On The Optimized Mixing of Order and Disorder in High-T c Ceramic Superconductors
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ON THE OPTIMIZED MIXING OF ORDER AND DISORDER IN HIGH-Tc CERAMIC SUPERCONDUCTORS E. MEZZEllI Politecnico di Torino - Dipartimento di Fisica C.so Duca degli Abruzzi, 24 - 10129 TORINO - ITALY
ABSTRACT Coherent excitations of localized states, represented by a selfsimilar Cayley tree, produce a gap in the density of states, which is deeper as the number of generations increases. The model can suitably describe systems in which an optimized concentration of "defects" allows self similarity of the state-tree and coherent coupling between clusters of localized states. A renormalization of the state-tree leads to a hierarchy of the energy eigenvalues.
INTRODUCTION The main common feature of the new high Tc oxides is that in such alloys some defect-states with optimized concentrations are introduced in a stoichiometric matrix. This condition, although not sufficient, is necessary in order to have high-Tc behaviour. Moreover in granular materials, such as sintered ceramics, we have a disordered array of single grains or single crystals; inside a grain we see coupled planes, and each plane consists of an array of domains [1]. Inside the domain microscopic mechanisms set up. A relevant question is how to bring together the two afore mentioned characteristics by a phenomenological synthesis. Without consideri'ng the details of the microscopic coupling mechanisms, we only observe that the "doping" defects yield to hopping behaviour due to electron or hole localization. In this paper we show that, if at least a percolative path sets up in the state-tree, which represents the states associated with the configurations of the system, a gap opens up in the density of these localized states . This gap becomes sharper as the number of involved states increases and its amplitude depends on the number of "brothers" in each family of the state-tree. Mat. Res. Soc. Symp. Proc. Vol. 195. 01990 Materials Research Society
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THEORETICAL MODEL In the configuration space we move one electron at a time, while all other electrons are frozen. The points representing the electron configuration are connected by lines, forming a graph. The resulting graph is a Cayley tree with a "brothers" for each node and N generations. Such a tree can also be seen as a self similar fractal structure, whose characteristics depend on the number of brothers for each node [2]. The transitions between different points of the graph correspond in the configurations space to hoppings between the nearest neighbouring atoms [3,4,5]. The hamiltonian H is a hopping hamiltonian [6], shaped in such a way that symmetric states
G
Wsj = ;-j/ 2 1 m=1
msj
(1) n
originate from all the projections such as En cn H In eq.(1) 'Tmsj is sth ensemble of the jth the state corresponding to the mth brother of the generation. Defining the vector (2)
D= F,j Y,S bj "' Ys j
the eigenvalue equation assumes the form H
=-uV@
(3)
where u is the energy in units of V. Taking into account also the antisymmetric states coming from the root of the tree, the hopping hamiltonian of eq. 3 be
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