Magnetization NormalizationMethods for

Landau-Lifshitz-GilbertDon Porter

Mike Donahue

Mathematical & Computational Sciences Division

Information Technology Laboratory

National Institute of Standards and Technology

Gaithersburg, Maryland

Introduction

• Exact solutions of LLG,

m =dm

dt=

γ

1 + α2m × Heff −

αγ

1 + α2m × Heff × m (1)

satisfy |m| = 1.

• Cartesian numerical solvers allow |m| 6= 1.

• Renormalization required to put solvers back on track.

• Different renormalization techniques influence results.

Example: Single Spin Undamped Precession

HeffTOP VIEW

SIDE VIEWPERSPECTIVE VIEW

m1 m2m1

m2

m1 m2

Heff

Exact solution

Euler step

Renormalized

m·Heff

Renormalization Artifiacts

• Traditional (naive) renormalization

– Keep direction

– Reset magnitude to 1.

– Nearest point on sphere.

• Produces error in m · Heff .

• Therefore, error in energy, dissipation rates, etc.

Single Spin, Euler Integration

-1.0

-0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.00.67

0.68

0.69

0.70

0.71

my

mz

Time (ns)

mz, rk2, ∆t = 10 psmz, ∆t = 0.5 psmz, ∆t = 1.0 ps

my

• Damping α = 0 ⇒ mz (= -energy) should be constant.

(rk2 is second order Runge-Kutta, others are 1st order Euler.)

Single Spin, Runge-Kutta Integration

-1.0

-0.5

0.0

0.5

1.0

0.0 1.0 2.0 3.0 4.0 5.00.707106

0.707108

0.707110

0.707112

mx

mz

Time (ns)

mz, rk4mz, rkf54

mx

• Similar (but smaller) errors. Time step = 10 ps.

(rk4 = 4th order; rkf54 = 5 + 4th order Runge-Kutta-Fehlberg.)

Micromagnetic Example: Instabilities

58 nm

30 n

m

Thickness = 2 nmPy material parametersSimulation cellsize = 2nm

Hpulse = e-200(t-1)2/9 x 1 mTHbias= 1 T

Damping α = 0.001

Renormalization Induced Instability

0.02594

0.02595

0.02596

0.02597

0.0 0.5 1.0 1.5 2.0-8

-4

0

4

8

Red

uced

mag

netiz

atio

n

Rel

ativ

e en

ergy

(J

x 10

-25 )

Time (ns)

CorrectEnergy

my, corner spin

• rkf54 method, variable stepsize.

Revised Example: Modified Normalization

HeffTOP VIEW

SIDE VIEWPERSPECTIVE VIEW

m1 m2m1

m2

m1 m2

Heff

Exact solution

Euler step

Renormalized

Revised Example: Modified Normalization

• Modified renormalization

– Adjust both direction and magnitude.

– Nearest point on “orbit of precession”.

– Generalized orbit: Nearest point on intersection of sphere and

plane through unnormalized value perpendicular to m1 × m2 .

– Generalized orbit accounts for non-zero damping and for depen-

dence of Heff on m.

• Greatly reduced errors.

Modified Normalization, Single Spin

-1.0

-0.5

0.0

0.5

1.0

0.0 1.0 2.0 3.0 4.0 5.00.707106

0.707108

0.707110

0.707112

mx

mz

Time (ns)

Euler, ∆t = 1 psrkf54, ∆t = 10 ps

• Revised normalization improves all integration techniques.

(Data points are subsampled.)

Modified Normalization, Stability

0.02594

0.02595

0.02596

0.02597

0.0 0.5 1.0 1.5 2.0-8

-4

0

4

8

Red

uced

mag

netiz

atio

n

Rel

ativ

e en

ergy

(J

x 10

-25 )

Time (ns)

Energymy, corner spin

• Revised normalization greatly reduces instability.

Revised Equation

m =γ

1 + α2m × Heff −

αγ

1 + α2m × Heff × m + u(|m| − 1)V (m) (2)

• u(·) is scalar weighting function, output from PID controller. Initially,

u(0) = 0.

• V (m) is vector in same direction as modified normalization.

• Exact solutions of (2) are same as exact solutions of (1).

• Correction term in the equation itself has advantages:

– More direct use by solvers with automatic step size control

– Multi-step solvers do not require resets at normalization points.

Modified LLG, Stability

0.02594

0.02595

0.02596

0.02597

0.0 0.5 1.0 1.5 2.0-8

-4

0

4

8

Red

uced

mag

netiz

atio

n

Rel

ativ

e en

ergy

(J

x 10

-25 )

Time (ns)

Energymy, corner spin

• No instability with modified LLG.

• Also fixes single spin precession (not shown).

Summary

• Cartesian solvers employ renormalization when solving LLG.

• Simple renormalization choice introduces artifacts.

– Energy calculation errors compared with analytical solution.

– Numerical instabilities in more complex problems.

• Modified renormalization techniques yield improved results

– Normalization to “orbit of precession”

– Modified equation that self-corrects to normalized values.