Unified theory of the photovoltaic Hall effect by field- and light-induced Berry curvatures

Abstract

Photovoltaic Hall effect, i.e., generation of a photocurrent perpendicular to the bias electric field, is an interesting platform of Berry curvature engineering by external fields. Floquet engineering aims at generation of light-induced Berry curvature associated with topological phase transition in solids, which may manifest itself as a light-induced anomalous Hall effect. However, recent studies have pointed out a larger contribution by momentum asymmetry of photocarriers, termed a field-induced circular photogalvanic effect. Except for numerical studies, the two mechanisms have been described by different theoretical frameworks, hindering a coherent understanding. Here, we develop a unified theory of the photovoltaic Hall effect capable of describing both mechanisms on an equal footing. We reveal that the bias electric field alters the interband transition dipole moment and transition energy, both contributing to the field-induced circular photogalvanic effect in nonmagnetic materials. These effects are governed by an electric field-induced Berry curvature and the shift vector coupled to bias field, respectively. A resonant enhancement of the transverse photocurrent is found in GaAs owing to the topological character of the valence band. We also clearly distinguish the anomalous Hall effect by light-dressed states within the density matrix calculation using the length gauge. Our theory unifies a number of nonlinear optical processes in a physically transparent way and presents a geometric picture of the third-order nonlinear response under light and bias fields, shedding new light on Berry curvature engineering.

Figures (6)

• Figure 1: (a) Band-resolved Berry curvature $\Delta\boldsymbol{\Omega}_{nm}$ generated by a bias electric field $\mathbf{E}_0$, which causes a momentum asymmetry of photoexcited carriers. (b) Field-induced energy shift of interband transitions, arising from the potential gradient corresponding to the shift vector $\mathbf{R}_{nm}(\mathbf{e})$. (c) Anomalous velocity of photoexcited carriers generated by the Berry curvature $\boldsymbol{\Omega}_n$.
• Figure 2: (a) Band structure of a massive Dirac electron system. (b) Field-induced correction to the $z$ component of the band-resolved Berry curvature in a massive Dirac electron system. A bias field $E_0 = 10$ kV/cm in the $x$ direction is assumed. (c) Band structure of GaAs described using an eight-band Kane model. (d) Field-induced correction to the $z$ component of the band-resolved Berry curvature between CB and HH in GaAs. A bias field $E_0 = 10$ kV/cm in the $x$ direction is assumed. $k_z$ is fixed at 0 nm$^{-1}$, and $k_x$ is varied as indicated in the figure. The curve for $k_x = 0$ nm$^{-1}$ diverges at $k_y = 0$ nm$^{-1}$.
• Figure 3: (a), (b) Shift vectors for left- and right-circularly polarized light, respectively, in a massive Dirac electron system. The arrow on the right side corresponds to a length of 0.2 nm. (c), (d) Magnitude of the transition energy shift for paths 1 and 2 caused by the bias field $E_0 = 10$ kV/cm along the $x$ axis. (e) Shift vectors for left-circularly polarized light in GaAs. The first and second quadrants resolve the two transition paths from HH to CB, and the third and fourth quadrants resolve those from LH to CB. (f) The $x$ component of the shift vectors at $\mathbf{k} = (0,k_y,0)$.
• Figure 4: (a) Three contributions to the field-induced CPGE in a massive Dirac electron system. The electric field amplitude of light is set to 100 kV/cm. (b) The sum of the three contributions in panel (a). The dashed line represents the real part of optical conductivity in equilibrium. (c) Three contributions to the field-induced CPGE in GaAs. $K_{\mathrm{dip}}^{yx}$ (orange) and $K_{\mathrm{ene}}^{yx}$ (green) almost overlap. The electric field amplitude of light is set to 100 kV/cm. (d) The sum of the three contributions in panel (c). The dashed line represents the real part of optical conductivity in equilibrium.
• Figure 5: Classification of photovoltaic Hall effect in nonmagnetic materials.