Simulation of Switching Properties of Ferroelectrics on the Basis of Dipole Lattice Model

ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII "AL.I.CUZA" DIN IAŞI Tomul XLI-XLII, s.I.b.fasc.2 Fizica Solidelor - Fizică Teoretică, 1995-1996

SIMULATION OF SWITCHING PROPERTIES OF FERROELECTRICS AND OF THE DIPOLE KINETICS

USING A DISCREET MODEL

BY

Dan Ricinschi♣ , Vasile Ţura♣, Liliana Mitoşeriu♣ and Cătălin Harnagea♣,

ABSTRACT: The field and temperature dependence of the switching parameters in ferroelectric media are obtained using a discreet lattice model developped by Ishibashi. Partial switching process has been simulated using both pulse frequency modulation (PFM) and pulse with modulation (PWM). A new interpretation of the differences between the switching and non-switching responses of the ferroelectric has been obtained on the basis of dipole kinetics described by the proposed model.

1. INTRODUCTION

The switching properties of ferroelectric media have been intensely investigated for quite a long time. The studies performed by a number of authors on the Barium Titanate (BT) single crystals have shown that the main mechanism of the polarization reversal is the nucleation of antiparallel domains, followed by their growth under the action of the applied electric field [1-4].

They showed that the domain dynamics essentially depend on the electric field strength. In weak fields the polarization reversal is achieved mainly by sideways growth of the few reversed domains that nucleated in this case, whereas is case of stronger fields the nucleation of many new domains takes over.

Following Kolmogorov [5] and Avrami's [6] work, Ishibashi and Takagi [7] were the first who consistently applied the concepts of first order type phase transitions for describing the polarization reversal in ferroelectrics. The advantage of considering the polarization switching in ferroelectrics as a gradual transition between two phases with opposite polarizations is that one

♣ "Alexandru Ioan Cuza" University, Faculty of Physics, IASI, 6600, ROMANIA Received 15 January 1995, in revised form 16 November 1996

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Dan RICINSCHI & al. 112

can preserve Avrami' s original nucleation-growth theory, supported by many experiments. Another approach than the initial nucleation-growth mechanism was proposed by Ishibashi [8], who described the ferroelectric system as a lattice of electric dipoles. Omura et. al. [9] used this model to perform some simulations of polarization reversal in ferroelectrics, computing switching parameters and the hysteresis loops. In the present paper, we used the Ishibashi' s lattice model in order to explain the field amplitude and thermal dependences of the switching currents and polarization, to simulate pulse switching experiments and the partial switching regime. A representation of the dipole kinetics during switching process is given and some new particularities of the switching behavior are discussed. 2. SIMULATION AND DISCUSSIONS

In the model proposed by Ishibashi [8], the ferroelectric is considered as a one dimensional lattice of interacting dipoles, described by the Landau-Ginzburg free energy. The time dependence of each dipole moment can be determined from a system of non-linear coupled differential equations, using the latent nucleation hypothesis. We used succesfully the lattice model to explain the grain size effects on the switching properties of BT ceramics [10].

First, the initial state corresponding to the negative remanent polarization is established. Applying an opposite electric field, the polarization reversal take place as a result of the gradual time variation of the dipole moments at each site, according to the following equation of motion [8,10]:

( ){ }γ α βdp

dtp p k p p p Enn n n n n= − + − − + −+ −

21 12 (1)

where pn is the dipole moment at position n, k the interaction coefficient between neighboring atoms, α and β are the coefficients of the Landau-Ginzburg [11] free energy, E is the applied electric field and γ is the viscosity coefficient. The temperature-dependent coefficient is α=c(T � T0), where c is the inverse of the Curie constant and T0 is the transition temperature. The polarization of the ferroelectric system and the switching transient are, respectively:

SIMULATION OF SWITCHING PROPERTIES OF... 113

PN

pn= ∑1 , j=dP

dt (2)

where N is the number of dipoles contained in the system. Solving the differential coupled equations (1) and computing the total polarization and switching current (2), one can simulate many switching experiments under step-field, pulsed signal, sinusoidal and triangular applied field and can obtain useful information about the field and temperature influence on the switching curves. 2.1 THE EFFECT OF FIELD AMPLITUDE AND TEMPERATURE ON THE SWITCHING PARAMETERS Many experiments aimed at measuring the switching currents and polarizations were performed using a circuit that contains a ferroelectric capacitor connected with a series linear resistor. One measures the transient induced by rectangular pulses, transient that flows through the resistor during the polarization reversal. The most popular electric field sequence used in order to obtain the switching characteristics of ferroelectric capacitors consists of double bipolar pulses, having variable amplitude and duration [12]. A pulse in the sequence following a pulse of opposite polarity induces the total switching response of the ferroelectric sample, which corresponds to the re-orientation of electric dipoles between the two states with opposite remanent polarizations. A pulse in the sequence coming after a pulse with identical polarity will lead to the non-switching response, associated with the reversible component of the switching behavior.


Fig. 1 The time variation of the Fig. 2 The time variation of the polarization switched under a switching current obtained under sequence of double alternating a sequence of double alternating rectangular pulses. rectangular pulses.

In order to obtain the pure switching response, we simulated the application of a double pulse sequence described above. Figs.1 and 2 contain the computed switching polarization and current that appear during application of two identical rectangular pulses following two other pulses, with opposite polarity ( the parameters have the values: E/Esat=0.56, N=200 dipoles, α=-1, β=0.1, λ=1, k=1, γ=1, t0=1, Esat.=18). One observes that the computed responses that correspond to the switching and non-switching pulses are indeed different, as showed by many experiments [12, 16,17]. In order to obtain pure switching responses, the non-switching part was subtracted from the total response If a similar pulse switching simulation using pulses of higher amplitudes is performed, one obtains the dependences of pure switching polarizations and currents depicted in Figs.3-4 (obtained for different amplitudes of the applied field: E/Esat:: (a) 1, (b) 0.89, (c) 0.78, (d) 0.67, (e) 0.56, where Esat.=18). Higher fields lead to higher currents and polarizations and the switching process take place faster.

Fig. 3 The time dependences of the Fig. 4 The time dependences of switched polarizations computed for the switching currents computed different fields. for various fields.

This model has the possibility to account for the temperature effect on the switching curves, by means of the thermal variation of Landau-Ginzburg coefficient α. Time dependences of the switching polarizations and currents,


computed for the same value of the applied field and different temperatures are presented in Figs 5-6 (for various ratios T/T0: (a) 0.75, (b) 0.8, (c) 0.85, (d) 0.9, where T0=4, E/Esat=0.67, Esat=18). One observes that higher temperatures decrease the amount of the polarization that is switched by a given field and decrease the peak of the switching transient also. The increase of temperature accelerates the polarization reversal, as can be seen in Fig. 6 by looking at the time when the switching current peaks. The effect of gradually suppressing the ferroelectric properties when temperature increases is also detectable by plotting the hysteresis loops computed in sinusoidal field, at different temperatures.

Fig. 5 Reversed polarizations induced Fig. 6 Switching currents induced by an identical pulsed sequence, at by an identical pulsed sequence, at different temperatures. various temperatures.

A valuable characteristic of this model is that both the switching current and polarization exhibit hysteretic behavior when the applied field is cycled during one period. One observes in Figs 7-8 that increasing temperature tends to decrease the saturation polarization and narrow the hysteresis loops, as a result of operating the ferroelectric sample in the vicinity of the Curie point. The effects of electric field amplitude and temperature on the switching curves predicted by the present discreet model are concordant with many switching experimental results [14]. 2.2 PARTIAL SWITCHING SIMULATIONS If the input pulse is shorter than the switching time of the ferroelectric sample, it can reverse only a small part of the polarization. However, many


short identical pulses can gradually reverse the polarization, but one expects the switching parameters in this case to be different from those obtained when total switching is induced by a single pulse. According to Tokumitsu et. al. [13], who experimentally studied the partial switching process in PZT thin films, there are two ways in which the polarization can be partially reversed. The first is the frequency modulation (PFM), where switching is induced by a sequence of short unipolar pulses, whose duration can be varied. The second alternative (called pulse width modulation or PWM) consists of applying a single pulse to the ferroelectric system, with pulse duration variable between zero and the time necessary to completely reverse the polarization of the sample. In order to simulate the partial switching regime, we �applied� to the ferroelectric lattice two sequences of pulses in accordance with the two alternatives described above.

Fig. 7 Hysteresis loops of the polarization Fig. 8 Hysteresis loops of the current computed for input sinusoidal signal, at computed for input sinusoidal signal, two temperatures T/T0: (a) 0.75, (b) 0.98, at the same temperatures and field. where Emax =10.

The results for the PFM simulations are presented in Figs.9-10. If many short pulses are applied, the polarization is gradually reversed, but the total switched polarization is smaller than the polarization reversed by a long single pulse (a). Moreover, the switching polarization and current are dependent on how long the input voltage keeps its maximum value +E (duration further referred as t1) and how long the input voltage applied to the sample is equal to zero (duration further referred as t2). The results depicted in Fig.9 were


obtained by increasing the duration of the each pulse, for various t1 (but keeping all the time t1=t2): (a) t1/t0=85 (b) t1/t0=3, (c) t1/t0=2, (d) t1/t0=1, where t0=0.2. The dependences shown in Fig. 10 have been obtained by increasing only t1 and maintaining t2 constant :(a) t1/t0=85 (b) t1/t0=3, (c) t1/t0=2, (d) t1/t0=1, where t0=0.2. It appears that pulses having a longer period constrain the system to respond differently, depending on the relation between t1 and t2. If t1=t2, although there are some pulse width-dependent initial changes in the polarization one obtains the same amount of the switched polarization, irrespective to pulses duration. On the other hand, if one increases the duration t1 of the pulses while keeping t2 constant, the switching polarization saturates at different values. Fig. 10 also contains the time variation of the polarization totally reversed by a single-pulse (a). The polarization obtained by partial switching is always smaller than the polarization totally reversed.

Fig. 9 The time dependence of the Fig. 10 The time dependence of the polarization induced by a sequence polarization induced by a sequence of pulses (PFM). of unipolar rectangular asymmetric pulses (PFM).

Fig.11 shows a switching current during a PFM process, whose amplitude modulation-type shape supports the term of modulation when referring to the partial switching variants PFM and PWM. Processes of PWM type were simulated using the lattice model, by applying a single pulse that partially switches the polarization of the ferroelectric system. When one varies the


duration of the single pulse of a given amplitude, the simulations reveal the existence of two critical times (having their normalized values further noted as τ′ , and τ″ , respectively) used as reference for the PWM switching behavior. If the applied pulse has a duration of positive voltage less than τ'=t1/t0=3.5, after reversing a width-dependent fraction of the switchable polarization, the system will return to its initial state (Fig.12). For pulses having duration between τ′ and τ″=t1/t0=5, the initial state is no longer conserved after switching and the polarization changes to a value dependent on the duration of the pulse. If the pulse is longer than the duration τ�� , the final state of the system after the partial switching does not depend anymore on pulse width, but still differs from the final state obtained after the polarization is totally reversed by a very long applied positive voltage.

Fig. 11 The time variation of a switching Fig. 12 The time dependence of the current computed during PFM type of polarization computed during PWM partial switching (t1/t0=2), where t0=0.2. type simulations of partial switching regime, for various fractions t1/t0. 2.3 SIMULATION AND REPRESENTATION OF DIPOLE KINETICS DURING SWITCHING The lattice-model of Ishibashi allows to obtain information about the switching behavior of individual electric dipoles when the polarization of the system is reversed. In the present work a dipole representation was performed as a first attempt to correlate some particularities of the computed integral


switching curves with local characteristics of dipole kinetics. This correlation has been used in order to explain the differences between the switching and non-switching responses of the ferroelectric system and also to clarify the grain size effects detected on the computed polarizations and currents. Both switching and non-switching responses were obtained by using the coupled rotating dipole and latent nucleation hypothesis. In the framework of these hypotheses, the time variation of electric dipoles has been regarded as a field induced gradual rotation of dipoles at every site of the lattice, excepting the latent nuclei's sites. Figs.13-14 show the electric dipoles of a one-dimensional ferroelectric system, corresponding to different moments during the switching and non-switching responses, respectively. The electric dipoles respond differently to the applied pulsed field, depending on their initial state and pulse polarity. The orientation of dipoles at the sites where latent nuclei are found remains unchanged during switching.

Fig. 13 Dipole kinetics representation performed during the switching response of a ferroelectric sample (E/Esat=0.56, T/T0=0.75, N=200 dipoles, ∆t/t0=1/20).


Fig. 14 Dipole kinetics representation performed during the non-switching response

of a ferroelectric sample, for the same value of field amplitude and temperature.

When a switching response is recorded (Fig.13), the dipoles leave their initial state (corresponding to the negative remanent polarization) and gradually rotate until they reach an equilibrium reversed state, depending on the width of the switching pulse. One observes that in our simulations the switching pulse does not completely reverse the polarization, although the widths of the write and read used pulses are equal. When the switching pulse terminates, the ferroelectric system relaxes and a positive remanent polarization state is reached. A different response of the ferroelectric system is recorded when a non-switching pulse is applied. As seen in Fig.14, in this case the initial state is the positive remanent polarization state established after the switching pulse. The non-switching pulse determines the dipoles to rotate again towards an equilibrium state, which in this case coincides with the totally reversed state. When the applied pulse is removed, the system reaches the same positive remanent polarization state and remains there until the following pulse with opposite polarity will switch it again. The above considerations apply to all dipoles except the latent dipoles that are not reversed by any strength of the applied field. Apart from reflecting the experimental evidence regarding the presence of the latent nuclei, preventing some dipoles from changing their original state is the only way one can obtain a


certain distribution of dipoles' orientation over the lattice in Ishibashi's model. A ferroelectric lattice with no latent nuclei would reverse its polarization by means of a synchrony motion of all dipoles. Therefore, the results of simulations carried out on two-dimensional of tri-dimensional dipole lattices were in poor agreement with the domain structure dynamics in a real ferroelectric sample. The main facts that are not possible to account for in the framework of this model are the existence of domains separated by 180° and 90° walls and the nucleation-growth switching mechanism, which are strongly supported by experimental findings. Even so, such a model of mainly simultaneous switching of electric dipoles might be appropriate for explaining processes that take place extremely fast, like in case of ferroelectric thin films. 3. CONCLUSIONS The present paper contains an application of Ishibashi's lattice model for explaining some new aspects of switching in ferroelectric media, important for design study of ferroelectric non-volatile memories. Simulations of the switching behaviour induced by a conventional pulse sequence used in ferroelectric studies are performed and the pure switching curves have been computed. The increasing of the electric field determines an increase in switching current�s density and polarization and a decrease of the switching time. Increasing the temperature while using the same applied electric field decreases the amount of switched polarization, decreases the maximum switching current density and shortens the switching time. In case of hysteresis loops, increasing temperature decreases the saturation polarization and narrows the loop. In case of Pulse Frequency Modulation (PFM) type simulations, the switching currents and polarizations were generally dependent on the lengths of pulse duration and zero field interval. An interesting result was that the switched polarization is independent on the duration of these intervals if they are of equal length. Simulations of Pulse Width Modulation (PWM) type showed the existence of two critical times for the pulse duration, τ' and τ". After applying a pulse whose duration is less than τ', the system will return to its original polarization


state. After pulses with duration greater than τ", the system will reach the same polarization state irrespective to the pulse length. The model accounts for grain size effects, also. Simulations performed considering the same applied electric field and different grain sizes showed that decreasing the grain size decreases the switching polarization and the maximum switching current density and shortens the switching time. Representation of dipole kinetics during switching showed that in fine grained ferroelectrics the moment of zero-polarization comes faster than in coarse grained ones, in agreement with the behaviour of integral currents recorded experimentally. ACKNOWLEDGEMENTS The authors wish to thank professor Constantin Păpuşoi for helpful discussions during preparing this paper and his interest in our work. REFERENCES [1] Merz W., � Phys. Rev. , 95 (1954) 3, J. Appl. Phys. , 27 (1956) 8. [2] Little E., � Phys. Rev. 98 (1955) 978. [3] Yakunin S. I. et. al., � Sov. Phys.-Solid State, 14 (1972) 2. [4] Shur V. Ya., et. al., � Integrated Ferroelectrics, 2 (1992) 51. [5] Kolmogorov A. N., � Izv. Akad. Nauk. SSSR, Ser. Math., 3 (1937) 355. [6] Avrami M., � J. Chem. Phys., 7 (1939) 1103, 8 (1940) 212, 9 (1941) 177. [7] Ishibashi .Y., Takagi Y., � J. Phys. Soc. Jpn., 31 (1971) 506. [8] Ishibashi Y., � J. Phys. Soc. Jpn. , 59 (1990) 11, 4148. [9] Omura M. et. al., � Jpn. J. Appl. Phys. 30 (9B) (1991) 2384. [10] Mitoseriu L., Ricinschi D., Harnagea C., Okuyama M., Tsukamoto T. and Tura V., �

Jpn. J. Appl. Phys., 35, part.1, (9B) (1996) 131. [11] Kanzig W., � Ferroelectrics and Antiferroelectrics, Academic Press, (1957) [12] Larsen P. K., � Proc. of NATO ARW on "Science and Technology of Electroceramic

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Simulation of Switching Properties of Ferroelectrics on the Basis of Dipole Lattice Model

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