Multipolar Contributions to the Nonlinear Optical Response of Crystalline Media

Within the electric dipole approximation, the second order nonlinear conductivity is forbidden in centrosymmetric materials. However, at the interface between two different materials, or by chemically doping, inversion symmetry is broken, and a second order conductivity is allowed. Further, even in centrosymmetric materials there exist electric quadrupole and magnetic dipole contributions that can give rise to processes such as second harmonic generation in the bulk. It has been shown that the surface and bulk contributions cannot be disentangled from each other in typical experiments. Experimental work has been done to examine the magnetic dipole and electric quadrupole like bulk contributions in various materials such as calcite, graphene and C60. However, theoretical work on crystalline systems generally does not treat the itinerant nature of Bloch electrons and a general band structure. We apply a microscopic theory of polarization and magnetization to this problem to obtain an expression for the bulk second order conductivity tensor that includes electric dipole, quadrupole, and magnetic dipole contributions, and is valid for general insulating materials. The response tensors depend only on the dispersion relation and Bloch function matrix elements such as the Berry connection. Further, the expressions are valid for arbitrary frequency mixing and polarization components.