µMAG Standard Problem #2 Results

See the problem specification.
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Solution directory


Summary Plots, µMAG Problem #2

The following plots compare independently calculated values of the remanent magnetization and coercivity over a range of d/lex.

Plot of M_x at zero field vs. d/lex
Remanent magnetization along the long axis of the particle as a function of d/lex.

Plot of M_y at zero field vs. d/lex
Remanent magnetization along the short axis of the particle as a function of d/lex.

Plot of coercivity vs. d/lex
Coercivity for fields applied along the (111) direction as a function of d/lex.


Submitted Solution

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. 8-10
             A-1040 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 5819-5821 (1999).

We studied standard problem #2 using a 3D-finite 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 lex becomes 5 nm.
A Gilbert-damping 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 FE-simulation of mumag standard problem #2" which will be presented at the 43rd annual conference on Magnetism & Magnetic Materials in Miami, November 9-12, 1998 (session FZ-05).

  1. For the coercive field and the remanence magnetization we find for varying particle extensions and the external field applied parallel to the [111]-direction:

  2.  

    d/lex Mrx/Ms  Mry/Ms  Mrz/Ms  Hc/Ms 
    0.999  0.029  -0.004 -0.056
    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)
     

  3. Images:

Submitted Solution

R. D. McMichael, M. J. Donahue, and D. G. Porter
National Institute of Standards and Technology, Gaithersburg, MD 20899
J. Eicke
Institute for Magnetics Research,
George Washington University, Washington, DC 20052

Contact: Bob McMichael

Reference: R. D. McMichael, M. J. Donahue, D. G. Porter and Jason Eicke, J. Appl. Phys, v. 85, pp 5816-5818 (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 8-neighbor 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/lex. These critical field values, extrapolated to zero cell size, are listed below in terms of Hc/Ms 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.

 
Hcrit/Ms
Remanence
d/lex end propagation end switch vortex formation Mx/Ms My/Ms Mz/Ms
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

Images


Submitted Solution

Luis Lopez Diaz, Luis Torres, Jose Ignacio Iniguez
Universidad de Salamanca, Dept. Fisica Aplicada, Salamanca, Spain
Oscar Alejos
Universidad de Valladolid, Dept. Electricidad y Electronia, Valladolid, Spain

Contact: Luis Lopez Diaz

Reference: L. Lopez-Diaz, O. Alejos, L. Torres and J. I. Iniguez J. Appl. Phys, v. 85, pp 5813-5815 (1999).

d/lex Hc/Ms M[111] Mrx Mry
0.8 0.0631 0.5773740 1.00000 7.00000E-05
1.0 0.0612 0.5773501 1.00000 7.00000E-05
2.0 0.0594 0.5773470 0.999996 4.27499E-05
3.0 0.0596 0.5773384 0.999981 -1.24466E-07
4.0 0.0594 0.5773172 0.999942 4.14687E-05
5.0 0.0592 0.5772781 0.999876 3.95000E-05
6.0 0.0585 0.5772195 0.999777 -7.30804E-08
7.0 0.0583 0.5771367 0.999596 8.22480E-08
8.0 0.0580 0.5770244 0.999439 3.25319E-05
9.0 0.0576 0.5769020 0.999191 6.88017E-05
10.0 0.0573 0.5956421 0.996272 3.54161E-02
11.0 0.0568 0.6069226 0.989998 6.12417E-02
12.0 0.0565 0.6106740 0.985207 7.25276E-02
13.0 0.0560 0.6120005 0.981448 7.85834E-02
14.0 0.0556 0.6122172 0.978436 8.19699E-02
15.0 0.0549 0.6118625 0.975986 8.38084E-02
16.0 0.0541 0.6112075 0.973947 8.47102E-02
17.0 0.0534 0.6103936    
18.0 0.0529 0.6095005 0.970768 8.49321E-02
20.0 0.0518 0.6076426    


Submitted Solution

M. J. Donahue, D. G. Porter, R. D. McMichael
National Institute of Standards and Technology, Gaithersburg, MD 20899
J. Eicke
Institute for Magnetics Research,
George Washington University, Washington, DC 20052

Contact: Mike Donahue (URL: http://math.nist.gov/mcsd/Staff/MDonahue/)

Reference: J. Appl. Phys, .v 87, pp. 5520-5522 (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/lex near 0). In this range the magnetization is nearly uniform, so we performed a 3D Stoner-Wohlfarth analysis, and found the theoretical value for the coercive field to be Hc/Ms = 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 (self-magnetostatic) energy calculated by OOMMF as a function of calculation cell size for the uniformly magnetized particle. It is observed that the sampled Hdemag 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 Hdemag that has been averaged over the calculation cell. The averaged Hdemag 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 Hdemag formulae allowed us to use calculation cells that were large compared to d. We could then run simulations at smaller values of d/lex 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 1-mi·mk (mi and mk are neighboring spins) with an expression of the form mi·(mi - mk).

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 Hs (the field at which the first irreversible change in magnetization occurs) is somewhat larger that the coercive field Hc (the field at which Happlied·M =0) for d/lex<15, as illustrated by this graph. We found that the theoretical value for the switching field in the small particle limit was Hs/Ms = 0.05714, also slightly higher than the coercive field.

The table below presents our results using OOMMF 1.1 with the changes outlined above. Hc is the coercive field, Hs is the switching field, Mx and My 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 non-equilibrium 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 Hc and Hs due to the applied field step size is less than ±0.0000138 Ms. In particular, the results at d/lex=0.125 agree to within this precision with the theoretical results for an infinitely small particle.

d/lex Cellsize/
lex
Hc/M Hs/Ms Mx/Ms My/Ms MaxAngle/
run (deg)
300.60 0.04594 0.04594 0.96270.0756 13.4
290.58 0.04630 0.04630 0.96320.0763 12.8
280.56 0.04666 0.04666 0.96380.0769 12.2
270.54 0.04704 0.04704 0.96440.0776 11.6
260.52 0.04740 0.04740 0.96500.0783 11.0
250.50 0.04779 0.04779 0.96570.0790 10.4
240.48 0.04817 0.04817 0.96640.0797 9.75
230.46 0.04859 0.04859 0.96720.0803 9.16
220.44 0.04903 0.04903 0.96800.0809 8.57
210.42 0.04947 0.04947 0.96900.0814 7.98
200.40 0.04991 0.04991 0.97010.0819 7.41
190.38 0.05041 0.05041 0.97130.0822 6.85
180.36 0.05090 0.05090 0.97270.0822 6.31
170.34 0.05145 0.05145 0.97430.0819 5.77
160.32 0.05200 0.05200 0.97620.0811 5.25
150.30 0.05258 0.05261 0.97840.0795 4.73
140.56 0.05309 0.05324 0.98120.0765 8.4
130.52 0.05352 0.05388 0.98450.0716 7.5
120.48 0.05389 0.05449 0.98860.0631 6.8
110.44 0.05423 0.05506 0.99380.0463 6.1
100.40 0.05454 0.05562 0.99890.0000 5.1
90.45 0.05480 0.05614 0.99920.0000 4.0
80.40 0.05515 0.05661 0.99950.0000 4.0
70.35 0.05546 0.05702 0.99960.0000 3.2
60.30 0.05579 0.05730 0.99980.0000 2.4
50.25 0.05612 0.05741 0.99990.0000 1.8
40.20 0.05646 0.05735 0.99990.0000 1.2
30.15 0.05674 0.05724 1.00000.0000 0.696
20.10 0.05694 0.05716 1.00000.0000 0.314
10.05 0.05704 0.05716 1.00000.0000 0.079
0.750.15 0.05705 0.05713 1.00000.0000 0.182
0.500.10 0.05706 0.05713 1.00000.0000 0.0813
0.250.05 0.05707 0.05713 1.00000.0000 0.0204
0.1250.03 0.05707 0.05713 1.00000.0000 0.0063
Theoryn/a 0.05707 0.05714 1.00000.0000 n/a


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26-DEC-2001