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This paper describes an electrostatic excited microcantilever sensor operating in static mode that is more sensitive than traditional microcantilevers. The proposed sensor comprises a simple microcantilever with electrostatic excitation ability and an optical or piezoresistive detector. Initially the microcantilever is excited by electrostatic force to near pull-in voltage. The nonlinear behavior of the microcantilever in near pull-in voltage

Micro- and nano-sensors, especially microcantilever sensors, have attracted considerable interest for recognition of target analytes in biological and chemical and force sensing because of their fast, ease of use and inexpensive properties [^{12}) level, even in highly optimized conditions; whereas the canine nose can work down to the parts per quad (ppq) levels. Consequently, trained dogs currently are the “gold standard” method for analyte detection [

To increase the sensitivity of microcantilever sensors, and therefore, to overcome many of these challenges, a number of methods have been developed [

Several groups have published reports on the best microcantilever shape in order to achieve maximum sensitivity. Louia and coworkers designed, fabricated, and tested five piezoresistive cantilever configurations to investigate the effect of shape and piezoresistor placement on the sensitivity of microcantilevers [

Conventionally, microcantilever sensors are fabricated on a silicon substrate [_{2}-based microcantilevers are good candidates having a higher sensitivity because they are made of materials with a lower Young’s modulus (57–70 GPa) than that of Si (170 GPa). For example, Li _{2}, resulting in lower electric noise.

The current detection methods in microcantilever biosensors include piezoelectric or piezoresistive detectors for tension sensing and optical or capacitive detectors for displacement measurement. Displacement detectors usually have a higher sensitivity and can respond to very weak input signals. However, the limitation of working in liquid media, which is essential for biological sensors, is the main drawback of displacement detectors. To address this problem, metal-oxide semiconductor field-effect transistors (MOSFET) have been used by Shekhawat and coworkers to achieve a higher sensitivity in microcantilever biosensors [

A successful method that has been used for increasing the biological force has been implemented in the force amplified biological sensor under development at the Naval Research Laboratory [

Conventional microcantilever sensors work in a linear mode of operation, but recently the nonlinear operation of sensors especially in resonator-based microdevice [

In this paper a novel microcantilever with electrostatic excitation that is more sensitive than traditional rectangular microcantilevers is proposed. The basic idea comes from the nonlinear electrostatic force:
_{0} = 8.854 × 10^{−12} C.N/m is the permittivity of vacuum,

In the following section, the nonlinear Euler-Bernoulli beam equations for the proposed microcantilever sensor have been obtained. The proposed model has been solved by Green’s function method, and the verification of results for pull-in voltage and displacement under electrostatic force has been performed. In Section 3, the numerical analysis and comparison of the sensitivity of traditional microcantilever sensors and the proposed electrostatic excited microcantilever sensor has been discussed. In addition, the influence of geometrical factors including the initial gap, width, length and thickness on the sensitivity of the microcantilever sensor has been explored. We close the paper with concluding remarks in Section 4.

An electrostatic excited microcantilever sensor is composed of a microcantilever beam separated by a dielectric spacer from a fixed ground plane (

For performance analysis of the proposed sensor, two different applications of microcantilevers are dealt with here. The tip force applied to the microcantilever in

To study the nonlinear behavior of the electrostatic excited microcantilever sensor, a beam model is derived for the microcantilever of length _{e}

For convenience, the model is formulated in a nondimensional form, by introducing the nondimensional variables:

The following nondimensional equation is obtained:

According to the definition of the nondimensional variables, physically meaningful solutions exist in the region

The coefficients _{i}_{i}

As the deflection of a microcantilever beam with concentrated load of unit strength at point ξ is:

Now, the derived Green’s function is used to construct the solution to our nonuniformly distributed loading problem. Multiplying

This is the integral representation of the nonlinear differential

The closed-form solution of the deflection of the microcantilever tip (

Substituting

Evaluating the integrals on the right side of

By solving _{0}

Because the applied tip force and distributed moment have similar influences on microcantilever displacement, as seen in

To ascertain the validity of the proposed model,

Based on the concept development in this paper, the external load applied on the microcantilever sensor in the presence of nonlinear electrostatic excitation should be amplified. To confirm the proposed idea, the amplification factor,

The amplification factor demonstrates the ratio of the proposed electrostatic pre-excited microcantilever deflection to simple microcantilever sensor deflection due to tip force or distributed moment. In _{es}_{es}_{0}_{0}_{es}

In order to increase the amplification factor, the applied voltage should be closer to the pull-in voltage.

This section has been devoted to studying the effect on the amplification factor of the proposed electrostatic excited microcantilever of the variation of four geometric parameter variations which include width, thickness, length and initial gap. First of all, for investigating the influence of initial gap on the amplification factor, numerical simulation has been done based on the data obtained from the reference microcantilever with an the initial gap that changes from 2 μm to 20 μm. The value of excitation voltage in simulation changes corresponding to the initial gap. ^{2}/g^{3}

In order to study the effect of microcantilever sensor thickness on amplification factor, simulations have been performed on the reference microcantilever with the thickness varying from 2 μm to 20 μm based on data of

As

We have presented a novel sensitive microcantilever force sensor with electrostatic excitation in a static mode operation. In order to study the performance of the proposed sensor, the governing equation of the microcantilever sensor subjected to the electrostatic forces is derived as a two-point boundary value problem (BVP). The equation is nonlinear due to the inherent nonlinearity of the electrostatic excitation. The nonlinear differential equation is transformed into a nonlinear integral equation using the Green’s function of the microcantilever. Assuming an appropriate shape function for the microcantilever deflection to evaluate the integrals, closed-form solutions are obtained. Then, the displacement of microcantilever tip and pull-in parameters were computed and compared with experimental and numerical methods. The results prove the validity of the modeling approach for the proposed microcantilever sensor. Using the developed theoretical model, we showed that the proposed microcantilever sensor compared with a traditional microcantilever sensor of the same dimensions can be 2 to 100 times more sensitive in the cases of force sensor or surface stress sensor.

Finally, the effects of width, length, thickness, and the initial gap of the microcantilever sensor on the sensor amplification factor have been studied. Increasing the initial gap, the thicknesses and the width increases the amplification factor. On the other hand, smaller microcantilever lengths generate bigger amplification factors.

_{2}-based piezoresistive MCLs

_{2}cantilever sensor for ultrasensitive detection of gaseous Chemicals

Schematic representation of an electrostatic excited microcantilever sensor.

Amplification factor

The relationship between amplification factor and the geometric parameters of proposed microcantilever sensor. The applied force is equal to 1 nN and the reference microcantilever data has been used for simulation. The excitation voltage is 1 mV below the pull-in voltage.

Comparison between analytical and experimental and the present work for microcantilevers deformation under electrostatic force.

20 | 90.2 | 90.5 | 90.2 | 0.3 |

40 | 84.3 | 84.6 | 84.3 | 0.3 |

60 | 71.5 | 70 | 70.8 | 0.8 |

65 | 67.5 | 64 | 64.3 | 0.3 |

67 | 65 | 59 | 60.4 | 1.5 |

Values of the parameters of the reference microcantilever sensor.

3.4 GPa | |

500 μm | |

100 μm | |

10 μm | |

10 μm | |

8.85 pF/m |

Amplification factor for various applied voltage.

0.1 nN | 1.0498 | 1.3759 | 1.645 | 1.947 | 2.2368 | 2.8352 | 5.9678 |

1 nN | 1.0498 | 1.3758 | 1.6447 | 1.9465 | 2.2358 | 2.8327 | 5.9355 |

10 nN | 1.0497 | 1.3749 | 1.6422 | 1.9411 | 2.2262 | 2.8087 | 5.6496 |

60 nN | 1.0494 | 1.3698 | 1.6289 | 1.9126 | 2.1764 | 2.6915 | 4.6951 |

100 nN | 1.0491 | 1.3658 | 1.6187 | 1.8916 | 2.1407 | 2.6132 | 4.264 |

Influence of geometric parameters on pull-in voltage and deformation of proposed microcantilever sensor with electrostatic excitation (_{0}_{st}

Initial gap (μm) | 2 | 3 | 4 | 5 | 7 | 10 | 15 | 20 |

_{st} × g |
1.1364 | 1.1364 | 1.1364 | 1.1364 | 1.1364 | 1.1364 | 1.1364 | 1.1364 |

_{0} × g |
1.076 | 1.6061 | 2.1357 | 2.665 | 3.723 | 5.3093 | 7.9519 | 10.5938 |

Pull-in Voltage (V) | 4.3999 | 8.0847 | 12.4485 | 17.3983 | 28.8224 | 49.2159 | 90.4189 | 139.2118 |

| ||||||||

Thickness (μm) | 2 | 4 | 6 | 8 | 10 | 14 | 16 | 20 |

_{st} × g |
142.045 | 17.756 | 5.2609 | 2.2195 | 1.1364 | 0.4141 | 0.1948 | 0.1420 |

_{0} × g |
5.3309 | 5.3318 | 5.3217 | 5.3144 | 5.3093 | 5.3027 | 5.2986 | 5.2971 |

Pull-in Voltage (V) | 4.3358 | 12.4288 | 22.8635 | 35.2120 | 49.2159 | 81.5324 | 118.8658 | 139.2181 |

| ||||||||

Length (μm) | 300 | 400 | 500 | 600 | 700 | 800 | 900 | 1000 |

_{st} × g |
0.2455 | 0.5818 | 1.1364 | 1.9636 | 3.1182 | 4.6546 | 6.6273 | 9.0909 |

_{0} × g |
5.2972 | 5.3033 | 5.3093 | 5.3151 | 5.3207 | 5.3262 | 5.3314 | 5.3364 |

Pull-in Voltage (V) | 136.7236 | 76.9043 | 49.2159 | 34.1747 | 25.1049 | 19.2178 | 15.1812 | 12.2936 |

| ||||||||

Width (μm) | 20 | 40 | 60 | 80 | 100 | 120 | 150 | 200 |

_{st} × g |
5.6818 | 2.8409 | 1.8939 | 1.4205 | 1.1364 | 0.947 | 0.7576 | 0.5682 |

_{0} × g |
5.3069 | 5.3084 | 5.3089 | 5.3091 | 5.3093 | 5.3094 | 5.3095 | 5.3096 |

Pull-in Voltage (V) | 49.1921 | 49.207 | 49.2119 | 49.2144 | 49.2159 | 49.2168 | 49.2178 | 49.2188 |