Localization in highly anisotropic systems.

The localization behavior of the Anderson model with anisotropic hopping integral t for weakly coupled planes and weakly coupled chains is investigated both numerically with the transfer-matrix method and analytically within the self-consistent theory of localization. It is found that the mobility edge is independent of the propagating direction. However, the correlation (localization) length in the extended (localized) side of the transition can be very different for the two directions. The critical disorder Wc is found to vary from t 1y4 for weakly coupled planes to t 1y2 for weakly coupled chains. [S0031-9007(96)00089-0] The problem of Anderson localization in anisotropic systems has attracted considerable attention [1‐5] in recent years, largely due to the fact that the high- Tc superconductors are highly anisotropic. Transport in the normal state is metallic in the a-b plane but appears semiconductorlike in the c axis [6]. The nature of the c-axis transport in high-Tc materials is still controversial and its understanding may have important consequences for the theories of the normal and the superconducting states. This paradoxical property has prompted the proposal [7] that a high-Tc material in the normal state is actually an insulator, appearing metallic only because the inelastic length in the plane is less than the localization length. It has also been argued [2,6] that a negative drydT in the c direction alone may signify anisotropic localization, with a metal-insulator transition depending on the propagating direction, in direct contradiction to the predictions of the scaling theory of localization [8]. A recent diagrammatic calculation [1] lent support to such a claim. Previous diagrammatic analysis [9], however, led to the conclusion that the scaling property in anisotropic systems remains the same as that of the isotropic systems with a simple substitution of the conductance by its geometric mean. Given the perturbative nature of all the previous work, it is important to carefully study the localization behavior of disordered anisotropic systems with reliable numerical techniques and to determine whether the scaling theory is valid for high anisotropic systems [8 ‐ 10]. In the present work, we have systematically studied the localization properties of a three-dimensional disordered anisotropic system described by a tight-binding Hamiltonian with random on-site energies and anisotropic hopping integrals. We calculate numerically both the conductance and the localization length of such systems with the transfer-matrix method [8]. By doing finite-size scaling analysis on sufficiently large systems, we find the metal-insulator transition occurs at the same critical disorder Wc in all directions. Remarkably, at criticality, the geometric mean of the ratio of the finite-size localization length to the width of the bar is independent of the anisotropy. This underlies the conformal invariance property at the Anderson transition. For systems with extremely weak plane coupling, the critical disorder Wc is found to vary with anisotropy t as t 1y4 , in good agreement with a self-consistent theory of localization. However, the correlation lengths in the extended regime, as well as the localization lengths in the localized regime, differ tremendously in different directions. As we will discuss later, this difference of the correlation lengths in the propagating directions may explain the c-axis transport of the high-Tc materials. The localization behavior of weakly coupled disordered chains, which simulate the 1D-to-3D behavior, is also studied. The critical disorder Wc is found to behave as t 1y2 .