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The dynamic characterization of a set of gold micro beams by electrostatic excitation in presence of residual stress gradient has been studied experimentally. A method to determine the micro-cantilever residual stress gradient by measuring the deflection and curvature and then identifying the residual stress model by means of frequency shift behaviour is presented. A comparison with different numerical FEM models and experimental results has been carried out, introducing in the model the residual stress of the structures, responsible for an initial upward curvature. Dynamic spectrum data are measured via optical interferometry and experimental frequency shift curves are obtained by increasing the dc voltage applied to the specimens. A good correspondence is pointed out between measures and numerical models so that the residual stress effect can be evaluated for different configurations.

Micro-electromechanical systems (RF-MEMS) have recently demonstrated advances in the field of radio-frequency in the realization of tuneable circuits including phase shifters, filters and matching networks. Although great improvements have been made in the reliability of these devices, significant gaps remain in the understanding of fundamental mechanical properties such as stress/strain relationships and the origin of residual stress within the thin-film metals used for the mechanical structure [

There has been a lot of research into controlling residual stress by controlling fabrication process parameters, for example by adjusting the bath compositions or selecting the seed layer material used for the growing of the suspended parts [

Residual stress causes the change of equilibrium configuration and variation of important system parameters such as resonant frequencies and pull-in tension. In recent literature residual stress in microbeams has been studied, being considered as unavoidable in surface micromachining techniques: in [

In coupled electromechanical systems the natural frequency values decrease as the voltage increases. This characteristic makes it possible, for example, to tune an operating frequency with an applied bias voltage [

In the present work a set of experimental tests was conducted to evaluate the frequency shift of curled cantilever gold beams. Typical RF-MEMS devices consist of either clamped beams or cantilevers. Due to its low electrical loss and chemical inertness, gold is the most common material for fabricating these structures. Despite these benefits, the high susceptibility to relaxation effects of gold often introduce much residual stress into these RF-MEMS structures [

The deflection profile of microcantilevers was obtained through a non-contact interferometric profilometry system using the fringe pattern generated by interference. Once the deflection profile is measured, the slope and curvature can be easily calculated. The models presented in this paper can thus uniquely determine the resultant bending moment due to residual stress gradients, furthermore it is possible to identify the dynamic behaviour of the specimen from zero dc voltage to pull-in.

The specimens used for this work were prepared at the ITC-IRST research center (Trento, Italy), using the

At first a 1000 nm thick thermal field oxide is grown at 975°C in a wet ambient on a silicon wafer, then a nitrogen annealing at the same temperature is performed. A 630 nm thick polysilicon layer employed for resistors and for actuation lines is deposited by LPCVD and subsequently patterned through dry etching.

A 300 nm thick silicon oxide is deposited at 718°C, also by LPCVD process and via-holes are opened through dry etching.

A multi-layer metal for signal lines is sputtered and subsequently patterned by dry etching. Temperature profile: Ti(30nm): 400°C; TiN(50 nm): 400°C; Al/Si+Ti(410/60 nm): ambient temperature; TiN(80 nm): 300°C. A 100 nm thick oxide layer is then deposited at 430°C. In order to define via-holes for opening contacts or to uncover the multimetal line oxide removal is defined with a mask by dry etching.

A 150 nm gold layer is deposited by PVD and patterned through wet etching. A 3 μm thick sacrificial photoresist layer is deposited and patterned.

A 1.3 μm thick gold layer is electroplated at 52°C employing a chromium-gold PVD adhesion layer, called seed layer. This is the suspended/movable part of the devices.

The last deposition step is another gold layer deposited at 52°C used to reinforce the anchors and the suspended parts of the structures. Finally, the structure release is obtained by ashing the sacrificial layer through plasma oxygen etching. At the end of the process a sintering is performed at 190°C.

The seed layer is employed to improve adhesion of gold with substrate; as an undesired consequence, poor thermal stability on gold film was verified because of the natural tendency of Cr to diffuse to the film surface at higher temperatures [_{2}O_{3}. This diffusion process can be quite extensive, with complete depletion of chromium adhesive layer and formation of channelled grain boundaries that are occupied with Cr_{2}O_{3} and eventually formation of single crystals of Cr_{2}O_{3} at surface. The chromium transport may manifest itself in development of undesirable characteristics, such as decrease in electrical conductivity and generation of internal stress. Residual stresses vary along the beam thickness because of the difference on percentage of diffused chromium.

Another feature considered as an origin of internal stresses within the gold beam is the difference in the coefficient of thermal expansion (CTE) between the gold beam and the photoresist sacrificial layer [

A stress characterization was made by Margesin [

The structures tested in this work were produced on two different wafers, in the first one (set 1, 2, 3) beams of increasing length are grouped in sets. In the second wafer (set 4, 5, 6) each set is composed by identical beams but thickness varies from one set to the other. Experimental images of the specimen were analyzed on the basis of the IRST fabrication process technical memo in order to extrapolate a simplified profile, being the real profile quite irregular (

Frequency shift measures were preceded by accurate profilometric measures in order to know with high precision the dimensions of the specimen. Both kinds of measures were performed with the

Experimental images were treated with the

In

Deflection towards the top was observed in almost all the specimens (

The vibration of the microbeams is obtained by the electromechanical action induced by an electric field applied between the suspended structure and the ground. The profiling system is equipped with a power supplier capable of provide up to 200 Volt; adjustable needles assure connection between power supplier and circuit. These are mounted on the

Tests were performed by applying a positive voltage _{G}

_{dc}_{ac}_{ac}_{dc}

With the

In order to evaluate the effect of residual stresses an analytical formulation was considered: the stress function _{0}_{z}_{0}_{z} represents the influence of the gradient stress anti-symmetric functions [

Stress gradient

When a micro-machined cantilever is fabricated by removing the supporting substrate of the film, traction at the film-substrate interface is removed, and the structure becomes free to deform out-of-plane following the relief of the internal stress: _{b}

On the assumption that the bending angle of cantilever is small, the linear differential equation that relates the bending moment to the vertical deflection ^{3}/12, c_{b}

The first assumption in _{1}= A_{1}(σ_{0},σ_{z}) [

The deflection of the end of the cantilever _{L}

Substituting

To obtain an analytical model of the cantilever bending, _{0}

Two different 2-D models were implemented with ANSYS™, assuming ideally clamped conditions in the microbeam. The constraint was applied at the beginning of

Particular attention to the fringing field effect of MEMS beam has been paid by authors in previous works, in particular extensively modelling and experimental identification of non-linearity of the electrostatic coupling has been done in the case of static behaviour of MEMS cantilevers [

The numerical modelling and the results included in the present paper refer only to the structural domain, in this case the dimension of the beam and the experimentally measured Q-factor allow disregarding the effect of dynamic fluid interaction around the structure. Specific investigation of fluid-structure coupling in MEMS using FEM multi-physics models and experimental measurements was carried out by authors in other works [_{0} obtained with the following formula:

For frequency behaviour simulation, the routine is first to proceed with a static analysis with the pre-stress option turned on and then perform a modal analysis. The included pre-stress is responsible for the effects of the applied voltage on the system frequency characteristic and, in the first model, for the residual stress inclusion. The program outputs are mechanical displacements and eigenfrequencies with incorporated electrostatic effects.

A first model was implemented by applying on the free tip of the cantilever a bending moment derived from

^{3} was used [

Stress gradients for each single specimen were calculated from

In

From

The Senturia-Osterberg formulation [

_{1}, γ_{2}, γ_{3} are fitting parameters depending on the constraints. In the case of cantilever γ_{1}=0.07, γ_{2}=1,γ_{3}=0.42.

Gupta's work [_{c}_{pi}_{pi,c}

Where _{pi}_{pi,c}_{c}

The ANSYS™ beam models used for frequency shift calculation were modified to implement an algorithm for the accurate calculation of the pull-in voltage value; the

The obtained pull-in values were used to complete the numerical frequency shift curves and were compared with analytical results from

In this study experimental frequency shift curves were obtained through optical interferometric measures on vibrating microcantilevers, the lowering of natural frequency with the increase of electrostatic force was detected.

The experimental results were compared with FEM solutions from reduced order models using a specific capacitive element of ANSYS™ to model the electrostatic field. The residual stress effect on the structures was included in the models being necessary to determine the initial upward curvature and the correct pull-in tension value. The presence of stress gradient strongly influenced the frequency shift curve and the pull-in. The structures tested in this work were produced on two different wafers, in the first one beams of increasing length are grouped in sets. In the second wafer each set is composed by identical beams but thickness varies from one set to the other. The adherence with experimental measures varied from specimen to specimen, the biggest discrepancies appeared near pull-in, due to the high variability of this parameter in measures.

In the studied cases of cantilever beams the stress gradient is taken into account in the FEM model simply by reproducing the measured initial curvature and thus obtaining frequency shift curves in good agreement with the measured curves. The stress gradient revealed by the curvature measurement slightly change in the same wafer due to the cantilever length. In the second wafer the stress gradient revealed by the curvature tends to zero by increasing the cantilever thickness.

In all these different cases the good agreement of the numerical versus experimental frequency shift graph shows how the stress gradient model works successfully. Among the causes of the stress gradient the experimental results of the present work may confirm a combined effect due to both chromium diffusion and the difference in the coefficient of thermal expansion (CTE) between the gold beam and the layer. The latter introduces a stress gradient for any temperature variation during the process that decreases with the thickness increase as confirmed in the frequency shift graph. The method presented in this paper can be useful both for identify the process residual stress gradient and to numerically evaluate the optimal thickness for the process.

This work was partially funded by the Italian Ministry of University, under grant PRIN-2005/2005091729. Specimens were built by ITC-IRST research center (Trento, Italy). Authors thank all above involved institutions.

curvature

elastic modulus

exitation frequency

_{c}

corrective coefficient for pull-in tension

_{n}

natural frequency

air-gap

_{0}

initial gap

_{m}

medium gap

inertia moment of beam section

effective length

_{c}

total length

mass

_{b}

bending moment

_{c}

curvature radius

surface

thickness

_{ac}

ac voltage

_{dc}

dc voltage

width

vertical beam deflection

vertical coordinate along the thickness

_{0}

dielectric constant

_{0}

planar constant stress

_{z}

coefficient of linear stress variation

stress gradient

The RF Switch (RFS) Surface Micromachining process.

Schematic cantilever beam profile with geometrical dimensional parameters.

Experimental set-up on the Fogale Zoomsurf 3D.

Specimen set: 3D image obtained with the Zoomsurf Fogale 3D profiler.

Experimental frequency spectrum of a microbeam with V_{ac}=4.5V and three different values of DC voltage: 0V: f_{n}=41640 Hz, Q= 43.4; 20V: f_{n}=41230 Hz, Q=38.2; 40V: f_{n}=40110, Q=35.9. Q factor was calculated with the half power method.

Sketch of the Ansys model.

Upper profile and frequency shift variation in cantilever beams of set 3.

Upper profile and frequency shift variation of cantilever beams of set 4 (nominal thickness 1.8 um), set 5 (nominal thickness 3 um) and set 6 (nominal thickness 4.8 um).

Experimental measures and stress gradient in first wafer.

g (μm) | t (μm) | L (μm) | u (μm) | c (μm^{-1}) |
Ω (MPa/μm) | |
---|---|---|---|---|---|---|

specimen1 | ||||||

set1 | 3.0 | 1.78 | 242.4 | 3.81 | 1.30E-04 | 12.8 |

set2 | 3.0 | 1.68 | 245.0 | 5.30 | 1.77E-04 | 17.4 |

set3 | 3.0 | 1.75 | 240.0 | 4.67 | 1.62E-04 | 16.0 |

| ||||||

mean value | 3.0 | 1.73±0.05 | 242.5±2.5 | 4.59±0.74 | 1.56E-04 | 15.4±2.3 |

| ||||||

specimen2 | ||||||

set1 | 3.0 | 1.84 | 288.5 | 13.43 | 3.23E-04 | 31.8 |

set2 | 3.0 | 1.71 | 289.4 | 13.61 | 3.25E-04 | 32.0 |

set3 | 3.0 | 1.72 | 293.3 | 12.23 | 2.84E-04 | 28.0 |

| ||||||

mean value | 3.0 | 1.76±0.06 | 290.6±2.7 | 13.09±0.69 | 3.11E-04 | 30.6±2.0 |

| ||||||

specimen3 | ||||||

set1 | 3.0 | 1.91 | 340.1 | 19.14 | 3.31E-04 | 32.6 |

set2 | 3.0 | 1.59 | 343.8 | 18.85 | 3.19E-04 | 31.4 |

set3 | 3.0 | 1.55 | 340.1 | 14.11 | 2.44E-04 | 24.0 |

| ||||||

mean value | 3.0 | 1.68±0.18 | 341.3±1.8 | 17.37±2.51 | 2.98E-04 | 29.3±4.3 |

Experimental measures and stress gradient in second wafer.

g (μm) | t (μm) | L (μm) | u (μm) | c (μm^{-1}) |
Ω (MPa/μm) | |
---|---|---|---|---|---|---|

specimen1 | ||||||

set4 | 3.0 | 1.78 | 192.8 | 4.16 | 2.23E-04 | 22.0 |

set5 | 3.0 | 1.77 | 191.4 | 4.65 | 2.53E-04 | 24.9 |

set6 | 3.0 | 1.74 | 189.6 | 4.23 | 2.35E-04 | 23.1 |

| ||||||

mean value | 3.0 | 1.76±0.02 | 191.3±1.6 | 4.35 | 2.37E-04 | 23.3 |

| ||||||

specimen2 | ||||||

set4 | 3.0 | 2.41 | 191.9 | 2.69 | 1.46E-04 | 14.4 |

set5 | 3.0 | 2.26 | 192.1 | 2.10 | 1.14E-04 | 11.2 |

set6 | 3.0 | 2,31 | 192.1 | 2.55 | 1.38E-04 | 13.6 |

| ||||||

mean value | 3.0 | 2.33±0.07 | 192.0±0.2 | 2.45 | 1.32E-04 | 13.7 |

| ||||||

specimen3 | ||||||

set4 | 3.0 | 4.31 | 189.9 | 0 | 0 | 0 |

set5 | 3.0 | 4.40 | 191.5 | 0 | 0 | 0 |

set6 | 3.0 | 4.37 | 190.5 | 0 | 0 | 0 |

| ||||||

mean value | 3.0 | 4.36±0.04 | 190.6±0.8 | 0 | 0 | 0 |