Analytical Results on Casimir Forces for Conductors with Edges and Tips

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; and dDepartment 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- in terms of a multipole expansion. Using these methods, we have mount to the design and operation of microelectromechanical been able to compute forces between various combinations of devices. However, these forces depend nontrivially on geometry, planes, spheres, and circular and parabolic cylinders (19–24) and existing analytical formulae and approximations cannot deal (see also refs. 25–27). Both perfect conductors and dielectrics with realistic micromachinery components with sharp edges and have been studied. However, with the notable exception of the tips. Here, we employ a novel approach to electromagnetic scatter- knife edge (23–28), which is a limit of the parabolic cylinder ing, appropriate to perfect conductors with sharp edges and tips, geometry, systems with sharp edges have not yet been studied † specifically wedges and cones. The Casimir interaction of these analytically. objects with a metal plate (and among themselves) is then com- In all the above cases, the object corresponds to a surface of puted systematically by a multiple-scattering series. For the wedge, constant radial coordinate. In this paper, we present results on we obtain analytical expressions for the interaction with a plate, the quantum and thermal EM Casimir forces between generically as functions of opening angle and tilt, which should provide a sharp shapes, such as a wedge and a cone, and a conducting plate. particularly useful tool for the design of microelectromechanical We accomplish this task by considering surfaces of constant an- devices. Our result for the Casimir interactions between conducting gular coordinate. Although the conceptual step—radial to angu- cones and plates applies directly to the force on the tip of a scan- lar—is simple, the practical computation of scattering properties ning tunneling probe. We find an unexpectedly large temperature is nontrivial, necessitating complex mathematical steps. Further- dependence of the force in the cone tip which is of immediate more, the inclusion of wedges and cones practically exhausts relevance to experiments. shapes for which the EM scattering amplitude can be treated analytically.‡ fluctuations ∣ quantum electrodynamics In what follows, we provide a summary of the approach and highlights of our results. The first section introduces the scatter- he inherent appeal of the Casimir force as a macroscopic ing formalism, whose main ingredients are scattering amplitudes Tmanifestation of quantum “zero-point” fluctuations has in- from individual objects. A multiple-scattering series then enables spired many studies over the decades that followed its discovery systematic exploration of Casimir interactions between any ar- (1). Casimir’s original result (2) for the force between perfectly rangements of such shapes. Edges are explored in the section reflecting mirrors separated by vacuum was quickly extended to devoted to wedge geometries, where for perfect conductors the include slabs of material with specified (frequency-dependent) EM field can be parameterized in terms of two scalar fields. The dielectric response (3). Quantitative experimental confirmation, corresponding scattering amplitudes in this case were previously however, had to await the advent of high-precision scanning known; their application to compute Casimir forces is developed

probes in the 1990s (4–7). Recent studies have aimed to reduce here. In particular, we obtain simple analytical expressions for the PHYSICS or reverse the attractive Casimir force for practical applications force between a knife edge and a plate from the first two terms of in micron-sized mechanical machines, where Casimir forces the multiple-scattering series, and verify (by numerical computa- may cause components to stick and the machine to fail. In the tion of higher order terms) that these expressions provide highly presence of an intervening fluid, experiments have indeed accurate estimates of the force. The limitations of common observed repulsion due to quantum (8) or critical thermal (9) approximation schemes can be illustrated by simple examples fluctuations. Metamaterials, fabricated designs of microcircuitry, in this geometry. Analogous computations relevant to a sharp have also been proposed as candidates for Casimir repulsion across vacuum (10). Author contributions: M.F.M., S.J.R., T.E., N.G., R.L.J., and M.K. performed research and Although there have been many studies of the role of materials wrote the paper. (dielectrics, conductors, etc.), the treatment of shapes and geo- The authors declare no conflict of interest. metry has remained comparatively less investigated. Interactions *This Direct Submission article had a prearranged editor. between nonplanar shapes are typically calculated via the proxi- †Knife-edge geometries have been studied for scalar fields obeying Dirichlet boundary mity force approximation (PFA), which sums over infinitesimal conditions in refs. 29 and 30. Wedges and related shapes have been studied in isolation segments treated as locally parallel plates (11). This approxima- (31), but computing interactions among such objects requires full scattering and tion represents a serious limitation because the majority of conversion matrices, which are synthesized in this paper. experiments measure the force between a sphere and a plate ‡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 with precision that is now sufficient to probe deviations from circular, elliptic, and parabolic), spherical, and conical. Surfaces on which one such PFA in this and other geometries (12, 13). Practical applications coordinate is constant are candidates for exact analysis. In addition to the plane and are likely to explore geometries further removed from parallel sphere discussed above, the circular (33) (bottom right of Fig. 1) and parabolic (23) plates. Several numerical schemes (14, –16), and even an analog 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 computer (17), have recently been developed for computing of the remaining coordinate systems (32) only leads to shapes such as cylinders and cones Casimir forces in general geometries. However, analytical formu- with elliptic cross-section, which are generically similar to their circular counterparts. lae for quick and reliable estimates remain highly desirable. 1To whom correspondence should be addressed. E-mail: [email protected]. The formalism recently implemented in refs. 18 and 19 enables This article contains supporting information online at www.pnas.org/lookup/suppl/ systematic computations of electromagnetic (EM) Casimir forces doi:10.1073/pnas.1018079108/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1018079108 PNAS ∣ April 26, 2011 ∣ vol. 108 ∣ no. 17 ∣ 6867–6871 Downloaded by guest on September 30, 2021 tip are carried out in the section on the cone. Scattering ampli- presented in SI Appendix.) The expression for the Casimir inter- tudes for the cone are obtained via a representation of the EM action energy, Green’s function and sketched briefly in Methods. The expressions for the interaction of a cone and a plate (both perfectly Z Z reflecting) are rather complex in general, but reduce to simple ℏc ∞ ℏc ∞ 1 E ¼ dκ tr ln½1 − N¼− dκ tr N þ tr N2 þ ⋯ ; forms in the limit of small opening angle (approaching a needle). 2π 0 2π 0 2 For the interaction between parallel metal plates, thermal correc- [1] 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 involves integration over the imaginary wave number κ, an impli- is considerably larger than at zero temperature (by roughly a fac- cit argument of the above matrices. For an object in front of an tor of 2 at micron separations). This difference should prove quite infinite plate, N ¼ UTobjectU†Tplate. An expansion of tr ln½1 − N important to experiments and designs involving sharp tips. in powers of N corresponds to multiple scatterings of quantum In particular, the prospects for experimental detection of the fluctuations of the EM field between the two objects; the trace force on the tip of an atomic force microscope are discussed in operation sums over the full spectrum of scattering channels Conclusion and Outlook. Detailed derivations, supporting for- (plane waves for example). The presence of both the speed of mulae, and graphs for each of these sections are provided in light c and Planck’s constant ℏ indicate that this energy is a result the SI Appendix. of relativistic quantum electrodynamics. This procedure can be generalized to multiple objects, with the material properties Casimir Forces via the Scattering Formalism and shape of each body encoded in its T matrix. Analytical results The conceptual foundations of the scattering approach can be – are restricted to objects for which EM scattering can be solved traced back to earlier multiple-scattering formalisms (7, 34 36), exactly in a multipole expansion, a familiar problem of mathema- but these were not sufficiently efficient to enable more compli- tical physics with classic applications to radar and optics. cated calculations. The ingredients in our method are depicted in Fig. 1. The Casimir energy associated with a specific geometry Edges via the Wedge depends on the way that the objects constrain the EM waves that For a perfectly reflecting wedge, translation symmetry makes it can bounce back and forth between them. The dependence on possible to decompose the EM field into two scalar components: the properties of the objects is completely encoded in the scatter- an E-polarization field that vanishes on its surface (Dirichlet ing amplitude or T matrix for EM waves. The T matrices are boundary condition) and an M-polarization field that has vanish- indexed in a coordinate basis suitable to each object. Fig. 1 sum- ing normal derivative (Neumann). In the cylindrical coordinate marizes the T matrices for perfectly reflecting planar, wedge, system ðr;ϕ;zÞ, a wedge has surfaces of constant ϕ ¼θ0. and conical geometries. Scattering from a plane mirror gives Whereas for describingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scattering from cylinders, a natural plate ¼1 ϕ ð1Þ 2 2 ð1Þ T for the two polarizations, irrespective of the wavevec- im ð κ þ Þ ikzz ~ ð Þ ðℓ Þ basis is e Hm i kz r e with Bessel-H functions tor k. Cylindrical m;kz and spherical ;m quantum numbers, ¼ 0 1 2 … indexed bypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim ; ; ; , for a wedge we must choose along with polarization, label appropriate bases for the cylinder μϕ ð1Þ 2 2 ð κ þ Þ ikzz μ ≥ 0 and sphere. As detailed in the following sections on the wedge e Hiμ i kz r e with real , corresponding to ima- and cone, imaginary angular momenta, labeled by μ for the wedge ginary angular momenta that are no longer quantized (see SI and λ for the cone, enumerate the possible scattering waves. The Appendix, Section I). The T matrices, diagonal in μ, take the cone’s scattering waves are also labeled by the real integer m, simple forms indicated in Fig. 1. Dimensional analysis indicates corresponding to the z component of angular momentum. The that the interaction energy of a wedge of edge length L at a 2 corresponding T matrices depend on the opening angle θ0. The separation d from a plate is E ¼ −ðℏcL∕d Þfðθ0;ϕ0Þ, where T matrix for the wedge (top right in Fig. 1) is independent of fðθ0;ϕ0Þ is a dimensionless function of the opening angle θ0 ϕ the axial wavevector kz. In addition to the usual polarizations, and inclination 0 to the plate. This geometry, and the corre- for the cone we needed to introduce an extra “ghost” field sponding function fðθ0;ϕ0Þ, are plotted in Fig. 2, Middle. (labeled Gh) and the corresponding T matrix (top left in Fig. 1). The limit θ0 → 0 corresponds to a knife edge which was We will also need the matrix U that captures the appropriate previously studied as a limiting form of a parabolic cylinder translations and rotations between the scattering bases for each (23, 28). The matrix N for the wedge simplifies in this limit, object. This matrix encodes the objects’ relative positions and enabling exact calculation of the first few terms in the expansion orientations. (The U matrices needed for our calculations are 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.

6868 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018079108 Maghrebi et al. Downloaded by guest on September 30, 2021 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 (θ0 ¼ 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 at θ0 ¼ 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 at π∕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 by ðℓ;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 II involves an additional field, similar to the ghost fields that appear elsewhere in quantum field theory. This representation of EM scattering can also be of use for describing reflection of ordinary EM waves from cones. Dimensional analysis indicates that, for a cone poised vertically at a distance d from a plate, the d Fig. 2. The Casimir interaction energy of a wedge at a distance above interaction energy scales as (ℏc∕d) times a function of its opening a plane, as a function of its semiopening angle θ0 and tilt ϕ0. The rescaled θ ϕ angle θ0. This arrangement and the resulting interaction energy energy as a function of 0 and 0 is shown in the middle panel. The symmetric 2 θ case, ϕ0 ¼ 0, is displayed in the top panel and the interesting case where are depicted in Fig. 3, Left, with the energy scaled by cos 0 to the back side of the wedge is hidden from the plane is shown in the bottom remove the divergence as the cone opens up to a plate for panel. See the text for further discussion. θ0 → π∕2. The PFA (11) (depicted by the dashed line) becomes exact in this limit, but it is progressively worse as θ0 decreases from π∕2. In particular, this approximation predicts that the en- 3 E ϕ 1 ϕ ϕ ð2ϕ − 2ϕ Þ θ → 0 PHYSICS − ¼ sec 0 þ þ csc 0 sec 0 0 sin 0 ergy vanishes linearly as 0 , although in fact it vanishes as ℏcL∕d2 16π2 192π3 256π3 ℏ 4 − 1 1 E ∼− c ln [3] þ ⋯: θ ; d 16π j ln 0 j [2] 2 where the logarithmic divergence is characteristic of the remnant The first square brackets corresponds to tr N (depicted by an or- N2∕2 line in this limit (33). The EM results shown as the bold blue ange line in Fig. 2) and the second to tr ; their sum (depicted curve in Fig. 3 are obtained by including two terms in the series by a red line) is in remarkable agreement with the full result (blue 1 N ϕ → π∕2 of Eq. , with constructed from the plate and cone T matrices surface). As 0 , the knife edge becomes parallel to the in Fig. 1, including the additional ghost field. We have also in- plate and the interaction energy diverges as it becomes propor- cluded the corresponding curves for scalar fields subject to tional to the area rather than L. For parallel plates, we know that Dirichlet and Neumann boundary conditions which are depicted ½Nn∕ the terms in the multiple-scattering series tr n are propor- as fine red curves where the top (bottom) one corresponds to the 1∕ 4 tional to n (1). Numerically, we find that the convergence Dirichlet (Neumann) boundary condition. The limit of θ0 → 0 is ϕ π∕2 is more rapid for 0 < , and that the first two terms in shown in the left panel of Fig. 3 as an orange dashed line. As the 2 Eq. are accurate to within 1%. Including more than three terms sharp tip is tilted by an angle β, the prefactor ðln 4 − 1Þ∕16π is in the series will not modify the curve at the level of accuracy for replaced by gðβÞ∕ cos β, where gðβÞ can be computed from inte- ’ this figure. Casimir s calculation for parallel plates gives an exact grals of trigonometric functions (see SI Appendix, Section II). We ϕ ¼ π∕2 × result at 0 , marked with an in Fig. 2, Middle. plot this quantity in the right panel of Fig. 3. The panels in Fig. 2 display some of the more interesting aspects of the wedge-plate geometry. In the middle panel, the Thermal Corrections energy, rescaled by an overall factor of ℏcL∕d2 and multiplied by For practical applications, the above results have to be corrected cosðθ0 þ ϕ0Þ, is plotted versus θ0 and ϕ0. The factor of cosðθ0 þ for finite temperature. These are easily incorporated by replacing ϕ Þ 1 κ ¼ 0 is introduced to remove the divergence as one face of the the integral in Eq. with a sum over Matsubara frequencies n ð2π ∕ℏ Þ wedge becomes parallel to the plate and the energy becomes kBT c n (37). For the knife edge, the first (single-scattering) proportional to the area rather than just the length of the wedge. term in the force resulting from Eq. 2 is modified to

Maghrebi et al. PNAS ∣ April 26, 2011 ∣ vol. 108 ∣ no. 17 ∣ 6869 Downloaded by guest on September 30, 2021 Fig. 3. Casimir interaction of a cone of semiopening angle θ0 a distance d above a plane, scaled by the dimensional factor of ℏc∕d. The left panel corresponds 2 to a vertical orientation, with the energy multiplied by cos θ0 to remove the divergence as the energy becomes proportional to the area for θ0 ¼ 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.

knife ℏcL 1 1 1 d 4 for a cone and have to be properly accounted for. Fortuitously, edge ¼ − 1 þ þ ⋯ [4] FEM 2 ϕ 3 45 λ ; for a (semiinfinite) needle, a shape-induced, drastic reduction of 8π cos 0 d T magnetic against electric polarizability limits the influence of imperfect conductivity, which for low frequencies can be easily ℏc where λ ¼ 2π is the thermal wavelength. At room temperature understood from the polarizability of a needle (40). In particular, T kBT λ ≈ 7 μ there is ongoing debate in the literature (41–44) regarding the T m, indicating that thermal corrections are insignificant at separations d ≈ 1 μm. (A similar correction appears in the first use of Drude or Plasma models of conductivity—the former does scattering term for parallel plates.) not shield low-frequency magnetic modes leading to decreased By contrast, the thermal corrections for a cone are quite sig- Casimir force at high temperatures (24). The magnetic modes nificant, and the force from Eq. 3 on a sharp cone is modified at actually penetrate the needle at all temperatures and do not con- low temperatures to 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- −ℏc ln 4 − 1 2 2d 0.810 Fneedle ∼ − ln þ þ ⋯ : [5] dependent dielectric response ϵðωÞ, we argue in the SI Appendix, EM 16πj θ0 j d2 3λ2 λ λ2 ln 2 T T T Section IV that for typical metals (e.g., Au or Al), even at zero temperature and for needles (the cases where these corrections More generally, for a sharp cone titled by an angle β, and at a are largest), the corrections due to imperfect conductivity are at temperature T, the result of Eq. 3 is multiplied by a (semi)analytic most around 5% for d ¼ 0.3 μm. ðβ ∕λ Þ∕ β Thus for separations 0.3 μm ≲ d ≲ 10 μm, relevant to MEM function g ;d T cos as reported in SI Appendix, Section III. ðβ ∕λ Þ β The function g ;d T is an increasing function of both and the devices and experiments, the perfect conductor results, with temperature T. We plot this function for d ¼ 1 μm in Fig. 3. the important finite temperature corrections, should suffice. Interestingly, the room temperature force is more than 100% For example, let us consider the tip of an atomic force micro- higher than the force for T ¼ 0. In contrast, the corresponding scope: At separations, d ≈ 0.3 μm, where the tip may be well increase for parallel plates at d ¼ 1 μm is only about 0.1%. This approximated as a metal cone, our results predict a force that enhanced role of thermal corrections for specific geometries has is a fraction of a piconewton. Such forces are at the limit of cur- been noted before (38, 39) and appears essential to the design of rent sensitivities (45) and will likely become accessible with future microelectromechanical (MEM) devices. improvements. Current experiments are performed on spheres of relatively large radius R, where the force is greater by a factor Conclusion and Outlook ∕ μ Construction of electromechanical devices at the micrometer of (R d) (a typical R is 100 m). A rounded wedge with radius of scale requires mastery of Casimir forces, which depend nontrivi- curvaturepffiffiffiffiffiffiffiffiffi R falls in an intermediate range, with forces larger (23) ∕ ally on geometry and shape. The few analytic solutions available by R d than the sharp case. The force can also be enhanced by previously apply generically to smooth objects such as cylinders using arrays of cones or wedges, at the cost of the difficulty of and spheres. Mechanical devices, however, can optimally manip- maintaining their alignment. ulate forces by adjusting the alignment and proximity of sharp components such as tips and edges. In this paper, we have pre- Methods The primary tool in finding the scattering matrix of an object is the (free) sented a synthesis of EM scattering results from wedges and Green’s function given in a coordinate system appropriate to the object’s cones to compute Casimir forces for perfect conductors with shape. The scalar Green’s function for the wedge and cone geometries were sharp edges and tips. The interaction of these objects with a metal previously known (46, 47). Although the translation symmetry of a perfect plate can then be computed systematically by a multiple-scatter- conducting wedge makes it possible to treat the EM field as a combination ing series, which provides an analytical expressions for the inter- of two scalar problems, the vector nature of the field is unavoidable in the action of a plate and a knife edge with arbitrary opening and tilt case of the cone. angles, and the singular behavior of the force on the tip of a sharp We obtain the EM Green’s function for the cone by analytic continuation scanning needle. of the standard representation in terms of spherical partial waves. Electro- In practice, the ideal Casimir results have to be corrected due magnetism, unlike the scalar field, does not allow monopole fluctuations, to imperfect conductivity and finite temperatures. We find that and this constraint introduces an additional subtlety. The full Green’s func- thermal fluctuations at room temperature are quite significant tion becomes

6870 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1018079108 Maghrebi et al. Downloaded by guest on September 30, 2021 Z κ ∞ ∞ 1 which have no analog in other representations of the Green’s function. The G ðr; r0 κÞ¼− λλ ðλπÞ 0 ;i d tanh 2 total angular momentum of the latter modes is zero, which might appear in 4π ∑ 0 λ þ 1∕4 m¼−∞ contradiction with the absence of the ℓ ¼ 0 channel in electromagnetism. × ðMout ðrÞMreg ðr0Þ Here, however, these modes cancel the unphysical contributions of the other iλ−1∕2;m iλ−1∕2;m two modes to this channel. For this reason, we designate them as ghost fields. − Nout ðrÞNreg ðr0ÞÞ This Green’s function is the primary tool for the result in Tips via the Cone. iλ−1∕2;m iλ−1∕2;m κ ∞ − Γðj jÞΓðj jþ1ÞRout ðrÞRregðr0Þ ACKNOWLEDGMENTS. We wish to thank Umar Mohideen for discussions on 4π ∑ m m 0;m 0;m ; the atomic force microscope tip. This work was supported by the National m¼−∞;m≠0 Science Foundation through Grants PHY08-55426 (to N.G.), DMR-08-03315 [6] (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 where, in addition to the usual magnetic (transverse electric) modes M and of Energy under cooperative research agreement DE-FG02-05ER41360 electric (transverse magnetic) modes N, we also find a set of discrete modes R, (to M.F.M., R.L.J.).

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