Theory of excitonic second-harmonic generation in monolayerMoS2

Abstract

Recent experimental results have demonstrated the ability of monolayer MoS2 to efficiently generate second harmonic fields with susceptibilities between 0.1 and 100 nm/V. However, no theoretical calculations exist with which to interpret these findings. In particular, it is of interest to theoretically estimate the modulus of the second harmonic response, since experimental reports on this differ by almost three orders of magnitude. Here, we present single-particle calculations of the second harmonic response based on a tight-binding band structure. We compare directly with recent experimental findings and include in the discussion also spectral features and the effects of multiple layers.The prediction and observation of a direct band gap in monolayer (ML) MoS 2 has revitalized the interest in the optical properties of this material 1-9 . Indeed, a substantial photoluminescence for 1H-MoS 2 has been observed 2,3 . Additionally, several papers 4-6 have recently demonstrated how second harmonic generation (SHG) microscopy can be used to extract important information regarding e.g. the number of layers and crystallographic orientation of few-layered MoS 2 platelets. Furthermore, exfoliated MoS 2 was shown experimentally to display a remarkably large second harmonic signal, with second harmonic susceptibilities on the order of ∼ 100 nm/V reported in Ref. 4 while Refs. 5 and 6 report only ∼ 0.1 nm/V (assuming a homogeneous susceptibility inside the monolayer of thickness ∼ 3Å when relating to sheet second harmonic susceptibilities). Regions covered with an odd number of 2H stacked layers were found to generate second harmonic fields with efficiencies decreasing slightly with the number of layers, while regions with an even number of layers displayed almost vanishing second harmonic signals as expected due to centro-symmetry. 4-6However, few-layered MoS 2 grown by chemical vapour deposition does not follow this trend 4 , possibly due the stacking order of CVD grown films deviating from 2H. In Ref. 5, second harmonic spectra of both ML and trilayer (TL) MoS 2 were presented, demonstrating an intense peak in the second harmonic spectrum at pump photon energies near 1.45 eV, with a slight redshift for TLs compared to MLs.There exists no theoretical work, with which to interpret the experimental findings mentioned above. Hence, in this letter we consider the microscopic origins of the second harmonic response of few-layered MoS 2 based on a tight-binding sp 3 d 5 band structure recently published 10 . At present, it is very computationally demanding to retain the k-space resolution needed to resolve delicate spectral features in a full exciton Bethe-Salpeter calculation. Moreover, the single particle results remain an important first step in the theoretical understanding of the nonlinear optical properties of MoS 2 . We have therefore chosen to omit excitonic effects in the present work and employ the single-particle second harmonic response formalism developed by Moss and Sipe 11,12 . We verify that the experimental me...

Observation of excitonic resonances in the second harmonic spectrum ofMoS2

“…Such ML simulations do not include effects of interlayer coupling, which are obviously present in the samples investigated here. To consider effects of multiple layers, we have extended our exciton model 22 to study SH generation in trilayer (TL) MoS 2 , which represents a minimal MoS 2 multilayer flake without inversion symmetry. We expand exciton states in a basis of singly excited Slater determinants, which are, in turn, constructed from tight-binding orbitals derived from the parameters of Ref.…”

“…While experimentally determined linear response functions of various TMDs, dominated by electron-hole pairs bound by several hundred meVs 5,6,20 , are reproduced reasonably well at the Bethe-Salpeter equation (BSE) level of complexity 6,[21][22][23] , similar comparisons for non-linear cases are lacking. Although several proposed theoretical models, based on both independent-particle 19,24 and excitonic 22,23 approaches, have been published, the SH spectrum of MoS 2 has been investigated experimentally 14 only in a narrow fundamental photon energy range between 1.2 and 1.7 eV, probing the so-called C resonance also known from linear optics.…”

“…Although several proposed theoretical models, based on both independent-particle 19,24 and excitonic 22,23 approaches, have been published, the SH spectrum of MoS 2 has been investigated experimentally 14 only in a narrow fundamental photon energy range between 1.2 and 1.7 eV, probing the so-called C resonance also known from linear optics. Theoretical models have so far been benchmarked by their ability to reproduce this single feature [22][23][24] . However, the parabolic dispersion, and the split valence bands near the K-points of the Brillouin zone, translates into bound electron-hole pairs in the BSE picture.…”

Review on Optical and Structural Characterization of 2d TMDC Semiconductors

“…SHG is used for the studies on the thickness of the materials and to find the different crystal structures [41] . The thickness dependence on the SHG is irregular.…”

Improvedab initiocalculation of surface second-harmonic generation from Si(111)(1×1):H

“…There are several theoretical formalisms that describe the SHG process for surfaces with different approximations and varying levels of difficulty [9,[13][14][15][16][17][18][19]. In this paper, we focus on a recent approach developed by us in Ref.…”