Stark-ladder resonances in ordered and disordered electrified chains.

Abstract

The electronic energy spectrum of a one-dimensional chain with periodic and disordered potentials in the presence of a constant electric field F is studied. Under certain conditions the spectrum shows the resonant states predicted by Wannier. These Stark-ladder resonances (SLR) are studied in detail for different potentials, amount of disorder, W, and length of the chains, L. Thermal population effects on the resonances are also considered. The different potentials correspond to rectangles with random widths and different heights, that include the extreme \ensuremath{\delta}-function limit. The Poincar\'e-map method is used to calculate the reflectivity and transmittivity of the chains. Use is made of different scattering theory criteria to characterize the resonances. For electrons incident on the chain with energies, E\ensuremath{\le}FL, the electrostatic potential energy produced by the field, Levinson's theorem is used to calculate the density of states from the derivatives of the phase shifts with respect to E. For energies E\ensuremath{\ge}FL, the transmission coefficient T is calculated as a function of E, and the SLR appear as equally spaced maxima of T as E is varied.From resonance and ensemble averages the following results are presented: (i) When L is fixed, there is a minimum value of F above which the resonances are clearly present. (ii) The effect of disorder is to perturb the mean properties of the resonances. The mean separation distance between resonances is affected by disorder, and it grows linearly with W, and the variance grows as well. The half width at half maximum (HWHM) of the resonances grows as ${W}^{2}$. (iii) In the periodic as in the disordered cases, the HWHM goes to zero as \ensuremath{\sim}${e}^{\mathrm{\ensuremath{-}}B(W}$,${V}_{0}$)/F. (iv) As the temperature \ensuremath{\tau} is varied, the height of the resonances decreases as ${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}1}$. As a function of the height of the potential barriers, ${V}_{0}$, it is found that the results qualitatively remain the same. However, quantitatively the results change, for instance, for fixed E, the HWHM increases as ${V}_{0}$ decreases. Finally, a critical discussion of the results is given to assess the possible experimental observation of these resonances in disordered quasi-one-dimensional devices.