Dynamical diffusion and renormalization group equation for the Fermi velocity in doped graphene

Physica B 452 (2014) 92–101

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Dynamical diffusion and renormalization group equation for the Fermi velocity in doped graphene J.S. Ardenghi n, P. Bechthold, P. Jasen, E. Gonzalez, A. Juan IFISUR, Departamento de Física (UNS-CONICET), Avenida Alem 1253, Bahía Blanca, Argentina

art ic l e i nf o

a b s t r a c t

Article history: Received 2 March 2014 Received in revised form 2 July 2014 Accepted 6 July 2014 Available online 15 July 2014

The aim of this work is to study the electron transport in graphene with impurities by introducing a generalization of linear response theory for linear dispersion relations and spinor wave functions. Current response and density response functions are derived and computed in the Boltzmann limit showing that in the former case a minimum conductivity appears in the no-disorder limit. In turn, from the generalization of both functions, an exact relation can be obtained that relates both. Combining this result with the relation given by the continuity equation it is possible to obtain general functional behavior of the diffusion pole. Finally, a dynamical diffusion is computed in the quasistatic limit using the definition of relaxation function. A lower cutoff must be introduced to regularize infrared divergences which allow us to obtain a full renormalization group equation for the Fermi velocity, which is solved up to order Oðℏ2 Þ. & 2014 Elsevier B.V. All rights reserved.

Keywords: Graphene Diffusion Sensor gas Fermi velocity

1. Introduction Graphene is a two-dimensional hexagonal lattice of carbon atoms and it is one of the most important topics in solid state physics due to the vast application in nano-electronics, optoelectronics, superconductivity and Josephson junctions [1–5]. The band structure shows that the conduction and the valence band touch at the Dirac point and the dispersion relation is approximately linear and isotropic [6]. This linear dispersion near the symmetry points have striking similarities with those of massless relativistic Dirac fermions [4]. This leads to a number of fascinating phenomena such as the half-quantized Hall effect [7,8] and minimum quantum conductivity in the limit of vanishing concentration of charge carriers [1]. Although this is an outstanding experimental result, there is no consensus about the theoretical value computed through different theoretical methods (see [9]), neither the physical reason for such minimum value (see [10]), where the minimum is due to the impurity resonance and is not related to the Dirac point. In particular, one of the theoretical methods used to compute response functions within the linear response theory is the Kubo formalism [11]. Deviations of charge and current densities from their equilibrium values are described by density and current response functions through the Kubo formulas using the same

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E-mail addresses: [email protected], [email protected] (J.S. Ardenghi). http://dx.doi.org/10.1016/j.physb.2014.07.013 0921-4526/& 2014 Elsevier B.V. All rights reserved.

two-particle Green function. Although, generally, it is unable to obtain exact relations between these response functions, several approximations can be obtained taking into account the dimensionality and dispersion relation of the system (see [12]). But these approximate relations are based on the continuity equation and Ward identities and are not clear if these assumptions are valid for linear dispersion relations and spinor wave functions. In turn, impurities in graphene can be considered in various types of forms: substitutional, where the site energy is different from those of carbon atoms, which generates resonances [13] and as adsorbates, that can be placed on various points in graphene: six-fold hollow site of a honeycomb lattice, two-fold bridge site of the two neighboring carbons or top site of a carbon atom [14]. Theoretical as well as experimental studies have indicated that substitutional doping of carbon materials can be used to tailor their physical and/or chemical properties [15–17]. In particular, nitrogen or boron dopants can be added to pristine graphene [18–21]. The detection and absorption of low levels of hydrogen becomes very important for sensor gas and hydrogen energy. Different methods of hydrogen detection are not entirely selective or it have a high cost of manufacture due to their complexity. Pddoped reduced graphene have a clear response to hydrogen and are very selective [22,23]. On the other side, the decoration of carbon support by transition metals can also be independently used to enhance the hydrogen storage of the specimens. Transition metals eliminate the hydrogen dissociation barrier altogether [24]. In this sense, the density and current response function of doped-graphene with low concentration of substitutional impurities

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

is of major importance for the consequences in the sensor effect [25,26]. In particular, the simplest graphene-based sensor detects the conductivity change upon adsorption of analyte molecules. The change of conductivity could be attributed to the changes of charge carrier concentration in the graphene induced by adsorbed gas molecules. It has been proposed that such device may be capable of detecting individual molecule [27]. These reactions release captured electrons in the interaction zone between the gas and the sensor, and increase their concentration in the conductivity zone. But the conductivity of electrons is based on the diffusion phenomena of charge carriers through the sample. Electrons moving in randomly distributed scatterers have a diffusive character, which is described at long distances by a diffusion equation. It has been shown that it is possible to suppress diffusion (see [28]), giving rise to localization phenomena, which will affect the sensor characteristics of the material. In turn, a dynamical generalization of the diffusion constant from the electron–hole correlation function cannot be linked to the frequency-dependent conductivity (see Eqs. (3.18) and (3.19) of [12]). In this sense, the aim of this work is two-fold: to introduce a generalization of the linear response theory for linear dispersion relation and spinor wave functions, to apply it to graphene, and the subsequent computation of minimal conductivity and dynamical diffusion, to analyze the general behavior of the system under local perturbations and the implications for sensor gas. This work will be organized as follows: In Section 2, the impurity averaged Green function will be computed. In Sections 3 and 4, a generalization of the conductivity tensor and response function will be computed using the current definition for relativistic Dirac fermions. In Section 5, different limit behavior of the current and density response functions are computed. The Boltzmann limit is introduced showing the minimal conductivity value. In Section 6, the dynamical diffusion will be computed through the relaxation function, showing how to obtain the full renormalization group equation for the Fermi velocity. Finally, the conclusion are presented. Appendices A and B are introduced for selfcontained lecture.

2. Impurity averaged Green function

93

momentum space1 G0RðAÞ ðq; EÞ ¼

1 1 E  vf ℏq 8 is e  iφq

eiφq

! ð5Þ

1

where the minus sign corresponds to the retarded Green function and the plus sign to the advanced Green function. The contribution to the second order in the perturbation expansion in the impurity potential reads (see [29, Eq. (3.31), p. 136]) 0

〈GR2 ðk; k ; EÞ〉i ¼ j ¼ δk;k0 ni ½GR0 ðk; EÞ2

Z

2

d k

0

ð2π Þ2

0

0

jV imp ðk k Þj2 IGR0 ðk ; EÞ ð6Þ 0 2

where ni is the impurity concentration and jV imp ðk  k Þj I is a diagonal matrix ! 0 0 jV imp ðk  k Þj2 0 2 ð7Þ jV imp ðk  k Þj I ¼ 0 0 jV imp ðk  k Þj2 and the angle brackets represent the configurational averaging that can be computed as Z N 〈A〉 ¼ ∏ dri Aðr1 ; r2 ; …; rN ÞPðr1 ; r2 ; …; rN Þ ð8Þ i¼1

where Pðr1 ; r2 ; …; rN Þ ¼ Pðr1 ÞPðr2 Þ…PðrN Þ and Pðri Þ is the probability density for having the impurity located around point ri .2 In Eq. (6), the Fourier transform of GR2 ðr; r0 ; EÞ has been taken first. Replacing the last equation and Eq. (5) in Eq. (6) the diagonal part of the averaged Green function reads Z

0

2 0 d k jV imp ðk  k Þj2 0 ð2π Þ2 E vf ℏk  is ! Z 2 0 0 E  vf ℏk d k s 0 2 jV imp ðk  k Þj þi ¼ 0 0 ð2π Þ2 ðE  vf ℏk Þ2 þ s2 ðE  vf ℏk Þ2 þ s2

ð9Þ If we consider for simplicity that the impurity potential is a Dirac delta potential, then3 Z Z 2 2 V imp ðkÞ ¼ d re  irq V imp ðrÞ ¼ d re  irq V 0 δðrÞ ¼ V 0 ð10Þ

The Hamiltonian of clean graphene in the K point in the Brillouin zone and in the long wavelength approximation reads (see [4]) ! 0 px þ ipy H ¼ vF ð1Þ px  ipy 0

Using the last result, the integral of Eq. (9) reads ! Z 2 0 0 E  vf ℏk d k s 2 þi V 0 lim 0 0 s-0 ð2π Þ2 ðE vf ℏk Þ2 þ s2 ðE  vf ℏk Þ2 þs2

where vF  106 m=s is the Fermi velocity. The eigenfunctions of this Hamiltonian read

where we have used that δðxÞ ¼ ð1=π Þlims-0 s=ðx2 þ s2 Þ and n(E) is the density of states at the Fermi energy.4 At this point it is important to notice that in clean graphene, the density of states n(E) at the Fermi energy is nðEF Þ ¼ 0 (see [30], Eq. (33)). Nevertheless, when impurities are introduced, the density of states at the Fermi energy is not zero (see [30], Fig. 3), which implies that the disorder introduces an imaginary term to the self-energy.

 eikr 2 λeiφk

1



ψ k ðrÞ ¼ pffiffiffi where

φk ¼ arctan

1

  ky kx

ð2Þ

Z

2

d k

0

ð2π Þ2

δðE  vf ℏk0 Þ ¼ iπ V 20 nðEF Þ

ð11Þ

ð3Þ

In turn, the eigenvalues read Eλ ðkÞ ¼ λvf ℏk

¼ iπ V 20

ð4Þ

where k ¼ jkj and where λ ¼1 are positive energy states (conduction band) and λ ¼ 1 are negative energy states (valence band). With the eigenfunctions of Eq. (2) we can compute the retarded and advanced Green functions for conduction electrons (λ ¼1) in

1 We assume that the valence-band states do not contribute to low temperature conductivity. 2 In this case we are assuming that the positions of the impurities are distributed independently. 3 In this case, the disorder introduced by the delta Dirac impurity potential is an on-site diagonal disorder. 4 In the last equation the real part is strictly not zero, but is a constant that does not depend on the momentum. In this sense, this value is arbitrary and has no observable consequences. From this we can assume that it is zero or redefine the reference for measuring energy.

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J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

Taking into account the direction of the current in index notation and to linear order in the electric field we obtain Z Z þ1 jα ðr; tÞ ¼ 〈jα ðr; tÞ〉0 þ ∑ dr0 Q αβ ðr; t; r0 ; t 0 ÞAβ ðr0 ; t 0 Þ ð21Þ

The non-diagonal term reads Z Z

2

0

V 20

d k 0 ð2π Þ2 E vf ℏk  is 0

0 0 kx þ iky 0

k

0

0

0

0

0

2 0 d k V 20 kx ðE  vf ℏk Þ  ky s skx þ ky ðE  vf ℏk Þ þi 0 0 0 ð2π Þ2 k ðE  vf ℏk Þ2 þ s2 ðE  vf ℏk Þ2 þ s2

! ð12Þ 0

0 kx

0

¼ k cos λ Introducing polar coordinates in the wave vector k , 0 0 and ky ¼ k sin λ, is not difficult to show that the last integral is zero due to the cosine and sine functions, which are integrated between 0 and 2π. Then, the averaged Green function at the second order in the perturbation expansion in the impurity potential reads ! iη 0 0 〈GR2 ðk; k ; EÞ〉i ¼ j ¼ δk;k0 ½GR0 ðk; EÞ2 ð13Þ 0 iη where η ¼ π ni V 20 nðEÞ. By introducing the one-particle irreducible propagator, which corresponds to all the diagrams which cannot be cut in two by cutting an internal line, the impurity averaged propagator can be written as a geometric series in terms of the self-energy (see [29, p. 141]) þ1

G ¼ G0 þG0 Σ G0 þ ⋯ ¼ G0 ∑ ðΣ G0 Þn ¼ G0 ðI  Σ G0 Þ  1

ð14Þ

n¼0

where Σ ¼ Σ 1 þ Σ 2 þ ⋯ contains the contributions at different orders in the perturbation expansion of the impurity concentration. With the computation done in Eq. (13) we finally obtain ! 1 1 eiφq RðAÞ 1 G ðqÞ ¼ G0 ðI  Σ G0 Þ ¼ ð15Þ E  vf ℏq 8 is  iη e  iφq 1 R

R

This last result is the impurity averaged Green function which takes into account the first contribution of the self-energy by comparing the last equation with Eq. (5). This is known as the full Born approximation, which includes electronic scattering from a single impurity. The diagonal part contains the shifted pole due to the imaginary part of the self-energy. The non-diagonal part contains the same contribution multiplied by a phase factor. The last result will be used in the following sections.

In this section, a generalization of the conductivity tensor for Dirac fermion systems, that is, linear dispersion relation and spinors wave functions, will be introduced. To do it we will follow the development introduced in [29] and take into account the differences introduced by Dirac systems. The Hamiltonian of Bloch electrons in the long wavelength approximation in a electric field and random impurities reads ð16Þ

where AðrÞ is the vector potential that is related to the electric field as E¼ 

∂A ∂t

ð17Þ

and where V imp ðrÞ is the impurity field. We can compute the current density to linear order in the external electric field (see Eq. (7.84) of [29]) Z i t jðr; tÞ ¼ Trðρ0 ðtÞjÞ  dt Trðρ0 ðt r Þ½jp ðr; tÞ; H A ðt Þ þ OðE2 Þ ð18Þ ℏ ti where the charge current density operator can be written as jp ¼ vF jr〉σ 〈rj

t

where Q αβ is the current response function. Taking into account that in linear response, each frequency contributes additively, it is only necessary to study what happens at one driving frequency Aðr; tÞ ¼ Aðr; ωÞe  iωt

ð22Þ

Then, the Fourier transform of the current reads Z jα ðr; ωÞ ¼ 〈jα ðr; ωÞ〉0 þ ∑ dr0 Q αβ ðr; r0 ; ωÞAβ ðr0 ; ωÞ

ð23Þ

β

where Q αβ ðr; r0 ; ωÞ ¼ K αβ ðr; r0 ; ωÞ  K αβ ðr; r0 ; 0Þ

ð24Þ

and

ρ λ  ρ λ0 0 0 〈λjjpα ðrÞjλ 〉〈λ jjpβ ðr0 Þjλ〉 0 þ ℏω þ is ϵ  ϵ λ λ λλ

K αβ ðr; r0 ; ωÞ ¼ ∑

ð25Þ

0

where jλ〉 are eigenstates of unperturbed Hamiltonian and ρλ is the mean occupation number for a energy level ϵλ . At this point, if we use the usual definition of current in non-relativistic quantum mechanics j¼

e fP; jr〉〈rjg 2m

ð26Þ

then 0

〈λjjpα ðrÞjλ 〉 ¼ 

iℏe n ! ! ½ψ 0 ðrÞ ∇ ψ λ ðrÞ  ψ λ ðrÞ ∇ ψ nλ0 ðrÞ 2m λ

ð27Þ

In the same line of thought, we can use the definition of relativistic Dirac current, then 0

vF 〈λ∣r〉σ 〈r∣λ 〉 ¼ vF ψ †λ ðrÞσψ λ0 ðrÞ

ð28Þ

and in the same way 0

〈λ jjpβ ðrÞjλ〉 ¼ vF ψ †λ0 ðr0 Þσψ λ ðr0 Þ

ð29Þ

Introducing Eqs. (28) and (29) into Eq. (25) and writing in index notation which allows us to move the functions ψ and the Pauli matrices we have

3. Current response function

H ¼ vf σ  ðp  eAÞ þV imp ðrÞ

β

ð19Þ

Z

K αβ ðr; r0 ; ωÞ ¼ e2 v2F

Z ¼ e2 v2F

dE 2π dE 2π

Z

0

0 0 dE ρðEÞ  ρðE0 Þ ψ λ ðr0 Þψ λi n ðrÞσ αij σ βkl ψ λj ðrÞψ λk n ðr0 Þ 2π E  E0 þ ℏω þ is l

Z

0

dE ρðEÞ  ρðE0 Þ β A ðr0 ; r; EÞσ αij σ kl Ajk ðr; r0 ; E0 Þ 2π E  E0 þ ℏω þ is li

ð30Þ where we have used the relation between the spectral weight Aðr; r0 ; EÞ and the wave functions (see Appendix A, Eqs. (118) and (119)). Finally, applying the relation between the spectral weight and the Green function we obtain Z Z 0 dE dE ρðEÞ  ρðE0 Þ K αβ ðr; r0 ; ωÞ ¼  e2 v2F 2π 2π E0  E þ ℏω þ is β

½GRli ðr0 ; r; EÞ  GAli ðr0 ; r; EÞσ αij σ kl ½GRjk ðr; r0 ; E0 Þ  GAjk ðr; r0 ; E0 Þ ð31Þ

which is the desired generalization of the current response function for Dirac fermion systems. In this case, the Pauli matrices play the role of momentum in Eq. (7.96) of [29]. In the momentum space, the current–current response function reads K αβ ðq; q0 ; ωÞ ¼ e2 v2F

Z

Z

2

d k ð2π Þ

2

d k

0

ð2π Þ

2

Z

2

Z

0

dE ρðEÞ  ρðE0 Þ 2π E0  E þ ℏω þ is

0

which is the usual definition of current in the relativistic Dirac system, where vF plays the role of velocity of light and

Trð½GR ðk þ q0 ; k þ q; EÞ

H A ðtÞ ¼ evf σ  A

 G ðk  q; k  q0 ; E Þσ β Þ

ð20Þ

dE 2π

 G ðk þ q ; k þ q; EÞσ α ½GR ðk  q; k  q0 ; E0 Þ 0

A A

0

0

0

0

ð32Þ

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

Introducing the impurity averaging of two Green functions (see [29, Eq. (8.3)]) 0

0

〈GRðAÞ ðk þ q0 ; k þ q; EÞGAðRÞ ðk  q; k q0 ; E0 Þ〉 0

0

¼ δq;q0 〈GRðAÞ ðk þ q; k þ q; EÞGAðRÞ ðk  q; k q; E0 Þ〉

ð33Þ

95

Introducing the impurity averaging of two Green functions Z 2 0 Z Z Z 2 0 d k d k dE dE ρðEÞ  ρðE0 Þ χ ðq; ωÞ ¼ 2 2 2π 2π E0  E þ ℏω þis ð2π Þ ð2π Þ 0 0 TrðAðk þ q; k þ q; EÞAðk  q; k  q; E0 ÞÞ

ð45Þ

computing one of the energy integrations and exploiting the analytical properties of the averaged Green function we obtain (see [29, p. 283])

and computing one of the energy integrations by exploiting the analytical properties of the averaged Green function we obtain

AA RR K αβ ðq; ωÞ ¼ K AR αβ ðq; ωÞ þ K αβ ðq; ωÞ þ K αβ ðq; ωÞ

χ ðq; ωÞ ¼ χ AR ðq; ωÞ þ χ AA ðq; ωÞ þ χ RR ðq; ωÞ

where β

2 2 α K RA αβ ðq; ωÞ ¼ e vF σ ij σ kl

β

2 2 α K AA αβ ðq; ωÞ ¼  e vF σ ij σ kl

β

2 2 α K RR αβ ðq; ωÞ ¼ e vF σ ij σ kl

where

Z

and

ð34Þ

dE AR ½ρðE þ ℏωÞ  ρðEÞΦlijk ðE þ ℏω; E; qÞ 2π i dE ρðE þ ℏωÞΦAA lijk ðE þ ℏω; E; qÞ 2π i

dE ρðEÞΦRR lijk ðE þℏω; E; qÞ 2π i

where the electron (hole)–electron (hole) correlation function reads

Φab lijk ðE þ ℏω; E; qÞ ¼

Z

2

d k

Z

2

d k

ð2π Þ

0

ð2π Þ2

2

0

χ AA ðq; ωÞ ¼ 

ð37Þ

χ RR ðq; ωÞ ¼

Φ

ab lijk

0

〈Gali ðk þ q; k þ q; E þ ℏωÞGbjk ðk  q; k  q; EÞ〉

The final conductivity tensor can be written in terms of the current response function K αβ ðq; ωÞ (see [29, Eq. (8.51)]) using the Kramer– Kronig relation K αβ ðq; ωÞ  K αβ ðq; 0Þ iω

ð39Þ

As we can see in Eq. (32), we have the multiplication of one matrix Green function with the Pauli matrix in the α direction and the other with the Pauli matrix in the β direction GRðAÞ ðk þ q; k þ q; E þℏωÞσ αij ¼ M ljRðAÞ ðk þq; k þ q; E þ ℏωÞ li

ð40Þ

0 β ðk  q; k  q; EÞ kl GRðAÞ jk

ð41Þ

0

0

σ ¼

Z

ð36Þ

ð38Þ

σ αβ ðq; ωÞ ¼

χ RA ðq; ωÞ ¼ and

Z

Z

ð35Þ

0 M RðAÞ ðk  q; k  q; EÞ jl

The two possible Pauli matrices are σ and σ and in particular if we choose the direction of the Pauli matrix in such a way that σ α is σ α ¼ σ x cos ϕex þ σ y sin ϕey and σ β ¼ σ x cos θex þ σ y sin θey , where ϕ and θ are angles in the real space, then ! ! 0 e  iϕ 0 e  iθ ; σβ ¼ ð42Þ σ α ¼ iϕ e 0 eiθ 0 x

y

At this point, we have to use the perturbation expansion of the product of two matrix Green functions in the impurity concentration which has been computed in last section.

4. Density response function In a similar way, we can generalize the density response function χ ðr; r0 ; ωÞ for linear dispersion and spinor wave functions, which is defined as (see Eq. (7.23) of [29])

ρ λ  ρ λ0 0 0 〈λ∣r〉〈r∣λ 〉〈λ ∣r0 〉〈r0 ∣λ〉 0 ϵ  λλ λ ϵλ þ ℏω þ is

χ ðr; r0 ; ωÞ ¼  ∑

ð43Þ

0

Using Eq. (119) and computing the Fourier transform we obtain for the density response function Z 2 0 Z Z Z 2 0 d k d k dE dE ρðEÞ  ρðE0 Þ χ ðq; q0 ; ωÞ ¼ 2 2 2π 2π E0  E þℏω þis ð2π Þ ð2π Þ 0

0

½GRij ðk þ q0 ; k þ q; EÞ  GAij ðk þ q0 ; k þq; EÞ 0 0 ½GRji ðk  q; k  q0 ; E0 Þ  GAji ðk  q; k q0 ; E0 Þ

ð44Þ

Z

dE AR ½ρðE þ ℏωÞ  ρðEÞΦlijk ðE þ ℏω; E; qÞ 2π i Z

dE ρðE þ ℏωÞΦAA lijk ðE þ ℏω; E; qÞ 2π i

dE ρðEÞΦRR lijk ðE þ ℏω; E; qÞ 2π i

ð46Þ

ð47Þ

ð48Þ

ð49Þ

With this result, the generalization of the density response function for linear dispersion relation and spinor wave function is obtained.

5. Current and density response relations With the generalization of the current and density response functions for linear dispersion relations and spinors, we can proceed to obtain a relation between those functions. Introducing a Kronecker delta product δij δkl in χ ðq; ωÞ and using Eq. (39) we can write iωσ αβ ðq; ωÞ þ e2 v2F χ ðq; ωÞ ¼ ξðq; ωÞ  K αβ ðq; 0Þ

ð50Þ

where

ξðq; ωÞ ¼ TrðΛΓ Þ ¼ Λαβ ijkl Γ ijkl ðq; ωÞ

ð51Þ

where 2 2 α β Λαβ ijkl ¼ e vF ðσ ij σ kl þ δij δkl Þ

ð52Þ

and

Γ ijkl ðq; ωÞ ¼

Z

2

d k

Z

ð2π Þ2

2

d k

0

ð2π Þ2

Z

dE 2π

Z

0

dE ρðEÞ  ρðE0 Þ 2π E0  E þ ℏω þ is

0 0 〈½GRli ðk þ q; k þ q; EÞ  GAli ðk þ q; k þ q; EÞ 0

0

½GRjk ðk q; k  q; E0 Þ  GAjk ðk  q; k  q; E0 Þ〉

ð53Þ

Relation Eq. (50) is analogous to the relation introduced in [12], Eq. (38), but in the former case, the relation obtained is not a definition as it occurs in [12]. The main difference is that in graphene and in general for spinor systems with linear dispersion relation, the space derivate is replaced by the Pauli matrix, then the Fourier transform does not introduce any momentum p. The non-appearance of the momentum in the current response function implies a different functional behavior, but the same diagrammatic expansion. In the other side, whenever there is a continuity equation, which expresses the charge conservation, it is possible to obtain a direct relation between isotropic conductivity and density response function

σ ðq; ωÞ ¼ 

ie2 ω χ ðq; ωÞ q2

ð54Þ

In the particular case of graphene, a continuity equation can be obtained, which is identical to continuity equation for quantum relativistic systems. Combining last equation and Eq. (50) we

96

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

obtain for the conductivity tensor σ αα ðq; ωÞ ¼ σ ðq; ωÞ

σ αβ ðq; ωÞ ¼

ω½ξðq; ωÞ  K αβ ðq; 0Þ iω2 þ iv2F q2

ð55Þ

The static limit of the homogeneous conductivity can be computed as (see Eq. (40a) of [12])

σ ¼ lim lim σ ðq; ωÞ ¼ lim

1

ξð0; ωÞ 

ω-0iω

ω-0 q-0

1 ∂n lim lim K ðq; ωÞ ¼ e2 D iωq-0 ω-0 αβ ∂μ ð56Þ

that relates the diffusion constant D with the static conductivity, known as the Einstein relation. At zero temperature, ∂n=∂μ ¼ nF , where nF is the density of states at the Fermi energy. On the other side, replacing Eq. (54) in Eq. (50) we obtain for the density response function

χ ðq; ωÞ ¼

q2 ½ξðq; ωÞ  K αα ðq; 0Þ e2 ω2 þ e2 v2F q2

ð57Þ

5.1. Current and density limits

AA RR Γ ijkl ðq; ωÞ ¼ Γ RA ijkl ðq; ωÞ þ Γ ijkl ðq; ωÞ þ Γ ijkl ðq; ωÞ

In particular, the ω-0 limit reads Z dE AA ρðEÞ½ΦRR lim Γ ijkl ðq; ωÞ ¼ lijk ðE; E; qÞ  Φlijk ðE; E; qÞ 2π i ω-0

ð58Þ

ð59Þ

R n

using that G ¼ ½G  we have AA RR ΦRR lijk ðE; E; qÞ  Φlijk ðE; E; qÞ ¼  2iI½Φlijk ðE; E; qÞ

ð60Þ

Finally, taking the q-0 limit and using the Ward identity (see [31]) we obtain   Z Z 2 dE d k ∂g R ðk; EÞ lim lim Γ ijkl ðq; ωÞ ¼ f li ðkÞf jk ðkÞ I ð61Þ ρðEÞ ∂E q-0ω-0 π ð2π Þ2 where

Because both contributions give the density of states at the Fermi αβ level when the tensor f li ðkÞf jk ðkÞ is contracted with Λijkl . The last equation and the result of Eq. (65) imply that the tensor Γ ijkl ðq; ωÞ is not analytical in the q-0 and ω-0 limit as it occurs in conventional systems. 5.2. Boltzmann limit and minimum conductivity The Boltzmann limit can be introduced by making the following approximation (see [29]) 0

RðAÞ ðk þ q; k þ q; EÞGjk ðk  q; k q; E0 Þ〉 〈GRðAÞ li 0

0

RðAÞ  〈GRðAÞ ðk þ q; k þq; EÞ〉〈Gjk ðk  q; k  q; E0 Þ〉 li

ðk þ q; EÞGRðAÞ ðk  q; E0 Þ ¼ δk;k0 GRðAÞ li jk

ð68Þ

where is the impurity averaged Green function computed in Section 2. Because in the ω-0 limit, the conductivity will depend on the electron–hole correlation function ΦRA lijk, we will RA compute Γ ijkl ðq; ωÞ. Introducing a shift E-E  ðℏω=2Þ we have      Z dE ℏω ℏω ρ  Γ RA ðq; ω ; η Þ ¼  i E þ ρ E  ijkl 2π 2 2     Z 2 d k R ℏω A ℏω  G ð69Þ G k  q; E þ k  q; E  li jk 2 2 ð2π Þ2 αα

Because we have to compute the trace Λijkl Γ ijkl ðq; ω; ηÞ, the only tensor elements that are not zero read αα αα αα RA RA RA RA ξðq; ωÞ ¼ e2 v2F ðΛαα 1212 Γ 1212 þ Λ1221 Γ 1221 þ Λ2112 Γ 2112 þ Λ2121 Γ 2121 αα αα αα αα RA RA RA RA þ Λ1111 Γ 1111 þ Λ1122 Γ 1122 þ Λ2211 Γ 2211 þ Λ2222 Γ 2222 Þ ð70Þ

where α α 2 2 2iϕ Λαα ; 1212 ¼ σ 12 σ 12 ¼ e vF e

α α 2 2  2iϕ Λαα 2121 ¼ σ 21 σ 21 ¼ e vF e

αα αα αα αα αα 2 2 Λαα 1221 ¼ Λ2112 ¼ Λ1111 ¼ Λ1122 ¼ Λ2211 ¼ Λ2222 ¼ e vF

ð71Þ ð72Þ

In turn

1 E  ℏvF k  iη  is

ð62Þ

RA RA RA RA Γ RA 1221 ¼ Γ 2112 ¼ Γ 1111 ¼ Γ 2222 ¼ g ðk; q; E; ωÞ

ð73Þ

and

and f ðkÞ ¼

ð67Þ

GliRðAÞ ðk þq; EÞ

To compute the two limits ω-0 and q-0 for the response functions it is only necessary to study the tensor Γ ijkl ðq; ωÞ that can be separated as

g R ðk; EÞ ¼

 ρðEÞðgA ðk; EÞ  g R ðk; EÞÞf li ðkðf jk ðkÞ ¼ 0

0

which is similar to Eq. (42) of [12], but in this case, this relation is exact.

A

On the other side, taking the q-0 limit and using the Ward identity we obtain Z Z 2 dE d k lim Γ ijkl ðq; ωÞ ¼ ½ρðE þ ℏωÞðg A ðk; E þ ℏωÞ  g R ðk; E þ ℏωÞÞ q-0 2π i ð2π Þ2

1

eiφk

e  iφk

1

! ¼

1

eiλ

e  iλ

1

! ð63Þ

where λ is the polar angle of the wave vector. Applying the chain rule in the derivate Z Z 2 dE ∂ρ d k lim lim Γ ijkl ðq; ωÞ ¼  I½g ðk; EÞf li ðkÞf jk ðkÞ ð64Þ q-0 ω-0 π ∂μ ð2π Þ2 R Those integral matrix elements that contain e 7 iλ will not contribute because the angular integration vanishes. Multiplying the last result with e2 v2F σ αij σ αkl and taking the T-0 limit we obtain RR 2 2 lim lim ðK AA αα ðq; ωÞ þ K αα ðq; ωÞÞ ¼  lim lim K αα ðq; ωÞ ¼  e vF nF ðηÞ

q-0 ω-0

q-0 ω-0

ð65Þ The longitudinal conductivity depends only on the electron–hole correlation function as it is expected: K RA ð0; ωÞ lim lim σ ðq; ωÞ ¼ lim αα iω q-0 ω-0 ω-0

ð66Þ

RA iðφk þ q þ φk  q Þ Γ RA ; 2121 ¼ g ðk; q; E; ωÞe

RA iðφk  q  φk þ q Þ Γ RA 1122 ¼ g ðk; q; E; ωÞe

RA iðφk þ q  φk  q Þ Γ RA ; 2211 ¼ g ðk; q; E; ωÞe

RA  iðφk þ q þ φk  q Þ Γ RA 1212 ¼ g ðk; q; E; ωÞe

ð74Þ where     ℏω ℏω g RA ðk; q; E; ωÞ ¼ g R k þ q; E þ g A k  q; E  2 2 1   ¼ ℏω ℏω E  ℏvF jk þ qj  iη  is  ℏvF jk  qj  iη þ is Eþ 2 2

ð75Þ For q ¼ 0 RA 2 2 ξRA ð0; ωÞ ¼ Λαα ijkl Γ ijkl ð0; ωÞ ¼  e vF

Z 

Z

     dE ℏω ℏω ρ E ρ Eþ 2 2 2π

d k e2iϕ e  2iφk þ e  2iϕ e2iφk þ 2    ℏω ð2π Þ2 E þ ℏω  ℏv k  iη  is E  ℏvF k  iη þ is F 2 2 2

ð76Þ

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

writing e 7 2iφk ¼ cos ¼

1



by applying Eq. (56)

     ky ky þ i sin 2arctg 2arctg kx kx

2

2

k

2

ðkx  ky 7 2ikx ky Þ

σ0 ¼ ð77Þ

using that kx ¼ k cos λ and ky ¼ k sin λ and computing the angular integration, we obtain RA 2 2 ξRA ð0; ωÞ ¼ Λαα ijkl Γ ijkl ð0; ωÞ ¼  2e vF

Z

Z

     dE ℏω ℏω ρ E ρ Eþ 2π 2 2

2

d k

1    ℏω ð2π Þ2 E þ ℏω  ℏv k  iη  is E  ℏvF k  iη þ is F 2 2



ð78Þ The small parameter is can be disregard because the self-energy iη RA moves the poles of ξ ð0; ωÞ away from the real line. Using a simplified version of the Ward identity, we can compute the integral in k as follows: Z 1=a 1 kdk    ℏω ℏω 2π 0  vf ℏk  iη  vf ℏk  iη Eþ E 2 2 0 1 Z 1=a 1 k k B C dk@  ¼ A ℏω ℏω 2π ℏω 0 vf ℏk  iη E   vf ℏk  iη Eþ 2 2 ð79Þ A special feature about graphene is the no disorder limit η-0. In this case, the density of states at the Fermi level is zero nðEF Þ ¼ 0 which implies that there are no charge carriers. Nevertheless, a minimal conductivity value can be found as follows: last equation can be separated in a real and imaginary part, but the principal part will not contribute to the conductivity because it vanishes since   ℏω  vf ℏk k Eþ 2  2 ℏω  vf ℏk þ η2 Eþ 2 is an odd function of E 7ℏω=2  vf ℏk, the width η is small and k is a slow varying function from 0 to 1/a, then 2 3 i η-02π ℏω

Z

1=a

lim

¼

0

6 7 kη kη 6 7 dk6  7 2 2 4 5 ℏω ℏ ω  vf ℏk þ η2  vf ℏk þ η2 Eþ E 2 2

i

ð80Þ

2v2F ℏ2

Using the last result in Eq. (78) and taking into account that   β ℏω     sinh ℏω ℏω 2   ð81Þ ρ E ¼ ρ Eþ 2 2 β ℏω cosh þ cosh ðβ EÞ 2 which behaves at low temperatures as ρðE þ ℏω=2Þ  ρðE  ℏω=2Þ  1 between  ℏω=2 and ℏω=2 and zero in the remaining energy values, then      Z Z ℏω=2 dE ℏω ℏω ρ E bðEÞ ¼ ρ Eþ bðEÞ dE ð82Þ 2π 2 2  ℏω=2 where b(E) is any function. Eq. (78) finally reads RA ξRA ð0; ωÞ ¼ Λαα ijkl Γ ijkl ð0; ω; ηÞ ¼

ie2 ℏω 2π ℏ2

97

ð83Þ

1 ie2 ℏω e2 ¼ 2 iω 2π ℏ 2π ℏ

ð84Þ

Although there is no disorder (η-0 limit) and in consequence no density of states at the Fermi energy, it is unusual to obtain a minimal conductivity. This result is in agreement with the result found in [30, Eq. (2.53)], but in disagreement with other results (see [9]).5 As we point before, we are using the Born approximation to treat impurity effects in graphene, which is valid only in the weak scattering regime. This impose conditions on the possible value of the impurity potential V0, in particular, it should be less than the bandwidth because we are in the linear dispersion regime. In turn, this approximation omits scatterings on pairs and larger groups of impurities, then it is expected to remain valid provided cluster effects are insignificant. On the other side, when impurity concentration is gradually increased, individual impurity states begin to overlap and the contribution from these states to the self-energy is becoming more pronounced in the vicinity of the impurity state energy and a spectrum rearrangement appears for a critical concentration (see [10]). This impose several restrictions to the possible values for the concentration of impurities and the potential V0 value (see [32,33]), which in turn impose several restrictions to the approximation used in this work, because it cannot be applied in a close vicinity of the Dirac point in the spectrum due to the increase in cluster scattering. Nevertheless, in [34,35], a EF -0 limit is taken on the average Green function and by using the Ioffe–Regel criterion (see [36]), one of the solutions of this limit implies that the self-consistent method is not applicable near the nodal point, which is equivalent to the conditions found in [32], but another low energy asymptotics solution exists, which impose more suitable conditions for the applicability of the Born approximation (see Eq. (9) of [35]). This point deserves more attention, because the low energy limit in the graphene Green function and correlation functions raise up a non-analytical behavior which produces different results (see [9]). Another important point is to compute minimum conductivity by taking into account the Velický–Ward identity, which introduce a two-particle irreducible vertex consistent with the coherent-potential approximation for the self-energy (see [37–39]). In particular, a Cooper pole could be computed in the two-particle irreducible vertex due to backscattering, which will dominate the low-energy behavior of the conductivity and this could give some insight for the minimum conductivity puzzle.

6. Dynamical diffusion A dynamical generalization of the diffusion constant from the electron–hole correlation function cannot be linked to the frequency-dependent conductivity (see Eq. (3.18) and (3.19) of [12]). For this, it is necessary to obtain a dynamical diffusion from a different procedure. The relaxation of a non-equilibrium particle density distribution can be studied through the diffusion equation ∂δn D∇2 δn ¼ 0 ∂t

ð85Þ

where the Fourier transformed solution reads

δnðq; ωÞ ¼

δnðt ¼ 0; qÞ iω  Dq2

ð86Þ

The induced non-equilibrium density variation that arose as a response to a weak inhomogeneous electric field, where this perturbation is first slowly switched on during the time interval 5 The conductivity of Eq. (84) must be multiplied by the degeneracy given by spin and valley K and K 0 . Then, the value would be 4σ0.

98

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

ð  1; 0Þ and then suddenly turned off at t¼0, reads (see [40])

δnðq; tÞ ¼ eVðqÞθðtÞ

Z

0

dt eϵt χ ðq; t  t 0 Þ ¼ eVðqÞϕðq; tÞ 0

1

0

ð87Þ

Eq. (94) can be written as   Z RR ∂2 Γ iill ðq; 0Þ dE ∂2 I½Φliil ðE; E; qÞ ¼ ρðEÞ     π ∂q2 ∂q2 q¼0

where VðqÞ is the Fourier transform of the scalar potential and ϕðq; tÞ is the relaxation function. The Fourier transform of last

q¼0

1 Sð0Þ ¼ Sð  1Þ þ

π

π

ð96Þ where we have to integrate by parts and use that ∂ρ=∂E ¼ δðEÞ in the T-0 limit. Taking into account Eq. (38) in the Boltzmann limit, the RR electron–electron correlation function Φliil ðE; E; qÞ can be written as

2

k  q2 1 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1=a Z 2π 2 2 ðk 2qk cos λ þ q2 Þðk þ 2qk cos λ þ q2 Þ dkk dλ    ΦRR qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi liil ðE; E; qÞ ¼ 2 2 2 ð2π Þ 0 0 E  ℏvF k  2kq cos λ þq2  iη E  ℏvF k þ 2kq cos λ þ q2  iη

equation gives a relation between δnðq; ωÞ and ϕðq; ωÞ, then ∂n ∂μ ϕðq; ωÞ ¼  iω þ DðωÞq2

ð88Þ

where

ð97Þ

where we have to put the q direction in the same direction as kx, then q  k ¼ qk cos λ where λ is the polar angle of k. In Appendix B we have computed  RR ∂2 I½Φliil ðE; E; qÞ ;   ∂q2 q¼0

∂n δnðt ¼ 0; qÞ ¼ ∂μ eVðqÞ

ð89Þ

From Eq. (88) we can obtain the dynamical diffusion  2  ∂n 2 ∂ ϕ 2 DðωÞ ¼ ω  ∂μ ∂q2 

In turn, from Eq. (87) we obtain a relation between the relaxation function ϕðω; qÞ and the response function χ ðω; qÞ ð91Þ

Using Eq. (57), we can obtain the dynamical diffusion in terms of ξðq; ωÞ without taking the η-0 limit   ∂n 2 ξð0; ωÞ  lim lim K αα ðq; ωÞ 2 Dðω; ηÞ ¼ 2 q-0 ω-0 ∂μ ie ω 0   ! iω ∂2 K αα ðq; 0Þ ∂2 ξðq; 0Þ ð92Þ þ 2 2@    q¼0  ∂q2 ∂q2  e vF q¼0

The dynamical diffusion will contain two contributions at order OðωÞ. The first one contains the diffusion pole 1/ω of the relaxation function and will not depend on disorder. The second term will be proportional to ω and the factor will be a η dependent function. From the last section, we found that ξð0; ωÞ ¼ 0 and that limq-0 limω-0 K αα ðq; ωÞ ¼ e2 v2F nF , then 2v2 nF ∂n ∂2 Γ iill ðq; 0Þ 2 DðωÞ ¼  F  iω q¼0 ∂μ iω ∂q2

ð93Þ

where we have used Eqs. (50)–(52). The η dependent factor will depend on the electron–hole correlation function, but in this case, we have taken into account the q dependence. Using Eq. (61), the last term of the r.h.s. of Eq. (59) can be written as h i  RR Z ∂2 I Φliil ðE; E; qÞ  ∂2 Γ iill ðq; 0Þ dE q ¼ 0 ρðEÞ ¼ ð94Þ   π ∂q2 ∂q2 q¼0

¼ q¼0

1

π

Z 0

1=a

  g 2 ðk; EÞ dk  ℏvF g 3R ðk; EÞ þ kℏ2 v2F g 4R ðk; EÞ  R k

ð98Þ The last integral will give a divergent result in the limit k-0, which is an infrared divergence due to the massless behavior of electrons. To isolate the divergence, we can expand the integral in powers of k before introducing the integral limits η  ! RR 10þ 6iarctg ∂2 ½Φliil ðE; E; qÞ 1 E2 þ η2 E þ ¼  ln  2  ∂q2 2π ðE  iηÞ2 6π ðE  iηÞ2 ℏ2 v2F k q¼0 þ

1 þ1 ∑b

ðℏvF kÞj

π j ¼ 1 j ðE  iηÞj þ 2

ð99Þ

where bj ¼

j3  3j2  10j  6 6j

Introducing a lower cutoff Λ, integral of Eq. (98) reads h i RR ∂2 Φliil ðE; E; qÞ  lnðaΛÞ 1 þ1 ðℏvF Þj 1 j ¼ þ ∑ bj ð Λ Þ  2 2 j þ 2 aj  π ∂q π ðE  i η Þ ðE  i η Þ j ¼ 1 q¼0

ð100Þ

ð101Þ

Taking the imaginary part of Eq. (101)  RR ∂2 ½Φliil ðE; E; qÞ 2Eη lnðaΛÞ ¼ I   ∂q2 π ðE2 þ η2 Þ2 q¼0

  η  1 1 þ1 j þ ∑ bj ðℏvF Þj ðE2 þ η2 Þ  ð1=2Þðj þ 2Þ sin ðj þ 2Þarctg  Λ πj ¼ 1 E aj ð102Þ

Integrating in E and taking the two limits of Eq. (96)    ∂2 Γ iill ðq; 0Þ lnðaΛÞ 1 þ 1 sin ð ð1 þ jÞ2π Þ ðℏvF Þj 1 j ∑ ¼ − − b − Λ  q ¼ 0 j  Πη Πj ¼ 1 jþ1 ∂q2 ηj þ 1 aj ð103Þ

Writing

 RR ∂2 I½Φliil ðE; E; qÞ   ∂q2

h i  ∂2 ΦRR liil ðE; E; qÞ    ∂q2

ð90Þ

q¼0

iωϕðq; ωÞ ¼ χ ðq; ωÞ  χ ðq; 0Þ

where the result reads

¼ q¼0

dS dE

ð95Þ

The last equation depends on the lower cutoff Λ, which is not desired. A correct procedure can be applied by assuming that the Fermi velocity vF will change with Λ.6 A renormalization group

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

equation can be obtained by assuming that the dynamical diffusion does not depend on Λ. Then 0 1  4nF vF dvF ∂ @∂2 Γ iill ðq; 0; η; Λ; vF ðΛÞÞ AdvF þ iω   ∂vF i ω dΛ dΛ ∂q2 q¼0 0 1  ∂ ∂2 Γ iill ðq; 0; η; Λ; vF ðΛÞÞ A¼0 ð104Þ þiω @   ∂Λ ∂q2 q¼0

Eq. (103) is suitable to compute different orders of ℏ to the renormalization group equation for vF. At order ℏ0 we obtain 

4vF nF dvF iω þ lnðaΛÞ ¼ 0 iω dΛ πη

ð105Þ

Dη

250

200

150

100

50

and the solution reads7 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v F ð ΛÞ ¼

v2F 0 

ω2

2πηnF

lnðΛÞ

0.005

ð106Þ

because we are in the approximation ω-0, the last equation reads v F ð ΛÞ ¼ v F 0 

ω2 lnðΛÞ 4π nF ηvð0Þ F

iv2F 0

ω

þ

iω lnðaÞ 2πηnF

ð110Þ

ð111Þ

In the cut off limit Λ-0, the Fermi velocity change as ð112Þ

which decreases with increasing order of ℏ. In turn, the Fermi velocity at higher orders of ℏ and in the no disorder limit do not 6

The Fermi velocity is one of the parameters of the Hamiltonian. In Eq. (106) Λ0 ¼ 1 as a low limit of the cutoff has been used. 8 If we introduce a upper cutoff k1 ¼ 1=Λ, the result of Eq. (106) follows the same behavior as other results. 7

η

Fig. 1. Dynamical diffusion as a function of disorder for different values of ω in arbitrary units. From red to black solid lines, ω increases. Dashed lines for dynamical diffusion at order Oðℏ2 Þ. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

8

6

j1

vF ¼ aða  j  1Þ1=j vF 0

0.020

ð108Þ

which is proportional to the impurity potential V0. This implies that at low resonance frequencies, a decrease in the diffusion can be expected. The full renormalization group equation can be computed in the no-disorder limit η-0, which gives the following differential equation at order Oðℏj Þ:

where the solution reads  1=j 1aj v F ð ΛÞ ¼ v F 0 Λj a  j

0.015

Dω

From the last equation, there is no real value for ω where DðωÞ ¼ 0, which is expected because suppression of diffusion can be achieved by taking into account maximally crossed diagrams in the perturbation expansion. Nevertheless, we can plot Dðω; ηÞ as a function of ω for different values of η and Dðω; ηÞ as a function of η for different values of ω. Both figures show the diffusion pole at ω ¼0 (see Figs. 1 and 2). Dynamical diffusion tends to v2F 0 =2iω for η-1. In turn, dynamical diffusion shows a minimum which corresponds to the following frequency: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2π v2F 0 ηnF ð109Þ ω¼ lnðaÞ

dvF Λ dΛ  ¼ 1 vF j  Λ aj

0.010

ð107Þ

which is similar to the results found in [41,42] and shows a singular behavior with the impurity factor η that is similar to the singular behavior of the Fermi velocity with impurities found in [43].8 Using Eq. (106), the dynamical diffusion at order ℏ0 reads DðωÞ ¼

99

4

2

2

4

6

8

10

ω

Fig. 2. Dynamical diffusion as a function of ω for different values of η in arbitrary units. From red to black solid lines, η increases. Dashed lines for dynamical diffusion at order Oðℏ2 Þ. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

depend on the frequency of the external perturbation. High impurity concentrations in graphene can lead to a diffusion suppression which would leave without effect the high performance of the material sample as a gas sensor. Weak localization of electrons in doped graphene implies to take into account higher orders in the diagrammatic perturbation expansion of the current response function. Some theoretical computations has been done (see [44,45]). For conventional impurities, the correction becomes positive and it leads to the fact that anti-localization is realized, which would enhance the gas sensor performance. In contrast, negative corrections for short-range impurities are expected from symmetry consideration. This suggest that the high sensitivity of graphene to detect individual dopants is highly dependent on the quantum corrections to the conductivity. Finally, taking into account the first quantum correction to the renormalization group equation for vF we obtain vF dvF ¼

5ωℏ2 v2F dΛ 1 dΛ  3πη3 LðΛ; ω; η; aÞ πη Λ2 LðΛ; ω; η; aÞ

ð113Þ

where  ! 5 ℏ2 1 2 þ Λ LðΛ; ω; η; aÞ ¼ Λ ω2 3π η3 a2 1 4nF

ð114Þ

100

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

Solution of Eq. (114) in an integral form reads Z Λ Z 0 0 0 2 dΛ 10ωℏ2 Λ v2F ðΛ Þ dΛ þ v2F ðΛÞ ¼ v2F 0  0 02 0 3 πη 1 Λ LðΛ ; ω; η; aÞ 3πη 1 LðΛ ; ω; η; aÞ

we can define the spectral weight as ð115Þ

at order Oðℏ2 Þ we need to compute the second term of r.h.s. of the last equation Z Λ 0 dΛ 3a2 πω2 η3 ¼ 02 0 2 2 10ℏ ω þ24a2 nF πη3 1 Λ LðΛ ; ω; η; aÞ "

 2# 2 2 2 2 5ℏ ω ða  1Þ  12a nF πη3 Λ ln ð116Þ 2 5ℏ2 ω2 ða2 Λ  1Þ  12a2 nF πη3 Introducing v2F ðΛÞ in Eq. (115) inside the integral of the r.h.s. in the same equation and using Eq. (116), the dynamical diffusion reads at order Oðℏ2 Þ DðωÞ ¼

v2F 0

ω

þ

2 2 iω lnðaÞ 5ω vF 0 ℏ þ 3 2πηnF 12π η nF a2

ð117Þ

The correction introduced at order Oðℏ2 Þ can be seen in both figures as dashed lines. In the case of dynamical diffusion in terms of frequency, the correction is small and only is appreciable for low values of η. In this sense, quantum corrections to the diffusion do not alter the behavior under local perturbations at linear order in ω.

Aij ðr; r0 ; EÞ ¼ i½GRij ðr; r0 ; EÞ  GAij ðr; r0 ; EÞ

If we integrate the spectral weight in the volume we obtain the density of states Z Z 2 d k 2 d rAij ðr; r; EÞ ¼ 2πδij δðE vf ℏkÞ ¼ 2π nðEÞδij ð120Þ ð2π Þ2 where n(E) is the density of states.

Appendix B. Electron–electron correlation function The electron–electron hole correlation function reads Z 1=a Z 2π dkk dλ ΦRR ϱðk; q; EÞ liil ðE; E; qÞ ¼ ð2π Þ2 0 0

In this work a generalization of linear response theory with the Kubo formula has been introduced for linear dispersion relations and spinor wave functions. A minimal conductivity can be found in the no disorder limit and the result is in discordance by a factor of 2 with other theoretical results, although there is no consensus of the physical reason of such value. Using the generalization introduced in the first sections, an exact relation between current and density response functions can be obtained. Combining this result with the relation obtained with the continuity equation, an exact functional form of response functions is obtained, where, in particular, a singular behavior appears at ω-0 and q-0 limit. Finally, dynamical diffusion is computed through the relaxation function at low order in ω. A regularization is introduced to avoid infrared divergences, which introduce a renormalization group equation for the Fermi velocity. Different contributions to this equation can be analyzed at different orders in ℏ. Different results are obtained which are of importance for local perturbations of graphene sample.

g R ðjk 7 qj; EÞ ¼

This paper was partially supported by grants of CONICET (Argentina National Research Council) and Universidad Nacional del Sur (UNS) and by ANPCyT through PICT 1770, and PIP-CONICET Nos. 114-200901-00272 and 114-200901-00068 research grants, as well as by SGCyT-UNS. E.A.G. and P.V.J. are members of CONICET. P.B. and J. S.A. are fellow researchers at this institution. The authors are extremely grateful to the referee, whose relevant observations have greatly improved the final version of this paper.

and 2

k  q2 jk þ qjjk  qj

ð123Þ

We can take the q derivate inside the integral in k. Taking into account that 7 qj ℏvF ∂jk∂q ∂g R ðjk 7 qj; EÞ ∂jk 7 qj 2 ¼ g R ðjk 7 qj; EÞ ¼  ℏvF ∂q ∂q ðE  ℏvF jk 7 qj iηÞ2 ð124Þ

then ∂ϱðk; q; EÞ ¼ ϱðk; q; EÞBðk; q; EÞ ∂q

ð125Þ

where ∂jk þ qj ∂jk  qj g R ðjk þ qj; EÞ ℏvF g R ðjk  qj; EÞ ∂q ∂q ∂αðk; qÞ 1 ð126Þ þ ∂q αðk; qÞ

Bðk; q; EÞ ¼  ℏvF

The second derivate reads   ∂2 ϱðk; q; EÞ ∂Bðk; q; EÞ ¼ ϱðk; q; EÞ B2 ðk; q; EÞ þ 2 ∂q ∂q

ð127Þ

" #  2 ∂Bðk; q; EÞ ∂2 jk þ qj 2 2 ∂jk þ qj þ ℏ v g ðjk þ qj; EÞ ¼ g R ðjk þ qj; EÞ  ℏvF R F ∂q ∂q ∂q2 " #  2 2 ∂ jk  qj 2 2 ∂jk  qj þ ℏ v g ðjk  qj; EÞ þ g R ðjk  qj; EÞ  ℏvF R F ∂q ∂q2 " #   2 1 ∂2 αðk; qÞ 1 ∂αðk; qÞ  ð128Þ þ αðk; qÞ αðk; qÞ ∂q ∂q2

Finally using that ∂jk 7qj q 7 k cos λ ¼ ∂q jk 7 qj ∂2 jk 7 qj 1 ðq 7 k cos λÞ2  ¼ jk 7qj ∂q2 jk 7 qj3

The Green function for Dirac fermion systems reads ðkÞ ðkÞ† 0 2 d k ψ i ðrÞψ j ðr Þ 2 E  v ℏkð 8 Þis ð2π Þ f

ð122Þ

ð129Þ

and that the second derivate reads

Appendix A. Spectral weight

Z

1 E  ℏvF jk 7 qj  iη

where

Acknowledgments

0 GRðAÞ ij ðr; r ; EÞ ¼

ð121Þ

where ϱðk; q; EÞ ¼ g R ðjk þ qj; EÞg R ðjk  qj; EÞαðk; qÞ

αðk; qÞ ¼ 1 þ 7. Conclusion

ð119Þ

ð118Þ

ð130Þ

Putting q ¼ 0 in Eq. (127)

ϱðk; 0; EÞ ¼ 2gR ðk; EÞgR ðk; EÞ

ð131Þ

J.S. Ardenghi et al. / Physica B 452 (2014) 92–101

and using that ∂αðk; qÞ=∂q∣q ¼ 0 ¼ 0 2

B ðk; 0; EÞ ¼ 0

ð132Þ

In turn,

" # 2 2 ∂Bðk; q; EÞ sin λ 2 sin λ 2 2 2 ¼ 2g R ðk; EÞ  ℏvF þ ℏ vF cos λg R ðk; EÞ  2 ∂q k k ð133Þ

where we have used that  2 ∂2 αðk; qÞ 4 sin λ ¼  2 ∂q2  k

ð134Þ

q¼0

Finally the second derivate of ϱðk; q; EÞ at q ¼ 0 reads  ∂2 ϱðk; q; EÞ   ∂q2

q¼0

"

"

¼ 4g 2R ðk; EÞ g R ðk; EÞ  ℏvF

# # 2 2 sin λ sin λ þ ℏ2 v2F cos 2 λg R ðk; EÞ  2 k k

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