The effect of a strong magnetic field H, perpendicular to the easy axis of a ferromagnetic, on the motion of a domain wall is investigated. It is shown that the peak ...
Physica B 202 (1994) 193-201
ELSEVIER
Dynamics of domain walls (solitons) in ferromagnets in a magnetic field perpendicular to the easy axis B.A. Ivanov*, N.E. Kulagin, K.A. Safaryan Institute of Metal Physics, Academy of Sciences of Ukraine, 36 Vernadsky Street, Kiev 180, 252680 Ukraine
Received 2 August 1993; revised 19 April 1994
Abstract
The effect of a strong magnetic field H, perpendicular to the easy axis of a ferromagnetic, on the motion of a domain wall is investigated. It is shown that the peak velocity of motion vc is defined by various scenarios at weak (H ,~ Ha) and strong ( H a - H ,~ Ha) magnetic fields, where Ha is the anisotropy field. Therefore, the peak velocity V¢ is a nonmonotonous function of the magnetic field, when H ranges from 0 to Ha. The character of the DW velocity dependence on the driving field Hz was studied.
The dynamics of domain walls (DW) in ferromagnetic material (FM) in a strong-in-plane magnetic field (i.e. perpendicular to the easy axis of FM) has attracted attention of a number of researchers (see the m o n o g r a p h y [1-1 and some recent publications I-2, 3]). Both the D W mobility and the limiting velocity are known to increase with increasing in-plane magnetic field. In comparatively weak fields, H ~< 4rtMs, where Ms is the saturation magnetization, this is caused by the disappearance of the effect of the D W torsion [1-1. At 4~tMs < H < Ha, where Ha is the anisotropy field (such a region exists only for F M with large quality factor q = HJ47tMs: we will consider only such a kind of magnets), D W may be considered onedimensional. Usual effects of rearrangement of a D W structure due to creation of horizontal Bloch lines etc. in our case are absent, and cannot change *Corresponding author.
the peak velocity. At the same time, experimental data give evidence [3-1 that in this region the limiting velocity increases with H. The theory of D W motion, and particularly the theory of limiting velocity for one-dimensional D W in F M under strong magnetic fields, was discussed in Refs. [4, 5, 6-1. Asymptotic analysis for the magnetization behaviour at some distance from D W shows [5-1 that at q ~> 1 two characteristic values for the velocity v~_~ and v~+~ exist: v(+) = c(1 + x/1 - (H/Ha)2), v~_~ --- c(1 - x / l - (H/Ha)2),
gmo~/~,
(1)
where c = ~ and fl are constants of the non-homogeneous exchange interaction and anisotropy (see Eq. (2) below) and q is the gyromagnetic ratio. When a transition form v < v~_~ to v~_) < v < v(+) occurs, then the monotonous
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change of the magnetization towards an equilibrium at a distance from the soliton solution (for example, the D W ) is replaced by a non-monotonous behaviour. At v > v~+~localized solutions of DW type become absolutely impossible because the value v~+) physically represents the spin-wave phase velocity. Schloemann [8] indicated that in the Walker model the value of vt_~ coincides with the Walker limiting value Vw. The value v~_) was associated with the D W limiting velocity [5] for H ¢: 0. According to the analysis of Ref. [6], at H -~ H , the magnetization vector dynamics is described using the Lorentz-invariant model, where the value of (v~+~ + v~_~)/2 (see Eq. (1)) represents the selected velocity c. Therefore, at H ~ Ha the limiting value of velocity approaches c, but results obtained in Ref. [5] do not permit us to describe the dependence of vc(H) at low but finite values of (Ha -- H)/Ha. According to experimental data [.3], the limiting D W velocity increases with in-plane magnetic field H, and reaches the value of 3.25 × 103m/s at H = 0.8 Ha. This value of the velocity exceeds not only the value of the Walker limit Vw or v¢_) or v~+) (for such magnets q = 40 and at H = 0.8Ha, vw ~ re_) - 1.5 x 103 m/s) but also that of c = 2.6 × 103 m/s. It means that the limiting value of the velocity must change according to a nonmonotonous law with H and reach the maximal value in the region of ( H ~ - H ) ~ H ~ Ha. N o t one of the existing theories can describe such a behaviour. Let us assume that q >> 1, then in the case of interest 4riMs ~ H < Ha we may neglect the energy of demagnetizing field and proceed from the following expression for the energy density in FM: W=~(VM) 2+
(M 2 + M 2 ) - H M x ,
(2)
where the axis z coincides with the easy axis in FM, the field H being parallel to the x-axis. In angular variables the magnetization can be expressed as M~ = M~ cos 0,
Mx = M~ sin 0 cos (p,
FM. The structure of the D W moving along the y-axis with the velocity of v is determined by the Landau-Lifshitz equation:
120" - sin 0 cos 0 [,1 + 12(¢p') 2] + h c o s 0 c o s q~ = lo(v/c) ~p' sin 0,
(3)
12(~p' sin 2 0)' - h sin 0 sin ~p = - lo(v/c) O' sin 0,
(4)
where l o = w / ~ , h = H / H a , 0 ' = d 0 / d ¢ , ~p'= &p/d~, ~ = y - vt. The first integral of this set of equations may be expressed by 12[,(0') 2 + (~o')2 sin 2 0] - sin 2 0 + 2h sin 0cos ~p = const.
(5)
Domain wall relaxation is described by taking into account the dissipation function. The usual Gilbert dissipative term can be described by the relativistic dissipation function Qr:
(Tr --
20 M~ 3f?MVdr, \ st )
16t
where 2 is a dimensionless relaxation constant (other terms contributing to the dissipative function of the F M in the in-plane magnetic field were discussed in more detail in Ref. [7]). The drag force F for the D W plane moving with constant velocity v is determined from the dissipative function Q using the following expression: F = - 2Q/vS, S being the D W area. For small values of 2, 2
