Starting from an empirical force constant model of valence interactions and calculating by Ewald’s method the ion-ion force constants, we derive the dynamical matrix for a monolayer crystal of hexagonal boron nitride (h-BN). The phonon dispersion relations are calculated. The interplay between valence and Coulomb forces is discussed. It is shown by analytical methods that the longitudinal and the transverse optical (LO and TO) phonon branches for in-plane motion are degenerate at the $\Gamma$ point of the Brillouin zone. Away from $\Gamma$, the LO branch exhibits pronounced overbending. It is found that the nonanalytic Coulomb contribution to the dynamical matrix causes a linear increase of the LO branch with increasing wave vector starting at $\Gamma$. This effect is general for two-dimensional (2D) ionic crystals. Performing a long-wavelength expansion of the dynamical matrix, we use Born’s perturbation method to calculate the elastic constants (tension coefficients). Since the crystal is noncentrosymmetric, internal displacements due to relative shifts between the two sublattices (B and N) contribute to the elastic constants. These internal displacements are responsible for piezoelectric and dielectric phenomena. The piezoelectric stress constant and the dielectric susceptibility of 2D h-BN are calculated.

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 E1u symmetry and a discontinuity of the A2u 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 N of layers. There a diverging bundle of 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 D6h) are not piezoelectric, multilayers with an uneven number of Nu layers (symmetry D3h) are piezoelectric; the piezoelectric coefficient e1,11 is inversely proportional to Nu.

We present a unified theory of the phonon dispersions and elastic properties of graphene, graphite, and graphene multilayer systems. Starting from a fifth-nearest-neighbor force-constant model derived from full in-plane phonon dispersions of graphite [Mohr et al., Phys. Rev. B 76, 035439 (2007)], we use Born’s long-wave method to calculate the tension and bending coefficients of graphene. Extending the model by interplanar interactions, we study the phonon dispersions and the elastic constants of graphite, and the phonon spectra of graphene multilayers. We find that the inner displacement terms due to sublattice shifts between inequivalent C atoms are quantitatively important in determining the elastomechanical properties of graphene and of graphite. The overall agreement between theory and experiment is very satisfactory. We investigate the evolution from graphene to graphite by studying the increase in the ${\omega }_{B_{2g_1}}^{\left(N\right)}$ rigid plane optical mode as a function of the number of layers $N$. At $N=10$ the graphite value ${\omega }_{B_{2g_1}}\approx 127{\text{cm}}^{-1}$ is attained within a few percent.

We present a tight-binding investigation of strained bilayer graphene within linear elasticity theory, focusing on the different environments experienced by the $A$ and $B$ carbon atoms of the different sublattices. We find that the inequivalence of the $A$ and $B$ atoms is enhanced by the application of perpendicular strain ${\varepsilon }_{zz}$, which provides a physical mechanism for opening a band gap, most effectively obtained when pulling the two graphene layers apart. In addition, perpendicular strain introduces electron-hole asymmetry and can result in linear electronic dispersion near the $K$ point. Our findings suggest experimental means for strain-engineered band gaps in bilayer graphene.