PHYSICAL REVIEW A 84, 032503 (2011) Finite-temperature Casimir force between perfectly metallic corrugated surfaces Jalal Sarabadani * Department of Physics, University of Isfahan, Isfahan 81746, Iran MirFaez Miri † Department of Physics, University of Tehran, P.O. Box 14395-547, Tehran, Iran and School of Physics, Institute for Research in Fundamental Sciences, (IPM) Tehran 19395-5531, Iran (Received 28 May 2011; published 9 September 2011) We study the Casimir force between two corrugated plates due to thermal fluctuations of a scalar field. For arbitrary corrugations and temperature T , we provide an analytical expression for the Casimir force, which is exact to second order in the corrugation amplitude. We study the specific case of two sinusoidally corrugated plates with corrugation wavelength λ, lateral displacement b, and mean separation H . We find that the lateral Casimir force is F l (T ,H ) sin(2πb/λ). In other words, at all temperatures, the lateral force is a sinusoidal function of the lateral shift. In the limit λ ≫ H, F l (T →∞,H ) ∝ k B TH −4 λ −1 . In the opposite limit λ ≪ H, F l (T → ∞,H ) ∝ k B TH −1/2 λ −9/2 e −2πH/λ . DOI: 10.1103/PhysRevA.84.032503 PACS number(s): 31.30.J−, 42.50.Lc, 03.70.+k I. INTRODUCTION In his seminal paper “On the attraction between two per- fectly conducting plates,” Casimir [1] predicted a force arising from the quantum fluctuations of the vacuum electromagnetic field. The Casimir force between two perfectly conducting flat plates at zero temperature is F (T = 0,H ) =− π 2 ¯ hcA 240H 4 , (1) where A is the area and H is the separation of the plates. Lifshitz [2] developed a more general theory of forces between two dielectric plates at a finite temperature T . He showed that the thermal fluctuations in an electromagnetic field give rise to a temperature-dependent force. As one of the direct macro- scopic manifestations of quantum theory, Casimir-Lifshitz interactions have gained much attention [3–7]. A variety of techniques, including the proximity force approximation (PFA) [8–10], perturbative expansion around ideal geometries [10–12], the world line method [13], and multiple scattering [14,15] have been developed to study the influence of geometry on the zero-temperature fluctuation- induced forces. The sphere and plate geometry is of great importance. The measurement of the Casimir-Lifshitz force encounters the difficulty of maintaining parallelism between two flat plates. Thus, most experiments measure the force between a sphere and a flat plate and compare their results with the predictions of PFA. However, PFA inherently assumes ad- ditivity of fluctuation-induced forces. Two corrugated surfaces, experiencing lateral Casimir force [16], is another geometry of interest. Direct measurements of the zero-temperature Casimir- Lifshitz force between various metallic and semiconducting materials are reported [17–19]. A recent experiment has shown that a long-range repulsive force can be observed between gold and silica particles that are separated by * j.sarabadani@phys.ui.ac.ir † miri@iasbs.ac.ir bromobenzene [20]. The measurements of the lateral Casimir force between corrugated surfaces are announced [21,22]. Recognizing electrostatic forces caused by potential patches on the plates’ surfaces [23–26], the precision of Casimir force measurements is improving. The magnitude of thermal Casimir-Lifshitz force is a subject of theoretical and experimental debate, cf. Refs. [4–6,27–35] and references therein. In Lifshitz theory, dielectric plates are characterized by a frequency-dependent permittivity. To describe the permittivity of the plates at low frequencies, the Drude model ǫ Drude (ω) = 1 − ω 2 p /(ω 2 + iγω) and the plasma model ǫ plasma (ω) = 1 − ω 2 p /ω 2 are widely used. Here, ω p is the plasma frequency, and γ is the dissipation rate. For large separations and high temperatures, the Drude model leads to a force, which is a factor of 2 smaller than the plasma model [27]. The use of the Drude model with γ → 0 as T → 0, leads to a finite zero-temperature entropy of the electromagnetic field [29]. References [30–33] elucidate this contradiction with the third law of thermodynamics. The experimental exclusions of the thermal effect predicted by the Drude model are announced [34]. A recent experiment of Sushkov et al. [35] agrees with the Casimir force calculated using the Drude model and demonstrates the existence of the T = 300 K thermal Casimir force in the separation range of 0.7–7 μm. We study the finite-temperature Casimir force between two corrugated surfaces. In a step-by-step approach to the reality, we consider perfectly metallic rather than dielectric boundaries. We go beyond the finite-temperature PFA [36]. For arbitrary temperature T and arbitrary height functions describing corrugated plates, we obtain analytical expressions for the normal and lateral Casimir forces. Our results are exact to second order in the corrugation amplitude. There is a good reason for our paper. The Casimir force influences the dynamics of miniaturized systems as Chan et al. demonstrated [37]. Recently, it has been proposed that normal and lateral Casimir forces may intermesh separate parts of submicrometer devices [38,39]. An example is a mechanical rectifier composed of one sinusoidally corrugated 032503-1 1050-2947/2011/84(3)/032503(9) ©2011 American Physical Society