µMAG Standard Problem #4 results

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Submitted Solution: Liliana Buda, Lucian Prejbeanu, Ursula Ebels and Kamel Ounadjela

Date:
November 27, 2000
From:
Liliana Buda, Lucian Prejbeanu, Ursula Ebels and Kamel Ounadjela
Institut de Physique et Chimie des Matériaux
Groupe d'Etude des Matériaux Métalliques
23, rue du Loess, 67037 Strasbourg Cedex, France
Contact:
L. D. Buda

A 3D code has been used for these numerical investigations. The ferromagnetic system is discretized in Nx × Ny × Nz identical tetrahedral cells having the size hx × hy × hz. The studies presented here implies a single cell across the thickness.

Inside each cell the magnetization is assumed to be uniform. The stray field is evaluated as the average over the magnetic cell [see A. J. Newell, W. Williams, and D. J. Dunlop, J. Geophys. Res. 98, 9551 (1993) and A. Hubert, R. Schäfer, Magnetic Domains, Springer-Berlin, p. 148 (1998)]. The demagnetization field and energy are computed using the FFT method (FFTW subroutines).

The exchange interaction approximation is the "7-point formula" applied to the 3D case [see Y. Nakatani, N. Uesaka and N. Hayashi, J. Appl. Phys. 28, 2485 (1989)]. On the free surfaces of the magnetic system the magnetization fullfills the Neumann boundary conditions.

For the integration of the LLG equation the implicit Crank-Nicholson backward scheme is used. Our algorithm uses a constant time step of less than 0.1 ps. After each iteration the magnetization is renormalized in order to satisfy the ferromganetic condition.

The convergence criteria imposes that the residual torque (the maximum value of the torque) to be less than 10-6.

Results:

Raw data:

Time series data contain 10 columns: time (ns), residual torque, demagnetization energy (erg/cc), exchange energy (erg/cc), anisotropy energy (erg/cc), Zeemann energy (erg/cc), total energy (erg/cc), m_x, m_y, m_z. Vector data has 6 columns: x, y, z coordinates, and m_x, m_y, m_z vector components.

References:

L. D. Buda, I. L. Prejbeanu, M. Demand, U. Ebels and K. Ounadjela, The 8th Joint MMM-Intermag Conference 2001 Proceedings

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21-DEC-2005