Optical Studies
of Carrier Transport and Fundamental Absorption
in 4H-SiC and Si
Paulius Grivickas
Laboratory of Material and Semiconductor Physics
Department of Microelectronics and Information Technology
Royal Institute of Technology (KTH)
Stockholm 2004
Cover illustration: Motion of electron and hole in a bound
excitonic state in analogy to the classical mechanics.
Optical Studies
of Carrier Transport and Fundamental Absorption
in 4H-SiC and Si
A dissertation submitted to the Royal Institute of Technology (KTH)
for the degree of Doctor of Philosophy.
© 2004 Paulius Grivickas
Royal Institute of Technology (KTH)
Department of Microelectronics and Information Technology
Laboratory of Material and Semiconductor Physics
Electrum 229, SE-16440
Sweden
ISRN KTH/FTE/FR-2004/1-SE
ISSN 0284-545
TRITA FTE
Forskningsrapport 2004:1
Printed in 200 copies by Universitetsservice US AB, Stockholm 2004
To tenacious Queen of my life…
To tender Gold of my soul…
To warbling Lark of my mind…
and…
To sunrise Dew of my heart…
Abstract
The Fourier transient grating (FTG) technique and a novel spectroscopic
technique, both based on free carrier absorption (FCA) probing, have been applied to
study the carrier diffusivity in 4H-SiC and the fundamental absorption edge in 4H-SiC
and Si, respectively.
FTG is a unique technique capable of detecting diffusion coefficient
dependence over a broad injection interval ranging from minority carrier diffusion to
the ambipolar case. In this work the technique is used for thin epitaxial 4H-SiC layers,
increasing the time- and spatial-resolution of the experimental setup by factors of ~100
and ~10, respectively, in comparison to the established Si measurements. It is found
that the diffusion coefficient within the detected excitation range in n-type 4H-SiC
appears to be lower than the analytical prediction from Hall-mobility data. To explain
this, it is suggested that the minority hole mobility is reduced with respect to that of
the majority one or that the hole mobility value is in general lower than previously
reported. Observed differences between the temperature dependency of the ambipolar
diffusion and the Hall-prediction, on the other hand, are attributed to the unknown Hall
factor for holes and the additional carrier-carrier scattering mechanism in Hall
measurements. Furthermore, at high excitations a substantial decrease in the ambipolar
diffusion is observed and additionally confirmed by the holographic transient grating
technique. It is shown that at least half of the decrease can be explained by
incorporating into the theoretical fitting procedure the calculated band-gap narrowing
effect, taken from the literature. Finally, it is demonstrated that numerical data
simulation can remove miscalculations in the analytical Fourier data analysis in the
presence of Auger recombination.
Measurements with variable excitation wavelength pump-probe are
established in this work as a novel spectroscopic technique for detecting the
fundamental band edge absorption in indirect band-gap semiconductors. It is shown
that the technique provides unique results at high carrier densities in doped or highly
excited material. In intrinsic epilayers of 4H-SiC, absorption data are obtained over a
wide absorption range, at different temperatures and at various polarizations with
respect to the c-axis. Experimental spectra are modeled using the indirect transition
theory, subsequently extracting the dominat phonon energies, the approximate
excitonic binding energy and the temperature induced band-gap narrowing (BGN)
effect in the material. Measurements in highly doped substrates, on the other hand,
provide the first experimental indication of the values of doping induced BGN in 4HSiC. The fundamental absorption edge is also detected in highly doped and excited Si
at carrier concentrations exceeding the excitonic Mott transition by several orders of
magnitude. In comparison to theoretical predictions representing the current
understanding of absorption behavior in dense carrier plasmas, a density dependent
excess absorption is revealed at 75 K. Summarizing the main features of the subtracted
absorption, it is concluded that an excitonic enhancement effect is present in Si.
Contents
List of appended papers
ii
Acknowledgements
v
1 Introduction
1
2 Indirect semiconductors SiC and Si
2.1 Basic properties
7
2.2 Band structure and vibrational spectra
12
2.3 Band gap narrowing
17
3 Carrier transport
3.1 Mobility and diffusion
25
3.2 Carrier recombination
34
4 Optical absorption
4.1 Excitons
43
4.2 Fundamental absorption
53
4.3 Free carrier absorption
68
5 Measurement methods
5.1 FCA probing
77
5.2 Free carrier gratings
83
5.3 Spectroscopic measurements
90
6 Summary
95
i
List of appended papers
I
Free carrier diffusion measurements in epitaxial 4H-SiC with a
Fourier transient grating technique: injection dependence
P. Grivickas, J. Linnros and V. Grivickas
Mater. Sci. Forum 338-342, 671(2000)
II
Free carrier diffusion in 4H-SiC
P. Grivickas, A. Martinez, I. Mikulskas, V. Grivickas, J. Linnros, U.
Lindefelt and R. Tomašiunas
Mater. Sci. Forum 353-356, 353 (2001)
III
Carrier lifetime investigation in 4H-SiC grown by CVD and
sublimation epitaxy
P. Grivickas, A. Galeckas, J. Linnros, M. Syväjärvi, R. Yakimova, V.
Grivickas and J. A. Tellefsen
Mater. Sci. in Semic. Proc. 4, 191 (2001)
IV
Carrier diffusion characterization in epitaxial 4H-SiC
P. Grivickas, J. Linnros and V. Grivickas
J. Mater. Res. 16, 524 (2001)
V
Temperature dependence of the absorption coefficient in 4H- and 6Hsilicon carbide at 355 nm laser pumping wavelength
A. Galeckas, P. Grivickas, V. Grivickas, V. Bikbajevas, and J. Linnros
Phys. Stat. Sol. (a) 191, 613 (2002)
VI
Characterization of 4H-SiC band-edge absorption properties by freecarrier absorption technique with a variable excitation spectrum
P. Grivickas, V. Grivickas, A. Galeckas and J. Linnros
Mater. Sci. Forum 389-393, 617(2002)
VII
Single- and two-photon band edge characterization in Si and 4H-SiC
by temporally excited free-carrier absorption as a novel technique
P. Grivickas, V. Grivickas, A. Galeckas and J. Linnros
in Physics of Semiconductors 2002, edited by A. R. Long and J. H.
Davies, Institute of Physics Conference Series Number 171, N3.1
(Institute of Physics Publishing, Bristol, UK, 2003)
VIII Excitonic absorption above Mott transition in Si
P. Grivickas, V. Grivickas and J. Linnros
Phys. Rev. Lett. 91, 246401 (2003)
IX
Intrinsic band-edge absorption in 4H-SiC
P. Grivickas, V. Grivickas, J. Linnros and A. Galeckas
(in manuscript, intended to Phys. Rev. B)
ii
Published work not appended
Relevance of the carrier transport change in porous silicon at the onset of
one excited pair per crystallite
V. Grivickas, J. Linnros, N. Lalic, J. A. Tellefsen and P. Grivickas
Thin Solid Films 297, 125 (1997)
Band edge absorption, carrier recombination and transport measurements
in 4H-SiC epilayers
V. Grivickas, J. Linnros, P. Grivickas and A. Galeckas
Mater. Sci. & Engin. B61-62, 197 (1999)
Determination of the polarization dependency of the free-carrierabsorption in 4H-SiC at high-level photoinjection
V. Grivickas, A. Galeckas, P. Grivickas and J. Linnros
Mater. Sci. Forum 338-342, 555(2000)
Thermopower measurements in 4H-SiC and theoretical calculations
considering the phonon drag effect
V. Grivickas, M. Stölzer, E. Velmre, A. Udal, P. Grivickas, M. Syväjärvi, R.
Yakimova and V. Bikbajevas
Mater. Sci. Forum 353-356, 491 (2001)
Excess free carrier optical excitation spectroscopy in indirect
semiconductors
V. Grivickas, A. Galeckas, P. Grivickas, J. Linnros and V. Bikbajevas
Mater. Sci. (ISSN 1392 - 1320 Medziagotyra) 7, 203 (2001)
Impact of phonon drag effect on Seebeck coefficient in p-6H-SIC:
experiment and simulation
V. Bikbajevas, V. Grivickas, M. Stölzer, E. Velmre, A. Udal, P. Grivickas, M.
Syväjärvi and R. Yakimova
Mater. Sci. Forum 433-436, 407 (2003)
Two-photon spectroscopy of 4H-SiC by using laser pulses at band-gap
frequencies
V. Grivickas, P. Grivickas, J. Linnros and A. Galeckas
Accepted in Mater. Sci. Forum (2004)
iii
Author’s contribution to the appended papers I-IX:
Planning, sample preparation, data acquisition, data analysis, theoretical
simulations and writing of the manuscript were done exclusively by the author
in all articles appended to the thesis except the following cases:
i) In the article V the author’s contribution consists of the following: sample
preparation (~30%), data acquisition (~30%), data analysis (~20%) and
manuscript writing (~20%).
ii) In the article VIII all enumerated contributions consist of 100% except ~20%
in the sample preparation and the data acquisition.
iv
Acknowledgements
First of all, I am very grateful to Jan Linnros for being my supervisor
and supporting me during the long and interesting PhD journey. He has shared
his scientific insights with me openly and has patiently discussed all of my new
and sometimes bizarre ideas. Thanks go also to Ulf Lindefelt and Bo Breitholtz,
who accepted me as a PhD student within the SiCEP program.
This work would have been almost impossible without my father,
Vytautas Grivickas. Only a few of us have the wonderful opportunity to get to
know the other side of our parents, where they are valued as professionals. As a
scientist I am lucky to have shared my time with one of the greatest researchers I
know, while as a son I am proud to have had some battles with my teacher in
life.
I also would like to thank Augustinas Galeckas for sharing with me
several precious things: homeland, coffee breaks, conference hotel rooms, lists
of coauthors and a laboratory. Many researchers say that high quality equipment
is not enough to be able to do high quality science. One also needs a person with
“golden arms” to run it, and I have had a good opportunity to see the truth of this
phrase.
I am also greatly in debt to Marianne Widing. All departments have
secretaries, but only a few have people who help you so much. None of my
everyday life problems during these years were allowed to become really big
problems, and it has been a pleasure to watch how one person can make life so
much brighter for so many of us.
I would like to send special thanks to Antonio Martinez for being my
colleague and coworker in this project (I miss your voice already), to Xavier
Badel for sharing corridors not only at work, but also in our flat, to Pascal
Kleimann for introducing me to the mysterious world of the clean-room, to
Patrick Leveque for discussing the strange world of physics, to Uwe
Zimmermann for honorably accepting my cookies in exchange of being my
computer mentor and to Nenad Lalic and Aliette Mouroux for being my guides
into the world of the PhD student. I am especially grateful to Christopher
Castleton who accepted the difficult task of turning the language of this thesis
into proper English. Many thanks also to the people who have kept me company
at the department: Robert, Jan, Marylene, Arturas, Leonardo, Martin, Jens,
Hanne, John, Paolo, Edouard, Andrej, Ilya, Maciej…
During these years I met a lot of people who have become good friends:
Rimas (for ever), Ausrius and Alvydas (I believe we share much more than just
being Lithuanian physics PhD students in Stockholm…), Giedre and Korkut,
Renata and Romanas, Daiva, Kestas 1 and Kestas 2, Paulius… Hi five to my
basketball teammates! I am pretty sure that I am forgetting someone, but it
doesn’t mean that your picture is not in my heart☺.
v
Finally, I would like to thank my mother and sister for accepting my
exile for so many years, and, even more, I would like to thank Rasa for
becoming my new inspiration, my sole mate and my new family. We already
have so many things to say to each other and I feel it is just the beginning of the
story.
The work summarized in this thesis has been done within the Silicon
Carbide Electronic Program (SiCEP) supported by the Swedish Strategic
Research Board (SSF). Most of the SiC samples were provided by the SiCEP
group in Linköping.
vi
1 Introduction
Since the middle of the last century semiconductors, as a group of
materials, have invaded our everyday life at an unimaginable speed. Even
though today many people are unfamiliar with words like diode, transistor or
semiconductor laser, it would be difficult to find someone unaware of such
things as computers, CD players or mobile phones, which provide the
foundations of our technological civilization. At the heart of this fancy
terminology lies the simple observation that some materials substantially change
their electric conductivity due to illumination, temperature variation or
incorporation of minute amounts of impurity atoms. Investigation of these
properties led to the current understanding that semiconductors represent a
group of solids in which covalently bound atoms share common electrons in the
completely occupied low energy valence band. If additional energy is provided
to the solid, the latter electrically active particles can be excited into the empty
conduction band. There, these induced free charges become carriers of electrical
current, creating an electrically conductive matter. Macroscopic behavior of the
carriers is described by a set of transport equations consisting of the Poisson’s
equation, the continuity equation, the drift-diffusion equation plus Fermi-Dirac
statistics for carrier concentration. In these equations the most important
parameters characterizing the semiconductor are the charge carrier lifetime,
mobility and diffusivity.
If one is only concerned with practical material questions for device
performance, the task would appear to be that of finding the numerical values of
these parameters. However, the true story is far more complicated. There are a
vast number of physical mechanisms, which interplay unpredictably under
different conditions to determine a particular parameter. In the case of the carrier
diffusivity, for example, the very incomplete physical model includes the
description of majority and minority carrier diffusivity at various temperatures
and injections, which in turn depend on the impurity-carrier, phonon-carrier or
electron-hole scattering, which in turn depend on the screening of the impurities
by the charge carriers, clustering of impurities or mutual carrier attraction due to
the Coulomb interaction and so on. Moreover, the parameters listed in textbooks
and manuals usually refer to bulk material properties or to the extracted values
under special experimental conditions, while practical device modeling contains
a number of constants and variable parameters referring to the sizes and
interfaces of the device. The full story should also account for the complexity of
modern processing technologies applied to the material, e.g. ion implantation,
surface passivation or internal gettering, which produce complex impurity
profiles and other structural non-linearities.
An answer to this complicated material challenge from the world of
physics was the invention of numerous characterization techniques, capable of
1
providing detailed answers in specific circumstances. Though only a few
techniques can be labeled as standard and not a single technique dominates, all
together they provide a rather consistent insight into the properties of semiconducting materials. Individual characterization strategies depend on the type
of specimen, ranging from bulk material to material structure or even the
complete device, and the particular group of the physical phenomena pursued.
This work focuses on optical characterization, which is favored both within the
scientific community and in industry due to its undestructive nature and fairly
simple data interpretation. More specifically, the results of this thesis were
attained using optical methods based on the selective probing of excess free
carrier concentration.
Pioneering investigations employing free-carrier absorption (FCA) as a
tool were done in the fifties in heavily doped Si and Ge [1] and the basic
absorption mechanisms were established at that time. However, the real break
through in the utilization of FCA came with the discovery of the laser, which
provided the means of generating instant excess carrier excitations by short,
highly energetic pulses [2]. Non-diverging beams of monochromatic and
coherent laser light together with sufficient instrumentation for the detection
system provided a powerful pump-probe technique capable of resolving carrier
dynamics within a crystal. The principle of perpendicular pump-probe
measurements is illustrated in Fig. 1(a). Light from the pump pulse
homogeneously excites the lateral side of the sample while a focused probe
beam monitors the time behavior of the created excess carrier concentration.
Employment of spatially resolved and calibrated FCA probing in such
measurements results in several unique methods to examine material properties.
Pump
b)
Sample
a)
hω probe
Probe
EG hω pump
Pump
Probe
Fig. 1 Principles of (a) the perpendicular pump-probe
measurement, (b) the FCA probing mechanism.
2
The idea of FCA probing originates from the fact that light with photon
energy below the material band-gap ( hω probe < EG ) is strongly absorbed by the
excited carriers. The latter are introduced by pump light with photon energy
above the material band-gap ( hω pump > E G ). The whole scenario is shown in Fig.
1(b). In the FCA process a single photon of light is annihilated with the
simultaneous transition of an electron or a hole to a higher energy state in the
conduction or valence band, respectively, which subsequently relaxes into the
initial state. While no additional excess carriers are introduced in the system, the
efficiency of the process is proportional to the instantaneous excess carrier
concentration present in the material. Therefore, looking at the intensity changes
in a weak probe beam passing through the excited volume of the specimen, it is
possible to correlate the signal to the absolute carrier concentration.
It is worth mentioning that a great deal of the development of FCA
methods was carried out by the two supervisors of this work: V. Grivickas and J.
Linnros [3,4]. They first used FCA probing several years ago to determine the
dependence of the bulk lifetime in Si on the injection density. Since the
fundamental work of Shockley, Read and Hall (SRH) [5] it was known that
recombination via localized impurities changes at different excess carrier
densities, and FCA probing was revealed to be a straight forward and extremely
accurate tool for studying this phenomenon. It becomes even more advantageous
at injected densities greater than 1017 cm-3, where nonlinear Augerrecombination becomes the leading recombination mechanism. Scanning
measurements into the depth of the sample at that moment provided first insights
into the diffusion processes of the free carriers and surface recombination rates.
As a consequence, in 1991 the two invented a novel Fourier Transient Grating
(FTG) technique [6], which stands out from other transient grating techniques,
like modulated diffraction or thermal deflection, due to its applicability over a
very wide excitation range. The technique resulted in detailed studies of
injection-dependent mechanisms of ambipolar diffusion and minority carrier
drift mobility in Si. Recently it was realized that FCA probing could also serve
as an instrument to detect fundamental absorption coefficients, while monitoring
the depth distribution of the excited carriers. In 2001 the idea was extended to
full spectroscopic measurements [7] when a laser excitation with a tunable
wavelength became available in the laboratory. As discussed in the rest of this
work, spectroscopy based on FCA detection shows exclusive applicability in the
case of complex material structures and reveals its unique capability to access
the fundamental absorption edge in the case of highly doped or excited
semiconductors.
In this work most of the enumerated advantages of the FCA probing
techniques were utilized to investigate a very ambitious, but still relatively
mysterious candidate among the indirect band gap semiconductors – silicon
carbide (SiC). This binary compound synthesized from the two well-established
3
representatives of the group IV of elements is famous for its extraordinary
mechanical, thermal and electrical properties. Starting its journey more than a
hundred years ago as a simple abrasive powder, it now challenges various fields
of applied technology ranging from fusion plasma devices and high temperature
sensors to light emitting diodes and high-power switches and rectifiers. The
turning point in history was marked 15 years ago, with the growth of high
quality epitaxial layers on commercial single crystalline SiC wafers. However,
despite the continuous achievements during recent years in material quality and
understanding of its basic properties, many unresolved questions remain. One of
them concerns the diffusion coefficients of free carriers in SiC, which has only
preliminary theoretical estimates from drift mobility values. The fundamental
absorption edge at all temperatures is only available in a few SiC polytypes,
while only a general assessment exists in the commercially attractive 4H-SiC
polytype. Even the complete analysis of material lifetime at different
temperatures is lacking. The list of the question marks can be continued with the
lack of data for band gap narrowing, the influence of anisotropy, unknown
exciton binding energy etc. This work attempts to provide at least some of the
answers to the mentioned directions.
As a final point, it should be noted that probably the most interesting and
weighty result of the thesis, however, was obtained in the most thoroughly
investigated of the indirect semiconductors – crystalline Si. The novel
fundamental absorption edge measurements were tested on the well-grounded
data of this material simultaneously extending them to the boundaries of high
doping and excitation. The previous understanding of the absorption constituents
in indirect semiconductors suggested that the pronounced bound excitonic
component should disappear at high plasma densities due to carrier-carrier
screening. Nevertheless, an unexpected excess excitonic-like absorption was
found at carrier concentrations far exceeding the critical Mott density for the
bound excitons. Though this effect, named excitonic enhancement, was
previously observed and established in direct gap semiconductors due to the
many-body interactions, it has never been confirmed for any indirect
semiconductor, because of the inherent limitations in conventional absorption
methods. Current verification of this carrier correlation effect should influence
explanation of some carrier transport properties as well as the modeling and
simulation of the device performance.
The thesis is organized as follows. Chapter 2 presents the crystal and
band gap structure of the two indirect semiconductors Si and SiC, also
introducing the concepts of lattice vibration and band gap narrowing. The data
are presented in a comparative manner with the main focus on SiC. Chapter 3
briefly overviews relevant carriers transport phenomena and recombination
mechanisms, providing the basic relations. Chapter 4 describes in detail all the
possible absorption mechanisms starting from the investigated fundamental edge
and ending with FCA, which both interferes with fundamental absorption
4
measurements and provides the probing technique. The concept of the exciton as
a correlated electron-hole particle together with the excitonic Mott transition and
the excitonic enhancement effect are also elucidated. Chapter 5 presents FCA as
a probe tool enumerating the spin-off techniques. Finally, the main conclusions
and future outlook are provided in the summary Chapter 6, including a step-bystep commentary on the appended papers.
Note that indexation of equations, figures and references in the work
starts from “1” in each subchapter (e.g. 2.1, 3.2), and they are referred in the text
in the following manner: Eq. (5) in Ch. 4.2, Fig. (7a) in Ch. 3.1 or Ref. 6 in Ch.
2.2. The list of references is provided after each subchapter. The nine appended
articles of the thesis are referred in the text according their indexation with
Roman numbers: II, VII. The following two abbreviations are commonly used in
the text: lhs – left hand side, rhs – right hand side.
References
[1] H. Y. Fan, Repts. Prog. Phys. 10, 107 (1956).
[2] D. K. Schroder, Semiconductor material and device characterization (Wiley
& Sons, New York, 1990).
[3] J. Linnros and V. Grivickas, Carrier lifetime: Free carrier absorption,
Photoluminescence and Photoconductivity, Ch. 5b.2 in Methods in Material
Research, edited. by E. N. Kaufman (John Wiley & Sons Inc., 2000).
[4] V. Grivickas et al., Ch. 13.1, 13.4, 13.5, 13.6 in Properties of Crystalline
Silicon (EMIS Data reviews Series No. 20, INSPEC), edited by R. Hull
(London, 1999).
[5] W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952); R. N. Hall, ibid.
87, 387 (1952).
[6] V. Grivickas and J. Linnros, Appl. Phys. Lett. 59, 72 (1991).
[7] V. Grivickas, A. Galeckas, P. Grivickas, J. Linnros and V. Bikbajevas,
Material science (ISSN 1392 - 1320 Medziagotyra) 7, 203 (2001).
5
6
2 Indirect Semiconductors SiC and Si
2.1 Basic properties
2.1.1 Crystal structure and polytypes of SiC
Pure Si crystallizes in the diamond structure, with an fcc Bravais lattice
and a two-atom basis. The chemical bond is solely covalent, each atom being
tetrahedrally coordinated. Its crystal properties are well investigated and
documented in textbooks [1]. SiC in this respect is more exotic material and will
be discussed in more details.
Both Si and C are elements of group IV of the periodic table. One would
expect, therefore, that their atoms would easily create covalent bonds resulting
in alloys of the SixC1-x form. However, a large bond-length mismatch appears
between the big Si and the much smaller C atoms, which is well illustrated by
the corresponding 2.35 Å and 1.55 Å values in their respective tetrahedral
diamond structures. As a consequence, a single composition exists between Si
and C, where each Si atom has a covalent bonding with four C atoms and vice
versa, providing an equal number of the two elements in the total crystal. The
expected crystal structure of such SiC material would be zincblende or wurtzite.
Nevertheless, early investigators were left in confusion after finding a variety of
stable SiC configurations ranging from cubic (C) to hexagonal (H) and
rhombohedral (R) categories. The result was explained by the polytypism
phenomenon in which several crystals exist with the same chemical composition
but different sequence of close packed atomic layers. Nowadays around 200
polytypes of SiC have been observed, while hundreds more are predicted
theoretically.
The stacking sequence of a particular SiC polytype is described using the
capital letters A, B and C. Each letter is associated with a single covalently bond
Si and C layer, distinguished by its position relative to the previous layer. Such a
notation system is explained in Fig. 1 for the four most common polytypes. The
simplest AB-AB-… repetition sequence results in the 2H polytype, where 2
stands for the number of layers in a single repetition and H signifies a hexagonal
crystal lattice. The ABC-ABC-… array produces the 3C polytype, the single
polytype in the cubic crystal system. Historically this polytype has also been
called β-SiC, whereas all other crystal types have been collectively referred to as
7
Fig. 1. The most common SiC polytypes represented by
their corresponding stacking sequences (adopted from [2]).
α-SiC. The ABCB-ABCB-… and ABCACB-ABCACB-… sequences
correspond, respectively, to the 4H and 6H polytypes in the hexagonal
configuration. From a crystallographical point of view, however, the latter
polytypes as well as all the rest are a mixture of cubic and hexagonal bonding
and form one-dimensional superlattice structures. For example, 4H-SiC consists
of an equal number of hexagonal and cubic bonds, while 1/3 hexagonal and 2/3
cubic bonds gives 6H-SiC. Quasi-cubic or quasi-hexagonal nature with respect
to the neighboring stacking plane produces inequivalent lattice sites in the
crystal, i.e. an atom in the same hexagonal or cubic site finds itself in a different
environment. Therefore, the same donor or acceptor atom placed at an
inequivalent site has a different activation energy. Note that in all polytypes the
interatomic distance, and thereby the density, is the same, it is only the sequence
of stacked layers that differs. Nevertheless, crystals with a shorter repeat
distance are generally easier to manufacture in large quantities without inducing
too many dislocations and defects. Finally, apart from the cubic 3C polytype, all
polytypes are expressed in the hexagonal coordinate system consisting of three
a-plane coordinates (oriented by 120 degree angles relative to each other) and a
8
Fig. 2. Principles of the CVD (top) and sublimation
(bottom) growth techniques (adopted from [3]).
perpendicular principal or c-axis. The latter lies in the stacking direction of the
hexagonally close packed layers.
2.1.2 SiC growth and doping
High quality low-doped SiC is usually grown by the low-pressure
chemical vapor deposition (CVD) technique, which is illustrated in the upper
part of Fig. 2: a highly-doped (~1019 cm-3) SiC substrate wafer is placed on a
graphite susceptor within a quartz tube. Gaseous precursors containing Si and C
are transported into the tube by a H2 carrier gas and cracked inside by heat from
an RF coil at typical temperatures of 1400 – 1800 oC. Separate atoms or radicals
produced by the cracking process nucleate at the atomic steps on the substrate
surface, which are intentionally created by polishing the growth surface by the
common 8 degrees in respect to the basal a-plane. Two types of the CVD system
can be distinguished: i) cold and ii) hot wall, which is more advantageous from
the growth rate point of view. In the former case the substrate is simply laid on
the susceptor, while in the latter it is enclosed within a tubular susceptor. For
growth speed it is also beneficial to build the whole system in a vertical (or
chimney) design [4]. Nevertheless, the typical growth rate by CVD technique is
9
< 5 µm/h, indicating that only thin < 100 µm epitaxial layers are typically
produced.
Epitaxial growth by sublimation has also been studied as a suitable choice
for high-growth rates such as ~ 100 µm/h [5]. This approach uses the same
principles as the modified basic Lely’s method, which is a standard technique
for growing large single crystal boules for SiC substrate wafers. As illustrated
by the schematic view of the process shown in the lower part of Fig. 2, SiC
source material is heated up to produce the vapor-phases of Si and C species
such as Si2C and SiC2. The latter are transported from the source to the
nucleating substrate by introducing a temperature gradient. Unfortunately,
material purity using this sublimation technique is highly affected by growth
conditions such as source material quality, temperature gradient control and
contamination from the graphite susceptor. In fact, sublimation epilayers in the
low-doped range are produced only by donor and acceptor compensation.
However, in the CVD technique the desired doping levels can be achieved by
well-controlled substitution of the doping gases into the main flow, as illustrated
in Fig. 2.
These days intrinsic SiC epilayers produced by the CVD technique have
unintentional doping levels reduced to ~1013 cm-3. In general, n-type
conductivity in SiC originates from N or P donors, which have ionization
energies varying between 40 and 140 meV [6], depending on polytype and type
of lattice site. The main “shallow” acceptors in the p-type SiC are B or Al,
exhibiting ~200 and ~300 meV levels above the valence band, respectively [6].
2.1.3 Comparison of 4H-SiC and Si
At this point it should be noted that of all the possible SiC polytypes, 4H
becomes the object of the current work as it is the common polytype of choice
for applications due to its superior transport parameter values. These, as well as
the following properties are summarized in Table 1 for 4H-SiC and Si: the
indirect band-gap EG, critical field at breakdown Ec, thermal conductivity σ,
saturated drift velocity vsat, mobilities for electrons and holes at room
temperature and low doping µe and µh, relative dielectric constant ε, density ρ
and lattice constant a (and c in 4H-SiC). The values provided can be weighted
using different figures of merit, but the basic guidelines are clear. 4H-SiC has
tremendous potential in power devices due to its higher critical field at
breakdown and thermal conductivity. It can be shown that specific onresistances of unipolar 4H-SiC devices can be as little as 1/300 for an equivalent
Si device. Already now, power-blocking voltages in 4H-SiC diodes exceed 5 kV
while more than 15 kV is expected theoretically. Moreover, SiC emerges as a
superior candidate for use in conditions of high temperature or in hash
environments.
10
At the same time, it is important to remember that α-SiC, due to its
hexagonality, has anisotropic crystal structure, i.e. measured parameters can
vary substantially in the directions parallel and perpendicular to the c-axis. The
degree of anisotropy in 4H-SiC, compared to other polytypes, is not very high.
For the carrier mobilities a ratio of 0.83 was found, while other constants vary
even less.
Table 1. Main properties of Si and 4H-SiC
Si
1.12
0.25
1.5
1
1350
471
11.6
2.3
5.43
EG (eV)
Ec (MV/cm)
σ (W/cm K)
νsat (107 cm/s)
µe (cm2/Vs)
µh (cm2/Vs)
εr
ρ (g/cm3)
a=/c= (Å)
4H-SiC
3.26
2.2
3.7
2
1000
115
10
3.2
3.08/10.0
References
[1] Properties of Crystalline Silicon, edited by R. Hull (INSPEC, London,
1999).
[2] K. Nassau, J. of Gemmology 26, 425 (1999).
[3] http://www.ifm.liu.se/Matephys/new_page/research/sic/.
[4] A. Ellison, Mater. Sci. Forum 338-342, 131 (1999).
[5] M. Syväjärvi, R. Yakimova, H. Jacobsson, M. Linnarsson, A. Henry and E.
Janzen, Proc. of ICSCRM’99 (in press).
[6] W. J. Choyke and G. Pensl, MRS Bulletin 22, 25 (1997).
11
2.2 Band structure and vibrational spectra
This section provides a basic description of the band structure of the two
semiconductors relevant for the optical transitions presented in later chapters.
The well-determined band structure of the crystallographically simpler Si serves
as the comparative foundation for the far more complicated and not yet
completely elucidated one in the 4H-SiC polytype. In the same manner, the
structure of the lattice vibrations in the two materials is summarized, focusing
on the phonon branches active in optical transitions.
2.2.1 Band structure
The lhs of Fig. 1 shows a segment of the Si band structure calculated
within the k⋅p method [1]. The high symmetry directions in the diagram are
indicated by the letters Λ and ∆, while the high symmetry points are labeled by
the letters L,Γ and X. The maximum of the valence band in Si appears at the Γpoint, which represents the center of the first Brillouin zone (k = (0,0,0)). The
maximum is six-fold degenerate and splits due to spin-orbit coupling into two
bands, as indicted schematically on the rhs of Fig. 1. Moreover, the top valence
band consists of the two branches, which have different energy dispersion and
are known as heavy- and light-hole bands. Their average effective masses are
mhh = 0.476m0 and mlh = 0.159m0, respectively [2], where m0 is the electron mass
in free space, which enter calculations of the total density of states mass mh
through the Luttinger parameters [2], providing mh = 0.527m0. The minimum of
the conduction band emerges in the ∆1 direction at a distance k0/kmax = 0.85 from
Fig. 1. Band structure of Si [1].
12
the Γ-point, where kmax indicates the wave vector size between Γ and X. The
minimum is parabolic, but has different effective masses in different directions
(effective mass anisotropy). Therefore, the density of states mass me* is
expressed as:
me * = (ml mt2 )1/ 3 = 0.321m0 ,
(1)
where ml = 0.916m0 is the longitudinal effective mass in the direction from Γ to
k0 and mt = 0.19m0 is the transverse effective mass in the two perpendicular
directions [2]. In addition, Eq. (1) has to be multiplied by a factor of g2/3, where
g = 6 represents the number of equivalent conduction band minima due to the
“multi-valley” structure of Si. Note that the density of states mass expressed by
Eq. (1) differs from the effective conductivity mass, which enters into carrier
transport equations. The latter is given by:
me =
3
= 0.259 m0 .
1/ ml + 1/ mt + 1/ mt
(2)
Energy differences between the various points in the band structure are indicated
on the rhs of Fig. 1.
The lhs of Fig. 2 shows the calculated band structure of 4H-SiC within the
local density approximation to density functional theory [3]. This approximation
is well known to underestimate the absolute fundamental band gap in
semiconductors by ~1.1 eV, but the shape of the bands is usually in agreement
with other approaches. The complexity of the diagram is evident in comparison
Fig. 2 Band structure of 4H-SiC [3].
13
E
4H-SiC
Si
0
π/8a π/4a
π/2a
π/a
k
Fig. 3 (lhs) Phonon branch formation in the form of reflections in the
k space due to the folding of the extended zones into the first Brillouin
zone. (rhs) Phonon dispersion in the ∆ direction of Si. The symbols
represent experimental results from neutron scattering [2].
to Si, although symmetry of the both materials is rather similar. The complexity
arises from the fact that the unit cell of 4H-SiC contains 8 atoms and is almost
twice as long in the hexagonal c-axis direction as the primitive unit cell of Si,
which contains 2 atoms. As a consequence, the first Brillouin zone of 4H-SiC is
only half as long in c-direction resulting from the back folded branches. This
process is more explicitly illustrated in Fig. 3 for the phonon branches. The
maximum of the valence band again appears at the Γ-point. However, due to the
spin-orbit coupling and the hexagonal crystal field it splits into the three
subbands, as schematically shown on the rhs of Fig. 2. Their average effective
masses are mh⎜⎢ = 1.6; 1.6 and 0.2m0 and mh⊥ = 0.6; 0.6 and 1.5m0 in the parallel
and perpendicular directions to the c-axis, respectively [3,4]. The total density of
states mass can be obtained through the parameterisation expressions for the
wurtzite crystal structure [3]. The Conduction band emerges in the g = 3
equivalent minima at the M point with the following effective masses: me⎜⎢ =
0.29m0, me⊥ = 0.42m0, me* = 0.37m0 and me = 0.36m0 [3,5]. Again, various
energy differences are shown on the rhs of Fig. 2.
14
Fig. 4 (top) Phonon dispersion in 4H-Si with symbols representing
experimental results from neutron scattering. (Table) Characteristic
energies at the M symmetry point for the 12 phonons with acoustic
nature (left side) and the 12 with optical nature (right side) [6].
2.2.2 Vibration spectra
The simplest representation of a solid is the linear-chain model of single
interacting atoms. Its equation of motion has a solution of the form
ω ∝ sin (ka / 2) , which is illustrated on the lhs of Fig. 3 by the line extending
through the k space of the first Brillouin zone from 0 to π/a, where a represents
the lattice constant. In such a model making the assumption of two equivalent
atoms per unit cell results in an increase in the unit cell length by a factor of two
and a reduction in the edge of the first Brillouin zone by one half to π/2a.
Consequently, the outer parts of the wave vectors in the previous solution are
15
reflected into the new zone creating an optical branch (anti-phase motion of the
neighboring atoms), while the previous branch is labeled as the acoustic one (in
phase motion), as illustrated by the gray line in Fig. 3. This simplified scenario,
nevertheless, predicts well the real phonon dispersion in Si, which is shown in
the ∆ crystallographic direction on the rhs in Fig. 3 [2]. The branches are
additionally split into the transverse (TA and TO) or longitudinal (LA and LO)
with respect to the wave propagation. It is phonons in the Si ∆ valley at the
k0/kmax = 0.85 value which complete the indirect interband transitions in the
presented earlier band structure. From Fig. 3 values of the dominating TA = 18.4
meV and TO = 57.8 meV phonons can be detected.
In the case of 4H-SiC containing 8 atoms per unit cell, the simplified
procedure of phonon branch formation from back reflected folds should be
repeated three times, as illustrated by the black line in the 0 to π/8a range of Fig.
3. The result gives an approximate guideline for the complicated vibration
spectra of the real crystal shown in Fig. 4 [6]. In the spectrum a distinct gap
appears between the optical and acoustical branches at the boundaries of the first
Brillouin zone due to the different masses of the vibrating Si and C atoms.
Phonon branches participating in the fundamental interband absorption lie at the
M symmetry point. In total 24 branches appear ([1L+2T]×23 = 24, where power
index 3 represents the number of reflections in the simplified model): 12 of
acoustic nature and 12 of optical. Their corresponding energies are summarized
on the left and right sides of the table in Fig. 4 [6]. Note that, due to the selection
rules in the anisotropic 4H-SiC, different phonons are active in the parallel and
perpendicular direction with respect to the c-axis, as shown by the symbols ⊥
and ⎜⎢ in the table.
References
[1] M. Cardona and F. H. Pollak, Phys. Rev. 142, 530 (1966).
[2] A. Dargys and J. Kundrotas, Handbook on Physical Properties of Ge, Si,
GaAs and InP (Science and Encyclopedia Publishers, Vilnius, 1994).
[3] C. Persson and U. Lindefelt, J. Appl. Phys. 82, 5496 (1997).
[4] N. T. Son et al., Phys. Rev. B 61, R10544 (2000).
[5] N. T. Son et al., Appl. Phys. Lett. 66, 1074 (1995).
[6] J. Serrano et al., Mat. Sc. For. 433-436, 257 (2003).
16
2.3 Band-gap narrowing
2.3.1 Band-gap changes with temperature
There are two basic contributions, both of a similar order, to the
temperature dependence of the band-gap, which can be expressed in the
following way:
dEG ⎛ dEG ⎞
⎛ dEG ⎞
=⎜
⎟ +⎜
⎟
dT ⎝ dT ⎠exp ⎝ dT ⎠ ph−el
(1)
The first term represents the change in the crystal volume with temperature
resulting from lattice expansion:
⎛ dEG ⎞ = ⎛ dV ⎞ ⎛ dP ⎞ ⎛ dEG ⎞
⎟ ⎜
⎟
⎜
⎟
⎜
⎟ ⎜
⎝ dT ⎠exp ⎝ dT ⎠ P ⎝ dV ⎠T ⎝ dP ⎠T
(2)
where 1/V(dV/dT)⏐P = 3αp, where αp is the linear expansion coefficient at
constant pressure, and V(dP/dV)⏐T = -βp, where βp is the isothermal bulk
modulus or the inverse of the isothermal compressibility [1]. The last term in
Eq. (2) represents the pressure coefficient of the band-gap. The parameters can
be easily evaluated once the equation of state for the solid is known. The second
term in Eq. (1) is the electron-phonon interaction at constant volume. In
theoretical studies it is separated into two different contributions, which are
referred as the Debye-Waller and self-energy terms. The first originates from the
smearing-out effect of the periodic potential, while the second represents the
mutual repulsion of intra-band electronic states through increased electronphonon coupling in second order perturbation theory [2]. Both these
contributions are much more complicated to obtain than the previously
introduced thermal lattice expansion and, therefore, are the primary target of
theoretical calculations. Nevertheless, it has been shown that all electron-phonon
interactions can be approximately taken into account by relating them to a single
parameter, namely the change in frequency of the phonons [3]. Lattice vibration
frequencies change from ωi(q) to ωf(q), where indexes i and f refer to the initial
or lower and final or upper states of the electron, respectively, when an electron
is excited across the bang-gap. In such a process the band-gap varies with
temperature according to:
EG (T ) ≈ EG (0) + k BT ∑ ln
q
17
ω f (q)
,
ωi ( q )
(3)
E
EG(0)
T
Fig. 1. Typical dependence of the band-gap change with temperature.
if k BT > hω (q) . Excitation of an electron weakens the atomic binding and thus
lowers the elastic restoring force. Hence ω f (q) < ωi (q) and so the logarithm in
Eq. (3) is negative and the gap is reduced as the temperature increases. Eq. (3)
correctly predicts linear dependency at high temperatures, as approximated by
the dashed line in Fig. 1. At lower temperatures, the band-gap tends to a
constant value, as shown by the solid line.
In practice, comprehensive EG(T) data sets are widely available from
various measurement techniques, and one is more concerned with the problem
of fitting them with some analytical expression. Technological applications
require that such expressions have as few empirical free fitting parameters as
possible. The most conventional and recognized in this respect is the threeparameter formula suggested by Varshni [4]:
EG (T ) = EG (0) − aT 2 / (b + T ) .
(4)
Here a represents the –dEG(T)/dT limit at T → ∞ and b is a temperature
parameter whose magnitude was believed to be comparable to the Debye
temperature θD. However, despite the common use of this expression, it was
concluded [5] that Varshni’s ad hoc model is generally incapable of providing
an accurate and, at the same time, physically adequate interpretation of the
available EG(T) data sets for most semiconductors. In fact, in some cases even
negative values of the b factor can be found. Moreover, at low temperatures in
the cryogenic region Eq. (4) predicts a quadratic dependence, whereas most
experiments find a more temperature independent behavior [6].
Among the proposed advanced models the most physically reasonable
basis was provided in the model of Passler [5]. It originates from noticing that
the electron-phonon interaction varies with lattice temperature just as the Bose-
18
Einstein occupation factor nB(Ep,T). Hence, performing a summation over all
contributions made by phonon modes with the same energy, it is possible to
replace Eq. (3) by:
EG (T ) = EG (0) − ∫ f ( E p )nB ( E p ,T )dE p ,
(5)
where f ( E p ) is a spectral function capable of incorporating thermal lattice
expansion as well. Subsequent expansion of the integral in Eq. (5) suggests the
following formula, with four fitting parameters:
3∆2
⎧ (1− 3∆2 )
⎫
⎪ exp(Θ / T ) −1 + 2 ×
⎪
⎪
⎪
EG (T ) = EG (0) − αΘ⎨
⎬ .(6)
2
3
4
6
2
2
⎛
⎛ 2T ⎞ + 3∆ −1 ⎛ 2T ⎞ + 8 ⎛ 2T ⎞ + ⎛ 2T ⎞ −1⎞⎟⎪
⎪⎜ 6 1+ π
⎜ ⎟
⎜
⎟
⎜ ⎟ ⎜ ⎟
⎟⎪
⎪⎜
3(1+ ∆2 ) ⎝ Θ ⎠
4 ⎝ Θ ⎠ 3⎝ Θ ⎠ ⎝ Θ ⎠
⎠⎭
⎩⎝
In this expression α represents the entropy of the material S (T ) = −
T → ∞ limit and Θ =
Ep
kB
dE (T )
in the
dT
is the average phonon temperature. The essential
difference is made by introducing the fourth fitting parameter, ∆, which
represents the degree of phonon dispersion in the material. Passler shows that in
comparison to such a framework the Varshni’s formula represents an unrealistic
approximation giving the case of extremely large degrees of phonon dispersion,
∆ > 1, whereas the upper limit for semiconductors is estimated to be at ∆ ≈ 0.75.
Though the values of EG(0), α, Θ and ∆ provided by Eq. (6) are very
informative for comparison between materials, sometimes it is convenient to use
a simpler phonon dispersion related model earlier proposed by the same author
(so called “p-representation” model) [7]:
[
]
EG (T ) = EG (0) − (αΘ p / 2) p 1+ (2T / Θ p ) −1 ,
p
(7)
where index p is related to the phonon dispersion by p ≈ 1 / ∆2 + 1 and Θp is
roughly comparable with the average phonon temperature.
2.3.2 Doping and plasma induced band-gap narrowing
The width of the band-gap also monotonically decreases with increasing
e-h density due to the exchange and correlation effects. Carriers of opposite
19
Fig. 2. Doping induced BGN in Si (left) and 4H-SiC (right) [13].
charge are attracted by the Coulomb interaction, while carriers of like charge are
repelled. The energies of the attraction and repulsion cancel each other if
electrons and holes are randomly distributed. However, the exchange interaction
of identical fermions (the Pauli principle) forbids two carriers with parallel spin
from occupying the same energy state. This fact increases the average distance
between the carriers and reduces the total repulsive energy. Moreover, the
repulsion of one type of carriers redistributes the system in such a way that in
the vicinity of some particular hole electrons are found with higher probability
than other holes and vice versa, increasing the total correlation energy. As a
result, the total energy of the interacting particles becomes lower, lowering the
effective band-gap.
Calculations of the doping induced band-gap narrowing (BGN) are
performed in the following way: in an unperturbed (intrinsic) semiconductor the
Hamiltonian H0 describes interactions between the valence electrons and ions of
the intrinsic crystal. Solution of the H0 eigenvalue problem results in singleparticle energies Ej0(k) for electron states with wave vector k in the jth energy
band. However, in an n-type semiconductor there are also donors, which above
the Mott transition [8] produce free carriers in the conduction-band. Thus,
additional interactions arise between electrons and electrons, electrons and
ionized impurities and, in polar material, electrons and optical phonons. For ptype material the situation is analogous if the appropriate substitutions for holes,
20
Fig. 3. Plasma induced BGN in Si (left) and 4H-SiC (right) [14].
acceptors and the valence-band are made. The interactions can be described
using the perturbation Hamiltonian H1 resulting in new single electron energies
Ej(k). (The total Hamiltonian is then H = H0 + H1). The energy shift Ej(k) - Ej0(k)
= Re{hΣ j [k , E 0j (k ) / h ]} is the real part of the self-energy of the electron state. This
self-energy originates from the electronic self-energy operator, which is a kind
of dynamic, non-local potential that represents the effect of the electronic
exchange and correlation on the motion of each individual electron. The
operator is expressed in terms of the unperturbed electron Green’s function
Gj0(k,ω) and the dielectric function ε(k’-k,ω). Calculations of the dielectric
screening are performed in the simplified single plasmon-pole approximation [9]
or in the more exact random phase approximation (RPA) [10,11], which is
capable of giving the exact energy dispersion of the band structure. These
models usually assume zero temperature formalism and neglect local
fluctuations in the density of impurities. Averaging over statistical fluctuations
may be performed, if necessary, at the end of the calculations leading to the
band tailing effect [12]. BGN calculated in n-type Si and 4H-SiC within the
RPA method is illustrated in Fig. 2 [13]. The solid lines represent fundamental
BGN ∆EG, while the dashed ones show BGN detected in the optical type of
measurements ∆EGopt = ∆EG + EF, where EF is the Fermi energy of the e-h
system. Note that the BGN values presented are the sum from almost equal
effects in both conduction and valence bands, though only one type of carriers is
present in the semiconductor.
The doping induced BGN is additionally explained on the top of Fig. 4,
where the horizontal axis represents the density of states. The constant shift of
21
E
N(E)
EF
Donor
EG
Mott
transition
Band
tailing
EGopt
EG-∆EG
EFe∗
EFe∗
EG
EG-∆EG
EGopt
EFh∗
EFh∗
Fig. 4 Schematic representation of the doping (top) and plasma
(bottom) induced BGN. The top picture also explains the Mott
transition and the band tailing effects.
the conduction and valence band edges towards each other with increasing
doping (the diagrams from left to right, respectively) stands for the theoretically
calculated rigid band-gap shift: EG - ∆EG. In the experiments, however, one
detects the optical band-gap, which appears above the Fermi energy: EGopt = EG ∆EG + EF. The Mott transition in the picture corresponds to the moment when
the spreading donor level merges with the conduction band. The band tailing
22
effect on the other hand represents the band states spreading below the rigid
band-gap EG - ∆EG.
Optical excitation of a specimen creates an e-h plasma consisting of equal
amounts of electrons and holes. The basic description of the BGN effect in this
case is, nevertheless, analogous to that in the doped material, except that the
major interactions now appear between electron-electron, hole-hole, electronhole and, in polar material, electron-optical phonon and hole-optical phonon.
BGN calculated in the e-h plasma of Si and 4H-SiC is illustrated in Fig. 3 [14],
where solid and dashed lines have the same meaning as in the case of the doped
material. The plasma induced rigid band-gap shift EG - ∆EG is also illustrated on
the bottom of Fig. 4. Contrary to the case of the doping induced BGN, however,
the Mott transition and the band tailing effect do nor appear in an excited
material and the optical band-gap is now determined by the quasi-Fermi levels
in the both conduction and valence bands: EGopt = EG - ∆EG + EFe*. + EFh*.
References
[1] K. J. Malloy and J. A. VanVechten, J. Vac. Sci. Technol. B 9 (4), 2212
(1991).
[2] M. L. Cohen and D. J. Chadi, in Semiconductor Handbook, edited by M.
Balkanski, Vol. 2, Chap. 4b (North-Holland, Amsterdam, 1980).
[3] V. Heine and J. A. VanVechten, Phys. Rev. B 13, 1622 (1976).
[4] Y.P. Varshni, Physica 34, 149 (1967).
[5] R. Passler, Phys. Rev. B 66, 085201 (2002).
[6] K. P. O’Donnell and X. Chen, Appl. Phys. Lett. 58 (25), 2924 (1991).
[7] R. Passler, Phys. Stat. Sol. (b) 216, 975 (1999).
[8] N. F. Mott, in Metal-Insulator Transitions (Taylor and Francis, London,
1974).
[9] J. C. Inkson, J. Phys. C: Solid State Phys. 9, 1177 (1976).
[10] R. A. Abram, J. Phys. C: Solid State Phys. 17, 6105 (1984).
[11] K. F. Berggren, Phys. Rev. B 24, 1971 (1981).
[12] P. Van Mieghem, Rev. Mod. Phys. 64, 755 (1992).
[13] C. Persson, Phys. Rev. B 60, 16479 (1999).
[14] C. Persson, Solid State El. 44, 471 (2000).
23
24
3 Carrier transport
3.1 Diffusion and mobility
The following is a short introduction to the diffusion coefficients for free
carriers and their injection dependence. The focus is on the high carriers
densities, where the unified motion of correlated electrons and holes results in
ambipolar diffusivity. Certain physical phenomena, which effect simplified
predictions based upon the continuity equation in this range, will be commented
on.
3.1.1 Transport equations and ambipolar diffusion
Carrier (electron is assumed unless otherwise specified) current density in
a semiconductor consists of both drift and diffusion components:
J = eµnE − eD∇n ,
(1)
where E and ∇n stand for the electric field and the gradient of the carrier
concentration with the corresponding proportionality constants µ, the carrier
mobility, and D, the carrier diffusion coefficient. Under equilibrium conditions J
= 0 and:
µnE = µn(−∇Φ ) = D∇n ,
(2)
where Φ is the electrostatic potential. In the case of non-degeneracy the carrier
concentration in a band obeys a classical exponential dependency:
n = const × exp(−eΦ / k BT ) ⇒
(3a)
∇n / n = −e∇Φ / k BT .
(3b)
Substituting ∇n from Eq. (3b) into Eq. (2) one obtains the Einstein relation for
the concentration of electrons (n) and holes (p):
25
Di =
k BT
µi ,
e
i = n, p .
(4)
In general form, Eq. (3) obeys the carrier Fermi-Dirac statistics:
n = NF1/ 2 (η ) ,
(5a)
where
∞
x m dx
Fm (η ) = ∫
,
exp( x −η ) +1
0
(5b)
is the Fermi integral. N represents the concentration of donors or acceptors and
η equals ( EFn − Ec ) / k BT for electrons and ( Ev − E Fp ) / k BT for holes, where EFn
and EFp are their respective quasi-Fermi levels and Ec and Ev are the edges of the
conduction and valence bands. Performing the same procedure one obtains
instead of Eq. (4):
Di =
F1/ 2 (η ) k BT
µi ,
F−1/ 2 (η ) e
where
F−1/ 2 =
∂F1/ 2 (η )
.
∂η
(6a)
(6b)
The Einstein relation as expected in Eq. (4) applies only to one type of
carrier, while in a real semiconductor both types are present and related through
the mass action law:
ni2 = n0 p0 ,
(7)
where n0 and p0 represent the electron and hole concentrations at thermal
equilibrium. In the case of inter-band excitation these concentrations change to n
= n0 + ∆n and p = p0 + ∆p (∆n = ∆p). The carrier distribution of each type,
nevertheless, maintains charge neutrality resulting in the continuity equation
∂n 1
+ ∇ J = G − R∆n ,
∂t e
(8)
where G is the generation term and R stands for the recombination rate. When
analyzing processes after an excitation (G = 0) Eq. (8) becomes [1]:
26
∂∆n
∂ 2 ∆n
∂∆n
∂E
= Dn
+ µn E
+ µn n − R∆n ;
2
∂t
∂x
∂x
∂x
2
∂∆p
∂ ∆p
∂∆p
∂E
= Dp
− µpE
−µp p
− R∆p .
2
∂t
∂x
∂x
∂x
(9a)
(9b)
The ∂E / ∂x terms can be canceled due to the Poisson equation:
∂E e(∆n − ∆p )
=
=0.
ε
∂x
(10)
Next multiplying Eq. (9a) by the conductivity σp and Eq. (9b) by σn and adding
the two equations, the result is the following:
∂∆n D pσ n + Dnσ p ∂ 2 ∆n µ pσ n − µ nσ p ∂∆n
=
−
E
− R∆n .
∂t
∂x 2
∂x
σ n +σ p
σ n +σ p
(11)
The coefficient in front of the second derivative is called the ambipolar diffusion
and in front of the first derivative – the ambipolar mobility. Using expressions
for electron and hole conductivities ( σ n = enµ n and σ p = epµ p , respectively)
together with the Einstein relation they can be expressed as:
Da =
n+ p
n
p ,
+
D p Dn
µa =
n− p
n
p .
+
µp
(12)
µn
Physically the ambipolar diffusivity or mobility arises from the fact that an
internal electric field develops between the fast moving electrons and the more
slowly moving holes tying them into a coupled or ambipolar motion. For an
intrinsic semiconductor (n0 = p0) or in conditions of high injection (n ≈ p ≈ ∆n)
Eq. (12) reduces to:
Da = 2
D p Dn
D p + Dn
.
(13)
producing Da = 2Dp if Dn >> Dp, as is normally the case. Interestingly, µam = 0 at
high injections and there is no drift in an external electric field. On the other
hand, in n-type semiconductors (n0 >> p0) Eq. (12) gives Da = Dp, while in ptype (p0 >> n0): Da = Dn. The example of the full injection dependence according
Eq. (12) is displayed in Fig. 1 for 4H-SiC assuming doping concentration n0 =
7×1014 cm-3 and Dn = 25 cm2/s; Dp, =2.5 cm2/s. It can be concluded that the
27
De
1000
p-type
10
400
Damb
Dh
n-type
n0,p0
2
1013
1015
Mobility (cm2/Vs)
Diffusivity (cm2/s)
40
100
1017
Injected concentration (cm-3)
Fig. 1 Dependence of the carrier diffusivity on excitation density
for n-type and p-type 4H-SiC.
minority carriers establish the diffusivity of low excited extrinsic
semiconductors, while the carriers with the lower diffusion coefficient dominate
the ambipolar case at high excitations.
3.1.2 Phenomenological diffusion analysis
The discussion of the origin of the diffusion coefficient is usually
readdressed to the derivation of the carrier mobility to which it is linked by the
Einstein relation. This mobility is also known as the diffusion mobility. For some
particular carrier type, it is introduced as the proportionality constant between
the carrier drift velocity and the electric field vd = µ E , and is expressed in the
following way:
µ=
e
τ ,
m*
(14)
where m* is the effective mass of the carrier, τ represents the drift momentum
brackets indicate averaging over the distribution
relaxation time and the
function of the carriers. Several different scattering mechanisms operate in a
crystal, however, resulting in the approximate net relaxation time:
1
τ
=∑
i
1
,
τ i ( E)
28
(15)
Fig. 2 The theoretical electron Hall mobility dependence on temperature
(lhs) and doping concentration (rhs) in n-type 4H-SiC [5].
where index i refers to the different scattering mechanisms. The main
mechanisms under low injection conditions involve lattice vibrations and
ionized or, to a less extent, neutral impurities. Their relaxation times, in general,
obey different power functions [2]: τi ~ Er, where r = -1/2 in case of the acoustic
phonons, r = +3/2 for the ionized impurities and r = 0 for the neutral impurities.
Subsequently, the average τ i values of the individual mechanisms are
calculated from the τi(E) dispersion employing the solution of the kinetic
Boltzmann equation [3]. At high injection conditions the carrier-carrier
scattering mechanism should be also included [4].
In practice, however, the straightforward comparison of the mobility and
diffusion coefficient is far more complicated. In measurements one obtains the
Hall mobility, which differs from the diffusion mobility by the Hall factor:
2
rH = τ 2 / τ
[2]. An illustration of Hall mobility calculations including the
various scattering mechanisms is provided in Fig. 2 and 3 for n-type [5] and ptype [6] 4H-SiC, respectively. The factor rH is usually close to unity. However
in case of p-type material rH can only be extracted from measurements in
magnetic field of ~100 T [7]. Such high magnetic fields are hardly attainable
and so the unknown rH values are believed to be the main source of discrepancy
between the temperature dependence of the experimental ambipolar diffusion
coefficient and that extracted from the Hall mobilities, as discussed in article 4
of this thesis.
It is also important to mention here the influence of the drag effect of the
minority carriers, when they are swept along by the flow of the opposing
majority carriers [8]. During the process frequent collisions between minority
and majority carriers lead to a significant net momentum transfer to the minority
carriers resulting in a reduction in their mobility or even reversal of their drift
29
Fig. 3 The theoretical hole Hall mobility dependence on temperature
(lhs) and doping concentration (rhs) in p-type 4H-SiC [6].
velocity. The degree of reduction depends on whether the majority carriers are
nearly static, as in diffusion situations, or whether they are drifting in the
opposite direction due to an external electric field. The approximate mobility of
the minority carriers in these two cases are [9]:
−1
⎛ 1
1 ⎞
⎟ ;
µdiff = ⎜⎜ 0 +
⎟
µ
µ
np ⎠
⎝ min
µ0 ⎞
0 ⎛
⎜1− maj ⎟ ,
µdrift = µmin
⎜ µ ⎟
np ⎠
⎝
(16)
(17)
where µ0min and µ0maj are the minority and majority carrier mobilities calculated
exclusively for the case of lattice and impurity scattering and µnp is the electronhole scattering mobility. Eq. (16) and (17) provide different results, indicating
that in this case the minority-carrier diffusion and the drift mobilities are no
longer related through the Einstein relation.
3.1.3 Factors affecting ambipolar diffusion
In both Si [10] and 4H-SiC [11] a substantial decrease in the ambipolar
diffusion has been detected at high injection concentrations. It is proposed that
the plasma induced BGN is the primary effect responsible for this. The
thermodynamic approach is the simplest way to introduce the effect into
diffusion coefficient calculations. The carrier current density can be expressed
as a function of the space gradient of the carrier quasi-Fermi energy [3]:
30
J n = µn n∇EFn .
(18)
Additionally, it is possible to write the derivative of Eq. (5a) in the following
way:
∇n = N
∂F1/ 2 (η ) ⎛ ∂η
∂η
∂η
⎞
∇EFn + 0 ∇Ec0 +
∇(eΦ ) ⎟ .
⎜
∂η ⎝ ∂EFn
∂Ec
∂ (eΦ )
⎠
(19)
It is assumed here that the edge of the conduction band Ec is effected by the
presence of the electric field:
Ec = Ec0 − eΦ .
(20)
0
Now, replacing N by n / F1/ 2 and remembering that η = ( EFn − Ec + eΦ ) / k BT , Eq.
(19) becomes:
∇n = n
F−1/ 2 1
(∇EFn − ∇Ec0 + ∇(eΦ) ) ⇒ ∇EFn = ∇n F1/ 2 k BT + ∇Ec0 − ∇(eΦ) ,(21)
F1/ 2 k BT
n F−1/ 2
where n was obtained by using Eq. (5a) again. Substituting Eq. (21) into Eq.
(18) one gets:
⎛ F1 / 2
n ∂Ec0 ⎞
⎟⎟∇n + eµ n nE ,
J = µ n k BT ⎜⎜
+
⎝ F−1 / 2 k BT ∂n ⎠
(22)
where
∇Ec0 =
∂Ec0
∇n
∂n
and
E = −∇Φ
(23)
are employed. Remembering the Einstein relation in Eq. (4) and using the
analogy of Eq. (22) to Eq. (1), one obtains an expression for the effective
electron diffusion coefficient in the presence of the BGN:
⎛ F
n ∂Ec0 ⎞
⎟⎟ .
Dneff = Dn ⎜⎜ 1 / 2 +
F
k
T
n
∂
B
⎠
⎝ −1 / 2
(24)
An analogous expression can be derived for holes. Again both of the effective
coefficients are related to the ambipolar diffusion through Eq. (12). Note that the
same result is also obtained in calculations using the random phase
approximation [12]. The carrier behavior in Eq. (24) is illustrated in Fig. 4 (b) in
31
a)
Dn
b)
Dn
Dneff
EG - ∆EG
EG
Dp
Dneff
Dpeff
Dp
Dpeff
Fig. 4 Ambipolar diffusion of carriers (a) at low excitation conditions;
(b) at conditions of high excitation, when the material band-gap EG is
effected by the plasma induced BGN ∆EG.
comparison to the simple prediction by Eq. (6a) shown in Fig. 4 (a). In Fig. 4(b)
the ∂Ec0 / ∂n ≈ −∆EG term strongly reduces the speed of the diffusion process.
It has also been pointed out that bound excitonic states can influence the
diffusion coefficient [13]. As discussed in the following chapter, the exciton is
an electrically neutral electron-hole complex, but it can contribute to the
diffusive motion. Extending the same idea to the scattered excitonic states above
the excitonic Mott transition and assuming a different diffusion coefficient Dex
for the excitons, the total effective ambipolar coefficient becomes:
⎛ dn ⎞
⎛ dn ⎞
Da,ef =Da ⎜1− ex ⎟ + Dex ⎜ ex ⎟ ,
⎝ dntot ⎠
⎝ dntot ⎠
(25)
where ntot = n + nex and nex is the concentration of excitonic complexes.
References
[1] R. A. Smith, Semiconductors (2nd edition, Cambridge university Press,
Cambridge, 1978).
[2] K. Seeger, Semiconductor Physics, (Speringer-Verlag, Vienna, 1973).
[3] M. Lundstrom, Fundamentals of Carrier Transport, (2nd edition, Cambridge
University Press, 2000).
32
[4] J. A. Meyer and M. Glickman, Phys. Rev. B 17, 3227 (1978).
[5] H. Iwata and K. M. Itoh, J. Appl. Phys. 89, 6228 (2001).
[6] A. Martinez et al., J. Appl. Phys. 91, 1359 (2002).
[7] G. Rutsch, J. Appl. Phys. 84, 2062 (1998).
[8] T. P. McLean and E. G. S. Paige, J. Phys. Chem. Sol. 16, 220 (1960).
[9] D. E. Kane and R. M. Swanson, IEEE Trans. El. Dev. 40, 1496 (1993).
[10] J. Linnros and V. Grivickas, Phys. Rev. B 50, 16943 (1994).
[11] Appended articles 1,2 and 4.
[12] J. F. Young and H. M. VanDriel, Phys. Rev. B 26, 2147 (1982).
[13] D. E. Kane and R. M. Swanson, J. Appl. Phys. 73, 1193 (1992).
33
3.2 Carrier recombination
The interpretation of pump-probe experiment relies upon an
understanding of the recombination processes of free carriers. Indeed, the free
carrier lifetime is one of the primary physical quantities extracted from the
excited free carrier concentration dynamics detected by a FCA probing
technique (see Ch. 5.1). This chapter describes the basic recombination
mechanisms and their relative importance at different excitation intensities.
Trapping effect for free carriers are also explained.
3.2.1 Carrier lifetime and recombination mechanisms
If the drift and diffusion of free carriers is neglected (i.e. no electric field,
homogeneous medium and excitation), then the following continuity equation
can be derived from the principle of detailed balance [1]:
−
d∆n
= R(np − ni2 ) = R(n0 ∆p + p0 ∆n + ∆n∆p) ,
dt
(1)
where R is the recombination rate. In the case of ∆n = ∆p << n0 , p0 , the solution
of Eq. (1) is a simple exponential decay of the carrier concentration:
d∆n
∫ ∆n
= −∫
dt
τm
⇒ ln ∆n = −
t
τm
+ c ⇒ ∆n = ∆n(0) exp(−t /τ m ) ,
(2)
where τ m =1/(n0 + p0 ) R is called the minority carrier recombination lifetime.
Conversely, when ∆n = ∆p >> n0 , p0 , the return to equilibrium follows the
equation:
d∆n
∫ ∆n
2
= − R ∫ dt ⇒
1
1
=
+ Rt ,
∆n ∆n ( 0 )
(3)
and recombination is said to be “bimolecular”. In the general case, Eq. (1) has
no simple analytical solution and results in an effective carrier lifetime, which is
dependent on the carrier concentration ( τ = τ (∆n) ):
∆n
∆n
d∆n
=−
⇒ τ =−
.
dt
τ
d∆n / dt
34
(4)
a)
b)
c)
d)
EC
ED
hω
ED
EV
Fig. 1 Schematic representation of the recombination mechanisms:
a) radiative, b) SRH, c) Auger and d) trap assisted Auger.
In analogy to Eq. (4), one can express “bimolecular” or high-level injection
lifetime as τ hl = 1/ ∆nR .
Recombination of the excess e-h pairs can proceed by several different
physical mechanisms. So far, we have assured recombination via the radiative
band-to-band mechanism (R = B usually signifies the purely radiative
coefficient), which is shown schematically in Fig. 1(a). This process results in
the emission of a photon with the characteristic energy of the band-gap.
Radiative recombination dominates in direct band-gap semiconductors, but it is
about four orders of magnitude smaller in indirect semiconductors [2]. In these
latter materials the most efficient mechanism is the Shockley-Reed-Hall (SRH)
[3] recombination illustrated in Fig 1(b). It assumes the involvement of deep
centers near the middle of the band-gap. In such processes an electron is first
trapped at a defect level and then a hole is captured at the same point,
recombining with the electron, or vice versa. In the classical interpretation the
excess energy in this process is released in form of phonons. The SRH lifetime
can be derived using Eq. (1) modified to include recombination and emission
through the deep center, and is given by the expression [4]
τ SRH =
τ p (n0 + ∆n + n1 ) +τ n ( p0 + ∆n + p1 )
p0 + n0 + ∆n
,
(5a)
where
τp =
1
and
σ p vth N D
35
τn =
1
,
σ n vth N D
(5b)
Fig. 2 The injection dependence of the SRH lifetime in n-type
Si (solid curves) with the trap energy level as parameter (τp =
100 µs, τn = 1 ms). The dashed lines show limitations due to
radiative and Auger recombination rates using generally
accepted values for their respective cross sections [5]. Note
that notation ET in the figure equals ED in this work.
where vth, ND, σp and σn denote thermal velocity, defect density and crosssections for hole and electron capture by the defect, respectively. The quantities
n1 = ni exp[(Ei-ED)/kBT] and p1 = ni exp[(ED-Ei)/kBT] are defined for ED, the
recombination center energy, with respect to the intrinsic Fermi energy Ei. Since
τp and τn are inversely proportional to the defect concentration ND, the
corresponding lifetime τSRH is a sensitive measure of the defect concentration in
a sample [5].
It is convenient to analyze Eq. (5) in the limiting cases of low-level (LL)
and high-level (HL) injections (i.e. at ∆n << n0 and ∆n >> n0 , assuming n-type
material):
⎡
⎛ E − EF
τ SRH (LL)= τ p ⎢1 + exp⎜ D
⎣
⎝ k BT
⎞⎤
⎛ 2 E − ED − EF ⎞
⎟⎥ + τ n exp⎜ i
⎟.
k BT
⎠⎦
⎝
⎠
τ SHR(HL) = τ p + τ n
36
(6a)
(6b)
where EF is the Fermi energy. It is evident that at LL injection conditions the
minority carrier lifetime τp dominates. Moreover, the LL lifetime depends a
great deal on the energy ED of the recombination center and the ratio of the
capture cross sections for electrons and holes: ξ = σn /σp [6]. As illustrated in
Fig. 2 for the case of n-type Si, the LL lifetime has its smallest value when the
recombination center lies close to the middle of the band-gap. If the center is
shifted towards either the conduction or the valence band edge, the LL lifetime
increases due to the decreasing probability of capturing holes or electrons,
respectively, and the competing processes of reemission of carriers to the bands.
When injections approach HL conditions, a constant HL lifetime independent of
the position of the recombination center or temperature is reached. Only the
dependence upon the capture cross-section ratio ξ remains.
The band-to-band Auger recombination, the third illustrated in Fig. 1(c),
is the recombination of an electron and a hole while energy is transferred to a
third carrier. The process can also be phonon assisted. According Eq. (4) it is
expected that the Auger lifetime follows the relation:
τ Auger =
∆n
,
γ h ( p n − p n ) + γ e ( pn 2 − p0 n02 )
2
2
0 0
(9)
where γe and γh are the Auger recombination coefficients for the e-e-h and h-h-e
combinations of particles during the process, respectively. In the LL minority
carrier regime Eq. (9) simplifies to
τ Auger =
1
;
γ e n02
∆n << n0 ;
(10a)
τ Auger =
1
;
γ h p02
∆n << p0 ;
(10b)
for highly doped n-type and p-type material, respectively. At HL injections in
intrinsic material the Auger lifetime becomes
τ Auger =
1
;
γ amb ∆n 2
∆n >> p0 ,n0 ;
(11)
with the ambipolar Auger coefficient γamb = γe + γh. Auger recombination
dominates only in highly doped samples or at very high injections, since the
reaction probability involves three carriers. Nevertheless, as indicated in Fig. 2,
this process, rather than radiative recombination, limits the carrier lifetime in
37
indirect semiconductors at high carrier densities. In addition, the concept of
enhanced “excitonic Auger” recombination has been introduced [7] to explain
an intrinsic upper limit to the carrier lifetime, which appears even at room
temperature.
The fourth recombination process shown in Fig. 1(d) is the trap-assisted
Auger recombination. This is similar to the SRH process, except that the energy
released when the electron (or hole) becomes trapped is transfered to a second
electron (or hole) in an Auger process, rather than to a phonon. Eq. (1) for the
recombination rate of the process, nevertheless, predicts a quadratic dependence
on the carrier concentration (i.e., the derivative is proportional to np, nn, or pp)
as in the case of radiative recombination; however, the rate constant R is now
proportional to the defect density NT. Radiative recombination and trap-assisted
Auger may be combined into a single radiative coefficient:
B = Brad + β Auger N D ,
(7)
where Brad signifies the purely radiative component and βAuger is a coefficient for
the trap-assisted process.
All the recombination mechanisms presented can be expected to occur in
a sample simultaneously. The different lifetime components can then be
summed in an inverse manner to give the total lifetime:
1
=
1
+
1
+
1
τ τ rad τ SRH τ Auger .
(12)
In this expression the shortest of the three lifetimes dominates.
Finally, excluding the effects of carrier diffusion, when constructing the
continuity equation in Eq. (1), means that our derived expressions are only
correct for the “bulk” lifetime in a material, i.e., they apply for a homogeneously
excited and doped sample with infinite boundaries. Such conditions are hardly
satisfied in many practical situations, where carrier gradients appear from
inhomogeneous excitation or from variations in the local recombination rate.
Moreover, experimental specimens usually possess many surfaces or interfaces
with an enhanced surface recombination rate. The latter is not an additional
recombination mechanism, but simply SRH recombination occurring at
numerous surface defects like, e.g., dangling bonds. In order to obtain a “bulk”
lifetime one has to solve the full diffusion equation [8]:
dn
∂ 2n
= D 2 − γ 1n − γ 2 n 2 − γ 3n3 ,
dt
∂x
38
(13)
with the boundary condition
D
∂n
= − Sn ,
∂x
(14)
where S is the surface recombination velocity expressed in units of cm/s.
Coefficients γ1, γ2 and γ3 in Eq. (13) specify monomolecular, bimolecular and
Auger recombination rates, respectively, which are related to the instantaneous
bulk lifetimes as:
1
τ
= γ 1n + γ 2 n 2 + γ 3n3 .
(15)
3.2.2 Carrier trapping
It has so far been assumed that ∆n = ∆p during the recombination process,
i.e., that there is a single type of defect center with a single trap level. This
condition is not satisfied, however, when the concentration ND of defects with
different capture cross-sections σp and σn is high relative to the doping or
injection densities or when a high concentration NT of shallow defect levels is
present in addition to the main recombination center. In the latter case, the
shallow level can act as a trap, into which a carrier can fall being thermally
released back into the band rather than the recombination process being
completed by the capture of an opposite charge carrier. The main condition for
this phenomenon is that the shallow level has to lie outside the energy region
between the quasi-Fermi levels at the particular excitation intensity [9], as
illustrated in Fig. 3(a), where the complete cycle between carrier generation and
recombination is shown. In general, such recombination processes can be
described using separate carrier emission e, capture c and generation G rates, as
illustrated in Fig. 3(b). Consequently, the electron concentration in the
conduction band n, in the trap nT, and at the recombination center nD, together
with the hole concentration in the valence band p, obey the following rate
equations:
dn
= −cnD ( N D − nD ) + enD nD − cnT ( NT − nT ) + enT nT + G ;
dt
(16a)
dnT
= cnT ( N T − nT ) − enT nT + eTp ( N T − nT ) − cTp nT ;
dt
(16b)
39
a)
b)
EFe
EFh
generation
EC
NT
ND
EV
recombination
trapping
G
cnT
enT
epT
cpT
cnD
enD
epD
cpD
Fig. 3 Schematic representation of the trapping mechanism.
dnD
= cnD ( N D − nD ) − enD nD + e Dp ( N D − nD ) − c pD nD ;
dt
(16c)
dp D
= e p ( N D − nD ) − c pD nD + eTp ( NT − nT ) − cTp nT − G ;
dt
(16d)
where
cnD ,T = σ nD ,T vn n ,
c Dp,T = σ pD ,T v p p
(17a)
are the capture rates for electrons and holes, respectively, and
⎛ E − ED ,T ⎞
⎛ E −E ⎞
enD ,T = σ nD ,T vn N C exp⎜ − C
⎟ , e pD ,T = σ pD ,T v p NV exp⎜ − D ,T V ⎟ (17b)
k BT ⎠
k BT ⎠
⎝
⎝
are their respective emission rates. These quantities differ between the deep
recombination center and the shallow trap only in the capture cross-section and
the position of the relevant energy level in the band-gap. Theoretical fits of Eq.
(16) to experimental carrier dynamics at different temperatures and at different
excitation conditions can provide values for the capture cross-section, the energy
position and concentration of traps, as has been done, for example, in the case of
3C-SiC [10.11].
References
40
[1] J. W. Orton and P. Blood, The Electrical Characterization of
Semiconductors: Measurement of Minority Carrier Properties (Academic Press,
London, 1990).
[2] J. Linnros and V. Grivickas, Carrier lifetime: Free carrier absorption,
Photoluminescence and Photoconductivity, Ch. 5b.2 in Methods in Material
Research, edited. by E. N. Kaufman (John Wiley & Sons Inc., 2000).
[3] W. Shockley and W. T. Read, Phys. Rev. 87, 835 (1952); R. N. Hall, ibid.
87, 387 (1952).
[4] R. A. Smith, Semiconductors (2nd edition, Cambridge university Press,
Cambridge, 1978).
[5] J. Linnros, J. Appl. Phys. 84, 275 (1998); ibid. 284 (1998).
[6] B. J. Baliga, Power Semiconductor Devices (PWS Publishing Company,
Boston, 1996).
[7] A. Hangleiter, Phys. Rev. Lett. 65, 215 (1990).
[8] A. Galeckas, Appl. Phys. Lett. 71, 3269 (1997).
[9] S. M. Ryvkin, Photoelectric Effects in Semiconductors (Consultants Bureau,
New York, 1964).
[10] M. Ichimura, Appl. Phys. Lett. 70, 1745 (1997).
[11] M. Ichimura, J. Appl. Phys. 84, 2727 (1998).
41
42
4 Optical absorption
4.1 Excitons
Absorption of a photon in a material corresponds to an optical transition
of one electron from the ground state (valence band) to the excited state
(conduction band). The transition, in fact, generates two oppositely charged
carriers, since simultaneously a hole is created in the valence band. The two
elementary charges physically emerge at the same part in space within the
crystal. Therefore, due to their mutual Coulomb attraction, electron and hole
engage into a correlated motion and appear in energy levels within the material
band-gap EG. This elementary excitation comprising correlated charged particles
is treated as a new particle: an exciton.
4.1.1 Quantum mechanical description
Though various levels of sophistication exist for the theoretical modeling,
the most simple and convenient way to describe an exciton is with the effective
mass approximation [1]. This approximation is valid if the interaction potential
between the carriers varies slowly over the dimensions of a unit cell, i.e. the
exciton dimension must be large compared to the lattice constant. The approach
is especially suited to semiconductor materials, where the Coulomb force
V (r ) = −e 2 /(4πε 0ε r re − rh ) between the excited carriers is strongly screened by
the valence electrons. Application of the effective mass approximation leads to a
hydrogen-like problem and an excitonic Bohr radius expressed by:
a =a
ex
0
H
B
4πε 0ε r h 2
εr =
;
µ
µ e2
m0
1
µ
=
1
1
+
,
me mh
(1)
where m0 is mass of the free particle and me and mh are effective masses of
electron and hole in the crystal, respectively. Substituting in Eq. (1) values of the
reduced mass µ and relative dielectric constant εr for typical semiconductors,
one obtains [2]:
ex
50 nm ≥ a0 ≥ 1 nm > alattice.
43
(2)
This means that orbits of the electron and hole around their common center of
mass (CM) average over many unit cells and this in turn justifies the effective
mass approximation in a self-consistent way. In general, excitons with Bohr
radius much larger than the lattice constant are called Wannier excitons.
To obtain the energy dispersion of the exciton one has to start from the
classical Schrödinger equation:
(H + V (r ))Ψ (re , rh ) = EΨ (re , rh ) ,
(3)
where Ψ(re,rh) represents the wavefunction of the exciton as a single particle,
which is a linear combination of the electron and hole Bloch functions:
Ψ (re , rh ) =
∑ C (k , k )ψ
e
h
ke , k h
ke
(re )ψ kh (rh ) .
(4)
This function can be expressed in terms of the Wannier functions of electron and
hole:
Ψ (re , rh ) = N −1 / 2
∑ Φ( R , R )ϕ
e
h
Re , Rh
Re
(re )ϕ Rh (rh ) ,
(5)
where N is the number of unit cells in the crystal. The major advantage is that
indexing the Wannier functions by the lattice vectors R in real space replaces the
Bloch function indexing by wave vectors k in reciprocal space. Wannier
functions are Fourier transforms of Bloch functions, but they are more
appropriate for representing localized states in the crystal. The function Φ(Re,Rh)
alone is sufficient for calculations in Eq. (3) if only properties of two particle
complex are sought.
In the effective mass approximation, kinetic energy operators for a free
particle replace the Hamiltonian Η of the perfect crystal and Eq. (3) can be
expressed as:
⎡ ⎛ h2
⎢− ⎜⎜
⎢⎣ ⎝ 2me
⎞
⎛ h2
⎟⎟∇ Re − ⎜⎜
⎠
⎝ 2mh
⎞
e2
⎟⎟∇ Rh −
ε 0ε r Re − Rh
⎠
⎤
⎥ Φ ( Re , Rh ) = EΦ ( Re , Rh ) ,
⎥⎦
(6)
where ∇Re and ∇Rh are Laplace operators. It is convenient to replace the lattice
coordinates of free particles Re and Rh by the coordinate R of the CM and the
relative internal coordinate r:
44
R=
me Re + mh Rh
;
me + mh
r = Re − Rh .
(7)
Now one can decouple the Schrödinger equation into separate motion of the CM
and relative motion of the charge carriers [3]:
⎛ h2 ⎞
⎟⎟∇ Rψ ( R ) = E Rψ ( R ) ;
⎜⎜ −
2
M
⎠
⎝
⎛ h2
e2 ⎞
⎜⎜ −
⎟⎟φ (r ) = Erφ (r ) ,
∇r −
2
µ
ε
ε
r
0 r ⎠
⎝
(8a)
(8b)
where M = me + mh is translative mass of exciton and µ is reduced mass of the
exciton introduced in Eq. (1). The total energy of the exciton E is simply the
sum of ER and Er and the product exciton wavefunction has the linear relation
Φ( Re , Rh ) = Φ( R, r ) = ψ ( R )φ ( r ) . Solutions for the eigenfunctions and energies in
Eq. (8a) and (8b) can be easily shown to be:
h2K 2
;
2M
E
Er (n) = Er (∞) − ex2 .
n
ψ K ( R) = N −1 / 2 exp(iKR) ;
ER =
φnlm (r ) = Rnl (r )Ylm (θ ,ϕ ) ;
(9a)
(9b)
Eq. (9a) indicates that an exciton can be described as a plane wave moving in
the crystal, the kinetic energy dispersion of its CM motion being analogous to
that of a free particle. In reciprocal space such motion can be represented as K =
ke + kh = kc − kv, with wave vectors kv and kc entering from the valence and
conduction band, respectively. At the same time, Eq. (9b) indicates that the
internal structure of the exciton is similar to that of a hydrogen atom, where
quantized relative particle motion is described by wavefunctions, expressed in
spherical coordinates Rnl(r) and Ylm(θ,ϕ). Here, index n is the principal quantum
number, l represents the angular momentum and m is the magnetic quantum
number. The description of these functions may be found in standard quantum
mechanics textbooks [4]. For practical convenience, the structure of the exciton
can be imagined in the real space where the light electron elliptically orbits
around the much heavier hole, as shown in Fig. 1(a). An analogy is possible in
classical mechanics, where an ellipse describes the movement of a body in an
attraction field proportional to 1/r2, if the averaged kinetic energy of the body is
less than the averaged potential energy. The arrow in the figure indicates the
orientation of the electron’s elliptical orbit, which is alternating in time. The
squares of the function’s modulus ⎟Rnl(r)⎟2 and ⎟Ylm(θ,ϕ)⎟2 in such picture
45
a)
b)
e
h
h
e
Fig. 1 Illustration of the discrete (a) and the continuum (b) excitonic
states in the real space in analogy to the classical mechanics. Gray
periodic background in the figure imitates lattice of the crystal.
acquire the meanings of the probability of finding the electron and hole at a
certain relative distance r from each other and in a certain direction in space,
respectively. For the lowest energy level n = 1, the maximum of ⎟Rnl(r)⎟2
ex
coincides with the excitonic Bohr radius a0 , given by Eq. (1). In general, the
function φ (r ) in Eq. (9b) is called the exciton envelope function, and it
determines the overall absorption probability into the excitonic states, as will be
shown in the next paragraph.
Eq. (9b) provides an energy spectrum of discrete excitonic states, which
depend on the principal quantum number n. The quantity Er(∞) is the minimum
energy of the continuum or unbound states, while Eex is called the exciton
binding energy, and it is related to the Rydberg energy of the hydrogen atom:
µ
Eex =
Ry H ;
2
m0ε r
e 4 m0
Ry =
= 13.6 eV.
32π 2 h 2ε 02
H
(10)
Combining Eq. (9a) and (9b), the total energy of the exciton is expressed as:
En ( K ) = EG −
Eex h 2 K 2
+
,
n2 2M
(11)
where the semiconductor band-gap EG replaces Er(∞). Since an exciton is a twoparticle system, its energy levels should be depicted in a separate diagram,
where the ground state (the absence of an electron and hole pair) is represented
by the origin. Such a diagram is shown in Fig. 2(a). The indexed solid curves are
46
a)
b)
E
E
n→∞
n=2
k
K
Eex
E(k∞)
potential
n=1
kinetic
EG
hω
K
0
Fig. 2 (a) Discrete (lines) and continuum (dashed area) states of an exciton versus
the CM wave vector K. Black arrows indicate the excitation of an exciton into a
continuum state. (b) Decomposition of the total continuum excitation energy
E(k∞) into the contributions from different wave vectors.
discrete excitonic states appearing below the band-gap EG and following the
h 2 K 2 / 2 M energy dispersion law.
The dashed area in Fig. 2(a) corresponds to the excitonic continuum. In
this energy range the solutions of the Schrödinger equation are delocalized
exciton functions Φ K ( R, r ) , which are, though, perturbed by the electron-hole
interaction in the space area around r → 0. This result can be understood as
correlated electron motion around the heavy hole in open hyperbolic orbits
shown in Fig. 1(b). The kinetic energy of the excited electron in the exciton
continuum exceeds the potential energy of the attracting hole and the electron
escapes from its closed orbit. Nevertheless, integration in time provides nonzero probability of correlated particle motion.
If one dares to draw a bizarre analogy between the excitonic continuum
and every day life, it is convenient to imagine young people dancing in a
discotheque. Though boys and girls are on average distributed fairly evenly over
the entire dance floor, the probability at any one time of finding them close to
one another is very big. Therefore, even though nobody dances in real waltz
couples nowadays, we still talk about the couples even at uproarious parties.
The total number of states per unit energy in the continuum is the same as
in the case of the free carrier approximation, although each continuum state is
degenerate due to the infinite number of possible combinations of the quantum
47
E
Eex
EGind
hω
0
k0 = kc
K
Fig. 3 Excitonic states in the indirect gap semiconductor. Dashed arrows
illustrate different phonons assisting the transition.
numbers l and m. Therefore, energy dispersion for the exciton continuum can be
derived from the total energy of the free electron and hole in the following way:
h 2 k 2 h 2 (k − K ) 2
h 2 ke2 h 2 kh2
h 2 K 2 h 2 k∞2
EG +
+
⇔ EG +
+
⇔ EG +
+
, (12a)
2me 2mh
2me
2mh
2M
2µ
providing
Econt ( K ) = EG ( K ) + E (k∞ ) ,
(12b)
where
k = ke ,
K = k e + k h , k ∞ = k − (me / M ) K .
(12c)
Here k∞ represents the exciton wave vector in the continuum. Eq. (12b) and
(12c) specify the main properties of the exciton continuum, stating that
excitation into an energy state of total value E(k∞) is possible with any K.
Changes in kinetic energy during the process are compensated by the potential
exciton energy represented by the relative electron motion wave vector k. This
scenario is illustrated in Fig. 2(b), where excitation into the arbitrary E(k∞) state
is composed of two parts controlled by the different wave vectors. Overall E(k∞)
dependency in the momentum space is steeper than the dispersion curves of the
discrete states (see Fig. 2(a)), due to the reduced mass µ present in the
denominator of Eq. (12a). It should be noted, however, that k in Eq. (12c) is the
48
electron wave vector in a one particle band picture and, therefore, k∞ is a
quantum number in the excitonic band representation and has nothing to do with
a real momentum [5].
The discussion has so far been for direct gap semiconductors. However,
the energy dispersion results obtained are equally applicable to indirect gap
material, for which the following expressions are acquired:
E (K ) = E
ind
n
ind
cont
E
ind
G
(K ) = E
ind
G
Eex h 2 ( K − kc ) 2
− 2 +
;
n
2M
(13)
h 2 ( K − kc ) 2 h 2 k∞2
+
+
.
2M
2µ
(14)
Here kc stands for the momentum difference between the conduction and
valence bands (kv = 0). Wave vector K now represents the momentum kq of the
phonon-assisted transition. The situation is shown in Fig. 3, where dashed
arrows illustrate different phonons transferring the exciton to bound or
continuum states. Note that even though the excitonic continuum in indirect
transitions is usually identified with continuum band states it has, in fact, the
same properties elucidated earlier.
4.1.2 Excitonic Mott transition and excitonic enhancement
So far, the discussion has been concerned with single e-h pairs. In the
high-density regime, however, many-particle effects considerably modify the
excitonic features presented. Indeed, bound excitonic states in the energy band
gap dissociate when the average distance between e-h pairs becomes comparable
( )
3
ex
ex
to the excitonic Bohr radius a0 of the bound exciton: n p a0 ≈ 1 , where np is
the density of e-h pairs. The main reason for this phase transition, known as the
excitonic Mott transition, is that free carriers screen the Coulomb interaction
between electrons and holes, transforming the potential into a Yukawa-type
potential:
−
e2
4πε 0ε r re − rh
⇒ −
⎛ r −r
1
exp⎜⎜ − e h
4πε 0ε r (n p ) re − rh
l
⎝
⎞
⎟,
⎟
⎠
(15)
where l is the screening length. This Yukawa-type potential does not have a
bound state if l falls below a certain critical value:
49
EG
ex.
enhancement
Eex
energy
ex. state
Mott
µ
T < TC
BGN
T > TC
np
Fig. 4 Schematic phase diagram according [2] showing the ground excitonic state, the
plasma induced BGN and the chemical potential µ (above and below critical
temperature TC) as a function of the concentration of e-h pairs. The upper dashed line
indicates the excitonic enhancement effect.
lc = aBex /1.19 .
(16)
Using classical Boltzmann statistics, lc equals the Debye-Huckel screening
length:
1/ 2
l DH
⎛ε ε k T ⎞
= ⎜ 0 r2 B ⎟
⎜ 4πe n ⎟
p ⎠
⎝
.
(17)
Combining Eq. (16) and (17) one obtains the expression for the Mott
concentration:
n M = (1.19)
2
ε 0ε r k BT
e 2 (a 0ex ) 2
.
(18)
Note that the metallic type screening described here, and mediated through free
e-h pairs, is only realized when the Mott transition is approached from the highdensity regime [6]. If excitonic states are excited resonantly, they encounter
nonmetallic or “dielectric” excitonic screening, which is caused by the existence
of the excitons themselves. It has been proven [7] that in this case the Mott
transition appears at concentrations over an order of magnitude higher.
Nevertheless, this type of screening occurs for relatively short times, because the
50
excitons are quickly ionized into a continuum of e-h pairs via inelastic
scattering.
A second process, which contributes to the Mott transition, is the phasespace filling effect. When an exciton is created, the substituting electron and
hole states in k-space become filled. Now that they have become occupied, these
states are blocked and cannot participate in the construction of additional
excitons. This is a consequence of the fermion character of the constituents of
the exciton. The process plays a more important role in quantum-confined than
bulk semiconductors where the screening of the Coulomb interaction is
substantially reduced.
In general, the excitonic Mott transition has the same origin as the
previously introduced impurity Mott transition (see Ch. 2.3). Only the excitonic
one usually appears at lower carrier concentrations due to the fact that excitonic
states in the band-gap lie above donor or acceptor levels. Fig. 4 illustrates
schematically the dependence of the ground excitonic state and the plasma
induced BGN on the e-h pair density np. In this representation the excitonic Mott
transition corresponds to the crossing of the two curves. With increasing np the
excitonic state remains at the same energy up to the Mott transition. This
stability is attributed to compensation between the decreasing exciton binding
energy Eex and the downwards-shifting band-gap EG. The figure also shows the
behavior of the chemical potential:
µ (n p ,T ) = Fe * + Fh * + EG ,
(19)
where Fe* and Fh* are the quasi-Fermi energies of electrons and holes,
respectively. At high densities, when the e-h plasma becomes degenerate (quasiFermi energies reach the corresponding bands), dependence of µ increases
steeply due to the Fermi statistics. At low densities, however, the dependency
may go through a maximum and a minimum below a certain critical temperature
TC, as shown by the corresponding dashed curve. In such conditions quasiequilibrium thermodynamics predict a first-order phase transition to an electronhole liquid [2]. Note, however, that all the results in this thesis were obtained
above this region, since TC = 24.5 K for Si and <60 K for 4H-SiC.
In absorption measurements for direct-gap semiconductors, excitonic
peaks (see the next paragraph) at increasing excitation intensity broaden and
eventually merge with the excitonic continuum. However, contrary to previous
expectations that the peaks would disappear above the excitonic Mott transition,
it was found that an enhanced absorption coefficient persists with growing
carrier density at the absorption edge for several orders of magnitude [8]. This
behavior is illustrated in Fig. 4 by the dashed line extending the excitonic state.
The effect, called excitonic enhancement (or the Fermi-edge singularity in quasi
two-dimensional systems), originates from the increased correlation between
electrons and holes in the many particle system resulting in enhanced oscillator
51
strength close to the Fermi level. Partial motion in the Bohr-orbits, nevertheless,
is possible only in the empty band states. The threshold for the effect, therefore,
follows the chemical potential µ at high-densities, as shown in the figure.
Optical nonlinearities at high e-h plasma densities, including absorption spectra,
have been calculated using a many-body theory [9], extended up to roomtemperature [10]. The same excitonic enhancement effect was also confirmed in
doped direct-gap semiconductors [11].
Finally, it is important to mention that excitons have integer spin and so
are bosons. At low temperatures one should theoretically, therefore, expect
spontaneous Bose-condensation of excitons [2].
References
[1] G. Dresselhaus, J. Phys. Chem. Solids 1, 14 (1956).
[2] C. F. Klingshirn, Semiconductor Optics (Speringer-Verlag, Berlin, 1997).
[3] P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (SperingerVerlag, Berlin, 1996).
[4] L. D. Landau and E. M. Lifshitz, Quantum mechanics (2nd edn., Pergamon
Press, Oxford, 1975).
[5] P. K Basu, Theory of Optical Processes in Semiconductors, (Clarendon
Press, Oxford, 1997).
[6] G. B. Norris, Phys. Rev. B 26, 6706 (1982).
[7] G. W. Fehrenbach, Phys. Rev. Lett. 49, 1281 (1982).
[8] J. Shah, Phys. Rev. B 16, 1577 (1977).
[9] J. P. Lowenau, Phys. Rev. Lett. 49, 1511 (1982).
[10] J. P. Lowenau, Phys. Rev. B 51, 4159 (1995).
[11] F. Fuchs, Phys. Rev. B 48, 7884 (1993).
52
4.2 Fundamental absorption
The term “fundamental absorption” refers to an optical transition in which
an electron from the valence band is excited to i) the conduction band or ii) an
excitonic state. The former case will be considered first, relating the spectral
shape of the absorption to the density of states in the crystal. The Afterwards,
appropriate modifications accounting for mutual particle interactions will be
introduced. It will be shown that the strongest modification due to excitonic
transitions occurs at the absorption edge near the energy gap of an intrinsic
semiconductor. Finally, possible analytical models will be provided for handling
practical absorption spectra. Most of the equations in this chapter are derived
from a set of textbooks [1,2,3,4,5,6].
4.2.1 Band-to-band absorption
Around the indirect band-gap EG in SiC and Si direct transitions
involving only the interaction between electrons and photons are impossible due
to the fact that wave vector k does not change in such type of transitions. In
these materials one observes indirect transitions, in which phonons have to
provide an additional wave vector kq to conserve the momentum of the excited
electron system. Nevertheless, the explanation of direct transitions will be
outlined first due to the phenomenological simplicity illustrating the quantummechanical nature of the optical transition process.
In general, the transition rate W between the initial i and final f states is
expressed by the Fermi golden rule:
W=
2
2π
f H fi i δ ( E f − Ei − hωυ ) ,
∑∑
h i f
(1)
where Hfi is the interaction Hamiltonian operator and the δ function indicates
that only the exact energy of the photon ensures conservation of energy. Hfi for
the direct optical transitions is treated using first-order perturbation theory
resulting in the electron-radiation interaction Hamiltonian Hυ, which has the
following form:
Hυ = −
Αυ =
e
Αυ ⋅ p ;
m0
A0 ⋅ aυ
{exp[i(kυ ⋅ r − ωυ t )] + exp[− i(kυ ⋅ r − ωυ t )]} ;
2
53
(2)
(3)
p fi (k ) = −ih ∫ u * f ,k (r )∇ui ,k (r ) d 3r .
(4)
unit
cell
Here Aυ is the electromagnetic vector potential corresponding to a wave
frequency ωυ, where aυ is a unit vector parallel to Aυ (direction of polarization)
and constant A0 is equal to:
A02 =
h
ε 0 n' ωυV
2
,
(5)
with n’ representing the refractive index and V being the volume of the crystal.
In Eq. (4) pfi(k) is the momentum operator, where the ui/f,k(r) is a function with
the same periodicity as the lattice .
To obtain the transition rate W one has to calculate the matrix element
c Aυ ⋅ p v , where indexes c and v for the conduction and valence bands
substitute indexes f and i, respectively. However, only the first term in the
brackets of Aυ has to be considered for the case of optical absorption. The
second term in Eq. (3) describes the photon emission (stimulated or
spontaneous) process, which is outside the scope of this work. Calculations
involve integration over time and space. The time integration concerns only the
exp[i(− ωυ t )] term in Aυ and gives the δ function in energy already included in
Eq. (1). The space integration provides the exp[i(kv − kc + kυ )r ] term resulting
from combination of Aυ with Bloch functions for electrons in the conduction
and valence bands. It supplies an additional δ (kv − kc + kυ ) function assuring the
conservation of momentum in direct transitions. Here, the wave vector of photon
kυ is of the order of the reciprocal lattice spacing and can be assumed to be
negligible with respect to kv and kc. The subsequent space integration over pcv(k)
is performed in the electric-dipole approximation. This is equivalent to
expanding the operator into a Taylor series:
∂
p cv (k ) = p cv (0) + k ⎡⎢ p cv ( k )⎤⎥ + ...
⎣ ∂k
⎦ k =0
(6)
and neglecting all k dependent terms in Eq. (6). Note that this approach is valid
only if pcv(0) ≠ 0, i.e. when transitions are said to be “allowed”. In case of pcv(0)
= 0 the transitions are said to be “forbidden” and are described instead by
subsequent terms in Eq. (6). These can be neglected in the semiconductors
analyzed in this work.
54
The resulting transition rate W contains the
p cv = c aυ ⋅ p cv (0) v and is independent of momentum:
matrix
2π e 2 A02
2
p cv ∑ δ ( Ec (k ) − Ev ( k ) − hωυ ) .
W=
2
h 4m0
k
element
(7)
Summation over the wave vector space can be converted into integration over
the joint density of states:
∑ ⇒ ∫ g (k )d k ⇒ ∫ g ( E
3
k
k
J
) dEJ ,
(8)
E
where
h 2k 2
EJ (k ) = Ec (k ) − Ev (k ) = EG +
,
2µ
(9)
providing
1 2µ
1/ 2
g ( EJ ) = 2 ⎛⎜ 2 ⎞⎟ ( E J − EG ) .
2π ⎝ h ⎠
3/ 2
(10)
µ in Eq. (9) is the reduced mass. Eq. (10) can easily be obtained following the
same arguments as are used in deriving the energy density of states.
In the final step the band-to-band absorption coefficient αbb at the photon
energy hωυ is obtained from the decrease in the photon flux F in the direction x
normal to the excited material surface:
F ( x) = F0 exp(−α bb x) ⇒ α bb = −
1 dF ( x)
.
F ( x) dx
(11a)
Using the continuity equation for the photon flux (dF(x)/dx = −dmυ/dt, where mυ
is the number of photons), the equation can be rewritten as:
α bb =
W
1 dmυ Wtot
=
= tot ,
F ( x) dt F ( x) vg mυ
(11b)
where Wtot = mυW represents the total transition rate for all photons and the
photon flux F is expressed as the energy flow velocity vg = c/n’. Hence, using
Eq. (7) and (10):
α bb (hωυ ) = A(hωυ − EG )1/ 2 ,
where
55
(12)
E
a)
b)
Ec = EG +
hω
h 2k 2
2me
α
hω − EG
EG
Ev = −
h2k 2
2mh
EG
k
hω
0
Fig. 1 (a) Schematic diagram showing direct optical transitions as vertical
transitions in the energy versus momentum space. (b) Corresponding band-to-band
absorption spectrum with the characteristic (hωυ − EG )1/ 2 dependency above EG.
⎛ 2µ ⎞
⎟
⎜⎜
A=
2
2 ⎟
⎝
2πm0 cε 0 n'ωυ h ⎠
e2
3/ 2
2
p cv .
(13)
Note that the δ function results in the substitution EJ = hωυ in Eq. (12).
The basic idea of the direct optical transitions is illustrated in Fig. 1. One
can see that they represent vertical transitions in the energy versus momentum
space, when a photon excites an electron from the full valence band to an empty
state in the conduction band conserving the momentum. The absorption
spectrum α bb (hωυ ) for direct transitions follows the dependency (hωυ − EG )1/ 2 of
the joint density of states above the band gap while being zero for hωυ < EG , as
shown in Fig. 1(b).
In contrast to direct transitions, the Hfi operator in Eq. (1) for indirect
transitions is treated using second-order perturbation theory:
f H fi i = ∑
n
f H1 n n H 2 i
= Sn ,
Ei − En
(14)
where Sn indicates the sum over intermediate virtual states n involved in the
absorption process. Operators H1 and H2 in Eq. (14) are either the electronradiation interaction Hamiltonian Hυ introduced previously or, in the case of
56
E
S1
f(k2)
n(k1)
hω
hω
n(k2)
i(k1)
S2
k
Fig. 2 Schematic diagram for indirect optical transitions occurring via virtual
states n. Indexes S represent corresponding transition operators treated in the
second-order perturbation theory.
phonons conserving the crystal momentum, the electron-phonon interaction
Hamiltonian Hq. Processes of two-photon ( Hυ active in both steps of Eq. (14))
or two-phonon (Hq active in both steps of Eq. (14)) absorption are not analyzed
in this work. As a consequence:
⎛ f (k 2 ) Hυ n(k2 ) n(k2 ) H q i (k1 ) ⎞
⎜
+⎟
EG (k 2 ) − hωυ
⎜
⎟
Sn = ∑⎜
= S 2 + S1
f (k2 ) H q n(k1 ) n(k1 ) Hυ i (k1 ) ⎟
n
⎜⎜
⎟⎟
hωυ − EG (k1 )
⎝
⎠
(15)
This scenario is illustrated in Fig. 2, where the matrix element S1 begins with the
optical perturbation Hυ inducing a transition from the initial state i(k1) to a
virtual state n(k1) (conserving crystal momentum but not energy) and completes
by the phonon perturbation Hq taking the system from n(k1) to the final state
f(k2) (conserving momentum and overall energy). Alternatively, in the matrix
element S2 the first step is a phonon mediated transition (a transition to a virtual
state n(k2)) and the second step is the optical transition. It is usually assumed
that in real semiconductors the most important contributions arise from the S1
term, because values of the direct band-gap EG(k1) (e.g. at Γ crystallographic
orientation in Si as shown in Fig. 1 in Ch. 2.2) are much smaller than EG(k2) (at
X crystallographic orientation in Si).
57
The optical matrix element n( k1 ) Hυ i ( k1 ) in Eq. (15) is identical to the
p cv obtained for direct transitions. Operator Hq in the second term S1 originates
from the weak interaction between the electrons and periodic vibrations of the
ions about their equilibrium positions in the lattice. It can be treated as a small
perturbation of the following form:
H q = −U q ⋅ Cq ;
Uq =
h
u q {z exp[i (kq ⋅ r − ωqt )]+ exp z *[− i(kq ⋅ r − ωqt )]}.
2 NMωq
(16)
(17)
Here Uq is the displacement potential corresponding to a phonon wave
frequency ωq, uq is a unit phonon polarization vector in the direction of the
lattice displacement, N is the number of unit cells in the periodic crystal. Index
M is the appropriate mass of the oscillator, i.e. the total mass of the unit cell in
the case of acoustic phonon modes or reduced mass 1/M = 1/M1 + 1/M2 in the
case of optical modes, where M1 and M2 are masses of the two different atoms.
In the case of an inter-valley scattering M in Eq. (17) becomes the total mass of
the unit cell and NM = DV, where D and V are the density and volume of the
crystal. The operators z and z* in Eq. (17) respectively annihilate or create
phonons and have the corresponding properties:
m − 1 z m = nB ( q ) ;
(18)
m + 1 z * m = nB ( q ) + 1 ,
(19)
where m is the number of phonons in the system and
nB ( q ) =
1
,
exp(hωq / k BT ) −1
(20)
is the Bose-Einstein phonon population factor. In general, operators z and z* and
constant h / 2Mωq in Eq. (17) enter from the harmonic oscillator wavefunctions
which, together with the one-electron Bloch functions, make up the total
wavefunctions of the interacting states. As in the case of the optical matrix
elements, the time integration of the exp[i(− ωqt )] terms in Uq gives
δ ( Ec (k ) − Ev (k ) ± hω q ) and the space integration provides a delta function
δ (kv − kc + kq ) assuring the momentum conserved for indirect transitions. Cq in
Eq. (16) is a coupling operator, which has the property that its matrix element
between two neighboring states in the same band with wave vectors k and k’ is
58
proportional to (k - k’), the constant of proportionality being the deformation
potential constant Dq. The most important feature of the Cq operator is that it
provides selection rules depending on the crystal symmetries during the space
integration.
The total transition rate W can now be written in the following way:
2π e 2 A02 h 2
W=
h 4m02 2 DV
∑∑ hω
p
k
X cvp (k )
p
q
2
( EG ± hωυ )
2
(nBp (q ) + 12 ± 12 ) ×
× δ ( Ec (k ) − Ev (k ) − hωυ ± hωqp ) , (21)
where
X cvp (k ) = ∑ c u q ⋅ D pq n n a ⋅ p nv (k ) v .
(22)
n
The index p here and in the rest of the thesis denotes the particular phonon
branch involved, while the upper and lower signs (±) correspond to phonon
p
emission and absorption, respectively. The combined transition operator X cv (k )
in Eq. (22) is again expanded into a Taylor series around k = kc, with all k
dependent terms being neglected, as was done in (Eq. (6)). Subsequently,
assuming kq to be constant in k space (i.e. hωq = hωk c = E p ), it is possible to
repeat for Eq. (21) the same transformations used in the case of direct
transitions, as follows:
∑⇒ ∫ g (k )d k ⇒ ∫ ∫ g ( E )g ( E )dE dE ,
3
v
k
k
v
c
c
v
(23)
c
Ev Ec
where
3/ 2
1 ⎛ 2mv ⎞
(− Ev )1/ 2 ;
2⎜
2 ⎟
2π ⎝ h ⎠
3/ 2
1 ⎛ 2mc * ⎞
1/ 2
g c ( Ec ) = 2 ⎜ 2 ⎟ ( Ec − EG ) .
2π ⎝ h ⎠
g v ( Ev ) =
(24)
(25)
Finally, the band-to-band absorption coefficient αbb for indirect transitions can
be derived:
α bb± (hωυ ) =
(nBp + 12 ± 12 ) A p
W
2
=∑
hωυ − ( EG ± E p ) , (26)
2
mυ
p hωυ E p ( EG ± hωυ )
[
]
where
A =
p
ge 2 (mc * mv )3/ 2
4πDm02cε 0 n'
59
2
X cvp (kc ) ,
(27)
a)
E
b)
α
nB (hω − ( EG + E p ))2
EG
(nB −1)(hω − ( EG − E p ))2
Ep
kq=kc
EG
k
hω
Fig. 3 (a) Schematic diagram for the indirect absorption process, where the momentum
difference kc between the two bands is matched by either emitting or absorbing a
phonon of the equal momentum kq (gray and black arrows, respectively). (b)
Corresponding band-to-band absorption spectrum with the characteristic
(hωυ − ( EG ± E p ))2 dependency initiated above and below EG by the phonon energy Ep.
and kc is assumed to be one of the set g of equivalent points in k-space
corresponding to the conduction band minimum.
The basic scheme for indirect optical transitions is illustrated in Fig. 3.
One can see that throughout the indirect absorption process the momentum
difference kc (kv = 0) between the two bands is conserved by either emitting or
absorbing a phonon of the equal momentum kq. The absorption spectrum
α bb± (hωυ ) for each of these transitions follows the dependency
(hωυ − ( EG ± E p ))2 , while their thresholds are displaced below and above the
indirect band gap EG by Ep, respectively, as shown in Fig. 3(b). Note that both
±
components α bb (hωυ ) are strongly temperature dependent in contrast to the
temperature independent spectrum α bb (hωυ ) of the direct transitions. This
dependence mostly originates from the Bose-Einstein phonon population factor
nB (Eq. (20)), which at low temperatures suppresses the component involving
phonon absorption, but which equalizes the two components involving phonon
absorption and emission at high temperatures. At arbitrary photon energy one
can obtain the following relation for some particular phonon branch:
α (em)
⎛ E ⎞
∝ exp⎜ p ⎟ .
α (abs)
⎝ k BT ⎠
60
(28)
In deriving Eq. (28) the small difference introduced by the ±Ep denominator in
Eq. (26) is disregarded.
4.2.2 Excitonic absorption
In deriving the absorption coefficient αbb, for both direct and indirect
transitions (Eq. (12) and (26), respectively), equal weight was given to all the
possible states of the electron-hole pair formed in the absorption process,
regardless of the relative particle motion in the final states. In reality, however,
when the pair is formed, the electron and hole appear at the same position in the
crystal, i.e. they are preferentially formed in excitonic states. Therefore, bound
exciton states at energies hωυ < EG or in the excitonic continuum above the
band-gap will affect the shape and magnitude of the fundamental band-to-band
absorption. These changes can be introduced by replacing the old momentum
dependent matrix element
2
f H fi i
2
= M cv (k ) in Eq. (1) by one dependent on
the new quantum number n:
M cv (n) = ∑Cn ,k M cv (k ) ,
(29)
k
where Cn,k are the same linear proportionality coefficients, which relate the
exciton envelope function introduced earlier to the plane wave parts of the
wavefunctions of the constituent particles (i.e. Bloch functions ignoring the cellperiodic part):
φn (r ) = ∑Cn ,k N −1/ 2 exp(ikr ) .
(30)
k
Combining Eq. (29) and (30) one obtains:
M cv (n) = M cv ( k )φn (r ) N 1/ 2 exp( −ikr ) .
(31)
M cv (n) = M cv (k )φn (0) N 1/ 2 ,
(32)
which can be reduced to:
since the probability is heavily weighted towards finding the electron and hole at
the same point in space. Eq. (32) indicates that the absorption coefficient for
61
excitonic transitions can be obtained by adding a factor of φ (0) into the
expressions for the transition rate W. The simpled derivation method outlined
here raises conceptual problems, however, because the initial particle states are
best described within a one-particle picture but the final states have a twoparticle nature. A full treatment requires the use of many-body techniques, in
which both the ground and excited states of the system are represented by
complex wavefunctions constructed from all possible wavefunctions in the k
space. Nevertheless, it was shown by Elliott [7] that such a treatment provides
basically the same result as the simplified presented model.
2
In the same work [7] solutions for φ (0) were calculated for the bound
and continuum excitonic states, respectively:
2
1
φn (0) 2 =
2
φk (0) =
;
πε 03n3
πz exp(πz )
∞
sinh(πz )
(33)
,
(34a)
where
z=
Eex
2µEex
=
.
2 2
h k∞
hωυ − EG
(34b)
Substituting these weighting factors into Eq. (7) for the direct transition rate W,
and remembering that in such processes K = 0 ( M cv (k ) ≈ M cv (0) ), one obtains the
following absorption coefficients αex for the bound and continuum excitonic
states, respectively:
3
⎛ µe 2 ⎞ 1
⎟
α ex ,n = ⎜
⎜ h 2ε ⎟ πn3 ;
⎝
0 ⎠
α ex ,con (hωυ ) = A
[
2π Eex
1− exp − 2π Eex (hωυ − EG )
(35)
],
(36)
where the constant A is the same as that in Eq. (13). The resulting absorption
coefficients are illustrated in Fig. 4(a). Eq. (35) generates a series of discrete
absorption lines at energies Eex/n2, n = 1, 2, 3… within the material band-gap EG.
Starting from the ground state n = 1, the amplitude of each line falls in
magnitude as n-3 merging into continuous absorption as n → ∞. Eq. (36)
corresponds to the absorption behavior shown by the solid line above EG. In the
limiting cases it approaches the following solutions: i) the non-zero value
62
α
ex, n=1
ex, n=2
a)
ex con
bb
Eex
hω
EG
α
~ E2
b)
bb ~ E2
ex cont ~ E?
ex, n=1
ex, n=2
Eex
hω
EG ± Ep
Fig. 4 (a) Excitonic absorption spectrum (solid lines) in a direct gap semiconductor. The
dashed line illustrates the equivalent non-excitonic absorption resulting from parabolic
bands. (b) Excitonic absorption spectrum in an indirect gap semiconductor: total (thick
solid line) and corresponding constituent contributions from bound and continuum
states (thin lines). The dashed line again illustrates non-excitonic absorption.
A2πEex1/ 2 when hωυ − EG → 0, and ii) A(hωυ − EG )1/ 2 when hωυ − EG >> Eex.
The latter result corresponds to the bare band-to-band absorption obtained in Eq.
(12) indicating that the effects of particle interaction become negligible at high
photon energies, as shown by the dashed line in Fig. 4(a). However, it is easy to
calculate that even at an energy of 20Eex above EG Coulomb enhancement is still
~1.5.
2
In the case of indirect transitions the weighting factors φ (0) are
substituted into Eq. (21) to obtain the transition rate W. The k space summation
now becomes integration over all possible final states with excitonic wave
63
vector K, yielding the following expressions for the bound and continuum
excitonic states:
W = C φn (0)
∫d
Eex
h2K 2
− hωυ ± hωq ) ;
2M
2 3
h 2 k∞2 h 2 K 2
3
+
− hωυ ± hωq ) .
W = C ∫ ∫ φk (0) d Kd k∞ ∂ ( EG +
k K
2µ
2M
2
3
K ∂ ( EG −
K
n2
+
∞
(37)
(38)
∞
The constant C here represents all the constants in Eq. (21). After appropriate
integration these equations can be reduced to the absorption coefficients:
1/ 2
(nBp + 12 ± 12 ) A1p
1 ⎡
E
α ex ,n (hωυ ) = ∑
hωυ − ⎛⎜ EG − ex2 ± E p ⎞⎟⎤⎥ ,
2
3 ⎢
n
⎝
⎠⎦
p hωυ E p ( EG ± hωυ ) n ⎣
where
2
ge 2 (mc * + mv )3/ 2 h p
p
A1 =
X cv (kc ) ;
4π 2 Dm02cε 04 n'
and
∆
(nBp + 12 ± 12 ) A2p
(∆ − z )1/ 2
2
E
α ex ,cont (hωυ ) = ∑
dz ,
2 ex ∫
1/ 2
1
−
exp(
−
2
/
z
)
p hωυ E p ( EG ± hωυ )
0
where
2
16π 2 ge 2 (mc * mv )3 / 2 p
p
A2 =
X cv (kc ) ,
4 Dm02cε 0 n'
∆=
1
π Eex
2
[hωυ − ( EG ± E p )].
(39a)
(39b)
(40a)
(40b)
(40c)
According to McLean [8], in the limiting cases Eq. (40a) yields the following
3/ 2
p 1/ 2
when
solutions for the excitonic continuum: i) B Eex [hωυ − ( EG ± E p )]
hωυ − ( EG ± E p ) → 0 (Bp summarizes all constants in Eq. (40a)), and ii) Eq. (26)
when hωυ − ( EG ± E p ) >> Eex.
The indirect absorption components due to the excitonic effects are
illustrated in Fig. 4(b) for the case of emission or absorption of some particular
phonon Ep. They behave quite differently from the ones in the direct transitions.
There is again a series of absorption contributions from the bound exciton states,
which begin at energies Eex/n2, n = 1, 2, 3… below the shifted material band-gap
EG ± Ep, and which also fall in magnitude as n-3. However, this time the
contributions are continuous in energy spectra varying as E1/2, as shown by the
64
lines marked with the index n. Subsequent absorption into the excitonic
continuum begins at the EG ± Ep threshold with an E3/2 energy dependence.
However, the energy range around EG is quite controversial from the theoretical
point of view. For example, Elliott [7] in his work claims that the integral in Eq.
(40a) has no analytical solution and probably has a linear energy dependence.
Moreover, at these energies a clear separation into contributions from the bound
and continuum states is doubtful, as represented by the dotted character of the n
indexed curves. The different authors only agree that the excitonic continuum
approaches E2 dependency when hωυ − ( EG ± E p ) >> Eex. Nevertheless, it is still
possible that the overall absorption dependence at all energies above EG
resembles E2, as shown by the thick black line. Such absorption would become
non-zero at EG and would approach the pure band-to-band absorption spectrum
at high energies (as shown by the dashed curve).
4.2.3 Analytical models for fitting the absorption spectra
Summarizing the absorption contributions for indirect transitions, it is
possible to propose several models to fit an experimental spectrum at arbitrary
temperature. The choice of a particular model depends on the spectral resolution
and absolute αbb errors. MacFarline at al [9] proposed that his explicit data at
the intrinsic absorption edge in Si, which are illustrated in Fig. 5 for various
temperatures, could be described by the following expression:
1/ 2
p
⎤
nB + 12 ± 12 ⎡ A1 [hω − ( EGX ± E p )] +
α bb =
⎢ p
⎥,
∑
hω
p ⎣
⎢ A2 β [hω − ( EGX ± E p + Eex *)]⎦⎥
(40)
where EGX is the excitonic band gap EG - Eex. The first term in the brackets of
Eq. (40) is intended to describe the absorption resulting in the creation of
excitons in the excitonic ground state while the second accounts for the
generation of the excited excitonic states (with Eex* representing the energy
difference between the n =1 and n = 2 excitonic states), the exciton continuum
and free e-h pairs. The β function was obtained empirically since the low
temperature data begin with a square root dependency and approach the square
form at high energies.
In the case of intrinsic absorption data in SiC with multi-phonon structure
[IX], we found that a satisfactory fitting result can be obtained with the model:
1/ 2
p
nB + 12 ± 12 ⎡ A1 [hω − ( EGX ± E p )] + ⎤
α bb =
∑p ⎢ A p [hω − ( E ± E + E )]2 ⎥ ,
hω
⎢⎣ 2
⎥⎦
GX
p
ex
65
(41)
Fig. 5 Indirect absorption spectra in intrinsic Si at various temperatures [9].
The label on the vertical scale: (αhω )1 / 2 , (cm −1 / 2 eV 1 / 2 ) .
accounting only for the ground exciton state and absorption coefficients of the
form of the “unperturbed” band-to-band absorption.
The indirect transition theory was first proposed in the mid-sixties and
was so convincing that it currently dominates the explanation of the absorption
shape of basically all indirect crystalline semiconductors. In fact, the cited
manuscript by G. G. MacFarline, T. P. McLean and coauthors [9] still appears
amongst the hundred most citied articles in the history of physical science.
However, the indirect transition theory in its present form (with various
approximations) does not provide a complete and totally satisfying picture of the
absorption edge. In the case of Si, for example, it does not explain the form of
the excess absorption below the n = 2 exciton threshold assuming the commonly
accepted exciton binging energy Eex = 14.7 meV [10]; it does not account for the
variation in the relative strengths Ap1/Ap2 of the excitonic and band-to-band
absorption components for different momentum conserving phonons; it does not
provide a consistent way to incorporate temperature broadening effects in
predictions of room-temperature absorption from the low-temperature spectrum.
It was indicated that at least some of these discrepancies can be removed by
including in calculations the omitted valence-band degeneracy or attributing
broadening effects to disorder theory rather than to inelastic scattering of
66
excitons by collisions with lattice atoms. The latter idea originates from the
study of amorphous materials, treating thermal fluctuations of the lattice atoms
using exponentially distributed band-tail states in the band-gap (Urbach edge).
References
[1] B. K. Ridley, Quantum Processes in Semiconductors (3rd ed., Oxford
University Press Inc., New York, 1993).
[2] K. Seeger, Semiconductor Physics (Speringer-Verlag, Vienna, 1973).
[3] P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (SperingerVerlag, Berlin, 1996).
[4] C. F. Klingshirn, Semiconductor Optics (Speringer-Verlag, Berlin, 1997).
[5] J. I. Pankove, Optical Processes in Semiconductors (Dover Publications,
New York, 1971).
[6] P. K. Basu, Theory of Optical Processes in Semiconductors / Bulk and
Microstructures (Clarendon Press, Oxford, 1997).
[7] R. J. Elliott, Phys. Rev. 108, 1384 (1957).
[8] T. P. McLean, Prog. Semicond. 5, 53 (1960).
[9] F. F. MacFarlane, Phys. Rev. 111, 1245 (1958).
[10] R. Corkish, J. Appl. Phys. 73, 3988 (1993).
67
4.3 Free carrier absorption
In the previous chapter the fundamental or interband absorption involving
states belonging to two (conduction and valence) bands was discussed. This
chapter presents intraband absorption processes involving occupied and
unoccupied states in a single (conduction or valence) band. In this type of
absorption, free carriers present in the band are excited into empty states within
the same band by relatively low energy photons. Alternatively the process can
be viewed as the acceleration of a free carrier in the high-frequency electric field
of the light, while carrier motion is restricted by the interaction of the carrier
with lattice vibrations. Therefore, free carrier absorption will first be described
using the classical Drude model where the problem is presented in terms of
harmonic oscillations with frequency ω and damping constant 1/τ. This
approach provides the principal dependence of the absorption on the free carrier
concentration and gives a rough description of the absorption spectral shape.
Afterwards, a quantum-mechanical treatment of the free carrier absorption will
be outlined, which also indicates the importance of direct transitions between
different branches of the same band.
4.3.1 Classical picture
The equation of motion of an electron in the presence of the electric field
of light E can be written in the form of classical harmonic oscillations [1]:
me
dv
+ meω0 v = eE = eε λ E0 exp[i (ωt − kr )] ,
dt
(1)
where v is the carrier velocity, ω0 is the damping constant, ελ is a unit vector in
the direction of polarization and E0 is the amplitude of the electric field. It is
convenient to express the solutions of this equation as a complex electrical
conductivity:
J ne v ne 2 ω0 + iω
σ= =
=
,
E
E
me ω02 + ω 2
(2)
where J is the current density and n is the carrier concentration. An additional
equation for the same electric field E can be derived from Maxwell’s equations:
ε
c
••
E+
2
σ
•
⎡⎛ ω ⎞ 2
iωσ ⎤
E
E
E = k 2E ,
=
−∇
×
∇
×
⇒
⎜ ⎟ ε+
⎢
2
2⎥
ε 0c
ε 0c ⎦
⎣⎝ c ⎠
68
(3)
where c is the speed of light in free space and ε is the permittivity of the crystal
lattice. Wave vector k in Eq. (1) is equal to ωnc’/c=(ω/c)(n’ + ini’), where nc’ is
the complex index of refraction, with the real part n’ corresponding to the
conventional refraction index and the imaginary part ni’ representing the
extinction coefficient. Therefore, from Eq. (2) one obtains:
(n'+ini ') 2 = n'2 + ni '2 +i 2n' ni ' = ε + i
and hence
2n' ni ' =
Re(σ )
ε 0ω
σ
,
ε 0ω
.
(4a)
(4b)
Using the relation between the absorption and extinction coefficients, α =
2ωni’/c, and extracting the real part of σ in Eq. (2), Eq. (4b) transforms into the
absorption of the free electron gas:
ne 2
τ
α FCA =
,
meε 0cn' 1+ ω 2τ 2
(5)
where the damping term is assumed to be inversely proportional to the carrier
relaxation time: ω0 = 1/τ. Note that τ in this case denotes the momentum
relaxation time (time between collisions) and is different from the carrier
lifetime as described in Ch. 3.2. The behaviour of αFCA in Eq. (5) is well
determined in the limiting cases of low and high frequencies:
α FCA =
ne 2
τ,
meε 0cn'
ne 2
1
α FCA =
,
2
meε 0cn'ω τ
when ω 2τ 2 << 1 ;
(6a)
2 2
when ω τ >> 1 .
(6b)
The latter approximation is well suited to all of the measurements presented in
this thesis, where spectral range for excitation was always above 1 eV,
corresponding to λ < ~1 µm. Assuming an approximate value of τ ~10-13s and
remembering that ω = 2πc/λ, this range corresponds to ω2τ2 > 104. The same
equation is also valid for the probe wavelength λ = 1.3µm exclusively employed
in the FCA probing technique in this work, as discussed in section 5.1.
Moreover, Eq. (6b) indicates that αFCA varies with energy as λ2 and increases
linearly with increasing carrier concentration.
69
All of the previous discussion has been based on the assumption that the
momentum relaxation time τ is an energy independent constant. In fact, τ is a
complicated function of energy determined by the various carrier scattering
mechanisms as discussed in Ch. 3, and in analogy to the Boltzmann transport
equation, Eq. (6b) should be re-written as:
ne 2λ2
1
α FCA = 2
,
3 ∑
4π meε 0c n' i τ i ( E )
(7)
where i stands for the particular scattering process. Therefore, detailed
calculations of αFCA are only possible using quantum mechanical models.
Nevertheless, it is interesting to consider the classical prediction for τ in, for
example, the case of deformation potential acoustic scattering [2]:
3 2 ne 2 De2 m1e / 2 k B3 / 2 3 / 2
α FCA =
T ,
4 π ε 0cn' ρh 4vs2ω 2
(8)
where De is the deformation potential constant, ρ is the crystal density and vs is
the velocity of sound. In addition to the λ2 spectral and the linear n concentration
dependencies, the free carrier absorption now also possesses a T3/2 temperature
behaviour.
4.3.2 Quantum mechanical picture
If an electron within a particular band absorbs a photon, it can conserve
both energy and momentum by either: i) making a transition to another branch
of the same band, or ii) making a transition to another state in the same branch
whilst simultaneously absorbing or emitting a phonon (or whilst scattering from
an impurity center; this case is not analyzed). The two processes resemble the
direct and indirect interband transitions introduced earlier and are illustrated in
Fig. 1 for electrons in the conduction band (equally to holes in the valence
band). The treatment of indirect transitions will be presented first, since their
quantum mechanical solution under the appropriate conditions can be reduced to
the classical result obtained in Eq. (8).
The transition rate W for indirect free carrier absorption is again
considered using second order perturbation theory. However, in this case the
sum of the transition operators introduced by Eq. (15) in Ch. 4.2 transforms to:
S k± =
1
hωυ
( k ± k q Hυ k ± k q
k ± k q H q k − k ± k q H q k ) k Hυ k ) ,
70
(9)
E
Ep2
Ep1
Di
In
kq2
kq1
k
Fig. 1 Free carrier absorption processes in a single (conduction) band:
“Di” illustrates a direct transition between different branches, while “In”
shows an indirect transition within the same branch. The wave vector in
the latter process is conserved by absorbing or emitting a phonon.
because the energy gaps appearing in the previous expression are zero. Initial,
virtual and final states are now specified by their corresponding wave vectors k
or k ± kq while the ± sign symbolizes emission or absorption of the phonon. Hυ
and Hq are the electron-radiation and scattering interactions, respectively. In
general, Eq. (9) describes four possible transition routes, which are illustrated by
the arrows in Fig. 1.
The optical matrix elements Hυ in Eq. (9) are modified by treating their
momentum component pfi in the effective mass approximation [3]:
k ± k q Hυ k ± k q − k H υ k =
eA0 p k ±k q − p k eA0 h (k ± k q − k ) eA0h
a⋅
=
a⋅
=
a ⋅ ±k q ,(10)
2
m0
2
me
2me
where the free electron mass m0 is replaced by the effective mass me in the
crystal and the wave vector kq is expressed in spherical coordinates. The
scattering matrix elements Hq in Eq. (9), on the other hand, are assumed to
contain no angular dependence when their effective average is taken. They obey
the simple power law [3]:
71
k ± kq H q k = As± kqr ,
(11)
where As± is a factor characteristic of the specific scattering process (acoustic,
optical, polar, piezoelectric or as introduced by impurities). Detailed expressions
for As± from deformation potentials, elastic constants and etc. together with
power indexes r can be found in textbooks [2,3]. Summarizing, the total
transition operator becomes:
S k± =
As± eA0h
a ⋅ kqr k ±q ,
hωυ 2me
(12)
and the transition rate becomes
2π ( As± ) 2 e 2 A02h 2
r ± 2
(
W =
a
⋅
k
k q ) δ ( Ek ±k q − Ek − hωυ ± hωq ) d 3kd 3kq . (13)
q
2
2
∫∫
h (hωυ ) 4me k q k
±
At this point it is reasonable to omit the complex integration procedures
over k and kq and simply state that W± = W±(Ek), i.e. the transition rate depends
on the electron energy in k space [2]. In the next step, the total rate for transition
involving the absorption of a photon of energy hωυ is obtained by summing over
all initial electron states:
∞
W = ∫W ± ( Ek ) f ( Ek ) g ( Ek )2VdEk ,
(14)
0
where g(Ek) is the density of states in the parabolic band:
3/ 2
1 2m
1/ 2
g ( Ek ) = 2 ⎛⎜ 2 e ⎞⎟ ( Ek ) ,
2π ⎝ h ⎠
(15)
while V stands for crystal volume and the factor of 2 arises from spin
degeneracy. The factor f(Ek) in Eq. (14) is the occupation probability of the band
states. Note that this factor was equal to one in the previously analyzed direct
and indirect interband transitions, where it was assumed that the valence band
was fully occupied and the conduction band was assumed ampty. In the case of
a non-degenerate sample, one can employ the Boltzmann distribution function:
72
f ( Ek ) =
n
⎛ E ⎞
exp⎜ − k ⎟ ,
Nc
⎝ k BT ⎠
(16a)
where
mk T
N c = 2⎛⎜ e B2 ⎞⎟
⎝ 2πh ⎠
3/ 2
,
(16b)
is the effective density of states in the conduction band.
Once integration in Eq. (14) has been done, the free carrier absorption
coefficient can be obtained using α = W / vg . For example, the result in the case
of acoustic deformation potential scattering (assuming hωq << hωυ ) is the
following [3]:
α FCA =
4 2 ne 2 De2 (me k BT )1/ 2
⎛ hωυ ⎞ ⎛ hωυ ⎞
exp⎜
⎟Κ ⎜
⎟,
3 2
3 π ε 0cn' ρh vs ωυ
⎝ 2k BT ⎠ ⎝ 2k BT ⎠
(17)
where K is a modified Bessel function. To finalize the expression, though, one
should also account for the induced or stimulated photon emission by the excited
free carriers. It can be shown that this process is derived in the same way as free
carrier absorption with just the sign of hωυ changed. This effects only the
exponential term in Eq. (17) (the argument of K is always positive) providing:
tot
α FCA
∝ e x − e − x = 2sinh ( x ) .
Apart from a small factor of 1.13, the quantum mechanical result obtained
matches that of Eq. (8) in the limit k BT >> hωυ . Therefore, the classical
approach presented previously is justified only in the narrow range
hωq << hωυ << k BT . At high energies ( k BT << hωυ ) the quantum mechanical
method gives instead a λ1.5 spectral dependency and a linear temperature
variation. In general, the dependencies are different from the predicted from the
classical arguments for all scattering mechanisms. In some cases (optical
deformation potential scattering) they are even non-linear [2].
In addition, many semiconductors possess spectral features or absorption
“bands”. They are present for E⊥c and EIIc polarizations in the case of n-type
4H-SiC, as illustrated on the rhs in Fig. 2. The structures are usually attributed to
inter-valley (branch to branch) transitions within the same band. Such transitions
are treated quantum mechanically in the same way as the inter-band direct
transitions described earlier:
2π e 2 A02
W=
p nl
h 4m02
2
∑∑ [ f
nl
k
n
(k ) − f l (k )]δ ( E n (k ) − El (k ) − hωυ ) .
73
(18)
Fig. 2 Free carrier absorption “bands” in 4H-SiC: (lhs) schematic diagram
of the possible transitions between the various conduction bands [4]; (rhs)
experimental spectra in n-type material at two polarizations [5].
where n and l now indicate the two branches involved and fn(k) and fl(k) are their
occupation factors (e.g. the Fermi distribution functions). In the case of 4H-SiC,
a good agreement between the peak positions of absorption bands and the
known energy levels in the unperturbed (undoped) conduction band is obtained,
as shown on the lhs in Fig. 2. Moreover, the absorption peaks at different
polarizations obey selection rules implying that transitions are allowed only
between branches of the same symmetry for EIIc and different symmetry for
E⊥c [4]. Nevertheless, the inter-valley transition theory strongly underestimates
the widths of all the peaks and is not capable of explaining the small width
shrinkage of the absorption bands with decreasing temperature. Several
explanations for this have been proposed, ranging from various broadening
mechanisms [6] to resonant impurity levels associated with the lowest branches
of the bands [7], but a puzzling set of questions remains.
When FCA is employed as a probe tool, the absorption bands are also
reflected in the FCA cross-section σeh = α/∆n, where ∆n = ∆p is the injected
carrier concentration of the sample. The open symbols on the lhs of Fig. 3 show
measured σeh values in Si from various published absorption measurements as a
function of wavelength [8]. The black squares are σeh data obtained by the
perpendicular pump-probe technique of this work. An obvious absorption band
appears at ∼2 µm in the σeh spectrum. Such behaviour (solid line) can be
74
Photon energy (eV)
2
Absorption cross-section (cm )
1.6
10
-16
10
-17
10
-18
0.2
0,5
E g=1.1 eV
σ eh
σh
σe
c-Si 297 K
1
3
10
Wavelength (µm)
Fig. 3 (lhs) Absorption cross-section σeh as a function of wavelength in Si.
Symbols correspond to the experimental data [8] while lines show different
fits as explained in the text. (rhs) Room-temperature absorption coefficient
for differently doped n-type Si samples [8].
predicted by decomposing the total absorption cross-section into separate
electron and hole components: σeh = σe +σh. The cross-section for electrons σe is
recalculated now from the FCA measurements in the doped n-type Si samples
(dash-dotted line), which are illustrated on the rhs of Fig. 3 [9]. The crosssection for holes, on the other hand, is approximated by the Drude model σh ∼ λ2
(dashed line).
References
[1] P. K. Basu, Theory of Optical Processes in Semiconductors / Bulk and
Microstructures (Clarendon Press, Oxford, 1997).
[2] K. Seeger, Semiconductor Physics, (Speringer-Verlag, Vienna, 1973).
[3] B. K. Ridley, Quantum Processes in Semiconductors (3rd ed., Oxford
University Press Inc., New York, 1993).
[4] W. R. L. Lambrecht, Mat. Sc. Forum 264-268, 271 (1998).
[5] E. Biedermann, Solid State Commun. 3, 343 (1965).
[6] S. Limpijumnong, Phys. Rev. B 59, 12890 (1999).
[7] S. G. Sridhara, Mat. Sc. Forum 338-342, 551 (2000).
[8] V. Grivickas, unpublished.
[9] W. Spitzer, Phys. Rev. 108, 268 (1957).
75
76
5 Measurement methods
5.1 FCA probing
5.1.1 Measurement setup
The experimental pump-probe setup used for FCA probing in this work is
shown in Fig. 1. The thin black line on the lhs of the figure represents a probe
beam generated by a 5mW continuous wave (cw) semiconductor laser at λ = 1.3
µm wavelength. The probe beam is focused by a lens onto the sample, which is
placed in an open cycle optical cryostat. The cryostat can operate over a 4 – 500
K temperature range and perform three dimensional motorized translations with
µm precision using a special mounting stage. The transmitted probe beam is
collected by a system of optical lenses and detected by a 0.5 ns rise-time photoreceiver, which is subsequently connected to a digital 1 GHz bandwidth
oscilloscope and a PC. The spatial resolution of the probe within a sample
depends on the probe wavelength, the focal length of the focusing lens and the
sample thickness along the probed path. In general, better resolution could be
obtained with shorter probe wavelengths. However, as indicated by the classical
Drude model (Eq. (7) in Ch. 4.3), the magnitude of αFCA scales as λ2 leading to a
lower signal to noise ratio for shorter wavelengths. In the case of low FCA
absorption within the sample one also has to avoid multiple reflections and
interference effects of the probe. The latter imperfections, however, are highly
suppressed if the polarized probe beam meets the surface at the Brewster angle
ϕB with respect to the surface normal. More details about the optimization of the
probe waist at different measurement conditions can be found in Ref. 1 and 2. In
most of the measurements in this work a probe diameter of around (3 – 4)×λprobe
was achieved.
The thick gray line in the middle of Fig. 1 illustrates the pump beam
generated by a pulsed Infinity laser, which is comprised of a specially amplified
Nd:YAG modulus. The laser pulse has a half-width of 2.5 ns, while its typical
energy is 1-10 mJ. If necessary, the beam intensity can be additionally damped
by neutral optical filters inserted into the beam path. The operation frequency of
the laser ranges from 10 to 100 Hz. A special feature of the Infinity laser beam is
its uniform and wide (~5 mm) intensity profile in all operation modes: I – 1.064
µm, II – 0.533 µm and III – 0.355 µm. As indicated in the figure, the appropriate
77
Oscillator
Flipping
mirror
Beam
selector
Amplifier
Spectrometer
OPO
Infinity pump laser
Polarizer
PC
Filters
Probe laser
(1.3 µm)
Aperture
Pyrrodetector
Polarizer
Sample
Digital
oscilloscope
PC
Grid
Lenses
Mirror
Photoreceiver
Lens
Optical cryostat with
3D translation
Fig. 1 Schematic illustration of the experimental pump-probe setup
utilizing FCA probing.
mode is selected at the laser output by a beam selector. Single laser modes are
used for excitation in measurements of the diffusion coefficient (Ch. 5.2) or the
lifetime. In the spectroscopic measurements (Ch. 5.3) the Infinity laser is
extended with an optical parameter oscillator (OPO) pumped by one of the
modes. OPO spectral ranges of 0.350 – 0.42 µm and 0.78 – 1.6 µm were
employed in the measurements of 4H-SiC and Si, respectively. The intensity of
the beam varies over the spectral ranges employed and is therefore calibrated
with a pyrroelectric detector. The latter monitors the photon density in a beam
fraction reflected from a glass plate. The spectral width of the OPO pulse is <1
nm at half-width, while its exact wavelength is verified using a spectrometer at
the output of the laser. Finally, the polarization of both probe and pump beams
78
a)
b)
Pump
Probe
Pump
Probe
Sample
Sample
Fig. 2 The two principal measurement
geometries: (a) orthogonal, (b) collinear.
can, if needed, be rotated using a Berek polarizer, which is capable of keeping
the polarization ratio greater than 1:100.
Due to the various possible orientations of the pump and probe beams
within the sample and the optical focusing of the probe beam, FCA based
methods are capable of accommodating many different sample structures [3].
The two principle measurement geometries are illustrated in Fig. 2: (a)
orthogonal, where the pump beam excites the front surface of the sample while
the probe beam is focused upon a perpendicular side surface; (b) collinear,
where both the pump and probe beams enter the sample from the same side. The
main advantage of the orthogonal geometry is the ability to perform depthresolved measurements, where the probe beam is scanned across the side
surface by moving the sample. Such a geometry is even capable of
characterizing an effective surface of the sample determined by the diameter of
the probe. For this geometry the samples can also be cut in such a way that the
probe beam path is rather extended. This yields a high sensitivity in terms of the
concentration detected. Moreover, in this type of orientation a uniform
excitation is ensured along the probe beam path. The orthogonal geometry is
also essential for the FCA based spectroscopic method developed in this work,
as discussed in Ch. 5.3. The collinear geometry, on the other hand, is
preferential when large areas of the sample should be scanned, for example in
lifetime mapping measurements, or when samples are analyzed under very high
excitation conditions. For example, this geometry was used to detect the
diffusion coefficient at high injection (Ch. 5.2).
79
5.1.2 σFCA calibration
Both the classical Drude model and the quantum-mechanical treatment in
Ch. 4.3 predicted linear FCA dependence on the carrier concentration (Eq. (8)
and (17) of chapter 4). This outcome can be expressed simply as:
α FCA (λ ) = nσ FCA (λ ) ,
(1)
where σFCA is the free carrier absorption cross section. Eq. (1) suggests that
once σFCA has been calibrated for some particular probe wavelength
measurements of the absorption coefficient at this wavelength can be used to
derive the absolute concentration of carriers.
In the case of negligible reflection the intensity of the probe beam within
the sample has a simple exponential dependence following Bouguer’s law:
I = I 0 exp(−α 0 d ) ,
(2)
where I0 and I are the incident and transmitted intensities respectively, and d is
the sample thickness along the path of the beam. After the excitation pulse
generates free carriers in the sample the quantities in Eq. (2) become time
dependent:
where
I (t ) = I 0 (0) exp(−α FCA (t )d ) ,
(3)
α FCA (t ) = α 0 + ∆α FCA (t ) .
(4)
Here α0 is the same as in Eq. (2), i.e., the time independent or background
absorption coefficient, which is mostly related to the doping-induced free
carriers. ∆αFCA(t) is then the dynamical absorption due to the excited free
carriers. If one is only interested in analyzing the transient processes of the
carrier system Eq. (3) can be rewritten as:
where
I (t ) = I 0 exp(−∆α FCA (t ) d ) ,
(5)
I 0 = I 0 (0) exp(−α 0 d )
(6)
is the transmitted probe beam intensity before excitation. Combining Eq. (1) and
(5) provides an expression for the induced carrier density:
80
4H-SiC λ = 1.3 µm
10
0
330
30
0,1
0
16
17
10
18
60
E⊥c
270
90
10
240
20
4H-SiC λ = 1.3 µm
E⊥c
E||c
10
E||c
300
10
-18
1
σFCA (10
-1
∆α (cm )
2
cm )
20
30
30
120
210
150
180
19
10
10
-3
Carrier concentration (cm )
Fig. 3 σFCA calibration for the λ = 1.3 µm probe beam in 4H-SiC [5]:
(lhs) ∆α dependency versus separately detected injected carrier
concentration at two different polarizations of the probe beam in respect
to the c axis. (rhs) Full σFCA polarization dependency.
∆n(t ) =
∆α FCA (t )
σ FCA
=
⎛ I ⎞
ln⎜ 0 ⎟ .
dσ FCA ⎝ I (t ) ⎠
1
(7)
If the carrier density is not homogeneous throughout the probed volume as, for
example in the collinear geometry measurements, Eq. (7) should be written as:
d
∫ ∆n( z, t )dz =
∆α
0
FCA
σ FCA
(t )
(8)
where⋅∆n(z,t) is the local excess carrier density in the depth z from the sample
surface [4].
Eq. (7) is used both to obtain excited carrier dynamics and to calibrate the
σFCA values. In the latter case the excited carrier concentration in the sample is
evaluated from a known absorption coefficient αbb at a particular excitation
wavelength and then determined separately from the photon density in the pump
beam. The results of such measurements in 4H-SiC are illustrated in Fig. 3 using
81
the III mode of the Infinity laser (αbb = 200 cm-1 at 296 K) [5]. The lhs of the
figure shows the ∆αFCA values obtained right after the excitation pulse versus the
evaluated concentration. This is shown for both perpendicular and parallel
polarizations of the λ = 1.3 µm probe beam (with respect to the c axis of the 4HSiC epilayer.) The fitting of these data by Eq. (7) provides the corresponding
σFCA values. The full polarization dependency of σFCA is shown on the rhs of
Fig. 3.The E⊥c probing polarization is found to be preferable in 4H-SiC due to
its higher σFCA values. Note that excited carriers are generated and recombine in
pairs (assuming negligible trapping): ∆n(t ) = ∆p (t ) . This means that the detected
σFCA is the sum of contributions from electrons and holes, even though their
individual absorption cross-sections σe and σh may be different.
References
[1] J. Linnros, J. Appl. Phys. 84, 275 (1998); ibid. 284 (1998).
[2] V. Grivickas, Mater. Sci. (ISSN 1392 - 1320 Medziagotyra) 7, 203 (2001).
[3] V. Grivickas, Thin Solid Films 364, 181 (2000).
[4] J. Linnros and V. Grivickas, Carrier lifetime: Free carrier absorption,
Photoluminescence and Photoconductivity, Ch. 5b.2 in Methods in Material
Research, edited. by E. N. Kaufman (John Wiley & Sons Inc., 2000).
[5] V. Grivickas, Mater. Sci. Forum 338-342, 555 (2000).
82
5.2 Free carrier gratings
5.2.1 Principles
Scanning the FCA probe along some particular direction within a sample
provides three dimensional (3D) data comprising the variation of the induced
carrier concentration in space and time. If carrier concentration gradients are
present just after excitation (or become induced by inhomogeneous
recombination rates), the time analysis of the data yields information about the
diffusion process governing the carrier motion. This idea has been developed to
give the powerful Fourier-transient-grating (FTG) method [1,2] for detecting the
diffusion coefficients of free carriers in indirect semiconductors. In these
measurements a special periodic free carrier grating is induced in the sample by
projecting the excitation beam through a semitransparent grid. The grid is
positioned across the beam in front of the sample as shown in Fig. 1 of the
previous subchapter. A more detailed schematic for the measurement is shown
below in Fig. 1 of this chapter in the orthogonal pump-probe geometry. It can be
seen that the pump beam is periodically blocked by a grid comprising horizontal
metallic stripes evaporated onto a quartz plate. Consequently, the injected carrier
concentration is modulated along the x-direction perpendicular to the probe
beam, although it is homogeneous along the beam. Stepwise translation of probe
beam along the x-direction (shown by the dashed arrows) then provides 3D data,
to detector
sample
Grid
x
z
y
translation
pump
beam
probe
beam
Fig. 1 Schematics of the FTG measurements in the orthogonal geometry.
83
as shown in Fig. 2.
The pump laser beam is diffracted around the edges of the nontransparent
areas of the grid. The excitation intensity thus ideally follows a sinusoidal
variation in the x-direction:
g ( x) = g1 sin(2πx / Λ + φ ) + g 0 ,
(1)
where Λ is the period of the grid and φ is an arbitrary phase angle. The excited
free carrier grating within the sample then has the same spatial variation at t = 0:
n( x, t = 0) = n1 (t = 0) sin( 2πx / Λ + φ ) + n0 (t = 0) ,
(2)
where n0(t=0) is the average carrier concentration independent of the xcoordinate and n1(t=0) is the modulation amplitude. Under real conditions the
sinusoidal shape of Eq. (2) is replaced by an arbitrary periodic function. Any
periodic function f(x) whose values are known at the arbitrary points xk = 2πk/m
(k = 0,1,2…,m-1) can, nevertheless, be expanded in a Fourier series:
m−1
f ( x) = ∑ c j exp(ijx ) ,
(3)
j =0
Λ=79 µm
-3
Carrier concentration (cm )
-3
Carrier concentration (cm )
Λ=50 µm
10
16
16
10
15
10
1
15
10
150
(µ 1,0
s)
0
50 sition
po
Grid
100
)
(µm
s)
(µ
Ti
m
e
e
m
Ti
0
240
2
160
80
m)
ion (µ
posit
id
r
G
Fig. 2 The 3D representation of the experimental FTG data in case
of low (lhs) and high (rhs) modulation of the free carrier grating.
84
with the Fourier components or harmonics:
cj =
ij 2π k
1 m−1 2π k
∑ f(
) exp(−
),
m k =0
m
m
(4)
where j = 0,1,2…,m-1. The components cj are now pure sinusoids, while their
periods decrease with increasing j: 2π/Λj = j2π/Λ. The first two components, c0
and c1, in Eq. (3) correspond to n0(t) and n1(t) in Eq. (2), respectively, when the
carrier concentration is analyzed.
When the time dependent n(x,t) expression given in Eq. (3) is inserted
into the linear diffusion equation (i.e., the continuity equation with no external
electric field), the following exponential solutions are obtained:
n0 (t ) = n0 (0) exp[ −t /τ ] ;
n1 (t ) = n1 (0) exp[−t / τ g ] ,
(5a)
(5b)
where the instantaneous lifetimes τ and τg are related by
4π 2 D
= +
.
τg τ
Λ2
1
1
(6)
Physically, the grating modulation amplitude n1(t) disappears after the
characteristic “erasure” time τg driven by lateral carrier diffusion, while the
average carrier density n0(t) decays with the time constant τ determined by the
recombination mechanisms described in Ch. 3.2. The diffusion coefficient D
now can be obtained directly from Eq. (6) by substituting the τg and τ
parameters available from the Fourier analysis of the 3D n(x,t) data.
The main advantage of the FTG technique is that it can be used over a
broad injection range determined by the sensitivity of the FCA probe and the
accuracy of the Fourier analysis. Also, it is relatively insensitive to the actual
shape of the grating. In principle, any spatial modulation of the carrier density
can be analyzed.
5.2.2 High injection measurements in 4H-SiC
In comparison to Si, FTG measurements in intrinsic 4H-SiC epilayers are
much more complex due to the shape of the samples available. A realistic
picture of such measurements is shown at the top of Fig. 3, with a magnification
of the FCA probe within the sample at the bottom of the figure. A typical 4H85
photoreceiver
Sample
Grid
1.3 µm
probe beam
0.355 µm
Infinity pulse
3 8 µm
980 µ
m
x
z
y
Probe Beam
Grid period:
60÷120 µm
Beam diameter
10 µm
Substrate
Epilayer
Fig. 3 FTG measurements in 4H-SiC samples (top) with the
magnification of the FCA probing along the epilayer (bottom).
SiC sample comprises a narrow strip sawed from a 4H-SiC wafer with the cross
cut walls polished to optical quality. In the orthogonal geometry the pump beam
excites a native epilayer surface, while the probe beam penetrates along the
epilayer. In order to increase the signal to noise ratio the sample thickness along
the probe beam was approaching mm dimensions, while initially the thickness of
epilayers was not exceeding 40 µm. As one can imagine, correctly aligning a
86
5,0
296 K
4,0
2
Diffusivity (cm /s)
4,5
3,5
3,0
2,5
2,0
1,5
n0
1,0
0,5
14
10
15
10
16
17
10
10
18
10
19
10
-3
Injected Carriers (cm )
Fig. 4 The dependency of the diffusion coefficient on carrier injection in n-type 4HSiC. The line shows the theoretical approximation according to the expression of the
ambipolar diffusion including the BGN effect (Eq. (12) and (24) in Ch. 3.2,
respectively). The dark symbols are the experimental data recalculated using the
numerical solution of the diffusion equation.
probe beam with a 10 µm waist along such a thin epilayer is quite a complicated
issue. The main guidelines used in the alignment procedure were the
interference patterns of the probe beam reflected from the native epilayer
surface and the epilayer-substrate interface. Although the carrier grating can be
scanned at any depth (along the x-direction) in the sample, scanning in the
middle of the epilayer is preferable due to the large bulk recombination lifetime
τ, which is the less affected by surface recombination [3] in the middle of the
epilayer.
Diffusion data collected in n-type epilayers of 4H-SiC at various injection
concentrations are shown by the grey symbols in Fig. 4. The line indicates their
approximate theoretical values according to the expression for ambipolar
diffusion (Eq. (12) in Ch. 3.2) including the BGN effect (Eq. (24) in Ch. 3.2).
The latter effect explains to a large extent the decrease of the ambipolar
coefficient observed at concentrations > 1016 cm-3. The subsequent steep
increase of the diffusion above 1018 cm-3 was interpreted in the appended article
II of this thesis as being due to the degeneracy of the quasi-Fermi levels.
However, such an interpretation was only possible assuming a much lower
effective density of states in the valence band than the 2.5×1019 cm-3 calculated
theoretically at room temperature [4]. Later it was suspected that the discrepancy
87
is due to the neglect of Auger recombination in the data analysis, which
becomes quite pronounced over the injection range [5] involved.
The analytical solution of the diffusion equation including the nonlinear
recombination exists only for free carrier gratings with low or high modulation
[6], which can be expressed through the contrast M:
M=
2∆n1 (0)
∆nmax − ∆nmin
=
,
∆nmax
∆n1 (0) + ∆n0 (0)
(7)
where ∆nmax and ∆nmin are, respectively, the maximum and minimum excess
carrier densities at the arrival time of the excitation pulse, and ∆n1 and ∆n0 are
the quantities in Eq. (2). Low and high contrast grids are illustrated on the lhs
and rhs of Fig. 2, respectively for the case of 4H-SiC. The corresponding
diffusion equation solutions are:
4π 2 D
2
= +
+ 3γnmax
,
2
τg τ
Λ
1
1
when M → 0;
(8)
when M → 1.
(9)
and
1
τg
=
1
τ
+
4π 2 D
2
+ γnmax
,
2
Λ
Both equations indicate that the neglect of Auger recombination overestimates
the diffusion coefficient. Application of these equations under real condition is
complicated though, because the diffusion of M → 0 profiles is too obscure to
detect, but M → 1 profiles are no longer related to a specific excitation range.
The ambipolar diffusion data at high concentrations were therefore
extracted from the experimental data using the full numerical calculations of the
diffusion equation:
∆nt + ∆t , x
∆t
∂
∂2
∆n
∆n = Damb 2 ∆n −
− γ (∆n) 3 ⇒
∂t
∂x
τ
∆n
D
= ∆nt , x + amb2 ( ∆nt , x + ∆x − 2∆nt , x + ∆nt , x − ∆x ) − t , x − γ (∆nt , x ) 3 .
( ∆x )
τ
(10a)
(10b)
The initial ∆n0,x data were fitted to the instantaneous experimental profiles with
a ∆x = 1 µm step size and their time evolution was calculated with a ∆t = 1 ns
time step using different values of Damb. The boundary conditions for the 3D
data were ∆nt,0 = ∆nt,s, where s is the length of the initial scan. The result of such
88
18
-3
Concentration (10 cm )
2,6
Simulation:
2
D = 0.5 cm /s
2
D = 1.5 cm /s
2
D = 2.5 cm /s
Measured data
2,4
2,2
0
10
20
30
40
50
Position (µm)
Fig. 5 The fitting of the numerically calculated diffusion profiles
to the experimental data 33 ns after the initial excitation.
calculations for the initial ∆n0,x profile with an average concentration of 5×1018
cm-3 is plotted (lines) in Fig. 5 and compared to the experimental data (symbols)
at 33 ns after the excitation. The best fit to the data is clearly obtained with Damb
= 1.5 cm2/s. This and other results at high carrier concentrations are shown in
Fig. 4 by the dark symbols at concentration values equal to the average carrier
concentration at the calculated time segment. The total Damb dependency regains
its constant decrease and becomes consistent with the theoretical calculations
and a realistic density of states is assumed for the valence band.
Further improvements in the perturbative method described here are
possible. This could be achieved through using injection level dependent values
of the Damb, τ and γ parameters and through solving the diffusion equation in two
dimensions, i.e., assuming carrier diffusion towards both the surface and the
interface of the epilayer. Nevertheless, it may be expected that the
improvements obtained would be small in comparison to the fluctuations in the
experimental data.
References
[1] V. Grivickas and J. Linnros, Appl. Phys. Lett. 59, 72 (1991).
[2] J. Linnros and V. Grivickas, Phys. Rev. B 50, 16943 (1994).
[3] A. Galeckas, Appl. Phys Lett. 79, 365 (2001).
[4] Y. Goldberg, in Properties of Advanced Semiconductor Materials GaN, AlN,
SiC, BN, SiC, SiGe, edited by M.E. Levinshtein, S.L. Rumyantsev, and M.S.
Shur, p. 93 (John Wiley & Sons Inc., New York, 2001).
[5] A. Galeckas, Appl. Phys Lett. 71, 3269 (1997).
[6] V. Grivickas, Semicond. Sci. Technol. 11, 1725 (1996).
89
5.3 Spectroscopic measurements
The novel spectroscopic method based on the pump-probe technique with
the FCA probing is presented in detail in the appended articles VII, VIII and IX
of this work. This chapter, therefore, focuses only on some important
measurement conditions, which were omitted from the papers for various
reasons.
Two main processes contribute to the absorption of the pump light within
a sample: i) fundamental inter-band transitions involving single photons and ii)
intra-band transitions involving photons plus extrinsic free-carriers. These
processes were analyzed explicitly in Ch. 4.2 and 4.3, respectively. Both
processes are proportional to the local light intensity Iex and within a semiinfinitive sample follow Burger’s law:
∆n( z ) = ∆n(0) exp(− α tot z ) ;
I (0)
α bb .
∆n(0) = (1 − R ) ex
A ⋅ hω
(1a)
(1b)
Here ∆n = ∆p is the instant concentration of excited carriers at a distance z from
the illuminated surface in the direction of light penetration. R is the reflectivity
coefficient and A is the cross sectional area of the pump beam. The overall
absorption coefficient is given by αtot = αbb + αfc, where αfc represents the
absorption related to doping-induced free carriers.
The general description of the absorption can also include a third process:
inter-band transitions involving two photons. In 4H-SiC this process is efficient
around the M-point in the band structure, where it occurs as a direct transition,
as shown schematically in Fig. 1 [1,VII]. Although the explicit spectra of
fundamental two photon absorption contain important information about the
material structure, in Ref. VI and IX absorption of two photons was treated as an
interference effect, which enters Eq. (1b) in the following way:
∆n(0) = (1 − R )
I ex (0)
I 2 (0)
α bb + (1 − R )2 ex 2 βbb ,
A ⋅ hω
A ⋅ hω
(2)
where β is the two-photon absorption coefficient expressed in units of cm/GW.
Since the second term in Eq. (2) is proportional to the square of the excitation
intensity Iex, its influence was substantially reduced by using low photon fluxes
in the measurements. In the appended article VIII high intensity pulses were
used in detecting the absorption spectra of excited Si. In this experiment the
90
Conduction bands
6
5
FCA
Energy (eV)
4
3
2
2 hω
hω
EGX
1
0
-1
-2
Valence bands
Γ
M
k-vector
Fig. 1 Direct two-photon absorption (dark arrows) in the
band structure of 4H-SiC.
two-photon absorption coefficient β was nevertheless rather small [VII] and the
process did not influence the measured αbb values.
In the case of low intensity measurements in intrinsic material there are
two ways to extract αbb according Eq. (1): i) from the exponential slope of the
excess carrier depth-distribution ∆n(z) using Eq. (1a) or ii) from the excess
carrier concentration ∆n(0) at the illuminated surface using Eq. (1b). Eq. (1) is,
however, derived for a semi-infinite sample along the propagation direction of
the excitation light, while real samples possess back surfaces which reflect the
pump beam. This is especially important in the case of small αbb values, i.e.,
when αbbd << 1, where d is the sample thickness along the direction of the pump
beam. In such conditions damping of the excitation intensity by the exponential
term in Eq. (1a) is small: exp(− α bb z ) ≈ 1 . This limitation can be reduced
substantially by introducing sharp angles between the back side and lateral
surfaces of the sample as shown in Fig. 2. The impact of such geometry is
illustrated in Fig. 3, which shows the top cross section of the sample. Fig. 3 (a)
shows the common case of a rectangular sample, where the effective probed
carrier concentration is generated by multiple reflections:
∆neff = ∆n[1 + R + R 2 + R3 + R 4 + ...] =
91
∆n
,
1− R
(3)
Probe
φB
z
Detector
Excitation
Fig. 2 Pump-probe measurements in the samples
with the sharp back side surfaces.
where ∆n is the carrier concentration injected by the initial laser pulse. Note that
the exponentials in front of each term are neglected due to the weakness of the
absorption. For comparison, Fig. 3 (b) shows the case where the back face of the
sample has been polished to give an angle of 20o - 25o. In this geometry the path
of the pump beam encounters several internal reflections and two partial
reflections (i.e., with intensity losses of 1 - R) at the back face of the sample.
The effective probed concentration near the front surface, therefore, becomes:
1 + 2R2
∆neff = ∆n[1 + R + R + R + R + ...] = ∆n
,
1 − R3
2
3
5
6
(4)
where R2, R5, R8… are reflections from the end of the sample while R3, R6, R9…
are from its front surface. Further reduction of the angle to 15o creates a
geometry in which three reflections of the pump beam occur at the back surface
of the sample as illustrated in Fig. 3 (c). The effective concentration then
becomes:
∆neff = ∆n[1 + R3 + R 4 + R 7 + R8 + ...] = ∆n
92
1 + R3
.
1 − R4
(5)
d
a)
pump
90°
b)
pump
20-25°
c)
pump
15°
probe
Fig. 3 Top view of the multiple reflections in the sample structures with
different angles between the back and lateral surfaces.
Now substituting the reflectivity value R ≈ 0.31 for Si (which is applicable for
absorption measurements around the material band-gap) into Eq. (3), (4) and (5)
one obtains the result that the initial ∆n result changes by 45%, 23% and 4%,
respectively. These values for the measurements in this work were even smaller,
because under real conditions exp(− α bbkd ) < 1 , where k refers to the number of
the reflection.
The next aspect of the experiments to be described here is the assumption
of a negligible probe beam waist. The experimentally detected values of ∆n(0)
are in fact reduced due to the variation of the carrier concentration across the
waist of the probe beam according to Eq. (1a). In evaluating this error it is
sufficient to assume z = lprobe = 5 µm in Eq. (1a), where lprobe is the diameter of
the probe beam, though a complete treatment should include an integration of
the intensity distribution across the probe beam. For the case of low intensity
measurements in intrinsic or doped material, substituting also in Eq. (1a) the
maximal value of αtot = 10 cm-1, one obtains an error in the range < 0.5%. A
more complication situation occurs under high excitation intensity conditions,
which were explored in the appended article VIII. In these measurements the
probe beam is also affected by the so called self-induced FCA effect upon the
pump beam. Here, a fraction of the photons at the end of the laser pulse are
93
absorbed by intra-band processes involving free carriers generated at the
beginning of the pulse. In Ref. 2 and 3 it was shown that the depth distribution
of the generated free carriers is then expressed as:
∆n(0) exp(− α bb z )
∆n( z ) =
1+
.
σ eh
∆n(0)[1 − exp(−α bb z )]
2α bb
(6)
Again neglecting the integration across the probe beam and substituting the
highest possible values of αtot = 10 cm-1, σeh = 5×10-18 cm2 and ∆n(0) = 1019 cm-3
into Eq. (6), the measurement error still does not exceed 2%.
One more aspect concerns variable excitation measurements: In the
appended article VIII, the analysis of the fundamental absorption results in Si at
different injected carrier concentrations was performed assuming that the
measured of ∆n(0) profiles were created with a constant absorption coefficient
α. However, a full analysis should separate ∆n(0) into parts created during the
excitation pulse before and after the creation of the excitonic Mott transition, i.e.
∆n(0) < nMott and ∆n(0) > nMott. In the first case ∆n(0) is produced with α = αex,
where αex is the fundamental absorption coefficient accounting for the excitonic
states as discussed in Ch. 4.2. In the second case ∆n(0) is produced only by α =
αbb, where αbb describes the “genuine” band-to-band absorption excluding the
excitonic states. Preliminary analysis of this effect shows that the extracted
excitonic enhancement effect in Si can be reduced up to 30% right above the
excitonic Mott transition. The percentage error, however, decreases quite rapidly
with increasing injection concentration due to the fact that the part of ∆n(0)
generated before reaching the Mott transition becomes relatively small.
References
[1] V. Grivickas, accepted in Mater. Sci. Forum (2004).
[2] K. G. Svantesson, J. Phys. D.: Appl. Phys. 12, 425 (1979).
[3] W. B. Gauster, J. Appl. Phys. 41, 3850 (1970).
94
6 Summary
The scope of the thesis extends from investigations of carrier transport
and recombination phenomena to studies of various absorption mechanisms in
4H-SiC and Si. This list of subject areas reflects also the chronological progress
of my PhD studies. Moreover, the appended papers are included in the
chronological order of publication. The following are step-by-step comments on
the papers, focusing on their relevance to the outlined theoretical background
and indicating the most important scientific contributions in each field. Finally,
a future outlook is provided including some unpublished data.
6.1 Comments on papers
Paper I
This is the first of three papers dealing with free carrier diffusion in 4HSiC. It reveals major difficulties in the straightforward application of the FTG
technique in 4H-SiC arising from the relatively short lifetime in the material and
the availability of appropriate samples only in the form of thin epitaxial layers.
In fact, the time- and spatial-resolution of the measurements had to be increased
by factors of ~100 and ~10, respectively, in comparison to Si. Nevertheless,
detailed experiment descriptions and data analysis show that accurate values of
the diffusion coefficient at different injections can be extracted. The main result
demonstrates that the diffusion coefficient values within the measured excitation
range appear to be lower than their analytical predictions from the Hall-mobility,
suggesting either that minority hole mobility is reduced with respect to majority
hole mobility or that the hole mobility value in general should be lower than
previously reported. Finally, first indications of an appreciable reduction in the
ambipolar diffusivity at high injections, similar to the one in Si, are reported.
Paper II
This paper substantially broadens the quantity and range of the diffusion
data reported in paper 1. It shows that data taken in the moderate excitation
regime from a thick 80 µm epilayer grown by high temperature CVD reproduce
well the previous results in 34 µm epilayer from conventional CVD. High95
injection values are extended up to the level of 5×1018 cm-3 by applying a
parallel measurement geometry in the FTG technique and utilizing pure eitaxial
layers. Moreover, these values are confirmed using the complementary
Holographic Transient Grating technique, which is briefly outlined in the paper.
Finally, initial results of the diffusion coefficient above room temperature are
presented using both techniques.
The paper also includes calculations for the diffusivity at two
temperatures, accounting for many-body interactions and changes in carrier
statistics. It is shown that the values of the theoretically calculated BGN effect
(adopted from literature) can explain at least half of the observed diffusion
decrease at high excitations. However, to explain the sharp increase at the
highest injections due to Fermi-Dirac statistics we had to use a density of states
in the valence band, which is an order of magnitude lower than that obtained
theoretically. This discrepancy, as explained in chapter 5, arises due to the
limitations in the analytical Fourier data analysis in the presence of Auger
recombination. It is removed, as shown in Fig. 3 of the same chapter, by
utilizing full numerical simulations of the three dimensional data sets. This last
result is not included in the appended paper.
Paper III
The carrier lifetime constitutes the backbone of all FCA techniques, but
more detailed investigations of its peculiarities in 4H-SiC are summarized in this
paper. Moreover, in contrast to the CVD grown epilayers explored in the rest of
the papers, this work focuses on samples grown by sublimation. This technique,
with a fast growth rate, has been suggested as a possible candidate to fulfill the
demand for thick epilayers in the market. However, growth conditions were
suspected of influencing material purity a great deal, so minority carrier lifetime
was studied as a sensitive characterization parameter.
Single exponential decays in the µs range are reported in the paper for
CVD epilayers. The overall dynamics of the excited carrier plasma is influenced
by diffusion towards the edges of the epilayer due to a pronounced
recombination via surface and interface defects. The compensated low-doped
sublimation material, on the other hand, exposed the non-exponential nature of
the carrier recombination, which possesses slow tails at low concentrations due
to carrier trapping. The trap density was correlated to the growth conditions,
while temperature measurements revealed the possible presence of a multi-trap
structure. These characteristics, which are similar to these of substrate material,
indicated that substantial improvements in the growth process are needed in
order to reach epitaxial material grade.
96
Paper IV
This is the last paper in the series about diffusion coefficients in 4H-SiC.
Chronologically, this journal article was submitted for publication before paper
2, and is based on the principal data obtained in paper 1. However, it also
presents new data for the dependence of the ambipolar diffusivity on
temperature, in the range from 296 to 523 K at a constant excitation level.
Significant differences in temperature behavior are shown between the
experimental data and theoretical predictions based on the Hall mobility result.
Possible reasons for the discrepancy are discussed.
Paper V
All the papers from here on are dedicated to fundamental absorption
measurements. This paper introduces the concept of αbb measurements from the
instant depth distribution of the excited carriers, employing FCA probing. The
absorption coefficient is detected in 4H and 6H-SiC polytypes at a single
wavelength using a Nd:YAG laser . The main focus is on the temperature
dependence of band-to-band absorption over a wide temperature range relevant
to SiC device operation. Results were obtained in highly doped substrates of
both polytypes and in an intrinsic 4H-SiC epilayer, and were modeled according
to the indirect transition theory. In all cases, it was concluded that the LA
phonon dominates, while appropriate results for the behavior of the band-gap
change with temperature were extracted using the Varshni formula.
Paper VI
This is the first paper presenting full spectroscopic measurements of the
αbb coefficient. Data are obtained for a 4H-SiC epilayer at several temperatures.
It is shown that our novel technique based on FCA probing shifts the inherent
absorption limit appearing in the conventional transmission techniques at
elevated temperatures by several orders of magnitude. Preliminary data analysis
confirms the dominance of the LA phonon in the spectra and provides
indications of the excitonic band-gap at elevated temperatures.
Paper VII
This is a review paper describing the spectroscopic measurements
employing FCA probing as a novel technique in indirect band-gap
semiconductors. The possibilities of studying fundamental band-to-band
97
absorption in bulk or epitaxial material are demonstrated under various different
carrier density, polarization and temperature conditions. Results for intrinsic Si
are compared with the abundant data from other techniques, while guidelines are
provided for measurements in doped and highly excited material. High spectral
resolution data are outlined in 4H-SiC for the first time. Moreover, approaches
to two-photon spectroscopy in both materials are suggested.
Paper VIII
This is the most advanced paper in this thesis with respect to new physical
effects, targeting current understanding of many body effects in indirect band
gap semiconductors at high carrier densities. The fundamental absorption edge
in highly doped and excited Si is detected using the novel technique introduced
in the previous three papers at carrier concentrations exceeding the excitonic
Mott transition threshold by several orders of magnitude. In comparison to
theoretical predictions, representing the current understanding of absorption
behavior at dense carrier plasmas, we find a density dependent excess absorption
at 75 K. Summarizing the main features of this excess absorption, we conclude
that an excitonic enhancement effect is present in Si. We attribute this to
enhanced oscillator strength above the Fermi level in the quasi-equilibrium of
the e-h plasma.
Paper IX
In this paper a summary of the fundamental absorption in both intrinsic
and doped 4H-SiC is provided over a wide absorption range (0.02 - 500 cm-1) at
temperatures from 75 K to 450 K. Experimental results are accurately
reproduced using indirect transition theory with an unique set of dominating
phonons which allow the conservation of crystal momentum. Their energies and
relative contributions are found to correlate with earlier findings using
differential absorption spectroscopy at 2 K. An exciton binding energy ranging
around 35 meV is derived from data fitting at 75 K. Differences in absorption
for light polarized perpendicular and parallel to the c principle axis are shown to
be consistent with the selection rules for the corresponding phonon branches.
The change in the energy gap with temperature is obtained from the absorption
data and explained by an analytical model related to the degree of phonon
dispersion. Finally, doping induced band-gap narrowing is extracted for material
above the impurity-Mott transition and is compared to theoretical predictions
calculated in the random phase approximation. A significant contribution from
the excitonic enhancement factor above the excitonic Mott transition is found,
98
which explains the shape of the fundamental absorption edge at high carrier
concentrations.
6.2 Future outlook
The application of the Fourier transient grating technique in 4H-SiC has
in this thesis been shown to be a difficult task due to the material properties.
Detection of the diffusion coefficient upon injection density within a satisfactory
error range was possible only through statistical averaging after tedious
repeating of data acquisition. The stability of the measurements become even
more problematic at lower temperatures, as illustrated in Fig. 1, although this
range is of particular interest: as can be seen from the Einstein relation, the
diffusion coefficient becomes larger at lower temperatures and it can, therefore,
be expected that the physical phenomena investigated at high injection densities
will become more pronounced there. Moreover, low temperatures would be
preferable for exploring the possible influence of exciton diffusion on the
ambipolar diffusion coefficient at high densities. Low temperature diffusion
measurements will, therefore, be very attractive in the future if pure SiC samples
approaching the bulk specimen dimensions become available. It is also worth to
mention that all diffusion measurements were performed in n-type 4H-SiC,
while in the future they should be completed in a p-type material, which is yet
7
T=200 K
2
Diffusivity (cm /s)
6
5
T=296 K
4
3
2
T=463 K
1
10
14
10
15
10
16
10
17
10
18
10
19
-3
Injected Carriers (cm )
Fig. 1 Free carrier diffusivity versus the excited carrier density in ntype 4H-SiC at different temperatures (FTG technique).
99
not available in a good quality.
Spectroscopic measurements involving FCA probing have been
established as a unique technique for the detection of the fundamental
absorption edge at high carrier densities. In contrast to Si, where both highexcitation and high-doping conditions were explored, in 4H-SiC only the latter
conditions were available due to the power limitations of the excitation system.
Such measurements using an improved experimental set-up as well as
application of the developed technique to other indirect semiconductors are
important in order to establish that the detected excitonic enhancement effect is
a fundamental property of indirect transitions. Moreover, only guidelines for
extension of the developed spectroscopic technique to two-photon fundamental
absorption measurements were given. Nevertheless, these spectra are shown to
be a substantial source of information about the band structure of the material.
They are currently lacking for indirect semiconductors due to similar technique
limitations to these for one-photon spectroscopy.
Finally, during the analysis of the experimental absorption data, I found
that indirect transition theory is commonly mistreated when fitting the intrinsic
sample data if they are compared to the results for the doped material. If this is
correct, it should result in an absolute error, which lowers the BGN values
extracted from the absorption spectra. In fact, BGN results extracted from
absorption measurements remain the only ones, which suggest a much smaller
BGN effect than that predicted theoretically or from other experimental
techniques. In my opinion, this subject requires careful scientific scrutiny.
100