Linear scaling calculation of an $n$-type GaAs quantum dot

A linear scale method for calculating electronic properties of large and complex systems is introduced within a local density approximation. The method is based on the Chebyshev polynomial expansion and the time-dependent method, which is tested on the calculation of the electronic structure of a model $n$-type GaAs quantum dot.

Figure 1

(Color online) (a) The electron density distribution obtained by CPE-TDM. 128 sets of random vectors are used at each self-consistent iteration procedure. (b) The electron density distribution obtained by CGM.

Figure 2

(Color online) (a) Cross-sectional views of the calculated Hartree potentials on the plane at the center of the quantum well layer obtained by CPE-TDM $\left[V_{\mathrm{H}}^{\mathrm{CPE}\text{−}\mathrm{TDM}}\left(\mathbf{r}\right)\right]$ with (i) 8, (ii) 16, (iii) 32, (iv) 64, and (v) 128 sets of random phase vectors for extracting $n\left(\mathbf{r}\right)$ at each self-consistent iteration procedure, and (vi) $V_H\left(\mathbf{x}\right)$ obtained by CGM. (b) Differences of obtained Hartree potentials $V_{\mathrm{H}}^{\mathrm{CPE}\text{−}\mathrm{TDM}}\left(\mathbf{r}\right)-V_{\mathrm{H}}^{\mathrm{CGM}}\left(\mathbf{r}\right)$ with (i) 8, (ii) 16, (iii) 32, (iv) 64, and (v) 128 sets of random phase vectors. (c) Standard deviations of the calculated Hartree potentials depending on the number of random phase vectors at each self consistent iteration procedure $j_{\max }$. The best fitted curve proportional to $1∕\sqrt{j_{\max }}$ is also shown.

Figure 3

(a) Density of states $\left[\rho \left(\omega \right)\right]$ obtained by CPE-TDM with $j_{\max }=k_{\max }=\left(i\right)$ 8, (ii) 16, (iii) 32, (iv) 64, and (v) 128 sets of random phase vectors for extracting $n\left(\mathbf{r}\right)$ at each self-consistent iteration procedure and for evaluating $\rho \left(\omega \right)$. (vi) $\rho \left(\omega \right)$ obtained by CPE-TDM with eight sets of random phase vectors for extracting $n\left(\mathbf{r}\right)$ and $k_{\max }=64$ sets of random phase vectors for evaluating $\rho \left(\omega \right)$. (vii) $\rho \left(\omega \right)$ obtained by CGM. (b) Standard deviations of the difference of the peak heights of the DOS obtained by CPE-TDM and by CGM depending on $j_{\max }$ (solid circles). Standard deviation for CPE-TDM with $j_{\max }=\left(8\right)$ at each self consistent iteration procedure and $k_{\max }=64$ for evaluating $\rho \left(\omega \right)$ is shown (open circle) The best fitted curve proportional to $1∕\sqrt{j_{\max }}$ is also shown. (c) Total number of electrons $\left(N_e\right)$ depending on $j_{\max }$. The horizontal line shows $N_e=77.1$ obtained by CGM.