Theory of Transport in Graphene with Long-Range Scatterers

The density of states and conductivity are calculated for scatterers with nonzero range in monolayer graphene within a self-consistent Born approximation. For scatterers with a Gaussian potential, the minimum conductivity at the Dirac point remains universal in the clean limit, but increases with disorder and becomes nonuniversal for long-range scatterers. For charged impurities, we use the Thomas-Fermi approximation for the screening effect. The conductivity increases in proportion to the electron concentration in agreement with experiments and the minimum conductivity becomes considerably larger than the universal value.


§1. Introduction
The graphene is a monolayer graphite sheet recently fabricated and has been attracting attentions theoretically and experimentally since the observation of the integer quantum Hall effect. 1,2)Actually, graphene has been a subject of theoretical study prior to the experimental realization because of the peculiar electronic structure also responsible for intriguing properties of carbon nanotubes. 3)The purpose of this paper is to study electronic and transport properties in the presence of scatterers with nonzero range including realistic charged impurities.
−11) An important feature is the presence of a topological singularity at k = 0.This singularity is the origin of the absence of backscattering in metallic carbon nanotubes. 12,13)It also leads to the presence of a Landau level at ε = 0, responsible for the singular diamagnetic susceptibility. 4,14)It is considered as the origin of the peculiar behavior in transport, such as the minimum conductivity, 15) the half-integer quantum Hall effect, 16) the dynamical conductivity, 17) and the special time reversal symmetry 18−20) leading to anti-localization behavior. 21,22)A massless Dirac system can also be realized in organic conductors. 23)n this paper, we extend self-consistent Born approximation previously used for short-range scatterers 15) and calculate the density of states and the conductivity in monolayer graphene.We first consider scatterers with a Gaussian potential to see explicit dependence on the potential range.We take into account effects of scattering from such impurities self-consistently by using Green's function technique within a self-consistent Born approximation.Because states are not localized in this system, this approximation is expected to give reliable results on transport properties.In the case of charged impurities, we use the Thomas-Fermi approximation for dielectric function determined in a self-consistent manner within the self-consistent Born approximation.
The paper is organized as follows: In §2, a brief review is given on the electronic states and the Boltzmann conductivity.The method of calculations of Green's function and static conductivity for long-range scatterers within the self-consistent Born approximation is presented.In §3 some examples of numerical results are presented for scatterers with a Gaussian potential and for charged impurities.Short discussion is made in §4.§2.Formulation

Effective-mass description
In a graphene sheet the conduction and valence bands consisting of π orbitals cross at K and K' points of the Brillouin zone, where the Fermi level is located. 24,25)lectronic states of the π-bands near a K point are described by the k•p equation: 3−9,10) with where σ = (σ x , σ y ) is the Pauli spin matrix, k = ( kx , ky ) = −i∇ is a wave-vector operator, and γ is a band parameter, given by γ = ( √ 3/2)aγ 0 with a being the lattice constant and γ 0 being the nearest-neighbor hopping integral.
The energy is given by 3) The density of states becomes where g v = 2 is the valley degeneracy associated with K and K' points and g s = 2 is the spin degeneracy.The electron concentration is given by The conductivity is written as where f (ε) is the Fermi distribution function.In the

Submitted to Journal of Physical Society of Japan
Boltzmann transport theory, the conductivity becomes where τ tr is the transport relaxation time defined as where U represents impurity potential and • • • denotes the average over impurity configurations.
Because of the final-state density of states in eq.(2.8), the relaxation time is usually proportional to the inverse of the density of states, i.e., τ tr ∝ D(ε As a result, the Boltzmann conductivity is essentially independent of the Fermi level and the electron concentration except for possible dependence of the effective scattering strength, showing that graphene is essentially a metal in stead of a "zero-gap semiconductor."

Self-Consistent Born Approximation
We consider scatterers with isotropic potential where v i (q)= v i (q) with q = |q|.Based on the symmetry, we assume the following form of Σ(k, ε) where n is the unit vector in the k direction, n = k/k.We further define (2.12) Then, the Green's function is written as In a self-consistent Born approximation diagrammatically represented in Fig. 1(a), the self-energy matrix becomes where n i is the impurity concentration.We have where X ≡ X(k , ε) and Y ≡ Y (k , ε).We separate n as where n and n ⊥ are the components of n parallel and perpendicular to n, respectively.We have where n ⊥ is a unit vector perpendicular to n.Then, the term proportional to (σ • n ⊥ ) vanishes after the integration over the k direction.We have where θ is the angle between k and k .Therefore, the self-consistency equations are The conductivity is written as where current vertex-part Ĵ satisfies the Bethe-Salpeter type equation, diagrammatically represented by Fig.
In order to proceed further, we first consider the integral for arbitrary function F (k).We separate n as in eq.

Scatterers with Gaussian Potential
In order to see the explicit dependence on the potential range, we first assume scatterers with a Gaussian potential with potential range d.This gives the Fourier transform We define a dimensionless parameter characterizing the scattering strength which is the same as that defined previously 26,27) for short-range scatterers with We introduce a cutoff energy and wave vector through and measure the corresponding quantities in units of ε c and k c in the following.The cutoff-energy is of the order of the half of the π-band width, i.e., ε c ≈ 3γ 0 , for which k c ≈ π/a.For numerical calculations, we discretize wave vector Then, the self-consistency equation becomes with Similar expressions can be written down for the current vertex part and the conductivity.For explicit numerical calculations in the following, we use j max = 1000 and with α ≈ 0 and β ≈ 1.The density of states clearly shows that the energy region affected strongly by the presence of scatterers is |ε| < ∼ γ/d.This is quite natural because the Gaussian potential becomes ineffective for wave vector k >d −1 .At the Dirac point, the conductivity is close to the universal value for short-range scatterers obtained previously 15) and increases rapidly with energy, becoming larger than the Boltzmann conductivity at sufficiently high energy depending on the potential range.Figure 3 for W = 0.2 shows the similar features in the density of states except that the enhancement in the density of states has become substantial particularly for short-range scatterers dk c = 2.The minimum conductivity at the Dirac point is slightly larger than eq.(3.11), but the dependence on range d is not so apparent.At higher energy, the deviation of the conductivity from the Boltzmann conductivity is larger.
In Fig. 4 for W = 0.5 (large disorder), the density of states for short-range case dk c = 2 has become completely different from the ideal graphene because of strong mixing with states up to the cutoff energy.The minimum conductivity is larger than that predicted by the Boltzmann theory and is larger for long-range scatterers, i.e., dk c ≥ 5.The enhancement of the conductivity in the high energy region has become further noticeable.
This enhancement of the conductivity over the Boltzmann conductivity in the high energy region can mainly be ascribed to level broadening effects.In fact, states at energy |ε| = γk have higher k components because of strong forward scattering even for long-range scatterers.These higher k states are weakly scattered in backward direction and therefore tend to have large contribution to the conductivity.This mixing effect increases in proportion to parameter W , giving rise to larger enhancement with the increase of W .
Figure 5 shows the minimum conductivity at the Dirac point.For very short-range case dk c < 1, the conductivity is nearly independent of W .For dk c = 1 and 2, the conductivity increases with W , takes a maximum at a certain value of W , and then starts to decrease.This decrease is likely to be due to strong deviation of the density of states from the ideal graphene, caused by strong mixings up to the energy cutoff.For dk c ≥ 10, such unphysical features are not present even for W = 2 because the density of states at high energy ε ∼ 0.5×ε c is close to that of the ideal graphene.This figure clearly shows that the conductivity at the Dirac point is not universal but depends on the degree of the disorder when the disorder is sufficiently large for scatterers with long-range potential.

Charged Scatterers
We consider charged impurities with screening effect included within a Thomas-Fermi approximation.The potential is where κ is the static dielectric constant, which is chosen to be 2.5 in the following, and q s is the Thomas-Fermi screening constant given by The scattering strength is a function of the density of states at the Fermi level and therefore should be determined in a self-consistent manner for each Fermi level.We have where n i is impurity concentration per unit area, εF is defined in eq.(3.13), and which roughly corresponds to the number of unit cells in a unit area.
Figure 6 shows some examples of calculated density of states and conductivity.The density of states at the Dirac point becomes nonzero and larger with the concentration of scatterers.The energy region where the disorder effect is considerable is limited to the vicinity of the Dirac point because the scattering effect decreases in proportion to ε −2 F .The conductivity at the Dirac point is slightly larger than 2 × σ 0 and its dependence on the impurity concentration is very weak.The Boltzmann conductivity at the Dirac point vanishes because of the infinite scattering strength caused by lack of screening due to the vanishing density of states.
Figure 7 shows the density of states and the conductivity as a function of the electron density at zero temperature.The conductivity is essentially the same as the Boltzmann conductivity and increases in proportion to n s except that it approaches a nonzero value slightly larger than 2 × σ 0 at the Dirac point.We should note that the electron density cannot be obtained by simple integration of the density of states shown in Fig. 6 (a), because the screening constant varies as a function of the Fermi level in this figure.§4.Discussion We have calculated the density of states and the conductivity in monolayer graphene containing scatterers with long-range potential within the self-consistent Born approximation.The self-energy part appearing in Green's function explicitly depends on the wave vector and becomes a (2,2) matrix.A self-consistent equation for the self-energy and the current vertex function have been numerically solved.The results for scatterers with Gaussian potential with range d show that both density of states and conductivity are sensitive to the potential range and to the effective scattering strength characterized by dimensionless parameter W .
Since the first experimental observation of the minimum conductivity, 1) which turned out to be larger than the theoretical prediction, 15) there have been various experimental 28−30) and theoretical works 31−37) on whether the minimum conductivity is really universal or not.The present results show within the self-consistent Born approximation that the minimum conductivity remains universal independent of the potential range only in the clean limit, but becomes nonuniversal and larger for scatterers with long-range Gaussian potential with the increase of the disorder.There remains some ambiguity in the accuracy of the self-consistent Born approximation and the results may not be quantitatively so correct.However, the qualitative feature is certainly valid that the minimum conductivity increases with disorder for long-range scatterers.
We have also considered the case of charged-impurity scattering within the Thomas-Fermi approximation for the screening effect, in which the polarization function is assumed to be independent of wave number q.−42) In ideal graphene, the polarization function is constant for q < 2k F and increases linearly with q for q > 2k F , showing that the Thomas-Fermi approximation is likely to slightly underestimate the screening effect.In order to make quantitatively accurate predictions, we should use full dielectric function in the present disordered system and perform a full self-consistent calculation.This is out of the scope of this paper and left for future.

Figure 2
shows calculated (a) density of states and (b) conductivity as a function of the energy for W = 0.1.In Fig. 2(a) the dashed line represents g s g v |ε|/2πγ 2 and in (b) the dotted lines represent the Boltzmann conductivity.Only the region of positive energy is shown because both density of states and conductivity are symmetric with respect to the Dirac point, ε = 0.

. 13 )
at zero temperature.Here, |ε F | is an effective Fermi energy defined by density of states D(ε F ) in the presence of disorder and is generally larger than |ε F |.

Fig. 1 2 (Fig. 3 (Fig. 4 (Fig. 5 (Fig. 7 (
Figure CaptionsFig.1 (a) A diagramatic representation of the matrix self-energy in the self-consistent Born approximation.(b) The vertex corrections for the velocity operator ∝ σ x .Fig. 2 (Color online) (a) Calculated density of states as a function of the energy for scatterers with Gaussian potential with range d in units of k −1 c ≈ 2π/a and dimensionless scattering strength W = n i v 2 i /4πγ 2 = 0.1.The dashed line represents g s g v |ε|/2πγ 2 .(b) Calculated conductivity versus energy.The dotted lines represent the Boltzmann conductivity.Fig. 3 (Color online) Calculated density of states (a) and conductivity for W = 0.2.Fig. 4 (Color online) Calculated density of states (a) and conductivity for W = 0.5.Fig. 5 (Color online) Calculated minimum conductivity at the Dirac point versus W for scatterers with Gaussian potential.Fig. 6 (Color online) Calculated density of states (a) and conductivity (b) versus the Fermi energy for charged scatterers screened in the Thomas Fermi approximation.Fig. 7 (Color online) Calculated density of states (a) and conductivity (b) versus the electron concentration for charged scatterers. n