Boundary resistance and "pseudolocalization" in one-dimensional periodic systems

The resistance of a one-dimensional periodic potential with large unit cell is analyzed as a function of Fermi energy and potential strength. We demonstrate that even in the allowed en-ergy regions the resistance can be nonzero due to the boundary contributions. The analysis of the wave functions shows that the probabilities of finding a particle in various regions within the same unit cell can differ by many orders of magnitude. Localization is known to be caused by elastic scattering from impurities which break the system’s translational invariance. But even in the absence of the impurities translational invariance of a periodic system can be broken by its boundaries. In the present paper we describe the effect of the boun-daries on the resistance of a one-dimensional periodic system. In the first part the energy spectrum is analyzed and the eigenvalues are identified. Then we consider the shape of the wave function and intro-duce the effect of "pseudolocalization. " Finally, we calculate the average resistance and demonstrate that it reaches a finite constant for the allowed values of energy in long-enough systems. Consider a quantum particle in a potential in the form of a frozen density wave with Schrodinger equa-tion 1 4" + k — V X5(x — x ") 4=0 n ! where the distance between the 5 functions a " =x " +t — x " =1+2 cos(2rrPn) is a periodic function with rational P (in our numerical studies A = 0.875 and P =0.04). The rationality of P makes the poten-tial commensurate with the spacing (a " =1) of the undisturbed lattice (A = 0) and therefore periodic. The quantity of interest is the dimensionless resis-tance, defined by Landauer’s formula’ R = T ’ — 1, where T is the transmissivity of the system. The ex-ponents of resistance were evaluated directly by the recurrence relations developed by the method of Az-bel and Soven’ (see Appendix for details). The energy spectrum can be represented by the resistance as a function of Fermi wave vector (Fig.