Atienza García, María Teresa

Domínguez Pumar, Manuel

Gorreta Mariné, Sergio

Pons Nin, Joan

IEEE Transactions on Industrial Electronics

English

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Characterization of dielectric charging in MEMS using Diffusive Representation

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© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 1Characterization of Dielectric Charging in MEMSUsing Diffusive RepresentationMaria-Teresa Atienza, Sergi Gorreta, Student Member, IEEE, Joan Pons-Nin and Manuel Dominguez-Pumar,Senior Member, IEEEAbstract—This paper introduces Diffusive Representation as anovel approach to characterize the dynamics of charge trappedin dielectric layers of microelectromechanical systems (MEMS).Diffusive Representation provides a computationally efficientmethod to achieve an arbitrary order state-space model of thecharging dynamics. This approach is particularly well suitedto analyze the dynamics of the dielectric charge under non-trivial controls, as in the case of sliding mode controllers. Thediffusive symbol of the experimental structure has been obtainedfrom open-loop measurements, in which Pseudo Random BinarySequences (PRBS) are applied to the device. The obtained modelexhibits good agreement with experimental data and also allowsto model the behaviour of the charge dynamics under excitationwith arbitrary binary signals.Index Terms—Diffusive Representation, MEMS, dielectricchargingI. INTRODUCTIONELECTROSTATICALLY actuated MEMS devices havebeen used as varactors, resonators and switches in a largeset of applications. Nevertheless, the parasitic charge trappedin the dielectric layers of these devices produces undesiredeffects such as pull-in voltage drift, capacitance-voltage C(V )curve shifts or even stiction of moveable parts. This seriousreliability problem still hinders the massive commercial use ofthese devices [1]. Accordingly, characterization of the chargingdynamics is an important issue that has been studied usingdifferent methods. For instance, in [2] iterative measurementsof pull-in voltage have been used to monitor the chargeevolution with time. However, this strategy alters the amountof trapped charge and is not compatible with normal operationof the devices. In [3], the application of bipolar voltagesallowed to monitor C(V ) shifts through quasi differentialcapacitance measurements. This method allows knowing thestate of the charge without distorting the measurement noraffecting the normal operation of the device.This paper improves the work introduced in [3], where thedynamics of the dielectric charge for the contactless case wasdescribed using a multi-exponential model. The parameters ofthis model were obtained with the unconstrained non-linearminimization Nelder–Mead algorithm. In this paper, however,the characterization method uses the Diffusive Representationtechnique, which is a suitable framework for any diffusiveManuscript received April 5, 2016; revised July 25, 2016; accepted August9, 2016. This work was supported in part by the Spanish Ministry MINECOunder Projects TEC2013-48102-C2-1-P and FPI grant BES-2012-057618.Micro and Nano Technologies Group, Universitat Politècnicade Catalunya, Barcelona, Spain. Corresponding author e-mail:maria.teresa.atienza@upc.edu.physical phenomenon [4], being even able to accurately des-cribe fractional order models. The use of this type of models isappropiate since diffusion processes have been associated withdielectric charging in MEMS [5]. Furthermore, there is a linkin [6] between fractional systems and the typically observedmultiexponential or stretched-time exponential type responses[2], [7] usually found in dielectric charging.These behavioural models are particularly useful in theanalysis of dielectric charging control using the tools of slidingmode controllers [8]. Diffusive Representation has been widelyused in thermal characterization [9] and dielectric polarization[4], [10], among others. Another key advantage of using Diffu-sive Representation is the reduction of the computational loadrespect other methods, as the model parameters are obtainedwith the linear least-squares method. This in particular allowsto increase the model order if necessary, due to the fact thatthe identification problem is computationally tolerable.II. CHARGE CHARACTERIZATION METHODThis paper uses the voltage waveforms, or symbols BIT0and BIT1, designed for charge characterization and controlin [3], [8], [11], shown in Fig. 1. In BIT0, a negative voltageV − is applied to the MEMS for a ’long’ time (1− δ)Ts, thenpositive voltage V + is applied for a ’short’ time δTs, beingδ < 1 and Ts the symbol duration. In BIT1, the same timingscheme but with opposite voltages is applied. During eachsymbol, the capacitance of the device is measured at times(1−δ)Ts and Ts. This allows obtaining a sample of the quasi-differential capacitance ∆C = C(V +)−C(V −) = C+−C−at the end of each symbol. ∆C is related to the voltage shift ofthe C(V ), commonly defined as Vsh = Qd/Cd, where Qd andCd are the total amount of charge trapped in the dielectric andthe dielectric capacitance respectively. For devices workingbelow pull-in a parabolic C(V ) shape can be considered. Inthis case the relationship between ∆C and Vsh is [11]:∆C(t) = α((V +)2 − (V −)2)− 2αVsh(t)(V + − V −) (1)V+V-(1-δ)Ts……δTs…BIT0 BIT1C+C-(1-δ)Ts δTsC-C+tFig. 1. Voltage waveforms, or symbols, for MEMS actuation and sensing.Capacitance measurements are performed at symbol times (1− δ)Ts and Ts.21 6 11 16 21 26 31 3600.511.522.533.544.555.56KRMS E [ f F]Fig. 2. Root Mean Square Error evolution of the fittings as a function of themodel order K.where α is the second order coefficient of the parabolicfunction.In the MEMS used in the experiments of this work, applyingBIT0 increases the amount of net dielectric charge Qd, andthus the voltage shift Vsh, whereas applying BIT1 decreasesboth Qd and Vsh.III. DIFFUSIVE REPRESENTATIONDiffusive Representation theory allows obtaining exact andapproximate state realizations of a wide class of integraloperators of rational or non-rational nature. This representationmethod, is an useful mathematical tool for any physicalphenomena based on diffusion [4].Given a non-rational transfer function, H(p), associatedwith a convolution causal operator denoted by H(∂t), thediffusive realization of this operator is expressed by thefollowing input (u) – output (y) state space realization ofu 7−→ y = H(∂t)u = h ∗ u of the form [10]:∂ψ(ξ,t)∂t = −ξψ(ξ, t) + u(t), ψ(ξ, 0) = 0y(t) =∫∞0η(ξ)ψ(ξ, t)dξ(2)where ξ ∈ R, η(ξ) is the diffusive symbol of H(∂t) and ψ(ξ, t)is called the diffusive representation of u(t) ∈ {V +, V −}. Thestate-variable ψ(ξ, t) is a time-frequency representation of theinput only and is the solution to the differential equation ofEq. (2) [10]. The diffusive symbol η(ξ) is a solution of Eq.(3) directly obtained from Laplace transform (with respect tot) [10].H(jω) =∫ ∞0η(ξ)jω + ξdξ ω ∈ R (3)The impulse response h := L−1H can also be expressedfrom η: [10]h(t) =∫ ∞0e−ξtη(ξ)dξ (4)and the diffusive symbol can be given also as the inverseLaplace transform of the impulse response: [10]η = L−1h (5)In this paper we will consider that the charging dynamicsassociated with the application of two voltages, V + and V −.This is described by the following equation that is equivalentto Eq. (2):∂ψ(ξ,t)∂t = −ξψ(ξ, t) + η(ξ)−+ (η(ξ)+ − η(ξ)−)uˆ(t), ψ(ξ, 0) = 0y(t) =∫∞0ψ(ξ, t)dξ(6)where uˆ(t) = u(t)−V−V +−V − ∈ {1, 0} is the normalized input signal.η(ξ)+ is the diffusive symbol associated with the actuation ofthe device with V + (when uˆ = 1), while η(ξ)− is the diffusivesymbol associated to V − (uˆ = 0).A discrete approximation of H(∂t) can be built discretizingthe continuous variable ξ into {ξk}1≤k≤K , where K is theorder of the discretized model. This leads to an input – outputapproximation u 7−→ y˜ ≈ y = H( ddt )u of the form:dψk(t)dt = −ξkψk(t) + η−k + ∆ηkuˆ(t), ψk(0) = 0y˜(t) =∑Kk ψk(t)(7)where ψk(t) = ψ(ξk, t), η±k = η(ξk)± and ∆ηk = (η+k − η−k )for k = 1, · · · ,K.If the first expression of Eq. (7) is splitted in two statevariables, Eq. (8) is reached:dψ0k(t)dt = −ξkψ0k(t) + 1, ψ0k(0) = 0dψ1k(t)dt = −ξkψ1k(t) + uˆ(t), ψ1k(0) = 0y˜(t) =∑Kk η−k ψ0k(t) +∑Kk ∆ηkψ1k(t)(8)ψ0k(t) and ψ1k(t) are obtained after resolving the first andsecond differential equations from Eq. (8).Given that the MEMS device can be charged at t = t0, somecoefficients, ak, are added to the output y˜(t) of the systemwhich allow to ’forget’ the initial conditions of the device.The third expression of Eq. (8) is then rewritten as:y˜(t) =K∑kη−k ψ0k(t) +K∑k∆ηkψ1k(t) +K∑kake−ξk(t−t0) (9)This step is necessary, since the device can be charged priorto starting the measurements and this may result in errors in thefitting procedure. The values of the coefficients are obtainedtogether with the values of η−k and ∆ηk in the fitting of theexperimental measurements.The goodness of the approximation will depend on the cho-sen frequency mesh {ξk}1≤k≤K , usually geometrically spacedin the band of interest and in concordance with the dynamiccharacteristics of the system [9]. The chosen bandwidth forξ goes from ξmin = 2pi/T to ξmax = pi/2Ts, where T isthe total duration of the measurements (long enough for thestabilization of the system), and Ts is the sampling period.Therefore, the experiment duration and the sampling period seta limit on the minimum and maximum frequency respectively.It must be noted that, with a sufficiently dense frequency meshit is possible to describe with arbitrary accuracy the responseof any fractional system, therefore other types of response3can be very well approximated with DR [4], for example,stretched-time exponentials. The same also applies to the effectof an initial charging in the device, as in Eq. (9).In particular, the objective of the identification problem is toinfer the finite order diffusive symbols η−k and η+k from experi-mental data. Let us assume a temporal mesh [tn]1≤n≤N ∈ R+,the matrix A = [[Fn,k] [Gn,k] [Hn,k]], where Fn,k = ψ0k(tn),Gn,k = ψ1k(tn) and Hn,k = e−ξk(tn−t0), and the vectorηˆT = [[η−k ]T [∆ηk]T [ak]T ]. Note that F,G,H ∈ RN×K andη,∆η, a ∈ RK . The solution to the identification problemis found solving the finite dimensional least-square problemformulated by [9]:minη∈RK||Aηˆ − Y||2 (10)where Y is the output vector of the measurements, YT =[y(t0) . . . y(tN )], being y(tn) = ∆C(tn) in our case. Thesolution to Eq. (10) is classically given by ηˆ = [A∗A]−1A∗Y.When resolving numerically the integral of Eq. (2), alogarithmic change of variable is necessary due to the factthat ξ variable is spaced geometrically. Specifically, ξ = 10xand dξ = (ln10)10xdx. Taking this into account, the secondexpression of the equation becomes:y(tn) =∫ x1x0η(10x)ψ(10x, t)ln(10)10xdx ≈K∑k=1ηkψk(tn)λk(11)The term λk divides the inferred diffusive symbols, ηk =ηˆkλk, where λk = ln(10)log(r)ξk and r is the ratio of thegeometric sequence of the mesh of ξ (ξk+1 = rξk).IV. EXPERIMENTAL RESULTSThe MEMS, fabricated with standard PolyMUMPS technol-ogy, is a two-parallel plate structure. The upper plate-electrodeis a polysilicon layer, suspended over 2.75 µm of air gap,followed below by a 0.6 µm thick silicon nitride layer and thedoped silicon substrate, which is the bottom electrode. The10−6 10−5 10−4 10−3 10−2 10−1 100−50−40−30−20−1001020304050frequency [Hz]d if f us i ve  s ymb ol  [ fF ] η−η+Fig. 3. 6-order diffusive symbols corresponding to the bipolar voltagesV − and V + resulting from pseudo-inversion of experimental measurements(model M1).72 78 84 90 96 102 108 114 120 126 132 138 144−40−30−20−1001020304050time [h]d if f er en ta l ca pa ci t an ce  [ f F] measurementsmodel output102 103 104 105 106 107 108−30−20−100102030d if f er en ta l ca pa ci t an ce  [ f F]102 103 104 105 106 107 10801time [h]P RB SFig. 4. Top: time evolution of the quasi differential capacitance from partII of the experiment (in blue) and the corresponding results obtained frommodel M1 from Fig. 3 (in red). Bottom: section between hours 102 and 108of the time evolution of the experimental and the obtained data with thecorresponding bit symbols applied to the MEMS in that period of time.plate dimensions are 515x515 µm2 and the pull-in voltage isaround 14V .The device is actuated with BIT0 and BIT1 symbols fol-lowing a specific pattern. The parameters of the symbols areTs = 1.5s, δ = 1/3, α = 1.4fF/V 2 and V + = −V − = 5V .The actuation pattern is a Pseudo Random Binary Sequence(PRBS), chosen because it has a wide spectrum and allows toimprove the quality of the fittings in presence of noise [12].A six-day experiment has been carried out. It is divided intwo consecutive three-day parts as follows:• Part I. For each PRBS symbol, ’0’s or ’1’s, either 40BIT0 or 40 BIT1 waveforms are applied to the device.The total length of the PRBS sequence is 4320 symbols.• Part II. For each PRBS symbol, 400 BIT0 or 400 BIT1waveforms are applied to the device. Now, the length ofthe PRBS sequence, different to that of part I, is 432symbols.The data from each experiment has been processed sepa-rately and then compared to see if making an inference fromthe data of part I or part II results in different charging models.In the fitting process, some parameters are set. As both partsof the experiment have identical duration T = 4320 min andsampling period Ts = 1.5s, then the set of frequencies, ξk,range from fmin = ξmin2pi ≈ 4µHz to fmax = ξmax2pi ≈ 0.1Hz,and are geometrically spaced. It must be noted that theexperiment should be adapted to the time scale desired in the472 78 84 90 96 102 108 114 120 126 132 138 1440246810time [h]ab so lu te  e rr or  [ fF ]Fig. 5. Time evolution of the absolute error between measured quasidifferential capacitance from part II of the experiment and the correspondingresults obtained with model M1.prediction capability of the model. This applies to the samplingfrequency and the duration of the experiment.The results of part I have been treated first. To deter-mine which is the minimum model order that provides goodadjustment to this experimental data, the evolution of theroot mean square error (RMSE) of the fittings as a functionof the model order has been analyzed. Figure 2 shows theevolution of the RMSE. As no substantial gain is achievedin terms of error beyond K = 6, this is the model orderchosen (model M1). It has been observed that there is a goodagreement between fittings and measurements. Figure 3 showsthe diffusive symbols corresponding to the bipolar voltagesapplied to the device.Secondly, the data of part II has been fitted using the sameprocedure as with data of part I to obtain model M2 andno noticeable difference is appreciated. The diffusive symbolsobtained from the pseudo-inversion of the experimental dataallow to model the dynamics of the charge trapped in thedielectric of the MEMS.Finally, the input signal applied in part II has been fed intothe model obtained from the measurements of part I (modelM1). The objective is to compare the part II measurementswith the results obtained with model M1. This is made inorder to analyze the real capability of the DR models todescribe the charging dynamics in real measurements. Figure(4) shows the quasi-differential measurements obtained duringthe experiment and the modeled data. The relationship between∆C(t) and the total charge trapped in the dielectric of theMEMS, Qd, is given in Eq. (1) [3], [8], [11]. To demonstratethe good matching between experimental and inferred data,Fig. (5) shows the time evolution of the absolute error,|∆Cmeas(t) − ∆Cmodel(t)|, along part II of the experiment.As it can be observed, the error is approximately constant overtime.V. CONCLUSIONA new method based on Diffusive Representation to char-acterize the dynamics of the parasitic charge trapped in thedielectric layer of MEMS devices has been presented. Adynamical finite-order model has been extracted from open-loop measurements using pseudo random binary sequences,which are useful in frequency domain system identification.The Diffusive Representation model agrees with the experi-mental data and has been successfully applied to model thebehaviour of the charge dynamics of a real device. These statespace models are very well suited to describe the behaviourof diffusive systems under non-trivial controls.REFERENCES[1] W. M. V. Spengen, "Capacitive RF MEMS switch dielectric chargingand reliability: a critical review with recommendations,” J. Micromech.Microeng., vol. 22, no. 7, p. 074001, Jun. 2012.[2] M. Kouteoureli and G. Papaioannou, "The discharge current through thedielectric film in MEMS capacitive switches,” in European MicrowaveIntegrated Circuits Conference (EuMIC), pp. 450–453, Oct. 2011.[3] M. Dominguez-Pumar, S. Gorreta, J. Pons-Nin, E. Blokhina, P. Giounan-lis, and O. Feely, ”Real-time characterization of dielectric charging incontactless capacitive MEMS,” Analog Integr. Circuits Signal Process.,vol. 82, no. 3, pp. 559–569, Mar. 2015.[4] G. Montseny, Representation diffusive. Lavoisier, 2005.[5] R. W. Herfst, P. G. Steeneken, J. Schmitz, A. J. G. Mank, and M. vanGils, ”Kelvin probe study of laterally inhomogeneous dielectric chargingand charge diffusion in rf mems capacitive switches,” in 2008 IEEEInternational Reliability Physics Symposium, pp. 492–495, Apr. 2008.[6] H. Hara and Y. Tamura, ”Dynamical process of complex systems andfractional differential equations,” Open Phys., vol. 11, no. 10, pp.1238–1245, Oct. 2013.[7] U. Zaghloul, G. J. Papaioannou, F. Coccetti, P. Pons, and R. 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Characterization of dielectric charging in MEMS using Diffusive Representation

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