Electronic band gaps from quantum Monte Carlo methods

We develop a method for calculating the fundamental electronic gap of semiconductors and insulators using grand canonical quantum Monte Carlo simulations. We discuss the origin of the bias introduced by supercell calculations of finite size and show how to correct the leading and subleading finite size errors either based on observables accessible in the finite-sized simulations or from density-functional theory calculations. Our procedure is applied to carbon, silicon, and molecular hydrogen crystals, and compared to experiment for carbon and silicon.

Figure 1

GCTABC analyses of the C2/c-24 structure of solid hydrogen at $r_s=1.38$ (234GPa). (a) The electron density, $n_e$, as a function of the chemical potential $\mu$ obtained from HSE functional in comparison to QMC; the inset illustrates the energy density, $e_0$, as a function of $\mu$ from HSE functional. (b) Energy density, $e_0$, as a function of $n_e$ using QMC; the inset shows the derivative discontinuity, where $\delta n_e$ is the change of the electronic density with respect to the insulating state. Size corrections, as discussed in the text, are included.

Figure 2

Change in the static structure factor as an electron (upper curves) or a hole (lower curves) is added to the insulating system with $N$ atoms. The lines are fits to the data points. The horizontal lines show the expected $k\to 0$ limit based on the experimental dielectric constants. We have used $c=0.41$ for C and $c=0.57$ for Si.

Figure 3

Upper bound to the inverse dielectric constant Eq. (7), where ${\mathrm{\Gamma }}_k\equiv \frac{2m{\omega }_p{S}_{N_e}\left(k\right)}{\hslash k^2}$. Lines are fits to the low-$k$ data. The horizontal lines mark experimental inverse dielectric constants.

Figure 4

Fundamental gap before and after finite-size corrections. ${\mathrm{\Delta }}_N$ is the DMC gap from a simulation with $N$ atoms in the supercell without any finite-size correction, $v_M/\epsilon$ is the leading-order Madelung correction using the experimental value of ${\epsilon }^{-1},\delta {\mathrm{\Delta }}_s^N$ is the next-to-leading-order density correction, which is related to the static part of the structure factor. The line is a fit to ${\mathrm{\Delta }}_N+\delta {\mathrm{\Delta }}_s^N$.

Figure 5

Density of states for carbon and silicon near the band edge. Each plot shows the derivative of the mean electron density with respect to the chemical potential. This is the electronic density of states (DOS) in DFT, so the gap appears as a depleted region. The calculated DOS is only valid near the band edge because only the two bands closest to the gap are considered within DFT and QMC. The DFT bands (done in a primitive cell) have been folded into the Brillouin zone (BZ) of the 64-atom supercell to allow comparison with QMC.