Repulsive Casimir forces produced in rectangular cavities: Possible measurements and applications

0.23 MB·English

by  A. Gusso

Repulsive Casimir forces produced in rectangular cavities: Possible measurements and
applications
A. Gusso∗
Departamento de F´ısica, Universidade Federal do Paran´a,
C.P. 19044, 81531-990 Curitiba-PR, Brazil
6 A. G. M. Schmidt†
0
Departamento de Ciˆencias Exatas e Tecnol´ogicas,
0
Universidade Estadual de Santa Cruz, CEP 45662-000, Ilh´eus-BA, Brazil
2

n Weperformatheoreticalanalysisofasetupintendedtomeasuretherepulsive(outward)Casimir
a forcespredictedtoexistinsideofperfectlyconductingrectangularcavities. Weconsidertherolesof
J the conductivity of the real metals, of the temperature and surface roughness. The possible use of

3 thisrepulsiveforcetoreducefrictionandwearinmicroandnanoelectromechanicalsystems(MEMS
1 and NEMS) is also considered.

] PACSnumbers: 85.85.+j,12.20.Fv
r
e Keywords: Microandnanoelectromechanical systems,CasimirEffect,Wear
h
t
o I. INTRODUCTION inside of an empty sphere [9] and an empty rectangular
.
t cavity [10, 11] with perfectly conducting walls, for the
a
caseofEuclideanspace. Suchrepulsiveforcesareproba-
m
CasimirforcesareawellknownpredictionofQuantum
blythemoststrikingexampleofthegeometrydependent
- Field Theory, and result whenever the quantum vacuum
nature of the Casimir effect. However no experiment
d
is subject to constraints. The Casimir forces are one as-
n was performed to measure RCFs. Only a weak depen-
pect of a broader subject usually referred to as Casimir
o dence on the geometry was tested measuring the force
c effect. Presently,theCasimireffectfindsapplicationsnot betweenaplate withsmallsinusoidalcorrugationsanda
[ only in Quantum Field Theory, but also in Condensed
large sphere [12]. The measurement of the RCF would
Matter Physics, Atomic and Molecular Physics, Grav-
2 be one of the most important probes of the nature of
itation and Cosmology, and in Mathematical Physics
v thequantumvacuumwithfarreachingimplications. Be-
8 [1, 2, 3], and its importance for practical applications causeRCFshavebeenpredictedconsistentlybydifferent
1 is now becoming more widely appreciated [4, 5]. Not quantum field theoretic techniques [9, 10, 11, 13] if they
2 withstanding its importance, the Casimir effect is elu-
are proved not to exist the physicists will be faced with
0 sive. The attractive Casimir forces (ACFs) predicted to
a new puzzle to be solved. Widely accepted Casimir en-
1
exist between electrically neutral bodies were measured
4 ergy renormalization and regularization procedures may
successfully only a few yearsago[1]. Presently,the ACF
0 need to be reviewed, as suggested by the only dissonant
between a sphere (or lens) above a flat disc coveredwith
/ resultpresentedinRef. [14],werenoRCFsarefoundfor
at metals is claimed to be measured with an experimental a rectangular piston.
m relativeerrorofapproximately0.27%ata95%confidence
level [6], and can be predicted with a theoretical uncer- The sign of the Casimir force is also predicted to de-
-
d tainty atthe levelof1%. The directmeasurementofthe pend uponelectric andmagnetic propertiesofmaterials.
n Casimirforcebetweentwoparallelconductingplates,the For instance, in Ref. [15] it is anticipated that a repul-
o originalsetup studied by H. B.G. Casimir [7] in 1948,is sive force will exist between two parallel plates if one
c even more challenging than for the sphere above a disc. is a perfect conductor and the other is perfectly perme-
:
v Forthat reasonit wasaccomplishedonly recentlywith a able. Morerecently,a repulsiveforce betweentwo paral-
Xi relatively poor precision of 15% [8]. lel plates made from dielectric materials with nontrivial
magnetic susceptibility was anticipated [16]. However,
r The measurementofsuchattractive forces sheds some
this effect, which could have interesting applications for
a
light on the question of the nature of the quantum elec-
MEMS and NEMS, has not been verifiedexperimentally
tromagnetic vacuum. However, very important predic-
and no dielectric material exists satisfying the require-
tionsbasedontheexistenceofthequantumvacuumhave
ments on the values of the magnetic susceptibility.
not received the same attention. This is the case of the
repulsive Casimir forces (RCFs). Such repulsive forces In spite of the fact that RCFs are predicted for sphere
(outward pressure on the walls) are predicted to exist andrectangularcavitywithperfectlyconductingwalls,it
is reasonable to expect, as for the case of parallelplates,
that they will also be present inside cavities made from
good conductors. For that reason, in this article we ad-
∗Electronicaddress: gusso@fisica.ufpr.br dress the most important practical aspects to be taken
†Electronicaddress: [email protected] into account in an experiment intended to measure the
2
force exerted on one of the walls of a rectangular cav- dimensions of the cavity a ,a and a it reads
1 2 3
ity: the finite conductivity and roughness of the walls
∞
and plate, andthe temperature. The choice of rectangu- E = −¯hca1a2a3 [(a l)2+(a m)2+(a n)2]−2
lar cavity insteadof the sphere is based primarilyon the C 16π2 1 2 3
l,m,n=−∞
factthatthe formercouldbe mosteasilyfabricatedwith X
theavailabletechniquesforthefabricationofMEMSand π 1 1 1
+h¯c + + . (1)
NEMS. We consider the experimental setup to measure 48 a a a
(cid:18) 1 2 3(cid:19)
the RCFs to be made of a series of microscopic metal-
The term with n = n = n = 0 is to be omitted from
lic rectangularcavities arrangedside by side, forming an 1 2 3
the summation. From the principle of virtual work the
array, with one of the walls open. The repulsive force is
force on the walls perpendicular to the direction of a is
thenmeasuredbybringingnearaplatewithaflatmetal- i
simply
lic surface. The forces on the plate are then measured.
The use of a series of cavities, instead of a single cav-
∂E
C
ity, generatesstrongerforces that can be measured most F =− , (2)
i
∂a
easily. ThissetupispresentedschematicallyinFig.1(a). i
WenotethatadifferentsetupwasconsideredinRef.[17]
and ranges from positive (outward) to negative (inward)
were a sphere is used instead of a flat plate. However, depending on the relative sizes of a ,a and a . The
1 2 3
it has to be mentioned that in Ref. [17] it is considered
Eqs. (1) and (2) allow one to search for the configura-
the case in which the radius of the sphere is comparable tion of the cavity resulting into the strongest outward
to the cavity length, implying that the ends of the cav-
forcesonthe walls. The numericalanalysisperformedin
ities are left essentially uncovered, and it is not clearly
Ref. [18], using the above expression for the energy, sug-
explained why in this case one still can expect the emer-
geststhattheforcesF andF arelarger(andpositive)in
2 3
gence of repulsive forces between the cavities and the
a configuration satisfying a ≪ a ≪ a , corresponding
1 2 3
sphere. Furthermore, the roughness of the walls and the
to an elongated parallelepiped. For such a configuration
roleofthetemperaturearenotconsideredintheanalysis
F is directed inward.
1
there presented. It is also not explained how the finite
Fortunately, whenevera ≪a ≪a for a rectangular
1 2 3
conductivityofthecavitywallsandthesphereweretaken
cavity we can use a simple analytical expression for the
into account in the calculation of the attractive and re- Casimir energy [1, 10]
pulsive Casimir forces.
Its worth to mention that to analyze the flat plate- π2a2a3 ζR(3)a3 π 1 1
E =−¯hc + − + ,
rectangularcavitiesconfigurationisspeciallyrelevantbe- C 720a3 16π a2 48 a a
(cid:20) 1 2 (cid:18) 1 2(cid:19)(cid:21)
causethemoveablepiecesofMEMSandNEMStypically (3)
involve flat surfaces (for example, the rotary pieces in where ζ denotes the Riemann zeta function. The ex-
R
micromotorsandgears),anditis naturalto askwhether pressions for the two outward forces are then calculated
metallic rectangularcavities could be used to make such using Eq. (2) and the result is
piecestolevitateor,atleast,tohavetheirweightorother
undesirable forces partially compensated by a repulsive π2a3 ζR(3)a3 π
F =h¯c − + , (4)
force. Forthatreason,followingtheanalysisontheRCF 2 720a3 8π a3 48a2
(cid:20) 1 2 2(cid:21)
measurement we present an analysis on the possible ap-
plicationofrepulsiveCasimirforcesinMEMSandNEMS and
to circumvent the problems resulting from friction and
π2a ζ (3) 1
wear. F =h¯c 2 + R . (5)
3 720a3 16π a2
(cid:20) 1 2(cid:21)
These formulas for the forces reproduce the results ob-
tainedfromEq.(1)tobetterthan1,andbecausethefirst
II. CASIMIR ENERGY AND FORCES
term dominates over the others the forces are positive.
Thereforethere aretwopossibleconfigurationsforour
In this section we argue that for a setup like that pre- system. In one configuration elongated cavities with
sented in Fig. 1 the resulting Casimir force on the plate height a are lying horizontally below the plate, like the
2
is given by the sum of two independent contributions, cavityinFig.2(a),withtheforceexertedontheplatecor-
namely, the RCF produced by the electromagnetic vac- responding to F . In the other configurationthe cavities
2
uum modesinside the cavityandthe attractiveforcebe- are standing vertically below the plate, like the cavity
tween the plate and the upper portion of the cavities. in Fig. 2(b), with the force exerted on the plate corre-
The renormalized Casimir energy inside a rectangular sponding to F . If there where no other forces acting
3
cavity with perfectly conducting and perfectly smooth on the plate, measuring the RCFs would be a relatively
wallsatzerotemperaturecanbederivedinvariousman- easytask. Theplatecouldbebroughtneartheopenwall
ners [1]. A simple expression suitable for numerical cal- closing it completely, thus assuring the existence of the
culationswasderivedin[11],andintermsoftheinternal electromagnetic vacuum modes that lead to the Casimir
3
ε a
1
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
δ (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
a
d
2
(a) (b)
FIG. 1: (a) A view of the setup including the rectangular cavities and the plate. The dots denote the possibility of having
more cavities arranged side by side. (b) The definitionsof thelengths of the cavities and walls shown in a side view.
modes to outside the cavity is kept small [19]. Now the
values of λ that give the most important contributionto
the Casimir energy for a cavity satisfying the condition
a ≪ a ≪ a , a geometry that closely resembles that
1 2 3
of two parallel plates (more on that in Section III), are
a
2 a those of the order of the smallest edge a . Thus, it is
3 1
reasonable to expect that F and F are constant up to
2 3
d<∼a1/2. For d>∼a1/2 the RCF will certainly decrease,
and for that reason the relevant distance in an actual
experiment is restricted to d<∼a1.
a3 Because d<∼a1, and a1 ≪a2 and a3, we can separate
a the total Casimir force between the plate and the cavi-
a 2
1 a ties into two components: the RCF, F2 and F3 and the
(a) 1 (b) ACF between the plate and the top of the walls. This
conclusion is not as trivial as it may seen to be. If the
plate were at relatively large distances from the cavities
FIG.2: (a)Cavitylyinghorizontallybelowaplate. (b)Cavity
the Casimir energy for the plate-cavities system should
standing vertically below a plate. Plate is in dark gray and
be calculate from first principles considering the whole
theopen wall is indicated in light gray.
intricategeometryofthe cavities. Thatmeans,the anal-
ysis should be similar to that carried on for periodically
energy Eq. (1). However, when the plate is close to the deformed objects in Ref. [20]. That would also be the
top of the walls there will be a resulting attractive force case whether a1 ∼ a2, that means in the case of shallow
that can surpass the repulsive force. For that reason, in cavities. For the deep cavities we are going to consider,
what follows we analyze which configurationis the more onlythe interactionbetweenthetopofthe wallsandthe
adequate for an experiment intended to measure RCFs, plate is responsible for the attractive force.
delivering the stronger repulsive force compared to the Because of the nontrivial geometry involved, in order
attractive forces between the plate and the cavity walls. to calculate the ACF between the plate and the cavity
Therepulsiveforcesexertedontheplate,F orF ,are walls we use the pairwise summation technique [21, 22].
2 3
expectedtodecreasewithincreasingd,thedistancefrom This technique was shown to give reliable results for the
theplatetothetopofthewalls[seeFig.1(a)]. However,a Casimir force between bodies of arbitrary shape. For
detailedestimationofthisdecreaseisbeyondthescopeof instance, for the force between a flat plate and a small
the present work. Instead we are going to assume in our bodyofarbitraryshapethemaximumpossibleerrorwas
calculations that the forces F and F do not depend on estimated to be 3.8 % [21] when compared to the exact
2 3
d. Weexpectthisisareasonableassumptionwheneverd results obtained by quantum field theoretic techniques.
is sufficiently small. In order not to disturb the electro- In the pairwise summation technique the Casimir en-
magnetic vacuum modes that give the most important ergy is given by
contributions to the Casimir energy. This expectation is
based on the fact that in the case the aperture at the Epw =−¯hcΨ(ǫ ) d3r d3r |r −r |−7, (6)
top of a cavity is smaller than λ the transmission of the C 20 1 2 2 1
ZV1 ZV2
4
where V1 and V2 are the volumes of the two interacting (a)
bodies, and Ψ(ǫ ) is a constant which depends on the
20
materials on V and V . This expression for the energy
1 2
does not take into account the fact that the pairwise in-
teractionbetweentheatomsinthevolumesV1andV2are δ
actually screened by the surrounding atoms. In order to
partiallycorrectforthisfactavoidingtooverestimatethe
attractive forces we do not integrate over the entire vol-
ume of the walls and the plate. Instead we consider that
δ
interactions are only relevant up to a distance δ inside
the metal. The resulting volumes of integration V and
1
V areshowninFig.3,highlightedinlightgray. Clearly,
2
from Eq. (6) it is not important to define which volume
corresponds to V and V since the variables are inter-
1 2
changeable. For the constant Ψ(ǫ20) we take its value in (b)
the limitofperfectconductorsΨ(ǫ )=π/24,andintro-
20
duce the corrections due to the finite conductivity later
(Section III).
In order to get results that are independent of the ex- δ
actnumberofcavitiesintheexperimentalsetup,making L
δ
the analysis more general, we employed a simple strat-
egy. We note that the top of the walls, highlighted in ε
light gray in Fig. 3(a), can be divided into a series of
parallelepipeds. Therefore, because of the additivity of FIG. 3: (a) The relevant volumes of integration highlighted
the Casimir energy in the context of the pairwise sum- in light gray. (b) Parallelepiped below a large plate.
mation technique, the final Casimir energy between the
plate and the walls is given by the sum of the individual
energy of each segment (a parallelepiped) and the plate. only one cavity, but the results are actually valid for a
Theonlyrestrictionisthatthe platemustbesufficiently large number of cavities as explained previously.
large to be considered infinite, making the calculations Similarly to the case of ACFs the RPCs are only ap-
independent of the actual location of the different seg- preciablewhenthedimensionsofthecavityareinthemi-
ments on the top of the walls. This condition can be crometerrange. Thisisthefirstpracticalaspectthathas
easilyfulfilled by aplate only slightlylargerthanthe ar- to be considered in any experiment. Presently, a struc-
rayofcavitiesbecausetheCasimirenergydecreasesvery turelikethatinFig.1canbemadefrommetalslikegold,
rapidly with the increasing distance. nickel, copper, and aluminum with the smallest features
We evaluated Epw analytically with the help of Math- with tens of nanometers, and structures like the cavity
C
ematica [23] for a parallelepiped with dimensions given walls can be made with high aspect-ratios[24, 25]. Con-
by ǫ,δ and arbitrary length L, below a plate with thick- sequently, the small dimensions of the cavities pose no
ness δ and lateral dimensions that ensure that the final problemifthey arekeptaboveafew tens ofnanometers.
resultiscloseenoughtothatforaninfinitely largeplate. The pressures caused by F and F , whenever a ≪
2 3 1
ThisarrangementisdepictedinFig.3(b). Anenergyper a ≪ a goes with 1/a2, as can be seen from Eqs. (4)
2 3 1
unityofareaisobtaineddividingEpw byǫ×L. Thefinal and (5). Consequently, the smaller the a the bigger the
C 1
Casimirenergy,includingallthecavitiesisthenobtained pressure, which is desirable. However, the smaller the
multiplying this energy per unit of area by the top sur- a the smaller the ratio A /S , for a given ǫ, therefore
1 i i
faceofthewalls,aprocedureequivalenttosummingover diminishing the ratio between the repulsive and the at-
the different segments. Moreover, to make our analysis tractive forces. The lower limit on a is then set by the
1
more general we assume that there is a sufficiently large lowerpracticallimitonǫ,becausethewallsmustbethick
numberofcavitiesthatwecancalculateEpw foronecav- enoughtoensuregoodreflectivitytotheelectromagnetic
C
ity and then simply multiply it by the total number of modes inside the cavity. Suchthickness is roughlydeter-
cavities. Insuchacasethecontributionoftheoutermost mined by the penetration depth of the electromagnetic
walls that are partially disregarded are negligible. The fieldδ =λ /(2π),withλ the plasmawavelengthofthe
0 p p
area that enters in the calculation of Epw is then the ef- metal. For aluminum(gold) λ ≈ 107(136) nm [27], im-
C p
fective “attractive area” per cavity S = (a +a +ǫ)ǫ, plyingδ ≈17(22)nm. Nowwenotethattheintensityof
i 1 i 0
where i = 2 or 3 depending on whether the cavity is theincidentelectromagneticwaveadistancexinsidethe
lying horizontally or standing vertically below the plate, metal decreases as I = I exp(−2x/δ ). We can ensure
0 0
respectively. Theareaoftheplateundertheactionofthe almost no transmission of the electromagnetic waves by
forcesF isA =a ×a . Inwhatfollowswealways making x = ǫ ≈ 2×δ . For that reason, we assume
2,3 2,3 1 3,2 max 0
calculate the repulsive and attractive Casimir forces for that the smallest possible thickness is ǫ = 30 nm. For
5
such an ǫ the smallest a is approximately 100 nm. As factors that range from approximately1 at large separa-
1
we will see next, another reason not to take a smaller tionstoapproximately0attheshortestdistances. These
1
than 100 nm is that for cavities made from real metals factors depend upon the materials on the walls through
the RCFs are predicted to decrease significantly when- their frequency dependent dielectric functions. In our
everthesmallersideofthecavityisbelowapproximately analysis we modeled the dielectric functions using the
100 nm. plasma model as done in Ref. [27]. In this context, since
We now point out that the configuration of the rect- ηE does not depend upon a2 the correction to the force
angular cavitythat can leadto the largestratio between F2 is the same as that for the energy. However,in order
the repulsive and attractive forces onthe plate is that of tobeconservativeweassumethatthe correctiontoF2 is
Fig.2(a). Thisissobecauseinspiteofthefactthatboth the same as that for F1 (ηF is slightly smaller than ηE).
configurationscandeliverthesameoutwardpressures,in In conclusion, we assume that F2 = ηF(a1)F20, where
practicethe ratioF2/S2 canbe madegreater(by oneor- ηF(a1)isplottedinFig.1ofRef.[27]andF20 is givenby
der of magnitude) than the ratio F /S . This results Eq. (4).
3 3
fromthefactthatthefabricationofaverticallystanding We have chosen to analyze two rectangular cavities
rectangular cavity with thin walls much higher than 1 with dimensions that have a good commitment with
µm would be very difficult. This implies that in general the need for strong RCF, to be approximate by parallel
a3 <∼ 1µm. Because for such a configuration a2 ≪ a3, plates, and to have reasonable aspect-ratios to meet the
the force F is highly constrained[see Eq.(5)] compared requirements of available fabrication techniques, namely
3
toF ,whichcanbemadearbitrarilylargesincea isnot
2 3
constrained. Forthatreason,inwhatfollowsweconsider a1 =0.1µm, a2 =0.5µm, a3 =5µm, (8)
onlythecasethecavitiesarelyinghorizontallybelowthe
and
plate. This configuration is exactly the one depicted in
Fig. 1(a).
a =0.2µm, a =1µm, a =5µm. (9)
1 2 3
We consider cavities made from aluminum, for its excel-
III. CONDUCTIVITY, ROUGHNESS AND
lent reflectivity in a wide range of frequencies, and gold,
TEMPERATURE CORRECTIONS
a metal widely employed for the fabrication of MEMS.
For the cavity with a = 0.1(0.2)µm: F = 2.1(0.27)
1 2
The first correction to be taken into account here is pN; the pressure is P =4.2(0.27) N m−2; for aluminum
2
that of finite conductivity, which alters both ACFs and η (a ) = 0.50(0.68); and for gold η (a ) = 0.44(0.62).
F 1 F 1
RCFs. To this date finite conductivity corrections were We stress that the energy inside the cavity for perfectly
calculated only for two simple geometries, namely, for conductingwallsdiffersfromthatforparallelplateswith
two plane parallel plates and for a sphere above a disc area a ×a by just 0.8(3)%, therefore justifying our as-
2 3
[1, 27]. Such calculations are quite involved, and similar sumptions.
calculations for a rectangular cavity and the interaction The finite conductivity correction for the ACF was
betweenthetopofthewallsandtheplatewouldbeeven taken to be the same as that for parallel plates, and the
moredemanding. Here,insteadofcalculatingthecorrec- forceobtainedfromtheuseofEq.(6)issimplymultiplied
tions fromfirstprinciples we adoptanotherstrategyand by η (d). This is certainly a good approximation when-
F
use the resultsalreadyobtainedforplane parallelplates. ever the separation d is small compared to ǫ, because in
In order to justify this approach we note that the such a case the top of the walls and the plate form a
Casimir energy for a rectangular cavity satisfying a1 ≪ system resembling two parallel plates. For d comparable
a2 ≪ a3 is approximately that in a region with dimen- or greater than ǫ we do not expect that this approxima-
sionsa2×a3betweentwoinfiniteparallelplatesseparated tion fails completely. This expectation relies on the fact
by a distance a1 that for larger distances the correctionfactor is nearly 1
andvaryatarelativelyslowpace,consequently,itis less
E0 =−¯hcπ2a2a3, (7) important. Another consequence of the finite conductiv-
C 720a3 ity is a rapid decay of electromagnetic fields inside the
1
metal. It was for that reason that we considered a finite
as can be inferred from Eq. (3). That means the modes δ in the calculation of the ACF. Based on the values of
inside the cavity are approximately the same as those δ for aluminum and gold we assume δ =50 nm.
0
between parallel plates a distance a apart. Hence it The second correction to the forces that we consider
1
is reasonable to assume that the corrections to the en- is that of surface roughness. What is relevant here is
ergy and F for the rectangular cavity are adequately thestochasticroughnessinboththe cavitywallsandthe
1
described by those to the energy and force between par- plate resulting from the fabrication process. As we did
allelmetallicplates. InthenotationofRef.[27]wewrite, for the finite conductivity corrections we use the results
E = η (a )E0 and F = η (a )F0, where E0 and F0 already derived for the case of two parallel plates. The
C E 1 C 1 F 1 1 C 1
are the energy and force for a cavity with perfectly con- corrected energy inside the cavity can be obtained from
ducting walls. The functions η (x) are the correction the expressionfor the correctedforce betweentwoplates
E,F
6
in Ref. [1] by simply integrating on the separation, re- imperfections on the walls can be as large as 5(10) nm.
sulting in Presently, by means of electron beam lithography a pre-
cision in the level of 1.3 nm has been obtained for the
δ 2 δ 4 fabrication of MEMS and NEMS [28]. However, most
Eroughness =E0 1+4 disp +60 disp ,
C C a a usualtechniquesarenotthataccurateandaprecisionat
" (cid:18) 1 (cid:19) (cid:18) 1 (cid:19) # the levelof 10nm is mostlikely to be foundin anexper-
(10)
iment [24]. As a first approximation the ACF could also
where δ is the dispersion (roughly the amplitude) of
disp receive the same correction expressed in Eq. (10).
the stochastic roughness. In this approximation there is
no dependence ofthe roughnesscorrectionon a anda . Finallyweaddresstheroleoftemperature. Anexpres-
2 3
Consequently, the force F is corrected by exactly the sion for the Casimir energy inside a rectangular cavity
2
samefactor asthe energy. To keepthe correctionsbelow with finite temperature was derived in Ref. [26]. It cor-
the 1(5)%levelitisrequiredthatδdisp/a1 <∼0.049(0.10). responds to adding the following terms to the energy in
That means for a cavity with a = 100 nm that the Eq. (1)
1
π2a a a π
Etemp = h¯c − 1 2 3 + (a +a +a )
C 45β4 12β2 1 2 3
(cid:18)
∞
a a a 1
1 2 3
−
π2 l,mX,n,p=1 (a1l)2+(a2m)2+(a3n)2+(β2p)2 2
∞ h i
1 a a a
1 2 3
+ + + , (11)
π [4(a l)2+(βp)2] [4(a l)2+(βp)2] [4(a l)2+(βp)2]
l,p=1(cid:26) 1 2 3 (cid:27)(cid:19)
X
where β = h¯c/k T, with k the Boltzmann’s constant As already mentioned in Section II in the analysis we
B B
and T the absolute temperature. For the cavity with did not model the expected decrease on the repulsive
a = 0.1(0.2)µm at a temperature of 300 K the energy force as a function of the separation d. However, it is
1
decreases significantly by 1.1(4.7) %, while the force F reasonable to assume that for small separations the re-
2
decreases just 0.08(0.2)%. The corrections are still very pulsive force can be well described by Eq. (4). For small
small at higher temperatures. separations we mean d small compared to a , because it
1
isthesmallercavitydimensionthat,afterall,determines
the the smallerfrequenciesallowedinside the cavity. For
smalldwecanexpectasmallperturbationonthemodes
IV. FORCE MEASUREMENT
inside the cavity in a large range of frequencies, hence-
forth ensuring the existence of the repulsive force.
In the present analysis of force measurement we dis-
Themostimportantinformationforanexperimentde-
regard both the temperature and roughness corrections.
signedtomeasureRCFsistheratiobetweentherepulsive
The former because the correction to the force is always
andattractiveforcesasafunctionoftheseparationd. In
much smaller than the expected experimental accuracy
Figs. 4(a) and 4(b) we present exactly this ratio for the
on the force measurement; of the order of a few percent
cavities with a = 0.1µm and 0.2µm, respectively. The
in any realistic scenario. The later because it could be 1
results are for the cavity and the plate made from gold,
made suitably small depending on the fabrication tech-
howeveressentiallythesameresultsareobtainedforalu-
nique. Yet, we consider the most important correction,
minum. Weconsideredfourdifferentǫ,fromthesmallest
that of the finite conductivity, that reduce the repulsive
possible value to one that could be most easily obtained
force produced by the cavity by half of its original value
by the presently available fabrication techniques.
andtheattractiveforcesbyevengreaterfactors[27]. Be-
sides being the most important correction, greatly ex- What we can infer from figure 4 is the smallness of
ceeding the expected corrections due to roughness, the the repulsive force comparedto the attractive one in the
conductivity will depend upon the material the cavity rangeof distances atwhich our calculationsare more re-
and the plate are made from and only marginally on the liable and precise (d <∼ a1/2). The ratios are larger for
fabrication technique. In this sense, the corrections due thecavitywitha =200nm, andford=a /2=100nm
1 1
to conductivity are universal, and do not depend upon theRCFamountsto30%oftheACF.Forthecavitywith
the specific fabrication technique that will be employed, a =100nmthe RCFamountstoonly10%atd=a /2.
1 1
henceforth justifying the present theoretical analysis. As a consequenceofthe smallnessof the ratiosF /F ,
rep at
7
1
1.4
0.5
1.2
0.2
€F€€€r€e€€p€€€€ 0.1 €€F€€€*€€€€ 1
Fat 0.05 Fat
0.8
0.02
0.6
0.01
20 40 60 80 100
(a) (a) 20 40 60 80 100
dHnmL
dHnmL
1
1.4
0.5
1.2
0.2
€F€€€r€e€€p€€€€ 0.1 €€F€€€*€€€€ 1
Fat 0.05 Fat
0.8
0.02
0.6
0.01
50 75 100 125 150 175 200
(b) (b) 40 60 80 100 120 140 160
dHnmL
dHnmL
FIG. 4: The ratio between repulsive (Frep) and attractive FIG. 5: Curves for the ratio defined in Eq. (12) for ∆ = 0
(Fat) Casimir forces for a rectangular cavity as a function of (continuous), ∆ = ±1 nm (dashed) and ∆ = ±3 nm (dot-
the separation d. From the upper to the lower curve ǫ =30,
dashed).Cavity dimensionsin (a) given byEq.(8) and in (b)
50,80,and100nm. Cavitydimensionsin(a)givenbyEq.(8)
byEq. (9).
and in (b) by Eq.(9).
surpass considerably those for the experiments already
any measurement of the force exerted on the top plate
carried out for the measurement of ACFs [1, 6, 8].
hastobeveryprecise. Forthestaticmeasurementofthe
force on the top plate a precise knowledge of the separa-
tiondisalsorequired. Thisfactcanbeillustratedbythe
V. APPLICATIONS
ratiobetweenthesumoftheattractiveforceattheactual
position and the repulsive force and the attractive force
at the distance d as determined from the experiment, As already mentioned in section I, repulsive forces
couldhaveinterestingapplicationsinMEMSandNEMS.
F∗ F (d+∆)+F Infact,suchforcescouldbethesolutionfortheproblems
at rep
= , (12)
F F (d) that are presently imposing severe restrictions on the
at at
functioning of MEMS with moveable parts, namely, fric-
where∆representstherelativedisplacementtothemea- tionandwear[29]. Theforcescausedbyfrictionareusu-
sured distance due to the uncertainties. The curves for ally verylargeatsmallscales [30] whencomparedto the
this ratio arepresented in Fig.5 for the two cavities and forces that can be delivered by the available driven sys-
for ∆ = 0,±1 and ±3 nm. The upper(lower) curves are tems in, e.g., micromotors and microactuators. Usually
for negative(positive) ∆. We note that even for ∆=±1 friction obeys Amonton’s law (frictional force depends
nm the errors are in the range 5−15% and are of the linearly on the load through the coefficient of friction),
orderoftheforcetheexperimentintendstomeasure(see however, at small scales friction turns out to be propor-
Fig. 4). Consequently, the distance has to be measured tional to the contact area between the surfaces [30]. For
with anaccuracybetter than 1 nm. In orderto estimate systems sufficiently large to obey Amonton’s law, repul-
the required accuracy we note that for a nominal sepa- siveforcescouldbeusedtoreducetheload. Forinstance,
ration d = 50 nm, an inaccuracy of 0.2 nm implies an therotarypieceofamicromotororgear(usuallywiththe
uncertainty in the force measurement of approximately shape of a disc) could be lifted by a bottom force that
±1.5% for both cavities, which is acceptable. couldpartiallyorcompletely compensates forits weight.
Our analysis leads to the conclusion that very strin- This force could be the RCF predicted in Ref. [16] or,
gentrequirementshaveto be satisfiedby the experimen- as we propose here, the force produced by a set of rect-
tal setup in order to allow for an adequate measurement angular cavities placed beneath the rotary piece of the
of the RCF in a rectangular cavity. Such requirements micromotoror gears. The firstoptionrequiresthe use of
8
TABLE I: The distances d0,d1 and d10 as defined on the TABLE II: The distances d0,d0.1 and d1 as defined on the
text for cavities made from Al(Au), and dimensions given in text for cavities made from Al(Au), and dimensions given in
Eq. (8). Eq.(9).
ǫ(nm) d0 (nm) d1 (nm) d10 (nm) ǫ (nm) d0 (nm) d0.1 (nm) d1 (nm)
30 88.2(88.1) 89.4(89.5) 107(112) 30 135(134) 137(136) 163(166)
50 100(100) 102(102) 126(135) 50 152(151) 154(153) 186(192)
suitablematerialsthatpresentlyarenotavailable,andis TABLE III: The distances d0,d1 and d10 as defined on the
still a matter of debate whether such forces could actu- textforcavitiesmadefromAl(Au),withthedimensionsgiven
in Eq. (8), and with additional top walls.
ally exist [31, 32]. The second option, the use of cavities
beneath the moveable pieces, could be a simple solution ǫ (nm) d0 (nm) d1 (nm) d10 (nm)
whenever these pieces were made from metals or could 30 78.4(78.3) 79.7(79.7) 96.6(102)
at least be covered with a thin metal layer. For smaller 50 82.0(82.0) 83.5(83.8) 107(116)
systems, where the load does not play the most impor-
tant role in the resulting frictional force, the repulsive
forces could be used in the same way to reduce the ef-
tively. For that reason the cavity with a = 0.2µm is
fective weight that has to be sustained by the rotating 1
the most adequate for investigations concerning the re-
pivotsorbearings. Consequently,thepivotsandbearings
ductionoffrictionandwear. Becaused isonlyslightly
could possibly be smaller, leading to a reduction in the 0.1
larger than d , levitation of thin metallic plates caused
frictional force and wear. Such a reduction is highly de- 0
by RCF is also likely to occur.
sirablesince wearis the mostimportantsource offailure
We also suggest the use of an artifice in order to re-
inMEMS,limitingtheircontinuousoperationlifetimeto
ducefurtherthedistancesatwhichrepulsiveforcescould
be of the order of seconds or minutes rather than hours
counterbalance the attractive forces: thiner walls with
or days [29].
short height built on top of the cavity walls. Thiner
To estimate the capability of the repulsive force pro-
walls may assure enough reflectivity for the electromag-
duced by the rectangular cavities to compensate for the
netic modes inside the cavity without further disturbing
weightofthemoveablepartsofMEMSandNEMSwede-
the modes if they are kept sufficiently short. The small
termined the distance d at which the RCF equates the
0 aspect-ratio further facilitates their fabrication. For in-
ACF and the distances required to the repulsive force to
stance,ifthe topwallswere15nm thickand45nmhigh
equatetheACFaddedtotheweightofaplatemadefrom
thedistancesbetweenthetopofthesewallsandtheplate
a metal with an intermediate density ρ = 8.9 g cm−3
arepredicted to be those presentedin TablesIII andIV.
(similar to that of nickel and cooper) and thickness of
In calculating those distances we summed over the con-
1 µm and 10 µm denoted d and d , respectively. At
1 10 tributions from the original wall and the additional top
this point we haveto note thatstructureswith thickness
wall. Thecontributionoftheoriginalwallissmallascan
ranging from 0.1 µm up to 10 µm are usually employed
beseenfromthesimilaritybetweentheresultsforǫ=30
inthefabricationofpartsofMEMSandNEMS[25]even
nm and 50 nm in Tables III and IV as compared to the
whentheotherdimensionsofthesepartsareoftheorder
results in Tables I and II that differ considerably.
ofafewmillimeters[33]. WepresentinTableId ,d and
0 1 It is clear that the introduction of the top walls can
d forthecavitywitha =0.1µm,madefromaluminum
10 1 considerably reduce the required separations. If the top
and gold, and for the thickness of the walls ǫ = 30 nm
walls can be made thiner and taller without further dis-
and 50 nm. In Table II the results are presented for the
turbing the modes inside the cavity is a subject that
cavity with a =0.2µm. In this case, because the repul-
1 deserves further theoretical and experimental investiga-
sive force is not strong enough to equate the weight of a
tion. Triangular structures are also worth of investiga-
plate 10 µm thick, we present the distance d required
0.1 tion. Anyhow, for the shorter distances thus obtained
to sustain a plate with thickness of 0.1 µm.
the assumption of a constant RCF as d varies is more
In order to better understand the implications of the
reliable, and therefore the results are self-consistent.
resultspresentedinTablesIandII wehavetoremember
thefactthattheRCFproducedbythecavityontheplate
is expected to decrease with the separation d. Actually,
VI. FINAL DISCUSSION AND CONCLUSIONS
from simple wave propagationarguments,the change on
theforceisexpectedtodependupontheratiod/a . Con-
1
sequently,wecanexpectasmallercorrection(smallerde- In this article we presented a realistic analysis of a
crease) to the force due to the separation for the cavity setup intended to measure the repulsive forces resulting
forwhichtheratiod/a issmaller. Wenownotethatfor from the geometrical constraints imposed on the quan-
1
ǫ = 30 nm d (d ) corresponds to 88(89)% and 68(82)% tumelectromagneticvacuum. Forrealisticwemeanthat
0 1
of the cavity width for a = 0.1µm and 0.2µm, respec- thenonidealityofthecavitywastakenintoaccountinthe
1
9
Thisfacthastobeconsideredinthedesignofanyactual
TABLE IV: The distances d0,d0.1 and d1 as defined on the experiment.
textforcavitiesmadefromAl(Au),withthedimensionsgiven
Theuseoftheplasmamodelinthecalculationofthefi-
in Eq.(9),and with additional top walls.
nite conductivity correctionsresults in correctionfactors
ǫ(nm) d0 (nm) d0.1 (nm) d1 (nm) ηF thatarefrom2%to10%smallerthanthosepredicted
30 125(124) 127(126) 153(157) using the tabulated data for the dielectric functions of
50 132(132) 134(134) 166(173) aluminum and gold for distances around 100 nm [27].
This fact along with our conservative assumption that
the force F is corrected by the factor η for the force
2 F
calculation of the RCF as well as the unavoidable ACF. F1, may imply that the actual repulsive forces delivered
We took advantageofthe similaritybetweena rectangu- bythecavitiesinanexperimentaregreaterthantheones
larcavitysatisfyingtheconditiona ≪a ≪a andtwo we predicted here by at most 20%. Such an increase in
1 2 3
plane parallelplates, considerablysimplifying the analy- the force does not significantly changes our results be-
sis. The results thus obtained are expected to be a very cause of the strong dependence of the attractive forces
good description of the reality for small ratios d/a and on the separationd. More precisely, the distances would
1
still reliable when the ratio is around 0.5. decrease no more than 5%.
From the results presented in section IV we conclude ThemostobvioususeoftheRCFinMEMSandNEMS
that for the smaller separations at which our approach is to levitate structures as we proposed here, prevent-
is more precise,attractive forces are alwaysconsiderably ing friction and wear. However,the applicability of such
greater than the attainable repulsive forces. This fact forces is conditioned to the actual decrease of the RCF
poses severe requirements for the experiment. For sep- with the separation between the cavities and the upper
arations larger than approximately a /2 a reduction of (plate-like) structure. As already mentioned the deter-
1
the repulsive force is expected and the curves in Figs. 4 mination of the actual repulsive force with the distance
and 5 are no longer precise. However, these curves in- is beyond the scope of the presentwork,and is expected
dicate that even under the more optimistic assumption to be quite involved, specially in the case that the fi-
that the decrease in the RCF is small and that the re- nite conductivity of the walls are taken into account.
liability of our results extends to larger separations, the Nonetheless, the results presented in section V, based
measurementmaybedifficult,unlessthecavitywallsare onthe extrapolationofa constantRCFto largersepara-
sufficiently thin. That this is specially true for the case tions,indicatethattheRCFproducedbytherectangular
thecavityhasa =0.1µm,canbeseenfromthefactthat cavityis potentially useful and the importance of the re-
1
for ǫ = 100 nm, at a separation d = a = 0.1µm the re- ductionofwearandfrictioninMEMSandNEMSmakes
1
pulsiveforceamountstoonly50%oftheattractiveforce. it worth of further investigation.
Fortunately, we have a better situation for the case of a
larger cavity since the RCF equates the ACF at shorter
distances, as can be seen in Fig. 4(b). Acknowledgments
It is worth to mention that there seems to be no ad-
vantageonthe use ofcavities with a much greaterthan TheauthorsgratefullyacknowledgeCNPqforresearch
1
200 nm. The reason for that is the fact that for cavi- fellowships. A. G. M. Schmidt was also partially sup-
ties with larger a the ratio F /F is essentially the ported by Fundac¸a˜o de Amparo `a Pesquisa do Es-
1 rep at
sameasthatfora =200nmwhenplottedasafunction tado da Bahia (FAPESB). A. Gusso. is thankful to
1
of d/a . Nevertheless, the RCF and ACF decrease sig- I. A. Hu¨mmelgen, D. H. Mosca and E. S. Silveira for
1
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