1 Linearized frequency domain Landau -Lifshitz-Gilbert equation formulat ion

2025-04-30 0 0 377.48KB 4 页 10玖币

Linearized frequency domain Landau-Lifshitz-Gilbert equation

formulation

Zhuonan Lin1,2,3 and Vitaliy Lomakin1,2

1)Department of Electrical and Computer Engineering, University of California San Diego, La Jolla, California 92093, USA

2)Center for Memory and Recording Research, La Jolla, California 92093, USA

3)Materials Science and Engineering Program, University of California San Diego, La Jolla, California 92093, USA

(*Electronic mail: vlomakin@eng.ucsd.edu)

(Dated: 3 October 2022)

We present a general finite element linearized Landau-Lifshitz-Gilbert equation (LLGE) solver for magnetic systems under weak

time-harmonic excitation field. The linearized LLGE is obtained by assuming a small deviation around the equilibrium state of the

magnetic system. Inserting such expansion into LLGE and keeping only first order terms gives the linearized LLGE, which gives a

frequency domain solution for the complex magnetization amplitudes under an external time-harmonic applied field of a given

frequency. We solve the linear system with an iterative solver using generalized minimal residual method. We construct a

preconditioner matrix to effectively solve the linear system. The validity, effectiveness, speed, and scalability of the linear solver are

demonstrated via numerical examples.

I. INTRODUCTION

The magnetization dynamics is described by the Landau-

Lifshitz-Gilbert equation (LLGE). LLGE is a time dependent

non-linear equation, and it describes the magnetization

dynamics in both linear and non-linear regimes. Solving

LLGE can be computationally expensive as it requires

obtaining solutions at many time steps and it is often

numerically stiff, which may require either small time steps or

assisting linear solvers. Under weak dynamics excitations,

however, the magnetization dynamic response may be linear.

Examples of such systems are spin wave excitations under

weak time-dependent applied fields or the initial dynamics

under spin torque excitations1,2. In such cases, LLGE can be

linearized in terms of a weak magnetization deviation around

the equilibrium state. Such linearization has been used to

obtain solutions in terms of the eigenstate representations3 and

it can be used to obtain solutions even for non-linear problems

4–6.

Here, we present a linear frequency domain LLGE (FD-

LLGE) solver. The FD-LLGE solver provides a linearized

magnetization solution as a response to a dynamic excitation,

e.g., the applied field, which is a characterized by a given

frequency. Considering a given frequency allows formulating

a time independent linear equation for a complex

magnetization amplitude, which can, then, be used to provide

linear time domain solutions. The FD-LLGE solver is

developed based on the finite element based micromagnetic

simulator FastMag7. The linear system is solved effectively

with an iterative solver. The number of iterations is

significantly reduced using a linear preconditioner. Solving

the FD-LLGE is much more efficient and provides a more

physically insight than solving the original LLGE.

II. FORMULATION

The magnetization dynamics is described by the LLGE,

which is, in its implicit form, is written as

()

eff a





= −  + + 



m H H m

, (1)

where

is the normalized magnetization,



is the

gyromagnetic ratio,



is the damping constant. The term

0a a a

=+H H h

is the applied field containing a static

component

and a dynamic component

, which is a

time harmonic excitation at a circular frequency



, i.e., it can

be written as

Re{ ( ) }



=h h r

, (2)

where

is the complex amplitude of the exciting magnetic

field, which represents its magnitude and phase. The term

eff

in LLGE (1) is the effective field. The effective field is a

function of

, and it is composed of several components,

including the magnetostatic field

, exchange field

and anisotropy field (assumed uniaxial)

ˆˆ

( ) .

eff ms ex an

ms s

an K

= + + 

 

= 

−

=

=



H H H H m

H k m k

(3)

Here,

is the saturation magnetization,

is the exchange

constant,

is the anisotropy field, and

is the uniaxial

anisotropy axis direction. The effective field is linear in

and, therefore,

is the linear field operator that is

independent of

The LLGE (1) is non-linear in

due to the presence of the

cross products and it describes the magnetization dynamics in

a broad range of situations, including linear and non-linear

effects. In various cases, however, the general LLGE can be

linearized. Such a linearization is allowed when the

magnetization varies insignificantly around its equilibrium

state. Weak magnetization variations can be due to weak

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摘要：

1LinearizedfrequencydomainLandau-Lifshitz-GilbertequationformulationZhuonanLin1,2,3andVitaliyLomakin1,21)DepartmentofElectricalandComputerEngineering,UniversityofCaliforniaSanDiego,LaJolla,California92093,USA2)CenterforMemoryandRecordingResearch,LaJolla,California92093,USA3)MaterialsScienceandEngine...

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1 Linearized frequency domain Landau -Lifshitz-Gilbert equation formulat ion

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