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Soliton instability in an easy plane ferromagnet
Physica 5D (1982)359-369 :: North-Holland Publishing Company
SOLITON INSTABILITY IN AN EASY PLANE ~ O M A G N E T Pradeep K U M A R Department of Physics, University of Florida Gainesville, Florida 32611, USA Received 15 December 1981
We report here an analysis of the limitations of a sine-Gordon like description of an easy plane ferromagnetic chain. We show that the sG description is valid only at low magnetic fields and low velocities of soliton motion. Even at low velocities, the soliton emits magnons. For fields larger than a critical field He(u), the soliton is unstable towards the formation of a distorted texture with large off-easy plane excursions for the spin. This instability is preceded by a split-off-localized mode (from the magnon continuum) that goes soft at the critical field. The distorted texture is found to be immobile in that no ~o = 0 translational mode exists. Instead we find oscillation frequencies for the texture. The sG soliton distorts its environment and gets trapped in it.
1. Introduction
Soliton-like excitations in one-dimensional magnets have been studied widely [I] in recent years. In one dimension problems, solitons constitute a degree of freedom in addition to the linear wave-like excitation and thus affect the thermodynamics and scattering properties. The more exciting experiments, in recent years have been those of neutron scatt,~ring. Here the neutrons can scatter inelastically and, in a linear magnet, excite the spin-waves (or magnons). In a non-linear magnet with a thermal distribution of solitons the neutrons can also scatter off the solitons. The latter being massive objects, give rise to a broad central peak corresponding to the transfer of momentum (translational energy) from the neutron to the soliton. The analysis of this peak was initiated by the predictions of Mikeska [2]. Kjems and Steiner [3] observed a linear chain with ferrothis peak in ~sr~w3, "-'"" magnetic exchange interaction and an easy plane anisotropy. Much of this work, including a careful reexamination of the relativistic effects associated with the soliton motion by Leung and Huber[4], has been reviewed by Steiner[Y]. There now exists a considerable body of
literature[6] containing results about the equilibrium as well as thermodynamic properties of these solitons. This paper deals with a specific class of linear chain magnets, namely those with ferromagnetic exchange interaction and an easy-plane anisotropy. The prototype here is CsNiF3. The objective of this paper is to develop mathematical techniques and physical understanding of this system in cases where simple sotiton concepts are not directly applicable. The need to do so ari,,'e~ when the simple soliton becomes unstable [7t~ as described in Section 3. The Hamiltonian for this system is described by E = ~ [ - J s , • g÷, + A(A~) 2 - S,. I/I.
(I)
i
Here the spins are measured in units of the Bohr magneton/~n. The sum over i refers to the lattice sites along the chain direct,on, the z-axis. The easy plane is the x-y plane with the magnetic field H along the x.direction. The exchange interaction is limited 'o the nearest neighbors. For CsNiF3, the pacameters take the values 3 ffi 23.6 K and A = 4.5 K. Much of the theoretica! work to date [6]
0167-2789/82/0000-00001502.75 © 1982 North-Holland
~¢~0
P. Kumarl Soliton instability in an easy plane ferromagnel
directly maps eq. (!) into a sine-Gordon (SG) equation [8] and proceeds to calculate the observable properties. Such a mapping restricts the spin motion to the (x, y) plane. This paper deals with effects associated with the out-ofp~ane motion. We find the out-of-plane motion to be negligible at low soliton velocity and at very low magnetic fields. However both of these perturbations alter the spin configuration [9]. A static sol iton at a critical field, 0.67 A, undergoes a di~:iorfion such that the high field texture has a substantial out of plane component in the spins. This new texture is lower in energy by as much as 30% compared to the sG soliton and half in x, idth. The energy of the distorted texture is smaller thar~ that of the sG soliton for magnetic fields larger than 0.414 A. The energy minima corresponding to ~he two solutions are separated by an energy barrier so that the sG soliton remains locally stable. Soliton motions also causes a finite out of plane component in the spin. The analysis here is complicated by the presence of magnetic field. The profile of the out of plane distortion due to motion is different from that due to the field instability. The variational problems then gets out of hand. Tl:e dynamics of a sine-Gordon soliton is also affected by the oral-of-plane motion I2,6]. The soli~on dynamics is largely the dynamics o! ,nplane motion, l'he om of plane motion has ~ts own normal modes and in contrast to the s 3 dynamics, their frequency is non-zero. Indeed this fr'~quency is a precursor to the instability and goes soft (¢o~0) as the magnetic field is increased to the critical value. But a further analysis including the damping shows the width of this peak to be comparable to the shift itself. This extra mode then only appears as an additional contribution to the width of the central peak. Because of the magnetic field dependence, this mode should be observable in the field dependence of the central peak. While it is possible to calculate the profile of a static distorted soliton, the fate of a moving sG soliton at fields larger than the critical value is
much more difficult to determine. It is however possible to study the dynamics of the distorted soliton by lookir, at the eigen frequencies of small fluctuations. Clearly, if the distorted soilton can move with a uniform velocity, there must be a small fluctuation eigen frequency at co = 0 [8]. Conversely, if the small fluctuation eigen frequencies are all real (the soliton would be unstable otherwise) and nonzero, the soliton cannot propagate freely. In this respect, the situation here is similar to that encountered in the A phase of superfluid 3He [10]. The sG soliton there distorts its environment and gets trapped in it. In the superfluid, the translational symmetry is broken spontaneously. In the linear chain ferromagnet, the magnetic field breaks the translational symmetry at the instability. in Section 2, we begin with the formalism used throughout the paper. Here, we describe the Hamiltonian and reduce it to the continuumc~assical form used in the rest of the analysis. We also reduce the equations for the spin dynamics, including a phenomenological damping terms in terms of the classical variables. Following that, we define an effective free energy whose minimum describes a uniformly moving soliton, in Section 3, we consider the sine-Gordon soliton, the limits of its applicability, the dynamic fluctuations and the instability which causes the distortions. Section 4 includes a calculation of the profile of a static distorted soliton and its energy. It also includes a discussion, of the fluctuation eigen values. Finally, Section 5 Summarizes our conclusions.
2. Preliminaries The calculations described here are classical. If the spins are expressed, in polar coordinates and assuming only slow variation over the chain sites, the spins at site i + I are Taylor expanded about spins at site i, the resulting Hamiltonian is
P. Kumar! $oliton instability in ca easy plane ferromagnet given by
361
is a minimum o f the energy E subject to the boundary conditions ~ - , ( 0 , 21r) as z - , (-oo, +oo). The analysis of a moving soliton is more subtle. Consider the kinetic energy of a spinning top [11], which is given by
- S H cos ck sin 0 ], S = S [ c o s ~ sin O, sin ~ sin O, c o s 0],
(2)
where a denotes the nearest neighbor distance and the subscripts imply differentiation. The dynamics of these spins is degcribed by the Bloch equation. In the following, we supplement the Bloch equation with a phenomenological damping term [ 11] 8E
a
s,=-~s,x 8-g s s x g '
!
- sin 0 0 = -Ja2(sin 2 0 ~ ) , + H sin 0 sin 4*
(4)
3'
_ / s i n 0~ = -Ja20.~:. + (~Ja2dp2,- A)sin 20 7 - H cos 0 cos ~ + a0.
(6)
where C is a constant (moment of inertia) and X is the angle of rotation about the top axis. Eqs. (4) and (5) are the Euler-Lagrange equations obtained from a Lagrangian L = T - H . The momentum conjugates to 0, ~ and X are given by
(3)
where -?,EI85~ is the effective magnetic field on site i. ,/= g/zn is the gyromagnetic ratio and a is the strength of the 9henomenologicai LandauGilbert type dissipation. In terms of angles 0 and d~, the equation of motion is given by (S = i)
+ '± sin 2 0 ~
C T = ~'(Xt + ,kt cos 0) 2,
(5)
These equations constitute the starting point for this paper. If we dispense with all the time dependence, the static equilibrium solutions are obtained. These are the minima of the energy function E. If eqs. (4) and (5) are linearized about the static solutions, the normal mode frequencies are obtained. These are simply related to the various curvatures of E. The lifetimes of the normal modes are found to be of order a. For a = 0, the sine-Gordon solitons are also the solations of eqs. (4) and (5). A static soliton
P~ = CO? + ~ cos 0)cos 0
(7)
and P~ = C(x + ~ cos 0).
Since P~ is a constant of motion, and can be identified with l/'y (to be consistent with eqs. (4) and (5)), the kinetic energy is a constant of motion (=112C'v2). The Hamiltonian then consists of the kinetic energy and potential energy E and each is separately a constant of motion. The problem of constructing a functional whose minimum is a moving soliton is then analogous to the situation in a superfluid. We construct a functional G=E-
f d-fiz ~,P~.
(8)
The minima of G are the uniformly moving s olitons. In a uniformly moving sG soliton, the only independent variable is z - ut so that ~kt = - u~z and
G : _ - E + u - f -dz~ ~ cos 0.
(9)
3'~2
P.
Kumarl Soliton instability in an easy plane ~erromagnet
In section 2, we study the minimum of the energy E. These are the static solutions. We show. by a stability analysis, how the low field sin,.~-Gordon soliton becomes unstable towards the for'.aation of a distorted soliton. The lip earization of E about the distorted solution shows no gapless fluctuations and indicates the absence of uniform translation motion.
3. Sine-Gordon limit, fluctuations and '~stability If 0 is uniform and a = 0, the right-hand side of eq. (4) is sine-Gordon-like with a length scale given by (Ja"/H)"'-. The first term in eq. (5), the d,~ term as well as the third term are all of order H. If H is small compared to A, eqs. (4) and (5) reduce to /
--" sin 0 d = - Ja2(sin 20cb:): + H sin 0 sin $, -y 1 -y
sin 0 6 = A sin 20,
(10)
which makes cos e of the order of d,. For a slowly movil~g soliton this can be taken small and the resulting equation is the sine-Gordon equation [21,
1 2-A~ ,(b - JaZ,b=: + H sin d, = 0.
-
v2/c2)'/2)1.
(12)
i-he effect of damping terms in eqs. (4) and (5) manifests itself in additional terms in eq. (11), ,
2-X:y" ( i + .-
15 + -
6,
i +
- Ja :'~h~.:+ H sin ,/, = 0.
cos,
l
Ot •
2 --z(l,,t~ + a2)d; +-4,3, - la2*= + H sin ~ = 0, (14) which has two major results. In the first case, we can calculate the width of the magnons, their frequency to and linewidth F are given by
(15)
(I + a')to 2 = 2AT:(H + Jaek2) - F2, F'-
(It
(1 + a "~)7A.
(16)
Eq. (16) can be used to determine F from the observed magnon linewidths. Secondly, the equation of motion of a slow moving soliton can be derived. If the soliton profile is given by $ ( z - R(t)), then (1 4-~") 2 A 7 ~ [ - 0:/~ +4,..(R)2I-~4~.Ry + ~-~0.~H = (17) The even symmetry of d,: leads to
(ll)
The corresponding solutions have the energy E(r)=8X/J/~/ [ t - , ' : / c 2 ] "2. the width .~= X/J/H ~ and the terminal velocity c " = 2AJo"~/:. The solution itself is given by sin 6/2 = sech[(z - t,t)/(.~(I
Clearly, except for the first term in the parenthesis, all other damFing terms are of o r d e r / / and therefore ne ligible in the limit H ,~A. in the latter limit the.~
]
c~ Ja"
- -.y - -
d,,= (13)
If the soliton is driven by an external force, the right-hand side is replaced b,, a weighted average of the driving force term. Eq. (18) implies a decay of the soliton velocity with a relaxation time given by 2 a A ¢ ( I + a 2). This has important consequences in the calculation of the ~trlietiiro
fntn,*-~icm •
,~sa~.l.a~'al
~hirh **
eea~,,aa
,~:11 h,~ vv
man
IJ~,
A;©f..t©eA,4 qbsmOq~.~g,
OO~qk8
separately [12]. The central approximations in the preceding calculation are (a) small field H ,~A; and (b) low velocity. To estimate the effect of larger fields, we do a stability analysis below. The physical reasons behind tile inst~;,ility [7] can be
363
P. Kamarl Soliton instability in an easy plane ferromagnel
gleaned from eq. (2). Clearly the anisotropy term prefers 0 = ~rl2. However the `6~ term prefers 0 # ,r/2. Also, in the center of the sG soliton, ,6 = ~r and the field term also prefers 0 # ~r/2. These t w o terms make t h e e n e r g y o f a distorted soliton (O # ¢ d 2 in the ne~hborhood of `6 ~ ~r) considerably lower than t h e sG soliton energy before the expected instability field A(~28 kG). To study the stability of the sG soliton, we expand the energy E in terms of `6t = ` 6 - `60(z) and 0 -- 0 - ¢r/2. For a static sG soliton E = Eo + f ~ [`6,L,`6, + q,L2qfl,
(19)
where Lt`6t = [-Ja20:: + H cos `6o]`61
and L20 = [ - JaZ#:~ + Ja2(`6o:) 2 - 2A - H cos `6o]01.
If the operators Lt and L2 are diagonalized so that the corresponding eigen values are ~, and tt, and the fluctuaVons `6~ and 0 are expanded in the complete sets of eigen functions, the energy E can be rewritten as U = no + ~ ,~.a 2 + ~ ~,,b ~, n
(20)
~o = 2A - 3 H , ~o -- Bo sech 2 z[~',
(21c)
ttl = 2.A, 01 -- Bt seth2 z/~" tanh z/£,
(21d)
p.~, = 2 A + H + ja2k 2,
Ok = Bk e ik' [ _ ( l +3k'2'~'')- ik~ tanh zl£ -, tanh2 zl ~ ], J
(21e)
where the A's and B's are the normalization constants. The eigenvalue ~o becomes negative for H > Hc = 23A • If A = 4.5 K, this corresponds to a field of 18 kG. T'ae soliton profile at the energy minimum they A acquires a finite amplitude for the wave function ~0. In the next section, we describe a variational calculation of the soliton profile. The instability discussed above is preceded by a soft mode in the spin dynamics. To ca?culate the frequency of this soft mode, we linearize eqs. (4) and (5) about the soliton solution, I_ 6 = - ]a'O.~ + S H cos `6od, + ~ ~,. 3'
(22)
m
where E0 is the energy of the sG soliton and the sums over n and m enumerate the space of the eigen functions. Clearly the stability requirements are A~ ~ 0 and it, ;B 0. If any one of these eigen values is negative, the energy c~n be lowered by increasing the corresponding amplitude, i.e. the system is unstable towards the growth of that eigen vector. The operators L t and L2 have eigenvalues and eigenfunctions given by
`61 = Ao sech z/!~,
(21a)
~,l = fl + JaCk 2 ,
d't = Ak et~:[-ik~ + tanh z/(,l,
(21b)
_ ! ~ = _ Ja%:~- (ja%2z- 2A - H cos ~0)~, 3'
+ ~ O. 3,
(23)
Liiniting ourselves to vitae vicinity of the ,o = 0 region, the primary eigenvalues are (a) cot- 0, the translational mode of the soliton. This mode gives rise to the central peak in the structure function S(k, co) in the neutron 5¢~ilI.Lg:l-ill[~ U A p ~ I
Illl~li~t
(b) ~02--aS'l, where a is a constant of order unity and 112= V 2 I t ( 2 A - 3 H ) . In the low field limit this frequency coincide.~ with f~,e bottom of the continuum of magnon frequencies. As the field increases, this mode shifts towards 00 = 0 and goes soft at the critical field. The cur-
~t,4
P. Kumarl Solilon insmbilily in an easy plane fen'omagnel
r,:sponding eigen vector is largely oscillations in 0,'z. tL If 1he external field is close to the critical field, the frequency (o. can be calculated by pe-turbation theory as described in appendix A. The value of the constant a = 27135. The interesting result is that the width of this •.node is also proportional to fl and therefore for a reasonable value of a this mode is merged into the central peak. The central peak width then decreases as the magnetic field is increased. An experimental search for this effect xviil considerably elucidate the validity of the model Hamiltonian in describing a linear chain ferro magnet. Finally we come to the instability in a moving sG soliton. The soliton motion causes the spins to swing out of :he x - y plane (eq. (10)). The instability due to the magnetic field is then aided bv the soliton motion. However there are problems in calculating the effects of this instability. T h e functional dependence of 0 in the magnetic fie~d soft mode is sech"(z/.£) while the curresponding Frofile due to the motion is sech(:/~). We calculate the instability field for a moving soliton assuming ( ~ the only effect on the ~ profile is to oarrow its width; and (b) the ~/J profile is proportional to seth z/(,. The former ignores any coupling to the propagating waves which are later shown to be important. The latter has the defect that the instability field for a static soliton is increased to 3A!4 (from 2A/3). However the calculation is considera01y simplified and should yield qualitatively correct results. The energy G can be written in a particularly convenient form. f d r F In 2
~1
]
--
We now assume that sin ~b,t2= sech(z - uOl£ and 0 = 0 - rr12 = a s, " h ( z - ut)l~,, where a and £ are treated as v a n ttional parameters. Substituting these expressions in eq. (24) and evaluating the various integrals (after expanding in ~ and retaining terms up to ~(04))9 we find G=_U_4___a_a(i _ 2
(25) where ~
f,(.=l-
--~4)
sin"
1
1
7
J
H ( 1 .... cos t/,) cos ~ + -" sin ~,~ !.
(24)
')o'
/3-3-(i 4 " "
J- f,(a) H f2(a) and 14 4a ),a
G -=
i -- a
+ Eo(f(c~)f,(a)) u2
( I - ~ a 2) + E . ( 1 + ( / 3 / 2 - 2 / 3 ) - ~ - - d ~ 4)
u4a
(26) where/3 = AIH and E0 = 8 X / f H is the energy of a static sG soliton. In eq. (26), we have not bothered to calculate the coefficient of a 4, d, exactly except to note that it is negative. The results indicate the effect of soliton dynamics to be catastrophic. Firstly, minimizing with respect t o ~ . the' m i n i m u m n f the ~ n ~ r o v r', ~,~,,,,~r~ ~t .
.
.
.
.
.
.
.
.
4
.+
4
Minimization of G with.respect to ~ yields
ao.....
8
f.,(a) = 1 + ( 2 / 3 - 1 / 3 ) ~ " / 4 -
.
L
a4jf,4,a)+~4Hf2(a~,~
z)
.
.
.
.
.
.
.
.
.
.
.
.,...~.,,
~
j
v
, ~ p.,. p
w
va E0(/3/2 ~- 2/3)"
(27)
In the limit of very low fi~lds, it agrees with eqs. (10) and (11). However as the field is increased it is considerably larger than low field
P. Kumar] ~;oliton instabilify in an easy plane ferromagnet value. The divergence in s0 at the critical field is a consequence of our retaining on'.y low order a terms. Finally if the critical field is identified from the zero of the curvature of G(c~) at oto; the critical field a t finite velocity satisfies
36~
fluctuations is the bottom of the magnon continuum and therefore the sG soliton, as it moves, excites the magnons.
4. Distorted soliton
[0c(U)- O¢(O)J3 = ~ \Eo/
"
(28)
The critical field decreases with u [9]. The value /3c(0)=413 is, as mentioned before, a consequence of the assumed O profile. We could have assumed Oa sech2(z - ut)l(, in which case 1Be(0) would be correct, but eq. (27) would have an extra numerical factor. In any case, the u 2/3 dependence in eq. (28) is independent of any ~uch approximations. The combined effect of the magnetic field and motion is to distort the sG soliton. So far we have suppressed all the effects on the 4, variable. It is easy to see that 4, profile is also distorted and in particular propagating waves are excited. To show that, let us expand G(4,, 0) in eq. (8) about a moving sG soliton. If 0 =: &o + &~ and 0 = 0o + 01, where Oo = (ul2A~) &0,;
The question remains; what happens to the soliton after it goes unstable. For a static soliton, it can be answered by a variational calculation. This is, however, possible only because a static soliton goes unstable in a single mode. A moving soliton goes unstable in a more complicated manner and a simple variational calculation is not possible. And yet, by looking at the fluctuation spectrum of the distorted soliton, we can draw some conclusions about the dynamic system as well. The distorted soliton profile is calculated by assuming sin 4,12 = s e c h zl¢; and sin 0/2 = a sechZzl(,, where cx and ~" are variational parameters. The choice of 0 profile originates from a simple generalization of the profile of the normal mode that goes unstable. If Eo and ~o refer to the energy and width of the sG soliton
[7], G(4), ~,) = Go(4,o, ~,o)+ G,~I + G?4,, +0(4, 2, #,2, ,~#,),
(29)
where G~ and G~ are integral operators. We need consider only G~;
EI Eo
=
(FI((~)F2((x)) '12
and ~/~o = (F1(a)IF~(c~)) '12,
G;q,' = - 3'a ~ f dy [ - q)o,, + sin q,o + Y--;=; 4,o2:,4,o,.v(3"/' - 1/2) ] 4, i,
where
(30)
where y = y ( z - u t ) l ¢ ; and ~/-~= I - u 2 1 c 2. The sG soliton satisfies 00, = sin 00. Hence G*~Ol is nonzero, the sG solution is intrinsically unstable. More specifically, since the kernel operating on 4,t is an odd function of y, the instability creates 4, distortion of that symmetry. The lowest odd symmetry mode in 4,
Fl(a)=l+2
[2 - ,/,-°c~
tan-I
l-a
- ~/' a ~' tanh-' ~ / 1 - ~ J 16
-2a: ]3
256 ,\ ~'" 315 /
and
F2(a) = I -
, )
,,
-~)0 c , : - ~
~a'.
(3~)
~
P. Kumarl Soliton instability in an easy plane ~erromagnet
A lower energy minimum is obtained if we take ?nstead sin 0/2 = a sech zig. Clearly the earlier choice is motivated by the profile of the unstable mode. H o w e v e r . at large a, the gradient energy contributions are m u c h too large. In as much as the changeover is discorMnuous [7], we expect the small ct region of E(t~) to be well described by eq. (31), while for large ,,~ we find I-a-'tan~/ F,(,~) : - . , _ / ~1 ,
'(~'~
"t t .':----~ -
8
,
a "+
32
61 and ~t are given ~' ..le
aE =
2 fd:r,LLv°:" (A ~
+ {Ja" cos: ~o6~: + H c o s 6 o c o s ~o6~12
+ { - J'a26o. sin 2 0 o 6 ~ 1
ot
- H sin q~o sin 6 o ~ t 6 1 } ]
=
+ (6,1L,16,) + (¢JlL,,[6).
(33)
(32)
The principal results are: (a) For /3 > 2.44 ( H < 11.5 kG), i.e. the sG sc'iton (a = 0) is the lowest energy minimum subject to the boundary conditions 6 ~ ( 0 , 2 ~ r ) as
200
Ja" H cos 6o c o s , o) } 2 6o: cos 2 o+ T
and F,(a) _ = 1• - ( 1 / 3 - 2/3)a" - :~/3a4.
cos
2 --~ ( - - ~ c ~c).
(b) For 2.44 > ~ > 1.5, the energy at a:~ 0 is lower than the one at c~ = 0. However the (~ = 0 state is stable agair~st small fluctuat;ons. The two min, ma are separated by a sma}, potential r~arrier. (c) For / 3 - - - 1 . 5 : a ~ 0 is the ~nly energy minimum. The off-xy plane excursion angle 6, clo,~e to tl~i~ instability is about 70 °. It increases with ~he magnetic field [7] and reaches zd2 at /3 ~ 0.5, at which time the solit,ons are unstable towards a uniform magnetization. (d) The width of the disterted soliton is approx;rnately half of the width of the sG soliton. The energy at/3 = !.5 is smaller than Eo by 13% and continues to decrease as H increases, We now look at the fluctuations about the di~orted soliton. If the distorted soliton has a translational mode (motiov at uniform velocity) the lowest eigenvalue in the fluctuations must be zero. The analysis reported here is variational. If E(& 0) in eq. (92) is expanded about 4) = ,5,,(:~ and 0 = ,'r/2 + 6dz). in terms of fluctuation fLnctions ,b,(z) and 6~(z), the terms quadratic in
It is possible to analyze eq. (33) in a two level approximation. The task then is to find the iov'est eigenvalues of operators L , and L,, say e, and e,,; calculate the matrix element of L**, the energy 8H is then given by e ~ , A 2 + e , , B ~ + AAB, where A and B are the normalization integrals. The eigenvalues of 8H are given by x. = ~[(~, + ~,,) + [(e~, - e,, )2 + A21,/21.
(34~
The stability condition that the eigenvalues be non-negative, translates into e,~,. ~ A2/4, the equality implies ,~_ = 0. For variational profiles of 01 and 61, we take ~i = A(sech z i g ) " and 01 = B(sech zig)" and minimize the fluctuation energy 8E with respect to p. and v. In as much as both functions are even functions of z and the cross-kernel L** is an odd function of z, the matrix element A = 0. The details of this calculation are shown in appendix B. The eigen values of e,, and e~ and the exponents :.t and v are shown in fig. !. For /3 = 2 for example, ¢,,/H = 0.683,
ta = 0.6t~
and c,IH = 0.304,
v = 0.36.
(35)
P. Kumar/ SoUton iastabi'ily in an easy plane ferromagnet
367
variant. With the distortion in O, this symmetry is broken in the distorted texture. |.O -
S. Summary •
o.e-
e@@@e
Oo,~/H
x •
o.6O
X 0 1
0
O0
0
X
X
I • Q
10
00000 °
X
%/H
0.2-
o0
x
I
X X O
0.4-
x
I
I
I
,.o
I
~
I
2.0
Fig. I. The fluctuation eigenvalues t . , and e.+. and the exponents ~t and v are shown as a function of # - A/H. The dots and the crosses are respectively e~s and/~. The trian$1es and circled crosses are respectively e.+. and v.
As shown in fig. I, the eigen value e~, increases with magnetic field slowly, reaching a maximum at /3 = 0.8 and then decreases. The ~-fluctuation frequency e~, on the other hand, continues to decrease with increasing magnetic field. Its field dependence is considerably weaker though and even at 0 = 0.5, e, = 0.23 H [13]. Thus the principal dynamic modes in a distorted soliton are oscillatory. The lowest frequency mode corresponds to the oscillations of qJ which, in the linear continuum case, are decoupled from the next hish~r frequency mode, the qJ fluctuations. A major distinction should be made here. In a sG soliton, ~he motion of the entire te~,ture is translationally uniform. This symmetry remains intact. The broken symmetry here corresponds to the relative motion of e and ~. The former being constant in sG case, the ~ motion is translationally in-
The results inthis paper [14] show the limitations of +the sG description of a linear chain, easy-plane ferromagnet. We find a static soliton to be unstable for fields H > 18 kG. However, in the field range 11.5 kG < H < 18 kG, the distorted texture is lower in energy:, compared to the sG soliton, In that sense, the instability resembles a first order transition as a function of the magnetic field, with associated hysteretic effects. The dynamics of the distorted soliton is studied by looking at the eige,values of the linear fluctuations. Th~,se in fact serve a dual purpose. On the one hand, they investigate the stability of the distorted texture about which the linear ltuctuations are studied. On the othe, hand, the frequencies associated with the spin dynamics are directly related to the fluctuatic~ frequencies. In particular, ff the distorted texture were free to move translationally, the lowest fluctuation eigen value would have to be zero. We find the fluctuation eigen wlues to be positive and non-zero, indicating (a) that the distorted soliton is stable and (b) it contains no translational mode, it oscillates with a characteristic frequency and remains fixed at a place. Clearly, the lattice discreteness could couple the translational and oscillatory modes. Finally, the instability of a moving soliton is explored briefly. The motion of a sG soliton causes the spins to move off the x-y plane and should help the instability. The limiting velocity depends on the magnetic field. Conversely, the critical field drops rather suddenly with the velocity of the sG soliton. At very small velocities, the sG soliton emits linear excitations, magnons and in all infinite chain its motion becomes dissipative. This is intrinsic dissipation, as opposed to any extrinsic dissipation that may be present in the system. The effects associated with the latter are also discussed.
le~
P. Kumart Solilon instability in an easy plane ferromalgnel
Acknowledgements Man Bishop has been a constant source of cotaments~ criticism and stimulating advice. Some of this work was done on a visit to Los Alamos National Lab and Bishop should be thanked for the hospitality. Comments by D. Huber and by H. Thomas are deeply appreciated. R. Joyce and V. Samalam must be thanked for help with the calculations and M. Bowick for a critical reading of the manuscript. This work was supported by NSF DMR-800o311 and by the Research Corporation.
Here the subscript k refers to the scattering states and the brackets imply an integral over z . The third term or the left-hand side represents the frequency of L e mode, proportional to/zo = (213- 3)H. The frequency is renormalized away from/~o due to the coupling with the plane wave states. The additional terms on the right-hand side represent higher order couplings. If they are ignored the frequency ~a~ and the lifetime F2 of this mode are given by the coefficients of A0 and Ao terms, respectively. Using the expressions for ~o and 6~ given in eqs. (21), co2 is found to be , 27 to~ = ~-¥ (yH)2(2/3- 3).
Appendix A
(A.I)
The numerical factor a is of order unity. That it is not exactly unity could be a consequence of the perturbative calculation. We suspect it is a real shirt but in an experiment, disguised by the width, it remains merged with the reagan continuum (co:= 2/3(7H)2). As the field H is increased, this mode gets split away and approaches zero near the instability.
(A.2)
Appendix B
In the Io~ field limit, the spin dynamics is described by eq~. (22) and (23). These are iinearizations of the Bloch equation about a sine-Gordon soliton. They can be rewritten formally as (1 4~ - ~ : - aLq/, + L~d,. 7 (1 + ),a".~ & = L~q~ + a l . , & .
(A.4)
where 1he operators L: and L. have been defined earlier in eq. (19). If we expand q~ = E., A.~,. and d, = E.,, B,.~bm whece q~. are the wavefm~ctions of L: and ~b~. belong to L~, separate :-quations can be derived for A0. the amplitude of the ~ bound state and B0, the amphtude of the & bound s~ate, t h e latter corresponds to a zero frequency mode, the translational mode of the: soliton. The former satisfies an equation
The stability analysis begins with eq. (33) and is restricted to a two level case. We assume 0, = A(sech zlO" and 4,t = B(sech zlOL Here Ix and v are the variational parameters. The eigenvalue e, is given by
H
if
dy
(¢,,,)2
+ [(2/3 - ~o2,) cos 200 + cos *o cos 4'0] x ~ ]
(S.I) Is
•
-"
,
i~ol
l
,
~t?
~k~l-
.~,, + V~,u,,Ao + V ~ hk~to (c!'~)(,lJo) Ao : - r" ÷
2L2~t + I +2/3 + 1 - ( 6 + ( 1 6 / 3 + 2 ) a ' ) 2/~+1 2P
xk 3#,0
+ (36a 2+ 16/3a 4)2 2-~+ 1
16.>].
(A.3)
2t~+3 2t~ (2tt + 2) (2t~ + 4)] - 32t~ (2~ + i) (2tt + 3) (2/z + 5)J'
(B.2)
P,./<:-marl Solilon inslabiUly in an easy plane ferromagnel
where we have substituted the distorted soliton profile for ~0 and qn0. Here y = z/G, ~/= Gd~ and G0 represents the width of the sG soliton. The integral important for the evaluation of eq. (B.2) is [10]
l(n) = (sech"z) -'- 2 "-I (F(n/2))2 F(n)
(B.3)
and n
l(n + 2)= n + I l(n).
(B.4)
Eq. (B.2) is then explicitly minimized for a given value of/3 and corresponding values of G and c~. Similar operation on e~ yields the various curves in fig. I.
References [I] M. Steiner, J. App. Phys. 50 (1979) 7395: and in Physics in One Dimension, J. Bernasconi and T. Schneidel, eds. (Springer, Berlin, 1981). [2] HJ. Mikeska, J. Phys. C1 I, (1978) L29. For a theoretical review see H.J. Mikeska, J. Appl. Phys. 15 (1981)
369
1950; and in Physics in One Dimension, J. Bernasconi and T. Schneider, eds. (Springer, Berlin, 1981). [3] J.K. Kjems and M. Steiner, Phys. Rev. Lett. 41 (1978) 1137. Also see M. Steiner, ref. 1. [4] K.M. Leung and D.L. Huber, Solid St. Comm. 32 (1981) 127. [51 Ref. !. [6] [so see K. ~ysics, V. VIII, D. Brewer, ed. CNorth-Holland, Amsterdam, 1981). [7] P. Kumar, Phys. Rev. B 15 (1982) 483. [8] A. R, Bishop, J. A. Krumhansl and S.E. Trullinger, Physica DI (1980) I. K. Maki, in ref. 6. [9] E. Magyad and H. Thomas, preprint (1981) have also arrived at this conclusion independently. [1o] K. Maki and P. Kumar, Phys. Rev. BI6 (1977) 182; K. Maki, in Solitons and Condensed Matter Physics, A. Bishop and T. Schneider, eds. (Springer, Berlin, 1978). [11] A.H. Morrish, The Physical Principles of Magnetism. (Wiley, New York, 1966). [12] P. Kumar and V.K. Samalam, University of Florida preprint (1981). [13] The variational ansatz must breakdown at large fields since it ignores any coupling to magnon excitations. We expect that if ~ approaches ~rl2, the system must be completely unstable for a homogeneous, essentially saturated state. At this stage, it is not possible to make a quantitative statement about this effect. [14] A brief summary of the results has been reported in P. Kumar and V.K. Samalam, in Proc. of 27th Annual Conf. on Magnetism and Magnetic materials, J. Appl. Phys. 53 (1982) 1856.
SOLITON INSTABILITY IN AN EASY PLANE ~ O M A G N E T Pradeep K U M A R Department of Physics, University of Florida Gainesville, Florida 32611, USA Received 15 December 1981
We report here an analysis of the limitations of a sine-Gordon like description of an easy plane ferromagnetic chain. We show that the sG description is valid only at low magnetic fields and low velocities of soliton motion. Even at low velocities, the soliton emits magnons. For fields larger than a critical field He(u), the soliton is unstable towards the formation of a distorted texture with large off-easy plane excursions for the spin. This instability is preceded by a split-off-localized mode (from the magnon continuum) that goes soft at the critical field. The distorted texture is found to be immobile in that no ~o = 0 translational mode exists. Instead we find oscillation frequencies for the texture. The sG soliton distorts its environment and gets trapped in it.
1. Introduction
Soliton-like excitations in one-dimensional magnets have been studied widely [I] in recent years. In one dimension problems, solitons constitute a degree of freedom in addition to the linear wave-like excitation and thus affect the thermodynamics and scattering properties. The more exciting experiments, in recent years have been those of neutron scatt,~ring. Here the neutrons can scatter inelastically and, in a linear magnet, excite the spin-waves (or magnons). In a non-linear magnet with a thermal distribution of solitons the neutrons can also scatter off the solitons. The latter being massive objects, give rise to a broad central peak corresponding to the transfer of momentum (translational energy) from the neutron to the soliton. The analysis of this peak was initiated by the predictions of Mikeska [2]. Kjems and Steiner [3] observed a linear chain with ferrothis peak in ~sr~w3, "-'"" magnetic exchange interaction and an easy plane anisotropy. Much of this work, including a careful reexamination of the relativistic effects associated with the soliton motion by Leung and Huber[4], has been reviewed by Steiner[Y]. There now exists a considerable body of
literature[6] containing results about the equilibrium as well as thermodynamic properties of these solitons. This paper deals with a specific class of linear chain magnets, namely those with ferromagnetic exchange interaction and an easy-plane anisotropy. The prototype here is CsNiF3. The objective of this paper is to develop mathematical techniques and physical understanding of this system in cases where simple sotiton concepts are not directly applicable. The need to do so ari,,'e~ when the simple soliton becomes unstable [7t~ as described in Section 3. The Hamiltonian for this system is described by E = ~ [ - J s , • g÷, + A(A~) 2 - S,. I/I.
(I)
i
Here the spins are measured in units of the Bohr magneton/~n. The sum over i refers to the lattice sites along the chain direct,on, the z-axis. The easy plane is the x-y plane with the magnetic field H along the x.direction. The exchange interaction is limited 'o the nearest neighbors. For CsNiF3, the pacameters take the values 3 ffi 23.6 K and A = 4.5 K. Much of the theoretica! work to date [6]
0167-2789/82/0000-00001502.75 © 1982 North-Holland
~¢~0
P. Kumarl Soliton instability in an easy plane ferromagnel
directly maps eq. (!) into a sine-Gordon (SG) equation [8] and proceeds to calculate the observable properties. Such a mapping restricts the spin motion to the (x, y) plane. This paper deals with effects associated with the out-ofp~ane motion. We find the out-of-plane motion to be negligible at low soliton velocity and at very low magnetic fields. However both of these perturbations alter the spin configuration [9]. A static sol iton at a critical field, 0.67 A, undergoes a di~:iorfion such that the high field texture has a substantial out of plane component in the spins. This new texture is lower in energy by as much as 30% compared to the sG soliton and half in x, idth. The energy of the distorted texture is smaller thar~ that of the sG soliton for magnetic fields larger than 0.414 A. The energy minima corresponding to ~he two solutions are separated by an energy barrier so that the sG soliton remains locally stable. Soliton motions also causes a finite out of plane component in the spin. The analysis here is complicated by the presence of magnetic field. The profile of the out of plane distortion due to motion is different from that due to the field instability. The variational problems then gets out of hand. Tl:e dynamics of a sine-Gordon soliton is also affected by the oral-of-plane motion I2,6]. The soli~on dynamics is largely the dynamics o! ,nplane motion, l'he om of plane motion has ~ts own normal modes and in contrast to the s 3 dynamics, their frequency is non-zero. Indeed this fr'~quency is a precursor to the instability and goes soft (¢o~0) as the magnetic field is increased to the critical value. But a further analysis including the damping shows the width of this peak to be comparable to the shift itself. This extra mode then only appears as an additional contribution to the width of the central peak. Because of the magnetic field dependence, this mode should be observable in the field dependence of the central peak. While it is possible to calculate the profile of a static distorted soliton, the fate of a moving sG soliton at fields larger than the critical value is
much more difficult to determine. It is however possible to study the dynamics of the distorted soliton by lookir, at the eigen frequencies of small fluctuations. Clearly, if the distorted soilton can move with a uniform velocity, there must be a small fluctuation eigen frequency at co = 0 [8]. Conversely, if the small fluctuation eigen frequencies are all real (the soliton would be unstable otherwise) and nonzero, the soliton cannot propagate freely. In this respect, the situation here is similar to that encountered in the A phase of superfluid 3He [10]. The sG soliton there distorts its environment and gets trapped in it. In the superfluid, the translational symmetry is broken spontaneously. In the linear chain ferromagnet, the magnetic field breaks the translational symmetry at the instability. in Section 2, we begin with the formalism used throughout the paper. Here, we describe the Hamiltonian and reduce it to the continuumc~assical form used in the rest of the analysis. We also reduce the equations for the spin dynamics, including a phenomenological damping terms in terms of the classical variables. Following that, we define an effective free energy whose minimum describes a uniformly moving soliton, in Section 3, we consider the sine-Gordon soliton, the limits of its applicability, the dynamic fluctuations and the instability which causes the distortions. Section 4 includes a calculation of the profile of a static distorted soliton and its energy. It also includes a discussion, of the fluctuation eigen values. Finally, Section 5 Summarizes our conclusions.
2. Preliminaries The calculations described here are classical. If the spins are expressed, in polar coordinates and assuming only slow variation over the chain sites, the spins at site i + I are Taylor expanded about spins at site i, the resulting Hamiltonian is
P. Kumar! $oliton instability in ca easy plane ferromagnet given by
361
is a minimum o f the energy E subject to the boundary conditions ~ - , ( 0 , 21r) as z - , (-oo, +oo). The analysis of a moving soliton is more subtle. Consider the kinetic energy of a spinning top [11], which is given by
- S H cos ck sin 0 ], S = S [ c o s ~ sin O, sin ~ sin O, c o s 0],
(2)
where a denotes the nearest neighbor distance and the subscripts imply differentiation. The dynamics of these spins is degcribed by the Bloch equation. In the following, we supplement the Bloch equation with a phenomenological damping term [ 11] 8E
a
s,=-~s,x 8-g s s x g '
!
- sin 0 0 = -Ja2(sin 2 0 ~ ) , + H sin 0 sin 4*
(4)
3'
_ / s i n 0~ = -Ja20.~:. + (~Ja2dp2,- A)sin 20 7 - H cos 0 cos ~ + a0.
(6)
where C is a constant (moment of inertia) and X is the angle of rotation about the top axis. Eqs. (4) and (5) are the Euler-Lagrange equations obtained from a Lagrangian L = T - H . The momentum conjugates to 0, ~ and X are given by
(3)
where -?,EI85~ is the effective magnetic field on site i. ,/= g/zn is the gyromagnetic ratio and a is the strength of the 9henomenologicai LandauGilbert type dissipation. In terms of angles 0 and d~, the equation of motion is given by (S = i)
+ '± sin 2 0 ~
C T = ~'(Xt + ,kt cos 0) 2,
(5)
These equations constitute the starting point for this paper. If we dispense with all the time dependence, the static equilibrium solutions are obtained. These are the minima of the energy function E. If eqs. (4) and (5) are linearized about the static solutions, the normal mode frequencies are obtained. These are simply related to the various curvatures of E. The lifetimes of the normal modes are found to be of order a. For a = 0, the sine-Gordon solitons are also the solations of eqs. (4) and (5). A static soliton
P~ = CO? + ~ cos 0)cos 0
(7)
and P~ = C(x + ~ cos 0).
Since P~ is a constant of motion, and can be identified with l/'y (to be consistent with eqs. (4) and (5)), the kinetic energy is a constant of motion (=112C'v2). The Hamiltonian then consists of the kinetic energy and potential energy E and each is separately a constant of motion. The problem of constructing a functional whose minimum is a moving soliton is then analogous to the situation in a superfluid. We construct a functional G=E-
f d-fiz ~,P~.
(8)
The minima of G are the uniformly moving s olitons. In a uniformly moving sG soliton, the only independent variable is z - ut so that ~kt = - u~z and
G : _ - E + u - f -dz~ ~ cos 0.
(9)
3'~2
P.
Kumarl Soliton instability in an easy plane ~erromagnet
In section 2, we study the minimum of the energy E. These are the static solutions. We show. by a stability analysis, how the low field sin,.~-Gordon soliton becomes unstable towards the for'.aation of a distorted soliton. The lip earization of E about the distorted solution shows no gapless fluctuations and indicates the absence of uniform translation motion.
3. Sine-Gordon limit, fluctuations and '~stability If 0 is uniform and a = 0, the right-hand side of eq. (4) is sine-Gordon-like with a length scale given by (Ja"/H)"'-. The first term in eq. (5), the d,~ term as well as the third term are all of order H. If H is small compared to A, eqs. (4) and (5) reduce to /
--" sin 0 d = - Ja2(sin 20cb:): + H sin 0 sin $, -y 1 -y
sin 0 6 = A sin 20,
(10)
which makes cos e of the order of d,. For a slowly movil~g soliton this can be taken small and the resulting equation is the sine-Gordon equation [21,
1 2-A~ ,(b - JaZ,b=: + H sin d, = 0.
-
v2/c2)'/2)1.
(12)
i-he effect of damping terms in eqs. (4) and (5) manifests itself in additional terms in eq. (11), ,
2-X:y" ( i + .-
15 + -
6,
i +
- Ja :'~h~.:+ H sin ,/, = 0.
cos,
l
Ot •
2 --z(l,,t~ + a2)d; +-4,3, - la2*= + H sin ~ = 0, (14) which has two major results. In the first case, we can calculate the width of the magnons, their frequency to and linewidth F are given by
(15)
(I + a')to 2 = 2AT:(H + Jaek2) - F2, F'-
(It
(1 + a "~)7A.
(16)
Eq. (16) can be used to determine F from the observed magnon linewidths. Secondly, the equation of motion of a slow moving soliton can be derived. If the soliton profile is given by $ ( z - R(t)), then (1 4-~") 2 A 7 ~ [ - 0:/~ +4,..(R)2I-~4~.Ry + ~-~0.~H = (17) The even symmetry of d,: leads to
(ll)
The corresponding solutions have the energy E(r)=8X/J/~/ [ t - , ' : / c 2 ] "2. the width .~= X/J/H ~ and the terminal velocity c " = 2AJo"~/:. The solution itself is given by sin 6/2 = sech[(z - t,t)/(.~(I
Clearly, except for the first term in the parenthesis, all other damFing terms are of o r d e r / / and therefore ne ligible in the limit H ,~A. in the latter limit the.~
]
c~ Ja"
- -.y - -
d,,= (13)
If the soliton is driven by an external force, the right-hand side is replaced b,, a weighted average of the driving force term. Eq. (18) implies a decay of the soliton velocity with a relaxation time given by 2 a A ¢ ( I + a 2). This has important consequences in the calculation of the ~trlietiiro
fntn,*-~icm •
,~sa~.l.a~'al
~hirh **
eea~,,aa
,~:11 h,~ vv
man
IJ~,
A;©f..t©eA,4 qbsmOq~.~g,
OO~qk8
separately [12]. The central approximations in the preceding calculation are (a) small field H ,~A; and (b) low velocity. To estimate the effect of larger fields, we do a stability analysis below. The physical reasons behind tile inst~;,ility [7] can be
363
P. Kamarl Soliton instability in an easy plane ferromagnel
gleaned from eq. (2). Clearly the anisotropy term prefers 0 = ~rl2. However the `6~ term prefers 0 # ,r/2. Also, in the center of the sG soliton, ,6 = ~r and the field term also prefers 0 # ~r/2. These t w o terms make t h e e n e r g y o f a distorted soliton (O # ¢ d 2 in the ne~hborhood of `6 ~ ~r) considerably lower than t h e sG soliton energy before the expected instability field A(~28 kG). To study the stability of the sG soliton, we expand the energy E in terms of `6t = ` 6 - `60(z) and 0 -- 0 - ¢r/2. For a static sG soliton E = Eo + f ~ [`6,L,`6, + q,L2qfl,
(19)
where Lt`6t = [-Ja20:: + H cos `6o]`61
and L20 = [ - JaZ#:~ + Ja2(`6o:) 2 - 2A - H cos `6o]01.
If the operators Lt and L2 are diagonalized so that the corresponding eigen values are ~, and tt, and the fluctuaVons `6~ and 0 are expanded in the complete sets of eigen functions, the energy E can be rewritten as U = no + ~ ,~.a 2 + ~ ~,,b ~, n
(20)
~o = 2A - 3 H , ~o -- Bo sech 2 z[~',
(21c)
ttl = 2.A, 01 -- Bt seth2 z/~" tanh z/£,
(21d)
p.~, = 2 A + H + ja2k 2,
Ok = Bk e ik' [ _ ( l +3k'2'~'')- ik~ tanh zl£ -, tanh2 zl ~ ], J
(21e)
where the A's and B's are the normalization constants. The eigenvalue ~o becomes negative for H > Hc = 23A • If A = 4.5 K, this corresponds to a field of 18 kG. T'ae soliton profile at the energy minimum they A acquires a finite amplitude for the wave function ~0. In the next section, we describe a variational calculation of the soliton profile. The instability discussed above is preceded by a soft mode in the spin dynamics. To ca?culate the frequency of this soft mode, we linearize eqs. (4) and (5) about the soliton solution, I_ 6 = - ]a'O.~ + S H cos `6od, + ~ ~,. 3'
(22)
m
where E0 is the energy of the sG soliton and the sums over n and m enumerate the space of the eigen functions. Clearly the stability requirements are A~ ~ 0 and it, ;B 0. If any one of these eigen values is negative, the energy c~n be lowered by increasing the corresponding amplitude, i.e. the system is unstable towards the growth of that eigen vector. The operators L t and L2 have eigenvalues and eigenfunctions given by
`61 = Ao sech z/!~,
(21a)
~,l = fl + JaCk 2 ,
d't = Ak et~:[-ik~ + tanh z/(,l,
(21b)
_ ! ~ = _ Ja%:~- (ja%2z- 2A - H cos ~0)~, 3'
+ ~ O. 3,
(23)
Liiniting ourselves to vitae vicinity of the ,o = 0 region, the primary eigenvalues are (a) cot- 0, the translational mode of the soliton. This mode gives rise to the central peak in the structure function S(k, co) in the neutron 5¢~ilI.Lg:l-ill[~ U A p ~ I
Illl~li~t
(b) ~02--aS'l, where a is a constant of order unity and 112= V 2 I t ( 2 A - 3 H ) . In the low field limit this frequency coincide.~ with f~,e bottom of the continuum of magnon frequencies. As the field increases, this mode shifts towards 00 = 0 and goes soft at the critical field. The cur-
~t,4
P. Kumarl Solilon insmbilily in an easy plane fen'omagnel
r,:sponding eigen vector is largely oscillations in 0,'z. tL If 1he external field is close to the critical field, the frequency (o. can be calculated by pe-turbation theory as described in appendix A. The value of the constant a = 27135. The interesting result is that the width of this •.node is also proportional to fl and therefore for a reasonable value of a this mode is merged into the central peak. The central peak width then decreases as the magnetic field is increased. An experimental search for this effect xviil considerably elucidate the validity of the model Hamiltonian in describing a linear chain ferro magnet. Finally we come to the instability in a moving sG soliton. The soliton motion causes the spins to swing out of :he x - y plane (eq. (10)). The instability due to the magnetic field is then aided bv the soliton motion. However there are problems in calculating the effects of this instability. T h e functional dependence of 0 in the magnetic fie~d soft mode is sech"(z/.£) while the curresponding Frofile due to the motion is sech(:/~). We calculate the instability field for a moving soliton assuming ( ~ the only effect on the ~ profile is to oarrow its width; and (b) the ~/J profile is proportional to seth z/(,. The former ignores any coupling to the propagating waves which are later shown to be important. The latter has the defect that the instability field for a static soliton is increased to 3A!4 (from 2A/3). However the calculation is considera01y simplified and should yield qualitatively correct results. The energy G can be written in a particularly convenient form. f d r F In 2
~1
]
--
We now assume that sin ~b,t2= sech(z - uOl£ and 0 = 0 - rr12 = a s, " h ( z - ut)l~,, where a and £ are treated as v a n ttional parameters. Substituting these expressions in eq. (24) and evaluating the various integrals (after expanding in ~ and retaining terms up to ~(04))9 we find G=_U_4___a_a(i _ 2
(25) where ~
f,(.=l-
--~4)
sin"
1
1
7
J
H ( 1 .... cos t/,) cos ~ + -" sin ~,~ !.
(24)
')o'
/3-3-(i 4 " "
J- f,(a) H f2(a) and 14 4a ),a
G -=
i -- a
+ Eo(f(c~)f,(a)) u2
( I - ~ a 2) + E . ( 1 + ( / 3 / 2 - 2 / 3 ) - ~ - - d ~ 4)
u4a
(26) where/3 = AIH and E0 = 8 X / f H is the energy of a static sG soliton. In eq. (26), we have not bothered to calculate the coefficient of a 4, d, exactly except to note that it is negative. The results indicate the effect of soliton dynamics to be catastrophic. Firstly, minimizing with respect t o ~ . the' m i n i m u m n f the ~ n ~ r o v r', ~,~,,,,~r~ ~t .
.
.
.
.
.
.
.
.
4
.+
4
Minimization of G with.respect to ~ yields
ao.....
8
f.,(a) = 1 + ( 2 / 3 - 1 / 3 ) ~ " / 4 -
.
L
a4jf,4,a)+~4Hf2(a~,~
z)
.
.
.
.
.
.
.
.
.
.
.
.,...~.,,
~
j
v
, ~ p.,. p
w
va E0(/3/2 ~- 2/3)"
(27)
In the limit of very low fi~lds, it agrees with eqs. (10) and (11). However as the field is increased it is considerably larger than low field
P. Kumar] ~;oliton instabilify in an easy plane ferromagnet value. The divergence in s0 at the critical field is a consequence of our retaining on'.y low order a terms. Finally if the critical field is identified from the zero of the curvature of G(c~) at oto; the critical field a t finite velocity satisfies
36~
fluctuations is the bottom of the magnon continuum and therefore the sG soliton, as it moves, excites the magnons.
4. Distorted soliton
[0c(U)- O¢(O)J3 = ~ \Eo/
"
(28)
The critical field decreases with u [9]. The value /3c(0)=413 is, as mentioned before, a consequence of the assumed O profile. We could have assumed Oa sech2(z - ut)l(, in which case 1Be(0) would be correct, but eq. (27) would have an extra numerical factor. In any case, the u 2/3 dependence in eq. (28) is independent of any ~uch approximations. The combined effect of the magnetic field and motion is to distort the sG soliton. So far we have suppressed all the effects on the 4, variable. It is easy to see that 4, profile is also distorted and in particular propagating waves are excited. To show that, let us expand G(4,, 0) in eq. (8) about a moving sG soliton. If 0 =: &o + &~ and 0 = 0o + 01, where Oo = (ul2A~) &0,;
The question remains; what happens to the soliton after it goes unstable. For a static soliton, it can be answered by a variational calculation. This is, however, possible only because a static soliton goes unstable in a single mode. A moving soliton goes unstable in a more complicated manner and a simple variational calculation is not possible. And yet, by looking at the fluctuation spectrum of the distorted soliton, we can draw some conclusions about the dynamic system as well. The distorted soliton profile is calculated by assuming sin 4,12 = s e c h zl¢; and sin 0/2 = a sechZzl(,, where cx and ~" are variational parameters. The choice of 0 profile originates from a simple generalization of the profile of the normal mode that goes unstable. If Eo and ~o refer to the energy and width of the sG soliton
[7], G(4), ~,) = Go(4,o, ~,o)+ G,~I + G?4,, +0(4, 2, #,2, ,~#,),
(29)
where G~ and G~ are integral operators. We need consider only G~;
EI Eo
=
(FI((~)F2((x)) '12
and ~/~o = (F1(a)IF~(c~)) '12,
G;q,' = - 3'a ~ f dy [ - q)o,, + sin q,o + Y--;=; 4,o2:,4,o,.v(3"/' - 1/2) ] 4, i,
where
(30)
where y = y ( z - u t ) l ¢ ; and ~/-~= I - u 2 1 c 2. The sG soliton satisfies 00, = sin 00. Hence G*~Ol is nonzero, the sG solution is intrinsically unstable. More specifically, since the kernel operating on 4,t is an odd function of y, the instability creates 4, distortion of that symmetry. The lowest odd symmetry mode in 4,
Fl(a)=l+2
[2 - ,/,-°c~
tan-I
l-a
- ~/' a ~' tanh-' ~ / 1 - ~ J 16
-2a: ]3
256 ,\ ~'" 315 /
and
F2(a) = I -
, )
,,
-~)0 c , : - ~
~a'.
(3~)
~
P. Kumarl Soliton instability in an easy plane ~erromagnet
A lower energy minimum is obtained if we take ?nstead sin 0/2 = a sech zig. Clearly the earlier choice is motivated by the profile of the unstable mode. H o w e v e r . at large a, the gradient energy contributions are m u c h too large. In as much as the changeover is discorMnuous [7], we expect the small ct region of E(t~) to be well described by eq. (31), while for large ,,~ we find I-a-'tan~/ F,(,~) : - . , _ / ~1 ,
'(~'~
"t t .':----~ -
8
,
a "+
32
61 and ~t are given ~' ..le
aE =
2 fd:r,LLv°:" (A ~
+ {Ja" cos: ~o6~: + H c o s 6 o c o s ~o6~12
+ { - J'a26o. sin 2 0 o 6 ~ 1
ot
- H sin q~o sin 6 o ~ t 6 1 } ]
=
+ (6,1L,16,) + (¢JlL,,[6).
(33)
(32)
The principal results are: (a) For /3 > 2.44 ( H < 11.5 kG), i.e. the sG sc'iton (a = 0) is the lowest energy minimum subject to the boundary conditions 6 ~ ( 0 , 2 ~ r ) as
200
Ja" H cos 6o c o s , o) } 2 6o: cos 2 o+ T
and F,(a) _ = 1• - ( 1 / 3 - 2/3)a" - :~/3a4.
cos
2 --~ ( - - ~ c ~c).
(b) For 2.44 > ~ > 1.5, the energy at a:~ 0 is lower than the one at c~ = 0. However the (~ = 0 state is stable agair~st small fluctuat;ons. The two min, ma are separated by a sma}, potential r~arrier. (c) For / 3 - - - 1 . 5 : a ~ 0 is the ~nly energy minimum. The off-xy plane excursion angle 6, clo,~e to tl~i~ instability is about 70 °. It increases with ~he magnetic field [7] and reaches zd2 at /3 ~ 0.5, at which time the solit,ons are unstable towards a uniform magnetization. (d) The width of the disterted soliton is approx;rnately half of the width of the sG soliton. The energy at/3 = !.5 is smaller than Eo by 13% and continues to decrease as H increases, We now look at the fluctuations about the di~orted soliton. If the distorted soliton has a translational mode (motiov at uniform velocity) the lowest eigenvalue in the fluctuations must be zero. The analysis reported here is variational. If E(& 0) in eq. (92) is expanded about 4) = ,5,,(:~ and 0 = ,'r/2 + 6dz). in terms of fluctuation fLnctions ,b,(z) and 6~(z), the terms quadratic in
It is possible to analyze eq. (33) in a two level approximation. The task then is to find the iov'est eigenvalues of operators L , and L,, say e, and e,,; calculate the matrix element of L**, the energy 8H is then given by e ~ , A 2 + e , , B ~ + AAB, where A and B are the normalization integrals. The eigenvalues of 8H are given by x. = ~[(~, + ~,,) + [(e~, - e,, )2 + A21,/21.
(34~
The stability condition that the eigenvalues be non-negative, translates into e,~,. ~ A2/4, the equality implies ,~_ = 0. For variational profiles of 01 and 61, we take ~i = A(sech z i g ) " and 01 = B(sech zig)" and minimize the fluctuation energy 8E with respect to p. and v. In as much as both functions are even functions of z and the cross-kernel L** is an odd function of z, the matrix element A = 0. The details of this calculation are shown in appendix B. The eigen values of e,, and e~ and the exponents :.t and v are shown in fig. !. For /3 = 2 for example, ¢,,/H = 0.683,
ta = 0.6t~
and c,IH = 0.304,
v = 0.36.
(35)
P. Kumar/ SoUton iastabi'ily in an easy plane ferromagnet
367
variant. With the distortion in O, this symmetry is broken in the distorted texture. |.O -
S. Summary •
o.e-
e@@@e
Oo,~/H
x •
o.6O
X 0 1
0
O0
0
X
X
I • Q
10
00000 °
X
%/H
0.2-
o0
x
I
X X O
0.4-
x
I
I
I
,.o
I
~
I
2.0
Fig. I. The fluctuation eigenvalues t . , and e.+. and the exponents ~t and v are shown as a function of # - A/H. The dots and the crosses are respectively e~s and/~. The trian$1es and circled crosses are respectively e.+. and v.
As shown in fig. I, the eigen value e~, increases with magnetic field slowly, reaching a maximum at /3 = 0.8 and then decreases. The ~-fluctuation frequency e~, on the other hand, continues to decrease with increasing magnetic field. Its field dependence is considerably weaker though and even at 0 = 0.5, e, = 0.23 H [13]. Thus the principal dynamic modes in a distorted soliton are oscillatory. The lowest frequency mode corresponds to the oscillations of qJ which, in the linear continuum case, are decoupled from the next hish~r frequency mode, the qJ fluctuations. A major distinction should be made here. In a sG soliton, ~he motion of the entire te~,ture is translationally uniform. This symmetry remains intact. The broken symmetry here corresponds to the relative motion of e and ~. The former being constant in sG case, the ~ motion is translationally in-
The results inthis paper [14] show the limitations of +the sG description of a linear chain, easy-plane ferromagnet. We find a static soliton to be unstable for fields H > 18 kG. However, in the field range 11.5 kG < H < 18 kG, the distorted texture is lower in energy:, compared to the sG soliton, In that sense, the instability resembles a first order transition as a function of the magnetic field, with associated hysteretic effects. The dynamics of the distorted soliton is studied by looking at the eige,values of the linear fluctuations. Th~,se in fact serve a dual purpose. On the one hand, they investigate the stability of the distorted texture about which the linear ltuctuations are studied. On the othe, hand, the frequencies associated with the spin dynamics are directly related to the fluctuatic~ frequencies. In particular, ff the distorted texture were free to move translationally, the lowest fluctuation eigen value would have to be zero. We find the fluctuation eigen wlues to be positive and non-zero, indicating (a) that the distorted soliton is stable and (b) it contains no translational mode, it oscillates with a characteristic frequency and remains fixed at a place. Clearly, the lattice discreteness could couple the translational and oscillatory modes. Finally, the instability of a moving soliton is explored briefly. The motion of a sG soliton causes the spins to move off the x-y plane and should help the instability. The limiting velocity depends on the magnetic field. Conversely, the critical field drops rather suddenly with the velocity of the sG soliton. At very small velocities, the sG soliton emits linear excitations, magnons and in all infinite chain its motion becomes dissipative. This is intrinsic dissipation, as opposed to any extrinsic dissipation that may be present in the system. The effects associated with the latter are also discussed.
le~
P. Kumart Solilon instability in an easy plane ferromalgnel
Acknowledgements Man Bishop has been a constant source of cotaments~ criticism and stimulating advice. Some of this work was done on a visit to Los Alamos National Lab and Bishop should be thanked for the hospitality. Comments by D. Huber and by H. Thomas are deeply appreciated. R. Joyce and V. Samalam must be thanked for help with the calculations and M. Bowick for a critical reading of the manuscript. This work was supported by NSF DMR-800o311 and by the Research Corporation.
Here the subscript k refers to the scattering states and the brackets imply an integral over z . The third term or the left-hand side represents the frequency of L e mode, proportional to/zo = (213- 3)H. The frequency is renormalized away from/~o due to the coupling with the plane wave states. The additional terms on the right-hand side represent higher order couplings. If they are ignored the frequency ~a~ and the lifetime F2 of this mode are given by the coefficients of A0 and Ao terms, respectively. Using the expressions for ~o and 6~ given in eqs. (21), co2 is found to be , 27 to~ = ~-¥ (yH)2(2/3- 3).
Appendix A
(A.I)
The numerical factor a is of order unity. That it is not exactly unity could be a consequence of the perturbative calculation. We suspect it is a real shirt but in an experiment, disguised by the width, it remains merged with the reagan continuum (co:= 2/3(7H)2). As the field H is increased, this mode gets split away and approaches zero near the instability.
(A.2)
Appendix B
In the Io~ field limit, the spin dynamics is described by eq~. (22) and (23). These are iinearizations of the Bloch equation about a sine-Gordon soliton. They can be rewritten formally as (1 4~ - ~ : - aLq/, + L~d,. 7 (1 + ),a".~ & = L~q~ + a l . , & .
(A.4)
where 1he operators L: and L. have been defined earlier in eq. (19). If we expand q~ = E., A.~,. and d, = E.,, B,.~bm whece q~. are the wavefm~ctions of L: and ~b~. belong to L~, separate :-quations can be derived for A0. the amplitude of the ~ bound state and B0, the amphtude of the & bound s~ate, t h e latter corresponds to a zero frequency mode, the translational mode of the: soliton. The former satisfies an equation
The stability analysis begins with eq. (33) and is restricted to a two level case. We assume 0, = A(sech zlO" and 4,t = B(sech zlOL Here Ix and v are the variational parameters. The eigenvalue e, is given by
H
if
dy
(¢,,,)2
+ [(2/3 - ~o2,) cos 200 + cos *o cos 4'0] x ~ ]
(S.I) Is
•
-"
,
i~ol
l
,
~t?
~k~l-
.~,, + V~,u,,Ao + V ~ hk~to (c!'~)(,lJo) Ao : - r" ÷
2L2~t + I +2/3 + 1 - ( 6 + ( 1 6 / 3 + 2 ) a ' ) 2/~+1 2P
xk 3#,0
+ (36a 2+ 16/3a 4)2 2-~+ 1
16.>].
(A.3)
2t~+3 2t~ (2tt + 2) (2t~ + 4)] - 32t~ (2~ + i) (2tt + 3) (2/z + 5)J'
(B.2)
P,./<:-marl Solilon inslabiUly in an easy plane ferromagnel
where we have substituted the distorted soliton profile for ~0 and qn0. Here y = z/G, ~/= Gd~ and G0 represents the width of the sG soliton. The integral important for the evaluation of eq. (B.2) is [10]
l(n) = (sech"z) -'- 2 "-I (F(n/2))2 F(n)
(B.3)
and n
l(n + 2)= n + I l(n).
(B.4)
Eq. (B.2) is then explicitly minimized for a given value of/3 and corresponding values of G and c~. Similar operation on e~ yields the various curves in fig. I.
References [I] M. Steiner, J. App. Phys. 50 (1979) 7395: and in Physics in One Dimension, J. Bernasconi and T. Schneidel, eds. (Springer, Berlin, 1981). [2] HJ. Mikeska, J. Phys. C1 I, (1978) L29. For a theoretical review see H.J. Mikeska, J. Appl. Phys. 15 (1981)
369
1950; and in Physics in One Dimension, J. Bernasconi and T. Schneider, eds. (Springer, Berlin, 1981). [3] J.K. Kjems and M. Steiner, Phys. Rev. Lett. 41 (1978) 1137. Also see M. Steiner, ref. 1. [4] K.M. Leung and D.L. Huber, Solid St. Comm. 32 (1981) 127. [51 Ref. !. [6] [so see K. ~ysics, V. VIII, D. Brewer, ed. CNorth-Holland, Amsterdam, 1981). [7] P. Kumar, Phys. Rev. B 15 (1982) 483. [8] A. R, Bishop, J. A. Krumhansl and S.E. Trullinger, Physica DI (1980) I. K. Maki, in ref. 6. [9] E. Magyad and H. Thomas, preprint (1981) have also arrived at this conclusion independently. [1o] K. Maki and P. Kumar, Phys. Rev. BI6 (1977) 182; K. Maki, in Solitons and Condensed Matter Physics, A. Bishop and T. Schneider, eds. (Springer, Berlin, 1978). [11] A.H. Morrish, The Physical Principles of Magnetism. (Wiley, New York, 1966). [12] P. Kumar and V.K. Samalam, University of Florida preprint (1981). [13] The variational ansatz must breakdown at large fields since it ignores any coupling to magnon excitations. We expect that if ~ approaches ~rl2, the system must be completely unstable for a homogeneous, essentially saturated state. At this stage, it is not possible to make a quantitative statement about this effect. [14] A brief summary of the results has been reported in P. Kumar and V.K. Samalam, in Proc. of 27th Annual Conf. on Magnetism and Magnetic materials, J. Appl. Phys. 53 (1982) 1856.