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Turbulence of domain walls in strong static fields
1015
Journal of Magnetism and Magnetic Materials 31-34 (1983) 1015-1016 TURBULENCE
OF DOMAIN
WALLS IN STRONG
STATIC FIELDS
F. W A L D N E R
Physik-Institut, Universityof Ziirich, CH-8001 Ziirich, Switzerland Domain walls in chains and square lattices reveal turbulent behavior when strong static fields are applied. These nonlinear phenomena were found by calculating the time evolution of the equations of motion for classical spins numerically.
To evaluate the dynamics of domain walls, a quantum Hamiltonian of spins coupled by exchange is usually approximated by: (i) treating the spins as classical vectors; and (ii) by expanding these vectors continuously [1-3]. By doing so, partial differential equations result with the exchange transformed into second derivatives. Unfortunately, these nonlinear equations can only be solved analytically for restricted initial conditions. More general numerical treatments are time consuming since a large number of data points is needed. Instead, without applying step (ii), the problem is described by N discrete classical vectors M~ governed by N equations of motion of the Landau-Lifshitz type
2I1 d m
flMiXHeff-°t~oMiX(M~×Heff)
where a denotes damping and fl is one unless fl = 0 forces pure relaxation into a stable state. The effective field writes for a linear chain with isotropic exchange and uniaxial anisotropy along z Heff = J ( M i _ ! + Mi+I) + D M z + H.
The exchange is n o w reduced to the evaluation of vector products without loss of nonlinearity. Therefore, N can be small. Starting from initial conditions M~(t = 0), the time evolution is calculated using finite time steps At. In addition, stable initial conditions such as a domain wall are easily found by letting a degenerate state (say a " w a l l " consisting of only one or two spins along the hard direction) relax with fl = 0. By using this method for walls only two lattice spacings wide, i.e. for 8D = J,
Fig. 1. Time evolution of the polar angle 0 along the chain azis x of a domain wall disturbed by a strong static field H. The parameters are for M 0 = 1 (arbitrary units): J = 20, 2D = 5, HdCm.= 2, H = ( - 3, 0, 9). Relaxation 5%. Calculated with 160 spins and 4000 time steps during 3 min. computing time, but only 120 spins plotted. 0304-8853/83/0000-0000/$03.00
© 1983 N o r t h - H o l l a n d
F. Waldner / Turbulence of domain walls"
1016
/
/
/ !
II
Fig. 3. The problem of fig. 1 is displayed after 3000 time steps. Here, instead of a "globe", a "map" is drawn with the polar angle as the vertical and the azimuthal angle as the horizontal axis, the "equator" is the center line. The points mark the directions of individual spins, and the line connects consecutive spins. Note again that excursions of both angles are of the same magnitude.
Fig. 2. The problem of fig. 1 is displayed after 1200 time steps. In order to show the orientations of the spins, the origins of all spins are moved to the center of the "globe". The arrows mark the directions of the domains, and the line on the surface of the "globe" connects the end-points of consecutive spins. (Further lines should give a 3d impression.) Note that the creation of a wall-antiwall pair also involves large excursions of the aximuthal angle. the wall energy 2 ( 2 D J ) 1/2 k n o w n from the c o n t i n u o u s model was f o u n d to within 1%. Further, the wall energy was the same whether there was a spin at the center or not. A Bloch wall 0 8 0 ° wall) in a linear chain was disturbed b y a strong static field H. Fig. 1 shows the time evolution of the polar angle 8 along the chain axis x. The d o m a i n s precess a r o u n d H which also has a c o m p o n e n t along x, a n d one d o m a i n is collapsing toward H due to the 5% relaxation. T h e wall is the source of spin waves emerging into the domains,. Surprizingly, the wall seems to create new w a l l - a n t i w a l l pairs. Figs. 2 a n d 3 illustrate in different ways that b o t h polar a n d a z i m u t h a l angles change in the wall regions giving patterns similar to t u r b u l e n t flow. Moreover, 360 ° kinks in easy-plane chains desintegrate into b r e a t h e r s for strong static fields, a n d the
time evolution might almost go t h r o u g h a " B l o c h p o i n t " (two opposed spins) [4,5]. Furthermore, in a square lattice with a d o m a i n wall c o n t a i n i n g Bloch lines, a strong field excites a n d couples J a n a k [6] modes. Preliminary runs indicate that the n o n l i n e a r b r e a k d o w n of wall velocity might be connected with the d y n a m i c creation of quasi-Bloch lines, i.e. with quasi-turbulent m o t i o n inside the wall. It is a pleasure to t h a n k F. Bloch, U. Enz, E. Magyari, P.F. Meier, T. Schneider, J.C. Slonczewski, M. Steiner, E. Stoll, H. T h o m a s and P.E. Wigen for m a n y helpful comments, a n d S. H u g e n t o b l e r for his graphical work.
References [1] F.H. de Leeuw, R. van den Doel and U. Enz. Rep. Prog. Phys. 43 (1980) 689 and refs. therein. [2] A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic, New York, 1979) and refs. therein. [3] P.E. Wigen, in: Proc. Varenna Summer Course LXX, ed. J.F. Dillon, Jr. (North-Holland, Amsterdam, 1978) p. 195 and refs. therein. [4] F. Waldner, Helv. Phys. Acta 54 (1981) 28. [5] E. Magyari and H. Thomas, Phys. Rev. B25 (1982) 531 and refs. therein. [6] J.F. Janak, Phys. Rev. 134 (1964) A 411.
Journal of Magnetism and Magnetic Materials 31-34 (1983) 1015-1016 TURBULENCE
OF DOMAIN
WALLS IN STRONG
STATIC FIELDS
F. W A L D N E R
Physik-Institut, Universityof Ziirich, CH-8001 Ziirich, Switzerland Domain walls in chains and square lattices reveal turbulent behavior when strong static fields are applied. These nonlinear phenomena were found by calculating the time evolution of the equations of motion for classical spins numerically.
To evaluate the dynamics of domain walls, a quantum Hamiltonian of spins coupled by exchange is usually approximated by: (i) treating the spins as classical vectors; and (ii) by expanding these vectors continuously [1-3]. By doing so, partial differential equations result with the exchange transformed into second derivatives. Unfortunately, these nonlinear equations can only be solved analytically for restricted initial conditions. More general numerical treatments are time consuming since a large number of data points is needed. Instead, without applying step (ii), the problem is described by N discrete classical vectors M~ governed by N equations of motion of the Landau-Lifshitz type
2I1 d m
flMiXHeff-°t~oMiX(M~×Heff)
where a denotes damping and fl is one unless fl = 0 forces pure relaxation into a stable state. The effective field writes for a linear chain with isotropic exchange and uniaxial anisotropy along z Heff = J ( M i _ ! + Mi+I) + D M z + H.
The exchange is n o w reduced to the evaluation of vector products without loss of nonlinearity. Therefore, N can be small. Starting from initial conditions M~(t = 0), the time evolution is calculated using finite time steps At. In addition, stable initial conditions such as a domain wall are easily found by letting a degenerate state (say a " w a l l " consisting of only one or two spins along the hard direction) relax with fl = 0. By using this method for walls only two lattice spacings wide, i.e. for 8D = J,
Fig. 1. Time evolution of the polar angle 0 along the chain azis x of a domain wall disturbed by a strong static field H. The parameters are for M 0 = 1 (arbitrary units): J = 20, 2D = 5, HdCm.= 2, H = ( - 3, 0, 9). Relaxation 5%. Calculated with 160 spins and 4000 time steps during 3 min. computing time, but only 120 spins plotted. 0304-8853/83/0000-0000/$03.00
© 1983 N o r t h - H o l l a n d
F. Waldner / Turbulence of domain walls"
1016
/
/
/ !
II
Fig. 3. The problem of fig. 1 is displayed after 3000 time steps. Here, instead of a "globe", a "map" is drawn with the polar angle as the vertical and the azimuthal angle as the horizontal axis, the "equator" is the center line. The points mark the directions of individual spins, and the line connects consecutive spins. Note again that excursions of both angles are of the same magnitude.
Fig. 2. The problem of fig. 1 is displayed after 1200 time steps. In order to show the orientations of the spins, the origins of all spins are moved to the center of the "globe". The arrows mark the directions of the domains, and the line on the surface of the "globe" connects the end-points of consecutive spins. (Further lines should give a 3d impression.) Note that the creation of a wall-antiwall pair also involves large excursions of the aximuthal angle. the wall energy 2 ( 2 D J ) 1/2 k n o w n from the c o n t i n u o u s model was f o u n d to within 1%. Further, the wall energy was the same whether there was a spin at the center or not. A Bloch wall 0 8 0 ° wall) in a linear chain was disturbed b y a strong static field H. Fig. 1 shows the time evolution of the polar angle 8 along the chain axis x. The d o m a i n s precess a r o u n d H which also has a c o m p o n e n t along x, a n d one d o m a i n is collapsing toward H due to the 5% relaxation. T h e wall is the source of spin waves emerging into the domains,. Surprizingly, the wall seems to create new w a l l - a n t i w a l l pairs. Figs. 2 a n d 3 illustrate in different ways that b o t h polar a n d a z i m u t h a l angles change in the wall regions giving patterns similar to t u r b u l e n t flow. Moreover, 360 ° kinks in easy-plane chains desintegrate into b r e a t h e r s for strong static fields, a n d the
time evolution might almost go t h r o u g h a " B l o c h p o i n t " (two opposed spins) [4,5]. Furthermore, in a square lattice with a d o m a i n wall c o n t a i n i n g Bloch lines, a strong field excites a n d couples J a n a k [6] modes. Preliminary runs indicate that the n o n l i n e a r b r e a k d o w n of wall velocity might be connected with the d y n a m i c creation of quasi-Bloch lines, i.e. with quasi-turbulent m o t i o n inside the wall. It is a pleasure to t h a n k F. Bloch, U. Enz, E. Magyari, P.F. Meier, T. Schneider, J.C. Slonczewski, M. Steiner, E. Stoll, H. T h o m a s and P.E. Wigen for m a n y helpful comments, a n d S. H u g e n t o b l e r for his graphical work.
References [1] F.H. de Leeuw, R. van den Doel and U. Enz. Rep. Prog. Phys. 43 (1980) 689 and refs. therein. [2] A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic, New York, 1979) and refs. therein. [3] P.E. Wigen, in: Proc. Varenna Summer Course LXX, ed. J.F. Dillon, Jr. (North-Holland, Amsterdam, 1978) p. 195 and refs. therein. [4] F. Waldner, Helv. Phys. Acta 54 (1981) 28. [5] E. Magyari and H. Thomas, Phys. Rev. B25 (1982) 531 and refs. therein. [6] J.F. Janak, Phys. Rev. 134 (1964) A 411.