Polar vibrational and dielectric properties of monolayer transition metal dichalcogenides from macroscopic equations

Abstract

Long wavelength polar vibrations in monolayer (ML) transition metal dichalcogenides (TMDs) are systematically studied for in-plane and out-of-plane motions, using two pairs of macroscopic equations deduced from a microscopic dipole lattice model accounting for local field effects (LFEs) and electronic polarization (EP). Longitudinal and transverse optical modes and out-of-plane modes are derived, and the analytical expressions describe previous first-principles calculations very well. Owing to the LFEs, the in-plane dielectric susceptibilities of ML TMDs are one order of magnitude greater than the out-of-plane susceptibilities. Furthermore, the effects of the dielectric environment on the polar vibrations are studied. Both EP and LFEs should be accounted for obtaining an accurate evaluation of dielectric susceptibility and key lattice-dynamical properties such as Born charge and phonon dispersion. A two-dimensional (2D) Lyddane–Sachs–Teller relation and a frequency–susceptibility relation are derived for in-plane and out-of-plane motions, relating the 2D dielectric functions or susceptibilities to the polar phonon frequencies. The results are also compared in detail with those of ML hexagonal boron nitride.

Lattice vibration modes and electron–phonon interactions in monolayer vs. bilayer of transition metal dichalcogenides

“…where c e is the out-of-plane electronic susceptibility 61 and p(z) is the polarization density out-of-plane as given by p(z) = 1/d; jzj < d/2 and 0 if jzj > d/2, d the thickness of the bilayer. From eqn (B1) and taking E z = E z (r,z) = −V4(r,z) it follows that V 2 4(r,z) = 4pV$P z (B3)…”

“…z = [a A 2u U ip (r) + c e E (r,0)]p(z), (B2)where c e is the out-of-plane electronic susceptibility61 and p(z) is the polarization density out-of-plane as given by p(z) = 1/d; jzj < d/2 and 0 if jzj > d/2, d the thickness of the bilayer. From eqn (B1) and takingE z = E z (r,z) = −V4(r,z) it follows that V 2 4(r,z) = 4pV$P z (B3) V$P z ¼ dpðzÞ dz  a A 2u U ip $e z þ c e E z $e z à : (B4)Poisson eqn (B3) points out that the electrostatic potential 4(r,z) is due to the polarization charge 4pV$P z of the polarization eld, eqn (B2).…”