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Retarded modes in layered magnetic structures
0038-1098/87 $3.00 + .00 Pergamon Journals Ltd.
Solid State Communications, Vol. 61, No. 6, pp. 405-408, 1987. ?rinted in Great Britain
RETARDED MODES 1N LAYERED MAGNETIC STRUCTURES J. Barna~
Institute of Physics, Technical University, 60-965 Poznafi, Piotrowo 3, Poland
(Received 16 August 1986 by M. Cardona) The dispersion equation for retarded modes in an infinite layered structure consisting of alternating ferromagnetic and non-magnetic films is derived from the classical Maxwell theory. The equation includes some peculiar cases discussed previously. The dispersion curves of waves propagating perpendicularly to the magnetization are calculated numerically and discussed. OWING TO THE RAPID progress in evaporation techniques there has recently been considerable interest in modulated and multilayer magnetic systems. Some features of the layered structures can be explained in terms of modified single-film properties. However, the excitation spectrum of such systems exhibits properties unique to the superlattice structure. Detailed theoretical accounts [1, 2] have been given of the magnetostatic modes in layered structures in which the magnetic and non-magnetic layers alternate. The wave spectrum of such systems is greatly modified in comparison with that of single layer or bulk system. For instance, the coupling between the surface magnetostatic (Damon-Eshbach) modes via the dipolar field generated by the precessing spins gives rise to the new collective bulk-type excitations [1, 2]. In recent years there has also been much interest in the exchange modes in superlattices in which both components are magnetically ordered [3-5]. The purpose of this paper is to explore the nature of the retarded-wave spectrum of magnetic layered systems composed of alternating ferromagnetic and non-magnetic layers. The retarded modes (referred to frequently as the magnon polaritons) in a single ferromagnetic layer have been studied theoretically in the recent literature [6-8]. As indicated in the inset in Fig. 1, the supeflattice under consideration here consists of alternating ferromagnetic films of thickness d~ and non-magnetic films of thickness d2. In each magnetic film the magnetization Mo (and applied magnetic field) is parallel to the film surfaces and parallel to the z direction. The layers are supposed to have infinite extent in the z and y directions. In what follows, the considerations are restricted to the infinite layered structure, n-+-+ ~' (n being an index of the bilayer - see Fig. 1). We shall also ignore the exchange interaction. 405
3 32
6
~
Ib ,1
2.8
2£ li[~
0.8
iQ
1 2 3 t, 5 6 7 8 9 10 ~ Fig. 1. Regions of the/c-63 plane, in which modes of the types I - I V discussed in the text below may propagate: I a and I b - - Ot (real), /3 (real); II a and 1 1 6 - c~ (real), /3 (imaginary); I I I - a (imaginary),/3 (real); W a and IV ° - c~ (imaginary),/3 (imaginary). The broken lines denote the dispersion curves of bulk magnon polaritons in ferromagnetic system (Voigt configuration) and the photon line. The dotted fine is defined as: o3 = &v, where d)v = [(2o(~o + 1)] t:2. The parameters used here are: 12o = 2 and /1 = 1.75. The inset shows geometry of the superlattice. First, consider the regions occupied by the magnetic material, i.e., the regions nL < x < nL + dl, where L = d l + d2 is the length of a "unit cell", and n = 0, +- 1, -+ 2 . . . . . The dynamical components h and m of the magnetic field and magnetization, respectively, fulfil the following equation [6] :
406
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
V2h(r;t)_cl[
u~(co) u~(co) = u , ( c o ) - ul(co) ---
[ b2h(r; t) + ~2m(r; t)] ~
Vol. 61, No. 6
Ot 2
--V[V'h(r;t)] = O,
(1)
where c is the light velocity in the vacuum. In the linear approximation m and h are related through the equation: mi(r;co) = ~ [/ai~(co)--~ij]hj(r; co); i = x , y , z (2) i where 6//= 1 for i = / and 6 i i = 0 for i:~/, whereas go(co) is the frequency-dependent linear magnetic permeability tensor which for uniaxial ferromagnetic systems has the following non-vanishing components:
(8)
The parameters B~ey)(n) and Bxey)(n) in equation (6) are constants which fulfil, according to the condition V • (h + m) = O, the following relations:
B°~(n) . (co~/c ~ ) u A c o ) - ok~3 B~(n) - t k2_(co2/c2)/a±(co ) ,
o = +,--,
where o on the right-hand side is understood as: o f = f for o = + and a f = - - f f o r o = --. Insertion of the identities #±(co) = 1 and #x(co) = 0 into the above equation gives the following expressions for the magnetic field in the non-magnetic layers:
aoam
h(x, k; co) = A+(n)e a~' + A-(n)e -~:'',
6o~2m
for dl < x ' = x - - ( n - - 1 ) L < L . defined as:
#xx(W) = /ay,(w) -= #±(co) = U 14-~2~-----~--2.),(3a) Ux,(W) = --Uyx(w) =- iUx(co) = iu a g -
co2' (3b)
(9)
(10)
The parameter a is
CCO2
u~(co) = u.
(3c)
Here ~2o and ~2rn have their usual meaning [6, 7] (we use the SI units) and /a denotes the high frequency permeability resulting from other magnetic excitations (e.g. from optic magnons). Since the system under consideration is translationally invariant in the y and z directions, one may write the fields in the form h, m ~ exp [i(kyy + kzz)] where ky and k, are the components of the twodimensional wave vector kll. In what follows, we consider only the case of kz -- 0 and write ky simply as k. Thus, one may write the magnetic field in the form: h(r; t) = h(x, k; w)e i(k:~-wO,
(4)
fornL < x < n L + d~. We will consider only the case when the magnetic field of the wave is perpendicular to the axis of spontaneous magnetization. The second polarization (when the field is parallel to the magnetization) is less interesting from the physical point of view. Thus, one may assume:
h~ (x, t,; ~o) = 0.
hi(x,k; a~) = B[(n)e &~' +Bi-(n)e-~X'; i = x , y ,
/3~ = I' ~
(.132
~ - u~(co),
~
k 2 ___
c2 ,
(11)
and the constants A°(n) and A~(n) (o = + , - - ) fulfil the relations:
ok A°(n) = - - i - - A ~ ( n ) ;
o = +,--
(12)
Ot
The remaining constants A;(n) and A;(n) vanish identically, A ~ ( n ) = A ; ( n ) = 0 (according to equation (5) which holds for all x). The magnetic field h and magnetization m satisfy the boundary continuity conditions for the tangential component of h and normal component of h + m. First, consider the interface inside the nth bilayer. The boundary conditions lead then to the following relations:
[
A~(n)] = ~ IBm(n)],
Ay(n)]
(13)
tB;(n)]
where the elements of the 2 x 2 matrix R are given by the expressions:
(5)
The remaining components of the amplitude h(x, k; co) can be found from equations (1) and (4) as:
for O < x ' = x - - ( n - - 1 ) L < d l . defined here as:
~2
(6)
R,1(22) =
71[ 1 + a k~/3"~(co)• kuAco) 'J] (---~2-/c2)tt±(--------,le +-~-~)a', (14a)
1 [1 -
~ /3.1(co) -+k.x(co) ]
The parameter /3 is
(14b)
(7)
in which the upper sign refers to the indices in parentheses. The boundary conditions applied to the interface between the n-th and ( n + l)-th brayer give the following relations:
where #v(w) is the magnetic permeability of the system in the Voigt configuration:
Vol. 61, No. 6
B;(n + 1)
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
(15)
[A~,(n)]
where the matrix S is given by: 0,9 2
'~rim(co) + '~/'ux (co) +/':
c5- re(co) e +-~L ' (1 6a)
=
SH (22)
2uri/aj.(co) (.0 2
~u±(co) + c~kux(co)- I, ~ + ~
S12(21)
U~(w) e~-aL '
2 ari/a±(co)
=
(16b) Equations (13) and (15) combine to give: (17) B~,(n + 1)
[B:j(n)]
where the "transfer matrix" Tis defined as: Y = gift.
(18)
The matrix T has the following important property: det T = 1.
(19)
Physical solutions have to fulfil the Bloch's theorem:
B~(n + 1)
(20)
[Bz~(n)j
where - 7r/L < Q <~ rr/L • Q is the component of the wave vector, which describes propagation perpendicular to the film interfaces. It results from equations (17) and (20) that the factor exp (iQL) in equation (20)is -+ an eigenvalue of the matrix T, i.e.:
tB;(n)J
_ e, L [B;(n) l [B;(n)J"
(21)
Taking into account the relation (19) one can write equation (21) in the form: cos (QL) : ½Tr(T),
(22)
where Tr(T) denotes the trace of matrix T; T r ( T ) = Txl + T:2. Finally, upon substitution of the explicit forms of T11 and T22 one gets the following equation: cos (QL) = cosh (rid1) cosh (ad2)
Equation (23) is just the gener-,d dispersion equation for waves propagating perpendicularly to the magnetization. The in-plane propagation (but perpendicular to the magnetization) is specified by k, whereas the propagation perpendicular to the interfaces is described by Q. It is worth to note that equation (23) is reduced to known results, if it is applied to some special cases. If, for instance, d2 = 0, equation (23) gives the known dispersion equation for bulk magnon polaritons in the Voigt geometry. When d2 ~ oo equation (23) reduces to the dispersion relation for waves propagating in the ferromagnetic film [6, 7] and when dl ~ oo and d2 ~ oo one gets from (23) the dispersion relation for waves propagating in the semiinfinite ferromagnetic system [9]. In the magnetostatic limit, c ~ , equation (23) gives the results derived in [1]. Consider now some general properties of the retarded waves in the systems under consideration. Let coo(k) be certain solution of equation (23). Spatial behaviour of the corresponding magnetic field (and also of the magnetization) is characterized by the parameters Oto(k) = a[k, Coo(k)] and/~o(k) = ri[k, wo(k)] (the k and co dependence of a and 13is written here explicitly). In the general case one can distinguish the following four situations: (i) Oto(k) and rio(k) are both real (and positive) for a wave number k. The wave is then exponentially localized at the interfaces, ile., the wave amplitude varies exponentially when one moves from any interface into the magnetic or non-magnetic layer. (ii) The parameter eto(k) is real (and positive) and rio(k) is imaginary. The solution coo(k) corresponds then to the wave which is exponentially localized at surfaces in the non-magnetic f'11ms and is oscillatory (extended) in the magnetic layers. (iii) C~o(k) is imaginary and rio(k) is real (positive). The corresponding wave is exponentially localized in the magnetic films and extended in the non-magnetic ones. (iv)cto(k) and rio(k) are both imaginary. The solution coo(k) corresponds to the wave extended in both magnetic and non-magnetic layers. The k-co plane can be then divided into regions where the modes of particular type can propagate. This is shown in Fig. 1. For further numerical purposes we have introduced there the following dimensionless parameters: ~o
+
2ariu±(co) sinh (rid I ) sinh (ad2).
=
-~2o -;
~2m
6O
¢h-
~2m '
23 (23)
407
ck ~2rn
k=--;
d-
~2rad c
408
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
Vol. 61, No. 6
"t
4.8
265
/+/.,
26°I
40
2.55!
~
~
36 250
32 28 24
2./.+5
...................................................... i
..........
2./40
i
0
08[ ,#/ o
.
.
.
.
.
Fig. 2. Dispersion curves of waves propagating parallel to the interfaces for ~ o = 2,/a = 1.75 and d l = d2 = 1. The broken and dotted lines have the same meaning as in Fig. 2.
&8 i I
/ /
L~.O
,'
36 32
0.8 O.Z, 0
8
I0
12
14
16
18
20
o f waves propagating parallel to the interfaces (Q = 0) and perpendicularly to the magnetization axis are shown in Fig. 2. The full spectrum of waves which can propagate perpendicularly to the magnetization, 0~< ]QLI % ~, is presented in Fig. 3. It is worthwhile noting that the modes of the type II (the regions II a and IIb in Fig. 1) result from the guided modes of separate magnetic layers [7]. Similarly, the modes which exist in the region I bresult from the interaction between the surface waves of separate magnetic layers. The spectrum of these waves is shown in Fig. 3 and also in Fig. 4. It is worth to note that the corresponding dispersion curves start at ke > O, whereas in the magnetostatic limit they start at k = 0 [1,21. The surface waves which exist in semi-infinite layered systems are not considered here and will be discussed elsewhere.
s
20 12
6
Acknowledgement - This work was carried out under the Research Project CPBP O1.06.
!
28L,., i. /
/
1.6
. . . . . . . . . . . . . . . . . . . . .
Fig. 4. Spectrum of waves which are localized at the interfaces. For ~2o = 2, t~ = 1.75 and d l = d2 = 0.1.
1.2~ //,,"
°.1/,
i
2 k~4
,/
REFERENCES 1.
/
2.
Ii¢,,' / /
3. 4.
1 2 3 4 5 6 7 8 9 I0 ~
Fig. 3. Spectrum of waves propagating perpendicularly to the magnetization axis for the same parameters as in Fig. 3. Quite similar definitions can be used for Q and L (QL = QL). Some numerical examples of the dispersion curves
5. 6. 7. 8. 9.
R.E. Camley, T.S. Rahman & D.L. Mills, Phys. Rev. B27,261 (1983). P. Grunberg & K. Mika, Phys. Rev. B27, 2955 (1983); P. Grunberg, J. Appl. Phys. 57, 3673 (1985). E.L. Albuquerque, P. Fulco, E.F. Sarmento & D.R. Tilley, Solid State Commun. 58, 41 (1986). L. Dobrzynski, B. Djafari-Rouhani & H. Puszkarski, Phys. Rev. B33, 3251 (1986). L.L. Hinchey & D.L. Mills, preprint. A.D. Karsono & D.R. Tilley, J. Phys. C l l , 3487 (1978). M. Marchand & A. Caille, Solid State Commun. 34, 827 (1980). N.P. Lima & F.A. Oliveira, Solid State Commun. 47,921 (1983). A. Hartstein, E. Burstein, A.A. Maradudin, R. Brewer & R.F. Wallis, J. Phys. C6, 1266 (1973).
Solid State Communications, Vol. 61, No. 6, pp. 405-408, 1987. ?rinted in Great Britain
RETARDED MODES 1N LAYERED MAGNETIC STRUCTURES J. Barna~
Institute of Physics, Technical University, 60-965 Poznafi, Piotrowo 3, Poland
(Received 16 August 1986 by M. Cardona) The dispersion equation for retarded modes in an infinite layered structure consisting of alternating ferromagnetic and non-magnetic films is derived from the classical Maxwell theory. The equation includes some peculiar cases discussed previously. The dispersion curves of waves propagating perpendicularly to the magnetization are calculated numerically and discussed. OWING TO THE RAPID progress in evaporation techniques there has recently been considerable interest in modulated and multilayer magnetic systems. Some features of the layered structures can be explained in terms of modified single-film properties. However, the excitation spectrum of such systems exhibits properties unique to the superlattice structure. Detailed theoretical accounts [1, 2] have been given of the magnetostatic modes in layered structures in which the magnetic and non-magnetic layers alternate. The wave spectrum of such systems is greatly modified in comparison with that of single layer or bulk system. For instance, the coupling between the surface magnetostatic (Damon-Eshbach) modes via the dipolar field generated by the precessing spins gives rise to the new collective bulk-type excitations [1, 2]. In recent years there has also been much interest in the exchange modes in superlattices in which both components are magnetically ordered [3-5]. The purpose of this paper is to explore the nature of the retarded-wave spectrum of magnetic layered systems composed of alternating ferromagnetic and non-magnetic layers. The retarded modes (referred to frequently as the magnon polaritons) in a single ferromagnetic layer have been studied theoretically in the recent literature [6-8]. As indicated in the inset in Fig. 1, the supeflattice under consideration here consists of alternating ferromagnetic films of thickness d~ and non-magnetic films of thickness d2. In each magnetic film the magnetization Mo (and applied magnetic field) is parallel to the film surfaces and parallel to the z direction. The layers are supposed to have infinite extent in the z and y directions. In what follows, the considerations are restricted to the infinite layered structure, n-+-+ ~' (n being an index of the bilayer - see Fig. 1). We shall also ignore the exchange interaction. 405
3 32
6
~
Ib ,1
2.8
2£ li[~
0.8
iQ
1 2 3 t, 5 6 7 8 9 10 ~ Fig. 1. Regions of the/c-63 plane, in which modes of the types I - I V discussed in the text below may propagate: I a and I b - - Ot (real), /3 (real); II a and 1 1 6 - c~ (real), /3 (imaginary); I I I - a (imaginary),/3 (real); W a and IV ° - c~ (imaginary),/3 (imaginary). The broken lines denote the dispersion curves of bulk magnon polaritons in ferromagnetic system (Voigt configuration) and the photon line. The dotted fine is defined as: o3 = &v, where d)v = [(2o(~o + 1)] t:2. The parameters used here are: 12o = 2 and /1 = 1.75. The inset shows geometry of the superlattice. First, consider the regions occupied by the magnetic material, i.e., the regions nL < x < nL + dl, where L = d l + d2 is the length of a "unit cell", and n = 0, +- 1, -+ 2 . . . . . The dynamical components h and m of the magnetic field and magnetization, respectively, fulfil the following equation [6] :
406
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
V2h(r;t)_cl[
u~(co) u~(co) = u , ( c o ) - ul(co) ---
[ b2h(r; t) + ~2m(r; t)] ~
Vol. 61, No. 6
Ot 2
--V[V'h(r;t)] = O,
(1)
where c is the light velocity in the vacuum. In the linear approximation m and h are related through the equation: mi(r;co) = ~ [/ai~(co)--~ij]hj(r; co); i = x , y , z (2) i where 6//= 1 for i = / and 6 i i = 0 for i:~/, whereas go(co) is the frequency-dependent linear magnetic permeability tensor which for uniaxial ferromagnetic systems has the following non-vanishing components:
(8)
The parameters B~ey)(n) and Bxey)(n) in equation (6) are constants which fulfil, according to the condition V • (h + m) = O, the following relations:
B°~(n) . (co~/c ~ ) u A c o ) - ok~3 B~(n) - t k2_(co2/c2)/a±(co ) ,
o = +,--,
where o on the right-hand side is understood as: o f = f for o = + and a f = - - f f o r o = --. Insertion of the identities #±(co) = 1 and #x(co) = 0 into the above equation gives the following expressions for the magnetic field in the non-magnetic layers:
aoam
h(x, k; co) = A+(n)e a~' + A-(n)e -~:'',
6o~2m
for dl < x ' = x - - ( n - - 1 ) L < L . defined as:
#xx(W) = /ay,(w) -= #±(co) = U 14-~2~-----~--2.),(3a) Ux,(W) = --Uyx(w) =- iUx(co) = iu a g -
co2' (3b)
(9)
(10)
The parameter a is
CCO2
u~(co) = u.
(3c)
Here ~2o and ~2rn have their usual meaning [6, 7] (we use the SI units) and /a denotes the high frequency permeability resulting from other magnetic excitations (e.g. from optic magnons). Since the system under consideration is translationally invariant in the y and z directions, one may write the fields in the form h, m ~ exp [i(kyy + kzz)] where ky and k, are the components of the twodimensional wave vector kll. In what follows, we consider only the case of kz -- 0 and write ky simply as k. Thus, one may write the magnetic field in the form: h(r; t) = h(x, k; w)e i(k:~-wO,
(4)
fornL < x < n L + d~. We will consider only the case when the magnetic field of the wave is perpendicular to the axis of spontaneous magnetization. The second polarization (when the field is parallel to the magnetization) is less interesting from the physical point of view. Thus, one may assume:
h~ (x, t,; ~o) = 0.
hi(x,k; a~) = B[(n)e &~' +Bi-(n)e-~X'; i = x , y ,
/3~ = I' ~
(.132
~ - u~(co),
~
k 2 ___
c2 ,
(11)
and the constants A°(n) and A~(n) (o = + , - - ) fulfil the relations:
ok A°(n) = - - i - - A ~ ( n ) ;
o = +,--
(12)
Ot
The remaining constants A;(n) and A;(n) vanish identically, A ~ ( n ) = A ; ( n ) = 0 (according to equation (5) which holds for all x). The magnetic field h and magnetization m satisfy the boundary continuity conditions for the tangential component of h and normal component of h + m. First, consider the interface inside the nth bilayer. The boundary conditions lead then to the following relations:
[
A~(n)] = ~ IBm(n)],
Ay(n)]
(13)
tB;(n)]
where the elements of the 2 x 2 matrix R are given by the expressions:
(5)
The remaining components of the amplitude h(x, k; co) can be found from equations (1) and (4) as:
for O < x ' = x - - ( n - - 1 ) L < d l . defined here as:
~2
(6)
R,1(22) =
71[ 1 + a k~/3"~(co)• kuAco) 'J] (---~2-/c2)tt±(--------,le +-~-~)a', (14a)
1 [1 -
~ /3.1(co) -+k.x(co) ]
The parameter /3 is
(14b)
(7)
in which the upper sign refers to the indices in parentheses. The boundary conditions applied to the interface between the n-th and ( n + l)-th brayer give the following relations:
where #v(w) is the magnetic permeability of the system in the Voigt configuration:
Vol. 61, No. 6
B;(n + 1)
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
(15)
[A~,(n)]
where the matrix S is given by: 0,9 2
'~rim(co) + '~/'ux (co) +/':
c5- re(co) e +-~L ' (1 6a)
=
SH (22)
2uri/aj.(co) (.0 2
~u±(co) + c~kux(co)- I, ~ + ~
S12(21)
U~(w) e~-aL '
2 ari/a±(co)
=
(16b) Equations (13) and (15) combine to give: (17) B~,(n + 1)
[B:j(n)]
where the "transfer matrix" Tis defined as: Y = gift.
(18)
The matrix T has the following important property: det T = 1.
(19)
Physical solutions have to fulfil the Bloch's theorem:
B~(n + 1)
(20)
[Bz~(n)j
where - 7r/L < Q <~ rr/L • Q is the component of the wave vector, which describes propagation perpendicular to the film interfaces. It results from equations (17) and (20) that the factor exp (iQL) in equation (20)is -+ an eigenvalue of the matrix T, i.e.:
tB;(n)J
_ e, L [B;(n) l [B;(n)J"
(21)
Taking into account the relation (19) one can write equation (21) in the form: cos (QL) : ½Tr(T),
(22)
where Tr(T) denotes the trace of matrix T; T r ( T ) = Txl + T:2. Finally, upon substitution of the explicit forms of T11 and T22 one gets the following equation: cos (QL) = cosh (rid1) cosh (ad2)
Equation (23) is just the gener-,d dispersion equation for waves propagating perpendicularly to the magnetization. The in-plane propagation (but perpendicular to the magnetization) is specified by k, whereas the propagation perpendicular to the interfaces is described by Q. It is worth to note that equation (23) is reduced to known results, if it is applied to some special cases. If, for instance, d2 = 0, equation (23) gives the known dispersion equation for bulk magnon polaritons in the Voigt geometry. When d2 ~ oo equation (23) reduces to the dispersion relation for waves propagating in the ferromagnetic film [6, 7] and when dl ~ oo and d2 ~ oo one gets from (23) the dispersion relation for waves propagating in the semiinfinite ferromagnetic system [9]. In the magnetostatic limit, c ~ , equation (23) gives the results derived in [1]. Consider now some general properties of the retarded waves in the systems under consideration. Let coo(k) be certain solution of equation (23). Spatial behaviour of the corresponding magnetic field (and also of the magnetization) is characterized by the parameters Oto(k) = a[k, Coo(k)] and/~o(k) = ri[k, wo(k)] (the k and co dependence of a and 13is written here explicitly). In the general case one can distinguish the following four situations: (i) Oto(k) and rio(k) are both real (and positive) for a wave number k. The wave is then exponentially localized at the interfaces, ile., the wave amplitude varies exponentially when one moves from any interface into the magnetic or non-magnetic layer. (ii) The parameter eto(k) is real (and positive) and rio(k) is imaginary. The solution coo(k) corresponds then to the wave which is exponentially localized at surfaces in the non-magnetic f'11ms and is oscillatory (extended) in the magnetic layers. (iii) C~o(k) is imaginary and rio(k) is real (positive). The corresponding wave is exponentially localized in the magnetic films and extended in the non-magnetic ones. (iv)cto(k) and rio(k) are both imaginary. The solution coo(k) corresponds to the wave extended in both magnetic and non-magnetic layers. The k-co plane can be then divided into regions where the modes of particular type can propagate. This is shown in Fig. 1. For further numerical purposes we have introduced there the following dimensionless parameters: ~o
+
2ariu±(co) sinh (rid I ) sinh (ad2).
=
-~2o -;
~2m
6O
¢h-
~2m '
23 (23)
407
ck ~2rn
k=--;
d-
~2rad c
408
RETARDED MODES IN LAYERED MAGNETIC STRUCTURES
Vol. 61, No. 6
"t
4.8
265
/+/.,
26°I
40
2.55!
~
~
36 250
32 28 24
2./.+5
...................................................... i
..........
2./40
i
0
08[ ,#/ o
.
.
.
.
.
Fig. 2. Dispersion curves of waves propagating parallel to the interfaces for ~ o = 2,/a = 1.75 and d l = d2 = 1. The broken and dotted lines have the same meaning as in Fig. 2.
&8 i I
/ /
L~.O
,'
36 32
0.8 O.Z, 0
8
I0
12
14
16
18
20
o f waves propagating parallel to the interfaces (Q = 0) and perpendicularly to the magnetization axis are shown in Fig. 2. The full spectrum of waves which can propagate perpendicularly to the magnetization, 0~< ]QLI % ~, is presented in Fig. 3. It is worthwhile noting that the modes of the type II (the regions II a and IIb in Fig. 1) result from the guided modes of separate magnetic layers [7]. Similarly, the modes which exist in the region I bresult from the interaction between the surface waves of separate magnetic layers. The spectrum of these waves is shown in Fig. 3 and also in Fig. 4. It is worth to note that the corresponding dispersion curves start at ke > O, whereas in the magnetostatic limit they start at k = 0 [1,21. The surface waves which exist in semi-infinite layered systems are not considered here and will be discussed elsewhere.
s
20 12
6
Acknowledgement - This work was carried out under the Research Project CPBP O1.06.
!
28L,., i. /
/
1.6
. . . . . . . . . . . . . . . . . . . . .
Fig. 4. Spectrum of waves which are localized at the interfaces. For ~2o = 2, t~ = 1.75 and d l = d2 = 0.1.
1.2~ //,,"
°.1/,
i
2 k~4
,/
REFERENCES 1.
/
2.
Ii¢,,' / /
3. 4.
1 2 3 4 5 6 7 8 9 I0 ~
Fig. 3. Spectrum of waves propagating perpendicularly to the magnetization axis for the same parameters as in Fig. 3. Quite similar definitions can be used for Q and L (QL = QL). Some numerical examples of the dispersion curves
5. 6. 7. 8. 9.
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