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Dynamics of interaction of domain walls in (2+1)-dimensional non-linear sigma-model
Section 14. Physics
Shokirov Farhod Shamsidinovich, Candidate of Physico-Mathematical Sciences,
S. U. Umarov Physical-Technical Institute, Academy of Sciences of the Republic of Tajikistan E-mail: farhod0475@gmail.com
Dynamics of interaction of domain walls in (2+1)-dimensional non-linear sigma-model
Abstract: By numerical simulation the dynamics of interactions of 180-degree Neel type domain walls in (2+1) dimensional O (3) vectorial nonlinear sigma model is investigated. Are obtained numerically: new solutions in the form of domain walls with the rotation of the vector of A3 field in isotopic space; long-range model of interaction of the domain walls; oscillating (bion) model of the bound states of the domain walls.
Keywords: bound state of domain walls — Neel-type domain wall — О (3) vectorial nonlinear sigma-model — long-range interactions of domain walls — dynamics of interaction.
Localized boundaries of magnetic domains (domain walls, topological solitons: kinks, antikinks) are an important element of the magnetic domain structures, primarily in terms of their practical application. Investigation of the properties of magnetic domain walls, attracts the attention of experts of the area due to their possible applications in a variety of modern technological processes, for example, in the spin electronics, while creating productive and reliable reading devices, recording, storing and processing digital data, where used the relationship of magnetization and electric polarization. Dynamics of domain walls is particularly relevant, for example, in the concept of the racetrack memory (magnetic racetrack memory, MRM) [1], based on the use of the spin current to move the domain walls in the limits of the magnetic nanowires.
In this paper we obtained a model of collision of new types of moving Neel type domain walls (with the rotation of the magnetization vector in isotopic space: q(x,у,t) + 0.0 ) in (2+l)-dimensional anisotropic O (3) vectorial nonlinear sigma model (VNSM). The Euler-Lagrange equations [2-3] of the studied model take the following form:
20^0 + sin 20(l -3^3^) = 0, (1)
2cos вдурди<р + sin вдуд^(р = 0, ^ = 0,1,...,D, D = 2, where Q(x,у,t) and y(x, y, t) are the Euler angles. Recall that equation (1) in the meridian section of the isotopic space (^(x,y,t) = 0 ) reduced [2-3] to the sine-Gordon equation of form:
2l з2в д 20 d 0 , 0
21 —2------------- I = 0 sm20.
dt2 dx2 dy2
In this paper, based on the given in [4] of the analytical form of the solution in the form of Neel-type domain-wall
z (x, y, t ) = 4arctg <
V
of the (2+1) -dimensional sine-Gordon equation of form:
A - kiTh - k2 Ay + w sin z = 0,
was obtained the numerical model of the domain wall in (2+1)-dimensional O (3) VNSM with rotation vector ofA3-field (p(x, y, t) 0) in the isotopic space (Fig. 1 abc) in the form:
(2)
0(x, y, t) = 2arctg
ф(х, y, t) = ат
Bl kx-yx 1+B2| — y-—y0
Fig. 1. The numerical model of the domain wall of the form (2) of model (1) with a = 0.3: a) energy density (DH). The dynamics of spins in the isotopic space at t = 0.0; b) kink; c) antikink; d) field of kink-antikink
151
Section 14. Physics
Fig. 2. The numerical model (DH, the dynamics of spins in the isotopic space) of the frontal collision of the domain walls the form (2), p(x,y,t) = or, rn = 0.3, uk(t0) = uak(t0)»0.1: a) t = 0.0; b) t = 16.5 — interaction; c) t = 18.0 — association; d) t = 19.5 — the passage of solitons the resonance zone; e) t = 30.0. The direction of movement: ^ — kink; <---------------------------------antikink
It is known that the real dynamics of soliton solutions, where fully manifested their specific, particle-like properties can be obtained by carrying out the study of the dynamics of their interactions [5].
In this paper, on the basis of our new dynamical numerical solutions (ris.ld) were developed the models of frontal collision of domain walls — solutions of kink-antikink of the form (2) of the (2+l)-dimensional O (3) VNSM (ф(х ^t) 0 ) (Fig.2).
Fig. 2 shows that when a frontal collision (Fig. 2 abc) topological solitons (2) (a kink-antikink) pass through each
other (Fig. 2 de), wherein kink proceeds to the antikink state and vice versa.
Next, we present a model of the long-range domain walls of the form (2) which we have obtained for the (2+1) -dimensional O (3) VNSM. Numerical simulations have shown that at the interaction of the pair ofdomain walls of the form (2), which differ from each other by the presence of rotation of vector of A3-field in the isotopic space (ф, (x,y,t) = 0, ф2 (x,y,t) > 0 ) (Fig. 3 a) appears the long-range effect. Fig. 3 shows an example of manifestation of long-range forces in the simulation of frontal collision domain walls of type (2) of the model (l).
Fig. 3. The numerical model (DH, the dynamics of spins in the isotopic space) of long-range interaction of domain walls form (2), ^ (x,y,t) = 0, p2(x,y,t) = 0.3r, uk(t0) = uak(t0) = 0.1: a) t = 0.0; b) t = 12.9 : uk ^ 0, uak ^ 0;
c) t = m : Uk > 0 , Uak ~ 0 ; d) t = 19.5 : Uk > 0 , Uak ~ 0
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Dynamics of interaction of domain walls in (2+1)-dimensional non-linear sigma-model
At this, the velocity of the domain wall (right) having a spin rotation in isotopic space (p2 (x, y, t) > 0) is reduced to almost zero ((t > 12.9) ^ 0) (Fig. 3 b), a kink (the left side) (Ф2 (x, y, t) = 0) after the interaction continues to move in the opposite direction (Fig. 3 cd).
The manifestation of long-range forces in the experiments is shown in Fig. 3 were detected at speeds of domain walls in the interval uk (t0 ) = vak (t0 )e(0.0,0.185) . In between uk (t0 ) = vak (t0 )e(0.345,0.86) numerical models showed the results described in Fig. 2, i. e. in this case the
solitons pass through each other. Qualitatively new results were obtained on the average interval of velocities of colliding domain walls form (2) of the model (l), which are described below.
At movement of domain walls at speeds uk (t0) = uak (t0) e (0.186,0.34)inmodel is shown in Fig. 3 were obtained new kinds of solutions of (2+l)-dimensional O (3) VNSM in the form of oscillating interconnec-ted (bion) state of the domain walls (2) (for ф (x,y,t) = 0 and (x У, *)> 0 ) (Fig. 4).
Fig. 4. The numerical model (DH, energy density) of the forming the interconnected (Bion) state of the kink-antikink at the collision of the domain walls form (2) with ф (x,y,t) = 0, y2 (x,y,t) = 0.3т , vk(t0) = uak(t0)» 0.29 : a) t e [0.0,10.2]; b) t e [5.0,24.0]; c) t e [27.6,36.0]; d) t e[39.6,46.8]; e) t e [50.4,57.9]; f) t = 60.0, the integral of energy of system at t e [0.0,60.0]. The direction of movement: ^ — kink; ^ — antikink
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