Establishing Composition Dependent k · p Parameters for (Al,Ga)N Alloys

Deep UV (Al,Ga)N-based light emitters exhibit very low quantum efficiencies when compared to UV emitters at longer wavelengths. To improve the efficiencies of e.g. (Al,Ga)N-based light emitting diodes, theory and simulation can help to guide the device design. The theoretical framework underlying device simulations is often based on drift-diffusion models coupled with a self-consistent Schrodinger-Poisson equation solver. To achieve accurate and predictive models, understanding the composition dependence of material input parameters is of central importance. We target the composition dependence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{k}\cdot \mathrm{p}$</tex> parameters in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{Al}_{\mathrm{x}}\text{Ga}_{(1-\mathrm{x})}\mathrm{N}$</tex> alloys by using density functional theory (DFT) to obtain effective band structures from alloy disordered supercells. Building on these effective band structures, a numerically efficient fitting scheme based on the Sobol-sequence method is employed to extract effective electron masses, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{m}_{\mathrm{e}}(\mathrm{x})$</tex>, Luttinger-like parameters <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{A}_{\mathrm{i}}(\mathrm{x})$</tex>, with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{i}=1\ldots 6$</tex>, and crystal field splitting energy, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Delta_{\text{CF}}$</tex>, as a function of Al content, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x$</tex>, in the system. Our calculations reveal that for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{m}_{\mathrm{e}}(\mathrm{x})$</tex> a linear interpolation of the GaN and AlN values provides a good description of the DFT data. However, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{A}_{\mathrm{i}}(\mathrm{x})$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Delta_{\text{CF}}(\mathrm{x})$</tex> this simple approximation breaks down and we find that composition dependent bowing parameters are required to describe effective DFT band structures accurately.