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

Abstract

Photovoltaic Hall effect 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 an important role of the bias electric field, which adds a field-induced circular photogalvanic effect to the photovoltaic Hall 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, transition energy, and intraband velocity, all contributing to the field-induced circular photogalvanic effect in nonmagnetic materials. The first process can be expressed as a manifestation of the electric field-induced Berry curvature. Shift vector plays an essential role in determining the transition energy shift. 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 reveals their geometric aspect.

Figures (4)

• Figure 1: Classification of photovoltaic Hall effect in nonmagnetic materials. Explanation of each component is given in Secs. \ref{['sec:FI-IC']} and \ref{['sec:LI-BC']}.
• Figure 2: (a), (b) Unperturbed transition matrix elements for a massive Dirac electron system excited by left-circularly polarized light, decomposed into the excitation paths 1 ($|\Downarrow\rangle\to|\Uparrow\rangle$) and 2 ($|\Uparrow\rangle\to|\Downarrow\rangle$), respectively. (c), (d) Field-induced changes in the transition matrix elements for the paths 1 and 2, respectively. A static electric field $E_0=10$ kV/cm along the $x$ axis is assumed.
• Figure 3: A simple picture of the electric field-induced energy change in a massive Dirac electron system. Dashed and solid lines present the unperturbed and modified dispersion relations, respectively, with the difference exaggerated for visibility. A bias electric field in the $x$ direction is assumed.
• Figure 4: (a) Before singular value decomposition (SVD), there are $g_{\nu}\times g_{\mu}$ excitation paths available for the interband transition from band $\mu$ to band $\nu$. The case of $g_\nu=g_\mu=2$ is shown. (b) SVD resolves mutually independent pairs of initial and final states, the number of which is less than $\operatorname{min}(g_\nu,g_\mu)$.