Design of a Pull-in Voltage Reference Structure

L. Rocha, E. Cretu* and R.F. WolffenbuttelDelft University of Technology, ITS/Et, DIMES,Mekelweg 4, 2628 CD Delft, The Netherlands

*Also at: Melexis, Transportstr. 1, Tessenderloo, BelgiumPhone +31 15 278 6287, Fax. +31 15 278 5755

e-mail: L.Rocha@its.tudelft.nl

Abstract

The pull-in voltage of a single side clampedfree-standing beam, under lateral deflection,has been investigated for application as a DCvoltage reference. A 2D energy-basedanalytical model for the static pull-in iscompared with measurements. Bifurcationdiagrams are computed numerically, based ona local continuation method. A left-right combdrive electrostatic structure is used to drivethe 100 µm long beam (by momentumactuation) beyond the stability border. Thedevice can be operated in feedback or as aseesaw, by using these two sets of electrodes.The single end clamping ensures that the pull-in voltage is insensitive to technology-inducedstresses.

1. Introduction

MEMS technology is gradually alsopenetrating into mainstream Instrumentation andMeasurement applications [1]. Although mostmicrosystems are in principle instrumentsdesigned for measuring a non-electrical quantity,and thus serving a measurement application, theI&M application area can be defined as the field,where the instrument is the purpose and not themeans. MEMS technology has a huge potentialfor contributing to critical components in aprofessional instrument, such as AC-to-DCconverter or internal DC reference.

The pull-in voltage of beams of differentdesigns has been investigated with the purposeof using the micromechanical structure as an on-chip voltage reference. Such a device may findapplication in integrated circuits, next to theconventionally used bandgap reference, and inmetrology, where Zener references are widelyused as so-called “transfer standard” [6]. Theoperation of a Zener diode is based on avalanchebreakdown and is associated with a high noiselevel.

As the electrostatic force in a vertical field isinversely proportional to the square of thedeflection and the restoring force of the beam is,in a first approximation, linear with deflection,an unstable system results in case of a deflection,v, beyond a critical value, vcrit. The pull-involtage, Vpi, is defined as the voltage that isrequired to obtain this critical deflection. For astable equilibrium deflection the secondderivative of the potential energy of the systemto deflection should be positive: ∂2Up/∂v2>0,thus Vpi results from ∂2Up/∂v2= 0. The potentialenergy is composed of the bending energy, Ub,the strain energy, Us and the potential energy ofthe excitation (the potential energy of theelectrostatic force, Uel).

2. Modeling the pull-in structure

The basic phenomena is the loss of stability

at the equilibrium position *x , where the

elastic forces equilibrate the electrostatic ones.

Two methodologies are generally used:(A). The dynamic system approach, in which

the electromechanical system is described by aset of differential equations, and an analysis ofthe stability of its equilibrium points is per-formed (indirect Lyapunov method)

(B). The variational approach, in which theequilibrium points and their stability aredetermined y studying the variations of the totalenergy ),()(),( VxUxUVxU electricelastic

��� += . The

equilibrium points eqx�

are given by )( eqxx

U �

�∂∂

;

these are stable if )( eqxU�

is a local minimum,

which is determined by the eigenvalues of

)(2

2

eqxx

U �

�∂∂

. The pull-in voltage is the value of

the applied voltage for which the physicalequilibrium point loses its stability.

Figure 1, Single-side clamped structure.

Specifics of the analytical modeling used:

1. Applied to elastic beam, clamped at one

end, and actuated by an electrostatic

momentum at the free end (Fig. 1).

Figure 2, Identification of the state variablesused in the model.

2. The energy balance is evaluated usingtwo parameters (w1, j1) to fully determine theconfiguration (Fig. 1 and Fig. 2).

The elastic energy is described by:

��

�

�

��

�

�

���

����

�+−

=��

���

�

∂∂= �

2

111

2

0

2

11

32

2

1),(

xxx

y

l

ybeam

l

w

l

w

l

EI

dxx

EIwUx

ϕϕ

ϕϕ

The electrostatic energy is described by:

[ ]201111

2

11

2),(),(2

1);,(

CwCwCV

VwU electric

−−+−

=

ϕϕ

ϕ

3. A local continuation method wasimplemented in Mathematica for tracing theequilibrium point coordinates at increasingvoltage. The approach used to solve theproblem is based on sweeping of the voltage,from the initial value V0 toward increasingpositive values. For each voltage value, thestability points are computed, by approximatingthe general potential in Taylor series around thepreviously computed equilibrium point,

]}1[],1[{ ,,1 −ϕ−= kkwx eqneqn . This makes it

possible to trace the evolution of theequilibrium point as function of V. For thecomputed values of eqeqw ,1,1 ,ϕ , the eigenvalues

of the associated Hessian at that point can becomputed. If any of them is <0, then theequilibrium point is unstable, and drawn with adifferent color. Otherwise, it is stable. Asshown in Fig. 3, the predicted pull-in voltage isat Vpi= 9.5V.

Figure 3, Variation of the equilibrium pointwith applied voltage.

3. Microstructure fabrication

An epi-poly process was used for thefabrication of 11µm thick single-side clamped200 µm long free-standing structures, as shownin Fig. 4. A differential sense capacitor,underneath the top mass of the horizontalbeam, was used for detecting the pull-involtage. The readout circuit is based on acapacitor bridge with two active arms.

Figure 4, Fabricated microstructure.

4. Experimental results

The measured pull-in voltage was about9.1V, which is near the predicted value fromMathematica model (Vpi= 9.5V). Thisdifference is mainly due to the simplifyingassumptions used, the process deviations anduncertainties in the value of the Young’smodulus used in the numerical computation.Figure 5 also indicates a profound qualitydifference between the devices. For properoperation as a voltage reference the pull-inshould be as abrupt as possible. Moreover, theeffect should be reproducible. Clearly, device#1 performs much better with respect to bothrequirements. The origins of these effectsrequire further analysis. Figure 6 shows theresponse of a selected device and Fig. 7indicates that some devices also exhibithysteresis. This property seriously complicatesoperation in a feedback loop.

Figure 5, Variation between several devices.

Figure 6, Experimental result of a selectedbeam.

Figure 7, Hysteresis in a pull-in microstructure.

5. Conclusions

The design and fabrication of a pull-inmicrostructure for application in metrology hasbeen described in this paper. The feasibility hasbeen demonstrated. First prototypes suffer fromlimited reproducibility and hysteresis.Experiments from another group on a metal-silicon electrode combination show a polaritydependent drift of about 0.5%/50hrs. [1, 6]. Thisis not acceptable in I&M applications. Thesource of drift has been identified as charging ofthe (native) oxide of the polysilicon beam. Aftercoating the silicon with a metal much betterresults are expected. The design of improveddevices is forthcoming and measurements are

scheduled to verify the long-termreproducibility.

6. References

[1] R.F. Wolffenbuttel, D.D.L. Wijngaardsand C.J. van Mullem, Proc. Symposiumon Microtechnology in Metrology andMetrology in Microsystems, Delft, TheNetherlands, August 31-Sept. 1, 2000.

[2] M. Klonz and T. Weimann, Accuratethin-film multi-junction thermalconverter on a silicon chip, IEEE Tr.ED, Vol. 38, No. 2, 1989, pp.335-337.

[3] M. Bartek, Z. Xiao, C.J. van Mullemand R.F. Wolffenbuttel, Bulkmicromachined electrostatic RMS-to-DC converter: design and fabrication,Proc. MME00, Uppsala, Sweden,October 1-3 2000.

[4] A.S. Oja, J. Kyynäräinen, H. Seppä andT. Lampola, A micromechanical DC-voltage reference, Conference digest ofCPEM2000, pp. 701.

[5] H.A.C. Tilmans, and R. Legtenberg,Electrostatically driven vacuum-encap-sulated polysilicon resonators, Part 2,Theory and performance, Sensors andAct. A 45(1994) 67-84.

[7] vdivde-it.de/mst/imsto/Europractice/Bosch/