Parametrization of the Optical Constants of AlAs_{x}Sb_{1x} Alloys in the Range 0.746.0 eV
 Author: Kim Tae Jung, Byun Jun Seok, Barange Nilesh, Park Han Gyeo, Kang Yu Ri, Park Jae Chan, Kim Young Dong
 Organization: Kim Tae Jung; Byun Jun Seok; Barange Nilesh; Park Han Gyeo; Kang Yu Ri; Park Jae Chan; Kim Young Dong
 Publish: Current Optics and Photonics Volume 18, Issue4, p359~364, 25 Aug 2014

ABSTRACT
We report parameters that allow the dielectric functions
ε =ε _{1} +iε _{2} of AlAs_{x}Sb_{1x} 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). Theε spectra were obtained previously by spectroscopic ellipsometry forx = 0, 0.119, 0.288, 0.681, 0.829, and 1. Theε data are successfully reconstructed and parameterized by six polynomials in excellent agreement with the data. We can determineε as a continuous function of As composition and energy over the ranges given above, andε 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 forin situ monitoring of AlAsSb film growth.

KEYWORD
AlAsSb , Dielectric function parametric model , Ellipsometry

I. INTRODUCTION
AlAsSbbased systems have been widely investigated for highspeed optoelectronic device applications such as ultrafast crossphase modulators [1], quantumcascade lasers [2], distributed Bragg reflectors [3], ultrafast alloptical switches using intersubband transitions [4], and photodetectors [5, 6]. The optical properties such as complex refractive index
ñ =n +ik , dielectric functionε =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_{1x} have been investigated using spectroscopic ellipsometry (SE) [10, 11], which is an excellent technique for determiningε data directly [12, 13]. However, no analytic representation of theseε spectra as a continuous function of Ascompositionx has been reported to date.Here we analytically determine
ε of AlAs_{x}Sb_{1x} as a continuous function ofx for 0 ≤x ≤ 1 in the energy range from 0.74 to 6 eV. The source data were obtained previously by rotatingcompensator SE [11]. In short, pseudodielectric function <ε > spectra of AlAs_{x}Sb_{1x} alloy films (of well beyond the critical thickness to have bulk optical properties) withx = 0.119, 0.288, 0.681, and 0.829 were obtained on semiinsulating (001) GaAs substrates using molecular beam epitaxy (MBE). We performedin 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 oxidefree 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, symmetriclineshape models such as Lorentz, harmonic, or Gaussian oscillators provide poor representations ofε , 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 representε analytically for the available spectra, then interpolate these parameters as a function of As compositionx .II. MODELING
The DFPM has the advantage of properly representing
ε data. In brief, in the DFPM the dielectric function is given as the sum ofm energybounded polynomials that represent CP contributions within the accessible spectral range plusP poles that represent outside contributions [17, 18]. The general expression iswhere
where
u (x ) is the unit step function. The use of pure Gaussian broadening in Eq. (2a) essentially prohibits closedform integration of Eq. (1). However, the equivalent expression shown in Eq. (2b) shows that onedimensional lookup tables as a function of ) can be constructed numerically for each order of polynomial required by Eq. (2c). The corresponding real part ofε is obtained by a KramersKronig transformation. Detailed information, including a program to calculate the DFPM, is given in Refs. [17] and [18].III. RESULTS AND DISCUSSION
We parameterized the
ε spectra of AlAs_{x}Sb_{1x} alloy using DFPM withx = 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 theε _{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 theε _{2} spectrum of theE _{1} CP component of AlSb. Using the conventional names of the DFPM,E_{C} is the CP energy, while the valuesE_{L} andE_{U} indicate the CP structure numbers whose CP energies (E_{C} ) are the lower and upper bounding energies, respectively.E_{LM} andE_{UM} are control points for establishing the asymmetric characteristics of the lineshape. The valuesE_{LM} andE_{UM} are not absolute energies but rather relative positions betweenE_{L} andE_{C} and betweenE_{U} andE_{C} , respectively.A_{LM} ,A , andA_{UM} are the respective amplitudes atE_{LM}, E_{C} , andE_{UM} . Also, the valuesA_{LM} andA_{UM} are amplitudes relative toA . ParameterB , not shown in Fig. 1(b), is the fullwidthhalfmaximum broadening parameter inherently embedded in harmonicoscillator lineshapes. To construct the lineshape of a CP, second and fourthorder 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 boundariesE_{L} andE_{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 ofE _{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.The reported <
ε > spectra of ternary alloys have oscillations below theE _{0} feature, which are interference effects involving light backreflected at the substratefilm interface, as a result of the films being transparent below their fundamental absorption edges. To extractε 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 forx = 0.119, 0.288, and 0.681, respectively. The fits of the DFPM (solid lines) agree well with the data (dots) forx = 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 forx = 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 ofin 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_{1x} alloy film forx = 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 fivephase 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 effectivemedium approximation [20] and Cauchy model, respectively. In principle, it is expected that the interface layer has absorption above itsE _{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_{1x} alloy film above theE _{0} CP.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 forx = 0.829 yielding similar fitting quality to that seen in Fig. 5. Forx = 0.829 the thicknesses of roughness, film, and interface layers are 0.81, 1940, and 11 nm, respectively.To construct numerical values of
ε for arbitrary compositions, we interpolated the results for all data. We represent thex dependences by the cubic equationThe bestfit parameters are given in Table 2. Figure 6 shows the Ascomposition 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 theE _{2} andE _{0}' CPs is also detected in our DFPM analysis, as in a previous bandcalculation study [11].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 calculateε for any AlAs_{x}Sb_{1x} 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 forx = 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 forx = 0.1.IV. CONCLUSION
We obtain excellent representations of the pseudodielectric functions of AlAs_{x}Sb_{1x} 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 highspeed optoelectronic applications andin situ monitoring of growth and deposition.

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[FIG. 1.] (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 E1 in the DFPM.

[TABLE 1.] DFPM parameters for AlSb. Parameters denoted by asterisks are assumed to be independent of composition

[FIG. 2.] Pseudodielectric function spectra (dots) for x = 0.119 taken from Ref. [11], together with the best fit (solid lines) using DFPM.

[FIG. 3.] Pseudodielectric function spectra (dots) for x = 0.288 taken from Ref. [11], together with the best fit (solid lines) using DFPM.

[FIG. 4.] Pseudodielectric function spectra (dots) for x = 0.681 taken from Ref. [11], together with the best fit (solid lines) using DFPM.

[FIG. 5.] (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.

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[Table 2.] Parameters for calculating the composition dependences by using Eq. (3)

[FIG. 6.] Ascomposition dependences of the energy parameters. The open dots are the parameters and the solid lines are fits using Eq. (3).

[FIG. 7.] Comparison of data (dots) to spectra (solid lines) reconstructed using the parameters of Table 2 for x = 0.288.

[FIG. 8.] (a) Real and (b) imaginary parts of the reconstructed dielectric functions of AlAsxSb1x for various x, as shown.

[FIG. 9.] (a) Refractive index n and (b) extinction coefficient k of the reconstructed dielectric functions of AlAsxSb1x for various x, as shown.