Transcript Vacuum-induced torque between corrugated metallic plates

Vacuum-induced torque
between corrugated metallic
plates
Robson B. Rodrigues
Federal University of Rio de Janeiro, Brazil
QFEXT07
University of Leipizig, Germany
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)
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.
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/
Motivations - Aplication in Mems/Nems
Examples of Actuation of Mems by the Casimir Force:
New schemes for actuation: Lateral Casimir force between corrugated
metallic plates : Chen, Mohideen, Klimchitskaya and Mostepanenko,
Phys. Rev. Lett 88, 101801 (2002) :
F ~ 0.3 pN for L ~ 200 nm, Corrugation amplitudes ~ 8 nm, 59 nm
F
a
L
F
Courtesy of S.
Reynaud
• “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].
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.

L
Courtesy of S. Reynaud
• Casimir torque : new mechanism of micromechanical control to be exploited in the
design of MEMS and NEMS.
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 nonplanar 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.
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) :
Basic Formalism - Principal References
We follow the approach of :
Scattering approach
(flat plates)
C. Genet, A. Lambrecht and S. Reynaud,
Phys. Rev. A67, 043811 (2003).
Applies for dissipative and/or magnetic media; Lifshitz
z
(1956) in a particular case (Lifshitz formula)
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).
Example:
mirror 1
Casimir energy obtained from non-specular reflection
coefficients; Perturbation expansion in powers of
, up
to second order
Z=0
Z
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.
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
: angle
between
and
(angular mismatch
between the two corrugations)
Casimir torque between corrugated plates
Corrugation
is imprinted on a very large plate :
Corrugation
centered at
is restricted to a section of area
b is the relative lateral displacement along the direction of
The energy does not depend on displacements perpendicular
to
and is invariant under
and
Casimir torque between corrugated plates
Limit of long corrugation lines:
Let us take
(
The scale of variation of
.
is negligible otherwise) and
is set by
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.
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
•Solid line:
(in this case
(maximizing
optimum torque for
.
).
).
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:
•Response function satisfies
or
.
PFA
Torque (PFA):
Perfect
reflectors
•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
Scaterring
approach
Conclusions
• Casimir torque between corrugated metallic
plates: may provide a new mechanism of micromechanical 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.