F.H. Pollak

K. Wei

M. Mu�oz

T. Holden

V.M. Asnin

English

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phys. stat. sol (b) 215, 33 (1999)
Subject classification: 78.20.Ci; 71.35.Cc; S7.12; S7.13; S8.13
Modeling the Optical Constants
of Diamond- and Zincblende-Type Semiconductors:
Discrete and Continuum Exciton Effects at E0 and E1
F.H. Pollak1), M. MunÄoz2), T. Holden, K. Wei, and V. M. Asnin
Physics Department and NY State Center for Advanced Technology in Ultrafast Photonic
Materials and Applications, Brooklyn College of CUNY, Brooklyn, NY 11210, USA
(Received March 30, 1999)
We present a comprehensive model dielectric function e(E) [= e1 + ie2] for diamond- and zinc-
blende-type semiconductors based on the energy-band structure near critical points (CPs) plus dis-
crete as well as continuum excitonic effects at the E0, E0 + D0,, E1, and E1 + D1 CPs. In addition to
the energies of these band-to-band CPs, our analysis also yields information about the binding
energies of not only the 3D exciton associated with E0 (R0), when resolved, but also the 2D exci-
ton related to the E1, E1 + D1 CPs (R1). This model has been applied to spectral ellipsometry meas-
urements of e1, e2 (0.3 eV < E < 5.5 eV) of ZnCdSe/InP, CdTe1± xSx, In0.66Ga0.34As, and GaSb and
a surface photovoltage spectroscopy determination of the absorption coefficient of GaAs near E0.
This work shows conclusively that even if the exciton at E0 is not resolved the lineshape is conti-
nuum exciton. The obtained values of R1 exhibit a trend which is in good agreement with effective
mass/k  p theory. Our analysis will be compared with the modeling of Adachi and the University
of Illinois-Chicago group, both of whom neglect exciton continuum effects and hence have not
evaluated R1. Our results, particularly for exciton continuum effects at E1, have considerable impli-
cations for recent first-principles band structure calculations which include exciton effects.
The optical constants associated with electronic transitions of diamond- and-zincblende-
type (DZB) materials have been measured by a number of methods. During the past
decade spectral ellipsometry (SE) has been used by number of investigators including
Aspnes and Studna [1], Cardona and coworkers [2,3], Adachi and coworkers [4], the
University of Illinois-Chicago group [5], etc. However, these workers either did not
model the optical constants [1 to 3] or used an incomplete single particle expression
[4, 5]. Cardona and coworkers fit derivative (with respect to photon energy) spectra
employing a two-dimensional density of states for the ``E1, E1 + D1º features; exciton
effects were taken into account by means of a phase angle (Slater-Koster ``contactº
potential) [2,3]. Recently, Holden et al. [6] have presented a comprehensive model di-
electric function e(E) [= e1 + ie2] for DZB semiconductors based on the energy-band
structure near critical points (CPs) plus discrete as well as band-to-band Coulomb en-
hanced (BBCE) effects, i.e., continuum excitons, at the E0, E0 + D0, E1, and E1 + D1
CPs. In addition to the energies of these band-to-band CPs, this analysis also yields
information about the binding energies of not only the 3D exciton associated with
F.H. Pollak et al.: Modeling the Optical Constants of Semiconductors 33
1) Corresponding author: Tel.: 718/951-5356; Fax: 718/951-4871;
e-mail: FHPBC@CUNYVM.CUNY.EDU
2) Permanent address: Departamento de Fisica, CINVESTAV, Mexico DF, Mexico.
e-mail: MARTIN@FIS.CINVESTAV.MX
E0 (R0), when resolved, but also the 2D exciton related to the E1, E1 + D1 CPs (R1).
This model has already been applied to SE measurements of e1, e2 (energy range
0.3 eV < E < 5.5 eV) of ZnCdSe/InP [6], CdTe1± xSx [7], and In0.66Ga0.34As (undoped as
well as n- and p-doped) [8].
In this paper we present an SE investigation of GaSb (in the photon energy range
0.3 to 5.3 eV) as well as a surface photovoltage spectroscopy (SPS) study of undoped
GaAs near E0. By fitting the SE data of GaSb to the Holden model we have evaluated
R1 in addition to the energies of the various CPs. The values of R1 for ZnCdSe/InP,
CdS, CdTe, In0.66Ga0.34As, and GaSb exhibit a trend which is in good agreement with
effective mass/k  p theory [9]. The ability to determine R1 has considerable implications
for recent first-principles band structure calculations which include exciton effects [10].
An analysis of the GaAs SPS data (measured in this work) and the CdTe SE [8] results
near E0 show conclusively that even if the exciton at this CP is not resolved the line-
shape is BBCE and not band-to-band single particle (BBSP) [4,5].
The sample consisted of bulk GaSb (001) (cut 4 towards [110]). The optical data in
the range 0.75 to 5.3 eV were taken using an Instruments SA variable angle ellips-
ometer while for the interval 0.3 to 0.8 eV a variable angle instrument which used a
Fourier transform infrared reflectometer as a light source was employed. Thus there
was some overlap between the two intervals. The SPS data were taken using a normal-
ized Kelvin probe technique [11].
The solid line in Fig. 1 is the experimental imaginary component e2(E) of the com-
plex dielectric function for GaSb. The obtained e1(E) spectrum is presented in Ref.
[12]. Our data are very similar to that ones reported in Refs. [1,2,13] in the appropriate
energy ranges. References [1,2] made measurements only in the range 1.5 to 5.5 eV.
There is an absorption edge around 0.7 eV, doublet peaks in the range 2.0 to 2.5 eV,
34 F.H. Pollak, M. MunÄoz, T. Holden, K. Wei, and V.M. Asnin
Fig. 1. The solid and dashed lines represent the experimental and fit values, respectively, of the
imaginary component (e2) of the complex dielectric function of GaSb
and a large feature with a peak around 4 eV with some structure on the low energy
side around 3.5 eV. There is also a weak feature at around 1.5 eV, not observed in
Refs. [1,2,13].
The data of Fig. 1 near the E0 band gap has been fit to a function which contains
Lorentzian broadened (a) discrete excitonic (DE) and (b) 3D M0 BBCE contributions,
i.e., continuum exciton effects. As will be demonstrated below, even if the E0 exciton is
not resolved, the Coulomb interaction still affects the band-to-band lineshape. There-
fore we can write [6 to 8, 12]
e2E  Im A
E2
2R0
E0 ÿ R0 ÿ Eÿ iGex0
 2R0E0 ÿ R0  E iGex0
 8<:
8<:

1
ÿ1
qE0 ÿ E0
1ÿ eÿ2pz1E0 ÿ
qÿE0 ÿ E0
1ÿ eÿ2pz2E0
 
d0E
E0 ÿ Eÿ iG0
9=;
9=; ;
where A is a constant, E0 is the energy of the direct gap, R0 is the effective Rydberg
energy [= (E0 ± Eex0 )], G
ex
0 is the broadening of the exciton, G0 is the broadening param-
eter for the BBCE transition, z1(E) = [R0/(E ± E0)]
1/2, z2(E) = [R0/(± E ± E0)]1=2 and
qx is the unit step function. For the E0 + D0 transition, Eq. (1) will have A! B,
E0! E0 + D0, R0! Rso and G! Gso.
For E1 we use [6 to 8, 12]
e2E  Im C1
E2
4R1
E1 ÿ R1 ÿ Eÿ iGE1
 4R1E1 ÿ R1  E iGE1
 8<:
8<:

1
ÿ1
qE0 ÿ E1
1 eÿ2pz3E0 ÿ
qÿE0 ÿ E1
1 eÿ2pz4E0
 
d0E
E0 ÿ Eÿ iGE1
9=;
9=; ;
where C1 is a constant, E1 is the energy of the gap, R1 is the 2D Rydberg energy, GE1 is
the broadening parameter for both the exciton and band-to-band transition,
z3(E) = [R1/4(E ± E1)]
1/2, and z4(E) = [R1/4(± E ± E1)]
1/2. For the E1 + D1 CP, Eq. (2)
has C1 ! C2 and E1 ! E1 + D1, GE1 ! GE1D1 , etc. The same 2D Rydberg (R1) was
used for both E1 and E1 + D1 CP features.
Because of their complexity, the E00; E
0
0  D00, and E2 features each have been described
by a damped harmonic oscillator term [6 to 8, 12]. The fact that Ref. [10] has found that
E2, like the E0/E0 + D0 and E1/E1 + D1 CP features, contains an excitonic component pro-
vides some justification in using a damped oscillator term for this feature.
Shown by the dotted curve in Fig. 1 is the fit to the experimental data using the above
expressions. Since the exciton at E0/E0 + D0 is not resolved we have taken R0 (= 1.6 meV)
from Ref. [14]. Because of the large number of fitting parameters, values for the various
gaps and their broadening parameters were initialized from values obtained by numeri-
cally taking the first-derivative of the dielectric functions with respect to energy. The de-
tails of this approach are given in Refs. [6 to 8,12]. The final obtained values of the var-
ious energies are indicated by arrows in the figures. All relevant parameters are listed in
Ref. [12]. Our values of the various energy gaps, i.e., E0, E0+D0, E1 ± R1, (E1 + D1) ± R1,
etc. are in good agreement with other works [1,2,13,14].
Modeling the Optical Constants of Semiconductors: Exciton Effects 35
It is important to note that due to the relatively large values of R1 (30 to
300 meV), as listed in Table 1, the optical structure associated with the E1, E1 + D1 CPs
in DZB semiconductors are actually mainly the excitonic features E1 ± R1,
(E1 + D1) ± R1, respectively, as denoted in Fig. 1. Almost all prior optical [1 to 5] and
modulated (see, for example, [15]) optical studies have incorrectly labelled these exci-
tonic features as ``E1, E1 + D1º.
Our value of R1 (32  3 meV) is in fairly good agreement with the effective mass/
k  p theory of Ref. [9]. According to the approach
R1 / m?=e211 ;
where m? is the perpendicular reduced interband effective mass and e1(1) is the high
frequency dielectric function. This calculation yields R1 = 25 meV, as listed in Table 1.
Table 1 also presents the experimental and calculated values of R1 for several materi-
als. The latter are based on the theory of Ref. [9]. The experimental numbers for R1
exhibit a trend which is in good agreement with effective mass/k  p theory. More reli-
able theoretical values can now be obtained from the recent first-principles band struc-
ture calculations which include exciton effects [10].
The solid lines in Fig. 2a and b are the experimental values of the numerical deriva-
tives de2/dE for CdTe and da/dE for GaAs, respectively, where a is the absorption
coefficient. The former is from Ref. [8] while the latter is our SPS study. It has been
shown that the SPS data in the vicinity of E0 are proportional to a (see, for example,
[11]). Both lineshapes are approximately Lorentzian with a slight asymmetric broaden-
ing on the high energy side. This indicates that the DE is not resolved since the first-
derivative of its Lorentzian lineshape would produce an ``Sº shaped component. The
dotted lines in Fig. 2 are least-square fits to an analytical form for the first derivative of
Eq. (1), as given in Ref. [6], where Gex0 has been taken sufficiently large so that the DE
is not resolved. The obtained values of E0 are indicated by arrows. The dashed lines
are fits to the first-derivative of a BBSP profile, i.e., Im (E ± E0 + iG0)
± 1/2. This pro-
duces a very asymmetric lineshape which does not fit the experimental data very well.
In Ref. [6] the DE at E0 was well resolved and it was evident that the band-to-band
lineshape was BBCE and not BBSP. The results of Fig. 2 clearly demonstrate that even
if the DE is not resolved the lineshape is the former and not the latter, an expression
36 F.H. Pollak, M. MunÄoz, T. Holden, K. Wei, and V.M. Asnin
Ta b l e 1
Experimental and calculated values for R1 for several different semiconductors.
The latter are based on the expressions in Ref. [9]
material experiment
R1 (meV)
effective mass/k  p theory
R1 (meV)
Zn0.53Cd0.47Se/InP 270  50a) 300
CdS 205  30b) 290
CdTe 145  50b) 150
In0.66Ga0.34As 92  15c) 55
GaSb 32  3d) 25
a) Ref. [6].
b) Ref. [8].
c) Ref. [7].
d) This work and Ref. [12].
that has been incorrectly employed in Refs. [4,5]. Also because Adachi and coworkers
and the University of Illinois-Chicago group used a BBSP expression at E1 they were
not able to extract R1.
Acknowledgements The authors F.H.P., M.M., T.H., and K.W. thank NSF grant
#DMR-9414209, PSC/BHE grant #666424 and the NY State Science and Technology
Foundation through its Centers for Advanced Technology program for support of this
project. M.M. also acknowledges support from CONACyT, Mexican Agency, through
Research Project #25135E.
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Modeling the Optical Constants of Semiconductors: Exciton Effects 37
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CdTe and b) da/dE of GaAs in the
region of E0. The dotted and
dashed lines are fits to the first de-
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not resolved) and BBSP expres-
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are indicated by arrows
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38 F. H. Pollak et al.: Modeling the Optical Constants of Semiconductors


Modeling the Optical Constants of Diamond- and Zincblende-Type Semiconductors: Discrete and Continuum Exciton Effects at E0 and E1

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Modeling the Optical Constants of Diamond- and Zincblende-Type Semiconductors: Discrete and Continuum Exciton Effects at E0 and E1

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