Miller 1990 Modeling Ferr Capacitors

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Device modeling of ferroelectric capacitorsS. L. Miller, R. D. Nasby, J. R. Schwank, M. S. Rodgers, and P. V. Dressendorfer

Citation: J. Appl. Phys. 68, 6463 (1990); doi: 10.1063/1.346845View online: http://dx.doi.org/10.1063/1.346845

View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v68/i12

Published by the American Institute of Physics.

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Device modeling of ferroelectric capacitorss L Miller, R D Nasby, J. R. Schwank, M S Rodgers, and P V DressendorferDivision 2144. Sandia National Laboratories P 0. Box 5800 Albuquerque New Mexico 87185(Received 30 July 1990; accepted for publication 31 August 1990)A physically based methodology is developed for modeling the behavior of electrical circuitscontaining nonideal ferroelectric capacitors. The methodology is illustrated by modeling thediscrete ferroelectric capacitor as a stacked dielectric structure, with switching ferroelectricand nonswitching dielectric layers. Electrical properties of a modified Sawyer-Tower circuitare predicted by the model. Distortions of hysteresis loops due to resistive losses as a functionof input signal frequency are accurately predicted by the model. The effect of signal amplitudevariations predicted by the model also agree with experimental data. The model is used as adiagnostic tool to demonstrate that cycling degradation , at least for the sample investigated,cannot be modeled by the formation of nons witching dielectric l ayer(s ) or the formation ofconductive regions near the electrodes, but is consistent with a spatially uniform reduction inthe number of switching dipoles.

I. INTRODUCTIONInterest in ferroelectric capaci tor stlUctures is increasing due to their recent utilization as the memory element of a

new type of integrated circuit nonvolatile memory. \-3 Electrical properties of ferroelectric capacitor structures aretypically determined by making electrical measurements using characterization circuits, such as the Sawyer-Tower testcircuit,4 which contains the capacitor as a circuit element.Unfortunately, the quantitative determination of basic ferroelectric capacitor device properties from typical measurement circuits is very difficult when the capacitor is notideal. s.6 Parasitic effects due to the characterization cir

cuit, as well as resulting from the ferroelectric circuit element itself, can make interpretation of the measurementsdifficult.Little has been published regarding quantitative modelling of the electrical behavior of circuits containing thin-filmferroelectric capacitors. Scott et al have investigated theswitching kinetics of potassium nitrate and lead zirconatetitanate thin films 7.8 successfully applying the theory ofIshibashi9 to describe the transient switching currents. Application of the results to model the behavior ofcircuits containing ferroelectric capacitor elements was apparently notpursued. The authors have recently become aware of a modeling effort by Evans and Bullington, 0 who model hysteresis loops by treating the dipole switching process as resultingfrom a statistical distribution of switching fields for eachdipole. Though this is a useful approach, it does not appearto be amenable to modelling effects resulting from trappedspace-charge, nonswitching dielectric layers, or other nonideal physical properties of the ferroelectric.

Motivated by the need to quantitatively characterizenon deal ferroelectric capacitors, as well as the need to accurately predict (not just fit) ferroelectric integrated circuitresponse, we have developed a technique to model ferroelectric capacitor and circuit properties, including hysteresis effects, which takes into account parasitic properties. The resulting modelling approach provides a means to gain insightinto the natureof degradation mechanisms. Fabrication pro-

cess optimization may also be facilitated, since variations inprocess parameters and techniques may be correlated withferroelectric capacitor device properties.

The approach we have taken to model the behavior offerroelectric capacitors in electrical circuits consists of a procedure with four major steps: (1) define a physical model ofthe ferroelectric capacitor, including parasitic features, (2)based on the physical model, derive equations relating thecharge on the capacitor electrodes to the voltage drop acrossthe capacitor (this results in the definition of a ferroelectriccapacitor circuit element or module ), (3) insert theferroelectric module into a circuit of interest, and solvethe circuit equations for desired parameters. such as currentsand voltages, and finally 4) experimentally verify the solutions, refining the original physical model if necessary. Depending on the accuracy required, the physical model mayneed to be iterated until the model prediCtions sufficientlymatch the experimentally observed results.

The first three steps of the procedure are illustrated inSec. II. The experimental verification of resulting solutionsis then presented in Sec. III. Application of the model as adiagnostic tool is illus trated in Sec. IV, with cycling degradation (fatigue) being the illustration vehicle. A discussionfollows in Sec. V in which refinements to the physical modelpresented are proposed. Applications of the modeling approach are also discussed.

II. THEORYA Ferroelectric capacitor moduleThere are a wide variety of physical features one might

wish to include in a model of a ferroelectric capacitor. Ourapproach is to begin by defining a fairly simple physicalmodel, to develop the mathematical techniques required forimplementing the model, and then to refine the physicalmodel as needed to be consistent with experimental results.

The physical model being considered is iIIustrated inFig. 1. The capacitor electrodes are separated by a distanceof df the area of each electrode is A The electrodes couldpossibly physically or electrically interact with the ferroelec-

6463 J. Appl. Phys. 68 (12), 15 December 1990 0021-8979/90/246463-09 03.00 1990 American Institute of Physics 6463



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vJ d p 121-at \T

P E)

L-a / Tt d p l2l

FIG. 1 Physical model of a ferroelectric capacitor circuit element containing a switching ferroelectric layer and nonswitching dielectric layers adjacent to the electrodes.

tric material; hence we include in our physical model twononswitching dielectric layers, one adjacent to each electrode, and each with a thickness dp /2 and dielectric constantEp For an ideal ferroelectric capacitor, dp is simply zero.The electrical behavior of the capacitor is independent ofwhether the nonswitching dielectric occurs as a single layeror as multiple layers, provided the to tal physical thickness isheld constant. The remaining dielectric consists of a switching ferroelectric whose dipole polarization is a function ofthe field and previous history, and whose linear dielectricconstant is designated by Ef . A voltage Vis applied across thecapacitor electrodes. Each electrode has a free surfacecharge per unit area whose magnitude is af . With these definitions, we proceed to develop the equations which definethe relationship between the applied voltage and the chargeon the capacitor electrodes.

The electric displacement vector D in the ferroelectriclayer is perpendicular to the plane of the electrodes, and isdefined by

D = EaEf + PlOtwhere Ef is the electric field in the ferroelectric and Pl t is thetota \ ferroelectric pohl.rization. The total polarization can bewritten as the sum of the linear contribution to the displacement and the contribution due to switching dipoles in theferroelectric film:

1)where Xf is the electric susceptibility of the ferroelectric,Ef = 1 + Xf is the linear dielectric constant of the ferroelectric, Eo is the permittivity offree space, and Pd is the contribution to the polariza tion due to switching dipoles only.

Utilizing Maxwell s first equation,VD=p, 2)

it can be shown that the electric field in the ferroelectric andthe magnitude of the charge on the capacitor electrodes aregiven by

E - 1 (v dpPd )f - dfYI - EoEp 3)and6464 J Appl. Phys Vol. 68. No. 12. 15 December 1990

af = ~ [EaEfV + Pd l - ~ ] ,dJ df 4)where 1 1 is defined by

1 1 = I- 1 _ J) .df Ep 5)The behavior of circuits containing hysteretic ferroelectric elements is dependent on the previous history of the cir

cuit parameters. Thus, the quantities of interest will be parameterized in terms of time. Once the circuit parameters aredefined at one point in time, circuit parameters at futuretimes are calculated by integrating from the initial boundaryconditions, i.e., by summing the incremental changes in theparameters. To facilitate subsequent computations, Eqs. 3)and 4) must be differentiated with respect to time. But first,we discuss a subtle issue associated with the process of differentiating these equations.

How one evaluates the total differential dPdldt depends on the parameters of which Pd is a function, i.e., thephysical nature of the polarization. The polarization is afunction of the electric field, which itself is a function of time.When an electric field is applied, the polarization does nottrack the field instantaneously, but lags slightly in time;hence the polarization is also an explicit function of time.Thus the derivative is written as

6)For application to the experimental examples considered later, it is assumed the polarization responds to the electric fieldwithin a time interval which is very short compared to thetime scale over which other relevant parameters change.Specificaily, the time constant for dipole switching is assumed to be much shorter than the time required for theapplied voltage or internal fields to change appreciably.Thus, the second term on the right-hand side will be omittedin subsequent equations. If this condition is not met, asmight be the case for ferroelectric elements in a high-speedintegrated circuit, the second term must be included.Differentiating Eqs. 3) and 4), utilizing Eq. 6), andeliminating the term aE lat, one obtains

7)and

daf = dV EaEf 1 + _1_ aPd E,t)dt dt df 1 2 Eo Ef aE

8)where 1 2 is defined by:

1 2 = 1 ~ [1_ E I +_l_aPd E,t ))] .df Ep EoEf aE

9)If the dipole polarization is known as a function of theelectric field, Eqs. 3) and 4) [or equivalently Eqs. 7) and

8) 1unambiguously define the relationship between the voltage applied across the ferroelectric capacitor and the freecharge on the capaci tor electrodes. The electric-field dependence of the dipole polarization will be discussed later in Sec.n D.

Miller et al. 6464



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_ ~ ~ ~ ~

B Sawyer-Tower circuitA common characterization circuit utilized to study fer

roelectric capacitors is the Sawyer-Tower circuit,4 or modifications thereof. A slightly modified version of the circuitshall be modeled, as illustrated in Fig. 2 A time-dependentinput voltage Vi t) is applied to the circuit, resulting in anoutput voltage Va t). The resistor RJ t) simulates the effectof current leakage through a ferroelectric capacitor with fi-nite resistivity. The resistor Rn t) might represent the inputimpedence of the instrument utilized to measure the outputvoltage, or perhaps current leakage through the integratingcapacitor. The goal is now to develop the circuit equationswhich give the output voltage as a function of the input voltage and the parameters associated with the circuit elements.

The charge at node in Fig. 2 is the sum of the charge onthe bottom and top electrodes of the ferroelectric and inte

I

where12)

and13)

Equation 11) now completes the set of equations required to describe the modified Sawyer-Tower circuit inFig. 2 containing the ferroelectric capacitor illustrated inFig. 1 A technique to solve the equations is described in thefollowing section.c Numerical Integration algorithm

For specified initial conditions, the output voltage as afunction of time can be determined by solving the coupled set

FE CAPACITORCIRCUIT ELEMENT

NORMALCAPACITOR

VI t)

---- , - - - -

FIG. 2 Modified Sawyer-Tower circuit with parasitic resistances.6465 J. Appl. Phys., Vol. 68, No. 12, 15 December 1990

grating capacitors, respectively. At any time, this totalcharge is equal to the charge due to current flowing throughthe resistors RJ t) and Rn t ), plus the total charge on thecapacitors which existed before any current flow occurred.This condition is defined mathemat ically by

10)

Taking the time derivative of Eq. 10), utilizing Eq.8), and noting that O n t)An = en v. t ) (where en andAnare the capacitance and area of the normal capacitor) one

arrives at

11)

rof algebraic and differential equations 3) and 11). I t isassumed the function aPdlaE is known as a function of thefield E (this function will be discussed in the next section),and the input voltage Vi t) is known as a function of time.After each increment in time, the following quantities arecalculated: Pd, aPdlaE, EJ Va t), and dVo t)ldt. A numerical integration algorithm for this calculation is now discussed. Although alternative computational procedures exist, the algorithm outlined below has been found to be easilyimplemented and to work over a wide range of parameters.

The initial conditions consist of the specification, atsome initial value of time, of the dipole polarization Pd , theoutput voltage Va the input vohage v. and the derivativedV;ldt. The field at that time is then calculated using

EJ = _1_ Vi _ Va _ dpPd).d Y op 14)Knowing the field, dPdldE can be computed. Finally,dValdt is calculated from Eq. 11). Once the initial valuesof these necessary functions are determined, the time is thenincremented. The procedure to compute the above quantities for the I th increment in time is now discussed.

The input voltage Vi and dV;ldt are known by definition. The output voltage at the I th increment in time is givenby

dVoV l = V l I t/ - t _ J . 15)dt/_ JThe field is calculated by observing that Eq. (14) can bewritten as

(16 )Solving Eq. 16) for the value of the field at the I th increment, we obtain

Miller et al. 6465



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The dipole polarization is calculated usingdPdPd = Pd + (EI - E I ).I I J dE _1 I I J 18)

The I th value of dPdldE is then calculated utilizing the appropriate algorithm to be discussed below). Finally, the I thvalue of dV ldt is calculated from Eq. ( II).

After the initial values of the functions are determined,the procedure defined in the above paragraph is applied repeatedly to determine the behavior of the circuit as a function of time.D Determination of dPdldE

The equations developed above can be solved only if thederivative of the dipole polarization with respect to the electric field is known. Before writing an explicit expression forthis quantity, we first s tate some basic physical properties ofthe dipole polarization, us well as some d ~ f i n i t i o n s As thefield is ramped between large absolute values opposite insign, the switching dipole polarization Pd must approach theasymptotic values of P where P is the spontaneous polarization. Ps is the polarization which results when all thedipoles are fully aligned. The resulting field-dependent polari:zation is referred to as the saturated polarization loopsaturated in the sense that the field is ramped between val

ues sufficiently large that essentially all the dipoles switch inboth directions). This behavior is illustrated in Fig. 3 Thesllturated polarization has zero magnitude at a value of theelectric field called the coercive field, E . At zero field, thevalue of the saturated polarization is Pr , the remanent polarization. We make the physical assumption that the twobranches of the saturated switching dipole polarizationcurve are symmetric. This assumption is expressed by

P d E ) = - P , ; - E ) , 19)where the plus ( + ) sign indicates the branch of the polar-

30p .

20 ,.,E ,. ,> Pr /U 10 I2; I,Z I0 ,i= ,"' ;-10a: Po E) - - - - ,"' ,0 ~ , I IfL -2Q-30

300 -200 -100 0 100 200 300ELECTRIC FIELD kV/cmjFIG. 3 Model for switched dipole polarization, computed from Eqs. ( 19)and 20). The polarization saturates at p ..6466 J. Appl. Phys., Vol. 68, No. 12,15 December 1990

ization for the positive-going field ramp and the minus ( - )sign indicates the branch for the negative-going field ramp.

In the absence of a fundamental physical theory whichpredicts dipole switching properties as a function of appliedfield, the satu rated polarization P / E) is defined utilizing aconvenient mathematical function whose behavior satisfiesthe physical requirements. A number of candidate functionsexist, among them being the error function an integral of aGaussian distribution), t the Langevin function encountered in magnetism,12 and the hyperbolic tangent function. 13 The hyperbolic tangent function is chosen due to itsconvenient mathematical properties. As will be shown later,this choice of function is consistent with experimental data.It is emphasized that the equations developed up to this

point are independent of the specific choice of the function todescribe the field dependent saturated d ipole polarization. )Specifically, the sa turated polarization loop is defined by

P J (E) = P, tanh [(E - Ee )/28],where

[l + P J P , ] ~ - 18 = Ee log PJP,

20)

(21 )and P P and Ee are taken as positive quantities. The constant 8 is defined such that P f 0) = - Pr. The negativebranch of the saturated switching dipole polarization is given by Eq. 19). Differentiating Eq. 20), one obtains

d P ~ ; E ) = P , [ 2 b C O S h 2 E ~ b E c ) ] (22)and

dP d (E) I = dP / (E) IdE E dE 23)E

The derivative of the switching dipole polarization withrespect to the ferroelectric electric field is given, for the casewhere the polarization is saturated, by Eqs. (22) and 23).What happens if the applied signal does not drive the polarization into saturation? Existing data indicate that the derivative of the polarization with respect to the field, evaluated a ta constant field, is independent of the amplitude of the applied signal, at least to first order. Hence, the assumption isnow made that the derivatives defined in Eqs. (22) and 23)are independent of the amplitude of the osciIlating electricfield. Figure 4 illustrates a family of polarization curves generated by Eqs. 22) and 23) for several different electricfield amplitudes. Note that the slope at a given field is thesame for each curveIII RESULTSA Experimental procedures

To examine the validity of the modelling approach discussed thus far, the circuit illustrated in Fig. 2 was assembled. A ferroelectric capacitor fabricated with the sol-gelprocess, of area 10 000 / lm2 and thickness 400 nm, was utilized. The normal integrating capacitor had a capacitanceof 10 nF. A high-speed digitizer was utilized to measure theinput and output voltages Vi t) and Vu t ); the input impedence was 1 Mil = Rn t). The input signal was a symmetric

Miller etal. 6466

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NE 20lJ:1Z 10o=

0ii:5o -10Q.WJ

-20Q

a.c.: EQUAL SLOPES ATEQUAL FIELDS

_ ~ L ~ __ __ ~ ~~ -200 -tOO 0 100 200ELECTRIC FIELD (kVlcm)

FIG. 4. Family of unsaturated switched dipole polarization hysteresisloops, computed from Eqs. (22) and (23). The slope of a given branch at agiven value of electric field is independen t of the amplitUdeof he symmetricperiodic field.

sine wave with adjustable frequency and amplitude. A givenset of data was obtained by first applying 500 cycles to thecircuit, then digitizing the input and output voltages for the501 st cycle.Though the resulting data may be graphically represented by plotting the voltages as a function of time, we choose toplot the output voltage as a function of the input voltage.Specifically, we plot on the x axis the quantity

XU) = Vj t)/dj 24)and on the y axis the quantity

(25)The resulting curve is a hysteresis loop; it is a structure

which is associated with the entire circuit, rather than simply being associated with the ferroelectric capacitor. Thefunction X t) is an approximation to the ferroelectric electric field. The function Y t) is an approximation to the totalferroelectric polarization (linear plus switching contribution). The quantities defined in Eqs. (24) and (25) are whatare commonly plotted when characterizing the hysteresis ofa ferroelectric capacitor. These approximate expressions forthe field and polarization become accurate only in the limitwhere the capacitance of the normal capacitor is large compared to that of the ferroelectric, when the current flowingthrough the resistors is small compared to the capacitor displacement currents, and if the capacitance of nonswitchinglayers is very large compared to that of the ferroelectric dielectric layer. Thus, the axes for hysteresis plots will be labeled Electric Field and Polarization, the quotes indicating that the values plotted do not necessarily correspondto the true ferroelectric field and the true polarization. Theparameters utilized in the following experiments span a sufficiently wide range that Eqs. (24) and (25) are not alwaysvalid approximations to the actual ferroelectric field and polarization.B Determination of model parameters

The next step in verifying the validity of he circuit equations for the circuit in Fig. 2 is to determine the parameters6467 J Appl. Phys., Vol. 68, No. 12 15 December 1990

Pr P Ee and EJ for the ferroelectric capacitor. A hysteresisloop was obtained utilizing a sine wave input with a to-Vamplitude (20 V peak to peak), driving the capacitor intosaturation. A frequency of 100kHz was utilized to minimizeeffects due to resistive losses. Values of P, = 14 /lC/cm2,Ps = 23 /lC/cm2, e = 45 kV /em and EJ = 600 were determined by fitting the model prediction to the hysteresis loop.(This value of EJ results in a computed value of 132 pF forthe linear capacitance of the ferroelectric capacitor.) Theseparameters, along with a value of dp = 0 and RJ t) 00were utilized as inputs to the model, and the solid curve inFig. 5 was generated using the computational algorithm discussed in Sec. II. The dots are the experimental data.

The good agreement between the model results and theexperimental data indicates that the physical description ofthe ferroelectric capacitor is reasonable, the computationalalgorithm works properly, and the physical parameters arecorrectly chosen. Next, the values of the circuit parametersare varied to see whether or not the resulting circuit behavioris accurately predicted. No further adjustments to the ferroelectric parameters are made; the only changes to themodel input parameters will be known parameters associated with the input signal and circuit elements.

c. Effects of digitizer impedanceThe finite digitizer impedance can significantly impact

the resulting hysteresis loop when the signal frequency issufficiently low. Distortion occurs when the charge flowingthrough the digitizer per input signal cycle is not negligiblecompared to the charge on the integrating capacitor. A hysteresis loop obtained at a frequency of 100 Hz is shown inFig. 6. Very good agreement is observed between the predicted hysteresis loop and the experimental data. The only model parameter which was changed from the set which generated the curve in Fig. 5 was the frequency.

A series of hysteresis loops was then obtained at different frequencies. A plot of the apparent remanent polarization, i.e., the total polarization at zero field, as a functionof signal frequency is shown in Fig. 7. Again, very good

MOOEL PREDICTION EXPERIMENTAL DATA

' E 20lJ:1

-200 -100 0 100 200 300ELECTRIC FIELD (kV/cm)FIG. 5. Hysteresis loops with model param eters extracted from experimental data. Resulting model calculaiions closely match the data.

Miller eta/. 6467



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40 ' - ~ - - ' - - ~ - ' I ~ - - - ~ - - ; I r - - - ~ - ~ ~~ - MODEL PRHllCnON frequency =100 Hz)

E X P E m ~ N T L DATA

E 20uU>

~ ~ o o ~ ~ _ ~ L ~ L ~ ~ L ~ L ~ ~ ~ ~-100 0 100 200 300ELECTRIC FIELD (kVlcm)FlG. 6. Characteristic hysteresis loop resulting from resistive leakagethrough R . The model (with no adjustable fitting parameters) accuratelypredicts the experimental data.

agreement between the model prediction and the experimental data is observed. The apparent reduction in the remanentpolarization is an artifact of the choice of circuit parameters;full switching of the ferroelectric still occurs. The true ferroelectric remanent polarization P is a property of the ferroelectric capacitor itself. and not a property of the characterization circuit. The distortion at low frequencies resultingfrom the finite impedence of the digitizer can also arise if thenormal capacitor has a finite resistive component to its impedence.D Effects of ferroelectric resistive leakage

The hysteresis loop can experience another type of frequency dependent distortion if the resistivity of the ferroelectric capacitor is not infinite_ Since we could not physicallychange the resistivity of the ferroelectric capacitor understudy, we simulated the existence of a finite resistivity by theaddition of a resistor in parallel with the ferroelectric capacitor. Its value was 1.1 M 1 = RJ(t), equivalent to a value of

20 , - - - - - - , - ~ ...... ... . ~ - - - - ; - - - - - - - - , - - - - ~

15

'Eui3 10>

o 1

~ ~ Mon L i f : r : ~ 7 1 0 N ~ ~ ~ 3 m z e r Impedance OSt .) E X ~ n : - M ; : : N T A l D T

2 3 4LOG Frequencyll Hz)

6

FIG 7. The apparent remanent polarization , as defined by Eq. (25) at7.ero field. decreases with decreasing frequency for finite resistance R . Thetrue remanent polarization differs from that plotted. The experimental re-sults are accurately predicted by the model.6468 J. Appl Phys . Vol 68. No. 12. 15 December 1990

3.X 106 ~ for the resistivity of the ferroelectric capacitordlelectnc_ Figure 8 shows resulting hysteresis loops obtainedat frequencies of 100 and 1 kHz_ The only model parameterswhich were changed to generate the solid theoretical curveswere the known values of the resistance Rf(t) and the frequency. As before, the experimental data are accurately predicted.A series of hysteresis loops was obtained at additional

f r e ~ u e ~ c i : ~ In Fig_ : a plot ?fthe apparent remanent polanzatlon as a functIOn of Signal frequency indicates goodagreement between the model prediction and the experimental data. This time, the apparent remanent polarizationincreases at lower frequencies, rathe r than decreases_ As before, this behavior is an artifact of the nature of the characterization circuit. Full switching of the ferroelectric still occurs_E Effects of signal amplitude

In addit ion to frequency, the amplitUde of the input signal affects the observed circuit hysteresis loops. Hysteresisloops were obtained for a range of amplitudes; the resultingcoercive field (value of the field at zero polarization) isplotted in Fig. 10 as a function of amplitude. The resistor

RI(t) was removed from the circuit for this series of experiments. The experimentally observed rapid increase and subsequent saturation of the coercive field is predicted by themodel. As before, no adjustable parameters were utilized toobtain the good fit; the solid curve was generated by themodel by changing only the value of the signal amplitUde.IV. USE OF MODEL AS A PHYSICAL DIAGNOSTIC TOOL

Up to this point, we have verified that the model accurately predicts the effects resulting from known changes inthe values of the circuit elements and input signal parameters. The good agreement between model predictions andexperimentally observed results supports the modelling approach as being correct. Now we are in a position to utilizethe model as a diagnostic tool, i.e., to infer information re-

I 1 I40

MODEL PREDICTION (R t = 1.1 Mil) EXPERIMENTAL DATA' E

20>zo 0

NE -20

-40

-300 -200 -100 0 100 200 300ELECTRIC FIELD (kV/cm)FIG. 8. C h a r ~ c t e r i s t i c h y ~ t e r e s i s loops resulting from resistive leakagethrough Rj' The model (with no adjustable fitting parameters) accuratelypredicts the experimental data.

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40

30~Eu0 20a 10

. MODELPREDfCTlON R, =1.1 MO) EXPERIMENTAL DATAO L ______ ________ ________ ________

2 3 .. 5 6LOG (Frequency/1 Hz)FIG. 9. Apparent remanent polarization, as defined by Eq. (25) at zerofield, increases with decreasing frequency for finite Rj The true remanentpolarization differs from that plotted. The experimental results are accurately predicted by the model.

garding the physical properties of the ferroelectric capacitorfrom measurements obtained from the test circuit.As an example of utilizing the model as a diagnostictool, we consider changes in the electrical response of the

ferroelectric capacitor resulting from repeated polarizationreversal cycling, often referred to as fatigue. The focus of thefollowing discussion is to illustrate the use of the model as aphysical diagnostic tool, and not to develop or establish acomprehensive microscopic theory of cycling degradation.Hysteresis loops before cycling degradation had occurred, and after 4 X 109 cycles, are shown in Fig. 11 (a).Model parameters were obtained from the predegradationdata, and were utilized to theoretically generate the solidcurve. After cycling, the apparent switched charge in theferroelectric capacitor is significantly reduced. The basicquestion is: what physically has changed in the capacitor toresult in the different hysteresis loop? The approach to answer this question is to make an assumption, and then seewhether or not the model prediction is consistent with the

I: 20

MODEL PREDICTION EXPERIMENTAL DATA

2 .. 6 8SIGNAL AMPLITUDE V) 10

FIG. 10. Observed coercive field, defined utilizing Eq. (24) at zero polarization, decreases with decreasing signal amplitude. The model predictionagrees with the experimental data.

6469 J. Appl. Phys. Vol. 68 No. 12 15 December 1990

(a)

fb)

~N

, 4110eCYCLES/ I

~ 3 O O L ~ _ 2 0 0 L ~ ~ ~ ~ ~ ~ ~ ~ ~ ~- 00 0 100 200 300ELECTRIC FIELD (kV/cm)


~ L ~ _ _ _ _ _ _ L _ _ _ _ _ _-300 zoo -100 0 100 200 300ELECTRIC FIELD (kV/cm)


E 20l)

zoo -100 0 100 200 300(e) ELECTRIC FIELD (kV/cm)FIG. 11. Physical effect of cycling degradation is examined utilizing themodel. (a) Pre- and post-cycling degradation hysteresis data. Physical parameters are determined from the predata. (b) Post-cycled data is not accurately predicted by assuming the presence of a nonswitching dielectric layer.(cl Post-cycled data is accurately predicted by assuming a spatially uniform reduction in the number of switching dipoles.

observed results. If the prediction is inconsistent, the assumption is wrong. If the prediction is consistent, the as-sumption may be correct.We begin by assuming that the cycling resulted in someinteraction between the electrodes and the ferroelectric material which caused the formation of a nonswitching regionof the ferroelectric near the electrodes. Several mechanisms

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have been proposed by which a nonswitching layer couldoccur. It has been proposed that cycling results in an increase in the space-charge region near the electrodes, causing a reduction in the switching of the dipoles in the affectedregions. 14 Another proposed effect of cycling is the formation of conductive dendrites growing from the electrodesinto the ferroelectric. 15. 16 A possible effect of dendriticstructures is to shield the domains under the dendriticbranches from the applied field, causing them not to switch.Independent ofa specific proposed mechanism which mightresult in a nonswitching dielectric layer, the solid curve inFig. 11 (b) was generated by adjusting the nons witching dielectric thickness dp to a value which gave the optimum fit tothe data. Rather than preserving the characteristic Sshape of the hysteresis loop, the existence of a nonswitchingdielectric layer collapses the hysteresis loop into a ratherstructureless oval. (This behavior was also experimentallyobserved when a normal capaci tor was inserted in series witha nondegraded ferroelectric capacitor, and was accuratelypredicted by the modeL) The lack of even qualitative agreement with the experimental post-cycling data indicates thatthe primary effect of cycling was not to produce a nonswitching dielectric layer (s), either at the electrodes, or elsewherein the structure.

The existence of conductive dendritic structures eminating from the electrodes could result in an additional type ofeffect. f the lateral spacing of the dendrite tips is less thanthe length of the dendrites, as is shown in Ref. 16 the resultwould be a reduction in the effective thickness of the ferroelectric capacitor. f the effective thickness varied laterally,the resulting switching behavior could be modeled simply bythe parallel connection of a large number of small-area capacitors with different thicknesses. The expected first-orderconsequence would be a scaling of the electric-field axis ofthe resulting hysteresis loop. The detailed form of he scalingfunction would depend on the distribution of thicknesses.The post -cycling data in Fig. 11 (a) cannot be obtained fromthe precycling data simply by scaling the field axis, even in anonlinear way.Next, we assume that the number of switching dipoles isuniformly reduced throughout the volume of the ferroelectric m a t e r i ~ a a result of repeated polarization reversals.This could occur ;; if, a result of pinning of domain walls. 17This effect would be simulated by scaling both P and P tobe smaller than their original values by the same scaling factor. The solid curve in Fig. 11 (c) was obtained using theprecycled parameters (including E, with the exceptionthat both P and P were scaled to values 30 the magnitudeof their original values. Observe that the resulting modelprediction matches the experimental data quite well over theentire hysteresis loop. The slight discrepancy at large valuesof the field may be due to losses resistive in nature. Oneconcludes that a uniform reduction in the number of switching dipoles is a plausible and consistent explanation for theobserved change in electrical behavior due to cycling degradation for the sample investigated.We have not yet verified whether or not these specificresults are also obtained with samples processed with different compositions or electrode interfaces. Thus, one must be6470 J Appl Phys . Vol 68 No 12 15 December 1990

cautious in making generic conclusions from this very limited data set regarding the physical effects resulting from cycling, and how those physical effects might impact the electrical properties. We have demonstrated the modelingapproach developed in this paper will certainly be of use inaddressing these issues.

V. DIS USSIONA conceptual and mathematical methodology for modelling the behavior of circuits containing at least one ferroe

lectric capacitor has been developed. This methodology hasbeen illustrated with a very simplified physical model of theferroelectric capacitor, which appears to work quite well inpredicting experimental results, at least to first order. However, there are a number of refinements one might wish tomake to the physical model. For example, one might wish tomake the dielectric constants frequency dependent, especially for high-frequency applications. For these applications,one may also need to include the explicit dependence of thedipole polarization on time, as was discussed in Sec. II A.The inclusion of space charge in the ferroelectric layer mightprovide insight into the response of ferroelectric capacitorsto ionizing radiation. Such modifications simply change therelationship between the charge on the ferroelectric capacitor electrodes and the voltage drop across the capacitor.Equations (3) and (4) are replaced by the appropriate expressions; the procedure required to set up and solve thecircuit equations is then the same as that developed in therest of Sec. II.

Another area which might require refinement is the definition of the derivative of the field dependent dipole polarization dP E) I dE Practical integrated circuit applicationsrequire the ability to model ferroelectric response to asymmetric and nonperiodic input signals. If the applied voltagedoes not result in a ferroelectric electric field which rangesbetween values equal in magnitude, Eqs. (22) and (23) arenot adequate. They can result in dipole polarizations whichexceed Ps in magnitude, a nonphysical result. A possibleapproach is to treat the polarization in a manner analogousto the magnetization hysteresis of ferromagnetic materials. 18

Independent of the specific physical model one utilizes,it is clear from the above examples that the nature of thecharacterization circuit and signal parameters can significantly impact the inferred results from simple electricalmeasurements such as hysteresis loops. The actual switchedcharge and coercive field may be very different from thevalues obtained by utilizing such simplified Eqs. (24) and(25) to calculate the ferroelectric electric field and switchedcharge. Thus, values of Pr P and Ec reported in the literature may be ambiguous unless the parameters associatedwith the test circuit elements, as well as the nature of theinput signal, are specified.

I t is possible to perform characterization measurementswhich are inherently immune to certain types of parasiticeffects. For example, regarding errors due to resistive parasitic effects, if the hysteresis loop does not change as a function of frequency over a range of at least one order of magni-

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tude, then the frequency is sufficiently high that parasiticresistive effects are negligible. In the other extreme, if theperiod of the applied signal is not large compared to theactual switching time of the dipoles, the resulting signal willbe convoluted with the effects of the finite switching time. Inthis regime, the second term in Eq. (6) must be included inthe mathematical analysis. Procedures to empirically determine whether or not other particular parasitic effects existcan be developed by utilizing the model. To do this, onesequentially varies given input parameters of the model andcomputes the effect on the electrical circuit parameter ofinterest.The ability to easily model ferroelectric circuits allowsone the flexibility to design circuits which might be moresensitive or immune to certain effects, or be more appropriate for a given measurement equipment set or application.For example, the conventional Sawyer-Tower characterization circuit typically consists of an integrating capacitorwhose capacitance is large compared to that of the ferroelectric capacitor. The equations developed inSec II can be usedto model a Sawyer-Tower circuit where the capacitance ofthe integrating capacitor is not large compared to that of theferroelectric capacitor. I f one prefers to characterize a fer-roelectric capacitor by numerically integrating the switchedcurrent passing through a resistor rather than using an integrating capacitor, one simply lets en Oin the above equations; the current is given by I t) = o t) I R n t). Variousmodifications to the standard Sawyer-Tower test circuit,such as compensated and virtual ground circuits, 6 can alsobe modeled with the approach described here. A ferroelectric circuit modeling capability is essential in order to accurately model the structures required in an integrated circuitmemory cell or sensing circuit.Additional applications of this modeling approach include understanding the effects of degradation phenomena.Experimentally observed results can be quantitatively related to basic physical properties of the ferroelectric capacitorsystem. This was illustrated by the above analysis of cyclinginduced degradation. One could gain further insight into thenature of cycling degradation by plotting the polarizationscaling factor as a function of the number of cycles. Utilizingthe model to extract such quantities as the remanent polarization or coercive field, one may be able to understand othereffects, such as the impact of the cycling amplitude or theconsequences of the thermal environment. This modelingapproach may also be implemented to optimize certain ferroelectric circuit fabrication procedures; electrical properties are directly related to physical properties which can thenbe related to fabrication procedures.VI SUMMARY

A physically based practical approach has been presented which can be utilized to model the behavior of electricalcircuits containing ferroelectric capacitors. The modelingapproach may be easily adapted to incorporate essentiallyany physical model of a ferroelectric capacitor element. The

6471 J. Appl. Phys. Vol. 68 No. 12,15 December 1990

model has been demonstrated for a Sawyer-Tower circuit,accurately predicting the experimentally observed distortions to hysteresis loops resulting from changes in circuitelement parameters and input signal frequency and amplitude. The standard approximations used for measurementsof the ferroelectric electric field and switched charge, listedin Eqs. (24) and (24), are often invalid. This can result inmisleading conclusions regarding the inferred properties of aferroelectric capacitor characterized with a given test circuit.The model can be used to provide further insight intothe physical mechanisms of ferroelectric operation and degradation. The utility of the model as a physical characterization tool was illustrated by examining the nature of cyclingdegradation. For the sample considered here, the observedcycling degradation was inconsistent with the developmentof nonswitching dielectric layers or an effective thinning ofthe capacitor; it was consistent with a uniform reduction inthe number of switching dipoles.Proper use of this modeling approach can facilitate accurate characterization of ferroelectric elements, and effi-cient optimization of design integration of ferroelectric devices into semiconductor integrated circuits.CKNOWLEDGMENTS

The authors are grateful to G. Goddard and K. Gutierrez for their technical assistance, and to N Godshall and ACampbelI for critically reviewing the manuscript.

I I. F. Scott and C. A. Araujo, Science 246, 1400 (1989); J. F. Scott and C.A. Araujo, in Molecular Electronics edite d by M. Borisson (World Scientific, Singapore, 1986), pp. 206-214.

2 J. T Evans and R. Womack, IEEEJ. Solid Sta te Circuits 23, 1171 (1988).'D. Bondurant, in The First Symposium on Integrated Ferroelectrics, University of Colorado, Colorado Springs, 1989, pp. 212-215 (unpublished).'c. B Sawyer and C. H. Tower, Phys. Rev. 35, 269 (1930).'M. E Lines and A. M. Glass, Principlesand ApplicationsofFerroelectricsand Related Materials (Oxford University Press, Oxford, 1977), p. 104.

J. T. Evans, M. D. hey and J. A. Bullington, in The First Symposium onIntegrated Ferroelectrics, University of Colorado, Colorado Springs,1989, pp. 217-221 (unpublished).

7 J. F. Scott, L. Kammerdiner, M. Parris, S Traynor, V Ottenbacher, A.Shawabkeh, and W F Oliver, 1 Appl. Phys. 64, 787 ( 1988). J. F. Scott, C. A. Araujo, H. B Meadows, L. D. McMillan, and A.Shawabkeh, 1. Appl. Phys. 66, 1444 (1989).y Ishibashi, Jpn. J Appl. Phys. 24, 126 (1986).

\0 J. A. Bullington and 1 T. Evans (private communication)./I G. Arfken, MathematicalMethodsfor Physicists (Academic, New York,1970), p 474.21. R. Reitz and F. J. Milford, Foundations of Electromagnetic Theory

(Addison-Wesley,Reading, MA, 1967), p 225;D.J. Craik, StructureandProperties ofMagnetic Materials (Pion, London, 1971 ). p 31.

\J W. H. Beyer, Ed., Re Standard Mathematical Tables 25th ed. (CRC,Boca Raton, FL, 1979), p 252.,. J. A. Bullington, M. D. Ivey, and J. T. Evans, in The First Symposium on

Integrated Ferroelectrics, University of Colorado, Colorado Springs,1989, pp. 201-211 (unpublis hed).

R. H. Plumlee, Sandia Laboratories Report No. SC-RR-67-730 (1967).' C. A. Araujo, L. D. McMillan, B M. Melnick, J. D. Cuchiaro, and J. F.Scott, Ferroelectrics 104, 241 (1990).171 F. Scott and B Pouligny,J. Appl. Phys. 64,1547 \W , ).D. C. Jiles, J. Magn. Magn. Mater. 61, 48 (1986).

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Miller 1990 Modeling Ferr Capacitors