The optical constants of Ga0.51In0.49P have been determined from 0.8 to 5.0 eV using variable-angle spectroscopic ellipsometry measurements at room temperature. The metal-organic vapor-phase-epitaxy-grown samples were x-ray analyzed to confirm lattice matching to the GaAs substrate. The effects of the native oxide were numerically removed from the data to determine the intrinsic optical constants. This is important because the optical constants reported become generally useful for modeling multiple-layer structures. A Kramers-Kronig analysis was used to reduce interference-related fluctuations in the below-gap refractive index. Near the band edge a mathematical form for excitonic absorption was included. Critical point energies were extracted using a numerical second-derivative fitting algorithm

Gottschalch, V.

Herzinger, Craig M.

Schubert, Mathias

Snyder, Paul G.

Woollam, John A.

Yao, Huade

English

UNL | Libraries

University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
Faculty Publications from the Department of 
Electrical and Computer Engineering 
Electrical & Computer Engineering, Department 
of 
4-1-1995 
Optical constants of GaxIn1-xP lattice matched to GaAs 
Mathias Schubert 
University of Nebraska-Lincoln, mschubert4@unl.edu 
V. Gottschalch 
University of Leipzig 
Craig M. Herzinger 
University of Nebraska-Lincoln 
Huade Yao 
University of Nebraska-Lincoln 
Paul G. Snyder 
University of Nebraska-Lincoln, psnyder1@unl.edu 
See next page for additional authors 
Follow this and additional works at: https://digitalcommons.unl.edu/electricalengineeringfacpub 
 Part of the Electrical and Computer Engineering Commons 
Schubert, Mathias; Gottschalch, V.; Herzinger, Craig M.; Yao, Huade; Snyder, Paul G.; and Woollam, John A., 
"Optical constants of GaxIn1-xP lattice matched to GaAs" (1995). Faculty Publications from the 
Department of Electrical and Computer Engineering. 60. 
https://digitalcommons.unl.edu/electricalengineeringfacpub/60 
This Article is brought to you for free and open access by the Electrical & Computer Engineering, Department of at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from 
the Department of Electrical and Computer Engineering by an authorized administrator of 
DigitalCommons@University of Nebraska - Lincoln. 
Authors 
Mathias Schubert, V. Gottschalch, Craig M. Herzinger, Huade Yao, Paul G. Snyder, and John A. Woollam 
This article is available at DigitalCommons@University of Nebraska - Lincoln: https://digitalcommons.unl.edu/
electricalengineeringfacpub/60 
Optical constants of GaJn, -,P lattice matched to GaAs 
Mathias Schubert 
University of Leipzig, Faculty of Physics and Geoscience, Department qf Semiconductor Physics, 
Linnbtrasse 5, 04103 Leipzig, Germany 
V. Gottschalch 
University of LeipGg, Faculty of Chemistry and Mineralogy, Department of Solid State Chemistry 
Linnkstrasse 3, 04103 Leipzig, Germany 
Craig M. Herzinger, Huacie Yao, Paul G. Snyder, and John A. Woollam 
University qf Nebraska, Cent&r for Microelectronic and Optical Material Research and Department of 
Electrical Engineering, Lincoln, Nebraska 68.588 
(Received 13 October 1993; accepted for publication 23 November 1994) 
The optical constants of GaeslInu4sP have been determined from 0.8 to 5.0 eV using variable-angle 
spectroscopic ellipsometry measurements at room temperature. The metal-organic 
vapor-phase-epitaxy-grown samples were x-ray analyzed to confirm lattice matching to the GaAs 
substrate. The effects of the native oxide were numerically removed from the data to determine the 
intrinsic optical constants. This is important because the optical constants reported become generally 
useful for modeling multiple-layer structures. A Kramers-Kronig analysis was used to reduce 
interference-related fluctuations in the below-gap refractive index. Near the band edge a 
mathematical form for excitonic absorption was included. Critical point energies were extracted 
using a numerical second-derivative fitting algorithm. 0 1995 American Institute of Physics. 
I. INTRODUCTION L. 
Because of the large band gap of Ga&n0.49P lattice 
matched to GaAs, this material is promising for use in visible 
optoelectronics. Ga,I.n,-,P can be grown with high purity 
and excellent morphology by metal-organic vapor-phase ep- 
itaxy (MOVPE) using a wide choice of growth precursors 
over a wide range of compositions. We report in this article 
fully analyzed optical constants of GaInP lattice matched to 
GaAs. Variable-angle spectroscopic ellipsometry (VASE) 
was used, including numerical oxide layer removal, and 
Kramers-Kronig (KK) analysis to reduce the fluctuations of 
the refractive index in the below-band-gtip region. We also 
determine 6, from the full KK analysis, where 6% means the 
dielectric function at frequencies sufficiently high that free- 
carrier effects are negligible. 
II. THEORY 
The complex reflectance ratio p=RJRB ,-where R, and 
R, are the reflection coefficients for light polarized parallel 
and perpendicular to the plane of incidence, respectively, can 
be determined accurately by spectroscopic ellipsometry. The 
ratio has been traditionally defined as 
p=tan w exp(iA), (14 
and is a function of Q and A, the amplitude and phase of the 
complex number. Results of ellipsometric measurements are 
often expressed as 
*jj=q(tiOi ,@j) 0) 
and 
A,j=A(~iwi,~j), Cl4 
where L.+ is the photon energy and ~j is the external angle 
of incidence relative to the sample normal. The pseudo- 
dielectric function (E) can be obtained from the ellipsometri- 
tally measured values of pij, based on a two-phase model 
(ambient/substrate),’ 
l-pij 2 
,= E, - 
I( 1 
1 +Pij 
sin* CDj tan2 cPj+ sin” “j 1 , 
where E, is the dielectric function of the ambient and’as- 
sumed to be constant. If the sample contains multiple layers 
(most samples have at least an oxide layer or surface,rough- 
ness) the SE data must be numerically fit. For this purpose; a 
model must be assumed and ~\I, and A, as in Eqs. (lb) and 
(lc) are generated and compared with measured data for the 
same wavelength and angle of incidence set. qjj and A, 
depend on a set of parameters, e.g., optical constants and 
thicknesses for each layer, interface qualities, surface prop- 
erties, etc. The model should include all necessary character- 
istics to realistically simulate the behavior of the actual 
sample. A regression analysis is used to vary the model pa- 
rameters until the generated and measured values match as 
closely as possible. This is done by minimizing the mean 
square error (MSE) function, defined as 
where i and j are indices for the wavelength and angle data 
sets, and the aij are the standard deviations of the measure- 
ments. Note that one should know as much as possible about 
the physical structure of a sample since the regression analy- 
sis can potentially yield ambiguous results. 
If a layer consists of an isotropic mixture of two mate- 
rials a and b, with the microstructure much less then a 
3416 J. Appl. Phys. 77 (7), 1 April 1995 0021-8979/95/77(7)13416/4/$6.00 Q 1995 American Institute of Physics 
Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
fraction of the wavelength of light, then the macroscopic 
dielectric function can be described with the Bruggemann 
effective medium approximation (EMA), 
E,- E 
f*,+,,+fb~=o, 
a - 
(4) 
where G, %, f, , and fb are the dielectric functions and the 
volume fractions of materials a and b, respective1y.a 
For the absorption spectra of semiconductors near the 
lowest direct band gap, it is well known that in addition to 
the atomic ionization continuum there are also broadened 
shoulders due to excitons. This can be described in simple 
form using coulomb forces, and a “dielectrically diluted” 
hydrogen atom model, where the binding energy is 
AE(exc.)(‘*)= - 8thcy Eo)2 -$ 
c +25X IO-“. VASE data, measured at room temperature, 
were taken on each sample at several angles of incidence and 
over the spectral range from E=0.8 to 5.0 eV. 
and where ,LL is the reduced effective mass given by IV. RESULTS AND DISCUSSIONS 
P -l=(m;‘+m;*). (6) 
en, Q, and h are the permittivity of free space, the static 
dielectric constant, and Plan&s constant, respectively. An 
analytic expression of the above gap absorption for a semi- 
conductor with the simple band model including excitonic 
effects is given by 
(7) 
where E, is the band-gap energy and A an arbitrary 
ampIitude.3 Since the extinction coefficient k is related to the 
absorption coefficient by the relation 
where c is the velocity of light in vacuum, a relation for k 
including excitonic effects for the band edge can be found. 
A detailed discussion of our VASE data and its analysis 
using this excitonic band edge model to smooth the fluctua- 
tions of the apparent refractive index below the band gap is 
presented in Sec. IV. 
III. EXPERIMENT 
Four samples of GaesiIn,-,.49P layers with nominal thick- 
nesses of 1600 nm and GaAs buffer layers -20 nm thick 
were epitaxially grown at 720 “C by MOVPE on liquid- 
encapsulated Czochralski GaAs substrates. The substrates 
were oriented along the [OOI] direction, with a tilt of 2” 
toward the [llO] direction. The samples were grown under 
different partial In vapor pressures to obtain different misfit 
lattice constant values. Partial pressures [I/(In+Ga)] of 
0.67, 0.66, 0.64, and 0.59 were recorded for the four 
samples. The corresponding perpendicular lattice mis- 
matches, determined from x-ray measurements, were 
Aala=+lxlO-4 (two samples), +2x 10-4, and 
3.8, I I I 
$ 3.4- 
-$ 3.2- 
$ 
3.0- 
1 - 
-Model Fit - 
---Exp 72.5’ 
212Wi”V 1.95 
Energy in eV 
PIG. 1. Experimental and generated (best-fit) data for (N) using reference 
data for Ga&n,,,49P and oxide optical constants. 
The two samples with the smallest perpendicular lattice 
mismatch (Aala = + 1 X IO-“) were independently used to 
determine the optical constants of the Gao,5,1na.49P layer and 
both results were found to be the same. First we assumed the 
refractive index N below the band gap, where k was set to 
zero, using Ada&i’s model for the optical constants of GaP 
and InP and linearly interpolating. between all model 
parameters.4 The oxide in the regression analysis was mod- 
eled using an appropriately weighted EMA of (51%) GaP 
oxide and {49%)InP oxide, using the method of Zollner to 
find the reference spectra for GaP oxide and InP oxide over a 
wide spectral range.’ 
The next step was to fit the generated data to the experi- 
mental data from 1.24 to 1.80 eV (below gap), using optical 
constants from Ada&i’s model and an oxide layer to deter- 
mine the thicknesses of all layers (see Fig. 1). The generated 
values closely follow all oscillations in (N) which are due to 
multiple internal reflections. Thicknesses were obtained for 
the Gacs11n,.4,P layer and the oxide layer, for both samples, 
as (1554.521.0) nm and (3.920.5) nm, (1540.721.5) nm 
and (4.4kO.8) nm, respectively. After fixing these thick- 
nesses the optical constants for the GacS,In0,4gP layer were 
extracted over the entire measured range from 0.8 to 5.0 eV. 
Note that the thicknesses obtained in this procedure are de- 
pendent on the initially assumed below-gap optical con- 
stants, and that the subsequently determined optical con- 
stants, over the full spectral range, are dependent on these 
thicknesses. Therefore, this procedure should produce in the 
end virtually the same refractive index as initially assumed in 
the below-gap region. 
Figure 2 shows the resulting N and k values. For com- 
parison, the above-gap pseudovalues from Eq. (2) are also 
shown. Above the gap, the Ga0.,,In0,4gP layer is optically 
thick, so the only difference between pseudo- and extracted 
values is the mathematical removal of the oxide layer in the 
latter. Note the large distorting effect that the overlayer has 
on the pseudovalues (the amount of distortion depends on the 
oxide thickness). The below-gap pseudovalues contain the 
J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. 3417 
Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
2.5 3.5 
Energy in eV 
-0.05 
1.70 1.80 1.90 2.00 2.10 
Energy in eV 
-0.2’ 
1.5 2.5 3.5 4.5 5.5 
Energy in eV 
FIG. 2. Refractive index N and extinction coefficient k for Gan,,,InO.,gP, 
Aalcx= 1 X 10m4, extracted directly from experimental data (short-dashed 
line) and after correction including KK analysis (solid line). Pseudovalues 
(long-dashed line) shown for comparison above the E, critical point. 
interference oscillations shown in Fig. 1. For these reasons 
pseudovalues are not generally useful for modeling multiple- 
layer structures (though they may be adequate for determin- 
ing critical point energies in a single layer, as in Ref. 11, if 
the oxide overlayer is not too thick). 
Since the surface and also the interfaces of real systems 
are not ideal, one will always find deviations between gen- 
erated and experimental values, especially below the band 
gap where the material is transparent and multiple reflections 
occur. Consequently many weak artificial oscillations appear 
in the transparent range of the extracted optical constants 
(Fig. 2). We used the following technique to correct the op- 
tical constants with knowledge of N and k above the band 
gap. We assume an excitonic model for the band edge for 
this material, using the analytic form from Eqs. (7) and (8). 
Thus, in the vicinity of the band edge, 
9.87X lo-” 
k(hw)=-A nw 
27rJm 
X 1 -exp{2rr[lAE(exc)(‘)l/(fiti-EG)]1’2}’ (9) 
1.3 1.4 1.5 1.7 1.8 
where all energies are expressed in eV. The values for the 
effective masses (m,,=O.63, m,=0.12) at the I‘ point in the 
Brillouin zone and the static dielectric constant (edc= 11.75) 
were taken from a simple interpolation between the values 
for GaP and InP.6*7 Thus, only two parameters, A and E,, 
were necessary to fit the analytical expression to the ex- 
Energy in eV 
FIG. 4. Refractive index before (short-dashed line) and after (solid line) KK 
correction, compared with Adachi mode1 (long-dashed line). 
FIG. 3. K extracted from measured values (dashed line: same as short- 
dashed line of Fig. 2), and fit (solid line) of the excitonic band-edge absorp- 
tion model to dashed line. 
tracted values for the extinction coefficient. Note that the 
excitonic energy shift IA E(exc)(‘)l is less than 0.01 eV which 
is below our measurement resolution. 
As a result of the fitting procedure (see Fig. 3) we ob- 
served a band-gap energy of E, = E. = 1.85 +0.02 eV and an 
amplitude of (2.5+0.05)X lo”, which corresponds to an os- 
cillator strength of roughly 5% for the excitonic transition if 
one compares these values to the quantum mechanically ex- 
pressed excitonic absorption coefficient given by Wolfe, Ho- 
lonyak, and Stillman.” In addition to the excitonic band-edge 
shape, we assumed that k is zero below the band-gap energy. 
The extinction coefficient spectra are then correct below and 
just above E. (up to 1.99 eV). Finally, we completed this 
spectral analysis at high energies by adding a fictitious oscil- 
lator to model the optical function above 5 eV. This con- 
trolled the numerically extracted values for N which were 
evaluated with a KK analysis over the entire spectral range. 
Note that N and k were transformed into the dielectric func- 
tion format for this purpose.- In Fig. 2, the original extracted 
and the KK calculated optical constants for G~&Q.~~P are 
shown in the spectral range from 0.8 to 5.0 eV. Figure 4 
shows the refractive index from 1.24 to 1.80 eV, before and 
after KK analysis, compared with the initial values for N 
assumed from Adachi’s model. The final below-gap N values 
are very close to the initially assumed values, as expected. 
3.60 I I I 
- - N af&er K-K analysis 
3.50- -----N extracted 
--.-N afler Adachi 
3418 J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. 
Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp
TABLE I. Critical point energies from present work, compared with other 
published values, and with the binary end-point values. 
Material 
IlllJ 
GaP 
GaJIl, -,P 
1.35” 
2.14” 
1.87?0.02= 
1.85d 
1.8@ 
1.87’ 
1.947g (x=0.49) 
El 
WI 
3.10a 
3.70” 
3.21so.02c 
3.2439 (x=0.51) 
Ea 
9.61b 
9.09b 
9.43+0.02’ 
9.34h 
‘Adachi (Ref. 4). 
hPalik [Ref. 13). 
‘Present work (x-0.51); Eu and E, from second-derivative line-shape 
analysis, e, from K analysis. 
dPresent work, excitonic band-edge fitting. 
‘Present work, photoluminescence (PL) measurement. 
rFouquet and co-workers (Ref. 13) results (x=0.50). 
sBiswas et al. (Ref. 11) (gas-source molecular-beam epitaxy): E, from PL, 
Et from a straight-line formula tit to SE results (without oxide removal) 
over a range of X. 
hInterpolation between GaP and InP (Ref. 13). 
Excellent agreement between extracted and calculated refrac- 
tive index was achieved. Note that do was found from the 
KK analysis to be 9.43kO.03, which compares well with a 
linear interpolation between the c, values for GaP and InP. l3 
We have also determined the critical point energies using 
a numerical algorithm fitting to the smoothed second deriva- 
tives of the optical constants.“” The results are shown in 
Table I, compared with values for GaP, InP, and results from 
earlier photoluminescence and ellipsometric measurements 
on Gaa511na,19P.“*‘2 Note that only E. and Et were found 
here, and not Es-!-A0 or El-l-At, which are very weak com- 
ponents at room temperature. The E, found here is in good 
agreement with the result above for the excitonic band-edge 
fitting procedure. Photoluminescence measurement of EO 
also compares well with the ellipsometrically determined 
value for the same sample. Fouquet and co-workers studied 
the E. gap energies from partially ordered Gao,$-rO.sP 
samples grown by metal-organic chemical-vapor deposition 
(MOCVD) at different temperatures. They found that the 
spectral position of the E. critical point transition determined 
by photohuninescence is very sensitive to growth conditions 
and varies from 1.86 to 1.99 eV.” Biswas et al. have deter- 
mined E, by ellipsometric analysis of samples grown by gas- 
source molecular-beam epitaxy over a range of 
compositions.” Although they did not account for the oxide 
layer, this probably had little effect on the extracted El val- 
ues; reasonable agreement with our result is obtained for 
x=0.51. 
V. SUMMARY 
We have presented the optical constants of G~,srIno.4gP 
on GaAs grown by MOVPE, determined by VASE in the 
range of 0.8-5.0 eV at room temperature. They differ sig- 
nificantly from previously reported pseudovalues, in that the 
effects due to a native oxide overlayer were removed math- 
ematically. A KK analysis and an excitonic band-edge ab- 
sorption mathematical model were used to reduce the optical 
constants fluctuations in the below-gap region. Finally the E, 
and El critical point energies were determined numerically 
using a smoothing derivative algorithm, for comparison with 
those from earlier work. 
ACKNOWLEDGMENTS 
We thank Dr. B. Reinlander of University Leipzig, Ger- 
many, for valuable discussions, and Dr. R. Schwabe for pho- 
toluminescence measurements. 
‘R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light 
(North-Holland, Amsterdam, 1977). 
zD. E. Aspnes and J. B. Theeten, Phys. Rev. B 20, 3292 (1979). 
3 K. Seeger, Semiconductor Physics (Springer, Heidelberg, 1989), Chap. 11. 
‘S. Adachi, Phys. Rev. B 35, 7454 (1987). 
‘S. ZBlner, Appl. Phys. Lett. 63, 2523 (1993). 
“S. Ada&i, J. Appl. Phys. 58, Rl (1985). 
7Semiconductors: Group IV Elements and III-V Compounds, edited by 0. 
Madelung (Springer, Berlin, 1991). 
‘C. M. Wolfe, N. Holonyak, Jr., and C. E. Stillman, Physical Properties of 
Semiconductors (Prentice-Hall, Englewood Cliffs, NJ, 1989). 
‘D. E. Aspnes, in Handbook on Semiconductors, edited by M. Balkanski 
(North-Holland. Amsterdam, 1980). Vol. 2, p. 109. 
“J W  Garland, C. Kim, H. Abad, and P. M. Raccah, Phys. Rev. B 41.7602 
&990). 
“D. Biswas, H. Lee, A. Salvador, M. V. Klein, and H. Morkof, J. Vat. Sci. 
Technol. B 10, 962 (1992). 
“J. E. Fouquet, V. M. Robbins, and S. J. Rosner, Appl. Phys. L&t. 57, 1566 
(1990). 
‘3Handbook ef Optical Constants of Solids, edited by E. D. Palik (Aca- 
demic, New York, 1985). 
J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. 3419 
Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp


Optical constants of Ga\u3csub\u3e\u3ci\u3ex\u3c/i\u3e\u3c/sub\u3eIn\u3csub\u3e1-\u3ci\u3ex\u3c/i\u3e\u3c/sub\u3eP lattice matched to GaAs

DigitalCommons@University of Nebraska

University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Faculty Publications from the Department of Electrical and Computer Engineering Electrical & Computer Engineering, Department of 4-1-1995 Optical constants of GaxIn1-xP lattice matched to GaAs Mathias Schubert University of Nebraska-Lincoln, mschubert4@unl.edu V. Gottschalch University of Leipzig Craig M. Herzinger University of Nebraska-Lincoln Huade Yao University of Nebraska-Lincoln Paul G. Snyder University of Nebraska-Lincoln, psnyder1@unl.edu See next page for additional authors Follow this and additional works at: https://digitalcommons.unl.edu/electricalengineeringfacpub  Part of the Electrical and Computer Engineering Commons Schubert, Mathias; Gottschalch, V.; Herzinger, Craig M.; Yao, Huade; Snyder, Paul G.; and Woollam, John A., "Optical constants of GaxIn1-xP lattice matched to GaAs" (1995). Faculty Publications from the Department of Electrical and Computer Engineering. 60. https://digitalcommons.unl.edu/electricalengineeringfacpub/60 This Article is brought to you for free and open access by the Electrical & Computer Engineering, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Faculty Publications from the Department of Electrical and Computer Engineering by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Authors Mathias Schubert, V. Gottschalch, Craig M. Herzinger, Huade Yao, Paul G. Snyder, and John A. Woollam This article is available at DigitalCommons@University of Nebraska - Lincoln: https://digitalcommons.unl.edu/electricalengineeringfacpub/60 Optical constants of GaJn, -,P lattice matched to GaAs Mathias Schubert University of Leipzig, Faculty of Physics and Geoscience, Department qf Semiconductor Physics, Linnbtrasse 5, 04103 Leipzig, Germany V. Gottschalch University of LeipGg, Faculty of Chemistry and Mineralogy, Department of Solid State Chemistry Linnkstrasse 3, 04103 Leipzig, Germany Craig M. Herzinger, Huacie Yao, Paul G. Snyder, and John A. Woollam University qf Nebraska, Cent&r for Microelectronic and Optical Material Research and Department of Electrical Engineering, Lincoln, Nebraska 68.588 (Received 13 October 1993; accepted for publication 23 November 1994) The optical constants of GaeslInu4sP have been determined from 0.8 to 5.0 eV using variable-angle spectroscopic ellipsometry measurements at room temperature. The metal-organic vapor-phase-epitaxy-grown samples were x-ray analyzed to confirm lattice matching to the GaAs substrate. The effects of the native oxide were numerically removed from the data to determine the intrinsic optical constants. This is important because the optical constants reported become generally useful for modeling multiple-layer structures. A Kramers-Kronig analysis was used to reduce interference-related fluctuations in the below-gap refractive index. Near the band edge a mathematical form for excitonic absorption was included. Critical point energies were extracted using a numerical second-derivative fitting algorithm. 0 1995 American Institute of Physics. I. INTRODUCTION L. Because of the large band gap of Ga&n0.49P lattice matched to GaAs, this material is promising for use in visible optoelectronics. Ga,I.n,-,P can be grown with high purity and excellent morphology by metal-organic vapor-phase ep- itaxy (MOVPE) using a wide choice of growth precursors over a wide range of compositions. We report in this article fully analyzed optical constants of GaInP lattice matched to GaAs. Variable-angle spectroscopic ellipsometry (VASE) was used, including numerical oxide layer removal, and Kramers-Kronig (KK) analysis to reduce the fluctuations of the refractive index in the below-band-gtip region. We also determine 6, from the full KK analysis, where 6% means the dielectric function at frequencies sufficiently high that free- carrier effects are negligible. II. THEORY The complex reflectance ratio p=RJRB ,-where R, and R, are the reflection coefficients for light polarized parallel and perpendicular to the plane of incidence, respectively, can be determined accurately by spectroscopic ellipsometry. The ratio has been traditionally defined as p=tan w exp(iA), (14 and is a function of Q and A, the amplitude and phase of the complex number. Results of ellipsometric measurements are often expressed as *jj=q(tiOi ,@j) 0) and A,j=A(~iwi,~j), Cl4 where L.+ is the photon energy and ~j is the external angle of incidence relative to the sample normal. The pseudo- dielectric function (E) can be obtained from the ellipsometri- tally measured values of pij, based on a two-phase model (ambient/substrate),’ l-pij 2 ,= E, - I( 1 1 +Pij sin* CDj tan2 cPj+ sin” “j 1 , where E, is the dielectric function of the ambient and’as- sumed to be constant. If the sample contains multiple layers (most samples have at least an oxide layer or surface,rough- ness) the SE data must be numerically fit. For this purpose; a model must be assumed and ~\I, and A, as in Eqs. (lb) and (lc) are generated and compared with measured data for the same wavelength and angle of incidence set. qjj and A, depend on a set of parameters, e.g., optical constants and thicknesses for each layer, interface qualities, surface prop- erties, etc. The model should include all necessary character- istics to realistically simulate the behavior of the actual sample. A regression analysis is used to vary the model pa- rameters until the generated and measured values match as closely as possible. This is done by minimizing the mean square error (MSE) function, defined as where i and j are indices for the wavelength and angle data sets, and the aij are the standard deviations of the measure- ments. Note that one should know as much as possible about the physical structure of a sample since the regression analy- sis can potentially yield ambiguous results. If a layer consists of an isotropic mixture of two mate- rials a and b, with the microstructure much less then a 3416 J. Appl. Phys. 77 (7), 1 April 1995 0021-8979/95/77(7)13416/4/$6.00 Q 1995 American Institute of Physics Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jspfraction of the wavelength of light, then the macroscopic dielectric function can be described with the Bruggemann effective medium approximation (EMA), E,- E f*,+,,+fb~=o, a - (4) where G, %, f, , and fb are the dielectric functions and the volume fractions of materials a and b, respective1y.a For the absorption spectra of semiconductors near the lowest direct band gap, it is well known that in addition to the atomic ionization continuum there are also broadened shoulders due to excitons. This can be described in simple form using coulomb forces, and a “dielectrically diluted” hydrogen atom model, where the binding energy is AE(exc.)(‘*)= - 8thcy Eo)2 -$ c +25X IO-“. VASE data, measured at room temperature, were taken on each sample at several angles of incidence and over the spectral range from E=0.8 to 5.0 eV. and where ,LL is the reduced effective mass given by IV. RESULTS AND DISCUSSIONS P -l=(m;‘+m;*). (6) en, Q, and h are the permittivity of free space, the static dielectric constant, and Plan&s constant, respectively. An analytic expression of the above gap absorption for a semi- conductor with the simple band model including excitonic effects is given by (7) where E, is the band-gap energy and A an arbitrary ampIitude.3 Since the extinction coefficient k is related to the absorption coefficient by the relation where c is the velocity of light in vacuum, a relation for k including excitonic effects for the band edge can be found. A detailed discussion of our VASE data and its analysis using this excitonic band edge model to smooth the fluctua- tions of the apparent refractive index below the band gap is presented in Sec. IV. III. EXPERIMENT Four samples of GaesiIn,-,.49P layers with nominal thick- nesses of 1600 nm and GaAs buffer layers -20 nm thick were epitaxially grown at 720 “C by MOVPE on liquid- encapsulated Czochralski GaAs substrates. The substrates were oriented along the [OOI] direction, with a tilt of 2” toward the [llO] direction. The samples were grown under different partial In vapor pressures to obtain different misfit lattice constant values. Partial pressures [I/(In+Ga)] of 0.67, 0.66, 0.64, and 0.59 were recorded for the four samples. The corresponding perpendicular lattice mis- matches, determined from x-ray measurements, were Aala=+lxlO-4 (two samples), +2x 10-4, and 3.8, I I I $ 3.4- -$ 3.2- $ 3.0- 1 - -Model Fit - ---Exp 72.5’ 212Wi”V 1.95 Energy in eV PIG. 1. Experimental and generated (best-fit) data for (N) using reference data for Ga&n,,,49P and oxide optical constants. The two samples with the smallest perpendicular lattice mismatch (Aala = + 1 X IO-“) were independently used to determine the optical constants of the Gao,5,1na.49P layer and both results were found to be the same. First we assumed the refractive index N below the band gap, where k was set to zero, using Ada&i’s model for the optical constants of GaP and InP and linearly interpolating. between all model parameters.4 The oxide in the regression analysis was mod- eled using an appropriately weighted EMA of (51%) GaP oxide and {49%)InP oxide, using the method of Zollner to find the reference spectra for GaP oxide and InP oxide over a wide spectral range.’ The next step was to fit the generated data to the experi- mental data from 1.24 to 1.80 eV (below gap), using optical constants from Ada&i’s model and an oxide layer to deter- mine the thicknesses of all layers (see Fig. 1). The generated values closely follow all oscillations in (N) which are due to multiple internal reflections. Thicknesses were obtained for the Gacs11n,.4,P layer and the oxide layer, for both samples, as (1554.521.0) nm and (3.920.5) nm, (1540.721.5) nm and (4.4kO.8) nm, respectively. After fixing these thick- nesses the optical constants for the GacS,In0,4gP layer were extracted over the entire measured range from 0.8 to 5.0 eV. Note that the thicknesses obtained in this procedure are de- pendent on the initially assumed below-gap optical con- stants, and that the subsequently determined optical con- stants, over the full spectral range, are dependent on these thicknesses. Therefore, this procedure should produce in the end virtually the same refractive index as initially assumed in the below-gap region. Figure 2 shows the resulting N and k values. For com- parison, the above-gap pseudovalues from Eq. (2) are also shown. Above the gap, the Ga0.,,In0,4gP layer is optically thick, so the only difference between pseudo- and extracted values is the mathematical removal of the oxide layer in the latter. Note the large distorting effect that the overlayer has on the pseudovalues (the amount of distortion depends on the oxide thickness). The below-gap pseudovalues contain the J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. 3417 Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp2.5 3.5 Energy in eV -0.05 1.70 1.80 1.90 2.00 2.10 Energy in eV -0.2’ 1.5 2.5 3.5 4.5 5.5 Energy in eV FIG. 2. Refractive index N and extinction coefficient k for Gan,,,InO.,gP, Aalcx= 1 X 10m4, extracted directly from experimental data (short-dashed line) and after correction including KK analysis (solid line). Pseudovalues (long-dashed line) shown for comparison above the E, critical point. interference oscillations shown in Fig. 1. For these reasons pseudovalues are not generally useful for modeling multiple- layer structures (though they may be adequate for determin- ing critical point energies in a single layer, as in Ref. 11, if the oxide overlayer is not too thick). Since the surface and also the interfaces of real systems are not ideal, one will always find deviations between gen- erated and experimental values, especially below the band gap where the material is transparent and multiple reflections occur. Consequently many weak artificial oscillations appear in the transparent range of the extracted optical constants (Fig. 2). We used the following technique to correct the op- tical constants with knowledge of N and k above the band gap. We assume an excitonic model for the band edge for this material, using the analytic form from Eqs. (7) and (8). Thus, in the vicinity of the band edge, 9.87X lo-” k(hw)=-A nw 27rJm X 1 -exp{2rr[lAE(exc)(‘)l/(fiti-EG)]1’2}’ (9) 1.3 1.4 1.5 1.7 1.8 where all energies are expressed in eV. The values for the effective masses (m,,=O.63, m,=0.12) at the I‘ point in the Brillouin zone and the static dielectric constant (edc= 11.75) were taken from a simple interpolation between the values for GaP and InP.6*7 Thus, only two parameters, A and E,, were necessary to fit the analytical expression to the ex- Energy in eV FIG. 4. Refractive index before (short-dashed line) and after (solid line) KK correction, compared with Adachi mode1 (long-dashed line). FIG. 3. K extracted from measured values (dashed line: same as short- dashed line of Fig. 2), and fit (solid line) of the excitonic band-edge absorp- tion model to dashed line. tracted values for the extinction coefficient. Note that the excitonic energy shift IA E(exc)(‘)l is less than 0.01 eV which is below our measurement resolution. As a result of the fitting procedure (see Fig. 3) we ob- served a band-gap energy of E, = E. = 1.85 +0.02 eV and an amplitude of (2.5+0.05)X lo”, which corresponds to an os- cillator strength of roughly 5% for the excitonic transition if one compares these values to the quantum mechanically ex- pressed excitonic absorption coefficient given by Wolfe, Ho- lonyak, and Stillman.” In addition to the excitonic band-edge shape, we assumed that k is zero below the band-gap energy. The extinction coefficient spectra are then correct below and just above E. (up to 1.99 eV). Finally, we completed this spectral analysis at high energies by adding a fictitious oscil- lator to model the optical function above 5 eV. This con- trolled the numerically extracted values for N which were evaluated with a KK analysis over the entire spectral range. Note that N and k were transformed into the dielectric func- tion format for this purpose.- In Fig. 2, the original extracted and the KK calculated optical constants for G~&Q.~~P are shown in the spectral range from 0.8 to 5.0 eV. Figure 4 shows the refractive index from 1.24 to 1.80 eV, before and after KK analysis, compared with the initial values for N assumed from Adachi’s model. The final below-gap N values are very close to the initially assumed values, as expected. 3.60 I I I - - N af&er K-K analysis 3.50- -----N extracted --.-N afler Adachi 3418 J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jspTABLE I. Critical point energies from present work, compared with other published values, and with the binary end-point values. Material IlllJ GaP GaJIl, -,P 1.35” 2.14” 1.87?0.02= 1.85d 1.8@ 1.87’ 1.947g (x=0.49) El WI 3.10a 3.70” 3.21so.02c 3.2439 (x=0.51) Ea 9.61b 9.09b 9.43+0.02’ 9.34h ‘Adachi (Ref. 4). hPalik [Ref. 13). ‘Present work (x-0.51); Eu and E, from second-derivative line-shape analysis, e, from K analysis. dPresent work, excitonic band-edge fitting. ‘Present work, photoluminescence (PL) measurement. rFouquet and co-workers (Ref. 13) results (x=0.50). sBiswas et al. (Ref. 11) (gas-source molecular-beam epitaxy): E, from PL, Et from a straight-line formula tit to SE results (without oxide removal) over a range of X. hInterpolation between GaP and InP (Ref. 13). Excellent agreement between extracted and calculated refrac- tive index was achieved. Note that do was found from the KK analysis to be 9.43kO.03, which compares well with a linear interpolation between the c, values for GaP and InP. l3 We have also determined the critical point energies using a numerical algorithm fitting to the smoothed second deriva- tives of the optical constants.“” The results are shown in Table I, compared with values for GaP, InP, and results from earlier photoluminescence and ellipsometric measurements on Gaa511na,19P.“*‘2 Note that only E. and Et were found here, and not Es-!-A0 or El-l-At, which are very weak com- ponents at room temperature. The E, found here is in good agreement with the result above for the excitonic band-edge fitting procedure. Photoluminescence measurement of EO also compares well with the ellipsometrically determined value for the same sample. Fouquet and co-workers studied the E. gap energies from partially ordered Gao,$-rO.sP samples grown by metal-organic chemical-vapor deposition (MOCVD) at different temperatures. They found that the spectral position of the E. critical point transition determined by photohuninescence is very sensitive to growth conditions and varies from 1.86 to 1.99 eV.” Biswas et al. have deter- mined E, by ellipsometric analysis of samples grown by gas- source molecular-beam epitaxy over a range of compositions.” Although they did not account for the oxide layer, this probably had little effect on the extracted El val- ues; reasonable agreement with our result is obtained for x=0.51. V. SUMMARY We have presented the optical constants of G~,srIno.4gP on GaAs grown by MOVPE, determined by VASE in the range of 0.8-5.0 eV at room temperature. They differ sig- nificantly from previously reported pseudovalues, in that the effects due to a native oxide overlayer were removed math- ematically. A KK analysis and an excitonic band-edge ab- sorption mathematical model were used to reduce the optical constants fluctuations in the below-gap region. Finally the E, and El critical point energies were determined numerically using a smoothing derivative algorithm, for comparison with those from earlier work. ACKNOWLEDGMENTS We thank Dr. B. Reinlander of University Leipzig, Ger- many, for valuable discussions, and Dr. R. Schwabe for pho- toluminescence measurements. ‘R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). zD. E. Aspnes and J. B. Theeten, Phys. Rev. B 20, 3292 (1979). 3 K. Seeger, Semiconductor Physics (Springer, Heidelberg, 1989), Chap. 11. ‘S. Adachi, Phys. Rev. B 35, 7454 (1987). ‘S. ZBlner, Appl. Phys. Lett. 63, 2523 (1993). “S. Ada&i, J. Appl. Phys. 58, Rl (1985). 7Semiconductors: Group IV Elements and III-V Compounds, edited by 0. Madelung (Springer, Berlin, 1991). ‘C. M. Wolfe, N. Holonyak, Jr., and C. E. Stillman, Physical Properties of Semiconductors (Prentice-Hall, Englewood Cliffs, NJ, 1989). ‘D. E. Aspnes, in Handbook on Semiconductors, edited by M. Balkanski (North-Holland. Amsterdam, 1980). Vol. 2, p. 109. “J W  Garland, C. Kim, H. Abad, and P. M. Raccah, Phys. Rev. B 41.7602 &990). “D. Biswas, H. Lee, A. Salvador, M. V. Klein, and H. Morkof, J. Vat. Sci. Technol. B 10, 962 (1992). “J. E. Fouquet, V. M. Robbins, and S. J. Rosner, Appl. Phys. L&t. 57, 1566 (1990). ‘3Handbook ef Optical Constants of Solids, edited by E. D. Palik (Aca- demic, New York, 1985). J. Appl. Phys., Vol. 77, No. 7, 1 April 1995 Schubert et al. 3419 Downloaded 04 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp

https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1059&amp;context=electricalengineeringfacpub

Optical constants of Ga\u3csub\u3e\u3ci\u3ex\u3c/i\u3e\u3c/sub\u3eIn\u3csub\u3e1-\u3ci\u3ex\u3c/i\u3e\u3c/sub\u3eP lattice matched to GaAs

Abstract

Similar works

Full text

Available Versions

UNL | Libraries

DigitalCommons@University of Nebraska