Gauge invariance of excitonic linear and nonlinear optical response

Abstract

We study the equivalence of four different approaches to calculate the excitonic linear and nonlinear optical response of multiband semiconductors. These four methods derive from two choices of gauge, i.e., length and velocity gauges, and two ways of computing the current density, i.e., direct evaluation and evaluation via the time-derivative of the polarization density. The linear and quadratic response functions are obtained for all methods by employing a perturbative density-matrix approach within the mean-field approximation. The equivalence of all four methods is shown rigorously, when a correct interaction Hamiltonian is employed for the velocity gauge approaches. The correct interaction is written as a series of commutators containing the unperturbed Hamiltonian and position operators, which becomes equivalent to the conventional velocity gauge interaction in the limit of infinite Coulomb screening and infinitely many bands. As a case study, the theory is applied to hexagonal boron nitride monolayers, and the linear and nonlinear optical response found in different approaches are compared.

Nonlinear optical selection rules of excitons in monolayer transition metal dichalcogenides

“…The approximation in Eq. (A2) is mainly because of the neglected terms for non-zero reciprocal vectors [30]. The equation of motion for ρ cvk is simply obtained by complex conjugating the first equation, since the density matrix operator is Hermitian.…”

“…The generalized derivative stems from the intraband part of the position operator [32,33]. Equations (2a) and (2b) have been reported in our previous work [30] while Eq. (2c) can be obtained in a similar manner by extending the general approach to the third order in the electric field.…”

“…Refs. [29,30]. In this approach, a set of coupled equations for the reduced density matrix ρ ijk = 0|ĉ † ikĉ jk |0 is derived, where |0 denotes the many-body ground state.…”

“…Note that linear and NLO susceptibilities, χ (N ) are related to their corresponding conductivities by σ (N ) = −iN ω 0 χ (N ) . The expressions of the first-, second-, and dominant part of the third-order conductivity tensors at vanishing temperature are given by [30] σ (1)…”

“…Although these expressions are obtained for a two-band semiconductor, they can be used for multi-band semiconductors by redefining the Q α nm matrix elements (see Ref. [30] for more details). Note that, with a minor modification, these expressions can also be employed for calculating the nonlinear spin conductivities [5].…”