High harmonic generation in crystals using maximally localized Wannier functions

In this work, the nonlinear optical response, and in particular, the high harmonic generation of semiconductors, is addressed by using the Wannier gauge. One of the main problems in the time evolution of the Semiconductor Bloch equations resides in the fact that the dipole couplings between different bands can diverge and have a random phase along the reciprocal space, and this leads to numerical instability. To address this problem, we propose the use of the maximally localized Wannier functions that provide a framework to map ab initio calculations to an effective tight-binding Hamiltonian with great accuracy. We show that working in the Wannier gauge, the basis set in which the Bloch functions are constructed directly from the Wannier functions, the dipole couplings become smooth along the reciprocal space, thus avoiding the problem of random phases. High harmonic generation is computed for a two-dimensional monolayer of hexagonal boron nitride as a numerical demonstration.

Figure 1

(a) Single layer of hexagonal boron nitride. The boron (nitrogen) atoms are in green (gray). The blue (yellow) arrow represents the laser polarization along $\mathrm{\Gamma }-\mathrm{M}\left(\mathrm{\Gamma }-\mathrm{K}\right)$. (b) Electronic band structure for the effective model (blue line) and for the ab initio model (red line).

Figure 2

(a) Transition dipole moment between the valence band and the conduction band, $\left|{\mathit{A}}_{vc}^{\left(H\right)}\left(\mathit{k}\right)\right|=\sqrt{{\left|A_{x,vc}^{\left(H\right)}\left(\mathit{k}\right)\right|}^2+{\left|A_{y,vc}^{\left(H\right)}\left(\mathit{k}\right)\right|}^2}$, for the effective model. (b) Same as in (a) but for ${\varepsilon }_B=-{\varepsilon }_N=0.2$ eV.

Figure 3

High harmonic generation spectrum computed for a single monolayer of hBN for the effective model (blue lines) and for the ab-initio model (red lines). (a,b,c) Harmonic spectrum for a laser pulse in the $\mathrm{\Gamma }-\mathrm{M}\left(\mathrm{\Gamma }-\mathrm{K},\mathrm{\Gamma }-\mathrm{K}\right)$ direction along its parallel (parallel, perpendicular) direction, respectively.