Semiconducting and other major properties of gallium arsenide
J. S. Blakemore
Oregon Graduate Center, Beaverton, Oregon 97006
(Received 20 November 1981; accepted for pUblication 27 May 1982)
This review provides numerical and graphical information about many (but by no means all) of the
physical and electronic properties of GaAs that are useful to those engaged in experimental
research and development on this material. The emphasis is on properties of GaAs itself, and the
host of effects associated with the presence of specific impurities and defects is excluded from
coverage. The geometry of the sphalerite lattice and of the first Brillouin zone of reciprocal space
are used to pave the way for material concerning elastic moduli, speeds of sound, and phonon
dispersion curves. A section on thermal properties includes material on the phase diagram and
liquidus curve, thermal expansion coefficient as a function of temperature, specific heat and
equivalent Debye temperature behavior, and thermal conduction. The discussion of optical
properties focusses on dispersion of the dielectric constant from low frequencies [Ko(300) = 12.85]
through the reststrahlen range to the intrinsic edge, and on the associated absorption and
reflectance behavior. Experimental information concerning the valence and conduction band
systems, and on the direct and indirect intrinsic gaps, is used to develop workable approximations
for the statistical weights Nv(T) and Nc(T), and for the intrinsic density. Experimental data
concerning mobilities of holes and electrons are briefly reviewed, as is also the vn (E) characteristic
for the conduction band system.
PACS numbers: 72.80.Ey, 78.20. - e, 62.20.Dc, 63.20. - e
I. INTRODUCTION
The material of this review article, with appropriate
graphs, tables, and equations, is intended to be of value to
experimental and applied physicists, materials scientists,
and engineers, whose work involves gallium arsenide as an
electronic material and as a device medium. The properties
listed in the sections that follow are primarily "intrinsic"
ones: not those found only in totally pure undoped GaAs,
but those dependent on the response of the lattice rather than
of specific impurity species.
The III-V family of semiconductors has been reviewed
in a number of books since the first published article appeared on their electronic properties. I Those prior reviews
include the books by Hilsum and Rose-Innes,2 Madelung,3
and Neuberger4 on the entire I1I-V family. Some key features oflII -V binaries, including GaAs, are summarized also
in appropriate sections of the books by Goryunova, 5 Kressel
and Butler,6 and Casey and Panish,? and in a host of review
articles. A lengthy 1965 review of electronic and related properties of GaAs itself by Hilsum 8 is still valuable. Two series
of volumes are noted now 9 • 10 that will be drawn upon frequently in following sections. The first is the Institute of
Physics series, 9 proceedings of the biennial conferences on
gallium arsenide and related compounds. The second is the
Academic Press "Semiconductors and Semimetals" series, 10
edited since 1965 by R. K. Willardson and A. C. Beer. Gallium arsenide is a feature of many volumes in that series.
However, most the above cited sources do provide information about several III-V semiconductors. In contrast,
the present compilation is limited to GaAs itself. Accordingly, much material has been extracted from the numerous
published papers on GaAs research. For some properties,
the best (or even the only!) available citation dates back to the
1960s or even 1950s. For some other parameters, successive
R123
J. Appl. Phys. 53(10). October 1982
measurements over the years have led to a steadily reduced
set of error limits concerning important numerical quantities. One can certainly expect that, of the numbers quoted
herein, some are already known to certain persons as being
outdated and erroneous. Other numbers will acquire that
status over the next few years. Thus the present attempt at
systematizing data for properties of GaAs is just one stage in
a continuing task. The significance of GaAs, as a medium for
research measurements, and as a material of technological
importance, justifies the present work, without waiting for
further enlightment that will not be total in any event.
A. Topics covered in this review
Section II of this article summarizes information about
the atomic arrangement for the sphalerite lattice of GaAs,
and the consequences for the Brillouin zone of reciprocal
space. Section III deals briefly with mechanical, elastic, and
vibrational properties of the lattice. The material of that Section bears strongly on the thermal properties discussed in
their own right in Sec. IV: specific heat and Debye temperature, expansion behavior, and thermal conductivity.
Section V review dielectric and optical properties, from
the surprisingly controversial subject of the static dielectric
constant and its temperature dependence, to the intrinsic
absorption behavior. Energy bands and gaps are dealt with
in Sec. VI, with an emphasis on bands close enough to the
intrinsic gap to provide a possible home for mobile holes or
electrons. Some aspect of carrier velocity and mobility are
reported in Sec. VII, but this does not pretend to treat the
complexities of carrier scattering in any great detail.
Even for those topics that are mentioned in this review
article, the treatment has usually been made selective and
subjective, rather than encyclopaedic. Thus the material
0021-8979/82/10R123-59$02.40
@ 1982 American Institute of Physics
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•
that is presented by no means exhausts the possibilities.
However, it does give the experimental worker a reasonable
selection of numbers, for various properties, that have considerable built-in consistency. These may not all be correct,
but they do tend to agree with each other.
1[010]
--- ~~-­
~
I
1
I
I
I
1
I
I
B. Topics omitted from this review
1
I
I
I
I
The reader should note that, among other topics which
might logically have been considered for inclusion in a GaAs
review article, two major deliberate omissions are of imp urity phenomena, and recombination phenomena. In both
cases, it was felt that a full and adequate coverage would be
incompatible with the overall length and balance of the other
sections.
Neuberger4 cited literature to about 1970 concerning
active foreign impurities in GaAs, including the bald "facts"
of apparent donor or acceptor ionization energy, and the
experimental methods used. The spectroscopy of shallow
"hydrogenic" donors has become well established through
the work of Stillman and coworkers. 11.12 Much information
concerning deep-level impurities in GaAs was systematized
in the 1973 book of Milnes. 13 Milnes has recently prepared a
comprehensive review article specifically on the experimental aspects of defects and impurities in gallium arsenide. 14 It
may also be noted that several chapters dealing with semiinsulating GaAs, and with the deep-level impurities associated with the semi-insulating condition, are scheduled to
appear in a forthcoming volume l5 of the "Semiconductors
and Semimetals" series.
Omission of recombination phenomena from the list of
topics to be covered here is not easily remedied by citation of
other reviews. Not that there is any lack ofliterature pertaining to trapping and recombination phenomena in GaAs. Far
from it! However, a single comprehensive source of experimental information concerning recombination processes
and carrier lifetimes in this semiconductor has eluded this
writer's search.
Direct radiative and Auger proc~sses are necessarily
important for a direct gap semiconductor such as GaAs. Stimulated emission in an injection diode laser has occasioned
an extensive literature, including several recent books. 6.7. 16
Spontaneous radiative recombination was discussed for
GaAs in some detail by Varshni, 17 from the detailed balance
standpoint of van Roosbroeck and Shockley,18 and Casey
and Stern 19 further analyzed the spontaneous radiative lifetime for P-type GaAs. Gershenzon 2o discussed radiative recombination in GaAs primarily from the luminescence
standpoint, both spontaneous and stimulated.
Landsberg and Adams 21 compared radiative and Auger processes for semiconductors including GaAs; and
Landsberg also reviewed 22 a host ofnonradiative decay phenomena, including Auger types. However, GaAs was not a
specific focus of that account. 22 Osbourn and Smith 23 concluded that an exciton bound to a shallow acceptor in GaAs
is much more likely to decay radiatively than by an Auger
process. Actual experimental data indicative of substantial
Auger participation in GaAs carrier decay are usually the
province of strong excitation 24 or of high temperatures. 25
Room temperature carrier lifetimes in GaAs usually indiR124
J. Appl. Phys. Vol. 53, No.1 0, October 1982
I
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1
I
I
I
I
I
I
I
I
I
.. ~
[OOlt
I
I
I
I
I
I
I
I
_~~
~
----------
I
1---- ~ J-'-~o].
---~
~
-----A -----
FIG. I. Conventional unit cube for GaAs, with a volume A 3 that is four
times larger than that of a primitive cell.
cate 25 - 27 mostly radiative recombination for strong doping,
and control assumed by defect levels for weaker doping.
Among other topics omitted from this review are numerous fairly specialized deformation-related parameters.
Those include piezoelectric, piezoresistive, elastoresistive
and elasto-optic coefficients, deformation potentials, etc.
The 1971 book compiled by Neuberger4 quoted available
numbers for many of these coefficients, and other relevant
literature is cited at appropriate points in the following Sections.
II. THE GALLIUM ARSENIDE LATTICE
Goldschmide s first created GaAs in the 1920s, and
found it to have the cubic sphalerite (zincblende) lattice. This
has fcc translational symmetry, with a basis of one GaAs
molecule, one atom at 000, and the other at !U of the
(nonprimitive) fcc unit cube. Thus the cube illustrated in Fig.
1 contains four GaAs molecules in a volume A 3. The nearestneighbor bond length is (v'3A /4) = ro, and such bonds
(Ga to four As neighbors, and As to four Ga neighbors) are
mutually separated by the tetrahedral bond angle
if> = cos-II - 1/3) = 109.47°.
X-ray diffraction measurements in the I 960s of the unit
cell size, by Straumanis and Kim 29 for the temperature range
5-65 °C, and by Pierron et al. 30 for T = 24°C, are both in
T ABLE I. Unit cell size, atomic density, and crystal density at T = 300 K.
for stoichiometric GaAs. a
Length of side of unit cube
A,oo
Nearest-neighbor distance
ro = v'3A /4
Unit cube volume
A3
Primitive cell volume
!A 3
Molecular density
N /2 V = 4/A 3
Atomic density
N /V = 8/A 3
Molecular weight
M = (69.720 + 74.922)
Calculated crystal density
P3CX)
a
5.65325
2.44793
1.80674 X 10- 22
4.51684X 10-2>
2.2139X 1022
4.4279 X 1022
144.642
5.3174
A
A
cm'
cm'
cm-'
cm- 3
amu
g/cm'
Based on x-ray diffraction measurements of Driscoll et af. (Ref. 31). as
supported by the earlier work of Straumanis and Kim (Ref. 29). and of
Pierron et al. (Ref. 30).
J. S. Blakemore
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excellent agreement with the 300 K value subsequently obtained by Driscoll et al. 31 for undoped melt-grown and epitaxial GaAs:
A300
= 5.65325 ± 0.00002 A.
(1)
That value carries connotations for unit cell volume and
crystal density as indicated in Table I. Note, for comparison,
that Straumanis and Kim 29 made direct weighing experiments which (converted to 300 K equivalence) indicated
P300 = 5.3174 ± 0.0026 g/cm3 for stoichiometric GaAs.
Driscoll et al. reported 31 that the GaAs lattice constant
A increases by up to 0.02% for large concentrations of dopants such as Sn or Te. Straumanis and Kim 29 found that
GaAs grown from the melt has a compositional range, narrow, but of nonzero extent. They reported the Ga-rich extremeat 50.002% Ga, with A then enlarged by 0.001 %. The
As-rich composition was found to extend to 50.009% As,
with A then decreased by 0.004%. That total range is too
small to be apparent in a conventional Ga-As binary phase
diagram,32-34 as reproduced later in Fig. 9 of Sec. IV.
GaAs cleaves most readily on ! 110 I family planes. This
can sometimes happen unexpectedly and inadvertently!
Goryunova 5 noted that GaAs can also cleave on ! 1111
planes, and between (Ill) and (011).
Figure 2 shows the unit cube of the previous figure, as
bisected by the (110) plane. Note that this type of plane contains both kinds of atom. For any atom in a ! 110 I plane, two
of its four nearest-neighbor bonds lie within that plane. That
this should be coupled with a cleavage propensity led Goryunova 5 to speculate that GaAs might be less ionic than
other sphalerite-structured solids. However, Phillips 35 deduced a fraction}; = 0.31 for ionic bonding in GaAs, near
average for III-V compounds,36 though naturally less than
in II-VI solids. Bonding, fractional ionicity, valence electron
distribution, etc., are further discussed in Sec. VI. A.
The ! 1111 family of planes is also an important one for
a sphalerite solid such as GaAs. Figure 3 shows the unit cube
as truncated by (111). The terminator plane in Fig. 3 contains
only one atomic species: the smaller, shaded, type of sphere.
The larger, unshaded, sphere lies below this plane in Fig. 3.
..
[001]
~-
FIG. 3. Truncation of the GaAs unit cube by the (III) plane. Note that a
plane of this family contains only one kind of atom.
Of the eight planes in the ! 1111 family for GaAs, four
are (lIlA) planes containing only gallium atoms. The other
four are (1IIB) planes comprised entirely of arsenic atoms.
Thus a <111) -oriented GaAs wafer with plane parallel faces
has a (lIlA) plane for one face, and a (lIlB) plane for the
other. These have different chemical activity and behavior. 37
Most of the chemical etching solutions that have been developed for chemical polishing of this semiconductor38 .39 work
poorly for the (lIlA) surface, with a slower rate of attack
and an irregular surface finish.
A. Reciprocal space: The Brillouin zone
The crystal structure of GaAs has, among its various
consequences, imposed bcc symmetry on reciprocal space,
the coordinate system needed for description of the dispersion of lattice vibrations and of electronic states. Figure 4
shows the first Brillouin zone (BZ) of reciprocal space for
GaAs. Since sphalerite has fcc translational symmetry, this
zone is the same shape as for an fcc solid. 40 The zone comd
_/
_/-
.: [010]
I
R'
-~--------~--------~
I
_- I
/-
_--
....... -
I
I
r"------
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
..
I
I
I
I
I
I
I
I
I
I
I
[001]~
~
I
I
I
I
II
I
I
-------..J
_....... ~
-
l~~______
C
~
.--------~~
\ [100]
---.,
FIG. 2. Bisection of the GaAs unit cube (Fig. I) by the (110) plane.
R125
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J. Appl. Phys. Vol. 53. No. 10. October 1982
.......".......- _--
________ 1///
R
b
G200-----
It.kl = (41T/A), It.ql = 2
FI G. 4. First Brillouin zone for the GaAs lattice: the same as for other solids
with sphalerite. diamond. or fcc lattices.
J. S. Blakemore
R125
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~ _________:;.U_~X~__,,_________~
,
I
I
I
I
I
I
6.
L'
I
tances between selected pairs of locations in dimensionless
"q-space" units. [In q-space, the distance Irx I = 1. Thus
from a distance listed in Table II in q-space units, the corresponding distance in wave-vector space (k-space) is obtained
by multiplying with the factor (211"/A) = 1.111 X 10K cm -I.]
I
I
I
L
I
I
I
I
I
R't
A
K'
cf>
r
I
I
I
I
L
I
R
III. MECHANICAL, ELASTIC, AND VIBRATIONAL
PROPERTIES
I
I
I
I
I
I
I
I
:
I
I
~---------
---------!
d
c
- - - - - - - - G oi2" - - - - - - - -
l6.ql
= Va
FIG. 5. Hexagonal cross section of the GaAs first Brillouin zone as intersected by the (Oil) plane. (Locations a, b, c, and d in Fig. 4.) Note that this is
not a regular hexagon.
prises a truncated octahedron, lying within a cube (shown
with dashed lines) with a wave-vector space (k-space) side
length (411"/A ) = 2.223 X 108 cm - I.
For the purposes of interpreting the vibrational spectrum of GaAs, or of describing the electronic energy band
structure of this semiconductor, the most important paths
through the Brillouin zone are those from the zone center
to the high symmetry points X, L, and K on the zone boundary. Figure 5 shows a (011) plane section through the zone
center. This figure illustrates that the paths t1X, AL, and
~K form one quadrant of the BZ cross section in the (011)
plane.
Note that locations U and K of Figs. 4 and 5 are equivalent (as also areX and R ), since in each case the separation is a
reciprocal lattice vector, namely Gill' Thus when the dispersion of the vibrational spectrum for GaAs is shown in
Fig. 7 of Sec. III, the v-q dispersion shown between K and R
is indicative also of that along the zone boundary, from U to
r
r
r
r
X.
Table II lists coordinates for some high symmetry locations on or near the zone boundary, and also shows the disTABLE II. Relative location and distances for high symmetry points of the
Brillouin zone for GaAs.
Location
Distance"
r=ooo
x= 100
=1
W=I~
=M
2
~
L=...!...l...!...
lUX I = IUWI =N8
1 ITU I
=,J978
IrL I
=.Jf74
K=ol~
4 4
~ ITK I
11LKI
u=l...!...l
4 4
222
R =011
a
~irRI
= ,J978
=M
={i
11KR I = Iux I = N8
Distances in dimensionless "q-space" units. To convert to wave-vector
space (k-space) dimensions and units, multiply by the factor (21'1
A)= 1.11IX108 cm-'.
R126
J. Appl. Phys. Vol. 53, No.1 0, October 1982
Gallium arsenide is of medium grey appearance, and
can be mechanically polished with moderate difficulty.
Chemical etching can produce a bright shiny appearance for
most low-order crystal planes,'9 excepting the (lIlA) gallium face. The hardness is moderate, between 4 and 5 on the
Mohs scale. 41 The surface microhardness has been reported 41,42 as 750 ± 40 kg/mm2, using Knoop's pyramid
test. As previously noted in Sec. II, cleavage occurs most
readily on !110 1family planes.
A. Elastic constants, and the speeds of sound
Of the many aspects of the elastic response of a solid,
this subsection deals in any detail only with the stress-strain
relation for a small applied stress, and the relevance to
acoustic wave speeds. The tensor relation of stress to strain
involves the second-order elastic moduli. Ultrasonic speed
of sound measurements for monocrystal GaAs in various
crystal directions have determined these moduli accurately.43-46
Third-order elastic constants affect wave propagation
in stressed GaAs,47 and the six third-order moduli have been
measured by Drabble and Brammer,46 and by McSkimin
and Andreatch. 48 The present account will not pursue that
further.
Extrapolation of elastic behavior to a large hydrostatic
pressure provides another specialized area, which has received attention for GaAs,49 among other III-V solids.
Among various equations of state for a solid, so the choice
appropriate for GaAs should be dictated by the second- and
third-order moduli (as noted by Drabble and Brammer46 ), or
equivalently from the pressure dependence of the bulk modulus, as used by McSkimin et al. 44 Figure 6 shows the latter's estimate for the pressure dependence of specific volume. The curve of Fig. 6 was based on a simplified
(Birch-type) equation of state, with bulk modulus assumed
linearly dependent on pressure. An upper limit of 250 kbar
was used for the modeling, since GaAs assumes the more
compact rocksalt structure above that pressure. 49
For the cubic lattice of GaAs, the small-stress adiabatic
elastic response tensorS I simplifies to only three independent
second-order moduli: CII' C 12 , and C 44 . (That is true provided
piezoelectric complications can be averted in the measurement. 52 ) Together with the crystal density p, the quantities
C II' C 12' and C 44 determine the speeds oflong wavelength (i.e.,
nondispersive) longitudinal and transverse acoustic waves,
for any direction and polarization. Table III lists expressions
in terms of these quantities for longitudinal and transverse
waves along the high symmetry directions [100], [110], and
[Ill]. These expressions are valid for any cubic solid,sl a
category that includes the sphalerite structure of GaAs.
J. S. Blakemore
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1.00 .---,--,----,---,--,-----,----,--,--...,---,
TABLE IV. Speeds for (nondispersive) long-wavelength sound waves in
GaAs for the three principal lattice directions!
Direction
of wave
propagation
0.96
0.92
~
">
Wave speed (in units of 10' cm/s)
Wave
character
T= 300 K
T=77K
VL
4.731
± 0.005
4.784
± 0.015
Vr
3.345
± 0.003
3.350
± 0.005
Vn
5.238
3.345
2.476
± 0.008
± 0.003
± 0.005
5.289 ± 0.015
3.350 ± 0.005
2.479 ± 0.012
v'I
5.397
± 0.008
5.447
± 0.015
v;
2.796
± 0.007
2.799
± 0.015
[100]
0.88
COMPRESSION
OF
GaAs
VI
[110)
0.84
0.80
o
[III)
25
50
75
100
125
150
175
200 225
250
utll
PRESSURE (KBAR)
FIG. 6. Pressure-volume relation for GaAs, as calculated by McSkimin et
a/44 from the bulk modulus for zero pressure, and the derivative (dB, / dP).
Thus speed of sound measurements for two or three
directions in GaAs single crystals can permit a deduction of
the three moduli. These can, in turn, be used to express all
other second-order adiabatic moduli (and compliances), and
related parameters such as Poisson's ratio, etc. The results of
such ultrasonic speed of sound measurements are expressed
in Table IV, and the concomitant elastic parameter set of
Table V.
The numbers expressed in Tables IV and V incorporate
small adjustments, in effecting a consensus among values
reported from the various speed of sound experiments. 43 -46
Garland and Park 45 made measurements through the range
77-300 K, and estimated extrapolations to zero temperature. Higher precision room temperature measurements
were made by McSkimin et af.44 and by Drabble and Brammer. 46 In making a consensus among the various reported
values, slight adjustments have been made for differences in
thermometry, and in assumptions about the crystal density.
It is quite clear from the quoted speeds of Table IV that
GaAs is not perfectly isotropic in its elastic properties. A
TABLE III. Acoustic wave speeds for major directions in the cubic sphalerite lattice of GaAs, as controlled by the three second-order elastic constants. a
Wave
propagation
direction
Direction
or plane of
particle motion
Expressions for wave speed, as a
function of crystal density p and
the elastic constants c", C'2' C44
[100]
vL = (c,,/p)112
(100) plane
v T = (C44 /P) '12
[I00}
[lID]
= [(c" + C'2 + 2C44 )12p1'/2
[110]
[001]
V,
[lTO]
V,, = [(c" - C'2)/2p] '12
[111]
v; = [(c, I
(111) plane
v; = [(c" - C'2
v'lI = v T = (C 44 /p)1I2
+ 2C l2 + 4c44 )13p)'12
[111]
a
+ c44 )13p} '/2
Among other standard sources, the development of these equations is provided by de Launay (Ref. 51).
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J. Appl. Phys. Vol. 53, NO.1 0, October 1982
a
The speeds for both 300 and 77 are consistent with the crystal densities and
elastic moduli quoted in Table V, using the equations of Table III. Garland
and Park (Ref. 45) measured [lID] and [Ill] speeds for both temperatures.
The 300 K values indicated above and in Table V are influenced by the
precise [100] and [110] speed measurements made by McSkimin etal. (Ref.
44) and by Drabble and Brammer (Ref. 46).
perfectly isotropic solid would be one for which constant
frequency surfaces in wave-vector space were spheres concentric upon the zone center r; and then there is just one
speed for longitudinal sound waves, and one for all transverse sound waves, regardless of direction. This requires that
Cli = CI2 + 2C44 for the solid (i.e., that C44 equal the shear
modulus c' of Table V, or equivalently that the isotropy ratio
S = I). Anisotropy of the sound speeds listed in Table IV,
and the fact that S = 0.55 as indicated in Table V, are equivalent manifestations of the lack of spherical symmetry in
the tetrahedral bonding of a sphalerite lattice.
A diamond lattice nearest-neighbor force constant
model developed in 1914 by Born (see Ref. 51) would require
that the "Born ratio," last item in Table V, be unity. This is
some 5% off the mark for GaAs, presumably because of
effects of more remote neighbors.
The Cauchy-Poisson relations indicate that Cl2 should
equal C44 for a cubic lattice with appropriate centrosymmetric character for the bonding and deformation properties.
The required conditions have often been interpreted as mandating only central (completely ionic) interatomic forces,
which mostly covalent GaAs does not satisfy. However,
Weiner53 has recently analyzed elastic moduli theory on the
basis of the HeIIman-Feynman theorem, and adduced a less
restrictive condition for the Cauchy relations that does not
automatically exclude covalent bonding. On that basis, the
value C12/C44 =0.95 in Table V implies that equal electrondensity contours in GaAs deform under stress in a way similar to that of the nuclear array deformation.
Values for Cll' C12 ' and C44 extrapolated to T=O permit
calculation of the zero-temperature "elastic" Debye temperature &~l. de Launay5l reported
&~l = (3h /k )(c44/p)1/2[3NJa/41TV(18
+ v'J)] 1/3
(2)
for cubic solids. The factorfa = I only for an isotropic solid,
in which S = (C'/C44 ) = 1. Since C44 > c' for GaAs, then
fa < 1; and de Launay provided tables for interpolation
J. S. Blakemore
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TABLE V. Second-order adiabatic elastic parameters for GaAs.·
Temperature
(A) Elastic moduli:
T= 300 K
T=77K
Tc::;O
119.0 ± 0.1
53.8 ± 0.1
59.5 ± 0.1
32.6 ± 0.2
75.5 ± 0.1
122.1 ± 0.3
56.6 ± 0.3
59.9±0.1
32.8 ± 0.5
78.4 ± 0.3
(extrapolated)
112.6
57.1
60.0
32.8
78.9
(units of 1010 dyn/cm2)
c"
C 12
C44
Shear modulus
Bulk modulus
c' = (c" - c 12 )/2
B, = (c" + 2C'2)13
Y. _ (c" + 2cd(c" - cd
[100] Young's modulus
85.5
0-
(c" +cn!
(B) Dimensionless ratios:
(100] Poisson ratio
Isotropy ratio
Cauchy ratio
Born ratio
a
86.2
C44 )
± 0.7
0.32
0.55
0.95
1.05
0.31
0.55
0.90
1.05
(70 = C'2/(C" +cd
S:=C'/C 44 = (C" - C ,2 )/2c44
C ,2 /C 44
(c" + C ,2 )2/4c"(c,, -
± 0.3
86.3
0.32
0.55
0.95
1.05
Room temperature data from Refs. 43-46, converted to 300 K equivalence using P,oo = 5.3173 g/cm'.
77 K data and extrapolation to T:;:::;O from Garland and Park (Ref. 45), usingpn = 5.3360 g/cm 3
based on the T-:::=.O values of C II , C 12 , C44 • This yields
fa = 0.679 for GaAs, and with other relevant substitutions
into Eq. (2) yields e~1 = 345 ± 3 K. That agrees well with
low temperature specific heat experiments, 54 which Holste
has analyzed 55 to yield a calorimetric Oebye temperature
~al = 344.6 ± 2 K. Specific heats and the associated calorimetric Oebye temperature behavior are further discussed in
Sec. IV C.
The physics of low temperature ballistic phonon motion in GaAs involves the speeds of fairly energetic acoustic
phonons, as well as the weakness of phonon scattering/annihilation mechanisms (i.e., the thermal conduction considerations of Sec. IV 0). Crandall56 reported ballistic motion of
such phonons (v~ 10 12 Hz, hv-4 meV, A~30 A, Iql-0.2)
through insulating GaAs at liquid helium temperatures. His
flight times were shorter than the 1 J-ls instrumental resolution. More recently, Narayanamurti et al. 57 have resolved
the separate arrival of LA and T A phonon pulses, on a sub-
microsecond time scale. As expected from Table IV, one T A
pulse speed was seen along [100] and [111], but two distant
TA species for [110] propagation.
B. Lattice vibrational spectrum: The phonon dispersion
relations
e
r
[::;
Conventional ultrasonic methods explore the variation
of acoustic wave frequency with wavelength (or wave vector)
only for a small portion of the Brillouin zone, nearest to the
zone center. In practice, the vibrational spectrum of GaAs
extends to nearly 7x 10 12 Hz for acoustic modes, and to
almost 9 X 10 12 Hz for intramolecular (optical) modes. (The
vibrational spectrum has mode branches for both acoustic
and optical phonons, since GaAs has a two-atom primitive
basis.)
As with many other crystalline solids, the inelastic scattering of slow neutrons has proved useful in GaAs for determining phonon frequency-momentum relationships.
r
XR
A
L
9
8
N
7
I
~
0
6
FIG. 7. Room temperature dispersion
curves for acoustic and optical branch
phonons in GaAs, obtained by Waugh
and Dolling'" by inelastic neutron
scattering. Labelled high symmetry
points of the zone are as marked in
Figs. 4 and S. The dashed lines have
slopes for the various speeds of sound,
as listed in Table IV.
L ..
5
;:::,
>- 4
T
u
Z
w 3
=>
a
w 2
a::
u..
o
0.2 0.4 0.6 0.8
[qOO]
1.0 1.0
0.8
0.6
0.4
[Oqq]
0.2
o
0 0.1 0.2 0.3 0.4 0.5
(qqq]
REDUCED (DIMENSIONLESS) WAVE-VECTOR,
R128
J. Appl. Phys. Vol. 53, No.1 0, October 1982
q
J. S. Blakemore
R128
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Waugh and Dolling58 used this experimental method in deducing the form of the various v-q phonon dispersion curves
for room temperature. Their data are reproduced in Fig. 7,
for q expressed along [100], [110], and [Ill] directions. As is
customary with this type of data presentation, that for q
along [110] is extended beyond the zone boundary. For, as
noted in Sec. II in connection with Figs. 4 and 5, the path in
reciprocal space K -+R is equivalent to that along the surface
of the zone boundary from U to X. Accordingly, the normal
mode frequencies for X = 100 and for R = 011 are the same.
The various curves in Fig. 7, some with solid lines and
others dotted, for v(q) of the acoustic and optical phonon
branches, represent the attempts by Waugh and Dolling58 to
fit their data with two different versions of a dipole approximation force constant model. 59 The straight dashed lines,
inclined upwards from the locations of q = 0, have been added to show the slopes corresponding to the various speeds of
sound in Table IV.
Table VI lists the phonon frequencies and energies reported by Waugh and Dolling 58 for four important highsymmetry q -space locations. Several of these frequencies are
prominent in Fig. 8. This shows an angular "curve" of g(v)
for the summation of all GaAs normal lattice modes, acoustic and optical branches combined, as calculated60 from the
v(q) data of Fig. 7. The density of states with respect to frequency, g(v), receives a major boost when Vq (v)-+O for one of
the various branches, and this happens predominantly when
q for that branch reaches a zone boundary. Consequently,
g(v) of Fig. 8 shows a first maximum near 2 X 10 12 Hz (8 meV)
TABLE VI. GaAs phonon frequencies and energies for Brillouin zone high
symmetry locations, deduced from 296 K neutron scattering!
Reciprocal
space location
r
(q = (00)
X(q = 1(0)
Mode
v
character (10 12 Hz)
hv
(meV)
LO
8.55 ± 0.2
35.4 ± 0.8
TO
8.02 ± 0.08
33.2 ± 0.3
TO
7.56 ± 0.08
31.3 ± 0.3
LO
7.22 ± 0.15
29.9 ± 0.6
LA
TA
6.80 ± 0.06
2.36 ± 0.Ql5
28.1 ± 0.25
9.75 ± 0.06
TO
LO
7.84 ± 0.12
7.15±0.07
32.4 ± 0.5
29.6 ± 0.3
LA
TA
6.26 ± 0.10
1.86 ± 0.02
25.9 ± 0.4
7.70 ± 0.08
TOil
TO
LO
7.90±0.15
7.SI±0.12
6.44 ± 0.12
32.7 ± 0.6
31.1±0.S
26.6 ± 0.5
and
R (q=OII)
L(q=+; +)
K(=rJ-~)
q 44
LA
TAil
TA
j
a
5.65 ± 0.12
3.48 ± 0.06
2.38 ± 0.04
23.4 ± 0.5
14.4 ± 0.25
9.58±0.15
From experiments of Waugh and Dolling (Ref. 58), as illustrated in Fig. 7.
R129
J. Appl. Phys. Vol. 53, No. 10, October 1982
hI'
0
;:>.
(meV)
10
20
30
T", 300 K
0'
IJl
W
~<[1Jl
~~
IJlz
lJ,.:J
0>>-0::
~<[
-0::
1Jl~
zwill
00::
<[
zz
0
0
I
0...
0
2
8
PHONON FREQUENCY V
Hz)
FIG. 8. Frequency dependence of the total spectral density g(v) of all lattice
vibrational modes (acoustic plus optical mode branches), for GaAs at 296
K, after Dolling and Cowley."" That work was based on the phonon dispersion curve data of Waugh and Dolling,'" as provided in Fig. 7.
associated with T A modes. Short wavelength LA modes
contribute smaller peaks near 5.7X 10 12 Hz (23 MeV) and
6.8 X 10 12 Hz (28 meV). The massive concentration of optical
phonons near 8x 10 12 Hz (33 meV) provides the largest
peak.
Multiphonon absorption of infrared photons (discussed
in Sec. V C) requires that any photon annihilation satisfy the
requisite optical selection rules, and is most prominent for a
maximum in thejoint density of phonon states adding up to
zero wave vector. Thus when data concerning phonon combination bands are presented in Sec. V B, it is not surprising
that the strongest of these bands lies just below 40 meV,
assisted by peaks in the T A and optical branch spectral densities.
The neutron scattering results of Table VI have error
limits of 1%-2%. Other experimental methods are hard
pressed to do better for most ofthese numbers. That is not so
long-wave optical phonons.
for the first two entries, for
Optical methods, further discussed in Sec. V, can potentially
provide VTO and VLO with several significant figures. These
tend to confirm vTO of Table VI, but indicate a VLO value
some 2% larger. The ratio VLO/V TO is interesting in the context of the Lyddane-Sachs-Teller relation,61 in relating dielectric constants below and above the reststrahlen range.
r,
IV. THERMAL PROPERTIES
A. Solid-liquid-vapor phase equilibria for GaAs
The general features of the "normal pressure" Ga-As
binary phase diagram were delineated some 25 years ago, by
Koster and Thoma,32 and by van den Boomgaard and
Schol. 33 The liquidus curve on the Ga-rich side was further
measured by Hall,62 and the phase diagram shown here as
Fig. 9 incorporates data points from both Refs. 32 and 62.
J. S. Blakemore
R129
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T m c::::::1513-3.5P
__--Q--~-----1513K
Liquid
Phase
:x:
Go As + Liquid
1200
1083 K
~
/
1000
GoAs + Liquid
W
0:::
::>
~
<t
en
800
<t
o
0:::
W
0-
::E
t!)
/
I
\
~
" 0.98
600
'-
w
~
/
/'~-~
T~'K·
1090 \
"'"---------"
I
iJ.Hm
/
GoAs
1-
0
0.4
0.6
0.8
1.0
ARSENIC FRACTION, X As
FIG. 9. Conventional metallurgical format for the Ga-As binary phase diagram, with arsenic fraction X A , as a linear abscissa scale. The inset shows
the extreme arsenic-rich end, with the eutectic freezing point depressed below that of pure As. Data points shown are: 0 Work of Koster and
Thoma"; 6. Measurements of Hall 62 for the gallium-rich end.
The normal melting temperature T m for stoichiometric
GaAs was reported to be 1511 K by Koster and Thoma,
1510 ± 3 K by van den Boomgaard and Schol, 1511 K by
Richman,63 and
Tm
1513
=
±1K
(3)
in the subsequent work of Lichter and Sommelet.
± 0.6 kcal/mole = 728 ± 17 Jig.
(5)
(6)
which is slightly larger than values estimated from analyses
of the Ga-As system liquidus curve.34.63.69
An interesting feature of the arsenic-rich end of the
Ga-As liquidus curve, shown as an inset in Fig. 9, is that a
Gao02As098 eutectic has a melting point some 7 K below
that of pure arsenic. The form of the liquidus curve on the
gallium-rich end has received far more attention, with thermodynamic analyses by Vieland,70 Thurmond,69 Arthur,71
Sirota,34 and Panish,72 among others. The underlying thermodynamics is reviewed extensively in the book by Casey
and Panish. 7 The Ga-As liquidus curve is shown again in
Fig. 10, this time with a logarithmic scale for x As so as to give
most attention to the gallium-rich end of the spectrum: a
region of immense technological importance for liquid phase
epitaxial growth of GaAs itself, and of ternary and quaternary materials which are lattice matched go GaAs.7.73-76
That topic lies outside the scope of the present review.
Since arsenic is much more volatile than either gallium
or GaAs, the three-phase equilibrium among solid GaAs, a
Ga-As melt, and its vapor mixture, is of importance. The
GoAs + Solid Go
0.2
25.2
iJ.Sm = 16.64 ± 0.40 cal/mole K,
Solid As
200
0
=
That corresponds to an entropy of fusion
400
303 K
(4)
Lichter and Sommelet64 measured the heat offusion for
stoichiometric GaAs by direct calorimetry, obtaining
I
1083 /
/
(0<P<45kbar).
From that derivative ( - aTm lap) and other thermodynamic properties, it was estimated by Sirota34 that the specific
volume of GaAs should contract by some 6.4% on melting.
That is comparable wtih the contraction of germanium 66 (of
the same lattice constant in solid crystal form) when that
element melts to a liquid phase of higher coordination number. 67.6R
1400
~
K
64
J ayaraman et al. 65 showed that hydrostatic pressure depresses the melting point of GaAs, with
1500
0.7
i
1400
t
1300
1
I
I
1
I
II 00
w
a::
<t:1
°1
1000 -
t?1
I
I
I
I
I
I
~
....W
0.9 ....
o
"
o
(1)/
::::>
900
10- 3
2
5
10- 2
2
5
ARSENIC FRACTION,
R130
J. Appl. Phys. Vol. 53, No.1 0, October 1982
10- 1
X AS
I
~
1
1200
~
a::
w
a.
O.S
o
1.0
1.1
FIG. 10. Liquidus curve for the Ga-As
binary system, using a logarithmic abscissa scale for the arsenic fraction x A, .
The identification of data points is the
same as for Fig. 9.
1
1.2
2
5
•
---l~
J. S. Blakemore
R130
Downloaded 21 Sep 2011 to 210.32.148.91. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
0
.
1
-I
-2
'.
/1RICHMAN DATA
~\J.
.
-3
-4
:2:.
I-
<J: -5
(L
(9
-6
0
-1
-7
-8
-9
-10
0.7
0.8
0.9
(IOOOIT)
1.0
1.1
(K- I )
-II
0.7
FIG. II. Equilibrium vapor pressures of monatomic Ga and As, and of As,
and As., along the binary liquidus curve of the Ga-As system, from the
work of Arthur. 7' The vapor pressures of AS 2 and As. over pure arsenic are
also shown. (Note the changes in the latter curves at the melting point of
arsenic, \090 K.)
early Boomgaard and Schol work on the GaAs phase diagram 32 included P-x As and P- T projections, from the normal
T m down to about 1050 K. (Subsequent work all shows that
the vapor pressure falls more rapidly on cooling than this
early study had indicated.) Measurements by Drowart and
Goldfinger,77 Gutbier,78 Lyons and Silvestri,79 and Richman 63 for various temperature ranges were all reviewed in
terms of the multi phase thermodynamics by Thurmond,69
who modeled the temperature dependences of the partial
pressure for the various species present (dominated by AS 4
and As 2), and for the resulting total pressure. Further measurements by Arthur 71 in the 900-1200 K range enabled the
latter to refine Thurmond's analysis. Figure 11 shows Arthur's result for the variation of equilibrium vapor pressures
for Ga, As, As 2, and AS 4 along the binary liquidus of Ga-As,
plotted versus reciprocal temperature. For AS 2 and As 4 , the
figure also indicates the pressure over pure arsenic (which is
solid below 1090 K, and liquid above that temperature). Except in a situation when free As is present, the vapor pressure
above GaAs is controlled by the lower portions of the AS 2
and AS4 curves. Thus Fig. 12 reproduces Arthur's display of
equilibrium pressures for Ga, AS 2 and AS4 over GaAs itself,
with data points from Richman's total pressure measurements 63 as well as from Arthur's mass spectrometry.7I
Arthur deduced an equilibrium atmosphere over stoichiometric GaAs at its normal melting point of 0.976 atmospheresP As, = 0.328 atm, andP As. = 0.648 atm. Thecontributions of both monatomic species, Ga and As, to the total
pressure are trivial for those circumstances.
B. Thermal expansion
The cubic nature of the sphalerite structures endows
GaAs with an isotropic expansivity. There the simplicity
R131
J. Appl. Phys. Vol. 53, No.1 0, October 1982
0.8
0.9
1.0
1.1
FIG. 12. Equilibrium pressure vs reciprocal temperature for Ga, As 2 • and
As. vapor constituents over GaAs. From Arthur," with his data points
obtained for the three species in the temperature range 900-1200 K by mass
spectrometry. The higher temperature total pressure data of Richman" is
also shown.
ends,
for
the
volume
expansion
coefficient
/3 = (l/V)(aV laT)p, and the linear expansion coefficient
a = (j3 13) = (l/L )(aL laT)p vary in sign and magnitude
with temperature in a complicated way. The physics of why
and how this happens has been reviewed by Novikova,80 and
brief comments should suffice here. The sign of the expansivityislinkedtothatof(dyldT), wherey = - (d In Id In V)
is the Griineisen parameter, influenced by anharmonicity
terms in lattice energy.
Thus a (or, equivalently, /3 = 3a) decreases from its
high temperature positive value on cooling, passing through
zero for T~56 K as measured by Novikova. 81 Figure 13
shows Novikova's raw data for aT' covering the temperature range 28-348 K. Those negative values of a T for T < 56
K were consistent with Blackman's modeling 82 of the rela-
e
7
-,
~
..
5
:.::
.,
'"0
3
:
1
0
-I
\
. .
..
'\ .
~O
,
80
120
f50
200
TEMPERATURE
2~0
680
.JZO
350
T (K)
FIG. 13. Raw data for the linear expansion coefficient of gallium arsenide vs
temperature in the range 28-348 K, as reported I;y Novikova.", The second
entry in Table VII indicates a smoothed representation of these data for
T> 120K.
J. S. Blakemore
R131
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20r-----------,------------n----------~~
I
I
z
o
(LO=826Cm)~1
15
(/)
Z
<1-
c..o«
10
X W.-J
/
a::<J
/
/
«
z
The negative character of a is not maintained to the
lowest temperatures, however. The third law of thermodynamics requires zero expansivity for T = 0, and Daniels R3
observed that the pressure dependencies of elastic moduli for
diamond-type lattices should be consistent with a positive
expansivity, aT a:. + T\ for T </)D' This has been confirmed
for GaAs by high-senSitivity measurements of thermal expansion and contraction at low temperatures. Sparks and
Swenson R4 used magnetic induction of a variable transformer to detect length changes of a GaAs rod, with 5 X 10 - 10 em
resolution. Smith and WhiteR, obtained similar sensitivity
with capacitance dilatometry. Those two investigations gave
modestly different values for the low temperature expansivity, a matter of concern for the specialized topic of calculationg how the Griineisen parameter y varies with temperature. [A curve for y(T) is provided by Smith and White. H5 ]
For less exotic and demanding purposes, the agreement
between these two investigations R4 •R5 in a difficult type of
experiment is admirable.
As a consequence of the two reversals of expansivity
with temperature, the quantities a and fJ are zero for three
temperatures: T = 0, 12, and 56 K. Similarly, the length for
T = 0 is regained at both 15 and 68 K. Figure 14 illustrates
the low temperature behavior as determined experimentally,
with the lowest temperature positive expansion data of
Sparks and Swenson R4 in Fig. 14(a), and a comparison of the
aT data from Refs. 84 and 85 in Fig. 14(b).
The linear thermal expansion behavior for the temperature range including room temperature and upwards has
been measured in a number of investigations, of which some
are tabulated in Table VII. The approximate magnitude of
a 300 has been known since the early citation by Welker and
Weiss,xo which was remarkably close to the "consensus" value implicit in Eq. (8) to be quoted below.
!
t:.L ~ 2.2 x 10- 12 T4 LO
I
I
I
\
w
I
SPARKS
e.
5
SWENSON
.-J
(0 )
OL-----~~~L_
o
_ _ _ _ _ _ _ _ _ _L__ _ _ _ _ _ _ _
~~
10
5
TEMPERATURE
15
T( K)
o
W
U
u.
u.
w
0_
\./SMITH
-5
u-;-
\,
z::<:
-,
WHITE
SPARKS
e. SWENSON
\
Ocv
e.
(
~ ~ -10
\\
<1-
~
.,
W
-15
a::
<I
(b)
W
Z
_20L-__
~
o
____L-__- L____L __ _
10
20
TEMPERATURE
~
_ _ _ _L __ __ L_ _
30
~
40
T(K)
FI G. 14. Low tern perature linear expansion/contraction behavior of GaAs,
as deduced in experiments of Sparks and Swenson,"4 and of Smith and
White"Sla) The (positive) linear expansion for T < 15 K. (b) Behavior of aT'
with emphasis on the region of T> 12 K for which this derivative is negative.
tion between vibrational energy and lattice volume for the
sphalerite structure.
TABLE VII. Investigations of GaAs linear expansion coefficient that extend to 300 K and above.
Reported by,
and year
Footnote
citation
Temperature
range, and
method used
(K- I )
Welker and Weiss
(1956)
86
unspecified
5.7 X 10- 0
Novikova
/1961)
81
28-348 K
Quartz
dilatometer
5.87x 10
Bernstein and Beals
(1961)
89
298-965 K
Interference
dilatometer
4.84 X 10- 0
Amick
(1963)
87
300--875 K
X-ray
a", =5.93 X 10- 0 average over range
Nan and Yi-huan
11965)
88
296-1008 K
X-ray
a"v=6.4X 10 " as average of all data
Straumanis and Kim
(1965)
29
280--340 K
X-ray
6_0 X 10- 0
Pierron et at.
(1966)
30
211-473K
X-ray
6.9 X 10'"
Feder and Light
11968)
90
307-610 K
Interference
dilatometer
5.53 X 10-- 1,
R132
J. Appl. Phys. Vol. 53, No.1 0, October 1982
at IK ')
lifnot synonymous with a",,)
a.wo
n
=-
ar
1.12X 10 -n
Ivalid above 120 K)
at
at
=
T
2.9X 10-'"
+ 4_1 X 10
+ 7.0x 10
= + 2.9):" 10- b+ 1.1/10
'T-5.9X1O- I' T'
"T - 1.8 X 10 "T'
'T -- 7.5X 10
"T'
J. S. Blakemore
R132
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Of the entries in Table VII, the x-ray measurements of
lattice constant by Straumanis and Kim,29 and by Pierron et
al. 30 were noted in Sec. II. Other x-ray measurements by
Amick,87 and by Nan and Yi-huan,88 extended to quite high
temperatures; but the former quoted only an average value
for a, while the results of the latter lacked the precision necessary to be informative about variation of a with T. That
variation had, of course, been amply demonstrated by Novikova's results 81 as shown in Fig. 13, and was further evidenced by optical dilatometry work of Bernstein and
Beals,89 and of Feder and Light.90
A
polynomial
representation of the form
aT = (A + BT - CT2) obviously cannot be correct for the
entire high temperature domain, since the physics of the situation requires that a level off for I)AJ D • Nonetheless, that
simple polynomial form provides a useful framework for simulation of a T over reasonably wide temperature ranges,
and the nonlinearity of thermal expansion is expressed in
that form for three of the entries in Table VII.
Several of the investigations noted in Table VII were
taken into account in a 1976 volume concerning thermal
expansion of various solids,91 in the recommendation of a
"consensus" polynomial form for the temperature variation
oflength [and thus of aT = (l/L )(oL loT)p] well above the
tricky region of negative expansion. The recommended consensus expression for length, when slightly adjusted to make
the balance point occur at 300 K rather than at 293 K as
elected by Touloukian and Buyco,91 is
LT = L 300 (0.99849 + 4.24X 10- 0 T
+ 2.91 X 10- 9 T" - 9.40 X 10- 13 T 3 )
(200 < T < 1000 K).
(7)
Accordingly, the "consensus" value for the linear expansion
coefficient in the same temperature range, 200-1000 K, is
aT = 4.24x 10- 6
+ 5.82X 10- 9 T
(8)
This gives a room temperature value a 300 = 5.73x 10- 6
K - I which is a remarkable reaffirmation of the value quoted
in 1956 by Welker and Weiss. 86 For a temperature regime
that extends rather below that of Eqs. (7) and (8), the form
1.12 X 10- 6 + 4.1 X 10- 8 T
-5.9XIO- 11 T 2 K- I
aT~ -
(120 < T < 350 K),
(9)
noted in Table VII as modeling the results of Novikova,81
does a better job of portraying the steep falloff of a T as one
cools towards the cryogenic range.
c. Specific heat and thermal Oebye temperature
The "classical" specific heat at constant volume for N
atoms of a solids ( = ! N molecules for diatomic GaAs) is
Cel = 3kN. Expressed per unit mass of solid, this amounts
for GaAs to Cel = 0.345 Jig K, in view of the molecular
weight and atomic density listed in Table I. Of course, the
specific heat which is actually measured for a solid is Cp
rather than C v ' but one can reasonably expect Cp to approximate Cel for temperatures comparable with or a little higher
R133
J. Appl. Phys. Vol. 53, No.1 0, October 1982
5
Z
>
..J
2
W
~
10')
::!E
<l
a::
l?
"W
..J
:::;)
0
5
2
J
a. 10- 2 U
W
a::
I
5
:::;)
if)
if)
I
w
a::
I
0..
I
I
I
I
I
I
I
I
I~( 47T4 C el /5 e~)T3
/
for 8 0 = 345 K
I
I
I
f<l
W
I
I
I
I
I
I
10'·
I
U
LL
5
U
W
0..
if)
20
50
100
TEMPERATURE
T( K)
FIG. IS. Specific heat Cp(T), using Cetas etal. data" up to 30 K, those of
Piesbergen"2 for the range 35-273 K, and of Lichter and SommeletM from
300 K to the melting point.
than the thermal Debye temperature eD' Figure 15 shows
that this approximate equality is achieved for temperatures
in the range 300-500 K.
The curve of Fig. 15 is derived from three experimental
calorimetric investigations: that of Cetas et al. 53 which covered the range 1-34 K, by Pies bergen92 for the range 12273 K, and by Lichter and Sommelet64 from 300 K into the
molten range above 1513 K. Piesbergen's low temperature
data does not mesh perfectly with that of the subsequent
highly organized thermometric study by Cetas et al., and an
extrapolation of Pies bergen's (dCpldT) upwards would miss
the Cp (300) data point of Lichter and Sommelet by some 2%;
thus some liberties have been taken in the overlap and bridging temperature regions in the construction of the curve in
Fig. 15 and in the accompanying Table VIII. (Interpolation
between entries of Table VIII will be of assistance for those
who need to actually use specific heat information.)
It can be expected that the measurable specific heat Cp
will exceed Cel for a temperature well above the Debye eD'
since Cp > C v due to thermal expansion. Piesbergen93 remarks that one can expect
J. S. Blakemore
R133
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TABLE VIII. Specific heat Cp : a consensus of values from colorimetric studies."
T
Cp
T
Cp
T
Cp
(K)
(mJ/g K)
(K)
(mJ/g K)
(K)
(mJ/gK)
4
6
8
10
12
14
16
18
20
22
24
26
28
0.043
0.156
0.422
1.00
2.12
3.96
6.62
10.07
14.2
18.8
23.7
28.9
34.1
30
35
40
45
50
60
70
80
90
100
110
120
130
39.4
52.4
65.4
78.1
90.4
114
138
160
181
199
216
230
243
140
150
160
170
180
190
200
210
220
240
260
273
280
254
264
272
279
285
292
298
304
308
315
320
323
325
T
(K)
Cp
(mJ/g K)
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
327
335
343
351
359
367
375
383
391
400
408
416
424
"Values for T(30 K from Cetas el al. (Ref. 54). Values for 1'",300 K from Lichter and Sommelet (Ref. 64). Values from Piesbergen (Ref. 92) between 35-200
K, and adjusted very slightly higher between 200-300 K so as to mesh with Ref. 64 data for the upper range.
Cp
Cv~9a2B, VT,
-
(10)
= 345 ± 3 K, and the good agreement with the calorimetric
limit e zal was remarked in Sec. III A. The Debye model for
lattice vibrational energy results in the statement
3
where Vis the specific volume (i.e., 0.188 cm /g for GaAs),
the bulk modulus B, ~ 7.5 X 1011 dyn/cm2 from Table IV,
and a=6.5 X 10- 6 K - I for T~! T m' Equation (10) would
thus lead one to expect (dCpldT)=5 J1J/g K2, an order of
magnitude smaller than the rate at which the upper temperature entries increase in Table VIII. The specific heat of intrinsically generated electron-hole pairs for high temperatures is far too small to account for the discrepancy; thus the
data concerning n i developed in Sec. VI would indicate a
total free carrier specific heat smaller than 20 J1J/g K at the
melting point. Thus the steady rise of Cp deduced by Lichter
and Sommelet 64 for high temperatures is open to question.
The Bose statistics of phonons dictate that Cp fall
progressively below Cel when the temperature is lowered
below the range of the Debye parameter eD' The zero-temperature limit of the "elastic constant" Debye temperature
was encountered by means of Eq. (2), as evaluated at e~I
:>c::
(11 )
Cp=C" = Cel F(eDIT),
where the function
l
y 3x4exdx
( 12)
F(y)= o y :I(ex - 1.)2
has been tabulated 94 to cover from the high temperature limit F(O) = 1 to the low temperature regime F( y)=(41T4/5y 3)
for y> 20, so that
Cp~(41T4Ccl/5)(T le}))3
}
T<e D !20.
= 26.9(T le D)3 Jig
(13)
K
The curve in Fig. 15 shows that for GaAs (as for many
other solids), Cp faUs/aster than aT 3 manner below about 20
- - - - 8ocal = 345 K
340
O~
ero
1-0
wQ)
:2w 320
FIG. 16. Temperature dependence of the effective calorimetric Debye variable (J'al. This is
generated by inversion of Eq. (II), based on
Cel = 0.345 Jig K, and with Cp taken from Table VIII and Fig. IS. (For the region above 70 K,
where the data of Piesbergen Q2 are used, the
very small differences between Cp and C" values were taken into account.) The zero-temperature asymptotic limit (J~al = 345 K is as deduced by Holste."
-er
er::J
01-
-1<1
<1er 300
Ow
we..
>:2
-w
t;1- 280
Ww
LL>LLcn
Ww
a
260
240
L-~_-L_l--J_-L
o
20
40
__~~_-L__~_L--k_~_L-~__-k~
60
80
TEMPERATURE
R134
J. Appl. Phys. Vol. 53, No.1 0, October 1982
100
120
140
160
T(K)
J. S. Blakemore
R134
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K, indicative of a Debye parameter which is then rising as T
falls. Figure 16 plots the temperature dependence ofthe effective calorimetric Debye variable (jCal which results from
inverting Eq. (11), with the Cp values of Table VIII and Fig.
15 as raw material. As previously remarked in Sec. III A,
Holste 55 has determined the zero-temperature limit of that
variable to be e ~al = 344.6 ± 2 K, from a detailed reanalysis
of the data of Cetas et al. 54 That agrees admirably with expectations from the elastic moduli. Above the temperature
range covered in Fig. 16, Piesbergen's data92 .93 indicate that
(jCal continues within ± 2% of 360 K.
It accordingly seems appropriate to use eD =360 K for
the many phenomena that occur in GaAs near and above
room temperature, for which the phonon population plays a
role. It should come as no surprise that eD =360 K is equivalent to (keD/h )= 7.9 X 10 12 Hz. That approximates the long
wave TO mode frequency, and is near the highest peaks of
the g(v) curve in Fig. 8.
1.0
:.::
:!:
HOLLAND "PURE" SAMPLE
3
"0'" 7.10 15 CM-
o.s
u
.....
~
0.6
..J
~
>-
I-
>
0.4
IU
=>
0
z
0
u
-.J
«
:!:
a::
w
r
0.2
AMITH et cL SAMPLES
I-
16
o
~
o
n o ",5xI0 CM·3
no'" 4.10 17 CM·3
no"'SxI0 18 CM- 3
«
[:,
Po'" 6. 10 19 CM- 3
w
u
'V
I-.J
D. Thermal conduction
Since GaAs has a cubic lattice, thermal conduction can
ordinarily be expected to be isotropic: whether by transport
primarily of phonons, of photons, or of electrons and/or
holes. Marucha et al. 95 were able to observe heat flow anisotropy in an inhomogeneously doped GaAs crystal, but this
was rather different from the conduction of bulk, homogeneous material. However, low temperature ballistic phonon
motion in GaAs,56.57 as excited by a heat pulse or hot electron pulse, is affected by focussing effects which are consequences 96 of the elastic moduli anisotropy. This can make
phonon ballistic propagation strongly dependent on direction and polarization. As with other solids (such as germanium) with a comparable elastic tensor, the consequences can
include magnification. 97
The intent of this review was stated at the outset as a
concentration on properties of the GaAs lattice itself rather
than on effects of specific defect and impurity species. However, it is perhaps worthy of note that phonon transportation
can facilitate study of obstacles created in the GaAs lattice.
Vuillermoz et al. have used low temperature thermal conduction to monitor phonon scattering by defect clusters98 or
dislocations 99 that they created by heat treatment of GaAs.
Challis et al. lOO .101 have used low temperature phonon scattering to assist them in finding out how many Cr impurities
were on various types of site, and in what states of charge.
Drabble and Goldsmid's classic book on thermal conduction in semiconductors lO2 appeared too early in the development phase of III-V family to be a complete source of
information on that phenomenon in GaAs. A few years later, Holland 103,104 and Carlson et al. 105 published major experimental studies of the lattice thermal conductivity KL (T)
for GaAs, from 300 K to the liquid helium range, and from
"high purity" to moderately strong doping on both N- and Ptype sides. Almost contemporaneously with those studies,
Amith et al. 106 described thermal conduction measurements
in GaAs for the 300-900 K range, Thus, the new material
could be incorporated in a 1966 review by Holland, 107 which
commented on the development of phonon transport theory
R135
J. Appl. Phys. Vol. 53, No.1 0, October 1982
0.1
200
T (K)
FIG. 17. Lattice conductivity K L (T) in the range 340-950 K, as measured by
Amith et al. 106 for three N- and one P-type GaAs monocrystal sample. Also,
at upper left, the higher temperature portion of KL (T) as measured by HoIland 104 for a lightly doped sample.
in a nonmetal, from Peierls' distinction of three-phonon Nand U-processes, to the cOLtributions of Casimir, Berman,
Klemens, Callaway, and others. The general framework of
understanding of thermal conduction processes in a nonmetal is presented much more fully in the book by Berman, 108
while the physics of what goes on especially at temperatures
comparable to or higher than D has recently been thoroughly reviewed for the sphalerite and several other crystal
structures by Slack. 109
Figure 17 displays the high temperature lattice thermal
conductivity K L for four variously doped GaAs monocrystal
samples, as deduced by Amith et al. 106 From the measured
values of total thermal conductance (obtained as ac thermal
diffusivity), KL was obtained by Amith et al. by subtraction
of the expected electronic thermal conduction terms (not a
major correction even for the more strongly doped of these
samples). The following points may be made concerning the
resulting curves:
(i) The curves of K L vs T for the two moderately doped
N-type samples show an enhancement for a temperature
range centered on about 800 K. Amith et al. attributed this
to the net flow of blackbody radiation (photons). That
mechanism had been proposed earlier by Genzel llO as observable for a semiconductor when phonon conduction is weak,
and so also is free carrier absorption for the blackbody peak.
(Ap =4,um for T~800 K.) Thus the photon transport effect
is inhibited by long wavelength opacity for lower temperatures, and by free carrier absorption for heavy doping and/or
higher temperatures.
(ii) The downwards path of K L (T) is depressed by heavy
doping: a common occurrence in a semiconductor, associat-
e
J. S. Blakemore
R135
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50
:.c
PHONON FREE PATH
~
U
Ap:O.OICM
"- 20
~
..J
10
~
>-
-f>
5
fU
=>
a
2
2
0
u
CARLSON etal.
-l
N - TYPE SAMPLES
no'" 10 16 CM-3
<{
•
~
c::
W
:r:
f-
W
U
f=
f-
'V
no: 1.4 x 10 16 CM- 3
[Tel: 9 x 10 17 CM- 3
0
[Se): 1.2 x 1018CM- 3
•
0.5
KL
P- TYPE SAMPLES
o [Zn] : 3 x 10 18 CM- 3
0.2
<{
"
-l
[Zn]: 1.2 x 1019 CM -3
0.1
3
10
30
TEMPERATURE
100
300
T (K)
FIG. 18. Lattice conductivity K LIT) from 300 K down, as measured by Carlson et al. "" for single-crystal GaAs samples doped as indicated. The dashed
line indicates K LIT) as appropriate for a phonon free path of 0.0 I em.
ed with scattering of phonons by free carriers. III
(iii) The temperature dependence of KL for the various
samples (once the photon contribution has been subtracted)
is appreciably steeper than the T - I dependence expected
from an Umklapp three-acoustic-phonon process above
Umklapp temperature 0U=WD' The temperature dependence, resembling T -5/4, co~ld nominally be explained by
higher-order processes such as four phonon ones, 112 including those assisted by long wavelength longitudinal phonons; 113 but it seems more likely that three-phonon processes
involving both acoustic and optical phonons are involved, 114
as discussed by Berman 108 and Slack. I ()9
Figure 18 shows how K L (T) can be expected to vary for
the range from room temperature downwards, as measured
in this case by Carlson et al. 105 for N- and P-doped single
crystal samples. These data agree well with those obtained
by Holland 103,104. 107 for doped monocrystal samples in a similar temperature range. The data at upper left in Fig. 17
(which appear consistent with Amith et al. 's higher temperature data) are the highest temperature points Holland reported 104 for his "purest" N-type sample,
As discussed by Berman,108 the lattice conductivity is
controlled primarily by resistive phonon scattering processes, of probability r Ii I, though Callaway showed 115 that
"normal" three-phonon processes have an indirect effect.
The resistive scattering probability should be assessed from
(14)
averaged over the phonon energy spectrum. Here, r b- I refers
to scattering at boundaries (of the sample itself, or at grain
boundaries for a polycrystalline sample), as described by CaR136
J. Appl. Phys. Vol. 53, No. 10, October 1982
simir. 116 The term r;- I accounts for isotope scattering, nontrivial for a pure sample at low temperatures. 103 The rate
r d- I associated with impurities, point and line defects, etc., is
the principal one for impure GaAs at low temperatures; and
r;;' I signifies the Umklapp scattering rate: a quantity which
decreases drastically as temperature falls below Ou ~40D'
and phonons of /q/ > 0.5 becomes unavailable.
The Debye model treatment of the phonon supply is
usually adopted 108, liS for T < 0 D; and some additional rigor
is imparted if the contributions of LA and T A phonons to
Tu 1 are assessed separately, as attempted by Tiwari et al.I 17
for KdT) in fairly pure GaAs up to room temperature. The
summation over resistive process probabilities can be used,
at the lowest level of sophistication, to define a phonon mean
free path Ap (T). In simple kinetic theory language, this is
related to the lattice conductivity by
=~pC
vA P'
3
P
(15)
where p is the density, and v = (kODlh) (41TV 13N)I/3 is a
mean of the speeds for the various participating phonons.
Thus for each of the curves in Fig. 18, the rise of KL on
cooling is a response to the decrease of Umklapp scattering
as a limitation on r R and A p , The subsequent steep fall of K L
for lower temperatures is required by the decrease of Cp once
Ap has become controlled by static entities.
For material pure enough so that r;;- I controls Eq. (14)
down to low temperatures, Peierls noted that 118 one should
expect KL to vary as exp(Ou IT), for T < Ou ~~OD' In sympathy with this trend, the weakly doped sample of highest conductivity in Fig. 18 hasK L =0.5 exp (160IT) W Icm K in the
range 50<1<; 150 K. As a more complete examination of
what controls phonon scattering in that sample for various
temperatures, Fig. 19 shows a curve of Ap = (3KL / pCp v) for
the complete range 3-300 K using the lattice conductivity
data of Fig. 18, and thermal data from Sec. IV C.
The dashed curve in Fig. 18 represents the upward
trend of KdT) from Eq. (15), with a phonon free path
Ap = 0.01 cm. That curve is very slightly perturbed from a
T) slope by the temperature dependence of the calorimetric
Debye variable (J"al (see Fig. 16) as influencing both Cp and v.
Thus it can be seen from Figs. 18 and 19 that all five of the
samples of Fig, 18 have A p ~ 300 A at room temperature,
and that this rises to a low temperature maximum of some
10- 3 cm for the P-type samples, and to 0.05 cm or more for
each of the N-type samples. 119
The investigations of Carlson et al. 105 and of Holland 1()4
both showed that acceptor impurities such as Zn and Cd are
much more effective than shallow donors in holding down
Ap for low temperatures (as is also true in other Group IV
and 111- V semiconductors). Holland 104 also reported the
more complex behavior of K L (T) for GaAs doped with manganese, associated with phonon coupling to carriers in the
Mn acceptor ground state. Chaudhuri et ai, 120 measured the
resonance scattering of phonons (as affecting K L ) for Crdoped GaAs, an impurity for which the measurements of
Challis et al. 100,]() I on phonon scattering were noted earlier in
this section. Thus there are several extrinsic considerations
which can affect the thermal conductivity KL (or, equivalentJ. S. Blakemore
R136
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knowledge from them may have influenced our current understanding of the topic on which this Section focusses. That
topic is the behavior of the complex dielectric constant, from
the far IR to the vacuum UV.
Several terminologies are in concurrent usage for dielectric and optical response function. That used in this Section is sketched here. Let E* (v) be the complex permittivity
for frequency v. Then the (dimensionless) complex dielectric
constant is written
10- 1
~
U
0.10- 2
<
:z:
~
<t
a..
10- 3
E*/Eo==-K*
W
W
= KI
-
iK 2.
(16)
The real and imaginary components, K I and K 2' can each be a
function of frequency. The complex refractive index is
a::
u...
z
4
<t 10-
(17)
W
~
Z
0
Z
0
with the real refractive index n and the extinction coefficient
k as its components, From Eqs. (16) and (17), it follows that
10- 5
KI
:z:
a..
=
n2 - k
2
,
(18)
K2 = 2nk,
and also that
10- 6
3
10
30
TEMPERATURE
100
300
T (K)
FIG. 19. Temperature variation of the phonon mean free pat~ Ap = (3KLI
pCp v). for the N-type GaAs sample of highest lattice conduct1Vlty and weakest doping in Fig. 18. (no"'" \016 cm- 3 .)
ly, the thermal diffusivity D = KL / pCp, which is often the
quantity experimentally measured) for a low temperature
situation.
From room temperature upwards, however, the effects
of impurity doping on the total thermal conductivity are of a
different character: a slight decrease of the phonon transport
itself, a tendency offree carriers to suppress any blackbody
transport, and the conduction (not remarkably strong) of
free carriers themselves. Unlike semiconductors of smaller
energy gap (such as Ge or Si 121), the intrinsic carrier pair
density in GaAs is never large enough to permit the bipolar
transport K bp of electron-hole pairs 102 [with energy
E bp "'(E; + 3kT)] to become a sizeable fraction of the entire
thermal conduction. The intrinsic carrier pair density for
GaAs has recently been reviewed elsewhere,122 and will be
discussed further in Sec. VI.
V_ DIELECTRIC AND OPTICAL PROPERTIES
Much work has been done concerning optical properties ofGaAs, as reported in numerous research papers and in
an extensive review literature. The optical properties for
hV':::::f.E; have naturally interested the authors of books on
GaAs lasers. 6 •7 The third volume of the "Semiconductors
and Semimetals" series was devoted to III-V compound optical properties,123 while other relevant chapters occur in
Volumes 8, 9,12, and 14 of that series. 10
With that wealth of original and review material readily
available, it did not seem useful for this Section to attempt an
encyclopaedic recapitulation of all that has been deduced
and published previously. Thus many more specialized optical phenomena are not recounted here in detail, even though
R137
J. Appl. Phys. Vol. 53, No.1 0, October 1982
n=(l/V'L)[(Ki +~)1/2+Kdl/2,
k = (1V'L) [(Ki + ~ )1/2 - Kd 1/2.
(19)
Thus the dielectric/optical response behavior can be described equally readily in terms of K I and K 2 , or in terms of n
andk.
The absorption which occurs whenever K2 (and thus k )
is nonzero can be characterized by the "extinction length"
(c/41TVk), the distance in which the energy transmitted decays by a factor of "e." More conveniently and commonly,
absorption is expressed in terms of the reciprocal of the extinction length: the absorption coefficient
a
=
41TVk /e.
(20)
Absorption coefficient data in this Section are expressed in
cm- I .
The normal incidence reflectance of a solid:vacuum (or
solid:air) interface is
R
=
n* - 112
In* + 1
=
(n - If + k
(n + 1)2 + k
2
(21)
2 •
This applies both entering and leaving the dense medium.
The reflectance simplifies to R ':::::f.[(n - 1)I(n + 1 when
k <1: tantamount to a requirement that a <105 cm - I for the
near infrared. For GaAs at photon energies just below the
intrinsic absorption edge, n':::::f.3.3 results thus in R':::::f.0.29.
However, when k becomes large for hv> E;, the reflectance
can be very informative about both the absorptive and dispersive aspects of n* and K*.
Figure 20 provides a rough sketch of how the components KI and K2 of K*(V) vary through the spectral range 10
meV to 10 eV. The real part of the dielectric constant asymptotically approaches the "static" or "low frequency" dielectric constant (K I-.Ko) below the reststrah1en range; the
imaginary part K2 becomes negligibly small, and stays that
way for all lower frequencies in nonconducting GaAs.
The strong dispersion/absorption/reflection properties
of GaAs in the reststrahlen region are indicated crudely in
Fig. 20, for hv"'35 meV. In that region, photons can be
annihilated by excitation of long wavelength optical mode
W
J. S. Blakemore
R137
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.
I
KO
---~",
o
I
I
+
I
---------- K I
---K2
/l
!
",'
. , ....-_ .... _------_ .. _----------_ .. --''''''
z
,/ I\V
~A
~'
\
I
«
! /'
':i:
I:
!J
I~
~
II
i .,---
II
I
IJ
I
\.'/i
!.
I!
OL---
I
0.01
K 00
'
I
0.03
:
I
::
•
I
"
0'
I
I
0.1
0.3
hv
oj
I
3
10
(eV)
FIG. 20. Symbolic representation ofthe real part (K ,) and imaginary part (K 2)
of the complex dielectric constant for GaAs, from the "low frequency" regime ofK ,->Ko well below the reststrahlen region of the infrared, through to
the highly absorbing visible and ultraviolet regions of interband transitions.
Artistic license has been used in drawing the dispersion (K,) and absorption
(K 2) curves in the reststrahlen range ten times wider (and with peaks and
valleys ten times reduced) than reality.
phonons. The curves of K I and K2 in Fig. 20 for that reststrahlen region have been drawn ten times broader than in reality,
and with the amplitudes of excursion correspondingly reduced, in order that the S shape of dispersion of K I and the
inverted V shape of absorption of K2 should be visible.
Weak phonon overtone and combination absorption
bands occur for GaAs in the range 35-105 meV. These are
far too weak to produce any notable departure of K2 from
zero in that spectral range, or to modify K I from the "high
frequency" or "infrared" value K r--....K = (n )2 that it enjoys for the middle to near infrared. Appropriately doped
GaAs samples also show free carrier absorption in the infrared, also a process of relatively low efficiency. All of the
above are discussed in the ensuing subsections.
For room temperature, the direct intrinsic absorption
edge occurs when hVj -1.4 eV (A -0.9 ,um). Various opportunities for direct electron-hole pair creation ensure that dispersion and absorption are strong (and the reflectance high)
throughout the visible, and well into the ultraviolet.
Measurements of the optical properties of GaAs have,
as for many other semiconductors, often entailed techniques
far beyond the (apparent) simplicity of refraction, absorption, and reflection. The restriction of intentions for this Section means that just about all of the more specialized optically related topics are omitted, though citations are given to
representative papers on many ofthem. A reader interested
in one of these more specialized topics might usefully check a
citation given in this Section against recent editions of
Science Citation Index. That procedure could well give
pointers towards more recent relevant experiments involving GaAs.
Among the various optically related topics that are not
provided with specific coverage in this Section are Raman
and Brillouin scattering. The literature on Raman scattering
. resonant Ra. GaAs 124-128'mc1u d es a1so papers concernmg
m
man scattering,129.130 and on how Raman phenomena are
00
R138
00
J. Appl. Phys. Vol. 53, No.1 0, October 1982
influenced by hydrostatic pressure 128 • 130 or by uniaxial
stress. 127 The scattering of photons by polaritons involving
acoustic phonons, in resonant Brillouin scattering, has also
been reported for GaAs.131.132
Many optical phenomena can be observed and measured with enhanced precision by modulation and derivative
methods. Cardona 133 discussed the status, as of about 1968,
of how one may usefully modulate wavelength, temperature,
stress, and/or electric field, in order to enrich the studies of
optical phenomena in semiconductors. Volume 9 of Reference 10,134 published some three years later, comprised six
detailed review chapters on aspects of modulated optical
phenomena.
Among the modulation-type optical experiments carried out for GaAs, one can note: wavelength modulated reflectance,135 piezoabsorption 136 and piezoreflectance, 137.138
thermo reflectance, 139 electroabsorption,140.141 electroreflectance, 142-152 and piezoelectroreflectance. 153 The application
of a modulated electric field and/or uniaxial stress, for the
spectral range hV>t;, has been of great value in elucidating
the orderings of and spacing of the GaAs valence and conduction band systems. 154 That information is drawn upon in
Secs. VI and VII, in connection with the thermal distribution of conduction electrons, and with the transport of those
groups of electrons.
The spectral range hv < t; is not lacking in significance
either, for the effects of stress upon the optical properties.
However, these also get no more than a mention here. Weinstein and Cardona 155 measured the effect of uniaxial stress of
the optical response of GaAs in the reststrahlen spectral region. Recently, Feldman and Waxler/ 56 have used uniaxial
stress to make GaAs piezobirefringent in the infrared. From
this experiment, they were able to deduce the components of
the photoelastic and piezo-optic tensors.
Just as the various stress-dependent optical properties
are noted here by literature citations, but not reported in
detail, a similar omission of detailed coverage must now be
admitted for the many and fascinating optical phenomena
that are created by a large applied magnetic field.157.158
Among these specialized magneto-optic effects, one can note
Faraday rotation,159-162 interband and intraband magnetoabsorption, 163-165 and magnetoreflectance. '66-'67 An additional complication of the latter experiment can be piezomagnetoreflectance, with a modulating stress superimposed. 158,166
Having furnished a long list of topics that will not receive detailed coverage in this Section, it is necessary to get
down to the declared business at hand: the spectral dependence of the complex dielectric response function. The first
topic to be examined is that of the low frequency or "static"
dielectric constant Ko. One might reasonably expect this to be
straightforward to the point of dullness. For GaAs, this has
not been the case at all.
A. The low frequency dielectric constant
The parameter Ko can nominally be measured anywhere
from dc to 10" Hz in the millimeter wave region. Reported
measurements included conventional capacitance ones at
frequencies from a few kHz to tens of MHz (with due regard
J. S, Blakemore
R138
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for fringing effects), and microwave methods using a microstrip or a cavity. As an alternative to citation of the entire
published literature, the reader is referred to an excellent
1977 review by Stillman et al. 168 This summarized the results
of a dozen room temperature determinations of Ko. It is probably wise to not take into account two anomalously low Ko
values Stillman et al. report (obtained by microstrip methods). The remaining values Stillman et al. review all lie in the
range Ko (300)~11.6 to 13.3. The breadth of that range is
quite discouraging, particularly in view of the fairly narrow
error bars that several of the original investigators had
placed on their Ko values as included in that survey. It becomes clear that measurement of Ko (even for the simplicity
of room temperature) is not all that easy!
After examining the evidence, Stillman et al. 168 ended
up with a recommended choice of Ko (300) = 12.9. A combination of dielectric and optical arguments was used by
Rode l69 in arriving at the very similar choice Ko
(300) = 12.91. It is worth noting that both of these choices
are substantially larger than the value Ko (300) = 12.5, which
Hambleton et al. 170 had deduced in 1961, from 1 MHz capacitance measurements. This comment is made since the
value from Ref. 170 has been widely, and sometimes uncritically, used in the past two decades.
In good support of the higher Ko (300) conclusions
reached by Stillman et al., 168 and by Rode, 169 it is a pleasure
to report on recent room temperature microwave measurements made by Neidert. 171 For these measurements, a semiinsulating GaAs sample, in the form of a rectangular parallelepiped, was completely metallized to form a resonator,
excepting only small areas at two opposite corners as input!
output ports. Using this techniques, Neidert found that
Ko (300) = 12.9 ± 0.07, over the frequency range 4-18 GHz.
Neidert concluded that this should be valid for all lower
frequencies also.
As may be seen from the summary in Table IX, of infrared and static dielectric response parameters, the present
reviewer is able to deduce a Ko (300) value which is slightly
smaller than Neidert's value as noted above-but to an extent ofless than one standard deviation in Neidert's conclusion. TheKo(T) expression in Table IX, with its consequences
for 300 K, and also for temperatures appreciably lower and
higher, was in fact obtained without using any of the direct Ko
determinations as primary data. This reviewer actually en-
joyed the singular pleasure of coming across Neidert's short
paper 171 some time after having made the analysis that produced Table IX.
The temperature-linearized expression for Ko (T) in Table IX was obtained by use of the Lyddane-Sachs-Teller
(LST)61 relation
KO =K""
(vLO/vTOf
(22)
It thus depends on the precision of available optical measuremen ts, of the infrared refraction index n 00 = vIK:, and of
the long-wavelength optical mode phonon frequencies VTO
and V LO ' The temperature-linearized expressions for the
reststrahlen parameters, as shown in Table IX, are discussed
further in Sec. V B. The primary data for the infrared refractive index is reproduced in Sec. V E.
It was shown in Sec. IV B that the thermal expansion of
GaAs is nonlinear with respect to temperature. Accordingly
it is not likely that quantities such as K o, K and n are truly
linearly dependent on temperature over any very wide range.
Despite this, the convenience of a linearized approximation
is considerable. Since the deduced temperature cofficients of
the various quantities are all small, it is not probable that
serious error could be incurred by the use of linearized approximations for low temperatures (where n data extends
down to 100 K, while VTO and VLO values exist down to 4 K),
or for temperatures above ambient up to (say) 600 K.
It happens that two direct explorations of Ko(T) have
been reported in the literature, as providing an apparently
linear temperature dependence. In one of these, Champlin
and coworkers 172.173 used microwave methods, for the
ranges 100-300 K and 300-600 K. Subsequently, Strzalkowski et al. 174 used capacitance methods, for v<; 1 MHz,
and for l00<;T<;300 K. Figure 21 shows the data of these
two investigations, and displays a dramatic difference in the
two assessments of temperature dependence.
Figure 21 also shows the room temperature result of
Neidert,171 and a dashed line obeying
00 ,
00
00
Ko(T) = 12.40(1
+ 1.2X 1O- 4 T).
(23)
That is the conclusion of the present reviewer, based on Eq.
(22), and on the various linearized expressions summarized
in Table IX.
The value of Ko (adjusted appropriately for temperature)
is needed for modeling the ionization energy, Bohr radius,
TABLE IX. Summary of GaAs dielectric response parameters for hv < E j •
300 K value
Linearized Form
Parameter
Static dielectric
constant
KO = 12.40(1
High frequency
dielectric constant
K~
Infrared refractive
index
(K~ )'/2
Long-wave TO phonon
energy
hVTO
= 33.81(1 - 5.5 X IO-'T) meV
33.25 meV
Long-wave LO phonon
energy
hVLO
= 36.57(1 - 4.0X IO-'T) meV
36.13 meV
Ratio
(KoIK~) = 1.170(1
R139
+ 1.20 X 1O- 4T)
12.85
+ 9.0X IO-'T)
10.88
= 10.60(1
= n~ = 3.255(1 + 4.5x IO-'T)
J. Appl. Phys. Vol. 53, NO.1 0, October 1982
+ 3.0X IO-'T)
3.299
1.181
J. S. Blakemore
R139
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13.6
r------,---,----.---,-------,--~
o
~
f-
z
e
oe CHAMPLIN et 01.
b. STRZALKOWSKI et 01.
c NEIDERT
13.4
<{
f-
en
sions of frequency. The relative narrowness of the absorption/dispersion region is scaled by the factor (vTO/yp). The
LST relation, Eq. (22), builds the value of
VLO = VTO yI(Ko/Koo) into this model, whereby
Z
o
u
u
a::
f-
u
w
132 -
K*(V) = (n - ik)2 = (KI - iK2) =
Koo)
Vro(Ko -
+
Vro - Y
+iypv
(24)
13.0
..J
W
o 12.8
>-
u
z
w 12.6
::J
o
w
a::
u..
~
o
..J
Koo
12.2
~_----L _
o
100
_-'---_ _-'---_----i_ _---'-_--.J
200
300
400
TEMPERATURE T
500
600
(K)
FIG. 21. Temperature dependence of the static (low frequency) dielectric
constant. The upper line shows the microwave data of Champlin and Glover 172 for the range 100-300 K; and as extended by that group 173 to the range
300-600 K. The much steeper line is that recommended by Strzalkowski el
al. 174 from low-frequency capacitance data. The room temperature point of
Neidert 171 is also shown. Error bars are shown as proposed by these various
authors. The dashed line shows the course ofEq. (23). as recommended in
the present review.
etc., for a shallow impurity in GaAs; and was so used by
Stillman et at. 168 in discussing the spectroscopy of and photoconductivity associated with shallow donors. For use of Ko,
the criterion is that the Bohr orbital frequency be small compared with V TO . The high frequency dielectric consant Kocis
needed, however, in modeling the small radius, high orbital
frequency, motion of a carrier trapped at a deep-level impurity.
B. The reststrahlen region
When Waugh and Dolling carried out inelastic neutron
scattering studies of GaAs,58 these provided among other
information the energies for long-wavelength (q = 0) optical
mode phonons: hVTO = 33.2 ± 0.3 meV, and hVLO = 35.4
± 0.7 meV. (See Table VI for the equivalent frequencies.)
Just prior to that work, a value hVLO ~35 meV had been
deduced by Hall and Racette 175 from electron tunneling experiments. However, the reststrahlen parameters can be provided more directly and accurately from several kinds of
optical experiment.
Reststrahlen absorption (with concomitant dispersion
and reflection) cons it utes an oscillation ofGa and As cations
and anions in responses to the transverse E-field of an electromagnetic wave, with a natural oscillation frequency V TO '
For over a century (cf. Helmholtz, 1874) it has been conventional to use a "single oscillator" model to simulate such
absorption/dispersion behavior. Such a model requires four
parameters: the low frequency dielectric constant Ko, the
high frequency (optical) dielectric constant I( the resonant
frequency VTO ' and a damping coefficient yp with dimen00 ,
R140
J. Appl. Phys. Vol. 53, No.1 0, October 1982
Here n, k, 1(1' andK2 are interrelated as described by Eqs. (16)
to (19).
Information about the parameters controlling Eq. (24)
can be gained from measurements of absorption, refraction
index dispersion, reflection at normal and oblique incidence,
and additionally from Raman scattering, plasmon-phonon
coupling, and other more complicated phenomena. It is not
wise to expect that the fit of experimental data to Eq. (24)
should be perfect, since this equation necessarily simplifies
the actual GaAs lattice response for that spectral region.
One complication over and above the provisions ofEq.
(24) that occurs even for an ideally pure GaAs lattice is the
ability of IR photons (some not much more energetic than
hVLOJ to be absorbed in the creation of phonon combinations. Cochran et al. 176 made some quite early observations
of the GaAs overtone and combination phonon absorption
bands, reported in the next subsection.
The response to electromagnetic radiation in the far- to
mid-IR is further complicated for a GaAs sample with a
conduction electron density no large enough to move the
electron plasma frequency
2
vp
= (nO
e2 /4rre
m t-'"
ty' w
(25)
)il2
into or above the reststrahlen range. The usual procedure for
modifying K*(V) is by subtraction of a term from what Eq.
(24) had provided:
K*(V) = K oc
V~O(K() - K",)
+ - 2 - - - 2 - - 'V TO - V + lYp V
V(V
+ iYe)
(26)
Here, the parameter Ye is proportional to the free electron
damping coefficient. 177 Kukharskii 178,179 showed that Eq.
(26) oversimplifies the coupling of plasmons and phonons,
and can lead principally to errors in deducing the damping
coefficients YP and Ye' However, Eq. (26) has been used as
the basis for sorting out plasmon-phonon effects, in a number of GaAs investigations. 125.145.180-184 An example of what
a large vp does to the reflectance spectrum will be provided
shortly, in part (b) of Fig. 26. It is prudent to deal first with
reststrahlen absorption for GaAs of small carrier density,
uncomplicated by plasmon effects.
Figure 22 shows the spectral variation of the real refractive index n, and the extinction coefficient k (both of these on
a logarithmic ordinate scale), for a numerical modeling of
the consequences of Eq. (24) intended to portray GaAs at
room temperature. The size of the dispersive oscillation for
n, and of the peak in k, permits an elucidation of the parameters through measurements of optical dispersion, optical
transmission (at normal and/or oblique incidence), and reflectance (also at normal and/or oblique incidence).
The imaginary part K2 ofthe dielectric constant reaches
for frequency 1'TO,
a maximum value (VTO/yp) (Ko - I(
00
)
J. S. Blakemore
R140
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20
10
n
.>C
S
cIZ
XW
wo~
2
z~
-~
--
"00
-.filO
n
WW
>0
_u
I- z
~o
0.5
0::;:::::
~u
Wz
I-
0::- 0.2
~x
<lw 0.1
W
k
0::"C
c:
0
0.05
0.02
28
30
hJ hJ
32
36
34
k
38
40
hv (meV)
FIG. 22. Spectral variation of the real refractive index and the extinction
coefficient for GaAs at room temperature, modeled by Eq. (24) with values
consistent with Table IX; hVTO = O.920hvLO = 33.25 meV, Ko = 12.85,
and (n ~ )2 = K ~ = 10.88. The additionally required parameter is the damping coefficient, hyp = 0.25 meV for these curves.
and drops to half of that peak value for frequencies (vTO
± !Yp). The frequency domain just above VTO is one in
which KI is driven strongly negative, which depresses n to
values well below unity. The curves of nand k cross again at
the frequency vw, when n=k=[YpKoK",/2vw(Ko
- K
1/2 and K 1 = 0, on its way back to positive behavior.
Only a little higher in frequency is the value
00 ) )
VR
= VTO[(Ko -l)/(Koo
-
lW /2
(27)
for which n = 1, to produce a mlOlmum R min = (1
+ 11k 2)-1 in the normal reflectance that is close to zero. As
frequency rises beyond that point, n-+n
and the absorp00 ,
tion related quantities K 2 , k, and a, drop towards insignificance.
The parameters used for the curves modeled in Fig. 22
were arrived at from a comparison of numbers reported in
published accounts based on various optical experiments.
Values of hVTO and hvw reported in six such investigations
are summarized in Table X. The early reflectivity measurements of Hass and Henvis l85 already provided higher accuracy than feasible with non optical techniques such as electron tunneling 175 or neutron scattering,58 and several subsequent optical investigations have improved on that accuracy. Cochran et al. 176 had attempted optical transmission of
thin samples through the reststrahlen range, but with results
for that specific spectral range that are no longer considered
viable. However, Iwasa et al. 186 obtained data, some at room
temperature and some for lower temperatures, using transmission on thin samples at normal and at oblique incidence.
They also obtained reflectance information, and interferometric data concerning the refractive index dispersion. Figure 23 reproduces some of the normal incidence transmission curves Iwasa et al. obtained for very thin GaAs platelets
at room temperature. On the short-wave side of (c/vTO )' the
fractional transmission
1110 = (1 - R )2e -
at
(1 - R 2e - 2U1) -
1
(28)
was inhibited both by the large absorption coefficient a, and
by the high reflectivity associated with that condition.
Of the remaining entries in Table X, Mooradian and
Wrightl24 deduced V TO and vLa from Raman effect, while
Kukharskii, 179 Holm et al.18~ and Chandrasekhar and Ramdas 184 all analyzed reflectance, including deconvolution of
coupled plasmon-phonon modes for N-type GaAs with
vp >VTO. As may be seen among the entries of Table X, determinations by two sets of investigators often differ by more
than the sums of their estimated error limits; but a consensus
can be arrived at with reasonable confidence. Moreover, the
trend of the results with temperature in Table X is accepta-
TABLE X. Long-wavelength optical phonon energies as deduced in some optical investigations.
T(K)
4.2
Reported by,
and year
Footnote
citation
VTO (meV)
VLO
Hass and Henvis
(1962).
Iwasa et al.
(1964)
Mooradian and Wright
(1966)
185
33.9 ± 0.2
36.9 ± 0.2
1.183
186
33.77 ± 0.06
36.48 ± 0.06
1.166
124
33.86 ± 0.04
36.75 ± 0.04
1.178
(meV)
(vLOlvTO )0
77
Iwasa et al.
(1964)
186
33.71 ± 0.06
36.44 ± 0.06
1.168
300
Iwasa et at.
(1964)
Mooradian and Wright
(1966)
Kukharskii
(1973)
Holmetal.
(1977)
Chandrasekhar and
Ramdas (1980)
186
33.25 ± 0.06
36.08 ± 0.06
1.177
124
33.30 ± 0.04
36.19 ± 0.04
1.181
179
33.41 ± 0.20
36.14 ± 0.15
1.170
183
33.25 ± 0.07
36.14 ± 0.07
1.181
184
33.19 ± 0.06
36.16 ± 0.06
1.187
R141
J. Appl. Phys. Vol. 53, No.1 0, October 1982
J. S. Blakemore
R141
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~ 5r-'---'---~----'----'----'----'
IZ
U 4
c::
-,
W
Q..
2
:::!:
~
3
z
o
~
5
T=300K
w
10
4
tI
1.9f-Lm
2
21
:::!:
en
z
«
Z
w
u
c::
I- OL-~~~~~~~~~~~~L-____L-__~
36
37
38
39
40
10
0
3
'J
Z
]"O<TA
5
:::>
A (fLm)
Lo
2
lJ...
lJ...
W
21f-Lm
WAVELENGTH
5
I-
5.3f-Lm
10.4f-Lm
I-
''\, ro.TA
Q..
FIG. 23. Optical transmission at room temperature through thin platelets
of GaAs, as measured at normal incidence by Iwasa et at. IX6 through the
reststrahlen spectral region. (Ordinate scale values have been reduced by a
factor of 10 with respect to the original figure ofIwasaet aI., so as to produce
conformity with the numbers they quote in their text.)
0::
0
2
~
100
en
B.. _
'\.
30
32
34
hI!
bly consistent with descriptions of V TO and VLO as linearized
functions of temperature. It is accordingly suggested that
hVTO =
hVLO =
K(/K=
33.81(1 - 5.5X 1O- 5 T)
5
36.57(1 - 4.0X 1O- T)
= (vLOlvTO)2 =
1.170(1
meV,
meV,
+ 3.0X 1O-
(29)
5
T),
serve as such consensus expressions, as already listed in Table IX.
In addition to the four parameters K o, K V TO , and VLO
[of which only three are independent, in view ofEq. (22)], the
damping coefficient YP must also be specified for Eq. (24), in
order to trace the routes ofn and k (as in Fig. 22), or ofK] and
K 2 , through the reststrahlen range. The quantity y can nom•
p
mally be deduced either from transmission or reflectance
data. Kukharski's analysis of reflectance 179 led him to indicate hyp ~O.S meV, but three other investigations yield
numbers only one half as large. (hyp = 0.29 meV as deduced
by Holm et al. IR2 ,IR3 and hyp = 0.23 meV in both the studies
ofIwasa of et al., IR6 and ofChandrasekhar and Ramdas. IR4 )
The mean of the three provides hyp = 0.25 meV, which has
been used here in the calculation of the curves for Figs. 22,
24, and 25.
Figure 24 shows the spectral dependence of the optical
absorption coefficient a = (41TVk Ie), in cm - I , with the
damping constant chosen as indicated above. The maximum
of the absorption coefficient is some 4 X 104 cm - I. Values of
a as deduced from applying Eq. (28) to the transmission of
samples with thickness of from 0.04 to 0.11 mm are shown
on the shoulders of the absorption maximum. These are affected by multi phonon absorption processes 176 on the high
energy side. As one progresses further from the central absorption peak, the transmission of a thin sample becomes
large enough so that spectrometer slits can be narrowed, and
interference modifications of Eq. (28) observed. Figure 25
shows curves of n vs hv for these more transparent wings of
the reststrahlen region (using the same parameters as for Fig.
22), with refractive index data superimposed from the interference measurements of Iwasa et aI. 186
Figure 26 shows room temperature reflectance data for
GaAs samples, as measured at normal incidence by Holm et
I
(
-
\
'j
36
38
40
42
(meV)
FIG. 24. Optical absorption coefficient a = (41TV/C) through the reststrahlen absorption peak, modeled (solid curve) for the room temperature parameters listed in the caption of Fig. 22. The dashed curve shows the added
absorption of the first few multiphonon combination bands, as measured by
Cochran et at. 176 Data points correspond with the results of transmission
measurements by Iwasa et al. 1K6 (Note: the values for a reported in Fig. 2 of
Ref. 186 have all been lowered by a factor of 10, to achieve consistency with
the statements in the text of that paper.)
00 ,
R142
J. Appl. Phys. Vol. 53, No.1 0, October 1982
aI. 183 Part (a) of the figure shows their reststrahlen reflectance data for a semi-insulating sample, compared with a
curve calculated on the basis of the single oscillator model
[Eq. (24)] and the reflectance expression Eq. (21). As can be
seen, the fit of the model to experiment is quite satisfactory in
this case. Holm et al. 182,183 showed how the reflectance spectrum could be distorted by an inappropriate surface pre para-
(meV)
22
28
30
32
6,--'--r--r~r--,-;~.-~~~~
T = 300 K
c
x
~
5
Z
w
>
I-
U 4
«
0::
----
lJ...-
~
-.J
«
noo
1-<0
3
w
0::
42
44
hI!
(meV)
46
48
FIG. 25. Room temperature refractive index data of Iwasa et aI., IX6 for the
outer skirts of the GaAs reststrah1en region. These were measured from
interference modulations of the transmittance for a O.09-mm-thick sample.
They are here compared with curves of n vs hv calculated from Eq. (24)
using the same set of parameters as with Fig. 22.
J. S. Blakemore
R142
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4~0________~3~8________~3T6~_________3,4__________--,32
1.0
0.8
0:
w
U
Z
<l:
I-
0.6
U
w 0.4
....J
ll...
W
0:
0.2
(0)
0.0
L..-_ _-L-_ _---I._ _ _l....-_ _---I._ _--'-_-=-_-'--_ _"'""'--:---'
31
32
33
34
35
PHOTON ENERGY hll
36
37
(meV)
FIG. 26. Normal incidence room temperature
reflectance spectra for GaAs samples measured
by Holm et al. ' " (a) The reststrahlen region for
a semi-insulating sample. The authors' curve
was fitted for hVTO = O.920hv LO = 33.25 meV,
hyp =O.29meV,andK~ = 11.1. (Note that the
latter is some 2 % larger than the consensus value proposed in Table IX.) (b) Reflectance for an
N-type
tellurium-doped
sample
with
n()~2.7X 10" cm--', to produce a plasma edge
at hvp ~67 meV.
1.0
0:
w
U
Z
<l:
IU
0.8
0.6
w 0.4
....J
ll...
w
0.2
0:
0.0
(b)
30
40
50
60
70
PHOTON ENERGY hll (meV) --+
tion. Part (b) of Fig. 26 shows a reflectance spectrum by
Holm et al. for an N-type doped sample, with the plasma
edge hvp moved above the optical phonon energies. Comparable reflectance spectra for the plasmon-phonon coupled
modes when hvp is comparable with or larger than hVTO
have been illustrated in a number of other published reports, 145.179-181,184 and the crossover characteristics discussed. Chandrasekhar and Ramdas 184 have shown that this
is affected by the nonparabolicity of the GaAs lowest conduction band, a band structure complication discussed in
Sec. VI.
C. Multiphonon lattice absorption
The high energy end of the optical absorption data in
Fig. 24 showed the first two of the numerous multiphonon
combination and overtone absorption bands. Cochran et
al. 176 reported on the spectrum of these weak absorption
bands from 38 to 100 meV. The three parts of Fig. 27 are
based on their curves of a(hv) for this spectral range.
Spitzer 187 reviewed the physics of multiphonon absorption in semiconductors with diamond and sphalerite lattices,
including GaAs. The annihilation of a photon in the creation
R143
J. Appl. Phys. Vol. 53, No.1 0, October 1982
of two or more phonons (with ~ q = 0) requires an interaction mechanism; and for GaAs this arises in part from anharmonicity in the crystal potential, 188 and in part from secondorder (or higher-order) multipole electric moments of atomic
displacements. 189
Two-phonon absorption is a continuum process, subject to selection rules, 190 and with a maximum for an allowed
transition corresponding to a peak in the combined density
of phonon states. A Van Hove singularity,191 Vq v-+O, is required for at least one of the phonons created. Thus absorption peaks in Fig. 27 are related to the phonon energies for
high symmetry zone locations (such as those listed in Table
VI); and it is not surprising that hv for the strongest band in
Fig. 27 is the sum of the phonon energies for prominent g(v)
peaks in Fig. 8.
Note the widely differing ordinate scales for the three
parts of Fig. 27. Multiphonon absorption strength in GaAs
falls off rapidly with increased hv. (That feature is also the
case for Ge, 192 and is much less so for Si 193 or GaP. 194) The
most energetic photon that GaAs can absorb in a two-photon overtone or combination process is hv= 71 me V, for creation of a pair of zone-center LO phonons. Figure 27(c)
J. S. Blakemore
R143
Downloaded 21 Sep 2011 to 210.32.148.91. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
300
-
TABLE XI. Prominent two-phonon absorption peaks in GaAs: and a set
of possible assignments. b
(a)
200
hv(meV)
Assignment
38
40
41
44
49
51
55
57
62
64
66
71
LO+TA
TO+TA
TO+TA
TO+TA
LO+LA
LO+LA
TO+LA
TO+LA
TO+LO
2TO
2TO
2LO
~
~
t:J
100
0
38
40
hv
42
(meV)
50
44
46
(b)
40
~
U
30
20
10
0
X,W
X
K
W
K
W
K
W
L,X
X
r,L
r
Absorption peaks as illustrated in Fig. 27.
Assigned in conformity with selection rules of Birman (Ref. 190), and in
accordance with the critical phonon energies listed in Table VI.
a
b
t:J
Brillouin zone
locations(5)
D. Free carrier absorption
45
hv (meV)
1.5
(e)
For GaAs, as with any other semiconductor, free carriers of finite mobility provide an ac conductivity aac that
impacts on the dielectric and optical response as an absorption coefficient
(30)
1.0
~
U
t:J
0.5
293K
/
17K
00
70
80
110
hll
FIG. 27. GaAs optical absorption in the mid-IR, showing phonon combination and overtone bands, as measured by Cochran et al. "" Measurements
were made on high resistivity GaAs, so as to preclude free carrier absorption competition.
shows that this process is not very efficient. All threephonon processes involving hv> 70 me V are even less efficient.
Cochran et al. 176 suggested phonon combination/overtone assignments for the various absorption maxima they
saw. Other assignment proposals have been made for GaAs
by Spitzer l95 and by Johnson. 196 Yet another assignment list
is offered here, in Table XI, based on critical phonon energies (see Table VI), and on Birman's selection rules for this
structure. 1'10 The assignments in Table XI agree mostly, but
not totally, with those that Cochran et al. had postulated
before the phonon dispersion curves of this solid were
known.
The strongest multiphonon band is at 39 meV, with
a max =200 cm - lover and above the coexisting reststrahlen
absorption tail (see Fig. 24). All parties agree that this is an
LO + T A combination, benefitting from Van Hove singularities of both phonon branches near the location X, U and
Won the Brillouin zone boundary (see Fig. 4).
R144
J. Appl. Phys. Vol. 53, NO.1 0, October 1982
(Here n denotes refractive index, not electron density.) It is
also possible for a photon with hv < tj to assist an intraband
transition of an electron in the conduction band system, or
hole in the valence band system.
It happens that such intravalence band transitions
dominate over any "free hole absorption" for P-type GaAs.
That was shown in the infrared absorption of Braunstein and
Kane, 197 as exemplified in Fig. 28 by their curves of a(hv) for
a P-type sample at various temperatures. Those data can be
accounted for by hole transitions between the light-hole and
heavy-hole bands for hv < 0.34 eV. These are supplemented
by transitions into the splitoff valence band (see Sec. VI B)
for hv;;;.L1,o = 0.34 eV.
Absorption over the range 1-30/-fm for N-doped GaAs
samples was measured by Spitzer and Whelan. 198 Figure 29
<10° 1 -,
200
-
-;-
::;!;
U
r
1
----r---1
i'\
R\
\~
-,
I
'\
100 l- , _
f
84 K
80 -
197 "
295 K
GO ~
370 K
t:J
40 [
I
~
~: I
0.05 0.1
0.15 0.20 0.25 0.30 0.35 00400.45 0.50 0.55 0.60
hv
(eV)
FIG. 28. Infrared optical absorption for P-type GaAs (Zn-doped, with
Po = 2.7 X 10" em -'I, showing absorption caused by holes in intra-valence-
band transitions, after Braunstein and Kane. 197
J. S. Blakemore
R144
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1000
40rr,--------,----,--.--,-.-ro-..
I
I
v
SYMBOL
SAMPLE
NO.
1
0
6
400
v
x
2
3
•
4
0
5
6
v
200
v
v
••
v
v
~
..
•
v
v
v
v
100-
)(
w
U
LL
LL
W
o
U
4
VV
'i)VVV
2
20
•
x
10
t-
-
o
z
e
0....
0::
4
CD
2
10
(no = 4.9 X 10 17 cm-') in the work of Spitzer and Whelan, I n for temperatures of 100 K (0), 297 K (LJ, and 443 K (e).
"X
!
6
""
It-
-
I>.
6
A
661>.""
o
E
"
I>.
"Au/i
0.4
I
2
4
10
20
A (fLm)
FIG. 29. Optical absorption at 296 K for N-type GaAs samples, as measured by Spitzer and Whelan. 19x Weakly doped sample no. I shows no absorption beyond the intrinsic edge, but samples 2-6 show (i) optically assisted intra-CB transitions for A < 3.5 /-lm, and (ii) free carrier absorption for
longer wavelengths. Conduction electron concentrations (in units of 10 17
cm -- ') are: 1.3, 4.9, 10.9, 11.2, and 54, for samples 2-6, respectively.
shows their room temperature results for six samples ofvarious conduction electron densities no. Other subsequent investigationsl99.2oo have confirmed the essence of what this
figure shows, that free electron absorption controlled by Eq.
(30) is present and dominant for long wavelengths, while opticallyassisted transitions toward upper conduction minima
dominate the absorption for A < 4 /-lm (i.e., hv> 0.3 eV).
Fan 20J has reviewed the physics of sub-band-gap free carrier
absorption processes in Group IV and III-V semiconductors, while Jordon 202 has recently discussed the specific case
ofGaAs in some detail. Jordan's account is recommended to
the interested reader for details of analytic and numerical
fitting of the various absorption mechanisms. Since free carR145
8
FIG. 30. Free' carrier absorption vs wavelength for sample no. 3
J..
o(f)
«
6
6
o
~
6
v
.V
Z
8
•
VV
v
-,
••
VV
40
10
•
•
v
v
v
v
v
v
v
~
•
••
v
U
20
•
•
v
v
v
v
J. Appl. Phys. Vol. 53, No.1 0, October 1982
rier absorption depends on electron scattering processes, the
contributions by acoustic and optical phonons, and by ionized and neutral impurities, all need to be taken into account.
The result in Eq. (30) thus depends on temperature and on
compensation, as well as on no and A.
Of the six samples illustrated by Fig. 29, weakly doped
Sample No.1 evidently showed negligible absorption related
to free electrons. Those kinds of absorption had some different characteristics for the most strongly doped sample (No.
6); but Samples 2-5, with no roughly from 10 17 to 10 18 cm- 3
showed a consistent pattern. For these samples, the longer
wave absorption varied approximately as
a=7.5XIO- 2 °n o A 3
cm-
I
(A>4/-lm)
(31)
(for no in cm - 3 and A in /-lm). A more exact rendition would
require an accounting of impurity compensation, and of the
various electron scattering machanisms, as done by Jordan. 202 Figure 30 compares absorption spectra at three temperatures for Sample No.3. It can be seen that the longer
wavelength part of the spectrum is relatively temperature
insensitive.
For A < 4/-lm, when the absorption ofEq. (31) weakens,
the actual absorption of N-type samples starts to rise again,
because of optically assisted transitions away from the lowest (T6) conduction minimum. The curves of Figs. 29 and 30
show that this has a spectral form dependent on both no and
T. For A~2 /-lm (hv=0.6 eV), this has a room temperature
strength
a ",-,6 X 10- 18 no cm -I.
(32)
This appears to hold even up to the doping level
(no = 5.4 X 10 '8 cm -3) of Spitzer and Whelan's degenerate
Sample No.6.
J. S. Blakemore
R145
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E. Near band-gap dispersion
There have been a number of determinations of the
GaAs refractive index for frequencies well above the reststrahlen range, yet well below the intrinsic band-gap edge.
That index is the quantity V K ~ n oc , for which the numerical magnitude recommended in the present subsection:
K"
X
W
35
0
s
= 10.60(1 + 9.0X lO- T),
n oc. =3.255(1
c
z
+4.5XlO- s T),
(33)
was given advance mention in Table IX. Experimental data
to be borne in mind in the elucidation of n include prism
refraction measurements by Hambleton et al. 170 and by Marple,203 combined transmission/reflection by Oswald and
Schade,204 polarized light oblique reflectance Barcus et al. 205
and others. The various results were discussed in the 1977
review by Stillman et al. loH
Seraphin and Bennett206 reported the primary literature for infrared measurements of n(4 ), and also provided
tables of optical "constants" for various temperatures. These
included: (i) the real and imaginary parts of n* = (n - ik) at
300 K for 0.05 <4 < 25 ,um; (ii) the normal incidence reflectivity R at 300 K for the range 0.05 <4 < 40 ,um; and (iii)
values of nand k near the band edge for T = 21, 90, 103, and
185 K. Seraph in and Bennett elected to quote data on a
"point by point" basis (with identification of the primary
author for each point), rather than any attempt at an averaging or weighting process. That does give all the data, but can
be confusing to read!
Sell et al. 207 used high precision double beam reflectance measurements in order to deduce the dispersion of n
for the spectral range 1.2-1.8 eV. These room temperature
(T = 297 K) measurements thus extended the direct measurements of n through the intrinsic edge. Figure 31 reproduces the result derived by Sell et al. from mesurements on a
high purity sample at 297 K. The portion of their curve for
hv < ti agreed well with the much earlier refraction measurements of Marple,203 and also confirmed the features elecidated by Eden 20H from a Kramers-Kronig analysis 209 of
the reflectance data of Philipp and Ehrenreich. 210
00
w
>
f
r
300 K.",
,
f-U
<t
0:
I.J...
34
w
0:
3_3L---~--~--
__~__~____L -__-L__~____~
14
hv
(eV)
FIG. 32. Infrared dispersion of the refractive index of GaAs (measured by
prism refraction) for three temperatures. From the work of Marple. 2<"
The curve of Fig. 31 shows a small but sharp and distinctive peak at the band-gap energy (ti = 1.424 eV at 297
K), since these data were for a high purity sample, with
no < 10 14 cm -4. Sell et aI. 19R also measured reflectance, in
order to derive the refractive index behavior, for N- and Ptype doped samples; and their sets of curves showed that this
peak becomes smeared out for doping of either type exceeding 10 16 cm- 3 •
Figures 32 and 33 show refractive index data for the
.'"'"
..,.
3.6
'" Marple
o
c
L:..
Komblelon et 01
• De Mels
!
X
w
I
I
i---,--r----,---
AS-~EASUR~D
I
------1---·--o
0
c/.
Z
-J_jj
.:
CORRECTED
a a a
d
W
>
-
f-
'"
u
<l:
0::
':;
3.5
~
3.61-----1
'".
.
.Q••'"
3.4
W
0::
f-
u
<l
0:
LL
w
3.5-
0:
T = 297 K
1.2
1.3
1.4
1.5
1.6
1.7
1.8
ENERGY, hv leV)
FIG. 31. Refractive index n for a high-purity GaAs sample (weakly N-type,
no=S X 1013 cm -3), as deduced by Sell et al. 207 from two-beam reflectance
measurements at 297 K. Data obtained by Marple 203 from refraction measurements are shown for comparison. Shown also are points calculated by
Eden,20" from Kramers-Kronig analysis.
R146
J. Appl. Phys. Vol. 53, NO.1 0, October 1982
300 K
ifiJ
3.3
3.4~~LUL--L__~~__~~__~_Jt__~I_
~r
•
0
1.0
0.5
hll
1_5
(eV)
FIG. 33. A comparison assembled by Stillman et al. loX of three room temperature investigations of dispersion in the infrared refractive index for
GaAs. The data of Marple203 are the same as in Fig. 32. Data of Hambleton
et al. 170 and of DeMeis 211 extend to lower photon energy.
J. S. Blakemore
R146
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range hv < E; . The prism refraction data of Marple 203 for
three temperatures is shown in Fig. 32, while Fig. 33 is taken
from the 1977 review of Stillman et al., 168 as a comparison of
three sets of room temperature data. The room temperature
results from the various investigations do not agree perfectly. However, all the data in Fig. 33 (which incorporates the
300 K curve from Fig. 32) are reasonably consistent with a
first-order Sellmeier type of equation
(n 2 -A) = (n: -A )I[I-B(hvf],
(34)
constrained by the three parameters A, B, and n"" . For hv
expressed in electron volts, the necessary 300 K parameters
are
A = 7.10,
B = 0.18
K""
=
(eV)-2,
n: = 10.88.
(35)
Those values correspond to a 300 K dispersion of the refractive index:
n
= VK = {7.1O + 3.78/[ 1 -
0.18(hv)2]}I!2
(T= 300 K).
(36)
The long wave limit of Eq. (36), [K"" = 10.88, n "" = 3.299]
has already been quoted in Table IX, and its temperature
dependence in Eq. (33). That information was used in tracing
a path back towards the static dielectric constant K o'
Equation (36) is intended principally to be a reliable
guide for n at photon energies well below E;. However, it is
interesting to find that Eq. (36) tracks the behavior of n fairly
well all the way to the intrinsic threshold. [Thus, Eq. (36)
gives n = 3.60 for hv = 1.40 eV, compared with an experimental value n = 3.61 for this energy according to Sell et
a/. 207] Of course, Eq. (36) is not applicable for hv > E, .
The Marple 203 dispersion curves for three temperatures, shown in Fig. 32, are consistent with a linear progression of n"" with temperature, at a rate (lin"" )(dn"" 1dT)
= + 4.5 X 10- 5 . That was incorporated into Eq. (33), as
also reproduced in Table IX. Yu and Cardona 212 have attempted to calculate the temperature coefficient of n for
semiconductors with the diamond and sphalerite lattices,
using the "Penn gap" model 213 for the electronic contribution to the dielectric constant. For GaAs, they used a Penn
gap of 4.9 eV (which they associated with the direct gap at
the X point), and thereby estimated (lin"" )(dn"" IdT)
~ + 5.4x 10- 5 • In view of the simplifications inherent in
such a model, the agreement with experiment appears quite
satisfactory.
Among the reasons for needing to understand, and to be
able to model, the optical properties of GaAs, diode lasers
and LEDs are prominent. Thus the optical properties for
hV-E; are of particular interest to the optoelectronics community. That motivated the measurements of Sell et al.,207
and led to much collateral material that was reviewed in the
subsequent book by Casey and Panish. 7 Mendoza-Alvarez et
al. 214 have recently attempted to model the effects of injected
free carriers on the refractive index ofGaAs, for the spectral
range 1.34-1.44 eV. That work indicates that n should increase very slightly (by less than 0.1 %) for an injected carrier
00
R147
J. Appl. Phys. Vol. 53, No.1 0, October 1982
pair density of up to t1.p- 3 X 10 17 cm -3. Making t1.p larger
than this causes n to decrease below its equilibrium value.
F.lntrinsic absorption edge
The refractive index dispersion for hV-E; that has just
been discussed, is on a modest scale compared with the steep
rise of intrinsic absorption for the same spectral range. This
optical absorption has characteristics that are influenced by
a number of interesting topics in semiconductor physics.
Thus, Oswald and Schade204 made some of the earliest
(-1954) measurements of intrinsic absorption for GaAs.
These measurements extended only to a -100 cm - I , yet
showed strong signs that GaAs is a direct gap semiconductor, with a fundamental transition that is allowed for hV>E;.
The simplest model for allowed transitions in a direct
gap semiconductor provides a spectral dependence of the
intrinsic absorption coefficient:
ao(hv)
= (5 Ihv)v(hv -
E;),
hV>E;.
(37)
aside from the additional small contributions of multiphonon absorption, free carrier absorption and intraband
transitions, etc. The coefficient 5 in Eq. (37) is controlled by
the effective masses mv and me of the valence and conduction bands, and by the interband matrix element.
However, Eq. (37) is not adequate for GaAs. Even for
the absorption edge of pure, unstrained GaAs, one nontrivial
complication that affects ao(hv) is nonparabolicity of the
bands: notably the lowest IF~) conduction band, and the
light-hole band. Those features are described in a number of
band models: for example, in the k.p model that Kane 215
developed, as discussed further in Sec. VI.
Equation (37) is more seriously deficient than that, in its
neglect of the Coulombic attraction between the electron
and hole created in each act of intrinsic absorption. The absorption for hV>E; is profoundly affected by that electronl
hole interaction. 216.2 17 Moreover, a set of exciton absorption
lines is then expected to make an appearance for hv very
slightly smaller than E;.
For a direct gap semiconductor with ideally simple
bands, an exciton of the large radius (Wannier,218 or "effective mass") type has a ground state Rydberg energy
Rxl = e4 /8h
2K 2
Ca (m v-
1+ m e- I).
(38)
Thus the set of photon energies that correspond to the creation of a free exciton,219 in the ground state (r = 1), or an
excited state (r = 2,3,4, ... ) is
(39)
The true complexities of the GaAs valence band system
vitiate the simplicity ofEqs. (38) and (39). However, the large
values of K and of m e- I ensure that Rxl <E;. Abe 220 used a
variational technique in calculating R x I = 4.4 meV for
GaAs. A perturbation calculation by Baldareschi and Lipari221 used the GaAs band parameters ofVrehen l65 in deducing that Rx I = 4.1 meV. Those calculated values compare quite well with Rx I numbers deduced from
experimental data. Sturge222 deduced Rx 1=3.4 meV from
his 1962 measurements of the intrinsic absorption edge, and
somewhat larger values were derived from subsequent work.
Thus, Gilleo et al. 223 deduced Rx I = 4.4 meV from photoluJ. S. Blakemore
R147
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minescence measurements at liquid helium temperatures,
while SeIf24 reported Rx I = 4.2 ± 0.2 meV from absorption
measurements made in 1972 using high purity VPE GaAs
layers.
With this in mind, let us return to Elliott's modef I6 .217
for intrinsic absorption when hV>Ei , and the electron/hole
attraction is included. An expression such as ao(hv) of Eq.
(37) then needs to be modified to
ae
T ~ 294 K
::;
u
8
90 K
'21 K
6
1.44
146
1.48
hv
150
152
154
\.56
leV)
FIG. 34. Upper portion of the GaAs intrinsic absorption edge, as observed
by Sturge m by transmission through a 6.5-jlm-thick semi-insulating polished slice. The bound exciton peak at hv = (~i - ~<.) is clearly apparent for
the three lowest temperatures, and vestigially so for the room temperature
data. These curves have been shifted upwards in energy by ~ 2 meV, as a
correction for the internal strains in this thin, freely supported sample.
R148
10'
u
I0.
IX:
10 2
(41)
Equation (40) provides that band to band absorption should
drop from a(E;) to zero, precipitately, as soon as hv falls
below E j •
The foregoing might lead us to expect that the combination of band to band absorption, and exciton absorption, for
GaAs which is pure, and free from internal strain, would
show a step function rise to strong absorption for
hV~(E, - Rx I)' with the absorption rising at a much less
drastic rate above that threshold. Exciton peaks for r = 1
(and possibly for r> 1) should be resolvable at a low temperature. Figure 34 shows that Sturge222 was able to resolve an
exciton peak for temperatures almost up to 200 K. SeW 24
was able to resolve both the r = 1 and r = 2 exciton absorption maxima for T~2 K. The residual effects of the r = 1
excitonic transition are detectable in the room temperature
refractive index curves {see Fig. 31) by Selletal.,207 and in the
2S
absorption characteristic measured by Casey et
for a
142
....4:w
0
0
[Rxi /(hv - E,W12·
185 K
w
G
'"ro
(42)
"'o
10 4
<t
Thus for a photon energy far enough above Eo Yx <1, which
makes a asymptotically approach a o. Of course, for GaAs
one must still elaborate a o from the (hv - E, ) I/2 dependence
of Eq. (37), in order to take band nonparabolicity into account.
The striking feature of Eq. (40) is, however, that as
hV-Ei from above, the absorption coefficient a remains
large. At the threshold energy hv = Ei itself, a "cliff-top"
value is sustained of
10
"
1-'
Z
9
(40)
where the dimensionless quantity
=
E
u
z
21TYx a o(hv)
a(hv) =
,
1 - exp( - 21TYx)
Yx
10 5
J. AppL Phys. Vol. 53. No.1 O. October 1982
hv
(eV)
E
u
"
tZ
W
<)
5o 103~----~---x~H~---------r­
u
z
o
:;: 10 2 ~-=--.(/
a:
o
'"
(ll
<t
15
IA
hv
1.6
leV)
FIG. 35. Intrinsic absorption edge at 297 K, as reported by Casey et a/. ns
for highly and deliberately doped GaAs. (a) Four stages of N-type doping,
compared with a high-purity sample of no~5X WI) cm -] (b) The same
pure sample, with six stages of P·type doping. For both parts of the figure, a
was deduced up to 1000 cm - 1 from transmission measurements. and by
Kramers-Kronig reflectance analysis 209 for a > 1000 cm - '.
high purity sample. The absorption spectrum for this high
purity sample is identified in both parts of Fig. 35. That
figure also shows the changes in a{hv) near the intrinsic
threshold as produced by doping of GaAs.
Before one proceeds with the effects of doping, there are
some aspects of the absorption edge for "pure" GaAs that
deserve some further comment. It can be seen from the "high
purity" curve in parts (a) and (b) of Fig. 35 that the intrinsic
edge is not, in fact, precipitously steep at room temperature.
From the top of the steeply sloping region, at a~8000 cm- I
and hv~ 1.42 eV, photon energy must be decreased by some
0.05 eV in order to bring a down to -10 cm- I.
This phenomenon, even for nominally pure GaAs, had
been observed long before the work of Casey et aI., in GaAs
absorption edge studies by Moss and Hawkins,226 Kudman
and Seidel,227 Sturge,222 Hill,228 Vrehen,16s and others. For
GaAs and other direct gap semiconductors, the edge gets
blunted. Urbach commented some 30 years ago 229 that many
nonmetallic solids tend to have an exponentially sloping
edge:
a~exp[A (hv - B))
(43)
for the major steep region, which would appear linear on the
J. S. Blakemore
R148
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~ 10 3
Z
W
o
u.
u.
~
o
10
2
Z
o
~
ll.
a:
oIf)
10
CD
<[
1.38
1.40
1.44
h If
1.52
(eV)
FIG. 36. Curves ofSturge 222 for the intrinsic absorption edge in semi-insulating GaAs at five temperatures, of which the curves for \0 and 21 K are
indistinguishable on this scale. The upper parts of these curves were shown
(with a linear ordinate scale) in Fig. 34. The dashed portions here show
Sturge's estimate of the effect of subtracting impurity related absorption.
semi log coordinate systems of Figs. 35 and 36. In the case of
"pure" GaAs, however, it can be seen in Fig. 35 that the
exponential steepness of a increases as one proceeds towards
the top of the edge. The curves of Fig. 36 shows that this
characteristic becomes more prominent for low temperatures.
What causes the exponential slope of an "Urbach edge"
rather than the precipice of Elliott's model: for GaAs among
other direct gap solids, and without the additional physics
that one could invoke from the presence of impurity doping?230 Redfield 231 has proposed that the intrinsic slope degradatiaon in a direct gap solid arises from the Franz-Keldysh effect.232.233 That is to say, the optical threshold energy
is a microscopic function oflocation within a sample, affected by microscopic electric fields that arise both from doping
inhomogeneities (if any), and from internal strains.
If microscopic internal strains in a GaAs crystal cause a
pattern oflocalized electric fields, and a smeared out absorption edge, what is the effects of a large, uniform, applied
electric field? It was argued by Callaway234 that absorption
should rise in a series of steps, each the consequence of a
Wannier level. Measurements by Koss and Lambert,235 with
a field of up to 1.6x 105 Vfcm applied to semi-insulating
GaAs, show some evidence of these Wannier levels at T~24
K. The evidence is less clearcut than one would prefer, since
the measured absorption is a superposition of two Wannier
staircases, in view of light-hole and heavy-hole processes.
The absorption curves in Fig. 35 for doped GaAs now
require attention. Figure 35(a) shows that N-type doping, by
shallow donors, moves the room temperature threshold
progressively towards larger hv. The doping also makes the
absorption edge less steep. These features had previously
been observed in work by Hill,228 Turner and Reese,236 Pankove,237 and others. The shift towards larger hv is as one
would expect from the Burstein-Moss effect,238.239 the consequence of degenerate conditions in the lowest conduction
band (a band with a small density of states).
R149
J. Appl. Phys. Vol. 53, No.1 0, October 1982
The effect of a sharply defined CF higher than cc' for a
low temperature degenerate conduction band condition,
should make the Burstein-Moss shift of the intrinsic edge to
(Ei - Ev) > Ej clear cut, and easy to recognize. This was the
case for data taken at 77 K by Hill,228 and for 5 K data of
Pankove. 237 However, at any temperature, strong doping is
apt to accentuate the possibilities for microscopic compositional and potential fluctuations that can lead to band tailing.19.237.240-242 Those fluctuations will also, through the
mechanism ofthe Franz-Keldysh effect, contribute to a lessening of the "Urbach effect" slope for the intrinsic absorption edge. And part (a) of Fig. 35 does show a continuous
decrease in edge steepness with increasing N-type doping.
Part (b) of Fig. 35 shows that P-type doping moves the
intrinsic absorption edge inexorably towards smaller hv: at
least, it does for room temperature, and for the range of Po
displayed there. Casey and Stern 19 modeled these data in
terms of band tailing. 240.24 I
Those shifts in the location and steepness of the edge
with P-type doping had been reported in a number of earlier
investigations; and the curves of Fig. 35(b) had forerunners
in the 300 K results of Kudman and Seidel,227 and in Hill's
data 228 for both 300 and 77 K. Pankove 237 measured the
intrinsic edge for P-type samples at 5 K, and up to very large
doping (po~ 1020 cm -3). He found that the edge moves back
towards larger hv again for Po> 10 19 cm - 3. This appears to
be the Burstein-Moss effecf38.239 again. It occurs much less
readily in P-type GaAs, because of the relatively large
(heavy-hole) valence band effective mass.
G. Optical properties above the intrinsic edge
The optical absorption coefficient increases less rapidly
once it has climbed to a~8000 cm - I at the top of the steep
edge; but it still has an additional factor of more than 100 to
climb before it finally peaks near hv,::::::5 eV. Transmission
measurements with extremely thin samples can penetrate
only the first part of that spectral region, and reliance on
reflectivity is essential for an evaluation of K I and K 2 , or
equivalently of n, k, and a = (41TVk Ie), throughout the visibleand UV.
Figure 37 illustrates results for a(hv) obtained by analysis of the transmission through very thin GaAs monocrystal
layers. Sturge's measurements 222 were made with polished
specimens 1.4 and 0.67 11m thick, supported on a glass backing, and cooled to 21 K. This experimental arrangement inevitably strained the specimens, and the dashed curve of Fig.
37 incorporates Sturge's estimate of the horizontal shift
needed to correct for strain. Sturge's 21 K data for these
specimens did extend all the way down to threshold, but
merges for all practical purposes with the 2 K curve of the
results Sell and Casey243 obtained eleven years later, and
shown as a solid line in Fig. 37.
Those results of Sell and Casey shown in Fig. 37 for
both 2 and 298 K were obtained with a 1.2-l1m-thick epitaxial GaAs layer sandwiched between layers of Alx Gal _ x As.
That sandwich structure permitted measurements of the
GaAs optical density, up to the threshold (hv,::::::2.2 eV) for
direct transitions in the Al x Ga I _ x As cladding layers. It
J. S. Blakemore
R149
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=-::;:,
14
..2 12
u
0.6
" 10
05
f-Z
w 8
u
u..
u..
w 6
0
u
z
2 4
a::
w
u 04
Z
c:(
298 K",
f-U
w 03
f-0-
0::
0
V)
.J
lJ..
2
W
0::
<Il
0.2
c:(
0 14
18
2.0
h~
2.2
2.4
2.6
2 8
(eV)
0.1
FIG. 37. Intrinsic absorption of GaAs, measured by optical transmission,
through the visible part of the spectrum. The solid curves, for 2 and 298 K,
are from the work of Sell and Casey. 24> The dashed curve shows 21 K data of
Sturge. 222 Since 10; changes by only I meV between 2 and 21 K, the two low
temperature curves are essentially the same, apart from differences in estimates of strain corrections.
should also have permitted something much closer to a
strain-free situation for the GaAs.
The low temperature data in Fig. 37, both ofSturge,z22
and of Sell and Casey,243 show clearly the onset of transitions
to the split-off valence band. This has a low temperature
threshold (E; + LiSD )~ 1.86 eV. The room temperature curve
of Sell and Casey also shows that threshold, albeit less clearly
delineated, for hv~ 1. 76 eV.
The optical properties of GaAs in the ultraviolet part of
the spectrum are preeminently the domain of reflectance
analysis, including various techniques of derivative reflectance measurements. For the experimentally more easily
accessible region up to 6 eV, measurements have been reported as assisted by modulation of wavelength, 135 temperature,139 strain,137.138 and electric field; the last-named form
offield modulation (electro reflectance) impressed by means
of a Schottky barrier contact, 149 an electrolyte, 148 or transverse electrodes. 144
These various ingenious modulation experiments have
provided interesting maxima and minima of the various derivatives; and these, in turn, have fueled various theories for
the energy band structure ofGaAs, as relevant to the material of Sec. VI. However, there seems little value in a review
paper of this type to present such derivative spectra as
figures, since the actual curve shapes-and even the exact
photon energies of maxima and minima-are as much artifacts of the way the experiment is carried out as they are of
the actual properties of GaAs. It is thus the plan in this
section to concentrate on the available data for K 1 and K2 as
functions of hv, and on the related data of the normal incidence reflectance R, and of absorption coefficient
a = (4m·k Ie), through the visible and UV.
It happens that important measurements of normal incidence reflectance for GaAs (and other III-V solids) were
made by Philipp and Ehrenreich 210 in the early 1960s. Those
data, extending as far as 25 eV, still stand as the primary
source of such information to date. Figure 38 shows the
R150
J. Appl. Phys. Vol. 53, No.1 0, October 1982
00
0
5
10
hll
15
(eV)
FIG. 38. Normal incidence reflectivity of GaAs, after Philipp and Ehrenreich,"o showing the features in the UV.
GaAs reflectance curve of Philipp and Ehrenreich, plotted
here as far as 15 eV.
Figure 39 shows the curves Philipp and Ehrenreich ded uced for the real and imaginary parts of the GaAs dielectric
constant, K 1and K 2' also plotted as far as 15 eV. These curves
were calculated from a Kramers-Kronig analysis 209 of the
normal reflectance data, the latter being used over the full
range from threshold to 25 eV.244
Philipp and Ehrenreich 210 commented on the two contrasting spectral regions in Figs. 38 and 39. Since we all know
25,----------,-----------,----------,
20
15
N
lC 10
Cl
Z
<!
lC
5
0
/(1
-5
-10
0
10
5
15
hv (eV)
FIG. 39. Real and imaginary parts K I and K, of the GaAs complex dielectric
constant, as deduced by Philipp and Ehrenreich'lO from Kramers-Kronig
analysis of reflectance data from the infrared to the far ultraviolet.
J. S. Blakemore
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a great deal more about the band structure of GaAs now
than when their remarks were made, their comments are
perhaps even more valuable now as insights into what things
one-electron band theory does well, and where it is less apt.
Thus, Philipp and Ehrenreich noted the sharp structures for
hv up to some 8 eV, as associated with valence to conduction
band vertical transitions in various parts of the Brillouin
zone. Comparable information has been obtained from photoemission experiments in the same photon energy
range,245.246 though here, as in the deliberate exaggeration
techniques of electroreflectance,144-149 the actual response
curve depends on the details of the measurement technique. 247
Note in Fig. 39 that KI is most strongly negative for
hv=5 eV, and is reapproaching positive status by hv=8 eV.
For the higher energy range above 8 eV, KI appears to progress monotonically towards its eventual destiny at a + 1
value. Meanwhile, K2 and R both decline in a comparably
featureless fashion. Philipp and Ehrenreich remarked that
this resembles the behavior of a plasma oscillation model, for
the response of quasi-unbound electrons (in, presumably,
what one normally thinks of as being the valence bands).
Thus these data suggest that, for this range of hv at any rate,
one-electron band ideas may be less appropriate than collective response concepts.
Disturbances do occur of the monotonic trends of K I' K 2'
and R in the spectral range from 8 to 20 eV, as has been
shown in that range with the higher sensitivity of electroreflectance. 15o However, these featuares are too small to be
visible on the scales of Figs. 38 and 39.
The optical properties of GaAs acquire a new set of
features for hv> 20 eV. Transitions then become possible to
the conduction bands from the gallium 3d core states, which
lie some 10 eV below the lowest valence band. Philipp and
Ehrenreich 210 had noted a small reflectance maximum (Rmax
-0.016) for hv=21 eV, and this and related features have
been studied in more detail in reflectivity investigations of
more recent years.248.249 In one of these, Skibowski et al. 249
used reflectivity derivative spectroscopy, plotting (d 2R / dv 2 )
vs hv, in demonstrating six features between 19.5 and 21.5
eV.
The same spectral region has also been studied actively
by means of electroreflectancel50-152 and photoemission.25O-253 All of these sources of information, and others,
were used by Aspnes 152 in determining the ordering of the
GaAs conduction bands-a subject taken up in Sec. VI.
As a complement to the curves of R in Fig. 38, and of K 1
and K2 in Fig. 39, a curve is provided in Fig. 40 of the optical
absorption coefficient, a = (41TVk / c). This is shown from the
intrinsic threshold to hv = 25 e V, and is taken from the work
of Casey et al. 225 The curve is based on transmission measurements of those authors for the intrinsic edge region,
combined with a Kramers-Kronig analysis 209 of reflectance
data. For the latter, reflectance data from Sell et al. 207 was
used in the spectral range 1.2-1.8 eV, and the traditional
Philipp and Ehrenreich data 210 was utilized for the range
1.8-25 eV.
The peaks of a in Fig. 40 for hv near 3, 5, and 6.5 eV, can
easily be recognized as having counterparts in K2 of Fig. 39.
R151
J. Appl. Phys. Vol. 53, No.1 0, October 1982
6
10
"7
~
2
u
...... 105
.>£
;:,.
t::
v
"
1:1 104
2
4
hv
6
8
10
20
30
(eV)
FIG. 40. Log-log plot of the absorption coefficient a(hv) from the intrinsic
edge to 25 eV, after Casey et al. 225 Aside from transmission data in the edge
region, this is based on Kramers-Kronig reflectance analysis, using data of
Sell et al.207 for the range 1.2-1.8 eV, and that of Philipp and Ehrenreich 210
for all larger photon energies.
Figure 40 shows graphically what was expressed in words in
the first sentence of this subsection: the absorption coefficient of GaAs approximates 106 cm - 1throughout the ultraviolet part of the spectrum.
VI. ENERGY BANDS AND GAPS
A historical curiosity ofGaAs and the other 111-V compounds is that their sphalerite crystal structure was known
in the 1920s,28 yet a demonstration of their semiconducting
character had to await Welker's work l a quarter century
later. In the last several years of that inverval, the semiconducting properties of Ge and Si had been vigorously explored, and the stage was set for for the unveiling, by Herman,254 of calculated band structures for these (indirect gap)
Group IV semiconducting elements. It did not escape notice
that GaAs is isoelectronic with Ge. Welker's appraisal 255 of
such a situation was that he expected the III-V binary to
have a wider gap and higher melting point than the element,
because of the partly ionic binding. Initially, Welker also
surmised that the admixture of homopolar (covalent) and
heteropolar (ionic) binding might even lead to larger carrier
mobilities. The mobilities to be reported in Sec. VII do not
support that additional conjecture. 256
A. Bond character and valence charge distribution
In order to report the status of models for bonding and
the energy bands of GaAs, one would prefer to avoid excessive entanglement in the large literature concerning bonding
in tetrahedrally coordinated solids. Some mention of this is
unavoidable, and there have been expressed differences of
opinion concerning the ionic:covalent balance in solids such
as GaAs.
J. S. Blakemore
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FIG. 4J. Total electron density distribution in the (\10) plane of GaAs, as
deduced by Sirota and Olekhnovich"" from x-ray structure analysis. Nearest-neighbor bonds extend in this figure from the three Ga atoms forming
the bottom row to the two As atoms one quarter of the way up.
FIG. 42. A comparison, after Chelikowsky and Cohen";" of the valence
electron density distributions ofGe and GaAs, projected on the (110) plane.
For these Isoelectronrc solids, there are in each case eight valence electrons
per primitive basis, for four valence bands. These data are consequences of
Chehkowsky and Cohen's nonloeal pseudo potential band model.
A partly ionic bond can be considered to resonate
between covalent and ionic charge distributions_ As Welker
noted, this gives a stronger bond than a strictly covaient one_
In principle, the ionicity fraction}; represented in the energy-minimized wavefunction can be calculated by a variational method, provided the wavefunctions for covalent
and ionic limits can be simulated properly. One approach to
this uses atomic orbitals, as in tight binding, and Coulson et
at. 257 deduced a charge transfer of OA6e from Ga to As in
solid GaAs by this linear combination of atomic orbitals
(LCAO) method. Sirota 34 inferred that the charge transfer
from Ga to As would have to be >036e; the excess depending on how the valence electron charge distribution was to be
interpreted.
What does the electronic distribution look like in practice? Sirota and Olekhnovich 258 used x-ray structure analysis in order to deduce the spatial distribution for the totality
of extra-nuclear electrons in GaAs. Figure 41 shows contour
lines they deduced for the (110) plane. Four nearest neighbor
Ga-As bonds lie in the lower quarter of this figure. (Compare
with Fig. 2.) Of course, regions near the nuclei are dominated
by the large electron densities associated with As and Ga
cores of closed K, L, and M shells_ However, Fig. 41 does
show contours for smaller electron density further from the
nuclei, indicative of nearest-neighbor "bridging bond" regions. These have some asymmetry, as to be expected.
The 4s and 4p subshelJs contribute eight electrons per
GaAs primitive basis, three from Ga and five from As. A
totally covalent bond, with the eight equally shared, would
thus be Ga - As -+- , with one electron transferred to Ga. The
ionic Ga -+- 3 As - 3 is the opposite extreme, and the truth lies in
between. As just noted, Coulson et al. 2)7 and Sirota J4 both
expected a small (electronic) charge transferrence from Ga
to As, resulting in Ga -+- x As - x with x~O.4. Related experimental evidence is puzzling and inconclusive. X-ray reflec260
tion analyses by DeMarco and Weiss 259 and by Colella
both indicated a bond slightly more ionic than the "neutral"
GaOAso situation, with x > O. However, x < 0 was indicated
by x-ray work of Attard et al. 261 and by piezoelectric studies
of Arlt and Quadfleig. 262 Polarization in GaAs has more
recently been reexamined by Martin and Kunc.26~
Figure 42 pictures the spatial charge distribution for the
eight valence electrons per primitive basis of the isoelectronic solids Ge and GaAs, projected (as in the case of Fig.
41) onto the (110) plane. The contours of Fig. 42 are not
experimental: they are calculated from the nonlocal empirical pseudopotential model (EPM) for band calculation, as
employed by Chelikowsky and Cohen. 264
The curves of Fig, 42 are based on experiment, to the
extent that known (or surmised) energy gaps were used in the
development of pseudopotential form factors. However,
such curves are less tangibly connected to experiment than
are the Fourier-transformed x-ray data of Fig. 41. The two
parts of Fig. 42 assist in visualizing the polarization foreseen
by an empirical pseudopotential method (EPM) calculatioil
for a Ga-As bond, compared (contrasted is too harsh a word)
with a Ge-Ge bond.
The contour lines of Fig. 42, from Chelikowsky and
Cohen, provide some updating of valence charge contours
that Walter and Cohen 26s had worked out some years earlier. The earlier work may be of interest to a reader who
wishes to dig deeper, in that this also gave contour plots for
each of the four GaAs valence bands separately (each filled
by two electrons per primitive basis), as well as one for the
complete valence ensemble. The interested reader may also
wish to compare Fig. 42 with GaAs valence charge contours
subsequently calculated by Ihm and Joannopoulos 266 from a
self-consistent pseudopotential method, or by Wang and
Klein 267 using a self-consistent linear combination of Gaussian orbitals (LCGO) method.
Before the work of Walter and Cohen 265 is left too far
behind by these more recent calculations, it may be noted
that this arrived at an ionicity fraction}; = 0.31 for GaAs.
This can be compared with the results of other procedures
R152
J. Appl. Phys. Vol. 53, No.1 0, October 1982
J. S. Blakemore
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for arriving at this quantity. Thus, a heat of formation approach was used by Linus Pauling 268 in the development of
his well-known empirical scales of atomic electronegativity.
The electronegativity difference (x A - x B ) between atoms A
and B then determines/; for an isolated A-B bonding situation. The eventual value of/; in a solid is further affected by
the valence (Z = 3 for GaAs), and by the coordination number (N = 4 for the sphalerite lattice). The Pauling procedure
accordingly resulted in/; = 0.28 for crystalline GaAs.
In the course of an extensive discussion of bonding in
semiconductors, Phillips35 deduced an ionicity fraction
/; = 0.31 for GaAs. Rather different criteria had been used
by Walter and Cohen265 in obtaining the same value. Phillips' approach to mixed covalent/ionic bonding, as also discussed elsewhere by Phillips and Van Vechten,269 has been
based on the difference in directionality (and hence in eigenvalue) between bonding and antibonding orbitals. This approach has led Phillips to the formalism of a complex energy
gap, with homopolar and heteropolar contributions. The result has been an electronegativity scale differing from Pauling's one. For GaAs, the differences are unremarkable:
(x As - x Ga ) = 0.4 according to Pauling, and 0.44 from Phillips' scale.
B. Energy band calculations and pertinent experimental
results
One of the earliest band calculations for GaAs was
made by Herman,270 using the orthogonalized plane wave
(OPW) method he had employed so successfully for silicon
and germanium. 254 Herman's approach benefited from the
gross similarities of the valence and conduction band systems for the diamond structure Group IV elements and the
sphalerite structure 111-V compounds in general, and of isoelectronic Ge and GaAs in particular. These features in common are certainly worth emphasizing, for in all of these solids, the three upper valence bands have maxima at the zone
center, r (000): the two uppermost bands (heavy-hole and
light-hole) with a degenerate maximum, and the third separated by the spin orbit splitting energy Ll so . For GaAs, that
splitting Llso = 341 meV, only 18% larger than in germanium. Moreover, the various solids provide an interesting
competition as to whether the lowest conduction minimum
will be a single one at the zone center, or a multiple set along
[l00] or [Ill] directions in the zone.
Experiment and theory during 1953-54 had shown that
the lowest Si conduction minima are a set of six ellipsoids
along [100], while for Ge the four L (m) minima are (slightly)
lower than the one at the center of the zone. Thus both Si and
Ge are indirect gap solids. In contrast, a direct gap status was
indicated quite early on for GaAs, from the intrinsic absorption 204 and reflectance 205 behavior. Moreover, a single minimum at the zone center would be consistent with the observed isotropy of piezoresistance27I and magnetoresistance 272 for N-type GaAs.
Gross similarities of the band structures, especially for
an isoelectronic pair of solids such as Ge and GaAs, continued to influence band calculations for GaAs in the 1950s and
1960s,273-279 and still does so. Two well-represented schools
of thought in work of the 1970s have been the EPM apR153
J. Appl. Phys. Vol. 53, No. 10, October 1982
4
..
>
>a::
w
z
C>
w
GaAs
-8
-10
-12
l
A
r
A
x
U,K
REDUCED WAVE VECTOR
L
r
q
FIG. 43. Electron energy vs reduced wave vector, for the four GaAs valence
bands, and the first several conduction bands, as calculated by Chelikowsky
and Cohen"'" from a nonlocal EPM approach. The top of the valence bands
€ v is zero on this scale. Generally similar forms for the €-k curves have been
calculated by local EPM,277 k.p,278 OPW,279 and LCGQ267 methods.
proach (elaborated from locaf 77 ,280 to nonlocaI264.28I,282
forms of pseudopotential), and the "bond orbital" approach
espoused by Harrison and coworkers.283-285 The latter is a
descendant of the LCAO method, and is useful mostly for
filled bands. Its validity even for that has been debated vigorously by Phillips and Harrison,286 a test it seems to have
withstood. Shevchik et al. 253 used a comparable approach in
analyzing gallium arsenide photoemission data.
Figure 43 shows the general features of the E-k curves
along high symmetry directions in the zone, for the four
valence bands and the first several conduction bands. These
curves are as deduced by Chelikowsky and Cohen 264 from
the nonlocal EPM approach. They provide more detail, and
cover a larger energy range, than in Cohen and Bergstresser's earlier EPM work. 277 The broad features of this
figure are comparable to those found in the k.p calculation of
Pollak et al.,278 the OPW calculation of Herman et al.,279 or
the self-consistent linear combination of Gaussian orbitals
(LCGO) calculation of Wang and Klein. 267 Despite some
differences on the smaller scale of energies (some of which
are important!), several kinds of band calculation-some
nearer to a "first principles" basis than others--encourage
an acceptance of the broad features of Fig. 43.
An energy band calculation that yields a set of E-k dispersion curves can, if asked, provide several other kinds of
related information. The valence charge density contours of
Fig. 42 are one example of an additional output from Chelikowsky and Cohen's calculation; one also provided in some
subsequent band calculations266.267 with different starting
criteria.
Figure 44 exemplifies a further product of a band calculation, by plotting g(E), the density of electron states with
respect to energy. The solid line isg(E) as calculated by CheliJ. S. Blakemore
R153
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T= 300 K
3
,
g 1.0
o
o
>
,
c: 2 L6
>
~
~
_
0.5
Iv
-12
-9
-6
-3
(E-E v ) (eV)
3
6
I
~
~O
>-
<.!)
FIG. 44. Density of states with respect to energy glE) for the lower part of the
conduction band system, and for the four valence bands. Solid curve, as
calculated by Chelikowsky and Cohen. 264 Dashed curve, experimental result of Ley et al.'" for the valence bands, obtained by x-ray photoelectron
spectroscopy.
0::
W
2
W_I
.---1
permitted them to detect GaAs core states. The Ga 3d core
states were seen as a narrow band of 1 eV FWHM width at
(C" - 19.5 eV), some 7 eV below the lowest valence band.
The As 3d band was of similar width, centered on
(£v - 40.5 eV).
Figures 43 and 44 report on calculated band features
over a 20 eV span of energy. However, some aspects of the
band structure are confined to a small fraction of an eV-yet
can be very important! One such for GaAs is the detailed
form of £-k at the Brillouin zone center, near energy £v' Only
dimly apparent from Fig. 43 is that two of the four valence
bands reach a common maximum here: this is the (nominal)
extremum for both the heavy-hole (VI) and light-hole (V2)
bands. That degenerate extremum of VI and V2 is shown in
a little more detail in Fig. 45. There are further aspects of this
complicated extremal behavior too small to show even on
the scale of Fig. 45, caused by the absence of inversion symmetry,287 and these are reviewed in Sec. VI D.
R154
J. Appl. Phys. Vol. 53, No. 10, October 1982
X6
6
r,
,I _____
'42
'
~~___J'v
(v I)
Heavy hale s
0.34 eV
Light holes
(V2)
Split-off band
(V3)
A
kowsky and Cohen, for the first 4.5 eV of the conduction
band system, and for the full span of the valence bands. Chelikowsky and Cohen draw that valence band total range as
12.5 eV, which can be compared with the 12.9 eV range
reported in the experiments of Grobman and Eastman. 25o A
g(c) curve was also provided from Wang and Klein's band
calculation. 267 This is similar to the solid line in Fig. 44 for
the valence bands-and extends an extra 2 eV in the conduction range.
The dashed line in Fig. 44 shows an experimentally obtained g(£) for the valence bands: x-ray photoemission spectroscopy data of Ley et al. 252 These authors estimated their
resolution as being 0.55 eV full width half maximum
(FWHM). The substantial areas of agreement between the
dashed and solid curves of Fig. 44 are pleasing, bearing in
mind the smoothing effect of the experimental resolution.
Other experimental curves for g(£) in GaAs have been obtained by methods such as photoemission. Early experiments 246 could obtain g(£) data for the upper part of the valence bands, and later measurements 25 0- 25 .1 with more
energetic photons secured the entire curve.
Incidentally, the 1487 eV photons used by Ley et al. m
Tr--
OAO eV
nOOO)
/::;
REDUCED WAVE VECTOR
X(IOO)
q
FIG. 45. Variation of energy with wave vector for the uppermost part of the
valence band system, and for the lowest sets of conduction band minima.
Energy gaps are shown as appropriate for room temperature.
A second "fine detail" of the GaAs bands-of large
importance-concerns the order of eigen energies of three
types of Brillouin zone location where the lowest conduction
band dips to a minimum. Over the years, some band calculations have found for T 6-X6-L6 as the order of increasing energy.27K,279 Others have indicated the alternative T6-L6-X6 order.264.273.277.282 Thus a definitive experimental decision was
long overdue when T-L-X was demonstrated by the 1976
electro reflectance experiments of Aspnes et al. 151
Prior to 1976, many GaAs studies had been based on an
assumption that the conduction band order was T-X-L. That
widely held presumption drew upon the successes of Ehrenreich's skillful 1960 analysis 288 of many observed GaAs properties. Ehrenreich assumed a single zone-center lowest conduction minimum, modeled by an adaptation of the k.p
model Kane 215 had used for InSb. However, Ehrenreich
took note also of the spectral character of free electron absorption in N-type GaAs,l98 and of high temperature Hall
effect anomalies in N-type GaAs. 2R9 These led him to predict
a set of secondary conduction minima only some 0.36 eV
higher than the first single minimum.
Callaway'S band calculations 273 had previously predicted [111] symmetry for the second lowest conduction band.
However, Ehrenreich's remarks suggest that he was more
influenced by trends of pressure coefficient data. In any
event, he ended up by suggesting [100] symmetry for the
second-lowest conduction minima. He was not emphatic
about that assignment, to the point ofplacing a sign (?) alongside it. Nevertheless, Ehrenreich's postulated assignment
was accepted in the interpretation of almost all GaAs experimental work of the ensuing 16 years. As one example, predictions of and modeling of the "Gunn effect" in
GaAs290-293 during the 1960s were always constrained by a
presumption of a T-X-L band sequence. 294
J. S. Blakemore
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Aspnes and Cardona 295 subsequently wryly remarked
that Ehrenreich's F-X proposal had been argued so well, that
it tended to discourage serious attention to any contrary indications, from either calculation or experiment. Not until
the 1976 measuremens of UV Schottky barrier electroreflectance by Aspnes et af. 151 did the true (one hopes) F-L-X ordering receive an unequivocal demonstration. The literature
on GaAs in subsequent years has made a rapid adjustment to
the abruptly altered perception of the conduction bands.
Figure 45 shows the F 6 , L 6 , and X6 minima in that sequence,
with the energy gaps above the top of the valence bands
shown as appropriate for room temperature and ordinary
pressures.
The band characteristics within the 5 eV total range of
Fig. 45 are the ones of most importance for most of the topics
reviewed in the remainder of this section, and also for the
various transport phenomena topics of Sec. VII.
The present section continues now, focussed on the features of Fig. 45, in discussing first the valence to conduction
energy gaps, followed by material concerning the characteristics (effective masses, densities of states, etc.) for the various
band extrema within the range of that figure.
c. The direct and indirect intrinsic gaps
1. The smallest (direct) intrinsic gap
A variety of optical experiments has provided information concerning the direct intrinsic gap and its temperature
dependence. The relevant measurements include: room temperature reftectance207 and absorption,225 absorption extended from room temperature upwards as far as 973 K,296
absorption222.297 and refiectance 29R extended from room
temperature downwards towards the liquid hydrogen and
liquid helium ranges, and several methods used at pumped
helium temperatures. The latter include exciton line absorption,224 reflectance,299 and photoluminescence. 223 ,229,300
As with many other semiconductors, a linearized form
€j=(€.o - aT) provides an inadequate representation of the
temperature dependence, especially if it is desired to model
€j(T) to well below room temperature. An empirical equation, requiring the choice of three parameters, was proposed
by Varshni 301 as an improvement over a linearized form.
Varhsi's expression was
(44)
and this has a quadratic dependence on temperature for
small T, changing smoothly towards a linear dependence for
1'>/3. Varshni speculated that the parameter /3 should be
comparable to the Debye temperature of the solid; but this is
an imperfect guide, with GaAs as with other semiconductors.
Pan ish and Casey296 fitted their own absorption edge
data (for 300 < T < 973 K), and lower temperature data from
other sources, to Eq. (44), and deduced thereby values for €j 0'
a, and /3. Their values were subjected to further refinement
in an analysis reported by Thurmond,302 which took additional experimental results into account. Thurmond accordingly arrived at the parameter set €jO = 1.519 eV,
a = 5,405x 10- 4 eV/K, and/3 = 204 K, for use in Eq. (44)
R155
J. Appl. Phys. Vol. 53, No.1 O. October 1982
in the temperature range 0 < T < 1000 K. He thus suggested
the numerical form
€j(T)
=
1.519 - 5.405 X 1O- 4 T 2/(T + 204) eV
(45)
for GaAs, and estimated a standard deviation of some 3 me V
over the above cited range. It may well be necessary to expect
deviations of that size for the highest temperatures, but Eq.
(45) does even better for both room temperature and the
cryogenic extreme.
Thus Eq. (45) is in perfect accordance with the result of
Sell et al. 299 that the r = 1 exciton line emission at T = 2 K
occurs at a transition energy Rxl = 1.515 eV; for the r = 1
exciton Rydberg energy Rxl = 0.004 eV must be added to
reach €j = (€xl + Rxl ) = 1.519 eV. Equation (45) also agrees
with the value €j(297) = 1.424 eV that was found by Sell et
al. 207
Subsequent experimental work (of which more diverse
data taken well above room temperature would be desirable)
may eventually lead to some small further refinement of the
parameters in Eq. (45). It seems unlikely that any error in
that expression could be a major one.
The lattice constant of GaAs decreases when a large
hydrostatic pressure is applied, as was seen in connection
with Fig. 6. Not surprisingly, this enforced contraction increases the width of the direct gap. A large effect of this type
was seen by Welber et al. 303 These workers applied pressure
up to 180 kbar at room temperature, while monitoring the
intrinsic edge by optical transmission. Their result for the
direct gap (at the zone center) was reported as
€r=1.45
+ 0.0126P -
3.77X 1O- 5 p
2
eV
(46)
for P in kilobars. Thus the initial derivative (J€r/JP)
= + 0.0126 eV/kbar, while Welber et al. noted that €r had
reached 2.5 eV by their high pressure limit of 180 kbar.
Those workers seemed concerned by the observation that €r
seemed to be (approximately) linearly dependent on lattice
constant; but was a nonlinear function of pressure-as indicated by Eq. (46). Tsay and Bendow 304 further analyzed and
modeled that nonlinear pressure dependence. However,
since the GaAs bulk modulus increases with pressure (as
shown by the curve of Fig. 6), thepresenceofaP 2 term in Eq.
(46)-with sign opposite to the linear term-should not be a
surprise.
The dependent variable in Eq. (46) is described as €r
rather than €j' since GaAs becomes an indirect gap semiconductor for P> 35 kbar, as shown by high pressure transport
data of Pitt and Lees. 305 Interestingly, the zone-center minimum is not supplanted by the L6 set of minima that are the
second-lowest at normal pressure, for these also rise with
pressure. Aspnes 152 suggests (a€ L I JP);::::: 0.0055 eVIkbar, so
these minima are still -0.2 eV above the F6 one for P;:::::35
kbar. However, the [100] oriented ellipsoids, marked as X6
(or ..1 5)306 in Fig. 44, are brought slightly downwards by pressure: (J€xIJP);::::: - 0.0015 eV Ikbar, and itis these ellipsoids
that form the high pressure conduction band. Welber et
al. 303 were able to detect both the indirect and direct absorption thresholds for P> 35 kbar.
The thermodynamic significance of €j (T), as described
by Eq. (45), is worth some emphasis. As the smallest photon
energy for creation of a valence band hole and a conduction
J. S. Blakemore
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1.60,-----,---.----,--,-_---,
>19
a:
2. The three smallest indirect intrinsic gaps
Many conduction minima can be seen in the various
conduction bands of Fig. 43. Consideration was reduced to
the four lowest in drawing Fig. 45. Of these, three represent
indirect intrinsic gaps, for which phonon absorption or emision is necessary in making an optical transition from the top
of the valence band.
Aspnes l52 has presented arguments in considerable detail as to the relative positions of the L 6 , X 6 , and X 7 sets of
minima. The interested reader is recommended to read his
account for the variety of detailed considerations that led to
his proposals. The analysis by Aspnes was based in part on
UV electroreflectance measurements in which he was involved,149-151 but also took account of other measurements
of optical properties under modulated conditions, 135-148
photoemission,245-247,250-253 intraconduction band absorption,308,309 high pressure transport,305 and high temperature
transport. 289 ,310
1.50
W
Z
w IA5
Z
o
f-
en
lAO
Z
<l
a:
f-
1.35
1300:----1-'0...,.0--2...L0-0--30LO--4....L0-0----.J500
TEMPERATURE
T(K)
FIG. 46. Variation with temperature of the direct intrinsic gap EdT), from
Eq. (45), and the enthalpy of that transition, JH (T) ofEq. (49). The dashed
lines show that a tangent to E, (T) at any temperature extrapolates back to
JH (T) on the ordinate scale, and that for GaAs this is invariably larger than
E, for any real temperature.
electron, the energy E; must create each with zero kinetic
energy. As pointed out by Thurmond,302 this is the Gibbs
free energy of the transition. Also associated with this transition are an enthalpy.t1H (T), and an entropy .t1S (T). The three
quantities are related by
E;
=.t1H - T.t1S.
(47)
The entropy can be obtained by differentiation of Eq. (44),
for the numerical specifics ofEq. (45):
.t1S (T) = ( - dE;ldT) = aT(T + 2/3 )/(T + /3 f,
= 5.405 X 1O- 4 T(T + 408)1(T + 204)2 eVIK. (48)
Similarly, the enthalpy of the intrinsic transition for GaAs
satisfies
+ T.t1S) = E;o + a/3T 2 /(T + /3 f
1.519 + 0.1103T 2 /(T + 204f eV.
.t1H(T) = (E;
=
(49)
Thus the enthalpy of the transition for any finite temperature is larger than E;o'
Figure 46 illustrates the variation with temperature of
E; from Eq. (45), and of.t1H = (E; + T.t1S) from Eq. (49), for
the range 0.;;;1.;;;500 K. The dashed lines illustrate that a
tangent to the E;(T) curve, extended back to the ordinate
scale, crosses this at the value of the enthalpy. For room
temperature, as illustrated, the difference between the two
curves is T.t1S~0.14 eV. Thus it makes a great deal of difference, when analyzing data for their information about the
gap width (in GaAs or any other semiconductor), to make a
clear distinction between .t1H and E;. Otherwise, an erroneously large E; can be deduced from intrinsic carrier population data, subjected to an Arrhenius plot. In that way, Whelan and Wheatley307 arrived at a supposition that E; ~ 1. 58
eV from high temperature Hall data Oust the value of.t1H for
T=600 K) in their important early study of transport in
GaAs. Thus it is necessary to identify and separate the various parts of .t1H in unraveling ambipolar conduction
data. 122
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J. Appl. Phys. Vol. 53, No.1 0, October 1982
It was assumed by Aspnes that the temperature dependence of the three smallest indirect gaps [from the valence
bands at r (000) to the L 6 , X 6 , and X 7 minima, respectively]
could be modeled with reasonable accuracy by three-parameter expressions analogous to Eq. (44). For the lack of better
and more specific information, he also assumed that Thurmond's choice 302 of /3 = 204 K for the direct gap would also
serve adequately for the three indirect gap expressions.
Of the three sets of conduction minima under consideration here, the L6 set is the most important for electron occupancy in thermal equilibrium at high temperatures. Aspnes
suggested that this smallest indirect gap be modeled by
EL
= (E; + .t1 n ) =
1.815 - 6.05X 1O- 4 T 2/(T + 204)
eV .
(50)
In view ofEq. (45) for E, itself, this means that the elevation
of the L6 minima with respect to the zone-center F6 minimum, denoted .t1 n, decreases fairly slowly with rising temperature:
.t1 n =O.296-6.45XI0- 5 T 2 /(T+204)
eV.
(51)
For the second-smallest indirect gap, that between the valence bands and the X6 minima, Aspnes proposed
Ex =(€; +.t1 rx ) = 1.981-4.60XlO- 4 T 2 /(T+204)
eV.
(52)
And so the excess elevation of this conduction band with
respect to the lowest is
.t1 rx =0.462+8.05XlO- 5 T 2 /(T+204)
eV,
(53)
which gradually increases with rising temperature.
As a consequence of the differing temperature dependences of.t1 nand.d rx indicated by Eqs. (51) and (53), everything is favorable for the L6 conduction band to acquire the
lion's share of the conduction electron popUlation at high
temperatures-approaching 80% of the total by the melting
point, as discussed in Sec. VI E. Things are much less favorable for thermal elevation of electrons to the X6 conduction
band, a situation aggravated by the increase of .t1 rx with
temperature.
Of the X 7 conduction band, electroreflectance measurements 149 appear to have improved on the accuracy of earlier
J. S. Blakemore
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absorption ones,30B in placing these minima 0.40 ± 0.01 eV
higher than the X6 ones (for low temperatures and ordinary
pressures). Such a placement for the X 7 band minimum is
high enough to assure that this band will never contain more
than an insignficant fraction of the total conduction electron
population, for any combination of temperature, pressure,
and/or electric field. The X 7 band is not considered further
in this section or in Sec. VII.
D. Characterization of the zone-center band extrema
Further comments are made in this subsection concerning approximation of the forms of c-k dependence to be expected near the major band maxima and minima of GaAs,
and how these relate to "effective mass" concepts. For an
ideal semiconductor, the latter task reduces simply to definition of a valence band mv and a conduction band me' That is
inadequate for GaAs, because of: (i) A degenerate maximum,
at the center of the zone and for energy c v' of two valence
bands: the VI "heavy-hole" band, and the V2 "light-hole"
band; (ii) Warping and nonparabolicity complications for
this pair of bands; (iii) The presence also of the V3 "splitoff"
band, with its maximum at energy (cv - .1 so ), some 0.34 eV
below Cv ; (iv) Nonparabolicity (and slight anisotropy) for the
r6 minimum of the lowest conduction band, denoted as Cl
in Fig. 47; (v) The presence also of sets of conduction minima
also at L6 and at X6 (or .1 5 ), each requiring longitudinal and
transverse mass components for its specification.
Even with the last of these complications deferred until
later in this section, the first four of the above enumerated
items offer ample complexity. Figure 47 shows some of their
attributes, a figure which plots energy versus k 2, out from the
zone center in [100] and [111] directions. This coordinate
system is used in Fig. 47, since a parabolic band (one that can
>III
0
>.,
-0.2
>
,
\0
\0
-0.4
- O. 6
~---'-_-"----L-'--_'-------'-'>'---"--_-'-----'
2
0
-[III]
k 2 (10 14 em- 2 )
2
[100]-
FIG. 47. An approximation for energy vs k 2 around the zone center for the
lowest conduction minimum (el) and the three highest valence bands (VIV3). [100] and [III] directions are visualized. The range covered here is
equivalent to l.Jkl to about one tenth of the Brillouin zone radius.
R157
J. Appl. Phys. Vol. 53, No.1 0, October 1982
be described by an effective mass tensor with energy-independent elements), would appear as straight lines, sloping
away from the external location. The figure shows several
departures from that!
Some complications of band extrema for solids (such as
GaAs) with the sphalerite structure, are also present for the
Group IV elements with the diamond lattice. Elliott 311
showed the significance of the relativistic spin-orbit interaction, in depressing maximum energy for band V3 by an
amount L1so compared with the common maximum of VI
and V2. That remaining V 1 and V2 extremal degeneracy
does not permit either of these bands to escape warping of
their constant energy surfaces from spherical form, as a consequence of interactions among the bands.
Dresselhaus et al. 312 derived an expression
c(k)"'-'c v
-
(/i 2/ 2m oHAk 2
± [B 2k 4 + C 2 (k ~k ~
+k~k;+k~k;)]'/2J
(54)
to characterize the warped heavy-hole and light-hole constant energy surfaces near Cv in Ge or Si. (The positive choice
corresponds to the light-hole band V2, and the negative
choice to VI.) This warping was measured for the Group IV
elements by such experiments as cyclotron resonance, in
work of the 1950s312-314 and thereafter.315-318 Warping of
the GaAs valence bands has also been detectable in cyclotron resonance measurements, 319.320 for comparison with estimates of valence band anisotropy from other kinds of experiment.136.165.321-323
The absence of inversion symmetry in the sphalerite
structure means that III-V and II-VI compounds with this
structure have additional possibilities for complexity of their
band structures, as compared with the Group IV elements of
the diamond structure. Dresselhaus 287 pointed out that,
with inversion symmetry absent, a twofold Kramers degeneracy of eigenvalues throughout the Brillouin zone is not
required. Thus terms linear in wave vector are not automatically of zero coefficient when the c-k behavior is expressed as
a polynomial expansion in powers of k around the valence
maximum. It is thus possible that the heavy-hole VI valence
band ofGaAs might have four "mini-maxima"; each close to
the zone center, but slightly removed from r (000) by a small
vector of [111] symmetry. One of these four hypothetical
mini-maxima 324 is visualized (on a probably exaggerated
scale) in the left half of Fig. 47, showing an energy just slightly higher than Cu for the actual zone center.
Vrehen's interband magnetoabsorption data '65 included some small anomalies, on a scale of 1-2 meV, which he
speCUlated could have been associated with transitions from
VI valence states just above c". However, Gilleo and Bailey325 estimated that any energy elevation of such minimaxima in GaAs could not be more than _10- 4 eV, and
most assessments of data on this subject for other III -V compounds also tend to show 32 f> that linear k terms will be minor
for GaAs. 327 Nonetheless, it is interesting that cyclotron resonance experiments 320 with P-type GaAs reveal a heavyhole mass tensor that has a [111] component 25% larger
than along [100). The existence of shallow mini-maxima
along [111] directions would tend to assist in, rather than
detract from, that observed anisotropy.
J. S. Blakemore
R157
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In addition to its illustration of this slightly bizarre possible complication of the maxima for the VI heavy hole
band, Fig. 37 also illustrates nonparabolicity of c-k for the
Cl, V2, and V3 extrema. Unlike Cl and V2, for which "effective mass" increases steadily with k, the initial tendency
in the splitoffband is for mso to decrease with k, according to
a k.p model. That trend is reversed as k becomes larger.
The term "light-hole band" is something of a misnomer
(for GaAs and for various other semiconductors) except for
energies only slightly below cu' For dispersion curves of the
V2 band along directions including [100] and [111] follow
paths generally parallel to those of the V 1 band, after quite
short initial sections of steeper slope. That behavior was, of
course, indicated in the curves of Fig. 45 for the upper 1.5 eV
of the valence bands, and in the curves of Fig. 43 for a coarser
energy scale still. As Fig. 43 shows, it is for [110] and nearby
directions that the c-k curves for bands VI and V2 are substantially divergent well out into the Brillouin zone.
Nonparabolicity of the light-hole band is thus a feature
that has to be taken into account, in describing the statistical
weight of the V 1-V2 combination for free holes, especially
when the hole density is large and/or the temperature rather
high. The need for that complication of V2 is mitigated, to
some extent, by the larger density of states (some 14 times
larger) of the VI band for energies slightly below cu'
For the lowest conduction band, nonparabolicity cannot be ignored in evaluating the statistical weight for free
electrons. The nonparabolicity for Cl (and also for V2 and
V3) can be treated in a useful approximate manner by the k.p
perturbation method, outlined below.
1. k.p modeling of GaAs at the zone center
A k.p perturbation approach towards making c-k expansions about high symmetry locations in the reduced zone
was taken by Dresselhaus,287 Parmenter/ 28 Kane,215 and
Pollak et al. 278 The Kane approach for describing InSb also
serves well for GaAs (as used by Ehrenreich 288 ), once approximations relevant to Cj > .1'0 (rather than the reverse)
are taken.
A k.p model can be made considerably more complicated than Kane's version, by allowing for the additional perturbations caused by other lower and higher bands.278.329-333
However, the "three level" Kane model, with c and.1 as
the only two gaps taken into account, is able to des~;ibe
features of the zone-center extrema for bands C I, V2, and V3
with respectable accuracy. This model describes the interactions among the four bands of Fig. 47 in terms of a momentum matrix element P, with dimensions of eV cm. Those interactions can equivalently be parametrized by a quantity
I
x=2moP 2/3fz2
(55)
with dimensions of energy.332
Kane showed that the secular determinant has a solution for three of the bands (Cl, V2, and V3) in the form of a
cubic equation 334
The energy variable in Eq. (56) is
c' =
C -
Cc -
fz 1k 2/2mO'
(57)
where mo denotes the "free space" ordinary electronic mass.
Since the conduction band effective mass is much smaller
than m o, the variable c' is just a few percent smaller than the
kinetic energy of a conduction electron in a state of energy c.
The three solutions ofEq. (56) can be identified with the
bands Cl, V2, or V3, respectively. Each solution reduces
towards parabolic form (energy varying as k 2) as a small-k
asymptotic limit. These asymptotic forms are:
e:::",.cc
+ (fz2 k 2/2mo)
XP+X[(2/ci)+(Cj+.1,o)~I]]
C~C" -
Wk 212mo)[(2X/cj) -
C~Cu
.1'0 -
1]
(Cl Band),
(58)
(V2 Band),
(59)
and
-
Wk 2/2m O)[X/(Cj + J so ) -
1]
(V3 Band).
(60)
However, note that there is no single value for the quantity
= (2moP 2/3fz2) that can produce the correct band edge
curvature effective mass
X
m* = Lim
k.o
fz2
I (d 2e1dk 2)
I
(61)
for more than anyone of the three bands modeled by Eqs.
Instead, experimentally measured effective mass
parameters must be used to quote individual values of X for
each of the three bands separately. These are so listed in
Table XII, based on the low temperature band-edge mass
values.
The parabolic dispersion of Eqs. (58)-(60) ceases to be
an acceptable approximation when Ikl is more than a small
fraction of a reciprocal lattice vector. That feature is demonstrated by the curves of Fig. 47, which depart markedly from
the dashed straight lines of the above mentioned equations.
Several stages of generalization of the k.p model were discussed by Kane 215 in a format appropriate for InSb. A variety of expressions suitable for the direct gap members of the
III-V compound family were recapitulated by Madelung.3
Both nonparabolicity and anisotropy can then be incorporated into the formalism, as desired.
The review will continue to comment on the presence of
anisotropy for the various zone-center extrema, though
usually without overt attempts at analytic description of
these complex phenomena. Nonparabolicity is another matter, which must be considered in detail. Thus, the lowest (Cl)
conduction minimum at r 6 has a small anisotropy (demonstrated by the calculated curves of Fig. 48), but a pronounced
nonparabolicity.
If Eq. (58) is construed as being the statement
(58)~(60).
C~Cc
+ (fz2k 212meo)
(k~),
(62)
in defining meo to be the conduction band-edge effective
mass, then this is related to Xc for that band by
(63)
c'(c'
+ Cj)(c' + Cj + .1 so ) =
k
2p
2
(c'
+ Cj + 2.150/3).
(56)
R158
J. Appl. Phys. Vol. 53, No.1 0, October 1982
Table XII includes the low temperature consensus value
J. S. Blakemore
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TABLE XII. Directionally averaged (density of states) band-edge curvature effective masses and related parameters, for the zone center extrema of band C I,
and bands VI through V3.
Band
extremum
Conduction
band CI
Heavy-hole
band VI
Light-hole
band V2
Splitoff
band V3
Band-edge density
of states effective
mass, for T-.o
k.p model matrix
element, expressed
as X = (2m"p2/31i2)
k.p model
k.pmodel
Nonparabolicity
parameter
equation
for m*(T)
Band-edge density
of states effective
mass, for T = 300 K
meo = 0. 067m o
Xc = 7.51 eV
a = - 0.824
Eq. (63)
mco = 0.063m"
m l =0.082mo
XI = 1O.0eV
{3= - 3.80
Eq. (68)
m", = 0.154mo
X", = 13.geV
Y=
+ 10.8
Eq. (72)
m. =0.5Im o
meo = 0.067mo that this review recommends for use as the
band-edge effective mass as T-0. (That value is based on
experimental results noted in Section VI E. Table XII also
showsXe = 7.51 eV as required for consistency with Eq. (63)
at low temperatures. As Trises, €; decreases, and accordingly so does meo' Table XII shows how much Eq. (63) predicts
meo to have decreased by 300 OK.
When Ik I is allowed to become a little larger, nonparabolicity must be allowed for, most innocuously through a k 4
term. This involves examination ofEq. (56) as a quadratic for
€' (dropping the term in €'3), and expanding the solution in
powers of k 2 as far as the k 4 terms. One can thereby write
e::::!.€e
+ (1i2k 2/2m eo ) + (a/€;)(1i2k2/2meo)2,
(64)
where the nonparabolicity coefficient a turns out to be negative:
a=
(1 - m eo /m o)2(3E7 + 4€;..::1so + 2..::1 ~o)
(€; + . ::1 so )(3€; + 2..::1 so )
[l - €Aso/(3€; + 2..::1 so )(€; + . ::1 so )]
(65)
[l +€;(€; +..::1 so )/xe(3€; + 2..::1 so )] 2
The low temperature value a = - 0.824 is also shown in
Table XII.
Since a < 0, energy rises less rapidly with k than would
have been the case for a perfectly parabolic conduction band
of energy-independent mass meo' That situation is commonly viewed as an increase of effective mass me with energy.
The curves of energy versus k 2 in Fig. 47 for the conduction
band illustrate that characteristic of nonparabolicity with
a<O.
The dimensionless quantity a is not quite temperature
independent for GaAs, in view of the €;(T) behavior modeled by Eq. (45). However, a is much less dependent on
temperature than meo' The linearized approximate form
a':::::!. - (0.824
+ 2.0X 1O- 5 T)
J. Appl. Phys. Vol. 53, No.1 0, October 1982
E':::::!.€v - Wk 2/2m/) -1j3/E;)(1i2k 2/2mIl 2
(67)
for the light hole (V2) valence band. Conformity with Eq.
(59) means that the density of states effective mass for the
0.5
r-----------------,---,
GALLIUM
- - EXACT TWO- BAND INCLUDING
HIGHER BANDS
- - - EXACT TWO-BAND
•••••• APPROX. TWO-BAND
_. PARABOLIC BAND
0.4
>
0.3
>(!)
a::
~
0.2
~
ARSENIDE
UJ
0.1
(66)
is quite faithful to Eq. (65) over a wide temperature range.
Thus a has progressed as far as - 0.854 by the melting
point, and the room temperature value is a':::::!. - 0.83, as remarked by Vrehen. 165
It must be emphasized that the k.p model is a perturbation approximation, and should not be expected to be reliable except quite close to an extremum. Thus Eq. (64), with
the nonparabolicity coefficient of Eqs. (65) and (66), will still
R159
be inadequate for anyone who wishes to know about the Cl
conduction band more than (say) 0.1 eV above the F6 minimum. Fortunately, that possible 0.1 eV range of validity for
Eq. (64) is enough to describe the contribution oftheF6 minimum towards the statistical weight of the entire conduction
band system, for any temperature of solid GaAs.
For those who are interested in how the nonparabolicity and the small anisotropy of the C 1 band appear up to
€':::::!.(€e + 0.4 eV), Fig. 48 reproduces curves from the work of
Rode, 169 based on unpublished 1969 calculations by Fawcett
and Ruch. The uppermost curve in Fig. 48 corresponds to
the upper dashed lines of Fig. 47-the meo parabolic assumption. It is a little reassuring to see from Fig. 48 that the
so-called "exact two-band including higher bands" model
predicts a quite mild degree of anisotropy.
As a companion to Eqs. (62)-(65), the k.p model permits
one to describe the €-k behavior for a small energy range
downwards from Ev by
o
2
4
6
8
10
FIG. 48. Energy vs wave vector for the first -0.4 eVofthe F. (el) GaAs
conduction band, from Rode. 169 These curves are based on unpublished
modeling (- 1969) by Fawcett and Ruch. Note that the uppermost curve is
the parabolic dispersion for a room temperature mass moo ~0.063mo. Sophistication in the modeling can provide nonparabolicity and anisotropy.
J. S. Blakemore
R159
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light holes must satisfy
(68)
m l = E;mol(2XI - E;).
Table XII provides the low temperature value of this bandedge mass for the light holes, and shows that this requires a
value XI = 10.0 eV that differs sharply from XC'
The last term in Eq. (67) can be evaluated, as in the
conduction band case, by expanding the appropriate root of
Eq. (56) as far as terms in k 4. Again it turns out that the
nonparabolicity parameter /3 is a negative quantity. Some
manipulation yields the result
/3=
-(I+EJ2L\so)/(I-Ej2X,)2.
(69)
As indicated by the entries of Table XII, the low temperature
value /3 = - 3.80 is several times larger than the corresponding conduction band nonparabolicity parameter. Thus
nonparabolicity takes a similar form for light holes-mass
increases as one goes away from the extremum-but the effect is more severe than with conduction electrons.
Equation (69) is less Obliging than Eq. (65) in its provisions for a temperature dependence of /3 in a simple linear
form useful for all temperatures. However, for a substantial
range including room temperature, the behavior of /3 can be
approximated by
/3~
- (3.87 -
T /1000),
200.;;;T.;;;I000 K.
(70)
The third solution of Eq. (56) is for holes in the splitoff
valence band V3. A perturbation expansion to terms in k 4
can be carried out here also. Thus if the E-k relation for the
uppermost part of that band is approximated by
E~E" - .1'0 - (f?k 2/2m so ) - (y/E;H/fk 2/2mso)2, (71)
then conformity with Eq. (60) requires that the band-edge
effective mass be described by
(72)
Table XII shows the low temperature mass m,o
perature dependence of y provided by Eq. (73) is not perfectly linear, but the linearized approximation
y~
+ (11.16 -
0.0043T),
200.;;;T.;;;800 K
(74)
is adequate within ± 0.02 limits. Thus y~ + 9.87 at room
temperature.
The large size of this positive quantity y is relatively
insensitive to the value of Xso used in Eq. (73); i.e" y is guaranteed to be large for any plausible value of mso. Thus the
effects ofnonparabolicity just below the top of this band are
marked, and are in the sense of making the curvature effective mass decrease with increasing Ik I. Any underestimate of
this tendency, in the experimental analysis by Reine et al., 166
could have led these workers towards an m 50 on the low
side-with a consequent error of 1'50 on the high side by use
ofEq. (72).
These comments on the splitoffband in the k.p perturbation model should conclude with a reminder that this is a
perturbation model. As Ikl increases further, the E-k behavior assuredly departs from that approximated by Eq. (71),
and the curvature mass certainly increases again. The beginning of that trend is indicated (not quantitatively) in the lowest part of Fig. 47.
E. Electron effective masses: Conduction band system
statistical weight
f. Effective mass for the r6 (ef) lowest conduction band
All definitions of effective mass lead to the same numerical value for a band that is both isotropic and parabolic.
As already noted, those provisions are inapplicable for the
lowest GaAs conduction band, with nonparabolicity more
serious than anisotropy. Thus the numbers quoted in this
subsection are all intended to be spherically averaged ones.
One important definition of effective mass is the band
curvature one:
= 0.154mo reported by Reine et al. 166 from piezomagneto-
(75)
reflectance measurements, and also the value Xso = 13.9 eV
accordingly required by Eq. (72). That value for 1"0 is nearly
twice as large as the value noted for Xc in the same Table.
This discrepancy hints that, if there was any error in the
experimental mso value of Reine et al. (they had estimated
± 10%), the true m,,, might be larger. Why? For a reason
~xpounded in the paragraph after next.
For phenomena such as Faraday rotation, magnetic susceptibility, etc., the important quantity is the "optical" or
"slope" effective mass
When the root of Eq. (65) appropriate for holes in the
splitoffband is developed as a quadratic equation for energy,
and the solution is expanded for terms in k 2 and in k 4 (omitting terms of higher powers of k ), then the nonparabolicity
parameter y of Eq. (71) is found to be a positive quantity:
y= +E,(2E;
+ .1so)/.1so(E,
+.1,o)[l-(E; +.1so)/X,or·
(73)
Not only is y> 0, it is also quite large. Thus, Table XII includes the value y~ 10.8 for low temperatures. And so the
uppermost region of the splitoff valence band is characterized by a curvature effective mass that decreases rather rapidly as one departs from the (Ev - .1 so ) extremum. This is
pictorialized by the downwards curvature of the E-k 2 lines
just below the upper limit of this band in Fig. 47. The temR160
J. Appl. Phys. Vol. 53, No.1 0, October 1982
(76)
Now if E-k for the band can be approximated by Eq. (64),
then both me and mOp! converge upon mco for the edge of the
band (E-->-Ec,k-D). For energies slightly higher than Ec [but
still within the plausible range of applicability for Eq. (64)],
then
(77)
Thus both versions of "effective mass" increase with energy
(since a < O), but the curvature mass at a rate three times
faster than the optical one.
These effects are far from trivial. As an example, consider N-type GaAs at room temperature, doped so that
18
no~2X 10 cm--" so that the Fermi energy EF~(Ec + 78
meV). Then at the Fermi energy, the curvature mass is
mc~1.27mco,~0.080mo, while the optical mass at that energy is mop, ~ 1.09mo~0.069mo. That numerical example
illustrates how important it is to know which of the various
J. S. Blakemore
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"effective mass" definitions is being used, when conduction
electron properties are being discussed. Thus, Cardona, 160
Piller,161 Raymond et al.,333 Ukhanov,335 and others report
on experimental data of mass versus energy in this band. The
various expressions used or implied need to be interpreted
with care.
Experimental values for the band-edge mass meo have
been deduced from a wide variety of experiments. Stillman et
al. 168 give a summary of 18 determinations by various
workers. These included the interband magnetoabsorption
measurements of Vrehen,165 Faraday rotation results by
Moss and Walton, 159 Cardona, 160 Piller, 161 and Ukhanov,335
magnetophonon resonance results of Stradling and
Wood,336 and cyclotron resonance measurements by Chamberlain et al.,337 Poehler,338 and others. That summary also
noted the quite accurate value meo/mo = 0.0665 ± 0.0005
that Stillman et al. 339 obtained from an analysis of the Zeeman spectroscopy of shallow donor levels at liquid helium
temperatures. A consensus of all the work reviewed by Stillman et al. 168 is consistent with adoption of a low temperature, band-edge value meo = 0.067m o, as displayed in Table
XII.
Indications from various of the experiments noted
above were that meo had decreased by several percent on
warming to ambient temperatures. The extent of the decrease (or increase) depended on exactly what was being
measured. Thus, Chandrasekhar and Ramdas 184 have recently noted a 2% increase of the "optical mass" when
doped GaAs (no = 7 X 10 18 cm -3) was warmed from the liquid nitrogen range to room temperature. In contrast, a decrease of the actual band-edge curvature mass meo is indicated by several of the above noted experimental investigations.
It is also implicit in the k.p model, through Eq. (63). Figure
49 shows how meo/mo should decrease from its low temperature value when T rises and €j falls. As noted in Table
XII, the low temperature combination of knowledge of meo
and €j (and Ll so ) determines Xc = 7.51 eV. The behavior of
o
E
"-
meo for higher temperatures is affected by a temperature
dependence for any of the quantities in the denominator of
Eq. (63), but it is assumed here that €j changes, while Xc and
Llso do not. (The experiments of Nishino et al. 340 found no
reason to modify Llso from its low temperature value of 0.341
eV, at least as far as 300 K.)
2. Effective density of states for the
r6 conduction band
Had this lowest GaAs conduction band been a parabolic one for which the band-edge mass meo was equally applicable at higher energies, then the dimensionless Fermi energy 7J [(€F-€c)lkT] could have been related to the
equilibrium conduction electron concentration no by
_
Loo
(21Y1T)t 1/2dt
(78)
N eo iJld7J)·
o 1 + exp(t - 7J)
Here, iJI/2(7J) is a member of the Fermi-Dirac integral family,341.342 and
Neo = 2(21TmeokT /h 2)3/ 2
(79)
no - Neo
is usually called the "statistical weight," or "effective density
of conduction band states" for temperature Tand mass meo'
However, the nonparabolicity of the GaAs conduction band
makes no> Neo iJI/2(7J) for any finite temperature. This is
particularly so for GaAs doped strongly enough N type to be
degenerate, but the influence of the nonparabolicity does not
disappear when conditions in the conduction band are nondegenerate-as happens in weakly N-type, semi-insulating,
intrinsic, or P-type GaAs. This writer has recently discussed
the topic apropos intrinsic GaAs.122 What follows emphasizes the aspects that Ref. 122 did not need to pursue.
The conduction band €-k behavior ofEq. (64) augments
the electron capacity of this band, for a given Tand €F' to the
extent
no = Neo [iJI/2(7J) - (15akT /4€;)iJ3/2(7J)].
The latter term is actually additive, since Eqs. (64) through
(66) show that a < O. One would like to be able to conform
with usual semiconductor terminology, in writing the relation between no and Fermi energy in a form
o
(81)
EO 0.065
Then this band minimum can be said to have an effective
density of states
Q
~
a: 0.060
Nc = 2(21TmeokT /h 211 1 - (15akT /4€j)[iJ3/2(7J)liJld7J)] j
= Nco 11 - (15akT /4€j)[iJ3/2(7J)liJld7J)] j.
(82)
(f)
(f)
<{
~
~ 0.055
o
w
I
o
Z
~ 0.050~----L---~L---~----~----~
o
200
400
600
800
1000
TEMPERATURE T(K)
FIG. 49. Variation of the conduction band-edge effective mass with temperature, as described by the k.p model of Eq. (63). This assumes
..:l= = 0.341 eV independent of temperature, and that meo = 0.067mo at
low temperatures when E j = 1.519 eV, from which Xc = (2moP 2j
2
311 ) = 7.51 eV as the strength of the momentum matrix element, as noted in
Table XII.
R161
(80)
J. Appl. Phys. Vol. 53, No.1 0, October 1982
The multiplying effect of the factor! ..... j depends on T,
and also on the relation of € F to €c' For nondegenerate conditions (€F < €c' no < Nco) all members of the Fermi-Dirac integral family reach a common asymptotic form: iJj (7J)-e'1.
That minimizes the multiplying factor, but it is still larger
than unity for any finite temperature.
Let us denote Nc as N; for such nondegenerate conditions, so that
N;/Nco = (I - 15akT /4€j).
(83)
The right side of this equation contains three temperaturedependent quantities: Titself, a ofEqs. (65) and (66), and €j
ofEq. (45). It is thus not possible to expressN ;/Neo in simple
J. S. Blakemore
R161
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factor in Eq. (84). The rate of decline is about half of
(Nco /T 3/2), and so the augmenting effects of nonparabolicity
offset about half of the decline in m~~2. The magnitude and
total temperature dependence of N ; is represented quite well
by
N
....
'",
8
~
'",
:::!' 7
U
N; ~8.63 X 1OI3T3/2(1 - 1.93 X 1O- 4 T - 4.19 X 1O- 8 T 2 ),
~
0
6
cm- 3
l.L
0
(100< T < 1200 K).
(f)
I-
5
z
:::>
4
100
300
500
700
900
TEMPERATURE
1100
1300 1500
T (K)
FIG. 50. Components of the temperature dependence for the effective density of states associated with the lowest GaAs conduction minimum. The
lower curve shows (Nco /T 3/2), where Nco is defined in Eq. (79). The decrease
is caused by the falloff of m eo , modeled by Eq. (63) with c, varying as in Eq.
(45). The upper curve shows (N ;/T3/2), enlarged by the mUltiplying factor
ofEq. (83), modeled numerically by Eq.184).
exact analytic form. However, the numerical behavior of this
ratio is tracked quite well by
(1 - 15akT /4E;)
4
H
2
= N;Nco~[ 1 + 1.73 X 1O- T + 3.80X 1O- T
3
+4.36xlO- " T ] (100.;;1<;1000 K).
(84)
Thus the multiplying factor for nondegenerate conditions
amounts to some 1.056 for room temperature, and to 1.255
by 1000 K.
The band-edge mass meo is steadily decreasing with rising temperature, as indicated by the curve of Fig. 49 if one
can rely on the k.p model Eq. (63) as a guide towards high
temperature behavior. Accordingly, Nco rises with temperature, but less rapidly than its explicit T3/2 factor. The lower
curve of Fig. 50 shows the decrease of(Neo/T3/2) with rising
.. f rom tee
h b h
' 0 f meo
3/2 .
temperature, ansmg
aVlOr
The upper curve of Fig. 50 shows that (N ;/T3/2) also
decreases with rising temperature, despite the effect of the
(85)
This is recommended for use as the statistical weight of the
Cl conduction band whenever conditions are nondegenerate. Thus, N;(3OO) = 4.21 X 10 17 cm- 3.
N-type GaAs is often doped strongly enough so that the
condition no < Nco is grossly violated. One must then contend with a relation between no and E F appropriate for partial or complete degeneracy, as described by Eqs. (81) and
(82). Those equations allow the interested reader to compute
what value of no is appropriate for any combination of T and
341 or
E F , with ;51/2(7]) and i"Y3/2(7]) provided from tabulations
from analytic approximations,34 1-343 as required.
Rather than attempting a presentation of families of
curves for all combinations of T and no, this subsection of the
review closes with notes on two important temperature regimes: for 300 K, and for the strong degeneracy limit of no
still large as T---'>0.
Figure 51 displays the results of Eqs. (81) and (82) for
room temperature, T = 300 K. The two abscissa scales show
no and the resulting Fermi energy, while the two ordinate
scales show the ratio (Ne/Neo ) and the actual value of N c.
The ordinate scale at the left ofthe figure demonstrates what
has already been noted for room temperature, that Nc = N;
for nondegenerate conditions is more than 5% larger than
Nco. However, the ratio (Nj Nco) has grown to more than 1.2
when N-type GaAs is made markedly degenerate by imposing no~ 10 19 cm -3 Of course, that has forced EF more than
200 meV above Ec , which has a variety of consequences, inckding a large Burstein-Moss shiff 38 .239 of the intrinsic absorption edge as demonstrated in Fig. 35(a). The rise of the
curve in Fig. 50 with increasing no is equivalent to the curves
(meV)
4.75
I
::2:
N- Type GaAs
<.)
T = 300 K
1.15
I<)
I"-
0
4.50
o
u
Z
........ 1.10
u
....N
Z
4.25 ~
~-----===~=------1.05
o
FIG. 51. Variation of the "effective density of
conduction states" with electron density no for
GaAs at 300 K. In addition to ordinate scales
for N, ofEq.182) and ofIN,/Neo )' an upper abscissa scale is shown oflcF - c,). Since this is for
T = 300 K. it may be noted that
meo = 0.0632mo to give Nco = 3.99 X 10 17
cm-'. and also that (-15akT/4c,)
= 0.05647.
C
'--'
"u
I •00
L _ - - - L_ _L:o-_-'--_ _L-_--L:-::---L_ _- L _ - - '
2
5
10
17
2
5
10
18
2
CONDUCTION ELECTRON CONCENTRATION
R162
J. Appl. Phys. Vol. 53, No.1 O. October 1982
5
10
4.00 Z
19
no (CM- 3 )
J. S. Blakemore
R162
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TABLE XIII. The low temperature degenerate limit for the lowest conduction band: electron density, Fermi energy, and two different definitions of
effective mass.
(CF - c,.)
(meV)
Optical mass
(cm-')
ratio (moor/mo)
Curvature mass
ratio (mJmo)
5 X 10 16
1 X 10 17
2X 10 17
5 X 10 17
I X 10"
2 X 10"
1
5X 10 "
I X 10 19
7.4
11.6
18.4
33.6
52.8
82.5
147
225
0.0675
0.0679
0.0683
0.0694
0.0708
0.0730
0.0770
0.0834
0.0686
0.0695
0.0710
0.0743
0.0785
0.0850
0.0991
0.1162
no
of Cardona, 160 Piller, 161 Stillman et al. 16M Raymond et al.,333
etc. for dependence of "effective mass" on no.
As a concluding comment on degeneracy in this band,
consider N-type GaAs at a very low temperature. Even a
relatively modest permanent electron population is then sufficient to produce strongly degenerate conditions. That situation, as first analyzed by Sommerfeld 344 for a metal, permits
use of the asymptotic approximation
(86)
Inserting iYld1]) and iY3/2(1]) in this fashion into Eq. (80), one
finds that a finite electron population no at a near-zero temperature will result in a Fermi energy EF such that
no~(81T/3)[2meo(EF - Ee)/h
2]3/2[ 1- (3a/2Ei)(EF - Ee)].
(87)
For N-type GaAs with T---+0, so that meo = 0.0670m o,
Ei = I.S19 eV, and a = - 0.824, this amounts numerically
to
no~2.S0X
IOI5(E F - Eef/2[ 1+ 8.2X IO- 4 (E F - Ee )]Cm- 3
(88)
for (EF - Ee) expressed in meV. Table XIII provides a short
tabulation of degenerate Fermi energies corresponding to a
range of low temperature no values, in accordance with Eq.
(88). The same table also shows the corresponding values of
the "optical" or "slope" mass m opt and of the "curvature"
mass me at these Fermi energies, both of the latter as defined
by Eq. (77). All of this emphasizes again how important it is
to know which definition of "mass" is applicable in a given
situation.
3. Mass parameters for the L6 and X6 upper conduction
valleys
r6
Conduction minima higher than the
(CI) one have
been of interest for many years, ever since the free electron
infrared absorption experiments of Spitzer and Whelan 198
and the high temperature Hall data of Aukerman and Willardson 289 at the close of the 19S0s. These experimental resuits indicative of conduction valleys above the first were,
naturally, compared with Callaway'S band model,272 and
were incorporated into Ehrenreich's model 288 of 1960. The
presumption in the latter that minima at or near X(I00)
would be the important second-lowest conduction band, was
remarked in Sec. VI B, as was the revelation by Aspnes et
R163
J. Appl. Phys. Vol. 53, No.1 0, October 1982
al. ISI that many otherwise puzzling and nonconforming
pieces of data fitted into place with a r-L-X conduction band
order instead. The following remarks are admittedly influenced by the intensive analysis that Aspnes 152 made of
data concerning these two sets of upper minima, at L6 and at
X6 (or perhaps a little inside the zone boundary, at ..1 5 ),
In the apparent absence of direct experimental evidence
for the longitudinal and transverse mass components, m{
and m r , associated with the four-ellipsoid band having minima at L (!H), Aspnes made a scaling transformation from the
Ge conduction band to estimate m {~ 1. 9m o. He also used
electroreflectance data l49 concerning interband reduced
mass to infer mr ~0.07S mo. Thus his summary estimate for
this band, including the density-of-states mass for all four
ellipsoids combined was
ml~1.9m()}
mr~0.075mo
mL
L6 Band,
T---+O.
(89)
= (16m{m~)1/3~0.S6mo
The latter of these can be compared with the low temperature density-of-states conduction band mass me = 0.5Smo
for germanium, as derivable34I from cyclotron resonance
measurements 345 of the longitudinal and transverse components for that four-ellipsoid band.
It was Aspnes' expectation that the decreases of Ei and
EL with rising temperature [as modeled by Eqs. (4S) and (SO)]
would be reflected by a modest decrease of m L also: to
-0.5Smo by room temperature, and to -0.52mo by 650 K.
That will be borne in mind quite shortly, in an assessment of
the large effect the L6 minima have on the total statistical
weight of the conduction band system for high temperatures.
Whereas it is natural to compare the L6 conduction
minima of GaAs with the lowest conduction band of its isoelectronic analog germanium, it is a somewhat larger extrapolation to model details of the GaAs minima at or near X6
upon the six-ellipsoid conduction bands of silicon or GaP.
Nevertheless, such arguments, along with interpolations
among the k.p model results for various solids by Pollak et
al.,278 were used by Conwell and VasselJ292 to surmise longitudinal and transverse mass components m{ ~1.3mo and mr
~0.23mo· Hence, a density-of-states mass mx = (9m{m;)1/3
~0.8Smo on the basis of three ellipsoids: i.e., the minima
actually on the zone boundary at X (100).
That density-of-states mass was about one third smaller
than the one Ehrenreich 288 had postulated (with generous
error limits) nearly ten years earlier in modeling high temperature Hall behavior of N-type GaAs. Thus, Gaylord and
Rabson 346 pointed out that a six-ellipsoid band (minima inside the zone boundary, as with Si) having m{ and mr as
noted above would give a density-of-states m x in line with
Ehrenreich's expectation. That line of reasoning was, of
course, based on the premise (which we now known to be
incorrect) that the anomalies of the high temperature Hall
effect in N-type materialZ89.31O.347 were attributable to the X6
rather than to the L6 set of minima.
The three-ellipsoid model was urged by Pitt and
Lees,305 on the basis of their high pressure Hall effect measurements, under conditions which made the X6 the lowest
conduction band. By extrapolation back to zero pressure,
J. S. Blakemore
R163
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they deduced that the set of X6 minima have a density of
states 45 times larger than that of the r6 (Cl) band, for mx
::::::::O.Smo. That appeared to put things back the way Conwell
and Vassell had conjectured. Aspnes 152 seemed to accept the
arguments of Pitt and Lees. In one of his analyses of the
GaAs conduction band system, Aspnes 34S suggested parameters for a three ellipsoid situation:
m l ::::::::1.9m o}
mt~0.19mo
X6 Band,
(90)
mx = (9mlm;)113~O.8Smo
which he surmised would not be particularly sensitive to
temperature.
The next subsection uses that density of states mass m x
= 0.S5mo for the purpose of assessing the impact of the X6
minima on the total statistical weight for the conduction
band system. Since this band has considerably smaller influence than the L6 band for high temperatures, some uncertainty in m x is tolerable. That remark is made in view of the
experimental results of Pinczuk et al. 128 that indicated m,
::::::::O.27m Q , some 50% higher than the value quoted in Eq.
(90). Their result was deduced from Raman measurements
as a function of hydrostatic pressure. As already noted,306
Pinczuk et at. went on to assert that there are six minima (i.e.,
of .:::15 symmetry) rather than three, lying some 10% inside
the zone boundary. They argued for m x ~ l. 2m o, and remarked that the data of Pitt and Lees could be reconciled
with that value.
Despite that interesting and provocative new twist, a
value m x = 0.S5mo will be used in what follows, as has been
used recently in the extensive high temperature transport
study by Nichols et aJ.3 47
4. Conduction electron thermal distribution among the three
lowest bands
Since.:::1 n ofEq. (51) is only a few kT for high temperatures, and m L ::::::::0. 52m o for such temperatures is an order of
magnitude larger than m"" of the zone-center minimum, one
can expect that a major fraction of the thermal conduction
electron population will make the r -+L transition as temperature rises. While m x is even larger than m L (even taking
the conservative estimate of Pitt and Lees, and of Aspnes),
the activation energy .:::1 rx of Eq. (53) represents a more formidable barrier, and one which, moreover, increases with
temperature. Thus one can expect thermal population of the
X6 minima at high temperatures, but to a lesser degree.
The effects of this thermal redistribution of conduction
electrons were seen early on in the study ofGaAs, in the 1955
Hall effect measurements of Folberth and Weiss. 349 These
phenomena were incorporated into Ehrenreich's band model,288 albeit with an X6 attribution. More complete and precise measurements of high temperature transport that are
affected by electron transfer to the L6 and X6 minima have
been reported by Blood,310 and by Nichols et al. 347
The total conduction electron population no should
thus be written as a sum of three thermal contributions
no
=
nx
+ n + nr .
L
(91)
It should be safe to assume that for almost any degree of NR164
J Appl. Phys. Vol. 53. NO.10, October 1982
type doping of GaAs, EF will remain low enough to permit an
essentially Boltzmann distribution in the upper two bands.
Then at any temperature T, the three terms on the right of
Eq. (91) should be:
nx
nL
= 2(21Tm x kT /h 2flZ exp(,., = 2(21TmLkT /h 2)312 exp(,., -
.:::1 rX /kT),}
.:::1 n
/kT),
(92)
nr = Nco [fr Il 2("') - (ISakT /4E;)fr313(1J)]'
The last of the three expressions in Eq. (92) re-expresses Eq.
(80\, and NN is as defined in Eq. (79\.
The ratio nx:n L :n r has to be expressed as a function of
both temperature and of degeneracy, when the N-type doping of GaAs is so strong that 1J==[(EF - Ec)/kT] >0 even
for the higher temperatures at which first n L and then nx
come into play. For a simplified view of the thermal distribution among the bands at high temperatures, let us suppose
that no < Nc ofEq. (Sl). That connotes a more modest N-type
doping, consistent with a Maxwell-Boltzmann distribution
in all three of the bands. When that condition is satisfied,
then nr-+N: exp(1J/kT), where the effective density of
states for the lowest band is N; = n co (1 - lSakT /4E;), as
expressed numerically by Eq. (S5).
In conjunction with that numerical expression, assume
(for simplicity) that m L ::::::::0. 52mo for all high temperatures,
and that mx=0.8Smo as in Eq. (90). Then the three quantities on the right of Eq. (91), add up to no in a way we can
describe as
no=N~exp[(EF-Ec)lkT],
no<N~,
(93)
where the total statistical weight of the conduction band system (for nondegenerate conditions) is
N*
=c 8.63 X lOlJT JI2[{l _ 1.93 X 10 -4T - 4.19 X lO-RT 2 \
+ 21 exp( - .:::1 n /kT) + 44 exp( - .:::1 rx lkT)] cm- 3.
(94)
The magnitude of N ~ for temperatures from ambient
upwards has been used in Fig. 52, in plotting (N: IT 312) vs T
4
....N
'?
::.::
..,
3
~
u
!
0
2
N
....
....
r-
......
>leU
I
Z
oL-__
300
~
____
500
~
____L -__
700
900
TEMPERATURE
~
____-L__
1100
~
1300
1500
T (K 1
FIG. 52. Temperature dependence of IN~ IT3/2), where N ~ i~ (for nondegenerate conditions) the total statistical weight of the three lowest conduction bands combined, as expressed by Eq. (94).
J. S. Blakemore
R164
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1.0
0.8
0.6
0.4
0
c
...J
<t
0.2
t-
O
t-
LL
0
z
0.1
0.08
0 0.06
t-
u
<t
0.04
(nx/no)
0::
LL
0.02
500
700
900
TEMPERATURE
II 00
1300
1500
T (K)
FIG. 53. Temperature dependence of the three-way split of conduction electronsamong the T6 (el), L b , andXb bands. It is assumed here, as in Eqs. (93)
and (94), that no < N;, so that conditions are nondegenerate in the lowest of
these three bands, as well as in the higher ones.
up to the melting point. This curve may be compared with
that for (N ;/T3/2) in Fig. 50. The two quantities are the
same (to within 0.05%) for room temperature, but the influence particularly of the L6 band starts to pull (N ~ /T 3/2)
noticeably upwards above 500 K.
Figure 53 displays the fractional contributions of
nr,n L , and nx towards the total no for the same temperature
range, and with the same supposition of non degenerate conditions even in the r 6 band. This figure shows that more than
half of all conduction electrons are thermally in the L6 band
for 1> 900 K, and that the proportion of electrons remaining
in the r 6 (C 1) conduction band has declined to less than 12%
by the melting point.
The phenomena illustrated by Figs. 52 and 53-the influences of the upper conduction bands on the total statistical weight of the conduction band system--obviously affect
the properties of N-doped GaAs at high temperatures. They
are also on a large enough scale to affect the high temperature intrinsic pair concentration 122 and Fermi level. This is
briefly reviewed in Sec. VI G. First, however, the density of
states presented by the valence bands is considered.
F. Hole effective masses: Valence band system
statistical weight
I. Masses for the heavy-hole (VI) and light-hole (V2) bands
Many observable properties of holes in GaAs can be
interpreted fairly well in terms of scalar (spherical equivalent) effective masses m h and m/ for bands VI and V2, respectively. (This despite the various complications of the E-k
relations for these two bands, as recounted in Sec. VI D.) A
density-of-states hole mass,
(95)
R165
J. Appl. Phys. Vol. 53, NO.1 O. October 1982
is then what one would like to be able to use in describing the
statistical weight for the valence band system. That is the
specific topic of the next subsection, and requires additional
considerations.
The quantities that contribute towards mu have been
measured in many ways, from the 1950s onwards. As a forerunner of the intra-valence-band absorption experiments 197
that were illustrated in Fig. 28, some early measurements of
Braunstein350 indicated that m h =.5.7m/. Among the subsequent measurements having a bearing on m/ and/or m h , we
may note:
(i) Interband magnetoabsorption measurements by Vrehen, 165 at 2,77, and 300 K. Vrehen concluded that his results
were consistent with isotropy of both the light-hole and
heavy-hole bands, with masses (m//mo) = 0.082 ± 0.006
and (mh/mO) = 0.45 ± 0.05 for both the low temperatures,
and slightly smaller values for room temperature. These values have been adopted for many analyses of P-type GaAs.
(ii) Piezoabsorption over the 4-100 K range by Balslev. 136 This work indicated a strongly anisotropic heavy-hole
band, with the component of the m h tensor some 50% larger
along [111] than along [100).
(iii) A much more modest heavy-hole anisotropy (about
10%), as indicated by the magneto-optical measurements of
Seisyan et aI.322
(iv) Directionally averaged values for m/ and m h as provided from the Faraday rotation experiments of Walton and
Mishra,35I and from the oscillatory photoconductivity work
ofShaw. 352
(v) Components of the two masses as functions of orientation, obtained by Eaves et al. 323 from magnetophonon resonance experiments.
(vi) Mass components versus direction obtained in the
cyclotron resonance work of Stradling and coworkers. 319.320
The latter work described two corrections that were
made to mass components, in order to obtain their "bare"
values for the band edge. Thus, a nonparabolicity correction
was made, in order to express the curvature for the band
~dge itself. A polaron correction,53 was additionally made,
In order to subtract the effect of optical phonons on the apparent masses for this partly ionic solid.
Skolnick et al.320 concluded that the light holes are essentially isotropic, but not the heavy holes. Their estimate of
the m h tensor was that the component along [111] is 25%
larger than along [100]. These directionally differentiated
effective mass data can be converted, for example by the
computational method of Lax and Mavroides, 3~4 into a scalar density-of-states value for each band.
As a consensus of the various experiments, but weighted by the results of Skolnick et al.,320 low temperature values,
(m//mo) = 0.082
+ 0.004}
(mhlmo) = 0.51 ;0.02
(T < 100 K),
(96)
are suggested here, as listed in Table XII. It can be noted that
the light-hole mass is exactly as reported both by Vrehenl6~
and by Skolnick et al. 320 The heavy-hole mass of Eq. (96) is
13% larger than the (isotropic) value Vrehen had elucidated,
but has error limits that overlap with those he quoted. More
J. S. Blakemore
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to the point, m h ofEq. (96) is 13% larger than Skolnick et al.
reported for the [100] direction band-edge "bare" mass, and
11 % smaller than their [111] direction mass component. For
the two values in Eq. (96), m h ~6.2 m{, not very far from the
ratio Braunstein 350 deduced in his early work.
The two numbers of Eq. (96) give a density-of-states
mass
m" = (mhl2
+ mV2f/3 =
0.53mo
(T < 100 K)
(97)
for the V 1-V2 combination of bands at low temperatures.
Now, m" is a quantity of interest in connection with the
effective density of valence band states (Nv of the next subsection). In order to cope with N v ' however, one needs to deal
with m h and m{ (the latter particularly) as functions oftemperature, and also with the nonparabolicity of the latter.
The value of m h must certainly depend to some extent
on temperature, but this can be expected to be much less
drastic than the temperature sensitivity of the conduction
band meo [modeled in the kp approximation by Eq. (63)], or
of the light-hole band m{ [described in that same model by
Eq. (68)]. Auvergne et al. 355 described results of piezoreflectance measurements that supported their expectation from
theory that a solid such as GaAs should have a valence band
structure which is rigidly preserved on heating. Such an expectation, carried to extreme, would require that mv not depend at all on temperature. However, Auvergne et al. were
considering temperature independence on the gross eigenvalue scale of the valence band system as a whole. That does
not automatically preclude changes of (d 2Eldk 2) with temperature right around the energy E v that is of especial importance for thermal holes. And from an experimental standpoint, Vrehen's data 165 did point towards a slight decrease of
both m h and m 1 on warming to room temperature.
Accordingly, Table XII does show a 300 K value
m h = 0.50m o, the reduction from the low temperature value
restrained to a token 2% in recognition of the arguments of
Auvergne et al. 355 However, m 1 decreases to a much more
drastic degree, if Eq. (68) of the k.p model is a reasonable
guide. The low temperature value m l = O.082mo requires XI
= 10.0 eV, as also listed in Table XII; and the operation of
Eq. (68) then mandates that m{ decreases as the temperature
rises and the intrinsic gap narrows.
The final column of Table XII includes the 300 K value
m{ = 0.076mo required by Eq. (68). Further application of
tht equation to higher temperatures would yield a curve for
(m{/mo) vs T looking generally similar to the (meo /mo) curve
of Fig. 49. Equation (68) thus predicts that the band-edge
value of m{ declines to only 0.056 mo by 1000 K. This does
not mean, however, that light holes dwindle to an insignificant minority at high temperatures.
2. Effective density of states for the VI- V2 combination
As a companion to Eq. (81), one would like to express
the relation between free hole density Po and Fermi energy E F
in a form
(98)
where 5 == [(Ev - EF)lkT] is the dimensionless expression
of EF relative to E,,, and N v is the effective density of states
R166
J. Appl. Phys. Vol. 53. No. 10, October 1982
provided by the heavy-hole (VI) and light-hole (V2) bands
combined. How does the quantity N v have to be represented?
If either or both of V 1 and V2 were anisotropic, yet both
could be adequately approximated by parabolic E-k relations
(i.e., by energy-independent effective masses), then the quantity m" of Eg. (95) would produce N v = 2(2rrm"kT /h 2)3/2.
Yet this does not suffice in practice, because of the appreciable nonparabolicity of the light-hole band.
Because of the absence of inversion symmetry, it will be
clear that E-k is not parabolic for the heavy-hole band,
viewed on a scale of things quite close to the extremum.
However, it appears likely that m h as quoted in Eq. (96) for
low temperatures, and as speculatively extrapolated to 300
K in Table XII, might serve to characterize the directionally
averaged curvature over the first 100 meV or so of that band.
That is manifestly not so for light holes, in view of Eqs.
(67)-(70) as a portrayal ofnonparabolicity that enhances the
statistical weight of this band, to an extent that depends on
both temperature and degeneracy. The same kinds of argument that led to Eq. (80) for the conduction band now lead to
Nu
=
2(2rrkT /h
X \ m~/2
2f/2
+ mt /2 [ 1 - (15{JkT /4E i HY3/2(s )/15In\t) 1J.
(99)
l2
We already know that mi decreases as temperature rises.
However, the factor [..... ] multiplying mi!2 increases with
temperature, since Eq. (69) shows that (J < O. The contribution of that factor [....1also depends on whether the free hole
density Po is larger or smaller than N v , since 153/2(5 )> 15 1 nls )
for a degenerate P-type situation of Po> N v ,5 > O.
Situations of strong P-type doping, and attendent degeneracy of the free hole population, certainly do occur in
GaAs. However, since N v (300) _10 19 cm -3 (about 25 times
degeneracy at room temperature and above
larger than N
is less commonly a complication for P-type GaAs than for Ntype materiaL
Accordingly, we examine here the consequences of Ptype doping which are weak enough to avoid degeneracy. In
that event,
n
Po=N~
exp[(Ev -EF)lkT], Po<N~,
(100)
where the effective density of valence states for these nondegenerate conditions is
N~ =2(21TkT/h2)3/2[m~l2+mt!2(1-l5{JkT/4Ei)]
=
2(21T'kT /h 2f12[ml12 + (m;f /2 1.
(101)
The quantity defined therein as m; thus incorporates both
the temperature dependence of m{ itself, and of the
(1 - 15{JkT /4£;) factor. By a happy chance, these opposing
tendencies cancel each other over a quite large temperature
range, and for the purpose of describing the statistical weight
of the light-hole band, one may use
(102)
If it is supposed that the density-of-states value m h
=O.05mo is valid for a major part of that temperature range,
then light holes constitute about 7% of the total hole population shared between these two bands, for room temperature
and above. A combination of m h = 0.05mo and m;
= 0.088mo makes the effective density of valence states,
J. S. Blakemore
R166
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N~~1.83X
10 15 T3/2
cm- 3 ,
to an adequate accuracy for virtually any temperature. Thus
while Eq. (103) is designed specifically to model the situation
for 1>200 K, it falls only some 2% below No ofEq. (99) for
the lowest temperaures.
3. The splitoff valence band
As will be seen below, thermal hole occupancy of the
splitoff (V3) valence band is negligibly small all the way up to
the melting point. However, optical transitions to this band
show up in a wide variety of experimental results. A zonecenter spin-orbit splitting Llso~0.33 eV was deduced by
Braunstein350 from the infrared limit of intravalence band
transitions. In subsequent work, the electroreflectance analysis of Aspnes and Studna 149 and the reflectance measurements of Sell et al. 299 indicated a low temperature splitting:
Llso = 0.341
± 0.002 eV
(T""'O).
(104)
How will Llso respond to rising temperature? No change
is predicted by the "rigid valence band" view of Auvergne et
al. 355 As experimental confirmation, Nishino et al. 340 saw no
change in Llso over the range 25-300 K. As with other properties of GaAs, data for T> 300 K are apparently unavailable.
A low temperature mass mso = 0.154m o was deduced
by Reine et al. 166 from stress modulated magneto reflectance.
That value was given in Table XII, as also was the momentum matrix element Xso = 13.9 eV thereby required by Eqs.
(60) and (72). Lawaetz 330 also concluded that mso ~0.15mo
in GaAs, from a five level k.p model applied to solids with
the diamond and sphalerite structures. It was remarked in
Sec. VI D 1 that any underestimate of mso in the Reine et al.
work (because of the strong nonparabolicity) would have
contributed to the large size ofXso compared withXe and Xl'
However, there does not appear to be any other experimental
evidence concerning mso that might validate such a conjecture.
In any event, the statistical weight of the splitoff band
does not increase as fast as T 312 , for two reasons: (a) The
decrease of mso as the band-edge curvature mass, as T rises.
(b) As a consequence of a nonparabolicity parameter y> 0
for the initial stages of departure from this band extremum.
In accordance with Eqs. (71) and (73), this makes the effective mass in the band decrease with increasing wave vector.
As previously noted in Sec. VI D 1, the initial effects of
that positive nonparabolicity parameter yare reversed by a
mass enlargement further from the extremum. Thus it would
be unrealistic to attempt any representation of the statistical
weight of the splitoff band in terms of a quantity
m:~2(1 - 15ykT /4.:;). For the numerical value of y [from
Eq. (74)] is large enough to force (1 - 15ykT /4.: i ) negative
for all T> 500 K, a clearly nonphysical result so far as this
band's statistical weight is concerned.
Even with total neglect of the nonparabolicity effects,
the ability of the splitoff band to attract holes in thermal
equilibrium is miserably small, even up to the GaAs melting
point. With the factor (1 - 15ykT /4.:i ) laid aside as unnecessary overkill, then the ratio of holes in the splitoff band
(V3) to those in the VI-V2 combination is
R167
J. Appl. Phys. Vol. 53, No. 10, October 1982
_=-P:::so_"", m:~2 exp( - Llso/kT)
Po - Pso
mhl2 + (m;)3/2
(103)
(105)
Thus with m h and m; as noted in connection with Eq. (103),
Llso = 0.341 eV, and mso in accordance with Eq. (72),
(Pso/Po)~0.0004 for a temperature of 1000 K, and barely
reaches 0.5% by 1500 K. For ordinary circumstances, the
impact of the splitoff band on the statistical weight of the
valence band system can be ignored. Thus the interesting
propensity of the splitoffband is for provision of electrons to
higher lying bands (and impurity states356 ) under optical
stimulation.
G. Intrinsic conditions in GaAs
This specific aspect of GaAs has recently been reviewed
by the present writer,122 and the comments here are brief.
With the splitoff valence band dismissed, the statistical
weight of the valence bands for intrinsic gallium arsenide is
fully represented by N ~ ofEq. (103). Similarly, the statistical
weight of the conduction band system is fully represented by
ofEq. (94), which in turn involves Eqs. (51) and (53) for
Ll rL and Ll rx'
All of these contribute towards the intrinsic carrier pair
density n i through the form
N:
n;(T) = (N:N~)1/2 exp( - .:J2kT).
(106)
Here, .:; (T) can be modeled by Eq. (45). The result for n; (T) is
displayed in Table XIV, for a series of temperatures from
250 to 1500 K. (The latter only a few degrees short of the
melting point.) As a homily, it must be remarked that these
values tend to be a few percent larger than those suggested by
this writer elsewhere, 122 the consequence of a reevaluation of
the heavy-hole mass. One may expect such minor readjustments to continue, as': i (T) and the various mass components
are pinned down with even higher precision.
Included also in Table XIV are values for the intrinsic
Fermi energy if;. Of possible interest to various readers are
the locations of this quantity with respect to the valence
band, the conduction band, and the actual midgap energy.
Thus.
if; =
':0
+ kTIn(N ~/ni)
=':e =':v
kTIn(N~/n;)
+ !.:; + !kTIn(N~/N~).
(107)
The last of these three definitions of if; is the most interesting:
the displacement from a midgap location as a consequence of
the imbalance of N~ and N~. As can be observed in the
entries of Table XIV, the elevation of if; above the midpoint
of the gap increases proportionately with temperature at
first: a consequence of the ratio (mh/mco)' That trend is arrested and finally reversed as temperature continues to rise,
as a result of the massive effect theL6 conduction band (with
minor assistance from the X6 conduction band) has in accelerating the increase of N~ for high temperatures. Less dramatically, both of the quantities (if; - ':v) and (':e - if;) decrease with increasing temperature, as natural consequences
of the decrease in total intrinsic gap width.
J. S. Blakemore
R167
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TABLE XIV. Intrinsic carrier pair density nj • and consequent intrinsic Fermi energy tf;. for gallium arsenide"
T
(K)
250
300
350
400
450
500
600
700
800
900
1050
1200
1350
1500
N~
to,
(meV)
.d n
(meV)
.d rx
(meV)
(cm- 3 )
N;
(cm- 3 )
(cm- 3 )
(tf;-E,)
(meV)
(tOe - tf;)
(meV)
(tf; - IOu - !to,)
(meV)
1446.7
1422.5
1399.5
1375.8
1351.6
1327.1
1277.0
1226.0
1174.5
1122.2
1043.8
964.6
885.1
805.3
287.1
284.5
281.7
278.9
276.0
273.1
267.1
261.0
254.9
248.7
239.3
229.8
220.4
210.8
473.1
476.1
479.8
483.3
486.9
490.6
498.0
505.6
513.3
521.1
532.8
544.6
556.4
568.3
3.238 E17
4.209 E17
5.251 E17
6.369 EI7
7.594 EI7
8.979 EI7
1.258 E18
1.809 E18
2.665 E18
3.951 El8
6.943 El8
1.152E19
1.793 E19
2.642 EI9
7.234 E18
9.509 E18
1.198 EI9
1.464EI9
1.747 E19
2.046 EI9
2.690 E19
3.389 E19
4.141 EI9
4.941 E19
6.226 E19
7.607 El9
9.077 EI9
1.063 E20
4.20 E3
2.25 E6
2.10E8
6.57 E9
9.83 EIO
8.78 Ell
2.52 El3
3.02 El4
2.IOE15
1.01 E16
6.50 E16
2.79 El7
8.99 E17
2.35 E18
756
752
747
742
737
731
718
701
682
659
621
580
537
493
689
671
653
634
615
596
559
525
493
463
423
385
348
313
34
40
47
54
61
67
79
88
95
98
99
98
94
90
"Using 10, from Eq (45);.d n ofEq. (51) • .d rx ofEq. (53) contributing towards
n,
N~
ofEq. (94); and N: ofEq. (103).
(110)
VII. ELECTRON AND HOLE TRANSPORT
A great deal has been written and published concerning
transport in GaAs and other 111-V compounds. From the
first work of Welker and his associates,86 the accounts have
included the books by Hilsum and Rose-Innes 2 and Madelung 3 ; and review papers such as those of Hilsum, 8 Stillman
et al., 168 Rode, 169 and Wiley. 357 Accordingly, this section is
kept brief, with the intention of pinpointing a few aspects of
carrier transport that the experimentalist commonly uses in
the analysis of raw data.
A. Hole mobility
The mobility of holes in GaAs was studied as part of the
Siemens laboratories work of the 1950's.86 Rosi et al. 358 reviewed data accumulated to the end of that decade. The 1975
review by Wiley 357 deals with hole transport in III-V compounds in considerable detail, including citations of a dozen
or more P-type GaAs transport papers that appeared subsequent to Rosi et al.'s review. Thus Wiley's account supercedes quite thoroughly the exposition of hole transport in
Madelung's 1964 book. 3 Two figures from Wiley's review
are used in this subsection.
It will be supposed here, based on the discussion in Sec.
VI F, that heavy holes in the VI valence band have an effective mass mh = 0.50mo, averaged over all directions, and for
temperatures around and above ambient. Thus the rms thermal speed of these heavy holes is
Vh (rms)
= (3kT Imh )1/2 =
1.65 X 107(T 1300)1/2
cm/sec.
(108)
For holes in the noticeably nonparabolic light-hole band, the
speed corresponding to three classical degrees of freedom
(KE = 3kT /2) is
v/(rms) = (3kT Im/)1/2[ 1 + (3/3kT IE;)] .
(109)
The factor (....] in Eq. (109) lowers the rms speed of light
holes, since/3ofEqs. (69) and (70) is a negative quantity. As a
result, v/(rms) does not conform particularly well to a TI/2
temperature dependence, though
R16S
J. Appl. Phys. Vol. 53. No.1 O. October 1982
should be serviceable enough close to the ambient range.
In view of the remarks made in connection with Eqs.
(101) and (102), one can thus think of mobile holes in (nondegenerate) P-type GaAs as being 93.1 % heavy holes with the
rms speed of Eq. (108), and 6.9% light holes with an rms
speed approximated by Eq. (110). That is equivalent to a
speed
(vp)(rms) = 1.77 X 107(T 1300)1/2
cmlsec
(Ill)
for the light and heavy holes combined.
Even for the heavy-hole component, the thermal speed
is apt to exceed the various speeds of sound (Table IV) by
more than a factor of ten, for temperatures from the liquid
nitrogen range upwards. Light holes can easily have a thermal speed 100 times faster than any of the speeds of sound.
Data reported in the literature for the mobility of holes
in P-type GaAs usually concerns the Hall mobility; derived
from a combination of measurements, of conductivity
a p = e/-l p Po, and of Hall coefficient RH = rHlepo' This combination of measurements yields the Hall mobility
/-lH
= aR H =
rH/-l p
(112)
.
Since for holes in P-type GaAs we have a two band
(light holes, heavy holes) situation, with band warping, etc.,
it is not easy to say a priori when the Hall factor
r H = (/-lHI/-l p ) might be larger than, or smaller than, unity.
The same problem has beset the analysis of hole transport in
the Group IV semiconducting elements. 359 Thus for hole
transport in GaAs, as in Si and Ge, r H is a complicated function of magnetic induction strength, temperature, doping
(and doping compensation). The latter pair enter the problem through the ratio oflattice scattering to ionized impurity
scattering.
As one relevant experimental contribution towards this
complicated situation, Mears and Stradling 319 made Hall coefficient measurements for a "high-purity" P-type GaAs
sample. This rather interesting sample seemed to have two
acceptor species present, adding up toNa _10 15 em -.3; and a
compensating donor density Nd -5 X 10 14 cm- 3 . Even that
J. S. Blakemore
R168
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4
g::::-o ___
1.0
"0....
0.9
...........
"0 ......
" ' - 293K
'Q
0
0.8
a>
"<>,
"
',.....
:I:
"'-
000
>
......-""'0
.......
0
---0
'o!.87 K
a:
"-
'lJ
~~
.'
0.7
al
~o
.
a:
0.6
b
0
".
4
J:
0.5
0
:l..
°
0
04
1
2
5
10
20
~
50
(Il
o
~
FIG. 54. Magnetic induction dependence of the Hall coefficients RH(B),
normalized by the weak-field limiting value RHo for each temperature, as
measured for a "high-purity" P-type GaAs sample by Mears and Stradling.·lIO
combination should have made impurity scattering rather
weak compared with the vast majority of P-doped GaAs
samples! Hall measurements were made over the temperature range 50<T<293 K, and the magnetic induction range
I <B<40 kG. Figure 54 shows their results for the field dependence of the ratio RH(B )/RHo for six temperatures.
Here, RHo signifies the Hall coefficient extrapolated to the
weak-field extreme.
Magnetotransport theory359 indicates that rH~1.00
for B~ 00, for any superposition of bands. Thus the uppermost curve in Fig. 54 indicates that rH ~ 1.25 for zero field at
room temperature. The various lower curves in Fig. 54 indicate that the weak-field value of rH is even larger for the
lower temperatures, in weakly doped P-type GaAs. Incidentally, the two solid curves in Fig. 54 give the result of calculations by Mears and Stradling319 for a simplified model of two
isotropic bands, with scattering times (7h) = 1.5 (7/ ). Thus
it appears that this model provides for a 50 K result of
rH~2.0 at zero field.
All of this has a non-negligible impact when the Hall
mobility f-l H = oR H is experimentally deduced and plotted
as a function of temperature. Figure 55 shows three examples, all of purportedly "high-purity" P-type GaAs. One is
the Mears and Stradling sample. 319 Another, for which f-lp
continues to move gloriously upwards on cooling to 20 K, is
from the work of Zschauer36o ; and a third sample from the
work of Hill 361 appears to be rather more affected by impurity scattering for the lower temperatures.
Just what magnetic field strength has been used to make
Hall measurements is often glossed over in reporting curves
of mobility versus temperature; however, one may reasonably suppose that magnetic inductions in the range 1-5 kG
were employed for the data of Fig. 55. And so the conditions
could well have been approaching the high field end of the
range for the lowest temperatures, but would have been close
to the weak-field RHo situation for the highest temperatures.
Wiley357 suggests that a consensus among the higher temperature portions of these data might be
(113)
J. Appl. Phys. Vol. 53, No.1 0, October 1982
2
.0
--1
B (KILOGAUSS)
R169
-
0;'0
J:
:I:
a:
-
HIGH-PURITY
P-TYPE GoAs
CD o
....... ......
"<l...,
I
.0
.
•
3
10 -
--1
--1
•ra
<[
I
-
.0
400
200 10
•
I
20
50
100
TEMPERATURE
200
500
(K)
FIG. 55. Temperature dependence of Hall mobility for three high-purity Ptype GaAs VPE samples, after Wiley. m These samples re~::s.en~ the work
of. Hill,'61 0 Mears and Stradling,Jl9 and 0 Zschauer, In increasing
order of low temperature mobility.
That probably means a room temperature conductivity mobility (a/epa) = f-lp (300) c::::: 320 cm 2/V s, in view of the apparent (RHo/RHoo) = 1.25 for temperatures near the ambient
range.
Wiley's review 357 goes into some detail on scattering
mechanisms for holes. He points out that acoustic phonon
and nonpolar optical mode scattering processes are of comparable importance for T> 100 K, with polar mode scattering probably less important. Ionized impurity scattering almost inevitably dominates for low temperatures, as seen to a
small extent in the roll off of mobility below 40 K for Hill's
sample in Fig. 55.
Ionized impurity scattering takes a much greater toll of
f-lp for strongly doped material, especially if also heavily
compensated. This scattering mechanism can assume control of f-l at room temperature when the doping is strong
enough, ~s exemplified by the curve and data points of Fig.
56. This figure, taken from Wiley's review, shows the weakfield 300 K Hall mobility f-lHO as a function of the apparent
362
PO·
The strength of ionized impurity scattering is commonly handled by an expression of the type
f-l[c:::::(A /N[)(T /300)3/2
(114)
as provided by Conwell-Weisskopf or Brooks-HerringDingle treatments of this scattering. 363 Here, N[ is the concentration of immobile charged scattering centers. One may
expect that N[ ""Po for weakly compensated material. Substantial compensation results in N[ > Po, and the mobility is
then smaller for a given T and Po.
The data of Fig. 56 came from experiments of Rosi et
al.,358 Hill,228.361 Vilms and Garrett,364 Rosztoczy et al. 365
J. S. Blakemore
R169
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-;;;
>
B. Electron velocity and mobility
500
"-
At first sight, a discussion of the motion of, and transport by, conduction electrons ought to be simpler than that
of valence band holes. For there is, nominally, a single reasonably isotropic minimum at the zone center, albeit with
some nonparabolicity. The rms speed of these conduction
electrons for a ~kT kinetic energy is
N
::!:
.s
T = 300 K
400
+
r
b
0
~p~
a:::
{;
300
•
>-
0
"0"
::1..
~
.
• C>
r
200
"
a
0
-l
iii
0
vc(rms)
.
100
,
::!:
-l
-l
«
:r
10 15
101 6
10
17
10 18
FREE HOLE DENSITY Po
0
10
0
0
a
19
10
20
(CM- 3 )
FIG. 56. Variation of 300 K Hall mobility with hole concentration, after
Wiley.357 The data points represent the work of 6, Rosi et al., m 0 Hill,22K.
Hill, '" I X Vilms and Garrett,'64 + Rosztoczy et al., '.50 Emel'yanenko et
al.,'66 and 'i1 Gasanli etal.'"?
Emel'yanenko et al.,366 and Gasanli et al. 367 Wiley commented that the curve in this figure was a calculated one,
using a combination of lattice scattering (set for f1L = 400
cm 2/V s), and Brooks-Herring-type ionized impurity scattering.
Now the crudest and simplest way to combine scattering contributions would be by an additive approach:
f1HO = [(lIf1L)
+ (1If11)] -\ ,
(115)
withf11 as given by Eq. (114), using an appropriate value for
A. However, no single value for A can generate the entire
curve in Fig. 56 with much fidelity, though a compromise
value A =2.5 X 1020 cm -\ V- \ S - \ does a tolerable job.
Thus for temperatures/aMy close to room temperature,
it might be reasonable to expect that
f1HO= [2.5 X 1O- 3(T 1300)23 + 4x 1O- 2I N / (300IT)LS] -I
cm 2 /V s
(116)
could serve as a reasonable expectation for the Hall mobility
measured under weak field conditions.
In order to do much better, it would be necessary for a
start to replace Eq. (114) with a more complete expression
including a screening adjustment term, as provided in different ways by the Conwell-Weisskopf and Brooks-Herring
formulations of impurity scattering. One should then also
generalize Eq. (115) to an integral solution of the Boltzmann
transport equation for the combination of scattering probabilities, performed for the light-hole and heavy-hole bands,
and considering intraband scattering, and so on.
Before getting too enthusiastic over these possibilities,
it should be cautioned that the value A=2.5 X 1020
cm -\ V - I S - I (which does make f1 H start to drop rapidly
for about the right range of Po in Fig. 56) seems to be about
three or four times too large for the usual expressions363 of
Conwell-Weisskopf or Brooks-Herring scattering by heavy
holes of mh ~0.5mo. Thus our ability to describe scattering
of holes by this combination of processes does not seem to
extend beyond an empirical level.
R170
(117)
ID
......
0
10 14
.
= (3kT Imco)I12[ 1 + 3akT lEi]
=4.4 X 107 (T /300)112 cmlsec,
J. Appl. Phys. Vol. 53, No.1 0, October 1982
the latter numerical approximation being suitable for temperatures fairly close to the ambient range. Electrons in that
small mass band minimum can enjoy a fairly high mobility,
and therein lies much of the attraction of GaAs for device
purposes.
However, the nonthermal redistribution of conduction
electrons in a substantial electric field is an important subject
that requires at least some mention here. As predicted by
Ridley and Watkins,290 and by Hilsum,291 field-heated electrons can transfer from the central valley to the large mass,
low mobility upper conduction valleys (ofGaAs, and of other binary, ternary, and quaternary solids of comparable band
structure). The consequences as first seen in GaAs, and commonly known as the Gunn effect,368 were at first (quite naturally) interpreted in terms of transitions to the X6 minima as
the first available band. 292,293 It was not surprising that such
calculations agreed qualitatively-but not fully quantitatively-with experimentally measured drift velocity versus
field data. 369 Since the promulgation of r-L-X conduction
band order, lSI Monte Carlo simulations of the velocity:field
characteristic to be thus expected have followed in short order.37°-372
A current pulse in N-type GaAs propagates at a speed
usually called the electron drift velocity vd • That name combines several physical phenomena, depending on the range of
the applied field E. For a small field, Vd = f1nE, where f1n
denotes the drift or conductivity mobility, f1n = (a/eno)' As
discussed in more detail later in this subsection,
f1n (300)=8000 cm 2IV s in lightly doped N-type GaAs. The
low field region at the left of Fig. 57 shows Vd rising with E at
that slope. As the field increases, and electrons in the reo
minimum start to warm up, the slope decreases-as it does
for any situation of lattice-scattering limited mobility in a
semiconductor.
However, the historically significant difference with
electrons in GaAs is the retrograde behavior of vd beyond a
threshold [E'h ,vd(max)] at which electron transfer to lower
mobility conduction minima becomes likely. The solid curve
in Fig. 57 shows Vd vs E as calculated by Pozela and Reklaitis 372 on an assumption of r-L-Xband ordering (so that electron transfer to the X6 minima would come into play only
well above the field range of this figure), and the figure shows
a comparison with four experimental investigations. 369,373-375
What are the values of E'h and vd(max), and how do
they vary with temperature? These are obviously important
quantities, in terms of the physics of GaAs, and as affecting
the design of transferred electron devices. The published liJ. S. Blakemore
R170
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is arranged to provide the room temperature value
7
Vd (max)=2 X 10 cm/sec shown within ± 10% limits by all
the curves in Fig. 57. A comparison with Eq. (117) shows
that this "ensemble peak drift velocity" is about one half of
the rms thermal speed for 300 K conduction electrons.
The retrograde behavior of Vd (E) for fields much larger
than E th has attracted several investigations.375.380-382 The
largest field range covered was in the room temperature microwave time of flight measurement of Smith et al.,3H2 whose
result is shown in Fig. 58. Figure 58 also shows, as a dashed
curve, the 300 K data of Houston and Evans,375 extending to
some 105 kV/cm.
Houston and Evans made measurements at various
temperatures in the range 130-400 K, and from - 20 k V/ cm
to various high-field limits from 65 to 110 kV/cm. Selecting
50 kV/ cm as a field strength for which all these temperatures
were represented, the trend of Vd with temperature can be
approximated by
2.0
(f)
"- 1.5
~
u
tO
---
1.0
-------.------.-------.
'0
>
0.5
Calculated
Ruch and Kino
Braslau and Houge (bulk)
Braslau and Houge (epitaxial)
Ashida et 01
Houston and Evans
2
4
FIELD E
6
(kV/CM)
10
8
FIG. 57. Drift velocity of conduction electrons in GaAs at 300 K, for the 010 k V/ cm range, after Pozela and Reklaitis.372 The solid curve was generated by Pozela and Reklaitis in a Monte Carlo calculation, and compared with
experimental data of Ruch and Kino,369 Braslau and Hauge,373 Ashida et
al,374 and Houston and Evans. 375
vd =(1.28-0.0015T)X107
(E=50 kV/cm).
(120)
6
That equation was arranged to provide Vd =8.3 X 10 cm/
sec at 300 K for E = 50 k V/ cm, some 40% of vd (max), Figure
58 shows a monotonic decline of the ensemble Vd throughout
the range 20<E <200 kV/cm. The Monte Carlo calculations of Shichijo and Hess 383 imply that vd(300)=6X 106
cm/sec throughout the ensuing field range to 500 kV/cm.
Electric fields that large are important for avalanche
breakdown, as encountered in a reverse-biased P-N junction,
and as so utilized in avalanche photodiodes 384 and in impactavalanche transit time (IMPATT) diodes. 385 Since the ionization coefficient of electrons in GaAs for avalanche multiplication is considerably smaller than that of holes for a
given field strength,386 the mean free path of energetic electrons is able to remain tolerably large even in a very high
field. The two parts of Fig. 59 show calculated estimates by
Shichijo and Hess 383 of how the mean energy (€ - Ec ) and the
mean free path An might vary with field with one set of starting assumptions,
The electron mobility in GaAs for a small electric field
has been discussed by many writers, usually in terms of the
Hall mobility,
terature does not give an unqualified answer. Thus deductions of E th (T) from the behavior ofGunn diodes 376,377 have
suggested that this threshold field decreases slightly on cooling, contrary to the trend
Eth"-'(4.7-T/215)
cm/sec
kV/cm
(118)
369
shown by the time-of-flight data ofRuch and Kino. Equation (118) gives the value E th (300) = 3,3 k V/ cm shown for
Ruch and Kino's curve in Fig. 57, and these authors 369 comment on the errors that can occur in measurement of E th if
there is any doping inhomogeneity in the sample.
That sensitivity of a measured threshold condition to
any inhomogeneity of doping or of geometry-induced field
distribution has also resulted in considerable scatter among
the various experimental investigations ofthe peak electron
drift velocity vd(max) with temperature. 269,377-379 Thus the
Ruch and Kino data conform approximately with
v (max)=(3.3 - O.OO4T) X 107 cm/sec
(119)
d
while some other types ofmeasuremene77.378 have indicated
a temperature coefficient up to twice as large. Equation (119)
1.0
(f)
"-
~
0.8
u
t-
0
GoA. 300 K
0.4
"0
>
0.2
a
#1
o
#2 LPE I 2.13/-,m
o. 0
LPE I 1.20/-,m
•
#3 LPE 2 3 16/-,m
A
#4 VPE
-.-
4.95/-,m
HOUSTON
a
EVANS
~--:'::--:-:'-~_J----'L----L_~_..l....-_L---J._--.J
o
20
40
60
80
100
FIELD E
R171
FIG. 58. Electron drift velocity Vd as a function
of field for (1 (0) N-type GaAs at 300 K, in the
higher field regime of strong electron transfer.
After Smith et al.,382 showing data for four of
their samples. The dashed curve reproduces the
result of Houston and Evans. 375
0.6
120
140
160
180
200 220
(kV/CM)
J. Appl. Phys. Vol. 53, NO.1 0, October 1982
J. S. Blakemore
R171
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1.0
;;-
10
0.8
~
6
~
0.6
<J)
>
.......
<)
'",
0'1
(\J
I",
0.2
:2:
U
5
10
FIELD E
50 100
500
(kV/CM)
10 5
I
::t.
.;;x
c:
>r--
( b)
300
-...J
OJ
-<
I
~
«
0-
200
0
100
c::(
z
10
:2:
W
W
...J
...J
0::
"-
>-
4
Z
I
«
W
:::;;
0
I
5
10
50
100
10 3
500
2
4
10
FIELD E (kV/CM)
20 40
100 200400 1000
TEMPERATURE
FIG. 59. Curves from the Monte Carlo simuation of conduction electrons in
GaAs at 300 K. by Shichijo and Hess.'"' (a) Average electron energy vs
electric field. (b) Electron mean free path An vs field above the Gunn effect
threshold.
(121)
rather than as a conductivity or drift mobility J1n' This
means that one would like to know the strength of the magnetic induction used. For, the Hall factor rH = 1 for
"strong-field" conditions (BJ1n >1 in the SI system of teslas
and m 2 IV s, BJ1n >108 for units of gauss and cm 2 IV s). Yet
rH is a function of B, T, doping and compensation for a
somewhat weaker magnetic field, not offering much hope of
simplification unless "weak-field" (BJ1n <108 ) conditions
can be assured. And such an assurance is hard to come by for
carriers whose mobility rises from a few thousand cm 2IV s
at room temperature and above, to the 104 _10 5 cm 2 /V s
range at liquid nitrogen temperatures.
The 1975 review by Rode l69 of electron transport in
Group IV, III-V, and II-VI materials, includes a useful bibliography of prior work on GaAs, and three figures from that
work are reproduced here. The first of these is Fig. 60, which
showsJ1H as a function of temperature (using a typical measuring induction of 5 kG), for various samples of "high-purity" N-type GaAs. The solid line in Fig. 60 of "lattice scattering" mobility J1L does rise on cooling, but not in the power
law form so beloved of the semiconductor community!
There are reasons for this, in terms of the three physical
processes contributing to lattice scattering, as further noted
below.
First, one can continue to observe in Fig. 60 the low
temperature falloff ofJ1 H below 40 K for the sample ofWolfe
and Stillman 387 ; and the expectation that ionized impurity
scattering363 would have similarly affected the sample of
Hicks and Manley 388 had their data been pursued to lower
temperatures. The dashed line in Fig. 60 shows Rode's calculation of how J1H would fall off as provided by Dingle's
formulation of ionized impurity scattering. The slightly better fit is based on the Brooks-Herring formulation. 363
R172
J. Appl. Phys, Vol. 53, No. 10, October 1982
T (K)
FIG. 60. Temperature dependence of the electron Hall mobility Il H, as measured for B~5 kG with rather pure N-type GaAs samples. After Rode. 169
Data identifications: 0 Wolfe and Stillman 3.'; • Hicks and Manl ey 3S"; ....
Chang (data by private communication); 6. Blood.310 Wolfe and Stillman
estimated Nd ~5 X 10 13 cm -3 and (No/Nd)~O.4 for the sample with data
plotted down to 4 K. which is the same as sample (A) of Fig. 61.
Figure 61 showsJ1 H (T) for three samples, as reported by
Stillman et al./ 89 with dashed lines indicating the expected
contributions of (a) ionized impurity scattering, (b) deformation potential scattering by acoustic phonons, and (c) polar
mode scattering by optical phonons. The first of these evi-
~
I/)
>
.......
(\J
10
6
DEFORMATION
POTENTIAL
MOBILITY
->
\V
1
'\
:!E
I
,,;;
::t..
/
/'
-d
\(
/
u
POLAR
MOBILITY
\
//1
\
~'\
/
\>
~ '\
\
5
>r-- 10
-'
-
CD
0
:!E
/
I
1
/ /
\
'\ 1
~
/
/
"
h
/-::i
IONIZED
IMPURITY
MOBILITY
...J
~
c::(
10
4
I
10
100
1000
TEMPERATURE T (K)
FIG. 61. Temperature dependence of the electron Hall mobility IlII (for
B = 5 kG) for three N-type GaAs samples, after Stillman et al. ,"9 They estimated donor densities of (A) 5 X 10" cm- J , (B) lO'~ cm -" and (e) 5 X 10"
cm- 3 for the three samples so identified, with (Na/Nd)~0.3 to 0.4 in each
case. Expected contributions of three major processes towards the scattering are shown.
J, S, Blakemore
R172
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dently does an efficient job in controlling and modeling /.l H
for low temperatures, while it is the third one (polar optical)
that dominates the course of the "lattice" mobility /.l L' The
acoustic phonon scattering that controls /.l L for semiconductors such as Ge and Si appears, according to Fig. 61, to provide only - 10% of the lattice scattering for electrons in
GaAs at room temperature. A similar conclusion was
reached by Pador and Nador.390
Several lesser scattering processes are omitted from the
set of dashed curves in Fig. 61, but should at least be mentioned here for completeness. Piezoelectrically active acoustic phonons can be involved in polar scattering, because of
the lack of inversion symmetry in the sphalerite lattice.
Wolfe et al. 391 deduced that this would be comparable in
strength to deformation potential scattering around room
temperature, with a /.lpiezo a: T -1/2 behavior. Thus it does
not have a major impact onJ-lL for reasonably high temperatures. However, Pador and Nador 390 show that /.lpiezo is the
limiting constraint on /.l L below about 50 K (at which point
/.lL > 106 cm 2/V s).
Figure 61 also does not show the expected strength of
neutral impurity scattering, which was estimated by Wolfe
et af.391 on the basis of the simple Erginsoy model. 392 As
usual, this was determined to contribute only a small fraction of the low temperature scattering.
It is all very well to discuss electron scattering in N-type
GaAs of fairly high purity and modest compensation. However, electron mobility is assuredly of interest to some
readers with fairly heavy doping, perhaps rather large compensation-and maybe not even N-type, for that matter.
Walukiewicz et al. 393 have examined the interesting differences between /.In (T) for N- and P-type GaAs-the latter
being of concern for injected minority electrons. A significant feature of that regime is the screening by heavy holes,
tending to shield electrons from ionized impurity scattering.
Whereas the combination of processes noted in connection with Fig. 61 appears to be quite adequate for N-type
samples that are relatively pure and modestly compensated,
they seem to overestimate the room temperature mobility for
highly compensated samples.395-398 This discrepancy appears accountable by some process for which /.In a: T -1/2.
Weisberg 395 suggested that this might be space-charge scattering caused by doping inhomogeneity, and Conwell and
VasselJ292 confirmed that small P-type islands would give
scattering with that temperature dependence. Katado and
Sugano,396 and Pador et al.,397 incorporated such a process
into their mobility modeling of epitaxial GaAs layers; and
Stringfellow and Kiinzel 398 remark that a scattering process
with the /.l- T - 1/2 dependence has been invoked for a number of other semiconductors. However, the latter authors go
on to argue the thesis that individual carbon atoms (CAs
= acceptor), or carbon related point complexes, are responsible for this scattering, rather than multi-acceptor spacecharge regions.
It seems to be generally accepted that the Hall factor
rH > 1 for medium or weak field measurements with N-type
GaAs. Figure 62 shows the calculated estimate of Rode l69
for the weak-field rHO = (/.lHolJ-ln) versus temperature, appropriate for weakly doped material in which impurity scatR173
J. Appl. Phys. Vol. 53, No. 10, October 1982
0
...J:
a::
0
f-
u
1.20
<!
lL.
...J
...J
<!
::c
0
...J
w 1.10
lL.
:.::
<!
w
~
1.00 L_---.L_~---L~_ _.L-_-----.l_-::-L-__L...J
20
40 60 100
200
400 600 1000
TEMPERATURE T (K)
FIG. 62. Theoretical weak-field Hall factor rHO = !JiHoifl.) as a function of
temperature. as calculated by Rode 10. for modestly doped N-type GaAs.
tering plays a minor role above 100 K. Rode comments that
the large lobe of rHO> 1 for T> 100 K arises from the dominance of polar mode optical phonon scattering; while acoustic phonon scattering (deformation potential and piezoelectric) and impurity scattering determine the lower
temperature course of events.
The temperature dependence of rHO is not large enough
to make the temperature dependences of the drift mobility
/.In and the weak-field Hall mobility appreciably different
from each other. As was noted in connection with Fig. 60, a
T - n type of dependence is not a good fit for any extensive
portion of the temperature range above 100 K, when polar
mode scattering is in control. Yet a power law type of expression is very useful, even if not exact. Accordingly it is suggested that
f-ln=8000(300/Tf3 }
/.lHO =9400(300/T)23
cm 2/V s
(122)
be used as crude guides to the drift mobility and the weakfield Hall mobility reasonably close to the room temperature
range.
Mobility as measured at 300 or 77 K (or preferably
both) has been suggested as a means for determination of the
concentrations of both shallow donors (Nd) and shallow
compensating acceptors (Na ) in N-type GaAs. The use in
particular of the 77 K mobility for this purpose was urged by
Stillman and coworkers,168.399 with curves and a procedure
to get from a measured Hall mobility (at a 5 kG magnetic
induction) to the value of (Ndl + Nal )=(n o + 2Nal )· The approach of Stillman et al. 168 is predicated on an assumption
that/.l H(77)=2.5 X 105 cm 2/V s before any impurity scattering is taken into account.
The same assumption concerning the 77 K Hall mobility was made in the analogous modeling of Rode, for which
the guiding curves are reproduced in Fig. 63 for both 300 and
77 K. Rode assumed a lattice-limited 300 K drift mobility
and Hall mobility as provided above in Eq. (122). With the
aid of these curves, one can expect to get a rough idea of the
J. S. Blakemore
R173
Downloaded 21 Sep 2011 to 210.32.148.91. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
-;n
>
,
T =300 K
10,000
U>
>
- - DRI FT MOBILITY
- - - HALL MOBILITY
N
::2:
(J
4
10
[',
"-
[][',
N
~
u
5
>-
::l
>-
I....J
£II
0
::2:
iIi
a
10 14
10 15
10 16
10 17
10 19
1018
(CM- 3 l
ELECTRON DENSITY no
,>
10 5
--- , , ,
~
>-
,
"-
"-
" "-"-,,- I
"-
10 13
10 14
ELECTRON DENSITY no
10 16
compensation ratio (Na/Nd) from 300 K data. If 77 K data is
also available, this will permit improved accuracy.
None of these procedures can be successful, of course, if
the GaAs has spatially inhomogeneQus doping. In the "Semiconductors and Semimetals" series of volumes, 10 articles
concerning strange apparent mobility effects caused by doping inhomogeneity have been published by Bate,400 and by
Wolfe and Stillman. 401 The latter of these was concerned
with the less common of the two directions mobility can go
in the presence of inhomogeneity: apparent mobility enhancement through some geometric forms of impurity segregation. When the doping conditions are conducive for such
an effect, it will be seen more particularly in liquid nitrogen
range measurements than at room temperature,
Transport in N-type GaAs up to and including the intrinsic range has been an objective of many investigations
over the years, from the 1950s reports of Folberth and
Weiss,349 Welker and Weiss, 86 Whelan and Wheatley, 307 and
Aukerman and Willardson 289 : all of which showed a temperature region of Hall coefficient "anomaly" above 500 K.
This phenomenon, the result of thermal electron transfer out
of the r6 minimum (first and most prominently to L 6 , and
later on a smaller scale to X 6 ) has been measured with higher
precision in the subsequent work ofBlood,31o and of Nichols
et al. 347
In the latter of these, Nichols et al. used N-type epitaxial
samples from which the substrate had been etched away.
J. Appl. Phys. Vol. 53, No.1 0, October 1982
400
600
800
1000
1200
(K)
10 17
(CM- 3 l
FIG. 63. Curves modeled by Rode for Pn and PHO (measured at 5 kG) for
electrons in N-type GaAs, at 300 and 77 K, vs electron density no and compensation. Supposing shallow impurities, no = (Nd; - N a;), while the scattering ion density is (Nd; + N a;) = (no + 2NuJ By use of these curves, mobility data for either temperature (or both) should permit an estimate of
compensation ratio (Na I N d ).
R174
aU
FIG. 64. Data of Nichols et al.'4? for Hall mobility (at 3.5 kG) vs T for Ntype GaAs epitaxial layers, above room temperature. The solid line shows
Rode's expectation 169 for high-purity N-type GaAs. Doping of the three
samples here is: 0 N d""4Na",, 1.2X 10" cm-'; 0 Nd",,4Na "" 10 16 cm-';
and 6, Nd ",,3Nu ",,2 X 10" cm-'.
-~IO
10 15
[J
"-
3
10
10 12
&1
TEMPERATURE
iii
l(Ndl +Nojl/nol =
08'
3
0
500
"-
- - DRIFT MOBILITY
- - - HALL MOBILITY
::2:
10
'6
,
o 1J
o 'b
o 'b
W
4
0
'b
o
w
"- "-
10
o
....J
::l
I..J
'Jt'l
o
u
,
::2:
u"b
00
2
z
T = 77 K
N
0
°Oy,
~
0:
I-
---
[',
0
0
-;n
[][',
00
,
00
00
0y,
5000 -
I..J
"
[b
::l
Figure 64 shows the Hall mobility (URH) they found for
these three samples, compared with a solid curve of Rode's
expectations l69 for a lattice-scattering Hall mobility. Thus
the empirical result of the work by Nichols et a/. supports a
conclusion reached by Blood31O in measurements up to 800
K, that the mobility (averaged over all places that electrons
choose to spend their time) decreases quite rapidly with rising temperature.
Nichols et al. 347 went on to analyze their combination
of Hall and conductivity data in an attempt to deduce mobility versus temperature for carriers associated with the various band extrema. They used equations similar to Eqs. (91)
and (92) for the thermal distribution of conduction electrons
among r, L, and X minima. Figure 65 shows what they deduced from this mUltiband am bipolar analysis of two of the
samples of Fig. 64: temperature dependences for fl- r, fl- L' and
for the hole mobility fl-p' (And so these estimates of fl-p take
over from the upper temperature limit of the hole mobility
data in Fig. 55.)
The various curves of Fig. 65 thus suggest, among other
things, that fl- r decreases at a modest rate as temperature
rises. The ensemble (and ambipolar) average represented by
the curves of Fig. 64 is thus forced downwards so rapidly by
the steep decline offl- L with rising temperature, aided at the
higher temperatures by the positive Hall contribution ofminority holes.
Both the magnitude and the temperature coefficient of
fl- L in Fig. 65 come as a slight surprise, in view of analogies
with other semiconductors (notably germanium) with Ltype conduction minima. Using such analogies, Aspnes l52
had anticipated fl- L (300)=900 cm 2IV s for weakly doped
GaAs, only one quarter as large as the values the fl-dT)
curves of Fig. 65 extrapolate towards. Incidentally, Aspnes
also surmised thatfl-x(300)=300 cm 2IV s, based onfl-n in Ntype silicon, GaP, and other analogies.
J. S. Blakemore
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10
U>
C. Electrical conductivity
4
The ambipolar conductivity of a semiconductor can be
written as
5
>
......
(123)
N
~
u
2
:t
10
>r-
3
...J
lD
500
0
~
a::
w 200
a::
a::
« 100
u
50
300 400 500 600 700 800 900
TEMPERATURE
(K)
FIG. 65. Estimates by Nichols et aU" of mobilities in three types of band
extremum: the To and Lo conduction minima for electrons, and flp for the
combination oflight and heavy holes. As in Fig. 64, the sample dopings are:
o Nd~4Na,:::dO'6 cm~3; and 0 Nd~4Na~1.2X 10 17 cm~3.
where I1n and I1p denote the drift (i.e., conductivity) mobilities. The 300 K values of these have already been quoted for
lightly doped GaAs. Thus, Eq. (122) quotes I1n (300) = 8000
cm 2 /V s, while the comments concerning Eq. (113) indicate
that I1p (300) = 320 cm 2 /V s. Both of these numbers are also
quoted in the 300 K summary information of Table XV, and
are used in Eq. (123) to calculate the a(300) curve of Fig. 66.
For GaAs doped predominantly with shallow donors
and/or acceptors, the ratio no/po is so far from unity at ordinary temperatures that conduction is unipolar. (Meanwhile,
Table XIV shows that nj exceeds 10 16 cm ~3 for 1'>900 K, so
high temperature conduction is another matter.) Doping
conditions are well known, however, which pin the Fermi
level CF near to its intrinsic location t/J, enforcing am bipolar
conduction at ordinary temperatures. The room temperature resistivity of such semi-insulating GaAs is in the range
105~1O9 n cm. That weak conduction, strongly activated,
was first found as an accident of crystal growth conditions. 307 Achievement of the semi-insulating condition
through creation of the so-called EL2 defect402 (tentatively
identified at the time of writing as the AS Ga antisite disor-
TABLE XV. Some room temperature (300 K) properties of gallium arsenide."
(a) Mechanical, thermal, and dielectric properties
3
Crystal density
P300 = 5.317 g/cm
Bulk modulus (compressibility~ ')
B, = 7.55 X 10" dyn/cm2
Shear modulus
c' = 3.26 X 1011 dyn/cm2
Linear expansion coefficient
a 300 = 5.73 X 1O~6 K~ I
Volume expansion coefficient
3a = r = 1.72 X IO~' K ~ I
Specific heat
Cp = 0.327 Jig K
Effective Debye temperature
()300 = 360 K
Lattice thermal conductivity
KL = 0.55 W Icm K
Static dielectric constant
KO = 12.85
(See also
noo = 3.299
Table IX)
Infrared refractive index
(b) Energy band separations and derivatives
Direct (zone center) intrinsic gap
Ei = 1.423 eV
Pressure derivative
(aE,Iap) = +0.0126eV/kbar
Temperature coefficient
(aE,IaT) = - 0.000452 eV IK
Direct exciton transition energy
Ex} = 1.419 eV
Spin-orbit splitting energy
..::1,0 = 0.341 eV
L6 conduction band gap
EL = 1.707 eV
Pressure derivative
(aELlap) = +0.0055 eV/kbar
(aELlaT) = - 0.000506 eV IK
Temperature coefficient
.d rL = 0.284 eV
Energy elevation (E L - E,) =
X6 conduction band gap
Ex = 1.899 eV
Pressure derivative
(aExlap) = - 0.0015 eVlkbar
Temperature coefficient
(aExlaT) = - 0.000385 eV/K
Energy elevation (Ex - E,) =
.d rx = 0.476 eV
X, - X6 band separation
::::O.40eV
(c) Intrinsic properties
ni =
Intrinsic carrier pair density
U =
Intrinsic electrical conductivity
i
Urnin =
Minimum conductivity (for Po = bno)
Intrinsic Fermi energy
(I/! - Eo) =
R175
2.25 X 106 cm ~3
3.0X 1O~9.a ~I Cm~1
1.15X 1O~9.a ~I Cm~1
0.752 eV
J. AppL Phys. Vol. 53. No.1 D, October 1982
(d) Parameters for lowest conduction band
Band-edge effective mass
moo =
Conduction electron rms speed
v,(rms) =
(Nondegen.) effective state density
N; =
Drift mobility (weak doping)
fln =
Hall mobility (weak doping. weak field) flHO =
rHO = V-tllolfln) =
Weak-field Hall factor
0.0632 mo
4.4 X 10' cm/s
4.21 X 10 17 cm ~}
8000 cm 2/V s
9400 cm 2 /V s
1.175
(e) Parameters for upper conduction bands
For L6 conduction band:
Density of states effective mass
m L =0.55 m"
Conductivity mobility
flL -2500 cm'/V s
Fraction of all conduction electrons
-0.0004
For X6 conduction band:
Density of states effective mass
mx = 0.85 ma
Conductivity mobility
flx-3OOcm2/Vs
< 1O~6
Fraction of all conduction electrons
(f) Parameters for valence bands
Heavy-hole density of states mass
m h = 0.50 mo
rms heavy-hole speed
Vh (rms) = 1.65 X 10' cm/s
Light-hole density of states mass
m, = 0.088 mo
rms light hole speed
v,(rms) = 3.4 X 10' cm/s
Light holes as fraction of total
0.069
Combined heavy:light state density
N; = 9.51 X 10 18 Cm~3
Drift mobility (weak doping)
f-lp = 320 cm 2 /V S
Hall mobility (weak doping, weak field) flHO = 400 cm 2/V S
Weak-field Hall factor
rHO = V-tHolflp) = 1.25
Splitoff band effect mass
m,o =0.15 ma
< 1O~8
Splitoff holes as fraction of total
Note that some properties are quite rapid functions of temperatures. The
numbers listed above are specific for 300 K.
a
J. S. Blakemore
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19
10
and XV show that n,(300) = 2.25x 106 cm--', and so
T = 300 K
,
I"l
~
U
c
10
17
10
~
-,
- 10
0
Z
0
~
I
0
0',(300)
However,
= en, f-lp(1
the
+ b) = 3.00X 10- 9
conductivity
is
n
-I
cm - I .
(124)
minimized
when
Po = vbn, = bn o, and then
O'rnin
=2en,(pnf-lp)I12= [20',I(b+ 1)]vb
= 0.3850', = 1.15x 10- 9
n- I cm- I
.
(125)
u
10
U
13
I
I
l-
\
lO"
I
0
-'
I
10
I
I
\
I
I
10- 5
I
10
-'
a::
_ 10- 7
--ni
-'
0
U
<l
7
W
~
U
I
I
0
Z
<l
I
IU
0
Z
1 /
I
W
10- 3
I I
I I
I I
I I
/ I
I
I
9
>
/1
\
I
I
a::
lU
W
Z
6
>-
I-
I
I I
I 1
a::
I
I
Z
1
Z
I
10- 1
I
if)
I
(/)
Z
W
0
..
u
0
~
15
(/)
W
3
.z
a.
l-
10
"-
ILl
0
"
0
.,.
IU
W
-'
W
0
I
10
(Ti-
5
IT"min0
0.2
OA
0.6
0.8
1.0
FERMI ENERGY (CF-c v )
1.2
10- 9
IA23
(eV)
FIG. 66. Variation at T = 300 K of the thermal carrier densities no and Po,
and the resulting electrical conductivity, with the Fermi energy location. In
conformity with numbers quoted in room temperature Table XV and elsewhere, assumed that n, = 2.25x 10" cm--" and that ftn = 25ftp = 8000
cm 2 /V s, with E, = 1423 meV, and Itb - E,) = 752 meV.
der 40 -,), presents an interesting combination of technical opportunities and problems. Chromium doping 404 -408 provides
an alternative route to near-intrinsic status for GaAs at
room temperature.
The semi-insulating condition of GaAs was recognized
but not understood in the early 1960s (some might say this is
still the case), when Hilsum 9 commented that it was then (as
now) fairly easy to make strongly N- or P-type GaAs, but
very hard to make weakly doped material reproducibly.
Whereas semi-insulating crystals by the "undoped" procedure could then and can now be achieved, it has always been
very difficult to produce bulk GaAs with a room temperature no or Po in the range from (say) 10 14 cm -3 down to
around 109 cm -- -'. The various branches of epitaxial growth
technology have helped, but it is inevitably hard to grow any
compound to be both highly pure and free from the various
kinds of native disorder. Thus some large ranges of carrier
density and conductivity are shown as dashed rather than
solid lines in Fig. 66, to show the ranges which are not readily accessible.
Figure 66 shows no, Po, and the resulting conductivity of
Eq. (123), for T= 300 K: as functions of the Fermi energy
(c F - c,.). The intrinsic condition c p = ¢ means that
no=po=n,. Since the mobility ratio b=(f-ln/f-lp)
= (8000/320) = 25 for modestly doped material-a value
considerably larger than unity-the lowest conductivity
O'rn," occurs appreciably on the P-type side of ¢. Tables XIV
R176
J. Appl. Phys. Vol. 53, No. 10, October 1982
For 300 K, the minimum conductivity situation occurs
when cp = [¢-!kTln(b)] =(c v +0.710 ev). That is
0.042 eV lower than ¢ itself.
GaAs near the O'rnin condition will have a negative Hall
coefficient even though Po> no. Not until Po> bn, > b 2n o
does the Hall coefficient become positive. Note that 0' = 0',
again for Po = bn" when cp = [¢ - kTln(b )] = (¢ - 0.083
eV) = (c u + 0.669 eV).
These principles of ambipolar conduction when
f-ln > f-lp are exactly those described 30 years ago for nearintrinsic conditions in germanium.409.410 For the analysis of
semi-insulating GaAs, Martin 411 has chosen 400 K rather
than 300 K as the temperature for plots of the various quantities versus c p, and the principles noted above still dictate
the conditions of the conductivity minimum, Hall reversal,
etc.
Far away from semi-insulating and intrinsic conditions,
the lines of Fig. 66 are curved at their upper ends, as CF
approaches a band edge. For Po and no, the data are corrected
for band occupancy degeneracy in the usual manner. 341 The
conductivity curve is additionally adjusted for the decline of
mobility with heavy doping, as indicated by Figs. 56 and 63
for holes and electrons, respectively. However, Fig. 66 could
not be drawn conveniently to incorporate the shrinking of
the band gap by band tailing when the shallow impurity density is large: as discussed, for example, by Casey and Stern. 19
In this, as for so many other topics, there are further detailed
ramifications for which a review such as this cannot provide
space.
ACKNOWLEDGMENTS
Information for this review accumulated in part as a
byproduct of semi-insulating GaAs research, supported by
the National Science Foundation through Grant DMR 7916454. Additional support from Tektronix, Inc. is also acknowledged. Useful advice and comments is gratefully acknowledged from a number of people, including P. K.
Bhattacharya, N. Holonyak, S. G. Johnson, R. Y. Koyama,
D. C. Look, J. W. McClure, G. M. Martin, S. Rahimi, B. K.
Ridley, G. E. Stillman, G. H. Wannier, and R. K. Willardson.
This writer must, however, be held responsible for errors of omission, commission, and misunderstanding. In
some cases, I have failed to understand important nuances of
reported results. In others, I have leaned (perhaps wisely,
perhaps not) towards one reported number rather than another. Nonetheless, I hope this material will assist the intended reader: the experimentalist who lacks time to read all
the original source papers.
J. S. Blakemore
R176
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306Figure 45 is drawn to be deliberately ambiguous as to whether the lowest
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J. Appl. Phys. Vol. 53, No.1 0, October 1982
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