Diffusion Monte Carlo study of a valley-degenerate electron gas and application to quantum dots

A many-flavor electron gas (MFEG) in a semiconductor with a valley degeneracy ranging between 6 and 24 was analyzed using diffusion Monte Carlo (DMC) calculations. The DMC results compare well to an analytic expression derived by one of us [Phys. Rev. B 78, 035111 (2008)] for the total energy to within $\pm 1\%$ over an order of magnitude range of density, which increases with valley degeneracy. For ${\text{Bi}}_2{\text{Te}}_3$ (sixfold valley degeneracy) the applicable charge-carrier densities are between $7\times {10}^{19}{\text{cm}}^{-3}$ and $2\times {10}^{20}{\text{cm}}^{-3}$. DMC calculations distinguished between an exact and a useful approximate expression for the 24-fold degenerate MFEG polarizability for wave numbers $2p_F<q<7p_F$. The analytical result for the MFEG is generalized to inhomogeneous systems by means of a gradient correction; the validity range of this approach is obtained. Employed within a density-functional theory calculation this approximation compares well to DMC results for a quantum dot.

Figure 1

The Ge band structure in the [111] direction calculated using a plane-wave pseudopotential method (Ref. 19). The Fermi energy is at $E=0\text{eV}$; below are valence bands with the holes centered around H, above are conduction bands. The first conduction band valley is highlighted in bold and low-lying conduction-band electrons are centered around $C$.

Figure 2

The lower panel shows the fractional difference of DMC interaction energy $E_{\text{DMC}}$ from the model $E_{\text{int}}$ with MFEG density $n$ (and Wigner-Seitz radius $r_s$) for different numbers of flavors; the dotted lines show $\pm 1\%$ disagreement. The central panel bars highlight the numerical region of applicability and the gray shaded area represents the analytically predicted region of $\pm 1\%$ applicability. The upper panel shows the total energy for 6, 12, 18, and 24 flavor electron gases.

Figure 3

The density-density response $1/ϵ$ versus the wave vector $\left|\mathbf{q}\right|$ of a MFEG with $\nu =24$ and $r_s=0.6a_0^{\ast }$. The solid curve shows the exact result $1/{ϵ}_{\text{exact}}$ [Eq. (2)] and the dotted curve the Eq. (3) approximation. The shaded gray region $\left|\mathbf{q}\right|/2<p_F$ is where the many-flavor limit breaks down. The points show the values for the permittivity calculated from QMC results, the circle is from modulated charge measurements (CM), the triangle from VMC energy, and the cross from DMC energy. The lower panel plots the actual response, the upper panel shows the deviation of response from the exact theoretical result, Eq. (2), with standard error bars.

Figure 4

In the upper plot the crosses and solid line show the difference between DFT $\left(E_{\text{DFT}}\right)$ and DMC $\left(E\right)$ energies with varying external potential strength $k$. If agreement were exact, points would lie along the horizontal dotted line. In the lower plot the circles and solid line show the maximum density gradient of the dots with varying $k$ and the squares and dashed line the gradient at which the theory breaks down.

Figure 5

The lower panel shows the density profile of quantum dots estimated using both DFT and DMC at external potential strengths of $k=1$ and $k=8$. The difference between the DFT and DMC results at $k=1$ and $k=8$ is shown in the upper panel. The DMC statistical error is less than the size of the points.

Figure 6

The upper panel shows, for dots with external potential strength $k=1$, the density profile maximum gradient as a function of $\xi$. The central panel shows the variation of external potential DFT energy based on the primary $y$-axis using square points and the solid line, the secondary $y$-axis shows electron-electron DFT energy using crosses and the dashed line. The lower panel solid line shows the variation of DFT ground-state energy with $\xi$ and the horizontal dotted line the ground-state energy predicted using DMC from the $\xi =1$ trial wave function. The DMC statistical error is less than the size of the points.

Figure 7

The upper panel shows the variation of the $\xi =0$ (dotted line, squares) and the $\xi =2$ (dashed line, crosses) DFT density profiles from the $\xi =1$ DFT density profile. The lower panel shows DFT density profiles for $\xi =0$, $\xi =1$, (solid line, triangles), and $\xi =2$ in a $k=1$ dot.