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International Journal of Modern Physics A
c© World Scientific Publishing Company
RECENT RESULTS ON THERMAL CASIMIR FORCE BETWEEN
DIELECTRICS AND RELATED PROBLEMS
B. GEYER, G. L. KLIMCHITSKAYA∗ and V. M. MOSTEPANENKO†
Center of Theoretical Studies and Institute for Theoretical Physics, Leipzig University,
Augustusplatz 10/11, 04109, Leipzig, Germany
Received 26 May 2006
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 low-temperature 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 the-
orem 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 low-temperature 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 thermo-
dynamics 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.
1. INTRODUCTION
The Casimir effect1 is the force and also the specific polarization of the vacuum
arising in restricted quantization volumes and originating from the zero-point oscil-
lations 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,
∗On leave from North-West Technical University, St.Petersburg, Russia.
†On leave from Noncommercial Partnership “Scientific Instruments”, Moscow, Russia.
http://arxiv.org/abs/0704.1040v1
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2 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
mathematical physics, and in nanotechnology (see monographs 2–4 and reviews
5–7). In multidimensional Kaluza-Klein supergravity the Casimir effect was used8
as a mechanism for spontaneous compactification of extra spatial dimensions and
for constraining the Yukawa-type corrections to Newtonian gravity.9,10,11,12,13 In
quantum chromodynamics the Casimir energy plays an important role in the bag
model of hadrons.3 In cavity quantum electrodynamics the Casimir interaction be-
tween 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.2 Both the van der Waals and
Casimir forces are used14,15 for the theoretical interpretation of recent experiments
on quantum reflection and Bose-Einstein 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.16 The Casimir force was used to actuate
nanoelectromechanical devices17 and to study the absorption of hydrogen atoms
by carbon nanotubes.18 Theoretical work on the calculation of the Casimir ener-
gies and forces stimulated important achievements in mathematical physics and in
the theory of renormalizations connected with the method of generalized zeta func-
tion and heat kernel expansion.6,19 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. 20–33). During this time
the agreement between experiment and theory on the level of 1-2% 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 be-
cause their surfaces avoid charging. In Refs. 34, 35, where the importance of the
Casimir effect for nanotechnology was pioneered, it was demonstrated that at sepa-
rations 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 be-
comes 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 theoreti-
cal 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 Lifshitz36,37,38 shortly after the publication of Casimir’s
paper. It describes the boundary bodies in terms of the frequency dependent dielec-
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Thermal Casimir force between dielectrics 3
tric permittivity ε(ω) at nonzero temperature T . 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 ob-
tained from the Lifshitz theory by using the so-called Schwinger’s prescription,39
i.e., that the limit |ε(ω)| → ∞ should be taken first and the static limit ω → 0
second. For ideal metals the same result, as follows from the Lifshitz theory com-
bined with the Schwinger’s prescription, was obtained independently in the frame-
work of thermal quantum field theory in Matsubara formulation.40 However, the
cases of real dielectrics and metals (which possess some nonzero dc conductivity
at T > 0 and finite dielectric permittivity at nonzero frequencies, respectively) re-
mained 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 con-
ductivity and arrived to controversial conclusions. In Ref. 41, using the dielectric
function of the Drude model, quite different results than for ideal metals were ob-
tained. According to Ref. 41, 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 microm-
eters (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”42,43). In Refs. 44, 45 the dielectric permittivity of the plasma model was
used to describe real metals and quite different results were obtained. At short sep-
arations 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. 41 was supported in Refs. 46,
47. The plasma model approach can be used at such separations a that the char-
acteristic frequency c/(2a) belongs to the region of infrared optics. Later a more
general framework, namely the impedance approach was suggested48,49 which is
applicable at any separation larger than the plasma wave length. It was supported
in Refs. 7, 50, 51. 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. 52, 53, the Drude model approach leads to the violation of the Nernst heat
theorem when applied to perfect metal crystal lattices with no impurities. This ap-
proach was also excluded by experiment at 99% confidence in the separation region
from 300 to 500nm and at 95% confidence in the wider separation region from 170
to 700 nm.30,32,33 On the contrary, the plasma model and impedance approaches
were shown to be in agreement with thermodynamics and consistent with exper-
iment. 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. 47, 54–56.
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4 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
These findings on the application of the Lifshitz theory to real metals have in-
spired 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. 57, 58 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,59 a phenomenon having so far no satisfactory theoretical explana-
tion. 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 de-
sirable to investigate the low-temperature 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. 60 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. 60 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 low-temperature 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. 61.
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
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Thermal Casimir force between dielectrics 5
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 high-temperature
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 appro-
priate to include the dc conductivity of a semiconductor into the model of dielectric
response. In doing so, the results of recent experiments62,63 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. 64
– 68) 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. 64 – 68, 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 frequency-
dependent dielectric permittivity ε(ω) and restricted by the parallel planes z = ±a/2
with a separation a between them, in thermal equilibrium at temperature T . The
Lifshitz formula for the free energy of the van der Waals and Casimir interaction
between the plates is given by2,3,4,5,6,36,37,38
F(a, T ) = kBT
1− δl0
k⊥dk⊥ (1)
1− r2‖(ξl, k⊥)e−2aql
1− r2⊥(ξl, k⊥)e−2aql
where the reflection coefficients for two independent polarizations of electromagnetic
field are defined as
r‖(ξl, k⊥) =
εlql − kl
εlql + kl
, r⊥(ξl, k⊥) =
kl − ql
kl + ql
. (2)
Here k⊥ is the magnitude of the wave vector in the plane of plates, ξl = 2πkBT l/~
are the Matsubara frequencies, kB is the Boltzmann constant, εl = ε(iξl), and
+ k2⊥, kl =
ε(iξl)
+ k2⊥. (3)
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6 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
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 metals40 it holds
r‖(0, k⊥) = r⊥(0, k⊥) = 1. (4)
• For real metals described by the dielectric function of the Drude model,
ε(iξl) = 1 +
ξl[ξl + ν(T )]
, (5)
where ωp is the plasma frequency and ν(T ) is the relaxation parameter, it
holds41
r‖(0, k⊥) = 1, r⊥(0, k⊥) = 0. (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.52,53,54,56
• For real metals described by the dielectric function of the plasma model,
ε(iξl) = 1 +
, (7)
from Eq. (2) it follows:44,45
r‖(0, k⊥) = 1, r⊥(0, k⊥) =
c2k2⊥ + ω
p − ck⊥
c2k2⊥ + ω
p + ck⊥
. (8)
Here, in the limit of ideal metals (ωp → ∞) the continuity is preserved
because r⊥(0, k⊥) in Eq. (8) goes to unity. The free energy (1) 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 Ninham-Parsegian
representation,69,70
ε(iξl) = 1 +
1 + ξ2l /ω
, (9)
where the parameters Cj are the absorption strengths satisfying the condi-
Cj = ε0 − 1 (10)
and ωj are the characteristic absorption frequencies. Here, the static dielec-
tric permittivity ε0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an
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Thermal Casimir force between dielectrics 7
approximate one, it gives a very accurate description for many materials.71
By the substitution of Eq. (9) in Eq. (2) one arrives at
r‖(0, k⊥) ≡ r0 =
ε0 − 1
ε0 + 1
, r⊥(0, k⊥) = 0. (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 (
ε0−1)/(
ε0+1). 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. (1)
at low temperatures. For convenience in calculations, we introduce the dimensionless
variables
= τl, y = 2aql, (12)
where ξc = c/(2a) is the characteristic frequency, τ = 4πkBaT/(~c), and ql was
defined in Eq. (3). Then the Lifshitz formula (1) takes the form
F(a, T ) = ~cτ
32π2a3
1− δl0
dy f(ζl, y) , (13)
where
f(ζ, y) = f‖(ζ, y) + f⊥(ζ, y), (14)
f‖,⊥(ζ, y) = y ln
1− r2‖,⊥(ζ, y)e−y
, (15)
and reflection coefficients (2), in terms of variables (12), being given by
r‖(ζl, y) =
εly −
y2 + ζ2l (εl − 1)
εly +
y2 + ζ2l (εl − 1)
, r⊥(ζl, y) =
y2 + ζ2l (εl − 1)− y
y2 + ζ2l (εl − 1) + y
. (16)
To separate the temperature independent contribution and thermal correction
in Eq. (13) we apply the Abel-Plana formula,3,6
1− δl0
F (l) =
F (t) dt+ i
F (it)− F (−it)
e2πt − 1
, (17)
where F (z) is an analytic function in the right half-plane. Here, taking it as
F (x) =
dy f(x, y) (18)
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8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
and using Eq. (17), we can identically rearrange Eq. (13) to the form
F(a, T ) = E(a) + ∆F(a, T ) , (19)
where E(a) is the energy of the van der Waals or Casimir interaction at zero tem-
perature,
E(a) =
32π2a3
dy f(ζ, y) , (20)
and ∆F(a, T ) is the thermal correction to this energy,
∆F(a, T ) =
32π2a3
F (iτt)− F (−iτt)
e2πt − 1
. (21)
Note that, in fact, Eq. (21) describes the dependence of the free energy on the tem-
perature arising from the dependence on temperature of the Matsubara frequencies.
Thus, ∆F(a, T ) in (21) coincides with the thermal correction to the energy, defined
as F(a, T )− F(a, 0), only for plate materials with temperature independent prop-
erties.
The asymptotic expressions for the energy E(a) at both short and large sep-
arations are well known.6,37,38 Below we find the asymptotic expressions for the
thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account
the definition of τ in Eq. (12), the asymptotic expressions at τ ≪ 1 are applicable
both at small and large separations if the temperature is sufficiently low.
We begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) – (16),
expand the function f(x, y) in powers of x = τt, and than integrate the obtained
expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and
F (ix)− F (−ix) in Eq. (21).
It is easy to check that f⊥(ζ, y) does not contribute to the leading, second, order
in the expansion of F (ix)− F (−ix) in powers of x. Thus, we can restrict ourselves
by the consideration of the expansion
f‖(x, y) = y ln
1− r20e−y
2ε0 r
ε0 + 1
y (ey − r20)
4ξ2c r
(ε0 + 1)ω
ey − r20
+O(x3), (22)
where r0 was defined in Eq. (11). Note that for simplicity we consider here only one
oscillator in Eq. (9) and put ωj = ω1. 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 x and sum up the obtained series. Thereby we obtain
the following expressions:
Z1(x) ≡
y dy ln
1− r20e−y
(1 + nx)e−nx
= −Li3(r20)−
ln(1− r20) + O(x3) , (23)
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Thermal Casimir force between dielectrics 9
Z2(x) ≡
ε0 + 1
y (ey − r20)
= − 2ε0x
ε0 + 1
r2n0 Ei(−nx) , (24)
Z3(x) ≡
4r20ξ
(ε0 + 1)ω
ey − r20
4ξ2cx
(ε0 + 1)ω
(1 + nx)e−nx
(ε0 + 1)ω
x2 Li2(r
1− r20
+O(x5)
, (25)
where Lin(z) is the polylogarithm function and Ei(z) is the exponential integral
function.
From these equations it follows
Z1(ix) − Z1(−ix) = O(x3) , Z3(ix)− Z3(−ix) = O(x5) , (26)
Z2(ix) − Z2(−ix) = 2iπ
ε0 + 1
1− r20
x2 +O(x3) , (27)
and, thus, Z1 and Z3 do not contribute to the leading order in the expansion of
F (ix)− F (−ix). The latter is determined by Z2 only. As a result, we arrive at
F (ix)− F (−ix) = iπ
(ε0 − 1)2
2(ε0 + 1)
x2 − iαx3 +O(x4), (28)
where r0 was substituted from Eq. (11) and α was introduced for the still unknown
real coefficient of the next to leading order resulting from Z1 and Z2 as well as,
possibly, from f⊥(ζ, y). At this stage it is difficult to determine the value of this
coefficient because all powers in the expansion of f(x, y) contribute to it. Remark-
ably, the two leading orders depend only on the static dielectric permittivity ε0 and
are not influenced by the dependence of the dielectric permittivity on the frequency
contained in Z3.
Substituting Eq. (28) in Eq. (21) and using Eq. (19), we obtain
F(a, T ) = E(a)− ~c
32π2a3
(ε0 − 1)2
ε0 + 1
τ3 − C4τ4 +O(τ5)
, (29)
where C4 ≡ α/240 and ζ(z) is the Riemann zeta function.
So far we have considered the free energy. The thermal pressure is obtained as
P (a, T ) = −∂F(a, T )
= P0(a)−
32π2a4
4 +O(τ5)
, (30)
where P0(a) = −∂E(a)/∂a is the Casimir pressure at zero temperature.
In order to determine the value of the coefficient C4 of the leading term, we
express the pressure directly through the Lifshitz formula
P (a, T ) = − ~cτ
32π2a4
1− δl0
(ζl, y)
ey − r2
(ζl, y)
r2⊥(ζl, y)
ey − r2⊥(ζl, y)
. (31)
Again, applying the Abel-Plana formula (17), we represent the pressure as follows,
P (a, T ) = P0(a) + ∆P (a, T ), (32)
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10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where the thermal correction to P0(a), the pressure at zero temperature, is
∆P (a, T ) = − i~cτ
32π2a4
Φ(iτt)− Φ(−iτt)
e2πt − 1
and the function Φ(x) is given by
Φ(x) ≡ Φ‖(x) + Φ⊥(x) , Φ‖,⊥(x) =
y2 r2
(x, y)
ey − r2
(x, y)
. (34)
First, we determine the leading term of the expansion of Φ⊥(x) in powers of
x. For this purpose, let us introduce the new variable v = y/x and note that the
reflection coefficient r⊥(x, v) depends on x only through the frequency dependence
of ε given by Eq. (9). Thus, we can rewrite and expand Eq. (34) as follows:
Φ⊥(x) = x
v2 r2⊥(x, v)
evx − r2⊥(x, v)
v2 r2⊥(v)
1− r2⊥(v)
+ O(x4), (35)
where, according to Eq. (16),
r⊥(v) ≡ r⊥(0, v) =
v2 + ε0 − 1− v√
v2 + ε0 − 1 + v
. (36)
Integration in Eq. (35) with account of Eq. (36) results in
Φ⊥(x) =
1− ε0(3− ε0)
+ O(x4) , (37)
from which it follows:
Φ⊥(ix)− Φ⊥(−ix) = −i
ε0 (3− ε0)
+ O(x5). (38)
The expansion of Φ‖(x) from Eq. (34) in powers of x is somewhat more cumber-
some. 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 F (ix)−F (−ix) starting from only the 5th power in x. Bearing in mind the con-
nection between free energy and pressure, we can conclude that the dependence on
the frequency contributes to Φ‖(ix)−Φ‖(−ix) starting from the 4th order. We are
looking for the lowest (third) order expansion term of Φ‖(ix)−Φ‖(−ix). Because of
this, it is permissible to disregard the frequency dependence of ε and describe the
dielectric by its static dielectric permittivity.
To begin with, we identically rearrange Φ‖(x) in Eq. (34) by subtracting and
adding the two first expansion terms of the function under the integral in powers
of x,
Φ‖(x) =
(x, y)
ey − r2
(x, y)
ey − r20
ε0 + 1
(1− r20e−y)
ey − r20
− x2 2ε0
ε0 + 1
(1− r20e−y)
, (39)
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Thermal Casimir force between dielectrics 11
and consider these three integrals separately. The first integral in terms of the new
variable v = y/x reads
Q1(x) ≡ x3
evx − r2
− v2 r
evx − r20
ε0 + 1
(1− r20e−vx)
, (40)
where, in accordance with Eq. (16),
r‖(v) ≡ r‖(0, v) =
ε0v −
v2 + ε0 − 1
ε0v +
v2 + ε0 − 1
. (41)
Expanding Q1(x) in powers of x and explicitly calculating the remaining inte-
grals for the lowest, third, power of x results in
Q1(x) = x
1− r2
1− r20
ε0 + 1
(1− r20)
+O(x4) (42)
1 + 3ε0 − 2 ε20
2 r20
1− r20
− 12 ε0
ε0 + 1
(1− r20)
+O(x4).
The second and third integrals on the right-hand side of Eq. (39) are simply deter-
mined with the following result:
Q2(x) ≡
ey − r20
(2 + 2nx+ n2x2)e−nx
= 3Li3(r
1− r20
+O(x4) , (43)
Q3(x) ≡ −
ε0 + 1
(1− r20e−y)
= − 2ε0x
ε0 + 1
1− r20e−x
= − 2ε0
ε0 + 1
1− r20
− x3 r
(1− r20)2
+O(x4) . (44)
Substituting Eqs. (42), (43) and (44) into Φ‖(x) = Q1(x) + Q2(x) +Q3(x), we
arrive at
Φ‖(ix)− Φ‖(−ix) = −i
2ε20 − 3ε0 − 1
+ O(x4). (45)
Then, by summing Eqs. (38) and (45), the result is obtained
Φ(ix)− Φ(−ix) = −i
2 + ε
0 − ε
0 − 2
+ O(x4). (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
P (a, T ) = P0(a)−
32π2a4
ε0 − 1
ε20 + ε
0 − 2
τ4 +O(τ5)
. (47)
By comparison with Eq. (30) the explicit form of the coefficient C4 is found as
ε0 − 1
ε20 + ε
0 − 2
/720 (48)
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12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
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 thermody-
namic 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
S(a, T ) = −
∂F(a, T )
3kBζ(3)(ε0 − 1)2
64π3a2(ε0 + 1)
τ2 (49)
2π2(ε0 + 1)
0 + 2ε0 + 2
ε0 + 2
135ζ(3)(
ε0 + 1)2
τ + O(τ2)
As is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions
to the entropy are of the second and the third powers in the small parameter τ .
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 metals40,72,73 and for real metals
described by the plasma model.45,52 For example, in the case of plates made of
ideal metal the entropy at low temperatures is given by40,72,73
S(a, T ) =
3kBζ(3)
32π3a2
1− 2π
135ζ(3)
τ + O(τ2)
. (50)
Note, however, that the expansion coefficients in Eq. (50) cannot be obtained as
a straightforward limit |ε0| → ∞ 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 τ → 0 and |ε0| → ∞ in the power expansions
of functions depending on ε0 as a parameter.
Remarkably, the low-temperature behavior of the free energy, pressure and en-
tropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is deter-
mined only by the static dielectric permittivity. The absorption bands included in
Eq. (9) do not influence the low-temperature behavior. A more simple derivation
of the results (29), (47)–(49) for dielectrics with constant ε is contained in Ref. 74.
As was demonstrated above, all these results remain unchanged if the dependence
of dielectric permittivity on frequency is taken into account.
In Ref. 60 more general results were obtained related to two dissimilar dielectric
plates with dielectric permittivities ε(1)(ω) and ε(2)(ω). For brevity here we present
only the final expressions for the low-temperature behavior of the Casimir free
energy, pressure and entropy between dissimilar plates. They are as follows:60
F(a, T ) = E(a)− ~c
32π2a3
0 + ε
0 + 2ε
0 + 1)(ε
0 + 1)
0 − 1)(ε
0 − 1)
0 + ε
− C4 τ +O(τ2)
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Thermal Casimir force between dielectrics 13
P (a, T ) = P0(a)−
32π2a4
4 +O(τ5)
, (51)
S(a, T ) =
3ζ(3)
0 + ε
0 + 2ε
0 + 1)(ε
0 + 1)
0 − 1)(ε
0 − 1)
0 + ε
− C4 τ +O(τ2)
Here ε
(1,2)
0 ≡ ε(1,2)(0) and the coefficient C4 is given by60
0 + ε
0 + ε
0 + 2ε
0 − ε
0 − 3ε
0 − 3ε
0 + 1
0 − ε
0 − ε
0 − 1
0 − 1
0 + ε
Artanh
0 + ε
0 − ε
0 − ε
It is easily seen that in the limit ε
0 = ε
0 = ε0 equations (51), (52) coincide with
equations (29), (47)–(49) having obtained above. Note that in the application region
of low-temperature 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 a = 400 nm
as functions of temperature in the configuration of two dissimilar plates made of
high-resistivity Si and SiO2. The dielectric permittivities of both materials along
the imaginary frequency axis were computed in Ref. 75 using the optical data of
Ref. 76. The precise thermal corrections computed by taking into account these
permittivities are shown by the solid lines and corrections computed by our analyt-
ical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines
indicate the results computed by the Lifshitz formula with constant dielectric per-
mittivities of Si and SiO2 equal to ε
0 = 11.67 and ε
0 = 3.84, respectively. As is
seen in Fig. 1a,b, at T < 60K the results obtained using the analytical asymptotic
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14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
25 50 75 100 125 150 175 200
T (K)
−∆F (pJ)
25 50 75 100 125 150 175 200
T (K)
−∆P (µPa)
Fig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration
of two plates, one made of Si and another one of SiO2, at a separation a = 400 nm as a function of
temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated
optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed
lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines).
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 τ ≫ 1, i.e., at high temperatures (large
separations). It is well known37,38,77 that in this case the approximation of static
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Thermal Casimir force between dielectrics 15
dielectric permittivity works good and the main contribution is given by the zero-
frequency term of the Lifshitz formula (13)
F(a, T ) = ~cτ
64π2a3
ydy ln
1− r20e−y
(the other terms being exponentially small). Performing the integration in Eq. (53)
we obtain
F(a, T ) = − kBT
16πa2
. (54)
In a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows
P (a, T ) = −
, S(a, T ) =
16πa2
. (55)
Equations (54) and (55) are simply generalized60 for the case of two dissimilar di-
electric plates by performing the replacement r20 → r
0 , where r
(1,2)
0 are defined
by Eq. (11) with the static dielectric permittivities of dissimilar plates ε
(1,2)
3. IS THE DC CONDUCTIVITY RELATED TO THE CASIMIR
INTERACTION BETWEEN DIELECTRICS?
As was discussed in the previous section, the zero-frequency term in formula (13),
i.e., the contribution with l = 0, is of prime importance and determines many of
the basic properties of the Casimir interaction. In the above consideration we have
described dielectric materials by Eq. (9) with finite static dielectric permittivity ε0.
This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger
and the other one is smaller than the physical value for real photons at normal
incidence. However, in the Lifshitz theory, the departure of both coefficients from
their physical values is coordinated in such a way that the Nernst heat theorem
remains valid.
It is common knowledge that at nonzero temperatures dielectric materials pos-
sess a negligibly small but not equal to zero dc conductivity. From physical intuition
it is reasonable to expect that the influence of this conductivity on the van der Waals
and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the
contrary, the inclusion of small dielectric dc conductivity in the model of dielectric
response leads to a large effect in dispersion forces. This raises the question if dc
conductivity is related to dispersion forces or if, on the contrary, the zero-frequency
contribution should be understood not literally but as an analytic continuation from
the region of high frequencies determining the physical phenomenon of dispersion
forces.
To illustrate this problem in more details, we consider the asymptotic behavior
of the free energy and entropy at low temperature with included dc conductivity.
What this means is that, instead of the dielectric permittivity ε(iξl) given in Eq. (9),
one uses57,58
ε̃(iξl) = ε(iξl) +
= εl(iξ) +
β(T )
, (56)
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16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where σ0 is the dc conductivity of the plate material and β(T ) = 2~σ0/(kBT ). The
conductivity of dielectrics depends on temperature as σ0 ∼ exp(−b/T ) where b is
determined by the energy gap ∆which differs for different materials.78 The small-
ness of the dc conductivity of dielectrics can be illustrated79 by the example of
SiO2 where at T = 300K it holds β ∼ 10−12. Thus, the role of the dc conductivity
is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease
of T and, as a consequence, remains negligible at any T . In spite of this, the substi-
tution of Eq. (56) into the reflection coefficients (16) leads to different result than
in Eq. (11):
r̃‖(0, y) = 1, r̃‖(0, y) = 0. (57)
Equation (57) is in some analogy to Eq. (6), obtained for metals described by the
Drude model, which leads to the violation of the Nernst heat theorem in the case
of perfect crystal lattices.
Now we substitute the dielectric permittivity ε̃l ≡ ε̃(ξl) in Eqs. (13) – (16)
instead of εl and find the Casimir free energy F̃(a, T ) with included dc conductivity.
For convenience, we separate the zero-frequency term, subtract and add the usual
zero-frequency contribution for dielectric without dc conductivity. The result is
F̃(a, T ) =
16πa2
1− e−y
1− r20e−y
16πa2
ydy ln
1− r20e−y
1− r̃2‖(ζl, y)e
1− r̃2⊥(ζl, y)e−y
where r0 was defined in Eq. (11). Let us expand the last, third, integral on the right-
hand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the
zero-order contribution in this expansion with the second integral on the right-
hand side of Eq. (58) and obtain the Casimir free energy F(a, T ) calculated with
the dielectric permittivities εl. The first integral on the right-hand side of Eq. (58)
is calculated explicitly. Then, Eq. (58) can be rewritten as
F̃(a, T ) = F(a, T )− kBT
16πa2
ζ(3)− Li3
+R(a, T ). (59)
Here, R(a, T ) is of order O(β/l). It represents the first and higher-order contribu-
tions in the expansion of the third integral on the right-hand side of Eq. (58) in
powers of β/l. Restricting its explicit form to the first order contribution we get
R(a, T ) = R1(a, T ) +O
(β/l)
, (60)
R1(a, T ) =
dy y2e−y
y2 + ζ2l (εl − 1)
(2− εl) ζ2l − 2y2
y2 + ζ2l (εl − 1) + εly
r‖(ζl, y)
1− r2
(ζl, y)e−y
y2 + ζ2l (εl − 1) + y
r⊥(ζl, y)
1− r2⊥(ζl, y)e−y
. (61)
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Thermal Casimir force between dielectrics 17
Calculating the entropy by the first equality in Eq. (49), we arrive at
S̃(a, T ) = S(a, T ) +
16πa2
ζ(3)− Li3
− ∂R(a, T )
, (62)
where S(a, T ) is the entropy for the plates with the dielectric permittivity εl given
by Eq. (49).
Let us now show that the quantity R(a, T ) exponentially goes to zero with the
decrease of T . First we consider only the integral in Eq. (61), expand the integrated
function in powers of τ (we recall that ζl = τl), restrict ourselves to the main
contribution, resulting for τ = 0, and rearrange it appropriately:
(ε0 + 1)2
1− r20e−y
= − 2
(ε20 − 1)
1− r20e−y
(ε20 − 1)
dy ye−ny = −
(ε20 − 1)
1 + nζl
e−nζl . (63)
Substituting this in Eq. (61), we find
R1(a, T ) = −
4πa2 (ε20 − 1)
e−nτl
e−nτl
= − kBTβ
4πa2 (ε20 − 1)
1− e−nτ
enτ − 1
kB Li2
2πa2 (ε20 − 1)
Tβ ln τ + TβO(τ0) . (64)
Here, the last line is obtained by using the equality
1− e−nτ
enτ − 1
= − ln τ + 1− lnn+O(τ2) , (65)
substituting only its leading term and observing the definition of the integral log-
arithm, Li2 (z) = (1/2)
zn/n2. Taking into account that β ∼ (1/T ) exp(−b/T ),
we get the conclusion that the temperature dependence of R1(a, T ) is given by
R1(a, T ) ∼ e−b/T lnT. (66)
Thus, both R1(a, T ) and its derivative with respect to T in Eqs. (59) and (62) go
to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even
faster than R1 when T → 0.
Finally, in the limit T → 0 from Eq. (62) it follows
S̃(a, 0) =
16πa2
ζ(3)− Li3
> 0. (67)
The right-hand side of this equation depends on the parameter of the system under
consideration (the separation distance a) and implies a violation of the Nernst heat
theorem. An analogous result was obtained60 in the case of two dissimilar dielectrics.
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18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
The violation of the Nernst heat theorem in the Casimir interaction for dielectrics
originates from the inclusion of the dc conductivity in the model of dielectric re-
sponse. This violation is, however, of different nature than the one discussed above
in the case of Drude metals. In the case of dielectrics the entropy at zero tempera-
ture is positive but in the case of Drude metals it is negative. In the case of metals
the violation is caused by the vanishing contribution from the transverse electric
mode at zero frequency whereas the other reflection coefficient takes the physical
value 1 [see Eq. (6)]. For dielectrics the situation is quite opposite. In this case the
transverse reflection coefficient at zero frequency is always equal to zero [compare
Eqs. (11) and (57) in the absence and in the presence of the contribution from
dc conductivity]. Here the violation occurs due to the unity value of the parallel
reflection coefficient at zero frequency in Eqs. (57) which departs from the value
r0 = (ε0 − 1)/(ε0 + 1) coordinated with the zero value of the transverse coefficient
in Eq. (11).
One can conclude that the dc conductivity of a dielectric is not related to the na-
ture of the van der Waals and Casimir forces and must not be included in the model
of dielectric response. Ignoring this rule results in a violation of thermodynamics.
Physically it is amply clear that there is no fluctuating field of zero frequency and
that for such high-frequency phenomena as the van der Waals and Casimir forces the
low-frequency behavior should be obtained by analytic continuation from the region
of high frequencies. This permits to conclude that the correct procedure consists in
the substitution of the finite static dielectric permittivities into the zero-frequency
term of the Lifshitz formula, as Lifshitz and his collaborators really did.36,37,38
4. THERMAL CASIMIR FORCE BETWEEN DIELECTRIC
AND METAL PLATES
The Casimir interaction between metal and dielectric plates suggests the interest-
ing opportunity to verify the thermodynamic consistency of the Lifshitz theory
with different models of the dielectric response. This configuration was first inves-
tigated in Ref. 61 where it was proved that the Casimir entropy is in accordance
with the demands of the Nernst heat theorem if the static permittivity of the di-
electric plate is finite. In Ref. 61, however, only the first leading terms in the low-
temperature asymptotic expressions for the free energy and entropy were obtained
and the Casimir pressure was derived only in the dilute approximation. Here we
derive the more precise low-temperature behavior for the Casimir free energy, pres-
sure and entropy in the configuration of one plate made of ideal metal and another
plate made of dielectric with any finite static dielectric permittivity.
For the configuration of metal and dielectric plates the Lifshitz formula takes
the form80 analogical to Eq. (13)
F(a, T ) = ~cτ
32π2a3
1− δl0
ydy (68)
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Thermal Casimir force between dielectrics 19
1− rM‖ (ζl, y) rD‖ (ζl, y) e−y
1− rM⊥ (ζl, y) rD⊥ (ζl, y) e−y
Here the reflection coefficients r
for metal and dielectric, respectively, are given
by Eqs. (16) where εl should be changed for ε
l = ε
M,D(iξl).
For an ideal metal rM‖,⊥(ζl, y) = 1 and Eq. (68) takes the more simple form
F(a, T ) =
32π2a3
ydy (69)
1− r‖(ζl, y)e−y
1− r⊥(ζl, y)e−y
(here and below we omit the index D near the reflection coefficient and permit-
tivity of a dielectric plate). We admit that the dielectric permittivity calculated at
Matsubara frequencies εl ≡ ε0, i.e., is equal to its static value and find the asymp-
totic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove
that the deviations of ε(iξl) from ε0 at high frequencies do not influence the low-
temperature behavior of the Casimir free energy, pressure and entropy. It can be
shown also that the results of this section are valid not only for ideal metal plate
but for plate made of real metal as well.] The free energy (69) can be represented
by Eqs. (18) – (21), with the function f(ζ, y) replaced by
f̂(ζ, y) = y ln
1− r‖(ζ, y)e−y
+ y ln
1− r⊥(ζ, y)e−y
≡ f̂‖(ζ, y) + f̂⊥(ζ, y). (70)
In the case of one dielectric and one metal plate both f̂‖ and f̂⊥ contribute to
F (ix)− F (−ix). The expansion of f̂(x, y) in powers of x takes the form
f̂(x, y) = y ln(1− r0e−y)−
ε0 − 1
e−y − ε0
ε0 + 1
x2 +O(x3). (71)
Now we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x).
The integral of the first term on the right-hand side of Eq. (71) is evaluated using
the new variable v = y − x:
ydy ln(1− r0e−y) =
vdv ln(1− r0e−v) + O(x2), (72)
where the coefficient near the first-order contribution in x vanishes. As a result, this
term could contribute to F (ix) − F (−ix) only starting from the third expansion
order. The integrals of the second-order terms on the right-hand side of Eq. (71)
are simply calculated using the formulas
= −Ei(−x),
= −Ei(−nx). (73)
Finally, we obtain
F (ix)− F (−ix) = iπ
(ε0 − 1)2
4 (ε0 + 1)
x2 − iγ x3 +O(x4), (74)
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20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where the unknown third order expansion coefficient is designated as γ.
Substituting Eq. (74) in Eq. (21) and using Eq. (19), we find the free energy in
the system metal-dielectric in the form
F(a, T ) = E(a)− ~c
32π2a3
(ε0 − 1)2
ε0 + 1
τ3 −K4 τ4 +O(τ5)
, (75)
where K4 ≡ γ/240.
The Casimir pressure in the configuration of metal and dielectric plates obtained
from Eq. (75) is equal to
P (a, T ) = P0(a)−
32π2a4
4 +O(τ5)
. (76)
The direct application of the Lifshitz formula gives the expression for the pressure
analogical to Eq. (31),
P (a, T ) = − ~cτ
32π2a4
1− δl0
r‖(ζl, y)
ey − r‖(ζl, y)
r⊥(ζl, y)
ey − r⊥(ζl, y)
. (77)
Using the Abel-Plana formula (17), Eq. (77) can be represented in the form of
Eqs. (32), (33) where
Φ‖,⊥(x) =
y2 r‖,⊥(x, y)
ey − r‖,⊥(x, y)
. (78)
Again, we deal first with Φ⊥(x). By adding and subtracting the asymptotic
behavior of the integrated function at small x,
y2r⊥(x, y)
ey − r⊥(x, y)
(ε0 − 1)x2e−y +O(x3) , (79)
and introducing the new variable v = y/x, the function Φ⊥(x) can be identically
rearranged and expanded in powers of x as follows:
Φ⊥(x) =
(ε0 − 1)x2e−x + x3
rn⊥(v)e
−nvx − 1
(ε0 − 1)e−vx
(ε0 − 1)x2(1− x) + x3
v2r⊥(v)
1− r⊥(v)
ε0 − 1
+O(x4). (80)
The integral on the right-hand side of Eq. (80) is converging and can be simply
calculated with the result
Φ⊥(x) =
ε0 − 1
x2 − 1
ε0 − 1)x3 +O(x4). (81)
To deal with Φ‖(x) we add and subtract in Eq. (78) the two first expansion
terms of the integrated function in powers of x,
Φ‖(x) =
ey − r0
− ε0r0e
y2(ε0 + 1) (1− r0e−y)2
r‖(x, y)
ey − r‖(x, y)
ey − r0
ε0r0e
y2(ε0 + 1) (1− r0e−y)2
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Thermal Casimir force between dielectrics 21
The asymptotic expansion of the first integral on the right-hand side of Eq. (82) is
given by
2Li3(r0)−
ε0(ε0 − 1)
2(ε0 + 1)
(ε0 − 1)(3ε0 − 2)x3 +O(x4), (83)
and of the second one by
ε0(ε0 − 1)−
ε0(ε0
ε0 − 1) +
ε0(ε0 − 1)
x3 +O(x4). (84)
By summing Eqs. (83) and (84) we find
Φ‖(x) = 2Li3(r0)−
ε0(ε0 − 1)
2(ε0 + 1)
x2 (85)
(ε0 − 1) + (ε0
ε0 − 1)− 3ε0(ε0 − 1)
x3 +O(x4).
Finally we add Eq. (85) to Eq. (81) and arrive at
Φ(ix) − Φ(−ix) = −i
1− 2ε0
ε0 + ε
x3 +O(x4). (86)
Substituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the
Casimir pressure is obtained
P (a, T ) = P0(a)−
32π2a4
1− 2ε0
ε0 + ε
τ4 +O(τ5)
. (87)
Comparing Eqs. (87) and (76), the explicit expression for the coefficient K4 reads
1− 2ε0
ε0 + ε
. (88)
Now we are in a position to find the asymptotic behavior of the entropy in the
limit of low temperatures in the configuration of two parallel plates one of which
is metallic and the other one dielectric. By calculating the negative derivative of
Eq. (75) with respect to temperature, one arrives at
S(a, T ) =
3kBζ(3)(ε0 − 1)2
128π3a2(ε0 + 1)
τ2 (89)
8π2(ε0 + 1)
1− 2ε0
ε0 + ε
135ζ(3)(ε0 − 1)2
τ +O(τ2)
This equation is in analogy to Eq. (49) obtained for the case of two dielectrics.
As is seen from Eq. (89), the entropy of the Casimir and van der Waals interactions
between metal and dielectric plates vanishes when the temperature goes to zero,
i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ2 on the
right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at
low temperatures the entropy goes to zero remaining positive. At the same time, as
was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may
take negative values. This interesting property distinguishes the configuration of
metal and dielectric plates from two dielectric plates. In the latter configuration the
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22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
negative entropy appears only for nonphysical dielectrics with anomalously large
and frequency independent dielectric permittivities.53
Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations).
In the same way as for two dielectric plates, here the main contribution to the free
energy is given by the term with l = 0 in Eq. (69),
F(a, T ) = ~cτ
64π2a3
ydy ln
1− r0e−y
. (90)
Performing the integration in Eq. (90), we obtain
F(a, T ) = − kBT
16πa2
Li3 (r0) . (91)
From this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows
P (a, T ) = − kBT
Li3 (r0) , S(a, T ) =
16πa2
Li3 (r0) . (92)
The results (91) and (92) are analogous to (54) and (55) for two dielectric plates.
5. THE PROBLEM ORIGINATING FROM THE ACCOUNT OF
DIELECTRIC DC CONDUCTIVITY
In the previous section it was supposed that the static dielectric permittivity ε0 of
the dielectric plate is finite. Now we will deal with the configuration of metal and
dielectric plates with included dc conductivity of the dielectric material. In doing
so the permittivity of the dielectric plate is given by
ε̃(iξl) = ε0 +
β(T )
, (93)
where all notations were introduced in and below Eq. (56). Thus, the reflection
coefficients at zero frequency satisfy Eq. (4) for a plate made of ideal metal and
Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find
the low-temperature behavior of the Casimir entropy and verify the consistency of
the Lifshitz theory with thermodynamics in this nonstandard situation.
For this purpose we substitute the dielectric permittivity (93) in Eq. (69) in-
stead of ε0 and find the Casimir energy F̃(a, T ) with included dc conductivity of
a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the
zero-frequency term of F̃(a, T ) and subtract and add the usual zero-frequency con-
tribution for metal-dielectric plates computed with the dielectric permittivity ε0,
F̃(a, T ) = kBT
16πa2
1− e−y
1− r0e−y
16πa2
ydy ln
1− r0e−y
1− r̃‖(ζl, y)e−y
1− r̃⊥(ζl, y)e−y
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Thermal Casimir force between dielectrics 23
Here the reflection coefficients r̃‖,⊥ are calculated with the permittivity (93). We
expand the third integral on the right-hand side of Eq. (94) in powers of the small
parameter β/l. The zero-order contribution in this expansion together with the
second integral of Eq. (94) form the Casimir free energy F(a, T ) calculated with
dielectric permittivity ε0. The first integral on the right-hand side of Eq. (94) is
calculated explicitly. As a result, Eq. (94) is rearranged to
F̃(a, T ) = F(a, T )− kBT
16πa2
[ζ(3)− Li3 (r0)] +Q(a, T ), (95)
where Q(a, T ) contains the first and higher-order contributions in the expansion of
the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit
form of the main first-order term in Q(a, T ) is the following:
Q1(a, T ) =
dy y2e−y
y2 + ζ2l (ε0 − 1)
(2− ε0)ζ2l − 2y2
y2 + ζ2l (ε0 − 1) + ε0y
1− r‖(ζl, y)e−y
y2 + ζ2l (ε0 − 1) + y
1− r⊥(ζl, y)e−y
. (96)
In the same way as in Sec. 3, we expand the integrated function in Eq. (96) in
powers of τ (bearing in mind that ζl = τl) and preserve only the main contribution
at τ = 0:
Q1(a, T ) = −
dy yr0e
(ε20 − 1) (1− r0e−y)
= − kBTβ
4πa2(ε20 − 1)
e−nτl
e−nτl
Dealing with this expression in the same way as with Eq. (64), we arrive at
Q1(a, T ) ∼ e−b/T lnT. (98)
The Casimir entropy in the configuration of metal and dielectric plates with
included dc conductivity of the dielectric plate is obtained as minus derivative of
Eq. (95) with respect to temperature,
S̃(a, T ) = S(a, T ) +
16πa2
[ζ(3)− Li3 (r0)]−
∂Q(a, T )
. (99)
Using Eq. (98), the calculation of the limiting value at T → 0 is straightforward:
S̃(a, 0) =
16πa2
[ζ(3)− Li3 (r0)] > 0. (100)
From this equation it follows that the inclusion of the dc conductivity of dielectric
plate in the configuration metal-dielectric results in a violation of the Nernst heat
theorem. In the above this result was obtained for a metallic plate made of ideal
metal. It can be shown that it remains valid for a metallic plate made of real metal
with finite conductivity.
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24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
Thus, both configurations (two dielectric plates or one metal plate and one
dielectric) lead to the same conclusion that when the dc conductivity is included
in the model of dielectric response of the dielectric plate, the Lifshitz theory loses
its consistency with thermodynamics. This confirms the conclusion made in Sec. 3
that the actual properties of dielectric materials at very low frequencies are in fact
not related to the van der Waals and Casimir forces.
6. QUALITATIVE DISCUSSION OF CASIMIR INTERACTION
BETWEEN METAL AND SEMICONDUCTOR
As was mentioned in the Introduction, semiconductors possess a wide variety of
electric and optical properties ranging from metallic to dielectric. This opens the
possibility to modulate the van der Waals and Casimir forces by changing the charge
carrier density. Bearing in mind the discussed above problems on the consistency
of the Lifshitz theory with thermodynamics, semiconductors can provide us with
a test for the validity of different approaches. Thus, if for good dielectric the dc
conductivity does not play any real role in the van der Waals and Casimir forces,
the question arises on how much it should be increased in order to become a relevant
factor in the description of dispersion forces.
In Ref. 81 the Casimir force acting between two Si plates was calculated using the
simple analytic expression for Si dielectric permittivity as a function of frequency.
The complete tabulated optical data of Si were used in Ref. 75 to calculate the van
der Waals interaction of different atoms with a Si wall. The first attempt to measure
the van der Waals force between a glass lens and a Si plate and to modify it by light
due to the change of carrier density was undertaken in Ref. 82. However, glass is a
dielectric and therefore the electric forces due to localized point charges could not
be controlled. This might explain that no force change occured on illumination at
small separations where the effect should be most pronounced.
The first measurements of the Casimir force between a gold coated sphere and
a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic
force microscope. The experiments used a p-type B doped Si plate of resistivity
ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate
between the resistivity of metals (which is usually two or three orders of magnitude
lower) and the resistivity of dielectrics (it can be by about a factor of 105 larger; for
instance, high-resistivity “dielectric” Si has the resistivity ρ0 = 1000 Ω cm). Thus,
the used Si plate had a relatively large absorption typical for semiconductors but it
was also enough conductive to avoid the accumulation of charges.63
In Fig. 2, taken from Ref. 63, the differences of the theoretical and mean ex-
perimental Casimir forces acting between Au sphere and Si plate are presented as
functions of separation. In Fig. 2a the theoretical force F theor is computed using
the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively
low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as
ξ−1 with decreasing frequency (see the solid line in Fig. 3). In Fig. 2b the theo-
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Thermal Casimir force between dielectrics 25
80 100 120 140
  (nm)
F theor − F̄ expt (pN)
80 100 120 140
 (nm)
F̃ theor − F̄ expt (pN)
Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation.
Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si.
Solid and dashed lines indicate 95 and 70% confidence intervals, respectively.
retical Casimir force F̃ theor is computed using the dielectric permittivity of the Si
plate made of “dielectric” Si with high resistivity ρ0. The dielectric permittivity of
high-resistivity Si is shown by the dashed line in Fig. 3. It is characterized by the
finite static value εSi(0) = 11.67. The solid and dashed lines in Fig. 2a,b indicate
the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2, the the-
oretical approach using the dielectric permittivity of high-resistivity “dielectric” Si
is excluded by experiment within the separation range from 60 to 110nm at 70%
confidence. At the same time, the theory using the dielectric permittivity of Si with
a low resistivity ρ is consistent with experiment.
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26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
13.5 14 14.5 15 15.5 16 16.5 17
Log10 [ξ (rad/s)]
ε (iξ)D
Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis
(solid line). Dashed line shows the dielectric permittivity of dielectric Si.
The above results suggest an approach on how to correctly determine the pos-
sible role of the low-frequency conductivity properties in dispersion forces. As is
seen from Fig. 3, the dielectric permittivity of low-resistivity Si (solid line) sig-
nificantly departs from the dielectric permittivity of “dielectric” Si in the region
around the important dimensional parameter of the problem, the characteristic fre-
quency c/(2a) ∼ 1014 − 1015 rad/s. That is why the low-resistivity sample cannot
be described at low frequencies by the static dielectric permittivity of Si equal to
εSi(0) = 11.67. To describe it, the term β(T )/l, like in Eq. (93), should be added
to εSi(0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with
l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter.
At the same time, for a high-resistivity sample the inclusion of the dc conduc-
tivity would lead to deviations from the dashed line in Fig. 3 only at frequencies
ξ < 108 rad/s, which are much less than the characteristic frequency. This compari-
son permits to make a conclusion in what experimental situations the conductivity
properties of semiconductors at low frequencies should be taken into account and
when they should be omitted as being not related to dispersion forces.
The future experiments on the modification of semiconductor charge carrier
density by laser light83 will bring a more clear understanding of this problem on
the connection between the low-frequency material properties and dispersion forces.
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Thermal Casimir force between dielectrics 27
7. DOES SPATIAL DISPERSION LEAD TO AN IMPORTANT
IMPACT ON THERMAL CASIMIR FORCE?
The presented above new analytic results on the low-temperature behavior of the
Casimir free energy, pressure and entropy between dielectrics or between dielectric
and metal are based on the conventional Lifshitz theory which describes dielectric
materials by means of the frequency dependent dielectric permittivity. In fact, the
assumption that the material of the plates possesses only the frequency dispersion
means that the components of electric displacement are connected with the compo-
nents of electric field by the relation
Dk(r, ω) = εkl(r, ω)El(r, ω). (101)
This equation is central in all different derivations of the Lifshitz formula (see, e.g.,
Refs. 2, 6, 36–38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion)
are in fact essential only at shortest separations between the plates comparable with
atomic dimensions and also for metals at sufficiently large separations (typically of
about 2–3µm) in the frequency region of the anomalous skin effect. The Casimir
force in the latter region was described by the Lifshitz theory reformulated in terms
of the Leontovich impedance.85
Recently the spatial dispersion came to the attention in connection with the
thermal Casimir force.64,65,66,67,68 In particular, it was claimed64,65 that for real
metals at any separation the account of spatial dispersion leads to practically the
same result (6) for the reflection coefficients at zero frequency as was obtained earlier
using the Drude model dielectric function (5). This conclusion, if it is correct, not
only returns us to the contradiction with experiment30,32,33 but also casts doubts
on all results obtained by means of the conventional Lifshitz theory accounting for
only the frequency dispersion. It is natural when the spatial dispersion contributes
a small fraction of a percent as it is generally believed in numerous applications of
the Lifshitz theory. It is, however, quite another matter when the account of the
spatial dispersion results in some “dramatic effects”,64 i.e., in several hundred times
larger thermal correction than is obtained in the local case. Below we demonstrate
that the conclusions of Refs. 64 – 68 are in fact not reliable because they use the
Lifshitz theory of dispersion forces outside of its application range.86
To find the electromagnetic modes associated with an empty gap between the
plates, Refs. 64 – 68 use the standard continuity boundary conditions,
E1t = E2t, B1n = B2n, D1n = D2n, B1t = B2t, (102)
which are commonly applied in the derivation of the Lifshitz formula for spatially
local nonmagnetic materials. Here B is the magnetic induction, n is the normal to
the boundary directed inside the medium, the subscripts n, t refer to the normal
and tangential components, respectively, the subscript 1 refers to the vacuum and
subscript 2 to the plate material. In Refs. 64 – 68 the spatial dispersion is described
by the longitudinal and transverse dielectric permittivities depending on the wave
vector and frequency: εkl = εkl(q, ω). However, as is shown below, in the theory
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28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
of the Casimir effect both the boundary conditions (102) and permittivities εkl =
εkl(q, ω) are inapplicable.
We start from the boundary conditions and recall the set of Maxwell equations
in a metal describing the Casimir effect,
rotE +
= 0, divD = 0, rotB − 1
= 0 divB = 0. (103)
Equations (103) do not contain any external, i.e., independent on E, D and B,
current or charge densities. The definition of the electric displacement is
+ 4πi, (104)
where the volume current i is induced by E and B and takes into account the
conduction electrons.
In electrodynamics with spatial dispersion the electric field and magnetic induc-
tion are finite at the boundary surfaces, whereas the electric displacement can tend
to infinity.87 Then, integrating Eqs. (103) over the thickness of the boundary layer
as is done in Ref. 88, we reproduce the first two conditions in Eq. (102) and arrive
at the modified third and fourth conditions,87,89
E1t = E2t, B1n = B2n, D2n −D1n = 4πσ, [n× (B2 −B1)] =
j, (105)
where the induced surface charge and current densities are given by
div[n× [D × n]]dl, j = 1
dl. (106)
Note that the boundary conditions (105), (106) are obtained from the macroscopic
Maxwell equations for physical fields. They should not be mixed with the boundary
conditions arising in perturbative theories and for the fictitious fields (see below).
In linear electrodynamics for a medium with time-independent properties with-
out spatial dispersion the material equation connecting the electric displacement
and electric field takes the form
Dk(r, t) =
ε̂kl(r, t− t′)El(r, t′)dt′. (107)
According to this equation, the electric displacement at a point r and moment t is
determined by the electric field at the same point r at different moments t′ ≤ t (the
spatial dispersion is absent but the temporal may be present). It is easily seen that
the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the
boundary conditions (105) coincide with the standard continuity conditions (102). It
is unjustified, however, to use conditions (102) in the presence of spatial dispersion.
In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ
nor j is equal to zero.
We now turn to a discussion of the use of dielectric permittivity εkl(q, ω) in the
theory of the Casimir effect with account of spatial dispersion. In the presence of
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Thermal Casimir force between dielectrics 29
only frequency dispersion, it is possible to perform the Fourier transformation of
the fields
E(r, t) =
E(r, ω)e−iωtdω, D(r, t) =
D(r, ω)e−iωtdω, (108)
in Eq. (107) and arrive at Eq. (101) where
εkl(r, ω) =
ε̂kl(r, τ)e
iωτdτ (109)
is the frequency-dependent dielectric permittivity and τ ≡ t− t′. In fact Eqs. (101),
(108) and (109) are used in parallel with the boundary conditions (102) in all
derivations of the Lifshitz formula.2,6,36,37,38,69,70,84
If the material of the plates is characterized not only by temporal but also spatial
dispersion, Eq. (107) is generalized to
Dk(r, t) =
dr′ε̂kl(r, r
′, t− t′)El(r′, t′). (110)
If the material medium were uniform in space (i.e., all points were equivalent), the
kernel ε̂ would not depend on r and r′ separately, as in Eq. (110), but on the
difference R ≡ r − r′. In this case, by performing the Fourier transformation,
E(r, t) =
dqE(q, ω)ei(qr−ωt),
D(r, t) =
dqD(q, ω)ei(qr−ωt), (111)
and substituting it in Eq. (110), one could introduce the dielectric permittivities
εkl(q, ω) =
dR ε̂kl(R, τ)e
−i(qR−ωτ), (112)
as Refs. 64–68 do, and rearrange Eq. (110) to the form
Dk(q, ω) = εkl(q, ω)El(q, ω). (113)
In the Casimir effect, however, the material medium is not uniform due to the
presence of a macroscopic gap between the two plates (half spaces). Because of this,
the assumption that the kernel ε̂ depends only on R and τ is wrong. As a result,
it is not possible to introduce the dielectric permittivity εkl(q, ω) depending on
the wave vector and frequency. In fact, for systems with spatial dispersion in the
presence of boundaries the kernel ε̂ depends not only on R and τ but also on the
distance from the boundary.87 In this complicated situation the following approx-
imate phenomenological approach is sometimes applicable.87 For electromagnetic
waves with a wavelength λ the kernel ε̂(r, r′, τ) in Eq. (110) differs essentially from
zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ
(for nonmetallic condensed media l is of the order of the lattice constant). Then it is
reasonable to assume that ε̂ is a function of R=r–r′, except for a layer of thickness
l adjacent to the boundary surface. If one is mostly interested in bulk phenomena
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30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
and neglects the role and influence of a subsurface layer, the quantity εkl(q, ω) may
be employed as a reasonable approximation.
This approximate phenomenological approach is widely applied in the theoretical
investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91
some kind of fictitious infinite system was introduced and electromagnetic fields in
this system are discontinuous on the boundary surface. This discontinuity should
not be confused with the discontinuity of physical fields of a real system in the
presence of spatial dispersion described by Eqs. (105) and (106). (There is also
another approach to the description of the anomalous skin effect in polycrystals
using the generalizations of the local Leontovich impedance which takes into account
the shape of Fermi surface92). The frequency and wave vector dependent dielectric
permittivity in the presence of boundaries is also approximately applied in the
theory of radiative heat transfer93 or in the study of electromagnetic interaction of
molecules with metal surfaces.94 In all these applications the boundary effects are
usually taken into account by the boundary conditions (105) supplemented by so
called “additional boundary conditions”.
It is unlikely, however, that the approximate phenomenological approach using
such quantity as εkl(q, ω) in the presence of boundaries would be applicable in the
theory of the Casimir force where the boundary effects on the zero-point electromag-
netic oscillations are of prime importance. It is notable also that this approach faces
serious theoretical difficulties including the violation of the law of conservation of
energy.95 It is then not surprising that the application of this approach in Refs. 64,
65 results in the contribution to the Casimir free energy from the transverse electric
mode which is in contradiction with experiment.30,32,33
One more shortcoming of Refs. 64–68 is that they substitute the dielectric per-
mittivity εkl(q, ω), depending on both wave vector and frequency, into the conven-
tional Lifshitz formula derived in the presence of only temporal dispersion. In the
famous review paper96 it has been noticed, however, that with the inclusion of spa-
tial dispersion the free energy of a fluctuating field takes the form F = FL +∆F ,
where FL is given by the conventional Lifshitz expression derived in a spatially
local case and written in terms of the Fresnel reflection coefficients, and ∆F is an
additional term which can be expressed in terms of the thermal Green’s function of
the electromagnetic field and polarization operator. Review96 calls as not reliable
the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity
εkl(q, ω), taking account of spatial dispersion, into the conventional Lifshitz for-
mula. It can be true that the Lifshitz formula written in terms of general reflection
coefficients is applicable in both spatially local and nonlocal situations. However,
as far as the exact reflection coefficients in a spatially nonlocal case are unknown,
the use of some approximate phenomenological models, elaborated in literature for
applications different than the Casimir effect, may lead to incorrect results for ∆F
and create inconsistencies with experiment.
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Thermal Casimir force between dielectrics 31
To conclude this section, the results of Refs. 64–68 on the influence of spatial
nonlocality on the Casimir interaction are shown to be not reliable. Although at
present there is no fundamental theory of the thermal Casimir force incorporating
spatial dispersion, there is no reason to expect that it can play any significant role
in the frequency region of infrared optics (experimental separations) or normal skin
effect (i.e., at separations between plates greater than 4–5µm).
8. CONCLUSIONS AND DISCUSSION
In the foregoing we have presented the derivation of analytic asymptotic expres-
sions for the free energy, pressure and entropy of the Casimir interaction between
two dielectric plates and between metal and dielectric plates at low and high temper-
atures. It was shown that the low-temperature behavior of the Casimir interaction
between dielectrics and between dielectric and metal is determined by the static
dielectric permittivities of nonpolar dielectrics. The obtained results were shown to
be in agreement with thermodynamics when the static dielectric permittivities of
dielectrics are finite. In particular, the entropy of the Casimir interaction goes to
zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This
demonstrates the consistency of the original Lifshitz’s approach to the van der Waals
forces between dielectrics which disregards the small conductivity of dielectrics at
constant current.
The second important result shown above is that the inclusion of the dc con-
ductivity of dielectrics into the model of dielectric response leads not to some small
corrections to the characteristics of the Casimir interaction, as one could expect,
but makes the Lifshitz theory inconsistent with thermodynamics leading to the vi-
olation of the Nernst heat theorem. This reveals that real material properties at
very low, quasistatic frequencies are in fact not related to the phenomenon of the
van der Waals and Casimir forces which is actually determined by sufficiently high
frequencies. In this case the zero-frequency contribution to the Casimir force should
be understood not literally but as analytic continuation to zero of the material
physical behavior in the region around the characteristic frequency.
The presented results provide a basis for the calculation of the van der Waals
and Casimir forces between real materials. Such calculations are much needed for
numerous applications of the Casimir force discussed in the Introduction, in par-
ticular for the applications in nanotechnology and for constraining predictions of
fundamental physical theories beyond the Standard Model. Bearing in mind that
semiconductors are the main constituent materials in nanotechnological devices, it
is a subject of high priority to understand the Casimir effect with semiconductor
boundaries. In this connection we have discussed new experimental results and the-
oretical ideas on the Casimir interaction between a metal sphere and semiconductor
plate. It was stressed that by changing the charge carrier density in the semicon-
ductor it is possible to bring it in different intermediate states between metallic
and dielectric. In this case the problem arises when the conductivity properties of
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32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
semiconductor are not related to dispersion forces and when they are becoming rel-
evant. A criterion for the resolution of this problem was formulated based on the
relation between the typical frequency at which the dc conductivity properties come
into play and the characteristic frequency of the Casimir effect. In fact the thermal
Casimir interaction between semiconductors remains an open question and much
work should be done both in experiment and theory to gain a better insight into
this subject.
The last major problem discussed in this review is whether or not the spatial
dispersion influences essentially the thermal Casimir force between real materials. In
the present literature there is no agreement on this subject. We adduced arguments
in favor of the statement that in the region of experimental separations the influence
of the spatial dispersion on the Casimir force is negligible small. The statements
on the opposite, contained in the literature, were shown to be not reliable because
they are obtained by the application of the Lifshitz theory outside of its application
range. At the same time it was ascertained that at the moment there is no consistent
fundamental theory of the van der Waals and Casimir forces taking the spatial
dispersion into account. This is the problem to solve in the foreseeable future.
ACKNOWLEDGMENTS
G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Insti-
tute for Theoretical Physics, Leipzig University for their kind hospitality. This work
was supported by Deutsche Forschungsgemeinschaft grant 436RUS113/789/0-2.
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  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 low-temperature 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 low-temperature
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.

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 metals40 it holds
r‖(0, k⊥) = r⊥(0, k⊥) = 1. (4)
• For real metals described by the dielectric function of the Drude model,
ε(iξl) = 1 +
ξl[ξl + ν(T )]
, (5)
where ωp is the plasma frequency and ν(T ) is the relaxation parameter, it
holds41
r‖(0, k⊥) = 1, r⊥(0, k⊥) = 0. (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.52,53,54,56
• For real metals described by the dielectric function of the plasma model,
ε(iξl) = 1 +
, (7)
from Eq. (2) it follows:44,45
r‖(0, k⊥) = 1, r⊥(0, k⊥) =
c2k2⊥ + ω
p − ck⊥
c2k2⊥ + ω
p + ck⊥
. (8)
Here, in the limit of ideal metals (ωp → ∞) the continuity is preserved
because r⊥(0, k⊥) in Eq. (8) goes to unity. The free energy (1) 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 Ninham-Parsegian
representation,69,70
ε(iξl) = 1 +
1 + ξ2l /ω
, (9)
where the parameters Cj are the absorption strengths satisfying the condi-
Cj = ε0 − 1 (10)
and ωj are the characteristic absorption frequencies. Here, the static dielec-
tric permittivity ε0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an
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Thermal Casimir force between dielectrics 7
approximate one, it gives a very accurate description for many materials.71
By the substitution of Eq. (9) in Eq. (2) one arrives at
r‖(0, k⊥) ≡ r0 =
ε0 − 1
ε0 + 1
, r⊥(0, k⊥) = 0. (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 (
ε0−1)/(
ε0+1). 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. (1)
at low temperatures. For convenience in calculations, we introduce the dimensionless
variables
= τl, y = 2aql, (12)
where ξc = c/(2a) is the characteristic frequency, τ = 4πkBaT/(~c), and ql was
defined in Eq. (3). Then the Lifshitz formula (1) takes the form
F(a, T ) = ~cτ
32π2a3
1− δl0
dy f(ζl, y) , (13)
where
f(ζ, y) = f‖(ζ, y) + f⊥(ζ, y), (14)
f‖,⊥(ζ, y) = y ln
1− r2‖,⊥(ζ, y)e−y
, (15)
and reflection coefficients (2), in terms of variables (12), being given by
r‖(ζl, y) =
εly −
y2 + ζ2l (εl − 1)
εly +
y2 + ζ2l (εl − 1)
, r⊥(ζl, y) =
y2 + ζ2l (εl − 1)− y
y2 + ζ2l (εl − 1) + y
. (16)
To separate the temperature independent contribution and thermal correction
in Eq. (13) we apply the Abel-Plana formula,3,6
1− δl0
F (l) =
F (t) dt+ i
F (it)− F (−it)
e2πt − 1
, (17)
where F (z) is an analytic function in the right half-plane. Here, taking it as
F (x) =
dy f(x, y) (18)
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8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
and using Eq. (17), we can identically rearrange Eq. (13) to the form
F(a, T ) = E(a) + ∆F(a, T ) , (19)
where E(a) is the energy of the van der Waals or Casimir interaction at zero tem-
perature,
E(a) =
32π2a3
dy f(ζ, y) , (20)
and ∆F(a, T ) is the thermal correction to this energy,
∆F(a, T ) =
32π2a3
F (iτt)− F (−iτt)
e2πt − 1
. (21)
Note that, in fact, Eq. (21) describes the dependence of the free energy on the tem-
perature arising from the dependence on temperature of the Matsubara frequencies.
Thus, ∆F(a, T ) in (21) coincides with the thermal correction to the energy, defined
as F(a, T )− F(a, 0), only for plate materials with temperature independent prop-
erties.
The asymptotic expressions for the energy E(a) at both short and large sep-
arations are well known.6,37,38 Below we find the asymptotic expressions for the
thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account
the definition of τ in Eq. (12), the asymptotic expressions at τ ≪ 1 are applicable
both at small and large separations if the temperature is sufficiently low.
We begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) – (16),
expand the function f(x, y) in powers of x = τt, and than integrate the obtained
expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and
F (ix)− F (−ix) in Eq. (21).
It is easy to check that f⊥(ζ, y) does not contribute to the leading, second, order
in the expansion of F (ix)− F (−ix) in powers of x. Thus, we can restrict ourselves
by the consideration of the expansion
f‖(x, y) = y ln
1− r20e−y
2ε0 r
ε0 + 1
y (ey − r20)
4ξ2c r
(ε0 + 1)ω
ey − r20
+O(x3), (22)
where r0 was defined in Eq. (11). Note that for simplicity we consider here only one
oscillator in Eq. (9) and put ωj = ω1. 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 x and sum up the obtained series. Thereby we obtain
the following expressions:
Z1(x) ≡
y dy ln
1− r20e−y
(1 + nx)e−nx
= −Li3(r20)−
ln(1− r20) + O(x3) , (23)
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Thermal Casimir force between dielectrics 9
Z2(x) ≡
ε0 + 1
y (ey − r20)
= − 2ε0x
ε0 + 1
r2n0 Ei(−nx) , (24)
Z3(x) ≡
4r20ξ
(ε0 + 1)ω
ey − r20
4ξ2cx
(ε0 + 1)ω
(1 + nx)e−nx
(ε0 + 1)ω
x2 Li2(r
1− r20
+O(x5)
, (25)
where Lin(z) is the polylogarithm function and Ei(z) is the exponential integral
function.
From these equations it follows
Z1(ix) − Z1(−ix) = O(x3) , Z3(ix)− Z3(−ix) = O(x5) , (26)
Z2(ix) − Z2(−ix) = 2iπ
ε0 + 1
1− r20
x2 +O(x3) , (27)
and, thus, Z1 and Z3 do not contribute to the leading order in the expansion of
F (ix)− F (−ix). The latter is determined by Z2 only. As a result, we arrive at
F (ix)− F (−ix) = iπ
(ε0 − 1)2
2(ε0 + 1)
x2 − iαx3 +O(x4), (28)
where r0 was substituted from Eq. (11) and α was introduced for the still unknown
real coefficient of the next to leading order resulting from Z1 and Z2 as well as,
possibly, from f⊥(ζ, y). At this stage it is difficult to determine the value of this
coefficient because all powers in the expansion of f(x, y) contribute to it. Remark-
ably, the two leading orders depend only on the static dielectric permittivity ε0 and
are not influenced by the dependence of the dielectric permittivity on the frequency
contained in Z3.
Substituting Eq. (28) in Eq. (21) and using Eq. (19), we obtain
F(a, T ) = E(a)− ~c
32π2a3
(ε0 − 1)2
ε0 + 1
τ3 − C4τ4 +O(τ5)
, (29)
where C4 ≡ α/240 and ζ(z) is the Riemann zeta function.
So far we have considered the free energy. The thermal pressure is obtained as
P (a, T ) = −∂F(a, T )
= P0(a)−
32π2a4
4 +O(τ5)
, (30)
where P0(a) = −∂E(a)/∂a is the Casimir pressure at zero temperature.
In order to determine the value of the coefficient C4 of the leading term, we
express the pressure directly through the Lifshitz formula
P (a, T ) = − ~cτ
32π2a4
1− δl0
(ζl, y)
ey − r2
(ζl, y)
r2⊥(ζl, y)
ey − r2⊥(ζl, y)
. (31)
Again, applying the Abel-Plana formula (17), we represent the pressure as follows,
P (a, T ) = P0(a) + ∆P (a, T ), (32)
October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr
10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where the thermal correction to P0(a), the pressure at zero temperature, is
∆P (a, T ) = − i~cτ
32π2a4
Φ(iτt)− Φ(−iτt)
e2πt − 1
and the function Φ(x) is given by
Φ(x) ≡ Φ‖(x) + Φ⊥(x) , Φ‖,⊥(x) =
y2 r2
(x, y)
ey − r2
(x, y)
. (34)
First, we determine the leading term of the expansion of Φ⊥(x) in powers of
x. For this purpose, let us introduce the new variable v = y/x and note that the
reflection coefficient r⊥(x, v) depends on x only through the frequency dependence
of ε given by Eq. (9). Thus, we can rewrite and expand Eq. (34) as follows:
Φ⊥(x) = x
v2 r2⊥(x, v)
evx − r2⊥(x, v)
v2 r2⊥(v)
1− r2⊥(v)
+ O(x4), (35)
where, according to Eq. (16),
r⊥(v) ≡ r⊥(0, v) =
v2 + ε0 − 1− v√
v2 + ε0 − 1 + v
. (36)
Integration in Eq. (35) with account of Eq. (36) results in
Φ⊥(x) =
1− ε0(3− ε0)
+ O(x4) , (37)
from which it follows:
Φ⊥(ix)− Φ⊥(−ix) = −i
ε0 (3− ε0)
+ O(x5). (38)
The expansion of Φ‖(x) from Eq. (34) in powers of x is somewhat more cumber-
some. 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 F (ix)−F (−ix) starting from only the 5th power in x. Bearing in mind the con-
nection between free energy and pressure, we can conclude that the dependence on
the frequency contributes to Φ‖(ix)−Φ‖(−ix) starting from the 4th order. We are
looking for the lowest (third) order expansion term of Φ‖(ix)−Φ‖(−ix). Because of
this, it is permissible to disregard the frequency dependence of ε and describe the
dielectric by its static dielectric permittivity.
To begin with, we identically rearrange Φ‖(x) in Eq. (34) by subtracting and
adding the two first expansion terms of the function under the integral in powers
of x,
Φ‖(x) =
(x, y)
ey − r2
(x, y)
ey − r20
ε0 + 1
(1− r20e−y)
ey − r20
− x2 2ε0
ε0 + 1
(1− r20e−y)
, (39)
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Thermal Casimir force between dielectrics 11
and consider these three integrals separately. The first integral in terms of the new
variable v = y/x reads
Q1(x) ≡ x3
evx − r2
− v2 r
evx − r20
ε0 + 1
(1− r20e−vx)
, (40)
where, in accordance with Eq. (16),
r‖(v) ≡ r‖(0, v) =
ε0v −
v2 + ε0 − 1
ε0v +
v2 + ε0 − 1
. (41)
Expanding Q1(x) in powers of x and explicitly calculating the remaining inte-
grals for the lowest, third, power of x results in
Q1(x) = x
1− r2
1− r20
ε0 + 1
(1− r20)
+O(x4) (42)
1 + 3ε0 − 2 ε20
2 r20
1− r20
− 12 ε0
ε0 + 1
(1− r20)
+O(x4).
The second and third integrals on the right-hand side of Eq. (39) are simply deter-
mined with the following result:
Q2(x) ≡
ey − r20
(2 + 2nx+ n2x2)e−nx
= 3Li3(r
1− r20
+O(x4) , (43)
Q3(x) ≡ −
ε0 + 1
(1− r20e−y)
= − 2ε0x
ε0 + 1
1− r20e−x
= − 2ε0
ε0 + 1
1− r20
− x3 r
(1− r20)2
+O(x4) . (44)
Substituting Eqs. (42), (43) and (44) into Φ‖(x) = Q1(x) + Q2(x) +Q3(x), we
arrive at
Φ‖(ix)− Φ‖(−ix) = −i
2ε20 − 3ε0 − 1
+ O(x4). (45)
Then, by summing Eqs. (38) and (45), the result is obtained
Φ(ix)− Φ(−ix) = −i
2 + ε
0 − ε
0 − 2
+ O(x4). (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
P (a, T ) = P0(a)−
32π2a4
ε0 − 1
ε20 + ε
0 − 2
τ4 +O(τ5)
. (47)
By comparison with Eq. (30) the explicit form of the coefficient C4 is found as
ε0 − 1
ε20 + ε
0 − 2
/720 (48)
October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr
12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
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 thermody-
namic 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
S(a, T ) = −
∂F(a, T )
3kBζ(3)(ε0 − 1)2
64π3a2(ε0 + 1)
τ2 (49)
2π2(ε0 + 1)
0 + 2ε0 + 2
ε0 + 2
135ζ(3)(
ε0 + 1)2
τ + O(τ2)
As is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions
to the entropy are of the second and the third powers in the small parameter τ .
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 metals40,72,73 and for real metals
described by the plasma model.45,52 For example, in the case of plates made of
ideal metal the entropy at low temperatures is given by40,72,73
S(a, T ) =
3kBζ(3)
32π3a2
1− 2π
135ζ(3)
τ + O(τ2)
. (50)
Note, however, that the expansion coefficients in Eq. (50) cannot be obtained as
a straightforward limit |ε0| → ∞ 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 τ → 0 and |ε0| → ∞ in the power expansions
of functions depending on ε0 as a parameter.
Remarkably, the low-temperature behavior of the free energy, pressure and en-
tropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is deter-
mined only by the static dielectric permittivity. The absorption bands included in
Eq. (9) do not influence the low-temperature behavior. A more simple derivation
of the results (29), (47)–(49) for dielectrics with constant ε is contained in Ref. 74.
As was demonstrated above, all these results remain unchanged if the dependence
of dielectric permittivity on frequency is taken into account.
In Ref. 60 more general results were obtained related to two dissimilar dielectric
plates with dielectric permittivities ε(1)(ω) and ε(2)(ω). For brevity here we present
only the final expressions for the low-temperature behavior of the Casimir free
energy, pressure and entropy between dissimilar plates. They are as follows:60
F(a, T ) = E(a)− ~c
32π2a3
0 + ε
0 + 2ε
0 + 1)(ε
0 + 1)
0 − 1)(ε
0 − 1)
0 + ε
− C4 τ +O(τ2)
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Thermal Casimir force between dielectrics 13
P (a, T ) = P0(a)−
32π2a4
4 +O(τ5)
, (51)
S(a, T ) =
3ζ(3)
0 + ε
0 + 2ε
0 + 1)(ε
0 + 1)
0 − 1)(ε
0 − 1)
0 + ε
− C4 τ +O(τ2)
Here ε
(1,2)
0 ≡ ε(1,2)(0) and the coefficient C4 is given by60
0 + ε
0 + ε
0 + 2ε
0 − ε
0 − 3ε
0 − 3ε
0 + 1
0 − ε
0 − ε
0 − 1
0 − 1
0 + ε
Artanh
0 + ε
0 − ε
0 − ε
It is easily seen that in the limit ε
0 = ε
0 = ε0 equations (51), (52) coincide with
equations (29), (47)–(49) having obtained above. Note that in the application region
of low-temperature 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 a = 400 nm
as functions of temperature in the configuration of two dissimilar plates made of
high-resistivity Si and SiO2. The dielectric permittivities of both materials along
the imaginary frequency axis were computed in Ref. 75 using the optical data of
Ref. 76. The precise thermal corrections computed by taking into account these
permittivities are shown by the solid lines and corrections computed by our analyt-
ical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines
indicate the results computed by the Lifshitz formula with constant dielectric per-
mittivities of Si and SiO2 equal to ε
0 = 11.67 and ε
0 = 3.84, respectively. As is
seen in Fig. 1a,b, at T < 60K the results obtained using the analytical asymptotic
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14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
25 50 75 100 125 150 175 200
T (K)
−∆F (pJ)
25 50 75 100 125 150 175 200
T (K)
−∆P (µPa)
Fig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration
of two plates, one made of Si and another one of SiO2, at a separation a = 400 nm as a function of
temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated
optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed
lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines).
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 τ ≫ 1, i.e., at high temperatures (large
separations). It is well known37,38,77 that in this case the approximation of static
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Thermal Casimir force between dielectrics 15
dielectric permittivity works good and the main contribution is given by the zero-
frequency term of the Lifshitz formula (13)
F(a, T ) = ~cτ
64π2a3
ydy ln
1− r20e−y
(the other terms being exponentially small). Performing the integration in Eq. (53)
we obtain
F(a, T ) = − kBT
16πa2
. (54)
In a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows
P (a, T ) = −
, S(a, T ) =
16πa2
. (55)
Equations (54) and (55) are simply generalized60 for the case of two dissimilar di-
electric plates by performing the replacement r20 → r
0 , where r
(1,2)
0 are defined
by Eq. (11) with the static dielectric permittivities of dissimilar plates ε
(1,2)
3. IS THE DC CONDUCTIVITY RELATED TO THE CASIMIR
INTERACTION BETWEEN DIELECTRICS?
As was discussed in the previous section, the zero-frequency term in formula (13),
i.e., the contribution with l = 0, is of prime importance and determines many of
the basic properties of the Casimir interaction. In the above consideration we have
described dielectric materials by Eq. (9) with finite static dielectric permittivity ε0.
This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger
and the other one is smaller than the physical value for real photons at normal
incidence. However, in the Lifshitz theory, the departure of both coefficients from
their physical values is coordinated in such a way that the Nernst heat theorem
remains valid.
It is common knowledge that at nonzero temperatures dielectric materials pos-
sess a negligibly small but not equal to zero dc conductivity. From physical intuition
it is reasonable to expect that the influence of this conductivity on the van der Waals
and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the
contrary, the inclusion of small dielectric dc conductivity in the model of dielectric
response leads to a large effect in dispersion forces. This raises the question if dc
conductivity is related to dispersion forces or if, on the contrary, the zero-frequency
contribution should be understood not literally but as an analytic continuation from
the region of high frequencies determining the physical phenomenon of dispersion
forces.
To illustrate this problem in more details, we consider the asymptotic behavior
of the free energy and entropy at low temperature with included dc conductivity.
What this means is that, instead of the dielectric permittivity ε(iξl) given in Eq. (9),
one uses57,58
ε̃(iξl) = ε(iξl) +
= εl(iξ) +
β(T )
, (56)
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16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where σ0 is the dc conductivity of the plate material and β(T ) = 2~σ0/(kBT ). The
conductivity of dielectrics depends on temperature as σ0 ∼ exp(−b/T ) where b is
determined by the energy gap ∆which differs for different materials.78 The small-
ness of the dc conductivity of dielectrics can be illustrated79 by the example of
SiO2 where at T = 300K it holds β ∼ 10−12. Thus, the role of the dc conductivity
is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease
of T and, as a consequence, remains negligible at any T . In spite of this, the substi-
tution of Eq. (56) into the reflection coefficients (16) leads to different result than
in Eq. (11):
r̃‖(0, y) = 1, r̃‖(0, y) = 0. (57)
Equation (57) is in some analogy to Eq. (6), obtained for metals described by the
Drude model, which leads to the violation of the Nernst heat theorem in the case
of perfect crystal lattices.
Now we substitute the dielectric permittivity ε̃l ≡ ε̃(ξl) in Eqs. (13) – (16)
instead of εl and find the Casimir free energy F̃(a, T ) with included dc conductivity.
For convenience, we separate the zero-frequency term, subtract and add the usual
zero-frequency contribution for dielectric without dc conductivity. The result is
F̃(a, T ) =
16πa2
1− e−y
1− r20e−y
16πa2
ydy ln
1− r20e−y
1− r̃2‖(ζl, y)e
1− r̃2⊥(ζl, y)e−y
where r0 was defined in Eq. (11). Let us expand the last, third, integral on the right-
hand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the
zero-order contribution in this expansion with the second integral on the right-
hand side of Eq. (58) and obtain the Casimir free energy F(a, T ) calculated with
the dielectric permittivities εl. The first integral on the right-hand side of Eq. (58)
is calculated explicitly. Then, Eq. (58) can be rewritten as
F̃(a, T ) = F(a, T )− kBT
16πa2
ζ(3)− Li3
+R(a, T ). (59)
Here, R(a, T ) is of order O(β/l). It represents the first and higher-order contribu-
tions in the expansion of the third integral on the right-hand side of Eq. (58) in
powers of β/l. Restricting its explicit form to the first order contribution we get
R(a, T ) = R1(a, T ) +O
(β/l)
, (60)
R1(a, T ) =
dy y2e−y
y2 + ζ2l (εl − 1)
(2− εl) ζ2l − 2y2
y2 + ζ2l (εl − 1) + εly
r‖(ζl, y)
1− r2
(ζl, y)e−y
y2 + ζ2l (εl − 1) + y
r⊥(ζl, y)
1− r2⊥(ζl, y)e−y
. (61)
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Thermal Casimir force between dielectrics 17
Calculating the entropy by the first equality in Eq. (49), we arrive at
S̃(a, T ) = S(a, T ) +
16πa2
ζ(3)− Li3
− ∂R(a, T )
, (62)
where S(a, T ) is the entropy for the plates with the dielectric permittivity εl given
by Eq. (49).
Let us now show that the quantity R(a, T ) exponentially goes to zero with the
decrease of T . First we consider only the integral in Eq. (61), expand the integrated
function in powers of τ (we recall that ζl = τl), restrict ourselves to the main
contribution, resulting for τ = 0, and rearrange it appropriately:
(ε0 + 1)2
1− r20e−y
= − 2
(ε20 − 1)
1− r20e−y
(ε20 − 1)
dy ye−ny = −
(ε20 − 1)
1 + nζl
e−nζl . (63)
Substituting this in Eq. (61), we find
R1(a, T ) = −
4πa2 (ε20 − 1)
e−nτl
e−nτl
= − kBTβ
4πa2 (ε20 − 1)
1− e−nτ
enτ − 1
kB Li2
2πa2 (ε20 − 1)
Tβ ln τ + TβO(τ0) . (64)
Here, the last line is obtained by using the equality
1− e−nτ
enτ − 1
= − ln τ + 1− lnn+O(τ2) , (65)
substituting only its leading term and observing the definition of the integral log-
arithm, Li2 (z) = (1/2)
zn/n2. Taking into account that β ∼ (1/T ) exp(−b/T ),
we get the conclusion that the temperature dependence of R1(a, T ) is given by
R1(a, T ) ∼ e−b/T lnT. (66)
Thus, both R1(a, T ) and its derivative with respect to T in Eqs. (59) and (62) go
to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even
faster than R1 when T → 0.
Finally, in the limit T → 0 from Eq. (62) it follows
S̃(a, 0) =
16πa2
ζ(3)− Li3
> 0. (67)
The right-hand side of this equation depends on the parameter of the system under
consideration (the separation distance a) and implies a violation of the Nernst heat
theorem. An analogous result was obtained60 in the case of two dissimilar dielectrics.
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18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
The violation of the Nernst heat theorem in the Casimir interaction for dielectrics
originates from the inclusion of the dc conductivity in the model of dielectric re-
sponse. This violation is, however, of different nature than the one discussed above
in the case of Drude metals. In the case of dielectrics the entropy at zero tempera-
ture is positive but in the case of Drude metals it is negative. In the case of metals
the violation is caused by the vanishing contribution from the transverse electric
mode at zero frequency whereas the other reflection coefficient takes the physical
value 1 [see Eq. (6)]. For dielectrics the situation is quite opposite. In this case the
transverse reflection coefficient at zero frequency is always equal to zero [compare
Eqs. (11) and (57) in the absence and in the presence of the contribution from
dc conductivity]. Here the violation occurs due to the unity value of the parallel
reflection coefficient at zero frequency in Eqs. (57) which departs from the value
r0 = (ε0 − 1)/(ε0 + 1) coordinated with the zero value of the transverse coefficient
in Eq. (11).
One can conclude that the dc conductivity of a dielectric is not related to the na-
ture of the van der Waals and Casimir forces and must not be included in the model
of dielectric response. Ignoring this rule results in a violation of thermodynamics.
Physically it is amply clear that there is no fluctuating field of zero frequency and
that for such high-frequency phenomena as the van der Waals and Casimir forces the
low-frequency behavior should be obtained by analytic continuation from the region
of high frequencies. This permits to conclude that the correct procedure consists in
the substitution of the finite static dielectric permittivities into the zero-frequency
term of the Lifshitz formula, as Lifshitz and his collaborators really did.36,37,38
4. THERMAL CASIMIR FORCE BETWEEN DIELECTRIC
AND METAL PLATES
The Casimir interaction between metal and dielectric plates suggests the interest-
ing opportunity to verify the thermodynamic consistency of the Lifshitz theory
with different models of the dielectric response. This configuration was first inves-
tigated in Ref. 61 where it was proved that the Casimir entropy is in accordance
with the demands of the Nernst heat theorem if the static permittivity of the di-
electric plate is finite. In Ref. 61, however, only the first leading terms in the low-
temperature asymptotic expressions for the free energy and entropy were obtained
and the Casimir pressure was derived only in the dilute approximation. Here we
derive the more precise low-temperature behavior for the Casimir free energy, pres-
sure and entropy in the configuration of one plate made of ideal metal and another
plate made of dielectric with any finite static dielectric permittivity.
For the configuration of metal and dielectric plates the Lifshitz formula takes
the form80 analogical to Eq. (13)
F(a, T ) = ~cτ
32π2a3
1− δl0
ydy (68)
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Thermal Casimir force between dielectrics 19
1− rM‖ (ζl, y) rD‖ (ζl, y) e−y
1− rM⊥ (ζl, y) rD⊥ (ζl, y) e−y
Here the reflection coefficients r
for metal and dielectric, respectively, are given
by Eqs. (16) where εl should be changed for ε
l = ε
M,D(iξl).
For an ideal metal rM‖,⊥(ζl, y) = 1 and Eq. (68) takes the more simple form
F(a, T ) =
32π2a3
ydy (69)
1− r‖(ζl, y)e−y
1− r⊥(ζl, y)e−y
(here and below we omit the index D near the reflection coefficient and permit-
tivity of a dielectric plate). We admit that the dielectric permittivity calculated at
Matsubara frequencies εl ≡ ε0, i.e., is equal to its static value and find the asymp-
totic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove
that the deviations of ε(iξl) from ε0 at high frequencies do not influence the low-
temperature behavior of the Casimir free energy, pressure and entropy. It can be
shown also that the results of this section are valid not only for ideal metal plate
but for plate made of real metal as well.] The free energy (69) can be represented
by Eqs. (18) – (21), with the function f(ζ, y) replaced by
f̂(ζ, y) = y ln
1− r‖(ζ, y)e−y
+ y ln
1− r⊥(ζ, y)e−y
≡ f̂‖(ζ, y) + f̂⊥(ζ, y). (70)
In the case of one dielectric and one metal plate both f̂‖ and f̂⊥ contribute to
F (ix)− F (−ix). The expansion of f̂(x, y) in powers of x takes the form
f̂(x, y) = y ln(1− r0e−y)−
ε0 − 1
e−y − ε0
ε0 + 1
x2 +O(x3). (71)
Now we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x).
The integral of the first term on the right-hand side of Eq. (71) is evaluated using
the new variable v = y − x:
ydy ln(1− r0e−y) =
vdv ln(1− r0e−v) + O(x2), (72)
where the coefficient near the first-order contribution in x vanishes. As a result, this
term could contribute to F (ix) − F (−ix) only starting from the third expansion
order. The integrals of the second-order terms on the right-hand side of Eq. (71)
are simply calculated using the formulas
= −Ei(−x),
= −Ei(−nx). (73)
Finally, we obtain
F (ix)− F (−ix) = iπ
(ε0 − 1)2
4 (ε0 + 1)
x2 − iγ x3 +O(x4), (74)
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20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
where the unknown third order expansion coefficient is designated as γ.
Substituting Eq. (74) in Eq. (21) and using Eq. (19), we find the free energy in
the system metal-dielectric in the form
F(a, T ) = E(a)− ~c
32π2a3
(ε0 − 1)2
ε0 + 1
τ3 −K4 τ4 +O(τ5)
, (75)
where K4 ≡ γ/240.
The Casimir pressure in the configuration of metal and dielectric plates obtained
from Eq. (75) is equal to
P (a, T ) = P0(a)−
32π2a4
4 +O(τ5)
. (76)
The direct application of the Lifshitz formula gives the expression for the pressure
analogical to Eq. (31),
P (a, T ) = − ~cτ
32π2a4
1− δl0
r‖(ζl, y)
ey − r‖(ζl, y)
r⊥(ζl, y)
ey − r⊥(ζl, y)
. (77)
Using the Abel-Plana formula (17), Eq. (77) can be represented in the form of
Eqs. (32), (33) where
Φ‖,⊥(x) =
y2 r‖,⊥(x, y)
ey − r‖,⊥(x, y)
. (78)
Again, we deal first with Φ⊥(x). By adding and subtracting the asymptotic
behavior of the integrated function at small x,
y2r⊥(x, y)
ey − r⊥(x, y)
(ε0 − 1)x2e−y +O(x3) , (79)
and introducing the new variable v = y/x, the function Φ⊥(x) can be identically
rearranged and expanded in powers of x as follows:
Φ⊥(x) =
(ε0 − 1)x2e−x + x3
rn⊥(v)e
−nvx − 1
(ε0 − 1)e−vx
(ε0 − 1)x2(1− x) + x3
v2r⊥(v)
1− r⊥(v)
ε0 − 1
+O(x4). (80)
The integral on the right-hand side of Eq. (80) is converging and can be simply
calculated with the result
Φ⊥(x) =
ε0 − 1
x2 − 1
ε0 − 1)x3 +O(x4). (81)
To deal with Φ‖(x) we add and subtract in Eq. (78) the two first expansion
terms of the integrated function in powers of x,
Φ‖(x) =
ey − r0
− ε0r0e
y2(ε0 + 1) (1− r0e−y)2
r‖(x, y)
ey − r‖(x, y)
ey − r0
ε0r0e
y2(ε0 + 1) (1− r0e−y)2
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Thermal Casimir force between dielectrics 21
The asymptotic expansion of the first integral on the right-hand side of Eq. (82) is
given by
2Li3(r0)−
ε0(ε0 − 1)
2(ε0 + 1)
(ε0 − 1)(3ε0 − 2)x3 +O(x4), (83)
and of the second one by
ε0(ε0 − 1)−
ε0(ε0
ε0 − 1) +
ε0(ε0 − 1)
x3 +O(x4). (84)
By summing Eqs. (83) and (84) we find
Φ‖(x) = 2Li3(r0)−
ε0(ε0 − 1)
2(ε0 + 1)
x2 (85)
(ε0 − 1) + (ε0
ε0 − 1)− 3ε0(ε0 − 1)
x3 +O(x4).
Finally we add Eq. (85) to Eq. (81) and arrive at
Φ(ix) − Φ(−ix) = −i
1− 2ε0
ε0 + ε
x3 +O(x4). (86)
Substituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the
Casimir pressure is obtained
P (a, T ) = P0(a)−
32π2a4
1− 2ε0
ε0 + ε
τ4 +O(τ5)
. (87)
Comparing Eqs. (87) and (76), the explicit expression for the coefficient K4 reads
1− 2ε0
ε0 + ε
. (88)
Now we are in a position to find the asymptotic behavior of the entropy in the
limit of low temperatures in the configuration of two parallel plates one of which
is metallic and the other one dielectric. By calculating the negative derivative of
Eq. (75) with respect to temperature, one arrives at
S(a, T ) =
3kBζ(3)(ε0 − 1)2
128π3a2(ε0 + 1)
τ2 (89)
8π2(ε0 + 1)
1− 2ε0
ε0 + ε
135ζ(3)(ε0 − 1)2
τ +O(τ2)
This equation is in analogy to Eq. (49) obtained for the case of two dielectrics.
As is seen from Eq. (89), the entropy of the Casimir and van der Waals interactions
between metal and dielectric plates vanishes when the temperature goes to zero,
i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ2 on the
right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at
low temperatures the entropy goes to zero remaining positive. At the same time, as
was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may
take negative values. This interesting property distinguishes the configuration of
metal and dielectric plates from two dielectric plates. In the latter configuration the
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22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
negative entropy appears only for nonphysical dielectrics with anomalously large
and frequency independent dielectric permittivities.53
Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations).
In the same way as for two dielectric plates, here the main contribution to the free
energy is given by the term with l = 0 in Eq. (69),
F(a, T ) = ~cτ
64π2a3
ydy ln
1− r0e−y
. (90)
Performing the integration in Eq. (90), we obtain
F(a, T ) = − kBT
16πa2
Li3 (r0) . (91)
From this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows
P (a, T ) = − kBT
Li3 (r0) , S(a, T ) =
16πa2
Li3 (r0) . (92)
The results (91) and (92) are analogous to (54) and (55) for two dielectric plates.
5. THE PROBLEM ORIGINATING FROM THE ACCOUNT OF
DIELECTRIC DC CONDUCTIVITY
In the previous section it was supposed that the static dielectric permittivity ε0 of
the dielectric plate is finite. Now we will deal with the configuration of metal and
dielectric plates with included dc conductivity of the dielectric material. In doing
so the permittivity of the dielectric plate is given by
ε̃(iξl) = ε0 +
β(T )
, (93)
where all notations were introduced in and below Eq. (56). Thus, the reflection
coefficients at zero frequency satisfy Eq. (4) for a plate made of ideal metal and
Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find
the low-temperature behavior of the Casimir entropy and verify the consistency of
the Lifshitz theory with thermodynamics in this nonstandard situation.
For this purpose we substitute the dielectric permittivity (93) in Eq. (69) in-
stead of ε0 and find the Casimir energy F̃(a, T ) with included dc conductivity of
a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the
zero-frequency term of F̃(a, T ) and subtract and add the usual zero-frequency con-
tribution for metal-dielectric plates computed with the dielectric permittivity ε0,
F̃(a, T ) = kBT
16πa2
1− e−y
1− r0e−y
16πa2
ydy ln
1− r0e−y
1− r̃‖(ζl, y)e−y
1− r̃⊥(ζl, y)e−y
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Thermal Casimir force between dielectrics 23
Here the reflection coefficients r̃‖,⊥ are calculated with the permittivity (93). We
expand the third integral on the right-hand side of Eq. (94) in powers of the small
parameter β/l. The zero-order contribution in this expansion together with the
second integral of Eq. (94) form the Casimir free energy F(a, T ) calculated with
dielectric permittivity ε0. The first integral on the right-hand side of Eq. (94) is
calculated explicitly. As a result, Eq. (94) is rearranged to
F̃(a, T ) = F(a, T )− kBT
16πa2
[ζ(3)− Li3 (r0)] +Q(a, T ), (95)
where Q(a, T ) contains the first and higher-order contributions in the expansion of
the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit
form of the main first-order term in Q(a, T ) is the following:
Q1(a, T ) =
dy y2e−y
y2 + ζ2l (ε0 − 1)
(2− ε0)ζ2l − 2y2
y2 + ζ2l (ε0 − 1) + ε0y
1− r‖(ζl, y)e−y
y2 + ζ2l (ε0 − 1) + y
1− r⊥(ζl, y)e−y
. (96)
In the same way as in Sec. 3, we expand the integrated function in Eq. (96) in
powers of τ (bearing in mind that ζl = τl) and preserve only the main contribution
at τ = 0:
Q1(a, T ) = −
dy yr0e
(ε20 − 1) (1− r0e−y)
= − kBTβ
4πa2(ε20 − 1)
e−nτl
e−nτl
Dealing with this expression in the same way as with Eq. (64), we arrive at
Q1(a, T ) ∼ e−b/T lnT. (98)
The Casimir entropy in the configuration of metal and dielectric plates with
included dc conductivity of the dielectric plate is obtained as minus derivative of
Eq. (95) with respect to temperature,
S̃(a, T ) = S(a, T ) +
16πa2
[ζ(3)− Li3 (r0)]−
∂Q(a, T )
. (99)
Using Eq. (98), the calculation of the limiting value at T → 0 is straightforward:
S̃(a, 0) =
16πa2
[ζ(3)− Li3 (r0)] > 0. (100)
From this equation it follows that the inclusion of the dc conductivity of dielectric
plate in the configuration metal-dielectric results in a violation of the Nernst heat
theorem. In the above this result was obtained for a metallic plate made of ideal
metal. It can be shown that it remains valid for a metallic plate made of real metal
with finite conductivity.
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24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
Thus, both configurations (two dielectric plates or one metal plate and one
dielectric) lead to the same conclusion that when the dc conductivity is included
in the model of dielectric response of the dielectric plate, the Lifshitz theory loses
its consistency with thermodynamics. This confirms the conclusion made in Sec. 3
that the actual properties of dielectric materials at very low frequencies are in fact
not related to the van der Waals and Casimir forces.
6. QUALITATIVE DISCUSSION OF CASIMIR INTERACTION
BETWEEN METAL AND SEMICONDUCTOR
As was mentioned in the Introduction, semiconductors possess a wide variety of
electric and optical properties ranging from metallic to dielectric. This opens the
possibility to modulate the van der Waals and Casimir forces by changing the charge
carrier density. Bearing in mind the discussed above problems on the consistency
of the Lifshitz theory with thermodynamics, semiconductors can provide us with
a test for the validity of different approaches. Thus, if for good dielectric the dc
conductivity does not play any real role in the van der Waals and Casimir forces,
the question arises on how much it should be increased in order to become a relevant
factor in the description of dispersion forces.
In Ref. 81 the Casimir force acting between two Si plates was calculated using the
simple analytic expression for Si dielectric permittivity as a function of frequency.
The complete tabulated optical data of Si were used in Ref. 75 to calculate the van
der Waals interaction of different atoms with a Si wall. The first attempt to measure
the van der Waals force between a glass lens and a Si plate and to modify it by light
due to the change of carrier density was undertaken in Ref. 82. However, glass is a
dielectric and therefore the electric forces due to localized point charges could not
be controlled. This might explain that no force change occured on illumination at
small separations where the effect should be most pronounced.
The first measurements of the Casimir force between a gold coated sphere and
a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic
force microscope. The experiments used a p-type B doped Si plate of resistivity
ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate
between the resistivity of metals (which is usually two or three orders of magnitude
lower) and the resistivity of dielectrics (it can be by about a factor of 105 larger; for
instance, high-resistivity “dielectric” Si has the resistivity ρ0 = 1000 Ω cm). Thus,
the used Si plate had a relatively large absorption typical for semiconductors but it
was also enough conductive to avoid the accumulation of charges.63
In Fig. 2, taken from Ref. 63, the differences of the theoretical and mean ex-
perimental Casimir forces acting between Au sphere and Si plate are presented as
functions of separation. In Fig. 2a the theoretical force F theor is computed using
the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively
low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as
ξ−1 with decreasing frequency (see the solid line in Fig. 3). In Fig. 2b the theo-
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Thermal Casimir force between dielectrics 25
80 100 120 140
  (nm)
F theor − F̄ expt (pN)
80 100 120 140
 (nm)
F̃ theor − F̄ expt (pN)
Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation.
Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si.
Solid and dashed lines indicate 95 and 70% confidence intervals, respectively.
retical Casimir force F̃ theor is computed using the dielectric permittivity of the Si
plate made of “dielectric” Si with high resistivity ρ0. The dielectric permittivity of
high-resistivity Si is shown by the dashed line in Fig. 3. It is characterized by the
finite static value εSi(0) = 11.67. The solid and dashed lines in Fig. 2a,b indicate
the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2, the the-
oretical approach using the dielectric permittivity of high-resistivity “dielectric” Si
is excluded by experiment within the separation range from 60 to 110nm at 70%
confidence. At the same time, the theory using the dielectric permittivity of Si with
a low resistivity ρ is consistent with experiment.
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26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
13.5 14 14.5 15 15.5 16 16.5 17
Log10 [ξ (rad/s)]
ε (iξ)D
Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis
(solid line). Dashed line shows the dielectric permittivity of dielectric Si.
The above results suggest an approach on how to correctly determine the pos-
sible role of the low-frequency conductivity properties in dispersion forces. As is
seen from Fig. 3, the dielectric permittivity of low-resistivity Si (solid line) sig-
nificantly departs from the dielectric permittivity of “dielectric” Si in the region
around the important dimensional parameter of the problem, the characteristic fre-
quency c/(2a) ∼ 1014 − 1015 rad/s. That is why the low-resistivity sample cannot
be described at low frequencies by the static dielectric permittivity of Si equal to
εSi(0) = 11.67. To describe it, the term β(T )/l, like in Eq. (93), should be added
to εSi(0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with
l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter.
At the same time, for a high-resistivity sample the inclusion of the dc conduc-
tivity would lead to deviations from the dashed line in Fig. 3 only at frequencies
ξ < 108 rad/s, which are much less than the characteristic frequency. This compari-
son permits to make a conclusion in what experimental situations the conductivity
properties of semiconductors at low frequencies should be taken into account and
when they should be omitted as being not related to dispersion forces.
The future experiments on the modification of semiconductor charge carrier
density by laser light83 will bring a more clear understanding of this problem on
the connection between the low-frequency material properties and dispersion forces.
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Thermal Casimir force between dielectrics 27
7. DOES SPATIAL DISPERSION LEAD TO AN IMPORTANT
IMPACT ON THERMAL CASIMIR FORCE?
The presented above new analytic results on the low-temperature behavior of the
Casimir free energy, pressure and entropy between dielectrics or between dielectric
and metal are based on the conventional Lifshitz theory which describes dielectric
materials by means of the frequency dependent dielectric permittivity. In fact, the
assumption that the material of the plates possesses only the frequency dispersion
means that the components of electric displacement are connected with the compo-
nents of electric field by the relation
Dk(r, ω) = εkl(r, ω)El(r, ω). (101)
This equation is central in all different derivations of the Lifshitz formula (see, e.g.,
Refs. 2, 6, 36–38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion)
are in fact essential only at shortest separations between the plates comparable with
atomic dimensions and also for metals at sufficiently large separations (typically of
about 2–3µm) in the frequency region of the anomalous skin effect. The Casimir
force in the latter region was described by the Lifshitz theory reformulated in terms
of the Leontovich impedance.85
Recently the spatial dispersion came to the attention in connection with the
thermal Casimir force.64,65,66,67,68 In particular, it was claimed64,65 that for real
metals at any separation the account of spatial dispersion leads to practically the
same result (6) for the reflection coefficients at zero frequency as was obtained earlier
using the Drude model dielectric function (5). This conclusion, if it is correct, not
only returns us to the contradiction with experiment30,32,33 but also casts doubts
on all results obtained by means of the conventional Lifshitz theory accounting for
only the frequency dispersion. It is natural when the spatial dispersion contributes
a small fraction of a percent as it is generally believed in numerous applications of
the Lifshitz theory. It is, however, quite another matter when the account of the
spatial dispersion results in some “dramatic effects”,64 i.e., in several hundred times
larger thermal correction than is obtained in the local case. Below we demonstrate
that the conclusions of Refs. 64 – 68 are in fact not reliable because they use the
Lifshitz theory of dispersion forces outside of its application range.86
To find the electromagnetic modes associated with an empty gap between the
plates, Refs. 64 – 68 use the standard continuity boundary conditions,
E1t = E2t, B1n = B2n, D1n = D2n, B1t = B2t, (102)
which are commonly applied in the derivation of the Lifshitz formula for spatially
local nonmagnetic materials. Here B is the magnetic induction, n is the normal to
the boundary directed inside the medium, the subscripts n, t refer to the normal
and tangential components, respectively, the subscript 1 refers to the vacuum and
subscript 2 to the plate material. In Refs. 64 – 68 the spatial dispersion is described
by the longitudinal and transverse dielectric permittivities depending on the wave
vector and frequency: εkl = εkl(q, ω). However, as is shown below, in the theory
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28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
of the Casimir effect both the boundary conditions (102) and permittivities εkl =
εkl(q, ω) are inapplicable.
We start from the boundary conditions and recall the set of Maxwell equations
in a metal describing the Casimir effect,
rotE +
= 0, divD = 0, rotB − 1
= 0 divB = 0. (103)
Equations (103) do not contain any external, i.e., independent on E, D and B,
current or charge densities. The definition of the electric displacement is
+ 4πi, (104)
where the volume current i is induced by E and B and takes into account the
conduction electrons.
In electrodynamics with spatial dispersion the electric field and magnetic induc-
tion are finite at the boundary surfaces, whereas the electric displacement can tend
to infinity.87 Then, integrating Eqs. (103) over the thickness of the boundary layer
as is done in Ref. 88, we reproduce the first two conditions in Eq. (102) and arrive
at the modified third and fourth conditions,87,89
E1t = E2t, B1n = B2n, D2n −D1n = 4πσ, [n× (B2 −B1)] =
j, (105)
where the induced surface charge and current densities are given by
div[n× [D × n]]dl, j = 1
dl. (106)
Note that the boundary conditions (105), (106) are obtained from the macroscopic
Maxwell equations for physical fields. They should not be mixed with the boundary
conditions arising in perturbative theories and for the fictitious fields (see below).
In linear electrodynamics for a medium with time-independent properties with-
out spatial dispersion the material equation connecting the electric displacement
and electric field takes the form
Dk(r, t) =
ε̂kl(r, t− t′)El(r, t′)dt′. (107)
According to this equation, the electric displacement at a point r and moment t is
determined by the electric field at the same point r at different moments t′ ≤ t (the
spatial dispersion is absent but the temporal may be present). It is easily seen that
the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the
boundary conditions (105) coincide with the standard continuity conditions (102). It
is unjustified, however, to use conditions (102) in the presence of spatial dispersion.
In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ
nor j is equal to zero.
We now turn to a discussion of the use of dielectric permittivity εkl(q, ω) in the
theory of the Casimir effect with account of spatial dispersion. In the presence of
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Thermal Casimir force between dielectrics 29
only frequency dispersion, it is possible to perform the Fourier transformation of
the fields
E(r, t) =
E(r, ω)e−iωtdω, D(r, t) =
D(r, ω)e−iωtdω, (108)
in Eq. (107) and arrive at Eq. (101) where
εkl(r, ω) =
ε̂kl(r, τ)e
iωτdτ (109)
is the frequency-dependent dielectric permittivity and τ ≡ t− t′. In fact Eqs. (101),
(108) and (109) are used in parallel with the boundary conditions (102) in all
derivations of the Lifshitz formula.2,6,36,37,38,69,70,84
If the material of the plates is characterized not only by temporal but also spatial
dispersion, Eq. (107) is generalized to
Dk(r, t) =
dr′ε̂kl(r, r
′, t− t′)El(r′, t′). (110)
If the material medium were uniform in space (i.e., all points were equivalent), the
kernel ε̂ would not depend on r and r′ separately, as in Eq. (110), but on the
difference R ≡ r − r′. In this case, by performing the Fourier transformation,
E(r, t) =
dqE(q, ω)ei(qr−ωt),
D(r, t) =
dqD(q, ω)ei(qr−ωt), (111)
and substituting it in Eq. (110), one could introduce the dielectric permittivities
εkl(q, ω) =
dR ε̂kl(R, τ)e
−i(qR−ωτ), (112)
as Refs. 64–68 do, and rearrange Eq. (110) to the form
Dk(q, ω) = εkl(q, ω)El(q, ω). (113)
In the Casimir effect, however, the material medium is not uniform due to the
presence of a macroscopic gap between the two plates (half spaces). Because of this,
the assumption that the kernel ε̂ depends only on R and τ is wrong. As a result,
it is not possible to introduce the dielectric permittivity εkl(q, ω) depending on
the wave vector and frequency. In fact, for systems with spatial dispersion in the
presence of boundaries the kernel ε̂ depends not only on R and τ but also on the
distance from the boundary.87 In this complicated situation the following approx-
imate phenomenological approach is sometimes applicable.87 For electromagnetic
waves with a wavelength λ the kernel ε̂(r, r′, τ) in Eq. (110) differs essentially from
zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ
(for nonmetallic condensed media l is of the order of the lattice constant). Then it is
reasonable to assume that ε̂ is a function of R=r–r′, except for a layer of thickness
l adjacent to the boundary surface. If one is mostly interested in bulk phenomena
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30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
and neglects the role and influence of a subsurface layer, the quantity εkl(q, ω) may
be employed as a reasonable approximation.
This approximate phenomenological approach is widely applied in the theoretical
investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91
some kind of fictitious infinite system was introduced and electromagnetic fields in
this system are discontinuous on the boundary surface. This discontinuity should
not be confused with the discontinuity of physical fields of a real system in the
presence of spatial dispersion described by Eqs. (105) and (106). (There is also
another approach to the description of the anomalous skin effect in polycrystals
using the generalizations of the local Leontovich impedance which takes into account
the shape of Fermi surface92). The frequency and wave vector dependent dielectric
permittivity in the presence of boundaries is also approximately applied in the
theory of radiative heat transfer93 or in the study of electromagnetic interaction of
molecules with metal surfaces.94 In all these applications the boundary effects are
usually taken into account by the boundary conditions (105) supplemented by so
called “additional boundary conditions”.
It is unlikely, however, that the approximate phenomenological approach using
such quantity as εkl(q, ω) in the presence of boundaries would be applicable in the
theory of the Casimir force where the boundary effects on the zero-point electromag-
netic oscillations are of prime importance. It is notable also that this approach faces
serious theoretical difficulties including the violation of the law of conservation of
energy.95 It is then not surprising that the application of this approach in Refs. 64,
65 results in the contribution to the Casimir free energy from the transverse electric
mode which is in contradiction with experiment.30,32,33
One more shortcoming of Refs. 64–68 is that they substitute the dielectric per-
mittivity εkl(q, ω), depending on both wave vector and frequency, into the conven-
tional Lifshitz formula derived in the presence of only temporal dispersion. In the
famous review paper96 it has been noticed, however, that with the inclusion of spa-
tial dispersion the free energy of a fluctuating field takes the form F = FL +∆F ,
where FL is given by the conventional Lifshitz expression derived in a spatially
local case and written in terms of the Fresnel reflection coefficients, and ∆F is an
additional term which can be expressed in terms of the thermal Green’s function of
the electromagnetic field and polarization operator. Review96 calls as not reliable
the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity
εkl(q, ω), taking account of spatial dispersion, into the conventional Lifshitz for-
mula. It can be true that the Lifshitz formula written in terms of general reflection
coefficients is applicable in both spatially local and nonlocal situations. However,
as far as the exact reflection coefficients in a spatially nonlocal case are unknown,
the use of some approximate phenomenological models, elaborated in literature for
applications different than the Casimir effect, may lead to incorrect results for ∆F
and create inconsistencies with experiment.
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Thermal Casimir force between dielectrics 31
To conclude this section, the results of Refs. 64–68 on the influence of spatial
nonlocality on the Casimir interaction are shown to be not reliable. Although at
present there is no fundamental theory of the thermal Casimir force incorporating
spatial dispersion, there is no reason to expect that it can play any significant role
in the frequency region of infrared optics (experimental separations) or normal skin
effect (i.e., at separations between plates greater than 4–5µm).
8. CONCLUSIONS AND DISCUSSION
In the foregoing we have presented the derivation of analytic asymptotic expres-
sions for the free energy, pressure and entropy of the Casimir interaction between
two dielectric plates and between metal and dielectric plates at low and high temper-
atures. It was shown that the low-temperature behavior of the Casimir interaction
between dielectrics and between dielectric and metal is determined by the static
dielectric permittivities of nonpolar dielectrics. The obtained results were shown to
be in agreement with thermodynamics when the static dielectric permittivities of
dielectrics are finite. In particular, the entropy of the Casimir interaction goes to
zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This
demonstrates the consistency of the original Lifshitz’s approach to the van der Waals
forces between dielectrics which disregards the small conductivity of dielectrics at
constant current.
The second important result shown above is that the inclusion of the dc con-
ductivity of dielectrics into the model of dielectric response leads not to some small
corrections to the characteristics of the Casimir interaction, as one could expect,
but makes the Lifshitz theory inconsistent with thermodynamics leading to the vi-
olation of the Nernst heat theorem. This reveals that real material properties at
very low, quasistatic frequencies are in fact not related to the phenomenon of the
van der Waals and Casimir forces which is actually determined by sufficiently high
frequencies. In this case the zero-frequency contribution to the Casimir force should
be understood not literally but as analytic continuation to zero of the material
physical behavior in the region around the characteristic frequency.
The presented results provide a basis for the calculation of the van der Waals
and Casimir forces between real materials. Such calculations are much needed for
numerous applications of the Casimir force discussed in the Introduction, in par-
ticular for the applications in nanotechnology and for constraining predictions of
fundamental physical theories beyond the Standard Model. Bearing in mind that
semiconductors are the main constituent materials in nanotechnological devices, it
is a subject of high priority to understand the Casimir effect with semiconductor
boundaries. In this connection we have discussed new experimental results and the-
oretical ideas on the Casimir interaction between a metal sphere and semiconductor
plate. It was stressed that by changing the charge carrier density in the semicon-
ductor it is possible to bring it in different intermediate states between metallic
and dielectric. In this case the problem arises when the conductivity properties of
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32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko
semiconductor are not related to dispersion forces and when they are becoming rel-
evant. A criterion for the resolution of this problem was formulated based on the
relation between the typical frequency at which the dc conductivity properties come
into play and the characteristic frequency of the Casimir effect. In fact the thermal
Casimir interaction between semiconductors remains an open question and much
work should be done both in experiment and theory to gain a better insight into
this subject.
The last major problem discussed in this review is whether or not the spatial
dispersion influences essentially the thermal Casimir force between real materials. In
the present literature there is no agreement on this subject. We adduced arguments
in favor of the statement that in the region of experimental separations the influence
of the spatial dispersion on the Casimir force is negligible small. The statements
on the opposite, contained in the literature, were shown to be not reliable because
they are obtained by the application of the Lifshitz theory outside of its application
range. At the same time it was ascertained that at the moment there is no consistent
fundamental theory of the van der Waals and Casimir forces taking the spatial
dispersion into account. This is the problem to solve in the foreseeable future.
ACKNOWLEDGMENTS
G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Insti-
tute for Theoretical Physics, Leipzig University for their kind hospitality. This work
was supported by Deutsche Forschungsgemeinschaft grant 436RUS113/789/0-2.
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