Cantilever surface stress sensors with single-crystalline silicon piezoresistorsP. A. Rasmussen, O. Hansen, and A. Boisen Citation: Applied Physics Letters 86, 203502 (2005); doi: 10.1063/1.1900299 View online: http://dx.doi.org/10.1063/1.1900299 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Note: Helical nanobelt force sensors Rev. Sci. Instrum. 83, 126102 (2012); 10.1063/1.4769757 A novel method of temperature compensation for piezoresistive microcantilever-based sensors Rev. Sci. Instrum. 83, 035002 (2012); 10.1063/1.3690380 Design optimization of piezoresistive cantilevers for force sensing in air and water J. Appl. Phys. 106, 064310 (2009); 10.1063/1.3224965 Four point bending setup for characterization of semiconductor piezoresistance Rev. Sci. Instrum. 79, 044703 (2008); 10.1063/1.2908428 Parylene cantilevers integrated with polycrystalline silicon piezoresistors for surface stress sensing Appl. Phys. Lett. 91, 083505 (2007); 10.1063/1.2772189

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Cantilever surface stress sensors with single-crystalline siliconpiezoresistors

P. A. Rasmussen,a! O. Hansen, and A. BoisenDepartment of Micro and Nanotechnology (MIC), Technical University of Denmark, Building 345E,2800 Kgs. Lyngby, Denmark

sReceived 8 July 2004; accepted 1 March 2005; published online 12 May 2005d

We present a cantilever with piezoresistive readout optimized for measuring the static deflection dueto isotropic surface stress on the surface of the cantileverfSens. Actuators B79s2–3d, 115s2001dg.To our knowledge nobody has addressed the difference in physical regimes, and its influence oncantilever sensors with integrated piezoresistive readout, that one finds between typical atomic forcemicroscopy measurements and the surface stress sensors used in, e.g., biochemical measurements.We have simulated the response from piezoresistive cantilevers as a function of resistor type andplacement for the two different regimes, i.e., surface stress measurements and force measurements.The model thus provides the means to specifically design piezoresistive cantilevers for surface stressmeasurements. ©2005 American Institute of Physics. fDOI: 10.1063/1.1900299g

Cantilevers are now widely used as chemical sensors,where the selective adsorption or immobilization of mol-ecules on one side of the cantilever creates a surface stressdifference between the two sides of the cantilever.1 This sur-face stress difference deforms the cantilever, and this defor-mation is the measured quantity. The detection principles areusually either the optical laser leverage method2 where thebending angle of the cantilever is measured or piezoresistivereadout where the generated strain is measured via an inte-grated resistor.3

When dealing with piezoresistive readout the two keydesign issues determining the final resolution of the sensorare the strain sensitivity of the resistor and the electricalnoise inherent in all resistors. We have previously reportedon optimized cantilevers with polysilicon piezoresistors,4,5

where we focused on signal–to–noise ratio and the strain inmultilayered cantilevers. In the present work we report onthe optimization work on a cantilever sensor with singlecrystalline silicon piezoresistors made on a silicon on insu-lator substrate. Compared to polysilicon, the single crystal-line silicon will enhance the sensitivity as its piezoresistancecoefficients typically are two-three times larger and it willalso improve the signal to noise ratio as the 1/ f noise islower.6

The optimization takes into account the different stressregimes encountered on the cantilever as a function ofclamping effects, and uses the results to specify optimumdesign, placement, and doping of the piezoresistors. Particu-larly, this model clearly pinpoints the difference betweencantilevers with integrated piezoresistors developed foratomic force microscopysAFMd and biosensing, respec-tively.

Combined analytical and finite elementsFEd work on thebending of cantilevers due to surface stress changes has beenpresented by Sader,7 where the effect of the clamping on thecantilever deflection is calculated for use in systems usingthe optical readout method. We, however, are interested inthe in-plane deformation of the cantilever due to surfacestress and its effect on the integrated resistor. This subject

has been the topic of recent works8,9 where a model for apiezoresistive AFM sensor is used, that is valid only for thef110g direction inp-type silicon ons100d wafers

DR

R~ pLssL − sTd. s1d

pL is the longitudinal piezoresistance coefficient andsL andsT are the longitudinal and transversal stress in the piezore-sistor, respectively. However, this model is not suitable fordescribing the sensitivity to the in-plane isotropic stress.

We will use a simple approximation to analytically esti-mate the strain in the cantilever close to the clamping, anduse a free bending model where the effect of the clampingvanishes. FE analysis will be used to check the validity ofthe analytical models and to calculate the precise solution toa realistic cantilever model.

When an isotropic surface stress is applied to a cantile-ver the unrestricted end will be subject to free bending,meaning that the strain in the cantilever is uniform and iso-tropic in the plane of the cantilever as long as edge effectsare neglected, as sketched in Fig. 1. For comparison a canti-lever with an applied forceF working on the apex, as foundin AFM, is also pictured. In AFM the stress will be concen-trated at the base of the cantilever, as the longitudinal stresssL is given bysL=zFsL−xd / I, whereI is the area moment ofinertia.10 The transversal stress for AFM is given by freePoisson contraction assT=−nsL wheren is Poisson’s ratio.If we consider a resistor placed in the cantilever with thecurrent running along the length of the cantilever, then therelative resistance change is written as the sum of a longitu-dinal and a transversal piezoresistive contribution

DR

R= pLsL + pTsT = pLsx + pTsy, s2d

which for the free bending model with isotropic surfacestress applied simplifies toadElectronic mail: pra@mic.dtu.dk

APPLIED PHYSICS LETTERS86, 203502s2005d

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DR

R free= sLspL + pTd, s3d

sincesL=sT. It is assumed that the thickness of the cantile-ver is much smaller than both width and length. Thus, freePoisson contraction on practically all of the cantilever alongthe z axis is allowed, leading to zero stress in this direction,since the effect of the clamping will only affect a region ofthe cantilever with a length comparable to that of the thick-ness. Withsz=0 it then follows thatsT=sy. Near the clamp-ing we assume the cantilever is restricted so that there is nostrain along the width of the cantilever, i.e., the strain«y=0,which givessy=nsx⇔sT=nsL leading to

DR

R clamp= sLspL + npTd. s4d

To check the validity of the analytical model it has beenplotted against a FE model of a cantilever with a simulatedsurface stress. The exemplified cantilever is 100mm long,

50 mm wide, and consists of a 1mm thick silicon layer witha 0.1mm gold layer on top. In the starting position beforeany strain is developed in the cantilever there is a stress of50 MPa in the gold layer. The stress and strain distribution inthe cantilever along its length axis is shown in Fig. 2. Thestrain«x for the free cantilever is calculated analytically for amultilayered cantilever from Ref. 5. It can be seen from Fig.2 how the analytical model fits well for distances more thanthe width s50 mmd away from the clamping. The model ob-viously does not give a precise solution near the clamping, asthe assumption of«y=0 only holds very close to the clamp.The graphs also very precisely show why optimizations us-ing Eq. s1d give the result that the resistor should be placedas close to the clamping point as possible; when movingaway from the clamp the longitudinal and transversalstresses converge and the signal becomes vanishing. Assum-ing we work with silicon piezoresistors ons100d siliconaligned along thek110l directions, the longitudinal and trans-versal piezoresistance coefficientspL and pT can be foundfrom pL= 1

2sp11+p12+p44d and pT= 12sp11+p12−p44d sRef.

11d using Smith’s12 values for the piezoresistance coeffi-cients for low dopedp- andn-type silicon. According to Ref.11, the piezoresistance coefficients decrease with increasingdoping concentration. However, since this trend does not dif-fer significantly betweenp- andn-type silicon, then the over-all observations done with the values from Ref. 12 hold forhigher doping concentrations as well. The longitudinal andtransversal piezoresistance coefficients are found and listedin the left part of Table I. If these values are applied in Eqs.s3d ands4d, then effective piezoresistance coefficients can be

TABLE I. Left: Longitudinal and transversal piezoresistance coefficients forthe f110g direction ons100d silicon. Right: Effective piezoresistance coeffi-cients near rigid clamping and on a free cantilever forp- andn-type silicon.A Poisson’s ration of 0.3 has been used.

pL pT

peff

ClampedpL+npT

FreepL+pT

p-type 72 −66 52 6n-type −31 −18 −36 −49

f10−11 Pa−1gFIG. 1. Top: Cantilever with surface stress. Bottom: Cantilever experiencinga force at the apex.

FIG. 2. StresssMPad and strain in the silicon just below the gold layer, inthe middle of and along the length of the cantilever.x, y, andz refer to Fig.1 salong length, width, and thickness, respectivelyd. The cantilever isclamped atx=0 and the apex is atx=100mm. The dashed lines representthe analytical solutions, with that of a clamped cantilever used forx=0–25mm sw/2d and that of a free cantilever from 25–100mm.

FIG. 3. DR/R from Eq. s2d with piezoresistance coefficients from Table Iand longitudinal and transversal stress from the FE results plotted in Fig. 2.DR/R is plotted against the distancex from the clamped end of the cantile-ver. The dashed lines represent the numerical value of the integrated signals,sx−x0d−1ex0=−5 mm

x uDR/Rudx, giving the average sensitivity for a resistorplaced from −5mm to x. The integrated signals cross atx=17 mm.

203502-2 Rasmussen, Hansen, and Boisen Appl. Phys. Lett. 86, 203502 ~2005!

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found, and these are calculated in the two columns to theright in Table I. At the clamped edge thep-type silicon givesthe highest sensitivity. However, as seen in Fig. 3, where FEresults for the stress in the resistor from Fig. 2 are combinedwith the general expression from Eq.s2d to give the sensi-tivity DR/R, the placement of the resistor is very critical. Itis seen how the sensitivity for thep-type sensor is low at thefree bending part of the cantilever, so the more the resistor isextended from the clamped edge of the cantilever, the lesssignal per length is obtained. The dashed lines show aweighed integrationsx−x0d−1ex0

x uDR/Rudx over the two sen-sitivity signals and for this specific example it is seen howthe n-type resistor will give the largest signal for 17mmresistors and longer. Below 17mm the p-type resistor willgive a signal that is marginally higher than that of then-typeresistor. For the free bending part of the cantilever the n-typeresistor is by far the most sensitive. Since the free bendingcondition is more easily fulfilled than the clamped, then forany practical device then-type silicon will be the first choicefor a sensitive cantilever surface stress sensor. It simply re-quires a long, slender beam for most of it to behave as freebending, whereas an optimum placement for thep-type re-sistor in the clamped region is delicate and most likely willresult in short, low volume resistors, which will increase theelectrical 1 /f noise.

In conclusion, it has been shown how the design of pi-ezoresistors in cantilevers for surface stress measurements in

general should be different than that for AFM probes. InAFM the stress is concentrated at the base of the cantileverand the stress is primarily directed along the length of thecantilever, whereas in biosensing the stress is uniform andisotropic. For the specific case withs100d silicon with resis-tors placed along thek110l directions, ann-type resistor willgive the most sensitive surface stress sensor.

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203502-3 Rasmussen, Hansen, and Boisen Appl. Phys. Lett. 86, 203502 ~2005!

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