In this work we explore the influence of random surface roughness on the
cantilever sensitivity to respond to curvature changes induced by changes in
surface stress. The roughness is characterized by the out-of-plane roughness
amplitude w, the lateral correlation length x, and the roughness or Hurst
exponent H (0<H<1). The cantilever sensitivity is found to decrease with
increasing roughness (decreasing H and/or increasing ratio w/x) or equivalently
increasing local surface slope. Finally, analytic expressions of the cantilever
sensitivity as a function of the parameters w, x, and H are derived in order to
allow direct implementation in sensing systems.Comment: 10 pages, 3 figure

Ergincan, O.

Kooi, B. J.

Palasantzas, G.

English

arXiv

In this work we explore the influence of random surface roughness on the cantilever sensitivity to respond to curvature changes induced by changes in surface stress. The roughness is characterized by the out-of-plane roughness amplitude w, the lateral correlation length xi, and the roughness or Hurst exponent H(0 <H <1). The cantilever sensitivity is found to decrease with increasing roughness (decreasing H and/or increasing ratio w/xi) or equivalently increasing local surface slope. Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, xi, and H are derived in order to allow direct implementation in sensing systems

NARCIS 

Influence of random roughness on cantilever curvature sensitivity

In this work we explore the influence of random surface roughness on the cantilever sensitivity to respond to curvature changes induced by changes in surface stress. The roughness is characterized by the out-of-plane roughness amplitude w, the lateral correlation length xi, and the roughness or Hurst exponent H(0 &lt;H &lt;1). The cantilever sensitivity is found to decrease with increasing roughness (decreasing H and/or increasing ratio w/xi) or equivalently increasing local surface slope. Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, xi, and H are derived in order to allow direct implementation in sensing systems

ARTS repository - University of Groningen

   University of GroningenInfluence of random roughness on cantilever curvature sensitivityErgincan, O.; Palasantzas, G.; Kooi, B. J.Published in:Applied Physics LettersDOI:10.1063/1.329971310.1063/1.3299713IMPORTANT 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):Ergincan, O., Palasantzas, G., & Kooi, B. J. (2010). Influence of random roughness on cantilever curvaturesensitivity. Applied Physics Letters, 96(4), 041912-1-041912-3. [041912].https://doi.org/10.1063/1.3299713, https://doi.org/10.1063/1.3299713CopyrightOther 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.Download date: 16-07-2023Influence of random roughness on cantilever curvature sensitivityO. Ergincan, G. Palasantzas,a and B. J. KooiZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 22 October 2009; accepted 5 January 2010; published online 29 January 2010In this work we explore the influence of random surface roughness on the cantilever sensitivity torespond to curvature changes induced by changes in surface stress. The roughness is characterizedby the out-of-plane roughness amplitude w, the lateral correlation length , and the roughness orHurst exponent H0H1. The cantilever sensitivity is found to decrease with increasingroughness decreasing H and/or increasing ratio w / or equivalently increasing local surface slope.Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, , and Hare derived in order to allow direct implementation in sensing systems. © 2010 American Instituteof Physics. doi:10.1063/1.3299713Microelectromechanical/Nanoelectromechanical systemsMEMS/NEMS are important devices that combine inmany cases the advantages of mechanical systems with thespeed and large scale integration of silicon based microelec-tronics. As a result a large number of groups are experiment-ing on various aspects to understand fully the properties andpotential of MEMS/NEMS.1–11 As a matter of fact, microme-chanical cantilevers allow mass resolution down to femto-grams in air environment.12 The sensitivity is determined bythe effective vibratory mass of the resonator determined bygeometry, configuration, and material properties of the reso-nant structure, and the stability of the device resonancefrequency.6Complete understanding of cantilever bending is crucialfor ultrasensitive applications including bending experimentsto monitor the evolution of material properties, monitoringthe surface stress during bending, adsorption studies, self-assembly, thin film deposition, and molecular-recognitionbased on-chip biomedical sensing devices.13–22 Such experi-ments exploit Stoney’s equation, which assumes planar ge-ometry, to relate the cantilever bending to the magnitude ofthe surface stress f change f .23 However, the surfaces ofreal cantilevers and in more general material systems; evenfor the most thoroughly polished surfaces have random sur-face roughness on different lateral length scales.Recently, it was shown that the response of the curvatureof cantilevers to changes in their surface stress depends sig-nificantly on the surface morphology.23 This dependence wasattributed to the transverse coupling between the out-of-plane and in-plane components of the surface-induced stress.Moreover, roughness corrections were introduced, which arehighly important for experiments measuring the surfacestress on nominally planar surfaces.23 However, calculationsof the roughness effects on the cantilever sensitivity as afunction of characteristic parameters of random rough sur-faces are still missing. This will be the topic of the presentletter.Initially, we will present the theory of cantilever bendingincluding the general roughness corrections from Ref. 23,and afterwards we will implement specifics of random self-affine roughness. It has been shown that the in-plane stress Tin a surface layer, which is required to undo the additionalin-plane strain components induced by the stress at equilib-rium in the layer if it was detached from the substrate andallowed to strain freely under the influence of the surfacestress while allowing for free relaxation along the normal isgiven by23T =1hLAroughAs − vL1 − vLs , 1with hL the mean thickness of a layer deposited on oneside of the cantilever surface see inset Fig. 1 and havinga rough surface with area Arough, and a Poisson’s ratio vL.We assume an isotropic and continuously differentiablerough profile hr with r= x ,r the in-plane position vectorhL= hr	 with . . ..	 denoting an ensemble average.The stress components s and s are, respectively, the in-plane and out-of-plane components. They are given, respec-tively, by23 s = A /Aroughf3+cos 2 /4 cos 	 and s= A /Aroughfsin2  /cos 	. f is the position dependent sca-lar surface stress. The angle  inset, Fig. 1 is defined astan =h  and A is the average flat surface area. Substitu-tion in Eq. 1 yields for the stress TaAuthor to whom correspondence should be addressed. Electronic mail:g.palasantzas@rug.nl.FIG. 1. Cantilever sensitivity T /To as a function of local surface slope rmswith a varying correlation lengths 10 nm500 nm, w=1 nm, H=0.5,and two different Poisson’s ratios vL=0.18 corresponding to Si111 andvL=0.28 corresponding to Si100. The inset shows a random rough sur-face of an overlayer on a cantilever surface,  is the inclination angle be-tween the normal vectors n and n̂.APPLIED PHYSICS LETTERS 96, 041912 20100003-6951/2010/964/041912/3/$30.00 © 2010 American Institute of Physics96, 041912-1Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspT =12hL f1 + vL1 − vLcos  +1 − 3vL1 − vLsec  . 2For weak roughness tan =h1 and hLw with w=h−hL2	 the rms roughness amplitude, s is onlyweakly affected by surface roughness, while the dominanteffect arises from the out-of-plane stress component s.Local values of f and  will be correlated due to dependenceof the surface properties on the surface crystallographicorientation.24Furthermore, if we assume a Gaussian height distribu-tion of the height profile which is reasonable in many casesof deposited overlayers and depending on the growthmode,25–29 the rough surface area in Eq. 1 is given by25Arough /A=0+	du1+rms2 ue−u with rms= h2	1/2 the rmslocal surface slope. The latter in terms of Fourier-transformanalysis is given by rms= oQcq2 hq2	d2q1/2 and hq2	the power roughness spectrum.26 Qc= /ao is the integrationlimit with ao the minimum lower roughness cut-off of theorder of atomic dimensions. For weak roughness rms1we obtain Arough /A1+ rms2 /2=1+ 2	 /2 since rms2	 rms=h 2	=tan2 . Note that the first orderperturbative expansion for the surface area ratio Arough /A isalso independent of the assumption of Gaussian height dis-tribution.A wide variety of surfaces exhibit a so-called self-affineroughness. This type of roughness is characterized, besidesthe rms roughness amplitude w, by the lateral correlationlength  indicating the average lateral feature size, andthe roughness exponent 0H1. Small values of H0corresponds to jagged surfaces, while large values of H1to a smooth hill valley morphology.26–29 For self-affineroughness, the power spectrum obeys the scaling behaviorhq2	q−2−2H if q1 and  hq2	constant ifq1.27,28 This scaling is satisfied by the analytic model28hq2	 =aw221 + q221+H. 3The parameter “a” in Eq. 3 is obtained by the normaliza-tion condition 0qQcq2 hq2	d2q=w2 yielding a= H / / 1− 1+Qc22−H.28 This is equivalent to the factthat the height-height correlation Cr= hrh0	= hq2	e−iq ·rd2q obeys the condition: Cr=0=w2. Fur-thermore, we obtain for the local slope rms the analytic ex-pressionrms =w a1 − H 1 + Qc221−H − 1 − 11/2. 4Analytic expressions in the limiting cases H=0 and 1 can beobtained using the identity lnx=limc→01 /BxB−1.Therefore, we obtain rms H=1= w /a ln1+Qc22−11/2with aH=1= 1 /1+ Qc−2, and rms H=0= w /aQc2−11/2 with aH=0=1 /  ln1+Qc22.For weak roughness rms2	1, by consideringthe Taylor expansions cos x1−x2 /2+. . . and sec x1+x2 /2+. . ., Eq. 2 yields after series expansion the simplerform13 Tf	S /hL1−vL / 1−vLrms2  where substitutionof Eq. 4 gives the analytic form for the cantilever sensitiv-ity, which is defined by the ratio T /To,TTo 1 − vL1 − vLw 2 a1 − H 1 + Qc221−H − 1 − 15with To= f	S /hL representing the effective stress for a planarsurface with the same surface stress as the rough surface.Figure 1 shows the cantilever sensitivity T /To as a func-tion of the local slope rms. In fact, Eq. 5 defines a limitingvalue of the local slope rms for which T=0, yieldingrms max=1−vL /vL. For Poisson’s ratios vL=0.18Si111 Ref. 30 and vL=0.28 Si100 Ref. 30 we ob-tain, respectively, rms/max=2.13 and 1.6. For a metallic over-layer, as gold widely used to coat cantilevers with vL=0.44,30 we obtain rms/max=1.12. These are relatively sig-nificant values for rms and the pertubative expansion of Eq.5 is valid only for local slopes rms1. It is clear e.g.,from Fig. 3 and Eq. 5 that the local surface slope influenceis minimized with decreasing Poisson’s ratio. The cantileversensitivity can decrease significantly with changing crystal-lographic structures corresponding to different Poisson’s ra-tios and the effect becomes more pronounced when the localslope increases or equivalently the surface roughness in-creases.In order to obtain a direct feeling of the influence of thecharacteristic roughness parameters w , ,H, we present inFig. 2 the cantilever sensitivity T /To as a function of the longwavelength roughness ratio w / for various roughness expo-nents H. The inset shows simultaneously the correspondinglocal slopes to ensure that our calculations are performed forrms1 and thus to ensure validity of the present formalism.It is clear that for lower roughness exponents H and/or largerroughness ratios w / the cantilever sensitivity decreasesrather drastically. A similar behavior is depicted in Fig. 3,where the cantilever sensitivity is depicted as a function ofw / for two different Poisson’s ratios vL. The inset, showinga typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23, indicates thatthe random roughness parameters used in this study are oftenmet in experimental coatings depending on material andpreparation conditions.In conclusion, random surface roughness influences thecantilever sensitivity T /To to respond to changes in the as-sociated surface stress when the cantilever sensor is usedFIG. 2. Cantilever sensitivity T /To as a function of long wavelength rough-ness ratio w / for w=1 nm and H=0.5, and two different Poisson’s ratiosvL=0.18 corresponding to Si111 and vL=0.28 corresponding toSi100.041912-2 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspwith a sensing layer on top. In more detail the cantileversensitivity T /To is found to decrease with increasing localsurface slope or equivalently increasing surface roughnessdecreasing H and/or increasing ratio w /. Even weak localsurface slopes are shown to have a significant effect on thecantilever sensitivity T /To. Finally, the analytic expressionsderived for the cantilever sensitivity T /To as a function ofcharacteristic roughness parameters allow direct implemen-tation in sensing systems if measurements of the character-istic roughness parameters of cantilever surfaces are per-formed and the corresponding roughness correction is takeninto account for the surface stress T. This is desirable sincecantilever bending studies are a major source for current ex-perimental data of surface stress T.We would like to acknowledge financial support by theSTW under Grant No. 10082.1A. B. Hutchinson, P. Truitt, L. Sekaric, J. M. Parpia, H. G. Craighead, J. E.Butler, and K. C. Schwab, Appl. Phys. Lett. 84, 972 2004.2J. Barnaś and G. Palasantzas, J. Appl. Phys. 82, 3950 1997.3A. N. Cleland and M. L. Roukes, Appl. Phys. Lett. 69, 2653 1996.4K. L. Ekinci and M. L. Roukes, Rev. Sci. Instrum. 76, 061101 2005.5K. L. Ekinci, Y. T. Yang, X. M. H. Huang, and M. L. Roukes, Appl. Phys.Lett. 81, 2253 2002.6K. L. Ekinci, Y. T. Yang, and M. L. Roukes, J. Appl. Phys. 95, 26822004.7K. C. Schwab and M. L. Roukes, Phys. Today 58, 36 2005.8V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T. A. Arias, and P. L.McEuen, Nature London 431, 284 2004.9K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B.C. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 2000.10D. W. Carr, S. Evoy, L. Sekaric, H. G. Craighead, and J. M. Parpia, Appl.Phys. Lett. 75, 920 1999.11S. Evoy, A. Olkhovets, L. Sekaric, J. M. Parpia, and H. G. Craighead,Appl. Phys. Lett. 77, 2397 2000.12N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, Rev. Sci. Instrum. 75,2229 2004.13R. Desikan, I. Lee, and T. Thundat, Ultramicroscopy 106, 795 2006.14M. Godin, P. J. Williams, V. Tabard-Cossa, O. Laroche, L. Y. Beaulieu, R.B. Lennox, and P. Grutter, Langmuir 20, 7090 2004.15C. W. Pao and D. J. Srolovitz, Phys. Rev. Lett. 96, 186103 2006.16C. Friesen and C. V. Thompson, Phys. Rev. Lett. 89, 126103 2002.17H. Ibach, Surf. Sci. Rep. 29, 195 1997.18M. Seo and M. Yamazaki, J. Solid State Electrochem. 11, 1365 2007.19M. J. Harrison, D. P. Woodruff, J. Robinson, D. Sander, W. Pan, and J.Kirschner, Phys. Rev. B 74, 165402 2006.20N. Vasiljevic, T. Trimble, N. Dimitrov, and K. Sieradzki, Langmuir 20,6639 2004.21J. A. Floro, E. Chason, R. C. Cammarata, and D. J. Srolovitz, MRS Bull.27, 19 2002.22F. Spaepen, Acta Mater. 48, 31 2000.23J. Weissmüller and H. L. Duan, Phys. Rev. Lett. 101, 146102 2008.24J. J. Métois, A. Saul, and P. Muller, Nature Mater. 4, 238 2005.25B. N. J. Persson and E. Tosatti, J. Chem. Phys. 115, 5597 2001.26G. Palasantzas, Phys. Rev. E 56, 1254 1997.27J. Krim and G. Palasantzas, Int. J. Mod. Phys. B 9, 599 1995; P. Meakin,Phys. Rep. 235, 1991 1994; Y. -P. Zhao, G. -C. Wang, and T. -M. Lu,Experimental Methods in the Physical Science Academic, New York,2001, Vol. 37.28G. Palasantzas, Phys. Rev. B 48, 14472 1993; 49, 5785 1994.29G. Palasantzas and J. Krim, Phys. Rev. Lett. 73, 3564 1994.30W. A. Brantley, J. Appl. Phys. 44, 534 1973.FIG. 3. Color online Cantilever sensitivity T /To as a function of longwavelength roughness ratio w / for w=1 nm, different roughness expo-nents H, and Poisson’s ratio vL=0.18 corresponding to Si111. The insetshows a typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23.041912-3 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

University of Groningen

   University of GroningenInfluence of random roughness on cantilever curvature sensitivityErgincan, O.; Palasantzas, G.; Kooi, B. J.Published in:Applied Physics LettersDOI:10.1063/1.329971310.1063/1.3299713IMPORTANT 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):Ergincan, O., Palasantzas, G., & Kooi, B. J. (2010). Influence of random roughness on cantilever curvaturesensitivity. Applied Physics Letters, 96(4), 041912-1-041912-3. [041912].https://doi.org/10.1063/1.3299713, https://doi.org/10.1063/1.3299713CopyrightOther 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.Download date: 26-04-2023Influence of random roughness on cantilever curvature sensitivityO. Ergincan, G. Palasantzas,a and B. J. KooiZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 22 October 2009; accepted 5 January 2010; published online 29 January 2010In this work we explore the influence of random surface roughness on the cantilever sensitivity torespond to curvature changes induced by changes in surface stress. The roughness is characterizedby the out-of-plane roughness amplitude w, the lateral correlation length , and the roughness orHurst exponent H0H1. The cantilever sensitivity is found to decrease with increasingroughness decreasing H and/or increasing ratio w / or equivalently increasing local surface slope.Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, , and Hare derived in order to allow direct implementation in sensing systems. © 2010 American Instituteof Physics. doi:10.1063/1.3299713Microelectromechanical/Nanoelectromechanical systemsMEMS/NEMS are important devices that combine inmany cases the advantages of mechanical systems with thespeed and large scale integration of silicon based microelec-tronics. As a result a large number of groups are experiment-ing on various aspects to understand fully the properties andpotential of MEMS/NEMS.1–11 As a matter of fact, microme-chanical cantilevers allow mass resolution down to femto-grams in air environment.12 The sensitivity is determined bythe effective vibratory mass of the resonator determined bygeometry, configuration, and material properties of the reso-nant structure, and the stability of the device resonancefrequency.6Complete understanding of cantilever bending is crucialfor ultrasensitive applications including bending experimentsto monitor the evolution of material properties, monitoringthe surface stress during bending, adsorption studies, self-assembly, thin film deposition, and molecular-recognitionbased on-chip biomedical sensing devices.13–22 Such experi-ments exploit Stoney’s equation, which assumes planar ge-ometry, to relate the cantilever bending to the magnitude ofthe surface stress f change f .23 However, the surfaces ofreal cantilevers and in more general material systems; evenfor the most thoroughly polished surfaces have random sur-face roughness on different lateral length scales.Recently, it was shown that the response of the curvatureof cantilevers to changes in their surface stress depends sig-nificantly on the surface morphology.23 This dependence wasattributed to the transverse coupling between the out-of-plane and in-plane components of the surface-induced stress.Moreover, roughness corrections were introduced, which arehighly important for experiments measuring the surfacestress on nominally planar surfaces.23 However, calculationsof the roughness effects on the cantilever sensitivity as afunction of characteristic parameters of random rough sur-faces are still missing. This will be the topic of the presentletter.Initially, we will present the theory of cantilever bendingincluding the general roughness corrections from Ref. 23,and afterwards we will implement specifics of random self-affine roughness. It has been shown that the in-plane stress Tin a surface layer, which is required to undo the additionalin-plane strain components induced by the stress at equilib-rium in the layer if it was detached from the substrate andallowed to strain freely under the influence of the surfacestress while allowing for free relaxation along the normal isgiven by23T =1hLAroughAs − vL1 − vLs , 1with hL the mean thickness of a layer deposited on oneside of the cantilever surface see inset Fig. 1 and havinga rough surface with area Arough, and a Poisson’s ratio vL.We assume an isotropic and continuously differentiablerough profile hr with r= x ,r the in-plane position vectorhL= hr	 with . . ..	 denoting an ensemble average.The stress components s and s are, respectively, the in-plane and out-of-plane components. They are given, respec-tively, by23 s = A /Aroughf3+cos 2 /4 cos 	 and s= A /Aroughfsin2  /cos 	. f is the position dependent sca-lar surface stress. The angle  inset, Fig. 1 is defined astan =h  and A is the average flat surface area. Substitu-tion in Eq. 1 yields for the stress TaAuthor to whom correspondence should be addressed. Electronic mail:g.palasantzas@rug.nl.FIG. 1. Cantilever sensitivity T /To as a function of local surface slope rmswith a varying correlation lengths 10 nm500 nm, w=1 nm, H=0.5,and two different Poisson’s ratios vL=0.18 corresponding to Si111 andvL=0.28 corresponding to Si100. The inset shows a random rough sur-face of an overlayer on a cantilever surface,  is the inclination angle be-tween the normal vectors n and n̂.APPLIED PHYSICS LETTERS 96, 041912 20100003-6951/2010/964/041912/3/$30.00 © 2010 American Institute of Physics96, 041912-1Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspT =12hL f1 + vL1 − vLcos  +1 − 3vL1 − vLsec  . 2For weak roughness tan =h1 and hLw with w=h−hL2	 the rms roughness amplitude, s is onlyweakly affected by surface roughness, while the dominanteffect arises from the out-of-plane stress component s.Local values of f and  will be correlated due to dependenceof the surface properties on the surface crystallographicorientation.24Furthermore, if we assume a Gaussian height distribu-tion of the height profile which is reasonable in many casesof deposited overlayers and depending on the growthmode,25–29 the rough surface area in Eq. 1 is given by25Arough /A=0+	du1+rms2 ue−u with rms= h2	1/2 the rmslocal surface slope. The latter in terms of Fourier-transformanalysis is given by rms= oQcq2 hq2	d2q1/2 and hq2	the power roughness spectrum.26 Qc= /ao is the integrationlimit with ao the minimum lower roughness cut-off of theorder of atomic dimensions. For weak roughness rms1we obtain Arough /A1+ rms2 /2=1+ 2	 /2 since rms2	 rms=h 2	=tan2 . Note that the first orderperturbative expansion for the surface area ratio Arough /A isalso independent of the assumption of Gaussian height dis-tribution.A wide variety of surfaces exhibit a so-called self-affineroughness. This type of roughness is characterized, besidesthe rms roughness amplitude w, by the lateral correlationlength  indicating the average lateral feature size, andthe roughness exponent 0H1. Small values of H0corresponds to jagged surfaces, while large values of H1to a smooth hill valley morphology.26–29 For self-affineroughness, the power spectrum obeys the scaling behaviorhq2	q−2−2H if q1 and  hq2	constant ifq1.27,28 This scaling is satisfied by the analytic model28hq2	 =aw221 + q221+H. 3The parameter “a” in Eq. 3 is obtained by the normaliza-tion condition 0qQcq2 hq2	d2q=w2 yielding a= H / / 1− 1+Qc22−H.28 This is equivalent to the factthat the height-height correlation Cr= hrh0	= hq2	e−iq ·rd2q obeys the condition: Cr=0=w2. Fur-thermore, we obtain for the local slope rms the analytic ex-pressionrms =w a1 − H 1 + Qc221−H − 1 − 11/2. 4Analytic expressions in the limiting cases H=0 and 1 can beobtained using the identity lnx=limc→01 /BxB−1.Therefore, we obtain rms H=1= w /a ln1+Qc22−11/2with aH=1= 1 /1+ Qc−2, and rms H=0= w /aQc2−11/2 with aH=0=1 /  ln1+Qc22.For weak roughness rms2	1, by consideringthe Taylor expansions cos x1−x2 /2+. . . and sec x1+x2 /2+. . ., Eq. 2 yields after series expansion the simplerform13 Tf	S /hL1−vL / 1−vLrms2  where substitutionof Eq. 4 gives the analytic form for the cantilever sensitiv-ity, which is defined by the ratio T /To,TTo 1 − vL1 − vLw 2 a1 − H 1 + Qc221−H − 1 − 15with To= f	S /hL representing the effective stress for a planarsurface with the same surface stress as the rough surface.Figure 1 shows the cantilever sensitivity T /To as a func-tion of the local slope rms. In fact, Eq. 5 defines a limitingvalue of the local slope rms for which T=0, yieldingrms max=1−vL /vL. For Poisson’s ratios vL=0.18Si111 Ref. 30 and vL=0.28 Si100 Ref. 30 we ob-tain, respectively, rms/max=2.13 and 1.6. For a metallic over-layer, as gold widely used to coat cantilevers with vL=0.44,30 we obtain rms/max=1.12. These are relatively sig-nificant values for rms and the pertubative expansion of Eq.5 is valid only for local slopes rms1. It is clear e.g.,from Fig. 3 and Eq. 5 that the local surface slope influenceis minimized with decreasing Poisson’s ratio. The cantileversensitivity can decrease significantly with changing crystal-lographic structures corresponding to different Poisson’s ra-tios and the effect becomes more pronounced when the localslope increases or equivalently the surface roughness in-creases.In order to obtain a direct feeling of the influence of thecharacteristic roughness parameters w , ,H, we present inFig. 2 the cantilever sensitivity T /To as a function of the longwavelength roughness ratio w / for various roughness expo-nents H. The inset shows simultaneously the correspondinglocal slopes to ensure that our calculations are performed forrms1 and thus to ensure validity of the present formalism.It is clear that for lower roughness exponents H and/or largerroughness ratios w / the cantilever sensitivity decreasesrather drastically. A similar behavior is depicted in Fig. 3,where the cantilever sensitivity is depicted as a function ofw / for two different Poisson’s ratios vL. The inset, showinga typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23, indicates thatthe random roughness parameters used in this study are oftenmet in experimental coatings depending on material andpreparation conditions.In conclusion, random surface roughness influences thecantilever sensitivity T /To to respond to changes in the as-sociated surface stress when the cantilever sensor is usedFIG. 2. Cantilever sensitivity T /To as a function of long wavelength rough-ness ratio w / for w=1 nm and H=0.5, and two different Poisson’s ratiosvL=0.18 corresponding to Si111 and vL=0.28 corresponding toSi100.041912-2 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspwith a sensing layer on top. In more detail the cantileversensitivity T /To is found to decrease with increasing localsurface slope or equivalently increasing surface roughnessdecreasing H and/or increasing ratio w /. Even weak localsurface slopes are shown to have a significant effect on thecantilever sensitivity T /To. Finally, the analytic expressionsderived for the cantilever sensitivity T /To as a function ofcharacteristic roughness parameters allow direct implemen-tation in sensing systems if measurements of the character-istic roughness parameters of cantilever surfaces are per-formed and the corresponding roughness correction is takeninto account for the surface stress T. This is desirable sincecantilever bending studies are a major source for current ex-perimental data of surface stress T.We would like to acknowledge financial support by theSTW under Grant No. 10082.1A. B. Hutchinson, P. Truitt, L. Sekaric, J. M. Parpia, H. G. Craighead, J. E.Butler, and K. C. Schwab, Appl. Phys. Lett. 84, 972 2004.2J. Barnaś and G. Palasantzas, J. Appl. Phys. 82, 3950 1997.3A. N. Cleland and M. L. Roukes, Appl. Phys. Lett. 69, 2653 1996.4K. L. Ekinci and M. L. Roukes, Rev. Sci. Instrum. 76, 061101 2005.5K. L. Ekinci, Y. T. Yang, X. M. H. Huang, and M. L. Roukes, Appl. Phys.Lett. 81, 2253 2002.6K. L. Ekinci, Y. T. Yang, and M. L. Roukes, J. Appl. Phys. 95, 26822004.7K. C. Schwab and M. L. Roukes, Phys. Today 58, 36 2005.8V. Sazonova, Y. Yaish, H. Ustunel, D. Roundy, T. A. Arias, and P. L.McEuen, Nature London 431, 284 2004.9K. Y. Yasumura, T. D. Stowe, E. M. Chow, T. Pfafman, T. W. Kenny, B.C. Stipe, and D. Rugar, J. Microelectromech. Syst. 9, 117 2000.10D. W. Carr, S. Evoy, L. Sekaric, H. G. Craighead, and J. M. Parpia, Appl.Phys. Lett. 75, 920 1999.11S. Evoy, A. Olkhovets, L. Sekaric, J. M. Parpia, and H. G. Craighead,Appl. Phys. Lett. 77, 2397 2000.12N. V. Lavrik, M. J. Sepaniak, and P. G. Datskos, Rev. Sci. Instrum. 75,2229 2004.13R. Desikan, I. Lee, and T. Thundat, Ultramicroscopy 106, 795 2006.14M. Godin, P. J. Williams, V. Tabard-Cossa, O. Laroche, L. Y. Beaulieu, R.B. Lennox, and P. Grutter, Langmuir 20, 7090 2004.15C. W. Pao and D. J. Srolovitz, Phys. Rev. Lett. 96, 186103 2006.16C. Friesen and C. V. Thompson, Phys. Rev. Lett. 89, 126103 2002.17H. Ibach, Surf. Sci. Rep. 29, 195 1997.18M. Seo and M. Yamazaki, J. Solid State Electrochem. 11, 1365 2007.19M. J. Harrison, D. P. Woodruff, J. Robinson, D. Sander, W. Pan, and J.Kirschner, Phys. Rev. B 74, 165402 2006.20N. Vasiljevic, T. Trimble, N. Dimitrov, and K. Sieradzki, Langmuir 20,6639 2004.21J. A. Floro, E. Chason, R. C. Cammarata, and D. J. Srolovitz, MRS Bull.27, 19 2002.22F. Spaepen, Acta Mater. 48, 31 2000.23J. Weissmüller and H. L. Duan, Phys. Rev. Lett. 101, 146102 2008.24J. J. Métois, A. Saul, and P. Muller, Nature Mater. 4, 238 2005.25B. N. J. Persson and E. Tosatti, J. Chem. Phys. 115, 5597 2001.26G. Palasantzas, Phys. Rev. E 56, 1254 1997.27J. Krim and G. Palasantzas, Int. J. Mod. Phys. B 9, 599 1995; P. Meakin,Phys. Rep. 235, 1991 1994; Y. -P. Zhao, G. -C. Wang, and T. -M. Lu,Experimental Methods in the Physical Science Academic, New York,2001, Vol. 37.28G. Palasantzas, Phys. Rev. B 48, 14472 1993; 49, 5785 1994.29G. Palasantzas and J. Krim, Phys. Rev. Lett. 73, 3564 1994.30W. A. Brantley, J. Appl. Phys. 44, 534 1973.FIG. 3. Color online Cantilever sensitivity T /To as a function of longwavelength roughness ratio w / for w=1 nm, different roughness expo-nents H, and Poisson’s ratio vL=0.18 corresponding to Si111. The insetshows a typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23.041912-3 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

In this work we explore the influence of random surface roughness on the cantilever sensitivity to respond to curvature changes induced by changes in surface stress. The roughness is characterized by the out-of-plane roughness amplitude w, the lateral correlation length ξ, and the roughness or Hurst exponent H(0<H<1). The cantilever sensitivity is found to decrease with increasing roughness (decreasing H and/or increasing ratio w/ξ) or equivalently increasing local surface slope. Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, ξ, and H are derived in order to allow direct implementation in sensing systems.

Ergincan, O.,

Palasantzas, G.,

Kooi, B.J.,

University of Groningen Digital Archive

Response to “Comment on ‘Inﬂuence of random roughness on cantilever
curvature sensitivity’ ” †Appl. Phys. Lett. 96, 226101 „2010…‡
O. Ergincan, G. Palasantzas,
a and B. J. Kooi
Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands
Received 26 April 2010; accepted 11 May 2010; published online 1 June 2010
doi:10.1063/1.3442494
In our paper, we state after Eq. 5, page 041912-2 Ref.
1 that “Fig. 1 shows the cantilever sensitivity T/To as a
function of the local slope rms. In fact, Eq. 5 deﬁnes a
limiting value of the local slope rms for which T=0, yielding
rmsmax=1−vL/vL. For Poisson ratios vL=0.18 Si111
Ref. 30 and vL=0.28 Si100 Ref. 30 we obtain, respec-
tively, rms/max=2.13 and rms/max=1.6. For a metallic over-
layer as gold widely used to coat cantilevers with vL
=0.44 Ref. 30 we obtain rms/max=1.12. These are relatively
signiﬁcant values for rms and the perturbative expansion of
Eq. 5 is valid only for local slopes rms1.….”
Therefore as we explain in our paper the validity of the
approximate formula is for roughness parameters that lead to
local slopes rms1. Although in a strict sense we must have
rms1, the expansion in powers of rms
2 multiplied by
vL/1−vL1 limits the contribution of higher order terms
rms
2n n1 signiﬁcantly. Around the regime rms1 or ef-
fectively 45° one has to consider higher order terms in
	2
 in the expansion of the generic Eq. 1 in the comment
or Eq. 2 in Ref. 1. In any case as stated in our paper, our
calculations were performed for local slopes 0rms1 cor-
responding effectively to inclinations tan−1 rms45°.
Moreover, as one can observe from Fig. 1 made from the
commenting authors,
2 for inclinations below 45° the
agreement between Eq. 5 in Ref. 1 and the full calculation
shown by the commenting authors
2 is reasonably good for
both Au and Si. Therefore, for inclinations 45° our ana-
lytic formula, as it is shown also by the commenting
authors,
2 is having the correct behavior, while any discussion
for angles 45° is not relevant to our paper since we do
not consider this regime. In any case, it came to our attention
that due to error in our original publication,
1 Figs. 2 and 3
are not the correct ones and for this reason we have submit-
ted an erratum.
1O. Ergincan, G. Palasantzas, and B. J. Kooi, Appl. Phys. Lett. 96, 041912
2010.
2Y. Wang, J. Weissmuller, and H. Duan, Appl. Phys. Lett. 96, 226101
2010.
aAuthor to whom correspondence should be addressed. Electronic mail:
g.palasantzas@rug.nl.
APPLIED PHYSICS LETTERS 96, 226102 2010
0003-6951/2010/9622/226102/1/$30.00 © 2010 American Institute of Physics 96, 226102-1
Downloaded 03 Jun 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

   University of GroningenInfluence of random roughness on cantilever curvature sensitivityErgincan, O.; Palasantzas, G.; Kooi, B. J.Published in:Applied Physics LettersDOI:10.1063/1.3299713IMPORTANT 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):Ergincan, O., Palasantzas, G., & Kooi, B. J. (2010). Influence of random roughness on cantilever curvaturesensitivity. Applied Physics Letters, 96(4), 041912-1-041912-3. [041912]. https://doi.org/10.1063/1.3299713CopyrightOther 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).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.Download date: 12-11-2019Erratum: “Influence of random roughness on cantilever curvaturesensitivity” †Appl. Phys. Lett. 96, 041912 „2010…‡O. Ergincan, G. Palasantzas,a and B. J. KooiZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 26 April 2010; accepted 27 April 2010; published online 1 June 2010doi:10.1063/1.3442492Due to error Figs. 2 and 3 and captions shown in the published paper are not the correct ones. Results, discussion, andconclusions of the paper remain unaltered by this correction.aAuthor to whom correspondence should be addressed. Electronic mail: g.palasantzas@rug.nl.FIG. 2. Cantilever sensitivity T /T0 as a function of long wavelength rough-ness ratio w / for w=1 nm, different roughness exponents H, and Poissonratio vL=0.18 corresponding to Si111. The inset shows the dependenceof the average local surface slope on the roughness ratio w / for differentroughness exponents H.FIG. 3. Color online Cantilever sensitivity T /T0 as a function of longwavelength roughness ratio w / for w=1 nm, H=0.5, and two differentPoisson ratios, vL=0.18 corresponding to Si111 and vL=0.28 corre-sponding to Si100. The inset shows a typical gold rough surface depositedonto Si with H=0.9, w=7 nm, and =30 nm yielding w /=0.23.APPLIED PHYSICS LETTERS 96, 229901 20100003-6951/2010/9622/229901/1/$30.00 © 2010 American Institute of Physics96, 229901-1Downloaded 03 Jun 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

University of Groningen Research Database

  
 University of Groningen
Influence of random roughness on cantilever curvature sensitivity
Ergincan, O.; Palasantzas, Georgios; Kooi, Bart
Published in:
Applied Physics Letters
DOI:
10.1063/1.3299713
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2010
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Ergincan, O., Palasantzas, G., & Kooi, B. J. (2010). Influence of random roughness on cantilever curvature
sensitivity. Applied Physics Letters, 96(4), 041912-1-041912-3. [041912]. DOI: 10.1063/1.3299713
Copyright
Other 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 the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 10-02-2018
Response to “Comment on ‘Influence of random roughness on cantilever
curvature sensitivity’ ” †Appl. Phys. Lett. 96, 226101 „2010…‡
O. Ergincan, G. Palasantzas,a and B. J. Kooi
Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands
Received 26 April 2010; accepted 11 May 2010; published online 1 June 2010
doi:10.1063/1.3442494
In our paper, we state after Eq. 5, page 041912-2 Ref.
1 that “Fig. 1 shows the cantilever sensitivity T /To as a
function of the local slope rms. In fact, Eq. 5 defines a
limiting value of the local slope rms for which T=0, yielding
rms max=1−vL /vL. For Poisson ratios vL=0.18 Si111
Ref. 30 and vL=0.28 Si100 Ref. 30 we obtain, respec-
tively, rms/max=2.13 and rms/max=1.6. For a metallic over-
layer as gold widely used to coat cantilevers with vL
=0.44 Ref. 30 we obtain rms/max=1.12. These are relatively
significant values for rms and the perturbative expansion of
Eq. 5 is valid only for local slopes rms1.….”
Therefore as we explain in our paper the validity of the
approximate formula is for roughness parameters that lead to
local slopes rms1. Although in a strict sense we must have
rms1, the expansion in powers of rms
2 multiplied by
vL / 1−vL1 limits the contribution of higher order terms
rms
2n n1 significantly. Around the regime rms1 or ef-
fectively 45° one has to consider higher order terms in
	2
 in the expansion of the generic Eq. 1 in the comment
or Eq. 2 in Ref. 1. In any case as stated in our paper, our
calculations were performed for local slopes 0rms1 cor-
responding effectively to inclinations tan−1 rms45°.
Moreover, as one can observe from Fig. 1 made from the
commenting authors,2 for inclinations below 45° the
agreement between Eq. 5 in Ref. 1 and the full calculation
shown by the commenting authors2 is reasonably good for
both Au and Si. Therefore, for inclinations 45° our ana-
lytic formula, as it is shown also by the commenting
authors,2 is having the correct behavior, while any discussion
for angles 45° is not relevant to our paper since we do
not consider this regime. In any case, it came to our attention
that due to error in our original publication,1 Figs. 2 and 3
are not the correct ones and for this reason we have submit-
ted an erratum.
1O. Ergincan, G. Palasantzas, and B. J. Kooi, Appl. Phys. Lett. 96, 041912
2010.
2Y. Wang, J. Weissmuller, and H. Duan, Appl. Phys. Lett. 96, 226101
2010.
aAuthor to whom correspondence should be addressed. Electronic mail:
g.palasantzas@rug.nl.
APPLIED PHYSICS LETTERS 96, 226102 2010
0003-6951/2010/9622/226102/1/$30.00 © 2010 American Institute of Physics96, 226102-1
Downloaded 03 Jun 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp


O. Ergincan

G. Palasantzas

B. J. Kooi

Crossref

   University of GroningenInfluence of random roughness on cantilever curvature sensitivityErgincan, O.; Palasantzas, G.; Kooi, B. J.Published in:Applied Physics LettersDOI:10.1063/1.329971310.1063/1.3299713IMPORTANT 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):Ergincan, O., Palasantzas, G., & Kooi, B. J. (2010). Influence of random roughness on cantilever curvaturesensitivity. Applied Physics Letters, 96(4), 041912-1-041912-3. Article 041912.https://doi.org/10.1063/1.3299713, https://doi.org/10.1063/1.3299713CopyrightOther 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.Download date: 09-09-2023Influence of random roughness on cantilever curvature sensitivityO. Ergincan, G. Palasantzas,a and B. J. KooiZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The NetherlandsReceived 22 October 2009; accepted 5 January 2010; published online 29 January 2010In this work we explore the influence of random surface roughness on the cantilever sensitivity torespond to curvature changes induced by changes in surface stress. The roughness is characterizedby the out-of-plane roughness amplitude w, the lateral correlation length , and the roughness orHurst exponent H0H1. The cantilever sensitivity is found to decrease with increasingroughness decreasing H and/or increasing ratio w / or equivalently increasing local surface slope.Finally, analytic expressions of the cantilever sensitivity as a function of the parameters w, , and Hare derived in order to allow direct implementation in sensing systems. © 2010 American Instituteof Physics. doi:10.1063/1.3299713Microelectromechanical/Nanoelectromechanical systemsMEMS/NEMS are important devices that combine inmany cases the advantages of mechanical systems with thespeed and large scale integration of silicon based microelec-tronics. As a result a large number of groups are experiment-ing on various aspects to understand fully the properties andpotential of MEMS/NEMS.1–11 As a matter of fact, microme-chanical cantilevers allow mass resolution down to femto-grams in air environment.12 The sensitivity is determined bythe effective vibratory mass of the resonator determined bygeometry, configuration, and material properties of the reso-nant structure, and the stability of the device resonancefrequency.6Complete understanding of cantilever bending is crucialfor ultrasensitive applications including bending experimentsto monitor the evolution of material properties, monitoringthe surface stress during bending, adsorption studies, self-assembly, thin film deposition, and molecular-recognitionbased on-chip biomedical sensing devices.13–22 Such experi-ments exploit Stoney’s equation, which assumes planar ge-ometry, to relate the cantilever bending to the magnitude ofthe surface stress f change f .23 However, the surfaces ofreal cantilevers and in more general material systems; evenfor the most thoroughly polished surfaces have random sur-face roughness on different lateral length scales.Recently, it was shown that the response of the curvatureof cantilevers to changes in their surface stress depends sig-nificantly on the surface morphology.23 This dependence wasattributed to the transverse coupling between the out-of-plane and in-plane components of the surface-induced stress.Moreover, roughness corrections were introduced, which arehighly important for experiments measuring the surfacestress on nominally planar surfaces.23 However, calculationsof the roughness effects on the cantilever sensitivity as afunction of characteristic parameters of random rough sur-faces are still missing. This will be the topic of the presentletter.Initially, we will present the theory of cantilever bendingincluding the general roughness corrections from Ref. 23,and afterwards we will implement specifics of random self-affine roughness. It has been shown that the in-plane stress Tin a surface layer, which is required to undo the additionalin-plane strain components induced by the stress at equilib-rium in the layer if it was detached from the substrate andallowed to strain freely under the influence of the surfacestress while allowing for free relaxation along the normal isgiven by23T =1hLAroughAs − vL1 − vLs , 1with hL the mean thickness of a layer deposited on oneside of the cantilever surface see inset Fig. 1 and havinga rough surface with area Arough, and a Poisson’s ratio vL.We assume an isotropic and continuously differentiablerough profile hr with r= x ,r the in-plane position vectorhL= hr	 with . . ..	 denoting an ensemble average.The stress components s and s are, respectively, the in-plane and out-of-plane components. They are given, respec-tively, by23 s = A /Aroughf3+cos 2 /4 cos 	 and s= A /Aroughfsin2  /cos 	. f is the position dependent sca-lar surface stress. The angle  inset, Fig. 1 is defined astan =h  and A is the average flat surface area. Substitu-tion in Eq. 1 yields for the stress TaAuthor to whom correspondence should be addressed. Electronic mail:g.palasantzas@rug.nl.FIG. 1. Cantilever sensitivity T /To as a function of local surface slope rmswith a varying correlation lengths 10 nm500 nm, w=1 nm, H=0.5,and two different Poisson’s ratios vL=0.18 corresponding to Si111 andvL=0.28 corresponding to Si100. The inset shows a random rough sur-face of an overlayer on a cantilever surface,  is the inclination angle be-tween the normal vectors n and n̂.APPLIED PHYSICS LETTERS 96, 041912 20100003-6951/2010/964/041912/3/$30.00 © 2010 American Institute of Physics96, 041912-1Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspT =12hL f1 + vL1 − vLcos  +1 − 3vL1 − vLsec  . 2For weak roughness tan =h1 and hLw with w=h−hL2	 the rms roughness amplitude, s is onlyweakly affected by surface roughness, while the dominanteffect arises from the out-of-plane stress component s.Local values of f and  will be correlated due to dependenceof the surface properties on the surface crystallographicorientation.24Furthermore, if we assume a Gaussian height distribu-tion of the height profile which is reasonable in many casesof deposited overlayers and depending on the growthmode,25–29 the rough surface area in Eq. 1 is given by25Arough /A=0+	du1+rms2 ue−u with rms= h2	1/2 the rmslocal surface slope. The latter in terms of Fourier-transformanalysis is given by rms= oQcq2 hq2	d2q1/2 and hq2	the power roughness spectrum.26 Qc= /ao is the integrationlimit with ao the minimum lower roughness cut-off of theorder of atomic dimensions. For weak roughness rms1we obtain Arough /A1+ rms2 /2=1+ 2	 /2 since rms2	 rms=h 2	=tan2 . Note that the first orderperturbative expansion for the surface area ratio Arough /A isalso independent of the assumption of Gaussian height dis-tribution.A wide variety of surfaces exhibit a so-called self-affineroughness. This type of roughness is characterized, besidesthe rms roughness amplitude w, by the lateral correlationlength  indicating the average lateral feature size, andthe roughness exponent 0H1. Small values of H0corresponds to jagged surfaces, while large values of H1to a smooth hill valley morphology.26–29 For self-affineroughness, the power spectrum obeys the scaling behaviorhq2	q−2−2H if q1 and  hq2	constant ifq1.27,28 This scaling is satisfied by the analytic model28hq2	 =aw221 + q221+H. 3The parameter “a” in Eq. 3 is obtained by the normaliza-tion condition 0qQcq2 hq2	d2q=w2 yielding a= H / / 1− 1+Qc22−H.28 This is equivalent to the factthat the height-height correlation Cr= hrh0	= hq2	e−iq ·rd2q obeys the condition: Cr=0=w2. Fur-thermore, we obtain for the local slope rms the analytic ex-pressionrms =w a1 − H 1 + Qc221−H − 1 − 11/2. 4Analytic expressions in the limiting cases H=0 and 1 can beobtained using the identity lnx=limc→01 /BxB−1.Therefore, we obtain rms H=1= w /a ln1+Qc22−11/2with aH=1= 1 /1+ Qc−2, and rms H=0= w /aQc2−11/2 with aH=0=1 /  ln1+Qc22.For weak roughness rms2	1, by consideringthe Taylor expansions cos x1−x2 /2+. . . and sec x1+x2 /2+. . ., Eq. 2 yields after series expansion the simplerform13 Tf	S /hL1−vL / 1−vLrms2  where substitutionof Eq. 4 gives the analytic form for the cantilever sensitiv-ity, which is defined by the ratio T /To,TTo 1 − vL1 − vLw 2 a1 − H 1 + Qc221−H − 1 − 15with To= f	S /hL representing the effective stress for a planarsurface with the same surface stress as the rough surface.Figure 1 shows the cantilever sensitivity T /To as a func-tion of the local slope rms. In fact, Eq. 5 defines a limitingvalue of the local slope rms for which T=0, yieldingrms max=1−vL /vL. For Poisson’s ratios vL=0.18Si111 Ref. 30 and vL=0.28 Si100 Ref. 30 we ob-tain, respectively, rms/max=2.13 and 1.6. For a metallic over-layer, as gold widely used to coat cantilevers with vL=0.44,30 we obtain rms/max=1.12. These are relatively sig-nificant values for rms and the pertubative expansion of Eq.5 is valid only for local slopes rms1. It is clear e.g.,from Fig. 3 and Eq. 5 that the local surface slope influenceis minimized with decreasing Poisson’s ratio. The cantileversensitivity can decrease significantly with changing crystal-lographic structures corresponding to different Poisson’s ra-tios and the effect becomes more pronounced when the localslope increases or equivalently the surface roughness in-creases.In order to obtain a direct feeling of the influence of thecharacteristic roughness parameters w , ,H, we present inFig. 2 the cantilever sensitivity T /To as a function of the longwavelength roughness ratio w / for various roughness expo-nents H. The inset shows simultaneously the correspondinglocal slopes to ensure that our calculations are performed forrms1 and thus to ensure validity of the present formalism.It is clear that for lower roughness exponents H and/or largerroughness ratios w / the cantilever sensitivity decreasesrather drastically. A similar behavior is depicted in Fig. 3,where the cantilever sensitivity is depicted as a function ofw / for two different Poisson’s ratios vL. The inset, showinga typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23, indicates thatthe random roughness parameters used in this study are oftenmet in experimental coatings depending on material andpreparation conditions.In conclusion, random surface roughness influences thecantilever sensitivity T /To to respond to changes in the as-sociated surface stress when the cantilever sensor is usedFIG. 2. Cantilever sensitivity T /To as a function of long wavelength rough-ness ratio w / for w=1 nm and H=0.5, and two different Poisson’s ratiosvL=0.18 corresponding to Si111 and vL=0.28 corresponding toSi100.041912-2 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspwith a sensing layer on top. In more detail the cantileversensitivity T /To is found to decrease with increasing localsurface slope or equivalently increasing surface roughnessdecreasing H and/or increasing ratio w /. Even weak localsurface slopes are shown to have a significant effect on thecantilever sensitivity T /To. Finally, the analytic expressionsderived for the cantilever sensitivity T /To as a function ofcharacteristic roughness parameters allow direct implemen-tation in sensing systems if measurements of the character-istic roughness parameters of cantilever surfaces are per-formed and the corresponding roughness correction is takeninto account for the surface stress T. This is desirable sincecantilever bending studies are a major source for current ex-perimental data of surface stress T.We would like to acknowledge financial support by theSTW under Grant No. 10082.1A. B. Hutchinson, P. Truitt, L. Sekaric, J. M. Parpia, H. G. Craighead, J. E.Butler, and K. C. Schwab, Appl. Phys. Lett. 84, 972 2004.2J. 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J. Mod. Phys. B 9, 599 1995; P. Meakin,Phys. Rep. 235, 1991 1994; Y. -P. Zhao, G. -C. Wang, and T. -M. Lu,Experimental Methods in the Physical Science Academic, New York,2001, Vol. 37.28G. Palasantzas, Phys. Rev. B 48, 14472 1993; 49, 5785 1994.29G. Palasantzas and J. Krim, Phys. Rev. Lett. 73, 3564 1994.30W. A. Brantley, J. Appl. Phys. 44, 534 1973.FIG. 3. Color online Cantilever sensitivity T /To as a function of longwavelength roughness ratio w / for w=1 nm, different roughness expo-nents H, and Poisson’s ratio vL=0.18 corresponding to Si111. The insetshows a typical gold rough surface deposited onto Si with H=0.9,w=7 nm, and =30 nm yielding w /=0.23.041912-3 Ergincan, Palasantzas, and Kooi Appl. Phys. Lett. 96, 041912 2010Downloaded 01 Feb 2010 to 129.125.63.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

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Influence of random roughness on cantilever curvature sensitivity

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