00381098/82/42092105 $03.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 44, No. 6, pp. 921925, 1982. Printed in Great Britain.
SPIN RELAXATION IN METALS AT VERY LOW TEMPERATURES F. Shibata Department of Physics, Faculty of Science, Ochanomizu University, Bunkyoku, Tokyo 112, Japan and Y. Hamano Department of Pure and Applied Sciences, College of General Education, University of Tokyo, Komaba, Tokyo 153, Japan
(Received 24June 1982 by Y. Toyozawa) A general theory of spin relaxation in metals is developed from a statistical mechanical point of view. The theory is valid for all temperature domain and the multipletime characteristics of the relaxation process are completely determined: The relaxation times are strongly dependent on the temperature and magnetic field. At very low temperatures, behaviours of the relaxation times are quite different from the usual ones showing a saturation effect. Temperature variations of the relaxation times for 1 t> 1 (1 the magnitude of spin) are qualitatively different from those for I = 1/2. Namely, in the former case, the largest relaxation time has a maximum as a function of inverse temperature. I. INTRODUCTION IN RECENT YEARS there have been considerable advances in obtaining very low temperature [1,2]. Therefore, at the present time, it is rather easy to keep a physical system in the milliKelvin temperature domain. While, in some cases existing theories cannot treat phenomena at very low temperature. For instance, in the theory of spin relaxation, most of the treatments [3, 4] are based on the socalled "high temperature approximation" which is characterized by a condition x "¢ 1, where x = hCOL[2kT (coL the Larmor frequency). However, as the temperature is lowered and/or the magnetic field is increased, the quantity x exceeds the value of unity. This is realized in the miUiKelvin temperature domain for NMR whereas the condition is fulfilled in a temperature range of a few Kelvins for ESR under an applied static magnetic field of ~ 104 G. These conditions for the temperature and the magnetic field are accessible quite easily so that the "high temperature theories" break down completely in this case. Then it is highly necessary to construct a theory which is free from such a restriction like x "~ 1. Indeed, from a nonequilibrium statistical mechanical viewpoint, we have already succeeded in making a theoretical framework [58] which is valid for any values o f x . When the narrowing condition is satisfied, an average value of spin operator Iu is represented by 2I
(Itz)t = Co+ ~, Cn e t/rn, rl=l
(la=x,Y,Z),
(1)
where I is the magnitude of spin. In our theory, explicit forms of the cn's and rn's are known as a function o f x . The/dependent character like equation (1) manifests itself strongly when x exceeds unity. Several aspects of the nonlinear spin relaxation process are fully discussed in [58]. An extension of the theory of arbitrary timescale taking into account the nonMarkoffian effect has also been performed recently [912]. In this note, we apply our general formalism to the relaxation phenomena in metals in order to make a close contact with experiments at very low temperatures. In certain experiments, even a condition x >> 1 is realized with the use of an internal magnetic field of ~ l0 s G [13, 14]. Thus our theory will afford a necessary tool for analysis of experiments at very low temperatures with a strong magnetic field. An extension of the relaxation theory in metals is also necessary from a practical reason. An absolute value of the temperature itself at very low temperature domain is frequently determined by measuring T1 (the longitudinal relaxation time) and with the use of the Korringa relation which is derived within the "high temperature approximation". In the following we develop a fully quantum statistical mechanical theory which is valid for all temperatures. 2. FORMULATION Our whole system is composed of A (nuclear spins) and B (conduction electrons) subsystems, and the latter 921
922
SPIN RELAXATION IN METALS AT VERY LOW TEMPERATURES
is assumed to be in thermal equilibrium with temperature T. Then the total Hamiltonian is decomposed into (2)
J( = ~fo+~f ', where Jfo is the sum of Jfa and ~B; J(' represents a hyperfine coupling between spins of conduction electrons S/s and a nuclear spin I located at the origin. Namely, we have Jr' : h a ~ ~(rj)Sj'I,
(3)
where we have specialized to the Fermi contact interaction for simplicity and put (4)
In a language of second quantization (3) becomes J£ ' : haS" I
1
c~ow,Ck, v,u~(O)Uk,(O),

Equation (7) together with equations (9)(12) completely determines the relaxation process of the nuclear spin 1.
In the following we confine ourselves to a simplified treatment where the narrowing condition is satisfied [58], namely, the correlation time of the conduction electron system is so short that we can replace I'(t) by I~(~) in equation (8). Then we introduce a notation like ¢+(wn) = ¢+_(t = ~).
(13)
(6)
~z)t      { ( l z ) t + T1
[•(•+ 1) ¢Jz2)t] thx},
(14)
where
where c ~ (c~) creates (annihilates) a conduction electron with wave number k and spin v, and Uk(r) is the periodic part of the Bloch function; o is the usual Pauli spin matrix. With the use of the timeconvolutionless projection formalism [ 1517], we ffmd an equation of the density matrix for the Asubsystem alone [18]: =
(12)
Moreover we will use a noninteracting conduction electron model in calculating ~'s. An equation for (lz)t is obtained from equation (7):
k, k' 1.',v'
WA(t)
~on = COn + a(Sz)B.
(5)
with S = ~ ~
Moreover the quantity wn is defmed by
3. BASIC FORMULAE
J
ha = 81rTe7nh2.
Vol. 44, No. 6
i~On [Iz, WA(t)] + {~+ (t) [I+ WA(t), I_]
1  = T1
2 Re ~+(Wn)
(15)
with
,16, 0
We also introduce the following quantity:
+ $_+(t)[I_ WA(t), I.1 + ePzz(t)[IzWA(t), Iz] + h.c.} 
{ i~OnIx + r ( t ) }
WA(t),
(7) (8)
where qL(t) = ~  o
dt e it°nt
(9)
,I,+_(~on) =i(~)fdte~'°nt<[~S+(t),~S_(O)]>n. (17) The quantities (16) and (17) are related with each other through the fluctuationdissipation theorem [ 19] and therefore we have 1
TI
and t
¢zz(t) = a 2 j dt (~Sz(t)~Sz(O))B,o.e.,
(10)

ha 2 ~
~on being the Zeeman frequency of the nuclear spin. The function $_+(t) is obtained from equation (9) by interchanging (+) with (). An equilibrium average over the Bsubsystem is represented by ( . . . ) B and the subscript "o.c." implies an ordered cumulant. Timeevolution of Su(t ) is determined by = e iJ£°t/h S ~ e iJ£°t/h.
(1 1)
(18)
With the simple model for the conduction electrons, we can calculate xP+_(wn) as
0
~Su(t)
2 coth (y/2) Im xP+_(~on).
/kt
fk',
xP+(~n) x lUk(0)l 2" lUk,(0)l 2,
(19)
where e~w = ek+½VheOe,
(v=+land1
fortand~),
with the kinetic energy ek and the Zeeman frequency We of the conduction electron system. The Fermi distribution function f ( e ~ ) is abbreviated as f ~ .
Vol. 44, No. 6
SPIN RELAXATION IN METALS AT VERY LOW TEMPERATURES
i
/
i
i
,
i
923
,/
/
I = 3/2
I =1/2
I
I
I
2
I
I
I
4
6
I
I
0
I
I
4
2
I
6
{"10J n/kT
~&/n/kT
Fig. 1. The longitudinal relaxation time T1 for I = 1/2 as a function o f y = hwn/kT. Unit of the ordinate is T ° .
Fig. 3. The longitudinal relaxation times ~h, T2 and ra for I = 3/2. 1
T:I = ½a2nh(luk(O)12)~F" P2(eF)hC°""
(22)
Similarly the shift in the resonance frequency (Knight shift [20]) is obtained from equation (12):
[=1
a
Ago ~ ~
.,g
 con = ~ ~ (fk*   f k * ) " luk(O)l 2 k
a
(23)
From equations (21)(23), we have a relation of the form I
I
2
I
I
I
I
4
6
I
TI (~~12hcon coth (hcon/2kT) = h
fi wn / k T
\ o% /
Fig. 2. The longitudinal relaxation times rl and ~'2 for I = 1. The ordinate is measured in the unit o f T °.
2rr'
(24)
which reduces to the usual Korringa relation [21 ] in the high temperature limit, x '¢ I.
From equations (18) and (19), it is straightforward to find
4. RELAXATION TIMES When I = 1/2, equation (14) reduces to
1
T1

½a2rrh(luk(O)12)2eF coth x
f
de {f(e)
(/z) =
{ho~ e
[(e
 n~%)}p(e  ½mo~)p(e  t o o . + ½h~o~), (20)
where p(e) is the density of states and is assumed to be a smooth function of e, while the function in the parentheses has a sharp peak around the Fermi surface. With these considerations in mind we can replace values of p(e)'s at the Fermi surface, and a remaining value of the integral is seen to be hco,,. Thus we have 1 T1

1
ToCOthx,
(x = h~On/2kT),
with the longitudinal relaxation time at T = 0:
(21)
1
   ((Iz)t L_ (Iz)e,7),
(25)
TI
where
(Iz)e~ =  ½ th x,
(26)
and can be solved to give (Iz) t = (Iz)ea +
((Iz)o  LIz)eq)
e t/T~ .
(27)
Indeed, in this case, T1 is the longitudinal relaxation time and is given by equation (21) or equation (24) [22]. However, for 1 t> 1 the Bloch type equation (25) no longer holds. We have already solved equation (7) rigorously and found the solution of the form given by equation (1). Using equation (21) in the previous formulae [5] we have for I = 1 :
924
SPIN RELAXATION IN METALS AT VERY LOW TEMPERATURES
1/rl/2 : (2 cosh x T 1)IT ° sinhx
(28)
where T O is given by equation (22). Similarly we obtain for I = 3/2 [6] : 1/rl/3 = ½(7 Tx/25  24 tanh2x) cothx/T °
(29)
and 3 1/r2 = T~coth x.
(30)
These longitudinal spin relaxation times (21), (28)(30) are shown in Figs. 1  3 as a function y = h~on/kT = 2x. One can see from the figures the following characteristics: (i) F o r / = 1/2, the TI correctly gives the longitudinal relaxation time which deviates from the Korringa relation when y ~> 0.4. It is a monotonously increasing function o f y . (ii) For 1/> 1, there are 2/relaxation times as is seen from equation (1). It is a remarkable feature that the largest relaxation time r~ for each ! has a maximum around y = 2 and the rl behaves similarly to the Korringa relation f o r y ¢ 1. The relaxation times other than r~ increase monotonously with increasing values o f y . (iii) Absolute values of the relaxation times become smaller as I increases. In order to give a qualitative measure, let us consider the largest relaxation time rl at T = 0, r°(/). This quantity behaves like
7°(1) = T°/(21).
(31)
In addition to these characteristics drawn from the relaxation times, we can also obtain useful information by calculating each ratio (Cn/Cl) in equation (1) for various initial condition. Our preliminary result is the following: (iv) Although the high temperature behaviour is essentially governed by rl, namely, (c,,/cl) ~ 0 for n 4: 1, y ~ 1, the ratio gradually becomes large as y increases. Thus especially for an intermediate range of y around 2, it is highly necessary to use the multitime relaxation formula (1).
Vol. 44, No. 6
data based on the formulae derived in this paper. Owing to the recent progresses in the low temperature physics, we can expect more precise measurements of the relaxation times with the use of usual method of NMR. It is also possible to use ESR because our results can also be true for a localized electron system coupled with conduction electrons (with replacement con ~ ~e). The multipletime character of the relaxation processes and the temperature variations of the cn's and Tn'S will be verified by these experiments. Moreover, we should emphasize that the absolute value of the temperature itself is to be determined according to our formulae for the range o f x > 1. More detailed account of the present work will be reported elsewhere [23].
Acknowledgements  The authors are indebted to Professor K. Ono and Professor S. Kobayashi for valuable discussions. We are also grateful to Professor A. Ito and Dr N. Nishida for their helpful advices and comments. REFERENCES 1.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
5. CONCLUDING REMARKS
12.
It is interesting to compare our findings with the experiments. In [13 ] and [ 14], several experimental data with the use of NMR/ON method are reported. Although their data were forced to fit with a single exponential form, in a certain case, the relaxation time (T~ in their terminology) shows a maximum as a function o f y . Moreover, the absolute values of the relaxation times for larger I become quite small compared with the one given by equation (21). These are indeed consistent with our results, (ii) and (iii). We hope more careful analyses of the experimental
13. 14. 15. 16. 17. 18.
K. Ono, S. Kobayashi, M. Shinohara, K. Asahi, H. Ishimoto, N. Nishida, M. Imaizumi, A. Nakaizumi, J. Ray, Y. Iseki, S. Takayanagi, K. Terui & T. Sugawara, J. Low Temp. Phys. 38, 737 (1980). R.M. Mueller, Chr. Buchal, H.R. Folle, M. Kubota & F. Pobell, Phys. Lett. 75A, 164 (1980). A. Abragam, The Principles of Nuclear Magnetism Clarendon Press, Oxford (1961). C.P. Slichter, Principles of Magnetic Resonance. Harper and Row, New York (1963). F. Shibata, J. Phys. Soc. Japan 49, 15 (1980). F. Shibata & M. Asou, J. Phys. Soc. Japan 49, 1234 (1980). M. Asou & F. Shibata, J. Phys. Soc. Japan 50, 1846 (1981). M. Asou & F. Shibata, J. Phys. Soc. Japan 50, 2481 (1981). Y. Hamano & F. Shibata, J. Phys. Soc. Japan 51, 1727 (1982). Y. Hamano & F. Shibata, J. Phys. Soc. Japan 51, 2085 (1982). Y. Hamano & F. Shibata, J. Phys. Soc. Japan 51, 2721 (1982). Y. Hamano & F. Shibata, J. Phys. Soc. Japan 51, 2728 (1982). W.D. Brewer, D.A. Shirley & J.E. Templeton, Phys. Lett. 27A, 81 (1968). F. Bacon, J.A. Barclay, W.D. Brewer, D.A. Shirley & J.E. Templeton, Phys. Rev. BS, 2397 (1972). F. Shibata, Y. Takahashi & N. Hashitsume, J. Star. Phys. 17,171 (1977). S. Chaturvedi & F. Shibata, Z. Phys. B35,297 (1979). F. Shibata & T. Arimitsu, J. Phys. Soc. Japan 49, 891 (1980). N. Hashitsume, F. Shibata & M. Shingu, J. Stat. Phys. 17, 155 (1977).
Vol. 44, No. 6 19. 20. 21. 22.
SPIN RELAXATION IN METALS AT VERY LOW TEMPERATURES
R. Kubo,Rept. Progr. Phys. 29,Part I, 255 (1966). W.D. Knight, Solid State Physics (Edited by F. Seiz & D. Turnbull), Vol. 2, p. 93. Academic Press, New York (1956). J. Korringa, Physica 16,601 (1950). Although a relation of the type (21) was found in
23.
925
[14], the derivation is based on the high temperature version of the spin temperature method and therefore the applicability is not confined to I = 1/2 in contrast to our derivation. F. Shibata & Y. Hamano, J. Phys. Soc. Japan (submitted).