The low-temperature expansion of the Casimir-Polder free energy of an atom with graphene
TThe low-temperature expansion of the Casimir-Polder free energy of anatom with graphene
Nail Khusnutdinov ∗ CMCC, UFABC, 09210-170 Santo Andr´e, SP, Braziland Regional Scientific and Educational Mathematical Center of Kazan Federal University,Kremlevskaya 18, Kazan, 420008, Russia
Natalia Emelianova † CMCC, UFABC, 09210-170 Santo Andr´e, SP, Brazil
February 7, 2021
Abstract
We consider the low-temperature expansion of the Casimir-Polder free energy for an atom andgraphene by using the Poisson representation of the free energy. We extend our previous analysis onthe different relations between chemical potential µ and mass gap parameter m . The key role playsthe dependence of graphene conductivities on the µ and m . For simplicity, we made the manifestcalculations for zero values of the Fermi velocity. For µ > m the thermal correction ∼ T and for µ < m we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction ∼ T . In the case of exact equality µ = m the correction ∼ T . This point is unstable and the systemfalls to the regime with µ > m or µ < m . The analytical calculations are illustrated by numericalevaluations for the Hydrogen atom/graphene system. The Casimir [1] and Casimir-Polder [2] dispersion forces play an important role in different phenomena[3, 4]. The Casimir-Polder force is usually referred to as the van der Waals force on large distances betweenmicro-particles and macro-objects when the retardation of interaction is taken into account. The Casimir-Polder force essentially depends on the material of the macro-objects, its dimension, shapes, conductivity,and temperature [3, 4], and it is important for the interaction of graphene with micro-particles [5–9]. TheCasimir-Polder force and torque for anisotropic molecules have been the subject of investigations in therecent years [10–14].The thermal corrections to the Casimir-Polder interaction for micro-particle/graphene were consideredin Refs. [15–21]. In the last few years, much attention is given to the low-temperature expansion of theCasimir-Polder free energy for the atom/graphene system [7, 19–21]. In the case of an atom/ideal plane,the low-temperature correction to the Casimir-Polder free energy is proportional to the fourth degree oftemperature ∼ T . As opposed to the ideal case, the conductivity of graphene depends on the chemicalpotential and temperature, and it has temporal and spatial dispersion [22–25]. The low-temperatureexpansion depends on the relations between these macro-parameters. ∗ email: [email protected] † email: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b t was shown in Ref. [19] that the low-temperature expansion reveals the unusual quadratic ∼ T behaviour. Next detail considerations [20, 21] showed a more rich picture of low-temperature expansiondepending on the relation between chemical potential µ and mass gap parameter m of the Dirac electron.For µ > m the same quadratic behaviour was confirmed but in the case, µ < m the ∼ T dependence wasobtained. In the case of the very specific exact relation µ = m the linear ∼ T dependence was observed.To obtain these results the authors of Refs. [20, 21] made the sophisticated treatment of the Matsubaraseries.In the present paper, we extend our analysis made in Ref. [19] in the framework of the Poissonrepresentation of the Matsubara series to all relations between µ and m and confirm results by numericalanalysis. In Ref. [19] we considered conductivity of graphene with cutting scattering rate as in Refs.[26, 27] where the Kubo approach was used. In this case, the conductivity has a constant value at zerofrequencies which depends on the scattering rate parameter γ . It leads to the low-temperature dependence ∼ T for any relation between chemical potential µ and mass gap m . In the framework of the polarizationtensor approach [23–25], there is no scattering parameter and the behaviour of the conductivity at zerofrequency strongly depends on the relation between µ and m . We show that for µ > m the temperaturecorrections to the free energy ∼ T and for µ < m we obtain correction ∼ T and in the point µ = m thelinear dependence ∼ T appears. Therefore, we confirm expansions obtained in Ref. [20] in the frameworkof our approach.Throughout the paper the units (cid:126) = c = k B = 1 are used. Taking into account the Poisson summation formula (see details in Ref. [19]), the free energy may berepresented in the following form F tm | te E ∞ CP = − ∞ (cid:88) l =0 (cid:48) (cid:90) ∞ z e − z dz (cid:90) dx cos (cid:18) zxlaT (cid:19) α ( λ ) α (0) { x − | x } r tm | te , (1)normalized to the E ∞ CP = − α (0) / πa – the Casimir-Polder ( CP ) energy for an ideal plane/atom at largedistance a . Here, the prime means factor 1 / l = 0 and we have to use λ = zxa , k = za √ − x forimaginary frequency ω = i λ and wave-vector k in conductivities η tm = 2 πσ tm and η te = 2 πσ te . The α isthe polarizability of atom or molecule at the imaginary frequency, and the refraction coefficients of TEand TM modes are r te = −
11 + xη te , r tm = 11 + xη tm . (2)The form of the zero terms, l = 0, in (1) coincides exactly with that obtained for zero temperaturesbut, in general, with temperature and chemical potential dependence through the conductivities η tm | te = η tm | te ( λ, k, µ, m, T ). We extract the zero, l = 0, term F = F tm + F te = F + ∆ F , (3)and consider the low-temperature expansion for ∆ F and F separately. We extract the temperaturecontribution from F . Then, F = F T =0 + ∆ F + ∆ F , (4)where ∆ F = F − F T =0 . Therefore, the total temperature correction, ∆ T F , consists of two parts,∆ T F = ∆ F + ∆ F . (5)In general (for the graphene case, for example), the F T =0 depends on the chemical potential.2he expansion crucially depends on the behaviour of the conductivities at zero frequencies. In Ref.[19] we considered in detail the different models of conductivities with a constant value of conductivityat zero frequencies. In particular, we have taken into account the graphen’s conductivity with finitescattering factor γ , which means that the η tm | te | ω → = η tm | te | ω = γ . As a result, the free energy has themain low-temperature term ∼ T for any relation between µ, m and T . With zero scattering factor, wehave to consider this expansion more carefully.In the framework of the polarization tensor approach [22], the conductivities of the TM and TE modesread [23–25] η i = η i + ∆ η i , ∆ η i = η gr (cid:90) ∞ m dyf i ( y )Ξ( y, µ, T ) , (6a)where η te η gr = 4 mπλ (cid:18) k F − m mk F arctan (cid:18) k F m (cid:19)(cid:19) , η tm η te = λ k F ,f te ( y ) = 8 πλ Re (4 m + q )( q k F + 4 m k v F ) − q k F λ r ( q k F + 4 m k v F + qλr ) , q = λ − y,f tm ( y ) = 8 π Re q ( q + k v F + 4 m ) − λrr ( r + qλ ) , Ξ = 1 e y + µT + 1 + 1 e y − µT + 1 ,k F = (cid:113) λ + v F k , r = (cid:113) k F ( q + k v F ) + 4 m k v F , (6b)and η gr = πσ gr c = πe (cid:126) c = 0 . σ gr = e (cid:126) being the graphene universal conductivity.Let us consider, for simplicity, the conductivity in the zero approximation over the Fermi velocity v F = 1 / (cid:28)
1. In this approximation, we obtain more simple expressions ( x = tm , te ) η x η gr = 4 mπλ (cid:18) λ − m mλ arctan (cid:18) λ m (cid:19)(cid:19) , f x ( y ) = 16 (cid:0) m + y (cid:1) πλ ( λ + 4 y ) , (7)and we observe that the conductivities have no dependence on k which should be the case because theFermi velocity and wave-vector come in the single combination kv F . ∆ F The sum with l ≥
1, the ∆ F , may be represented in the following form [19]:∆ FE ∞ CP = 83 Re ∞ (cid:88) l =1 (cid:90) ∞ dze i Λ l z ( Y tm + Y te ) , (8)where Y tm ( z ) = α ( λ ) α (0) (cid:90) ∞ z e − s s (2 s − z ) s + z/η tm ds,Y te ( z ) = α ( λ ) α (0) (cid:90) ∞ z e − s z z + s/η te ds, (9)and Λ l = l/aT, λ = z/a , and k = √ s − z /a . This representation is suitable for T → l → ∞ . 3n the case v F →
0, the conductivities do not depend on k , and therefore we can calculate integralover s : Y tm ( z ) = α (cid:0) za (cid:1) e − z α (0)2 η tm (cid:8) η tm (cid:0) η tm + (cid:0) η tm − η tm + 2 (cid:1) z + η tm (2 η tm − z (cid:1) − (cid:0) η tm − (cid:1) z e ( η − tm ) z Ei (cid:2) − z (cid:0) η − tm (cid:1)(cid:3)(cid:111) ,Y te ( z ) = − α (cid:0) za (cid:1) α (0) η te z e η te z Ei [ − z (1 + η te )] , (10)where Ei[ x ] is the exponential logarithm function. The function Ei[ x ] has the following representation asa series Ei( x ) = γ E + ln( − x ) + (cid:88) n ≥ x n n · n ! , (11)that contains polynomials as well as logarithmic contributions.Then we make expansion over z , Y x = (cid:88) n ≥ A x n z n + (cid:88) n ≥ B x n z n ln z, (12)and use the Lemmas Erd´elyi (see Ref. [28, Eqs. 1.13 and 1.35] and Ref. [19]) to calculate asymptoticΛ l → ∞ for integral over z in Eq. (8) for each term of series. The manifest form of the coefficients dependson the specific model of conductivity. We obtain the series (12) in which we have to make replacements z n → n ! e i π ( n +1) Λ − n − l ,z n ln z → n ! e i π ( n +1) (cid:18) i π ψ ( n + 1) − ln Λ l (cid:19) Λ − n − l , where ψ ( x ) is the digamma function, and then we take the real part (see Eq. (8)) and obtain the followingreplacements z n +1 → ( − n +1 (2 n + 1)!Λ − n − l ,z n → ,z n +1 ln z → ( − n +1 (2 n + 1)!( ψ ( n + 1) − ln Λ l )Λ − n − l ,z n ln z → ( − n +1 (2 n )! π − n − l . (13)Then we make a summation over l ≥ FE ∞ CP = 83 ( X tm + X te ) , (14)where X x = (cid:88) n ≥ A x n +1 ( − n +1 (2 n + 1)! ζ R (2 n + 2)( aT ) n +2 + (cid:88) n ≥ B x n ( − n +1 (2 n )! π ζ R (2 n + 1)( aT ) n +1 + (cid:88) n ≥ B x n +1 ( − n +1 (2 n + 1)!([ ψ ( n + 1) + ln( aT )] ζ R (2 n + 2) − ζ (cid:48) R (2 n + 2))( aT ) n +2 , These lemmas are sometimes called etalon integrals in the asymptotic methods of the stationary phase ζ R is the Riemann zeta function. We observe that the polynomial contributions come from odd A x n +1 and B x n . The logarithmic contributions come from odd B i n +1 . The main contribution reads∆ F x E ∞ CP = − π (cid:20) A x − B x (cid:18) γ E + 6 ζ (cid:48) R (2) π (cid:19)(cid:21) ( aT ) + 8 πζ R (3)3 B x ( aT ) + 8 π (cid:20) A x + B x (cid:18) − γ E − ζ (cid:48) R (4) π (cid:19)(cid:21) ( aT ) − πζ R (5) B x ( aT ) − π (cid:20) A x + B x (cid:18) − γ E − ζ (cid:48) R (6) π (cid:19)(cid:21) ( aT ) + . . . + ln( aT ) (cid:26) − π B x ( aT ) + 8 π B x ( aT ) − π B x ( aT ) + . . . (cid:27) , (15)where γ E is the Euler constant. Note, that in general the coefficients A x n and B x n depend on m, µ and T .For the constant conductivity case B x = B x = 0 and A te = 0 , A tm = − / (2 η tm ). Therefore, the maincontribution comes from TM mode and reads [19]∆ FE ∞ CP = − (2 πaT ) A tm = 2 π ( aT ) η tm . (16)For graphene case, the conductivities are expanded in the following series over z : η x η gr = 1 z ( b + z b + z b + . . . ) , (17)where the coefficients b n are functions of m, µ and T : b = 4 aπ (cid:90) ∞ m dyy ( m + y )Ξ ,b = 43 πma − πa (cid:90) ∞ m dyy ( m + y )Ξ ,b = − πm a + 14 πa (cid:90) ∞ m dyy ( m + y )Ξ . (18)The zero term b crucially depends on m and µ for low temperatures:I. µ > m, T (cid:28) µ − m b = (cid:90) µm g ( y ) dy + π T g (cid:48) ( µ ) + 7 π T g (cid:48)(cid:48)(cid:48) ( µ ) + O ( e − µ − mT ) . (19a)II. µ = m, T (cid:28) mb = T ln 2 g ( m ) + π T g (cid:48) ( m ) + 34 ζ R (3) T g (cid:48)(cid:48) ( m ) + 7 π T g (cid:48)(cid:48)(cid:48) ( m ) + O ( e − mT ) , (19b)III. µ < m, T (cid:28) m − µ b = O ( e − m − µT ) . (19c)Here, g ( y ) = aπ m + y y . These asymptotic are illustrated by numerical evaluation in Figure 1.In the first case (19a) with µ > m , the expansion b over T starts from the constant term (see Figure 1,right panel) and we obtain the following non-zero coefficients A tm = 13 , A te = − ,A tm = α (cid:48)(cid:48) (0)6 a α (0) + 23 η gr b − η gr b − , A te = − α (cid:48)(cid:48) (0)2 a α (0) − η gr b − η gr b − ,B tm = − η gr b , B tm = − α (cid:48)(cid:48) (0)2 a α (0) η gr b + b − η gr b + 2 η gr b . (20)5
100 200 300 40001234 · − T (K) b µ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 eV · − T (K) b µ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eV Figure 1: The functions b for a = 100 nm, m = 0 . µ . For µ < m the function b ∼ µ = m it is linear and for µ > m it starts fromconstant values in agreement with Eqs. (19).The main contribution to the b comes from the first term of expansion in Eq. (19a): b = 4 aπ µ − m µ . (21)In the last case (19c) with µ < m , the b is exponentially small (see Figure 1, left panel) and we set itzero and the conductivities are expanded as the following η x η gr = z ( b + z b + . . . ) , (22)and the firsts non-zero coefficients read A tm = 115 η gr b ( η gr b + 2) , A te = 2 η gr b , B te = − η gr b . (23)For the specific case (19b), when µ = m , we consider the main contribution for T → b = 0 as in the case (19c). Therefore, for all cases, we have the following expansion up to T :∆ FE ∞ CP = − π
135 ( aT ) + 8 π ζ R (3) µaη gr ( µ − m ) ( aT ) + . . . , ( µ > m, T (cid:28) µ − m ) , ∆ FE ∞ CP = 128 η gr ζ R (5)3 am ( aT ) + . . . , ( µ = m, T (cid:28) m ) , ∆ FE ∞ CP = 128 η gr ζ R (5)3 am ( aT ) + . . . , ( µ < m, T (cid:28) m − µ ) . (24) F The zero term reads F tm | te E ∞ CP = 43 (cid:90) ∞ z e − z dz (cid:90) dx α ( zxa ) α (0) (cid:40) − x xη tm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x xη te (cid:41) . (25)The conductivities (6) depend on the temperature and ∆ η x have the following low-temperature expansions[19] 6. µ > m, T (cid:28) µ − m ∆ η x η gr = (cid:90) µm f x ( y ) dy + π T f (cid:48) x ( µ ) + 7 π T f (cid:48)(cid:48)(cid:48) x ( µ ) + O ( e − µ − mT ) . (26a)II. µ = m, T (cid:28) m ∆ η x η gr = T ln 2 f x ( m ) + π T f (cid:48) x ( m ) + 34 ζ R (3) T f (cid:48)(cid:48) x ( m ) + 7 π T f (cid:48)(cid:48)(cid:48) x ( m ) + O ( e − mT ) , (26b)III. µ < m, T (cid:28) m − µ ∆ η x η gr = O ( e − m − µT ) . (26c)Taking these expansions into account we obtain the following low-temperature corrections to zero term∆ F x E ∞ CP = ( aT ) G x , ( µ > m, T (cid:28) µ − m ) , ∆ F x E ∞ CP = ( aT ) H x , ( µ = m, T (cid:28) m ) , ∆ F x E ∞ CP = 0 , ( µ < m, T (cid:28) µ − m ) , (27)where G tm | te = 2 π a (cid:90) ∞ z e − z dz (cid:90) dx α ( zxa ) α (0) (cid:40) (2 − x ) xf (cid:48) tm ( µ )( x + η tm + ∆ η tm ) (cid:12)(cid:12)(cid:12)(cid:12) x f (cid:48) te ( µ )(1 + x (cid:0) η te + ∆ η te (cid:1) ) (cid:41) ,H tm | te = 4 ln 23 a (cid:90) ∞ z e − z dz (cid:90) dx α ( zxa ) α (0) (cid:26) (2 − x ) xf tm ( m )( x + η tm ) (cid:12)(cid:12)(cid:12)(cid:12) x f te ( m )(1 + xη te ) (cid:27) . (28)The functions f x are given by Eq. (6) and∆ η x = η gr (cid:90) µm f x ( y ) dy. (29)Therefore, taking into account (24) we obtain the low-temperature expansion of the free energy∆ T FE ∞ CP = ( aT ) ( G tm + G te ) , ( µ > m, T (cid:28) µ − m ) , (30a)∆ T FE ∞ CP = ( aT )( H tm + H te ) , ( µ = m, T (cid:28) m ) , (30b)∆ T FE ∞ CP = 128 η gr ζ R (5)3 am ( aT ) + . . . . ( µ < m, T (cid:28) m − µ ) . (30c) We evaluated numerically the total temperature correction (5) by using the expression for the free energyin the form of Matsubara sum. We considered the Hydrogen atom at distance a = 100 nm from thegraphene sheet. The polarizability of the Hydrogen atom in the single oscillator approximation may befound in Ref. [8], for example. The graphene conductivities are given by Eqs. (6). We used the Fermivelocity v F = 1 /
300 and the mass gap m = 0 . µ dependence we madecalculations for the value of µ close to m = 0 . E ∞ CP = − α (0) / πa – the CP energy for an ideal plane/atom at large distance a .7 . . . . . . . . . . · − µ (eV) F T =0 / E ∞ CP Figure 2: The zero temperature free energy, F T =0 , as a function of chemical potential µ .The zero-temperature free energy, F T =0 , depends on the chemical potential µ and this dependence isshown in Figure 2. For µ ≤ m it has the constant value, and it grows up starting with mass gap µ > m .We proceed now to the consideration of the temperature correction, ∆ T F to the free energy. First ofall let us consider the functions G x which define low temperature expansion for µ > m case (30a). Theyare plotted in Figure 3. We observe that they are negative and contribution from the TE mode is 100times smaller. . .
11 0 .
12 0 .
13 0 .
14 0 . − − − − − µ (eV) G tm . .
11 0 .
12 0 .
13 0 .
14 0 . − − − − − · − µ (eV) G te Figure 3: The functions G x (30a) for different values of the chemical potential µ > m .The numerical evaluation of H x ( a ) for a = 100 nm gives the following values H tm = 3 · and H te = 0 .
18. Again the main contribution comes from TM mode. The value of H tm strongly depends onthe value of the Fermi velocity. For v F → H tm → ∞ .The temperature contribution for µ ≥ m is shown in Figure 4. We observe that for low temperaturesthe free energy has the form of parabola, ∼ G tm ( aT ) , in agreement with (30) with negative parameter G tm (see Figure 3).The closer µ > m to m , the smaller domain of temperature T where this approximation valid, and8
50 100 150 20000 . . · − T (K)∆ T F / E ∞ CP · − T (K)∆ T F / E ∞ CP µ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eV Figure 4: The temperature contribution to the free energy in the different intervals of temperatures and µ ≥ m .the greater value of the parameter of parabola G tm , in agreement with Figure 3. If µ = m this domainbecomes zero and the free energy changes drastically its form. If µ →
0, the part of the curve which isout of this domain (the vertical part of the green curve, for example) goes to free energy for this veryspecial position with µ = m (black curve). Therefore, for any infinitely small difference µ − m (cid:54) = 0 thederivative of free energy with respect temperature T is zero for T = 0 and the Nernst theorem is valid.The experimental realization of the exact equality µ = m can not be realized, and we conclude that theNernst theorem is valid for this system.The temperature contribution for µ ≤ m is shown in Figure 5. We observe the completely differentdependence of the energy on the µ . . . · − T (K)∆ T F / E ∞ CP · − T (K)∆ T F / E ∞ CP µ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 . eVµ = 0 eV Figure 5: The temperature contribution to the free energy in the different intervals of temperatures and µ ≤ m .For zero chemical potential (brown curve), the temperature correction is, in fact, zero for the largedomain of temperatures. The closer µ to m , the smaller domain in which temperature correction is zero.According to (30b) ∆ T FE ∞ CP = 128 η gr ζ R (5)3 am ( aT ) = 10 − T ( K ) , (31)9n this domain.The Figure 6 shows the temperature contribution to the free energy as a function of chemical potential.The function has a very sharp form with the maximum for µ = m with a different slope at the left andthe right of this point. .
04 0 .
06 0 .
08 0 . .
12 0 . . . . . · − µ (eV)∆ T F / E ∞ CP T = 60K T = 30K Figure 6: The temperature contribution to the free energy as a function of the chemical potential fordistance between Hydrogen atom and graphene a = 100nm.The point µ = m looks like phase transition point between different regimes from ∆ T F ∼ T to∆ T F ∼ T . In fact, it is an unstable point – infinitely small deviation µ from m changes regime.From the Figures 4,5 and relations (30) we observe the different signs of the entropy, S , for µ < m and µ > m . The entropy is the negative derivative of the free energy with respect of temperature.Therefore, S µ
0. The negative entropy of the dispersion forces has already been observed in Refs. [15, 29]for plain and spherical configurations in the framework of plasma model and also was discussed recentlyin Ref. [30].
We considered the low-temperature correction to the Casimir-Polder free energy for atom/graphene systemby using the Poisson representation of the free energy, which is more suitable for low-temperature analysis.The analysis is naturally broken into three different regions: i) µ > m , ii) µ = m and iii) µ < m forchemical potential. This division is the consequence of the same regions for the conductivity of graphene(see Eq. (26)). The conductivities have completely different expansion in these regions. It starts fromthe constant in the first region, linear on the temperature in the second one, and exponential small in thethird region.The free energy may be divided into the two parts (3). The first one, F , has the form of the freeenergy at zero temperature but with µ, m and T dependence via the conductivity dependence on theseparameters. The main contribution in the low-temperature expansion in the first (i) and second (ii)regimes comes from this first term, and it is quadratic and linear on the temperature correspondingly (seeEq. (30)). In the third (iii) regime, the main contribution ∼ T comes from the rest part ∆ F .10 cknowledgments We are grateful to Dmitri Vassilevich, Galina Klimchitskaya, and Vladimir Mostepanenko for fruitfuldiscussions. The NK was supported in part by the grants 2019/10719-9, 2016/03319-6 of S˜ao PauloResearch Foundation (FAPESP) and by the Russian Foundation for Basic Research Grant No. 19-02-00496-a.
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