Xue-Meng Nan(南雪萌), Chuan Qu(屈川), Peng-Bin He(賀鵬斌), and Zai-Dong Li(李再東),?

1Tianjin Key Laboratory of Quantum Optics and Intelligent Photonics,School of Science,Tianjin University of Technology,Tianjin 300384,China

2School of Physics and Electronics,Hunan University,Changsha 410082,China

Keywords: inertial effect,minimum magnetic field,ultrafast magnetism

1.Introduction

Understanding how magnetization moves and how it is manipulated on femtosecond time scales is of great significance for ultrafast and efficient data processing and storage applications.The magnetization dynamics in the writing process is described by the Landau-Lifhitz-Gilbert(LLG)equation, which correctly simulates the magnetization reversal on the nanosecond time scale.[1,2]The study of these fast magnetization reversals has great potential applications to the future development of high-speed information industry.[3]Until 20 years ago, it was believed that all relevant magnetization dynamics were included in this equation, and the optimization of storage devices was based on it alone.In recent years, the study of magnetization dynamics in the nanoscale systems has become very important.Hot topics include current-induced domain wall motion and the demagnetization effect of femtosecond laser pulses.[4-7]

However, the pioneering experiment of Bigotet al.in 1996 revealed the occurrence of spin dynamics on subpicosecond scales,[1,8]and Ciornai and F¨ahnle[9,10]et al., and other theorists[4]also pointed out that in the magnetization dynamics of very short time scale, the lack of inertial is questionable.[11]Since then, many studies have also begun to focus on the inertial effect in the dynamics of ultrafast magnetization,[12-14]which was revealed by the ultrafast optical techniques[15]in 2016.The existence of the inertial effect will give birth to the field of ultrafast magnetism,which could not be described by the LLG equation.The LLG equation only describes precession and relaxation,but does not include nutation.[4]Recently, the LLG equation was reformulated to include a term to obtain a physically correct inertial response that does not exist in the original formula.[16-22]With the discovery of ultrafast magnetism,[23]people’s interest in ultrafast magnetization on femtosecond scale has greatly increased.

In the future,the rapid growth of information will require rapid processing of information, which in principle depends on the ultrafast reversal of magnetization.The magnetization reversal is how to change the magnetization from one state to another,i.e.,how to turn the magnetic moment from bit“1”to bit“0”(here,the bit“1”represents that the magnetic moment is upward and the bit“0”represents that the magnetic moment is down).The driven reversal force can be a laser, a spinpolarized current,or a magnetic field.Many reversal schemes have been proposed and tested in fast magnetism.[24-29]Also,for the issue of a minimal reversal field the classical result is the famous Stoner-Wohlfarth limit.[30]However, in ultrafast magnetism, the research on the magnetization reversal field has not been well explored.In this paper, the inertial LLG equation is analyzed in detail, and simulated numerically by the fourth-order Runge-Kutta algorithm.The limit of the minimum field for the magnetization reversal under inertial effect is obtained[16,17]by a tedious calculation.Compared with the magnetization reversal field in fast magnetism,a smaller value of reversal field has the more advantages in ultrafast magnetism owing to the inertial effect.

2.Effective field form of the magnetic inertial effect

Until recently,with the discovery of ultrafast magnetization dynamics, the inertial effect has been added into the dimensionless LLG equation,[31-35]which takes the form

wheremis the normalized magnetization,m ≡M/Ms,Mis the localized magnetization, andMsis the saturation magnetization.In Eq.(1), ˙mrepresents the derivative with respect to time, and the time is measured in unitsγMswithγbeing the gyromagnetic ratio.The effective field?effincludes both external and internal fields, and its form is given below.The parameterαis the Gilbert damping factor.The last term on the right side in Eq.(1)denotes the inertial effect, in whichτis the angular momentum relaxation time,[12,36]and we found that in Refs.[9,33] the approximate estimation result of the theoretically derived inertial parameterτis about 10 fs-100 fs.In Ref.[1],the fundamental value ofτcollected experimentally was extracted from the resonance linewidth by approximately 10 ps.In Ref.[37],the estimated time-resolved magneto-optical measurements of Co thin filmsτvalue is approximately 1 ps.It is obvious that the first term on the right side of Eq.(1) denotes the magnetization precession and the second term represents the damping effect of precession.From Eq.(1)we have ¨m=-(1/ατ)(m× ˙m+m×(m×?eff))-α˙m/τ-| ˙m|2m.It is to say that under the action of the inertial effect,i.e.,the second derivative of magnetization ¨mon the femtosecond time scale,this will include the trend toward the direction ofm.That is exactly what causes magnetization nutation.In this short time scale, the precession motion of magnetization is superimposed with the rotating circuit,and the inertial effect becomes very important.For ultrafast switching,the state space of the possible paths is different from the one in the absence of inertial effect-it will produce nutation.It is pointed out that the existence of inertia-driving magnetization dynamics opens up a way to surpass the limit of precession for ultrafast magnetic switch.[38,39]By iterating Eq.(1)once,we have

where?′eff=?eff-ατ¨m.Equation(2)clearly shows that the inertia effect not only affects the precession of magnetization,but also acts as an effective damping.It also implies that the motion of magnetization can be driven by the inertial effect in some special conditions.

3.Inertial effect on the magnetization reversal field

For simplicity, we consider only the uniaxial anisotropy model, in which thezaxis is selected as the easy axis.Then,the effective field has the expression as follows,?eff=?+hiez,where?is the applied external field,hi=kmzdenotes the internal field due to the magnetic anisotropy.In the spherical coordinates{er,eθ,e?}, i.e.,m=(1,0,0), we can simplify Eq.(2)into the form The above equation clearly clarifies how the inertial parameterτdetermines the theoretical limit value of the magnetization reversal field with the different dampingα,and anisotropyk.

In Eq.(6), we can see that whenτ=0, which is the influence of inertia effect is not considered,the results are consistent with that of the mechanism of fast magnetization dynamics in Ref.[24],h1c=kα/(2√α1),which has a linear relationship with anisotropykand is directly proportional toα.Also, whenαis very large, whether it is ultrafast magnetism or fast magnetism,hcgradually tend to the same value and this value is close to the result of Stoner-Wohlfarth.[30]Theoretical analysis of Eq.(6),we can know that under the influence of the inertia effect, the relationship betweenhcand the anisotropic parameterkis no longer a simple linear relationship, and the relationship withαbecomes more complex.From the simple analysis, we can see that compared with the fast magnetization dynamics,the theoretical limit value of the magnetization reversal field in ultrafast magnetism has changed significantly.So does this change make sense, that is can we get a better magnetization reversal field in ultrafast magnetism than that of fast magnetism?

From the analytical results of Eq.(6), we can find that there is a critical value for the inertia parameter,i.e.,when the value of the external magnetic field generated is equal to the theoretical limit value of the magnetization reversal magnetic field under the fast magnetic mechanism, the corresponding angular momentum relaxation time

which is affected by the dampingαand anisotropy parameterkof the system.When the inertial parameter factorτis greater than this critical value,i.e.,τ>τc,the theoretical limit value of the magnetization reversal field under the ultrafast magnetic mechanism is greater than that of the fast magnetic mechanism,i.e.,hc>h1c.When the inertial parameter factor is less than this critical value, i.e.,τ<τc, we havehcτc/2, the situation is just the opposite.Also, whenτ=τc/2, we get the smallest limit value of magnetization reversal field under the ultrafast magnetic mechanism

this equation indicates that whenτ=τc/2, the limit valuehcof the magnetization inversion field is only related to Gilbert damping and anisotropy, which is illustrated in Fig.2.From Fig.2 we find that the smallest magnetization reversal fieldhcminhas a linear relationship with anisotropykandhcminis directly proportional toαfirstly and then inversely proportional.

Fig.1.The variation of the limit value of the magnetization reversal field hc depends on the inertia parameter τ.The parameters are α =0.2,0.3,and k=0.25,respectively.Inset: hc vs. τ in a small range.

Fig.2.The smallest limit value of the magnetization reversal field hcmin changes with damping α and the anisotropy k,in which(a)k=0.1,0.2,0.3,and(b)α =0.1,0.2,0.3,respectively.

We also find that the limit value of the magnetization reversal fieldhcand the inertia factorτare in a quadratic parabolic relationship under the condition|τ-τc/2|?2α1/kwith the help of Eq.(6).However, a larger reversal field will be generated and the limit value of the magnetization reversal fieldhcincreases linearly with the inertial factorτunder the the conditionτ ?τc/2+2α1/k.It is very similar to the case that the limit value of the magnetization reversal fieldhcincreases linearly with the anisotropykwith the absence of inertial effectτ=0.This result shows that under the ultrafast magnetic mechanism,the inertial effect profoundly affects the limit value of the magnetization reversal field.As can be seen from Eq.(2), the inertial effect can be considered as part of the modified effective field, which will affect the magnitude of the critical magnetic field value.Therefore,the relationship between the reversal field and the inertial parameter is nonmonotonic.In the experiment, the minimum magnetic field can be driven to flip the magnetization by adjusting the inertia effect under different materials and damping,which has great energy-saving significance in the field of information processing.

From Eq.(6) we see that the limit value of the magnetization reversal field increases linearly with the anisotropykin the fast magnetic mechanism,i.e.,τ=0.However,under the ultrafast magnetic mechanism, the dependence of limit value of the magnetization reversal field on the anisotropy of the system material has some characteristics,as shown in Fig.3.The critical valueτcchanges with the anisotropykis plotted in Fig.3(a).Also, in this ultrafast magnetic mechanism it satisfies thathc>h1c, asτ>τc, andhc

Fig.3.The graphical representation of the limit value of the magnetization reversal field hc with the anisotropy k.The parameters are α =0.5,(a) the variation of critical inertial value τc changes with the anisotropy k,(b)the inertial parameter factor is less,i.e.,τ =2,5,and is larger,i.e.,τ =26,30,respectively.

Under the influence of different inertial parameter factorsτ,the effect of damping on the limit value of magnetization reversal field is different.In order to fully observe the relationship between the magnetic moment reversal field and damping,we have adopted a wide range of damping parameter values because the damping value is theoretically unlimited.For comparison with the results in Ref.[24], we chose the largerα, as shown in Fig.4.The critical valueτcchanges with the damping factorα,as plotted in Fig.4(a).It can still be proved that when the inertial parameter factor isτless than critical valueτc,the limit value of the magnetization reversal field under the ultrafast magnetic mechanism is less than that of the fast magnetic mechanism, i.e.,hch1c.These results are shown by the four lines above and below the blue solid line in Fig.4(b), in which the blue solid line denotes the case ofτc.We also found when the inertial parameter factor is smaller, the magnetization reversal fieldhcincreases almost proportional with the damping factorα.However, when the inertial parameter is larger, thehcincreases first, then decreases, and then increases finally.Both cases eventually increase until it is close to the result of the Stoner-Wohlfarth.[3]

Fig.4.The illustration of the limit value of the magnetization reversal field hc with the damping α (theoretically, α can take a larger value).The parameters are k=0.3.(a)The variation of critical inertial value τc changes with damping α.(b) The inertial factor τ is less, i.e., τ =3, 6,and is larger,i.e.,τ =13,16,respectively.

4.Conclusion

In summary, we investigate the magnetization reversal field for the ultrafast (femtosecond time) processes, which is described by the inertial LLG equation.The rich properties of the limit value of the magnetization reversal field are discussed in detail.The most important findings are that the inertial effect greatly affects the magnetization reversal field.In the ultrafast magnetism, there is a critical valueτcfor the inertia parameter, which determines that the limit value of the magnetization reversal field,whenτ<τc,the minimum magnetic field theoretical limit value less than the fast magnetic mechanism will be generated.Whenτ=τc/2,we get the smallest limit value of the magnetization reversal field under the ultrafast magnetic mechanism.We expect that these results will have potential energy-saving significance in the field of information processing in the future.

Appendix A:Magnetization dynamics under inertial effect

By iterating Eq.(1)once,we have

Appendix B:Inertial LLG equation in spherical coordinates

In spherical coordinates,the second derivative of the magnetization can be written as

The equilibrium condition in our simulation is defined asθ=π/2,?=π/2, ˙θ=0, ˙?=0.Then,by using the fourth-order Runge-Kutta algorithm we numerically integrate Eqs.(B2)and (B3).The diagram of the numerical inertial dynamics has been shown in Fig.B1(a), and the schematic diagram in Fig.B1(b) can be used to better understand the inertial magnetization dynamics.

Fig.B1.The diagram of numerical inertial dynamics and schematic diagram.(a)Time dependence of θ for the inertial LLG equation.(b)The red solid dashed curve denotes schematically the magnetization nutation by the superposition effect of Gilbert damping and inertia.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No.61774001), the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, China (Grant No.KF202203), the NSF of Changsha City (Grant No.kq2208008), and the NSF of Hunan Province(Grant No.2023JJ30116).