Berry curvature and dynamics of a magnetic bubble

Magnetic bubbles have been the subject of intensive studies aiming to investigate their applications to memory devices. A bubble can be regarded as the closed domain wall and is characterized by the winding number of the in-plane components or the skyrmion number Nsk, which are related to the number of Bloch lines (BLs). For the magnetic bubbles without BLs, the Thiele equation assuming no internal distortion describes the center-of-mass motion of the bubbles very well. For the magnetic bubbles with BLs, on the other hand, their dynamics is affected seriously by that of BLs along the domain wall. Here we show theoretically, that the distribution of the Berry curvature bz, i.e., the solid angle formed by the magnetization vectors, in the bubble plays the key role in the dynamics of a bubble with N sk = 0 in a dipolar magnet. In this case, the integral of bz over the space is zero, while the nonuniform distribution of bz and associated Magnus force induce several nontrivial coupled dynamics of the internal deformation and center-of-mass motion as explicitly demonstrated by numerical simulations of Landau–Lifshitz-Gilbert equation. These findings give an alternative view and will pave a new route to design the bubble dynamics.

Magnetic textures in ferromagnets play vital roles in determining their physical properties. Domain walls (DWs) and bubbles are the representative examples of magnetic textures, and their dynamics is a keen issue from the viewpoints of both fundamental science and applications [1,2]. The DWs are classified into Bloch wall and Néel wall, where the direction of the in-plane magnetization at the wall is parallel or perpendicular to the wall itself, respectively. From the viewpoint of the dipolar energy, Bloch wall is preferable since the magnetic charge - M · (M : magnetization) is zero. However, it often happens that the Bloch lines (BLs) are inserted into the Bloch wall. At the BL, the local Néel wall is introduced to separate the two Bloch wall regions of opposite magnetic directions.
When the DW forms a cylinder, it becomes a magnetic bubble. In this case, BL has a topological meaning because it is related to the skyrmion number N sk of the bubble [3,4]. Let N win be the vorticity of the bubble defined as the winding number of the in-plane components of the magnetization direction along the wall. Let f F( ) be the angle of the in-plane magnetization along the wall measured from the x-axis, i.e.,    ) at the core. When we put n into equations (1) and (2), one obtains = -N N sk win . When the DW of the bubble is always the Bloch wall (see figure 1(a)), i.e., f p F =  2, the winding number = N 1 win , and the bubble is topologically the same as the skyrmion. On one hand, when the DW has the mixed winding segments with Bloch and Néel wall structure, the magnetic bubble can take different values of N sk . For example, the magnetic bubble shown in figure 1 which is opposite sign to the skyrmion, so that it is called antiskyrmion. As seen in the textbooks [1,2] the response in magnetic textures to the external force(s) (the spatial gradient of the magnetic field in particular) was extensively studied. Once the Hamiltonian  to describe the magnetic system is given, the dynamics of the magnetic textures is determined by the Landau-Lifshitz-Gilbert (LLG) where γ is the gyromagnetic ratio and α is the Gilbert damping constant. Starting with the LLG equation, Thiele [6][7][8] derived an equation of motion for the DWs, and the magnetic bubble motion is expressed by the equation, Here (X, Y) is the centre-of-mass coordinate of the bubble, = v X Y , d (˙˙) is its velocity and v s represents the spin current by the spin wave or the spin polarized electron current in the metallic system. Recent studies [9,10] have revealed that the mass m of the skyrmion emerges with the deformation of the magnetic texture compared with the static solution. The gyrovector ) (e z : unit vector along z-direction) relates the skew motion of the bubble and the bubble topology. The dyadic  is composed of the dimensionless diagonal components being order of unity, β represents the coefficient of the non-adiabatic effect, U is the potential and -U is the force acting on the magnetic bubble, e.g., those from the boundary, gradient of the magnetic field, and the impurities.
The gradient of the magnetic field drives the magnetic bubble motion. However, just after turnoff of the driving force, motion of bubble which is not always understood by an inertia effect is often observed. To explain this interesting bubble motion beyond the Thiele equation (equation (4)), the BLs' dynamics induced mechanism has been studied [1,2,[11][12][13][14][15][16][17]. The generalized Thiele equation has been formulated by Slonczewski [17] to describe the motion of the DW of generalized shape and the effect of the gyrovector density. However, to solve Slonczewski equations is very difficult in general, and the discussion in these past is mostly in the high-Q limit where Q is the quality factor defined as p = Q K M 2 2 (in CGS units) with uniaxial anisotropy K. In this limit, the BL is considered to be the one-dimensional topological defects on the two-dimensional DW and the line dynamics on the cylinder plays a crucial role for the bubble motion.
In many magnets [18], the DWs and the BLs have substantial thickness and width compared with the bubble sizes. In those cases, the dynamics of the magnetic texture with spatial extent becomes an important issue to discuss the bubble motion. To characterize the local magnetic texture, we define the Berry curvature  (2), and also important for the transport properties in the metallic magnetic systems [3].) Because , b z is the origin of the Magnus force (see equation (4)). Therefore, a local Magnus force is naturally expected by the local gyrovector e b 2 z z . In the present paper, we focus on the spatial distribution of b z , which is more basic quantity and directly linked to the local Magnus force acting on each part of the magnetic bubble.

Model and Simulation
The model Hamiltonian for the dipolar magnets defined on the two-dimensional square lattice is given by where, I dip and K represent the dipole interaction and the uniaxial anisotropy, respectively. The wavenumber q of the helix structure is given by @ q I J dip in the present case [19]. The evolution in magnetic texture is experimentally observed [18], i.e., the helix ground state is seen at zero magnetic field and the magnetic bubbles emerge as the magnetic field is increased.
Using this model Hamiltonian equation (6), we solve the LLG equation (equation (3)) numerically and study the real-time dynamics of the skyrmion, the antiskyrmion and the type-II magnetic bubbles (see figure 1). We use gJ 1 ( )for the unit of time t. Typically~-J 10 3 eV and the unit gJ 1 ( )becomes ∼0.7 ps for  g m = g s B (g s : electron spin g-factor, m B : Bohr magneton). Figure 2 shows the time evolution in the magnetic texture of the system which initially involves skyrmion and antiskyrmion, i.e, in figure 2 [20][21][22][23] which is well described by the Thiele equation, and hence the skyrmion and antiskyrmion move upward direction along the system edge (see figures 2(a)  (b)). As a result, the skyrmion and antiskyrmion collide and are merged into a type-II magnetic bubble (see figures 2(b)  (c)). Note that in this dynamics, the skyrmion number N sk is conserved in total. The resulting type-II magnetic bubble in figure 2(c) is topologically the same as the bubble shown in figure 1(c), i.e., the left (right) half of the DW structure is the same as that of figure 1(a) ((b)). Later on, the type-II magnetic bubble shows the bounce dynamics at the edge of the system for a while like a pinball in a bowl and eventually, the bubble is stabilized at the center of the system.
It is noted that the Thiele equation (equation (4)) is not enough to understand the dynamics of type-II bubble observed in our numerical simulation. Instead, the domain structure of the distribution of b z , and the Magnus effect acting on each domain induced by both the external force and internal force are the key to analyze the complex behavior of type-II bubble with the zero net b z . For example, Thiele equation (equation (4)) predicts that the motion of the type-II bubble is expected to be parallel to the force because = N 0 sk and Magnus effect is absent. The bounce back from the upper edge shown in figures 2(c)  (d)  (e) is apparently consistent with Thiele equation (equation (4)). However, it passes through the center of the system, and moves to the lower edge of the system against the confining force from the lower edge, and again shows the bounce behavior at the lower edge as shown in figures 2(e)  (f)  (g). This behavior is beyond the description of Thiele equation (equation (4)), and indicates that the internal force plays the essential role on the center-of-mass motion.
To investigate the dynamics in more details, we set the initial configuration of type-II magnetic bubble as shown in figure 3(b). (See also the supplementary movie.) To prepare this, we first stabilize a magnetic skyrmion at the center of the system ( figure 3(a)), and apply an operation n n y y to the magnetic texture on the righthalf of the system. The spatial distribution of b z is topologically the same as the bubble shown in figure 2(c) where the singly connected > b 0 ) segment at the DW is on the right-half (left-half) of the magnetic bubble. The spatial distribution of b z represents the mixed winding segments with Bloch and Néel wall structures, where the latter reverses the direction of the winding. The initial state shown in figure 3(b) is not stable, and hence  in left (right) side of the bubble. When this exchange of b z segments is achieved, the bubble makes a turn and moves in the lower direction (see figures 3(f)  (g)). Such bounce behavior is seen during the relaxation dynamics, and finally, the bubble is stabilized at the center of the system with the fourfold pattern of b z ( figure 3(h)). Note that the singly connected > b 0 z and < b 0 z segments are on the right and left hand sides to the moving direction of the center-of-mass. Now we discuss the relation between the center-of-mass motion and distribution of b z . During the relaxation dynamics, the size of the magnetic bubble shrinks (see the difference in the bubble size between figures 3(c) and (h)). In other words, the internal shrinking force to the center of the bubble is acting on the DW at the perimeter. The Thiele equation generalized to the local region tells us that the Magnus effect appears on the magnetic texture perpendicular to the acting internal or external force when ¹ b 0 z , which explains the bubble dynamics described above. In the early stage (see figures 3(c)  (d)), the positive (negative) b z segment is arranged in right (left) side of the type-II magnetic bubble. Therefore, the positive and negative b z segments have the drift velocity in the same upper direction due to the Magnus effect by the internal shrinking force as seen in figures 3(c)  (d). As approaching to the upper edge of the system, the effect of the confining force in the lower direction from the edge becomes strong. In the case figures 3(d)  (e), the Magnus force due to the confining force appears in opposite direction, i.e., it is left (right) for the positive (negative) b z segment of the type-II magnetic bubble. This leads to the change in the arrangement of b z from figures 3(d) to3(f). Note that during the change in the arrangement of b z , the fourfold pattern appears once (see the supplementary movie). The separation and merge of the b z segments determines the dynamics of BLs in the type-II magnetic bubble. In the fourfold pattern of b z , the net Magnus effect vanishes due to the cancellation between four segments, and hence the meta-stable final configuration shown in figure 3(h) is achieved at the center of the sample.
Next, we examine the bubble dynamics starting with the fourfold pattern as shown in figures 4(a) and (b). (See also the supplementary movie.) Note that it is similar to that in figure 3(h), but the bubble size is the same as that in figure 3(a). Namely, we apply the operations, n n n n , , ) for p f p < < 5 4 7 4, and  -n n n n , , x y x y ( ) ( ) for f p < 4 | | (mod p 2 ) to the magnetic skyrmion shown in figure 3(a).
Since the size of the bubble is larger than that of the meta-stable state, the shrinking force causes the Magnus force in the circumferential (f) direction, and the sign of b z determines its direction, i.e., it is clockwise