Nonlinear optical response of a two-dimensional atomic crystal

Abstract

The theory of Bloembergen and Pershan for the light waves at the boundary of nonlinear media is extended to a nonlinear two-dimensional atomic crystal, i.e. a single planar atomic lattice, placed in between linear bulk media. The crystal is treated as a zero-thickness interface, a real two-dimensional system. Harmonic waves emanate from it. Generalization of the laws of reflection and refraction give the direction and the intensity of the harmonic waves. As a particular case that contains all the essential physical features, second order harmonic generation is considered. The theory, due to its simplicity that stems from the special character of a single planar atomic lattice, is able to elucidate and to explain the rich experimental details of harmonic generation from a two-dimensional atomic crystal.A two-dimensional (2D) atomic crystal consists of a single planar atomic lattice. These materials, were shown to exist and to be stable by marvelous experiments [1,2]. Few-layer materials, consisting of a few planar atomic lattices, can be created as well and they are also interesting. Anyway single layer materials are particularly special. For instance the overlapping between the valence and the conduction band in graphene is exactly zero, while it is finite in graphite. Single-layer transition metal dichalcogenides are direct band semiconductors while the bulk materials (or the few-layer ones) have an indirect band gap [3].Also the linear optical properties of a 2D atomic crystal are remarkable. The fine structure constant defines the optical transparency of graphene [4]. Thanks to an enhanced optical contrast, 2D atomic crystals can be well visualized if deposited on top of suitable substrates [5][6][7]. In two recent papers it was shown that for light a single-layer material has no thickness [8,9]. In practice for optics it appears as a real 2D system. This result is by no means obvious. When 2D atomic crystals are deposited on a substrate, atomic force microscopy tips can be used to measure their thickness [1,2]. But a model treating them as three dimensional (3D) slabs with a certain thickness fails to explain the overall experiments on linear light matter interaction (absorption for instance) [8]. Instead a model considering the 2D atomic crystal as part of the interface plus the right boundary conditions, turns out to be the successful approach to explain its linear optical properties [8].Nonlinear optical properties of single layer and few layer atomic crystals have been investigated recently and they exhibit an extremely rich variety of physical phenomena. Single-layer MoS 2 [10-13] and BN [12] are non centrosymmetric materials, while their bilayers and bulk counterparts are expected to exhibit inversion symmetry [11][12][13]. More specifically slices with an even number of layers belong to the centrosymmetric D 3d space group, while slices with an odd number of layers belong to the non-centrosymmetric D 3h space group. Strong second harmonic generation (SHG) from materials with an odd number of layers wa...

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