Effects of damping on micromagnetic dynamics P. X. Fang Supervisors: Second reader: Dr. M. Titov Dr. S. Wiedmann R. Sokolewicz M.Sc Bachelor Thesis Theory of Condensed Matter Radboud University Nijmegen July 26, 2018 Contents 1. Introduction 3 2. Micromagnetic Interactions 4 2.1. Overview . 4 2.1.1. Exchange Energy . 4 2.1.2. Magnetic Anisotropy . 5 2.1.3. Zeeman Energy . 5 2.2. Micromagnetic Dynamics . 6 2.2.1. Effective field . 6 2.2.2. Precession and Damping . 6 2.2.3. Spin-Transfer Torque . 7 3. Magnetic Domains 9 3.1. Overview . 9 3.2. Domain Walls . 9 3.2.1. Bloch walls . 9 3.2.2. Neel walls . 10 4. Methods 11 4.1. Program . 11 4.2. Systems . 11 4.2.1. Overview . 11 4.2.2. Spontaneous Domain Formation . 11 4.2.3. 2D Domain Movement . 12 4.2.4. Switching . 12 5. Results and Discussion 13 5.1. Spontaneous Domain Formation . 13 5.2. Domain Wall Movement . 14 5.2.1. Isotropic . 14 5.2.2. Anisotropic . 17 5.3. Switching . 19 6. Conclusion 23 7. Outlook & Discussion 24 7.1. Domain Wall Movement . 24 7.2. Switching . 24 References 25 A. Code changes 26 2 1. Introduction Magnetic materials have been the subject of many studies due to their fascinating and useful properties, which has lead to their widespread application. In particular, ferromagnetism is present in the commonly used iron. A key characteristic of ferromagnetic materials is the pres- ence of regions of magnetic moments with similar directions, which we call magnetic domains. The walls separating domains with different magnetic orientations are the focus of this work. Through application of external forces through for example either a magnetic field or a current it is possible to modify the magnetic properties of these domains and metals. A closer look at this mechanism might yield some more insight in designing new materials. The model we will use to describe these domains is micromagnetics, which is an approximation of magnetic systems on the level of nano- to micrometers. At this scale it is possible to describe the properties and dynamics of the domain walls. The dynamics of these walls are important for a description of the magnetic properties of a material. In particular, it is relevant for storing and transfering information. In this work we visualize the micromagnetic dynamics based on the Landau-Lifshitz-Gilbert (LLG) equation with spin-transfer torque (STT) terms. We study the effect of damping on these dynamics through simulating the systems using the Object Oriented Micromagnetic Modular Framework (OOMMF)[1]. In particular, we are interested in the movement and formation of domain walls, and spin-transfer torque. In chapter 2 we will discuss the micromagnetic interactions and dynamics present in the simulated systems. These interactions determine the physical properties of the system. The LL equations govern the movement of the magnetic moments and are highly dependent on the aforementioned interactions. We will then define magnetic domains and domain walls in chapter 3. Chapter 4 shall describe the methodology for simulating the systems by discussing the parameters and initial conditions, as well as any predictions of the results. The results are presented and discussed in chapter 5, where we also will reflect upon the predictions as well as model. Finally, we will conclude this piece in chapter 6 and 7 with a recap and discussion. 3 2. Micromagnetic Interactions 2.1. Overview We consider the micromagnetic energy, which consists of the exchange interaction, magne- tocristalline anisotropy and an (external) Zeeman energy. This energy is given by: E = Eext + Eani + Eexc (2.1) where Eext, Eani and Eexc are the energy contributions from external magnetic fields, magne- tocrystalline anisotropy interaction and exchange interaction respectively[2]. Z 3 Eext = −µ0 d rH~ ext · ~m(~r) (2.2a) Z Ku 3 2 Eani = − 2 d r(^n · ~m(~r)) (2.2b) Ms Z Aex X @ E = d3r ( ~m)2 (2.2c) exc M 2 @r s α α −7 where µ0 is the magnetic constant or 4π × 10 T·m/A, H~ ext is the external magnetic field 2 in A/m, ~m(~r) is the magnetic moment at position ~r in A·m , Ku is the magnetocrystalline 3 anisotropy constant in J/m , Ms is the saturation magnetization in A/m,n ^ is the magnetocrys- talline anisotropy direction and Aex is the exchange energy constant in J/m. Note that we do not consider demagnetization energy as well as the Dzyaloshinskii-Moriya energy, as the point of these simulations is to show the effect of anisotropic damping on the proposed systems by using a minimal set of effects necessary for domain walls. The micromag- netic energy is used to derive the effective field H~ eff , which is used in describing the dynamics of the moments in the later sections. In the following sections, Eext, Eani and Eexc given are in the form of a discrete micromag- netic hamiltonian. However, the energies as seen in Eqs. (2.2a) to (2.2c) are continuous. The transformation from the hamiltonian to this continuous model should be fairly self-evident, as it is simply a matter of applying the thermodynamic limit. 2.1.1. Exchange Energy The exchange interaction is a direct consequence of the exchange symmetry of wavefunctions for non-indistinguishable particles. As we consider a fermionic system, the overall wavefunction is anti-symmetric which leads to an anti-symmetry in position space and symmetry in spin space or vice-versa. The energy difference between the two configuration caused by the exchange symmetry is called the exchange energy. As was shown independently by Heisenberg and Dirac, this exchange energy can be written in the form: X m^ i · (m ^ i − m^ j) H = A (2.3) exc;i ex ∆2 <i;j> ij . Herem ^ i andm ^ j are the directions of the magnetic moments in cells i and j respectively, and ∆ij is the distance between two discrete cells in m. The summation iterates over nearest neighbors. 4 The sign of Aex determines if the system is ferromagnetic or anti-ferromagnetic. For positive Aex the energy is at a minimum whenm ^ i =m ^ j. This is the case for ferromagnetic systems. For negative Aex we have an anti-ferromagnetic system, where the energy is minimized when m^ i = −m^ j. 2.1.2. Magnetic Anisotropy In certain materials a preferred direction of magnetization can be found. This direction is called an easy axis, and the energy is given by: 2 Hani;i = Ku(1 − (m ^ i · n^) ) (2.4a) This equation holds only for positive Ku. In some materials Ku is negative and the interaction describes an easy plane, written as: 2 Hani;i = −Ku(m ^ i · n^) (2.4b) Figure 2.1.: Anisotropy energy of cell i as a function of angle θ between the spin direction of the moment ~mi and the anisotropy directionn ^. Note that Ku;axis > 0 for the easy axis situation and Ku;plane = −Ku;axis. Fig. 2.1 shows that the moment naturally tends towards an angle of 0 (parallel) or π (anti- parallel) in an easy axis situation, as the lowest energy state is the most stable one in general. In the case of an easy plane the preferred angle is 0.5π in which case any direction within the entire plane orthogonal ton ^ is the preferred direction. 2.1.3. Zeeman Energy An external magnetic field induces a torque on the magnetic moments and causes the spin to align with the field. This is called the Zeeman energy and is given by: @ ~m H = −µ ( i · H~ ) (2.5) zee;i 0 @V ext The external field causes a torque in ~mi which in turn causes it to align with H~ ext. 5 2.2. Micromagnetic Dynamics The dynamics of magnetic system is described by a phenomological non-linear differential equa- tion, introduced by Landau and Lifshitz in 1953. This equation is called the Landau-Lifshitz (LL) equation and is given by[3]: _ ~m = −|γ¯j~m × H~ eff − jγ¯jα~m × (~m × H~ eff ) (2.6) where ~m is the reduced magnetization per unit volume ~m = M~ and jγ¯j is the gyromagnetic ratio Ms in m/(A·s). This equation was later reformulated by Gilbert leading to the Landau-Lifshitz- Gilbert (LLG) equation, written as[4]: _ _ ~m = −|γj~m × H~ eff + α~m × ~m (2.7) γ This is equivalent to Eq. (2.6) under the relation given by jγ¯j = j 1+α2 j. 2.2.1. Effective field The effective field is derived from the aforementioned total free energy given by Eqs. (2.1) and (2.2a) to (2.2c). The total effective field is given by the functional derivative of the free energy to the magnetization[2]: 1 δE H~ = − eff µ δ ~m(~r) 0 (2.8) ~ 2Ku 2Aexc 2 = Hext + 2 (^n · ~m(~r))^n + 2 r ~m µ0Ms µ0Ms This effective field is a magnetic field representing the preferred orientation of the magnetic moments at a position ~r. 2.2.2. Precession and Damping Isotropic _ ~m = −|γ¯j~m × H~ eff − jγ¯jα~m × (~m × H~ eff ) (2.9a) | {z } | {z } precession damping The LL equation consists of two parts which we call the precession and damping term. The first term describes the precession around H~ eff (see Fig. 2.2), whereas the damping term forces the direction of the movement towards H~ eff . The movement of this moment is visualized in Fig. 2.3b which shows that the moment tries to align with H~ eff .