Numerical analysis of the Landau-Lifshitz-Gilbert equation with inertial effects

Ruggeri, Michele

doi:10.48550/arxiv.2103.09888

Cited by 3 publications

(5 citation statements)

References 26 publications

(80 reference statements)

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“…The realization of such short and intense magnetic fields have been suggested recently designing magnetic-field enhancing metamaterials 32 , by the use of vector laser beams 33 or thanks to ultrafast electronic switches 34 . Finally, we anticipate that our findings will be relevant to much of the recent works in the magnetism community aimed at the understanding of inertial spin dynamics [35][36][37][38][39][40][41][42][43][44][45][46][47][48] . defined by a set of three unit vectors {e r , e θ , e φ }.…”

Section: Discussionmentioning

confidence: 99%

Magnetization switching in the inertial regime

Neeraj,

Pancaldi,

Scalera

et al. 2021

Preprint

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We have numerically solved the Landau-Lifshitz-Gilbert (LLG) equation in its standard and inertial forms to study the magnetization switching dynamics in a 3d thin film ferromagnet. The dynamics is triggered by ultrashort magnetic field pulses of varying width and amplitude in the picosecond and Tesla range. We have compared the solutions of the two equations in terms of switching characteristic, speed and energy analysis. Both equations return qualitatively similar switching dynamics, characterized by regions of slower precessional behavior and faster ballistic motion. In case of inertial dynamics, ballistic switching is found in a 25% wider region in the parameter space given by the magnetic field amplitude and width. The energy analysis of the dynamics is qualitatively different for the standard and inertial LLG equations. In the latter case, an extra energy channel, interpreted as the kinetic energy of the system, is available. Such extra channel is responsible for a resonant energy absorption at THz frequencies, consistent with the occurence of spin nutation.

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Section: Discussionmentioning

confidence: 99%

Magnetization switching in the inertial regime

Neeraj,

Pancaldi,

Scalera

et al. 2021

Preprint

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“…According to the convergence theory of GMRES [25], the number of iterations is proportional to the condition number (20) in 1D or (21) in 3D, respectively. For smaller α and τ , less number of iterations is needed in GMRES.…”

Section: A Second-order Semi-implicit Finite Difference Schemementioning

confidence: 99%

“…1 when is set to be 1. The initial magnetization is chosen as the uniform state m 0 = e 1 and a magnetic pulse with high frequency H e = µ 0 M s F (t)e 2 is applied along the e 2 direction, where F (t) = 0.01 sin(2πf t)χ 0≤t≤2.0×10 −12 and f = 500GHz [21]. Furthermore, following the study of hysteresis loops, we simulate magnetization dynamics using the iLLG equation with two sets of damping parameters and inertial parameters; see Fig.…”

Section: Micromagnetics Simulationsmentioning

confidence: 99%

“…Numerical methods for the iLLG equation are rarely studied. In [21], introducing V = ∂ t M, W = M × ∂ t M, and using the property M • V = 0 for any x and t, the author designed the tangent plane scheme (TPS) with first-order accuracy and angular momentum method (AMM) with second-order accuracy. In these schemes, M and V in TPS, or M and W in AMM are treated as unknown fields to be approximated and thus the number of unknowns is doubled.…”

Section: Introductionmentioning

confidence: 99%

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A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

Li¹,

Yang

Lan

et al. 2021

Preprint

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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 widely been used in micromagnetics simulations over decades. Recent theoretical and experimental advances show that the inertia of magnetization emerges at sub-picoseconds and contributes 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 like a hyperbolic system at the sub-picosecond scale while behaves like a parabolic system at larger timescales. Such hybrid behaviors impose additional difficulties on designing numerical methods for the iLLG equation. In this work, we propose a second-order semi-implicit scheme to solve the iLLG equation. The second temporal derivative of magnetization is approximated by the standard centered difference scheme and the first derivative is approximated by the midpoint scheme involving three time steps. The nonlinear terms are treated semi-implicitly using one-sided interpolation with the second-order accuracy. At each step, the unconditionally unique solvability of the unsymmetric linear system of equations in the proposed method is proved with a detailed discussion on the condition number. Numerically, the second-order accuracy in both time and space is verified. Using the proposed method, the inertial effect of ferromagnetics is observed in micromagnetics simulations at small timescales, in consistency with the hyperbolic property of the model at sub-picoseconds. For long time simulations, 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 at larger timescales.

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“…The experimental confirmation has been achieved only recently and detected nutation dynamics in the ∼ 1 THz range [24]. Despite this novel experimental evidence, and the significant theoretical progress to understand the microscopic origin of inertia [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], a complete picture of how different material parameters affect the nutation dynamics is still missing. * stefano.bonetti@fysik.su.se…”

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confidence: 99%

Inertial spin dynamics in epitaxial cobalt films

Unikandanunni¹,

Medapalli²,

Asa³

et al. 2021

Preprint

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We investigate the spin dynamics driven by terahertz magnetic fields in epitaxial thin films of cobalt in its three crystalline phases. The terahertz magnetic field generates a torque on the magnetization which causes it to precess for about 1 ps, with a sub-picosecond temporal lag from the driving force. Then, the magnetization undergoes natural damped THz oscillations at a frequency characteristic of the crystalline phase. We describe the experimental observations solving the inertial Landau-Lifshitz-Gilbert equation. Using the results from the relativistic theory of magnetic inertia, we find that the angular momentum relaxation time η is the only material parameter needed to describe all the experimental evidence. Our experiments suggest a proportionality between η and the strength of the magneto-crystalline anisotropy.

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Numerical analysis of the Landau-Lifshitz-Gilbert equation with inertial effects

Cited by 3 publications

References 26 publications

Magnetization switching in the inertial regime

Magnetization switching in the inertial regime

A second-order semi-implicit method for the inertial Landau-Lifshitz-Gilbert equation

Inertial spin dynamics in epitaxial cobalt films

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