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 210093, China and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China
共Received 24 March 2006; revised manuscript received 6 February 2007; published 31 July 2007兲 The magnetic hysteresis and its area for two-dimensional nanomagnets with precessional magnetization reversal, driven by ac magnetic field of frequencyfand amplitudeH0, are investigated by numerically solving the Landau-Lifshitz-Gilbert equation. Irregular hysteresis loops are observed when f is low andH0is high, indicating the significant contribution of nonadiabatic precession to the magnetization reversal. The frequency dispersion of hysteresis areaA共f兲shows the double-peaked pattern with the low-fpeak caused by the preces- sional 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⬀H0f with exponents
= 0.60 and= 0.50, is observed in the low-frange limit. We present the one-parameter dynamic scaling on the low-fhysteresis dispersions over a broad range ofH0, 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 amplitudeH0.
DOI:10.1103/PhysRevB.76.014439 PACS number共s兲: 75.60.⫺d, 75.40.Gb, 78.20.Bh, 75.70.⫺i
I. INTRODUCTION
When a ferromagnet, which is a cooperatively interacting many-body system, is placed in an oscillating external per- turbation共hereafter we refer to an ac magnetic fieldHextwith frequencyf and amplitudeH0兲at a temperatureTwell below the system’s Curie point TC, the system generally cannot respond instantaneously to the perturbation. The dynamic de- lay gives rise to a magnetization共M兲-Hext hysteresis whose nonvanishing area represents the loss of magnetic energy in one circulation of the ac field. We understand that the dy- namics of magnetization reversal or the dynamic hysteresis not only is of great importance for technical applications such as developing memory storage devices but also repre- sents a typical example for understanding the magnetization reversal in many-body systems. One typical example to il- lustrate the significance of dynamic hysteresis are magnetic memory devices via the spin reversal mechanism. It is usu- ally 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 hyster- esis in response to different frequencies and amplitudes of Hext.
Many theories concerning this dynamic were developed in the last decades and most of them1–6 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⬀H0f is followed, whereAis the hysteresis loop area andandare 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 model3presents us with an analytical solu- tion 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 informa- tion, one may refer to the comprehensive review article of Chakrabarti and Acharyya7 and references therein.
More recently, the need for smaller and faster storage devices8as well as the interest in quantum computing9have brought the dynamics of nanoscale magnets 共nanomagnets兲 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.10–13 This type of preces- sional magnetization reversal was first discussed in the pio- neering work of Stoner and Wohlfarth.14 The time depen- dence of magnetizationMdriven by an effective fieldHef fis described by the Landau-Lifshitz-Gilbert共LLG兲equation:15
dM
dt = −␥M⫻Hef f+ ␣ Ms
M⫻dM
dt , 共1兲
where t is time, ␥ is the gyromagnetic ratio, and ␣ is the damping factor. The effective field Hef f is defined via the variational derivative of free energyEwith respect toM. The first term on right-hand side of Eq.共1兲takes into account the precession torque effect, which enables rotation ofMaround Hef fand 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 Hef f through the precession and eventually enforcesMto be aligned along Hef f. Thus, the equilibrium condition of the LLG equation is M⫻Hef f= 0. Interestingly, the torque term M⫻Hef f 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.16This du- ality makes the dynamics of spin response toHef f, thus the hysteresis dynamics, quite complicated.
So far, no much work on the dynamic hysteresis of nano- magnets with precessional magnetization reversal following
1098-0121/2007/76共1兲/014439共9兲 014439-1 ©2007 The American Physical Society
the LLG equation has been reported. Earlier works mainly concentrated on the microwave oscillation17 and unified ul- trafast magnetization dynamics.18–21The LLG equation pro- vides 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 disper- sion, namely, the hysteresis areaAas a function of f andH0, given the system parameters and temperatureT. The scalabil- ity of A共f,H0兲 was well demonstrated for the continuum model and Ising-like models where the spin reversal is acti- vated 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 re- versal 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 hyster- esis, 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共2D兲square lattice共5⫻5,x-y plane兲 with periodic boundary conditions applied. This lattice is employed as an approach to ultrathin film consisting of nanoparticles. The external field Hext is applied almost perpendicularly to the lattice, and the ratio of the three components of Hext 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: 兩M兩
=Ms, where Ms is the saturated value of M. In a three- dimensional coordinate system, magnetization M can be written as
M=Ms共sincosex+ sinsiney+ cosez兲, 共2兲 whereis the azimuthal angle between magnetizationMand effective fieldHef f, is the angle between the projection of Mon theex-eyplane and the directionex, whileex,ey, andez
are the three unit vectors in the coordinates. We set the initial values ofandas0= 7°,0= 45°, although such a setting is somehow arbitrary. In fact, different values ofandmay 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. 共1兲 numerically comes from effective field Hef f,22 which usually includes external fieldHext, dipole field Hdip, random thermal fieldHther, de- magnetizating fieldHdem, and anisotropy fieldHanis共we take the z axis as an easy axis兲, in addition to exchange field Hexch:
Hef f=Hext+Hexch+Hanis+Hdip+Hther. 共3兲 The first three components ofHef f can be written as22
Hext=H0cos共2ft兲, 共4兲
Hexch=2Ax⌬M
Ms2 , 共5兲
Hanis=
冉
0,0,2KMuMs2 z冊
, 共6兲whereAx is the stiffness constant,Ku is the strength of the magnetic anisotropy,⌬M is the magnetization difference of the nearest neighbors, andMzis thez-axis component ofM.
The dipole fieldHdiporiginates from the long-range inter- action of spins in nanomagnets. For the ith nanomagnet at sitei in the lattice, the dipole field is given by
Hdipi = −
兺
具j典
冉
Mrij3j− 3共Mjrij兲rijrij5
冊
, 共7兲whereMj is the magnetic moment of the jth nanomagnet at site jin the lattice and rijis the relative separation between siteiand its neighboring site j.具j典marks a summation over all sites within a circle with a truncated radiusRTcentered at sitei. In a strict sense, radiusRTshould be infinite because of the essence of long-range interaction. However, it was shown that a finite truncation atRT艋8 is already precise enough for a reliable numerical simulation.23 The last term ofHef f, i.e., Hther, takes into account the effect of nonzero temperatureT, whose Cartesian components are randomly chosen from a normal distribution with the variance chosen so that the sys- tem relaxes to the Boltzmann distribution at the equilibrium,24 specifically
具Htheri 共t兲Htherj 共t
⬘
兲典= 2KBT␣␥V0Ms␦ij␦共t−t
⬘
兲, i,j=x,y,z, 共8兲 wheretandt⬘
are times,KBis the Boltzmann constant,Vis the volume of an individual mesh, and0 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/Ms and effective field hef f
=Hef f/Hk, whereHk= 2Ku/Ms. The driving frequency is also redefined by f⬅f/fre, where freis the resonant frequency at h0= 2. We typically use a time step⌬t= 0.005/f in units of 共␥Hk兲−1, and much extended simulations based on a smaller time step ⌬t= 0.001/f indicate no distinct differences be- tween them.
Assuming that the lattice is placed in a highhextand a low T, we calculate the dynamic hysteresis over a wide 共h0,f兲 range ofh0= 0.02– 20 andf= 10−4– 50. Our simulation is not performed for the purpose of a particular experimental veri- fication, but rather for a deep study on magnetization rever- sal controlled by the precession and energy dissipation based on the LLG equation. Hence, for the purpose of simplifica- tion, we set parameters a共lattice constant兲,0, andV all to 1.0 andHK= 5000. The correspondence between the chosen
magnitudes here for simulations and the realistic magnitudes forh0andf 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 fieldhther= 0.01, corresponding to relatively low thermal fluctuations with respect to h0共0.02⬍h0⬍20兲. We are particularly interested in the effect of damping factor ␣ on the dynamic hysteresis, which is allowed for a large range under different experimental envi- ronments. 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 byhdipwhich is long ranged.25,26In this paper we focus on the dynamic hysteresis in low driving frequencyf⬍1 and high external fieldh0⬎1, under which the simulated hyster- esis is of central symmetry. Therefore, among all the compo- nents ofhef f,hextis the dominant field to drive the magneti- zation reverse, namely, the contribution of other components includinghdipis 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 dy- namic 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 highh0do not show significant difference from those obtained in the 5⫻5 lattice, while such a large lattice calculation seems to require a huge computational ca- pacity.
III. RESULTS AND DISCUSSION A. Shape evolution of hysteresis loop
With the parameters given above, we calculate the hyster- esis loops at different f andh0. Some typical loops are plot- ted in Fig.1, wheremzis thezcomponent ofmandhzis the z component of hext. Figures 1共a兲–1共c兲 show the loops ob- tained at three different values ofh0: 0.02, 1.0, and 2.0, re- spectively, with damping factor ␣= 10 and frequency f
= 0.05. It is clearly shown that there exists a minimal field in order to fully reversemto form a well-saturated loop. Ifh0is smaller than such a critical value, the loop loses its symmetry around the origin and shows a bias, as seen in Fig. 1共a兲. Whenh0increases to an intermediate value共h0⬃1兲, the loop becomes saturated and exhibits a well-symmetrical shape, as shown in Fig. 1共b兲. Further increasing of h0 generates the irregular hysteresis consisting of a symmetrical loop around lowhzrange and irregular fluctuations in the high hz range, as shown in Fig. 1共c兲 with h0= 2, which gradually evolves
into the shape similar to that shown in Fig.1共d兲or1共h兲under the even higher h0. It is easily understood that the well- saturated and symmetrical loop is ascribed to the precession ofmwith the steady statemz= 1 ormz= −1 in response to the variation of hz, while the fluctuations correspond to the ul- trafast precessional state where the trajectory of m around hef fis somewhat chaotic.16,27This irregular phenomenon im- plies 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 共hext= 0.5兲 and a fixed large field 共hext= 5兲. One mode finally tends toward a fixed point mz
= 1共or −1兲because of the effect of the damping term共here- after called the adiabatic mode, square dots in Fig.2兲, and the other would never converge to a fixed direction but show back and forth transfers betweenmz= 1 andmz= −1 共hence- forth called the nonadiabatic mode, cycle dots in Fig. 2兲.
(b)
(d) (a)
(c)
(e)
(g)
(h)
(i) (f)
FIG. 1. 共Color online兲 Hysteresis loops with different sets of external parameters 共h0,f,␣兲. 关共a兲–共c兲兴 ␣= 10, f= 0.05, h0
= 0.02, 1.0, 2.0.关共d兲–共f兲兴␣= 10,h0= 2, f= 0.025, 0.25, 1.5. 关共g兲–共i兲兴 f= 0.05,h0= 2.0,␣= 0.01, 1.0, 100.
FIG. 2. 共Color online兲 Two modes of the magnetization re- sponse: the adiabatic mode and nonadiabatic mode. Square: The adiabatic mode exists under a fixed 共dc兲 low external field hext
= 0.5. Circle: The nonadiabatic mode appears under a fixed 共dc兲 large fieldhext= 5.
Figure1共c兲 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共d兲 and 1共h兲 whose spatial average of mz
slightly fluctuates around thexaxis.
In order to understand the origin of the irregular loop, we consider the stability of the state兩mz兩= 1. By vector multiply- ing both sides of Eq.共1兲by mand remembering the vector identity a⫻共b⫻c兲=b共a·c兲−c共a·b兲, and observing that m·共dm/dt兲= 0, one obtains
m⫻dm
dt = −m⫻共m⫻hef f兲−␣ms
dm
dt . 共9兲 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 = − 1
1 +␣2m⫻hef f−
␣
共1 +␣2兲m⫻共m⫻hef f兲. 共10兲 For the purpose of simplification,hef f only includes field componenthz applied along thez axis; thus, we can derive the following equation:28
dmz
dt = − ␣
1 +␣2hz共1 −mz2兲. 共11兲 It is clear that magnetization reversal from the state mz
= 1 to the statemz= −1 共or vice versa兲 is driven exclusively by damping. It seems from Eq. 共11兲 that no magnetization reversal is possible if the magnetization is in equilibrium state 兩mz兩= 1. However, in a realistic case, ifhef f is not ex- actly parallel to thezaxis关the ratio betweenhzandhx共orhy兲 is ⬃100 here兴, then the in-plane 共x-y plane兲 field would changemzby the torque. The changes of mx andmywould also have a significant effect onmz. On the other hand, due to thermal effects, the magnetization m randomly fluctuates around the above equilibrium state. By the mechanisms dis- cussed above, the equilibrium state兩mz兩= 1 is possibly broken and the magnetization never converges to a final fixed direc- tion, as seen in Fig.2. Taking −dmz/dt as a measure of the instability of state 兩mz兩= 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 共nonadiabatic mode兲 is greater when the external field is higher, which explains why irregular fluctuations appear whenhzis high.
Given the value ofh0, we investigate the patterns of hys- teresis at different f. Three examples are presented in Figs.
1共d兲–1共f兲with␣= 10,h0= 2, and f= 0.025, 0.25, and 1.5, re- spectively, noting that h0= 2 is an intermediate value. It is seen that the well-saturated but irregular loop obtained at f
= 2.5⫻10−3 关Fig.1共d兲兴evolves into a fat rhombic loop pat- tern atf= 0.25关Fig.1共e兲兴. Asf= 1.5, the loop loses its central symmetry around the origin and becomes positively biased 关Fig.1共f兲兴. What should be pointed out is that a lower fre- quency favors the occurrence of irregular loops关Fig.1共d兲兴, owing to the nonadiabatic mode. It is not sufficient to acti- vate 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 con- venient to investigate the irregular loops at low frequency.
Secondly, the frequency dependence of hysteresis can be ex- plained in the classical framework of magnetization reversal.
Keeping in mind the simple assumption that the magnetiza- tion 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
andf−1, whereas timeis essentially determined byh0and T. When f is low, Ⰶf−1, the rate of precession for the re- versal can keep up with the variance rate ofhext. Therefore, the loop in Fig. 1共d兲 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共e兲, because the adiabatic mode would be close to a resonating state with hext. If Ⰷf−1, the symmetry of the hysteresis loop is broken since the magnetization reversal by precession cannot catch up with the variation of hext. Here, we may define an order parameterQ=f养Mzdtto 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共g兲–1共i兲, with f= 0.05, h0= 2, and ␣
= 0.01, 1.0, and 100, respectively. It is observed that either weak damping共␣= 0.01兲or strong damping共␣= 100兲would result in the formation of fat and well-saturated foursquare loop, as shown in Figs.1共g兲 and1共i兲. No much difference between the two loops in terms of loop shape can be identi- fied although␣is very different from each other. In fact, our simulation shows that for a smaller␣, the largerh0is needed to switch the magnetization reversal for 10−3⬍␣⬍1, while for a smaller␣, the smallerh0is needed for 1⬍␣⬍500. In such cases, no essential contribution from the nonadiabatic mode to the hysteresis is observed. However, as␣ takes an intermediate value共␣= 1.0兲, the hysteresis becomes irregular fluctuations and we observe a significant effect of the nona- diabatic mode. This peculiar effect is argued to ascribe to the competition between the adiabatic and nonadiabatic modes.
One can easily understand from Eq.共11兲thatdmz/dtis not a monotonic function of factor ␣ and the maximal value of dmz/dt appears at the intermediate value ␣⬃1.0. Because dmz/dt is a measure of the instability of state 兩mz兩= 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 hys- teresis may originate from both the adiabatic and nonadia- batic modes driven by fieldhext. However, the fluctuations of the nonadiabatic magnetization reversal can hardly form loops. Given the values ofh0,␣, andT, hysteresis areaAas a function of f, i.e., the hysteresis dispersion, will reach its maximal value once the steady precessional mode resonates
withhext. Therefore, we may expect a single-peaked hyster- esis dispersionA共f兲. We define the hysteresis areaA as the work done in one cycle:
A=
冖
mzdhz. 共12兲Figure 3共a兲 presents the calculated curves A共f兲 at several values ofh0 as␣= 10. Indeed, we observe thatA共f兲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-fside is observed. We name the low- f 共left兲one peak-I and the right one peak-II. Although such a double-peak dispersion was reported earlier,32 the mecha- nism responsible for the present double-peak dispersion is different. Peak-I here, in fact, corresponds to the competition between the precession andhext, as mentioned above, and it is time independent, as evidenced in Fig.3共b兲where several A共f兲curves were calculated from cycleN= 50, 100, and 400 withNthe 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共b兲. 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 fieldhdipin our simulation. Its magnitude can be as large as 10−2h0–h0.
Consequently, quasichaotic oscillations ofA共f兲 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共c兲. In the sections presented below, we shall focus onA共f兲in the low-f range共peak-I兲, and detailed discussion on the chaotic behaviors of A共f兲 in the high-f range共peak-II兲will be reported elsewhere.33
In Fig.3, one notes that peak-I moves to a higher f with increasingh0. 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 increasingT, which is understandable because shorter characteristic time and lower coercivity are expected at higherT.
C. Power-law scaling of hysteresis area
The power-law scaling behavior of the hysteresis area as a function of f and h0 has been extensively studied for the continuum model and Ising-like models.1,34Also, quite a lot of experimental data are accumulated to support the power- law 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-h0 range at lowT, so that the generated hysteresis is well satu- rated, namely, order parameter兩Q兩⬍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 areaA is 4h0 becausems= 1. It is found thatAscales with h0 by
A⬀h0, 共13兲
where= 0.60± 0.05. Figure4共a兲shows two examples of the scaling for ␣= 5 and f= 0.25 as well as ␣= 10 and f= 0.5.
Exponentis, as far as we can tell, independent off,T, and
␣. A similar value was found for the continuum 共⌽2兲3 model.34Furthermore, we normalizeAby 4h0and present all of the data within the low-f range in Fig. 4共b兲. Except the data for lowh0共h0⬍1兲, we observe
A⬀h0f, 共14兲 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 forA共f,h0兲also works for the nanomag- netic model with precessional magnetization reversal, al- though exponentsandmay be different from those origi- nating from other reversal mechanisms or from other models, such as= 2 / 3 and= 1 / 3 for the共⌽2兲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 h0, area A decays exponentially with increasing f in this frequency range, but the frequency exponent cannot be reliably evalu- ated when peak-II exists. If we exclude the dipole interaction fromhef f, peak-II disappears and the power-law behavior for A共f,h0兲 over the high-frequency range 共f⬎fre兲 appears, as shown in Fig.5. AreaArapidly increases withh0and decays with f, as indicated by the two exponents in Eq. 共14兲: FIG. 3. 共Color online兲 Frequency dispersion of the loop area
A共f兲 with ␣= 10. 共a兲 Two resonant frequencies are observed. 共b兲 Temporal evolution of the double-peak dispersion共h0= 2兲.共c兲Two peaks appear for considering the dipole interaction. Peak-II would disappear when there are no dipole interactions.
= 8.2± 0.5 and= −7.2± 0.5. The absolute values of both ex- ponents are markedly larger than the values from共⌽2兲2and 共⌽2兲3theories where= 2 and= −1. Therefore, the magne- tization response toh0 andf under the precessional magne- tization 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 h0 the hysteresis dispersion originating from the
precessional mode of magnetization reversal also exhibits a single-peak 共peak-I兲 pattern if no transverse field effect which contributes to peak-II of the dispersion curve is in- cluded. The peak pattern is basically due to the resonation of precessional magnetization reversal in response to the ac field mext. 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 timeas an average of the distri- bution. 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 scaling36 on the hysteresis dispersion, as we did earlier on both the continuum 共⌽2兲2 model and Ising-like models.37,38
Following our earlier work,37 we define several scaling parameters below:
␥= log10共f0兲, Sn共h0兲=
冕
−⬁+⬁
␥nA共␥,h0兲d␥, n= 0,1,2, . . . ,
␥n共h0兲=Sn共h0兲/S0共h0兲,
n2共h0兲=S2共h0兲S0共h0兲/S12共h0兲, 共15兲 where␥ is the modified frequency, 0 is the time constant chosen arbitrarily共2⫻10−4is used兲,Snis thenth moment of the hysteresis dispersion, ␥n is the nth characteristic fre- quency, andn2is the scaling factor. These scaling parameters are mathematically definable, because integralSnconverges to finite value as long asn is finite. On the other hand, the relative uncertainties using the data overf= 5⫻10−4– 50 in- stead of −⬁⬍f⬍⬁ to evaluate these parameters is less than 0.01. The evaluated data represent the averaging over 100 cycles ofhextand a longer measuring time shows a tiny difference.
Figure6presents these scaling parameters as functions of h0with different damping factors. A perfect linearity is ob- tained apart from the cases where h0 is very small 共h0
⬍0.02兲 and very large 共h0⬎20兲. Parameter ␥1 shows a gradual growth with increasingh0; however, the scaling fac- torn2remains unchanged within the calculation uncertainty, which is almost equal to 1. The same conclusions can be obtained for other temperatures. The independence ofn2 on h0 over a wide range indicates the possibility that all the dispersion curves at different h0 can be scaled by a one- parameter scaling function. However, in the strict sense, a tiny deviation of n2 from 1 can be observed and is more evident in the high-field region or with an intermediate damping factor共␣= 1兲, because for this case the nonadiabatic precessional motion becomes significant with respect to the cases of ␣Ⰷ1 and ␣Ⰶ1. Consequently, the validity of the one-parameter scaling function may not be as good as that for the single-loop hysteresis where factorn2 is very close to 1.
Construction of such a one-parameter scaling function is based on the assumption that there exists a unique character- FIG. 4.共Color online兲Scaling of the loop areaAwithin the low-
f range by a power law.共a兲 A as a function of the amplitudeh0, where the exponent= 0.6.共b兲A as a function ofh0and f, where the exponents= 0.6 and= 0.5 are found to be independent of␣ andT. The scaling is valid for intermediateh0and low f.
FIG. 5. 共Color online兲 Scaling of loop area A over the high-f range by a power law. The exponents ofAas a function ofh0andf are= 8.2 and= −7.2, respectively, which are both independent of
␣andT. Note that theyaxis is the normalized areaAdivided 4h0 andf decreases with the increasing abscissa value.
istic timeto scale a resonant precession for a given h0. If the scaling behavior is approved, this assumption becomes applicable for the nanomagnetic model, or at least for the LLG model. The scaling hypothesis36 suggests that the evo- lution of the hysteresis dispersionA共f,h0兲 can be written as the following form, as long as characteristic time 1 is unique:
W共兲=共1/0兲dA共␥,h0兲,
1−d⬃h0, 共16a兲 whered 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 scal- ing function can be defined as
W共兲=1/0A共␥,h0兲,
= log10共f1兲, 共16b兲 with
1=0⫻10−␥1 共17兲 as the scaling variable共i.e., the inverse scaling frequency or effective characteristic time兲. Correspondingly, we can define the characteristic frequencyf1=0/1and rewrite Eq. 共16兲:
W共兲=f1−1A共␥,h0兲. 共18兲 The validity of the scaling function in Eq. 共18兲 for ␣
= 10 and ␣= 1 is exhibited in Figs. 7共a兲 and 7共b兲, respec-
tively. For␣= 10, all dispersion curves almost fall into the same curve, taking the numerical uncertainties, as long ash0 is not very small or very large. For too lowh0共⬍0.2兲and too highh0共⬎8兲, the data show slight deviations from the scaling curve, while over the whole h0 range, the peak position is nearly immovable. However, when ␣= 1, the intermediate value, where the nonadiabatic motion has the significant con- tribution to the dynamic hysteresis, the one-parameter scal- ing does not apply, as observed in Fig.7共b兲, 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 nona- diabatic magnetization reversal at highh0or intermediate␣. The existence of the nonadiabatic magnetization reversal ob- viously conflicts with the assumption that an adiabatic mag- netization reversal is used to construct the scaling function.
Further analysis is required to determine whether this con- flict 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 h0 in double-logarithmic scale when h0 is within the intermediate range共0.1⬍h0⬍1兲. According to the scaling hypothesis, time1 should bear an exponential rela- tionship toh0once 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␣ andT. The significant deviation of the data from the exponential relation occurs whenh0is very high. However, if the scaling function FIG. 6. 共Color online兲 Scaling variables S1, ␥1, and n2 as a
function of the amplitudeh0with different damping factors. FIG. 7. 共Color online兲 Scaling functionW共兲 as evaluated by scaling transform, Eqs.共17兲and共18兲, applied to all hysteresis dis- persion curvesA共f兲 for␣= 10 and␣= 1.0, respectively.
in Eq. 共18兲 applies, one must have a perfect inverse linear relationship between1 andh0because the scaling exponent that appeared in Eq.共16兲is 1共Refs.37–40兲:
1⬀h0−1. 共19兲 The error in the exponent is thought to come from the nonadiabatic mode. For the scaling function in Eq.共15兲, the as-generated fluctuations by this mode may have character- istic time different from1. To exclude this error, we adopt Eq. 共19兲 instead of Eq. 共17兲 to fit the data over the whole range of h0 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 ifh0reaches 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 nanomag- netic LLG model allows us to argue that the characteristic time for the precession spin reversal is unique, if the adia- batic mode is dominant over the nonadiabatic mode. In fact, if one excludes those chaotic fluctuations induced by the nonadiabatic mode, Eq. 共19兲 is followed and the one- parameter scaling works quite well.
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 sim- ply 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 applica- bility 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. Al-
though 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 re- versal 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 orienta- tions due to the external field at finite temperature. In spite of the different mechanisms responsible for the pattern of hys- teresis 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 pre- sented in the future.
IV. CONCLUSION
In conclusion, we have investigated the dynamic hyster- esis 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共low f, high h0兲 brings about some irregular hysteresis loops, FIG. 8. 共Color online兲 Characteristic time 1as a function of
amplitudeh0with different damping factors.
FIG. 9. 共Color online兲 Scaling functionW共兲 as evaluated by scaling transform, Eqs.共17兲and共19兲, applied to all hysteresis dis- persion curvesA共f兲 for␣= 10 and␣= 1.0, respectively.
which corresponds to a nonadiabatic precessional state in the magnetization reversal, and this behavior becomes more evi- dent for a higherh0or an intermediate damping factor␣. The frequency dispersion exhibits a double-peak pattern. The low-f peak could be understood by the spin precessional re- versal 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 hys- teresis area as a function of frequencyf and amplitudeh0in the low-f region:A⬀h0f. The one-parameter scaling analy-
sis indicates the existence of a unique characteristic time for spin precession, and this characteristic time is an inverse linear function of amplitudeh0over a wide range.
ACKNOWLEDGMENTS
This work was supported by the Natural Science Founda- tion of China 共Nos. 10674061, 50332020, and 10464039兲 and the National Key Projects for Basic Research of China 共Nos. 2002CB613303 and 2006CB921802兲.
*Corresponding author. [email protected]
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