Analytic approach to the thermal Casimir force between metal and dielectric

# Analytic approach to the thermal Casimir force between metal and dielectric

B. Geyer, G. L. Klimchitskaya, V. M. Mostepanenko Center of Theoretical Studies and Institute for Theoretical Physics,
Leipzig University, Augustusplatz 10/11, D-04109 Leipzig, Germany
North-West Technical University, Millionnaya St. 5, St.Petersburg, 191065, Russia Noncommercial Partnership “Scientific Instruments”, Tverskaya St. 11, Moscow, 103905, Russia
###### Abstract

The analytic asymptotic expressions for the Casimir free energy, pressure and entropy at low temperature in the configuration of one metal and one dielectric plate are obtained. For this purpose we develop the perturbation theory in a small parameter proportional to the product of the separation between the plates and the temperature. This is done using both the simplified model of an ideal metal and of a dielectric with constant dielectric permittivity and for the realistic case of the metal and dielectric with frequency-dependent dielectric permittivities. The analytic expressions for all related physical quantities at high temperature are also provided. The obtained analytic results are compared with numerical computations and good agreement is found. We demonstrate for the first time that the Lifshitz theory, when applied to the configuration of metal-dielectric, satisfies the requirements of thermodynamics if the static dielectric permittivity of a dielectric plate is finite. If it is infinitely large, the Lifshitz formula is shown to violate the Nernst heat theorem. The implications of these results for the thermal quantum field theory in Matsubara formulation and for the recent measurements of the Casimir force between metal and semiconductor surfaces are discussed.

###### keywords:
Casimir force; Thermal corrections; Lifshitz formula; Nernst heat theorem;
###### Pacs:
03.70.+k, 11.10.Wx, 12.20.-m, 12.20.Ds
Corresponding author. Fax: +49 341 9732548

## 1. Introduction

The Casimir effect [1] is the direct manifectation of zero-point oscillations of quantized fields. It finds multidisciplinary applications in quantum field theory, gravitation and cosmology, atomic physics, condensed matter and, most recently, in nanotechnology (see, e.g., the monographs [2, 3, 4, 5] and reviews [6, 7, 8, 9]). According to Casimir’s prediction, the existence of zero-point oscillations leads to the polarization of vacuum in quantization volumes restricted by material boundaries and in spaces with non-Euclidean topology. This is accompanied by forces acting on the boundary surfaces (the so called Casimir force). The Casimir force acts between electrically neutral closely spaced surfaces. It is a pure quantum phenomenon (there is no such a force in the framework of classical electrodynamics) being the generalization of the well known van der Waals force for the case of relatively large separations where relativistic effects become essential.

The theoretical basis for the description of both the van der Waals and Casimir forces is given by the Lifshitz theory [10, 11, 12]. The main formulas of the Lifshitz theory express the free energy and pressure of the van der Waals and Casimir interaction between two plane parallel plates as some functionals of the frequency-dependent dielectric permittivities of plate materials. These formulas can be derived in many different theoretical schemes [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In particular, they were obtained in the framework of thermal quantum field theory in the Matsubara formulation [8]. During the last few years the Lifshitz theory was successfully applied to the interpretation of many measurements of the Casimir force between metal surfaces [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] and between metal and semiconductor [25, 26, 27, 28].

A complicated problem of the Lifshitz theory is how to describe the Casimir interaction between real metals at nonzero temperature. The most convenient form of the Lifshitz formulas exploits the dielectric permittivity along the imaginary frequency axis. The latter is obtained from the tabulated optical data for the complex index of refraction by means of the Kramers-Kronig relations. The available data are, however, insufficient and must be extrapolated in some way to lower frequencies. In this respect the contribution from the zero frequency is of most concern. The point is that in Matsubara thermal field theory the zero-frequency term becomes dominant at large separations (high temperatures) whereas the contributions from all other Matsubara frequencies being exponentially small. In [29, 30, 31] the zero-frequency term of the Lifshitz formula was obtained by using the dielectric function of the Drude model. This results in a violation of the Nernst heat theorem in the case of perfect crystal lattices [32, 33] and is in contradiction with experiments at separations below 1m [22, 23, 24, 34]. The asymptotic value of the Casimir force at large separations predicted in [29, 30, 31] is equal to one half of the value predicted for ideal metals, i.e., to one half of the so called classical limit [35, 36].

Another approach [37, 38] uses the dielectric permittivity of the plasma model to determine the zero-frequency term of the Lifshitz formula. This approach was shown to be in agreement with thermodynamics [32, 33] and consistent with experiment [23]. It predicts the magnitudes of the Casimir force at short separations in qualitative agreement with the case of ideal metals. At large separations the predicted force magnitude is practically equal to that for ideal metals. Very similar results, which are also in agreement with the requirements of thermodynamics and consistent with experiment, are predicted by the surface impedance approach [39, 40]. The controversies among different approaches to the thermal Casimir force between metals are detailly discussed in [31, 33, 40, 41, 42, 43, 44].

Recently it was demonstrated [45, 46, 47] that even the traditional application of the Lifshitz formula to the case of two dielectric semispaces presents problems. In [45, 46, 47] the analytic asymptotic expressions for the Casimir free energy, pressure and entropy at low temperatures (short separations) were found for two dielectrics. It was shown that if the dielectric materials possess finite static dielectric permittivities the theory is self-consistent and in agreement with thermodynamics. If, however, a nonzero dc conductivity of dielectrics is taken into account (any dielectric at nonzero temperature is characterized by some nonzero dc conductivity which is many orders of magnitude lower than for metals), this leads to a qualitative enhancement of the Casimir force and a simultaneous violation of the Nernst heat theorem. (Note that the dc conductivity of dielectrics was taken into account in [48] to explain the large observed effect in noncontact friction [49].) In [45, 46, 47] the phenomenological prescription was proposed that the dc conductivity of dielectrics is not related to the Casimir interaction, and to avoid contradictions with thermodynamics it should not be included in the model of dielectric response.

The difficulties which were met in the application of the Lifshitz theory to two metal and two dielectric plates attracted attention to the case when one plate is metallic and another one dielectric. This configuration was first considered in [50]. It presents the interesting opportunity to investigate the Casimir force in the case when different plates are described by quite different models of the dielectric response. In [50], however, only the first leading terms in the low-temperature asymptotic expressions for the Casimir free energy and entropy were obtained, and the pressure was derived only in the dilute approximation. In the analytical derivations in [50] (see also the review [47]) it was supposed that the metallic plate is made of ideal metal and the dielectric of the other plate is described by the frequency independent dielectric permittivity. These suppositions narrow the applicability of the obtained results. Also, the role of the dc conductivity of a dielectric plate was not investigated for plates with frequency-dependent dielectric permittivities.

In the present paper we develop the analytic approach to the thermal Casimir force acting between metal and dielectric permitting to find several expansion terms in the asymptotic expressions for all physical quantities at low temperature. This approach is applied not only to the configuration of ideal metal and dielectric with frequency independent dielectric permittivity but also to real metal and dielectric described by the dielectric permittivities depending on frequency. We pioneer in derivation of the low-temperature asymptotic expressions for the Casimir free energy, entropy and pressure between real metal and dielectric. The asymptotic behavior of all physical quantities at high temperatures (large separations) is also provided. What is more, the obtained representation for the Casimir free energy permits to find the low-temperature behavior of the Casimir force acting between a metal sphere and a dielectric plate (or, alternatively, dielectric sphere above a metal plate). This can be done with the help of the proximity force theorem [51]. The configuration of a sphere above a plate is most topical in experiments on the measurement of the Casimir force [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. (Note that for the experimental parameters the error introduced by the use of the proximity force theorem was recently shown to be less than 0.1% [52, 53, 54, 55].) Thus, our results will find immediate utility in experiment. The analytic expressions for the Casimir interaction between metal and dielectric at zero temperature are also found here for the first time. We determine the applicability region of the obtained analytic formulas by compairing them with numerical computations using the tabulated optical data for metallic and dielectric materials. The fundamental conclusion following from our results is that the Lifshitz theory, applied to the configuration of a metal and a dielectric plate, is in agreement with the Nernst heat theorem if the static dielectric permittivity of a dielectric plate is finite. Note that this conclusion could not be achieved by using the numerical computations which inevitably identify zero with all nonzero numbers in the limits of a computational error. If, however, the dc conductivity of a dielectric plate is included in the model of dielectric response, we show that the Nernst heat theorem is violated. This is in analogy to the same conclusion in [45] obtained for the configuration of two dielectric plates and confirms our phenomenological prescription that the dc conductivity is not related to the van der Waals and Casimir forces and should not be included in the model of dielectric response. Recently this prescription was confirmed experimentally [28].

The paper is organized as follows. In Section 2 we summarize the main formulas of the Lifshitz theory for the configuration of one plate made of metal and another one made of dielectric. Section 3 is devoted to the simplified model where the metal is an ideal one and dielectric is described by a constant dielectric permittivity. In the framework of this model a perturbation formalism applicable at low temperatures (short separations) is developed. In Section 4 the realistic case is considered when the dielectric permittivities of both metal and dielectric plates depend on the frequency. The analytic asymptotic expressions for the free energy, entropy and pressure of the Casimir interaction at both low and high temperatures are obtained. Section 5 contains the comparison between the analytical results and numerical computations using the tabulated optical data for plate materials. The application region of the derived asymptotic expressions is determined. In Section 6 it is shown that the inclusion of the dc conductivity in the description of dielectric plate leads to a violation of the Nernst heat theorem. Section 7 contains our conclusions and discussion.

## 2. Lifshitz formula in the configuration of metal and dielectric plates

We consider two thick parallel plates (semispaces) at temperature in thermal equilibrium separated by the empty gap of width . One plate is made of metal with the dielectric permittivity and another of dielectric with permittivity . The free energy of the van der Waals and Casimir interaction between the plates per unit area is given by the Lifshitz formula [10, 11, 12, 45, 50]

 F(a,T)=kBT2π∞∑l=0(1−12δ0l)∫∞0k⊥dk⊥ (1) aaaaa×{ln[1−rM∥(ξl,k⊥)rD∥(ξl,k⊥)e−2aql] aaaaaaaa+ln[1−rM⊥(ξl,k⊥)rD⊥(ξl,k⊥)e−2aql]}.

Here the plates are perpendicular to the axis, is the magnitude of the wave vector in the plane of plates, are the Matsubara frequencies, and is the Boltzmann constant. are the reflection coefficients for metal () and dielectric () plates for the two independent polarizations of electromagnetic field calculated along the imaginary frequency axis. Index stands for the electric field parallel to the plane formed by and the axis (transverse magnetic field), and index stands for the electric field perpendicular to this plane (transverse electric field). The explicit expressions for the reflection coefficients are [45, 50]

 rM,D∥(ξl,k⊥)=εM,Dlql−kM,DlεM,Dlql+kM,Dl,rM,D⊥(ξl,k⊥)=kM,Dl−qlkM,Dl+ql, (2)

where

 ql=√ξ2lc2+k2⊥,kM,Dl=√εM,Dlξ2lc2+k2⊥, (3)

and

 εM,Dl=εM,D(iξl). (4)

The pressure of the van der Waals and Casimir interaction between metal and dielectric (i.e., the force per unit area of plates) is obtained from

 P(a,T)=−∂F(a,T)∂a. (5)

Using Eq. (1) we arrive at

 P(a,T)=−kBTπ∞∑l=0(1−12δ0l)∫∞0k⊥dk⊥ql (6) aaa×⎡⎣rM∥(ξl,k⊥)rD∥(ξl,k⊥)e2aql−rM∥(ξl,k⊥)rD∥(ξl,k⊥)+rM⊥(ξl,k⊥)rD⊥(ξl,k⊥)e2aql−rM⊥(ξl,k⊥)rD⊥(ξl,k⊥)⎤⎦.

Using the proximity force theorem [51], one can obtain from Eq. (1) the approximate expression for the Casimir force acting between a sphere and a plate

 F(a,T)=2πRF(a,T). (7)

This equation is widely used for the interpretation of measurements of the Casimir force [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Recently both exact analytic and numerical results for the Casimir force in the configuration of a cylinder above a plate (electromagnetic case) and for a sphere above a plate (scalar case) were obtained [52, 53, 54, 55]. It was shown that the error introduced by the use of Eq. (7) for the experimental parameters in already performed experiments is less than 0.1%. Using Eq. (7), the analytical results derived below for metal and dielectric plates, can be immediately applied for the interpretation of measurements of the Casimir force between Au coated sphere and Si plate [25, 26, 27, 28].

The analytic perturbation expansions in Eqs. (1) and (6) can be conveniently performed by using the dimensionless variables and

 ζl=ξlωc=2aξlc=τl,y=2aql, (8)

where is the characteristic frequency of the Casimir effect and . In terms of these variables the free energy (1) takes the form

 F(a,T)=ℏcτ32π2a3∞∑l=0(1−12δ0l)∫∞ζlydy{ln[1−rM∥(ζl,y)rD∥(ζl,y)e−y] aaaaaaaaaaaaaaaaa+ln[1−rM⊥(ζl,y)rD⊥(ζl,y)e−y]}. (9)

Using the variables (8), the reflection coefficients (2) are

 rM,D∥(ζl,y)=εM,Dly−√y2+ζ2l(εM,Dl−1)εM,Dly+√y2+ζ2l(εM,Dl−1), rM,D⊥(ζl,y)=√y2+ζ2l(εM,Dl−1)−y√y2+ζ2l(εM,Dl−1)+y, (10)

where in accordance with Eq. (4) .

The pressure (6) is rearranged as follows:

 P(a,T)=−ℏcτ32π2a4∞∑l=0(1−12δ0l)∫∞ζly2dy (11) aaa×⎡⎣rM∥(ζl,y)rD∥(ζl,y)ey−rM∥(ζl,y)rD∥(ζl,y)+rM⊥(ζl,y)rD⊥(ζl,y)ey−rM⊥(ζl,y)rD⊥(ζl,y)⎤⎦.

The other important characteristic of the van der Waals and Casimir interaction is the entropy

 S(a,T)=−∂F(a,T)∂T. (12)

In [32, 33] the behavior of the Casimir entropy at was used as a phenomenological constraint on the selection of theoretically consistent models of the dielectric response for real metals at low frequencies. It was proposed that all consistent models should satisfy the thermodynamic condition , i.e., be in agreement with the Nernst heat theorem. In [45] it was demonstrated that this condition is respected for two dielectric plates with the finite static dielectric permittivities. The new analytic expressions for the free energy obtained in the present paper permit investigate the behavior of entropy in the configuration of one metal and one dielectric plate and find when it vanishes with vanishing temperature.

## 3. Model of ideal metal and dielectric with constant dielectric permittivity

To find the analytic expressions for the free energy, pressure and entropy of the Casimir interaction between metal and dielectric, we start from a simplified model when the metal is an ideal one and the dielectric possesses some finite dielectric permittivity independent on the frequency. Such modeling is widely used in Casimir physics (see, e.g., [2, 3, 5, 6, 8, 10, 11, 12, 56]). It provides rather good description of real metals and dielectrics at sufficiently large separations between the interacting surfaces. For an ideal metal it holds at all frequencies and from Eq. (10) one obtains

 rM∥(ζl,y)=1,rM⊥(ζl,y)=1,l≥0. (13)

Using Eq. (13), the free energy (9) and pressure (11) are represented in a more simple form,

 F(a,T)=ℏcτ32π2a3∞∑l=0(1−12δ0l)∫∞ζlydy aaa×{ln[1−rD∥(ζl,y)e−y]+ln[1−rD⊥(ζl,y)e−y]}, (14) P(a,T)=−ℏcτ32π2a4∞∑l=0(1−12δ0l)∫∞ζly2dy aaa×⎡⎣rD∥(ζl,y)ey−rD∥(ζl,y)+rD⊥(ζl,y)ey−rD⊥(ζl,y)⎤⎦.

Notice that in the framework of our model in Eq. (10) it holds , i.e., the dielectric permittivities computed at different imaginary Matzubara frequencies do not depend on . In particular, at it follows:

 rD∥(0,y)=εD0−1εD0+1≡r0,rD⊥(0,y)=0. (15)

In fact Eq. (15) is valid not only for our simplified model but for any dielectric with a finite static dielectric permittivity . Usually for nonpolar dielectrics in the frequency region from up to rather high frequencies of about rad/s and for higher frequencies decreases to unity. The simplified model does not take the latter into account (in the next section we show that this does not influence the first terms in the asymptotic behavior of the free energy, entropy and pressure at low temperature).

Now we proceed with the derivation of the asymptotic behavior of the Casimir free energy at low temperature (). Using the Abel-Plana formula [3, 8]

 ∞∑l=0(1−12δl0)F(l)=∫∞0F(t)dt+i∫∞0dtF(it)−F(−it)e2πt−1, (16)

where is an analytic function in the right-plane, we can rearrange Eq. (14) to the form

 F(a,T)=E(a)+ΔF(a,T). (17)

Here,

 E(a)=ℏc32π2a3∫∞0dζ∫∞ζf(ζ,y)dy (18)

is the energy of the Casimir interaction at zero temperature,

 ΔF(a,T)=iℏcτ32π2a3∫∞0dtF(iτt)−F(−iτt)e2πt−1 (19)

is the thermal correction to it, and the following notations are introduced,

 f(ζ,y)=yln[1−rD∥(ζ,y)e−y]+yln[1−rD⊥(ζ,y)e−y], F(x)=∫∞xdyf(x,y). (20)

The expansion of in powers of takes the form

 f(x,y)=yln(1−r0e−y) (21) aaaaaaa−x2(εD0−14ye−y−εD0εD0+1∞∑n=1rn0e−nyy)+O(x3).

To find in Eq. (20) we integrate the right-hand side of Eq. (21) with respect to . Notice that the first term on the right-hand side of Eq. (21) does not contribute to the first expansion orders of which is in fact the quantity of our interest. This is because in the expression

 ∫∞xydyln(1−r0e−y)=∫∞0vdvln(1−r0e−v)+O(x2), (22)

where the new variable was introduced, the first-order in contribution vanishes. Thus, this term could contribute to only starting from the third expansion order. Integrating the second term on the right-hand side of Eq. (21) using the formula

 ∫∞xdye−nyy=−Ei(−nx), (23)

where Ei is the exponential integral function, we finally obtain

 F(ix)−F(−ix)=iπ(εD0−1)24(εD0+1)x2−iγx3+O(x4), (24)

where the third order real expansion coefficient cannot be determined at this stage of our calculations because all powers in the expansion of in powers of contribute to it.

Now we substitude Eq. (24) in Eq. (19) and find the free energy (17)

 F(a,T)=E(a)−ℏc32π2a3[ζ(3)16π2(εD0−1)2εD0+1τ3−K4τ4+O(τ5)], (25)

where and is the Riemann zeta function.

The Casimir pressure is obtained from Eqs. (5) and (25). It is equal to

 P(a,T)=P0(a)−ℏc32π2a4[K4τ4+O(τ5)], (26)

where .

In order to determine the coefficients , we now start from the Lifshitz representation of the pressure in Eq. (14). Using the Abel-Plana formula (16), we rearrange Eq. (14) to the form analogical to (17)–(19),

 P(a,T)=P0(a)+ΔP(a,T), P0(a)=3ℏc32π2a4∫∞0dζ∫∞ζf(ζ,y)dy, (27) ΔP(a,T)=−iℏcτ32π2a4∫∞0dtΦ(iτt)−Φ(−iτt)e2πt−1.

Here is the Casimir pressure at zero temperature, is the thermal correction to it and the following notation is introduced:

 Φ∥,⊥(x)=∫∞xdyy2r∥,⊥(x,y)ey−r∥,⊥(x,y). (28)

To find the expansion of in powers of , we first deal with . By adding and subtracting the asymptotic behavior of the intergrand function at small ,

 y2r⊥(x,y)ey−r⊥(x,y)=14(εD0−1)x2e−y+O(x3), (29)

under the integral in Eq. (28) and introducing the new variable , the function can be identically rearranged and expanded in powers of as follows:

 Φ⊥(x)=14(εD0−1)x2e−x+x3∫∞1dv[v2∞∑n=1rn⊥(v)e−nvx−14(εD0−1)e−vx] Φ⊥(x)=14(εD0−1)x2(1−x) (30) aa+x3∫∞1dv[v2r⊥(v)1−r⊥(v)−εD0−14]+O(x4).

Calculating the integral on the right-hand side of Eq. (30), we arrive at the result

 Φ⊥(x)=εD0−14x2−16(εD0√εD0−1)x3+O(x4). (31)

To deal with , we add and subtract under the integral in Eq. (28) the two first expansion terms of the integrated function in powers of ,

 Φ∥(x)=∫∞xy2dy[r0ey−r0−εD0r0e−yx2y2(εD0+1)(1−r0e−y)2] Φ∥(x)+∫∞xy2dy[r∥(x,y)ey−r∥(x,y)−r0ey−r0+εD0r0e−yx2y2(εD0+1)(1−r0e−y)2]. (32)

The asymptotic expansions of the first and second integrals on the right-hand side of Eq. (32) are

 2Li3(r0)−εD0(εD0−1)2(εD0+1)x2+112(εD0−1)(3εD0−2)x3+O(x4), (33) [−14εD0(εD0−1)−16εD0(εD0√εD0−1)+12εD0(εD0−1)√εD0]x3 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa+O(x4), (34)

respectively, where Li is the polylogarithm function. By summing Eqs. (33) and (34) we find

 Φ∥(x)=2Li3(r0)−εD0(εD0−1)2(εD0+1)x2 (35) Φ∥(x)−16[(εD0−1)+(εD0√εD0−1)−3εD0(εD0−1)√εD0]x3+O(x4).

After summing Eqs. (31) and (35) the following result is obtained:

 Φ(ix)−Φ(−ix)=−2i3(1−2εD0√εD0+(εD0)2√εD0)x3+O(x4). (36)

Substituting this in Eq. (27) and integrating we arrive at the asymptotic expression for the Casimir pressure in the limit of small ,

 P(a,T)=P0(a)−ℏc32π2a4⎡⎢ ⎢⎣1−2εD0√εD0+(εD0)2√εD0360τ4+O(τ5)⎤⎥ ⎥⎦. (37)

The comparison of this equation with Eq. (26) leads to the explicit expression for the coefficient :

 K4=1360(1−2εD0√εD0+(εD0)2√εD0) (38)

and, thus, the explicit asymptotic expression (25) for the free energy is also fully determined.

Notice that the energy and pressure at zero temperature [ and defined in Eqs. (18) and (27), respectively] depend on the separation through only the factors and in front of the integrals. They can be conveniently presented in the form

 E(a)=−π2720ℏca3ψDM(εD0), (39) P0(a)=−π2240ℏca4ψDM(εD0),

where the function is defined as

 ψDM(εD0)=−452π4∫∞0dζ∫∞ζf(ζ,y)dy. (40)

In fact, in Eqs. (39), (40) is the correction factor to the famous Casimir result [1] obtained for two ideal metals. It is equal to the function introduced in [12], multipled by .

The function in Eq. (40) can be presented in a more simple analytical form as follows. Presenting the logarithms in Eq. (20) as series and changing the order of integrations, one obtains

 ψDM(εD0)=452π4∞∑n=11n∫∞0ydye−ny (41) ψDM(εD0)×∫y0dζ{[rD∥(ζ,y)]n+[rD⊥(ζ,y)]n}.

Introducing the new variable , we rearrange Eq. (41) to the form

 ψDM(εD0)=452π4∞∑n=11n∫∞0y2dye−ny (42)

where

 rD∥(w)=εD0−√1+(εD0−1)w2εD0+√1+(εD0−1)w2, rD⊥(w)=√1+(εD0−1)w2−1√1+(εD0−1)w2+1. (43)

Calculating the integral in and performing the summation with respect to in Eq. (42) one arrives at

 ψDM(εD0)=45π4∫∞0dw{Li4[rD∥(w)]+Li4[rD⊥(w)]}. (44)

In Fig. 1 the function (44) is plotted versus as a solid line (when it goes to zero and when it goes to unity reproducing the limit of ideal metals).

It is notable that the model under consideration represents correctly the Casimir energy and pressure (39) at in only the retarded regime (i.e., at sufficiently large separations). As to the thermal corrections in Eqs. (25) and (37), the obtained expressions are valid also at short separations in a nonretarded regime under the condition that the parameter is sufficiently small due to sufficiently low temperature.

From Eqs. (12) and (25) the asymptotic behavior of the Casimir entropy in the limit of small is given by

 S(a,T)=3kBζ(3)(εD0−1)2128π3a2(εD0+1)τ2 (45) S(a,T)×⎡⎢ ⎢ ⎢ ⎢⎣1−8π2(εD0+1)(1−2εD0√εD0+(εD0)2√εD0)135ζ(3)(εD0−1)2τ+O(τ2)⎤⎥ ⎥ ⎥ ⎥⎦.

As is seen from Eq. (45), the entropy of the Casimir interaction between metal and dielectric plates vanishes with vanishing temperature as is required by the Nernst heat theorem (note that the first term of order in Eq. (45) was obtained in [50]). The important property of the perturbation expansions in powers of in Eqs. (25), (37) and (45) is that it is impermissible to consider the limiting case in order to obtain the case of two ideal metals like it was discussed above in application to Eq. (39). The mathematical reason is that in the power expansion of functions depending on as a parameter the limiting transitions and are not interchangeable. Of great importance is the possibility to apply Eq. (45) at as small as is wished. This is the principal advantage of analytical calculations as compared to numerical ones.

Now we consider the opposite limiting case , i.e., the limit of high temperatures (large separations). Here the main contribution to the free energy (14) is given by the term with whereas all terms with are exponentially small [8],

 F(a,T)=ℏcτ64π2a3∫∞0ydyln(1−r0e−y). (46)

By integrating in Eq. (46) we obtain

 F(a,T)=−kBT16πa2Li3(r0). (47)

For the Casimir pressure and entropy at from Eqs. (5), (12) and (47) it follows

 P(a,T)=−kBT8πa3Li3(r0),S(a,T)=kB16πa2Li3(r0). (48)

## 4. Thermal Casimir force between metal and dielectric with frequency-dependent dielectric permittivities

In this section we obtain the analytic expressions for the low-temperature behavior of the Casimir interaction between metal and dielectric plates taking into account the dependence of their dielectric permittivities on the frequency. The metal plate is described by the dielectric permittivity of the plasma model,

 εM(iξl)=1+ω2pξ2l, (49)

where is the plasma frequency, and is the plasma wavelength. In the theory of the thermal Casimir force this description was first used in [37, 38] and was shown to work good at separations between plates greater than the plasma wavelength. At such separations the characteristic frequency of the Casimir effect belongs to the region of infrared optics where the relaxation processes do not play any role [57].

For dielectric plate we use the Ninham-Parsegian representation of the dielectric permittivity along the imaginary frequency axis [58, 59],

 εD(iξl)=1+∑jCj1+ξ2lω2j. (50)

Here are the absorption strengths satisfying the condition

 ∑jCj=εD0−1, (51)

and are the characteristic absorption frequencies [recall that now ]. Eq. (50) gives a very accurate approximate description of the dielectric properties for many dielectrics. It has been successfully used by many authors for the comparison of experimental data with theory [60].

From Eq. (50) we return to the same values (15) of the reflection coefficients of the dielectric plate at zero frequency as were obtained in the simplified model of the frequency-independent dielectric permittivity. Thus, due to the zero value of , the transverse electric mode at zero frequency does not contribute to the free energy (9) of the Casimir interaction between metal and dielectric regardless of the value of for a metal. As was told in the Introduction, there are different approaches on how to correctly calculate the transverse electric coefficient at zero frequency, , for a plate made of real metal. In the configuration of metal and dielectric this problem, however, does not influence the result. Note that if we would use instead of Eq. (49) the Drude model, taking relaxation into account, the prime perturbation orders in all results below remain unchanged for metals with perfect crystal lattices. The role of impurities in the validity of the Nernst heat theorem in the case of two metal plates is discussed in [31, 33, 41, 42, 43, 44, 61].

We start from Eq. (9) for the free energy. Once again, using the Abel-Plana formula, Eq. (9) can be represented by Eq. (17) as the sum of in Eq. (18) and in Eqs. (19) and Eq. (20), where we mark by a hat all quantities related to real metal and dielectric. The single difference is that the function in Eq. (20) should be replaced by

 ^f(x,y)≡^f∥(x,y)+^f⊥(x,y), (52) ^f∥,⊥(x,y)=yln[1−^rM∥,⊥(x,y)^rD∥,⊥(x,y)e−y].

It is notable that for real metal and dielectric and in Eq. (17) may lose the obvious meaning of the energy at zero temperature and the thermal correction to it. In fact, this meaning is preserved only in the case when the dielectric permittivities do not depend on the temperature as a parameter like it was in Section 3. In the latter case it holds

 ΔF(a,T)=F(a,T)−F(a,0)=F(a,T)−E(a) (53)

in accordance with the intuitive definition of the thermal correction. If, however, or or both depend on the temperature as a parameter, Eq. (53) is violated. In this case defined in Eq. (19) takes into account only the part of temperature dependence of the free energy originating from the Matsubara frequencies and is not equal to . Moreover, in this case in Eqs. (17) and (18) is in fact temperature-dependent and it would be more correct to use the notation .

To obtain the analytic expressions of our interest we develop the perturbation theory in two small parameters and , where is the penetration depth of the electromagnetic oscillations into a metal. For the sake of simplicity we will consider dielectrics which can be described by Eq. (50) with only one oscillator, i.e., with . The high-resistivity Si is a typical example of such materials. The function in Eq. (20) can be conveniently presented in the form

 ^F(x)=^F∥(x)+^F⊥(x),^F∥,⊥(x)=∫∞xdy^f∥,⊥(x,y). (54)

As a first step we perform the expansion with respect to the powers of small parameter . This results in:

 ^F∥(x)=∫∞xydyln[1−^rD∥(x,y)e−y]+2x2η∫∞xdy^rD∥(x,y)ey−^rD∥(x,y) ^F∥(x)−2x4η2∫∞xdyey^rD∥(x,y)y[ey−^rD∥(x,y)]2+O(η3), (55) ^F⊥(x)=∫∞xydyln[1−^rD⊥(x,y)e−y]+2η∫∞xy2dy^rD⊥(x,y)ey−^rD⊥(x,y) ^F∥(x)−2η2∫∞xy3dyey^rD⊥(x,y)[ey−^rD⊥(x,y)]2+O(η3).

The dielectric reflection coefficients in Eq. (55) are obtained after the substitution of Eqs. (50) and (51) with in Eq. (10):

 ^rD∥(x,y)=(1+εD0−11+b2x2)y−√y2+εD0−11+b2x2x2(1+εD0−11+b2x2)y+√y2+εD0−11+b2x2x2, ^rD⊥(x,y)=√y2+εD0−11+b2x2x2−y√y2+εD0−11+b2x2x2+y, (56)

where .

Let us consecutively consider the contributions to from the terms of order , and in Eq. (55). As to the terms of order [the first and fourth lines in Eq. (55)], the expansion in powers of small (small ) performed using Eq. (56) leads to

 ^Fη0