- Email: [email protected]
Nonlinear interface spin waves in magnets
cm .__
i!tl
AA lournal of metism
B%
B
ELSEVIER
Journal
of Magnetismand
metic irlals Magnetic Materials
164 (1996) 105-l 10
Nonlinear interface spin waves in magnets A.L. Sukstanskii
*, S.V. Tarasenko, E.Yu. Melikhov Received 6 March 1996
Abstract A new type of nonlinear magnetic soliton-like excitation (nonlinear interface spin wave), localized in the vicinity of the interface between two semi-infinite ferromagnets, is investigated in the framework of the phenomenological approach. The excitations are shown to have no linear analogy. The results are generalized for the two-sublattice model of ferrimagnets. Kr,~or&t
Magnetic
soliton; Interface
1. Introduction In recent years much experimental and theoretical has been devoted to multilayered magnetic structures with properties differing considerably from the properties of the magnets of which they are composed. Among other things, layered structures exhibit specific wave branches, which are known as intrinsic (or interface) spin waves (ISW), localized in the vicinity of the interface between layers. Such excitations, as well as their special case - surface spin waves. which are widely used in technical devices - are usually considered in the linear approximation, while nonlinear dynamic excitations such as magnetic solitons are mainly studied in unbounded magnetically ordered crystals (see, e.g., Ref. [I]). Naturally, one should expect the existence work
~ Corresponding
author. Email: [email protected].
0304.8853/96/$15.00 Copyright P/I SO304-8853(96)00374-5
of nonlinear surface excitations with properties that differ from those of their linear analog, just as bulk magnetic solitons differ from linear spin waves. In Ref. [2], the existence of a new type of surface excitation, so-called surface spin waves (surface solitons), in a semi-infinite ferromagnet, was first predicted and analyzed, and the essential difference between these excitations and their linear analogs was demonstrated. Moreover, it was shown that, under certain conditions. such linear analogs do not exist at all. For these reasons, it is of considerable interest to study nonlinear excitations in various multilayered magnetic structures in which intrinsic solitons having no linear limit may also exist. In the present paper. nonlinear interface spin waves (NISW), i.e. interface solitons, are studied in the simplest two-layered structures, which consist of two semi-infinite ferromagnets with different magnetic parameters (Section 2). In Section 3 the results are generalized to the case of a structure consisting of two-sublattice ferrimagnets.
0 1996 Elsevier Science B.V. All rights reserved.
2. NISW in a ferromagnetic
structure
angles 0 + and cp+ parametrize vectors M,
Linear surface spin waves are known to be caused by long-range magnetostatic [3] or magnetoelastic [4,5] interactions. If one takes into account the short-range exchange interaction only, then such excitations may exist in models of finite magnets, including a specific surface layer in which some parameters (saturation magnetization, anisotropy or exchange constants, gyromagnetic ratio, etc.) differ from those in the volume [6,7]. If the thickness of the layer is much smaller than the characteristic length of the inhomogeneity in the magnetization distribution, the surface properties of the magnet may be considered in the framework of the dynamic Landau-Lifshitz equations by means of introducing phenomenological boundary conditions (see. for example, Refs. [4,8]). In our investigation, we use this phenomenological approach in simple models of uniaxial semi-infinite single-sublattice ferromagnets and two-sublattice ferrimagnets. Restricting ourselves to the investigation of spin waves of exchange origin only, we will not take into consideration magnetostatic and magnetoelastic interactions. We assume that the interface between magnets coincides with the plane z = 0 and that each magnet is an easy-axis ferromagnet with an anisotropy axis aligned with the normal to the interface. The coupling between magnets can be associated with different interactions: exchange, magnetostatic, magnetoelastic, etc. Here again we restrict ourselves to the local exchange interaction on the interface. Therefore, the total energy of the system can be written as the sum of the energies of the magnets, and their interaction can be taken into account through the boundary conditions on the interface [9]. The energy of each ferromagnet can be written in the form
W,=M,‘, -
p,
( VoBi)’ + sir&,( j”+dV,( %r
Icos’o
VP*)?]
-1 +
(hereafter the indices ‘+ ’ and ‘- ’ correspond to parameters of the magnets occupying the upper ( z > 0) and lower (z < 0) half-spaces, respectively). The
M==M,+(sin
0,
cos p+,sin _
the magnetization
O_ sin cp,,cos
M,j+= IG,
O,), (2)
and cr* and p f > 0 are the constants of the nonuniform exchange interaction and anisotropy, respectively. The solution of the equation of motion (LandauLifshitz equations) in each half-space will be sought in the form CJJ+= cp_= Rt-kr., 02= O_(z). The functions 0 *(z) satisfy the equation PO, .a-? ~ + w+ sin OI - sin 0 + cos 0 i = 0, I i)z’
(3)
where
and g+ is the gyromagnetic
ratio. The-ground state of the system is determined by the type of exchange interaction between the magnets. The latter can be of two different types: ferromagnetic or antiferromagnetic. Obviously, in the first case (ferromagnetic coupling) the minimum of the energy of the system corresponds to the parallel orientation of the magnetization vectors M, and M_ in the magnets, and boundary conditions at i.- + 5~ for solutions localized in the vicinity of the interface are ‘ferromagnetic’: M;, ( + x) = MC,_, M,_ ( - m> = M,,_. In the opposite case (antiferromagnetic coupling) the minimum of the energy corresponds to the antiparallel orientation of the magnetization vectors M, and M_ in the ground state, therefore M,+(+m) = M,)+, MC-(-x) = -M,... Boundary conditions on the interface ; = 0 are also defined by the character of coupling between magnets. If this coupling is strong enough, the magnetization vectors are parallel to each other on the interface for ferromagnetic coupling and antiparallel for antiferromagnetic coupling, not only in the ground state but also in excited states. In the case of ferromagnetic coupling, the boundary conditions to Eq. (3) have the form [9]: X = KM,;_ a+M,2+ ~ a,_ :=+” O+( +0) = O_( -0).
8_ -az
;= _o’
(4)
A.L. Szrkstnnskii ct al. /Jo~wnal of Mqnetisnz
Besides, for the solutions localized in the vicinity of the interface, in which we are interested, we must put e(+m> = e(-,=) = 0. In the case of antiferromagnetic coupling between magnets it is convenient to change the parametrization of the magnetization vector M_ by redefining the angles H_ and cp_: H_+r-0_, (p_+ -cp_. In terms of these new variables, Eq. (3) and the boundary conditions (Eq. (4)) are as before. That is why we shall discuss below only the case of ferromagnetic coupling, so the results for antiferromagnetic coupling can be obtained by means of trivial redesignation. Eq. (3) has been analyzed in detail (see. e.g., Refs. [ 1,101). Its localized solutions were shown to exist only for w f < 1 and have the form = 2arctan cosh[K&--Z&X+]
w,>o, A*
w+
0i(z)=2arctan
sinh[K+(Z-zi)/Xk]
where
-q)“‘,
$=(I
A,=
i
+$
a+&+
K+
=
1
X-
K-;-/X_)
’
where a = A+/A-, p = a+M~+x~/a_M~~_.r+, and the signs of the parameters Z+ and Z_ in Eq. (7) must be chosen to be identical. Hence, the magnetization distribution in NISW has only one maximum. For example, in the case z i > 0, this maximum 0,, = 0+(; = Z+ > = 2 arctan A+ is at the point z = Z, > 0, whereas the magnetization in the half-space z < 0 is described by the exponential ‘tail’ of the function 0_(z). If z + < 0, then a similar maximum is situated in the lower half-space at z = z_ < 0. The energy of NISW at R > 0 can be readily obtained by substituting the magnetization distribution (Eq. (5)) into Eq. (1):
X+
+s)
=D+w+-
U>
1,
pK+<
Kp
Or
Cl <
p> 1, f2+
R
K_
(6)
s
(8)
’
1,
pK+>
K_.
(9)
These inequalities impose certain restrictions both on the parameters of the ferromagnets for which NISW exist:
p< 1,
Or
i
*
frequency
= (P?-W+fip*fi-
(10)
l2+> R-,
and on the precession from above:
i
Ifs
where D,=~M,?+K~[LY~(P~+ a+k2)]‘/’ and s = PK+/K(both expressions for E are equal to one another). It can readily be shown that solutions (Eq. (7)) exist if
i
i
KM;_
cosh(
K+Z_
tanh
x+ =
z+ = f K+cosh-’ K+
A-
cosh( K+Z+/X,)
107
The solvability of Eq. (6) with respect to Z, is a necessary condition for the existence of NISI (Eq. (5)) under R > 0. Eq. (6) proves to have two solutions:
E=D_w_(l
It should be noted that the functional dependencies of the solutions in the upper and lower halfspaces are the same because the signs of the dimensionless frequencies w+ and o_ always coincide. The integration constants z + are defined by means of the boundary conditions (Eq. (4)). Substituting the solutions (Eq. (5)) in the case of positive frequencies R > 0 (w ~ > 0) into Eq. (41, one obtains the system of equations for Z__ A+
164 (1996) 105-110
1’ 1’ (5)
A, 19,(z)
nnd Magnetic Marerids
0,
which
is limited
(11)
108
A.L. Sukstanskii
et al. /Journal
If fi < 0 (w ~ < O), the boundary in the following equations for zt:
conditions
result
A-
A+ sinh(
of Magnetism
=
K+z+/x+)
sinh( K_:_/.X._)
’
and Mtrprtic
Materials
164 (19961 105-I
IO
any value of frequency within the interval of existence of the wave, and hence NISW have no linear analog: their localization is determined completely by nonlinearity (in analogy with surface magnetic solitons [2] and surface shear waves in the theory of elasticity [ 1 11, which also do not have linear analogs).
‘~+Mo?t K+ x+ 3. NISW in two-sublattice
=
Eq. (12) also has two solutions
(the signs of z, and Z_ must be chosen to be identical). The energy of NISW at 0 < 0 can also be represented in the form of Eq. (8) with the substitution w~+lo~I. Solutions of Eq. (13) exist under the conditions: Kp
Or
UK_
K_.
These inequalities also result in certain tions, both on the magnets parameters p> 1, i
pYLn+.
ferrimagnets
(‘2)
p< 1,
Or i p%>
fl,,
(14) restric-
(‘5)
and on the precession frequency which is now limited from below: - 0, < 0 < 0. Thus, NISW, described by Eq. (51, represents a nonlinear magnetic excitation propagating along the interface between the magnets and localized near this surface. In our model, such excitations exist in the finite interval of precession frequencies 0, - fl, < n< +n*. For ]0\+0,, the parameters I z *I + x, and NISW degenerates. Taking into account that the dimensionless frequencies w+ and w_ at R = + 0, are less than 1, one can make the important conclusion that the amplitude 0, of NISW remains finite at
In the previous sections we analyzed NISW in the model of single-sublattice ferromagnets. However, the theory developed above can easily be generalized for more complicated magnet models. As an example, we consider NISW in a two-layered system consisting of semi-infinite two-sublattice ferrimagnets with easy-axis magnetic anisotropy. As shown in Ref. [ 121, nonlinear dynamics in two-sublattice magnets with non-equivalent sublattices in the main approximation with respect to the small parameters /3/S -K 1 and w/GgM,,==KI (6 is the homogeneous exchange interaction constant. M,f M,,,= IM,,21,and M,,, are the = (Mf + Mt)/2. sublattice magnetization vectors). can be studied by means of the effective Lagrange function L = LIZ) written in terms of the unit antiferromagnetic vector 1 = L/I Ll, L = M, - M2 [ 121. In terms of the angle variables f?* and cp+, parametrizing the vectors I + in the ferrimagnets just as in Eq. (2) for the vectors M,. the Lagrange function for each uniaxial ferrimagnet under consideration can be written as:
where ui= CM,+- M, )/2M,+ . " ‘I,at-g the charact~r~~trc = &+M,+)(a$,)"' locities coinciding with the minimal phase velocities of the spin waves, and the ‘potential’ energies W,. written in terms of the angle variables 0 + and p +,
A.L. Sukstanskii
et al./Jountal
qf‘Magnetism
have the same form as in Eq. (1). The equations motion for these variables have the form:
---0,
6W, 69,
of
(17)
It is noteworthy (see details in Ref. [12]) that, in the limit u + 0, Eq. (17) describes the nonlinear dynamics in antiferromagnets, and in the limit 6 -+ x (c - m> they coincide with the Landau-Lifshitz equations for ferromagnets. Assuming as above p+=q_=Ll-krL,O,= 0 +( z), one obtains from Eq. (17) that the functions 0 k ( Z> satisfy Eq. (3) with the coefficients x k and w+ _ redefined:
and Mugmtic
Materials
164 (10%)
The boundary conditions to the equations for the functions 0 + ( Z> also depend on the character of the exchange interaction between the magnets. It is easy to show that, in the case under consideration, a rather strong ‘ferromagnetic’ coupling leads to the parallel orientation of the unit vectors Zi on the interface z = 0 both in the ground state and for dynamic excitation, whereas ‘antiferromagnetic’ coupling results in the antiparallel orientation. This is why the boundary conditions for localized solutions can be written in the form of Eq. (4). Thus, for a structure consisting of two ferrimagnets, we have the same equations and the same boundary conditions as in the previous section. Consequently, the results of the previous section (localized solutions describing NISW (Eq. (5)), expressions for the parameters z + (Eqs. (7) and (1311, and the inequalities (Eqs. (9) and (14)) defining the region of NISW existence) are relevant for the ferrmagnetic system. The energy of the NISW in a ferrimagnetic structure has a far more complicated form than that in a ferromagnetic structure: E = 2M,‘+
cY+W+L12 K,X+
[
K,C;
x(
(18)
109
105-110
K,
tanh [+I]
tanhhi
K++ tanh-’
1
+ (1 + v+)( P++ a+k2) a_w_n2
+ 24_
tanhh
KpXp
’K_
-
tanh-
’
K-C!
Localized solutions of Eq. (3), as mentioned above, exist under w + < 1. In the case of ferromagnets, this condition bounds the precession frequency R from above (0 < 0 ~ 1. In the case of ferrimagnets, together with the inequality w + < 1, there is one more condition under which localized solutions exist: (p&-t a+k’a,f12/c:)> 0. These inequalities are shown to lead to the limitation of the precession frequency not only from above but from below as well: fly)< 0 < Q’,‘), fi(‘.‘,_ * -$[-&+[[&!l
X(K_
tanht-)]
+(1
-rl_)(P-+aPk2) i
(20) at R > 0, and E = 2M,2+ K+X+
cY+W+R2 2
+ 2 MC;- K-XP++
‘y+k’)
1’2 (19)
2
c+ Note that Ry)<
I
0 and R’:_‘> 0.
1.
K++
coth-’
x (K, coth {+)I + (1 + ++)( P++ a+k2) a_w_02 K_ C?
+ 4a,(
[coth-’
K+C+
x( K_ coth t_)]
[
cothh’ K_-coth-’
+ (1 - qp)(
pp+
a-k’) I
(21)
A.L.
110
Sukstanskii
et al. / Journal
of Magrwtism
Muterids
164 (19961
10%110
References
at 0 < 0. Here sinh2,5,
‘I
’* = 2(cosh25
and Magnetic
k + A?,) ’ ‘*=
sinh2l, 2(Sinh2*iiA,4ZL)
’
and z, are defined by Eqs. (7) and (13) for the cases 0 > 0 and R < 0, respectively. In the limit c i --) m, Eqs. (20) and (2 1) reduce to Eq. (81. Naturally, the inequalities Eqs. (9) and (141, written in terms of the initial parameters of the magnets, are rather cumbersome, and therefore are not given here. We only emphasize that, in the case under consideration, NISW also exist in the finite frequency interval 10 1< fl,,,, as it does in ferromagnetic systems, 1(1,,, lying inside the interval (a(‘) RC2’) and w ~ being less than 1 at I i2 + n ,,,,:I Ckequently, in the ferrimagnetic structure, NISW have no linear analog as for the ferromagnetic structure, and the localization of NISW is entirely due to nonlinearity.
[I] A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Nonlinear Magnetization Waves. Dynamic and Topological Solitons (Naukova Dumka, Kiev. 1983) (in Russian). [2] A.L. Sukstanskii and S.V. Tarasenko, Fiz. Tverd. Tela 35 (1993) 270 [Sov. Phys. Solid State 35 (1993) 1361. [3] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19 (1961) 308: Phys. Rev. 118 (1960) 1208. [4] A.I. Akhiezer, V.G. Bar’yakhtar and S.V. Peletminskii. Spin Waves (Nauka, Moscow. 1967). [5] H.K. Tiersten, J. Appl. Phys. 36 (1965) 2250. [6] B.N. Filippov. Fiz. Tverd. Tela 9 (1967) 1339. [7] T. Wolfram and R.E. De Wames, Phys. Rev. I X5 (1969) 762. [S] G. Rado and I. Weertman, Phys. Rev. 94 (1954) 1386. [9] V.A. Ignatchenko, Fiz. Met. Metalloved. 36 (1973) 12 19. [IO] K.A. Long and A.R. Bishop, J. Phys. A 12 (1979) 1325. [I I] V.G. Mozhaev, Phys. Lett. A 139 (1989) 333. [12] B.A. Ivanov and A.L. Sukatanskii. Zh. Eksp. Teor. Fiz. 84 (1983) 370 [Sov. Phys. JETP 57 (1983) 2141.
i!tl
AA lournal of metism
B%
B
ELSEVIER
Journal
of Magnetismand
metic irlals Magnetic Materials
164 (1996) 105-l 10
Nonlinear interface spin waves in magnets A.L. Sukstanskii
*, S.V. Tarasenko, E.Yu. Melikhov Received 6 March 1996
Abstract A new type of nonlinear magnetic soliton-like excitation (nonlinear interface spin wave), localized in the vicinity of the interface between two semi-infinite ferromagnets, is investigated in the framework of the phenomenological approach. The excitations are shown to have no linear analogy. The results are generalized for the two-sublattice model of ferrimagnets. Kr,~or&t
Magnetic
soliton; Interface
1. Introduction In recent years much experimental and theoretical has been devoted to multilayered magnetic structures with properties differing considerably from the properties of the magnets of which they are composed. Among other things, layered structures exhibit specific wave branches, which are known as intrinsic (or interface) spin waves (ISW), localized in the vicinity of the interface between layers. Such excitations, as well as their special case - surface spin waves. which are widely used in technical devices - are usually considered in the linear approximation, while nonlinear dynamic excitations such as magnetic solitons are mainly studied in unbounded magnetically ordered crystals (see, e.g., Ref. [I]). Naturally, one should expect the existence work
~ Corresponding
author. Email: [email protected].
0304.8853/96/$15.00 Copyright P/I SO304-8853(96)00374-5
of nonlinear surface excitations with properties that differ from those of their linear analog, just as bulk magnetic solitons differ from linear spin waves. In Ref. [2], the existence of a new type of surface excitation, so-called surface spin waves (surface solitons), in a semi-infinite ferromagnet, was first predicted and analyzed, and the essential difference between these excitations and their linear analogs was demonstrated. Moreover, it was shown that, under certain conditions. such linear analogs do not exist at all. For these reasons, it is of considerable interest to study nonlinear excitations in various multilayered magnetic structures in which intrinsic solitons having no linear limit may also exist. In the present paper. nonlinear interface spin waves (NISW), i.e. interface solitons, are studied in the simplest two-layered structures, which consist of two semi-infinite ferromagnets with different magnetic parameters (Section 2). In Section 3 the results are generalized to the case of a structure consisting of two-sublattice ferrimagnets.
0 1996 Elsevier Science B.V. All rights reserved.
2. NISW in a ferromagnetic
structure
angles 0 + and cp+ parametrize vectors M,
Linear surface spin waves are known to be caused by long-range magnetostatic [3] or magnetoelastic [4,5] interactions. If one takes into account the short-range exchange interaction only, then such excitations may exist in models of finite magnets, including a specific surface layer in which some parameters (saturation magnetization, anisotropy or exchange constants, gyromagnetic ratio, etc.) differ from those in the volume [6,7]. If the thickness of the layer is much smaller than the characteristic length of the inhomogeneity in the magnetization distribution, the surface properties of the magnet may be considered in the framework of the dynamic Landau-Lifshitz equations by means of introducing phenomenological boundary conditions (see. for example, Refs. [4,8]). In our investigation, we use this phenomenological approach in simple models of uniaxial semi-infinite single-sublattice ferromagnets and two-sublattice ferrimagnets. Restricting ourselves to the investigation of spin waves of exchange origin only, we will not take into consideration magnetostatic and magnetoelastic interactions. We assume that the interface between magnets coincides with the plane z = 0 and that each magnet is an easy-axis ferromagnet with an anisotropy axis aligned with the normal to the interface. The coupling between magnets can be associated with different interactions: exchange, magnetostatic, magnetoelastic, etc. Here again we restrict ourselves to the local exchange interaction on the interface. Therefore, the total energy of the system can be written as the sum of the energies of the magnets, and their interaction can be taken into account through the boundary conditions on the interface [9]. The energy of each ferromagnet can be written in the form
W,=M,‘, -
p,
( VoBi)’ + sir&,( j”+dV,( %r
Icos’o
VP*)?]
-1 +
(hereafter the indices ‘+ ’ and ‘- ’ correspond to parameters of the magnets occupying the upper ( z > 0) and lower (z < 0) half-spaces, respectively). The
M==M,+(sin
0,
cos p+,sin _
the magnetization
O_ sin cp,,cos
M,j+= IG,
O,), (2)
and cr* and p f > 0 are the constants of the nonuniform exchange interaction and anisotropy, respectively. The solution of the equation of motion (LandauLifshitz equations) in each half-space will be sought in the form CJJ+= cp_= Rt-kr., 02= O_(z). The functions 0 *(z) satisfy the equation PO, .a-? ~ + w+ sin OI - sin 0 + cos 0 i = 0, I i)z’
(3)
where
and g+ is the gyromagnetic
ratio. The-ground state of the system is determined by the type of exchange interaction between the magnets. The latter can be of two different types: ferromagnetic or antiferromagnetic. Obviously, in the first case (ferromagnetic coupling) the minimum of the energy of the system corresponds to the parallel orientation of the magnetization vectors M, and M_ in the magnets, and boundary conditions at i.- + 5~ for solutions localized in the vicinity of the interface are ‘ferromagnetic’: M;, ( + x) = MC,_, M,_ ( - m> = M,,_. In the opposite case (antiferromagnetic coupling) the minimum of the energy corresponds to the antiparallel orientation of the magnetization vectors M, and M_ in the ground state, therefore M,+(+m) = M,)+, MC-(-x) = -M,... Boundary conditions on the interface ; = 0 are also defined by the character of coupling between magnets. If this coupling is strong enough, the magnetization vectors are parallel to each other on the interface for ferromagnetic coupling and antiparallel for antiferromagnetic coupling, not only in the ground state but also in excited states. In the case of ferromagnetic coupling, the boundary conditions to Eq. (3) have the form [9]: X = KM,;_ a+M,2+ ~ a,_ :=+” O+( +0) = O_( -0).
8_ -az
;= _o’
(4)
A.L. Szrkstnnskii ct al. /Jo~wnal of Mqnetisnz
Besides, for the solutions localized in the vicinity of the interface, in which we are interested, we must put e(+m> = e(-,=) = 0. In the case of antiferromagnetic coupling between magnets it is convenient to change the parametrization of the magnetization vector M_ by redefining the angles H_ and cp_: H_+r-0_, (p_+ -cp_. In terms of these new variables, Eq. (3) and the boundary conditions (Eq. (4)) are as before. That is why we shall discuss below only the case of ferromagnetic coupling, so the results for antiferromagnetic coupling can be obtained by means of trivial redesignation. Eq. (3) has been analyzed in detail (see. e.g., Refs. [ 1,101). Its localized solutions were shown to exist only for w f < 1 and have the form = 2arctan cosh[K&--Z&X+]
w,>o, A*
w+
0i(z)=2arctan
sinh[K+(Z-zi)/Xk]
where
-q)“‘,
$=(I
A,=
i
+$
a+&+
K+
=
1
X-
K-;-/X_)
’
where a = A+/A-, p = a+M~+x~/a_M~~_.r+, and the signs of the parameters Z+ and Z_ in Eq. (7) must be chosen to be identical. Hence, the magnetization distribution in NISW has only one maximum. For example, in the case z i > 0, this maximum 0,, = 0+(; = Z+ > = 2 arctan A+ is at the point z = Z, > 0, whereas the magnetization in the half-space z < 0 is described by the exponential ‘tail’ of the function 0_(z). If z + < 0, then a similar maximum is situated in the lower half-space at z = z_ < 0. The energy of NISW at R > 0 can be readily obtained by substituting the magnetization distribution (Eq. (5)) into Eq. (1):
X+
+s)
=D+w+-
U>
1,
pK+<
Kp
Or
Cl <
p> 1, f2+
R
K_
(6)
s
(8)
’
1,
pK+>
K_.
(9)
These inequalities impose certain restrictions both on the parameters of the ferromagnets for which NISW exist:
p< 1,
Or
i
*
frequency
= (P?-W+fip*fi-
(10)
l2+> R-,
and on the precession from above:
i
Ifs
where D,=~M,?+K~[LY~(P~+ a+k2)]‘/’ and s = PK+/K(both expressions for E are equal to one another). It can readily be shown that solutions (Eq. (7)) exist if
i
i
KM;_
cosh(
K+Z_
tanh
x+ =
z+ = f K+cosh-’ K+
A-
cosh( K+Z+/X,)
107
The solvability of Eq. (6) with respect to Z, is a necessary condition for the existence of NISI (Eq. (5)) under R > 0. Eq. (6) proves to have two solutions:
E=D_w_(l
It should be noted that the functional dependencies of the solutions in the upper and lower halfspaces are the same because the signs of the dimensionless frequencies w+ and o_ always coincide. The integration constants z + are defined by means of the boundary conditions (Eq. (4)). Substituting the solutions (Eq. (5)) in the case of positive frequencies R > 0 (w ~ > 0) into Eq. (41, one obtains the system of equations for Z__ A+
164 (1996) 105-110
1’ 1’ (5)
A, 19,(z)
nnd Magnetic Marerids
0,
which
is limited
(11)
108
A.L. Sukstanskii
et al. /Journal
If fi < 0 (w ~ < O), the boundary in the following equations for zt:
conditions
result
A-
A+ sinh(
of Magnetism
=
K+z+/x+)
sinh( K_:_/.X._)
’
and Mtrprtic
Materials
164 (19961 105-I
IO
any value of frequency within the interval of existence of the wave, and hence NISW have no linear analog: their localization is determined completely by nonlinearity (in analogy with surface magnetic solitons [2] and surface shear waves in the theory of elasticity [ 1 11, which also do not have linear analogs).
‘~+Mo?t K+ x+ 3. NISW in two-sublattice
=
Eq. (12) also has two solutions
(the signs of z, and Z_ must be chosen to be identical). The energy of NISW at 0 < 0 can also be represented in the form of Eq. (8) with the substitution w~+lo~I. Solutions of Eq. (13) exist under the conditions: Kp
Or
UK_
K_.
These inequalities also result in certain tions, both on the magnets parameters p> 1, i
pYLn+.
ferrimagnets
(‘2)
p< 1,
Or i p%>
fl,,
(14) restric-
(‘5)
and on the precession frequency which is now limited from below: - 0, < 0 < 0. Thus, NISW, described by Eq. (51, represents a nonlinear magnetic excitation propagating along the interface between the magnets and localized near this surface. In our model, such excitations exist in the finite interval of precession frequencies 0, - fl, < n< +n*. For ]0\+0,, the parameters I z *I + x, and NISW degenerates. Taking into account that the dimensionless frequencies w+ and w_ at R = + 0, are less than 1, one can make the important conclusion that the amplitude 0, of NISW remains finite at
In the previous sections we analyzed NISW in the model of single-sublattice ferromagnets. However, the theory developed above can easily be generalized for more complicated magnet models. As an example, we consider NISW in a two-layered system consisting of semi-infinite two-sublattice ferrimagnets with easy-axis magnetic anisotropy. As shown in Ref. [ 121, nonlinear dynamics in two-sublattice magnets with non-equivalent sublattices in the main approximation with respect to the small parameters /3/S -K 1 and w/GgM,,==KI (6 is the homogeneous exchange interaction constant. M,f M,,,= IM,,21,and M,,, are the = (Mf + Mt)/2. sublattice magnetization vectors). can be studied by means of the effective Lagrange function L = LIZ) written in terms of the unit antiferromagnetic vector 1 = L/I Ll, L = M, - M2 [ 121. In terms of the angle variables f?* and cp+, parametrizing the vectors I + in the ferrimagnets just as in Eq. (2) for the vectors M,. the Lagrange function for each uniaxial ferrimagnet under consideration can be written as:
where ui= CM,+- M, )/2M,+ . " ‘I,at-g the charact~r~~trc = &+M,+)(a$,)"' locities coinciding with the minimal phase velocities of the spin waves, and the ‘potential’ energies W,. written in terms of the angle variables 0 + and p +,
A.L. Sukstanskii
et al./Jountal
qf‘Magnetism
have the same form as in Eq. (1). The equations motion for these variables have the form:
---0,
6W, 69,
of
(17)
It is noteworthy (see details in Ref. [12]) that, in the limit u + 0, Eq. (17) describes the nonlinear dynamics in antiferromagnets, and in the limit 6 -+ x (c - m> they coincide with the Landau-Lifshitz equations for ferromagnets. Assuming as above p+=q_=Ll-krL,O,= 0 +( z), one obtains from Eq. (17) that the functions 0 k ( Z> satisfy Eq. (3) with the coefficients x k and w+ _ redefined:
and Mugmtic
Materials
164 (10%)
The boundary conditions to the equations for the functions 0 + ( Z> also depend on the character of the exchange interaction between the magnets. It is easy to show that, in the case under consideration, a rather strong ‘ferromagnetic’ coupling leads to the parallel orientation of the unit vectors Zi on the interface z = 0 both in the ground state and for dynamic excitation, whereas ‘antiferromagnetic’ coupling results in the antiparallel orientation. This is why the boundary conditions for localized solutions can be written in the form of Eq. (4). Thus, for a structure consisting of two ferrimagnets, we have the same equations and the same boundary conditions as in the previous section. Consequently, the results of the previous section (localized solutions describing NISW (Eq. (5)), expressions for the parameters z + (Eqs. (7) and (1311, and the inequalities (Eqs. (9) and (14)) defining the region of NISW existence) are relevant for the ferrmagnetic system. The energy of the NISW in a ferrimagnetic structure has a far more complicated form than that in a ferromagnetic structure: E = 2M,‘+
cY+W+L12 K,X+
[
K,C;
x(
(18)
109
105-110
K,
tanh [+I]
tanhhi
K++ tanh-’
1
+ (1 + v+)( P++ a+k2) a_w_n2
+ 24_
tanhh
KpXp
’K_
-
tanh-
’
K-C!
Localized solutions of Eq. (3), as mentioned above, exist under w + < 1. In the case of ferromagnets, this condition bounds the precession frequency R from above (0 < 0 ~ 1. In the case of ferrimagnets, together with the inequality w + < 1, there is one more condition under which localized solutions exist: (p&-t a+k’a,f12/c:)> 0. These inequalities are shown to lead to the limitation of the precession frequency not only from above but from below as well: fly)< 0 < Q’,‘), fi(‘.‘,_ * -$[-&+[[&!l
X(K_
tanht-)]
+(1
-rl_)(P-+aPk2) i
(20) at R > 0, and E = 2M,2+ K+X+
cY+W+R2 2
+ 2 MC;- K-XP++
‘y+k’)
1’2 (19)
2
c+ Note that Ry)<
I
0 and R’:_‘> 0.
1.
K++
coth-’
x (K, coth {+)I + (1 + ++)( P++ a+k2) a_w_02 K_ C?
+ 4a,(
[coth-’
K+C+
x( K_ coth t_)]
[
cothh’ K_-coth-’
+ (1 - qp)(
pp+
a-k’) I
(21)
A.L.
110
Sukstanskii
et al. / Journal
of Magrwtism
Muterids
164 (19961
10%110
References
at 0 < 0. Here sinh2,5,
‘I
’* = 2(cosh25
and Magnetic
k + A?,) ’ ‘*=
sinh2l, 2(Sinh2*iiA,4ZL)
’
and z, are defined by Eqs. (7) and (13) for the cases 0 > 0 and R < 0, respectively. In the limit c i --) m, Eqs. (20) and (2 1) reduce to Eq. (81. Naturally, the inequalities Eqs. (9) and (141, written in terms of the initial parameters of the magnets, are rather cumbersome, and therefore are not given here. We only emphasize that, in the case under consideration, NISW also exist in the finite frequency interval 10 1< fl,,,, as it does in ferromagnetic systems, 1(1,,, lying inside the interval (a(‘) RC2’) and w ~ being less than 1 at I i2 + n ,,,,:I Ckequently, in the ferrimagnetic structure, NISW have no linear analog as for the ferromagnetic structure, and the localization of NISW is entirely due to nonlinearity.
[I] A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Nonlinear Magnetization Waves. Dynamic and Topological Solitons (Naukova Dumka, Kiev. 1983) (in Russian). [2] A.L. Sukstanskii and S.V. Tarasenko, Fiz. Tverd. Tela 35 (1993) 270 [Sov. Phys. Solid State 35 (1993) 1361. [3] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19 (1961) 308: Phys. Rev. 118 (1960) 1208. [4] A.I. Akhiezer, V.G. Bar’yakhtar and S.V. Peletminskii. Spin Waves (Nauka, Moscow. 1967). [5] H.K. Tiersten, J. Appl. Phys. 36 (1965) 2250. [6] B.N. Filippov. Fiz. Tverd. Tela 9 (1967) 1339. [7] T. Wolfram and R.E. De Wames, Phys. Rev. I X5 (1969) 762. [S] G. Rado and I. Weertman, Phys. Rev. 94 (1954) 1386. [9] V.A. Ignatchenko, Fiz. Met. Metalloved. 36 (1973) 12 19. [IO] K.A. Long and A.R. Bishop, J. Phys. A 12 (1979) 1325. [I I] V.G. Mozhaev, Phys. Lett. A 139 (1989) 333. [12] B.A. Ivanov and A.L. Sukatanskii. Zh. Eksp. Teor. Fiz. 84 (1983) 370 [Sov. Phys. JETP 57 (1983) 2141.