Nonsteady dynamic properties of a domain wall for the creep state under an alternating driving field
With Monte Carlo simulations, the nonsteady dynamics properties of a domain wall have been systematically investigated for the thermally activated creep state under an alternating driving field. Taking the driven random-field Ising model in two dimensions as an example, two distinct growth stages of the domain interface are identified with both the correlation length and roughness function. One stage belongs to the universality class of the random depositions, and the other to that of the quenched Edwards-Wilkinson equation. In the latter case, due to the dynamic effect of overhangs, the domain interface may exhibit an intrinsic anomalous scaling behavior, different from that of the quenched Edwards-Wilkinson equation.
pacs:64.60.Ht, 05.10.Ln, 75.60.Ch
In recent years, the domain-wall motion has become a source of much experimental and theoretical research Tatara and Kohno (2004); Kleemann (2007); Martinez et al. (2007); Metaxas et al. (2007); Kim et al. (2009); Schütze and Nattermann (2011); Miron et al. (2011); Lee et al. (2011). The dynamics under an alternating driving field has attracted extensive interests in vortex lattices Dolz et al. (2010); Cao et al. (2012), liquid crystals Jeżewski et al. (2008), ferromagnetic/ferroelectric materials Kagawa et al. (2011); Steinke et al. (2012); Zhernenkov et al. (2013) and crystalline solids Laurson and Alava (2012). In particular, the magnetic domain-wall dynamics is an important topic in nanomaterials, thin films and semiconductors, because of its potential technological applications including magnetic random access memories and logic devices Yamanouchi et al. (2007); Hayashi et al. (2008); Parkin et al. (2008). In the experiments of ultrathin ferromagnetic and ferroelectric films, considerable attention is devoted to the complex susceptibility () Venimadhav et al. (2013); Braun et al. (2005); Kleemann et al. (2007), which depicts the domain-wall motion. Four dynamic states are observed in the Cole-Cole diagram of vs , which are relaxation, creep, sliding and switching. Recently, experimental evidences of the relaxation-to-creep dynamic transition have been found, not only in ultrathin ferromagnetic trilayers and ferroelectric films Braun et al. (2005); Kleemann et al. (2007, 2006); Yang et al. (2010), but also in liquid crystals, ferroelastic materials and molecular ferrimagnets Harrison et al. (2004); Jeżewski et al. (2008); Mushenok et al. (2011). However, theoretical understanding of the transition is limited, especially for the growth process of the correlation length Nattermann et al. (2001); Glatz et al. (2003).
For the creep state at low temperatures and low frequencies, one observes an inverse power-law behavior for the complex susceptibility Kleemann (2007). Here denotes the bulk background susceptibility when the frequency , is the characteristic relaxation time, and is the creep exponent. According to scaling arguments, a theoretical value is expected with the roughness exponent and dynamic exponent Kleemann et al. (2007). While experimental values of vary from to in ultrathin ferromagnetic and ferroelectric films Braun et al. (2005); Kleemann et al. (2006, 2007). Hence, it remains much challenging to understand the creep exponent.
Up to date, theoretical tools for describing the domain-wall motion are typically based on the Edwards-Wilkinson equation with quenched disorder (QEW) Petracic et al. (2004); Kolton et al. (2006); Duemmer and Krauth (2005); Ferrero et al. (2013); Schütze and Nattermann (2011); Schütze (2010). With this equation, the dynamics properties for the creep state under a constant driving field or zero field have been well understood Kolton et al. (2005a, 2009); Chauve et al. (2000). It can be viewed as a thermally activated hopping movement from one local energy minimum to the next, dominated by the energy barrier that must be overcome. The energy barrier grows as a power law , responsible for the logarithmical growth of the correlation length Monthus and Garel (2008a, b). An effective energy barrier exponent has been numerically measured Kolton et al. (2005b); Monthus and Garel (2008c), and the roughness exponent has also been estimated from the kinetic roughening of the domain wall Kolton et al. (2005a, 2009). However, few works deal with the creep dynamics under an alternating driving field. Moreover, detailed microscopic structures and interactions of real materials are not concerned in the phenomenological QEW equation Petracic et al. (2004).
To further understand the creep dynamics from a more fundamental viewpoint, we should build lattice models which allow a closer comparison between theory and experiment. The driven random-field Ising model (DRFIM) is a candidate, which has been used to understand the dynamic transitions in ferroic systems Colaiori et al. (2006); Zhou et al. (2009, 2010). Despite not including all interactions in real materials, it at least captures robust features of the domain-wall motion. Very recently, the creep motion of a domain wall driven by a constant field has been numerically investigated with the DRFIM model, and the results are comparable with experiments Dong et al. (2012a).
Taking the two-dimensional () DRFIM model as an example, we conduct a comprehensive study on the creep dynamics under an alternating driving field. With Monte Carlo simulations, we accurately determine the scaling exponents and , and identify the universality classes, in comparison with those of the QEW equation and experiments. In Sec. II, the model and scaling analysis are described, and in Sec. III, the numerical results are presented. Finally, Sec. IV includes the conclusions.
Ii model and scaling analysis
The DRFIM model is defined by the following Hamiltonian
where is a Ising spin at site of the lattice, denotes a nearest-neighbor pair of spins, and is a quenched random field uniformly distributed within an interval . We use a homogeneous alternating driving field , and set the coupling constant Zhou et al. (2010). In order to make sure that the dynamic evolution of spins occurs at or around the domain wall, we restrict the temperature , the disorder strength and the driving field . Simulations are performed on a rectangular lattice with the antiperiodic and periodic boundary conditions along the and directions, respectively.
The initial state is a semiordered state with a perfect domain wall in the direction. To eliminate the pinning effect irrelevant for the disorder, we rotate the lattice such that the initial domain wall orients in the direction Zhou et al. (2009); Zhou and Zheng (2010). After preparing the initial state, we update spins with the heat-bath algorithm Zheng (1998). As time evolves, the domain wall moves and roughens, while the bulk remains unchanged. Therefore, the domain wall can also be called a domain interface N.J. Zhou and B. Zheng (2007); Zhou and Zheng (2008). Main results of numerical simulations are presented with the lattice size and , up to Monte Carlo steps (MCS). Here MCS is defined by single-spin-flips attempts. For each set of model parameters (), more than samples are performed for average. Errors are estimated by dividing the samples into three subgroups. If the fluctuation of the curve in the time direction is comparable with or larger than the statistical error, it will also be taken into account. Additional simulations with are performed to confirm that the finite-size effect is negligible.
Denoting a spin at site by , we first introduce the height function
and then define the position of the domain interface
Here represents not only the statistic average over Monte Carlo samples, but also the average in the direction. After the stationary magnetic hysteresis loop is obtained at , the complex susceptibility can be calculated by Petracic et al. (2004); Zhou et al. (2010),
where is the time period of the alternating driving field.
With the height function at hand, the roughness function and the correlation function are defined respectively by
and describe the roughening of the domain interface in the direction and the growth of the spatial correlation in the direction, respectively. To reveal the characteristics of the thermally activated creep dynamics, we introduce the creep susceptibility
and the pure roughness function
To detect overhangs generated in the creep motion, another two definitions of the height functions, and , are introduced by the envelopes of the positive and negative spins, respectively Zhou and Zheng (2010). It is believed that the difference describes the average size of overhangs.
According to the phenomenological scaling arguments Fedorenko and Stepanow (2005), a power-law dispersion of the creep susceptibility is obtained for the creep dynamics under an alternating driving field,
For the -length domain-wall segments, a certain hopping time is required to overcome the energy barrier Kolton et al. (2005b). Assuming that the energy barrier scales as , one may deduce
Here is the so-called creep correlation length, and is the energy barrier exponent.
where is the scaling function with , and is the roughness function defined in Eq. (5). In the kinetic roughening of the domain interface, a power-law scaling behavior of the pure roughness function is expected with the roughness exponent ,
Meanwhile, one may determine the local roughness exponent by fitting with an empirical scaling form Jost and Usadel (1996),
Iii Monte Carlo simulations
iii.1 Numerical results
In Fig. 1, the spectrum of the creep susceptibility defined in Eq. (7) is plotted on a log-log scale at the temperature , the strength of the disorder and the driving field . To obtain stationary results, the data in a waiting time periods are skipped in the computation of . A power-law behavior is observed but with certain corrections to scaling. A direct measurement from the slope yields the exponent , and the correction in the form extends the fitting to the early times but with the same value of within the error bar. For comparison, the creep exponents at other temperatures , , , and are measured. As shown in the inset, the result jumps suddenly to around . In order to understand the underlying dynamic mechanism, we investigate the nonsteady dynamics in the following.
Taking the set of model parameters ( and Hz ) as an example, the correlation function is displayed as a function of in Fig. 2(a). According to Eq. (13), a perfect fitting to the numerical data is observed, and the local roughness exponent is measured. Based on the scaling form of in Eq. (11), numerical data of different time nicely collapse to the curve at MCS by rescaling to and to . With the data-collapse technique Dong et al. (2012a), we extract the nonequilibrium correlation length from the correlation function . In the inset, the dynamic evolution of is displayed on a log-log scale. The significant deviation from the power-law behavior indicates that the correlation length does not obey the usual growth law Kolton et al. (2005b); Dong et al. (2012b). Additionally, a time-independent correlation length is observed at for the relaxation state, irrelevant to the creep dynamics. Therefore, we define the creep correlation length by subtracting the relaxation length , and introduce a dimensionless correlation length .
In Fig. 2(b), the creep correlation length is displayed as a function of at different on a log-log scale. Power-law behaviors are observed, and the effective energy barrier exponents and are estimated from the slopes of the curves for and , respectively. To confirm the scaling form in Eq. (10), vs is plotted in Fig. 3(a) on a double-log scale. Data of different nicely collapse to a master curve, and two distinct scaling regimes are detected with the slopes and . Between the two regimes, a dynamic crossover occurs at and . Then we extract the characteristic of the energy barrier
in the small- and large- scaling regimes, respectively.
With the creep correlation length at hand, we measure the roughness exponent in Eq. (12). Since the amplitude of the alternating field is much smaller than the depinning field Zhou et al. (2009), the roughness exponent is actually the equilibrium exponent, though the equilibrium state is not yet reached Kolton et al. (2005b). In Fig. 3(b), is plotted as a function of on a log-log scale. Similarly, data collapse of different is displayed with different symbols. In the small- regime, the slope of the master curve yields the roughness exponent , close to . It suggests that the domain interface belongs to the universality class of the random depositions Yang and Lu (1995); Sarma et al. (1996); Fedorenko et al. (2004); Agoritsas et al. (2013). In the large- regime, is estimated, in good agreement with the equilibrium value of the QEW equation Kolton et al. (2005b); Monthus and Garel (2008a); Kolton et al. (2009); Iguain et al. (2009). Hence, it belongs to the universality class of the QEW equation. In the inset, however, a noticeable increase of is observed at the high temperature , and the asymptotic value is .
To understand the unexpectedly large roughness exponent at , we examine the existence of overhangs in the creep motion Zhou et al. (2009); Zhou and Zheng (2010). As shown in Fig. 4(a), black and red lines represent the time evolutions of the height functions and , respectively. The coincidence and noncoincidence of the curves in the upper and lower panels suggest that the contribution of overhangs is negligible at and important at . Besides, the snapshots of the domain walls at the time MCS are also shown in the insets. Overhangs can be observed directly in the lower panel but not in the upper panel. Consequently, it is convincing that the overhangs affect the dynamic evolution of the spin configuration and play an essential role in the increase of the roughness exponent.
Due to the existence of overhangs, the position of the domain interface is not single-valued and the definition of the height function is not unique. As shown in Fig. 4(b), and at are displayed for the domain interface defined with the envelop of the negative spins, in comparison with those for the domain interface defined with the magnetization in Eq. (3). The roughness exponent and the energy barrier exponent are measured, consistent with and for the domain interface defined with the magnetization. These results again support that the definition of the height function with the magnetization is reasonable.
Besides the temperature, the effects of the quenched disorder and driving field are also investigated. In Fig. 5(a), the creep correlation length is displayed as a function of at on a log-log scale. Taking as input, data collapse of different disorders and is demonstrated, and is determined, close to in Fig. 3(a). According to the scaling relation , one may derive the scaling form of in the large- regime,
In the inset, the scaling function vs is plotted on a log-log scale. Data of different nicely collapse together with the parameter as input. Similarly, the abnormal increase of the roughness exponent from to is also induced by the dynamic effect of overhangs.
In Fig. 5(b), the creep dynamics of the domain wall for different frequencies is presented at the driving field on a log-log scale. If the frequency is sufficiently low, e.g., Hz, a power-law behavior of can be observed with the exponent , somewhat larger than the one at . Additional simulations at and show that the energy barrier exponent is -dependent, and data of different are unlikely to collapse together. For a high frequency, e.g., Hz corresponding to the relaxation state Zhou et al. (2010), drops obviously at the tail of the curve. It suggests that the nonsteady dynamics properties of the relaxation state are different from those of the creep state.
The measurements of scaling exponents at different are summarized for and in Table. 1. As increases, the creep exponent changes from to , compatible with experimental results in the ferromagnetic and ferroelectric films Kleemann et al. (2006), e.g., in the ultrathin Pt/Co(nm)/Pt trilayer and in the periodically poled KTiOPO Braun et al. (2005); Kleemann et al. (2007). The exponent measured at the highest temperature is consistent with the prediction of the scaling relation Kleemann et al. (2007). According to the general scaling arguments Nattermann et al. (1990), we propose the complex susceptibility with an effective coefficient . Then a novel scaling relation is obtained. As shown in Table. 1, holds for the whole temperature range. With this scaling relation, one can predict the creep exponent by only measuring and from the nonsteady dynamics.
Two distinct growth stages of the creep correlation length are found with the scaling exponents in the small- scaling regime and in the large- scaling regime. The results indicate that the former belongs to the universality class of the random depositions, while the later belongs to the universality class of the QEW equation. The two universality classes are separated by the so-called Larkin length at which the effects of the quenched disorder and domain-wall elasticity are of the same order Tanguy and Vettorel (2004); Fedorenko and Stepanow (2005); Nogawa et al. (2008); Kagawa et al. (2011). According to Eq. (14), is estimated for the creep dynamics. Now let us recall the growth process of the creep correlation length. At the beginning, the elasticity is dominant. The kinetic roughening of the domain interface is then dominated by thermal fluctuations with Duemmer and Krauth (2005); Kolton et al. (2005a), and the energy barrier is linear with . After reaches , the quenched disorder overcomes the elasticity. Then the domain-wall motion can be described by the QEW equation with the nontrivial exponents and Kolton et al. (2005b); Iguain et al. (2009).
In Table. 2, the effects of , and are uncovered in the large- regime with the fixed sets of model parameters (Hz), (Hz) and (), respectively. For a moderate disorder, i.e., , a robust value is determined, close to of the QEW equation. According to Eq. (15), the hopping time is derived with Monthus and Garel (2008a); Kagawa et al. (2011), consistent with the ones in Refs.Kolton et al. (2005b); Corberi et al. (2011). The factor shows that the hopping process is determined by the competition between the quenched disorder and thermal noise Kolton et al. (2005a). Since increases monotonically with and the curve drops at the tail, further studies are needed to derive the exact functional form on and .
For the kinetic roughening of the domain interface, a robust value is determined in the large- scaling regime at different , , and . The scaling relation indicates that the domain interfaces belongs to the Family-Vicsek universality class Ramasco et al. (2000). When exceeds a certain threshold, however, differs from by more than percent not only at a higher temperature , but also at weaker quenched disorders , stronger driving fields and higher frequencies Hz, Hz. A similar phenomenon has also been observed in Ref.Kolton et al. (2009) where a crossover of the roughness exponent from to is obtained for different driving fields. It suggests that in this case the domain interface is no longer single-valued and one-dimensional. As a consequence, the domain interface belongs to a new universality class with intrinsic anomalous scaling and spatial multiscaling Zhou et al. (2009); Ramasco et al. (2000).
With Monte Carlo simulations, we have explored the nonsteady dynamics properties of a domain wall for the creep state under an alternating driving field. Since the phenomenological QEW equation contains little microscopic information, lattice models based on microscopic structures and interactions are considered. Taking the D DRFIM model as an example, two distinct growth stages of the domain interface are identified with both the creep correlation length and the pure roughness function . The small- regime corresponds to the universality class of the random depositions with the scaling exponents and , while the large- one belongs to the universality class of the QEW equation with and .
However, due to the dynamic effect of overhangs, the roughness exponent may significantly deviate from of the QEW equation at either higher temperatures, weaker quenched disorders, stronger driving fields or higher frequencies, but comparable with experiments Lemerle et al. (1998); Lee et al. (2009). The result indicates that the domain interface belongs to a new universality class with intrinsic anomalous scaling and spatial multiscaling.
In addition, as the temperature increases, the creep exponent measured from the stationary magnetic hysteresis loops changes from to , compatible with the experimental measurements Braun et al. (2005); Kleemann et al. (2007, 2006). The temperature-independent scaling relation is then observed with the coefficient . With this scaling relation, one can predict the creep exponent by only measuring and from the nonsteady dynamics.
Acknowledgements: This work was supported in part by the National Natural Science Foundation of China (under Grant Nos. 11205043, 11375149 and 11304072), the Zhejiang Provincial Natural Science Foundation (under Grants No. LQ12A05002), and the funds from Hangzhou City for supporting Hangzhou-City Quantum Information and Quantum Optics Innovation Research Team.
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