Theory of the nuclear relaxation rates

216

Journal of Magnetism and Magnetic Materials 63 & 64 (1987) 216-218 North-Holland, Amsterdam

T H E O R Y OF T H E N U C L E A R R E L A X A T I O N R A T E S M CRI~AN Department of Phystcs, Umvers~ty of CluI, :J400 CluI, Rumt~ma The nuclear relaxation rates as functmn of temperature have been calculated m the Varma model for the heavy-fermlon matermls The results are compared with the calculation performed recently which neglected the spin fluctuatmns

1. Introduction One of the most important methods to study heavy fermlons in the normal and superconducting state IS the m e a s u r e m e n t of the relaxation rates T~ l and T j ~ as function of temperature T h e majority of these materials present an unusual behavlour [l, 2] for these quantities and different mechanisms have been proposed to explain T~ ~ (l = l, 2) temperature dependence Overhauser and Appel [3] used the "clothing of the conductton electrons by the plasmons" in order to explain the temperature behav~our for T l I In U B I ~ Recently, Nguyen et al [4] cons~dered the influence of the f-electrons on the s-electrons and showed that the generahzed suscept~bdtty of the s-electrons is strongly affected by the " f " - e l e c t r o n s U s m g this idea the author [5] showed that the F r a d m [6] formula for the relaxation rate TrY(T) can be generahzed for the heavy-fermion superconductors T h e temperature dependence obtained in ref [5] for the relaxation rate Is

T~'(T) = T ~ ( T ) [ I + 2N(O)C(q)],

(1)

where T ~ is the Koringa rate, N(0) the density of the conduction electrons, and

C(q) = 21I(q)12xf(T)

(2)

In (2), I(q) is the " s - f " mteractlon, and xf(T) is the susceptlbtllty of the electrons m the superconducting state An accurate analysts on the experimental data obtained in materials which exhtbit h e a v y - l e t talon behaviour, recently p e r f o r m e d by V a r m a

[7] showed that heavy fermlons arise from ordinary conduction electrons exchanging spin fluctuations In the following considerations we will determine the t e m p e r a t u r e dependence of T~-1 which is determined by the electron-spin fluctuanon matrix element T h e simple case Tj = T2 discussed m [7] is analysed

2. Spin fluctuation model T h e Anderson H a m d t o n t a n can be transformed [5] in an " s - f " exchange Hamtltonlan

H = ~ e(k)a~,~ak,~ ko; -

~. lo(k~kz)a~k,,~o',~oak,k~S(k2),

(3)

kj k~

where 10 is the effective exchange between conduction electrons and the spin S of the selectrons The second term in (3) gives rise to the scattermg with rime reversal spin, and following the standard procedure we use

1 CkBT(iqodq I M k k+ql2X~ T~ = q [VkEik=k,/k~k,,

(4)

where the angle brackets denote the average over the Fermi surface This equation has been obtained taking for the spin fluctuations the dispersion law which satisfies t o ( q ) = t o ( - q ) = aq 2 In the low temperature limit qo = qM= kaT/ha and if the matrix element M k k+l oc q (4) gives

1/T~ T ~ = constant

(1304-8853/87/$03 50 © Elsevier Science Pubhshers B V (North-Holland Phystcs Pubhshmg DtVlSlOn)

(5)

M Cn~an / Theory of the nuclear relaxanon rates If the matrix element Mk,k+q oc q2 lS proporUonal t o = q 2 the relaxation rate T1 IS

1 / T I T = constant

(6)

T 4

3. The effective Hamiltonian A more realistic model will contain a Hamfltonlan of the form

H = Ho+ H',

(7)

where Ho is the hamlltonlan for the conduction electrons and H ' contains the following contnbutlons

general formula

~(~' = X

(8)

H~ = ~ A S d t - 2 ( g - 1) ~ J(I r, - R,I)~r,S~ I

l,

+u. goH~XSL

Y. X In, m, ~lH'ln', m', ~'>l ~ (9)

an equation which can be applied if the ground state n = 0 is a slnglet or ff we have the states as fv2-F7 doublet with the wave functions given in ref [3] In both cases we will be interested by the terms which gtve rise to the transmons M---~ M ± 1 (where M is the quantum number of S) We denote these matrix elements by 2

Ao=A.,,~,; E.-E.

A±=A H ' = HI + H2 + H3,

'

Z I(nIS~ln')l: .,.;

(10)

E . - E.,

where the state n has to be gtven In this case using (9) we get He1~ = - 2( g - 1) Io N

I

/42 = 2/~H~ ~ trf +/ZNgN ~. I 7 , i

217

l

/-/3 = a ~/~r,6(r,), I,l

1[_lko~ ~ k , ], a k

?

k k']

+ (A+ + A_)(t[a~, t ak, + I T a - ~ , ak t )] (1l)

where A, J, a are the couphng constants, g the Land6 factor, Hz the external magnetic field S - t h e spin operator for the f-electrons, I the operators for the nuclear spins and o" the Pauh matrices In order to calculate the relaxation rate T~ we have to obtain an effective Hamlltonlan descnblng the interaction between nuclear spins and the conduction electrons taking into cons~deratlon the effect of f-electrons In this point we consider the following approximation the f-electrons can be treated as a perturbation for the system consisting from conduction electrons and nuclear spins Using the perturbation theory up to the second order we can obtain an effective " I - s " Hamlltonlan if we know the wave function in the zero order The calculations can be performed using the

The external magnetic field will be changed due to the f-electrons and it is useful to consider an effecnve field Ho=Hz( 1

Io(g-1)

)

Nge t~ Xf

(12)

4. The relaxation rate The relaxanon rate T ] q for the nuclear spin interacting with the conduction electrons is defined as

M Cn~an / Theory of the nuclear relaxanon rates

218

where W , . . is the transition probabdlty from the state m in the state n T h e interaction (11) gives rise to the transitions m - + m q : l (m. the quantum number of the nuclear spm) and has the general expression 2wr E

[V,.k~ .,:~, ko-[2

[(Eke)),

(14)

where E is the energy difference between the nuclear levels m and m :t: 1, f ( E ) the dlstnbunon funcnon for the conduction electrons with the energy E The matrix element if the Hamlltonmn (11) denoted by V can be calculated and we get for (14)

Wm ~TI-

8'rr ( g -- 1) 2 h N2 I2(A++ A )2

x G + N(O)I%T

)

dx ~(x, ~)

(15)

where • (x,7:[~) = [l + e x p ( x ~ -

z)] ~

x [1-(exp(x + [~-z)+

1/T~ = CI~,Ho cotgh I ~ H o / ku T

1)~],

(16)

with

(18)

where C is a constant g w e n as C-

x 6(Ek,, - E k ,~' + ~ ) f ( E k ,, ) x (1 -

we get

16(g-l) h ~N

2 I(2)(A + ~+)2

(19)

We have to mention that in (18) for ~ H o - - > 0 we get the K o r m g a result I/T~ T = Ck

(20)

5. Discussion T h e temperature dependence of the spin-lattice relaxation time m the heavy-fermlon systems seems to be essentially determined by the presence of the f-electrons which strongly affect the interact~on between conduction electrons and nuclear spms From the results obtained m the spin-fluctuation model (section 2) and m the effective Hamlltoman approximation we see that the f-electrons c o n t n b u t m n ~s ~mportant m the temperature dependence of T1 Eq (18) is in agreement with the experimental results obtained m ref [1] for UB~3 References

---- H o ~ e / kB T ,

z =

Ef/ kB T

The upper hmlt m (15) can be considered - ~ and the integral becomes

l(q:~) = f ~ dx ~ ( x , q:~) = e±~-z/3(1, 1)2Fl(l, 1, 1 - e 2n)

(17)

where /3(1, 1) is the beta funcnon and 2G is the h y p e r g e o m e m c function Using now 2F1(1, 1 , 2 , - z ) =In(1 + z ) / z

[1] W G Clark, Z Flsk, K Glover, M D Lan, D E MacLaughhn, J L Smith and Cheng Tten, m Proc 17th Intern Conf on Low Temp Phys Eckern, eds A Schmld, W Weber and H Wuhl (North-Holland, Amsterdam, 1984) p 227 [2] M J Lysak and D E MacLaughhn, Phys Rev ql (1984) 6963 [3] A W Owerhauser and J Appel, Phys Rev 31 (1985) 193 [4] T N Nguyen, L C Lopez and B Coqbhn, J Magn Magn Mat 47 & 48 (1985) 136 [5] M Cn~an, Phys Lett 115 (1986) 69 [6] F Y Fradm, J Phys Chem Sohds31 (1970) 271~ [7] C M Varma, Phys Rev Lett, 24 (1985) 272q [8] Y Yafet, m Sohd State Physics, Vol 14, eds F Seltz and D Furnbull (Academic Press, New York 1963) p 2