Pressure-induced diamond to $\beta$-tin transition in bulk silicon: A quantum Monte Carlo study

The pressure-induced structural phase transition from diamond to $\beta$ tin in silicon is an excellent test for theoretical total-energy methods. The transition pressure provides a sensitive measure of small relative energy changes between the two phases (one a semiconductor and the other a semimetal). Experimentally, the transition pressure is well characterized. Density-functional results exhibit sensitivity to the particular form of the exchange-correlation functional. Even the generally much more accurate diffusion Monte Carlo method has shown a noticeable fixed-node error. We use the recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method to calculate the relative energy differences in the two phases. In this method, all but the error due to the phaseless constraint can be controlled systematically and driven to zero. In both structural phases we were able to benchmark the error of the phaseless constraint by carrying out exact unconstrained AFQMC calculations for small supercells. Comparison between the two shows that the systematic error in the absolute total energies due to the phaseless constraint is well within $0.5\text{m}{E}_{\text{h}}/\text{atom}$. Consistent with these internal benchmarks, the transition pressure obtained by the phaseless AFQMC from large supercells is in very good agreement with experiment.

Figure 1

(Color online) Equation of state for the Si diamond and $\beta$-tin phases, comparing all-electron (solid lines) and pseudopotential (dashed lines) DFT-LDA results. The inset shows the Gibbs energy as a function of the pressure.

Figure 2

(Color online) AFQMC and LDA Si $\beta$-tin energies at nine randomly chosen $\mathbf{k}$ points in the Brillouin zone of $3\times 3\times 3$ supercell. The dashed horizontal line is the converged LDA energy using a dense $\left(8\times 8\times 8\right)$ Monkhorst-Pack grid. The AFQMC and LDA one-body FS errors are correlated, and the correction in Eq. (24) reduces the variation in the QMC data. The two-body FS error is significant even with a 54-atom supercell but is essentially independent of $\mathbf{k}$ points.

Figure 3

(Color online) Comparison of phaseless AFQMC with exact free projection. The calculations shown are for the primitive cell in the diamond structure at the experimental equilibrium lattice constant $\left(10.264a_0\right)$. The calculations used $L$ as the $\mathbf{k}$ point, and a time step size of $\tau =0.025E_{\text{h}}^{-1}$.

Figure 4

(Color online) Trotter time step $\tau$ extrapolation for the diamond structure, as in Fig. 3, comparing local-energy and hybrid phaseless methods with free projection.

Figure 5

(Color online) Trotter time step $\tau$ extrapolation for the $\beta$-tin structure, comparing the local-energy phaseless method with free projection. Calculations are for the primitive unit cell with volume of $30{\text{Å}}^3$ and $c/a=0.552$ at the reduced $\mathbf{k}=\left(0.25,0.25,0.25\right)$.