.
Assuming that surface relaxation is negligible, Noguera and co-workers have developed a
criteria for the stability of polar and “weakly polar” surfaces. It is based on the fact that each bilayer contains a constant electrostatic field such that the electrostatic potential increases
monotonically with the thickness of the crystal. The total dipole moment is directly proportional
to the number of bi-layers so that an infinite energy is associated with macroscopic crystals and
accounts for the instability of the surface. This macroscopic dipole moment, however, is found
to be completely removed if the magnitudes of the surface σs and subsurface σss charges are
modified to fulfill the following condition:
σ ss − σ s =
σb
,
2
[1.4]
where σb is the magnitude of the charge on the bulk (001) planes. This is referred to as the
electrostatic condition for the stability of a polar surface.
26
It is straightforward to show that in the absence of a modification of surface/ subsurface
intraplane covalency (i.e., σss = σb), the stability condition is met for the (001) termination of
SrTiO3 simply as a result of the bond breaking mechanism in the formation of the SrO and TiO2
surfaces [50]. Therefore, significant modification in the electronic structure is not expected. The
reduction of the surface ionic charges with respect to the bulk has been verified by X-ray
photoemission studies [4,51].
The latter has also given evidence, however, in support of
enhanced covalency of the (001) surface due to a decrease in the Madelung potentials as
compared with the bulk. When this decrease is significantly large, relaxation will be observed.
Relaxation can be anticipated from the results of the ionic model. The general approach
requires a minimization of the full expression for the lattice energy (i.e., [1.1] plus a short-range
repulsive term) to determine the equilibrium inter-atomic separation, rijeq . The well-known result
shows that rijeq is an increasing function of the coordination number and inversely proportional to
the Madelung constant (where the Madelung constant varies more slowly than the coordination
number). Since both the SrO and TiO2 terminations necessarily contain under coordinated
atoms, one might expect a contraction for the outer layers of both surface terminations. Detailed
theoretical calculations [52] and a detailed LEED study [11] both confirm an inward relaxation
of the SrO-terminated plane. The former study also predicted an inward relaxation of the TiO2
plane, while the latter study observed an outward relaxation of this plane (at T = 120K). Yet
another group reported an outward relaxation on both terminations [12]. Whatever the actual
case may be, it is clear that the equilibrium structure is influenced by a balance between shortrange (intra-atomic) and long-range (inter-atomic) Coulombic forces.
The former favors
increased ionicity while the latter favors increased covalency. This competition is found to
explain and predict a variety of surface effects in SrTiO3 including a planar ferroelectric domain
27
structure on the SrO face [52], as well as the surface electronic structure for the stoichiometric
and oxygen deficient TiO2-terminated face.
Other structural modifications at the (001) surface may be attributed to differences in the
properties between the surface ions. The large polarizability of the anions as compared to the
cations, for example, is effective in shielding the often significantly large field associated with
the surface dipole moment. This field exerts a greater force on the cations than on the anions
giving rise to surface rumpling. This effect can be thought of as an inhomogeneous inward
relaxation of the surface where the cations are relaxed more strongly than the anions. Evidence
of rumpling was observed in the LEED investigations by Bickel et al. [11].
Understanding the crystallography and morphology of the real SrTiO3-x (001) surface is
critical to the interpretation of the observed electrical/optical properties and the associated
chemical reactivity. It can never be too overstated that the observed structure and properties are
strongly dependent on the thermal history of the sample. It has become clear that the resulting
chemistry and morphology vary considerably with temperature, oxygen partial pressure,
annealing time, as well as cooling rate [53–62]. Many of the conflicting results reported to date
have been argued to arise due to varying sample preparation recipes from one research group to
the next. Often at the heart of debate is whether or not the observed structure represents an
equilibrium structure. The most comprehensive study should combine the available tools of
modern surface science to examine chemistry, crystallography, morphology, and electronic
structure.
Moreover, comparing the analysis of the various data must draw consistent
conclusions that are well-substantiated on theoretical grounds. Below are brief discussions of
two views currently proposed to describe the real surface of (001) SrTiO3-x. The first argues in
28
favor of a phase separation or demixing phenomenon, while the second argues in favor of defect
ordering.
The observation of a variation in cation:cation stoichiometry, facilitated by modification
of the surface Sr content, has led to the idea that surface restructuring occurs by the formation of
new oxides [53,54]. The mechanism of this restructuring in some cases is attributed to migration
of Sr between the surface and the bulk, and its affect has been reported to extend up to 200 [54]
to 1000 [53] atomic planes beneath the surface. The two most comprehensive studies of this
phenomenon, however, do not agree on the details of the resulting new surface oxide. In one
case it was determined that upon heating in a reducing atmosphere the surface restructures to
form various orders (n) of the sub-oxide Srn+1TinO3n+1 known as Ruddlesden-Popper (R-P)
phases, where n = 1 or 2 depending on the extent of local reduction [53]. In the other case, it was
determined that reducing the surface results in the formation of TiO-rich phases, known as
Magneli phases, with R-P phases forming in the sub-surface region [54]. The latter study also
reported that the order of this bi-layer structure reversed upon high temperature equilibration in
an oxidizing atmosphere.
One might expect that the formation of new surface oxides cannot proceed without a
simultaneous modification of the electronic structure and associated electrical/optical properties.
The three to five times increase in the unit cell upon the formation of R-P phases (for example)
not only reduces the size of the Broullouin zone, but the reduction in lattice symmetry should
modify the band structure in terms of lifting energy degeneracies in the Bloch functions. No
study, theoretical or experimental, has yet reported a modification in electronic structure with the
formation of these new surface phases.
29
The other view of surface restructuring explains the mechanism in terms of oxygen
vacancy ordering.
Again, a simple electrostatic argument is sufficient for a qualitative
description of this effect. The introduction of a single oxygen vacancy on the TiO2 termination
redistributes charge q such that a quadrupole (co-planar with the surface) forms with -q on the
two linearly coordinated Ti ions and +2q on the vacancy site. Numerical studies suggest that
aligned quadrupoles on the TiO2 surface have a large attractive interaction while the interaction
is repulsive if two quadrupoles are normal to each other [50]. It can therefore be expected that
large densities of Ti-Vo-Ti quadrupoles will preferentially order to form a parallel row structure.
This structure maximizes the attractive energy while minimizing the repulsive energy. It should
be noted, however, that this simple explanation ignores relaxation effects. Indeed the literature
reports the formation of a variety of superstructures observed with LEED, RHEED and STM
including 2 × 1, 2 × 2 , c( 4 × 2 ) , c( 6 × 2) and
5 × 5 − R26.6 o [53–61]. Interestingly, atomic
scale STM images showing features 8.7Å apart and aligned in parallel rows have been ascribed
to oxygen vacancies on the TiO2-terminated surface [57]. These images were acquired with a
tunneling microscope reported to be biased such that the features reflect the occupied density of
states in the surface band gap which are attributed to oxygen vacancies or oxygen vacancyrelated defects. A similar structure was reported to be observed on the reduced (001) surface of
BaTiO3 [63].
Oxygen vacancies may also be considered to form on the SrO-terminated surface.
Following a similar electrostatic argument, however, it is clear that this does not occur without
tending to destabilize the surface. The charge redistribution (neglecting relaxation effects) for a
single oxygen vacancy shifts electron density towards the Ti located in the subsurface plane
introducing a dipole oriented normal to the surface. If planar SrO1-x surfaces were able to form, it
30
can be expected that, for sufficiently large x, a superstructure will develop due to repulsion from
adjacent (parallel) dipoles. As previously discussed, however, surfaces sustaining a net dipole
moment are associated with large energies. In fact, the oxygen defect formation energy on a SrO
surface is larger than that on the TiO2 surface. This is supported by the absence of surface states
on SrO-terminated samples annealed under UHV conditions [59].
It was previously mentioned that the interplay between inter- and intra-atomic Coulombic
forces influence many of the observed surface properties including the electronic structure.
Early theoretical determinations of the surface electronic structure for the perfect (001) TiO2terminated plane predicted the existence of band gap states of pure d-orbital character extending
from the center of the gap to the conduction band edge with a density of about 1015 electrons per
cm2 [64,65]. As such states were not observed experimentally, the theory was refined to include
electron correlation effects which pushed the previously predicted states up towards the
conduction band. The enhancement of covalency at the surface, due to a decrease in the
Madelung potential of the ions, transfers charge from the oxygens to the titaniums. This lowers
the energies of the surface d-bands into the gap. The effect of the intra-atomic Coulombic force,
however, tends to shift these states up in energy so that surface resonances in the conduction
band are expected. Such resonances are said to be difficult to distinguish experimentally from
bulk conduction band states [31]. The introduction of oxygen vacancy defects is a mechanism
by which the inter-atomic force can dominate the intra-atomic force, as a result of a further
reduction of the titanium Madelung potentials, and thus shift the d-states into the band gap region
[66].
31
A
Titanium
Oxygen
B
Figure 1.5
TiO2 termination of (001) SrTiO3 showing titanium adatoms at (A) a terrace site
and at (B) a step edge. The B adatom is symmetrical with the corner Ti as defined by the shaded
mirror plane.
Recent advanced calculations based on tight binding [67] and first-principles
pseudopotential methods [68] qualitatively predict similar surface electronic structures. It is
worth noting that in the former study [67] no gap states were derived upon the introduction of
oxygen vacancies until relaxation of the titaniums towards the center of the defect was
considered. It was found, however, that no degree of relaxation was sufficient to explain the
experimentally observed deep defect levels (see discussion below). It was therefore assumed
that upon significant removal of oxygen ions, titanium adatoms may be present on TiO2 terraces
or adsorbed on step sites, as schematically shown in Figure 1.5. Theoretical treatment of these
configurations indeed derived band gap defect states in reasonable agreement with experiment.
In the case of a terrace adatom (A) two levels were derived at 2.44 eV and 3.01 eV above the top
32
of the valence band edge. When cation adatoms appear at step sites (B) a state is found to lie 1.3
eV above the valence band edge. This deeper lying state is a direct consequence of the greater
Ti–Ti interaction at corner sites as opposed to terrace sites. Indeed, step sites on vicinal (001)
SrTiO3 are believed to be active in the dissociative adsorption of H2O, whereas the latter adsorbs
molecularly upon terraces [4].
Table 1.4: Theoretical and observed defect-induced ionization energies in SrTiO3-x.
Energy (eV)
Proposed origin
Reference
1.30
Argon ion sputtering
56
1.77
single electron excitation to CB
18,43,44
1.90
Ti adatom at step site
67
2.14
possible divacancies
44
2.40
interband excitation (CB to CB)
18,43,45
2.90
saturable upon reduction
18,42,43
The surface electronic structure has also been studied experimentally using photoelectron
spectroscopy (PES), resonant photoemission (RESPE), and inverse photoemission spectroscopy
(IPES) for surfaces prepared by several methods such as vacuum fracturing, Ar-ionbombardment, and polishing followed by vacuum annealing [31,55,56,69].
The general
observations may be summarized as follows: a) vacuum fractured surfaces do not show band gap
emission; b) fractured (or Ar-bombarded) and annealed surfaces show band gap emission of
~1013 electrons per cm2 centered at E F − 0.7 eV and E F − 1.3eV attributable to extrinsic states,
where the former is associated with a bulk state and both defect states show strong Ti-3d
character; c) surface enhanced covalency is confirmed by observed Ti resonance over the entire
width of the valence band; and d) the introduction of oxygen vacancy defects reduces the density
33
of states near the top of the valence band, as might be expected since charge is transferred to the
defect states. Table 1.4 summarizes defect ionization energies believed associated with oxygen
non-stoichiometry as observed by bulk or surface probes or as determined by first principles
calculations.
1.3
PHOTO-ASSISTED TUNNELING SPECTROSCOPY
1.3.1 Introduction to PATS
The effects of coupling light to the STM junction have been investigated in many experimental
and theoretical studies over the past decade [70–84]. There are various methods of coupling
light to the junction as well as a variety of signals generated at the junction. The first, and
perhaps simplest, application of photo-STM utilized the generation of a local surface
photovoltage (LSPV) to increase the carrier density in semi-insulating GaAs to facilitate image
acquisition [70]. The bulk of the literature to date report using the LSPV effect to either image
topography [71] or to generate LSPV maps [72–76]. The latter gives information regarding
spatial variations in materials properties such as band structure, doping density, and surface
defect structure (see section 1.3.2). The LSPV effect in tunneling spectra has been observed in a
number of investigations on both narrow- and wide-band gap semiconductors [77–80]. The
photo-induced tunneling current at constant voltage has also been measured as a function of
photon energy (from below to above the fundamental absorption edge) to characterize the
efficiency of photocell material [81].
In addition to photovoltage or photocurrent measurements, the signal generated at the
junction may arise from: a) thermal expansion of the tip and sample due to local heating; b) a
thermoelectric potential due to differential heating of the tip — i.e., the Thompson effect; c)
excitation of tip-induced plasmons on a metallic surface; or d) enhancement of the incident
34
electric field vector due to the geometry of the tunneling junction (antenna effect). Depending
on the details of the experimental method chosen for photo-STM, one or more of these effects
may have significant influence on the observed data and may obscure the particular phenomenon
under investigation. Care must be taken to minimize unwanted contributions. In some more
elaborate experimental schemes these effects may be exploited to generate photo-induced
signals. Nonlinearities in the thermal expansion (or surface polarization), for example, have
been used to generate harmonics in the tunneling current (or a displacement current) due to the
difference-frequency signal (kilohertz to megahertz range) generated by illuminating the junction
with two monochromatic lasers of different wavelengths [82]. When the difference frequency
generated is in the gigahertz range, the inherent nonlinearity of the tunneling junction currentvoltage characteristic results in a rectification of the tunneling current proportional to the square
root of the product of the incident laser powers [83]. The former effect has been utilized for
imaging and local conductivity characterization [84].
Ebi
eVB' < eVB
eVB
Ec
Ef
--------
Eg
+++
++++
+++
depletion width
Figure 1.6
Ev
Ec
Ef
-------++
+++
++
Ev
depletion width
Surface photovoltage effect upon illuminating a n-type depletion semiconductor
with energies equal to or greater than the band gap energy, Eg. The left sketch shows the band
35
structure at equilibrium (dark); the right sketch shows the band structure upon injection of excess
carriers. Note the decrease in both the surface potential and width of the depletion region.
1.3.2 Three basic photon absorption mechanisms
The experimental geometry and method chosen for the application of PATS is described in
Chapter 2. This section describes the three primary charge excitation mechanisms that may be
detected in the tunneling spectra using such a method. These include: 1) generation of a surface
photovoltage; 2) decrease of surface charge via surface and/or bulk photoabsorption; and 3)
increase of the surface charge via bulk photoabsorption. In principle, one or more of these
mechanisms may be activated (if allowed) depending on the energy and intensity of the incident
light.
Figure 1.6 illustrates the generation of a surface photovoltage for a n-type semiconductor
in depletion. At equilibrium the Fermi energy is assumed to be determined by the energy of the
acceptor-type surface states. The negative charge on the surface is compensated by a positive
space charge consisting of a uniform distribution of ionized donor defects. The latter is referred
to as the depletion region and its width is defined by the point in the bulk where the sample is
charge neutral. The displacement of charge at the surface of the semiconductor is compensated
by a built-in electric field, Ebi, that opposes further drift of electrons to the surface states. If the
sample absorbs light energy equal to or greater than the band gap, excess carriers are introduced
in the form of mobile electrons in the conduction band and holes (with considerably less
mobility) in the valence band. If the excitation occurs within the depletion width, the electrons
are swept into the bulk by the built-in field and the holes are swept to the surface where they
recombine with the charge in occupied surface states. At steady state, a constant supply of holes
to the surface may be facilitated by electron flow from the metal tip (not shown in the figure).
36
Therefore, at zero external bias, a net current is induced by the absorption of band gap light. The
decrease in the depletion width is a direct consequence of the decrease in the surface charge and
an induced field within the tip-sample gap accompanies the displacement of the Fermi levels.
The resulting steady state LSPV is thus the manifestation of a balance between minority carrier
injection at the surface and recombination via surface states.
eVB' < eVB
eVB' > eVB
Ec
Ef
--------
Ev
++
+++
++
depletion width
Ec
Ef
--------
Ev
++
+++
++
depletion width
b
a
Figure 1.7
Other photoabsorption mechanisms that (if allowed) may accompany illumination
of a n-type depletion semiconductor with photon energies less than the band gap energy. The
surface potential and depletion width decrease in (a); the surface potential is expected to increase
in (b) with little or no change in the depletion width.
Sub-band gap light may be absorbed via excitation of charge from occupied surface or
bulk states to the conduction band or via excitation of charge from the valance band to acceptortype
states
within
the
depletion
region.
These
mechanisms
are
sketched
in
Figure 1.7. Similar to the LSPV effect, both the surface charge and depletion width are reduced
as shown in the case of Figure 1.7a. In contrast to the LSPV effect, a zero bias current is not
sustained; only a redistribution of charge to accommodate a reduced surface potential. This
37
effect has profound influence on the characteristics of tunneling spectra (see section 3.3). A
different absorption mechanism, illustrated in Figure 1.7b, will result in an increase in the
surface potential due to an increase in the positive space charge. In oxygen deficient SrTiO3,
where the oxygen vacancies are singly ionized at room temperature, such a mechanism may
occur if sufficiently energetic light induces excitation from deep lying oxygen vacancy or
vacancy-related trap states in the depletion region. In reality it is not too unreasonable to expect
several surface and/or bulk absorption mechanisms to occur simultaneously under the
appropriate conditions.
1.3.3 Limitations of PATS
A few comments are in order regarding the potential contribution of additional signals generated
at the tunneling junction, as discussed in section 1.3.1. In the present application of PATS, the
most probable origins of extraneous signals include tip (or sample) expansion due to local
heating, thermovoltages due to differential tip and/or junction heating, and rectification due to
field enhancement at the tip. These effects have been studied in detail by various groups [85–92]
and the general findings are summarized below.
Tip and sample heating due to junction illumination is simply unavoidable.
Some
researchers have resorted to back-illumination [78] or laser modulation [85] techniques to
minimize this effect. In the latter case, it has been shown that there exists a cutoff frequency, fc,
dependent on the thermal properties of the sample and tip, above which the effect is rectified
[86,87]. The high frequency roll-off of current pre-amplifiers, however, may present difficulties
in obtaining desired photo-induced signals using modulation techniques. Constant illumination
is applied in the present implementation of PATS. If the thermal conductivity of the metal tip is
much larger than that of the material under investigation, then the initial response upon
38
illumination will be dominated by the response of the tip. For a tungsten tip, the tunneling signal
increases to a maximum within 18µs after a 20ns laser pulse (~6.4µJ) due to tip expansion [89].
One might conclude that an alternative method for minimizing expansion effects is to use low
incident power and acquire the tunneling signal rapidly (in ≤ 18µs). This, however, must be
balanced against the time to average multiple spectra (typically 5 to 10 per acquisition) and
coupling enough power to the junction to generate a response above the noise floor of the
electronics.
Figure 1.8
The laser induced thermovoltage versus irradiance for two different cone angles
resulting from surface heating only [ref. 89].
39
Thermovoltages are induced within a metal rod as a result of a temperature gradient along
the rod (the Thomson effect). Additionally, a heated metal/semiconductor junction may be
cooled by the induction of a current flow from semiconductor to metal (the Peltier effect). In a
STM junction, the latter effect has been found to be negligible compared to the former [89].
Figure 1.8 shows a theoretical determination of induced thermovoltage due to surface heating as
a function of incident power density. Thermovoltages on the order of a few tens of millivolts are
predicted for sharper tips (half-cone angle = 15º) and a few millivolts for blunter tips (half-cone
angle = 30º) when illuminated with irradiance in the range 1–2 MW/cm2. For tunneling
spectroscopy on wide band gap systems, which typically requires a voltage sweep over several
volts, this effect may be lost within the noise floor of the electronics.
Under suitable
circumstances, however, thermovoltage effects can be important. Indeed several studies have
used this effect for chemical differentiation in STM images, to study the effects of surface defect
structure on the behavior of a two-dimensional electron gas, as well as to obtain average
thermoelectric coefficients for surface structures with better than 1nm lateral resolution [90].
40
Figure 1.9
Calculated electric field intensity between tip and sample versus irradiance for
illumination of λ = 10 µm light. The half-cone angle used was 15° and the field was evaluated at
a distance 2000 Å from the tip apex [ref. 91].
Finally, the junction comprised of a sharp tip and conductive sample is similar to the
well-studied metal-metal point contact diode. Rigorous calculations have been performed to
describe the nonlinear I-V characteristics of the latter [91]. It has been established that the
geometry of the junction is such that when incident radiation has a wavelength larger than the
dimensions of the tip apex (i.e., infrared radiation), oscillating currents are induced on the tip
surface as if it were a wire antenna. These oscillating currents induce charges at the tip apex and
mirror charges on the conducting sample giving rise to an oscillating voltage at the junction. The
resulting electric fields in the junction have been shown to be up to a thousand times larger than
that in the incident radiation. As shown in Figure 1.9, the magnitudes of these fields can be of
the same order as those induced by the applied bias and in some cases sufficient to induce field
41
emission (rather than tunneling) of electrons from the metal surface. These effects are not
expected to be significant for light in the visible region. On the other hand, results from recent
theoretical calculations for 488 nm and 850 nm light suggest that field enhancement is greatest
when the angle of incidence (measured with respect to the surface normal) is less than 10–20
degrees and it is larger for sharper tips [92] owing to an inverse relationship between the optical
electric field at the tip apex and the radius of curvature of the tip. Furthermore, the field
enhancement is greatest when the electric field vector lies in the plane of incidence (the plane
containing the k vector of the incident wave and the axis of the tip).
A non-zero field
enhancement for E-fields polarized normal to the plane of incidence may arise due to
irregularities in the tip shape (i.e., deviation from ideal conical symmetry), a condition more
likely close to reality.
It is concluded that the aforementioned light-induced effects may be minimized to
varying degrees of success by proper experimental design. Specifically, one should consider
illumination power and angle of incidence, tunneling tip material and shape, and finally junction
illumination duration versus data acquisition mode/rate.
Note that the bias applied at the
tunneling junction, with a typical gap width of 5–10 Å, induces fields of the order 107 volts per
cm. The experimental flux data (see Figure 2.2) indicates that the irradiance of the light source
used is of the order 1.64 mW per cm2. These powers are off to the left of Figure 1.9 so that the
expected electric field induced by the light source in the present application is at least three
orders of magnitude less than the field induced by the applied bias and thus should not present an
experimental difficulty.
42
1.4 THESIS OBJECTIVES
The previous sections have presented an overview of the progress made in increasing the
understanding of bulk and surface structure and properties of electron-doped strontium titanate.
The various, and at times conflicting, observations of the surface structure highlights the need for
continued and focused research on the subject.
Specific unresolved issues regard the
microscopic nature of the reactivity or optical response of the defective (001) surface. Is it
surface or subsurface mediated?
Are acceptor-type surface states associated with oxygen
vacancies, titanium adatoms, or other undetermined defects such as surface dislocations or a new
oxide? One objective of this thesis is to contribute an unambiguous piece to the growing puzzle
of information, and/or to establish a relationship amongst the existing pieces of the puzzle,
through a study of the optical activity of (001) surfaces of electron-doped strontium titanate.
This is accomplished by another objective of this thesis which is to demonstrate the utility of the
technique of photo-assisted tunneling microscopy and spectroscopy for identification and
characterization of local charge transfer mechanisms at surfaces. The investigation seeks to
identify the local origins of surface (or subsurface) optical responsivity associated with
previously identified energies as outlined in Table 1.4. The quantitative assessment of the
electronic behavior is derived through PATS combined with theoretical modeling. The details of
the experimental design are described in Chapter 2 and the theoretical background of tunneling
spectroscopic analysis is presented in Chapter 3.
REFERENCES
1.
D. P. Woodruff and T. A. Delchar Modern Techniques of Surface Science 2nd ed.
Cambridge University Press, Cambridge (1994).
43
2.
V. E. Henrich and P. A. Cox The Surface Science of Metal Oxides Cambridge
University Press, Cambridge (1994).
3.
J. P. Zielger, E. K. Lesniewski and J. C. Hemminger J. Appl. Phys. 61 [8] (1987)
3099; J. P. Zieger and J. C. Hemminger J. Electrochem. Soc. 134 (1987) 358.
4.
N. B. Brookes, F. M. Quinn and G. Thornton Physica Scripta. 36 (1987) 711
5.
P. Salvador, C. Gutiérrez, G. Campet and P. Hagenmüller J. Electrochem. Soc.
131 [3] (1984) 550
6.
E. B. Várhegyi, I. V. Perczel, J. Gerblinger, M. Fleischer, H. Meixner and J. Giber
Sensors and Actuators B 18–19 (1994) 569
7.
D. M. Hill, H. M. Meyer III and J. H. Weaver J. Appl. Phys. 65 [12] (1989) 4943
8.
M. Copel, P. R. Duncomber, D. A. Neumayer, T. M. Shaw and R. M. Tromp Appl.
Phys. Lett. 70 (1997) 3227
9.
See Appendix A for a discussion on the optical resolution limit realizedwhen electronic
transitions are coupled to vibrational modes of the lattice.
10. E. Schaar-Gabriel, N. Alonso-Vante and H. Tributsch Surface Science 366
(1996) 508
11. N. Bickel, G. Schmidt, K. Heinz and K. Müller Phys. Rev. Lett. 62 [17]
(1989) 2009
12. T. Hikita, T. Hanada, M. Kudo and M. Kawai Surface Science 287 (1993) 377
13. U. Balachandran and N. G. Eror J. Mat. Sci. 17 (1982) 2133
14. S. Witek, D. M. Smyth and H. Pickup J. Am. Ceram. Soc. 67 [5] (1984) 372
15. W. A. Harrison Electronic Structure and the Properties of Solids W. H. Freeman and
Company, San Francisco (1980) 468
44
16. M. J. Akhtar, Z. Akhtar, R. A. Jackson and C. R. A. Catlow J. Am. Ceram. Soc.
78 [2] (1995) 421
17. CRC Handbook of Chemistry and Physics 72nd ed., David R. Lide, Ed.
(1991–1992)
18. H. Yamada and G. R. Miller J. Solid State Chem. 6 (1973) 169
19. A. H. Kahn and A. J. Leyendecker Phys. Rev. 135 [5A] (1964) A1321
20. L. F. Mattheiss Phys. Rev. B 6 [12] (1972) 4718
21. T. F. Soules, E. J. Kelly, D. M. Vaught and J. W. Richardson Phys. Rev. B 6 [4]
(1972) 1519
22. J. C. Slater and G. F. Koster Phys. Rev. 94 (1954) 1498
23. H. P. R. Frederikse, W. R. Thurber and W. R. Hosler Phys. Rev. 134 [2A]
(1964) A442
24. M. Cardona Phys. Rev. 140 [2A] (1965) A651
25. R. C. Casella Phys. Rev 154 [3] (1967) 743
26. M. J. Lax Symmetry Principles in Solid State and Molecular Physics Wiley,
New York (1974)
27. L. Bouckaert, R. Smoluchowski and E. Wigner Phys. Rev. 50 (1936) 58
28. P. A. Cox Transition Metal Oxides: An Introduction to Their Electronic Structure
and Properties Oxford University Press, New York (1992)
29. P. G. Perkins and D. M. Winter J. Phys. C: Solid State Phys. 16 (1983) 3481
30. F. P. Koffyberg, K. Dwight and A. Wold Solid State Comm. 30 (1979) 433
31. B. Reihl, J. G. Bednorz, K. A. Müller, Y. Jugnet, G. Landgren and J. F. Morar
Phys. Rev. B 30 [2] (1984) 803
45
32. J. Blanc and D. L. Staebler Phys. Rev. B 4 [10] (1971) 3548
33. D. M. Smyth Prog. Solid St. Chem. 15 (1984) 145
34. S. Ruddlesden and R. Popper Acta Crystallogr. 11 (1958) 54
35. K. R. Poeppelmeier, M. E. Leonowicz and J. M. Longo J. Solid St. Chem. 44
(1982) 89
36. A. Reller, J. M. Thomas, F. R. S.,D.A. Jefferson, and M. K. Uppal Proc. R. Soc.
Lond. A 394 (1984) 223
37. M. O. Selme and P. Pêcheur J. Phys C: Solid State Phys. 16 (1983) 2559
38. C. Lee, J. Yahia and J. L. Brebner Phys. Rev. B 3 [8] (1971) 2525
39. N. Shanthi and D. D. Sarma Phys. Rev. B 57 [4] (1998) 2153
40. Levin, Field, Plock and Merker J. Opt. Soc. Am. 45 (1955) 737
41. J. A. Noland Phys. Rev. 94 (1954) 724
42. H. W. Gandy Phys. Rev. 113 [3] (1959) 795
43. C. Lee, J. Destry and J. L. Brebner Phys. Rev. B 11 [6] (1975) 2299
44. G. Perluzzo and J. Destry Can. J. Phys. 56 (1978) 453
45. R. L. Wild, E. M. Rockar and J. C. Smith Phys. Rev. B 8 [8] (1973) 3828
46. K. V. Yumashev, P. V. Prokoshin, A. M. Malyarevich and V. P. Mikhailov J. Opt.
Soc. Am. B 14 [2] (1997) 415
47. J. Y. Chang, M. H. Garrett, H. P. Jenssen and C. Warde Appl. Phys. Lett. 63 [26]
(1993) 3598
48. M. A. Butler, M. Abramovich, F. Decker and J. F. Juliao J. Elctrochem Soc.: SolidState Science and Technology 128 [1] (1981) 200
49. P. W. Tasker J. Phys. C: Solid State Physics 12 (1979) 4977
46
50. J. Goniakowski and C. Noguera Surface Science 365 (1996) L657; C. Noguera,
A. Pojani, F. Finocchi and J. Goniakowski (unpublished)
51. R. Courths, J. Noffke, H. Wern and R. Heise Phys. Rev. B 42 [14] (1990) 9127
52. V. Ravikumar, D. Wolf and V. P. Dravid Phys. Rev. Lett. 74 [6] (1995) 960
53. Y. Liang and D. A. Bonnell Surface Science Lett. 285 (1993) L510; Y. Liang and
D. Bonnell J. Am. Ceram. Soc. 78 [10] (1995) 2633
54. K. Szot, W. Speier, J. Herion and Ch. Freiburg Appl. Phys. A. 64 (1997) 55; K.
Szot and W. Speier (unpublished)
55. V. E. Henrich, G. Dresselhaus and H. J. Zeiger Phys. Rev. B 17 [12] (1978) 4908
56. B. Cord and R. Courths Surface Science 162 (1985) 34
57. H. Tanaka, T. Matsumoto, T. Kawai and S. Kawai Jpn. J. Appl. Phys. Pt.1 32
[3B] (1993) 1405; T. Matsumoto, H. Tanaka, T. Kawai and S. Kawai Surface
Science Lett. 278 (1992) L153
58. Q. Jiang and J. Zegenhagen Surface Science 367 (1996) L42; Q. D. Jiang and
J. Zegenhagen Surface Science 425 (1999) 343
59. A. Hirata, A. Ando, K. Saiki and A. Koma Surface Science 310 (1994) 89
60. N. Ikemiya, A. Kitamura, and S. Hara J. of Crystal Growth 160 (1996) 104
61. M. Naito and H. Sato Physica C 229 (1994) 1
62. B. Stäuble-Pümpin, B. Ilge, V. C. Matijasevic, P. M. L. O. Scholte, A. J. Steinfort
and F. Tuinstra Surface Science 369 (1996) 313
63. H. Bando, T. Shimitzu, Y. Aiura, Y. Haruyama, K. Oka and Y. Nishihara J. Vac.
Sci. Technol. B 14 [2] (1996) 1060
64. F. J. Morin and T. Wolfram Phys. Rev. Lett. 30 [24] (1973) 1214
47
65. T. Wolfram, E. A. Kraut and F. J. Morin Phys. Rev. B 7 [4] (1973) 1677
66. S. Ellialtioglu and T. Wolfram Phys. Rev. B 18 [8] (1978) 4509
67. G. Toussaint, M. O. Selme and P. Pecheur Phys. Rev. B 36 [11] (1987) 6135
68. S. Kimura, J. Yamauchi, M. Tsukada and S. Watanabe Phys. Rev. B 51 [16]
(1995) 11,049
69. R. Courths, B. Cord and H. Saalfeld Solid State Comm. 70 [11] (1989) 1047
70. G. F. A. Van De Walle, H. van Kempen, P. Wyder and P. Davidsson Surface
Science 181 (1987) 356
71. T. W. Mercer, D. L. Carroll, Y. Liang, N. J. DiNardo and D. A. Bonnell J. Appl.
Phys. 75 [12] (1994) 8225
72. R. J. Hamers and K. Markert Phys. Rev. Lett. 64 [9] (1990) 1051
73. R. J. Hamers and K. Markert J. Vac. Sci. Technol. A 8 [4] (1990) 3524
74. Y. Kuk, R. S. Becker, P. J. Silverman and G. P. Kochanski Phys. Rev. Lett. 65
[4] (1990) 456
75. D. G. Cahill and R. J. Hamers J. Vac. Sci. Tenchol. B 9 [2] (1991) 564
76. G. P. Kochanski and R. F. Bell Surf. Sci.Lett. 273 (1992) L435
77. L. Q. Qian and B. W. Wessels Appl. Phys. Lett. 58 [12] (1991) 1295
78. F. F. Fan and A. J. Bard J. Phys. Chem 97 (1993) 1431
79. G. S. Rohrer, D. A. Bonnell and R. H. French J. Am. Ceram. Soc. 73 [11]
(1990) 3257
80. H. Sugimura, T. Uchida, N. Shimo, N. Kitamura and H. Masuhara Mol. Cryst. Liq.
Cryst. 253 (1994) 205
48
81. S. Akari, M. Ch. Lux-Steiner, K. Glöckler, T. Schill, R. Heitkamp, B. Koslowski
and K. Dransfeld Ann. Physik 2 (1993) 141
82. L. Arnold, W. Krieger and H. Walther J. Vac. Sci. Technol. A 6 (1988) 466
83. M. Völcker in Scanning Probe Microscopy: Analytical Methods R. Wiesendanger,ed.
Springer-Verlag, New York (1998) 135
84. M. Völcker, W. Krieger and H. Walther J. Vac. Sci. Technol. B 12 [3]
(1994) 2129
85. S. Graftström, J. Kowalski, R. Neumann, O. Probst and M. Wörtge J. Vac. Sci
Technol. B 9 (1991) 568
86. E. Oesterschulze, M. Stopka and R. Kassing Microelectronic Engineering 24
(1994) 107
87. A. V. Bragas, S. M. Landi, J. A. Coy and O. E. Martínez J. Appl. Phys. 82 [9]
(1997) 4153
88. I. Lyubinetsky, Z. Dohnálek, V. A. Ukraintsev and J. T. Yates, Jr. J. Appl. Phys.
82 [8] (1997) 4115
89. S. H. Park, N. M. Miskovsky, P. H. Cutler, E. Kazes and T. E. Sullivan Surface
Science 266 (1992) 265
90. A. Rettenberger, C. Baur, K. Läuger, D. Hoffmann, J. Y. Grand and R. Möller
Appl. Phys. Lett. 67 [9] (1995) 1217; D. H. Hoffmann, A. Rettenberger, J. Y.
Grand, K. Läuger, P. Leiderer, K. Dransfeld and R. Möller Thin Solid Films 264
(1995) 223; D. Hoffmann, J. Y. Grand, R. Möller, A. Rettenberger, K. Läuger
Phys. Rev. B 52 [19] (1995) 13,796
91. T. E. Sullivan, P. H. Cutler and A. A. Lucas Surface Science 62 (1977) 455
49
92. M. J. Hagmann J. Vac. Sci. Technol. B 15 [3] (1997) 597
50
Chapter 2: Experimental Method
The details of the experimental instrumentation are described and sample preparation techniques
are discussed. A brief introduction to tunneling microscopy is also presented with distinctions
identified for the approach to PATS.
2.1
PHOTO-ASSISTED TUNNELING SPECTROSCOPY
2.1.1 Experimental arrangement
A schematic diagram of the experimental arrangement appears in Figure 2.1. Unpolarized white
light from an Oriel 400 W xenon arc lamp was focused onto the 2 mm width entrance slit of a
Fastie-Ebert-type grating monochromator. After passing through a 0.25 mm width exit slit and a
25 mm diameter, 150 mm focal length (F.L.) spherical lens, the monochromatic light was
focused onto the surface of the sample by a 68 mm F.L. cylindrical mirror mounted above the
tunneling microscope inside a vacuum chamber. This configuration was chosen to focus the
image of the exit slit onto the STM junction (magnification = −0.6) without excessive throughput
loss due to beam reflection and divergence.
The monochromator contained two reflection
gratings blazed for 600 nm (low blaze) and 1200 nm (high blaze), respectively. The former
grating is most efficient for wavelengths in the range 400–900 nm, while the latter is most
efficient in the range 800–1800 nm. The linear dispersion for the low blaze grating is 3.3 nm per
mm. Therefore, an exit slit width of 0.25 mm combined with a counter accuracy of ±1 nm gives
a total monochromator-limited wavelength uncertainty of ±1.83 nm (i.e., an energy uncertainty
of ±14 meV). Finally, a filter was used at the exit slit of the monochromator in order to
minimize unwanted energies contributed from higher order diffraction. It should be noted,
however, that the monochromator design maximizes first order diffraction from the low blaze
42
grating. This grating has a large groove density (1180 g/mm) which gives it a relatively small
angular dispersion. Overlap between first and second order diffraction is therefore assumed to
be negligible.
incident light
sample holder
θ ≈ 42°
sample
vacuum chamber
mirror
lens
400W
Xe
arc
lamp
monochromator
ion pump
sample
computer
Figure 2.1
STM stage
STM
electronics
Experimental arrangement for photo-assisted tunneling spectroscopy.
Since the incident light necessarily interacted with several optical components (including
passage through the vacuum chamber window), a characterization of the spectral response of the
optical set-up was performed by placing a Si pn-type photodiode at the position of the tip apex
and measuring the total flux reflected from the mirror. Using a pre-amp and the photo-sensitivity
data furnished by Hamamatsu Photonics, the current from the photodiode was converted to total
incidence flux as shown in Figure 2.2. It can be seen that the behavior reflects the efficiency
expected of the reflection gratings — i.e., the low blaze grating is more efficient in the near UV
43
range, while the high blaze grating is more efficient in the near IR range. It should be noted that
the particular photodiode used for these measurements had a suppressed sensitivity for
wavelengths greater than 750 nm; therefore, the actual flux in this region is expected to be
greater
than
that
shown
in
Figure 2.2. The average flux is of the order 1013 photons per second. It is assumed that this
photon density is sufficient to produce a measurable photo-response, if the cross section for
charge carrier excitation is close to unity.
The active area for the photodiode was 0.012 cm2. One watt of incident monochromatic
power is equivalent to λ (5.03 × 1015 ) photons per second, where λ is in nanometers, such that the
irradiance of the light source on the junction was of the order of a few mW per square
centimeter. As illustrated in Figure 2.1, the angle of incidence with respect to the axis of the tip
was approximately 42 degrees. These last two points, as well as the fact that the experiment was
confined to a bandwidth of 628 nm from 327 to 955 nm, suggest that the data should not be
significantly skewed by antenna effects, as discussed in section 1.3.3.
44
1.2E+14
1E+14
high blaze
8E+13
low blaze
6E+13
4E+13
2E+13
960
920
880
840
800
760
720
680
640
600
560
520
480
440
400
360
320
280
240
200
0
Wavelength (nm)
Figure 2.2
Spectral response of experimental optics measured by a Si photodiode.
The STM used was built in-house based on a concentric piezoelectric tube design [1]
where the inner tube served to provide linear motion of the tip in three mutually orthogonal
directions over the surface being imaged. The outer tube served to dynamically translate the
sample and to stabilize the STM against thermal drift due to thermal expansion of the piezo tubes
[2]. All movement and data acquisition were controlled and monitored by computer via the STM
electronics.
The latter consisted of a differential feedback unit and a high voltage triple
amplifier. The STM was housed inside a vacuum chamber where all measurements were
performed at base pressures in the low 10-10 Torr (ultra-high vacuum, UHV) range.
The chamber plus ion pump were mounted on a T-shaped steel bracket which was
suspended upon three pneumatic legs providing isolation from building (acoustic) vibrations.
Additional mechanical stability was provided by mounting the STM upon a stack of viton-spaced
steel plates, with springs placed between the bottom two plates to improve vibration damping.
45
Perhaps the most significant contribution to the mechanical stability of the STM is attributed to
materials selection. All components were constructed from Macor® glass ceramic except for the
piezo tubes (which were PZT), the rails upon which the sample holder was translated (tungstencarbide), and parts of the sample holder itself (stainless steel). These metals (as well as a few
nuts and bolts) served primarily to provide electrical connections to carry the tunneling signal.
The rigidity of the glass ceramic as well as PZT tubes ensures a high resonance frequency of the
STM assembly, well above that which may be excited by the scanning mechanism during
operation of the microscope [3]. A low-pass filter is often used to remove high frequency
(electronic) noise from images. This was observed to be problematical when attempting to
rapidly acquire spectra since it contributed additional capacitance to the circuit. A filter was thus
not used. Other than the STM, the chamber was equipped only with an ion gauge and two linear
feedthroughs. One feedthrough served to translate the sample to and from the chamber. The
other served to translate the sample to and from the STM as well as providing the electrical
connection for in-situ cleaning of the sample surface by joule heating of a tantalum foil. The
absence of several analytical peripherals on the chamber was beneficial to the extent that
additional acoustical coupling was absent.
The piezoelectric constant of the tube scanner was measured by the double piezoelectric
response technique [4] to be 44.7 Å/V. The maximum voltage applied to the electrodes was
±100 V, giving a maximum scanning area of 8940 × 8940 Å2. The edge of such a scan is less
than one hundredths the width of a fine human hair (which is about 100 microns in diameter).
Hence this device was designed for high resolution imaging.
46
2.1.2 Experimental method
All tunneling images acquired in this thesis represent constant current images.
These are
generally obtained by specifying a current/voltage combination (i.e., a set point) in the control
electronics. As the tip is scanned across the sample, a differential op-amp provides a voltage to
the z piezo that is proportional to the difference between the set point and the monitored
tunneling current. Variation in the opacity of the tunneling barrier gives rise to variations in the
tunneling current. The op-amp output voltage serves to restore the tunneling current to the set
point value. Knowing the scanning piezo constant, a map of the restoring voltage as a function
of tip position constructs the STM image. Often the latter is interpreted as a map of constant
surface state density. On the other hand, a local variation in work function (perhaps as a result of
local variation in surface charge) will vary the opacity of the tunneling barrier and also give rise
to image features. Therefore, STM images obtained in this way are also often interpreted as
maps of constant surface charge density.
In general, contrast in STM images may result from variations in surface state density,
surface charge, morphology, or any combination thereof. To appreciate image interpretation, it
is important to recall that the feedback circuit functions to maintain a constant current. If it may
be assumed that the active area of the tunneling tip does not change in the course of the
measurement, the STM produces images of constant current density. Therefore, one should have
a sufficient understanding of the charge transport mechanism near the surface of the material
under investigation.
Near surface transport for metals is different than that for traditional
semiconductors which can be different than that for low mobility semiconducting oxides. In the
presence of a Schottky barrier, the dominant mechanism can vary depending on the position of
the equilibrium Fermi energy, EF (see section 3.2.5). A degenerate n-type semiconductor (i.e.,
47
such that EF lies above the conduction band minimum) at forward bias will generate images of
constant local state density modified by topography. A non-degenerate semiconductor (EF lies
below the conduction band minimum), however, will generate images of constant surface charge
modified by topography. The difference is due only to the dominant transport mechanism in the
depletion region of the semiconductor. In the former case tunneling dominates; in the latter case
emission dominates.
The usual sample-and-hold scheme for measuring tunneling spectra was employed with a
slight modification for PATS. Upon identifying a surface feature of interest, the feedback unit
was disengaged momentarily while the bias applied to the sample was varied over a fixed
voltage range and the variation in current was recorded. Ten such acquisitions were averaged
together for a given spectrum. During a conventional measurement, the feedback unit would be
re-engaged briefly between each of the ten acquisitions in order to minimize thermal drift effects.
For the PATS application, however, re-engaging the feedback can possibly introduce an
unwanted variation in the tip-sample distance, s, if a photocurrent is induced at the set point
voltage. Such a variation in s may be misinterpreted as a charge transition photoresponse. To
avoid this problem the feedback was not re-engaged between each of the ten acquisitions of a
given spectrum. Thermal drift effects were minimized by acquiring the spectra rapidly — a
typical spectrum was acquired on the order of a few tenths of a second. The rate varied
depending on voltage sweep range and voltage step size (together determining the number of
data points) and the delay between each voltage increment.
Proper selection of a probe tip is critical to successful STM and STS measurements.
Both materials selection and tip geometry can have a significant impact on the quality of the
images and the interpretation of the spectra. Mechanically clipped Pt-Ir wire is most often used
48
as a probe because the material is inert in an ambient environment and often the mechanical
forming results in a small cluster of atoms near the apex that can facilitate atomic resolution
imaging. Unfortunately, these tips often do not appear microscopically sharp and thus can be
limited to surfaces that are “smooth” on the scale of a few tenths of a micron squared. Many of
the surfaces studied in this thesis may be characterized as microscopically rough (i.e., see Figure
2.8 on page 55) and hence required the use of tips prepared by chemical etching. Tungsten tips
were thus prepared by the dc drop off method [4] using 5M KOH solution and 12 volts between
anode (tip) and cathode. The tips were rinsed with ethanol and deionized water followed by
annealing in a reducing environment at ≥ 725 ºC overnight to remove residual oxides formed
during the etching process.
2.1.3 Experimental noise
In addition to the potential artifacts associated with the optical measurement as discussed in
section 1.3.3, there are certain aspects of conventional STM experiments that may contribute
noise to the phenomena being observed. The most significant source of electronic noise results
from stray capacitance throughout the STM circuitry that precedes the current preamp. Indeed
the decision to apply Pt electrodes to the samples was primarily to minimize the capacitance
between the sample and the clips of the sample holder.
A typical characterization of the experimental noise inherent to the measurements
presented in this thesis is shown in Figure 2.3 as a plot of the sample standard deviation in the
tunneling current versus sample bias.
This plot was generated by acquiring ten separate
tunneling spectra and calculating the variance for each I/V coordinate from this set. The result
suggests that the accuracy of the measurement decreases for larger absolute values of the applied
bias. The rapid decrease in variance observed beyond roughly ±3 volts is a consequence of the
49
saturation of the current preamp and should not be interpreted as an improvement in
reproducibility. Since the set point for the spectra obtained to generate Figure 2.3 was 500 pA at
-3 V, it is rather interesting that the variance near -3V is about five times the set point current
itself! This behavior should be considered when analyzing experimental tunneling spectra such
that one might be cautious about interpreting spectral features far from the equilibrium Fermi
level. It should also be kept in mind, however, that Figure 2.3 represents a worst-case scenario
given the manner in which the data was acquired. In practice, the spectra are acquired in rapid
succession such that phenomena like thermal drifting has negligible effects on a given spectrum.
3
2.5
2
1.5
1
0.5
4.01
3.68
3.35
3.02
2.69
2.35
2.02
1.69
1.36
1.03
0.69
0.36
0.02
-0.31
-0.64
-0.97
-1.30
-1.63
-1.97
-2.30
-2.63
-2.96
-3.29
-3.63
-3.93
0
Sample Bias (V)
Figure 2.3
Quantification of current variance inherent to the tunneling measurements obtained
in this study.
50
2.2 SAMPLE PREPARATION AND CHARACTERIZATION
2.2.1 Sample processing history
Initial attempts to prepare clean and “flat” (001) surfaces of single crystal SrTiO3 did not proceed
without considerable difficulty. The process began by orienting a single crystal boule (obtained
from Atomergic Chemetals Corp.) such that the 〈001〉 direction was aligned normal to the
surface.
An observed 4-fold rotational symmetry by back-reflection Laue verified this
orientation, as shown in Figure 2.4. The boule was then wafered using a 0.015" diameter
diamond wire blade. The wafers were polished using diamond impregnated lapping films down
to 0.1µm and finally to 0.05µm using alumina paste. Note that these last two steps could not be
controlled to absolute precision such that preservation of the original orientation could not
always be assumed. Variation in surface misorientation will result in a variation in step size and
density. For this study, it was only important that steps be present and accessible within the
STM maximum scanning range.
The crystals were then heat treated in air by a slow temperature ramp (3 degrees per
minute) to 1000 °C, held for 10 hours, and subsequently furnace cooled (5 degrees per minute).
Platinum electrodes were added to the samples using Engelhard #6926 unfritted paste. An
additional vacuum heating was necessary to reduce the undoped crystals and to activate the Nbdoped crystal (see discussion in section 4.1.1).
The air anneal resulted in flat surfaces
characterized by plateaus separated by straight-edged (and kinked) multiple unit cell steps, as
observed by AFM (Figure 2.5). After vacuum reducing, however, no surface yielded a LEED
pattern until further heat treatment in the analysis (high vacuum) chamber at ≥1000 °C for 3–5
minutes. The LEED patterns on these surfaces contained relatively broad spots and significant
background intensity. These surfaces were also extremely difficult to image with STM. Figure
51
2.6 is a representative STM image obtained from one of these crystals. It can be seen from the
left image that the surface was not uniformly clean and flat. A magnification of the region
circled in the left image appears in the right image. This stepped region reveals the nascent
ordering that probably accounted for the observed (albeit poor quality) 1×1 LEED patterns.
Figure 2.4
Laue back-reflection photograph showing 〈001〉 orientation — i.e., a 4-fold rotation
axis is centered in and normal to the plane of the photograph.
52
Figure 2.5
Atomic force microscope (AFM) image showing the stepped surface of SrTiO3
after polishing and air annealing for up to 10 hours at 1000 °C.
Figure 2.6
Scanning tunneling microscope (STM) images showing a stepped surface of SrTiO3
after a similar preparation as the sample in Figure 2.5 with an additional vacuum anneal at 1000
°C for 30 minutes. The image to the right is a magnification of the circled region appearing in
the image to the left.
53
It was subsequently determined that the reduction anneal deposited carbon upon the
surface, most likely due to contamination in the vacuum furnace. This required extended heat
treatments in the analysis chamber for surface cleaning which resulted in heavily reduced
samples. Auger electron spectroscopy (AES) measurements also revealed a growing sulfur peak
with extended vacuum annealing. The initial vacuum treatment could not be eliminated because
it was necessary for the formation of good Pt electrodes. To solve this problem, chemical
etching and an additional air anneal were introduced into the preparation procedure.
After polishing, all samples were etched using a buffered NH4–HF (BHF) solution for
1.25 minutes. The pH was determined to be in the range 4.5-4.7 using colorpHast® indicator
strips. This solution is expected to preferentially attack the more basic (i.e., the SrO) planes
leaving flat surfaces predominately TiO2-terminated [5]. After the initial vacuum annealing, the
samples were briefly re-annealed in air to remove carbon contamination and restore them to a
common insulating state. The resulting surfaces appear quite similar to those without etching
except for the occasional addition of deep trenches and square etch pits due to chemical attack
upon residual polishing damage. Figure 2.7 shows an AFM image of a typical sample prepared
with the BHF solution before additional vacuum annealing. Figure 2.8 shows AFM images of a
typical sample similarly prepared after being heavily reduced by vacuum annealing. Preparing
the samples in this way resulted in clean (i.e., carbon free) surfaces that could be reduced in a
controlled manner to obtain samples with varying conductivities and surface morphologies.
54
Figure 2.7
A 500 nm × 500 nm AFM image showing a stepped surface of SrTiO3 after a
similar preparation as the sample in Figure 2.5 with an additional BHF etching and re-annealing
in air.
Figure 2.8
AFM images of heavily reduced SrTiO3 (001) showing morphological development
as compared to the air-annealed sample in Figure 2.7. Left: 3 µm × 3µm scan. Right: 1 µm ×
1µm scan of lower right edge of the image to the left.
55
2.2.2 Methods of characterization
In addition to SPM (scanning probe microscopy), other methods were used to characterize the
developing structure, chemistry and physical properties of the samples. The cleanliness of the
surfaces was verified by Auger electron spectroscopy (AES) and surface ordering was verified
using low energy electron diffraction (LEED). The bulk optical constants (index of refraction
and extinction coefficient) were characterized using ellipsometric and transmission
spectroscopies, and bulk majority carrier density/mobility was determined by Hall
measurements.
REFERENCES
1.
J. W. Lyding, S. Skala, J. S. Hubacek, R. Brockenbrough and G. Gammie
Rev. Sci. Instrum. 59 [9] (1988) 1897
2.
D. W. Pohl Rev. Sci. Instrum. 58 [1] (1987) 54
3.
D. A. Bonnell in Scanning Tunneling Microscopy and Spectroscopy: Theory,
Techniques and Applications, D.A. Bonnell, ed. VCH Publishers Inc.,
New York (1993)
4.
J. C. Chen Introduction to Scanning Tunneling Microscopy Oxford University
Press, Oxford (1993)
5.
M. Kawasaki, K. Takahashi, T. Maeda, R. Tsuchiya, M. Shinohara, O. Ishiyama,
T. Yonezawa, M. Yoshimoto and H. Koinuma Science 266 (1994) 1540
56
Chapter 3: Tunneling Spectroscopy Theory
Tunneling spectra are simulated for the purpose of interpreting the changes observed in
experimental data. The approach to the simulations follows from the contributions of various
authors over the past ten years or so. A brief background in the concepts of 1D tunneling is
presented, followed by the details of the mathematical formulas used to generate the simulated
points. Additional details are found in Appendix B.
3.1 INTRODUCTION
3.1.1 Quantum mechanical tunneling and the WKB approximation
Before discussing the theory developed for interpretation of experimental tunneling spectra, a
brief review of quantum mechanical tunneling is presented in this section. Detailed treatment of
particle scattering by various one-dimensional potentials may be found in any textbook on
quantum physics. Only the salient features of quantum transmission through a potential barrier
are discussed here.
The problem presented is sketched in Figure 3.1. A particle of energy E, propagating in
the positive x direction, is expected to be totally reflected by the potential barrier by the laws of
classical physics. The principles of quantum physics, however, permit the incident particle to be
treated as an incident wave described by a suitable wave function ψ which exactly solves the
time-independent wave equation — i.e., Schrödinger’s eigenvalue equation (S-E) given by
d 2 ψ (x)
− κ2 ψ (x ) = 0 ,
2
dx
where
κ=
2m (E − V(x))
.
h2
58
The most general solution to the S-E for a piecewise constant potential function
(Figure 3.1a) consists of a linear sum of plane waves, ψ = Ceiκx , propagating in both the +x and x directions, where C is the amplitude of the wave.
The central problem amounts to a
determination of the ratio of the transmitted amplitude (at x > +a) to the incident amplitude (at x
< -a). In other words, one is interested in determining the transmission probability, |T|2, of the
potential barrier. For a particularly opaque barrier (i.e., where κa is large) the well-known
solution is of the form
2
T ∝e
−4 κa
.
The transmission probability is thus finite and strongly dependent on the width of the barrier as
well as the magnitude of the kinetic energy associated with the particle (which is negative in the
classically forbidden region, |x| < a). Note that this extreme sensitivity to κa is responsible for
the intrinsically high vertical spatial resolution of scanning tunneling microscopy. (The lateral
resolution depends on the experimental conditions and may be limited by the physical properties
of the electrodes.)
59
V(x)
E
-a
+a
x
a
V(x)
E
b
b′
x
b
Figure 3.1
A particle wave of energy E propagating (a) within a piecewise constant potential
or (b) within a continuous potential function.
Real barriers are often not as simple as depicted in Figure 3.1a, so that one must find
suitable solutions to the S.E. and determine the transmission probability for a potential function
of arbitrary shape as shown in Figure 3.1b. This problem has not yet been treated with exact
solutions. Instead, an approach known as the Wentzel-Kramers-Brillouin (WKB) method is
usually taken. It is assumed that the eigenfunctions for the potential in Figure 3.1b will be very
nearly in form to those for the potential in Figure 3.1a if the variation in V(x) within one de
Broglie wavelength is small compared to the kinetic energy of the particle. This condition is
easily satisfied for typical barrier functions except near the boundaries labeled b and b′ in Figure
60
3.1b. These are known as the classical turning points and the WKB method continues with the
derivation of a connection rule that establishes continuity between the appropriate WKB wave
functions far from, and on both sides of, these points.
The derivation of connection rules requires an approximation regarding the nature of the
potential in the vicinity of the turning points [1]. Simply stated, the kinetic energy is assumed to
vary linearly over the distance where the WKB functions are invalid, and the
S-E is solved exactly over this region. Two conditions are identified where the WKB method
contributes greater error to the determination of |T|2 for real tunneling barriers. The first is when
the kinetic energy approaches zero such that the incident energy is near the top of the tunneling
barrier. In this case, the connection rules are invalid because the slope of the potential varies
rapidly and a linear approximation to the behavior of the kinetic energy is not possible. The
second condition is when the barrier is very narrow such that the classical turning points
approach each other. In this case, a greater portion of the total potential function is treated by the
(artificial) straight-line approximation.
3.1.2 The purpose of modeling tunneling spectra
The system defined by a metal tip, a vacuum gap, and a semiconductor may be viewed as a MIS
(metal-insulator-semiconductor) junction. The theories concerning the behavior of planar MIS
junctions and the associated Schottky barrier are well-developed [2–4]. Similarly, the study of
MIM (metal-insulator-metal) as well as MIS tunnel junctions has resulted in spectroscopic
methods for the determination of surface and interface electronic structure that corresponds well
with experimental data [5–10]. It is common practice to model the effects of the junction
dynamics on experimentally obtained tunneling spectra using the tools of MIS and tunneling
61
theories [11–13]. This is necessary since the electric field at the junction is known to perturb the
electronic structure of both the tip and the material under investigation [9,14].
Ultimately, one must be able to distinguish between the electrical characteristics of the
tip-sample tunnel junction as a MIS diode (i.e., device behavior) and the effects due purely to
surface states — the two are not independent. Therefore, this chapter discusses a method by
which the tunneling current between a metal probe and a semiconducting sample (separated by
free space) may be calculated as a function of the potential difference applied across the junction.
The discussion begins with the development of Harrison’s formula [15] for one dimensional
tunneling across a planar barrier which, when applied to experimental tunneling spectroscopy
(TS), correctly describes the transport of charge across both the tunneling gap and (given the
appropriate conditions) the semiconductor depletion region (section 3.2.1). Following will be a
discussion of the treatment of dynamic band bending (section 3.2.4), and finally the relative
importance of tunneling versus thermionic emission as a means by which charge carriers
negotiate the semiconductor diffusion barrier (section 3.2.5).
The focus throughout the discussion is on the application of this method as an aid in the
study of Nb-doped or reduced strontium titanate (SrTiO3) (001) surfaces. Therefore, the Fermi
statistics appropriate for monovalent as well as divalent defect centers is considered (see
Appendix B). A parametric study of the model will be discussed for the purposes of establishing
a fitting procedure and a context in which to interpret experimental data (section 3.3.2).
62
3.2
THE TUNNELING MODEL
3.2.1 One-dimensional quantum transmission
Walter A. Harrison [15] developed an expression for one-dimensional tunneling of an
independent electron where the initial and final states were considered to be single particle
solutions to the Schrödinger equation for two distinct Hamiltonians separated by a planar
junction. This approach is similar to the many-body description of tunneling based on an
extension of first-order time-dependent perturbation theory [16], where Bardeen [17]
demonstrated their equivalence given the appropriate evaluation of the tunneling matrix element,
Mab. The model therefore begins with a fundamental result of time-dependent perturbation
theory known as the Golden Rule which gives the probability per unit time that an electron
makes a transition from state a to state b as
2
 2π
Pab =   M ab ρb ,
h
[3.1a]
where |Mab| is the transition matrix element (analogous to the transmission probability |T| of
section 3.1.1), ρb is the density of final states per unit energy, and O is the reduced Planck
constant.
Including the density of initial states per unit energy as well as the probability
functions, f(E), for the occupancy of the initial and final states obtains
2
 2π
Pab =   M ab ρa ρb fa (1− fb ) .
h
[3.1b]
Two restrictions imposed on the states defining the system are that: 1) the initial and final state
energies must be equal, and 2) the transverse wave vector, kt, of the tunneling electron must be
conserved. In other words, the model is restricted to elastic tunneling (1) obtained by specular
transmission (2) through the energy barrier.
63
A summation is performed over the allowed energy range (as determined by the applied
bias) for fixed kt, as well as over the appropriate range of allowed kt (see section 3.2.2). Adding
a factor of 2 for spin degeneracy and e for the fundamental unit of charge, the total current
density from electrode a to electrode b is
2
 4πe 
Mab ρaρb fa (1− fb )dE k t .
j a→b = 
∑
∫

h kt
[3.2a]
The current density from electrode b to electrode a is similarly given by
2
 4πe 
M
j b→a = 
ρaρb fb (1− fa )dE k t ,
∑
ab
∫
hk
[3.2b]
t
so that the net one-dimensional current density is given by the difference
2
 4πe 
j z = j a→b − jb→ a = 
Mab ρ a ρb (fa − fb )dE k t .
∑
∫

h kt
[3.2c]
Following Bardeen [17], as long as the system has the property of separability (i.e., the
Hamiltonians for each electrode do not overlap in configuration space) [18], the tunneling matrix
element may be evaluated as
 h  * ∂ψ b
∂ψ a *  
M ab = −ihˆJab = − ih
ψ
−
ψ
.
b
 2mi  a ∂z
∂z  
[3.3]
It is assumed that the potential varies slowly in the barrier region permitting use of the WKB
method to find the initial (ψa) and final (ψb) state solutions to the Schrödinger equation. The
details of obtaining such solutions are beyond the scope of this text and will not be included
here; however, the results of the solution [15] gives
M ab
2
2
 zb

 h2   m   m  ∂E a ∂E b

=
exp
−2
k
dz
∫ z  ,
 2m   h2 L a   h2 L b  ∂k z ∂k z
 za
where the 1D density of states, ρ, for a particle in a “box” of length L is given by
64
[3.4]
1 ∂E  π 
.
≡
ρ ∂k z  L 
Using [3.4] and [3.5], and replacing
∑
kt
by
∫
1
4π2
[3.5]
d 2 k t [3.2c] becomes
 e
 1
j z =   ∫ dE(fa − fb )∫ d 2 k t  2  exp(−η)
4π
hπ
[3.6a]
zb
η = 2 ∫ k z dz
[3.6b]
za
The wave vector of the tunneling electron is given by k z =
2 me ϕ
h2
, me is the free electron mass,
and ϕ is the average tunneling barrier height.
Equation [3.6] is the formula derived by Harrison for one-dimensional single particle
tunneling through a planar junction. The limits of the integral in [3.6b] are the classical turning
points which must be determined as a function of both the electron energy and the applied bias.
They are simply given by the roots of the equation ϕ (z, V, E) = 0. The limits of the total energy
integral in [3.6a] are determined by the width of the energy window for elastic tunneling as also
determined by the applied bias. Lastly, the limits of the wave vector integral in [3.6a] are
determined by the overlap of the projections of the constant energy surfaces of both electrodes
onto the plane of the tunneling barrier. It is important to note that it is by this latter restriction
that some vestige of the dependence on the density of states is recovered since an explicit
dependence vanishes upon making the necessary replacements to derive [3.6]. A more direct
dependence on the band structure of either electrode is represented by the density of states
(DOS) effective mass for the tunneling electron which is introduced by transforming the wave
vector integral into one over the longitudinal (normal) energy [19].
65
This may be done simply by a Jacobian transform to polar coordinates, so that [3.6] may
be re-written as
 4πem * 
dEΘ ± [E − E i ]∫ dE L exp(− η),
jz = 
h3  ∫
[3.7]
where m* is the DOS effective mass of the tunneling electron, and Ei is the band edge for band i
(i.e., i = CB or VB). The positive sign corresponds to the conduction band (CB), whereas the
negative sign corresponds to the valence band (VB). It has been assumed that the shape of the
energy bands are approximately parabolic as well as isotropic. The latter means that the constant
energy surfaces are assumed to be concentric spheres in k space over the energy range where the
assumption of an isotropic band structure is valid. The value for the effective mass is determined
by the electrode with the larger E(k) curvature (i.e., smaller sphere) for a given total energy since
this electrode will also determine the limits of the normal energy integral in [3.7]. In a MIS
structure,
where
S
represents
a
n-type semiconductor in depletion, if it may be assumed that the electron effective mass in the
metal is approximately equal to the free electron mass, then (in absence of other limiting factors)
the tunneling current at small forward and moderate reverse bias will be limited predominately
by the electronic structure of the semiconductor. With a sufficient magnitude of the applied
voltage, the tunneling current will tend to be limited by the metal, depending on the curvature of
the energy bands. (This point will be restated and demonstrated schematically in section 3.2.2.)
This does not inhibit the semiconductor band structure from further influencing the features of
the tunneling spectrum. For now, however, it should be appreciated that the tunneling spectrum
is generally a complex convolution of the electronic properties of both the metal and the
semiconductor [7,8].
66
In writing [3.7] the Fermi distribution functions were assumed to take their absolute zero
forms so that (fa - fb) was replaced by the step function Θ (also known as the Heaviside
function) defined by
0, z ≤ 0
.
Θ(z ) = 
1, z > 0
[3.8]
This replacement is justified given that the energy resolution of experimental TS is not expected
to be much better than several times kT [20]. It should be observed that Θ , as defined in this
way, has the effect of completely ignoring tunneling events between the metal and energy states
within the band gap of the semiconductor.
The task ahead consists of the following: a) determining the limits of integration in [3.7],
and b) determining the average tunneling barrier height as a function of the applied bias. There
are two barrier functions which must be considered — the insulator (i.e., vacuum gap) barrier
and the semiconductor Schottky-type barrier. The determination of both of these functions will
be discussed following a consideration of the correct distribution of the applied bias, Va, across
the MIS structure. For convenience, the equilibrium semiconductor Fermi level is taken as the
reference energy, E SC
F = 0 . Furthermore, the semiconductor is held at ground potential so that
the bias is applied to the tip. The bulk conduction and valence band edges are thus simply given
by
B
E Bcb − E SC
F = E cb = ξ
E Bvb = E cbB − E g = ξ − E g
where Eg is the semiconductor band gap and ξ is a function of the temperature and/or the
properties of the material. The functional form of ξ depends on whether the majority carrier
67
density (n) is exceeded by or exceeds the majority carrier band effective density of states ( N )
[21,22]. The former case refers to a non-degenerate semiconductor and ξ is given by
kTln
n
,
N
[3.9a]
while the latter case refers to a degenerate semiconductor and ξ is given by
 3n 
 π
2
3
h2
∗ .
8m
[3.9b]
3.2.2 Effects of specular transmission
Figure 3.2 is an example of a simple construction that may be used to illustrate the effects of
specular transmission on the limits of integration in [3.7].
To this end, it also serves to
qualitatively demonstrate the effects of the electrode band structures on the characteristics of the
calculated tunneling spectra. Shown plotted on an energy versus wave vector scale are the
conduction band of the metal displaced by −eVa with respect to the valence band of the
semiconductor. In this non-equilibrium configuration, a current is expected to result due to
electrons transporting from the occupied states of the valence band to the unoccupied states of
the metal conduction band. The bold line highlights the energy window over which transport
occurs elastically. Therefore, it can be immediately seen that the limits of the total energy
integral will be given by the energy of the valence band edge E V and the Fermi energy of the
metal, E M
F . Since Θ ignores band gap energies, these limits may simply be chosen as the Fermi
energies of the two electrodes. In the present example
68
E
SC
EF
EV
eVa
E′
M
EF
Eo
SC
kt,E′
Figure 3.2
M
k
kt,E′
Schematic representation of the energy band structure of a metal with respect to a
semiconductor in a non-equilibrium (Va ≠ 0) configuration. For a tunneling electron with total
energy E′, the transmission factor is integrated from the zone center to E′, where the effective
mass in [3.7] is given by the curvature of the valence band.
EM
F
E SC
F −eVa
∫ dE → ∫
E SC
F
E SC
F
− eVa
dE →
∫ dEΘ − [E − E ].
B
vb
[3.10]
0
For a given total energy, shown as E′ in Figure 3.2, the limits of the normal energy integral
reflect the limits of the integral over d2kt in [3.6a]. Indicated along the k axis are the values of
the maximum transverse wave vectors for an electron with energy E′ in the metal ( kM
t ,E ′ ) and in
SC
M
the semiconductor ( kSC
t ,E ′ ). Since kt ,E ′ < k t, E ′ , the current is limited by the semiconductor at this
69
energy and the integration over the normal energy extends from the center of the zone to E′.
This gives
E′
∫ dE
E (k =0)
E′
L
→
∫ dE
L
exp(−η),
[3.11]
E Bvb
and the effective mass in [3.7] will be given by the curvature of the valence band. A similar
construction for the conduction band gives identical results except for a sign change in the
argument for the Heaviside function and E Bvb being replaced by E Bcb in [3.10] and [3.11]. Note
that the selection of these limits does not depend on the nature of band bending which
contributes only to the depletion region transmission factor integral, ηsc.
It is instructive to examine the overlap of the band structures of the metal and the
semiconductor in a manner similar to Figure 3.2 for both valence and conduction bands and for
bands with varying curvature. The following conclusions regarding the effects of specular
transmission into the conduction band may be drawn from such an exercise:
1) When the curvature of the semiconductor conduction band (CB) is identical to the
curvature of the metal CB, the current will be limited by the semiconductor until the bottom of
the metal conduction band increases above the bottom of the semiconductor CB. At this point
the current will be limited entirely by the metal.
2) When the curvature of the semiconductor CB is smaller than that of the metal CB,
there will be an energy, Eo, at which the two dispersion curves intersect for a given applied bias.
When Eo lies above the metal Fermi level, the current is always limited by the semiconductor.
(Note that this is opposite to the case for tunneling from the valence band as can be seen in
Figure 3.2.) When the bias reaches some critical value, Va′, so that Eo < E M
F , the current will be
partially limited by the semiconductor and partially limited by the metal. As the bias continues
70
to exceed Va′, the metal assumes an increasing dominance over the current until, as before, the
current is limited entirely by the metal.
3) When the curvature of the semiconductor CB is larger than that of the metal CB, the
current is always limited by the semiconductor until the bottom of the metal CB increases above
the semiconductor CB edge. At this point the current is partially limited by the metal which
assumes an increasing dominance with voltage (as before) until the current is entirely metallimited.
A similar exercise with the valence band draws similar conclusions except that the
current is never limited completely by the metal. When the curvature of the valence band and
the metal CB are identical, the current is semiconductor limited until a critical value of the bias
Va′ is reached (where Eo > E M
F ) and the current becomes partially limited by the metal. When
the valence band curvature is smaller than the curvature of the metal CB, Va′ is reduced in
magnitude. The reverse is true when the valence band curvature is larger. Since the Fermi
energy in a metal is determined primarily by its atomic density, the greater the density of the
metal, the less it will influence the current for a given bias. Most metals have quite large
densities (owing to their efficient packing) so that as a general rule of thumb, the smaller the
curvature of the semiconductor bands, the greater the influence of the metal on limiting the
tunneling current.
Recall that these effects arise as a consequence of the restriction to specular transmission;
a condition that is not guaranteed to exist in every experimental situation.
however,
that
the
assumption
of
specular
transmission
It is believed,
is
reasonably
valid for surface roughness comparable to the wavelength of the incident electrons [3]. Recall
71
the de Broglie relation λ = 150.4 E , where E is in eV, and λ is in Å. For electrons with low
energy, such as those tunneling at low biases, λ is ∼100Å. At ±2V, the high-energy electrons
will have a wavelength of 8.7Å. At ±4V, λ is 6.1Å. Therefore, higher applied biases require
successively smoother surfaces for the assumption of specular transmission to be valid. One
might consider, however, the meaning of “roughness” on the scale of the tunneling junction
cross-section (usually assumed to be ∼25Å2). A roughness corresponding to atomic corrugations
(i.e., ≤1.5Å) extends the valid voltage range out to ±67V. A roughness on the order of a unit cell
step edge on (001) SrTiO3 (i.e., ∼4Å) extends the valid voltage range to ±9.4V, well beyond
typical experimental voltage sweeps.
72
Vi
Va
RM≈0
Vs
RG≈∞
a
e(∆-Vi)
χ
(ψs,o-Vs)
φM
EC
SC
B
b
s
Figure 3.3
ξ
eVa
M
EF
+++
++++
+++
EF
EV
ω
a) Equivalent circuit for b) a metal-vacuum-semiconductor tunnel junction at
forward bias. (See text for symbol definitions.)
3.2.3 The potential distribution functions
Equation [3.7] correctly describes the transport of charge across a classically impenetrable
barrier. The barriers in a STM may be represented by a resistor in series with a diode, as shown
schematically in Figure 3.3a. Any applied bias, Va, will be divided between the resistor RG (free
space gap) and the diode (semiconductor depletion region). These components of Va will be
referred to as Vi and Vs, respectively. Figure 3.3b schematically shows the spatial behavior of
the energy bands for a forward biased junction. The symbols of the figure have the following
meanings:
73
φM
work function of the metal
χ
electron affinity of the semiconductor
∆
equilibrium (“contact”) potential across the insulating region
ψs,o
equilibrium surface potential
ξB
bulk value of the conduction band edge with respect to E SC
F
s
metal-to-semiconductor separation
ω
width of the depletion region
From inspection of Figure 3.3b one can immediately write
φ M = eVa + ξ + e(ψ s,o − Vs )+ χ + e(∆ − Vi ).
B
[3.12]
At thermal equilibrium, it is assumed that there is an initial negative surface charge density of
magnitude QSS, which is compensated by a positive charge density of magnitude QSC in the
depletion region and a positive charge density of magnitude QM induced on the metal,
so that
QM = −(−QSS + QSC ).
[3.13]
Gauss’ law gives
∆=−
sQ M s (−QSS + QSC )
,
=
εi
εi
[3.14]
where ε i is the permittivity of the insulator (or free space in this case). Also from Gauss’ law,
the field at the surface of the semiconductor at equilibrium and non-equilibrium are respectively
given by
Fs,o = −
Qsc,o
εs
and
Fs,a = −
Q sc,a
,
εs
[3.15]
where the difference between Fs,o and Fs,a is proportional to the component of the applied bias
that falls across the semiconductor. Using [3.14] and [3.15] one may write
74
∆o =
s(−Qss,o − εs Fs,o )
εi
and
∆a =
s (−Q ss,a − ε sFs,a )
εi
so that the component of the applied bias that drops across the insulator is
Vi = ∆ a − ∆ o =
s(−Qss,a − ε s Fs,a ) s(−Qss, o − ε sFs,o )
s(ε sFs, a − εs Fs, o ) s(Qss,a − Qss,o )
−
=−
−
εi
εi
εi
εi
or
Vi = −
s(∆Qsc + ∆Q ss )
.
εi
[3.16]
Therefore, the development of the component of the external field which drops across the
insulator is accompanied by a change in both the space charge and the surface charge. Assuming
an abrupt junction approximation, from semiconductor theory [2]


Qsc, o =  2eε sN D  ψ s,o

1
kT  2
−  
e 
[3.17]
and
1
Qsc,a
kT


2
=  2eε sN D  ψ s,o −
− Vs   ,
e


[3.18]
where ND is the density of ionized single electron donor defects. The equilibrium band bending
ψ s, o may be selected as an initial condition. This value will be close to φ M − χ in the absence of
a large density of surface states; otherwise, ψ s, o will be independent of φ M − χ . From [3.17] and
[3.18] it is clear that ∆Qsc is determined by Vs as stated earlier. Since ∆Qss is not known a
priori, an assumption is made regarding its behavior. From [3.12] and Figure 3.3b it is clear that
75
both the “contact” potential and the surface band bending change linearly with applied bias so
that it seems reasonable to assume that Q ss (which constitutes the discontinuity of the field
across the insulator-semiconductor junction) also varies linearly with applied bias. This permits
writing
∆Qss (Va ) = −βVa ,
[3.19]
where β is an adjustable parameter and the minus sign indicates that a positive bias decreases the
depletion region charge density and is thus expected to decrease the surface charge density. This
parameter shall be refered to as the capacitance constant; it has units of C/V cm2. Combining
[3.16]–[3.19], and using the identity Va = Vs + Vi , gives
1
1
−1


2
2
 sβ 
s
kT
kT






− Vs  −  2eεs ND  ψ s,o −     . [3.20]
Va = 1−   Vs −  2eε sN D  ψ s,o −



εi  
εi  
e
e  
By varying Va and numerically solving for Vs, and using Va = Vs + Vi to find Vi, the potential
distribution functions Vs(Va) and Vi(Va) are completely determined to be used further to
determine the bias-dependent behavior of the average tunneling barrier heights. The equilibrium
band bending is determined by the equilibrium surface charge. It is also assumed that most of
the compensation for surface charge occurs in the depletion region of the semiconductor (i.e.,
changes in surface charge on the metal are assumed to be negligible) so that from [3.17]
1
′ − Q sc,o
∆Qss, o = Qsc,o
1
kT   2 
kT   2



=  2eε sND  ψ ′s, o −
−
2eε
N
ψ
−
 s D  s,o e   , [3.21]

e  
where ψ s,′ o and ψ s, o are the required fitting parameters.
76
2
2
a
1.5
1
Sample Bias, Vs (V)
1
Sample Bias, Vs (V)
b
1.5
0.5
0
-0.5
-1
-1.5
0.5
0
-0.5
-1
-1.5
-2
-2
-3
Figure 3.4
-2
-1
0
1
Tip Bias, Va (V)
2
3
4
-3
-2
-1
0
1
Tip Bias, Va (V)
2
3
4
Calculated potential across the sample, Vs, as a function of the total applied bias,
Va. The effect of variation of donor density, ND, is demonstrated: solid line =1018; dashed line =
1019; dot-dashed line = 1020 cm-3; a) β = 0; b) β = 0.002.
Figure 3.4 shows the results of using the above procedure to calculate the voltage drop
across the sample, Vs, as a function of the bias applied to the tip, Va, for a metal tip–vacuum
gap–semiconductor junction. The effect of increasing the donor density is to decrease |Vs| for all
non-zero values of Va. This effect can be expected since increasing the free carrier density
increases the screening strength in the semiconductor and thus a smaller potential is expected to
drop across the diode in Figure 3.3a. It should be noted that increasing the static dielectric
constant, which also increases screening, has a similar effect. A comparison of Figure 3.4a and
3.4b demonstrates the effect of a bias-dependent variation of the surface charge — i.e., β ≠ 0 in
77
Figure 3.4b. For a given tip bias, |Vs| decreases with decreasing Qss. It can also be seen that this
effect is greater at larger biases. The main point of Figure 3.4 is to illustrate how the potential
distribution curves depend on the experimental variables and is most useful when fitting
experimental data.
Values which give unreasonable potential distribution and/or surface
potential curves are assumed unphysical.
φM-φS-eVi
χ
φM
φS
EC
M
F
SC
E
E′
EF
s
Figure 3.5
+++
++++
+++
EV
ω
An equilibrium (Va = 0) configuration for a metal-vacuum-semiconductor tunnel
junction separated by a gap of width s.
78
3.2.4 The potential barrier functions
An arbitrary electron at the surface of the metal must overcome a barrier determined by the sum
of the surface work function, φm, and the difference in energy between the metal Fermi level and
the energy of the electron, E′. This is shown schematically on the left hand side (z=s) of the
barrier in Figure 3.5. The barrier is thus given by
φ m + (E F − E ′) = (E F − E ′)+ (E cb − E F
M
SC
S
SC
)+ χ + (φ
m
− φ s − eVi ),
[3.22]
where E Scb is the value of the conduction band edge at the surface of the semiconductor due to
band banding. On the semiconductor side (z=0) of the junction, an electron at the same energy
must overcome a barrier given by
χ + (E cb − E F
S
SC
)+ (E
SC
F
− E ′).
[3.23]
Notice that χ + (E Scb − E SC
F ) may be replaced by the semiconductor work function φ s so that both
[3.22] and [3.23] may be written as
SC
E F + φm − eVi − E ′
(z = s)
E SC
F + φs − E ′
(z = 0)
from which a trapezoidal function is constructed to describe the vacuum barrier. Recalling that
SC
EF ≡ 0 ,
z
z

ϕ trap (z,Vi , E′ ) = (φ m − eVi − E ′)  + (φ s − E ′ ) 1 −  .
s
s
[3.24]
Following Simmons [23], a term is included to account for the image force lowering of the
barrier given by
ϕ image ( z) = −
0.4e 2  s 
,
8πκ op ε o  z(s − z )
79
[3.25]
where κ op is the optical dielectric constant of the barrier region (assumed to be unity for a
vacuum barrier). The total vacuum barrier is the sum of [3.24] and [3.25]. Figure 3.6a shows
the calculated barrier “seen” by an electron at the Fermi energy for a junction
at equilibrium for several values of the tip-sample separation, s.
It can be seen
that the average barrier height decreases and becomes more symmetrical as s decreases. Figure
3.6b shows the effect of a junction under non-equilibrium conditions for a constant tip-sample
separation of 9 Å. The bold curve corresponds to equilibrium — i.e., Vi = 0. Increasing in the
forward bias (Vi > 0) direction decreases the average barrier height and there is a substantial
decrease
in
the
width
of
the
barrier.
Increasing
in
the
reverse
bias
(Vi < 0) direction, however, increases the average barrier height and the barrier width increases
only slightly.
80
Potential (eV)
3.5
a
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
Gap width (Å)
Potential (eV)
6
b
5
4
3
2
1
0
0
2
4
6
8
Gap width (Å)
Figure 3.6
Calculated spatial (a) and voltage (b) dependent vacuum potential barrier (in eV).
The abscissa is measured with respect to the electrode with the smaller work function. In (a) the
arrow indicates the direction of decreasing tip-sample separation, s. In (b) the top and bottom
arrows indicate the direction of increasing reverse and forward bias, respectively, in voltage
steps of 0.9 eV.
81
A second potential barrier often exists in the form of band bending within the depletion
region of the semiconductor (see Figure 3.3 and Figure 3.5).
Feenstra and Stroscio [11]
demonstrated the importance of including voltage-dependent (i.e., dynamic) band bending to
improve agreement between experimental and theoretical tunneling spectra. Their approach
consisted of calculating the equilibrium surface potential φs of the semiconductor as a function of
the applied bias. This potential is assumed to decrease parabolically from the surface (z = 0)
towards the bulk, reaching a value defined as the equilibrium bulk potential φb at the depletion
edge (z = w). The resulting energy profile φbb is added to both the valence and conduction band
edges to model the band bending. The parabolic potential profile in the depletion region can be
written as
φ bb (z, Vs ) =
(ω − z)2
ω
2
φsbb (Vs ),
[3.26]
where ω is the width of the depletion region given by
ω=
2εs φsbb (Vs )
.
e2 ND
[3.27]
Therefore, the potential barrier associated with band bending in the semiconductor depletion
region (for an electron with energy E ′ ) is given by
ϕ sc (z,Vs, E′ ) = φ bb (z,Vs ) + E cb − E ′ .
[3.28]
A similar approach is employed here to determine φ sbb (Vs ) , which begins with a single
integration of the one-dimensional form of Poisson’s equation (in MKS units):
∇ ⋅ D z = ρz ,
[3.29]
where D z = ε s Fz is the z component of the electric displacement vector. Below are defined the
dimensionless parameters that are usually introduced for convenience [24,25]. Since the band
82
bending at the surface is given by the difference between the potential at the surface and the
potential in the bulk,
φ sbb = (φ b − φ s )
[3.30]
where
φz ≡
1
(E F − E I, z ).
e
[3.31]
In [3.31] E I, z is called the intrinsic Fermi level which maintains a constant relation with respect
to the conduction and valence band edges, and is given by
3
 m *vb  4
1
E I = (E vb + E cb ) + kT ln *  .
2
 m cb 
[3.32]
Therefore, [3.31] may be referred to as the depletion region potential function and with [3.30]
gives the surface band bending (φsbb) when z = 0.
( uz =
By writing [3.31] in a reduced form
eφ z
dφ
), and noting that Fz = − z , Poisson’s equation may be written as:
kT
dz
eρ z
d 2 uz
.
2 = −
dz
κεokT
[3.33]
Here κ is the static dielectric constant of the semiconductor.
The equilibrium charge
distribution function ρ z = ρ D + ρ A + e (p z − n z ) is found employing Fermi statistics. The free
carrier densities, in the degenerate limit, are given by the usual relations [26]
3
 m * kT  2
 E − E cb 
= 2 cb 2
n z = Ncb F 1 F
F 1 (uz − wcb,I )
2 
 2πh  2
kT 
3
2
 m kT 
 E vb − E F 
F1 (w − uz )
= 2
2
 2πh2  2 vb,I
kT 
*
vb
pz = Nvb F 1
83
[3.34]
where w1,2 =
E1 − E 2
for any two energies, Ni is the effective density of states for energy band i,
kT
and F12 is the Fermi-Dirac (F-D) integral of order
1
2
.
The latter is one of a family of
dimensionless integrals [21] defined by
∞
ε jdε
1
F j (η/ ) =
.
Γ ( j + 1) ∫0 1 + exp (ε − η/ )
[3.35]
These integrals cannot be solved analytically. Extensive tabulations exist (see for example
Appendix B of reference 21) for F-D integrals of various orders and for both positive and
negative values of the argument, η
/ . In the non-degenerate limit, these integrals may be replaced
in [3.34] by the “classical” approximation Fj (η/ ) ≈ exp(η/ ).
The ionized donor ρ D and acceptor ρ A charge densities are also determined by
equilibrium Fermi statistics. (In the present formalism it is assumed that ρ A = 0.) For ionized
single donor defect centers (i.e., Nb•Ti or VO• ), the contribution to the space charge density is
given by (see Appendix B)


1
ρ D = eND 
.
 1 + 2 exp[uz − wD1, I ]
[B.5]
Assuming donor defects that may accommodate two electrons, the charge density due to ionized
donor sites (i.e., both VO• and VO• • ) can be shown to be given by
 f −f f 
ρ D = eND  E ′ E ′ D1 
 1 − fD1 + fE ′fD1 
[B.22]
where the functions fE ′ and fD1 are similar to the bracketed term in [B.5] and are derived in
Appendix B.
Poisson’s equation can be rewritten using [B.5] or [B.22] combined
with [3.34].
84
As an example, consider the case of a degenerate semiconductor with divalent donor
defects. Equation [3.33] thus takes the form

d 2 uz
Ncb
1   fE ′ − fE ′fD1  Nvb
+
=
−
F
w
−
u
−
F
u
−
w
1
(
)
1
(
)
z
cb,I 
dz 2
L2D   1− fD1 + fE ′ fD1  N D 2 vb,I
ND 2 z

[3.36]
where the extrinsic Debeye length LD is defined by
LD ≡
εs kT
.
e2N D
It is convenient to make a change of variables [2,3] by writing F/ = −
that F/ dF/ =
[3.37]
du z
, from which it follows
dz
d 2 uz
du . Integrating from the bulk to the surface, observing the boundary condition
dz2 z
 du z 
= 0 (i.e., zero electric field in the charge neutral region) gives
 dz  b
F/ s
1
∫ F/ dF/ = 2 F/
2
s
F/b =0
 1  f − f f

 E ′ E ′ D1  + Nvb F 1 (wvb,I − uz )− Ncb F 1 (uz − wcb,I ) duz [3.38]
−

2
∫ LD  1− fD1 + fE ′ fD1  N D 2

ND 2

ub
us
=
The first part of the integrand on the RHS of [3.38] integrates as




exp(ub ) + exp(wE ′, I )
1


2exp
w
arctan
( E ′,I )


2
LD 
 exp(wE ′,I + wD1,I )− exp(2wE ′, I )




exp(us ) + exp(wE ′,I )
− arctan 

 exp (wE ′,I + wD1,I )− exp (2wE ′, I ) 
85


1
×
.
+
w
−
exp
2w
exp
w
( E ′,I D1,I ) ( E ′,I ) 

The F-D integrals are integrated using the formula [27]
dF j (η)
= jF j −1 (η),
dη
so that the remainder of the integrand on the RHS of [3.38] integrates as
2
1  Ncb  2
F
F 3 (u − wcb,I )
3 (us − wcb, I ) −
2 
3 2 b
LD  ND  3 2
−
2
Nvb  2
F
F 3 (w − ub ) .
3 (wvb, I − us )−
3 2 vb ,I
ND  3 2
The latter two solutions (sans the factor L−2
D ) are referred to below as sol1(us) and sol2(us),
respectively, so that for the electric field at the surface of the semiconductor
Fs =
1
kT
kT  duz 
kT 2
2
F/ s = −
=
±
sol1(u
)
+
sol2(u
)
[
s
s ] .
eLD
e
e  dz  s
[3.39]
Recalling [3.15], the net space charge per unit surface area, Qsc,a, as a function of the reduced
surface band bending, us, is written as
1
Qsc,a = eND LD 2 [sol1(us ) + sol2(us )]2 .
[3.40]
Equations [3.18] and [3.40] together give a means by which to model the band bending behavior.
Varying us and numerically solving for Vs obtains a sample bias-dependent surface band bending
function, φsbb(Vs).
86
3.5
a
3
Surface Potential (eV)
Surface Potential (eV)
3
3.5
2.5
2
1.5
1
0.5
b
2.5
2
1.5
1
0.5
0
0
-0.5
-0.5
-3
Figure 3.7
-2
-1
0
1
2
Sample voltage,Vs (V)
3
-3
4
-2
-1
0
1
2
3
4
Sample voltage,Vs (V)
Calculated surface potential (i.e., band bending) as a function of the voltage
component across the sample.
a) Variation of defect density ND: solid = 1018 cm-3;
dashed = 1019 cm-3; dot-dashed = 1020 cm-3.
b) Variation of initial band bending ψ s, o :
solid = 0.3eV; dashed = 1.5eV; dot-dashed = 2.4eV
Figure 3.7 shows calculated surface potential curves for the monovalent donor case and
their dependence on ND and ψ s, o . As can be anticipated, decreasing ψ s, o shifts the curves
uniformly towards more positive sample biases (Figure 3.7b). It is interesting to observe that
increasing ND increases the range of band bending at positive sample bias (Figure 3.7a). There
are three important points to consider on these plots. 1) At zero applied bias (Vs=0) they show
the equilibrium surface band bending, or Schottky barrier height, associated with the “contact”
potential. 2) At forward bias (Vs<0) ∆Qss is negative (see [3.19]) and the majority carrier density
increases at the surface of the semiconductor thus decreasing both φsbb and ω. When sufficient
bias is applied, the surface potential and depletion width fall to zero. The sample voltage at
87
which this occurs is called the flat band voltage and it marks the onset of majority carrier
accumulation at the surface of the semiconductor. Figure 3.7 shows this to be given by the
voltage at which the surface potential reaches the onset of the lower plateau. 3) In the reverse
bias (Vs>0) case, the surface potential and depletion width both increase. This moves the sample
Fermi level towards the valence band edge in the near surface region. The bias at which the
Fermi energy (EF) falls below the intrinsic Fermi energy (EI) at the surface defines the onset of
inversion [2]. This is given by the voltage at which the surface potential reaches the onset of the
upper plateau.
One may also compare calculated band bending for monovalent and divalent donor defect
semiconductors. In the latter case an additional occupied donor level resides a few eV below the
conduction band edge. At sufficient reverse bias, the Fermi energy will approach this level near
the surface and thus the state is expected to become unoccupied, assuming there exists a
mechanism for this process to occur. Applying the present formalism, the effect on band
bending appears to shift the onset of inversion to a lower applied bias, depending on the energy
level
of
the
deep
donor
state,
as
shown
in
Figure 3.8. Note that the rather abrupt transition to inversion as shown is believed to be an
artifact resulting from use of the abrupt junction approximation. Hence Figure 3.8 demonstrates
a possible limitation of simplified semiconductor device theory when applied to semiconductors
with multivalent defect centers. The true behavior is likely to lie somewhere between the solid
curve and the broken curves — i.e., a gradual increase in slope giving rise to an earlier onset of
inversion.
88
3.5
Surface Potential (eV)
3
2.5
2
1.5
1
0.5
0
-0.5
-3
-2
-1
0
1
2
3
4
Sample voltage,Vs (V)
Figure 3.8
Comparison of equilibrium band bending for monovalent (solid) and divalent
donors in a semiconductor with band gap energy Eg = 3.2 eV. The defects lie at 1.7 (dashed), 1.3
(dot-dashed), and 0.9 eV (dotted) below the conduction band minimum.
Note that the curves in Figures 3.7 and 3.8 are calculated in what is referred to as a quasiequilibrium formalism, where the surface of the semiconductor is always assumed to be in
thermal equilibrium with the bulk. One consequence of this assumption is that the difference
between the surface potential energies corresponding to the upper and lower plateaus is given by
the semiconductor band gap. There is both experimental [11] and theoretical [28] evidence
which support the view that non-equilibrium conditions via minority carrier extraction prevail in
89
practice due to a sufficiently large tunneling current.
The absence of spectral features
corresponding to the onset of inversion has been interpreted in terms of non-equilibrium
conditions which pin the majority carrier quasi-Fermi level to the Fermi level of the metal [28].
Consequently, band bending is no longer limited to the width of the forbidden energy gap [13].
Considering that oxides tend to have large band gap energies, the intrinsic carrier density is
negligible. Hence, non-equilibrium via minority carrier extraction for oxides may be assumed a
non-measurable effect. This justifies adopting the quasi-equilibrium approach described above.
An absence of inversion for wide band gap (i.e., large dielectric constant) systems may be
expected by considering the potential distribution curves in Figure 3.4. Depending on the
experimental conditions, a relatively small fraction of the applied voltage may fall across the
semiconductor depletion region due to screening effects. This suggests that, for wide band gap
semiconductors, experimental band bending will be confined to a narrow voltage range about the
origin in Figures 3.7 and 3.8.
3.2.5 Determination of the defect-induced current
In defect semiconductors, there is a current density that is induced by the contribution of free
carriers from the defect states. There are three ways in which this current density may be treated
in the model. At the appropriate density of defects, there may be a defect band introduced into
the band gap of the solid. This band can be treated in the same manner as the valence and
conduction bands — i.e., sum the tunneling probability over the energy window corresponding to
the width of the band. If the conditions are such that the semiconductor is degenerate, then the
Fermi level will lie a few meV above the conduction band edge. The current at small forward
bias will generally include two components due to the (defect-induced) carriers in the conduction
90
band — a tunneling component and an emission component. The tunneling component is
automatically accounted for by the limits of the integral in [3.10].
The standard Richardson-Schottky (R-S) equation is derived based on the assumption
that the electrons in the conduction band behave approximately as an ideal gas. Consequently,
the electron energy is assumed to be completely kinetic in nature and the emission current is
given as
 eφ
j DI = A* T 2 exp − b  exp (β E 0 ),
kT
where E0 is the electric field at the semiconductor surface,
[3.41]
4πem* k 2
A =
h3
*
and
1
e 2
 e 
 . In semiconducting insulators, however, the behavior of the conduction
β =   
kT  4πε o κ op 
band electrons cannot always be described by free electron theory. This is true a fortiori for
oxides wherein strong electron-lattice coupling is believed to determine the mechanism of charge
transport at low carrier densities. Relaxation of the ideal gas assumption results in a modified RS equation as
3
 m*  2
 eφ
j DI = αT E 0 µ   exp  − b  exp(β E 0 ),
kT
 me 
3
2
where α = 3 × 10 −4 As cm3 K
3
2
[3.42]
and µ is the electron mobility in the semiconductor. This relation
was first derived by Simmons [29] and recently shown to demonstrate improved consistency
with experimental mobility data for Ba1-xSrxTiO3 (BST) thin films [30].
In the present
application, the forward-biased defect current is assumed to be dominated by the emission
process (i.e., depletion region diffusion currents are neglected) and emission from metal to
91
semiconductor is ignored. The net current is thus assumed to be given by [3.42] modified by a
vacuum barrier transmission factor and including the term
1
4
 e 3 N  
kT  
D

 ψs −
∆ φ′o = 
2
2

e  
 8π ε sε op 
[3.43]
for image force lowering of the diffusion barrier.
Most insulating semiconductors are non-degenerate at low to moderate defect densities.
The Fermi level thus will lie below the conduction band edge and the defect-induced current
density may be assumed to be given entirely by [3.41] or [3.42].
The vacuum barrier
transmission factor is calculated exactly as before (equation [3.6b]) assuming all of the electrons
“see” the barrier as measured from the bottom of the conduction band at the surface.
In the degenerate semiconductor case, the relative importance of tunneling versus
thermionic emission may be considered in terms of the ratio [3]
Ξ=
kT
,
eE 00
[3.44]
where
1
E 00
and α = m
∗
me
1
eh N  2
N 2
=  *D  = 18.5 × 10 −15  D  ,
2  m εs 
ακ
[3.45]
. As a rough guide, tunneling is expected to dominate when Ξ is much less than
unity, thermionic-field emission is expected for Ξ close to unity, and thermionic emission
dominates when Ξ is much greater than unity. It can be seen from [3.45] that a larger carrier
density favors tunneling, whereas a larger effective mass and dielectric constant favor thermionic
emission as a means to negotiate the semiconductor diffusion barrier. Assuming α = 12 and κ =
300, for tunneling to be considered dominant, [3.44] and [3.45] require N D >> 6.57 × 1021 cm-3,
92
or approximate stoichiometry SrTiO2.625 for a reduced crystal. For carrier densities much less
than this value, it is reasonable to use [3.41] or [3.42] in all current calculations.
3.3
SAMPLE CALCULATIONS
3.3.1 Simulated vs experimental spectra
The current density due to tunneling between the conduction band (jCB) and the valence band
(jVB) states are calculated independently using [3.7] and together with [3.42] the total current is
given by
I = j total A = (jCB + jVB + jDI )π r2 ,
[3.46]
where r is the radius of curvature of the metal tip. A sample calculation was performed using the
parameters in Table 3.1. Figure 3.9a shows the spectrum on a linear scale, while Figure 3.9b is a
semi-log plot of the same result.
Table 3.1: Parameters used to calculate the tunneling spectrum in Figure 3.9.
tip-to-sample separation
radius of curvature of tip
conduction band effective mass
valence band effective mass
static dielectric constant
optical dielectric constant
equilibrium surface potential
donor defect density
electron mobility
capacitance constant
band gap energy
metal work function
sample electron affinity
s
r
9Å
5Å
12
1
210
5
m *cb
m *vb
κ sc
κ op
ψ s, o
ND
µe
0.30 eV
5 × 1019 cm−3
2
30cm Vs
0.0001
3.2 eV
5.65 eV
3.0 eV
β
Eg
φm
χ
93
10
7.5
5
a
-4
-3
2.5
-2
-1
-2.5
1
2
3
4
-5
-7.5
-10
Sample voltage,Va (V)
2
1.5
b
Log|I| (nA)
1
0.5
0
-0.5
-1
-1.5
-2
-3
-2
-1
0
1
2
3
4
Sample voltage,Va (V)
Figure 3.9
Calculated tunneling spectrum using the parameters in Table 3.1.
94
Experimental limitations may truncate the measured spectra. The upper dotted line in
Figure 3.9b marks the point at which the current pre-amp voltage saturates, while the lower
dotted line marks a typical upper limit to the noise floor of the electronics. The spectrum in
Figure 3.9b is replotted in Figure 3.10 and compared to an experimentally obtained spectrum.
No effort was made to “fit” the calculated curve to the experimental one; however, the agreement
between
the
qualitative
features
of
the
curves
quite remarkable.
1
Log|I| (nA)
0.5
0
-0.5
-1
-3
-2
-1
0
1
2
3
4
Sample voltage,Va (V)
Figure 3.10
Comparison of calculated (solid) and experimental (dashed) spectra.
95
is
The modeled spectrum may be characterized by an asymmetry in the slope of the current
as a function of applied bias. The current at small forward bias rises more slowly compared to
reverse bias. This current is due to the finite density of extrinsic carriers in the conduction band
which must overcome a decreasing surface potential as well as a similarly evolving vacuum
barrier (Figure 3.6b). The rate at which this current rises is strongly dependent on the strength of
dynamic band bending, which itself depends strongly on the potential distribution functions.
That is, a larger potential drop in the semiconductor gives a greater degree of dynamic band
bending. A rapid drop in the surface potential corresponds to a rapid increase in surface electron
density and hence a steep slope in the curve at negative sample bias. Hence, the relative slope in
the spectra at forward bias is a measure of the relative degree of dynamic band bending.
At reverse bias, however, there is a much faster rise in current at the onset voltage. In
this case, the current is determined by the density of states of the electrodes and a large density
of electrons from the metal tunnel into a large “volume” of momentum space provided by the
unoccupied states in the oxide conduction band. In contrast to negative biases, the current slope
at positive bias is less dependent on dynamic band bending. The dominant portion of the
tunneling current involves exchange between states near the top of the diffusion barrier where
the width varies most slowly.
At sufficient negative sample bias, the metal Fermi level crosses the valence band edge
so that a large increase in current due to electrons tunneling from the valence band is expected.
This may be observed as a kink or slope change. Since the valence band current is superimposed
on the defect-induced current, the occurrence of such feature depends on the initial surface
potential as well as the strength of dynamic band bending.
96
In most cases for insulating
semiconductors, as in Figure 3.9, the current at forward bias is dominated by the defect-induced
current density.
3.3.2 Parametric study of tunneling model
The behavior of the calculated curves may be studied as a function of several experimental and
material parameters. The goals of such a parametric study are 1) to establish a fitting procedure,
and 2) to facilitate identification of experimental artifacts due to changes in tip-sample separation
or sample permittivity, for example. Figure 3.11 shows the results of varying the specified
parameters about their initial values which appear in Table 3.1. The magnitude of the specified
parameters increases from the solid curve to the dot-dashed curve. The dashed curve in all of the
plots of Figure 3.11 is identical to the calculated curve appearing in Figures 3.9 and 3.10.
It may be immediately observed that variations in carrier density, surface potential, and
sample permittivity all strongly affect the behavior at forward bias.
Only a very weak
dependence at reverse bias is predicted. Varying the effective mass is predicted to have a weak
effect on the curve at all applied biases. And finally, varying the tip-sample separation or the
electron affinity is predicted to strongly affect the curve at reverse bias while only weakly
affecting it at forward bias.
The dependence upon each of these parameters is discussed
separately.
Carrier density: A larger carrier density increases the image force lowering term in the
R-S emission relation [3.43], thus increasing the carrier density at the semiconductor surface for
a given applied bias. The screening efficiency also increases with increasing free carrier density
such that less voltage is expected to drop within the semiconductor. These two effects are
opposing and when the latter outweighs the former, the calculated current decreases with
increasing carrier density.
97
Surface potential: Modifying the surface potential is a more direct means to modify the
surface carrier density. The latter constitutes the fraction of mobile electrons with sufficient
kinetic energy in the direction of the surface to overcome the diffusion potential. The surface
carrier
density
thus
naturally
falls
98
with
increasing
surface
potential.
1
Log|I| (nA)
0.5
0
-0.5
-1
a
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
1
Log|I| (nA)
0.5
0
-0.5
-1
b
-3
Sample voltage,Va (V)
Figure 3.11a,b
a) Increasing carrier density: 1, 5, and 10 × 1019 cm-3. b) Increasing surface
potential: 0.25, 0.30, and 0.35 eV.
99
1
Log|I| (nA)
0.5
0
-0.5
-1
c
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
1
Log|I| (nA)
0.5
0
-0.5
-1
d
-3
Sample voltage,Va (V)
Figure
3.11c,d
c)
Increasing
static
dielectric
d) Increasing effective mass: 5, 12, and 50.
100
constant:
100,
210,
and
300.
1
Log|I| (nA)
0.5
0
-0.5
-1
e
-3
-2
-1
0
1
2
3
4
-2
-1
0
1
2
3
4
1
Log|I| (nA)
0.5
0
-0.5
-1
f
-3
Sample voltage,Va (V)
Figure 3.11e,f
e) Increasing tunneling gap: 8, 9, and 10 Å. f) Increasing electron affinity: 2.6,
3.0, and 3.4 eV.
101
This effect is due solely to emission over the Schottky barrier and is not observed at reverse bias
where the transport mechanism is described completely by tunneling. Only a weak shift in the
current onset is observed, attributable to the dependence of the depletion width on the surface
potential. As previously discussed, the dominant portion of the tunneling current at reverse bias
occurs near the top of the diffusion barrier where the width varies most slowly, hence the weak
dependence.
Dielectric constant: The total polarizability of a medium is represented by the complex
dielectric constant. A larger value thus corresponds to greater screening strength and thus
(similar to decreasing resistivity) reduces dynamic band bending.
Effective mass: The absence of a significant dependence on the DOS effective mass is
not surprising considering that it only enters as a term in the pre-exponential factors of [3.7] and
[3.42]. When the semiconductor is degenerate, however, the Fermi energy, measured with
respect to the conduction band edge, has an inverse dependence on the effective mass.
Vacuum gap: An exponential dependence on the vacuum gap is expected from [3.6].
The asymmetry in Figure 3.11e thus cannot be explained in a straighforward manner. Perhaps it
may be understood by considering the development of the vacuum barrier as shown in Figure
3.6b. At forward bias the barrier “seen” by electrons at the surface of the semiconductor
decreases significantly in width as well as in height. Furthermore, the majority of the “hot”
electrons are near the top of the barrier where the WKB treatment begins to break down (see
section 3.1.1). To a first approximation, it may be concluded that the voltage-induced narrowing
of the barrier at forward bias very nearly compensates for the gap increase.
Electron affinity: The vacuum barrier is determined by the average value of the work
functions of the two electrodes. Figure 3.11f is simply a reflection of an increasing vacuum
102
barrier potential.
The weaker dependence at forward bias may be explained by the same
argument used to explain the weak dependence on the vacuum gap.
Figures 3.11b and 3.11f are a bit unrealistic to the extent that surface potential and
electron affinity are both directly dependent on the surface charge density. Thus the variations in
these figures may be associated with an increase in surface charge such that the true variation in
the spectra might appear as shown in Figure 3.12. It may be concluded that a variation in surface
charge is predicted to have an asymmetric effect on the tunneling spectra, where the effect is
stronger at forward bias than at reverse bias.
1
Log|I| (nA)
0.5
0
-0.5
-1
-3
Figure 3.12
-2 -1
0
1
2
3
Sample voltage,Va (V)
4
The predicted effect of increasing surface charge density in steps of ∼1.40×10-7
coulombs per cm2.
103
3.3.3 Discussion
Figure 3.10 must be regarded with caution because the physical properties of the sample
were not verified to match those used in Table 3.1. The values chosen, however, are average
values for semiconducting SrTiO3 and, except for the electron mobility, they are not expected to
deviate by orders of magnitude from the actual values of the sample in question. In practice,
materials parameters measured by the methods described in section 2.2.2 give average
macroscopic values; scanning tunneling spectroscopy measures local surface properties which
may deviate significantly from their average values. Other experimental parameters, such as the
tip-sample distance, are impossible to know precisely. Figure 3.10 thus serves as a reasonable
test case for the reliability of the model. The greatest error is introduced by parameters that give
rise to big effects for small to moderate variations in their values.
The predictions of any theoretical model strongly depend on the how the model is
constructed. Several assumptions, approximations, and restrictions have been made throughout
this chapter which potentially may separately or collectively steer the predictions away from the
true behavior of the system.
In some cases the assumptions are justified for the right
experimental conditions or rather limit interpretation to a specified experimental window. This,
for example, is true for the conditions of elastic/specular transmission as well as assuming the
abrupt junction approximation. In other cases, however, the assumptions are a more significant
deviation from reality and their use is justified only by the resulting simplicity that they facilitate
in the computations. An example of this is the assumption of an isotropic band structure for the
semiconducting electrode.
Inspection of Figure 1.2 immediately demonstrates that this
assumption is significantly challenged. The band structure influences the calculated results
through the DOS effective mass and through the transverse wave vector integral. As Figure
104
3.11d shows, error in the effective mass has negligible effects on the results. The isotropic
assumption, on the other hand, can possibly introduce more profound error through the limits of
the integral in [3.6a] since the shapes of the Broullion zone constant energy surfaces projected
onto the tunneling junction are not ideally circular.
It has been assumed that the materials parameters do not change during the course of a
measurement. It has already been argued, however, that the surface charge varies with applied
bias and that the semiconductor electron affinity varies with surface charge. One might therefore
expect a voltage-dependent electron affinity to appear in [3.24], and its value should increase
with increasing reverse bias. Qualitatively, this will cause the current to rise more slowly at
larger reverse bias and improve the agreement between experiment and theory in Figure 3.10.
It has also been recently observed that the dielectric constant decreases for applied
electric fields greater than 300 kV/cm [32]. The average depletion width in semiconducting
SrTiO3 is of the order 102 Å. For biases of the order of a volt, this gives electric fields across the
depletion width of the order 103 kV/cm. For fields of this magnitude the dielectric constant
decreases by nearly a factor of two. This decrease in screening strength will result in greater
dynamic band bending, and it can be anticipated from Figure 3.4 that this effect will be most
apparent at forward sample bias. Qualitatively, this will cause the current to rise more quickly at
larger forward bias, worsening the agreement between experiment and theory in Figure 3.10. It
should be noted, however, that the value of the electron mobility was chosen arbitrarily and can
be up to an order of magnitude smaller in reality. To improve the fit between experiment and
theory, it may be necessary to adjust parameters such as the electron mobility or the capacitance
constant β which decreases band bending with increasing magnitude (see Figure 3.4).
105
Finally, it has been implicitly assumed that electrons with positive kinetic energy (i.e.,
those with energies greater than the potential barriers) will have a transmission probability of
unity. The phenomenon of quantum mechanical reflection (QMR) has not yet been treated
rigorously in this context. Crowell and Sze used a numerical approach to determine QMR of
electrons at metal-semiconductor Schottky barriers [33] and their results give some insight into
what one might expect in the case of a MIS junction.
Although their treatment is only
approximate, it is consistent with the present model to the extent that it is a one-dimensional
isotropic effective mass approach. The general results predict that QMR can be up to 60%–70%
when the incident energy is equal to the barrier height for “abrupt” potentials, and decreases
slowly with energy in excess of the barrier maximum. Moreover, QMR increases a) with the
electric field at the surface of the semiconductor, and b) when the semiconductor effective mass
is larger than that of the metal. Both points a and b are significant to the present work. In
practice, the electric field across the depletion region is typically of the order 106 V/cm, and the
effective mass of SrTiO3 is several times larger than that of the metal electrode.
Despite these challenges to the accuracy of the model, the agreement between experiment
and theory represented in Figure 3.10 is sufficient for the present application for the following
reason.
As discussed in section 1.4.2, the photoinduced effects are expected to involve
modification of the surface charge.
Therefore the effect will be mainly observed as a
modification of band bending. Exact matching of the experimental current densities over the
entire width of the voltage sweep is not necessary for a quantitative assessment of the change in
surface charge. A “good fit” might be characterized by a matching of the apparent surface gap as
well as the slope of the curves at the current onset voltages. A suitable fitting procedure might
therefore consist of adjusting the unknown parameters until a good fit is obtained for the “dark”
106
spectra, then adjusting the surface potential (and electron affinity) until a good fit is obtained for
the “illuminated” spectra. The photo-induced change in the surface charge is then given by
[3.21]. This approach will be employed to interpret the data presented in Chapter 5.
REFERENCES
1.
D. Bohm Quantum Theory Prentice-Hall, New York (1951)
2.
S. M. Sze Physics of Semiconductor Devices 2nd ed. John Wiley & Sons,
New York (1981)
3.
E. H. Rhoderick and R. H. Williams Metal-Semiconductor Contacts Clarendon
Press, Oxford (1988)
4.
H. K. Henisch Semiconductor Contacts Clarendon Press, Oxford (1984)
5.
J. Shewchun, A. Waxman and G. Warfield Solid-State Electronics 10
(1967) 1165
6.
E. Wolf Electron Tunneling Spectroscopy Oxford University Press, Oxford (1986)
7.
J. A. Stroscio and W. J. Kaiser, Eds. Scanning Tunneling Microscopy Academic
Press, Inc., San Diego (1993)
8.
D. A. Bonnell, Ed. Scanning Tunneling Microscopy and Spectroscopy: Theory,
Techniques and Applications VCH Publishers, Inc., New York (1993)
9.
W. J. Kaiser, L. D. Bell, M. H. Hecht and F. J. Grunthaner J. Vac. Sci. Technol.
A 6 [2] (1988) 519
10. E. Burstein and S. Lundqvist, Eds. Tunneling Phenomena in Solids Plenum
Press, New York (1969)
11. R. M. Feenstra and J. A. Stroscio J. Vac. Sci. Technol. B 5 [4] (1987) 923
12. R. M. Silver, J. A. Dagata, and W. Tseng J. Appl. Phys. 76 [9] (1994) 5122
107
13. Ch. Sommerhalter, Th. W. Matthes, J. Boneberg, P. Leiderer, and M. Ch.
Lux-Steiner J. Vac. Sci. Technol. B 15 [6] (1997) 1876
14. M. McEllistrem, G. Haase, D. Chen, and R. J. Hamers Phys. Rev. Lett. 70 [16]
(1993) 2471
15. W. A. Harrison Phys. Rev. 123 [1] (1961) 85
16. J. R. Oppenheimer Phys. Rev. 31 (1928) 66
17. J. Bardeen Phys. Rev. Lett. 6 [2] (1961) 57
18. C. B. Duke Tunneling In Solids Academic Press, New York 1969
19. J. Bono and R. H. Good, Jr. Surface Science 175 (1986) 415
20. P. Hansma Tunneling Spectroscopy: Capabilities, Applications, and New
Techniques Plenum Press, New York (1982)
21. J. S. Blakemore Semiconductor Statistics Pergamon Press, New York (1962)
22. E. Spenke Electronic Semiconductors McGraw-Hill, New York (1958)
23. J. G. Simmons J. Appl. Phys. 34 [9] (1963) 2581
24. R. H. Kingston and S. F. Neustadler J. Appl. Phys. 26 [6] (1955) 718
25. R. Seiwatz and M. Green J. Appl. Phys. 29 [7] (1958) 1034
26. K. Seeger Semiconductor Physics 4th ed. Springer-Verlag, New York (1989)
27. J. McDougall and E. C. Stoner Trans. Roy. Soc. London A 237 (1938) 67
28. M. A. Green, F. D. King and J. Shewchun Solid-State Electronics 17
(1974) 551
29. J. G. Simmons Phys. Rev. Lett. 15 [25] (1965) 967
30. S. Zafar, R. E. Jones, B. Jiang, B. White, V. Kaushik, and S. Gillespie Appl.
Phys. Lett. 73 [24] (1998) 3533
108
31. Wolfram Research, Inc., Mathematica, Version 3.0, Champaign, IL (1996)
32. R. A. van der Berg, P. W. M. Blom, J. F. M. Cillessen, and R. M. Wolf Appl.
Phys. Lett. 66 [6] (1995) 697
33. C. R. Crowell and S. M. Sze J. Appl. Phys. 37 [7] (1966) 2683
109
Chapter 4: Characterization of The Bulk
All ellipsometry and transmission measurements were performed and analyzed with technical
assistance at the DuPont experimental station in Wilmington Delaware. Hall measurements were
performed in the standard configuration with the exception that a bipolar power source,
modulated at 1kHz, supplied an AC current to the sample and a lock-in amplifier was used to
detect the Hall voltage. The magnetic field was 9.5 kGauss and the currents used were between
35 and 85 mA. Materials parameters (i.e., the free carrier density and optical dielectric constant)
obtained from these measurements were used to model the tunneling spectra as discussed in
Chapter 3.
4.1
BULK PROPERTIES OF REDUCED SrTiO3
4.1.1 Hall /resistivity measurements
The samples were heat treated by one of two methods: 1) vacuum annealing by joule heating; or
2) furnace annealing in a hydrogen atmosphere.
In the former case, the temperature was
monitored by an optical pyrometer. Upon cooling, the temperature was observed to drop below
800–850 ºC (the “freeze-in” point for oxygen vacancies) within 50 to 100 seconds,
corresponding to an initial quench rate of approximately 150 º/min. In the latter case, the
cooling rate was not easily verified since the samples were simply extracted from the hot zone of
the furnace to initiate cooling. The quench rate in this case is thus expected to have been much
slower. All samples appeared uniform in color indicating a uniform distribution of point defects.
Table 4.1 summarizes the results of the Hall and resistivity measurements performed on
four out of the six samples studied. The results for the hydrogen reduced samples (the H series)
are also presented graphically in Figure 4.1. The geometries of samples V–930 and V–1100 (the
108
V series) were not compatible with the geometry of the apparatus designed to make these
measurements as a result of cracking during thermal treatment. Therefore, all conclusions
regarding the oxygen vacancy density dependence are restricted to the H series.
Table 4.1: Thermal history and Hall/resistivity measurements.
Sample ID
Thermal history
VH (mV)
ρ (Ω-cm)
n (cm-3)
V–930
930 °C, 10 min / vacuum
—
—
—
V–1100
1100 °C, 12 min / vacuum
—
—
—
V–930Nb
930 °C, 10 min / vacuum
2.1
0.042 ± 0.001
8.8 × 10+17
H–700
700 °C, 2 hrs / H2 furnace
2.0
0.430 ± 0.008
3.3 ± 0.7 × 10+17
H–850
850 °C, 2 hrs / H2 furnace
2.4
0.069 ± 0.002
3.2 ± 1.1 × 10+17
H–1000
1000 °C, 2 hrs / H2 furnace
3.2
0.026 ± 0.003
1.3 ± 0.5 × 10+17
109
4
0.45
Resistivity (ohm-cm)
3.5
0.35
3
0.3
2.5
0.25
2
0.2
1.5
0.15
1
0.1
0.5
0.05
Carrier Density x 10+17 (cm-3)
0.4
0
0
700
850
1000
Temperature (°C)
Figure 4.1
Resistivity and carrier density of undoped single crystal SrTiO3 as a function of
annealing temperature.
These samples show a decrease in resistivity with increasing degree of reduction (i.e.,
increasing annealing temperature). The measured carrier density, however, apparently also
decreases with reduction. Note the larger error in measurement of the carrier density for the H850 sample. The lower end of the variance (2.16 × 10+17 cm-3) lies mid-way between H–700 and
H–1000 to give a reasonable trend, although not exactly in concert with the observed resistivity
trend as shown in Figure 4.1.
The simple theory developed to relate the Hall voltage (VH) to the carrier density, (n),
necessarily assumes that the transport properties of the material may be adequately described by
the Drude model or the semiclassical model — i.e., charge carriers behave as a free electron gas.
The carrier density is simply related to the Hall coefficient (RH) and can be written in terms of
measurable parameters as
110
n=
IH z
,
VHe t
[4.1]
where I is the measured current, t is the thickness of the sample, and Hz is the applied magnetic
field.
Furthermore,
for
n-type
extrinsic
materials,
the
Hall
mobility
µH
is
given by
µH =
RH
1
=
,
ρ
en ρ
[4.2]
and is very nearly equivalent to the true carrier drift mobility when acoustic phonon scattering is
the predominant scattering mechanism [1]. Since electron-lattice coupling is often strong in
transmission metal oxides (giving rise to small or large polaron formation), it is not obvious that
[4.1] and [4.2] will give accurate results.
It was anticipated that sample V–930Nb would be a good case to test the relative
accuracy of the Hall measurement since this sample was commercially prepared with a specified
doping of 0.17 weight percent Nb2O5. This means that in one gram of doped crystal one may
expect 1.28 × 10-5 moles of added impurity atoms. Niobium substitutes on the Ti sublattice and
is known to introduce shallow (i.e., hydrogenic) donor states. If the crystal were stoichiometric,
then
the
formula
SrTi1-xNbxO3
describes
the
crystal
where
x ≈ 0.00235. This corresponds to a carrier density of 3.95 × 1019 cm-3, much larger than that
determined by the Hall measurement. On the other hand, this estimated value for n predicts a
Hall voltage of 0.047 mV whereas the observed experimental noise was 0.1 mV. The observed
Hall voltage was well reproduced and the behavior was not consistent with a magnetoresistance
effect.
Using [4.2] and assuming a 1019 carrier density gives a mobility of 3.73 cm2/Vs, whereas
the measured carrier density gives 168.34 cm2/Vs. The former estimate is close to tabulated
111
values for “small” x in SrTiO3-x [2]. The latter value, however, is not too unreasonable and is
still an order of magnitude less than the electron mobilities in Si (1500 cm2/Vs), Ge (3900
cm2/Vs), or GaAs (8500 cm2/Vs). The reported purity of this sample was 10 ppma (i.e, an
accidental impurity density of 1017 cm-3). Therefore, accidental acceptor impurity compensation
for the donor impurities is not an acceptable explanation for the discrepancy in n. It is possible
that the sample was unintentionally under doped.
An alternative explanation for such observations was reported by Perluzzo and Destry
[3]. These authors also observed discrepancies in the ratio of estimated carrier densities to
measured carrier densities for Nb-doped single crystals until a thermal “activation” process
restored the ratio to approximate unity. It should be noted that, as described in Chapter 2, all
samples were processed with similar thermal histories prior to vacuum or hydrogen annealing.
The doped crystal was observed to become insulating (although it remained blue in color) after
the air anneal and therefore required further processing to “activate” the carriers. Since it was
desirable to yield a surface structure comparable to the reduced crystals, the activation procedure
recommended by Perluzzo and Destry could not be applied. Thus, a possible explanation of the
low observed carrier density could be that a small fraction of the defects were re-activated by the
thermal treatment described in Table 4.1. It will be shown in the following chapter that the
assumed carrier density does not present difficulty in obtaining good agreement between the
experimental and calculated tunneling spectra.
4.1.2 Optical measurements
It was mentioned earlier that all samples appeared uniform in color.
Both V–930 and
H–700 appeared clear, both V–930Nb and H–850 transmitted in the blue region of the visible
spectrum, and both V–1100 and H–1000 appeared black. All samples can be expected to
112
strongly absorb light with energies above approximately 3.2 eV where electronic interband
transitions are excited between the highest occupied valence band and the lowest unoccupied
conduction band. The changing color of the crystals with increasing defect density suggests
increases in absorption at energies throughout the infrared and visible regions of the spectrum.
This interpretation is confirmed by the optical measurements presented below.
Spectroscopic ellipsometry is a technique that yields the optical constants of a material
from measurements of light reflected from its surface. The technique generally consists of
illuminating the surface with circularly- or elliptically-polarized monochromated light and
detecting changes in amplitude and phase upon reflection. One component of the incident light
is linearly polarized in the plane of incidence (the p wave) and another component is linearly
polarized normal to the plane of incidence (the s wave). The actual parameters measured by an
ellipsometer are the angles Ψ (Psi) and ∆ (Del), where the former is a measure of the ellipticity
introduced to the reflected wave due to a relative change in amplitude between the p and s
waves, and the latter is a measure of the change in phase difference between the p and s waves
upon reflection. Subsequently, an appropriate model must be assumed for the reflecting system
upon solving an equation of the form [4]
i∆
tan Ψe =
r p (nˆ ,φ)
,
s
r (nˆ ,φ )
[4.3]
where r p and r s are called the Fresnel reflection coefficients. They are functions of known
experimental parameters (such as the angle of incidence, φ , and the refractive index of air) and
the (unknown) complex index of refraction, nˆ , of the material under investigation.
The
appropriate forms of r p and r s depend on the assumed model of the reflecting system. In the
present case, a simple model with two reflecting surfaces (the top and bottom of a single crystal)
113
was sufficient to fit the experimental data. Optical transmission measurements were performed
to obtain better accuracy in the optical constants in spectral regions where absorption is very
small
and
thus
the
ellipsometer
greater error.
2.5
a
0.01
Extinction coefficient
0.008
2
0.006
0.004
1.5
0.002
0
1.52
1.54
1.56
1.58
1.6
1.62
1
0.5
0
2
2.5
3
3.5
4
4.5
5
Energy (eV)
4
b
Palik
oxidized
H-700
H-850
H-1000
Refraction index
3.5
3
2.5
2
1.5
2
2.5
3
3.5
4
4.5
Energy (eV)
114
5
produced
data
with
Figure 4.2 (previous page) The dispersion curves for the optical constants (n and k) of
SrTiO3. The extinction coefficient (a) and index of refraction (b) are shown for oxidized and
reduced single crystals as indicated by the legend. These are also compared to literature values
measured over the same spectral range (solid curve). The inset in (a) shows an absorption tail for
H-850 near the “red” region of the visible spectrum.
Figure 4.2 shows the measured dispersion of the extinction coefficient (k) and the
refractive index (n) for each sample in the H series. Also included are results of measurements
from an oxidized sample of the same boule as well as the tabulated values for undoped SrTiO3 at
room temperature [5]. All curves in Figure 4.2b show an increase in refractive index with
increasing photon energy up to some energy where n reaches a maximum. This is called normal
dispersion. At energies above 3.8 eV to 4.1 eV n decreases with increasing photon energy. This
is called anomalous dispersion. A shoulder appears above the peak energy for all samples. This
is indicative of two overlapping dissipating processes.
Except for a finite spectral range from 2.0 to 3.0 eV, all tabulated values appearing in
Figure 4.2b are compiled from a single reference. The variation (i.e., ∆n) in the tabulated values
from 2.0 to 3.0 eV are in the range 0.043 to 0.125; variations between the Palik values and those
from the oxidized sample are in the range 0.12 to 0.24 over the same spectral window. (At
higher energies |∆n| is as large as 0.4 near 3.5 eV and 4.8 eV.) Therefore, the deviation between
the oxidized crystal data and the tabulated values is not too unreasonable. The results of Figure
4.2a, however, suggest a significant difference in the band structure which is observed to evolve
upon reduction.
To see this, the data in Figure 4.2a are used to calculate the absorption coefficient
(α) which is related to the extinction coefficient by
115
α=
4πk
,
λ
[4.4]
where λ is the wavelength of the light. The results are shown in Figure 4.3a. Within the
independent electron approximation, the absorption coefficient near the fundamental absorption
edge is assumed to be approximated by [6]
a
0.4
(αEph)2
1
0.3
0.25
0.2
0.15
0.1
6
Absorption coeff. x 10 (1/cm)
0.35
0.8
0.05
0
3.2
3.4
3.6
3.8
4
4.2
0.4
(αEph)0.5
0.35
0.6
0.3
0.25
0.2
0.15
0.1
0.4
0.05
0
3.2
3.4
3.6
3.8
4
4.2
0.2
0
2
2.5
3
3.5
4
4.5
5
Energy (eV)
400
b
Absorption depth (nm)
350
300
250
200
150
100
Palik
oxidized
H-700
H-850
H-1000
50
0
2
2.5
3
3.5
Energy (eV)
116
4
4.5
5
Figure 4.3
(a) The dispersion curves for the absorption coefficient of SrTiO3. The insets
demonstrate that this measurement gives evidence of both direct and indirect transition
mechanisms. The transition energies are deduced by linear fitting near the absorption edge. (b)
The inverse of the curves in (a).
Table 4.2: Optical transition energies deduced from Figure 4.3a.
Sample ID
Direct Gap Energy (eV)
Indirect Gap Energy (eV)
Palik
3.52
3.28
oxidized
3.79
3.59
H–700
3.88
3.77
H–850
3.90
3.68
H–1000
3.58
3.00
(hω − E ) ,
α~
/n
g
hω
[4.5]
where hω = Eph is the energy of the incident photon, and the value of n/ depends on the
transition mechanism. For indirect transitions n/ = 2; for direct transitions n/ = 0.5.
1
Plotting (αE ph )n/ versus Eph near the absorption edge obtains a linear fit when the
appropriate transition mechanism is assumed, and the intercept with the abscissa gives the energy
of the transition. Table 4.2 lists the absorption edge transition energies deduced in this manner
from the insets in Figure 4.3a. Both direct and indirect transition energies are found. In
addition, the energies for the oxidized sample are greater than the Palik values. The trend
amongst the second to fourth entries in the table show an increase in the direct gap energy. The
heavily reduced sample, however, shows a substantial decrease in this transition energy which
agrees better with the Palik value. The lowest energy transitions are indirect and follow a similar
117
trend with reduction as observed for the direct gap energies except that the energy decrease has
already begun with H–850.
Figures 4.2a and 4.3a also show a general trend in the H series of decreasing absorption
with reduction in the UV region (i.e., Eph > 4.5 eV). There is, however, an initial increase in
absorption from the oxidized sample to H–700. These trends are reversed for lower photon
energies (i.e., Eph < 4.0 eV). H–1000 is observed to cross H–850 near 4.5 eV; it crosses H–700
near 4.0 eV. Below the latter energy the absorption increases with reduction, with an initial
decrease from the oxidized sample to H–700. The strongest absorption in the visible region is
observed for sample H–1000 as shown in Figure 4.3b. This is a plot of the inverse of the
absorption coefficient and thus determines the depth to which the transmitted light intensity falls
below approximately 37% of its incident value. It can be seen that the bulk of the intensity drops
within approximately 50 to 375 nm of the crystal surface for energies well below the onset of
interband transitions. All other crystals remain transparent in the visible, becoming opaque only
in the UV where the bulk of the transmitted intensity drops within 10 nm of the crystal surface.
118
6
band gap (eV)
oxidized
H-700
H-850
H-1000
Dielectric constant
5.8
3.79
3.88
3.90
3.58
5.6
5.4
5.2
5
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Energy (eV)
Figure 4.4
Dielectric function, εˆ = n2 − k 2 , of SrTiO3 below anomalous dispersion.
4.1.3 Discussion
The data presented in this chapter was acquired for two primary reasons. To determine the
values of physical parameters to be used in tunneling calculations (such as the carrier density and
optical dielectric constant) and to characterize changes in the physical properties of the bulk
upon reduction to facilitate interpretation of observed surface properties which strongly depend
on bulk properties. Examples of the latter are the surface potential and electrostatic energy
which
both
have
an
inverse
dependence
on
the
density
of
ionized defects.
If the structure at the surface does not strongly disrupt the Ti-O coordination, then the
surface electronic structure may not deviate strongly from the bulk electronic structure. Thus the
optical data is a good starting point to characterize changes in the band structure induced by
reduction annealing. The allowed direct transitions in cubic SrTiO3 were determined by Casella
119
[7], based on band structure calculations and the dipole selection rule, to be: Γ15 → Γ2 5′ ;
∆ 5 → ∆ 2 ′ ; ∆ 5 → ∆ 5 ; Χ5 ′ → Χ3 ; Χ4 ′ → Χ5 ; and Χ5 ′ → Χ 5 , where the last two transitions were
assumed possible in the presence of a [001] uniaxial stress. Based on the observed splitting (0.86
eV) of the primary absorption peak from reflectivity data, however, Cardona concluded that both
Χ4 ′ → Χ5 and Χ5 ′ → Χ 5 transitions accounted for the observed dispersion in the refractive
index
[8].
A
similar
splitting
of
0.8 eV is observed at the onset of anomalous dispersion for the oxidized curve in
Figure 4.2b. The decreasing intensity in both n and k among the H series in this spectral region
can thus be interpreted as a decrease in the joint density of states for these transitions. It has
been established by photoemission studies that this is caused by a decrease in the density of
states of the upper valence band upon reduction [9].
There is an apparent increase in the joint density of states for these transitions
(accompanied by a slight decrease in the splitting to ∼ 0.6 eV) within the early stages of
reduction that, to the author’s knowledge, has not been previously reported. The H series differs
from the oxidized crystal only by the additional heating in 100% flowing hydrogen. It will be
shown in the following chapter that extended vacuum annealing resulted in the appearance of a
sulfur peak in Auger spectra. Galbraith analysis determined a bulk sulfur concentration of less
than 0.08 ppm by weight (or ≈ 0.46 ppm based on anion sites). This is equivalent to ∼ 7.7 × 1015
cm-3 sulfur impurities. If it is assumed that sulfur diffuses to the surface with vacuum heat
treatment, such a low density could hardly account for a large change in optical properties upon
removal from the lattice. Moreover, it is implausible that introducing a small amount of sulfur to
the oxygen sublattice depletes states either from the top of the valence band or from the bottom
120
of the conduction band in a manner reminiscent of strong nonstoichiometry. The origin of the
initial increase thus remains undetermined.
Consequently, the classical Lorentz oscillator model perhaps does not appropriately
describe the behavior of SrTiO3 at the early stages of nonstoichiometry. This model predicts that
the dielectric function εˆ = n2 − k 2 should decrease with increasing band gap if the dispersion
may be approximately described as the response of a single harmonic oscillator [10]. Figure 4.4
shows that this behavior is observed only between the H–700 and H–850 samples. The observed
decrease in εˆ with decreasing band gap from H–850 to H–1000 is perhaps a consequence of the
fact that a single oscillator cannot be assumed for the heavily reduced sample such that the band
gap is rather ill-defined. The electronic structure of heavily reduced SrTiO3 thus can not be
interpreted in terms of a rigid band model. Indeed, the broad absorption throughout the visible
suggests a broadening in the static dielectric constant. Local variations in nonstoichiometry
might give rise to local variations in κst and thus a spread in hydrogenic binding energies. The
following discussion of this last point leads to a possible interpretation of the resistivity and
carrier density results.
It is perhaps customary to interpret nonstoichiometric crystal properties in terms of a
random distribution of point defects. From a thermodynamic point of view, however, if the
enthalpy of formation of associated defects (i.e., oxygen vacancy pairs) is large and negative,
then the free energy of the crystal will be minimized by the formation of defect clusters. Recent
theoretical work suggests that oxygen vacancy clustering is required in order to form localized
states in the band gap of SrTiO3 [11]. Such states will have the potential to depopulate the
conduction band by free carrier trapping. It is reasonable to expect defect clustering in crystals
with larger nonstoichiometry. It should be emphasized, however, that within the context of the
121
theory, a larger overall defect density is neither a necessary nor sufficient condition for the
formation of band gap states. Rather than a random distribution, it may be more appropriate to
think in terms of a statistical distribution of point defects with local density variations. The static
dielectric constant tends to decrease with increasing nonstoichiometry [12] such that if the defect
state energies are assumed to be approximated by [1.3], then a distribution of binding energies
will accompany a distribution of defect densities. This can lead to optical absorption over a
broad spectral range in the visible (assuming defect to conduction band transitions are symmetry
allowed) as observed in Figure 4.3b.
The above interpretation involves the removal of free carriers from the conduction band
as the oxygen vacancy density (and especially clustering) increases. Hence, a decrease in carrier
density as measured by a Hall probe would be expected. This was indeed observed as shown in
Figure 4.1. The foregoing qualitative argument is supported only with data obtained by a
method which has been shown to inadequately describe temperature-dependent transport
properties in semiconducting transition metal oxides [13]. The values for n in Table 4.1 are,
however, within the range observed in previous studies on reduced single crystal SrTiO3 [14,15].
The
predicted
Hall
mobilities
for
H–700,
H–850, and H–1000 are 44, 281, and 1,945 cm2/V s, respectively. Except for the last value,
these are reasonably smaller than carrier mobilities in traditional semiconductors as discussed
above,
although
much
larger
than
values
expected
for
systems
with
a
tendency for large polaron formation. The average value obtained by previous authors is 6.07
cm2/V s [15]. If the carrier mobility is assumed to be independent of carrier density and
clustering, and given by the latter value, one can work backwards using sample resistivities in
Table
4.1
to
predict
the
carrier
densities
122
to
be
2.45,
0.24,
1.5,
and
3.96 × 1019 cm-3, for V–930Nb, H–700, H–850, and H–1000, respectively. Now the carrier
density for the doped crystal agrees better with the estimated value based on the doping
specification. It should be noted, however, that a carrier density of 2.4 × 1018 cm-3 for sample H700 is difficult to reconcile with the fact that the sample appeared transparent and colorless!
4.1.4 Conclusions
The measured optical and transport properties of single crystal SrTiO3 give results that are in
reasonable agreement with tabulated and other reported values. Evidence of decreasing valence
to conduction band transitions is provided by decreasing n and k values in the UV region
supporting the view that oxygen nonstoichiometry depletes densities of states in the upper
valence band in agreement with previous photoemission studies. Both direct and indirect band
gap energies were determined, the values of which apparently depend upon the degree of
reduction. Although the resistivities of the samples decrease with reduction as expected, the
carrier density also decreases suggesting a significant increase in electron mobility. Given this
result, there may be some skepticism in the accuracy of the Hall measurements; however,
alternative explanations are not consistent with experimental observations. The results are thus
assumed acceptable. The validity of this assumption is supported by the successful matching of
experimental tunneling spectra as presented in Chapter 5.
REFERENCES
1.
J. P. McKelvey Solid State and Semiconductor Physics Krieger Pub. Co.,
Florida (1982)
2.
P. A. Cox Transition Metal Oxides: An Introduction to Their Electronic Structure
and Properties Oxford University Press, New York (1992)
3.
G. Perluzzo and J. Destry Can. J. Phys. 56 (1978) 453
123
4.
H. G. Tompkins A User’s Gride to Ellipsometry Academic Press, Inc.,
Boston (1993)
5.
Handbook of optical constants of solids II Edward D. Palik, Ed., Academic Press,
Inc., San Diego, CA (1991)
6.
S. M. Sze Physics of Semiconductor Devices 2nd ed. John Wiley & Sons,
New York (1981)
7.
R. C. Casella Phys. Rev 154 [3] (1967) 743
8.
M. Cardona Phys. Rev. 140 [2A] (1965) A651
9.
V. E. Henrich, G. Dresselhaus and H. J. Zeiger Phys. Rev. B 17 [12] (1978) 4908
10. F. Wooten Optical Properties of Solids Academic Press, San Diego, CA (1972)
11. N. Shanthi and D. D. Sarma Phys. Rev. B 57 [4] (1998) 2153
12. H. B. Lal Indian J. Pure Appl. Phys. 8 (1970) 81
13. S. Fu Field Effect Study of the Transport Properties of TiO2, Ph.D. Thesis,
University of Pennsylvania (1998)
14. W. S. Baer Phys. Rev. 144 [2] (1966) 734
15. H. Yamada and G. R. Miller J. Solid State Chem. 6 (1973) 169
124
Chapter 5: Properties of Vicinal SrTiO3 (001)
The surfaces of several electron doped samples are characterized in terms of their
crystallographic structure and morphology using measurements primarily from low energy
electron diffraction (LEED) and STM.
The local electrical and optical properties are
characterized by STS and PATS.
5.1 STRUCTURE AND CHEMISTRY OF REDUCED SrTiO3 (001)
5.1.1 LEED/Auger observations
Auger measurements were performed primarily to verify the cleanliness of the surfaces. A
separate vacuum chamber equipped with an Omicron retarding field rear view LEED was used
and the results displayed on a chart recorder. Prominent peaks for Sr (106 eV), Ti (391 and 420
eV) and O (515 eV) were observed in all spectra. Figure 5.1 shows two typical spectra obtained
from a vacuum annealed sample cut from the same boule and processed in the same manner as
all samples studied in this thesis. Weak carbon (273 eV) and sulfur (153 eV) peaks were
observed when the samples were subjected to low annealing conditions (i.e., 600 °C for a few
minutes)
Figure 5.1.
as
shown
in
the
top
curve
of
When subjected to more extensive annealing conditions (i.e., 1000 °C for
6 hours), reduction in the carbon peak and an increase in a sulfur peak (to varying intensities)
was often observed, as shown in the bottom curve of Figure 5.1.
126
C
Ti
O
S
Sr
Figure 5.1
Chart recorder traces showing AES spectra of vacuum reduced SrTiO3 (001)
surface. Top: 600 °C for a few minutes; bottom: 1000 °C fro 6 hours.
LEED patterns similarly showed evidence of surface crystallographic and morphological
development. Two of the most frequently observed patterns are shown in Figure 5.2, where both
patterns were obtained with a primary beam energy of 100 eV. Figure 5.2a shows the square
symmetry as observed for sample V–930. This is usually interpreted as a 1 × 1 pattern (using the
notation of E. A. Wood) suggesting that the surface symmetry is equivalent to that expected from
a projection of the bulk lattice onto the surface plane. The pattern shown in Figure 5.2b was
observed for sample V–1100 as well as other samples subjected to extended thermal treatment.
127
a
b
Figure 5.2
Two distinct LEED patterns from SrTiO3-x (001) vicinal surfaces, both obtained
with an incident beam energy of 100 eV.
Comparison of these two LEED patterns also suggests an increase in the roughness of the
surface as indicated by a slight increase in background intensity. This intensity is due to diffuse
scattering, rather than diffraction, of the primary beam. STM images confirm an increase in
roughness as will be shown in the following section.
To better understand the changes in surface structure and chemistry with thermal history
and to establish a “recipe” for preparing flat surfaces, one sample (designated
STO–8) was cut from the same boule as all other undoped samples and successively heat treated
in vacuum at 500, 1000, 1100 and 1200 °C for five minutes each. A qualitative assessment of
128
the AES results, shown in Figure 5.3, suggests that the surface sulfur content did not change
significantly as compared to Figure 5.1. A rather strong carbon peak is seen in Figure 5.3 that
also did not change significantly with heat treatment. The latter observation can be interpreted in
one of two ways: a) The complex geometry of the vacuum chamber precluded a visual inspection
of the primary beam incident upon the surface of the sample such that focusing the primary beam
was difficult. A sufficiently broad beam can generate Auger signals from other parts of the
sample holder which contained carbon contamination (i.e., the Ta peak in Figure 5.3 was
generated by the foil used to heat the sample). b) If surface carbides were present as a result of
carbon bonding to surface cations (i.e, Ti), removing them would require heating at substantially
higher temperatures than was accessible in the present experimental configuration.
Also interesting is the former observation of the apparent constant sulfur peak. This also
may be interpreted in one of two ways: a) A similar broad beam argument suggests that the
signal may have originated from a source other than the surface of the sample. b) If the sulfur
indeed originated from the bulk of the sample, then it appears as if some sort of surface
saturation occurred which slowed or stopped further accumulation. It should be noted that
sample V-930Nb also showed a sulfur peak similar in intensity to those observed in Figure 5.3.
Since this crystal did not originate from the same boule as the H series, this raises the prospect
that the observed peaks had a common source unrelated to the oxide crystals. Recalling that the
Galbraith analysis measured a sulfur content less than 0.08 ppm by weight in the bulk, the
surface structures are not expected to be strongly influenced by impurity segregation or
precipitation.
129
Ti
O
C
S
Ta
Sr
Figure 5.3
AES spectra of sample STO–8 heat treated successively (from top to bottom) at:
500, 1000, 1100, and 1200 °C for 5 minutes each. The arrows indicate baseline shifting due to
instability of the chart recorder.
LEED patterns were also acquired for STO–8 with the following results: a) the
500 °C anneal produced the pattern in Figure 5.2a that was observed independent of sample
orientation with respect to the incident electron beam; b) the 1000 °C anneal produced the
pattern in Figure 5.2b, also independent of orientation; c) both the 1100 and 1200 °C anneals
produced both patterns in Figure 5.2 depending on orientation of the sample with respect to the
incident electron beam. LEED patterns that varied as the sample was rotated on its axis were
associated with samples that showed non-uniformity in color, suggesting non-uniformity in
surface defect density and/or surface structure. It should be noted that this sample was left in the
vacuum chamber for up to nine days following the 1200 °C thermal anneal. The same LEED
130
pattern as described in (c) was subsequently reproduced, indicating that the surface was very
stable in the 10-9 Torr environment for up to nine days.
The surface ordering can be determined directly from inspection of Figure 5.2. Since the
real space primitive mesh vectors are parallel to the reciprocal space primitive mesh vectors,
where the magnitude of the latter is given by
b∗ =
∗
2π a
2π
=
=
,
b
2
2a
[5.1]
the real space surface ordering is readily derived as shown in Figure 5.4. It is determined that
Figure
5.2b
corresponds
to
a
2 × 2R45o
superstructure.
As
discussed
in
Chapter 1, there is some belief that the observed superstructures on reduced SrTiO3 (001) are due
to ordering of oxygen vacancies. The latter may be represented as shaded spheres in Figure 5.4.
It is common practice to describe this superstructure as c( 2 × 2) in order to use a notation
analogous to that used to describe the 1 × 1 structure. Note, however, that a centered square unit
mesh (indicated by the dotted line in Figure 5.4) is not one of the five unique two dimensional
Bravais lattices.
131
a*
b
b*
[01]
(0,0)
[10]
Figure 5.4
The
2 × 2R45o (sometimes called c( 2 × 2) ) superstructure (left) corresponding
to the LEED pattern (right) of Figure 5.2b. The open circles are the anions; the small circles are
the cations; the shaded circles indicate anion vacancies.
5.2
MORPHOLOGICAL STRUCTURE BY STM
The images that are presented in this section demonstrate the changing morphology of
undoped/reduced SrTiO3. For comparison, the surface images of niobium-doped SrTiO3 are also
presented. All images were acquired in constant current mode. The set points are indicated in
the text of the figures with the bias specified as that applied to the sample with respect to the tip.
The results were processed only by a flattening procedure to reduced the tilt that is often present
to varying degrees between the tip axis and the surface normal.
5.2.1 Surface morphology of V–930
The surface appears clean and flat with crystallographically aligned step edges as shown in
Figure 5.5. It is believed that these steps are aligned along 〈01〉 [1]. All step heights are
multiples of the unit cell edge (a = 3.9 Å). In this image the step heights are 4a and 8a. The scan
size is roughly one fifth the size of the image which appears in Figure 2.7. At this higher
132
resolution the “texture” of the surface can be resolved. There does not appear to be an ordering
of surface features. Therefore, the long range order observed in Figure 5.2a has not been
resolved. Analysis on the large terrace gives a z range (i.e., black-to-white scale) of 61 Å and a
calculated rms roughness of 6.4 Å. (The rms roughness is defined as the standard deviation of
the image z values within the area being analyzed.) The higher intensity near the edge of this
terrace is most likely due to a tilt in the surface normal with respect to the axis of the STM tip.
In other cases (i.e., heavily reduced samples) there may actually be material build up or a change
in electronic properties near these edges (see section 5.2.2).
Figure 5.6 compares two images taken with (right) and without (left) illumination of the
surface with greater than band gap light. The primary difference between the two is due to a
southwest shift as a result of thermal drifting. This is commonplace in STM imaging and the
effect can be very small or very large. Other than the latter effect, the images are virtually
identical. No photo-induced effects are apparent. In the upper left of both images is resolved a
much smaller step edge which was determined to be equivalent to a single unit cell height. This
was the only such observation for this surface. All other step edges were multiples of the unit
cell edge as in Figure 5.5.
133
Figure 5.5
Multiple unit cell high step edges observed on V–930. The image scan size is
1,084 Å × 1,084 Å taken with a set point of -2.0 V and 0.25 nA.
light
dark
Figure 5.6
Comparison of dark and illuminated surfaces, with incident photon energy of 3.6
eV. The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.0 V and 0.25 nA. No
photo-induced effects are observed.
134
dark
Figure 5.7
light
Comparison of dark and illuminated surfaces, with incident photon energy of 1.9
eV. The image scan size is 2,167 Å × 2,167 Å taken with a set point of -2.0 V and 0.25 nA. No
photo-induced effects are observed.
Another comparison is shown in Figure 5.7. The energy of the incident light was 1.9 eV,
the energy expected to excite charge from states localized on step edges. Again, no photoinduced effects are evident in the right image as compared to the left. Thermal drifting is in a
southeast direction in this case. Figure 5.8 shows an image acquired while illuminating the
surface with 2.4 eV light for the first 100 of 200 scan lines. Recall from Table 1.4 that this
energy was believed to give rise to interband electronic transitions between lower and upper
conduction bands. Assuming that a photo-induced effect would be uniform, and that “hot
carriers” significantly increased the surface conductivity, the bottom half of the image would be
darker than the top, indicating an inward displacement of the tip towards the surface. No such
shift was observed, although it would be difficult to distinguish from a tip expansion due to local
heating. Therefore, Figure 5.8 demonstrates at least that a tip expansion did not occur at this
135
photon energy. This was also found to be true, however, for all other photon energies used in
this study.
light
dark
Figure 5.8
Comparison of dark (bottom) versus illuminated (top) sections of a single surface.
Incident photon energy = 2.4 eV. The image scan size is 2,167 Å × 2,167 Å taken with a set
point of -2.0 V and 0.25 nA.
136
Figure 5.9
Step with apparent holes along the edge (20 Å deep) and at the kink (depth
undetermined). The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.0 V and
0.25 nA.
Figures 5.6 and 5.8 also show the presence of surface clusters near step edges and upon
terraces. These clusters range in height from as small as 7 Å to as high as 35 Å. The diameters
of these features (measured at half height) range from 38 Å to 80 Å. In addition to these
protruding features, depressions were occasionally observed near step edges or step kinks, as
indicated by the arrows in Figure 5.9. The origin of these types of apparent depressions will be
discussed in section 5.3.1. They are distinguished, however, from much larger holes that are
most likely due to local chemical attack. An example is shown in Figure 5.10 where a ∼300 Å
deep pit is observed to have a square-shaped bottom and clear ∼28 Å step edges are resolved
proceeding down into the hole. Comparison with images taken by AFM on heavily annealed
samples, as in Figure 2.8, suggests that these pits are rounded out as a result of reshaping of step
edges. This will be further demonstrated in the following section.
137
Figure 5.10
Surface hole formed by local chemical attack. The depth of the hole is ∼300 Å;
the step sizes are ∼28 Å. The image scan size is 1,084 Å × 1,084 Å taken with a set point of -2.5
V and 1.0 nA.
5.2.2 Surface morphology of V–1100
Heavy vacuum annealing results in extensive development of the (001) surface.
Figures 5.11 through 5.15 immediately show striking deviations from the structure of the
surfaces shown in Figures 5.5 through 5.10. Step edges are no longer crystallographically
aligned but appear to wander randomly. Figure 5.11a and 511b show several steps edges with
similar curvature — i.e., concave with respect to the terraces. The curvature of the edges can be
very large at some points giving step edge bulging along the surface as shown in Figure 5.11b.
The step heights are difficult to extract in Figure 5.11a; in Figure 5.11b they are twelve times the
unit cell height.
138
a
Figure 5.11
b
Surface morphology of V–1100 showing wandering step edges. The image scan
sizes are (a) 5,418 Å × 5,418 Å and (b) 3,386 Å × 3,386 Å, taken with set points -2 V at (a) 0.5 nA
and (b) 1.0 nA.
139
Figure 5.12
Apparent cluster-free surface with concave and convex step edges. Large 22a
deep holes suggest possible healing of chemical etch pits. The image scan size is 5,418 Å ×
5,418 Å taken with a set point of -2.0 V and 0.5 nA.
In some cases both concave and convex step edges were observed as shown in Figure
5.12. The step height in the center of the image measures approximately 6a. The large hole in
the upper left has a depth of 22a. This is significantly shallower than the depth of the etch pit
measured in Figure 5.10, suggesting that the vacuum anneal may redistribute material in order to
close these pits. The step heights are about the same as that observed for sample V–930. This is
also observed in Figure 5.13a which shows a series of convex steps separated by ∼20 Å.
140
a
b
Figure 5.13
(a) Series of convex step edges separated by ∼20 Å. Local ordering of surface
features normal to the step edges is apparent. The image scan size is 1,355 Å × 1,355 Å taken
with a set point of -2.0 V and 1.0 nA. (b) Local terrace structure showing apparent ordering of
clusters. The image scan size is 542 Å × 542 Å taken with a set point of -2.5 V and 0.5 nA.
The terraces of these steps appear to have some local alignment of clusters in a direction
normal
to
the
step
edge.
A
magnified
view
of
this
ordering
appears
in
Figure 5.13b. This image bares some resemblance to previously observed surface ordering;
however, the scale of these “row structures” is much larger than those formerly reported [2]. The
rows are separated by approximately 76 Å (i.e., ∼ 20a). This type of ordering was frequently
observed on this surface, although there were several regions where the local structure was less
well-defined, as shown in Figure 5.14. The cluster heights range from 10 Å to 20 Å with an
average
of
15
Å.
The
z
range
in
Figure
5.14
is
90 Å and the calculated rms roughness is about 6.8 Å, only slightly larger than that observed for
sample V–930.
141
Figure 5.14
Terrace cluster structure of heavily reduced SrTiO3 (001). Local ordering is
apparent but less well-defined than in other areas as shown in Figure 5.13. The image scan size
is 542 Å × 542 Å taken with a set point of -2.5 V and 0.5 nA.
142
light
dark
Figure 5.15
eV.
Comparison of dark and illuminated surfaces, with incident photon energy of 2.95
The image scan size is 542 Å × 542 Å taken with a set point of -2.0 V and
1.0 nA.
Finally, Figure 5.15 demonstrates again that photo-induced effects were not observed in
the STM images. The photon energy used in this measurement was 2.95 eV (i.e., λ = 420 nm).
This energy is in the region where the measured flux from the light source (see Figure 2.2) was
maximized and thus any possible photo-induced effects were expected to also be maximized.
The absence of photo effects in the images is surprising given that tunneling spectra show strong
effects at certain energies, as will be shown in section 5.3.
5.2.3 Surface morphologies of the H series
The images presented in the previous sections completely characterize the surface structures
observed on all samples processed under mild and heavy vacuum reducing conditions. A few
images are presented in this section for the H series to highlight some differences that may have
occurred due to annealing in a hydrogen atmosphere or cooling at a different rate. It should be
143
noted that all of the samples in this series were probed with the same tunneling tip. Continued
scanning with one tip tends to blunt the tip end as a result of random tip crashes. Blunt tips often
can produce cleaner, well-reproduced spectra but usually the quality of images diminish or the
images are completely featureless. This was the case for all images acquired on sample H–850
and all but one acquired on sample H–700. Only discernible images are presented. It is assumed
that the surface structures of all the H samples may be described by the images presented in this
section, which are not very different than those presented in the last section.
As an example, Figure 5.16 is an image acquired from H–700. The terrace contains
clusters similar to those observed on V–1100. A possible step edge is apparent towards the
upper left corner of the image.
Step height analysis suggests that this step is
2 Å (i.e., half the unit cell height); however, this may be inconclusive given the extensive cluster
structure which complicated the analysis. The z range is ∼40 Å and calculated rms roughness is
4 Å.
144
Figure 5.16
Terrace and step edge morphology of H–700. The image scan size is 800 Å × 800
Å taken with a set point of -2.0 V and 0.25 nA.
Similar structures were observed for H–1000 as shown in Figure 5.17. This figure also
shows that the step edges are straighter than those observed for V–1100. It is assumed that these
edges
are
crystallographically
aligned
similar
to
those
observed
for
V–930. Two step edges can be resolved in Figure 5.17 with 45 degrees between them. If one
edge is assumed to be parallel to 〈01〉, the other is necessarily parallel to 〈11〉. This was not
observed for V–930. These step heights are 6 Å and 12 Å, respectively. The z range is 44 Å and
the calculated rms roughness is 5 Å.
Figures 5.18 and 5.19 are the results of measurements for optical effects. Single and
multiple unit cell steps are observed in Figure 5.18, with step sizes not more than 3a. There is no
well-defined order to the surface clusters although there may be a slight preference for
accumulation
at
step
ledges
as
seen
in
Figure
5.18b.
The
images
in
Figure 5.18 were acquired with illumination during the bottom halves of the scans with
145
2.4 eV (Figure 5.18a) and 2.14 eV (Figure 5.18b) light. The comparison in Figure 5.19 sought to
show the effect using more energetic photons at 3.8 eV. Light-induced effects were not apparent
in any of the images.
45°
Figure 5.17
Terrace and step edge morphology of H–1000. The bottom half was illuminated
with 2.05 eV light. The image scan size is 800 Å × 800 Å taken with a set point of +2.0 V and
0.25 nA.
146
a
Figure 5.18
b
Local terrace cluster structure. Illuminated with (a) 2.4 and (b) 2.14 eV light
during the bottom half of each scan. The image scan sizes are (a) 600 Å × 600 Å and (b) 800 Å ×
800 Å. Both images were acquired with a set point of +2.0 V and 0.25 nA.
light
dark
Figure 5.19
Comparison of dark and illuminated surfaces, with incident photon energy of 3.8
eV. The image scan size is 1,000 Å × 1,000 Å taken with a set point of +2.0 V and 0.25 nA.
147
5.2.4 Surface morphology of V-930Nb
The surface of the niobium-doped crystal was also distinct from the vacuum-annealed undoped
crystals. A large area scan is shown in Figure 5.20. The overall appearance is that of a severely
damaged surface as a result of chemical or thermal etching. The former is more probable since
this sample was prepared commercially. The rugged step edges in Figure 5.20 are characteristic
of SrTiO3 (001) prepared by a standard Syton polish. This sample was further etched with BHF
and subsequently air annealed. It is well-documented that annealing single crystal SrTiO3 in an
oxygen-rich environment tends to repair the surface and produce clean steps with
crystallographically aligned edges [3,4]. A significant density of oxygen vacancies may ensure
that the mobility of the anion is large enough that sufficient material redistribution is possible as
the crystal is oxidizing. The absence of oxygen vacancies in the doped crystal may therefore
explain why the surface for V-930Nb does not resemble V-930, despite the fact that their thermal
treatments were identical.
148
Figure 5.20 (previous page)
Terrace and step edge morphology of V–930Nb. The image
scan size is 2,000 Å × 2,000 Å taken with a set point of +2.0 V and 0.25 nA. Notice the presence
of holes near step edges. A particularly large hole is indicated by the arrow.
Except for the rugged contours of the step edges, this surface has a closer resemblance to
V–1100 as well as H–1000. This can be seen in Figure 5.21. The terraces are extensively
covered with clusters. There is an apparent increase in intensity near the step edges and many
“holes” can be seen near step corners. Some of the latter are indicated by arrows in Figures 5.20
and 5.21. Step height analysis is difficult when the step edge is not straight; however, the
majority
of
the
steps
in
Figures
5.20
and
5.21
are
apparently
4 Å, with very few measuring close to 2 Å. The clusters range in size from 4 Å to 17 Å, where
the larger sizes are seen accumulated near the bottom part of the image in
Figure 5.21. Consistent with the observations on the undoped samples, photo-induced effects
were not observed in images for any of the photon energies used. Examples are shown in
Figures
5.21
and
5.22
where
the
surface
was
illuminated
2.14 eV light, respectively, during the bottom halves of each scan.
with
2.05
eV
and
No photo-effects are
observed on the terrace or near the step edge. The latter measures 4 Å high; the cluster-covered
terrace has a z range of 126 Å and a calculated rms roughness of 6 Å.
149
Figure 5.21
Terrace and step edge morphology of V–930Nb. The bottom half was illuminated
with 2.05 eV light. The image scan size is 1,000 Å × 1,000 Å taken with a set point of +2.0 V
and 0.25 nA. The arrows indicate apparent holes near step edges.
a
Figure 5.22
dark
b
light
Local terrace cluster structure of V–930Nb. Illuminated with (a) 2.4 eV and (b)
2.14 eV light during bottom half of each scan.
The image scan sizes are 800 Å ×
800 Å. Both images were acquired with a set point of +2.0 V and 0.25 nA. The arrows locate a
resolved unit cell high step edge.
150
5.2.5 Summary of Observed Morphologies
The morphological structures of reduced and niobium-doped SrTiO3 surfaces have been
characterized on a nanometer scale. This work presents the first surface images of optically
transparent (i.e., within the very early stages of oxygen nonstoichiometry) SrTiO3 (001) obtained
by STM. The structure appeared similar to that observed by AFM. The terraces were flat with
crystallographically aligned step edges.
It has been observed that material redistribution
proceeded upon vacuum reduction resulting in apparent hole formation, wandering step edges,
and/or locally ordered cluster coverage. The “row structures” observed were separated by ∼20a;
this represents an ordering on a larger scale than previously reported. By comparison, no “row
structures” were observed on the surface of the niobium-doped crystal. Hydrogen annealing also
resulted in cluster formation, although ordering was not clearly observed on these samples. The
step edges on the hydrogen-annealed samples appeared straighter than the vacuum annealed
samples; they appeared more rugged on the niobium-doped sample, due to prior chemical
treatment. Dark regions were apparent on the more heavily reduced crystals, as well as the
niobium-doped crystal, mostly near step corners and kinks. In some cases hole formation was
apparent. Roughness calculations on terraces yielded similar results for all samples imaged —
rms values ranged from 4 Å to 6.8 Å. Photo-induced effects were not observed in any of the
images acquired.
5.3
SURFACE ELECTRONIC PROPERTIES BY STS AND PATS
The local electrical and optical properties are demonstrated by measurements obtained from the
samples in the H series as well as the niobium-doped sample.
Only these samples were
characterized in terms of their bulk properties in Chapter 4. The similarity in STM images
between vacuum-annealed and hydrogen-annealed surfaces suggests that the electronic
151
properties are not significantly different. (Recall that a STM image is an image of electronic
information as well as structural information.) Therefore, the conclusions drawn from the study
of the H series are likely to be applicable to the V series.
A notable limitation of the study is that spectra could not be easily correlated with surface
features. This has already been made clear by the images presented in the previous section
which do not show the photo-induced effects that are readily apparent in the tunneling spectra.
A well-known rule of thumb in tunneling microscopy is that sharper tips give improved image
quality (particularly for larger area scans on “rough” surfaces), while blunt tips are required for
reproducible spectra. This was observed to be true a fortiori for imaging on these oxide surfaces.
In most cases when tunneling spectra were well-reproduced, the images appeared featureless.
Similarly, the spectra were usually completely random when images revealed sharp features.
The series of spectra presented in section 5.3.2, which represent the reproducible photo-induced
effects, were all obtained from apparently flat (i.e., terrace) regions of the samples.
The
electronic structure of a step is quite unique as shown in the following section, and thus could be
easily distinguished if thermal drift were to bring the STM tip into its vicinity. Consequently, the
results are not likely to reflect the optical activity of surface steps. All spectra are presented
without smoothing and only corrected for the junction capacitance.
5.3.1 Terrace and step edge electronic properties by STS
A comparison of the tunneling spectra acquired from terrace regions of the samples in the H
series appears in Figure 5.23.
All spectra show the characteristics expected of a n-type
semiconductor in depletion. The observed trend is a decreasing conductance for increasing
degree of reduction. This is indicated by an increase in the conductance well, where the latter is
defined as the region between the current onsets at opposite sides of the equilibrium Fermi level.
152
A noteworthy result is the “metallic” behavior of the spectrum for sample H–700.
Identical set point conditions were used during the establishment of tunneling on all of these
samples. If the tip–sample distance, s, is assumed to be determined by the bulk conductivity,
then since the latter increases from H–700 to H–1000, s is expected to increase in kind. If, on the
other hand, s is assumed to be determined more by surface properties, such as the surface
potential barrier, then an increasing surface potential (and hence overall increase in the tunneling
barrier opacity) will result in a decreasing s. Both mechanisms cannot simultaneously account
for the observed trend in Figure 5.23 since they suggest a change in s with opposite signs. Bulk
properties are known to have profound effects on tunneling spectra; however, the differences
observed in Figure 5.23 are believed to be due predominately to variations in surface properties
(i.e., the surface potential). Note that a larger carrier density increases screening and is expected
to reduce dynamic band bending effects. Recall, however, that the Hall measurements suggest a
slight decrease in carrier density with vacuum annealing. This can induce greater dynamic band
bending and contribute to increasing the width of the conductance well.
153
2
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
-1.5
a
-2
-3
-2
-1
0
1
2
3
-2
-1
0
1
2
3
1
Log|I| (nA)
0.5
0
-0.5
-1
b
-1.5
-3
Sample voltage,Va (V)
Figure 5.23
Linear (a) and semi-log (b) plots comparing the terrace electronic properties in the
H series. An increasing conductance well is observed with increasing reduction, consistent with
an increasing surface potential. Solid = H–700; large dash = H–850; small dash = H–1000
154
solid spectrum
dotted spectrum
a
Figure 5.24a
Step edge on V–930Nb where local electronic structure is observed to vary as
shown by the tunneling spectra in Figure 5.24b.
A similar “metallic” spectrum was observed upon moving the tunneling tip near a step
edge imaged on sample V–930Nb as shown in Figure 5.24a. The comparison between terrace
and step edge spectra are shown in Figure 5.24b. Again, the STM feedback maintains the set
point such that the tip–sample gap is not expected to be constant between the two spectra. The
tip will be displaced due to a change in the local tunneling barrier. Recall the many dark regions
near the corners of the steps in the STM images. This suggests that the local barrier increases
near step edges resulting in an inward displacement of the tunneling tip. A similar displacement
is not observed in images obtained by AFM.
The most consistent explanation for these
observations is that the increase in conductance observed in the spectra is due predominately to a
decrease in s as a result of an increase in the surface potential.
155
2
Tunneling Current (nA)
1.5
b
1
0.5
0
-0.5
-1
-1.5
-2
-3
-2
-1
0
1
0
0.5
2
3
10
c
Tunneling Current (nA)
7.5
5
2.5
0
-2.5
-5
-7.5
-10
-1
-0.5
1
1.5
Sample voltage,Va (V)
Figure 5.24b,c Terrace versus step edge electronic behavior. (b) Solid curve acquired on the
terrace; dashed curve acquired near the step. (c) The dashed curve on a larger xy scale. The
NDC feature near +0.7 V is associated with an increase in local state density.
156
Note that this explanation is opposite to the case discussed for Figure 5.23 where s is
assumed to have decreased, but the conductance well increased due to the more dominant
influence of an increasing surface potential as well as a possible decrease in screening strength.
On the other hand, many of the images that contained dark regions in the step corners also
showed increased intensity just above on the edge of the step. It is possible that the spectra in
Figure 5.24b reflect the properties of this region rather than the dark corners of the step. The
bright features at the step edge mean that the tip was displaced away from the surface. The
observed increase in conductance can only be explained in terms of a local decrease in the
tunneling opacity due to a local decrease in the surface potential barrier. The latter involves the
local trapped charge density.
The foregoing arguments are based on the strong dependence of the tunneling current on
s and the tunneling barrier height. As can be seen from equation [3.7], an exponential prefactor
contains contributions from the local density of states. Variation in the latter is expected to have
a small influence on the tunneling current. The following observations, however, are offered in
support of an additional explanation of the observed changes in the spectrum.
An increase in the local density of states will provide a proportionally greater tunneling
probability (see equation [3.1b]). The filling of these states at reverse bias is expected to result
in two well-known effects. Since these states do not extend into the bulk to form a band in
which charge can propagate (i.e., step-related states are localized on the surface), tunneling
electrons from the metal tip initially populate these states giving rise to a measurable capacitance
current. This is manifested as a finite current over the voltage range where the conductance well
is expected. This charging current must be removed when plotting results on a semi-log scale so
that the apparent equilibrium Fermi level always appears at zero applied bias. The changes in
157
the current necessary to shift the spectra in Figure 5.24a were 0.005 nA and 0.011 nA for the
terrace and step edge spectra, respectively. Therefore, an increase in local capacitance has been
observed.
The other effect is related to the increased local capacitance. Since the local surface
charge increases as these states populate, a Coulomb field is believed to induce a blockade to
additional tunneling electrons such that the current is expected to increase more slowly with
increasing reverse bias and tend to flatten the current-voltage curves or give decreasing
conductance with increasing applied bias. This effect is referred to as negative differential
conductance (NDC) since the derivative of the spectra has a negative value in the voltage range
of these features. This effect is not strongly observed in Figure 5.24b. The spectra for the step
edge is replotted on a larger xy scale in Figure 5.24c where a shoulder in the conductance is
apparent near +0.7 V and is likely associated with a NDC effect. These two latter observations
are in support of an increase in local density of states near the step edge.
It is likely that the differences observed in Figure 5.24b are due to both variations in local
density of states and variations in local surface potential. This is reasonable since a larger state
density can trap a larger charge density. From the preceding arguments it can be deduced that
the “metallic” spectrum in Figure 5.24b reflect the dark regions near the step corners and can
thus be used to distinguish this feature from others on surface terraces when the tip conditions
are such that images are not particularly coherent.
5.3.2 Terrace optical responsivity by PATS
All tunneling spectra were acquired using the modified sample-and-hold technique (as described
in Chapter 2) where the feedback was momentarily disengaged, the surface illuminated with
mono-energetic light, then the voltage varied from -4 V to +4 V before the feedback was re158
engaged. The measurement was fast so that all of the features in one spectrum correlate with one
region on the surface. Thermal drift, however, is always present to varying degrees such that
two sequentially acquired spectra can appear completely different if the tunneling tip wanders
over a region with spatially varying electronic structures. A more insidious problem can occur
due to instabilities in the feedback electronics.
During the stage when the feedback is
disengaged, even a small instability in the amplifier voltage output (i.e., millivolts) can cause an
unwanted tip displacement and hence place the measurement on a new current/voltage curve.
Random fluctuations of this kind can account for the error observed in the standard deviation
curve plotted in Figure 2.3.
In principle, if the tunneling gap narrows significantly, the large electric field at the tip
can continuously redistribute atoms near the tip apex during a single voltage sweep [5]. The
resulting spectrum can be a composite of several different spectra and thus not be useful for
interpreting the constant properties of the surface.
All of these mechanisms (thermal drift, circuit instabilities, and tunneling tip instabilities)
can separately or collectively result in non-reproducibility in the features of the spectra. This
difficulty was encountered frequently and it was often necessary to allow the STM to “settle”
before taking additional spectra. Careful documentation of frequently observed characteristics
distinguished “good data” from misleading data. It was C. B. Duke that said, “Tunneling is an
art, not a science” [6]. It may also be said that scanning tunneling spectroscopy requires certain
experimental artistry to recognize the science.
Figures 5.25 through 5.40 represent the science observed from the application of PATS to
vicinal SrTiO3. It is believed that this data reflects the photo-activity of only the terrace regions.
Figure 5.25 shows representative spectra obtained from H–700. The solid curve was obtained
159
without illumination. The dot-dashed curve was obtained while illuminating with 3.8 eV light.
This figure is identical to the results observed at all energies used in the study. Thus, no
evidence has been found to support photo-activity on the surface of H–700.
160
2
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-1
-0.5
0
0.5
1
1.5
-1
-0.5
0
0.5
1
1.5
1
Log|I| (nA)
0.5
0
-0.5
-1
-1.5
Sample voltage,Va (V)
Figure 5.25
Dark versus light spectra for H–700 illuminated with 3.8 eV light.
No photo-induced effects are observed, consistent with the absence of effects in
the images.
161
The plots in Figures 5.26 to 5.40 demonstrate optically-induced changes in the electrical
behavior and also show the corresponding modeled spectra. The observed noise in the tunneling
current signal was typically 25 mV. The data are all truncated in the semilog plots at the same
upper and lower limits set at 1 and -1.6, respectively. The dark spectra were modeled initially to
match the experimental dark spectra. Three parameters requiring adjustment to match the photoinduced data were the equilibrium surface potential (ψ), the capacitance constant (β), and the
electron affinity (χ). All depend on the surface charge, which is the single dominant property
believed to vary to induce the observed effect. The required values of ∆ψ, ∆β and ∆χ are noted
within the text of each figure. The parameters required to determine the change in surface
charge are the values of the surface potential before and after illumination. The changes in
surface charge were calculated using [3.21] and the results plotted in Figure 5.41a.
It can be expected that a spectral variation in the light intensity, as demonstrated in Figure
2.2, will be reflected in the measured changes in tunneling spectra. The light flux at the
tunneling junction was not measured simultaneously with the spectra so that exact quantification
of materials properties can not be assumed. Approximate values can be determined, however,
using the data in Figure 2.2 assuming that the spectral response of the optics did not change
significantly from the original measurement. Dividing the data in Figure 5.41a by the product of
the photon flux (from Figure 2.2) and the fundamental unit of charge, and applying the
appropriate conversion factors to express the results in electrons per mW, gives the data as
shown in Figure 5.41b. The term “responsivity” is used here similarly to its use to describe
photonic devices [7]. Processed in this way the data better represents the response of the
material to incident light in terms of the number of single electron transitions per unit incident
power.
162
log
linear
2
H-850
2.90 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.26
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.040 eV; ∆β = 3.5 × 10-4 C/V; ∆χ = 0.80 eV.
163
log
linear
1
2
H-850
2.05 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
1.5
0.5
1
Log|I| (nA)
Tunneling Current (nA)
0
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.27
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.016 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV.
164
log
linear
1
2
H-850
1.77 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.28
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.023 eV; ∆β = - 3.3 × 10-4 C/V; ∆χ = - 0.65 eV.
165
log
linear
1
2
H-1000
3.80 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.29
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.035 eV; ∆β = 0 C/V; ∆χ = 0.40 eV.
166
log
linear
1
2
H-1000
2.90 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
1.5
0.5
1
Log|I| (nA)
Tunneling Current (nA)
0
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.30
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.145 eV; ∆β = 1.0 × 10-4 C/V; ∆χ = 0.70 eV.
167
log
linear
1
2
H-1000
2.82 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
1.5
0.5
1
Log|I| (nA)
Tunneling Current (nA)
0
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.31
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.040 eV; ∆β = - 6.0 × 10-4 C/V; ∆χ = 0.20 eV.
168
log
linear
1
2
H-1000
2.40 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
1.5
0.5
1
Log|I| (nA)
Tunneling Current (nA)
0
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.32
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.150 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV.
169
log
linear
1
2
H-1000
2.14 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
1.5
0.5
1
Log|I| (nA)
Tunneling Current (nA)
0
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.33
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.170 eV; ∆β = - 6.3 × 10-4 C/V; ∆χ = 0.40 eV.
170
log
linear
1
2
H-1000
2.05 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
-3
3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.34
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.054 eV; ∆β = 0 C/V; ∆χ = 0.30 eV.
171
log
linear
2
H-1000
1.90 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.35
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.090 eV; ∆β = 0 C/V; ∆χ = 0.10 eV.
172
log
linear
2
H-1000
1.77 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.36
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.140 eV; ∆β = 3.0 × 10-4 C/V; ∆χ = - 0.70 eV.
173
log
linear
2
H-1000
1.30 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.37
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.018 eV; ∆β = 0 C/V; ∆χ = 0.26 eV.
174
log
linear
2
V-930Nb
3.80 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.38
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = - 0.043 eV; ∆β = - 1.5 × 10-4 C/V; ∆χ = - 0.50 eV.
175
log
linear
2
V-930Nb
2.90 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
2
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
2
-2
-1
0
1
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
4
-3
Sample voltage,Va (V)
Figure 5.39
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.032 eV; ∆β = 1.5 × 10-4 C/V; ∆χ = 0.50 eV.
176
log
linear
2
V-930Nb
2.40 eV
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
1
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1
2
3
4
1
2
3
4
1
2
dark
dark
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
-2
-1
0
1
2
light
light
0.5
1
Log|I| (nA)
Tunneling Current (nA)
1.5
0.5
0
-0.5
-1
0
-0.5
-1
-1.5
-1.5
-2
-3
-2
-1
0
1
2
3
-3
4
Sample voltage,Va (V)
Figure 5.40
-2
-1
0
Sample voltage,Va (V)
Surface photo-effect matched with the following parameter variations:
∆ψ = 0.014 eV; ∆β = 9.6 × 10-4 C/V; ∆χ = 0.40 eV.
177
15
V-930Nb
10
H-850
H-1000
5
0
1.3
1.77
1.9
2.05
2.14
2.4
2.82
2.9
3.8
2.4
2.82
2.9
3.8
-5
-10
-15
a
Photon Energy (eV)
10
V-930Nb
H-850
H-1000
5
0
1.3
1.77
1.9
2.05
2.14
-5
-10
-15
b
Photon Energy (eV)
Figure 5.41
(a) The photo-induced change in surface charge determined by modeling the
observed changes in the tunneling spectra. (b) The responsivity of the surfaces determined by
normalizing for the spectral response of the experimental optics.
178
Except at energies 1.77, 1.90, and 3.8 eV light, most observed photo-induced responses
may be described as a decrease in tunneling conductance. Photo-induced effects were observed
at only three different energies for the H–850 sample and the V–930Nb sample. The H–1000
sample showed photo-effects at all incident energies, similar to the broad band bulk absorption
observed
from
the
ellipsometry
measurements.
Only
the
V–930Nb sample showed an increase in conductance at 3.8 eV light.
The qualitative behavior of the spectra for V–930Nb is similar to H–850 in terms of the
average width of their conductance wells.
This implies a similarity in surface electronic
properties so that the electronic contribution to terrace images acquired from V–930Nb might
well represent the electronic contribution to terrace images acquired from H–850.
The modeled spectra were seen to fit the experimental data rather closely within
reasonable experimental error (see discussion in section 2.1.3). In some cases the fit was nearly
exact. All spectra appear to be interpretable in terms of a change in surface charge. The
behavior, however, was not always as predicted in Chapter 3. Recall that modifying surface
charge was predicted to modify the width of the conductance well, with the shift at forward bias
being stronger than that at reverse bias. In at least one case (Figure 5.29) the opposite was
observed. In another case a strong effect was observed at forward bias only (Figure 5.35). The
model was observed to systematically underestimate the current at larger forward biases for
samples H–850 and V–930Nb; the possible reasons for this were discussed in Chapter 3.
On the other hand, the model systematically overestimated the current at larger forward
biases for H–1000 which is believed to be due to the effects of band gap states. The model does
not treat gap states directly. A particular demonstration of this fact was observed for the H–1000
spectra which showed possible additional photo-induced effects. The semilog plots of Figures
179
5.32 and 5.33 contain a change in the slope of the current near -1.5 eV giving an apparent
conductance tail which decreases upon illumination with 2.4 eV and 2.14 eV light.
The values of ∆β reported along with the figures are small or zero, consistent with the
fact that surface states are not anticipated to depend on the presence, or energy, of incident light.
The values of ∆χ are also reported with the figures. The latter are given mainly for the purpose
of demonstrating the possible limitations of the model.
It should be noted that ∆χ was
systematically larger than ∆ψ, often by more than an order of magnitude. Also, the absolute
values of χ that were necessary for a “good fit” at reverse bias deviated from the tabulated values
by up to ∼ 50% of the latter. This does not negate the validity of the conclusions derived from
the results since they are based only on the values of ∆ψ which are believed to accurately
describe the variations of the surface charge.
The results are summarized in the bar plots of Figure 5.41, where Figure 5.41b is a closer
representation of the spectroscopic response of the material to incident light. It is clear that the
strongest effect was observed on H–1000 and occurs near 1.77 eV light where the responsivity is
more than an order of magnitude larger than at any other energy. This absorption was also
observed for H–850 but with much lower response. The second strongest responses from H–
1000 occurred near 1.9 eV and 2.4 eV, where the effects are due to decreasing and increasing
electronic charge, respectively. All three samples showed similar responsivity at 2.9 eV, where
an increase was observed from H–850 to H–1000, suggesting a correlation to carrier density or
trap density. Finally, the weakest response was observed to occur at 2.82 eV, and only for H–
1000.
This energy does not appear in Table 1.4; it is included because absorption was
consistently observed at this energy.
180
5.3.3 Summary of Observed Optical Responsivity
Evidence of optically active on the terrace sites of reduced SrTiO3 was observed
spectroscopically as increases and decreases in the width of the conductance well in the
tunneling spectra. The largest effect, after normalizing for the contribution of the experimental
optics, occured for absorption of 1.77 eV monochromatic light. The smallest effect occurs near
2.8 eV monochromatic light. Broad absorption effects were observed over the entire energy
range of interest for the H-1000 sample. The H-700 sample did not display absorption effects at
any of the energies, and H-850 showed absorption at only three energies. The effect may be
described as an increase in surface electron density or a decrease in surface electron density
where the average order of magnitude in responsivity was 1011 electrons/mW.
REFERENCES
1.
N. Ikemiya, A. Kitamura, and S. Hara J. of Crystal Growth 160 (1996) 104
2.
Y. Liang and D. A. Bonnell Surface Science Lett. 285 (1993) L510; Y. Liang and
D. Bonnell J. Am. Ceram. Soc. 78 [10] (1995) 2633
3.
M. Naito and H. Sato Physica C 229 (1994) 1
4.
B. Stäuble-Pümpin, B. Ilge, V. C. Matijasevic, P. M. L. O. Scholte, A. J. Steinfort
and F. Tuinstra Surface Science 369 (1996) 313
5.
J. C. Chen Introduction to Scanning Tunneling Microscopy Oxford University
Press, Oxford (1993)
6.
C. B. Duke Tunneling In Solids Academic Press, New York 1969
7.
B. E. A. Saleh and M. C. Teich Fundamentals of Photonics John Wiley and Sons,
Inc., New York (1991)
181
Chapter 6: Discussion and Conclusions
6.1
DISCUSSION OF RESULTS
6.1.1 Photo-assisted tunneling microscopy and spectroscopy
One of the goals of this thesis was to explore the feasibility of enhancing the detectability of
defect-induced band gap states by activating charge-transfer transitions near the surface. By
modifying local surface charge densities, tunneling spectra are expected to show characteristic
variations largely in the form of an increase or decrease in the width of the conductance well.
Since images obtained by STM are essentially a map of the surface electronic properties (or
electronic structure) modulated by the surface topography, one might expect photo-induced
effects to enable spatially resolved imaging of surface defects. In the event that several different
transitions are optically active, which may be associated with different types of defects, the
prospect of selectively imaging defect structures, to facilitate distinction between the local
environments responsible for the resulting defect states, is intriguing. This would be a valuable
metrological tool for the study of photocatalytic processes.
Evidence of optical activity was not observed in any of the STM images acquired. This
is despite the fact that the tunneling spectra showed strong photo-induced effects on three out of
the four samples studied. One implication of this observation is that STM images are not
influenced by variations in electronic properties and thus show only topography similar to AFM
imaging. That this is an unacceptable conclusion has been demonstrated by the difference in step
edge structure observed by AFM (i.e., Figure 2.7) as compared to that observed by STM (i.e.,
Figure 5.5). The latter contains additional intensity on top of step ledges and decreases in
intensity near step corners. These regions are where tunneling spectroscopy has indicated a
182
variation in electronic properties, such as the existence of a local density of states associated with
the step edge structure.
It is possible that the “features” associated with the photo-effects simply have not been
resolved. The lateral resolution of the STM is determined by the properties of both electrodes at
the junction. The images invariably required sharp tips in order to resolve the details of step
edges and terrace clusters. Blunt tips (necessary for reproducibility in measured spectra) do not
always generate coherent images. On the other hand, the large dielectric constant of SrTiO3,
combined with the condition of low carrier density, results in a relatively large depletion width,
usually on the order of several hundred angstroms or larger. Depending on the set point
conditions, this large depletion width can “spread out” the contribution of the electronic
properties to an image [1]. The half-illuminated images (such as Figure 5.18), however, should
still contain at least a uniform displacement due to the extreme sensitivity of the STM to changes
in the tunneling barrier function. The images and the spectra are thus in disagreement regarding
the electronic properties of the (001) surface of SrTiO3. This apparent paradox remains presently
unresolved, a situation that represents a unique opportunity for continued progress.
6.1.2 Surface structures and morphologies
The surface structures and morphologies of oxygen deficient SrTiO3 were studied in order to
identify a correlation between surface structure and observed optical activity. In the early stages
of reduction, STM images of the (001) surface appear similar to the AFM images of fully
oxidized crystals. This implies that the electronic properties of this surface are uniform within
the (lateral) resolution limit of the tunneling microscope. Mild thermal treatment (such as short
time anneals or annealing below the “freeze-in” temperature for oxygen vacancies) apparently
does
not
result
in
significant
183
surface
restructuring
and/or
that oxygen vacancies are distributed randomly such that sharp 1 × 1 LEED patterns
are obtainable.
Upon further reduction the surface morphology undergoes extensive restructuring.
Roughening of the surface is likely to be due mostly to the formation of holes. This is supported
by the fact that the calculated rms roughness was similar for all of the terraces imaged from
different samples. It has been suggested that hole formation is evidence of pre-existing line
defects (such as dislocations) or microcracks which may form during thermal processing or
mechanical polishing.
The local stress fields near such defects can cause preferential
sublimation during high temperature vacuum annealing [2]. This seems reasonable since hole
formation was not observed for high temperature air annealing, where the large abundance of
oxygen is likely to accelerate defect repair. In vacuum annealing, the formation of holes is
probably driven by the reduction in strain energy associated with mechanically or thermally
induced defects.
Wandering step edges with local ordering of clusters on terraces are the manifestations of
this system attempting to lower the excess free energy of the surface, which for ionic solids has a
large electrostatic component. These changes have been documented in terms of long range
defect ordering or a demixing phenomena giving rise to the formation of new surface phases
[3,4]. The LEED patterns observed for vacuum-reduced samples suggest a tendency to form a
2 × 2R45o superstructure.
It is significant that higher temperature annealing sometimes
returned the structure to 1 × 1 ordering. Although not shown in Chapter 5, it should be noted that
upon one occasion a two domain 2 × 1 superstructure was also observed after UHV heating at
900 °C for a few minutes.
A perusal of the literature shows a rich variety of annealing
temperatures and annealing times. (Unfortunately, the sample cooling rates were not carefully
184
documented.) There is also a corresponding richness in observed superstructures. These facts
imply that the crystallographic structure (and perhaps also morphology) is ultimately controlled
by kinetic effects and supports the view that mobile species such as oxygen vacancies are the
mechanism by which the surface structure is obtained (see discussion below). This also implies
that there should be a direct correlation between the structure and the electrical/optical properties
of the surface that depend on the oxygen vacancy density.
A look at the structure/property relationship of the niobium-doped sample compared to
the vacuum-reduced samples draws implications regarding the roll of oxygen vacancies versus
the doping of electrons into the conduction band on determining the electronic properties and
perhaps surface structure. The apparent terrace morphology of the vicinal surface appears to
have a greater dependence on the type (and density) of defect than on the density of carriers in
the conduction band. Recall, however, that the Hall measurements show the free carrier density
in V–930Nb to be larger than those for the H series, but still of the same order of magnitude.
The vacuum anneal for this sample was identical to that for V–930; however, the LEED pattern
indicated that it formed a
2 × 2R45o superstructure similar to V–1100. On the other hand, no
row-type structures were observed in the images of V–930Nb. The absence of the latter might
very well be correlated with the absence of optical responsivity near 1.77 eV as was observed for
the reduced samples in Figure 5.41b.
The important point is that it is now more difficult to assign the observed superstructure
to the ordering of oxygen vacancies when the latter is not expected to be present in large
abundance in the niobium-doped crystal. Furthermore, the observed LEED pattern for V–1100
suggests a surface described by C4 symmetry, while the apparent surface morphology suggests
C2 symmetry. These observations indicate that the “structure” observed by a LEED probe is not
185
the same as the “structure” observed by a STM probe.
This is not unreasonable since a
diffraction probe detects order in the lattice on the scale of the coherence width of the incident
electron beam. The STM, on the other hand, generally has a more convoluted signal. In the
event that tip morphology is not influencing the scans, and the surface is relatively flat, the STM
probe detects order in the electronic behavior or electronic structure. Therefore, even in the
event that the surface has a well-defined crystallographic order in its top-most layer, a different
order in the local environment of perhaps the subsurface plane means that the surface cations
have a different order in their local Madelung potentials. The STM will be sensitive to the
variations in the local surface potentials. Therefore, the difference in apparent morphologies
between say V–930Nb and V–1100 must be correlated to the difference in electronic properties
as demonstrated by the difference in their local surface optical responsivities.
6.1.3 Defect-induced electronic properties
The correlation between surface structure and surface photo-activity is inferred from
Figure 5.41 as an overall increase in absorption across the experimental spectral window with
increasing reduction.
A characteristic decrease is observed in the spectral region
(i.e., 2.82 eV) corresponding to the “color” of the crystal. These properties should be expected
since a similar broad band increase in absorption has been observed by ellipsometry as a bulk
response to increased reduction, and transmission measurements similarly showed an increase
near 2.82 eV. The bulk measurements are interpreted as an increase in the density of occupied
band gap states which supply electronic transitions to unoccupied conduction band states.
Reduction also varies the bulk electronic structure in terms of variations in the band gap energy
and a decrease in the density of states at the top of the valence band. The variation in the band
186
gap, however, is not expected to have a strong influence on the characteristics of the observed
tunneling spectra and thus is not relevant to the optical responsivity observed by PATS.
All previous STM studies on oxygen deficient strontium titanate report annealing the
crystal in a reducing environment until it became “dark blue” or “black,” at which point it was
assumed that the crystal was “sufficiently conductive” for tunneling to be established. This work
is the first STM study to report tunneling images and spectra obtained from a reduced crystal that
remained colorless and transparent. Therefore, the series of spectra in Figure 5.23 represent a
unique look at the changing electronic behavior of the surface from the early stages of reduction
to strong reduction. The results indicate an increasing surface potential which, for a n-type
semiconducting oxide in depletion, is associated with an increasing surface charge density. This
must follow since it is counterintuitive (and contrary to the resistivity measurements) that the
“clear” sample is more conductive than the “black” sample. The increase in surface charge
density manifests also as an increase in surface optical responsivity.
The surface is most responsive at 1.77 eV incident light. The number of electrons excited
at this energy grows with increasing reduction and a similar effect is not observed for the
niobium-doped crystal.
This suggests a correlation with the oxygen vacancy defect.
The
increase in the response observed indicates either an increase in the density of the defect or a
growth in the size of domains in which these defects are ordered. Since atomic scale resolution
had not been achieved in these experiments and (more importantly) photo-induced effects were
not imaged, it is presently not possible to assign a causation relation between the observed
behavior and the structure of the defect sites. Some comments can be made, however, regarding
the nature of the defect at this energy.
187
Firstly, the mechanism of the charge transfer can be deduced by considering the fact that
the tunneling measurement can only detect photo-responses that result in redistribution of charge
between the surface and the bulk. An increase in tunneling conductance corresponds to a
decreasing surface potential as a result of electron excitation from surface trap states to the
conduction band where they can be swept into the bulk by the built-in electric field.
Alternatively, electrons from the valence band might be excited to bulk trap states and the
surface charge decreased by recombination of holes with surface electrons (see Figure 1.7a).
The former mechanism is more likely since the low mobility of holes make it more probable that
the latter mechanism will be suppressed by bulk recombination effects. It is therefore believed
that the mechanism is described by the depopulation of surface trap states resulting in the
negative value of the surface responsivity in Figure 5.41b. Consequently, the mechanisms
associated with the broad absorption at higher energies must involve the transfer of electrons to
the surface to increase the surface potential and thus give positive values to the surface
responsivity. A plausible mechanism is illustrated in Figure 1.7b, where a bulk transition
involves excitation from singly ionized oxygen vacancies, and the electrons charging the surface
are supplied by the metal STM tip.
Secondly, if the defect state is described by non-itinerant wave functions, then it will not
exhibit dispersion when represented in a band diagram — i.e., it will be represented as a straight
line across the Broullouin zone. It might therefore be expected that excitations are possible at all
energies greater than the threshold energy to induce defect-to-conduction band (CB) transitions.
Figure 5.41, however, does not suggest the same type of transition behavior for all energies
above 1.77 eV so that the observed behavior might be attributed to the narrow width of the
lowest CB. The strong response suggests that the transition is symmetry-allowed, which requires
188
that the defect states have opposite parity to the lower CB states. If transitions are allowed
between lower and upper CB states (which is believed to account for absorption at 2.4 eV [5-7]),
then these states must also have opposite parity. It is thus clear that the defect states must have
the same parity as the upper conduction band states, the latter of which are composed of Ti–4p
and O–3p orbitals and thus have odd parity. It is suggested that the defect state may also be
described by functions of odd parity.
Thirdly, there is a clear correlation between the apparent surface ordering and the
increase in band gap absorption at 1.77 eV.
The least reduced sample showed sufficient
conductivity that is clearly associated with its oxygen vacancy density; however, the absence of
defect ordering resulted in a 1 × 1 surface structure and no observed optical responsivity in the
tunneling spectra. It follows that the surface defect state is not merely associated with the
oxygen vacancy, but rather with the ordered arrangement of oxygen vacancies. Moreover, the
ordering of oxygen vacancies responsible for the state may not be confined to the surface.
(Recall that the 1.77 eV absorption band was also observed by transmission studies [5,6].) The
electrostatic fields of bulk vacancies may modify the surface potentials and give rise to an
ordering observed by STM that is distinct from that observed by LEED. Note that a similar
argument was recently used to explain the atomically resolved long range order observed on
oxygen deficient SrTiO3 by STM [8].
Lastly, the determined responsivity suggests a relatively low occupancy for the surface
state. This may actually be a consequence of low incidence flux upon the junction. For
example, the calculated change in surface charge for the 1.77 eV peak can be used
to approximate the cross section for transitions at this energy.
The value of ∆Qss in
Figure 5.41a at 1.77 eV corresponds to a change in surface electron density of 0.00076 electrons
189
per unit mesh, where the unit mesh is defined by the area 0.39 nm × 0.39 nm. The average flux
from Figure 2.2 was of order 1013 photons per second. Considering the active area of the
photodiode
and
converting
to
nanometer
units,
this
corresponds
to
8.3 × 10-2 photons per second per square nanometer, or 0.013 photons per second per unit mesh.
Assuming an absorption cross section of unity, and an experimental measuring time of ∼ 0.1 s
(see section 2.1.2), gives an expected response of 0.0013 electrons per unit mesh. Therefore, the
actual cross section for excitation is closer to 60%, close enough to unity to describe a surface
with strong optical responsivity. This implies that the use of solid state lasers, which can supply
up to 100 mW or more in power, might induce a greater response and even drive the surface to
flat band conditions by completely depopulating the gap state.
This can lead to a direct
experimental quantification of the density of states associated with the oxygen vacancy defect.
Considering the manner in which the experiment was performed, the coarse view of the
results is not surprising.
Therefore, it is difficult to assess how the resolution of the
monochromator (0.014 eV) may have influenced the results. It is nevertheless interesting that the
strongest optical effect at the surface occurred at the same energy as oxygen vacancy-related
absorption observed in the bulk [5,6]. This suggests that the mere creation of the surface does
not strongly modify the surface electronic structure from that of the bulk, as predicted by recent
theoretical models [9].
One final speculation is suggested when considering the difference between the
responsivity of the niobium-doped sample and the reduced samples. Figure 1.6 illustrates a
mechanism by which surface charge may be reduced when greater than band gap energy is
incident upon the surface. Only the niobium-doped sample showed a decrease in surface charge
consistent with this type of mechanism. Figure 4.3 clearly shows that the majority of the light is
190
absorbed within 50 nm of the surface plane for energies well above the band gap in all of the
hydrogen-reduced samples. This strong absorption is due primarily to interband transitions
between the valence band and the conduction band. On the other hand, midgap states are
believed to act as efficient recombination centers. These excitations may not be detectable in the
reduced crystals due to efficient recombination via the bulk oxygen vacancy states within the
depletion region. For the niobium-doped crystal, however, the impurity places a hydrogenic
(i.e., shallow) state near the conduction band edge. These are not efficient recombination centers
and thus charge redistribution as illustrated in Figure 1.6 may explain the observed decrease in
surface potential at 3.8 eV. The origin of the trapped surface charge on V–930Nb is not
expected to be the same as that on the reduced crystals. Other than step sites, which were not
probed by the PATS study, the results do not offer an origin for acceptor type defect states on the
terraces of the niobium doped crystal.
6.1.4 Conclusions
This experimental work has demonstrated the successful application of the recently developed
technique of photo-assisted tunneling spectroscopy (PATS) to the study of electron-doped
SrTiO3 (001) surfaces. The technique combined with theoretical modeling has been shown to
characterize surface charge transfer transitions in terms of threshold energy and absorption
mechanism. The related technique of photo-assisted tunneling microscopy (PATM) did not
identify the origins of surface optical responsivity in STM images. This is an unexpected result
given the close relationship between tunneling spectra and tunneling images which identifies a
potential opportunity for future progress.
The quantification of surface properties, such as the magnitude of the change in the
surface charge, relies on the accuracy of the theoretical model used to match the experimental
191
data. The model developed in this thesis produced tunneling spectra that closely matched the
experimental data using values for the electron affinity which often deviated from the
tabulations, highlighting the weakness of the theory under reverse bias conditions.
Under
forward bias conditions, however, the model is less sensitive to the absolute value of the electron
affinity. The behavior of the surface charge is directly modeled as changes in the surface
potential
which
is
treated
accurately
under
forward
bias conditions.
Strong absorption at 1.77 eV is ascribed to the association of oxygen vacancies as
measured on terrace sites under heavy reducing conditions. This association may also determine
the row-type surface structure observed by STM and it is likely to involve oxygen vacancy
ordering in the depletion region. The scale of the ordering observed in this work is larger than
that previously reported. Given the sensitivity of the surface structure to thermal history and
quality of the crystal, the reason for this difference would be no more than speculation. It is,
however, clear that the observed structure and the observed optical responsivity are correlated.
Evidence of optically active step sites was not observed; however, conventional tunneling
spectroscopy indicate the presence of an occupied localized density of states at step sites. The
existence of optically active step sites is therefore not ruled out.
REFERENCES
1.
D. A. Bonnell, I. Solomon, G. S. Rohrer and C. Warner Acta. Metall. Mater. 40
Suppl. (1992) S161
2.
B. Stäuble-Pümpin, B. Ilge, V. C. Matijasevic, P. M. L. O. Scholte, A. J. Steinfort
and F. Tuinstra Surface Science 369 (1996) 313
192
3.
Y. Liang and D. A. Bonnell Surface Science Lett. 285 (1993) L510; Y. Liang and
D. Bonnell J. Am. Ceram. Soc. 78 [10] (1995) 2633
4.
K. Szot, W. Speier, J. Herion and Ch. Freiburg Appl. Phys. A. 64 (1997) 55; K.
Szot and W. Speier (unpublished)
5.
H. Yamada and G. R. Miller J. Solid State Chem. 6 (1973) 169
6.
C. Lee, J. Destry and J. L. Brebner Phys. Rev. B 11 [6] (1975) 2299
7.
R. L. Wild, E. M. Rockar and J. C. Smith Phys. Rev. B 8 [8] (1973) 3828
8.
Q. D. Jiang and J. Zegenhagen Surface Science 425 (1999) 343
9.
J. Goniakowski and C. Noguera Surface Science 365 (1996) L657
193
Chapter 7: Summary of Dissertation
The use of the (001) surface of strontium titanate (SrTiO3) as a catalyst, as a substrate for
epitaxial growth of oxide superconductors, as well as a photoelectrode for solar energy
conversion, demonstrates the versatility of the perovskite oxides in widespread commercial
applications today.
A deeper understanding of the microscopic (or nanoscopic)
structure/property relations is therefore vital to several industries
Many of the important applications of SrTiO3 are possible due to occupied defect states
energetically located within the band gap.
Most bulk and surface sensitive probes have
identified this state; however, the details of the structural origin are often left to first principles
calculations or even to speculation. In particular, agreement has not yet been reached regarding
the physical nature of the oxygen vacancy-related band gap states.
The objective of this thesis work was to develop a technique, with theoretical
understanding, utilizing the combined methods of tunneling spectroscopy and optical
spectroscopy, to enable the identification and characterization of optically active surface defect
structure. This technique could then be applied to study the (001) vicinal surface of oxygen
deficient SrTiO3 in order to characterize the physical origins of the observed optically active
deep level defects.
The technique successfully identifies surface and subsurface charge transfer transitions
giving both the energy of an optically active surface state and the mechanism by which the state
is populated or depopulated. The mechanism is modeled as a change in the surface charge. A
model has been developed to simulate tunneling spectra on wide band gap (i.e., large dielectric
constant) oxides which has been shown to accurately match the experimental behavior at forward
194
biases. The theory-experiment fit at reverse bias was made possible only by assuming large
deviations in the electron affinity from its tabulated values, therefore identifying the need for
continued refinements in the model.
It has been determined that reduction annealing introduces deep level band gap states in
SrTiO3 that are optically active and are correlated with oxygen vacancy association on the terrace
sites of the (001) surface.
This work represents the first successful application of spectroscopic PATS to the study
of a wide band gap oxide material. The results demonstrate the strong potential for its use as a
metrological tool to study adsorption modes and other surface mediated processes that occur via
surface or subsurface gap states.
195
APPENDIX A: FRANCK-CONDON PRINCIPLE AND THE SPECTROSCOPIC
RESOLUTION
The strength of optically induced charge transitions in solids, and the characteristics of observed
absorption bands, in principle may be limited by the properties of the crystal matrix. The most
fundamental restriction on the strength of any transition between some initial state, ψi, to some
final state, ψf, may be derived from a consideration of the transition rate. This requires the
evaluation of an appropriate transition matrix element such as [1]
+∞
M fi =
∫ψ
∗
f
(τ)g(τ)ψ i (τ )dτ ,
[A.1]
−∞
where the form of the operator g( τ) depends on the nature of the particular transition process,
and the volume element τ includes both electronic and nuclear coordinates. A finite result is
obtained from [A.1] only when the integrand is an even function. Therefore, for a given
transition process where the parity of g( τ) is fixed, a selection rule is established regarding the
parities of ψi and ψf.
The matrix element for a radiative transition, when the wavelength of the radiation is
r
large relative to the size of the absorbing atom/molecule, is given by [A.1] where g( τ) = µ e [2].
The latter is the electric dipole operator — an odd function. The integral may be separated based
on the electronic (dτe) and nuclear (dτn) coordinates such that [A.1] becomes [3]
+∞
+∞
r
e
M fi = ∫ θ φ µ eθ iφ idτ e dτ n = M fi ∫ φ∗f φ i dτn ,
∗
f
∗
f
−∞
[A.2a]
−∞
+∞
where
r
M = ∫ θ ∗f µe θi dτ e .
e
fi
−∞
196
[A.2b]
One can see from [A.2b] that the selection rule for a radiative transition requires θf and θi to have
opposite parities. The parities of the state functions in a solid are primarily determined by the
lattice symmetry. Therefore, a radiative transition is said to be symmetry-forbidden if the initial
and final states are described by functions of like parity. When such a transition is symmetryallowed, the magnitude of Mfi is determined by the second integral in [A.2a]. The latter is also
known as the Franck-Condon (F-C) factor, where φi and φf correspond to different vibrational
states of the system.
Figure A.1
Illustration of molecular energy as a function of internuclear distance R for two
different electronic states of a diatomic molecule [from reference 5].
197
The Franck-Condon principle is based on the fact that the characteristic time for an
electronic transition is several orders of magnitude shorter than the period associated with
nuclear vibrations [4]. The consequence of this fact is demonstrated in Figure A.1 which
illustrates transitions between some ground state and an excited state of a diatomic molecule.
(Note that rotational energy levels do not exist in a lattice; however, the molecular vibrational
levels are analogous to the normal modes of a lattice.) The transitions are shown as vertical lines
in accordance with a rapid absorption or fluorescence process. If the equilibrium geometry of
the initial and final states are the same, the F-C factor is maximized when the vibrational
momentum is conserved; otherwise the F-C factor will be maximized for a transition involving a
finite change in vibrational momentum (as exemplified in Figure A.1).
In a lattice, this
corresponds to the creation or annihilation of a phonon.
ECB
EVB
ED
ED
EA
EA
a
Figure A.2
b
a) Scheme to depict defect thermal ionization energies; b) scheme to depict the
same defect spectroscopic energies. The Gaussian distribution about each energy results from
phonon interaction in accordance with the Franck-Condon principle.
The usual scheme used to depict donor and acceptor ionization energies is shown in
Figure A.2a. These levels are assumed to describe thermal excitation processes. A scheme
198
suggested to be more appropriate to depict spectroscopic energies is shown in Figure A.2b,
which acknowledges the superposition of a F-C envelope when charge-transfer excitations are
accompanied by phonon interaction [6]. Based on this discussion, it may be appreciated that the
nature of optical absorption bands in oxides is determined by the density of initial and final states
that satisfy the electric dipole selection rule, as well as the allowed phonon spectrum which gives
rise to a F-C spectral distribution. The latter imposes a fundamental limit on the resolution of
spectroscopic energies determined experimentally. To the author’s knowledge, there are no
reported attempts to deconvolute the F-C envelope from existing absorption spectra for transition
metal oxides. It is therefore difficult to estimate the F-C limited spectral resolution for a given
material
such
as SrTiO3. It is interesting to note, however, that measured absorption bands associated with
defect-induced absorption in SrTiO3 can have a typical FWHM bandwidth of approximately
1.5eV.
REFERENCES
1.
J. P. Elliott and P. G. Dawber Symmetry In Physics Oxford University Press,
Oxford 1979
2.
A. T. Fromhold, Jr. Quantum Mechanics for Applied Physics and Engineering
Dover Publications, New York 1991
3.
L. S. Forsler in Concepts of Inorganic Photochemistry A. W. Adamson and
P. D. Fleischauer, Eds. Robert E. Krieger Publishing, Inc., Florida 1984
4.
K. Nassau The Physics and Chemistry of Color Wiley, New York 1983.
5.
R. Eisberg and R. Resnick Quantum Physics of Atoms, Molecules, Solids, Nuclei,
and Particles 2nd ed. Wiley, New York 1985
199
6.
P. A. Cox Transition Metal Oxides: An Introduction To Their Electronic Structure and
Properties Oxford University Press, Oxford 1995
200
APPENDIX B: SEMICONDUCTOR DEFECT STATISTICS
One of the problems central to statistical thermodynamics is the determination of the occupation
probability of a set of states with energy degeneracy Z in order to predict the observable
properties of a material. The fundamental postulate of statistical thermodynamics states that all
possible microstates for a closed and isolated assembly of N particles in a given macrostate are
equally probable. For the states in the conduction band of a metal or semiconductor, the number
of possibilities for the occupation of the ith level is assumed to be independent of the state of
occupancy of the (i+1) level; the occupancy of the conduction band for a given macrostate is thus
derived from the product
∞
Zi !
,
i − N i )!
∏ N !(Z
i=1
i
[B.1]
also known as the thermodynamic probability. The numerator gives the total number of ways to
permute the energy states of level i; the denominator gives the total number of ways to permute
the energy states that do not result in unique permutations, since the particles being considered
are assumed indistinguishable. Maximizing [B.1], subject to the constraints of conserved mass
and energy, gives the well-known Fermi function
Ni
1
= f(E i ) =
.
Ei − EF
exp (
Zi
kT) + 1
[B.2]
In single electron donor semiconductors, an expression similar to [B.1] is also written as
(2 )N !(NN −! N )!
N
×
D
D
D×
D
[B.3]
D×
where N D × is the density of occupied (and hence charge neutral) defect states. From [B.1] and
[B.3] one derives
201
N D×
ND
= f (E D ) =
1
2
exp(
1
E D −E F
kT
)+ 1
.
[B.4]
The pre-exponential factor in the denominator of [B.4] is associated with the factor 2
N
D
×
in [B.3]
and is a consequence of electron spin degeneracy.
When solving Poisson’s equation, the quantity of interest is actually N D + , the density of
ionized defect states. This may be obtained by a bit of manipulation of [B.4] as follows. By first
taking the inverse of [B.4] and adding -1 to both sides obtains
N D − ND ×
ND ×
=
1
 E − EF 
exp D
.
 kT 
2
Next, taking the inverse again and adding +1 to both sides obtains
N D× + N D+
N D+
= 1 + 2exp
The numerator in this expression is simply ND.
 EF − ED 
.
 kT 
Taking another inverse and expressing
the energy terms in a reduced form (see section 3.2.4), the desired expression may be written as
N D+
ND
=
1
.
1 + 2exp[uz − wD1,I ]
[B.5]
In double electron donor semiconductors, a complication arises from the fact that the
occupancies for the singly-ionized states and for the neutral states are mutually dependent upon
each other [1]. The problem now requires the derivation of a Fermi function that not only
determines how the existing states will be filled, but also which states exist to be filled. Figure
B.1 illustrates this point and serves as the working hypothesis of the following derivation.
202
ECB
ED2
ED1
a
Figure B.1
b
a) A defect-related state occupied by two electrons of opposite spin; b) the same
defect-related state with one electron removed to the conduction band. The second electron
requires considerably more energy to also make a transition to the conduction band.
Figure B.1 represents the conditions presented by the presence of oxygen vacancies in
oxides such as SrTiO3 or TiO2. It is assumed that these defect states can accept two electrons to
satisfy their valency. The first electron can be accepted as either spin up or spin down into a
state described by ionization energy ED1. The second electron can be accepted with only spin up
or
only
spin
down
into
a
state
described
by
ionization
energy
ED2 (< ED1). The total number of defect-associated electrons N are accounted for by the number
contained in the conduction band
∞
∑ N , the number contained in singly-ionized states N
i =1
i
the number contained in neutral states 2ND × , so that
∞
N = 2N D× + N D + + ∑ Ni .
i =1
Another conservation rule is
203
[B.6]
D+
, and
∞
U = 2N D× E D2 + ND + E D1 + ∑ Ni E i ,
[B.7]
i =1
and the total number of donor states, ND, consists of the sum of neutral, singly-ionized and
doubly-ionized states; i.e.,
N D = ND × + ND + + ND ++ .
[B.8]
Now, the total number of different ways to occupy ND states with a single electron is given by
(2 )N !(NN −! N )! .
N
+
D
D
D+
[B.9]
D+
D
The total number of different ways to occupy N D − ND + states with an electron pair is given by
(N − N )! .
!(N − N − N )!
D
N D×
D+
D+
D
[B.10]
D×
Therefore, the thermodynamic probability for a double-donor semiconductor is written as
D
+
(
(
)

ND − N D+ !

 ∞
Z i!
N D!


. [B.11]

∏
 N D + ! N D − ND + !  ND × ! ND − N D+ − N D× ! i =1 Ni !(Z i − Ni )!
( )
N
W= 2
(
)
)
Maximizing [B.11] subject to the constraints of [B.6] and [B.7] yields the desired distribution
functions. That is, one must solve ∂H ∂X = 0, where Xi are the variables N i , N D + , N D × , λ, and
i
µ, and H = ln W . Using Stirling’s formula, ln A!≈ A ln A − A , so that
d
dA
ln A!≈ ln A , one must
solve the five separate equations
∂H
∂H
∂H
∂H = ∂H = 0 .
∂Ni =
∂ND + =
∂ND × =
∂λ
∂µ
The last two equations just return the conservation rules [B.6] and [B.7]. The first equation gives
[B.2]. The second and third equations give
ND +
N D − ND ×
=
1
2
exp(
1
E D1 − E F
204
kT
)+1
[B.12]
and
ND ×
N D − ND +
=
[
exp 2(
1
E D2 −E F
kT
)]+ 1
,
[B.13]
respectively. Using [B.8], the LHS of both [B.12] and [B.13] may be re-written as
and
ND ×
N D ++ + ND ×
ND +
N D ++ + ND +
, respectively, so that taking the inverse of both gives
N D ++
ND +
=
1
 E − EF 
exp D1
 kT 
2
[B.14a]
and
N D ++
ND ×
= exp
 2E D 2 − 2E F 
.


kT
[B.14b]
Dividing [B.14b] by [B.14a], adding +1 and taking the inverse gives
f (E D2 ) =
ND ×
ND + + N D×
=
1
2exp (
2E D2 − E D1 −E F
kT
)+ 1
.
[B.15]
This is the occupation probability of neutral states. The occupation probability of singly-ionized
states is given by [B.12] and is re-written below as
f (E D1 ) =
N D+
ND ++ + ND +
=
1
2
1
exp(
E D1 − E F
kT
)+1
.
[B.16]
The re-casting of [B.12] and [B.13] in the respective forms [B.15] and [B.16] was done for the
convenience of application to Poisson’s equation, in which case the quantities of interest are N D +
and N D ++ . For convenience let E ′ = 2E D 2 − E D1 . Also let
fD1 =
N D+ +
N D − ND ×
=
2 exp(
1
E F − E D1
and
205
kT
)+1
[B.17]
fE ′ =
N D+
ND − N D ++
=
1
2
exp(
1
EF − E′
kT
)+ 1
,
[B.18]
where [B.17] and [B.18] are obtained in a straightforward manner from [B.15] and [B.16]. In a
similar way, it can be easily shown that
N D× =
1
ND + exp (E F −E ′ kT).
2
[B.19]
Substituting [B.19] into [B.17] gives a relation that, together with [B.18], provide two
independent equations with two unknowns — namely, N D + and N D ++ . The results of the
necessary algebraic manipulations give

1 − fD1

N D + = N DfE ′ 
 1 − fD1 + fE ′fD1 
[B.20]

fE ′ fD1

N D ++ = ND 
.
 1 − fD1 + fE ′fD1 
[B.21]
and
Equations [B.20] and [B.21] are used in Poisson’s equation for determining the relation between
band bending and space charge per unit surface area. In [3.33], the donor charge density is given
by
ρ D = 2eN D++ + eN D +
or
 f −f f 
ρ D = eND  E ′ E ′ D1  .
 1 − fD1 + fE ′fD1 
REFERENCES
1.
Dr. E. Spenke Elektronische Halbleiter Springer-Verlag, Berlin 1965
206
[B.22]
APPENDIX C:
MATHEMATICA CODE FOR MODELED TUNNELING SPECTRA
The following code was written to generate a simulation of the tunneling current-voltage
behavior of a n-type semiconducting material in depletion probed by a metal counter–electrode.
It will execute successfully on a PC or workstation running Mathematica version 2.2.2 or 3.0.
All modeling of experimental spectra in this thesis were executed on a PC running version 3.0 at
400 MHz. The calculations were completed within 10 minutes to 2 hours, depending on the
average magnitude of the tunneling current. Larger currents result in longer calculation times.
The output of this code gives: a) the total dynamic band bending function; b) the potential
distribution functions; c) the predicted experimental dynamic band bending function; d) a linear
current-voltage plot of the result; and e) a semi-log current-voltage plot of the result.
(* Output control *)
Off[General::spell1];
(* turn off similar spelling warning *)
Off[NIntegrate::precw];
(* turn off less than precision warning *)
Off[NIntegrate::slwcon];
(* turn off slow convergence warning *)
Off[NIntegrate::ncvb];
(* turn off non-convergence warning *)
Off[FindRoot::cvnwt];
(* turn off non-convergence warning *)
Off[FindRoot::precw];
(* turn off less than precision warning *)
Off[InterpolatingFunction::dmwarn];
(* turn off domain size warning *)
Off[Plot::plnr];
SetDirectory["user files:Asa:mathematica:c results"];
208
(* define physical constants *)
jeV = 1.60217733 * 10^-19;
(* joules to electron volts conversion factor *)
k = (1.380658 * 10^-23/jeV);
(* Boltzmann's constant; eV/K *)
hbar = (1.05457266 10^-34/jeV);
(* reduced Planck's constant; eV s *)
h = 2 Pi hbar;
(* Planck's constant; eV s *)
m0 = 9.1093897 * 10^-31/jeV;
e = 1.60217733 * 10^-19;
(* free electron mass; eV s2/m2*)
(* fundamental unit of charge; C *)
ep0 = 8.854187817 * 10^-12;
(* permittivity of free space; F/m = C/(V m) *)
(* define variables *)
temp = 300;
(* temperature; Kelvins *)
s = 9;
(* tunneling gap width; angstroms *)
r = 5;
(* radius of curvature of tip; angstroms *)
mceff = 12;
(* conduction band relative effective mass *)
mveff = 0.99;
(* valence band relative effective mass *)
mmeff = 0.99;
(* metal electrode relative effective mass *)
kappasc = 300;
(* semiconductor static dielectric constant *)
kappaschf = 5;
(* semiconductor high frequency dielectric constant *)
kappain = 1;
kappainhf = 1;
nd = 1.0 * 10^19;
(* insulator static dielectric constant *)
(* insulator high frequency dielectric constant *)
(* free carrier density; cm-3 *)
initBB = 0.30;
beta = 0.002;
(* initial surface potenial; eV *)
(* adjustable parameter for potential distribution *)
209
eGap = 3.2;
(* semiconductor band gap; eV *)
scfermi = 0;
(* semiconductor Fermi level; eV *)
wfM = 4.55;
(* tungsten metal work function; eV *)
eAff = 3.0;
emobil = 30;
(* semiconductor electron affinity; eV *)
(* semiconductor electron mobility; cm2/V s *)
numPoints = 100;
(* number of points in IV curve *)
vaMin = -4;
(* starting applied tip bias; V *)
vaMax = 4;
(* ending applied tip bias; V *)
eDC1 = 0.0012;
(* ionization energy of donor defect; eV *)
(* derive constants *)
ceffm := If[mceff>mmeff, mmeff, mceff];
(* eff. masses for integral limits *)
veffm := If[mveff>mmeff, mmeff, mveff];
barNc = 2 *
(* conduction band effective dos; m-3 *)
((mceff m0 k temp)/(2 Pi (hbar^2)))^1.5;
barNv = 2 *
(* valence band effective dos; m-3 *)
((mveff m0 k temp)/(2 Pi (hbar^2)))^1.5;
eConB := N[scfermi - ((3 (nd * 10^6))/Pi)^(2/3) * (* conduction band edge; eV *)
(h^2/(8 mceff m0))] /; nd >= (barNc * 10^-6);
eConB := N[scfermi - (k temp Log[(nd * 10^6)/barNc])] /; nd < (barNc * 10^-6);
wfSC = eAff + eConB;
(* semiconductor work function; eV *)
eValB = eConB - eGap;
(* valence band edge; eV *)
eIntB := N[0.5(eConB + eValB) +
210
(k temp Log[(mveff/mceff)^(3/4)])];
(* intrinsic Fermi level; eV *)
eD1 = eConB - eDC1;
(* energy of donor state; eV *)
debeye = Sqrt[(kappasc ep0 jeV k temp)/(e^2 (nd * 10^6))]; (* Debeye length; m *)
ub = (scfermi - eIntB)/(k temp);
hevi[t_] := If[t>0,1,0];
(* Heaviside function *)
(* derive variables *)
wdep[vs_] := Sqrt[(2 kappasc ep0 jeV Abs[bbFunc[vs]])/(nd * 10^6 e^2)];
(* depletion layer width *)
phicSC[z_,vs_,longE_] :=
N[(((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eConB - longE];
phivSC[z_,vs_,longE_] :=
N[-1*((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eValB - longE)];
phiVac[z_,vi_,longE_] := N[(((wfM - vi - longE) * (z/(s * 10^-10)))
+ ((wfSC - longE) * (1-(z/(s 10^-10))))
- (((0.4 e^2)/(8 Pi kappainhf jeV ep0)) * ((s 10^-10)/(z ((s 10^-10) - z)))))];
phiczSC[vs_,longE_] :=
N[((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eConB - longE)];
phivzSC[vs_,longE_] :=
N[-1*((((wdep[vs] - z)^2/wdep[vs]^2) * bbFunc[vs]) + eValB - longE)];
phizVac[vi_,longE_] := N[(((wfM - vi - longE) * (z/(s * 10^-10)))
+ ((wfSC - longE) * (1-(z/(s 10^-10))))
211
- (((0.4 e^2)/(8 Pi kappainhf jeV ep0)) * ((s 10^-10)/(z ((s 10^-10) - z)))))];
(* generate equilibrium band bending function -> bbFunc[] *)
w[p_,q_] := (p - q)/(k temp);
fD1 = (1 + 2 Exp[x-w[eD1,eIntB]])^-1;
donorint := NIntegrate[fD1, {x,ub,us}];
xval = (eValB-energy)/(k temp);
fdival[j_,eta_]:= Re[(j!^-1)*NIntegrate[(xval^j)/(Exp[xval-eta]+1),
{energy,-30,0}, WorkingPrecision->15, AccuracyGoal->10]];
xcond = (energy-eConB)/(k temp);
fdicond[j_,eta_]:= Re[(j!^-1)*NIntegrate[(xcond^j)/(Exp[xcond-eta]+1),
{energy,0,30}, WorkingPrecision->15, AccuracyGoal->10]];
elecintdeg := (((2 barNc)/(3 nd * 10^6))*
((fdicond[1.5,us-w[eConB,eIntB]])-(fdicond[1.5,ub-w[eConB,eIntB]])));
holeintdeg := (((2 barNv)/(3 nd * 10^6))*
((fdival[1.5,w[eValB,eIntB]-us])-(fdival[1.5,w[eValB,eIntB]-ub])));
elecintnondeg =
N[(barNc/(nd * 10^6)) * (Exp[us-w[eConB,eIntB]]-Exp[ub-w[eConB,eIntB]])];
holeintnondeg =
N[(barNv/(nd * 10^6)) * (Exp[w[eValB,eIntB]-us]-Exp[w[eValB,eIntB]-ub])];
elecint := elecintdeg /; nd >= (barNc * 10^-6);
elecint := elecintnondeg /; nd < (barNc * 10^-6);
holeint := holeintdeg /; nd >= (barNc * 10^-6);
212
holeint := holeintnondeg /; nd < (barNc * 10^-6);
tvs := N[initBB - (k temp) - (((e debeye^2 nd 10^6)/(kappasc ep0)) *
(-1 donorint - elecint + holeint))];
phibbs := N[(ub - us) * (k temp)];
bbCurve = Table[{-tvs,phibbs}, {us,-73,73,1.0}];
bbFunc = Interpolation[bbCurve,InterpolationOrder->1];
bbPlot = ListPlot[bbCurve, PlotJoined -> True, Frame -> True,
FrameLabel -> {" Sample voltage,Vs (V)",None}, AspectRatio -> 1,
PlotRange ->{{-vaMax,-vaMin},{(-eGap+initBB),(eGap+0.5)}},
AxesLabel -> {None, "Equilibrium Surface Potential (eV)"}];
Clear[bbCurve,us,tvs];
" = extrinsic Debeye length (meters)" debeye; " = reduced bulk potential" ub
" = equilibrium depletion width (meters)" wdep[0]
" = depletion width to Debeye length" wdep[0]/debeye
(* define limits for barrier integrals *)
zcbtemp[vs_,longE_] := NSolve[phiczSC[vs,longE] == 0,z];
deltazcb[vs_,longE_] := z /. zcbtemp[vs,longE][[1]];
zccb[vs_,longE_] := If[Re[deltazcb[vs,longE]] > 0,Re[deltazcb[vs,longE]],0];
zvbtemp[vs_,longE_] := NSolve[phivzSC[vs,longE] == 0,z];
deltazvb[vs_,longE_] := z /. zvbtemp[vs,longE][[1]];
zcvb[vs_,longE_] := If[Re[deltazvb[vs,longE]] > 0,Re[deltazvb[vs,longE]],0];
213
zvactemp[vi_,longE_] := NSolve[phizVac[vi,longE] == 0,z];
zsoltemp[vi_,longE_] := z /. zvactemp[vi,longE][[1]];
vsol[vi_,longE_] := If[Re[zsoltemp[vi,longE]]<0,2,1];
za[vi_,longE_] := z /. zvactemp[vi,longE][[vsol[vi,longE]]];
zb[vi_,longE_] := z /. zvactemp[vi,longE][[(vsol[vi,longE]+1)]];
(* define transmission factor integrals *)
phibarVac[z_,vi_,longE_] := If[0<phiVac[z,vi,longE], phiVac[z,vi,longE], 0];
phibarcSC[z_,vs_,longE_] := If[0<phicSC[z,vs,longE], phicSC[z,vs,longE], 0];
phibarvSC[z_,vs_,longE_] := If[0<phivSC[z,vs,longE], phivSC[z,vs,longE], 0];
etaVac[vi_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) *
NIntegrate[Sqrt[phibarVac[z,vi,longE]],{z,Re[za[vi,longE]],Re[zb[vi,longE]]},
WorkingPrecision -> 10, AccuracyGoal -> 8];
etacSC[vs_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) *
NIntegrate[Sqrt[phibarcSC[z,vs,longE]],{z,0,zccb[vs,longE]},
WorkingPrecision -> 10, AccuracyGoal -> 8];
etavSC[vs_,longE_] := (2 * Sqrt[(2 m0)/(hbar^2)]) *
NIntegrate[Sqrt[phibarvSC[z,vs,longE]],{z,0,zcvb[vs,longE]},
WorkingPrecision -> 10, AccuracyGoal -> 8];
214
(* define energy band integrals *)
dVac[vi_,longE_] := Exp[-1 etaVac[vi,longE]];
dcSC[vs_,longE_] := Exp[-1 etacSC[vs,longE]];
dvSC[vs_,longE_] := Exp[-1 etavSC[vs,longE]];
eValBs[va_] := If[bbFunc[(-1*vsFunc[va])] > 0,
bbFunc[(-1*vsFunc[va])] + eValB, eValB];
jCB[va_] := N[(4 Pi e m0 ceffm)/h^3] *
NIntegrate[hevi[+1 (totE - eConB)]*dVac[viFunc[va],w]
*dcSC[(-1*vsFunc[va]),w], {totE,0,-va},
{w,eConB,totE}, WorkingPrecision -> 10, AccuracyGoal -> 8];
215
jVB[va_] := N[-(4 Pi e m0 veffm)/h^3] *
NIntegrate[hevi[-1 (totE - eValBs[va])]*dVac[viFunc[va],w]
*dvSC[(-1*vsFunc[va]),w], {totE,eValBs[va],-va},
{w,eValBs[va],totE}, WorkingPrecision -> 10, AccuracyGoal -> 8];
(* define defect induced current *)
alpha = 3 * 10^-4;
surfE[va_] := N[(-1*vsFunc[va])/wdep[0]];
sbl[va_] := N[(((e^3 nd 10^6)/(8 Pi^2 ep0^3 kappasc kappaschf^2)) *
(bbFunc[(-1*vsFunc[va])] + (k temp)))^0.25]; (* barrier lowering *)
dI[va_] := N[1*(alpha * temp^1.5 * surfE[va] * emobil * mceff^1.5 *
(etaVac[viFunc[va],bbFunc[(-1*vsFunc[va])]]) *
Exp[(-1*(bbFunc[(-1*vsFunc[va])] + Abs[eConB] - sbl[va]))/(k temp)] *
Exp[(e/(k temp)) * (e/(4 Pi ep0 kappaschf))^0.5 * ((surfE[va])^0.5)])];
jDI[va_] := If[va>=0,dI[va],0];
(* generate potential distribution functions -> viFunc[] and vsFunc[] *)
vaFunc := N[((1 - (s 10^-10 kappain^-1 ep0^-1 beta))^-1) *
(vs - ((s 10^-10 kappain^-1 ep0^-1) *
(Sqrt[(2 e kappasc ep0 nd 10^6) * (initBB - (k temp) - vs)] 216
Sqrt[(2 e kappasc ep0 nd 10^6) * (initBB - (k temp))])))];
vss[va_] := FindRoot[vaFunc == va, {vs,10^1}];
vs[va_] := Re[vs /. vss[va]];
vi[va_] := va - vs[va];
viCurve = Table[{va,vi[va]}, {va,vaMin,vaMax,0.5}];
viFunc = Interpolation[viCurve,InterpolationOrder -> 3];
viPlot = Plot[viFunc[x], {x,vaMin,vaMax}, Frame -> True,
PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}},
FrameLabel -> {"Tip Bias, Va (V)",None}, AspectRatio ->1,
AxesLabel -> {None, "Insulator voltage, Vi (V)"}];
vsCurve = Table[{va,vs[va]}, {va,vaMin,vaMax,0.5}];
vsFunc = Interpolation[vsCurve,InterpolationOrder -> 3];
vsPlot = Plot[vsFunc[x], {x,vaMin,vaMax}, Frame -> True,
PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}},
FrameLabel -> {"Tip Bias, Va (V)",None}, AspectRatio ->1,
AxesLabel -> {None, "Sample voltage, Vs (V)"}];
Clear[vsCurve,viCurve];
Show[vsPlot,viPlot, PlotRange -> {{vaMin,vaMax},{vaMin,vaMax}},
AxesLabel -> {None,None},AspectRatio -> 1];
surfPot = Table[{-va,bbFunc[(-1*vsFunc[va])]}, {va,vaMin,vaMax,0.05}];
surfPotPlot = ListPlot[surfPot, PlotJoined -> True, Frame->True,
FrameLabel -> {"Sample Bias, Va (V)", None},
PlotRange -> {{-vaMax,-vaMin},{(-eGap + initBB),(eGap + 0.3)}},
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AxesLabel -> {None, "Surface Potential (eV)"},AspectRatio->1];
(* define total current density *)
current[va_] := (N[jCB[va]] + N[jVB[va]] + N[jDI[va]]) N[Pi] r^2 10^-20;
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(* generate iv curve *)
i[v_] := Re[current[v]];
ivCurve = Table[{(-1*v),i[v]}, {v,vaMin,vaMax,((vaMax - vaMin)/numPoints)}];
ivCurve >> sro2_22;
(* name of file to save results*)
Clear[ivCurve];
rawdata = << sro2_22;
linearspectra=Interpolation[rawdata];
vmin = (-vaMax); vmax = (-vaMin);
Plot[(linearspectra[x]*10^9),{x,vmin,vmax},PlotRange->{{vmin,vmax},{-10,+10}}];
Plot[Log[10,Abs[(linearspectra[x]*10^9)]],{x,vmin,vmax},AxesOrigin->{vmin,-1.6},
PlotRange->{{vmin,vmax},{-1.6,1}},Frame->True,AspectRatio -> 1];
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