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[Kosuke Minami](https://orcid.org/0000-0003-4145-1118)

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[Topology optimization for piezoresistive nanomechanical surface stress sensors in anisotropic 111> orientations.](https://mdr.nims.go.jp/datasets/f647ad07-7a62-4ca7-9a67-792067e6c7b4)

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Topology optimization for piezoresistive nanomechanical surface stress sensors in anisotropic 〈111〉 orientationsNano ExpressPAPER • OPEN ACCESSTopology optimization for piezoresistivenanomechanical surface stress sensors inanisotropic 〈111〉 orientationsTo cite this article: Chao Zhuang et al 2023 Nano Ex. 4 035007 View the article online for updates and enhancements.You may also likePreferred growth direction of III–Vnanowires on differently oriented SisubstratesHaotian Zeng, Xuezhe Yu, H AruniFonseka et al.-Influence of carbon content on cold rollingand recrystallization texture inpolycrystalline 3% Si-FeM Takenaka, Y Shingaki, T Imamura et al.-A numerical modeling approach toestimate the piezoresistance of diffusedresistors with experimental validationK V Meena, Ribu Mathew and A RaviSankar-This content was downloaded from IP address 144.213.253.16 on 01/09/2023 at 06:10https://doi.org/10.1088/2632-959X/acef44/article/10.1088/1361-6528/abafd7/article/10.1088/1361-6528/abafd7/article/10.1088/1361-6528/abafd7/article/10.1088/1757-899X/82/1/012042/article/10.1088/1757-899X/82/1/012042/article/10.1088/1757-899X/82/1/012042/article/10.1088/1361-6439/ac848b/article/10.1088/1361-6439/ac848b/article/10.1088/1361-6439/ac848bNano Express 4 (2023) 035007 https://doi.org/10.1088/2632-959X/acef44PAPERTopology optimization for piezoresistive nanomechanical surfacestress sensors in anisotropic 〈111〉 orientationsChaoZhuang1,2,∗ , KosukeMinami2 , Kota Shiba2 andGenki Yoshikawa1,2,∗1 Materials Science and Engineering, Graduate School of Pure and Applied Science, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,Ibaraki, 305-8571 Japan2 Research Center for Macromolecules and Biomaterials (RCMB), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba,Ibaraki 305-0044 Japan∗ Authors towhomany correspondence should be addressed.E-mail: zhuang.chao@nims.go.jp and yoshikawa.genki@nims.go.jpKeywords: piezoresistivemicrocantilever, surface stress, topology optimization, nanomechanical sensors, anisotropicmaterialsSupplementarymaterial for this article is available onlineAbstractMicroelectromechanical systems (MEMS)-based piezoresistive nanomechanical sensors are compactsensing platformswidely employed in vapor sensing, environmentalmonitoring, and biosensing.Despite their extensive utility, their lower sensitivity relative to their optical readout counterparts hasbeen a limiting factor, constraining thewider application of this technology. Prior research hassuggested that alternative silicon orientations, such as 〈111〉 orientations in (110)wafers, cansignificantly improve the sensitivity of piezoresistive sensors. However, the complexity of optimizingtwo-dimensional stress distribution and handling anisotropic elasticity hasmade device design aformidable task, leaving this promising avenue largely unexplored. To address this challenge, weemploy density-based topology optimization to generate a series of optimized designs for piezo-resistive nanomechanical sensorsmanufactured along 〈111〉 orientations. The properties of theimmobilization layer—the functional coating on the sensor—are parametrically varied to exploreoptimal designs. Our study reveals a transition in optimized designs from a double-cantileverconfiguration to a suspended platform configuration, dictated by the stiffness ratio between theimmobilization layer and the silicon layer. This transition is attributed to the shift in the neutral planeand the prevailing stress relaxationmechanism. In addition, we scrutinize the effects of piezoresistorgeometry andfind that the optimized designs depend asymmetrically on the piezoresistor position, acharacteristic stemming from the anisotropic elasticity in 〈111〉orientations. These optimized designs,verified byfinite element analysis (FEA), demonstrate a notable improvement in sensitivity ofmorethan 20%when benchmarked against traditional rectangular designs and equivalent optimizeddesigns in conventional orientations, thereby validating the effectiveness of the presentmodel. Thisstudy provides crucial knowledge for the design of piezoresistive biosensors, facilitatingmore efficientgeometric design in future sensor development.1. IntroductionThe advancement of nanotechnology has catalyzed the development of variousmicroelectromechanical systems(MEMS)-based sensors adept at detecting chemicals and biomolecules with exceptional sensitivity andspecificity. Among different types ofMEMS-based sensors, nanomechanical sensors have garnered significantattention due to their availability and high sensitivity [1–3]. Compared to the conventional optical readout,piezoresistive readout-based nanomechanical sensors demonstrate promising capabilities in numerousbiosensing applications, as they eliminate the need for laser alignment and function in opaque liquidenvironments, facilitating in situ detection ofDNA [4], RNA [5], and proteins [6, 7]. Nevertheless, despite theirOPEN ACCESSRECEIVED23May 2023REVISED2August 2023ACCEPTED FOR PUBLICATION11August 2023PUBLISHED31August 2023Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 4.0licence.Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.© 2023TheAuthor(s). Published by IOPPublishing Ltdhttps://doi.org/10.1088/2632-959X/acef44https://orcid.org/0000-0002-4469-5364https://orcid.org/0000-0002-4469-5364https://orcid.org/0000-0003-4145-1118https://orcid.org/0000-0003-4145-1118https://orcid.org/0000-0001-7775-0318https://orcid.org/0000-0001-7775-0318https://orcid.org/0000-0002-9136-8964https://orcid.org/0000-0002-9136-8964mailto:zhuang.chao@nims.go.jpmailto:yoshikawa.genki@nims.go.jphttps://doi.org/10.1088/2632-959X/acef44https://crossmark.crossref.org/dialog/?doi=10.1088/2632-959X/acef44&domain=pdf&date_stamp=2023-08-31https://crossmark.crossref.org/dialog/?doi=10.1088/2632-959X/acef44&domain=pdf&date_stamp=2023-08-31http://creativecommons.org/licenses/by/4.0http://creativecommons.org/licenses/by/4.0http://creativecommons.org/licenses/by/4.0compactness, piezoresistive readout suffers from a low signal-to-noise ratio attributable to limited sensitivityand elevated electronic noise. Prior studies have proposed various strategies tomitigate electronic noise,includingminiaturization [8] and optimizing boron doping [9, 10], yet low sensitivity continues to obstruct thebroader application of this technology.The sensitivity of piezoresistive nanomechanical sensors can be improved throughmaterial design. Forexample, substituting siliconwith softermaterials such as SU-8 can reduce stiffness and consequently enhancesensitivity [11, 12]. However, these softmaterials are prone to age effects, and their performance is heavilyinfluenced by environmental conditions [13], rendering them less robust compared to conventionalmaterials,such as silicon.In the context of silicon-based piezoresistive sensors, oneway to enhance the sensitivity is by usingalternative silicon orientations with higher piezoresistance coefficients. It has been recommended in previousliterature that silicon in 〈111〉 orientations is an effective alternative to conventional 〈110〉 orientations forpiezoresistive diaphragmpressure sensors [14]. However, the elevated piezoresistance coefficients concurrentlyintroduce anisotropic elasticity, complicating sensor design and hindering the identification of the optimalgeometry that best capitalizes on high sensing performance.Designing a sensor geometry that accommodates anisotropicmaterial properties calls for a strategy thatsurpasses traditional optimization based on dimensional reduction, such as beam theory [15–18] and platetheory [19–21]. The response of a piezoresistivemicrocantilever to surface stress loading is dictated by the tensorproduct of stress and piezoresistance averaged at the piezoresistor. Given that the piezoresistance coefficients ofamaterial can be either positive or negative,maximizing sensitivity entails stressmaximization in somedirections andminimization in others—a task too complex for simplemodels with a limited number of degreesof freedom. Consequently, numerical studies utilizing finite element analysis (FEA) have been conducted toestablish optimal principles for cantilevers with elementary geometries, such as rectangles and triangles [22–25].However, these studies have focused on cantilevers with rudimentary shapes and isotropicmaterials, leaving avast design space involving complex boundaries and asymmetric designs uncharted. A systematic designapproach is therefore essential to identify sensor geometries that exhibit the highest sensitivity and optimallyexploit anisotropic orientations.Density-based topology optimization is a computer-aided inverse designmethod employed across variousengineering domains to address intricate optimization challenges. By discretizing a design into thousands ofvariables and resolving themulti-dimensional optimization problem, this algorithm can yield non-intuitiveoptimal designs thatmaximize orminimize specific objectives [26]. Pederson [27] employed this approach togenerate a series of optimized designs under diverse conditions, such as varying sizes and placements of thepiezoresistor and anisotropic silicon orientations. However, the optimized designs produced at the time sufferedfroma numerical instability known as the one-node connection problem,whereinmaterials in the designwerelinked by elements with vanishingly low density. This numerical defect severely compromised themanufacturability of the optimized design, rendering it challenging to verify optimization results using FEA andexperiments, and casting doubt on the effectiveness of the optimization algorithm.In this study, we investigate the optimal sensor design for piezoresistive biosensors fabricated alonganisotropic 〈111〉 orientations. By integrating topology optimizationwith robust formulation, a numericaltechnique developed recently to efficiently address the one-node connection problemby imposing a robustlength scale on the optimization problem,we successfully generate a series of optimized designs under variousmaterial properties and geometric constraints. Specifically, wemodify thematerial properties of the sensor toexplore the optimal design in different application scenarios, such as vapor sensing and biosensing, and identifythemost efficient designs under anisotropic silicon orientations. Further, we vary the shape and position of thepiezoresistor to examine their combined impacts on the optimized designs. All optimized designs are verified byFEA, and they exhibitmarkedly improved sensitivity and distinct stress behaviors when benchmarked againstconventional rectangular designs under identical conditions. To comprehend the significance of variousgeometrical features in the optimized design, we develop an FEAmodel to scrutinize the origin of high sensitivityand the role of anisotropy, offering insights into the rational design of piezoresistive nanomechanical sensors.Through a comprehensive investigation of designs based on anisotropicmaterials, this study provides crucialknowledge for the design of piezoresistive biosensors, enablingmore efficient geometric design in future sensordevelopment.2.Methods2.1. Piezoresistive sensorThe bending of the piezoresistivemicrocantilever ismodeled in 3D as a bimorph consisting of animmobilization layer and a silicon structural layer under a fixed-free boundary condition. The cross-section2Nano Express 4 (2023) 035007 CZhuang et alalong the x-z plane is shown infigure 1. The surface stress in the immobilization layer is caused bymolecularinteractions, and it ismodeled by a biaxial stress bs (in units ofN·m–2) relating to the surface stress (in units ofN·m–1) by t ,s b is s= with ti being the thickness of the immobilization layer. The bending deformation caused bythe surface stress is calculated by solving:( )Ku F , 1=s swhere Fs denotes the force vector containing the bendingmoment load induced by the surface stress, K is thedesign-dependent stiffnessmatrix, and us is the deformation vector under surface stress. The constitutiverelation yields the stress in the cantilever:[( ) ] ( )C u u12, 2Ts =  + s swhere C is the stiffnessmatrix of thematerial [28].The piezoresistor ismodeled at the clamped end and the surface of the silicon, and it is assumed to have zerothickness to simplify themodeling. The resistance change of the piezoresistor with area A is determined by theaveraged tensor product between stress and piezoresistance under unit surface stress loading, which can besimplified into the following formby assuming plane stress conditions [29]:( ) ( )R RAA1d , 3l l t tò ò ps p sD = +where ,lp ,tp ls and ts are the piezoresistance coefficients and stresses along the longitudinal and transversedirections, respectively. The values of piezoresistance coefficients in these two directions depend on thecrystallographic axes of the piezoresistor in the silicon substrate, and they are computed by rotating thepiezoresistance tensorwith respect to the standard 〈100〉 directions:( ) ( ) ( )l m m n n l2 4l 11 11 12 44 121212121212p p p p p= - - - ´ + +and( ) ( ) ( )l l m m n n , 5t 12 11 12 44 122212221222p p p p p= + - - ´ + +where l, m, and n are known as direction cosines and they are given by the Euler angles ( ,f ,q y) of thecorrespondingwafer plane and orientation. ,11p ,12p and 44p are piezoresistance coefficients of p-type silicon in〈100〉 directions according to theVoigtmatrix notation as listed in table 1. Sincewemodel the piezoresistivemicrocantilever in away that the longitudinal direction alignswith the x-axis and the transverse direction alignswith the y-axis, ls corresponds to xxs while ts corresponds to .yys Therefore, the expression for the sensitivity ofthe piezoresistor (i.e., the resistance change) becomes( ) ( )R RAA1d , 6l xx t yyò ò ps p sD = +When the piezoresistor is orientated along 〈110〉 directions that are conventionally used for piezoresistivesensing, neglecting small piezoresistance coefficients (i.e., 11p and 12p ), equations (4) and (5) lead to a simplifiedformof the sensitivity in equation (6) as( ) ( )R RA AA1d , 7xx yy44ò òps sD = +where the sensitivity ismaximizedwhen the difference between stresses in the x and y directions ismaximized[30]. In these directions, explicit calculation of piezoresistance coefficients using Euler angles of (0,0, /4p ) yields71.8 10l11p = ´ - [Pa–1] and 66.3 10t11p = - ´ - [Pa–1], which have approximately the same absoluteFigure 1.A schematic of piezoresistivemicrocantilever. The immobilization layer interacts with external stimuli, resulting in a surfacestress ss that deforms the cantilever and piezoresistive signals.3Nano Express 4 (2023) 035007 CZhuang et almagnitude, justifying the simplification of /244p in the previous literature. However, for the alternative 〈111〉orientations considered in this work, for example [ ̅ ̅1 11], which is associatedwith Euler angles of ( /2,p /4,p( )/cos 2 31- ), the piezoresistance coefficients in the longitudinal and transverse directions are significantlydifferent: 93.5 10l11p = ´ - [Pa–1] and 44.6 10t11p = - ´ - [Pa–1]. Therefore, the sensor output can no longerbe simply treated as the difference between stresses in the two directions, but aweighted difference defined bythe piezoresistance coefficients.The previously used Euler angles are used to construct a 6 by 6 rotationmatrix a to rotate the stiffnessmatrixto account for the anisotropic elasticity of silicon at 〈111〉 orientations according to [31]:( )C C , 8c Ta a=where the superscript c denotes the stiffnessmatrix elements in the crystallographic coordinate as listed intable 1.2.2. Topology optimizationTopology optimization is typically formulated in a predetermined design domainΩwith a given set of loadingand boundary conditions, inwhich the algorithm finds thematerial distribution that optimizes a given objectivefunction f [34, 35]. Specifically, the design domain is discretized into finite elements where thematerialdistribution is described by an element-wise constant density vector .jr The density of each element can takevalues between 1 and 0, representing the presence and absence ofmaterial, respectively. Intermediate densitiesare allowed such that the optimization variable is continuous. Thematerial properties and the biaxial stressrepresenting surface stress are weighted by the element density raised to the power of p,which is a penalizationfactor based on themethod of Simple IsotropicMaterial with Penalization (SIMP) [36, 37] that guarantees thealgorithm to generate discrete designs with only 0 and 1 densities. p is typically 3 depending on the Poisson’sratio of thematerial [38].The optimization is carried out through the following steps: (1)Evaluate the objective function of the designusing FEA; (2)Evaluate the sensitivity of objectives and constraints with respect to the density variation of eachelement based on FEA and associated adjoint solutions; (3)Update the densities of all elements based on thesensitivity analysis using themethod ofmoving asymptotes (MMA); (4)Repeat steps (1)–(3) until a convergencecriterion ismet or amaximumnumber of iterations is reached [26, 39].2.2.1. Filters and projectionsDensity filtering based on aHelmholtz filter [40] is used to prevent thewell-knownnumerical instability ofcheckerboarding [41] andmesh dependence [42]. It is implemented by calculating the elemental density as aweighted average of all the neighboring elements within a radius of rmin at each iteration:⎜ ⎟⎛⎝⎞⎠( )rx y, 9jj jjmin22222 rr rr-¶¶+¶¶=where jr and jr are the original and filtered density of element j, respectively.Since the density filter introduces transition zones between solid and void regions that are not physicallymeaningful, aHeaviside projection function is further implemented to eliminate intermediate densities [43, 44].The projected density is given by [45]̅( ) ( ( ))( ) ( ( ))( )tanh tanhtanh tanh 1, 10jjrbh b r hbh b h=+ -+ -Table 1.The stiffness and piezoresistancecoefficients of p-type silicon in 〈100〉orientations used for tensor rotation.PiezoresistanceCoeff. ( 10 11´ - Pa–1) aStiffness Coeff.(GPa) bπ11 6.6 c11 165.7π12 −1.1 c12 63.9π44 138.1 c44 79.6a The piezoresistance coefficients are obtainedfrom reference [32].b The stiffness coefficients are obtained fromreference [33].4Nano Express 4 (2023) 035007 CZhuang et alwhere j̅r represents the projected density at element j, b represents the projection slope, and h represents theprojection threshold, which determines the density at which the projection function applies.2.2.2. Robust formulationTo enforce aminimum feature size for solid and void regions, robust formulation evaluates the objectivefunctions of dilated, intermediate, and eroded designs during each iteration, which correspond to0.5,h h= <* 0.5,h = and 1 ,h h= - * respectively. TheMMA solver will automatically optimize theperformance of theworst-performing design among the three. By solving thisminimax problem, aminimumlength scale is enforced to the optimized design, which effectively eliminates one-node connection problems andimproves the robustness of the design to fabrication errors [46–48].2.2.3. Optimization problemThe design domain W is defined as the square interface between the silicon and immobilization layer asillustrated infigure 2(a), where the interface, top, and bottom surfaces of the bending bimorph are delineated bysolid and dashed lines. During the optimization process, thematerial distribution in the design domain evolvestomaximize the objective function as illustrated infigure 2(b). The surface stress andmaterial properties of thedesign are weighted by the projected density j̅r via a general extrusion function, whichmeans that the presenceof an element simultaneously represents the presence of the immobilization layer and siliconwith predefinedthicknesses aswell as local surface stress.With this treatment, the thicknesses of the immobilization layer and thesilicon are assumed to be constant everywhere in the design domain. The piezoresistor is regarded as a passiveareawith a density of 1. The optimization objective is the sensitivity of the sensor, that is, the fractional change inresistance at the piezoresistor divided by unit surface stress :ss( ˜ )∣ ∣( )fR R. 11srs=DThe optimization problem can be described by the followingminimax problem:{ ( ̅ ( )) ( ̅ ( )) ( ̅ ( ))} ( )Max f f f: min , , , 12e i d  r r r r r rrwhere the superscripts of e, i, and d denote eroded, intermediate, and dilated designs. The volume constraint isapplied to the dilated design for design regularization. Such a constraint is a commonpractice in solvingoptimization problems as it regularizes the final design and avoidsfloatingmaterials that are not physicallymeaningful [49].Figure 2.A schematic of the topology optimizationmodel. (a)The geometry of the present design problem. The optimization isperformed in the 2DdomainΩ, which is the interface between the silicon layer and the immobilization layer. The solid line representsthe interface between two layers of the optimized structure, and the dashed lines represent the top and bottom surfaces. (b)Theiteration of sensor geometries during the optimization process.5Nano Express 4 (2023) 035007 CZhuang et al2.2.4. Computational implementationThe topology optimization is implemented in the commercial FEA softwareCOMSOLMultiphysics 6.1. Thestructural deformation of the sensor is simulated using the structuralmechanicsmodule, wherein a coupledinterface of shell and solid is employed tomodel the stressed immobilization layer and the silicon structurallayer, with the immobilization layermodeled by the shell and the silicon layermodeled by the solid. Themodel isdiscretized by linear quadrilateral elements with 50 elements per side in the in-plane directions and threeelements in the thickness direction to prevent shear locking. The optimization is performedwith theoptimizationmodule using the globally convergent version ofMMA. The adjoint problem is solvedautomatically inCOMSOL to obtain the sensitivities for the objective and constraint functions. COMSOL isinterfacedwithMATLAB to automate the parameter sweep. To ensure the optimization reaches the globaloptimum, a continuationmethod involving ramping up the projection slope b from1 to 16 by doubling every20 iterations is employed [39, 46, 50]. For the robust formulation, the projection threshold h is set to 0.3, 0.5, and0.7 for dilated, intermediate, and eroded designs, respectively. The initial volume and the volume constraint areset to 0.6. The intermediate design is considered the final design. Othermodel parameters remain constantduring the optimization, as shown in table 2.To evaluate the sensing performance and stress distribution in the optimized designs, the optimizedsolutions from topology optimization are reimported,meshed, and evaluated inCOMSOLusing two timesdenser quadratic triangular elements compared to the optimizationmodel to ensure accuracy. The resultsobtained from these verification simulations are used to analyze the sensor performance and stress distribution.3. Results and discussionTo examine the optimal sensor designsmade along 〈111〉 silicon under various application scenarios and designconstraints, the topology optimizationmodel is employed to generate optimized designs under variousimmobilization layermaterials, shapes, and positions of the piezoresistor. The default values of these parametersare listed in table 3 unless explicitly specified.To illustrate a typical optimization result, an optimized design generated by default settings is shown infigure 3, with the optimization history compiled infigure S1 of the supplementarymaterials. The small square atthe left boundary represents the piezoresistor. The optimized design features an asymmetric ‘double-cantilever’configuration, where the piezoresistor is located at the junction between the two combined cantilevers. Theasymmetry is in contrast to the symmetric designs obtained in previous reports where isotropic orientationswere considered [51]. The double-cantilever design leads to a concentrated stress profile at the piezoresistor, asshown infigure 3(b), the averaged value of which, according to equation (6), constitutes the signal output of thesensor. The concentrated stress profile stems from a synergy of high stress in the x-direction and low stress in they-direction at the piezoresistor, as demonstrated infigures 3(c)–(d). The synergized stress distribution, andTable 2.Model and optimization constants.Constants Value UnitSilicon thickness, ts 3 μmSilicon Poisson’s ratio 0.27 1Surface stress, ss −1 N·m–1Immobilization layer Poisson’s ratio 0.42 1Piezoresistor area 1600 μm2Design domainwidth 300 μmMaximummesh size 6 μmFilter radius, rmin 7.2 μmTable 3.Modelling parameters that are used in this workunless explicitly stated otherwise.Parameters Value UnitImmobile layer Young’smodulus, Ei 80 GPaImmobile layer thickness, ti 50 nmPiezoresistor aspect ratio 1 1Piezoresistor offset 0 μm6Nano Express 4 (2023) 035007 CZhuang et alconsequently the high sensitivity, is a result of the non-intuitive arrangement ofmaterials enabled by topologyoptimization.3.1.Material of the immobilization layerAs an essential component of a biosensor, the immobilization layer provides the chemical specificity and thedriving force for the transduction of chemical signals. Different applications require distinct types ofimmobilizationmaterials, and the geometry and elastic properties of thematerial govern the design of thesensor. To better understand the impact of immobilizationmaterials on the design of piezoresistive sensors andenable application-specific sensor designs, we systematically varied Young’smodulus and thickness of theimmobilization layermaterials and compared their optimized designs obtained through topology optimization.Young’smodulus Ei and thickness ti of the immobilization layer are varied from10 to 1000 GPa and from0.01to 3 μm, respectively. This extensive range of parameters encompasses practically relevant immobilization layermaterials, such as gold (with Young’smodulus of 80 GPa and a thickness in the tens of nanometers) andpolymericmaterials (with Young’smodulus of 10 GPa and a thickness of severalmicrometers) [4–7, 52].Weomitted lower Young’smoduli fromour study, as optimization results exhibitedminimal alteration at such lowvalues.The optimization results underscore a potent dependence of optimized piezoresistive sensor designs on thegeometry and elastic properties of the immobilization layer, as exemplified by a selected set of optimized designsinfigure 4. The full gamut of optimized designs is demonstrated infigure S2–S5.When the immobilization layerpossesses a relatively lowYoung’smodulus and thickness, the double-cantilever configuration appears to beoptimal, whereas a suspended platform configuration is optimal for higher Young’smodulus and thickness, asdisplayed in the upper left and lower right corners offigure 4. The origin of this transition in optimalconfigurations is not fully understood. It is clear, however, that this transition is associatedwith the increasedrigidity of the immobilization layer and the shifted neutral plane of the bilayer structure of the immobilizationlayer and silicon, as implied by the diminishing bending deformation and piezoresistive signal of the sensor(figure S6).To investigate this phenomenon, we plotted averaged bending displacement of the piezoresistor and twoother key attributes of optimized designs against the stiffness ratio (r) between the immobilization layer andsilicon as shown infigure 5. The stiffness ratio is defined as /r E t E t ,i i s sº where Es and ts are the Young’smodulus and thickness of the silicon, respectively. The product of E and t represents the in-plane stiffness of alayer. This concept comes from the calculation of uniform elongation/contraction strain c of a bimorph, asopposed to the thickness-dependent bending strain, under surface stress loading [53]:( )cE tE t E t, 13i ii i s s=+where  represents the biaxial strain of the immobilization layer. Equation (13) relates the uniform strain of theentire bimorphwith the strain in the immobilization layer, with a proportional constant being the ratio of the in-plane stiffness of the immobilization layer and the total stiffness of the bimorph. By substituting in the definitionof stiffness ratio r, the equation (13) can bewritten as:( )crr 1, 14=+where ( )/r r1 + is the transformation ratio that scales from0 to 1 asymptotically as r increases from0 toinfinity.When this ratio is small, only a small portion of the strain transforms into the uniform strain.When theratio is 1, all strain transforms into uniform strain.Figure 3.A typical optimized design generated by topology optimization. (a)The density distribution of the optimized design. (b)Theintensity of piezoresistive signals at the sensor surface. (c), (d) Stresses in the x direction (c) and the y direction (d).7Nano Express 4 (2023) 035007 CZhuang et alFigure 4.The intensity of piezoresistive signals of optimized designs obtained under various thickness andYoung’smoduli of theimmobilization layer.Figure 5.The averaged bending displacement of piezoresistor, the position of neutral axis t ,n and the transformation ratio ( )/r r 1+as functions of the stiffness ratio between the immobilization layer and silicon. The inset figures denote the optimized designs at thecorresponding stiffness ratio.8Nano Express 4 (2023) 035007 CZhuang et alThe neutral plane is the plane that experiences zero strain during bending deformation, and its position tnrelative to the bilayer interface is calculated using [53].( )( )tE t E tE t E t2. 15ni i s si i s s2 2=-+We take Young’smodulus of silicon Es to be 175.1 GPa, which is the average elasticmoduli of thefirst threediagonal elements in the stiffnessmatrix of silicon in 〈111〉 orientations.The results presented infigure 5 demonstrate that the optimal design of piezoresistive sensors stronglydepends on the stiffness ratio between the immobilization layer and silicon. Specifically, the double-cantileverconfiguration is optimal when the immobilization layer has a negligible stiffness, that is, r = 1. At this point, theneutral plane is located below the immobilization layer/silicon interface and at themid-plane of the siliconlayer, with a position tn of approximately –1.5 μm.Moreover, the transformation ratio ( )/r r 1+ is close to 0,meaning there is barely any uniformdeformation. Since there are only two possiblemechanisms, uniformelongation/contraction and bending [16], that can relax the strainmismatch between two layers and reachequilibrium, a near-zero transformation ratio thenmeans that the stress relaxation is dominated by bendingdeformation. As the stiffness ratio r increases and approaches 0.1, the immobilization layer can no longer beignored. This leads to a rapidly diminishing bending displacement, a rising ( )/r r 1 ,+ an up-shifting neutralplane, and the transition in optimal configuration: the gap between the two cantilevers deforms and develops anL-shape configuration.As the stiffness ratio r continues to increase and exceeds 1, the offset between the bilayer interface andneutral plane vanishes and then becomes positive, indicating that the neutral plane now resides within theimmobilization layer.Moreover, the transformation ratio is now getting closer to 1,meaningmost of the strainin the immobilization layer becomes uniform strain in the bimorph. The dominating stress relaxationmechanism shifts frombending deformation to uniform elongation/contraction, which accompanies theemergence of the suspended platform configuration. Therefore, the drastic transition of optimal configurationcould be a result of the shifting stress relaxationmechanism as the suspended platform configurationmay bemore efficient in generating piezoresistive signals than the double-cantilever in the regimewhere stretchingdeformation, instead of bending deformation, dominates.Since practically important immobilization layersmade from gold and polymericmaterials have a typicalbending stiffness E ti i of the order of∼103 and∼104, respectively, while the bending stiffness of the silicon E ts sconsidered here is∼105. This leads to a stiffness ratio below 0.1,making the double-cantilever configurationoptimal in these cases. Therefore, wewill omit the L-shape gap and suspended platform configurations in thefollowing discussion.However, it is important to note thatwhen the thickness of the silicon is reduced tonanometer length scales where the stiffness of the immobilization layer becomes significant, furtherconsideration on these designsmay be required as they could potentially provide better sensing performancethan the double-cantilever design.In the present study, geometric nonlinearity has been omitted due to its associated high computational cost.The optimization process involves only linear analysis; as a result, variations in themagnitudes of surface stress( ss ) and biaxial stress ( bs )do not influence the resulting optimized designs. Consequently, any adjustments to bstomaintain a constant ss when varying ti will not affect the optimized designs, indicating that transitions ofoptimized designs are purely geometric. It is important to note that nonlinearity often arises from the clampedboundary condition under compressive surface stress loading, including the fixed-free boundary conditionconsidered in this study [54, 55]. However, this omission of nonlinearity here can be justified by findings by Buhlet al [56], who reported onlymarginal improvements in designs upon incorporating geometric nonlinearitiescompared to those generated using linear analysis. Nonetheless, in scenarioswhere the applied surface stress islarge, the silicon layer is thin, or the problem is formulated undermore stringent boundary conditions, such as afully-fixed boundary condition [57], nonlinearity will become significant enough to alter the structure’s stiffnessand even lead to elastic instability [58]. In such instances, nonlinear analysis becomes essential, as it is knownthat snap-throughmotions during instability can result in substantial changes in the optimizationoutcomes [59].3.2. The shape of piezoresistorIn order to design a piezoresistive sensor with a high signal-to-noise ratio, it is essential to optimize the dopingprofile, placement, and geometry of the piezoresistor, in addition to a sensitive immobilization layer as discussedin the previous section. In this section, we scrutinize the impact of the piezoresistor shape on the optimizeddesign and explore non-intuitive design schemes thatmay achieve better performances.Wefirst study the effectof the piezoresistor shape on the optimized design by varying its aspect ratio between 1/9 and 9, with theoptimization results depicted infigure 6. The stress distribution in the x and y directions is illustrated infigures9Nano Express 4 (2023) 035007 CZhuang et alS7–S9. The total area is kept constant tomaintain a consistent signal-to-noise ratio, as the noise level iscontingent upon the doping area [9, 10].For low aspect ratios (from1/9 to 1/7), in contrast to the double-cantilever demonstrated infigure 3, atriple-cantilever configuration emerges. As the aspect ratio increases, the central cantilever disappears, and theoptimized design reverts to a double-cantilever configuration. The disappearance of the central cantilever can berationalized by the fact that while the double-cantilever configuration is an efficientmechanism for generatinghigh-sensitivity regions, it has a limited area of influence, necessitating the low aspect ratio design to producetwo such configurations to compensate for its widened sensing area.Interestingly, when the aspect ratio exceeds 4, an oblique hollow feature begins to develop from the clampedend. This feature does not enhance sensitivity; instead, it emerges as an artifact caused by the volume constraint.As the aspect ratio increases, the cantilevers need to elongate tomaintain the concave double-cantilevergeometry.However, this growth is hindered by the scarcity ofmaterials, i.e., the imposed volume constraint. As aresult, these designsmust hollow out themselves to conservematerials for cantilever extension, leading todegradation in design quality at high piezoresistor aspect ratios.To validate the efficacy of topology optimization, the sensing performances of optimized designs arebenchmarked against conventional rectangular cantilevers via FEAmodeling under the same piezoresistorshape. The conventional rectangular cantilevers aremodeledwith a similarfixed-free boundary condition, andtheir length andwidth are adjusted tomaximize the piezoresistive sensitivity according to the principlessuggested by previous studies [22–25]. Specifically, the cantilever length should beminimized and set to onemesh longer than the piezoresistor to avoid the high-stress region near the free edge. Secondly, the cantileverwidth should bemaximized and set to be the same as the design domain, as depicted infigure 7. Thewidth of thedesign domain and themesh densitymirror those of the topology optimization to facilitate a fair comparisonbetween the two design approaches. The simulation results, including the distribution of piezoresistive signalsand stresses in the x and y directions, are displayed infigures S10–S12.The sensitivities and associated stresses of optimized designs and rectangular benchmarks are assessed andpresented infigure 8, which illustrates the pronounced dependency of sensitivity on the piezoresistor shapeFigure 6.The intensity of piezoresistive signals of optimized designs obtained under various piezoresistor aspect ratios.Figure 7.The geometry of the rectangular benchmarkmodel used for validation.10Nano Express 4 (2023) 035007 CZhuang et alwithin both design strategies. As the aspect ratio increases, the sensitivity of the optimized designs initiallyascends and subsequently descends, attaining amaximum sensitivity of 13.7 10 4´ - (N·m–1) –1 at an aspectratio of 1/2.5 (figure 8(a)). This is 22.3%higher than themaximum sensitivity of rectangular benchmarks,which is11.2 10 4´ - (N·m–1) –1 at an aspect ratio of 1/7 (figure 8(b)).Elevated sensitivity should result from themaximization of xxs and theminimization of ,yys as underscoredin section 2.1. Indeed, in optimized designs, the average stress in the x-direction xx ave,s peaks at an aspect ratio of1/3, while yy ave,s reaches itsminimumat an aspect ratio of 2. The interplay between these two stresses yields theoptimal aspect ratio of 1/2.5. Conversely, the sensitivities of rectangular benchmarks exhibit a plateauedsensitivity at low aspect ratios and consistently trail those of the optimized designs. The sensitivity of therectangular benchmark is amonotonically decreasing function of the piezoresistor aspect ratio, signifying adeficiency of design flexibility in this conventionalmodel.3.3. Piezoresistor positionAs illustrated in the preceding section, the optimal aspect ratio of the piezoresistor for optimized designs liesaround 1/2.5, driven by the extremization of stress along the x and y directions. However, the asymmetryevident in the designs obtained thus far necessitates further scrutiny, as they suggest thatmaterials in the upperand lower halves of the design domainmay contribute to sensitivity with varying efficacies. Therefore, displacingthe piezoresistor from the centermightmodify the optimal aspect ratio and engender alternative optimaldesigns.To explore alternative designs with off-center piezoresistors, we adjust the position of the piezoresistor fromthe lower to the upper of the design domain.We define the offset of a piezoresistor as the distance between themidpoints of the piezoresistor and the clamped boundary. Thus, with a design domain having an edge length of300 μm, a square piezoresistor with an edge length of 40μmsituated at the lower edge equates to an offset of−130 μm,while the upper edge corresponds to 130 μm.We investigate various positionswith offsets rangingfrom−110 to 110 μmwhile the piezoresistor aspect ratio varies from1/4 to 4. A selected set of optimizationresults is displayed infigure 9, and a comprehensive depiction is provided infigures S13–S16. Thefiguresunderline the pronounced dependency of optimized designs on the placement of piezoresistors. As thepiezoresistormigrates away from the center, the double-cantilever configuration deforms in tandemwithincreasing piezoresistor aspect ratio, and the degree of deformation is contingent upon the direction of theoffset.When the piezoresistor is proximal to the lower or upper edges, intricate new configurations emerge,characterized by holes at the clamped end andwithin the cantilever. These designs leverage the stressconcentration at the corners tomaximize sensitivity.To further investigate the interplay between piezoresistor position and its shape, and their subsequentimpact on sensor sensitivity, we conducted a comprehensive evaluation of the two parameters. Our evaluation,as depicted infigure 10(a), indicates that the sensitivity of optimized designs is a nonlinear function of aspectratio.Moreover, the specific dependency appears to depend stochastically on the piezoresistor position. Theoptimal aspect ratio of 1/2.5 at zero-offset becomes suboptimal at other piezoresistor positions. The seeminglyrandompatterns infigure 10(a) could be attributed to the solutions being trapped at localminima due to themultidimensional nature and the non-convexity inherent in the current optimization problem.Nevertheless,some general trends are discernable. Specifically, a positive offset typically yields higher sensitivity compared to anegative offset, irrespective of the piezoresistor shape. Furthermore, sensitivity generally decreases withFigure 8.The intensity of piezoresistive signals, averaged stresses in x and y directions as functions of piezoresistor aspect ratio for theoptimized designs (a) and rectangular benchmarks (b).11Nano Express 4 (2023) 035007 CZhuang et alincreasing piezoresistor aspect ratios, likely due to the degrading design quality as pointed out in section 3.2.Therefore, a piezoresistor that is short andwide, positioned at the upper corner, will generally lead to optimizeddesigns with superior sensitivity.For comparative purposes, the sensitivity of rectangular benchmarks with the same piezoresistorconfiguration is evaluated infigure 10(b). The distribution of piezoresistive signals is shown infigure S17. Thesensitivity of these rectangular benchmarks exhibits a distinct dependence on the shape and position of thepiezoresistor. Notably, the highest sensitivity is attained at zero offsets, as opposed to the 110 μmoffset infigure 10(a). The sensitivity diminishes as the piezoresistor aspect ratio increases, and the decreasing trendsexhibit farmore regularity compared to the previous case. It is worth noting that an optimal aspect ratio of 1/3 isobservedwhen the offset is−110 and 110 μm, likely due to the optimal alignment of the piezoresistor with thestress profile in bending cantilevers, as can be observed infigure S17.figures S18 and S19 present a detailedbreakdownof sensitivity as theweighted sumof stresses in the x and y directions, depending on the position andaspect ratio of the piezoresistor, thereby illustrating the significantly different stress behaviors in optimizeddesigns and rectangular benchmarks.Figure 9.The intensity of piezoresistive signals of optimized designs obtained under various piezoresistor positions and aspect ratios.Figure 10.The sensitivity of optimized designs (a) and rectangular benchmarks (b) under various piezoresistor positions and aspectratios.12Nano Express 4 (2023) 035007 CZhuang et al3.4. A simplifiedmodelThe inherent complexity of topology optimizationmakes it challenging to comprehensively analyze thefunctional consequences of individual geometric components. To investigate the specific contributions fromvarious geometric features—particularly the asymmetric designs observed in the optimized structures—wehavedeveloped an FEAmodel with simplified geometry abstracted from the optimized designs. Thismodel, depictedinfigure 11, consists of two rectangular cantilevers joined by a broad base, with the piezoresistor situated at thecenter. Themodel ismeshedwith the same density of quadratic elements as the benchmarkmodels from theprevious sections, ensuring comparable accuracy. Utilizing this simplifiedmodel, we can systematicallymanipulate the geometric parameters to examine the origins of high sensitivity and asymmetry resulting fromtopology optimization. The aspect ratio of the piezoresistor, the lengths of the upper (Lupper) and lower (Llower)cantilevers, and the gap size (Wgap) serve as control parameters in this investigation.3.4.1. Effects of gap sizeBeing a key component in the double-cantilever configuration, wefirst investigate the influence of the gapbetween two cantilevers and the dependency of sensitivity on the gap sizeW .gap WevaryWgap from0 to 150 μm,maintaining the aspect ratio of the piezoresistor at 1/8 and 1/2.5 for comparison, and setting Lupper and Llower atthemaximum length of 300 μm.The distributions of piezoresistive signals for a series of geometries are depictedinfigure 12, with the stress distributions shown infigures S20 and S21.Without a gap, the stress profilemirrorsthat of a cantilever under biaxial surface stress, as demonstrated in prior studies [23]. The introduction of a gap,effectively divides one cantilever into two smaller ones, creating a common corner that facilitates a high-sensitivity region—an ideal location for the piezoresistor. As the gap expands, the stress concentration lessens,leading to a decrease in sensitivity. Importantly, the shape of the piezoresistor significantly influences sensitivityby determining the extent towhich stress concentration generated by the gap is converted into piezoresistivesignals. Hence, the relative size of the piezoresistor and the gapmay be a critical factor inmaximizing sensitivity.The dependence of sensitivity on gap size varies according to the piezoresistor aspect ratio. In an effort toreproduce the optimized gap sizes in a series of optimized designs showcased infigure 6, we vary both the gapsize and the aspect ratio of the piezoresistor to quantitatively investigate their interaction. Figure 13 illustrates amap of sensitivity as a function of gap size and piezoresistor aspect ratio, whilefigure S22 depicts thecorresponding stresses in the x and y directions. In alignment with the previous remarks onfigure 8, thesensitivity of the simplifiedmodelfirst increases and then decreases, reaching peak sensitivity at an aspect ratio of1/2.5 as the piezoresistor aspect ratio increases.Moreover, an optimal gap size exists for all piezoresistor shapes.For lower aspect ratios, the optimal gap size extends up to 80 μm.However, as the aspect ratio increases, theoptimal gap size rapidly shrinks and stabilizes around 30 μm, coincidingwith the gap sizesmeasured inoptimized designs generated via topology optimization (figure 6), asmarked by the black squares infigure 13. Itis noteworthy that the dependencies of sensitivity and stresses in different directions in 〈111〉 orientationsmirror those in 〈110〉 orientations, albeit with differentmagnitudes, as demonstrated infigures S23 and S24.Figure 11.The geometry of the simplifiedmodel developed to investigate the functions of individual geometric components.13Nano Express 4 (2023) 035007 CZhuang et alFurthermore, to evaluate how accurately the simplifiedmodel replicates the sensing behavior of optimizeddesigns obtained from topology optimization, we assess the sensitivity of the simplifiedmodel using the samegap sizes as in the optimized designs. These results are presented infigure 14, with the distributions ofpiezoresistive signals and stresses illustrated infigures S25–S27. Figure 14 highlights the close resemblancebetween the performance of the simplifiedmodel and the optimized designs, barring those that with high-aspect-ratio piezoresistors. The discrepancy can be accounted for by the degraded design quality at highpiezoresistor aspect ratios, as discussed in section 3.2, which results in increased xxs and ,yys as depicted infigureS28. Interestingly, the simplifiedmodel outperforms the optimized designs at lowpiezoresistor aspect ratios.This superior performancemay be ascribed to the sharp corners in the simplifiedmodel, as theminimumfeature size in topology optimization is constrained by the dimensions of themesh and filter. This phenomenonreduces the xxs whilst barely influencing ,yys as indicated infigure S28. Regardless, the strong correspondencebetween the simplifiedmodel and the optimized designs attests to the efficacy of the simplifiedmodel incapturing the essential features of optimized designs, thereby enabling further exploration of the impacts ofother geometric features.3.4.2. Effects of asymmetryThe preceding sections have demonstrated that the optimized designs obtained via topology optimizationexhibit asymmetrical geometries, with the placement of the piezoresistor significantly influencing the optimizedFigure 12.The intensity of piezoresistive signals of the simplifiedmodel under fixed cantilever lengths and varying piezoresistor aspectratios and gap sizes.Figure 13.The sensitivity of the simplifiedmodel under different gap sizes and piezoresistor shapes. For each piezoresistor shape,there exist optimal gap sizes thatmaximize sensitivity, which coincidewell with gap sizesmeasured fromoptimized designs (n 3= )obtained from topology optimization (TO) asmarked by black squares.14Nano Express 4 (2023) 035007 CZhuang et alstructures. The asymmetry in optimized designs suggests that the structuralmaterials in the upper and lowerhalves of the design domainmay contribute differently to the sensor performance.To explore this effect, wemaintain the piezoresistor aspect ratio at 1/2.5 and the gap size at 30 μm, and varythe length of one cantilever (e.g., the upper cantilever Lupper)while keeping the other (e.g., Llower) constant. Theresulting distributions of piezoresistive signals and stresses are illustrated infigure 15 andfigures S29–S30,respectively. This parametric study is conducted in both 〈111〉 and 〈110〉 orientations, with a quantitativeevaluation of sensitivity depicted and compared infigure 16. The sensitivity appears to rise, then plateau, inrelation to the length of cantilevers. The saturation sensitivity reaches 13.7 10 4´ - (N·m–1)–1, coinciding withthe sensitivity of the optimized design at the optimal aspect ratio, as indicated by the dotted horizontal line in thefigure. This sensitivity is 23.4%higher than that of the 〈110〉 orientations, which stands at 11.1 10 4´ -(N·m–1)–1, thereby demonstrating the superior performance of 〈111〉 orientations even in this optimizedgeometry.The saturation length depends on the position of the piezoresistor and alignswell with the average cantileverlengthmeasured in the optimized designwith a piezoresistor aspect ratio of 1/2.5 (figure 6), as indicated by thevertical lines. The existence of a saturation length elucidates the phenomenonwhereby optimized designs with adouble-cantilever configuration tend to extend the cantilevers until the sensitivity gain no longercounterbalances the penalty imposed by volume constraints, as discussed in section 3.2.Figure 14.The sensitivity of optimized designs and the simplifiedmodel under various piezoresistor aspect ratio. The gap sizes used inthe simplifiedmodel aremeasured fromoptimized designwith the corresponding piezoresistor aspect ratio.Figure 15.The intensity of piezoresistive signals of simplifiedmodel underfixed cantilever length on one sidewhile varying that of theother. The piezoresistor aspect ratio and gap size are heldfixed.15Nano Express 4 (2023) 035007 CZhuang et alAnother key observation from figure 16 is that when the Lupper isfixed and Llower is adjusted, the initialsensitivity is lower, but it reaches a higher saturated sensitivity compared to the inverse case. This featureindicates that the lower cantilever contributesmore significantly to sensitivity than the upper one, consistentwith the asymmetrical designs obtained thus far and the superior performance of designs that position thepiezoresistor at the upper edge of the design domain. This asymmetry is exclusive to 〈111〉 orientations, while itis absent in the conventional 〈110〉 orientations, as evidenced by the completely overlapping curves infigure 16.Furthermore, the asymmetry of sensitivity contribution between the two cantilevers is exacerbated forpiezoresistors with high aspect ratios, as illustrated infigure S31.It should be noted that the preference for the lower cantilever could be trivially switched to the uppercantilever by reversing the third Euler angle used for tensor rotation, specifically, from ( /2,p /4,p( )/cos 2 31- ) used here to ( /2,p /4,p ( )/cos 2 31- - ). This change does not alter the piezoresistancecoefficients but doesmodify the sign of certain shear components in the stiffnessmatrix, leading to optimizeddesigns that favor the upper cantilever. The shift of symmetry under tensor rotation allows us to conclude thatthe asymmetry observed in various optimized designs and the preference for a positive piezoresistor offset areconsequences of the anisotropic elasticity of 〈111〉 orientations.4. ConclusionIn this study, we leverage topology optimization to explore the optimal design of piezoresistive nanomechanicalsensors in highly sensitive 〈111〉 orientations. The efficacy of these designs is rigorously evaluated andbenchmarked against traditional rectangular designs, affirming the validity of our approach.Moreover, wedevelop a simplified FEAmodel to investigate the functional roles of various geometrical attributes and elucidatethe origin of the prevalent asymmetric designs found in optimized designs.Key findings fromour investigation are twofold: Firstly, the optimal design is significantly influenced by theimmobilization layermaterials. For instance, a double-cantilever design is optimal for thin, soft immobilizationlayers, whereas a suspended platformdesign ismost suitable for thick, stiff immobilization layers. This transitionin optimal design is attributed to the shift in the neutral plane and the dominating stress relaxationmechanism.However, within a technically important parameter range, the double-cantilever designwith a lowpiezoresistoraspect ratio emerges as themost performant design. This design is 22.3% and 23.4%more sensitive than therectangular benchmarks in 〈111〉 orientations and the same double-cantilever design in 〈110〉 orientations,respectively. This result underscores the superiority of both the design strategy and thematerial choice.Secondly, the piezoresistor geometry plays a consequential role in defining the optimized designs and theaccompanying sensitivity enhancement. The optimal piezoresistor aspect ratios are found to be asymmetricallyposition-dependent as a consequence of anisotropic elasticity in 〈111〉 orientations. Hence. the sensitivityFigure 16.The sensitivity of the simplifiedmodel under a fixed length of upper/lower cantilever while the length of the othercantilever is varied. The simulation is run twice for 〈110〉 and 〈111〉 orientations. The optimized cantilever lengthsmeasured fromtopology optimization (aspect ratio 1/2.5 infigure 6) aremarked by vertical dashed lines. The sensitivity of the correspondingoptimized design ismarked by the horizontal dotted line.16Nano Express 4 (2023) 035007 CZhuang et alenhancements obtained in the study are of exemplary value. The actual benefit derived fromoptimized designsand alternative silicon orientations would depend on the specific application and the fabrication process of thepiezoresistive sensor.Crucially, while the theoretical implications of ourfindings are profound, it is worth noting the practicalchallenges associatedwith their implementation. One notable difficulty in fabricatingMEMSdevices from 〈111〉orientations on (110) silicon compared to (100) silicon is the anisotropic etching of silicon. This process, whichis dependent on the crystallographic orientation of the siliconwafer, results in different etch rates and shapes fordifferent planes. The {111} planes, for instance, are the slowest etching plane in all anisotropic etchants, and assuch, prolonged etching invariably leads to the appearance of {111} facets at the sidewalls of the fabricatedstructures [60]. This occurrence can influence the performance ofMEMSdevices that rely on accurategeometries and dimensions.Despite the potential challenges with fabrication, this study provides invaluable guidelines for future designconsiderations utilizing alternative silicon orientations. This work can potentially improve the capabilities ofpiezoresistive nanomechanical sensors across awide array of disciplines, including but not limited to healthcare,agriculture, and homeland security.AcknowledgmentsCZ thanksNIMS Joint Graduate School Program,NIMS. This workwas financially supported in part by theJSPSKAKENHIGrantNumbers 20K20554; 21H01971; 21K18859; 22K05324; JP19KK0141,MEXT, Japan; andthe Public/Private R&D Investment Strategic Expansion Program (PRISM), CabinetOffice, Japan.Data availability statementThe data cannot bemade publicly available upon publication because they are not available in a format that issufficiently accessible or reusable by other researchers. The data that support the findings of this study areavailable upon reasonable request from the authors.Declaration of competing interestThere are no conflicts of interest to declare.ORCID iDsChaoZhuang https://orcid.org/0000-0002-4469-5364KosukeMinami https://orcid.org/0000-0003-4145-1118Kota Shiba https://orcid.org/0000-0001-7775-0318Genki Yoshikawa https://orcid.org/0000-0002-9136-8964References[1] Fritz J, BallerMK, LangHP, RothuizenH,Vettiger P,Meyer E, GüntherodtH J, Gerber C andGimzewski J K 2000Translatingbiomolecular recognition into nanomechanics Science 288 316–8[2] Berger R,Delamarche E, LangHP,Gerber C, Gimzewski J K,Meyer E andGüntherodtH J 1997 Surface stress in the self-assembly ofalkanethiols on gold Science 276 2021–4[3] Ruz J J,MalvarO,Gil-Santos E, RamosD,CallejaM andTamayo J 2021A review on theory andmodelling of nanomechanical sensorsfor biological applications Processes 9 164[4] Rasmussen PA, Thaysen J,HansenO, Eriksen SC andBoisenA 2003Optimised cantilever biosensor with piezoresistive read-outUltramicroscopy 97 371–6[5] Huber F, LangHP,Heller S, Bielicki J A,Gerber C,Meyer E and Egli A 2022Rapid bacteria detection frompatients’ blood bypassingclassical bacterial culturingBiosensors 12 994[6] Kooser A,Manygoats K, EastmanMP andPorter T L 2003 Investigation of the antigen antibody reaction between anti-bovine serumalbumin (a-BSA) and bovine serum albumin (BSA) using piezoresistivemicrocantilever based sensorsBiosens. 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Material of the immobilization layer 3.2. The shape of piezoresistor 3.3. Piezoresistor position 3.4. A simplified model 3.4.1. Effects of gap size 3.4.2. Effects of asymmetry 4. Conclusion Acknowledgments Data availability statement Declaration of competing interest References