C. Wang, K. Xia*  Abstract Full Text Html PDF w/ Links Export Citation Figures 
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NanoMicro Letters, , Volume 1, Issue 1, pp 3439 

Publication Date (Web): December 20, 2009 (Article) 

DOI:10.5101/nml.v1i1.p3439 

*Corresponding author. Email: kexia@bnu.edu.cn 
Abstract
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Introduction
The application of spintronics [1] requires effective ways of spin manipulations in nanoscale devices. Recently, growing interest has been focused on the current induced magnetic dynamics. Electric currents can be spinpolarized by passing through magnetic domains, and the polarized spins of the conducting electrons can exert spin transfer torque (STT) to the localized magnetizations when going through a magnetic domain wall. On the other hand, the spins of electrons will be altered by the localized magnetizations in return [2,3]. Beautiful experiments have been conducted using STT effect in devices [47]. However, the dynamics of magnetic domain walls (DWs) in the presence of electric current are still not well understood, especially for the components of spin transfer torques perpendicular to the plane where the local magnetization’s directions change. Shibata et al [8,9], studied the DW motion based on the sd electron model and collective coordinates of the DW. They found that the DW motion will be dominated by the effective force on the DW, which is rooted from the reflection of the electrons incident on the DW. On the other hand, S. Zhang and Z. Li [10,11] pointed out that the outofplane STT was crucial for the stead current induced DWrelaxation mechanism. Another nonadiabatic STT was proposed by J. Xiao [12], which is due to the misalignment of nonequilibrium spin density and local effective field. The outplane spin transfer torques can stem from either spin relaxation or nonlocal nonadiabatic effects [9]. However, the link between the microscopic STT and the effective force on the DW is missing.
In this paper, we start from the LandauLifshitzGilbert (LLG) equation, including the microscopic STT Terms from our numerical calculation, theoretically analyze the motion of the DW. The relation between the effective forces on DW and STT are established. The connections between the different components of STT and the effective forces and STTs on domain wall are established. Then we discuss the relationships between our results and other models in the end of Section II. In Section III the STT in DW is studied numerically for free electron model. Our calculations of outplane torques show that the parameter c_{j} describing the nonadiabatic torque in LLG equation is not constant but position dependent, implying its nonlocal characters. The DW width dependence of the effective force is obtained.
Experimental
EFFECTIVE FORCE AND OUTOFPLANE STT
The currentdriven domain wall dynamics is studied based on the LandauLifshitzGilbert equation, the connection between effective force and outofplane STT on each site is obtained. The LLG equation reads,
where the magnetizationcan be written as the function of polar angles as indicated in Fig. 1:
We assume that the magnitude of the magnetization will not change when the DW moves. We can rewrite the on sites LLG equations Eq. (1) with sphere polar angle coordinates (θ_{i}, φ_{i}) and then apply the Walker ansatz so that Euler angles are treated as functions of position and time.
Figure 1 Sphere polar coordinates describing the direction of localized magnetization. M represent onsite local magnetization, the blue circle illustrate the plane in which magnetizations alter their directions. In the simplest model (ϕ = 0), the plane reduces to xz plane. 
The DW is parallel to the z direction, θ and ϕ are the function of position and time. θ=θ(z−X(t)), ϕ=ϕ(z,t), X is defined as the wallcenter’s position within the “rigidwall” approximation, and ϕ(z,t) is simply reduced to ϕ(t) in the model used in following derivation. The variable X and ϕ can serve as collective coordinates to describe the wall motion. Namely, we can get the dynamic information of the domain wall if we know the time evolution of (X(t),ϕ(t)). (In the simplest picture, we may let ϕ constant so that the localized spins change their orientations within aplain). The effective field on each local magnetization
contains respective contributions from exchange interaction, anisotropy, demagnetization field, and external field where subscript i denotes different sites in the wall. The nearest neighboring approximation and assumption of constant magnitude of M were applied. The unit vectors in Cartesian coordination can be transformed into polar coordination. And spin transfer torques can be explicitly expressed as,
Putting all the above terms into the Eq. (1), we eventually obtain the LLG equation written in polar coordination with variable (θ,ϕ) as,
The above two sets of equations can be solved numerically for arbitrary STT form and effective field. However, when current and applied magnetic fields are not very large, rigid wall is a good approximation to study the DW motion. Applying Walker’s ansatz [16], the localized spins in Neel domain wall structure (Shown in Fig. 2) expressed in spin polar coordination satisfies
Figure 2 Illustration of Neel domain wall structure and Cartesian coordinates. The direction of currentI in CIP structure used in our calculation is also shown. 
Considering the magnetization of the domain wall and the localized macroscopic spin satisfy the relationship: , we can express this relationship by pole angles：
And we can see the time dependent behavior of θ is completely determined by the wall(center) position’s dynamics property:
For the convenience inconnecting to our first principles calculation, we use the →Sin the following of the paper. Using the relationship of magnetic moment and spin angular momentum, the Eq. (5) (6) could be expressed by localized macroscopic spins. Summing all the equations on each site together, we obtained the corresponding equations for the collective coordinates. Replacing the summation by integration for the simplicity, the region of integration is set as where is the wall length, d is the distance between neighboring sites, and N is the total number of sites in a domain wall which can be extended to ∞.Since cosθ varies from 1 to 1 in a symmetric way, the sum over position is ∫cosθ (z, t)dz = 0.Considering the two collective coordinations (X,ϕ) are independent on position, the sum of equations on all the sites along the wall results in
So the equations portraying the dynamic behavior of the rigid wall are obtained. Recalling the expressions of spin torque in LLG Eq. (4), the spin torque exerted on localized spins in the right side of equations above can be simplified by
With the obtained results, we can immediately obtain the velocity, effective force and torques exerted on the domain wall in the following. If we regard b_{j} and c_{j} are independent on position (we will discuss the validity of this simplification with our calculations section III), the currentdriven domain wall motion velocity, as the time derivative of wall position X, can be expressed by b_{j} and c_{j} using Eq. (8) (9) (10):
Our results Eq.(9) and (10) also imply that fort→∞ in which, the magnitude of the domain wall ultimate velocity is determined by c_{j} while the initial speed is mainly determined by b_{j}. The expression of velocities agrees well with other groups’ previous outcome [10]. In order to further clarify the roles of different components of STT, we combine our results with what obtained fromsdmodel Hamiltonian. We canimmediately come to some interesting results after comparing Eq. (8) (9) with effective forces and effective torques [9]. It turns out that the outplane spin torque, which appears in Eq. (9) corresponds to the force on domain wall, while the inplane spin torque contributes to the z component of torque on the wall as a whole:
Eq. (12) tells that force exerted on the rigid domain wall is also originated from the STT (but only the perpendicular element), while Eq. (13) shows that the summed effective spin transfer torque along z is determined by integration involving different positions’ and . After we apply Eq. (10) and defineand as the average values of the two parameters, the two integrations in Eq. (12) and Eq. (13) become
Although it is widely accepted that different components of STTs can be written in the form of Eq. (4), the origin of outofplane STT could be stem from different physics. In our calculation, we focused on the nonadiabatic term pointed out by J. Xiao and M. Stiles [12].
Results and discussion
NUMERICAL CALCULATION AND DISCUSSION
In the following, numerical calculation is used to investigate the torque as the function of domain wall width λ. After introducing the calculation methods we used, we subsequently investigate the inplane and outplane STTs. The obtained calculations are in accordance with our model and can be helpful to the discussion about effective force.
A. Calculation of STT in Domain Wall
The structure used in the calculation is current in plane (CIP) domain walls in ferromagnetic materials. We apply free electron model in our calculations where the current is along (110) direction in fcc structure using the lattice constant of Cobalt a_{Co} = 3.549 Å. And we set the energy split between majority and minority spins in free electron energy band as 1.69eV, which equals the value of exchange energy split of bulk Cobalt. Our numerical approach is based on the Tightbinding linear muffintin orbital formulism [13,14]. Scattering wave function is obtained by the wave function matching method [15]. The rigid potential approximation is employed to simulate the DW. Here STTs in our study can be defined as the difference between the incoming and outgoing spin current at R site.
Here summation over the 2D Brillouin zone is performed. Combining with the expression of current passing through Rsite, the STT per unit current is obtained：
Given a specific current density (~10^{10} A/m^{2}), we can calculate corresponding torque. The calculated spin transfer torques along the DW is presented in different components in Fig. 3. From the Eq. (12) and Eq. (13) the force and zcomponent’s effective torque can be calculated.
Figure 3 Position dependence of different components of spin transfer torque in the domain wall structure. 
B. Outplane STT and Effective Force
Let us focus on the outplane component of STTs. From our numerical calculations, we find a decaying oscillation for outofplane STT as shown in Fig. 4. From the inset in Fig. 4 we can see the decaying oscillation of at the sites away from DW centers. We can see that deviate significantly from zero only within a certain distance from wall center. We can fit these STTs with Eq. (10) to obtain the on site parameter c_{j} =λ/(S sinθ). Numerical difficulty comes from the position far away from the DW center, where sin θ (z, t) is very small and outofplane STTs oscillate as the function of position. Considering that the STT mainly dominated by the DW near the center where the angle of local magnetization changes significantly, a cut off is applied so that our focus is concentrated to regions near the wall centers. The calculated c_{j} as a function of position is presented in Fig. 5. From Fig. 5 we can see that c_{j }obtained from our model are position dependent rather than constant. This is reasonable because the nonadiabatic torques is nonlocal and has vibrating behaviors. Combining the calculated results, we will discuss the DW velocities expressed in Eq. (11). When the rigid wall still holds, it is suitable to express the terminal velocities v(t→∞)=c_{j}/a.
Figure 4 Distribution of outplane STT components in DW with λ=6ML. The inset shows the decaying oscillations of Ty at sites away from DW centers. 
Considering c_{j} is not constant but oscillating, we may use the average to express the terminal velocity. When the wall is thicker, the increase of inertia or wall mass makes it harder to move, thus the velocities get smaller and consequently larger driving currents are required. In our model, the velocities will indeed drop significantly when the out plane torques (c_{j} terms) vanish in thicker walls, as depicted in Fig. 6. However, we need to keep in mind that the Walker’s model we use to describe rigid walls will break down under high external field [16,17] or high current density [9]. In that case, the domain wall would be oscillating so the definition of DW velocities is no longer available.
Figure 5 The values of cj near the wall center. We make a cutoff and only consider the contribution of sites not far from the wall center. 
The forces at different wall structures are calculated from integration within the narrowed region of c_{j}(z) using and presented in Fig. 6. We can see from the result that the forces exerted on the walls decrease significantly as the walls get thicker, implying that the nonadiabatic effect is obvious in thin domain walls. And the forces or the integrated nonadiabatic torques decay approximately parabolic exponentially with the increase of DW width.
Figure 6 The force is calculated in wall structures with different widths. Our average cj only take account of the contributions from sites near the wall center where the value of sinθ(z, t) is nonzero. The red line is a parabolic exponential curve. 
Conclusion
In sum, domain wall motion under electrical current is studied focusing on spin transfer torque effect. Under the approximation of rigid wall the relationships between the force, velocity and different components of STT are clarified. And the results of our calculation successfully testify the validity of adiabatic limit. For outplane component, the calculated torques show oscillating and decaying behaviors. The calculated nonadiabatic torque dependence on wall width is in good accordance with the physical illustration that an extremely thick wall would become completely adiabatic and thus the forces and terminal velocities origined from nonadiabatic terms vanish.
Acknowledgement
The work is supported by NSF of China (Grant No.60825405) and MOST of China (2006CB933000). We are grateful to Shuai Wang for useful discussions and his codes that we used for calculating spin transfer torques.
References
[1] S. A. Wolf, D. D. Awschalom, R.A. Buhrman, J. M. Daughton and S. Von Molnar, Science 294, 1488 (2001); I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76, 2 (2004). [2] Berger, Phys. Rev. B 54, 9353 (1996). http://dx.doi.org/10.1103/PhysRevB.54.9353 [3] J. Slonczewski and J. Magn. Magn. Mater. 159, L1 (1996). http://dx.doi.org/10.1016/03048853(96)000625 [4] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi and J. L. Erskine, Phys. Rev. Lett. 97, 057203 (2006).http://dx.doi.org/10.1103/PhysRevLett.97.057203 [5] M. Tsoi, Nature Phys. 4, 17 (2008). http://dx.doi.org/10.1038/nphys822 [6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman and D. C. Ralph, Phys. Rev. Lett. 96, 227601 (2006). http://dx.doi.org/10.1103/PhysRevLett.96.227601 [7] S. Stuart and P. Parkin, Science 320, 190 (2008). http://dx.doi.org/10.1126/science.1145799 [8] J. Shibata, G. Tarara and H. Kohno, Phys. Rev. Lett. 94, 076601 (2005). http://dx.doi.org/10.1103/PhysRevLett.94.076601 [9] G. Tatara, H. Kohno and J. Shibata, Phys. Rep. 468, 6 (2008). http://dx.doi.org/10.1016/j.physrep.2008.07.003 [10] Z. Li and S. Zhang, Phys. Rev. Lett. 93, 127204 (2004). http://dx.doi.org/10.1103/PhysRevLett.93.127204 [11] S. Zhang and Z. Li, Phys. Rev. Lett 92, 207203 (2004). http://dx.doi.org/10.1103/PhysRevLett.92.207203 [12] J. Xiao, A. Zangwill and M. D. Stiles, Phys. Rev. B. 73, 054428 (2006). http://dx.doi.org/10.1103/PhysRevB.73.054428 [13] I. Turek, V. Drchal, J. Kudrnovsky, M. Sob and P. Weinberger, Electronic Structure of Disordered Alloys: Surface sand Interfaces (Kluwer, BostonLondon Dordrecht, 1997). [14] O. K.Andersen, O. Jepsen and D. Glotzel, in Highlights in Condensed Matter Theory, edited by F. Bassani,Fumi and M. P. Tosi (NorthHolland, Amsterdam, 1985), p. 59. [15] S. Wang, Y. Xu and K. Xia, Phys. Rev. B 77, 184430 (2008). http://dx.doi.org/10.1103/PhysRevB.77.184430 [16] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). http://dx.doi.org/10.1063/1.1663252 [17] X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. 324, 8 (2009). http://dx.doi.org/10.1016/j.aop.2009.05.004 
References
[1] S. A. Wolf, D. D. Awschalom, R.A. Buhrman, J. M. Daughton and S. Von Molnar, Science 294, 1488 (2001); I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76, 2 (2004).
[2] Berger, Phys. Rev. B 54, 9353 (1996). http://dx.doi.org/10.1103/PhysRevB.54.9353
[3] J. Slonczewski and J. Magn. Magn. Mater. 159, L1 (1996). http://dx.doi.org/10.1016/03048853(96)000625
[4] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi and J. L. Erskine, Phys. Rev. Lett. 97, 057203 (2006).http://dx.doi.org/10.1103/PhysRevLett.97.057203
[5] M. Tsoi, Nature Phys. 4, 17 (2008). http://dx.doi.org/10.1038/nphys822
[6] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman and D. C. Ralph, Phys. Rev. Lett. 96, 227601 (2006). http://dx.doi.org/10.1103/PhysRevLett.96.227601
[7] S. Stuart and P. Parkin, Science 320, 190 (2008). http://dx.doi.org/10.1126/science.1145799
[8] J. Shibata, G. Tarara and H. Kohno, Phys. Rev. Lett. 94, 076601 (2005). http://dx.doi.org/10.1103/PhysRevLett.94.076601
[9] G. Tatara, H. Kohno and J. Shibata, Phys. Rep. 468, 6 (2008). http://dx.doi.org/10.1016/j.physrep.2008.07.003
[10] Z. Li and S. Zhang, Phys. Rev. Lett. 93, 127204 (2004). http://dx.doi.org/10.1103/PhysRevLett.93.127204
[11] S. Zhang and Z. Li, Phys. Rev. Lett 92, 207203 (2004). http://dx.doi.org/10.1103/PhysRevLett.92.207203
[12] J. Xiao, A. Zangwill and M. D. Stiles, Phys. Rev. B. 73, 054428 (2006). http://dx.doi.org/10.1103/PhysRevB.73.054428
[13] I. Turek, V. Drchal, J. Kudrnovsky, M. Sob and P. Weinberger, Electronic Structure of Disordered Alloys: Surface sand Interfaces (Kluwer, BostonLondon Dordrecht, 1997).
[14] O. K.Andersen, O. Jepsen and D. Glotzel, in Highlights in Condensed Matter Theory, edited by F. Bassani,Fumi and M. P. Tosi (NorthHolland, Amsterdam, 1985), p. 59.
[15] S. Wang, Y. Xu and K. Xia, Phys. Rev. B 77, 184430 (2008). http://dx.doi.org/10.1103/PhysRevB.77.184430
[16] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). http://dx.doi.org/10.1063/1.1663252
[17] X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. 324, 8 (2009). http://dx.doi.org/10.1016/j.aop.2009.05.004
Citation Information
C. Wang and K. Xia, Ballistic Current induced Effective Force on Magnetic Domain Wall. NanoMicro Lett. 1, 3439 (2009). doi: 10.5101/nml.v1i1.p3439
History
Received 24 September 2009; accepted 5 December 2009; published online 20 December 2009.