Recent Results on Thermal Casimir Force Between Dielectrics and Related Problems
Abstract
We review recent results obtained in the physics of the thermal Casimir force acting between two dielectrics, dielectric and metal, and between metal and semiconductor. The detailed derivation for the lowtemperature behavior of the Casimir free energy, pressure and entropy in the configuration of two real dielectric plates is presented. For dielectrics with finite static dielectric permittivity it is shown that the Nernst heat theorem is satisfied. Hence, the Lifshitz theory of the van der Waals and Casimir forces is demonstrated to be consistent with thermodynamics. The nonzero dc conductivity of dielectric plates is proved to lead to a violation of the Nernst heat theorem and, thus, is not related to the phenomenon of dispersion forces. The lowtemperature asymptotics of the Casimir free energy, pressure and entropy are derived also in the configuration of one metal and one dielectric plate. The results are shown to be consistent with thermodynamics if the dielectric plate possesses a finite static dielectric permittivity. If the dc conductivity of a dielectric plate is taken into account this results in the violation of the Nernst heat theorem. We discuss both the experimental and theoretical results related to the Casimir interaction between metal and semiconductor with different charge carrier density. Discussions in the literature on the possible influence of spatial dispersion on the thermal Casimir force are analyzed. In conclusion, the conventional Lifshitz theory taking into account only the frequency dispersion remains the reliable foundation for the interpretation of all present experiments.
Keywords: Casimir force; Lifshitz theory; thermal corrections.
Received 26 May 2006
1 Introduction
The Casimir effect is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zeropoint oscillations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or molecules. During more than fifty years, passed after the discovery of the Casimir effect, it has attracted much theoretical attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, mathematical physics, and in nanotechnology (see monographs ?–? and reviews ?–?). In multidimensional KaluzaKlein supergravity the Casimir effect was used as a mechanism for spontaneous compactification of extra spatial dimensions and for constraining the Yukawatype corrections to Newtonian gravity. In quantum chromodynamics the Casimir energy plays an important role in the bag model of hadrons. In cavity quantum electrodynamics the Casimir interaction between an isolated atom and a cavity wall leads to the level shifts of atomic electrons depending on the position of the atom near the wall. Both the van der Waals and Casimir forces are used for the theoretical interpretation of recent experiments on quantum reflection and BoseEinstein condensation of ultracold atoms on or near the cavity wall of different nature. In condensed matter physics the Casimir effect turned out to be important for interaction of thin films, in wetting processes, and in the theory of colloids and lattice defects. The Casimir force was used to actuate nanoelectromechanical devices and to study the absorption of hydrogen atoms by carbon nanotubes. Theoretical work on the calculation of the Casimir energies and forces stimulated important achievements in mathematical physics and in the theory of renormalizations connected with the method of generalized zeta function and heat kernel expansion. All this made the Casimir effect the subject of general interdisciplinary interest and attracted permanently much attention in the scientific literature.
The last ten years were marked by the intensive experimental investigation of the Casimir force between metallic test bodies (see Refs. ?–?). During this time the agreement between experiment and theory on the level of 12% of the measured force was achieved. This has become possible due to the use of modern laboratory techniques, in particular, of atomic force microscopes and micromechanical torsional oscillators. Metallic test bodies provide advantage in comparison with dielectrics because their surfaces avoid charging. In Refs. ?, ?, where the importance of the Casimir effect for nanotechnology was pioneered, it was demonstrated that at separations below 100 nm the Casimir force becomes larger than the typical electrostatic forces acting between the elements of microelectromechanical systems. Bearing in mind that the miniaturization is the main tendency in modern technology, it becomes clear that the creation of new generation of nanotechnological devices with further decreased elements and separations between them would become impossible without careful account and calculation of the Casimir force.
Successful developments of nanotechnologies based on the Casimir effect calls for more sophisticated calculation methods of the Casimir forces. Most of theoretical output produced during the first decades after Casimir’s discovery did not take into account experimental conditions and real material properties of the boundary bodies, such as surface roughness, finite conductivity and nonzero temperature. The basic theory giving the unified description of both the van der Waals and Casimir forces was elaborated by Lifshitz shortly after the publication of Casimir’s paper. It describes the boundary bodies in terms of the frequency dependent dielectric permittivity at nonzero temperature . In the applications of the Lifshitz theory to dielectrics it was supposed that the static dielectric permittivity (i.e., the dielectric permittivity at zero frequency) is finite. The case of ideal metals was obtained from the Lifshitz theory by using the socalled Schwinger’s prescription, i.e., that the limit should be taken first and the static limit second. For ideal metals the same result, as follows from the Lifshitz theory combined with the Schwinger’s prescription, was obtained independently in the framework of thermal quantum field theory in Matsubara formulation. However, the cases of real dielectrics and metals (which possess some nonzero dc conductivity at and finite dielectric permittivity at nonzero frequencies, respectively) remained practically unexplored for a long time. The case of semiconductor boundary bodies was also unexplored despite of the crucial role of semiconductor materials in nanotechnology.
Starting in 2000, several theoretical groups in different countries attempted to describe the Casimir interaction between real metals at nonzero temperature in the framework of Lifshitz theory. They have used different models of the metal conductivity and arrived to controversial conclusions. In Ref. ?, using the dielectric function of the Drude model, quite different results than for ideal metals were obtained. According to Ref. ?, at short separations (low temperatures) the thermal correction to the Casimir force acting between real metals is several hundred times larger than between ideal ones. In addition, at large separations of a few micrometers (high temperatures) a two times smaller magnitude of the thermal Casimir force was found than between ideal metals (the latter is known as “the classical limit”). In Refs. ?, ? the dielectric permittivity of the plasma model was used to describe real metals and quite different results were obtained. At short separations the thermal correction appeared to be small in qualitative agreement with the case of ideal metals. At large separations for real metals the familiar classical limit was reproduced. Later the approach of Ref. ? was supported in Refs. ?, ?. The plasma model approach can be used at such separations that the characteristic frequency belongs to the region of infrared optics. Later a more general framework, namely the impedance approach was suggested which is applicable at any separation larger than the plasma wave length. It was supported in Refs. ?, ?, ?. In the region of the infrared optics, the impedance approach leads to practically the same results as the plasma model approach. As was shown in Refs. ?, ?, the Drude model approach leads to the violation of the Nernst heat theorem when applied to perfect metal crystal lattices with no impurities. This approach was also excluded by experiment at 99% confidence in the separation region from 300 to 500 nm and at 95% confidence in the wider separation region from 170 to 700 nm. On the contrary, the plasma model and impedance approaches were shown to be in agreement with thermodynamics and consistent with experiment. The polemic between different theoretical approaches to the description of the thermal Casimir force in the case of real metals can be found in Refs. ?, ?–?.
These findings on the application of the Lifshitz theory to real metals have inspired a renewed interest in the Casimir force between dielectrics. As was mentioned above, at nonzero temperature dielectrics possess an although small but not equal to zero dc conductivity. In Ref. ?, ? the van der Waals force arising from the dc conductivity of a dielectric plate was shown to lead to large effect in noncontact atomic friction, a phenomenon having so far no satisfactory theoretical explanation. This brings up the question: Is it necessary or possible to take into account the dc conductivity of dielectrics in the Lifshitz theory? Recall that in the case of a positive answer the static dielectric permittivity of a dielectric material would be infinitely large. It is amply clear that the resolution of the above issue should be in accordance with the fundamentals of thermodynamics. For this reason, it is desirable to investigate the lowtemperature behavior of the Casimir free energy and entropy for two dielectric plates both with neglected and included effects of the dc conductivity.
A major breakthrough in the investigation of this problem was achieved in the year 2005. In Ref. ? a new variant of perturbation theory was developed in a small parameter proportional to the product of the separation distance between the plates and the temperature. As a result, the behavior of the Casimir free energy, entropy and pressure at low temperatures was found analytically. If the static dielectric permittivity is finite, the thermal correction was demonstrated to be in accordance with thermodynamics. This solves positively the fundamental problem about the agreement between the Lifshitz theory and thermodynamics for the case of two dielectric plates. In Ref. ? it was shown that, on the contrary, the formal inclusion of a small conductivity of dielectric plates at low frequencies into the model of their dielectric response leads to a violation of the Nernst heat theorem. This result gives an important guidance on how to extrapolate the tabulated optical data for the complex refractive index to low frequencies in numerous applications of the van der Waals and Casimir forces. All these problems and related ones arising for semiconductor materials are discussed in this review.
In Sec. 2 we derive the analytical behavior of the Casimir free energy, entropy and pressure in the configuration of two parallel dielectric plates at both low and high temperature. It is demonstrated that if the static dielectric permittivity is finite the Lifshitz theory is in agreement with thermodynamics. Sec. 3 contains the derivation of the lowtemperature behavior for the Casimir free energy and entropy between two dielectric plates with included dc conductivity. In this case the Lifshitz theory is found to be in contradiction with the Nernst heat theorem. The conclusion is made that the conductivity properties of a dielectric material at a constant current are unrelated to the van der Waals and Casimir forces and must not be included into the model of dielectric response. In Sec. 4 we consider the thermal Casimir force between metal and dielectric. This problem was first investigated in Ref. ?. It was found that for dielectric plate with finite static dielectric permittivity the Nernst heat theorem is satisfied but the Casimir entropy may take negative values. Here we not only reproduce an analytical proof of the Nernst heat theorem but also find the next perturbation orders in the expansion of the Casimir free energy and entropy in powers of a small parameter. The results obtained in hightemperature limit are also provided. Sec. 5 is devoted to the Casimir interaction between metal and dielectric plates with included dc conductivity of the dielectric material. We demonstrate that in this case the Nernst heat theorem is violated. Sec. 6 contains the discussion of semiconductors which present a wide variety of electric properties varying from metallic to dielectric. We consider the Casimir interaction between metal and semiconductor test bodies and formulate the criterion when it is appropriate to include the dc conductivity of a semiconductor into the model of dielectric response. In doing so, the results of recent experiments on the measurement of the Casimir force between metal and semiconductor test bodies are taken into account. To this point the assumption has been made that metal, dielectric and semiconductor materials of the Casimir plates possess only temporal dispersion, i.e., can be described by the dielectric permittivity depending only on frequency. In Sec. 7 we discuss recent controversial results by different authors (see, e.g., Refs. ? – ?) attempting to take into account also spatial dispersion. As is shown in this section, the way of inclusion of spatial dispersion into the Lifshitz theory, used in Refs. ? – ?, is unjustified. We argue that the account of spatial dispersion cannot influence theoretical results obtained with the help of usual, spatially local, Lifshitz theory within presently used ranges of experimental separations. Sec. 8 contains our conclusions and discussion.
2 New Analytical Results for the Thermal Casimir Force Between Dielectrics
We consider two thick dielectric plates (semispaces) described by the frequencydependent dielectric permittivity and restricted by the parallel planes with a separation between them, in thermal equilibrium at temperature . The Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the plates is given by
(1)  
where the reflection coefficients for two independent polarizations of electromagnetic field are defined as
(2) 
Here is the magnitude of the wave vector in the plane of plates, are the Matsubara frequencies, is the Boltzmann constant, , and
(3) 
The problems in the application of the Lifshitz theory to real materials discussed in the Introduction are closely connected with the values of the reflection coefficients at zero Matsubara frequency. For later use we discuss it for the various cases.

For ideal metals it holds
(4) 
For real metals described by the dielectric function of the Drude model,
(5) where is the plasma frequency and is the relaxation parameter, it holds
(6) Eq. (6) results in the discontinuity between the cases of ideal and real metals and leads to the violation of the Nernst heat theorem for metallic plates having perfect crystal lattices.

For real metals described by the dielectric function of the plasma model,
(7) from Eq. (2) it follows:
(8) Here, in the limit of ideal metals () the continuity is preserved because in Eq. (8) goes to unity. The free energy (2) calculated with the permittivity (8) is also consistent with thermodynamics.

For dielectrics and semiconductors the dielectric permittivities at the imaginary Matsubara frequencies are given by the NinhamParsegian representation,
(9) where the parameters are the absorption strengths satisfying the condition
(10) and are the characteristic absorption frequencies. Here, the static dielectric permittivity is supposed to be finite. Although Eq. (10) is an approximate one, it gives a very accurate description for many materials. By the substitution of Eq. (9) in Eq. (2) one arrives at
(11)
Note that the vanishing of the transverse reflection coefficient for dielectrics at zero frequency in Eq. (11) has another meaning than for the Drude metals in Eq. (6). For Drude metal the parallel reflection coefficient is equal to the physical value for real photons at normal incidence, i.e., to unity, and the transverse one vanishes instead of taking unity, its physical value. This results in the violation of the Nernst heat theorem for perfect crystal lattices. In the case of dielectrics both reflection coefficients at zero frequency in Eq. (11) depart from the physical value for real photons which is equal to . In this case, however, one of them is larger and the other one is smaller than the physical value. As we will see below, this leads to the preservation of Nernst’s heat theorem confirming that Eq. (9), despite being approximate, describes the material properties of dielectric and semiconductor plates in a thermodynamic consistent way.
Now we derive the analytic representation for the Casimir free energy in Eq. (2) at low temperatures. For convenience in calculations, we introduce the dimensionless variables
(12) 
where is the characteristic frequency, , and was defined in Eq. (3). Then the Lifshitz formula (2) takes the form
(13) 
where
(14)  
(15) 
and reflection coefficients (2), in terms of variables (12), being given by
(16) 
To separate the temperature independent contribution and thermal correction in Eq. (13) we apply the AbelPlana formula,
(17) 
where is an analytic function in the right halfplane. Here, taking it as
(18) 
and using Eq. (17), we can identically rearrange Eq. (13) to the form
(19) 
where is the energy of the van der Waals or Casimir interaction at zero temperature,
(20) 
and is the thermal correction to this energy,
(21) 
Note that, in fact, Eq. (21) describes the dependence of the free energy on the temperature arising from the dependence on temperature of the Matsubara frequencies. Thus, in (21) coincides with the thermal correction to the energy, defined as , only for plate materials with temperature independent properties.
The asymptotic expressions for the energy at both short and large separations are well known. Below we find the asymptotic expressions for the thermal correction (21) under the conditions and . Taking into account the definition of in Eq. (12), the asymptotic expressions at are applicable both at small and large separations if the temperature is sufficiently low.
We begin with condition . Let us substitute Eq. (9) in Eqs. (14) – (16), expand the function in powers of , and than integrate the obtained expansion with respect to from to infinity in order to find in Eq. (18) and in Eq. (21).
It is easy to check that does not contribute to the leading, second, order in the expansion of in powers of . Thus, we can restrict ourselves by the consideration of the expansion
(22) 
where was defined in Eq. (11). Note that for simplicity we consider here only one oscillator in Eq. (9) and put . The case of several oscillator modes can be considered in an analogous way.
As a next step, we integrate Eq. (22) term by term according to Eq. (18), expand the partial results in powers of and sum up the obtained series. Thereby we obtain the following expressions:
(23)  
(24)  
(25) 
where Li is the polylogarithm function and Ei is the exponential integral function.
From these equations it follows
(26)  
(27) 
and, thus, and do not contribute to the leading order in the expansion of . The latter is determined by only. As a result, we arrive at
(28) 
where was substituted from Eq. (11) and was introduced for the still unknown real coefficient of the next to leading order resulting from and as well as, possibly, from . At this stage it is difficult to determine the value of this coefficient because all powers in the expansion of contribute to it. Remarkably, the two leading orders depend only on the static dielectric permittivity and are not influenced by the dependence of the dielectric permittivity on the frequency contained in .
Substituting Eq. (28) in Eq. (21) and using Eq. (19), we obtain
(29) 
where and is the Riemann zeta function.
So far we have considered the free energy. The thermal pressure is obtained as
(30) 
where is the Casimir pressure at zero temperature.
In order to determine the value of the coefficient of the leading term, we express the pressure directly through the Lifshitz formula
(31) 
Again, applying the AbelPlana formula (17), we represent the pressure as follows,
(32) 
where the thermal correction to , the pressure at zero temperature, is
(33) 
and the function is given by
(34) 
First, we determine the leading term of the expansion of in powers of . For this purpose, let us introduce the new variable and note that the reflection coefficient depends on only through the frequency dependence of given by Eq. (9). Thus, we can rewrite and expand Eq. (34) as follows:
(35) 
where, according to Eq. (16),
(36) 
Integration in Eq. (35) with account of Eq. (36) results in
(37) 
from which it follows:
(38) 
The expansion of from Eq. (34) in powers of is somewhat more cumbersome. It can be performed in the following way. As is seen from the second equality in Eq. (26), the dependence of the dielectric permittivity on frequency contributes to starting from only the 5th power in . Bearing in mind the connection between free energy and pressure, we can conclude that the dependence on the frequency contributes to starting from the 4th order. We are looking for the lowest (third) order expansion term of . Because of this, it is permissible to disregard the frequency dependence of and describe the dielectric by its static dielectric permittivity.
To begin with, we identically rearrange in Eq. (34) by subtracting and adding the two first expansion terms of the function under the integral in powers of ,
(39) 
and consider these three integrals separately. The first integral in terms of the new variable reads
(40) 
where, in accordance with Eq. (16),
(41) 
Expanding in powers of and explicitly calculating the remaining integrals for the lowest, third, power of results in
The second and third integrals on the righthand side of Eq. (39) are simply determined with the following result:
(43)  
(44) 
Substituting Eqs. (2), (43) and (44) into , we arrive at
(45) 
Then, by summing Eqs. (38) and (45), the result is obtained
(46) 
Now we substitute Eq. (46) in Eq. (33) and perform integration. Finally, from Eq. (32) the desired expression for the Casimir pressure is derived
(47) 
By comparison with Eq. (30) the explicit form of the coefficient is found as
(48) 
and, thus, both two first perturbation orders in the expansion for the free energy (29) are determined.
Equations (29), (47) and (48) solve the fundamental problem of the thermodynamic consistency of the Lifshitz theory in the case of two dielectric plates. From Eqs. (29) and (48) the entropy of the van der Waals and Casimir interaction between plates takes the form
(49)  
As is seen from Eq. (49), in the limit () the lower order contributions to the entropy are of the second and the third powers in the small parameter . Thus, the entropy vanishes when the temperature goes to zero as it must be in accordance with the third law of thermodynamics (the Nernst heat theorem).
A similar behavior was obtained for ideal metals and for real metals described by the plasma model. For example, in the case of plates made of ideal metal the entropy at low temperatures is given by
(50) 
Note, however, that the expansion coefficients in Eq. (50) cannot be obtained as a straightforward limit in Eq. (49) and the above equations for the free energy and pressure. The mathematical reason is that it is impermissible to interchange the limiting transitions and in the power expansions of functions depending on as a parameter.
Remarkably, the lowtemperature behavior of the free energy, pressure and entropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is determined only by the static dielectric permittivity. The absorption bands included in Eq. (9) do not influence the lowtemperature behavior. A more simple derivation of the results (29), (47)–(49) for dielectrics with constant is contained in Ref. ?. As was demonstrated above, all these results remain unchanged if the dependence of dielectric permittivity on frequency is taken into account.
In Ref. ? more general results were obtained related to two dissimilar dielectric plates with dielectric permittivities and . For brevity here we present only the final expressions for the lowtemperature behavior of the Casimir free energy, pressure and entropy between dissimilar plates. They are as follows:
(51)  
Here and the coefficient is given by
(52)  
It is easily seen that in the limit equations (51), (52) coincide with equations (29), (47)–(49) having obtained above. Note that in the application region of lowtemperature asymptotic expressions the entropy of the Casimir interaction between dielectric plates is nonnegative.
The obtained analytic behavior of the free energy, pressure and entropy at low temperatures can be compared with the results of numerical computations using the Lifshitz formula. Dielectric properties of the plates can be described by the static dielectric permittivity or more precisely using the optical tabulated data for the complex index of refraction. As an example, in Fig. 1 we present the thermal corrections to the Casimir energy (a) and pressure (b) at a separation nm as functions of temperature in the configuration of two dissimilar plates made of highresistivity Si and SiO. The dielectric permittivities of both materials along the imaginary frequency axis were computed in Ref. ? using the optical data of Ref. ?. The precise thermal corrections computed by taking into account these permittivities are shown by the solid lines and corrections computed by our analytical asymptotic expressions are shown by the longdashed lines. Shortdashed lines indicate the results computed by the Lifshitz formula with constant dielectric permittivities of Si and SiO equal to and , respectively. As is seen in Fig. 1a,b, at K the results obtained using the analytical asymptotic expressions practically coincide with the solid lines computed using the tabulated optical data for the materials of the plates.
Now we return to the case of two similar dielectric plates and consider the asymptotic expressions under the condition , i.e., at high temperatures (large separations). It is well known that in this case the approximation of static dielectric permittivity works good and the main contribution is given by the zerofrequency term of the Lifshitz formula (13)
(53) 
(the other terms being exponentially small). Performing the integration in Eq. (53) we obtain
(54) 
In a similar manner for the Casimir pressure and entropy at it follows