Abstract
Interaction between local magnetization and conduction electrons is responsible for a variety of phenomena in magnetic materials. It has been recently shown that spin current and associated electric voltage can be induced by magnetization that depends on both time and space. This effect, called spinmotive force, provides for a powerful tool for exploring the dynamics and the nature of magnetic textures, as well as a new source for electromotive force. Here we theoretically demonstrate the generation of electric voltages in magnetic bubble array systems subjected to a magnetic field gradient. It is shown by deriving expressions for the electric voltages that the present system offers a direct measure of phenomenological parameter β that describes nonadiabaticity in the current induced magnetization dynamics. This spinmotive force opens a door for new types of spintronic devices that exploit the fieldgradient.
Introduction
Spinmotive force (SMF) refers to the generation of spin current, which is accompanied by an electric voltage, as a result of dynamical magnetic textures in conducting ferromagnets^{1,2,3}. This is due to the exchange coupling between conduction electrons and the local magnetization. SMF reflects the temporal and spatialdependence of the local magnetization^{4,5,6,7,8,9}, and thus it offers a powerful method to probe and explore the dynamics and the nature of magnetic textures. In addition, SMF can be a new source of electromotive force, directly converting the magnetic energy into the electric energy of conduction electrons. While the classical electromagnetism tells us that the conventional inductive electromotive force requires a timevarying magnetic flux, it has been reported that an electromotive force can be generated by a static and uniform magnetic field via the SMF mechanism^{10,11,12}.
In the last few years, more attention has been focusing on topologically nontrivial magnetic structures such as magnetic vortices in soft ferromagnetic nanodiscs^{13,14} and skyrmion lattices in chiral magnetic thin films^{15,16}. The SMF offers some insights into and gains benefit from such magnetic systems; the polarity of a moving magnetic vortex core can be electrically detected^{17,18}, and arbitrarilylarge ac SMF was predicted due to skyrmion lattice motion^{19}. To the best of our knowledge, however, there has been no work on SMF induced in systems that contain magnetic bubble domains. Magnetic bubbles are observed in ferromagnetic films with outofplane anisotropy as spotlike closed domains, where the magnetization is oriented to the opposite direction to the one outside the bubbles. The structures of magnetic bubbles are similar to those of skyrmions and vortices in the sense that bubbles carry a topological number called skyrmion number. Since their first observation in the 1960s, magnetic bubbles have been showing distinctive and interesting behaviors^{20,21,22,23}.
In this paper, SMF due to the motion of magnetic bubble arrays is theoretically investigated based on the steadymotion model. As the bubble motion may be induced by a spatiallyvarying magnetic field, our work reveals that a magnetic field gradient can be exploited to generate a spin current and an associated electric voltage. By deriving expressions for the electric voltages, we demonstrate their cumulative nature, i.e., they can be proportional to the number of involved bubbles. An important fact to be stressed is that the present system can confirm the SMF originating from the nonadiabatic dynamics of the electron spin, leading to a direct measurement of the controversial, socalled, β parameter.
Results
Steadymotion model for magnetic bubble
Let us begin by reviewing the collectivecoordinate model for magnetic bubble dynamics. We consider a cylindrical bubble domain with radius R in a thin film. This type of magnetic structure can be stabilized by applying bias magnetic field in the outofplane with appropriate magnitude. The distribution of the magnetization direction m = (sin θ cos ψ, sin θ sin ψ, cos θ) is assumed to be two dimensional and described as^{20} [see Fig. 1]where (r, χ, z) is the cylindrical coordinate measured from the bubble center, Δ is the wall width parameter, ψ_{0} is a constant, Q is the topological parameter defined asand S is the winding number:where is the contour integral taken counterclockwise around the circumference. The domain wall separating the bubble and the outside, in general, can contain vertical Bloch lines, i.e., there are many possibilities in the way of distributing the azimuthal angle ψ along the perimeter. In Eq. (2) we assumed the linear dependence of ψ on χ, as we focus on the following two cases. First, when the magnetization rotates one full turn around the wall of the bubble with no Bloch line, ψ = χ ± π/2, that is, S = 1 and ψ_{0} = ± π/2, with the sign + (−) corresponding to left (right) handed chirality. Eq. (2) is also a good approximation when the Bloch lines are packed so closely that the distance between the adjacent Bloch lines is comparable to the wall width, i.e., . If one considers a small number of Bloch lines, the distribution of ψ would be no longer as simple as Eq. (2).
When a magnetic field is applied in the z direction with its magnitude varying in the xy plane, a bubble is driven to move in the film seeking positions with lower Zeeman energy. In the following the steady motion of the bubble under the constant gradient ∇H_{z} is assumed, i.e., during its motion with constant velocity v the bubble stays rigidly cylindrical with constant radius R and the ψdistribution does not change with respect to the coordinate frame moving with the bubble:
Assuming Eq. (5) and that the magnetization dynamics obeys LandauLifshitzGilbert equation, the equation of motion for the bubble is given by^{20} (the derivation is shown in Methods)where γ is the gyromagnetic ratio and α is the dimensionless Gilbert damping parameter. represents the unit vector along the zdirection. An assumption that was made when deriving Eq. (6) is that the spin transfer with the conduction electrons are negligible. Remarkably, the bubble is deflected away from the direction of the field gradient at an angle that is determined by the material parameters (see Methods for the detail). The extension of the above discussion to multiplebubble problem is straightforward.
Spinmotive force due to bubble motion
Let us examine the SMF induced by the steady motion of bubble indicated by Eq. (5). We assume that a conduction electron in the ferromagnetic film is described by a onebody Hamiltonianwhere m_{e} is the electron's mass. The second term represents the exchange interaction between the electron spin and the magnetization, with J_{ex} being the exchange coupling energy. According to theory of SMF^{4,5,6}, dynamical magnetization exerts an effective electric field on the electrons via the exchange coupling, which is called spin electric field since its sign depends on the electron spin (see Methods):with
The upper (lower) signs in Eq. (8) correspond to the electron with majority (minority) spin. and are referred to as adiabatic and nonadiabatic spin electric fields, respectively, as β = ħ/2J_{ex}τ_{sf} is the dimensionless parameter describing the nonadiabaticity in the electron spin dynamics^{5,6,8,9}, with τ_{sf} the relaxation time for the electron spin flip. The spin electric fields (9) and (10) require both time and spatial dependences of the magnetization, and this condition is satisfied around the perimeter of moving bubbles.
The spin electric field (8) induces a spin current and a charge current in the sample, with the electric conductivity for the majority (minority) electrons. These currents generate byproducts such as the charge redistribution, the spin accumulation, and the charge/spin diffusion current. In an open circuit system, the electric field E_{ind} = −∇ϕ−∂A/∂t appears to keep the total charge current zero, i.e., , where ϕ and A are the electromagnetic scalar and vector potentials. An electric voltage between two given points x_{a} and x_{b}, which is given by the difference in the electric potential ϕ(x_{b}) − ϕ(x_{a}), enables one to detect the spinmotive force electrically. To determine the gauge potentials one has to fix the gauge, and here let us adopt the Coulomb gauge, ∇ · A = 0. From the above equation for the open circuit condition, one obtains the Poisson equationHere is the conservative electric field induced by the electric potential distribution, where is the spin polarization of the conduction electrons. In the above argument, we have neglected the contribution from the diffusive current to the total charge current as we have metals in our mind as the samples; once the effective U(1) electric field is given, the problem of computing the electric voltage induced by the electric field falls within the classical electromagnetism and the established transport theory, and it is known that in metals the induced diffusion potential is mostly negligible compared to the electric potential, unlike in semiconductors. Technically, the diffusive current can be taken into account by replacing the electric potential −eϕ by the electrochemical potential , where is the Fermi energy. ( may be dependent on the space and the spin electric field in complex ways.)
Eq. (11) can be applied to systems with arbitrary sample geometry and magnetic texture. In Fig. 2, we show an example of electric potential distribution by numerically solving the Poisson equation (11) with spin electric fields (9) and (10), where the steady motion of eight identical bubbles in a squareshape thin film is assumed [see Methods for the numerics]. Here the coordinate system is set that the bubble array flows along the x direction. It is seen that the potential drop occurs at the position of the bubbles, as is expected. Notice that the adiabatic field gives rise to the net potential drops only in the y direction (perpendicular to the bubble flow), while the nonadiabatic one only to the x direction (along the bubble flow), indicating that in this setup the two contributions can be separately identified by longitudinal and perpendicular voltage measurements. While S = 1 is assumed in Fig. 2, qualitatively much the same profiles are also obtained in the case of S = R/Δ, but with height of each potential drop being different (not shown). The S dependence of the electric voltage will be discussed later.
Next, let us adopt a “quasionedimensional” approximation for the electric voltage and derive its analytic expression. We limit ourselves to a rectangular thin film where bubbles move along either of two sides of sample, as is the case in Fig. 2. Assuming a bubble moving in the positive x direction, we estimate the electric voltage V_{x} along the motion of the bubble by where L_{x}_{(y)} is the side length of the sample along the x(y) direction. The yintegral operation and the appearance of the factor 1/L_{y} are for spatialaveraging of the electric potential along the yaxis. Eqs. (1), (2), (5), (6) and (8) have been used to carry out the integrals. Eq. (12) gives the exact solution for V_{x} in onedimensional systems: . In twodimensional systems as is the present case, Eq. (12) is still a good approximation when effects on V_{x} from ∂ϕ/∂y and the detailed profile of ∂ϕ/∂x can be ignored. This is often the case if there are some appropriate symmetries in the system, and the divergence of the electric field, i.e., the moving bubble, keeps away from the electrodes. Similar concept for the electric voltage V_{y} measured in the perpendicular direction to the motion of the bubble leads to
The dc electric voltages appear both in the x and y directions, being proportional to the field gradient for both magnetic configurations S = 1 and . It is seen from Eq. (12) that V_{x} for S = 1 is larger than that for S = R/Δ under the same applied field, because R > Δ. With the parameters shown in Fig. 2, V_{x} for S = 1 is about an order of magnitude greater compared to that for S = R/Δ. On the other hand, V_{y} has little dependence on S.
When there are multiple bubbles, the net electric potential distribution is given by the superposition of all individual electric potentials produced by each bubble. If N identical bubbles move in the x direction, the simplest extensions, and , of the above expressions for the electric voltages, V_{x} and V_{y}, respectively, may be
Figure 3 compares Eqs. (14) with the electric voltages obtained by numerically solving the Poisson equation (11), showing good agreement between them.
Discussion
Eqs. (12), (13) and (14) indicate key features of this SMF. is proportional to N/L_{y}, and thus, roughly speaking, depends on the “density” of bubbles along the y axis and the “number” of bubbles along the x axis, which contribute to, respectively, the height of each potential drop and the number of occurrence of potential drop along the x axis. The dependence of on the number of bubbles and the sample geometry is demonstrated in Fig. 3(a). Comparing the configurations i and ii, which share the same sample shape, the slope of V_{x} is twice larger for ii because the configuration ii contains twice as many bubbles, i.e., the sites where the potential drop occurs. It is not necessarily that the same value of N leads to the same magnitude of electric voltage; the configuration iii provides twice larger than ii does under the same applied field because of the difference in the factor 1/L_{y}. Similar discussion is applied for , see Fig. 3(b). Eq. (14) indicates that one may control the dc electric voltages by adjusting the sample geometry and the number of bubbles.
Remarkably, only the nonadiabatic field contributes to , while only the adiabatic one to . Since in most magnetic materials β is believed to be smaller than unity and thus , it is difficult to identify the contribution of the nonadiabatic field in the conventional systems. In fact, there has been no experimental confirmation of its effects. Now, by the measurement of one can detect free from the larger adiabatic contribution. This can lead to direct and unambiguous measurement of the phenomenological parameter β; as both V_{x} and V_{y} contain in their expressions P, which may be another uncertain material parameter, one can be free from P by measuring the ratio of V_{x} and V_{y}.
In the present study, we assumed the specific and simple profile of magnetization, Eqs. (1), (2) and (5), to derive the analytic expressions for the electric voltages. Although the investigation of the effects of the bubble shape distortion, local disorder, and etc. must be interesting, it will require systematic study based on numerical approaches to the magnetization dynamics, which is out of reach of this paper. The assumption made in the present study is reasonable when the gradient of applied field is sufficiently moderate^{22}, and our analytical model will work well in this field range.
In conclusion, We have shown for the first time that a magnetic field gradient can generate a SMF, i.e., spin currents and associated electric voltages, by driving the motion of magnetic bubbles. Based on the steadymotion model, expressions for the dc electric voltages in longitudinal and perpendicular to the bubble motion are derived, which turned out to be controllable by tuning the sample geometry and the number of involved bubbles. An important implication of our result is that the present system offers an experimental determination of the phenomenological parameter β that describes nonadiabaticity in the electron spin dynamics. This SMF can lead to a new route for basic study of the electronmagnetization interaction as well as a new concept in spintronic devices, exploiting the gradient of magnetic fields.
Methods
Steadymotion of bubble
In order to get to the equation of motion for the bubble, consider the increment of the stored magnetic energy , with the magnetic energy density w, due to the variations δθ and δψ:where h is the film thickness and M_{S} is the saturation magnetization. δw/δθ and δw/δψ have been expressed in terms of ∂θ/∂t and ∂ψ/∂t by using LandauLifshitzGilbert equation of motion without spintransfertorque effect, and the time derivatives have been further expressed in terms of r and χ assuming Eqs. (1), (2) and (5). The change δU in the internal energy is supposed to be balanced by the external pressure on the bubble due to ∇H_{z}:where X denotes the position of the center of the bubble. Rearranging the above equation, Eq. (6) is obtained. The net mobility of the bubble is found by solving Eq. (6) for v in terms of R∇H_{z} as
The angle ρ of deflection of the bubble away from the field gradient may be defined asWith S = 1, Q = 1, α = 0.02, Δ = 2 nm and R = 50 nm, corresponding to the calculation in Fig. 2, one obtains .
Derivation of spin electric fields
Under the Hamiltonian (7), the Heisenberg equation of motion for the electron is given bywith r and denoting the operators for the electron's position and the force acting on the electron, respectively. The actual motion of the electron is obtained by determining the expectation value 〈σ〉_{↑↓} of the electron spin with majority (↑) and minority (↓) states. Notice that m · ∇m = 0 and thus the component of 〈σ〉_{↑↓} that is (anti)parallel to m does not contribute to the force. Let us decompose the electron spin as , where the upper (lower) sign corresponds to the majority (minority) spin and δm_{↑↓} represents a slight deviation from . The expectation value of the force is written as ; what causes nonzero force due to the exchange coupling is a misalignment between the electron spin and the magnetization. Assume that the electron spin dynamics is described by
The first term on the righthand side represents the Larmor precession about the magnetization, and the damping motion toward the magnetization is phenomenologically introduced by the second term, which describes the nonadiabaticity in the electron spin dynamics, with τ_{sf} the relaxation time for the electron spin flip. By substituting the above expression for 〈σ〉_{↑↓} into Eq. (20), δm_{↑↓} is expressed in terms of m, by which one obtains with given by Eq. (8).
We have considered an open circuit condition. More generally Eq. (20) should include the divergence of the spin current carried by conduction electrons, leading to appearance of the spin magnetic field. While we have followed Ref. 9 here, essentially the same result was obtained by a different approach where Onsager's reciprocal relation is taken into account^{5,6}.
Numerical approach to the Poisson equation
In the numerical calculations, the sample is divided into number of meshes, and a single magnetization vector m_{i} is assigned to each mesh, where i is the index of the meshes. The time evolution of {m_{i}(t)} (i = 1, 2, …, N_{m}), where N_{m} is the number of the meshes, is given based on Eqs. (1), (2), and (5). As the spin electric field is updated at each mesh by Eq. (8), the induced electric potential distribution {ϕ_{i}(t)} satisfies the Poisson equation [see Eq. (11)]
In twodimensional discrete systems, the above equation is equivalent towhere (i, j) are the indices for the meshes in two dimension, and ΔxΔy is the area of the mesh. The potential distribution is obtained by solving the above equation selfconsistently with Neumann boundary condition, where the spatial derivative of the electric potential is zero at the sample edge.
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Acknowledgements
The authors would like to thank Cristian Cernov for valuable discussion and making the image for Fig. 1. This research was supported by Research Fellowship for Young Scientists from Japan Society for the Promotion of Science, GrantinAid for Scientific Research (No. 24740247 and No. 26247063) from MEXT, Japan, and Alexander von Humboldt Foundation.
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Affiliations
Institut für Physik, Johannes Gutenberg Universität Mainz, 55128 Mainz, Germany
 Yuta Yamane
 & Jairo Sinova
Department of Physics, Texas A&M University, College Station, Texas 778434242, USA
 Shayan Hemmatiyan
 & Jairo Sinova
Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 3191195, Japan
 Jun'ichi Ieda
 & Sadamichi Maekawa
Spin Phenomena Interdisciplinary Center (SPICE), Johannes Gutenberg Universität Mainz, 55128 Mainz, Germany
 Jairo Sinova
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Contributions
Y.Y. carried out theory calculation and wrote the manuscript. S.H. analyzed the data. J.I. and S.M. conceived the problem. J.I., S.M. and J.S. supervised the study. All authors discussed the results and commented on the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Yuta Yamane.
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