Polarizability and impurity screening for phosphorene

Using a tight-binding Hamiltonian for phosphorene, we have calculated the real part of the polarizability and the corresponding dielectric function, Re$[\epsilon(\textbf{q},\omega)]$, at zero temperature (T = 0) with free carrier density $10^{13}$/ $cm^2$. We present results showing the real part of dielectric function in different directions of the transferred momentum $\bf{q}$. When $q$ is larger than a particular value which is twice the Fermi momentum $k_F$, Re$[\epsilon(\textbf{q},\omega)]$ becomes strongly dependent on the direction of $\bf{q}$. We also discuss the case at room temperature (T = 300K). 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.

Of these which have been successfully synthesized by various experimental techniques and which have been extensively investigated by various experimental techniques, few-layer black phosphorus (phosphorene) or BP has been produced by using mechanical cleavage [20,21], liquid exfoliation [7,22,23], and mineralizer-assisted short-way transport reaction [24][25][26].
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 Γ point, thereby being quite different from the narrow or zero gaps of group-IV systems. Specifically, experimental measurements have shown that the BP-based field effect transistor has an on/off ratio of 105 and a carrier mobility at room temperature as large as 103 cm 2 /Vs. We note that BP is expected to play an important role in the next-generation of electronic devices [20,21]. Phosphorene exhibits a puckered structure related to the sp 3 hybridization of (3s, 3p x , 3p y , 3p z ) orbitals. The deformed hexagonal lattice of monolayer BP has four atoms [28], while the group-IV honeycomb lattice includes two atoms. The low-lying energy dispersions, which are dominated by 3p z orbitals, can be described by a four-band model with complicated multi-hopping integrals [28]. The low-lying energy bands are highly anisotropic, e.g., the linear and parabolic dispersions near the Fermi energy E F , respectively, along the k x and k y directions. The anisotropic behaviors are further reflected in other physical properties, as verified by recent measurements on optical and excitonic spectra [27] as well as transport properties [20,29].
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=0K and room temperature (T=300K). 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 Sec. II, 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 Sec. III. This is employed in our calculations of the energy bands and eigenfunctions.
Section IV. 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 Sec. V and we summarize our important results in Sec. VI.

II. SURFACE RESPONSE FUNCTION FOR A PAIR OF 2D LAYERS
The external potential will give rise to an induced potential which, outside the structure, can be written as This equation defines the surface response function g(q, ω). It has been implicitly assumed that the external potential φ ext is so weak that the medium responds linearly to it.
The quantity Im[g(q, ω)] can be identified with the power absorption in the structure due to electron excitation induced by the external potential. The total potential in the vicinity of the surface (z ≈ 0), is given by which takes account of nonlocal screening of the external potential.

A. Model for phosphorene layer
In this section, we present the surface repsonse 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 (ω), with thickness d 1 , and 2 (ω), of semi-infinite thickness. Calculation shows that the surface response function is given by [30,31] where and In this notation, q is the in-plane wave vector, ω is the frequency and χ 1 (q, ω) and χ 2 (q, ω) are the 2D layer susceptibilities.
When we take the limit d 1 → 0, i.e., the separation between the two layer is small, the 1 drops out and we have the following result for the surface response function corresponding to the structure in Figure 2 g(q, ω) = 1 − 1 Here, the dispersion equation which is given by the zeros of the denominator (q, ω) of the second term is expressed in terms of the 'average' susceptibility for the two layers.
Clearly, this dispersion equation is that for a 2D layer of the Stern form where we make the identification χ → e 2 Π (0) in terms of the polarizability. This result in Eq. (6) clearly illustrates that for the buckled BP structure shown in Figure 3, the dielectric function can be treated as that for a single layer whose susceptibility arises from a 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 b ) which corresponds to the free-standing situation which we consider below. For this, we have (q, ω) = b − e 2 /(2 0 q)Π (0) (q, ω), expressed in terms of the 2D layer polarizability Π (0) (q, ω).

III. MODEL HAMILTONIAN
Phosphorene is treated as a single layer of phosphorus atoms arranged in a puckered orthorhombic lattice, as shown in Fig. 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 (A 1 , A 2 , B 1 , B 2 ), of the form Here, we consider up to five nearest atomic interactions through five independent terms of T i with i = 1, 2, 3, 4, 5. These terms are given by the following expressions.
The valence and conduction energy bands present strong anisotropic behaviors, as illustrated by the energy bands in Fig. 3(b) and the constant-energy loops in Figs. 3(c) and 3(d).
As a result, the polarizability and dielectric function are shown to be strongly dependent on the direction of the transferred momentum q. The values of 2k F for different θ are given in (d).

IV. 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 [37] Here, the π-electronic excitations are described in terms of the transferred momentum q and (1/Å)) transferred momentum remains unchanged. In general, Π (0) (0, q) falls off rapidly beyond a critical value of q (2k F ) which depends on θ. For increasing θ from 0 to 90 • , the specific values are getting larger, as shown in Fig. 4(a). This means that the polarizability is stronger for 0.2 ≤ q ≤ 0.7 (1/Å). The main features of the polarizability for BP are quite similar to those for the 2D electron gas, but different with those for graphene. Temperature has an effect on the polarization function which is demonstrated in Fig. 4

VI. 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 y directions for a range of doping concentrations. The Re[ (q, ω = 0)] of the static dielectric function for BP also reveals some interesting characteristics. At zero temperature (T = 0) and with free carrier density corresponding to chosen Fermi energy E F , we have presented numerical results for Re[ (q, ω = 0)] in different directions of the transferred momentum q.
When q is larger than a critical value which is twice the Fermi momentum k F , our calculations show that Re[ (q, ω = 0)] becomes substantially dependent on the direction of q. We also discuss the case at room temperature (T = 300K). 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.

Conflict of interest
All the authors declare that they have no conflict of interest.