Spectroscopic ellipsometry is used to measure the dielectric function of heavily doped p-type GaAs for wave numbers from 100 to 2000 cm-1. Due to partial filling of the heavy- and light-hole valence bands, heavy holes as well as light holes form a multiple-component plasma coupled with longitudinal optical phonons. Line-shape analysis of the infrared response allows differentiating between light- and heavy-hole contributions to the carrier plasma, and the results observed suggest nonparabolicity effects of the heavy- and light-hole valence bands in GaAs

Schubert, Mathias

Thompson, Daniel W.

Woollam, John A.

Zangooie, S.

English

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University of Nebraska - Lincoln 
DigitalCommons@University of Nebraska - Lincoln 
Faculty Publications from the Department of 
Electrical and Computer Engineering 
Electrical & Computer Engineering, Department 
of 
2-12-2001 
Infrared response of multiple-component free-carrier plasma in 
heavily doped p-type GaAs 
S. Zangooie 
University of Nebraska-Lincoln 
Mathias Schubert 
University of Nebraska-Lincoln, mschubert4@unl.edu 
Daniel W. Thompson 
University of Nebraska-Lincoln, dthompson2@unl.edu 
John A. Woollam 
University of Nebraska-Lincoln, jwoollam1@unl.edu 
Follow this and additional works at: https://digitalcommons.unl.edu/electricalengineeringfacpub 
 Part of the Electrical and Computer Engineering Commons 
Zangooie, S.; Schubert, Mathias; Thompson, Daniel W.; and Woollam, John A., "Infrared response of 
multiple-component free-carrier plasma in heavily doped p-type GaAs" (2001). Faculty Publications from 
the Department of Electrical and Computer Engineering. 3. 
https://digitalcommons.unl.edu/electricalengineeringfacpub/3 
This Article is brought to you for free and open access by the Electrical & Computer Engineering, Department of at 
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DigitalCommons@University of Nebraska - Lincoln. 
Infrared response of multiple-component free-carrier plasma
in heavily doped p-type GaAs
S. Zangooie, M. Schubert,a) D. W. Thompson, and J. A. Woollamb)
Center for Microelectronic and Optical Materials Research and Department of Electrical Engineering,
University of Nebraska, Lincoln, Nebraska 68588-0511
~Received 24 August 2000; accepted for publication 20 November 2000!
Spectroscopic ellipsometry is used to measure the dielectric function of heavily dopedp-type GaAs
for wave numbers from 100 to 2000 cm21. Due to partial filling of the heavy- and light-hole valence
bands, heavy holes as well as light holes form a multiple-component plasma coupled with
longitudinal optical phonons. Line-shape analysis of the infrared response allows differentiating
between light- and heavy-hole contributions to the carrier plasma, and the results observed suggest
nonparabolicity effects of the heavy- and light-hole valence bands in GaAs. ©2001 American
Institute of Physics.@DOI: 10.1063/1.1343490#
Infrared optical properties of semiconductors are influ-
enced significantly by the free-carrier response.1,2 Optical
characteristics ofp-type GaAs is complicated by penetration
of the Fermi level into the valence bands at theG point of the
Brillouin zone, providing contributions from heavy holes
~hhs! as well as light holes~lhs! to the free-carrier plasma.3,4
In addition, an accurate characterization of GaAs surfaces
requires consideration of further complexities such as a de-
pleted surface layer caused by Fermi-level pinning to mid-
band-gap surface states.4,5 Infrared spectroscopic ellipsom-
etry ~IRSE! is nondestructive and intrinsically sensitive to
thin-film optical properties. IRSE has been employed previ-
ously for characterization of carrier-depleted surface layers
in n-type GaAs~Ref. 5! andn-type GaN.6 Electrical proper-
ties of doped semiconductors are traditionally measured us-
ing the electrical Hall effect. However, complexities nor-
mally accompanied with electrical Hall measurements of
p-type GaAs, such as surface contamination due to electrical
contacts, or non-Ohmic contact behavior,4 motivates em-
ployment of additional nondestructive and accurate charac-
terization techniques. In this work, IRSE is used to simulta-
neously determine the real and imaginary parts of the
dielectric function of heavily dopedp-type GaAs. It is shown
that IRSE can be used to differentiate between contributions
from different carrier types in multiple-component plasma
interactions, i.e., between hh and lh plasma components as
well as between their contributions to the dielectric function.
Here, carrier concentration and mobility parameters are of
particular interest. Accurate knowledge of these parameters
is crucial in design of device structures that utilize heavily
dopedp-type GaAs. For data analysis it is assumed that the
bands are parabolic, i.e., the effective masses are assumed to
be the same as those from the top of the valence bands at the
G point of the Brillouin zone, and do not depend on the
Fermi-level energy. The results obtained here, however, sug-
gest nonparabolicity effects of the valence bands.
IRSE measurements at room temperature were carried
out for wave numbers from 500 to 2000 cm21, with 2 cm21
resolution, using a commercially available Fourier-
transform-based spectroscopic ellipsometer at incidence
angles of 50° and 70°. Data in the 100–500 cm21 spectral
region ~far-IR! were acquired using a prototype IRSE
equipped with a He-cooled bolometer detector unit. The
far-IR measurements are necessary for accurate determina-
tion of the GaAs phonon-mode frequencies and the free-
carrier parameters. Zn-dopedp-type GaAs wafers with~100!
crystal orientation were obtained from the Materials Tech-
nology International~MTI ! Corporation. The carrier concen-
tration and carrier mobility, reported by the manufacturer,
are 1.431019cm23 and 70 cm2 V21 s21, respectively.
The measured quantity in ellipsometry7 is the so-called
complex reflectance ratio, which is expressed as
r5
Rp
Rs
5tan~c!exp~ iD!. ~1!
Rp andRs are the complex reflection coefficients forp- and
s-polarized light, respectively. The amplitude ratio tan(c)
and the relative difference in phase changeD for the p- and
-polarized light components upon reflection from the
sample surface are the standard ellipsometry parameters.
From experimentally determinedr, information can be ob-
tained about the optical properties of the material under
study in terms of the complex dielectric function«5«1
1 i«2 . Optical properties of thep-type GaAs wafers in terms
of the « are modeled using the following expression:8
«~v!5«`1
«`~vLO
2 2vTO
2 !
vTO
2 2v22 ivg
2
«`vphh
2
v~v1 iGhh!
2
«`vplh
2
v~v1 iG lh!
.
~2!
«` andg are the high-frequency dielectric function and the
transverse-optical~TO! phonon-damping parameters, respec-
tively. vTO and vLO represent the IR-active TO and
longitudinal-optical~LO! phonon frequencies, respectively.
The third and fourth terms in Eq.~2! are the Drude expres-
sions, which are introduced to consider effects of the
a!Present address: Universita¨t Leipzig, Fakulta¨t für Physik und Geowissen-
schaften, Arbeitsgruppe Festko¨rperoptik, Linnéstrasse 5, Leipzig 04103,
Germany.
b!Author to whom correspondence should be addressed; electronic mail:
jwoollam@unl.edu
APPLIED PHYSICS LETTERS VOLUME 78, NUMBER 7 12 FEBRUARY 2001
9370003-6951/2001/78(7)/937/3/$18.00 © 2001 American Institute of Physics
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multiple-component free-carrier plasma, which consists of
heavy hole and light hole components in the highly doped
semiconductor. The screened plasmon frequencies (vp, lh,hh)
and the plasmon broadening parameters (G lh,hh) are related to
the carrier concentrations (plh,hh) and mobilities (m lh,hh) ac-
cording to the following relations:
vp j5A pje2«0«`mj* , ~3!
G j5
e
m* m j
, j 5 lh,hh. ~4!
«0 , e, and mj* are the vacuum permitivity, electrical unity
charge, and the effective mass of the free carriers, respec-
tively. IRSE is only sensitive to (p/m* ) and 1/(mm* ) be-
cause these ratios parametrize the dielectric response of a
multiple-component carrier plasma. Thus, to calculate the
concentration and mobility of the carriers from«, the param-
etersmj* must be known. For the calculations here the aver-
aged hh effective mass of 0.56m0 , and the averaged lh
effective mass of 0.08m0 are used~m0 is the free-electron
mass!.3,4 Note that one could not differentiate betweenplh
andphh ~or vplh andvphh! if both carrier species would have
the same broadening parameters given by the mass–mobility
product. It is the different strength of absorption of both
types that allows us to separate their different contributions
to «. Furthermore, depending on the choice ofmj* , the
Drude expressions in Eq.~2! provide two different sets of
solutions for each species. Here, the solution with larger car-
rier concentration is assumed to represent the heavy holes
@see, e.g., Eq.~6!#. Note that in Zn-doped GaAs the ionized-
impurity scattering was found as the most significant scatter-
ing mechanism.4 We further note that Kukharskii’s10 factor-
ized form of«, which allows for independent broadening of
TO and LO branches in polar materials, did not improve the
fits to the experimental data and was, therefore, abandoned.
This model was used by Kukharskii to explain IR-reflectivity
spectra of heavily dopedn-type GaAs.10
The difference in the free-carrier concentration between
the depleted layer and the underlying bulk material yields
static electric-field gradients influencing the long-range Cou-
lomb forces. Hence, different LO phonon frequencies are
encountered in the depleted layer and the underlying heavily
doped GaAs. The phenomenon results in a characteristic sig-
nature within the IRSE data above the TO phonon frequency
and near to the LO frequency of intrinsic GaAs. The signa-
ture, which is distinguished as a minimum inc spectra, can
be utilized for determination of the thicknessd of the de-
pleted layer. The phenomenon was first observed by Berre-
man in thin LiF films on silver.11 The Berreman effect is also
reported for SiC homostructures consisting of low-doped ep-
itaxial layers on highly doped substrates.5,12 It is often as-
sumed that at a distance~ qual tod! away from the surface
the carrier concentrationp abruptly rises to the bulk value,
which yields for the depletion depthd5A(2«I s«0f0 /p/e)
where«s512.5 is the static dielectric constant for GaAs, and
ef050.5 eV is the electrostatic potential energy.
4,8 For car-
rier concentrations 1018 and 1019cm23, d equals 26.3 and 8.3
nm, respectively. Note, however, that Eq.~5! overestimates
the depletion layer depth.4 The large wavelength near
vLO(l;26mm!, and negligible changes of the polarization
state of the reflected light by the very thin depletion layer in
the sample studied here, explain lack of a clear depletion-
layer-related signature in our IRSE data. This layer is, there-
fore, not considered in the analysis.
Figure 1 presents experimental and calculated IRSE
spectra in terms ofc and D for the heavily dopedp-type
GaAs wafer. For clarity, data are shown for the 50° angle of
incidence only. Note the excellent agreement between mea-
sured and calculated spectra. Experimental data above the
TO phonon frequency and near the position of the plasma
minimum @«1(v;470 cm
21);1; see, also, Fig. 2# provide
sensitivity to the free-carrier parameters. The best-fit calcu-
lation parameter values and the corresponding confidence in-
tervals are given in Table I. Values of«` , vTO, vLO , andg
for heavily doped GaAs were also reported in Ref. 4, and
compare well to those found here~Table I!. Moreover, the
FIG. 1. Experimental~solid circles! and calculated@solid lines; after Eq.~7!#
c andD spectra of the heavily dopedp-type GaAs sample at a 50° angle of
incidence. The dotted lines represent the best-fit model calculation assuming
(phh /plh)
25(mhh /mlh)
3. The derivative-like structure inc and D at v
;267 cm21 is due to resonant excitation of the GaAs TO mode.
FIG. 2. Real~a! and imaginary part~b! of the dielectric function of GaAs
with a multiple-component plasma using the parameter values given in
Table I ~solid lines!, and with the lattice contribution considered only
~dashed lines!. Separate contributions from heavy~dotted lines! and light
holes~dash-dotted lines! to «1 and«2 are also shown.
938 Appl. Phys. Lett., Vol. 78, No. 7, 12 February 2001 Zangooie et al.
Downloaded 03 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
effective hole mobility, assuming decoupled valence
bands,3,9
meff5
phhm lh1plhm lh
phh1plh
, ~5!
is 76 cm2 V21 s21, which is in good agreement with the Hall
measurements performed on samples reported in Ref. 4 with
similar carrier concentrations, and with 70 cm2/~V s! reported
by the supplier. For a parabolic band,p can be expressed as9
p52S 2pkTm*h2 D
3/2
Nn expS En2EfkT D , ~6!
where En , Ef , k, h, and T are energy at the top of the
valence band, Fermi-level energy, Boltzmann’s and Planck’s
constants, and temperature, respectively. Note that Eq.~6! is
valid when Ef.En14kT. According to Eq. ~6!, in our
sample the position of the Fermi level is above the split-off
~so! valence band ~spin-orbit splitting parameterdSO
50.36 eV!. Consideration of the so contribution to«(mSO
50.165 m0) did not improve the fit to the experimental
data. A consequence of the parabolic-band assumption is that
phh andplh fulfill the following relation:
3,4
phh
plh
5S mhhmlh D
3/2
. ~7!
Using this constraint during analysis deteriorated the agree-
ment between the best-fit parameter calculation and the ex-
perimental data~dotted lines in Fig. 1!. Parametersphh and
plh obtained here assumingmhh50.56 m0 and mlh
50.08 m0 are inconsistent with Eq.~7!. Hence, the
parabolic-band assumption may not hold for large carrier
concentrations. Nonparabolicities of the density-of-states ef-
fective masses for the hh, lh, and so valence bands in GaAs
were predicted from fullk•p theory by Lowney and Kahn.13
A better approach for IRSE data analysis would be to allow
for the nonlinear dependenciesmhh(phh) and mlh(plh), e.g.,
as predicted by Lowney and Kahn, instead of assumingmj*
independent ofpj . However, because IRSE determines the
productpmj* , and notmj* andpj independently, the results
for pj obtained thereby would still rely on the input functions
mj* (pj ). Magneto-optic ellipsometry experiments may
present an alternative to obtain more independent parameter
information.
Figures 2~a! («1) and 2~b! («2) show plasma and lattice
contributions to«, according to Eq.~2! using parameter val-
ues in Table I. For wave numbers below the TO phonon
mode, coupling of hhs to the LO phonons dominates the IR
response, whereas for wave numbers above the TO phonon
frequency« is dominated by lh contributions. In Fig. 2~b! the
light-hole carriers generate most of the IR absorption,
whereas the heavy-hole plasma interaction dominates the ab-
sorption for short wavelengths. Note that a single-component
plasma, employingptot ,meff given in Table I, assumingm*
50.43 m0 ,
4 fails to mimic « of the highly degenerate
p-GaAs wafer measured here. For completeness, Fig. 2 also
contains the experimentalC andD values converted directly
into the so-called pseudodielectric function̂«& ~solid
circles!.14 Because no overlayer was considered during data
analysis^«& equals«, which is sufficiently satisfied here ex-
cept for increasing data noise at the long-wavelength end.
In conclusion, light holes play a more significant role in
the absorption mechanism above the TO phonon frequency,
whereas for lower frequencies carriers with heavier effective
masses dominate the dielectric function line shape. As a con-
sequence, heavy holes and light holes can be separated by an
appropriate IRSE measurement and analysis. For highly
dopedp-type GaAs the IRSE data analysis revealed nonpa-
rabolicity effects.
Financial support for this study was provided in part by
the Swedish Foundation for International Cooperation in Re-
search and Higher Education~STINT!, CMOMR at UNL,
and NSF Contract No. DMI-9901510.
1T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen, R. B. Gregory, T.
Wetteroth, S. R. Wilson, and A. P. Powell, Phys. Rev. B60, 11464
~1999!.
2H. Nakano, T. Sakamoto, and K. Taniguchi, J. Appl. Phys.83, 1384
~1998!.
3S. Adachi,GaAs and Related Materials: Bulk Semiconducting and Super-
lattice Properties~World Scientific, Singapore, 1994!.
4G. Irmer, M. Wenzel, and J. Monecke, Phys. Rev. B56, 9524~1997!.
5J. Humliček, R. Henn, and M. Cardona, Appl. Phys. Lett.69, 2581~1996!.
6A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, Phys.
Rev. B62, 7365~2000!.
7R. M. A. Azzam and N. M. Bashara,Ellipsometry and Polarized Light
~North-Holland, New York, 1987!.
8P. Y. Yu and M. Cardona,Fundamentals of Semiconductors: Physics and
Materials Properties~Springer, Berlin, 1996!.
9C. M. Wolfe, N. Holonyak, Jr., and G. E. Stillman,Physical Properties of
Semiconductors~Prentice-Hall, Englewood Cliffs, NJ, 1989!.
10A. A. Kukharskii, Solid State Commun.13, 1761~1973!.
11D. W. Berreman, Phys. Rev.130, 2193~1963!.
12S. Zollner, J. P. Carrejo, T. E. Tiwald, and J. A. Woollam, Phys. Status
Solidi A 208, R3 ~1998!.
13J. R. Lowney and A. H. Kahn, J. Appl. Phys.64, 447 ~1988!.
14D. E. Aspnes and A. A. Studna, Phys. Rev. B27, 985 ~1983!.
TABLE I. Best-fit parameters including 90% confidence intervals.mhh
50.56m0 andmlh50.08m0 are used during IRSE data analysis. Parameters
obtained by Raman scattering in Ref. 4, for similar carrier concentrations
and mobility, are shown for comparison. The nominal hole concentration
and mobility reported by the sample supplier are 1.431019 cm23 and 70
cm2/V s.
Model parameters This work Ref. 4
«` 10.5260.02 ~10.6!
a
vTO ~cm
21! 267.460.1 268.1
vLO ~cm
21! 291.560.2 291.4
g ~cm21! 3.160.1 2
phh ~cm
23! (1.2660.03)31019 1.5231019
plh ~cm
23! (5.8960.02)31018 8.2031017
ptot5phh1plh ~cm
23! 1.8531019 1.6031019
mhh ~cm
2 V21 s21! 8561 62
m lh ~cm
2 V21 s21! 5761 163
meff ~cm
2 V21 s21! 76 67
aReference 14.
939Appl. Phys. Lett., Vol. 78, No. 7, 12 February 2001 Zangooie et al.
Downloaded 03 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp


Infrared response of multiple-component free-carrier plasma in heavily doped \u3ci\u3ep\u3c/i\u3e-type 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 2-12-2001 Infrared response of multiple-component free-carrier plasma in heavily doped p-type GaAs S. Zangooie University of Nebraska-Lincoln Mathias Schubert University of Nebraska-Lincoln, mschubert4@unl.edu Daniel W. Thompson University of Nebraska-Lincoln, dthompson2@unl.edu John A. Woollam University of Nebraska-Lincoln, jwoollam1@unl.edu Follow this and additional works at: https://digitalcommons.unl.edu/electricalengineeringfacpub  Part of the Electrical and Computer Engineering Commons Zangooie, S.; Schubert, Mathias; Thompson, Daniel W.; and Woollam, John A., "Infrared response of multiple-component free-carrier plasma in heavily doped p-type GaAs" (2001). Faculty Publications from the Department of Electrical and Computer Engineering. 3. https://digitalcommons.unl.edu/electricalengineeringfacpub/3 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. Infrared response of multiple-component free-carrier plasmain heavily doped p-type GaAsS. Zangooie, M. Schubert,a) D. W. Thompson, and J. A. Woollamb)Center for Microelectronic and Optical Materials Research and Department of Electrical Engineering,University of Nebraska, Lincoln, Nebraska 68588-0511~Received 24 August 2000; accepted for publication 20 November 2000!Spectroscopic ellipsometry is used to measure the dielectric function of heavily dopedp-type GaAsfor wave numbers from 100 to 2000 cm21. Due to partial filling of the heavy- and light-hole valencebands, heavy holes as well as light holes form a multiple-component plasma coupled withlongitudinal optical phonons. Line-shape analysis of the infrared response allows differentiatingbetween light- and heavy-hole contributions to the carrier plasma, and the results observed suggestnonparabolicity effects of the heavy- and light-hole valence bands in GaAs. ©2001 AmericanInstitute of Physics.@DOI: 10.1063/1.1343490#Infrared optical properties of semiconductors are influ-enced significantly by the free-carrier response.1,2 Opticalcharacteristics ofp-type GaAs is complicated by penetrationof the Fermi level into the valence bands at theG point of theBrillouin zone, providing contributions from heavy holes~hhs! as well as light holes~lhs! to the free-carrier plasma.3,4In addition, an accurate characterization of GaAs surfacesrequires consideration of further complexities such as a de-pleted surface layer caused by Fermi-level pinning to mid-band-gap surface states.4,5 Infrared spectroscopic ellipsom-etry ~IRSE! is nondestructive and intrinsically sensitive tothin-film optical properties. IRSE has been employed previ-ously for characterization of carrier-depleted surface layersin n-type GaAs~Ref. 5! andn-type GaN.6 Electrical proper-ties of doped semiconductors are traditionally measured us-ing the electrical Hall effect. However, complexities nor-mally accompanied with electrical Hall measurements ofp-type GaAs, such as surface contamination due to electricalcontacts, or non-Ohmic contact behavior,4 motivates em-ployment of additional nondestructive and accurate charac-terization techniques. In this work, IRSE is used to simulta-neously determine the real and imaginary parts of thedielectric function of heavily dopedp-type GaAs. It is shownthat IRSE can be used to differentiate between contributionsfrom different carrier types in multiple-component plasmainteractions, i.e., between hh and lh plasma components aswell as between their contributions to the dielectric function.Here, carrier concentration and mobility parameters are ofparticular interest. Accurate knowledge of these parametersis crucial in design of device structures that utilize heavilydopedp-type GaAs. For data analysis it is assumed that thebands are parabolic, i.e., the effective masses are assumed tobe the same as those from the top of the valence bands at theG point of the Brillouin zone, and do not depend on theFermi-level energy. The results obtained here, however, sug-gest nonparabolicity effects of the valence bands.IRSE measurements at room temperature were carriedout for wave numbers from 500 to 2000 cm21, with 2 cm21resolution, using a commercially available Fourier-transform-based spectroscopic ellipsometer at incidenceangles of 50° and 70°. Data in the 100–500 cm21 spectralregion ~far-IR! were acquired using a prototype IRSEequipped with a He-cooled bolometer detector unit. Thefar-IR measurements are necessary for accurate determina-tion of the GaAs phonon-mode frequencies and the free-carrier parameters. Zn-dopedp-type GaAs wafers with~100!crystal orientation were obtained from the Materials Tech-nology International~MTI ! Corporation. The carrier concen-tration and carrier mobility, reported by the manufacturer,are 1.431019cm23 and 70 cm2 V21 s21, respectively.The measured quantity in ellipsometry7 is the so-calledcomplex reflectance ratio, which is expressed asr5RpRs5tan~c!exp~ iD!. ~1!Rp andRs are the complex reflection coefficients forp- ands-polarized light, respectively. The amplitude ratio tan(c)and the relative difference in phase changeD for the p- and-polarized light components upon reflection from thesample surface are the standard ellipsometry parameters.From experimentally determinedr, information can be ob-tained about the optical properties of the material understudy in terms of the complex dielectric function«5«11 i«2 . Optical properties of thep-type GaAs wafers in termsof the « are modeled using the following expression:8«~v!5«`1«`~vLO2 2vTO2 !vTO2 2v22 ivg2«`vphh2v~v1 iGhh!2«`vplh2v~v1 iG lh!.~2!«` andg are the high-frequency dielectric function and thetransverse-optical~TO! phonon-damping parameters, respec-tively. vTO and vLO represent the IR-active TO andlongitudinal-optical~LO! phonon frequencies, respectively.The third and fourth terms in Eq.~2! are the Drude expres-sions, which are introduced to consider effects of thea!Present address: Universita¨t Leipzig, Fakulta¨t für Physik und Geowissen-schaften, Arbeitsgruppe Festko¨rperoptik, Linnéstrasse 5, Leipzig 04103,Germany.b!Author to whom correspondence should be addressed; electronic mail:jwoollam@unl.eduAPPLIED PHYSICS LETTERS VOLUME 78, NUMBER 7 12 FEBRUARY 20019370003-6951/2001/78(7)/937/3/$18.00 © 2001 American Institute of PhysicsDownloaded 03 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspmultiple-component free-carrier plasma, which consists ofheavy hole and light hole components in the highly dopedsemiconductor. The screened plasmon frequencies (vp, lh,hh)and the plasmon broadening parameters (G lh,hh) are related tothe carrier concentrations (plh,hh) and mobilities (m lh,hh) ac-cording to the following relations:vp j5A pje2«0«`mj* , ~3!G j5em* m j, j 5 lh,hh. ~4!«0 , e, and mj* are the vacuum permitivity, electrical unitycharge, and the effective mass of the free carriers, respec-tively. IRSE is only sensitive to (p/m* ) and 1/(mm* ) be-cause these ratios parametrize the dielectric response of amultiple-component carrier plasma. Thus, to calculate theconcentration and mobility of the carriers from«, the param-etersmj* must be known. For the calculations here the aver-aged hh effective mass of 0.56m0 , and the averaged lheffective mass of 0.08m0 are used~m0 is the free-electronmass!.3,4 Note that one could not differentiate betweenplhandphh ~or vplh andvphh! if both carrier species would havethe same broadening parameters given by the mass–mobilityproduct. It is the different strength of absorption of bothtypes that allows us to separate their different contributionsto «. Furthermore, depending on the choice ofmj* , theDrude expressions in Eq.~2! provide two different sets ofsolutions for each species. Here, the solution with larger car-rier concentration is assumed to represent the heavy holes@see, e.g., Eq.~6!#. Note that in Zn-doped GaAs the ionized-impurity scattering was found as the most significant scatter-ing mechanism.4 We further note that Kukharskii’s10 factor-ized form of«, which allows for independent broadening ofTO and LO branches in polar materials, did not improve thefits to the experimental data and was, therefore, abandoned.This model was used by Kukharskii to explain IR-reflectivityspectra of heavily dopedn-type GaAs.10The difference in the free-carrier concentration betweenthe depleted layer and the underlying bulk material yieldsstatic electric-field gradients influencing the long-range Cou-lomb forces. Hence, different LO phonon frequencies areencountered in the depleted layer and the underlying heavilydoped GaAs. The phenomenon results in a characteristic sig-nature within the IRSE data above the TO phonon frequencyand near to the LO frequency of intrinsic GaAs. The signa-ture, which is distinguished as a minimum inc spectra, canbe utilized for determination of the thicknessd of the de-pleted layer. The phenomenon was first observed by Berre-man in thin LiF films on silver.11 The Berreman effect is alsoreported for SiC homostructures consisting of low-doped ep-itaxial layers on highly doped substrates.5,12 It is often as-sumed that at a distance~ qual tod! away from the surfacethe carrier concentrationp abruptly rises to the bulk value,which yields for the depletion depthd5A(2«I s«0f0 /p/e)where«s512.5 is the static dielectric constant for GaAs, andef050.5 eV is the electrostatic potential energy.4,8 For car-rier concentrations 1018 and 1019cm23, d equals 26.3 and 8.3nm, respectively. Note, however, that Eq.~5! overestimatesthe depletion layer depth.4 The large wavelength nearvLO(l;26mm!, and negligible changes of the polarizationstate of the reflected light by the very thin depletion layer inthe sample studied here, explain lack of a clear depletion-layer-related signature in our IRSE data. This layer is, there-fore, not considered in the analysis.Figure 1 presents experimental and calculated IRSEspectra in terms ofc and D for the heavily dopedp-typeGaAs wafer. For clarity, data are shown for the 50° angle ofincidence only. Note the excellent agreement between mea-sured and calculated spectra. Experimental data above theTO phonon frequency and near the position of the plasmaminimum @«1(v;470 cm21);1; see, also, Fig. 2# providesensitivity to the free-carrier parameters. The best-fit calcu-lation parameter values and the corresponding confidence in-tervals are given in Table I. Values of«` , vTO, vLO , andgfor heavily doped GaAs were also reported in Ref. 4, andcompare well to those found here~Table I!. Moreover, theFIG. 1. Experimental~solid circles! and calculated@solid lines; after Eq.~7!#c andD spectra of the heavily dopedp-type GaAs sample at a 50° angle ofincidence. The dotted lines represent the best-fit model calculation assuming(phh /plh)25(mhh /mlh)3. The derivative-like structure inc and D at v;267 cm21 is due to resonant excitation of the GaAs TO mode.FIG. 2. Real~a! and imaginary part~b! of the dielectric function of GaAswith a multiple-component plasma using the parameter values given inTable I ~solid lines!, and with the lattice contribution considered only~dashed lines!. Separate contributions from heavy~dotted lines! and lightholes~dash-dotted lines! to «1 and«2 are also shown.938 Appl. Phys. Lett., Vol. 78, No. 7, 12 February 2001 Zangooie et al.Downloaded 03 Dec 2007 to 129.93.17.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspeffective hole mobility, assuming decoupled valencebands,3,9meff5phhm lh1plhm lhphh1plh, ~5!is 76 cm2 V21 s21, which is in good agreement with the Hallmeasurements performed on samples reported in Ref. 4 withsimilar carrier concentrations, and with 70 cm2/~V s! reportedby the supplier. For a parabolic band,p can be expressed as9p52S 2pkTm*h2 D3/2Nn expS En2EfkT D , ~6!where En , Ef , k, h, and T are energy at the top of thevalence band, Fermi-level energy, Boltzmann’s and Planck’sconstants, and temperature, respectively. Note that Eq.~6! isvalid when Ef.En14kT. According to Eq. ~6!, in oursample the position of the Fermi level is above the split-off~so! valence band ~spin-orbit splitting parameterdSO50.36 eV!. Consideration of the so contribution to«(mSO50.165 m0) did not improve the fit to the experimentaldata. A consequence of the parabolic-band assumption is thatphh andplh fulfill the following relation:3,4phhplh5S mhhmlh D3/2. ~7!Using this constraint during analysis deteriorated the agree-ment between the best-fit parameter calculation and the ex-perimental data~dotted lines in Fig. 1!. Parametersphh andplh obtained here assumingmhh50.56 m0 and mlh50.08 m0 are inconsistent with Eq.~7!. Hence, theparabolic-band assumption may not hold for large carrierconcentrations. Nonparabolicities of the density-of-states ef-fective masses for the hh, lh, and so valence bands in GaAswere predicted from fullk•p theory by Lowney and Kahn.13A better approach for IRSE data analysis would be to allowfor the nonlinear dependenciesmhh(phh) and mlh(plh), e.g.,as predicted by Lowney and Kahn, instead of assumingmj*independent ofpj . However, because IRSE determines theproductpmj* , and notmj* andpj independently, the resultsfor pj obtained thereby would still rely on the input functionsmj* (pj ). Magneto-optic ellipsometry experiments maypresent an alternative to obtain more independent parameterinformation.Figures 2~a! («1) and 2~b! («2) show plasma and latticecontributions to«, according to Eq.~2! using parameter val-ues in Table I. For wave numbers below the TO phononmode, coupling of hhs to the LO phonons dominates the IRresponse, whereas for wave numbers above the TO phononfrequency« is dominated by lh contributions. In Fig. 2~b! thelight-hole carriers generate most of the IR absorption,whereas the heavy-hole plasma interaction dominates the ab-sorption for short wavelengths. Note that a single-componentplasma, employingptot ,meff given in Table I, assumingm*50.43 m0 ,4 fails to mimic « of the highly degeneratep-GaAs wafer measured here. For completeness, Fig. 2 alsocontains the experimentalC andD values converted directlyinto the so-called pseudodielectric function̂«& ~solidcircles!.14 Because no overlayer was considered during dataanalysis^«& equals«, which is sufficiently satisfied here ex-cept for increasing data noise at the long-wavelength end.In conclusion, light holes play a more significant role inthe absorption mechanism above the TO phonon frequency,whereas for lower frequencies carriers with heavier effectivemasses dominate the dielectric function line shape. As a con-sequence, heavy holes and light holes can be separated by anappropriate IRSE measurement and analysis. For highlydopedp-type GaAs the IRSE data analysis revealed nonpa-rabolicity effects.Financial support for this study was provided in part bythe Swedish Foundation for International Cooperation in Re-search and Higher Education~STINT!, CMOMR at UNL,and NSF Contract No. DMI-9901510.1T. E. Tiwald, J. A. Woollam, S. Zollner, J. Christiansen, R. B. Gregory, T.Wetteroth, S. R. Wilson, and A. P. Powell, Phys. Rev. B60, 11464~1999!.2H. Nakano, T. Sakamoto, and K. Taniguchi, J. Appl. Phys.83, 1384~1998!.3S. Adachi,GaAs and Related Materials: Bulk Semiconducting and Super-lattice Properties~World Scientific, Singapore, 1994!.4G. Irmer, M. Wenzel, and J. Monecke, Phys. Rev. B56, 9524~1997!.5J. Humliček, R. Henn, and M. Cardona, Appl. Phys. Lett.69, 2581~1996!.6A. Kasic, M. Schubert, S. Einfeldt, D. Hommel, and T. E. Tiwald, Phys.Rev. B62, 7365~2000!.7R. M. A. Azzam and N. M. Bashara,Ellipsometry and Polarized Light~North-Holland, New York, 1987!.8P. Y. Yu and M. Cardona,Fundamentals of Semiconductors: Physics andMaterials Properties~Springer, Berlin, 1996!.9C. M. Wolfe, N. Holonyak, Jr., and G. E. Stillman,Physical Properties ofSemiconductors~Prentice-Hall, Englewood Cliffs, NJ, 1989!.10A. A. Kukharskii, Solid State Commun.13, 1761~1973!.11D. W. Berreman, Phys. Rev.130, 2193~1963!.12S. Zollner, J. P. Carrejo, T. E. Tiwald, and J. A. Woollam, Phys. StatusSolidi A 208, R3 ~1998!.13J. R. Lowney and A. H. Kahn, J. Appl. Phys.64, 447 ~1988!.14D. E. Aspnes and A. A. Studna, Phys. Rev. B27, 985 ~1983!.TABLE I. Best-fit parameters including 90% confidence intervals.mhh50.56m0 andmlh50.08m0 are used during IRSE data analysis. Parametersobtained by Raman scattering in Ref. 4, for similar carrier concentrationsand mobility, are shown for comparison. The nominal hole concentrationand mobility reported by the sample supplier are 1.431019 cm23 and 70cm2/V s.Model parameters This work Ref. 4«` 10.5260.02 ~10.6!avTO ~cm21! 267.460.1 268.1vLO ~cm21! 291.560.2 291.4g ~cm21! 3.160.1 2phh ~cm23! (1.2660.03)31019 1.5231019plh ~cm23! (5.8960.02)31018 8.2031017ptot5phh1plh ~cm23! 1.8531019 1.6031019mhh ~cm2 V21 s21! 8561 62m lh ~cm2 V21 s21! 5761 163meff ~cm2 V21 s21! 76 67aReference 14.939Appl. Phys. Lett., Vol. 78, No. 7, 12 February 2001 Zangooie et al.Downloaded 03 Dec 2007 to 129.93.17.223. 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Infrared response of multiple-component free-carrier plasma in heavily doped \u3ci\u3ep\u3c/i\u3e-type GaAs

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