Localization landscape theory of disorder in semiconductors. I. Theory and modeling

We present here a model of carrier distribution and transport in semiconductor alloys accounting for quantum localization effects in disordered materials. This model is based on the recent development of a mathematical theory of quantum localization which introduces for each type of carrier a spatial function called localization landscape. These landscapes allow us to predict the localization regions of electron and hole quantum states, their corresponding energies, and the local densities of states. We show how the various outputs of these landscapes can be directly implemented into a drift-diffusion model of carrier transport and into the calculation of absorption/emission transitions. This creates a new computational model which accounts for disorder localization effects while also capturing two major effects of quantum mechanics, namely, the reduction of barrier height (tunneling effect) and the raising of energy ground states (quantum confinement effect), without having to solve the Schrödinger equation. Finally, this model is applied to several one-dimensional structures such as single quantum wells, ordered and disordered superlattices, or multiquantum wells, where comparisons with exact Schrödinger calculations demonstrate the excellent accuracy of the approximation provided by the landscape theory.

Figure 1

The localization landscape theory: (a) 3D representation of the original 2D disordered potential $V$; (b) 3D representation of the landscape $u$ solving Eq. (2). (c) The valley lines of the landscape $u$ (black lines) delimit the various localization regions. (d) Effective localization potential ${\displaystyle W\equiv u^{-1}}$. The localization subregions outlined in (c) are also the basins of $W$.

Figure 2

Local density of states (LDOS) deduced from the effective potential ${\displaystyle W_c\equiv \frac{1}{u_e}}$.

Figure 3

Schematic structure of the self-consistent Poisson-landscape model allowing us to bypass solving the Schrödinger equation.

Figure 4

Flow chart solving the Poisson and drift-diffusion equations by applying the localization landscape theory.

Figure 5

Overlap regions are defined as intersections between electron and hole localization subregions. (Left) Superimposed electron and hole landscapes. (Right) Example of one overlapping region defined as the intersection between two electron and hole localization subregions.

Figure 6

QW structures used to test the LL theory. (Left) SQW well with $m$-plane orientation of the well material. (Right) 3QW structure with $c$-plane orientation. The well width and the barrier thickness are 3 and 7 nm, respectively.

Figure 7

Band structures for the $m$-plane single QW and the $c$-plane 3QW. The localization potentials $W_c$ (blue) and $W_v$ (green) are superimposed over the band edges $E_c$ (black) and $E_v$ (red). The integration regions ${\mathrm{\Omega }}_e$ and ${\mathrm{\Omega }}_h$ are indicated on each frame.

Figure 8

Simulation results for the $m$-plane SQW (left structure in Fig. 6). The energies computed by solving directly the Schrödinger equation are compared to the energies computed using our Poisson-LL model. Top frames display comparisons for electrons (left) and holes (right), respectively. Bottom left frame displays the energy difference, i.e., the smallest energy for a radiative transition. Bottom right frame shows the overlap integral between fundamental electron and hole states, computed from the Schrödinger equations (dashed line) and from the estimate using the LLs (plain line).

Figure 9

Simulation results for the $c$-plane 3QW (right structure in Fig. 6). The energies computed by solving directly the Schrödinger equation are compared to the energies computed using our Poisson-LL model. Top frames display comparisons for electrons (left) and holes (right), respectively. Bottom left frame displays the energy difference, i.e., the smallest energy for a radiative transition. Bottom right frame shows the overlap integral between fundamental electron and hole states of the central well, computed from the Schrödinger equations (dashed line) and from the estimate using the LLs (plain line).

Figure 10

(a) Schematic structure of 20-period ${\mathrm{Al}}_{0.4}{\mathrm{Ga}}_{0.6}\mathrm{N}$/GaN SLs. (b), (c) The conduction band potential $\left(E_c\right)$ and effective quantum confining potential $\left(W_c\right)$ for 5 nm/5 nm and 1 nm/1 nm SLs.

Figure 11

(a), (b) The counted states for 5 nm/5 nm and 1 nm/1 nm SLs.

Figure 12

(a), (c) The conduction band potential $\left(E_c\right)$ and effective quantum confining potential $\left(W_c\right)$ for $m$-plane (top) and $c$-plane (bottom) double-period SLs. (b), (d) The corresponding integrated density of states (IDOS).

Figure 13

(a), (c) The conduction band potential $\left(E_c\right)$ and effective quantum confining potential $\left(W_c\right)$ for $m$-plane (top) and $c$-plane (bottom) 20-pair disordered SL. (b), (d) The corresponding IDOS.

Figure 14

(a), (c) The conduction band potential $\left(E_c\right)$ and effective quantum confining potential $\left(W_c\right)$ for $m$-plane (top) and $c$-plane (bottom) 200-pair disordered SL. (b), (d) The corresponding IDOS.

Figure 15

Comparison of the LIDOS of a $c$-plane 200-pair disordered SL structure computed using on one hand a coupled Poisson-Schrödinger approach, and on the other hand our coupled Poisson-LL model. The structure is divided into four regions (top frame). Comparisons of the IDOS of each region computed using both methods are displayed in the four bottom frames.

Figure 16

(Top) Light absorption spectra (in linear and log scales) for the $c$-plane 20-pair disordered SL of Fig. 13. The absorption spectrum computed using the Schrödinger equation is displayed with a red dashed line while the spectrum computed with the landscape approach is displayed with a black continuous line. (Bottom) Light absorption spectrum (in linear and log scales) for the $c$-plane 200-pair disordered SL of Fig. 14.

Figure 17

(a) Schematic structure of the single ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}/\mathrm{GaN}/{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QW, where $x$ is variable. (b) Potential energy and carrier distribution for $m$ plane and $x=0.2$. (c) Potential energy and carrier distribution for $c$ plane and $x=0.2$. (d) Potential energy and carrier distribution for $m$ plane and $x=0.4$. The blue, black, and red lines are solved by the classical Poisson equation, Poisson-LL model, and Poisson-Schrödinger equation, respectively. The Fermi level is located at zero energy as the reference.

Figure 18

Relative difference between the predictions of the peak carrier distribution by the Poisson-LL and Poisson-Schödinger models for various Al compositions of the ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}/\mathrm{GaN}$ $m$-plane SQW structure.