E05 Resonator Design

Transcription

1 POLITECNICO DI MILANO MSC COURSE - MEMS AND MICROSENSORS /2019 E05 Resonator Design Giorgio Mussi 16/10/2018 In this class we will learn how an in-plane MEMS resonator handles process variabilities, we will exploit electrostatic spring softening to tune the resonance frequency down to a certain desired value, we will design the springs and the mass to target a certain resonance frequency, we will evaluate the quality factor of the resonator, and we will finally infer its RLC-series equivalent electrical model. PROBLEM We are asked to design a MEMS resonator, based on the so-called Tang structure. The device has to fit into a layout area of 200 µm x 200 µm. The required resonance frequency, f 0, is khz. The equivalent resistance of the electrical model of the resonator, noted as R eq (equivalent) or R m (motional), should be lower than 10 MΩ. The mechanical structure, visible in Fig. 1, is inherited from a previously designed frame, whose moving mass has an equivalent area of 150 µm x 85 µm; the central shuttle has a y-dimension of 10 µm. The device has 8 differential parallel plates cells for electrostatic tuning. The fabrication technology has some rules and characteristic dimensions; they are listed in Table What do etch spread and thickness spread mean? Due to process variability, the dimensions of the real device may be different from the designed (layouted) one: etch spread indicates that the PolySi layer can be wider or narrower than the designed dimension; thickness spread refers to the thickness of the device (the process height), that can be thicker or thinner with respect to the nominal value. Process variabilities can be evaluated with gaussian probability density functions (PDFs): the most probable case is the one of null overetch, and the values reported in Table 1, i.e., 3σ x,h, refer to the the 3σ value of the gaussian probability density function. 1

2 Symbol Value Young s Modulus E 160 GPa Density ρ 2320 kg/m 3 Process thickness h 15 µm Minimum PolySi width w min 1.5 µm Minimum comb fingers gap g min 1 µm Minimum distance from chip edge d min 2.5 µm Tuning stators gap g tun 2 µm Etch spread ±3σ x ±0.05 µm Thickness spread ±3σ h ±1 µm Normalized damping coefficient b area 6 N s/m 3 Table 1: Technological constraints and rules. We are asked to Calculate the variability of the resonance frequency of the device, given the process tolerances (3σ x and 3σ h ). 2. Knowing (i) that the final resonance frequency of the device has to match the target frequency, f 0, and (ii) that there is the possibility to exploit electrostatic tuning to solve issues related to over- or under-etch of the process, find a clever target natural frequency, f r, of the device and choose the dimensions of the springs and of the proof mass, aiming at a small footprint device. 3. Calculate the maximum voltage that should be applied to the tuning electrodes to bring all the devices down to the target resonance frequency, f Calculate the quality factor of the device. 5. Find the minimum value of the DC voltage, V DC, applied to the rotor in order to comply with the requirement on R eq, and extract the electrical model of the resonator, specifying the values of all equivalent electrical components. INTRODUCTION The Tang resonator is a common structure for MEMS resonators. It can be made very small with respect to the typical dimensions of MEMS inertial sensors, e.g., accelerometers or gyroscopes. Provided that the springs are all of the same dimensions, this structure shows a large compliance and a great capability for relief of built-in residual strain in the PolySi layer. In addition, temperature stress and stresses induced in the structure by large displacements are well rejected. Tang resonators employ comb-finger structures for both actuation and sensing. Why do we use them instead of parallel-plate structures? There are several drawbacks when using the 2

3 Figure 1: Structure of the MEMS resonator and some characteristic dimensions. Blue: moving mass; yellow: tuning electrodes; grey: anchored mass; green: drive and sense electrodes. parallel plates: the electrostatic force is non-linear unless the amplitude of vibration is limited to a very small fraction of the gap. In addition, the quality factor, Q, is usually lower (with respect to comb-finger-only structures), because of squeeze-film damping in the micronsized capacitor gap. On the other hand, with comb-finger structures, the drive capacitance is linear with the displacement of the structure, resulting in a force which is independent of the vibration amplitude. Moreover, the vibration amplitude can be as high as 10 µm for certain comb designs, with huge benefits in terms of signal-to-noise ratio (SNR). Why a resonance frequency of khz? This frequency value is commonly used because it is a power of 2 (32768 = 2 15 ). One can easily obtain a precise 1-Hz clock using binary frequency dividers (flip-flop chains used as binary counters). For this reason, this frequency value is the industry standard for timing application where one needs a real-time clock. 3

4 Figure 2: Generic, non folded, guided-end spring. QUESTION 1 Let us analyze the effect of the variation of the thickness of the PolySi layer on the resonance frequency. For a generic non-folded, guided-end spring, as the one shown in Fig. 2, the spring constant can be estimated as k = E w 3 h L 3, where E is Young s modulus, w is the width of the spring, h is its height, and L is the length. Note that this expression is valid for displacements occurring along the w-axis of the spring. The mass of a suspended structure also depends from the thickness of the PolySi layer: m = Ahρ, where A is the area of the structure, h is its thickness, ρ is its density. The resonance frequency of such a structure can be easily evaluated as f r = 1 k 2π m = 1 Ew 3 h 2π AhρL 3 = 1 Ew 3 2π AρL 3, which is independent from the thickness of the device. This is a general rule for all the structures that have an in-plane mode. Note that the dependance is different if the device is a torsional structure: in this case, the spread on the process height becomes a critical parameter for the determination of its resonance frequency. On the contrary, the effect of the over- or under-etch, i.e., of the variability of the in-plane dimensions, is very important for both in-plane and out-of-plane structures. Indeed, the stiffness of a spring has a cubic dependence on its width, hence a small change of this dimension has a huge effect on the spring constant, hence on the resonance frequency (while the mass is usually affected in a negligible way). In order to numerically evaluate the effects of this over-/under-etch, it is common practice to differentiate the expression of the resonance 4

5 frequency with respect to the spring constant: f r = 1 k 2π m f r k = 1 k 4π km = f r k 2k f r = 1 k f r 2 k. We can then differentiate the expression of the stiffness with respect to its width k = E w 3 h L 3 k w = 3Ew2 h L 3 = 3 Ew 3 h w L 3 = 3 w k k k = 3 w w. By combining the last two expressions, the resonance frequency variation due to a width variation is easily calculated: f r = 3 w f r 2 w. In other terms, f r = f r 0 + f r = f r f r 0 }{{}}{{}}{{} 2 w w. actual resonance frequency nominal resonance frequency resonance frequency variation The maximum resonance frequency variation is obtained for minimum nominal spring width, w min, and for maximum over-etch, w max, i.e., 2 times 2 the 3σ value of the spread s probability density function (PDF): f r,max = 3 2 w max f r = 3 2 3σ x f 0 = µm w min 2 w min µm f r = Hz. This is the unilateral maximum variation of the resonance frequency of the device due to under-/over-etch of the process. This means that such a device can randomly resonate from f 0 f r,max = Hz to f 0 + f r,max = Hz. QUESTION 2 Tang resonators are usually used for applications that require very precise oscillation frequencies. It is therefore mandatory to obtain the target resonance frequency value in any case, e.g., independently from process spreads. As we learned in other classes, one can exploit electrostatic spring softening (sometimes referred to also as frequency tuning) to change the resonance frequency of a mechanical structure. We know that resonance frequency tuning is capable to move the resonance frequency only toward lower values with respect to the nominal (un-tuned) one. For this reason, a clever design requires a higher (than khz) target average (with no over-/under-etch) resonance frequency, f r. In particular, one can design the structure so that, for maximum over-etch, the obtained frequency is anyway higher than khz. 2 We consider two times the over-/under-etch because a spring has two over-/under-etched edges, as shown in Fig. 3. 5

6 Figure 3: Over-/under-etch effect on the spring width. Referring to Fig. 4, we can choose the new target frequency as the frequency for which the maximum over-etch makes the resonator to resonate at f 0 = khz. Starting from the relationship obtained above, for a certain resonance frequency, f r, the maximum frequency variation has to be f r,max = f r f 0. Hence: where f r is the equation unknown. Hence, f r,max = f r f 0 = 3 w max = 3 2 3σ x, f r f r 2 w min 2 w min f 0 f r = σ x 2 w min = khz. With a resonance frequency of khz, the maximum unilateral frequency variation due to under-/over-etch is f r,max = 3 2 3σ x f r = 3641 Hz. 2 w min In this way, assuming a linear dependence between spring width and resonance frequency, the minimum obtainable frequency for the resonator structure turns out to be exactly f r,min = f r f r = f 0 = khz, while the maximum one f r,max = f r + f max = khz, as shown in Fig. 5. It should be here underlined that, as we introduced parallel-plate structures for frequency tuning, aforementioned comb-finger advantages (such as high displacement, and high Q), are now strongly reduced! We have now to dimension the springs and the mass of our device to target f r. Note that we decided to use the minimum spring width, i.e., a low stiffness. As a general rule, this is a good approach... as long as it can handle to bigger effects process spreads on the resonance frequency value. Given a certain resonance frequency, a lower spring constant 6

7 Figure 4: Schematic representation of the final (f 0 ) and design (f r ) target frequency for the device, evidencing the spread due to under-/over-etch. requires a lower mass, hence, a lower device area, as required by the problem (minimum footprint comment). Following this approach, the width of the springs will be minimized, w sp = w min = 1.5 µm, while the length will be maximized, i.e., we will choose the maximum value that complies with area constraints. Following the fabrication rules listed in Table 1, it is mandatory to have a gap of at least d min = 2.5 µm between the edge of the chip and one spring end; we can assume that PolySi links between the springs are of minimum width, w min ; the mass in the middle part of the device has a dimension of 10 µm, as shown in Fig. 1. From these observations, the maximum length of one spring can be: L sp = L chi p L mass 2 L spacing 2 L links µm 10 µm µm µm = = 91µm. 2 With the Tang resonator structure shown in Fig. 1, the springs can be well approximated to four beams in parallel, which, in turn, are in series with other four parallel beams. The proof mass is connected with 4 beams; assuming that the mass moves by an amount equal to x, each end of those 4 springs moves by an amount equal to x/2 (assuming that all the beams are equal): this means that the four beams are in parallel. The same approach can be used to demonstrate that also the 4 inner beams are in parallel. And one can further demonstrate that each parallel is connected in series 3. Given these considerations, the total stiffness of our device can be thus expressed as k = k sb 1 2 4, 3 Remember: when two springs are connected in parallel, they stretch by the same amount, and elastic forces sum up; when two springs are connected in series, the same force acts on both springs and the displacements sum up. 7

8 Figure 5: Real (blue curve) behavior of the resonance frequency for different over-/underetch values and its linear fitting (red-dashed curve). where k sb is the spring constant of a single beam; the factor 1/2 accounts for the series connection, while the factor 4 indicates the parallel between the four springs. Fig. 6 tries to generalize the calculation of the total spring constant of a structure. With the chosen values of width, w sp = w min, and length, L sp, the spring constant of the structure can be thus evaluated as k = 2k sb = 2 Ew3 min h L 3 sp = 21.5 N/m. Given the target resonance frequency, f r, and the spring constant, we can calculate the required mass: f r = 1 k 2π m m = 1 k 4π 2 = 0.41 nkg. We should now make a comparison between the required mass, m, and the mass we effectively have (from the geometry), and then somehow act to match the requirements. With the data provided, the total area that constitutes the mass of the device is A eq = 150 µm x 85 µm, that corresponds to a mass of f 2 r m f ull = A eq hρ = 0.44 nkg. 8

9 Figure 6: Calculation of the spring constant, given the number of springs in parallel, N, and the number of folded beams of each spring, M. To obtain the new target frequency, f r, we should reduce the mass of the resonator. As we learned in previous classes, in order to get the release of the structure during the device fabrication process, the proof mass must be holed! One can cleverly exploit the sizing of the holes to target a desired mass value. We can calculate the hole factor of our mass, m m f ull = 0.93, that corresponds, for example, to holes of 2.6 x 2.7 µm, with a pitch of 10 µm between two holes. These are common dimensions for the holing of a moving mass in a typical MEMS industrial process. QUESTION 3 Given the resonance frequency, f r, and its variability due to process spreads, the maximum possible resonant frequency, f r,max, can be found, and it is possible to calculate the maximum voltage that has to be applied to the tuning stators in order to tune the resonance frequency down to f 0. As shown in Fig. 1, there are 8 differential parallel plates tuning cells, 4 in the upper part of the structure and 4 in the lower part. The differential cells are all composed by two stators and one rotor. In order to calculate the voltage to be applied to the tuning stators, we have to calculate the contribution to the stiffness given by the tuning process, i.e., the electrostatic stiffness. Starting from the maximum frequency, we can calculate the maximum mechanical stiffness 9

10 of our device: f r,max = 1 k max 2π m k max = f 2 r,max 4π2 m = 26 N/m. In order to tune the resonance frequency down to f 0 = khz even in this worst case, the needed electrostatic stiffness, k el, can be evaluated from the following equation: f 0 = 1 k max k el,max 2π m k el,max = 4π 2 f 2 0 m t ar g et k max = 8.6 N/m. The expression of the electrostatic stiffness for a differential configuration of tuning electrodes is k el = 2Vtun 2 C 0 N tun gtun 2 = 2Vtun 2 ɛ 0 hl tun N tun, where V tun is the tuning voltage, i.e., the voltage difference between the rotor and the tuning electrodes, C 0 is the rest capacitance of a single tuning electrode, g tun is the tuning-electrode gap, L tun is the length of each tuning electrode, N tun is the number of tuning electrodes. As the length of the electrodes, L tun, is not yet decided, we choose to make it as long as possible, so that to minimize the tuning voltage. Assuming the same distance from the moving parts, d min, for both the upper and the lower part of the electrodes, its maximum length can be easily evaluated as L tun = L sp 2 d min = 88 µm. The gap between tuning electrodes and springs is specified by technological rules (g tun = 2 µm), the number of differential cells is given (N tun = 8). The maximum voltage to be applied to the tuning electrode can thus be found: k el,max = 2Vtun,max 2 ɛ 0 hl tun N tun gtun 3 g 3 tun k el gtun 3 V tun,max = = 19.2 V. 2ɛ 0 hl tun N tun Note that, since k el,max < k max, the obtained tuning voltage is low enough to prevent pull-in effects. QUESTION 4 The quality factor, Q, is one of the most important parameters for a resonator. It is responsible for the amplification of the movement of the moving mass at resonance: the bigger, the better, as the structure needs less driving force to achieve the same displacement amplitude, or, said in other words, given a certain actuation voltage, the corresponding displacement is higher: Q = ( ) X j ωr ( ) F j ωr ( ) = X j 0 ( ) F j 0 ( ) X j ωr ( ) F j ωr 1 k. 10

11 Given the mass-spring-damper model of a resonator, the quality factor can be calculated as Q = ω r m b = 2πf r m. b To calculate Q, we need en estimation of the damping coefficient, b. It is quite easy to estimate b when damping is limited by squeeze-film damping, as it happens in those devices where parallel-plate electrodes are present. In order to minimize b (and maximize Q), the pressure inside the package where the MEMS is fabricated is usually kept as low as possible (0.2-2 mbar). At these pressures, the number of gas molecules inside the package is so small that interactions between molecules can be neglected, and the only interactions that have to be taken into account are the one between gas molecules and device walls. Said in other words, the parallel-plate configuration basically pushes gas molecules toward the wall of the stator facing the rotor, and vice-versa, increasing the number of collisions and so making more difficult the movement of the rotor. This phenomena is commonly referred to as squeeze-film damping. It is quite intuitive that this contribution is more relevant as the facing area increases. Usually, squeeze-film damping is modeled by the following equation: b = b area A tot, where b area is a parameter that depends on the pressure and on the gap between the electrodes, while A tot is the total facing area of the parallel-plate electrodes. The evaluation of the damping coefficient, b, for the structure of this example is: b = b area A tot = b area (2 N tun L tun h ) = 126 nn/(m/s), where the factor 2 takes into account that the walls of the rotor that collide with the particles are 2 (one for each electrode of the differential tuning configuration). With this value of damping coefficient, the quality factor turns out to be: Q = 2πf r m b = 667, a common value for devices where parallel-plate electrodes are presents. Note that when parallel plates are absent, as in a Tang resonator with no tuning capability, Q-factors are usually higher, ranging from few thousands up to QUESTION 5 Let s infer the electrical model of a resonator in the Laplace domain. The output (or motional or sense) current, I, of a resonator can be evaluated as: I (s) = η s Ẋ (s), where η s = V DC C s x 11

12 is the sense-electrode electromechanical transduction coefficient, and Ẋ is the velocity of the proof mass 4. The velocity is the derivative of the displacement, Ẋ (s) = sx (s). Hence, I (s) = η s sx (s). The displacement, X is related with the actuation force through the well-known second-order mechanical transfer function of a MEMS resonator: X (s) = 1 k + bs + ms 2 F (s). Assuming a small-signal actuation voltage, one can linearize the actuation voltage-vs-force relationship as F = η a V, where η a = V DC C a x is the actuation-electrode (sometimes referred to as drive) electro-mechanical transduction coefficient. One thus obtains s I (s) = η s k + bs + ms 2 η av (s). We can now define the transfer function between actuation (driving) voltage and motional (sense) current as Y (s) = I (s) V (s) = η s s k + bs + ms 2 η a, which has the dimensions of a physical admittance. Its inverse, Z = 1/Y has the expression of an impedance, and can be referred to as the motional impedance of a MEMS resonator, where motional refers to the fact that models the moving, dynamic section of the resonator. Assuming, for simplicity, equal drive- and sense- electro-mechanical transduction coefficients, η a = η s = η, the motional impedance can be expressed as where k + bs + ms2 Z (s) = s 1 η 2 = 1 s η2 k + b η 2 + m η 2 s = 1 sc m + R m + L m s, R m = b η 2, L m = m η 2, C m = η2 k, are, respectively, the motional resistance, motional inductance, and motional capacitance of the series electrical model of a MEMS resonator, shown in Fig Remember that this is not the only current that may flow through the sense port! There may be also another current due to an ac voltage applied across a fixed capacitor: I (s) = sc 0 V (s). 12

13 Figure 7: Series-RLC electrical model of a MEMS resonator. As inferred from the electrical model of the resonator, the DC voltage applied to the moving mass, V DC, is directly related to its motional resistance, R m, R m = b = 1 2πf r m η a η s η a η s Q. Note that, the higher is the quality factor, i.e., the lower is the damping coefficient, the lower is the equivalent resistance. In order to evaluate the needed DC bias voltage, we first need to evaluate the electromechanical transduction coefficient of our resonator. The rotor-stator capacitance of a generic comb electrode can be expressed as C = 2 ɛ 0h (x + L ov ), g where L ov is the rotor-stator overlap length at rest (i.e., with x = 0), x is the displacement of the rotor, g is the gap between the rotor and the stator; the factor 2 takes into account both faces of the comb. Last equation stands for a single comb structure; assuming N CF comb structures, the capacitance is simply C = 2 ɛ 0h (x + L ov ) N CF g The capacitance variation per unit displacement of a comb-structure is thus: C x = 2ɛ 0hN CF, g which is independent from the displacement. As mentioned, this is a huge benefit of combfinger electrodes, when compared to parallel-plate electrodes. For a symmetric resonator, as the one considered in this example, where the number of drive- and sense-comb structures are the same, η a and η s coefficients are equal, as C a / x = C s / x. The equivalent resistance can be thus expressed as R m = 1 2πf r m ( ) C 2 Q. V DC x. 13

14 Figure 8: Single comb-finger structure. The equivalent resistance depends on the inverse of the square of V DC : the higher is the DC voltage, the lower is R eq. In order to calculate the DC voltage that guarantees a motional resistance lower than 10 MΩ, we need to dimension the comb structures, in particular the number of combs. It is convenient to maximize the number of comb-fingers, so as to allow low-voltage operation. The maximum number of comb-fingers is given by a combination of area and technological constraints. First of all, it is useful to determine the pitch of a single comb-finger structure; it will then be easy to evaluate the maximum number of structures that fits within the allowed dimensions. Looking at Fig. 8, the technological rules that have to be taken into account are the minimum gap constraint and the minimum dimension of a PolySi structure, w min. The elementary cell is composed by a central finger with the minimum width and two half fingers, all separated by the minimum gap. The pitch of such a structure is thus w CF,cell = w min 2 + g min + w min + g min + w min 2 = 5 µm. Considering the minimum PolySi-to-PolySi distance, d min, and considering that we should consider an additional w min /2 for both the first and the last fingers, the maximum number of comb structures for each port is N CF = L chi p 2d min w min µm = w CF,cell 5 µm = 38.7 N CF = 38. Once the number of comb fingers is calculated, we have all the parameters needed to calculate V DC : R m = b η 2 = 1 C V 2 DC x 2πf r m = Q V 2 DC 1 2ɛ 0 hn CF g 2πf r m < 10 MΩ V DC > 11.2 V. Q Be careful! The voltage applied to the tuning electrodes has now to be increased! Remember that the spring softening effect depends on the voltage difference between the rotor and the tuning electrodes: the value for the tuning voltage we found above is not the absolute tuning voltage, but it s the voltage difference that has to be applied between rotor and tuning electrodes. 14

15 Figure 9: Magnitude of the admittance of the resonator, for different V DC values. Once this value is calculated, it is easy to evaluate η and so the values of the components of the equivalent electrical model of the resonator, whose admittance is depicted in Fig. 9 for different V DC values. Using the value found above, V DC = 11.2 V, and the total stiffness relative to the final frequency f 0, i.e., k tot = 4π 2 f0 2 m = 17.4 N/m, one gets R m = b = 10 MΩ, η2 L m = m = 32 kh, η2 C m = η2 = 0.72 ff. k These values are typical numbers that apply for MEMS resonators. Note that there is a very important analogy between the mechanical and electrical domain from the dissipation point of view. In fact the only dissipative term in the electrical domain, due to the Joule effect, is the resistance, while in the mechanical domain the only dissipative term is the damping coefficient. In the modeling of the mechanical system to its electrical equivalent the only component that contains the damping coefficient is exactly the equivalent resistance, and the bigger is the dissipation in the mechanical domain, the higher is the dissipation in the electrical one. 15