- Email: [email protected]
Scaling behavior of dynamic hysteresis in multi-domain spin systems
January 2002
Materials Letters 52 Ž2002. 213–219 www.elsevier.comrlocatermatlet
Scaling behavior of dynamic hysteresis in multi-domain spin systems J.-M. Liu a,b,c,) , H.L. Chan b, C.L. Choy b a
Laboratory of Laser Technologies, Huazhong UniÕersity of Science and technology, Wuhan 430074, China b Department of Applied Physics, The Hong Kong Polytechnic UniÕersity, Hong Kong, China c Laboratory of Solid State Microstructures, Nanjing UniÕersity, Nanjing 210093, China Received 29 January 2001; accepted 5 February 2001
Abstract The dynamic hysteresis in two-dimensional multi-domain spin systems is simulated using Monte-Carlo method based on Q-state planar Potts model. The pattern and scaling property of the hysteresis as a function of the oscillating time-varying external field are studied and compared qualitatively with the experimental data on PZT ferroelectric films and theoretical predictions. While the scaling of the hysteresis area with frequency f and amplitude E0 of the external field at a form of f 1r3 E02r3 over the low-f range is well-established, the high-f range scaling relations as derived from the simulation, experiment and theoretical model differ from one and another. q 2002 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ej; 77.80.Dj; 05.10.Ln Keywords: Hysteresis; Scaling behavior; Monte-Carlo simulation
The problem of the first-order phase transitions in condensed matters, both the early stage kinetics Žnucleation-and-growth or spinodal decomposition. and the late stage domain growth, has been studied extensively over the last decades w1x. Our understanding of this problem from both theoretical and experimental aspects has been advanced, referring to a number of comprehensive review articles in this field w2x. However, not much attention to the response dynamics of the first-order phase transition against the external field Žmagnetic field for ferromagnetics and electrical field for ferroelectrics., i.e. dynamic hysteresis, has been paid. When a spin system far )
Corresponding author. Laboratory of Laser Technologies, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail address: [email protected] ŽJ.-M. Liu..
below the Curie point Tc is submitted to the timevarying field E, a hysteresis loop is observed if plotting the system order parameter P against E, due to the dynamic relaxation of the order parameter through domain switching and outward movement of the boundaries of new domains. Therefore, the loop shape at a fixed temperature T depends essentially on both amplitude E0 and frequency f of the field. An experimental interest in this phenomenon can be dated as far back as the end of the 19th century w3x, mainly focusing on ferromagnetic materials. The experimental verifications of the Steinmetz law w4x, i.e. ² A: ; B01.6 , where ² A: is the loop area and B0 the maximum induction have occasionally been available in literature w5,6x, although they show quite big scattering, including the experiment on the dynamic hysteresis of ferroelectric PZT thin film capacitors w7x. The theoretical studies started from the two-state
00167-577Xr02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X Ž 0 1 . 0 0 3 9 6 - 2
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
214
Ising model and emphasized more or less the relaxation behaviours in the vicinity of Tc w8,9x. The static scaling property of the order parameter and the loop area with respect to T was approached mainly from the mean-field scheme. A theoretical study on the dynamic hysteresis had not been done until the work by Rao et al. w10x on both continuous and lattice spin systems w10–13x. Two scaling relations applicable to the low-f limit and high-f limit, respectively, were proposed. It is our aim in this paper to study the dependence of the dynamic hysteresis in a multi-domain spin system on the oscillating time-varying field. Most of the technologically important systems are multi-domained but neither analytical nor simulation approach to this problem is available up to date. In this aspect, the Q-state Potts model can be an appropriate scheme. Furthermore, the scaling relations as derived from the ŽF 2 . 2 continuous model w10x has not yet received much attention. It is of interest to start from the Q-state Potts model in order to establish a Monte-Carlo ŽMC. algorithm so that a close checking of the scaling relations can be performed. Our experiment data on the ferroelectric hysteresis w7x also provides us possibility to question on the scaling argument. On the other hand, the high-frequency performance of the spin systems, such as ferroelectric RAM operating at ultra-high frequency Ž100 MHz and over. brings big challenge to a reliable measurement of the hysteresis w6x. It is technologically helpful to understand the scaling behaviour so that the ultra-high frequency property of the hysteresis can be predicted. The dynamic hysteresis in the spin systems is understood by investigating the response of an Ncomponent ŽF 2 . 2 model with O Ž N . symmetry to an oscillating field E w10x. The system order parameter set F is non-conserved and its evolution with time t obeys the Langevin equation EFa Et
s yG
dF dFa
q ha
Ž 1.
with the Gaussian white noise ha as ²ha Ž x ,t . : s 0 ²ha Ž x ,t . P hb Ž xX ,tX . : s 2 G dab d Ž x y xX . d Ž t y tX .
Ž 2.
where a , b s 1,2, . . . , N, respectively; x is the spatial coordinate and G is the mobility for the spinlattice relaxation which is of ; 10 7 Hz on the order of magnitude for real solids, F is the free-energy function ŽŽF 2 . 2 type. 1
F s d3 x
H
u q 4N
2
J Ž =Fa . Ž =Fa . q
r 2
Ž FaFa .
2 Ž FaFa . y 'N EaFa
Ž 3.
where F is an N-component vector and J is the interaction between two-components; r s T y TcTF where TcTF is the mean-field critical temperature; u is the pre-factor and counts the contribution of the second order nonlinear interaction and u s y2p 2 ŽTc y TcTF .. Since Fa Fa scales as N, each term in the bracket scales as N, therefore does the free energy. We assume the external field Ea s Eda ,1 s E0 sinŽ2p f P t . da ,1 , pointing to axis a s 1. Eq. Ž1. is equivalent to an infinite hierarchy of differential equations for the cumulants of Fa . In the N ´ ` limit, this infinite hierarchy of differential equations can be solved numerically and detailed results were reported by Rao et al. w10x. Two scaling relations were obtained, independent of T once T - Tc is satisfied ² A: s
2p
H0
P Ž t . P E0 P sin u d u ,
² A: A f 1r3 P E02r3 , ² A: A f y1 P E02 ,
u s 2p f P t
at low-f limit at high-f limit.
Ž 4.
It is of special interest to question on the validity of the two scaling relations through simulation or experiments. In fact, this theory takes only account of the second-order interaction besides the linear term and couples the N ´ ` limit, i.e. ŽF 2 . 2-model. It is basically a Landau phenomenological scheme within the mean field framework. Therefore, performing the simulation and experiments and comparing them with the model-predictions come to be an essential effort along this line. The dynamic hysteresis for a 2-d multi-domain L = L squared lattice with periodic boundary condi-
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
tions is simulated using the Q-state Potts model. Each site in the lattice is imposed with one of the Q spin states, i.e. q s 1,2, . . . ,Q Ž Q G 2.. All spins have the same magnitude P0 . The field E s E0 sinŽ2p ft . points to spin q s 1. The lattice is therefore occupied with finite-sized and randomly distributed domains. In each of the domains all sites have the same spin state q, whereas those sites located at domain boundaries may have the spin orientation different from their neighbours. The Hamiltonian of the lattice is written as w14x H s HP q H E HP s y
J 2
Ý cos ² ij :
HE s y Ý P P E
ž
ayb 2p
Q
/ Ž 5.
²i:
where HP and HE are the Potts interaction energy and field-induced static energy; J Ž) 0. the Potts interaction factor; ² ij : represents that over the nearest-neighbours is summed once and ² i : denotes site i; a s q at site i and b s q ate site j. Here we use the planar Potts description for our simulation whereas the standard Potts model deals with the spin distribution in a Q y 1 dimensional space w15x. In fact, the case of Q s 2 is equivalent to the Ising model. The dynamic relaxation of the system as described by Eq. Ž5. is realized through the MC spin-flip kinetics. In our model, two types of possible spin-flip sequences for any site are considered in order to achieve the same spin-flip event. One is the flip induced by the field E, i.e. spin reversal. The spin q is permitted to take itself or spin q q Qr2 if q q Qr2 F Q or q y Qr2 if q q Qr2 ) Q. What should be pointed out here is that we consider only the 1808 spin flip inside any domain, although the Potts model permits spin-flip from the present state to any of the Q-states. In general, PZT-like ferroelectrics produce 1808, 908 or even 458 domains. The 1808 reversal is the dominant sequence since the 908 and 458 domain creates significant strain energy. Our previous simulation didn’t not reveal essential difference for the 1808 spin-flip event Žtwo equivalent directions. and
215
the 908 one Žfour equivalent directions., unless the lattice is occupied with only a few domains. In our simulation, the lattice is occupied with ; 160 domains of different spin orientations. An overall averaging of data from so many domains makes the 1808-flip-restriction no longer artificial. The other type of flip is domain switching through movement of domain boundaries ŽDBs., driven by the associated excess energy. Following the previous consideration, the spin is permitted to flip from its present state to one of its four neighbours. The Metropolis algorithm for MC simulation is utilized here. The possible two types of spin flips are chosen at random. For an initial configuration of the lattice, one defines the spin for each domain. The average domain size R is taken as required. At any time t, site i in the lattice is chosen at random and then the spin-reversal event or domain-switching one is approved by a random number RX compared with the Metropolis probability. This process is repeated until a given number of events have been completed. The kinetics of the process is scaled in units of the MC step Žmcs.. One mcs scales completed L = L events. The initial multi-domain lattice is constructed by randomly choosing a series of square-like domains until all sites are occupied. This configuration may be constructed via other ways, such as depositing triangle, circle or hexagon domains, however, the dynamic hysteresis shows no substantial difference. Because the domains are distributed randomly, the initial lattice shows a homogeneous order parameter, i.e. P s 0. In our simulation L s 256, Q s 24, R s 20 and P0 s 0.6 Ž P0 E0 takes a unit of kT here.. JrkT s 1.20 is chosen from which TrTc ; 0.367 is estimated if we assume the critical point for the Potts lattice is not very different from that of the Ising lattice w15x. The dependence of the P–E loop shape on f and E0 is clearly identified in Fig. 1a where a series of loops at E0 s 2.0 but various f are presented. At quite low f, the loop is thin rhombic Žloop a. with saturated tips at E ; E0 . The rhombic pattern becomes fat with increasing f Žloops b and c. and the saturated tip is replaced with rounded corner, until an elliptical shape is formed Žloop d.. Further increased f results in shrink of the elliptical pattern along P-axis and eventually collapse into a line a little
216
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
established, although the measured loops show bigger inclination with respect to E-axis. The loop evolution as shown above can be easily explained by the response retardation of the spin-flips to the field E. At quite low f, the retardation is so weak that the complete spin-flips can be achieved. The saturated P is exactly defined by P s 2 P0rp s 0.382.
Ž 6.
With increasing f, the retardation is enhanced, resulting in flip-prohibition of spins in the lattice. The loops show shrink along the P-axis. The loop area ² A: as a function of f and E0 is evaluated too, as shown in Fig. 2. A single-peaked pattern for ² A: against f with the peak and its position shifting upward and rightward, respectively,
Fig. 1. Ža. Simulated hysteresis loops for the multi-domain lattice at various values of f Ž a: 7.81=10y5 , b: 3.125=10y4 , c: 1.25=10y3 , d: 5=10y3 , e: 0.02, f : 0.1, unit: mcsy1 . and Žb. measured loops for PZT thin films at four frequencies.
inclined to E-axis at a ultra-high f. Take a note that at the high frequency, the loop no longer keeps the two-fold symmetry with respect to zero-point. It shifts a little upward from the E-axis. Such an evolution of loop pattern with varying f is consistent with that as predicted by the ŽF 2 . 2 model. As a comparison, we present in Fig. 1b the measured loops for PZT thin films at various frequencies w7x. A qualitative one-to-one correspondence in pattern between the simulated loops and measured ones can be
Fig. 2. Simulated hysteresis area ² A: as a function of f at various E0 Ža. and as a function of E0 at three frequencies Žb..
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
Fig. 3. Scaling of the simulated loop area ² A: with respect to f 1r 3 E02r3 in the low-f limit Ža. and with respect to f y1 E03 in the high-f limit Žb..
is identified. Refer to ² A: as a function of E0 , the dependence is related to E0 - Ec Ž f . and E0 ) Ec Ž f ., where Ec is the coercivity A: ; E0m
as E0 - Ec Ž f .
² A: ; E0n
as E0 ) Ec Ž f .
²
Ž 7.
wih exponent m and n being f-dependent. For example, m s 4 and n s 1r3 at f s 7.8 = 10y5 mcsy1 , and m s 5 and n s 3 at f s 10y1 mcsy1 . Now, check the scaling relation Eq. Ž4.. In Fig. 3a, we plot all of the simulated data over the low-f range and find the scaling at low f works quite well. For the high-f limit, instead of the high-f limit scaling given in Eq. Ž4., one observes ² A: A f y1 E03
Ž 8.
217
within the simulating uncertainty. Therefore, the scaling is different in E0-term from that proposed by the ŽF 2 . 2-model w10x. The former produces an exponent of 3 but the latter give an exponent of 2. An explanation of such a difference may come from the spin-interaction terms as considered differently in the ŽF 2 . 2-model and in the Potts model. In the former case, the spin-flip just has one contribution, i.e. spin-reversal, which requires overcoming higher energy barrier, whereas in our simulation the spin orientation in various domains is not anti-parallel to direction of E. These domains meet lower energy barrier. This barrier difference is not important for the low-f range. However, for the case of high frequency, the lower energy barrier allows a faster spin-flip kinetics, which is very critical for a more complete spin-flip overall the lattice. A high exponent for the E0 term is expected from the Potts model rather than the ŽF 2 . 2-model. An experimental confirm of this explanation is given below. The dynamic hysteresis for PZT thin films deposited on SrTiO 3 substrates with YBCO layers as electrodes were measured w7x. The films are of multi-domain configuration. Several examples of the hysteresis are shown in Fig. 1b. The measured loop area ² A: as a function of f shows a single-peaked pattern too, very similar to that shown in Fig. 3a. Here, we are concerning with the scaling behaviors at both low-f and high-f limit. The results are presented in Fig. 4a and b. While the low-f limit data obey as predicted the scaling, Eq. Ž4., within the measuring uncertainty, however, the high-f scaling takes a different form, reading ² A: A f y1r3 E03
Ž 9.
noting that the E0-term remains the same as the simulated form. However, the f-term shows an exponent of y1r3, smaller in absolute value than that given by the ŽF 2 . 2-model and Potts model Žexponent is y1.. To explain this difference, one may need to consider the dielectric response of PZT at the high frequency range. At a frequency of 10y5 –10y7 Hz, the ionic-type contribution besides the domain switching to the dynamic hysteresis may no longer be negligible. The ionic type contribution will play a
218
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
proach to this high frequency part seems not easy, however, it is certain that an additional term taking into account of this contribution is needed. The comparison presented above shows that the proposed scaling laws over the low-f range from the ŽF 2 . 2-model, the Potts lattice simulation and the experiment remain the same, because the spin-flip is the only dominating mechanism responsible for the dynamic hysteresis. However, the as-derived high-f scaling laws are different from one and another. Although a qualitative explanation of the difference has been presented, a quantitative approach of the problem is essential before a reliable prediction of the high-frequency scaling can be given. The present model which only takes into account of spin-interaction and field-induced static energy seems to be a little over-simplified and still has space to be improved.
Acknowledgements The authors would like to acknowledge the financial support from the Centre for Smart Materials of the Hong Kong Polytechnic University, the National Natural Science Foundation of China and the National Key Project for Basic Research of China. JML is a Ke-Li fellow.
References
Fig. 4. Scaling of the measured loop area ² A: for PZT thin films with respect to f 1r 3 E02r3 in the low-f limit Ža. and with respect to f y1r 3 E03 in the high-f limit Žb..
more important role at a higher f. Therefore, the as-measured hysteresis area must show a relatively weaker dependence on f than that predicted by the purely spin-flip based models. A quantitative ap-
w1x J.D. Gunton, M. San Miguel, P.S. Sahni, in: C. Domb, J.L. Lebowitz ŽEds.., Phase Transitions and Critical Phenomena, vol. 8, Academic Press, London, 1983. w2x R.W. Cahn, P. Haasen, E.J. Kramer ŽEds.., Mater. Sci. Technol.: Compr. Treat., vol. 5, VCH Publishers, Weinheim, 1991. w3x J.A. Ewing, H.G. Klassen, Philos. Trans. R. Soc. London, Ser. A 184 Ž1893. 985. w4x C.P. Steinmetz, Trans. Am. Inst. Electr. Eng. 9 Ž1892. 3. w5x R.M. Bozorth, Ferromagnetism. Van Nostrand, New York, 1951. w6x J.F. Scott, in: R. Ramesh ŽEd.., Thin Film Ferroelectric Materials and Devices. Kluwer Academic Publishing, Boston, 1997, p. 115. w7x J.-M. Liu, H.P. Li, C.K. Ong, L.C. Lim, J. Appl. Phys. 86 Ž1999. 5198. w8x M. Acharyya, Phys. Rev. E 58 Ž1998. 174.
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219 w9x G.P. Zheng, J.X. Zhang, J. Phys.: Condens. Matter. 10 Ž1998. 275. w10x M. Rao, H.R. Krishnamurthy, R. Pandit, Phys. Rev. B 42 Ž1990. 856. w11x Q. Jiang, H.N. Yang, G.C. Wang, Phys. Rev. B 52 Ž1995. 14911.
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w12x M. Acharyya, B.K. Chakrabarti, Phys. Rev. B 52 Ž1995. 6550. w13x S.W. Sides, R.A. Ramos, P.A. Rikvold, M.A. Novotny, J. Appl. Phys. 79 Ž1996. 6482. w14x J.-M. Liu, Z.G. Liu, Appl. Phys. A 70 Ž2000. 113. w15x F.Y. Wu, Rev. Mod. Phys. 54 Ž1982. 235.
Materials Letters 52 Ž2002. 213–219 www.elsevier.comrlocatermatlet
Scaling behavior of dynamic hysteresis in multi-domain spin systems J.-M. Liu a,b,c,) , H.L. Chan b, C.L. Choy b a
Laboratory of Laser Technologies, Huazhong UniÕersity of Science and technology, Wuhan 430074, China b Department of Applied Physics, The Hong Kong Polytechnic UniÕersity, Hong Kong, China c Laboratory of Solid State Microstructures, Nanjing UniÕersity, Nanjing 210093, China Received 29 January 2001; accepted 5 February 2001
Abstract The dynamic hysteresis in two-dimensional multi-domain spin systems is simulated using Monte-Carlo method based on Q-state planar Potts model. The pattern and scaling property of the hysteresis as a function of the oscillating time-varying external field are studied and compared qualitatively with the experimental data on PZT ferroelectric films and theoretical predictions. While the scaling of the hysteresis area with frequency f and amplitude E0 of the external field at a form of f 1r3 E02r3 over the low-f range is well-established, the high-f range scaling relations as derived from the simulation, experiment and theoretical model differ from one and another. q 2002 Elsevier Science B.V. All rights reserved. PACS: 75.60.Ej; 77.80.Dj; 05.10.Ln Keywords: Hysteresis; Scaling behavior; Monte-Carlo simulation
The problem of the first-order phase transitions in condensed matters, both the early stage kinetics Žnucleation-and-growth or spinodal decomposition. and the late stage domain growth, has been studied extensively over the last decades w1x. Our understanding of this problem from both theoretical and experimental aspects has been advanced, referring to a number of comprehensive review articles in this field w2x. However, not much attention to the response dynamics of the first-order phase transition against the external field Žmagnetic field for ferromagnetics and electrical field for ferroelectrics., i.e. dynamic hysteresis, has been paid. When a spin system far )
Corresponding author. Laboratory of Laser Technologies, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail address: [email protected] ŽJ.-M. Liu..
below the Curie point Tc is submitted to the timevarying field E, a hysteresis loop is observed if plotting the system order parameter P against E, due to the dynamic relaxation of the order parameter through domain switching and outward movement of the boundaries of new domains. Therefore, the loop shape at a fixed temperature T depends essentially on both amplitude E0 and frequency f of the field. An experimental interest in this phenomenon can be dated as far back as the end of the 19th century w3x, mainly focusing on ferromagnetic materials. The experimental verifications of the Steinmetz law w4x, i.e. ² A: ; B01.6 , where ² A: is the loop area and B0 the maximum induction have occasionally been available in literature w5,6x, although they show quite big scattering, including the experiment on the dynamic hysteresis of ferroelectric PZT thin film capacitors w7x. The theoretical studies started from the two-state
00167-577Xr02r$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 5 7 7 X Ž 0 1 . 0 0 3 9 6 - 2
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
214
Ising model and emphasized more or less the relaxation behaviours in the vicinity of Tc w8,9x. The static scaling property of the order parameter and the loop area with respect to T was approached mainly from the mean-field scheme. A theoretical study on the dynamic hysteresis had not been done until the work by Rao et al. w10x on both continuous and lattice spin systems w10–13x. Two scaling relations applicable to the low-f limit and high-f limit, respectively, were proposed. It is our aim in this paper to study the dependence of the dynamic hysteresis in a multi-domain spin system on the oscillating time-varying field. Most of the technologically important systems are multi-domained but neither analytical nor simulation approach to this problem is available up to date. In this aspect, the Q-state Potts model can be an appropriate scheme. Furthermore, the scaling relations as derived from the ŽF 2 . 2 continuous model w10x has not yet received much attention. It is of interest to start from the Q-state Potts model in order to establish a Monte-Carlo ŽMC. algorithm so that a close checking of the scaling relations can be performed. Our experiment data on the ferroelectric hysteresis w7x also provides us possibility to question on the scaling argument. On the other hand, the high-frequency performance of the spin systems, such as ferroelectric RAM operating at ultra-high frequency Ž100 MHz and over. brings big challenge to a reliable measurement of the hysteresis w6x. It is technologically helpful to understand the scaling behaviour so that the ultra-high frequency property of the hysteresis can be predicted. The dynamic hysteresis in the spin systems is understood by investigating the response of an Ncomponent ŽF 2 . 2 model with O Ž N . symmetry to an oscillating field E w10x. The system order parameter set F is non-conserved and its evolution with time t obeys the Langevin equation EFa Et
s yG
dF dFa
q ha
Ž 1.
with the Gaussian white noise ha as ²ha Ž x ,t . : s 0 ²ha Ž x ,t . P hb Ž xX ,tX . : s 2 G dab d Ž x y xX . d Ž t y tX .
Ž 2.
where a , b s 1,2, . . . , N, respectively; x is the spatial coordinate and G is the mobility for the spinlattice relaxation which is of ; 10 7 Hz on the order of magnitude for real solids, F is the free-energy function ŽŽF 2 . 2 type. 1
F s d3 x
H
u q 4N
2
J Ž =Fa . Ž =Fa . q
r 2
Ž FaFa .
2 Ž FaFa . y 'N EaFa
Ž 3.
where F is an N-component vector and J is the interaction between two-components; r s T y TcTF where TcTF is the mean-field critical temperature; u is the pre-factor and counts the contribution of the second order nonlinear interaction and u s y2p 2 ŽTc y TcTF .. Since Fa Fa scales as N, each term in the bracket scales as N, therefore does the free energy. We assume the external field Ea s Eda ,1 s E0 sinŽ2p f P t . da ,1 , pointing to axis a s 1. Eq. Ž1. is equivalent to an infinite hierarchy of differential equations for the cumulants of Fa . In the N ´ ` limit, this infinite hierarchy of differential equations can be solved numerically and detailed results were reported by Rao et al. w10x. Two scaling relations were obtained, independent of T once T - Tc is satisfied ² A: s
2p
H0
P Ž t . P E0 P sin u d u ,
² A: A f 1r3 P E02r3 , ² A: A f y1 P E02 ,
u s 2p f P t
at low-f limit at high-f limit.
Ž 4.
It is of special interest to question on the validity of the two scaling relations through simulation or experiments. In fact, this theory takes only account of the second-order interaction besides the linear term and couples the N ´ ` limit, i.e. ŽF 2 . 2-model. It is basically a Landau phenomenological scheme within the mean field framework. Therefore, performing the simulation and experiments and comparing them with the model-predictions come to be an essential effort along this line. The dynamic hysteresis for a 2-d multi-domain L = L squared lattice with periodic boundary condi-
J.-M. Liu et al.r Materials Letters 52 (2002) 213–219
tions is simulated using the Q-state Potts model. Each site in the lattice is imposed with one of the Q spin states, i.e. q s 1,2, . . . ,Q Ž Q G 2.. All spins have the same magnitude P0 . The field E s E0 sinŽ2p ft . points to spin q s 1. The lattice is therefore occupied with finite-sized and randomly distributed domains. In each of the domains all sites have the same spin state q, whereas those sites located at domain boundaries may have the spin orientation different from their neighbours. The Hamiltonian of the lattice is written as w14x H s HP q H E HP s y
J 2
Ý cos ² ij :
HE s y Ý P P E
ž
ayb 2p
Q
/ Ž 5.
²i:
where HP and HE are the Potts interaction energy and field-induced static energy; J Ž) 0. the Potts interaction factor; ² ij : represents that over the nearest-neighbours is summed once and ² i : denotes site i; a s q at site i and b s q ate site j. Here we use the planar Potts description for our simulation whereas the standard Potts model deals with the spin distribution in a Q y 1 dimensional space w15x. In fact, the case of Q s 2 is equivalent to the Ising model. The dynamic relaxation of the system as described by Eq. Ž5. is realized through the MC spin-flip kinetics. In our model, two types of possible spin-flip sequences for any site are considered in order to achieve the same spin-flip event. One is the flip induced by the field E, i.e. spin reversal. The spin q is permitted to take itself or spin q q Qr2 if q q Qr2 F Q or q y Qr2 if q q Qr2 ) Q. What should be pointed out here is that we consider only the 1808 spin flip inside any domain, although the Potts model permits spin-flip from the present state to any of the Q-states. In general, PZT-like ferroelectrics produce 1808, 908 or even 458 domains. The 1808 reversal is the dominant sequence since the 908 and 458 domain creates significant strain energy. Our previous simulation didn’t not reveal essential difference for the 1808 spin-flip event Žtwo equivalent directions. and
215
the 908 one Žfour equivalent directions., unless the lattice is occupied with only a few domains. In our simulation, the lattice is occupied with ; 160 domains of different spin orientations. An overall averaging of data from so many domains makes the 1808-flip-restriction no longer artificial. The other type of flip is domain switching through movement of domain boundaries ŽDBs., driven by the associated excess energy. Following the previous consideration, the spin is permitted to flip from its present state to one of its four neighbours. The Metropolis algorithm for MC simulation is utilized here. The possible two types of spin flips are chosen at random. For an initial configuration of the lattice, one defines the spin for each domain. The average domain size R is taken as required. At any time t, site i in the lattice is chosen at random and then the spin-reversal event or domain-switching one is approved by a random number RX compared with the Metropolis probability. This process is repeated until a given number of events have been completed. The kinetics of the process is scaled in units of the MC step Žmcs.. One mcs scales completed L = L events. The initial multi-domain lattice is constructed by randomly choosing a series of square-like domains until all sites are occupied. This configuration may be constructed via other ways, such as depositing triangle, circle or hexagon domains, however, the dynamic hysteresis shows no substantial difference. Because the domains are distributed randomly, the initial lattice shows a homogeneous order parameter, i.e. P s 0. In our simulation L s 256, Q s 24, R s 20 and P0 s 0.6 Ž P0 E0 takes a unit of kT here.. JrkT s 1.20 is chosen from which TrTc ; 0.367 is estimated if we assume the critical point for the Potts lattice is not very different from that of the Ising lattice w15x. The dependence of the P–E loop shape on f and E0 is clearly identified in Fig. 1a where a series of loops at E0 s 2.0 but various f are presented. At quite low f, the loop is thin rhombic Žloop a. with saturated tips at E ; E0 . The rhombic pattern becomes fat with increasing f Žloops b and c. and the saturated tip is replaced with rounded corner, until an elliptical shape is formed Žloop d.. Further increased f results in shrink of the elliptical pattern along P-axis and eventually collapse into a line a little
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established, although the measured loops show bigger inclination with respect to E-axis. The loop evolution as shown above can be easily explained by the response retardation of the spin-flips to the field E. At quite low f, the retardation is so weak that the complete spin-flips can be achieved. The saturated P is exactly defined by P s 2 P0rp s 0.382.
Ž 6.
With increasing f, the retardation is enhanced, resulting in flip-prohibition of spins in the lattice. The loops show shrink along the P-axis. The loop area ² A: as a function of f and E0 is evaluated too, as shown in Fig. 2. A single-peaked pattern for ² A: against f with the peak and its position shifting upward and rightward, respectively,
Fig. 1. Ža. Simulated hysteresis loops for the multi-domain lattice at various values of f Ž a: 7.81=10y5 , b: 3.125=10y4 , c: 1.25=10y3 , d: 5=10y3 , e: 0.02, f : 0.1, unit: mcsy1 . and Žb. measured loops for PZT thin films at four frequencies.
inclined to E-axis at a ultra-high f. Take a note that at the high frequency, the loop no longer keeps the two-fold symmetry with respect to zero-point. It shifts a little upward from the E-axis. Such an evolution of loop pattern with varying f is consistent with that as predicted by the ŽF 2 . 2 model. As a comparison, we present in Fig. 1b the measured loops for PZT thin films at various frequencies w7x. A qualitative one-to-one correspondence in pattern between the simulated loops and measured ones can be
Fig. 2. Simulated hysteresis area ² A: as a function of f at various E0 Ža. and as a function of E0 at three frequencies Žb..
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Fig. 3. Scaling of the simulated loop area ² A: with respect to f 1r 3 E02r3 in the low-f limit Ža. and with respect to f y1 E03 in the high-f limit Žb..
is identified. Refer to ² A: as a function of E0 , the dependence is related to E0 - Ec Ž f . and E0 ) Ec Ž f ., where Ec is the coercivity A: ; E0m
as E0 - Ec Ž f .
² A: ; E0n
as E0 ) Ec Ž f .
²
Ž 7.
wih exponent m and n being f-dependent. For example, m s 4 and n s 1r3 at f s 7.8 = 10y5 mcsy1 , and m s 5 and n s 3 at f s 10y1 mcsy1 . Now, check the scaling relation Eq. Ž4.. In Fig. 3a, we plot all of the simulated data over the low-f range and find the scaling at low f works quite well. For the high-f limit, instead of the high-f limit scaling given in Eq. Ž4., one observes ² A: A f y1 E03
Ž 8.
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within the simulating uncertainty. Therefore, the scaling is different in E0-term from that proposed by the ŽF 2 . 2-model w10x. The former produces an exponent of 3 but the latter give an exponent of 2. An explanation of such a difference may come from the spin-interaction terms as considered differently in the ŽF 2 . 2-model and in the Potts model. In the former case, the spin-flip just has one contribution, i.e. spin-reversal, which requires overcoming higher energy barrier, whereas in our simulation the spin orientation in various domains is not anti-parallel to direction of E. These domains meet lower energy barrier. This barrier difference is not important for the low-f range. However, for the case of high frequency, the lower energy barrier allows a faster spin-flip kinetics, which is very critical for a more complete spin-flip overall the lattice. A high exponent for the E0 term is expected from the Potts model rather than the ŽF 2 . 2-model. An experimental confirm of this explanation is given below. The dynamic hysteresis for PZT thin films deposited on SrTiO 3 substrates with YBCO layers as electrodes were measured w7x. The films are of multi-domain configuration. Several examples of the hysteresis are shown in Fig. 1b. The measured loop area ² A: as a function of f shows a single-peaked pattern too, very similar to that shown in Fig. 3a. Here, we are concerning with the scaling behaviors at both low-f and high-f limit. The results are presented in Fig. 4a and b. While the low-f limit data obey as predicted the scaling, Eq. Ž4., within the measuring uncertainty, however, the high-f scaling takes a different form, reading ² A: A f y1r3 E03
Ž 9.
noting that the E0-term remains the same as the simulated form. However, the f-term shows an exponent of y1r3, smaller in absolute value than that given by the ŽF 2 . 2-model and Potts model Žexponent is y1.. To explain this difference, one may need to consider the dielectric response of PZT at the high frequency range. At a frequency of 10y5 –10y7 Hz, the ionic-type contribution besides the domain switching to the dynamic hysteresis may no longer be negligible. The ionic type contribution will play a
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proach to this high frequency part seems not easy, however, it is certain that an additional term taking into account of this contribution is needed. The comparison presented above shows that the proposed scaling laws over the low-f range from the ŽF 2 . 2-model, the Potts lattice simulation and the experiment remain the same, because the spin-flip is the only dominating mechanism responsible for the dynamic hysteresis. However, the as-derived high-f scaling laws are different from one and another. Although a qualitative explanation of the difference has been presented, a quantitative approach of the problem is essential before a reliable prediction of the high-frequency scaling can be given. The present model which only takes into account of spin-interaction and field-induced static energy seems to be a little over-simplified and still has space to be improved.
Acknowledgements The authors would like to acknowledge the financial support from the Centre for Smart Materials of the Hong Kong Polytechnic University, the National Natural Science Foundation of China and the National Key Project for Basic Research of China. JML is a Ke-Li fellow.
References
Fig. 4. Scaling of the measured loop area ² A: for PZT thin films with respect to f 1r 3 E02r3 in the low-f limit Ža. and with respect to f y1r 3 E03 in the high-f limit Žb..
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