Photovoltaic sum rule, polarization distribution and the band geometry behind them

The bulk photovoltaic effect (BPVE) is a second-order optical DC current in materials with broken inversion symmetry that is important in some designs of solar cells. The connection of BPVE to spontaneous polarization has been long recognized but the precise relation has only been established in special cases. In this work, we clarify this relation for general multi-band free-electronic materials. To this end, we derive explicitly gauge-invariant expressions for both the BPVE and polarization distribution that also give an advantage for numerical calculations. We then use these expressions to identify the part of the frequency averaged BPVE given by the skewness of electronic polarization and the remaining part determined by more detailed properties of band states that we also discuss.The developed theory is illustrated for a tight-binding model of a two-dimensional dichalcogenide.