Properties of microscopic superconducting glasses

Physica C 153-155 (1988) 711-712 North-Holland, Amsterdam

PROPERTIES OF MICROSCOPIC SUPERCONDUCTING GLASSES

Reinhold OPPERMANN* Service de Physique Theorique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex and Ecole Normale Superieure, 24, rue Lhomond, F-75231 Paris Cedex 05

Complete or large partial disruption of phase order of the superconducting order parameter on an atomic scale defines microscopic superconducting glass phases. Their properties can provide an explanation for the non-BCS-like behaviour of High T c oxide superconductors, if one supposes e.g. that the microscopic mechanism for the attractive interaction is linked to randomly placed oxygen vacancies in CuO-planes.

i. INTRODUCTION Cooper pair delocalization by breaking of particle hole symmetry (doping) can occur in a superconducting glass state. For half-filled band the state is called locally superconducting glass (LSCG), since the Cooper pairs ~hich start to form below a critical temperature TSCG>TBC S do not change the global normal metallic or insulating behaviour. For small deviations from half-filling we predict here a globally superconducting glass (GSCG) which is stabilized and feeded by the local order at a critical temperature only slightly below TSC G. The picture developed here bears similarities to Bosecondensed superconductivity in strong coupling limits (1,2), but it is in fact obtained from the weak coupling limit for the intermediate coupling regime. This will be discussed in §3 and 4 together with earlier results on properties of superconducting glasses (3-5). It will be evident that these results could explain experiments on High Tcoxide superconductors showing linear low temperature behaviour with 'normal' amplitudes in the specific heat, in coexistence with a wide distribution of gaps, a specific heat jump at criticality, absence of paramsgnetic pairbreaking effect, band filling dependence, and slso the irreversible behaviour in an external magnetic field (for references see (4,5)). Some of the equations given below were worked out in a 'local gauge invariant' formalism (6), which I find useful to describe the multiple crossover between LSCG, GSCG, the ideally phase coherent superconductivity in narrow bands, and possibly the strong interaction limit. The qualitative picture is more general and holds beyond local gauge invariance. 2. SOME DETAILS OF 'LOCAL GAUGE INVARIANT' SUPERCONDUCTIVITY IN NARROW BANDS In (6) I have introduced a new description of narrow band superconductivity by using real space pairing for dirty systems with approx-

imately zero electron mean free path 11 but finite two-particle mean free path 12~ The condition that particles can hop only in pairs was due to disorder and hence not instantaneous in time as it would be in presence of large negative U-Hubbard interaction (i). In our first case we still find a BCS-type coherent state, if one is not too close to the low-density regions near the band edges and if 12 is large enough to justify the assumption of a homogeneous order parameter, while in the latter case Cooper-pairs behave like Bose-particles which, if the density is not too high, can move as such or by virtual ionization. The crossover between these cases including a proper treatment of the Mort-Anderson transition remains a difficult problem for the future. Consider an ii=0 , 12~0 electronic tightbinding model ~trr,C~oer,o-(~c~+e~+h.e.) where t is random but random dependence of A on the center of mass coordinate is neglected in this section. I report here two results concerning the calculation of the global supercurrent <> and the coherence length ~. In ll=O models non-cancellation of the diamagnetic part of j is nontrivial, since the ensembleaveraged Green's functions are local in real space. At half-filled band, #=0, and in meanfield approximation we can use the exact result =-izm/= 2izm(l-(~E~/(z~+/ ~ < t ~ > and to I(A)-I(O) where theZreal i~egral I=fa~(G2(z)+F2(z)) can be evaluated exactly in terms of hypergeometric functions. We find I(O) approximately 50% is left, while for Eo=IOO this is only 5%. Deviations from halffilling further decrease the percentage, but increasing i I above zero should invert the effect.

*Heisenberg fellow of the Deutsche Forschungsgemeinschaft

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R. Oppermann / Microscopic superconducting glasses

The spatial variation of the averaged order parameter is given in continuum limit by the Ginzburg-Landau expansion ~ V 2 + T - 78-~(3)(~kBTc)-21

12=O

which is valid near Tc(=TBC S for weak coupling) and the rhs neglects disorder correlations to be treated below. The reduced temperature is denoted by T , the GL-coherence length diverges as (Tc-T)-i/2but with an amplitude

(~(p:O,11:O,12)--mEol~/(48kBT c) (1) which d i f f e r s from the conventional d i r t y l i m i t coherence length given by ~ = ~ V F l l / ( 2 4 k B T c) for 12=O and f i n i t e i 1. The coherence length of Eq. (1) decreases with increasing IpJ/E o. 3. THE SPIN-GLASS ANALOG SUPERCONDUCTINGGLASS (LSCG) The 12superconductor described before was i n v a r i a n t under c ~ e x p ( i X r o ) C r o w i t h Xro=-Xr_ ca r e a l c-number. This led to A r r , = & r ~ r r , . Now we a l l o w a d d i t i o n a l l y phase c a n c e l l a t i o n for A at each s i t e (see eg ( 3 ) ) . Such a s i t u a t i o n may be met in High Tc superconductors, i f , as r e c e n t l y worked out by Halley and Shore (7), the random oxygen vacancies in CuO-planes are responsible for local and random strong correlation causing superconductivity. By comparing & with a spin S we arrive at the possibility of spin-glassanalog 'superconducting order'. One way to obtain such order is from the selfconsistent equation < !Al2>=, =O, (2) which has under certain conditions even a weakcoupling critical point (see (3) and (4)) at TSCG=0.39 ~ exp(-i/(4p~. Note that weak coupling JPF is possible for certain concentrations of randomly placed strong coupling centers VrP F. Disregarding complications of renormalization in Eq.(2) I had already discussed in (4) the help of a small electron diffusion constant and in (3) that of long-ranged v in order to reach the maximal TSC G. The former is obviously related to short coherence length due to small 1 2 and an almost localized situation. Supposing now that TSCG>TBCS, no matter from which mechanism, I report the new result found in one loop order and in linear response to an external field that no global supercurrent and no Meissner effect exist below TSC G if =O. This was left as an open question in (5), in particular since the depletion of states due to glassy order resulted in a pseudogap of the global density of states near the Fermi level. However, as far as the mentioned approximation can be trusted, the Cooper pairs are localized if
4. GLOBAL SUPERCONDUCTIVITY STABILIZED BY GLASSY ORDER (GSCG) The behaviour of superconducting glasses can well be compared with the situation found in High TcSuperconductors , when one realizes that the correlation delocalizes the Cooper pairs if the band is not exactly half-filled. Then, =O is no more possible, while at halffilling nothing changes. A remarkable fact is that finite is not generated by a BCS-like

one p a r t i c l e s e l f e o n s i s t e n t equation, which could not have a TO other than a TBCS even depressed by the glass order parameters. Instead it is proportional to the -correlator, to Z, and to the deviation IPl from the band center. Thus the glassy order localized at half filling feeds which delocalizes the Cooper pairs as the particle hole symmetry is broken. If IPl does not become too large, the Tcfor <&> should therefore be almost as high as TSCG, and moreover the physical properties of the GSCG, except for supercurrent and Meissner effect, are still similar to those of the LSCG. In this sense the results earlier obtained in (3-5) can now be summarized for the 'LSCGstabilized GSCG'.Severa] mean-field results for observables in SCG-phases are unstable against quantum fluctuations, which were taken into account by means of loop expansions in (4,5). One finds i) a soft pseudogap Aeff(not to be confused with ) in the averaged density of states around E F with roughly AeFF/TscG=I.3 for weak coupling. A sharp BCS-type gap does not occur for 2~PFI 2> > , ii) spin-spin correlations (Pauli-paramagnetic contribution) are singularly depressed. Meanfield results remain however unchanged for iii) charge-charge correlations, and e.g. iv) the electronic specific heat, which at low temperatures b#haves linearly like CeI=2~2pMF(O)k~T/3 where the global mean-field density of states at the Fermi level is not depressed by glassy order. The pseudogap effect is compensated by low-energy two-particle excitations. At the onset of glassy order the s~ecific heat has a universal jump of order kB/~Z for weak coupling. Results on coexistence with local moments and on upper critical fields are contained in (4,5). REFERENCES (i) l.O.Kulik and A.G.Pedan, Zh.Eksp. Teor.Fiz.

(2) (3) (4) (5) (6) (7) (8)

79,1469 (1980) (Sov.Phys. JETP52,742(1980)) P.W.Anderson, Varenna-lectures (1987) R.Oppermann, Z.Phys.B63, 33 (1986) R.Oppermann, Z.Phys.B70, 43 and 49 (1988) RoOppermann, Sol. St.Commun. xxx(1988) R.Oppermann, J.Phys. Soc.Japan 52,3554(1983) J.W.Halley and H.B.Shore, Phys. Rev. B37, 525 (1988) R.Oppermann, Nucl.Phys.B280 [FSI8], 753 (]987)