Breakdown of Optical Phonons’ Splitting in Two-Dimensional Materials

Abstract

We investigate the long-wavelength dispersion of longitudinal and transverse optical phonon modes in polar two-dimensional materials, multilayers, and their heterostructures. Using analytical models and density-functional perturbation theory in a two-dimensional framework, we show that, at variance with the three-dimensional case, these modes are degenerate at the zone center but the macroscopic electric field associated with the longitudinal-optical modes gives rise to a finite slope at the zone center in their corresponding phonon dispersions. This slope increases linearly with the number of layers and it is determined solely by the Born effective charges of the material and the dielectric properties of the surrounding media. Screening from the environment can greatly reduce the slope splitting between the longitudinal and transverse optical modes and can be seen in the experimentally relevant case of boron nitride-graphene heterostructures. As the phonon momentum increases, the intrinsic screening properties of the two-dimensional material dictate the transition to a momentum-independent splitting similar to that of three-dimensional materials. These considerations are essential to understand electrical transport and optical coupling in two-dimensional systems.PACS numbers: 77.22.Ej Van der Waals heterostructures 1 will assuredly play a key role in future electronic and optoelectronic devices 2-5 . Many of the potential candidates to build those next-generation devices are polar two-dimensional (2D) materials, including transition-metal dichalcogenides (TMDs) and hexagonal boron nitride (h-BN). A significant consequence of polarity in these materials is the generation of long-ranged electric fields by the polarization density associated with their longitudinal optical (LO) phonon modes. These fields strongly influence the transport properties of the monolayers as well as their heterostructures, via the Fröhlich electron-phonon interaction 6-8 . Importantly, they lead to additional dipoledipole interaction terms affecting the dispersion characteristics of optical phonons. This leads to a splitting between the LO and transverse optical (TO) modes, driven by the long-ranged Coulomb interactions and electronic screening. Here, we show in detail how this LO-TO splitting is drastically affected by dimensionality. In 3D, it is well understood how this splitting is independent of the norm of the phonon momentum and how it lifts the degeneracy of the LO and TO phonon modes in the long wavelength limit. We show here that in 2D, the splitting depends on the norm of the phonon momentum, vanishes at the zone center and leads to a discontinuity in the derivative of the LO phonon dispersion. First-principles computations of this phenomenon are delicate independently of dimensionality, due to the long-ranged nature of the dipole-dipole interactions leading to electric fields that are not compatible with periodic boundary conditions. In 3D materials, however, the development of analytical models based on the Born effectiv...

Phonon Polaritonics in Two-Dimensional Materials

“…This is consistent with the result of Ref. 29 Given these results, we now re-express the conductivity explicitly in terms of the 2D phonon dispersion, and derive a universal form for the conductivity in the local (q → 0) limit specified in terms of three parameters: the LO phonon frequency at the Γ point (i.e., ω TO ), the group velocity of the LO phonon at the Γ point, and the damping rate. From Equation (7), we can immediately write the conductivity as…”

“…25 Concerning the optical response, the transition from three-dimensional (3D) to two-dimensional (2D) polar materials is nontrivial, however, since in a polar monolayer, the LO-TO splitting that gives rise to phonon polaritons in 3D is absent at the Γ point. [26][27][28][29] This raises a fundamental question about the nature of electromagnetic modes in polar monolayers.…”

Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene

“…In 2D, as shown in Ref. 10, the splitting vanishes in the zero momentum limit, but the dispersion of the LO mode displays a finite slope at the Γ point. The implementation of the 2D cutoff in DFPT as detailed above guarantees the correct treatment of the LO-TO splitting.…”

“…When it is of the order of the distance between periodic images, there is some spurious interactions. This issue is critical when simulating the screening properties 9 of the material as well as its response to phonon perturbations 10,11 .…”

Phonon-assisted emission and absorption of individual color centers in hexagonal boron nitride

“…This could be the case for a local strain field, which naturally changes the phonon dispersion. But also the dielectric environment might alter the LO phonon energies, as it affects in particular the LO-TO splitting [39]. Figure 2(c) focusses on the LE phonon side bands by plotting the ratio between the first side band weight A LE 1 and the ZPL weight A ZPL as a function of the ZPL energy.…”