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On electromagnetic properties of superconductors in the “correlated hopping” model
PHYSICA
Physica C 215 (1993) 97-104 North-Holland
On electromagnetic properties of superconductors in the "correlated hopping" model T a d e u s z D o m a r i s k i a n d K a r o l I. Wysokifiski Institute of Physics, Maria Curie - Sklodowska University, 20-031 Lublin, Poland Received 2 April 1993 Revised manuscript received 9 June 1993
We study the eleetrodynamic properties of quasi-three-dimensional superconductors described by the "correlated hopping" model. Finite band effects, energy dependence of the gap, carrier velocity and realistic density of states are shown to be responsible for some unusual properties of superconductors. The finite bandwidth together with energy dependence of the gap strongly affect the tunneling characteristics and lead to large asymmetries in the voltage-dependent conductance of the junctions. The NMR relaxation rate is practically not influenced by the dependence of materials characteristics but is sensitive to inelastic scattering processes. On the other hand, deviations of AC conductivity al (to, T) from BCS model predictions can be traced back to the energy dependence of the carrier velocity.
1. Introduction Experimental studies ofelectrodynamic properties of classical superconductors have clarified their phenomenological picture [ 1 ] by providing detailed information on the superconducting parameters: the value of the gap, its temperature dependence, etc. This is still not the case for high-temperature superconductors ( H T S ) because of: ( 1 ) difficulties in obtaining high quality (without grains, twin boundaries, etc.) single crystals, (2) the complicated structure o f materials (variable number of planes per unit cell, presence of chains in the mostly studied YBa2Cu307_6 material, possible ordering in the anion subsystem), (3) problems with interpretation o f the data (partially connected with the overflow of theoretical models [2] ). HTS all possess very short coherence lengths ~ and are quite disordered (on a metallic standard). This complicates the proper identification o f intrinsic effects and subsequent comparison with theory. These problems have been extensively discussed and partially clarified by Halbritter [ 3]. On the theoretical side the important task is to perform a thorough analysis o f the various models
(the old BCS one included) with the real metal effects (anisotropy of interactions, impurities, wave vector and energy dependence of matrix elements etc.) taken into account. In our opinion such calculations are necessary for the proper identification o f real departures of HTS behaviour from classical models. The purpose of this work is the calculation of tunneling characteristics, N M R relaxation rate and AC conductivity in a so-called "hole superconductivity" [ 4 ] or "correlated hopping" [ 5 ] model. This model belongs to the family o f extended Hubbard models. The hamiltonian is written as /~= ~ + 0 + / ~ ,
(1)
where the kinetic term ~ = - Yo (to + I ~ o ) c ~+c ~ (with/t being the chemical potential), Coulomb repulsion U = U ~ nit n~, and a correlated hopping term I£= ~ Kijc+ cj,,(ni_,,+nj_,,) .
(2)
Oa
It has been proposed [4,5] that the hamiltonian ( 1 ) may serve as a model describing high-To superconductors. The two-dimensional version o f this model and the properties of materials described by it have been intensively studied by Hirsch and co-
0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
98
T. Domanski, K.I. Wysokihski / Electromagnetic propertws in the "correlated hopping" model
workers [6,7], as well as by others [5,8,9]. There is a growing opinion that the model as it stands captures a lot o f the physics of C u - O layered cuprate superconductors. The main features of the model are: ( 1 ) A strong dependence o f the superconducting properties on the charge carrier concentration; (2) The energy dependence of the superconducting gap which is a source of some interesting behaviours of these materials. In particular, the funneling characteristics are unconventional, there is a difference between emission and reverse photo emission spectra [ 10], and normal non-magnetic impurities strongly reduce critical temperature T~ and may lead to the disappearance of the energy gap [ 9 ]. It is a difficult task to accurately estimate values o f parameters entering the hamiltonian. Some estimates [ 11 ] point at the renormalised values lying in the range t ~ (0.05-0.1) eV, U ~ 5 eV, K ~ ( 1 - 3 ) eV. Expressed in units of the two-dimensional energy bandwidth D they become K ~ 10D, U ~ 2 O D . Mosl of the results obtained up to now have been calculated for U and K in this range, where the HartreeFock-Bogoliubov approximation may not be valid. In this work we extend the previous studies [6,7] of the electromagnetic characteristics of superconductors by taking into account the quasi-three-dimensional character of energy bands and using exact (energy-dependent) forms of the density of states and velocities.
2. The model and approach We specify the model (1) for a quasi-three-dimensional system by assuming K , j = K t o and postulating [ 12 ]
{i to =
alneben enaetn b°ursin t
for i, j being the nearest neighbours from adjacent planes, elsewhere, (3)
with 0 < p < l . When eq. (1) is transformed to kspace, we get the reduced BCS hamiltonian which reads [4 ]
H~°d= Z ( t ~ - l l ) c L c ~ ko
+ ~ l'kk C;~C+k~C_k~CU~,
(4)
kk'
where ek = - 2t (cos k~a~ + cos k~.a~,) + 2ptcos k=a=,
!5 )
K I~k. = ( % 4t ( ~ k + e k ) .
(6)
The standard procedure applied to eq. (4) gives the following condition for the energy gap [4,12 ]: A,=-~
l'kk, Fmk tgh
,
{7)
with Ek = ~,~( ek --/z ) 2 + A2 which allows the extended s-wave type fo the energy gap, - 4t+c
Ak=A,,
.
(8)
as a possible solution of eq. (7). The system of lwo equations determines parameters Am and c as functions of actual concentration n and temperature. Wc have solved them numerically together with the condition for carrier concentration 1
n = 1 -- N ~ T
t,-p
Ek
tgh( Ek \2kB T/
"
(9 )
There are important differences between the present BCS-like model and the standard one based on electron-phonon interaction. We mention here the lack of energy cut-offin eq. (7) different from bandwidth D and the strong energy dependence of the gap function Ak. As we shall see, these features lead to some new effects in the tunneling characteristics.
3. Single particle tunneling In this section we shall show how the finite bandwidth, together with the position of the gap relative to the bottom of the band, affects the current voltage functions for both N - I - S and S - I - S junctions. We adopt here standard procedure for calculation of the tunneling current flowing through the narrow insulating layer [ 13 ]. A single particle current reads
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "'correlated hopping" model
7
dco
l ( V ) = 2 e ~~, IT~l = j ~-~ AL~, o9+eV)
N-I-S
99
n=0.08
--oo
>(AR(k, o9)[f(o9, T)-f(og+eV, T ) ] ,
(10)
where e is the absolute value of the electron charge, V is the external potential set on the junction, T~ denotes the matrix element of hopping through the barrier and f(o9, T) = (exp(og/kaT) + 1 ) - l . AL and AR are spectral densities of the left- and right-handside systems. The spectral density is related to the Green's function via
A(k, to) = - 2 I m [G(k, o9+i0 + ) ] .
.6
g. ]0.0
( 11 )
i
0"0-5.0
I
i
-3.0
-1.0
For a normal metal of bandwidth DN it reads
(12) while for a superconductor with finite energy band
-D/2
where
u~=~1 ( 1+ e ,E- /kt ~] = l _ v Z .
3.0
e VIA o
A(k, o9) = 27t~(o9- (~k - # ) ) ' O ( - [ek[ + DN/2) ,
xO(-l~k[+D/2) ,
1.0
(14)
Below we discuss three various types of tunneling junction. We will be mostly interested in the systems with DN ~ D and with low concentration of carriers. In this situation the Fermi level lies near the edge of the band and an interesting structure in tunneling conductance may appear. The normal conductor is taken here with a flat density of states extending to high energies (see inset to fig. 1 ). In fig. 1 the conductance d u d V of a N I-S junction is plotted as a function of the applied voltage in units of Ao/e, for different temperatures T = 0 K, T=0.5Tc and T=Tc. Note that the distance between the band edge and the Fermi level is of the order of Ao. It corresponds to n = 0.08 with the model parameters k = 1, u = 1 a n d p = 0 (where k - K I D and
u - U/D). It is the energy dependence of the superconducting energy gap which leads to the asymmetry of tunneling characteristics and reflects itself in the different heights of the plotted peaks. This has already been
Fig. 1. The tunneling conductance of a N - I - S junction vs. the normalized voltage for three temperatures 0, Td2 and T¢. Following Hirsch and Marsiglio we use here the notation Ao=/I ( p ) / a, where a = x/1 + ( ~ , , / ( D / 2 ) )2 and Ao has the meaning of the effective gap at zero temperature. The other model parameters are: u= 1, k = 1, n=0.08. Inset: Schematic diagram of the N - I - S junction, where the dashed line denotes chemical potential level and the shaded area are the occupied (at zero temperature) states.
noted previously [12]. In fig. 1 there appears another kind of asymmetry. It is due to the low concentration of carriers and is seen as the vanishing of dl/d V for eV> 2.5Ao. In contrast, the dI/d V stays almost constant for negative voltages, as expected for wide bands. In fig. 2 we show how the conductance of the junction N ' - I - S changes in a voltage range of a few A0 around EF. Here N' denotes the material whose normal state properties are described by the model ( 1 ). In addition to the previously mentioned asymmetry of conductance, we observe the appearance of negative conductance in the system. This negative conductance seen in the figure around V= -3Ao arises when the external voltage V puts all occupied states of the normal conductor N' above the gap in the superconductor spectrum (see inset to fig. 2). The corresponding condition for this is
-eV>-Ao + p - ( - D / 2
).
(15)
The finite band effects can also be seen if T = Tc (fig. 2). The conductance possesses a maximum at zero voltage and decreases for higher [ V[. For wide band materials the conductance in the normal state
1O0
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "correlated hopping" model
n=0.08
N'-I-S
T:O "~
L
.5 g.
7"~
.--n=0.2
~5
I T=O.O
)/A
n=0.08.
<
S"
i
0.0 ~
i
,J/ //'
~
i I -5
1J3
-
-
tl . . . .
J 1
[ 3
0.0
S/
0
2
e V/A o Fig. 2. The tunneling conductance ofa N'-l-Sjunctionas a function of the normalised voltage. The description is as in fig 1, Inset: Diagram of the N'-I-S junction when the external voltage V= ( - 1)/e(Ao+p+D/2) is applied, causing the negative conductance. is constant. This behaviour is easy to understand by looking at the corresponding energy diagrams, and noting that the chemical potential is a few A0's above the b a n d edges of both materials. For completeness we show the I ( V ) characteristics of the S - I - S j u n c t i o n between two identical superconductors. Both superconductors have finite energy bands, the same carrier concentration and energy gap. Such assumptions make the problem anti-symmetrical with respect to the sign of the voltage when the current I ( V ) is taken into account. In fig. 3 we plot I ( V ) at zero temperature (solid lines) and at T = T ~ (dashed curves) for two concentrations, n = 0 . 0 8 and 0.2. As usual there is no current below the critical voltage Vc=2Ao/e. For I" somewhat above Vc we observe the slower increase ( n = 0 . 2 ) or even decrease ( N = 0 . 0 8 ) of the tunneling current. The effects are stronger for smaller concentrations n. For very small values of n one expects finite band effects even in the normal state (the dashed line in fig. 3 for n = 0.08). It is interesting to ask whether the predicted effects can be (or have b e e n ) observed experimentally. It is well known that t u n n e l i n g measurements on high-Tc superconductors are quite difficult for a n u m b e r of reasons. Besides general problems men-
4
6
8
e V/Ao Fig. 3. The single particle current for a S-I-S junction vs. the normalised voltage. The function is asymmetric with respect to the change of sign of the voltage, so only the region of positive I" is shown. Concentrations n=0.08 and n=0.2 are chosen to show that in a dilute region (see n=0.08 lines in fig. 3) the finite band effects are stronger than for larger concentrations, when the chemical potential lies deep within the band. tioned in the Introduction there are additional ones specific to this technique. It is a preparation of the j u n c t i o n and connected pressure effects, Coulomb blockade, charging effects, etc. [14]. The recent progress in technology of HTS makes possible the production of good quality single crystals and thin films which may allow the observation of predicted finite band effects. It may well be possible that the described effects have already been observed [15], but we have not enough information to be sure.
4. N M R
relaxation rate
The nuclear spin relaxation rate can be derived from the standard expression k B T lim ~ [ A ( q ) l 2 Z ' ' ( q ' ° 9 ) 1/T1 - 4fi2ti2 . . . . . L (o
(16)
where the d y n a m i c spin susceptibility is given by
1
z ( q , k o ) = ~ Y" ~ [ G ( k + q , i a ) , + i o 2 ) G ( k , ico,) i ogn k
T. Domariski, K.I. Wysokifiski / Electromagnetic properties in the "'correlated hopping" model
-F(k+q, ioJ. +ito)F(k,
io~n) ] .
(17)
In our case Green's functions G and F are the usual BCS ones with the only difference being the energy dependence of the gap (8), so we get D/2
1
101
2.0
~=1, k = l p=O.O
n=O.Oa
I
D/2
IAI 2
T-4h31t2t_OL ~ --D/2
X[I+
d~N(~)
~ d~'N(~')
1.0
--D/2
(e--,u)(e'--/.Q+A(,E)A(e')]E(~.)E(U) _]
X [f(E, T)-f(E+~L, T ) ] g ( E ' - E - t O L ) ,
(18)
where o)L is the Zeeman frequency and N ( E ) = 1/N Z ~ ( ¢ - % ) . We put tOE#0 in order to remove the unintegrable singularity of the integrand. l / T, has previously been investigated [ 7 ] in the two -dimensional version of the model ( 1 ) with the additional assumption of constant density of states N(¢). It turns out that for the carrier concentration of interest (n > 0.2 ) the density of states corresponding to the quasi-three-dimensional model ( p > 0 ) practically does not change the result. We have investigated the temperature dependence of the relaxation rate T i-1 for several sets of parameters u, k and p. The results are qualitatively the same for various parameters, so we present in figs. 4(a) and (b), 1/T~calculated for u = l and k = 1 w i t h p = 0 and p=0.4, respectively. The most pronounced coherence peak appears for such a value of concentration nm,x when Tc becomes maximal (nm,x = 0.08 for p = 0 , nm,x=0.1 for p--0.4). With increase of concentration (n > nm~) the Hebel-Slichter peak slightly diminishes but does not disappear. On the other hand, decrease of concentration in comparison to nm~x considerably reduces the coherence peak. For a dilute region, when at critical temperature T¢ the chemical potential falls below the lower edge of the band, the peak disappears. Both real two-dimensional (fig. 4 ( a ) ) and quasi-three-dimensional (fig. 4 ( b ) ) systems exhibit similar tendencies with only quantitative differences. A few reasons have already been discussed [ 16 ] as to why the model may not be directly applicable to the experimental situation. The most serious objection is the neglect of Cu states and the hybridisation. Even a two- or three-band [ 17 ] model is perhaps not realistic enough for the description of HTS. Here we
0.0
0.5
0.0
1.0
T/T~ 2.0
zt=l, k = l p=0.4
n=0.1 I
00
0.0
0.5
1.0
T/T~ Fig. 4. The temperature-dependent N M R relaxation rate for u = 1, k = l in a two - d i m e n s i o n a l system ( a ) a n d a quasi-three-dimensional one with p = 0 . 4 (b). Different curves correspond to various concentrations.
add that, because HTS are quite disordered, the effect of impurities has to be taken into account. This is an important issue due to the observation [ 9,18] that normal impurities lead to pair breaking. The pair breaking appears in the model because the gap depends on energy [9 ], and it is even more effective in a system with narrow bands [ 18 ]. For the purpose of demonstration we consider the effect of quasiparticle lifetime in a semi-classical way [19]. One makes the replacement in the dynamic susceptibility ( 17 ) after analytic continuation from i~o to co+i0 +,
72 Domanski, K.I. Wysokihski / Electromagnetic properties in the "correlated hopping" model
102
z(q, ¢o+0+ )~z(q, ~o+ i F ) ,
(19)
where F = 1/2z is the inelastic scattering rate. Simple calculation leads to an expression for 1/T~ analogous to eq. (18), which contains the F parameter. In fig. 5 we show numerical results of the relaxation rate for several values of F for a quasi-three-dimensional system with n=0.1. F reduces the coherence peak considerably and, if strong enough, causes it to disappear. Such inelastic scattering has been recently discussed [20] for disordered superconductors and pointed out as the main factor responsible for the absence of a coherence peak.
which directly relates to the complex conductivity in the dirty limit ac(~o) = (~rc(q, oJ) )q through
K(q, co)
1
ac(Og)=~,q~V~(q) (-i~o) '
(21)
where t'x(q)=0eq/0qx. In this work v2(q) will be calculated exactly. It is the inverse parabolic function of energy %. The absorptive pan of the conductivity cq = Re (0,,) we are interested in, reads D/2
D/2
'f
~r, = --
O9
X[I
5. AC conductivity
d~v2(E)
--D/2
f
d(N(()
--D/2
+ (~-/I)((-//)7--~A({)A(()]E(~)E(~) J
× [/'(E, T)-f(E+oa, T ) ] . 6 ( E ' - E - ¢ o ) , In this section we present calculations of the (T, oJ)-dependent conductivity of a superconductor described by the correlated hopping model. The formalism for the calculation of a~(q, ~o) is similar to that used for the relaxation rate. We define the electromagnetic kernel
K(q, io;)= 1
[ G(k +q, i~, + i{o)G(k, i{on)
f l iCOn
+ F(k +q, ioo. + io~)F(k, ioJn) ] .
(20)
(22)
with
(0%YO(e-%).
(23)
z'2(~-)=N~q \Oq,]
The temperature dependence of a~ of a quasi-threedimensional system with p = 0.4 for several frequencies 09 is plotted in fig. 6. Solid lines correspond to n=O.l, dashed ones to n=0.2. Behaviour is of the BCS type for both concentra3.0
2.0
7
~=1
~=0. !
/c=l
p=0.4 ~=0.1 r'/T¢=0
w/A°= 1 0 - a ' ( / ~
I 0.01
"',\'i
2.0
l
b .5
1.0
1.0 .0
0.0
0.0
0.5
1.0
T/T~ Fig. 5. The normalised relaxation rate of the quasi-three-dimensional system (p = 0.4 ) with actual concentration n = 0.1 for several values of the inelastic scattering rate, introduced here by the Fparameter. F i s measured in Tc units.
0.0
L,S
u_
0.0
.... 0.5
1.0
T/To Fig. 6. The temperature dependence of optical conductivity of the quasi-three-dimensional superconductor (p = 0.4 ) for several frequencies oJ expressed in Ao units. The solid lines are plotted for n = 0.2 and the dashed ones for n = 0. I.
T. Domahski, K.I. Wysokitiski / Electromagnetic properties in the "correlated hopping" model
tions. At low frequencies the peak in a~ (T) arises below T¢. It has the same origin as the Hebel-Slichter one in 1/T~ versus T, and is due to the coherence factors. Frequency-dependent al(to) in optical region shows considerable deviations from the BCS model predictions. First of, all in the very dilute region ( n = 0 . 0 0 5 , p = 0 . 4 ) (fig. 7 ( a ) ), it shows Drude-like to dependence for temperatures close to Tc up to 0.6 T¢. Monotonically decreasing with to, a~ hardly reflects the presence of the energy gap in the superconducting system.
n=O. O05
t.0
b
O.5
0.0
1.o
0
l
2
c /Ao
3
.................................
,,,
~7 0 . 5
0
1
2
103
Even for larger concentrations (n = 0.1 in fig. 7 ( b ) ) where the finite band effects are expected to be unimportant, a~(to) is different from the usual BCS results. The energy dependence of the superconducting gap produces broad minima at temperature T < Tc which change their positions in (a~, to) coordinates, depending on temperature. For comparison, we have plotted in fig. 7 (b) the predictions of the BCS model (dashed lines). One can see sharp kinks in a~ (co) changing their positions with temperature. The frequency at which a kink appears directly gives a value of the BCS gap parameter A for a given temperature. In the present model the mentioned kinks are much less pronounced. At temperatures close to Tc they are almost invisible. Such effects are caused by the energy dependence of the gap A~ and the presence of the v2 factor in eq. (21), usually taken as a constant value in the BCS model and also in previous studies of this model [ 7], for p = 0. For concentrations larger than nmax=0.1 one finds characteristics similar to those shown in fig. 7(b). The most important finding in our opinion is that due to energy dependences o f the parameters: gap Ak, density of states N(¢) and velocity v 2 (¢), there is no clear sign o f the gap and its Tdependence in the a~ (09, T) function. This adds to the possible interpretation of experimental data on a~ (to) [ 21,22 ]. In our previous studies [23], we have shown that those experimental data can be explained by means of a BCSlike model with gap anisotropy taken into account. The dependence o f the gap on directions in the Brillouin zone also leads to smearing o f the kinks in a~(o9) and to their apparent temperature independence. Here we do not discuss the important question of the dependence ofa~ (T) coherence peaks on the inelastic scattering rate F. It follows from the work o f Marsiglio [24] that the effect will be the same as on the N M R realaxation rate. The narrow peak observed in a~ (T) in some experiments near T¢ seems to be of different origin.
3
w/Ao Fig. 7. The optical conductivity vs. normalised frequency ~o of the quasi-three-dimensional system in a dilute region with n = 0.005 (a) and for larger concentration n = 0.1 (b). The dashed curves in fig. 7(b) have been calculated within the BCS model. As previously, u= l, k= 1.
6. Conclusions
We have studied the tunneling characteristics of N - I - S and S - I - S junctions, N M R relaxation rate 1/ T1 and AC conductivity a~(to, T) o f superconduc-
104
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "'correlated hopping" model
tors d e s c r i b e d by the "'correlated h o p p i n g " model. O n e m a i n o b j e c t i v e has b e e n to e x t e n d the p r e v i o u s calculations ( d o n e for a t w o - d i m e n s i o n a l system and c o n s t a n t m a t e r i a l p a r a m e t e r s ) by using realistic, and exact for the d e n s i t y o f states o f the present m o d e l gap f u n c t i o n a n d carrier velocity. T h e energy d e p e n d e n c e has b e e n f o u n d to severely m o d i f y the l o w - f r e q u e n c y c o n d u c t i v i t y al (co) o f the system. T h e a j ( ¢ o ) c u r v e s calculated for v a r i o u s t e m p e r a t u r e s hardly s h o w a sign o f the energy gap, e v e n t h o u g h the T d e p e n d e n c e o f the gap is in the p r e s e n t m o d e l o f the BCS type. T h e t u n n e l i n g characteristics s h o w v a r i o u s a s y m m e t r i e s related to the finite b a n d effects a n d energy d e p e n d e n c e o f the gap. In contrast, the N M R r e l a x a t i o n rate is r a t h e r ins e n s i t i v e to finite b a n d effects a n d energy d e p e n dence. 1/T~ is s h o w n to be strongly i n f l u e n c e d by the inelastic scattering processes a n d this is also true for o'1 ( T ) . T h e p r e s e n t e d results allow us to say that p r o p e r i n t e r p r e t a t i o n fo the e x p e r i m e n t s in high-T~ superc o n d u c t o r s is r a t h e r a d e l i c a t e m a t t e r , as v a r i o u s real m e t a l effects can strongly affect the characteristics o f s u p e r c o n d u c t o r s . O n the t h e o r e t i c a l side the realistic t r e a t m e n t o f n o r m a l state p a r a m e t e r s is necessary b e f o r e c o m p a r i s o n s w i t h e x p e r i m e n t s are m a d e .
Acknowledgement T h e w o r k has b e e n partially s u p p o r t e d by the Polish C o m m i t t e e for Scientific R e s e a r c h ( K B N ) .
References [ 1] J.R. Schrieffer, Theory of Superconductivity (Benjamin, New York, 1964).
[2] P.W. Anderson, Frontiers and Borderlines in Many-Particle Physics, eds. R.A. Broglia and J.R. Schrieffer, (NorthHolland, Amsterdam, 1988) p. 1. [ 3 ] J. HalbriUer, J. Appl. Phys. 68 (1990) 6315; J. Halbritter, J. Supercond. 5 ( 1992 ) 171. [4] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39 (1989) 11515: F. Marsiglio and J.E. Hirsch, Physica C 165 (1990) 71. [5] R. Micnas, J. Ranninger and S. Robaszkiewiez, Phys. Rev. B 39 (1989) 11 653. [6] J.E. Hirsch, Physica C 201 (1992) 347; J.E. Hirsch and F. Marsiglio, Physica C 195 (1992) 355: J.E. Hirsch and F. Marsiglio, Phys. Rev. B 45 (1992) 4807. [ 7 ] F. Marsiglio and J.E. Hirsch, Phys. Rev. B 44 ( 1991 ) 11960. [ 8 ] S. Das and N.C. Das, Solid State Commun. 81 ( 1992 ) 867. [9] K.I. Wysokifiski, Physica C 198 (1992) 87. [ 10] J.E. Hirsch, Physica C 182 (1992) 277. [ 11 ] J.E. Hirsch and F. Marsiglio, Phys. Lett. A 140 (1989) 122. [12] F. Marsiglio and J.E. Hirsch, Phys. Rev. B41 (1990) 6435. [ 13 ] G.D. Mahan, Many - Particle Physics, (Plenum, New York, 1981) pp. 798-815. [ t4] H. van Kempen et al, Proc. European Conf. on High-l~ Thin Films and Single Crystals, eds. W. Gorzkowski, M. Gutowski, A. Reich and H. Szymczak (World Scientific, Singapore, 1989), vol. 24, pp. 147-159. [ 15 ] G. Alvarez, K.N. Taylor, A. Bailey and G.J. Russel, Physica C185-189(1991)2541. [ 16] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 43 ( 1991 ) 424. [ 17] J. Spalek and J.M. Honig, Studies of High-Temperature Superconductors, ed. A. Narlikar (Nova Science, New York, 1991 ) vol. 8, pp. 1-67; J. Dutka, M. Kamifiski and A.M. Oleg, J. Magn. Magn. Mater. 104-107 (1992) 257. [ 18] F. Marsiglio, Phys. Rev. B 45 (1992) 45 956. [ 19 ] N. Bulut and D.J. Scalapino, Phys. Rev. Lett. 68 (1992) 706. [20] T.P. Devereaux, Z. Phys. B 90 (1993) 65. [ 21 ] R.T. Collins et al., Phys. Rev. B 43 ( 1991 ) 8701. [22] D. van der Marel et al., Phys. Rev. B 43 ( 1991 ) 8606. [23] K.I. Wysokiflski and T. Domafiski, Phys. Rev. B 45 (1992) 5005. [24] F. Marsiglio, Phys. Rev. B 44 ( 1991 ) 5373.
Physica C 215 (1993) 97-104 North-Holland
On electromagnetic properties of superconductors in the "correlated hopping" model T a d e u s z D o m a r i s k i a n d K a r o l I. Wysokifiski Institute of Physics, Maria Curie - Sklodowska University, 20-031 Lublin, Poland Received 2 April 1993 Revised manuscript received 9 June 1993
We study the eleetrodynamic properties of quasi-three-dimensional superconductors described by the "correlated hopping" model. Finite band effects, energy dependence of the gap, carrier velocity and realistic density of states are shown to be responsible for some unusual properties of superconductors. The finite bandwidth together with energy dependence of the gap strongly affect the tunneling characteristics and lead to large asymmetries in the voltage-dependent conductance of the junctions. The NMR relaxation rate is practically not influenced by the dependence of materials characteristics but is sensitive to inelastic scattering processes. On the other hand, deviations of AC conductivity al (to, T) from BCS model predictions can be traced back to the energy dependence of the carrier velocity.
1. Introduction Experimental studies ofelectrodynamic properties of classical superconductors have clarified their phenomenological picture [ 1 ] by providing detailed information on the superconducting parameters: the value of the gap, its temperature dependence, etc. This is still not the case for high-temperature superconductors ( H T S ) because of: ( 1 ) difficulties in obtaining high quality (without grains, twin boundaries, etc.) single crystals, (2) the complicated structure o f materials (variable number of planes per unit cell, presence of chains in the mostly studied YBa2Cu307_6 material, possible ordering in the anion subsystem), (3) problems with interpretation o f the data (partially connected with the overflow of theoretical models [2] ). HTS all possess very short coherence lengths ~ and are quite disordered (on a metallic standard). This complicates the proper identification o f intrinsic effects and subsequent comparison with theory. These problems have been extensively discussed and partially clarified by Halbritter [ 3]. On the theoretical side the important task is to perform a thorough analysis o f the various models
(the old BCS one included) with the real metal effects (anisotropy of interactions, impurities, wave vector and energy dependence of matrix elements etc.) taken into account. In our opinion such calculations are necessary for the proper identification o f real departures of HTS behaviour from classical models. The purpose of this work is the calculation of tunneling characteristics, N M R relaxation rate and AC conductivity in a so-called "hole superconductivity" [ 4 ] or "correlated hopping" [ 5 ] model. This model belongs to the family o f extended Hubbard models. The hamiltonian is written as /~= ~ + 0 + / ~ ,
(1)
where the kinetic term ~ = - Yo (to + I ~ o ) c ~+c ~ (with/t being the chemical potential), Coulomb repulsion U = U ~ nit n~, and a correlated hopping term I£= ~ Kijc+ cj,,(ni_,,+nj_,,) .
(2)
Oa
It has been proposed [4,5] that the hamiltonian ( 1 ) may serve as a model describing high-To superconductors. The two-dimensional version o f this model and the properties of materials described by it have been intensively studied by Hirsch and co-
0921-4534/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
98
T. Domanski, K.I. Wysokihski / Electromagnetic propertws in the "correlated hopping" model
workers [6,7], as well as by others [5,8,9]. There is a growing opinion that the model as it stands captures a lot o f the physics of C u - O layered cuprate superconductors. The main features of the model are: ( 1 ) A strong dependence o f the superconducting properties on the charge carrier concentration; (2) The energy dependence of the superconducting gap which is a source of some interesting behaviours of these materials. In particular, the funneling characteristics are unconventional, there is a difference between emission and reverse photo emission spectra [ 10], and normal non-magnetic impurities strongly reduce critical temperature T~ and may lead to the disappearance of the energy gap [ 9 ]. It is a difficult task to accurately estimate values o f parameters entering the hamiltonian. Some estimates [ 11 ] point at the renormalised values lying in the range t ~ (0.05-0.1) eV, U ~ 5 eV, K ~ ( 1 - 3 ) eV. Expressed in units of the two-dimensional energy bandwidth D they become K ~ 10D, U ~ 2 O D . Mosl of the results obtained up to now have been calculated for U and K in this range, where the HartreeFock-Bogoliubov approximation may not be valid. In this work we extend the previous studies [6,7] of the electromagnetic characteristics of superconductors by taking into account the quasi-three-dimensional character of energy bands and using exact (energy-dependent) forms of the density of states and velocities.
2. The model and approach We specify the model (1) for a quasi-three-dimensional system by assuming K , j = K t o and postulating [ 12 ]
{i to =
alneben enaetn b°ursin t
for i, j being the nearest neighbours from adjacent planes, elsewhere, (3)
with 0 < p < l . When eq. (1) is transformed to kspace, we get the reduced BCS hamiltonian which reads [4 ]
H~°d= Z ( t ~ - l l ) c L c ~ ko
+ ~ l'kk C;~C+k~C_k~CU~,
(4)
kk'
where ek = - 2t (cos k~a~ + cos k~.a~,) + 2ptcos k=a=,
!5 )
K I~k. = ( % 4t ( ~ k + e k ) .
(6)
The standard procedure applied to eq. (4) gives the following condition for the energy gap [4,12 ]: A,=-~
l'kk, Fmk tgh
,
{7)
with Ek = ~,~( ek --/z ) 2 + A2 which allows the extended s-wave type fo the energy gap, - 4t+c
Ak=A,,
.
(8)
as a possible solution of eq. (7). The system of lwo equations determines parameters Am and c as functions of actual concentration n and temperature. Wc have solved them numerically together with the condition for carrier concentration 1
n = 1 -- N ~ T
t,-p
Ek
tgh( Ek \2kB T/
"
(9 )
There are important differences between the present BCS-like model and the standard one based on electron-phonon interaction. We mention here the lack of energy cut-offin eq. (7) different from bandwidth D and the strong energy dependence of the gap function Ak. As we shall see, these features lead to some new effects in the tunneling characteristics.
3. Single particle tunneling In this section we shall show how the finite bandwidth, together with the position of the gap relative to the bottom of the band, affects the current voltage functions for both N - I - S and S - I - S junctions. We adopt here standard procedure for calculation of the tunneling current flowing through the narrow insulating layer [ 13 ]. A single particle current reads
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "'correlated hopping" model
7
dco
l ( V ) = 2 e ~~, IT~l = j ~-~ AL~, o9+eV)
N-I-S
99
n=0.08
--oo
>(AR(k, o9)[f(o9, T)-f(og+eV, T ) ] ,
(10)
where e is the absolute value of the electron charge, V is the external potential set on the junction, T~ denotes the matrix element of hopping through the barrier and f(o9, T) = (exp(og/kaT) + 1 ) - l . AL and AR are spectral densities of the left- and right-handside systems. The spectral density is related to the Green's function via
A(k, to) = - 2 I m [G(k, o9+i0 + ) ] .
.6
g. ]0.0
( 11 )
i
0"0-5.0
I
i
-3.0
-1.0
For a normal metal of bandwidth DN it reads
(12) while for a superconductor with finite energy band
-D/2
where
u~=~1 ( 1+ e ,E- /kt ~] = l _ v Z .
3.0
e VIA o
A(k, o9) = 27t~(o9- (~k - # ) ) ' O ( - [ek[ + DN/2) ,
xO(-l~k[+D/2) ,
1.0
(14)
Below we discuss three various types of tunneling junction. We will be mostly interested in the systems with DN ~ D and with low concentration of carriers. In this situation the Fermi level lies near the edge of the band and an interesting structure in tunneling conductance may appear. The normal conductor is taken here with a flat density of states extending to high energies (see inset to fig. 1 ). In fig. 1 the conductance d u d V of a N I-S junction is plotted as a function of the applied voltage in units of Ao/e, for different temperatures T = 0 K, T=0.5Tc and T=Tc. Note that the distance between the band edge and the Fermi level is of the order of Ao. It corresponds to n = 0.08 with the model parameters k = 1, u = 1 a n d p = 0 (where k - K I D and
u - U/D). It is the energy dependence of the superconducting energy gap which leads to the asymmetry of tunneling characteristics and reflects itself in the different heights of the plotted peaks. This has already been
Fig. 1. The tunneling conductance of a N - I - S junction vs. the normalized voltage for three temperatures 0, Td2 and T¢. Following Hirsch and Marsiglio we use here the notation Ao=/I ( p ) / a, where a = x/1 + ( ~ , , / ( D / 2 ) )2 and Ao has the meaning of the effective gap at zero temperature. The other model parameters are: u= 1, k = 1, n=0.08. Inset: Schematic diagram of the N - I - S junction, where the dashed line denotes chemical potential level and the shaded area are the occupied (at zero temperature) states.
noted previously [12]. In fig. 1 there appears another kind of asymmetry. It is due to the low concentration of carriers and is seen as the vanishing of dl/d V for eV> 2.5Ao. In contrast, the dI/d V stays almost constant for negative voltages, as expected for wide bands. In fig. 2 we show how the conductance of the junction N ' - I - S changes in a voltage range of a few A0 around EF. Here N' denotes the material whose normal state properties are described by the model ( 1 ). In addition to the previously mentioned asymmetry of conductance, we observe the appearance of negative conductance in the system. This negative conductance seen in the figure around V= -3Ao arises when the external voltage V puts all occupied states of the normal conductor N' above the gap in the superconductor spectrum (see inset to fig. 2). The corresponding condition for this is
-eV>-Ao + p - ( - D / 2
).
(15)
The finite band effects can also be seen if T = Tc (fig. 2). The conductance possesses a maximum at zero voltage and decreases for higher [ V[. For wide band materials the conductance in the normal state
1O0
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "correlated hopping" model
n=0.08
N'-I-S
T:O "~
L
.5 g.
7"~
.--n=0.2
~5
I T=O.O
)/A
n=0.08.
<
S"
i
0.0 ~
i
,J/ //'
~
i I -5
1J3
-
-
tl . . . .
J 1
[ 3
0.0
S/
0
2
e V/A o Fig. 2. The tunneling conductance ofa N'-l-Sjunctionas a function of the normalised voltage. The description is as in fig 1, Inset: Diagram of the N'-I-S junction when the external voltage V= ( - 1)/e(Ao+p+D/2) is applied, causing the negative conductance. is constant. This behaviour is easy to understand by looking at the corresponding energy diagrams, and noting that the chemical potential is a few A0's above the b a n d edges of both materials. For completeness we show the I ( V ) characteristics of the S - I - S j u n c t i o n between two identical superconductors. Both superconductors have finite energy bands, the same carrier concentration and energy gap. Such assumptions make the problem anti-symmetrical with respect to the sign of the voltage when the current I ( V ) is taken into account. In fig. 3 we plot I ( V ) at zero temperature (solid lines) and at T = T ~ (dashed curves) for two concentrations, n = 0 . 0 8 and 0.2. As usual there is no current below the critical voltage Vc=2Ao/e. For I" somewhat above Vc we observe the slower increase ( n = 0 . 2 ) or even decrease ( N = 0 . 0 8 ) of the tunneling current. The effects are stronger for smaller concentrations n. For very small values of n one expects finite band effects even in the normal state (the dashed line in fig. 3 for n = 0.08). It is interesting to ask whether the predicted effects can be (or have b e e n ) observed experimentally. It is well known that t u n n e l i n g measurements on high-Tc superconductors are quite difficult for a n u m b e r of reasons. Besides general problems men-
4
6
8
e V/Ao Fig. 3. The single particle current for a S-I-S junction vs. the normalised voltage. The function is asymmetric with respect to the change of sign of the voltage, so only the region of positive I" is shown. Concentrations n=0.08 and n=0.2 are chosen to show that in a dilute region (see n=0.08 lines in fig. 3) the finite band effects are stronger than for larger concentrations, when the chemical potential lies deep within the band. tioned in the Introduction there are additional ones specific to this technique. It is a preparation of the j u n c t i o n and connected pressure effects, Coulomb blockade, charging effects, etc. [14]. The recent progress in technology of HTS makes possible the production of good quality single crystals and thin films which may allow the observation of predicted finite band effects. It may well be possible that the described effects have already been observed [15], but we have not enough information to be sure.
4. N M R
relaxation rate
The nuclear spin relaxation rate can be derived from the standard expression k B T lim ~ [ A ( q ) l 2 Z ' ' ( q ' ° 9 ) 1/T1 - 4fi2ti2 . . . . . L (o
(16)
where the d y n a m i c spin susceptibility is given by
1
z ( q , k o ) = ~ Y" ~ [ G ( k + q , i a ) , + i o 2 ) G ( k , ico,) i ogn k
T. Domariski, K.I. Wysokifiski / Electromagnetic properties in the "'correlated hopping" model
-F(k+q, ioJ. +ito)F(k,
io~n) ] .
(17)
In our case Green's functions G and F are the usual BCS ones with the only difference being the energy dependence of the gap (8), so we get D/2
1
101
2.0
~=1, k = l p=O.O
n=O.Oa
I
D/2
IAI 2
T-4h31t2t_OL ~ --D/2
X[I+
d~N(~)
~ d~'N(~')
1.0
--D/2
(e--,u)(e'--/.Q+A(,E)A(e')]E(~.)E(U) _]
X [f(E, T)-f(E+~L, T ) ] g ( E ' - E - t O L ) ,
(18)
where o)L is the Zeeman frequency and N ( E ) = 1/N Z ~ ( ¢ - % ) . We put tOE#0 in order to remove the unintegrable singularity of the integrand. l / T, has previously been investigated [ 7 ] in the two -dimensional version of the model ( 1 ) with the additional assumption of constant density of states N(¢). It turns out that for the carrier concentration of interest (n > 0.2 ) the density of states corresponding to the quasi-three-dimensional model ( p > 0 ) practically does not change the result. We have investigated the temperature dependence of the relaxation rate T i-1 for several sets of parameters u, k and p. The results are qualitatively the same for various parameters, so we present in figs. 4(a) and (b), 1/T~calculated for u = l and k = 1 w i t h p = 0 and p=0.4, respectively. The most pronounced coherence peak appears for such a value of concentration nm,x when Tc becomes maximal (nm,x = 0.08 for p = 0 , nm,x=0.1 for p--0.4). With increase of concentration (n > nm~) the Hebel-Slichter peak slightly diminishes but does not disappear. On the other hand, decrease of concentration in comparison to nm~x considerably reduces the coherence peak. For a dilute region, when at critical temperature T¢ the chemical potential falls below the lower edge of the band, the peak disappears. Both real two-dimensional (fig. 4 ( a ) ) and quasi-three-dimensional (fig. 4 ( b ) ) systems exhibit similar tendencies with only quantitative differences. A few reasons have already been discussed [ 16 ] as to why the model may not be directly applicable to the experimental situation. The most serious objection is the neglect of Cu states and the hybridisation. Even a two- or three-band [ 17 ] model is perhaps not realistic enough for the description of HTS. Here we
0.0
0.5
0.0
1.0
T/T~ 2.0
zt=l, k = l p=0.4
n=0.1 I
00
0.0
0.5
1.0
T/T~ Fig. 4. The temperature-dependent N M R relaxation rate for u = 1, k = l in a two - d i m e n s i o n a l system ( a ) a n d a quasi-three-dimensional one with p = 0 . 4 (b). Different curves correspond to various concentrations.
add that, because HTS are quite disordered, the effect of impurities has to be taken into account. This is an important issue due to the observation [ 9,18] that normal impurities lead to pair breaking. The pair breaking appears in the model because the gap depends on energy [9 ], and it is even more effective in a system with narrow bands [ 18 ]. For the purpose of demonstration we consider the effect of quasiparticle lifetime in a semi-classical way [19]. One makes the replacement in the dynamic susceptibility ( 17 ) after analytic continuation from i~o to co+i0 +,
72 Domanski, K.I. Wysokihski / Electromagnetic properties in the "correlated hopping" model
102
z(q, ¢o+0+ )~z(q, ~o+ i F ) ,
(19)
where F = 1/2z is the inelastic scattering rate. Simple calculation leads to an expression for 1/T~ analogous to eq. (18), which contains the F parameter. In fig. 5 we show numerical results of the relaxation rate for several values of F for a quasi-three-dimensional system with n=0.1. F reduces the coherence peak considerably and, if strong enough, causes it to disappear. Such inelastic scattering has been recently discussed [20] for disordered superconductors and pointed out as the main factor responsible for the absence of a coherence peak.
which directly relates to the complex conductivity in the dirty limit ac(~o) = (~rc(q, oJ) )q through
K(q, co)
1
ac(Og)=~,q~V~(q) (-i~o) '
(21)
where t'x(q)=0eq/0qx. In this work v2(q) will be calculated exactly. It is the inverse parabolic function of energy %. The absorptive pan of the conductivity cq = Re (0,,) we are interested in, reads D/2
D/2
'f
~r, = --
O9
X[I
5. AC conductivity
d~v2(E)
--D/2
f
d(N(()
--D/2
+ (~-/I)((-//)7--~A({)A(()]E(~)E(~) J
× [/'(E, T)-f(E+oa, T ) ] . 6 ( E ' - E - ¢ o ) , In this section we present calculations of the (T, oJ)-dependent conductivity of a superconductor described by the correlated hopping model. The formalism for the calculation of a~(q, ~o) is similar to that used for the relaxation rate. We define the electromagnetic kernel
K(q, io;)= 1
[ G(k +q, i~, + i{o)G(k, i{on)
f l iCOn
+ F(k +q, ioo. + io~)F(k, ioJn) ] .
(20)
(22)
with
(0%YO(e-%).
(23)
z'2(~-)=N~q \Oq,]
The temperature dependence of a~ of a quasi-threedimensional system with p = 0.4 for several frequencies 09 is plotted in fig. 6. Solid lines correspond to n=O.l, dashed ones to n=0.2. Behaviour is of the BCS type for both concentra3.0
2.0
7
~=1
~=0. !
/c=l
p=0.4 ~=0.1 r'/T¢=0
w/A°= 1 0 - a ' ( / ~
I 0.01
"',\'i
2.0
l
b .5
1.0
1.0 .0
0.0
0.0
0.5
1.0
T/T~ Fig. 5. The normalised relaxation rate of the quasi-three-dimensional system (p = 0.4 ) with actual concentration n = 0.1 for several values of the inelastic scattering rate, introduced here by the Fparameter. F i s measured in Tc units.
0.0
L,S
u_
0.0
.... 0.5
1.0
T/To Fig. 6. The temperature dependence of optical conductivity of the quasi-three-dimensional superconductor (p = 0.4 ) for several frequencies oJ expressed in Ao units. The solid lines are plotted for n = 0.2 and the dashed ones for n = 0. I.
T. Domahski, K.I. Wysokitiski / Electromagnetic properties in the "correlated hopping" model
tions. At low frequencies the peak in a~ (T) arises below T¢. It has the same origin as the Hebel-Slichter one in 1/T~ versus T, and is due to the coherence factors. Frequency-dependent al(to) in optical region shows considerable deviations from the BCS model predictions. First of, all in the very dilute region ( n = 0 . 0 0 5 , p = 0 . 4 ) (fig. 7 ( a ) ), it shows Drude-like to dependence for temperatures close to Tc up to 0.6 T¢. Monotonically decreasing with to, a~ hardly reflects the presence of the energy gap in the superconducting system.
n=O. O05
t.0
b
O.5
0.0
1.o
0
l
2
c /Ao
3
.................................
,,,
~7 0 . 5
0
1
2
103
Even for larger concentrations (n = 0.1 in fig. 7 ( b ) ) where the finite band effects are expected to be unimportant, a~(to) is different from the usual BCS results. The energy dependence of the superconducting gap produces broad minima at temperature T < Tc which change their positions in (a~, to) coordinates, depending on temperature. For comparison, we have plotted in fig. 7 (b) the predictions of the BCS model (dashed lines). One can see sharp kinks in a~ (co) changing their positions with temperature. The frequency at which a kink appears directly gives a value of the BCS gap parameter A for a given temperature. In the present model the mentioned kinks are much less pronounced. At temperatures close to Tc they are almost invisible. Such effects are caused by the energy dependence of the gap A~ and the presence of the v2 factor in eq. (21), usually taken as a constant value in the BCS model and also in previous studies of this model [ 7], for p = 0. For concentrations larger than nmax=0.1 one finds characteristics similar to those shown in fig. 7(b). The most important finding in our opinion is that due to energy dependences o f the parameters: gap Ak, density of states N(¢) and velocity v 2 (¢), there is no clear sign o f the gap and its Tdependence in the a~ (09, T) function. This adds to the possible interpretation of experimental data on a~ (to) [ 21,22 ]. In our previous studies [23], we have shown that those experimental data can be explained by means of a BCSlike model with gap anisotropy taken into account. The dependence o f the gap on directions in the Brillouin zone also leads to smearing o f the kinks in a~(o9) and to their apparent temperature independence. Here we do not discuss the important question of the dependence ofa~ (T) coherence peaks on the inelastic scattering rate F. It follows from the work o f Marsiglio [24] that the effect will be the same as on the N M R realaxation rate. The narrow peak observed in a~ (T) in some experiments near T¢ seems to be of different origin.
3
w/Ao Fig. 7. The optical conductivity vs. normalised frequency ~o of the quasi-three-dimensional system in a dilute region with n = 0.005 (a) and for larger concentration n = 0.1 (b). The dashed curves in fig. 7(b) have been calculated within the BCS model. As previously, u= l, k= 1.
6. Conclusions
We have studied the tunneling characteristics of N - I - S and S - I - S junctions, N M R relaxation rate 1/ T1 and AC conductivity a~(to, T) o f superconduc-
104
T. Domahski, K.I. Wysokihski / Electromagnetic properties in the "'correlated hopping" model
tors d e s c r i b e d by the "'correlated h o p p i n g " model. O n e m a i n o b j e c t i v e has b e e n to e x t e n d the p r e v i o u s calculations ( d o n e for a t w o - d i m e n s i o n a l system and c o n s t a n t m a t e r i a l p a r a m e t e r s ) by using realistic, and exact for the d e n s i t y o f states o f the present m o d e l gap f u n c t i o n a n d carrier velocity. T h e energy d e p e n d e n c e has b e e n f o u n d to severely m o d i f y the l o w - f r e q u e n c y c o n d u c t i v i t y al (co) o f the system. T h e a j ( ¢ o ) c u r v e s calculated for v a r i o u s t e m p e r a t u r e s hardly s h o w a sign o f the energy gap, e v e n t h o u g h the T d e p e n d e n c e o f the gap is in the p r e s e n t m o d e l o f the BCS type. T h e t u n n e l i n g characteristics s h o w v a r i o u s a s y m m e t r i e s related to the finite b a n d effects a n d energy d e p e n d e n c e o f the gap. In contrast, the N M R r e l a x a t i o n rate is r a t h e r ins e n s i t i v e to finite b a n d effects a n d energy d e p e n dence. 1/T~ is s h o w n to be strongly i n f l u e n c e d by the inelastic scattering processes a n d this is also true for o'1 ( T ) . T h e p r e s e n t e d results allow us to say that p r o p e r i n t e r p r e t a t i o n fo the e x p e r i m e n t s in high-T~ superc o n d u c t o r s is r a t h e r a d e l i c a t e m a t t e r , as v a r i o u s real m e t a l effects can strongly affect the characteristics o f s u p e r c o n d u c t o r s . O n the t h e o r e t i c a l side the realistic t r e a t m e n t o f n o r m a l state p a r a m e t e r s is necessary b e f o r e c o m p a r i s o n s w i t h e x p e r i m e n t s are m a d e .
Acknowledgement T h e w o r k has b e e n partially s u p p o r t e d by the Polish C o m m i t t e e for Scientific R e s e a r c h ( K B N ) .
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