Second-order optical response in TR-symmetry breaking insulators

We present a general formalism for investigating the second order optical susceptibility of a clean cold semiconductor in the independent particle approximation. Our derivation naturally decomposes the response into electric and magnetic multipole moment contributions and a free current contribution that avoids unphysical divergences of the response at zero frequency. However, true divergences do exist, like the “injection current”, also known as the “photogalvanic effect”. In the past, derivations have assumed time-reversal symmetry, however interest in nonlinear optical response of materials that break time-reversal symmetry is increasing and as such we allow for time-reversal symmetry breaking and consider how this extends the formulas. We investigate then the time-even and time-odd contributions to phenomena like the shift current, injection current, and second harmonic generation. For example, an injection current can be induced by linearly polarized light (where previously only circularly polarized light could do so), and a new `real divergence’ at zero frequency arises in the tensor describing second harmonic generation. We then examine these effects in the Haldane model as it breaks the requisite inversion and TR symmetry.