Quantum field theoretical description of the Casimir effect between two real graphene sheets and thermodynamics
aa r X i v : . [ c ond - m a t . o t h e r] J u l Quantum field theoretical description of the Casimir effectbetween two real graphene sheets and thermodynamics
G. L. Klimchitskaya
1, 2 and V. M. Mostepanenko
1, 2, 3 Central Astronomical Observatory at Pulkovo of the RussianAcademy 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 analytic asymptotic expressions for the Casimir free energy and entropy for two parallelgraphene sheets possessing nonzero energy gap ∆ and chemical potential µ are derived at arbitrarilylow temperature. Graphene is described in the framework of thermal quantum field theory inthe Matsubara formulation by means of the polarization tensor in (2+1)-dimensional space-time.Different asymptotic expressions are found under the conditions ∆ > µ , ∆ = 2 µ , and ∆ < µ taking into account both the implicit temperature dependence due to a summation over theMatsubara frequencies and the explicit one caused by a dependence of the polarization tensoron temperature as a parameter. It is shown that for both ∆ > µ and ∆ < µ the Casimirentropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for ∆ = 2 µ this fundamental requirement is violated. The physical meaning of the discovered anomaly isconsidered in the context of thermodynamic properties of the Casimir effect between metallic anddielectric bodies. . INTRODUCTION The Casimir effect was discovered [1] as an attractive force which arises between twoparallel uncharged ideal metal planes in vacuum and depends only on the Planck constant ~ , speed of light c , and interplane distance a . At zero temperature of the planes this effectis entirely caused by the zero-point oscillations of the quantized electromagnetic field whosespectrum is altered by the presence of boundary conditions on the planes as compared tothe free Minkowski space. More recently, the Casimir effect was generalized to the caseof metallic or dielectric plates kept at arbitrary temperature T . In the framework of theLifshitz theory, the free energy and force of the Casimir interaction between real-materialplates are represented as some functionals of the reflection coefficients expressed via thefrequency-dependent dielectric permittivities of plate materials. Detailed information oncalculation of the Casimir free energies and forces using the Lifshitz theory, as well as abouta comparison between experiment and theory, can be found in the monograph [2]. Thereare also generalizations of the Lifshitz theory for bodies of arbitrary shape and alternativederivations of the Casimir interaction in the literature (see, e.g., Refs. [2–5]).During the last few years, much attention is given to graphene which is a one-atom-thick layer of carbon atoms possessing unusual physical properties [6]. It has been shownthat at energies below 1–2 eV graphene is well described by the Dirac model as a set ofmassless or very light electronic quasiparticles. The corresponding fermion field satisfies therelativistic Dirac equation in (2+1)-dimensions where the speed of light c is replaced withthe Fermi velocity v F ≈ c /300 [6, 7]. This allowed application of the methods developedearlier in planar quantum electrodynamics [8–11] for investigation of various quantum effectsin graphene systems [12–18].One of these effects is the Casimir attraction between two parallel graphene sheets whichcan be calculated using the Lifshitz theory [2]. For this purpose, one should know theresponse function of graphene to the electromagnetic field which does not reduce to thestandard dielectric permittivities of metallic and dielectric materials. It is important to keepin mind that the permittivities of ordinary materials are usually derived using the kinetictheory or Kubo formula under several assumptions which are not universally applicable[19]. These ones and some other theoretical approaches have been used in approximatecalculations of the response functions and the Casimir force in graphene systems [20–37].2n the framework of the Dirac model, however, the dielectric response of graphene can bedescribed exactly by means of its polarization tensor found on the basis of first principles ofthermal quantum field theory.Although the polarization tensor of graphene was considered in many papers (see, e.g.,Ref. [38] and literature therein), the exact expression for it at zero temperature, as well asthe corresponding formulas for the reflection coefficients, have been found in Ref. [39]. Thepolarization tensor of gapped graphene (the energy gap ∆ arises for quasiparticles of nonzeromass) at any temperature was derived in Ref. [40]. The expressions of Ref. [40] are valid atthe pure imaginary Matsubara frequencies and were used to investigate the Casimir effectin many graphene systems [40–50]. In Ref. [51] another form for the polarization tensorof graphene at nonzero temperature was derived valid over the entire plane of complexfrequencies. It was generalized for the case of nonzero chemical potential µ in Ref. [52].This form of the polarization tensor was also successfully used in calculations of the Casimirforce in various graphene systems [52–57] , as well as for investigation of the reflectivity andconductivity properties of graphene [58–61].An interest to the thermodynamic aspects of the Lifshitz theory in application to graphenesystems arose from the so-called Casimir puzzle. It turned out that the theoretical predic-tions for the Casimir force between both metallic and dielectric test bodies are excluded bythe measurement data if one takes into account in calculations the dissipation of free electronsand the conductivity at a constant current, respectively (see the reviews in Refs. [2, 62, 63]and the most recent experiments [64–67]). As to thermodynamics, it was found that anaccount of dissipation of free electrons for metals with perfect crystal lattices and the dcconductivity for dielectrics results in a violation of the third law of thermodynamics whichis also known as the Nernst heat theorem (see the reviews in Refs. [2, 62] and the mostrecent results in Refs. [68–73]). In the single experiment on measuring the Casimir interac-tion in graphene systems performed up to date [74], the measurement data were found ingood agreement with theoretical predictions using the polarization tensor [75]. Taking intoconsideration that the polarization tensor of graphene results in two spatially nonlocal di-electric permittivities, the longitudinal one and the transverse one, each of which is complexand takes dissipation into account, the question arises whether the Casimir free energy andentropy of graphene systems is consistent with the requirements of thermodynamics.To answer this question, the low-temperature behavior of the Casimir free energy and3ntropy between two sheets of pristine graphene with ∆ = µ = 0 was found in Ref. [76].It was shown that in this case the Casimir entropy vanishes with vanishing temperature,i.e., the Nernst heat theorem is satisfied. The same result was obtained for the Casimir-Polder entropy of an atom interacting with a sheet of a pristine graphene [77]. For anatom interacting with real graphene sheet possessing nonzero ∆ and µ it was shown thatthe Nernst heat theorem is followed for ∆ > µ [78] and ∆ < µ [78, 79]. As to the case∆ = 2 µ , the nonzero value of the Casimir-Polder entropy at zero temperature was foundin this case depending on the parameters of a system, i.e., an entropic anomaly [79] (thelow-temperature behavior of the Casimir-Polder free energy for ∆ , µ =0 was also consideredin Ref. [80]).In this paper, we derive the low-temperature analytic asymptotic expressions for theCasimir free energy and entropy of two real graphene sheets possessing nonzero values of ∆and µ . This is a more complicated problem than for an atom interacting with real graphenesheet because the free energy of an atom-graphene interaction is the linear function of thereflection coefficients, which is not the case for two parallel graphene sheets. The Casimir freeenergy is presented by the Lifshitz formula where the reflection coefficients are expressed viathe polarization tensor of graphene in (2+1)-dimensional space-time. The thermal correctionto the Casimir energy at zero temperature is separated in two contributions. In the firstof them, the temperature dependence is determined exclusively by a summation over theMatsubara frequencies, whereas the polarization tensor is defined at zero temperature. Thetemperature dependence of the second contribution is determined by an explicit dependenceof the polarization tensor on temperature as a parameter.We find the asymptotic behaviors at low temperature for each of these contributionsunder different relationships between ∆ and 2 µ . It is shown that the leading terms deter-mining the low-temperature behavior of the total Casimir free energy originate from the firstcontribution to the thermal correction for both ∆ > µ and ∆ < µ and from the secondcontribution for ∆ = 2 µ . As a result, for ∆ > µ and ∆ < µ the Nernst heat theoremis satisfied, whereas for ∆ = 2 µ it is violated. The physical meaning of this anomaly isdiscussed in the context of problems considered earlier in the literature on the Casimir effectbetween metals and dielectrics.The paper is organized as follows. In Sec. II, we briefly summarize the necessary formalismof the polarization tensor. Section III is devoted to the perturbation expansion of the Lifshitz4ormula at low temperature. In Secs. IV, V, and VI, the derivation of the asymptoticexpressions for the Casimir free energy and entropy at low temperature is presented for thecases ∆ > µ , ∆ = 2 µ , and ∆ < µ , respectively. Section VII contains our conclusions anda discussion. In the Appendix, the reader will find some calculation details. II. THE POLARIZATION TENSOR OF GRAPHENE AND THE REFLECTIONCOEFFICIENTS
We consider two parallel graphene sheets separated by a distance a at temperature T in thermal equilibrium with the environment. The electronic quasiparticles in grapheneconsidered in the framework of the Dirac model [6, 7] are characterized by some small butnonzero mass which results in the energy gap ∆ taking the typical value 0.1–0.2 eV. Theenergy gap arises due to an impact of the defects of structure, interelectron interactions andinteraction with a substrate if any [38, 81]. We also assume that the graphene sheets underconsideration possess some value of the chemical potential µ which depends on the dopingconcentration [82] (for a pristine graphene ∆ = µ = 0).The polarization tensor of graphene describes its response to an external electromagneticfield in the one-loop approximation. The values of this tensor at the pure imaginary Mat-subara frequencies ξ l = 2 πk B T l/ ~ (where k B is the Boltzmann constant and l = 0 , , . . . )are usually notated as Π mn (i ξ l , k ⊥ , T, ∆ , µ ) ≡ Π mn,l ( k ⊥ , T, ∆ , µ ) , (1)where m, n = 0 , , k ⊥ is the magnitude of the wave vectorprojection on the plane of graphene. Below it is convenient to consider the dimensionlesspolarization tensor, frequencies and the wave vector projection defined as˜Π mn.l = 2 a ~ Π mn,l , ζ l = ξ l ω c , ω c ≡ c a , y = 2 a (cid:18) k ⊥ + ξ l c (cid:19) / . (2)In fact only the two components of the polarization tensor are the independent quan-tities. As an example, the 00 component ˜Π and the trace ˜Π mm are often used for a fullcharacterization of this tensor [40]. For our purposes it is more convenient to use ˜Π andthe following linear combination of the 00 component and the trace:˜Π l ≡ ˜Π(i ζ l , y, T, ∆ , µ ) = ( y − ζ l ) ˜Π mm (i ζ l , y, T, ∆ , µ ) − y ˜Π (i ζ l , y, T, ∆ , µ ) . (3)5he reason is that the reflection coefficients on graphene sheets for the transverse mag-netic (TM) and transverse electric (TE) polarizations of the electromagnetic waves take thefollowing simple form [39, 40, 51, 52]: r TM (i ζ l , y, T ) = y ˜Π ,l ( y, T, ∆ , µ ) y ˜Π ,l ( y, T, ∆ , µ ) + 2( y − ζ l ) , (4) r TE (i ζ l , y, T ) = − ˜Π l ( y, T, ∆ , µ )˜Π l ( y, T, ∆ , µ ) + 2 y ( y − ζ l ) , where we omitted the parameters ∆ and µ in the notations of the reflection coefficients forthe sake of brevity.Now we present the exact expressions for ˜Π ,l and ˜Π l obtained in the literature. Firstof all, it is convenient to present them as the respective quantity defined at T = 0 plus thethermal correction to it˜Π ,l ( y, T, ∆ , µ ) = ˜Π ,l ( y, , ∆ , µ ) + δ T ˜Π ,l ( y, T, ∆ , µ ) , ˜Π l ( y, T, ∆ , µ ) = ˜Π l ( y, , ∆ , µ ) + δ T ˜Π l ( y, T, ∆ , µ ) . (5)It is also useful to present ˜Π ,l and ˜Π l as the sums of contributions which do not dependand, quite the reverse, depend on µ and T [57]˜Π ,l ( y, T, ∆ , µ ) = ˜Π (0)00 ,l ( y, ∆) + ˜Π (1)00 ,l ( y, T, ∆ , µ ) , ˜Π l ( y, T, ∆ , µ ) = ˜Π (0) l ( y, ∆) + ˜Π (1) l ( y, T, ∆ , µ ) . (6)As the first contributions on the right-hand side of Eq. (6) we choose the 00 componentand the combination (3) for the polarization tensor of gapped (∆ = 0) but undoped ( µ = 0)graphene defined at zero temperature [39, 57]˜Π (0)00 ,l ( y, ∆) = α y − ζ l p l Ψ (cid:18) Dp l (cid:19) , (7)˜Π (0) l ( y, ∆) = α ( y − ζ l ) p l Ψ (cid:18) Dp l (cid:19) , where α = e / ( ~ c ) is the fine structure constant, D ≡ ∆ / ( ~ ω c ), and the following notationsare introducedΨ( x ) = 2 (cid:2) x + (1 − x ) arctan( x − ) (cid:3) , p l = (cid:2) ˜ v F y + (1 − ˜ v F ) ζ l (cid:3) / , ˜ v F = v F c . (8)6n accordance to our choice, ˜Π (0)00 ,l ( y, ∆) = ˜Π ,l ( y, , ∆ , , ˜Π (0) l ( y, ∆) = ˜Π l ( y, , ∆ , . (9)In so doing, ˜Π (1)00 ,l and ˜Π (1) l acquire a meaning of the thermal corrections to the polarizationtensor of undoped graphene defined at T = 0:˜Π (1)00 ,l ( y, T, ∆ ,
0) = δ T ˜Π ,l ( y, T, ∆ , , ˜Π (1) l ( y, T, ∆ ,
0) = δ T ˜Π l ( y, T, ∆ , . (10)These corrections vanish in the limit of zero temperature.The second contributions on the right-hand side of Eq. (6) can be explicitly presented inthe form [57, 79] ˜Π (1)00 ,l ( y, T, ∆ , µ ) = 4 αD ˜ v F Z ∞ dtw ( t, T, ∆ , µ ) X ,l ( t, y, D ) , (11)˜Π (1) l ( y, T, ∆ , µ ) = − αD ˜ v F Z ∞ dtw ( t, T, ∆ , µ ) X l ( t, y, D ) , where the µ -dependent factor is given by w ( t, T, ∆ , µ ) = (cid:16) e t ∆+2 µ kBT + 1 (cid:17) − + (cid:16) e t ∆ − µ kBT + 1 (cid:17) − , (12)and the functions X ,l and X l are defined as follows: X ,l ( t, y, D ) = 1 − Re p l − D t + 2i ζ l Dt [ p l − p l D t + ˜ v F ( y − ζ l ) D + 2i ζ l p l Dt ] / , (13) X l ( t, y, D ) = ζ l − Re ζ l p l − p l D t + ˜ v F ( y − ζ l ) D + 2i ζ l p l Dt [ p l − p l D t + ˜ v F ( y − ζ l ) D + 2i ζ l p l Dt ] / . It has been shown [56, 57] that for a doped and gapped graphene satisfying the condition∆ > µ the polarization tensor at T = 0 also does not depend on µ . As a result, one obtainsthe equalities similar to those in Eqs. (9) and (10)˜Π ,l ( y, , ∆ , µ ) = ˜Π (0)00 ,l ( y, ∆) , ˜Π l ( y, , ∆ , µ ) = ˜Π (0) l ( y, ∆) , (14)and δ T ˜Π ,l ( y, T, ∆ , µ ) = ˜Π (1)00 ,l ( y, T, ∆ , µ ) , δ T ˜Π l ( y, T, ∆ , µ ) = ˜Π (1) l ( y, T, ∆ , µ ) , (15)7here the thermal corrections vanish with vanishing temperature.It is significant that under the condition ∆ < µ the polarization tensor of doped andgapped graphene at T = 0 depends both on ∆ and µ , and Eqs. (14) and (15) are not validany more. In this case, the 00 component of the polarization tensor at T = 0 and thecombination of its components (3) are given by [56]˜Π ,l ( y, , ∆ , µ ) = 8 αµ ˜ v F ~ ω c − α ( y − ζ l ) p l (cid:26) ( p l + D )Im (cid:18) z l q z l (cid:19) +( p l − D ) (cid:20) Im ln (cid:18) z l + q z l (cid:19) − π (cid:21)(cid:27) , (16)˜Π l ( y, , ∆ , µ ) = − αµζ l ˜ v F ~ ω c + 2 α ( y − ζ l ) p l (cid:26) ( p l + D )Im (cid:18) z l q z l (cid:19) − ( p l − D ) (cid:20) Im ln (cid:18) z l + q z l (cid:19) − π (cid:21)(cid:27) , where z l ≡ z l ( y, ∆ , µ ) = p l ˜ v F p p l + D p y − ζ l (cid:18) ζ l + i 2 µ ~ ω c (cid:19) . (17)The thermal corrections to the polarization tensor of graphene satisfying the condition∆ < µ are immediately obtained from Eqs. (5) and (6) δ T ˜Π ,l ( y, T, ∆ , µ ) = ˜Π ,l ( y, T, ∆ , µ ) − ˜Π ,l ( y, , ∆ , µ )= ˜Π (1)00 ,l ( y, T, ∆ , µ ) − ˜Π (1)00 ,l ( y, , ∆ , µ ) , (18) δ T ˜Π l ( y, T, ∆ , µ ) = ˜Π l ( y, T, ∆ , µ ) − ˜Π l ( y, , ∆ , µ )= ˜Π (1) l ( y, T, ∆ , µ ) − ˜Π (1) l ( y, , ∆ , µ ) . As to the case of an exact equality ∆ = 2 µ , it is considered in Sec. V. III. PERTURBATION EXPANSION OF THE LIFSHITZ FORMULA AT LOWTEMPERATURE
Using the reflection coefficients (4) expressed above via the polarization tensor, one canrepresent the Casimir free energy per unit area of graphene sheets by means of the Lifshitz8ormula [2, 83] F ( a, T ) = k B T πa ∞ X l =0 ′ Z ∞ ζ l ydy X λ ln (cid:2) − r λ (i ζ l , y, T ) e − y (cid:3) , (19)where the prime on the summation sign divides the term with l = 0 by 2, and the sum in λ is over two polarizations of the electromagnetic field, transverse magnetic and transverseelectric ( λ = TM , TE).We are in fact interested not in the total Casimir free energy but in its temperature-dependent part, i.e., in the thermal correction to the Casimir energy defined as δ T F ( a, T ) = F ( a, T ) − E ( a ) , (20)where the Casimir energy at zero temperature is given by [2, 83] E ( a ) = ~ c π a Z ∞ dζ Z ∞ ζ ydy X λ ln (cid:2) − r λ (i ζ , y, e − y (cid:3) . (21)Here, the reflection coefficients are expressed by Eq. (4) in which one should replace theMatsubara frequencies with a continuous frequency ζ and put T = 0 r TM (i ζ , y,
0) = y ˜Π (i ζ , y, , ∆ , µ ) y ˜Π (i ζ , y, , ∆ , µ ) + 2( y − ζ ) , (22) r TE (i ζ , y,
0) = − ˜Π(i ζ , y, , ∆ , µ )˜Π(i ζ , y, , ∆ , µ ) + 2 y ( y − ζ ) . Note that both the propagating waves, which are on the mass shell, and the evanescentwaves off the mass shell contribute to Eqs. (19) and (21).In the case ∆ > µ , following Eq. (14), one should substitute to Eq. (22) the expressionsfor the ˜Π and ˜Π defined in Eq. (7) making there the above replacement ζ l → ζ . If, however,the condition ∆ < µ is fulfilled, it is necessary to substitute in Eq. (22) the quantities (16)with the same replacement.Now we identically rearrange Eq. (20) to the form δ T F ( a, T ) = δ impl T F ( a, T ) + δ expl T F ( a, T ) , (23)where 9 impl T F ( a, T ) = k B T πa ∞ X l =0 ′ Z ∞ ζ l ydy X λ ln (cid:2) − r λ (i ζ l , y, e − y (cid:3) − E ( a ) (24)and δ expl T F ( a, T ) = F ( a, T ) − k B T πa ∞ X l =0 ′ Z ∞ ζ l ydy X λ ln (cid:2) − r λ (i ζ l , y, e − y (cid:3) . (25)As is seen from Eqs. (23)–(25), we have simply added and subtracted from Eq. (20) thequantity having the same form as the Casimir free energy in Eq. (19) but containing thereflection coefficients (4) taken at T = 0.An advantage of Eq. (23) is that the implicit temperature dependence of the first term, δ impl T F , is entirely determined by a summation on the Matsubara frequencies, whereas thepolarization tensor is taken at T = 0. As to the second term, δ expl T F , it simply vanishes forthe temperature-independent polarization tensors. Thus, the dependence of this term on T can be called explicit.We turn our attention to the perturbation expansion of the Casimir free energy at lowtemperature. Taking into account that the thermal corrections δ T ˜Π ,l and δ T ˜Π l go to zerowith vanishing T , we substitute Eq. (5) in Eq. (4), expand up to the first order of smallparameters δ T ˜Π ,l ( y, T, ∆ , µ )˜Π ,l ( y, , ∆ , µ ) ≪ , δ T ˜Π l ( y, T, ∆ , µ )˜Π l ( y, , ∆ , µ ) ≪ r TM(TE) (i ζ l , y, T ) = r TM(TE) (i ζ l , y,
0) + δ T r TM(TE) (i ζ l , y, T ) , (27)where the first contributions are given by Eq. (4) taken at T = 0 and the thermal correctionsto the reflection coefficients are given by δ T r TM (i ζ l , y, T ) = 2 y ( y − ζ l ) δ T ˜Π ,l ( y, T, ∆ , µ )[ y ˜Π ,l ( y, , ∆ , µ ) + 2( y − ζ l )] , (28) δ T r TE (i ζ l , y, T ) = − y ( y − ζ l ) δ T ˜Π l ( y, T, ∆ , µ )[ ˜Π l ( y, , ∆ , µ ) + 2 y ( y − ζ l )] . This approach is applicable under the conditions ˜Π ,l ( y, , ∆ , µ ) = 0 and ˜Π l ( y, , ∆ , µ ) = 0which are valid for the cases ∆ > µ considered in Secs. IV and V. For the case ∆ < µ ,however, one cannot use the perturbation theory in the parameters (26) for the contributionof the Matsubara term with l = 0 (see Sec. VI).10he implicit thermal correction δ impl T F defined in Eq. (24) is the difference between thesum in l and the integral (21) with respect to ζ . From Eq. (2) it is seen that ζ l = τ l where τ ≡ πk B T a/ ( ~ c ). By replacing the integration variable ζ in Eq. (21) with t = ζ /τ , one canbring Eq. (24) to the form δ impl T F ( a, T ) = k B T πa " ∞ X l =0 ′ Φ( τ l ) − Z ∞ dt Φ( τ t ) , (29)where Φ( x ) = Z ∞ x ydy X λ ln (cid:2) − r λ (i x, y, e − y (cid:3) . (30)By applying the Abel-Plana formula [2, 84], Eq. (29) can be rewritten as δ impl T F ( a, T ) = i k B T πa Z ∞ dte πt − τ t ) − Φ( − i τ t )] . (31)In the next sections, Eq. (31) is used to find the asymptotic behavior of δ impl T F at arbitrarilylow T .In order to determine the low-temperature behavior of the second thermal correction tothe Casimir energy, δ expl T F , we substitute Eq. (27) into its definition (25) and use the identityln (cid:8) − [ r λ (i ζ l , y,
0) + δ T r λ (i ζ l , y, T )] e − y (cid:9) − ln (cid:2) − r λ (i ζ l , y, e − y (cid:3) = ln (cid:26) − r λ (i ζ l , y, δ T r λ (i ζ l , y, T ) + [ δ T r λ (i ζ l , y, T )] − r λ (i ζ l , y, e − y e − y (cid:27) . (32)Then, expanding the logarithm up to the first power of a small parameter and preservingonly the term of the first power in δ T r λ (i ζ l , y, T ), one arrives at δ expl T F ( a, T ) = − k B T πa ∞ X l =0 ′ Z ∞ ζ l ydye − y X λ r λ (i ζ l , y, δ T r λ (i ζ l , y, T )1 − r λ (i ζ l , y, e − y . (33)This equation valid under a condition that ˜Π ,l and ˜Π l are nonzero at T = 0 and, thus, r λ (i ζ l , y, = 0 is used below to determine the behavior of δ expl T F at low temperature. IV. LOW-TEMPERATURE BEHAVIOR OF THE CASIMIR FREE ENERGYAND ENTROPY FOR GRAPHENE SHEETS WITH ∆ > µ We assume that the graphene sheets under consideration in this section satisfy the con-dition ∆ > µ and start with the thermal correction δ impl T F ( a, T ) to the Casimir energy11efined in Eq. (24) and expressed by Eqs. (29) and (31). In accordance to Eq. (30) thefunction Φ entering Eq. (29) is defined as the sum of contributions from the TM and TEmodes Φ( x ) = Φ TM ( x ) + Φ TE ( x ) . (34)As a result, δ impl T F ( a, T ) becomes the sum of δ impl T F TM ( a, T ) and δ impl T F TE ( a, T ).Under the condition ∆ > µ , the polarization tensor at T = 0 is given by Eq. (7). Byreplacing ζ l with x in Eq. (7) and substituting the obtained expressions in Eq. (22) where ζ is also replaced with x , one obtains r TM (i x, y,
0) = αy Ψ( Dp − ) αy Ψ( Dp − ) + 2 p ( x, y ) , (35) r TE (i x, y,
0) = − αp ( x, y )Ψ( Dp − ) αp ( x, y )Ψ( Dp − ) + 2 y , where the quantity p is defined as p ≡ p ( x, y ) = [˜ v F y + (1 − ˜ v F ) x ] / . (36)In the analytic asymptotic expressions here and below we use the condition ∆ > ~ ω c (i.e., D >
1) which is satisfied at not too small separations between the graphene sheets. Underthis condition, at sufficiently small x (low T ) one can safely use the inequality D ≫ p ( x, y )because the dominant contribution to the integrals in Eq. (30) is given by y ∼ λ = TM. By expanding in Eq. (30) in Taylor series around x = 0 with the help of the first formula in Eq. (35) and above condition, we findΦ TM ( x ) = Φ TM (0) + 4 α D x + 16 α (8 α + 3 D )135 D x + O ( x ) ≈ Φ TM (0) + 4 α D x + 16 α D x + O ( x ) . (37)The first two terms on the right-hand side of this equation do not contribute to Eq. (31),whereas the third term leads toΦ TM (i τ t ) − Φ TM ( − i τ t ) = i 32 α D τ t . (38)Substituting this result in Eq. (31), one arrives at δ impl T F TM ( a, T ) = − α π a ( k B T ) ( ~ c ) . (39)12e continue with the case λ = TE. The function Φ TE ( x ) cannot be expanded in Taylorseries around the point x = 0. Because of this, we substitute the second line of Eq. (35)in Eq. (30), expand the integrand in powers of x and integrate with respect to y thereafter.The result isΦ TE ( x ) = (cid:18) α D (cid:19) (cid:20) − v F − v F (1 − ˜ v F ) x + ˜ v F (cid:18) −
34 ˜ v F (cid:19) x + (1 − ˜ v F ) x Ei( − x ) − v F (cid:18) −
710 ˜ v F (cid:19) x + O ( x ) (cid:21) , (40)where Ei( z ) in the exponential integral.The first three terms on the right-hand side of this expression do not contribute toEq. (31). The dominant contribution is given by the term containing the exponential integralwhich leads to Φ TE (i τ t ) − Φ TE ( − i τ t ) = i π (cid:18) α D (cid:19) τ t . (41)Substituting this equation in Eq. (31) and integrating, one arrives at the result δ impl T F TE ( a, T ) = − ζ (5) α ( k B T ) π ∆ ( ~ c ) . (42)Comparing this with Eq. (39), we conclude that the dominant term in the asymptoticbehavior of δ impl T F at low T is given by Eq. (42) and determined by the contribution of theTE mode, i.e., δ impl T F ( a, T ) = δ impl T F TE ( a, T ) ∼ − α ( k B T ) ∆ ( ~ c ) . (43)We are now coming to the asymptotic behavior of the second thermal correction, δ expl T F ,at low T which takes into account an explicit dependence of the polarization tensor ontemperature as a parameter. This correction is presented in Eq. (33). It is convenient toexpress δ expl T F as a sum of two contributions δ expl T F ( a, T ) = δ expl T, l =0 F ( a, T ) + δ expl T, l > F ( a, T ) , (44)where the first one contains the term of Eq. (33) with l = 0 and the second one — all termswith l > l = 0 are13btained from Eq. (35) by putting x = 0 r TM (0 , y,
0) = α Ψ( D ˜ v − F y − ) α Ψ( D ˜ v − F y − ) + 2˜ v F , (45) r TE (0 , y,
0) = − α ˜ v F Ψ( D ˜ v − F y − ) α ˜ v F Ψ( D ˜ v − F y − ) + 2 , Taking into account that for y ∼ v F y ≪ D , we expand the function Ψ in powersof the small parameter ˜ v F y/D and obtainΨ( D ˜ v − F y − ) ≈
83 ˜ v F yD . (46)As a result, Eq. (45) reduces to r TM (0 , y, ≈ αyαy + D ≈ αy D , (47) r TE (0 , y, ≈ − α ˜ v F yα ˜ v F y + D ≈ − α ˜ v F y D .
From Eq. (47) it is seen that r TE (0 , y, ≈ − ˜ v F r TM (0 , y, , (48)i.e., the magnitude of the TE reflection coefficient taken at zero frequency and temperatureis negligibly small as compared to the TM one.Next, we consider the thermal corrections to the reflection coefficients (47) enteringEq. (33). By putting l = 0 in Eq. (28), one obtains δ T r TM (0 , y, T ) = 2 yδ T ˜Π , ( y, T, ∆ , µ )[ ˜Π , ( y, , ∆ , µ ) + 2 y ] , (49) δ T r TE (0 , y, T ) = − y δ T ˜Π ( y, T, ∆ , µ )[ ˜Π ( y, , ∆ , µ ) + 2 y ] . Under the condition ∆ > µ we can use Eq. (15) and, thus, the quantities δ T ˜Π , and δ T ˜Π can be obtained from Eq. (11) taken at l = 0. Taking into account that under the condition∆ > µ the first contribution to Eq. (12) leads to an additional exponentially small factorexp[ − µ/ ( k B T )], one can preserve only the second contribution. As a result, we have δ T ˜Π , ( y, T, ∆ , µ ) = 4 αD ˜ v F (cid:20) I (1)00 , + 1˜ v F y I (2)00 , (cid:21) , (50)14here I (1)00 , = Z ∞ dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − , (51) I (2)00 , = Z f ( y,D )1 dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − D t − ˜ v F y [˜ v F y − D ( t − / and the function f ( y, D ) is defined as f ( y, D ) = r v F y D . (52)For the thermal correction δ T ˜Π from the second line in Eq. (11) one obtains δ T ˜Π ( y, T, ∆ , µ ) = − αD y ˜ v F Z f ( y,D )1 dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − t − v F y − D ( t − / . (53)Since we consider arbitrarily low T , we can use the condition ∆ − µ ≫ k B T . Under thiscondition the quantity I (1)00 , in Eq. (51) takes an especially simple form I (1)00 , ≈ k B T ∆ e − ∆ − µ kBT . (54). The quantity I (2)00 , defined in Eq. (51) is calculated at low temperature in the Appendix.According to Eq. (A3), the asymptotic behavior of I (2)00 , is given by I (2)00 , ∼ k B T ˜ v F ∆( ~ ω c ) e − ∆ − µ kBT . (55)Then, from Eqs. (50), (54), and (55) we can conclude that δ T ˜Π , ( y, T, ∆ , µ ) ∼ αk B T ~ ω c e − ∆ − µ kBT (cid:18) C + C y (cid:19) , (56)where C ∼ ˜ v − F and C ∼ ˜ v − F are the constants.The integral with respect to t in Eq. (53) for δ T ˜Π can be estimated similar to Eqs. (A2)and (A3). For this purpose, using Eq. (52), we replace t − v F y /D and obtain δ T ˜Π ( y, T, ∆ , µ ) ∼ − αk B T ~ ω c C e − ∆ − µ kBT , (57)where C ∼ ˜ v F .Substituting Eqs. (7), (46), (56) and (57) in Eq. (49), one finds δ T r TM (0 , y, T ) = δ T ˜Π , ( y, T, ∆ , µ )2 y (cid:0) αy D + 1 (cid:1) ≈ δ T ˜Π , ( y, T, ∆ , µ )2 y ∼ αk B T ~ ω c e − ∆ − µ kBT (cid:18) C y + C y (cid:19) ,δ T r TE (0 , y, T ) = − δ T ˜Π ( y, T, ∆ , µ )2 y (cid:0) α ˜ v F y D + 1 (cid:1) ≈ − δ T ˜Π ( y, T, ∆ , µ )2 y ∼ αk B T ~ ω c y C e − ∆ − µ kBT . (58)15rom these equations, we obtain δ T r TE (0 , y, T ) ∼ ˜ v F δ T r TM (0 , y, T ) , (59)i.e., similar to Eq. (48), thermal correction to the TE reflection coefficient at zero Matsubarafrequency is negligibly small comparing to the TM one.Now we substitute the first lines of Eqs. (47) and (58) in the term of Eq. (33) with l = 0and obtain δ expl T, l =0 F ( a, T ) ≈ δ expl T, l =0 F TM ( a, T ) ∼ − α ( k B T ) a ∆ e − ∆ − µ kBT Z ∞ dy e − y C y + C − (cid:0) αy D (cid:1) e − y . (60)Taking into consideration that the integral in this equation converges, the final result is δ expl T, l =0 F ( a, T ) ∼ − α ( k B T ) a ∆ e − ∆ − µ kBT (61)We are passing now to a consideration of the correction δ expl T, l > F which is equal to thesum of all terms with l > x = ζ l , using anapproximate equality Ψ (cid:18) Dp l (cid:19) ≈ p l D (62)similar to Eq. (46), we find r TM (i ζ l , y, ≈ αyαy + D ≈ αy D , (63) r TE (i ζ l , y, ≈ − αp l αp l + Dy ≈ − αp l Dy ≈ − α ˜ v F y D .
Here we have used that for y ∼
1, giving the dominant contribution to Eq. (33), D ≫ αy andconsidered p l ≈ ˜ v F y at τ →
0. From Eq. (63) it is seen that similar to Eq. (48) relationship r TE (i ζ l , y, ≈ − ˜ v F r TM (i ζ l , y, , (64)holds at any ζ l .Using Eq. (28), in the same approximation as in Eq. (58) one obtains δ T r TM (i ζ l , y, T ) ≈ yδ T ˜Π ,l ( y, T, ∆ , µ )2( y − ζ l ) ,δ T r TE (i ζ l , y, T ) ≈ − δ T ˜Π l ( y, T, ∆ , µ )2 y ( y − ζ l ) . (65)16rom Eqs. (11), (13) and (15) one can make sure that δ T ˜Π ,l ( y, T, ∆ , µ ) (cid:12)(cid:12)(cid:12) y = ζ l = δ T ˜Π l ( y, T, ∆ , µ ) (cid:12)(cid:12)(cid:12) y = ζ l = 0 . (66)Because of this, the integrals with respect to y in Eq. (33) are convergent at the low inte-gration limit for all l >
1. Since the dominant contribution in Eq. (33) is given by y ∼ τ → ζ l = τ l . For the order of magnitude estimation of the asymptotic behavior at T →
0, it willsuffice to consider the lowest expansion order. In this way, from Eqs. (33), (56) and (65) wefind δ expl T, l > F TM ( a, T ) ∼ − k B Ta ∞ X l =1 Z ∞ ζ l ydye − y r TM (0 , y, − r (0 , y, e − y δ T ˜Π , ( y, T, ∆ , µ ) y ∼ − α ( k B T ) ~ ca e − ∆ − µ kBT ∞ X l =1 Z ∞ ζ l dye − y r TM (0 , y, − r (0 , y, e − y (cid:18) C + C y (cid:19) . (67)By introducing the variable v = y/ζ l and using Eq. (63), it is seen that in the asymptoticlimit τ → δ expl T, l > F TM ( a, T ) ∼ − α ( k B T ) ~ ca e − ∆ − µ kBT ∞ X l =1 ζ l Z ∞ ζ l vdve − ζ l v (cid:18) C + C ζ l v (cid:19) (68)= − α ( k B T ) ~ ca e − ∆ − µ kBT ∞ X l =1 [ C (1 + ζ l ) + C ] e − ζ l ∼ − α ( k B T ) ~ ca e − ∆ − µ kBT τ ∼ − α k B Ta e − ∆ − µ kBT . Similar estimation shows that the contribution of the TE mode to Eq. (33) is againnegligibly small δ expl T, l > F TE ( a, T ) ∼ ˜ v F δ expl T, l > F TM ( a, T ) . (69)Because of this, the result is δ expl T, l > F ( a, T ) ∼ δ expl T, l > F TM ( a, T ) ∼ − α k B Ta e − ∆ − µ kBT . (70)Comparing Eqs. (61) and (70), we notice that a summation over the nonzero Matsubarafrequencies decreases by one the power of temperature in front of the main exponentialfactor. Note also that Eqs. (43), (61), and (70) are obtained under the condition ∆ > ~ ω c and, thus, one cannot put there ∆ = 0. These equations, however, are well applicable forgraphene with µ = 0. 17ow we can find the dominant asymptotic behavior of the total thermal correction tothe Casimir energy at zero temperature δ T F in the limit of low temperature. Taking intoaccount that in accordance to Eqs. (23) and (44) δ T F is given by the sum of Eqs. (43),(61), and (70), one concludes that under a condition ∆ > µ its leading behavior is givenby Eq. (43), i.e., δ T F ( a, T ) ∼ − α ( k B T ) ∆ ( ~ c ) , (71)and is determined by the TE contribution to the implicit temperature dependence.This result gives the possibility to find the low-temperature behavior of the Casimirentropy per unit area of the graphene sheets defined as S ( a, T ) = − ∂ F ( a, T ) ∂T = − ∂δ T F ( a, T ) ∂T . (72)Using Eq. (71), one finds S ( a, T ) ∼ α k B T ∆ ( ~ c ) , (73)which vanishes with vanishing temperature in agreement with the third law of thermody-namics (the Nernst heat theorem) [85, 86]. This means that the Lifshitz theory using theresponse function of graphene with ∆ > µ expressed in terms of the polarization tensor isthermodynamically consistent.To summarize the application region of the obtained results, in this section we used theconditions k B T ≪ ~ v F a ≪ ~ c a < ∆ , k B T ≪ ∆ − µ (74)and made the asymptotic expansions in three small parameters τ ≡ πk B T a ~ c ≪ , ~ v F a ∆ ≪ , e − ∆ − µ kBT ≪ . (75)The last parameter was used in finding the low-temperature behavior of δ expl T F . It is possible,however, to dispense with this parameter (see the next section). V. LOW-TEMPERATURE BEHAVIOR OF THE CASIMIR FREE ENERGY ANDENTROPY FOR GRAPHENE SHEETS WITH ∆ = 2 µ As was stated in Sec. II, Eqs. (14) and (15) preserve their validity in the case ∆ = 2 µ .Because of this, all the results for δ impl T F obtained in Sec. III for the graphene sheets with18 > µ remain valid in the case ∆ = 2 µ . Specifically, the low-temperature behavior of δ impl T F is again determined by the TE mode and is given by Eq. (43).An explicit temperature dependence, however, leads to a radically different results. Al-though Eqs. (44)–(53) remain valid in the case ∆ = 2 µ , the subsequent equations obtainedunder a condition ∆ − µ ≫ k B T are not applicable. Thus, instead of Eq. (54), from thefirst line of Eq. (51) we obtain I (1)00 , = 2 k B T ∆ ln 2 . (76)A more exact calculation of the integral I (2)00 , defined in Eqs. (51) and (52) in the case∆ = 2 µ (see Appendix) in accordance to Eq. (A6) results in I (2)00 , ∼ k B T ˜ v F ∆( ~ ω c ) ln 2 . (77)As is seen from the comparison of Eqs. (76) and (77) with Eqs. (54) and (55), respectively,the values of I (1)00 , and I (2)00 , in the cases ∆ > µ and ∆ = 2 µ differ only by the missingexponential factor and by an occurrence of the factor ln 2 in the latter case. This allows toconclude that, similar to the case ∆ > µ considered in Sec. IV, the dominant contribution tothe thermal correction δ expl T, l =0 F is determined by the TM mode. Up to an order of magnitudeestimation of this contribution for the case ∆ = 2 µ , in accordance to Eq. (61), is given by δ expl T, l =0 F ( a, T ) ∼ − α ( k B T ) a ∆ . (78)In a similar way, by repeating the derivation in Eqs. (62)–(70), one arrives at a conclusionthat for ∆ = 2 µ the contribution δ expl T, l > F to the thermal correction at low temperature isestimated by Eq. (70) where the exponential factor is replaced with unity δ expl T, l > F ( a, T ) ∼ − α k B Ta . (79)From the comparison of Eq. (43) for an implicit contribution to the thermal correction,which is valid also for the case ∆ = 2 µ , with the explicit contributions (78) and (79), oneconcludes that in this case the low-temperature behavior of the total thermal correction isgiven by δ T F ( a, T ) ∼ − α k B Ta , (80)which originates from the TM mode in an explicit temperature dependence. In the case∆ = 2 µ , Eqs. (43) and (78)–(80) are obtained under the first set of inequalities in Eq. (74),19.e., do not using the condition k B T ≪ ∆ − µ . They employ only the first two smallparameters indicated in Eq. (75) and are valid for graphene with ∆ = 0 and µ = 0.The result (80) leads to problems. The point is that, in accordance to Eq. (72), therespective Casimir entropy per unit area of the graphene sheets at low temperature behavesas S ( a, T ) ∼ α k B a . (81)Thus, the Casimir entropy at zero temperature is the nonzero (positive) constant depend-ing on the volume of a system in violation of the Nernst heat theorem [85, 86]. As discussedin Sec. I, the same situation holds for metals with perfect crystal lattices described by thedielectric permittivity of the Drude model which, as opposed to the polarization tensor ofgraphene, is not derived from the first principles of quantum field theory. It should be takeninto consideration, however, that for a real graphene sheet the values of ∆ and µ cannot beknown precisely. Thus, from the practical standpoint, the equality ∆ = 2 µ can be consid-ered as some singular point (see further discussion in Sec. VII). It is only important whatare the properties of the Casimir free energy and entropy at low temperatures for graphenesheets with ∆ < µ . This question is answered in the next section. VI. LOW-TEMPERATURE BEHAVIOR OF THE CASIMIR FREE ENERGYAND ENTROPY FOR GRAPHENE SHEETS WITH ∆ < µ Here, we consider the last possibility when the chemical potential is relatively large byexceeding the half of the energy gap. As in two preceding sections, we begin with consid-eration of the implicit contribution to the thermal correction given by Eq. (31), where thefunction Φ( x ) is expressed via the reflection coefficients at zero temperature by Eq. (30).In order to find these reflection coefficients, we consider the polarization tensor (16) and(17) found in the case ∆ < µ , replace ζ l with x in Eqs. (16) and (17) and expand the resultsup to the first power in x under the condition p µ − ∆ > ~ ω c which is satisfied at nottoo small separations between the graphene sheets. The result is˜Π ( x, y, , ∆ , µ ) = Q − Q xy , ˜Π( x, y, , ∆ , µ ) = Q yx, (82)where the following notations are introduced Q = 4 α ˜ v F µ ~ ω c , Q = 16 αµ ˜ v F ~ ω c p µ − ∆ , Q = 4 α p µ − ∆ ˜ v F ~ ω c . (83)20t is easily seen that under the used conditions Q ≫ x ) in Eqs. (30) and (34) andexpand it up to the first power in small x Φ TM ( x ) = Φ TM (0) + x Φ ′ TM (0) . (84)Substituting Eq. (82) in the first line of Eq. (22), where ζ is replaced with x , one obtains r TM ( x, y,
0) = yQ − Q xyQ − Q x + 2( y − x ) ,r TM (0 , y,
0) = Q Q + 2 y . (85)From Eq. (30) at λ = TM, using Eq. (85), it is easily seen that the quantity Φ TM ( x ) at x = 0 is represented by a converging integral. Calculating the first derivative of Φ TM ( x ),one obtains Φ ′ TM ( x ) = − x ln(1 − e − x ) − Z ∞ x ydy r TM ( x, y, e − y − r ( x, y, e − y ∂r TM ( x, y, ∂x . (86)By differentiating the first equality in Eq. (85), one finds ∂r TM ( x, y, ∂x (cid:12)(cid:12)(cid:12)(cid:12) x =0 = − Q ( Q + 2 y ) . (87)Then, substituting Eq. (87) in Eq. (86), we haveΦ ′ TM (0) = 4 Q Z ∞ dy y ( Q + 2 y ) r TM (0 , y, e − y − r (0 , y, e − y . (88)Taking into account that Q ≫ y ∼
1, one finds from the second equality in Eq. (85) that r TM (0 , y, ≈
1. In such amanner, Eq. (88) reduces toΦ ′ TM (0) ≈ Q Q Z ∞ y dye y − π Q Q . (89)Substituting this equation in Eq. (84), one obtainsΦ TM (i τ t ) − Φ TM ( − i τ t ) = i 4 π Q Q τ T. (90)Now we consider the contribution of the TE mode in Eqs. (30) and (34). In this case thereflection coefficient is obtained by substituting Eq. (82) in the second line of Eq. (22) r TE ( x, y,
0) = − Q xQ x + 2( y − x ) . (91)21s is seen from this equation, r TE ( x, y,
0) goes to zero with vanishing x .Using the first expansion term in the powers of r TE ( x, y,
0) in Eq. (30), we findΦ TE ( x ) ≈ − Z ∞ x ydyr ( x, y, e − y . (92)Substituting here Eq. (91), one obtainsΦ TE ( x ) ≈ − Q x Z ∞ x dy y e − y [ Q x + 2( y − x )] ≈ − Q x Z ∞ x dy e − y y = Q x (cid:20) Ei( − x ) − e − x (1 − x ) x (cid:21) ≈ − Q (cid:2) − x + x ln x + O ( x ) (cid:3) . (93)From this equation, the difference of our interest is given byΦ TE (i τ t ) − Φ TE ( − i τ t ) = i Q τ t. (94)Comparing the difference in Eq. (90) with that in Eq. (94), one finds that the latter islarger than the former by the factor3 Q Q π Q = 24 π α p µ − ∆ ˜ v F ~ ω c ! ≫ . (95)Thus, one can approximately putΦ(i τ t ) − Φ( − i τ t ) ≈ Φ TE (i τ t ) − Φ TE ( − i τ t ) . (96)Finally, substituting Eqs. (94) and (96) in Eq. (31), one arrives at the result δ impl T F ( a, T ) ≈ − k B T πa Q τ Z ∞ t dte πt − − α a ( k B T ) (4 µ − ∆ )3˜ v F ( ~ c ) . (97)This result is obtained under a condition p µ − ∆ > ~ ω c and, thus, µ = 0. However,∆ = 0 is allowed.Now we consider the explicit contributions to the thermal correction in the case ∆ < µ starting with δ expl T, l =0 F . We again use the condition p µ − ∆ > ~ ω c . Under this condi-tion, in accordance with Eq. (82), ˜Π , ( y, , ∆ , µ ) = Q = 0 and the reflection coefficient r TM (0 , y,
0) is given by the second expression in Eq. (85) and, thus, is not equal to zero.Because of this, for calculating the TM contribution to δ expl T, l =0 F one can use the term with l = 0 in Eq. (33). 22he TE contribution to δ expl T, l =0 F is a different matter. Here, in accordance to the secondformula in Eq. (82), ˜Π ( y, , ∆ , µ ) = 0 and, due to Eq. (91), r TE (0 , y,
0) = 0. Because ofthis, Eq. (33) is not applicable in this case and one should calculate δ expl T, l =0 F TE using itsdefinition as the term with l = 0 in Eq. (25). Taking into account that due to the equality r TE (0 , y,
0) = 0 one has r TE (0 , y, T ) = δ T r TE (0 , y, T ), Eq. (25) leads to δ expl T, l =0 F TE ( a, T ) = k B T πa Z ∞ y dy ln (cid:8) − [ δ T r TE (0 , y, T )] e − y (cid:9) ≈ − k B T πa Z ∞ ydy [ δ T r TE (0 , y, T )] e − y , (98)where the last transformation is valid at sufficiently low T .The thermal correction to the TM reflection coefficient in Eq. (33), in accordance toEqs. (49) and (82) taken at x = 0, is given by δ T r TM (0 , y, T ) = 2 yδ T ˜Π , ( y, T, ∆ , µ )( Q + 2 y ) . (99)For obtaining δ T r TE , Eq. (49) is not applicable, so that it is found using Eq. (4) taken at l = 0 with account of the equalities ˜Π = δ T ˜Π and r TE (0 , y, T ) = δ T r TE (0 , y, T ) δ T r TE (0 , y, T ) = − δ T ˜Π ( y, T, ∆ , µ ) δ T ˜Π ( y, T, ∆ , µ ) + 2 y ≈ − δ T ˜Π ( y, T, ∆ , µ )2 y . (100)In the last transformation we have taken into account that the dominant contribution toEq. (98) is given by y ∼ δ T ˜Π goes to zero with vanishing T .In the case ∆ < µ under consideration now, the quantities δ T ˜Π , and δ T ˜Π , enteringEqs. (99) and (100), can be found from Eqs. (11) and (18) δ T ˜Π , ( y, T, ∆ , µ ) = 4 αD ˜ v F "Z ∞ dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − X , ( t, y, D ) − Z µ/ ∆1 dtX , ( t, y, D ) , (101) δ T ˜Π ( y, T, ∆ , µ ) = 4 αD ˜ v F "Z ∞ dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − X ( t, y, D ) − Z µ/ ∆1 dtX ( t, y, D ) . Here, similar to Eqs. (50) and (51), we have omitted the first contribution to Eq. (12) leadingto an additional exponentially small factor.The quantities X , and X in Eq. (101) are defined by Eq. (13) where one should put23 = 0 X , ( t, y, D ) = 1 + 1˜ v F y Re D t − ˜ v F y p ˜ v F y − D t + D ,X ( t, y, D ) = ˜ v F yD Re t − p ˜ v F y − D t + D . (102)Note that here the real part is not equal to zero only for t f ( y, D ), where f ( y, D ) isdefined in Eq. (52). It is easily seen that f ( y, D ) < µ/ ∆ [the upper integration limit in thesecond contributions in Eq. (101)] if y satisfies the inequality y < p µ − ∆ ˜ v F ~ ω c . (103)Under the condition p µ − ∆ > ~ ω c , accepted above, this inequality is satisfied withlarge safety margin over the entire range of y giving the major contribution to Eqs. (33)and (98). Because of this, the upper integration limits of the integrals with respect to t inEq. (101), containing the real parts indicated in Eq. (102), should be replaced with f ( y, D ).Taking into account also that D >
1, i.e., D ≫ ˜ v F y , and t − < ˜ v F y /D over the entiredomain of integration, from Eqs. (101) and (102) in the asymptotic limit k B T ≪ µ − ∆one obtains δ T ˜Π , ( y, T, ∆ , µ ) = 4 αD ˜ v F × "Z ∞ dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − − Z µ/ ∆1 dt + 4 αD ˜ v F y Y ( y, T, ∆ , µ ) ,δ T ˜Π ( y, T, ∆ , µ ) = 4 α ˜ v F y D Y ( y, T, ∆ , µ ) , (104)where the following notation is introduced Y ( y, T, ∆ , µ ) ≡ Z f ( y,D )1 dt (cid:20)(cid:16) e t ∆ − µ kBT + 1 (cid:17) − − (cid:21) v F y − D ( t − / . (105)The first contribution to δ T ˜Π , in Eq. (104) is easily calculated4 αD ˜ v F "Z ∞ dt (cid:16) e t ∆ − µ kBT + 1 (cid:17) − − Z µ/ ∆1 dt = 8 α ˜ v F ~ ω c k B T ln (cid:16) e ∆ − µ kBT (cid:17) (cid:16) e µkBT (cid:17) e − µkBT − µ ≈ α ˜ v F ~ ω c e − µ − ∆2 kBT . (106)The low-temperature behavior of the integral Y defined in Eq. (105) is found in theAppendix. According to Eq. (A9) one has Y ( y, T, ∆ , µ ) ≈ − ˜ v F yD e − µ − ∆2 kBT . (107)24ubstituting Eqs. (106) and (107) in Eq. (104), one obtains δ T ˜Π , ( y, T, ∆ , µ ) ≈ α ˜ v F (cid:18) k B T ~ c − D (cid:19) e − µ − ∆2 kBT ∼ − α ∆ ~ ω c e − µ − ∆2 kBT ,δ T ˜Π ( y, T, ∆ , µ ) ≈ − α ˜ v F y D e − µ − ∆2 kBT . (108)We note that according to Eq. (98) δ expl T, l =0 F TE is of the order of ( δ T r TE ) , i.e., ∼ ( δ T ˜Π ) ∼ exp[ − µ − ∆) / (2 k B T )] and, thus, contains an additional exponentially small factor. Be-cause of this, we have δ expl T, l =0 F ( a, T ) ≈ δ expl T, l =0 F TM ( a, T ) . (109)Substituting Eqs. (85), (99), and the first equality in Eq. (108) in the TM term of Eq. (33)with l = 0, one finally finds δ expl T, l =0 F TM ( a, T ) ∼ k B T Q α ∆ a ~ ω c e − µ − ∆2 kBT Z ∞ y dy e − y ( Q + 2 y ) − Q ( Q + 2 y ) e − y . (110)Taking into account Eq. (109), the convergence of the integral which is of the order of Q − ,and substituting the definition of Q given in Eq. (83), the up to an order of magnitudebehavior of δ expl T, l =0 F at low temperature is δ expl T, l =0 F ( a, T ) ∼ k B T ~ c ∆ αa µ e − µ − ∆2 kBT . (111)We recall that this asymptotic behavior is derived under the conditions D >
1, i.e.,∆ > ~ ω c and p µ − ∆ > ~ ω c which are satisfied at sufficiently large separations betweengraphene sheets with nonzero ∆ and µ .It only remains to find the low-temperature behavior of the last contribution to thethermal correction δ expl T, l > F . We note that for l > ,l ( y, , ∆ , µ ) = 0and ˜Π l ( y, , ∆ , µ ) = 0 so that δ expl T, l > F is given by sum of all terms with l > τ ( ζ l = τ l ) only in the lower integrationlimits of all integrals in Eq. (33) and substitute the integrands in the lowest perturbationorder in τ . For the TM mode, this means that one should use in Eq. (33) the second line inEq. (85), Eq. (99), and the first line in Eq. (108). For the TE mode, according to Eq. (91), r TE (0 , y,
0) = 0. Because of this, r TE (i ζ l , y,
0) should be taken in the first perturbation orderin τ as given by Eq. (91), whereas the thermal correction to the TE reflection coefficient isgiven by Eq. (100) and by the second line in Eq. (108).25s a result, for the contribution of the TM mode one obtains δ expl T, l > F TM ( a, T ) ∼ k B T ~ c ∆ αµ a e − µ − ∆2 kBT ∞ X l =1 Z ∞ ζ l y dye y − , (112)where we have used that y giving the major contribution to the integral satisfies the condition y ≪ Q .For the sum of integrals in Eq. (112) we have ∞ X l =1 Z ∞ ζ l y dye y − ∞ X n =1 n Z ∞ nζ l dxx e − x = ∞ X n =1 (cid:20) n e τn − τn e τn ( e τn − + τ n e τn (1 + e τn )( e τn − (cid:21) ∼ τ ∞ X n =1 (cid:20) n + 2 n + 1 n (cid:21) ∼ τ . (113)Substituting this to Eq. (112), we arrive at δ expl T, l > F TM ( a, T ) ∼ ( ~ c ) ∆ αµ a e − µ − ∆2 kBT . (114)The contribution of the TE mode is obtained by substituting Eqs. (91), (100), and (108)in Eq. (33) at low Tδ expl T, l > F TE ( a, T ) ∼ αk B Ta Q D e − µ − ∆2 kBT τ ∞ X l =1 l Z ∞ ζ l dye − y ∼ α p µ − ∆ ( ~ c ) ∆ a e − µ − ∆2 kBT . (115)It is easily seen that the quantities in Eqs. (114) and (115) can be of the same order ofmagnitude. Thus, for the total contribution δ expl T, l > F we obtain δ expl T, l > F ( a, T ) ∼ ( ~ c ) a ∆ αµ + α ~ c p µ − ∆ a ∆ ! e − µ − ∆2 kBT . (116)This result is derived for µ > > < µ is determinedby the TE mode in the implicit contribution given by Eq. (97). Substituting Eq. (97) inEq. (72) one arrives at the Casimir entropy at low temperature S ( a, T ) ∼ α a (4 µ − ∆ ) k B T ( ~ c ) . (117)In the limit of vanishing temperature, the Casimir entropy (117) goes to zero in agreementwith the Nernst heat theorem. 26he results of this section were derived under the conditions k B T ≪ ~ v F a ≪ ~ c a < ∆ , k B T ≪ µ − ∆ . (118)Thus, although the first two expansion parameters in Eq. (75) remain the same, the thirdone is replaced with e − µ − ∆2 kBT ≪ . (119)One more condition used in the derivation of expressions (82) for the polarization tensor is ~ c a < p µ − ∆ . (120)These application conditions are discussed in Sec. VII. VII. CONCLUSIONS AND DISCUSSION
In this paper, we have found the low-temperature behavior of the Casimir free energyand entropy of two real graphene sheets possessing the nonzero energy gap and chemicalpotential. This problem is solved analytically in the framework of the Dirac model. Theresponse of graphene to the electromagnetic field is described on the basis of first principlesof thermal quantum field theory by means of the polarization tensor in (2+1)-dimensionalspace-time. The thermal correction to the Casimir energy of two parallel graphene sheetsat zero temperature is presented as a sum of two contributions. The first of them, calledimplicit, contains the polarization tensor at zero temperature, and the dependence of thiscontribution on temperature is determined by a summation over the Matsubara frequencies.The temperature dependence of the second contribution, called explicit, is determined bythe thermal correction to the polarization tensor. The low-temperature behaviors of bothcontributions were found for different relationships between the energy gap and chemicalpotential of graphene sheets, i.e., for ∆ > µ , ∆ = 2 µ , and ∆ < µ , and turned out to beessentially different.According to the results of Sec. IV, which are repeated here by presenting only thedimensional quantities, the low-temperature behavior of the Casimir free energy and entropyfor graphene sheets with ∆ > µ is eventually determined by the TE mode in an implicitcontribution to the thermal correction δ T F ( a, T ) ∼ − ( k B T ) ( ~ c ) ∆ , S ( a, T ) ∼ k B T ( ~ c ) ∆ , (121)27nd it does not depend on the chemical potential.In Sec. V it is shown that for graphene sheets with ∆ = 2 µ the eventual low-temperaturebehavior of the Casimir free energy and entropy is determined by the TM mode in an explicitcontribution to the thermal correction δ T F ( a, T ) ∼ − k B Ta , S ( a, T ) ∼ k B a , (122)Finally, as shown in Sec. VI, for the case ∆ < µ the low-temperature behavior of theCasimir free energy and entropy is governed by the TE mode in an implicit contribution tothe thermal correction given by δ T F ( a, T ) ∼ − a (4 µ − ∆ )( k B T ) ( ~ c ) , S ( a, T ) ∼ a (4 µ − ∆ ) k B T ( ~ c ) . (123)It is interesting to compare these results with the case of a pristine graphene with ∆ = µ = 0 where [76] δ T F ( a, T ) ∼ ( k B T ) ( ~ c ) ln ak B T ~ c , S ( a, T ) ∼ − k B ( k B T ) ( ~ c ) ln ak B T ~ c . (124)As is seen from the comparison of Eqs. (121)–(123) with Eq. (124), for real graphenesheets there is a nontrivial interplay between the values of ∆ and µ which leads to differentbehaviors of the Casimir energy and entropy with vanishing temperature, especially in thecase ∆ < µ where the polarization tensor at T = 0 depends on µ .From Eqs. (121) and (123) one concludes that the Casimir entropy is positive and vanisheswith vanishing temperature, i.e., for graphene with ∆ > µ and ∆ < µ the Nernst heattheorem is satisfied and, thus, the Lifshitz theory of the Casimir interaction is consistent withthe requirements of thermodynamics (the same holds for a pristine graphene). According toEq. (122), this is, however, not so for graphene with ∆ = 2 µ = 0 where the Casimir entropyat zero temperature is not equal to zero and its value depends on the parameter of a system(volume). As discussed in Sec. V, however, this anomaly is not relevant to any physicalsituation because for real graphene samples the exact equality ∆ = 2 µ is not realizable. Wenote that the real part of the electrical conductivity of graphene as a function of frequencyalso experiences a qualitative change when the energy gap ∆ decreases from ∆ > µ to∆ < µ [60].It should be noted that the asymptotic expressions (121) and (123) are not applicable tographene sheets with too small values of ∆ − µ and 2 µ − ∆, respectively. The point is that28f the values of ∆ and 2 µ are too close to each other the exponentially small parametersin Eqs. (75) and (119) lose their meaning and cannot be used. Taking into account thatthe polarization tensor is a continuous function of ∆ at the point ∆ = 2 µ , the possibilityexists that an apparent discontinuity of the obtained asymptotic formulas at ∆ = 2 µ maybe an artifact of the expansion in small parameters at the crossover region. For a compre-hensive resolution of this question, it would be desirable to find the more exact asymptoticexpressions applicable for the values of 2 µ arbitrarily close to ∆ from the left and fromthe right. In future it is also interesting to investigate the case of two dissimilar graphenesheets with different values of the energy gap and chemical potential. The configuration ofa graphene sheet interacting with an ideal metal plane (it has been known that for two idealmetal planes the Casimir entropy satisfies the Nernst heat theorem [87]) or a plate made ofconventional metallic or dielectric materials.According to Sec. I, theoretical description of the Lifshitz theory using the polarizationtensor of graphene [75] is in good agreement with the experiment on measuring the Casimirinteraction in graphene system [74]. Taking into consideration that the polarization tensor ofgraphene results in two spatially nonlocal, complex dielectric permittivities (the longitudinalone and the transverse one [50]), it may be suggested that a more fundamental theoreticaldescription of the dielectric response of metals admits a similar approach. In applicationto metals, the nonlocal dielectric permittivities of this kind could lead to almost the sameresults, as the dissipative Drude model, for the propagating waves on the mass shell, butdeviate from them significantly for the evanescent fields off the mass shell (in contrast to thenonlocal dielectric functions describing the anomalous skin effect [88]). In such a mannergraphene might point the way for resolution of the Casimir puzzle which remains unresolvedfor already 20 years. Acknowledgments
The work of G.L.K. and V.M.M. was partially supported by the Peter the Great SaintPetersburg Polytechnic University in the framework of the Program “5–100–2020”. Thework of V.M.M. was partially funded by the Russian Foundation for Basic Research, GrantNo. 19-02-00453 A. His work was also partially supported by the Russian GovernmentProgram of Competitive Growth of Kazan Federal University.29 ppendix A
Here, we derive the low-temperature behavior of two integrals used in the main text. Webegin with the integral I (2)00 , defined in Eqs. (51) and (52). To calculate the quantity I (2)00 , inthe case ∆ > µ we introduce the integration variable v = t − I (2)00 , ≈ e − ∆ − µ kBT Z f ( y,D ) − dve − v ∆2 kBT D ( v + 1) [˜ v F y − D v ( v + 2)] / , (A1)where we have omitted the negligibly small quantity ˜ v F y taking into account that thedominant contribution to Eq. (33) is given by y ∼
1. Under this condition f (1 , D ) − ≪ v ≪
1. Then the asymptotic behavior of Eq. (A1) at low T can be estimated as I (2)00 , ∼ e − ∆ − µ kBT Z f (1 ,D ) − dve − v ∆2 kBT D (˜ v F − D v ) / (A2)= D k B T ∆ e − ∆ − µ kBT Z U ( D,T )0 du e − u (cid:16) ˜ v F D − k B T ∆ u (cid:17) / , where u = v ∆ / (2 k B T ) is the integration variable introduced in place of v , and U ( D, T ) ≡ ∆[ f (1 , D ) − / (2 k B T ). In view of the fact that 4 k B T u/ ∆ goes to zero when T vanishes andthe main contribution to the integral is given by u ∼
1, we find I (2)00 , ∼ D ˜ v F k B T ∆ e − ∆ − µ kBT Z ∞ due − u = k B T ˜ v F ∆( ~ ω c ) e − ∆ − µ kBT . (A3)Now we consider the same integral but for graphene with ∆ = 2 µ . For this purpose, weagain begin from Eq. (51), where now ∆ = 2 µ , and substitute there the identity (cid:20) e ( t − kBT + 1 (cid:21) − = ∞ X n =1 ( − n − e − n ( t − ∆2 kBT . (A4)Then, after introducing the integration variable v = t −
1, one obtains instead of Eq. (A2) I (2)00 , ∼ ∞ X n =1 ( − n − Z f (1 ,D ) − dve − nv ∆2 kBT D (˜ v F − D v ) / (A5)= D k B T ∆ ∞ X n =1 ( − n − n Z nU ( D,T )0 du e − u (cid:16) ˜ v F D − k B Tn ∆ u (cid:17) / , where u = nv ∆ / (2 k B T ). For arbitrarily small T this equation can be rearranged as I (2)00 , ∼ D ˜ v F k B T ∆ ∞ X n =1 ( − n − n Z ∞ due − u = k B T ˜ v F ∆( ~ ω c ) ln 2 . (A6)30ow we find the low-temperature behavior of the integral Y defined in Eq. (105). Thepower of exponent in Eq. (105) is negative over the entire integration range. Because of this,one can use the following expansion Y ( y, T, ∆ , µ ) = Z f ( y,D )1 dt " ∞ X n =1 ( − n − e t ∆ − µ kBT ( n − − v F y − D ( t − / = − ∞ X k =1 ( − k − Z f ( y,D )1 dte t ∆ − µ kBT k v F y − D ( t − / . (A7)Now we replace the integration variable t with v = t − y ∼ f ( y, D ) − ≈ ˜ v F y D ≪ . (A8)For this reason, one can neglect by v as compared to unity in the power of exponent and alsoin the denominator of Eq. (A7). In the sum, we can restrict ourselves by only the first termbecause all other terms contain additional exponentially small factors as compared with it.The result is Y ( y, T, ∆ , µ ) ≈ − e − µ − ∆2 kBT Z f ( y,D ) − dv (˜ v F y − D v ) / ≈ − ˜ v F yD e − µ − ∆2 kBT , (A9)where we have used the condition (A8). [1] H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Ned.Akad. Wet. B , 793 (1948).[2] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Advances in theCasimir Effect (Oxford University Press, Oxford, 2015).[3] T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Casimir forces between compact objects:The scalar case, Phys. Rev. D , 025005 (2008).[4] S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar, Scattering theory approach toelectromagnetic Casimir forces, Phys. Rev. D , 085021 (2009).[5] R. Passante, L. Rizzuto, S. Spagnolo, S. Tanaka, and T. Y. Petrosky, Harmonic oscillatormodel for the atom-surface Casimir-Polder interaction energy, Phys. Rev. A , 062109 (2012).[6] M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University Press, Cam-bridge, 2012).
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