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This study presents the deflection, resonant frequency and stress results of rectangular, triangular, and step profile microcantilevers subject to surface stress. These cantilevers can be used as the sensing element in microcantilever biosensors. To increase the overall sensitivity of microcantilever biosensors, both the deflection and the resonant frequency of the cantilever should be increased. The effect of the cantilever profile change and the cantilever cross-section shape change is first investigated separately and then together. A finite element code ANSYS Multiphysics is used and solid finite elements cantilever models are solved. A surface stress of 0.05 N/m was applied to the top surface of the cantilevers. The cantilevers are made of silicon with elastic modulus 130 GPa and Poisson’s ratio 0.28. To show the conformity of this study, the numerical results are compared against their analytical ones. Results show that triangular and step cantilevers have better deflection and frequency characteristics than rectangular ones.

The ability of label-free detection, scalability to allow massive parallelization, and sensitivity of the detection range applicable to

The overall sensitivity of a microcantilever biosensor depends on the design sensitivity of the cantilever and the measurement sensitivity of the deflection measurement system. A sensitive cantilever design should efficiently convert the biomolecular stimulus into a large cantilever deflection, whereas the measurement sensitivity should ensure that the deflections measured are only induced because of the biomolecular stimulus and not due to some ambient disturbance source. The design sensitivity of the cantilever can be improved by changing the cantilever design in such a way that for a given surface stress larger deflections can occur. This scheme can be realized by reducing the bending stiffness of the cantilever [

To improve the design sensitivity of cantilevers various designs and schemes have been reported. Silicon microcantilevers are commonly used in biosensors. However, due to high elastic modulus silicon cantilevers exhibit extremely low deflections for a given surface stress change. Therefore, to increase the deflections polymer cantilevers can be used. Since the elastic modulus of polymers is generally much lower than silicon, the deflections induced are magnified many folds. Polymer cantilevers, however, have a major drawback in being very temperature sensitive, because of the thermal bimetallic effects. Thermal induced deflections exceeding the surface-stress induced deflections are not uncommon. Hence, polymer cantilevers require a fine control of the surrounding. The other way to improve design sensitivity is to increase the cantilever deflection by changing the shape of the cantilever. By reducing the moment of inertia of a cantilever its bending stiffness can be reduced, which results in higher deflection.

With the objective of increasing the deflection and resonant frequency at the same time, this paper investigates the deflection and vibration analysis of rectangular, triangular, and step profile microcantilevers having basic and modified shapes. The surface-stress induced deflection in the microcantilever is modelled by an equivalent in-plane tensile force acting on the top surface of the cantilever, in the length direction. A commercial finite element method (FEM) software ANSYS Multiphysics is used in this analysis. All the cantilevers are investigated for deflection, fundamental resonant frequency and stress.

The surface stresses, in general, are generated either by the redistribution of the electronic charge at the surface, due to the change in the equilibrium positions of the atoms near the surface, or by the adsorbtion of foreign atoms onto its surface to saturate the dangling bonds [

For a rectangular profile microcantilever (_{0} and _{0} are the length and thickness of the cantilever, and _{0}) for a rectangular profile cantilever of mass density (

As can be seen from _{0}) term as:

It should be, however, noticed in _{0} values is a better way to compare the performance of a particular microcantilever design, because depending on this value appropriate cantilever dimensions and characteristics can be selected. To select the best cantilever model, we should choose one that has higher Δ_{0} value, more inclined towards increased deflection. Therefore, in this study we calculated and compared the sensitivity values of all the cantilever models. For achieving higher deflection, we should choose longer cantilevers (

Since it is not possible to achieve zero tip thickness for a cantilever, the triangular profile can be approximated as a trapezoidal profile, of the form _{l} + (_{0} – _{l})

The Stoney Equation for a triangular profile cantilever can be given as [_{0} and _{l} are the thicknesses of the cantilever at the fixed and free ends. Hoffman and Wertheimer [

By using Euler beam theory and principle of superposition for nonprismatic beams [_{0} and

The surface-stress induced deflection in a microcantilever can be modelled by applying a lengthwise in-plane tensile force at the free end of the top surface of the cantilever [^{−6} m = 5 × 10^{−6} N was applied to the top free edge of all the six models.

_{0}/2 and _{0}. In triangular cantilevers, the free end thickness is one-tenth the fixed end. Simulations used FEM software ANSYS Multiphysics to calculate the deflection, fundamental resonant frequency and stress induced in all the six designs. The simulations were performed on three-dimensional FE models of the cantilevers, under linear and static conditions. In simulations, we used micrometre as unit of length and newton of force (

_{0}), overall sensitivity (Δ_{0}), and maximum stress induced (_{max}) for all the six models shown in

The deflections are observed to increase in all the cases whether the profile is changed, cross-section shape is changed, or both are changed (

The major difference between #R1, #T1, and #S1 designs is that in #R1 the bending stiffness is constant along the cantilever length, whereas in both #T1 and #S1 it is not. When we changed the cantilever profile, we basically changed the thickness of the cantilever towards the free end (i.e., #T1 and #S1), which reduced the bending stiffness. It should be noticed that in #T1 the thickness is reduced continuously along the cantilever length, and therefore, the bending stiffness is reduced continuously along the length. In case of step profile (#S1), since both the thick and thin sections have constant thickness, the stiffness is constant in both sections. Since the thickness of thin section is half the thick section, its stiffness is lower and therefore larger deflection will occur in thin section. Among the three basic profiles, since #T1 has the least thickness at free end, it shows the maximum tip deflection, see

As discussed before, the dynamic properties of microcantilevers used in biosensors are critical in accurate measurement of deflections. In practical applications, there can be thermal, structural, or flow induced excitations that can interfere with and hence produce noise in the signals. Therefore, it is vital to eliminate or isolate the noise in the signal, and to insure that the deflections induced are solely due to the surface stress change. To prevent noise, a cantilever should have high signal-to-noise ratio, which can be achieved by making the resonant frequency of the cantilever as high as possible. The higher the resonant frequency, the higher the measurement sensitivity will be. In

The improved resonant frequencies due to profile change observed in _{0} value.

In _{0}) values of the basic cantilever designs are improved by 368% for #T1 and 113% for #S1 from conventional design #R1. However, the reduction in cross-sectional area has mixed effect on the sensitivity values. For instance, the area change improved the sensitivity by 17% for #R2, but reduced by 15% for #T2. In case of #S2 the area change has almost no effect on the overall sensitivity. Since in case of #T2 the sensitivity is reduced by 15%, mainly because of the reduction in frequency, we can conclude that #T2 has no significant advantage over #T1. Therefore, the basic triangular design should be preferred. If we compare the rectangular and step designs, we observe that #S2 has big advantage over #R1 and #R2 in terms of both greater deflection and higher sensitivity. And therefore, #S2 design should be preferred.

^{6} MPa). A comparison between the stress values of #R1, #T1, and #S1 shows that profile change alone increased the stresses from 0.41 to 3.85 and 0.75 MPa, respectively (

Thus, we see that both #T1 and #S2 show both greater deflection and higher sensitivity than the rest designs, and therefore should be the preferred microcantilever designs for improving the biosensor performance. However, in terms of practical applications, #S2 has big advantage over #T1, because the latter is very difficult to fabricate. For instance, microfabrication of a triangular profile cantilever of taper 1 μm to 0.1 μm is extremely difficult. In addition, ensuring the structural and functional reliability of such a thin cantilever is very challenging. On the other hand, the dimensions of #S2 analyzed in this study can be easily fabricated in silicon using deep reactive ion etching (DRIE). Therefore, we may conclude that though the performance of #T1 seems much improved than #S2, for practical application #S2 is better.

Arrays of microcantilevers are increasingly being used as physical, biological, and chemical sensors in various applications. In this work, we investigated improving the overall sensitivity of the microcantilevers that can be used in biosensors by increasing their deflection and frequency characteristics of the cantilevers. To improve the sensitivity we studied basic and modified design rectangular, triangular, and step profile microcantilevers. The overall sensitivity of microcantilever depends on both the deflection and the resonant frequency of the cantilever. The simulation results obtained in this study correspond well to their analytical models, validating the conformity of the analyses. The surface stress was successfully modelled by an in-plane tensile force applied to the top surface of the cantilevers. We found that by changing the profile from rectangular to triangular and step, the cantilever deflections are improved by 257% and 79%, and frequencies by 31% and 19%, respectively. Further, for each cantilever type, the cross-section shape change by reducing fixed-end area increased the deflection by 89%, 11%, and 50%, but reduced the frequencies 38%, 24%, and 37%, respectively. The overall sensitivity values of all the cantilevers are improved, however. Though the triangular profile cantilevers showed better deflection and frequency characteristics, fabrication and structural integrity constraints suggest that a step cantilever (#S2) is more practical as the sensing element of the biosensor. We also found that compared to the excellent mechanical properties of silicon, the maximum stress induced in the designs are negligible.

This study was supported by Inha University.

Schematic designs of (a) rectangular, (b) triangular, and (c) step profile cantilevers.

Comparison between basic and modified rectangular (R), triangular (T), and step (S) profile cantilevers. All the models have equal length, width, and fixed-end thickness (unit: micrometre).

Results showing deflection, frequency, stress, and sensitivity values for the basic and modified design rectangular, triangular, and step profile cantilevers.

Stress distributions in the rectangular (#R1, #R2), triangular (#T1, #T2), and step (#S1, #S2) profile microcantilevers.

Comparison between analytical and simulation results for basic rectangular, triangular, and step cantilevers.

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#R1 | 0.28 | 0.28 | 4.79 | 4.91 |

#T1 | 0.96 | 1.00 | 7.30 | 6.44 |

#S1 | 0.53 | 0.50 | 5.53 | 5.84 |

Comparison between normalized values for maximum deflection, fundamental resonant frequency, sensitivity and maximum induced stress.

_{0} |
_{0} |
_{max} | ||
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#R1 | 1 | 1 | 1 | 1 |

#R2 | 1.89 | 0.62 | 1.17 | 2.76 |

#T1 | 3.57 | 1.31 | 4.68 | 9.39 |

#T2 | 3.96 | 0.99 | 3.92 | 8.65 |

#S1 | 1.79 | 1.19 | 2.13 | 1.83 |

#S2 | 2.68 | 0.79 | 2.12 | 4.83 |