Solid State Communications,Vol. 73, No. 9, PP. 633635, 1990. Printed in Great Britain.
00381098/90$3.00+.00 Pergamon Press plc
Masafumi EHIRAI, Naoshi SUZUKIand Kazuko MOTIZUKI Department of Material Physics, Faculty of tigineering Science, Osaka University, Toyonaka 560, Japan ( Received 19 December
1989 by J. Ksnamori )
Tunneling spectra of 13axK, BiO (BKB) are studied on the basis of the strong coupling theory of &on0a mechanism for superconductivity. By using the electronlatticecoupling, which is calculated microscopically by the use of the realistic electronic tid structure andzthe renormalised phonons, we have calculated the spectral function c1F(U). The gap function A(E) is obtained by solving the Eliashberg equation at T=O K. The superconductingenergy gap A0 is evaluated to be 4.8 meV for x=0.7 and 1.0 meV for x=0.5. Differential conductance dI/dV and its derivative d21/d$spe ctra show apparent deviations from those predicted bythe BCS weak coupling theory. The value of A0 and the line shapes of dI/dV and d21/dV2 are in good agreement with the recent observations.
Oxide superconductorBaxK,_xBiO (BKB) with a cubic perovskitetype structure ha2 been found to have the highest transition temperature (T 2 28Katx=0.7)amongCufree superconductors[l?I. Since BKB contains no transitionmetalelement, the magnetic mechanism may not be expected for the superconductivityin BKB. In fact any magnetic cder has not been observed in BKB by the uon spin rotation experiments (41 and the magnetic susceptibility in the normal state in BKB shows a Pauli paramagnetic behavior [5,61. We have intensively made microscopic study on the electronlatticeinteraction and the lattice dynamics of BKB and its related c6mpound BaPb,_xBixO by using the realistic electronic bands of &BiO (7111. Furthremore, we have evaluated T b3 solving the linearized Eliashberg equati%n with use of the calculated spectral function LX%(W) [lo111. The magnitude and the isotope shift of T calculated by us are in good agreement with esperimental data for BKB Our microscopic study has clarified t3,6,121. that the origin for high T of BKB can be ascribed to the strong elect&nphonon coupling
caused by vibrations of 0 atoms toward Bi atoms. The purpose of the present report is to study the effects of the strong electronphonon coupling upon the tunneling spectra. Tunneling measurement is one of powerful methods to observe directly the superconducting energy gap. It can also provide direct information about the electronphononspectral function (131. The differential conductance dI/dV through a junction between a superconductorand a normal metal is approximately proportional to the electronic density of states N3(~) in the superconducting state [141: d1
a I eV=E
N k) .%S=Re N(EF)
ICI I k2  A(E)']"~
I, (1)
where N(EF) denotes the electronic density of states at the Fermi level EFin the normal state, and A(E) represents the energy dependent gap function at T=O K, which is determined by solving the Eliashbsrg equation for T=O K [13,14):
= d&'Re II AO
(3)
633
THEORETICAL STUDY OF TUNNELING SPECTRA
634
where S(E) is the electronic selfenergy of the normal state, Z(E) is called the mass renormalization function, and A, denotes the superconducting energy gap defined by AO=A(AO). The Coulomb interaction is treated in term of an effective parameter u* which is about 0.1 in usual metals. The spectral function c?F(w) is defined by
Vol. 73, No. 9
binding approximation have been described elsewhere [15]. In the present calculation we use the same values of the transfer integrals and their derivatives as adopted in ref.11. Furthermore we use the rigidband approximation, i.e. we assume that as x increases the conduction band is filled without changing its dispersion. Figs.l(a) and Z(a) show the spectral func
cr2F(w)=
where wi is the phonon frequency of mode y, denotes the conduction band energy and VY(k,k+q) represents the electronphononcoupling coefficient defined by V'(k,k+q) =I ' uo 3
EYtUo
(q) <(k,k+q) .
(5)
is the mass of the uth atom, E Y&o(q) 4 d¬es the phonon polarization vector and gu(k,ktq) represents the coefficient of coupling
Here
between two conduction band states k and k+q caused by displacement of the uth atom along the CLdirection. In the tight binding approximation g;(k,k+q) is expressed in terms of transfer integrals end their derivatives. Details of calculation of the electronlattice coupling and the lattice dynamics on the basis of the tight&40.3
A
tion a2F(w) and the phonon density of states F(w) calculated for x=0.7 (Bag. 7K0. 3Bi03) and x=
0.5 @a0 5K0 5BiO3), respectively. It is clearly seen that the frerequency dependence of 02F(w) is entirely different from2that of F(W). For smaller value of x (cO.3) (rF(W) has some prominent structures only in the frequency range of 0 stretching/breathingmode around 60 meV. 2With increasing x, however, some main peaks in CLF(W) shift to lower frequency side and the magnitude of u2F(W)increases in the whole frequency range up to6OmeV as shown2in Fig.l(a) or 2(a). This drastic change in u F(w) is caused by frequency renormalization of the 0 stretching/breathing mode. Once the spectral function a2F(u) and I1'+ are given, c(E) and A(a) are calculated by utilizing eqs.(2) and (3) in a selfconsistentmanner. In actual calculations the value of U*=O.l
Bad0.5 Bi03
Bi 03 (a) a’F(w)
:~~ o 80
2,’
2.
4i;‘
. . .._
__
6;
80
w [m&l 1.04

(b) dr dV
A, 40 0.98
W[mevl
I
I
60
I
!
80
VImVI
80
Vtmvl

20
40
v’
60 ‘
’
’
’
(c) fi 0.02
Figure 1.
(a) a2F(w) ( full curves) and F(w)
(broken curves), (b) dI/dV, and (c) d21/dV2 calculated for x=0.7. The broken curves in (b) and (c) represent the BCS results.

Figure 2.
dV*
(a) 02F(w) ( full curves) and F(w)
(broken curves), (b) dI/dV, and (c) d21/dV2 calculated for x=0.5. The broken curves in (b) and (c) represent the BCS results.
THEORETICAL STUDY OF TUNNELING SPECTRA
Vol. 73, No. 9
has been used. Further the cutoff energy E has been taken to be 200 meV and then A(E) h& sufficiently converged in iteration of several In result, the superconducting energy times. gap A0 is found to be 4.8 meV for x=0.7 and 1.0 meV for x=0.5. The result for x=0.7 is in good agreement with the recent experimental data A = 4.3 _ meV determined bv outical measurements Por Since Tc has ( TCZ jOKj [161. Ba0.6K0.4Bi03 been evaiuatkd to be 31.3 K for x=0.7, the ratio 2Ao/kBTc is found to be about 3.6, which is accidentally close to that predicted by the BCS theory. However, the gap function A(E) exhibits the remarkable energy dependence which is typical in the strong coupling superconductor, and such prominent structures in A(E) manifest themselves in the tunneling spectra. We have calculated the differential conThe results for ductance dI/dV from eq.(l). x=0.7 and x=0.5 are shown by the full curves in Figs.l(b) and 2(b), respectively. The broken curves represent the BCS results which are given by N (E) s z Re N(EF)
IsI 2 l/2 2 [E  AC1 I.
Deviations from the BCS results are clearly seen in Figs.l(b) and 2(b). We have calculated also the second derivative spectra, d21/dV2. The results are shown in Figs. l(c) and 2(c). In general d21/dV2 gives direct information about 02F(w), i.e. negative peaks (dips) in d21/dV2
635
correspond to peaks in u2F(w) [131. By comparing Figs. l(c) and l(a) it is clear that our results show certainly such correspondences. It should be noted here that the fine structures in the d21/dV2 spectra are drastically reduced when This is due to degoing from x=0.7 to x=0.5. crease of the magnitude of the gap function A(E) as x changes from 0.7 to 0.5. Recently, tunneling spectroscopy measurements with high resolution have been performed (Tc=29 K) [171. The obOn Ba0.625K0.375Bi03 " * served curves of dI/dV and dLI/dVL have line shapes similar to those shown in Fig. l(b) and (cl, respectively. In particular, the dips in d21/dV2 observed below 60 meV show good correspondence to the negative peaks in Fig. l(c). Very recently, tunneling measurements have been performed also on Bao.5K0.5Bi03 ( Tc=13K) (181 and a drastic reduction of prominent structures in d21/dV2 spectra hasbeen observed aspredicted by our calculation. In conclusion, superconducting properties in BKB, such as the energy gap and the tunneling spectra as well as the magnitude and the isotope shift of Tc, can be understood in the framework of the strong coupling theory of the phononmediated pairing mechanism. Acknowledgements We would like to thank Dr. J.F. Zasadzinski for showing us his experimental data of tunneling measurements before publication. This work is supported by a GrantinAid of Scientific Research on Priority Areas (No.01631005) from the Ministry of Education, Science and Culture.
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