1. Introduction Radio frequency applications, such as bandpass filters or electronic oscillators, rely on vibrating elements with high quality factors to maximize performance. Quartz crystals and surface

acoustic wave resonators are widely used in these applications, but thin-film micromechanical resonators have been seen as a promising alternative solution, due to their high documented quality factors (105 – 106) [1]. In addition, mass, force or biological sensing are some of other possible applications for the microresonators. Regardless of the application, one of the most exciting potential of thin-film micromechanical resonators is the possibility to monolithically integrate them with electronics [2]. While MEMS transducers sense or control some physical quantity of interest, the adjacent circuits typically provide functionalities related with signal processing and communication with the outside [3]. The co-fabrication of complementary metal-oxide-semiconductor (CMOS) electronics and microelectromechanical systems (MEMS), such as resonators, is technically feasible, although the specific requirements of each technology frequently constrain the process sequence, material selection or thermal budget of the other. In one of the possible approaches, MEMS and integrated circuits (IC) components are manufactured on separate substrates using dedicated MEMS and IC processes and are subsequently joined together in the final system. But a more integrated strategy, where the MEMS and IC components are fabricated on the same substrate using consecutive or interlaced processing schemes [3], would be ideally preferable. Although there are a number of possible approaches towards integrating MEMS with CMOS [3], the most practical strategy is the fabrication of the MEMS as part of the backend processing on previously processed CMOS wafers. Stacking the MEMS on top of the electronic circuit allows improving interconnects and minimizing the parasitics, resulting in lower power consumption and better sensitivity [4]. However, the processes to deposit the structural material and to release the sacrificial material must be compatible with the CMOS substrate. After the deposition of the aluminum metallization of a CMOS process, the temperature is limited to 400-450 ºC for the micromachining steps, to avoid degrading the silicon-aluminum contacts and the electronic performance [5]. Polycrystalline silicon, a common MEMS material, is usually deposited at temperatures of ~600 ºC in low pressure chemical vapor deposition (LPCVD) furnace and often requires a thermal annealing step at more than 900 ºC to reduce the residual stresses [6], which makes it incompatible with standard aluminum metallization. Metal-based and piezoelectric structural materials, such as nickel and aluminum nitride, respectively, or polycrystalline silicongermanium alloys are some of the most attractive solutions due to their lower deposition temperatures [4]. Nevertheless, metals can suffer from long-term instabilities in their mechanical

properties and piezoelectric materials require a controlled and oriented growth [4]. An alternative material that can be used to fabricate the thin-film resonators is hydrogenated amorphous silicon (a-Si:H). The plasma-enhanced chemical vapor deposition (PECVD) process provides uniform growth of either an intrinsic or doped semiconductor thin-film over large areas and low temperatures (< 300 ºC). The possibility of using a wide variety of substrates, such as glass, plastic or steel, has made a-Si:H the material of choice in a number of commercial applications including solar cells [7], flat panel displays [8] and digital medical imagers [9]. Although the electronic properties of a-Si:H are inferior to single crystalline silicon or polysilicon, its mechanical properties are similar, making it a good candidate for MEMS applications [10]. The low deposition temperatures make this material a particular interesting choice for post-processed MEMS resonators integrated to CMOS ICs [11]. The electromechanical properties of the material can be finely controlled through the deposition conditions [12], [13]. In a capacitive transduction method, the motion of the resonant structure causes a change in the gap of the device, which results in an ac electrical current that can be sensed. A clear advantage of capacitive methods is that they can be easily integrated on-chip and do not require bulky or external apparatus that cannot be miniaturized. Also, the mechanical characteristics and the electromechanical coupling of the system are defined by the properties of the material and by the design of the resonators. The drawbacks are related with difficulties in operating the devices at increasingly high frequencies due to intrinsic parasitic elements that often dominate the measurements [14], [15]. The presence of these parasitic currents presents a challenge to CMOS integration and readout. Another serious concern is the typically high equivalent impedances, or motional resistance, Rm, of the microresonators. This parameter governs the relationship between the input voltage and the impedance of the system at resonance and can be responsible for reflecting back part of the waves that carry the electrical signals between elements of the circuit. In electronics, impedance matching is very important to maximize the power transfer or minimize signal reflection from the load. Generally, existing systems require an impedance matching of 50 Ω and oscillators require small resistances for low power consumption [11]. In this work, the electrical characterization of flexural thin-film silicon resonators is used to address some important questions that arise when thinking on a possible integration of these

devices with electronics. Hydrogenated amorphous silicon flexural resonators are fabricated using a simple surface micromachining fabrication process, at a maximum temperature of 175 ºC, compatible with CMOS processing. The motion of the resonators is electrostatically actuated and capacitively sensed using a one-port measurement scheme. The linear mechanical motion of the resonators is described by an electrical equivalent model [16] from where the electrical motional parameters of the resonator are extracted and studied as function of the geometry and applied dc voltage. A design rule to reduce the motional resistance of a resonator working in linear regime is developed and discussed. Finally, a strategy to extract the linear mechanical spring constant term of the system is applied to the experimental results. Nonlinearities are easily encountered in the resonant motion of MEMS resonators when driven into high amplitude motion. This phenomenon can lead to hysteresis or instabilities, which limits the operating conditions, geometric design or energy that can be stored in the device. Knowing the range of linear vibrations allows to identify the dynamic range for resonators. The nonlinear regime of motion of the resonators is shown to occur either by mechanical stiffening or electrical softening and a strategy to extract the cubic mechanical spring constant coefficient is presented and applied to experimental results.

2. EXPERIMENTAL PART

2.1. Fabrication of the microresonators In this work, three batches of resonators were fabricated in parallel using the same fabrication process. The only difference between the batches is the deposited thickness of the aluminum sacrificial layer, which is used to define the height of the transduction gap. The microfabrication process starts with the deposition, by dc magnetron sputtering, of a 70 nm thick TiW(N) gate electrode that is patterned using photolithography and reactive ion etching (RIE). The aluminum sacrificial layer is then deposited on top of the gate electrode layer by dc magnetron sputtering. Different thicknesses of the aluminum layer were used in each batch of resonators: 250 nm, 400 nm and 600 nm. The aluminum layer is then patterned by photolithography and etched by a commercial Al wet etchant. The structural material of the microbridges consists of a bilayer of TiW (100 nm) and n-type doped hydrogenated amorphous silicon deposited at 175 °C by the PECVD technique discussed elsewhere [12], using a mixture of silane (SiH4), hydrogen (H2) and phosphine (PH3) as source gases. The mechanical and electrical properties of the a-Si:H material can be modified by adjusting the deposition conditions to optimize device performance [12]. In

this case, a 2-µm structural silicon thin-film with near-zero stress was chosen, to avoid extraneous mechanical phenomena induced by the mechanical stress of the layer [13]. The structural layer is patterned by RIE. In the final step, the aluminum sacrificial layer is selectively removed using a commercial Al wet etching solution, leaving an air gap between the bridge and the gate. Finally, the samples are mounted and wirebonded to DIP packages. Fig. 1 show scanning electron micrographs (SEM) of two of the microbridges with sub-micron gaps fabricated using this process. The dimensions of each resonator are shown in the figures. Resonators with gaps (g) of 250, 400 and 600 nm, variable lengths (L), from 50 to 150 μm, and variable widths (w), from 20 to 50 μm, were fabricated. These dimensions are easily attained using this fabrication process, avoiding stiction problems. All flexural resonators fabricated are clamped-clamped microbridges.

2.2. Exciting and detecting the resonant motion The flexural resonators can be considered a parallel plate capacitor. An ac voltage, 𝑉𝑎𝑐 , applied between the electrodes, drives the motion at a specific frequency. By adding a dc voltage, 𝑉𝑑𝑐 , the force component at the excitation frequency is amplified. The electrostatic force established between the two plates of the capacitor is always attractive and given by:

F

2 1 dC  x  Vdc  Vac cos t   , (1)  2 dx

where Vdc  Vac cos t   Vt  t  is the potential applied between the electrodes at frequency  ,

C  x 

0 A

g  x t 

is the capacitance of the device, with A and g , respectively, the area and the

air-gap between the bottom electrode and the resonator, x the displacement of the resonator and

 0 the vacuum permittivity. Inserting the expression for C  x  into eq. (1), and expanding the square of Vt  t  , the electrostatic force can be written: 2  0 A   2 Vac 2 Vac cos  2t   1 F  V  2 V V cos  t      dc  . (2) dc ac 2   g  x  t  2   2 2   

Eq. (2) shows that the electrostatic driving force has a dc component that causes a static deflection of the structure, a component at the frequency of the ac signal,  , and a component at twice that frequency, 2  . The component at the frequency  is amplified by the dc voltage, while the component at twice the driving frequency is amplified by the ac voltage.

In cases where Vdc

Vac , the second harmonic of the driving force can usually be neglected and

the electrostatic force has the same frequency as the input ac signal. The resulting mechanical motion of the resonator translates into a time-varying capacitance that can be detected as a displacement current flowing through the resonator [17], [18]. The total current through the device is given by:

i0  t  

  C  x Vt  t   t

 Vt  t 

C  x  t

 C  x

Vt  t  t

. (3)

Considering small displacements of the resonator, x

g , implies that C  x   C0 

0 A g

, where

C0 is the static capacitance of the capacitor formed between the drive electrode and the structure. In addition, using small ac voltages, Vdc

Vac , causes Vt  t   Vdc . Under these assumptions eq.

(3) simplifies to:

i0  t   Vdc

C  x  x x

t

 C0

 Vac cos t   t

 im  t   i p  t  , (4)

Thus, the total output current consists of two terms. The first term is proportional to the dc voltage and originates from the time variation of the capacitance of the device. It is usually called motional current, im  t  . The second term, which is proportional to the static capacitance of the device, C0 , is generated by the ac excitation voltage and is called the parasitic current, i p  t  , since it does not reflect the motion of the resonator [19]. To detect capacitively the motion

of the resonators, the samples are mounted on a printed circuit board (PCB) and placed inside a vacuum chamber at pressures ~1 Pa. Electrical tracks on the PCB are directly wirebonded to the electrodes and resonator on the MEMS chip. Fig. 2-A shows the experimental setup for a one-port measurement scheme. In this setup, the network analyzer generates an ac signal with excitation frequency  through the source, port-S. This signal is split to the port-R, which is used as a reference, and to the bias-T, where it is added to the dc voltage coming from a power supply, before being connected to the resonator. The output signal generated by the motion of the resonator at frequency  is addressed to the test port, port-T, where its amplitude and phase are compared with the reference. Fig. 2-B shows the electrical equivalent of this experimental setup scheme, where ZDUT refers to the impedance of the resonator (device under test).

The resonant motion of the device (in linear regime) in the mechanical domain can be described by an equivalent series RLC electrical circuit. Both the mechanical and electrical systems are described by the same second order differential equation and all the properties of one system have an analogous equivalent property in the other. In addition, to account for the parasitic effects, a capacitor is added in parallel to the circuit. This electrical circuit that models the resonant motion of the device in linear regime is shown in Fig. 2-C, where the subscript m stands for motional. In the measurement setup shown in Fig. 2-A, both ac and dc voltages are used and the electrostatic force is described by eq. (2), where, typically, the second harmonic of the force can be neglected. This means that the electrostatic force has the same frequency, 𝜔, as the ac input signal and the detected output current. Therefore, a large feedthrough capacitance is present between the input and output signals. These parasitic effects that degrade the output current consist of capacitances intrinsic to the MEMS device, such as the static capacitance, 𝐶0 , or those coming from the measurement setup, such as the electrical interconnects, substrate or cables, which allow the current to leak through several distinct paths.

3. RESULTS AND DISCUSSION

3.1. Linear Regime

3.1.1. Experimental admittance spectra and extraction of the motional parameters The transmission spectrum measured by the network analyzer in the setup presented in Fig. 2, Tr , can be related to the admittance of the device by:

Tr 

V RL i0 VT RL  out   , (5) VR Vin  RL  Z DUT  i0 RL  Y  j 1

where VR and VT are, respectively, the signal entering ports-R and T, RL is the 50 Ω characteristic impedance of the test port, i0 is the displacement current flowing through the circuit (and resonator) and Z DUT and Y  j  are the impedance and the admittance of the resonator, respectively. This equation shows that the power delivered to the measurement setup is reduced if the impedance of the resonator is large [2]. At resonance, the motional inductance and capacitance, Lm and Cm, respectively, cancel each other and the system is purely resistive. In this case, the admittance of the system depends only on the motional resistance (and on the feedthrough capacitance) of the device.

Knowing that the resonance frequency of the electrical Rm LmCm series is given by

0  1/ LmCm and that the quality factor can be written as Q  Lm / Cm / Rm , the admittance and phase response of the electrical equivalent circuit presented in Fig. 2-C are:

Y  j   jC ft 

jCm 2

  j 1       0  Q  0 

. (6)

 Im Ym   Φ  arctan  , (7)  Re Y   m   where Re Ym  and Im Ym  are the real and imaginary parts of the admittance, respectively. Using eq. (5), the transmission measured experimentally using the setup shown in Fig. 2 is converted to the experimental admittance spectra of the device. Then, the admittance of the electrical equivalent model, given by eq. (6), is fitted to the experimental admittance results. The result of this procedure is shown in Fig. 3. The fitting parameters of eq. (6) are the resonance frequency of the device, 0 , the quality factor, Q, the motional capacitance, Cm, and the feedthrough capacitance, Cft. From the values of these parameters, the values of Rm and Lm can be calculated, using the relations:





Lm  1/ Cm0 2 .

Rm  Lm / Cm / Q . Fig. 3-A

(8) (9)

shows the experimental transmission curve measured using the setup described in Fig. 2.

Fig. 3-B shows the corresponding admittance curve (determined using eq. (5)) and associated phase of a resonator with moderante gap distance (L = 100 µm, g = 400 nm and f0 = 0.776 MHz). The open symbols represent the experimental data and the solid lines represent the fit of the electrical model, given by eq. (6). The feedthrough capacitance obtained from fitting the model to the resonator is 273 fF. This value is higher than the static capacitance of the device,

C0 

0 A g

, determined from its geometry (88.5 fF). This result shows that the total feedthrough

capacitance, C ft , of the electrical circuit includes more than just the static capacitance term of the resonators, C0 . Different sources may contribute to this feedthrough capacitance, such as the

metallic lines on the PCB, the coaxial cables or interconnects. Some of these sources can be minimized, for example, by separating the cables that carry the input and output signals. In the admittance plot shown in Fig. 3-B, an antiresonance peak at frequencies above the series resonance frequency is observed. The presence of this parallel resonance is a direct consequence of the high feedthrough capacitance [20], [21], Cft, which is generally two orders of magnitude larger than the motional capacitance Cm of the flexural resonators used in this work. Thus, the parallel resonance frequency, f p , is very close to the series resonant frequency, f 0 . As the value of the static/ feedthrough capacitances increases, the parallel resonance approaches the series resonance peak. The presence of the feedthrough capacitance is reflected in the phase signal by the additional transition back to the original 90º at the parallel resonance frequency. If the two resonance peaks are too close together, the phase may not pass through 0º, which can present a challenge when designing a MEMS-based oscillator, since this transition is a requirement for the oscillation condition.

3.1.2. Comparing the extracted electrical motional parameters with the parameters calculated from the mechanical domain Among the elements in the equivalent electrical circuit, the motional resistance, Rm, is probably the most important, since it determines the interface with control electronics, the impedance matching between components and the power consumption. At resonance, the motional inductor, Lm, and capacitor, Cm, are equal and out of phase, and the impedance of the circuit depends only on the motional resistance, Rm. Recalling the term of the motional current, im  t  shown in eq. (4), the electromechanical coupling factor term  can be introduced:

im  t   Vdc

C  x  x x

t



x , t

(10)

Considering small displacements, x  t 

  Vdc

0 A

 g  x t 

2

   Vdc

0 A g2

,

g , and  is given by:

(11)

Nguyen et al. derived equations for the equivalent circuit elements using the electromechanical coupling factor and the geometry and material properties of the resonator. The strategy consisted of determining the effective impedance of the single element seen looking into the resonator port [22], [23]. These expressions are:

 kr mr  Rm  2  Q   2 ,  Cm  kr   m  Lm  2r  

(12)

where k r and mr are, respectively, the effective stiffness and mass on the center of the resonator beam. These expressions can be rearranged by substituting the natural angular frequency of the resonator, given by 0  2π f0  kr / mr  kr  02  mr , and the effective mass of the fundamental mode of the flexural resonators, given by mr  0.4  Lwh [23], [24], where  is the density of the material, and L, w and h are, respectively, the length, width and thickness of the resonator:

0.4 2π f 0  g 4   h Rm  , Vdc 2 0 2 L wQ Cm 

(13)

Vdc 2 0 2 Lw

0.4 g 4  h  2π f 0 

2

,

0.4 g 4  h Lm  2 2 . Vdc  0 L w

(14)

(15)

Eq. (13) implies that Rm can be greatly reduced by decreasing the electrode-to-resonator gap spacing g, since the motional resistance depends on this parameter to the fourth power. It is also possible to reduce Rm by increasing the dc voltage, Vdc, the electrode area (Lw), the quality factor Q or reducing the thickness of the resonator, h. Fig. 4 shows the dependence of the motional resistance on the applied dc voltage and the gap distance for different resonators. The dashed-dotted lines are plots of eq. (13), representing the theoretical motional resistance of each resonator. These curves depend on the geometrical parameters of the resonator, the density of the hydrogenated amorphous silicon structural layer (ρ = 2330 kg/ m3) [13] and the applied dc voltage. The sweeps of the dc voltages were done for constant ac voltages. The symbols represent the motional resistance extracted from fitting the admittance of the equivalent electrical model (eq. (6)) to the experimental admittance curve in

linear regime of motion, measured for different dc voltages. The measurements were performed until the onset of nonlinear regime of motion, where the electrical equivalent model is no longer valid. In general, the measured motional resistances lie between ~85 kΩ and a few MΩ, following the theoretical predictions. From the graph, no dependence on the ac voltage is evident, as predicted by eq. (13). The motional resistances depend on geometrical parameters, namely the gap distance, g, the area between electrodes (Lw) and the quality factor, Q. The resonator represented by the red circles (L = 80 µm, g = 250 nm, w = 25 µm, t = 2 µm, Q = 700) presents the lowest motional resistance (~85 kΩ), mainly due to the narrow actuation gap. The blue stars show a resonator (L = 150 µm, g = 600 nm, w = 20 µm, t = 2 µm, Q = 1600) that compensates the effect of the relatively large gap with a high quality factor and high actuation area, achieving a limit motional resistance of about ~170 kΩ. The motional resistance of the resonator represented by the orange triangles (L = 120 µm, g = 600 nm, w = 20 µm, t = 2 µm, Q = 700) is the highest, ~400 kΩ, due to the relatively high gap of 600 nm and lack of an advantageous feature, such as a high quality factor, that might compensate it. Finally, the green diamonds show a resonator (L = 100 µm, g = 400 nm, w = 40 µm, t = 1.1 µm, Q = 80) with a limit motional resistance of ~200 kΩ, due to the large area, small gap and lower thickness, despite having a low quality factor. Note that eqs. (13) – (15) do not consider variations in the gap distance as the displacement of the center of the beam increases with the increase of the applied dc voltage. A more accurate model is presented in [25], in which the motional capacitance of the device is calculated considering shape factors of the fundamental mode that account for variation of gap along the beam. This model is represented by the orange thin-solid lines added to Figs. 4 and 5 and shows that is reasonable to use these simpler equations in cases of small displacements (in linear regime) and far from the pull-in voltage of the resonators. One cannot also neglect the presence of high capacitive feedthrough on this chip, as shown in Fig. 3-B by the presence of the antiresonance peak. For example, considering the resonator showed in Fig. 3, the estimated impedance of the parallel capacitor is ~0.750 M at its resonance frequency, which is comparable of the measured motional resistance of ~0.700 M. This suggests that the parallel capacitor creates a viable alternative path for the current to leak. In [23], the authors measure motional resistances of 8.27 k and 340  for flexural beams with different dimensions. Very thin actuation gaps (100 nm), which result from a demanding

fabrication process, can explain the extremely low values of motional resistance, but it is equally important to note the absence of the antiresonance peak in the presented resonance spectra, which indicates the absence of parasitic capacitances. Different strategies to eliminate the feedthrough current can be adopted, such as using the second harmonic of the electrostatic force to actuate the motion of the resonator [26], [27], or using differential measurement setups [19], [28]. The same type of curves is shown in Fig. 5 for the motional inductance and capacitance. The description of the symbols is equivalent to that made for motional resistance in Fig. 4. The analogy between the mechanical and electrical domains allows the association of the motional capacitance with the amount of energy stored in the mechanical spring, through its elongation or compression. The physical interpretation of the motional inductance is related to the presence of mass and inertia in the system. Fig. 5 shows that the experimental values extracted from the electrical model follow the values predicted theoretically from the mechanical domain, given by eqs. (14) and (15). As the applied dc voltage increases, the amplitude of the motion and the electromechanical couple factor also increase, which means that the motional capacitance (Fig. 5-A) (stored energy in the spring/ capacitor) increases to a maximum value of ~1 fF, and the motional inductance (Fig. 5-B) (resistance to the change of state of motion/ flow of current) decreases to ~3 H.

3.1.3. Hysteresis criterion If the distance between the ends of the beam is rigidly fixed, large amplitude deflections of the beam will develop a tensile axial force at the anchors that increase the resonance frequency of the device [29]. This is known as the hard-spring effect. There are analytical solutions for the resonance frequencies of beams subjected to axial forces applied at the anchors but solving these systems of equations is difficult and often no closed-form solutions are obtained [30]–[32]. For large displacements of the beam, approximated solutions for the natural frequencies can be found by employing Rayleigh’s energy method. The resonance frequency,  , which depends on the deflection of the beam, can be written as [29]: 2   2  wmax     1  0.53 1    ,    h     2

2 0

(16)

with 0 the natural angular frequency of the device, wmax the maximum vibration amplitude, h the thickness of the resonator and  is the Poisson’s ratio. Eq. (16) strictly describes the phenomena of mechanical stiffening, where the resonance frequency increases. A consequence

of this equation is that beyond a critical amplitude, wc , the resonance curve becomes triple valued within a range of frequencies. This frequency interval defines the region of hysteresis, where three stables solutions for the beam motion are found [33]. Gui et al. derived an approximated hysteresis criterion, based on the idea that it is possible to avoid hysteresis in the motion of the resonator if the maximum amplitude of vibration is inferior to the critical amplitude, wmax  wc . This criterion depends on the quality factor of the resonant mode, Q, the dc voltage, Vdc, and the ac sinusoidal voltage, Vac, applied to the device [34]:

Vdc Vac  Q3/2  Khc ,

(17)

where

 E K hc  45    1  2  0



4    h  g2 1/2    L  

(18)



is a critical hysteresis constant in which E  E / 1  2  is the plate modulus, E is the Young’s modulus of the material, L is the beam length and g is the gap distance between the bottom electrode and the resonator. Eq. (17) represents a necessary condition for hysteresis-free operation of a resonant microbridge. In general, observing the expression for K hc in eq. (19), the chance of vibration hysteresis can be lowered by choosing a smaller beam aspect ratio or by increasing the gap spacing. Axial tensile stress is not contemplated in eq. (16) and will make the beam stiffer, increasing K hc . A few important notes on the derivation of this criterion: the critical amplitude for the first flexural mode of vibration, wc , is derived by Landau in [35] and given by





1/2

wc  h / 0.53Q 1  2 

and describes the amplitude of the resonance curve when it becomes

triple valued. The maximum vibration amplitude for the first mode of a resonator electrostatically driven by a dc polarization and an oscillating voltage is derived in [25] and 1/2

given by wmax  0.37 gQVdcVac / Vpi 2 , where Vpi  3.48  Eh3 g 3 /  0 

/ L2 is the static pull-in

voltage of resonator with a uniform electrode, equally derived in [25]. The criterion presented in eq. (17) results from substituting these relations into the expression wmax  wc . Note that this

criterion considers the static pull-in voltage of the resonator and, therefore, the static mass or spring constant of the device must be used in the calculations. Fig. 6 exemplifies the application of the hysteresis criterion presented in eq. (17) for a resonator where mechanical stiffening occurs before showing hysteresis. Values of E = 150 GPa and υ = 0.3, typical for hydrogenated amorphous silicon [13], were used to calculate K hc of eq. (18). According to the criterion presented in eq. (17), on the left side of the green curve the resonator will operate without hysteresis. On the right side, hysteresis will occur. The orange line indicates the dc voltage that was progressively applied to the resonator, while keeping the ac voltage constant at 0.1 V. The admittance curves of the resonator, measured capacitively as previously described, are shown as an inset in the right bottom corner. For each applied dc voltage, the curves of experimental admittance and phase are shown as blue points, while the attempts to fit the admittance of the electrical equivalent circuit to the experimental results are shown as solid red lines. As theoretically predicted by the criterion of eq. (17), the hysteretic regime is experimentally observed to start at ~1.5 V of applied dc voltage, as shown in Fig. 6. At this point, the electrical equivalent model for the motion of the resonators, and consequently the concept of motional resistance itself, is no longer valid and does not fit the experimental results.

3.1.4. Design rules for minimum attainable motional resistance, Rm, in linear regime As was discussed in previous sections, among the parameters in the equivalent electrical circuit, the motional resistance, Rm, is probably the most important. Hence, design rules that allow reducing this parameter are sought. Using a purely geometric point of view, and according to eq. (13), it appears that the ideal structure for decreasing the motional resistance is a long and thin bridge (consequently with low resonance frequency), with high quality factor and a very small gap distance. However, two drawbacks can limit this approach: first, decreasing the transduction gap will likely require a more complex fabrication process; and second, actuation voltages imposed in the circuit can rapidly drive a long and floppy structure to hysteresis. Therefore, to decrease the motional resistance, it might be of interest to keep the resonator working in the linear regime at higher applied dc voltages. From the criterion of eq. (17), the maximum dc voltage that can be used before hysteresis occurs is given by:

Vdc 

8.53 1023 h4 g 2 , Vac L4 Q3/2

(19)

where the typical values for amorphous silicon (E = 150 GPa,   0.3 ) [13] were used in the expression of K hc of eq. (18), in SI units. On the brink of hysteresis, the maximum dc voltage can be used and the equality stands. By increasing the dc voltage applied to the resonator, the motional resistance of the system decreases, as shown in eq. (13). Therefore, using the maximum dc voltage before the system enters in hysteresis, will minimize the motional resistance. Replacing eq. (19) into eq. (13):

Rm 

2π g 4   h  f 0 2

 8.53 1023 h 4 g 2  2    0 L wQ 4 3/2  Vac L Q 

.

(20)

Finally, considering that the resonance frequency of the fundamental mode of a stress-free fixed bridge is given by f 0 

h L2

E



[27], substituting numerical values for the vacuum permittivity

and parameters of amorphous silicon (   2330 Kg / m3 ) in SI units and simplifying the expression, one gets: 5

Rm  2.06 1018

Vac 2 Q 2  L    . hw  h 

(21)

Using SI units in the parameters of eq. (21) allows obtaining the motional resistance in ohms. Note that the single assumption behind the derivation of this equation is that the resonators can be driven at any arbitrary dc voltage, just before the onset of hysteretic behavior resulting from mechanical stiffening. This design rule can be useful when trying to choose the geometry of the device that allows minimizing the motional resistance, within the constraints of any imposed external actuation conditions. The result expressed by eq. (21) shows that the motional resistance does not depend on the distance of the transduction gap, which alleviates the emphasis on the fabrication process. Decreasing the aspect ratio of the structure is shown to be crucial, since the motional resistance depends on this parameter to the fifth power. Another interesting observation is that the motional resistance decreases when the structures present low quality factors (reducing the sensitivity of the sensor) or that, when actuating the motion of the device, low ac voltages should be used and compensated with high dc voltages. Nevertheless, reducing the ac voltage will markedly reduce the applied force at frequency of excitation, which decreases the signal-to-noise ratio.

In summary, when trying to decrease the motional resistance of the resonator, a short and thick plate-type structure, with lower quality factor, should be electrostatically actuated with a largearea bottom electrode at arbitrarily high dc voltages. This result is consistent with what was experimentally observed by Nguyen et al. in [23]. Eq. (21) can be used to predict that a possible 40 x 40 µm square plate with a thickness of 2 µm, a quality factor of 250, and actuated with an ac voltage of 0.1 V will have a motional resistance of ~33 Ω when driven to just before the onset of hysteresis by an arbitrarily high dc voltage (540 V predicted by eq. (19)). Of course, in practice, there are limits to how high this dc voltage can be, creating a complex interplay between all the parameters involved and the need to contemplate tradeoffs in the design of the system. Yet a final example, considering a typical silicon beam of 40 x 4 µm with a thickness of 2 µm, a quality factor of 1000 and an actuation gap of 200 nm, the maximum dc voltage before getting into nonlinear regime is 67.4 V (eq. (19)), which will allows obtaining a motional resistance of ~8.25 k (eq. (21)). In this structure, the pull-in voltage is 75.1 V, calculated using the expression presented above and derived in [25]. Note that this is the static pull-in voltage, obtained using the value of static mass or static spring constant.

3.1.5. Extraction of linear mechanical spring constant, k1m The general single-degree-of-freedom equation of motion of resonators electrostatically actuated using a dc voltage and an ac voltage at excitation frequency ω is given by [36], [37]:

mr x''  t    x  t   k1m x  t   k3m x  t   3

2 1 C  x  Vdc  Vac cos t    2 x

(22)

where the prime represents the time derivative, x  t  is the displacement of the center of the beam, mr is the effective mass of the resonator, Vdc and Vac are the components of the voltage applied at frequency

 between the electrodes, C  x  is the capacitance of the parallel plate capacitor,  is the linear viscous damping coefficient and k1m and k3m are the linear and cubic mechanical coefficients of the spring constant, respectively. The right side of the equation is the electrostatic force between the two plates of the capacitor, as shown in eq. (1). For symmetric resonator structures the quadratic mechanical coefficient spring constant term can be ignored [35].

Taking the derivative of the capacitance with respect to the gap distance,

    C  x    0 A 1  0 A 1    2 , and expanding it using the Taylor series in the limit x  t   g  x  t  2 x x  g 1   1   g   g   of small displacements ( x  t   0) , results in

C  x  x



0 A g2

 x t    n  1   . Substituting the first  g n 0   n



terms of the Taylor series and the expansion of the squared voltage term (eq. 2) in eq. (22), and assuming that Vdc

Vac and x

g:

     A 2 A  A 3 mr x''  t    x  t    k1m  0 3 Vdc 2  x  t    k3m  05 Vdc 2  x  t   0 2 VdcVac cos t  , g g g    

(23)

where the quadratic nonlinear spring constant terms were ignored with the assumption of a symmetric resonator structure. Eq. (23) shows that the linear and cubic terms of the spring constant have contributions of the mechanical and electrical domains, where

k1  k1m  k1e  k1m 

0 A g

3

Vdc 2 and k3  k3m  k3e  k3m 

2 0 A 2 Vdc . The mechanical components g5

of the spring constant depend on the geometry and material properties of the resonator and are fixed after fabrication. The electrostatic terms depend on the applied dc voltage and the negative sign means that the force acts in the direction of the displacement. Hence, the resonance frequency of the device can be tuned by changing the applied dc voltage. The natural frequency of the forced harmonic oscillator is given by 0  kr / mr . In the case of small actuation forces and displacements, the effective spring constant consists solely on the mechanical term, kr  k1m . Adding the contribution of the electrostatic term, the resonance frequency of the device is given by [38], [39]:



k1m  k1e  mr

0 A 2 Vdc g3 , mr

k1m  

(24)

where  is a scaling factor parameter added to absorb all the geometrical uncertainties of the microfabrication process and the approximation of the ideal parallel plate capacitor. Eq. (24) shows that a way to extract the linear mechanical term of the spring constant, k1m , and the effective mass of the resonant mode, mr , is to fix Vac and change the applied dc voltage, Vdc , to obtain the relation between the resonance frequencies and the applied voltage, while operating the resonator under linear vibration regime conditions. The results of this procedure are shown in Fig. 7. Fig. 7-A shown the transmission spectra of a resonator with L = 80 µm, w = 25 µm, t = 2 µm and

g = 250 nm, measured with a constant ac voltage of 0.1 V and different applied dc voltages (from 1 to 6 V). The measurements are performed until the onset of nonlinear regime. Fig. 7-B shows the frequency tuning effect as the applied dc voltage increases for two different resonators, using two different ac voltages. By fitting the experimental data with eq. (24), the linear mechanical spring constant of each resonator is found. In the case of this fit, the effective mass was used as a fixed parameter, since there is a well-known analytical expression of the fundamental resonant mode shape of the flexural resonators [30], which, after integration, enables to determine that

mr  0.4  Lwh [23]. In cases where the mode shape is not known, this strategy can be very useful to extract the effective mass of the mode, together with the linear mechanical spring constant. Eq. (24) proved to be highly sensitive to variation on the geometry of the resonator and the parameter  was used in the fit of eq. (24) to the experimental results.

3.2. Nonlinear behavior

3.2.1. Extraction of cubic mechanical spring constant, k3m Increasing either the dc or ac voltages, Vdc or Vac, will increase the electrostatic force and drive the resonator into high amplitude motion and eventually into nonlinear regime. MEMS resonators exhibit mechanical and electrical nonlinearities. Mechanical nonlinearities of the resonators arise both from geometrical effects (area and volume change of the resonator), and material effects (nonlinear stress-strain relation for high amplitude deformations). Since both parameters are determined by the intrinsic device properties, mechanical nonlinearities can differ significantly for different types of resonators. Electrical nonlinearities are induced by the nonlinear electrostatic force. As it was seen in the previous section, increasing Vdc magnifies capacitive nonlinear effects. Therefore, to decouple the effect of mechanical and electrical nonlinearities it is common to fix the applied dc voltage, Vdc, and increase the ac voltage, Vac.

When the capacitive electrostatic forces are weak, due, for example, to large actuation gaps, increasing the amplitude of motion of the resonator by increasing the actuation voltage will likely cause a mechanical stiffening effect. This causes the resonance peak to shift and bend towards higher frequencies, as it was seen in the inset of Fig. 6. When the nonlinearities in the electrostatic forces are dominant, for example due to a very narrow gap, increasing the actuation voltages will cause an electrically induced softening effect, which causes the peak to shift and bend towards low frequencies, as observed in Fig. 7-A. The interaction between the opposing mechanical stiffening and electrical softening determines the behavior of the beam in the nonlinear regime [40]. Regardless of which mechanism causes the nonlinearities, the power handling capacity of MEMS devices is limited by these nonlinear effects. Often, it is important to investigate the maximum usable drive voltage of the microresonator to avoid damaging the device or to understand how much energy it can store [40]. The existence of nonlinearities on the resonator induces a shift on its resonance frequency that depends on the square of the amplitude of vibration. This relation was derived by Landau [35], [41]:

0'  0    KX 02 , 0 0

(25)

3k3 5k22  . 8k1 12k12

(26)

where K 

' In eq. (25), 0 is the natural angular frequency of the resonator, while 0 is the resonance frequency of

the resonator in nonlinear regime. X 0 is the absolute amplitude of vibration and K is a proportionality factor that contains the first, quadratic and cubic spring constant coefficients, k1  k1m  k1e ,

k2  k2m  k2e and k3  k3m  k3e . For symmetric structures, the quadratic coefficient can be neglected and K reduces to the first term. The motional current that flows through the device can be used as a good indicator of the amplitude of vibration of the resonator. Therefore, an equivalent expression can be written [38], [40]:

0'  0     im2 , 0 0

(27)

Where im is the motional current that flows in the device at resonance frequency f 0' and  is a proportionality constant. Relating eqs. (25) and (27),  can be expressed as:



KX 02 . im2

(28)

The term of the motional current shown in eq. (4) is given by im  t   Vdc

C  x  x x

t

. Expanding the

derivative of the capacitance as showed in eq. (23), the motional current can be written as: 2  x  x t   x t  Vdc 0 A    , im  t   1  2  3     t g2  g g    

(29)

and can be reduced to the fundamental term if the displacements of the center of the beam are small when compared with the gap distance, x  t 

g . This assumption is supported with the good agreement

between eqs. (13)-(15) and the experimental results presented in Figs. 4 and 5, which shows that the resonators have a small displacement when enter in the nonlinear regime of motion. If this were not the case, the motional current would not be a proper indicator of the amplitude of motion of the beam, and an alternative measurement method should be used. Finally, eq. (29) can be written:

im  t   Vdc

 0 A x g 2 t

.

(30)

Replacing eqs. (26) and (30) into eq. (28) one obtains:

3  k3m  k3e   g 2   t  2      X0 . 8  k1m  k1e   Vdc 0 A   x  2

2

(31)

 2  , and that k   2 m ,  t  Considering that at the resonance frequency   X 02  1m 0 r 02  x  2

2

k1e  



0 A g

3

Vdc 2 and k3e   3k3m g 4  2 

2 0 A 2 Vdc , eq. (31) can be expanded to: g5 6  2 

2

   A 2 8  mr02  0 3 Vdc 2  Vdc 0 A g  



2 0

2

   A 8  mr02  0 3 Vdc 2  g 0 A g  

.

(32)

2 0

The first and second terms of  are attributed to the cubic mechanical and electrostatic spring constant coefficients, respectively. The second term is always negative and contributes to decrease the resonance frequency, whereas the first term is either positive or negative, depending on the signal of k3m, fixed upon fabrication. A positive  means that the resonance frequency increases with amplitude of vibration, showing the hard spring effect, whereas a negative  shows the spring softening behavior. When the two terms are equal in magnitude, the nonlinearities cancel and this condition leads to a maximum amplitude of motion (or motional current) of the resonator [40]. If the dc actuation voltage is fixed, both terms of eq. (32) are constant and k3m is the only unknown. Fig. 8 illustrates the procedure followed to extract the cubic mechanical spring constant term, k3m. Fig. 8-A shows the measured transmission spectra of a resonator driven into nonlinear regime by increasing Vac, while keeping Vdc constant at 6 V. The red arrow shows the shift in the resonance frequency of the resonance spectra when the ac actuation is increased, f 0' , and the natural frequency of the resonator, f 0 , measured in the limit of low actuation voltages. The corresponding motional current is estimated from the admittance spectra (after subtracting the parasitic current, or the baseline obtained for Vdc = 0), using Ohm’s law, and plotted against the shift in resonance frequencies in Fig. 8-B. Eq. (27) is then fitted to the experimental results and k3m is directly calculated from the slope  using eq. (32) [38], [39]. The value of the parameter  is -1.78 x 107 Hz/ m2, as indicated in Fig. 8-B, from where a value of k3m  2.49 x1014 N / m3 is calculated. Fig. 8-B shows also the same procedure for the case where Vdc = 7 V (these spectra are not shown in Fig. 8-A), to illustrate the fact that the value of

k3m calculated should be similar and independent of the experimental conditions. Using the purely geometrical approximation proposed by Kaajakari in [24], k3m  5.30 x1013 N / m3 . For this set of parameters, the cubic electrostatic spring constant component is

k3e  

2 0 A 2 Vdc  3.39 x1014 N / m3 . It can then be concluded that both the electrical and 5 g

mechanical terms have similar magnitudes and both contribute evenly to the cubic spring constant of the system. As a final note, the apparent truncation of the peak shown in Fig. 8-A can be the result of a peak with double hysteresis. To explain this phenomenon, higher terms of nonlinearities should be considered in the equation of motion, which would give rise to several possible solutions for the amplitude of the beam, instead of the three solutions characteristic of the Duffing equation [33], [35].

4. CONCLUSIONS This work presents the detailed electrical characterization of flexural resonators made of amorphous hydrogenated silicon. The resonators were fabricated on glass substrates with a maximum processing temperature of 175 ºC and with transduction gaps heights ranging from 250 nm to 600 nm. The resonators are electrostatically actuated and their motion is capacitively sensed. The resonant motion of the mechanical structure in the linear regime is modeled by an RLC circuit with a parasitic parallel capacitance. The experimental results for the equivalent electrical motional parameters agree with the theoretical values for low actuation voltages and a minimum motional resistance, Rm, of ~85 k was measured for a narrow gap resonator (250 nm). A hysteresis criterion for microbridges that show mechanical stiffening, based on the design of the resonator and on the properties of the material, is presented and applied to the experimental results. It is shown that this criterion accurately predicts the onset of hysteresis in the motion of the resonator and can be used to define design criteria to minimize motional resistance and to avoid nonlinear effects. A short, wide and thick structure, with low quality factor is preferred in order to achieve low motional resistances if the resonator can be excited by an arbitrarily high dc voltage. A strategy to extract the linear mechanical spring constant and effective mass of a resonator working in linear regime is discussed and applied to the experimental results. At high applied dc voltages, the resonators enter a nonlinear regime, either through mechanical stiffening or electrical softening effects. To maximize the current flowing through the device it is shown that is possible to balance the electrical and mechanical contributions of the spring constant of the system. A strategy to extract the cubic mechanical spring constant of a resonator

working in nonlinear regime is discussed and applied to the experimental results. Acknowledgments J. Mouro acknowledges the Fundação para a Ciência e Tecnologia (FCT) for a PhD grant SFRH/ BD/ 73698/ 2010. The work was supported by FCT through the research unit IN - Institute of Nanoscience and Nanotechnology and the research project EXCL/CTM-NAN/0441/2012. Biographies João Mouro received the Masters degree in Chemical Engineering from Instituto Superior Técnico, Lisbon, Portugal, in 2010. He is currently attending to the PhD degree program of Technological Physics Engineering in Instituto Superior Técnico and is affiliated with INESC-MN Research Center. His PhD dissertation is on fabrication, characterization, modeling and application of thin-film NEMS/ MEMS. His current interests include design, fabrication and modeling of thin-film MEMS resonators. Laura Teagno received her Master Degree in Electronic Engineering from the Polytechnic of Turin, Italy, in 2014. Her thesis topic was related to the research of sub-micron gaps thin-film amorphous silicon microelectromechanical resonators and was developed at INESC Microsystems and Nanotechnologies (INESC-MN), of Lisbon, in particular with the collaboration of the Thin-film MEMS group. She is currently working at INESC-MN on the electronic detection of microelectromechanical resonators. Alexandra Gualdino received the Licenciatura degree in technological physics engineering from Instituto Superior Técnico, Lisbon, Portugal, in 2004. She is currently pursuing a PhD degree at Department of Materials Engineering of Instituto Superior Técnico and is affiliated with the INESC-MN Research Center. Her PhD dissertation is on the development of low-temperature thin-film resonators and her research interests include design, fabrication and characterization of RF-MEMS, thin-film mechanical characterization and nanoscale fabrication. Virginia Chu received her Ph.D. in Electrical Engineering in 1989 from Princeton University. Following her degree, she was a post-doctoral fellow at LPICM, Ecole Polytechnique, Palaiseau, France. Since 1990, she has been part of the Research Staff of INESC MN where she is a coDirector and co-responsible of the MEMS and BioMEMS research group. Her present research interests include thin film microelectromechanical systems (MEMS) for sensors and actuators, large area electronics and integration of thin film technology to biological applications. João Pedro Conde received his Ph.D in Electrical Engineering from Princeton University in 1989. His thesis topic involved the study of the optoelectronic properties of amorphous silicon-silicon germanium multilayers. Between 1989 and 1990 he was an IBM postdoctoral fellow at Yorktown Heights, where he developed a low temperature process for the deposition of amorphous silicon for which a patent was awarded. Since 1990 he has been at the Instituto Superior Tecnico where he is presently a full professor in the Department of Bioengineering. He is a co-responsible for the Thin Film MEMS and BioMEMS research group of INESC Microsystems and Nanotechnologies. His current research interests include novel thin film devices such as thin film MEMS and sensors, low temperature deposition of thin film semiconductors and electronic devices and micro- and nanotechnologies applied to Lab-on-aChip devices. References [1] A. Gualdino, J. Gaspar, V. Chu and J. P. Conde,;1; ``Sub-micron gap in-plane micromechanical resonators based on low-temperature amorphous silicon thin-films on glass substrates'', J. Micromech. Microeng. vol. 25, pp. 075026, 2015.

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Fig. 1. SEM micrographs of fabricated flexural resonators with gaps of 250 nm and 400 nm.

Fig. 2. A) Experimental setup for one-port capacitive measurement. B) Electrical equivalent of the experimental measurement scheme. The impedance of the device under test, ZDUT, refers to the electrical impedance of the resonating device. C) Electrical equivalent circuit to describe the mechanical motion of the device. The ac input signal has the same frequency as the detected output current, 𝜔, and the electrical path between these signals is represented by the feedthrough capacitor. The series RLC represents the mechanical motion of the resonator.

Fig. 3. A) Experimental transmission curve of a resonator with L = 100 µm, g = 400 nm, w = 40 µm, t = 1.1 µm, and f0 = 0.776 MHz, measured using the setup shown in Fig. 2. B) Experimental admittance curve (open symbols) and fit of the admittance of the electrical equivalent circuit (solid lines). The parameters extracted from the fit are shown in the legend of the figure. The motional resistance and inductance are calculated from eqs. (8) and (9).

Fig. 4. Motional resistance, Rm, as function of the applied dc voltage, Vdc, for four different resonators with actuation gaps of 250, 400 and 600 nm. Dashed-dotted lines represent the theoretical motional resistance, calculated from the mechanical domain using eq. (13). The symbols represent the experimental motional resistance, determined from fitting the admittance of the electrical equivalent circuit, given by eq. (6), to the experimental data. The measurements were performed until the onset of nonlinear regime of motion was observed, where the electrical equivalent model is no longer valid. The quality factors indicated were used to fit eq. (13) and correspond to the values measured for the maximum Vdc used. The thin-solid orange line is a fit of a more rigorous model presented in [25].

Fig. 5. Electrical motional parameters of different resonators as function of the applied dc voltage. A) Motional capacitance; B) Motional inductance. The dashed-dotted lines represent eqs. (14) and (15) and the symbols are the values of the parameters extracted from fitting the electrical model of eq. (6) to the experimental admittance curve. Thin-solid orange lines are fits of the more rigorous model presented in [25].

Fig. 6. Hysteresis criterion given by eq. (17) (green line) applied to a resonator with L = 150 µm and g = 600 nm, showing mechanical stiffening. On the left side, the resonator works in linear regime and the electrical model fits the experimental admittance and phase curves. The electrical model, and consequently the value of the motional resistance, Rm, is not valid when hysteresis occurs.

Fig. 7. A) Measured transmission spectra of a resonator with L = 80 µm, w = 25 µm, t = 2 µm and g = 250 nm actuated with fixed Vac = 0.1 V and variable Vdc in the linear regime, showing the resonance frequency tuning as the applied dc voltage increases. B) Fits of eq. (24) to the experimental results of two distinct resonators, from where the linear mechanical spring constant parameter, k1m, is extracted.

Fig. 8. A) Measured transmission spectra of a resonator with L = 100 µm, w = 40 µm, t = 1.1 µm and g = 400 nm actuated with fixed Vdc = 6 V and variable Vac (from 0.1 V to 0.3 V) in nonlinear regime. The red arrow indicates the resonance frequencies, f0’, of the measured spectra, while the dotted spectrum shows the natural frequency of the resonator, f0, for unloaded conditions (very small actuation voltages). B) Fits of the resonance frequency shift of the measured spectra against the squared motional current at resonance, from where the slope 𝛼 is extracted (eq. (27)) and k3m calculated using eq. (32). The circles represent the case shown in A), using a fixed voltage of Vdc = 6 V. The stars show the same procedure for spectra measured using a dc voltage of 7 V, which allows obtaining a similar value for k3m. TDENDOFDOCTD