Parametrization of the Optical Constants of AlAs_{x}Sb_{1-x} Alloys in the Range 0.74-6.0 eV

Tae Jung Kim
,
Jun Seok Byun
,
Nilesh Barange
,
Han Gyeo Park
,
Yu Ri Kang
,
Jae Chan Park
,
Young Dong Kim

Journal of the Optical Society of Korea.
2014.
Aug,
18(4):
359-364

- Received : July 08, 2014
- Accepted : July 28, 2014
- Published : August 25, 2014

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We report parameters that allow the dielectric functions
ε
=
ε
_{1}
+
iε
_{2}
of AlAs
_{x}
Sb
_{1-x}
alloys to be calculated analytically over the entire composition range 0 ≤
x
≤ 1 in the spectral energy range from 0.74 to 6.0 eV by using the dielectric function parametric model (DFPM). The
ε
spectra were obtained previously by spectroscopic ellipsometry for
x
= 0, 0.119, 0.288, 0.681, 0.829, and 1. The
ε
data are successfully reconstructed and parameterized by six polynomials in excellent agreement with the data. We can determine
ε
as a continuous function of As composition and energy over the ranges given above, and
ε
can be converted to complex refractive indices using a simple relationship. We expect these results to be useful for the design of optoelectronic devices and also for
in situ
monitoring of AlAsSb film growth.
ñ
=
n
+
ik
, dielectric function
ε
=
ñ
^{2}
=
ε
_{1}
+
iε
_{2}
, and interband transitions including the band gap of AlAsSb are needed for further device optimization
[7]
, such as designs for distributed feedback grating waveguides
[4]
and simulations to predict the performance of solar cells
[8
,
9]
. As a result, the dielectric functions and critical point (CP) energies of AlAs
_{x}
Sb
_{1-x}
have been investigated using spectroscopic ellipsometry (SE)
[10
,
11]
, which is an excellent technique for determining
ε
data directly
[12
,
13]
. However, no analytic representation of these
ε
spectra as a continuous function of As-composition
x
has been reported to date.
Here we analytically determine
ε
of AlAs
_{x}
Sb
_{1-x}
as a continuous function of
x
for 0 ≤
x
≤ 1 in the energy range from 0.74 to 6 eV. The source data were obtained previously by rotating-compensator SE
[11]
. In short, pseudodielectric function <
ε
> spectra of AlAs
_{x}
Sb
_{1-x}
alloy films (of well beyond the critical thickness to have bulk optical properties) with
x
= 0.119, 0.288, 0.681, and 0.829 were obtained on semi-insulating (001) GaAs substrates using molecular beam epitaxy (MBE). We performed
in situ
measurements directly on films grown by MBE, where the films were prepared and maintained in ultrahigh vacuum, preventing rapid oxidization of Al and ensuring oxide-free data. The endpoint data for AlAs and AlSb were taken from Refs.
[14]
and
[15]
, respectively.
The CPs generate asymmetric features in
ε
[16]
. Therefore, symmetric-lineshape models such as Lorentz, harmonic, or Gaussian oscillators provide poor representations of
ε
, requiring additional CP oscillators that have no physical basis. We circumvent this difficulty through the use of the dielectric function parametric model (DFPM)
[17
,
18]
, which can treat asymmetric characteristics appropriately while avoiding unnecessary additions. Here, we determine the parameters needed to represent
ε
analytically for the available spectra, then interpolate these parameters as a function of As composition
x
.
ε
data. In brief, in the DFPM the dielectric function is given as the sum of
m
energy-bounded polynomials that represent CP contributions within the accessible spectral range plus
P
poles that represent outside contributions
[17
,
18]
. The general expression is
where
where
u
(
x
) is the unit step function. The use of pure Gaussian broadening in Eq. (2a) essentially prohibits closed-form integration of Eq. (1). However, the equivalent expression shown in Eq. (2b) shows that one-dimensional lookup tables as a function of
) can be constructed numerically for each order of polynomial required by Eq. (2c). The corresponding real part of
ε
is obtained by a Kramers-Kronig transformation. Detailed information, including a program to calculate the DFPM, is given in Refs.
[17]
and
[18]
.
ε
spectra of AlAs
_{x}
Sb
_{1-x}
alloy using DFPM with
x
= 0 (AlSb), 0.119, 0.288, 0.681, 0.829, and 1 (AlAs). The source data for ternary alloys are obtained from our previous report
[11]
, while data for AlAs and AlSb were taken from Refs.
[14]
and
[15]
, respectively. As an example,
Fig. 1(a)
shows how the component CP structures combine to generate the
ε
_{2}
spectrum of AlSb. The open dots are the data from Ref.
[15]
while the solid line is the fit of the DFPM to these data. The dashed lines show the contributions of the six individual CPs. To show the quality of the fits better, we reduced the number of data points appropriately. The parameters obtained from
Fig. 1(a)
are listed in
Table 1
. In the present work each CP is described by nine parameters, which are depicted in
Fig. 1(b)
. This is the
ε
_{2}
spectrum of the
E
_{1}
CP component of AlSb. Using the conventional names of the DFPM,
E_{C}
is the CP energy, while the values
E_{L}
and
E_{U}
indicate the CP structure numbers whose CP energies (
E_{C}
) are the lower and upper bounding energies, respectively.
E_{LM}
and
E_{UM}
are control points for establishing the asymmetric characteristics of the lineshape. The values
E_{LM}
and
E_{UM}
are not absolute energies but rather relative positions between
E_{L}
and
E_{C}
and between
E_{U}
and
E_{C}
, respectively.
A_{LM}
,
A
, and
A_{UM}
are the respective amplitudes at
E_{LM}, E_{C}
, and
E_{UM}
. Also, the values
A_{LM}
and
A_{UM}
are amplitudes relative to
A
. Parameter
B
, not shown in
Fig. 1(b)
, is the full-width-half-maximum broadening parameter inherently embedded in harmonic-oscillator lineshapes. To construct the lineshape of a CP, second- and fourth-order polynomials were used for the energy regions (I, IV) and (II, III), respectively, with the constraint that the lines are connected smoothly and the values forced to zero at the boundaries
E_{L}
and
E_{U}
. Parameters that are held constant independent of composition are indicated by asterisks in
Table 1
. Here, No. 0 (
E
_{0}
indirect) and No. 7 are not CPs but are the lower bounding energy of
E
_{0}
(direct) and upper bounding energy of
, respectively. The analysis was repeated for all AlAsSb spectra, and the fitting quality was similar to that seen for AlSb.
(a) Dielectric function (open dots) of AlSb, together with the DFPM reconstruction (solid line) using six CP components (dashed lines). (b) Schematic of a single CP structure of E _{1} in the DFPM.
DFPM parameters for AlSb. Parameters denoted by asterisks are assumed to be independent of composition
The reported <
ε
> spectra of ternary alloys have oscillations below the
E
_{0}
feature, which are interference effects involving light back-reflected at the substrate-film interface, as a result of the films being transparent below their fundamental absorption edges. To extract
ε
in the oscillation region we performed the DFPM calculation using a multilayer model (ambient/AlAsSb alloy film/GaAs substrate). The fitting results for both real and imaginary parts of <
ε
> are shown in
Figs. 2
,
3
, and
4
for
x
= 0.119, 0.288, and 0.681, respectively. The fits of the DFPM (solid lines) agree well with the data (dots) for
x
= 0.119 and 0.288, with thicknesses 1835 and 1837 nm respectively. To show the quality of the fits better, we reduced the number of data points appropriately. However, we note that the fit for
x
= 0.681 cannot follow the experimental data of the sharp interference oscillation patterns below 3.5 eV, as shown in
Fig. 4
. In our previous study of
in situ
monitoring of the growth of AlSb on GaAs by SE, we detected imperfect growth caused by glomerulate Al before perfect laminar growth of the film
[19]
. Accordingly, we assumed that our AlAs
_{x}
Sb
_{1-x}
alloy film for
x
= 0.681 has an intermediate layer on GaAs substrate and a surfaceroughness layer on the top of the film caused by the remaining effect of roughness from the agglomeration of Al. Therefore, we used a five-phase multilayer model with ambient/surface roughness/AlAsSb alloy film/interface layer/ GaAs substrate. The dielectric functions of the surfaceroughness and interface layers were calculated using the effective-medium approximation
[20]
and Cauchy model, respectively. In principle, it is expected that the interface layer has absorption above its
E
_{0}
CP. However, we claim that the transparent Cauchy model is sufficient, because the probe beam cannot reach the interface layer, interrupted by the absorption of the AlAs
_{x}
Sb
_{1-x}
alloy film above the
E
_{0}
CP.
Pseudodielectric function spectra (dots) for x = 0.119 taken from Ref. [11] , together with the best fit (solid lines) using DFPM.
Pseudodielectric function spectra (dots) for x = 0.288 taken from Ref. [11] , together with the best fit (solid lines) using DFPM.
Pseudodielectric function spectra (dots) for x = 0.681 taken from Ref. [11] , together with the best fit (solid lines) using DFPM.
Figure 5
shows the DFPM spectrum (solid line) obtained as a best fit to the data (open dots) for
x
= 0.681 with excellent agreement, demonstrating the validity of our analysis. The dashed lines show the contributions of the six individual CPs in
Fig. 5(a)
. To show the quality of the fits better, we reduced the number of data points appropriately here also. The obtained thicknesses of roughness, film, and interface layers are 0.71, 1687, and 28 nm respectively. The same analysis was repeated for
x
= 0.829 yielding similar fitting quality to that seen in
Fig. 5
. For
x
= 0.829 the thicknesses of roughness, film, and interface layers are 0.81, 1940, and 11 nm, respectively.
(a) Dielectric function (open dots) for x = 0.681, together with the DFPM reconstruction (solid line) using six CP components (dashed lines). (b) Data (open dots) and fit (solid line) to both real and imaginary parts of <ε > in the region of oscillations.
To construct numerical values of
ε
for arbitrary compositions, we interpolated the results for all data. We represent the
x
dependences by the cubic equation
The best-fit parameters are given in
Table 2
.
Figure 6
shows the As-composition dependences of
E_{C}
as an example. The open dots are parameterized values for each temperature, and the solid lines are the best fits to Eq. (3). The crossing of the
E
_{2}
and
E
_{0}
' CPs is also detected in our DFPM analysis, as in a previous band-calculation study
[11]
.
Parameters for calculating the composition dependences by using Eq. (3)
As-composition dependences of the energy parameters. The open dots are the parameters and the solid lines are fits using Eq. (3).
In
Fig. 7
we compare the original data to the reconstruction for the representative composition
x
= 0.288. The dots and solid lines are measured and reconstructed spectra, respectively. The reconstructions are in excellent agreement with the data on this scale. Using these results we could calculate
ε
for any AlAs
_{x}
Sb
_{1-x}
alloy, as shown in
Fig. 8
, where panels (a) and (b) show the real and imaginary parts respectively. The spectra are offset by increments of 20 relative to that for
x
= 0.1. For convenient application in device design, we also show the complex refractive index in
Fig. 9
. The spectra are offset by increments of 2 relative to that for
x
= 0.1.
Comparison of data (dots) to spectra (solid lines) reconstructed using the parameters of Table 2 for x = 0.288.
(a) Real and (b) imaginary parts of the reconstructed dielectric functions of AlAs_{x}Sb_{1-x} for various x , as shown.
(a) Refractive index n and (b) extinction coefficient k of the reconstructed dielectric functions of AlAs_{x}Sb_{1-x} for various x , as shown.
_{x}
Sb
_{1-x}
from 0.74 to 6.0 eV for arbitrary As composition
x
. Agreement is achieved with reasonable parameter values. The optical properties reported here will be useful in device design for high-speed optoelectronic applications and
in situ
monitoring of growth and deposition.

I. INTRODUCTION

AlAsSb-based systems have been widely investigated for high-speed optoelectronic device applications such as ultrafast cross-phase modulators
[1]
, quantum-cascade lasers
[2]
, distributed Bragg reflectors
[3]
, ultrafast all-optical switches using intersubband transitions
[4]
, and photodetectors
[5
,
6]
. The optical properties such as complex refractive index
II. MODELING

The DFPM has the advantage of properly representing
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III. RESULTS AND DISCUSSION

We parameterized the
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DFPM parameters for AlSb. Parameters denoted by asterisks are assumed to be independent of composition

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Parameters for calculating the composition dependences by using Eq. (3)

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IV. CONCLUSION

We obtain excellent representations of the pseudodielectric functions of AlAs
Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2013-016297).

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Citing 'Parametrization of the Optical Constants of AlAs_{x}Sb_{1-x} Alloys in the Range 0.74-6.0 eV
'

@article{ E1OSAB_2014_v18n4_359}
,title={Parametrization of the Optical Constants of AlAs_{x}Sb_{1-x} Alloys in the Range 0.74-6.0 eV}
,volume={4}
, url={http://dx.doi.org/10.3807/JOSK.2014.18.4.359}, DOI={10.3807/JOSK.2014.18.4.359}
, number= {4}
, journal={Journal of the Optical Society of Korea}
, publisher={Optical Society of Korea}
, author={Kim, Tae Jung
and
Byun, Jun Seok
and
Barange, Nilesh
and
Park, Han Gyeo
and
Kang, Yu Ri
and
Park, Jae Chan
and
Kim, Young Dong}
, year={2014}
, month={Aug}