Sensors and Actuators A 123–124 (2005) 555–562

Empirical and theoretical characterisation of electrostatically driven MEMS structures with stress gradients J. De Coster a,∗ , H.A.C. Tilmans b , J.M.J. den Toonder c , J.T.M. van Beek c , Th.G.S.M. Rijks c , P.G. Steeneken c , R. Puers a a

KULeuven, Department ESAT-MICAS, Kasteelpark Arenberg 10, B-3001 Heverlee, Belgium b IMEC, Kapeldreef 75, B-3001 Leuven, Belgium c Philips Research, Prof. Holstlaan 4, 5656AA Eindhoven, The Netherlands

Received 13 September 2004; received in revised form 18 March 2005; accepted 31 March 2005 Available online 22 June 2005

Abstract This paper investigates the influence of residual stress on the characteristics of electrostatic actuators. This is first done empirically by mechanical characterisation of a set of RF-MEMS switches with varying geometries. The mechanical measurements are performed on a Dektak surface profilometer. In addition, low-frequency electrical characterisation is performed. The measurement results allow for finetuning of the analytical relationship between pull-in voltage and geometrical parameters. This generally applicable method therefore reduces the need for more advanced but time-consuming electromechanical simulations to predict the electrostatic switching behaviour of a large range of (RF-)MEMS devices with various geometries. Next, an analytical formula is derived, relating the deformation of the actuator due to stress and stress gradients to the electrostatic pull-in voltage. The results obtained with this formula are in good agreement with the measurements. © 2005 Elsevier B.V. All rights reserved. Keywords: MEMS actuators; Electrostatic pull-in; Thin-film testing; Stress gradient

1. Introduction Electrostatic actuation is widely used in MEMS structures. Its simplicity and low power consumption make it a versatile actuation mechanism that can be applied in many micromachined devices. Moreover, the electrostatic instability that is encountered in devices with parallel-plate electrodes can be taken advantage of in such devices as RF and optical switches. Essentially, a parallel-plate switch can be considered a lumped spring-mass system: the mechanical compliance is entirely attributed to the suspension, whereas the moving electrode is considered infinitely rigid. For such a lumped system as represented in Fig. 1, the electrostatic

∗

Corresponding author. Tel.: +32 16 321716/077; fax: +32 16 321975. E-mail address: [email protected] (J. De Coster).

0924-4247/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2005.03.075

switching voltage VPI is given by [1]:
3 8kgN VPI = 27ε0 A

(1)

where k is the mechanical spring constant, gN the nominal zero-voltage air gap and A is the overlapping electrode area as indicated in Fig. 1a. The switch may also be considered a distributed system, in which case other tools are required to determine VPI : finite difference schemes can be implemented in circuit simulators such as Spice or Simulink [2–4] or a coupled electrostatic-mechanical finite element (FE) code can be used [5,6]. Unfortunately, these tools lack the flexibility to be used as a quick and indicative tool for design evaluation. An expression is, therefore, sought that captures some of the complexity offered by these schemes, yet is simple enough to allow for rapid evaluation. Section 2 introduces a generic empirical method to quantify the mismatch

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Fig. 1. (a) Schematical view and (b–d) optical photographs of capacitive switches with various suspension geometries [9]. The top electrodes are highlighted by the dashed lines and the suspensions are indicated by the solid lines in the photographs.

between measurement data and predictions according to (1). It is shown that Eq. (1) becomes inadequate when stresses and – more importantly – stress gradients are present in the structural material. FE simulation results are compared to the empirical data from Section 2 and the stress assumed for the FE simulations is validated by stress measurements on dedicated material test structures [7]. A modified formula for VPI is proposed in Section 4, taking the deformations due to stress and stress gradients into account.

2. Empirical verification In order to verify the validity of (1), a number of RFMEMS switches were selected and all their relevant parameters were measured individually. Dimensional analysis of these parameters yields a convenient way of representing the mismatch between the measured and predicted results and an empirically modified version of (1) is derived. 2.1. Geometries Fig. 1 shows a number of capacitive switches with various suspensions, which were fabricated in Philips’ 5 ␮m-Alu PassiTM process [8]. For each of the geometries shown, an array of devices was implemented: the area A of the top electrode ranges from 75 × 103 ␮m2 to 200 × 103 ␮m2 , while the

mechanical stiffness of the suspension beams varies between 30 N/m and 150 N/m. Since the spring constant of the suspended electrode is about one order of magnitude higher than the stiffness of the suspension beams, the overall spring constant of the devices is dominated by the suspension beams and the suspended electrode can be considered rigid, as schematically represented in Fig. 1a. 2.2. Measurement conditions Two types of measurements were carried out to characterise the electrostatic actuators. First, a Dektak 3030 surface profile measuring system was used to apply mechanical loads Fa [10] to the movable plates of the switches and to measure the profile of the membranes under varying loads. An example of such a trace is depicted in Fig. 2. Repeated measurements with different values of Fa were carried out on each of the switches. The spring constant k of the device is then defined as the slope of the least-squares straight line fitted through the measured points of the force–displacement curve. This is shown in Fig. 3 for the devices depicted in Fig. 1b. The vertical axis shows values of z that are relative to the nominal electrode height gN as indicated in Fig. 1a. Moreover, z is an averaged value of the measured profile of the top electrode. Additionally, the zero-voltage air gap g0 can be derived from these measurements as the intercept of the least-squares

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2.3. Dimensional analysis In order to reduce the number of independent variables in (1) and to obtain a more convenient way of presenting the measurement data, a dimensional analysis of the quantities involved is performed. The Buckingham Π theorem [11] states that the five quantities, which are expressed in terms of three independent units (N, m, and C), may be grouped into 5 − 3 = 2 independent dimensionless groups and that relation (1) may be rewritten in terms of these dimensionless groups. The dimensionless groups (or ‘Buckingham Π parameters’) are found by selecting a set of primary variables. These primary variables should be chosen such that all units are represented, and the dimensional matrix of the set must be of full rank. For instance, k, A, and VPI can be chosen as primary variables. Their dimensional matrix is as follows:

Fig. 2. Measured topology along the trace indicated in the photograph.

C N m

line, bearing in mind that the z values are relative to the nominal gap height gN (which is defined by the process). The actual gap height of the device, g0 , is then found as g0 = gN + hd with hd as indicated in Fig. 3. For the specific device that is shown in this figure, the spring constant A amounts to 88.5 N/m while the air gap g0 is 1.57 ␮m and the nominal air gap gN for the process is 1.43 ␮m. After the Dektak measurement, low-frequency capacitance-versus-voltage (CV) measurements were carried out in order to determine the pull-in voltages VPI . In the next paragraph, the measured values of g0 , k, and VPI will be compared against the expected relationship (1). Finally, the area A of the switches was determined using a microscope with an eyepiece measuring scale.



k

0   1 −1

A 0 0 2

VPI

 −1  1 

(2)

1

which has rank 3. The Π parameters are found by solving the following set of dimensional equations: x3,i Πi = [kx1,i Ax2,i VPI Pi ] = 1,

i = 1, 2

(3)

for the coefficients xj,i , where Pi , are the variables from Table 1 that do not belong to the set of primary variables. The notation with square brackets is used to designate ‘has the dimension of’. Solving the dimensional equations yields the following dimensionless quantities: 2ε VPI 0 Π1 = √ , k A

g0 Π2 = √ A

(4)

This allows Eq. (1) to be re-written in terms of Π 1 and Π 2 :  2ε VPI 8 g0 3 0 √ = √ (5) ⇒ Π1 = 0.296Π23 27 k A A 2.4. Measurement data analysis The measured data can now be plotted in the (Π 1 , Π 2 )plane: indeed, Π 1 and Π 2 can be calculated for every device from the measurement results as explained in Section 2.2, and filling in the measured values of k, g0 , A, and VPI in expression (4). For instance, the device for which Table 1 Variables used in Eq. (1) and their dimensions

Fig. 3. Measured deflection z versus applied force Fa for switch shown in Fig. 1.

Symbol (units)

Quantity

VPI (N m/C) g0 (m) k (N/m) A (m2 ) ε0 (C2 /(Nm2 ))

Actuation voltage Zero-voltage air gap Spring constant Overlapping electrode area Permittivity of free space

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Fig. 5. Geometry used for the FE calculations.

The method outlined above is therefore applicable to a much wider range of phenomena than just the pull-in instability [12]. Fig. 4. Data of Dektak and CV measurements yields the locus of devices in the (Π 1 , Π 2 )-plane.

measurements are shown in Fig. 3 has the following values: k = 88.5 N/m, g0 = 1.57 ␮m, A = 1.59 × 105 ␮m2 , VPI = 12 V. In terms of Π parameters, this yields: Π 1 = 3.62 × 10−8 and Π 2 = 3.5 × 10−3 . These values are shown along with the results for more than 30 other devices in Fig. 4 where Eq. (5) is represented by the solid line and the measured devices are indicated by dots. In order to assess the agreement of the measured data with the original formula, a least-squares curve of the form y = axn was retrofitted on the measured data. This retrofitted least-squares curve is shown as the dashed line in Fig. 4; its coefficients are given by: Π1 = aΠ2n = 0.187Π22.73

(6)

Fig. 4 clearly reveals a significant mismatch between (5) and (6), which becomes larger as the value of Π 2 increases. This means that the operation of the devices is affected by an unknown quantity that was not listed in Table 1, and that the devices with a large Π 1 , and Π 2 are particularly affected by this. Eq. (5) does not take this unknown effect into account, whereas (6) – intrinsically – does. In order to illustrate this, a comparison between the measurements and FE calculations with residual stress are presented in the next section. The subsequent section investigates the influence of electrode curvature due to a stress and/or stress gradient in the material of the movable electrode. From a more pragmatic point of view, one can consider the parameters a and n in (6) to be process-specific parameters; once their values are determined, the retrofitted curve can be applied for a variety of devices. This is borne out by the fact that the measurements presented in Fig. 4 were carried out on two wafers which were processed in two slightly different varieties of the PASSITM process flow. No clustering of the measurement points of the two wafers can be observed in the (Π 1 , Π 2 )plane. In addition, this illustrates how dimensional analysis can reveal relationships between a set of quantities, even if a closed-form analytical expression is not known beforehand.

3. FE calculations with residual stress In order to numerically investigate the influence of material stress, a number of FE calculations were made. The studied geometry is depicted in Fig. 5. By varying the lengths ls of the suspension beams and zp of the perforated top electrode, a parameter sweep in the (Π 1 , Π 2 )-plane is made. Using Coventorware [13] FE software, the same ‘experiments’ can be carried out on the test geometry as on the real devices, i.e. determining the mechanical stiffness of the devices for a concentrated point load (mimicking the Dektak measurement) and coupled electromechanical simulations to determine the actuation voltage (mimicking the low-frequency CV measurement). These simulations were performed for four devices with varying sizes and with residual stress σ; the resulting points in the (Π 1 , Π 2 )-plane are shown in Fig. 6 along with the retrofitted curve that was obtained from the measurement data in Fig. 4. Two sets of calculated points are shown in the figure: the + signs designate simulations with zero residual stress in the

Fig. 6. FE simulation results for σ = 0 MPa and σ = 100 MPa.

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Fig. 8. Electrostatic actuator with deformed electrode. Fig. 7. Profile of ring beams of various sizes; the first and second largest beams are buckled.

structural layer, whereas the triangles correspond to simulations with a uniform tensile stress of 100 MPa in the structural material. This value of σ matches rather well with the stress that was measured on dedicated ring-beam test structures [7], an array of which is shown in Fig. 7. The critical length for buckling is a measure for the tensile stress, which amounts to roughly 125 MPa according to these structures. Two conclusions can be drawn from the FE simulation results that are shown in Fig. 6. First, the figure clearly demonstrates that the uniform stress does indeed have a significant influence on the operation of the electrostatic actuator. Secondly, it suggests that the uniform stress alone is not sufficient to account for the mismatch that was observed in Fig. 4, as there is still a significant mismatch between the triangles and Eq. (6). These conclusions imply that the influence of stress on the electrostatic actuator is two-fold: in the first place, the spring constant k is affected by the uniform stress. Secondly, the suspended electrode is deformed due to the stress gradient (and to a lesser extent due to the uniform stress) and the gap spacing g0 is affected accordingly. The first effect is already included in the measurements shown in Fig. 4, since the values of k were determined by Dektak measurements. It is also included in the simulations that were shown in Fig. 6. The second effect, i.e. the curvature of the electrode, is not included in the simulation results from Fig. 6. In the next paragraph, a detailed analysis of this effect will be made. Afterwards, the measurement results will be re-evaluated in the light of this analysis and a modified version of Eq. (1) will be derived.

amount of curvature in the electrode is h. A cross-section of this configuration is shown in Fig. 9. The topology of the membrane is specified as h(x, y). Due to the curvature, the capacitance of the actuator can be expressed as the series-equivalent of two hypothetical capacitances C1 and C2 . C2 = ␧0 A/(g0 + z) is the regular parallelplate capacitance, whereas C1 is introduced by the electrode curvature; as h approaches zero, C1 goes to infinity. The total capacitance CT is given by:

 CT =

A

ε0 g0 + z + h(x, y)

=

C 1 C2 C1 + C 2

(7)

The electrostatic force Fel when applying a voltage V across the actuator and the mechanical restoring force Fr can then be written as: Fel =

δ δz



CT V 2 2

=

ε0 AV 2 2(g0 + z)2



C1 C1 + C 2

2 ,

Fr = kz

(8)

These forces are represented in Fig. 10 for different values of V. The difference Fel − Fr vanishes at pull-in, and the same applies to the sum of the inverted forces, Fel −1 − Fr −1 , as depicted in Fig. 11. Using the inverse of the electrostatic and restoring forces rather than the forces themselves, simplifies the expressions below. In addition, the derivative of Fel −1 − Fr −1 becomes zero as well at pull-in. This yields the

4. Theoretical analysis for stress-deformed electrodes 4.1. Pull-in voltage of a deformed electrode Stress and stress gradients can cause the free-standing electrode to deform, as schematically illustrated in Fig. 8. In the analysis below, the overlapping electrode area will be supposed to stretch from x = −a to x = a and from y = −b to y = b. The minimum air gap is designated by g0 and the

Fig. 9. Cross-section of the actuator from Fig. 8.

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of each is retained. This yields the following result:  g0 g0 ε0 A (0) (1) zPI ≈ − zPI = − + 3 3 2C1

(11)

The corrected displacement z(1) is then introduced in the second condition in (9). Combining this with (8) finally yields the value of the pull-in voltage VPI :      32g02 8g03  (12) VPI ≈ k + 27ε0 A 9C1

Fig. 10. Electrostatic and restoring forces in the actuator.

following condition for the pull-in voltage and displacement:   δ 1 1    =0 −  V = VPI δz Fel Fr ⇔ (9)  1 1 z = zPI   − =0 Fel Fr Combining the first condition in (9) with (8), a fourth-order polynomial in z is obtained: f (z) = −4

(g0 + z)2 (g0 + z) ε0 A g0 + z − 12 −8 2 2− =0 ε0 AV 2 C1 V 2 kz2 C1 V (10)

In order to solve this equation, one Newton–Raphson step is performed with an initial value z(0) = −g0 /3, which is the solution to (9) in case there is no electrode curvature. The Newton–Raphson method requires the values of f(z(0) ) and f(z(0) ) to be calculated; since the symbolic expressions for these values are rather lengthy, only the most significant term

Expression (12) is a generalised form of (1); in case there is no electrode deformation, C1 goes to infinity and the expression reduces to its classical form. In the next paragraph, the value of C1 will be calculated from the curvature of the membrane. The two-fold influence of the stresses on the electrostatic actuator, are represented in (12) by k and C1 . 4.2. Capacitance of the deformed actuator As indicated in Fig. 9, the topology of the membrane is specified by h(x, y). Since the idea is to obtain a generic expression for VPI , the topology h(x, y) is approximated by an elliptic paraboloid—the Cartesian equation of which can be expressed in terms of only three geometrical quantities: a, b and h:  x2 + y 2 (13) h(x, y) = h 1 − 2 a + b2 The total capacitance of the actuator can then be found by combining (13) and (7). In order to simplify the integration, the integration along x is done first, after which the integrand is approximated by a second order series expansion about the origin y = 0. This finally leads to an expression for CT : CT = (2Y1 + Y2 )

b 3

(14)

where

  Q + a P   ln Y1 = 2Q  Q − a 

   Q2 − b 2 + a    Y2 =  ln    2 Q2 − b 2  Q2 − b 2 − a  P

4ε0 (a2 + b2 ) h  g 0 + 1 (a2 + b2 ) Q= h P=

Fig. 11. Conditions for pull-in of the movable electrode.

(15)

(16)

(17) (18)

CT can now be calculated from the deformation h by combining Eqs. (14)–(18). C1 is then found from CT and C2 using Eq. (7). If this result is introduced in (12), the switching voltage VPI is found.

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mation that occurs in the devices. Since an extra variable C1 was introduced in (12), an extra Π parameter has to be added to the dimensional analysis from Section 2.3. The convenient graphical representation of the data in the (Π 1 , Π 2 )-plane is therefore no longer valid. Fig. 13 presents an alternative way of plotting the same data: the calculated values of VPI according to (12) are shown, together with the values obtained using the initial formula (1). The triangles approach much better the target locus indicated by the diagonal in the figure. This means that an estimate of the electrode’s h is sufficient to obtain a reasonably accurate value of the switching voltage VPI . The h can be derived from a single mechanical FE calculation with the appropriate stress and stress gradient, rather than a fully coupled electromechanical FE simulation.

Fig. 12. Comparison between measured topology and calculated topology with FE model incorporating stress and stress gradient.

4.3. Comparison of calculated and measured results The devices that were measured and located in the (Π 1 , Π 2 )-plane will now be re-evaluated using Eqs. (12) and (14). The Dektak measurement results are now used to determine the h of the movable electrodes and the minimum gap spacing g0 in accordance with Figs. 8 and 9. For instance, re-examining the device for which the measurement results are shown in Fig. 3, yields the following values: g0 = 1.4 ␮m and h = 0.32 ␮m. The values of a and b are 212.5 ␮m for this device, resulting in a capacitance C1 = 5.25 pF. The resulting pull-in voltage VPI according to (12) is then 12.94 V, while a value of 12 V was measured and 7.1 V was predicted by (1). For one of the measured devices, Fig. 12 shows how the measured Dektak profile compares to the FE calculated topology when a stress as well as a stress gradient is used in the calculations. This confirms that the h which is measured with the Dektak is indeed consistent with the actual defor-

5. Conclusions The influence of stress and deformations on the operation of electrostatic actuators was empirically and analytically studied. Empirical data was collected for a large number of devices and compared with elementary 1D theory. The mismatch between both was studied by running a few FE experiments with stress levels as measured on dedicated test structures. The simulated results were in rather good agreement with the measurements, albeit at the cost of lengthy electromechanical FE calculations. In order to quantify the effect of deformations in a more direct way, an analytical formula relating the pull-in voltage of an actuator to the deformation of the electrodes was presented, and the calculated results showed a good agreement with the measurement data.

Acknowledgement This work was carried out with the financial support of the European Commission, IST-project MEMS2TUNE (IST2000-28231).

References

Fig. 13. Measured switching voltages and calculated values using (1) and (12).

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Biographies Jeroen De Coster was born in Leuven, Belgium, in 1978. He received the MS degree in electrical power engineering from the Katholieke Universiteit Leuven in 2001. He is currently a research assistant at the MICAS microelectronics laboratory of the electrical engineering department ESAT at the KULeuven, where he is working on the design, modelling and characterisation of MEMS devices and RF-MEMS in particular. Harrie A.C. Tilmans was born in Elsloo (The Netherlands) in 1957. He received the MSc degree in electrical engineering from the University of Twente, Enschede, The Netherlands, in May 1984, and the PhD degree in electrical engineering from the same university in January 1993. Dr. Tilmans has held MEMS R&D positions at the University of Twente, Boston University, the University of Wisconsin-Madison, Catholic University of Leuven, Johnson Controls Inc., and CP Clare Inc. His research covered micromechanical resonators, low-range differential resonant pressure sensors, resonating force sensors, MEMS-CMOS process integration technology, mechanical properties of MEMS thin films and microrelays. In 1995–1998, he was project manager for the Esprit-MIRS project involving the development of a micromachined electromagnetic relay. Since September 1999 he is with IMEC, Leuven, Belgium where he is section head of RF-MEMS components and systems. Dr. Tilmans has (co)authored over 120 papers and issued 6 patents in the area of MEMS. Dr. Tilmans is a member of the IEEE’s Components, Packaging and Manufacturing Technology Society (CPMT), of the IEEE Microwave Theory and Techniques Society (MTT-S) (MTT-S), of the International Microelectronics and Packaging Society (IMAPS) and of the European Microwave Association (EuMA). In 2001 he received the Eurosensors XV Fellow award for his pioneering work on microresonators.

Jaap den Toonder studied at the Delft University of Technology and got his masters degree in applied mathematics in 1991. He received a PhD degree from the same university in 1996 after a numerical/theoretical and experimental study of drag reduction in turbulent flows by polymer additives. His advisor was professor Frans Nieuwstadt. In 1995, he joined the Philips Research Laboratories in Eindhoven, The Netherlands, where he started working in the field of the mechanics of solid materials. He has worked on a wide variety of applications, such as ceramic capacitors, optical storage systems, IC low-k materials, RF MEMS, soft electronics, and polymer MEMS. He is now head of the research cluster ’Thin Film Mechanics’ of Philips Research. Since 2004, he combines his work as a scientist at Philips Research with a part-time professorship at the Materials Technology group of the University of Technology of Eindhoven. His current main research interests are: • small-scale characterisation of mechanical properties of materials, using nano-indentation and other advanced experimental methods; • length scale effects in small structures and thin films; • polymer and metal Micro-ElectroMechanical Systems (MEMS); • soft electronics. Theo G.S.M. Rijks (born 1968) received his MS degree in technical physics from Eindhoven University of Technology (The Netherlands) in 1992. In Oct. 1996 he received his PhD degree from the same university, studying thin-film multilayers for GMR-based magnetic sensors. Since then he has been employed as senior research scientist at Philips Research in The Netherlands, working on the preparation and characterisation of nanocrystalline permanent magnet materials (1996–1998) and on RF design of integrated passive circuits (1998–2001). Presently he is working on design and characterisation of RF MEMS switches and tunable capacitors. P.G. Steeneken was born in Groningen, The Netherlands, in 1974. From 1996 to 2002 he studied strongly correlated electron materials using electron spectroscopy. For this work he received the M.Sc. (with honors) and Ph.D. degrees in experimental physics from the University of Groningen, The Netherlands. In 2002 he joined the Philips Research Laboratories in Eindhoven, The Netherlands, where his current interests include the design, simulation, application and characterization of RF MEMS switches and resonators. Robert Puers was born in Antwerp, Belgium, 1953. He recieved his BS degree in electrical engineering in Ghent in 1974, and his MS degree at the Katholieke Universiteit te Leuven in 1977, where he obtained his PhD in 1986. From 1980, he was employed as a research assistant at the Laboratory ESAT at KULeuven. In 1986, he became Director (NFWO) of the clean room facilities for silicon and hybrid circuit technology at the ESAT-MICAS laboratories of the same University. He was a pioneer in the European research efforts in silicon micromachined sensors, MEMS and packaging techniques, for biomedical implantable systems as well as for industrial devices. In addition, his general interest in low power telemetry systems, with the emphasis on low power intelligent interface circuits and on inductive power and communication links has promoted the research of the ESAT-MICAS labortary to international recognition.At present, he is as a full professor at the KULeuven, teaching courses in ‘microsystems and sensors’, in ‘biomedical instrumentation and stimulation’ and a basic course in ‘electronics, system control and information technology’. He is the author or co-author of more than 250 papers on biotelemetry, sensors, MEMS and packaging in reviewed journals or international conferences. He is a Fellow of the Institute of Physics (UK), council member of the International Microelectronics and Packaging Society (IMAPS), member of the Electron Device Society (EDS) and many others. He is editor-inchief of the IOP Journal of Micromechanics and Microengineering.