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The NSW interaction with nuclear local modes in antiferromagnets
~
Solid State Communications, Vol.53,No.10, pp.867-872, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
THE N8W INTERACTION WITH NUCLEAR LOCAL MODES IN ANTIFERROMAGNETS
D.A.Garanin, V.B.Lutovinov Institute of Radioengineering, Electronics and Automation,
117~.54-, Moscow, USSR
(Received 28 December 1984 by V.~.Agranovlch) The nuclear spin wave (NSW) relaxation rates due to the interaction with nuclear impurity local modes (NILM> in antiferromagnets are considered in the framework of the Keldysh formalism based on the spin operator diagram technique. It is shown that the NSW relaxation frequency due to the scattering on the fluctuations of impurity nuclear spins is of resonance character. The NSW damping due to the resonance absorption by NILM depends strongly on the NSW amplitude N, which accounts for the "hard" excitation of NSW's in parallel pumping experiments /6/. The NSW relaxation rates due to the processes involving two NSW's and one impurity nuclear excitation are also calculated.
I. Introduction. The indirect nuclear spin-spin interaction in magnetically ordered materials via eleatronic magnons (the Suhl-Nakamura one /I/) gives rise to nuclear spin waves (NSW) which are the subject of numerous investigations, both theoretical / 2 - ~ / a n d experimental /5,6/. NSW's exist in the range ~f temperatures T ~ T ' a n d wavevectors k ~ k ~ k ~ /~/ where their dispersion is proflounced, ag~K/$K ~ ~ ( ~ a and ~K are the NBW frequency and damping). In the easy plane antiferromagnets (AFEP) the NSW region is especially wide d u e to the exr~ o~ange e n h ~ e m e n t effect: T , ~ ( ~ / 6 o ) ~ , k~k^(T/T )~, k~=k^(T~/T)~S.=~He@e ~ m i~ un~ulled NMR ~req~ency, ~ E S J is exchange frequency, 5 ~ = m k ^ is AFMR frequency, s is the elect~oni~ magnon velocity. In the temperature range T g ~ (particularly for helium ones) the math source of the NSW relaxation in ideal crystals is the NBW scattering on the thermal fluctuations of the nuclear spin's longitudinal components, whereas the other contributions into the NSW damping (such as due to the NSW scattering on each ~ther) contain additional powers of (T/TW) m /~/. In non-ideal crystals one has to take into account various many-stage indirect interactions involving de~rees of a freedom associated with impurities. Recently the interaction of nuclear subsystem with electronic impurity spins in antiferromagnets has been considered in ref./7/. It was shown that the NSW scattering on the fluctuations of electronic spins of impurities and the relaxational absorption of NSW's are the leading sources of the NSW damping in this case. 867
One more groupe of improper mechanisms of the NSW relaxation arises from the interaction of nuclear subsystem with nuclear impurity spins. One expects that such processes are of great importance if the NSW f r e q u e n c y ~ is close to the nuclear impurity local mode (NILM) one ~ . This is especially plausible for AFEP where the NSW pulling may be made large. The consideration of the NSW interaction with NILM is the subject of the present paper. The NSW relaxation frequencies due to the processes I) of the resonance absorption N S W ~ N I L M ; 2) of the improper fluctuational scattering; 3) of the type 2 NSW-~NILM are calculated in the framework of quantum kinetic equations based on the spin operator diagram technique. It is shown that the NSW damping due to the improper fluctuational scattering increase resonantly when ~ $ ~ . The rescmance absorption process is easily saturated with the increase of the NSW amplitude. This nonlinear behavior of the NSW damping accounts for the "hard" parametric excitation of NSW's in parallel pumping experiments /6/. 2. The hamiltonian. The hamiltonian of the system under consideration contains th~ exchange interactions of ~he ~ype SS, S ~ and_ the hyperfine ones IS, ~i, where S and I are the electronic and nuclear spins of a matrix, ~ a n d ~ are those of an impurity. ~enerally, the equilibrium directions of S and may be noncollinear~ ~n this~c~se the terms of the form B g~ and S~g appear in the hamiltonian. Since the energies of nuclear degrees of a freedom are much less
868
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
than electronic ones, one may construct an effective hamiltonian containing indirect interactions between host and impurity nuclear spins through virtual electronic magnons and impurity local modes. Suc~ #ff~ct~ve nuclear hamiltonian reads H = ~ + H i + H n i where -
_
(I)
describes the host-host nuclear spin interaction (complete form of ~ containing multispin ones see in ref./@/~, H',
=
-{" "~-f._~ ''<
-
Vol. 53, NO. I0
Fig.3. The amplitude U x. The w a v ~ line corresponds to the interaction S 0-~ (the l-st term) and to the one 8 ± 6 ~ (the 2-nd and the 3-rd terms). The host part of the h-mlltonian may be diagonalized via the unitary transformation /8,@/
+= exp(~.) H exp(-P,)
(21
(6)
describs~the interaction of impurity nuclear spins with each other and
H,<~
where ~-ln(~Jn/~k) , ~)k is the NSW spect~
~' ~
-~'
(5) .
:
describes the host-impurity~uclear spin one. In eqs. (1)-(5) m~= I~+l~;~=+,z; ~A and I~ are the nuclear spin operators belonging to A and B sublattices (each determined in its own coordinate system). The impurity nuclear spin operators i'are determined in a similar way (with proper occupation probabilities). The amplitudes in eqs. (1)-(5) are of the form
-
4 V.lW.4
is the nuclear spin polarization. result one gets
_~'!W~'+
'-
-
J_
"'
+--
- IV~'~-~ u~'~lrL~"r[L~k<) -~+~, ~ ~'~-Q
.
7F x - 6 ~
As a
(8)
~)~'% where
Here ~.~---('£~+ ~iK~71is the dispersion law of lowfrequency electronic magnons,A] (~=I,2,5) are numerical factors depending on the relative direction of quantization ax~s of~electronic and impurity spins (if ~S>ll~>, one has ~ : ~ = l , ~s=O). The diagrammatic representation of the interaction vertices above is given by figs.
Fig.1. The amplitudes V, ~ i and ~ . The straight line represents the electronic magnon Green's function (GF)) the broken one denotes the electronic ILM GF.
Fig.2. The amplitude U z. The wavy line represents the hyperfine coupling between (longitudinal components of) electronic and nuclear spins.
The n o n r e s o n a n t terms o f t h e t y p e ( i + i + + h.c.) are o m i t t e d i n eq. (8~ since their contributions are small as a r ( C is the impurity concentration). The ~-th term of the hamiltonian eq.(8) causes a weak pulling of impurity excitations (proportional to C ) which here will be neglected. The transformed nuclear hamiltonian eq. (8) allows one to calculate the ISW relaxation frequencies due to the interaction with paramagnetic impurities in terms of diagonal Green's functions. 5- The Keldysh formalism for spin systems. The kinetic properties of nuclear subsystem are considered here within the Keldysh formalism /9/ generalized for the spin systems in ref./IO/. The excitations of a spin system are described by transverse GF of the form
Vol. 53, No.
where t is the impurity concentration, b i is the polarization of nuclear impurity spins (at equilibrium bi-bi°- iBi (~Di/T)), I h i ~ L 0 ) is the NILM lineshape. After the averaging over the distribution of the NILM frequencies ~ ( ~ ) o n e obtains
where
.1.
-
+ ,
+
-
I
+
/,
-
~'~_
i
~
-
+
(16)
s c
(~) -
869
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
I0
,
Here we take into account that in magnetically ordered materials the dispersion ~ ; of the distribution ~P(~) is usually much and ~ are chronological and antichronogreater than the width PL o f I ~ G ~ (AG)~IOMHz, logical operators, ~ (t) and ~+(t') are F[ ~ q ~ 6 0 ~ - N ,~60~-N~1 MHz for the manganese the Heisenberg spin ones. As it was shown ions). Dependence o f ~ ~ on the NSW frequin ref /9/, there are two independent com- ency a)~ can be measured experimentally in binations of GF's introduced above: G+(t,~ ) order to extract the distribution function ~(~[). The temperature dependence of . ~ associated with the generalized distribuis determined by the p o l a r i z a t i o n 3 ~ of tion function and retarded GF G~(t,t ' )= co<6~ G~-G~= G--G c describing the dynamic proper- electronic impurity spins: ~ o ~ ties of the system with given distribution. Note that the resonance increase of the NSW damping observed by Weber and Seavey The generalized kinetic equation for /11/ in RbMnFs at the magnetic field H . spin system is written in the form 20KOe may be explained by the process under consideration. ~4 The 6-th term of the hamiltonian eq. (8) where the Fourier transformation with res- together with the 5-th one in the higher order of a perturbatiob theory give rise pect to the time variable t-t' is carried to the NSW scattering on the thermal flucout, ~ (4,~) and Z~(~, ~)) a~e appropriate tuations of the impurity nuclear spin lonelements of matrix self energy part. The gitudinal components. The corresponding retarded GF G z obeys the Dyson equation relaxation frequency is given by the expression (the diagrammatic representation of the self energy part see in ref./~a/) -
where ~,% = ~ a _ ~ + = ~ - _ ~ c . The concrete form of ~ in eqs.(12),(1~) depends on the interaction amplitudes in the system under consideration (see below). Reduction of eq.(~2) to usual kinetic equation may be realized by the integration over frequency ~ if the spectral dSs~ribution @+ satisfies the condition y ~ G ~ ) ~ <
where W is the transition probability under the influence of pumping. For the nuclear subsystem at equilibrium b=IBr(I~)n/T) , B (x) is the Brillouin function. The NSW spec@rum is determined from the Dyson equation (15) of the chain type /5/ involving the 5-rd term of the hamiltonian eq.(8) wich results in eq. (7).
•
I where b i &~ i(i+~)/~ and
" : 4 e (v < v: v,
) ,
<
)
is the fluctuational scattering amplitude. Carrying out the integration in eq.(~7) one gets
wher@ ~ . i s structure dependent factor (~-27~.3~/£ for s.c. lattice and ~ = 2 for b.~c.c, one), cg~ is the transverse relaxation time of NILM' s. Eq. (19a) corresponds to the case l~)k-U31l<<~U)£ , whereas eq.(19b) describes the..op2osite one. The resonance behavior o f ~ is due to the second term ~. The NSW relaxation. in eq.(18). It should be noted that T, k . The 5-th term of the hamiltonian eq.(8) and U3K dependence of the f o r m T K ' ( ~ - ~ ) -~ is responsible for the resonance absorption was experimentally observed in ref./8/ in antiferromagnetically ordered RbMnF~. In of NS~'s by nuclear impurity local modes. this connection the NMR investigation of The NBW relaxation frequency due to this the nuclear local modes in materials of process is given by the formula such a type would be of great interest. The last terms of the hamiltonian eq. (8) give rise to the processes of Raman NSW scattering with absorption or emission
87O
Vol. 53, No. 10
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
of NILM I ~ / and the process of confluence of two NSW's into NILM. Corresponding NSW relaxation rates are of the form Fig.4. The income term in the kinetic equation for b i. The double solid line represents the NSW GF, the double broken one represents the NILM GF. and reads
and
r<,- oo
-
In eq. (20) the processes of absorption and emission of NILM are allowed in the regions G)~ - ~ a & UJa-~[ and~o+~;~)~--~ ~ correspondingly. These two processes give nonzero contribution into the NSW relaxation rate if the NSW pulling is sufficiently large: ~)o -------~-~)~ • The confluence process (eq.(2~)) is allowed for the NSW energies u)~-~Ja~_~)~&~);-~ and the NILM frequencies P%Oe ~ ~)~ -~ ~ ) ~ The integration in eq. (20).with the use
of
>>i
The stationary solution of the kinetic equation is of the form
where ~ £ is the NILM longitudinal relaxation time. For the Lorenzian lineshape from eqs.(~5),(25),(24) one gets
yields where
.
+
_
.
~;b = ( l q ) - i a n d
_
where t h e q u a n t i t i e s %A and 9m a r e d e f i n e d One can see t h a t t h e a b s o r p t i o n o f NSW b y by the conditions ~ @ I I ~ + ~ Z a n d ~ = ~ - ~ ) ~ NILM causes effective broadening of NILM In the l i m i t ~ ; < < ~ t h e Raman processes be-and, as a result, the decrease of the NSW come quasielastic which results in % { ~ ~ K . damping due to this process. Averaging of Eq.(22) should be compared with eq[.m(99~) ~ eq.(25) over the distribution~(~£) gives containing a resonance factor ( q - ~ / ~ z ~ . An analogous calculation in eq.(2J) yields one the terms in e q . ( 2 2 ) w i t h : (27) ~9~$where ~ is determined b y ~ ¢ = ~ - ~ . C~mparisoh 9~ improper contributions where ~ { 0 ) ~ ~ is given by eq. (16). ~:~ and ~ c with proper one due to The nonlinear behavior of the NBW damthe NSW scattering on the host nuclear ping considered above leads to the "hard" spin fluctuations /4/ excitation of NSW's in parallel pumping experiments. Indeed, if the number of parametrically excited NSW's beyond the threshold h c (h is the amplitude of the r.f. field) is limited by the phase mechanism, shows t h a t f o r ~ i ~ m t h e former are subone gets /q4/ stantial in the range & @ ( 4 a l ~ ) 5 ( K I K o J
of
5. Nonlinear NSW relaxation and the "hard" mechanism of the NSW excitation. if the nuclear subsystem of a magnet undergoes parallel pumping /5,6/ there is a NSW of a finite amplitude N (N is a number of nuclear magnons per unit cell) travelling through the crystal. The resonance interaction of such NSW with NILM may drive impurity nuclear polarization b i away from its equilibrium value. Under such conditions the NSW damping due to the resonance absorption by NILM is given by eq.(15) with b i determined from the kinetic equation of the type eq.(q4). The income term in this equation is represented in fig.4
~.-,~V¢ut
0..o._
'/u.
lS u) = C v) r Cu),
where S~ is four-NSW scattering amplitude, V is th4 coupling between the r.f. field and the nuclear subsystem and P(N) is the total NSW relaxation rate which may be represented as
F(u) :
+
(29)
(see eq.(27)). One can see that the dependence ~ a ~ ) results in the hysteresis in the d6pendence N(h). Whereas the upper crltical amplitude hcl reads
Vol. 53, No.
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
10
871
function of the electronic impurity excitation is "splitted" into a sum of GF's (corresponding to the transitions Im~In) the lower one hc~ is determined by the equ- between nonequidi~tant energy levels) of the form (~m~-~)with proper weights. ations If the transition frequency ~ol between the ground and the first excited s t a t e s ( ~ ) is much less than the others (~0l~m~-----~) the corresponding GF G °i gives the dominant contribution into indirect interactions between host and impurity nuclear JN + = Bimce in the cases of interest the quantity spins..As a result, the indirect amplitudes vn~,v ~ and uZ,~ in eqs.(2),(~) are replaced by ~ V n~, ~ V ~ and ~ U Z , x where
(
.dr(NI
+ ff( v i
o
is large ( Z ~ 1 0 9 - 1 0 4 ) , eqs.(tq) may be solved analitically, which yields the value of the "hardness" ~-(hc±-hc~)/hcl__
(34)
(Concrete evaluation of indirect interactions for particular configurations of crystal field may be carried out with the - 4-#/(aa#'i use of the universal basis operator diagram technique /15-17/). The quantity ~h;beyond~ The results obtained show the parametric excitation thre- is proportional to 1/T in the temperature range ~ o i ~ T ~ where the NILM frequenshold the channel of resonance NSW abcy ~ ~ o < ~ > ~ / T ~ is practically consorption by NILM is practically saturated. stant, in accordance with the results of ref./6/. The final expressions for the 6. Discussion. NSW relaxation frequencies are of the form The main result of the present paper is the mechanism of the "hard" parametric excitation of NSW's based on the nonlinear behavior of the NSW damping due to the remo is given by eqs.(25),(27)) and ~ = sonance absorption process. It seems plau- ( ~ ~)~ for the fluctuational, scattering sible that such mechanism is responsible for the resonance increase of the hardness and confluence processes (see eqs.(qt), at the frequency f - 400 MHz that was ob- (22)). In this paper the hardness S of paraserved experimentally on CsMnF~ in ref. /6/. The corresponding value o f ~ ( 0 ) c a l c u - metric excitation of NSW's has been calculated according to e q s . ( ~ O ) , ( ~ ) turns out lated in resonance ( l ~ K - ~ i l ~ ) . The full to be temperature independent for T ~ 2 K frequency dependence of ~ may be calculated in a similar way with the use of eq. and decreasing in T for T ~ 2 K, which qualitatively agrees with the d e p e n d e n c e ~ ( 0 ) (25). One must remember that the linear o~<~>~obtained above (see eq.(16)). HowNBW relaxation frequency ~o . $ n , e q . - ~ ) contains the resonance par~ ~ - ~ J ~ eq.(19). ever, as has been pointed out b$ A. Yu. YaOne may conceive one more type of expekubovsky,(private communication), the reriments concerning the "hard" excitation sonance frequency ~i corresponding to the of NSW's. If the polarization of nuclear maximum of the hardness S measured in impurity spins b i is driven to zero by ref./6/ remains practically unchanged in this temperature range, although according means of phonon or perpendicular r.f. pumping on the frequency ~[ , the process of to the present theory ~ L c o & g ~ varies as the resonance NSW absorption by NILM is The discrepancy mentioned above may be cut off, and the hardness N disappears. The authors thank V.M.Agranovich, M.A. explained by the influence of the crystal field on the energy levels of electronic Bavchenko and A.Yu.Yakubovsky for the impurity spins. In this case the Green,s discussion of the results of the paper.
<
REFERENCES I. H.Suhl, Phys.Rev.,109,606 (1958); T. Nakamura, Progr. Theor.Phys., 20,542 (I 958) • 2. P.G.de Gennes,P.A.Pincus,F. Hartmann-Boutron,J.M.Winter, Phys.Rev. ,129,1105 (1963). 3- P.M.Richards, Phys.Rev.,l_~,581 (1968). 4. D.A.Garanin,V.S.Lutovinov, Zh.Exp.Teor.Fiz.,85,2060 (1983); V.S.Lutovinov,M.A.Savchenko,D.A. Garanin, I E ~ Trans. ~ MAG-19,q97@ (1983). 5- V.A. Tulin, Fiz.Niz.Temper.,~,965 (1969). 6. A.V. Andrienko, Thesis, I.V.Kurchatov Institute, Moscow, 1983. 7- V.S.Lutovinov, Phys.Lett. ,97A,357 (1983).
872
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
Vol. 53, No. |0
8. D.A.Garanin,V.S.Lutovinov, Solid State Comm.,~u~,1359 (1982). 9. L.V.Keldysh, Zh.Exp.Teor.Fiz.,~7,1515 (196~).-10. D.A.Garanin,V.S.Lutovinov, I n : - ~ l Union Conf. on Magnetic Phenomena, U ~ , Tula, 9 sept. 1983, p.184. 11. R.Weber,M.H.Seavey, Solid State Comm.,7,619 (1969). 12. S.A. Govorkov, V. A. Tulin, Zh °Exp. Teor. Fi z., 7~, 389 (I 968 ). 13. V.S.Lutovinov, In: Intern. Conf. on Low Temperature Physics, Bulgaria, Varna, 1983 , o. I66 . I@. V.E.Zakharcv, ~.S.L'vov,S.S.StaroDinewz, Usp.Fiz.Nauk~10@,609 (197@). 15. D.A.Garamin,V.S.Lutovinov, Teor.Mat.Fiz.,55,106 (1983~. 16. D.A.Garanim,V.B.Lutovinov, Teor.Mat.Fiz.,~,133 (198@). 17. D.A.Garamin,V.S.Lutovinov, Phys.Stat.Sol.,--T~___~,~133(198@).
Solid State Communications, Vol.53,No.10, pp.867-872, 1985. Printed in Great Britain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
THE N8W INTERACTION WITH NUCLEAR LOCAL MODES IN ANTIFERROMAGNETS
D.A.Garanin, V.B.Lutovinov Institute of Radioengineering, Electronics and Automation,
117~.54-, Moscow, USSR
(Received 28 December 1984 by V.~.Agranovlch) The nuclear spin wave (NSW) relaxation rates due to the interaction with nuclear impurity local modes (NILM> in antiferromagnets are considered in the framework of the Keldysh formalism based on the spin operator diagram technique. It is shown that the NSW relaxation frequency due to the scattering on the fluctuations of impurity nuclear spins is of resonance character. The NSW damping due to the resonance absorption by NILM depends strongly on the NSW amplitude N, which accounts for the "hard" excitation of NSW's in parallel pumping experiments /6/. The NSW relaxation rates due to the processes involving two NSW's and one impurity nuclear excitation are also calculated.
I. Introduction. The indirect nuclear spin-spin interaction in magnetically ordered materials via eleatronic magnons (the Suhl-Nakamura one /I/) gives rise to nuclear spin waves (NSW) which are the subject of numerous investigations, both theoretical / 2 - ~ / a n d experimental /5,6/. NSW's exist in the range ~f temperatures T ~ T ' a n d wavevectors k ~ k ~ k ~ /~/ where their dispersion is proflounced, ag~K/$K ~ ~ ( ~ a and ~K are the NBW frequency and damping). In the easy plane antiferromagnets (AFEP) the NSW region is especially wide d u e to the exr~ o~ange e n h ~ e m e n t effect: T , ~ ( ~ / 6 o ) ~ , k~k^(T/T )~, k~=k^(T~/T)~S.=~He@e ~ m i~ un~ulled NMR ~req~ency, ~ E S J is exchange frequency, 5 ~ = m k ^ is AFMR frequency, s is the elect~oni~ magnon velocity. In the temperature range T g ~ (particularly for helium ones) the math source of the NSW relaxation in ideal crystals is the NBW scattering on the thermal fluctuations of the nuclear spin's longitudinal components, whereas the other contributions into the NSW damping (such as due to the NSW scattering on each ~ther) contain additional powers of (T/TW) m /~/. In non-ideal crystals one has to take into account various many-stage indirect interactions involving de~rees of a freedom associated with impurities. Recently the interaction of nuclear subsystem with electronic impurity spins in antiferromagnets has been considered in ref./7/. It was shown that the NSW scattering on the fluctuations of electronic spins of impurities and the relaxational absorption of NSW's are the leading sources of the NSW damping in this case. 867
One more groupe of improper mechanisms of the NSW relaxation arises from the interaction of nuclear subsystem with nuclear impurity spins. One expects that such processes are of great importance if the NSW f r e q u e n c y ~ is close to the nuclear impurity local mode (NILM) one ~ . This is especially plausible for AFEP where the NSW pulling may be made large. The consideration of the NSW interaction with NILM is the subject of the present paper. The NSW relaxation frequencies due to the processes I) of the resonance absorption N S W ~ N I L M ; 2) of the improper fluctuational scattering; 3) of the type 2 NSW-~NILM are calculated in the framework of quantum kinetic equations based on the spin operator diagram technique. It is shown that the NSW damping due to the improper fluctuational scattering increase resonantly when ~ $ ~ . The rescmance absorption process is easily saturated with the increase of the NSW amplitude. This nonlinear behavior of the NSW damping accounts for the "hard" parametric excitation of NSW's in parallel pumping experiments /6/. 2. The hamiltonian. The hamiltonian of the system under consideration contains th~ exchange interactions of ~he ~ype SS, S ~ and_ the hyperfine ones IS, ~i, where S and I are the electronic and nuclear spins of a matrix, ~ a n d ~ are those of an impurity. ~enerally, the equilibrium directions of S and may be noncollinear~ ~n this~c~se the terms of the form B g~ and S~g appear in the hamiltonian. Since the energies of nuclear degrees of a freedom are much less
868
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
than electronic ones, one may construct an effective hamiltonian containing indirect interactions between host and impurity nuclear spins through virtual electronic magnons and impurity local modes. Suc~ #ff~ct~ve nuclear hamiltonian reads H = ~ + H i + H n i where -
_
(I)
describes the host-host nuclear spin interaction (complete form of ~ containing multispin ones see in ref./@/~, H',
=
-{" "~-f._~ ''<
-
Vol. 53, NO. I0
Fig.3. The amplitude U x. The w a v ~ line corresponds to the interaction S 0-~ (the l-st term) and to the one 8 ± 6 ~ (the 2-nd and the 3-rd terms). The host part of the h-mlltonian may be diagonalized via the unitary transformation /8,@/
+= exp(~.) H exp(-P,)
(21
(6)
describs~the interaction of impurity nuclear spins with each other and
H,<~
where ~-ln(~Jn/~k) , ~)k is the NSW spect~
~' ~
-~'
(5) .
:
describes the host-impurity~uclear spin one. In eqs. (1)-(5) m~= I~+l~;~=+,z; ~A and I~ are the nuclear spin operators belonging to A and B sublattices (each determined in its own coordinate system). The impurity nuclear spin operators i'are determined in a similar way (with proper occupation probabilities). The amplitudes in eqs. (1)-(5) are of the form
-
4 V.lW.4
is the nuclear spin polarization. result one gets
_~'!W~'+
'-
-
J_
"'
+--
- IV~'~-~ u~'~lrL~"r[L~k<) -~+~, ~ ~'~-Q
.
7F x - 6 ~
As a
(8)
~)~'% where
Here ~.~---('£~+ ~iK~71is the dispersion law of lowfrequency electronic magnons,A] (~=I,2,5) are numerical factors depending on the relative direction of quantization ax~s of~electronic and impurity spins (if ~S>ll~>, one has ~ : ~ = l , ~s=O). The diagrammatic representation of the interaction vertices above is given by figs.
Fig.1. The amplitudes V, ~ i and ~ . The straight line represents the electronic magnon Green's function (GF)) the broken one denotes the electronic ILM GF.
Fig.2. The amplitude U z. The wavy line represents the hyperfine coupling between (longitudinal components of) electronic and nuclear spins.
The n o n r e s o n a n t terms o f t h e t y p e ( i + i + + h.c.) are o m i t t e d i n eq. (8~ since their contributions are small as a r ( C is the impurity concentration). The ~-th term of the hamiltonian eq.(8) causes a weak pulling of impurity excitations (proportional to C ) which here will be neglected. The transformed nuclear hamiltonian eq. (8) allows one to calculate the ISW relaxation frequencies due to the interaction with paramagnetic impurities in terms of diagonal Green's functions. 5- The Keldysh formalism for spin systems. The kinetic properties of nuclear subsystem are considered here within the Keldysh formalism /9/ generalized for the spin systems in ref./IO/. The excitations of a spin system are described by transverse GF of the form
Vol. 53, No.
where t is the impurity concentration, b i is the polarization of nuclear impurity spins (at equilibrium bi-bi°- iBi (~Di/T)), I h i ~ L 0 ) is the NILM lineshape. After the averaging over the distribution of the NILM frequencies ~ ( ~ ) o n e obtains
where
.1.
-
+ ,
+
-
I
+
/,
-
~'~_
i
~
-
+
(16)
s c
(~) -
869
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
I0
,
Here we take into account that in magnetically ordered materials the dispersion ~ ; of the distribution ~P(~) is usually much and ~ are chronological and antichronogreater than the width PL o f I ~ G ~ (AG)~IOMHz, logical operators, ~ (t) and ~+(t') are F[ ~ q ~ 6 0 ~ - N ,~60~-N~1 MHz for the manganese the Heisenberg spin ones. As it was shown ions). Dependence o f ~ ~ on the NSW frequin ref /9/, there are two independent com- ency a)~ can be measured experimentally in binations of GF's introduced above: G+(t,~ ) order to extract the distribution function ~(~[). The temperature dependence of . ~ associated with the generalized distribuis determined by the p o l a r i z a t i o n 3 ~ of tion function and retarded GF G~(t,t ' )= co<6~ G~-G~= G--G c describing the dynamic proper- electronic impurity spins: ~ o ~ ties of the system with given distribution. Note that the resonance increase of the NSW damping observed by Weber and Seavey The generalized kinetic equation for /11/ in RbMnFs at the magnetic field H . spin system is written in the form 20KOe may be explained by the process under consideration. ~4 The 6-th term of the hamiltonian eq. (8) where the Fourier transformation with res- together with the 5-th one in the higher order of a perturbatiob theory give rise pect to the time variable t-t' is carried to the NSW scattering on the thermal flucout, ~ (4,~) and Z~(~, ~)) a~e appropriate tuations of the impurity nuclear spin lonelements of matrix self energy part. The gitudinal components. The corresponding retarded GF G z obeys the Dyson equation relaxation frequency is given by the expression (the diagrammatic representation of the self energy part see in ref./~a/) -
where ~,% = ~ a _ ~ + = ~ - _ ~ c . The concrete form of ~ in eqs.(12),(1~) depends on the interaction amplitudes in the system under consideration (see below). Reduction of eq.(~2) to usual kinetic equation may be realized by the integration over frequency ~ if the spectral dSs~ribution @+ satisfies the condition y ~ G ~ ) ~ <
where W is the transition probability under the influence of pumping. For the nuclear subsystem at equilibrium b=IBr(I~)n/T) , B (x) is the Brillouin function. The NSW spec@rum is determined from the Dyson equation (15) of the chain type /5/ involving the 5-rd term of the hamiltonian eq.(8) wich results in eq. (7).
•
I where b i &~ i(i+~)/~ and
" : 4 e (v < v: v,
) ,
<
)
is the fluctuational scattering amplitude. Carrying out the integration in eq.(~7) one gets
wher@ ~ . i s structure dependent factor (~-27~.3~/£ for s.c. lattice and ~ = 2 for b.~c.c, one), cg~ is the transverse relaxation time of NILM' s. Eq. (19a) corresponds to the case l~)k-U31l<<~U)£ , whereas eq.(19b) describes the..op2osite one. The resonance behavior o f ~ is due to the second term ~. The NSW relaxation. in eq.(18). It should be noted that T, k . The 5-th term of the hamiltonian eq.(8) and U3K dependence of the f o r m T K ' ( ~ - ~ ) -~ is responsible for the resonance absorption was experimentally observed in ref./8/ in antiferromagnetically ordered RbMnF~. In of NS~'s by nuclear impurity local modes. this connection the NMR investigation of The NBW relaxation frequency due to this the nuclear local modes in materials of process is given by the formula such a type would be of great interest. The last terms of the hamiltonian eq. (8) give rise to the processes of Raman NSW scattering with absorption or emission
87O
Vol. 53, No. 10
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
of NILM I ~ / and the process of confluence of two NSW's into NILM. Corresponding NSW relaxation rates are of the form Fig.4. The income term in the kinetic equation for b i. The double solid line represents the NSW GF, the double broken one represents the NILM GF. and reads
and
r<,- oo
-
In eq. (20) the processes of absorption and emission of NILM are allowed in the regions G)~ - ~ a & UJa-~[ and~o+~;~)~--~ ~ correspondingly. These two processes give nonzero contribution into the NSW relaxation rate if the NSW pulling is sufficiently large: ~)o -------~-~)~ • The confluence process (eq.(2~)) is allowed for the NSW energies u)~-~Ja~_~)~&~);-~ and the NILM frequencies P%Oe ~ ~)~ -~ ~ ) ~ The integration in eq. (20).with the use
of
>>i
The stationary solution of the kinetic equation is of the form
where ~ £ is the NILM longitudinal relaxation time. For the Lorenzian lineshape from eqs.(~5),(25),(24) one gets
yields where
.
+
_
.
~;b = ( l q ) - i a n d
_
where t h e q u a n t i t i e s %A and 9m a r e d e f i n e d One can see t h a t t h e a b s o r p t i o n o f NSW b y by the conditions ~ @ I I ~ + ~ Z a n d ~ = ~ - ~ ) ~ NILM causes effective broadening of NILM In the l i m i t ~ ; < < ~ t h e Raman processes be-and, as a result, the decrease of the NSW come quasielastic which results in % { ~ ~ K . damping due to this process. Averaging of Eq.(22) should be compared with eq[.m(99~) ~ eq.(25) over the distribution~(~£) gives containing a resonance factor ( q - ~ / ~ z ~ . An analogous calculation in eq.(2J) yields one the terms in e q . ( 2 2 ) w i t h : (27) ~9~$where ~ is determined b y ~ ¢ = ~ - ~ . C~mparisoh 9~ improper contributions where ~ { 0 ) ~ ~ is given by eq. (16). ~:~ and ~ c with proper one due to The nonlinear behavior of the NBW damthe NSW scattering on the host nuclear ping considered above leads to the "hard" spin fluctuations /4/ excitation of NSW's in parallel pumping experiments. Indeed, if the number of parametrically excited NSW's beyond the threshold h c (h is the amplitude of the r.f. field) is limited by the phase mechanism, shows t h a t f o r ~ i ~ m t h e former are subone gets /q4/ stantial in the range & @ ( 4 a l ~ ) 5 ( K I K o J
of
5. Nonlinear NSW relaxation and the "hard" mechanism of the NSW excitation. if the nuclear subsystem of a magnet undergoes parallel pumping /5,6/ there is a NSW of a finite amplitude N (N is a number of nuclear magnons per unit cell) travelling through the crystal. The resonance interaction of such NSW with NILM may drive impurity nuclear polarization b i away from its equilibrium value. Under such conditions the NSW damping due to the resonance absorption by NILM is given by eq.(15) with b i determined from the kinetic equation of the type eq.(q4). The income term in this equation is represented in fig.4
~.-,~V¢ut
0..o._
'/u.
lS u) = C v) r Cu),
where S~ is four-NSW scattering amplitude, V is th4 coupling between the r.f. field and the nuclear subsystem and P(N) is the total NSW relaxation rate which may be represented as
F(u) :
+
(29)
(see eq.(27)). One can see that the dependence ~ a ~ ) results in the hysteresis in the d6pendence N(h). Whereas the upper crltical amplitude hcl reads
Vol. 53, No.
THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
10
871
function of the electronic impurity excitation is "splitted" into a sum of GF's (corresponding to the transitions Im~In) the lower one hc~ is determined by the equ- between nonequidi~tant energy levels) of the form (~m~-~)with proper weights. ations If the transition frequency ~ol between the ground and the first excited s t a t e s ( ~ ) is much less than the others (~0l~m~-----~) the corresponding GF G °i gives the dominant contribution into indirect interactions between host and impurity nuclear JN + = Bimce in the cases of interest the quantity spins..As a result, the indirect amplitudes vn~,v ~ and uZ,~ in eqs.(2),(~) are replaced by ~ V n~, ~ V ~ and ~ U Z , x where
(
.dr(NI
+ ff( v i
o
is large ( Z ~ 1 0 9 - 1 0 4 ) , eqs.(tq) may be solved analitically, which yields the value of the "hardness" ~-(hc±-hc~)/hcl__
(34)
(Concrete evaluation of indirect interactions for particular configurations of crystal field may be carried out with the - 4-#/(aa#'i use of the universal basis operator diagram technique /15-17/). The quantity ~h;beyond~ The results obtained show the parametric excitation thre- is proportional to 1/T in the temperature range ~ o i ~ T ~ where the NILM frequenshold the channel of resonance NSW abcy ~ ~ o < ~ > ~ / T ~ is practically consorption by NILM is practically saturated. stant, in accordance with the results of ref./6/. The final expressions for the 6. Discussion. NSW relaxation frequencies are of the form The main result of the present paper is the mechanism of the "hard" parametric excitation of NSW's based on the nonlinear behavior of the NSW damping due to the remo is given by eqs.(25),(27)) and ~ = sonance absorption process. It seems plau- ( ~ ~)~ for the fluctuational, scattering sible that such mechanism is responsible for the resonance increase of the hardness and confluence processes (see eqs.(qt), at the frequency f - 400 MHz that was ob- (22)). In this paper the hardness S of paraserved experimentally on CsMnF~ in ref. /6/. The corresponding value o f ~ ( 0 ) c a l c u - metric excitation of NSW's has been calculated according to e q s . ( ~ O ) , ( ~ ) turns out lated in resonance ( l ~ K - ~ i l ~ ) . The full to be temperature independent for T ~ 2 K frequency dependence of ~ may be calculated in a similar way with the use of eq. and decreasing in T for T ~ 2 K, which qualitatively agrees with the d e p e n d e n c e ~ ( 0 ) (25). One must remember that the linear o~<~>~obtained above (see eq.(16)). HowNBW relaxation frequency ~o . $ n , e q . - ~ ) contains the resonance par~ ~ - ~ J ~ eq.(19). ever, as has been pointed out b$ A. Yu. YaOne may conceive one more type of expekubovsky,(private communication), the reriments concerning the "hard" excitation sonance frequency ~i corresponding to the of NSW's. If the polarization of nuclear maximum of the hardness S measured in impurity spins b i is driven to zero by ref./6/ remains practically unchanged in this temperature range, although according means of phonon or perpendicular r.f. pumping on the frequency ~[ , the process of to the present theory ~ L c o & g ~ varies as the resonance NSW absorption by NILM is The discrepancy mentioned above may be cut off, and the hardness N disappears. The authors thank V.M.Agranovich, M.A. explained by the influence of the crystal field on the energy levels of electronic Bavchenko and A.Yu.Yakubovsky for the impurity spins. In this case the Green,s discussion of the results of the paper.
<
REFERENCES I. H.Suhl, Phys.Rev.,109,606 (1958); T. Nakamura, Progr. Theor.Phys., 20,542 (I 958) • 2. P.G.de Gennes,P.A.Pincus,F. Hartmann-Boutron,J.M.Winter, Phys.Rev. ,129,1105 (1963). 3- P.M.Richards, Phys.Rev.,l_~,581 (1968). 4. D.A.Garanin,V.S.Lutovinov, Zh.Exp.Teor.Fiz.,85,2060 (1983); V.S.Lutovinov,M.A.Savchenko,D.A. Garanin, I E ~ Trans. ~ MAG-19,q97@ (1983). 5- V.A. Tulin, Fiz.Niz.Temper.,~,965 (1969). 6. A.V. Andrienko, Thesis, I.V.Kurchatov Institute, Moscow, 1983. 7- V.S.Lutovinov, Phys.Lett. ,97A,357 (1983).
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THE NSW INTERACTION WITH NUCLEAR LOCAL MODES
Vol. 53, No. |0
8. D.A.Garanin,V.S.Lutovinov, Solid State Comm.,~u~,1359 (1982). 9. L.V.Keldysh, Zh.Exp.Teor.Fiz.,~7,1515 (196~).-10. D.A.Garanin,V.S.Lutovinov, I n : - ~ l Union Conf. on Magnetic Phenomena, U ~ , Tula, 9 sept. 1983, p.184. 11. R.Weber,M.H.Seavey, Solid State Comm.,7,619 (1969). 12. S.A. Govorkov, V. A. Tulin, Zh °Exp. Teor. Fi z., 7~, 389 (I 968 ). 13. V.S.Lutovinov, In: Intern. Conf. on Low Temperature Physics, Bulgaria, Varna, 1983 , o. I66 . I@. V.E.Zakharcv, ~.S.L'vov,S.S.StaroDinewz, Usp.Fiz.Nauk~10@,609 (197@). 15. D.A.Garamin,V.S.Lutovinov, Teor.Mat.Fiz.,55,106 (1983~. 16. D.A.Garanim,V.B.Lutovinov, Teor.Mat.Fiz.,~,133 (198@). 17. D.A.Garamin,V.S.Lutovinov, Phys.Stat.Sol.,--T~___~,~133(198@).