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Due to the absence of interlayer coupling and inversion symmetry, transitionmetal dichalcogenide (MX$_2$) semiconductor monolayers exhibit novel propertiesthat are distinctly different from their bulk crystals such as direct opticalband gaps, large band spin splittings, spin-valley coupling, piezoelectric andnonlinear optical responses, and thus have promising applications in, e.g.,opto-electronic and spintronic devices. Here we have performed a systematicfirst-principles study of the second-order nonlinear optical properties ofMX$_2$ (M = Mo, W; X = S, Se) monolayers and trilayers within the densityfunctional theory with the generalized gradient approximation plus scissorscorrection. We find that all the four MX$_2$ monolayers possess largesecond-order optical susceptibility $\chi^{(2)}$ in the optical frequency rangeand significant linear electro-optical coefficients in low frequency limit,thus indicating their potential applications in non-linear optical devices andelectric optical switches. The $\chi^{(2)}$ spectra of the MX$_2$ trilayers areoverall similar to the corresponding MX$_2$ monolayers, {\it albeit} with themagnitude reduced by roughly a factor of 3. The prominent features in the$\chi^{(2)}$ spectra of the MX$_2$ multilayers are analyzed in terms of theunderlying band structures and optical dielectric function, and also comparedwith available experiments.

journal of physical chemistry. c./journal of physical chemistry. c

Nonlinear Optical Properties of Transition Metal Dichalcogenide MX$_2$  (M = Mo, W; X = S, Se) Monolayers and Trilayers from First-principles  Calculations