A review on in situ stiffness adjustment methods in MEMS

Two capacitive plates, separated by gap g₀, a potential difference V and an overlapping surface $A=\left({{a}_{0}}+x\right){{b}_{0}}$, are shown in figure 2. The bottom plate is fixed, and the top plate can move in both x and y directions. The capacitance between two electrodes is defined as:

$$\begin{eqnarray}&&C=\frac{\epsilon \left({{a}_{0}}+x\right){{b}_{0}}}{{{g}_{0}}-y}.\end{eqnarray} \tag{ 4 }$$

Zoom In Zoom Out Reset image size

The amount of energy U, stored between capacitive plates can be calculated with:

$$\begin{eqnarray}&&U=\frac{1}{2}C{{V}^{2}}.\end{eqnarray} \tag{ 5 }$$

The force on the top plate can be found by taking the derivative of the energy with respect to the direction of interest. The stiffness is found by taking the derivative of this force.

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{F}_{\text{x}}} & =\frac{\partial U(x)}{\partial x}{{|}_{y}}=\frac{1}{2}\frac{\epsilon {{b}_{0}}}{{{g}_{0}}}{{V}^{2}} \end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{K}_{\text{x}}}=\frac{\partial {{F}_{x}}(x)}{\partial x}=0 \end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{F}_{\text{y}}}(\,y) & = & \frac{\partial U(\,y)}{\partial y}{{|}_{x}}= & \frac{1}{2}\frac{\epsilon {{a}_{0}}{{b}_{0}}}{{{\left({{g}_{0}}-y\right)}^{2}}}{{V}^{2}} \end{array}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{K}_{\text{y}}}(\,y) & =\frac{\partial {{F}_{y}}(\,y)}{\partial y}=\frac{\epsilon {{a}_{0}}{{b}_{0}}}{{{\left({{g}_{0}}-y\right)}^{3}}}{{V}^{2}} \end{array}\end{eqnarray} \tag{ 9 }$$

When a voltage is applied to the system, the parallel plates will move such that there is an equilibrium between the electrostatic attraction and mechanical restoring force. Around this new equilibrium position the stiffness is determined by sum of the mechanical and the electrostatic stiffness. The two capacitive plates have a (non-linear) stiffness in the y-direction, but have zero stiffness in x-direction. This electrostatic stiffness in the y-direction is used in parallel plate devices. Because the electrostatic stiffness is in the direction opposite that of the mechanical restoring force it is commonly called a 'negative'-stiffness. (For a positive displacement in y, the electrostatic force increases positively, while the mechanical restoring force increases negatively).

The amounts of electrostatic force and stiffness depend on the surface area, as shown in equations (6)–(9). So in order to have a significant effect without the need to further increase the voltage, the surface of the electrodes is often increased by applying a comb-like design as shown in figure 3.

Zoom In Zoom Out Reset image size

Parallel plate electrostatic tuning was reported by Adams et al [6]. The device is shown in figure 4. A tuning voltage is applied between the stationary and moving plates of the capacitor. With this design, a device with a resonance frequency of 21 kHz was reduced to 7.7% of its nominal value. The estimated mass of the system is $1.0\times {{10}^{-9}}$ kg. So the mechanical stiffness of the system can be calculated by using equation (1). This gives a stiffness of 17.41 N m⁻¹. The measured electrostatic stiffness coefficient $k_{e}^{0}$ is $-7.9\times {{10}^{-3}}$ N m⁻¹ V⁻², so for a tuning voltage of approximately 46 V (estimated from figure 5), this gives an electrostatic stiffness of:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{k}_{\text{elec}}} & =k_{e}^{0}\centerdot {{V}^{2}} \\ {} & =-16.59~\text{N} ~{{\text{m}}^{-1}}. \end{array}\end{eqnarray} \tag{ 10 }$$

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

More devices were presented in this paper and are discussed later in this section, in sections 3.1.2 and 3.1.3. Each device is separately mentioned in table 2. The parallel plate design is marked with (a). The measurement results for these four devices can be found in figure 5.

A similar device was presented by Horsley et al [7]. A parallel plate comb drive adds an electrostatic stiffness of $-9.4$ N m⁻¹ to the mechanical suspension of 29 N m⁻¹. Parallel plate tuning was also applied by Torun et al [8], to tune the stiffness of a membrane that is used for force spectroscopy applications. The stiffness was reduced from 24.4 N m⁻¹ to 11 N m⁻¹.

The parallel plate tuning mechanism of figure 4 has a limited range of motion. When the gap between the stationary and moving electrodes becomes too small, the electrostatic attraction forces will become larger than the mechanical restoring force. The plates will collide and destroy the device. In order to increase the range of motion, Adams et al [6] applied parallel plate tuning with a branched finger design instead of having straight fingers, as shown in figure 6. When the position of the mass changes, the suspended combs move into the entrance region of the stationary combs. In order to achieve a linear behavior in positive and negative moving directions, a symmetric setup is used, such that the comb fingers will penetrate an entrance region in both directions of motion. The range of motion is increased from 0.8 μm to 1.3 μm compared to the previous design, while the efficiency remains approximately equal. The untuned resonance frequency is 7.5 kHz and the mass is $2.3\times {{10}^{-9}}$ kg. So the mechanical stiffness of the system is 5.1 N m⁻¹ according to equation (1). The experimental electrostatic stiffness is $-6.3\times {{10}^{-3}}$ N m⁻¹ V⁻², resulting in a change in stiffness of $-3.2$ N m⁻¹ for 500 V⁻² which is estimated from figure 5. A similar design was presented by Park et al [9]. The untuned and tuned resonance frequencies are 5.12 kHz and 0.950 kHz respectively. The theoretical proof mass weight is 1μg, so this yields an untuned stiffness of 1.03 N m⁻¹ and a tuned stiffness of 0.04 N m⁻¹ for a tuning voltage of 10 V.

Zoom In Zoom Out Reset image size

Parallel plate tuning has been applied in a MEMS gyroscope by Sonmezglu et al [26]. The gyroscope has two vibrating modes (in x- and y-direction, in plane) and the resonance frequencies of these modes should be as close as possible. But due to manufacturing errors or temperature influences, there is usually a mismatch between these resonance frequencies. So by tuning the stiffness in one of these modes, the frequency can be tuned, such that it matches the other mode. A normalized change in stiffness of 1.30 is achieved, according to equation (3).

Parallel plate stiffness tuning is also applied without the commonly used comb structures. Yao et al [27] and Evoy et al [28] used flat electrodes under a suspended out-of-plane resonator to apply electrostatic softening. Zhao et al [29] developed a micromirror that has two torsional modes and one translational mode of which the stiffness can be reduced with electrostatic softening. The device is shown in figure 7. The stiffness in z-direction is 0.1338 N m⁻¹ and has a corresponding resonance frequency of 1150 Hz. This can be reduced to approximately 1000 Hz, which corresponds to 0.10 N m⁻¹, according to equation (3). The torsional stiffness of the device can be tuned using the same tuning electrodes.

Zoom In Zoom Out Reset image size

Nano-resonators.

Nano resonators are very small MEMS devices that often operate in the order of megahertz (MHz), or even gigahertz (GHz). These devices are being applied, for example, in the field of ultra-sensitive mass sensing [30–32] and radio frequency signal processing [33]. The resonators are usually doubly clamped structures, as shown in figure 8. The resonating structure can either be a micro-fabricated beam or an even smaller structure such as a carbon nanotube or graphene flake. Some of these nano resonators also have stiffness tuning capabilities. The common tuning method for this class of devices is parallel plate tuning. But instead of having dedicated structures that cover the stiffness tuning as discussed in the previous paragraph, the resonator itself serves as tuning electrode. The substrate of the device is commonly used as the tuning electrode. Just like the devices discussed in section 3.1.1, an electrostatic (negative) stiffness can be added to the system. One device is discussed as an example for these type of devices. Other devices that use the same tuning method were reported by Schwab et al [11], Kwon et al [34], Lopez et al [35], Yan et al [36], Stiller et al [37] and Wu et al [38].

Zoom In Zoom Out Reset image size

Piazza et al [39] reported on quality factor enhancement and capacitive fine tuning of resonators. The device is intended as a high quality factor MEMS resonator for on-chip filtering and frequency reference. The resonance frequency of the device can be tuned by applying a voltage between the substrate and the device layer. The ZnO layer is a piezoelectric layer. By applying a voltage between the drive electrode and the device layer, a vibration can be induced. The vibration of the device can be sensed by measuring the voltage on the sense electrode. The configuration of the device is shown in figure 8. The untuned frequency of the device is 719 kHz and for a tuning voltage of 20 V the resonance frequency decreases with 6 kHz. There is insufficient data available to determine the stiffness of the system.

Some of these nano resonators have the capability to tune the stiffness both positively as negatively in the same device. However, the positive stiffness tuning is for the resonance mode perpendicular to the mode of the negative stiffness tuning. An electrode is placed along the side of the resonator, in the same plane. When a voltage is applied between this electrode and the resonator, a negative stiffness is added to the in-plane motion. This voltage also has the effect that the resonator is pulled into the direction of the electrode, due to the electrostatic forces. This force induces a tensile stress in the resonator. When the beam vibrates in the out-of-plane mode, this tensile stress causes a stiffening effect. This effect is explained in section 3.4. This method of both positive and negative stiffness tuning has been applied by Fung et al [40], Pandey et al [41], Rieger et al [42], Solanki et al [43], Fardindoost et al [44] and Kozinsky et al [45]. The last of these is discussed in more detail as an example.

Kozinsky et al [45] presented a device with an electrostatic mechanism to tune the nonlinearity of a resonator, increase the dynamical range and tune the resonance frequency. A theoretical model was developed that can serve as a design guideline, and a device was fabricated. This section only goes into the details of the frequency tuning capabilities. The device consists of a clamped-clamped beam with dimensions of $150~\text{nm}\times 100~\text{nm}\times 15$ μm. A gate electrode is placed 400 nm from this beam and covers almost the entire length as shown in figure 9. This electrode can apply a DC bias voltage and an AC actuation voltage. The resonance frequencies of both in-plane and out-of-plane motion were measured. When the gate electrode exerts a DC voltage to the resonator, both the in-plane and out-of-plane frequencies change. Two different mechanisms are responsible for the change in stiffness. For the in-plane resonating mode the resonance frequency will decrease, because the electrostatic force will add a negative stiffness. The untuned resonance frequency of the in-plane motion is 8.78 MHz and can decrease by 6% for a tuning voltage of approximately 28 V (estimated from figure 9). This corresponds to an increase of 12% in stiffness according to equation (3). For the out of plane mode the resonance frequency will increase as a result of induced stress in the beam. The gate electrode can exert a force on the resonator, resulting in tensile stress in the beam. The untuned resonance frequency is 7.60 MHz and for a tuning voltage of approximately 30 V the resonance frequency increases by 4%. This corresponds to an increase of 8% in stiffness according to equation (3).

Zoom In Zoom Out Reset image size

The concepts in this section are similar to the parallel plate devices of section 3.1.1. The comb structure is commonly applied and the stiffness is added by applying a voltage between a moving and a stationary electrode. But the direction of motion is perpendicular compared to the parallel plate devices (x-direction instead of y, for figure 2). If we look at equation (7), it was shown that no electrode stiffness can be added in this direction for two flat, parallel electrodes. However, by having a gap g(x) that is a function of the displacement in x, a stiffness can be added to the system. This is illustrated in figure 10. The equation governing the stiffness can be derived as:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \text{d}C=\frac{\epsilon \text{d}A}{g(x)}=\frac{\epsilon {{b}_{0}}\text{d}x}{g(x)} \end{array}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} C=\epsilon {{b}_{0}}{\int}_{0}^{{{a}_{0}}+x}\frac{\text{d}x}{g(x)} \end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{F}_{\text{x}}}(x) & =\frac{\partial U(x)}{\partial x}{{|}_{y}}=\frac{\epsilon {{b}_{0}}{{V}^{2}}}{2}\frac{\partial}{\partial x}{\int}_{0}^{{{a}_{0}}+x}\frac{\text{d}x}{g(x)}=\frac{\epsilon {{b}_{0}}{{V}^{2}}}{2g\left({{a}_{0}}+x\right)} \end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{K}_{\text{x}}}(x) & =\frac{\partial {{F}_{\text{x}}}(x)}{\partial x}=\frac{\epsilon {{b}_{0}}{{V}^{2}}}{2}\frac{\partial g\left({{a}_{0}}+x\right)}{\partial x} \end{array}.\end{eqnarray} \tag{ 14 }$$

Zoom In Zoom Out Reset image size

Because the capacitance now is a function of the motion in x-direction, the derivative of the force term does not become zero and yield a non-zero stiffness. By tailoring the gap shape, the force-deflection relation (stiffness) can be chosen by design.

This method was applied to a number of devices. The first notion of a variable gap comb drive was in the context of actuators [46–48]. Later this was applied to stiffness tuning mechanisms. Jensen et al [14] applied the method to a comb-finger structure. Seven shapes of comb fingers were designed, of which two were fabricated and tested. The shape of the fingers was different, but applied in the same device. One of the fabricated designs has a shape such that the system becomes stiffer ((a) in table 2) under an applied voltage, the other fabricated one becomes more pliant ((b)) in table 2). The schematic drawings of the stiffening (a) and weakening (b) comb shapes can be seen in figure 11. When the gap becomes smaller for a displacement x, $\frac{\partial g(x)}{\partial x}$ is negative, resulting in a negative stiffness. The opposite applies to gaps that become larger for a displacement x. The stiffness was tuned from 0.47 N m⁻¹ to 0 N m⁻¹ with the weakening fingers for 80 V and to 0.75 N m⁻¹ with the stiffening fingers for 90 V.

Zoom In Zoom Out Reset image size

The same concept was applied by Engelen et al [12], who developed a musical instrument using MEMS technology with stiffness tuning capabilities. The device consists of several similar resonators, that differ in mass and spring constant and thus (untuned) resonance frequency. Because the softest device has the largest normalized change in stiffness, this device is included in table 2 and figure 1. The theoretical stiffness is 0.6 N m⁻¹. The resonance frequency of this device decreases approximately 5% for a tuning voltage of 29 V, so the normalized change in stiffness is 1.11 according to equation (3).

The device of Lee et al [15] applies the same method. The nominal stiffness of the device is 2.64 N m⁻¹. By applying a tuning voltage up to 150 V, this stiffness can be decreased by 80% to 0.53 N m⁻¹.

Varying shape electrodes were similarly applied by Guo et al [13]. The untuned stiffness is 17.3 N m⁻¹ and the linear electrostatic stiffness is $-2.8\times {{10}^{-4}}$ N m⁻¹ V⁻². For a tuning voltage of 60 V this results in a change of $-1$ N m⁻¹.

In equation (7) it was shown that it was not possible to generate a stiffness in the x-direction for flat parallel plates. In that case, the second derivative of the energy U(x) with respect to x will be equal to zero (so there is no electrostatic stiffness). But when a comb drive is designed such that the overlapping surface A has a higher order dependency on the displacement (xⁿ with $n\geqslant 2$), the second derivative of the energy with respect to x will not be equal to zero. There will be an electrostatic stiffness in the x-direction. The equations for the capacitance, electrostatic force and stiffness yield:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} C(x)=\frac{\epsilon A(x)}{{{g}_{0}}} \end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{F}_{\text{x}}}(x) & =\frac{\partial U(x)}{\partial x}{{|}_{y}}=\frac{\epsilon {{V}^{2}}}{2{{g}_{0}}}\frac{\partial}{\partial x}A(x) \end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{k}_{\text{x}}}(x)=\frac{\partial {{F}_{\text{x}}}(x)}{\partial x} \end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} =\frac{\epsilon {{V}^{2}}}{2{{g}_{0}}}\frac{{{\partial}^{2}}}{\partial {{x}^{2}}}A(x) \end{array}.\end{eqnarray} \tag{ 18 }$$

So if ${{\partial}^{2}}A(x)/\partial {{x}^{2}}\ne 0$, there is an electrostatic stiffness in the x-direction. This non-linearly varying overlapping electrode surface can be achieved by using a comb drive with fingers of varying length. This concept was first applied by Lee et al [18]. Later, Dai et al [16], Scheibner et al [20, 49], Shmulevich et al [50] and Kao et al [17] used the same method. This type of tuning comb is shown in figures 12 and 13. The electrostatic stiffness ${{k}_{\text{el}}}$ as presented by Kao et al can be calculated as:

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{k}_{\text{el}}} & =\frac{N\epsilon H{{t}_{h}}(b+x)}{2Bp\text{d}x}{{V}^{2}} \end{array}\end{eqnarray} \tag{ 19 }$$

with N the number of combs, the permittivity, t_h the thickness of the device, and H, b, B, p and d geometric properties that can be found in figure 12. A device with a theoretical untuned stiffness of 17 N m⁻¹ was fabricated. Under a tuning voltage of 30 V the stiffness increased with 21% to 20.6 N m⁻¹.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Scheibner et al [20, 49] used varying length comb fingers in a 'wide range tunable resonator'. The purpose of the device is to recognize the wear state in machinery, by checking the mechanical vibrations. It consists of an array of eight cells, such that a total range of 1 kHz– 10 kHz is covered. Each cell in the array has a fixed base frequency, which can be decreased by applying a tuning voltage. The tuning ranges of each subsequent cell are overlapping, so that no gap will occur in the total bandwidth. In table 2 cell one (a) and eight (b) are included as separate devices. Cell one has a theoretical untuned stiffness of 2.09 N m⁻¹, the theoretical stiffness of cell eight is 17.59 N m⁻¹. To approximate the tuned stiffness of the two cells the ratio between the measured tuned and theoretical untuned resonance frequencies is used (as described in equation (3)). This results in a tuned stiffness of 0.2 N m⁻¹ and 14.7 N m⁻¹ for cell one and eight respectively. These two extremes are included in table 2 and figure 1 as (a) and (b) respectively.

Dai et al [16] presented a similar device with a (theoretical) untuned stiffness of 0.76 N m⁻¹. For a tuning voltage of 40 V, the stiffness drops to 0.4 N m⁻¹. Shmulevich et al [50] achieved a broad linear range of stiffness tuning with a similar device. The stiffness of the device was not presented, but the resonance frequency was changed from 957 Hz–173 Hz for a tuning voltage of 81 V, resulting in a normalized change in stiffness of 30.6 according to equation (3). Lee et al [18], who were the first to develop this tuning method, achieved tuning from 0.3 N m⁻¹ to 0.28 N m⁻¹.

A different approach was chosen by Morgan et al [19]. Instead of having varying comb finger lengths, the vertical dimension of the comb fingers was varied. This is illustrated in figure 14. These structures could be manufactured due to 'gray-scale' technology, which allows the fabrication of varying height structures with a single lithography and dry etch step. Both positive and negative tuning can be achieved, by designing the combs such that the overlapping surface decreases or increases respectively. Several designs were presented and those with the largest positive (a) and negative (b) tuning range were added to table 2. The largest positive change in stiffness is from 6.1 N m⁻¹ to 7.55 N m⁻¹ for 120 V and the largest negative stiffness tuning was from 4.3 N m⁻¹ to 3.3 N m⁻¹ for 90 V.

Zoom In Zoom Out Reset image size

A varying overlapping electrode surface was applied to a MEMS gyroscope by Hu et al [51]. A proof-mass is brought into resonance in the x-direction. The resonance frequency in the y-direction should be the same as in the x-direction, such that a rotation of the device results in a transfer of energy from the actuated resonance mode (x-direction) into the sensing direction (y-direction) due to the Coriolis effect. But due to manufacturing errors the resonance frequencies of the in-plane modes do not match. Stiffness tuning is used to change the resonance frequency in the y-direction such that it matches the frequency of the mode in the x-direction. The tuning electrodes are placed below the electrodes that are connected to the proof mass. The tuning electrodes are triangularly shaped, as shown in figure 15, such that the overlapping surface changes non-linearly when the top electrode moves in the y-direction. The stiffness is determined by the shape of the electrodes. The resonance frequency is tuned from 1.984 kHz to 2.005 kHz for a tuning voltage of 17.5 V.

Zoom In Zoom Out Reset image size

A varying overlapping electrode surface was applied in an angular vertical comb drive by Eun et al [52]. A torsional micro mirror is suspended by two torsional springs. Perpendicular to the springs a comb structure is applied, which overlaps with stationary electrodes. When the micro mirror rotates, the overlapping surface between the moving and stationary electrodes changes non-linearly. This adds a torsional stiffness to the system. The architecture of the device can be found in figure 16. The unmodified resonance frequency of 3176 Hz was tuned to 3066 Hz for a tuning voltage of 30 V.

Zoom In Zoom Out Reset image size

The previous electrostatic tuning mechanisms of sections 3.1.1–3.1.3 were based on overlapping comb fingers. It is possible though to use non-interdigitated comb fingers for stiffness tuning. These comb fingers do not overlap and use fringe fields to exert a displacement-dependent force on each other. Due to the complexity of these fringe fields, it is not possible to derive an analytical equation for the electrostatic stiffness. Finite element modeling is needed to find this electrostatic stiffness. Both positive and negative stiffness tuning can be achieved, depending on the initial alignment of the fingers.

This method of stiffness tuning was applied by Adams et al [6, 21]. The device is shown in figure 17. For these transverse non-overlapping comb drives it is possible to have either a positive or a negative electrostatic stiffness, depending on the initial alignment of the comb fingers. When the initial alignment of the fingers is such that a moving finger is in between two stationary fingers, the restoring force will decrease with a displacement. This adds a negative stiffness to the system (denoted as (b) in table and graph). When the fingers of the stationary and moving parts are aligned, a relative motion will cause a restoring force. This is a positive electrostatic stiffness (c). Both these configurations are applied in a similar device; only the initial alignment is different. These initial configurations are illustrated in figure 18. This device has also been used by Zhang et al [53] to research nonlinearity effects on an auto-parametric amplification. The theoretical untuned stiffness of the mechanical mechanism is 2.6 N m⁻¹. The experimental electrostatic stiffnesses are $-0.84\times {{10}^{-3}}$ N m⁻¹ V⁻² and $1.0\times {{10}^{-3}}$ N m⁻¹ V⁻² for the reduction and augmentation system respectively. The tuning voltage squared is estimated from figure 5 to be 3200 V⁻² and 5200 V⁻² respectively, resulting in a change of stiffness of $-2.69$ N m⁻¹ and 5.2 N m⁻¹. DeMartini et al [54] applied the same tuning method for both positive and negative stiffness tuning, but insufficient data was presented to determine the change in stiffness. The device was based on earlier work of Rhoads et al [55].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Kafumbe et al [56] used the pull-in of a cantilever beam to change the effective length and tune the stiffness of the system. The configuration of the design is shown in figure 20. An electrode is placed close to a cantilever. By applying a voltage between the cantilever and electrode, the cantilever will start bending toward the electrode. This corresponds to state 1 in figure 20. At a critical point, the cantilever will snap in towards the electrode and a dielectric layer on top of the electrode prevents a short circuit. The snap in occurs when the electrostatic forces exceed the mechanical restoring forces of the cantilever. The cantilever is now in a stable clamped-pinned state (state 2). In this state the stiffness decreases for an increasing voltage. This results in a second unstable state after which the cantilever will move into the clamped-clamped configuration, which corresponds with state 3 in figure 20. If the voltage is further increased, the contact area between the cantilever and insulative layer will start to increase. This results in a change in effective length, so that the stiffness is changed. When the tuning voltage is decreased once the cantilever is snapped-in, there will be a hysteresis in the system, due to stiction between the cantilever and dielectric layer. Several mechanisms are responsible for the change in stiffness: adding electrostatic stiffness (states 1 and 2), change of boundary conditions (from state 1 to state 2 and state 2 to state 3) and change in effective length (state 3). The device has a long, linear operational range in state 3. The device is intended to work in this state. The change in normalized resonance frequency for the applied normalized voltage for the different stages is shown in figure 21.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Another device that uses the principle of length change for stiffness adjustment is presented by Zalalutdinov et al [57]. A Scanning Tunneling Microscope (STM) is used to locally actuate and constrain a silicon nitride cantilever. The vertical motion of the STM is actuated by an AC voltage signal on the piezo drive, which is used to drive the cantilever. The force between the STM tip and cantilever provides an actuation and motion constraint. By changing the position where the STM tip engages the cantilever, the effective length is changed. This results in a stiffness and resonance frequency change. A change in resonance frequency of 300% is reported. The engagement of the STM tip with the cantilever can be seen for two different spots on the cantilever in figure 22. The first image (a) is actuated at the base of the cantilever, the second (b) at 45 μm from the base. The resulting change in resonance frequency can be seen in part (c) of the image. There is insufficient data provided to calculate the stiffness of this system.

Zoom In Zoom Out Reset image size

Zine-El-Abidine et al [22] developed an electrostatic comb resonator with adjustable stiffness by using the change in effective length. The device is shown in figure 23. This device moves in-plane, and its position can be determined by measuring the capacitance between the moving and stationary fingers. By changing the effective length of the suspension beams the stiffness of the system changes. This change in effective length is achieved by electrostatic attraction of the suspension beams along a curved electrode as shown in figure 24. By applying a sufficiently large voltage between the electrode and the beam, the beam will be pulled in. In order to prevent a short circuit between the suspension beams and electrodes, silicon dioxide stoppers are used. These stoppers are insulated from the electrodes and will be the only contact points with the beam. There are three states in which the system can operate: zero sets activated, one set activated and two sets activated (In order to keep the system symmetrical it is only possible to actuate an entire set of actuators and not just one beam). The bias voltage that was used is 240 V. When an actuator is activated the effective length changes from 300 μm to 160 μm. The thickness of the structure is 25 μm and the width of the beams is 2 μm. Simple beams on the same die have been used to test the Young's modulus (107 GPa). The stiffness has been calculated using equation (21). The stiffness of the system in the three different configurations can be found in table 3. The maximum change in stiffness is used in table 2 and figure 1.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Mueller-Falcke et al [23, 58] proposed a mechanical way to add stiffness to a system. The application of this device is scanning probe microscopy. Usually, a cantilever is used in scanning probe applications, but instead of using a cantilever that moves out of plane, an in plane motion was used. The device has two operating modes; a soft and a stiff mode. In the soft mode, the probe is suspended by two pairs of flexure beams (flexure 1 and 3). By applying a voltage of 130 V a third pair of flexure beams (flexure 2) can be engaged, see figures 25 and 26. When these flexures are engaged, the probe is in stiff mode. The device covers an area of $500~\mu \text{m}\times 650$ μm. The proposed design has an unadjusted stiffness of 0.01 N m⁻¹ and an adjusted stiffness of 0.1 N m⁻¹.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Piezoelectric materials show mechanical deformation when an electric field is applied [71]. This deformation can be used to apply a stress to change the stiffness of a system.

Karabalin et al [72] developed a new model to predict stiffness changes in micro- and nanocantilever beams due to surface stress. The validity of this model has been confirmed with measurements. The device is a single chip with series of both cantilevers with a free end and doubly clamped beams as shown in figure 28. All have the same width of 900 nm and have the same stack of materials: 20 nm aluminum nitride, 100 nm molybdenum, 100 nm aluminum nitride, and 100 nm molybdenum. The lengths are 6, 8 and 10 μm. A voltage is applied between the molybdenum layers, such that the aluminum nitride layer will apply a stress to the stack due to its piezoelectric properties. Both a compressive and a tensile stress can be applied. This stress causes the beam to bend, because it is applied above the neutral axis of the stack. An AC voltage to the same piezo stack is used for actuation. Interferometric measurements are used to detect the resonance frequency. In table 2 only the 10 μm doubly clamped beam is included, as two separate devices; one for positive tuning voltage (a) and one for negative tuning voltage (b). Similar experiments, with an off-center piezoelectric stack on a silicon nitride doubly clamped beam were done by Olivares et al [73].

Zoom In Zoom Out Reset image size

Most materials expand when the temperature is increased. This is a result of the increase in kinetic energy in the molecules. When a doubly clamped beam is subjected to an increase in temperature, the expansion of the material will lead to an increased internal stress, which can be used for stiffness tuning.

Two devices that made use of stressing effects by thermal expansion were presented by Syms et al [24]. A one degree- (figure 29) and two degree of freedom device were made. The first device is explained in more detail. A mass is suspended by folded flexures and is tunable in the y-direction. This is done by applying a DC-voltage between the anchors of the flexures such that a current will start to flow. Due to Joule heating, these flexures will heat up. Whether compressive or tensile stress will arise depends on the ambient pressure. This is a result of the dominant cooling mechanism; under high pressure this is convection, while thermal conduction dominates at low pressures. When the cooling is dominated by convection, the temperature will be higher in the flexures than in the suspended part, because the surface of the latter is much greater, which results in a faster cooling. The thermal expansion will be larger in the flexures than in the suspended part, so compressive stress will arise. When the pressure is low, the convective cooling will be negligible and conduction to the bulk of the chip will be the dominant cooling mechanism. The flexures and the suspended parts will have the same uniform temperature in steady state. Now the thermal expansion of the suspended part will be larger, since it is longer than the flexures; the flexures will be under tensile stress. For a compressive stress, the stiffness will decrease, while a tensile stress will result in an increased stiffness. The unadjusted stiffness of the device is not mentioned in the paper, but can be derived from the geometry. The stiffness for such a set of doubly clamped beams is shown in equation (21). Assuming that the silicon has a Young's modulus of 169 GPa [74], this results in a stiffness of ${{k}_{\text{sys}}}=0.2$ N m⁻¹. For the lowest pressure of 10 mTorr, the increase in resonance frequency is almost 50%, resulting in 0.45 N m⁻¹. For the highest pressure of 500 mTorr, the decrease in resonance frequency is almost 10%, resulting in 0.16 N m⁻¹. In table 2 the device is shown as two separate devices; one for the increase in stiffness(a), the other for the decrease in stiffness (b). The two degree of freedom device has a different flexure geometry. The mass is suspended by two sets of beams instead of one and the length of the beams is smaller compared to the other device. By using equation (21) we get $k=0.69$ N m⁻¹. The resonance frequency in the y-direction changes from 1.56 kHz to 2.29 kHz for 3 mW of tuning power. By applying equation (3), a tuned stiffness of 1.49 N m⁻¹ is found. This device can be found in table 2 as device (c).

Zoom In Zoom Out Reset image size

The stiffness of the device designed by Remtema et al [75] is tuned by thermally induced stressing effects. The configuration of the device can be seen in figure 30. A mass is suspended by 'beam 1' and by a folded flexure mechanism that consists of 'beam 2' and 'beam 3'. The folded flexure mechanism results in a high stiffness in the x direction and a low (linear) stiffness in the w direction. By applying a current to the suspension beams, the temperature will increase due to Joule heating and the beams will expand. The expansion of 'beam 2' and 'beam 3' is in opposite directions, so no compressive stress is developed. 'Beam 1' however, will be stressed due to the thermal expansion. This stress results in the change in stiffness. On top of the stressing effect there is a second mechanism that decreases the stiffness of the structure. The Young's modulus of silicon has a negative temperature dependency, as described in section 3.5; by increasing the temperature in the suspension beams both the compressive stress as the decrease in Young's modulus will decrease the stiffness. (Because the compressive stress has the dominant effect, the device is placed in this category). The results of the change in resonance frequency of the device can be seen in figure 31. The maximum change in frequency is 14%, so according to equation (3), this is an increase of 30% in stiffness.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Previously discussed devices use uniform heating and uniformly thermal expansion by using the suspension beam itself as the heater. This requires an electrically conductive suspension beam. Sviličić et al [76, 77] and Mastropaolo et al [78] used non-uniform thermal expansion, by using a separate electro-thermal electrode on top of the structural element that supplies the thermal energy. So thermally induced stress can also be applied to non-conductive material. This method has been applied to a doubly clamped beam [76], a cantilever beam [77–81] and a disk [78]. In the case of a cantilever beam, the expansion is not fully restricted. The thermal stress is induced due to the difference in thermal expansion coefficient between the electrode and the support material. This results in a stress gradient and out-of-plane bending of the structure.

Thermally induced stressing effects have also been applied on nano resonators. Jun et al [82] used 12 μm long doubly clamped composite beams consisting of 30 nm 3C-SiC and 30–195 nm aluminum. A current was applied to the resonator itself, resulting in Joule heating and thermally induced stress. Mei et al [83] used carbon nanotubes in similar experiments.

Cabuz et al [84] presented a MEMS resonator of which the stiffness can be tuned by using stressing effects. This stress is applied by using electrostatic attraction. A silicon structure with a thin, doubly clamped cantilever resonator is installed in a glass package, as shown in figure 32. One of the sides of the structure is clamped in, the other side is suspended by a torsion bar. Electrodes are situated close to the top and bottom of the free hanging part of the silicon structure. These electrodes can exert a force on the structure, such that the structure can rotate around the torsion bar. An axial force will be induced to the resonator. A third electrode is placed close to the resonator to detect the deflection of the resonator by capacitive measurement. The applied voltage attracts the bottom of the free end. This induces a tensile stress in the resonator, resulting in an increase of resonance frequency. For 15 V the frequency increased with 14.5 Hz. Tuning with the upper electrode is not demonstrated, but a similar change in frequency, in opposite direction may be expected. The untuned resonance frequency is not mentioned in the paper. Yao et al [27] mentioned the use of electrostatic actuators for applying an axial force for stiffness tuning. No experimental data was provided though.

Zoom In Zoom Out Reset image size