Repulsive Casimir Forces Produced in Rectangular Cavities: Possible Measurements and Applications

Transcription

1 68 Brzilin Journl of Physics, vol. 36, no. B, Mrch, 006 Repulsive simir Forces Produced in Rectngulr vities: Possible Mesurements nd Applictions A. Gusso Deprtmento de Físic, Universidde Federl do Prná,.P. 9044, uritib-pr, Brzil nd A. G. M. Schmidt Deprtmento de iêncis Exts e Tecnológics, Universidde Estdul de Snt ruz, EP , Ilhéus-BA, Brzil Received on 0 August, 005 We perform theoreticl nlysis of setup intended to mesure the repulsive (outwrd) simir forces predicted to exist inside of perfectly conducting rectngulr cvities. We consider the roles of the conductivity of the rel metls, of the temperture nd surfce roughness. The possible use of this repulsive force to reduce friction nd wer in micro nd nnoelectromechnicl systems (MEMS nd NEMS) is lso considered. Keywords: Micro nd nnoelectromechnicl systems; simir effect I. INTRODUTION simir forces re well known prediction of Quntum Field Theory, nd result whenever the quntum vcuum is subject to constrints. The simir forces re one spect of broder subject usully referred to s simir effect. Presently, the simir effect finds pplictions not only in Quntum Field Theory, but lso in ondensed Mtter Physics, Atomic nd Moleculr Physics, Grvittion nd osmology, nd in Mthemticl Physics [ 3], nd its importnce for prcticl pplictions is now becoming more widely pprecited [4, 5]. Not withstnding its importnce, the simir effect is elusive. The ttrctive simir forces (AFs) predicted to exist between electriclly neutrl bodies were mesured successfully only few yers go []. Presently, the AF between sphere (or lens) bove flt disc covered with metls is climed to be mesured with n experimentl reltive error of pproximtely 0.7% t 95% confidence level [6], nd cn be predicted with theoreticl uncertinty t the level of %. The direct mesurement of the simir force between two prllel conducting pltes, the originl setup studied by H. B. G. simir [7] in 948, is even more chllenging thn for the sphere bove disc. For tht reson it ws ccomplished only recently with reltively poor precision of 5% [8]. The mesurement of such ttrctive forces sheds some light on the question of the nture of the quntum electromgnetic vcuum. However, very importnt predictions bsed on the existence of the quntum vcuum hve not received the sme ttention. This is the cse of the repulsive simir forces (RFs). Such repulsive forces (outwrd pressure on the wlls) re predicted to exist inside of n empty sphere [9] nd n empty rectngulr cvity [0, ] with perfectly conducting wlls, for the cse of Eucliden spce. Such repulsive forces re probbly the most striking exmple of the geometry dependent nture of the simir effect. However no experiment ws performed to mesure RFs. Only wek dependence on the geometry ws tested mesuring the force between plte with smll sinusoidl corrugtions nd lrge sphere []. The mesurement of the RF would be one of the most importnt probes of the nture of the quntum vcuum with fr reching implictions. Becuse RFs hve been predicted consistently by different quntum field theoretic techniques [9, 3] if they re proved not to exist the physicists will be fced with new puzzle to be solved. Widely ccepted simir energy renormliztion nd regulriztion procedures my need to be reviewed, s suggested by the only dissonnt result presented in Ref. [4], were no RFs re found for rectngulr piston. The sign of the simir force is lso predicted to depend upon electric nd mgnetic properties of mterils. For instnce, in Ref. [5] it is nticipted tht repulsive force will exist between two prllel pltes if one is perfect conductor nd the other is perfectly permeble. More recently, repulsive force between two prllel pltes mde from dielectric mterils with nontrivil mgnetic susceptibility ws nticipted [6]. However, this effect, which could hve interesting pplictions for MEMS nd NEMS, hs not been verified experimentlly nd no dielectric mteril exists stisfying the requirements on the vlues of the mgnetic susceptibility. In spite of the fct tht RFs re predicted for sphere nd rectngulr cvity with perfectly conducting wlls, it is resonble to expect, s for the cse of prllel pltes, tht they will lso be present inside cvities mde from good conductors. For tht reson, in this rticle we ddress the most importnt prcticl spects to be tken into ccount in n experiment intended to mesure the force exerted on one of the wlls of rectngulr cvity: the finite conductivity nd roughness of the wlls nd plte, nd the temperture. The choice of rectngulr cvity insted of the sphere is bsed primrily on the fct tht the former could be most esily fbricted with the vilble techniques for the fbriction of MEMS nd NEMS. We consider the experimentl setup to mesure the RFs to be mde of series of microscopic metllic rectngulr cvities rrnged side by side, forming n rry, with one of the wlls open. The repulsive force is then mesured by bringing ner plte with flt metllic surfce. The forces on the plte re then mesured. The use of tht cn be mesured most esily. This setup is presented schemticlly in Fig. (). We note tht different setup ws considered in Ref. [7] were sphere

2 A. Gusso nd A. G. M. Schmidt 69 is used insted of flt plte. However, it hs to be mentioned tht in Ref. [7] it is considered the cse in which the rdius of the sphere is comprble to the cvity length, implying tht the ends of the cvities re left essentilly uncovered, nd it is not clerly explined why in this cse one still cn expect the emergence of repulsive forces between the cvities nd the sphere. Furthermore, the roughness of the wlls nd the role of the temperture re not considered in the nlysis there presented. It is lso not explined how the finite conductivity of the cvity wlls nd the sphere were tken into ccount in the clcultion of the ttrctive nd repulsive simir forces. Its worth to mention tht to nlyze the flt plterectngulr cvities configurtion is specilly relevnt becuse the moveble pieces of MEMS nd NEMS typiclly involve flt surfces (for exmple, the rotry pieces in micromotors nd gers), nd it is nturl to sk whether metllic rectngulr cvities could be used to mke such pieces to levitte or, t lest, to hve their weight or other undesirble forces prtilly compensted by repulsive force. For tht reson, following the nlysis on the RF mesurement we present n nlysis on the possible ppliction of repulsive simir forces in MEMS nd NEMS to circumvent the problems resulting from friction nd wer. II. ASIMIR ENERGY AND FORES In this section we rgue tht for setup like tht presented in Fig. the resulting simir force on the plte is given by the sum of two independent contributions, nmely, the RF produced by the electromgnetic vcuum modes inside the cvity nd the ttrctive force between the plte nd the upper portion of the cvities. The renormlized simir energy inside rectngulr cvity with perfectly conducting nd perfectly smooth wlls t zero temperture cn be derived in vrious mnners []. A simple expression suitble for numericl clcultions ws derived in [], nd in terms of the internl dimensions of the cvity, nd 3 it reds E = c 3 6π + c π 48 l,m,n= ( [( l) + ( m) + ( 3 n) ] ). () The term with n = n = n 3 = 0 is to be omitted from the summtion. From the principle of virtul work the force on the wlls perpendiculr to the direction of i is simply F i = E i, () nd rnges from positive (outwrd) to negtive (inwrd) depending on the reltive sizes of, nd 3. The Eqs. () nd () llow one to serch for the configurtion of the cvity resulting into the strongest outwrd forces on the wlls. The numericl nlysis performed in Ref. [8], using the bove expression for the energy, suggests tht the forces F nd F 3 re lrger (nd positive) in configurtion stisfying 3, corresponding to n elongted prllelepiped. For such configurtion F is directed inwrd. Fortuntely, whenever 3 for rectngulr cvity we cn use simple nlyticl expression for the simir energy [, 0] [ π 3 E = c ζ R(3) 3 6π π ( + )], (3) 48 where ζ R denotes the Riemnn zet function. The expressions for the two outwrd forces re then clculted using Eq. () nd the result is nd [ π 3 F = c 70 3 [ π F 3 = c 70 3 ζ R(3) 3 8π 3 + π ] 48, (4) + ζ R(3) 6π ]. (5) These formuls for the forces reproduce the results obtined from Eq. () to better thn %, nd becuse the first term domintes over the others the forces re positive. Therefore there re two possible configurtions for our system. In one configurtion elongted cvities with height re lying horizontlly below the plte, like the cvity in Fig. (), with the force exerted on the plte corresponding to F. In the other configurtion the cvities re stnding verticlly below the plte, like the cvity in Fig., with the force exerted on the plte corresponding to F 3. If there where no other forces cting on the plte, mesuring the RFs would be reltively esy tsk. The plte could be brought ner the open wll closing it completely, thus ssuring the existence of the electromgnetic vcuum modes tht led to the simir energy Eq. (). However, when the plte is close to the top of the wlls there will be resulting ttrctive force tht cn surpss the repulsive force. For tht reson, in wht follows we nlyze which configurtion is the more dequte for n experiment intended to mesure RFs, delivering the stronger repulsive force compred to the ttrctive forces between the plte nd the cvity wlls. The repulsive forces exerted on the plte, F or F 3, re expected to decrese with incresing d, the distnce from the plte to the top of the wlls [see Fig. ()]. However, detiled estimtion of this decrese is beyond the scope of the present work. Insted we re going to ssume in our clcultions tht the forces F nd F 3 do not depend on d. We expect this is resonble ssumption whenever d is sufficiently smll in order not to disturb the electromgnetic vcuum modes tht give the most importnt contributions to the simir energy. This expecttion is bsed on the fct tht in the cse the perture t the top of cvity is smller thn λ the trnsmission of the modes to outside the cvity is kept smll [9]. Now the vlues of λ tht give the most importnt contribution to the simir energy for cvity stisfying the condition 3,

3 70 Brzilin Journl of Physics, vol. 36, no. B, Mrch, 006 ε d () FIG. : () A view of the setup including the rectngulr cvities nd the plte. The dots denote the possibility of hving more cvities rrnged side by side. The definitions of the lengths of the cvities nd wlls shown in side view. () 3 FIG. : () vity lying horizontlly below plte. vity stnding verticlly below plte. Plte is in drk gry nd the open wll is indicted in light gry. geometry tht closely resembles tht of two prllel pltes (more on tht in Section III), re those of the order of the smllest edge. Thus, it is resonble to expect tht F nd F 3 re constnt up to d /. For d / the RF will certinly decrese, nd for tht reson the relevnt distnce in n ctul experiment is restricted to d. Becuse d, nd nd 3, we cn seprte the totl simir force between the plte nd the cvities into two components: the RF, F nd F 3 nd the AF between the plte nd the top of the wlls. This conclusion is not s trivil s it my seen to be. If the plte were t reltively lrge distnces from the cvities the simir energy for the pltecvities system should be clculte from first principles considering the whole intricte geometry of the cvities. Tht mens, the nlysis should be similr to tht crried on for periodiclly deformed objects in Ref. [0]. Tht would lso be the cse whether, tht mens in the cse of shllow cvities. For the deep cvities we re going to consider, only the interction between the top of the wlls nd the plte is responsible for the ttrctive force. 3 Becuse of the nontrivil geometry involved, in order to clculte the AF between the plte nd the cvity wlls we use the pirwise summtion technique [, ]. This technique ws shown to give relible results for the simir force between bodies of rbitrry shpe. For instnce, for the force between flt plte nd smll body of rbitrry shpe the mximum possible error ws estimted to be 3.8% [] when compred to the exct results obtined by quntum field theoretic techniques. In the pirwise summtion technique the simir energy is given by E pw = cψ(ε 0) d 3 r d 3 r r r 7, (6) V V where V nd V re the volumes of the two intercting bodies, nd Ψ(ε 0 ) is constnt which depends on the mterils on V nd V. This expression for the energy does not tke into ccount the fct tht the pirwise interction between the toms in the volumes V nd V re ctully screened by the surrounding toms. In order to prtilly correct for this fct voiding to overestimte the ttrctive forces we do not integrte over the entire volume of the wlls nd the plte. Insted we consider tht interctions re only relevnt up to distnce inside the metl. The resulting volumes of integrtion V nd V re shown in Fig. 3, highlighted in light gry. lerly, from Eq. (6) it is not importnt to define which volume corresponds to V nd V since the vribles re interchngeble. For the constnt Ψ(ε 0 ) we tke its vlue in the limit of perfect conductors Ψ(ε 0 ) = π/4, nd introduce the corrections due to the finite conductivity lter (Section III). In order to get results tht re independent of the exct number of cvities in the experimentl setup, mking the nlysis more generl, we employed simple strtegy. We note tht the top of the wlls, highlighted in light gry in Fig. 3(), cn be divided into series of prllelepipeds. Therefore, becuse of the dditivity of the simir energy in the context of the pirwise summtion technique, the finl simir energy between the plte nd the wlls is given by the sum of the individul energy of ech segment ( prllelepiped) nd the plte. The

4 A. Gusso nd A. G. M. Schmidt 7 () ε FIG. 3: () The relevnt volumes of integrtion highlighted in light gry. Prllelepiped below lrge plte. L only restriction is tht the plte must be sufficiently lrge to be considered infinite, mking the clcultions independent of the ctul loction of the different segments on the top of the wlls. This condition cn be esily fulfilled by plte only slightly lrger thn the rry of cvities becuse the simir energy decreses very rpidly with the incresing distnce. We evluted E pw nlyticlly with the help of Mthemtic [3] for prllelepiped with dimensions given by ε, nd rbitrry length L, below plte with thickness nd lterl dimensions tht ensure tht the finl result is close enough to tht for n infinitely lrge plte. This rrngement is depicted in Fig. 3. An energy per unity of re is obtined dividing E pw by ε L. The finl simir energy, including ll the cvities is then obtined multiplying this energy per unit of re by the top surfce of the wlls, procedure equivlent to summing over the different segments. Moreover, to mke our nlysis more generl we ssume tht there is sufficiently lrge number of cvities tht we cn clculte E pw for one cvity nd then simply multiply it by the totl number of cvities. In such cse the contribution of the outermost wlls tht re prtilly disregrded re negligible. The re tht enters in the clcultion of E pw is then the effective ttrctive re per cvity S i = ( + i + ε)ε, where i = or 3 depending on whether the cvity is lying horizontlly or stnding verticlly below the plte, respectively. The re of the plte under the ction of the forces F,3 is A,3 = 3,. In wht follows we lwys clculte the repulsive nd ttrctive simir forces for only one cvity, but the results re ctully vlid for lrge number of cvities s explined previously. Similrly to the cse of AFs the RPs re only pprecible when the dimensions of the cvity re in the micrometer rnge. This is the first prcticl spect tht hs to be considered in ny experiment. Presently, structure like tht in Fig. cn be mde from metls like gold, nickel, copper, nd luminum with the smllest fetures with tens of nnometers, nd structures like the cvity wlls cn be mde with high spect-rtios [4, 5]. onsequently, the smll dimensions of the cvities pose no problem if they re kept bove few tens of nnometers. The pressures cused by F nd F 3, whenever 3 goes with /, s cn be seen from Eqs. (4) nd (5). onsequently, the smller the the bigger the pressure, which is desirble. However, the smller the the smller the rtio A i /S i, for given ε, therefore diminishing the rtio between the repulsive nd the ttrctive forces. The lower limit on is then set by the lower prcticl limit on ε, becuse the wlls must be thick enough to ensure good reflectivity to the electromgnetic modes inside the cvity. Such thickness is roughly determined by the penetrtion depth of the electromgnetic field 0 = λ p /(π), with λ p the plsm wvelength of the metl. For luminum(gold) λ p 07(36) nm [7], implying 0 7() nm. Now we note tht the intensity of the incident electromgnetic wve distnce x inside the metl decreses s I = I 0 exp( x/ 0 ). We cn ensure lmost no trnsmission of the electromgnetic wves by mking x mx = ε 0. For tht reson, we ssume tht the smllest possible thickness is ε = 30 nm. For such n ε the smllest is pproximtely 00 nm. As we will see next, nother reson not to tke smller thn 00 nm is tht for cvities mde from rel metls the RFs re predicted to decrese significntly whenever the smller side of the cvity is below pproximtely 00 nm. We now point out tht the configurtion of the rectngulr cvity tht cn led to the lrgest rtio between the repulsive nd ttrctive forces on the plte is tht of Fig. (). This is so becuse in spite of the fct tht both configurtions cn deliver the sme outwrd pressures, in prctice the rtio F /S cn be mde greter (by one order of mgnitude) thn the rtio F 3 /S 3. This results from the fct tht the fbriction of verticlly stnding rectngulr cvity with thin wlls much higher thn µm would be very difficult. This implies tht in generl 3 µm. Becuse for such configurtion 3, the force F 3 is highly constrined [see Eq. (5)] compred to F, which cn be mde rbitrrily lrge since 3 is not constrined. For tht reson, in wht follows we consider only the cse the cvities re lying horizontlly below the plte. This configurtion is exctly the one depicted in Fig. (). III. ONDUTIVITY, ROUGHNESS AND TEMPERATURE ORRETIONS The first correction to be tken into ccount here is tht of finite conductivity, which lters both AFs nd RFs. To this dte finite conductivity corrections were clculted only for two simple geometries, nmely, for two plne prllel pltes nd for sphere bove disc [, 7]. Such clcultions re quite involved, nd similr clcultions for rectngulr cvity nd the interction between the top of the wlls nd the plte would be even more demnding. Here, insted of clculting the corrections from first principles we dopt nother strtegy nd use the results lredy obtined for plne prllel pltes.

5 7 Brzilin Journl of Physics, vol. 36, no. B, Mrch, 006 In order to justify this pproch we note tht the simir energy for rectngulr cvity stisfying 3 is pproximtely tht in region with dimensions 3 between two infinite prllel pltes seprted by distnce E 0 = 3 cπ 70 3, (7) s cn be inferred from Eq. (3). Tht mens the modes inside the cvity re pproximtely the sme s those between prllel pltes distnce prt. Hence it is resonble to ssume tht the corrections to the energy nd F for the rectngulr cvity re dequtely described by those to the energy nd force between prllel metllic pltes. In the nottion of Ref. [7] we write, E = η E ( )E 0 nd F = η F ( )F 0, where E 0 nd F0 re the energy nd force for cvity with perfectly conducting wlls. The functions η E,F (x) re the correction fctors tht rnge from pproximtely t lrge seprtions to pproximtely 0 t the shortest distnces. These fctors depend upon the mterils on the wlls through their frequency dependent dielectric functions. In our nlysis we modeled the dielectric functions using the plsm model s done in Ref. [7]. In this context, since η E does not depend upon the correction to the force F is the sme s tht for the energy. However, in order to be conservtive we ssume tht the correction to F is the sme s tht for F (η F is slightly smller thn η E ). In conclusion, we ssume tht F = η F ( )F 0, where η F( ) is plotted in Fig. of Ref. [7] nd F 0 is given by Eq. (4). We hve chosen to nlyze two rectngulr cvities with dimensions tht hve good commitment with the need for strong RF, to be pproximte by prllel pltes, nd to hve resonble spect-rtios to meet the requirements of vilble fbriction techniques, nmely nd = 0.µm, = 0.5µm, 3 = 5µm, (8) = 0.µm, = µm, 3 = 5µm. (9) We consider cvities mde from luminum, for its excellent reflectivity in wide rnge of frequencies, nd gold, metl widely employed for the fbriction of MEMS. For the cvity with = 0.(0.)µm: F =.(0.7) pn; the pressure is P = 4.(0.7) N m ; for luminum η F ( ) = 0.50(0.68); E temp ( = c π 3 45β 4 + π β ( ) 3 π + π l,p= l,m,n,p= { nd for gold η F ( ) = 0.44(0.6). We stress tht the energy inside the cvity for perfectly conducting wlls differs from tht for prllel pltes with re 3 by just 0.8(3)%, therefore justifying our ssumptions. The finite conductivity correction for the AF ws tken to be the sme s tht for prllel pltes, nd the force obtined from the use of Eq. (6) is simply multiplied by η F (d). This is certinly good pproximtion whenever the seprtion d is smll compred to ε, becuse in such cse the top of the wlls nd the plte form system resembling two prllel pltes. For d comprble or greter thn ε we do not expect tht this pproximtion fils completely. This expecttion relies on the fct tht for lrger distnces the correction fctor is nerly nd vry t reltively slow pce, consequently, it is less importnt. Another consequence of the finite conductivity is rpid decy of electromgnetic fields inside the metl. It ws for tht reson tht we considered finite in the clcultion of the AF. Bsed on the vlues of 0 for luminum nd gold we ssume = 50 nm. The second correction to the forces tht we consider is tht of surfce roughness. Wht is relevnt here is the stochstic roughness in both the cvity wlls nd the plte resulting from the fbriction process. As we did for the finite conductivity corrections we use the results lredy derived for the cse of two prllel pltes. The corrected energy inside the cvity cn be obtined from the expression for the corrected force between two pltes in Ref. [] by simply integrting on the seprtion, resulting in [ ( ) ( ) ] 4 = E 0 disp disp , (0) E roughness where disp is the dispersion (roughly the mplitude) of the stochstic roughness. In this pproximtion there is no dependence of the roughness correction on nd 3. onsequently, the force F is corrected by exctly the sme fctor s the energy. To keep the corrections below the (5)% level it is required tht disp / 0.049(0.0). Tht mens for cvity with = 00 nm tht the imperfections on the wlls cn be s lrge s 5(0) nm. Presently, by mens of electron bem lithogrphy precision in the level of.3 nm hs been obtined for the fbriction of MEMS nd NEMS [8]. However, most usul techniques re not tht ccurte nd precision t the level of 0 nm is most likely to be found in n experiment [4]. As first pproximtion the AF could lso receive the sme correction expressed in Eq. (0). Finlly we ddress the role of temperture. An expression for the simir energy inside rectngulr cvity with finite temperture ws derived in Ref. [6]. It corresponds to dding the following terms to the energy in Eq. () ] [( l) + ( m) + ( 3 n) + ( β p) }) [4( l) + (βp) ] + [4( l) + (βp) ] + 3 [4( 3 l) + (βp), () ]

6 A. Gusso nd A. G. M. Schmidt 73 where β = c/k B T, with k B the Boltzmnn s constnt nd T the bsolute temperture. For the cvity with = 0.(0.)µm t temperture of 300 K the energy decreses significntly by.(4.7)%, while the force F decreses just 0.08(0.)%. The corrections re still very smll t higher tempertures. IV. FORE MEASUREMENT In the present nlysis of force mesurement we disregrd both the temperture nd roughness corrections. The former becuse the correction to the force is lwys much smller thn the expected experimentl ccurcy on the force mesurement; of the order of few percent in ny relistic scenrio. The lter becuse it could be mde suitbly smll depending on the fbriction technique. Yet, we consider the most importnt correction, tht of the finite conductivity, tht reduce the repulsive force produced by the cvity by hlf of its originl vlue nd the ttrctive forces by even greter fctors [7]. Besides being the most importnt correction, gretly exceeding the expected corrections due to roughness, the conductivity will depend upon the mteril the cvity nd the plte re mde from nd only mrginlly on the fbriction technique. In this sense, the corrections due to conductivity re universl, nd do not depend upon the specific fbriction technique tht will be employed, henceforth justifying the present theoreticl nlysis. As lredy mentioned in Section II in the nlysis we did not model the expected decrese on the repulsive force s function of the seprtion d. However, it is resonble to ssume tht for smll seprtions the repulsive force cn be well described by Eq. (4). For smll seprtions we men d smll compred to, becuse it is the smller cvity dimension tht, fter ll, determines the the smller frequencies llowed inside the cvity. For smll d we cn expect smll perturbtion on the modes inside the cvity in lrge rnge of frequencies, henceforth ensuring the existence of the repulsive force. The most importnt informtion for n experiment designed to mesure RFs is the rtio between the repulsive nd ttrctive forces s function of the seprtion d. In Figs. 4() nd 4 we present exctly this rtio for the cvities with = 0.µm nd 0.µm, respectively. The results re for the cvity nd the plte mde from gold, however essentilly the sme results re obtined for luminum. We considered four different ε, from the smllest possible vlue to one tht could be most esily obtined by the presently vilble fbriction techniques. Wht we cn infer from Fig. 4 is the smllness of the repulsive force compred to the ttrctive one in the rnge of distnces t which our clcultions re more relible nd precise (d /). The rtios re lrger for the cvity with = 00 nm, nd for d = / = 00 nm the RF mounts to 30% of the AF. For the cvity with = 00 nm the RF mounts to only 0% t d = /. As consequence of the smllness of the rtios F rep /F t, ny mesurement of the force exerted on the top plte hs to be very precise. For the sttic mesurement of the force on the top plte precise knowledge of the F rep F t F rep F t () d nm d nm FIG. 4: The rtio between repulsive (F rep ) nd ttrctive (F t ) simir forces for rectngulr cvity s function of the seprtion d. From the upper to the lower curve ε =30, 50, 80, nd 00 nm. vity dimensions in () given by Eq. (8) nd in by Eq. (9). seprtion d is lso required. This fct cn be illustrted by the rtio between the sum of the ttrctive force t the ctul position nd the repulsive force nd the ttrctive force t the distnce d s determined from the experiment, F = F t(d + ) + F rep, () F t F t (d) where represents the reltive displcement to the mesured distnce due to the uncertinties. The curves for this rtio re presented in Fig. 5 for the two cvities nd for = 0,± nd ±3 nm. The upper(lower) curves re for negtive(positive). We note tht even for = ± nm the errors re in the rnge 5 5% nd re of the order of the force the experiment intends to mesure (see Fig. 4). onsequently, the distnce hs to be mesured with n ccurcy better thn nm. In order to estimte the required ccurcy we note tht for nominl seprtion d = 50 nm, n inccurcy of 0. nm implies n uncertinty in the force mesurement of pproximtely ±.5% for both cvities, which is cceptble. Our nlysis leds to the conclusion tht very stringent requirements hve to be stisfied by the experimentl setup in order to llow for n dequte mesurement of the RF in rectngulr cvity. Such requirements surpss considerbly those for the experiments lredy crried out for the mesurement of AFs [, 6, 8]. V. APPLIATIONS As lredy mentioned in section I, repulsive forces could hve interesting pplictions in MEMS nd NEMS. In fct,

7 74 Brzilin Journl of Physics, vol. 36, no. B, Mrch, 006 F F t F F t () d nm d nm FIG. 5: urves for the rtio defined in Eq. () for = 0 (continuous), = ± nm (dshed) nd = ±3 nm (dot-dshed).vity dimensions in () given by Eq. (8) nd in by Eq. (9). such forces could be the solution for the problems tht re presently imposing severe restrictions on the functioning of MEMS with moveble prts, nmely, friction nd wer [9]. The forces cused by friction re usully very lrge t smll scles [30] when compred to the forces tht cn be delivered by the vilble driven systems in, e.g., micromotors nd microctutors. Usully friction obeys Amonton s lw (frictionl force depends linerly on the lod through the coefficient of friction), however, t smll scles friction turns out to be proportionl to the contct re between the surfces [30]. For systems sufficiently lrge to obey Amonton s lw, repulsive forces could be used to reduce the lod. For instnce, the rotry piece of micromotor or ger (usully with the shpe of disc) could be lifted by bottom force tht could prtilly or completely compenstes for its weight. This force could be the RF predicted in Ref. [6] or, s we propose here, the force produced by set of rectngulr cvities plced beneth the rotry piece of the micromotor or gers. The first option requires the use of suitble mterils tht presently re not vilble, nd is still mtter of debte whether such forces could ctully exist [3, 3]. The second option, the use of cvities beneth the moveble pieces, could be simple solution whenever these pieces were mde from metls or could t lest be covered with thin metl lyer. For smller systems, where the lod does not ply the most importnt role in the resulting frictionl force, the repulsive forces could be used in the sme wy to reduce the effective weight tht hs to be sustined by the rotting pivots or berings. onsequently, the pivots nd berings could possibly be smller, leding to reduction in the frictionl force nd wer. Such reduction is highly desirble since wer is the most importnt source of filure in MEMS, limiting their continuous opertion lifetime TABLE I: The distnces d 0,d nd d 0 s defined on the text for cvities mde from Al(Au), nd dimensions given in Eq. (8). ε (nm) d 0 (nm) d (nm) d 0 (nm) (88.) 89.4(89.5) 07() 50 00(00) 0(0) 6(35) TABLE II: The distnces d 0,d 0. nd d s defined on the text for cvities mde from Al(Au), nd dimensions given in Eq. (9). ε (nm) d 0 (nm) d 0. (nm) d (nm) 30 35(34) 37(36) 63(66) 50 5(5) 54(53) 86(9) to be of the order of seconds or minutes rther thn hours or dys [9]. To estimte the cpbility of the repulsive force produced by the rectngulr cvities to compenste for the weight of the moveble prts of MEMS nd NEMS we determined the distnce d 0 t which the RF equtes the AF nd the distnces required to the repulsive force to equte the AF dded to the weight of plte mde from metl with n intermedite density ρ = 8.9 g cm 3 (similr to tht of nickel nd cooper) nd thickness of µm nd 0 µm denoted d nd d 0, respectively. At this point we hve to note tht structures with thickness rnging from 0. µm up to 0 µm re usully employed in the fbriction of prts of MEMS nd NEMS [5] even when the other dimensions of these prts re of the order of few millimeters [33]. We present in Tble I d 0,d nd d 0 for the cvity with = 0.µm, mde from luminum nd gold, nd for the thickness of the wlls ε = 30 nm nd 50 nm. In Tble II the results re presented for the cvity with = 0.µm. In this cse, becuse the repulsive force is not strong enough to equte the weight of plte 0 µm thick, we present the distnce d 0. required to sustin plte with thickness of 0. µm. In order to better understnd the implictions of the results presented in Tbles I nd II we hve to remember the fct tht the RF produced by the cvity on the plte is expected to decrese with the seprtion d. Actully, from simple wve propgtion rguments, the chnge on the force is expected to depend upon the rtio d/. onsequently, we cn expect smller correction (smller decrese) to the force due to the seprtion for the cvity for which the rtio d/ is smller. We now note tht for ε = 30 nm d 0 (d ) corresponds to 88(89)% nd 68(8)% of the cvity width for = 0.µm nd 0. µm, respectively. For tht reson the cvity with = 0.µm is the most dequte for investigtions concerning the reduction of friction nd wer. Becuse d 0. is only slightly lrger thn d 0, levittion of thin metllic pltes cused by RF is lso likely to occur. We lso suggest the use of n rtifice in order to reduce further the distnces t which repulsive forces could counterblnce the ttrctive forces: thiner wlls with short height built on top of the cvity wlls. Thiner wlls my ssure enough reflectivity for the electromgnetic modes inside the cvity with-

8 A. Gusso nd A. G. M. Schmidt 75 TABLE III: The distnces d 0,d nd d 0 s defined on the text for cvities mde from Al(Au), with the dimensions given in Eq. (8), nd with dditionl top wlls. ε (nm) d 0 (nm) d (nm) d 0 (nm) (78.3) 79.7(79.7) 96.6(0) (8.0) 83.5(83.8) 07(6) TABLE IV: The distnces d 0,d 0. nd d s defined on the text for cvities mde from Al(Au), with the dimensions given in Eq. (9), nd with dditionl top wlls. ε (nm) d 0 (nm) d 0. (nm) d (nm) 30 5(4) 7(6) 53(57) 50 3(3) 34(34) 66(73) out further disturbing the modes if they re kept sufficiently short. The smll spect-rtio further fcilittes their fbriction. For instnce, if the top wlls were 5 nm thick nd 45 nm high the distnces between the top of these wlls nd the plte re predicted to be those presented in Tbles III nd IV. In clculting those distnces we summed over the contributions from the originl wll nd the dditionl top wll. The contribution of the originl wll is smll s cn be seen from the similrity between the results for ε = 30 nm nd 50 nm in Tbles III nd IV s compred to the results in Tbles I nd II tht differ considerbly. It is cler tht the introduction of the top wlls cn considerbly reduce the required seprtions. If the top wlls cn be mde thiner nd tller without further disturbing the modes inside the cvity is subject tht deserves further theoreticl nd experimentl investigtion. Tringulr structures re lso worth of investigtion. Anyhow, for the shorter distnces thus obtined the ssumption of constnt RF s d vries is more relible, nd therefore the results re self-consistent. VI. FINAL DISUSSION AND ONLUSIONS In this rticle we presented relistic nlysis of setup intended to mesure the repulsive forces resulting from the geometricl constrints imposed on the quntum electromgnetic vcuum. For relistic we men tht the nonidelity of the cvity ws tken into ccount in the clcultion of the RF s well s the unvoidble AF. We took dvntge of the similrity between rectngulr cvity stisfying the condition 3 nd two plne prllel pltes, considerbly simplifying the nlysis. The results thus obtined re expected to be very good description of the relity for smll rtios d/ nd still relible when the rtio is round 0.5. From the results presented in section IV we conclude tht for the smller seprtions t which our pproch is more precise, ttrctive forces re lwys considerbly greter thn the ttinble repulsive forces. This fct poses severe requirements for the experiment. For seprtions lrger thn pproximtely / reduction of the repulsive force is expected nd the curves in Figs. 4 nd 5 re no longer precise. However, these curves indicte tht even under the more optimistic ssumption tht the decrese in the RF is smll nd tht the relibility of our results extends to lrger seprtions, the mesurement my be difficult, unless the cvity wlls re sufficiently thin. Tht this is specilly true for the cse the cvity hs = 0.µm, cn be seen from the fct tht for ε = 00 nm, t seprtion d = = 0.µm the repulsive force mounts to only 50% of the ttrctive force. Fortuntely, we hve better sitution for the cse of lrger cvity since the RF equtes the AF t shorter distnces, s cn be seen in Fig. 4. It is worth to mention tht there seems to be no dvntge on the use of cvities with much greter thn 00 nm. The reson for tht is the fct tht for cvities with lrger the rtio F rep /F t is essentilly the sme s tht for = 00 nm when plotted s function of d/. Nevertheless, the RF nd AF decrese significntly, possibly mking its mesurement less precise. This fct hs to be considered in the design of ny ctul experiment. The use of the plsm model in the clcultion of the finite conductivity corrections results in correction fctors η F tht re from % to 0% smller thn those predicted using the tbulted dt for the dielectric functions of luminum nd gold for distnces round 00 nm [7]. This fct long with our conservtive ssumption tht the force F is corrected by the fctor η F for the force F, my imply tht the ctul repulsive forces delivered by the cvities in n experiment re greter thn the ones we predicted here by t most 0%. Such n increse in the force does not significntly chnges our results becuse of the strong dependence of the ttrctive forces on the seprtion d. More precisely, the distnces would decrese no more thn 5%. The most obvious use of the RF in MEMS nd NEMS is to levitte structures s we proposed here, preventing friction nd wer. However, the pplicbility of such forces is conditioned to the ctul decrese of the RF with the seprtion between the cvities nd the upper (plte-like) structure. As lredy mentioned the determintion of the ctul repulsive force with the distnce is beyond the scope of the present work, nd is expected to be quite involved, specilly in the cse tht the finite conductivity of the wlls re tken into ccount. Nonetheless, the results presented in section V, bsed on the extrpoltion of constnt RF to lrger seprtions, indicte tht the RF produced by the rectngulr cvity is potentilly useful nd the importnce of the reduction of wer nd friction in MEMS nd NEMS mkes it worth of further investigtion. Acknowledgments The uthors grtefully cknowledge NPq for reserch fellowships. A. G. M. Schmidt ws lso prtilly supported by Fundção de Ampro à Pesquis do Estdo d Bhi (FAPESB). A. Gusso is thnkful to I. A. Hümmelgen, D. H. Mosc nd E. S. Silveir for useful converstions.

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