- Email: [email protected]
Domain wall dynamics in ferrites in an external oscillating magnetic field
Journalof magnetism ~ 4 and magnetic ~ l ~ materials ELSEVIER
Journal of Magnetism and Magnetic Materials 146 (1995) 323 334
Domain wall dynamics in ferrites in an external oscillating magnetic field V.S. Gerasimchuk
a9 A . L . S u k s t a n s k i i
b,,
~' Donbass Engineering and Construction Institute, 339023, Makeeuka, Ukraine h Donetsk Physieo-Technical Institute, Ukrainian Academy of Sciences, 340114 Donetsk, Ukraine
Received 5 August 1994
Abstract
Domain-wall drift motion in ferrimagnets with two non-equivalent sublattices induced by an external oscillating magnetic field has been studied, The dependences of the drift velocity on the amplitude, frequency and polarization of the field have been obtained. The possibility of the drift of a stripe domain structure is considered.
1. Introduction
There is steadily increasing interest in investigations of the dynamic properties of magnetic inhomogeneities (domain walls (DW), magnetic solitons, etc,) in magnetically ordered crystals. The most adequate theory has been developed for two types of magnets: one-sublattice ferromagnets [1-3] and two-sublattice antiferromagnets (and weak ferromagnets) with two equivalent sublattices [3-5]. Both theoretical and experimental investigations have revealed some important differences between the dynamic properties of nonlinear excitations for these two types of magnets. For example, a limit steady motion velocity of DW in ferromagnets is small and determined by constants of relativistic interactions, whereas in antiferromagnets this is connected only with constants of exchange origin and achieves 2 X 10 6 c m / s [5]. There exists one other very important class of magnets: ferrimagnets, or ferrites, e.g. magnets with some nonequivalent magnetic sublattices. Practically, all nonmetallic magnets with a magnetic momentum of purely exchange origin are ferrites. The theoretical description usually used to explain experimental results in ferrites (linear spin wave spectrum, DW dynamics, magnetic bubbles, etc.) is based on the model of an effective ferromagnet with a constant modulus of a magnetization vector, M~ = I M~ + M2I = const. (M1, 2 are the sublattice magnetization vectors). But such a model is known to fail in the vicinity of a compensation point where I M~ - M= I << ML2. A more adequate theory of nonlinear dynamics in two-sublattice ferrites has been
* Corresponding author. Email: [email protected]. 0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights rescrved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 6 1 0 - 4
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
324
proposed in Ref. [6]. In this paper, in particular, the criterion of the effective ferromagnet model usability was obtained: v=
Ml,z
>>
~
10-2_10
1
(1)
where /3 and 6 are the anisotropy and homogeneous exchange interaction constants, respectively. If criterion (1) is fulfilled, the effective ferromagnet model is adequate, but in the opposite case when the parameter v is small enough, the dynamic properties of the magnet under consideration are similar to those of the antiferromagnet. In Ref. [6], the steady motion of a DW induced by a constant external magnetic field was considered and the field dependence of the DW velocity was obtained. The purpose of the present paper is to present the theory of the dynamic properties of DWs in ferrites under the influence of an external oscillating magnetic field, main attention being paid to the so-called drift motion of the DW, i.e. the onset of a constant DW velocity component on the background of a DW vibration. DW drift motion in a ferromagnet (FM) was first predicted in Refs. [7,8] and was experimentally observed in Refs. [9,10]. A more consistent analysis of DW drift in FMs, based on the solution of the equations of motion averaged over the period of the oscillations, was carried out in Ref. [11] and an analogous method was used in Ref. [12] to analyze the drift of Bloch lines in DWS. The results in Ref. [11,12], however, are valid only for field frequencies to substantially higher than the ferromagnetic resonance frequency. The most adequate theoretical approach for the description of DW drift motion was proposed in Ref. [13] for ferromagnets and in Ref. [14] for two-sublattice weak ferro- and antiferromagnets. In the present paper, we use an analogous method for the theoretical investigation of the DW drift motion in a two-sublattice ferrite with two nonequivalent sublattices.
2. General equations: domain walls According to Ref. [6], the macroscopic dynamics of a two-sublattice ferrite can be described on the basis of a Lagrange function L expressed in terms of the antiferromagnetism unit vector 1, 12= 1. Parametrizing the vector ! by means of two angle variables 0 and ~,
l z + il x = sin 0 exp(iq~),
ly = cos 0,
(2)
the Lagrange function for the ferrite with a rhombic magnetic anisotropy may be written in the form
L{O, ~} = M 2 2c----S(02+sinZOgpZ)-~[(VO) z +sin20(Vq~) z] 1:
/3, sin20 sine~p_ /3:, cos20 + - - ~ b ( l
2
2
gM o
- cos 0)
+ 2 v [ h x sin 0 sin ~#+hx cos O+h z sin 0 cos ~p] 4
+ 8gM--~o[hx((~ sin 0 cos 0 sin ~ p - 0 cos ~) +hz(~b sin 0 cos 0 cos ~ + 0 sin q~) - h y ( ~ s i n 2 0 ) ] - ~2 ( h , sin 0 sin ¢ + h y cos O+hz sin 0 cos ¢)~,} ,
(3)
where M0z = 2 ( M ( + M2Z), g is the gyromagnetic ratio, ce is the inhomogeneous exchange constant, c = gM o
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
325
(c~6)~/e/2 is the characteristic velocity, h = H / M o, H = H(t) is the external magnetic field, /31, /32 are the anisotropy constants, and a superior dot marks a derivative with respect to time. The dynamic stopping of the DW, due to dissipative processes, is taken into account by using the dissipative function Q, Q =
AM° 12 2g
a M , /}2 2g ( + sinZ0~b2)
(4)
where A is the relaxation constant. The equations of motion for the angle variables 0 and q~, allowing for the relaxation terms, take the lorm:
t 't
c~ : . . 1 0 - 7 0
I(
+ s i n 0 c o s 0 c~ 7
t
- ( Vq~)e - / 3 ' sin2~+/32
]
u
4
+ gM,(p sin 0 + 2~,(h. cos 0 sin q ~ - h y sin O+h: cos 0 cos q~) - 7 ( h , cos 0 O+h~ cos 0 cos (#)
+h~ sin 0 sin q~+h. sin 0 cos ~ ) ( h x cos 0 sin q ~ - h , sin
+ 6gM,----~4[h~ cos q ~ - h . sino ~ # - 2 ~ b s i n 2 0 ( h . sin q~+h. cos ~)-h,.(Dsin2O] A -
gMo
(},
(5) OL
cgV(sin20(V'q~)) - --(sin20~b)
c2
•
p
-/31 sin20 sin ~p cos q~-
+ 2 v ( h x sin 0 cos ~ p - h . sin 8 sin ~,) +
+2(} sin20(h,sin (# + hzcos ~) + J*~.sin2O+
4 -agM,, -[
gMq~
(} sin 0
"sin 2O(h, sin ~# + h. cos ~#) 4
h,.(} sin 20] - ~ ( h , . c o s 0 a
+ h , sin 0 sin ~#+h z sin 0 cos ~#)(h, sin 0 cos ~ - h .
sin 0 sin (#) = - -
" sin20
gM, (#
(6) "
Eqs. (5) and (6) generalize equations used to describe the nonlinear dynamics of one-sublattice ferromagnets and two-sublattice antiferromagnets. If M~ =M 2, i.e. M~ = 0, one obtains the equations describing the dynamics of antiferromagnets [3-5], whereas, neglecting the noncollinearity of the sublattices (formally, in the limit 6--* ~ or c 2--+ ~), Eqs. (5) and (6) transform into the well-known Landau-Lifshitz equations for a one-sublattice ferromagnet with magnetization M = M~l. So, studying different solutions of Eqs. (5) and (6). 'an antiferromagnetic limit' (u ~ 0) and 'a ferromagnetic limit' (6 --+ zc) may be analyzed. If /3~, /32 > 0, the vector l in the absence of an external magnetic field is collinear with the Z-axis in the homogeneous ground state, and two types of 180 ° DW can exist. The vector l rotates in the XZ plane in one of them, and in the YZ plane in the other. If/32 >/3~ > 0, the stable DW is the one with l rotating in the XZ plane [4]. This DW corresponds to 0 = 00 = 'rr/2, and the angle variable q~ = q~0(Y) satisfies the equation: c~q~'(',-/31 sin q~0 cos q~0 = 0,
(7)
(we assume that the magnetization distribution in the DW is not uniform along the Y-axis; a prime denotes
326
KS. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
differentiation with respect to this coordinate). A static 180° DW in which the function ~Po(Y) satisfies the boundary conditions ~Po(- oo) = 0, q~o(+ ~) = rr, q~0(+ oo) = 0, are described by the relations q~o'=--Sinyo q~o = Yo
1Yo ,
cos ~ o ( Y ) =
-tanh
(8)
where Yo -- ( a / f l l ) 1/2 is the wall thickness.
3. Perturbation theory: first approximation We now consider the solution of the equations of motion in an external magnetic field. A DW moves in a constant field of definite orientation (in our case along the Z-axis) with a fixed velocity determined by the balance of the magnetic pressure acting on the the DW and the dynamic stopping force [6]. In an oscillating field, the wall oscillates at the field frequency, and we shall show below that its center drifts with a certain definite velocity. In addition, the presence of the field distorts the shape of the DW. Assuming the external field amplitude to be small enough, we determine the drift velocity and the distortion of its form, following Refs. [13,14], by one of the perturbation-theory versions for solitons. To this end we introduce a collective variable Y(t), which has the meaning of the coordinate of the DW at the instant t, and seek solutions of Eqs. (5) and (6) in the form: O=Tr/2+O(~,t),
q~= q~0(~:) + ~ ( ~ , t),
(9)
where sc = y - Y(t). The function ~o(~c) describes the motion of an undistorted DW (the structure of ~o(~:) is the same as that of ~0(Y) in the static solution (8)). The wall drift velocity is defined as the instantaneous DW velocity v ( t ) = l?(t) averaged over the oscillation period, COt = c,(t) (the bar denotes averaging over the external field oscillation period). We represent the functions O(~:, t) and ~( ~:, t), describing the distortion of the shape of the DW, as well as the wall velocity v(t), by series in powers of the field amplitude, recognizing that we are interested only in the stimulated DW motion: 0 ( ~, t) = 0,( ~, t) + 02( ~,
t)
+...
~ ( ~ , t) = ~1(~c, t ) + ~b2(~, t ) + ... U ~'UI
(10)
-~'- U2 + . . . ,
where the subscripts n = 1, 2 , . . . , denote the order of smallness of the quantity relative to the field amplitude, We substitute the expansions (10) in Eqs. (5) and (6) and separate terms of different orders of smallness. Obviously, in the zeroth approximation we obtain Eq. (7), which describes a DW at rest. The first-order perturbation theory equations can be written in the form / ~ + o r + og~-dt--~ +
ogr d o92dtt O,
o)~ d~l _ o92 dt
~o~ L,I sin q~o o9~ Yo g M 0 ,. + - - ~ (h~ cos ~ o - h ~ sin
1 d2 tor d ) w,, dO 1 L + w---~-----' dt Y + o92 ~ t ~b, + o92 at
2v
~o)--~lhy ,
(ll)
b 1 + vlo9~ . yotor sin q~o gM o . 2v + ~hy + N ( hx cos ¢'o - hz sin ~,,),
(12)
V.S. Gerasimchuk,A.L. Sukstanskii/Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
327
where tol = C/yo = gMo(/318)1/2/2 is the activation frequency of the lower spin-wave mode, % - - A 6 g M J 4 is the characteristic relaxation frequency, wv = v6gMo/4, and o-= (/32 -/31)//31' The operator L, takes the form of a Schr6dinger operator with a nonreflecting potential: d2 2
L = -Y
'dU + 1
(13)
ch2~/yo
The spectrum and the eigenfunctions of the operator/~ (13) are well known. It has one discrete level with the eigenvalue A0 = 0 corresponding to the localized wave function: 1
f o ( ~ ) = (2y(,),/2 ch~/y,)'
(14)
and also a continuous spectrum A, = 1 + kZYo, corresponding to the eigenfunctions
f,(~)
b,L,/2 th
-iky(,
e ik~,
(15)
where b k = (1 + k2y~) I/2, and L is the crystal length• The functions {f(), f,} form a complete orthonormalized set, and it is natural to seek the first-approximation solutions of Eqs. (11) and (12) in the form of an expansion in this set. For a monochromatic external field of frequency m, we put 0,( sc, t ) = R e { [ y" ckfk(k ~ ) + c ° f " ( ~:)] e'~') '
(16)
One important remark is in order here. The first-approximation equations (11) and (12) describe the excitation of linear spin waves against a DW background. The last term in the expansion of the function ~0~(~, t) corresponds to the Goldstone mode, i.e. to DW motion as a whole• The corresponding degree of freedom of the system, however, has already been taken into account by introducing the collective coordinate Y(t) into the definition of the variable ~. The Goldstone mode should therefore be left out of the expansion (17), i.e. one must put d o = 0 (for a detailed discussion of this question see Rajaraman's book [15]. The coefficients ck, c 0 and d k in the expansions (16) and (17) can be found by in standard fashion multiplying the right-hand sides of (11) and (12) by fk* and fo* and integrating with respect to the variable ~. For a monochromatic field of frequency o9, with all three components different from zero and with arbitrary phase shifts, h , = h x o cos wt,
h>.=hy o c o s ( t o t + g l ) ,
hz=h:o c o s ( t o t + X ) ,
(18)
we obtain from Eqs. (11) and (12): 01( ~:, t) = 2 Re{a,(t) sin ~0(sc) + a2(t) cos q~o(~c) + a3(t)Gl(~)}, ~bl(~, t) = 2 Re{a4(t) cos ¢ 0 ( sc) + as(t)G2( c)}. We have introduced here the notations
i togM o al(t)--
a3(t ) =
to (
~v 2/3=Tr3hy(t)__
r h (t)-
v ~lh>(t)
v a4(t ) = ~ i r 4 h x ( t )
Q-2K
r~ = 2 [ Q ( o . _ Q ) + K Q 1 ]
ae(t)
(19)
i togM() to~ r2hx(t)
as(t )
i to gM o to~ h y ( t )
1 -Q+2K
,
r2= 2[(I_Q)(I+cr_Q)_KQI]
,
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
328
2Q - Q1 r 3 = 2 [ Q ( o ' - Q) + K Q , ] '
2 ( l + o - - Q ) +Q1 r4 = 2[(1 - Q)(1 + o r - Q) - KQ1 ] '
[tanh(~/Yo) YOf+~ -~ dk
G , ( s¢ ) = ~ - -
sin k ~ - ky0 cos k~](A k - Q + Q1/2)
Ak[( A k - Q ) ( Ak + O--Q) - KQ1 ] s i n h ( - ~ )
y0f?
G2(~ ) = ~
_
dk
'
[tanh(~/Yo) sin k ~ - ky0 cos ksC](Ak + o - - Q + 2K)
( "~kyo)
Ak[(A k - Q)(A k + o ' - Q) - KOl ] sinh T
where Q = Q1 - iQ2, Q1 = ( ° ) / w l ) 2, Q2 = (O')Okr//0)2), and K = 6v2/4~1. It follows from (19) that the external field components h x and h z excite bulk oscillations only with k = 0, whereas the presence of the field components hy m a k e s possible the excitation of bulk spin waves with k 4: 0. The condition of absence of the Goldstone mode in the expansion (17) in turn leads to the requirement that the right-hand side of (12) be orthogonal to the function f0, which determines in turn the equation for the DW velocity vl(t) in the approximation linear to the field:
(o--Q+K)bl
+Wr(o--Q)v,
vyohz(t) [ 2 ( o - - Q ) w 2 + 092] fl, - ~ry°gM°hy(t) ( o ' - Q + 2K). 2
(20)
The solution of Eq. (20) can be written in the form (21)
v,( t) = Zwy o Xm -~i hz( t)rs - -~a5( t)r 6 , where 2( o- - Q) + Q1 r5 = 2 [ Q ( o ' - Q) + KQ1 ] '
o--Q+2r r6= 2 [ 0 ( o ' - - 0 ) +KQ1]"
It is easy to show that Eq. (20) generalizes the corresponding equation for one-sublattice ferromagnets and two-sublattice antiferromagnets. Really, in the ferromagnetic limit, Eq. (20) reduces to the form
) ~l + Ao-OOoVl 2°-yot°~hz(t ~m
7ryot°o hy(t),
~1
Wo=fllgMo,
which coincides (within the accuracy of the notation) with the corresponding result of Refs. [1,13]. In the antiferromagnetic limit one obtains from Eq. (20): U1 "~ WrU1 =
~ry° gM° hy( t ) . 2
The latter equation coincides with the analogous equation obtained in Ref. [16] (neglecting the Dzyaloshinskii interaction). The solution (21) describes the DW vibrations in ferrites, induced by an oscillating external field and, as can be easily seen, does not lead to a DW drift, i.e. vl(t ) = O.
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
329
4. Second approximation: DW drift We now analyze the second-order equations for the amplitude of the external field. We shall not write down the pertinent equations in general form, since they are extremely cumbersome, but only equations averaged over the period of the oscillations, which from Eqs. (5) and (6): =
_
( [ + , r ) O : ( ~ , t)
W~
't
-~2(~q~,,, so)?2 + K(sC, t),
(22)
~;(~)-~,+U(~, t),
Lq,2(~, t)=
(23)
where the functions K(~, t) and N(~, t) are defined as K ( ¢ . t) = ~ ( i , ~ + o~/,,)0; +2y~O16'~9'o + 6~01 sin 2,¢,,- ~ " , 6 ; O) 1
O9 i
4 hy ) (h x sin q~,,+ h z cos q~,,) + gM°[w---~2hxc1¢' o sin q~,, - (2iv O, - -~-i
-(h.,t),+2hx(hl)sin
q~o+2hzc,q~'ocos
~,,-(h:tp,+2h~(bl)cos
1
(%
O) I
tO ~
q~,,],
(24)
2 t t "~ t N( ~:, t) = --2( i:, + WrC 1) ~1r - 2y,,O~O, q~,,+ qff sin 2q~,, + ~ t',O~
1 y ( l o ) i c l°q ~ ~ 2
,
cos
2u
4,,(hx sin
+ hz cos
4
cos 2¢,,
+ 12(h~- h2~)sin 2 % ] + g~'=[(i~xO 1 + 2h,01) sin q~,,+ (/):0, + 2h:O) cos q:"l"
(25)
Just as in the first-approximation equations, we must stipulate that the expansion of the function ~02( ~:, t) in terms of the eigenfunctions of the operator L contains no shear mode. This yields an expression for the DW drift velocity l,d~= c,: "9C
t~,,~ = - y,,o)~_j2 (.or - ~. dsCN( ~:' t) q~'°( ~:)"
(26)
Substituting the functions Ol(sc, t) and Ol(~c, t) (19) calculated in the previous section in (25), and integrating over the oscillation period in (26), we obtain for the drift velocity t,a~:
Cd~ = fix.( to; x)H,,~Hoz + rl~y ( w; x,)H,,~H,,~.,
(27)
where r/~(w; X) and rlxy(W; Xl) are certain functions of the frequency and of the phase shifts, which we shall call nonlinear mobilities (NM) of the domain wall (their structure will be given below). It follows from (27) that the DW drift occurs only if at least two components of the magnetic field differ from zero - - either h x and h~ or h~ and hy. This fact may be interpreted in the following manner: the z or y components of the field, as follows from (19), cause DW oscillations, while the x component ensures different values of the wall's mobility as it moves in the positive and negative y directions. If, however, the field is oriented in the yz plane, there is no DW drift. We consider next DW drift in a field polarized separately in the xz or xy plane.
V.S. Gerasimchuk, A.L, Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
330
4.1. Field in the xz plane The nonlinear mobility "0,z which determines the drift velocity in an oscillation field polarized (in general, elliptically) in the xz plane (see (18) for hy = 0) is of the form: r/xz(o9;X) =2rh, Re{[A(w; X ) + F(o9; X)] e-ix},
(28)
where r/o = 'rrg2yo/4ogr, A(o9; X ) = 7Q,I [r 2 - r,* +2r2r,* ],
(29)
F(w; X) = to[r4 - rs(Q, r2 + Qr4)]It is easy to see that the NM r/,z(w; X) demonstrates a typical resonance or resonance-antiresonance behavior at the characteristic frequencies which coincide with activation frequencies of two bulk spin-wave branches and of one localized on the DW (the spin-wave spectrum in a ferrimagnet on the background of DW is calculated in Ref. [6]). Expressions (28) and (29) testify to the fact that the NM r/xz is determined by two terms of different nature: an 'antiferromagnetic' term (A) and a 'ferromagnetic' one (F). Really, the first term A(w; X) tends to zero in the ferromagnetic limit (6 ~ 2) and defines fizz in the antiferromagnetic limit (v ~ 0),
r/:~(o9; X ) = "0,A sin )~ [(o9o9,)2 + (o92
+cos
o92)2][(o9o9~)2 + (0)32 _ o92)2]
O94O9~O9r x [t(ogo9 ~
.+_(£O2
2+
} 1o 32 121
,
where o92 = 0-o9~, o92 = (1 + 0-)o9~, r/2 = 7ryog2/% is the characteristic NM in antiferromagnets. Expression
~A
t/A
J"
70
J\
tO"
a
a)
J
ilm
8)
Fig. 1. Frequency dependence of the nonlinear mobility '0~z(w; X) in the antiferromagnetic limit (v = 0). (a) X = 0; (b) X = w / 2 .
V.S. Gerasimchuk, A.L. Sukstanskii / Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
~F
~
331
F
M i
Fig. 2. Frequency dependence of the nonlinear mobility r h :(w; X) in the ferromagnetic limit (6 ~ ~, ~, = 1). (a) X = 0; (b) X = 7r/2.
(30) coincides with the analogous expression obtained by the authors [14] for weak Dzyaloshinskii interaction). The frequency dependences of the NM r/A(o3; X) at shown schematically in Fig. 1. The second term F(w; X) in (28) is proportional to the parameter K (i.e. to antiferromagnetic limit, providing the NM r/xz in the ferromagnetic limit (6 ~ ~,
03,o0{o,2~ - 03 "~ )(0-030- 2 + r/:;.(03; X) = rt(~ sin )c (0-2t7)~ q_ 0 3 2 ) [ ( t 7 ) 2 -
+cos X
~
-2
v) and tends to zero in the v = 1),
O32 )
032)2 q_ 0326~ff(2 + 0 . ) 2 ]
032_r,°__24 ( .+_ 2 -'~
ferromagnets (neglecting the different phase shifts X are
(0-03;+03~)[(0so-03")
~ 2
+ 12_+
)_ t
~_-~
2 J'
+W-03r(2+0-)]
(31)
where &o = (1 + 0-)03o, &r = ao3o, rt(~ = "rry(,g2/o3o is the characteristic NM in ferromagnets. Expression (31) coincides (within the accuracy of the notations) with the main result obtained in Ref. [13] for a one-sublattice ferromagnet. The frequency dependences of the NM "q~vz(03; X) at different phase shifts X are shown schematically in Fig. 2. These dependences demonstrate a resonance behavior at the frequency &0It should be underlined that there exists a strong dependence of the NM rt, z(w; X) on the phase shift X (see Figs. 1 and 2). In particular, in the low-frequency region (w << wo), rtxVz(03; w / 2 ) ~ 03 and tends to zero under to ~ 0, whereas r/xz(to, V . 0) 4:0 under o 9 = 0 . In the high-frequency interval (w>>03j, or 03>>w 0) r/x~(03; " r r / 2 ) ~ 03 t, whereas rlxz(W; 0 ) ~ to 2. It is easily shown that at intermediate values of X, the frequency dependence of the NM rb~ is similar to the case of X = r r / 2 (Figs. lb, 2b), and the transition to the case X = 0 (Figs. la, 2a) takes place at small X of the order A1/2 << 1. To demonstrate the dependence of the NM on the parameter u in the vicinity of the compensation point, in Fig. 3 we plot the function rh~(to; X) at X = 'rr/4 for various values of the parameter K, K = 0.1, 1 and 10 corresponding to u = (4K [ 3 1 / 6 ) 1/2 ~ 2 × 10 -3, 6 × 10 -3 and 2 × 10 2 (we consider [3]/6 10-4). ~
332
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
A f'-
|lw
3
OY w,
A
to
It
oY
©
Fig. 3. Nonlinear mobility r/x.(O~; )() at X = ' r r / 4 and various values of the parameter K. (a) K = 0.1; (b) K = 1; (c) K = 10.
Let us estimate numerically the nonlinear mobility and the drift velocity of D W in the ferro- and antiferromagnetic limits. In the antiferromagnetic limit we use the values of the parameters indicative of typical ferrites in the compensation point (for instance, Gd3FesO12) [17]: Y0 ~ 1 0 - 4 cm, g = 2 × 107 (S O e ) - 1 , and 01 ~ 1011 s - i This yields the value of the characteristic NM ~7A = 1.2 c m / ( s Oe2), and for an oscillating field of amplitude ~ 10 Oe the drift velocity reaches a value of the order of 10 2 c m / s . Using the typical values of the parameters for ferrites far from the compensation point (in the region within which the effective ferromagnetic model is adequate) [1,17]: Yo ~ 10 6 cm, too ~ 3 × 109 S I g = 1.7 × 10 7 (S O e ) - ~, we obtain an estimate of the characteristic NM %v in the ferromagnetic limit: ~7(F ~ 0.3 c m / s Oe2.
v.s. Gerasimchuk,A.L. Sukstanskii/Journal of Magnetism and Magnetic Materials 146 (I995) 323-334
333
4.2. Field in the xy plane The nonlinear mobility r/xy in the case of an oscillating field (18) at h~ = 0 is given by r/x,.(~o; X,) = -'Oo--~
Im{[2r6(Q,r; + Q ' r 4 )
-2(O,(w)
+r3(1 -2r2 )
+ D2(w)G* ) - P,(oJ) - P 2 ( w ) r ; ] eiX'},
(32)
where the functions Dn(w), P,,(~o), n = 1, 2, are defined by the integrals 1
~
P,,(w): ~f,
[2(l+xZ-O)+Q,]x
[(l+x2_Ol(l+x2+o.
1
~
D,,(~o)=-~f,
z"
Q)_KQ,](l+xZ)sinh2(iTX/2
(l+x2+cr-Q+2K)x [(l+x 2-O)(l+x
dx ),
2" d x
2+o'-Q)-KQ,](I+x
2) s i n h 2 ( r r x / 2 ) '
From a comparison of expressions (28), (29) and (32), it follows that the structure of the NM r/,~. is considerably different from that of r/x~. First, "Oxy~ v, i.e, r/xv = 0 at the compensation point (in the antiferromagnetic limit). This statement confirms a conclusion o f Ref. [14] that an external field with the polarization under consideration does not result in DW drift in antiferromagnets. Besides, the NM r/~,. is proportional to o~ and, consequently, turns to zero in the low-frequency limit w ~ 0. The integrals P,,(w) and D,,(w) in (32) may be approximated by the formulas: P,, ----p,, r 7 and D,,-~ p,,r s, where 2(1-Q)
+Q1
r7= 2[(l_O)(l+cr_Q)_KQi],
1 +o'-Q+2K r~= 2 [ ( 1 - O ) ( l + ~ r - O ) - K O ~ ] '
and p,, are certain constants of the order of unity. Using these approximations, it is easy to show that the frequency dependence of the drift velocity in the xy field is similar to that in the previously considered field in the xz plane for the 'ferromagnetic' term in (28).
5. Drift of a stripe-domain structure
We now consider the possibility of drift in an external alternating field with a stripe-domain structure (DS), consisting of 180° DWs. It must be borne in mind here that neighboring DWs in the DS have opposite topological charges determined by the boundary conditions of Eq. (7). In addition, the rotation of the vector l in various DWs can be positive or negative about the z-axis. These two factors determine the DW drift direction in a field of fixed frequency w and a phase shift X (or Xl)- A DS drift is possible, naturally, only when neighboring DW move in one and the same direction. We define the topological charge R = + 1 of the DW and the parameter p = + 1 that describes the rotation of the vector ! in a DW as follows: 1=(+~) = ~ R ,
Ix(y=0) = +p.
(33)
The domain walls considered in Section 4, having a magnetization distribution (8), correspond to R = p = 1. In the general case we have in lieu of (8): ¢'o=--R Yo
sin q ~ o = - - R p s e c h , Yo \ Yo !
cos q%= - R t a n h
Y .
(34)
334
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
Analysis shows that in the general case the drift velocity of a DW with the given values of the parameters R and p is determined by an equation similar to (27): Udr
=
Rprlxz( w; X)HoxHoz + g~?xy( w; Xl)HoxHor,
(35)
where the NM r/xz and r/xy are described as before by (28)-(30). Thus we see from (35) that DS drift in a field polarized in the xy plane is altogether impossible, since the topological charges of neighboring DWs are different. In a field polarized in the xz plane, DS drift is possible, but provided that neighboring DWs have different values of the parameter p and of the topological charge, i.e. the rotations of the vector ! in neighboring DWs must be in opposite directions. A similar situation (possibility of DS drift in a field polarized in the DW plane) also obtains in ferromagnets [13] and weak ferromagnets [14].
Acknowledgements The present research was financed by a grant from the Ukraine State Committee for Science and Technics No. 2.3/670.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). A. Hubert, Theorie der Domanenwande in geordneten Medien (Springer-Verlag, Berlin, 1974). A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Nonlinear Magnetization Waves (Naukova Dumka, Kiev, 1988) (in Russian), V.G. Bar'yakhtar, B.A. Ivanov and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 78 (1980) 15. V.G. Bar'yakhtar, B.A. lvanov and M.V. Chetkin, Usp. Fiz. Nauk 146 (1985) 417. B.A. Ivanov and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz., 84 (1983) 370. E. Schlomann and J.D. Miln, IEEE Trans. Magn. 10 (1974) 791. E. Schlomann, IEEE Trans. Magn. 11 (1975) 1051. Yu. Dragoshanskii, E.B. Khan and V.A. Zaikova, Fiz. Met. Metallogr. 39 (1975) 289. V.K. Vlasko-Vlasov, L.M. Dedukh, V.I. Nikitenko ann L.S. Uspenskaya, Fiz. Tverd. Tela 24 (1982) 1255. V.M. Eleonskii, A.K. Zvezdin and V.G. Red'ko, Fiz. Met. MetaUov. 43 (1977) 7. V.S. Gerasimchuk, Fiz. Tverd. Tela 32 (1990) 391. V.G. Bar'yakhtar, Yu.l. Gorobets and S.I. Denisov, Zh. Eksp. Teor. Fiz. 98 (1990) 1345. V.S. Gerasimchuk and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 103 (1993) 151. R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). V.G. Bar'yakhtar, B.A. Ivanov, V.F. Lapchenko and V.A. L'vov, Fiz. Nizkh. Temp. 12 (1987) 551. S. Krupicka, Physik der Ferrite (Academia Verlag, Prague, 1973).
Journal of Magnetism and Magnetic Materials 146 (1995) 323 334
Domain wall dynamics in ferrites in an external oscillating magnetic field V.S. Gerasimchuk
a9 A . L . S u k s t a n s k i i
b,,
~' Donbass Engineering and Construction Institute, 339023, Makeeuka, Ukraine h Donetsk Physieo-Technical Institute, Ukrainian Academy of Sciences, 340114 Donetsk, Ukraine
Received 5 August 1994
Abstract
Domain-wall drift motion in ferrimagnets with two non-equivalent sublattices induced by an external oscillating magnetic field has been studied, The dependences of the drift velocity on the amplitude, frequency and polarization of the field have been obtained. The possibility of the drift of a stripe domain structure is considered.
1. Introduction
There is steadily increasing interest in investigations of the dynamic properties of magnetic inhomogeneities (domain walls (DW), magnetic solitons, etc,) in magnetically ordered crystals. The most adequate theory has been developed for two types of magnets: one-sublattice ferromagnets [1-3] and two-sublattice antiferromagnets (and weak ferromagnets) with two equivalent sublattices [3-5]. Both theoretical and experimental investigations have revealed some important differences between the dynamic properties of nonlinear excitations for these two types of magnets. For example, a limit steady motion velocity of DW in ferromagnets is small and determined by constants of relativistic interactions, whereas in antiferromagnets this is connected only with constants of exchange origin and achieves 2 X 10 6 c m / s [5]. There exists one other very important class of magnets: ferrimagnets, or ferrites, e.g. magnets with some nonequivalent magnetic sublattices. Practically, all nonmetallic magnets with a magnetic momentum of purely exchange origin are ferrites. The theoretical description usually used to explain experimental results in ferrites (linear spin wave spectrum, DW dynamics, magnetic bubbles, etc.) is based on the model of an effective ferromagnet with a constant modulus of a magnetization vector, M~ = I M~ + M2I = const. (M1, 2 are the sublattice magnetization vectors). But such a model is known to fail in the vicinity of a compensation point where I M~ - M= I << ML2. A more adequate theory of nonlinear dynamics in two-sublattice ferrites has been
* Corresponding author. Email: [email protected]. 0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights rescrved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 6 1 0 - 4
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
324
proposed in Ref. [6]. In this paper, in particular, the criterion of the effective ferromagnet model usability was obtained: v=
Ml,z
>>
~
10-2_10
1
(1)
where /3 and 6 are the anisotropy and homogeneous exchange interaction constants, respectively. If criterion (1) is fulfilled, the effective ferromagnet model is adequate, but in the opposite case when the parameter v is small enough, the dynamic properties of the magnet under consideration are similar to those of the antiferromagnet. In Ref. [6], the steady motion of a DW induced by a constant external magnetic field was considered and the field dependence of the DW velocity was obtained. The purpose of the present paper is to present the theory of the dynamic properties of DWs in ferrites under the influence of an external oscillating magnetic field, main attention being paid to the so-called drift motion of the DW, i.e. the onset of a constant DW velocity component on the background of a DW vibration. DW drift motion in a ferromagnet (FM) was first predicted in Refs. [7,8] and was experimentally observed in Refs. [9,10]. A more consistent analysis of DW drift in FMs, based on the solution of the equations of motion averaged over the period of the oscillations, was carried out in Ref. [11] and an analogous method was used in Ref. [12] to analyze the drift of Bloch lines in DWS. The results in Ref. [11,12], however, are valid only for field frequencies to substantially higher than the ferromagnetic resonance frequency. The most adequate theoretical approach for the description of DW drift motion was proposed in Ref. [13] for ferromagnets and in Ref. [14] for two-sublattice weak ferro- and antiferromagnets. In the present paper, we use an analogous method for the theoretical investigation of the DW drift motion in a two-sublattice ferrite with two nonequivalent sublattices.
2. General equations: domain walls According to Ref. [6], the macroscopic dynamics of a two-sublattice ferrite can be described on the basis of a Lagrange function L expressed in terms of the antiferromagnetism unit vector 1, 12= 1. Parametrizing the vector ! by means of two angle variables 0 and ~,
l z + il x = sin 0 exp(iq~),
ly = cos 0,
(2)
the Lagrange function for the ferrite with a rhombic magnetic anisotropy may be written in the form
L{O, ~} = M 2 2c----S(02+sinZOgpZ)-~[(VO) z +sin20(Vq~) z] 1:
/3, sin20 sine~p_ /3:, cos20 + - - ~ b ( l
2
2
gM o
- cos 0)
+ 2 v [ h x sin 0 sin ~#+hx cos O+h z sin 0 cos ~p] 4
+ 8gM--~o[hx((~ sin 0 cos 0 sin ~ p - 0 cos ~) +hz(~b sin 0 cos 0 cos ~ + 0 sin q~) - h y ( ~ s i n 2 0 ) ] - ~2 ( h , sin 0 sin ¢ + h y cos O+hz sin 0 cos ¢)~,} ,
(3)
where M0z = 2 ( M ( + M2Z), g is the gyromagnetic ratio, ce is the inhomogeneous exchange constant, c = gM o
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
325
(c~6)~/e/2 is the characteristic velocity, h = H / M o, H = H(t) is the external magnetic field, /31, /32 are the anisotropy constants, and a superior dot marks a derivative with respect to time. The dynamic stopping of the DW, due to dissipative processes, is taken into account by using the dissipative function Q, Q =
AM° 12 2g
a M , /}2 2g ( + sinZ0~b2)
(4)
where A is the relaxation constant. The equations of motion for the angle variables 0 and q~, allowing for the relaxation terms, take the lorm:
t 't
c~ : . . 1 0 - 7 0
I(
+ s i n 0 c o s 0 c~ 7
t
- ( Vq~)e - / 3 ' sin2~+/32
]
u
4
+ gM,(p sin 0 + 2~,(h. cos 0 sin q ~ - h y sin O+h: cos 0 cos q~) - 7 ( h , cos 0 O+h~ cos 0 cos (#)
+h~ sin 0 sin q~+h. sin 0 cos ~ ) ( h x cos 0 sin q ~ - h , sin
+ 6gM,----~4[h~ cos q ~ - h . sino ~ # - 2 ~ b s i n 2 0 ( h . sin q~+h. cos ~)-h,.(Dsin2O] A -
gMo
(},
(5) OL
cgV(sin20(V'q~)) - --(sin20~b)
c2
•
p
-/31 sin20 sin ~p cos q~-
+ 2 v ( h x sin 0 cos ~ p - h . sin 8 sin ~,) +
+2(} sin20(h,sin (# + hzcos ~) + J*~.sin2O+
4 -agM,, -[
gMq~
(} sin 0
"sin 2O(h, sin ~# + h. cos ~#) 4
h,.(} sin 20] - ~ ( h , . c o s 0 a
+ h , sin 0 sin ~#+h z sin 0 cos ~#)(h, sin 0 cos ~ - h .
sin 0 sin (#) = - -
" sin20
gM, (#
(6) "
Eqs. (5) and (6) generalize equations used to describe the nonlinear dynamics of one-sublattice ferromagnets and two-sublattice antiferromagnets. If M~ =M 2, i.e. M~ = 0, one obtains the equations describing the dynamics of antiferromagnets [3-5], whereas, neglecting the noncollinearity of the sublattices (formally, in the limit 6--* ~ or c 2--+ ~), Eqs. (5) and (6) transform into the well-known Landau-Lifshitz equations for a one-sublattice ferromagnet with magnetization M = M~l. So, studying different solutions of Eqs. (5) and (6). 'an antiferromagnetic limit' (u ~ 0) and 'a ferromagnetic limit' (6 --+ zc) may be analyzed. If /3~, /32 > 0, the vector l in the absence of an external magnetic field is collinear with the Z-axis in the homogeneous ground state, and two types of 180 ° DW can exist. The vector l rotates in the XZ plane in one of them, and in the YZ plane in the other. If/32 >/3~ > 0, the stable DW is the one with l rotating in the XZ plane [4]. This DW corresponds to 0 = 00 = 'rr/2, and the angle variable q~ = q~0(Y) satisfies the equation: c~q~'(',-/31 sin q~0 cos q~0 = 0,
(7)
(we assume that the magnetization distribution in the DW is not uniform along the Y-axis; a prime denotes
326
KS. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
differentiation with respect to this coordinate). A static 180° DW in which the function ~Po(Y) satisfies the boundary conditions ~Po(- oo) = 0, q~o(+ ~) = rr, q~0(+ oo) = 0, are described by the relations q~o'=--Sinyo q~o = Yo
1Yo ,
cos ~ o ( Y ) =
-tanh
(8)
where Yo -- ( a / f l l ) 1/2 is the wall thickness.
3. Perturbation theory: first approximation We now consider the solution of the equations of motion in an external magnetic field. A DW moves in a constant field of definite orientation (in our case along the Z-axis) with a fixed velocity determined by the balance of the magnetic pressure acting on the the DW and the dynamic stopping force [6]. In an oscillating field, the wall oscillates at the field frequency, and we shall show below that its center drifts with a certain definite velocity. In addition, the presence of the field distorts the shape of the DW. Assuming the external field amplitude to be small enough, we determine the drift velocity and the distortion of its form, following Refs. [13,14], by one of the perturbation-theory versions for solitons. To this end we introduce a collective variable Y(t), which has the meaning of the coordinate of the DW at the instant t, and seek solutions of Eqs. (5) and (6) in the form: O=Tr/2+O(~,t),
q~= q~0(~:) + ~ ( ~ , t),
(9)
where sc = y - Y(t). The function ~o(~c) describes the motion of an undistorted DW (the structure of ~o(~:) is the same as that of ~0(Y) in the static solution (8)). The wall drift velocity is defined as the instantaneous DW velocity v ( t ) = l?(t) averaged over the oscillation period, COt = c,(t) (the bar denotes averaging over the external field oscillation period). We represent the functions O(~:, t) and ~( ~:, t), describing the distortion of the shape of the DW, as well as the wall velocity v(t), by series in powers of the field amplitude, recognizing that we are interested only in the stimulated DW motion: 0 ( ~, t) = 0,( ~, t) + 02( ~,
t)
+...
~ ( ~ , t) = ~1(~c, t ) + ~b2(~, t ) + ... U ~'UI
(10)
-~'- U2 + . . . ,
where the subscripts n = 1, 2 , . . . , denote the order of smallness of the quantity relative to the field amplitude, We substitute the expansions (10) in Eqs. (5) and (6) and separate terms of different orders of smallness. Obviously, in the zeroth approximation we obtain Eq. (7), which describes a DW at rest. The first-order perturbation theory equations can be written in the form / ~ + o r + og~-dt--~ +
ogr d o92dtt O,
o)~ d~l _ o92 dt
~o~ L,I sin q~o o9~ Yo g M 0 ,. + - - ~ (h~ cos ~ o - h ~ sin
1 d2 tor d ) w,, dO 1 L + w---~-----' dt Y + o92 ~ t ~b, + o92 at
2v
~o)--~lhy ,
(ll)
b 1 + vlo9~ . yotor sin q~o gM o . 2v + ~hy + N ( hx cos ¢'o - hz sin ~,,),
(12)
V.S. Gerasimchuk,A.L. Sukstanskii/Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
327
where tol = C/yo = gMo(/318)1/2/2 is the activation frequency of the lower spin-wave mode, % - - A 6 g M J 4 is the characteristic relaxation frequency, wv = v6gMo/4, and o-= (/32 -/31)//31' The operator L, takes the form of a Schr6dinger operator with a nonreflecting potential: d2 2
L = -Y
'dU + 1
(13)
ch2~/yo
The spectrum and the eigenfunctions of the operator/~ (13) are well known. It has one discrete level with the eigenvalue A0 = 0 corresponding to the localized wave function: 1
f o ( ~ ) = (2y(,),/2 ch~/y,)'
(14)
and also a continuous spectrum A, = 1 + kZYo, corresponding to the eigenfunctions
f,(~)
b,L,/2 th
-iky(,
e ik~,
(15)
where b k = (1 + k2y~) I/2, and L is the crystal length• The functions {f(), f,} form a complete orthonormalized set, and it is natural to seek the first-approximation solutions of Eqs. (11) and (12) in the form of an expansion in this set. For a monochromatic external field of frequency m, we put 0,( sc, t ) = R e { [ y" ckfk(k ~ ) + c ° f " ( ~:)] e'~') '
(16)
One important remark is in order here. The first-approximation equations (11) and (12) describe the excitation of linear spin waves against a DW background. The last term in the expansion of the function ~0~(~, t) corresponds to the Goldstone mode, i.e. to DW motion as a whole• The corresponding degree of freedom of the system, however, has already been taken into account by introducing the collective coordinate Y(t) into the definition of the variable ~. The Goldstone mode should therefore be left out of the expansion (17), i.e. one must put d o = 0 (for a detailed discussion of this question see Rajaraman's book [15]. The coefficients ck, c 0 and d k in the expansions (16) and (17) can be found by in standard fashion multiplying the right-hand sides of (11) and (12) by fk* and fo* and integrating with respect to the variable ~. For a monochromatic field of frequency o9, with all three components different from zero and with arbitrary phase shifts, h , = h x o cos wt,
h>.=hy o c o s ( t o t + g l ) ,
hz=h:o c o s ( t o t + X ) ,
(18)
we obtain from Eqs. (11) and (12): 01( ~:, t) = 2 Re{a,(t) sin ~0(sc) + a2(t) cos q~o(~c) + a3(t)Gl(~)}, ~bl(~, t) = 2 Re{a4(t) cos ¢ 0 ( sc) + as(t)G2( c)}. We have introduced here the notations
i togM o al(t)--
a3(t ) =
to (
~v 2/3=Tr3hy(t)__
r h (t)-
v ~lh>(t)
v a4(t ) = ~ i r 4 h x ( t )
Q-2K
r~ = 2 [ Q ( o . _ Q ) + K Q 1 ]
ae(t)
(19)
i togM() to~ r2hx(t)
as(t )
i to gM o to~ h y ( t )
1 -Q+2K
,
r2= 2[(I_Q)(I+cr_Q)_KQI]
,
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
328
2Q - Q1 r 3 = 2 [ Q ( o ' - Q) + K Q , ] '
2 ( l + o - - Q ) +Q1 r4 = 2[(1 - Q)(1 + o r - Q) - KQ1 ] '
[tanh(~/Yo) YOf+~ -~ dk
G , ( s¢ ) = ~ - -
sin k ~ - ky0 cos k~](A k - Q + Q1/2)
Ak[( A k - Q ) ( Ak + O--Q) - KQ1 ] s i n h ( - ~ )
y0f?
G2(~ ) = ~
_
dk
'
[tanh(~/Yo) sin k ~ - ky0 cos ksC](Ak + o - - Q + 2K)
( "~kyo)
Ak[(A k - Q)(A k + o ' - Q) - KOl ] sinh T
where Q = Q1 - iQ2, Q1 = ( ° ) / w l ) 2, Q2 = (O')Okr//0)2), and K = 6v2/4~1. It follows from (19) that the external field components h x and h z excite bulk oscillations only with k = 0, whereas the presence of the field components hy m a k e s possible the excitation of bulk spin waves with k 4: 0. The condition of absence of the Goldstone mode in the expansion (17) in turn leads to the requirement that the right-hand side of (12) be orthogonal to the function f0, which determines in turn the equation for the DW velocity vl(t) in the approximation linear to the field:
(o--Q+K)bl
+Wr(o--Q)v,
vyohz(t) [ 2 ( o - - Q ) w 2 + 092] fl, - ~ry°gM°hy(t) ( o ' - Q + 2K). 2
(20)
The solution of Eq. (20) can be written in the form (21)
v,( t) = Zwy o Xm -~i hz( t)rs - -~a5( t)r 6 , where 2( o- - Q) + Q1 r5 = 2 [ Q ( o ' - Q) + KQ1 ] '
o--Q+2r r6= 2 [ 0 ( o ' - - 0 ) +KQ1]"
It is easy to show that Eq. (20) generalizes the corresponding equation for one-sublattice ferromagnets and two-sublattice antiferromagnets. Really, in the ferromagnetic limit, Eq. (20) reduces to the form
) ~l + Ao-OOoVl 2°-yot°~hz(t ~m
7ryot°o hy(t),
~1
Wo=fllgMo,
which coincides (within the accuracy of the notation) with the corresponding result of Refs. [1,13]. In the antiferromagnetic limit one obtains from Eq. (20): U1 "~ WrU1 =
~ry° gM° hy( t ) . 2
The latter equation coincides with the analogous equation obtained in Ref. [16] (neglecting the Dzyaloshinskii interaction). The solution (21) describes the DW vibrations in ferrites, induced by an oscillating external field and, as can be easily seen, does not lead to a DW drift, i.e. vl(t ) = O.
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
329
4. Second approximation: DW drift We now analyze the second-order equations for the amplitude of the external field. We shall not write down the pertinent equations in general form, since they are extremely cumbersome, but only equations averaged over the period of the oscillations, which from Eqs. (5) and (6): =
_
( [ + , r ) O : ( ~ , t)
W~
't
-~2(~q~,,, so)?2 + K(sC, t),
(22)
~;(~)-~,+U(~, t),
Lq,2(~, t)=
(23)
where the functions K(~, t) and N(~, t) are defined as K ( ¢ . t) = ~ ( i , ~ + o~/,,)0; +2y~O16'~9'o + 6~01 sin 2,¢,,- ~ " , 6 ; O) 1
O9 i
4 hy ) (h x sin q~,,+ h z cos q~,,) + gM°[w---~2hxc1¢' o sin q~,, - (2iv O, - -~-i
-(h.,t),+2hx(hl)sin
q~o+2hzc,q~'ocos
~,,-(h:tp,+2h~(bl)cos
1
(%
O) I
tO ~
q~,,],
(24)
2 t t "~ t N( ~:, t) = --2( i:, + WrC 1) ~1r - 2y,,O~O, q~,,+ qff sin 2q~,, + ~ t',O~
1 y ( l o ) i c l°q ~ ~ 2
,
cos
2u
4,,(hx sin
+ hz cos
4
cos 2¢,,
+ 12(h~- h2~)sin 2 % ] + g~'=[(i~xO 1 + 2h,01) sin q~,,+ (/):0, + 2h:O) cos q:"l"
(25)
Just as in the first-approximation equations, we must stipulate that the expansion of the function ~02( ~:, t) in terms of the eigenfunctions of the operator L contains no shear mode. This yields an expression for the DW drift velocity l,d~= c,: "9C
t~,,~ = - y,,o)~_j2 (.or - ~. dsCN( ~:' t) q~'°( ~:)"
(26)
Substituting the functions Ol(sc, t) and Ol(~c, t) (19) calculated in the previous section in (25), and integrating over the oscillation period in (26), we obtain for the drift velocity t,a~:
Cd~ = fix.( to; x)H,,~Hoz + rl~y ( w; x,)H,,~H,,~.,
(27)
where r/~(w; X) and rlxy(W; Xl) are certain functions of the frequency and of the phase shifts, which we shall call nonlinear mobilities (NM) of the domain wall (their structure will be given below). It follows from (27) that the DW drift occurs only if at least two components of the magnetic field differ from zero - - either h x and h~ or h~ and hy. This fact may be interpreted in the following manner: the z or y components of the field, as follows from (19), cause DW oscillations, while the x component ensures different values of the wall's mobility as it moves in the positive and negative y directions. If, however, the field is oriented in the yz plane, there is no DW drift. We consider next DW drift in a field polarized separately in the xz or xy plane.
V.S. Gerasimchuk, A.L, Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
330
4.1. Field in the xz plane The nonlinear mobility "0,z which determines the drift velocity in an oscillation field polarized (in general, elliptically) in the xz plane (see (18) for hy = 0) is of the form: r/xz(o9;X) =2rh, Re{[A(w; X ) + F(o9; X)] e-ix},
(28)
where r/o = 'rrg2yo/4ogr, A(o9; X ) = 7Q,I [r 2 - r,* +2r2r,* ],
(29)
F(w; X) = to[r4 - rs(Q, r2 + Qr4)]It is easy to see that the NM r/,z(w; X) demonstrates a typical resonance or resonance-antiresonance behavior at the characteristic frequencies which coincide with activation frequencies of two bulk spin-wave branches and of one localized on the DW (the spin-wave spectrum in a ferrimagnet on the background of DW is calculated in Ref. [6]). Expressions (28) and (29) testify to the fact that the NM r/xz is determined by two terms of different nature: an 'antiferromagnetic' term (A) and a 'ferromagnetic' one (F). Really, the first term A(w; X) tends to zero in the ferromagnetic limit (6 ~ 2) and defines fizz in the antiferromagnetic limit (v ~ 0),
r/:~(o9; X ) = "0,A sin )~ [(o9o9,)2 + (o92
+cos
o92)2][(o9o9~)2 + (0)32 _ o92)2]
O94O9~O9r x [t(ogo9 ~
.+_(£O2
2+
} 1o 32 121
,
where o92 = 0-o9~, o92 = (1 + 0-)o9~, r/2 = 7ryog2/% is the characteristic NM in antiferromagnets. Expression
~A
t/A
J"
70
J\
tO"
a
a)
J
ilm
8)
Fig. 1. Frequency dependence of the nonlinear mobility '0~z(w; X) in the antiferromagnetic limit (v = 0). (a) X = 0; (b) X = w / 2 .
V.S. Gerasimchuk, A.L. Sukstanskii / Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
~F
~
331
F
M i
Fig. 2. Frequency dependence of the nonlinear mobility r h :(w; X) in the ferromagnetic limit (6 ~ ~, ~, = 1). (a) X = 0; (b) X = 7r/2.
(30) coincides with the analogous expression obtained by the authors [14] for weak Dzyaloshinskii interaction). The frequency dependences of the NM r/A(o3; X) at shown schematically in Fig. 1. The second term F(w; X) in (28) is proportional to the parameter K (i.e. to antiferromagnetic limit, providing the NM r/xz in the ferromagnetic limit (6 ~ ~,
03,o0{o,2~ - 03 "~ )(0-030- 2 + r/:;.(03; X) = rt(~ sin )c (0-2t7)~ q_ 0 3 2 ) [ ( t 7 ) 2 -
+cos X
~
-2
v) and tends to zero in the v = 1),
O32 )
032)2 q_ 0326~ff(2 + 0 . ) 2 ]
032_r,°__24 ( .+_ 2 -'~
ferromagnets (neglecting the different phase shifts X are
(0-03;+03~)[(0so-03")
~ 2
+ 12_+
)_ t
~_-~
2 J'
+W-03r(2+0-)]
(31)
where &o = (1 + 0-)03o, &r = ao3o, rt(~ = "rry(,g2/o3o is the characteristic NM in ferromagnets. Expression (31) coincides (within the accuracy of the notations) with the main result obtained in Ref. [13] for a one-sublattice ferromagnet. The frequency dependences of the NM "q~vz(03; X) at different phase shifts X are shown schematically in Fig. 2. These dependences demonstrate a resonance behavior at the frequency &0It should be underlined that there exists a strong dependence of the NM rt, z(w; X) on the phase shift X (see Figs. 1 and 2). In particular, in the low-frequency region (w << wo), rtxVz(03; w / 2 ) ~ 03 and tends to zero under to ~ 0, whereas r/xz(to, V . 0) 4:0 under o 9 = 0 . In the high-frequency interval (w>>03j, or 03>>w 0) r/x~(03; " r r / 2 ) ~ 03 t, whereas rlxz(W; 0 ) ~ to 2. It is easily shown that at intermediate values of X, the frequency dependence of the NM rb~ is similar to the case of X = r r / 2 (Figs. lb, 2b), and the transition to the case X = 0 (Figs. la, 2a) takes place at small X of the order A1/2 << 1. To demonstrate the dependence of the NM on the parameter u in the vicinity of the compensation point, in Fig. 3 we plot the function rh~(to; X) at X = 'rr/4 for various values of the parameter K, K = 0.1, 1 and 10 corresponding to u = (4K [ 3 1 / 6 ) 1/2 ~ 2 × 10 -3, 6 × 10 -3 and 2 × 10 2 (we consider [3]/6 10-4). ~
332
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
A f'-
|lw
3
OY w,
A
to
It
oY
©
Fig. 3. Nonlinear mobility r/x.(O~; )() at X = ' r r / 4 and various values of the parameter K. (a) K = 0.1; (b) K = 1; (c) K = 10.
Let us estimate numerically the nonlinear mobility and the drift velocity of D W in the ferro- and antiferromagnetic limits. In the antiferromagnetic limit we use the values of the parameters indicative of typical ferrites in the compensation point (for instance, Gd3FesO12) [17]: Y0 ~ 1 0 - 4 cm, g = 2 × 107 (S O e ) - 1 , and 01 ~ 1011 s - i This yields the value of the characteristic NM ~7A = 1.2 c m / ( s Oe2), and for an oscillating field of amplitude ~ 10 Oe the drift velocity reaches a value of the order of 10 2 c m / s . Using the typical values of the parameters for ferrites far from the compensation point (in the region within which the effective ferromagnetic model is adequate) [1,17]: Yo ~ 10 6 cm, too ~ 3 × 109 S I g = 1.7 × 10 7 (S O e ) - ~, we obtain an estimate of the characteristic NM %v in the ferromagnetic limit: ~7(F ~ 0.3 c m / s Oe2.
v.s. Gerasimchuk,A.L. Sukstanskii/Journal of Magnetism and Magnetic Materials 146 (I995) 323-334
333
4.2. Field in the xy plane The nonlinear mobility r/xy in the case of an oscillating field (18) at h~ = 0 is given by r/x,.(~o; X,) = -'Oo--~
Im{[2r6(Q,r; + Q ' r 4 )
-2(O,(w)
+r3(1 -2r2 )
+ D2(w)G* ) - P,(oJ) - P 2 ( w ) r ; ] eiX'},
(32)
where the functions Dn(w), P,,(~o), n = 1, 2, are defined by the integrals 1
~
P,,(w): ~f,
[2(l+xZ-O)+Q,]x
[(l+x2_Ol(l+x2+o.
1
~
D,,(~o)=-~f,
z"
Q)_KQ,](l+xZ)sinh2(iTX/2
(l+x2+cr-Q+2K)x [(l+x 2-O)(l+x
dx ),
2" d x
2+o'-Q)-KQ,](I+x
2) s i n h 2 ( r r x / 2 ) '
From a comparison of expressions (28), (29) and (32), it follows that the structure of the NM r/,~. is considerably different from that of r/x~. First, "Oxy~ v, i.e, r/xv = 0 at the compensation point (in the antiferromagnetic limit). This statement confirms a conclusion o f Ref. [14] that an external field with the polarization under consideration does not result in DW drift in antiferromagnets. Besides, the NM r/~,. is proportional to o~ and, consequently, turns to zero in the low-frequency limit w ~ 0. The integrals P,,(w) and D,,(w) in (32) may be approximated by the formulas: P,, ----p,, r 7 and D,,-~ p,,r s, where 2(1-Q)
+Q1
r7= 2[(l_O)(l+cr_Q)_KQi],
1 +o'-Q+2K r~= 2 [ ( 1 - O ) ( l + ~ r - O ) - K O ~ ] '
and p,, are certain constants of the order of unity. Using these approximations, it is easy to show that the frequency dependence of the drift velocity in the xy field is similar to that in the previously considered field in the xz plane for the 'ferromagnetic' term in (28).
5. Drift of a stripe-domain structure
We now consider the possibility of drift in an external alternating field with a stripe-domain structure (DS), consisting of 180° DWs. It must be borne in mind here that neighboring DWs in the DS have opposite topological charges determined by the boundary conditions of Eq. (7). In addition, the rotation of the vector l in various DWs can be positive or negative about the z-axis. These two factors determine the DW drift direction in a field of fixed frequency w and a phase shift X (or Xl)- A DS drift is possible, naturally, only when neighboring DW move in one and the same direction. We define the topological charge R = + 1 of the DW and the parameter p = + 1 that describes the rotation of the vector ! in a DW as follows: 1=(+~) = ~ R ,
Ix(y=0) = +p.
(33)
The domain walls considered in Section 4, having a magnetization distribution (8), correspond to R = p = 1. In the general case we have in lieu of (8): ¢'o=--R Yo
sin q ~ o = - - R p s e c h , Yo \ Yo !
cos q%= - R t a n h
Y .
(34)
334
V.S. Gerasimchuk, A.L. Sukstanskii /Journal of Magnetism and Magnetic Materials 146 (1995) 323-334
Analysis shows that in the general case the drift velocity of a DW with the given values of the parameters R and p is determined by an equation similar to (27): Udr
=
Rprlxz( w; X)HoxHoz + g~?xy( w; Xl)HoxHor,
(35)
where the NM r/xz and r/xy are described as before by (28)-(30). Thus we see from (35) that DS drift in a field polarized in the xy plane is altogether impossible, since the topological charges of neighboring DWs are different. In a field polarized in the xz plane, DS drift is possible, but provided that neighboring DWs have different values of the parameter p and of the topological charge, i.e. the rotations of the vector ! in neighboring DWs must be in opposite directions. A similar situation (possibility of DS drift in a field polarized in the DW plane) also obtains in ferromagnets [13] and weak ferromagnets [14].
Acknowledgements The present research was financed by a grant from the Ukraine State Committee for Science and Technics No. 2.3/670.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
A.P. Malozemoff and J.C. Slonczewski, Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). A. Hubert, Theorie der Domanenwande in geordneten Medien (Springer-Verlag, Berlin, 1974). A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Nonlinear Magnetization Waves (Naukova Dumka, Kiev, 1988) (in Russian), V.G. Bar'yakhtar, B.A. Ivanov and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 78 (1980) 15. V.G. Bar'yakhtar, B.A. lvanov and M.V. Chetkin, Usp. Fiz. Nauk 146 (1985) 417. B.A. Ivanov and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz., 84 (1983) 370. E. Schlomann and J.D. Miln, IEEE Trans. Magn. 10 (1974) 791. E. Schlomann, IEEE Trans. Magn. 11 (1975) 1051. Yu. Dragoshanskii, E.B. Khan and V.A. Zaikova, Fiz. Met. Metallogr. 39 (1975) 289. V.K. Vlasko-Vlasov, L.M. Dedukh, V.I. Nikitenko ann L.S. Uspenskaya, Fiz. Tverd. Tela 24 (1982) 1255. V.M. Eleonskii, A.K. Zvezdin and V.G. Red'ko, Fiz. Met. MetaUov. 43 (1977) 7. V.S. Gerasimchuk, Fiz. Tverd. Tela 32 (1990) 391. V.G. Bar'yakhtar, Yu.l. Gorobets and S.I. Denisov, Zh. Eksp. Teor. Fiz. 98 (1990) 1345. V.S. Gerasimchuk and A.L. Sukstanskii, Zh. Eksp. Teor. Fiz. 103 (1993) 151. R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1982). V.G. Bar'yakhtar, B.A. Ivanov, V.F. Lapchenko and V.A. L'vov, Fiz. Nizkh. Temp. 12 (1987) 551. S. Krupicka, Physik der Ferrite (Academia Verlag, Prague, 1973).