Wigner-Weyl description of light absorption in disordered semiconductor alloys using the localization landscape theory

The presence of disorder in semiconductors can dramatically change their physical properties. Yet, models faithfully accounting for it are still scarce and computationally inefficient. We present a mathematical and computational model able to simulate the optoelectronic response of semiconductor alloys of several tens of nanometer sidelength, while at the same time accounting for the quantum localization effects induced by the compositional disorder at the nanoscale. The model is based on a Wigner-Weyl analysis of the structure of electron and hole eigenstates in phase space made possible by the localization landscape theory. After validation against eigenstate-based computations in 1D and 2D, our model is applied to the computation of light absorption in 3D InGaN alloys of different compositions. We obtain the detailed structures of the absorption tail below the average band gap and the Urbach energies of all simulated compositions. Moreover, the Wigner-Weyl formalism allows us to define and compute 3D maps of the effective locally absorbed power at all frequencies. Finally, the proposed approach opens the way to generalize this method to all energy-exchange processes such as radiative and nonradiative recombination in realistic devices.

Figure 1

Realization of a two-dimensional ${\mathrm{In}}_{0.05}{\mathrm{Ga}}_{0.95}\mathrm{N}$ alloy. (a) Atomic configuration. Open blue circles denote Ga atoms and red disks denote In atoms. (b) Conduction and (c) valence potentials obtained from Eqs. (6) and (7). The smearing length was set to $\sigma =2a\approx 6.4$ Å.

Figure 2

Integrated density of states in phase space, i.e., the sum of the Wigner transforms $W_{{\chi }_{\mu }^{\left(c\right)}}$ of eigenstates whose eigenenergy lie below a given energy $E$ [see Eq. (27)] for a one-dimensional alloy with In concentration $x=5$% and $L=100$ nm. The smearing length was set to $\sigma =2a$. (a) $E=3.06$ eV (first two states), (b) $E=3.22$ eV (first 14 states), (c) $E=3.46$ eV (first 20 states), and (d) $E=3.84$ eV (first 50 states). The dashed lines are the contour $H_c(x_1,k_1)=E$ and the solid lines are the contour $H_c^{\left(\mathrm{eff}\right)}(x_1,k_1)=E$. The color scale is held fixed for the sake of visibility.

Figure 3

Average spectral coupling density per unit length, $\mathbb{E}[\mathcal{C}/|\mathrm{\Omega }\left|\right]$, for one-dimensional ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys. (a) In concentration fixed $x=5\%$, and smearing length fixed $\sigma =2a$. (b)–(d) Varying In concentration $x\in \{5\%,10\%,15\%\}$ and fixed smearing length (b) $\sigma =a$, (c) $\sigma =2a$, and (d) $\sigma =3a$. The results were obtained by using the eigenstate-based expression (eig.), Eq. (17), the Wigner-Weyl expression (WW), Eq. (38), and the Wigner-Weyl localization landscape expression (WWL), Eq. (43), averaged over $N=100$ realizations of the alloy chain of length $L=200$ nm. The shaded areas correspond to one standard deviation around the average.

Figure 4

Average spectral coupling density per unit area, $\mathbb{E}[\mathcal{C}/|\mathrm{\Omega }\left|\right]$, for two-dimensional ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ alloys. (a)–(c) Comparison between the eigenstates based formula [eig., Eq. (17)] and the Wigner-Weyl law based on the localization landscape [WWL, Eq. (43)] for varying In concentration $x\in \{5\%,10\%,15\%\}$ and fixed smearing length (a) $\sigma =a$, (b) $\sigma =2a$, and (c) $\sigma =3a$. (d) Comparison between the usual Wigner-Weyl law [WW, Eq. (38)] and the Wigner-Weyl law based on the localization landscape [WWL, Eq. (43)] for a fixed In-concentration $x=10\%$, and varying smearing length $\sigma \in \{a,2a,3a\}$. The results were obtained by averaging over $N=100$ realizations of the alloy of area $L\times L=40$ nm $\times 40$ nm. The shaded areas correspond to one standard deviation around the average. The vertical dashed lines indexed with a percentage corresponds to the band gap energy obtained with the bowing formula, Eq. (5), for $X$ set to the average concentration $x$.

Figure 5

(a)–(c) Absolute value of the reduced absorbed power density obtained with Eq. (61) for $\varepsilon =20$ meV and (d)–(f) reduced absorbed approximated by the Wigner-Weyl-localization landscape model, Eq. (62), for a two-dimensional ${\mathrm{In}}_{0.15}{\mathrm{Ga}}_{0.85}\mathrm{N}$ alloy, with $\sigma =2a$. The maps are shown for different values of the photon energy, (a,d) $\hslash \omega =2.7$ eV, (b,e) $\hslash \omega =2.85$ eV, (c), (f) $\hslash \omega =3.0$ eV.

Figure 6

(a) Absorption coefficient as a function of photon energy $\hslash \omega$ for different values of the average indium concentration $x$ and of the smearing length $\sigma$. The shaded area correspond to $\pm 2\sqrt{\mathbb{V}\mathrm{ar}\left[\alpha \right]/N}$. (b) Urbach energy as a function of $x$ for different values of $\sigma$. The data are obtained based on the Wigner-Weyl localization landscape approach.

Figure 7

Absorbed power density ${\mathcal{P}}_{\mathrm{WWL}}$ for a realization of the alloy for three values of $\hslash \omega$: (a) $\hslash \omega =2.8$ eV, (b) $\hslash \omega =2.85$ eV, (c) $\hslash \omega =2.9$ eV. The results were obtained with Eq. (55) for a size of the computational domain $L\times L\times L=50$ nm $\times 50$ nm $\times 50$ nm, with element size $\mathrm{\Delta }x=3$ Å, average indium concentration $x=15\%$ and smearing length $\sigma =2a$. The eighth top front corner cube is removed to help visualize the inside of the volume. The color scale is common for the three values of $\hslash \omega$ to ease the comparison.