A General and Simplified Method for Rapid Evaluation of Electrostatic Switches with Varying Geometries J. De Coster1, H.A.C. Tilmans2, J.T.M. van Beek3, Th. G.S.M. Rijks3, P.G. Steeneken3, R. Puers1 1

KULeuven, EE Department, Kasteelpark Arenberg 10, B-3001 Heverlee, Belgium 2 IMEC, Kapeldreef 75, B-3001 Leuven, Belgium 3 Philips Research, Prof. Holstlaan 4, 5656AA Eindhoven, the Netherlands

Summary: A set of RF MEMS switches with varying geometries is mechanically and electrically characterised. The measurement results allow to fine-tune the analytical relationship between pull-in voltage and geometrical parameters. This reduces the need for more advanced but time-consuming simulations to predict the electrostatic switching behaviour of a large range of RF MEMS devices. Keywords: RF MEMS switches, mechanical characterisation

1. Introduction Several methods exist for determining the switching voltage of electrostatic RF MEMS switches. In the simplest case, the switch is considered as a lumped system: all the mechanical compliance is attributed to the suspension beams, while the top electrode is considered infinitely rigid. In this 1-dimensional case, the electrostatic switching voltage is given by:

VPI =

8 k g 03 27 ε 0 A

(1)

yields a modified, empirical version of (1). In addition, FEM simulation results are presented in section 6. The residual stress in the structural material is used as a degree of freedom in order to make the simulation results and the measurement data correspond. The residual stress that is thereby obtained, fits rather well with the value that was extracted from measurements on dedicated material characterisation structures such as Guckel rings [7].

where k is the mechanical stiffness of the suspension beams, g0 is the zero-voltage air gap and A is the overlapping electrode area. The switch may also be considered as a distributed system, in which case other tools are required: finite difference schemes can be implemented in circuit simulators such as Spice or Simulink [1], [2], [3], or a coupled electrostatic-mechanical finite element (FEM) code can be used [4], [5]. In the following paragraphs, the validity of the simple 1dimensional approach will be investigated for a number of devices with varying geometries.

(a)

(b)

(c) [8]

(d)

2. Geometries Fig. 1 shows a number of capacitive shunt switches with varying suspension beam geometries, which were fabricated in Philips’ 5µm-Alu PassiTM process [6]. For each of the geometries shown in Fig. 1, an array of devices was implemented: the mechanical stiffness of the suspension beams is varied between 30N/m and 150N/m, while the area of the top electrode ranges from 75x103µm2 to 200x103µm2. In order to assess the validity of (1) for these devices, all variables that appear in the formula were measured individually on a mix of devices as shown in Fig. 1. Next, the measurement results are combined as explained in sections 3 and 4. This

Fig. 1. Optical photographs of capacitive shunt switches with varying geometries of suspension beams.

Table 1. Variables and dimensions in equation (1).

Symbol VPI g0 k A ε0

Quantity Actuation voltage Nominal air gap Stiffness constant Electrode area Permittivity of free space

Units Nm/C m N/m m2 C2/(Nm2)

3. Dimensional Analysis In order to reduce the number of independent variables in eq. (1) and to find a more convenient way to present the measurement data, a dimensional analysis is performed. The variables and their units are listed in Table 1. The Buckingham Π theorem [9] states that the 5 variables listed in Table 1 may be grouped into 5 – 3 independent dimensionless groups and that relation (1) may be rewritten in terms of these dimensionless groups. In order to identify the dimensionless groups (also called Buckingham Π parameters), a set of primary variables has to be selected first. The 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:

k A VPI C N m

⎡ 0 0 −1⎤ ⎢ 1 0 1⎥ ⎢ ⎥ ⎣ −1 2 1⎦

(2)

which has rank 3. The Π parameters are found by solving the following set of dimensional equations:

⎡ k x1,i Ax2,i VPIx3,i Pi ⎤ = 1, i = 1, 2 ⎣ ⎦

Fig. 2. Example of Dektak trace: scan path (bottom) and surface profile (top).

Next, a low-frequency CV measurement was carried out in order to determine the actuation voltage VPI. Together, these measurements yield all the required values to evaluate the Π parameters given in (4). In order to reveal the relationship between the dimensionless groups, the data is plotted in Fig. 3. The data presented in this figure was obtained from a set of devices as shown in Fig. 1, and comprises measurements from 2 wafers. −6

(3)

1

x 10

Original, untuned curve Retrofitted curve Measurement

0.9

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:

Π1 =

VPI2 ε 0 k

A

,

Π2 =

g0

(4)

A

In the next section, measurements will be presented in order to determine the values of (Π1,Π2) for a selection of devices.

4. Measurements Two types of measurements were carried out to characterise the electrostatic switching behaviour. First, a Dektak surface profile measuring system was used to apply predetermined mechanical loads to the movable plates (cf. Fig. 2). The stiffness constant k is then defined as the slope of the leastsquares straight line fitted through several Dektak measurement points obtained with different mechanical loads applied. The zero-voltage air gap g0 can be derived from these measurements as the intercept of the leastsquares straight line. It can also be extracted from the measured capacitance obtained by an RF Sparameter measurement.

0.8 0.7 0.6

P 10.5 0.4 0.3 0.2 0.1 0 2

4

6

8

P

10

12 −3

x 10

2

Fig. 3. Data of Dektak and CV measurements yielding the locus of devices in the (Π1,Π2)-plane.

The dots indicate the characteristics of devices in the (Π1,Π2)-plane. Relation (1) can be rewritten as:

8 ⎛ g0 ⎞ = ⎜ ⎟ A 27 ⎝ A ⎠

VPI ε 0 2

3

⇒

Π 1 = 0.296Π 2 3

(5) k This equation is represented by the solid line in Fig. 3. In order to assess the agreement of the measured data with the original, untuned 1-dimensional formula (5), a least-squares curve of the form y=a·xn was retrofitted on the measured data. This retrofitted least-squares curve is shown as the

dotted line in Fig. 3, and its coefficients are given by:

Π1 = 0.183 ⋅Π 2.73 2

Comparing (5) and (6), two observations are made. It is seen that the theoretically predicted functional relationship y=a·xn fits well with the measurement data. However, the absolute values of the coefficient and exponent deviate substantially from the theoretical values. Rewriting eq. (6) in terms of the standard variables, yields:

4.9 k g 02.7 VPI = 27 ε 0 A0.87

zp

(6)

ls Fig. 4: Geometry studied in FEM simulations.

−6

1

(7)

As Fig. 3 suggests, this formula yields higher actuation voltages for given values of k, g0 and A than eq. (1). Intuitively, however, one would expect lower actuation voltages because of the bending of the movable plate during electrostatic actuation. The reason for this is twofold, as will be confirmed by the FEM simulation results in the next section. In the first place, the value of k was defined as the maximum deflection of the structure when applying a concentrated load. This results in an underestimation of the mechanical stiffness of the device. Secondly, eq. (1) does not take the effect of residual stresses into account whereas eq. (7) – intrinsically – does. From a more pragmatic point of view, one can consider the parameters a and n in (6) as processspecific parameters; once their values are determined, the retrofitted 1-dimensional approach is adequate for a large variety of designs. This is further illustrated by the fact that the measurements presented in Fig. 3 were carried out on 2 wafers which were processed in 2 varieties of the PASSITM process flow. No clustering of the measurement points of the 2 wafers can be observed in the (Π1,Π2)-plane.

5. FEM simulations The studied geometry is presented in Fig. 4. The influence of two geometrical parameters on the plate stiffness and actuation voltage is studied: the lengths ls of the suspension beams and of the perforated plate, zp. Using Coventorware [10] as FEM software, the same ‘experiments’ can be carried out as on the real devices, i.e. determining the mechanical stiffness of the devices for a concentrated point load (mimicking the Dektak measurement) and electromechanical simulations to determine the actuation voltage (mimicking the CV measurement). These simulations were performed for 4 devices with varying sizes, and the resulting points in the (Π1,Π2)-plane are shown in Fig. 5 along with the retrofitted curve that was obtained from the measurement data in Fig. 3.

x 10

0.9 0.8

Original curve Retrofitted curve Coventorware s = 0 MPa Coventorware s = 100 MPa

0.7 0.6

P1

0.5 0.4 0.3 0.2 0.1 0 2

4

6

8

P

10

12 −3

2

x 10

Fig. 5: FEM simulation results for σ = 0 MPa and σ = 100 MPa. The dotted line was fitted on the measurements shown in Fig. 3.

Two sets of points are shown in the figure: the ‘+’ signs designate simulations with zero residual stress in the structural layer, whereas the triangles correspond to devices with a residual tensile stress of 100 MPa in the structural material. For σ = 100 MPa, the fit between the simulations and the measurements is better than for σ = 0 MPa, especially for the devices with large Π2 values. These are the smallest devices, where stress relaxation by in-plane deformation of the suspension beams is only very limited. Therefore, the influence of the residual stress on the actuation voltage is more important for small devices than for the larger ones (with lower Π2 values). The residual stress in the structural layer has also been characterised by means of dedicated test structures, such as double clamped beams and Guckel rings [7] Fout! Verwijzingsbron niet gevonden.. These rings allow determining the tensile stress magnitude by observing the critical length for buckling. Optical inspection of the rings (cf. Fig. 6) revealed a critical buckling length corresponding to a tensile stress of about 125Mpa, which suggests that the fit in Fig. 5 could be further improved by increasing the stress level used in the simulations. As mentioned in section 4, the mismatch between the original and fitted curves in Fig. 3 can be

largely attributed to two phenomena: the bending of the movable plates during actuation and the residual stress. The relative importance of these can be derived from Fig. 5. For small devices (large Π2), the influence of the residual stress (i.e. the vertical distance between the triangle and the corresponding ‘+’ sign) is comparatively much larger than the influence of the bending of the movable plate (i.e. the vertical distance between the ‘+’ signs and the original curve). For large devices (low Π2), the bending of the plate is dominant whereas the residual stress is almost entirely relaxed by in-plane deformation of the structures.

Fig. 6: Surface profile of Guckel rings of varying sizes, only the first and second largest rings are buckled.

6. Conclusions A measurement-based tuning of the simple 1dimensional approach can be used to predict the electrostatic switching behaviour of a large variety of mechanical geometries. A given technology can be characterised by two parameters, which allow to fine-tune the 1-D theory, thereby eliminating the

need for complex and time-consuming coupled electromechanical FEM simulations.

7. Acknowledgement This work was carried out with the financial support of the European Commission, IST-project MEMS2TUNE (IST-2000-28231).

8. References [1] Tinttunen T et al, Int’l Conference on Modeling and Simulation of Microsystems, MSM (San Juan, Puerto Rico, US, 22-25 Apr) pp 166-169, 2002 [2] Fedder GK and Jing Qi, IEEE Int’l Workshop on Behavioral Modeling and Simulation, Orlando, FL, US, 27-28 Oct), 1998 [3] Clark JV et al, Microscale systems: Mechanics & Measurement Symposium, Orlando, FL, US, 8 June) pp 68-75, 2000 [4] Gilbert et al, IEEE MEMS Workshop, Amsterdam, the Netherlands, 29 Jan – 2 Feb) pp 122 – 127, 1995 [5] De Coster et al, International Symposium on Electromagnetic Fields in Electrical Engineering, ISEF (Maribor, Slovenia, 18-20 Sept) pp 577-580, 2003 [6] van Beek JTM et al, Int’l Microelectronics And Packaging Society, IMAPS (Baltimore, US, 911 Oct) pp 467-470, 2001 [7] Guckel H et al, J. Micromech. Microeng. 2 pp 86-95, 1992 [8] Nieminen H et al, US patent no. 6557413B2 [9] Curtis WD et al J. Lin. Alg. Appl. 47 pp 117126, 1982 [10] CoventorwareTM, http://www.coventor.com