Screening of Coulomb Impurities in Graphene

Abstract

We calculate exactly the vacuum polarization charge density in the field of a subcritical Coulomb impurity, Z|e|/r, in graphene. Our analysis is based on the exact electron Green's function, obtained by using the operator method, and leads to results that are exact in the parameter Zα, where α is the "fine structure constant" of graphene. Taking into account also electron-electron interactions in the Hartree approximation, we solve the problem self-consistently in the subcritical regime, where the impurity has an effective charge Z eff , determined by the localized induced charge. We find that an impurity with bare charge Z = 1 remains subcritical, Z eff α < 1/2, for any α, while impurities with Z = 2, 3 and higher can become supercritical at certain values of α.PACS numbers: 81.05. Uw, 73.43.Cd It has been known for a long time that the single electron dynamics in a monolayer of graphite (graphene) is described by a massless two-component Dirac equation [1,2,3]. A surge of interest in the problem was caused by the recent successful fabrication of graphene [4] and measurements of transport properties [5,6,7,8,9,10], including an unconventional form of the quantum Hall effect. Due to the Coulomb interaction between electrons, graphene represents a peculiar two-dimensional (2D) version of massless Quantum Electrodynamics (QED) [3]. It appears to be much simpler than conventional QED because the interaction is described by the instantaneous 1/r Coulomb's law. On the other hand the Fermi velocity v F ≈ 10 6 m/s ≈ c/300 (c is the velocity of light), and therefore the "fine structure constant" α = e 2 / v F ∼ 1, leading to a strong-coupling version of QED. Below we set = v F = 1. Screening of a charged nucleus due to vacuum polarization is an effect of fundamental importance in QED. This problem was investigated in detail both in the subcritical and supercritical regimes [11,12,13,14]. The problem of charged impurity screening in graphene, which also can be treated in terms of vacuum polarization, has recently received a lot of attention [15,16,17,18,19,20,21,22,23], due to the importance of the problem for transport properties involving charged impurities, as well as for our general understanding of the theory of graphene.To leading order in the weak coupling expansion, Zα ≪ 1, the induced charge is negative and localized at the impurity position, ρ ind = −|e| π 2 (Zα)δ(r), which leads to screening of the impurity potential [20,21,22,24]. We denote by Z|e| the impurity charge, and e = −|e| is the effective electron charge; from now on we refer to Z as the impurity charge with the understanding that it is measured in units of |e|. In graphene, the strong-coupling problem Zα ∼ 1 was recently addressed [20], and it was found that the supercritical regime occurs for Zα > 1/2, where a 1/r 2 tail appears in the induced charge density, while in the subcritical regime Zα < 1/2, the induced charge is always localized at the impurity site. Analytical results were also supplemented by numerical lattice calculations [21], lea...

Strong-coupling phases of 3D Dirac and Weyl semimetals. A renormalization group approach

“…This leads to expect relevant manybody effects as, for instance, the imperfect screening of impurities with sufficiently large charge on top of graphene [11][12][13][14]. Another important property is the scale dependence of physical observables, like that predicted theoretically for the Fermi velocity in 2D Dirac semimetals [15,16] and measured recently in suspended graphene samples at very low doping levels [17].…”

Excitonic effects in two-dimensional massless Dirac fermions

“…The wavefunctions for this problem are essentially exactly calculable [5][6][7][8][9][10] . One finds that the m-th circular component of an electron wavefunction has the short distance form ψ m (r) ∼ r √ (m+1/2) 2 −Z 2 β 2 −1/2 at a distance r from the impurity.…”

Electron-electron interactions in the vacuum polarization of graphene

“…For example measurements of the compressibility 13 have not detected electron correlation effects. In addition, screening of external charged impurities introduced in graphene is also expected to be sensitive to interaction effects, at least on theoretical level, 14,15,16,17 and could be relevant for interpretation of recent experiments on charged impurity scattering. 18 It is thus generally important to investigate the problem of how the correlations affect the effective charge of the carriers in graphene, which is determined by the vacuum polarization.…”