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Spin fluctuation damping of magnons in itinerant electron magnets
Physics Letters A 177 (1993) 362-366 North-Holland
PHYSICS
LETTERS
A
Spin fluctuation damping of magnons in itinerant electron magnets AZ. Solontsov
and A.N. Vasil’ev
Theoretical Laboratory, A.A. Bochvar Institute for Inorganic Materials, 123060 Moscow, Russian Federation
Received 15 March 1993; accepted for publication 25 April 1993 Communicated by V.M. Agranovich
The lifetime of magnons in itinerant magnets due to three-mode scattering processes by overdamped spin fluctuations, or paramagnons, is analysed on the basis of the nonlinear time-dependent Ginzburg-Landau equations for the magnetic order parameter.
The magnon damping in itinerant magnets qualitatively differs from that in Heisenberg systems due to effects of the conducting electrons, the role of which is twofold. First, they directly affect the magnon damping via the electron-magnon scattering processes. Among them one-magnon scattering by spin-flip electron excitations resulting in the Landau damping of spin waves in the Stoner continuum [ 1 ] and two-magnon processes yielding the magnon relaxation outside the Stoner continuum [ 2,3] are now well understood. Besides that, itinerant electrons give rise to overdamped collective modes - spin fluctuations (SF), or paramagnons, playing a predominant role in the thermodynamics of weak itinerant magnets close to a magnetic instability [ 451. One may expect that SF would also strongly affect the damping of magnons in these systems. The fluctuation mechanism of the magnon relaxation is supported by the recent neutron scattering experiments in the Invar alloys [ 6 1, where an anomalously high magnon damping was reported [ 7-9 1. In this paper we analyse three-mode processes of magnon damping by SF which are the lowest order modes contributing to the magnon damping. Here we neglect the effects of electron-magnon interactions affecting higher order processes and discuss the kinetics of magnons and overdamped SF solely in terms of the collective variables describing the fluctuating magnetic order parameter. Assuming the major role of low frequency long wavelength fluctua362
tions we also neglect the time and spatial dispersion of the SF interactions and use the Ginzburg-Landau model [ lo] to account for the mode-mode coupling effects in magnetic dynamics. It was recently shown by Lonzarich and Taillefer that the quantum Ginzburg- Landau model yields a surprisingly accurate quantitative account of SF effects in the thermodynamics of weak ferromagnets [ 5,111 and we expect it to provide a useful basis for the description of SF kinetics as well. We start with the conventional Ginzburg-Landau effective Hamiltonian for an isotropic ferromagnet (cf. ref. [lo]), &r=t x
~Xo1(WW)12+$~ k kl+kz+kj+b=O
[M(k, )-M(k*)
c
IT
I [M~,)~W~.4) (1)
where M(k) is the Fourier transform of a slowly varying magnetic order parameter, y is the mode coupling constant and x0(k) is the static inhomogeneous susceptibility in the absence of coupling. To describe the relaxational magnetic dynamics we use the time dependent Ginzburg-Landau equations (cf. ref. [ 12 ] ) written in the spin-invariant form, 1 -T(k)
dM(k) at
=-
W,, 6M(k)’
where T(k) characterizes the relaxation we adopt a long wavelength approximation
rate. Here which for
Elsevier Science Publishers B.V.
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177, number
PHYSICS
4,5
itinerant ferromagnets in the collisionless yields the expansions (see e.g. ref. [ 111) Xg’(k)=Xg’
+cP+...
)
T(k)=Tlkl
-t... .
Introducing
SF amplitudes,
m(k) =
dt exp(iwt)
regime
(3)
[M(k)
-M]
,
we write (2) in the explicit form
LETTERS
21 June 1993
A
gions they may be microscopically derived for weak itinerant magnets starting from the nonlinear equations of motion for the density matrix of an electron Fermi liquid formulated in refs. [ 3,5] retaining only the leading contributions to the low frequency and long wavelength expansions for the mode-mode coupling. This procedure will be described elsewhere. Confining ourselves to the analysis of the magnon relaxation we first derive the kinetic equations for the correlation functions, (d)k
=gexp[i(wr-k’*r)](m,(k+$k’)m,(k-fk’))
-y
c
m,(h)
=2 ImxAk)
[%(~2)?2(~3)
kl+kz+k3=k
(4)
-y
c
kl+kz+k,=k
+m(k,)m:(
m,(h)
-k3)
[w?(k2)m?(k3)
1>
(5)
where
k= (k, 0) > x~‘(k)=xo’(k)+3yMZ-io/T(k) and x;‘(k)= xg ’ (k) + yM2 - io/T( k) are the inverse longitudinal and transverse dynamical susceptibilities without taking account of SF, M is the spontaneous magnetization. The phenomenological equations (4) and ( 5 ) describe the dynamics of relaxational fluctuations known in weak itinerant magnets as paramagnons arising in the Stoner continuum. For the transverse excitations in the low frequency limit it begins at wavevectors k above the maximum magnon wavevector b, whereas the region Ik 1
lNkcr,
t)+tl
,
(6)
or, equivalently, for the occupation numbers Nk of SF, which are slowly time and space dependent. Here ( > designates the statistical average and at equilibrium yields the fluctuation dissipation equality with the Bose factor Nk= [exp(w/T)l]-‘No (below we use the frequency o and temperature T in energy units). In the lowest order approximation retaining only terms bilinear in the correlation functions (6) we obtain from eqs. (4) and (5) the following dynamical equation for the transverse correlation function,
[X;‘(k)+ji(%Y$$ _ %-$&._C)](m:)k =-I’5
~(m:)k,(mp2)k--k,[1-4YM2X,(k')l
+2(m:)k(m:)k’[1-yM2xp(k-k’)l -4rM’X:(k)(m:)k,(m~)k--k.}.
(7)
The real part of the r.h.s. of eq. (7) in the low frequency and long wavelength limit yields the SF corrections to the inverse susceptibility being replaced by X;‘(O)-*X;‘(O)+y(26m:+36m,2)~x;‘,
(8)
where 6m& = & (6m$)k are the average squared amplitudes. An analogous correction may be obtained for the inverse longitudinal susceptibility, x[‘(0)-+x<‘(O)+y+M
(26mz+36mp2)rx,’ (9) 363
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PHYSICS
starting from the dynamical equation for the correlation function ( mi)k. Assuming that x, and xa equal the corresponding thermodynamical susceptibilities one arrives at the magnetic equation of state previously obtained by other means [ 5,11,14]. It should be mentioned that eqs. (8) and (9) for the magnetic susceptibilities result from formulae (4)-( 7) in the lowest order approximation in the spin anharmonicity parameter [ 15 1,
g-~x,(k)(mt,)k
(v,f=t,P),
LETTERS
for
A
21 June 1993
magnons
o=o,(k)
with
wavevector
k and
frequency
=2pBA4ck2. Here
T-‘(k)=r,‘,+,(k)+T,~,+,(k)+T,:,,,(k) (14) is the inverse lifetime of magnons, written of the following contributions. The term
7,‘rn+,
(k)=(bM2C
k’
(10)
as a sum
IW,,p121mxa(k-k’)
x (No,, -Nw,-,L,=tiwj and thus hold only for the weakly anharmonic magnets. Taking account of equalities (6) the imaginary part of the r.h.s. of eq. (7) may be written in the form of the collision integral 4 ,,+z=,
I W&k
xImxP(k”)
[Nk(Nk. + l)(N,.,
-(Nk+l)Nk.Nk..]
xImxn(k-k’)
+ 1) (11)
describing three-mode scattering processes with emission or absorption of a longitudinal SF. Here I+‘,& is the matrix element of the scattering, in the here adopted approximation W,,, = 2 yM, being wavevector and frequency independent. For hypothetical undamped modes, when Irn~~,~ may be approximated by &functions, formula ( 11) reduces to the conventional Boltzmann form for the three-wave collision integral similar e.g. to the well-known theory in nonlinear plasma physics [ 161. Expression (11) generalizes the latter, taking account of the scattering processes involving strongly damped modes. Here we use eqs. (7) and ( 11) to describe the damping of magnons due to scattering by SF. Linearizing eq. (11) with respect to Nk and taking account of the expression x,(k) =
2pBM w,,(k) --w
(12)
for the transverse susceptibility in the spin wave region, we arrive at the kinetic equation
= -
364
y&(Nk-No,,,~w),
is due to the emission (for CO’> CO)or absorption o’ CO) of a longitudinal SF, rIGJ+t+p(k)=
k’, k” ) I2 Im x,(k) Im x,(k’ )
(13)
(15)
$MF
7 dw’
0
(for
I W,,, I2 Imxt(k’)
(N,, -NW._,)
(16)
describes a transformation of a magnon into a transverse SF with emission (or absorption) of a longitudinal SF and r&-P
(k)=
xImxdk+k’)
;PBMF
7
0
do’IWtp121mxt(k’)
(N,, -NW,+,)
(17)
accounts for annihilation of a magnon and transverse SF with emission of a longitudinal SF. The scattering processes ( 15 )- ( 17 ) exhaust the possible three-mode relaxation channels persisting in the ordered state of an isotropic itinerant ferromagnet. Taking account of the explicit expressions ~;‘(k)=~;‘+ck2-i~/~~k~, xp’(k)=xp’
+ck2-ico/r(kj
(18)
for the dynamical magnetic susceptibilities, following from eqs. (3 ), (8 ) and (9) in the paramagnon region, we can easily evaluate the integrals in the contributions ( 15 )-( 17 ) to the magnon lifetime. Estimating o,,, - okaX _ 2,uu,MIx,, where CO,.,,,, = Tck;, to be of the same order of w,(ko)> &Lx = magnitude, for the low frequency magnons o,(k) -=xCO,,, we obtain rIn-~,+a(k)=ro’(k)Z~(o,(k),
T) >
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PHYSICS LETTERS A
4,5
The magnon lifetime defined by eqs. ( 14), ( 19)(25 ) should be compared with the magnon relaxation time r+, arising via the electron-magnon scattering which for an isotropic Fermi surface yields [ 31
T rA+dk)=7cT1(k)
Tm:,-~(k)=~R’(k)~r,zm(wm(k),
T)
3
(19) where 1 To’(k)=
Im(w,
PB
T-x w,(k),
ok(k) (20)
~MF>
n=&
X-
T-=z co,
>
w cc Tc
w,,
T w,(k)
T X w,(k)
,
T In w,(k) WITlaX In w,(k) ’
w,(k)
w,,,
-=z Tea,,,,
-=+< T,
(26) T wnax = o In 0,
CO,,, << T,
(21)
TKO,
Z:(wT)=$, =-
27~ T2 , 3 00
w
(22)
2
Z;(w,T)=;
0 =-
5
,
2rc T2 , 3 0w
Tew, w << T<< w,_,
(23)
and Z:(w,
& =G,, T)=2jwz 1
T)=Z;(w,
(24)
in the high temperature limit T>> Tckl. Here ?j, and & are dimensionless constants of order unity approximately given by Wlco
1 r1(4+5V) z---1 3 2 1+?j k/k0
+ 5 q3j2 tan-‘(l/Jq)
,
dx
1 =ln
x(l+?jx~)(1+2~x2)
(l+?J2 v(l+rl)
’
with rl=xgks, k, is a cutoff wavevector be much larger than k,.
(25) assumed
to
where pF is the Fermi momentum. It follows from eqs. ( 19)-(26) that in the relatively wide temperature range Te wmaxthe contribution r~:,+~ (k) due to the emission (or absorption) of longitudinal SF dominates the magnon lifetime. The term T;!.,+* (k) accounting for a magnon transformation into a transverse SF is estimated as {max[w,(k), T~/w,~~}T,&,,+P~G,L+~, and is much less than the former. The term z~!+,.,,+~ also dominates over the annihilation contribution 7&:t_p which is negligible in the low temperature limit T-=xw,,,(k) and is of the order ( T/w,,,)~&,,+~ e T;!,~+~ for cold magnons w,(k) & T. In the high temperature limit T>> w,,,,,, Tck2, the contributions r;L,+p and z;:,,~ to the magnon lifetime (14) may be comparable with the r;!+,+* term. The electron-magnon term r& is at all temperatures smaller than T;!,,+~ by a factor (ko/k)2> 1. The here calculated SF damping of magnons qualitatively accounts for the reported temperature dependent linewidths of magnons for the Invar alloys which follow the relation [ 6-81 _ Tak2, where (YZ 1 for Fe,Pt and a< 1 for Fee5Ni,,. This relation, obtained for relatively high temperatures Ta w, (k) may be attributed to the t~!,,,,+~ contribution to the lifetime of cold magnons, thus supporting the recent suggestion of ref. [ 61 concerning the important role of longitudinal SF in the magnon relaxation of Invar alloys. The interpretation of the magnon damping in terms of the scattering by longitudinal SF would also suggest that both longitudinal and transverse SF contribute to the effects of the so-called “hidden exci365
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PHYSICS
tations” [6,7,9] introduced for the Invar alloys to account for the observed thermal properties, though the intrinsic intensities of SF may be too small to be directly measured by inelastic neutron scattering. The here proposed relaxation mechanism via longitudinal SF may also be responsible for the appreciable damping of magnons reported in ref. [ 17 ] for the weak itinerant ferromagnet ZrZnz though for an accurate comparison additional neutron scattering data are desirable. We would like to thank G. Lonzarich and V.P. Silin for useful discussions.
References [ 1 ] C. Herring, in: Magnetism, Vol. IV, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1966). [2] E.D. Thompson, J. Appl. Phys. 36 (1965) 1333. [3] V.P. Silin and A.Z. Solontsov, Fiz. Met. Metalloved. 52 ( 198 1) 23 1; in: Proc. P.N. Lebedev Physical Institute, Vol. 139 (Nauka, Moscow, 1982) p. 121.
366
LETTERS
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[ 41 T. Moriya, Spin fluctuations in itinerant electron magnetism (Springer, Berlin, 1985). [ 51 G. Lonzarich and L. Taillefer, J. Phys. C 18 ( 1985) 4339. [ 61 E.F. Wassermann, in: Ferromangetic materials, Vol. 5, eds. K.H. Buchow and E.P. Wohlfarth (North-Holland, Amsterdam, 1990). [ 71 Y. Ishikawa, S. Onodera and K. Tajima, J. Magn. Magn. Mater. 10 (1979) 183. [8] S. Onodera, Y. Ishikawa and K. Tajima, J. Phys. Sot. Japan 50 (1981) 1513. [ 91 Y. Ishikawa, Y. Noda, K.R.A. Ziebeck and D. Givord, Solid StateCommun. 57 (1986) 531. [lo] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, Oxford, 1980). [ 111 G. Lonzarich, J. Magn. Magn. Mater. 54-57 (1986) 612. [ 121 A.P. Levanyuk, Sov. Phys. JETP 32 (1964) 901. [ 131 V.P. Silin and A.Z. Solontsov, Sov. Phys. JETP 62 (1985) 836. [ 141 I.E. Dzyaloshinsky and P.S. Kondratenko, Sov. Phys. JETP 43 (1976) 1036. [ 151 A.Z. Solontsov, Phys. Met. Metallogr. 75 ( 1993) 1; in: Proc. Int. Conf. on Physics of transition metals, Darmstadt, 2024 July (1993), to be published. [ 161 V.V. Pustovalov and V.P. Silin, in: Proc. P.N. Lebedev Physical Institute, Vol. 6 1 (Nauka, Moscow, 1972) p. 42. [ 17 ] N.R. Bernhoeft, S.A. Law, G.G. Lonzarich and D. McK. Paul, Phys. Ser. 38 ( 1988) 19 1.
PHYSICS
LETTERS
A
Spin fluctuation damping of magnons in itinerant electron magnets AZ. Solontsov
and A.N. Vasil’ev
Theoretical Laboratory, A.A. Bochvar Institute for Inorganic Materials, 123060 Moscow, Russian Federation
Received 15 March 1993; accepted for publication 25 April 1993 Communicated by V.M. Agranovich
The lifetime of magnons in itinerant magnets due to three-mode scattering processes by overdamped spin fluctuations, or paramagnons, is analysed on the basis of the nonlinear time-dependent Ginzburg-Landau equations for the magnetic order parameter.
The magnon damping in itinerant magnets qualitatively differs from that in Heisenberg systems due to effects of the conducting electrons, the role of which is twofold. First, they directly affect the magnon damping via the electron-magnon scattering processes. Among them one-magnon scattering by spin-flip electron excitations resulting in the Landau damping of spin waves in the Stoner continuum [ 1 ] and two-magnon processes yielding the magnon relaxation outside the Stoner continuum [ 2,3] are now well understood. Besides that, itinerant electrons give rise to overdamped collective modes - spin fluctuations (SF), or paramagnons, playing a predominant role in the thermodynamics of weak itinerant magnets close to a magnetic instability [ 451. One may expect that SF would also strongly affect the damping of magnons in these systems. The fluctuation mechanism of the magnon relaxation is supported by the recent neutron scattering experiments in the Invar alloys [ 6 1, where an anomalously high magnon damping was reported [ 7-9 1. In this paper we analyse three-mode processes of magnon damping by SF which are the lowest order modes contributing to the magnon damping. Here we neglect the effects of electron-magnon interactions affecting higher order processes and discuss the kinetics of magnons and overdamped SF solely in terms of the collective variables describing the fluctuating magnetic order parameter. Assuming the major role of low frequency long wavelength fluctua362
tions we also neglect the time and spatial dispersion of the SF interactions and use the Ginzburg-Landau model [ lo] to account for the mode-mode coupling effects in magnetic dynamics. It was recently shown by Lonzarich and Taillefer that the quantum Ginzburg- Landau model yields a surprisingly accurate quantitative account of SF effects in the thermodynamics of weak ferromagnets [ 5,111 and we expect it to provide a useful basis for the description of SF kinetics as well. We start with the conventional Ginzburg-Landau effective Hamiltonian for an isotropic ferromagnet (cf. ref. [lo]), &r=t x
~Xo1(WW)12+$~ k kl+kz+kj+b=O
[M(k, )-M(k*)
c
IT
I [M~,)~W~.4) (1)
where M(k) is the Fourier transform of a slowly varying magnetic order parameter, y is the mode coupling constant and x0(k) is the static inhomogeneous susceptibility in the absence of coupling. To describe the relaxational magnetic dynamics we use the time dependent Ginzburg-Landau equations (cf. ref. [ 12 ] ) written in the spin-invariant form, 1 -T(k)
dM(k) at
=-
W,, 6M(k)’
where T(k) characterizes the relaxation we adopt a long wavelength approximation
rate. Here which for
Elsevier Science Publishers B.V.
Volume
177, number
PHYSICS
4,5
itinerant ferromagnets in the collisionless yields the expansions (see e.g. ref. [ 111) Xg’(k)=Xg’
+cP+...
)
T(k)=Tlkl
-t... .
Introducing
SF amplitudes,
m(k) =
dt exp(iwt)
regime
(3)
[M(k)
-M]
,
we write (2) in the explicit form
LETTERS
21 June 1993
A
gions they may be microscopically derived for weak itinerant magnets starting from the nonlinear equations of motion for the density matrix of an electron Fermi liquid formulated in refs. [ 3,5] retaining only the leading contributions to the low frequency and long wavelength expansions for the mode-mode coupling. This procedure will be described elsewhere. Confining ourselves to the analysis of the magnon relaxation we first derive the kinetic equations for the correlation functions, (d)k
=gexp[i(wr-k’*r)](m,(k+$k’)m,(k-fk’))
-y
c
m,(h)
=2 ImxAk)
[%(~2)?2(~3)
kl+kz+k3=k
(4)
-y
c
kl+kz+k,=k
+m(k,)m:(
m,(h)
-k3)
[w?(k2)m?(k3)
1>
(5)
where
k= (k, 0) > x~‘(k)=xo’(k)+3yMZ-io/T(k) and x;‘(k)= xg ’ (k) + yM2 - io/T( k) are the inverse longitudinal and transverse dynamical susceptibilities without taking account of SF, M is the spontaneous magnetization. The phenomenological equations (4) and ( 5 ) describe the dynamics of relaxational fluctuations known in weak itinerant magnets as paramagnons arising in the Stoner continuum. For the transverse excitations in the low frequency limit it begins at wavevectors k above the maximum magnon wavevector b, whereas the region Ik 1
lNkcr,
t)+tl
,
(6)
or, equivalently, for the occupation numbers Nk of SF, which are slowly time and space dependent. Here ( > designates the statistical average and at equilibrium yields the fluctuation dissipation equality with the Bose factor Nk= [exp(w/T)l]-‘No (below we use the frequency o and temperature T in energy units). In the lowest order approximation retaining only terms bilinear in the correlation functions (6) we obtain from eqs. (4) and (5) the following dynamical equation for the transverse correlation function,
[X;‘(k)+ji(%Y$$ _ %-$&._C)](m:)k =-I’5
~(m:)k,(mp2)k--k,[1-4YM2X,(k')l
+2(m:)k(m:)k’[1-yM2xp(k-k’)l -4rM’X:(k)(m:)k,(m~)k--k.}.
(7)
The real part of the r.h.s. of eq. (7) in the low frequency and long wavelength limit yields the SF corrections to the inverse susceptibility being replaced by X;‘(O)-*X;‘(O)+y(26m:+36m,2)~x;‘,
(8)
where 6m& = & (6m$)k are the average squared amplitudes. An analogous correction may be obtained for the inverse longitudinal susceptibility, x[‘(0)-+x<‘(O)+y+M
(26mz+36mp2)rx,’ (9) 363
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PHYSICS
starting from the dynamical equation for the correlation function ( mi)k. Assuming that x, and xa equal the corresponding thermodynamical susceptibilities one arrives at the magnetic equation of state previously obtained by other means [ 5,11,14]. It should be mentioned that eqs. (8) and (9) for the magnetic susceptibilities result from formulae (4)-( 7) in the lowest order approximation in the spin anharmonicity parameter [ 15 1,
g-~x,(k)(mt,)k
(v,f=t,P),
LETTERS
for
A
21 June 1993
magnons
o=o,(k)
with
wavevector
k and
frequency
=2pBA4ck2. Here
T-‘(k)=r,‘,+,(k)+T,~,+,(k)+T,:,,,(k) (14) is the inverse lifetime of magnons, written of the following contributions. The term
7,‘rn+,
(k)=(bM2C
k’
(10)
as a sum
IW,,p121mxa(k-k’)
x (No,, -Nw,-,L,=tiwj and thus hold only for the weakly anharmonic magnets. Taking account of equalities (6) the imaginary part of the r.h.s. of eq. (7) may be written in the form of the collision integral 4 ,,+z=,
I W&k
xImxP(k”)
[Nk(Nk. + l)(N,.,
-(Nk+l)Nk.Nk..]
xImxn(k-k’)
+ 1) (11)
describing three-mode scattering processes with emission or absorption of a longitudinal SF. Here I+‘,& is the matrix element of the scattering, in the here adopted approximation W,,, = 2 yM, being wavevector and frequency independent. For hypothetical undamped modes, when Irn~~,~ may be approximated by &functions, formula ( 11) reduces to the conventional Boltzmann form for the three-wave collision integral similar e.g. to the well-known theory in nonlinear plasma physics [ 161. Expression (11) generalizes the latter, taking account of the scattering processes involving strongly damped modes. Here we use eqs. (7) and ( 11) to describe the damping of magnons due to scattering by SF. Linearizing eq. (11) with respect to Nk and taking account of the expression x,(k) =
2pBM w,,(k) --w
(12)
for the transverse susceptibility in the spin wave region, we arrive at the kinetic equation
= -
364
y&(Nk-No,,,~w),
is due to the emission (for CO’> CO)or absorption o’ CO) of a longitudinal SF, rIGJ+t+p(k)=
k’, k” ) I2 Im x,(k) Im x,(k’ )
(13)
(15)
$MF
7 dw’
0
(for
I W,,, I2 Imxt(k’)
(N,, -NW._,)
(16)
describes a transformation of a magnon into a transverse SF with emission (or absorption) of a longitudinal SF and r&-P
(k)=
xImxdk+k’)
;PBMF
7
0
do’IWtp121mxt(k’)
(N,, -NW,+,)
(17)
accounts for annihilation of a magnon and transverse SF with emission of a longitudinal SF. The scattering processes ( 15 )- ( 17 ) exhaust the possible three-mode relaxation channels persisting in the ordered state of an isotropic itinerant ferromagnet. Taking account of the explicit expressions ~;‘(k)=~;‘+ck2-i~/~~k~, xp’(k)=xp’
+ck2-ico/r(kj
(18)
for the dynamical magnetic susceptibilities, following from eqs. (3 ), (8 ) and (9) in the paramagnon region, we can easily evaluate the integrals in the contributions ( 15 )-( 17 ) to the magnon lifetime. Estimating o,,, - okaX _ 2,uu,MIx,, where CO,.,,,, = Tck;, to be of the same order of w,(ko)> &Lx = magnitude, for the low frequency magnons o,(k) -=xCO,,, we obtain rIn-~,+a(k)=ro’(k)Z~(o,(k),
T) >
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PHYSICS LETTERS A
4,5
The magnon lifetime defined by eqs. ( 14), ( 19)(25 ) should be compared with the magnon relaxation time r+, arising via the electron-magnon scattering which for an isotropic Fermi surface yields [ 31
T rA+dk)=7cT1(k)
Tm:,-~(k)=~R’(k)~r,zm(wm(k),
T)
3
(19) where 1 To’(k)=
Im(w,
PB
T-x w,(k),
ok(k) (20)
~MF>
n=&
X-
T-=z co,
>
w cc Tc
w,,
T w,(k)
T X w,(k)
,
T In w,(k) WITlaX In w,(k) ’
w,(k)
w,,,
-=z Tea,,,,
-=+< T,
(26) T wnax = o In 0,
CO,,, << T,
(21)
TKO,
Z:(wT)=$, =-
27~ T2 , 3 00
w
(22)
2
Z;(w,T)=;
0 =-
5
,
2rc T2 , 3 0w
Tew, w << T<< w,_,
(23)
and Z:(w,
& =G,, T)=2jwz 1
T)=Z;(w,
(24)
in the high temperature limit T>> Tckl. Here ?j, and & are dimensionless constants of order unity approximately given by Wlco
1 r1(4+5V) z---1 3 2 1+?j k/k0
+ 5 q3j2 tan-‘(l/Jq)
,
dx
1 =ln
x(l+?jx~)(1+2~x2)
(l+?J2 v(l+rl)
’
with rl=xgks, k, is a cutoff wavevector be much larger than k,.
(25) assumed
to
where pF is the Fermi momentum. It follows from eqs. ( 19)-(26) that in the relatively wide temperature range Te wmaxthe contribution r~:,+~ (k) due to the emission (or absorption) of longitudinal SF dominates the magnon lifetime. The term T;!.,+* (k) accounting for a magnon transformation into a transverse SF is estimated as {max[w,(k), T~/w,~~}T,&,,+P~G,L+~, and is much less than the former. The term z~!+,.,,+~ also dominates over the annihilation contribution 7&:t_p which is negligible in the low temperature limit T-=xw,,,(k) and is of the order ( T/w,,,)~&,,+~ e T;!,~+~ for cold magnons w,(k) & T. In the high temperature limit T>> w,,,,,, Tck2, the contributions r;L,+p and z;:,,~ to the magnon lifetime (14) may be comparable with the r;!+,+* term. The electron-magnon term r& is at all temperatures smaller than T;!,,+~ by a factor (ko/k)2> 1. The here calculated SF damping of magnons qualitatively accounts for the reported temperature dependent linewidths of magnons for the Invar alloys which follow the relation [ 6-81 _ Tak2, where (YZ 1 for Fe,Pt and a< 1 for Fee5Ni,,. This relation, obtained for relatively high temperatures Ta w, (k) may be attributed to the t~!,,,,+~ contribution to the lifetime of cold magnons, thus supporting the recent suggestion of ref. [ 61 concerning the important role of longitudinal SF in the magnon relaxation of Invar alloys. The interpretation of the magnon damping in terms of the scattering by longitudinal SF would also suggest that both longitudinal and transverse SF contribute to the effects of the so-called “hidden exci365
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PHYSICS
tations” [6,7,9] introduced for the Invar alloys to account for the observed thermal properties, though the intrinsic intensities of SF may be too small to be directly measured by inelastic neutron scattering. The here proposed relaxation mechanism via longitudinal SF may also be responsible for the appreciable damping of magnons reported in ref. [ 17 ] for the weak itinerant ferromagnet ZrZnz though for an accurate comparison additional neutron scattering data are desirable. We would like to thank G. Lonzarich and V.P. Silin for useful discussions.
References [ 1 ] C. Herring, in: Magnetism, Vol. IV, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1966). [2] E.D. Thompson, J. Appl. Phys. 36 (1965) 1333. [3] V.P. Silin and A.Z. Solontsov, Fiz. Met. Metalloved. 52 ( 198 1) 23 1; in: Proc. P.N. Lebedev Physical Institute, Vol. 139 (Nauka, Moscow, 1982) p. 121.
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21 June 1993
[ 41 T. Moriya, Spin fluctuations in itinerant electron magnetism (Springer, Berlin, 1985). [ 51 G. Lonzarich and L. Taillefer, J. Phys. C 18 ( 1985) 4339. [ 61 E.F. Wassermann, in: Ferromangetic materials, Vol. 5, eds. K.H. Buchow and E.P. Wohlfarth (North-Holland, Amsterdam, 1990). [ 71 Y. Ishikawa, S. Onodera and K. Tajima, J. Magn. Magn. Mater. 10 (1979) 183. [8] S. Onodera, Y. Ishikawa and K. Tajima, J. Phys. Sot. Japan 50 (1981) 1513. [ 91 Y. Ishikawa, Y. Noda, K.R.A. Ziebeck and D. Givord, Solid StateCommun. 57 (1986) 531. [lo] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, Oxford, 1980). [ 111 G. Lonzarich, J. Magn. Magn. Mater. 54-57 (1986) 612. [ 121 A.P. Levanyuk, Sov. Phys. JETP 32 (1964) 901. [ 131 V.P. Silin and A.Z. Solontsov, Sov. Phys. JETP 62 (1985) 836. [ 141 I.E. Dzyaloshinsky and P.S. Kondratenko, Sov. Phys. JETP 43 (1976) 1036. [ 151 A.Z. Solontsov, Phys. Met. Metallogr. 75 ( 1993) 1; in: Proc. Int. Conf. on Physics of transition metals, Darmstadt, 2024 July (1993), to be published. [ 161 V.V. Pustovalov and V.P. Silin, in: Proc. P.N. Lebedev Physical Institute, Vol. 6 1 (Nauka, Moscow, 1972) p. 42. [ 17 ] N.R. Bernhoeft, S.A. Law, G.G. Lonzarich and D. McK. Paul, Phys. Ser. 38 ( 1988) 19 1.