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
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
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
where
and
In this notation,
When we take the limit
Here, the dispersion equation which is given by the zeros of the denominator
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 (
Here, we consider up to five nearest atomic interactions through five independent terms of
In this notation,
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
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
Figure 4(a) and (b) shows the directional/
Plots of the static dielectric function of BP for various values of
5. Impurity shielding
Starting with Eq. (2), we obtain the static screening of the potential on the surface at
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
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
Acknowledgments
G.G. would like to acknowledge the support from the Air Force Research Laboratory (AFRL) through Grant #12530960.
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