Dynamics and scaling of low-frequency hysteresis loops in nanomagnets

Transcription

1 Dynamics and scaling of low-frequency hysteresis loops in nanomagnets Zhihuai Zhu, Yanjun Sun, Qi Zhang, and Jun-Ming Liu* Laboratory of Solid State Microstructures, Nanjing University, Nanjing , China and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang , China Received 24 March 2006; revised manuscript received 6 February 2007; published 31 July 2007 The magnetic hysteresis and its area for two-dimensional nanomagnets with precessional magnetization reversal, driven by ac magnetic field of frequency f and amplitude H 0, are investigated by numerically solving the Landau-Lifshitz-Gilbert equation. Irregular hysteresis loops are observed when f is low and H 0 is high, indicating the significant contribution of nonadiabatic precession to the magnetization reversal. The frequency dispersion of hysteresis area A f shows the double-peaked pattern with the low-f peak caused by the precessional magnetization reversal and the high-f peak originating from the quasichaotic oscillations of the spin precession. The power-law scaling behavior of the hysteresis dispersion, i.e., A H 0 f with exponents =0.60 and =0.50, is observed in the low-f range limit. We present the one-parameter dynamic scaling on the low-f hysteresis dispersions over a broad range of H 0, demonstrating the scalability of the hysteresis dispersion and thus the existence of the unique characteristic time for magnetization reversal process in nanomagnets, given the field amplitude H 0. DOI: /PhysRevB PACS number s : d, Gb, Bh, i I. INTRODUCTION When a ferromagnet, which is a cooperatively interacting many-body system, is placed in an oscillating external perturbation hereafter we refer to an ac magnetic field H ext with frequency f and amplitude H 0 at a temperature T well below the system s Curie point T C, the system generally cannot respond instantaneously to the perturbation. The dynamic delay gives rise to a magnetization M -H ext hysteresis whose nonvanishing area represents the loss of magnetic energy in one circulation of the ac field. We understand that the dynamics of magnetization reversal or the dynamic hysteresis not only is of great importance for technical applications such as developing memory storage devices but also represents a typical example for understanding the magnetization reversal in many-body systems. One typical example to illustrate the significance of dynamic hysteresis are magnetic memory devices via the spin reversal mechanism. It is usually believed that the time of spin reversal for a spin system, given the external field magnitude, is finite and follows a time-domain distribution. If the periodicity of the ac external field is even shorter than this time, the spin reversal may no longer be able to happen. This distribution of spin reversal time can be evaluated by investigating the dynamic hysteresis in response to different frequencies and amplitudes of H ext. Many theories concerning this dynamic were developed in the last decades and most of them 1 6 have based their discussions on two different models: 2,3 the continuum spin model and the Ising-like model. It was demonstrated that for ferromagnetic systems, a power-law scaling of A H 0 f is followed, where A is the hysteresis loop area and and are the scaling exponents. For the continuum spin model, =1/3 and =2/3 are often reported in low frequency and =2, = 1 in high frequency is identified, whereas the mean-field Ising model 3 presents us with an analytical solution with = =2/3. Although different values of the two exponents for various systems were reported, the power-law scaling is of general significance and has been confirmed extensively by a number of experiments. For more information, one may refer to the comprehensive review article of Chakrabarti and Acharyya 7 and references therein. More recently, the need for smaller and faster storage devices 8 as well as the interest in quantum computing 9 have brought the dynamics of nanoscale magnets nanomagnets into focus. Achieving these technological goals thus requires an understanding of the dynamic magnetization reversal at nanosecond time scale and the magnetization switching of single-domain nanoparticles through precessional modes to overcome the reversal time limit This type of precessional magnetization reversal was first discussed in the pioneering work of Stoner and Wohlfarth. 14 The time dependence of magnetization M driven by an effective field H eff is described by the Landau-Lifshitz-Gilbert LLG equation: 15 dm = M H eff + M dm dt M s dt, where t is time, is the gyromagnetic ratio, and is the damping factor. The effective field H eff is defined via the variational derivative of free energy E with respect to M. The first term on right-hand side of Eq. 1 takes into account the precession torque effect, which enables rotation of M around H eff and maintains a fixed angle between them in the absence of the second term. The second term is a phenomenological damping term, which makes M spiral toward H eff through the precession and eventually enforces M to be aligned along H eff. Thus, the equilibrium condition of the LLG equation is M H eff =0. Interestingly, the torque term M H eff plays a dual role in the magnetization process: either conserving the magnetic energy as the effective field does or dissipating the magnetic energy as the oscillation damping does. 16 This duality makes the dynamics of spin response to H eff, thus the hysteresis dynamics, quite complicated. So far, no much work on the dynamic hysteresis of nanomagnets with precessional magnetization reversal following /2007/76 1 / The American Physical Society

2 ZHU et al. the LLG equation has been reported. Earlier works mainly concentrated on the microwave oscillation 17 and unified ultrafast magnetization dynamics The LLG equation provides us a possibility to investigate the dynamic hysteresis in nanomagnets. In this paper, we shall examine the dynamic hysteresis and the scaling behaviors of the hysteresis dispersion, namely, the hysteresis area A as a function of f and H 0, given the system parameters and temperature T. The scalability of A f,h 0 was well demonstrated for the continuum model and Ising-like models where the spin reversal is activated by thermal spin flip mechanism instead of precessional process. It would be interesting to examine whether or not similar scaling applies to the spin precession dominated reversal in nanomagnets. This paper is organized as follows: In Sec. II, we present a brief description of the effective field associated with the LLG equation and the numeral simulation to be performed. The detailed results and discussions on the dynamic hysteresis, power-law scaling behavior, as well as the dynamic scaling analysis on the hysteresis dispersion will be given in Sec. III. We end with a brief conclusion in IV. II. MODEL AND NUMERICAL SIMULATION A. Model description Our simulation is based on a two-dimensional 2D square lattice 5 5, x-y plane with periodic boundary conditions applied. This lattice is employed as an approach to ultrathin film consisting of nanoparticles. The external field H ext is applied almost perpendicularly to the lattice, and the ratio of the three components of H ext is 0.01:0.01:1.00. The initial configuration is set by imposing all of the lattice sites with the identical magnetic moment in the same direction: M =M s, where M s is the saturated value of M. In a threedimensional coordinate system, magnetization M can be written as M = M s sin cos e x + sin sin e y + cos e z, where is the azimuthal angle between magnetization M and effective field H eff, is the angle between the projection of M on the e x -e y plane and the direction e x, while e x, e y, and e z are the three unit vectors in the coordinates. We set the initial values of and as 0 =7, 0 =45, although such a setting is somehow arbitrary. In fact, different values of and may be taken but no identifiable difference referring to the steady state hysteresis can be observed. The only difference is that it may take longer or shorter time to reach the steady state. The complexity of solving Eq. 1 numerically comes from effective field H eff, 22 which usually includes external field H ext, dipole field H dip, random thermal field H ther, demagnetizating field H dem, and anisotropy field H anis we take the z axis as an easy axis, in addition to exchange field H exch : H eff = H ext + H exch + H anis + H dip + H ther. The first three components of H eff can be written as 22 H ext = H 0 cos 2 ft, H exch = 2A x M M s 2, 5 H anis = 0,0, 2K um z M s 2, 6 where A x is the stiffness constant, K u is the strength of the magnetic anisotropy, M is the magnetization difference of the nearest neighbors, and M z is the z-axis component of M. The dipole field H dip originates from the long-range interaction of spins in nanomagnets. For the ith nanomagnet at site i in the lattice, the dipole field is given by i H dip = j M j r ij 3 3 M jr ij r ij 5, 7 where M j is the magnetic moment of the jth nanomagnet at site j in the lattice and r ij is the relative separation between site i and its neighboring site j. j marks a summation over all sites within a circle with a truncated radius R T centered at site i. In a strict sense, radius R T should be infinite because of the essence of long-range interaction. However, it was shown that a finite truncation at R T 8 is already precise enough for a reliable numerical simulation. 23 The last term of H eff, i.e., H ther, takes into account the effect of nonzero temperature T, whose Cartesian components are randomly chosen from a normal distribution with the variance chosen so that the system relaxes to the Boltzmann distribution at the equilibrium, 24 specifically i j H ther t H ther t = 2K BT ij t t, i, j = x,y,z, V 0 M s 8 where t and t are times, K B is the Boltzmann constant, V is the volume of an individual mesh, and 0 is the magnetic conductivity. B. Numerical simulation Given the 2D lattice and system parameters, the LLG equation is numerically solved using the fourth-order Runge- Kutta method. To simplify the parameters in terms of unitless parameters, we define m=m /M s and effective field h eff =H eff /H k, where H k =2K u /M s. The driving frequency is also redefined by f f / f re, where f re is the resonant frequency at h 0 =2. We typically use a time step t=0.005/ f in units of H k 1, and much extended simulations based on a smaller time step t=0.001/ f indicate no distinct differences between them. Assuming that the lattice is placed in a high h ext and a low T, we calculate the dynamic hysteresis over a wide h 0, f range of h 0 = and f = Our simulation is not performed for the purpose of a particular experimental verification, but rather for a deep study on magnetization reversal controlled by the precession and energy dissipation based on the LLG equation. Hence, for the purpose of simplification, we set parameters a lattice constant, 0, and V all to 1.0 and H K =5000. The correspondence between the chosen r ij

3 DYNAMICS AND SCALING OF LOW-FREQUENCY magnitudes here for simulations and the realistic magnitudes for h 0 and f is determined by these parameters. Fluctuations of these parameters within finite ranges do not change the qualitative features of the calculated dynamic hysteresis. Consequently, we set thermal field h ther =0.01, corresponding to relatively low thermal fluctuations with respect to h h We are particularly interested in the effect of damping factor on the dynamic hysteresis, which is allowed for a large range under different experimental environments. Because of our inability to obtain a reasonable and realistic approximation for, we run simulation over a wide range of. Therefore, the value of indicates the weight of the damping term compared to the torque term in the LLG equation. Moreover, we do not impose the value of given by earlier experiments because of other probably unrealistic parameters. In our calculation, the value of is changed from 0.01 to 100. In the simulations, the data of the initial 100 loops are discarded in order to exclude the effect of the initial configuration of the lattice. What should be mentioned here is the finite size effect of the lattice, noting that a small 5 5 lattice is used in our calculation. The size effect of the system is mainly brought up by h dip which is long ranged. 25,26 In this paper we focus on the dynamic hysteresis in low driving frequency f 1 and high external field h 0 1, under which the simulated hysteresis is of central symmetry. Therefore, among all the components of h eff, h ext is the dominant field to drive the magnetization reverse, namely, the contribution of other components including h dip is relatively small. In addition, we can obtain similar results even if we calculate a one-site lattice with an in-plane anisotropy. The reason why we take the dipole field in our simulation is to show its significant effect on the dynamic hysteresis in the high-f range, which will be seen in Sec. III B. In fact, we performed extensive calculation in a large lattice of 32 32, and the data on the dynamics in the range of low-f and high h 0 do not show significant difference from those obtained in the 5 5 lattice, while such a large lattice calculation seems to require a huge computational capacity. III. RESULTS AND DISCUSSION A. Shape evolution of hysteresis loop With the parameters given above, we calculate the hysteresis loops at different f and h 0. Some typical loops are plotted in Fig. 1, where m z is the z component of m and h z is the z component of h ext. Figures 1 a 1 c show the loops obtained at three different values of h 0 : 0.02, 1.0, and 2.0, respectively, with damping factor =10 and frequency f =0.05. It is clearly shown that there exists a minimal field in order to fully reverse m to form a well-saturated loop. If h 0 is smaller than such a critical value, the loop loses its symmetry around the origin and shows a bias, as seen in Fig. 1 a. When h 0 increases to an intermediate value h 0 1, the loop becomes saturated and exhibits a well-symmetrical shape, as shown in Fig. 1 b. Further increasing of h 0 generates the irregular hysteresis consisting of a symmetrical loop around low h z range and irregular fluctuations in the high h z range, as shown in Fig. 1 c with h 0 =2, which gradually evolves (a) (b) (c) (d) (e) (f) FIG. 1. Color online Hysteresis loops with different sets of external parameters h 0, f,. a c =10, f =0.05, h 0 =0.02,1.0,2.0. d f =10, h 0 =2, f =0.025,0.25,1.5. g i f =0.05, h 0 =2.0, =0.01,1.0,100. into the shape similar to that shown in Fig. 1 d or 1 h under the even higher h 0. It is easily understood that the wellsaturated and symmetrical loop is ascribed to the precession of m with the steady state m z =1 or m z = 1 in response to the variation of h z, while the fluctuations correspond to the ultrafast precessional state where the trajectory of m around h eff is somewhat chaotic. 16,27 This irregular phenomenon implies the coexistence of two types of precessional modes which require different threshold fields to activate. As shown in Fig. 2, the evolution of magnetization shows two modes under a fixed low field h ext =0.5 and a fixed large field h ext =5. One mode finally tends toward a fixed point m z =1 or 1 because of the effect of the damping term hereafter called the adiabatic mode, square dots in Fig. 2, and the other would never converge to a fixed direction but show back and forth transfers between m z =1 and m z = 1 henceforth called the nonadiabatic mode, cycle dots in Fig. 2. FIG. 2. Color online Two modes of the magnetization response: the adiabatic mode and nonadiabatic mode. Square: The adiabatic mode exists under a fixed dc low external field h ext =0.5. Circle: The nonadiabatic mode appears under a fixed dc large field h ext =5. (g) (h) (i)

4 ZHU et al. Figure 1 c shows the transition state of the hysteresis loop between these two precessional modes, and the irregular loops mainly controlled by the nonadiabatic mode can be seen in Figs. 1 d and 1 h whose spatial average of m z slightly fluctuates around the x axis. In order to understand the origin of the irregular loop, we consider the stability of the state m z =1. By vector multiplying both sides of Eq. 1 by m and remembering the vector identity a b c =b a c c a b, and observing that m dm/dt =0, one obtains m dm dt = m m h dm eff m s dt. 9 By substituting the latter equation in the right-hand side of the LLG equation, the equation can be appropriately recast to obtain the following expression: dm dt = m h eff 1+ 2 m m h eff. 10 For the purpose of simplification, h eff only includes field component h z applied along the z axis; thus, we can derive the following equation: 28 dm z dt = 1+ 2h z 1 m z It is clear that magnetization reversal from the state m z =1 to the state m z = 1 or vice versa is driven exclusively by damping. It seems from Eq. 11 that no magnetization reversal is possible if the magnetization is in equilibrium state m z =1. However, in a realistic case, if h eff is not exactly parallel to the z axis the ratio between h z and h x or h y is 100 here, then the in-plane x-y plane field would change m z by the torque. The changes of m x and m y would also have a significant effect on m z. On the other hand, due to thermal effects, the magnetization m randomly fluctuates around the above equilibrium state. By the mechanisms discussed above, the equilibrium state m z =1 is possibly broken and the magnetization never converges to a final fixed direction, as seen in Fig. 2. Taking dm z /dt as a measure of the instability of state m z =1, we understand that the higher external field, the more unstable this state. In other words, the contribution of the precession without a fixed end point nonadiabatic mode is greater when the external field is higher, which explains why irregular fluctuations appear when h z is high. Given the value of h 0, we investigate the patterns of hysteresis at different f. Three examples are presented in Figs. 1 d 1 f with =10, h 0 =2, and f =0.025, 0.25, and 1.5, respectively, noting that h 0 =2 is an intermediate value. It is seen that the well-saturated but irregular loop obtained at f = Fig. 1 d evolves into a fat rhombic loop pattern at f =0.25 Fig. 1 e.asf =1.5, the loop loses its central symmetry around the origin and becomes positively biased Fig. 1 f. What should be pointed out is that a lower frequency favors the occurrence of irregular loops Fig. 1 d, owing to the nonadiabatic mode. It is not sufficient to activate this mode only by applying a high field. The frequency f has to be low so that the frequency of the precession in nonadiabatic mode is much higher than f. Otherwise, the magnetization reversal via this nonadiabatic mode cannot be achieved on a timely basis. Therefore, it would be more convenient to investigate the irregular loops at low frequency. Secondly, the frequency dependence of hysteresis can be explained in the classical framework of magnetization reversal. Keeping in mind the simple assumption that the magnetization reversal can be characterized by a characteristic time for the adiabatic mode, one understands that the loop shape and area are determined by the relative relationship between and f 1, whereas time is essentially determined by h 0 and T. When f is low, f 1, the rate of precession for the reversal can keep up with the variance rate of h ext. Therefore, the loop in Fig. 1 d is symmetric around the origin with additional contribution from nonadiabatic mode. When f 1, the loop becomes quite fat and foursquare, as seen in Fig. 1 e, because the adiabatic mode would be close to a resonating state with h ext.if f 1, the symmetry of the hysteresis loop is broken since the magnetization reversal by precession cannot catch up with the variation of h ext. Here, we may define an order parameter Q= f M z dt to describe the symmetric property of the loop. A nonzero Q indicates a broken symmetry of the loop with respect to the origin, and detailed theoretical analysis on this symmetry breaking was given earlier. 7,29 32 We also observe the significant effect of damping factor on the dynamic hysteresis, as shown by several examples presented in Figs. 1 g 1 i, with f =0.05, h 0 =2, and = 0.01, 1.0, and 100, respectively. It is observed that either weak damping =0.01 or strong damping =100 would result in the formation of fat and well-saturated foursquare loop, as shown in Figs. 1 g and 1 i. No much difference between the two loops in terms of loop shape can be identified although is very different from each other. In fact, our simulation shows that for a smaller, the larger h 0 is needed to switch the magnetization reversal for , while for a smaller, the smaller h 0 is needed for In such cases, no essential contribution from the nonadiabatic mode to the hysteresis is observed. However, as takes an intermediate value =1.0, the hysteresis becomes irregular fluctuations and we observe a significant effect of the nonadiabatic mode. This peculiar effect is argued to ascribe to the competition between the adiabatic and nonadiabatic modes. One can easily understand from Eq. 11 that dm z /dt is not a monotonic function of factor and the maximal value of dm z /dt appears at the intermediate value 1.0. Because dm z /dt is a measure of the instability of state m z =1, the influence of damping factor on the shape of the hysteresis loop would be significant as 1.0. B. Hysteresis dispersion The above results allow us to argue that the dynamic hysteresis may originate from both the adiabatic and nonadiabatic modes driven by field h ext. However, the fluctuations of the nonadiabatic magnetization reversal can hardly form loops. Given the values of h 0,, and T, hysteresis area A as a function of f, i.e., the hysteresis dispersion, will reach its maximal value once the steady precessional mode resonates

5 DYNAMICS AND SCALING OF LOW-FREQUENCY Consequently, quasichaotic oscillations of A f in the high-f region with time become possible. If we consider a case with no dipole interactions, peak-ii will disappear, as seen in the inset plot of Fig. 3 c. In the sections presented below, we shall focus on A f in the low-f range peak-i, and detailed discussion on the chaotic behaviors of A f in the high-f range peak-ii will be reported elsewhere. 33 In Fig. 3, one notes that peak-i moves to a higher f with increasing h 0. The time independence of peak-i reflects that the hysteresis dynamics associated with peak-i is steady. A careful comparison of the dispersion curves at different T tells us that peak-i shifts rightward with increasing T, which is understandable because shorter characteristic time and lower coercivity are expected at higher T. FIG. 3. Color online Frequency dispersion of the loop area A f with =10. a Two resonant frequencies are observed. b Temporal evolution of the double-peak dispersion h 0 =2. c Two peaks appear for considering the dipole interaction. Peak-II would disappear when there are no dipole interactions. with h ext. Therefore, we may expect a single-peaked hysteresis dispersion A f. We define the hysteresis area A as the work done in one cycle: A = m z dh z. 12 Figure 3 a presents the calculated curves A f at several values of h 0 as =10. Indeed, we observe that A f reaches its maximal value and then decays gradually with increasing f from the low-f side. However, what is interesting is that a second peak in the high-f side is observed. We name the lowf left one peak-i and the right one peak-ii. Although such a double-peak dispersion was reported earlier, 32 the mechanism responsible for the present double-peak dispersion is different. Peak-I here, in fact, corresponds to the competition between the precession and h ext, as mentioned above, and it is time independent, as evidenced in Fig. 3 b where several A f curves were calculated from cycle N=50, 100, and 400 with N the cycle number from the beginning. Nevertheless, peak-ii appears in the high-f range and its position is time dependent, i.e., gradually shifts rightward to higher f with time evolution, eventually reaching a steady position, while the peak height remains constant, as shown in Fig. 3 b. It should be noted that this time-dependent behavior is intrinsic instead of numerical errors. The detailed calculation by including different interaction terms in sequence suggests that peak-ii originates from the effect of nonzero mean transverse field. This transverse field is induced by the in-plane component of dipole field h dip in our simulation. Its magnitude can be as large as 10 2 h 0 h 0. C. Power-law scaling of hysteresis area The power-law scaling behavior of the hysteresis area as a function of f and h 0 has been extensively studied for the continuum model and Ising-like models. 1,34 Also, quite a lot of experimental data are accumulated to support the powerlaw scaling. 6,35 It would be of interest to check this scaling for the present nanomagnet model. Here, we perform the power-law fitting on the data over the low-f and high-h 0 range at low T, so that the generated hysteresis is well saturated, namely, order parameter Q 0.05 and the trajectory of M is steady. It is mentioned again that the effect of the nonadiabatic mode on loop area is negligible. In our calculation, the upper limit of hysteresis area A is 4h 0 because m s =1. It is found that A scales with h 0 by A h 0, 13 where =0.60±0.05. Figure 4 a shows two examples of the scaling for =5 and f =0.25 as well as =10 and f =0.5. Exponent is, as far as we can tell, independent of f, T, and. A similar value was found for the continuum 2 3 model. 34 Furthermore, we normalize A by 4h 0 and present all of the data within the low-f range in Fig. 4 b. Except the data for low h 0 h 0 1, we observe A h 0 f, 14 where =0.60±0.05 and =0.50±0.05. Similarly, the two exponents are independent of T and too. Therefore, the power-law behavior for A f,h 0 also works for the nanomagnetic model with precessional magnetization reversal, although exponents and may be different from those originating from other reversal mechanisms or from other models, such as =2/3 and =1/3 for the 2 2 model. 1 Unfortunately, we cannot perform a reliable scaling on the dispersion above peak-i because of the appearance of peak- II. Usually, it is found that given the value of h 0, area A decays exponentially with increasing f in this frequency range, but the frequency exponent cannot be reliably evaluated when peak-ii exists. If we exclude the dipole interaction from h eff, peak-ii disappears and the power-law behavior for A f,h 0 over the high-frequency range f f re appears, as shown in Fig. 5. Area A rapidly increases with h 0 and decays with f, as indicated by the two exponents in Eq. 14 :

6 ZHU et al. precessional mode of magnetization reversal also exhibits a single-peak peak-i pattern if no transverse field effect which contributes to peak-ii of the dispersion curve is included. The peak pattern is basically due to the resonation of precessional magnetization reversal in response to the ac field m ext. This indicates that the precession of spin has a cycle time. The spins here are identical and therefore the distribution of cycle times in lattice should be homogeneous. We define a characteristic time as an average of the distribution. Since the distribution is monotonic, one can argue that this characteristic time should be unique. To check the uniqueness of this characteristic time, we can perform the one-variable dynamic scaling 36 on the hysteresis dispersion, as we did earlier on both the continuum 2 2 model and Ising-like models. 37,38 Following our earlier work, 37 we define several scaling parameters below: + = log 10 f 0, S n h 0 = n A,h 0 d, n = 0,1,2,..., FIG. 4. Color online Scaling of the loop area A within the lowf range by a power law. a A as a function of the amplitude h 0, where the exponent =0.6. b A as a function of h 0 and f, where the exponents =0.6 and =0.5 are found to be independent of and T. The scaling is valid for intermediate h 0 and low f. =8.2±0.5 and = 7.2±0.5. The absolute values of both exponents are markedly larger than the values from 2 2 and 2 3 theories where =2 and = 1. Therefore, the magnetization response to h 0 and f under the precessional magnetization reversal is more sensitive than the response under the conventionally dualistic spin flips. D. Dynamic scaling analysis From the above results, we understand that given the value of h 0 the hysteresis dispersion originating from the FIG. 5. Color online Scaling of loop area A over the high-f range by a power law. The exponents of A as a function of h 0 and f are =8.2 and = 7.2, respectively, which are both independent of and T. Note that the y axis is the normalized area A divided 4h 0 and f decreases with the increasing abscissa value. n h 0 = S n h 0 /S 0 h 0, n 2 h 0 = S 2 h 0 S 0 h 0 /S 2 1 h 0, 15 where is the modified frequency, 0 is the time constant chosen arbitrarily is used, S n is the nth moment of the hysteresis dispersion, n is the nth characteristic frequency, and n 2 is the scaling factor. These scaling parameters are mathematically definable, because integral S n converges to finite value as long as n is finite. On the other hand, the relative uncertainties using the data over f = instead of f to evaluate these parameters is less than The evaluated data represent the averaging over 100 cycles of h ext and a longer measuring time shows a tiny difference. Figure 6 presents these scaling parameters as functions of h 0 with different damping factors. A perfect linearity is obtained apart from the cases where h 0 is very small h and very large h Parameter 1 shows a gradual growth with increasing h 0 ; however, the scaling factor n 2 remains unchanged within the calculation uncertainty, which is almost equal to 1. The same conclusions can be obtained for other temperatures. The independence of n 2 on h 0 over a wide range indicates the possibility that all the dispersion curves at different h 0 can be scaled by a oneparameter scaling function. However, in the strict sense, a tiny deviation of n 2 from 1 can be observed and is more evident in the high-field region or with an intermediate damping factor =1, because for this case the nonadiabatic precessional motion becomes significant with respect to the cases of 1 and 1. Consequently, the validity of the one-parameter scaling function may not be as good as that for the single-loop hysteresis where factor n 2 is very close to 1. Construction of such a one-parameter scaling function is based on the assumption that there exists a unique character

7 DYNAMICS AND SCALING OF LOW-FREQUENCY FIG. 6. Color online Scaling variables S 1, 1, and n 2 as a function of the amplitude h 0 with different damping factors. istic time to scale a resonant precession for a given h 0.If the scaling behavior is approved, this assumption becomes applicable for the nanomagnetic model, or at least for the LLG model. The scaling hypothesis 36 suggests that the evolution of the hysteresis dispersion A f,h 0 can be written as the following form, as long as characteristic time 1 is unique: W = 1 / 0 d A,h 0, d 1 h 0, 16a where d is the dimension of variable 1. Because the time scale is only definable in one-dimensional space, i.e., the only possible scaling exponent for time is 1, hence the scaling function can be defined as W = 1 / 0 A,h 0, FIG. 7. Color online Scaling function W as evaluated by scaling transform, Eqs. 17 and 18, applied to all hysteresis dispersion curves A f for =10 and =1.0, respectively. = log 10 f 1, 16b with 1 = as the scaling variable i.e., the inverse scaling frequency or effective characteristic time. Correspondingly, we can define the characteristic frequency f 1 = 0 / 1 and rewrite Eq. 16 : W = f 1 1 A,h The validity of the scaling function in Eq. 18 for =10 and =1 is exhibited in Figs. 7 a and 7 b, respectively. For =10, all dispersion curves almost fall into the same curve, taking the numerical uncertainties, as long as h 0 is not very small or very large. For too low h and too high h 0 8, the data show slight deviations from the scaling curve, while over the whole h 0 range, the peak position is nearly immovable. However, when =1, the intermediate value, where the nonadiabatic motion has the significant contribution to the dynamic hysteresis, the one-parameter scaling does not apply, as observed in Fig. 7 b, which is similar to previous work. 38 To explicate the possible cause of the invalidity of the scaling, we consider the irregular behavior led by the nonadiabatic magnetization reversal at high h 0 or intermediate. The existence of the nonadiabatic magnetization reversal obviously conflicts with the assumption that an adiabatic magnetization reversal is used to construct the scaling function. Further analysis is required to determine whether this conflict leads to an error, or the dynamic scaling hypothesis in origin. In Fig. 8, we present the evaluated characteristic time 1 as a function of h 0 in double-logarithmic scale when h 0 is within the intermediate range 0.1 h 0 1. According to the scaling hypothesis, time 1 should bear an exponential relationship to h 0 once the dispersion reaches the scaling state or vice versa. The evaluated exponent turns out to be 0.82±0.03 in our simulation, independent of and T. The significant deviation of the data from the exponential relation occurs when h 0 is very high. However, if the scaling function

8 ZHU et al. FIG. 8. Color online Characteristic time 1 as a function of amplitude h 0 with different damping factors. in Eq. 18 applies, one must have a perfect inverse linear relationship between 1 and h 0 because the scaling exponent that appeared in Eq. 16 is 1 Refs : 1 h The error in the exponent is thought to come from the nonadiabatic mode. For the scaling function in Eq. 15, the as-generated fluctuations by this mode may have characteristic time different from 1. To exclude this error, we adopt Eq. 19 instead of Eq. 17 to fit the data over the whole range of h 0 and rescale all of the dispersion curves. The results are shown in Fig. 9. It is shown that for both large and small values, all the dispersion curves fall satisfactorily onto the same curves, even if h 0 reaches 10, demonstrating the one-parameter scalability of the dynamic hysteresis for the precessional magnetization reversal. On the other hand, the scalability of the hysteresis dispersion for the nanomagnetic LLG model allows us to argue that the characteristic time for the precession spin reversal is unique, if the adiabatic mode is dominant over the nonadiabatic mode. In fact, if one excludes those chaotic fluctuations induced by the nonadiabatic mode, Eq. 19 is followed and the oneparameter scaling works quite well. FIG. 9. Color online Scaling function W as evaluated by scaling transform, Eqs. 17 and 19, applied to all hysteresis dispersion curves A f for =10 and =1.0, respectively. E. Remarks and discussion The problem of magnetization reversal in nanomagnets is not so simple as described well by the LLG equation. In such sense, the above results apply only to those nanomagnetic systems in which the precessional mode is dominant, or simply say, to the LLG equation. Furthermore, one understands that we do not establish any correspondence between the present model lattice and realistic nanomagnetic system. Therefore, we cannot claim any realistic reasonability for the present calculation. So far, we also have not found reliable experimental data which can be used to testify the applicability of the power-law scaling of the hysteresis area and the dynamic scaling of the hysteresis dispersion. Even so, the present work represents a comprehensive approach to the spin reversal issue in nanomagnets where spin precession contributes to the magnetization reversal. Although spin precession is much shorter in time scale than the thermally activated spin flip, the above results suggest that those scaling behaviors proposed for conventional spin reversal mechanisms seem applicable to the precessional mode. We observe that there are two peaks in the hysteresis dispersion, one from the resonation of the precessional mode itself and the other from the transverse field due to the dipole field. In fact, for the continuum model and Ising-like models, it was reported that the hysteresis dispersion exhibits two peaks too, one from the spin flip resonation and the other from the spin fluctuations around the equilibrium orientations due to the external field at finite temperature. In spite of the different mechanisms responsible for the pattern of hysteresis dispersion, the LLG equation and the conventional models show similar scaling behaviors, a very interesting finding. As for the second peak in the hysteresis dispersion, the chaotic features of the dynamic response will be presented in the future. IV. CONCLUSION In conclusion, we have investigated the dynamic hysteresis of two-dimensional nanomagnets through precessional magnetization reversal by numerically solving the LLG equation. In studying the hysteresis shape evolution, it is shown that a certain combination of external parameters low f, high h 0 brings about some irregular hysteresis loops,

9 DYNAMICS AND SCALING OF LOW-FREQUENCY which corresponds to a nonadiabatic precessional state in the magnetization reversal, and this behavior becomes more evident for a higher h 0 or an intermediate damping factor. The frequency dispersion exhibits a double-peak pattern. The low-f peak could be understood by the spin precessional reversal with a synchronous response to the ac field and the high-f peak derives from a quasichaotic oscillation of the spin precession when the loop symmetry is lost. We have also demonstrated the power-law scaling behavior of the hysteresis area as a function of frequency f and amplitude h 0 in the low-f region: A h 0 f. The one-parameter scaling analysis indicates the existence of a unique characteristic time for spin precession, and this characteristic time is an inverse linear function of amplitude h 0 over a wide range. ACKNOWLEDGMENTS This work was supported by the Natural Science Foundation of China Nos , , and and the National Key Projects for Basic Research of China Nos. 2002CB and 2006CB *Corresponding author. liujm@nju.edu.cn 1 M. Rao, H. R. Krishnamurthy, and R. Pandit, Phys. Rev. B 42, ; J. Phys.: Condens. Matter 1, ; P. Jung, G. Gray, R. Roy, and P. Mandel, Phys. Rev. Lett. 68, D. Dhar and P. B. Thomas, J. Phys. A 25, ; Europhys. Lett. 21, ; P. B. Thomas and D. Dhar, J. Phys. A 26, P. Jung, G. Gray, R. Roy, and P. Mandel, Phys. Rev. Lett. 65, Q. Jiang, H.-N. Yang, and G.-C. Wang, Phys. Rev. B 52, S. Q. Yang and J. L. Erskine, Phys. Rev. B 72, C. Nistor, E. Faraggi, and J. L. Erskine, Phys. Rev. B 72, B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys. 71, T. R. Koehler and M. L. Williams, IEEE Trans. Magn. 31, M. N. Leuenberger and D. Loss, Nature London 410, C. Thirion, W. Wernsodorfer, and D. Mailly, Nat. Mater. 2, C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Bär, and Th. Rasing, Nature London 418, Schumacher, H. W. Chappert, C. Sousa, R. C. Freitas, and P. P. Miltat, Phys. Rev. Lett. 90, E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Statistical Physics, 3rd ed. Pergamon, Oxford, 1980, Pt Z. Li and S. Zhang, Phys. Rev. B 68, S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Palph, Nature London 425, B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge, Phys. Rev. Lett. 95, M. Vomir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Phys. Rev. Lett. 94, V. V. Kruglyak, A. Barman, R. J. Hicken, J. R. Childress, and J. A. Katine, Phys. Rev. B 71, R G. Ju, A. Vertikov, A. V. Nurmikko, C. Canady, Gang Xiao, R. F. C. Farrow, and A. Cebollada, Phys. Rev. B 57, R J. M. Deutsch, Trieu Mai, and Onuttom Narayan, Phys. Rev. E 71, B. L. Li, X. P. Liu, F. Fang, J. L. Zhu, and J.-M. Liu, Phys. Rev. B 73, W. F. Brown, Jr., Phys. Rev. 130, Y. Takagaki and K. H. Ploog, Phys. Rev. B 71, M. A. Kayali and W. M. Saslow, Phys. Rev. B 70, D. Berkov and N. Gorn, Phys. Rev. B 71, ; 72, G. Bertotti, I. Mayergoyz, C. Serpico, and M. Dimian, J. Appl. Phys. 93, G. Korniss, P. A. Rikvold, and M. A. Novotny, Phys. Rev. E 66, T. Tome and M. J. de Oliveira, Phys. Rev. A 41, M. Acharyya and B. K. Chakrabarti, Phys. Rev. B 52, H. Zhu, S. Dong, and J.-M. Liu, Phys. Rev. B 70, Z. H. Zhu and J.-M. Liu unpublished. 34 M. Rao and R. Pandit, Phys. Rev. B 43, Y.-L. He and G.-C. Wang, Phys. Rev. Lett. 70, A. Craevich and J. M. Sanchez, Phys. Rev. Lett. 47, , and references therein. 37 J.-M. Liu, H. L. W. Chan, C. L. Choy, and C. K. Ong, Phys. Rev. B 65, L. F. Wang and J.-M. Liu, J. Appl. Phys. 98, B. Pan, H. Yu, D. Wu, X. H. Zhou, and J.-M. Liu, Appl. Phys. Lett. 83, J. F. Scott, Ferroelectr. Rev. 1,