Phonon dispersions and piezoelectricity in bulk and multilayers of hexagonal boron nitride

A unified theory of phonon dispersions and piezoelectricity in bulk and multilayers of hexagonal boron nitride (h-BN) is derived. The dynamical matrix is calculated on the basis of an empirical force constant model of intralayer valence and interlayer van der Waals interactions. Coulomb interactions are calculated by Ewald’s method, adapted for the three-dimensional (3D) and the multilayer case. The deformation of the ionic charge distribution with long-wave lattice displacements is taken into account. Special attention is devoted to the nonanalytic long-range Coulomb contribution to the dynamical matrix which is different for the 3D crystal and the multilayer case. Consequently there is a splitting of the transverse optical (TO) and longitudinal optical (LO) phonon branches of $E_{1u}$ symmetry and a discontinuity of the $A_{2u}$ branch at the $\Gamma$ point in 3D h-BN. No such splitting and discontinuity at $\Gamma$ are present in multilayer crystals with a finite number $\mathcal{N}$ of layers. There a diverging bundle of $\mathcal{N}$ overbending optical phonon branches emerges from $\Gamma$. Born’s long-wave theory is applied and extended for the study of piezoelectricity in layered crystals. While 3D h-BN and h-BN multilayers with an even number of layers (symmetry $D_{6h}$) are not piezoelectric, multilayers with an uneven number of ${\mathcal{N}}_u$ layers (symmetry $D_{3h}$) are piezoelectric; the piezoelectric coefficient $e_{1,11}$ is inversely proportional to ${\mathcal{N}}_u$.

Figure 1

Excerpt of two neighboring h-BN planes.

Figure 2

Brillouin zone of 3D h-BN; plane containing $\Gamma MK$ is BZ of 2D h-BN.

Figure 3

Phonon dispersions of 3D h-BN (a) without Coulomb forces and (b) with Coulomb forces included.

Figure 4

Prismatic unit cell for a three-layer h-BN slab.

Figure 5

Phonon dispersions of multilayers, Coulomb forces included.

Figure 6

Zoom-in of the highest optical phonon branches at the $\Gamma$ point for (a) $50$ and (b) $100$ h-BN multilayers.

Figure 7

Detailed highest optical phonon dispersions near $\Gamma$ for the bilayer.

Figure 8

Fractional change of $e_{1,11}$ with number of layers $\mathcal{N}$.