Analytical results on Casimir forces
for conductors with edges and tips
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Maghrebi, M. F. et al. “Analytical results on Casimir forces for
conductors with edges and tips.” Proceedings of the National
Academy of Sciences 108 (2011): 6867-6871. Web. 9 Nov. 2011. ©
2011 National Academy of Sciences
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http://dx.doi.org/10.1073/pnas.1018079108
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National Academy of Sciences
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http://hdl.handle.net/1721.1/66993
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Analytical results on Casimir forces for
conductors with edges and tips
Mohammad F. Maghrebia,1, Sahand Jamal Rahia,b, Thorsten Emigc, Noah Grahamd, Robert L. Jaffea, and Mehran Kardara
aDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139;bCenter for Studies in Physics and Biology, The Rockefeller
University, 1230 York Street, New York, NY 10065;cLaboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, 91405 Orsay, France;
anddDepartment of Physics, Middlebury College, Middlebury, VT 05753
Edited* by John D. Joannopoulos, Massachusetts Institute of Technology, Cambridge, MA, and approved March 5, 2011 (received for review December 3, 2010)
Casimir forces between conductors at the submicron scale are para-mount to the design and operation of microelectromechanical devices. However, these forces depend nontrivially on geometry, and existing analytical formulae and approximations cannot deal with realistic micromachinery components with sharp edges and tips. Here, we employ a novel approach to electromagnetic scatter-ing, appropriate to perfect conductors with sharp edges and tips, specifically wedges and cones. The Casimir interaction of these objects with a metal plate (and among themselves) is then com-puted systematically by a multiple-scattering series. For the wedge, we obtain analytical expressions for the interaction with a plate, as functions of opening angle and tilt, which should provide a particularly useful tool for the design of microelectromechanical devices. Our result for the Casimir interactions between conducting cones and plates applies directly to the force on the tip of a scan-ning tunneling probe. We find an unexpectedly large temperature dependence of the force in the cone tip which is of immediate relevance to experiments.
fluctuations∣ quantum electrodynamics
T
he inherent appeal of the Casimir force as a macroscopic manifestation of quantum “zero-point” fluctuations has in-spired many studies over the decades that followed its discovery (1). Casimir’s original result (2) for the force between perfectly reflecting mirrors separated by vacuum was quickly extended to include slabs of material with specified (frequency-dependent) dielectric response (3). Quantitative experimental confirmation, however, had to await the advent of high-precision scanning probes in the 1990s (4–7). Recent studies have aimed to reduce or reverse the attractive Casimir force for practical applications in micron-sized mechanical machines, where Casimir forces may cause components to stick and the machine to fail. In the presence of an intervening fluid, experiments have indeed observed repulsion due to quantum (8) or critical thermal (9) fluctuations. Metamaterials, fabricated designs of microcircuitry, have also been proposed as candidates for Casimir repulsion across vacuum (10).Although there have been many studies of the role of materials (dielectrics, conductors, etc.), the treatment of shapes and geo-metry has remained comparatively less investigated. Interactions between nonplanar shapes are typically calculated via the proxi-mity force approximation (PFA), which sums over infinitesimal segments treated as locally parallel plates (11). This approxima-tion represents a serious limitaapproxima-tion because the majority of experiments measure the force between a sphere and a plate with precision that is now sufficient to probe deviations from PFA in this and other geometries (12, 13). Practical applications are likely to explore geometries further removed from parallel plates. Several numerical schemes (14,–16), and even an analog computer (17), have recently been developed for computing Casimir forces in general geometries. However, analytical formu-lae for quick and reliable estimates remain highly desirable.
The formalism recently implemented in refs. 18 and 19 enables systematic computations of electromagnetic (EM) Casimir forces
in terms of a multipole expansion. Using these methods, we have been able to compute forces between various combinations of planes, spheres, and circular and parabolic cylinders (19–24) (see also refs. 25–27). Both perfect conductors and dielectrics have been studied. However, with the notable exception of the knife edge (23–28), which is a limit of the parabolic cylinder geometry, systems with sharp edges have not yet been studied analytically.†
In all the above cases, the object corresponds to a surface of constant radial coordinate. In this paper, we present results on the quantum and thermal EM Casimir forces between generically sharp shapes, such as a wedge and a cone, and a conducting plate. We accomplish this task by considering surfaces of constant an-gular coordinate. Although the conceptual step—radial to angu-lar—is simple, the practical computation of scattering properties is nontrivial, necessitating complex mathematical steps. Further-more, the inclusion of wedges and cones practically exhausts shapes for which the EM scattering amplitude can be treated analytically.‡
In what follows, we provide a summary of the approach and highlights of our results. The first section introduces the scatter-ing formalism, whose main scatter-ingredients are scatterscatter-ing amplitudes from individual objects. A multiple-scattering series then enables systematic exploration of Casimir interactions between any ar-rangements of such shapes. Edges are explored in the section devoted to wedge geometries, where for perfect conductors the EM field can be parameterized in terms of two scalar fields. The corresponding scattering amplitudes in this case were previously known; their application to compute Casimir forces is developed here. In particular, we obtain simple analytical expressions for the force between a knife edge and a plate from the first two terms of the multiple-scattering series, and verify (by numerical computa-tion of higher order terms) that these expressions provide highly accurate estimates of the force. The limitations of common approximation schemes can be illustrated by simple examples in this geometry. Analogous computations relevant to a sharp Author contributions: M.F.M., S.J.R., T.E., N.G., R.L.J., and M.K. performed research and wrote the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
†Knife-edge geometries have been studied for scalar fields obeying Dirichlet boundary conditions in refs. 29 and 30. Wedges and related shapes have been studied in isolation (31), but computing interactions among such objects requires full scattering and conversion matrices, which are synthesized in this paper.
‡Morse and Feshbach (32) enumerate six coordinate systems in which the vector Helmholtz equation for EM waves is generically solvable in this way: planar, cylindrical (comprising circular, elliptic, and parabolic), spherical, and conical. Surfaces on which one such coordinate is constant are candidates for exact analysis. In addition to the plane and sphere discussed above, the circular (33) (bottom right of Fig. 1) and parabolic (23) cylinder have already been studied. These shapes are generically smooth, with the extreme limit of the parabolic cylinder (a“knife edge”) a notable exception. A survey of the remaining coordinate systems (32) only leads to shapes such as cylinders and cones with elliptic cross-section, which are generically similar to their circular counterparts. 1To whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/ doi:10.1073/pnas.1018079108/-/DCSupplemental. P H Y S ICS
tip are carried out in the section on the cone. Scattering ampli-tudes for the cone are obtained via a representation of the EM Green’s function and sketched briefly in Methods. The expres-sions for the interaction of a cone and a plate (both perfectly reflecting) are rather complex in general, but reduce to simple forms in the limit of small opening angle (approaching a needle). For the interaction between parallel metal plates, thermal correc-tions at room temperature are known to be practically negligible (at micron separations). Although these corrections are small for the wedge, we find that the room temperature result for the cone is considerably larger than at zero temperature (by roughly a fac-tor of 2 at micron separations). This difference should prove quite important to experiments and designs involving sharp tips. In particular, the prospects for experimental detection of the force on the tip of an atomic force microscope are discussed in Conclusion and Outlook. Detailed derivations, supporting for-mulae, and graphs for each of these sections are provided in theSI Appendix.
Casimir Forces via the Scattering Formalism
The conceptual foundations of the scattering approach can be traced back to earlier multiple-scattering formalisms (7, 34–36), but these were not sufficiently efficient to enable more compli-cated calculations. The ingredients in our method are depicted in Fig. 1. The Casimir energy associated with a specific geometry depends on the way that the objects constrain the EM waves that can bounce back and forth between them. The dependence on the properties of the objects is completely encoded in the scatter-ing amplitude or T matrix for EM waves. The T matrices are indexed in a coordinate basis suitable to each object. Fig. 1 sum-marizes the T matrices for perfectly reflecting planar, wedge, and conical geometries. Scattering from a plane mirror gives
Tplate¼ 1 for the two polarizations, irrespective of the
wavevec-tor ~k. Cylindricalðm;kzÞ and spherical ðℓ;mÞ quantum numbers, along with polarization, label appropriate bases for the cylinder and sphere. As detailed in the following sections on the wedge and cone, imaginary angular momenta, labeled byμ for the wedge andλ for the cone, enumerate the possible scattering waves. The cone’s scattering waves are also labeled by the real integer m, corresponding to the z component of angular momentum. The corresponding T matrices depend on the opening angleθ0. The T matrix for the wedge (top right in Fig. 1) is independent of the axial wavevector kz. In addition to the usual polarizations, for the cone we needed to introduce an extra “ghost” field (labeled Gh) and the corresponding T matrix (top left in Fig. 1). We will also need the matrixU that captures the appropriate translations and rotations between the scattering bases for each object. This matrix encodes the objects’ relative positions and orientations. (The U matrices needed for our calculations are
presented inSI Appendix.) The expression for the Casimir inter-action energy, E ¼ℏc 2π Z ∞ 0 dκ tr ln½1 − N ¼ − ℏc 2π Z ∞ 0 dκ trN þ1 2trN2þ ⋯ ; [1] involves integration over the imaginary wave numberκ, an impli-cit argument of the above matrices. For an object in front of an infinite plate,N ¼ UTobjectU†Tplate. An expansion of tr ln½1 − N in powers ofN corresponds to multiple scatterings of quantum fluctuations of the EM field between the two objects; the trace operation sums over the full spectrum of scattering channels (plane waves for example). The presence of both the speed of light c and Planck’s constant ℏ indicate that this energy is a result of relativistic quantum electrodynamics. This procedure can be generalized to multiple objects, with the material properties and shape of each body encoded in its T matrix. Analytical results are restricted to objects for which EM scattering can be solved exactly in a multipole expansion, a familiar problem of mathema-tical physics with classic applications to radar and optics. Edges via the Wedge
For a perfectly reflecting wedge, translation symmetry makes it possible to decompose the EM field into two scalar components: an E-polarization field that vanishes on its surface (Dirichlet boundary condition) and an M-polarization field that has vanish-ing normal derivative (Neumann). In the cylindrical coordinate system ðr;ϕ;zÞ, a wedge has surfaces of constant ϕ ¼ θ0. Whereas for describing scattering from cylinders, a natural basis is eimϕHð1Þmði
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ2þ k2 z p
rÞeikzz with Bessel-Hð1Þ functions indexed by m¼ 0;1;2;…, for a wedge we must choose eμϕHð1Þiμði ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiκ2þ k2z
p
rÞeikzz with realμ ≥ 0, corresponding to ima-ginary angular momenta that are no longer quantized (see SI Appendix, Section I). The T matrices, diagonal in μ, take the simple forms indicated in Fig. 1. Dimensional analysis indicates that the interaction energy of a wedge of edge length L at a separation d from a plate is E ¼ −ðℏcL∕d2Þfðθ0;ϕ0Þ, where fðθ0;ϕ0Þ is a dimensionless function of the opening angle θ0 and inclination ϕ0 to the plate. This geometry, and the corre-sponding function fðθ0;ϕ0Þ, are plotted in Fig. 2, Middle.
The limit θ0→ 0 corresponds to a knife edge which was previously studied as a limiting form of a parabolic cylinder (23, 28). The matrix N for the wedge simplifies in this limit, enabling exact calculation of the first few terms in the expansion of tr ln½1 − N in Eq. 1,
Fig. 1. Ingredients in the scattering theory approach to EM Casimir forces. See the text for further discussion.
− E ℏcL∕d2¼ secϕ0 16π2 þ 1 192π3þ csc3ϕ0secϕ0ð2ϕ0− sin 2ϕ0Þ 256π3 þ ⋯: [2] The first square brackets corresponds to trN (depicted by an or-ange line in Fig. 2) and the second to trN2∕2; their sum (depicted by a red line) is in remarkable agreement with the full result (blue surface). Asϕ0→ π∕2, the knife edge becomes parallel to the plate and the interaction energy diverges as it becomes propor-tional to the area rather than L. For parallel plates, we know that the terms in the multiple-scattering series tr½Nn∕n are propor-tional to 1∕n4 (1). Numerically, we find that the convergence is more rapid for ϕ0<π∕2, and that the first two terms in Eq.2 are accurate to within 1%. Including more than three terms in the series will not modify the curve at the level of accuracy for this figure. Casimir’s calculation for parallel plates gives an exact result atϕ0¼ π∕2, marked with an × in Fig. 2, Middle.
The panels in Fig. 2 display some of the more interesting aspects of the wedge-plate geometry. In the middle panel, the energy, rescaled by an overall factor ofℏcL∕d2and multiplied by cosðθ0þ ϕ0Þ, is plotted versus θ0andϕ0. The factor of cosðθ0þ ϕ0Þ is introduced to remove the divergence as one face of the wedge becomes parallel to the plate and the energy becomes proportional to the area rather than just the length of the wedge.
The blue surface is obtained by numerical evaluation of the first three terms in the multiple-scattering series of Eq. 1, with N constructed in terms of the plate and wedge T matrices given in Fig. 1. The front curve (θ0¼ 0) corresponds to a knife edge as previously mentioned. The top panel depicts the“butterfly” configuration where the wedge is aligned symmetrically with respect to the normal to the plate. As the wings open up to a full plate atθ0¼ π∕2, the energy again approaches the classic parallel plate result, also marked by an ×. Finally, and perhaps most interestingly from a qualitative point of view, the bottom panel depicts the case where one wing is fixed atπ∕4, and the other opens up by ψ ¼ 2θ0. This case displays the sensitivity of the Casimir energy to the back side of the wedge, which is“hidden” from the plate. In the proximity force approximation, the energy is independent of the orientation of the back side of the wedge, and thus the PFA result (solid line) is constant until the back surface becomes visible to the plate. The correct result (dotted line) varies continuously with the opening angle and differs from the proximity force estimate by nearly a factor of 2, showing that the effects responsible for the Casimir energy are more subtle than can be captured by the PFA.
Tips via the Cone
Computations for a cone—the surface of constant θ in spherical coordinates ðr;θ;ϕÞ—require a similar passage from spherical waves labeled byðℓ;mÞ to complex angular momentum, ℓ → iλ− 1∕2. In this case λ ≥ 0 is real, whereas m remains quantized to integer values. However, unlike the wedge case, the EM field can no longer be separated into two scalar parts; the more com-plicated representation that we report in the SI Appendix, Section IIinvolves an additional field, similar to the ghost fields that appear elsewhere in quantum field theory. This representa-tion of EM scattering can also be of use for describing reflecrepresenta-tion of ordinary EM waves from cones. Dimensional analysis indicates that, for a cone poised vertically at a distance d from a plate, the interaction energy scales as (ℏc∕d) times a function of its opening angleθ0. This arrangement and the resulting interaction energy are depicted in Fig. 3, Left, with the energy scaled by cos2θ0to remove the divergence as the cone opens up to a plate for θ0→ π∕2. The PFA (11) (depicted by the dashed line) becomes exact in this limit, but it is progressively worse as θ0decreases fromπ∕2. In particular, this approximation predicts that the en-ergy vanishes linearly asθ0→ 0, although in fact it vanishes as
E ∼ −ℏc d ln4 − 1 16π 1 j lnθ0 2j ; [3]
where the logarithmic divergence is characteristic of the remnant line in this limit (33). The EM results shown as the bold blue curve in Fig. 3 are obtained by including two terms in the series of Eq.1, with N constructed from the plate and cone T matrices in Fig. 1, including the additional ghost field. We have also in-cluded the corresponding curves for scalar fields subject to Dirichlet and Neumann boundary conditions which are depicted as fine red curves where the top (bottom) one corresponds to the Dirichlet (Neumann) boundary condition. The limit ofθ0→ 0 is shown in the left panel of Fig. 3 as an orange dashed line. As the sharp tip is tilted by an angle β, the prefactor ðln 4 − 1Þ∕16π is replaced by gðβÞ∕ cos β, where gðβÞ can be computed from inte-grals of trigonometric functions (seeSI Appendix, Section II). We plot this quantity in the right panel of Fig. 3.
Thermal Corrections
For practical applications, the above results have to be corrected for finite temperature. These are easily incorporated by replacing the integral in Eq.1 with a sum over Matsubara frequencies κn¼ ð2πkBT∕ℏcÞn (37). For the knife edge, the first (single-scattering) term in the force resulting from Eq.2 is modified to
Fig. 2. The Casimir interaction energy of a wedge at a distance d above a plane, as a function of its semiopening angleθ0and tiltϕ0. The rescaled
energy as a function ofθ0andϕ0is shown in the middle panel. The symmetric
case,ϕ0¼ 0, is displayed in the top panel and the interesting case where
the back side of the wedge is hidden from the plane is shown in the bottom panel. See the text for further discussion.
P
H
Y
S
F knife edge EM¼ − ℏcL 8π2cos1ϕ0d13 1 þ451 d λT 4 þ ⋯ ; [4] whereλT¼2πkℏc
BTis the thermal wavelength. At room temperature λT≈ 7 μm, indicating that thermal corrections are insignificant at separations d≈ 1 μm. (A similar correction appears in the first scattering term for parallel plates.)
By contrast, the thermal corrections for a cone are quite sig-nificant, and the force from Eq.3 on a sharp cone is modified at low temperatures to Fneedle EM ∼ −ℏc 16πj lnθ0 2j ln4 − 1 d2 − 2 3λ2 T ln2d λTþ 0.810 λ2 T þ ⋯ : [5]
More generally, for a sharp cone titled by an angleβ, and at a temperature T, the result of Eq.3 is multiplied by a (semi)analytic function gðβ;d∕λTÞ∕ cos β as reported inSI Appendix, Section III. The function gðβ;d∕λTÞ is an increasing function of both β and the temperature T. We plot this function for d¼ 1 μm in Fig. 3. Interestingly, the room temperature force is more than 100% higher than the force for T¼ 0. In contrast, the corresponding increase for parallel plates at d¼ 1 μm is only about 0.1%. This enhanced role of thermal corrections for specific geometries has been noted before (38, 39) and appears essential to the design of microelectromechanical (MEM) devices.
Conclusion and Outlook
Construction of electromechanical devices at the micrometer scale requires mastery of Casimir forces, which depend nontrivi-ally on geometry and shape. The few analytic solutions available previously apply generically to smooth objects such as cylinders and spheres. Mechanical devices, however, can optimally manip-ulate forces by adjusting the alignment and proximity of sharp components such as tips and edges. In this paper, we have pre-sented a synthesis of EM scattering results from wedges and cones to compute Casimir forces for perfect conductors with sharp edges and tips. The interaction of these objects with a metal plate can then be computed systematically by a multiple-scatter-ing series, which provides an analytical expressions for the inter-action of a plate and a knife edge with arbitrary opening and tilt angles, and the singular behavior of the force on the tip of a sharp scanning needle.
In practice, the ideal Casimir results have to be corrected due to imperfect conductivity and finite temperatures. We find that thermal fluctuations at room temperature are quite significant
for a cone and have to be properly accounted for. Fortuitously, for a (semiinfinite) needle, a shape-induced, drastic reduction of magnetic against electric polarizability limits the influence of imperfect conductivity, which for low frequencies can be easily understood from the polarizability of a needle (40). In particular, there is ongoing debate in the literature (41–44) regarding the use of Drude or Plasma models of conductivity—the former does not shield low-frequency magnetic modes leading to decreased Casimir force at high temperatures (24). The magnetic modes actually penetrate the needle at all temperatures and do not con-tribute to the Casimir force plotted in Fig. 3, Right. Imperfect conductivity can also modify the electric modes. However, although we do not compute forces for a general frequency-dependent dielectric responseϵðωÞ, we argue in theSI Appendix, Section IVthat for typical metals (e.g., Au or Al), even at zero temperature and for needles (the cases where these corrections are largest), the corrections due to imperfect conductivity are at most around 5% for d¼ 0.3 μm.
Thus for separations0.3 μm ≲ d ≲ 10 μm, relevant to MEM devices and experiments, the perfect conductor results, with the important finite temperature corrections, should suffice. For example, let us consider the tip of an atomic force micro-scope: At separations, d≈ 0.3 μm, where the tip may be well approximated as a metal cone, our results predict a force that is a fraction of a piconewton. Such forces are at the limit of cur-rent sensitivities (45) and will likely become accessible with future improvements. Current experiments are performed on spheres of relatively large radius R, where the force is greater by a factor of (R∕d) (a typical R is 100 μm). A rounded wedge with radius of curvature R falls in an intermediate range, with forces larger (23) bypffiffiffiffiffiffiffiffiffiR∕dthan the sharp case. The force can also be enhanced by using arrays of cones or wedges, at the cost of the difficulty of maintaining their alignment.
Methods
The primary tool in finding the scattering matrix of an object is the (free) Green’s function given in a coordinate system appropriate to the object’s shape. The scalar Green’s function for the wedge and cone geometries were previously known (46, 47). Although the translation symmetry of a perfect conducting wedge makes it possible to treat the EM field as a combination of two scalar problems, the vector nature of the field is unavoidable in the case of the cone.
We obtain the EM Green’s function for the cone by analytic continuation of the standard representation in terms of spherical partial waves. Electro-magnetism, unlike the scalar field, does not allow monopole fluctuations, and this constraint introduces an additional subtlety. The full Green’s func-tion becomes
Fig. 3. Casimir interaction of a cone of semiopening angleθ0a distance d above a plane, scaled by the dimensional factor of ℏc∕d. The left panel corresponds
to a vertical orientation, with the energy multiplied by cos2θ0to remove the divergence as the energy becomes proportional to the area forθ0¼ π∕2. The right
panel shows the force, F, suitably scaled, for a tilted, sharp cone (θ0→ 0, evocative of an atomic force microscope tip) as a function of tilt angle β and
temperatures T ¼ 300, 80, and 0 K (top to bottom), at a separation of 1 μm. See the text for further discussion.
G0ðr; r0;iκÞ ¼ −4π ∑κ ∞ m¼−∞ Z ∞ 0 dλλ tanhðλπÞ 1 λ2þ 1∕4 × ðMout
iλ−1∕2;mðrÞMregiλ−1∕2;mðr0Þ
− Nout
iλ−1∕2;mðrÞNregiλ−1∕2;mðr0ÞÞ
−4πκ ∑∞ m¼−∞;m≠0
ΓðjmjÞΓðjmj þ 1ÞRout
0;mðrÞRreg0;mðr0Þ;
[6]
where, in addition to the usual magnetic (transverse electric) modes M and electric (transverse magnetic) modes N, we also find a set of discrete modes R,
which have no analog in other representations of the Green’s function. The total angular momentum of the latter modes is zero, which might appear in contradiction with the absence of theℓ ¼ 0 channel in electromagnetism. Here, however, these modes cancel the unphysical contributions of the other two modes to this channel. For this reason, we designate them as ghost fields. This Green’s function is the primary tool for the result in Tips via the Cone. ACKNOWLEDGMENTS. We wish to thank Umar Mohideen for discussions on the atomic force microscope tip. This work was supported by the National Science Foundation through Grants PHY08-55426 (to N.G.), DMR-08-03315 (to S.J.R. and M.K.), Defense Advanced Research Projects Agency Contract S-000354 (to M.F.M., S.J.R., M.K., and T.E.), by the Deutsche Forschungsge-meinschaft through Grant EM70/3 (to T.E.), and by the US Department of Energy under cooperative research agreement DE-FG02-05ER41360 (to M.F.M., R.L.J.).
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