Localization as a breakdown of extended states

HORACIO M. PASTAWSKI; CLAUDIA M. SLUTZKY; JUAN F. WEISZ

PHYSICAL REVIEW B - SOLID STATE

American Physical Society

Lugar: Ridge N.Y.; Año: 1985 vol. 32 p. 3642 - 3653

0556-2805

Phys. Rev. B 32, 3642 - 3653 (1985) Localization as a breakdown of extended states A way of understanding localization as a breakdown of extended states is presented by considering a three-dimensional disordered cluster of length L1 and finite cross section L2L3, which is repeated periodically along the L1 direction. The exact Green¡¯s function is calculated in the tight-binding scheme and the complex self-energy ¥Ò(¥å)=¥Ä(¥å)+i¥Ã(¥å) is studied as a function of the cell length; this procedure of finding ¥Ã guarantees that one is always looking at the properties coming from the eigenvalues. The singular spectrum of an infinite isolated cell is then obtained as a limit from the absolutely continuous spectrum of periodic systems with finite cell length. For the repeating cell the measure of the spectrum support approaches zero as L1 increases and this is reflected by the law ¡´¥ÃL1(¥å)¡µav¡­e-L1 ¥ë for the ensemble average. Theoretical arguments allow us to show that the localization length ¥ë of the eigenfunction of the isolated cells is the same as that calculated by previous authors from the exponential decay of the transmittance and by ourselves from convergence of self-energies in the realm of real numbers. A numerical calculation of the dependence of ¥ë on cross section and disorder parameter W shows a behavior similar to that found by MacKinnon and Kramer.