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The influence of electron-electron interaction on the temperature dependence of the magnon damping in disordered ferromagnetic metals
Solid State Communications,Vol. 69, No. 1, pp.l-6, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .OO Pergamon Press plc
THE INFLUENCE OF ELECTRON-ELECTRON INTERACTION ON THE TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING IN DISORDERED FERROMAGNETIC METALS V.S.Lutovinov. O.E.Molodykh Institute
of Radio Engineering, Electronics and Automation USSR, Moscow, 117454, pr. Vernadskogo. 78 (Received 3 October 1988 by G.S.Zhdanov)
The temperature dependent part of maenon damping in disordered ferromagnetic metals is investigated within the framework of the spin operator diagram technioue. The interference between the magnon electron interaction and the Coulomb one as welkhp the ele+.ron-ims~;;;y ryhFttter~z considered. are caloulations contribution to the magnon relaxation rate’ due to this interference mechanism is of the hydrodynamic type Y,* kzwk pends on temperature as fl for d=3, log(T) for d=2 which and 11% for d=i Cd is the space dimensionality).
The
relaxation of’ 1orgwav~;g;~;; magnetical 1y since fFy%$ls i!: of great interest, exactly this region of magnon spectrum is extensive1 y treated experimental 1y via the use of microwave techniques wave resonance, ( f erromagnetio spin 1ongitudinal parametric resonance, pumping). In ure ferromagnetic metals (FM1 mapon aamping is mainly provided by the interaotion of localized magnetio moments with conduotion electrons. The presenoe of inhomogeneities of different origin in FMannodlfg.ss thelea;eottog propagation essential renormalization ’ of the elec-
1 owe&g
of
the
space
FM the interaotion between localized magnetio moments leading to a spatial dispersion of the susceptibility is realized via the oonduction eleotrons (RKKY interaotion). and. hence. the soattering oftheele&rons by impurities infl uenoes magnetio subsystem dynamics, namely magnon speotrum and rel axation frequency. In the resent paper we investi ate the temperuI:ure dependent local iza7,ion
corrections to the magnon relaxation rate due to the interference between the impurity scattering electronand interaotion. We -electron (Coulomb)
consider magnons as the whereas ‘“rg: dimensional excitations, -. with dimensional ity eff eotive may be respect to s!%tron diffusion different. The latter is determined by the relation between the sample sizes Li 31 and the charaFco~ristti& (i=l 2 s&ale lg”q-i. diffusive interaction Coulomb renormal ized diffusion (Lbv2dd is the el e:tron constant. d is the space dimensionais l ity). Thus, if the film thickness space smaller than Z , the effective dimensional ity is6 equal to 2, while for ihe wires with the diameter ,l;;$ t$,” , d-l. From the other m”agnons are three-dimensional if their wavelength A is smaller than the sample size (A < L < lb). We will interestead in the relaxation of the 1ongwav;l~n$h magnons with waveveotors c)’
dimensional itY- In 1
2
TEMPERATURE
DEPENDENCE
ko=2bJ/ur (J is the s-d-exchange b=). The inequality ky< oonstant, < tc is eauivalent to T/To < its and. henoe, is fullfilled in the magnetioa!ly ordered ;l&e. This s>rong inequal;;; for gives op ortumty r@ ization of P011owing relationship < A < L < Zc oorresponding to the k are magnons srtuation when the el eotron whil f! the 3;dimefsion$, d$&us~~~ 1s low dlmenslonal Cd=1 or ’ The hamiltonian of the system under oonsidzration is, (1) H = Ho l Hsd + ii, ’ Himp . Here (2) iO=+l +I;~L~ - gHSt
corresponds p ’ t” the kiineio thy oonduotion el eotrons !Yeeman Of of looalized spins in a magnetio T?f?r Jf: “=,O,a~.. are the oreation and anihil ation operators of l oonduotion eleotrons with momentum p and spin projection c (v=+,+); sT=6: -bJ-HG * s”=s,+bJ+H- E (31 “p i; i; P the eleo&n are energies 0: two subbands. The hamiltonian
OF THE MAGNON
DAMPING
Vol. 69, No. 1
Since the energy subbands of conduction electrons are split by the quantity 2bJ (see eq.(3)>. the onemagnon process t magnon emission or absorption accompanied by conduction electron spin flip) is forbidden by the conservation laws in an ideal FM in the range of magnon wavevectors k < kO= 2bJ/up Cur is the Fermi velocity). In a disordered FM, the magnon-electr-on interaction is renormalized due to the electron-impurity rescattet-ing taking place without momentum conservation. and, thus, the combined magnon-electronimpurity process is allowed in the range k < kO. The contribution of this interference mechanism to the magnon damping is given by the expression *
i2b.lJ2 --_----__ .3b
-
C2bA2+i/r2
i (
hJ
8)
T&P k and does notIndergyd on tempera+-e at. T c T c* S caSe the 1mpurit.v renormalization of the one-magnon vertex function is orinciually imoortant for the occurence- of the -hydrddynamic-type dependence of y, with respect t.o the magnon wavevector. In eq. (8),_ r.
f
=
?j
d ne cif-cl
,$ r
takes into ,a~~cour$ ttce translationally invariant s-d-exchanee interaction , Sz Cv=t,z) are the Fourier transformed operators of localized spins: 6s: = Sa - N~A(%: b=
and the last term in eq.Cl). Eimpf Lli;.;;‘)ir&p..a J Q.P,U
describes the atomic potential
(6)
electron scattering by inhomoaeneities. Here pn eir;-;Yk” &$~,= ; , (7) Z R where u is the a&litude of the impurity scattering, p-=1 for lattice sites being occupied by impurity atoms, and p,=O otherwise.
where T is the electron lifet.ime due to c is atomic inhomogeneity scattering. the impurity concentration. n, is the number of electron per atom, E~,L) are the Fermi enera and momentum, and ‘11~ is the magnon &rzquency. The renormal+zation of spectrum * relatively small andma??‘irst order witi respect to the weak localization parameter (TE 1-l (see ref.a). 0k
where w;O’=DkZ,
(10)
D=$b/8)
LJ2/EpJ I>;*
(IL)
is the magnon srjectrum in an ideal FM jn the longwavelength region. The temperature independence of the magnon damping due to one-magnon process renormilized by impurities is due to the fact that the energy deficit in the one-magnon process is larg.e, 1ti..J--r-‘~~T. Thus. t. he impurity renormalizat.ion c,f the maenon-electron vertex functioln does not lead to the appearance of diffusive poles. in oontr-ast. to what haopens for electron-electron and electron-~llc~nc~~l interactions. Temper-a\ye dependent auantum corrections magon the damping arise if one takes into account.. the interfet;ence betwee!! ilntiut-ity scattering and maenon-elect ran interaction as well as tile other irlelastic processes. We treat the most efficient
Vol. 69. No. 1
of
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
3
these mechanisms.namely the Coulomb
process.
The “Coulomb” contribution to the selfenergy (;;;t of the, magr;ir Gr;;; accountine. function electron-impurity rescattering is shown in Fig.1 and Fig.2. The diagrams shown in Fig.1 contain Coulomb interactions of the Fock type, and the diagrams in Fig.2 coupling of electron-electron contain the Hartree type. the It sho,uld be noted that :iagrry ;lmlla; to that , shown f;; Figs. responsible the temperature dependent corrections to the conductivity of disordered nonmagnetic case under metals ’ . In the the one-magnon vertex consideration ;;~;~ions A s”,“la_.. edes of fermlon henc?, are are Impurity renormalized really ‘by scattering in contrast to the case of where the analogous the conductivity, positions are occ;yd by the vector of electron-photon vertices The gaphio equatio;er[;; interaction. renormalized one-magnon the function A is given in Fig. 3. Its solution is of the form
1 A++A_ _ 1 *+ 11=-2qc . Lz=f * 7 , L,=rloeq +-A_ tand . *+ = f kvF+2bJ-iwL-i/r Here ioL -and icn are the Matsubara frequencies of the incoming maqnon GF and the electron GF respectively: 8(x)=1 for x>O and @(x)=0 otherwise. For the definiteness we suppose that o),>O. In the longwavelength region kccko, WITCH. the dependence on w\ in eq.(13) is negligible and as is L”“( k,y) follows kv 2 The other elements of the diagrams, such as the renormalized Coulomb coupling as
FigA. The selfenergy part of the mwon Green function taking into account the electron-im urity rescattering and the Coulomb in0eraotion of the Fuck type. The light arrow correspoqds to the electron spin direction W= one corresponds to cr+. T6e %ra?%tk line with a small arrow represents fhe pagnon GF. the wavy line 4s the Coulomb interaction. t..iee cycle ,wlt_h the cry.s rerjresents impurity scattering amvlitude the’ dotted line shows the configurational averaging.
4
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
Hartree-type contribution se1 f energy part.
Vol. 69, No. 1
to
_k+Prr,+“,
F9.3, The Rraphio . equation, for _ the el eotron-mafznon vertex being renormal ized due to %ztion impurity rescattering. well as the impurity “ladders” (the latters are represented in FigA and 2 by cross-hatched rectan&s). satisfy similar to that shown in equations Fig.3. In the case of the renormalized Coulomb interaction only the electron lines of the same subbands are connected by the f;J_ted impurity lines repreaveragng conflguratlonal senting effeotive the (see F&da). Hence, pole in the coup1 in9 has a diffusive region of small momentum and frequency w~T<
The accounting of this element eaUall Y of with the impurity renormal izations responsimagnon-electron vertices is ble for the formation of the momentum dependence of rel axation frequency of 1ongwavel en&h magnons. The analysis shows that all the six diagrams of the Hartree type (see Fig.2) are canoelled out by the first six diagrams of the Took ty e represented in the two uppekliines o P F1g.l. ,F;o; tltng reason, we cal cul ate diarzramsTh~hoybsi.;.e th;f third line , of oorresponding FigA. “cancel1 it-g” diagrams in the Hartree set of graphs is due to a specific feature of the interference between eleotronmagnon and Coulomb interactions: the
On the okporwn hand, thFg41iagl-ammatic in element el eotron 1ines of dif fer’ent ~I?$~~~ conneoted by the dotted impurity lines, and is given by
s in flip, former occurs with electron while the latter one occurs wit Rout spin flip. Thus, extra Hartree-type the the oonserdiaqams are forbidden b vation of the total spin o r the system.
Vol.
69,
No.
1
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
i&n (a) Fig.4. The containing 1izations.
(b) eiients,
The magnon ,relaxation rate is determined ,if the imaginary part of the represented magnon energy, diagrammatical 1Y in Fig.1. Using the expressions (121, (15) and (16) for t-helements various diagrammatio oarring out the integration over the one fermion momenta and frequency, obtaines after analytio oontinuation the foi%;ng oontribution to the magnon
where
Here, nw=(exv(cJ/T1-i)-‘. It should be stressed that eo.(17) corresponds t”, tte , contribution of the range gl<
of , the impurity
diagrams renorma-
so it is valid only at low temperatures, T~ctl. In writi down of ecls.(17) and (18) we have s aken into aocount the q-dependence only in the most divergent factors, neglecting it otherwise. Using for the quantity LCd’C k) its longwavelength representation (14), and carring out the integrations one gets in the limit o,_
5
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
This answer for the magnon damping is Of the usual hydrodynamic form and temperature dependent in contrast iz eq.(8). The straightforward comparison of eqs.(l9) and (81 demonstrates that the interference mechanism of magnon relaxation under consideration is more effective ’ low-dimensio k cases. Thus, for 223 ay(TJ/yk~ J" T~/(TE~Y<
d=i
Hence, the temperature dependekt contribution to the magnon relaxation rate in disordered FM may be of order of the the temperature independent one (8) in low-dimensional cases. The vhysical origin of the nontrivial T-behavior of magnon dam&g (see a.6 may be recognized follows. lonawavelenfzth diffusive modes which ren6rmali~;e- the Coulomb interaction have square-law dispersion equation (see the last factor in eq. (17)). The characteristic value of the frequency of this excitations is determined by th,e boson factors in the square byfckets in eq.(17) andSilEeof the temperature. the order integration ove~~_f~,ziri eq.(17) gives the resulting depenJlence o the yn;gUbu;<$pd,,fo th~~_m~grlo; $..~&g is * _ w for $1.3 and log(o)-log(T) for d=2. In the range uk>>T, Trctl the dependence of the relaxation rate on the magnon frequency is stronger a~(TI/y~--i/h?.
eq%?t
Vol. 69, No. 1
where A”-1/8V%:
A;=-l/8&
A~=3V?/(64flb)
OU1‘ noticed that It should be for 011lv valid are considerations and, magnons (k < k) longwavelength hence. the dependences given by0 eq.(20) are valid only in the low temperature Tc(J/~r)Z region T
REFERENCES B.L.Altshuler. AGAronov. In: Modern Froblems in Condensed Matter Science. Vol.lO:Electron-Electron Interactions in Diso;poE;k Systems. , Ed. A,L&FL;;; and Elsevler Publishers B.V.,’ 1985. 2. A.~~edyay~, V.S.Lutovinov. In: Seminar Amorphous Magn’etism. Krasnoyarsk, $80. 117. 1.
3. V.S.Lutovinov. O.E.Molodykh, A.V.Vedyayev. Sol. State Comm., 2985, 5.4 4.
Phys.
5. I.G.Cullis, .Heath. 1980. j&Q,309!
Sot.
J.
.Jr,n., 1980, Ftiys.
F,
0038-1098/89 $3.00 + .OO Pergamon Press plc
THE INFLUENCE OF ELECTRON-ELECTRON INTERACTION ON THE TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING IN DISORDERED FERROMAGNETIC METALS V.S.Lutovinov. O.E.Molodykh Institute
of Radio Engineering, Electronics and Automation USSR, Moscow, 117454, pr. Vernadskogo. 78 (Received 3 October 1988 by G.S.Zhdanov)
The temperature dependent part of maenon damping in disordered ferromagnetic metals is investigated within the framework of the spin operator diagram technioue. The interference between the magnon electron interaction and the Coulomb one as welkhp the ele+.ron-ims~;;;y ryhFttter~z considered. are caloulations contribution to the magnon relaxation rate’ due to this interference mechanism is of the hydrodynamic type Y,* kzwk pends on temperature as fl for d=3, log(T) for d=2 which and 11% for d=i Cd is the space dimensionality).
The
relaxation of’ 1orgwav~;g;~;; magnetical 1y since fFy%$ls i!: of great interest, exactly this region of magnon spectrum is extensive1 y treated experimental 1y via the use of microwave techniques wave resonance, ( f erromagnetio spin 1ongitudinal parametric resonance, pumping). In ure ferromagnetic metals (FM1 mapon aamping is mainly provided by the interaotion of localized magnetio moments with conduotion electrons. The presenoe of inhomogeneities of different origin in FMannodlfg.ss thelea;eottog propagation essential renormalization ’ of the elec-
1 owe&g
of
the
space
FM the interaotion between localized magnetio moments leading to a spatial dispersion of the susceptibility is realized via the oonduction eleotrons (RKKY interaotion). and. hence. the soattering oftheele&rons by impurities infl uenoes magnetio subsystem dynamics, namely magnon speotrum and rel axation frequency. In the resent paper we investi ate the temperuI:ure dependent local iza7,ion
corrections to the magnon relaxation rate due to the interference between the impurity scattering electronand interaotion. We -electron (Coulomb)
consider magnons as the whereas ‘“rg: dimensional excitations, -. with dimensional ity eff eotive may be respect to s!%tron diffusion different. The latter is determined by the relation between the sample sizes Li 31 and the charaFco~ristti& (i=l 2 s&ale lg”q-i. diffusive interaction Coulomb renormal ized diffusion (Lbv2dd is the el e:tron constant. d is the space dimensionais l ity). Thus, if the film thickness space smaller than Z , the effective dimensional ity is6 equal to 2, while for ihe wires with the diameter ,l;;$ t$,” , d-l. From the other m”agnons are three-dimensional if their wavelength A is smaller than the sample size (A < L < lb). We will interestead in the relaxation of the 1ongwav;l~n$h magnons with waveveotors c)’
dimensional itY- In 1
2
TEMPERATURE
DEPENDENCE
ko=2bJ/ur (J is the s-d-exchange b=
corresponds p ’ t” the kiineio thy oonduotion el eotrons !Yeeman Of of looalized spins in a magnetio T?f?r Jf: “=,O,a~.. are the oreation and anihil ation operators of l oonduotion eleotrons with momentum p and spin projection c (v=+,+); sT=6: -bJ-HG * s”=s,+bJ+H- E (31 “p i; i; P the eleo&n are energies 0: two subbands. The hamiltonian
OF THE MAGNON
DAMPING
Vol. 69, No. 1
Since the energy subbands of conduction electrons are split by the quantity 2bJ (see eq.(3)>. the onemagnon process t magnon emission or absorption accompanied by conduction electron spin flip) is forbidden by the conservation laws in an ideal FM in the range of magnon wavevectors k < kO= 2bJ/up Cur is the Fermi velocity). In a disordered FM, the magnon-electr-on interaction is renormalized due to the electron-impurity rescattet-ing taking place without momentum conservation. and, thus, the combined magnon-electronimpurity process is allowed in the range k < kO. The contribution of this interference mechanism to the magnon damping is given by the expression *
i2b.lJ2 --_----__ .3b
-
C2bA2+i/r2
i (
hJ
8)
T&P k and does notIndergyd on tempera+-e at. T c T c* S caSe the 1mpurit.v renormalization of the one-magnon vertex function is orinciually imoortant for the occurence- of the -hydrddynamic-type dependence of y, with respect t.o the magnon wavevector. In eq. (8),_ r.
f
=
?j
d ne cif-cl
,$ r
takes into ,a~~cour$ ttce translationally invariant s-d-exchanee interaction , Sz Cv=t,z) are the Fourier transformed operators of localized spins: 6s: = Sa - N~A(%: b=
and the last term in eq.Cl). Eimpf Lli;.;;‘)ir&p..a J Q.P,U
describes the atomic potential
(6)
electron scattering by inhomoaeneities. Here pn eir;-;Yk” &$~,= ; , (7) Z R where u is the a&litude of the impurity scattering, p-=1 for lattice sites being occupied by impurity atoms, and p,=O otherwise.
where T is the electron lifet.ime due to c is atomic inhomogeneity scattering. the impurity concentration. n, is the number of electron per atom, E~,L) are the Fermi enera and momentum, and ‘11~ is the magnon &rzquency. The renormal+zation of spectrum * relatively small andma??‘irst order witi respect to the weak localization parameter (TE 1-l (see ref.a). 0k
where w;O’=DkZ,
(10)
D=$b/8)
LJ2/EpJ I>;*
(IL)
is the magnon srjectrum in an ideal FM jn the longwavelength region. The temperature independence of the magnon damping due to one-magnon process renormilized by impurities is due to the fact that the energy deficit in the one-magnon process is larg.e, 1ti..J--r-‘~~T. Thus. t. he impurity renormalizat.ion c,f the maenon-electron vertex functioln does not lead to the appearance of diffusive poles. in oontr-ast. to what haopens for electron-electron and electron-~llc~nc~~l interactions. Temper-a\ye dependent auantum corrections magon the damping arise if one takes into account.. the interfet;ence betwee!! ilntiut-ity scattering and maenon-elect ran interaction as well as tile other irlelastic processes. We treat the most efficient
Vol. 69. No. 1
of
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
3
these mechanisms.namely the Coulomb
process.
The “Coulomb” contribution to the selfenergy (;;;t of the, magr;ir Gr;;; accountine. function electron-impurity rescattering is shown in Fig.1 and Fig.2. The diagrams shown in Fig.1 contain Coulomb interactions of the Fock type, and the diagrams in Fig.2 coupling of electron-electron contain the Hartree type. the It sho,uld be noted that :iagrry ;lmlla; to that , shown f;; Figs. responsible the temperature dependent corrections to the conductivity of disordered nonmagnetic case under metals ’ . In the the one-magnon vertex consideration ;;~;~ions A s”,“la_.. edes of fermlon henc?, are are Impurity renormalized really ‘by scattering in contrast to the case of where the analogous the conductivity, positions are occ;yd by the vector of electron-photon vertices The gaphio equatio;er[;; interaction. renormalized one-magnon the function A is given in Fig. 3. Its solution is of the form
1 A++A_ _ 1 *+ 11=-2qc . Lz=f * 7 , L,=rloeq +-A_ tand . *+ = f kvF+2bJ-iwL-i/r Here ioL -and icn are the Matsubara frequencies of the incoming maqnon GF and the electron GF respectively: 8(x)=1 for x>O and @(x)=0 otherwise. For the definiteness we suppose that o),>O. In the longwavelength region kccko, WITCH. the dependence on w\ in eq.(13) is negligible and as is L”“( k,y) follows kv 2 The other elements of the diagrams, such as the renormalized Coulomb coupling as
FigA. The selfenergy part of the mwon Green function taking into account the electron-im urity rescattering and the Coulomb in0eraotion of the Fuck type. The light arrow correspoqds to the electron spin direction W= one corresponds to cr+. T6e %ra?%tk line with a small arrow represents fhe pagnon GF. the wavy line 4s the Coulomb interaction. t..iee cycle ,wlt_h the cry.s rerjresents impurity scattering amvlitude the’ dotted line shows the configurational averaging.
4
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
Hartree-type contribution se1 f energy part.
Vol. 69, No. 1
to
_k+Prr,+“,
F9.3, The Rraphio . equation, for _ the el eotron-mafznon vertex being renormal ized due to %ztion impurity rescattering. well as the impurity “ladders” (the latters are represented in FigA and 2 by cross-hatched rectan&s). satisfy similar to that shown in equations Fig.3. In the case of the renormalized Coulomb interaction only the electron lines of the same subbands are connected by the f;J_ted impurity lines repreaveragng conflguratlonal senting effeotive the (see F&da). Hence, pole in the coup1 in9 has a diffusive region of small momentum and frequency w~T<
The accounting of this element eaUall Y of with the impurity renormal izations responsimagnon-electron vertices is ble for the formation of the momentum dependence of rel axation frequency of 1ongwavel en&h magnons. The analysis shows that all the six diagrams of the Hartree type (see Fig.2) are canoelled out by the first six diagrams of the Took ty e represented in the two uppekliines o P F1g.l. ,F;o; tltng reason, we cal cul ate diarzramsTh~hoybsi.;.e th;f third line , of oorresponding FigA. “cancel1 it-g” diagrams in the Hartree set of graphs is due to a specific feature of the interference between eleotronmagnon and Coulomb interactions: the
On the okporwn hand, thFg41iagl-ammatic in element el eotron 1ines of dif fer’ent ~I?$~~~ conneoted by the dotted impurity lines, and is given by
s in flip, former occurs with electron while the latter one occurs wit Rout spin flip. Thus, extra Hartree-type the the oonserdiaqams are forbidden b vation of the total spin o r the system.
Vol.
69,
No.
1
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
i&n (a) Fig.4. The containing 1izations.
(b) eiients,
The magnon ,relaxation rate is determined ,if the imaginary part of the represented magnon energy, diagrammatical 1Y in Fig.1. Using the expressions (121, (15) and (16) for t-helements various diagrammatio oarring out the integration over the one fermion momenta and frequency, obtaines after analytio oontinuation the foi%;ng oontribution to the magnon
where
Here, nw=(exv(cJ/T1-i)-‘. It should be stressed that eo.(17) corresponds t”, tte , contribution of the range gl<
of , the impurity
diagrams renorma-
so it is valid only at low temperatures, T~ctl. In writi down of ecls.(17) and (18) we have s aken into aocount the q-dependence only in the most divergent factors, neglecting it otherwise. Using for the quantity LCd’C k) its longwavelength representation (14), and carring out the integrations one gets in the limit o,_
5
TEMPERATURE DEPENDENCE OF THE MAGNON DAMPING
This answer for the magnon damping is Of the usual hydrodynamic form and temperature dependent in contrast iz eq.(8). The straightforward comparison of eqs.(l9) and (81 demonstrates that the interference mechanism of magnon relaxation under consideration is more effective ’ low-dimensio k cases. Thus, for 223 ay(TJ/yk~ J" T~/(TE~Y<
d=i
Hence, the temperature dependekt contribution to the magnon relaxation rate in disordered FM may be of order of the the temperature independent one (8) in low-dimensional cases. The vhysical origin of the nontrivial T-behavior of magnon dam&g (see a.6 may be recognized follows. lonawavelenfzth diffusive modes which ren6rmali~;e- the Coulomb interaction have square-law dispersion equation (see the last factor in eq. (17)). The characteristic value of the frequency of this excitations is determined by th,e boson factors in the square byfckets in eq.(17) andSilEeof the temperature. the order integration ove~~_f~,ziri eq.(17) gives the resulting depenJlence o the yn;gUbu;<$pd,,fo th~~_m~grlo; $..~&g is * _ w for $1.3 and log(o)-log(T) for d=2. In the range uk>>T, Trctl the dependence of the relaxation rate on the magnon frequency is stronger a~(TI/y~--i/h?.
eq%?t
Vol. 69, No. 1
where A”-1/8V%:
A;=-l/8&
A~=3V?/(64flb)
OU1‘ noticed that It should be for 011lv valid are considerations and, magnons (k < k) longwavelength hence. the dependences given by0 eq.(20) are valid only in the low temperature Tc(J/~r)Z region T
REFERENCES B.L.Altshuler. AGAronov. In: Modern Froblems in Condensed Matter Science. Vol.lO:Electron-Electron Interactions in Diso;poE;k Systems. , Ed. A,L&FL;;; and Elsevler Publishers B.V.,’ 1985. 2. A.~~edyay~, V.S.Lutovinov. In: Seminar Amorphous Magn’etism. Krasnoyarsk, $80. 117. 1.
3. V.S.Lutovinov. O.E.Molodykh, A.V.Vedyayev. Sol. State Comm., 2985, 5.4 4.
Phys.
5. I.G.Cullis, .Heath. 1980. j&Q,309!
Sot.
J.
.Jr,n., 1980, Ftiys.
F,