IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006
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Hybrid FE-FFTM Algorithm for Micromagnetic Modeling of Perpendicular SOMA Media H. H. Long1 , E. T. Ong2 , P. Y. Xiao3 , Z. J. Liu2 , and K. L. Huang4 Hitachi Global Storage Technology, Singapore 417939, Singapore. MRC Division, A*Star Data Storage Institute, Singapore 117608, Singapore A*Star Data Storage Institute, Singapore 117608, Singapore Net and Computer Center Teaching and Research Section, Huazhong University of Science and Technology, Wuhan 430074, China Abstract—This paper presents numerical simulation of the write and read process based on self-organized magnetic array (SOMA) media to study the particle-array misalignment effects on recording performances. The simulation is based on a three-dimensional finite element micromagnetic model of the media and an analytical model of the single pole perpendicular recording write field. The physical problem is modeled with a hybrid finite-element method–boundary-element method formulation, and the evaluation is accelerated using a new fast Fourier transform on multipoles method. The results show the lowest signal and signal-to-noise ratio occur at the best alignment situation, which is to be accepted as the intrinsic characteristic for SOMA media. Index Terms—Fast Fourier transform on multipoles, hybrid FEM-BEM, micromagnetic modeling, SOMA.
I. INTRODUCTION
R
ECENTLY, theoretical studies of perpendicular magnetic recording systems with an areal density of 1 Tbit/in have been reported. It is envisioned that even higher areal densities are possible by using an assisted assembly process e.g., of chemically synthesized FePt nanoparticles. Such self-organized magnetic array (SOMA) is composed of grains with a uniform size distribution and a non-magnetic coating to reduce exchange coupling. The large magnetocrystalline anisotropy erg/cc) allows grains as small as nm to be thermally ( stable over typical data storage periods of 10 years [1]. SOMA media can serve as 1) conventional granular media with better grain uniformity; 2) patterned media with bit-transitions defined by rows of grains; 3) single-particle-per-bit recording.[2] In this paper, a three-dimensional (3-D) finite-element micromagnetic model is used to simulate the switch dynamics of SOMA recording system. Our studies investigate the effect of different particle-array alignments on the signal to noise ratio (SNR). The fluctuations of signal and noise are investigated under the condition that the transition location is shifted in a square assembled array. The micromagnetic model is developed on the basis of hybrid finite element method and boundary element (Hybrid FEM/BEM) formulation [3] and the demagnetization field is evaluated using a new fast Fourier transform on multipoles (FFTM) method. II. SIMULATION MODEL
A. Model Description The magnetic recording layer is modeled by spherical particles assembled in a 32 32 square grid. The nanoparticles are
Digital Object Identifier 10.1109/TMAG.2006.880109
FePt-based, with diameters of 4 nm and a distance of 2 nm between each other. The finite element micromagnetic model is based on the LLG equation as follows:
(1) where is the gyromagnetic ratio, the effective field contributed by exchange, anisotropy, magnetostatic and external fields, the damping constant, and M the magnetization. In the medium layer, the easy axis of grains is oriented towards perpendicular direction and the anisotropy constant of erg/cm . The saturation magnetization emu/cm , and the intragrain exchange erg/cm. In combination with the micromagnetic modeling, a modified 3-D analytical solution [4] is applied to describe the write field for perpendicular magnetic recording system in which the flux path in softmagnetic underlayer (SUL) between the write head and the return pole is considered. The effect of time dependence of the field profile is not included in the following simulation. Fig. 1 shows the field distribution for horizontal and perpendicular components along the recording direction at the medium center. The write head covers 20 particles in cross-track direction and 6 particles in the down-track direction. In the simulation, transitions are perpendicularly recorded in a 32 32 particle array. The particles of the array are subdivided into tetrahedral finite elements. The variations in magnetization states are modeled during simulation. The array alignment is described by the angle between the trailing edge of the head and the x axis of the array (Fig. 2). Because of the symmetry, we need only consider angles in the range 0 to 45 . The transition location is defined as shown in Fig. 2(b). When the particle-array alignment is 0 , the transition might be recorded exactly along a column of particles (at the center of the particle site), or between two neighboring columns, which are separated by the particle pitch . By definition, the center of the particle site is denoted as , where and the middle positions , respectively. between particles are denoted as
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006
Fig. 2. Transition position in SOMA array with head-media alignments at: 1) 0 ; 2) 10 ; (b) transition location definition in SOMA array. Fig. 1. Head field distribution calculated from analytical head model: Horizontal (Hy) and perpendicular (Hz) field component at media center. (Color version available online at http://ieeexplore.ieee.org.)
B. Demagnetization Field Calculation Hybrid FEM-BEM [3] is a common approach for calculating the demagnetizing potential by splitting it into two components, i.e., . The potential component is governed by the Poisson equation with Neumann boundary conditions, which can be easily solved using standard FEM. The other potential component is defined by the following boundary integral equation: (2) where is the Green’s function, and is the number of boundary elements. It is noted here that evaluating (2) is often the computational bottleneck in the Hybrid FEM-BEM method, especially in analysis of SOMA media. This is because SOMA problems tend to result in large number of boundary elements. On the other hand, solving the FEM part for is relatively efficient due to the decoupling nature of SOMA media, i.e., the individual particles can be analyzed independently using FEM. Hence, highly efficient algorithm is needed to resolve this computation issue. Here, the FFTM algorithm [5] is used to speedup this computation via the multipole expansion approximation which is briefly described as follows: Consider two well-separated groups of boundary elements, say group and , that are clustered and bounded within two spheres centered at point and , respectively. Now, the potential at a given node in due to the boundary elements in can be approximated by the following expansions: (3) where
is the order at which the expansion is truncated, is the relative position of with respect to in spherical co-ordinates, and the local expansion coefficients are derived using (4)
with
and is the spherical coordinates of the vector The multipole moments in (4) are defined as
.
(5) where is the number of elements in and is the in the coordinates of with respect to . Note that above equations is the spherical harmonics function of degree and order . FFTM algorithm utilized the discrete convolution nature of (4) to speedup this part of the calculation via FFT algorithms. This is essentially the main feature in the FFTM algorithm. [5] Fig. 3 compares the computational efficiency of evaluating (2) directly using BEM, and that accelerated with FFTM with multipole expansion order , which can provide 3-digits accuracy when measured in the norm sense [5]. The setup CPU time and memory storage correspond to that required for forming and storing the various matrices, respectively. And the CPU time per solve refers to that required to compute at each time step. It is obvious that FFTM is significantly more efficient than the direct approach. More importantly, the computational complexities for FFTM scale approximately linearly, which makes it a good candidate for solving large scale problems.
III. COMPUTATION RESULTS The signal-to-noise ratio (SNR) of an isolated transition for the SOMA recording system with respect to different particlearray alignments is simulated using the combination of the micromagnetic modeling and the analytical write field. Repeated recordings are performed, each with a different spatial arrangement of the grain crystalline easy axes under the same statistical parameter, mimicking recording at different locations of a medium.
LONG et al.: HYBRID FE-FFTM ALGORITHM FOR MICROMAGNETIC MODELING OF PERPENDICULAR SOMA MEDIA
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Fig. 3. Complexity plots of direct (dashed-lines) and FFTM methods (solid-lines).
The isolated output pulse is obtained by differentiating of GMR head analytical model[6] (6) Here, is the reciprocity potential expression in the recording media, which is determined by GMR head geometry of free layer thickness and shield-to-shield distance [6]. Typically, the noise power within a bit length nm is studied
(7) The SNR is expressed as [6] (8) microtrack To ignore the track edge effect, only centered is read. The particle-array alignments of 0, 10, 30, and 45 are investigated. In each alignment case, a series of transitions are simulated when the write head moves in the down track direction at a step of . Signal and noise are calculated from 50 repeated recordings, each with a different Gaussian distribution of perpendicular easy axes with 5 deviation angle. Fig. 4 shows signal and noise fluctuations inside the transition position shifting range-one grain pitch in x direction. For 0 alignment, the coincidence of signal minimum and noise maximum corresponds to the situation that transition center is located at the grain site. For 45 alignment, the amplitudes of fluctuation become smaller, and the fluctuation frequency is two times of 0 alignment case, which is due to the array periodical characteristics in 0 and 45 directions. For other degree alignments, the fluctuations are even smaller and non-periodical. IV. CONCLUSION A numerical technique making use of 3-D finite element micromagnetic modeling and analytical write field solution is
Fig. 4. (a) Noise power; (b) signal and noise fluctuations of different particle-array alignments in perpendicular oriented SOMA media (transition location defined in Fig. 2)
applied to study the particle-array alignment effect on SNR of SOMA recording media. The lowest SNR occur at the best alignment situation, which is to be accepted as the intrinsic characteristic for SOMA media. The micromagnetic model is developed on the basis of Hybrid FEM-BEM formulation. It is worth noting that the study shows a fast algorithm FFTM is effective for rapid calculation of demagnetization field. REFERENCES [1] M. L. Plumer, J. van Ek, and D. Weller, The Physics of Ultra-High-Density Magnetic Recording. Berlin, Germany: Springer-Verlag, 2001. [2] T. Klemmer and D. Weller, SOMA and Nanomagnetics for Ultra-high Density Storage, 2004. 02pC-04, PMRC. [3] T. R. Koehler, “Hybrid FEM-BEM method for fast micro-magnetic calculations,” Phys. B. Condens. Matter., vol. 233, no. 4, pp. 302–307, 1997. [4] Z. J. Liu, J. T. Li, and H. H. Long, “Sensitivity analysis of write field with respect to design parameters for perpendicular recording heads,” J. Appl. Phys., vol. 97, no. 10, pp. 515–517, 2005. [5] H. H. Long, E. T. Ong, Z. J. Liu, and E. P. Li, “Fast Fourier transform on multipoles for rapid calculation of magnetostatic fields,” IEEE Trans. Magn., vol. 42, no. 2, pp. 295–300, 2006. [6] H. Zhou, “Micromagnetic analyses of effects of intergranular magnetostatic and exchange interactions on signal-to-noise ratio and thermal stability in hard disk drive recording media,” Ph.D. Dissertation, Univ. California, San Diego, CA, Feb. 2001. Manuscript received March 13, 2006; revised May 22, 2006 (e-mail:
[email protected];
[email protected]).
