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Physica B 354 (2004) 266–270 www.elsevier.com/locate/physb
Magnetic normal modes in nano-particles M. Grimsditcha,, L. Giovanninib, F. Montoncellob, F. Nizzolib, G. Leafc, H. Kaperc, D. Karpeevc a Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA Dipartimento di Fisica, Universita di Ferrara and Istituto Nazionale per la Fisica della Materia, Via del Paradiso 12, I-44100 Ferrara, Italy c Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
b
Abstract We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-particle. One of the methods is based on a conventional micromagnetic approach in which the time evolution of the magnetization of each cell is monitored. After filtering in frequency domain, the magnetic normal modes can be reconstructed. The second method is based on solving the same micromagneitc system in a dynamical matrix formulation. The results of the two methods, applied to a rectangular parallelepiped of Fe, will be presented and compared. r 2004 Elsevier B.V. All rights reserved. PACS: 75.75.+a; 75.30.Ds; 75.40.Mg Keywords: Nano-magnetism; Magnetic normal modes; Spin waves
The calculation of the magnetic normal modes of small particles, which include dipolar and exchange contributions, is a complex problem. Ignoring exchange contributions, the problem was addressed by Walker [1,2] who succeeded in obtaining solutions for a few, relatively simple, geometrical shapes. Analytical approaches have recently had some success but still require certain Corresponding
author. Tel.: +1 630 252 5544; fax: +1 630 252 7777. E-mail address:
[email protected] (M. Grimsditch).
assumptions to be made regarding the mode profiles [3,4]. Micromagnetic simulation approaches have also provided insight into a small subset of the normal modes of small magnetic particles [5]. Here we will first describe a micromagnetic approach that relies on simulation codes originally devised for the calculation of the static ground state of magnetic particles [6]. With available freeware programs it is possible to track the time evolution of the average magnetization of a particle and it is found that its Fourier transform
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ARTICLE IN PRESS M. Grimsditch et al. / Physica B 354 (2004) 266–270
contains information about some of the normal modes. The only modes observed in these simulations are those with non-zero average magnetization. By extending the simulation approach [7] to a code that independently tracks the time evolution of all the spins in a particle, it is possible to reconstruct the normal modes from the Fourier transforms of each individual spin. To show that the technique is viable we calculated the normal modes of a 116 60 20 nm3 Fe parallelepiped magnetized along its long axis. For this simple shape normal modes with a standing spin wave nature are expected. Consistent with this expectation Fig. 1 shows a few selected mode profiles obtained in our simulations that are reminiscent of standing spin wave modes. As shown in Fig. 2, the frequencies of these modes are also in good agreement with those predicted by existing spin wave approximations. The dashed and dashed–dotted lines are approximations including only exchange and dipolar terms, respectively. The full line includes the effect of both exchange and dipolar terms. The dependence of the frequencies on magnetic field is also well reproduced. More interesting is the appearance of modes that do not resemble standing waves. In Fig. 3 we show symmetric and antisymmetric ‘end’ modes that resulted from our simulations. These types of modes do not fit into a conventional sanding wave picture and their frequencies lie well below the bulk-like excitations in Fig. 1. We believe they are closely related to the magnetic instability that occurs near magnetization reversal. In a conventional micromagnetic simulation the sample is divided into cells, the magnetization is assumed to be uniform in each cell, and that each ‘spin’ precesses about its equilibrium direction under the influence of the external field and the dipolar and exchange forces due to all the spins in the system. Noting the similarity between this and conventional molecular dynamics, where each atom obeys Newton’s laws in the potential of the surrounding atoms, it seemed reasonable that it should be possible to solve the normal mode problem in a manner similar to the dynamical matrix approach used for atomic vibrations. A
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Fig. 1. Four symmetric standing-wave-like normal modes with frequencies at 49, 52, 57 and 89 GHz, respectively.
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similar approach has been used in the past in the case of magnetic multilayers [8,9]. As a starting point we recall that a uniformly magnetized film, can be treated in terms of the motion of a single dipole. The dynamical matrix is obtained by writing the total energy as a function of the orientation of the dipole in terms of the polar angles f and y; the torque is then obtained from the second derivative of the energy density (E) with respect to the polar angles. It is known that the precession frequency (O) is given by the eigen-values of the matrix [10]: E ff =sin y E fy =sin y iMO=g ¼ 0; E =sin y þ iMO=g E yy =sin y yf Fig. 2. Circles: calculated frequencies vs. wave vector at 10 kOe. The dashed, dashed–dotted and full lines are calculated for an infinite plate using: only exchange, only dipolar, and both contributions, respectivley.
(1) where M is the saturation magnetization and g the gyromagnetic ratio. This method can be generalized to the case of a particle divided into N cells [11]. With a reference frame with the z-axis along the particle normal and the x and y-axis along the particle sides. We define the polar angles that describe the orientation of magnetization in the nth cell to be yn and fn : The (unitary) vector specifying the magnetization is then given by mn ¼ ðsin yn cos fn ; sin yn sin fn ; cos yn Þ:
Fig. 3. Symmetric and antisymmetric end modes.
(2)
The eigenvectors of the generalization of Eq. (1) can then be interpreted in terms of the magnetization and thus the magnetic normal modes of the particle. Although the eigenvectors define the normal mode amplitude in each cell it must be noted that in general the eigenvectors will be complex so that an instantaneous snapshot may not be a time-invariant property. This becomes particularly important in cases where chiral modes are present. The energy density can be written as a sum of Zeeman, exchange and dipolar energies. The first two terms are straightforward and are equivalent to what is used in all micromagnetic simulations. The dipolar energy is by far the most difficult to treat. In the full simulations discussed above the dipolar field was obtained from a complete solution of the Poisson equation at every step [12]. Such an approach is not well suited to yield the derivatives needed for the torque matrix. An alternative approach, also sometimes adopted in
ARTICLE IN PRESS M. Grimsditch et al. / Physica B 354 (2004) 266–270 Table 1 Frequencies in GHz of the normal modes 116 20 60 nm3 Fe particle in a 10 kOe field Method
of
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a
Fundamental End mode 6-nodes 10-nodes
Full simulation 52.9 Dynamical matrix 52.9
30.2 28.3
56.6 54.8
89.3 85.2
Fig. 4. Mode profiles of two DE-like modes, 1-node mode with frequency (a) 58.6 GHz, and (b) a 6-node mode with frequency 138.2 GHz. We plot the real part mz as a function of the cell position. Fig. 5. Mode profiles of three end-modes, 1-node mode with frequency (a) 37.5 GHz, (b) a 2-node mode with frequency 46.3 GHz, and (c) a 6-node mode with frequency 122.0 GHz.
micromagnetics, is to write the dipolar energy as a sum of the interactions between the magnetic moments; either as a simple dipole interaction or in the more accurate approach developed by Newell [13].
For a system with a small number of cells (typically less than 100) the above equations can be directly solved using a symbolic mathematic
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software. In this way the second derivatives need not to be evaluated analytically and the solutions of the determinant equation can be obtained with a few lines of code. However, in order to treat realistic cases, it is necessary to develop a computer code based on the analytical formulae. In this format we are able to treat particles with up to 2000 cells with computation times less than 1 h on a standard PC. As already anticipated, we have calculated the magnetic excitations of the same particle discussed above. Also, as expected, the mode profiles and frequencies of the modes investigated in that study are well reproduced. This agreement is summarized in Table 1. However, contrary to the previous method where only a subset of all the normal modes are excited (depending on the perturbation), the dynamical matrix method yields all the normal modes of the system. In Fig. 4 we show two of the modes with wave vectors perpendicular to the applied field and in Fig. 5 three additional modes in the family of ‘end’ modes. To summarize, a full micromagnetic simulation that keeps track of all individual spins, and a dynamical matrix approach have been shown to yield the correct magnetic normal modes—frequencies and profiles—of a magnetized particle. The dynamical matrix method requires considerably less computational power than the micromagnetic approach but it does not allow nonlinear effects to be probed nor does it yield information
regarding which modes are excited in pulsed field experiments. We thank R. Stamps and R. Zivieri for useful discussions. Work at ANL was supported by DOE-BES under contract W-31-109-ENG-38.
References [1] L.R. Walker, Phys. Rev. 105 (1957) 390. [2] L.R. Walker, J. Appl. Phys. 29 (1958) 318. [3] K. Yu. Guslienko, A.N. Slavin, J. Appl. Phys. 87 (2000) 6337. [4] B.A. Ivanov, C.E. Zaspel, Appl. Phys. Lett. 81 (2002) 1261. [5] V. Novosad, M. Grimsditch, K. Yu. Guslienko, P. Vavassori, Y. Otani, S.D. Bader, Phys. Rev. B 66 (2002) 052407. [6] M. Donahue, D. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376. National Institute of Standards and Technology, Gaithersburg, MD, 1999. [7] M. Grimsditch, G.K. Leaf, H.G. Kaper, D.A. Karpeev, R.E. Camley, Phys. Rev. B 69 (2004) 174428. [8] M. Grimsditch, S. Kumar, E.E. Fullerton, Phys. Rev. B 54 (1996) 3385. [9] R. Zivieri, L. Giovannini, F. Nizzoli, Phys. Rev. B 62 (2000) 14950. [10] J. Smit, H.G. Beljers, Philips Res. Rep. 10 (1955) 113. [11] M. Grimsditch, L. Giovannini, F. Montoncello, F. Nizzoli, G. Leaf, H. Kaper, Phys. Rev. B 70 (2004) 054409. [12] D.R. Fredkin, T.R. Koehler, IEEE Trans. Magn. 26 (1990) 415. [13] A.J. Newell, W. Williams, D.J. Dunlop, J. Geophys. Res. 98 (1993) 9551.
