Upper limit on nonlinear optical processes: shift current and second harmonic generation in extended systems

The response functions of a material characterize its behavior under external stimuli, such as electromagnetic radiation. Such responses may grow linearly with the amplitude of the incident radiation, as is the case of absorption, or may be nonlinear. The latter category includes a diverse set of phenomena such as second harmonic generation (SHG), shift current, sum frequency generation, and excited state absorption, among others. Despite decades of research into nonlinear response theory, and the occasional discovery of materials with large nonlinear responses, there has been no systematic investigation into the maximum amount of nonlinear optical response attainable in solid-state materials. In this work, we present an upper bound on the second-order response functions of materials, which controls the SHG and shift current responses. We show that this bound depends on the band gap, band width, and geometrical properties of the material in question. We find that Kuzyk's bound for the maximum SHG of isolated molecules can be exceeded by conjugation or condensation of molecules to form molecular solids, and that strongly coupled systems generally have larger responses than weakly coupled or isolated ones. As a proof of principle, we perform first-principles calculations of the response tensors of a wide variety of materials, finding that the materials in our database do not yet saturate the upper bound. This suggests that new large SHG and shift current materials will likely be discovered by future materials research guided by the factors mentioned in this work.