Anomalous localization in the aperiodic Kronig–Penney model

Abstract

We analyse the anomalous properties of specific electronic states in the Kronig-Penney model with weak compositional and structural disorder. Using the Hamiltonian map approach, we show that the localisation length of the electronic states exhibits a resonant effect close to the band centre and anomalous scaling at the band edges. These anomalies are akin to the corresponding ones found in the Anderson model with diagonal disorder. We also discuss how specific cross-correlations between compositional and structural disorder can generate an anomalously localised state near the middle of the energy band. The tails of this state decay with the same stretched-exponential law which characterises the band-centre state in the Anderson model with purely off-diagonal disorder.

Transmission in waveguides with compositional and structural disorder: experimental effects of disorder cross-correlations

“…The weak-disorder case is identified by the conditions [18][19][20] with n = a n+1 − a n representing the relative displacement of two contiguous barriers. If disorder is weak, one can obtain an analytical expression for the localization length l loc of the electronic states ψ of the model (1) [18][19][20]. Following the Hamiltonian map approach, one can show that, within the second-order approximation, the electronic states have an inverse localization length equal to…”

“…Since slabs of both types can have random widths, each structure is characterized not by one but by two random sequences which, in addition to self-correlations, can exhibit cross-correlations. Another model exhibiting two types of disorder is the aperiodic Kronig-Penney model with both compositional and structural disorder [18][19][20]. For this model, it was possible to work out a perturbative expression for the localization length of the quantum states valid for weak disorder with any kind of selfand cross-correlations [18].…”

Localisation and transport in bidimensional random models with separable Hamiltonians

“…We also observe that in figure 4 the Lyapunov exponent drops in the middle of the localisation window. We interpret this decrease as a manifestation of the anomaly which appears when the Bloch wavevector takes the value k=π/2, as shown in [54]. It is known that correlations can enhance the anomaly in the Anderson model [58]; the numerical data suggest that similar effects occur in the Kronig-Penney model.…”

“…[52]). An analytical expression for the inverse localisation length (22) was derived for the case of weak disorder in [53][54][55]. As shown in [53], disorder can be considered weak provided that…”

“…As can be seen from figure 14, the numerical data generally match the theoretical predictions for the Lyapunov exponent in a satisfactory way. A discrepancy occurs for k 2 p = , where the Kronig-Penney model exhibits an anomaly which is the equivalent of the band-centre anomaly in the Anderson model, and where the expression (24) for the Lyapunov exponent fails as Thouless formula does at the band centre [54]. The band structure also corresponds to the theoretical expectations: this can be seen by considering that the points where the numerically computed Lyapunov exponent starts a rapid increase coincide with the band limits obtained by solving (20) for the 1D Kronig-Penney model (21).…”