Ferroelectrics, 1972, Vol. 3, pp. 177-182 IEEE Trans. Sonics & Ultrasonics, 1972, SU-19, pp. 177-182

@ INSTITUTE OF ELECTRICAL AND ELECTRONIC ENGINEERS Printed in Norwich, England

EQUIVALENT CIRCUIT ANALYSIS OF A FERROELECTRIC-PHOTOCONDUCTOR MEMORY DEVICE

J. T. JACOBS, B. D. SILVERMAN and I . P. BATRA ZBM Research Laboratory. San Jose, Calfornia 951 14, USA

(Received 7 June 1971)

An analysis of an equivalent circuit representing a ferroelectric photoconductor memory device has been performed. The purpose of this work is to provide a basis for describing the transient switching characteristics of the ferroelectric- photoconductor memory element. Two models describing the polarization reversal in the ferroelectric have been investigated. The first model assumes that the polarization charge is a nonlinear function of the voltage that is applied to the ferroelectric. This represents a simple generalization of an RC model for which the ferroelectric is assumed to have a well defined threshold field and allows one to qualitatively describe the partial switching of the ferroelectric. The second model of polarization reversal assumes that domain wall motion is activated. described adequately by the sidewise motion of domains, and dependent upon the instantaneous electric field in the ferroelectric domain. Domain statistics involving domain overlap as well as polarization reversal conditions are imposed to realistically describe the state of the ferroelectric as it traverses a hysteresis loop. The time dependence of the polari7ation, current, and voltage is examined and voltage is examined and attention focussed on polarization switching time as an important device parameter. Non-

saturated switching increases this speed and the feasibility of operating the device in this mode is discussed.

I . INTRODUCTION

Several schemes have been proposed which would achieve protective addressing of ferroelectric mem- ories.’-4 One such scheme utilizes a ferroelectric- photoconductor sandwich (Figure 1) operating as a capacitive voltage divider. Unaddressed bits therefore experience a voltage that is decreased by the ratio of the photoconductor to ferroelectric capacitance and hence the disturb voltage is significantly reduced from the voltage associated with a half disturb. The present paper is devoted to an investigation of the switching characteristics of a ferroelectric-photoconductor sandwich. The time dependence of the polarization, current, and voltage is examined and attention focussed on the import- ant device parameter, polarization switching time.

In the following analysis, the ferroelectric and photoconductor are characterized independently 2nd hence the designation ‘equivalent circuit’ has been used. The ferroelectric-photoconductor inter- face is known to play a significant role in the sandwich characteristics and it can only be hoped that properties resulting from the interface are treated implicitly. The switching behavior of the sandwich is essentially determined by the current

through the photoconductor or voltage across the ferroelectric. The current is assumed to be a function of the voltage across the photoconductor. Linear behavior will be generally assumed with some discussion of the effects resulting from deviations from linear behavior. The ferroelectric circuit behavior has in the past been characterized by a charge voltage characteristic given by a square hysteresis loop. This will be generalized in two ways. First retaining the assumption that the equivalent ferroelectric capacitance is a function of voltage, the effects of shearing or degrading the loop will be investigated. It will be shown, as one might expect, that the device switching time increases. The square loop case, where polarization charge is accumulated at constant voltage leads to the shortest time for switching most of the charge. Second, the ferroelectric will be characterized in terms of a model proposed to describe sidewise domain motion during switching. For this model the effective capacitance is not a function but a functional of the voltage. For sandwich switching time comparable to intrinsic ferroelectric switching time the increased effective ferroelectric capacitance during switching can result in a decrease in voltage across the ferroelectric. This effect has been

177

J. T. JACOBS. B. D. SILVERMAN AND 1. P. BATRA 178

Fe Pc Device

TranspGent Electrode

Equivalent Circuit

4 %

L Electrode

-

P

FIGURE 1 (Top) FEjPC memory element. (Bottom) Equiv- alent circuit of an FE/PC memory element.

observed and will be discussed. It should be pointed out that such nonmonotonic behavior of thevoltage with time isnot predicted by a model which assumed that the effective ferroelectric capacitance is a function of voltage.

For the equivalent circuit model that we will consider, the photoconductor will be represented by an RC combination as in Figure l . At time t = 0, the voltage is applied at the same time illumination is initiated and the voltage division is therefore capacitive at this time. The generated carriers lead to a lowering of the photoconductor resistance and the current which starts to flow is continuous across the entire structure.

11. CAPACITIVE FERROELECTRIC MODEL

The development in this section will proceed by an examination of the microscopic quantities associ- ated with the photoconductor. The results that will

be derived are exactly the same as one could have obtained by performing an RC circuit analysis. It is hoped that the present treatment more clearly indicates just what generalizations might be re- quired to treat deviations from Ohmic behavior.

The ferroelectric is characterized by a capacitance which is a function of voltage. This is given by the hysteresis loop shown in Figure 2. It is assumed that the voltage is applied across the structure at the instant the light is incident on the photocon- ductor. At this time the voltage division is capacitive. After ihis time, the current flow results in a transfer of voltage from the photoconductor to the ferro- electric and ferroelectric switching is therefore initiated.

Hysterssls Loop - Parallelogram Model

P

FIGURE 2 Idealized hysteresis loop of a ferroelectric.

The equations describing the photoconductor are

The total current density is given by Jp( t ) . It is assumed that there is an equal density of electrons, n(x, t ) , and holes p ( x . f), maintaining local space charge neutrality. The holes are also assumed to be immobile. The hole and electron densities are also independent of x since it is assumed that the photoconductor is illuminated with light that is absorbed uniformly throughout. Furthermore, q is the electronic charge, p the mobility, Ep(x , t ) the field in the photoconductor, K, the relative dielectric constant of the photoconductor and is the permittivity of free space.

Since there is no space charge in the ferroelectric,

EQUIVALENT CIRCUIT ANALYSIS OF A FERROELECTRIC-PHOTOCONDUCTOR MEMORY DEVICE 179

the total current is a displacement current in the reaches the value V: can be written ferroelectric and can be written

where P is the ferroelectric polarization and d , is the thickness of the ferroelectric layer.

The preceding equations are solved subject to the conditions that the voltage is constant across the structure a t all times. At t = 0 there is capacitive division.

and

0 < V,(O) < v; ( 5 )

The capacitances per unit area C, and C;, are given by

and RP is the resistance per unit area of the photo- conductor. In the limit of a square hysteresis loop, for which all the charge is switched at the constant voltage V,, Eq. (6) reduces to the well known limiting expression' for switching at constant current.

which is required to prevent the ferroelectric from 2PsR, switching prematurely. Here V,(O) is the voltage (7) across the ferroelectric at time t = 0, V: V,( t = t') and V : = VF(t = L''). The constant current expression is valid since the

The switching time, t", which is defined to be the voltage across the photoconductor is constant, i.e. time for which the voltage across the ferroelectric V, - V , .

t" = ~

V0 - v,

Ferroelectric Voltage vs Time

t 1

P >" --.

I I I I I I I I I 1 l *

0 1 2 3 4 5 6 7 8 9 10

Ferroelectric Voltage vs Time A

.o -

.8 -

I I I I I I I I 1 l *

0 1 2 3 4 5 6 7 8 9 10

FIGURE 3 Normalized voltage across the ferroelectric vs. normalized time during switching for the parallelogram of a ferroelectric.

180 J . T. JACOBS. B. D. SILVERMAN AND I . P. BATRA

Comparison of Eqs. (6 ) and (7 ) shows that one effect of a non-square loop material is to increase the switching time over the Anderson limit.

In Figure 3, the ferroelectric voltage has been plotted as a function of time. It is to be noted that during ferroelectric switching the time rate change of voltage transfer is slowed down due to the increasein thecapacitanceassociated withthe ferro- electric and subsequent increase in RC switching time constant. The voltage still increases monot- onically with time. In the next section it will be shown that a more realistic description of the ferroelectric during switching, leads to nonmono- tonic behavior of the voltage versus time. Such behavior has been observed, and will be the subject of discussion in the next section.

111. DOMAIN REVERSAL MODEL

In this section the ferroelectric will be treated in a more realistic manner. The model treated in the previous section assumed that the switched charge was simply a function of the voltage across the ferroelectric. Constant voltage across the ferro- electric would therefore result in no change in polarization charge on the ferroelectric. It is known, however, that due to the absence of a 'true coercive field' for ferroelectric materials, any field is sufficient to switch the ferroelectric. Therefore, even though switching times increase rapidly as the voltage decreases, the switched charge will continue to increase under constant or decreasing field con- ditions. It will be seen that with the domain reversal model we are able to explain the experimentally observed 'voltage turn around'.

FIGURE 4 Ferroelectric domains (cylinders) switching by sidewise domain wall motion.

It will be assumed that the ferroelectric switches due to sidewise domain movement6 (Figure 4). Previous calculations7*' have considered the impli- cations of this for ferroelectric switching. The present calculation involves representing the ferro- electric in this manner when it is incorporated into an FE/PC structure.

It is assumed that a fixed number of domains are switched during the reversal of the ferroelectric polarization. The extended area of thcse domains is given by

t

A,&) = .nnlr(t,) + .c L', exp ( - w ( 4 ) d 4 2 (8) 10

where n is the number of domains, r(t,) is the initial radius of each domain, 6 is the activation field for domain wall motion, urn is the maximum lateral domain wall velocity and E is the instantaneous field applied to the ferroelectric. Hence domain wall motion is described by a Miller-type relation assumed valid for instantaneous field.

The polarization and current can be written'

P(t) = 2P, [(I - ] (9)

J ( t ) = 4nP,r(t)v(t)n(l -. nr(t)2)n- (10)

subject to a constant voltage V, across the FE/PC structure

V, = U t ) + V,(t), (1 1)

V,@) = J(t)R (12)

and

where V,(t) and VF(t) are the voltage drops across the photoconductors and ferroelectric respectively.

To find the current transient, Eqs. (8) through (12) must be solved simultaneously. This involves the solution of a set of integrodifferential equations with non-constant coeffients. A computer program was used to solve this set of equations numerically.

The values of no, and 6 were determined by fitting the bismuth titanate data of Taylor. The choice of 6 = 47 kV/cm and nu, = 3 x l oL2 (cm-sec)- gave excellent agreement between computed and experimental switching times' over four decades in time. These parameters were used to compute typical switching transients.

In Figure 5, typical computed voltage, current and polarization transients are shown. One sig- nificant feature of these results is the drop in the ferroelectric voltage as the ferroelectric begins to

EQUIVALENT CIRCUIT ANALYSIS OF A FERROELECTRIC-PHOTOCONDUCTOR MEMORY DEVICE l81

0.0 I I I l I l *

0 .47

- 1 0.0 I I I l +

-PS / COMPUTED TIME FE/PC ( p sec1 TRANSIENTS

FIGURE 5 Computed transients of the FEjPC sandwich using the (Miller) model for ferroelectric switching.

switch. This is easily understood if one considers that during switching a larger effective capacitance shunts the ferroelectric. The original charge on the ferroelectric is now redistributed over both capaci- tors which results in a reduced voltage across the ferroelectric. One expects such behavior to be prominant when the R$, time is shorter than the time required for the switching current to reach its maximum value. Such nonmonotonic behavior of voltage with time was not exhibited by the model treated in Section I1 since for that the effective ferroelectric capacitance was chosen to be a function of the voltage.

Figure 6 shows experimental evidence for the voltage turn around. The turn around does not appear if the circuit parameters are chosen such that the voltage rise time is longer than the intrinsic ferroelectric switching time. For this case the switching voltage shows only a slow rise rather than a turn around.

In Figure 7 a comparison of switching times is made for threedifferent photoconductor resistances. The reciprocal of the switching times were plotted versus the steady state field across the ferroelectric.

L VF= 40 Volts

SWITCHING TRANSIENTS Bi,Ti,O, with RP= 5300 R

FIGURE 6 Experimental voltage and current for the ferro- electric with resistance per unit area of 5300 Ohms-cm'.

c I 1 l I

1 o3

1/E (cm/kV) X 10

SWITCHING SPEEDS OF Bi4Ti30,, WITH SERIES RESISTANCE

FIGURE 7 Reciprocal of the switching time to the 90% level vs. reciprocal applied steady-state field for Bismuth Titanate in series with a photoconductor of varying resistivity.

182 J. T. JACOBS. B. D. SILVERMAN AND I . P. BATRA

The same computer program used to obtain the results shown in Figure 5 was used to calculate the time for switching 90 7; of 2 P . The curves illustrate how.the speed of the device is limited by the series resistance. As an illustration, we estimate the switching speed of Bi,Ti,OI2 in series with a 0.6 pm thick film of CdSe" to be approximately 1.5 psec.

The voltage drop across the photoconductor decreases rapidly as the applied field is decreased. Consequently the switching speed is not severely limited by the series resistance at the lower switch- ing voltages. This behavior is illustrated in Figure 7 in the manner that a curve for a given resistance approaches the intrinsic switching curve at lower fields.

It is interesting to note that the computations give longer switching times than for a square hysteresis loop. The non-square hysteresis loop is responsible for this reduction because this causes the current not to be at a maximum throughout the entire switching cycle. The computer program has been generalized to use any power law type ofphoto- conductor voltage-current relationship. The results of these computations indicate that the major restriction on the switching time is the time required to get the charge through the photoconductor. Methods of reducing this transfer time result in decreased switching time until the intrinsic ferro- electric switching time is reached. By going to non- saturated switching the amount of charge necessary for switching is reduced. One observes in Figure 7

an additional decrease in switching time is achieved because the initial risetime, i.e., the time required for the domains to reach a significant size, is eliminated. When non-saturated switching is em- ployed the domains are always large and switching is rapid. The price paid for this increased speed is a decrease in the signal to noise ratio.

ACKNOWLEDGMENT

The authors wish to thank D. Horne for his help in building special electronic equipment for measuring the switching transients.

REFERENCES

I . R. M. Schaffert. U.S. Patent 3. 148. 354issued Sept. 8. 1964.

2. D. W. Chapman, Proc. IEEE Computer Group Conf..

3. G. W. Taylor and A. Miller, Proc. IEEE 58, 1220 i 1970). 4. A. A. Snaper. U.S. Patent 3. 348. 217 issued Oct. 17, 1967. 5. J. R. Anderson. IRE Trans. on Electronic Computers EC-5.

6. R. C. Miller and A. Savage, Phys. Rev. 115, 1176 (1959). 7. J. T. Jacobs and B. D. Silverman, Ferroelectrics 1, 265

8. N. W. Dalton. J. T. Jacobs. and B. D. Silverman, Ferro-

9. G. W. Taylor, Ferroelectrics. 1, 79 (1970).

assigned to IBM Corporation.

p. 56. June 1970. Washington. D.C.

184 (1956).

( 1970).

electrics 2, 21 (1971).

IO. B. S. Sharma and R. R. Mehta, Ferroelectrics. 3, this issue. (1971).