Physica C 235-240 (1994) 2385-2386 North-Holland
PHYSICA
'/'he Influence o f t h e T e m p e r a t u r e - D e p e n d e n t Inelastic E l e c t r o n Scattering o n t h e Thermodynamieal and Transport Properties of Superconductors A.M.Gabovich and A.I.Voitenko Crystal Physics Department, Institute of Physics, Ukrainian National Academy of Sciences, prospekt Nauki 46, 252650 Kiev-22 GSP, Ukraine The temperature, T, dependence of the superconducting order parameter A was calculated serf-consistently taking into account the inelastic electron scattering by thermal excitations. The superconducting phase transition becomes of the first kind. The obtained functions Aft) are almost rectangular for large scattering amplitudes and describe well the tunnel and optical experimental data for high-T¢ oxides. The T-dependences of the stationary critical Josephson current, the nuclear spin-lattice relaxation rate R~ and other transport properties were also calculated. Rs(T) reveals the Hebel-Slichter peak near T~ for small inelastic scattering amplitude and the monotonous behaviour for strong scattering, the latter being true for high-Tc oxide ceramics. The s~ane origin of superconductivity, i.e., the pairing (although it is still unknown whether it is here s- or d-wave case), both in highT c oxides and conventional low temperature superconductors res!dts in the similarity of many their properties. On the other hand, there are definite characteristics inherent mostly to high-Tc oxides: i) the almost rectangular shape of the temperature, T, dependence of the superconducting order parameter A or/and the superconducting gap As; ii) the absence of the Hebel-Slichter peak below T c in the temperature dependence of the nuclear spin-lattice relaxation rate Rs; iii) the non-BCS-like magnitude and shape of the specific heat jump near T~ (the only exception is YBa2Cu307 but here the jump may be caused by the structural transition at T s > T~ [ 1]. Here we show that these peculiarities of oxides can be explained by the influence of the thermal Bose excitations: phonons, magnons, etc. Their role is greatly enhanced for high temperatures changing the surru2-,rct~ndnating pro~_a_ies in the near-To region. The account of this phenomenon needs a selfconsistent treatment. The electrons (holes) are scattered inelastically by thermal excitations mentioned above. The inverse scattering time x-~ is determined by various processes. It is a sum of terms some of which depend on A (see, e.g., [2])
Cooper
v(t) = (rT~0) -~ = A t " e x p ( - 8 ( t ) / t )
(1)
or A8
v(t) = A t " e xp(-Sg(t)/t).
Here n lies in the interval 1 to 3 [3], t = T/T~o, 8 = ~rI'~o, 8g = Ag/'l'~o, h = k B = 1, and A is a constant. We consider the expressions (1) or (2) to be the dominant term in the sum and neglect hereafter the other processes. The functions 8(0 and 8g(t) depend on v which is the T-dependent pair-breaking factor in the spirit of [4] in close analogy with the classical Abrikosov-Gor'kov theory. The serf-consistent calculations of 8(t,v) for the region of the non-zero gap 8g and eq.(1) or (2) lead to the following results (see Fig. la and b). Here we have chosen the more suitable for experimentalists dimensionless variable 0 = T/T c. Curves 2 and 3 correspond to eqs.(1) and (2), respectively, with A = n = 1. The curves 1 describe the order parameter and energy gap when the self-consistency is not taken into account, i.e., v(t) = t. The dashed curves l g b t l l l . '" , U l l b t t l l a l L l .l k./ l.t 3. 3 1!" I U.... ~'~ are the ordinary BCS ..... "U .l . . .I .k , , d'"-"^': that self-consistent dependences 8(0 and 8~(t) become double-valued. Only the upper branches of the curves are thermodynamically stable. The unstable branches are not displayed in Fig. 1. Thus, the superconducting transition becomes the phase transition of the first kind and the straight vertical lines describe this phenomenon. The shapes of the curves 2 and 3 resemble ones for the tunnel,
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A.M. Gabovich, A.I. Voitenko/ Physica C 235-240 (1994) 2385-2386
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infrared, and Raman experiments for various highTo oxides. The change in the phase transition order can explain instabilities and relatively small specific heat jumps for these substances. Another consequence is the sharp T-dependence of the stationary critical Josephson current near T o.
2.0 "
1.5
I~,,~
1.00.5
suppressed and at last disappears for the strong enough inelastic thermal electron scattering. We note that the disappearance occurs for the state with non-zero gap 5g. Earlier [6] the peak elimination was obtained in the framework of the Eliashberg theory for the gapless state, the latter being the consequence of the non-zero imaginary part of the order parameter A. We have also calculated the T-dependences of the electron thermal conductivity and the ultrasound attenuation below Te They are monotonous and in agreement with experiment fall rapidly near T o. These d e tcharacteristics a i l e l will s ebewconsidered h e r ein .
P
1 1.5. -
0 0
°O.0
0.5
~.0
1.0
-
Figure la.
2.0
0.5 -
0.0 1.0
0.0
0.5
1.0
Figure 2. REFERENCES
0.5 O.0
5
'
'
I
I
]
O.5
,
,
,
,'~
1.0 ~9
Figure lb. The ratio p(0) = I~/Rn for the nuclear spin!article .-~.lov~"^-...~..~....~,~.~.....o in o. . . . . . . "~''""" and normal states calculated in the usual manner [5] is shown in Fig.2 for the model (2), n - 1, and _~ -0.1, 0.2, 0.5, 1, and 1.5 (curves 1 to 5, respectively). As is readily seen, the Hebel-Slichter peak is
1. A.I.Sokolov, in: High-Tc Superconductivity, Experiment and Theory, A.S.Davydov and V.M.Loktev (eds), Springer, Berlin (1992), p. 194. 2. S.B.Kaplan et al., Phys. Rev. B,14(1976)4854. 3. P.B.AUen, Comments Cond.Mat.Phys., 15(1992)327 4. J.Appel, Phys. Rev. Lett., 21(1968)1164. 5. K.Maki, in" Superconductivity, R.D.Parks (ed.), Dekker, New York (1969), vol.2, p. 1035. 6. P.B.AUen and DRainer, Nature, 349 (1991) 396.