Micromagnetic Simulations

One major part of the thesis is to investigate the magnetization dynamics by using micromagnetic simulations. In this chapter the basic description of the object oriented micromagnetic framework and how to solve the partial differential equation (LLG equation) by using the finite difference method will be explained.

For example, first and forth order Runge-Kutta methods will be discussed. As a basic simulation, the static vortex state and applied field dependent vortex core displacements will be explained. Lastly, numerical generated spin-waves on a magnetic nanowire will be introduced.

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Figure 2.15: (a) The first order Runge-Kutta method. (b) Runge-Kutta method of fourth order [RuKu].

2.5.1 Object Oriented MicroMagnetic Framework: OOMMF

The object oriented micromagnetic framework (OOMMF) has been developed by Mike Donahue and Don Porter in the National Institute of Standard and Technology. The program is based on C++ with Tcl/Tk. In order to explain OOMMF, the LLG equation (see Chapter 1.4.1) is recalled:

dM

dt =−γM×H− α MS

M×dM

dt . (2.5)

In this equation both left and right side, there are the time derivative terms of magnetization (dM/dt). Since the LLG equation is a nonlinear differential equa-tion, this LLG equation analytically cannot be solved for most geometries. From this point of view, the finite difference method (FDM) with equivalent rectan-gular or cubic cells and a certain discrete time step are employed. OOMMF is based on the FDM. There are various algorithms to solve the LLG equation nu-merically. Now, a Runge-Kutta method of fourth order as a general algorithm is introduced [Kius05].

A Runge-Kutta algorithm is shown in Figure 2.15 [RuKu, Drews09]. The first order of Runge-Kutta method, which is called Euler algorithm is introduced in Figure 2.15(a). This method is based on the linear extrapolation by the deriva-tive of the function between the initial state (0) and the next step (1) with time step h. The next simulation step is expressed by y_n+1 = y_n +hf(t_i, y_n). Fig-ure 2.15(b) shows the fourth order Runge-Kutta method with time step h. In

order to calculate the next simulation step (4), two trial middle points (1 and 2) and a trial end point (3) are considered by using a Taylor expansion. The expansion is numerically given by:

y(ti+h) = y(ti) + 1

6(K1+ 2K2+ 2K3 +K4), (2.6) where K₁, K₂, K₃, and K₄ correspond to the initial point (0), the two estimated middle points (1 and 2), and the estimated end point (3), respectively. Therefore, K₁ =h×f(t₁, y(t_i)), K₂ =h×f(t_i + ¹₂, y_ti+ ^K₂¹), K₃ =h×f(t_i+ ¹₂, y_ti +^K₂²), and K₄ = h×f(t_i+h, y(t_i) +K₃) are obtained. The time step h is able to be controlled for the total simulation times and a maximum step size with a given error limit. The numerical simulations always have a limit for time and length scale. OOMMF has this limits as well. In the point of view, OOMMF based on the LLG equation is not an appropriate tool to explain the quantum mechanical ferromagnetism since the limit of the length scale is about few nanometer and the limit of time scale is about pico-second. In micromagnetic simulation, to obtain the reliable simulation results, the cell size should be smaller than the exchange lengthl_ex=p

2A/µ₀M_S². For the case of Permalloy, the exchange length is given by [HHK03]: Here, the magnetic permeability of free spaceµ₀ = 4π×10⁻⁷ N/A², the saturation magnetization M_S = 8 ×10⁵ A/m, and the exchange stiffness constant A = 1.3×10⁻¹¹ J/mare used as typical material parameters of Py.

2.5.2 Static Simulations: Magnetic Vortex State and Do-main Wall

Now, the numerically calculated magnetic vortex state with static applied field and the magnetization structure of domain walls is introduced. The magnetic vortex states in a Py disc (diameter: 1µm and thickness: 35 nm) is shown in Fig-ure 2.16. Due to the symmetry of the circular shape and strong shape anisotropy, there are two kinds of in-plane curling magnetization structures (chirality: clock-wise and counter-clockclock-wise). Moreover, due to the geometrical frustration in the center of the disc, there is a vortex core, which has an out-of-plane magnetization

Figure 2.16: magnetic vortex states with an external fields. ((a)-(c)): the magnetic vortex states with an external field −15 mT, 0 mT, and +15 mT along thex direction, respectively. Yellow arrows show the direction and the amplitude of the external field.

(b) indicates the in-plane curling magnetization (chirality: counter-clockwise). ((d)-(f)):

the out-of-plane magnetization components (polarity: pointing up) with the external fields −15mT, 0 mT, + 15 mT along thex direction, respectively.

(polarity: up or down to the plane). Figure 2.16(b) shows the in-plane curling magnetization, which is called "chirality". In this case, the chrality is counter-clockwise. Figure 2.16(a and c) indicate the simulated magnetization structures with an external fields ∓15 mT along the x direction, respectively. The displace-ments of magnetization structure due to the external applied field depend on the chirality. On the other hand, the out-of-plane component (polarities) as a func-tion of the external field are shown in Figure 2.16(d and f). In this case, the red dot indicates the pointing up direction out of the plane. The effective diameter of the vortex core is about 10 nm [UsPe93]. Moreover, it is able to understand that the displacements of the vortex core does not depend on the polarity of the vortex core.

Figure 2.17 shows the simulations of two-dimensional spin structures, a trans-verse domain wall (TW: see Figure 2.17(a)) and a vortex wall (VW: see Fig-ure 2.17(b)). For the case of narrow and thin structFig-ures, the large stray field is dominant so that a transverse domain wall is nucleated between two magnetic domains. For the case of thick and wide structures, the large exchange energy is energetically favourable so that a vortex domain wall is nucleated [Klaeui08].

Figure 2.17: The simulations of a two-dimensional head-to-head (a) transverse domain wall. (width: 100 nm and thickness: 5 nm) (b) vortex wall (width: 200 nm and thickness:

30 nm).

2.5.3 Dynamic Simulations: a Propagating Spin-Wave

This section briefly introduces how to numerically generate propagating spin-waves in a magnetic nanowire [SKKim10]. A Py nanowire (width: 4000 nm and thickness: 5 nm) is considered. In order to generate spin waves, a localized oscillating applied field (Bosc) is used, which is given by:

B_osc =B₀sin(2πf_Ht)u_y, (2.8) where, B0 is the amplitude of the oscillating field, fH is the spin wave frequency, and uy is an unit vector of the oscillating field. Figure 2.18 shows the snapshots of the propagating spin waves as a function of time. The sinusoidal oscillating field along theydirection acts at x= 1500 nm from the left edge of the nanowire.

B0 = 200 mT is used to generate a spin-wave. The spin wave propagates in both direction along the ±x directions. The propagation speed is about 1.1 km/s.

Figure 2.18: The snapshots of the propagating spin-waves as a function of simulation time (0, 1, and 2 ns). The oscillating field (see Equation 2.8) is along the y direction.

The spin wave source is located atx= 1500 nm from the left edge.