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Study of mechanical properties of RF MEMS switches by nanoindentation: characterization and modeling of electro-mechanical behavior
Sensors and Actuators A 163 (2010) 205–212
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Study of mechanical properties of RF MEMS switches by nanoindentation: characterization and modeling of electro-mechanical behavior Adam Koszewski ∗ , Diane Levy, Frédéric Souchon 1 CEA-LETI-MINATEC, Grenoble, France
a r t i c l e
i n f o
Article history: Received 4 December 2009 Received in revised form 30 March 2010 Accepted 13 May 2010
Keywords: MEMS switch nanoindentation electrostatic mechanical clamped-clamped model
a b s t r a c t This paper presents a thorough methodology to evaluate the mechanical properties of RF MEMS switches. It includes an analytical electro-mechanical model and nanoindentation characterizations which allow explaining the electrostatic behavior of the MEMS switches. Experimental and theoretical aspects are compared through two runs of ohmic electrostatic switches which are different from geometric and process flow standpoint. The switch is modeled as a clamped-clamped beam with mobile vertical loads, two stretching axial forces and two reaction moments in the anchors. The mechanical properties are characterized by nanoindentation technique; specific test protocol for measuring very low stiffness has been specially implemented with an adapted continuous stiffness measurement method. The measured values of the stiffness at the membrane center are respectively 79 and 137 N/m for the two runs, and they remain in good agreement with the theoretical model. The difference between these two runs is explained by the difference of the tensile residual stress. Moreover, the nanoindentation data turn out to be useful to evaluate the contact quality: dimple deformation due to twisting of the beam (contact in multiple steps), or inhomogeneous surface (stiffness is increasing while in contact). The fitted model is also used to approximate the restoring and contact forces and to point out the influence of the mechanical properties on the electrostatic behavior. A simple model with two parallel-plate capacitors in parallel, used for calculation of the up- and down-state capacitances, shows no discrepancy with experimental values. Finally, it may be concluded that the electrostatic behavior of the switches (pull-in voltage, up- and down-state capacitances, and probability of stiction) can be better understood thanks to the mechanical characterizations made by nanoindentation. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Over the past years CEA-LETI has been developing electrostatic RF MEMS ohmic switches. Their key parameters, such as pull-in voltage, contact and restoring forces and up- and down-state capacitances depend on the stiffness of the switch and its geometry. Among the standard testing techniques used to characterize RF MEMS switches there are electrical tests for measuring the pullin and pull-out voltages and the up- and down-state capacitances, AFM and SEM observations for assessing the contact quality and laser vibrometry for determining the gap heights and the dynamic mechanical behavior.
∗ Corresponding author. 17, rue des Martyrs, 38054 Grenoble Cedex 9, France. Tel.: +33 4 38 78 18 53; fax: +33 4 38 78 51 40. E-mail addresses: [email protected] (A. Koszewski), [email protected] (F. Souchon). 1 Tel.: +33 4 38 78 28 68. 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.05.011
In this paper the nanoindentation technique is used for the evaluation of the behavior of RF MEMS switches. The nanoindentation technique allows not only to measure the membrane stiffness and the gap heights, but also to evaluate the contact quality. Moreover, the nanoindentation measurements can be coupled with an electro-mechanical model in order to improve the understanding of the electrostatic behavior of the RF MEMS switches. 2. Experimental details 2.1. Switch description The ohmic electrostatic switch, which is used in this study, is designed for RF applications and is implemented on a coplanar wave guide (CPW) using a full wave analysis [1]. The series switch (Fig. 1) is a silicon nitride clamped-clamped beam with patterned metallic contacts and two symmetrical side electrodes, which are located inside the beam and are used for
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Figure 3. Position of the indenter tip for the nanoindentation experiments.
2.2. Experimental setup
Figure 1. Top view of an ohmic electrostatic series switch manufactured by CEALETI.
actuation. When a biasing voltage is applied between these electrodes and the CPW ground plan, the membrane is pulled down and the transmission line is short-circuited by the two dimples of the metallic mobile contact. The main fabrication steps of the switch (Fig. 2) comprise: - Thermal oxidation of a silicon wafer, followed by etching steps of a cavity with bumps, - Fabrication of the CPW: deposition and patterning of a gold layer in order to define the CPW and the RF lines, - Spin coating and patterning of a photoresist sacrificial layer, - Fabrication of the beam: deposition of the first silicon nitride layer (beam), followed by deposition of a TiN layer (electrodes); patterning of the electrodes and deposition of the second silicon nitride layer, - Fabrication of the contact: silicon nitride etching and deposition of a gold layer followed by deposition of the last (third) silicon nitride layer, - Opening of the pads and the beam, stripping of the sacrificial layer in dry oxygen plasma. Up to now various types of switches have been designed and manufactured; they are different from a geometric and process flow standpoint. This paper investigates two types of switches denoted as A-type and B-type coming from two manufacturing runs. The main differences between these two runs concern:
The mechanical properties of the switches are characterized using the nanoindentation technique. In this technique a vertical concentrated load is applied to the indenter tip. The tip is approaching the sample surface, and when in contact, it exerts a quasi-static load on it. This load and the tip displacement are recorded simultaneously. The stiffness is the slope of a recorded force-displacement curve (Figs. 6-9). Apart from the quasi-static measurement, a continuous stiffness measurement mode (CSM) is also used. In this mode, the stiffness between the indenter tip and the beam is continuously measured by superimposing simultaneously an oscillating force. Here it is used to detect each change in the measured stiffness, which may be due to a contact in multiple steps or a deformation of the dimples, thus it is used to evaluate the quality of the contact. In order to keep the switch working after the tests, a maximum stiffness limit is set for each measurement. If the measured stiffness exceeds this limit the measurement is stopped. The nanoindentation experiments (Fig. 3) are performed at the membrane center (geometrical center of the beam) and at the electrode center (geometrical center of the electrode, which is around 1/4 of the beam length). The nanoindentation at the membrane center gives the free membrane stiffness and the contact gap height (distance between the dimples of the mobile contact and the transmission line).The nanoindentation at the electrode center gives the stiffness before and after the contact in the central part and the electrode gap height (distance between the electrode and the CPW). Functional characterization of the switches (standard C-V and R-V sweep tests) are carried out using a dedicated test-bench [2] equipped with a HP4284 impedance meter and Keithley 2400 source meter. The measurements are recorded through a voltage ramp, usually from 0 V to 40 V with a voltage step of 1 V. 3. Coupled electro-mechanical model 3.1. Mechanical model
- Beam length: longer beam for B-type, - Residual stress in the membrane: lower for B-type, - Gap heights: larger electrode gap and smaller contact gap for Atype (these gaps are managed by controlling the etching depths), - Contact: different process flows have been used to realize the mobile contact (modification of the process of etching of the silicon nitride and the sacrificial layer before deposition of the gold layer).
Figure 2. Schematic stack of the ohmic electrostatic series switch.
For the mechanical model, the switch (Fig. 4e) is approximated by a clamped-clamped beam with two lateral mobile loads, two stretching axial forces and two reaction moments at the ends [3]. The mobile lateral loads F1 and F2 correspond to the electrostatic actuation forces when they are applied at the geometrical center of the electrodes. The axial forces S represent the stretching effect of the residual stress which is unavoidable in micro fabrication process and has a significant influence on the mechanical properties of the structure [4,5]. The axial stretching component due to deflection of the beam is omitted as the typical maximum deflection of the beam does not exceed its thickness [4]. The couples M1 and M2 are the reaction moments, which cancel the deflection in the built-in ends. The most significant difference between our model and those presented in the literature is the fact that the deflection curve here, is calculated for one lateral load applied at any point along the beam, and not only in the beam center [4,6]. This is the partial solution, which is then used to obtain the deflection curve of a beam with two lateral loads, which correspond to electrostatic forces in a real switch.
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lateral position, l is the length of the beam, w is its width and t is its thickness. For the left part of the beam, it is sufficient to replace the l by (l-d1 ) and x by (l-x) in the corresponding equations. After necessary transformations, the solution of eq. 1 for the right part is: z=−
F1 sinh pd1 F1 d1 sinh px + x Sp sinh pl Sl
(3)
where: S EIz
p2 =
(4)
From the eq.3 the slopes of the deflection curve at both ends can be calculated, as the first derivative of z for x = 0 and x = l, and at right end the slope is equal to:
F1 d1 dz F1 sinh pd1 = − Sp sinh pl + Sl dx x=0
1 =
(5)
In case of the clamped-clamped beam, there is no deflection in both built-in ends, thus the slopes calculated from (eq.5) are compensated by the reaction moments M1 and M2 . To find the M1 and M2 the deformation of a clamped-free beam with two reaction moments M1 and M2 and two axial forces S, in the absence of the lateral load F1 is calculated (Fig. 4c). The slopes at both ends, calculated for these conditions, have to be opposite to those given by (eq.5). The deflection curve for a single moment M may be obtained immediately from the eq. 3 by making the following substitution: F1 d1 = M and sinhpd1 = pd1 [3], and it is: z=
M S
x
−
l
sinh px sinh pl
(6)
The total deformation is a sum of the deformations caused by each moment (the reaction moments at each end) which is: z=
M1 S
x l
−
sinh px sinh pl
+
M2 S
l − x l
−
sinh p(l − x) sinh pl
And the deflection at the right end, is:
Figure 4. Mechanical model: schematic representation of the RF MEMS swudy of mechanical properties of RThe stages of deriving the mechanical model (a)-(d).
1 =
dx dz
x=0
=
M1 S
= M1 ˛ + M2 −ˇ The problem of a clamped-clamped beam with the loading conditions as in (Fig. 4e) is statically indeterminable. To find the deflection of this overrigid beam the same approach as in [3] is used. In this approach the partial solutions for simplified loading conditions (Fig. 4a-d) are found and the general solution is a superposition of these partial solutions. First, the deflection for a clamped-free beam with one mobile vertical load F1 and two axial forces S (Fig. 4a) is calculated by solving a set of Euler-Bernoulli beam equations. The deflection curve (eq. 1) at any point depends only on the magnitude of the bending moments at that point. In our case the bending moments come from the lateral force F1 and axial force S (the right hand side of the eq. 1), thus the equation can be written in the following form, for the right part of the beam: EIz
d2 z F1 d1 x = Sz − l dx2
(1)
where Iz is the moment of inertia of the beam: Iz =
wt 3 12
(2)
and E is the Young’s modulus, z is the deformation of the beam, S is the axial force originating from the residual stress, F1 is the mobile lateral load applied at the distance d1 from the end, x is the current
1 l
−
p sinh pl
−
M2 S
1 l
−
p cosh pl sinh pl
(7)
(8)
where: ˛=
1 p − l sinh pl
ˇ=
1 p cosh pl − l sinh pl
(9) (10)
As it was mentioned before, there is no deflection at the built-in ends, thus the deflection due to the lateral force eq. 5 (Fig. 4b) have to be compensated by the deflection due to the reaction moments eq. 8 (Fig. 4c):
1 = −1
(11)
By solving the eq.11, the reaction moments M1 and M2 can be calculated, and the general solution is given by the set of following equations: M1 =
˛1 − ˇ2 ˇ2 − ˛2
(12)
M2 =
ˇ1 − ˛2 ˇ2 − ˛2
(13)
In the last step, the total deformation of the clamped-clamped beam as in Fig. 4d is calculated as a superposition of the partial solutions, that is the deflection curves given by the eq. 3 (Fig. 4a) and
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eq. 7 (Fig. 4c). For the right part of the beam it takes the following form: z (di ) = − +
Fi sinh pdi Fd M1 sinh px + i i x + S Sp sinh pl Sl
M2 l − x S
l
−
sinh p(l − x)
x l
−
sinh px sinh pl
(14)
sinh pl
In case, two lateral loads are applied to the beam (Fig. 4e), the superposition method is used again. The total deformation is considered to be the sum of the deformations of the beam produced by each load separately: ztotal =
zi (di , Fi )
(15)
i
For the electrostatic model [7,8], the switch is approximated by two parallel-plate capacitors formed between the top mobile electrodes and the CPW with an air-gap and a dielectric layer inbetween. Directly from this model it is possible to calculate the up-state and down-state capacitances of the switch provided that the electrode gaps and the thickness of the dielectric are known. The electrostatic force acting on the membrane is then given by the equation: ∂W (z) 1 = 2 ∂z
ε0 A Go − z +
td εr
2 V
2
(16)
where W is the energy stored in the capacitor, ε0 is the vacuum permittivity, A is the area of the electrode, G0 is the electrode gap, td is the thickness of the dielectric and εr is the relative permittivity of the dielectric. Because of a small gap between the electrodes (<1 m) and large electrodes’ area (>4000 m2 each) the fringing field correction can be omitted [9]. When a voltage is applied, the electrostatic force acts on the top membrane which is balanced by the mechanical force proportional to membrane deflection Fmech. = kz, where k is the beam stiffness calculated from the following equation: k=
F1 + F2 z1 (d1 , F1 ) + z2 (d2 , F2 )
(17)
where x1 and x2 are the lateral positions of the electrodes’ centers, where the concentrated electrostatic forces F1 and F2 are applied. The eq. 16 can be rewritten in the form which gives the relation between the voltage and the electrode gap height: V (z) =
t G0 − z + d εr
kz
ε0 A
Fr = Gc kr
(19)
where kr is the free membrane stiffness at its center (i.e. before contact) and Gc is the contact gap height. The contact force Fc is exerted when the deflection of the membrane is larger than the contact gap height Gc , that is after the transmission line is short-circuited. When the contact is closed, the stiffness at the electrode center increases, as there is no further displacement possible at the membrane center. This stiffness kc is than calculated using the equation (14) for a half-length beam and one vertical force applied at the center of the electrode. The Fc is then calculated as a sum of the reaction forces in the contact part coming from the two deformed side electrodes, hence it is equal to: Fc =
3.2. Electro-mechanical model
Fel. (z) = −
The restoring force Fr is given by the equation:
2 (G0 − Gc ) kc d1 l
(20)
And the value of Gc where the restoring and contact force are equal is: Gc = G0
˛ 2
1 −1
(21)
where: ˛=
kr l kc d1
(22)
4. Results and discussion 4.1. Discussion of mechanical model The nominal values of the length, width, thickness and residual stress used in this study are 340 m, 50 m, 600 nm and 300MPa, respectively. One mobile vertical load is applied at the center of the beam and the free stiffness is calculated as kf = F1 /z(d1 ,F1 ). The nominal free membrane stiffness at the beam center calculated with the mechanical model is then around 112N/m which is similar to the value obtained from FEM simulation [10]. It is worth mentioning that in case the residual stress is neglected in the model, the calculated value of the stiffness drops to 1N/m. The influence of thickness, width and residual stress on the nominal value of the stiffness is assessed using the mechanical model. It turns out that all of these parameters have a linear effect on the stiffness and none of them is dominating within the limits of 0.5 to 1.5 of the nominal values (Fig. 5). As none of these parameters has a stronger influence on the stiffness it is the residual stress which is arbitrary chosen as a fitting
(18)
The maximum value of the voltage given by this equation in function of the beam deformation z is the capacitive pull-in voltage. 3.3. Contact and restoring forces For ohmic RF MEMS switches the RF signal passes through the transmission line which is short-circuited by the dimples of the mobile contact. The signal loss depends on the resistance of this contact, which may depend, among other things, on the contact force at which the top contact is pressed to the transmission line. Acting in opposite direction there is a restoring force which allows releasing the switch when no voltage is applied. It is important to keep a good balance between these two forces as a low contact force may decrease the resistance while an insufficient restoring force may result in permanent stiction.
Figure 5. Stiffness variation at the membrane center in function of process parameters.
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Figure 6. Results of the nanoindentation measurement of A-type switch at the membrane center.
Figure 8. Results of the nanoindentation measurement of B-type switch at the membrane center.
parameter. The reason is that the geometry of the switch is very well controlled during the fabrication process and it remains stable. The residual stress is measured during the deposition phase and can be adjusted easily by managing the Si/N ratio. The deposited silicon nitride layer is then submitted to further fabrication steps which cause stress relaxation. After all, the real value of the resiudal stress is lower than the one measured after deposition (typically it is 300MPa for standard conditions).
Table 1 Summary of the nanoindentation results of A-type switch.
Stiffness [N/m]
Centre 2-3
137
Displacement [nm]
2-3
140
Electrode 2-3 3-4 2-3 3-4
174 244 220 420
The results of the nanoindentation at the membrane and at the electrode centers for A-type and B-type switches are presented and discussed here. The figures (Figs. 6-9) show the quasistatic force-displacement and the harmonic stiffness-displacement (CSM) curves. The stiffness obtained from these two methods is comparable but for CSM the noise is higher. The stiffness of the switch is calculated from the slope of the force-displacement curve, while the harmonic stiffness is used to detect any changes of its value. The A-type switch has a usual mechanical behavior as shown in Fig. 6 (membrane center) and Fig. 7 (electrode center). The membrane center stiffness presents one step 2-3 (Fig. 6) which corresponds to the membrane free stiffness before the dimples of the metallic mobile contact and the transmission line get into contact. The presence of only one step and the good stability of the stiffness suggest a good contact quality, as both dimples short-circuit the transmission line simultaneously which means that neither deformation of the contact nor twisting of the membrane are observed. The electrode center stiffness (Fig. 7) presents
2 steps: the first segment 2-3 before contact at the membrane center and the second one 3-4 after contact at the membrane center (Fig. 7). The summarized results of the nanoindentation for A-type switch are presented in Table 1. The measured value of the stiffness in the central part (137N/m) is higher than the nominal one (112N/m) which is probably due to a higher residual stress in the silicon nitride. In addition, a small contact gap (140 nm) compared to the electrode gap (640 nm) is measured which will result in a contact force significantly higher than the restoring one. The B-type switch does not show the same behavior as the Atype (Figs. 8 and 9). The membrane center stiffness (Fig. 8) shows an additional step (3-4). The first step corresponds to the membrane free stiffness (2-3). The (3-4) segment indicates either a two phase contact between the metallic contact and the transmission line (each dimple of the metallic contact could make contact successively–twisting of the membrane) or the deformation of the dimple. The behavior at the electrode center (Fig. 9) is similar to the Atype switch with 2 steps: one before and another one after contact at the membrane center.
Figure 7. Results of the nanoindentation measurement of A-type switch at the electrode center.
Figure 9. Results of the nanoindentation measurement of B-type switch at the electrode center.
4.2. Nanoindentation results
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Table 2 Summary of the nanoindentation results of B-type switch. Centre Stiffness [N/m] Displacement [nm]
2-3 3-4 2-3 3-4
Electrode 79 380 500 220
2-3
102
2-3
750
Experimental stiffness [N/m] Theoretical fitted stiffness [N/m] Fit inaccuracy [%] Fitted residual stress [MPa]
Center A type 137 141 3 390
B type 79 77 2 250
Electrode A type 174 170 3 390
B type 102 103 1 250
difference can be linked with the modification of the process flow for manufacturing the contact part as it is mentioned in paragraph 2.1. 4.3. Fitting of the mechanical model
Figure 10. SEM image of the surface of the membrane dimple of A-type switch – homogenous surface.
The summarized results of the nanoindentation for B-type switch are presented in Table 2. The free stiffness at the membrane center (79 N/m) is lower than for A-type (137 N/m) due to a lower residual stress in the silicon nitride layer. Moreover, the difference between both gaps does not exceed 50-100 nm which will decrease the difference between the contact and restoring forces. The hypothesis of a two phase contact between the dimples of the metallic contact and the transmission line is confirmed by a failure analysis. The SEM observations (Figs. 10 and 11) show that the dimples for A-type (Fig. 10) have a smooth and quite homogeneous surface, while for B-type switches (Fig. 11) they are less homogeneous with high peaks which can explain the stiffness trend. It is assumed that after the first contact on the highest peak, the membrane could twist to put the second one in contact also, which results in an increase of the stiffness as observed in Fig. 9. The deformation of the edges of the dimple may increase the stiffness. This
The experimental results are fitted with the mechanical model from the paragraph 3.1. A mobile lateral load is applied at the same position as the tip of the nanoindenter and the residual stress is used as the fitting parameter. The stiffness is calculated as in the paragraph 4.1. The experimental and fitted results for the free membrane stiffness are summarized in the Table 3 and show a good agreement with a maximum error below 3%. The difference of the residual stress between the two runs of switches is confirmed. The residual stress is respectively 390 and 250 N/m for A-type and B-type switches which is in agreement with the residual stress measured at wafer level after deposition. 4.4. Estimation of the restoring and the contact force The calculated contact and restoring forces are presented in Table 4. The two switches show completely different behavior as the etching depths are different. In case of A-type switch, the contact gap is relatively small when compared to the electrode gap which results in a high contact force and a low restoring force. In case of B-type, the difference between both gaps is small which results in similar contact and restoring forces. A low restoring force may cause stiction problems which can be emphasized by the homogeneity of the surface of the dimple which may increase the adhesion forces. This is in agreement with what is observed from preliminary bipolar cycling test, which showed that A-type switch is more likely to stick than B-type, where both forces are better balanced. When these two forces are plotted as a function of the contact gap (Fig. 12), it is clear that a small ratio between the contact and electrode gap results in a contact force which is much larger than the restoring force. The figure illustrates also that the contact force is more sensible to the contact gap than the restoring force. This trend is explained by the fact that the contact force depends on the membrane stiffness at the electrode center kc after the contact in the central part, which is significantly higher than the free membrane stiffness kr at the membrane center used to evaluate the restoring force. 4.5. Electrostatic results Table 5 summarizes the electrostatic results obtained for the two types of switches, where Cup is the capacitance of the actuator Table 4 Contact and restoring forces with the values of stiffness used for the calculations.
Figure 11. SEM image of the surface of the membrane dimple of B-type switch – particles on the surface and high edge profile.
A type B type
kc [N/m]
kr [N/m]
Fc [N]
Fr [N]
362 306
141 77
200 65
20 45
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Figure 14. Single sweeping test for B-type switch. Figure 12. Estimation of the contact and restoring forces for A-type and B-type switches as a function of the contact gap (the gray vertical line indicates the measured contact gap height).
Table 5 Up-state and down-state capacitance given by the electrical measurements and calculated from the electro-mechanical model.
A type B type
Measured Cdn [pF] 0.02 0.67
Cfdn [pF] 0.95 1.07
Theoretical Cdn [pF] 0.04 0.43
model, that is 15 V and 13 V for A-type and B-type switches, respectively. 5. Conclusions
Cfdn [pF] 0.94 1.07
It is showed in this paper that the nanoindentation technique may be successfully used for analyzing the behavior of the RF MEMS switches. Thanks to an analytical electro-mechanical model it is possible to fit the measured data and to predict the electrostatic behavior of the switches. The discrepancy between the measured and fitted stiffness values are below 3%. The stiffness varies from 79 up to 137N/m between the runs, which is explained by the difference of the tensile residual stress. Moreover, it is possible to evaluate the quality of the contact from the run of the CSM and force-displacement curves. Further analysis of the mechanical model allows predicting the contact and restoring forces, which may be useful for the prediction of the probability of stiction. To conclude, the nanoindentation technique turns out to be useful for evaluating the electrostatic behavior of RF MEMS switches. References
Figure 13. Single sweeping test for A-type switch.
in the up-state position, Cdn is the capacitance of the actuator in the down-state position for contact switching, C = Cdn -Cup and Cfdn is the capacitance of the actuator with the electrodes in contact. The capacitances calculated by using the gap heights from the nanoindentation experiments show no discrepancy compared to the ones measured by C-V sweep tests. Thus the pertinence of the measurements of the gap heights by the nanoindentation is confirmed. The complete C-V and R-V sweeping tests (Figs. 13 and 14) illustrate how the two gap heights and the stiffness manage the electrostatic behavior. A-type switch (Fig. 13) has a significant shift between the ohmic (8 V) and capacitive (16 V) responses due to different gap heights. For the B-type switch (Fig. 14), there is no significant shift between the two responses (14 V) due to similar gap heights. In practice, A-type and B-type switches have similar pull-in voltages for the capacitive response. In fact, the larger electrode gap of B-type is compensated by its lower stiffness which is the reason that the capacitive pull-in voltages are similar for both types. Comparable pull-in voltages are obtained from the electro-mechanical
[1] D. Mercier, P.L. Charvet, P. Berruyer, C. Zanchi, L. Lapierre, O. Vendier, J.L. Cazaux, P. Blondy, A DC to 100 GHz high performance ohmic shunt switch, Microwave Symposium Digest, IEEE MTT-S International Volume 3 (2004) 1931–1934. [2] F. Souchon, P. Charvet, C. Maeder-Pachurka, M. Audoin, Dielectric Charging Sensitivity on MEMS Switches, Proc. Transducers ‘07 (2007) 363–366. [3] S. Timoshenko, Strength of Materials, Krieger Pub Co, 1976. [4] L.X. Zhang, Y.-P. Zhao, Electromechanical model of RF MEMS switches, Microsys. Tech 9 (2003) 420–426. [5] X. Wang, L.P.B. Katehi, D. Peroulis, AC actuation of fixed-fixed beam MEMS switches, in: Proc. 6th Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems SiRF ‘06, 2006, pp. 24–27. [6] B. Choi, E.G. Lowell, Improved analysis of microbeams under mechanical and electrostatic loads, J. Micromech. Microeng. 7 (1997) 24–29. [7] J.E. Massad, H. Sumali, D.S. Epp, C.W. Dyck, Modeling, simulation and testing of the mechanical dynamics of an RF MEMS switch, in: Proc. Inter. Conf. on MEMS, NANO and Smart Systems ICMENS’05, 2005, pp. 237–240. [8] G.M. Rebeiz, RF MEMS: Theory, Design, and Technology, Wiley-Interscience, New Jersey, 2002. [9] S. Chowdhury, M. Ahmadi, W.C. Miller, Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model, J. Microelec-tromech. Sys. 15 (2006) 639–651. [10] A. Koszewski, F. Souchon, Mechanical modeling of electrostatically actuated RF MEMS ohmic switch with two side electrodes–analysis of various loading conditions, in: Proc. of WCCM8/ECCOMAS 2008, 2008.
Biographies Frederic Souchon got his Engineering Degree and PhD from Institut National des Sciences Appliquées in Lyon respectively in 1993 and 1996. Between 1996 and 2004,
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he was involved in several MEMS start-ups: at SILMAG, as Test Engineer in charge of reliability studies, at ALDITECH, as Reliability and Test Manager, and at VARIOPTIC, as Reliability Manager. Since 2004, he has been working as Reliability Expert at CEALETI in the Characterization and Reliability Assessment Laboratory. More precisely, he has been leading all the reliability studies on MEMS switches. Adam Koszewski was born in 1983 in Warsaw, Poland. He received his M.Sc. degree in mechanical engineering (micromechanics) from Warsaw University of Technol-
ogy (WUT), Poland in 2007. From 2005 until 2007 he was with WUT, where he was involved in testing mechanical and tribological properties of materials for MEMS and polymeric resists for the Nanoimprint Litography (NIL). From 2007 he is with CEA-LETI, where he is currently working toward PhD. His research on reliability of electrostatically actuated MEMS switches aim in better understanding of their failure mechanisms, in particular, the dielectric charging phenomenon and contact degradation.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
Study of mechanical properties of RF MEMS switches by nanoindentation: characterization and modeling of electro-mechanical behavior Adam Koszewski ∗ , Diane Levy, Frédéric Souchon 1 CEA-LETI-MINATEC, Grenoble, France
a r t i c l e
i n f o
Article history: Received 4 December 2009 Received in revised form 30 March 2010 Accepted 13 May 2010
Keywords: MEMS switch nanoindentation electrostatic mechanical clamped-clamped model
a b s t r a c t This paper presents a thorough methodology to evaluate the mechanical properties of RF MEMS switches. It includes an analytical electro-mechanical model and nanoindentation characterizations which allow explaining the electrostatic behavior of the MEMS switches. Experimental and theoretical aspects are compared through two runs of ohmic electrostatic switches which are different from geometric and process flow standpoint. The switch is modeled as a clamped-clamped beam with mobile vertical loads, two stretching axial forces and two reaction moments in the anchors. The mechanical properties are characterized by nanoindentation technique; specific test protocol for measuring very low stiffness has been specially implemented with an adapted continuous stiffness measurement method. The measured values of the stiffness at the membrane center are respectively 79 and 137 N/m for the two runs, and they remain in good agreement with the theoretical model. The difference between these two runs is explained by the difference of the tensile residual stress. Moreover, the nanoindentation data turn out to be useful to evaluate the contact quality: dimple deformation due to twisting of the beam (contact in multiple steps), or inhomogeneous surface (stiffness is increasing while in contact). The fitted model is also used to approximate the restoring and contact forces and to point out the influence of the mechanical properties on the electrostatic behavior. A simple model with two parallel-plate capacitors in parallel, used for calculation of the up- and down-state capacitances, shows no discrepancy with experimental values. Finally, it may be concluded that the electrostatic behavior of the switches (pull-in voltage, up- and down-state capacitances, and probability of stiction) can be better understood thanks to the mechanical characterizations made by nanoindentation. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Over the past years CEA-LETI has been developing electrostatic RF MEMS ohmic switches. Their key parameters, such as pull-in voltage, contact and restoring forces and up- and down-state capacitances depend on the stiffness of the switch and its geometry. Among the standard testing techniques used to characterize RF MEMS switches there are electrical tests for measuring the pullin and pull-out voltages and the up- and down-state capacitances, AFM and SEM observations for assessing the contact quality and laser vibrometry for determining the gap heights and the dynamic mechanical behavior.
∗ Corresponding author. 17, rue des Martyrs, 38054 Grenoble Cedex 9, France. Tel.: +33 4 38 78 18 53; fax: +33 4 38 78 51 40. E-mail addresses: [email protected] (A. Koszewski), [email protected] (F. Souchon). 1 Tel.: +33 4 38 78 28 68. 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.05.011
In this paper the nanoindentation technique is used for the evaluation of the behavior of RF MEMS switches. The nanoindentation technique allows not only to measure the membrane stiffness and the gap heights, but also to evaluate the contact quality. Moreover, the nanoindentation measurements can be coupled with an electro-mechanical model in order to improve the understanding of the electrostatic behavior of the RF MEMS switches. 2. Experimental details 2.1. Switch description The ohmic electrostatic switch, which is used in this study, is designed for RF applications and is implemented on a coplanar wave guide (CPW) using a full wave analysis [1]. The series switch (Fig. 1) is a silicon nitride clamped-clamped beam with patterned metallic contacts and two symmetrical side electrodes, which are located inside the beam and are used for
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Figure 3. Position of the indenter tip for the nanoindentation experiments.
2.2. Experimental setup
Figure 1. Top view of an ohmic electrostatic series switch manufactured by CEALETI.
actuation. When a biasing voltage is applied between these electrodes and the CPW ground plan, the membrane is pulled down and the transmission line is short-circuited by the two dimples of the metallic mobile contact. The main fabrication steps of the switch (Fig. 2) comprise: - Thermal oxidation of a silicon wafer, followed by etching steps of a cavity with bumps, - Fabrication of the CPW: deposition and patterning of a gold layer in order to define the CPW and the RF lines, - Spin coating and patterning of a photoresist sacrificial layer, - Fabrication of the beam: deposition of the first silicon nitride layer (beam), followed by deposition of a TiN layer (electrodes); patterning of the electrodes and deposition of the second silicon nitride layer, - Fabrication of the contact: silicon nitride etching and deposition of a gold layer followed by deposition of the last (third) silicon nitride layer, - Opening of the pads and the beam, stripping of the sacrificial layer in dry oxygen plasma. Up to now various types of switches have been designed and manufactured; they are different from a geometric and process flow standpoint. This paper investigates two types of switches denoted as A-type and B-type coming from two manufacturing runs. The main differences between these two runs concern:
The mechanical properties of the switches are characterized using the nanoindentation technique. In this technique a vertical concentrated load is applied to the indenter tip. The tip is approaching the sample surface, and when in contact, it exerts a quasi-static load on it. This load and the tip displacement are recorded simultaneously. The stiffness is the slope of a recorded force-displacement curve (Figs. 6-9). Apart from the quasi-static measurement, a continuous stiffness measurement mode (CSM) is also used. In this mode, the stiffness between the indenter tip and the beam is continuously measured by superimposing simultaneously an oscillating force. Here it is used to detect each change in the measured stiffness, which may be due to a contact in multiple steps or a deformation of the dimples, thus it is used to evaluate the quality of the contact. In order to keep the switch working after the tests, a maximum stiffness limit is set for each measurement. If the measured stiffness exceeds this limit the measurement is stopped. The nanoindentation experiments (Fig. 3) are performed at the membrane center (geometrical center of the beam) and at the electrode center (geometrical center of the electrode, which is around 1/4 of the beam length). The nanoindentation at the membrane center gives the free membrane stiffness and the contact gap height (distance between the dimples of the mobile contact and the transmission line).The nanoindentation at the electrode center gives the stiffness before and after the contact in the central part and the electrode gap height (distance between the electrode and the CPW). Functional characterization of the switches (standard C-V and R-V sweep tests) are carried out using a dedicated test-bench [2] equipped with a HP4284 impedance meter and Keithley 2400 source meter. The measurements are recorded through a voltage ramp, usually from 0 V to 40 V with a voltage step of 1 V. 3. Coupled electro-mechanical model 3.1. Mechanical model
- Beam length: longer beam for B-type, - Residual stress in the membrane: lower for B-type, - Gap heights: larger electrode gap and smaller contact gap for Atype (these gaps are managed by controlling the etching depths), - Contact: different process flows have been used to realize the mobile contact (modification of the process of etching of the silicon nitride and the sacrificial layer before deposition of the gold layer).
Figure 2. Schematic stack of the ohmic electrostatic series switch.
For the mechanical model, the switch (Fig. 4e) is approximated by a clamped-clamped beam with two lateral mobile loads, two stretching axial forces and two reaction moments at the ends [3]. The mobile lateral loads F1 and F2 correspond to the electrostatic actuation forces when they are applied at the geometrical center of the electrodes. The axial forces S represent the stretching effect of the residual stress which is unavoidable in micro fabrication process and has a significant influence on the mechanical properties of the structure [4,5]. The axial stretching component due to deflection of the beam is omitted as the typical maximum deflection of the beam does not exceed its thickness [4]. The couples M1 and M2 are the reaction moments, which cancel the deflection in the built-in ends. The most significant difference between our model and those presented in the literature is the fact that the deflection curve here, is calculated for one lateral load applied at any point along the beam, and not only in the beam center [4,6]. This is the partial solution, which is then used to obtain the deflection curve of a beam with two lateral loads, which correspond to electrostatic forces in a real switch.
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lateral position, l is the length of the beam, w is its width and t is its thickness. For the left part of the beam, it is sufficient to replace the l by (l-d1 ) and x by (l-x) in the corresponding equations. After necessary transformations, the solution of eq. 1 for the right part is: z=−
F1 sinh pd1 F1 d1 sinh px + x Sp sinh pl Sl
(3)
where: S EIz
p2 =
(4)
From the eq.3 the slopes of the deflection curve at both ends can be calculated, as the first derivative of z for x = 0 and x = l, and at right end the slope is equal to:
F1 d1 dz F1 sinh pd1 = − Sp sinh pl + Sl dx x=0
1 =
(5)
In case of the clamped-clamped beam, there is no deflection in both built-in ends, thus the slopes calculated from (eq.5) are compensated by the reaction moments M1 and M2 . To find the M1 and M2 the deformation of a clamped-free beam with two reaction moments M1 and M2 and two axial forces S, in the absence of the lateral load F1 is calculated (Fig. 4c). The slopes at both ends, calculated for these conditions, have to be opposite to those given by (eq.5). The deflection curve for a single moment M may be obtained immediately from the eq. 3 by making the following substitution: F1 d1 = M and sinhpd1 = pd1 [3], and it is: z=
M S
x
−
l
sinh px sinh pl
(6)
The total deformation is a sum of the deformations caused by each moment (the reaction moments at each end) which is: z=
M1 S
x l
−
sinh px sinh pl
+
M2 S
l − x l
−
sinh p(l − x) sinh pl
And the deflection at the right end, is:
Figure 4. Mechanical model: schematic representation of the RF MEMS swudy of mechanical properties of RThe stages of deriving the mechanical model (a)-(d).
1 =
dx dz
x=0
=
M1 S
= M1 ˛ + M2 −ˇ The problem of a clamped-clamped beam with the loading conditions as in (Fig. 4e) is statically indeterminable. To find the deflection of this overrigid beam the same approach as in [3] is used. In this approach the partial solutions for simplified loading conditions (Fig. 4a-d) are found and the general solution is a superposition of these partial solutions. First, the deflection for a clamped-free beam with one mobile vertical load F1 and two axial forces S (Fig. 4a) is calculated by solving a set of Euler-Bernoulli beam equations. The deflection curve (eq. 1) at any point depends only on the magnitude of the bending moments at that point. In our case the bending moments come from the lateral force F1 and axial force S (the right hand side of the eq. 1), thus the equation can be written in the following form, for the right part of the beam: EIz
d2 z F1 d1 x = Sz − l dx2
(1)
where Iz is the moment of inertia of the beam: Iz =
wt 3 12
(2)
and E is the Young’s modulus, z is the deformation of the beam, S is the axial force originating from the residual stress, F1 is the mobile lateral load applied at the distance d1 from the end, x is the current
1 l
−
p sinh pl
−
M2 S
1 l
−
p cosh pl sinh pl
(7)
(8)
where: ˛=
1 p − l sinh pl
ˇ=
1 p cosh pl − l sinh pl
(9) (10)
As it was mentioned before, there is no deflection at the built-in ends, thus the deflection due to the lateral force eq. 5 (Fig. 4b) have to be compensated by the deflection due to the reaction moments eq. 8 (Fig. 4c):
1 = −1
(11)
By solving the eq.11, the reaction moments M1 and M2 can be calculated, and the general solution is given by the set of following equations: M1 =
˛1 − ˇ2 ˇ2 − ˛2
(12)
M2 =
ˇ1 − ˛2 ˇ2 − ˛2
(13)
In the last step, the total deformation of the clamped-clamped beam as in Fig. 4d is calculated as a superposition of the partial solutions, that is the deflection curves given by the eq. 3 (Fig. 4a) and
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eq. 7 (Fig. 4c). For the right part of the beam it takes the following form: z (di ) = − +
Fi sinh pdi Fd M1 sinh px + i i x + S Sp sinh pl Sl
M2 l − x S
l
−
sinh p(l − x)
x l
−
sinh px sinh pl
(14)
sinh pl
In case, two lateral loads are applied to the beam (Fig. 4e), the superposition method is used again. The total deformation is considered to be the sum of the deformations of the beam produced by each load separately: ztotal =
zi (di , Fi )
(15)
i
For the electrostatic model [7,8], the switch is approximated by two parallel-plate capacitors formed between the top mobile electrodes and the CPW with an air-gap and a dielectric layer inbetween. Directly from this model it is possible to calculate the up-state and down-state capacitances of the switch provided that the electrode gaps and the thickness of the dielectric are known. The electrostatic force acting on the membrane is then given by the equation: ∂W (z) 1 = 2 ∂z
ε0 A Go − z +
td εr
2 V
2
(16)
where W is the energy stored in the capacitor, ε0 is the vacuum permittivity, A is the area of the electrode, G0 is the electrode gap, td is the thickness of the dielectric and εr is the relative permittivity of the dielectric. Because of a small gap between the electrodes (<1 m) and large electrodes’ area (>4000 m2 each) the fringing field correction can be omitted [9]. When a voltage is applied, the electrostatic force acts on the top membrane which is balanced by the mechanical force proportional to membrane deflection Fmech. = kz, where k is the beam stiffness calculated from the following equation: k=
F1 + F2 z1 (d1 , F1 ) + z2 (d2 , F2 )
(17)
where x1 and x2 are the lateral positions of the electrodes’ centers, where the concentrated electrostatic forces F1 and F2 are applied. The eq. 16 can be rewritten in the form which gives the relation between the voltage and the electrode gap height: V (z) =
t G0 − z + d εr
kz
ε0 A
Fr = Gc kr
(19)
where kr is the free membrane stiffness at its center (i.e. before contact) and Gc is the contact gap height. The contact force Fc is exerted when the deflection of the membrane is larger than the contact gap height Gc , that is after the transmission line is short-circuited. When the contact is closed, the stiffness at the electrode center increases, as there is no further displacement possible at the membrane center. This stiffness kc is than calculated using the equation (14) for a half-length beam and one vertical force applied at the center of the electrode. The Fc is then calculated as a sum of the reaction forces in the contact part coming from the two deformed side electrodes, hence it is equal to: Fc =
3.2. Electro-mechanical model
Fel. (z) = −
The restoring force Fr is given by the equation:
2 (G0 − Gc ) kc d1 l
(20)
And the value of Gc where the restoring and contact force are equal is: Gc = G0
˛ 2
1 −1
(21)
where: ˛=
kr l kc d1
(22)
4. Results and discussion 4.1. Discussion of mechanical model The nominal values of the length, width, thickness and residual stress used in this study are 340 m, 50 m, 600 nm and 300MPa, respectively. One mobile vertical load is applied at the center of the beam and the free stiffness is calculated as kf = F1 /z(d1 ,F1 ). The nominal free membrane stiffness at the beam center calculated with the mechanical model is then around 112N/m which is similar to the value obtained from FEM simulation [10]. It is worth mentioning that in case the residual stress is neglected in the model, the calculated value of the stiffness drops to 1N/m. The influence of thickness, width and residual stress on the nominal value of the stiffness is assessed using the mechanical model. It turns out that all of these parameters have a linear effect on the stiffness and none of them is dominating within the limits of 0.5 to 1.5 of the nominal values (Fig. 5). As none of these parameters has a stronger influence on the stiffness it is the residual stress which is arbitrary chosen as a fitting
(18)
The maximum value of the voltage given by this equation in function of the beam deformation z is the capacitive pull-in voltage. 3.3. Contact and restoring forces For ohmic RF MEMS switches the RF signal passes through the transmission line which is short-circuited by the dimples of the mobile contact. The signal loss depends on the resistance of this contact, which may depend, among other things, on the contact force at which the top contact is pressed to the transmission line. Acting in opposite direction there is a restoring force which allows releasing the switch when no voltage is applied. It is important to keep a good balance between these two forces as a low contact force may decrease the resistance while an insufficient restoring force may result in permanent stiction.
Figure 5. Stiffness variation at the membrane center in function of process parameters.
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Figure 6. Results of the nanoindentation measurement of A-type switch at the membrane center.
Figure 8. Results of the nanoindentation measurement of B-type switch at the membrane center.
parameter. The reason is that the geometry of the switch is very well controlled during the fabrication process and it remains stable. The residual stress is measured during the deposition phase and can be adjusted easily by managing the Si/N ratio. The deposited silicon nitride layer is then submitted to further fabrication steps which cause stress relaxation. After all, the real value of the resiudal stress is lower than the one measured after deposition (typically it is 300MPa for standard conditions).
Table 1 Summary of the nanoindentation results of A-type switch.
Stiffness [N/m]
Centre 2-3
137
Displacement [nm]
2-3
140
Electrode 2-3 3-4 2-3 3-4
174 244 220 420
The results of the nanoindentation at the membrane and at the electrode centers for A-type and B-type switches are presented and discussed here. The figures (Figs. 6-9) show the quasistatic force-displacement and the harmonic stiffness-displacement (CSM) curves. The stiffness obtained from these two methods is comparable but for CSM the noise is higher. The stiffness of the switch is calculated from the slope of the force-displacement curve, while the harmonic stiffness is used to detect any changes of its value. The A-type switch has a usual mechanical behavior as shown in Fig. 6 (membrane center) and Fig. 7 (electrode center). The membrane center stiffness presents one step 2-3 (Fig. 6) which corresponds to the membrane free stiffness before the dimples of the metallic mobile contact and the transmission line get into contact. The presence of only one step and the good stability of the stiffness suggest a good contact quality, as both dimples short-circuit the transmission line simultaneously which means that neither deformation of the contact nor twisting of the membrane are observed. The electrode center stiffness (Fig. 7) presents
2 steps: the first segment 2-3 before contact at the membrane center and the second one 3-4 after contact at the membrane center (Fig. 7). The summarized results of the nanoindentation for A-type switch are presented in Table 1. The measured value of the stiffness in the central part (137N/m) is higher than the nominal one (112N/m) which is probably due to a higher residual stress in the silicon nitride. In addition, a small contact gap (140 nm) compared to the electrode gap (640 nm) is measured which will result in a contact force significantly higher than the restoring one. The B-type switch does not show the same behavior as the Atype (Figs. 8 and 9). The membrane center stiffness (Fig. 8) shows an additional step (3-4). The first step corresponds to the membrane free stiffness (2-3). The (3-4) segment indicates either a two phase contact between the metallic contact and the transmission line (each dimple of the metallic contact could make contact successively–twisting of the membrane) or the deformation of the dimple. The behavior at the electrode center (Fig. 9) is similar to the Atype switch with 2 steps: one before and another one after contact at the membrane center.
Figure 7. Results of the nanoindentation measurement of A-type switch at the electrode center.
Figure 9. Results of the nanoindentation measurement of B-type switch at the electrode center.
4.2. Nanoindentation results
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Table 2 Summary of the nanoindentation results of B-type switch. Centre Stiffness [N/m] Displacement [nm]
2-3 3-4 2-3 3-4
Electrode 79 380 500 220
2-3
102
2-3
750
Experimental stiffness [N/m] Theoretical fitted stiffness [N/m] Fit inaccuracy [%] Fitted residual stress [MPa]
Center A type 137 141 3 390
B type 79 77 2 250
Electrode A type 174 170 3 390
B type 102 103 1 250
difference can be linked with the modification of the process flow for manufacturing the contact part as it is mentioned in paragraph 2.1. 4.3. Fitting of the mechanical model
Figure 10. SEM image of the surface of the membrane dimple of A-type switch – homogenous surface.
The summarized results of the nanoindentation for B-type switch are presented in Table 2. The free stiffness at the membrane center (79 N/m) is lower than for A-type (137 N/m) due to a lower residual stress in the silicon nitride layer. Moreover, the difference between both gaps does not exceed 50-100 nm which will decrease the difference between the contact and restoring forces. The hypothesis of a two phase contact between the dimples of the metallic contact and the transmission line is confirmed by a failure analysis. The SEM observations (Figs. 10 and 11) show that the dimples for A-type (Fig. 10) have a smooth and quite homogeneous surface, while for B-type switches (Fig. 11) they are less homogeneous with high peaks which can explain the stiffness trend. It is assumed that after the first contact on the highest peak, the membrane could twist to put the second one in contact also, which results in an increase of the stiffness as observed in Fig. 9. The deformation of the edges of the dimple may increase the stiffness. This
The experimental results are fitted with the mechanical model from the paragraph 3.1. A mobile lateral load is applied at the same position as the tip of the nanoindenter and the residual stress is used as the fitting parameter. The stiffness is calculated as in the paragraph 4.1. The experimental and fitted results for the free membrane stiffness are summarized in the Table 3 and show a good agreement with a maximum error below 3%. The difference of the residual stress between the two runs of switches is confirmed. The residual stress is respectively 390 and 250 N/m for A-type and B-type switches which is in agreement with the residual stress measured at wafer level after deposition. 4.4. Estimation of the restoring and the contact force The calculated contact and restoring forces are presented in Table 4. The two switches show completely different behavior as the etching depths are different. In case of A-type switch, the contact gap is relatively small when compared to the electrode gap which results in a high contact force and a low restoring force. In case of B-type, the difference between both gaps is small which results in similar contact and restoring forces. A low restoring force may cause stiction problems which can be emphasized by the homogeneity of the surface of the dimple which may increase the adhesion forces. This is in agreement with what is observed from preliminary bipolar cycling test, which showed that A-type switch is more likely to stick than B-type, where both forces are better balanced. When these two forces are plotted as a function of the contact gap (Fig. 12), it is clear that a small ratio between the contact and electrode gap results in a contact force which is much larger than the restoring force. The figure illustrates also that the contact force is more sensible to the contact gap than the restoring force. This trend is explained by the fact that the contact force depends on the membrane stiffness at the electrode center kc after the contact in the central part, which is significantly higher than the free membrane stiffness kr at the membrane center used to evaluate the restoring force. 4.5. Electrostatic results Table 5 summarizes the electrostatic results obtained for the two types of switches, where Cup is the capacitance of the actuator Table 4 Contact and restoring forces with the values of stiffness used for the calculations.
Figure 11. SEM image of the surface of the membrane dimple of B-type switch – particles on the surface and high edge profile.
A type B type
kc [N/m]
kr [N/m]
Fc [N]
Fr [N]
362 306
141 77
200 65
20 45
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Figure 14. Single sweeping test for B-type switch. Figure 12. Estimation of the contact and restoring forces for A-type and B-type switches as a function of the contact gap (the gray vertical line indicates the measured contact gap height).
Table 5 Up-state and down-state capacitance given by the electrical measurements and calculated from the electro-mechanical model.
A type B type
Measured Cdn [pF] 0.02 0.67
Cfdn [pF] 0.95 1.07
Theoretical Cdn [pF] 0.04 0.43
model, that is 15 V and 13 V for A-type and B-type switches, respectively. 5. Conclusions
Cfdn [pF] 0.94 1.07
It is showed in this paper that the nanoindentation technique may be successfully used for analyzing the behavior of the RF MEMS switches. Thanks to an analytical electro-mechanical model it is possible to fit the measured data and to predict the electrostatic behavior of the switches. The discrepancy between the measured and fitted stiffness values are below 3%. The stiffness varies from 79 up to 137N/m between the runs, which is explained by the difference of the tensile residual stress. Moreover, it is possible to evaluate the quality of the contact from the run of the CSM and force-displacement curves. Further analysis of the mechanical model allows predicting the contact and restoring forces, which may be useful for the prediction of the probability of stiction. To conclude, the nanoindentation technique turns out to be useful for evaluating the electrostatic behavior of RF MEMS switches. References
Figure 13. Single sweeping test for A-type switch.
in the up-state position, Cdn is the capacitance of the actuator in the down-state position for contact switching, C = Cdn -Cup and Cfdn is the capacitance of the actuator with the electrodes in contact. The capacitances calculated by using the gap heights from the nanoindentation experiments show no discrepancy compared to the ones measured by C-V sweep tests. Thus the pertinence of the measurements of the gap heights by the nanoindentation is confirmed. The complete C-V and R-V sweeping tests (Figs. 13 and 14) illustrate how the two gap heights and the stiffness manage the electrostatic behavior. A-type switch (Fig. 13) has a significant shift between the ohmic (8 V) and capacitive (16 V) responses due to different gap heights. For the B-type switch (Fig. 14), there is no significant shift between the two responses (14 V) due to similar gap heights. In practice, A-type and B-type switches have similar pull-in voltages for the capacitive response. In fact, the larger electrode gap of B-type is compensated by its lower stiffness which is the reason that the capacitive pull-in voltages are similar for both types. Comparable pull-in voltages are obtained from the electro-mechanical
[1] D. Mercier, P.L. Charvet, P. Berruyer, C. Zanchi, L. Lapierre, O. Vendier, J.L. Cazaux, P. Blondy, A DC to 100 GHz high performance ohmic shunt switch, Microwave Symposium Digest, IEEE MTT-S International Volume 3 (2004) 1931–1934. [2] F. Souchon, P. Charvet, C. Maeder-Pachurka, M. Audoin, Dielectric Charging Sensitivity on MEMS Switches, Proc. Transducers ‘07 (2007) 363–366. [3] S. Timoshenko, Strength of Materials, Krieger Pub Co, 1976. [4] L.X. Zhang, Y.-P. Zhao, Electromechanical model of RF MEMS switches, Microsys. Tech 9 (2003) 420–426. [5] X. Wang, L.P.B. Katehi, D. Peroulis, AC actuation of fixed-fixed beam MEMS switches, in: Proc. 6th Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems SiRF ‘06, 2006, pp. 24–27. [6] B. Choi, E.G. Lowell, Improved analysis of microbeams under mechanical and electrostatic loads, J. Micromech. Microeng. 7 (1997) 24–29. [7] J.E. Massad, H. Sumali, D.S. Epp, C.W. Dyck, Modeling, simulation and testing of the mechanical dynamics of an RF MEMS switch, in: Proc. Inter. Conf. on MEMS, NANO and Smart Systems ICMENS’05, 2005, pp. 237–240. [8] G.M. Rebeiz, RF MEMS: Theory, Design, and Technology, Wiley-Interscience, New Jersey, 2002. [9] S. Chowdhury, M. Ahmadi, W.C. Miller, Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model, J. Microelec-tromech. Sys. 15 (2006) 639–651. [10] A. Koszewski, F. Souchon, Mechanical modeling of electrostatically actuated RF MEMS ohmic switch with two side electrodes–analysis of various loading conditions, in: Proc. of WCCM8/ECCOMAS 2008, 2008.
Biographies Frederic Souchon got his Engineering Degree and PhD from Institut National des Sciences Appliquées in Lyon respectively in 1993 and 1996. Between 1996 and 2004,
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he was involved in several MEMS start-ups: at SILMAG, as Test Engineer in charge of reliability studies, at ALDITECH, as Reliability and Test Manager, and at VARIOPTIC, as Reliability Manager. Since 2004, he has been working as Reliability Expert at CEALETI in the Characterization and Reliability Assessment Laboratory. More precisely, he has been leading all the reliability studies on MEMS switches. Adam Koszewski was born in 1983 in Warsaw, Poland. He received his M.Sc. degree in mechanical engineering (micromechanics) from Warsaw University of Technol-
ogy (WUT), Poland in 2007. From 2005 until 2007 he was with WUT, where he was involved in testing mechanical and tribological properties of materials for MEMS and polymeric resists for the Nanoimprint Litography (NIL). From 2007 he is with CEA-LETI, where he is currently working toward PhD. His research on reliability of electrostatically actuated MEMS switches aim in better understanding of their failure mechanisms, in particular, the dielectric charging phenomenon and contact degradation.