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Spin wave excitations in amorphous itinerant ferromagnet
24 March 1975
PHYSICS LETTERS
Volume 5 1A, number 5
SPIN WAVE EXCITATIONS IN AMORPHOUS ITINERANT FERROMAGNET R. KISHORE Physics Department, University of Roorkee, Roorkee, India Received 5 Februari 1975 The Izuyama, Kim and Kubo theory of spin waves in crystalline itinerant ferromagnets is extended to the amorphous itinerant ferromagnetic materials. The qualitative features of this theory are similar to that of the generalized Landau-Lifshitz theory discussed by Henderson and de Graaf.
Recently the Landau-Lifshitz theory of spin waves in crystalline ferromagnets is generalized to amorphous ferromagnets [ 11. It predicts the existence of both localized and extended spin waves and thereby explains the width of spin wave resonance lines in amorphous Ni [2]. A microscopic derivation of this theory was presented by Henderson and de Graaf [3] using Heisenberg Hamiltonian which unfortunately is not appropriate for Ni because of the itinerant nature of d electrons. And therefore, it is natural to ask whether this itinerant nature of electrons leads to any effects on the spin waves. We have answered this question by developing a microscopic theory of spin waves using Hubbard Hamiltonian [4] with random distribution of intraatomic interaction. Let us consider an amorphous material described by the Hubbard Hamiltonian [4] H=CTd~gai, ijo
+ CZitIi+“ii i
3
(1)
where aia, a& are the annihilation and creation operators for an electron of spin u at the site i; nio = U~oUia; Tit is the hopping integral assumed to be translationally invariant; and the intraatomic interaction Zi varies from site to site in a random fashion. If we define the fourier transform of the operator aio as ako =-$ ‘c
lZio exp
(ik*Ri) ,
(2)
then the equation of motion for the operator +$+e+ within the random phase approximation for the Hamiltonian (1) is [5]
(3) where angular brackets denote the thermal average, N is the total number of Sites, 3, Zk, and n, are defined by Tii =Y$ F
Ek exp [ik-(Ri-Ri)]
Zi’~~Zk
eXp(ik'Ri),
no =$~(&ak,).
,
(4) (5)
(6)
Now on multiplying eq. (3) by exp (iq .Ri)and then summing over the variables k and q we get, Si’= ~ Xii(O)Si+ )
(7)
where $=a&ai,
,
(8)
and
For crystalline ferromagnets (i.e. where Zi does not depend on the site i), Xii is the translationally invariant and eq. (1) admits the extended wave solutions of the form $o:exp(iq-Rr). The energy of the spin waves is given by the Izuyama, Kim and Kubo equation [S] 1t)b(w>=0,
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Volume 5 1A, number 5
24 March 1975
PHYSICS LETTERS
where q( w ) is t h e f ourier transform of xii(w). In case of amorphous ferromagnet, because of the random variation of fi from site to site, the solutions of eq. (7) are more complicated. We solve it for a situation when the ferromagnet is excited slightly above the ground state so that the spin S’ varies slowly from one site to the next although this variation is random, In this case it is possible to expand S,? on the right hand side about the site i in Taylor’s series. After retaining the terms upto the second order, we get
volumes. It can be shown that the solution of equation (11) are the localized waves of the form $ a exp (-ik,xwhere K, = U3(0)),/2(C(O)),
=Bi(O)‘(V
‘S+)i+ Ci(O)‘(VVS+)i
(11)
where Ai
(124
= 1 - C Xii(O) ) i
Bi(a)
= C i
Xi&W) @j-Ri)
7
C~(W) = ~ C Xii(W) (R~-Ri) (Ri- Ri) . i
(12b) (124
Eq. (11) can be considered as the generalized Landau Lifshitz equation for spin waves in discrete space. It differs from the usual Landau-Lifshitz equation for crystalline ferromagnets in the sense that the first term on the right hand side is nonzero and all the coefficients A&U), B&w), and C&o) are frequency dependent. It is not possible to solve equation (11) directly because of these unknown random varying coefficients. However, as shown by Henderson and de Graaf [3] some information about its solutions can be obtained by replacing the discrete variable i by the continous variable r, dividing the volume V of the ferromagnetic sample into smaller volumes a (Q29 A3, where h is the wavelength of the spin waves to be considered), and then taking the average of both sides over each of these
294
,
and the spin wave energies are given by
w=A-&{L-4(0)), t4s X
Ai(o)Si+
l&x I) ,
X
+c(O),k;).
(13)
Here A’(0) is the first derivative of A(w) with respect to o at o = 0; suffu x denotes the x component. It is assumed for simplicity that the spin S+(r) varies only in the x direction. Eq. (13) is of the same form as that of Henderson and de Graaf [3] and therefore, the qualitative predictions of both the theories will be also of the same nature. Thus qualitatively there is no difference between localized, and itinerant electron models of amorphous ferromagnets as far as the study of spin waves is concerned. We are grateful to Prof. S.K. Joshi for his encouragements and CSIR, New Delhi for financial support.
References [l] R.G. Henderson and A.M. de Graaf, Proc. of the International Symposium on Amorphous Magnetism (Plenum New York, 1973, p. 331). [2] Y. Ajiro, K. Tamura and H. Endo, Phys. Letters 35A (1971) 275. [3] R.G. Henderson and A.M. de Graaf (Preprint). [4] J. Hubbard, Proc. Roy. Sot. A276 (1963) 238. [5] T. Izuyama, D. Kim and R. Kubo, 1. Phys. Sot. Japan 18 (1963) 1025.
PHYSICS LETTERS
Volume 5 1A, number 5
SPIN WAVE EXCITATIONS IN AMORPHOUS ITINERANT FERROMAGNET R. KISHORE Physics Department, University of Roorkee, Roorkee, India Received 5 Februari 1975 The Izuyama, Kim and Kubo theory of spin waves in crystalline itinerant ferromagnets is extended to the amorphous itinerant ferromagnetic materials. The qualitative features of this theory are similar to that of the generalized Landau-Lifshitz theory discussed by Henderson and de Graaf.
Recently the Landau-Lifshitz theory of spin waves in crystalline ferromagnets is generalized to amorphous ferromagnets [ 11. It predicts the existence of both localized and extended spin waves and thereby explains the width of spin wave resonance lines in amorphous Ni [2]. A microscopic derivation of this theory was presented by Henderson and de Graaf [3] using Heisenberg Hamiltonian which unfortunately is not appropriate for Ni because of the itinerant nature of d electrons. And therefore, it is natural to ask whether this itinerant nature of electrons leads to any effects on the spin waves. We have answered this question by developing a microscopic theory of spin waves using Hubbard Hamiltonian [4] with random distribution of intraatomic interaction. Let us consider an amorphous material described by the Hubbard Hamiltonian [4] H=CTd~gai, ijo
+ CZitIi+“ii i
3
(1)
where aia, a& are the annihilation and creation operators for an electron of spin u at the site i; nio = U~oUia; Tit is the hopping integral assumed to be translationally invariant; and the intraatomic interaction Zi varies from site to site in a random fashion. If we define the fourier transform of the operator aio as ako =-$ ‘c
lZio exp
(ik*Ri) ,
(2)
then the equation of motion for the operator +$+e+ within the random phase approximation for the Hamiltonian (1) is [5]
(3) where angular brackets denote the thermal average, N is the total number of Sites, 3, Zk, and n, are defined by Tii =Y$ F
Ek exp [ik-(Ri-Ri)]
Zi’~~Zk
eXp(ik'Ri),
no =$~(&ak,).
,
(4) (5)
(6)
Now on multiplying eq. (3) by exp (iq .Ri)and then summing over the variables k and q we get, Si’= ~ Xii(O)Si+ )
(7)
where $=a&ai,
,
(8)
and
For crystalline ferromagnets (i.e. where Zi does not depend on the site i), Xii is the translationally invariant and eq. (1) admits the extended wave solutions of the form $o:exp(iq-Rr). The energy of the spin waves is given by the Izuyama, Kim and Kubo equation [S] 1t)b(w>=0,
(10) 293
Volume 5 1A, number 5
24 March 1975
PHYSICS LETTERS
where q( w ) is t h e f ourier transform of xii(w). In case of amorphous ferromagnet, because of the random variation of fi from site to site, the solutions of eq. (7) are more complicated. We solve it for a situation when the ferromagnet is excited slightly above the ground state so that the spin S’ varies slowly from one site to the next although this variation is random, In this case it is possible to expand S,? on the right hand side about the site i in Taylor’s series. After retaining the terms upto the second order, we get
volumes. It can be shown that the solution of equation (11) are the localized waves of the form $ a exp (-ik,xwhere K, = U3(0)),/2(C(O)),
=Bi(O)‘(V
‘S+)i+ Ci(O)‘(VVS+)i
(11)
where Ai
(124
= 1 - C Xii(O) ) i
Bi(a)
= C i
Xi&W) @j-Ri)
7
C~(W) = ~ C Xii(W) (R~-Ri) (Ri- Ri) . i
(12b) (124
Eq. (11) can be considered as the generalized Landau Lifshitz equation for spin waves in discrete space. It differs from the usual Landau-Lifshitz equation for crystalline ferromagnets in the sense that the first term on the right hand side is nonzero and all the coefficients A&U), B&w), and C&o) are frequency dependent. It is not possible to solve equation (11) directly because of these unknown random varying coefficients. However, as shown by Henderson and de Graaf [3] some information about its solutions can be obtained by replacing the discrete variable i by the continous variable r, dividing the volume V of the ferromagnetic sample into smaller volumes a (Q29 A3, where h is the wavelength of the spin waves to be considered), and then taking the average of both sides over each of these
294
,
and the spin wave energies are given by
w=A-&{L-4(0)), t4s X
Ai(o)Si+
l&x I) ,
X
+c(O),k;).
(13)
Here A’(0) is the first derivative of A(w) with respect to o at o = 0; suffu x denotes the x component. It is assumed for simplicity that the spin S+(r) varies only in the x direction. Eq. (13) is of the same form as that of Henderson and de Graaf [3] and therefore, the qualitative predictions of both the theories will be also of the same nature. Thus qualitatively there is no difference between localized, and itinerant electron models of amorphous ferromagnets as far as the study of spin waves is concerned. We are grateful to Prof. S.K. Joshi for his encouragements and CSIR, New Delhi for financial support.
References [l] R.G. Henderson and A.M. de Graaf, Proc. of the International Symposium on Amorphous Magnetism (Plenum New York, 1973, p. 331). [2] Y. Ajiro, K. Tamura and H. Endo, Phys. Letters 35A (1971) 275. [3] R.G. Henderson and A.M. de Graaf (Preprint). [4] J. Hubbard, Proc. Roy. Sot. A276 (1963) 238. [5] T. Izuyama, D. Kim and R. Kubo, 1. Phys. Sot. Japan 18 (1963) 1025.