μMAG Standard Problem #5

Adapted from proposed problem published by Najafi, et al. [1].

Please send comments to donald.porter@nist.gov and join the μMAG discussion e-mail list for ongoing discussion.

Specification

This problem serves as a test of proper basic functioning of those micromagnetic solvers that include the effects of spin momentum transfer between in-plane polarized current and spatial patterns of magnetism. The movement of a magnetic vortex in response to a constant current flowing in the plane of magnetic material is computed. The final vortex position and its trajectory serve as results to be compared among different solvers. The problem is to be solved for a sequence of values for the degree of non-adiabaticity to detect any divergence of results among solvers over the range.

Geometry

A rectangular film of magnetic material with dimensions 100 nm × 100 nm × 10 nm, aligned with the x, y, z axes of a Cartesian coordinate system, with origin at the center of the film.

Material Parameters

Similar to Permalloy: Saturation magnetization: Ms = 8.0e5 A/m Exchange constant: A = 1.3e-11 J/m Anisotropy energy density: K = 0 J/m3 Gilbert damping parameter: α = 0.1 Gilbert gyromagnetic ratio: γ0 = 2.211e5 m/C [ = (rad/s)/(A/m)]

With these values, the exchange length is 5.7 nm.

Initial State

At each point xyz within the magnetic material, set the initial magnetization direction so that m = M / M s = f / f where f x , y , z = -y , x , R T and R= 10  nm. From that initial state, solve the Landau-Lifshitz-Gilbert dynamics d m d t = - γ 0 m × H + α m × d m d t to reach an equilibrium configuration. This magnetization vortex pattern is the initial state for all the remaining simulations.

Applied Current

To the initial magnetization vortex pattern, an in-plane current is applied in the x direction, with the product of the polarization rate P and current density J given by P J = 10 12 x ^ in units of A/m2, where x ^ is the unit vector in the positive x direction.

Boundary Conditions

Outside of the magnetic material, the applied current is unpolarized. This places conditions on the boundaries where current enters and leaves the magnetic material. At those surfaces care must be taken with the calculation such that the applied current delivers zero spin torque to the magnetization there.

Spin Torque Dynamics

Equation (3) of Thiaville, et al. [2] describes the magnetization dynamics including spin torque with an extended LLG equation.

d m d t = γ 0 H × m + α m × d m d t - u T m + β m × u T m (1)
where the new terms introduce a velocity vector (in m/s)
u T = - P g μ B J 2 e M s (2)
and a dimensionless quantity β representing the degree of non-adiabaticity.

In the definition of the velocity vector, we take g to be 2 and we have the physical constants Bohr magneton: μB = 9.274e-24 J/T Elementary charge: e = 1.6022e-19 C Given these constants and the material parameters already specified, the magnitude and direction of u T is determined by the applied current alone, independent of the value of β. For this problem we have u T = - 72.35 x ^ in units of m/s.

Equation (1) of Najafi, et al. [1] describes the magnetization dynamics with a different extended LLG equation adapted from Equation (11) of Zhang and Li [3],

d M d t = - γ 0 M × H + α M s M × d M d t - b M s 2 M × M × J M - ξ b M s M × J M. (3)

Here the new terms also introduce a dimensionless quantity ξ representing the degree of non-adiabaticity, as well as a factor

b = P μ B e M s 1 + ξ 2 (4)
with units of m3/C.

The functional dependence of b on ξ observed in (4) but not in (2) makes clear that equations (1) and (3) above do not predict the same variation of dynamics over a range of inputs describing different physical scenarios. They are equivalent for only the specific case of ξ=0 (and because of our assumption that g=2 ). As the degree of non-adiabaticity is varied, the two models diverge in their calculations.

Equations (1) and (3) may be reconciled into a common form

d m d t = - γ 0 m × H + α m × d m d t - u m × m x × m + ξ u m × m x (5)
where we have also simplified the expression using knowledge of the direction of the applied current. The same equation can be expressed in explicit Landau-Lifshitz form as
1 + α 2 d m d t = - γ 0 m × H + α γ 0 m × H × m - u 1 + α ξ m × m x × m + u ξ - α m × m x . (6)

In order to contribute a solution to this standard problem, a solver needs to be able to compute solutions of (5) [equivalently (6)] for at least some values of u and ξ under control of user inputs.

Problem Sets

The originally published standard problem from [1] explicitly demands that equation (3) be solved, and it sets the value of ξ =0.05. Translated into the common form, we seek solutions of (5) for ξ =0.05 and u = - b J = u T 1 + ξ 2 = - 72.17 m/s. If your solver permits direct entry of the quantities u and ξ, this is a simple matter. If not, it is necessary to determine what values of what available inputs are needed to produce the same results. For a solver that has the assumptions of equation (1) deeply embedded in it, this may involve entry of a value of applied current scaled by the factor ( 1+ ξ 2 ) .

Then, to explore and verify solver operations over other values of ξ, we ask that the solutions of (3) be computed for ξ set to the values 0, 0.1 (= α), and 0.5. This corresponds to solutions of (5) for u set to the values -72.35 m/s, -71.64 m/s, and -57.88 m/s. The aim is to cover interesting example cases to verify proper solver functioning.

Submitters are also invited to submit whatever solutions their solver deems to be an appropriate solution for the physical parameters specified without any attempt to force a solution of (3) from your software if it prefers something else, also for the range of values for ξ set to 0, 0.05, 0.1 (= α), and 0.5.

When ξ=0, equations (1) and (3) compute precisely the same dynamics. This is an important baseline case. Divergent solutions here indicate likely program errors.

When ξ=0.05, the computed results can be compared with the results found in [1].

When ξ=0.1=α, the final term of the Landau-Lifshitz form of the dynamics (6) vanishes. Testing this case is useful for detecting whether that leads to any trouble with the implementation.

Finally, when ξ=0.5, the different solutions of (1) and (3) are great enough to observe with certainty. There may be no material for which this calculation is meaningful, but there is value as a unit of software testing.

To summarize, submissions should include computed solutions of one of the three equivalent sets of problems:

  1. Solve (1) for: u T = - 72.35 x ^ m/s and β = 0 u T = - 72.17 x ^ m/s and β = 0.05 u T = - 71.64 x ^ m/s and β = 0.1 u T = - 57.88 x ^ m/s and β = 0.5
  2. Solve (3) for: b J = 72.35 m/s and ξ = 0 b J = 72.17 m/s and ξ = 0.05 b J = 71.64 m/s and ξ = 0.1 b J = 57.88 m/s and ξ = 0.5
  3. Solve (5) for: u = - 72.35 m/s and ξ = 0 u = - 72.17 m/s and ξ = 0.05 u = - 71.64 m/s and ξ = 0.1 u = - 57.88 m/s and ξ = 0.5

In addition, submitters may submit whatever solutions they believe more properly compute the physical problem posed with degree of non-adiabaticity from the set 0 0.05 0.1 0.5 .

Submissions

So long as the single vortex structure of the magnetization is not lost during the simulation, the spatially averaged components of magnetization Mx and My are a good proxy for the vortex location, and are easy to compute.

For each completed simulation, a text file should be submitted with three columns of numeric values. The first column should report the simulation time value in seconds. The second column should report the spatially averaged magnetization component Mx computed for that instant in time in A/m. The third column should report the spatially averaged magnetization component My computed for that instant in time in A/m. Lines beginning with the # character can be used to make comments that will not be processed as data. The time resolution of the data should be fine enough to represent the vortex trajectory well. A suggested minimum time sampling for submission is one data line for every 100 ps of simulation time. The data need not be evenly sampled. The trajectories should continue until the submitter is satisfied an equilibrium location of the vortex has been reached. The results in [1] suggest this means a period of about 14 ns.

For each submitted file that represents a solution of (5), the values of u and ξ should be identified. Other submissions should include whatever other description will make clear the problem solution they represent.

All solution files may be gathered up into any common archive format and transmitted to donald.porter@nist.gov for archiving and publication on the μMAG Standard Problem pages. Simultaneous reporting to the μMAG mailing list is optional. Submissions should identify the solver tested, and any other details needed for others to replicate the results.


References

[1] Proposal for a standard problem for micromagnetic simulations including spin-transfer torque, M. Najafi, B. Krüger, S. Bohlens, M. Franchin, H. Fangohr, A. Vanhaverbeke, R. Allenspach, M. Bolte, U. Merkt, D. Pfannkuche, D.P.F. Möller, and G. Meier, Journal of Applied Physics, 105, 113914 (8 pages) (Jun 2009).

[2] Micromagnetic understanding of current-driven domain wall motion in patterned nanowires, A. Thiaville, Y. Nakatani, J. Miltat, Y. Suzuki, Europhysics Letters, 69, 990 (7 pages) (March 2005).

[3] Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets, S. Zhang and Z. Li, Physical Review Letters, 93, 127204 (4 pages) (September 2004)


Please send comments to donald.porter@nist.gov and join the µMAG discussion e-mail list.


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29-SEP-2014