Optimized structure and electronic band gap of monolayer GeSe from quantum Monte Carlo methods

We have used highly accurate quantum Monte Carlo methods to determine the chemical structure and electronic band gaps of monolayer GeSe. Two-dimensional (2D) monolayer GeSe has received a great deal of attention due to its unique thermoelectric, electronic, and optoelectronic properties with a wide range of potential applications. Density functional theory (DFT) methods have usually been applied to obtain optical and structural properties of bulk and 2D GeSe. For the monolayer, DFT typically yields a larger band-gap energy than for bulk GeSe but cannot conclusively determine if the monolayer has a direct or indirect gap. Moreover, the DFT-optimized lattice parameters and atomic coordinates for monolayer GeSe depend strongly on the choice of approximation for the exchange-correlation functional, which makes the ideal structure—and its electronic properties—unclear. In order to obtain accurate lattice parameters and atomic coordinates for the monolayer, we use a surrogate Hessian-based parallel line search within diffusion Monte Carlo to fully optimize the GeSe monolayer structure. The DMC-optimized structure is different from those obtained using DFT, as are calculated band gaps. The potential energy surface has a shallow minimum at the optimal structure. This, combined with the sensitivity of the electronic structure to strain, suggests that the optical properties of monolayer GeSe are highly tunable by strain.

Figure 1

(a) DMC total energy of bulk GeSe as function of $N^{-1}$, where $N$ is the total number of atoms in the supercell. The dotted line indicates a simple linear regression fit. (b) Extrapolated DMC total energy of GeSe as function of unit-cell volume. The dotted line represents a Vinet fit.

Figure 2

(a) PBE band structure of bulk GeSe. (b) DMC excitonic gap and quasiparticle gap for bulk GeSe as function of $N^{-1}$, where $N$ is the total number of atoms in the supercell. The dotted line indicates a simple linear regression fit.

Figure 3

(a) Top and (b) side view of GeSe monolayer. Parameters in the geometry are taken from Ref. [14].

Figure 4

Total energy contour plots for (a) PBE(HSE06) and (b) DMC(HSE06) energies as functions of lattice parameters $a$ and $b$ in atomic units. Crosses indicate global minima of the energy contours.

Figure 5

PBE monolayer GeSe band structure for (a) PBE and (b) DMC geometry. Blue lines represent candidates for direct and indirect gap.

Figure 6

DMC quasiparticle and excitonic gaps for monolayer GeSe as function of $N^{-1}$, where $N$ represents the total number of atoms in the supercell. The dotted line indicates a simple linear regression fit.

Figure 7

PBE total energy per formula unit GeSe for bulk GeSe as a function of $k$-point mesh (left panel) and as a function of kinetic energy cutoff (right panel).

Figure 8

DMC total energy per GeSe formula unit for bulk GeSe as function of number of twists in TABC.

Figure 9

DMC total energy for GeSe monolayer in lattice parameter (a) $a=4.05$, (b) 4.15, (c) 4.26, and (d) 4.47 Å as a function of DFT XC functional used for atomic coordinates relaxation. Note that lattice parameter $b$ is the same as 3.95 Å.

Figure 10

DFT band structure using the PBE (black) and PBE0 (red) XC functionals. While the gap is different from that of PBE, as expected, and the direct and indirect gaps occur at the same $k$ points as for PBE, indicated by the arrows.