design of an integrated electrostatic stepper motor with axial field

Sensors and Actuators A, 25-27 (1991) 597-603

Design of an Integrated Electrostatic Stepper Motor with Axial Field

L. PARATTE, G.-A. RACINE and N. F. DE ROOIJ

Institute of Microlechnology, Universi?y of Neuch&el, Rue A.-L. Breguer 2, CH-2069 Neuchcjrel (Switzerland)

E. BORNAND

Asulab S.A., R +D Laboratories of SMH Group, Passage Max-Meuron 6, CH-2001 NeuchBtel (Switzerland)

597

Abstract

A variable capacitance, top-drive synchronous, electrostatic stepper motor model is presented. A surface micromachining technique is proposed for its fabrication. Simple theoretical modelling predicts that the Cou- lomb dry friction torque dominates the viscous and inertial torques over a large speed range. Dry friction coefficient values between thin-film-covered silicon chips have been measured by means of a pin-on-disc tribometer machine from CSEM S.A. Values between 0.3 and 0.4 have been found for LPCVD silicon nitride on LPCVD polysilicon, polysilicon on itself and silicon nitride on itself, in agreement with previous work. This result shows that a material pair with better friction behaviour still has to be found.

Introduction

Polysilicon film deposition over a sacrificial silicon dioxide layer appears to be a promising micromachining technique for building motors having disc-shaped rotors with typical dimensions of hundreds of pm in diameter of l-2 pm thickness [2,3,5]. When motors are shrunk down in size, electrostatic is preferred to electromagnetic drive because the heat losses due to current in the coils increase [ 11. Moreover, micrometer-sized gaps sustain high field values (Paschen’s law), as reported elsewhere [2] and confirmed by measurements made on movable aluminium structures. This motor is a variable capaci-

0924-4247/91/%3.50

tance (VC), synchronous stepper motor with stator electrodes deposited underneath the rotor (top drive). Field lines in the air gaps are globally axially oriented. In this motor type, the sacrificial layer technique provides more uniform and smaller air gaps, and the planar geometry leads to larger VC facing areas. This results in an increased capacitance change rate, and hence an .increased motor torque. The absence of peripheral stators makes adjacent coupling to a driven mecha- nism easy. But under electrostatic attraction, the rotor teeth will bend downwards, and even switch down to contact the base plane if a certain ‘pull-in’ voltage is applied, thus preventing any rotation from taking place. Moreover this attraction force becomes larger as the rotor and stator align, causing the dry friction torque to increase drastically. In the following study, a three-polysilicon-layer fabrication process for this motor will be described first, followed by a simple analytical model and a numerical example which takes into account the most significant parameters. The importance of the dry dynamic friction coefficient will be outlined. Finally, a friction measurement set-up and measurement results will be presented.

Fabrication

The fabrication is based on the surface micromachining technique available in our laboratory and documented in ref. 10. The fabrication steps are shown for a half-motor cross section in Fig. 1, the dashed line being

0 Elsevier Sequoia/Printed in The Netherlands

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rotor axis 2nd SiO2 th si (n) 1st SiO2 th I I knd plane) I

(4

I 2nd LPCVD 1st LPCVD Poly-Si 1st LPCVD

I SiN (stator electmdes) SiN

;b) 2nd LPCVD Poly-Si 1st PSG

3rd LPCVD Poly-Si 2nd PSG I I I

(4

I I Aluminium I (bonding) I I I

I I

(e)

Fig. I. Simplified fabrication process sequence of the electrostatic motor with axial field.

the rotation axis. The aspect ratio of the cross section has been modified for presenta- tion purposes. (a) The silicon wafers (3”, 100 orientation, p-type, resistivity = 5 -7 Sz cm) are oxidized, and a ground plane region act- ing as an electrostatic shield [6] is opened with BHF. N-doping of the region is per-

formed through phosphorus-doped oxide deposition (CVD) and 1100 “C drive-in. After removal of that doping layer, electrical isola- tion to the substrate is obtained with a second thermal oxide. (b) A first silicon nitride layer is deposited (LPCVD), followed by a first polysilicon layer (LPCVD) which is doped in the same way as for the bulk silicon under (a). The stator electrodes are then defined with a dry SF6/02 plasma etching of the polysilicon with etch-stop on the first nitride. A second nitride layer is deposited. (c) A first sacrificial PSG (phosphosilicate glass) layer is deposited and reflowed at 1100 “C for surface smoothing, unless the sharp steps of the stator electrodes will repro- duce their image in the rotor. The rotor bushing spacer negatives are opened, then ‘refilled’ with a small CVD oxide, otherwise the rotor would stick to the second nitride when released. The rotor is defined with plasma etching, and remaining open-surface PSG is removed by wet etching. (d) A second sacrificial PSG layer is deposited and the openings to the ground planes are wet and plasma etched. A third polysilicon layer is deposited and the rotor bearing is patterned with plasma etching. At that stage a drive-in from the sacrificial PSG is performed for the doping of the two polysilicon layers. (e) The open-surface sacrificial PSG is removed by wet etching, and the metal contacts to stator lands are opened with plasma etching. Metal- lization and its patterning is performed, and finally the rotor is released through long wet etching of the remaining PSG.

Modelling

A geometrical motor model is shown in Fig. 2(a). If a voltage V is applied across one phase (one pair of diametrically opposed stator teeth), the rotor teeth tend to align to it. Continuous rotation of the rotor is obtained by switching V to the next adjacent phase, causing the rotor to turn in the same or opposite direction, depending on the number and width of the stator and rotor teeth. The

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A-A y+ .g Top view

(4

W (4

Fig. 2. (a) Geometrical motor model. (b) Motor and friction torques vs. rotor position. (c) Tooth flexure characteristics.

following assumptions will be made for the modelling: (1) The rotor is electrically con- ductive. (2) The current at the rotor due to charge transfer is negligible. (3) The viscous torque is mainly due to the laminar Couette flow of the air between the rotor and the base plane [4]. (4) No significant charges are gen- erated through friction. (5) The friction forces between two surfaces in contact are only proportional to the normal contact force (Coulomb friction). (6) All calculations are made for one phase. (7) The relative dielec- tric constant of the air is assumed to be one. The Newton motion law, 1, J2~/8t2 = C M with I, being the rotor inertia and I: M the total torque, in our case where a constant voltage is applied (see Fig. 2) becomes

where C, &p/& = A4, is the viscous friction torque due to air flow. The dry friction torque Mf and motor torque M,,, can be calculated by taking the derivative of the capacitance energy W = CV2/2 stored in the air gaps with relation to axial and angular displacements:

(2)

M,,, = -sgn(cp)$ V2 2 (R,2 - R:) (3)

where p is the dynamic (we assume decelera- tion until standstill) dry friction coefficient between the rotor and stator, co is the permit- tivity in vacuum and all other parameters are as shown in Fig. 2. In fact, the practical motor torque will not show an abrupt change at cp = 0 because the fringing effects near that aligned position cause the real capacitance to decrease as described in ref. 1 and numeri- cally simulated in ref. 3 (Fig. 2(b)). Typically this effect will appear when the teeth align to twice the air gap distance. Because the rotor teeth edges are not parallel to the stator ones if not aligned, we calculate an average angle qd near cp = 0 where the motor torque begins to decrease:

qd = 4d/(& + 4 1 (4)

Hence, the motor torque between -(Pi and (Pd can be considered as a linear function of cp and is expressed as

M,= -~ ~~(Rt-R’)(R.+Ri)‘p (5)

The viscous coefficient C, can be calculated from the laminar Couette air flow model between two adjacent planes over the rotor geometry by integration [4]:

1 (6)

where qair is the viscous coefficient for air. By replacing eqns. (2), (3) and (5) in eqn. (I), the general solution for the step motion q(t)

600

is found to be

q(t) = exp( - C, t/21,)[A exp( KU)

+B exp(-Rt)] + C

with

where A, B and C are integration constants which depend on the starting position of the rotor. We have k = k, - k2 for 0 < cp <(Pi (linear motor torque region) and k = -k2 for (Pd < q < 0, (constant motor torque region) with the following relations taken from eqns. (2) and (3):

1 k, = 32 24J V d’ (R: - R?)(R, + Ri)

(motor spring constant) (9)

k,=;pRJ’$(R:-Rf)

(dry friction constant) (10)

If the root argument of eqn. (8) is >O, we have an over-damped motion. The term in square brackets in eqn. (7) becomes a hyper- bolic sine function of Qt. If the root argument is negative, we have an under-damped, sinu- soidal motion. The first condition is in fact always satisfied, because when the tooth is still in the constant torque domain, we have k = -k, so the root argument of eqn. (8) is always positive. When entering the linear torque region, the motor effective torque M,,, - Mf decreases even faster. It follows that the motion is completely overdamped by the dry friction term. When the rotor has a continuous rotation it is possible to estimate a speed LJq/at,-,it where the viscous friction equals the dry friction by simply writing

G/&t = Mn/C” (11)

For operating speeds below a user-determined fraction of the speed given by eqn. (ll), we can describe the rotor equilibrium position cp = (Pi when solving the simplified eqn. (1) M, = Mf with eqns. (1) and (2):

This shows that the rotor-stator electrode overlap angle is independent of the tooth width &. This indicates that one has to re- duce the tooth width to obtain a better align- ment, but also has to multiply the phase number. A remarkable fact is that the equilibrium position does not depend upon phase voltage or tooth radii R, and Ri. The angle (P~ has to be small enough in order that the next adjacent phase just ‘catches’ the rotor, and that there is no reversal of the motion. These two conditions of proper step-by-step opera- tion can be expressed in terms of (P_ tooth width 8, and rotor and stator teeth numbers n, and n, as

(13)

(14)

We want to prevent the teeth ends from touching the substrate when the phase is en- ergized and for cp = 0. In general, the solving of the elastic curve differential equation for electrostatically deformed planar elements is rather complex, because the distributed electrostatic force is a function of the deflection. The analytical solution has already been in- vestigated [9]. Here we will consider a simpler model. First, the deflection is symmetrical so one tooth of the two can be modelled, as shown in cross section A-A of Fig. 2. We assume that the rotor ring part (R, to Ri) is rigid. We have here the case of a simply fixed beam (cantilever) with variable section width. We will consider a rectangular beam of length R, - Ri and width erRi. The tooth is supposed to be uniformly loaded with a distributed electrostatic charge having a re- sultant of Fz/2 in its middle. This approxima- tion can be made here considering that in reality the higher electrostatic force due to the inclination of the tooth down to a maximum of a half gap is compensated by the greater mechanical restoration force F, due to the tooth width variation. We will consider the displacement d at the end of the tooth, which we assume to be about three times bigger (after the drawing of the elastic curve

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under distributed load [ 111) than at its middle, in order to compare with the maximum end deflection d,, - d given for the mechanical restoration force. We have the following ex- pressions for the resultants at the middle of the tooth as a function of the end deflection d and for cp = 0:

F 1 1=-- Eo 2 16 vz($d,,+$d)2

(RZ - Rf)& (15)

F, = 3 (;‘lR;;, (do -d) I

(16)

where E is the Young’s modulus of polysilicon. Figure 2(c) shows eqns. ( 15) and (16) as a function of d, with the phase voltage V as parameter. The intersections show stable and unstable equilibrium points. When the voltage is increased, the hyperbola Fz/2(d) moves up until it is a tangent to F*(d). At this point, the tooth will suddenly clip down in contact with the substrate. This voltage will be called the pull-in voltage VPi.

Numerical Example

We choose R, = 100 pm. Currently deposited sacrificial layer thicknesses lead to air gaps of typically d = 1.5 pm. A reasonable d.c. phase voltage is V = 50 V so we have a nominal electrostatic field of 16.7 V/pm (six times below the maximum of 100 V/pm, see [2] and measurement results). We choose to have a 12:8 stator:rotor number, where the pole width is 0, = 20”. p is assumed to be between 0.2 and 0.5. The layout design con- strains the minimum rotor hole radius Ri, to about 30 pm. In order to still have a good rigidity of the ring part (R, to Ri), we are bound to have Ri = 60 pm. The bushing radius is between Ri and &: R, = 50 pm.

First, the critical speed at which the viscous and dry friction torques are equal is calculated with eqns. (3), (6), ( 11) and is found to be only J(P/dtc,i, = 280000 rpm, the viscosity of the air at ambient T and P being qair = 1.8 x 10e5 kg/m s. The equilibrium position rp, is calculated with eqn. (12) as shown

0 0.5

Air gap d (pmf 1.5

(b)

Fig. 3. Numerical motor example: (a) rotor static equilibrium olihet angle, (b) axial deflection equilibrium position for a 200 pm diameter rotor.

in Fig. 3(a), and is 11.4” for p = 0.2 and 16.6 for ~1 =0.5. This motor does not fulfil the forward operating condition of eqn. ( 13) where (Pi should be <5”, nor the no-reversal condition of eqn. (14) where (Pi should be Q lo”. The following maximum ratings are calculated for the aligned position rp = 0, hence the phase capacitance is C,,, = 3.3 fF, the maximum motor torque is M,,, = 11.8 pN m, the axial force is F= = 2.7 pN, the maximum dry friction torque is M,= 27.4 pN m for p = 0.2. The axial equilibrium deflections of the tooth extremity at 50 V and 200 V are calculated with eqns. ( 15) and ( 16) and plotted in Fig. 3(b), the Young’s modulus for polysilicon being E = .1.7 x 10” N/ m* [7]. It can be observed that the teeth are

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still very stiff under 50 V, and that no pull-in voltage seems to exist from that structure below a phase voltage of at least 250 V.

Dry Friction Measurements

Silicon wafers have been processed in the same way as for the motor contact layers (see Fabrication). Thin films of LPCVD silicon nitride and polysilicon have been deposited, then chips have been cut to appropriate dimensions. A pin-on-disc machine from CSEM S.A. (see Fig. 4) has been used for the friction measurement. Previous measurements have also been made with the same principle [S]. A vertical force of N = 1 New- ton is applied on a 4.7 x 4.7 mm’ silicon chip (material l), corresponding to approximately double the pressure calculated for the proto-

motorised lower wafer “ppzr wafer upper sample turntable with holder

t F,=pN

Fig. 4. Schematics of the dry friction coefficient measurement set-up using a pin-on-disc tribometer from CSEM.

type motor (four bushings of 8 pm x 8 ,um each). The sample is offset from the tum- table centre by Y = 15 mm. Two-thirds of a 3” silicon wafer (material 2) is glued to the turntable. The lower sample is spun from 0 to 2 rpm, corresponding to a rotational speed of the numerical example motor of 0 to 600 rpm. The friction force Ff causes the arm to bend laterally, which deformation is sensed with an induction coil. Appropriate electron- ics deliver a voltage proportional to the force, which is recorded on an x-t plotter. Initially, we have a peak value due to higher static friction. During rotation, the speed is increased to 2 rpm. All wafer parts and chips have been rinsed in acetone and isopropanol, and dried with nitrogen prior to testing. We counted every run which did not exhibit vis- ible wear scars on the chips, indicating that we measured proper dry friction and not some unknown grinding. There are not enough data to permit a better distinction between the various materials. Nevertheless, we can already state that one material on itself is not a good solution. The nitride was harder because it left scars on the other sam- ples. Sometimes the static coefficient was lower than the dynamic one, as indicated clearly by the absence of stick-slip. A strong variation of the coefficient within the same material category, even if issued from the same wafer, has been observed. Dynamic friction coefficients of 0.35 f 0.10 for polysilicon on nitride, 0.40 & 0.10 for polysilicon on itself and 0.30 + 0.10 for nitride on itself has been measured over approximately five runs. Pre- vious work [5] reported friction coefficient values of 0.21 and 0.38 for polysilicon on itself from measurements and modelling on a side-drive VC motor. This can be in agreement with the value found here: the difference may be explained by the effect of a smoother contact surface in the case of the motor.

Conclusions

Theoretical considerations show that for this electrostatic motor architecture, the dry

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friction dominates the other effects (inertia, viscous drag) to the point that no oscillation around the equilibrium position is expected at any rotating speed or voltage. A minimum set of orthogonal values (0,, rzsr n,, R,, p, d) determines completely the rotor behaviour, which is more static than dynamic. The measurement of a high dynamic friction coefficient value of up to 0.5 shows that the motor according to the numerical example would not turn. Either the motor geometry has to be optimized with relation to the equations developed, or a pair of low-friction materials needs to be found. The simple method presented for the measurement of the friction coefficient is interesting for a quick comparison between thin-film covered chips.

Acknowledgements

The authors would like to thank M. Mail- lat from the tribology laboratory of CSEM, Switzerland, for assistance with the friction measurements on the tribometer, and C. Borgeaud from EPF-Lausanne, Switzerland, for interesting and helpful discussions about dry friction. S. Jeanneret, P.-A. Clerc and C. Linder from our IMT group provided techni- cal assistance and discussion of some fabrication process steps. R. Vuilleumier from CSEM is acknowledged for the measurements of the electrostatic breakdown voltage of air on aluminium movable structures. We finally acknowledge the financial support of the Committee for the Promotion of Applied Scientific Research, Switzerland.

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design of an integrated electrostatic stepper motor with axial field

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