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Spin wave dynamics in antiferromagnets with hyperfine interactions
Physica 108B (1981) 1071-1072 North-Holland Publishing Company
RA 4
SPIN WAVE DYNAMICS
IN ANTIFERROMAGNETS
WITH HYPERFINE
INTERACTIONS
V. S. Lutovinov
Moscow Institute of Radio Engineering, Electronics Moscow 117454, USSR
and Automation
The diagrammatic spin-operator technique has been used to treat the kinetic and relaxation properties of spin wave system caused by the interaction with nuclear subsystem (NS) in antiferromagnets. The interaction between spin waves and both long wavelength collective excitations of a NS and short wavelength paramagnetic states is considered. The spin wave relaxation frequencies are calculated for ideal magnets as well as for magnets with paramagnetic impurities. It is shown that three wave processes with the participation of nuclear spin waves are responsible for the instability of spin wave system at high excitation levels in antiferromagnets.
The correlation effects due to the Suhl-Nakamura interaction in magnetic dielectrics are essential in the low temperature region T < T (J0/£0) , ~ = 3/4 for ferromagnets and ~ = 3/2 n for antiferromagnets. As a result the well defined long wavelength collective excitations-nuclear spin waves (NSW)--are realized in a nuclear subsystem [i]. The indirect coupling between nuclear spins is ceased as the wavevector increases, and for short wavelength states K > K* = r c ~ a-l(Wn/T)2/3
(r c is a correlation
radius) the space dispersion in the NS is negligible: the behavior of this state is quite paramagnetic [2]. Thus for the temperature interval w < T < T*, collective (wavelike) as well as n paramagnetic (localized) features are inherent in the NS simultaneously. These features are revealed most strongly in easy plane antiferromagnets (EP AF) where the hyperfine interaction is enhanced by exchange. The influence of the collective (K < K*) and localized (K > K*) states of a nuclear subsystem on the spin wave (SW) kinetic properties in an EP AF is investigated in the present note. The processes of elastic SW scattering tuations of the longitudinal component clear spin are most pronounced in the wavelength part of the phase space of subsystem [3] (see fig. i) 2 2 rz = b~' • ~n Jo SK 2m 4~ 8N ON2 e k
by flucof a nushort a nuclear
(i)
s~ectrum, S ~s the magnon velocity, E0 2 = H(H+H~)+H^ is the squared SW spectrum gap, H is t h e ~ e x t ~ r n a l field, H D is the Dzjaloshlnskll field, HA 2 = 2bJ0~ , J0 is the exchange integral, mnaiS the un~ulled NMR frequency, b(~n/T) is the Brillouin
function,
and b'(y) = db/dy.
As can be seen from (i), the relaxation frequency due to the fluctuating longitudinal component
0378-4363/81/0000-0000/$02.50 © Norlh-Holland PublishingCompany
(b)
(o)
Figure i : Diagrams illustrate spin wave scattering and the linear transformation of the spin wave into a phonon by fluctuations of a nuclear spin. of a nuclear spin is proportional to the spin wave wavevector and is temperature independent when T >> w . By taking the linear magneton elastic coupling into account, one can obtain the process of the linear transformation of a spin wave into a phonon (see fig. ib). In contrast to the expression (i) the relaxation frequency due to this combined magnon-phononfluctuation relaxation process is nonvanishing when K + 0. It was shown in ref. [5] that the process of linear transformation of a phonon into a NSW by the fluctuating nuclear spin is an efficient excitation mechanism of the NS in antiferromagnets. The analysis of the SW interaction with collective excitations of the NS shows that the strongest three wave processes are SW and NSW confluence into a phonon and SW splitting in a NSW and a phonon. For these processes F±
where @N = S/a, ek = (e0 2 + $2K2) I/2 is the SW
m.ph
4J(J+l) 3z
@2 Mv 2
Jo 3 3 WD
~n 2 2 Ek
'
(2)
where 2SKS/v << £k' @ = Ba3' B is the magnetoelastic constant, M is a mass of a unit cell, ~D = v/a, and v is the sound velocity. In (2) we use the relation b = J(J+I)~ /3T which is n valid for T >> ~ . The three wave processes mentioned above nare realized when both the hyperfine interaction and nonlinear magnetoelastic couplings [4] are taken into account (Fig. 2) 1071
1072 4 F imp n2m
(cB')2 B' T 28
g
o n Wn2gkS K
2 o
+F
, (4)
Co2_HA2+F
where F = ek2Wn2/(SK) 4 and (SK/~k) 2 >> 2Wn/~ k.
Figure 2 : Diagram for the three Wave SW - NSW phonon process. The straight line represents the magnon, the wavey line - the phonon and the double line the excitation in the NS. Besides the processes considered above, the SW interaction with the longitudinal component of an impurity spin may be essential in a real EP AF. The linear interaction between the magnon amplitude and an impurity spin [6] leads to the so called "slow" SW relaxation (fig. 3a)
The nonlinear effects in the SW system are also considered. The splitting of a SW into a phonon and NSW is responsible for the finite SW amplitude instability. In particular, the threshold field of the parametric instability under perpendicular pumping in an EP AF is the following: 2 ~h
= 2A~(QnOph)i/2
c
•
sl
cB' =
~o
2 3 ~Jo " 6k2
~nk
~n
Eo x ~HA
(5)
Ck~l (gkTi)2 + 1
,
(3)
where c is the impurity concentration, ~0 is the impurity precession frequency in a molecular field, B' = dB/dy0 is the derivative of the appropriate Brillouin function, ~ % i, and T is I the spin-lattice relaxation time for an impurity. The analogous expression for J. = 1/2 was obtained in ref. [7] at gkTl >> i. Imp
where Aw is the AFMR linewidth,
REFERENCES:
[3]
[4] [5] [6] [7] Figure 3 : (a) Diagram for the "slow" SW relaxation mechanism by the paramagnetic impurity. (b) The effective anharmonic interaction between spin waves and transverse components of nuclear spins. The black points represent the z-component of an impurity spin. By using the substitution b'w 2 + cB,w02 one can n easily obtain from (i) the expresslon for the SW relaxation rate due to the elastic SW scattering by an impurity. The interference of the two impurity mechanisms indicated above leads to an effective three wave interaction between the NSW and the SW (fig. 3b). The SW relaxation frequency in these processes is proportional to the temperature of the NS and depends on field
~nk is
The theoretical results obtained are compared with the experimental data of ref. [8].
[2]
(b)
Qn and Qph are
the nuclear and the subsystem qualities, the NSW spectrum.
[i]
(e)
u(H+HD)O
Mv2 1/2 i ×[7o]
2 Fimp
Ek
[8]
De Gennes, P., Pincus, P., Hartman-Boutron, F., Winter, J., Phys. Rev. 129 (1963) 1105-1115. Lutovinov, V., Saphonov, V., Fiz. Tverd Tela, 21 (1979) 2772-2783. Lutovinov, V., Abstr. 21st National Conf. Low Temp. Phys., USSR, Kharkov, part II (1980) 202-203. Lutovinov, V., Preobrajenskii, V., Semin, S., Zh. ETP, 74 (1978) 1159-1169. Bunkov, Yu., Lutovinov, V., Monakhov, A., Zh. ETP, 80 (1981) 1449-1468. Hartman-Boutron, F., Phys. Kondens. Mater., 2 (1964) 80-98. Mikhailov, A., Farzetdinova, R., Abstr. 21st National Conf. Low Temp. Phys., USSR, Kharkov, part II (1980) 212-213. Andrienko, A., Prozorova, L., Zh. EPT, 78 (1980) 2411-2418.
RA 4
SPIN WAVE DYNAMICS
IN ANTIFERROMAGNETS
WITH HYPERFINE
INTERACTIONS
V. S. Lutovinov
Moscow Institute of Radio Engineering, Electronics Moscow 117454, USSR
and Automation
The diagrammatic spin-operator technique has been used to treat the kinetic and relaxation properties of spin wave system caused by the interaction with nuclear subsystem (NS) in antiferromagnets. The interaction between spin waves and both long wavelength collective excitations of a NS and short wavelength paramagnetic states is considered. The spin wave relaxation frequencies are calculated for ideal magnets as well as for magnets with paramagnetic impurities. It is shown that three wave processes with the participation of nuclear spin waves are responsible for the instability of spin wave system at high excitation levels in antiferromagnets.
The correlation effects due to the Suhl-Nakamura interaction in magnetic dielectrics are essential in the low temperature region T < T (J0/£0) , ~ = 3/4 for ferromagnets and ~ = 3/2 n for antiferromagnets. As a result the well defined long wavelength collective excitations-nuclear spin waves (NSW)--are realized in a nuclear subsystem [i]. The indirect coupling between nuclear spins is ceased as the wavevector increases, and for short wavelength states K > K* = r c ~ a-l(Wn/T)2/3
(r c is a correlation
radius) the space dispersion in the NS is negligible: the behavior of this state is quite paramagnetic [2]. Thus for the temperature interval w < T < T*, collective (wavelike) as well as n paramagnetic (localized) features are inherent in the NS simultaneously. These features are revealed most strongly in easy plane antiferromagnets (EP AF) where the hyperfine interaction is enhanced by exchange. The influence of the collective (K < K*) and localized (K > K*) states of a nuclear subsystem on the spin wave (SW) kinetic properties in an EP AF is investigated in the present note. The processes of elastic SW scattering tuations of the longitudinal component clear spin are most pronounced in the wavelength part of the phase space of subsystem [3] (see fig. i) 2 2 rz = b~' • ~n Jo SK 2m 4~ 8N ON2 e k
by flucof a nushort a nuclear
(i)
s~ectrum, S ~s the magnon velocity, E0 2 = H(H+H~)+H^ is the squared SW spectrum gap, H is t h e ~ e x t ~ r n a l field, H D is the Dzjaloshlnskll field, HA 2 = 2bJ0~ , J0 is the exchange integral, mnaiS the un~ulled NMR frequency, b(~n/T) is the Brillouin
function,
and b'(y) = db/dy.
As can be seen from (i), the relaxation frequency due to the fluctuating longitudinal component
0378-4363/81/0000-0000/$02.50 © Norlh-Holland PublishingCompany
(b)
(o)
Figure i : Diagrams illustrate spin wave scattering and the linear transformation of the spin wave into a phonon by fluctuations of a nuclear spin. of a nuclear spin is proportional to the spin wave wavevector and is temperature independent when T >> w . By taking the linear magneton elastic coupling into account, one can obtain the process of the linear transformation of a spin wave into a phonon (see fig. ib). In contrast to the expression (i) the relaxation frequency due to this combined magnon-phononfluctuation relaxation process is nonvanishing when K + 0. It was shown in ref. [5] that the process of linear transformation of a phonon into a NSW by the fluctuating nuclear spin is an efficient excitation mechanism of the NS in antiferromagnets. The analysis of the SW interaction with collective excitations of the NS shows that the strongest three wave processes are SW and NSW confluence into a phonon and SW splitting in a NSW and a phonon. For these processes F±
where @N = S/a, ek = (e0 2 + $2K2) I/2 is the SW
m.ph
4J(J+l) 3z
@2 Mv 2
Jo 3 3 WD
~n 2 2 Ek
'
(2)
where 2SKS/v << £k' @ = Ba3' B is the magnetoelastic constant, M is a mass of a unit cell, ~D = v/a, and v is the sound velocity. In (2) we use the relation b = J(J+I)~ /3T which is n valid for T >> ~ . The three wave processes mentioned above nare realized when both the hyperfine interaction and nonlinear magnetoelastic couplings [4] are taken into account (Fig. 2) 1071
1072 4 F imp n2m
(cB')2 B' T 28
g
o n Wn2gkS K
2 o
+F
, (4)
Co2_HA2+F
where F = ek2Wn2/(SK) 4 and (SK/~k) 2 >> 2Wn/~ k.
Figure 2 : Diagram for the three Wave SW - NSW phonon process. The straight line represents the magnon, the wavey line - the phonon and the double line the excitation in the NS. Besides the processes considered above, the SW interaction with the longitudinal component of an impurity spin may be essential in a real EP AF. The linear interaction between the magnon amplitude and an impurity spin [6] leads to the so called "slow" SW relaxation (fig. 3a)
The nonlinear effects in the SW system are also considered. The splitting of a SW into a phonon and NSW is responsible for the finite SW amplitude instability. In particular, the threshold field of the parametric instability under perpendicular pumping in an EP AF is the following: 2 ~h
= 2A~(QnOph)i/2
c
•
sl
cB' =
~o
2 3 ~Jo " 6k2
~nk
~n
Eo x ~HA
(5)
Ck~l (gkTi)2 + 1
,
(3)
where c is the impurity concentration, ~0 is the impurity precession frequency in a molecular field, B' = dB/dy0 is the derivative of the appropriate Brillouin function, ~ % i, and T is I the spin-lattice relaxation time for an impurity. The analogous expression for J. = 1/2 was obtained in ref. [7] at gkTl >> i. Imp
where Aw is the AFMR linewidth,
REFERENCES:
[3]
[4] [5] [6] [7] Figure 3 : (a) Diagram for the "slow" SW relaxation mechanism by the paramagnetic impurity. (b) The effective anharmonic interaction between spin waves and transverse components of nuclear spins. The black points represent the z-component of an impurity spin. By using the substitution b'w 2 + cB,w02 one can n easily obtain from (i) the expresslon for the SW relaxation rate due to the elastic SW scattering by an impurity. The interference of the two impurity mechanisms indicated above leads to an effective three wave interaction between the NSW and the SW (fig. 3b). The SW relaxation frequency in these processes is proportional to the temperature of the NS and depends on field
~nk is
The theoretical results obtained are compared with the experimental data of ref. [8].
[2]
(b)
Qn and Qph are
the nuclear and the subsystem qualities, the NSW spectrum.
[i]
(e)
u(H+HD)O
Mv2 1/2 i ×[7o]
2 Fimp
Ek
[8]
De Gennes, P., Pincus, P., Hartman-Boutron, F., Winter, J., Phys. Rev. 129 (1963) 1105-1115. Lutovinov, V., Saphonov, V., Fiz. Tverd Tela, 21 (1979) 2772-2783. Lutovinov, V., Abstr. 21st National Conf. Low Temp. Phys., USSR, Kharkov, part II (1980) 202-203. Lutovinov, V., Preobrajenskii, V., Semin, S., Zh. ETP, 74 (1978) 1159-1169. Bunkov, Yu., Lutovinov, V., Monakhov, A., Zh. ETP, 80 (1981) 1449-1468. Hartman-Boutron, F., Phys. Kondens. Mater., 2 (1964) 80-98. Mikhailov, A., Farzetdinova, R., Abstr. 21st National Conf. Low Temp. Phys., USSR, Kharkov, part II (1980) 212-213. Andrienko, A., Prozorova, L., Zh. EPT, 78 (1980) 2411-2418.