michael levin, alexander p. mccauley, alejandro w. rodriguez, m. t. homer reid and steven g....

8/3/2019 Michael Levin, Alexander P. McCauley, Alejandro W. Rodriguez, M. T. Homer Reid and Steven G. Johnson- Casimir r

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Casimir repulsion between metallic objects in vacuum

Michael Levin,1 Alexander P. McCauley,2 Alejandro W. Rodriguez,2 M. T. Homer Reid,2 and Steven G. Johnson3

1Department of Physics, Harvard University, Cambridge MA 021382Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139

3Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

We give an example of a geometry in which two metallic objects in vacuum experience a repulsiveCasimir force. The geometry consists of an elongated metal particle centered above a metal platewith a hole. We prove that this geometry has a repulsive regime using a symmetry argument andconfirm it with numerical calculations for both perfect and realistic metals. The system does notsupport stable levitation, as the particle is unstable to displacements away from the symmetry axis.

Introduction: The Casimir force between two parallelmetal plates in vacuum is always attractive. A long-standing question is whether this is generally true formetallic/dielectric objects in vacuum, or whether the signof the force can be changed by geometry alone. Moreprecisely, can the force between non-interleaved metal-lic/dielectric bodies in vacuumthat is, bodies that lieon opposite sides of an imaginary separating planeever

be repulsive? In this paper, we answer this question inthe affirmative by showing that a small elongated metalparticle centered above a thin metal plate with a hole, de-picted in Fig. 1(a), is repelled from the plate in vacuumwhen the particle is close to the plate. The particle isunstable to displacements away from the symmetry axis,so that the system does not support stable levitation,consistent with the theorem of Ref. 1. We establish ourresult using a symmetry argument for an idealized caseand by brute-force numerical calculations for more re-alistic geometries and materials. We also show that thisgeometry is closely related to an unusual electrostatic sys-tem in which a neutral metallic object repels an electric

dipole (in fact, one can even obtain electrostatic repul-sion for the case of a point charge [2]). Anisotropic parti-cles are essential here; a spherical particle above a perfo-rated plate is always attracted, although non-monotoniceffects in an isotropic case have been suggested for thenull-energy condition rather than the Casimir energy [3].

Casimir repulsion is known to be impossible for1d/multilayer [4] or mirror-symmetric [5] metal-lic/dielectric geometries in vacuum. Interleaved zippergeometries can combine attractive interactions to yielda separating repulsive force [6], but the sign of theforce is ambiguous in such geometries. (In contrast, in

this paper the objects lie on opposite sides of a separat-ing z = 0+ plane and the interaction is unambiguouslyrepulsive.) Repulsive forces also arise for fluid-separatedgeometries [7] or magnetic [8, 9] or magnetoelectric mate-rials [10, 11]. A repulsive Casimir pressure was predictedwithin a hollow metallic sphere [12, 13], but this is contro-versial as it does not correspond to a rigid-body motion,is intrinsically cutoff-dependent [14], and the repulsiondisappears if the sphere is cut in half [5]. Another pro-posal is to use metamaterials formed of metals and di-

z

W

z0

rep

ulsiv

e attrac

tive

U(z

)

(

)

(a)

(b) (c)

FIG. 1: (color online) (a) Schematic geometry achievingCasimir repulsion: an elongated metal particle above a thinmetal plate with a hole. The idealized version is the limit of aninfinitesimal particle polarizable only in the z direction. (b)At z = 0, vacuum-dipole field lines are perpendicular to theplate by symmetry, and so dipole fluctuations are unaffectedby the plate (for any , shown here for = 0). (c) Schematicparticleplate interaction energy U(z)U(): zero at z = 0and at z , and attractive for z W, so there must beCasimir repulsion (negative slope) close to the plate.

electrics arranged into complex microstructures [10, 1517]. However, no specific metamaterial geometries thatexhibit Casimir repulsion have been proposed, and thetheoretical result [1] indicates that repulsion in the meta-material limit (separations microstructure) is impos-sible for parallel plate geometries.

Symmetry argument: We begin by establishing repul-sion in an idealized geometry: an infinitesimal particlecentered above an infinitesimally thin perfect-metal platewith a hole. We assume the particle is electrically polar-

izable only in the z direction and is not magneticallypolarizable at all (the limit of an infinitesimal metallicneedle) and the plate lies in the z = 0 plane [Fig. 1(a)].The Casimir(Polder) energy for such a particle at posi-tion x is given by [18]

U(x) = 1

2

0

zz(i)Ez(x)Ez(x)id. (1)

Here zz is the electric polarizability of the particle inthe z direction and EzEzi is the mean-square z com-

a

rXiv:1003.3487v2

[cond-mat.stat-mech]6Aug2010


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ponent of the electric-field fluctuations at position x andimaginary frequency = i. This expectation value isevaluated in a geometry without the particle (i.e. a geom-etry consisting of only the perforated plate in vacuum).Conventionally, it is renormalized by subtracting the (for-mally infinite) mean-square fluctuations in vacuum. Oneway to compute the expectation value is to note thatit is related to a classical electromagnetic Greens func-

tion via the fluctuation-dissipation theorem. More specif-ically, Ez(x)Ez(x

) is proportional to the electric fieldEz(x

)eit produced by an oscillating z-directed electricdipole p = pz ze

it at position x.

The key idea for establishing repulsion is simple: wefind a pointx such that the classical field of an oscillatingz-directed electric dipole atx is unaffected by the presenceof the metallic plate with a hole. It then follows thatU(x) = U(), implying that the energy U must varynon-monotonically between x and and hence must berepulsive at some intermediate points. While in most ge-ometries no such x exists, in the perforated plate geom-etry this condition is achieved by symmetry at x = 0. Ifa z-directed electric dipole is placed at z = 0 in the hole,then the electric-field lines of the dipole in vacuum arealready perpendicular to the plate by symmetry, as illus-trated in Fig. 1(b); thus, the vacuum dipole field solvesMaxwells equations with the correct boundary condi-tions in the presence of the plate, and U(z = 0) = U().Note that this is true by symmetry at every frequency (real or imaginary), because the dipole moment p atz = 0 is antisymmetric with respect to the z = 0 mir-ror plane. Intuitively, the basic point is that the electricdipole fluctuations of the particle do not couple to theplate at all when z = 0.

For large zthat is, z much larger than the hole diam-eter Wthe presence of the hole in the plate is negligible,and we must have the usual attractive CasimirPolder in-teraction. So, as schematically depicted in Fig. 1(c), weexpect the interaction energy U(z)U() to be zero atz = 0, decrease to negative values for small z > 0 (lead-ing to a repulsive force) then increase to zero for largez (leading to an attractive force). If the hole is circular,then by symmetry the force is purely in the z directionand the point of minimum U is an equilibrium position,stable under z perturbations; however, both the theoremof Ref. 1 and explicit calculations show that this equilib-rium point is unstable under lateral (xy) perturbations of

the particle position. In fact, numerical calculations (notshown) indicate that the particle is unstable to lateralperturbations and tilting at all separations z.

Electrostatics : Strictly speaking, this symmetry argu-ment only shows that the force must be repulsive at somez = 0: U could conceivably have multiple oscillations. Todefinitively rule out this possibility, we rely on the ex-plicit numerical calculations described below. However,on an intuitive level, the basic behavior of the force canbe understood from electrostatic considerations.

+

+++ + + + ++++ ++

+

z

r

(a) (b)

FIG. 2: (a) Schematic electrostatic interaction of a dipolewith a neutral perforated plate (side view), depicting thecharge density induced on the plate. Since is positivefor small r and negative for large r, can be constructed outof a superposition of dipoles in the z = 0 plane, oriented radi-ally inward about the z axis. A simple calculation then showsthat the interaction is repulsive for small z. (b) In contrast,a dipole oriented parallel to the plate is always attracted, asone can see from the induced charge density shown above.

To see this, let us focus on the = i = 0 contributionto the Casimir energy (1); we expect the contributionfrom nonzero imaginary frequencies to be qualitativelysimilar (though this expectation can sometimes be vio-lated, as in the inset of Fig. 4). The = 0 contributionis proportional to the electrostatic energy of a z-directed

electric dipole in the presence of a neutral metal plate.By the same arguments as above, such an electrostaticdipole must be repelled from the plate for some z > 0.To see this explicitly, suppose there is a static dipole atsome position (0, 0, z), and consider the induced chargeson the plate. In the limit where the plate is infinitesi-mally thin, we can combine the charges on the two sidesof the plate into a single surface charge density . On aqualitative level, we expect this total charge density tobe of the form shown in Fig. 2(a), with positive forsmall r and negative for large r. In particular, can beconstructed out of a superposition of dipoles in the z = 0plane, oriented radially inward about the z axis. A sim-

ple calculation shows that vertical force on a dipole at(0, 0, z) from a horizontal dipole at distance r from the zaxis is repulsive if r > 2z and attractive if r < 2z. Thus,if the hole is circular with diameter W and z < W/4,all the dipoles will exert a repulsive force and the totalforce is necessarily repulsive. On the other hand, whenz W, most of the dipoles will exert an attractive force,so the total force is attractive. In contrast, a dipole ori-ented parallel to the plate is always attracted, as one cansee from the induced charge density schematically shownin Fig. 2(b). This explains why an elongated shape isnecessary for the repulsive effect (see Fig. 4): dipole fluc-tuations parallel to the plate give rise to an attractive

Casimir force.We can confirm this picture by solving this electro-

statics problem exactly in the two-dimensional (2d) case,where the metal plate with a hole is replaced by a metalline with a gap of width W. Assuming a 2d Coulombforce F(r12) = q1q2/r12, and a z-directed dipole moment

pz, we find

Uelectrostatic(z) = p2z

2z2

(W2 + 4z2)2. (2)


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0 100 200 300 400 500 600-15

-10

-5

0

5

10

15

Center-center separation d(nm)

Casimir

ForceF

z(aN)

Perfect Metal, t= 0Gold, t= 20 nm

Perfect Metal, t= 20 nm

Perfect MetalSphere, t= 20 nm

t1m

320 nm

20 nm

Not to scale

d

Separating Plane for t= 20nm

FIG. 3: (color online) Exact Casimir force for cylinderplategeometry (inset) for perfect metals (computed with BEM)and gold (computed with FDTD); positive (shaded) force isrepulsive. For d 300 nm and d > 170 nm (vertical dashedline), the force is unambiguously repulsive as the cylinder isentirely above the plate. In contrast, a perfect-metal sphere(diameter 60 nm) is always attracted to the plate.

The force is indeed repulsive for small z, with a signchange occurring at z = W/2. A similar calculation fora y-directed dipole yields a uniformly attractive force.

The repulsion in the z-directed case is quite unusual,even in electrostatics: in almost all cases, the electro-static interaction between an electric dipole and a neu-tral metal object is attractive, not repulsive. Indeed, onan intuitive level, it seems almost inevitable that a dipole

will induce a dipole moment in the metal object orientedso that the force is attractive. More rigorously, one canprove that this interaction is attractive in several differentlimits. For example, if a dipole is very far away (z )or very close to the surface of a metal object, the inter-action is always attractive. One can also prove that theforce is attractive if the metal object is replaced by a di-electric material with a permittivity /0 = 1 + where0 < 1, using a perturbative expansion in . Clearly,a special geometry is necessary to obtain a repulsive forcein electrostatics, and arguably in Casimir interactions aswell by extension to = 0 along the imaginary frequencyaxis (see concluding remarks below).

Numerical demonstration: Moving beyond the ideal-ized geometry, we expect the repulsion to be robustunder small perturbations, such as finite particle size,plate thickness, and permittivity. This expectation isvalidated by the explicit calculations described below.We utilize two recent numerical methods, evaluating theCasimir force at zero temperature. First, we use a finite-difference time-domain (FDTD) technique that computesthe Casimir stress tensor via the Greens function [19, 20],with a free-software implementation [21]. Second, we use

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Displacement d / W

ay

az

d

= az/ay

(not to scale)

W t-1

-0.5

0

0.5

1

=2, t=0

Force(TE)

(unitsof10-7hc/W2) t=0

=4, t=0

=4, t=0.025 W

=4, t=0.1 W

0.5

0

Im (units ofc/W)

= 2

= 4

Force/d

10-7

h/W t= 0

0 1 2 3 4 5

0

0.25

0 1 2 3 4 5

= 4t= 0

t= 0.1 W

Im (c/W)

=1.25, t=0

Frequency integrand for d= 0.2 W

FIG. 4: (color online) Exact (BEM) 2d Casimir force forellipseline geometry (lower inset) with perfect metals and TEpolarization (in-plane electric field); positive force (shaded) isrepulsive. The effects of both particle width ay and line thick-ness t are shown, for fixed az = 0.002W. As the ellipticity = az/ay decreases or t increases, the repulsive force di-

minishes. Upper-left inset: frequency-resolved force F(Im )at fixed separation d = 0.2W and t = 0+: as decreases,attractive contributions arise from small Im . Upper-rightinset: F(Im ) at fixed d = 0.2W and = 4: as t increases,attractive contributions arise from large Im .

a boundary-element method (BEM) that can solve ei-ther for the stress tensor or directly for the Casimir en-ergy/force via a path-integral expression [22].

Figure 3 shows the Casimir force for a finite-sizecylindrical metal particle (20 320 nm) above a finite-thickness (t = 20 nm) plate with a circular hole of di-

ameter 1 m, considering both perfect metals and finite-permittivity gold, along with the force on a perfect-metal sphere (diameter 60 nm) for comparison. The per-fect metal results were computed with BEM; the oth-ers were computed by our FDTD technique in cylindri-cal coordinates, with the gold permittivity described by( = i) = 1 + 2p/

2, where p = 1.37 1016 rad/sec(the omission of the loss term, which is convenient forFDTD [19], does not significantly affect our results). Theforce on the sphere is always attractive, while the forceon the cylindrical particle is repulsive for a centercenterseparation d 300 nm. Because of the finite sizes, whend < 170 nm the tip of the particle intrudes into the

hole. However, there is a range of about 130 nm ford > 170 nm where the force is unambiguously repulsive:the two objects lie on opposite sides of an imaginary sep-arating plane. Similar behavior is seen for an infinites-imally thick (t = 0) plate (Figure 3). Somewhat sur-prisingly, the finite-permittivity gold exhibits a strongerrepulsive force than the perfect-metal case of the samegeometry; this is explained below as a consequence ofthe finite thickness of the plate.

In order to better understand the dependence on ge-


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ometry, we use BEM to explore the parameter space of asimplified 2d (yz) version of the problem: a metal ellipti-cal particle above a metal line with a gap of width W. Wecompute the Casimir force in this setup for perfect metalsand 2d electromagnetism, with the standard conventionthat the electric field is in the plane, as in a TE mode.(This system is equivalent to a scalar field with Neumannboundary conditions on the two objects). In Figure 4, we

explore how both the ellipticity = az/ay of the parti-cle and the line thickness t affect the force, for a fixedwidth W and particle length az = 0.002W. As 1,the elliptical particle becomes increasing circular, and re-pulsion diminishes due to the attractive force associatedwith dipole fluctuations in the y direction. We find thatthe repulsive force disappears for 1.25, when t = 0.Similarly, as t becomes larger, one can no longer makethe approximation that the metal line does not affect thefield of a z-directed dipole at d = 0, and the repulsiveeffect disappears by t 0.1W for = 4.

Further insight can be gained from the contributionof each imaginary frequency = i to the force for afixed particleline separation d = 0.2W (roughly maxi-mum repulsion). The upper-left inset to Fig. 4 shows thatas decreases, the attractive contributions first appearat small , eventually making the overall force attrac-tive. The right half of the inset shows that, in contrast,a nonzero t gives rise to attractive force contributions atlarge . This may explain the larger repulsive force ofreal gold compared to perfect metal in Fig. 3: the finiteskin depth of gold cuts off the large- contributions, re-ducing the attractive effects of the finite plate thickness(which, in this case, dominate the attractive effects of thefinite particle size and ellipticity).

Concluding remarks: In this paper, we have shownthat the sign of the Casimir force in vacuum can bechanged by geometry alone, without cheating by inter-leaving the bodies as in Ref. 6. Consistent with Ref. 1,the geometry described here does not support stable levi-tation since the particle is unstable with respect to lateral(xy) translation and tilting (as shown by additional 3dBEM calculations). As for the question of experimen-tal realizations, we leave this to future work, though wenote that one approach would be to anchor the particleto a substrate plate made of a low permittivity materialusing a pillar made of the same kind of material. Assum-ing a periodic array of such pillars and a complementary

array of holes with a unit cell of area 10(m)2

, and esti-mating the force per unit cell as the maximum repulsiveforce calculated in Fig. 3, one obtains a repulsive pres-sure of about 106 Pa. This is 2 or 3 orders of magnitudesmaller than typical experimental sensitivities [23], butthe repulsion could be increased further by shrinking oroptimizing the geometry.

This geometry was motivated by the electrostatic ana-logue shown in Fig. 2, where a qualitatively similar

effect is observed. Previously, another interesting non-monotonic Casimir effect was also seen to have an elec-trostatic analogue [24]. Mathematically, the mostly non-oscillatory exponential decay of the Casimir-force con-tributions for imaginary frequencies = i tends tomake the total force qualitatively similar to the 0+

contribution (in fact, this similarity becomes an exactproportionality in the unretarded, van der Waals limit).

This suggests that one approach for discovering interest-ing geometric Casimir effects is to first find an interest-ing electrostatic interaction, and then seek an analogousCasimir system.

This work was supported in part by the Harvard Soci-ety of Fellows, the US DOE grant DE-FG02-97ER25308,and the DARPA contract N66001-09-1-2070-DOD.

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michael levin, alexander p. mccauley, alejandro w. rodriguez, m. t. homer reid and steven g....

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