First generation set of prototypes

The first generation of designed and fabricated devices corresponds to the Run 2015 set.

Table 6.2 summarizes the dimensions of the fabricated resonators corresponding to this first generation, whose desing in Cadence is shown in figure C-1 (appendix C). Figure 6-5 shows an optical image of a C4 resonator corresponding to the Run 2015 set.

Table 6.2: Desing parameters of the fabricated resonators corresponding to the Run 2015 set.

Metal 4 Polysilicon

Parameters C1 C2 C3 C4 C5 C6

l (µm) 73.75 63.85 52.1 63.85 54.6 63

w (µm) 0.60 0.6 0.6 0.6 0.35 0.35

s (µm) 1.72 1.5 1.45 2.07 0.9 0.9

freq. (MHz) 0.75 1 1.5 1 1 0.75

Figure 6-5: Optical image of a fabricated C4 resonator corresponding to the Run 2015 set.

Extensive experimental measurements of the frequency response of the fabricated CMOS cc-beam resonators have been performed under different thermal and biasing conditions.

The Setup#1 (see figure 6-1) is used in this experimental procedure to reduce the parasitic feedthrough current, allowing an accurate measurement of the system frequency response.

Figure 6-6a) shows that the power of the primary resonance peak grows with the applied bias voltage, as does the attenuation of the antiresonance, while the primary resonance frequency is decreasing. On the other hand, in figure 6-6b) the shapes of the experimental resonance frequency dependence on the bias voltage show a very good agreement with the theoretical curves provided by equation (6.2), demonstrating the accuracy of the approach, and fulfilling the first step to the model validation by means of experimental data. Moreover, such experimental data have been used to find second order parameters as well as the respectiveV_pi0values (these values of bias voltage correspond with the lower bias boundary value if the geometric condition (s/w) for bistability is attained, otherwise they imply the collapse of the towards one of the electrodes beam because of the pull-in effect). The average value of the fringing field constant is obtained from the slope of each curvef_rT² =m·V_DC² +n.

From equation (6.2), the slopem(which can be numerically obtained from the experimental points) is found to be

m=−k_FF 0lth

2π²m_effN s³

N

X

n=1

P_n² (6.3)

and from this expression and the definition of k_FF , given in equation (5.15), the α_FF

con-stant can be calculated. From the natural frequency at room temperature, the equation of k1σT (5.26) provides the value of the residual stress (σeff) not modified by the thermal effect.

The experimentalV_pi0 values are obtained as the intersection of the curves of equation (6.2) given by the experimental points with the frT = 0 axis; Finally the thermal constantαT is obtained from the variation of the natural frequency (f_rT(V_DC = 0)) caused by the change in the temperature: given the respective definitions ofET, andσT (provided by (5.24)) and equation (6.2) forV_DC = 0, theα_T with best fitting of the experimental curve is numerically obtained, as shown in figure 6-7. All these values are summarized in tables 6.3, 6.4 and 6.5, as well as the minimum gap value attaining the geometric condition for bistablility and its associated lower bias boundary value V_pi0 when the resonator is subjected to the room temperature (RT=20^◦C by default) and when the resonator is subjected to an ”operational temperature” (OT=120^◦C by default). The second order nonlinear parameters are shown in figures 6-12 and 6-13. The reasonable similarity (keeping in mind the fabrication tolerances and the uncertainties in the chemical attack releasing procedure) between the respective val-ues of all these parameters corresponding to the different experimentally measured devices belonging to the Run 2015 set, constitutes a prove of the models accuracy.

The results tables include the wet etching time using the different wet etchants explained in section 3.2.3; in these tables the acronym HMS stands for the home-made hydrofluoric solution.

Figure 6-6: From resonator C4(15)-k15 a) experimental frequency response of a metal 4 (aluminium and titanium nitride) cc-beam resonator for a temperature of 110^◦C under different bias voltage values; b)representation of the experimental maximum response power frequency of this metal 4 cc-beam resonator with respect to the applied bias voltage for different temperature values (20^◦C, 40^◦C, 60^◦C, 80^◦C and 100^◦C), and their respective theoretical counterparts (in dotted lines).

Figure 6-7: Experimental natural frequencies (frT (VDC = 0)) values for different tempera-tures, and the analytical fitting function from which the value of theα_T constant is obatined, from the resonator C4(15)-k15.

Table 6.3: Experimentally measured parameters corresponding to Metal 4 resonators be-longing to the Run 2015 set.( * OT=130^◦C ,** OT=110^◦C ).

METAL4 Run2015 Chip4 C1 Chip4 C4 Chip14 C4 Chip15 C4 Wet etching time 10’ (HMS) 10’ (HMS) 10’ (HMS) 10’ (HMS)

f₀ (MHz) at RT 1.919 2.232 2.223 2.409

σ_eff (GPa) 0.163 0.156 0.155 0.1885

α_T (K⁻¹) 8.71e-06 8.99e-06 9.73e-06 1.096e-05

αFF (∅) 0.172 0.150 0.265 0.235

smin(µm) at RT 2.130 1.857 1.850 2.005 Vpi0(V) at RT 232.17 339.59 328.15 359.69 smin(µm) at OT 1.298* 1.247 1.128 1.355**

Vpi0(V) at OT 152.58* 236.33 206.17 265.27**

Table 6.4: Experimentally measured parameters corresponding to Metal 4 resonators be-longing to the Run 2015 set. (* OT=130^◦C ,** OT=110^◦C).

METAL4 Run2015 Chip16 C4 Chip17 C4 Chip34 C4 Chip43 C4 Wet etching time 10’ (HMS) 20’ (HMS) 10’ (Silox) 20’ (ALPAD)

f0 (MHz) at RT 2.31 1.951 2.143 1.886

σ_eff (GPa) 0.170 0.110 0.141 0.100

α_T (K⁻¹) 1.13e-05 8.35e-06 9.76e-06 5.84e-06

α_FF (∅) 0.236 0.652 0.2638 0.382

s_min (µm) at RT — 1.623 — 1.569

Vpi0(V) at RT — 241.35 — 248.12

smin (µm) at OT 1.209** 0.918 1.040 1.122*

Vpi0(V) at OT 228.65** 136.83 215.37 182.02*

Table 6.5: Experimentally measured parameters corresponding to Metal 4 resonators be-longing to the Run 2015 set. (* OT=130^◦C ,*** RT=25^◦C).

METAL4 Run2015 Chip52 C4 Chip53 C4 Chip57 C4 Chip59 C4 Wet etching time 10’ (HMS) 10’ (HMS) 10’ (Silox) 10’ (Silox)

f₀ (MHz) at RT 1.857 1.681 2.169 1.982

σ_eff (GPa) 0.096 0.0716 0.145 0.115

α_T (K⁻¹) 3.49e-06 4.55e-06 9.07e-06 8.74e-06

αFF (∅) 0.519 0.689 0.310 0.227

smin (µm) at RT 1.5450 1.3985 1.8049*** 1.649***

Vpi0(V) at RT 248.60 203.26 307.48*** 294.96***

smin (µm) at OT 1.288* — 1.168 0.944

Vpi0(V) at OT 196.39* — 209.90 182.31

These experimental results provided the information about the order of magnitude of the residual stress generated from the fabrication procedure. Given the experimental values of the fabrication residual stress, although the geometric condition for bistability is attained by the most of the resonators at room temperature, the desired margin of 10% between the upper and the lower bias boundary value does not occur at room temperature. In practical application, driving the resonator into the bistable regime is not recommended without a margin wide enough, because it would eventually lead to the resonator collapse as a consequence of the pull in effect. Thus, for the resonators belonging to the Run 2015 set, the operating temperature must be applied in order to compensate the tensile residual stress before the bistable region is reached.