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Running oscillatory motion of solitons in a driven damped anisotropic magnetic chain
Journal of Magnetism and Magnetic Materials 54-57 (1986) 865 866 RUNNING OSCILLATORY MAGNETIC CHAIN J.A. H O L Y S T ,
MOTION
A. S U K I E N N I C K I
865
OF SOLITONS
IN A DRIVEN
DAMPED
ANISOTROPIC
a n d J.J. Z E B R O W S K I
Institute of Physics, Warsaw Technical University, 00-662 Warsaw, Poland
Undercritical and overcritical motion of solitons in a driven damped magnetic chain with two local anisotropies is considered. The undercritical stationary motion results from balance of drive and damping. The overcritical motion is a superposition of the running and oscillatory motions. An analytical solution for a limiting case is presented.
1. I n t r o d u c t i o n
respect to x, a dot - the time derivative, and a - the G i l b e r t d a m p i n g p a r a m e t e r (0 < a << 1).
It is a well-known fact that in the one-dimensional n o n l i n e a r systems large a m p l i t u d e solitary waves in the form of pulses, kinks or d o m a i n walls can p r o p a g a t e with c o n s t a n t velocities a n d c o n s t a n t shapes if b o t h d a m p i n g a n d external forces are neglected. One can expect, however, that some interesting features can appear if d a m p i n g a n d external forces are taken into account. Here, the m o t i o n of kinks in the classical Heisenberg chain with a special double local anisotropy a n d with b o t h the Gilbert d a m p i n g as well as the c o n s t a n t external field included is considered. This p r o b l e m appears to be formally equivalent to the very well-known p r o b lem of the d o m a i n wall dynamics in bulk ferromagnetic materials [1-4]. Thus the solutions proposed for d o m a i n wall dynamics can be easily m a p p e d to our situation. Only the most i m p o r t a n t results are presented here; the full version will be p u b l i s h e d elsewhere.
2. E q u a t i o n s
3. U n d e r c r i t i e a l s o l u t i o n s
Exactly as in the case of the Walker solution for d o m a i n wall dynamics in bulk ferromagnetic materials [1], the soliton solution in the form of "~-kinks with cp = cp0 = const and O(x, t) = O(x - vt) can be found if I BI < B~ = a A [5]. Namely, there are two solutions ~V
0 =
•
where the minus sign corresponds to a stable solution a n d the plus to an unstable solution (in the sense that for a small p e r t u r b a t i o n of this angle the time evolution leads to results far from it). The velocity of the stable solution is equal to
v: + B{J/a2[A(1-
~I - ( B / a A ) 2
) + 2 C ] } '/2, (5)
a n d the distribution of the angle 0 is given by
of motion
cos 0 ( x , t ) = - tanh[ +_( x - v t ) / 8 ], A classical spin chain with the H a m i l t o n i a n
-E[-ssi.si+l +a(s:)2-v(s;f -ybs:] i
-
b sin ~ - a ~ v sin 2 0 = J ( O / O x ) (
sin 2 0q0x)
- 2 A sin 2 0 sin cp cos qv, --c~ sin 0 - a ~ =
where
(1)
is considered with j, a, c > O , y = - g [ l l n [ / h . The spin coordinates are written in the form S i = S[cos 0 i, sin 0 i sin cpi, sin 0 i cos cpi ] a n d the cont i n u u m a p p r o x i m a t i o n is taken by c h a n g i n g Oi(t), cpi(t ) to O(x, t), qo(x, t). The classical equations of m o t i o n with the Gilbert d a m p i n g included are:
(2)
- J O ~ + J sin ~ cos 0 (Cpx) 2
+ 2 A sin 0 cos 0 cos2cp + 2 C sin 0 cos 0 + B sin 0 ,
(3) where J = j S l 2, A = aS, C = cS, B = by, l is the lattice constant, the index x denotes the partial derivative with 0304-8853/86/$03.50
(6)
-
+
.
(7)
The formula (6) shows that our soliton corresponds to a v-kink or v-antikink (with the width p a r a m e t e r 6) propagating with a c o n s t a n t velocity v given by (5). One can see that this solution corresponds to the well-known soliton for the completely integrable system without any d a m p i n g and external force [6], but in our case the velocity results from a balance of the drive a n d d a m p ing. One can also see that for B = 0 the only soliton is a static (v = 0) kink or antikink with cp0 = ½"~(2n + 1), n - integer. 4. O v e r c r i t i c a l s o l u t i o n s
The most interesting is the character of solutions which one obtains if [ B[ > B~ = aA. N o motion with a c o n s t a n t q~ is possible in this case. Numerical analysis
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
J.A. Hob,st et aL / Solitons in a magnetic" chain
866
made by Schryer and Walker [2] for the dynamics of domain walls in bulk ferromagnetic materials, mapped to our situation shows that for I B I > B,. the solutions with cost~(x, t)=
-tanh{[x-q(t)]/6(t)},
9)=9)(t)
(s) can be found, but in this case the motion of v-kinks consists of a forward or running motion and an oscillatory motion superimposed. For example, after turning on the external field at t = t 0, the kink velocity (which was assumed to be zero for t < to) quickly increases to a maximum value, simultaneously the azimuthal angle 9) 1 changes monotonically to the value ~'~ + n (n - integer) and the width parameter decreases. Next, the kink velocity begins slowly to decrease which is accompanied by a further monotonical change of 9) and a further decrease of the width parameter. In the following phase of motion the velocity decreases rapidly reaching zero and even negative values down to the minimum value. In the last phase the kink moves in the direction opposite to that in the first phase of the motion ! After that the kink velocity starts to increase again with an increase of the width parameter and a quick change of the angle 9). The motion of the kink has therefore, to some extent, an oscillatory character however a net forward motion results. The physical reason for such a motion lies, of course, in the interplay between the external field effect, both anisotropies considered and the damping. It is interesting that the very peculiar oscillatory behaviour of domain walls in magnetic bubble materials for I BI > B c was observed experimentally [7] although in later literature indications may be found that the backward motion may be caused by a more complicated wall structure. 5. T h e c a s e o f C >> A
The character of motion described above may be understood much better if one considers a special case for which an approximate analytical solution can be found. Namely, if the condition C / A >> 1 is fulfilled, a nonstationary motion of the kink may be described approximately by the following equations [2,3]:
2t + aSogp = A6 o sin 29), -3o• + a~) = B60,
(9) (10)
where 9)= 9)(0 is the azimuthal angle assumed to be independent of x, 3 o ~ VFJ/2C is the approximative value of 8 and q = q(t) represents the position of the
middle of the kink. In this approximation the kink is treated as a rigid object with a constant width parameter 80. Direct integration of equations (9) and (10) leads to the results (for I B I > Be)
q( t ) = q( to) + 6o[ Bt + 9)( t ) l / a , 9)(t)=tg-l[b
l+(1-b
"r=(l+a2)/[Bc(b2-1)l/2],
(11)
2)'/2ctg(t
t0)/q-],
b = B / B c.
(121
The solution (11), (12) represents just the running oscillatory motion described in section 4. The mean velocity in our case is equal:
(,80/a)- [A80/(1 +
2-1]'<(13/
The equations (9), (10) allow to calculate also the velocity of the kink for the whole range of the external field B. One obtains for I B I < B c a linear increase of the velocity with B, but for I B I > Bc the mean velocity decreases sharply and then slowly increases with the value of B. An interesting limiting case is for the damping parameter a tending to zero. The solution of (9), (10) for a ~ 0
q( t ) = q( to ) - ( ASo/2 B ) cos 2 ( 9 ) 0 - Bt ), 9)(t)=9)o
Bt
(14)
corresponds to pure oscillations of the kink with no net forward motion 6. Conclusions A very peculiar behaviour of kink-solitons in the classical Heisenberg chain with the double local anisotropy is found in the presence of damping and the constant external field. Stationary motion for fields below a critical value is in a clear contrast to the overcritical motion being a nontrival superposition of forward translational and oscillatory motions. The important role of damping is shown. In the case of zero damping only pure oscillations are obtained. [1] L.R. Walker (unpublished). An account of this work was given by J.F. Dillon, Jr., in: Magnetism, vol. III, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963) p. 450 453. [2] N.L. Schryer and L.R. Walker, J. Appl. Phys. 45 (1974) 5406. [3] J.C. Slonczewski, Intern. J. Magn. 2 (1972) 85. [4] K.A. Long and A.R. Bishop, J.P. Phys. A 12 (1979) 1325. [5] E. Magyari and H. Thomas, Z. Phys. B 59 (1985). [6] C. Etrich and H.J. Mikeska, J. Phys. C 16 (1983) 4889. [7] G.P. Vella-Coleiro, Appl. Phys. Lett. 29 (1976) 445.
MOTION
A. S U K I E N N I C K I
865
OF SOLITONS
IN A DRIVEN
DAMPED
ANISOTROPIC
a n d J.J. Z E B R O W S K I
Institute of Physics, Warsaw Technical University, 00-662 Warsaw, Poland
Undercritical and overcritical motion of solitons in a driven damped magnetic chain with two local anisotropies is considered. The undercritical stationary motion results from balance of drive and damping. The overcritical motion is a superposition of the running and oscillatory motions. An analytical solution for a limiting case is presented.
1. I n t r o d u c t i o n
respect to x, a dot - the time derivative, and a - the G i l b e r t d a m p i n g p a r a m e t e r (0 < a << 1).
It is a well-known fact that in the one-dimensional n o n l i n e a r systems large a m p l i t u d e solitary waves in the form of pulses, kinks or d o m a i n walls can p r o p a g a t e with c o n s t a n t velocities a n d c o n s t a n t shapes if b o t h d a m p i n g a n d external forces are neglected. One can expect, however, that some interesting features can appear if d a m p i n g a n d external forces are taken into account. Here, the m o t i o n of kinks in the classical Heisenberg chain with a special double local anisotropy a n d with b o t h the Gilbert d a m p i n g as well as the c o n s t a n t external field included is considered. This p r o b l e m appears to be formally equivalent to the very well-known p r o b lem of the d o m a i n wall dynamics in bulk ferromagnetic materials [1-4]. Thus the solutions proposed for d o m a i n wall dynamics can be easily m a p p e d to our situation. Only the most i m p o r t a n t results are presented here; the full version will be p u b l i s h e d elsewhere.
2. E q u a t i o n s
3. U n d e r c r i t i e a l s o l u t i o n s
Exactly as in the case of the Walker solution for d o m a i n wall dynamics in bulk ferromagnetic materials [1], the soliton solution in the form of "~-kinks with cp = cp0 = const and O(x, t) = O(x - vt) can be found if I BI < B~ = a A [5]. Namely, there are two solutions ~V
0 =
•
where the minus sign corresponds to a stable solution a n d the plus to an unstable solution (in the sense that for a small p e r t u r b a t i o n of this angle the time evolution leads to results far from it). The velocity of the stable solution is equal to
v: + B{J/a2[A(1-
~I - ( B / a A ) 2
) + 2 C ] } '/2, (5)
a n d the distribution of the angle 0 is given by
of motion
cos 0 ( x , t ) = - tanh[ +_( x - v t ) / 8 ], A classical spin chain with the H a m i l t o n i a n
-E[-ssi.si+l +a(s:)2-v(s;f -ybs:] i
-
b sin ~ - a ~ v sin 2 0 = J ( O / O x ) (
sin 2 0q0x)
- 2 A sin 2 0 sin cp cos qv, --c~ sin 0 - a ~ =
where
(1)
is considered with j, a, c > O , y = - g [ l l n [ / h . The spin coordinates are written in the form S i = S[cos 0 i, sin 0 i sin cpi, sin 0 i cos cpi ] a n d the cont i n u u m a p p r o x i m a t i o n is taken by c h a n g i n g Oi(t), cpi(t ) to O(x, t), qo(x, t). The classical equations of m o t i o n with the Gilbert d a m p i n g included are:
(2)
- J O ~ + J sin ~ cos 0 (Cpx) 2
+ 2 A sin 0 cos 0 cos2cp + 2 C sin 0 cos 0 + B sin 0 ,
(3) where J = j S l 2, A = aS, C = cS, B = by, l is the lattice constant, the index x denotes the partial derivative with 0304-8853/86/$03.50
(6)
-
+
.
(7)
The formula (6) shows that our soliton corresponds to a v-kink or v-antikink (with the width p a r a m e t e r 6) propagating with a c o n s t a n t velocity v given by (5). One can see that this solution corresponds to the well-known soliton for the completely integrable system without any d a m p i n g and external force [6], but in our case the velocity results from a balance of the drive a n d d a m p ing. One can also see that for B = 0 the only soliton is a static (v = 0) kink or antikink with cp0 = ½"~(2n + 1), n - integer. 4. O v e r c r i t i c a l s o l u t i o n s
The most interesting is the character of solutions which one obtains if [ B[ > B~ = aA. N o motion with a c o n s t a n t q~ is possible in this case. Numerical analysis
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
J.A. Hob,st et aL / Solitons in a magnetic" chain
866
made by Schryer and Walker [2] for the dynamics of domain walls in bulk ferromagnetic materials, mapped to our situation shows that for I B I > B,. the solutions with cost~(x, t)=
-tanh{[x-q(t)]/6(t)},
9)=9)(t)
(s) can be found, but in this case the motion of v-kinks consists of a forward or running motion and an oscillatory motion superimposed. For example, after turning on the external field at t = t 0, the kink velocity (which was assumed to be zero for t < to) quickly increases to a maximum value, simultaneously the azimuthal angle 9) 1 changes monotonically to the value ~'~ + n (n - integer) and the width parameter decreases. Next, the kink velocity begins slowly to decrease which is accompanied by a further monotonical change of 9) and a further decrease of the width parameter. In the following phase of motion the velocity decreases rapidly reaching zero and even negative values down to the minimum value. In the last phase the kink moves in the direction opposite to that in the first phase of the motion ! After that the kink velocity starts to increase again with an increase of the width parameter and a quick change of the angle 9). The motion of the kink has therefore, to some extent, an oscillatory character however a net forward motion results. The physical reason for such a motion lies, of course, in the interplay between the external field effect, both anisotropies considered and the damping. It is interesting that the very peculiar oscillatory behaviour of domain walls in magnetic bubble materials for I BI > B c was observed experimentally [7] although in later literature indications may be found that the backward motion may be caused by a more complicated wall structure. 5. T h e c a s e o f C >> A
The character of motion described above may be understood much better if one considers a special case for which an approximate analytical solution can be found. Namely, if the condition C / A >> 1 is fulfilled, a nonstationary motion of the kink may be described approximately by the following equations [2,3]:
2t + aSogp = A6 o sin 29), -3o• + a~) = B60,
(9) (10)
where 9)= 9)(0 is the azimuthal angle assumed to be independent of x, 3 o ~ VFJ/2C is the approximative value of 8 and q = q(t) represents the position of the
middle of the kink. In this approximation the kink is treated as a rigid object with a constant width parameter 80. Direct integration of equations (9) and (10) leads to the results (for I B I > Be)
q( t ) = q( to) + 6o[ Bt + 9)( t ) l / a , 9)(t)=tg-l[b
l+(1-b
"r=(l+a2)/[Bc(b2-1)l/2],
(11)
2)'/2ctg(t
t0)/q-],
b = B / B c.
(121
The solution (11), (12) represents just the running oscillatory motion described in section 4. The mean velocity in our case is equal:
(,80/a)- [A80/(1 +
2-1]'<(13/
The equations (9), (10) allow to calculate also the velocity of the kink for the whole range of the external field B. One obtains for I B I < B c a linear increase of the velocity with B, but for I B I > Bc the mean velocity decreases sharply and then slowly increases with the value of B. An interesting limiting case is for the damping parameter a tending to zero. The solution of (9), (10) for a ~ 0
q( t ) = q( to ) - ( ASo/2 B ) cos 2 ( 9 ) 0 - Bt ), 9)(t)=9)o
Bt
(14)
corresponds to pure oscillations of the kink with no net forward motion 6. Conclusions A very peculiar behaviour of kink-solitons in the classical Heisenberg chain with the double local anisotropy is found in the presence of damping and the constant external field. Stationary motion for fields below a critical value is in a clear contrast to the overcritical motion being a nontrival superposition of forward translational and oscillatory motions. The important role of damping is shown. In the case of zero damping only pure oscillations are obtained. [1] L.R. Walker (unpublished). An account of this work was given by J.F. Dillon, Jr., in: Magnetism, vol. III, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1963) p. 450 453. [2] N.L. Schryer and L.R. Walker, J. Appl. Phys. 45 (1974) 5406. [3] J.C. Slonczewski, Intern. J. Magn. 2 (1972) 85. [4] K.A. Long and A.R. Bishop, J.P. Phys. A 12 (1979) 1325. [5] E. Magyari and H. Thomas, Z. Phys. B 59 (1985). [6] C. Etrich and H.J. Mikeska, J. Phys. C 16 (1983) 4889. [7] G.P. Vella-Coleiro, Appl. Phys. Lett. 29 (1976) 445.