J Phyr. Chrm
Saldr.
1977. Vol 38. pp 1321-1326.
Perpmmon Press.
Pnnkd in Grta Bnkin
A SIMPLE THEORY OF SPIN-WAVE RELAXATION
IN FERROMAGNETIC METALS? L. BERGER Carnegie-Mellon University,
Physics Department,
Pittsburgh, PA 15213. U.S.A.
21 July 1976; accepted 25 March
(Received
1977)
Abstract-The anisotropic exchange interaction between localized spins and conduction electrons is described by an appropriate spin hamiltonian. This is used to calculate the lifetime of magnons for arbitrary values of h,q, where A, is the electron mean free path and q the magnon wavevector. At AA ID I. this lifetime depends on the angle between q and the saturation magnetization. The antisymmetric part of anisotropic exchange (DzialoshinskyMoriya interaction) may dominate the relaxation of spin-waves of large q. The complicated band structure of transition metals gives rise lo a magnon lifetime independent of A,. The contribution of isotropic exchange is also considered.
1. INTRODUCTION
Electron-magnon
collisions
of spin relaxation tropic
in metallic
exchange
in which
are an important
interaction
a magnon
mechanism
ferromagnets[l].
The iso-
causes “spin-flip”
is created
variables,
fermion
character
while
still retaining
Other interactions,
and an
experimentally
observable
elementary
system refer to these new “clothed”
to a sheet of the opposite
the bare ones.
papers[2]
k,i
spin. Spin angular momentum
in such a process.
The first
theoretical
which
exists
between
the spin-up
magnon
and spin down
symmetry
Fermi surfaces, and treated the h,q %=1 limit, where A, is the electron wavenumber. account.
mean
free
path,
Later contributions[3-S]
Then
Ar(kF*
-k;)
q the
and a
more
- k,)
variables
of the
variables, and not to between clothed
is
limited
only
This is the well-known
131, used extensively
by
“spin
to describe the low-
lying energy levels of magnetic salts. For magnon-electron
relevant
parameter than A.q. and they treated the A,(k;
electron
considerations.
hamiltonian”[
magnon
took the gap into
is
and
scheme. The
excitations
The form of the residual interaction
gap kF* -
on this topic ignored the momentum
such as spin-
orbit, can be included in this diagonalization
electron is scattered from one sheet of the Fermi surface is conserved
their boson and
and still labeling their states by well-
defined wave vectors.
collisions
or annihilated
electron
scattering,
the important
part of the spin hamil-
tonian is the general anisotropic
exchange:
s
I case. If interactions introduced,
which do not conserve the total spin are
“non-flip”
collisions
become
possible[69].
i labels the lattice sites, Si is the (clothed) local
The electron is scattered between two states of the same
where
spin. One such interaction
spin operator
A classical
exchange. developed
by
spin-orbit
mechanical
of the AA 4
sists in writing
Korenman
and
problem[8]
for arbitrary
similar to the magnetic dipole
In a cubic crystal,
type. This
Another
simplification
where Jr is a measure of the isotropic dipole) part. In a cubic crystal,
states have an uncertain
momentum,
i.e.
values of AA. The purpose of the present
to is:
cubic symmetry uniaxial
z.EFFEcTIvEsFlNEAMlLToNL4N It is well known [ IO-121 that the intratomic causes a strong interaction This interaction redefining(l21
spin-wave
(pseudo-
the presence of k in the wave function
and thus to allow
bands such as the 3d band. According uniaxial
almost
axis is everywhere
(I into a
the anisotropic
the periodic factor ur(x)
which is very kdependent parallel
in degenerate to eqn (2). the
to k, which
is the
simplest possibility.
and one
As in other areas of the field of magnetism, the use of a spin hamiltonian
tWork supported by the U.S. National Science
part of intratomic
for the anisotropic
of the electronic
symmetry,
in Q = ~&)e’~,
states of an itinerant
can be diagonalized the
I,
exchange to exist. This involves exchange
between the bare spin-waves
modes and the bare one-electron
and similarly
second term of eqn (2) is necessary in order to lower the
paper is to present these ideas in more detail.
by
the simplest form
but
comes from
the electronic
completely
and
is for-
interaction[6],
exchange
ferromagnet.
si is the
electron,
At first, we will assume the exchange matrix J&k) be symmetric.
simple
using the Fermi golden rule in a form suitable even when for arbitrary
a spin wave,
of an itinerant
in the form of an effective
of the pseudo-dipole
larger in magnitude.
with
a and B both represent the x, y, .z coordinates.
Prange
of a very
associated
(clothed) local spin operator
was
1 and A.q % 1 limits, which con-
the interaction
spin hamiltonian
with the
for the Ad 4 I limit
We gave a short description[9]
treatment
mally
theory
Kambersky[7].
solved the quantum Ad.
combines
Foundation.
example, 1321
avoids unessential complications.
all irrelevant
orbital
variables
For
are eliminated.
1322
L. BERCER
Also, the problem of properly combining spin-orbit and exchange interaction is already solved, and the solution already incorporated in the anisotropic exchange hamtltonian of eqn (1). 3. NON-FLIP PROCESS
Among the elements of the symmetric exchange matrix J,+ only J,, and JYz cause electron-magnon collisions where si is not flipped. Keeping these only, and transforming to the usual spin-wave creation operators b*, and electron creation operators of both spins at+* and air-*, we obtain from eqns (1) and (2): I-z=-(2s) &+k,-kb*,(a:Tak’
t a;yak-) t C.C.
(3)
where N is the total number of atoms. The exact electronic stationary states + of energy E* in an alloy are linear combinations of Bloch waves having many different k directions: t+h= V-“*
C, C&.(x)e'". t
sumed to correspond to only one well-defined wave vector q and one energy fto = E(q). Consider a process in which a magnon is created and an electron scattered without spin-flip. According to the Fermi “golden rule”, the rate of change of the magnon distribution function n(q) due to the anisotropic exchange interaction (eqn 3) is: y
=
n(s) f 1’
On the other hand, magnon normal modes are still as-
n(q)lHI$‘,
9’
$
I’f(vw - f(tw(E& -
%’ -
e-l))
(5)
where f($) is the electronic distribution function for the state 9 having energy eigenvalue Q. We combine eqns (3)-(5). For simplicity of notation, we assume that electrons of one spin only are present at the Fermi level, and we omit any electronic spin index. This restriction is not essential. Because of the random nature of the alloy, we can assume the average (CtC,_,) to vanish. Then we introduce the spectral density function de>k) =
(4)
Because of the Heisenberg uncertainty principle, the finite electronic mean free path A, implies an uncertainty Ak = l/A, in the magnitude of k (Fig. 1). Thus, for a given value of Q,, the C, are large in the finite range:
c 2 ;I <44
T
S(E- c,,)(C’,~’
and the spectral distribution functions:
f(e k) = da k)/pic, k) where we define:
Like the C,, .P(E,k) is large onlv in a narrow strip of width Ak around [k[= (2~m)“‘lh. Equation (5) gives finally:
dn(q)
-= dt
V
(n(q)+ l)-
xda
ded3k12(k)
k)f(c k)p(e -
E(q), k - q)( 1
(6)
- f(~. - E(q), k - q)) where I*(k) =
9J
12 k?kx’f k,‘) A 8N
kZ
.
(7)
Considering the scattering process in which a magnon is annihilated and an electron scattered without spin-flip, we obtain in a similar fashion:
Wq) -=dt spots b Degenerate Fig. I. Electron-magnon scattering through the non-flip process. In the limit A.q s 1,only the electrons having a k located in the . . eifectlve zone perpendtcular to q can scatter a magnon.
V
-“‘9’(2,,‘r
de d3kZ2(k)
xda Wf(ck)p(e + E(q), - f(e f E(q), k f q)).
k + q)( 1 (8)
We assume the magnons to be out of equilibrium, and
A simple theory of spin-wave relaxation in ferromagnetic metals
the electrons to be in an equilibrium distribution:
L= 7,
n(q) = ndr) + n&q) u = (E - +)/keT; fo(u) = (exp (u) + I)-‘;
r = E(q)/kBT n&)
= (exp (r)-
(9) I)-‘.
We add the contributions of eqn (6) and eqn (8) to dn(q)/dt, after substituting the eqns (9). If we make the assumption of low magnon speed
we can factorize the integrations over k and over B, and we obtain finally:
dnt
dt- - n,lrm
lhm =
$$ldr)I d’kP(Bw k)ph
k +q)!*(k)
~~r)=f+-du(fdu-I)-Wu))=r. -_
(10)
Thus the magnon relaxation time 7,,, depends on the degree of “overlap” in k-space between the spectral density functions for the initial and final states. The anisotropicexchange factor Z*(k) depends on the angle between k and the z axis (by definition parallel to S). We will replace I’(k) by its average over k directions involved in collisions. In the limit Aa 6 I, all points of the Fermi surface are involved equally (see Section 4). For a qherical Fermi surface, the average is then found to be 1 =O.l5OSJ~‘/N. When .I\& * 1, only the points located on the great circle normal to q (“effective zone”) are involved (see Section 4 and Fig. I). Then 7 depends on the angle between q and S. Equation (7) gives 7 = 0 if q@. so that magnon-electron scattering vanishes. However, if Is, n, and if the Fermi surface is spherical, we find I = 0.140s1,2/N, which differs from the AA 4 1 result by only 7%. As an approximation, we will assume this value to hold even for A& = 1. Hence, eqn (IO) becomes in this case qlS, for arbitrary A&:
1 = 0.150 7,
SL* “E(q) 7 m
d’kp(ri;
k)P(% k + qh
(11) At first, we assume a lorentzian shape for the spectral density functions &F,
W =
Ai\dvdr 4A,2( k - kd
+ I’
(12)
Then the “overlap” integral in k-space can be calculated, using polar coordinates, and eqn (11) becomes,
o.l5osJ~*
I323 (13)
This expression agrees with the results of Ref. 181. Our derivation is very simple, because of the use of a spin hamiltonian. Note that it is the electron effective mass m rather than the density of states which appears in eqn (13). It is also interesting that the temperature does not appear explicitly. In the clean limit AA * 1, eqn (13) gives[S. 91 I/T,,, x E(q)/q 01 u/q. In the dirty limit AA
In the “dirty limit” AA 4 I, the uncertainty in electron wavenumber is larger than q, so that momentum conservation plays no role in restricting possible non-flip processes. However, in the “clean limit” (or anomalous limit) AA @ 1, the combination of momentum and energy conservation allows a low-energy magnon to collide only with electrons having v(k)lq. where v(k) is the electron group velocity. For a spherical Fermi surface, this reduces to klq (Fig. I ). The larger A,q, the narrower this “effective zone” on the Fermi surface. This zone contributes dominantly to the overlap integral in eqns (10) and (11). The concept of effective zone is already familiar in the anomalous skin effect problem, and in the acoustic attenuation problem. Tbe Fermi surface of nickel has a few points where orbital degeneraciesIl41 are present. At these points, the effects of spin-orbit interaction are considerably enhanced. It is easy to show[lS] that even a few localized degeneracies can make a dominant contribution to certain macroscopic properties of the sample, if these properties depend on a high power n of the spin-orbit parameter. This has been confirmed experimentally [ 161 in the case of the magnetocrystalline anisotropy (n = 4). Since n = 2 for the constant .I,,, our eqn (13) also gives n =4 for l/7,,,. Therefore, orbital degeneracies are also likely to have a dominant effect on the value of I/r,,,, in the case of this non-flip process. This idea was first proposed by Kambersky[7]. In the clean limit (Fig. l), for certain directions of q the effective zone will pass through the degenerate spots of the Fermi surface. When this happens, I/T,,, should increase considerably. It should be possible to observe this effect by performing ferromagnetic resonance experiments at 4K in nickel single crystals of various orientations. This would give new information on the nature and location of the degeneracies. These sharp
L.
1324
BERGER
directional variations of l/7,,, are added to the smooth ones mentioned in the last section. 5. INFLUENCE OF UNIFORM BACKGROUND IN k SPACE The main defect of the calculation in Section 3 is that it considers one lorentzian spectral density function only (eqn 12), representing one sheet of the Fermi surface. Actually, in nickel and cobalt, where orbital degeneracies are present, a given sheet is approached at arbitrarily small distances by other sheets of the same spin. The simplest way to describe this “background” of sheets located at all possible distances from the first lorentzian sheet is by the addition of a constant C:
p(cF,
k) =
(14)
’
Substituting eqn (14) into eqn (1 l), we obtain now, instead of eqn (13): $=O.i5O~J~‘&!$
Atiarctan(A.q) C
I
E(q)
value of J, and JA in eqn (2), but not change the form of the equation. Finally, since Ja = Jr((g - 2)/2)’ = lo-‘Jr, we see that our spin hamiltonian is too weak to generate vertex corrections on its own. 7. NON-FLIP PROCESSESTHROUGH THE ANTlSYMMETRlC EXCHANGE
So far, we have considered only the symmetric part of the matrix .J+ It is well known[21] that an antisymmetric part can exist if no center of inversion is present between the two interacting spins. Usually, there is a sufficient lack of orbital symmetry between a localized spin Si and an itinerant electron spin si passing on the nearest-neighbor site, that this would apply here. Since this so-called Dzialoshinsky-Moriya interaction HDM is first order in the spin-orbit interaction parameters, it can be much more important than the second-order symmetric exchange. It is of the form
HDM = 2
2
,
(15) where A is another constant, independent of A,, q, o or
,
dij . (Si x si)
(16)
dii = - dii.
‘p
1.
We see that the inclusion in p(eF, k) of a constant term, representing a statistically uniform background of other sheets of the Fermi surface, leads to a new and constant term A in l/r,,, or in a. Such a constant magnon relaxation rate, independent of A,, has been observed by ferromagnetic resonance [ 171in nickel above 200 K. The arctan term dominates [ 171 below 200 K in pure nickel and cobalt, on the other hand, and is already theoretically familiar[8]. Our “uniform background” idea is somewhat similar to some ideas developed by Kambersky [ 181.
The magnitude of dii is of order [dii(= J$(g - 2)/2[= lo-‘.I;, where Ji is an interatomic isotropic-exchange integral. The simplest choice of direction is to assume dii to be parallel to the vector joining the sites i and j. We neglect all terms except those between nearest neighbors. Keeping only the matrix elements which do not flip sir and transforming to the spin-wave creation operators and electron creation operators as in Section 3, we obtain for a simple cubic lattice: ,,* N-l’*
(sin (qxa)- i sin (q#))
H,,M= (2s) ,-dz ‘ 6. EFFECT OF YERTEX CORRECTIONS
Several authors [19,20] have pointed out that intratomic exchange is too strong to be treated by finiteorder perturbation theory, since it is as large as the 3d band width. In particular, vertex corrections cannot be neglected. However, Korenman[20] has argued that these vertex corrections do not affect the relaxation rate l/r,,, as calculated by Korenman and Prange[8]. Since their expression for l/7, has the same form as our eqn (13) their mechanism is probably identical to ours; this is also suggested by the fact that their mechanism for l/7,,, involves a combination of spin-orbit and of i&atomic exchange interaction, and so does our spin hamiltonian. Therefore, we conclude that vertex corrections should not affect our l/7,,, expression either. The observation of propagating rather than diffusive behavior for spin waves in metals suggests that vertex corrections are unimportant. And so does the substantial agreement between observed and calculated Fermi surfaces. Note that our theory predicts for l/r,,, an o and q dependence which agrees with existing data. This suggests that vertex corrections would at most change the
9
4+w&$(u;*a~++ a;f*ar-) t xccs /G k’
C.C.
(17)
where d is the magnitude of dii between nearest neighbors. Though similar to eqn (3), this contains a new factor which vanishes for q+O. Hence this antisymmetric exchange is probably important for spin relaxation only for high-q magnons, such as thermal magnons at temperatures of the order of the Curie point or above, where it may lead to an increased relaxation rate. 8. SPIN-FLIP PROCESS
Finally, we consider another kind of magnon-electron collision, where the electron spin is flipped. Here, the simple isotropic exchange (first term of eqn 2) is sufficient. Nevertheless, spin-orbit interaction plays a role in the electron spin relaxation mentioned later. We arrive at an equation essentially idential to eqn (ll), containing the overlap integral
rd’b’b,
k)p-(e, k + q)
(18)
1325
A simple theory of spin-wave relaxation in ferromagnetic metals
where p+ and p- are the spectral density functions for up spin and down spin. By analogy to eqn (14), we assume:
IL/D&r
p’k~. k) = 4A.2(k - k;)2 + 1
+f (191
Ml&r +CP-(EF, k) = 4h,z(k - kF-jZ+ 1 where C’, C- represent backgrounds of other sheets, located at all possible distances from the lorentzian sheet of the other spin. Assuming I&+- k~-f%q, l/A,, the overlap integral (eqn 18) is evaluated, and this gives ;
01[A + A’IA,lE(q)
where A and A’ are certain constants, independent of A, q, o or ‘I. Then the Gilbert parameter a is independent of frequency. The term in A’, caused by the overlap of the two lorentzians, agrees with existing theoretical results[3-51 for the spin-flip process, and may have been observed[l7] by ferromagnetic resonance in iron and cobalt at high temperatures. The constant term in A is theoretically new, and also seems to be observed in iron and cobalt [ 171. Since A’ is inversely proportional to I/&- kF-}, and thus to the saturation magnetization M3, the term in A’ should dominate over other terms and over other mechanisms in materials with small M, and small A, (NiCu[9], and Ni-Fe of invar type[22]). As we have assumed the electrons to have an equilibrium distribution, the present results are valid only if the spin relaxation time rS of the conduction electrons for collisions with impurities or phonons is sufficiently short. This is realized in alloys at least, where rS = 10-‘2sec. Because of the momentum gap, the parameter A,q is irrelevant for spin-flip processes. And so is the concept of effective zone.
9. WHY IS TEJ% GILBERTPARAMETERBEG
c
Fig. 2. A magnon of energy L scatters an electron from a filled state of energy e to an empty state of energy t t ho. Only the electrons in a layer of thickness Bcubelow the Fermi level g can be scattered, at T = 0.
the magnon can collide. If the matrix element of the scattering hamiltonian is independent of q, and if momentum conservation does not further restrict possible collisions (non-flip process at A,q 6 1, or spin-flip process at &+ - k~-)lq B l), we get therefore l/r,,, 0: o. Actually, this result holds even at T#O, as eqn (13) shows.
1. Article by Turov E. A., In ~e~omagnerjc Resonance (Edited by S. V. Vonsovskii), pp. 119-125.Israel program for scientific translations, Jerusalem (1964). 2. Turov E. A., Izv. Akad. Nauk SSSR, Ser. Fiz. 19.462 (1955) [Bull. Acad.
3. 4.
OF FREQUENCY?
The Gilbert parameter a = l/207, is convenient because it is dimensionless. Physically, it is much more meaningful than the Landau-Lifshitz parameter A = ~M,(Y, which contains the magnetization in a completely arbitrary manner. We show how the result L/T~ a o (i.e. a = const) is a simple consequence of energy conservation and of the Pauli exclusion principle. For simplicity we assume the electron system to be at T=O. In order for a magnon of energy 1Roto be annihilated, it must take an electron from a filled state of energy c below the Fermi level EF (Fig. 2), and scatter it into an empty state of energy e + ho above eF. Thus only the electrons contained in a thin layer of thickness &SJ below eF can possibly collide with the magnon. The larger o, the larger the number of electrons with which
5. 6.
Sci.
U.S.S.R..
Phw.
Ser. 19. 414 11955%
kfitcbell A. H., Phys. Reu. iO$, 1439(1957). ’ -‘l’ See Ref. [II, p. 132. Heinrich B., F&ova D. and Kambersky V., Phys. Status Solidi 23, 501 (1%7); Phys. Letters 23, 26 (1966). Fraitova D. and Dvorak V., Czech. 1. Phys. B22.413 (1972). Abrabams E., Phvs. Reo. 98. 387 11955). See his Section IV.
7. KamberskyV., &n. 1. Phyi. 48, bO6i1970). 8. Korenman V. and Prange R. E., Phys. Rev. B6, 2769 (1972); Praage R. E. and Korenman V., J. Magn. Resonance 6,274
(1972). 9. Yelon W. B. and Berger L., Phys. Rev. B6, 1974,1981(1972). 10. Herring C., Phvs. Reu. 85. 1003(1952):87. 60 (1952). Il. BergerL., Phyi. Rev. 137, A220; A22j (1965).’ -’ 12. Hirst L. L., Phys. Rev. 141, 503 (1966); Prange R. E. and Korenman V., AIP Con& Pm. No. 24 (Edited by C. D. Grabam, G. H. Lander and J. J. Rhyne), p. 325. Am. Inst. Physics, New York (1975). 13. Article by Stevens K. W. H., In Magnetism (Edited by G. T. Rado and H. Suhl), Vol. 1, p. 1. Academic press, New York (1963). 14. Hodges L., Stone P. R. and Gold A. V., Phys. Rev. Letters 19, 655 (1%7). 15.
BergerL., unpublished calculations.
16. Franse J. J. M., J. Phys. (Paris) 32-C& 186 (1971).
1326
L. BERGER
17. Bhagat S. M. and Lubitz P.. Phys. Reu. BIO. 179 (1974). See their Fig. 7. 18. Kambersky V., In Proc. Intent. Conf. on Magnetism, ~/OSc‘aw: 1973. Vol. 5, p. 124. Nauka. Moscow (1974). 19. Fulde P. and Luther A.. Phys. Rev. 170, 570 (1968); Levin K.. Hertz J. and Beal-Monod M. T.. A.I.P. hoc. No. 29 (Edited
by J. 1. Becker, G. H. Lander and J. J. Rhyne), Inst. Physics, New York (1976). 20. Korenman V., Phys. Rev. B9, 3147 (1974). 21. See Ref. [13], p. 85. 22. Turov E. A., to be published.
p. 323. Am.