Microscopic quantum description of second-order nonlinearities in two-dimensional hexagonal nanostructures beyond the Dirac cone approximation

H K Avetissian,K G Batrakov,G F Mkrtchian,S A Maksimenko

doi:10.1103/physrevb.102.165406

H K Avetissian, K G Batrakov + Show 2 more

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https://doi.org/10.1103/physrevb.102.165406

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Abstract

Single layers of hexagonal two-dimensional nanostructures such as graphene, silicene, and germanene exhibit large carrier Fermi velocities and, consequently, large light-matter coupling strength making these materials promising elements for nano-opto-electronics. Although these materials are centrosymmetric, the spatial dispersion turns out to be quite large allowing the second-order nonlinear response of such materials to be comparable to the non-centrosymmetric 2D ones. The second-order response of massless Dirac fermions has been extensively studied, however a general approach correct over the full Brillouin zone is lacking so far. To complete this gap, in the current paper we develop a general quantum-mechanical theory of the in-plane second-order nonlinear response beyond the Dirac cone approximation and applicable to the full Brillouin zone of the hexagonal tight-binding nanostructures. We present explicit calculation of the nonlinear susceptibility tensor of 2D hexagonal nanostructures applicable to arbitrary three-wave mixing processes.

Highlights

In the last decade, graphene[1,2] and its analogs silicene,[3,4,5] germanene,[6,7] and stanene[8] have attracted enormous interest due to their unique electronic and optical properties
To clear up the deviations of the secondorder susceptibility tensor calculated for the full Brillouin zone (FBZ) from the Dirac cone approximation (DCA), in Fig. 7 we plot the ratios of the absolute values of susceptibility tensor components for the process of the second-harmonic generation calculated with and without DCA as a function of the Fermi energy scaled to the nearest-neighbor hopping energy γ0
We have developed a microscopic quantum ansatz for analytical and numerical calculation of the secondorder nonlinear response of hexagonal 2D nanostructures beyond the Dirac cone approximation, which is applicable to the excitations in the full Brillouin zone

Summary

INTRODUCTION

Graphene[1,2] and its analogs silicene,[3,4,5] germanene,[6,7] and stanene[8] have attracted enormous interest due to their unique electronic and optical properties. [42,43] the full quantum-mechanical theory of the in-plane second-order nonlinear response beyond the electric dipole approximation has been developed for graphene-like nanostructures considering the low-energy dynamics in the K+ and K− valleys. At visible and deep UV frequencies of driving waves for graphene and even more for silicene, germanene, and stanene one should have microscopic theory describing nonlinear interaction beyond the DCA and applicable to the full Brillouin zone (FBZ) of the hexagonal nanostructure with tight-binding electronic states. Note that spatial dispersion induced second-order nonlinear response is nonzero for doped system and at sufficiently high doping > 0.2 eV one can omit spin-orbit coupling in silicene, germanene, and stanene considering those as gapless hexagonal nanostructures with corresponding lattice spacing a and hopping transfer energy γ0. The Dirac delta function in Eqs. (25) and (26) expresses conservation law for momentum

SECOND ORDER NONLINEAR RESPONSE OF 2D HEXAGONAL NANOSTRUCTURE

Sum-frequency generation process in 2D hexagonal nanostructure

Difference frequency generation processes: generation of plasmons

CONCLUSION