Kinetic approach to fatigue phenomenon in ferroelectrics

JOURNAL OF APPLIED PHYSICS

VOLUME 90, NUMBER 12

15 DECEMBER 2001

Kinetic approach to fatigue phenomenon in ferroelectrics Vladimir Ya. Shur,a) Evgenii L. Rumyantsev, Ekaterina V. Nikolaeva, Eugene I. Shishkin, and Ivan S. Baturin Institute of Physics and Applied Mathematics, Ural State University, Ekaterinburg 620083, Russia

共Received 14 June 2001; accepted for publication 11 September 2001兲 We propose an approach to the explanation of the fatigue effect as an evolution of the switching area during cyclic switching as a result of self-organized domain kinetics due to retardation of bulk screening of the depolarization field. The formation of spatially nonuniform internal bias field during cycling 共kinetic imprint effect兲 slows the domain kinetics in some regions leading to formation of the kinetically frozen domains. Presented fatigue and rejuvenation experimental data measured in sol-gel PbZrx Ti1⫺x O3 thin films are in accordance with the results of computer simulation. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1418008兴 I. INTRODUCTION

II. MODEL

It has been shown in classical works1,2 that ferroelectric materials demonstrate essential decreasing of the switching charge during ac voltage cycling known as fatigue phenomenon. In recent years this effect has been intensively studied in thin films 共mostly in PbZrx Ti1⫺x O3兲, which are widely discussed for application in ferroelectric memory devices.3–18 The knowledge of the fatigue origin is of principal importance for improving the film endurance, which is essential for ferroelectric nonvolatile memories 共FeRAM兲. Several alternative scenarios for the suppression of the switching polarization have been proposed. Bulk-locking mechanism is based on the assumption that the charged domain walls are pinning with the help of electric charge trapping centers.12 The interface scenario supposes the inhibition of domain growth due to inhibition of nucleation at the electrode interfaces.7–9 The crucial role of electromigration of oxygen vacancies is discussed.16,17 The growth of the passive surface layer with damaged ferroelectric properties during cycling is proposed recently14 and must be stressed that the fatigue process is always considered without relation with domain structure evolution during cycling, though it was shown by direct observations using scanning probe microscope that fatigue in thin films manifests itself through arising and growth of complicated structure of nonswitched regions 共‘‘frozen’’ domains兲.11,19 In this work we propose an approach, which allows us to describe an evolution of switching area during cycling. We point out the crucial role of retardation effect during bulk screening of the depolarization field. During cycling this effect leads to self-organized formation of spatially nonuniform internal bias field 共kinetic imprint兲, which suppresses arising/nucleating of new domains and slows the polarization reversal. As a result the switching becomes incomplete due to inhibition of nucleation in the regions with maximum value of internal bias field. The main model predictions such as the existence of rejuvenation stage and the evolution of the geometry of switching area are in accord with the experiment.

It is known that in the ferroelectric capacitor after polarization reversal the external screening through charge redistribution at the electrodes 共switching current兲 rapidly compensates the depolarization field E dep . Nevertheless the residual depolarization field E rd ⫽E dep⫺E ex.scr remains due to the existing intrinsic surface dielectric gap.20,21 Slow compensation of E rd by bulk screening20–24 leads to the arising of the internal bias field E b observed as a loop shift 共imprint effect25,26兲 for ac cyclic switching with period T much shorter than the bulk screening time constant ␶.21,22 It was shown that E b can be spatially nonuniform and can change during long-enough cycling.21–23 Three groups of bulk screening mechanisms are revealed and widely discussed: 共1兲 reorientation of the defect dipoles,27,28 共2兲 redistribution of bulk charges/carriers,20–23 and 共3兲 injection from the electrodes.7–9 In our model we account for the fact that the domain kinetics during cycling is a self-organized process, because the spatial distribution of E b depends on the previous domain evolution and in turn defines subsequent domain kinetics. The local value of E b tends to the field value defined by the relative difference of stay duration of the given sample region in the states with opposite polarization directions 关Fig. 1共b兲兴. Thus both sign and value of E b become spatially nonuniform and E b distribution function varies with the number of cycles. Our simulation of domain kinetics shows that this kinetic imprint effect leads to arising and growth of kinetically frozen regions. The domain kinetics during periodical switching in rectangular pulses in uniaxial thin ferroelectric was simulated on 2D matrix L⫻L. We use the classical model of polarization reversal,29,30 where the domain kinetics is achieved through domain nucleation and domain wall motion by generation and growth of steps. Thus we have distinguished three nucleation processes: 共1兲 step growth by nucleation at the step edge, 共2兲 step generation by nucleation at the wall, 共3兲 new domain formation by appearance of isolated nucleus 关Fig. 1共a兲兴. The nucleation probability at a given point 共i,j兲 during considered switching cycle N is defined by the local field E loc(i, j,N):

a兲

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FIG. 1. 共a兲 Nucleation types considered during simulation: 共1兲 nuclei at the step end, 共2兲 nuclei at the wall, 共3兲 isolated nuclei. 共b兲 A schematic of time dependence of spontaneous polarization at the given point during cycling.

p k 共 i, j,N 兲 ⬃exp兵 ⫺E ack / 关 E loc共 i, j,N 兲 ⫺E thk 兴其 ,

共1兲

where E ack and E th k are the activation and threshold fields for given nucleation process k. It is known that the probability of nucleation at the wall exceeds essentially the bulk one as E ac 1 ⬍E ac 2 ⬍E ac 3 . 21 Thus the increase of the domain boundary length facilitates the switching. The local field is a sum of uniform external field E ex , residual depolarization field E rd , and local value of internal bias field E b (i, j,N) formed by the end of previous cycle: E loc共 i, j,N 兲 ⫽E ex⫹E rd ⫹E b 共 i, j,N 兲 .

共2兲

For slow bulk screening, when ␶ ⰇT, which is typical for fatigue experiments, E b (i, j,N) relaxes compensating the sum of external and residual depolarization fields averaged over the switching cycle 具 E ex ⫹E rd 典 . For symmetric rectangular pulses 具 E ex 典 ⫽0. The local value of 具 E rd 典 ⫽E rd (T ⫹ ⫺T ⫺ )/T, where T ⫹ and T ⫺ -time of given point residence in the states with opposite sign of spontaneous polarization during the switching cycle 关Fig. 1共b兲兴. It is clear that the values of T ⫹ and T ⫺ being local nevertheless can be obtained only by simulation of the domain kinetics in the whole matrix. We understand that in real ferroelectrics the bulk screening is a complicated process due to competition of several above-

FIG. 2. Simulated sequence of instantaneous domain patterns during one switching: 共a兲 before and 共b兲 after fatigue cycling. Black and white: switching domains of opposite sign, light and dark gray: frozen domains of opposite sign. Initial state with zero internal bias field.

FIG. 3. Simulated internal bias field distribution functions for different N fitted by Gaussian. Initial state with zero internal bias field.

mentioned mechanisms. Within our approach it is difficult to favor any of them, because in our model ␶ is the only parameter characterizing the screening kinetics. After each simulation of domain kinetics during the cycle we recalculate the spatial distribution of the internal bias field: E b 共 i, j,N 兲 ⫽E b 共 i, j,N⫺1 兲共 1⫺T/ ␶ 兲 ⫺ 具 E rd共 i, j,N⫺1 兲典 T/ ␶ .

共3兲

We have considered two variants of the initial spatial distribution of the internal bias field E b (i, j,1): 共1兲 uniform zero internal bias and 共2兲 completely screened multidomain initial state. The first corresponds to idealized initial state without bulk screening. In the second one E b takes the equal positive value at domains of one sign and the negative one elsewhere. III. RESULTS AND DISCUSSION A. Initial state with zero internal bias field

The set of patterns corresponding to one switching cycle at the different fatigue stages simulated for the initial state with zero internal bias 关 E b (i, j,1)⫽0 兴 are presented in Fig. 2. It is seen that initially the switching occurs in the whole area 关Fig. 2共a兲兴. After fatigue cycling the switching occurs mainly in narrow regions surrounding the frozen domains

FIG. 4. Simulated switching current in triangular pulse for N⫽50:(䊉) Preisach theory, 共䊏兲 modified approach 共for ⌬E/w⫽3.2兲, 共Ref. 28兲 in inset: dependence of j max on ⌬E/w fitted by Eq. 共5兲.


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FIG. 5. Simulated suppression of 共a兲 switching charge and 共b兲 current maximum during cycling. Initial state with zero internal bias field.

关Fig. 2共b兲兴. Such evolution of the switching area geometry demonstrates the qualitative change of domain kinetics. In the initial state the process is governed by 2D growth of isolated domains 关Fig. 2共a兲兴. Increasing of the input of 1D wall motion follows arising and growth of the frozen regions. For strong fatigue, the domain kinetics is defined mostly by reversible parallel translation of the walls 关Fig. 2共b兲兴. It is important to point out that such geometry change has been observed directly in fatigued PZT films.11 We characterize the nonuniform E b at a given cycle by internal bias field distribution function f 共 E b ,N 兲 ⫽L ⫺2 ⌺ ␦ 关 E b ⫺E b 共 i, j,N 兲兴 .

共4兲

The simulated f (E b ,N) is well fitted by the Gaussian 共Fig. 3兲 and fatigue induced evolution shows the smearing of initial narrow distribution with dispersion w increasing during cycling. The formation of two peaks at E b ⫽⫾E rd corresponds to the arising of frozen domains 共Fig. 3兲. According to the Preisach theory this field distribution function defines the voltage dependence of switching charge and current31,32 for testing in triangular pulses. The broadening of f (E b ,N) during cycling leads to decrease of the switching charge due to termination of polarization reversal in the regions where E loc is lower than threshold field E th . Our modification of this approach33 accounting for the difference between the threshold fields for nucleation and growth allows us to simulate the dependence of the switching current maximum j max on w, which is well-fitted by the following formula 共Fig. 4兲: j max共 1/w 兲 ⫽ j max共 0 兲 ⫹J 关 exp共 a/w 兲 ⫺1 兴 ,

共5兲

where J and a are constants. It is important that the fatigue diminishing of current maximum is more pronounced than that for switching charge 共Fig. 5兲.

FIG. 6. Rejuvenation and fatigue of switching charge. 共a兲 Simulation for completely screened multidomain initial state and 共b兲 experimental data for cycling in PZT films.

FIG. 7. Rejuvenation and fatigue of current shape. 共a兲 Simulation for completely screened multidomain initial state and 共b兲 experimental data for cycling in PZT films.

B. Completely screened multidomain initial state

This demonstrates the existence of additional rejuvenation stage 共increasing of the switching charge兲 preceding the fatigue process 关Fig. 6共a兲兴. The rejuvenation after the fatigue stage has been experimentally discovered in bulk ferroelectrics and thin films.3,34 We found that rejuvenation and fatigue stages differ by geometry of switching area 共Fig. 7兲. The formation of complicated maze structure during rejuvenation is accompanied by the increase of switching area width. During fatigue stage the self-consistent flattering and simplification of maze pattern is observed. Simulation shows that contrary to fatigue from the initial state with zero internal bias, the unipolar growth of the frozen domains of only one sign is usually obtained. The sign of unipolarity is defined by geometry of the initial domain structure. Such tendency is in accord with published experimental results.15 Evolution of internal bias field distribution function differs qualitatively from the one for zero internal bias initial state. Two peaks corresponding to initial domain areas with opposite sign of polarization smear and merge during rejuvenation thus forming one wide peak 关Fig. 8共a兲兴. Its subsequent behavior follows the above discussed evolution of f (E b ,N) 关Fig. 8共b兲兴. At the rejuvenation stage the switching current is a sum of two inputs corresponding to the switching in the regions with different sign of internal bias field 关Fig. 9共a兲兴. C. Experiment

The fatigue/rejuvenation cycling was carried out in Pb共Zr0.2Ti0.8兲O3 films 共100–200 nm thick兲 grown by chemical solution deposition on Pt/Ti/SiO2 /Si substrates.13 Pt top electrodes have been used for applying a bipolar rectangular

FIG. 8. Simulated evolution of the switching domain area during cycling for completely screened multidomain initial state. Black: switching domains, light and dark gray: frozen domains of opposite sign.


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of endurance on the geometry of initial domain structure. The discussion of the dependence of fatigue behavior on the cycling frequency and geometry of the initial domain structure will be presented in a subsequent article. ACKNOWLEDGMENTS

FIG. 9. Simulated evolution of the internal bias field distribution function during cycling for completely screened multidomain initial state: 共a兲 rejuvenation and 共b兲 fatigue stages.

pulse trains 共100% duty cycle, the amplitude U cyc⫽5 – 8 V, the cycling frequency f cyc⫽10 Hz– 100 kHz兲. The hysteresis loops and switching currents were measured in triangular waveform 共f m ⫽10– 100 Hz, U m ⫽5 – 7 V兲 after a certain number of switching cycles. Cycling and measurements have been done at room temperature. The fatigue-induced evolution of the switching charge 关Fig. 6共b兲兴 and current 关Fig. 9共b兲兴 pulses for PZT films confirms the existence of the predicted rejuvenation stage, which is most pronounced for j max 共Fig. 10兲. The evolution of the shape of switching current pulse during cycling is in accord with simulated behavior 关Figs. 6共a兲 and 9共a兲兴. IV. CONCLUSIONS

The proposed model of self-organized evolution of local internal bias field 共kinetic imprint effect兲 allows us to describe the change of the geometry of switching area during cycling and predicts the existence of rejuvenation stage. The simulation shows that the evolution of the internal bias field distribution function and switching current are correlated. Thus the switching current data provide the essential information on fatigue kinetics. The comparison with experimental data for PZT thin films confirms the validity of our model. Within our model the nonswitched regions, formed during self-organized domain evolution, qualitatively differs from the seeds discussed by Tagantsev et al.7–9 The regions are kinetically frozen because the ac field amplitude and period are insufficient for complete switching. As a result the considered fatigue effect is reversible and partial rejuvenation occurs after increasing the field amplitude in accordance with experiment34 and in contrast with Ref. 14. The recently observed fatigue anisotropy35 can be attributed to dependence

FIG. 10. Increase of current maximum during rejuvenation stage: 共a兲 simulation for completely screened multidomain initial state and 共b兲 experimental data for cycling in PZT films.

The authors are grateful to T. Schneller and R. Gerhardt, who fabricated the investigated PZT films and to O. Lohse for technical assistance. The research was made possible in part by a Grant of the Ministry of Education of the Russian Federation, by the Program ‘‘Basic Research in Russian Universities’’ and by Award No. REC-005 of the CRDF. D. S. Campbell, Philos. Mag. 7, 1157 共1962兲. J. C. Burfoot and G. W. Taylor, Polar Dielectrics and Their Applications 共Macmillan, New York, 1979兲. 3 H. M. Duiker, P. D. Beale, J. F. Scott, C.-A. Paz de Araujo, B. M. Melnick, J. D. Cuchiaro, and L. D. McMillan, J. Appl. Phys. 68, 5783 共1990兲. 4 C. J. Brennan, Integr. Ferroelectr. 2, 73 共1992兲. 5 J. Lee, S. Esayan, A. Safari, and R. Ramesh, Appl. Phys. Lett. 65, 254 共1994兲. 6 G. Arlt and U. Robels, Integr. Ferroelectr. 3, 343 共1993兲. 7 E. Colla, D. Taylor, A. Tagantsev, and N. Setter, Appl. Phys. Lett. 72, 2478 共1998兲. 8 I. Stolichnov, A. Tagantsev, E. Colla, and N. Setter, Appl. Phys. Lett. 73, 1361 共1998兲. 9 A. K. Tagantsev, I. Stolichnov, E. L. Colla, and N. Setter, J. Appl. Phys. 90, 1387 共2001兲. 10 V. Ya. Shur, S. Makarov, N. Yu. Ponomarev, I. L. Sorkin, E. V. Nikolaeva, E. I. Shishkin, L. Suslov, N. Salashchenko, and E. Kluenkov, J. Korean Phys. Soc. 32, S1714 共1998兲. 11 E. Colla, S. Hong, D. Taylor, A. Tagantsev, N. Setter, and K. No, Appl. Phys. Lett. 72, 2763 共1998兲. 12 W. Warren, D. Dimos, B. Tutle, R. Nashby, and G. Pike, Appl. Phys. Lett. 65, 1018 共1995兲. 13 M. Grossmann, D. Bolten, O. Lohse, U. Boettger, R. Waser, and S. Tiedke, Appl. Phys. Lett. 77, 1894 共2000兲. 14 M. Bratkovsky and A. P. Levanyuk, Phys. Rev. Lett. 84, 3177 共2000兲. 15 A. Kholkin, E. Colla, A. Tagantsev, D. Taylor, and N. Setter, Appl. Phys. Lett. 68, 2577 共1996兲. 16 J. F. Scott and M. Dawber, Appl. Phys. Lett. 76, 1060 共2000兲. 17 J. F. Scott and M. Dawber, Appl. Phys. Lett. 76, 3655 共2000兲. 18 R. Ramesh, W. K. Chan, B. Wilkens, T. Sands, J. M. Tarascon, V. G. Keramidas, and J. T. Evans, Integr. Ferroelectr. 1, 1 共1992兲. 19 A. Gruverman, O. Auciello, and H. Tokumoto, Appl. Phys. Lett. 69, 3191 共1996兲. 20 V. M. Fridkin, Ferroelectrics Semiconductors 共Consulting Bureau, New York, 1980兲. 21 V. Ya. Shur, in Ferroelectric Thin Films: Synthesis and Basic Properties 共Gordon and Breach, New York, 1996兲, Vol. 10, Chap. 6. 22 V. Ya. Shur, Yu. A. Popov, and N. V. Korovina, Sov. Phys. Solid State 26, 471 共1984兲. 23 V. Ya. Shur, Phase Transitions 65, 49 共1998兲. 24 V. Ya. Shur and E. L. Rumyantsev, Ferroelectrics 191, 319 共1997兲. 25 W. L. Warren, D. Dimos, G. E. Pike, B. A. Tuttle, and M. V. Raymond, Appl. Phys. Lett. 67, 866 共1995兲. 26 M. Grossmann et al., Appl. Phys. Lett. 76, 363 共2000兲. 27 G. Arlt and U. Robels, Integr. Ferroelectr. 3, 343 共1993兲. 28 P. V. Lambeck and G. H. Jonker, Ferroelectrics 22, 729 共1978兲. 29 R. C. Miller and G. Weinreich, Phys. Rev. 117, 1460 共1960兲. 30 M. Hayashi, J. Phys. Soc. Jpn. 33, 616 共1972兲. 31 G. Robert, D. Damjanovic, and N. Setter, Appl. Phys. Lett. 77, 4413 共2000兲. 32 A. Bartic, D. Wouters, H. Maes, J. Rickes, and R. Waser, J. Appl. Phys. 89, 3420 共2001兲. 33 V. Ya. Shur, E. L. Rumyantsev, E. V. Nikolaeva, E. I. Shishkin, I. S. Baturin, and M. V. Kalinina 共unpublished兲. 34 J. F. Scott and B. Pouligny, J. Appl. Phys. 64, 1547 共1988兲. 35 V. Bornand, S. Trolier-McKinstry, K. Takemura, and C. A. Randall, J. Appl. Phys. 87, 3965 共2000兲. 1 2
