Remanent magnetization along the long axis of the particle as a
function of d/l_{ex}.
Remanent magnetization along the short axis of the particle as
a function of d/l_{ex}.
Coercivity for fields applied along the (111) direction
as a function of d/l_{ex}.
Solution to standard problem #2
From: B Streibl, T Schrefl and J Fidler
Institute of Applied and Technical Physics
Vienna University of Technology
Wiedner Hauptstr. 810
A1040 Vienna, Austria
Contact: T Schrefl
URL:
http://magnet.atp.tuwien.ac.at/
Reference: B. Streibl, T. Schrefl and J. Fidler, J. Appl. Phys, v. 85, pp 58195821 (1999).
We studied standard problem #2 using a 3Dfinite element simulation
based on the solution of the Gilbert equation. Asymptotic boundary conditions
were imposed in order to compute demagnetizing fields. The direction cosines
of the magnetization and the magnetic scalar potential were interpolated
with piecewise linear and quadratic polynomials on hexahedral finite elements,
respectively. The finest mesh used for the calculations contains 500 elements
within the magnetic thin film and 4200 in the exterior space. The values
of Permalloy were assumed for the magnetization and the exchange
constant, so that the exchange length l_{ex} becomes 5 nm.
A Gilbertdamping parameter (alpha=1) was used to drive the system towards
equilibrium. For more details on the calculation method we refer to our
forthcoming paper "Dynamic FEsimulation of mumag standard problem
#2" which will be presented at the 43rd annual conference on Magnetism
& Magnetic Materials in Miami, November 912, 1998 (session FZ05).
d/l_{ex}  M_{rx}/M_{s}  M_{ry}/M_{s}  M_{rz}/M_{s}  H_{c}/M_{s} 
1  0.999  0.029  0.004  0.056 
5  0.999  0.006  0.000  0.056 
10  0.998  0.020  0.000  0.054 
20  0.973  0.081  0.000  0.050 
30  0.963  0.078  0.000  0.046 
(The coordinate axis are oriented as proposed in the specification
of the standard problem)
(l_{ex} denotes the exchange length)
(M denotes the component of the magnetization parallel to the field)
Reference: R. D. McMichael, M. J. Donahue, D. G. Porter and Jason Eicke, J. Appl. Phys, v. 85, pp 58165818 (1999).
We used the OOMMF public code to compute solutions to standard problem #2. The mesh was a 2D square grid with 3D spins interacting through an 8neighbor exchange representation and magnetostatic fields. Computations were done for a number of mesh sizes, using up to 12500 cells, using energy minimization (precession neglected). We obtained similar results for two methods of calculating the magnetostatic fields, the constant magnetization method, and the constant charge method.
We find three types of critical fields, end domain switching, end domain propagation, and vortex formation, depending on the value of d/l_{ex}. These critical field values, extrapolated to zero cell size, are listed below in terms of H_{c}/M_{s} with an uncertainty of plus/minus 0.0014, corresponding to the field step size. The remanent magnetization values, calculated by relaxing from saturation in the (111) direction are also listed.



d/l_{ex}  end propagation  end switch  vortex formation  M_{x}/M_{s}  M_{y}/M_{s}  M_{z}/M_{s} 
3.16  0.0600  1.0000  0.0007  0.0000  
10  0.0586  0.0007  0.9959  0 .0376  0.0000  
14  0.0558  0.0145  0.9783  0.0822  0.0000  
17.8  0.0531  0.0278  0.9711  0.0851  0.0000  
23  0.0503  0.0393  0.9657  0.0823  0.0000  
27  0.0489  0.0448  0.9631  0.0792  0.0000  
31.6  0.0517  0.9609  0.0760  0.0000  
35  0.0553  0.9595  0.0739  0.0000  
40  0.0532  0.9577  0.0710  0.0000  
56  0.0408  0.9527  0.0638  0.0000  
75  0.0200  0.9478  0.0572  0.0003 
Contact: Luis Lopez Diaz
Reference: L. LopezDiaz, O. Alejos, L. Torres and J. I. Iniguez J. Appl. Phys, v. 85, pp 58135815 (1999).
d/l_{ex}  H_{c}/M_{s}  M[111]  M_{rx}  M_{ry} 
0.8  0.0631  0.5773740  1.00000  7.00000E05 
1.0  0.0612  0.5773501  1.00000  7.00000E05 
2.0  0.0594  0.5773470  0.999996  4.27499E05 
3.0  0.0596  0.5773384  0.999981  1.24466E07 
4.0  0.0594  0.5773172  0.999942  4.14687E05 
5.0  0.0592  0.5772781  0.999876  3.95000E05 
6.0  0.0585  0.5772195  0.999777  7.30804E08 
7.0  0.0583  0.5771367  0.999596  8.22480E08 
8.0  0.0580  0.5770244  0.999439  3.25319E05 
9.0  0.0576  0.5769020  0.999191  6.88017E05 
10.0  0.0573  0.5956421  0.996272  3.54161E02 
11.0  0.0568  0.6069226  0.989998  6.12417E02 
12.0  0.0565  0.6106740  0.985207  7.25276E02 
13.0  0.0560  0.6120005  0.981448  7.85834E02 
14.0  0.0556  0.6122172  0.978436  8.19699E02 
15.0  0.0549  0.6118625  0.975986  8.38084E02 
16.0  0.0541  0.6112075  0.973947  8.47102E02 
17.0  0.0534  0.6103936  
18.0  0.0529  0.6095005  0.970768  8.49321E02 
20.0  0.0518  0.6076426 
Contact: Mike Donahue
(URL: http://math.nist.gov/mcsd/Staff/MDonahue/)
Reference: J. Appl. Phys, .v 87, pp. 55205522 (2000).
An examination of the coercive fields from first three submissions (Streibl et al., McMichael et al., Diaz et al.) reveals discrepancies in the small particle limit (d/l_{ex} near 0). In this range the magnetization is nearly uniform, so we performed a 3D StonerWohlfarth analysis, and found the theoretical value for the coercive field to be H_{c}/M_{s} = 0.05707. This result is in close agreement with the Streibl submission, but is removed from our first calculation (McMichael), which was based on version 1.0 of the OOMMF public code.
In the uniformly magnetized setting, only magnetostatic energies are active. We therefore examined the demagnetization (selfmagnetostatic) energy calculated by OOMMF as a function of calculation cell size for the uniformly magnetized particle. It is observed that the sampled H_{demag} model used in OOMMF 1.0 for demagnetization energy calculations is inaccurate in this case.
The magnetostatic energy between two uniformly magnetized rectangular prisms can be evaluated analytically (A. J. Newell, W. Williams, and D. J. Dunlop, J. Geophysical Research, 98, 9551 (1993)). From the standpoint of calculating demagnetization energy, these formulae can be viewed as providing a value for H_{demag} that has been averaged over the calculation cell. The averaged H_{demag} model is also completely accurate for uniformly magnetized particles (as seen in the aforementioned graph).
In the small particle range, the accuracy of the averaged H_{demag} formulae allowed us to use calculation cells that were large compared to d. We could then run simulations at smaller values of d/l_{ex} than we had in our earlier submission. We discovered, however, that numerical errors in the calculation of the exchange energy produced an artificial stiffening of the simulation. We were able to reduce this problem by replacing our base exchange energy term 1m_{i}·m_{k} (m_{i} and m_{k} are neighboring spins) with an expression of the form m_{i}·(m_{i}  m_{k}).
These changes have been incorporated into version 1.1 of OOMMF, which was used to produce this submission. From the summary coercive field plot above, we see that OOMMF 1.1 closely tracks the Streibl results, and agrees with the theoretical result for the infinitely small particle case.
It is worth noting that the switching field H_{s} (the field at which the first irreversible change in magnetization occurs) is somewhat larger that the coercive field H_{c} (the field at which H_{applied}·M =0) for d/l_{ex}<15, as illustrated by this graph. We found that the theoretical value for the switching field in the small particle limit was H_{s}/M_{s} = 0.05714, also slightly higher than the coercive field.
The table below presents our results using OOMMF 1.1 with the changes outlined above. H_{c} is the coercive field, H_{s} is the switching field, M_{x} and M_{y} are the average x and y magnetizations at remanence. MaxAngle/run is the largest angle between neighboring spins (i.e., calculation elements) that occurred throughout the simulation, including nonequilibrium states; this is an indication of the ability of the sample mesh to represent the variation in the magnetization. The uncertainty in the field values H_{c} and H_{s} due to the applied field step size is less than ±0.0000138 M_{s}. In particular, the results at d/l_{ex}=0.125 agree to within this precision with the theoretical results for an infinitely small particle.
d/l_{ex}  Cellsize/
l_{ex} 
H_{c}/M_{s }  H_{s}/M_{s}  M_{x}/M_{s}  M_{y}/M_{s}  MaxAngle/
run (deg) 
30  0.60  0.04594  0.04594  0.9627  0.0756  13.4 
29  0.58  0.04630  0.04630  0.9632  0.0763  12.8 
28  0.56  0.04666  0.04666  0.9638  0.0769  12.2 
27  0.54  0.04704  0.04704  0.9644  0.0776  11.6 
26  0.52  0.04740  0.04740  0.9650  0.0783  11.0 
25  0.50  0.04779  0.04779  0.9657  0.0790  10.4 
24  0.48  0.04817  0.04817  0.9664  0.0797  9.75 
23  0.46  0.04859  0.04859  0.9672  0.0803  9.16 
22  0.44  0.04903  0.04903  0.9680  0.0809  8.57 
21  0.42  0.04947  0.04947  0.9690  0.0814  7.98 
20  0.40  0.04991  0.04991  0.9701  0.0819  7.41 
19  0.38  0.05041  0.05041  0.9713  0.0822  6.85 
18  0.36  0.05090  0.05090  0.9727  0.0822  6.31 
17  0.34  0.05145  0.05145  0.9743  0.0819  5.77 
16  0.32  0.05200  0.05200  0.9762  0.0811  5.25 
15  0.30  0.05258  0.05261  0.9784  0.0795  4.73 
14  0.56  0.05309  0.05324  0.9812  0.0765  8.4 
13  0.52  0.05352  0.05388  0.9845  0.0716  7.5 
12  0.48  0.05389  0.05449  0.9886  0.0631  6.8 
11  0.44  0.05423  0.05506  0.9938  0.0463  6.1 
10  0.40  0.05454  0.05562  0.9989  0.0000  5.1 
9  0.45  0.05480  0.05614  0.9992  0.0000  4.0 
8  0.40  0.05515  0.05661  0.9995  0.0000  4.0 
7  0.35  0.05546  0.05702  0.9996  0.0000  3.2 
6  0.30  0.05579  0.05730  0.9998  0.0000  2.4 
5  0.25  0.05612  0.05741  0.9999  0.0000  1.8 
4  0.20  0.05646  0.05735  0.9999  0.0000  1.2 
3  0.15  0.05674  0.05724  1.0000  0.0000  0.696 
2  0.10  0.05694  0.05716  1.0000  0.0000  0.314 
1  0.05  0.05704  0.05716  1.0000  0.0000  0.079 
0.75  0.15  0.05705  0.05713  1.0000  0.0000  0.182 
0.50  0.10  0.05706  0.05713  1.0000  0.0000  0.0813 
0.25  0.05  0.05707  0.05713  1.0000  0.0000  0.0204 
0.125  0.03  0.05707  0.05713  1.0000  0.0000  0.0063 
Theory  n/a  0.05707  0.05714  1.0000  0.0000  n/a 
Contact: Rasmus Bjørk
(URL: http://www.magtense.org)
Reference: J. Magn. Magn. Mater., .v 535, pp. 168057 (2021), DOI: https://doi.org/10.1016/j.jmmm.2021.168057
We used the MagTense open source micromagnetic simulation framework to compute solutions to standard problem #2. The mesh was a 3D square grid with 3D spins. A Cartesian mesh with 100x20x3 rectangular cuboid tiles in the x, y and zdirections respectively were utilized. The hysteresis loop for each individual value of d/l_{ex} was computed by starting with an applied field of 0.08 H/M_{s} and then decreasing the field in 4000 steps to a value of 0.08 H/M_{s}. The micromagnetic damping parameter is taken to be 4000, close to the value for mumag standard problem 4, and the precession parameter is taken to be 0. The demagnetization field was calculated using an analytical approach, assuming only a constant magnetization within each grid cell. The exchange field was calculated using a finite difference approach.
We find the results as tabulated below
d/l_{ex}  H_{c}/M_{s}  M_{rx}  M_{ry} 
3.00  0.05705  1.000000  8.69376e4 
4.93  0.05586  0.999885  3.55424e4 
6.86  0.05507  0.999667  2.90060e4 
8.79  0.05441  0.999290  7.83946e4 
10.71  0.05378  0.996211  3.40159e2 
12.64  0.05310  0.986592  6.72743e2 
14.57  0.05229  0.980252  7.71206e2 
16.50  0.05133  0.975886  8.05364e2 
18.43  0.05032  0.972714  8.13575e2 
20.36  0.04938  0.970298  8.09996e2 
22.29  0.04851  0.968385  8.00821e2 
24.21  0.04772  0.966817  7.89040e2 
26.14  0.04698  0.965493  7.76162e2 
28.07  0.04627  0.964346  7.62986e2 
30.00  0.04558  0.963332  7.49929e2 