Table Of ContentOPTICAL PROPERTIES
AND BAND STRUCTURE
OF SEMICONDUCTORS
BY
DAVID L. GREENAWAY
AND
GÜNTHER HARBEKE
LABORATORIES R.C.A. LTD.,
ZURICH, SWITZERLAND
Φ
TNf OUttil S AWAHO
TO INOUSTRY It··
PERGAMON PRESS
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Copyright ©1968
D. L. Greenaway and G. Harbeke
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First edition 1968
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PREFACE
THIS monograph is concerned with experimental studies of the
fundamental energy band structure of semiconductors and in
sulators. Research in this area of solid state physics has under
gone rapid expansion during recent years, and this expansion has
been due to extensive work on the intrinsic optical properties,
particularly the measurement of reflectance spectra, and to the
emergence of conceptionally new techniques — both experimental
and theoretical—for the determination of the relevant band para
meters. These approaches have led to a well-established body of
knowledge about the structure of energy bands in real materials.
It is felt that the time is now suitable for the presentation of a
more comprehensive account of this field than has so far been
available. This monograph should thus be of interest not only to
those who have intimate acquaintance with the specific problems
of band structure determination, but to solid state physicists
generally. The treatment is primarily from an experimental view
point, but for the sake of completeness, a number of the basic
theoretical concepts—for example, the theory of interband tran
sitions, group-theoretical classification of electronic states, and
different methods of band structure calculation —are briefly
dealt with.
Considerable effort has been directed to making this monograph
serve as a reference work for those with interest in the electron
and optical properties of semiconductors and insulators. Besides
providing an overall picture of the current state of this field,
detailed information is given of the available measurement tech
niques and results for a large number of both cubic and non-cubic
materials.
ACKNOWLEDGEMENTS
WE WOULD like to record our gratitude to those who have
helped, particularly in the later stages, with the production of this
Monograph. In particular we thank Professor F. Bassani, Pro
fessor M. Cardona, Dr. R. Klein and Dr. W. Rehwald for
their labour in critically reading the manuscript, and suggest
ing a number of modifications. We would also like to thank
Dr. B. Seraphin for his comments on the section on electro-
reflectance, together with Professor H. Ehrenreich and Pro
fessor J. C. Phillips for permission to include a not inconsiderable
number of their results in the text. It may be taken for granted
that we are also indebted to all the people, who cannot be
named separately, whose results we have quoted and repro
duced in the following chapters.
CHAPTER 1
INTRODUCTION
1.1. Historical
The purpose of this monograph is to present an overall descrip
tion of the techniques and results in a field that has received a
great deal of attention during recent years: the study of the funda
mental band structure of semiconductors. The wealth of datanow
available on this subject stems essentially from two factors. First,
it was realized that the measurement of absorption or reflection
coefficient in the visible and ultraviolet spectral region could
provide much information about the position and nature of the
energy bands in a large number of materials. The interpretation
of much of the data is aided by the existence of definite familial
relationships between materials having similar crystal structures
and outer electronic configurations. The behaviour of the energy
bands in isoelectronic sequences of diamond and zincblende
semiconductors, and the consistent variation of energy gaps as a
function of lattice parameter for the lead salts PbS, PbSe and
PbTe are two examples of this.
Secondly, the last few years have witnessed the emergence of
theoretical techniques for the calculation of the band structures
of real solids with an accuracy commensurate with that of the
available experimental data. Clearly it would be desirable to
perform such a calculation by starting with an accurate knowledge
of the atomic potentials and to compute the band structure entirely
from first principles. A number of such calculations have been
made on semiconductors in recent years but in general have not
been able to predict the detailed structure of the valence and
conduction bands. Thus while a first principles calculation is
obviously to be preferred on fundamental grounds, for the time
being more realistic approaches have found greater utility.
The great success of recent band calculations has been due
to the use of pseudopotential methods in which one constructs a
1
OPTICAL PROPERTIES AND BAND STRUCTURE OF SEMICONDUCTORS
potential having a small number of adjustable parameters. These
parameters are chosen to fit a few of the basic well-established
features of the band structure (such as the optical gap or spin-
orbit splitting) and the momentum versus energy curves are then
calculated. This pseudopotential approach has been applied with
great success to the elemental semiconductors Si and Ge, and to
heavy materials such as the lead salts where relativistic corrections
become extremely large.
1.2. Optical properties
In this monograph the accent is on the experimental deter
mination of band structure; here we are concerned with the
measurement and evaluation of the optical constants of semi
conductors and insulators in their fundamental region, i.e. from
the absorption edgfc out to some higher energy where no more
structure is discernible in the optical constants. We concentrate
on this region since the lower energy aspects of the band structure
have been treated extensively by other authors. Thus we do not
deal with free carrier effects and the evaluation of detailed
extremal band parameters, and optical or magneto-optical effects
in the region of the lowest absorption edge. The present treatment
is concerned with a more general approach to the overall band
structure; while our aim is primarily to investigate the higher
energy bands, in many cases the results also provide a basis for
a better understanding of the low energy results.
The extremely high absorption coefficients encountered
throughout the fundamental absorption region make transmission
measurements impractical in many cases, thus a major fraction
of the results discussed in this monograph involve the analysis
of normal incidence reflectance data. Such data can be processed
using the Kramers-Kronig relations to yield the optical constants
n and k and the real and imaginary parts of the dielectric constant
e and e . It is shown that for many purposes it is possible to
1? 2
make good energy band assignments from a direct evaluation
of primary data rather than having to compute either e or a
2
density of states function for each material.
1.3. Contents
The relationships between the optical constants and their
evaluation from primary experimental data using the Kramers-
2
INTRODUCTION
Kronig transforms are treated in Chapter 2. We then discuss the
available instrumentation and experimental techniques for the
measurement of absorption and reflectivity over the energy range
from the near infrared to the vacuum ultraviolet. Chapter 4 gives
a short treatment of the quantum theory of interband transitions
contributing to the optical constants and an introduction to some
of the group theoretical concepts which form an integral part of
the band structure models dealt with in subsequent chapters.
There follow two sections dealing with the experimental evalu
ation of band structure: the first considers cubic materials and
includes the diamond and zincblende semiconductors, the lead
salts and allied materials, and the alkali halides. In the second,
results for anisotropic materials are discussed, in particular the
hexagonal wurtzite semiconductors, where the anisotropy can be
regarded as a small perturbation to a cubic system, and the rhom-
bohedral Bi Te type materials. In this section the effect of the
2 3
selection rules for optical transitions is studied in some detail.
Four shorter chapters follow, dealing briefly with some addi
tional methods of obtaining information about the band structure:
deformation (pressure and strain) phenomena, exciton effects
with some emphasis on recent results on metastable excitons,
electro-optic and photoemissive effects, and valence electron
plasma effects and electron energy loss phenomena. Finally a
number of reference tables have been included in the text which
give the relevant physical constants and detailed energy band
parameters of the materials which have been discussed.
3
CHAPTER 2
OPTICAL CONSTANTS AND
DISPERSION RELATIONS
2.1. Plane waves in isotropic media
The dispersion and absorption of a plane electromagnetic
wave is described by the complex index of refraction N = n + ik,
where n is the ordinary refractive index and k the extinction
coefficient. The plane wave can be represented by
E = E e/(K'r-wr) (2.1)
0
and
H = Ηοβ''«"·^ (2.2)
where κ is the wave propagation vector, κ will in general be
complex. κ = ΚχΛ-ίκ^, and the imaginary part governs the
attenuation of the wave. Equations (2.1) and (2.2) form a solution
of Maxwell's equations for an uncharged medium of magnetic
permeability μ = 1(1) if
K- K = μ,ο^ο^ = —£- · (2.3)
Here e is the complex dielectric constant which includes the
effects of the displacement and conduction currents and is
defined by:
e = d + ie = €, +1 (2.4)
2
ω€
0
We also define the complex refractive index N by e = N2, giving
z = r?-k\ (2.5)
x
€ = 2nk = (2.6)
2
ω€
0
4
DISPERSION RELATIONS
If Κχ and κ are parallel (homogeneous plane wave) the two
2
fields and the direction of propagation are mutually perpendicular
and we find from equation (2.3)
\K\=— #c = —· (2·7)
l 2
c c
In this case the time-averaged energy flow is given by
5 = ϊψ(Ε*·Έ)κ , (2.8)
Ε
where κ is a unit vector in the direction of Kj and κ . With
Ε 2
the expression for E from equation (2.1) we obtain
S= ^ψ{Ε% · Eo)K e-^·'. (2.9)
Ä
It is seen that the energy flow decreases by a factor e-21*2^
over the distance d. The absorption coefficient K, defined by
the relative decrease of energy flow per unit distance in the direc
tion of propagation, is thus
K = 2\ K \ = = —- * (2.10)
2
C A
λ being the wavelength in vacuum.
2.2. Reflection and transmission of plane waves
2.2.1. General formulae
The amplitudes of reflected and refracted waves of a plane
wave incident upon a plane boundary, say z = 0, between two
media, are given by the boundary conditions requiring the
continuity of the x and y components of E and H. We consider
separately the cases for the components of the electric field E
n
normal and E parallel to the plane of incidence. The boundary
p
conditions lead to FresneFs formulae for the amplitude reflection
coefficients r and r of a wave incident at an angle φ from a
n v
transparent medium of refractive index n on the boundary of an
x
absorbing medium of complex refractive index N :
2
_ n cos φ — (N 2 — Mi2 sin2<fr)m
x 2
Tn ~ n cos φ 4- (iV 2 - AZJ2 sin2</>)m ' (2.11)
x 2
5
OPTICAL PROPERTIES AND BAND STRUCTURE OF SEMICONDUCTORS
_ N 2 cos φ - (N W- n * sin2(/>)
2 2 x (2.12)
Vv ~ N 2 cos φ + (N W - nS sin2(/>)
2 2
The reflection coefficient R is defined as the ratio of the time-
averaged energy flow reflected from the surface to the incident
flow. Thus R = r * · r ; R = r * · r .
n n n p p p
If the second medium is also transparent (N = n ) Snell's
2 2
law sin φ · n = sin ψ · η , where ψ is the angle of refraction, holds
x 2
and we obtain:
sin2(i// —φ)
Rn sin2(i//-h$) (2.13)
^ι*η2(ψ-φ)
(2.14)
p tan2(ψ+φ)
Equation (2.14) shows that R goes to zero if ψ + φ = π/2. The
p
angle of incidence is then called the Brewster angle φ and
Β
Snell's law gives tanφ = n /n . R is different from zero at the
ß 2 1 n
Brewster angle, hence the reflected light is totally polarized.
Figure 2.1 shows R and R as a function of φ for n = 3, n = 1.
n p 2 1
Rn.R,
FIG. 2.1. Reflection coefficients /?„ and 7? plotted against the angle of
p
incidence for n = 3 and n = 1. Upper curve /?„, lower curve /? .
2 x p
It is seen that at all angles of incidence except φ = 0 or π/2
i? > 7?p. Hence, if the incident light is unpolarized the reflected
n
6
