- Email: [email protected]
Spin waves spectrum and damping in quasi-periodic multilayers
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1947-1948
~
,~
ELSEVIER
|nllrnal c[
magnetism and magnetic m~erlals
Spin waves spectrum and damping in quasi-periodic multilayers V.A. Ignatchenko *, R.S. Iskhakov, Yu.I. Mankov Kirensky Institute of Physics, Krasnoyarsk 660036, Russian Federation Abstract Spectrum and damping of spin waves are studied theoretically for the model of periodic multilayer structure with randomly modulated period. It is shown that the energy gap in the spectrum, which is characteristic for periodical structures, transforms into an inflexion of the dispersion curve with increasing random modulation. The results are compared with spin-wave resonance experimental data.
It is well known that the spectrum of waves in periodic multilayer media (one-dimensional superlattices) has the band structure which is characterized by the reciprocal superlattice vector q = 2~r/l (I is the superlatice period). Degeneration removing and formation of the gap A takes place at k = q/2. This problem was studied in many of papers (for magnetic excitations see Refs. [1-4]). However, a strict periodicity in real multilayer media can be realized only approximately. There are always random deviations from the periodicity caused by technological reasons or formed intentionally. This circumstance stimulated theoretical study of the influence of the interfaces positions randomization on the spectrum and other properties of the system. Such theory for some special models of quasi-periodic multilayers has been developed successfully recently [5,6]. The aim of this paper is to suggest another approach to this problem which is appropriate only in the case of small differences of the material parameters in the adjusting layers (small percentage modulation), but for a wide class of random functions. We shall describe the dynamics of a ferromagnet by the Landau-Lifshitz equation:
M = -g
MX
- a--M + Ox a ( a M / a x )
'
(1)
with an energy density U in the form of
la(~)Mt 2 U= -~ -~x I - ½ / 3 ( M n ) 2 - M H .
+39-12-438923;
agMV2m + [ w - too- ep( x ) ]m = O,
(2)
email:
(3)
where e = A/3gM, w o is the FMR frequency. Representing eigen values of this equation in the form of the Wigner-Brillouin expansion, averaging them, and changing summation by integration we obtain in the first approximation to
For simplicity we neglect here magnetodipole fields. In an inhomogeneous medium all parameters of this expression:
* Corresponding author. Fax: [email protected].
exchange parameter a, value of anisotropy/3, direction of anisotropy axis n and the value of the magnetization M are random or regular functions of coordinates. But to make our approach clear we begin with the model where only the value of anisotropy/3(x) is a function of coordinates, while all the other parameters are regarded as constants. Let us write this parameter in the form /3(x) =/3 + A/3 p(x) where /3 is the average value of /3(x), A/3 is the mean-square-root fluctuation of /3(x) and p ( x ) is a centralized ( ( p ) = 0) and normalized ( ( p 2 ) = 1) function of coordinates which can be periodic, stochastic or mixed. The angle brackets denote here in the general case the spatial averaging which coincides, in consequence with the ergodicity principle, with the averaging over stochastic realizations if p ( x ) is a pure random function. For circular projection of the magnetization m ~ exp(itot) we obtain from Eqs. (1) and (2) in the linear approximation the equation
tok =
e2 cJ S( k - k l ) d k 1 t O - tOkl
,
(4)
Here tok = too + ag Mk2 is the initial dispersion law, a function S(k) is the Fourier-transform of the correlation function of the inhomogeneities K(r) = ( p ( x ) p(x + r)). The function p(x) can be represented by the Fourier expansion for a periodic multilayer medium. We shall restrict our consideration to the first harmonic of this expansion. Stochastization in the interfaces positions corresponds to the random modulation of the multistructure
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01539-2
V.A. lgnatchenko et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1947-1948
1948
period l. For the harmonic function p(x) this problem is analogous to the well-known problem of oscillations with a stochastically modulated frequency [7]; hence it follows that K(r) and S(k) can be taken in the form:
K ( r ) = exp( - k c r ) cos qr,
1 S(k) = -~
(5)
1
k 2 + ( k - q)2 + k 2 + ( k + q ) 2
] ,
(6)
where k c is a correlation wave number (kc 1 is a correlation radius of inhomogeneities). Upon integrating Eq. (4) with spectral density (6) we obtain:
v- k 2 2 1 2U E 2 2(agM) s= l (~f'v --ikc) - [ k + ( - 1 ) S q ] 2 £2
=
(7) where v = ( / 2 - too)/agM, u = (vl-v - i k c ) / v/-v. For kc--* 0 this equation describes the spin-wave dispersion law for the periodic function p ( z ) = v ~ c o s qz. For k~ 4:0 the frequency becomes complex: O = to + i~ and Eq. (7) describes both the modified dispersion law to(k) and the relaxation ~(k). The system of equations for to and ~, which follows from Eq. (7), is complicated enough. Thus we shall use for its analysis in the vicinity of k = q / 2 the approximation method used in Ref. [8] for the case of stochastic coupling between two waves of different nature. Following Ref. [8] we rewrite Eq. (7) in the form: ( O - ,ok) ( O - ,ok-q - i F ) = A2/4.
(8)
Here F = 2k~(ctgM)l/2(,o- too) 1/2 is a damping parameter, A is the coupling parameter. Comparing Eqs. (8) and (7) one can see that A is a complex value. However we shall consider here A as real neglecting small values proportional to A2kc . To satisfy the conditions of the perturbation theory both A and F must be small in comparision with to but an arbitrary relation can be between them. The behavior of the solution of Eq. (8) is determined by this relation at the resonance frequency tot = `oo + agMq2/4" If Ar > F r the degeneration removing and the appearance of a gap A between different branches of the solution of Eq. (8) take place at the resonance point k = q/2: A= ~
- F~2 ,
(9)
where Ar = x/2-~ = ¢~gMa/3, Fr = agMqk~. Relaxations of both branches of the solution at this point have the same values: Ca = ~ 2 = F J 2 . If Ar < Fr the dispersion curves of coherent spin waves (,o = ,ok) and fluctuation spin waves (to--~ tok-q) cross at k = k r. In this case a characteristic modification of the coherent spin waves dispersion curve with inflexion point
at k = k r takes place. More detailed analysis of the behavior of the solutions of Eq. (8) can be found in Ref. [8]. We are coming to the model where only the value of exchange is a function of coordinates: a ( x ) = a + A a p(x), where a is an average value and A a is a meansquare-root fluctuation of the exchange parameter. For this model instead of (7) we obtain
,,-k~(1-
3' ~)
~2 2 1 =--vu~_, 2 2 s=l ( ~ ~ - i k c ) -- [ k l - ( - 1 ) S q ]
2'
(10)
where 3' = A a / a . The solutions of this equation differ in some details from the solutions of Eq. (7). However, their general properties are the same. Behavior of the solution depends now on the relation between new coupling parameter Ar = A a g M q 2 / 2 v ~ and the same parameter F~. In conclusion we compare the theoretical results with experimental data. Previously [9] both dispersion and relaxation of spin waves were studied experimentally in multilayer films of C o - P / N i - P alloys for different values of the period l = 120 and 160 A. The inflexion of the dispersion curve which is characteristic for the conditions Ar < F r was observed in the vicinity of some wave number k u. A sharp increase of the spin-waves resonance linewidth was observed at the same point too and k u was identified with the resonance wave number k r = q / 2 = ~r/l. However, closer examination of these results showed that the peculiarities were really observed at k u = kr/3. In subsequent experiments in a multilayer film with l = 500 A the resonance modifications were observed at k u = k r. By this means it was established that modifications in the form of the inflexion must arise not only at k r, corresponding to the Bragg reflection conditions, but also at the third subharmonic of this wave number. Acknowledgements: This work was supported in part by a Sloan Foundation Grant awarded by the American Physical Society.
References [1] P. Griinberg and K. Mika, Phys. Rev. B 27 (1983) 2955. [2] Yu.V. Gribkova and M.I. Kaganov, Pis'ma v JETF 47 (1988) 588. [3] B. Hillebrands, Phys. Rev. B 41 (1990) 530. [4] Yu. I. Gorobets et al., Fiz. Tverd. Tela 34 (1992) 1486. [5] M. Kohmoto, B. Sutherland and Chao Tang. Phys. Rev. B 35 (1987) 1020. [6] G.D. Pang and F. Ch. Pu, Phys. Rev. B 17 (1988) 12649. [7] S.M. Ritov, Vvedenie v Statistieh. Radiofiz. I (Nauka, Moscow, 1976). [8] V.A. Ignatchenko and L.I. Deich, Phys. Rev. B 50 (1994) 16364. [9] R.S. Iskhakov, A.S. Chekanov and L.A. Chekanova, Fizika Tverdogo Tela 32 (1990) 441.
~
,~
ELSEVIER
|nllrnal c[
magnetism and magnetic m~erlals
Spin waves spectrum and damping in quasi-periodic multilayers V.A. Ignatchenko *, R.S. Iskhakov, Yu.I. Mankov Kirensky Institute of Physics, Krasnoyarsk 660036, Russian Federation Abstract Spectrum and damping of spin waves are studied theoretically for the model of periodic multilayer structure with randomly modulated period. It is shown that the energy gap in the spectrum, which is characteristic for periodical structures, transforms into an inflexion of the dispersion curve with increasing random modulation. The results are compared with spin-wave resonance experimental data.
It is well known that the spectrum of waves in periodic multilayer media (one-dimensional superlattices) has the band structure which is characterized by the reciprocal superlattice vector q = 2~r/l (I is the superlatice period). Degeneration removing and formation of the gap A takes place at k = q/2. This problem was studied in many of papers (for magnetic excitations see Refs. [1-4]). However, a strict periodicity in real multilayer media can be realized only approximately. There are always random deviations from the periodicity caused by technological reasons or formed intentionally. This circumstance stimulated theoretical study of the influence of the interfaces positions randomization on the spectrum and other properties of the system. Such theory for some special models of quasi-periodic multilayers has been developed successfully recently [5,6]. The aim of this paper is to suggest another approach to this problem which is appropriate only in the case of small differences of the material parameters in the adjusting layers (small percentage modulation), but for a wide class of random functions. We shall describe the dynamics of a ferromagnet by the Landau-Lifshitz equation:
M = -g
MX
- a--M + Ox a ( a M / a x )
'
(1)
with an energy density U in the form of
la(~)Mt 2 U= -~ -~x I - ½ / 3 ( M n ) 2 - M H .
+39-12-438923;
agMV2m + [ w - too- ep( x ) ]m = O,
(2)
email:
(3)
where e = A/3gM, w o is the FMR frequency. Representing eigen values of this equation in the form of the Wigner-Brillouin expansion, averaging them, and changing summation by integration we obtain in the first approximation to
For simplicity we neglect here magnetodipole fields. In an inhomogeneous medium all parameters of this expression:
* Corresponding author. Fax: [email protected].
exchange parameter a, value of anisotropy/3, direction of anisotropy axis n and the value of the magnetization M are random or regular functions of coordinates. But to make our approach clear we begin with the model where only the value of anisotropy/3(x) is a function of coordinates, while all the other parameters are regarded as constants. Let us write this parameter in the form /3(x) =/3 + A/3 p(x) where /3 is the average value of /3(x), A/3 is the mean-square-root fluctuation of /3(x) and p ( x ) is a centralized ( ( p ) = 0) and normalized ( ( p 2 ) = 1) function of coordinates which can be periodic, stochastic or mixed. The angle brackets denote here in the general case the spatial averaging which coincides, in consequence with the ergodicity principle, with the averaging over stochastic realizations if p ( x ) is a pure random function. For circular projection of the magnetization m ~ exp(itot) we obtain from Eqs. (1) and (2) in the linear approximation the equation
tok =
e2 cJ S( k - k l ) d k 1 t O - tOkl
,
(4)
Here tok = too + ag Mk2 is the initial dispersion law, a function S(k) is the Fourier-transform of the correlation function of the inhomogeneities K(r) = ( p ( x ) p(x + r)). The function p(x) can be represented by the Fourier expansion for a periodic multilayer medium. We shall restrict our consideration to the first harmonic of this expansion. Stochastization in the interfaces positions corresponds to the random modulation of the multistructure
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01539-2
V.A. lgnatchenko et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1947-1948
1948
period l. For the harmonic function p(x) this problem is analogous to the well-known problem of oscillations with a stochastically modulated frequency [7]; hence it follows that K(r) and S(k) can be taken in the form:
K ( r ) = exp( - k c r ) cos qr,
1 S(k) = -~
(5)
1
k 2 + ( k - q)2 + k 2 + ( k + q ) 2
] ,
(6)
where k c is a correlation wave number (kc 1 is a correlation radius of inhomogeneities). Upon integrating Eq. (4) with spectral density (6) we obtain:
v- k 2 2 1 2U E 2 2(agM) s= l (~f'v --ikc) - [ k + ( - 1 ) S q ] 2 £2
=
(7) where v = ( / 2 - too)/agM, u = (vl-v - i k c ) / v/-v. For kc--* 0 this equation describes the spin-wave dispersion law for the periodic function p ( z ) = v ~ c o s qz. For k~ 4:0 the frequency becomes complex: O = to + i~ and Eq. (7) describes both the modified dispersion law to(k) and the relaxation ~(k). The system of equations for to and ~, which follows from Eq. (7), is complicated enough. Thus we shall use for its analysis in the vicinity of k = q / 2 the approximation method used in Ref. [8] for the case of stochastic coupling between two waves of different nature. Following Ref. [8] we rewrite Eq. (7) in the form: ( O - ,ok) ( O - ,ok-q - i F ) = A2/4.
(8)
Here F = 2k~(ctgM)l/2(,o- too) 1/2 is a damping parameter, A is the coupling parameter. Comparing Eqs. (8) and (7) one can see that A is a complex value. However we shall consider here A as real neglecting small values proportional to A2kc . To satisfy the conditions of the perturbation theory both A and F must be small in comparision with to but an arbitrary relation can be between them. The behavior of the solution of Eq. (8) is determined by this relation at the resonance frequency tot = `oo + agMq2/4" If Ar > F r the degeneration removing and the appearance of a gap A between different branches of the solution of Eq. (8) take place at the resonance point k = q/2: A= ~
- F~2 ,
(9)
where Ar = x/2-~ = ¢~gMa/3, Fr = agMqk~. Relaxations of both branches of the solution at this point have the same values: Ca = ~ 2 = F J 2 . If Ar < Fr the dispersion curves of coherent spin waves (,o = ,ok) and fluctuation spin waves (to--~ tok-q) cross at k = k r. In this case a characteristic modification of the coherent spin waves dispersion curve with inflexion point
at k = k r takes place. More detailed analysis of the behavior of the solutions of Eq. (8) can be found in Ref. [8]. We are coming to the model where only the value of exchange is a function of coordinates: a ( x ) = a + A a p(x), where a is an average value and A a is a meansquare-root fluctuation of the exchange parameter. For this model instead of (7) we obtain
,,-k~(1-
3' ~)
~2 2 1 =--vu~_, 2 2 s=l ( ~ ~ - i k c ) -- [ k l - ( - 1 ) S q ]
2'
(10)
where 3' = A a / a . The solutions of this equation differ in some details from the solutions of Eq. (7). However, their general properties are the same. Behavior of the solution depends now on the relation between new coupling parameter Ar = A a g M q 2 / 2 v ~ and the same parameter F~. In conclusion we compare the theoretical results with experimental data. Previously [9] both dispersion and relaxation of spin waves were studied experimentally in multilayer films of C o - P / N i - P alloys for different values of the period l = 120 and 160 A. The inflexion of the dispersion curve which is characteristic for the conditions Ar < F r was observed in the vicinity of some wave number k u. A sharp increase of the spin-waves resonance linewidth was observed at the same point too and k u was identified with the resonance wave number k r = q / 2 = ~r/l. However, closer examination of these results showed that the peculiarities were really observed at k u = kr/3. In subsequent experiments in a multilayer film with l = 500 A the resonance modifications were observed at k u = k r. By this means it was established that modifications in the form of the inflexion must arise not only at k r, corresponding to the Bragg reflection conditions, but also at the third subharmonic of this wave number. Acknowledgements: This work was supported in part by a Sloan Foundation Grant awarded by the American Physical Society.
References [1] P. Griinberg and K. Mika, Phys. Rev. B 27 (1983) 2955. [2] Yu.V. Gribkova and M.I. Kaganov, Pis'ma v JETF 47 (1988) 588. [3] B. Hillebrands, Phys. Rev. B 41 (1990) 530. [4] Yu. I. Gorobets et al., Fiz. Tverd. Tela 34 (1992) 1486. [5] M. Kohmoto, B. Sutherland and Chao Tang. Phys. Rev. B 35 (1987) 1020. [6] G.D. Pang and F. Ch. Pu, Phys. Rev. B 17 (1988) 12649. [7] S.M. Ritov, Vvedenie v Statistieh. Radiofiz. I (Nauka, Moscow, 1976). [8] V.A. Ignatchenko and L.I. Deich, Phys. Rev. B 50 (1994) 16364. [9] R.S. Iskhakov, A.S. Chekanov and L.A. Chekanova, Fizika Tverdogo Tela 32 (1990) 441.