Low-temperature behavior of the Casimir free energy and entropy of metallic films
aa r X i v : . [ qu a n t - ph ] F e b Low-temperature behavior of the Casimir free energy andentropy of metallic films
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
We derive an analytic behavior of the Casimir free energy, entropy and pressure of metallicfilms in vacuum at low temperature. It is shown that this behavior differs significantly dependingon whether the plasma or the Drude model is used to describe the dielectric properties of filmmetal. For metallic films described by the lossless plasma model the thermal corrections to theCasimir energy and pressure drop to zero exponentially fast with increasing film thickness. Thereis no classical limit in this case. The Casimir entropy satisfies the Nernst heat theorem. Formetallic films with perfect crystal lattices described by the Drude model the Casimir entropy atzero temperature takes a nonzero value depending on the parameters of a film, i.e., the Nernstheat theorem is violated. The Casimir entropy at zero temperature is positive, as opposed to thecase of two metallic plates separated with a vacuum gap, where it is negative if the Drude modelis used. Possible applications of the obtained results in investigations of stability of thin films arediscussed.
PACS numbers: 12.20.Ds, 42.50.Ct, 78.20.-e . INTRODUCTION During the last few years the van der Waals and Casimir interactions have attractedwidespread interest due to important role they play in many physical phenomena [1, 2]. Inmost cases, however, the emphasis has been made on the forces acting between two closelyspaced bodies, be it two atoms or molecules, an atom or a molecule and a macroscopicsurface, or two macroscopic surfaces. It is common knowledge that the van der Waals andCasimir forces are caused by the zero-point and thermal fluctuations of the electromagneticfield and are described by the Lifshitz theory of dispersion forces [3]. At the moment theseforces are actively investigated not only theoretically, but also experimentally (see Refs.[4, 5] for a review) and are used in technological applications [6–8].Another important role of dispersion interactions is that they contribute to the freeenergy of free-standing material films and films deposited on some material plates. Theformulation of this problem goes back to Derjaguin who took into account the dispersion-force contribution in studies of stability of thin films and introduced the concept of disjoiningpressure (see Refs. [9, 10] for a review). During a few decades this contribution to the freeenergy, which depends on the film thickness, was estimated using the power-type force lawand the Hamaker constant.In the present state of the art, the question of the Casimir energy for a free-standing orsandwiched between two dielectric plates metallic film was raised in Ref. [11]. Then, theCasimir energy of a free-standing in vacuum metallic film was considered in Refs. [12, 13]. Indoing so, the dielectric properties of metal were described by either the Drude or the plasmamodel. When employing the plasma model, the Lifshitz theory at nonzero temperaturehas been used in calculations. However, all calculations employing the Drude model havebeen performed at zero temperature. This did not allow to reveal significant differencesin theoretical results for the free energy of metallic films predicted by the Lifshitz theorycombined with either the Drude or the plasma model.Full investigation of the Casimir free energy and pressure for metallic films in the frame-work of the Lifshitz theory at nonzero temperature was performed in Refs. [14–16]. Thecases of a free-standing or sandwiched between two dielectric plates [14], deposited on ametal plate [15] or made of magnetic metal [16] metallic films have been considered. Thedielectric properties of metals were described by using the optical data for the complex index2f refraction extrapolated to zero frequency by the Drude or plasma models. It was shownthat magnitudes of the free energy of metallic films of less than 150 nm thickness differby up to a factor of 1000 depending on the calculation approach used [14–16]. So greatdifference is explained by the fact that the Casimir free energy of metallic films drops tozero exponentially fast when the plasma model is used for extrapolation and goes to theclassical limit when the optical data are extrapolated by the Drude model [14–16]. Thislimit is already reached for the film of 150 nm thickness.Here we note that although routinely it is quite natural to use the Drude model forextrapolation of the optical data to lower frequencies because it takes into account therelaxation properties of conduction electrons, there are also strong reasons for using thelossless plasma model for this purpose in the case of fluctuating fields. The point is that themeasurement data of all precise experiments on measuring the Casimir interaction betweentwo material bodies separated with a vacuum gap exclude theoretical predictions of theLifshitz theory combined with the Drude model and are consistent with predictions of thesame theory using the plasma model [17–23]. For the gap width below 1 µ m, used inthese experiments, the variation in theoretical predictions of both approaches is below a fewpercent. Recently, however, the differential force measurement scheme has been proposed[24–26], where this variation is by up to a factor 1000. The results of one of these experiments,already performed [27, 28], exclude with certainty the predictions of the Drude model andare consistent with the plasma model. Basing on this, it was hypothesized that reaction ofa physical system to real and fluctuating electromagnetic fields (having a nonzero and zeroexpectation values, respectively) might be different [16, 29].On theoretical side, it was shown [30, 31] that for two metallic plates, separated by morethan 6 µ m distance, the classical statistical physics predicts the same Casimir force as doesthe Lifshitz theory combined with the plasma model. By contrast, for metals with perfectcrystal lattices the Lifshitz theory was shown to violate the third low of thermodynamics(the Nernst heat theorem) when the Drude model is used [32–36]. In this respect, onemay guess that even at separations exceeding 6 µ m, where the major contribution to theCasimir force between two parallel plates becomes classical, the quantum effects still remainimportant and make the classical treatment inapplicable.In view of the above problem, which is often called “the Casimir puzzle”, it is desirable topresent additional arguments regarding an applicability of the Drude and plasma models in3alculations of the Casimir free energy of metallic films. Here, the calculation results differgreatly, and the subject is not of only an academic character because the obtained valuesshould be taken into account in the conditions of film stability.In this paper, we derive the asymptotic expressions at low temperature for thermal cor-rections to the Casimir free energy and pressure of metallic films described by the plasmamodel. The asymptotic behavior of the Casimir entropy is also obtained. Unlike the familiarcase of two parallel plates separated with a gap, all these quantities decrease exponentiallyfast with increasing film thickness and do not have the classical limit by depending on ~ atarbitrarily large film thicknesses. It is shown that the Casimir entropy of a film preservesthe positive values and, in the limiting case of zero temperature, goes to zero. Thus, it isproved that the Casimir entropy of metallic films described by the plasma model satisfiesthe Nernst heat theorem, i.e., the Lifshitz theory is thermodynamically consistent.Then, the low-temperature behavior of the Casimir free energy and entropy for metallicfilms described by the Drude model is considered. We show that in the limiting case of zerotemperature the Casimir entropy goes to a positive value depending on the parameters of afilm. Therefore, the Nernst heat theorem is violated [38, 39]. Furthermore, it is demonstratedthat in this case the Casimir free energy does not go to zero in the limiting case of idealmetal film, which is in contradiction to the fact that electromagnetic oscillations cannotpenetrate in an interior of ideal metal. Thus, the description of a film metal by the Drudemodel in the Lifshitz theory results in violation of basic thermodynamic demands. Becauseof this, the dispersion-force contribution to the free energy of metallic films might need areconsideration taking into account that the low-frequency behavior of the film metal isdescribed by the plasma model.The paper is organized as follows. In Sec. II, we present general formalism and derive thelow-temperature behavior of the Casimir free energy, pressure and entropy for metallic filmsdescribed by the plasma model. In Sec. III, we consider the low-temperature behavior of theCasimir free energy and entropy of metallic films with perfect crystal lattices described bythe Drude model and demonstrate violation of the Nernst heat theorem. Section IV containsour conclusions and discussion. In Appendix, some details of the mathematical derivationsare presented. 4 I. METALS DESCRIBED BY THE PLASMA MODEL
The free energy per unit area of a free-standing metallic film of thickness a in vacuum attemperature T in thermal equilibrium with an environment is given by the Lifshitz formula[2, 3] F ( a, T ) = k B T π ∞ X l =0 ′ Z ∞ k ⊥ dk ⊥ (1) × X α ln (cid:2) − r α ( iξ l , k ⊥ ) e − ak ( iξ l ,k ⊥ ) (cid:3) . Here, k B is the Boltzmann constant, k ⊥ is the magnitude of the projection of the wave vectoron the film plane, ξ l = 2 πk B T l/ ~ , l = 0 , , , . . . are the Matsubara frequencies, the primeon the summation sign multiplies the term with l = 0 by 1/2, and k ( iξ l , k ⊥ ) = r k ⊥ + ε l ξ l c , (2)where ε l ≡ ε ( iξ l ) is the frequency-dependent dielectric permittivity of film metal calculatedat the pure imaginary Matsubara frequencies.The reflection coefficients for two independent polarizations of the electromagnetic field,transverse magnetic ( α = TM) and transverse electric ( α = TE), are given by r TM ( iξ l , k ⊥ ) = k ( iξ l , k ⊥ ) − ε l q ( iξ l , k ⊥ ) k ( iξ l , k ⊥ ) + ε l q ( iξ l , k ⊥ ) ,r TE ( iξ l , k ⊥ ) = k ( iξ l , k ⊥ ) − q ( iξ l , k ⊥ ) k ( iξ l , k ⊥ ) + q ( iξ l , k ⊥ ) , (3)where q ( iξ l , k ⊥ ) = r k ⊥ + ξ l c . (4)Equation (1) is obtained [14] from the standard Lifshitz formula for a three-layer system[40–42], where the metallic plate is sandwiched between two vacuum semispaces. Note thatthe reflection coefficients (3) have the opposite sign, as compared to the case of two platesseparated by the vacuum gap [2]. The reason is that here an incident wave inside thefilm material goes to its boundary plane with a vacuum, and not from the vacuum gap tothe material boundary. Another distinctive feature of Eq. (1) from the standard Lifshitzformula is that here the dielectric permittivity of metal enters the power of the exponent [inthe standard case this exponent contains the quantity q defined in Eq. (4)]. This makes the5roperties of the free energy (1) quite different from those in the case of two parallel platesseparated by a vacuum gap.It is convenient to introduce the dimensionless integration variable y = 2 aq ( iξ l , k ⊥ ) . (5)Using the characteristic frequency ω c ≡ c/ (2 a ), we also pass on the dimensionless Matsubarafrequencies ζ l = ξ l ω c = 4 π k B T a ~ c l ≡ τ l. (6)Then, the Casimir free energy (1) takes the form F ( a, T ) = k B T πa ∞ X l =0 ′ Z ∞ ζ l y dy (7) × X α ln h − r α ( iζ l , y ) e − √ y +( ε l − ζ l i . In terms of the quantities (5) and (6), the reflection coefficients (3) are given by r TM ( iζ l , y ) = p y + ( ε l − ζ l − ε l y p y + ( ε l − ζ l + ε l y ,r TE ( iζ l , y ) = p y + ( ε l − ζ l − y p y + ( ε l − ζ l + y . (8)Now we assume that at the imaginary Matsubara frequencies the film metal is describedby the lossless plasma model ε l,p = 1 + ω p ξ l , (9)where ω p is the plasma frequency. In terms of dimensionless frequencies (6), the dielectricpermittivity (9) takes the form ε l,p = 1 + ˜ ω p ζ l , ˜ ω p ≡ ω p ω c = 2 aω p c . (10)Substituting Eq. (10) in Eq. (8), one obtains the reflection coefficients in the case whenthe plasma model is used r TM ,p ( iζ l , y ) = ζ l ( p y + ˜ ω p − y ) − ˜ ω p yζ l ( p y + ˜ ω p + y ) + ˜ ω p y ,r TE ,p ( iζ l , y ) = r TE ,p ( y ) = p y + ˜ ω p − y p y + ˜ ω p + y . (11)6or the film described by the plasma model, it is convenient to rewrite the Casimir freeenergy (7) as F p ( a, T ) = k B T πa ∞ X l =0 ′ Φ( ζ l ) = k B T πa ∞ X l =0 ′ [Φ TM ( ζ l ) + Φ TE ( ζ l )] , (12)where Φ TM(TE) ( x ) = Z ∞ x y dy ln h − r ,p ( ix, y ) e − √ y +˜ ω p i . (13)It is well known that the Casimir free energy can be presented in the form F p ( a, T ) = E p ( a, T ) + ∆ T F p ( a, T ) , (14)where the Casimir energy per unit area at zero temperature is given by [2, 3] E p ( a, T ) = ~ c π a Z ∞ dζ Φ( ζ ) (15)and ∆ T F p is the thermal correction to it.Applying the Abel-Plana formula to Eq. (12) and taking into account that ζ l = τ l , onearrives at ∆ T F p ( a, T ) = i k B T πa Z ∞ dt Φ( iτ t ) − Φ( − iτ t ) e πt − . (16)It is evident that the low-temperature behavior of the Casimir free energy of thin metallicfilms can be found from the perturbation expansion of Eq. (16) under the condition τ t ≪ x due to the second equality in Eq. (11). This mean thatthe total dependence of Φ TE ( x ) on x is determined by only the lower integration limit inEq. (13).Now we expand the function Φ TE ( x ) in a series in powers of x . The first term in thisseries is Φ TE (0) = Z ∞ ydy ln[1 − r ,p ( y ) e − √ y +˜ ω p ] . (17)This is a converging integral, which does not contribute to the difference∆Φ TE ≡ Φ TE ( iτ t ) − Φ TE ( − iτ t ) , (18)7ntering Eq. (16).Then, calculating the first and second derivatives of Eq. (13), one findsΦ ′ TE (0) = 0 , Φ ′′ TE (0) = − ln(1 − e − ˜ ω p ) . (19)The respective terms of the power series again do not contribute to the difference (18).Finally, we find Φ ′′′ TE (0) = − ω p e ˜ ω p − TE ( x ) = Φ TE (0) − x − e − ˜ ω p ) − ω p x e ˜ ω p − O ( x ) , (21)where Φ TE (0) is defined in Eq. (17).Restricting ourselves by the third perturbation order, Eqs. (18) and (21) result in∆Φ TE ≈ i ω p τ t e ˜ ω p − . (22)We are coming now to the contribution of the TM mode to the quantity (16). This caseis more complicated because both the lower integration limit and the function under theintegral in Eq. (13) depend on x .By calculating several first derivatives of Eq. (13), where the reflection coefficient isdefined by the first equality in Eq. (11), one findsΦ TM (0) = Z ∞ ydy ln(1 − e − √ y +˜ ω p ) , Φ ′ TM (0) = 0 , (23)Φ ′′ TM (0) = 8˜ ω p Z ∞ dy p y + ˜ ω p e √ y +˜ ω p − − ln(1 − e − ˜ ω p ) , Φ ′′′ TM (0) = − ω p e ˜ ω p − . It is evident that the first two terms in the power series, defined by Eq. (23),Φ TM ( x ) = Φ TM (0) + Φ ′′ TM (0) x − ω p x e ˜ ω p − O ( x ) , (24)do not contribute to the quantity∆Φ TM ≡ Φ TM ( iτ t ) − Φ TM ( − iτ t ) . (25)Then, restricting ourselves by the third perturbation order, we arrive at∆Φ TM ≈ i ω p τ t e ˜ ω p − . (26)8y summing up Eqs. (22) and (26), one obtainsΦ( iτ t ) − Φ( − iτ t ) = ∆Φ TM + ∆Φ TM ≈ i ω p τ t e ˜ ω p − . (27)Substituting this result in Eq. (16), integrating with respect to t and returning to thedimensional variables, we find the behavior of the thermal correction to the Casimir energyof metallic film at low temperature∆ T F p ( a, T ) = − π ( k B T ) ~ c ω p ( e aω p /c − . (28)The respective thermal correction to the Casimir pressure of a free-standing metallic filmat low T takes the form∆ T P p ( a, T ) = − ∂ F p ( a, T ) ∂a = − π ( k B T ) ~ c e aω p /c ( e aω p /c − . (29)An interesting feature of Eqs. (28) and (29) is that the thermal corrections to the Casimirenergy and pressure of metallic film, calculated using the plasma model, go to zero expo-nentially fast with increasing film thickness a . Thus, there is no classical limit in this case.Another important point is that for fixed film thickness the Casimir free energy andpressure of the film go to zero in the limiting case ω p → ∞ . This is true for both thethermal corrections (28) and (29) and for the zero-temperature quantities E ( a ) and P ( a ).Note that for ω p → ∞ the magnitudes of both the TM and TE reflection coefficients (11) goto unity, i.e., the film becomes perfectly reflecting. One can conclude that when the plasmamodel is used in calculations an ideal metal film is characterized by the zero Casimir energyand pressure, as it should be because the electromagnetic fluctuations cannot penetrate inan interior of ideal metal.From Eq. (28) one can also obtain the low-temperature behavior of the Casimir entropyof metallic film S p ( a, T ) = − ∂ F p ( a, T ) ∂T = 8 π k B ( k B T ) ~ c ω p ( e aω p /c − . (30)It is seen that the Casimir entropy of a film is positive. When the temperature vanishes,one has from Eq. (30) S p ( a, T ) → , (31)i.e., the Casimir entropy of metallic film calculated using the plasma model satisfies theNernst heat theorem. 9n the end of this section, we discuss the application region of asymptotic Eqs. (28)–(30), which were derived under a condition x ≪
1, i.e., τ t ≪
1. Taking into account thatthe dominant contribution to the integral (16) is given by t ∼ / (2 π ) and considering thedefinition of τ in Eq. (6), one rewrites the application condition in the form k B T ≪ ~ c a = ~ ω c . (32)For a typical film thickness a = 100 nm, this inequality results in T ≪ T a = 1 µ m these equations are applicable at T
100 K.
III. METALS DESCRIBED BY THE DRUDE MODEL
Now we describe metal of the film by the Drude model which takes into account therelaxation properties of conduction electrons. At the pure imaginary Matsubara frequenciesthe dielectric permittivity of the Drude metal takes the form ε l,D = 1 + ω p ξ l [ ξ l + γ ( T )] , (33)where γ ( T ) is the temperature-dependent relaxation parameter.Using the dimensionless variables (6) and (10) and introducing the dimensionless relax-ation parameter, ˜ γ ( T ) = γ ( T ) ω c , (34)Eq. (33) can be rewritten as ε l,D = 1 + ˜ ω p ζ l [ ζ l + ˜ γ ( T )] . (35)It is convenient also to introduce one more dimensionless parameter δ l ( T ) = ˜ γ ( T ) ζ l = γ ( T ) ξ l = ~ γ ( T )2 πk B T l , (36)where l > δ l ( T ) ≪ , (37)10nd becomes progressively smaller with decreasing temperature. Thus, at T = 300 K forgood metals we have γ ∼ rad/s (for Au γ = 5 . × rad/s), whereas ξ = 2 . × rad/s. In the temperature region T D / < T <
300 K, where T D is the Debye temperature(for Au we have T D = 165 K [43]), it holds γ ( T ) ∼ T , i.e., the value of δ l remains unchanged.In the region from T D / γ ( T ) ∼ T in accordance tothe Bloch-Gr¨uneisen law [44] and at lower temperatures γ ( T ) ∼ T for metals with perfectcrystal lattices [43]. As a result, even the quantity δ ( T ) and, all the more, δ l ( T ) go to zerowhen T vanishes. For example, for Au at T = 30 and 10 K one has δ ≈ × − and2 × − , respectively.Now we express the permittivity (35) in terms of the small parameter (37) ε l,D = 1 + ˜ ω p ζ l [1 + δ l ( T )] (38)and, in the first perturbation order in this parameter, obtain ε l,D ≈ ε l,p − ˜ ω p ζ l δ l ( T ) . (39)We next use the following identical representation for the Casimir free energy of metallicfilm calculated using the Drude model: F D ( a, T ) = F p ( a, T ) + F (0) D ( a, T ) − F (0) p ( a, T ) + F ( γ ) ( a, T ) . (40)Here, F p is the free energy (12) calculated using the plasma model and F (0) p is its zero-frequency term F (0) p ( a, T ) = k B T πa Z ∞ ydy n ln (cid:16) − e − √ y +˜ ω p (cid:17) + ln h − r ,p ( y ) e − √ y +˜ ω p io , (41)where the reflection coefficient r TE ,p is defined in the second line of Eq. (11).The quantity F (0) D in Eq. (40) is the zero-frequency term in the Casimir free energy of afilm when the Drude model is used in calculations. From Eqs. (7) and (8) one obtains F (0) D ( a, T ) = k B T πa Z ∞ ydy ln (cid:0) − e − y (cid:1) = − k B T πa ζ (3) , (42)where ζ ( z ) is the Riemann zeta function. 11inally, the quantity F ( γ ) in Eq. (40) is the difference of all nonzero-frequency Matsubaraterms in the Casimir free energy (7) calculated using the Drude and plasma models F ( γ ) ( a, T ) = k B T πa ∞ X l =1 Z ∞ ζ l ydy (43) × n ln h − r ,D ( iζ l , y ) e − √ y +˜ ω p (1 − δ l ) i + ln h − r ,D ( iζ l , y ) e − √ y +˜ ω p (1 − δ l ) i − ln h − r ,p ( iζ l , y ) e − √ y +˜ ω p i − ln h − r ,p ( iζ l , y ) e − √ y +˜ ω p io , As shown in Appendix,lim T → F ( γ ) ( a, T ) = 0 , lim T → ∂ F ( γ ) ( a, T ) ∂T = 0 . (44)Because of this, we concentrate our attention on the other contributions to the right-handside of Eq. (40).The quantity F p is already found in Eqs. (14) and (28), and the quantity F (0) D is presentedin Eq. (42). Here, we calculate the quantity F (0) p defined in Eq. (41). Let us start with theintegral I (˜ ω p ) ≡ Z ∞ y dy ln (cid:16) − e − √ y +˜ ω p (cid:17) . (45)Expanding the logarithm in power series and introducing the new integration variable t = n q y + ˜ ω p , (46)one obtains from Eq. (45) I (˜ ω p ) = − ∞ X n =1 n Z ∞ n ˜ ω p tdte t (47)= − ∞ X n =1 n (1 + n ˜ ω p ) e − n ˜ ω p . After a summation, Eq. (47) results in I (˜ ω p ) = − (cid:2) Li ( e − ˜ ω p ) + ˜ ω p Li ( e − ˜ ω p ) (cid:3) , (48)where Li k ( z ) is the polylogarithm function.12ow we consider the second integral entering Eq. (41), i.e., I (˜ ω p ) ≡ Z ∞ y dy ln h − r ,p ( y ) e − √ y +˜ ω p i , (49)where the reflection coefficient r TE ,p is defined in Eq. (11). Note that for physical valuesof ˜ ω p the quantity subtracted from unity under the logarithm in Eq. (49) is much smallerthan unity. The reason is that if ˜ ω p is not large the squared reflection coefficient r ,p israther small. Then, one can expand the logarithm up to the first power of this parameterand obtain I (˜ ω p ) ≈ − Z ∞ y dy r ,p ( y ) e − √ y +˜ ω p . (50)Numerical computations show that Eqs. (49) and (50) lead to nearly coincident results for˜ ω p > .
5. Taking into account the definition of ˜ ω p in Eq. (10), this results in the condition a > . ω p = 1 . × rad/s. This is quite sufficientfor our purposes because here we consider metallic films of more than 7 nm thickness, whichcan be described by the isotropic dielectric permittivity [45] (for thinner Au films the effectof anisotropy should be taken into account [46]).Now we introduce the variable t = y/ ˜ ω p and, using Eq. (11), identically represent thequantity r ,p in the form r ,p ( y ) = 1 + 8 t + 8 t − t √ t − t √ t . (51)Introducing the integration variable t in Eq. (50), one finds I (˜ ω p ) ≈ − ˜ ω p Z ∞ t dte − ˜ ω p √ t (52) × (1 + 8 t + 8 t − t √ t − t √ t ) . Calculating all the five integrals in Eq. (52) [47], we arrive at I (˜ ω p ) ≈ − (cid:18) ˜ ω p + 17 + 112˜ ω p + 432˜ ω p + 960˜ ω p + 960˜ ω p (cid:19) e − ˜ ω p + 4 (cid:20) ˜ ω p K (˜ ω p ) + 9 K (˜ ω p ) + 30˜ ω p K (˜ ω p ) (cid:21) . (53)As a result, the Casimir free energy (40), calculated using the Drude model, can berewritten in the form F D ( a, T ) = F p ( a, T ) + F ( γ ) ( a, T ) (54) − k B T πa (cid:20) ζ (3) + I (cid:18) aω p c (cid:19) + I (cid:18) aω p c (cid:19)(cid:21) , F p and F ( γ ) are presented in Eqs. (14), (28), (43), and I and I are found in Eqs. (48),(53).Now we calculate the negative derivative of Eq. (54) with respect to T and find thelimiting value of this derivative when T goes to zero using Eqs. (28) and (44). The result is S D ( a,
0) = k B πa (cid:20) ζ (3) + I (cid:18) aω p c (cid:19) + I (cid:18) aω p c (cid:19)(cid:21) . (55)As is seen in Eq. (55), the Casimir entropy of metallic film at zero temperature, calculatedusing the Drude model, is not equal to zero and depends on the parameters of a film (thethickness a and the plasma frequency ω p ). Thus, in this case the Nernst heat theorem isviolated [38, 39].Calculations using Eqs. (48) and (53) show that S D ( a, > . (56)Thus, for ˜ ω p = 1 (i.e., for a Au film of approximately 11 nm thickness) one has I = − . I = − . C = 0 . ω p = 5 ( a = 55 nm) the respective results are: I = − . I = − . C = 1 . ω p = 15 ( a = 165 nm) I = − . × − , I = − . × − ,and C = 1 . I and I become negligibly small, as compared with ζ (3). IV. CONCLUSIONS AND DISCUSSION
In the foregoing, we have considered the low-temperature behavior of the Casimir freeenergy, entropy and pressure of metallic films in vacuum. It was shown that the calculationresults are quite different depending on whether the plasma or the Drude model is usedto describe the dielectric response of a film metal. If the lossless plasma model is used, asis suggested by the results of several precise experiments on measuring the Casimir force,we have obtained explicit analytic expressions for the thermal corrections to the Casimirenergy and pressure and for the Casimir entropy of a film, which are applicable over thewide temperature region down to zero temperature. These expressions do not have a classicallimit and go to zero when the film material becomes perfectly reflecting. The Casimir entropyis shown to be positive and satisfying the Nernst heat theorem, i.e., it goes to zero in thelimiting case of zero temperature. 14f the film metal is described by the Drude model taking into account the relaxation prop-erties of conduction electrons at low frequencies, the calculation results are quite different,both qualitatively and quantitatively. In accordance to what was shown in previous work[14–16], the Casimir free energy and pressure reach the classical limit for rather thin metallicfilms of approximately 150 nm thickness. However, in contradiction to physical intuition,the Casimir free energy does not go to zero in the limiting case of ideal metal film.We have found analytically the Casimir entropy of metallic films with perfect crystallattices, described by the Drude model, at zero temperature. It is demonstrated that thisquantity takes a positive value depending on the parameters of a film, i.e., the Nernst heattheorem is violated. Thus, the case of a free-standing film is different from the case of twononmagnetic metal plates described by the Drude model interacting through a vacuum gap.In the latter case the Nernst heat theorem is also violated if the Drude model is used incalculations, but the Casimir entropy takes a negative value at T =0 [32–34].The obtained results raise a problem on what is the proper way to calculate the dispersion-force contribution to the free energy of metallic films. As discussed in Sec. I, the resolutionof this problem is important for investigations of stability of thin films. Previous preciseexperiments on measuring the Casimir force between metallic test bodies [17–23, 27, 28]have always been found in agreement with theoretical predictions of the thermodynamicallyconsistent approach using the plasma model and excluded the theoretical predictions ob-tained using the Drude model. Recently it was shown [48] that theoretical description ofthe Casimir interaction in graphene systems by means of the polarization tensor, which isin agreement [49] with the experimental data [50], also satisfies the Nernst heat theorem.Thus, there is good reason to suppose that the contribution of dispersion forces to the freeenergy of metallic films should also be calculated in a thermodynamically consistent way,i.e., using the plasma model. An experimental confirmation to this hypothesis might beexpected within the next few years. Acknowledgments
The work of V.M.M. was partially supported by the Russian Government Program ofCompetitive Growth of Kazan Federal University.15 ppendix A
Here, we investigate the low-temperature behavior of the quantity F ( γ ) defined in Eq. (43)and prove Eq. (44) used in Sec. III. For this purpose we expand F ( γ ) up to the first orderin small parameter δ l ( T ) defined in Eq. (36). According to the results of Sec. III, for metalswith perfect crystal lattices this parameter becomes progressively smaller with decreasing T . The reflection coefficients in the case when the Drude model is used can be obtained bysubstituting Eq. (39) in Eq. (8) r TM ,D ( iζ l , y ) ≈ q y + ( ε l,p − ζ l − ˜ ω p δ l − ε l,p y + ˜ ω p yζ l δ l q y + ( ε l,p − ζ l − ˜ ω p δ l + ε l,p y − ˜ ω p yζ l δ l ,r TE ,D ( iζ l , y ) ≈ q y + ( ε l,p − ζ l − ˜ ω p δ l − y q y + ( ε l,p − ζ l − ˜ ω p δ l + y . (A1)Expanding the second powers of these coefficients up to the first order of δ l = δ l ( T ), oneobtains r ,D ( iζ l , y ) ≈ r ,p ( iζ l , y ) − δ l ( T ) R TM ( iζ l , y ) ,r ,D ( iζ l , y ) ≈ r ,p ( iζ l , y ) − δ l ( T ) R TE ( iζ l , y ) , (A2)where the quantities R TM and R TE are given by R TM ( iζ l , y ) = 2˜ ω p ζ l y (˜ ω p + 2 y − ζ l )(˜ ω p y + ζ l y − ζ l p y + ˜ ω p ) p y + ˜ ω p (˜ ω p y + ζ l y + ζ l p y + ˜ ω p ) ,R TE ( iζ l , y ) = R TE ( y ) = 2˜ ω p y ( p y + ˜ ω p − y ) p y + ˜ ω p ( p y + ˜ ω p + y ) . (A3)It is easily seen that for any y > ζ l it holds R TM > R TE > δ l , this factor can be presented in the form e − √ y +˜ ω p (1 − δ l ) = e − √ y +˜ ω p s − δl ˜ ω py ω p ≈ e − √ y +˜ ω p (cid:20) − δl ˜ ω p y ω p ) (cid:21) . (A4)Next we use the fact that not only δ l , but also δ l ˜ ω p / δ ˜ ω p γξ aω p c . (A5)16or Au at T = 10 K we have γ/ξ ≈ × − , so that the quantity (A5) does not exceed 0.2for film thicknesses a µ m. At T = 5 K the parameter (A5) does not exceed 0.2 for Aufilms with a µ m thickness.Expanding the right-hand side of Eq. (A4) up to the first order in parameter δ l ˜ ω p /
2, weobtain e − √ y +˜ ω p (1 − δ l ) ≈ e − √ y +˜ ω p δ l ˜ ω p p y + ˜ ω p ! . (A6)Substituting Eqs. (A2) and (A6) in Eq. (43), expanding the first two logarithms in powersof δ l and preserving only the terms of the first order, one arrives at F ( γ ) ( a, T ) ≈ − k B T πa ∞ X l =1 δ l ( t ) Z ∞ ζ l y dy × Q TM ( iζ l , y ) e √ y +˜ ω p − r ,p ( iζ l , y )+ Q TE ( iζ l , y ) e √ y +˜ ω p − r ,p ( iζ l , y ) . (A7)Here, we have introduced the notations Q TM ( iζ l , y ) = ˜ ω p p y + ˜ ω p − R TM ( iζ l , y ) , (A8) Q TE ( iζ l , y ) = Q TE ( y ) = ˜ ω p p y + ˜ ω p − R TE ( y ) , and the quantities R TM and R TE are defined in Eq. (A3).It is easily seen that Q TM > Q TE >
0, so that F ( γ ) ( a, T ) <
0. This is becausethe magnitude of the Casimir free energy of a film described by the Drude model is largerthan that of a film described by the plasma model (as opposed to the case of metallic platesseparated with a vacuum gap [34]).Equation (A7) can be used to prove the validity of Eq. (44). For this purpose we increasethe magnitude of the right-hand side of Eq. (A7) by replacing r ,p with unities in thedenominators, and by omitting the quantities R TM(TE) in Eq. (A8) for the numerators. Usingalso the definition of δ l in Eq. (36), and the definition of ˜ ω p from Eq. (10) in the prefactor,one obtains |F ( γ ) ( a, T ) | < ~ γ ( T ) ω p π c ∞ X l =1 l Z ∞ ζ l y dy p y + ˜ ω p e √ y +˜ ω p − . (A9)17ow we introduce the new variable t = p y + ˜ ω p and expanding in powers of e − t find |F ( γ ) ( a, T ) | < ~ γ ( T ) ω p π c ∞ X n =1 ∞ X l =1 l Z ∞ √ ζ l +˜ ω p dt e − nt . (A10)Calculating the integral and using the inequality ζ l + ˜ ω p √ < q ζ l + ˜ ω p , (A11)we arrive at |F ( γ ) ( a, T ) | < ~ γ ( T ) ω p π c ∞ X n =1 n ∞ X l =1 l e − n ˜ ωp + ζl √ . (A12)Taking into account that ζ l = τ l , we perform a summation with respect to l and obtain |F ( γ ) ( a, T ) | < − ~ γ ( T ) ω p π c ∞ X n =1 n e − n ˜ ωp √ ln (cid:16) − e − n τ √ (cid:17) , (A13)where, due to a smallness of τ ,ln (cid:16) − e − n τ √ (cid:17) ≈ ln (cid:18) n τ √ (cid:19) = ln τ + ln n −
12 ln 2 . (A14)Substituting Eq. (A13) in Eq. (A12), we represent the final results in the form |F ( γ ) ( a, T ) | < X ( a, T ) , (A15)where X ( a, T ) = ~ γ ( T ) ω p π c (cid:18) C ln 4 πk B T a ~ c − C (cid:19) , (A16)and the following independent on T coefficients are introduced C = − ∞ X n =1 n e − n ˜ ωp √ = ln (cid:16) − e − ˜ ωp √ (cid:17) ,C = ∞ X n =1 n − ln 22 n e − n ˜ ωp √ . (A17)Note that the second series is converging, as well as the first one.Taking into account that for metals with perfect crystal lattices at very low temperature γ ( T ) ∼ T (see Sec. III), one concludes from Eq. (A16) that X ( a, T ) → T → X ( a,
0) = 0, but ∂X ( a, T ) ∂T (cid:12)(cid:12)(cid:12)(cid:12) T =0 = 0 (A18)18s well. Using Eqs. (A15) and (A18), one easily proves that the second equality in Eq. (44)is valid.In the end it is pertinent to note that the above results, including Eq. (44), are alsovalid under a slower vanishing of the relaxation parameter with temperature according to γ ( T ) ∼ T β where β > [1] V. A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, andPhysicists (Cambridge University Press, Cambridge, 2005).[2] M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko,
Advances in theCasimir Effect (Oxford University Press, Oxford, 2015).[3] E. M. Lifshitz and L. P. Pitaevskii,
Statistical Physics , Part II (Pergamon, Oxford, 1980).[4] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Rev. Mod. Phys. , 1827(2009).[5] A. W. Rodrigues, F. Capasso, and S. G. Johnson, Nature Photonica , 211 (2011).[6] R. H. French, V. A. Parsegian, R. Podgornik, et al., Rev. Mod. Phys. , 1887 (2010).[7] R. Esquivel-Sirvent and R. P´erez-Pascual, Eur. Phys. J. B , 467 (2013).[8] J. Zou, Z. Marcet, A. W. Rodriguez, M. T. H. Reid, A. P. McCauley, I. I. Kravchenko, T. Lu,Y. Bao, S. G. Johnson, and H. B. Chan, Nature Commun. , 1845 (2013).[9] A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces (Wiley, New York, 1997).[10] L. Boinovich and A. Emelyanenko, Adv. Colloid Interface Sci. , 60 (2011).[11] Y. Imry, Phys. Rev. Lett. , 080404 (2005).[12] A. Benassi and C. Calandra, J. Phys. A: Math. Theor. , 13453 (2007).[13] A. Benassi and C. Calandra, J. Phys. A: Math. Theor. , 175401 (2008).[14] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A , 042109 (2015).[15] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A , 042508 (2016).[16] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B , 045404 (2016).[17] R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. L´opez, and V. M. Mostepa-nenko, Phys. Rev. D , 116003 (2003).[18] R. S. Decca, D. L´opez, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepa-nenko, Ann. Phys. (N.Y.) , 37 (2005).
19] R. S. Decca, D. L´opez, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepa-nenko, Phys. Rev. D , 077101 (2007).[20] R. S. Decca, D. L´opez, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, and V. M. Mostepa-nenko, Eur. Phys. J. C , 963 (2007).[21] C.-C. Chang, A. A. Banishev, R. Castillo-Garza, G. L. Klimchitskaya, V. M. Mostepanenko,and U. Mohideen, Phys. Rev. B , 165443 (2012).[22] A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev.Lett. , 137401 (2013).[23] A. A. Banishev, G. L. Klimchitskaya, V. M. Mostepanenko, and U. Mohideen, Phys. Rev. B , 155410 (2013).[24] G. Bimonte, Phys. Rev. Lett. , 240401 (2014).[25] G. Bimonte, Phys. Rev. Lett. , 240405 (2014).[26] G. Bimonte, Phys. Rev. B , 205443 (2015).[27] G. Bimonte, D. L´opez, and R. S. Decca, arXiv:1509.05349v2.[28] R. S. Decca, Int. J. Mod. Phys. A , 1641024 (2016).[29] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. E , 026101 (2016).[30] P. R. Buenzli and P. A. Martin, Phys. Rev. E , 011114 (2008).[31] G. Bimonte, Phys. Rev. A , 042107 (2009).[32] V. B. Bezerra, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. A , 052113(2002).[33] V. B. Bezerra, G. L. Klimchitskaya, and V. M. Mostepanenko, Phys. Rev. A , 062112(2002).[34] V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko, and C. Romero, Phys. Rev. A ,022119 (2004).[35] M. Bordag and I. Pirozhenko, Phys. Rev. D , 125016 (2010).[36] G. L. Klimchitskaya and C. C. Korikov, Phys. Rev. A , 032119 (2015); , 029902(E)(2015).[37] M. Bordag, Adv. Math. Phys. 981586 (2014).[38] L. D. Landau and E. M. Lifshitz, Statistical Physics , Part I (Pergamon, Oxford, 1980).[39] Yu. B. Rumer and M. Sh. Ryvkin,
Thermodynamics, Statistical Physics, and Kinetics (Mir,Moscow, 1980).
40] M. S. Tomaˇs, Phys. Rev. A , 052103 (2002).[41] S. Raabe, L. Kn¨oll, and D.-G. Welsch, Phys. Rev. A , 033810 (2003).[42] S. Y. Buhmann and D.-G. Welsch, Prog. Quantum Electron. , 51 (2007).[43] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1996).[44] N. W. Ashcroft and N. D. Mermin,
Solid State Physics (Saunders College, Philadelphia, 1976).[45] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B , 205410 (2015).[46] M. Bostr¨om, C. Persson, and Bo E. Sernelius, Eur. Phys. J. B, , 43 (2013).[47] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series , Vol. 1 (Gordon& Breach, New York, 1986).[48] G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A , 042501 (2016).[49] G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko, Phys. Rev. B , 115419 (2014).[50] A. A. Banishev, H. Wen, J. Xu, R. K. Kawakami, G. L. Klimchitskaya, V. M. Mostepanenko,and U. Mohideen, Phys. Rev. B , 205433 (2013)., 205433 (2013).