Optical lattice vibrations in monolayer hexagonal boron nitride from macroscopic equations

Abstract

Using two pairs of macroscopic equations deduced from a dipole lattice model including electronic polarization (EP) of ions and local field effects (LFEs) self-consistently, optical lattice vibrations in monolayer hexagonal boron nitride (BN) are studied theoretically for both in-plane and out-of-plane motions. Longitudinal and transverse optical (LO and TO) modes and out-of-plane (ZO) modes are derived, and explicit expressions are obtained for phonon dispersion, group velocity and density of states. The analytical phonon dispersion relations describe previous numerical results very well; the LO phonon dispersion is identical to the analytical expression of Sohier et al, which shows the degeneracy of the LO and TO modes at Γ and the splitting at finite wavevectors due to the long-range macroscopic field. Whilst relating to microscopic quantities, the linear coefficients of the lattice equations are determined by first-principles calculated quantities (such as macroscopic susceptibilities). Therefore the EP and LFEs on the vibrational properties are studied. With no EP or LFEs, all the phonon frequencies are overestimated significantly. Including EP, the LFEs increase (decrease) the in-plane (out-of-plane) dielectric susceptibility by a factor of 2.5-3.5. Both ionic EP and LFEs should be included to obtain an accurate description of the lattice dynamics.

Two-dimensional phonon polaritons in multilayers of hexagonal boron nitride from a macroscopic phonon model

“…The LO modes [Eq. ( 3)] have the minimum frequency ω 0 at k=0 [13,14,18]. For PhPs, however the same minimum frequency occurs not at Γ but at a finite wavevector k 0 = ω 0 /c ≈ 8.62 × 10 −5 / Å (Appendix A); accordingly β=1.…”

“…In a latest study [14], the long wavelength in-plane optical vibrations of ML hBN are described by a pair of macroscopic equations, ẅ = a 11 w + a 12 E ρ , and P = a 21 w + a 22 E ρ , where w is the optical displacement between the positive and negative ions weighted by m/s, m being their reduced mass and s the unit-cell area, P is the macroscopic polarization (average dipole moment per unit area), and E ρ the in-plane (ρ) component of the macroscopic field in the ML. All in-plane vectors w, P and E ρ depend on time t and in-plane position vector ρ.…”

“…Thus the LO phonon group velocity v l =37.54 km/s, and the two dimensionless quantities α ≈ 0.0001 and η ≈ 0.38. In the PhP dispersion calculations the wavevectors are restricted to a small-k region, k < 0.1|Γ − K| (|Γ − K| is the distance between points Γ and K in the Brillouin zone), where the macroscopic description of the polar modes is found to be accurate [13,14]. These parameters are used throughout this paper.…”

“…in the two Reststrahlen bands [4,7], where the permittivity tensor has negative in-plane (x − y) or out-of-plane (z) component(s). In a 2D polar monolayer (ML) such as ML hBN there are two major changes to its properties: (i) while the TO phonons are dispersionless at long wavelengths (with frequency ω 0 ), the LO phonons have dispersion, which are degenerate with the TO modes at the Brillouin zone * Electronic address: phyjzzhang@jlu.edu.cn centre [11][12][13][14]; (ii) the dielectric function (DF) depends linearly on the norm of in-plane wavevector k (leading to nonlocal screening) [15], e.g., the high-frequency DF ǫ ∞ (k) = 1 + 2πχ e k for an isolated ML, where χ e the ML's electronic susceptibility (i.e. high-frequency susceptibility).…”

“…Recently a macroscopic model including electronic polarization of ions (i.e. HFS) and local field effects has been used to study optical phonons in ML hBN [14]. In this paper, we extend this model to N layers of hBN to study PhPs due to the in-phase optical vibrations (i.e.…”