Collaborators: Paulo A. Maia Neto Federal University of Rio de Janeiro, Brazil and

1. Vacuum-induced torque between corrugated metallic platesRobson B. Rodrigues Federal University of Rio de Janeiro, BrazilQFEXT07University of Leipizig, Germany

2. Collaborators: Paulo A. Maia Neto Federal University of Rio de Janeiro, Brazil and Astrid Lambrecht and Serge Reynaud Laboratoire Kastler Brossel, CNRS, ENS, UPMC, Paris, France Based on the paper: Europhys. Lett., 76 (5), pp. 822–828 (2006)

3. Motivations - Aplication in Mems/Nems • The Casimir effect plays a major role in micro- and nano-electromechanical systems (MEMS and NEMS). • The Casimir force could lead to stiction of Mems/Nems components that results in permanent adhesion of nearby surface elements… • ...but can also be used to actuate components of small devices without contact.

4. Motivations - Aplication in Mems/Nems Examples of Actuation of Mems by the Casimir Force: • Actuator powered by the normal Casimir force between a flat plate and a sphere : Chan, Aksyuk, Kleiman, Bishop, Capasso, Science 291, 1943 (2001). Figure from www.seas.harvard.edu/capasso/

5. Motivations - Aplication in Mems/Nems Examples of Actuation of Mems by the Casimir Force: • “Non-contact rack and pinion powered by the lateral Casimir force”: Ashourvan, Miri, Golestanian, Phys. Rev. Lett. 98, 140801 (2007) [1]. (talk by Golestanian). • “Casimir force driven ratchets”: T. Emig, Phys. Rev. Lett. 98, 160801 (2007) [2]. • Figures from [1] and [2]. New schemes for actuation: Lateral Casimir force between corrugated metallic plates : Chen, Mohideen, Klimchitskaya and Mostepanenko, Phys. Rev. Lett88, 101801 (2002) : F ~ 0.3 pN for L ~ 200 nm, Corrugation amplitudes ~ 8 nm, 59 nm F a L F Courtesy of S. Reynaud

6. Motivations - Aplication in Mems/Nems • When the corrugations are not aligned, the rotational symmetry is broken and a Casimir torque arise to minimize the system’s energy. Courtesy of S. Reynaud • Casimir torque : new mechanism of micro-mechanical control to be exploited in the design of MEMS and NEMS.

7. Motivations - Investigation of the non-trivial geometry dependence of the Casimir energy • The Proximity Force Approximation (PFA) connect plane-plane (PP) and curved geometries. Basic idea: replace the curved surface by a set of differential planes, and add contributions of each plane. Valid for corrugation wavelengths>>plate separation. • (Talk by Reynaud, Talk by Dalvit): Scattering approach for non-planar surfaces is a theoretical model beyond the PFA. Perturbative approach requires but allows arbitrary corrugation wavelengths with respect to plate separation. • Others approachs beyond PFA: worldline numerics,Gies, Klingmuller, Phys.Rev.Lett. (2006); talk by Bordag.

8. Other proposal: torque between birefringent plates • J N Munday, D Iannuzzi and F Capasso, New Journal of Physics 8 (2006) 244 [3]. • Y. Barash, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 12, 1637 (1978). Figure from [3] • The Casimir energy depend on the relative orientation of the optical axes of the materials. • This leads to a torque that tends to align two of the principal axes of the materials in order to minimize the system’s energy. • Maximum torque per unit area (L=100nm) :

9. Basic Formalism - Principal References Scattering approach (flat plates) C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A67, 043811 (2003).Applies for dissipative and/or magnetic media;Lifshitz (1956) in a particular case (Lifshitz formula) We follow the approach of : z Extension to rough mirrors (perturbative theory) Scattering approach with rough mirrors (Stochastic roughness correction)P. A. Maia Neto, A. Lambrecht and S. Reynaud, Phys. Rev. A72, 012115 (2005).Casimir energy obtained from non-specular reflection coefficients; Perturbation expansion in powers of , up to second order Example: mirror 1 Z=0 Z

10. Casimir energy between corrugated plates Lateral Casimir force beyond the PFA: R. B. Rodrigues, P. A. Maia Neto, A. Lambrecht and S. Reynaud Phys. Rev. Lett. 96, 100402 (2006); Phys. Rev. A75, 062108 (2007). (Talk by Reynaud) Casimir energy: Perturbation theory holds for : Second-order correction to the Casimir energy • is the Fourier transform of . • The response function is a function of the specular and non-specular reflection coefficients and does not depend on the direction of the corrugation wave vector K.

11. Casimir torque between corrugated plates Conventions: The corrugations have sinusoidal shapes: Corrugation wave vectors have the same modulus: : lateral displacements with respect to the configuration with a line of maximum height at the origin : anglebetween and (angular mismatch between the two corrugations)

12. Casimir torque between corrugated plates Corrugation is imprinted on a very large plate : Corrugation is restricted to a section of area centered at b is the relative lateral displacement along the direction of The energy does not depend on displacements perpendicular toand is invariant under and

13. Casimir torque between corrugated plates Limit of long corrugation lines: . Let us take( is negligible otherwise) and The scale of variation ofis set by

14. Casimir torque between corrugated plates • The Casimir energy is minimum at and • and and (shallow wells) • For , the plate is attracted back to without sliding laterally. • For , its motion will be a combination rotation and lateral displacement.

15. Casimir torque between corrugated plates Torque = (optimum) It is maximum at , where it is given by: • We take , , . • Dotted line : . At L = 100nm, • (3 orders of magnitude larger than the torque/area for anisotropic plates). • Dashed line: optimum torque; occurs at (maximizing ). • Solid line: optimum torque for • (in this case . ).

16. Casimir torque between corrugated plates Under optimum conditions, the torque probes a non-trivial geometry dependence of the Casimir energy • PFA holds for smooth surfaces: or . • Response function satisfies PFA Torque (PFA): Perfect reflectors Scaterring approach • Solid line: Torque as a function of k for L = 1μm • Dashed line: model with perfect reflectors. Overestimates the torque by 16% near the • peak region. • Dotted line: PFA. Overestimates the torque by 103% at the peak value k = 2.6/L

17. Conclusions • Casimir torque between corrugated metallic plates: may provide a new mechanism of micro-mechanical control to be exploited in the design of MEMS and NEMS. • The torque is up to three orders of magnitude larger than the torque between anisotropic dielectric plates for comparable distance and area. • An experimental observation of the Casimir torque with seems feasible. • The PFA grossly overestimates the optimum torque by a factor of the order of 2.