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Für meine Eltern
List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I J. Isberg, M. Gabrysch, A. Tajani, and D. J. Twitchen, Transient current
electric field profiling of single crystal CVD diamond, Semiconductor
Science and Technology 21 (8), 1193–1195 (2006).
II J. Isberg, M. Gabrysch, A. Tajani, and D. J. Twitchen, High-field Electrical Transport in Single Crystal CVD Diamond Diodes, Advances in
Science and Technology 48, 73–76 (2006).
III M. Gabrysch, E. Marklund, J. Hajdu, D. J. Twitchen, J. Rudati,
A. M. Lindenberg, C. Caleman, R. W. Falcone, T. Tschentscher,
K. Moffat, P. H. Bucksbaum, J. Als-Nielsen, A. J. Nelson, D. P.
Siddons, P. J. Emma, P. Krejcik, H. Schlarb, J. Arthur, S. Brennan,
J. Hastings, and J. Isberg, Formation of secondary electron cascades
in single-crystalline plasma-deposited diamond upon exposure to
femtosecond x-ray pulses, Journal of Applied Physics 103 (6), 064909
(2008).
IV M. Gabrysch, S. Majdi, A. Hallén, M. Linnarsson, A. Schöner,
D. J. Twitchen, and J. Isberg, Compensation in boron-doped CVD
diamond, Physica Status Solidi (a) 205 (9), 2190-2194 (2008),
presented at the Diamond Workshop 2008, SBDD XIII, Hasselt
(Belgium).
V J. Isberg, S. Majdi, M. Gabrysch, I. Friel, and R. S. Balmer, A lateral
time-of-flight system for charge transport studies, Diamond & Related
Materials 18, 1163–1166 (2009).
VI J. Isberg, M. Gabrysch, S. Majdi, and D. J. Twitchen, Negative differential electron mobility and single valley transport in diamond, submitted to Nature Materials, April 2010.
VII M. Gabrysch, S. Majdi, D. J. Twitchen, and J. Isberg, Electron and hole
drift velocity in CVD diamond, submitted to Physical Review B, April
2010.
Reprints were made with permission from the publishers.
The author has contributed to the following papers which are not included in the
thesis.
VIII S. Majdi, M. Gabrysch, R. S. Balmer, D. J. Twitchen, and J. Isberg, Characterization by Internal Photoemission Spectroscopy of Single-Crystal
CVD Diamond Schottky Barrier Diodes, accepted for publication in
Journal of Electronic Materials, DOI: 10.1007/s11664-010-1255-8,
April 2010.
IX C. Caleman, C. Ortiz, E. Marklund, F. Bultmark, M. Gabrysch,
F. G. Parak, J. Hajdu, M. Klintenberg, and N. Tîmneanu, Radiation
damage in biological material: electronic properties and electron
impact ionization in urea, Europhysics Letters 85, 18005 (2009).
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Common allotropes of carbon . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Natural diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Synthetic diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Diamond properties and applications . . . . . . . . . . . . . . . . . . . . .
1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Diamond as a semiconductor material . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Semiconductor materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Electrical properties of semiconductors . . . . . . . . . . . . . . . . . . . .
2.2.1 Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Intrinsic carrier concentration . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Advantages of the semiconductor diamond . . . . . . . . . . . . . . . .
2.4 CVD diamond synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Doping diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Future diamond devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Principles of the time-of-flight technique . . . . . . . . . . . . . . . . . . . . . .
3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Mobility measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Low injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 High injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Data acquisition and evaluation . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Free charge carrier transport in diamond . . . . . . . . . . . . . . . . . . . . . .
4.1 Drift-diffusion equations from the BTE . . . . . . . . . . . . . . . . . . . .
4.1.1 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . .
4.1.2 Equilibrium distribution function for a Fermi gas . . . . . . .
4.1.3 Uniform electric field with RTA . . . . . . . . . . . . . . . . . . . . . . .
4.2 Charge transport in 1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Fundamental transport equations . . . . . . . . . . . . . . . . . . . .
4.2.2 Carrier generation by laser, low injection . . . . . . . . . . . . . . .
4.2.3 Carrier transit with homogenous space charge density
and trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Carrier diffusion during transit . . . . . . . . . . . . . . . . . . . . . . .
4.2.5 Carrier extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6 Full transit signal and fast processes . . . . . . . . . . . . . . . . . .
11
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12
13
14
17
17
18
18
19
20
22
24
25
26
27
29
29
30
30
33
34
37
37
37
38
39
42
42
44
45
47
47
48
4.3 Electrical field profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Carrier transit simulations in 2-D . . . . . . . . . . . . . . . . . . . . . . . . .
5 Electron cascades in diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Diamond as a detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Pair-creation from ionising radiation . . . . . . . . . . . . . . . . . . . . . .
5.3 Impact ionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Summary of results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1 Investigation of δ-doped structures . . . . . . . . . . . . . . . . . . . . . . .
9.2 High-voltage low-loss converters . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Time-resolved study of electron cascades . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Nomenclature and abbreviations
|
|X
magnitude of X
e
electron (as index)
[X ]
concentration of element X
electrical potential
(external) force
〈...〉
crystallographic direction
φ
F
∼
approximately
F
Fermi-Dirac distribution
α
ionisation rate
f
distribution function
αn/p
electron/hole ionisation rate
f0
equilibrium distr. function
a p /b p
hole ionisation coefficients
g
generation rate
β
correction factor for SCL case
ga
spin degeneracy factor
C
capacitance
ħ
reduced Planck’s constant
c
speed of light
h
Planck’s constant
∂
partial derivative
h
hole (as index)
D
diffusion constant
I
current
d
differential
i
intrinsic (as index)
d
sample thickness
j
current density
ε0
vacuum permittivity
k
Boltzmann’s constant
εr
relative permittivity
λ
wavelength
pc
E
ap
E
average pair-creation energy
μ
(drift) mobility
electric field
m∗
effective mass
applied electric field
∇
del operator
sc
E
space charge electric field
NA
acceptor doping concentration
E
energy
NC
effective DOS in conduc. band
EA
acceptor ionisation energy
ND
donor doping concentration
EC
E at bottom of conduc. band
NV
effective DOS in valence band
EF
Fermi level
n
electron density
E F
quasi-Fermi level
ni
intrinsic carrier concentration
EV
E at the top of valence band
p
momentum
Eg
bandgap energy
p
hole density
E kin
kinetic energy
Q
total charge
E pot
potential energy
q
elementary charge
E th
thermal energy
ρc
carrier concentration
E tot
total energy
ρ sc
space charge concentration
ê x
unity-vector in x-direction
r
recombination rate
9
10
σ
variance
V
voltage
τ
time-of-flight
vd
drift velocity
τf
relaxation time
v sat
saturation (drift) velocity
T
(absolute) temperature
v th
thermal velocity
U
(bias) voltage
Z
atomic number
1-D
one-dimensional
NDM
neg. differential mobility
AC
alternating current
RT
room temperature
BTE
Boltzmann transport eq.
RTA
relaxation time
C
carbon / diamond
CVD
chemical vapour deposition
SC
single-crystalline
DC
direct current
SCL
space charge limited
DOS
density-of-states
SIMS
Secondary Ion Mass
FEM
finite element method
FWHM
full width at half maximum
TCT
transient-current technique
GUI
graphical user interface
ToF
time-of-flight
HPHT
high-pressure high-temp.
UV
ultraviolet
IR
infrared
VUV
vacuum ultraviolet
MC
Monte Carlo
XFEL
X-ray free-electron laser
approximation
Spectrometry
1. Introduction
Diamonds have been known as gemstones for several thousand years and
were recognised by various early cultures for their religious or industrial
uses [1]. The word “diamond” has its origin in the Ancient Greek word “adámas / ἀδάμας” meaning invincible. Besides the property of being the hardest known natural material, diamond is mainly appreciated as a gemstone
because of its optical properties: the high refractive index and large colour
dispersion result in a unique brilliance.
1.1 Common allotropes of carbon
Carbon is the lightest Group IV element in the periodic table, having
a half-filled valence shell with the electronic configuration s 2 p 2 . In the
case of diamond these s- and p-states hybridise and form the extremely
strong tetrahedral sp 3-bonds. Together with the special three-dimensional
arrangement of the atoms in the lattice, the so-called diamond structure,
they make diamond so exceptionally hard and also lead to other amazing
intrinsic properties such as a high refractive index, extremely high thermal
conductivity, and a high melting point.
Besides diamond, pure carbon can also be formed as graphite, amorphous carbon, graphene, fullerenes (e.g. buckyballs, nanotubes, nanowires)
or lonsdaleite (hexagonal crystal lattice) just to mention some of the best
known.
Graphite is the most common form of pure carbon on Earth and in
contrast to diamond, each carbon shares one electron with two of its
neighbours, and two electrons with the third neighbour. The atoms all
bond in planes (two-dimensional hexagonal lattice) which are stacked on
top of each other resulting in quite weak forces between different planes.
Graphite is therefore a rather soft material and differs also considerably in
other physical properties from diamond.
At normal temperatures and pressures, graphite is thermodynamically
favoured, as can be seen from the phase diagram for carbon. The fact that
diamond exists at all is due to the very large activation barrier for conversion between the two. In the absence of an easy interconversion mechanism, a phase transition would require almost as much energy as destroying the entire lattice and rebuilding it. Because the barrier is too high, once
11
formed, diamond cannot reconvert to graphite. That is why diamond is said
to be metastable: it is kinetically stable, but not thermodynamically stable [2].
1.2 Natural diamond
Diamond can form naturally at depths greater than 150 km in the upper
mantle of the Earth [3]. Under conditions of extreme pressure and temperature it is the most stable form of carbon. Over a period of millions of years
carbonaceous deposits slowly crystallise into single crystal diamond gemstones. Together with magma the crystals are then brought up to the Earth’s
crust by kimberlite or lamproite volcanic eruptions. Since this transfer happens within just a few hours, the conversion to graphite does not occur and
it is possible to mine diamond in the volcanic pipes containing material
that was transported toward the surface by volcanic action, but was not
ejected before the volcanic activity ceased.
Natural diamond generally contains a significant degree of impurities
such as nitrogen or boron. In order to classify the purity of diamond the
following scheme is used [1]:
– Type Ia diamond contains nitrogen in fairly substantial amounts (in the
order of 0.1%). The majority of natural diamonds are of this type.
– Type Ib diamond also contains nitrogen but in dispersed substitutional
form. Almost all synthetic diamonds are of this type.
– Type IIa diamond is effectively free of nitrogen. Diamonds of this type
have enhanced optical and thermal properties but are rare in nature.
– Type IIb diamond is a very pure type which has p-type semiconducting
properties. It is extremely rare in nature and has (uncompensated) boron
acceptor impurities which result in a light blue colour.
1.3 Synthetic diamond
Diamond synthesis can be achieved through several routes. The two main
methods are high-pressure high-temperature synthesis (HPHT) and chemical vapour deposition (CVD).
HPHT diamond
Ever since the discovery that diamond was pure carbon in 1797, many attempts were made to convert inexpensive graphite into gemstones. However, it was not before the 1950s that this could be achieved in a reproducible and verifiable way [4–6]. Both Allmänna Svenska Elektriska Aktiebolaget (ASEA) in Sweden and General Electric in the United States developed
a technique to raise the pressure in the reaction chamber to more than
12
8 GPa and the temperature to above 2000 ℃ in order to reproduce the conditions under which natural diamond forms inside the Earth. (Without the
use of suitable catalysts conditions must be even more extreme.)
This catalytic high-pressure high-temperature method is the most widely
used technique for diamond synthesis today, because of its relatively low
cost. The amount of annually produced HPHT diamond exceeds the one of
mined natural diamond by approximately a factor of four.
Today, HPHT diamond is commonly used in many industrial applications such as cutting, drilling, thermal management etc. However, it invariably contains many crystal defects and impurities. The yellow colour indicates a relatively high nitrogen concentration.
CVD diamond
The second method, using chemical vapour deposition, was first applied in
the 1980s, and basically creates a carbon plasma on top of a substrate onto
which the carbon atoms deposit to form diamond [6].
In contrast to HPHT diamond, it is possible to grow CVD diamond under conditions of high purity resulting in fewer defects and impurities. In
the beginning, the deposition conditions have invariably resulted in polycrystalline material. Only since the last couple of years it is also possible to
grow thick free-standing plates of high-purity single-crystalline CVD (SCCVD) diamond by homoepitaxy [7]. The CVD process is discussed in more
detail in Sec. 2.4.
1.4 Diamond properties and applications
Diamond is an outstanding material, as Table 1.1 shows. It is the hardest
known material, has the lowest coefficient of thermal expansion, is chemically inert and wear-resistant, offers low friction, has the highest thermal
conductivity, is electrically insulating and optically transparent from the
ultra-violet (UV) to the far infra-red (IR) region (> 70% transmission for
λ > 1μm but 60% at ∼ 5μm). Because of these properties, diamond is already used in many diverse applications besides its appreciation as a gemstone. The most common are:
– Mechanical applications: abrasive and wear-resistant coatings for cutting tools such as drills, saws, knives, glass cutting and wire dies
– Optical applications: lenses, windows for high power lasers and diffractive optical elements
– Thermal applications: heat sinks for power transistors and
semiconductor laser arrays
– Detector applications: “solar blind” photodetectors, radiation-hard
and/or chemically inert detectors
13
Table 1.1: Some of the outstanding properties of diamond, taken from [8].
Property
Description
Extreme mechanical hardness
∼ 90 GPa
Highest bulk modulus
1.2 × 1012 N/m2
8.3 × 10−13 m2 /N
Lowest compressibility
Lowest thermal expansion coefficient at RT
a)
0.8 × 10−6 K−1
Highest thermal conductivity at RT
24 W/(cm K)
Good electrical insulator
R 1016 Ω cm
Semiconductor if doped
R 10–106 Ω cm
Wide bandgap
5.47 eV at RT
Broad optical transparency
from deep UV to far IR
Biologically compatible
nontoxic and tissue equivalent
Low electron affinity
even negative in some cases
Very low coefficient of friction
∼ 0.001 in water
High resistance to wear and chemical corrosion
a) RT stands for room temperature.
Given its many unique properties, it is clearly possible to imagine numerous other potential applications for diamond as an engineering material.
However, progress in implementing many such ideas has been constricted
by the comparative shortage of (natural) diamond. Most of the electrical
applications of diamond are just at their infancy because only for the last
couple of years have electronic grade SC-CVD diamond become available
for device design and development [7].
1.5 Outline of the thesis
This doctoral thesis deals with several aspects of the electronic properties
of single-crystalline CVD diamond, with a special focus on charge
transport. It is organised as follows: Chapter 2 is a recap of diamond as
a semiconductor material including the CVD process, diamond doping
and devices. Chapter 3 focuses on mobility measurements (both with low
and high injection) by using the time-of-flight (ToF) technique. Chapter 4
studies charge transport both analytically and using finite element
simulation tools. In the same context the electric field profiling method is
explained. Chapter 5 is about electron cascades in diamond initiated by
ionising radiation such as X-rays or α-particles. Besides the study of CVD
diamond as a detector material and model compound, focus is also put
on the investigation of important mechanisms such as pair-creation and
14
impact ionisation that can lead to avalanche breakdown in devices. The
last chapters include a summary of the papers the author has contributed
to (Chapter 6), a summary of results and their discussion (Chapter 7), the
conclusion (Chapter 8) and suggestions for future work (Chapter 9).
15
2. Diamond as a semiconductor
material
In this chapter some of the basic properties of semiconductor materials in
general and those of diamond in particular are discussed. The overview of
semiconductor materials and some of their electrical characteristics such
as energy bands, intrinsic carrier concentration and mobility is followed
by a compilation of the advantages of diamond over other semiconductor
materials. A description of the single-crystalline CVD diamond growth
technique is given before briefly discussing doping diamond and the
importance of having low compensation. The chapter is completed by an
overview of possible future diamond devices.
2.1 Semiconductor materials
Solid-state materials can be grouped into three classes: conductors, semiconductors, and insulators [9]. Conductors have high conductivities (i.e.
low resistances), typically on the order of 104 –106 S/cm; and insulators have
very low conductivities, ranging from 10−18 to 10−8 S/cm. The conductivity of semiconductors lies in between those of conductors and insulators
and is in general sensitive to temperature, illumination, magnetic field, and
minimal amounts of impurity atoms (∼ 1–100 ppm). It is this sensitivity in
conductivity which makes semiconductors so important for electronics.
Group II
III
IV
V
VI
B
C
N
O
Al
Si
P
S
Zn
Ga
Ge
As
Se
Cd
In
Sn
Sb
Te
Hg
Tl
Pb
Bi
Po
Metals
Non-metals
Figure 2.1: Semiconductor related
elements of the periodic table. Metals are shown in orange and nonmetals in yellow. Elements with intermediate colours show intermediate properties.
17
Figure 2.1 shows the part of the periodic table which is related to
semiconductors. The element semiconductors can be found in Group IV
and are composed of just a single element such as diamond (C), silicon (Si)
or germanium (Ge). If two or more elements from the periodic table are
combined, one talks about a compound semiconductor. For example, if the
Group III element gallium (Ga) and the Group V element arsenic (As) are
combined, the binary III–V compound gallium arsenide (GaAs) is formed.
Furthermore, there are also II–VI compounds such as zinc oxide (ZnO) and
IV–IV compounds such as silicon carbide (SiC). Ternary and quaternary
compounds are formed of three and four elements, respectively, but
involve relatively complex processes to prepare them in single-crystalline
form. A list of common semiconductors is given below:
element semiconductors:
C, Si, Ge
IV–IV semiconductors:
SiGe, SiC
III–V semiconductors:
GaAs, InSb, GaN, AlN, GaP, AlAs, InP, ...
II–VI semiconductors:
ZnO, CdS, CdSe, CdTe, ...
With the advent of semiconductor electronics in the 1950s, germanium
was the major semiconductor material, but silicon has virtually supplanted
it since the 1960s because of its superior properties at room temperature
(RT) and lower costs. Nowadays, silicon is one of the most studied elements
in the periodic table. Device-grade silicon costs much less than any other
semiconductor material and silicon technology is by far the most advanced
among all semiconductor technologies [9]. However, besides these
advantages, there are also major drawbacks of silicon-based electronics.
This is the motivation for establishing alternative semiconductor materials
such as gallium arsenide, silicon carbide, gallium nitride or – diamond, the
object of interest of this thesis.
2.2 Electrical properties of semiconductors
2.2.1 Energy bands
The electrical properties of solid state materials can be analysed by looking at their band structure [9, 10]. Energy bands form as a quantum mechanical consequence (Pauli’s exclusion principle) when isolated atoms are
brought together forming a crystal. One distinguishes between the valence
band, which consists of the electrons forming the chemical bonds, and the
conduction band, which consists of electrons of higher energies which can
move freely across the crystal. Electronic conduction can only take place
within these bands, i.e. at least one band should be partially populated by
electrons.
18
In the case of conductors/metals, electrons are free to move with only
a small electrical field applied since there exist many unoccupied energy
states close to the occupied ones at all temperatures. This is because the
conduction band is either partially filled or overlaps the valence band.
The case for semiconductors and insulators is different. At zero absolute temperature, electrons can only occupy the lowest energy states. This
means that all states in the lower band are occupied and all states in the
upper band are unoccupied and thus no conduction can take place.
The bottom of the conduction band is called E C , and the top of the valence band is called E V . The bandgap energy E g = E C − E V between these
two levels is the width of the forbidden energy gap. It is the amount of energy required to break a bond in the crystal, i.e. to free an electron to the
conduction band and to leave a hole in the valence band.
In an insulator, the valence electrons form strong bonds between neighbouring atoms and consequently these bonds are difficult to break. Thus,
the bandgap is large and there are no free electrons to participate in current conduction at or near room temperature.
For a semiconductor the situation is similar but the energy gap is much
smaller, on the order of a few electronvolts. This leads to poor conduction
at low temperatures, but when the thermal energy E = kT constitutes a
larger fraction of the bandgap energy, an appreciable number of electrons
are thermally excited from the valence to the conduction band. This excitation does not necessarily have to be thermal. For example, it can also happen by illumination, ionising radiation or strong electric fields. These cases
will be discussed in Chapters 3, 4 and 5.
The conduction at lower temperatures can be improved by doping – the
process of intensionally introducing impurity levels into the bandgap. If
these levels are shallow, i.e. close to E C or E V , they can be activated thermally even at low temperatures. Sec. 2.5 will deal with the doping of diamond.
2.2.2 Intrinsic carrier concentration
An intrinsic semiconductor is one that contains relatively small amounts of
impurities compared to the thermally generated electrons and holes [9]. In
order to obtain the electron concentration (the number of electrons per unit
volume) in an intrinsic semiconductor, one has to integrate the electron
density n(E ) in an incremental energy range dE from the bottom to the top
of the conduction band. The density n(E ) is the product of the Fermi-Dirac
distribution F (E ) and the density of allowed energy states per energy range
per unit volume N (E ). The Fermi-Dirac distribution
e −(E −E F )/kT
for (E − E F ) > 3kT
1
F (E ) =
1 + e (E −E F )/kT
1 − e −(E −E F )/kT for (E − E F ) < 3kT
19
gives the probability at the absolute temperature T that an electron occupies an electronic state with energy E . The Fermi level E F is by definition
the energy at which this probability is one-half and k is the Boltzmann constant. The integral returns the electron density in the conduction band
n = NC e −(E C −E F )/kT
(2.1)
where NC is the effective density-of-states (DOS) in the conduction band.
With m e∗ being the density-of-states effective electron mass and h Planck’s
constant, it reads in the case of diamond:
2π m e∗ kT 3/2
.
(2.2)
NC = 2
h2
The hole density p in the valence band can be obtained in a similar way:
p = NV e −(E F −E V )/kT
(2.3)
where NV is the effective DOS in the valence band. For diamond it reads:
2π m h∗ kT 3/2
(2.4)
NV = 2
h2
with m h∗ being the density-of-states effective hole mass.
For an intrinsic semiconductor, the number of electrons per unit volume
in the conduction band equals the number of holes per unit volume in the
valence band, and the intrinsic carrier concentration n i can be calculated
from the mass action law which reads np = n i 2 . By using Eqs. (2.1) and (2.3)
it follows that
(2.5)
n i 2 = np = NC NV e −E g /kT .
The intrinsic carrier concentration in diamond is shown in Fig. 2.2
as a function of temperature. It can be clearly seen that, because of the
wide bandgap, the intrinsic charge carrier concentration is very low and
becomes only significant for temperatures exceeding approx. 1000 ℃.
2.2.3 Mobility
The electrons in the conduction band and the resulting holes in the valence
band can be considered as free particles that follow the classical equation
of motion but have an effective mass that differs from the free electron mass
in order to incorporate the fact that they move in a crystal lattice [9]. For
finite temperatures, the charge carriers move rapidly and completely randomly in all directions because of thermal excitation. When applying an
they will experience a force −q E
and become accelerated in
electric field E
that direction until they collide with lattice atoms, impurity atoms or other
20
1018
1017
1016
-3
ni (cm )
1015
1014
1013
1012
1011
1010
109
108
1000
1500
2000
2500
T (°C)
Figure 2.2: Intrinsic carrier concentration in diamond as a function of temperature.
scattering centres. The velocity of the charge carriers will then consist of the
thermal velocity v th and the additional drift velocity vd.
For small electric fields the drift velocity is much lower than the thermal
:
velocity and v d is proportional to E
v d = μE
(2.6)
where the proportionality constant μ is called mobility. Since electrons and
holes have different properties, such as effective masses, one distinguishes
between electron mobility μe and hole mobility μh .
When increasing the electric field, the drift velocity |
v d | starts to saturate. For diamond, silicon and germanium, it increases monotonically at
RT and converges towards v sat , the so-called saturation (drift) velocity. This
behaviour is well described by the empirical model [9, 11]:
vd =
μE
1+
|
μ|E
v sat
,
(2.7)
plotted in Fig 2.3. For diamond, the electron saturation velocity at room
temperature is ∼ 2 × 107 cm/s – twice as high as for silicon.
Mobility and saturation velocity are important parameters for carrier
transport because they describe how strongly the motion of the charge
carrier is influenced by an applied electric field. At high impurity or defect
concentration mobility is limited by (impurity) scattering [12]. Therefore, a
high mobility is also an indicator of low impurity or defect concentration.
21
Drift velocity vd (cm/s)
20x106
15x106
10x106
chosen parameters:
vsat = 2 x 107 cm/s
5x106
μ = 4500 cm2/Vs
0
0
10
20
30
40
50
Electric field E (kV/cm)
|. The gray straight line
Figure 2.3: Drift velocity |
v d | versus applied electric field |E
is a fit to weak electric fields and therefore reflects Eq. (2.6).
2.3 Advantages of the semiconductor diamond
After this recap of general semiconductor properties, the advantages of SCCVD diamond as a semiconductor material will now be discussed.
A selection of electronic and thermal properties [7, 13, 14] of some common semiconductor materials is shown in Fig. 2.4 and Tab. 2.1. Having a
closer look shows clearly that diamond has many superior intrinsic properties compared to silicon, the state-of-the-art semiconductor.
GaAs
Ge
InP
Diamond
2
Electron + Hole mobility at 300K (cm /Vs)
10000
GaN
Si
AlAs
1000
SiC
AlSb
ZnSe
GaP
ZnO
CdS
InN
ZnS
100
AlP
AlN
10
0
1
2
3
4
5
6
7
Bandgap (eV)
Figure 2.4: Added electron and hole mobility versus bandgap for different semiconductors. The area of the circles is proportional to the thermal conductivity.
22
Table 2.1: Electronic and thermal properties [7, 13, 14] of some common semiconductor materials in comparison to diamond (C). The data refer to measurements
at room temperature.
Ge
Si
GaAs
4H-SiC
GaN
C
Bandgap
0.7
1.1
1.4
3.2
3.4
5.5
eV
Breakdown field
0.1
0.3
0.4
3
5
20
MV/cm
Electron mobility
3900
1450
8500
900
2000
4500
cm2 /Vs
Hole mobility
1900
480
400
120
200
3800
cm2 /Vs
Thermal conductivity
0.58
1.5
0.55
3.7
1.3
24
W/cmK
The mobilities of diamond are very high and the electron mobility at RT
is only exceeded by a few materials, e.g. GaAs or InP. High carrier mobilities are desirable for fast-response and high-frequency electronic devices.
GaAs is mainly used for high-speed electronics, mobile phones and satellite communication [9], but it is considered highly toxic and carcinogenic.
It has also the drawback of a low hole mobility, which makes it unsuitable
for certain applications.
High thermal conductivity is very suitable for power electronics where
devices suffer from a high generation of heat [15]. For traditional power devices heat sinks have to be included in the structure to prevent the device
from malfunction. For diamond devices this is not necessary because diamond has the highest thermal conductivity and it is already used as a heat
sink for power transistors and semiconductor laser arrays [16].
As a wide bandgap semiconductor, diamond offers the benefit of thermal
stability and a high breakdown field – the maximal field strength a material
can withstand intrinsically without breaking down. Breakdown is discussed
in more detail in Section 5.3.
Diamond offers many advantages to other wide bandgap materials such
as silicon carbide (SiC) or gallium nitride (GaN). Although diamond epitaxy
is still in its infancy, growing epitaxial layers of diamond in a CVD process
is in many ways simpler than growing other wide bandgap semiconductor
materials. This is due to the simpler structure of diamond, consisting of carbon atoms only. Polyatomic materials such as SiC or GaN require careful
control of the stoichiometry and they exist in many hundred different crystalline structures. Single-crystalline configurations of carbon, on the other
hand, exist only in a few forms. Epitaxy of SiC is riddled with a particular problem, the formation of tubular channels, called micropipes, during
growth [17]. This problem does not exist for diamond. Another advantage
associated with growing diamond in a CVD process is that the raw materials are cheap and naturally abundant gases: methane and hydrogen, while
epitaxy of SiC and GaN involves extremely toxic substances.
23
Of all wide bandgap semiconductor materials, diamond clearly has by far
the most intriguing and extreme properties: mechanical, optical, thermal,
as well as electrical. Diamond exhibits the highest breakdown field strength
and thermal conductivity of any material and has the highest carrier mobilities of any wide bandgap semiconductor. Therefore, it enables the development of electronic devices with superior performance regarding power efficiency, power density, high frequency properties, power loss and cooling.
See Sec. 2.7 for future application areas for diamond electronic devices.
2.4 CVD diamond synthesis
This section focuses on the process to grow diamond on a substrate. The
schematic setup of the CVD synthesis is shown in Fig. 2.5. The substrate that
shall be overgrown by diamond is placed in the reaction chamber. Typically
for this type of synthesis, the temperature on the substrate is around 800 ℃
and the pressure is kept below 10 kPa [6]. Varying amounts of gases, mostly
hydrogen but sometimes also argon (∼ 10%), and a carbon source such as
methane (∼ 5%), are fed into the chamber and energised by microwave
power. The chemical bounds are broken down and a plasma of highly reactive atoms is formed. The result is both diamond and graphite growth on
the substrate. Atomic hydrogen in the plasma removesany graphite phase
formed and eventually only diamond stays on the substrate.
In contrast to HPHT diamond, it is possible to grow CVD diamond under
conditions of high purity, resulting in fewer defects and impurities. Other
advantages are the possibilities to grow diamond over larger areas, to deposit it on a substrate, and to control the properties of the diamond produced. This allows the addition of many of diamond’s important qualities to
other materials: Coated valve rings or cutting tools benefit from diamond’s
hardness and wear resistance. On extensive heat-producing components a
diamond coating can function as a heat sink [16], as mentioned before.
In the beginning, deposition conditions have invariably resulted in polycrystalline material. Only during the last couple of years has it become possible to grow thick free-standing plates of high-purity SC-CVD diamond by
homoepitaxy. This was first achieved by DeBeers Industrial Diamonds1 in
England [7]. Besides requiring a very careful control of the growth process,
for single-crystalline diamond growth, the substrate itself has to be a single
crystal diamond. Most commonly a specially prepared type Ib HPHT sample of high quality is taken2 . At the end of the process, the SC-CVD layer is
separated from the substrate by a laser cutting technique and polished in
order to obtain a free-standing plate.
1
Today known as Element Six Ltd. They provided the samples investigated in Papers I-VIII.
Other substrates, such as silicon, are also possible, but due to the different lattice spacing
only polycrystalline growth is possible then.
2
24
Microwave
source
Microwave
cavity
Quartz
bell jar
Magnetron
Substrate
H2
Gas in CH4
(Ag)
Plasma
To vacuum
pump
Water cooling
Figure 2.5: Schematic setup of the CVD synthesis of diamond.
2.5 Doping diamond
As mentioned previously, diamond has a very low concentration of intrinsic charge carriers at temperatures below 1000 ℃. If a significant room temperature conductivity is desired, it is necessary to dope diamond. When a
semiconductor is doped with impurity atoms, it becomes extrinsic and impurity energy levels are introduced [9].
Unfortunately, there are no shallow dopants known for diamond. As
shown in Fig. 2.6 both for p-type dopant boron (B) and for n-type dopants
phosphorus (P) and nitrogen (N) the dopant levels are rather deep and
result in low thermal excitation of free charge carriers at room temperature
1
(E th = 40
eV).
EC
conduction band
P 0.52eV
N 1.7eV
5.47 eV
B 0.37eV
EV
valence band
Figure 2.6: Activation energies for some common dopants in diamond.
25
Type IIb natural diamond is p-type but extremely rare in nature. Since
doping diamond through diffusion is not achievable, the first intentional
doping of (natural) diamond was done by ion implantation at end of the
1960s [18]. However, with this technique a considerably amount of damage
is done to the crystal lattice which cannot be reversed by annealing.
With the advent of the diamond CVD process in the 1980s it became possible to add the dopants to the gas phase during growth. Only a few years
later several groups demonstrated p-type boron doping of the diamond
films from the gas phase [19–21]. This type of doping can, for instance, be
achieved by a diborane B2 H6 addition to the H2 /CH4 /Ar source gas mixture [22]. This leads to a very homogenous distribution in the bulk and allows rather sharp interfaces by quickly changing the gas-phase boron concentration during the growth process (see e.g. Paper IV).
n-type diamond does not exist in nature at all and it is even more difficult to form it artificially. However, it is also desirable to have efficient
n-type diamond doping for electronic applications such as cold cathode
electron emitters, UV photodetection and UV light emission diodes. The
first success to form n-type diamond thin films by phosphorus doping with
PH3 , CH4 , and H2 gas mixtures during the growth was reported in 1997 [23].
Four years later a UV light emission diamond diode with a pn-junction was
demonstrated [24].
There are different approaches of how to circumvent the problem of low
thermal excitation of free charge carriers at room temperature. One idea is
to use two different layers. One layer is very highly doped and thus has a
low mobility but a high number of free charge carriers which diffuse into a
second (intrinsic) layer. This combines a high number of carriers with the
excellent mobility in the intrinsic layer leading to desirable properties. The
concept of the so-called pulse- or δ-doped structure is described in more
detail in Sec. 9.1.
2.6 Compensation
In order to achieve successful diamond devices the ability to grow doped
diamond films with low concentrations of defects and residual impurities
is of very high importance. The presence of deep-level impurities in doped
semiconductors causes compensation effects, i.e. electrons (holes), which
in the absence of the impurity would have been emitted to the conduction
(valence) band, are instead trapped, leading to a reduced free carrier concentration. For this reason, the compensation ratio, i.e. the ratio between
the dopant and the compensating defect concentration, should be kept to
a minimum.
In diamond, known dopants are only partially thermally activated at
room temperature. Because of this, very low compensation ratios are
26
necessary to achieve reasonable carrier densities. As an example, consider
boron-doped diamond with a concentration of [B] = 1018 cm−3 . Without
compensation, a room-temperature hole density of 2 × 1015 cm−3 is
expected. For a compensation ratio ND /NA = 1% the hole concentration
drops by almost a decade.
Deep impurity states can also be undesirable in devices for other reasons.
For example, a high concentration of impurities reduces carrier mobility,
mainly by ionised-impurity scattering [12].
In a p-type semiconductor, normal band-conduction occurs in the
valence band through the transport of holes, which result from the
ionisation of acceptors at an energy E A above the valence-band edge. The
hole concentration p in the valence band of a non-degenerate p-type
semiconductor is the solution to the equation
NV −E A /kT
p(p + ND ) − n i 2
=
e
≡ NV
2
NA − ND − p − n i /p
ga
(2.8)
assuming that both acceptors and compensating donors are present with
concentrations NA and ND , respectively, and with NA > ND , see e.g. [25]. In
the above equation, g a is the spin degeneracy factor for the valence band
and E A the acceptor ionisation energy. Because of diamond’s large bandgap
(E g = 5.47eV) one can neglect the intrinsic carrier concentration and obtains for the hole concentration
N + ND 2
N + ND
V
,
(2.9)
+ NV
(NA − ND ) − V
p=
2
2
which is a function of temperature. By measuring the hole concentration at
different temperatures and fitting the data to the equation above, the ionisation energy of the dopant and the impurity concentrations can be obtained. This was done in Paper IV.
2.7 Future diamond devices
This section gives an overview of possible future application areas for diamond electronic devices summarised in Tab. 2.2.
Present-day power electronics, mainly based on silicon devices, exhibit a
number of serious problems and limitations. Many of these limitations are
inherent in the material itself and the only way to make significant progress
is to switch to other semiconductor materials. Some of the most significant
problems are: high losses, the difficulty to meet high switching frequencies,
and limitations on voltages that can be attained [9]. Diamond power devices have the potential to increase efficiencies and reduce total system
27
costs due to diamond’s ability to operate successfully at higher voltages
than silicon-based devices or even other wide bandgap materials such as
SiC or GaN.
Diamond power devices also play an important role for a faster introduction of hybrid vehicles. With present technology, separate cooling systems
for both the combustion engine and the electro-motor’s converter system
are necessary. Using just one (combined) cooling system would result in
significantly smaller and lighter motor blocks. The combination of the two
systems is possible with diamond power devices because they can operate
at such high temperatures due to diamond’s wide bandgap. In this context
it is worth noting that more efficient converters also means higher total efficiency of the vehicle and reduced emissions.
Table 2.2: Future application areas for diamond electronics.
Area/device
related properties
Power electronics
high breakdown field, high thermal conductivity
HV diodes, Power MESFETS
RF and Microwave
HF MESFETS
high saturation velocity, high breakdown field
high thermal conductivity
High Temperature Electronics
wide bandgap, chemical inertness
Photoconductive Switches
high breakdown field, high carrier mobilities
for Pulsed Power applications
Radiation and UV detectors
28
radiation hardness, solar blindness
3. Principles of the time-of-flight
technique
The time-of-flight (ToF) method1 is of great importance for the study of
sample properties such as low-field drift mobilities and charge trapping
[26]. This chapter deals with the basic principles by first describing the experimental setup and then explaining how drift mobility and saturation velocity can be measured for the low and high injection regime.
3.1 Experimental setup
The idea behind how to study low-field drift mobility by ToF measurements
is quite simple. A sample with semitransparent contacts is illuminated with
photons of high enough energy in order to create electron-hole pairs [7, 27,
28]
hc
.
(3.1)
λ<
Eg
In the case of diamond, the bandgap E g is 5.47 eV and hence the wavelength
has to be below 226 nm, i.e., we need UV or X-ray photons. Alternatively,
other sources of ionising radiation such as α-particles [29], β-particles [30]
or pulsed electron beams [31] can be utilised for electron-hole pair creation.
When applying an electric field across the sample, the created charge carriers experience a force and will drift. In order to measure the velocity of the
charge carriers, a pulsed source with a pulse length much shorter than the
time-of-flight is needed. The source should preferably provide a trigger signal to the oscilloscope where the induced current is measured. A second
constraint for measuring the velocity is that the distance travelled by the
charge carriers is known which means that all charge should be created at
the close proximity of the illuminated contact. This is fulfilled for just above
bandgap radiation that has a penetration depth of only ∼ 5 μm in diamond.
Compared to the standard sample thickness of around ∼ 500 μm this can
mostly be neglected.
1
An alternative name in the literature is “transient-current technique” (TCT).
29
The semitransparent contacts allow both illumination and the application of a rather homogenous field across the thickness of the sample. Titanium (Ti) is commonly used as injecting contact metal since it can form a
carbide phase at the interface2 during contact annealing at around 450 ℃
[34]. The Ti layer has to be covered by a less oxidising metal such as gold
(Au) or aluminum (Al). Figures 3.1 and 3.2 show a patterned diamond sample with a magnification of the mesh contact and the setup for the ToF measurements presented in Papers I,V,VI and VII.
3.2 Mobility measurements
When measuring the low-field drift mobilities of the charge carriers by creating electron-hole pairs of total charge Q within a sample of capacitance C
and determining their drift velocity under an applied bias voltage of known
magnitude U , three cases are distinguished:
– low injection regime: Q C ·U and therefore the applied voltage results
| = |U |/d (if charge trapping can be
in an homogenous electric field |E
neglected and d is the sample thickness).
– high injection regime: Q C · U implies that the electric field is inhomogeneously distributed in the sample which has to be corrected for by
a factor β (see below) that can be obtained from simulations.
– intermediate regime: Q ≈ C · U and precise knowledge of the created
charge is necessary in order to compare to simulations. This makes studies rather complicated and that is why this case is mostly avoided in experiments.
3.2.1 Low injection
The mobilities of the carriers can be determined for low injection in the
following way. A bias voltage U synchronised with the illumination source is
applied for a short time (∼1 ms) across the sample in order to keep charging
effects to a minimum. If the illuminated contact is negatively charged, the
created holes get immediately annihilated at this contact but the created
electrons travel through the whole thickness d of the sample. In the absence
of trapping, the applied electric field is given by
| =
|E
|U |
.
d
(3.2)
The measured current is constant when the electron cloud travels at constant drift velocity v d through the sample. With the cloud arriving at the
2
The surface termination of the diamond also has an influence on the electrical transport
properties [32, 33].
30
Figure 3.1: Patterned diamond sample with a semitransparent 40 μm Ti/Al mesh.
1064
Nd-YAG
Laser 5ns
532+
2Z
nm
4Z
1064
266+
532+
1064
Halogen
or
Xe lamp
(4+1)Z
10 Hz trigger
attenuator
Power Supply
&
Q-switch trigger
21
3n
m
interference
filters
SP150
Monochromator
shutter
trig
LN-cryostat
Pulse
generator
10m
koax
Sample
Temperature
controller
Computer
trig
data acqusition
Heater
Oscilloscope
TDS 640
5 Gs/s
Figure 3.2: Setup for the ToF measurements presented in Papers I,V,VI and VII.
16
14
-11.3V
-30.1V
-89.5V
current (μA)
12
10
8
6
4
2
0
-2
0
20
40
60
80
100
120
140
160
time (ns)
Figure 3.3: Typical ToF curves for the low injection regime. The contribution from
immediately annihilated charge carriers cannot be seen due to limited bandwidth.
31
electron time-of-flight (ns)
100
80
sample thickness:
d = 490μm
fit coefficients:
slope = a = 1095.7
offset = b = 9.00
60
40
electron mobility:
μe = d2 / a = 2190 cm2/Vs
20
0
0.00
0.02
0.04
0.06
0.08
1 / |bias voltage| (V-1)
Figure 3.4: Typical plot of electron ToF τe versus the inverse applied voltage |U |−1 .
The mobility can be extracted from the linear best fit as discussed in the text.
positively charged contact, the current drops (see Fig. 3.3). The full width at
half maximum (FWHM) time interval with respect to the current plateau is
the time-of-flight τe of the electrons. It follows from Eqs. (2.7) and (3.2)
|
μ |E
1 + vesat
d
d
d
=d
τe =
=
+
|
| v sat
|
vd|
μe |E
μe |E
=
d
d2
+
μe |U | v sat
(3.3)
Applying a set of negative bias voltages and plotting τe versus the inverse
applied voltage |U |−1 should theoretically lead to a straight line with the
slope a only depending on the sample thickness and the electron mobility
and the offset b only depending on the sample thickness and the saturation
velocity. A typical plot of a medium quality sample is shown in Fig. 3.4.
A completely analogous consideration for holes leads to the hole mobility
and one can write for both cases
τe/h = a e/h |U |−1 + b e/h
(3.4)
with the following identifications made
d2
,
a e/h
d
e/h
v sat
=
.
b e/h
μe/h =
32
(3.5)
(3.6)
0.74
0.73
0.72
0.71
0.70
E
0.69
0.68
0.67
0.66
0.65
0.64
0.63
0.62
200
300
400
500
600
700
800
900
1000
thickness (μm)
Figure 3.5: Correction factor β obtained from simulations in case of high injection.
3.2.2 High injection
In the case of high injection, the carrier transport is space charge limited
(SCL) and the electric field has an inhomogeneous distribution because of
the non-negligible charge in the carrier cloud [35, 36].
A reservoir of carriers is created near the illuminated electrode [7], which
thermalises quickly. A first current peak results from the screening of the
electrical field in this electron-hole plasma and the second from the arrival
of the first charge carriers at the opposite electrode. The charge carriers do
not travel at a constant drift velocity like in the low injection case. Instead,
they experience an increasing electric field during the space charge limited
transit through the sample. To compensate for that effect, a correction factor β is introduced, which is a slowly varying function of the sample thickness as shown in Fig. 3.5. This factor can be obtained from simulations. Just
as before, the polarity of the bias determines whether electron or hole drift
is observed.
In this case, the electric field in Eq. (3.2) has to be divided by the correction factor β. Thus, the mobility and the saturation velocity can be extracted
from the linear fit with slope a e/h and offset b e/h by the identifications
βd 2
,
a e/h
βd
e/h
v sat
=
.
b e/h
μe/h =
(3.7)
(3.8)
33
3.3 Data acquisition and evaluation
Mobility measurements by the time-of-flight technique involve the
recording of several current transits with different bias voltages applied, as
pointed out in the previous section. The ToF setups for both vertical and
lateral are shown in Fig. 3.6.
The author of this thesis was involved in developing a fully automated
measurement software of which a screenshot is shown in Fig. 3.7. The program was written in TESTPOINT and controls the temperature, the bias
voltage and also, if desired, a magnetic field that can be applied perpendicular to the electrical field. The current signals are sampled by a digital
oscilloscope and stored on the harddrive of the PC. The communication
with the equipment is via the GPIB interface (also known as IEEE-488).
For the evaluation of the recorded data a graphical user interface (GUI)
was written in MATLAB. The software allows to extract the time-of-flight
from the current transit curves not only by marking the plateau and the
related FWHM manually but also by fitting the data to Eq. (4.37) – to be derived in the next chapter. Error bars can be obtained in this way, which can
be taken into account (by weighting) when extracting the low-drift mobility
from ToF versus 1/U plots. The GUI is shown in Fig. 3.8.
vertical ToF
lateral ToF
Figure 3.6: The time-of-flight setup with the cryostat used for vertical measurements (on the left) and the holder used for lateral measurements (on the right).
34
Figure 3.7: Screenshot of the program controlling the measurement hardware via
the GPIB bus. The software was written in TESTPOINT.
Figure 3.8: Screenshot of the evaluation GUI. The software was written in MATLAB.
35
4. Free charge carrier transport in
diamond
This chapter deals with carrier transport in more detail. The aim is to show
how the different stages between carrier generation by a short laser pulse
and collection at the opposite contact influence the current transient. Or
the other way round, to show what conclusions we can draw from the
observed current transient. In this context, the electrical field profiling
method presented in Paper I will be explained.
4.1 Drift-diffusion equations from the BTE
In this section, the fundamental drift-diffusion equations will be derived by
solving the Boltzmann transport equation (BTE) in the case of a Fermi gas
with an applied electric field using the relaxation time approximation. For
a more detailed discussion see e.g. [37]. Our objectives in this section are to
understand the origin of the drift-diffusion equation and the assumptions
that limit its validity1 .
4.1.1 The Boltzmann transport equation
The Boltzmann transport equation2 is a particle continuity equation describing particle flow in the six-dimensional (6-D) phase space
∂f
∂ f df
· ∇p f =
.
(4.1)
=
+
v · ∇x f + F
dt
∂t
∂t coll
, t ) is a function of time and the
The distribution function f (
x, p
six-dimensional (6-D) phase space which is normalised in such a way that
it describes the probability of finding a carrier with crystal momentum
, at location p
x , at time t . The carrier’s velocity is connected to the
momentum by
(4.2)
v = ∇p E kin
1
To establish the limitations of the drift-diffusion equation is especially important since it
serves as the cornerstone of semiconductor device analysis.
2
The BTE is an approximation because it is a single particle description of a many particle
system of carriers. But correlations between carriers are not treated, even though carrier
interact through their electric field.
37
is the external force which is experienced by the particles described
and F
by f . The term on the right hand side (RHS) is added to describe the effect
of collisions between particles; if it is zero then the particles do not collide.
This term is also called the collision integral and is in general a higher dimensional integral by which f is connected non-linearly. The explicit form
of this integral depends on the type and interactions of the particles we are
investigating and has to be determined from the microscopic theory (e.g.
quantum mechanics). That is why the BTE is an integro-differential equation and cannot be solved by standard means of classical mechanics.
4.1.2 Equilibrium distribution function for a Fermi gas
In equilibrium nothing changes with time and the BTE reads
· ∇p f = 0 .
v · ∇x f + F
(4.3)
The solution is the equilibrium distribution function commonly denoted by
f 0 which in the case of a Fermi gas is the previously mentioned Fermi-Dirac
distribution F (E )
) = F (E ) ≡
f 0 (
x, p
1
1 + eθ
with θ =
) − E F
E tot (
x, p
,
kT
(4.4)
with Boltzmann’s constant k, the (lattice) temperature T , the Fermi level E F
and the carrier’s total energy E tot which is the sum of potential and kinetic
) = E pot (
energy E tot (
x, p
x ) + E kin (
p ).
In this case we can write Eq. (4.3) as
v·
∂ f0
∂f v
· 0
∇x θ + F
=0
∂θ
∂θ kT
(4.5)
v /(kTL ) because of Eq. (4.2). If we permit E pot , E F and T to vary
since ∇p θ = with position Eq. (4.3) becomes
E pot + E kin − E F
F
+
0 = ∇x
T
T
E
1
F
1
pot + E kin − E F
∇x
+
0 = ∇x E pot − E F +
T
T
T
T
E pot + E kin − E F
1
1
0 = − ∇x (E F ) +
∇x
T
T
T
is a conservative field with F
= −∇x E pot . This equation has to hold for
if F
so each of the terms has to vanish independently and we can conany p
clude that ∇x E F = ∇x T = 0 meaning that both the Fermi-level and temperature are constant in equilibrium.
38
4.1.3 Uniform electric field with RTA
In order to obtain an analytical solution of the BTE we have to make further
3
simplifications. In the case of an applied electric field E
= qE
F
(4.6)
we will use the relaxation time approximation (RTA)
f − f0
∂f
· ∇p f = −
+
v · ∇x f + q E
.
∂t
τf
(4.7)
) depends only on the nature of the scattering
The relaxation time τf (
x, p
process and is the characteristic time describing how the system relaxes.
The RTA assumes that the collision probability during the time interval dt
) is equal to dt /τf and that some
for an electron at phase space point (
x, p
time after scattering has occurred, the electron distribution does not depend on the non-equilibrium distribution just before the scattering. This
means that the information about the non-equilibrium state is completely
lost due to the scattering processes. A more detailed study of the RTA shows
(see e.g. [37]) that Eq. (4.7) is a good approximation of the BTE in the case
of low fields when the scattering is elastic and/or isotropic.
The quasi-equilibrium distribution function f 0 has the same form as
F (E ) but with the Fermi-level E F replaced by the quasi-Fermi level E F
) =
f 0 (
x, p
1
1 + eθ
with θ =
x ) + E kin (
p ) − E F (
x)
E pot (
kT (
x)
.
(4.8)
The function f 0 cannot be the solution to the BTE because it is symmetric
in momentum , so the average velocity is zero, and no current flows. But we
expect the solution to be f 0 plus a correction term and make the guess
q ∂ f0
∂ f0
.
+
v · ∇x f 0 +
v ·E
(4.9)
f = f 0 − τf
∂t
kT ∂θ
Inserting our guess into the BTE yields
∂
q ∂ f0
∂ f0
· ∇p f 0 − τf
+
v · ∇x + q E
+
v · ∇x f 0 +
v ·E
∂t
∂t
kT ∂θ
∂ f0
q ∂ f0
=
+
v · ∇x f 0 +
v ·E
∂t
kT ∂θ
q ∂ f0
∂
∂ f0
⇔
+
v · ∇x + q E · ∇p −τf
+
v · ∇x f 0 +
v ·E
∂t
∂t
kT ∂θ
q ∂ f0
− qE
· ∇p f 0 .
=
v ·E
kT ∂θ
denotes the electric field while the energy is given by the
Throughout Sec. 4.1 the field E
scalar E .
3
39
Using Eq. (4.9) we see that our guess is correct if
∂
q ∂ f0
− qE
· ∇p f 0 .
· ∇p f − f 0 =
+
v · ∇x + q E
v ·E
∂t
kT ∂θ
(4.10)
only through E kin
The right side of Eq. (4.10) vanishes since θ depends on p
· ∇p f 0 = q
qE
∂ f0
∂f 1
q ∂ f0
· ∇p θ = q 0
· ∇p E kin =
·
E
E
E
v
∂θ
∂θ kT
kT ∂θ
which means that the left side of Eq. (4.10) has to vanish, too.
Assuming that changes in carrier concentration are much slower than
the typical time between two scattering events and that the concentration
does not vary significantly over the mean free path, the left side of Eq. (4.10)
can be approximated by
q ∂ f0
∂
∂ f0
· ∇p f − f 0 ≈ q E
· ∇p −τf
+
v · ∇x + q E
+
v · ∇x f 0 +
v ·E
∂t
∂t
kT ∂θ
q ∂ f0
· ∇p τf
v ·E
≈ −q E
kT ∂θ
2
|
= 0 + O |E
and it vanishes under the assumptions above if we are dealing with small
|.
electric fields, i.e. we can neglect second and higher order terms in |E
4
The carrier concentration in our semi-classical approach is the momentum integral of the distribution function divided by twice the unit volume
(2πħ)3 since it can be occupied by two fermions
1
1
3
x , t ) = 3 3 f d p ≈ 3 3 f 0 d3p ,
(4.11)
ρ c (
4π ħ
4π ħ
and the approximation is valid in our case considering small perturbations.
For the current density we can write
1
q
j(
x , t ) = 3 3 q
v ( f − f 0 )d3p ,
(4.12)
v f d3p = 3 3 4π ħ
4π ħ
since the integral of v f 0 vanishes. Using our guess Eq. (4.9) we obtain
q
q ∂ f0
∂ f0
j(
x, t ) = − 3 3 +
v · ∇x f 0 +
v · E d3p
v τf
4π ħ
∂t
kT ∂θ
q ∂ f0
q
v · E d3p
v · ∇x f 0 +
v τf =− 3 3 4π ħ
kT ∂θ
where the last equal sign holds because f 0 is time-independent.
4
The approach is semi-classical because carriers are treated as classical particles obeying
Newton’s laws. Quantum mechanics is used only to describe the collisions.
40
Under non-degenerate conditions, i.e. θ > 3, we can approximate f 0 by
∂f
f 0 = e −θ which yields ∂θ0 = − f 0 . Thus, we can write
3
q
q
q
d3p ,
j = − 3 3
v · ∇x f 0 d p + 3 3
v ·E
v τf f 0 v τf 4π ħ
4π ħ kT
or in component form (with α, β, γ ∈ {1, 2, 3} and summation convention)
q
q
∂ f0 3
q
jα = − 3 3
v α τf f 0 v β E β d3p .
v α τf v β
d p+ 3 3
4π ħ
∂x β
4π ħ kT
If we further assume spherical, parabolic bands and τf = τf (p), we can define the (scalar) mobility μ using the Kronecker symbol δαβ
τf v α v β f 0 d3p
3
1
q
q
, (4.13)
δαβ μ ≡ 3 3
τf v α v β f 0 d p =
4π ħ kT ρ c
kT
f 0 d3p
which is a constant since the dependence on the quasi-Fermi level and the
potential energy drops out. With help of the definition, Eq. (4.13), we can
write the current density in the compact form
j α = qρ c μ δαβ E β −
kT ∂ ∂ρ c
,
qρ c μ δαβ = ρ c μ E α − kT μ
q ∂x β
∂x α
which corresponds in vector notation to
− kT μ∇x ρ c ≡ qρ c μ E
− qD ∇x ρ c .
j = qρ c μ E
(4.14)
We have finally obtained the drift-diffusion equation and Einstein’s relation
D = kT μ/q. Since electrons and holes respond differently to an electric field
but in the same way to a concentration gradient, the equations read
+ kT μn ∇x n ≡ qnμn E
+ qD n ∇x n
x , t ) = qnμn E
j n (
(4.15)
− kT μp ∇x p ≡ q pμp E
− qD p ∇x p .
j p (
x , t ) = q pμp E
(4.16)
So we have seen under which assumptions the drift-diffusion equation
can be derived from the BTE. In the case of low electric fields the transport parameter, μ and D can be assumed material-dependent but deviceindependent.
If we had solved the BTE and found f , we would have learned how electrons (or holes) are distributed in momentum space as a function of location within the device. By solving the derived drift-diffusion equation, we
only learn how the current density depends on the carrier concentration.
This information, however, often suffices for analysing the performance of
a device, and it is far simpler to solve the drift-diffusion equation than it is
to solve the BTE directly.
41
4.2 Charge transport in 1-D
This section deals with modelling the transport of free charge carriers in
a one-dimensional (1-D) geometry. This simplification is a good approximation if the material properties such as mobility, dielectric permittivity,
magnetic permittivity and the diffusion constants can be assumed as homogenous and isotropic. The derived (analytic) solution can then be fitted
to the measured current density. An example curve including the different
stages of carrier generation, transit and extraction is shown in Fig. 4.1.
4.2.1 Fundamental transport equations
In a 1-D model of a semiconductor, the total current density j (x, t ) is the
sum of the current densities for both carrier types and Maxwell’s displacement current:
∂E
,
(4.17)
j = jn + j p + ε
∂t
with ε = ε0 εr being the semiconductor’s dielectric permittivity and E (x, t )
the electric field strength5 .
The current densities j n/p are given by the drift-diffusion equations derived in the previous section which read in 1-D as follows:
∂n
∂x
∂p
j p ≡ j p (x, t ) = q pμp E − qD p
.
∂x
j n ≡ j n (x, t ) = qnμn E + qD n
(4.18)
(4.19)
As before, q is the elementary charge and the electron and hole concentrations are given by n(x, t ) and p(x, t ), respectively. We have seen that the diffusion constants D n/p are linked to the (drift) mobilities μn/p by Einstein’s
relations:
kT μn
(4.20)
Dn =
q
kT μq
Dq =
.
(4.21)
q
The relation between free charges and electric field strength is given by
Poisson’s equation:
q
∂E ρ c
(4.22)
=
=− n−p +N ,
∂x
ε
ε
with ρ c being the charge density and N ≡ NA− − ND+ the (net) ionised impurity concentration.
5
Note: For the rest of this chapter E denotes the electric field strength and not the energy.
42
The continuity equations are the governing equations describing the
overall effect when drift-diffusion occurs simultaneously at generation rate
g and recombination rate r :
1 ∂ jn
∂n
= gn − rn +
∂t
q ∂x
1 ∂ jp
∂p
= gp − rp −
.
∂t
q ∂x
(4.23)
(4.24)
By taking the divergence of Ampere’s law in 3-D
∇×
B
= j ,
μ0 μr
(4.25)
and using Eq. (4.17) together with ∇ · (∇ × A) = 0 for any vector field A we
note that the total current has no x-dependence in our 1-D case:
∂
∂
∂E
j=
jn + j p + ε
⇒ j (x, t ) = j (t ) .
0=
∂x
∂x
∂t
∂φ
Using E = − ∂x and assuming that a constant bias voltage U = φ(d )−φ(0)
is applied across the semiconductor of thickness d , the expression for the
current density
∂
j (t ) = j n (x, t ) + j p (x, t ) + ε E (x, t )
∂t
can be simplified further:
1 d
j (t )dx
j (t ) ≡
d 0
q d
q d
∂n
∂p
ε ∂ d ∂φ
=
− Dp
dx −
dx
μn n + μp p E dx +
Dn
d 0
d 0
∂x
∂x
d ∂t 0 ∂x
qD n
q d
=
μn n + μp p E dx +
(n(d ) − n(0))
d 0
d
ε ∂ qD p p(d ) − p(0) −
φ(d ) − φ(0) .
−
d
d ∂t
=0 assumingU =const
Using Einstein’s relations and writing all space and time dependence explicitly the equation for the current density reads:
q d
μn n(x, t ) + μp p(x, t ) E (x, t ) dx
j (t ) =
d 0
kT
kT
μn (n(d , t ) − n(0, t )) −
μp p(d , t ) − p(0, t ) .
(4.26)
+
d
d
43
4.2.2 Carrier generation by laser, low injection
If we consider the case of low injection (see Sec. 3.2), i.e. the case where the
applied electric field is just minimally perturbed by the free charges, we can
assume a constant electric field which means that ∂E
∂x = 0. In this case, the
continuity together with the drift-diffusion equation and Einstein’s relation
reads for electrons
1 ∂ jn
1 ∂
∂n
∂n
= gn − rn +
= gn − rn +
qnμn E + kT μn
∂t
q ∂x
q ∂x
∂x
= g n − r n + μn E
∂n kT μn ∂2 n
+
,
∂x
q ∂x 2
(4.27)
and for holes
1 ∂ jp
1 ∂
∂p
∂p
= gp − rp −
= gp − rp −
q pμp E − kT μp
∂t
q ∂x
q ∂x
∂x
2
∂p kT μp ∂ p
+
,
= g p − r p − μp E
∂x
q ∂x 2
(4.28)
where the four terms on the right represent generation, recombination,
drift and diffusion of carriers.
If we assume that electron-hole pairs of charge Q inj are generated at t = t 0
by a completely absorbed laser pulse that has a Gaussian temporal profile
and the spatial absorption profile χ(x), the generation rates for electrons
and holes are equal g ≡ g n = g p
χ(x)
−(t − t 0 )2
g (x, t ) = Q inj exp
,
(4.29)
2σ20
2πσ0
d ∞
such that 0 −∞ g (x, t ) dxdt = Q inj because the absorption profile has to
d
fulfil 0 χ(x)dx = 1 in case of complete absorption.
For a short generation time the diffusion and recombination terms can
be neglected and we can write
∂n
= g + μn E
∂t
∂p
= g − μp E
∂t
∂n
∂x
∂p
,
∂x
and simply integrate over the sample’s thickness. For electrons this yields
d
d
∂n
∂ d
n dx =
g dx +
μn E
dx
∂t 0
∂x
0
0
Q inj
−(t − t 0 )2
exp
=
+ μn E [n(d ) − n(0)] .
2σ20
2πσ0
44
And further
∂
∂t
d
0
Q inj
−(t − t 0 )2
n dx = exp
,
2σ20
2πσ0
where the last equal sign holds if the majority of the photons from the laser
pulse get absorbed close but not at the very surface of the sample’s illuminated side such that n(d , t ) = 0 and n(0, t ) ≈ 0 for t ≈ t 0 .
The differential equation has the solution
d
t − t0
1
1 + erf n dx = Q inj
.
2
0
2σ0
Analogously we find for holes
d
1
t − t0
p dx = Q inj
1 + erf ,
2
0
2σ0
and using Eq. (4.26) we obtain for the current density for small t − t 0
qE d j (t ) =
μn n + μp p dx
d 0
qEQ inj
t − t0
(μn + μp ) 1 + erf =
.
(4.30)
2d
2σ0
4.2.3 Carrier transit with homogenous space charge density
and trapping
The injected charge carriers start to drift and diffuse until they reach the
surface of the semiconductor. During the transit the two types of carriers
get separated. One type travels the distance d through the bulk to the back
side, the other type only a short distance to the illuminated front side. This
clearly depends on the orientation of the applied electric field.
The following discussion is for the low injection regime, i.e. the injected
charge does not significantly perturb E (x) = E ap + E sc (x), and made under
the assumption that a negative voltage is applied to the front contact, such
that holes drift to the illuminated contact and electrons reach the back contact. We assume further that the electric field is the sum of the applied (constant) field E ap and the field arising from homogenously distributed space
charges ρ sc (x) = ρ sc = q N , so that Poisson’s equation (4.22) can be integrated easily6
d
ρ sc
x−
.
(4.31)
E sc (x) =
ε
2
6
For a homogenous distribution, it follows from symmetry considerations that the space
charge electric field must vanish in the middle of the sample.
45
The contribution to the overall current by the electrons follows from the
time derivative of Eq. (4.26) and g = 0
d
∂
q
∂n
j = μn
dx
E
∂t
d
∂t
0
d
q
ρ sc
μn ∂
d
∂n
= μn
x−
−r n +
qnE + kT
dx
E ap +
d
ε
2
q ∂x
∂x
0
d d
ρ sc
q
x−
r n dx
= − μn
E ap +
d
ε
2
0
d
μn ∂
∂n
q
ρ sc
x
qnE + kT
dx
+ μn
d
ε
q ∂x
∂x
0
where the last equal sign holds since n(0, t ) ≈ n(d , t ) ≈ 0 and ∂n
∂x ≈ 0 at the
contacts during transit.
In the case of linear trapping r n is proportional to the amount of conduction electrons r n = κn n, where κn denotes the rate constant, and we can
write
d ∂
q
ρ sc
d
j = − μn κn
x−
n dx
E ap +
∂t
d
ε
2
0
d μn ρ sc
q
∂n d
∂n
−
+ μn
x qnE + kT
dx
qnE + kT
d
qε
∂x 0
∂x
0
μn ρ sc q d
q d
μn n E dx −
μn n E dx
= −κn
d 0
ε d 0
μn ρ sc q d
= − κn +
μn n E dx
ε
d
0 μn ρ sc j (t ) ,
= − κn +
ε
= j (t )
where the last equality follows from Eq. (4.26). The solution to this differential equation is
μn ρ sc (4.32)
j (t ) = j 0 e − κn + ε (t −t0 ) ,
i.e., we obtain an exponential variation (in time) of the current density during carrier transit.
In absence of space charges ρ sc = 0 and without trapping κn = 0 we find
that the current density is constant and from Eq. (4.30) that
j (t ) = j 0 ≡
46
qμn E
Q inj .
d
(4.33)
4.2.4 Carrier diffusion during transit
If we first assume, for simplicity, that n(x, t ) = δ(x 0 ) with x 0 > 0 but x 0 ≈ 0
for the distribution of the charge carriers after laser generation at time t = 0,
then it is a known result from the theory of partial differential equations (see
e.g. [38]) that the fundamental solution to the diffusion equation
∂n
∂2 n
= Dn 2
∂t
∂x
is a Gaussian distribution for t > 0
−(x − x 0 )2
exp
n̂(x, t ) = .
4D n t
4πD n t
1
If the distribution is not singular but given by the function χ(x), then the
solution to the diffusion equation is the convolution of n̂ and χ
∞
n̂(x − y, t )χ(y)dy .
n(x, t ) =
−∞
In the case of laser illumination with a Gaussian temporal profile and variance σ0 (Eq. (4.29)), we create charge with a Gaussian spatial profile χ(x)
0 = σ0 /(μn E ) after laser illumination if we assume that
which has variance σ
drift dominates during carrier generation. Thus, the convolution of χ(x)
and n̂(x, t ) is also Gaussian
Q inj
−(x − x 0 )2
n(x, t ) = ,
(4.34)
exp
2
σ2 (t )
2π
σ(t )
(t ) =
where the variance σ
20 + 2D n t now becomes time dependent.
σ
4.2.5 Carrier extraction
In the previous section, we have neglected the drift of the charge carriers
in the applied electric field. With both drift and diffusion present, the peak
of the charge cloud will drift with velocity μn E and the induced current is
proportional to the moving injected charge between x 1 = 0 and x 2 = d since
from Eq. (4.26) we have
qE d
μn n(x − μn E t , t ) dx
j (t ) =
d 0
if we assume a constant electric field.
It is now easiest to consider the charge cloud peak like in Eq. (4.34) as
fixed in space and instead regard the back contact at x 2 = d as moving with
time such that x 2 = d − μn E t . With this picture in mind and assuming that
47
all injected charge gets collected, it is obvious that
d −μn E t
qE
n(x, t ) dx
j (t ) =
μn
d
−μn E t
d −μn E t
Q inj
−(x − x 0 )2
qE
exp
=
μn
dx
d
2
σ2 (t )
−∞
2π
σ(t )
where we have extended the lower integration bound to infinity which is
allowed if the electrons get extracted at the back contact, i.e. drift is dominating over diffusion. And further
Q inj
qE
d − μn E t − x 0
j (t ) =
+1
μn
erf
d
2
2
σ(t )
qEQ inj
t1 − t
=
μn erf +1 ,
2d
2σ(t )
with t 1 = (d − x 0 )/(μn E ) being the time-of-flight and σ the “temporal” variance which reads
20 + 2D n t
σ
(t )
σ
2D n
2kT
2
=
= σ0 +
σ(t ) =
t = σ20 +
t.
2
μn E
μn E
(μn E )
qμn E 2
If we neglect diffusion during carrier extraction at t ≈ t 1 we can write
qEQ inj
t1 − t
μn 1 + erf j (t ) =
,
(4.35)
2d
2σ1
with
σ1 ≈ σ(t 1 ) =
σ20 +
2kT
t1 =
qμn E 2
σ20 +
2kT (d − x 0 )
qμ2n E 3
.
(4.36)
Written in this way, we can easily compare Eq. (4.30) with Eq. (4.35) and see
that they can smoothly be connected.
4.2.6 Full transit signal and fast processes
The current signal of the complete electron transient (under the assumptions made in the previous sections) has all the features involving generation, trapping, space charges, broadening by diffusion, and carrier extraction. Clearly, before generation and after extraction the current is zero. A
function combining all these features is
μ ρ qμn E
t − t0
t1 − t
− κn + nε sc (t −t 0 )
j (t ) =
1 + erf Q inj e
1 + erf .
4d
2σ0
2σ1
(4.37)
48
In the previous sections 4.2.3 to 4.2.5 we have only focused on electron
transit and the case of a negative voltage applied to the front contact. But
the same discussion can be done for hole transit in a completely analogous manner. For example, the hole transit with trapping and homogenous
space charge density is
j (t ) =
qμp E
d
Q inj e
μp ρ sc
− κp − ε
t
,
(4.38)
such that for the full transit we obtain (with t 1 and σ1 now depending on μp
instead of μn )
μp ρ sc
qμp E
t − t0
t1 − t
− κn − ε
(t −t 0 )
j (t ) =
1 + erf Q inj e
1 + erf .
4d
2σ0
2σ1
(4.39)
If the charge does not get extracted at the back contact but at the front
contact, then the path travelled is not d − x 0 but only x 0 . For just-abovebandgap photons, x 0 is only a few micrometres and it would contribute to
the current signal by a narrow peak only for t t 0 .
Other fast processes which contribute to the current signal (if the
bandwidth of the amplifier chain allows it) are the photoelectric effect and
plasma relaxation. These processes occur at a timescale much shorter than
nanoseconds and therefore give rise to a current that follows the laser
intensity, which we assumed as Gaussian. If we want to incorporate these
fast processes into our model, we see from Eq. (4.29) that a Gaussian term
with variance σ0 and centred at t 0 has to be added. An example curve of
the full transit including fast processes is shown in Fig. 4.1.
-4
Measurement
Fit
-3
Current (μA)
fast processes
(Gaussian shape)
carrier generation
-2
carrier transit
extraction of
diffusion-broadened
carrier sheet
-1
time-of-flight
0
0
200
400
600
800
Time (ns)
Figure 4.1: Example curve of full transit including fast processes.
49
4.3 Electrical field profiling
In this section it will be shown how a good estimate of the space charge conx ) can be probed even if it is not homogeneously distributed
centration ρ sc (
as we had assumed in Sec. 4.2.3. The idea of this so-called “electrical field
profiling” is to translate the time-varying current induced at the contacts
into the spatial profile of the electric field.
For low injection we can assume that the injected carriers form a thin
sheet, i.e., we neglect the initial width of the carrier cloud and also carrier
diffusion. For a homogenous sample and a low electric field applied in
the x-direction, the charge transport can be (like before) considered
one-dimensional, and the mobility can be assumed as constant. If trapping
can be neglected, the time-dependent current I measured at the contacts
equals then
Q v d (t )
,
(4.40)
I (t ) =
d
with Q and v d being the total injected charge and v d (t ) the now
time-dependent drift velocity of the carrier sheet, respectively. Using
Eq. (2.6) one can write
I (t ) d
(x) =
ê x .
(4.41)
E
μQ
By simply integrating the drift velocity, time can be translated into space
and we obtain the distance x travelled by the sheet in time t
t
d t
v d (t )dt =
I (t )dt .
(4.42)
x=
Q 0
0
The total charge Q can be obtained by integrating the measured current
over the full transit and the mobility μ follows from the ToF as before.
versus x using Eqs. (4.41) and (4.42) the electric field
Finally, by plotting E
as a function of the distance from the illuminated contact is obtained. It is
ap and the field arising from trapped space
the sum of the applied field E
sc . The space charge concentration ρ sc can be determined from
charges E
Gauss’ law
ρ (x)
sc (x) = sc
∇·E
(4.43)
εr ε0
with εr = 5.7 the (relative) dielectric constant for diamond.
In the case of a homogenous distribution ρ sc (x) = ρ sc , the space charge
electric field E sc is given by Eq. (4.31) and it turns out that (under the assumptions made) the slopes of the (tilted) plateaus in Fig. 4.2 have to be
multiplied by the absolute permittivity in order to obtain the corresponding homogenous space charge concentrations.
50
700
- 0.83 nC/cm3
0.00 nC/cm3
+0.91 nC/cm3
E (V/cm)
600
500
400
300
200
0
100
200
300
400
500
600
700
x (μm)
Figure 4.2: Electric field as a function of the distance from the illuminated contact
for three different space charge concentrations.
4.4 Carrier transit simulations in 2-D
The previous one-dimensional simplification was motivated by the wish to
derive analytical solutions. A more realistic scenario has to take additional
dimensions into account. If the conditions vary in the direction perpendicular to the applied electric field, the 1-D model breaks down and we have
to solve the coupled Poisson’s and drift-diffusion equations using numerical tools such as the finite element method (FEM) or simulate the carrier
interactions with, for instance, the Monte Carlo (MC) approach.
This section presents two examples of FEM simulations of the carrier
transit using the modelling software COMSOL Multiphysics for a
two-dimensional geometry with d = 500 μm. The two previously discussed
cases (see Sec. 3.2) of low and high injection are shown in Figs. 4.3 and 4.4,
respectively.
The sample is illuminated from the left side with a short pulse of above
bandgap UV light at time t = 0 and a constant bias voltage of 30 V is applied (at contacts not covering the full side). The hole concentration and
equipotential lines are shown for six successive times in both cases. Since
the injected charge differs by three orders of magnitude, different colour
scales are used for the two figures.
It can be seen that for low injection the electric field is not changed during the transit of the charge cloud. However, for high injection the equipotential lines are clearly affected by the big charge reservoir created at the
illuminated contact. This reservoir is still present long after the first carriers
have passed through the sample.
51
t =0
t = 4.4 ns
t = 8.8 ns
t = 13 ns
t = 17 ns
t = 22 ns
Figure 4.3: Hole transit obtained from simulation for the case of low injection. The
colours in the surface plots correspond to the hole concentrations (black for low,
red for medium and yellow for high concentrations). The green curves are equipotential lines.
52
t =0
t = 2.2 ns
t = 6.6 ns
t = 11 ns
t = 16 ns
t = 22 ns
Figure 4.4: Hole transit obtained from simulation for the case of high injection presented in the same way as in the previous plot. However, a different colour scale
was used since the injected charge is a thousand times higher.
53
5. Electron cascades in diamond
A brief description of diamond in comparison to silicon as a detector material will be given in the first section of this chapter. The second section deals
with the average pair-creation energy in diamond, a material property of
which a detailed knowledge is both essential for detector applications and
for biomolecular imaging by ultra-short X-ray pulses. The last section discusses impact ionisation and avalanche breakdown at high electric fields.
5.1 Diamond as a detector
Diamond has many attractive properties as a radiation detector because
of its high radiation hardness, low leakage current, high-temperature operation and high chemical resistance [39]. The strong sp 3 carbon-carbon
bonds making diamond the hardest of all materials also result in a very high
damage threshold for ionising radiation such as X-rays, heavy ions, α- or
β-particles [40]. That is why diamond films are highly suitable for particle
detector applications, especially for those with strong particle fluxes or if
fast response and long device lifetime is crucial.
Diamond detectors have several advantages over the standard devices
made of silicon. Table 2.1 shows that the material properties of diamond
for high voltages and high temperatures surpass those of silicon. Most notably, the electric field breakdown strength is more than 60 times higher and
the thermal conductivity more than 15 times higher than the corresponding values for silicon. Diamond’s wide bandgap gives rise to a maximum
operating temperature above 1000 ℃ compared to a maximum of 200 ℃
for silicon.
The high thermal conductivity and maximum operating temperature allows diamond to be used in hot environments and for strong particle fluxes.
Since diamond is chemically inert, it does not have to be protected from
contamination and can be used in hostile environments.
A high breakdown field allows a strong electric field across the detector. Together with SC-CVD diamond’s high saturation drift velocity, high
carrier mobilities [7], long carrier lifetime and high charge collection efficiency [41], this leads to the possibility of rapid and complete extraction of
the generated charge, i.e. a radiation sensor with ultra-fast response.
55
Diamond’s large bandgap of 5.47 eV results in solar blindness and low
leakage currents from thermal excitation during room temperature operation. The trade-off is the comparatively small signal from an incident particle, since less ionisation is caused than in narrow bandgap semiconductors.
Silicon devices show a significant degradation in performance from radiation damage after a much shorter operation period than diamond devices [42]. Approximately twice the number of vacancies are created in silicon compared to diamond. At room temperature, vacancies in silicon migrate to form complex defects with dopants, whereas in diamond, these vacancies are immobile. Therefore, the effect of the radiation damage in diamond is much less severe than equivalent damage would be in a silicon
device [43].
Diamond has the highest density of atoms of any material. However, carbon has also a low atomic number, and the cross-section for interaction
with neutrons is very low. A particle will therefore have a long path length
in diamond without being scattered significantly as it passes through. Even
if the track of the particle can be determined, it will not have been deflected
much from its original path. That is why diamond is very suitable as a tracking detector material, e.g. in high energy physics experiments and beam
monitoring [44].
A further advantage of diamond follows from its atomic number which is
nearly tissue equivalent (Z = 6 against Z = 7.4 for biological tissue). Therefore, it could be used for applications in the field of dosimetry, when in vivo
measurements and a signal directly proportional to the absorbed dose rate
are required [45, 46].
5.2 Pair-creation from ionising radiation
When a semiconductor is exposed to high-energy radiation, electrons
are excited from the valence to the conduction band leaving holes which
also contribute to the conductivity. The average energy to create such
an electron-hole pair is called the pair-creation energy pc which can be
determined experimentally by exposing the sample to a pulsed radiation
source of known energy and measuring the created charge per event. This
is either done directly or indirectly by comparison to a simultaneously
exposed semiconductor of known pair-creation energy.
Numerous measurements have shown that the number of created pairs
per particle N is strictly proportional to the energy ΔE lost by an ionising
agent in the medium [47]. So the introduction of the pair-creation energy
〉
by pc = 〈 ΔE
N is well-defined and thus measures the average amount of energy given up by the incident radiation in the process for generating a single
electron-hole pair.
56
So far, for a given semiconductor and a given temperature pc has not
shown a large dependence on the type and the energy of the ionising radiation in experiments.1 Most spread in data might have come from poor sample quality and experimental setups with large systematical errors. Once
high-quality samples are available and accurate measurements can be carried out, the pair-creation values seem to match quite well (for silicon see
e.g. [49]).
It is rather amazing that pc does not seem to reflect whether the source
is a massive heavy particle (α-particle, heavy ion), a light massive particle
(β-particle), or a high-energetic photon (X-ray, γ-quantum). Since there is
no compelling reason for this phenomenon, it is important to measure pc
for all kinds of excitation sources and to compare these values. This was one
of the motivations to carry out the experiment described in Paper III.
A remarkable correlation exists between the pair-creation and the
bandgap energy [50]. Many common semiconductors, such as Ge, Si, and
GaAs obey the relation pc (T ) = 14
5 E g (T ) + 0.6 eV. At the time this was
discovered, there were only phenomenological models available, to explain
this relation, since few details of the processes through which incoming
radiation creates electron-hole pairs were known [51–54]. A fundamental
assumption of these models is that the average amount of radiation energy
consumed per pair can be accounted for by a sum of three contributions.
Besides the loss of energy from excitation over the bandgap, there is also
the thermalisation loss from residual kinetic energy and the loss from
optical phonons, which is generally small. For diamond the equation
above yields 15.9 eV.
Models describing the electron cascades in diamond have been developed over the last decade (see e.g. [55, 56] and references therein). The spatiotemporal evolution of the ionising cascades can be obtained from Monte
Carlo simulations, which explicitly include diamond’s band structure, elastic and inelastic scattering of secondary electrons (e.g. Auger electrons or
photoelectrons) with energies over a wide energy range. The average paircreation value obtained from simulations built on the WTPP-2 model [56]
is 12 eV. This value could be confirmed by the results of Paper III.
The molecular imaging community has a big interest in the investigation
of how well simulations describe electron cascades in biological materials.
Diamond is considered a model-compound for studying the tempo-spatial
investigation of secondary electron cascades after exposure to ultra-short
X-ray pulses in a nearly tissue equivalent material. If predictions for diamond turn out correct, the models underlying the simulations can most
likely be applied to biomolecules such as proteins. The aim is to image biological materials by X-ray diffraction at the free-electron laser (XFEL) facil1
This is only true if a certain threshold energy is exceeded. For example, E > 50 eV is needed
in the case of silicon at room temperature in order to obtain the constant value of 3.6 eV [48].
For smaller values, pc varies between 2.5 and 4.4 eV.
57
ities currently under construction. The maximum resolution today is limited by damage. Therefore, a detailed knowledge of the Coulomb explosion
of molecules by secondary electron cascades is crucial.
5.3 Impact ionisation
Electron cascades in a semiconductor can not only be generated by ionising
radiation but also by acceleration of charge carriers in high electric fields.
If the field is sufficiently high, electrons and holes attain enough energy to
create new electron-hole-pairs by what is called impact ionisation.
Impact ionisation is the process in a material by which an energetic
charge carrier can lose kinetic energy through the creation of other charge
carriers. It can only occur when the particle gains (at least) the threshold
energy for ionisation from the electrical field. The ionisation rate α is
defined as the number of electron-hole pairs generated by a carrier per
| and different for electrons (αn )
unit distance travelled. It is a function of |E
and holes (αp ).
If impact ionisation occurs in a region of high electrical field, it then can
result in a process called avalanche breakdown. At breakdown, the electric
field frees bound electrons, and if the electric field is sufficiently high, free
electrons may become accelerated to velocities that can liberate additional
electrons during collisions with neutral atoms or molecules. A simulation
of breakdown in diamond is shown in Fig. 5.1.
This is the mechanism that causes breakdown in devices at high fields
but that can also be used to increase the current in diodes and switches by
orders of magnitude. In an avalanche photodiode, for instance, the original
charge carrier is created by the absorption of a photon. This small optical
signal is then amplified by avalanche breakdown before entering an external electronic circuit. Since breakdown occurs quite abruptly (typically in
nanoseconds), resulting in the formation of an electrically conductive path,
it is possible to exploit this process for ultra-fast optical switches.
Regarding device design, it is of great importance to know precisely at
what field strength avalanche breakdown sets in. For diamond, values in
the literature range from 0.1–3.0 × 107 V/cm for certain structures made of
natural [57] and SC-CVD diamond [58]. However, no detailed information
on impact ionisation coefficients in diamond is presently available
because several difficulties have to be mastered when measuring these
material constants. Besides the high electric field necessary for breakdown
which makes measurements complicated, there is also still the problem
of a too high defect density in presently grown SC-CVD samples which
decisively facilitates breakdown.
The setup from Paper II which was used to measure the avalanche carrier
multiplication is shown in Fig. 5.2. Here, an α-source is brought in direct
58
t = 0.1 ns
t = 0.3 ns
t = 0.4 ns
Figure 5.1: Simulated hole transit in the case of a high electric field. The plots of the
hole concentration for three consecutive times were obtained such as in Figs. 4.3
and 4.4. The sample thickness is 50 μm and the applied bias 9 kV. The multiplication of charge carriers due to avalanche breakdown can be observed.
contact with a SC-CVD i -p diode structure that can be strongly reversed
biased by applying a high positive voltage to the metal probe which has
the α-source incorporated. Electron-hole pairs are then created along the
path of the particle and the collected charge per α-event is recorded using
a charge sensitive amplifier. At high bias, additional electron-hole pairs are
created by impact ionisation and the coefficients can be extracted from a
Q-V (collected charge versus bias) plot.
241
Am
Au
to amplifier
Au
D
i
oil
EHPs
p+
Ti/Al anode
Figure 5.2: Setup to measure impact ionisation coefficients in diamond.
59
6. Summary of papers
The research presented in the seven papers covers different aspects of diamond as a semiconductor material with a focus on charge transport. What
they have in common is that they deal with experimental investigation of
diamond’s electronic properties, which are important to know for design
and development of devices.
Paper I presents how the time-of-flight method can be applied not only
to study the charge carrier mobilities but also to gain information about the
electric field distribution in the sample. Space charge concentrations down
to approx. 1 nC/cm3 can be detected with this technique. It is important
to have detailed knowledge about trapped charge concentration and thus
obtain spatial information about bulk defects in order to improve the CVD
growth process of single-crystalline diamond. Optical methods by luminescence measurements usually incorporate surface and near-surface defects
from polishing, such that knowledge of crystal quality along the growth direction is rather limited. The information about bulk quality which can be
obtained by electric field profiling is essential for all type of devices. Defects generally downgrade the excellent electrical properties of diamond
such as high breakdown field, high saturation velocity, long carrier lifetime
and high carrier mobilities.
The author of this thesis made only minor contributions to this paper
since he had just started his PhD. Published in Semiconductor Science and
Technology.
Paper II investigates the high-field electrical transport properties in SCCVD diamond. The breakdown field strength ultimately determines the upper voltage limit for high voltage devices and the upper cut-off frequency in
high-frequency field effect transistors. Detailed knowledge is also essential
for devices where breakdown at a certain voltage is desired such as fast optical switches or high-voltage diamond Zener diodes. The paper describes a
working setup using α-particle injection and reports the observation of carrier multiplication. However, due to a relatively high leakage current, a low
signal-to-noise ratio is obtained and hence the ionisation coefficients could
not be precisely determined. This should be possible by using structures of
ultra-high quality with an effective edge termination and thus increasing
the signal-to-noise ratio by reducing the charge injection at surface defects
and edges.
61
The author has contributed to this paper by setting up the experiment,
writing the data acquisition program and performing the measurements.
Published in Advances in Science and Technology.
Paper III resulted from a large international cooperation including the
Molecular Biophysics group at the Biomedical Centre in Uppsala. The aim
was to measure the average pair-creation energy in diamond for direct
comparison with simulations. For ionisation, the Sub-Picosecond Pulsed
Source at the Stanford Linear Accelerator Center was used, providing
femtosecond X-ray pulses (∼ 100fs, 8.9 keV) with approx. 106 photons
per shot. The measurement was done indirectly using a silicon detector
of known pair-creation energy. Furthermore, the carrier drift mobilities
in diamond were measured by a time-of-flight setup. The obtained
experimental result (
pc = 12 eV) is in good agreement with Monte Carlo
simulations for secondary electron cascades in single-crystalline diamond.
Showing reliable predictions for diamond, the models which underlie the
simulations can most likely be applied to biomolecules, too.
The author of this thesis has been deeply involved in the experimental
setup, the measurements at the Stanford Linear Accelerator Center, the data
analysis, preparing the figures and writing the paper. Published in the Journal of Applied Physics.
Paper IV investigates the compensation ratio in boron-doped CVD
diamond samples. Low compensation is very desirable for diamond
devices in order to obtain a reasonable trade-off between reduced mobility
and increased free charge carrier densities. Record-low compensation
ratios (ND /NA < 10−4 ) are observed by a best fit of measured hole
concentration data as a function of temperature from Hall-effect
measurements over a temperature interval to theoretical predictions.
The obtained values are verified by results from Secondary Ion Mass
Spectrometry (SIMS) and Capacitance-Voltage (C -V ) measurements on
rectifying Schottky junctions.
The author has contributed to this paper by performing the measurements on the Schottky junctions and analysing the data from Hall-effect,
SIMS and C -V measurements. He has done most of the writing and prepared all the figures. The paper was presented by the author at the Diamond Workshop 2008 (SBDD XIII) in Hasselt. Published in Physica Status
Solidi (a).
Paper V presents a measurement system for lateral ToF charge carrier
transport studies in intrinsic diamond which is particularly suited for studies of thin layers and the influence of different surface conditions on transport dynamics. The use of reflective UV-optics allows for imaging the sample in UV or visible light without any dispersion, which has proven useful
for sample positioning and aligning the line focus in parallel with the con62
tacts. A clear hole transit signal was observed in one sample which showed
a near-surface hole drift mobility of about 860 cm2 /Vs.
The author has been involved in the setup of the measurement system,
the measurements and the data evaluation. He has contributed to preparing both the text and the figures. Published in Diamond & Related Materials.
Paper VI is a study of electron transport in isolated conduction band valleys across macroscopic distances, which is possible due to the very low
scattering cross section for intervalley scattering in SC-CVD diamond at
70 K. The ratio between longitudinal and transverse conduction band effective masses is found to be m l∗ /m t∗ = 5.2. For higher temperatures, in
the range 110–140 K, a negative differential mobility (NDM) is observed for
electrons with the electric field parallel to the crystallographic 〈100〉 direction. The NDM can be explained in terms of valley repopulation effects between the equivalent energy conduction band minima.
The author of this thesis has made major contributions to building the
experimental setup, has written the data acquisition program, performed
the measurements and evaluated the data. He was also involved in writing
the paper which was submitted to Nature Materials.
Paper VII presents electron and hole drift velocities for the temperature
interval 83 K ≤ T ≤ 460 K and electric fields between 90 and 4 × 103 V/cm
which were applied in the crystallographic 〈100〉 direction. The study is performed in the low-injection regime in order to perturb the applied electric field only minimally. The temperature dependence of electron and hole
low-field drift mobilities indicates that acoustic phonon scattering is the
limiting mechanism for T ≤ 280 K because the mobility follows the typical
T −3/2 dependence.
The author has been responsible for carrying out the measurements,
evaluating the data, preparing the figures and writing the paper. Submitted
to Physical Review B.
63
7. Summary of results and discussion
This section briefly summarises the results of the experiments described in
detail above, and discusses their implications
Electric field profiling
It was possible to extend the utility of the time-of-flight method for investigating both the drift mobilities and the profile of the electric field distribution in a single measurement. Space charge concentrations down to
∼ 1 nC/cm3 were detected by this method. This charge density corresponds
to an ionised impurity concentration of 6×109 cm−3 , and the obtained resolution of the field profile is approximately the actual thickness of the carrier
sheet (in the cases studied around 50–100 μm).
It was found that successive hole transits do not appreciably affect the
electric field distribution within the sample. Transits of holes can therefore
be used to probe the electric field distribution and also the spatial distribution of trapped charge – at least in the samples investigated so far. Electron
transits, on the other hand, cause an accumulation of negative charge.
Impact ionisation coefficients
An experimental setup for measuring the impact ionisation coefficients in
diamond was proven to work in principle. The method uses α-particles to
induce electron-hole pairs in Schottky m-i -p + (SMIP) diodes in order to
make charge multiplication measurements.
In the reversed biased diode, electron-hole pairs are created along the
path of the particle, and the collected charge per α-event is recorded using
a charge sensitive amplifier. At high bias, additional electron-hole pairs are
created by impact ionisation (see Fig. 2 in Paper II) and information about
this process can be extracted by measuring the collected charge versus applied bias. Assuming that the holes dominate the impact ionisation and fit |) [59] possible ionisation
ting to Chynoweth’s equation αp = a p exp(−b p /|E
coefficients are obtained. However, a reliable result could not be obtained1 ,
most likely because of charge injection at the contacts and the affiliated
low signal-to-noise ratio. This has to be further improved by using samples
where special care has been taken to keep defects from surface treatment
to an absolute minimum.
For example both the pairs a p = 4.0×106 cm−1 , b p = 1.1×107 V/cm and a p = 6.0×105 cm−1 ,
b p = 0.8 × 107 V/cm, are consistent with the measurements.
1
65
Average pair creation energy in diamond (eV)_
26
24.5
24
HTHP
natural diamond
CVD
theory
22
20
18.5
18
16
13.1
14
13.2
12.8
13.6
13.1
12.9
12.2
12.5
11.8
12.0
11.0
12
10.8
10
P-2
TP
)
88
(19
)
83
19
g(
Ali
)
80
19
g(
Ali
)
05
20
a,
lph
i (a
rsk
mo
)
Po
96
19
a,
ph
(al
ko
ne
Ka
)
79
19
ta,
be
li (
na
Ca
)
79
,19
ha
alp
li (
na
Ca
)
75
19
ta,
be
v(
zlo
Ko
5)
97
,1
ha
alp
v(
)
zlo
64
Ko
19
a,
ph
(al
ale
-M
an
)
De
59
19
ta,
be
y(
ed
nn
Ke
Figure 7.1: Comparison of previously reported values of the average pair-creation
energy in diamond. Values on the left were obtained from experiments and values
on the right from calulations/simulations. This figure is taken from Paper III.
Average pair creation energy and mobility
The average pair creation in diamond could be measured by using a femtosecond pulsed X-ray source in order to create electron-hole pairs. The obtained value was around pc = 12 eV, which is lower than previously reported values (see Fig. 7.1) but matches very well with theoretical predictions from Monte Carlo simulations [56].
Most likely we obtained this relatively low value of pc because of the excellent sample quality with very high charge collection efficiency. Poor sample quality results in short carrier lifetime, and prevents complete extraction of all charge created in the sample. As a consequence, the pair creation
energy appears to be higher. Previously examined samples in the literature
were mainly natural or HPHT diamond.
The measurement was done indirectly by comparison to a silicon detector. For obtaining a smaller relative error for the result of diamond, more
data should have been acquired, especially for the reference measurement
with silicon. This has not been done because of limited beam-time.
When the low field drift mobility was determined from time-of-flight
measurements, values lower than those in the space-charge limited case
with UV-excitation [7] were found2 . These values should be seen as lower
bounds because of the lack of knowledge about the amount of trapped
charge.
2
Around 2750 cm2 /V s for both holes and electrons compared to 3800 and 4500 cm2 /V s for
holes and electrons, respectively.
66
Table 7.1: Summary of the results from Paper IV with 95% confidence intervals.
Hall-effect measurement
C-V
SIMS
Sample
EA
in eV
NA
in cm−3
Comp.
ratio
NA -ND
in cm−3
[B]
in cm−3
1
0.37±0.02
4.8±3.7×1018
< 10−4
7±1×1017
1.5± 0.3×1018
2
0.34±0.01
1.5±0.5×1018
< 0.09
2.4±0.3×1018
–
0.36±0.01
18
−4
18
3
2.9±0.9×10
< 10
1.8±0.2×10
3.9± 0.8×1018
Compensation in boron-doped diamond
The dependence of hole concentration on temperature was determined
from Hall measurements (see Fig. 1 in Paper IV). The carrier concentrations
NA and ND and the acceptor ionisation energy E A in different samples were
obtained for best fits to Eq. 2.9. Record-low compensation ratios showing
ND /NA < 10−4 were observed in two SC-CVD diamond samples.
These values are in agreement with the total boron concentration measured by Secondary Ion Mass Spectrometry (SIMS) and suggested that a
fraction of the boron is passivated by some mechanism. This interpretation
was confirmed by Capacitance-Voltage (C-V ) measurements on rectifying
Schottky junctions. As can be seen from Table 7.1 the values of the activation energy lie between 0.34 and 0.37 eV. They exhibit a dependency on
boron incorporation in agreement with [60].
Lateral time-of-flight measurements
The near surface hole drift mobility of an intrinsic SC-CVD layer could be
measured across a spacing of 0.3 mm using the lateral time-of-flight measurement system sketched in Fig. 7.2. The expression describing 1-D charge
transport in the low injection regime which was derived in Chapter 4 was
fitted to both FEM device simulations and the measurements and showed
excellent agreement with the data, see Fig. 6 of Paper V.
This means that even though this expression was derived in 1-D under
many assumptions, it describes the different mechanisms during charge
transport induced by a short laser pulse very well. This has also been confirmed for vertical ToF measurements, and thus fitting Eq. (4.37) to the data
is an accurate way of extracting the time-of-flight from the curve.
The measured near-surface hole drift mobility of about 860 cm2 /Vs was
relatively low compared to previous vertical measurements with material
of similar quality [7]. This could result from scattering effects at the surface
where the assumption of a periodic crystal lattice clearly is not fulfilled. Also
scattering due to defects in the growth direction, which is perpendicular to
the carrier movement, could play a role.
67
Nd-YAG
Laser 5xw
attenuator
interference
filters
21
3n m
m
q
Cylindrical
lens
Semitransparent
mirror
CCD
Reflective
focusing
optics
DSO
Preamp.
Bias
Sample
x-y-z
stage
Figure 7.2: Schematics of the lateral ToF setup. This figure is taken from Paper V.
Negative differential mobility and single valley transport
The time-of-flight technique was used to investigate the electron drift velocity in SC-CVD diamond for temperatures below 140 K. Negative differential mobility (NDM) for electric fields in the range 300-600 V/cm was found
(see Fig. 1 in Paper VI).
In element semiconductors like silicon and diamond, the low-lying valleys in the conduction band are equivalent and NDM can occur when the
field is aligned in certain crystallographic directions. If we apply the electric
field, for instance, along the [100] axis, electrons in the two parallel valleys
will respond to the field with the longitudinal effective mass m l∗ and electrons in the other four perpendicular valleys will respond with the transversal effective mass m t∗ . The difference between longitudinal and transversal
effective masses results in a heating of the electrons in the four perpendicular (hot) valleys, which are then depleted while the two parallel (cool)
valleys are enriched. In the case of very low scattering cross section for intervalley scattering and low electric fields, the heated valleys will not be
completely depopulated and the transit signal of pure (hot) valley transport
could be observed. The related electron drift velocities at 70 K for both hot
and cold conduction band valleys are shown in Fig. 2 of Paper VI. From the
values of the mobilities related to transversal and longitudinal transport,
the ratio of the effective masses could be determined to m l∗ /m t∗ = 5.2 ± 0.1.
This was the first time that electron transport in isolated conduction
band valleys across macroscopic distances has been observed. Further,
NDM due to equivalent valley repopulation has not been previously
reported at temperatures above 77 K, or observed in diamond before.
68
Electron and hole drift velocity measurements for a wide temperature range
Electron and hole drift velocities in SC-CVD diamond were studied for temperatures in the range of 83-460 K. These measurements should be understood as reference curves because they give an accurate overview of a wide
temperature range and for electric fields over two orders of magnitude, as
can be seen in Fig. 2 of Paper VII.
The three samples investigated show very little spread, which indicates
that the deposition process of the SC-CVD diamond layers has come to
a stage where it is possible to keep defects and impurities to an absolute
minimum, such that carrier transport is dominantly limited by unavoidable mechanism such as acoustic and optical phonon scattering. Also the
choice of contact metal – Ti/Al or Ni – did not have an impact on the results.
In addition to the drift velocities, the low field drift mobility was investigated. It showed the expected behaviour μn ∼ T −3/2 for temperatures below 280 K. For higher temperatures the slope became steeper indicating the
onset of intervalley phonon scattering. This is in good agreement with previous findings [7, 61–63]. Also the measured hole drift mobilities agree very
well with the theoretical calculations of Reggiani et al. [63] – shown as the
dashed curve in Fig. 3 of Paper VII.
69
8. Conclusion
In the last decade, the diamond deposition process using CVD has constantly been improved, resulting in millimetre-thick layers of high-purity
single-crystalline diamond. The quality of the SC-CVD diamond samples
was studied in this thesis by time-of-flight measurements, breakdown
measurements, secondary electron cascades initiated by X-rays, Hall-effect
measurements as well as C -V and SIMS measurements. Excellent electrical
properties of some of the examined samples were confirmed by several
different indicators, such as high charge carrier mobilities, high breakdown
fields and efficient p-doping with low compensation ratios.
Further, it was possible to observe electron transport in isolated conduction band valleys across macroscopic distances. This is most likely due to
the very low cross section for intervalley scattering in single-crystalline CVD
diamond and indicates very low defect concentrations.
A measurement system for lateral time-of-flight charge carrier transport
studies has been constructed and tested successfully. In this setup the carriers travel close to the surface and are therefore more influenced by the
surface properties as compared to the vertical configuration where they
travel through the bulk of the sample. This makes the lateral setup suitable
to study transport in thin layers and the effects of different surface conditions on carrier transport.
Charge transport upon laser illumination on diamond could be successfully modelled using both an analytic solution in the low injection case and
the more general finite element method approach.
Despite the recent advances, some major steps for diamond device design
still have to be taken:
– The low activation at room temperature due to deep dopant levels might
be circumvented by so-called δ-layer structures1 .
– For high voltage-diodes a more efficient way to prevent the device from
breakdown at the edge of the Schottky contact has to be developed.
– The surface treatment for free-standing samples has to be improved in
order to reduce the defect density just below the surface.
1
An ultra-thin layer of highly doped material is situated under an intrinsic layer. By diffusion
of charge carriers into the intrinsic layer, a high mobility conduction channel is created. This
structure could be the solution for designing diamond transistors.
71
9. Suggestions for future work
Our groups plans or already conducts several projects addressing unsolved
problems for diamond device design.
9.1 Investigation of δ-doped structures
The intention is to design lateral p-channel MESFET devices on layered
diamond structures, developed especially for this purpose. The simplest
type of such a transistor consists of a thin layer of boron-doped diamond
grown epitaxially on intrinsic or near-intrinsic diamond as shown in
Fig. 9.1. The boron is introduced during CVD epitaxy and is activated by
high-temperature annealing. This technique gives a doped layer with good
activation and high Hall mobility [64]. Such MESFETs are expected to
show reasonable high frequency performance and it should be possible to
achieve a cut-off frequency f T ∼ 20 GHz from this type of device.
++
++
pp++
source
i
gate
drain
Nd > 1020 cm-3
d ~ 2 nm
intrinsic
Figure 9.1: Schematic sketch of a δ-doped diamond MESFET structure.
With the above described δ-doped devices, much better roomtemperature performance is achievable. Therefore, the main focus of this
activity is to investigate such devices consisting of intrinsic diamond with
an extremely thin heavily boron-doped layer. With doping levels above the
Mott transition in the thin layer, almost complete activation of the dopant
can in principle be achieved. Conduction takes place in a thin intrinsic
layer, where the hole mobility is extremely high.
Significant progress in making pulse-doped layers in diamond has already been accomplished and lateral time-of-flight measurement of prototype δ-doped samples have been performed by injection created from
UV-laser pulses.
73
9.2 High-voltage low-loss converters
In order to demonstrate the potential of CVD diamond rectifiers, another
project will initially focus on the development of a 1.7kV-50A Schottky
diode using innovative device structures specifically designed for diamond
devices. This includes ramp termination that significantly reduces the
electrical field at the edges of the Schottky contacts in order to prevent the
device from early breakdown. Once the successful operation of the 1.7 kV
device has been demonstrated, the boundaries of power semiconductor
technology will be pushed further, ultimately aiming to technically validate
a 20 kV Schottky diode.
Diamond diodes have significantly smaller losses and less cooling problems than those based on conventional semiconductor technology. This is
especially desirable in high-power applications but also for power plants
using renewable energy sources.
At the Division for Electricity at Uppsala University, we have several ongoing projects for developing and constructing power plants driven by renewable energy sources such as waves, underwater currents or wind. This
research resulted for instance in a successful installation of several wave
power generators which represent the start of a test farm currently under
construction. Since the output of such a generator varies both in frequency
and amplitude, it cannot be fed directly into the electrical grid which is restricted to 50 Hz [65]. Instead, the alternating current (AC) signals must first
be converted to direct current (DC) signals before finally being converted to
the desired grid compatible AC parameters. All these conversions cannot be
achieved without a certain amount of losses at each step. This means that
the addition of many small distributed and to some extent unpredictable
sources to the grid will not be possible or economically feasible without the
development of efficient converters (AC-AC, AC-DC and DC-DC) with very
low losses and built-in intelligence. Such converters are not only necessary
in renewable energy conversion. Other applications include electric or hybrid vehicles and high voltage DC transmission.
9.3 Time-resolved study of electron cascades
Furthermore, experiments at the Lund Laser Centre (LLC) and at the FLASH
facility of the Deutsches Elektronen Synchrotron (DESY) in Hamburg are
planned. The intention is to study the temporal evolution of electron cascades at femtosecond and picosecond timescales in order to test present
theoretical models of cascade evolution [56]. In the experiment underlying Paper III, the model was only tested for the total number of generated
charge, but not the explicit temporal evolution of the cascade.
74
m
fs probe
laser
Delay
m
Dt
m
Detector
m
fs pump
target
Etrans
Figure 9.2: Setup for time-resolved study of electron cascades in diamond.
The idea is use a pump-probe setup with a short (∼ 50 fs) probe pulse
800 nm optical laser that is synchronised with the VUV pump pulse. A possible experimental configuration is depicted schematically in Fig. 9.2. A thin
diamond target is mounted on an x-y-stage in order to align it with the
pump beam and with the probe laser incident at the Brewster angle. The
absorbed X-ray photons create electron-hole pairs in the diamond detector which changes its optical properties. The shift in reflectivity and transmission can be observed by using the synchronised optical laser with adjustable time-delay. The charge created by the pump pulse can also be collected at metal electrodes deposited on the sample in order to measure the
total collected charge. An alternative approach is to use the VUV pump
pulse in order to create an optical grating from which the probe pulse is
diffracted.
Model calculations of the shift in reflectance due to the fully developed
electron cascade indicate that a clear shift in optical properties is readily detectable for intensities above ∼ 1–10 mJ/cm2 . This is clearly below the damage threshold, which is extremely high for diamond. By varying the time delay between the pump pulse and the optical probe beam, and by comparing
to model calculations, the temporal evolution of the electron cascade and
the recombination rates can be probed experimentally.
75
Acknowledgements
First of all I would like to thank my supervisor, Assoc. Professor Jan Isberg,
for giving me the opportunity to conduct this work under his excellent guidance and for his great support in all situations. Many thanks also to my assistant supervisor, Professor Mats Leijon, the head of the Division for Electricity, for providing a stimulating and encouraging atmosphere to work in.
My room-mates, Saman Majdi, Florian Burmeister and Kiran Kumar
Kovi, are gratefully acknowledged for excellent teamwork in the diamond
project.
Professor Janos Hajdu from the Molecular Biophysics group is acknowledged for making it possible to carry out an experiment at the SPPS in Stanford. At the same time, I would like to thank Erik Marklund, Carl Caleman
and Magnus Bergh for good collaboration in this project.
Many thanks also to the staff of the Ångström Microstructure Laboratory,
especially Rimantas Bručas, for being a source of great help and advice with
cleanroom processing.
Special thanks furthermore to Gunnel Ivarsson, Christina Wolf, Thomas
Götschl and Ulf Ring for their competent help with administrative, computer and construction issues.
And a big Thank You! to all the colleagues at the division for always being
helpful and making this such a warm and inspiring place.
I would also like to thank Vetenskapsrådet, Ångpanneföreningens
Forskningsstiftelse, Stiftelsen för Miljöstrategisk Forskning, Element Six
Ltd. and the Ericsson Research Foundation for their financial support. Die
Studienstiftung des Deutschen Volkes is also acknowledged.
Many thanks to all my friends for being very understanding and caring under the time of writing this thesis.
Lastly, I am deeply thankful to my partens and sister for their love, encouragement and great support. Ich danke euch von ganzem Herzen für alles!
77
Summary in Swedish
Laddningstransport i enkristallin CVD diamant
Diamant är, som de flesta känner till, det hårdaste materialet och används
förutom som ädelstenar även i en del industriella applikationer, t.ex. i
borrkronor och vid glasslipning. Mindre känt är att diamant också har en
mängd andra extrema egenskaper som gör det till ett unikt och intressant
material i elektroniska tillämpningar. Speciellt gäller det komponenter
som är avsedda för så extremt höga frekvenser, höga elektriska spänningar
eller strålningsdoser där andra material, t.ex. kisel, inte duger.
Under de senaste åren har det skett en snabb utveckling av den
syntetiska diamantframställningen, vilket möjliggör tillverkning av
millimetertjocka skivor av högkvalitativ enkristallin (Single-Crystalline
= SC) diamant i en så kallad kemisk ångdeponerings (Chemical Vapor
Deposition = CVD) process. I denna process växer man diamantskivor
på ett substrat av enkristallin diamant i en vätgas- och metanatmosfär.
Sedan separeras skivorna från substratet som då kan återanvändas.
Syntetisk diamant tillverkad på detta sätt får en mycket högre renhet och
kristallkvalitet än både naturlig och syntetisk diamant framställd med
andra metoder. Detta är viktigt eftersom materialet måste ha en extremt
hög renhet för att kunna användas i elektroniska tillämpningar. För att det i
praktiken ska bli möjligt att utveckla fungerande komponenter av diamant
krävs dock mycket bättre kännedom om diamantens grundläggande
elektriska egenskaper än vad vi har i dag.
I och med att högkvalitativt material har blivit tillgängligt kan
materialets egenskaper studeras bättre än tidigare. Inget material kan
dock tillverkas helt utan kristalldefekter och störämnen. Dessutom kan
man avsiktligt tillsätta störämnen för att påverka materialets elektriska
egenskaper; detta kallas för dopning av materialet. För att kunna ta fram
elektroniska komponenter i ett material måste man förstå exakt hur olika
defekter och störämnen påverkar materialets egenskaper.
Time-of-flight (ToF) tekniken, där man skapar fria laddningar med hjälp
av en pulsad strålkälla, är en metod för att studera laddningstransport i
halvledarmaterial. Principen är ganska enkel: En spänning läggs på över
provet och elektron-hål par skapas i närheten av en av två kontakter, t.ex.
med en UV laser. Sen mäts den inducerade strömmen när laddningarna rör
sig i materialet. Formen av den tidsvarierande strömmen ger information
om ett antal elektriska egenskaper hos provet.
79
Denna doktorsavhandlingens mål är att ta fram ny kunskap om hur elektriska laddningar skapas och rör sig i diamant. Genom att mäta egenskaper
såsom driftmobilitet, kompensationsförhållande eller den genomsnittliga
energin för att skapa ett elektron-hål par och jämföra dem med teoretiska
förutsägelser från simuleringar möjliggör kontroll av dessa modeller och
förbättring av diamant tillväxtprocessen.
Från de studier som presenteras i avhandlingen kan några slutsatser dras.
Dessa presenteras nedan i punktform
• Under det senaste decenniet har diamant tillväxtprocessen med CVD
ständigt förbättrats vilket har resulterat i millimetertjocka lager av högrent enkristallin diamant. Också effektiv p-dopning med låg kompensationsförhållande kunde visas.
• Kvaliteten på SC-CVD diamantprover studerades bl.a. med time-offlight mätningar, Hall-effekt mätningar samt I -V och C -V mätningar.
Utmärkta elektriska egenskaper hos vissa av de undersökta proverna
bekräftades av flera olika indikatorer, t.ex. hög laddningsbärarmobilitet
och låg kompensationsförhållande.
• Laddningstransport i diamant (injicerat med UV laser belysning) modellerades för svaga elektriska fält och liten injicerad laddning genom att
härleda en analytisk lösning till det system av differentialekvationer som
beskriver drift, diffusion och det elektriska fältet.
• I det generella fallet är det inte möjligt att lösa ekvationssystemet analytiskt men den s.k. finita element metoden (FEM) kunde användas för
att få laddningskoncentrationen som funktion av tid.
• Resultat från modelleringar överensstämde med mätningar. Det betyder
att dessa modeller beskriver de fysikaliska fenomenen väl.
• Mätningar av elektroners drifthastighet för låga temperaturer
(T < 140 K) visade negativt differentiell mobilitet (NDM) som är
en effekt inte observerats tidigare för diamant. Denna effekt är
välkänt för III-V halvledare och används i s.k. Gunn dioder för
mikrovågsapplikationer.
• För ännu lägre temperaturer (T < 80 K) kunde observeras elektrontransport i isolerade ledningsbandsdalar över makroskopiska distanser
p.g.a. ett mycket lågt tvärsnitt för spridning mellan olika dalar i
SC-CVD diamant. Med hjälp av detta fenomen kunde kvoten mellan
longitudinella och transversella effektiva massor noggrant bestämmas
till m l∗ /m t∗ = 5.2 ± 0.1.
80
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