## Abstract

Using a tight-binding Hamiltonian for phosphorene, we have calculated the real part of the polarizability and the corresponding dielectric function, Re[ ϵ q ω ], at absolute zero temperature (T = 0 K) with free carrier density 10 13 / cm 2 . We present results showing Re[ ϵ q ω ] in different directions of the transferred momentum q. When q is larger than a particular value which is twice the Fermi momentum kF, Re[ ϵ q ω ] becomes strongly dependent on the direction of q . We also discuss the case at room temperature (T = 300 K). These results which are similar to those previously reported by other authors are then employed to determine the static shielding of an impurity in the vicinity of phosphorene.

### Keywords

- phosphorene
- polarizability
- impurity screening

## 1. Introduction

Emerging phenomena in physics and quantum information technology have relied extensively on the collective properties of low-dimensional materials such as two-dimensional (2D) and few-layer structures with nanoscale thickness. There, the Coulomb and/or atomic interactions play a crucial role in these complexes which include doped as well as undoped graphene [1, 2, 3], silicene [4, 5], phosphorene [6, 7], germanene [8, 9], antimonene [10, 11], tinene [12], bismuthene [13, 14, 15, 16, 17, 18] and most recently the 2D pseudospin-1

Unlike graphene, phosphorus inherently has an appreciable band gap. The observed photoluminescence peak of single-layer phosphorus in the visible optical range shows that its band gap is larger than that for bulk. Furthermore, BP has a middle energy gap (∼1.5–2 eV) at the
^{2}/Vs. We note that BP is expected to play an important role in the next-generation of electronic devices [6, 20]. Phosphorene exhibits a puckered structure related to the ^{3} hybridization of (
* p* orbitals, can be described by a four-band model with complicated multi-hopping integrals [26]. The low-lying energy bands are highly anisotropic, e.g., the linear and parabolic dispersions near the Fermi energy

_{z}

*, respectively, along the*E

_{F}

In this work, we have examined the anisotropic behavior of the static polarizability and shielded potential of an impurity for BP. The calculations for the polarizability were executed at T = 0 K and room temperature (T = 300 K). We treat the buckled BP structure as a 2D sheet in our formalism. Consequently, we present an algebraic expression for the surface response function of a pair of 2D layers with arbitrary separation and which are embedded in dielectric media. We then adapt this result to the case when the layer separation is very small to model a free-standing buckled BP structure.

The outline of the rest of our presentation is as follows. In Section 2, we present the surface response function for a pair of 2D layers embedded in background dielectric media. We then simplify this result for a pair of planar sheets which are infinitesimally close to each other and use this for buckled BP. The tight-binding model Hamiltonian for BP is presented in Section 3. This is employed in our calculations of the energy bands and eigenfunctions. Section 4 is devoted to the calculation of the polarizability and dielectric function of BP showing its temperature dependence and their anisotropic properties as a consequence of its band structure. Impurity shielding by BP is discussed in Section 5 and we summarize our important results in Section 6.

## 2. Surface response function for a pair of 2D layers

Let us consider a heterostructure whose surface is in the * xy*-plane and suppose that

*, an external potential*t

*and frequency*q

*will give rise to an induced potential which, outside the structure, can be written as*ω

This equation defines the surface response function

The quantity Im[g(

which takes account of nonlocal screening of the external potential.

### 2.1. Model for phosphorene layer

In this section, we present the surface response function we calculated for a structure which consists of a pair of 2D layers in contact with a dielectric medium, as shown in Figure 1. One of the 2D layers is at the top and the other is encapsulated by materials with dielectric constants
_{1}, and

where

and

In this notation, ** q** is the in-plane wave vector,

*is the frequency and*ω

When we take the limit

Here, the dispersion equation which is given by the zeros of the denominator
* combination* of two rows of atoms making up the layer. Our calculation can easily be generalized to the case when the monolayer is embedded above and below by the same thick dielectric material (dielectric constant

## 3. Model Hamiltonian

Phosphorene is treated as a single layer of phosphorus atoms arranged in a puckered orthorhombic lattice, as shown in Figure 3(a). It contains two atomic layers of A and B atoms and two kinds of bonds for in-plane and inter-plane P–P connections with different bond lengths. The low-lying electronic structure can be described by a tight-binding Hamiltonian, which is a 4 × 4 matrix within the basis (_{1}, _{2}, _{1}, _{2}), of the form

Here, we consider up to five nearest atomic interactions through five independent terms of * T* with

_{i}

In this notation, * t* (

_{m}

where

The valence and conduction energy bands present strong anisotropic behaviors, as illustrated by the energy bands in Figure 3(b) and the constant-energy loops in Figure 3(c) and (d). As a result, the polarizability and dielectric function are shown to be strongly dependent on the direction of the transferred momentum ** q**.

## 4. Dielectric function

When monolayer BP is perturbed by an external time-dependent Coulomb potential, all the valence and conduction electrons will screen this field and therefore create the charge redistribution. The effective potential between two charges is the sum of the external potential and the induced potential due to screening charges. The dynamical dielectric function, within the random-phase approximation (RPA), is given by [36].

Here, the * π*-electronic excitations are described in terms of the transferred momentum

**and the excitation frequency**q

*.*ω

*is the Boltzmann constant and*k

_{B}

*the chemical potential corresponding to the highest occupied state energy (middle energy of band gap) in the (semiconducting) metallic systems at T = 0 K.*μ

Figure 4(a) and (b) shows the directional/* θ*-dependence of the static polarization function

*defines the angle between the direction of*θ

**and the unit vector**q

*, the polarization function at lower (*θ

*(2*q

*) which depends on*k

_{F}

*. For increasing*θ

*from 0 to 90°, the specific values are getting larger, as shown in Figure 4(a). This means that the polarizability is stronger for 0.2 ≤*θ

*≤ 0.7 (*q

*instead of step-like structure at T = 0.*q

Plots of the static dielectric function of BP for various values of * θ* are presented in Figure 4(c) and (d) at absolute zero and room temperatures, respectively. In the range of 0.2 ≤

*≤ 0.5 (*q

*. The Re*q

*. The introduction of finite temperature smoothens the*θ

*-dependent Re*q

## 5. Impurity shielding

Starting with Eq. (2), we obtain the static screening of the potential on the surface at
_{0} above the surface of BP as

By employing the generalized form of Eq. (6) for free-standing BP in Eq. (10), we have computed the screened impurity potential. The screened potentials for various _{0}’s are shown in Figure 5 at absolute zero temperature and Fermi energy * E* = 1.0 eV. There exist Friedel oscillations for sufficiently small

_{F}

z

_{0}. Such oscillations might be smeared out for larger

z

_{0}, e.g., the green and red curves. It is noticed that for

*= 1.0 eV, the room temperature of 300 K which is much smaller than the Fermi temperature (10,000 K) does not have significant effect on the screened potential. Apparently,*E

_{F}

## 6. Concluding remarks and summary

The energy band structure of BP, calculated using the tight-binding method, is anisotropic and so are its polarizability, dielectric function and screened potential. To illustrate these facts, we have presented numerical results for the polarizability in the * x* and

*directions for a range of doping concentrations. The Re[*y

*, we have presented numerical results for Re[*E

_{F}

**. When**q

*is larger than a critical value which is twice the Fermi momentum*q

*, our calculations show that Re[*k

_{F}

**. We also discuss the case at room temperature (T = 300 K). These results are in agreement with those reported by other authors. We employ our data to determine the static shielding of an impurity in the vicinity of phosphorene.**q

## Acknowledgments

G.G. would like to acknowledge the support from the Air Force Research Laboratory (AFRL) through Grant #12530960.

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