Quantum electrodynamic approach to the conductivity of gapped graphene

# Quantum electrodynamic approach to the conductivity of gapped graphene

G. L. Klimchitskaya Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, Saint Petersburg, 196140, Russia Institute of Physics, Nanotechnology and Telecommunications, Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, 195251, Russia    V. M. Mostepanenko Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, Saint Petersburg, 196140, Russia Institute of Physics, Nanotechnology and Telecommunications, Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, 195251, Russia Kazan Federal University, Kazan, 420008, Russia
###### Abstract

The electrical conductivity of graphene with a nonzero mass-gap parameter is investigated starting from the first principles of quantum electrodynamics in (2+1)-dimensional space-time at any temperature. The formalism of the polarization tensor defined over the entire plane of complex frequency is used. At zero temperature we reproduce the results for both real and imaginary parts of the conductivity, obtained previously in the local approximation, and generalize them taking into account the effects of nonlocality. At nonzero temperature the exact analytic expressions for real and imaginary parts of the longitudinal and transverse conductivities of gapped graphene are derived, as well as their local limits and approximate expressions in several asymptotic regimes. Specifically, a simple local result for the real part of conductivity of gapped graphene valid at any temperature is obtained. According to our results, the real part of the conductivity is not equal to zero for frequencies exceeding the width of the gap and goes to the universal conductivity with increasing frequency. The imaginary part of conductivity of gapped graphene varies from infinity at zero frequency to minus infinity at the frequency defined by the gap parameter and then goes to zero with further increase of frequency. The analytic expressions are accompanied by the results of numerical computations. Possible future generalization of the used formalism is discussed.

###### pacs:
72.80.Vp, 73.63.-b, 65.80.Ck

## I Introduction

During the last few years, a two-dimensional hexagonal lattice of carbon atoms known as graphene has attracted widespread attention owing to its unusual physical properties and promising applications 1 (). Unlike conventional condensed matter systems, the quasiparticles in graphene are massless or very light particles which at low energies obey the relativistic Dirac equation, where the speed of light is replaced with the Fermi velocity (see the monograph 1 () and review papers 2 (); 3 (); 4 ()). This makes the electronic properties of graphene highly nontrivial, and, specifically, results in the existence 5 () of a universal (up to small nonlocal corrections) frequency-independent conductivity in the limit of zero temperature.

The electrical conductivity of graphene was investigated in many theoretical papers using the current-current correlation functions in the random phase approximation, the two-dimensional Drude model, the quasiclassical approach, the Kubo formula, and Boltzmann’s transport theory (see, e.g., Refs. 6 (); 7 (); 8 (); 9 (); 10 (); 11 (); 12 (); 13 (); 14 (); 15 (); 16 (); 17 (); 18 (); 19 (); 20 (); 21 (); 22 (); 23 (); 24 (); 25 (); 26 (); 27 (); 28 (); 29 (); 30 ()). At first, a few distinct values for the universal conductivity have been obtained by different authors (see Refs. 4 (); 9 (); 10 (); 21 () for a review) depending on the order of limiting transitions in the used theoretical approach. At a later time, a consensus was achieved on the value 5 (); 6 (); 13 (); 18 (); 21 (); 27 (); 30 (). However, up to now there is no complete agreement between the results of the different approximate and phenomenological approaches to the conductivity of graphene at nonzero temperature, in the presence of nonzero mass-gap parameter and chemical potential. In doing so, the experimental results 31 (); 32 (); 33 (); 34 (); 35 () are not of sufficient precision for a clear discrimination between the various theoretical predictions.

Quantum electrodynamics at nonzero temperature provides a regular way for describing the response of a physical system to the electromagnetic field by means of polarization tensor. For graphene with an arbitrary mass-gap parameter an explicit expression for the polarization tensor in (2+1)-dimensional space-time was found in Ref. 36 () at zero temperature. The temperature correction to the polarization tensor of graphene was derived in Ref. 37 () at the pure imaginary Matsubara frequencies. The results of Refs. 36 (); 37 () have been used to calculate the Casimir and Casimir-Polder forces 37a (); 37b () in many real physical systems incorporating graphene sheets 38 (); 39 (); 40 (); 41 (); 42 (); 43 (); 44 (); 45 () (previously such calculations were performed by means of the density-density correlation functions of graphene, the spatially nonlocal dielectric permittivities, phenomenological Drude model, etc. 46 (); 47 (); 48 (); 49 (); 50 (); 51 ()). The polarization tensor of graphene was also used 41 (); 52 () to compare with theory the experimental data on measuring the Casimir interaction in graphene systems 53 (), and very good agreement was obtained.

Another representation for the polarization tensor of graphene valid over the entire plane of complex frequency was derived in Ref. 54 (). It was applied to investigate the origin of large thermal effect in the Casimir interaction between two graphene sheets 55 (); 55a () and to calculate the reflectivity properties of gapless graphene 54 (). In Ref. 57 () the same polarization tensor allowed calculation of the reflectivity properties of graphene-coated substrates.

The derivation of the polarization tensor valid along the real frequency axis made it possible to investigate the electrical conductivity of graphene on the basis of first principles of quantum electrodynamics at nonzero temperature. Note that in Ref. 54 () explicit expressions for the polarization tensor at real frequencies were obtained only for a pure (pristine) graphene having a zero mass-gap parameter, whereas for a gapped graphene the respective expressions were presented in an implicit form. Because of this, as the starting point, the conductivity of pure graphene was investigated 58 (). In doing so, a few results obtained previously using different theoretical approaches have been reproduced, and several novel analytic asymptotic expressions and numerical results (especially for the imaginary part of conductivity) have been obtained.

In this paper we apply the formalism of the polarization tensor to systematically investigate the electrical conductivity of gapped graphene on the solid foundation of quantum electrodynamics. For this purpose, an explicit form of the results of Ref. 54 () related to the case of nonzero mass-gap parameter is employed, which was found in Ref. 56 (). It is known that quasiparticles in graphene may acquire some small mass under the influence of impurities, electron-electron interactions, and in the presence of substrates 2 (); 59 (); 60 (); 61 (). The conductivity of gapped graphene was studied in Ref. 20 () using the two-band model and in Ref. 29 () using the static polarization function. At zero temperature some analytic expressions for the conductivity of gapped graphene were obtained in Refs. 16 (); 17 (), whereas Ref. 27 () contains the final results for both real and imaginary parts of conductivity in the local approximation. We reproduce the zero-temperature analytic results of Ref. 27 () within the formalism of the polarization tensor and generalize them taking into account the effects of nonlocality. Specifically, it is shown that at zero temperature the real part of the conductivity of gapped graphene is equal to zero in the frequency region from zero to the characteristic frequency, which is slightly larger than due to nonlocal effects. With increasing frequency the real part of conductivity decreases monotonously from to .

According to our results, nonzero temperature has a dramatic impact on the conductivity of gapped graphene. Thus, the real part of conductivity for frequencies larger than the characteristic frequency may be either decreasing or increasing function depending on the relationship between the mass-gap parameter and the temperature. The imaginary part of the conductivity of gapped graphene varies from plus to minus infinity when frequency increases from zero to the characteristic frequency (at zero temperature it varies from zero to minus infinity in the same frequency region). We obtain the analytic asymptotic expressions and perform numerical computations for both real and imaginary parts of the conductivity of gapped graphene at nonzero temperature.

The paper is organized as follows. In Sec. II, we present explicit expressions for the polarization tensor of gapped graphene along the real frequency axis. In Sec. III, both real and imaginary parts of the conductivity of gapped graphene are derived at zero temperature. Sections IV and V investigate the impact of nonzero temperature on real and imaginary parts of the conductivity of gapped graphene, respectively. Our conclusions and discussion are contained in Sec. VI.

## Ii Polarization tensor of gapped graphene

Here, we present analytic results for the polarization tensor of gapped graphene along the real frequency axis derived in Refs. 54 (); 56 (). We also demonstrate how real and imaginary parts of the conductivity of graphene are expressed via imaginary and real parts of the polarization tensor, respectively. Both the longitudinal (along the graphene surface) and transverse (normal to the graphene surface) conductivities are considered.

The description of graphene using the polarization tensor in the one-loop approximation and the density-density correlation functions in the random phase approximation are equivalent 42 (). The former method is, however, somewhat advantageous because it relies on well understood formalism of quantum field theory at nonzero temperature. This allowed complete derivation of the polarization tensor of gapped graphene valid over the entire plane of complex frequency 54 (). In Ref. 56 () a more convenient form of this tensor especially along the real frequency axis has been obtained for a gapped graphene.

As usual, we notate the polarization tensor as , where , is the frequency of an electromagnetic wave, is the magnitude of the wave vector component parallel to graphene, and is the temperature. We assume real photons on a mass-shell, so that the inequality is satisfied. It has been shown 36 (); 37 () that only the two components of the polarization tensor of graphene are independent. It is conventional to use and as the two independent quantities. For our purposes, however, it is more convenient to use the following combination

 Π(ω,k,T)≡k2Πtr(ω,k,T)+(ω2c2−k2)Π00(ω,k,T) (1)

as the second quantity.

We present and as the sums of zero-temperature contributions and temperature corrections

 Π00(ω,k,T)=Π(0)00(ω,k)+ΔTΠ00(ω,k,T), Π(ω,k,T)=Π(0)(ω,k)+ΔTΠ(ω,k,T). (2)

The thermal corrections and vanish in the limit of zero temperature.

The analytic expressions for the polarization tensor of gapped graphene at were derived in Ref. 36 (). They result in

 Π(0)00(ω,k)=−αk2c2ω2η2Φ(ω,k), Π(0)(ω,k)=αk2Φ(ω,k), (3)

where is the fine structure constant, the quantity is defined as

 η≡η(ω,k)=√1−v2Fk2ω2 (4)

and the function along the real frequency axis is given by 54 ()

 Φ(ω,k)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩2Δc−2ℏωηc[1+(Δℏωη)2]arctanhℏωηΔ,ℏω<Δη,2Δc−2ℏωηc[1+(Δℏωη)2](arctanhΔℏωη+iπ2),ℏω≥Δη. (5)

Here, the energy gap and is the mass of a quasiparticle (according to some estimations 36 () may achieve 0.1 eV).

As is seen from Eqs. (3)–(5), under the condition the quantities and are real. If the condition is satisfied, they have both real and imaginary parts.

We now continue by considering the thermal corrections in Eq. (2). The imaginary parts of both and are also different from zero only under the condition 54 (); 56 (). The explicit forms are the following 54 (); 56 ():

 ImΔTΠ00(ω,k,T)=8αℏc2v2F∫u(+)u(−)dueβu+1(2cu−ω)2−v2Fk2ωη√v2Fk2A(ω,k)−(2cu−ω)2, (6) ImΔTΠ(ω,k,T)=8αℏηωv2F∫u(+)u(−)dueβu+1(2cu−ω)2+v2Fk2[1−A(ω,k)]√v2Fk2A(ω,k)−(2cu−ω)2,

where

 β≡ℏckBT,A(ω,k)≡1−(Δℏωη)2 (7)

and the integration limits are

 u(±)≡u(±)(ω,k)=12c[ω±vFk√A(ω,k)]. (8)

The real parts of thermal corrections in Eq. (2) are different from zero at all frequencies. For the frequencies satisfying the condition the results are 56 ()

 ReΔTΠ00(ω,k,T)=16αℏc2v2F∫∞Δ/(2ℏc)dueβu+1⎧⎪ ⎪⎨⎪ ⎪⎩1−12ωη⎡⎢ ⎢⎣(2cu+ω)2−v2Fk2√(2cu+ω)2−v2Fk2A(ω,k) −(2cu−ω)2−v2Fk2√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭, (9) ReΔTΠ(ω,k,T)=16αℏω2v2F∫∞Δ/(2ℏc)dueβu+1⎧⎪ ⎪⎨⎪ ⎪⎩1−η2ω⎡⎢ ⎢⎣(2cu+ω)2+v2Fk2[1−A(ω,k)]√(2cu+ω)2−v2Fk2A(ω,k) −(2cu−ω)2+v2Fk2[1−A(ω,k)]√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭.

Within the frequency region explicit representations for the quantities and are somewhat more complicated. Thus, according to Ref. 56 (), the most convenient expression for in this region is the following:

 ReΔTΠ00(ω,k,T)=16αℏc2v2F(I1+I2+I3), (10)

where the integrals are defined as

 I1≡∫u(−)Δ/(2ℏc)dueβu+1{1−12ωη[(2cu+ω)2−v2Fk2√(2cu+ω)2−v2Fk2A(ω,k) +(2cu−ω)2−v2Fk2√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭, (11) −(2cu−ω)2−v2Fk2√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭.

In a similar way, the most convenient expression for in the frequency region is given by 56 ()

 ReΔTΠ(ω,k,T)=16αℏω2v2F(J1+J2+J3), (12)

where the integrals are the following

 J1≡∫u(−)Δ/(2ℏc)dueβu+1⎧⎪ ⎪⎨⎪ ⎪⎩1−η2ω⎡⎢ ⎢⎣(2cu+ω)2+v2Fk2[1−A(ω,k)]√(2cu+ω)2−v2Fk2A(ω,k) +(2cu−ω)2+v2Fk2[1−A(ω,k)]√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭, (13) −(2cu−ω)2+v2Fk2[1−A(ω,k)]√(2cu−ω)2−v2Fk2A(ω,k)⎤⎥ ⎥⎦⎫⎪ ⎪⎬⎪ ⎪⎭.

The above expressions (2)–(13) provide an exact representation for the polarization tensor of graphene at arbitrary frequency, wave vector, temperature and mass-gap parameter in the application region of the Dirac model. This opens up possibility for a comprehensive investigation of the conductivity of gapped graphene on the basis of first principles of quantum electrodynamics. The point is that the longitudinal and transverse conductivities of graphene are directly connected with the respective density-density correlation functions 50 (); 62 (). The latter are expressed via the components of the polarization tensor 42 (). As a result, the longitudinal and transverse conductivities of graphene take the form 58 (); 56 ()

 σ∥(ω,k,T)=−iω4πℏk2Π00(ω,k,T), (14) σ⊥(ω,k,T)=ic24πℏk2ωΠ(ω,k,T).

In the next sections, using these equations, the conductivities of gapped graphene are treated at both zero and nonzero temperature.

## Iii Conductivity of gapped graphene at zero temperature

At the polarization tensor of gapped graphene is characterized by the quantities and which are presented in Eqs. (3)–(5). We consider separately the real and imaginary parts of the conductivity.

### iii.1 Real part of conductivity

According to Eq. (14), the real parts of graphene conductivities at zero temperature are given by

 Reσ(0)∥(ω,k)=ω4πℏk2ImΠ(0)00(ω,k), (15) Reσ(0)⊥(ω,k)=−c24πℏk2ωImΠ(0)(ω,k).

From Eqs. (3)–(5) one obtains

 ImΠ(0)00(ω,k)=απℏck2ωη[1+(Δℏωη)2]θ(ℏωη−Δ), (16) ImΠ(0)(ω,k)=−απℏk2ωηc[1+(Δℏωη)2]θ(ℏωη−Δ),

where is the step function.

Substituting Eq. (16) in Eq. (15) and taking into account that , one arrives at

 Reσ(0)∥(ω,k)=σ0η[1+(Δℏωη)2]θ(ℏωη−Δ), (17) Reσ(0)⊥(ω,k)=σ0η[1+(Δℏωη)2]θ(ℏωη−Δ),

where the universal conductivity of graphene is

 σ0=e24ℏ. (18)

As is seen from Eq. (17), in the frequency region satisfying the condition the real parts of the conductivities of gapped graphene are equal to zero. Note that the vanishing of the conductivity of gapped graphene at low frequencies was noted in many papers using various formalisms (see, for instance, Refs. 17 (); 20 (); 27 ()).

Taking into account that , one obtains

 v2Fk2ω2≤v2Fc2≈1.1×10−5. (19)

Using Eq. (4), we expand Eq. (17) in powers of the small parameter (19) and in the lowest order find

 Reσ(0)∥(ω,k)=σ0{1+(Δℏω)2+12(vFkω)2[1+3(Δℏω)2]}θ(ℏωη−Δ), (20) Reσ(0)⊥(ω,k)=σ0{1+(Δℏω)2−12(vFkω)2[1−(Δℏω)2]}θ(ℏωη−Δ).

As is seen from Eq. (20), and differ only in the first perturbation order in the small parameter (19) and, thus, can be considered as equal for all practical purposes. They are normalized to and plotted in Fig. 1 as functions of . Note that in this and all the following figures computational results for the longitudinal and transverse conductivities are indistinguishable in the used scales. Because of this, they are always presented by a single line. In the region the conductivities of graphene are the monotonous functions decreasing from to when the frequency increases from to infinity.

In the local limit, where the results are independent on the wave vector, Eq. (20) leads to

 Reσ(0)∥(ω,0)=Reσ(0)⊥(ω,0)=σ0(ℏω)2+Δ2(ℏω)2θ(ℏω−Δ). (21)

Note that in our case the same results are obtained from Eq. (20) taken at . They coincide with the results presented in Ref. 27 () obtained in the framework of another approach.

### iii.2 Imaginary part of conductivity

Using Eq. (14), for the imaginary part of conductivity of graphene at zero temperature we obtain

 Imσ(0)∥(ω,k)=−ω4πℏk2ReΠ(0)00(ω,k), (22) Imσ(0)⊥(ω,k)=c24πℏk2ωReΠ(0)(ω,k).

For the real part of the polarization tensor from Eqs. (3)–(5) one finds

 ReΠ(0)00(ω,k)=−αk2c2ω2η2ReΦ(ω,k), ReΠ(0)(ω,k)=αk2ReΦ(ω,k), (23)

where

 (24)

Using the identity 63 ()

 arctanhz=12ln1+z1−z, (25)

Eq. (24) can be rewritten in the form

 ReΦ(ω,k)=2Δc−ℏωηc[1+(Δℏωη)2]ln∣∣∣Δ+ℏωηΔ−ℏωη∣∣∣. (26)

Substituting Eqs. (23) and (26) in Eq. (22), one arrives at

 Imσ(0)∥(ω,k)=σ0πη[2Δℏωη−(ℏωη)2+Δ2(ℏωη)2ln∣∣∣Δ+ℏωηΔ−ℏωη∣∣∣], (27) Imσ(0)⊥(ω,k)=σ0ηπ[2Δℏωη−(ℏωη)2+Δ2(ℏωη)2ln∣∣∣Δ+ℏωηΔ−ℏωη∣∣∣].

From Eq. (27) it is seen that the nonlocal effects play equally small role in the imaginary parts of conductivity of graphene as in its real part. In the local limit we have and Eq. (27) reduces to

 Imσ(0)∥(ω,0)=Imσ(0)⊥(ω,0)=σ0π[2Δℏω−(ℏω)2+Δ2(ℏω)2ln∣∣∣Δ+ℏωΔ−ℏω∣∣∣]. (28)

This coincides with the result presented in Ref. 27 ().

In Fig. 2 we plot the imaginary part of the conductivity of graphene normalized to at as a function of the ratio of to the width of the gap . As is seen in Fig. 2, for any fixed the imaginary part of conductivity decreases monotonously from zero to minus infinity when frequency increases from 0 to and then increases to zero with further increase of frequency.

## Iv Real part of conductivity of gapped graphene at nonzero temperature

We start from the real parts of thermal corrections to the conductivities of graphene. According to Eqs. (2) and (14), they are given by

 ReΔTσ∥(ω,k,T)=ω4πℏk2ImΔTΠ00(ω,k,T), (29) ReΔTσ⊥(ω,k,T)=−c24πℏk2ωImΔTΠ(ω,k,T).

Recall that the quantities and (and, thus, and ) are equal to zero under the condition . Under the opposite condition they are given by Eq. (6).

It is convenient to introduce another integration variable in Eq. (6)

 τ=2cu−ωvFk√A(ω,k), (30)

where is defined in Eq. (7). Then Eq. (6) takes the form

 ImΔTΠ00(ω,k,T)=−4αℏck2ωη∫1−1dτexp(ℏω2kBT+γτ)+1[1−A(ω,k)√1−τ2+A(ω,k)√1−τ2], (31) ImΔTΠ(ω,k,T)=4αℏωηk2c∫1−1dτexp(ℏω2kBT+γτ)+1[1√1−τ2−A(ω,k)√1−τ2].

Here, the parameter is defined as

 γ≡γ(ω,k,T)=vFkωℏω2kBT√A(ω,k). (32)

It is equal to zero in the local limit.

Expanding the exponent-containing fraction in the first formula of Eq. (31) in a series and calculating the integrals 63 () one obtains

 ImΔTΠ00(ω,k,T)=−4αℏck2ωη∞∑n=1(−1)n−1e−nℏω2kBT∫1−1dτe−nγτ[1−A(ω,k)√1−τ2+A(ω,k)√1−τ2] (33) =−4αℏck2ωη∞∑n=1(−1)n−1e−nℏω2kBT{[1−A(ω,k)]I0(nγ)+A(ω,k)nγI1(nγ)},

where are the Bessel functions of an imaginary argument.

In a similar manner, the second formula of Eq. (31) can be rewritten as

 ImΔTΠ(ω,k,T)=4αℏωηk2c∞∑n=1(−1)n−1e−nℏω2kBT∫1−1dτe−nγτ[1√1−τ2−A(ω,k)√1−τ2] (34) =4αℏωηk2c∞∑n=1(−1)n−1e−nℏω2kBT[I0(nγ)−A(ω,k)nγI1(nγ)].

Substituting Eqs. (33) and (34) with added -functions in Eq. (29), we obtain the exact expressions for the real parts of thermal corrections to longitudinal and transverse conductivities of gapped graphene at nonzero temperature

 ReΔTσ∥(ω,k,T)=−4σ0ηθ(ℏωη−Δ)∞∑n=1(−1)n−1e−nℏω2kBT{[1−A(ω,k)]I0(nγ)+A(ω,k)nγI1(nγ)}, (35) ReΔTσ⊥(ω,k,T)=−4σ0ηθ(ℏωη−Δ)∞∑n=1(−1)n−1e−nℏω2kBT[I0(nγ)−A(ω,k)nγI1(nγ)].

To determinate the role of nonlocal effects we expand the right-hand sides of both expressions in Eq. (35) up to the first order in the small parameter (19) and find

 ReΔTσ∥(ω,k,T)=−σ0θ(ℏωη−Δ) ×{2[1+(Δℏω)2]+(vFkω)2[1+3(Δℏω)2]∞∑n=1(−1)n−1e−nℏω2kBT (36) ReΔTσ⊥(ω,k,T)=−σ0θ(ℏωη−Δ) ×{2[1+(Δℏω)2]−12(vFkω)2[1−(Δℏω)2]∞∑n=1(−1)n−1e−nℏω2kBT +14[1−(Δℏω)2][3+(Δℏω)2](vFkω)2(ℏω2kBT)2∞∑n=1(−1)n−1n2e−nℏω2kBT}.

Performing the summation according to Ref. 63 ()

 ∞∑n=1(−1)n−1e−nℏω2kBT=1eℏω2kBT+1, (37) ∞∑n=1(−1)n−1n2e−nℏω2kBT=eℏω2kBT(eℏω2kBT−1)(eℏω2kBT+1)3,

after identical transformations Eq. (36) can be written in the following form:

 ReΔTσ∥(ω,k,T)=−2σ0θ(ℏωη−Δ)eℏω2kBT+1⎧⎪⎨⎪⎩⎡⎢⎣tanhℏω4kBT1+e−ℏω2kBT⎤⎥⎦1+(Δℏω)2+12(vFkω)2 ×[1+3(Δℏω)2]⎡⎢⎣1+(ℏω)2−Δ216(kBT)2tanhℏω4kBT1+e−ℏω2kBT⎤⎥⎦⎫⎪⎬⎪⎭, (38) ReΔTσ⊥(ω,k,T)=−2σ0θ(ℏωη−Δ)eℏω2kBT+1⎧⎪⎨⎪⎩⎡⎢⎣tanhℏω4kBT1+e−ℏω2kBT⎤⎥⎦1+(Δℏω)2−12(vFkω)2 ×[1−(Δℏω)2]⎡⎢⎣1−3(ℏω)2+Δ216(kBT)2tanhℏω4kBT1+e−ℏω2kBT⎤⎥⎦⎫⎪⎬⎪⎭.

As is seen from Eq. (38), the real parts of thermal corrections to the conductivity of gapped graphene are mostly determined by the local contributions

 ReΔTσ∥(ω,0,T)=ReΔTσ⊥(ω,0,T)=−2σ0eℏω2kBT+1(ℏω)2+Δ2(ℏω)2θ(ℏω−Δ). (39)

The nonlocal contributions in Eq. (38) become dominant only at very low temperature when the complete thermal correction is exponentially small.

By adding Eqs. (21) and (39), one arrives at the final expression for the real part of conductivity of gapped graphene at nonzero temperature in the local approximation

 Reσ∥(⊥)(ω,0,T)=σ0tanhℏω4kBT(ℏω)2+Δ2(ℏω)2θ(ℏωη−Δ). (40)

In Fig. 3(a) we plot the real part of the normalized to longitudinal and transverse conductivities of graphene with the gap parameter eV as a function of frequency (measured in eV) at different temperatures. The lines from top to bottom correspond to , 100, and 300 K, respectively (note that K corresponds to eV). To gain a better understanding, the region of very low frequencies is shown in Fig. 3(b) on an enlarged scale. According to Fig. 3(a,b), the real part of conductivity decreases with increasing . This is because the thermal correction to conductivity presented in Eq. (39) is negative. As is seen in Fig. 3(a,b), the real part of the conductivity of graphene increases with increasing frequency and goes to . This is, however, not a universal property, but is determined by chosen (small) value of the gap parameter.

To illustrate the latter statement, In Fig. 4 we plot the real parts of the conductivities of graphene with the larger gap parameters equal to (a) eV and (b) eV as the functions of frequency. The lines from top to bottom are plotted for , 100, and 300 K, respectively. As is seen in Fig. 4(a), at and 100 K the real part of conductivity increases, but at K it decreases with increasing frequency. Figure 4(b) illustrates what is happening with an increase of the gap parameter to eV. In this case the real part of conductivity at and 100 K decreases with increasing frequency, whereas at K it becomes nonmonotonous due to an interplay of two factors in Eq. (40). With further increase of the gap parameter (up to eV or larger) the real part of conductivity becomes monotonously decreasing at all considered temperatures, whereas the values of at and 100 K become indistinguishable.

## V Imaginary part of conductivity of gapped graphene at nonzero temperature

The imaginary parts of thermal corrections to the conductivity of graphene are obtained from Eqs. (2) and (14)

 ImΔTσ∥(ω,k,T)=−ω4πℏk2ReΔTΠ00(ω,k,T), (41) ImΔTσ⊥(ω,k,T)=c24πℏk2ωReΔTΠ(ω,k,T).

At first we consider the frequency region . Here, the real parts of the temperature corrections to the polarization tensor are presented by Eq. (9). Thus, the exact expressions for and in this region are given by Eqs. (9) and (41). The total imaginary part of the conductivity is given by

 Imσ∥(⊥)(ω,k,T)=Imσ(0)∥(⊥)(ω,k)+ImΔTσ∥(⊥)(ω,k,T), (42)

where is defined in Eq. (27).

We expand Eq. (9) up to the first order in a small parameter (19) and find

 ReΔTΠ00(ω,k,T)=−8αℏc2k2ω2∫∞Δ/(2ℏc)dueβu+1[1+Δ2+(ℏω)24(ℏcu)2−(ℏω)2], (43) ReΔTΠ(ω,k,T)=8αℏk2∫∞Δ/(2ℏc)dueβu+1[1+Δ2+(ℏω)24(ℏcu)2−(ℏω)2].

Substituting these equations in Eq. (41), one arrives at equal results for and in the local approximation

 ImΔTσ∥(⊥)(ω,0,T)=8σ0cπω∫∞Δ/(2ℏc)dueβu+1[1+Δ2+(ℏω)24(ℏcu)2−(ℏω)2]. (44)

Similar to Sec. IV, it can be shown that the nonlocal corrections to are negligibly small.

The total imaginary part of the conductivity of graphene in the local approximation is given by

 Imσ∥(⊥)(ω,0,T)=Imσ(0)∥(⊥)(ω,0)+ImΔTσ∥(⊥)(ω,0,T), (45)

where and are given by Eqs. (28) and (44), respectively.

Note that under the condition we can obtain simple asymptotic expression for