Modeling, Simulation, And Design Guidelines for Piezoresistive Affinity Cantilevers

774 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 20, NO. 3, JUNE 2011

Modeling, Simulation, and Design Guidelines forPiezoresistive Affinity Cantilevers

Manoj Joshi, Prasanna S. Gandhi, Member, IEEE, Rakesh Lal,V. Ramgopal Rao, Senior Member, IEEE, and Soumyo Mukherji

Abstract—This paper presents design guidelines for piezoresis-tive affinity cantilevers for operation in liquid environments. Forthe first time, we consider the interdependence of various func-tional elements (such as biological, mechanical, and electrical) ofthe cantilever, their dependence on material choice, microfabrica-tion processes, and geometry, and the resultant effects on the me-chanical and electrical sensitivities of the cantilever. The cantileverdesign guidelines that include material selection as well as deter-mination of geometrical dimensions are proposed. As an example,we have designed and simulated a multilayer piezoresistive siliconnitride affinity cantilever for performance in a liquid environmentunder constraints imposed by microfabrication and electrical andmechanical considerations. Systematic steps toward optimizationof geometrical dimensions include initial analytical estimates ofgeometrical dimensions, followed by finite-element modeling andanalysis of such cantilevers under the applied surface stress. Simu-lation studies brought forth the limitation on maximum obtainableΔR/R as well as the nonlinear behavior of the cantilever whichwas not observed in analytical estimates. [2010-0116]

Index Terms—Affinity cantilever, nonlinear, piezoresistive.

I. INTRODUCTION

P IEZORESISTIVE affinity cantilevers have great potentialin microsystems used for sensing biomolecules, such as in

point-of-care systems. Such prototype systems using piezore-sistive cantilevers in Wheatstone bridge configurations withintegrated microfludics have been demonstrated [1]–[3]. Thesecantilevers have a multilayer structure in which the strain-sensitive layer is sandwiched between a structural layer and anencapsulation layer. Selective immobilization of biomoleculeson either top or bottom surface of the cantilever generates dif-ferential surface stresses on the opposite faces of the cantileverwhich leads to a bending of the cantilever, thereby inducing achange in resistance of the strain-sensitive layer incorporatedwithin it. For maximum sensitivity of cantilever biosensors,the selective immobilization of biomolecules is a prerequisite,since immobilization on both faces is expected to elicit a weakerresponse [4]. Studies have concentrated on the mechanical and

Manuscript received April 25, 2010; revised February 28, 2011; acceptedMarch 5, 2011. Date of publication April 25, 2011; date of current versionJune 2, 2011. Subject Editor A. Seshia.

M. Joshi is with R&D, Taiwan Semiconductor Manufacturing CompanyLimited, Hsinchu 300-77, Taiwan (e-mail: [email protected]).

P. S. Gandhi, V. R. Rao, and S. Mukherji are with the Indian Instituteof Technology Bombay, Mumbai 400076, India (e-mail: [email protected];[email protected]; [email protected]).

R. Lal, retired, was with the Indian Institute of Technology Bombay, Mumbai400076, India. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JMEMS.2011.2140353

electrical design of multilayer piezoresistive cantilevers, suchas those for atomic force microscope cantilevers [5] as well asaffinity cantilevers [6], [7]. For those affinity cantilevers, theelectrical sensitivity, i.e., change in resistance due to appliedsurface stress (ΔR/R) is less than one part per million (1 ×10−6–1 × 10−7) which can get suppressed by electrical noise.Also, previous work does not comprehensively discuss engi-neering figures of merit such as signal-to-noise ratio (SNR),power dissipation, and mechanical stability in liquid media,thus posing practical limitations in the realization of cantileversensors.

The interdependence of various functional elements (bio-logical, mechanical, and electrical) of piezoresistive affinitycantilevers, their dependence on choice of materials, process-ing, and geometry, and the resultant effects on mechanicalsensitivity (change of deflection with applied stress) and elec-trical sensitivity (change of resistance with applied stress) makethe design of affinity cantilevers challenging. This is furthercomplicated by the fact that these cantilevers are expectedto work in a liquid environment. Selective immobilization ofbiomolecules, packaging, and testing are other important con-siderations in the overall design of such systems [8]. In spiteof the complexity in designing such cantilevers, there are nostandard guidelines available in the literature.

In this paper, we report a design of piezoresistive affinitycantilevers based on a combination of analytic formulas andnumerical simulations. We did not intend to optimize thesensor for maximum biosensitivity, e.g., detection of lowestpossible analyte concentration. However, for a chosen surfacestress value, our aim was to maximize electrical sensitivity byobtaining the highest ΔR/R and SNR. Mechanical sensitivityparameters such as resonant frequency and spring constant wereused as design constraints for stable and noise-free operationof the cantilever. The design guidelines considering the inter-dependence of various functional domains of the cantilever, aswell as the criteria for material selection and determination ofgeometrical dimensions of the cantilever, are proposed. To illus-trate the proposed design, an example of a piezoresistive siliconnitride cantilever used for the detection of antibody–antigenbinding on the cantilever surface is presented. Surface stressgenerated due to the antibody–antigen interaction was modeledusing a MEMS simulator, and the cantilever response wasobtained. In order to optimize the cantilever geometry, simpleanalytical equations were used for initial estimates of can-tilever dimensions and subsequently fine-tuned using numericalsimulations. Simulation studies demonstrated the limitation onmaximum achievable electrical sensitivity that was many folds

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JOSHI et al.: MODELING, SIMULATION, AND DESIGN GUIDELINES FOR AFFINITY CANTILEVERS 775

higher than its counterparts reported in [6] and [7]. It alsodemonstrated nonlinear behavior of the cantilever which wasnot seen in analytical estimates.

II. DESIGN GUIDELINES

A piezoresistive affinity cantilever has four functional layers:1) an immobilization layer; 2) an anchor layer to which the im-mobilization layer is attached; 3) a piezoresistive layer; and 4) astructural layer. We assume immobilization of biomolecules onthe top surface of the cantilever. The tip deflection and changein resistance (ΔR/R) of the piezoresistor are measures of themechanical and electrical sensitivities, respectively.

The design process is iterative with material properties andelectromechanical effects impacting each other. However, wecan notionally divide the design process into material selectionand geometry determination.

A. Selection of Materials

1) Target–Probe Biomolecule Pair: Depending on the tar-get biomolecules (which have to be detected), an appropriateaffinity probe needs to be identified. Immobilization of suchbiological affinity probes on solid substrates can be achievedby adsorption, covalent bonds, entrapment in gels, etc. [9]. Theimmobilization using covalent bonds can allow a thin layerof biomolecules [10] with lesser impact on the mechanicalproperties of the cantilever. The biomolecule layer immobilizedusing this technique should be as less susceptible as possible tochanges in pH, temperature, or ionic strength of the surroundingmedium [9], [11] and should show low desorption of biomole-cules from the surface of the cantilever. It does not requirea diffusion barrier like a membrane, gel, or polymer matrixbetween the cantilever surface and the immobilized molecule;the analyte can react with the sensor surface more readily. Thisimproves the sensor kinetics as well as reduces the amountof biomolecules required for sensor fabrication. Thus, covalentattachment is a preferred choice for the immobilization.

2) Surface Layer for Immobilization: Preferably, the can-tilever should be incubated in an appropriately designed liquidcell or flow channel leading to selective immobilization oneither the top or the bottom surface. This can be achievedwith a surface of immobilization layer chemically different(isoelectric points) than structural layers such as gold againstsilicon nitride. Another alternative is chosen: The immobiliza-tion technique should allow the selective immobilization (e.g.,selective functionalization of SiO2 using aminosilanization asagainst the Si3N4 surface) [12], [13]. Ideally, such an anchorlayer film should be infinitesimally thin and have a low Young’smodulus. Preferably, it should be an insulator for thermal andelectrical isolations between the piezoresistive layer and im-mobilized biomolecules. However, the selected immobilizationprotocol affects the selection of material for the anchor layer.Gold [14], silicon dioxide [15], and silicon nitride films [16]have been widely used as an anchor layer. Gold film can beused as a reflecting surface for laser light and is often used forimmobilization in affinity cantilevers for optical detection ofdeflection. However, in the case of piezoresistive cantilevers, if

gold film is used for immobilization, an additional insulatinglayer (e.g., SiO2 and Si3N4) between the piezoresistive layerand the gold film is essential. Such an additional insulator layerhas an adverse impact on the sensitivity of the cantilever.

3) Strain-Sensitive Layer: The strain-sensitive layer may bemade of semiconductor or metal with the former providinghigher sensitivity due to high gauge factor. Although dopedsingle crystal silicon has higher sensitivity to strain than dopedpolysilicon, the latter offers the advantage of surface micro-machining and is preferred as a strain-sensitive layer. It alsoexhibits a lower temperature coefficient of resistance (0.04%per degree Celsius) than single crystal silicon (0.14% perdegree Celsius), which improves thermal stability in liquid en-vironment [17]. (111)-oriented P-type boron-doped polysiliconwith dopant concentration about 1019 cm−3 offers a high gaugefactor and is preferred as the piezoresistive layer [17]. SNR is akey performance index for minimum detectable stress changeswithin the piezoresistive layer and can be optimized basedon piezoresistor geometry, doping concentration, and annealtemperature [18]. The internal stresses within the polysiliconfilm can be minimized by controlling the deposition processparameters [18].

4) Structural Layer: The material for this layer has to bechosen such that biomolecules do not get attached to this layerdue to the selected immobilization protocol. For example, onecan use silicon nitride as a structural layer when a gold (chem-ically different surface) or silicon dioxide (e.g., silanizationprotocol for selective surface functionalization) film is used asan immobilization layer, thus allowing cantilever incubation inone solution [13], [14]. Cantilever sensors with hydrophobicsurfaces are prone to stiction in liquid environments and canbe susceptible to nonspecific adsorption of biomolecules. Agood inert surface for biosensors can be obtained by selectivesurface modification. For example, polyethyleneglycol-coveredsurfaces are hydrophilic in nature and known to prevent adsorp-tion of biomolecules.

The electrical sensitivity (ΔR/R) of the piezoresistive affin-ity cantilever is a function of the Young’s modulus of thematerials used, the distance between the piezoresistive layer,the neutral axis of the cantilever, as well as the thickness [(13),Appendix B]. Higher sensitivity can be achieved by increasingthe distance between the piezoresistive layer and the neutralaxis, which can be obtained by selecting a structural layer ma-terial having a higher Young’s modulus than the piezoresistivelayer. For example, if p-type polysilicon (E = ∼150 GPa) isused as a piezoresistive layer, silicon nitride (E = ∼280 GPa)is a better choice than silicon dioxide (E = ∼75 GPa) as astructural layer material.

B. Determination of Geometrical Dimensions

1) Mechanical Effects of Target–Probe Interaction:Antigen–antibody interactions on the cantilever surface pro-duce surface stress [19]. In order to determine the geometricaldimensions of an affinity cantilever, the approximate magnitudeand nature of the stress (compressive or tensile) need to beknown. For certain classes of probe–target interactions, theapproximate values may be obtained from literature. For

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Fig. 1. (a) Conversion of surface stress into surface force used for surfaceboundary conditions. Surface force = Surface stress (in newtons per meter) ×Width (in meter). (b) Change in the direction of surface force after bending ofthe cantilever.

example, the approximate surface stress generated on acantilever due to antibody–antigen binding is ∼5 mN/m [4]and that due to the surface stress induced by the adsorptionof thiolated DNA molecules on gold-coated microcantileversensors is on the order of a few newtons per meter [20].

The deflection of affinity cantilevers is normally muchsmaller than the length of the cantilever [1]. For such a lowrange of deflections, nonlinearity arrived due to cantilevergeometry and material is neglected. However, considerationof nonlinearity arrived due to changing simulation boundaryconditions [changing surface force direction in Fig. 1(b)] leadsto higher simulation accuracy [21]. Thus, nonlinear simulationsbased on finite-element analysis predict the mechanical andelectrical responses of the cantilever well.

2) Cantilever Shape: Piezoresistive affinity cantilever de-signs of T-shape, inverse T-shape, narrow as well as broadrectangular shapes, V-shape, and long- as well as short-basedU-shapes have been reported in [22]. A systematic performancecomparison of these cantilevers using constant mass, thickness,and surface area is discussed. Among those cantilevers, inverseT-shape and long-based U-shape cantilevers show the highestsurface stress and deflection sensitivity.

Final release of cantilevers often employs wet silicon etchingat elevated temperatures. For U-shape cantilevers, faster releaseis possible due to simultaneous etching from the outer and inneredges of the cantilever. The reduced time of exposure to a wetetchant results in a lower probability of damage to the ultrathincantilevers, thereby increasing the process yield. Therefore, along-based U-shape cantilever [Fig. 1(a)] is a preferred choice.

3) Cantilever Thickness: From the analytical formulas[(13), Appendix B], thickness is the only geometrical parameterwhich decides the electrical sensitivity of piezoresistive affin-ity cantilevers. The minimum thickness of the immobilizationlayer may be limited by considerations of chemical, physical,and biological stability and functionality requirements. Formaximum electrical sensitivity, the piezoresistive layer must beinfinitesimally thin and smooth and must be located close to thesurface where maximum surface stress is developed. However,to get the required piezoresistive properties, the minimumthickness is limited by the locally optimized deposition processparameters and noise considerations. Based on such limitationsdescribed and taking into account the desired mechanical andelectrical sensitivities, analytical calculations yield initial esti-mates of the desired thickness of the structural layer.

4) Cantilever Width: Determination of the width of affinitycantilevers is governed by conflicting requirements in termsof uniformity/reproducibility of immobilization versus elec-trical power dissipation and mechanical stability. Cantileversof smaller widths will have lower planar areas for immobi-lization with edge effects significantly affecting the unifor-mity of immobilization and quantitative reproducibility of thetarget–probe interaction. However, during the release of can-tilevers using wet silicon etching, the cantilevers with largerwidths need higher etching times, thereby impacting on theprocess yield. Furthermore, for a fixed measurement bias, asthe width of the cantilever increases, SNR increases along withelectrical power dissipation. Hence, the width of the cantileverhas to be determined based on biological, microfabrication, andelectrical performances.

5) Cantilever Length: In the microfluidic environment, themechanical stability of longer cantilevers gets affected dueto turbulence caused by circular velocity vectors near thecantilever tip [23]. However, in longer cantilevers, one maynotice lower electrical power dissipation and higher SNR in thepiezoresistive layer. Thus, the length of affinity cantilevers hasto be optimized under considerations of mechanical stabilityand performance parameters such as power dissipation andSNR.

6) Mechanical and Electrical Constraints in DeterminingGeometric Dimensions: The performance of piezoresistiveaffinity cantilevers is governed by both mechanical and elec-trical sensitivities. Mechanical parameters like spring constantand resonant frequency need to be used to design cantilevers ofdesired mechanical sensitivity and stability. The spring constantof affinity cantilevers has been reported to be in the range of1 mN/m [24]. In this exercise, it was chosen to be approxi-mately 1 mN/m. A cantilever with such low spring constant ishighly susceptible to external mechanical noise. Therefore, it ispossible to minimize the noise by mechanical shielding and/ordesigning the cantilever with adequate resonant frequency. Inthis exercise, it was chosen to be 5 kHz [24].

Electrical power dissipation and SNR are two importantconstraints under which the cantilever design should be op-timized. Increase in temperature due to the electrical powerdissipation can destroy the cantilever and/or may impact thebinding constant of the molecular interaction with target mole-cules; however, such studies are yet not reported. Assuming anexcitation voltage of 5 V, in liquid medium, such cantilevers cansafely handle power up to 2 mW [24]. Hence, power dissipationwas limited to this value. Higher SNRs can be achieved withreduction in internal noise by controlling design and processparameters as well [25]. In this paper, the desired figure of meritof SNR was chosen as 3 and above.

The cantilever under design uses selective immobiliza-tion [13] on the top surface. Thus, surface stress due toantibody–antigen binding is expected only on the top surfaceof the cantilever. The thickness of the biolayer (includingsurface modifier, linker, and probe molecule) is on the order of10–20 nm with a Young’s modulus of ∼1 MPa [26], whichare much lower than the cantilever thickness and the Young’smodulus of the cantilever material. Furthermore, it is unlikelythat the biolayer contributes in any way toward increasing


TABLE IMICROSCALE MATERIAL PROPERTIES USED IN SIMULATION [17], [18], [27], [28]

the stiffness in the direction of bending. Hence, the stiffnesscontributed by the biolayer can be neglected and only its effectas a surface force that occurs due to the surface stress wasconsidered. In this paper, the surface stress of 5 mN/m [4] wasused for design purposes.

III. INITIAL DESIGN OF A PIEZORESISTIVE

AFFINITY CANTILEVER

The considerations of design presented in the previous sec-tion are being illustrated with an example of a multilayer sili-con nitride piezoresistive affinity cantilever. In this cantilever,silicon nitride is the structural layer and p-type polysilicon isthe piezoresistive layer. A thin layer of silicon nitride is usedas surface layer for immobilization on top of the polysiliconlayer. Such a silicon nitride layer can undergo through oxygenplasma treatment followed by aminosilanization and antibodyimmobilization [10]. Gold contact pads are used for the electri-cal probing of the piezoresistive layer. According to the Ashbyapproach for material selection, material properties of macro-(bulk) structures can be different than the microstructures [27].Such properties are functions of length scale and details ofprocessing techniques employed. The correlation of macro-and microstructure properties in [27] was used to choose themicroscale properties of the cantilever material. Important ma-terial properties of silicon nitride, polysilicon, and gold filmsused for analytical calculations and for simulation studies areas shown in Table I.

The analytical calculations using formulas listed in Appen-dixes A and B were performed using Microsoft Excel. Theestimation process comprised of an initial selection of geo-metrical dimensions followed by iterative adjustments underthe constraints mentioned earlier. The final estimates at thisstage for the length and width of the cantilever were 200 and40 μm, respectively. The thicknesses of silicon nitride, polysili-con, and oxynitride layers were estimated at 100, 45, and 5 nm,respectively.

IV. SIMULATION AND DESIGN MODIFICATIONS

The aim of the simulation study was to model the effects ofstress generated due to probe–target biomolecular interactionon the cantilever surface and to determine cantilever response.Such simulations aided in deciding on the exact geometricaldimensions (thickness, width, and length) of the cantilever.

Fig. 2. Schematic of piezoresistive silicon nitride cantilever used insimulation.

A. Modeling of the Cantilever Structureand Boundary Conditions

Using the parameters estimated in the previous section, thecantilever structure modeled within the CoventorWare MEMSsimulator is shown in Fig. 2.

The length of the piezoresistor in this U-shaped cantilever is2 ∗ Lleg − W/2, where Lleg = L − W/2. The cantilever struc-ture was meshed using Manhattan bricks with parabolic-ordermesh elements by choosing the aspect ratio of cantilever meshelements the same as cantilever geometry.

The compressive surface stress of 5 mN/m (simulating anincrease in repulsive forces on the surface due to the interactionof the biomolecules) was converted into surface force (Fig. 1)and applied on the top surface of the cantilever. The cantilevermeshed model was simulated in the nonlinear mechanical andpiezoresistive domains in order to obtain endpoint deflectionand ΔR/R, respectively.

B. Bending of Cantilever and Strain DistributionWithin the Cantilever

It can be seen from Fig. 3 that the bending of the cantilevernear the fixed end is almost negligible and is more noticeablefrom the middle of the length of the cantilever. The deflectedprofile of the bent cantilever is different for a point force appliedto the free end compared with having surface stress distributedover the length of the cantilever. For the cantilever understudy, the bending curvature (Fig. 4) is maximum betweenthe fixed end and middle of the cantilever. The bent profileof the cantilever bears a good qualitative similarity with theexperimental results reported in [29], thereby validating the


Fig. 3. End-point deflection of cantilever showing predominant bending be-gins close to the middle of its length.

Fig. 4. Stress distribution within the cantilever showing unequal and opposite-in-nature stress near the top and bottom surfaces of the cantilever. (Photographis magnified by 100 times in Z direction).

modeling of the affinity cantilever. It can also be seen that, inaffinity cantilevers, the stress developed near the top surface,on which surface stress was applied, is much higher than thestress developed at the bottom surface. This is in concurrencewith reported findings [2], which mentions that, although thesimple Bernoulli–Euler model of cantilever bending (smalldeflections) would predict that the surface strains generatedat the opposite faces of a cantilever would have the samemagnitude, they are, in fact, different. Hence, to get maximumstress sensitivity, the piezoresistive layer must be located at alevel within the cantilever where maximum stress is expected.

C. Thickness Optimization of Affinity Cantilever

The constraints in processing stages limit our ability toactually get required functionality from piezoresistive layerswhich are less than ∼45 nm thick. As a result, the optimizationof the thickness of the total cantilever structure can only be doneby varying the thickness of the structural layer.

As shown in Fig. 5, ΔR/R increases with the reduction oftotal thickness and it attains a maximum at a certain value ofthickness below which it falls. The difference between the max-imum sensitivities obtained from numerical simulations and an-alytical calculations is attributed to the nonlinear behavior dueto changing surface force direction, which was not consideredfor the first-order estimations in analytical calculations. Thedecrease in ΔR/R after the maximum with reducing thickness ismainly due to the shifting of the neutral axis toward the piezore-sistive layer. Accordingly, a cantilever thickness of 100 nmmay be selected for maximum realistic ΔR/R. However, at this

Fig. 5. Thickness optimization of silicon nitride cantilever for maximumΔR/R.

thickness, the values of the mechanical performance parameterssuch as spring constant [(3), Appendix A], resonance frequency[(4), Appendix A], and electrical performance parameters suchas SNR [(14) and (20), Appendix B] need to be estimated. At athickness of 100 nm, the spring constant and resonant frequencyof the cantilever are 0.23 mN/m and 3.26 kHz, respectively,which are lower than the design constraints discussed earlier(k ∼ 5 mN/m, f ∼ 5 kHz, SNR > 3, and PD < 2 mW). Itmay be possible to achieve the desired performance parametersby retaining the thickness of 100 nm and reducing the lengthor increasing the width of the cantilever. However, such astrategy may cause degradation in SNR and higher powerdissipation [(16), Appendix B] within the cantilever. Therefore,the thickness of the structural layer has to be increased at thecost of a slight decrease in ΔR/R. It was observed that, at atotal thickness of 150 nm and at assumed length and widthof 200 and 40 μm, respectively, the value of spring constantwas 0.83 mN/m, resonant frequency was 4.89 kHz, and SNRwas 4.03. Since the SNR is not particularly low, one can settlewith 150 nm as the total thickness of the cantilever. The springconstant and resonant frequency can be improved further whileanalyzing the issues related to the width and the length of thecantilever.

D. Width Determination of Affinity Cantilever

The reproducibility of the immobilization process of bio-molecules on a cantilever surface and, thereby, the reproducibil-ity of its response increase with the increase in its width.Experimental results (e.g., Fig. 6) show that the immobilizationdensity is high and irregular near the edges of the cantilever.

Our further investigations suggest that it is mainly due to therounding of cantilever edges during the wet release of the can-tilever and is independent of its width. The biomolecules on therounded edges can have undesirable orientations and may notcontribute in a proportionate fashion toward the developmentof surface stresses along the cantilever top surface. Thus, forimmobilization on cantilevers of lower width, the ratio of planarsurface area to the area of rounded edges decreases, therebyhaving an adverse effect on its reproducibility. It is therefore


Fig. 6. Micrograph of FITC-tagged goat antihuman IgG immobilized silicon cantilever observed under an (a) optical microscope and (b) fluorescent microscope,showing edge effect.

Fig. 7. Width versus (a) deflection and spring constant and (b) ΔR/R of silicon nitride cantilever.

logical to conclude that the influence of this “edge effect” inwider cantilevers will be lower than the narrower ones.

The width of the cantilever impacts on power dissipationand SNR. From the analytical formulas, it may be presumedthat the deflection of the cantilever is independent of the width.Simulations were performed to determine the best width underthe discussed conditions of operation and constraints. Thethickness (150 nm) determined in the earlier section was usedalong with a length of 200 μm.

The cantilever width was varied from 20 to 60 μm. Simula-tions show a small decrease in the deflection with the increasein the width of the cantilever [Fig. 7(a)]. This departure fromthe earlier assumption is due to the fact that the cantilever notonly bends downward but wide and thin cantilevers also curvedownward along their widths near the free end [30]. This causesan increase in stiffness of the cantilever and, in turn, reduce theexpected deflection and ΔR/R.

Fig. 7(b) shows that, as the cantilever width increases, theelectrical power dissipation and SNR within the cantileverincrease. Higher power dissipation is due to reducing the elec-trical resistance of the piezoresistor. Since the volume of thepiezoresistor is proportional to the cantilever width, a largerwidth causes an increase in the number of free carriers withinthe piezoresistor and, hence, an increase in SNR [(17)–(20),Appendix B]. In this paper, the width determined earlier(40 μm) was found to be optimal under the conditions specified.

Fig. 8. Length versus ΔR/R and deflection of silicon nitride cantilever.

E. Length Determination of Affinity Cantilever

The analytical formulas predict that ΔR/R is independent ofthe length of the piezoresistive cantilever although this may af-fect other electrical and mechanical characteristics. Cantileversof varying length (100 to 200 μm) were simulated, keeping thethickness and width constant at values decided in earlier sec-tions. As shown in Fig. 8, deflection of the cantilever increaseswith an increase in cantilever length. As in the case of thewidth, the cantilever shows a slight decrease in ΔR/R as well.


TABLE IIOPTIMIZED GEOMETRICAL DIMENSIONS OF

SILICON NITRIDE CANTILEVER

TABLE IIIPERFORMANCE PARAMETERS OF OPTIMIZED

SILICON NITRIDE CANTILEVERS

The deviation is consistent with the results published in [30]and is due to the fact that, for longer cantilevers, change in thedirection of the surface force, as the cantilever bends, is morepronounced.

This causes reduction in expected deflection, thereby re-ducing ΔR/R. The mechanical (spring constant and resonantfrequency) and electrical performance (power dissipation andSNR) parameters for various lengths were estimated. Thoseparameters at the length of 180 μm closely fit (Table III) tothe desired performance parameters and was the best choice forthe length of the cantilever.

The geometrical dimensions of the optimized silicon nitridecantilever are summarized in Table II and their performanceparameters are summarized in Table III.

F. Performance of Optimized Piezoresistive Cantilever

The performance of the designed cantilever under differentsurface stress conditions (approximating possible variationsdue to the nature of immobilization and biomolecular inter-action) was investigated using simulation tools. Fig. 9 showsthat the deflection and ΔR/R of the cantilever increase with theapplied surface stress. The electrical sensitivity of the cantileverwas many folds higher than those of the simulation studiesreported earlier [6], [7]. This demonstrates that the designapproach considering the interdependence of various func-tional elements (e.g., biological, mechanical, and electrical) ofthe piezoresistive cantilever and their dependence on materialchoice, microfabrication processes, and geometry improved thesensitivity of the cantilever. From the analytical equations ofaffinity cantilevers, these parameters are linear functions of thesurface stress applied on the cantilever. However, simulation

studies show that they do not increase linearly with the ap-plied surface stress. One possible reason is the aforementionedchange in the surface force directions, which is addressed in thefinite-element analysis performed during simulation but not inthe analytical solutions.

V. CONCLUSION

The design guidelines of piezoresistive affinity cantileverssupported by analytical calculations and numerical simulationsfor performance in a liquid environment under the constraintsimposed by microfabrication and electrical and mechanicalconsiderations have been reported. The effect of stress gener-ated on a cantilever surface due to antigen–antibody interac-tion was modeled using a MEMS simulator. The simulationresults obtained were in concurrence with the experimental andpostulated characteristics of stress-sensitive affinity cantileverspublished by other investigators. It may be concluded that thethickness of the piezoresistive affinity cantilever is the principalgeometrical dimension which decides the electrical sensitiv-ity of the cantilever. Contrary to expectations from analyticalformulas, the width and length of the cantilever also affectelectrical sensitivity, albeit to a lesser degree. This obviouslyimproved the accuracy of the analyses although it brought in(expected) mismatches with the analytically derived parametersand solutions (Table IV).

APPENDIX AMECHANICAL PERFORMANCE PARAMETERS

The position of the neutral axis in the cantilever is givenby [2]

ZN =∑

EiZihi∑Eihi

(1)

where Ei is the Young’s modulus of the ith layer, Zi is theposition of the ith layer from the neutral axis, and hi isthe thickness of the ith layer. The effective stiffness of thecomposite cantilever is given by [6]

EI = W∑

i

(Ei

(h3

i

12+ hi(Zi − ZN )2

))(2)

where (Zi − ZN ) is the distance between the center of the ithlayer and the neutral axis.

The effective spring constant keff of the multilayer cantileveris given by [6]

keff =3W (EI)

L3(3)

where the value of (EI) can be calculated from (2).The total mass mTotal of the cantilever can be estimated as

mTotal = LW∑

i

(ρdihi). (4)

The resonant frequency of the multilayer cantilever is givenby [22]

fres = 0.32√

keff

mTotal. (5)


Fig. 9. Applied surface stress versus (a) deflection and (b) ΔR/R of optimized silicon nitride cantilever.

The approximate end-point deflection of the cantilever isgiven by [2]

δ =12βL2 (6)

and β is calculated by [2]

β =σsZT∑

i

Eihi

((ZT −

∑ij=0 hj + hi

2

)2

+ 13

(hi

2

)2) (7)

where ZT is the position of the top surface layer with respect tothe neutral axis and is given by [2]

ZT =

∑i

Eihi

(∑ij=0 hi − hi

2

)∑i

Eihi. (8)

The minimum detectable end-point deflection of the can-tilever in liquid medium is given by [24]

δmin =

√4KBΔf

2πQkfres. (9)

The approximate minimum detectable surface stress can beestimated by [24]

σsmin =43

√KBTΔf(ρ1)1/2(E1)1/2L

QW(10)

where ρ1 and E1 are the material density and Young’s modulusof the structural layer (assuming infinitely thin piezoresistivelayer).

APPENDIX BELECTRICAL PERFORMANCE PARAMETERS

The resistivity of the P-type piezoresistive polysilicon layercan be estimated as

ρR =1

μpqP. (11)

The mobility of holes in polysilicon material is highlyprocess parameter dependent. At the doping concentrationnear 1019 cm−3, the mobility of holes in polysilicon is ap-proximately one-third of the mobility of holes in silicon[31], [32].

The electrical resistance of the piezoresistive layer can beestimated using (11)

Rpiezo =2(Lleg + W

2

)ρR

W2 hpoly

. (12)

The change in the piezoresistance due to the surface stressdeveloped on top of the cantilever is given by [3]

ΔR

R=−K

ZT

(ZT −

∑Rj=0 hj + hR

2

)∑

i Eihi

((ZT −

∑ij=0 hj + hi

2

)2

+ 13

(hi

2

)2)

⎞⎟⎟⎠σs

− K1∑

i Eihiσs. (13)

The affinity cantilever under study is expected to be of usein on-chip Wheatstone bridge configuration. In that case, thechange in output voltage of the Wheatstone bridge can beestimated as

Vout =14

ΔR

RVB . (14)

However, for the design and simulation study demonstratedin this paper, the bias voltage is applied between the two padsof the cantilever whose prototype is shown in Fig. 1. Hence,all the electrical performance parameters demonstrated in thispaper are only for a single piezoresistive cantilever, and someof them will improve greatly when this cantilever is employedin the Wheatstone bridge. Current through the piezoresistor canbe estimated as

Ipiezo =VB

Rpiezo. (15)


TABLE IVTABLE OF VARIABLES

Power dissipation in the piezoresistor can be estimated as

PD = (Ipiezo)2Rpiezo. (16)

SNR is one of the important performance parameters of thepiezoresistive cantilever. The external noise generated from thesurrounding can be minimized by performing the experimentsin the controlled environment. However, the internal noisewithin the cantilever needs to be controlled using the processparameters [25]. The internal noise sources considered for thedesign purpose are Johnson noise and Hooge noise, whichare generated within the piezoresistive layer of the cantilever.Johnson or white noise power 〈v2

J 〉 due to the thermal fluc-tuation of charge carriers within the piezoresistor is givenby [2]

⟨v2

J

⟩= 4KBTΔfRpiezo. (17)

The Hooge noise voltage power in the range of frequencyfrom fmin to fmax is given by [2]

⟨v2

H

⟩=

αV 2B

Nln

(fmax

fmin

)(18)

where “α” is a material constant of piezoresistive polysiliconand has strong dependence on the annealing temperature andtime [25]. The total number of carriers in the piezoresistor canbe estimated as

N = PLlegWhpiezo. (19)

The total internal noise voltage power 〈v2N 〉 in the piezoresis-

tive cantilever is the sum of the Hooge noise power and Johnsonnoise power.

SNR is the figure of merit for the piezoresistive cantilevers.Neglecting the external noise and the vibration noise, SNR canbe estimated as

S

N=

√V 2

out

v2N

. (20)

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[11] R. A. Williams and H. W. Blanch, “Covalent immobilization of proteinmonolayers for biosensor application,” Biosens. Bioelectron., vol. 9, no. 2,pp. 159–167, 1994.

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Manoj Joshi received the M.Tech. degree in electri-cal engineering and the Ph.D. degree in bio-MEMSfrom the Indian Institute of Technology Bombay,Mumbai, India, in 2001 and 2007, respectively.

Since 2007, he has been a Principal Engineerand Member of Technical Staff with R&D, TaiwanSemiconductor Manufacturing Company Limited,Hsinchu, Taiwan. He is currently working on 28-nmnode high-κ and metal-gate CMOS device develop-ment. He has authored 24 peer-reviewed journal andconference papers, along with U.S., Chinese, and

Indian patents. His research interest includes device physics, layout-dependenteffects in advanced CMOS technologies, and bio-MEMS.

Prasanna S. Gandhi (M’99) received the B.Eng. de-gree in mechanical engineering from the Universityof Bombay, Mumbai, India, in 1994, the M.Tech.degree in mechanical engineering from the IndianInstitute of Technology Bombay, Mumbai, in 1996,and the Ph.D. degree in mechanical engineering fromRice University, Houston, TX, in 2001.

Since 2001, he has been a Faculty Member, cur-rently an Associate Professor, in the Department ofMechanical Engineering, Indian Institute of Technol-ogy Bombay. He has coordinated and set up a new

laboratory, Suman Mashruwala Microengineering Laboratory, for research inmicrodomain and has successfully completed several research projects spon-sored by the government and private sector. He has been a qualified teacher ofstress-relieving and life-enhancing techniques of the Art of Living Foundationby Sri Sri Ravishankar. He has authored over 50 peer-reviewed conferenceand journal papers, along with one U.S. patent and three Indian patents(pending). His research interests are in the areas of MEMS and microsystems,mechatronics, and nonlinear dynamical systems and control.

Dr. Gandhi was a recipient of the 2006 BOYSCAST fellowship from theGovernment of India and the Prof. J. R. Issac Fellowship.

Rakesh Lal received the B.Tech. degree in electron-ics and communication engineering from the IndianInstitute of Technology (IIT) Kharagpur, Kharagpur,India, the M.D. degree in electronics from NUFFIC,The Hague, The Netherlands, and the Ph.D. degreein electrical engineering from IIT Kanpur, Kanpur,India.

He was a faculty member at IIT Bombay, Mumbai,India, and retired as a Professor from the Electri-cal Engineering Department. He has also been aconsultant to industry for designing computer-aided

measurement systems for motors and transmission components and to the Gov-ernment of India on electronics and computer policies. His research interestsinclude physics and modeling of semiconductor devices, radiation and high-field effects in MOS and bipolar devices, and instrumentation for device char-acterization. He has also worked extensively on a variety of chemical, radiation,and biosensors, many as part of an interdisciplinary group encompassing theDepartments of Chemistry, Chemical Engineering, and Materials Science.


V. Ramgopal Rao (M’98–SM’02) received theM.Tech. degree from the Indian Institute of Technol-ogy (IIT) Bombay, Mumbai, India, in 1991, and theDr.Ing. degree from the Universitaet der BundeswehrMunich, Munich, Germany, in 1997.

From 1997 to 1998 and again in 2001, he wasa Visiting Scholar in the Department of ElectricalEngineering, University of California, Los Angeles.He is currently a Professor in the Department ofElectrical Engineering and the Chief Investigator forthe Centre of Excellence in Nanoelectronics at IIT

Bombay. He has over 280 publications in the area of electron devices andnanoelectronics in refereed international journals and conference proceedingsand is the holder of 15 patents issued or pending. He serves on the editorialboards of various other international journals.

Dr. Rao is a Fellow of the Indian National Academy of Engineering, theIndian Academy of Sciences, and the National Academy of Sciences in India.He is an Editor for the IEEE TRANSACTIONS ON ELECTRON DEVICES inthe CMOS Devices and Technology area. He is a Distinguished Lecturer ofthe IEEE Electron Devices Society and has served on the program/organizingcommittees of a large number of international conferences in the area ofelectron devices. He was Chairman of the IEEE AP/ED Bombay Chapterduring 2002–2003 and currently serves on the executive committee of theIEEE Bombay Section besides being the Vice-Chair of the IEEE Asia-PacificRegions/Chapters Subcommittee. He was the recipient of the coveted ShantiSwarup Bhatnagar Prize in Engineering Sciences awarded by the HonorablePrime Minister of the Government of India in 2005 for his work on electrondevices. He is also a recipient of the 2004 Swarnajayanti Fellowship from theDepartment of Science and Technology, the 2007 IBM Faculty Award, the 2008Materials Research Society of India Annual Prize, and the 2009 TechnoMentoraward from the Indian Semiconductor Association.

Soumyo Mukherji received the B.Tech. degree ininstrumentation engineering from the Indian Instituteof Technology (IIT) Kharagpur, Kharagpur, India,in 1989, the M.S. degree in mechanical engineer-ing from Colorado State University, Fort Collins, in1992, and the Ph.D. degree in biomedical engineer-ing from the University of North Carolina, ChapelHill, in 1997.

He is currently a Professor in the Departmentof Biosciences and Bioengineering, IIT Bombay,Mumbai, India. His research interests include physi-

cal, chemical, and biological sensing systems (macro and micro) for medical/biological applications and telemedicine systems.

Documents

Modeling, Simulation, And Design Guidelines for Piezoresistive Affinity Cantilevers