AlGaAs Bragg reflectors

Optical properties of Al_xGa_1-xAs Bragg reflectors, part 1

Bragg reflectors can be realized using optical Al_xGa_1-xAs superlattices. Since layers of different composition parameter x differ in their refractive index, multiple repitition of a double layer of low and high Al content can lead to pronounced constructive interference of partially reflected waves. For certain wavelengths (which depend on the refractive indices and the thicknesses of the layers) a very high reflectivity can be achieved. Such structures can be used as reflectors in AlGaAs laser structures. To predict the optical performance of Al_xGa_1-xAs multi-layer structures one has to know the optical constants for the whole range of composition x. This can be done with our thin film programs SCOUT and Coating Designer in a very elegant and efficient way which is described in this section.

Optical constant model

Optical constants of III-V semiconductors like GaAs and AlAs can be modeled quite well using the Tauc-Lorentz interband transition model (see e.g. G.E.Jellison, Jr., 'Spectroscopic ellipsometry data analysis: measured versus calculated quantities', Thin Solid Films 313-314 (1998), 33-39). The imaginary part of the dielectric function is computed as a function of four parameters: The gap energy, the resonance frequency, the transition strength and a damping constant. To cover a broad spectral range one has to superimpose several of such terms to take into account various band to band transitions. Since the Tauc-Lorentz model gives an expression for the imaginary part of the dielectric function only one has to construct the real part making use of a Kramers-Kronig transformation. This can be done in our software products very efficiently applying two fast Fourier transforms (FFT): One computes from the known imaginary part of the dielectric function in the frequency domain the response function in the time domain. After some modifications (causality conditions) the second one takes the response function and goes back to the frequency domain. Now real and imaginary part of the dielectric function are known. On modern desktop computers Kramers-Kronig transformations via two FFTs 'cost' only a few milliseconds and can be applied even for parameter fitting routines. In addition to Tauc-Lorentz terms a constant dielectric background is used which represents high energy transitions outside the spectral range of interest.

Optical constants for composition variations

To describe the optical constants for the whole composition range the following strategy is followed: Determine the model parameters for several selected compositions and express the variation of the model parameters with composition by suitable polynomials. If the polynomials are 'smooth' enough they can be used to interpolate between the investigated composition values and parameter values for any arbitrary choice of the aluminum content can be obtained. Finally the optical constants can be computed applying the interpolated values of the model parameters.

For this method experimental data for several compositions must be known. Here we have used experimentally determined dielectric functions obtained by ellipsometry (D.E Aspnes, S.M. Kelso, R.A. Logan and R.Bhat, J.Appl.Phys. 60, 754, (1986)) for the compositions x=0, 0.1, 0.2, 0.32, 0.42, 0.49, 0.57, 0.7, 0.8 and 1.0. The model is based on 8 Tauc-Lorentz interband transitions and a constant dielectric background, i.e. 33 model parameters have been determined in the range 1.0 to 5.5 eV. As an example of parameter variation with composition the polynomial describing the shift of the gap energy of one of the Tauc-Lorentz terms is shown in the following graph:

Similarly, all 33 model parameters are controlled by the composition parameter x. The following graphs show the results for the model dielectric function for x=0, 0.2, 0.8 and 1.0:

x=0.0

x=0.2

x=0.8

x=1.0

The next graph shows the variation of the imaginary part of the dielectric function with x:

Making use of this flexible dielectric function model optical reflectance spectra of multi-layer stacks with layers of different compositions can be computed easily. See a screenshot example of a SCOUT configuration predicting the reflectance of a superlattice.

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