^{*}

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Microfabrication limitations are of concern especially for suspended Micro-Electro-Mechanical-Systems (MEMS) microstructures such as cantilevers. The static and dynamic qualities of such microscale devices are directly related to the invariant and variant properties of the microsystem. Among the invariant properties, microfabrication limitations can be quantified only after the fabrication of the device through testing. However, MEMS are batch fabricated in large numbers where individual testing is neither possible nor cost effective. Hence, a suitable test algorithm needs to be developed where the test results obtained for a few devices can be applied to the whole fabrication batch, and also to the foundry process in general. In this regard, this paper proposes a method to test MEMS cantilevers under variant electro-thermal influences in order to quantify the effective boundary support condition obtained for a foundry process. A non-contact optical sensing approach is employed for the dynamic testing. The Rayleigh-Ritz energy method using boundary characteristic orthogonal polynomials is employed for the modeling and theoretical analysis.

Boundary supported suspended microstructures, such as MEMS cantilevers, are currently used in various microengineering sensor/actuator fields. Their relatively simple geometries make them very advantageous both from a design and microfabrication point of view. The wide range of applications include, for example, medical [

Microfabrication methods and limitations can lead to boundary support conditions for suspended microstructures that are not rigidly clamped [

The term boundary conditioning refers to the integrated influences of material property, device geometry, boundary support, and operating conditions on the elastic characteristics of a suspended microstructure. This paper presents an experimental approach to quantify the support boundary condition of AFM cantilevers through electro-thermo-mechanical testing. To apply the proposed experimental method, the boundary support condition for AFM microcantilevers provided by MikroMasch [

A scanning electron microscope (SEM) image of three AFM cantilevers and a close up of the non-classical boundary support are illustrated in _{T}_{R}

For the case of an electro-thermally actuated AFM probe, as shown in _{E}(x)

The thermal effect is modeled through changes in the cantilever geometry and material property [

The dimensions, (^{9} Pa and 2330 kgm^{-3} for all of the probes tested. The dimensions presented are average values for a given cantilever and were measured using SEM and optical microscope images.

Given in

The non-linear experimental and non-classical boundary support resonant frequency variations, at different temperatures, as a function of the applied DC voltage for the

From the results obtained it can be clearly seen that there is a non-classical boundary support condition present due to the limitations of the microfabrication process. In this regard, for these two test methods on the _{R}^{*}_{R}^{*}_{E}

The other AFM cantilevers,

From the rotational stiffness values calculated in _{R}^{E}

The non-dimensionalized rotational stiffness _{R}^{*}_{R}^{E}

The results obtained for the individual AFM cantilever boundary supports clearly show the influence of microfabrication influences, as can be seen in the _{R}^{*}_{R}^{*}_{R}^{E}_{R}^{E}^{-9} N·m resulting in standard deviation of 1.55 × 10^{-9} N·m for the experimental test results obtained, and in this regard, demonstrate the foundry influence in the microfabrication process. Hence, with this experimental approach, which is based on measuring the mechanical characteristics of a few AFM cantilevers under variant applied conditions, it is possible to extract the mechanical characteristics of similar devices manufactured using the same foundry microfabrication process. Furthermore, the experimental results obtained demonstrate the non-classical boundary support nature of AFM cantilevers due to microfabrication processes and limitations.

A comparison with 5 ideal nominal cantilevers is presented below. In this regard, the natural frequency results given in _{R}^{*}_{R}^{E}^{-9} N·m, and the respective natural frequencies are given in

For the cantilevers with the nominal _{R}^{*}_{R}^{*}

It can be seen that the measured data fits very well with the fitted curve, as shown in ^{-9} N·m for the effective rotational stiffness _{R}^{E}

A quantitative experimental approach for the non-classical boundary characterization of AFM cantilevers by electro-thermal-mechanical testing has been presented. This approach allows for the extraction of mechanical properties of many different microcantilevers based on the mechanical characteristics of only a few cantilevers subjected to variant applied influences, and where all the devices tested were manufactured using the same microfabrication foundry process. This approach may be applied to other suspended MEMS structures and other microfabrication foundry processes. Support boundaries for suspended microstructures in general, are non-classical in nature and need to be evaluated experimentally. In this regard, the boundary support condition can be extracted from the effective rotational stiffness _{R}^{E}

A non-contact laser based optical method [

This approach brings significant versatility to the test environment in the sense that it allows one to readily incorporate electrostatic and thermal influences onto the microstructure platform which would otherwise be difficult in a confined environment. The base excitation for the AFM cantilevers was provided by a small amplitude sinusoidal AC voltage or mechanical shaking using acoustic energy [

The qualitative boundary support conditioning approach presented in this work consisted of testing one AFM cantilever (_{R}^{E}_{R}^{E}

In order to compare the experimental results with theoretical values, the Rayleigh-Ritz energy method is employed to model the dynamic electro-thermal system [_{T}^{*}_{R}^{*}_{T}^{*}^{10}) so that at the very least _{R}^{*}_{R}^{E}

The high response of the fundamental frequency at resonance to a forced excitation makes it very suitable for analyzing the dynamic property of a vibrating system. In this regard, this method may be employed to estimate the natural frequencies of flexible structures such as AFM cantilevers. In this method the mechanical property of the system is a function of its potential and kinetic energies, and where the static (_{S}_{F}_{i}_{i}

The Rayleigh quotient is defined as,
_{M}_{M}^{(T)}_{AP}^{(T)}_{BSP}^{(T)}_{E}

The AFM probe portion of the maximum potential energy is given by,
^{(T)}^{(T)}^{(T)}^{(T)}^{(T)}_{F}″(x)_{F}(x)

The influence of the boundary support of the cantilever on the potential energy is given by,
_{F}(0)_{F}′(0)_{R}_{T}_{R}^{*}_{T}^{*}

The electrostatic potential energy, _{E}^{(T)}_{E}_{S}^{(T)}_{E}(x)^{(T)}^{(T)}_{E}

The eigensystem defining the flexural deflection, _{F}

The eigenvalues and mode shapes given by _{R}^{*}

a) AFM chip with 3 microcantilevers. b) CAD drawing of a microcantilever boundary support. c) Close up image of the non-classical boundary support

The non-classical boundary support modeled by artificial translational, _{T}_{R}

Environmental influences on an AFM cantilever modeled with artificial springs.

Resonant frequency map obtained for the

The resonant frequencies, at different temperatures, of the

a) The normalized variation of length, Young's modulus of elasticity and moment of inertia for an AFM cantilever as a function of temperature. b) The change in the _{R}^{*}

Calculated (from experiment and measured values) and fitted curve changes in the rotational stiffness _{R}^{*}

Top left: HeNe laser and lenses mounted on an optical bench. Top right: Digital image taken through a microscope of the laser spot on the AFM chip. Bottom: Sample responses obtained with the experimental method used in this work.

Top: Microscope image of an AFM chip and three AFM cantilevers. Bottom: Schematic side view of a cantilever in an applied electro-thermal environment as carried out in this work.

The variation of the first and second normalized eigenvalues of a boundary supported structure as a function of the rotational stiffness _{R}^{*}

The geometry of the 12 cantilevers tested in this work. The geometrical dimensions are in micrometers.

351 | 299 | 254 | 251 | 251 | 300 | 302 | 353 | 250 | 302 | 304 | 355 | |

35 | 35 | 36 | 35 | 35 | 36 | 35 | 36 | 36 | 35 | 36 | 36 | |

0.95 | 0.96 | 0.93 | 0.92 | 1.00 | 1.05 | 1.10 | 0.94 | 0.90 | 0.94 | 1.93 | 1.89 |

Electro-thermal dependence of the resonant frequency for the

10440 | 10440 | 10413 | 10424 | 8598 | 8608 | 5583 | 5615 | |

10437 | 10434 | 10409 | 10420 | 8592 | 8601 | 5565 | 5605 | |

10432 | 10430 | 10404 | 10415 | 8584 | 8593 | 5545 | 5581 | |

10427 | 10424 | 10399 | 10410 | 8577 | 8585 | 5524 | 5558 | |

10422 | 10421 | 10394 | 10406 | 8570 | 8577 | 5504 | 5534 | |

10418 | 10418 | 10390 | 10400 | 8562 | 8569 | 5483 | 5511 | |

10413 | 10414 | 10385 | 10396 | 8555 | 8561 | 5463 | 5487 | |

10409 | 10410 | 10381 | 10393 | 8550 | 8555 | 5447 | 5469 | |

10401 | 10401 | 10373 | 10384 | 8537 | 8541 | 5410 | 5426 | |

10380 | 10382 | 10351 | 10363 | 8504 | 8504 | 5313 | 5313 | |

10370 | 10373 | 10342 | 10354 | 8489 | 8488 | 5269 | 5262 | |

10368 | 10370 | 10340 | 10351 | 8485 | 8484 | 5257 | 5249 |

The experimental (_{R}^{*}

10.4 | 14.5 | 19.4 | 19.6 | 21.2 | 15.6 | 16.0 | 10.2 | 19.4 | 13.9 | 24.5 | 18.1 | |

_{R}^{*} |
108 | 90 | 83 | 84 | 65.5 | 68 | 60 | 109 | 87 | 96 | 10.85 | 13.5 |

A comparison of the rotational stiffness _{R}^{*}_{R}^{E} values for the 12 AFM cantilevers tested. # Indicates cantilevers with tip.

_{R}^{*} |
_{R}^{E}^{-9}) | |
---|---|---|

108 | 130.42 | |

90 | 131.66 | |

^{#} |
83 | 133.65 |

^{#} |
84 | 128.83 |

^{#} |
65.5 | 129.01 |

^{#} |
68 | 133.43 |

^{#} |
60 | 130.73 |

^{#} |
109 | 130.41 |

^{#} |
87 | 129.00 |

^{#} |
96 | 130.93 |

10.85 | 130.47 | |

13.5 | 130.55 |

Nominal cantilevers with ideal geometries. The associated tested cantilevers _{R}^{*}_{R}^{E}

_{R}^{*} |
_{R}^{E}^{-9}) |
|||||
---|---|---|---|---|---|---|

300 | 35 | 1 | 79.35 | 130.76 | 14940 | |

250 | 35 | 1 | 66.13 | 130.76 | 21410 | |

300 | 35 | 2 | 9.92 | 130.76 | 25820 | |

350 | 35 | 2 | 11.57 | 130.76 | 19375 | |

350 | 35 | 1 | 92.58 | 130.76 | 11014 |