Localization of Electronic States in One-Dimensional Disordered Systems

The localization of electronic states in one-dimensional disordered systems is examined in terms of the reflection and transmission coefficients. The transfer-matrix method is used. The main body of the work deals with a one-dimensional liquid model in which the central part of the potential remains the same in all cells, and only the lengths of the flat arms vary from cell to cell. It is found that the contribution of the initial phase of a wave at the zeroth cell to the phase at the $n\mathrm{th}$ cell is reduced by a factor $\frac{\mathrm{}(1-|\mathrm{}\mathcal{r}\mathrm{}\left|\right)\mathrm{}}{\mathrm{}(1+|\mathrm{}\mathcal{r}\mathrm{}\left|\right)\mathrm{}}$ every time in passing through a cell. When the phase memory is completely lost, ${\Phi }_j\sim {\phi }_j$, where the reflection coefficient of the $j\mathrm{th}$ cell is ${\mathcal{r}}_j=\left|\mathcal{r}\right|e^{i{\phi }_j}$. If ${\Phi }_j$ obeys a uniform or nearly uniform probability distribution, the wave function always grows exponentially. It is shown that in most cases, especially when cell size distribution has a wide spread, $P\left(\Phi \right)$ is nearly always uniform. All wave functions are localized in a completely disordered system, but in the one-dimensional liquid model nonlocalized states do exist.