Effects of Damping on Micromagnetic Dynamics

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 presence 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, magnetocrystalline 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(ˆn · ~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 ˆ is the magnetocrystalline 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 micromagnetic 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 micromagnetic 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 mˆ i · (m ˆ i − mˆ j) H = A (2.3) exc,i ex ∆2 ij

. Herem ˆ i andm ˆ 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 ˆ i =m ˆ j. This is the case for ferromagnetic systems. For negative Aex we have an anti-ferromagnetic system, where the energy is minimized when mˆ i = −mˆ 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 ˆ i · nˆ) ) (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 ˆ i · nˆ) (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 ˆ. 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 ˆ 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 equation, introduced by Landau and Lifshitz in 1953. This equation is called the Landau-Lifshitz (LL) equation and is given by[3]: ˙ ~m = −|γ¯|~m × H~ eff − |γ¯|α~m × (~m × H~ eff ) (2.6) where ~m is the reduced magnetization per unit volume ~m = M~ and |γ¯| 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 = −|γ|~m × H~ eff + α~m × ~m (2.7)

γ This is equivalent to Eq. (2.6) under the relation given by |γ¯| = | 1+α2 |.

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 (ˆn · ~m(~r))ˆn + 2 ∇ ~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¯|~m × H~ eff − |γ¯|α~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 .

Figure 2.2.: A visual representation of the terms in the LL-equation as well as a STT-term (from Ralph & Stiles, 2008)[2].

6 Figure 2.3.: a: Initial direction of the magnetization. b: Visualization of the trajectory of the magnetic moment for a weak current relative to the effective field. The moment is allowed to align parallel to the applied field. c: The anti-damping equalizes the damping, causing the spin to precess at the point where the two (anti-)damping terms balance eachother out. d: Application of a current much stronger than the magnetic field causes the moment to align anti-parallel to the applied field, as the STT anti-damping is much stronger than the damping term. (Modified, from Ralph & Stiles, 2008)

Anisotropic Usually the damping is taken to be a scalar, but it could also be expressed as a tensor[5, 6, 7, ˙ ˙ 8]: α~m × ~m → αi ~m × ~mi. In this work we consider a modiﬁed version of the LL-equation where the isotropic damping constants are split into two anisotropic constants, which diﬀer depending on the direction of ~m × H~ . ˙ ~m = −|γ¯|~m × H~ eff − |γ¯|[α1 ~m × (~m × H~ eff )x,z + α2 ~m × (~m × H~ eff )y] (2.9b) The constants and variables are the same as above, with the exception of both the damping term α and the cross terms ~m × (~m × H~ ), both of which have been split up in an out-of-plane and in-plane part.

2.2.3. Spin-Transfer Torque Slonczewsk and Bergeri proposed that a current gets spin-polarized in one layer and the mis- match between the direction of the conducting electrons’ spin and the magnetic moment in the second layer induces a torque. This torque is responsible for the transfer of angular momentum from the spin current to the 2nd magnetic layer and is called spin-transfer torque (STT)[9, 10]. In the simulation it is assumed that the current is already spin-polarized before reaching the layer in the model, and the LLG equation can be supplemented with thel STT term proposed by Slonczewski as follows[1]: ˙ ˙ 0 ~m = −|γ|~m × H~ eff + α(~m × ~m) + |γ|β~m × (~mp × ~m) − |γ|β ~m × ~mp (2.10a) | {z } | {z } | {z } | {z } precession damping anti-damping ﬁeld-like

Compare with Eq. (2.7), with the addition of β, the normalized electron polarization vector ~mp, a spin-transfer term and a secondary spin-transfer term 0. Furthermore, β is described as ¯h J β = | | µ0e tMs whereh ¯ is the reduced Planck constant in J·s, e is the charge of an electron in C, J is the current density in A/m2 and t the thickness of the layer in m. is given by P Λ2 = 2 2 (Λ + 1) + (Λ − 1)(~m · ~mp)

7 where P is the polarization strength and Λ a factor determining the dependence of on ~m · ~mp. The first part of the spin-torque is odd under time-reversal and is described as anti- damping, while the second part is even and is called the field-like term. The influence of the anti-damping term on the moment’s movement depends the direction and strength of ~mp: It either is overpowered by the damping term, counteracts the damping, or overpowers it entirely (see Fig. 2.3b-d respectively). The last case is called switching, due to the moment ’switching’ to the other direction as a result of the strength of the STT. The field-like term can be compared with the precession term with a different axis. We add anisotropic damping to Eq. (2.10a) by taking the LL-form and splitting the damping term like we did with Eq. (2.9b).

˙ ~m = −|γ¯|~m × H~ eff − |γ¯|[α1 ~m × (~m × H~ eff )x,z + α2 ~m × (~m × H~ eff )y] 0 + |γ¯|β~m × (~mp × ~m) − |γ¯|β ~m × ~mp (2.10b)

8 3. Magnetic Domains

3.1. Overview

A system may assume a semi-stable state during a simulation. This quasi-equilibrium manifests itself as localized pockets of aligned spins, which are called domains. The formation of these domains arises from a conflict between the exchange, external field and magneto-crystalline interactions each favoring a configuration of the magnetic moments. We identify three different states to describe these configurations:

• Homogenized, where all the moments have the same orientation;

• Domains, with the aforementioned localized pockets;

• Disordered, where there is no systematic ordering to the moments.

3.2. Domain Walls

A domain wall (DW) is the transition between two domains. The exact properties of these walls depend on the interactions present in the system, and the presence of a wall could indicate the system being in a (semi-)stable state.

Figure 3.1.: An example of a 1D system at its ground state, assuming an external ﬁeld or magnetic anisotropy pointing up. Although the complete system is in a 3D space, the moments are placed in a 1D (linear) fashion, hence it can be considered a 1D system.

The following systems (Figs. 3.2 and 3.3) are at a local minimum. Given time, the two domain wall types in the following sections should in principle resolve over time to the situation as depicted in Fig. 3.1.

3.2.1. Bloch walls Bloch walls are one of the two ways to categorize domain walls. We deﬁne this wall based on the orientation of the spins within the domain wall relative to the direction of the transition. In the case of the Bloch wall, the direction of the spins in the wall are all parallel to the plane separating the two domains. Fig. 3.2 shows a Bloch wall as viewed from the side.

Figure 3.2.: An example of a Bloch domain wall in 1D. Note that the moments in the middle of the wall do have a direction: they are pointing out of the visible plane.

9 3.2.2. Neel walls Just like the Bloch wall, Neel walls form a transition between two magnetic domains. Unlike Bloch walls however, these walls are oriented diﬀerently. Instead of going parallel to the plane separating the domains, the magnetic moments point from one domain to the other. An example of the Neel wall is shown in Fig. 3.3.

Figure 3.3.: An example of a Neel wall in 1D. All of the moments are pointing towards a direction within the visible plane.

10 4. Methods

In order to explore the effects of anisotropic damping a number of different simulation conditions were used. In this section, the changes to the code as well as the specific conditions of every system will be discussed.

4.1. Program

The following simulations were done using a modified version of the Object Oriented Micro- magnetic Framework (OOMMF) software, which has been used in scientific context[1, 11]. The changes made to the code are shown in Appendix A. These modifications were necessary to accommodate the anisotropy present in Eqs. (2.9b) and (2.10b), since neither are supported within the software. In the modified code they ˆ-direction is differentiated from the orthogonal xz-plane for the purposes of introducing anisotropy to the damping constant. Henceforth, unless otherwise specified, they ˆ-direction is taken to be the out-of-plane direction whilst thex ˆ andz ˆ directions are defined as in-plane. One iteration corresponds to a value in the range of 10−19 seconds simulation time, the exact value depending on the state of the simulation.

4.2. Systems

4.2.1. Overview As the various systems each have attributes that do not overlap, the properties will be noted for each system below. However, an overview of the common parameters can be found in Table 4.1.

Test Damping type Grid layout Exchange const.(J/m) Anisotropy const.(J/m3) Spon. Domain Formation Isotropic 100 × 1 × 100 3 · 105 3 · 105 2D Movement Isotropic 50 × 1 × 100 1 · 10−12[11] 3.5 · 104 Anisotropic 50 × 1 × 100 1 · 10−12 3.5 · 104 Switching Anisotropic 10 × 1 × 10 − 3.5 · 104 Table 4.1.: An overview of the common properties of the simulated systems. A more detailed description of each simulation will be presented in its respective subsection. Note that the grids are in an xyz-format and that the cells are 1x1x1 m (for the ﬁrst system), and 1x1x1 nm (for the rest of the systems).

It is worth mentioning that the systems stretch out in the xz-directions. Since they ˆ-direction is taken to be the out-of-plane direction, the xz-plane was thus chosen to represent the 2D plane. Additionally, since every system has a positive exchange interaction constant, all of them are ferromagnetic.

4.2.2. Spontaneous Domain Formation To better understand the formation of a domain wall, a simple simulation was set up. Starting with a system with all the moments in a random direction orthogonal to the y-axis, this system was allowed to evolve in time. With an easy axis pointing in they ˆ-direction as well as the exchange interaction being involved, the moments should prefer the ±y-direction.

11 Iso DWM Aniso DWM Switching

Fixed α1 Fixed α2 α α1 α2 α1 α2 α1 α2 1.0 1.0 0 0.0 0.0 0.0 0.0 0.9 0.9 0.1 1.0 0.0 0.0 1.0 0.83 0.83 0.17 1.0 0.17 0.17 1.0 0.67 0.67 0.33 1.0 0.33 0.33 1.0 0.5 0.5 0.5 1.0 0.5 0.5 1.0 0.33 0.33 0.67 1.0 0.67 0.67 1.0 0.17 0.17 0.83 1.0 0.83 0.83 1.0 0 0 1.0 1.0 1.0 1.0 1.0 Table 4.2.: An overview of the values for α, α1 and α2 used for the diﬀerent systems.

Although the exchange constant in Table 4.1 may seem to diﬀer nearly an order 1018 from the constant for the other systems, the resulting exchange interaction is comparable since according 2 to Eq. (2.3) it is dependent on the distance between two cells squared. In this situation, ∆ij. With ∆ij = 1 m for the random system and ∆ij = 1 nm for the others. Therefore the factor discrepancy of 1018 is accounted for properly. There is also a diﬀerence in magnetocrystalline anisotropy constant. In all the systems the contribution from exchange energy to the total energy is multiple orders of magnitude higher than the contribution from the magnetic anisotropy, so the anisotropy should be negligible.

4.2.3. 2D Domain Movement We study the effects of anisotropic damping on the dynamics of domain wall movement (DWM) by extending the systems portrayed in Figs. 3.2 and 3.3 to 2D, with these systems starting out 6 in either with a Bloch or Neel wall. In addition, we take an external field By = −10 T, with the easy axis parallel to this field aswell. Therefore H~ eff should mostly point in the -ˆy-direction with H~ exc causing a deviation in the direction. Furthermore, the α values used in the simulations can be found in Table 4.2. The relation between the isotropic and anisotropic α values is given by αiso = αani,1 = 1 − αani,2.

4.2.4. Switching We study the response of individual spins to an external field and STT for various sets of anisotropic damping constants. In particular we wish to parametrize when switching occurs, where we define switching as the points at which the spins align. We start off with a configuration of magnetic moments similar to the one found in the Spontaneous Domain Formation simulation and disabled the exchange interaction. Due to the application of the current this system uses Eq. (2.10b) instead of Eq. (2.9b) for the movement of the moments. ”Switching” refers to the system flipping, or the magnetic moments aligning in one direction, when certain conditions, such as the external field reaching a sufficient strength, are met. Each simulation runs only for 1000 iterations before stopping, to keep simulation times at a reasonable level. We varied the field strength B and the current density J between the range 11 11 2 [-10 , 10 ] mT and A/m respectively. We chose B~ = B0xˆ,m ˆ p =z ˆ and an easy axis in the yˆ-direction. Both ends of this range were chosen based on the system resolving in the iteration limit. This was done for a number of α1, α2 combinations, which are displayed in Table 4.2.

12 5. Results and Discussion

5.1. Spontaneous Domain Formation

We ran the simulation for 20000 iterations. From Fig. 5.1 we can obtain a general picture of what happens during the formation and consequent growth of the domain walls from a random system, assuming the moments start orthogonally to the z-axis (left to right). First small domains form rapidly as the individual moments align, with the orientation parallel to the y-axis determined by the y-component the moment starts with. Afterwards, these clusters slowly grow out to become large domains, which start swallowing the smaller domains with the opposite orientation.

(a) Initial condition (b) After 500 iterations

(c) After 1000 iterations (d) After 3000 iterations Figure 5.1.: Snapshots of the evolution of a random system depicting spontaneous formation of domains. The colors in the images correspond to the average magnetization in they ˆ-direction at that position, with red marking areas for positive y, blue for negative y and white for no y-component respectively.

In Fig. 5.2 the sharp drop in energy up until the dotted line coincides with the formation of the small pockets of similar direction. At this point the dynamics of the system is likely dictated by the exchange interaction forcing the moments to cluster, with the magnetocrystalline interaction forcing a preferred axis for the moments to align with.

13 Figure 5.2.: Total energy as a function of iterations for the same system as in Fig. 5.1. The energy drops sharply up until the dotted line at around 1000 iterations. With the very slow change in the energy curve towards the latter half of the simulation it may be considered in a semi-stable state.

5.2. Domain Wall Movement

5.2.1. Isotropic The development of a system starting out with a domain wall and at diﬀerent values for the isotropic damping can be brieﬂy explained with Fig. 5.3.

Figure 5.3.: A comparison of the average magnetization and energy for α = 1. The ﬁgure is divided into four distinct regions where the movement changes according to the energy curve. (I) The spins relax from the initial imposed orientations. (II) Domain wall movement causing a near-linear change inm ¯ y as well as E. (III) The domain wall reaches the edge of the system. This is accompanied by a sharp decline inm ¯ y, as well as E due to one of the domains disappearing, in this case the +y one. (IV) The system is homogenized afterm ¯ x andm ¯ z settles.

Both the total energy as well as averagem ˆ y reach a stable state faster for higher values of α. This trend holds for both the Neel as well as the Bloch case. Aside from this, there is a striking similarity between the two wall types in Figs. 5.4 and 5.5, in that:

14 • The Blochm ¯ x is (mirrored through them ¯ x = 0-axis) the same as the Neelm ¯ z,

• Both the Bloch and Neelm ¯ y graphs are the same and,

• Blochm ¯ z is the same as Neelm ¯ x. In the description of the two walls it was noted that the two are identical save for a rotation in the plane separating the two domains. There is therefore a symmetry between the Bloch and  0 0 1 Neel situation if ~m → Γ~m and H~ → ΓH~ , given that Γ =  0 1 0, leaves the LL equation −1 0 0 invariant. We can prove by direct calculation as follows:           mx Hx mx mx Hx ˙ ~m = −|γ¯| my × Hy − |γ¯|α my × (my × Hy) mz Hz mz mz Hz (5.1)  2    α(mx − 1) αmxmy − mz αmxmz + my Hx 2 ~ = −|γ¯| αmxmy + mz α(my − 1) αmymz − mx Hy = −|γ¯|Λ(~m)Heff 2 αmxmz − my αmymz + mx α(mz − 1) Hz

The Bloch situation at t=0 has nom ¯ x component, thereby reducing the contribution of the exchange interaction to the effective field to onlyz ˆ-component. Combined with the external field in they ˆ-direction, we obtain:   −Hymz + Hzmy ˙ 2 ~m = −|γ¯| Hyα(my − 1) + Hzαmymz (5.2) 2 Hyαmymz + Hzα(mz − 1)

In the case of a Neel wall, them ¯ z component is missing instead, and therefore also Hz. Thus this LLG equation can be written as follows:

 2  Hxα(mx − 1) + Hyαmxmy ˙ 2 ~m = −|γ¯| Hxαmxmy + Hyα(my − 1) (5.3) −Hxmy + Hymx   mz ~ Now suppose we perform a transformation such that m˜ = Γ~m =  my . Assuming then that −mx the transformed system is in a bloch configuration, we get:   −H˜ym˜ z + H˜zm˜ y ~˙ ˜ 2 ˜ m˜ = −|γ| Hyα(m ˜ y − 1) + Hzαm˜ ym˜ z (5.4) ˜ ˜ 2 Hyαm˜ym˜ z + Hzα(m ˜ z − 1) ~ where H˜x,y,z are the transformed components of the effective field for H˜eff = ΓH~ eff as well. Writing out these transformed elements gives:     m˙ z Hymx − Hxmy 2  m˙ y  = −|γ|  Hyα(my − 1) + Hxαmymx  (5.5) 2 −m˙x −Hyαmymx − Hxα(mx − 1) which is the same as Eq. (5.3). Therefore, rotating a Neel system by 90◦ through the y-axis nets a Bloch system, which has the exact same dynamics as the rotation preserves the properties of the systems.

15 Figure 5.4.: Time-evolution of the average magnetization for a Bloch (left column) and Neel (right) domain wall with isotropic damping for varying values of α. Note how the graphs for the Blochm ¯ z and Neelm ¯ x as well asm ¯ y starts above 0, due to the starting orientations of the moments.

16 Figure 5.5.: Energy as a function of time corresponding to the same system as in Fig. 5.4.

(a) Initial condition (b) After 1000 iterations (c) After 3000 iterations Figure 5.6.: Snapshots of the evolution of a domain wall starting out in a Neel conﬁguration for α1, α2 = 0.33, 0.67. The colors in the images correspond to the average magnetization in they ˆ- direction at that position, with red marking areas for positive y, blue for negative y and white for no y-component respectively.

Next, we consider the group-velocity of the domain walls as a function of α. A domain wall with magnetization m moving in thex ˆ-direction can be written as m(x-vt). If we take the ∂m ∂m time-derivative of m(x-vt) we obtain:m ˙ = −v ∂x , where v is the group speed. We assume ∂x to be constant for all values α, as the domain walls start out like Figs. 3.2 and 3.3 and moves via translation towards one of the edges within the linear regions in them ¯ y-plots in Fig. 5.4. This movement can be seen in Fig. 5.6. Fig. 5.9 therefore suggests a non-linear relation between the group speed and α.

5.2.2. Anisotropic We attempt to break the symmetry present between the two cases by introducing an anisotropic damping term. However, even though the perpendicular and orthogonal parts of M × H were split up, the results are identical to the isotropic situation in the case α1 = αiso, as can be seen from Fig. 5.7.

17 Figure 5.7.: Time-evolution of the average magnetization for a Bloch and Neel domain wall with anisotropic damping. Compare with Fig. 5.4.

Fig. 5.8 does appear to diﬀer from Fig. 5.5, but considering the dynamics of the system otherwise appear to be identical, it is more likely that this is due to a bug caused by the modiﬁcation of the code.

18 Figure 5.8.: Time-evolution of the energy with anisotropic damping. Compare with Fig. 5.5.

3 Figure 5.9.:m ¯ y diﬀerentiated over simulation time at t=10 iterations, plotted over diﬀerent values of alpha. This plot holds for both Bloch as well as Neel, as earlier it was proven that my is identical in both cases.

As in the isotropic case, the same symmetry between the Bloch and the Neel case can be found. This similarity might be caused by to the symmetry in the behaviour of the angular moments in both situations. Combining Eqs. (2.9b) and (5.1) we obtain:  2 2    −α1my − α2mz α1mxmy − mz α2mxmz + my Hx ˙ 2 ~m = −|γ¯|  α1mxmy + mz −α1(my − 1) α1mymz − mx  Hy (5.6) 2 2 α1mxmz − my α2mymz + mx −α1my − α2mx Hz

From this equation it should be quite clear why α1 is the main damping constant. The entirety ˙ of ~my is determined by α1, therefore inﬂuencing the main driving force behind the domain wall dynamics. It is also worth noting that ~m is normalized, which means that any multiplication of its components is likely to net a smaller contribution to the movement equation. We speculate this is the reason why the anisotropic eﬀect is negligible, as the precession provides most of the movement.

5.3. Switching

We show the extreme cases for α1 and α2 in Fig. 5.10. The contour plots show the average magnetization in they ˆ-direction depending on the applied external field B and current density strength J. More tests varying the damping constants between the ones in the shown plots were done but their effects are better described by the slices shown in Figs. 5.11 and 5.12. Fig. 5.10 shows that with an increasing α1 the edges of the diamond shift outwards, while less so for a change in α2. The slices in Fig. 5.11 offer a better view of this shift, as the resulting band

19 ofm ¯ y for a changing α2 is much tighter most notably after the peak, while it is signiﬁcantly broader when varying α1. In that sense these results are quite similar to the ones in the previous section, where α1 contribution a signiﬁcant portion to the dynamics of the magnetic moments and α2 less so. The movements of the moments is described by Eq. (2.10b). Grouping similar terms together, we rewrite it as follows:

˙ 0 ~m = −|γ¯|~m × (H~ eff + β ~mp) − |γ¯|~m × [α1(~m × H~ eff )x,z + α2(~m × H~ eff )y + β(~mp × ~m)] (5.7)

The contour plots show a distinct diamond shape suggesting that the relation between switching and the B and J values are likely linear. We prove this linear relationship arises from the conflict between the damping and the anti-damping term in the movement equation. More specifically, a value for the critical field strength is obtained when the two terms are equal. Supposing that H~ eff ≈ B~ largely due to the rather large field strengths chosen for the simulations, we obtain for the relation between Bcrit and J:

 0   m  β β0 y  mz  B = − (~mp × ~m) = − J −mx (5.8) α1 α1 −my 0

0 where we used β = β0 × J. This set of linear equations therefore gives B = − β mz J. crit α1 mx However, this statement is highly suspect due when looking at the bottom graph in Fig. 5.11, as it does not appear to show a shift in the peak for diﬀerent α1 values.

Figure 5.10.: Four contour plot detailing the response of the system to the strength of J and B for diﬀerent values of anisotropic damping constants. The dotted line at J=1e11 indicates where the slices displayed in Figs. 5.11 and 5.12 are located.

In the previous section we concluded that the α = 0 situation resulted in no dynamics. This is not exactly the case when we apply a current in thez ˆ-direction, as should be evident from

20 Fig. 5.12. Rewriting Eq. (5.7) for a system without damping gives us:

˙ 0 ~m = −|γ¯|~m × (H~ eff + β ~mp) − |γ¯|β~m × (~mp × ~m) (5.9)

In the absence of the damping term, it appears that the stable state is determined entirely by anti-damping term. Them ¯ y-component of the moment remains at zero as the damping term is absent. Instead, the anti-damping term forces the moments to align (anti-)parallel to ~mp. The dependence ofm ¯ x on B then likely means that the precession term is still relevant, as it does not allow the magnetic moments to settle (anti-)parallel to ~mp.

Figure 5.11.: Slices of the contour plots at a fixed J-value indicated in Fig. 5.10. The top figure shows the effect of keeping α1 constant while varying α2, with the inverse for the bottom figure. The external field strength associated with the peaks are suspected to be the critical value at which switching occurs.

̄ − ̄ ̄ − ̄ ̄ ̄ − ̄ ̄ − − − − − B

̄ − ̄ ̄ − ̄ ̄ ̄ − ̄ ̄ − − − − − B Figure 5.12.: Slices of the contour plots at the same place as in Fig. 5.11. Note the discontinuity at B = ±0.5 · 1011 T, which corresponds to the critical values in Fig. 5.11.

22 6. Conclusion

We found that there is no difference between the Bloch and Neel systems due to the inherent symmetry in the chosen system in the case of Domain Wall Movement. Furthermore, the movement of the domain walls can be described as a wave with a group velocity with a non- linear dependence on the damping constant α. We tried to break the symmetry in the previous test by breaking up the damping term into anisotropic terms, but found that this had no effect on the system as it does not appear to break this symmetry, likely due to the chosen system. In the case of switching, we conclude that the effect of α2 is much smaller than α1, but still is present. We have also obtained a critical field strength independent of α.

23 7. Outlook & Discussion

7.1. Domain Wall Movement

Is it then possible to break the symmetry at all? As the system likely is symmetryical due to the choice of directions for the interactions, the obvious solution would be to choose the directions such that both the α1 as well as α2 terms in Eq. (5.6) are applicable, such as ensuring that H~ eff points in thex ˆ orz ˆ direction. Currently H~ eff is mostly determined by Hy due to the magnetocrystalline anisotropy interaction and external field, with only a minor contribution from the Hx and Hz terms from the exchange interaction. This places a large emphasis on the middle column of the matrix, and changing the direction of the external field then should shift it significantly to the other columns, which do contain more significant anisotropic damping terms. Alternatively, if the directions of the anisotropic damping are changed it is likely to contribute towards breaking the symmetry.

7.2. Switching

The chosen values for both the external field and current density are likely both unrealistic in the context of real-life applications. It may be possible to obtain the same results over a larger time-scale with smaller numbers, however. Regardless, Fig. 5.11 remains puzzling: Why doesm ¯ y tend back towards 0 after B reaches Bcrit? According to Fig. 2.3 and Eq. (2.10b), a sufficiently high Heff will lead to the moments aligning with Heff . Yet, what we see in Fig. 5.11 are two peaks at most, which is most troubling. Perhaps it is worth exploring deeper to find a cause?

24 References

[1] M. J. Donahue and D. G. Porter. OOMMF User’s Guide, Version 1.0. Tech. rep. Gaithers- burg, MD: National Institute of Standards and Technology, Sept. 1999. [2] D. C. Ralph and M. D. Stiles. “Spin Transfer Torques”. In: J. Magn. Magn. Mater. 320 (Nov. 2008), pp. 1190–1216. [3] M Lakshmanan. “The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview”. In: Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 369.1939 (2011), pp. 1280–1300. [4] T. L. Gilbert. “A phenomenological theory of damping in ferromagnetic materials”. In: IEEE Transactions on Magnetics 40.6 (2004), pp. 3443–3449. issn: 0018-9464. doi: 10. 1109/TMAG.2004.836740. [5] C. Vittoria and S. D. Yoon. “Relaxation mechanism for ordered magnetic materials”. In: Phys. Rev. B: Condens. Matter Mater. Phys. 81.014412 (Jan. 2010). [6] K. Gilmore et al. “Anisotropic damping of the magnetization dynamics in Ni, Co, and Fe”. In: Phys. Rev. B: Condens. Matter Mater. Phys. 81.174414 (2010). [7] J. Seib, D. Steiauf, and M. Fähnle.“Local and nonlocal atomic contributions to unit-cell damping in near-adiabatic collinear magnetization dynamics”. In: Phys. Rev. B: Condens. Matter Mater. Phys. 79.064419 (Feb. 2009). [8] Jonas Seib, Daniel Steiauf, and Manfred Fähnle.“Linewidth of ferromagnetic resonance for systems with anisotropic damping”. In: Phys. Rev. B: Condens. Matter Mater. Phys. 79.092418 (Mar. 2009). [9] J.C. Slonczewski. “Current-driven excitation of magnetic multilayers”. In: J. Magn. Magn. Mater. (1996). [10] Z. Li and S. Zhang. “Magnetization dynamics with a spin-transfer torque”. In: Phys. Rev. B: Condens. Matter Mater. Phys. (2003). [11] Chun-Yeol Yu. “Dependence of the Spin Transfer Torque Switching Current Density on the Exchange Stiffness Constant”. In: Appl. Phys. Express 5.10 (Sept. 2012).

25 A. Code changes alphaevolve.cc Aside from every mention of EulerEvolve being replaced with AlphaEvolve instead, the following modiﬁcations were made: ... //Define anisotropic alpha a l p h a ort = GetRealInitValue(”alpha1” ,0.5); alpha par = GetRealInitValue(”alpha2” ,0.5); ... i f (HasInitValue(”gamma G”) && HasInitValue(”gamma LL” ) ) { throw Oxs ExtError ( this , ” I n v a l i d S p e c i f y block ; ” ” both gamma G and gamma LL specified.”); } else i f (HasInitValue(”gamma G”)) { gamma = GetRealInitValue(”gamma G”)/ (1+(( a l p h a o r t+alpha par ) / 2 ) ∗ ( ( a l p h a o r t+alpha par ) / 2 ) ) ; } else i f (HasInitValue(”gamma LL” ) ) { gamma = GetRealInitValue(”gamma LL” ) ; } else { gamma = 2.211e5/(1+alpha o r t ∗ alpha par ) ; } ... void Oxs AlphaEvolve :: Calculate dm dt (....) { .... OC REAL8m c o e f 2 = −a l p h a o r t ; OC REAL8m c o e f 3 = −alpha par ; ... //Split mxH into parallel and orthogonal part ThreeVector mxH ort = Oxs ThreeVector(0, scratch.y, 0); ThreeVector mxH par = Oxs ThreeVector(scratch.x, 0, scratch.z); //Orthogonal mxmxH mxH ort ˆ= s p i n [ i ] ; mxH ort ∗= c o e f 2 ; //Parallel mxmxH dm dt [ i ] += mxH ort ; mxH par ˆ= s p i n [ i ] ; mxH par ∗= c o e f 3 ; dm dt [ i ] += mxH par ; ... } alphaspinxferevolve.cc Like before, every mention of SpinXFerEvolve was replaced with AlphaSpinXFerEvolve, as well as the following modiﬁcation:

26 ... i f (HasInitValue(”eta1”)) { OXS GET INIT EXT OBJECT(”eta1” ,Oxs ScalarField , eta1 i n i t ) ; } else { e t a 1 init .SetAsOwner(...); } i f (HasInitValue(”eta2”)) { OXS GET INIT EXT OBJECT(”eta2” ,Oxs ScalarField , eta2 i n i t ) ; } else { e t a 2 init .SetAsOwner(...); } ... void Oxs AlphaSpinXferEvolve :: Calculate dm dt (...) { ... ThreeVector scratchA1 = Oxs ThreeVector(mxH.x, 0, mxH.z); ThreeVector scratchA2 = Oxs ThreeVector(0, mxH.y, 0); scratchA1 ∗= c e l l e t a 1 ; scratchA2 ∗= c e l l e t a 2 ; ThreeVector scratchA = (scratchA1 + scratchA2); //Normalize mxH scratchA ∗= 1/( c e l l e t a 1 + c e l l e t a 2 ) ; scratchA ∗= c e l l a l p h a ; ThreeVector scratchB = mxp; scratchB ∗= c o e f 2 ; scratchA += scratchB; }