Effect of disorder in specific realizations of multibarrier random systems Gasto ´ n Garcı ´ a-Caldero ´ n Instituto de Fı ´sica, Universidad Nacional Auto ´noma de Me ´xico, Apartado Postal 20-364, 01000 Me ´xico, Distrito Federal, Mexico Roberto Romo and Alberto Rubio Facultad de Ciencias, Universidad Auto ´noma de Baja California, Apartado Postal 1880, Ensenada, Baja California, Mexico Received 24 February 1997 A resonance formalism is used to study the effect of disorder in specific realizations of multibarrier random systems. We solve the periodic case and introduce disorder by allowing random values for the well widths. We analyze the motion of the complex poles of the S matrix on the energy plane and calculate the resonant states for systems of fixed length as a function of the disorder strength. Our analysis of the eigenfunctions, the decay widths, and the Thouless criterion allows us to distinguish in general three different types of states: quasilo- calized, intermediate, and border states. S0163-18299708331-8 I. INTRODUCTION Since the seminal work by Anderson, 1 almost four de- cades ago, localization in disordered systems has been the subject of numerous investigations. 2 One-dimensional sys- tems have been convenient models for both theoretical and numerical investigations on the properties of localized states, 3,4 as, for example, in studies on the localization length. In recent years technical improvements in the fabri- cation of semiconductor heterostructures have allowed the possibility of designing multibarrier potential profiles almost at will. 5,6 This provides the opportunity to study the proper- ties of electron propagation in multibarrier systems. Erdo ¨ s and Herndon pointed out some time ago 7 the con- venience of having a link between the transmission ampli- tude, i.e., the scattering properties, and the one-electron Green function that relates to the eigenfunctions and eigen- values of the problem. Our approach establishes this link. Previous works on specific systems considered either the eigenfunction or the scattering approaches. Among the former, one finds works where the wave function vanishes at the boundaries of the system, 8 implying that the system is closed and hence that a connection to the transmission prob- lem is not possible. On the other hand, one finds works based on the properties of resonant tunneling, where the transmis- sion coefficient and wave function as functions of the energy are studied. 9 These approaches lacked a definition for the resonance eigenfunctions and eigenvalues associated with the disordered potential. Here it is worth mentioning recent work involving resonant tunneling in connection with quan- tum dots. In this case, however, Coulomb interaction effects become relevant for an appropriate description of the prob- lem, and the corresponding treatment becomes more involved. 10 On the other hand, it is also appropriate to refer to a number of recent works that addressed the effect of correlated disorder on the properties of localization in one dimension. 11 These works showed that the prevalent notion that in one dimension all states are localized for any amount of disorder does not hold in general. The purpose of this work is to study the onset and prop- erties of localization in a specific realization of a multibarrier potential profile generated in a random manner. Our ap- proach is based on a resonance formalism that considers the multibarrier system as an open system. This leads to a com- plex eigenvalue problem, and allows one to connect the problem of transmission scattering with the resonant states and eigenvalues of the system. In our approach we consider a specific system of length L to study it as a function of disorder. Hence the notion of localization length, that arises from statistical considerations involving an ensemble of systems, does not seem appropriate to characterize the properties of a particular system. Trans- mission scattering probes the resonant states of a multibarrier system. As is well known, this is exhibited as peaks in a plot of the transmission coefficient versus energy. These peaks reflect the existence of resonant states of the system, and their position in energy is related to the real part of the com- plex poles of the transmission amplitude. Since the system has a finite length, eventually an electron seated on one of these states decays out of the system with a time scale pro- portional to the inverse of the imaginary part of the complex pole. Since the transmission amplitude is an element of the S matrix, the above complex poles are precisely the complex poles of the S matrix of the problem. A very important as- pect of our approach is that one may associate a resonant eigenfunction with each of the transmission levels. A conve- nient way to do this is by exploiting the analytical properties of the outgoing Green function of the problem. Our formal- ism establishes a connection between the wave solution and the outgoing Green function of the problem along the inter- nal region of the system that includes, as a special case, a connection with the transmission amplitude. It is well known that the effect of disorder on the transmission coefficient causes irregular fluctuations as a function of energy. 2 The effect of disorder on the S -matrix poles in one-dimensional chains also modifies their distribution on the complex energy plane. 12 However to our knowledge no treatments have been reported in the literature on the effect of disorder on resonant eigenfunctions and the connection of these functions with the complex poles. It is well known from numerical calculations of transmis- PHYSICAL REVIEW B 15 AUGUST 1997-II VOLUME 56, NUMBER 8 56 0163-1829/97/568/48458/$10.00 4845 © 1997 The American Physical Society