Thickness and electric-field-dependent polarizability and dielectric constant in phosphorene

Based on extensive first-principles calculations, we explore the thickness-dependent effective dielectric constant and slab polarizability of few-layer black phosphorene. We find that the dielectric constant in ultrathin phosphorene is thickness-dependent and it can be further tuned by applying an out-of-plane electric field. The decreasing dielectric constant with reducing number of layers of phosphorene is a direct consequence of the lower permittivity of the outer layers and the increasing surface-to-volume ratio. We also show that the slab polarizability depends linearly on the number of layers, implying a nearly constant polarizability per phosphorus atom. Our calculation of the thickness- and electric-field-dependent dielectric properties will be useful for designing and interpreting transport experiments in gated phosphorene devices, wherever electrostatic effects such as capacitance and charge screening are important.

Figure 1

(a) Top view and side view of bilayer phosphorene (BP) for AB stacking of layers, calculated using optB88-vdW. (b) Band structure and (c) density of states of BP, in the absence (red curve) and presence (blue curve) of a transverse electric field of 1.0 V/Å. The red curve clearly shows an intrinsic direct band gap of 0.48 eV at the $\mathrm{\Gamma }$ point. Note that in the presence of a large transverse electric field (blue curve), the band gap reduces to zero signifying a semiconductor-to-metal transition which is also known to be a topological phase transition [22]. Structural properties for bilayer AB-stacked phosphorene are consistent with those reported earlier [11].

Figure 2

(a) Variation of the (GGA) band gap with the applied electric field for multilayered (two to six layers) phosphorene. (b) Total energy of bilayer phosphorene, calculated with reference to its energy at equilibrium interlayer spacing of 3.2 Å and zero external field, plotted as a function of interlayer distance for different $E_{\mathrm{ext}}$. The $\mathsf{U}$-shaped curves with energy minima (up to 1.8 V/Å) are due to some interlayer binding, while beyond this electric field the energy changes monotonically as a function of interlayer distance and no energy minimum is observed, suggesting no interlayer binding. Reduced stability of bilayer AB-stacked phosphorene suggests that it can be more easily exfoliated at higher electric fields.

Figure 3

The orbital projected density of states for bilayer phosphorene (a) before band gap closing at $E_{\mathrm{ext}}=0.8$ V/Å (solid lines) and (b) after band gap closing (solid lines) at $E_{\mathrm{ext}}=1.1$ V/Å. In both panels, the dashed lines correspond to the PDOS without any external fields. Evidently, the $p_z$ orbitals dominate the conduction and valance bands for bilayer phosphorene (for $E_{\mathrm{ext}}=0$), and the $p_z$ orbitals get affected the most on application of a transverse electric field, and this leads to the closing of the band gap.

Figure 4

(a) The induced charge density for bilayer phosphorene, within the supercell, at two different $E_{\mathrm{ext}}$. The two extreme peaks become more asymmetric at higher electric field as depicted in the figure. (b) Variation of the effective electric field inside bilayer phosphorene at different $E_{\mathrm{ext}}$. (c) The effective electric field inside few-layer phosphorene as a function of distance along the $z$ axis for 2 to 8 layers at a fixed $E_{\mathrm{ext}}=0.08$ V/Å.

Figure 5

(a) Induced charge density (without macroscopic smoothing out) and induced Hartree potential, $V_H\left(z\right)$, due to the external electric field, (b) polarization with the vertical lines marking the slab boundaries where the polarization falls to 1% of the nearest peak values [see Eq. (6)], (c) effective electric field, and (d) the relative permittivity, as a function of the distance along the $z$ axis. All panels are for six-layer phosphorene for a transverse applied field of $E_{\mathrm{ext}}=0.1\mathrm{V}/\AA$ and except for panel (a) all other panels show Gaussian-filtered quantities.

Figure 6

(a) Variation of relative permittivity with applied electric field for varying number of layers. (b) Variation of relative permittivity with number of layers at fixed external electric field ($E_{\mathrm{ext}}=0.02\mathrm{V}$/Å). The red circles represent the DFT-calculated values, and the blue line represents the predicted values for any number of layers as per Eq. (9), which should yield good results for low electric fields where the electronic polarization is expected to vary linearly with the field, and the ionic contribution to the effective field is negligible.

Figure 7

Slab polarizability as a function of the number of layers (red squared), which fits very well with a straight line (black). The linear behavior of the slab polarizability with number of layers for low electric fields implies that the polarizability per phosphorus atom is nearly constant and it has a value of $1.42\times 4\pi {\epsilon }_0{\AA }^3$. For a comparison, the polarizability of a carbon atom in a multilayered graphene sheet has a value [26] of $0.5\times 4\pi {\epsilon }_0{\AA }^3$. Here $E_{\mathrm{ext}}=0.02\mathrm{V}$/Å.