Kamimura H., Ushio H., Matsuno S., Hamada T.Theory of Copper Oxide Superconductors

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14.6 Calculated Results of the Superconducting Transition Temperature 191

in ρ

(ε) while the decrease of T

in the overdoped region with increasing x is

due to the pair-breaking eﬀect related to the decrease of the local AF order by

the destruction of eﬀective superexchange interactions in the AF-correlated

region.

From the empirical value of γ = 5, we can say that a metallic region is

spread over 25 nm for the optimum doping of LSCO. This means that the

superconducting regions are inhomogeneous in cuprates, at least for LSCO.

Recently Lang et al. [209] reported the granular structure of high-T

super-

conductivity in underdoped Bi2212 by using scanning tunneling microscopy

(STM). Their STM studies have revealed an apparent segregation of the elec-

tronic structure into superconducting domains that are ∼ 3 nm in size. Since

STM studies concern with a much higher energy region over the Fermi en-

ergy, the discrepancy between the size of an empirically-determined metallic

region and the size determined by STM might be due to the diﬀerence in the

time scales caused from the diﬀerent energy regions with which the present

theory and STM experiments are concerned.

14.6.3 Isotope Eﬀects

Next we show the calculated result of the isotope eﬀect in Fig. 14.12 [26].

The isotope eﬀect is measured by a constant α which is deﬁned by

α = −

dlnT

dlnM

, (14.86)

where M denotes the mass of constituent atoms. In the BCS theory, T

just proportional to the Debye frequency ω

,andsinceω

generally has

M-dependence of ω

∝ M

−0.5

,thuswehaveα =0.5 for weak coupled

superconductors. But in the strong coupling regime, we do not have such

(K)

hole concentration x

0 0.1 0.2 0.3

1.0

2.0

Isotope effect α

Fig. 14.12. The calculated x-dependence α(x) of the isotope eﬀect. Here, the solid

line represents α(x). The dashed line represents the calculated T

(x), shown for

reference

192 14 Mechanism of High Temperature Superconductivity

a simple relation, even in the conventional superconductors. In the present

calculation we also use ω

∝ M

−0.5

with regard to the M-dependence of

. The calculated results are shown in Fig. 14.12. As seen from the ﬁgure,

the isotope eﬀect α becomes very large when x is small, while around the

optimally doped region α is 0.39, which is slightly smaller than that of the

BCS value, 0.5. In the present calculation, however, the mass of all constituent

atoms has been changed by the ratio of the mass of

Otothatof

Therefore, the calculated value of 0.39 is overestimated.

As we have seen, in T

-equation (14.77), T

depends on the density of

states ρ through the factor ρ = ρ(Z(0)ω)whereZ(0) > 1. Since the den-

sity of the states (DOS) of the K–S model ρ

(ε) has a signiﬁcant energy-

dependence, the isotope eﬀect becomes dependent on the hole-concentration

x critically. In particular α becomes large in a low-concentration region where

is small and thus T

is also small. This peculiar x-dependence of isotope

eﬀect is one of the characteristic features of the isotope eﬀect, obtained from

the K–S model. In Fig. 14.2, the calculated results are compared with ex-

perimental results of LSCO by Crawford et al. [210, 211], who made the

substitution of

Ofor

O in LSCO. In the ﬁgure the experimental results

are shown by solid squares and diamonds. We can see that there are good

agreement between the calculated results of the K–S model and the experi-

mental results shown by the solid squares in the underdoped and overdoped

region below 0.2 (x<0.2), where the K–S model holds.

14.7 Final Remarks

Recently a number of papers related to the phonon mechanism have been

published. Although the phonon mechanisms in these papers are diﬀerent

from the mechanisms of superconductivity due to the K–S model with two-

component scenario described in Chap. 13 and the present chapter, in closing

this chapter we would like to mention some of the papers as references [212,

213, 214, 215, 216, 217, 218, 219, 220].

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