Unified ab initio description of Fröhlich electron-phonon interactions in two-dimensional and three-dimensional materials

Ab initio calculations of electron-phonon interactions including the polar Fröhlich coupling have advanced considerably in recent years. The Fröhlich electron-phonon matrix element is by now well understood in the case of bulk three-dimensional (3D) materials. In the case of two-dimensional (2D) materials, the standard procedure to include Fröhlich coupling is to employ Coulomb truncation, so as to eliminate artificial interactions between periodic images of the 2D layer. While these techniques are well established, the transition of the Fröhlich coupling from three to two dimensions has not been investigated. Furthermore, it remains unclear what error one makes when describing 2D systems using the standard bulk formalism in a periodic supercell geometry. Here we generalize previous work on the ab initio Fröhlich electron-phonon matrix element in bulk materials by investigating the electrostatic potential of atomic dipoles in a periodic supercell consisting of a 2D material and a continuum dielectric slab. We obtain a unified expression for the matrix element, which reduces to the existing formulas for three-dimensional and 2D systems when the interlayer separation tends to zero or infinity, respectively. This expression enables an accurate description of the Fröhlich matrix element in 2D systems without resorting to Coulomb truncation. We validate our approach by direct ab initio density-functional perturbation theory calculations for monolayer BN and ${\mathrm{MoS}}_2$, and we provide a simple expression for the 2D Fröhlich matrix element that can be used in model Hamiltonian approaches. The formalism outlined in this work may find applications in calculations of polarons, quasiparticle renormalization, transport coefficients, and superconductivity, in 2D and quasi-2D materials.

Figure 1

Schematic representation of a superlattice consisting of a periodic stack of two layers alternating along the $z$ direction. The material under consideration is the layer of thickness $d$ and dielectric constant ${\epsilon }_{\infty ,1}$. The the other layer, of thickness $D$ and permittivity ${\epsilon }_{\infty ,2}$, is a dielectric continuum. The thick solid line denotes the boundary of the primitive unit cell; the thick dashed line indicates the boundary of the BvK supercell. Equation (18) gives the electrostatic potential generated by a point charge located in one of the layers (indicated by the disk $•$). To obtain a solution that is periodic in the BvK supercell, we superimpose the potential of all periodic images (indicated by the circle $\circ$). From the displacements of these charges we obtain the dipole potential. Our Fröhlich matrix element in Eq. (34) is then obtained by summing over the dipoles associated with every atomic displacement.

Figure 2

(a, b) Calculated high-frequency dielectric constants of h-BN (a) and ${\mathrm{MoS}}_2$ (b) monolayers in a supercell geometry, as a function of cell length $c=d+D$ along the $z$ direction. See Fig. 1 for details of the supercell construction. The dashed straight lines are guides to the eye and show that the supercell permittivity scale linearly with $1/c$. (c, d) Dielectric constants of monolayer h-BN (c) and ${\mathrm{MoS}}_2$ (d), as extracted from Eqs. (64) and (65), as a function of the dielectric thickness $d$. The crossing of the curves for the parallel and the perpendicular dielectric constants identify the effective dielectric thickness and permittivity of each slab. The symbols $\parallel$ and $\perp$ indicate dielectric constant parallel and perpendicular to the slab surface, respectively.

Figure 3

(a) Electron-phonon matrix elements of monolayer h-BN calculated along a high-symmetry path in the 2D Brillouin zone. We report the results of explicit DPFT calculations (blue disks) and calculations using the matrix element in Eq. (34) (pink lines). In both cases we show the average of $|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel }){|}^2$ over the TO and LO modes to avoid the discontinuity resulting from crossing phonon bands, $\left|g\right|=[{\sum }_{\nu }|g_{mn\nu }(\mathbf{k},\mathbf{q}){{|}^2]}^{1/2}$, for $m,n,\mathbf{k}$ corresponding to the valence band maximum at the $K$ point. The calculations were performed by using a supercell with a monolayer of h-BN and a vacuum buffer, with a total size $c=20\AA$ along the $z$ direction. (b) Same raw data as in (a), but this time the matrix element is scaled by the phase-space volume element in three dimensions, $4\pi |{\mathbf{q}}_{\parallel }{|}^2{\left|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel })\right|}^2$. The vertical dashed lines indicate wave vectors such that $|{\mathbf{q}}_{\parallel }|=\pi /c$.

Figure 4

(a) Electron-phonon matrix elements of monolayer h-BN calculated along a high-symmetry path in the 2D Brillouin zone, for infinite interlayer separation. We report the results of explicit DPFT calculations employing 2D Coulomb truncation (blue disks) and calculations using the matrix element in Eq. (34) with 2D kernel [Eq. (37)] (pink lines). In both cases we show the average of $|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel }){|}^2$ over the TO and LO modes to avoid the discontinuity resulting from crossing phonon bands, $\left|g\right|=[{\sum }_{\nu }|g_{mn\nu }(\mathbf{k},\mathbf{q}){{|}^2]}^{1/2}$. In this expression, $m,n,\mathbf{k}$ correspond to the valence band maximum at the $K$ point. (b) Same as in (a), but for monolayer ${\mathrm{MoS}}_2$ and infinite interlayer separation.

Figure 5

Comparison between various models of the Fröhlich matrix elements for monolayer h-BN, in the limit of infinite vacuum size. We show the modulus of the matrix element, $|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel })|$, for $m,n,\mathbf{k}$ corresponding to the valence band maximum at the $K$ point, and $\nu$ corresponding to the LO mode. The gray disks are the reference DFPT calculations using 2D Coulomb truncation. The data calculated using the method of Ref. [26] are shown as blue squares. The data obtained with our exact matrix element Eq. (34) are in magenta. The simplified model of Eq. (62), using the parameters in Table 1, is shown as yellow triangles.

Figure 6

(a) Modulus of the long-range part of the electron-phonon matrix element, $|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel })|$, for $m,n,\nu$ corresponding to the valence band top of h-BN at the $K$ point and the LO mode. We consider phonon wave vectors along the $\mathrm{\Gamma }M$ path. These data were calculated using Eq. (34), for various supercell sizes $c$ along the $z$ direction: $c=20\AA$(green), 40 Å (yellow), 140 Å (orange), and $c\to \infty$ (blue). (b) Same raw data as in (a), but this time plotted as $2\pi |{\mathbf{q}}_{\parallel }\left|\right|g_{mn\nu }(\mathbf{k},{\mathbf{q}}_{\parallel }){|}^2$.