Surface stress effects on the resonance properties of cantilever sensors

Pin Lu,1 H. P. Lee,1 C. Lu,1 and S. J. O’Shea2

1Institute of High Performance Computing, 1 Science Park Road, Science Park II, Singapore 117528, Singapore2Institute of Materials Research & Engineering, 3 Research Link, Singapore 117602, Singapore

�Received 16 March 2005; published 1 August 2005�

The effect of surface stress on the resonance frequency of a cantilever sensor is modeled analytically byincorporating strain-dependent surface stress terms into the equations of motion. This mechanistic approachcan be equated with a corresponding thermodynamic description, allowing basic equations to be derived thatlink the analysis to experimentally determined parameters. Examples are shown for the cases of a pure surfacestress and an adsorption-induced surface stress, and indicate that frequency measurements may be useful forfundamental understanding of surface and adsorption-induced stresses on metals, semiconductors, and nanos-cale structures. Application to biomolecular adsorption sensors appears unlikely.

DOI: 10.1103/PhysRevB.72.085405 PACS number�s�: 68.35.Gy, 62.40.�i, 68.37.Ps, 85.85.�j

Microfabricated cantilever structures have been demon-strated to be extremely versatile sensors1–3 and have potentialapplications in physical, chemical, and biologicalsciences.1–5 Adsorption on a functionalized surface of a mi-crocantilever may induce mass, damping, and stress changesof the cantilever response. One cantilever sensor technique isto monitor changes in the cantilever resonance frequency,and this method seems best suited for use in gaseous orvacuum environments to measure adsorption-induced massloading of the cantilever. For work in liquid environments,measurement of the static bending of a cantilever arisingfrom surface stress was proposed as the most appropriatemethod for detecting biomolecular adsorption6,7,10 in liquids,and this indeed appears to be the case from a practicalviewpoint.4,8 However, there is minimal understanding ofhow surface stress influences the resonance frequency of acantilever. As we show below, this problem is of interest inthe quantitative measurement of mass and stress loads foundusing resonant frequency sensors; as a method to measurefundamental properties of surface stress; and of increasingimportance to mechanical design as structures scale towardnanometer dimensions, including the development of poten-tial nanoscale stress sensors.

Both theoretical and experimental studies9–12 have beenundertaken to investigate surface stress effects in microcan-tilevers. In this study we give a simple analytical descriptionof surface stress effects and show how frequency measure-ments may be used to evaluate surface stresses. Previoustheoretical analyses on the problem of adsorption-inducedsurface stress changes in cantilever resonance followed thetreatment given by Chen et al.,10 in which the differentialsurface stress induced by adsorption is simplified to externalaxial forces exerted on the cantilever. In this way, the prob-lem of a self-balanced cantilever deformation due to mis-match stress without any external forces has been replacedby a problem of bending or vibration of the cantilever underan applied force. Although a taunt string model10 and a beamwith axial force model11,12 have been suggested based on thissimplification, neither approach represents the correct physi-cal model. The effective external forces are considerablyoverestimated in these models because in the real situationthe cantilever has a free end to allow deformation or bending

to relieve the stress. For a cantilever with a free end, changesin cantilever resonance frequency due to small changes incurvature �i.e., bending� have been shown to be negligible.13

In what follows we consider changes in mechanical reso-nance frequency that arise from a surface stress, not changesin the bulk elastic properties of the cantilever that may ariseon adsorption, e.g., by alloying,4 material changes,10 ordeposition14 of relatively thick films to create compositebeam structures. Further, we use the term surface stress todenote stress in the neighborhood of the surface of the bulk.This includes pure surface stress �i.e., of a free surface�, in-terfacial stress, and stress within a very thin adsorbed mate-rial.

Consider a cantilever used as a sensor. The experimentalquantity measured is the surface stress difference, ��=�u−�l, where �u and �l are the surface stresses on the upper andthe lower surfaces, respectively �see Fig. 1�. The stress termscan be described in a full tensor form. However, for clarityand with no loss of applicability, we simplify the problem byassuming a one-dimensional system with scalar, isotropicmaterial properties. In this case, and to first order in thestrain, the surface stresses may be written,15,16

�u = au + bu�u, �l = al + bl�l, �1�

where a is the strain-independent surface stress, b is a con-stant associated with the surface strain, � is the surface strainmeasured from the prestressed configuration, and the sub-scripts u and l always refer to the upper and lower surface,respectively. The surface stress difference can be writtenfrom Eq. �1� as

�� = ��0 + ��1 �2�

with ��0=au−al and ��1=bu�u−bl�l. The influence of thesurface stress on the resonance frequency of the cantileversensor can be investigated by considering the influences dueto ��0 and ��1. This approach follows from that of Gurtinet al.17 who described a cantilever with equal stress on topand bottom surfaces15 �see Fig. 2�a��. This work has beenoverlooked in more recent research and it seems timely toreintroduce their idea, which we extend to include the effects

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of surface adsorption and unequal stresses on top and bottomsurfaces �see Fig. 2�b��.

We first consider the strain independent term, ��0, whichcan be converted to a static deflection or a static internalstrain of the cantilever �similar to a temperature-inducedthermal strain�. If the relationship between ��0 and the in-ternal prestrain can be obtained, the influence of ��0 on thedynamic properties of the cantilever sensor can be estimatedusing the model of a prestrained cantilever vibration.

An end-part free body diagram cut in any section of thecantilever is drawn as shown in Fig. 1. If au and al are notequal, the cantilever undergoes not only axial but also bend-ing deformations, which is more complicated than the situa-tion discussed by Gurtin et al.17 The strain induced in thecantilever can be written as

�0�z� = �z − hn�/R , �3�

where R is the radius of curvature and hn the distance of theneutral plane measured from the lower surface. Except forantisymmetric surface stresses, the neutral plane is generallynot in the center of the sheet.18,19 The bulk stress �0�z� canbe expressed as

�0�z� = E�z − hn�/R , �4�

where E is Young’s modulus. Note that if the system has aplanar geometry, E is replaced by an isotropic biaxial modu-lus, E / �1−��, where � is Poisson’s ratio. The constants hn

and R can be determined from the equilibrium relations,

�0

h

�0�z�dz + au + al = 0 and �0

h

�0�z�zdz + auh = 0,

to give Stoney’s equation,20

hn =1

EhR���0 + 2al� +

h

2,

1

R= −

6

Eh2��0. �5�

The residual strain due to the surface stress is then obtainedas

�0�z� = −2

Eh�al + �3

hz − 1���0 . �6�

Equation �6� expresses how the influence of ��0 on the can-tilever mechanical deformation has been converted to thecoupling of the bending extension with the equivalent pre-strain, �0�z�.

We now consider the cantilever vibration. The elasticstress-strain relation of the cantilever in vibration can be ex-pressed as ��x ,z , t�=E���x ,z , t�−�0�z��, where � is the totalelastic strain in the cantilever and is given by

� = − �z − h/2��2w�x,t�/�x2, �7�

where w is the central plane deflection of the cantilever. Theresultant moment M on any cross section of the cantilever is

M�x,t� = �0

h �z −h

2��dz = − EI

�2w

�x2 +h

2��0, �8�

where I=h3 /12 is the moment of inertia. Note that ��0 pro-duces a constant bending moment on the cross section. Usingthe resultant moment given in Eq. �8� the beam bendingequation is

EI�4w/�x4 + �A�2w/�t2 = 0, �9�

where �A is the mass per unit length. Since the equation ofmotion is unchanged21 by ��0, the strain-independent part ofthe surface stress has no effect on the resonance frequenciesof the cantilever. This is the same conclusion stated by Gur-tin et al.17 for identical upper and lower surface stresses�au=al� but contradicts more recent modeling of theproblem,10–12 which leads to changed equations of motiondependant on the constant value �au ,al� of the surface stress.

We now consider the strain-dependent stress term, ��1.By replacing ��0 in Eq. �8� with the total surface stress dif-ference �� defined in Eq. �2�, the bending moment can beobtained as

M�x,t� = − �1 +3

Eh�bl + bu�EI

�2w

�x2 +h

2��0. �10�

Thus the effective modulus �E*I� to be used in the equationsof motion is

E*I = �1 + 3bl + bu

EhEI . �11�

From Eq. �10� the change in fundamental resonance fre-quency due to surface stress ���stress� can be found as

�stress2 − �0

2

�02 =

E* − E

E= 3

bl + bu

Eh 2

��stress

�0, �12�

where �0 is the fundamental resonance frequency with nosurface stress and �stress �=�0+��stress� is the new resonancefrequency with surface stresses acting. Note that the effect ofsurface stress depends on the size �h� of the structure22 and

FIG. 1. Schematic view of the free-body diagram of the end partof a cantilever.

FIG. 2. Schematic showing the general cases considered for �a�a cantilever with uniform surface stress on upper and lower surfacesand no adsorption; �b� the cantilever of �a� with adsorption on oneside of the cantilever, causing a surface stress difference betweenthe upper and lower surfaces.

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��stress can be positive or negative depending on the sign ofb.

Equation �12� has been left in a simple format as theparameters a and b can take various forms depending onspecific models, experiments, and nomenclature. We now il-lustrate with some specific cases.

Case (i): Thermodynamic expression for b. Equation �1� isa mechanistic description of the surface stress. When canti-levers are used as sensors, a thermodynamic approach is gen-erally more useful for which Equation �1�, for the upper sur-face, can be equated to16

�uL = � �u

L

��u�

�=0+ Cu�u, �13�

where C is the surface elastic constant and is the surfacefree energy. The superscript L indicates that the Lagrangecoordinate system is used in which the surface area does notchange during stretching. This definition shows most clearlythe definition of the parameters a and b and is particularlyuseful for atomistic studies of surface stress. The deforma-tion of the surface by a bulk strain ��� introduces an addi-tional thermodynamic excess, namely the surface elastic con-stant.

However, in cantilever experiments, we measure the ac-tual, strained area of surface for which the Euler coordinatesystem �denoted by superscript E� needs to be used and Eq.�1� is written16

�E = ��=0E + �C − ��=0

E ��

=�0 +�E

���

�=0+ �C − �0 −

�E

���

�=0� , �14�

where 0 and ��=0E are the surface free energy and surface

stress, respectively, for the undeformed surface. For claritythe subscripts u and l are removed.

Equations �13� and �14� describe a free surface, i.e., a“pure” surface stress. Adsorption typically generates an ad-ditional interface and a successful approach to extend themodeling in this case is to separate the free surface stress �F�and the interface stresses �H ,G� as23,24

F = +�

��; H = � +

��

��; G = � +

��

�e, �15�

where � is an interfacial surface energy; e is a strain associ-ated with stretching the adsorbed phase with respect to thesubstrate; and F, G, and H are given for an undeformedsurface. The stress terms F and H can be inserted into Eqs.�1�, �12�, and �14� to account for surface stress changes infrequency of a cantilever beam. In particular, Eq. �14� can bewritten

� = a + b� = F + H + �C − F − H�� . �16�

We assume the interface stress term G accounting for slip-ping between the adsorbed film and the substrate has negli-gible effect on the cantilever frequency because by definitionthis term is separated from any strain ��� of the cantilever,i.e., for a frequency change to be measured the physical pa-

rameter changing must couple to the bulk strain of the can-tilever.

Case (ii): Pure surface stress. We first consider whenthere is no adsorption and the top and bottom surfaces areidentical, as shown in Fig. 2�a�. This case represents a puresurface stress of a single material. There is no bending ���=��0=��1=0� and bu=bl. Equation �12� is then identical tothe expression obtained by Gurtin et al.,17 namely,

2��stress

�0= 6

bl

Eh=

6

Eh�C − F� . �17�

To estimate the magnitude of ��stress we consider a sili-con cantilever. Atomistic calculations for the Si �100� surfaceshow that C−11.5 N/m,22 and F−0.5 N/m.25 Using E=130 GPa we find b /E−110−10, which appears to betypical for crystalline silicon and aluminum beams.17,22 Thus,for a Si cantilever of thickness h�1 �m, the value of��stress /�0 is �10−4 or �100 ppm. This frequency shift issmall but measurable. If a cantilever has h�10 nm �e.g., ananowire� then ��stress /�0�10−2, which is a significant ef-fect. Clearly, the extent of the stress-induced resonance fre-quency shift ���stress� is influenced by the cantilever size.Due to the large surface-bulk ratio, the effects of surfacestress on the elastic modulus26–29 and resonance properties ofnanosized structure elements are significant.22

Conversely, the use of nanosized resonators would allowfor considerable improvement in the sensitivity and usage ofsurface stress sensors. For example, Eq. �17� shows is that itmay be possible to find the surface parameter �C-F� of singlecrystal surfaces by measuring frequency changes of ultrathincantilevers of varying thickness.

Case (iii): Adsorption. Given the considerable difficultyin measuring absolute values of surface stress ��u ,�l�, oneinvariably measures only a change in surface stress as thematerial properties of the surfaces are modified. We illustratethis by considering adsorption onto the cantilever giving achange in stress measured before adsorption �Fig. 2�a�� andafter adsorption �Fig. 2�b��. Using Equation �1�, the surfacestress after adsorption is

�u = au + �au + �bu + �bu��u; �l = al + �al + �bl + �bl��l,

�18�

where �a and �b represent the adsorption-induced change inthe stress parameters. Equation �18� assumes a linear varia-tion of b with strain since all our considerations are for elas-tic deformations. The change in fundamental resonance fre-quency �Eq. �12�� can be expressed as

2��stress

�0=

3

Eh�bu + bl� +

3

Eh��bu + �bl� , �19�

with the second term showing the effect of adsorption-induced surface stress.

Interestingly, Eq. �19� shows that surface stress willchange the resonance even if adsorption occurs identicallyon both upper and lower surfaces ��bu=�bl�, whereas nostatic bending would be observed �since ��0=0 in Eq. �5��.However, for most sensor applications, adsorption is con-strained to occur on one surface only. Assuming adsorption

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only occurs on the upper cantilever surface �see Fig. 2�b��then the resonance frequency shift arising solely from theadsorption is

�2��stress

�0�

adsorption=

3�bu

Eh=

3

Eh��C − �F − �H� �20�

with �C, �F, and �H representing changes in C, F, and Hon the upper surface.

Data are not readily available for the elasticity change�C. Nevertheless, we can use known values of the surfacestress terms �F and �H to estimate ���stress /�0�adsorption.For semiconductor surfaces the stress can change greatly onadsorption. For example, on a clean Si �100� surface thecalculated surface stress ��F+�H� changes by �0.1 to�1 N/m depending on the adsorbed species �e.g., As, Ga,Ge� and the surface microstructure.25 Taking typical valuesfor a silicon microcantilever of E=130 GPa and h=1 �m,one finds ���stress /�0�adsorption�10 ppm. This is a measur-able shift using high quality factor cantilevers. The fre-quency shifts will become more pronounced with the use ofultrathin cantilevers. Indeed, there are indications that suchsurface stress effects have been observed in the recent data ofWang et al.30 in which UHV clean �50 nm thick Si �100�and Si �111� cantilevers were exposed to molecular oxygen.The fundamental resonance frequency rapidly increased byapproximately �� /�0�5000 ppm on a low exposure �8Langmuir� to O2. It is highly unlikely that this large fre-quency shift is due to other adsorption effects because �a� anincrease in mass loading would decrease the value of�� /�0, and �b� a change in the bulk stiffness of the cantile-ver by formation of an oxide could increase �� /�0 on ad-sorption but at the low temperatures used by Wang et al. anO2 exposure of 8L will only result in submonolayer coverageof the silicon surface.31

One can also estimate that large stress-induced frequencyshifts will occur for monolayer adsorption of metals.24,25

However, the application of the resonator method for biosen-sors is more problematic because typical surface stresschanges on biomolecular adsorption are usually�0.01 N/m.5,8 The estimated frequency shifts, even for verythin cantilevers, would be �1 ppm. This level of sensitivitycannot be measured at present given that such cantilever bio-sensors must operate in ambient or liquid environments andthis results in high viscous damping of the cantilever �qualityfactor Q 1000�. A high Q factor is essential for measure-ment of surface stress using the cantilever resonator method.

Case (iv): Comparison with other adsorption effects. Wenow compare surface stress effects with other adsorp-tion-induced frequency changes. Equation �8� is the basisof computing changes in resonance frequency due to massloading and stiffness �i.e., effective inertia� changes arisingfrom adsorption.2,14 It is readily shown to first order thatmass and stiffness changes on adsorption shift the fundamen-tal cantilever resonance frequency ��0� by an amount ��mass

given by

2��mass

�0 �3Ef/E − � f/��

hf

h, �21�

where h is thickness, E is Young’s modulus, and � is materialdensity, with the subscript f representing the adsorbed film.

This assumes a uniform coverage on one side of the cantile-ver and a simple composite beam description to model theeffective cantilever stiffness.14 Both conditions can be gen-eralized.

For practical purposes the adsorbed and cantilever mate-rial properties are similar and hence ��mass /�0�hf /h. For atypical cantilever h�1 �m and a monolayer adsorption �i.e.,hf �0.1 nm� gives ��mass /�0�100 ppm. We note that �a�mass loading effects are the same order of magnitude orlarger than surface stress effects, and �b� the sensitivity ofEq. �21� also scales with 1/h. Therefore, for semiconductorand clean metal surfaces, care will be required to distinguishbetween ��stress and ��mass during adsorption. In biomo-lecular systems, adsorption-induced stress effects will causenegligible frequency shift in comparison with mass loading.

Finally, note the difference between the Young’s modulusof the adsorbed film �Ef in Eq. �21�� and the surface elasticity�C�. The elasticity C is a surface excess value, i.e., the dif-ference between the elasticity of the surface and the bulksubstrate. The modulus Ef is a bulk value for the adsorbedfilm. For example, if the adsorbed surface material were en-tirely identical to the bulk, then �C=0 and surface elasticitydoes not change the resonance frequency. In contrast for thisexample Ef =E, and Eq. �21� shows that the stiffness changewould lead to a frequency shift of ��mass /�0=hf /h, whichreflects the change in inertia of the cantilever beam.

To summarize, we have reinvestigated17 the effects of sur-face stress on the fundamental resonance frequency of a can-tilever sensor. Our approach differs from more recent inves-tigations of this problem,10–12 which use a strain-independent�i.e., constant� surface stress, which actually has no effect onthe resonance frequency. The resonance is only influenced bythe strain-dependent surface stresses, which in turn can berelated to a thermodynamic description to allow a compari-son to be made between the analysis and experimental data.

A major conclusion is that frequency measurements maybe used to evaluate surface stresses and such data can com-pliment static beam bending experiments. This may be usefulto study surface stress of semiconductors and metals, al-though the experiments will be challenging as it is difficult todisentangle all of the stress terms. A particularly interestingproblem will be the study of the absolute surface stress ofclean, single crystal surfaces in UHV.30

If the size of the cantilever is reduced to the nano-meter range, the surface stress effects on the mechanicalproperties become considerable, which may be detrimentalor beneficial depending on the application. For example, can-tilever stress sensors for adsorption typically measure a staticbending of the lever but measurement of the resonant fre-quency shift of a nanocantilever would offer some practicalbenefit, such as less instrumental drift. A high quality factoris still required for applications and this appears to rule outthe use of the method for biomolecular sensors requiringliquid environments.

ACKNOWLEDGMENT

We would like to thank David Srolovitz for discussionsand valuable suggestions.

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