Casimir-Lifshitz Interaction between Dielectric Heterostructures
aa r X i v : . [ qu a n t - ph ] J u l Casimir-Lifshitz Interaction between Dielectric Heterostructures
Arash Azari, Himadri S. Samanta, and Ramin Golestanian
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom (Dated: January 5, 2019)The interaction between arbitrary dielectric heterostructures is studied within the framework ofa recently developed dielectric contrast perturbation theory. It is shown that periodically patterneddielectric or metallic structures lead to oscillatory lateral Casimir-Lifshitz forces, as well as modu-lations in the normal force as they are displaced with respect to one another. The strength of theseoscillatory contributions increases with decreasing gap size and increasing contrast in the dielectricproperties of the materials used in the heterostructures.
PACS numbers: 05.40.-a, 81.07.-b, 03.70.+k, 77.22.-d
I. INTRODUCTION
In light of the ongoing miniaturization of mechanical devices and the recent developments in Casimir-Lifshitzinteractions [1, 2, 3, 4, 5], there has been some recent interest in the effect of these interactions between the componentsof small mechanical devices [6]. Since these interaction are particularly strong at small distances, it will be interestingto know how they can be utilized for designing novel mechanical systems that could work without physical contactand could potentially help solve the wear problem [7].In the past few years there have been a surge of interest in developing techniques that can be used to study theCasimir-Lifshitz interaction in non-ideal geometries, including geometry perturbation theories [8, 9, 10], semiclassical[11] and classical ray optics [12] approximations, multiple scattering and multipole expansions [13, 14, 15, 16, 17],world-line method [18] and exact numerical diagonalization methods [19, 20], as well as the numerical Green’s functioncalculation method [21]. These methods have been used in studying the Casimir force in a variety of differentgeometries, which have improved significantly our understanding of the nontrivial geometry dependence of this effect.The effect of non-ideal geometry has been shown to lead to a number interesting effects. For example, it hasbeen suggested that corrugated surfaces opposite one another can experience an oscillatory lateral Casimir force[8], which was subsequently observed experimentally [4]. A recent experiment probing the normal Casimir forcebetween a smooth surface and surface with tall rectangular corrugations also revealed further evidence on the non-additive nature of the Casimir force [5]. Here, we study the Casimir-Lifshitz interaction between arbitrary dielectricheterostructures within the framework of a recently developed formalism [23, 27]. We derive a closed form expressionfor the Casimir-Lifshitz energy between two dielectric heterostructures (such as the example depicted in Fig. 1) upto the second order in the perturbation theory and show that a coherent coupling between the different modes of thespectrum of the dielectric pattern takes place across the gap. As a special example, we consider unidirectional periodicheterostructures (see Fig. 1) and calculate the lateral and normal Casimir-Lifshitz force between them within thesame order in the perturbation theory. We find that coupling between modes with identical wavevectors of the patternstructures between the different objects can lead to modulations in the normal force and can give rise to oscillatorylater forces, reminiscent of the lateral Casimir force that appears due to coupling between geometrical features suchas corrugations [4, 8].This paper is organized as follows. Section II sketches the dielectric contrast perturbation theory, and Sec. IIIelaborates on how it can be used for dielectric heterostructures giving closed form expressions for the second orderterm in the perturbation theory. Section IV gives the results for the lateral and normal Casimir-Lifshitz force for anumber of choices of materials, and Sec. V contains some discussions and concluding remarks.
II. THEORETICAL FORMULATION
To calculate the Casimir-Lifshitz interaction we need to quantize the electromagnetic field in a background thatincludes the dielectric or metallic objects that modify the quantum fluctuations of the field. Describing a generalassortment of dielectric and metallic objects in space via a frequency dependent dielectric profile ǫ ( ω, r ), we can writea general expression for the Casimir-Lifshitz energy as [27] E = ~ Z ∞ dζ π tr ln [ K ij ( ζ ; r , r ′ )] , (1) l l f H a l z x ε (cid:1) ( ω ) ε (cid:2) ( ω ) FIG. 1: Schematic representation of two identical semi-infinite and periodic objects made of intercalated layers of high and low dielectric functions, occupying the fractions of f and 1 − f , respectively. Here H is the separation between them, a is adimensionless lateral displacement, and λ is the wavelength of the periodic structure. where K ij = (cid:20) ζ c ǫ ( iζ, r ) δ ij + ∂ i ∂ j − ∂ k ∂ k δ ij (cid:21) δ ( r − r ′ ) . (2)We can consider the dielectric function as ǫ ( iζ, r ) = 1 + δǫ ( iζ, r ), and expand Eq. (1) in powers of the dielectriccontrast. A similar approach has been the subject of a few recent studies [22, 23, 24, 25, 26].The expansion leads to the decomposition of K ij into a diagonal part K ,ij , corresponding to the empty space, anda perturbation part δ K ij , namely K ij ( ζ ; q , q ′ ) = K ,ij ( ζ, q )(2 π ) δ ( q + q ′ ) + δ K ij ( ζ ; q , q ′ ) , (3)where K ,ij ( ζ, q ) = ζ c δ ij + q δ ij − q i q j , (4)and δ K ij ( ζ ; q , q ′ ) = ζ c δ ij δ ˜ ǫ ( iζ, q + q ′ ) . (5)This yields an expansion tr ln[ K ] = tr ln[ K ] + ∞ X n =1 ( − n − n tr[( K − δ K ) n ] , (6)where K − ,ij ( ζ, q ) = ζ c δ ij + q i q jζ c h ζ c + q i . (7)The first term is the vacuum energy in the absence of the objects, and the terms in the series take account of theireffect in a perturbative scheme. The n -th order term in Eq. (6) takes on the explicit formtr[( K − δ K ) n ] = Z d q (1) (2 π ) · · · d q ( n ) (2 π ) [ ζ c δ i i + q (1) i q (1) i ] · · · [ ζ c δ i n i + q ( n ) i n q ( n ) i ][ ζ c + q (1)2 ] · · · [ ζ c + q ( n )2 ] δ ˜ ǫ ( iζ, − q (1) + q (2) ) · · · δ ˜ ǫ ( iζ, − q ( n ) + q (1) ) , (8)which involves the Fourier transform of the dielectric contrast profile. Going to real space, we can rewrite the energyof the system as [27] E = ~ Z ∞ dζ π ∞ X n =1 ( − n − n Z d r · · · d r n A i i ( r − r ) · · · A i n i ( r n − r ) (cid:20) δǫ ( iζ, r )1 + δǫ ( iζ, r ) (cid:21) · · · (cid:20) δǫ ( iζ, r n )1 + δǫ ( iζ, r n ) (cid:21) , (9)where A ij ( ζ, r ) = ζ c e − ζr/c πr (cid:20) δ ij (cid:18) cζr + c ζ r (cid:19) − r i r j r (cid:18) cζr + 3 c ζ r (cid:19)(cid:21) , (10)We now use this formulation to study the Casimir-Lifshitz interaction between structures with inhomogeneous orpatterned dielectric properties. III. DIELECTRIC HETEROSTRUCTURES
Let us now consider a configuration similar to the one depicted in Fig. 1, namely two dielectric heterostructuresthat are placed parallel to each other at a separation H . Using the definition r = ( x , z ), the dielectric profile can bewritten as ǫ ( iζ, r ) = ǫ u ( iζ, x ) , H ≤ z < + ∞ , , − H < z < H ,ǫ d ( iζ, x ) , −∞ < z ≤ − H , (11)using the labels u and d for the “up” and “down” bodies respectively.To keep the calculations tractable, we now focus on the second order term in the series expansion in Eq. (9). Forsuch two semi-infinite bodies, the second order interaction term between the bodies can be written as E = − ~ π c Z ∞ dζ ζ Z d x d x ′ Z d Q (2 π ) e i Q · ( x − x ′ ) E ( Q ) (cid:20) δǫ u ( iζ, x )1 + δǫ u ( iζ, x ) (cid:21) (cid:20) δǫ d ( iζ, x ′ )1 + δǫ d ( iζ, x ′ ) (cid:21) , (12)for any lateral dielectric function profile, where E ( Q ) = Z ∞ dp [2 p − p + 1][4 p + ( cQ/ζ ) ] / e − ζHc √ p +( cQ/ζ ) , (13)and δǫ u,d ( iζ, x ) = ǫ u,d ( iζ, x ) − high and low dielectric functions. We can use the periodic properties ofthe dielectrics and write them in Fourier series expansion. As Fig. 1 shows, we can define the dielectric profile of the d -object as ǫ d ( iζ, x ) = ǫ l ( iζ ) , − λ + sλ ≤ x ≤ − fλ + sλ,ǫ h ( iζ ) , − fλ + sλ < x < fλ + sλ,ǫ l ( iζ ) , fλ + sλ ≤ x ≤ λ + sλ, (14)where s is an integer number. We define the Fourier series as δǫ d ( iζ, x )1 + δǫ d ( iζ, x ) = ∞ X m = −∞ C m ( iζ ) e i πmx/λ , (15)where C m ( iζ ) = sin mπfmπ (cid:20) δǫ h ( iζ )1 + δǫ h ( iζ ) − δǫ l ( iζ )1 + δǫ l ( iζ ) (cid:21) , (16)for m = 0, and C ( iζ ) = f (cid:20) δǫ h ( iζ )1 + δǫ h ( iζ ) (cid:21) + (1 − f ) (cid:20) δǫ l ( iζ )1 + δǫ l ( iζ ) (cid:21) . (17)We can find the corresponding expansion for the u -object by changing x → x + aλ .Using the Fourier series expansion, one can find the Casimir-Lifshitz energy between two dielectric heterostructuresas depicted in Fig. 1 [up to second order in the Clausius-Mossotti expansion of Eq. (9)] as E pp = − ~ A π c ∞ X m =0 ′ Z ∞ dζ ζ E (cid:18) πmλ (cid:19) C m ( iζ ) cos(2 πma ) , (18)where the prime on the summation sign indicates that the m = 0 term is counted with half the weight, and the pp index means the energy calculated for the plate-plate geometry. This result shows that similar to the case of twocorrugated surfaces, two patterned dielectric heterostructures also couple to each other at the leading order whenthe two wavelengths of the modulations are equal [8]. Moreover, higher harmonics contribute to the Casimir-Lifshitzenergy with exponentially decaying contributions, such that at large separations only the fundamental mode (lowestharmonic) will survive [19]. IV. THE NORMAL AND LATERAL FORCES
We now use Eq. (18) to calculate the normal and lateral forces between different types of dielectric and metallicheterostructures. We look at three different types of materials as examples, namely, gold, silicon, and air/vacuum,and consider layered materials made of gold-silicon, silicon-air, and gold-air. We describe the dielectric function ofgold using a plasma model, namely, ǫ ( iζ ) = 1 + ω p ζ , where ω p is the plasma frequency, which is given as ω p (Au) =1 . × rad/s [28]. For silicon we use the Drude-Lorentz form ǫ ( iζ ) = 1 + ω p ζ + ω , where ω p (Si) = 3 . ω (Si) and ω (Si) = 6 . × rad/s [28]. Finally, for air/vacuum we use ǫ ( iζ ) = 1.Due to difficulties in keeping the surfaces of the objects parallel to each other, most experiments are performedin plate-sphere geometry. To perform the calculation of the forces for the plate-sphere configuration, we can use theDerjaguin Approximation [29], where we replace one of the semi-infinite objects with a planar surface with a spherewith radius R . The approximation is valid provided that the radius of sphere is much larger than the distance betweenthe dielectric heterostructures, namely, R ≫ H . Using this approximation we can find the normal force between asemi-infinite dielectric heterostructure and a sphere of the same material composition as [29] F norps = 2 πR (cid:18) E pp A (cid:19) . (19)Using this result, we can find the Casimir-Lifshitz energy for plate-sphere configuration as E ps = − Z ∞ H dH ′ F norps ( H ′ ) , (20)which we can now use to calculate the lateral Casimir force as F latps = − λ ∂E ps ∂a . (21)Substituting Eqs. (19) and (20) into Eq. (21), it reads F latps = 2 πRλ ∂∂a Z ∞ H dH ′ (cid:18) E pp ( H ′ ) A (cid:19) . (22)The above equations are the basis of the results that will be presented below.Figures 2a-c show the normal Casimir-Lifshitz force between two unidirectional (layered) dielectric heterostructuresas shown in Fig. 1 when laterally displaced with respect to one another by aλ . It corresponds to the symmetric casewith f = 0 .
5, and the corrugation wavelength of λ = 1 µ m. Three different compositions of gold-silicon, silicon-air, - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (a) - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (b) - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (c) - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (d) - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (e) - - 𝐹 𝑛𝑜𝑟𝑝𝑠 𝐹 𝑝𝑠 𝑎 (f) FIG. 2: Normal Casimir-Lifshitz force between layered dielectric heterostructures as shown in Fig. 1 in the plate-spheregeometry, corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with f = 0 .
5, and to (d) gold-silicon, (e) silicon-air,and (f) gold-air, with f = 0 .
2. The numerical value of corrugation wavelength used is λ = 1 µ m. Different curves correspondto different gap sizes of H = 100 nm, H = 300 nm, and H = 600 nm. The forces are normalized to F ps that corresponds tothe normal force when the laterally-averaged dielectric profile is used. and gold-air are considered each at three different gap sizes of H = 100 nm, H = 300 nm, and H = 600 nm. Thenormal forces are normalized using the normal force F ps that corresponds to the Casimir-Lifshitz force calculatedwithin the same scheme but with laterally averaged dielectric profile, which corresponds to the m = 0 term in theexpansion in Eq. (18). The normal force is found to oscillate as a function of the lateral displacement, having themaximum value when the regions of high dielectric constant from both sides are exactly opposite one another, andthe minimum value when in the staggered configuration where regions of higher dielectric constant face regions oflower dielectric constant. The amplitude of the oscillations increases by decreasing the gap size, and the effect isprogressively stronger when the contrast between the dielectric properties of the two regions is more pronounced,with a maximum relative change of 0.7 % for gold-silicon, 7 % for silicon-air, and 65 % for gold-air, at the closestseparation of H = 100 nm.In Figs. 2d-f the normal Casimir-Lifshitz forces between the same types of structures as above are presented, for theasymmetric case of f = 0 .
2. One can see two noticeable differences with the symmetric case. First, the oscillations arenow asymmetric, as enforced by the asymmetry of the dielectric profile, although the asymmetry weakens as the gapssize increases and eventually disappears—i.e. the oscillations become symmetric and harmonic—at sufficiently largeseparations. This is consistent with the picture that different harmonics of the dielectric contrast profile in Eq. (18)couple with each other via an exponential terms that decays with the corresponding wavelengths of each harmonicand as a result any asymmetry caused by higher harmonics will die out at large gap sizes. The second new featureis the significant enhancement of the amplitude of the oscillatory behavior as a function of the lateral displacement.While it is still the case that this amplitude increases with increasing contrast between the dielectric properties of thetwo materials used in the layered structure, the maximum relative change is 0.4 % for gold-silicon, 6 % for silicon-air,and 200 % for gold-air, at the closest separation of H = 100 nm.The lateral Casimir-Lifshitz forces for the same layered structures as above are shown in Figs. 3a-c for the symmetriccase with f = 0 .
5. In this case, we have assumed R = 180 µ m and λ = 1 µ m. Similar to the previous study, threedifferent compositions of gold-silicon, silicon-air, and gold-air are considered each at three different gap sizes of H = 100nm, H = 200 nm, and H = 400 nm. The lateral force is found to oscillate as a function of the lateral displacement,reminiscent of the lateral Casimir force that is induced by geometrical corrugations [4, 8]. The shape of the oscillatoryfunction approaches a sinusoidal behavior as the gap size increases, consistent with the fact that higher harmonics donot contribute to the force in that limit as also seen in geometrical lateral Casimir effect [19]. The amplitude of theoscillations increases by decreasing the gap size as well as the contrast between the dielectric properties of the tworegions. Numerically, we find an amplitude of 0.5 pN for gold-silicon, 8 pN for silicon-air, and 12 pN for gold-air, atthe closest separation of H = 100 nm.Figure 3d-f show the lateral Casimir-Lifshitz forces between the same types of structures as above, for the asymmetriccase of f = 0 .
2. Similarly, the profiles of the lateral force are noticeably asymmetric, with the asymmetry weakening - - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (a) - - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (b) - - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (c) - - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (d) - - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (e) - - - 𝐹 𝑙𝑎𝑡𝑝𝑠 (pN) 𝑎 (f ) FIG. 3: Lateral Casimir-Lifshitz force between layered dielectric heterostructures as shown in Fig. 1 in the plate-spheregeometry, corresponding to (a) gold-silicon, (b) silicon-air, and (c) gold-air, with f = 0 .
5, and to (d) gold-silicon, (e) silicon-air,and (f) gold-air, with f = 0 .
2. The numerical values used in these graphs are λ = 1 µ m and R = 180 µ m. Different curvescorrespond to different gap sizes of H = 100 nm, H = 200 nm, and H = 400 nm. as the gap size is increased and the shape of the profile approaches that of a sinusoidal function (single harmonic).We also see comparatively more significant enhancement of the amplitude of the oscillatory behavior as a function ofthe lateral displacement. The amplitude of the oscillations is found as 0.3 pN for gold-silicon, 5 pN for silicon-air,and 7 pN for gold-air, at the closest separation of H = 100 nm. V. DISCUSSION
In this paper, we have proposed a mechanism by which it is possible to create a lateral Casimir-Lifshitz force aswell as controlled modulations in the normal Casimir-Lifshitz force without geometrical corrugations. A couplingsimilar to what exists in the case of corrugated surfaces gives rise to these oscillatory forces, namely identical modesof the dielectric patterns couple across the gap to generate a macroscopic coherence in the fluctuations. The genericfeatures of these oscillatory forces are very similar to those of the forces caused by corrugations; the effect is strongerand involves more harmonics at closer separations, while it weakens and only involves the lowest mode of the patternin the dielectric contrast at larger separations.While the difference in the dielectric properties of the materials controls the general strength of the above results,comparison between Fig. 2 and 3 shows that the modulations in the normal force are more strongly affected by thecontrast in the dielectric properties. The choice of air/vacuum as one component also allows us to make predictionsabout geometrical features with large corrugation amplitudes, which provides an approximation scheme for the non-perturbative geometrical regime.In the present calculations we have only used the second order terms in the dielectric contrast perturbative series.Higher order terms shown in Eq. (8) will introduce coupling between different modes of the dielectric pattern in asystematic way, as imposed by the overall conservation of the sum of all wavevectors (momenta). While the presentis aimed at showing in terms of tractable calculations, one can in principle carry out the calculation of the Casimir-Lifshitz interaction in such dielectric heterostructures using numerical diagonalization methods [20].Controlled interactions between dielectric heterostructures with smooth outer surfaces could be very useful inpractical applications because it will help avoid the complications of bringing surfaces with geometrical protrusionsclose to each other while avoiding contact between them and controlling their separations. Moreover, it is much easierto pattern dielectric properties of materials in a controlled way than it is to shape them with the high precision thatis needed for Casimir effect type experiments.
Acknowledgments
The authors thank the ESF Research Network CASIMIR for providing excellent opportunities for discussion on theCasimir effect and related topics. This work was supported by EPSRC under Grant EP/E024076/1. [1] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. , 793 (1948).[2] E.M. Lifshitz, Sov. Phys. JETP , 73 (1956); I.E. Dzyaloshinskii, E.M. Lifshitz, and L.P. Pitaevskii, Adv. Phys. , 165(1961).[3] S.K. Lamoreaux, Phys. Rev. Lett. , 5 (1997); U. Mohideen and A. Roy, Phys. Rev. Lett. , 4549 (1998); A. Roy andU. Mohideen, Phys. Rev. Lett. , 4380 (1999); H.B. Chan et al , Science , 1941 (2001); G. Bressi et al , Phys. Rev.Lett. , 041804 (2002); R.S. Decca et al Phys. Rev. Lett. , 050402 (2003); D.E. Krause, Phys. Rev. Lett. , 050403(2007).[4] F. Chen, U. Mohideen, G.L. Klimchitskaya, and V.M. Mostepanenko, Phys. Rev. Lett. , 101801 (2002); Phys. Rev. A , 032113 (2002);[5] H.B. Chan et al , Phys. Rev. Lett. , 030401 (2008).[6] F.M. Serry et al , J. Microelectromech. Syst. , 193 (1995); E. Buks and M.L. Roukes, Phys. Rev. B , 033402 (2001);H.B. Chan et al , Phys. Rev. Lett. , 211801 (2001); G. Palasantzas and J.Th.M. De Hosson, Phys. Rev. B , 121409(R) (2005).[7] A. Ashourvan, M.F. Miri, and R. Golestanian, Phys. Rev. Lett. , 140801 (2007); Phys. Rev. E , 040103 (R) (2007);T. Emig, Phys. Rev. Lett. , 160801 (2007); M. Miri and R. Golestanian, Appl. Phys. Lett. , 113103 (2008); F.C.Lombardo et al. , J. Phys. A et al. , Phys. Rev. D , 3421 (1997); Phys. Rev. A , 1713 (1998); M. Kardar and R.Golestanian, Rev. Mod. Phys. , 1233 (1999).[9] T. Emig, A. Hanke, R. Golestanian, and M. Kardar, Phys. Rev. Lett. , 260402 (2001); Phys. Rev. A , 022114 (2003).[10] P.A. Maia Neto, A. Lambrecht, and S. Reynaud, Phys. Rev. A , 012115 (2005); R.B. Rodrigues, P.A. Maia Neto, A.Lambrecht, and S. Reynaud, Phys. Rev. Lett. , 100402 (2006).[11] M. Schaden and L. Spruch, Phys. Rev. A , 935 (1998).[12] R.L. Jaffe and A. Scardicchio, Phys. Rev. Lett. , 070402 (2004).[13] R. Balian and B. Duplantier, Ann. Phys. (New York) , 300 (1977); , 165 (1978).[14] O. Kenneth and I. Klich, Phys. Rev. Lett. , 160401 (2006)[15] R. Golestanian, Phys. Rev. E , 5242 (2000).[16] T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys. Rev. Lett. , 170403 (2007).[17] T. Emig, N. Graham, R.L. Jaffe, and M. Kardar, Phys. Rev. A , 054901 (2009).[18] H. Gies and K. Klingmuller, Phys. Rev. Lett. , 220401 (2006); Phys. Rev. Lett. , 220405 (2006).[19] T. Emig, Europhys. Lett. , 466 (2003); R. B¨uscher and T. Emig, Phys. Rev. Lett. , 133901 (2005).[20] A. Lambrecht and V.N. Marachevsky, Phys. Rev. Lett. , 160403 (2008).[21] A. Rodriguez, M. Ibanescu, D. Iannuzzi, F. Capasso, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. Lett. , 080401(2007); A. Rodriguez, M. Ibanescu, D. Iannuzzi, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. A , 032106 (2007);A. Rodriguez, J.D. Joannopoulos, and S.G. Johnson, Phys. Rev. A , 062107 (2008).[22] G. Barton, J. Phys. A , 4083 (2001).[23] R. Golestanian, Phys. Rev. Lett. , 230601 (2005).[24] S.Y. Buhmann and D.-G. Welsch, Appl. Phys. B , 2, 189 (2006).[25] G. Veble and R. Podgornik, Eur. Phys. J. E , 275279 (2007).[26] K.A. Milton, P. Parashar, and J. Wagner, Phys. Rev. Lett. , 160402 (2008).[27] R. Golestanian, Phys. Rev. A, in press (2009) [arXiv:0905.1046].[28] E. D. Palik, Handbook of Optical Constants of Solids , edited by E. D. Palik (Academic Press, New York, 1985).[29] J.N. Israelachvili,