Temperature correction to the Casimir force in cryogenic range and anomalous skin effect

12 Pages·0.19 MB·English

by  V. B. Svetovoy

Temperature correction to the Casimir force in cryogenic range
and anomalous skin effect
∗
V.B. Svetovoy and M.V. Lokhanin
Department of Physics, Yaroslavl State University and
Institute of Microelectronics RAS
Sovetskaya 14, Yaroslavl 150000, Russia
(February 1, 2008)
Temperature correction to the Casimir force is considered for real metals at low temperatures.
With the temperature decrease the mean free path for electrons becomes larger than the field
penetration depth. In this condition description of metals with the impedance of anomalous skin
effect is shown to be more appropriate than with the permittivity. The effect is crucial for the
temperaturecorrection. Itisdemonstratedthatinthezerofrequencylimitthereflectioncoefficients
3
should coincide with those of ideal metal if we demand the entropy to be zero at T = 0. All the
0
otherprescriptionsdiscussedintheliteratureforthen=0termintheLifshitzformulagivenegative
0
entropy. It is shown that the temperature correction in the region of anomalous skin effect is not
2
suppressed as it happens in the plasma model. This correction will be important in the future

n cryogenic measurements of theCasimir force.
a
J 12.20.Ds, 11.10.Wx, 12.20.Fv, 42.50.Lc

9

1
v
I. INTRODUCTION
5
3
0 Attraction between parallel metallic plates predicted by Casimir in 1948 [1] (see [2] for a recent review) has been
1 measuredinrecentexperiments[3,4,5,6,7,8]withhighprecision. Tocalculatetheforcewiththesameprecision,onehas
0 to take into account different corrections to the original Casimir force [2]. The corrections due to finite conductivity
3 of the plates and roughness of their surfaces can be large but there is no principal problems with them and the
0
discussion is going around the material properties [9,10,11,12,13,14,15]. However, the correction connected with
/
h the finite temperature raised debate in the literature [16,17,18,19,20,21,22,23,24,25,13,15]. The agreement between
p different authors has not been reached yet.
-
Inthe earlyperiodthetemperaturecorrectionwasfoundforidealmetal. Ithasbeendiscoveredthatthe correction
t
n following from the Lifshitz theory [26] of electromagnetic fluctuations in nonhomogeneous media did not agree with
a
that found with different methods [27,28]. The problem originatesfromthe n=0 termin the Lifshitz formula giving
u
the classical contribution in the Casimir force from long wavelength fluctuations. Special prescription for this term
q
: was proposed by Schwinger, DeRaad, and Milton [29] to reconcile different approaches. Namely, one has to take the
v
limit of infinite permittivity before allowing the frequency go to zero.
i
X The problem arose againwhen researchestried to find this correctionfor real metals. It has been realized that the
plasmaandthe Drudemodelsforthe metaldielectricfunctiongavedifferentcorrections,whichdidnotcoincidewhen
r
a theDruderelaxationfrequencyω isgoingtozero. DirectapplicationoftheLifshitzformulawiththeDrudedielectric
τ
function gives large linear in temperature correction [16], which contradicts to the torsion pendulum experiment [3].
Negligiblefortheexperimentscorrectionwasfoundusingtheplasmamodel[17,18]. However,theDrudemodelworks
much better for real metals and there is no a satisfactory reason to justify the use of the plasma model.
It became obvious that something wrong with the n=0 term in the Lifshitz formula and while a solid theoretical
reasonfor modificationof this termis not found one has to use a reasonableprescriptionto define the n=0 term. It
was proposed [13] to apply for real metals the same prescription as for ideal metal [29]. It was motivated by the fact
that in the static limit the boundary conditions onthe metalsurface did not depend on any materialproperty. Later
it has been shown [19] that the classical term can be constructed from very general principles using the dimensional
analysis, low frequency limit for the permittivity, and the known result for ideal metal. The same conclusion was
made in a recent paper [25], where the authors stressed difference between boundary conditions on dielectrics and
metals in the zero frequency limit. In this case the linear in temperature correction survives but it is suppressed by
small additional factor. The correctionis unobservable [13] in the conditions of torsion pendulum experiment [3] but
∗E-mail: [email protected]
1
3 times larger [15] than the experimental errors in the atomic force microscope (AFM) experiment [5]. Note that it
makes agreement between theory and experiment better.
Quite different prescription was proposed for real metals by Klimchitskaya and Mostepanenko [20]. The driving
idea was to modify the reflection coefficients in the n = 0 term in such a way to make them coincide with that of
plasma model in the limit ω 0. The reason for that was based on the property of the reflection coefficient for
τ
→
perpendicular polarization r (ζ,q) considered as a function of two variables: continuous imaginary frequency ζ and
1
the absolute value of the momentum along the plate q. It has integrated discontinuity in the point ζ = 0, q = 0
which authors [20] consider as unacceptable. The resulting temperature dependence does not include the linear in
temperature term and negligible in conditions of the torsion pendulum and AFM experiments.
Thus,atthemomenttherearethreedifferentapproachestotheproblemoftemperaturedependenceoftheCasimir
force. Recently an interesting proposition [30] to check the Nernst heat theorem for different prescriptions discussed
in the literature was made. According to the third law of thermodynamics the entropy S of the closed system in
equilibrium must go to zero in the limit T 0. Based on the analytical result for the plasma model and numerical
→
calculations for the Drude model a conclusion was made that the prescription of Schwinger, Deraad, and Milton
cannotbe appliedtorealmetals becausethe entropystillfinite inthe T 0limit. Onthe contrary,itwasconcluded
→
that for the prescription proposed by Klimchitskaya and Mostepanenko S 0 when T 0.
→ →
Inthispaperweintendedtodemonstratethattherealsituationinthelowtemperaturelimitisdifferent. Thereason
for this is that with the temperature decreasethe meanfree pathfor electrons l increasesand atsome temperature it
inevitably becomes larger than the field penetration depth δ. It is the range of anomalous skin effect when the space
dispersion becomes important. Taking this effect into account modifies the temperature correction, which does not
coincide with that found in Ref. [30]. Independent interestto the correctionatlow temperatures is stimulatedby the
progress in experiments which are going to cryogenic temperatures to improve the precision [5].
The paper is organized as follows. In Section II we give definition of the Casimir free energy and explain the
problem with the n=0 term. This term is written out for all prescriptions discussed in the literature. In Section III
it is demonstrated that neither plasma nor Drude models can be used for real metals at low temperatures. The way
to calculate the temperature correction using the impedance of anomalous skin effect is described. In Section IV the
actual calculations are presented. The results are applied to analyse the entropy behavior at T 0 and to find the
→
value of temperature correction in cryogenic range in Section V. Our conclusions are collected in the last section.
II. CASIMIR FREE ENERGY
We start from the Lifshitz expression for the force per unit area between two parallel plates separated by the
distance a at temperature T [31]. Simple transformation of this formula allows to write the free energy in the form
[2]
∞ ′ ∞
kT
(a,T)= dyy ln(1 r2(ξ ,y)e−y)+r r , (1)
F 8πa2 − 1 n 1 → 2
n=0 Z
X ξn (cid:2) (cid:3)
where the prime over the summation sign indicates that the first n=0 term has to be taken with the factor 1/2 and
2πT ¯hc
ξ =nτ, τ = , kT = ¯hω . (2)
n eff a
T 2a ≡
eff
Here ξ are the dimensionless Matsubara frequencies ξ =ζ /ω , where ζ =2πkTn/¯h. In Eq.(1) r2 (ζ ,q) are the
n n n a n 1,2 n
reflection coefficients for orthogonal (r ) or parallel (r ) polarizations. These coefficients depend on the material via
1 2
the dielectric function ε(iζ ) at imaginary frequencies. In the Drude model ε(iζ ) has the form
n n
ω2
p
ε(iζ)=1+ (3)
ζ(ζ+ω )
τ
and is defined by two parameters which are the plasma frequency ω and the relaxation frequency ω . This model
p τ
nicely fits good metals in the whole frequency range excluding the interband absorption region ω ω , which does
p
∼
not give considerable contribution especially at low temperatures.
Considerfirsttheproblematicn=0term. Expression(1)wasfoundbysolvingtheequationfortheGreenfunction
in nonhomogeneous media at finite temperature [31]. The explicit form of the reflection coefficients r2 depends on
1,2
the boundary conditions for the Green function at the metal-vacuum interface. Note that these conditions one can
2
put separately for any Matsubara components. Typically the continuity of tangential components of electric and
magnetic fields at the interface is demanded. The Drude model gives for the amplitudes r
1,2
R2+y2 y R2+y2 1+R2/ξ2 y
r (ξ,y)= − , r (ξ,y)= − , (4)
1 2
pR2+y2+y pR2+y2+(cid:0)(1+R2/ξ2)(cid:1)y
p p
where the function R(ξ) is defined as
ω ξ
p
R(ξ)= . (5)
ωasξ+ωτ/ωa
In the n=0 term we have to put ξ =0 and the reflection coefficients are
0
r2(0,y)=0, r2(0,y)=1. (6)
1 2
while for ideal metal both of these coefficients are equal to unit r2 = r2 = 1. There is no way to reconcile the value
1 2
of r with that of ideal metal since it does not depend on any parameters. This is why one needs to introduce a
1
prescriptionfor the n=0 term. The coefficients (6) wereusedto calculate the temperature correctionto the Casimir
force in Ref. [16].
It was proposed [13] to use the extension of the Schwinger, DeRaad, and Milton prescription to real metals. Since
in the static field limit the boundary condition on the metal surface does not depend on the characteristics of a
particular metal, the reflection coefficients were changed with that of ideal metal
r2(0,y) 1, r2(0,y)=1. (7)
1 → 2
Different prescription was introduced by Klimchitskaya and Mostepanenko [20] who proposed to modify r2 in such a
1
way that in the limit ω 0 it coincided with the plasma model result
τ
→
2
R2(y)+y2 y
r2(0,y) r2(y,y)= − , r2(0,q)=1. (8)
1 → 1 pR2(y)+y2+y! 2
p
The motivation of(7)and (8)was briefly describedaboveandwe arenot goingto the details referringto the original
papers [19,20].
Let us denote the n = 0 term in (1) as (a,T). Calculating the corresponding integral with the reflection
0
F
coefficients (6-8) one gets
kT
(a,T)= α ζ(3), (9)
F0 − 8πa2
where ζ(x) is the zeta function. The coefficient α has the following values for the prescriptions (6), (7), and (8),
respectively:
1
α(i) = , α(ii) =1,
2
ω ω 2 ω ω2
α(iii) =1 4 a τ I τ +O a . (10)
− ω − ω ζ(3) 2 ω ω2
p p (cid:18) a(cid:19) (cid:18) p(cid:19)
The function I (x) varies slowly with x and is given by Eq.(68) [20].
2
III. ANOMALOUS SKIN EFFECT
Letusconsidernowthen 1termsin(1)whenthetemperatureisgoingdown. Theplasmafrequencyω isdefined
p
≥
by the electron density in metal and practically does not depend on temperature. On the contrary, the relaxation
frequency ω changeswithT significantly. IfT <T , whereT is the Debye temperature for agivenmetal,then the
τ D D
dependence ω (T) can be presented as (see, for example, [32])
τ
3
ω (T)=ω (0)+C T2+C T5 (11)
τ τ e ph
with the parameters ω (0), C , and C . Here the first term is connected with the scattering on lattice irregularities
τ e ph
and impurities, the second one describes scattering on electrons, and the third term corresponds to the scattering
on phonons. The first and second terms can dominate only at very low temperatures. The first term needs special
discussion. The relaxationfrequency is proportionalto the material resistivity which disappears atT 0 for perfect
→
monocrystals. Ifwearegoingtocheckthethirdlawofthermodynamicsforoursystem,wehavetotakethisequilibrium
state and choose ω (0)=0. To describe the temperature behavior of real material used in the experiment, it can be
τ
wrong. For evaporated metallic films the residual resistivity can be significant due to large density of defects but it
is much smaller than the resistivity at room temperature.
Onecaneasilyseethe resultofRef.[30]withoutanycalculations. Atlowtemperatureω /ω in(5)isgoingtozero
τ a
faster than ξ T. Therefore, for all n = 0 terms one can use the plasma model. The reflection coefficients in the
n
∼ 6
n = 0 term (8) were prescribed to reproduce the plasma model at ω 0. Therefore, the plasma model completely
τ
→
describes the situation at low temperatures. It is well known [17] that the leading temperature correction in this
modelbehavesasT3 andthe entropywillgotozeroasS T2. AnyotherprescriptionwillnotagreewiththeNernst
∼
theoremsincethen=0termdoesnotcoincidewiththatfortheplasmamodel. For(6)theentropybecomesnegative
and for (7) it is positive but both of them are finite at T =0. No doubt that at low temperature and for equilibrium
state the relaxation frequency becomes negligible. However, it is wrong to think that the reflection coefficients (4)
for n = 0 will be the same at low temperatures. This is because the mean free path for electrons increases with
6
the temperature decrease according to the relation l = v /ω (T), where v is the Fermi velocity. At some small
F τ F
temperature inevitably the relation l δ will be fulfilled, where δ = c/ω is the field penetration depth1. Then the
p
≫
condition l δ is equivalent to the following
≫
v
F
ω (T) ω Ω. (12)
τ p
≪ c ≡
The frequency Ω=(v /c)ω is often used as a characteristic frequency of anomalous skin effect.
F p
Thus,thelocalconnectionbetweenthecurrentandelectricfield,whichistrueforl<δ,isbrokenatlowtemperature
and space dispersionbecomes important. This is the range of anomalousskin effect [33], when the dielectric function
ε(ω) cannot be used any more for description of the metal. Instead the interaction with the field is defined by the
surface impedance Z(ω) [34,33]. The relation Z(ω) ε−1/2(ω), which holds for the normal skin effect, is broken.
∼
When the metal is described by the impedance the boundary condition on its surface will be [34]
E =Z(ω)(H n), (13)
t t
×
where E ,H arethe tangentialcomponents ofelectric andmagnetic fields, n is the unit vectornormalto the surface
t t
and directed inside of the metal. It holds true while the impedance is small. If this boundary condition is used for
the Matsubara components (n 1) of the Green function, then the reflection coefficients can be written as follows
≥
[35,36]
ξ yZ(iζ ) y ξ Z(iζ )
n n n n
r = − , r = − . (14)
1 2
ξ +yZ(iζ ) y+ξ Z(iζ )
n n n n
Impedance in the range of anomalous skin effect one can find solving the kinetic equation for not in equilibrium
distribution function of electrons in the electromagnetic wave [33] or using simple qualitative analysis [32].
In the frequency range ω <Ω (strong anomalous skin effect) the impedance is [33,32]
vω2 1/3
Z(ω)=e−iπ/3 . (15)
cω2
(cid:18) p(cid:19)
Herev =βv ,whereβ isafactor 1,whichisdefinedbythestructureoftheFermisurface. Notethattherelaxation
F
∼
frequency falls out from the impedance at all. This is because the field interacts mainly with electrons, which are
moving in the surface layer of thickness δ. For this reason significant is the effective conductivity σ =σδ/l, which
eff
1Strictly speaking δ depends on frequency. We assumed that ω =ζn ≫ωτ(T) (n6=0). If it is not the case, we have to take
δ=(c/ωp) 2ωτ/ω but thequalitative conclusion will not change.
p
4
does not depend on ω . Actually we are interesting in the impedance at imaginary frequencies. For the strong
τ
anomalous skin effect it will be
vω2 1/3
Z(iζ )= aξ2 . (16)
n cω2 n
(cid:18) p (cid:19)
This expressionis true for ξ <Ω/ω while the rangeof important n in the sum (1) is given by the conditionξ 1.
n a n
∼
For small Ω/ω the relation (16) cannot be used for all important n. However, as we will see below the temperature
a
dependent part of free energy always can be described by (16).
The anomalous skin effect will change not only the temperature correctionbut the main temperature independent
partof the Casimirforce or free energy. To find this change one has to know the impedance in the whole rangeof ξ .
n
It is very important for comparison with the experimental data in cryogenic range but out of the scope of this paper
and will be discussed elsewhere.
IV. CALCULATION OF THE FREE ENERGY
Now we are ready to find the sum of all n 1 terms in (1). This sum can be transformed using the Abel-Plana
≥
formula [37]. To avoid inconvenient integrals we first change the summation index to m = n 1. In this way the
−
problematic coefficient r (0,y) will not appear in the formula as it usually happens [17]. The resulting expression is
1
longer than usual but simpler for calculations. Separating the temperature independent part one finds
kT 1
(a,T) (a,T)= (a,0)+ I(1)+I(2) I(1)+I(2) + I(1)+I(2) . (17)
F −F0 F 8πa2 2 1 1 − 2 2 3 3
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)(cid:21)
Here the temperature independent term (a,0) is
F
∞ ∞
¯hc
(a,0)= dξ dy (ξ+y)ln 1 r2(ξ,y+ξ)e−y−ξ +r r (18)
F 16πa3 − 1 1 → 2
Z Z
0 0 (cid:2) (cid:0) (cid:1) (cid:3)
and the integrals I(j) (j =1,2; k =1,2,3) are defined as
k
∞
I(j) = dyyln 1 r2(τ,y)e−y ,
1 − j
Z
τ (cid:2) (cid:3)
1 ∞
I(j) = dt dyyln 1 r2(τt,y)e−y ,
2 − j
Z Z
0 τt (cid:2) (cid:3)
∞ ∞
dt
I(j) =2Im dy(y+iτt)ln 1 r2(τ +iτt,y+iτt)e−y−iτt . (19)
3 e2πt 1 − j
Z0 − Zτ (cid:2) (cid:3)
Inthetemperatureindependentterm(18)theimportantrangeofcontinuousvariableξisξ 1and,aswasmentioned
∼
above, the impedance (16) cannot be used to cover all the range of ξ. But this term is not interesting for us. In the
temperature dependent terms ξ =τt τ as one can see from (19). We can use (16) in the integrals (19) if τ <Ω/ω
a
∼
or equivalently 2πkT <¯hΩ. This relation together with T <T will be supposed to be true.
D
In all the integrals (19) the lower limit in y can be changed into zero. Really, the error connected with such a
replacement will be of the order of τ2lnτ. It will be considered as small for analytical calculations but will be taken
into account in the numerical procedure. Then the integrals I(1) depending on the reflection coefficient r2 (14) will
k 1
be defined only by the parameter
1/3
cω2
p
A= τ . (20)
vω2
a !
5
Even for small τ the typical value of this parameter is large A 1. Only for very low temperature kT
(ω /ω )3¯hΩ/2π it becomes small (A 1). It is in contrast with≫the parameter B which defines the integra≪ls
a p
≪
I(2)
k
vω2 1/3
B = aτ5 . (21)
cω2
(cid:18) p (cid:19)
This parameter is always small in the range of interest τ 1.
≪
A. Free energy in the limit A≪1
Let us analyse now the analyticaldependence of the integrals (19) from the parametersA, B at very low tempera-
tures when A 1. We start from I(1). Neglecting the correction τ2lnτ this integral can be written
≪ 1 ∼
∞ ∞
I(1) = dyyln 1 e−y + dyyln 1+ 4Ay (ey 1)−1 . (22)
1 − " (y+A)2 − #
Z Z
0 (cid:0) (cid:1) 0
The first integral here is equal ζ(3). It describes contribution of the ideal metal. For the second integral one can
−
find the leading terms in the limit A 0. The important contribution gives the range y <∼ 1 but the main one is
→
collected near y √A. Then I(1) can be presented as
∼ 1
1
4A
I(1) ζ(3)+ dyyln 1+ 1 .
1 ≈− y2
Z (cid:18) (cid:19)
0
The result is the following
I(1) = ζ(3) 2A(lnA+2ln2 1)+O A2 . (23)
1 − − −
(cid:0) (cid:1)
The integralI(1) can be estimated in the same way but one has to change A At1/3 and then integrate it overt. It
2 →
gives
3 5
I(1) = ζ(3) A lnA+2ln2 +O A2 . (24)
2 − − 2 − 4
(cid:18) (cid:19)
(cid:0) (cid:1)
The same procedure can be applied for I(1) but realization is more complicated. Contribution of the ideal metal
3
(zero impedance) is negligible in our approximationsince the leading term is known [27,28]to be τ2. Correction due
to nonzero impedance depends on A. In the same approximation as for (23) and (24) one has
∞ 1
dt 4A (1+it)1/3
I(1) =2Im dy(y+iτt)ln 1+ .
3 e2πt 1 y2
!
Z − Z
0 0
The inner integral is calculated analytically. The contribution from the term proportional to iτt will be of the order
of τ√A which is small in comparison with the leading terms. All the rest can be written as
I(1) =4A(q lnA+q )+O τ√A . (25)
3 1 2
(cid:16) (cid:17)
The constants q here are
1,2
∞ ∞
dt 1+t2 1/6sinϑ dt 1+t2 1/6
q = , q = sinϑ ln4 1+t2 1/6 1 +ϑcosϑ , (26)
1 e2πt 1 2 e2πt 1 −
Z0 (cid:0) (cid:1)− Z0 (cid:0) −(cid:1) h (cid:16) (cid:0) (cid:1) (cid:17) i
6
where tan3ϑ=t. Numerically these constants are q =0.0137,q =0.0191.
1 2
The integrals I(2) can be calculated quite similar. Since B is always much smaller than A one can completely
k
neglect the terms containing B and for I(2) one has
k
I(2) = ζ(3), I(2) = ζ(3), I(2) =0. (27)
1 − 2 − 3
Substituting(23-25,27)into(17)andtakingintoaccountexplicitexpression(9)for (a,T)onefindsthetemperature
0
F
correction to the free energy in the limit A 0
→
kT 1 7
∆ (a,T)= (a,T) (a,0)= (1 α)ζ(3)+A +4q lnA+ln2 +4q . (28)
F F −F 8πa2 − 2 1 − 8 2
(cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:21)
This expression is true for very low temperature (see Eq.(20)) and is appropriate only to check the Nerst theorem.
It will be discussed in the next section. For realistic low temperatures, which will be explored in the near future
experiments, the typical value of A is large even for small τ and we have to analyse the opposite limit A 1.
≫
B. Free energy in the limit A≫1
(1)
To calculate the integrals I in the temperature range when the parameter A is large but the temperature still
k
small (τ 1), it will be convenient again to separate the contribution of ideal metal and neglect the correction
≪
τ2lnτ due to change ofthe low limit inintegrals into zero. Then for I(1) one has the same representation(22)and
∼ 1
similar for the other integrals. Because A is large one can expand the logarithm in series and collect the coefficient
at different powers of 1/A. For I(1) this procedure can be done straightforwardand we find for the first two terms
1
1 6
(1)
I = ζ(3)+8ζ(3) . (29)
1 − A − A2
(cid:18) (cid:19)
In case of I(1) the role of A plays At1/3. In this integral after logarithm expansion one has to make first the
2
integration over t not to run onto divergencies. The result is the following
1 12
I(1) = ζ(3)+12ζ(3) . (30)
2 − A − A2
(cid:18) (cid:19)
In the third integral instead of A appears A(1+it)1/3. In our approximation one can completely neglect iτt in
comparison with A or with y. Making the same procedures one gets
p 6p
I(1) =16ζ(3) 1 2 , (31)
3 A − A2
(cid:18) (cid:19)
where the coefficients are defined as
∞ ∞
sinϑ sin2ϑ
p = dt , p = dt . (32)
1 (e2πt 1)(1+t2)1/6 2 (e2πt 1)(1+t2)1/3
Z0 − Z0 −
Here the angle ϑ is defined as in (26). Numerically the coefficients are p =0.0133, p =0.0262.
1 2
The integrals I(2)giving the contribution of parallel (r ) polarization depends on the parameter B which is small
k 2
while τ 1. Therefore, they give the same result (27) as in the case of small A. Collecting all together one finds the
≪
temperature correction to the free energy in the limit A 1
≫
kT 1 2p 15 12p
1 2
∆ (a,T)= ζ(3) (1 α) 8 − − . (33)
F 8πa2 − − A − A2
(cid:20) (cid:18) (cid:19)(cid:21)
The temperature correctionto the Casimir force can be easily found via ∆ . Correctionto the force between sphere
F
and plate ∆F is directly proportional to ∆ if we adopt the proximity force theorem [38]
sp
F
∆F (a,T)=2πR∆ (a,T), (34)
sp
F
7
where R is the sphere radius. The temperature correction for the force between two plates ∆F is
pp
∂ kT 31 2p 15 12p
1 2
∆F (a,T)= ∆ (a,T)= ζ(3) (1 α) 8 − 2 − (35)
pp −∂a F 4πa3 − − 2 A − A2
(cid:20) (cid:18) (cid:19)(cid:21)
Thus, we found the asymptotics for ∆ (a,T) in the limits A 0 (28) and A (33). Now let us calculate
F → → ∞
numerically this function in the transition region.
C. Numerical calculation of the free energy
For numerical calculation we parametrize the temperature correction to the free energy in the following way
kT
∆ (a,T)= [(1 α)ζ(3) G(A,τ)]. (36)
F 8πa2 − −
The absolute scale of the correction is given by the factor kT/8πa2 then G(A,τ) can be treated as the relative
temperature correction if α = 1. The function G(A,τ) is calculated numerically as a function of A for a few given
values of τ. There is no need to separate B as an independent parameter since it is connected with the other two
as B = τ2/A. Alternatively at fixed material parameters ω and v it can be found as a function of T at a fixed a.
p
Actual calculation has been done using the relation
1
G(A,τ)= I(1)+I(2) 2ζ(3) + I(1)+I(2) 2ζ(3) I(1)+I(2) .
−2 1 1 − 2 2 − − 3 3
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
The integralswerecalculatedaccordingto (19)with the absolute precisionof10−6. Fig.1 showsthe function G(A,0)
and its asymptotics at A 0 and A , which can be extracted from (28) and (33). One can see that there is
→ → ∞
good agreement of numerical calculation and analytical asymptotics. If we consider nonzero but small τ, the result
will change only slightly due to correction τ2lnτ. For ω =1.37 1016 rad/s and v =1.5 108 cm/s (gold) G(a,T)
p
∼ · ·
as a function of T is shown in Fig.2 at a few values of a. This figure shows that the temperature correctionis always
significant in the interesting range of temperatures and distances between plates.
V. DISCUSSION OF THE RESULTS
A. Entropy in the limit T →0
We already saw that the anomalous skin effect changes behavior of the free energy at low temperatures. Now we
areabletoanswerthequestionwhichoftheprescriptions(6),(7)or(8)agreeswiththethirdlawofthermodynamics.
In the low temperature limit entropy can be calculated from (28)
∂ k 4 1 3
S = F = (α 1)ζ(3) A +4q lnA+ln2 +4q +q . (37)
−∂T 8πa2 − − 3 2 1 − 4 2 1
(cid:20) (cid:18)(cid:18) (cid:19) (cid:19)(cid:21)
Parameterαcorrespondingtodifferentprescriptionsusedforthen=0termintheLifshitzformulaisgivenbyEq.(10).
If this term is calculated as it appears in the Lifshitz formula [16] without any modification, then α=α(i) =1/2. In
this case the entropy is finite at T = 0 (A = 0) and, moreover, it is negative. This conclusion coincides with that
made in Ref. [30], where at low temperatures the authors describedthe plate materialwith the plasma model. It can
beconsideredasanadditionalphysicalargumentthatunmodifiedLifshitzformulacannotbeusedtogettemperature
behavior of the Casimir force. The previous argument [19,20] was that in this approach the n = 0 term in (1) does
not depend on any material parameter (see (9) with α = α(i) ) and for this reason it cannot be reconciled with the
ideal metal result, which is two times larger. These arguments show that the Lifshitz formula is really in trouble. It
obviouslyhas to be modified but after a few yearsof active discussionstill there is no solid theoreticalunderstanding
what is wrong and how one can do this.
PrescriptionproposedbyKlimchitskayaandMostepanenko[20]wasdesignedtoreproducethe plasmamodelresult
for the n = 0 term in the limit ω 0. This limit was supposed to be realized at low temperatures but, as was
τ
→
explained above, the material has to be described in this case by the impedance of anomalous skin effect rather than
the plasma model permittivity. For this reasonthe entropy for α=α(iii) is not going to zero when T 0. It is easy
→
8
to see from (37) and (10) that S is finite and negative at T = 0. Therefore, this prescription is also does not obey
the Nernsttheorem. At highertemperatures,when the Drude modelis valid, the n=0 termgivenby (9,10)depends
separately from ω and ω . Physically it is unacceptable since at low frequencies the only parameter characterizing
p τ
a metal in respect to electromagnetic field is the conductivity σ ω2/ω [34].
∼ p τ
If we use the prescription (7) [13] then α = α(ii) = 1. In this case the temperature independent term in (37) is
canceled completely and entropy is going to zero with the temperature as T1/3lnT. It is the only prescriptionwhich
is in agreement with the third law of thermodynamics. This prescription was criticized [17,20] on the basis that the
n =0 term does not depend on any material parameter (see (9)). We already explained [19] that in the static (long
wavelength)limit the boundary condition for metals do not include a particular metal parameters. More specifically,
from dimensional analysis it follows that α in (9) can be a function of the only variable [19]
ω ω
τ a
α=α . (38)
ω2
(cid:18) p (cid:19)
The characteristicfrequency which appearshere is huge ω2/ω 1018 rad/s (for goodmetals at roomtemperature).
p τ ∼
Of course, one can expand α in a series
ω ω
′ τ a
α α(0)+α (0)
≈ ω2
(cid:18) p (cid:19)
but the correction to α(0) becomes important only at microscopic distances between the plates a 10−8 cm where
∼
the macroscopic Casimir force is not defined. The value of α(0) should coincide with that for ideal metal which
is known to be 1. In this way the prescription (7) is reproduced without direct reference to the Drude reflection
coefficients (4).
B. Temperature correction to the Casimir force
The results of previous section for the temperature correction to the free energy or equivalently to the force (see
(34) and (35)) show that for α = 1 the correction always increases the absolute value of the force (the correction is
negative so as the attractive force). The correction is relatively large kT/8πa2. It is unlike to the plasma model
where additional small factors τ2 [17] or ω /ω [19] appear if one uses∼prescriptions (8) or (7), respectively. In this
a p
sense one can say that in the range of anomalous skin effect the temperature correction becomes large. The relative
correction G(a,T) in (36) is maximal ( 0.53) at the temperature
≈
¯hω vω2
kT 18 a a . (39)
m ≈ 2π cω2
(cid:18) p(cid:19)
In the ideal metal limit ω this temperature is going to zero and instead of large correctionappears usual ideal
p
→∞
metal correction [27,28]given by
τ 2 π3 τ 3
G(a,T)=ζ(3) . (40)
2π − 45 2π
(cid:16) (cid:17) (cid:16) (cid:17)
Let us describe the temperature rangewhere the resultofthis work will be applicable. Consider firstthe condition
l δ which guarantees that anomalous skin effect plays important role. It can give us the upper limit on the
≫
temperature. We approximate ω (T) with the Bloch-Gru¨neisen formula
τ
T/TD
ω (T) T 5 F (T/T ) x5
τ 5 D
= , F (T/T )= dx , (41)
ω (T ) T F (T /T ) 5 D (ex 1)(1 e−x)
τ 0 (cid:18) 0(cid:19) 5 0 D Z − −
0
◦
where T is some fixed temperature, for example, T = 0 C. This formula does not take into account scattering on
0 0
the defects and electrons, which can be important at very low temperatures, but it is good to find the upper limit
on T. The value of ω (T ) can be fixed via the material resistivity ρ using the relation ω = ε ω2ρ, where ε is
τ 0 τ 0 p 0
the permittivity of vacuum. For gold parameters ω = 1.37 1016 rad/s, v = 1.4 108 cm/s, ρ(T ) = 2.06 µΩ cm
p F 0
◦ · ◦ · ·
we found l/δ > 5 for T < 113 K and l/δ > 10 for T < 67 K. The other condition 2πkT < ¯hΩ ensures that a
◦
specific expressionfor the impedance (16) will be true. It restricts the temperature by the value T <77.5 K (gold).
Therefore, already for liquid nitrogen our result for the temperature correction is applicable.
9
VI. CONCLUSION
We considered the temperature correction to the Casimir free energy (force) in the low temperature range. The
aims of this analysis were to investigate the behavior of entropy in the limit T 0 and find out if it is possible to
→
neglect the temperature correction in the future low temperature experiments. The main observation of this work
is that at low temperature T <T (Debye temperature) neither plasma nor Drude models for material permittivity
D
are good for real metals. The reasonis that with the temperature decrease the mean free path for electrons becomes
larger than the penetration depth for electromagnetic field in metal. This relation between parameters is realized for
the anomalousskineffectforwhichdescriptionofmetalswithimpedance ismoreappropriatethanwiththedielectric
function. This change in the description is important for the temperature correction.
Itisknownthatthefirstn=0termintheLifshitzformulafortheCasimirforceiscontroversial. Alotofdiscussion
in the literature is goingonaroundthis term. The problemishow one candefine this termcorrectly. At the moment
there are three different approaches corresponding to three different results for the correction. It was proposed to
use the third law of thermodynamics to choose one of the approach [30]. We analysed the entropy behavior in the
limit T 0 fordiscussedinthe literature prescriptionsandcame to a conclusionthatone hasto define the reflection
→
coefficients in the static field limit as for the ideal metal r2 1, r2 = 1 to get agreement with the third law of
1 → 2
thermodynamics. It is in contrast with the conclusion of Ref. [30], where the plasma model was used for metals at
low temperatures.
To reduce the noise, the future experiments on precise measurement of the Casimir force will explore the low
temperature range. Previously metals at low temperatures were described by the plasma model because the Drude
relaxationfrequency decreasesfastwith temperature. The plasma modelpredictsa negligibletemperature correction
to the force. We demonstrated that the anomalous skin effect makes drastic change. In all interesting from the
experimentalpointofviewrangeoftemperaturesanddistancesbetweenbodies the correctionis notnegligibleonthe
precision level of modern experiments.
We do not consider our result as final solution of the problem with the temperature correction. This is because a
realsolutionshould not be basedon any prescription. The problemindicates that something wrongwith the Lifshitz
formula and efforts have to be directed on the careful analysis of this formula. It becomes more and more clear that
theessenceoftheproblemliesintheboundaryconditionsonthemetalsurface. Theimpedancecondition(13)applied
to the n =0 term reproduced the right result for the reflection coefficients r2 =r2 = 1 [25]. However, it is not clear
1 2
why there is no a smooth transition between continuity of tangential components of E and H and the impedance
boundary condition.
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