We described ferromagnetic film and bilayer films composed of two ferromagnetic layers coupled through antiferromagnetic interfacial interaction by classical Heisenberg model and simulated their magnetization state,ma...We described ferromagnetic film and bilayer films composed of two ferromagnetic layers coupled through antiferromagnetic interfacial interaction by classical Heisenberg model and simulated their magnetization state,magnetic permeability,and Faraday effect at zero and finite temperature by using the Landau–Lifshitz–Gilbert(LLG)equation.The results indicate that in a microwave field with positive circular polarization,the ferromagnetic film has one resonance peak while the bilayer film has two resonance peaks.However,the resonance peak disappears in ferromagnetic film,and only one resonance peak emerges in bilayer film in the negative circularly polarized microwave field.When the microwave field’s frequency exceeds the film’s resonance frequency,the Faraday rotation angle of the ferromagnetic film is the greatest,and it decreases when the thickness of the two halves of the bilayer is reduced.When the microwave field’s frequency remains constant,the Faraday rotation angle fluctuates with temperature in the same manner as spontaneous magnetization does.When a DC magnetic field is applied in the direction of the anisotropic axis of the film,the Faraday rotation angle varies with the DC magnetic field and shows a similar shape of the hysteresis loop.展开更多
Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component ma...Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component magnetization is solved analytically under the condition of H=nH_(k)(n=3,1 and 0).It is found that with an increase of H or a decrease of the initial polar angle of magnetization,the relaxation time decreases and the angular frequency of magnetization increases.For comparison,the analytical solution for H_(k)=0 is also given.When the magnetization becomes stable,the angular frequency is proportional to the total effective field acting on the magnetization.The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization,but also can be used as a standard model to test the numerical calculation of LLG equation.展开更多
Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to na...Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at nanosecond timescales,the results of the iLLG model are in nice agreements with those of the LLG model,in consistency with the parabolic feature of the iLLG model.展开更多
The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many ...The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many mathematical results in stochastic partial differential equations (SPDEs) cannot be applied. The aim of this work is to introduce the difference method to handle SPDEs and prove the existence of regular martingale solutions in dimension two. Some blow-up phenomena are presented, which are drastically different from the deterministic case. Finally, to yield correct thermal-equilibrium properties, Stratonovitch integral is used instead of Ito integral.展开更多
We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping paramet...We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.展开更多
With the development of spintronics,spin-transfer torque control of magnetic properties receives considerable attention.In this paper the Landau-Lifshitz-Gilbert equation including the torque term is used to investiga...With the development of spintronics,spin-transfer torque control of magnetic properties receives considerable attention.In this paper the Landau-Lifshitz-Gilbert equation including the torque term is used to investigate the magnetic moment dynamics in the free layer of the ferromagnet/non-magnetic/ferromagnet（FM1/N/FM2） structures.It is found that the reverse critical time τ_c decreases with the current increasing.The critical time τ_c as a function of current for the perpendicular and parallel easy magnetic axes are the same.The critical time τ_c increases with the damping factor α increasing.In the case of large current the influence of the damping factor α is smaller,but in the case of little torque the critical time τ_c increases greatly with the damping increasing.The direction of the magnetization in the fixed layer influences the critical time,when the angle between the magnetization and the z direction changes from 0.1π to 0.4π,the critical time τ_c decreases from 26.7 to 15.6.展开更多
The voltage controlled magnetic switching effect is verified experimentally. The Landau–Lifshitz–Gilbert（LLG）equation is used to study the voltage controlled magnetic switching. It is found that the initial values...The voltage controlled magnetic switching effect is verified experimentally. The Landau–Lifshitz–Gilbert（LLG）equation is used to study the voltage controlled magnetic switching. It is found that the initial values of magnetic moment components are critical for the switching effect, which should satisfy a definite condition. The external magnetic field which affects only the oscillation period should be comparable to the internal magnetic field. If the external magnetic field is too small, the switching effect will disappear. The precessions of mx and my are the best for the tilt angle of the external magnetic field θt = 0？, i.e., the field is perpendicular to the sample plane.展开更多
Based on the findings of skyrmion nature of stripes and the metastability of a state of an arbitrary number of skyrmions,precisely controlled manipulation of stripes of skyrmion number 1 in pre-designed structures and...Based on the findings of skyrmion nature of stripes and the metastability of a state of an arbitrary number of skyrmions,precisely controlled manipulation of stripes of skyrmion number 1 in pre-designed structures and mutual transformation between helical states and skyrmion crystals(Sk Xs)are demonstrated in chiral magnetic films.As a proof of the concept,we show how to use patterned magnetic fields and spin-transfer torques(STTs)to generate nematic and smectic stripe phases,as well as“UST”mosaic from three curved stripes.Cutting one stripe into many pieces and coalescing several skyrmions into one by various external fields are good ways to transform helical states and Sk Xs from each other.展开更多
基金the Research Program of Shenyang Institute of Science and Technology(Grant No.ZD-2024-05).
文摘We described ferromagnetic film and bilayer films composed of two ferromagnetic layers coupled through antiferromagnetic interfacial interaction by classical Heisenberg model and simulated their magnetization state,magnetic permeability,and Faraday effect at zero and finite temperature by using the Landau–Lifshitz–Gilbert(LLG)equation.The results indicate that in a microwave field with positive circular polarization,the ferromagnetic film has one resonance peak while the bilayer film has two resonance peaks.However,the resonance peak disappears in ferromagnetic film,and only one resonance peak emerges in bilayer film in the negative circularly polarized microwave field.When the microwave field’s frequency exceeds the film’s resonance frequency,the Faraday rotation angle of the ferromagnetic film is the greatest,and it decreases when the thickness of the two halves of the bilayer is reduced.When the microwave field’s frequency remains constant,the Faraday rotation angle fluctuates with temperature in the same manner as spontaneous magnetization does.When a DC magnetic field is applied in the direction of the anisotropic axis of the film,the Faraday rotation angle varies with the DC magnetic field and shows a similar shape of the hysteresis loop.
基金Project supported by the National Key R&D Program of China (Grant No.2021YFB3501300)the National Natural Science Foundation of China (Grant Nos.91963201 and 12174163)the 111 Project (Grant No.B20063)。
文摘Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component magnetization is solved analytically under the condition of H=nH_(k)(n=3,1 and 0).It is found that with an increase of H or a decrease of the initial polar angle of magnetization,the relaxation time decreases and the angular frequency of magnetization increases.For comparison,the analytical solution for H_(k)=0 is also given.When the magnetization becomes stable,the angular frequency is proportional to the total effective field acting on the magnetization.The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization,but also can be used as a standard model to test the numerical calculation of LLG equation.
基金P.Li is supported by the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX202711)L.Yang is supported by the Science and Technology Development Fund,Macao SAR(Grant No.0070/2019/A2)+4 种基金the National Natural Science Foundation of China(NSFC)(Grant No.11701598)J.Lan is supported by NSFC(Grant No.11904260)the Natural Science Foundation of Tianjin(Grant No.20JCQNJC02020)R.Du was supported by NSFC(Grant No.11501399)J.Chen is supported by NSFC(Grant No.11971021).
文摘Electron spins in magnetic materials have preferred orientations collectively and generate the macroscopic magnetization.Its dynamics spans over a wide range of timescales from femtosecond to picosecond,and then to nanosecond.The Landau-Lifshitz-Gilbert(LLG)equation has been widely used in micromagnetics simulations over decades.Recent theoretical and experimental advances have shown that the inertia of magnetization emerges at sub-picosecond timescales and contributes significantly to the ultrafast magnetization dynamics,which cannot be captured intrinsically by the LLG equation.Therefore,as a generalization,the inertial LLG(iLLG)equation is proposed to model the ultrafast magnetization dynamics.Mathematically,the LLG equation is a nonlinear system of parabolic type with(possible)degeneracy.However,the iLLG equation is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy,and exhibits more complicated structures.It behaves as a hyperbolic system at sub-picosecond timescales,while behaves as a parabolic system at larger timescales spanning from picosecond to nanosecond.Such hybrid behaviors impose additional difficulties on designing efficient numerical methods for the iLLG equation.In this work,we propose a second-order semiimplicit scheme to solve the iLLG equation.The second-order temporal derivative of magnetization is approximated by the standard centered difference scheme,and the first-order temporal derivative is approximated by the midpoint scheme involving three time steps.The nonlinear terms are treated semi-implicitly using one-sided interpolation with second-order accuracy.At each time step,the unconditionally unique solvability of the unsymmetric linear system is proved with detailed discussions on the condition number.Numerically,the second-order accuracy of the proposed method in both time and space is verified.At sub-picosecond timescales,the inertial effect of ferromagnetics is observed in micromagnetics simulations,in consistency with the hyperbolic property of the iLLG model;at nanosecond timescales,the results of the iLLG model are in nice agreements with those of the LLG model,in consistency with the parabolic feature of the iLLG model.
基金supported by National Natural Science Foundation of China (Grant No. 11001285)
文摘The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many mathematical results in stochastic partial differential equations (SPDEs) cannot be applied. The aim of this work is to introduce the difference method to handle SPDEs and prove the existence of regular martingale solutions in dimension two. Some blow-up phenomena are presented, which are drastically different from the deterministic case. Finally, to yield correct thermal-equilibrium properties, Stratonovitch integral is used instead of Ito integral.
基金supported by Australian Research Council Discovery Project (Grant No. DP170101060)National Natural Science Foundation of China (Grant No. 11201498)the China Scholarship Council (Grant No. 201606495010)
文摘We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.
文摘With the development of spintronics,spin-transfer torque control of magnetic properties receives considerable attention.In this paper the Landau-Lifshitz-Gilbert equation including the torque term is used to investigate the magnetic moment dynamics in the free layer of the ferromagnet/non-magnetic/ferromagnet（FM1/N/FM2） structures.It is found that the reverse critical time τ_c decreases with the current increasing.The critical time τ_c as a function of current for the perpendicular and parallel easy magnetic axes are the same.The critical time τ_c increases with the damping factor α increasing.In the case of large current the influence of the damping factor α is smaller,but in the case of little torque the critical time τ_c increases greatly with the damping increasing.The direction of the magnetization in the fixed layer influences the critical time,when the angle between the magnetization and the z direction changes from 0.1π to 0.4π,the critical time τ_c decreases from 26.7 to 15.6.
基金supported by the Advanced Research Plan of the Chinese Academy of Sciences(Grant No.QYZDY-SSW-JSC015)
文摘The voltage controlled magnetic switching effect is verified experimentally. The Landau–Lifshitz–Gilbert（LLG）equation is used to study the voltage controlled magnetic switching. It is found that the initial values of magnetic moment components are critical for the switching effect, which should satisfy a definite condition. The external magnetic field which affects only the oscillation period should be comparable to the internal magnetic field. If the external magnetic field is too small, the switching effect will disappear. The precessions of mx and my are the best for the tilt angle of the external magnetic field θt = 0？, i.e., the field is perpendicular to the sample plane.
基金supported by the National Key Research and Development Program of China(Grant Nos.2018YFB0407600,and 2020YFA0309600)the National Natural Science Foundation of China(Grant Nos.11974296+2 种基金11774296)Hong Kong Research Grant Council(Grant Nos.16301518,and 16301619)。
文摘Based on the findings of skyrmion nature of stripes and the metastability of a state of an arbitrary number of skyrmions,precisely controlled manipulation of stripes of skyrmion number 1 in pre-designed structures and mutual transformation between helical states and skyrmion crystals(Sk Xs)are demonstrated in chiral magnetic films.As a proof of the concept,we show how to use patterned magnetic fields and spin-transfer torques(STTs)to generate nematic and smectic stripe phases,as well as“UST”mosaic from three curved stripes.Cutting one stripe into many pieces and coalescing several skyrmions into one by various external fields are good ways to transform helical states and Sk Xs from each other.
