Quantum states in disordered media. II. Spatial charge carrier distribution

The space- and temperature-dependent electron distribution $n(\mathbf{r},T)$ is essential for the theoretical description of the optoelectronic properties of disordered semiconductors. We present two powerful techniques to access $n(\mathbf{r},T)$ without solving the Schrödinger equation. First, we derive the density for nondegenerate electrons by applying the Hamiltonian recursively to random wave functions (RWF). Second, we obtain a temperature-dependent effective potential from the application of a universal low-pass filter (ULF) to the random potential acting on the charge carriers in disordered media. Thereby, the full quantum-mechanical problem is reduced to the quasiclassical description of $n(\mathbf{r},T)$ in an effective potential. We numerically verify both approaches by comparison with the exact quantum-mechanical solution. Both approaches prove superior to the widely used localization landscape theory (LLT) when we compare our approximate results for the charge carrier density and mobility at elevated temperatures obtained by RWF, ULF, and LLT with those from the exact solution of the Schrödinger equation.

Figure 1

Reduced electron density $\tilde{n}(x,T)=e^{-\beta \mu }n(x,T)$ as a function of coordinate $x$ in a sample with one-dimensional white-noise potential. Solid orange line: exact density calculated using Eq. (10). Other lines: density calculated by the random-wave-function algorithm with $N_{\mathrm{R}}=20$ (green dashed line) and $N_{\mathrm{R}}=1000$ (black dotted line) realizations. Dimensionless units defined in Eqs. (43, 44, 45) are used. The sample size is equal to 30 dimensionless units, the discretization parameter is $a=0.1$, the temperature is $T=1$.

Figure 2

Distribution of the relative error $\mathrm{\Delta }$ of the random-wave-function algorithm with $N_{\mathrm{R}}=1000$ in a sample with a one-dimensional white-noise potential. The quantity $\mathrm{\Delta }$ is defined in Eq. (33). Blue bars: histogram of the numerically obtained distribution. Red line: Gaussian distribution with expectation value zero and variance $2/N_{\mathrm{R}}$. Parameters are the same as in Fig. 1, except for the sample size that equals 500 dimensionless units.

Figure 3

Comparison between exact (solid lines) and approximate (symbols) reduced electron density $\tilde{n}(x,T)=e^{-\beta \mu }n(x,T)$ in a one-dimensional white-noise potential at different temperatures $T$ and $N_{\mathrm{R}}=1000$ iterations. All other parameters are the same as in Fig. 1, except for the temperature. For clarity, the lowest three curves are scaled down by a multiplication by ${10}^{-3}$, ${10}^{-2}$, and ${10}^{-1}$, as indicated in the legend.

Figure 4

Reduced electron density $\tilde{n}(x,y,T)$ in a two-dimensional sample with white-noise potential. We compare the exact reduced density (solid orange line) with the approximate reduced density obtained from the random-wave-function algorithm (dashed green line for $N_{\mathrm{R}}=20$ and dotted black line for $N_{\mathrm{R}}=1000$). Profiles along the $x$ axis with different values of coordinate $y$ are shown (see inset). The sample size is $5\times 5$ dimensionless units, the discretization grid parameter is $a=0.1$, the temperature is $T=1$. Periodic boundary conditions apply. For clarity, profiles are multiplied by different coefficients, as indicated in the plot.

Figure 5

Reduced electron density $\tilde{n}(x,y,z,T)$ in a three-dimensional sample with white-noise potential. Compared are the exact density (solid orange line) with that obtained by the random-wave-function algorithm (dashed green line for $N_{\mathrm{R}}=20$ and dotted black line for $N_{\mathrm{R}}=1000$). Profiles along the $x$ axis with different values of coordinates $y$ and $z$ are shown (see inset). Sample size is $2\times 2\times 2$ dimensionless units, the discretization grid parameter is $a=0.1$, the temperature is $T=1$. Periodic boundary conditions apply. For clarity, profiles are multiplied by different coefficients, as indicated in the plot.

Figure 6

Left panel: parameter $C\left(T\right)$ as function of the temperature $T$ for systems with white-noise disorder in one, two, and three dimensions. The effective potential is defined by Eq. (50). The parameter $C$ is the average of the effective potential, $C={\langle W_{\mathrm{R}}\rangle }_{\mathrm{R}}$ and used in Eq. (62). Dimensionless units ($\hslash =k_B=m=S=1$) introduced in Sec. 2d are used. Grid spacing $a=0.1$. Right panel: parameter function ${\tilde{c}}_d(a,T)$ defined in Eq. (68) as a function of $a/ƛ$. The limit $a/ƛ\to 0$ defines the constants $c_d$.

Figure 7

Variance of the effective potential $W_{\mathrm{R}}\left(T\right)$, var $W_{\mathrm{R}}={\langle W_{\mathrm{R}}^2\rangle }_{\mathrm{R}}-{\langle W_{\mathrm{R}}\rangle }_{\mathrm{R}}^2$, as function of the temperature $T$ for systems with white-noise disorder in one, two, and three dimensions. Dimensionless units ($\hslash =k_B=m=S=1$) introduced in Sec. 2d are used. Grid spacing $a=0.1$.

Figure 8

Filter functions $\mathrm{\Gamma }\left(r\right)$ at different temperatures $T$ in 1D systems with white-noise potential as obtained from the optimization procedure. The dots indicate values of $\mathrm{\Gamma }\left(r\right)$ that serve as fitting parameters. Dimensionless units ($\hslash =k_B=m=S=1$) introduced in Sec. 2d are used.

Figure 9

Comparison between the exact effective potential $W_{\mathrm{R}}\left(x\right)$ (solid orange lines) and the filtered potential, see Eq. (62) (dashed blue lines) for a one-dimensional sample with while-noise potential. Filter functions at different temperatures $T$ are obtained by the optimization procedure. Dimensionless units ($\hslash =k_B=m=S=1$) are used. The sample size is equal to 30 dimensionless units, the discretization parameter is $a=0.1$. For clarity, curves for $T=0.5$, $T=1$, and $T=2$ are shifted upwards by 3, 6, and 9 units, respectively.

Figure 10

Relative error $\mathfrak{E}$ of the effective potential $W$ obtained from convolution of the potential with the filter function, Eq. (62), at different temperatures $T$. Circles are related to the filter function obtained from the optimization procedure, lines correspond to the universal filter function considered in Sec. 3d. The values of relative error $\mathfrak{E}$ are obtained from Eq. (71). Dimensionless units ($\hslash =k_B=m=S=1$) are used. Discretization parameter $a=0.1$.

Figure 11

Radial filter functions ${\mathrm{\Gamma }}_W^{\left(d\right)}(r/ƛ)$ ($d=1,2,3$ from top to bottom), obtained from the optimization procedure in Sec. 3c, plotted versus $r/ƛ$, in comparison with the universal filter function ${\mathrm{\Gamma }}_{\text{uni}}$ (thick violet line). Dimensionless units ($\hslash =k_B=m=S=1$) are used. $ƛ=1/\sqrt{2T}$ is the reduced de-Broglie wavelength at kinetic energy equal to $T$. The discretization parameter is $a=0.1$.

Figure 12

Electron density $n(x,T)$ in a one-dimensional white-noise potential at different temperatures $T$: exact (solid orange lines) and ULF (dashed blue lines). The parameters are the same as in Fig. 1, except for the temperature $T$. For clarity, the curves related to $T=0.2$, $T=1$ and $T=2$ are multiplied by ${10}^{-4}$, ${10}^2$, and ${10}^3$, respectively, as indicated in the plot; particle concentration $n=0.01$.

Figure 13

Comparison between the exact electron density $n(x,y,T)$ (upper part) and that obtained by using the universal filter function (lower part) in a two-dimensional white-noise potential. The sample size is $5\times 5$ dimensionless units, the discretization parameter is $a=0.1$, the temperature is $T=1$, the particle concentration is $n=0.01$. Periodic boundary conditions apply.

Figure 14

Comparison between the exact electron density $n(x,y,z=0,T)$ (upper part) and that obtained by using the universal filter function (lower part) in a three-dimensional white-noise potential. The sample size is $10\times 10\times 10$ dimensionless units, the discretization parameter is $a=0.1$, the temperature is $T=2$, the particle concentration is $n=0.01$. Periodic boundary conditions apply.

Figure 15

Comparison between the exact electron density $n(x,T)$ (solid orange lines) and that obtained using the ULF function (solid blue lines) and the LLT (dashed black line) in a one-dimensional white-noise potential at different temperatures $T$. The parameters are the same as in Fig. 1, except for the temperature $T$. For clarity, the curves related to $T=0.2$, $T=0.5$, and $T=1$ are multiplied by ${10}^{-5}$, ${10}^{-3}$, and ${10}^{-1}$, respectively, as indicated in the plot. The particle concentration is $n=0.01$.

Figure 16

Electron density $n(x,y,T)$ obtained using the LLT approach with different parameters $\sigma$ in units $a$ and $K$ in units ${\varepsilon }_0$, as described in the text. (a) $\sigma =0$, $K=18.2$; (b) $\sigma =1.5$, $K=5.1$; (c) $\sigma =1.5$, $K=18.2$. Other parameters are the same as in Fig. 13.

Figure 17

Temperature-dependent charge carrier mobility in one dimension. We compare exact results with those from the RWF, ULF, and LLT approximations to calculate the average of the local density $n(x,T)$ and of its inverse. $N_{\mathrm{R}}$ denotes the number of realizations used for averaging in the RWF algorithm.

Figure 18

Temperature-dependent charge carrier mobility in two dimensions. We compare exact results from percolation theory with those from the RWF, ULF, and LLT approximations to calculate the average of the local density $n(x,T)$ and of its median. $N_{\mathrm{R}}$ denotes the number of realizations used for averaging in the RWF algorithm. LLT results with and without Gaussian smoothing.