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F RINGING F IELD EFFECT IN ELECTROSTATIC ACTUATORS
by
Vitaly Leus and David Elata
T ECHNICAL R EPORT ETR-2004-2 May 2004
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Fringing Field effect in electrostatic actuators Technical report by Vitaly Leus and David Elata Faculty of mechanical engineering Technion - Israel Institute of Technology, Haifa 32000, Israel
1. Introduction Electrostatic actuation is a prevalent method of driving MEMS devices because it is compatible with microfabrication technology and it provides high power density. Electrostatic actuation is achieved by applying a voltage difference between opposite electrodes of deformable capacitors. The induced electrostatic forces deform the capacitor until they are balanced by the restoring mechanical forces. To facilitate the design and analysis of electrostatic actuators, explicit formulae for computing the capacitance as function of the geometrical parameters are required. In many electrostatic actuators that are fabricated by current micromachining processes, the nominal gap between the electrodes is not negligible relative to the lateral dimensions of the deformable capacitor. Therefore, fringing fields are considerable and must be accounted for when modeling the electrostatic forces. Many different formulae for computing fringing fields appear in literature. The purpose of this report is to summarize these formulae, compare their predictions to Finite Element simulations, and suggest an improved formula for the fringing field.
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2. Models of fringing field capacitance presented in the literature In this survey we only consider two-dimensional models of electrostatic capacitors. 2.1. Parallel-plate capacitor with zero thickness plates We begin by considering capacitors with negligible plate thickness, schematically described in Fig. 1. The capacitor electrodes have a width w and a nominal gap of h. Assuming that the electrostatic field between the two electrodes is parallel, the capacitance per unit length is given by C0
w h
(1)
However, due to the finite width of the electrodes, fringing field develop. Two different capacitance approximations are presented in literature for this system. V h
w Figure 1. Two zero-thickness metal strips in parallel.
2.1.1. H. B. Palmer 1927 [1] Using the Schwartz-Christoffel conformal mapping transformation, Palmer evaluated the parallel-plate capacitance per unit length, and derived the approximate analytical formula C
w 1 h
h w
h 2 w ln w h
(2)
2.1.2. R.S. Elliot 1966 [2] In 1966 Elliot calculated the fringing field capacitance of the same system using the SchwarzCristoffel conformal mapping approach. The capacitance per unit length according to Elliot is C
w 1 h
h w ln w h
(3)
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The two analytic results are compared with a finite elements simulation (all the simulations in this report were performed using Ansys 7.0). Figure 2 shows the fringing field capacitance ~ w / h . The parameters selected normalized by C (1) versus the normalized plate width w 0
for the simulation are w
50 , 1 h 10 .
Elliot Palmer Ansys
0.3 0.25
C
0.2
f
0.15 0.1 0.05 0
10
20
30
40
50
w Figure 2a. Comparison between two approximations respective to finite elements.
Relative error, %
0
-2
-4
-6
Elliot error Palmer error -8
-10
10
20
30
40
50
w Figure 2b. Relative error versus the normalized plate width. As shown in Fig. 2, the capacitance according to Palmer approximation (2) is much more accurate, such that the maximal error respective to finite elements simulation is about 1.3% .
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2.2. Parallel-plate capacitor with finite plate thickness In this chapter we consider three types of capacitors in which the plate thickness is not negligible. These models are schematically described in Fig. 3. w
w
V
t
V
t h
h
h
(a)
(b)
d t
w V
h
(c) Figure 3. Three types of electrostatic capacitors. a) Rectangular metal line over a conducting ground plane. b) Two identical rectangular metal lines in parallel. c) Rectangular metal line with two conducting ground planes.
We begin by considering the model presented in Fig. 3a. This geometry is the most prevalent in literature, since the line-to-ground capacitance problem is relevant to VLSI technology. We examine four different capacitance approximations that have been found in the literature and compare them to the finite elements simulations. 2.2.1. Chang 1976 [3] Chang used the Schwartz-Christoffel conformal transformation to derive relatively simple equations and proposed the following approximation
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C
2
2 Rb Ra
ln
(4a)
w 2h
ln( Ra )
1
Rb
p 1 ln 2
p
w 2h
p 1 tanh 1 1 / p
p
ln
p 1 4p
(4b)
(4c)
p 1 4 1 ln p 1 2 p
2 tanh
1
1 p
(4d)
max , p p
2B2 1
(4e) 2B2 1
2
1
(4f) (4g)
B 1 t/h
Provided that w / h 1 , Chang reports that the accuracy of this formula is within 1 percent of the exact value as computed by finite elements.
2.2.2. Yuan and Trick 1982 [4] In 1982 Yuan and Trick presented another simple analytic approximation for the geometry presented in Fig. 3a, which has a direct physical interpretation. They replaced the rectangular line profile with an “oval” one, composed of a rectangle and two half cylinders (Fig. 4).
t /
Figure 4. Metal line over a conducting plane – Yuan and Trick approximation.
The resulting capacitance can be calculated as the sum of a parallel plate capacitor with width w t / 2 and a cylindrical one with r
t / 2 . The resulting capacity is given by
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C
w t/2 h
Provided that w
2 2h ln 1 t
(5) 2h 2h 2 t t
t / 2 and that t
h , the maximal error with respect to finite elements
simulation is reported to be 10 percent.
2.2.3. Sakurai and Tamaru 1983 [5] Sakurai and Tamaru considered the geometry of Fig. 3a, and proposed an improved approximation by evaluating numeric solutions and deriving a correlation in the form
C
w 1.15 h
t 2.80 h
0.22
(6)
The first term represents the bottom and top surfaces of the interconnection line (top rectangle) and the second one relates to the side walls. Within ranges of 0.3 w / h 30 and 0.3 t / h 30 , an accuracy better than 6 percent is reported.
2.2.4. Van de Meijs and Fokkema 1984 [6] In 1984 Van de Meijs and Fokkema improved Sakurai’s approach by extending the empirical expression and simultaneously reducing the range of validity. The first term of their formula describes the parallel-plate capacitor and the other three allow for all side effects:
C
w h
w 0.77 1.06 h
0.25
t 1.06 h
0.5
(7)
The maximum deviation from numerical methods is reported to be 2 percent when w / h 1 , 0 .1 t / h
4 , and 6 percent when w / h 0.3 , 4 t / h 10 .
To simulate the above model in Ansys we selected the following parameters: w
50 , t
5,
1 h 10 . It can be seen from comparison presented in Fig. 5, that Chang's approximation
(4) is the most accurate, and the errors are about 0.2% only respective to finite elements.
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Meijs and Fokema's correlation formula (7) is also accurate with maximal relative error less ~. then 1%. Two other approximations are imprecise with large relative errors lengthwise w
Yuan and Trick Sakurai and Tamaru Meijs and Fokkema Chang Ansys
0.6 0.5
C
f 0.4 0.3 0.2 0.1 0
10
20
30
40
50
w Figure 5a. Comparison between the four approximations respective to finite elements. 15
Relative error, %
10
5
0
Yuan and Trick Sakurai and Tamaru Meijs and Fokkema Chang
-5
-10
10
20
30
40
50
w Figure 5b. Relative error versus the normalized plate width.
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The second geometry that we consider is schematically described in Fig. 3c. This capacitor is not typical in VLSI technology and only one approximation has been found in literature. 2.2.5. Chang 1976 [3] Chang derived the most accurate formula for the line-to-ground capacitance as presented earlier. In the same publication, he also used a Schwartz-Christoffel conformal mapping approach, to derive the capacitance per unit length of the geometry in Fig. 3c in the form C
2
ln
2 Rb Ra
(8a)
ln( Ra )
w 2" tanh 2h
ln( Rb ) !
1
" !
h d
p q 1 q p
1
w 2" tanh 2h
1
1 q p q
! ln
1
p 1 4
2 tanh
t /h
q2 /! 2
q
1 2 " 2
ln
1
1 p
4p p 1
(8b)
(8c) (8d) (8e)
d /h
p
1 p
2! tanh
(8f)
!2 1
Provided that for w / h
"2 ! 2 1
2
0.5 and that d / h
4! 2
(8g)
0.5 , Chang reports that the errors are less then 1
percent when compared with finite element simulation.
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For comparison, we simulated the problem in Ansys with the following parameters: d w
50 , t
h,
5 , 1 h 10 . Figure 6 shows the high accuracy results for the capacitance
calculations by analytical approximation (8) of Chang. The maximum error relative to Ansys simulations is less then 0.2% only.
0.3
Ansys Chang
0.25
C
0.2
f 0.15
0.1
0.05
10
20
30
40
50
w Figure 6a. Chang approximation respective to finite elements. 0.2
Relative error, %
0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2
10
20
30
40
50
w Figure 6b. Relative error versus the normalized plate width. -10-
Finally, we consider the capacitor presented in Fig. 3b. The system described in Fig. 1, namely a capacitor with two zero thickness plates, is a special case of the capacitor presented in Fig. 3b. 2.2.6. H. Yang 2000 [7] Yang presented an analytical approximation that is referenced in “Handbook of sensors and actuators”[8]
C
w 2h w 1 ln h w h
2h 2t t t2 ln 1 2 w h h h2
(9)
The right hand side of (9) includes three terms: the first term defines the capacitance of the simple parallel-plates capacitor, the second term is the fringing field due to the finite dimensions of the plates, and the third term is the fringing field associated with the thickness of the plates. Comparison to finite elements simulations shows that the capacitance predicted by (9) is not accurate. In the following we propose a modified approximation that corrects this inaccuracy.
3. A new improved approximation
Considering Yang's equation (9), it is easy to see that for zero-thickness plates, the third term vanishes and the functional form of the remaining terms is similar to the form of Elliot's approximation (3) except for a factor of 2. It is assumed that this factor 2 is the reason for the inaccuracy. In addition to this, we can see from Fig. 2, that Palmer's approximation (2) is much more accurate than the one proposed by Elliot (3). Accordingly, we replace the terms related to Elliot's approximation in (9) by the relevant terms in Palmer's expression (2). In light of these changes, we propose a new modified formula in the form
C
w 1 h
h w
h 2 w ln w h
h 2t t ln 1 2 w h h
Yang/2
Palmer
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t2 h2
(10)
Figure 7 present a comparison between the approximations (10), (9) and Ansys simulations for the following parameters: w
50 , 1 h 10 , and t
5 . As shown, the modified
approximation (10) is much more accurate than Yang's formula (9) and the accuracy is within 0.3% relative to finite elements.
0.6
Yang Modified formula Ansys
0.5
C
f 0.4 0.3 0.2 0.1 0
10
20
30
40
50
w Figure 7a. Comparison between two approximations respective to finite elements. 12
Yang Modified formula
Relative error, %
10 8 6 4 2 0 -2
10
20
30
40
50
w Figure 7b. Relative error versus the normalized plate width.
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Moreover, Eq. (10) can be implemented to solve the line-to-ground capacitance problem presented in Fig. 3a, that was discussed above. As shown in Fig. 5 the best analytical approximation of the plate over a ground capacitance is given by Chang (4), but it has a small disadvantage in the complexity of the approximation formulae. In contrast, Eq. (10) has a simpler functional form. The relation between the line-to-ground capacitance and parallel-plate capacitor is clarified by Fig. 8. Application of the voltages as illustrated in Fig. 8b results in zero voltage at the geometrical symmetry plane, and therefore the charge per unit length on the line (Fig. 8a) and top plate (Fig. 8b), is identical. w
w
V
t
V
t h
h
h -V (a)
(b)
Figure 8. a) Line-to-ground capacitor. b) Parallel-plate capacitor. Figure 9 presents a comparison of the capacitance per unit length of Fig. 8a, calculated by (10), and the previous results.
0.6
Yuan and Trick Sakurai and Tamaru Meijs and Fokkema Chang Modified formula Ansys
0.5
C
f
0.4 0.3 0.2 0.1 0
10
20
30
w (a)
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40
50
15
Relative error, %
10
5
0 Yuan and Trick Sakurai and Tamaru Meijs and Fokkema Chang Modified formula
-5
-10
10
20
30
40
50
w (b) Figure 9. a) Comparison of the new modified formula results to the previous approximations and to the finite elements. b) Relative error versus the normalized plate width.
Figure 9 shows that for the range w / h # 10 the error of modified formula (10) relative to the finite elements simulation small, and for 5
w / h 10 the error is less than 2%.
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References [1] H. B. Palmer, "Capacitance of a parallel-plate capacitor by the Schwartz-Christoffel transformation," Trans. AIEE, Vol. 56, pp. 363, March 1927. [2] R. S. Elliott, "Electromagnetics(book)," pp. 182-189, 1966. [3] W. H. Chang, "Analytic IC-metal-line capacitance formulas," IEEE Trans. Microwave Theory Tech., Vol. MTT-24, pp. 608-611, 1976; also vol. MTT-25, p. 712, 1977. [4] C. P. Yuan and T. N. Trick, "A simple formula for the estimation of the capacitance of two-dimensional interconnects in VLSI circuits," IEEE Electron Device Lett., Vol. EDL-3, pp. 391-393, 1982. [5] T. Sakurai and K. Tamaru, "Simple formulas for two- and three-dimensional capacitances," IEEE Trans. Electron Devices, Vol. ED-30, pp. 183-185, 1983. [6] N. Van de Meijs and J. T. Fokkema, "VLSI circuit reconstruction from mask topology," Integration, Vol. 2, pp. 85-119, 1984. [7] H. Yang, "Microgyroscope and microdynamics," Ph. D. Dissertation, December, 2000. [8] M. H. Bao, "Handbook of sensors and actuators," Vol. 8, pp. 144-145, 2001.
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