Nonlinear optical conductivity of a two-band crystal I

Abstract

The structure of the electronic nonlinear optical conductivity is elucidated in a detailed study of the time-reversal symmetric two-band model. The nonlinear conductivity is decomposed as a sum of contributions related with different regions of the first Brillouin zone, defined by single or multiphoton resonances. All contributions are written in terms of the same integrals, which contain all information specific to the particular model under study. In this way, ready-to-use formulas are provided that reduce the often tedious calculations of the second and third order optical conductivity to the evaluation of a small set of similar integrals. In the scenario where charge carriers are present prior to optical excitation, Fermi surface contributions must also be considered and are shown to have an universal frequency dependence, tunable by doping. General characteristics are made evident in this type of resonance-based analysis: the existence of step functions that determine the chemical potential dependence of electron-hole symmetric insulators; the determination of the imaginary part by Hilbert transforms, simpler than those of the nonlinear Kramers-Krönig relations; the absence of Drude peaks in the diagonal elements of the second order conductivity, among others. As examples, analytical expressions are derived for the nonlinear conductivities of some simple systems: a very basic model of direct gap semiconductors and the Dirac fermions of monolayer graphene.

Third-order nonlinear Hall effect in two-dimensional Dirac systems

“…xyxx (3ω) = 0. The results are in agreement with [24][25][26]. In a realistic system, there are always two Dirac points.…”

“…The above equation is similar to the third-order nonlinear optical effects in [32]. And the third harmonic generation in graphene without the Berry curvature has been extensively studied in many works [24][25][26].…”

“…The third harmonic current j N in the absence of the Berry curvature has been extensively studied in [24][25][26], one obtains…”

“…From equations ( 26) and ( 28), the nonlinear Hall current can be easily distinguished from the third harmonic current in the absence of the Berry curvature contributions [19,[24][25][26].…”

“…Based on the single-band Boltzmann transport equation, it is shown that the third-order nonlinear Hall response can survive in the systems without time-reversal symmetry, and is proportional to the second derivative of the Berry curvature on k [18][19][20]. Nevertheless, the nonlinear response usually involves mixed interband-intraband transitions for the multiband systems related to the Berry connection [21][22][23][24][25][26]. Such processes for the second-order nonlinear Hall effect have been theoretically verified [12,13].…”