Palasantzas, G.

van Zwol, P. J.

English

Atomic force microscopy measurements and force theory calculations using the Lifshitz theory show that van der Waals/Casimir dispersive forces have a strong dependence on surface roughness and material optical properties. It is found that at separations below 100 nm the roughness effect is manifested through a strong deviation from the normal scaling of the force with separation distance. Moreover, knowledge of precise optical properties of metals is shown to be very important for accurate force predictions rather than referring to idealized defect free material models. Finally, we compare the van der Waals/Casimir forces to capillary adhesive forces in order to illustrate their significance in stiction problems.</p

Proceedings - University of Groningen

   University of GroningenKinetic Roughening and Material Optical Properties Influence on Van der Waals/CasimirForcesvan Zwol, P. J.; Palasantzas, G.Published in:Acta Physica Polonica AIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2010Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):van Zwol, P. J., & Palasantzas, G. (2010). Kinetic Roughening and Material Optical Properties Influence onVan der Waals/Casimir Forces. Acta Physica Polonica A, 117(2), 379-383.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Vol. 117 (2010) ACTA PHYSICA POLONICA A No. 2Proceedings of the XLIVth Zakopane School of Physics, Zakopane, May 18–23, 2009Kinetic Roughening and Material Optical Properties Influenceon Van der Waals/Casimir ForcesP.J. van Zwol and G. Palasantzas∗Materials innovation institute M2i and Zernike Institute for Advanced MaterialsUniversity of Groningen, 9747 AG Groningen, NetherlandsAtomic force microscopy measurements and force theory calculations using the Lifshitz theory show thatvan der Waals/Casimir dispersive forces have a strong dependence on surface roughness and material opticalproperties. It is found that at separations below 100 nm the roughness effect is manifested through a strongdeviation from the normal scaling of the force with separation distance. Moreover, knowledge of precise opticalproperties of metals is shown to be very important for accurate force predictions rather than referring to idealizeddefect free material models. Finally, we compare the van der Waals/Casimir forces to capillary adhesive forces inorder to illustrate their significance in stiction problems.PACS numbers: 68.55.Jk, 68.37.Ps, 85.85.+j, 78.68.+m, 85.85.+j, 68.55.–a, 47.55.nb1. IntroductionWhen the proximity between material objects becomesof the order of nanometers up to a few microns, a regimeis entered where forces quantum mechanical in nature,namely, van der Waals (vdW) and Casimir (retardedvdW) forces, become operative [1]. Historically, theCasimir force has been considered to result from theperturbation of vacuum zero point fluctuations (ZPF)by conducting plates. Indeed, quantum theory predictsthat the vacuum of space in the universe is filled withlow-energy electromagnetic waves, random in phase andamplitude and propagating in all possible directions [1].Because of its relatively short range, the Casimir forceis now starting to take on technological importance inthe operation of micro/nano-electromechanical systems(MEMS/NEMS) at separations less than 200 nm, e.g.,micro oscillator devices, micro/nano switches or actua-tors [2]. Moreover, from a fundamental point of viewthe Casimir force plays important role in the search ofhypothetical forces beyond the standard model [2, 3].Although the first high accuracy measurements byLamoreaux [4] initiated further detailed investigationsat close separations including atomic force microscopy(AFM; see Fig. 1) [5, 6], it remains still a challenge ofhow well we understand the concept of zero point vac-uum fluctuations. Indeed, their energy density is givenby [1, 4]∗ corresponding author; e-mail: G.Palasantzas@rug.nlFig. 1. AFM set-up for force measurements. Thesphere is usually coated by Au (typically ∼ 100 nmthick).ρZPF =~c2π∫ Kmax0q3 dq (1)with Kmax = 2π/Lm and Lm a minimum length scaleof vacuum fluctuations. If we consider as a minimumlength scale Lm the Planck length LPl =√~G/c3 =1.6 × 10−33 cm (the space itself is thought to break upinto a kind of quantum foam beyond this length) withG the Gravitational constant, ~ the Planck’s constant,and c the velocity of light), then we obtain ρZPF =1095 g/cm3 [4]. If we consider as a more reasonable valuefor Lm the electron radius or re = 2.82×10−13 cm then weobtain ρZPF = 1014 g/cm3 [4]. Nevertheless, both valuesof the energy density ρZPF are still very large. Indeed,(379)380 P.J. van Zwol, G. Palasantzasdark energy density estimations yield an experimentaldensity value ρDark ≈ 10−29 g/cm [4]. If we compareit to zero point fluctuations of the vacuum we have anextremely large discrepancy ρZPF/ρDark ≈ 1043 → 10124representing the biggest discrepancy in physics [4]. If onthe other hand we fix Lm so that ρZPF ≈ ρDark thenthe measured Casimir force would be much smaller thanactual values obtained experimentally [4]. This issue re-mains still under investigation and it will lead researchdeep into the future in combination with challenges toNewtonian gravitation [3].In any case, van der Waals/Casimir forces between realmaterials can be strongly influenced by surface morphol-ogy and material optical properties as it will be shown inthe following. These are important issues if the influenceof these dispersive forces in MEMS/NEMS is under con-sideration. Moreover, comparison of the dispersive forcesin real materials to other forces, as for example capillaryforces, that influence device motion, is also necessary.2. Lifshitz theory and real materialsThe van der Waals/Casimir force between real materi-als is calculated using the Lifshitz theory [6]. For realparallel flat mirrors of area A, separation distance L,and reflection coefficient r(Φ) (with Φ the imaginary fre-quency of the electromagnetic wave), the van der Waals/Casimir energy Epp is given by (assuming absolute tem-perature T = 0 K and short separations, e.g., L < 1 µm,where thermal corrections are negligible) [4]Epp =− ~A∑v∫d2k4π2∫ ∞0dΦ2πln[1−rv(k,Φ)2 e−2κL]. (2)The index v denotes the sum over the s and p polariza-tion. Since for experimental force measurements by AFMa sphere-plate geometry is often used to avoid plate-platealignment problems (Fig. 1) [7–10], the Casimir forceFC is given by FC = (2πR/A)Epp (assuming R À L).The material optical properties enter the Lifshitz formulavia the frequency dependent dielectric function ε(ω). InEq. (2) the reflection coefficients for s and p polarizationand the function ε(iζ) are given, respectively, byrs =ko − k1ko + k1and rp =ε(iζ)ko − k1ε(iζ)ko + k1,ε(iζ) = 1 + (2/π)∫ ∞0ωε”(ω)ω2 + ζ2dω, (3)with ko =√ζ2/c2 + k2, k1 =√ε(iζ)ζ2/c2 + k2, andε′′(ω) the imaginary part of ε(ω).3. Experimental procedurePolysterene spheres with a diameter of 100 µm and a1.4%-deviation from sphericity were glued on a 450 µmlong Au coated cantilever (Fig. 1). A relatively stiff can-tilever is used to reduce the jump to contact, cantileverbending due to Au evaporation, and errors in the deflec-tion correction. AFM was used to determine the sphereroughness (prior to Au coating), which gave a 1.2 nmRMS roughness amplitude over an area of 25 µm2. Fur-ther, the spheres were plasma sputtered for electricalcontact with the cantilever, and then were coated with100 nm Au at a rate of 0.6 nm/sec in an evaporatorkept at a base pressure of 10−6 mbar. Si-oxide waferswere used as substrates and coated by Au layers of thick-nesses between 100–1600 nm, and under identical growthconditions as the Au coating on the sphere (see Fig. 2).Fig. 2. Increasing roughness amplitude w with increas-ing film thickness d. The inset shows topography exam-ples of two films 100 nm and 400 nm thick, respectively.4. Van der Waals/Casimir force measurements:effect of surface roughness and optical propertiesThe Lifshitz formula accounts for real optical proper-ties of the materials, and for finite temperature effects.Corrections to the force due to optical properties canbe very large, especially at small distances (< 100 nm)between the bodies [7, 8]. An additional source of correc-tions to the force is the surface roughness [9, 10]. Rough-ness corrections δEpp to the Casimir energy (and thus tothe force) within the scattering formalism [9] are formu-lated in terms of a roughness response function G(k) andthe roughness power spectrum σ(k):δEpp =∫ [d2k/4π2]G(k)σ(k), (4)where G(k) is derived in [9], yielding for the total energyEpp,rough = Epp+δEpp. The theory is valid under the as-sumptions: first, the lateral dimensions of the roughnessmust be much smaller than the system size, i.e., plateor sphere, which is usually the case; second, the RMSroughness w must be small compared to the separationdistance L (w ¿ L); and third, the local surface slope ofthe film must be small (ρrms ¿ 1) [11].A wide variety of surfaces exhibits the so-calledself-affine roughness [16], which is characterized forKinetic Roughening and Material Optical Properties Influence . . . 381isotropic surfaces by the RMS roughness amplitude w =〈[h(r)]2〉1/2 (〈h〉 = 0), the lateral correlation length ξ(indicating the lateral feature size), and the roughnessexponent 0 < H < 1. Small values of H ≈ 0 correspondto jagged surfaces, while large values H ≈ 1 to a smoothhill–valley morphology [12]. For self-affine roughness thespectrum σ(k) scales as σ(k) ∝ k−2−2H if kξ À 1, andσ(k) ∝ const if kξ ¿ 1 [12]. This scaling is satisfied bythe analytic model [13]σ(k) =(AHw2ξ2) / (1 + k2ξ2)1+H (5)with A = 2/[1 − (1 + k2cξ2)−H ], kc is a lower roughnesscut-off (≈ 1 nm−1). The parameters w, ξ and H canbe determined by direct measurement of the height dif-ference correlation function H(r) = 〈[h(r)− h(0)]2〉 with〈. . .〉 denoting the ensemble average over multiple surfacescans [13].Fig. 3. Force measurements with increasing roughnessamplitude w (see Fig. 2) as the arrow shows. The rough-ness influence becomes evident for sphere-plate separa-tions L < 70 nm.Furthermore, the Picoforce AFM from Veeco [10] wasused for the Casimir force measurements (Fig. 3) fol-lowing the procedure outlined in [5]. Moreover, aver-aging over 30 force curves was used when the Casimirforce was measured. The plate-sphere separation L =Dpiezo+d0−ddefl was measured with respect to the pointof contact with the surface, where Dpiezo is the piezomovement, d0 is the distance upon contact due to sub-strate and sphere roughness, and ddeff is the cantileverdeflection correction [5, 10].The cantilever spring constant k was determined elec-trostatically with an error of 3% [10], and it was recal-ibrated for all films without any measurable roughnesseffect on the electrostatic force. The residual or contactpotential Vc on the grounded sphere was determined elec-trostatically, and it was found to be Vc ≤ 25 mV (error≤ 5 mV) [10]. Once k and Vc are known, d0 is determinedalso electrostatically [10]. Indeed, the d0 was found tovary almost linearly with the sum of the RMS rough-ness amplitudes of the sphere (wsph) and substrate (w):d0 ≈ c(w + wsph) with c = 3.7± 0.3 [10]. The magnitudeof the measured roughness effect on the Casimir forceis of the order of 100% for thicker films (which are alsorougher films) at short separations (< 80 nm), while atlarger separations the scaling law is recovered and agree-ment with theory is restored.Fig. 4. Ellipsometry measurements of the dielectricfunction where it is shown its imaginary part. The latteris also relevant for the force measurement with the IR(infrared) — regime indicated behind the dotted line.At high frequences data were extrapolated from hand-book data (this regime does not play any significant rolein force calculations).Besides surface roughness the influence of the materialoptical properties is very crucial. The optical data wereobtained from ellipsometry measurements by Woolam forwavelengths 137 nm to 33 µm (see Fig. 4). At low fre-quences, where the dielectric function is large and givesthe highest contribution to the force (Eq. (3)), it was ob-tained by extrapolation using the Drude model [7, 8]ε(ω) = 1− ω2p [ω(ω + jωt)]−1 (6)with ωp the plasma frequency and ωt the relaxation fre-quency. Furthermore, Fig. 5 shows the theoretical forcecalculations using the sphere-plate configuration and theoptical data of Fig. 4. The data were calibrated withrespect to the perfect gold films (single crystal withoutdefects) with ωp = 9.0 eV and ωt = 0.035 eV. The valueof ωp = 9.0 eV was calculated using ωp =√Ne2/εome[7, 9] with N the number of conduction electrons per unitvolume (computed from its bulk density), e the electroncharge, and me the effective electron mass.However, the films used for measurement of theCasimir force can contain defects (grain boundaries, im-purities etc.), which are responsible for the reduction ofωp (< 9.0 eV) [7, 8]. There is significant difference be-tween this reference force curve and those that corre-sponds to actual Au films as it is shown in Fig. 5. Thedeviations remain significant even for separations as largeas 1000 nm (À λp = ωp/c; with λp the plasma wave-length).382 P.J. van Zwol, G. PalasantzasFig. 5. Force calculations using the optical data fromFig. 4 and Eq. (2). The vertical axis shows the forcerelative difference (in %) with respect to the force forperfect Au films.5. Capillary forcesFinally, we will compare the measured vdW/Casimirforces to capillary adhesive forces. Adhesive forces weremeasured by retracting a sphere attached on the can-tilever from the surface. The total adhesion force can bedivided into a capillary force and an interfacial tensionforce due to surface tension acting tangentially to the in-terface along the contact line with the solid body [14].For the relatively smooth films, the theory prediction forflat surfaces gives an approximate upper limit for theforce Fup = 4πγRs cos(θ) [14] (Fup = S∆σ with S =4πRsRk cos(θ) the meniscus surface area and ∆σ = γ/Rkthe Laplace pressure, while ignoring contributions fromsurface tension).For the contact angle of water onto Au surfaces ofapproximately 70 degrees and Rs = 50 µm, we obtainthe adhesive force Fup = 1.5 × 104 nN [15]. With in-creasing roughness, surfaces can be more hydrophobic[16] leading to reduced capilary force. In addition, thecontact area between sphere and plate is highly reduced.If we consider the lower limit for the force in terms ofcontact onto a single asperity [15], where the capillarymeniscus is formed only between this asperity and thesurface of the sphere, we obtain a force of magnitudeFlow = 4πγξ cos(θ) = 15 nN (where we used as an effec-tive asperity radius the roughness correlation length ξ).From Fig. 6 it appears that the smooth limit is reachedfor the Au/mica film. For the roughest films the valuesfound are up to ten times higher than that of a singleasperity indicating a capillary interaction of a multitudeof asperities. In any case, as Fig. 6 indicates, the effect ofcapillary forces can be by orders of magnitude larger thanthat of vdW/Casimir forces. Nevertheless, they have alsoa high sensitivity on changes of surface roughness.Fig. 6. Adhesion force vs. RMS roughness of thesphere and plate added together. The upper and lowerhorizontal lines represent the theoretical values for cap-illary interaction, in case of a perfectly smooth spherewith a perfectly smooth plate, and in case of a single as-perity with lateral size 50 nm. The inset shows complete(upper) and multiple asperity (lower) meniscus forma-tion.6. ConclusionsIn conclusion, surface functionality with respect to itsinfluence on the Casimir force is strongly affected by sur-face morphology and material optical properties. Theeffect of capillary forces can be by orders of magnitudelarger than that of vdW/Casimir forces. Nevertheless,they have also a high sensitivity on changes of surfaceroughness. Therefore, realistic implementations of theseforces in MEMS/NEMS should take carefully into ac-count the associated morphology and material function-ality.AcknowledgmentsThe research was carried out under project numberMC3.05242 in the framework of the Strategic Researchprogramme of the Materials innovation institute M2i(the former Netherlands Institute for Metals Researchor NIMR). Financial support from the M2i is gratefullyacknowledged. We would like to acknowledge useful dis-cussions with V.B. Svetovoy.References[1] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793(1948). For initial measurements of the Casimir effectsee: M.J. Sparnaay, Physica (Utrecht) 24, 751 (1958);P.H.G.M. van Blockland, J.T.G. Overbeek, J. Chem.Soc. Faraday Trans. 74, 2637 (1978).Kinetic Roughening and Material Optical Properties Influence . . . 383[2] F.M. Serry, D. Walliser, G.J. Maclay, J. Appl.Phys. 84, 2501 (1997); H.B. Chan, V.A. Aksyuk,R.N. Kleiman, D.J. Bishop, F. Capasso, Science 291,1941 (2001); G. Palasantzas, J.Th.M. De Hosson,Phys. Rev. B 72, 115426 (2005); ibid Phys. Rev. B72, 121409 (2005); G. Palasantzas, J.Th.M. De Hos-son, Surf. Sci. 600, 1450 (2006).[3] R. Onofrio, New J. Phys. 8, 237 (2006).[4] S.K. Lamoreaux, Phys. Rev. Lett 78, 5 (1997);S.K. Lamoreaux, Phys. Rev. Lett. 83, 3340 (1999);S.K. Lamoreaux, Phys. Today, (2007).[5] B.W. Harris, F. Chen, U. Mohideen, Phys. Rev. A62, 052109 (2000).[6] E.M. Lifshitz, Zh. Eksp. Teor. Fiz. 29, 94 (1956)[Sov. Phys. JETP 2, 73 (1956)]; I.E. Dzyaloshinskii,E.M. Lifshitz, L.P. Pitaevskii, Advances in Physics38, 165 (1961); E.M. Lifshitz, L.P. Pitaevskii, Statis-tical Physics, Part 2, Pergamon Press, Oxford 1980.[7] I. Pirozhenko, A. Lambrecht, V.B. Svetovoy, New J.Phys. 8, 238 (2006).[8] V.B. Svetovoy, P.J. van Zwol, G. Palasantzas,J.Th.M. De Hosson, Phys. Rev. B 77, 035439 (2008);P.J. van Zwol, G. Palasantzas, J.Th.M. DeHosson,Appl. Phys. Lett. 91, 144108 (2007).[9] P.A. Maia-Neto, A. Lambrecht, S. Reynaud, Phys.Rev. A 72, 012115 (2005); P.A. Maia-Neto, A. Lam-brecht, S. Reynoud, Europhys. Lett. 69, 924 (2005).[10] P.J. van Zwol, G. Palasantzas, J.Th.M. De Hosson,to appear in Phys. Rev. B (2008); P.J. van Zwol,G. Palasantzas, M. van de Schootbrugge, J.Th.M. DeHosson, Appl. Phys. Lett. 92, 054101 (2008).[11] G. Palasantzas, Phys. Rev. E 56, 1254 (1997).[12] P. Meakin, Phys. Rep. 235, 1991 (1994); J. Krim,G. Palasantzas, Int. J. Mod. Phys. B 9, 599 (1995).[13] G. Palasantzas, Phys. Rev. B 48, 14472 (1993); 49,5785 (1994); G. Palasantzas, J. Krim, Phys. Rev. Lett.73, 3564 (1994).[14] A. Ata, Y.I. Rabinovich, R.K. Singh, J. Adhesion,Sci. Technol. 16, 337 (2002).[15] P.J. van Zwol, G. Palasantzas, J.Th.M. de Hosson,Phys. Rev. E 78, 031606 (2008); P.J. van Zwol,G. Palasantzas, J.Th.M. DeHosson, Appl. Phys. Lett.91, 101905 (2007).[16] M.E. Abdelsalam, P.N. Bartlett, T. Kelf, J. Baum-berg, Langmuir 21, 1753 (2005).

Kinetic Roughening and Material Optical Properties Influence on Van der Waals/Casimir Forces

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Kinetic Roughening and Material Optical Properties Influence on Van der Waals/Casimir Forces

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