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New approach to the thermal Casimir force between real metals

arXiv:0801.3445v1 [quant-ph] 22 Jan 2008

V. M. Mostepanenko1,2 and B. Geyer1

Center of Theoretical Studies and Institute for Theoretical Physics, Leipzig University, D-04009, Leipzig, Germany 2 Noncommercial Partnership “Scienti?c Instruments”, Tverskaya St. 11, Moscow, 103905, Russia Abstract. The new approach to the theoretical description of the thermal Casimir force between real metals is presented. It uses the plasma-like dielectric permittivity that takes into account the interband transitions of core electrons. This permittivity precisely satis?es the Kramers-Kronig relations. The respective Casimir entropy is positive and vanishes at zero temperature in accordance with the Nernst heat theorem. The physical reasons why the Drude dielectric function, when substituted in the Lifshitz formula, is inconsistent with electrodynamics are elucidated. The proposed approach is the single one consistent with all measurements of the Casimir force performed up to date. The application of this approach to metal-type semiconductors is considered.

1

PACS numbers: 05.30.-d, 77.22.Ch, 12.20.Ds

1. Introduction The Casimir force [1] acts between two parallel electrically neutral metal plates in vacuum. This e?ect is entirely quantum. There is no such force in classical electrodynamics. In accordance to quantum ?eld theory, there are zero-point oscillations of the electromagnetic ?eld in the vacuum state. The Casimir e?ect arises due to the boundary conditions imposed on the electromagnetic ?eld on metal surfaces. The spectra of zero-point oscillations in the presence and in the absence of plates are di?erent. Casimir was the ?rst who found the ?nite di?erence between respective in?nite vacuum energies. The negative derivative of this di?erence with respect to the separation between the plates is just what is referred to as the Casimir force. In his famous paper [1] Casimir considered ideal metal plates at zero temperature. Modern progress in measurements of the Casimir force (see early stages in review [2] and later experiments [3, 4, 5, 6, 7, 8, 9]) demand consideration of realistic plates of ?nite conductivity at nonzero temperature. This is also of much importance for the applications of the Casimir e?ect in nanotechnology [10, 11]. The basic theory of both the van der Waals and Casimir force taking into account the e?ects of ?nite conductivity and nonzero temperature was developed by Lifshitz [12]. It describes material properties

2 by means of the frequency dependent dielectric permittivity. First applications of this theory at nonzero temperature using the Drude [13] and plasma [14, 15] models for the dielectric permittivity have led, however, to contradictory results. In particular, thermal Casimir force if calculated using the Drude model was found to be in qualitative disagreement with the case of ideal metals. The large thermal correction arising at short separation distances within this approach was excluded experimentally [5, 6, 7]. In addition, the Casimir entropy calculated using the Drude model violates the third law of thermodynamics (the Nernst heat theorem) for perfect crystal lattices with no impurities [16, 17]. As to the nondissipative plasma model, it leads to the thermal Casimir force in qualitative agreement with the case of ideal metals and satis?es the Nernst theorem. It was found to be consistent with the data of relatively large separation experiments on the measurement of the Casimir force [5, 6, 7]. However, as is demonstrated below in Section 2, it is in contradiction with the results of short separation experiment [18]. This can be explained by the fact that the plasma model completely disregards interband transitions of core electrons. Another approach to the thermal Casimir force is based on the use of the Leontovich surface impedance instead of dielectric permittivity [19]. This approach is consistent with thermodynamics and large separation experiments, but it is not applicable at short separations. In this paper we present and further elaborate a new theoretical approach to the thermal Casimir force based on the use of generalized plasma-like dielectric permittivity [20]. We demonstrate that this approach is consistent with all available experimental results. Physical reasons why the Drude dielectric function is not compatible with the Lifshitz theory in the case of ?nite plates [21] are discussed. We demonstrate that results obtained in [21] have far reaching consequences not only for metals but also for semiconductors of metallic type with su?ciently high dopant concentration. In Section 2, on the basis of fundamentals of statistical physics and all available experimental data we explain why the new approach to the thermal Casimir force is much needed. Section 3 explains the e?ect of ?nite plates, i.e., that the Drude dielectric function is not applicable when the front of an incident wave has much larger extension than the size of the plate. Section 4 contains the formulation of the generalized plasmalike dielectric permittivity. The results of a recent 6-oscillator ?t to its parameters (see Ref. [7]) using the tabulated optical data for Au [22] are also presented. In Section 5 we compare the generalized Kramers-Kronig relations valid for the plasma-like permittivity with the standard ones valid for dielectrics and for Drude metals. Section 6 brie?y presents the thermodynamic test for the generalized plasma-like permittivity. In Section 7 we apply the developed approach to the case of semiconductors with relatively high concentration of charge carriers. Section 8 contains our conclusions and discussion. 2. Why a new approach is needed? We start with the Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the two parallel metallic plates of thickness d at a separation

3 distance a at temperature T in thermal equilibrium F (a, T ) = kB T 2π

∞ l=0

1 1 ? δ0l 2

∞ 0

k⊥ dk⊥ . (1)

2 2 (iξl , k⊥ )e?2aql × ln 1 ? rTM (iξl , k⊥ )e?2aql + ln 1 ? rTE

Here kB is the Boltzmann constant and ξl = 2πkB T l/h ? with l = 0, 1, 2, . . . are the Matsubara frequencies (k⊥ = |k⊥ | is the projection of a wave vector on the plane of the plates). The re?ection coe?cients for the two independent polarizations of the electromagnetic ?eld (transverse magnetic and transverse electric) are given by rTM (iξl , k⊥ ) =

2 2 ε2 l ql ? kl , 2 2 ε2 l ql + kl + 2ql kl εl coth(kl d) kl2 ? ql2 , rTE (iξl , k⊥ ) = 2 ql + kl2 + 2ql kl coth(kl d)

(2)

where ξl2 ξl2 2 (3) ql = + 2 , kl = k⊥ + εl 2 , εl = ε(iξl ) c c and ε(ω ) is the dielectric permittivity of the plate material. As was already discussed in the Introduction, in the framework of the Lifshitz theory the calculation results strongly depend on the model of a metal used. The source of discrepances are di?erent contributions from zero frequency [the term with l = 0 in Eq. (1)]. For ideal metal plates it holds |ε| = ∞ at any frequencies, including ξ0 = 0, and from (2) one obtains

2 k⊥

rTM (iξl , k⊥ ) = rTE (iξl , k⊥ ) = 1.

(4)

For metals with ε ? 1/ξ when ξ → 0 (this includes but not reduces to the Drude model) from (2) it follows [13, 23] rTM (0, k⊥ ) = 1, rTE (0, k⊥ ) = 0. (5)

2 For metals with ε ? ωp /ξ 2 when ξ → 0 , where ωp is the plasma frequency, (2) leads to a qualitatively di?erent result for the transverse electric re?ection coe?cient [14, 15],

rTM (0, k⊥ ) = 1, rTE (0, k⊥ ) =

2 ωp 2 + 2k 2 c2 + 2k c k 2 c2 + ω 2 coth ωp ⊥ p ⊥ ⊥ d c 2 2 2 k⊥ c + ωp

(6) .

As is seen from the comparison of (4) and (5), there is a qualitative disagreement in the values of rTE (0, k⊥ ). This results in hundreds times larger thermal corrections at short separation if one uses (5) instead of (4). At large separations the magnitudes of the Casimir free energy and pressure obtained by using (5) are one half of those when using (4). At the same time all results obtained from (4) and (6) are in qualitative agreement. This is guaranteed by the fact that (6) smoothly approaches (4) when ωp → ∞.

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1 0.8 0.6 0.4 0.2

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1 0.8 0.6 0.4 0.2

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Figure 1. The Casimir entropy for two plates made of an ideal metal (a) and metal described by the plasma model with ωp = 9.0 eV (b) at a separation 300 nm.

The crucial question for any model used is its consistency with the fundamental physical principles. In our case the considered models of dielectric permittivity can be tested thermodynamically by the behavior of the Casimir entropy, ? F (a, T ) , (7) ?T at low temperatures. In ?gure 1(a) we plot the Casimir entropy as a function of 2 temperature for ideal metals [24, 25] and in ?gure 1(b) for metals with ε ≈ ωp /ξ 2 when ξ → 0 [16, 17]. In both cases it holds S (a, T ) ≥ 0 and S (a, T ) → 0 when T vanishes. This means that the Nernst heat theorem is satis?ed. Quite di?erent situation holds for metals with ε ? 1/ξ when ξ → 0. The Casimir 2 entropy for ε = 1 + ωp /[ξ (ξ + γ )], where γ is the relaxation parameter, is plotted in ?gure 2(a) [16]. As is seen in this ?gure, the entropy becomes negative at T of about several hundred K and remains negative S (a, T ) = ? S (a, 0) = ? c c kB ζ (3) 1?4 + 12 2 16πa ωp a ωp a

2

?··· < 0

(8)

at T = 0 [here ζ (z ) is the Riemann zeta function]. Thus, the Nernst heat theorem is violated, suggesting that the model used is inapplicable. Figure 2(a) is plotted for perfect crystal lattice with no impurities when γ → 0 with T → 0. In [26, 27] it was argued that the presence of impurities can remedy this situation. However, in [27] the dependence of the relaxation parameter on the temperature was not taken into account. As a result, the coe?cients of the obtained asymptotic expressions were determined incorrectly up to factors of several orders of magnitude [28]. For typical realistic concentrations of impurities the behavior of the entropy as a function of temperature remains the same, as in ?gure 2(a), up to as low temperatures as 10?3 ? 10?4 K. At lower temperatures, however, the Casimir entropy depends on T in a di?erent way, as is shown in ?gure 2(b) for a typical residual resistivity equal to 10?4 of the resistivity at room temperature. Although from ?gure 2(b) it is seen that, at least formally, the Nernst heat theorem is preserved for lattices with impurities, this does not solve the contradiction between the models with ε ? 1/ξ and thermodynamics. The point is that, according to quantum

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0 - 0.5 -1 - 1.5 -2

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?? ??? ?

50

100 150 200 250 300

0.005

0.01

0.015

0.02

? ?

? ???

? ?

? ???

Figure 2. The Casimir entropy for two plates made of Drude metal with perfect crystal lattice (a) and of Drude metal with impurities (b) at a separation 1 ?m (γ = 0.035 eV).

statistical physics, the Nernst heat theorem must be valid for perfect crystal lattice which has a nondegenerate dynamic state of lowest energy. Thus, any model that violates this rule is thermodynamically inacceptable. Another crucial question for any model is consistency with experiment. As is shown in [5, 6, 7], both the plasma model and the impedance approach are consistent with the results of most precise measurements of the Casimir pressure at separations a ≥ 160 nm. The same measurement results are, however, inconsistent with the Drude model approach. In ?gure 3(a) we plot the di?erences between the theoretical Casimir pressures in the con?guration of two parallel plates calculated using the Drude model and tabulated optical data and mean experimental pressures as a function of separation [7]. It is seen that all di?erences are outside of the 95% con?dence intervals whose boundaries generate a solid line. Within a wide separation region from 210 to 620 nm they are also outside of the 99.9% con?dence intervals indicated by the dashed line. Thus, the Drude model approach is experimentally excluded. Note that in theoretical computations in ?gure 3(a) the tabulated optical data were extrapolated to low frequencies by the Drude model with the plasma frequency ωp = 8.9 eV and the relaxation parameter γ = 0.0357 eV. Importantly, variations of γ practically do not in?uence the magnitudes of the Casimir pressure. For example, a descrease of γ until 0.02 eV would lead to th | at a = 200 nm and to 0.26% increase at a = 650 nm. only 0.29% increase of |PD This is because the value of γ does not in?uence the zero-frequency term of the Lifshitz formula. As a result, the width of separation intervals, where the Drude model approach is excluded, practically does not depend on the value of γ . If the smaller values of ωp are used (as suggested in [29]), the Drude model approach is excluded at 99.9% con?dence level within even wider separation interval. The plasma model approach, although it agrees with the measurements of [5, 6, 7], also cannot be considered as a universal. In ?gure 3(b) we plot the di?erences of the theoretical Casimir forces between a plate and a sphere which are calculated using the plasma model and mean experimental Casimir forces [18] versus separation. It is

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60 40 20 0 - 20 - 40 - 60 200 300 400 500 600 700

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Figure 3. Di?erences between the theoretical and mean experimental Casimir pressures (a) and forces (b) versus separation. Theoretical quantities are computed using the Drude model and tabulated optical data (a) and plasma model (b). Solid lines and dashed line show the con?dence intervals with 95% and 99.9% con?dence levels, respectively. In computations the values of the plasma frequency ωp = 8.9 eV and the relaxation parameter γ = 0.0357 eV were used, as determined in [7] for Au ?lms deposited on the test bodies from the resistivity measurements.

seen that at separations below 80 nm the plasma model approach is excluded by the experimental data. Bearing in mind that at so short separations (below the plasma wavelength) the impedance approach is not applicable, it may be concluded that until recently there was no theoretical approach to the thermal Casimir force consistent with both long-separation and short-separation experiments. Such an approach based on the generalized plasma-like permittivity was ?rst proposed in [20, 21] and used in [7]. Below we discuss the main points of this approach and apply it to semiconductors of metallic type. 3. Drude model and the e?ect of ?nite plates Before considering the generalized plasma-like permittivity, we brie?y discuss the physical reasons why the Drude model dielectric permittivity,

2 ωp , ε D (ω ) = 1 ? ω (ω + iγ )

(9)

fails to provide an adequate description of the thermal Casimir force. The idea of this explanation belongs to Parsegian [30] who noticed that the Drude model is derived from the Maxwell equations in an in?nite metallic medium (semispace) with no external sources, zero induced charge density and with nonzero induced current j = σ0 E (σ0 is the conductivity at a constant current). In such a medium there are no walls limiting the ?ow of charges. Physically the condition that the semispace is in?nite means that its extension is much longer than the extension of the wave front (recently the role of ?nite size of the conductors was also discussed in [32] in the case of two wires interacting through the inductive coupling between Johnson currents).

7

? ? ? ? ?

· · · ·

Figure 4. The electromagnetic plane wave of a vanishing frequency with a wave vector k is incident on a metal plate of ?nite size leading to the accumulation of charges on its back sides.

For real metal plates, however, the applicability conditions of the Drude model are violated. The extension of the wavefront of a plane wave is much longer than of any conceivable metal plate. For plane waves of very low frequency, the electric ?eld E i inside the plate is practically constant and it is parallel to the boundary surface (see ?gure 4 where k and ki are the wave vectors outside and inside the plate, respectively). Constant electric ?eld E i creates a short-lived current of conduction electrons leading to the formation of practically constant charge densities ±ρ on the opposite sides of the plate (see ?gure 4). As a result, both the electric ?eld and the current inside the plate vanish. The ?eld outside the plate becomes equal to the superposition of the incident ?eld E and the ?eld E ρ produced by the charge densities ±ρ [31]. This process takes place in a very short time interval of about 10?18 s [21]. One can conclude that a ?nite metal plate exposed to a plane wave of very low frequency is characterized by zero current of conduction electrons and nonzero induced current density. Thus, it cannot be described by the Drude dielectric function (9). As to the plasma dielectric permittivity obtained from (9) by putting γ = 0, it leads to zero real current of conduction electrons and admits only a displacement current. Because of this, the plasma model does not allow the accumulation of charges on the sides of a ?nite plate in the electromagnetic wave of low frequency. Note also that for the plane waves of su?ciently high frequency there is no problem in the application of the Drude dielectric function. Thus, at T = 300 K the ?rst Matsubara frequency ξ1 = 2.47 × 1014 rad/s and for plane waves with ω ≥ ξ1 the electric ?eld E i changes its direction so quickly that the average charge densities on the plate sides are equal to zero, in accordance with the applicability conditions of the Drude model. 4. Generalized plasma-like permittivity The generalized plasma-like dielectric permittivity disregards relaxation of conduction electrons (as does the usual plasma model) but takes into account relaxation processes

8 of core electrons. It is given by

2 K ωp fj , ε (ω ) = 1 ? 2 + 2 2 ω j =1 ωj ? ω ? igj ω

(10)

where ωj = 0 are the resonant frequencies of core electrons, gj are their relaxation parameters, and fj are the oscillator strengths. The generalized plasma-like permittivity was applied to describe the thermal Casimir force in [20]. As the usual plasma model, permittivity (10) admits only a displacement current and does not allow the accumulation of charges on the sides of a ?nite plate. It also leads to the same values (6) of the re?ection coe?cients at zero frequency as the usual plasma model. The values of parameters fj , ωj and gj can be found by ?tting the imaginary part of ε(ω ) in (10) to the tabulated optical data for the complex index of refraction. For example, for Au the results of a 3-oscillator ?t (K = 3) can be found in [30]. Using the complete data in [22], the more exact 6-oscillator ?t (K = 6) for Au was performed in [7]. The resulting values of the oscillator parameters are presented in table 1.

Table 1. The oscillator parameters for Au found from the 6-oscillator ?t to the tabulated optical data.

j 1 2 3 4 5 6

ωj (eV) 3.05 4.15 5.4 8.5 13.5 21.5

gj (eV) 0.75 1.85 1.0 7.0 6.0 9.0

fj (eV2 ) 7.091 41.46 2.700 154.7 44.55 309.6

Equation (10) and table 1 were used together with the Lifshitz formula (1) to calculate the theoretical Casimir pressure in the con?guration of two parallel plates and Casimir force in the con?guration of a sphere above a plate. The results were compared with the measurement data of [6, 7] and of [18], respectively. The di?erences of the theoretical and mean experimental Casimir pressures versus separation are shown in ?gure 5(a) as dots. The di?erences of the theoretical and mean experimental Casimir forces are shown as a function of separation in ?gure 5(b). In both ?gures solid lines represent the borders of the 95% con?dence intervals. As is seen in ?gures 5(a) and 5(b), all dots are well inside the con?dence intervals. Thus, the generalized plasmalike dielectric permittivity (10) combined with the Lifshitz formula is consistent with the measurement data of both long- and short-separation measurements of the Casimir force performed up to date.

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Figure 5. Di?erences between the theoretical and experimental Casimir pressures (a) and forces (b) versus separation. Theoretical quantities are computed using the generalized plasma-like permittivity (10). Solid lines show the con?dence intervals with 95% con?dence.

5. Kramers-Kronig relations and their generalizations An important advantage of the plasma-like dielectric permittivity (10) is that it precisely satis?es the Kramers-Kronig relations. There is some confusion in the literature concerning the Kramers-Kronig relations in the case of usual plasma model which is characterized by entirely real permittivity. In fact the form of Kramers-Kronig relations is di?erent depending on the analytic properties of the considered dielectric permittivity. If ε(ω ) = ε′ (ω )+iε′′ (ω ) is regular at ω = 0, the Kramers-Kronig relations take its simplest form [31],

∞ ε′′ (ξ ) ∞ ε ′ (ξ ) 1 1 P dξ, ε′′ (ω ) = ? P dξ, (11) π ?∞ ξ ? ω π ?∞ ξ ? ω where the integrals are understood as a principal value. However, if the dielectric permittivity has a simple pole at ω = 0, ε(ω ) ≈ 4π iσ0 /ω , the form of the Kramers-Kronig relations is di?erent [33],

ε ′ (ω ) = 1 +

∞ ε′′ (ξ ) ∞ ε ′ (ξ ) 1 1 4πσ0 P dξ, ε′′ (ω ) = ? P dξ + . (12) π ?∞ ξ ? ω π ?∞ ξ ? ω ω In both cases of being regular and having a simple pole at ω = 0 dielectric permittivity the third dispersion relation, expressing the dielectric permittivity along the imaginary frequency axis, is common:

ε ′ (ω ) = 1 +

ε(iω ) = 1 +

1 P π

∞ ?∞

ξε′′(ξ ) dξ. ξ2 + ω2

(13)

The standard derivation procedure [31, 33], when applied to the dielectric permittivities having a second-order pole at zero frequency [i.e., with an asymptotic

10

2 behavior ε(ω ) ≈ ?ωp /ω 2 when ω → 0] leads to another form of Kramers-Kronig relations [20],

1 ε ′ (ω ) = 1 + P π

∞ ?∞

2 ωp ε′′ (ξ ) dξ ? 2 , ξ?ω ω

1 ε′′ (ω ) = ? P π

∞ ?∞

ε ′ (ξ ) + ξ?ω

2 ωp ξ2

dξ.

(14)

In this case the third Kramers-Kronig relation (13) also is replaced with [20]

2 ∞ ξε′′ (ξ ) ωp 1 P dξ + , (15) ε(iω ) = 1 + π ?∞ ξ 2 + ω 2 ω2 i.e., it acquires an additional term. It is easily seen that both the usual nondissipative plasma model [given by (9) with γ = 0] and the generalized plasma-like dielectric permittivity (10) satisfy the KramersKronig relations (14) and (15) precisely.

6. Thermodynamic test for the generalized plasma-like permittivity As was noted in the Introduction, thermodynamics provides an important test for the used model of dielectric properties. The Casimir entropy computed by using the Lifshitz formula (1) must vanish when the temperature vanishes, i.e., the Nernst heat theorem must be satis?ed. This test was used to demonstrate the incompatibility of the Drude model with the Lifshitz formula [16, 17]. The physical reasons for this incompatibility were discussed in Section 3. In [21] the thermodynamic test was applied to the plasmalike dielectric permittivity (10). The low temperature behavior of the Casimir entropy (7) was found analytically under the conditions λp h ?c , α≡ ? 1, 2akB 4πa where λp = 2πc/ωp is the plasma wavelength. The Casimir entropy is given by T ? Te? = 3ζ (3)kB S (a, T ) = 8πa2

?

(16)

T Te?

2

? ? ?

1 + 4α

?

K K 4π 3 T ? 1 + 8α + 6ζ (3)α3 Cj δj ? 96ζ (3)α4 Cj δj ? ? 135ζ (3) Te? j =1 j =1

40ζ (5) T ? 3ζ (3) Te? fj , 2 ωj

2

Here, the quantities Cj and δj are expressed in terms of the oscillator parameters Cj = δj = cgj . 2 2aωj (18)

α 2 ? 1 + 3α ?

?

?

K

Cj

j =1

?? ? + 2? ? 12α2 ? . ? ?

(17)

Note that in [21] the values of numerical coe?cients in the third lines of (38) and (41) are indicated incorrectly. Correct values are obtained by the replacement of all π 2 with 1/12. In the last term on the right-hand side of (33) in [21] 6/π 2 should be replaced with 1/(2π 4 ).

11 The Casimir entropy de?ned in (17) is nonnegative. It is seen that S (a, T ) → 0 when T → 0, (19)

i.e., the Nernst heat theorem is satis?ed. Thus, the plasma-like dielectric permittivity is not only consistent with all experiments performed up to date, but it also withstands the thermodynamic test. 7. Metal-type semiconductors The above results are of importance not only for metals but also for metal-type semiconductors. In more detail, the thermal Casimir force between dielectrics and semiconductors is considered in [34]. Here, we brie?y discuss only one point, i.e., how to account for the in?uence of free charge carriers in metal-type semiconductors where the density of these carriers is relatively high. It is common [22] to include the role of free charge carriers in semiconductors by considering the dielectric permittivity of the form

2 ωp . ε (ω ) = ε d (ω ) ? ω (ω + iγ )

(20)

Here, εd (ω ) is the permittivity of high resistivity (dielectric) semiconductor such that εd (0) < ∞. This approach was used for the interpretation of most precise recent experiment on the measurement of the Casimir force between metallic sphere and semiconductor plate by means of an atomic force microscope [9]. In the measurement set under consideration the density of charge carriers in a Si membrane was changed from 5 × 1014 cm?3 to 2.1 × 1019 cm?3 through the absorption of photons from laser pulses. In this di?erential experiment only the di?erence of the Casimir forces, ?F exp , in the presence and in the absence of laser pulse was measured. The experimental data on mean di?erence forces, ?F exp , was compared with the theoretical di?erence forces, ?F th , computed using the Lifshitz theory. In ?gure 6(a) dots labeled 1 show the quantity ?F th ? ?F exp versus separation, where ?F th is computed under the assumption that in the absence of laser pulse high resistivity Si is described by εd (ω ), i.e., the e?ect of dc conductivity is disregarded. Dots labeled 2 show the same quantity, where ?F th is computed taking into account the dc conductivity of high resistivity Si in the absence of laser pulse. In both cases the dielectric permittivity (20) with appropriate values of ωp and γ is used when the laser pulse is on. Solid lines indicate the borders of 95% con?dence intervals. As is seen in ?gure 6(a), the Lifshitz theory taking the dc conductivity of high resistivity Si into account is experimentally excluded. The physical explanation for this result can be found in [34]. It is notable also that, as was shown in [35], the inclusion of the dc conductivity of a dielectric in the Lifshitz theory results in the violation of the Nernst heat theorem. Bearing in mind that the Si plate has ?nite size, as was discussed in Section 3, a question arises whether the use of the Drude-type dielectric permittivity (20) in the presence of laser pulses for the calculation of the Casimir force is warranted. To

12 check this point we have recalculated the values of ?F th versus separation by using the dielectric permittivity εd (ω ) of high resistivity dielectric Si in the absence of laser pulse and the plasma-like permittivity

2 ωp (21) ω2 ? th ? ?F exp is shown in ?gure in the presence of pulse. The resulting quantity ?F 6(b) as dots labeled 1. As is seen from the comparison of dots labeled 1 in ?gures 6(a) and 6(b), the use of the generalized plasma-like permittivity (21) leads to a bit better agreement with data than the use of the Drude-type permittivity (20). However, it is not possible to give a statistically meaningful preference to one of the models on these grounds because in both cases most of the dots [of about 95% in ?gure 6(a) and 100% in ?gure 6(b)] are inside the con?dence intervals. Thus, although for metals (Au) we already have a decisive con?rmation of the fact that the generalized plasma-like permittivity is consistent with experiments on measuring the Casimir force and that the Drude model is excluded, for metal-type semiconductors such con?rmation is still lacking. It can be obtained in the proposed, more precise experiments on measuring the di?erence Casimir force between two sections of a patterned Si plate of di?erent dopant concentration [36].

ε ?(ω ) = εd (ω ) ?

8. Conclusions and discussion To conclude, we have demonstrated that the use of the Drude dielectric function in the Lifshitz formula is inconsistent with electrodynamics in the case of ?nite plates. Instead, to calculate the thermal Casimir force, one should use the generalized plasmalike permittivity that disregards relaxation of free electrons but takes into account

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????

Figure 6. Theoretical minus experimental di?erences of the Casimir forces versus separation. In the absence of laser pulse the theoretical results for dots labeled 1 are computed using εd (ω ) and for dots labeled 2 taking the dc conductivity of high resistivity Si into account. In both cases ε(ω ) from (20) is used when the laser pulse is on (a). The same di?erences labeled 1 calculated using εd (ω ) when the pulse is o? and ε ?(ω ) from (21) when the pulse is on are shown in (b). Solid lines show the con?dence interval with 95% con?dence.

13 relaxation due to interband transitions of core electrons. This permittivity is the only one that is consistent with both short- and long-separation measurements of the Casimir force at T = 300 K. The generalized plasma-like permittivity satis?es precisely the Kramers-Kronig relations. The use of this permittivity also leads to a positive Casimir entropy which vanishes at zero temperature in accordance with the Nernst heat theorem. We have also demonstrated that the inclusion of dc conductivity of high resistivity semiconductors is inconsistent with recent experiment on the measurement of the di?erence Casimir force between metal sphere and Si plate illuminated with laser pulses. We have presented two theoretical descriptions for the dielectric properties of low resistivity Si in the presence of laser pulse by means of the Drude-type and plasmatype permittivities and concluded that available experimental data is not of su?cient precision to discriminate between them. This problem will be solved in the future using results of the proposed experiment [36]. A more fundamental approach to the thermal Casimir force between metals and metal-type semiconductors would require the consideration of ?nite plates and a more sophisticated description of conduction electrons that is far beyond the scope of the Lifshitz theory. Acknowledgments The authors are grateful to G Bimonte, R S Decca, E Fischbach, G L Klimchitskaya, D E Krause, K A Milton and U Mohideen for helpful discussions. VMM is grateful to the Center of Theoretical Studies and Institute for Theoretical Physics, Leipzig University for kind hospitality. This work was supported by Deutsche Forschungsgemeinschaft, Grant No. 436 RUS 113/789/0–3. References

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[36]

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