Ab initio calculation of second harmonic generation in solids

Abstract

An ab initio formalism for the calculation of macroscopic second order responses within the time dependent density functional theory (TDDFT) framework is presented. The linear Dyson-like equation of TDDFT is generalized to second order and the resulting microscopic response is related to the macroscopic non-linear susceptibility. The macroscopic relations are obtained considering Maxwell's equations and the non-linear relation between the microscopic and macroscopic field. Some approximations have to be made to account for many body effects. The impact of the several levels of approximation on the second harmonic spectrum is illustrated for the example of bulk AlAs in the zincblende structure. This success motivates expectations for an equally successful first principles description of non-linear optical processes. However, the basic requirement of such a description, is a comprehensive understanding of the non-linear microscopic physical mechanisms in the second order response function, and how they are related to the macroscopic quantities. This is a formidable task and considerable difficulties have delayed accurate calculations for many years [4-8], even for the lowest level of approximation: independent particle approximation (IPA) [9]. Within IPA, the absolute value of the second order susceptibility does not at all agree with the experimental measurements [10]. To go beyond the independent particle description, we need to carefully consider the second order susceptibility x ð2Þ defined as

Second-order response Bethe-Salpeter equation

“…It is the underlying structure of the Dyson equation that both have in common and makes this comparison possible. In the same way, there is a correspondence of the structure between the known secondorder TDDFT Dyson-like equation [16,17] and a second-order generalization of the BSE. Such a correspondence can not only guide the derivation of such an equation but also can serve to yield an expression for g xc in terms of many-body quantities.…”

“…The structure of Eq. (9) can be seen as the second-order equivalent of the Dyson equation and it is the same encountered in second-order response TDDFT [16,17].…”

“…A second-order response equivalent of the TDDFT Dyson equation is given in Refs. [16,17,21] and reads (20) and yields exactly the same structure as the second-order BSE, Eq. (9).…”

“…In this framework there does exist a relation that can be considered the second-order generalization of the Dyson equation. It relates the interacting second-order density response to its noninteracting one via the exchange and correlation kernels f xc and g xc , which are the first-and second-order variations of the exchange and correlation potential of TDDFT [16,17]. The exact form of the linear interaction kernel f xc is, however, unknown and finding good approximations to them remains one of the major challenges of the field [18][19][20].…”

“…The second-order kernel g xc , being the variation of the linear one, is even less known and it has been virtually never used in practical calculations so far. The second-order TDDFT Dyson-like equation where this kernel features has been published some time ago [16] but has also never been applied until recently [17,21,22], arguably because of the lack of knowledge about g xc .…”