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Casimir force measurements in Au-Au and Au-Si cavities at low temperature J. Laurent, H. Sellier, A. Mosset, S. Huant, and J. Chevrier Institut N´eel, CNRS et Universit´e Joseph Fourier, B.P. 166, F-38042 Grenoble, Cedex 9, France (Dated: January 16, 2012) WereportonmeasurementsoftheCasimirforceinasphere-planegeometryusingacryogenicforce microscope to move the force probe in situ over different materials. We show how the electrostatic environmentoftheinteractingsurfacesplaysanimportantroleinweakforcemeasurementsandcan overcometheCasimirforceatlargedistance. Afterminimizingtheseparasiticforces,wemeasurethe Casimirforcebetweenagold-coatedsphereandeitheragold-coatedoraheavilydopedsiliconsurface in the 100–400nm distance range. We compare the experimental data with theoretical predictions 2 and discuss the consequence of a systematic error in the scanner calibration on the agreement 1 between experiment and theory. The relative force over the two surfaces compares favorably with 0 theory at short distance, showing that this Casimir force experiment is sensitive to the dielectric 2 properties of the interacting surfaces. n PACSnumbers: 07.10.Pz,12.20.Fv,73.40.Cg,78.20.Ci a J 3 I. INTRODUCTION the experiments has been continuously improved, using 1 metallic or semiconductor materials like Au-Au,7,9,14–19 ] Nanoelectromechanical systems (NEMS) are used in a Al-Al,8,12 Cr-Cr,11 Au-Cu,10 Au-Ge,20 Au-Si,21–23 Au- l Si grating,24 Ge-Ge,25 and Au-indium tin oxide.26 In l broadening range of applications, such as actuators, sen- a parallel, the influences of layer thickness,27,28 surface h sors, resonators, or modern nanocharacterization tools. roughness,29gratingstructure,30andmaterialconductiv- - Size reduction not only allows for shrinking the energy s ity31–33havebeenstudiedtheoreticallytoprovidemodels consumption and for shortening the response time, but e for comparison or stimulate new experiments. m it also allows integrating a broader range of functionali- ties on a single chip.1 However, quantum physics comes In this paper, we report on a detailed study of the . t into play at the nanoscale and can affect NEMS behav- Casimir force between a gold-coated sphere and a doped a m ior.2 Understanding these effects, and possibly control- siliconsubstrateatliquidheliumtemperature(4.2K)and ling them, is a necessary prerequisite to optimize NEMS compare it in situ with the case of a gold-coated surface. - d design. In turn, new applications driven by quantum ef- The use of the sphere-plane geometry avoids the chal- n fects can emerge, in particular in the field of ultra-high- lenge of controlling with high accuracy the parallelism o sensitivity force or displacement detection. of two flat surfaces separated by a submicron gap. Our c TheCasimirforce,discoveredin1948,isthearchetypi- aim is to reveal the dependence of the force on the ma- [ calforceinthisframework.3Itspurelyquantumoriginre- terials properties and to compare it quantitatively with 1 sults from the zero-point fluctuations in the electromag- theory. Thermalization at low temperature provides an v netic field. Since its theoretical prediction, the Casimir exceptional mechanical stability of the interacting sur- 5 force has attracted the interest of a large community of faces, which is highly beneficial for long-term force mea- 8 scientists ranging from cosmologists4 to NEMS design- surements. In principle, it should also improve the force 7 2 ers5 through solid-state physicists. Experimentally, the sensitivity because of a reduced Brownian motion of the . first confirmation6 of the Casimir effect was reported as cantilever,butothereffects,suchasoptomechanicalcou- 1 early as 1958, but the first quantitative study7 of the plings,degradetheexpectedincreaseinsensitivityinour 0 2 Casimirforceusingatorsionpendulumwasreportednot experiment. Cooling down the experiment34 also sup- 1 before 1997. Soon after, an important activity has been presses the thermal contribution of the Casimir force,35 : triggered thanks to the use of atomic force microscopes allowing a direct comparison with the zero-temperature v (AFM)8 ormicroelectromechanicalsystems(MEMS).9,10 theoretical calculations. i X Most of the experiments have been carried out with a After a complete description of our calibration pro- r cavity in the sphere-plane geometry and very few in the cedure, we show that parasitic forces can perturb sig- a plane-plane geometry11,12 that requires highly parallel nificantly the Casimir force measurements and that the surfaces. setup environment can be modified to suppress this ar- ThelimitednumberofgroupsworkingonCasimirforce tifact. We then discuss the variations of the minimizing measurements confirms how difficult these experiments potentialwithdistancebyconsideringfirstthepatchpo- are.13 This is explained by the small magnitude of this tential effect,36 and then a simple electrostatic model37 force as compared to the electrostatic force, and by the that reproduces the data. Finally, we present relative stronger distance dependence scaling like the inverse of measurements of the Casimir force in the 100–400nm the fourth (third) power of the distance in plane-plane distance range obtained by changing in situ the sample (sphere-plane) geometry. In order to check the validity surface from gold to silicon. The relative force between oftheoriesdescribingfundamentalforces,theprecisionof thetwomaterialsisinqualitativeagreementwiththeory, 2 sphere side and 80nm on the backside) to provide an electriccontacttothesphere,whichisoneoftheCasimir mirror. Theroot-mean-squareroughnessofthegoldsur- faces is around 3nm, as measured by AFM. The probes havetypicallyaresonancefrequencyf about40kHzand 0 a spring constant k about 10N/m. The cantilever chip is glued with silver paint on a holder (G) made of an- odized aluminum and then fixed on a long holder (H). The cantilever is mechanically excited at resonance by a piezoelectric dither (I). The sample (B) is mounted with silver paint on a holder (D) that is separated from the piezoelectric z-scanner (F) by a grounded aluminum plate (E) for electrostatic screening of the high voltages applied to the scanner. The cantilever motion is mea- sured with a compact optical detection compatible with the severe space constraints of cryogenics,39 using the interferometric cavity formed by the flexible cantilever and the extremity of a single-mode optical fiber (C) an- chored to the holder (J). The fiber is positioned above the end of the lever with a set of XYZ cryogenic inertial FIG. 1: Drawing and photograph of the microscope head at motors (M1) and adjusted such as to obtain an interfer- the bottom part of the cage structure. The functional parts ometric cavity with good displacement sensitivity. The of the microscope, labeled by capital letters, are described in sampleisapproachedbelowtheforceprobewithanother the text. set of motors (M2) and the scanner (F) is used to finely tune the gap between the two surfaces. The scanner has been calibrated by interferometry and the hysteresis has but the absolute values of the force show a systematic beendeterminedfor definedscannerextensions. Itcould error with respect to the theoretical predictions that are be, however, thatthiscalibrationslightlyevolvesintime tentatively attributed to an aging of the scanner calibra- after successive thermal cycles as discussed later in the tion. analysis of the results. The paper is organized as follows. Section II describes Themicroscopeandtheentirecagestructurearesealed the homemade low-temperature force microscope. Sec- into a 2-in.-diameter stainless steel tube evacuated to a tion III explains the force measurements and data anal- secondaryvacuumandflushedwithheliumgas. Thetube ysis. SectionIVdiscussestheoriginandsuppressionofa is then filled with a low pressure of helium exchange gas long-rangeresidualforceduetotheelectrostaticenviron- (10mbaratroomtemperature)andimmersedinaliquid ment. Section V presents the results on the minimizing helium cryostat. During cooling down, it is necessary potential and Casimir force obtained for gold and silicon to continuously readjust the optical cavity with the M1 surfaces. Conclusions are drawn in Section VI. motors to compensate for thermal contractions. Measurements at low temperature have the advan- tage to benefit from strongly reduced thermal drifts and II. LOW-TEMPERATURE FORCE thermomechanical noises that usually limit the room- MICROSCOPE temperature experiments. For instance, position drifts of about 1nm/min at 300K are found to be reduced to We developed a new force microscope working in a less than 1nm/h at 4K. This is of particular importance cryogenic environment at 4.2K, as an evolution of the in the present study because the Casimir force strongly room-temperature instrument developed by Jourdan et depends on distance. In the same way, the frequency al.18 The structure of the low-temperature instrument drift of the cantilever resonance is strongly suppressed takes the shape of a 50-mm-diameter and 120-cm-long from 3mHz/min at 300K down to a negligible value at modular system based on a tubular cage.38 This cage 4K.Finally,anotheradvantageofcryogenictemperature structure links the top of the instrument (that bears all is to strongly suppress the cantilever Brownian motion the electrical and optical connections) to the microscope induced by thermomechanical force fluctuations. head located at the very bottom. The main parts of In such cryogenic conditions, the force detection sen- the microscope (marked by capital letters in Fig. 1) are sitivity is essentially limited by the optical readout of described below. the cantilever position. The intensity fluctuations of the The force probe is based on an AFM cantilever with a laser beam are here the main source of noise, well above 40-µm-diameter polystyrene sphere fixed at the extrem- thenoiseofthephotodiodeanditsamplifier. Inparticu- ity with standard epoxy glue (A). The sphere and can- lar,optomechanicalcouplingslikeradiationpressureand tilever are coated with gold (more than 200nm on the photothermalstressconvertthisintensitynoiseintocan- 3 nance. The lever is excited with a piezoelectric dither at (cid:1) its mechanical resonance and the lever vibration is mea- (a ) suredbyinterferometrywiththeopticalfiber. Theoscil- 0 lation amplitude and phase are recorded with a lock-in (cid:7)(cid:3)-1 andaphase-lockedlooptrackstheresonancefrequencyf (cid:5) (cid:1)(cid:2)f (cid:1) whentheprobeissubmittedtoaforcegradient. Theres- D -2 onance frequency shift is defined by ∆f =f −f , where 0 -3 (cid:16)(cid:30)(cid:31)"(cid:22)(cid:31)(cid:24) (cid:14)(cid:22)(cid:23)(cid:27)"(cid:22)(cid:31)(cid:24) f0 is the free resonance frequency. Since the zero sphere-plane distance cannot be de- 3 6 9 1 2 1 5 1 8 (cid:18)(cid:26)(cid:28)(cid:25)(cid:1)(cid:20)(cid:28)(cid:26)(cid:29)(cid:21) (cid:1) (cid:1) termined by bringing the sample into contact with the 0 (b ) 0 (c ) sphere, which would irreversibly damage the gold coat- -1 -5 0 Df(cid:1)(cid:2)(cid:5)(cid:7)(cid:3) ----5432 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)z(cid:1)(cid:1)(cid:9)(cid:8) (cid:23)(cid:6)(cid:6)(cid:22)(cid:12)(cid:6)(cid:29)(cid:1)(cid:1)(cid:29)(cid:29)(cid:28)(cid:28) (cid:1)C(cid:1)(cid:2)(cid:5)(cid:7)(cid:4)(cid:6)(cid:8)(cid:3)---211 050 000 b (cid:1)(cid:1)(cid:13)(cid:1)(cid:7)(cid:5)(cid:11)(cid:10)(cid:1)(cid:1)(cid:1)(cid:15)(cid:16)(cid:17)(cid:26)(cid:22)!#!(cid:1)(cid:22)$(cid:28)(cid:8)(cid:1)(cid:19)(cid:1)(cid:4)(cid:8) (cid:1) ibmnyegaeosluefcrttehrmoesestnuattrsif,cactcheasel,ibstarhametpiaolbnes.oislDuatuperpidnriogsatacahnsecedequitseondtceheteeorfsmpfhionerecrdee --76 (cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:1)(cid:9)(cid:12)(cid:10)(cid:6)(cid:1)(cid:1)(cid:29)(cid:29)(cid:28)(cid:28) --32 05 00 z(cid:6)(cid:1)(cid:13)(cid:1)(cid:4)(cid:8)(cid:10)(cid:1)(cid:29)(cid:28) rbeysosnmaanlclestferpeqs,ueanncdyfoshrifetac∆hfsciasnmneeraspuorseidtiofnorzsdciaffne,rtehnet -2 0 0 -1 0 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 biasvoltagesV appliedtothesamplewithrespecttothe V (cid:1)(cid:2)(cid:28)(cid:19)(cid:3) z (cid:1)(cid:2)(cid:29)(cid:28)(cid:3) (cid:23)(cid:22)(cid:29) grounded sphere [Fig. 2(a)]. The voltage is varied typ- ically over ±200mV around the potential V , which min minimizes the electrostatic force between the probe and FIG.2: (a)Frequencyshift∆f ofthecantileverresonanceasa functionofthetimeelapsedduringaforward(left)andback- sample. We obtain a series of ∆f(V) curves [Fig. 2(b)], ward (right) scan of the sample toward the sphere fixed on which are fitted by the second-order polynomial: thecantilever. Forwardandbackwardscansarenotperfectly symmetricbecauseofthescannerhysteresis. (b)Examplesof ∆f =C(V −V )2+∆f (1) min min ∆f(V)parabolasrecordedbysweepingtheprobe-surfacebias V fordifferentscannerpositionszscan. (c)Fitoftheparabola where C, Vmin, and ∆fmin are three adjustable parame- curvaturesC versusscannerpositionzscan usedtoobtainthe ters. The first term on the right-hand side corresponds force calibration factor β and the position z of the contact. 0 to the capacitive force, and the second term is the fre- quency shift due to the remaining forces, including the Casimir force, obtained for V = V at the summit of min the parabola. The curvature coefficient C is plotted as a tilever displacement noise, called backaction noise. As function of the scanner position z and fitted with scan a consequence, the low-temperature force sensitivity is found to be√of the same or√der as the room-temperature β sensitivity S ≈ 10fN/ Hz. This situation could be C = (2) FF (z −z )2 scan 0 improved by a broadband stabilization of the laser beam intensity or by the coherent coupling of laser noise and to obtain the sphere-sample contact position z and the 0 backaction noise, as demonstrated recently in our inter- force probe calibration parameter β [Fig. 2(c)]. The ab- ferometric setup.40 Therefore the only, but very reward- solute distance between sphere and sample is then given ing, advantage of the low temperature turns out to be by z =z −z . We have additionally checked that the scan 0 the exceptional mechanical stability of the microscope. cantilever static deflection generated by the electrostatic These retarded optomechanical forces also modify the force and the Casimir force is negligible in the studied resonance frequency and damping rate of the microlever separation range.49 through an optical spring effect induced by the inter- The theoretical expression of the sphere-plane capaci- ferometric process.41–43 Depending on the optical cavity tiveforcegradientgivesβ =−f0Rπ(cid:15)0,wheref isthefree detuning, this effect can induce self-oscillations44,45 or resonance frequency, R the spher2ekradius, (cid:15) t0he vacuum 0 provide self-cooling of the thermal noise.46–48 The detec- permittivity, and k the cantilever stiffness. The exper- tionconditionshavethusbeenoptimizedbychoosingthe imental value of β extracted from the fit can therefore cooling side of the detuning and by adjusting the laser be used to transform the measured frequency shift ∆f power. into a “reduced force gradient” G/R without any other parameter:18 G 2k π(cid:15) III. DATA ACQUISITION AND CALIBRATION =− ∆f = 0∆f (3) R f R β 0 Instead of measuring directly the electrostatic or Thiselectrostaticcalibrationoftheforceprobeusingthe Casimir force F(z) in static mode, we measure its force only parameter β is more relevant than the precise mea- gradient G(z)= dF in dynamic mode, which is given by surement of R and k. The traditional determination of dz the frequency shift ∆f = −f0G of the cantilever reso- k based on the thermal noise spectral density and the 2k 4 1 0 0 1 0 0 D a ta ( a ) T h e o ry 1 /z 3 1 0 (cid:7)(cid:3) 1 0 1 /z 4 (cid:6) (cid:1) M o d e l (cid:1)(cid:2)(cid:5)(cid:4) 1 /z 1.8 (cid:1) 1 D e fe c St a? m p le 1 , p o s itio n 1 R(cid:4) 1 S a m p le 1 , p o s itio n 2 0 .1 S a m p le 2 G 1 0 0 ( b ) 3 0 0 K 0 .1 1 0 1 0 0 1 0 0 0 z ( n m ) (cid:1) (cid:1) 1 (cid:7)(cid:3) P re s s u re ? (cid:6) FIG. 3: Reduced force gradient at V versus distance be- V a c u u m (0 .0 4 m b a r, 3 0 0 K ) tween two gold surfaces in sphere-planmeingeometry (measured (cid:5)(cid:4) 0 .1 H e liu m g a s (1 0 m b a r, 4 K ) (cid:2) at 300K). Experimental data are compared with the theo- (cid:1)1 0 0 ( c ) retical prediction of the real Casimir force and with a model R(cid:4) containing an additional long-range contribution scaling like G 1/z1.8. 1 0 (cid:1) C h a rg e ? (cid:1) 1 B e fo re m o d ific a tio n P ie z o s c re e n e d equipartition theorem is indeed not possible at 4K be- F ib e r m e ta liz e d causeofthedominantdetectionandbackactionnoises.40 0 .1 O x id e re m o v e d The measurement of G/R in sphere-plane geometry 1 0 0 ( d ) allows a direct comparison between experiment and the- ory within the so-called proximity force approximation 1 0 (PFA):50 (cid:1) (cid:1) G 1 dF F 1 ≡ =2π pp for z (cid:28)R (4) R R dz A S o lu tio n : B e fo re m o d ific a tio n where F /A is the force per unit area in plane-plane 0 .1 W ith a ll m o d ific a tio n s pp configuration, which is the quantity usually calculated 0 .1 1 z ( n m ) by theory. The determination of the free resonance frequency f 0 is a difficult but important issue, since it defines the ori- FIG.4: Analysisofthelong-rangeresidualforceatV show- gin of the frequency shift. This determination cannot be min ingthereducedforcegradientG/Rbetweentwogoldsurfaces done when the sample is further away from the probe measured at 300K in vacuum: (a) at two positions located than the scanner range (1.5µm), because using the step 1mm apart and on a different sample with the same probe; motor would cause slight changes of f due to mechani- 0 (b)atdifferenttemperaturesandgaspressures;(c)afterasin- cal vibrations that modify the system. In practice, f0 is gle change in the environment, either Faraday cages around determined just before starting the force measurements, piezoelectric elements, or gold coating of the optical fiber, or at the maximum scanner distance, and subsequently, we removal of the oxide layer covering the anodized chip holder; slightly adjust this value during the post-experimental (d)withthethreeabovemodificationsimplementedtogether. analysis to get a residual force going to zero at infinity. This small adjustment does not affect significantly the data below 300nm. searched-for Casimir force. The reduced force gradient G/Rcorrespondingto∆f (measuredat300K)isplot- min IV. SUPPRESSION OF THE LONG-RANGE ted in Fig. 3 as a function of the sphere-sample distance RESIDUAL FORCE z for two gold surfaces. The data are compared with the theoreticalpredictionforAu-AusurfacesusingtheDrude At the minimizing potential V , the residual fre- model.27 The force gradient in sphere-plane geometry is min quency shift ∆f should correspond a priori to the predicted to change from the 1/z3 short-range regime min 5 (van der Waals) to the 1/z4 long-range regime (Casimir) around the plasma wavelength of gold (136nm). It is clearly seen in the figure that the power law of the ex- perimentalforcegradientchangesinaverydifferentway, since the exponent is decreasing with distance instead of increasing. There is obviously an additional “parasitic” force, which overcomes the Casimir force at large dis- tance. By fitting the difference between data and theory at large distance using a power law with the exponent as a free parameter, we obtain a dependence scaling like 1/z1.8. The origin of this parasitic force could be the inho- mogeneity of the surface potential, which has been first identified by Speake and Tenkel36 as a source of resid- FIG.5: Photographshowingtheforceprobe,theopticalfiber, andthesamplemadeofahighlydopedsiliconsubstratepartly ual electrostatic force, which is not compensated at the covered by 150nm of gold (a third region is covered with a minimizing potential V . This inhomogeneity origi- min layer of silicon dioxide). nates from the random grain orientation of polycrys- talline films, with different contact potential on differ- ent crystal faces, or from an inhomogeneous layer of na- tive oxide, adsorbed contaminants, or chemical impuri- sult of the optomechanical noise discussed above in the ties. Anexperimenthasreportedsuchalong-rangeresid- paper. Finally, we analyzed the influence of the electro- ual force that could be attributed to this patch poten- staticenvironmentbytestingseparatelyafewchangesto tialeffect.25,51,52 Anotherexperiment,however,withtwo the setup, like covering the piezoelectric elements (scan- aluminum surfaces, could not explain the observed addi- neranddither)withgroundedFaradaycages,coatingthe tional force by the spatial distribution of the contact po- cladding of the optical fiber with gold, or removing the tential that was measured directly by Kelvin-probe force oxide layer of the anodized aluminum parts [Fig. 4(c)]. microscopy.53 In our case, the 1.8 power-law exponent Each change has only a small impact on the parasitic is larger than the value 1.44 obtained in Ref. 25 (0.72 force and none of them is able to cancel the parasitic for force gives 1.44 for force gradient), but is close to 2 force alone. After implementing all three changes si- as expected for a patch potential with small grains. In multaneously, the parasitic force has finally disappeared thisregime,theroot-mean-squarefluctuationsofthegold [Fig. 4(d)]. We therefore conclude that the origin of this contact potential in a granular film (Vrms = 90mV)36 forcewasprobablynotapatchpotentialeffect,butmore couldberesponsibleforanelectrostaticforcegradientas likelyaforceappliedbyresidualchargesindifferentparts large as G/R = π(cid:15)0Vr2ms/z2 = 0.2N/m2 for z = 1µm. ofthemicroscopehead. TheCasimirforcemeasurements This order of magnitude is compatible with the long- described in the next section have been performed in range force visible in Fig. 3 above 300nm. these conditions with a clean electrostatic environment. However, we discovered that this parasitic force could besuppressedafterseveralmodificationsofthemeasure- mentsetupandismoreprobablyduetotheelectrostatic V. RESULTS FOR GOLD-GOLD AND environment of the force probe. This conclusion is the GOLD-SILICON CAVITIES result of a detailed analysis of many experiments carried out in different situations as reported in Fig. 4. First, We now present our experimental results obtained at we checked the reproducibility of this parasitic force by 4.2Kwithagold-coatedforceprobeonasiliconsubstrate comparingtheresultsobtainedattwodifferentlocations partly covered with 150nm of gold. The objective is to on the same gold sample, and on a second gold sam- compare the Casimir force gradient measured with the ple [Fig. 4(a)]. Only small differences are visible be- same sphere on two different materials. We compare a tweenallthreecurves,withthesamelong-rangeresidual metalwithasemiconductorbecausethesematerialshave force, therefore ruling out sample specific artifacts, like very different electronic properties. The sample is made defects or inhomogeneities. Then, we tested the influ- of a heavily doped silicon substrate (1.5×1019At/cm3 ence of temperature and exchange gas used for cooling phosphorusdopingand4.2mΩcmresistivity)inorderto the microscope head, because the gas confinement be- keep the surface conducting at low temperature.57 A re- tween the sphere and surface could have produced an gion of the surface is then covered with 150nm of gold additional distance-dependent dissipation.54–56 By com- (e-beamevaporation)withasharptransitionwiththere- paringcurvesat300Kinvacuumwithcurvesat4.2Kin mainingpartofthesiliconsubstrate(Fig.5). Thetrans- helium gas [Fig. 4(b)], both showing the same additional lation stage (M2) is used to move the sample and place long-range component, we can rule out any significant the selected region in front of the sphere. The Casimir effect of temperature and surrounding gas. Note that forcecanthereforebemeasuredin situ onthetwomate- the larger noise on the low-temperature data is the re- rials, using a single force probe in a single environment 6 largedistance,theinteractionisaveragedonalargenum- ber of grains. Another explanation can be the pres- -5 0 ence of a smooth gradient of material work function D a ta A u -A u along the film.52 In this context, the relation V (z) = -5 5 D a ta A u -S i min M o d e l A u -A u a1log(z)+a2 was found to mimic the logarithm trend M o d e l A u -S i observed for two germanium surfaces25,51 and two gold -6 0 surfaces.58 Here,thefitofourAu-Audatawiththisrela- tion (dashed line on Fig. 6) is, however, not satisfactory -6 5 and we propose another model. ) Casimir force experiments are not the only ones to ev- V -7 0 m idence a distance dependence of the minimizing poten- (in tial. This effect is also observed in Kelvin-probe force m -7 5 V microscopy and a model was developed in this context byHadjadjet al.,37 whichtakesintoaccounttheinterac- -8 0 tionoftheprobewithitsentireenvironment. Byusinga -5 5 5 simple electrostatic model, these authors found that the presenceofmetallicobjectsinthesurroundinginfluences -5 6 0 the minimizing potential according to the relation b z -5 6 5 V (z)=V + 1 (5) 1 0 0 1 0 0 0 min c b +z z ( n m ) 2 where b and b are related to the electrostatic poten- 1 2 tial and capacitance of the environment, and V is the c FIG.6: EvolutionwithdistanceofthepotentialVmin applied contact potential obtained when z tends to zero. By fit- at 4.2K on the sample surface to minimize the electrostatic tingourAu-Audatawiththismodel,asshowninFig.6, force on the Au-Au and Au-Si sphere-plane cavities. Experi- we obtain V = −20±12mV, b = −60±12mV, and mentaldataarefittedbytheelectrostaticmodelEq.(5)(solid c 1 b = 40±13nm. This simple electrostatic model repro- lines) and by a logarithmic function (dashed line). 2 duces very well our experimental data and the contact potential is found very close to zero (considering the er- rorbar)asexpectedfortwoidenticalgoldsurfaces. This analysis demonstrates the influence played by the envi- (gasandtemperature),inordertocomparethedatawith ronment of the force probe on the minimizing potential better confidence than in separate experimental runs. and shows that V can usually not be assimilated to min the contact potential V at finite distance. c The same analysis has been applied on the data ob- A. Minimizing potential tained on the silicon surface and we obtain V =−565± c 4mV,b =−6±2mV,andb =300±800nm. Although 1 2 The potential V applied on the sample to minimize wecouldhaveexpectedasimilardependenceondistance min the electrostatic force is plotted in Fig. 6 as a function than for Au-Au because of the same environment, the of the sphere-plane separation. These values are stable minimizing potential is found to be almost constant for intimeanddonotdependonthepositionalongthesur- Au-Si. Anexplanationmightbethatthesamplehasbeen face. V is almost independent of the distance for the translated by a few millimeters to switch from gold to min silicon surface, around -560mV, but varies strongly with silicon, thereby slightly changing the environment. The distance for the gold surface, with an asymptote around constant V for Au-Si confirms that the variations ob- min -80mV at large distance. Since the contact potential V served above for Au-Au are not due to contact potential c is expected to be zero for identical surfaces like in the fluctuations, because we should also observe such varia- Au-Aucase,theseresultsshowthattheinterpretationof tions here, not due to the silicon surface, which is mono- V is more subtle. crystalline, but due to the contact potential fluctuations min Variations of the minimizing potential with distance over the gold-coated sphere. The microstructure of the have been observed previously in other Casimir force ex- goldfilmscouldbe,however,differentonthepolystyrene periments.16,25,26 This effect can be explained by the in- sphere and on the silicon substrate, making a definite homogeneous surface potential (called patch potential) conclusion difficult. induced by the random distribution of crystal orienta- tions in gold films made of interconnected grains, sev- eral tens of nanometers in diameter, with work function B. Casimir force fluctuations V ≈ 90mV.36 When the probe is close rms to the surface, the interaction area is small and more The Casimir force measured at 4.2K on the gold and sensitive to the local crystalline orientation, whereas at siliconregionsispresentedinFig.7togetherwiththethe- 7 1 0 0 1 .0 D a ta A u -A u D a ta u D a ta A u -S i T h e o ry T h e o ry A u -A u -A 0 .8 u (cid:6)(cid:7)(cid:3) 1 0 TP he erfoe rcyt Am uir-rSoirs i / A 0 .6 (cid:5)(cid:4) 1 0 0 -S GR(cid:4)(cid:1)(cid:2) 1 (cid:1)(cid:2)(cid:5)(cid:4)(cid:6)(cid:7)(cid:3) (cid:1) tio : Au 0 .4 r (%)100 (cid:4)GR ra 0 .2 Erro 1 4 K R/ -10 1 0 0 0 .9 z (n m ) 2 0 0 G 100 150 200 0 .1 0 .0 1 0 0 2 0 0 3 0 0 4 0 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 z ( n m ) z ( n m ) FIG.7: ReducedCasimirforcegradientG/Rversusdistance FIG.8: RatiobetweentheforcegradientsmeasuredforAu-Si z measuredat4.2Kbetweenagoldsphereandagoldsurface andAu-Aucavities,comparedwiththetheoreticalprediction. oradopedsiliconsurface. Dataarecomparedwiththeoretical Inset: relative error between experiment and theory. predictionsforAu-AuandAu-Sicavities. Thecaseofperfect mirrors is shown by a dashed line. Inset: same data G/R plotted for distances artificially reduced by a factor 0.9 to show that a systematic error in the calibration might be the origin of the discrepancy between experiment and theory. this hypothesis could not be checked at the time of the experiment, we stop here the discussion on the absolute comparisonbetweenexperimentandtheory,andnowdis- oretical predictions calculated for these specific sample- cuss the relative value obtained between gold and silicon probe configurations using the Drude model (see the surfaces. Appendix).27,31,32 It is clearly seen that the measured Casimir force is weaker on doped silicon than on gold, TheratiooftheAu-SiovertheAu-Auforcegradientis as predicted by theory. The experimental data are, how- plottedinFig.8foraseriesofdistanceswheretheexper- ever,abovethetheoreticalcurvesby50%,i.e.,muchmore imentalforcegradientshavebeendeterminedbyinterpo- than the error in the force calibration factor β, which is lation. The ratio is lower than unity as expected for a better than 1%. Recently, computations of the Casimir cavitywithasemiconductorplate,whichisopticallyless force59,60 have emphasized the sensitivity of the results reflecting than gold. The ratio decreases progressively to the choice of the materials optical data61,62 used in withdistanceasalsoexpectedfromtheory,32 withacor- the calculations: for gold mirrors, the uncertainty is, rection factor η , which saturates at large distance to a F however, only of about 5%. The validity of the PFA is lowervalueforAu-SithanforAu-Au. Quantitatively,the another important assumption in the theory-experiment experimentalratioisofthesameorderasthetheoretical comparison:63,64theerrorshouldbesmallerthan1%here value at short distance, with an error less than 10% in sincez/R<1%.22 Thelargediscrepancybetweentheory the 100–200nm distance range, but the ratio decreases and experiment regarding the absolute value of the force faster with distance than predicted by theory. These re- gradient requires, therefore, another explanation. sults show that, although the absolute comparison with A possible source of error being the calibration of the theory is not possible here, the material dependence of scanner extension, we found that multiplying the dis- theCasimirforceisclearlyevidencedwhenthesurfaceis tance z of each data point by a factor 0.9 translates the changed from gold to silicon. data points onto the theoretical curves as shown in the inset of Fig. 7. Since the force calibration is dependent To improve this experiment in the future, the scan- on the scanner calibration, it is in fact a factor 0.85 that ner extension should be measured by interferometry in should be applied on the relative distance (before the situ during the measurement to avoid the effect of ther- determination of β) in order to shift the data onto the mal cycles on the scanner piezoelectric coefficient. The theoretical curves. The piezoelectric z-scanner was cal- detectionsensitivitycouldbealsoimprovedbystabiliza- ibrated by interferometry nine months before the force tion of the laser intensity down to the shot noise level, measurements reported here and it could be that the inordertominimizethedetectionandbackactionnoises, scanner extension has been progressively reduced after and take advantage of the strongly suppressed thermo- successive thermal cycles between 300 and 4.2K. Since mechanical noise at 4.2K. 8 VI. CONCLUSION to this work and for helpful discussion. This work has benefited from fruitful discussions with P. Andreucci, L. From an instrumental point of view, we have shown Duraffourg, J-J. Greffet, G. Jourdan, and A. Lambrecht. that the presence of a long-range parasitic force at the The samples and probes were prepared in the Nanofab minimizing potential can be related to the electrostatic clean-room facility. This research was supported by the environment of the force probe. Precise measurements PNANO 2006 program of the Agence Nationale de la of the Casimir force therefore require an accurate con- Recherche under the project name MONACO. trol of the environment, like screening of every insulat- ing part close to the probe: chip holder, optical fiber, and piezoelectric actuators. Regarding the minimizing potential, the variations with distance observed for the Au-Au cavity could be explained by a model taking into Appendix account the electrostatic potential of the environment, and the absence of variation for the Au-Si cavity indi- cates that there is no patch potential effect on the gold- The Casimir force is computed for parallel plates of coated sphere. Finally, we have shown that our in situ infinitethicknessesbytakingintoaccounttherealmate- measurementoftheCasimirforceusingasinglespherical rialpropertiesasexplainedinRefs.31,32. Thedielectric probe(gold)andtwodifferentsurfaces(goldordopedsil- constant (cid:15)r is modeled by a plasma frequency ωp and icon) gives qualitatively the correct value for the relative a Drude relaxation parameter γp, plus a Lorentz func- force gradient, although the absolute values are not cor- tion with resonance frequency ω0, relaxation parameter rect due to a systematic error that might be attributed γ0,andsusceptibilityχ0,describinginterbandtransitions to the scanner calibration. for gold and intrinsic response for silicon: The sensitivity of the Casimir force to material prop- erties, as demonstrated here, could be used for surface ω2 χ ω2 characterization in a new type of non-contact scanning (cid:15) (iω)=1+ p + 0 0 (6) force microscopy. Such a technique would be the exten- r ω(ω+γp) ω2+ω02+ωγ0 sion of the near-field van der Waals force microscopy to the retarded Casimir regime at large separation. For a TheparametersusedinthecomputationarelistedinTa- given force sensitivity, measuring at large distances im- ble I. We checked that our computation algorithm gives plies the use of micron-size spherical probes with a lower the correct results for the well-known Au-Au cavity, be- spatial resolution than the sharp tips used in atomic fore computing the force for our specific Au-Si cavity. force microscopy, but provides information on the op- tical properties of the materials. With the Casimir force being obtained at the minimizing electrostatic potential, this technique would be complementary to the measure- TABLEI:Parametersofthedielectricfunctionforgold(Au) ment of the contact potential by Kelvin-probe force mi- and silicon (Si). croscopy. ω γ ω γ χ p p 0 0 0 Acknowledgments (1015rad/s) (1015rad/s) (1015rad/s) (1015rad/s) Au 13.7 0.05 20 25 5 The low-temperature AFM was designed and built by Si 0.37 0.052 6.6 0 10.87 J-F.Motte. WethankT.Ouisseforanearlycontribution 1 K.L. 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Casimir force measurements in Au-Au and Au-Si cavities at low temperature PDF

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Casimir force measurements in Au-Au and Au-Si cavities at low temperature J. Laurent, H. Sellier, A. Mosset, S. Huant, and J. Chevrier Institut N´eel, CNRS et Universit´e Joseph Fourier, B.P. 166, F-38042 Grenoble, Cedex 9, France (Dated: January 16, 2012) WereportonmeasurementsoftheCasimirforceinasphere-planegeometryusingacryogenicforce microscope to move the force probe in situ over different materials. We show how the electrostatic environmentoftheinteractingsurfacesplaysanimportantroleinweakforcemeasurementsandcan overcometheCasimirforceatlargedistance. Afterminimizingtheseparasiticforces,wemeasurethe Casimirforcebetweenagold-coatedsphereandeitheragold-coatedoraheavilydopedsiliconsurface in the 100–400nm distance range. We compare the experimental data with theoretical predictions 2 and discuss the consequence of a systematic error in the scanner calibration on the agreement 1 between experiment and theory. The relative force over the two surfaces compares favorably with 0 theory at short distance, showing that this Casimir force experiment is sensitive to the dielectric 2 properties of the interacting surfaces. n PACSnumbers: 07.10.Pz,12.20.Fv,73.40.Cg,78.20.Ci a J 3 I. INTRODUCTION the experiments has been continuously improved, using 1 metallic or semiconductor materials like Au-Au,7,9,14–19 ] Nanoelectromechanical systems (NEMS) are used in a Al-Al,8,12 Cr-Cr,11 Au-Cu,10 Au-Ge,20 Au-Si,21–23 Au- l Si grating,24 Ge-Ge,25 and Au-indium tin oxide.26 In l broadening range of applications, such as actuators, sen- a parallel, the influences of layer thickness,27,28 surface h sors, resonators, or modern nanocharacterization tools. roughness,29gratingstructure,30andmaterialconductiv- - Size reduction not only allows for shrinking the energy s ity31–33havebeenstudiedtheoreticallytoprovidemodels consumption and for shortening the response time, but e for comparison or stimulate new experiments. m it also allows integrating a broader range of functionali- ties on a single chip.1 However, quantum physics comes In this paper, we report on a detailed study of the . t into play at the nanoscale and can affect NEMS behav- Casimir force between a gold-coated sphere and a doped a m ior.2 Understanding these effects, and possibly control- siliconsubstrateatliquidheliumtemperature(4.2K)and ling them, is a necessary prerequisite to optimize NEMS compare it in situ with the case of a gold-coated surface. - d design. In turn, new applications driven by quantum ef- The use of the sphere-plane geometry avoids the chal- n fects can emerge, in particular in the field of ultra-high- lenge of controlling with high accuracy the parallelism o sensitivity force or displacement detection. of two flat surfaces separated by a submicron gap. Our c TheCasimirforce,discoveredin1948,isthearchetypi- aim is to reveal the dependence of the force on the ma- [ calforceinthisframework.3Itspurelyquantumoriginre- terials properties and to compare it quantitatively with 1 sults from the zero-point fluctuations in the electromag- theory. Thermalization at low temperature provides an v netic field. Since its theoretical prediction, the Casimir exceptional mechanical stability of the interacting sur- 5 force has attracted the interest of a large community of faces, which is highly beneficial for long-term force mea- 8 scientists ranging from cosmologists4 to NEMS design- surements. In principle, it should also improve the force 7 2 ers5 through solid-state physicists. Experimentally, the sensitivity because of a reduced Brownian motion of the . first confirmation6 of the Casimir effect was reported as cantilever,butothereffects,suchasoptomechanicalcou- 1 early as 1958, but the first quantitative study7 of the plings,degradetheexpectedincreaseinsensitivityinour 0 2 Casimirforceusingatorsionpendulumwasreportednot experiment. Cooling down the experiment34 also sup- 1 before 1997. Soon after, an important activity has been presses the thermal contribution of the Casimir force,35 : triggered thanks to the use of atomic force microscopes allowing a direct comparison with the zero-temperature v (AFM)8 ormicroelectromechanicalsystems(MEMS).9,10 theoretical calculations. i X Most of the experiments have been carried out with a After a complete description of our calibration pro- r cavity in the sphere-plane geometry and very few in the cedure, we show that parasitic forces can perturb sig- a plane-plane geometry11,12 that requires highly parallel nificantly the Casimir force measurements and that the surfaces. setup environment can be modified to suppress this ar- ThelimitednumberofgroupsworkingonCasimirforce tifact. We then discuss the variations of the minimizing measurements confirms how difficult these experiments potentialwithdistancebyconsideringfirstthepatchpo- are.13 This is explained by the small magnitude of this tential effect,36 and then a simple electrostatic model37 force as compared to the electrostatic force, and by the that reproduces the data. Finally, we present relative stronger distance dependence scaling like the inverse of measurements of the Casimir force in the 100–400nm the fourth (third) power of the distance in plane-plane distance range obtained by changing in situ the sample (sphere-plane) geometry. In order to check the validity surface from gold to silicon. The relative force between oftheoriesdescribingfundamentalforces,theprecisionof thetwomaterialsisinqualitativeagreementwiththeory, 2 sphere side and 80nm on the backside) to provide an electriccontacttothesphere,whichisoneoftheCasimir mirror. Theroot-mean-squareroughnessofthegoldsur- faces is around 3nm, as measured by AFM. The probes havetypicallyaresonancefrequencyf about40kHzand 0 a spring constant k about 10N/m. The cantilever chip is glued with silver paint on a holder (G) made of an- odized aluminum and then fixed on a long holder (H). The cantilever is mechanically excited at resonance by a piezoelectric dither (I). The sample (B) is mounted with silver paint on a holder (D) that is separated from the piezoelectric z-scanner (F) by a grounded aluminum plate (E) for electrostatic screening of the high voltages applied to the scanner. The cantilever motion is mea- sured with a compact optical detection compatible with the severe space constraints of cryogenics,39 using the interferometric cavity formed by the flexible cantilever and the extremity of a single-mode optical fiber (C) an- chored to the holder (J). The fiber is positioned above the end of the lever with a set of XYZ cryogenic inertial FIG. 1: Drawing and photograph of the microscope head at motors (M1) and adjusted such as to obtain an interfer- the bottom part of the cage structure. The functional parts ometric cavity with good displacement sensitivity. The of the microscope, labeled by capital letters, are described in sampleisapproachedbelowtheforceprobewithanother the text. set of motors (M2) and the scanner (F) is used to finely tune the gap between the two surfaces. The scanner has been calibrated by interferometry and the hysteresis has but the absolute values of the force show a systematic beendeterminedfor definedscannerextensions. Itcould error with respect to the theoretical predictions that are be, however, thatthiscalibrationslightlyevolvesintime tentatively attributed to an aging of the scanner calibra- after successive thermal cycles as discussed later in the tion. analysis of the results. The paper is organized as follows. Section II describes Themicroscopeandtheentirecagestructurearesealed the homemade low-temperature force microscope. Sec- into a 2-in.-diameter stainless steel tube evacuated to a tion III explains the force measurements and data anal- secondaryvacuumandflushedwithheliumgas. Thetube ysis. SectionIVdiscussestheoriginandsuppressionofa is then filled with a low pressure of helium exchange gas long-rangeresidualforceduetotheelectrostaticenviron- (10mbaratroomtemperature)andimmersedinaliquid ment. Section V presents the results on the minimizing helium cryostat. During cooling down, it is necessary potential and Casimir force obtained for gold and silicon to continuously readjust the optical cavity with the M1 surfaces. Conclusions are drawn in Section VI. motors to compensate for thermal contractions. Measurements at low temperature have the advan- tage to benefit from strongly reduced thermal drifts and II. LOW-TEMPERATURE FORCE thermomechanical noises that usually limit the room- MICROSCOPE temperature experiments. For instance, position drifts of about 1nm/min at 300K are found to be reduced to We developed a new force microscope working in a less than 1nm/h at 4K. This is of particular importance cryogenic environment at 4.2K, as an evolution of the in the present study because the Casimir force strongly room-temperature instrument developed by Jourdan et depends on distance. In the same way, the frequency al.18 The structure of the low-temperature instrument drift of the cantilever resonance is strongly suppressed takes the shape of a 50-mm-diameter and 120-cm-long from 3mHz/min at 300K down to a negligible value at modular system based on a tubular cage.38 This cage 4K.Finally,anotheradvantageofcryogenictemperature structure links the top of the instrument (that bears all is to strongly suppress the cantilever Brownian motion the electrical and optical connections) to the microscope induced by thermomechanical force fluctuations. head located at the very bottom. The main parts of In such cryogenic conditions, the force detection sen- the microscope (marked by capital letters in Fig. 1) are sitivity is essentially limited by the optical readout of described below. the cantilever position. The intensity fluctuations of the The force probe is based on an AFM cantilever with a laser beam are here the main source of noise, well above 40-µm-diameter polystyrene sphere fixed at the extrem- thenoiseofthephotodiodeanditsamplifier. Inparticu- ity with standard epoxy glue (A). The sphere and can- lar,optomechanicalcouplingslikeradiationpressureand tilever are coated with gold (more than 200nm on the photothermalstressconvertthisintensitynoiseintocan- 3 nance. The lever is excited with a piezoelectric dither at (cid:1) its mechanical resonance and the lever vibration is mea- (a ) suredbyinterferometrywiththeopticalfiber. Theoscil- 0 lation amplitude and phase are recorded with a lock-in (cid:7)(cid:3)-1 andaphase-lockedlooptrackstheresonancefrequencyf (cid:5) (cid:1)(cid:2)f (cid:1) whentheprobeissubmittedtoaforcegradient. Theres- D -2 onance frequency shift is defined by ∆f =f −f , where 0 -3 (cid:16)(cid:30)(cid:31)"(cid:22)(cid:31)(cid:24) (cid:14)(cid:22)(cid:23)(cid:27)"(cid:22)(cid:31)(cid:24) f0 is the free resonance frequency. Since the zero sphere-plane distance cannot be de- 3 6 9 1 2 1 5 1 8 (cid:18)(cid:26)(cid:28)(cid:25)(cid:1)(cid:20)(cid:28)(cid:26)(cid:29)(cid:21) (cid:1) (cid:1) termined by bringing the sample into contact with the 0 (b ) 0 (c ) sphere, which would irreversibly damage the gold coat- -1 -5 0 Df(cid:1)(cid:2)(cid:5)(cid:7)(cid:3) ----5432 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)z(cid:1)(cid:1)(cid:9)(cid:8) (cid:23)(cid:6)(cid:6)(cid:22)(cid:12)(cid:6)(cid:29)(cid:1)(cid:1)(cid:29)(cid:29)(cid:28)(cid:28) (cid:1)C(cid:1)(cid:2)(cid:5)(cid:7)(cid:4)(cid:6)(cid:8)(cid:3)---211 050 000 b (cid:1)(cid:1)(cid:13)(cid:1)(cid:7)(cid:5)(cid:11)(cid:10)(cid:1)(cid:1)(cid:1)(cid:15)(cid:16)(cid:17)(cid:26)(cid:22)!#!(cid:1)(cid:22)$(cid:28)(cid:8)(cid:1)(cid:19)(cid:1)(cid:4)(cid:8) (cid:1) ibmnyegaeosluefcrttehrmoesestnuattrsif,cactcheasel,ibstarhametpiaolbnes.oislDuatuperpidnriogsatacahnsecedequitseondtceheteeorfsmpfhionerecrdee --76 (cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:1)(cid:9)(cid:12)(cid:10)(cid:6)(cid:1)(cid:1)(cid:29)(cid:29)(cid:28)(cid:28) --32 05 00 z(cid:6)(cid:1)(cid:13)(cid:1)(cid:4)(cid:8)(cid:10)(cid:1)(cid:29)(cid:28) rbeysosnmaanlclestferpeqs,ueanncdyfoshrifetac∆hfsciasnmneeraspuorseidtiofnorzsdciaffne,rtehnet -2 0 0 -1 0 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 biasvoltagesV appliedtothesamplewithrespecttothe V (cid:1)(cid:2)(cid:28)(cid:19)(cid:3) z (cid:1)(cid:2)(cid:29)(cid:28)(cid:3) (cid:23)(cid:22)(cid:29) grounded sphere [Fig. 2(a)]. The voltage is varied typ- ically over ±200mV around the potential V , which min minimizes the electrostatic force between the probe and FIG.2: (a)Frequencyshift∆f ofthecantileverresonanceasa functionofthetimeelapsedduringaforward(left)andback- sample. We obtain a series of ∆f(V) curves [Fig. 2(b)], ward (right) scan of the sample toward the sphere fixed on which are fitted by the second-order polynomial: thecantilever. Forwardandbackwardscansarenotperfectly symmetricbecauseofthescannerhysteresis. (b)Examplesof ∆f =C(V −V )2+∆f (1) min min ∆f(V)parabolasrecordedbysweepingtheprobe-surfacebias V fordifferentscannerpositionszscan. (c)Fitoftheparabola where C, Vmin, and ∆fmin are three adjustable parame- curvaturesC versusscannerpositionzscan usedtoobtainthe ters. The first term on the right-hand side corresponds force calibration factor β and the position z of the contact. 0 to the capacitive force, and the second term is the fre- quency shift due to the remaining forces, including the Casimir force, obtained for V = V at the summit of min the parabola. The curvature coefficient C is plotted as a tilever displacement noise, called backaction noise. As function of the scanner position z and fitted with scan a consequence, the low-temperature force sensitivity is found to be√of the same or√der as the room-temperature β sensitivity S ≈ 10fN/ Hz. This situation could be C = (2) FF (z −z )2 scan 0 improved by a broadband stabilization of the laser beam intensity or by the coherent coupling of laser noise and to obtain the sphere-sample contact position z and the 0 backaction noise, as demonstrated recently in our inter- force probe calibration parameter β [Fig. 2(c)]. The ab- ferometric setup.40 Therefore the only, but very reward- solute distance between sphere and sample is then given ing, advantage of the low temperature turns out to be by z =z −z . We have additionally checked that the scan 0 the exceptional mechanical stability of the microscope. cantilever static deflection generated by the electrostatic These retarded optomechanical forces also modify the force and the Casimir force is negligible in the studied resonance frequency and damping rate of the microlever separation range.49 through an optical spring effect induced by the inter- The theoretical expression of the sphere-plane capaci- ferometric process.41–43 Depending on the optical cavity tiveforcegradientgivesβ =−f0Rπ(cid:15)0,wheref isthefree detuning, this effect can induce self-oscillations44,45 or resonance frequency, R the spher2ekradius, (cid:15) t0he vacuum 0 provide self-cooling of the thermal noise.46–48 The detec- permittivity, and k the cantilever stiffness. The exper- tionconditionshavethusbeenoptimizedbychoosingthe imental value of β extracted from the fit can therefore cooling side of the detuning and by adjusting the laser be used to transform the measured frequency shift ∆f power. into a “reduced force gradient” G/R without any other parameter:18 G 2k π(cid:15) III. DATA ACQUISITION AND CALIBRATION =− ∆f = 0∆f (3) R f R β 0 Instead of measuring directly the electrostatic or Thiselectrostaticcalibrationoftheforceprobeusingthe Casimir force F(z) in static mode, we measure its force only parameter β is more relevant than the precise mea- gradient G(z)= dF in dynamic mode, which is given by surement of R and k. The traditional determination of dz the frequency shift ∆f = −f0G of the cantilever reso- k based on the thermal noise spectral density and the 2k 4 1 0 0 1 0 0 D a ta ( a ) T h e o ry 1 /z 3 1 0 (cid:7)(cid:3) 1 0 1 /z 4 (cid:6) (cid:1) M o d e l (cid:1)(cid:2)(cid:5)(cid:4) 1 /z 1.8 (cid:1) 1 D e fe c St a? m p le 1 , p o s itio n 1 R(cid:4) 1 S a m p le 1 , p o s itio n 2 0 .1 S a m p le 2 G 1 0 0 ( b ) 3 0 0 K 0 .1 1 0 1 0 0 1 0 0 0 z ( n m ) (cid:1) (cid:1) 1 (cid:7)(cid:3) P re s s u re ? (cid:6) FIG. 3: Reduced force gradient at V versus distance be- V a c u u m (0 .0 4 m b a r, 3 0 0 K ) tween two gold surfaces in sphere-planmeingeometry (measured (cid:5)(cid:4) 0 .1 H e liu m g a s (1 0 m b a r, 4 K ) (cid:2) at 300K). Experimental data are compared with the theo- (cid:1)1 0 0 ( c ) retical prediction of the real Casimir force and with a model R(cid:4) containing an additional long-range contribution scaling like G 1/z1.8. 1 0 (cid:1) C h a rg e ? (cid:1) 1 B e fo re m o d ific a tio n P ie z o s c re e n e d equipartition theorem is indeed not possible at 4K be- F ib e r m e ta liz e d causeofthedominantdetectionandbackactionnoises.40 0 .1 O x id e re m o v e d The measurement of G/R in sphere-plane geometry 1 0 0 ( d ) allows a direct comparison between experiment and the- ory within the so-called proximity force approximation 1 0 (PFA):50 (cid:1) (cid:1) G 1 dF F 1 ≡ =2π pp for z (cid:28)R (4) R R dz A S o lu tio n : B e fo re m o d ific a tio n where F /A is the force per unit area in plane-plane 0 .1 W ith a ll m o d ific a tio n s pp configuration, which is the quantity usually calculated 0 .1 1 z ( n m ) by theory. The determination of the free resonance frequency f 0 is a difficult but important issue, since it defines the ori- FIG.4: Analysisofthelong-rangeresidualforceatV show- gin of the frequency shift. This determination cannot be min ingthereducedforcegradientG/Rbetweentwogoldsurfaces done when the sample is further away from the probe measured at 300K in vacuum: (a) at two positions located than the scanner range (1.5µm), because using the step 1mm apart and on a different sample with the same probe; motor would cause slight changes of f due to mechani- 0 (b)atdifferenttemperaturesandgaspressures;(c)afterasin- cal vibrations that modify the system. In practice, f0 is gle change in the environment, either Faraday cages around determined just before starting the force measurements, piezoelectric elements, or gold coating of the optical fiber, or at the maximum scanner distance, and subsequently, we removal of the oxide layer covering the anodized chip holder; slightly adjust this value during the post-experimental (d)withthethreeabovemodificationsimplementedtogether. analysis to get a residual force going to zero at infinity. This small adjustment does not affect significantly the data below 300nm. searched-for Casimir force. The reduced force gradient G/Rcorrespondingto∆f (measuredat300K)isplot- min IV. SUPPRESSION OF THE LONG-RANGE ted in Fig. 3 as a function of the sphere-sample distance RESIDUAL FORCE z for two gold surfaces. The data are compared with the theoreticalpredictionforAu-AusurfacesusingtheDrude At the minimizing potential V , the residual fre- model.27 The force gradient in sphere-plane geometry is min quency shift ∆f should correspond a priori to the predicted to change from the 1/z3 short-range regime min 5 (van der Waals) to the 1/z4 long-range regime (Casimir) around the plasma wavelength of gold (136nm). It is clearly seen in the figure that the power law of the ex- perimentalforcegradientchangesinaverydifferentway, since the exponent is decreasing with distance instead of increasing. There is obviously an additional “parasitic” force, which overcomes the Casimir force at large dis- tance. By fitting the difference between data and theory at large distance using a power law with the exponent as a free parameter, we obtain a dependence scaling like 1/z1.8. The origin of this parasitic force could be the inho- mogeneity of the surface potential, which has been first identified by Speake and Tenkel36 as a source of resid- FIG.5: Photographshowingtheforceprobe,theopticalfiber, andthesamplemadeofahighlydopedsiliconsubstratepartly ual electrostatic force, which is not compensated at the covered by 150nm of gold (a third region is covered with a minimizing potential V . This inhomogeneity origi- min layer of silicon dioxide). nates from the random grain orientation of polycrys- talline films, with different contact potential on differ- ent crystal faces, or from an inhomogeneous layer of na- tive oxide, adsorbed contaminants, or chemical impuri- sult of the optomechanical noise discussed above in the ties. Anexperimenthasreportedsuchalong-rangeresid- paper. Finally, we analyzed the influence of the electro- ual force that could be attributed to this patch poten- staticenvironmentbytestingseparatelyafewchangesto tialeffect.25,51,52 Anotherexperiment,however,withtwo the setup, like covering the piezoelectric elements (scan- aluminum surfaces, could not explain the observed addi- neranddither)withgroundedFaradaycages,coatingthe tional force by the spatial distribution of the contact po- cladding of the optical fiber with gold, or removing the tential that was measured directly by Kelvin-probe force oxide layer of the anodized aluminum parts [Fig. 4(c)]. microscopy.53 In our case, the 1.8 power-law exponent Each change has only a small impact on the parasitic is larger than the value 1.44 obtained in Ref. 25 (0.72 force and none of them is able to cancel the parasitic for force gives 1.44 for force gradient), but is close to 2 force alone. After implementing all three changes si- as expected for a patch potential with small grains. In multaneously, the parasitic force has finally disappeared thisregime,theroot-mean-squarefluctuationsofthegold [Fig. 4(d)]. We therefore conclude that the origin of this contact potential in a granular film (Vrms = 90mV)36 forcewasprobablynotapatchpotentialeffect,butmore couldberesponsibleforanelectrostaticforcegradientas likelyaforceappliedbyresidualchargesindifferentparts large as G/R = π(cid:15)0Vr2ms/z2 = 0.2N/m2 for z = 1µm. ofthemicroscopehead. TheCasimirforcemeasurements This order of magnitude is compatible with the long- described in the next section have been performed in range force visible in Fig. 3 above 300nm. these conditions with a clean electrostatic environment. However, we discovered that this parasitic force could besuppressedafterseveralmodificationsofthemeasure- mentsetupandismoreprobablyduetotheelectrostatic V. RESULTS FOR GOLD-GOLD AND environment of the force probe. This conclusion is the GOLD-SILICON CAVITIES result of a detailed analysis of many experiments carried out in different situations as reported in Fig. 4. First, We now present our experimental results obtained at we checked the reproducibility of this parasitic force by 4.2Kwithagold-coatedforceprobeonasiliconsubstrate comparingtheresultsobtainedattwodifferentlocations partly covered with 150nm of gold. The objective is to on the same gold sample, and on a second gold sam- compare the Casimir force gradient measured with the ple [Fig. 4(a)]. Only small differences are visible be- same sphere on two different materials. We compare a tweenallthreecurves,withthesamelong-rangeresidual metalwithasemiconductorbecausethesematerialshave force, therefore ruling out sample specific artifacts, like very different electronic properties. The sample is made defects or inhomogeneities. Then, we tested the influ- of a heavily doped silicon substrate (1.5×1019At/cm3 ence of temperature and exchange gas used for cooling phosphorusdopingand4.2mΩcmresistivity)inorderto the microscope head, because the gas confinement be- keep the surface conducting at low temperature.57 A re- tween the sphere and surface could have produced an gion of the surface is then covered with 150nm of gold additional distance-dependent dissipation.54–56 By com- (e-beamevaporation)withasharptransitionwiththere- paringcurvesat300Kinvacuumwithcurvesat4.2Kin mainingpartofthesiliconsubstrate(Fig.5). Thetrans- helium gas [Fig. 4(b)], both showing the same additional lation stage (M2) is used to move the sample and place long-range component, we can rule out any significant the selected region in front of the sphere. The Casimir effect of temperature and surrounding gas. Note that forcecanthereforebemeasuredin situ onthetwomate- the larger noise on the low-temperature data is the re- rials, using a single force probe in a single environment 6 largedistance,theinteractionisaveragedonalargenum- ber of grains. Another explanation can be the pres- -5 0 ence of a smooth gradient of material work function D a ta A u -A u along the film.52 In this context, the relation V (z) = -5 5 D a ta A u -S i min M o d e l A u -A u a1log(z)+a2 was found to mimic the logarithm trend M o d e l A u -S i observed for two germanium surfaces25,51 and two gold -6 0 surfaces.58 Here,thefitofourAu-Audatawiththisrela- tion (dashed line on Fig. 6) is, however, not satisfactory -6 5 and we propose another model. ) Casimir force experiments are not the only ones to ev- V -7 0 m idence a distance dependence of the minimizing poten- (in tial. This effect is also observed in Kelvin-probe force m -7 5 V microscopy and a model was developed in this context byHadjadjet al.,37 whichtakesintoaccounttheinterac- -8 0 tionoftheprobewithitsentireenvironment. Byusinga -5 5 5 simple electrostatic model, these authors found that the presenceofmetallicobjectsinthesurroundinginfluences -5 6 0 the minimizing potential according to the relation b z -5 6 5 V (z)=V + 1 (5) 1 0 0 1 0 0 0 min c b +z z ( n m ) 2 where b and b are related to the electrostatic poten- 1 2 tial and capacitance of the environment, and V is the c FIG.6: EvolutionwithdistanceofthepotentialVmin applied contact potential obtained when z tends to zero. By fit- at 4.2K on the sample surface to minimize the electrostatic tingourAu-Audatawiththismodel,asshowninFig.6, force on the Au-Au and Au-Si sphere-plane cavities. Experi- we obtain V = −20±12mV, b = −60±12mV, and mentaldataarefittedbytheelectrostaticmodelEq.(5)(solid c 1 b = 40±13nm. This simple electrostatic model repro- lines) and by a logarithmic function (dashed line). 2 duces very well our experimental data and the contact potential is found very close to zero (considering the er- rorbar)asexpectedfortwoidenticalgoldsurfaces. This analysis demonstrates the influence played by the envi- (gasandtemperature),inordertocomparethedatawith ronment of the force probe on the minimizing potential better confidence than in separate experimental runs. and shows that V can usually not be assimilated to min the contact potential V at finite distance. c The same analysis has been applied on the data ob- A. Minimizing potential tained on the silicon surface and we obtain V =−565± c 4mV,b =−6±2mV,andb =300±800nm. Although 1 2 The potential V applied on the sample to minimize wecouldhaveexpectedasimilardependenceondistance min the electrostatic force is plotted in Fig. 6 as a function than for Au-Au because of the same environment, the of the sphere-plane separation. These values are stable minimizing potential is found to be almost constant for intimeanddonotdependonthepositionalongthesur- Au-Si. Anexplanationmightbethatthesamplehasbeen face. V is almost independent of the distance for the translated by a few millimeters to switch from gold to min silicon surface, around -560mV, but varies strongly with silicon, thereby slightly changing the environment. The distance for the gold surface, with an asymptote around constant V for Au-Si confirms that the variations ob- min -80mV at large distance. Since the contact potential V served above for Au-Au are not due to contact potential c is expected to be zero for identical surfaces like in the fluctuations, because we should also observe such varia- Au-Aucase,theseresultsshowthattheinterpretationof tions here, not due to the silicon surface, which is mono- V is more subtle. crystalline, but due to the contact potential fluctuations min Variations of the minimizing potential with distance over the gold-coated sphere. The microstructure of the have been observed previously in other Casimir force ex- goldfilmscouldbe,however,differentonthepolystyrene periments.16,25,26 This effect can be explained by the in- sphere and on the silicon substrate, making a definite homogeneous surface potential (called patch potential) conclusion difficult. induced by the random distribution of crystal orienta- tions in gold films made of interconnected grains, sev- eral tens of nanometers in diameter, with work function B. Casimir force fluctuations V ≈ 90mV.36 When the probe is close rms to the surface, the interaction area is small and more The Casimir force measured at 4.2K on the gold and sensitive to the local crystalline orientation, whereas at siliconregionsispresentedinFig.7togetherwiththethe- 7 1 0 0 1 .0 D a ta A u -A u D a ta u D a ta A u -S i T h e o ry T h e o ry A u -A u -A 0 .8 u (cid:6)(cid:7)(cid:3) 1 0 TP he erfoe rcyt Am uir-rSoirs i / A 0 .6 (cid:5)(cid:4) 1 0 0 -S GR(cid:4)(cid:1)(cid:2) 1 (cid:1)(cid:2)(cid:5)(cid:4)(cid:6)(cid:7)(cid:3) (cid:1) tio : Au 0 .4 r (%)100 (cid:4)GR ra 0 .2 Erro 1 4 K R/ -10 1 0 0 0 .9 z (n m ) 2 0 0 G 100 150 200 0 .1 0 .0 1 0 0 2 0 0 3 0 0 4 0 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 z ( n m ) z ( n m ) FIG.7: ReducedCasimirforcegradientG/Rversusdistance FIG.8: RatiobetweentheforcegradientsmeasuredforAu-Si z measuredat4.2Kbetweenagoldsphereandagoldsurface andAu-Aucavities,comparedwiththetheoreticalprediction. oradopedsiliconsurface. Dataarecomparedwiththeoretical Inset: relative error between experiment and theory. predictionsforAu-AuandAu-Sicavities. Thecaseofperfect mirrors is shown by a dashed line. Inset: same data G/R plotted for distances artificially reduced by a factor 0.9 to show that a systematic error in the calibration might be the origin of the discrepancy between experiment and theory. this hypothesis could not be checked at the time of the experiment, we stop here the discussion on the absolute comparisonbetweenexperimentandtheory,andnowdis- oretical predictions calculated for these specific sample- cuss the relative value obtained between gold and silicon probe configurations using the Drude model (see the surfaces. Appendix).27,31,32 It is clearly seen that the measured Casimir force is weaker on doped silicon than on gold, TheratiooftheAu-SiovertheAu-Auforcegradientis as predicted by theory. The experimental data are, how- plottedinFig.8foraseriesofdistanceswheretheexper- ever,abovethetheoreticalcurvesby50%,i.e.,muchmore imentalforcegradientshavebeendeterminedbyinterpo- than the error in the force calibration factor β, which is lation. The ratio is lower than unity as expected for a better than 1%. Recently, computations of the Casimir cavitywithasemiconductorplate,whichisopticallyless force59,60 have emphasized the sensitivity of the results reflecting than gold. The ratio decreases progressively to the choice of the materials optical data61,62 used in withdistanceasalsoexpectedfromtheory,32 withacor- the calculations: for gold mirrors, the uncertainty is, rection factor η , which saturates at large distance to a F however, only of about 5%. The validity of the PFA is lowervalueforAu-SithanforAu-Au. Quantitatively,the another important assumption in the theory-experiment experimentalratioisofthesameorderasthetheoretical comparison:63,64theerrorshouldbesmallerthan1%here value at short distance, with an error less than 10% in sincez/R<1%.22 Thelargediscrepancybetweentheory the 100–200nm distance range, but the ratio decreases and experiment regarding the absolute value of the force faster with distance than predicted by theory. These re- gradient requires, therefore, another explanation. sults show that, although the absolute comparison with A possible source of error being the calibration of the theory is not possible here, the material dependence of scanner extension, we found that multiplying the dis- theCasimirforceisclearlyevidencedwhenthesurfaceis tance z of each data point by a factor 0.9 translates the changed from gold to silicon. data points onto the theoretical curves as shown in the inset of Fig. 7. Since the force calibration is dependent To improve this experiment in the future, the scan- on the scanner calibration, it is in fact a factor 0.85 that ner extension should be measured by interferometry in should be applied on the relative distance (before the situ during the measurement to avoid the effect of ther- determination of β) in order to shift the data onto the mal cycles on the scanner piezoelectric coefficient. The theoretical curves. The piezoelectric z-scanner was cal- detectionsensitivitycouldbealsoimprovedbystabiliza- ibrated by interferometry nine months before the force tion of the laser intensity down to the shot noise level, measurements reported here and it could be that the inordertominimizethedetectionandbackactionnoises, scanner extension has been progressively reduced after and take advantage of the strongly suppressed thermo- successive thermal cycles between 300 and 4.2K. Since mechanical noise at 4.2K. 8 VI. CONCLUSION to this work and for helpful discussion. This work has benefited from fruitful discussions with P. Andreucci, L. From an instrumental point of view, we have shown Duraffourg, J-J. Greffet, G. Jourdan, and A. Lambrecht. that the presence of a long-range parasitic force at the The samples and probes were prepared in the Nanofab minimizing potential can be related to the electrostatic clean-room facility. This research was supported by the environment of the force probe. Precise measurements PNANO 2006 program of the Agence Nationale de la of the Casimir force therefore require an accurate con- Recherche under the project name MONACO. trol of the environment, like screening of every insulat- ing part close to the probe: chip holder, optical fiber, and piezoelectric actuators. Regarding the minimizing potential, the variations with distance observed for the Au-Au cavity could be explained by a model taking into Appendix account the electrostatic potential of the environment, and the absence of variation for the Au-Si cavity indi- cates that there is no patch potential effect on the gold- The Casimir force is computed for parallel plates of coated sphere. Finally, we have shown that our in situ infinitethicknessesbytakingintoaccounttherealmate- measurementoftheCasimirforceusingasinglespherical rialpropertiesasexplainedinRefs.31,32. Thedielectric probe(gold)andtwodifferentsurfaces(goldordopedsil- constant (cid:15)r is modeled by a plasma frequency ωp and icon) gives qualitatively the correct value for the relative a Drude relaxation parameter γp, plus a Lorentz func- force gradient, although the absolute values are not cor- tion with resonance frequency ω0, relaxation parameter rect due to a systematic error that might be attributed γ0,andsusceptibilityχ0,describinginterbandtransitions to the scanner calibration. for gold and intrinsic response for silicon: The sensitivity of the Casimir force to material prop- erties, as demonstrated here, could be used for surface ω2 χ ω2 characterization in a new type of non-contact scanning (cid:15) (iω)=1+ p + 0 0 (6) force microscopy. Such a technique would be the exten- r ω(ω+γp) ω2+ω02+ωγ0 sion of the near-field van der Waals force microscopy to the retarded Casimir regime at large separation. For a TheparametersusedinthecomputationarelistedinTa- given force sensitivity, measuring at large distances im- ble I. We checked that our computation algorithm gives plies the use of micron-size spherical probes with a lower the correct results for the well-known Au-Au cavity, be- spatial resolution than the sharp tips used in atomic fore computing the force for our specific Au-Si cavity. force microscopy, but provides information on the op- tical properties of the materials. With the Casimir force being obtained at the minimizing electrostatic potential, this technique would be complementary to the measure- TABLEI:Parametersofthedielectricfunctionforgold(Au) ment of the contact potential by Kelvin-probe force mi- and silicon (Si). croscopy. ω γ ω γ χ p p 0 0 0 Acknowledgments (1015rad/s) (1015rad/s) (1015rad/s) (1015rad/s) Au 13.7 0.05 20 25 5 The low-temperature AFM was designed and built by Si 0.37 0.052 6.6 0 10.87 J-F.Motte. WethankT.Ouisseforanearlycontribution 1 K.L. 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