Lévy statistics and anomalous transport in quantum-dot arrays

A model of transport is proposed to explain power-law current transients and memory phenomena observed in partially ordered arrays of semiconducting nanocrystals. The model describes electron transport by a stationary Lévy process of transmission events and thereby requires no time dependence of system properties. The waiting time distribution with a characteristic long tail gives rise to a nonstationary response in the presence of a voltage pulse. We report on noise measurements that agree well with the predicted non-Poissonian fluctuations in current, and discuss possible mechanisms leading to this behavior.

Figure 1

(Color online) (a) Time dependence of the net transmitted charge $Q\left(t\right)$ in a single channel (red line) and charge $⟨Q\left(t\right)⟩$ averaged over $N=100\text{channels}$ (blue line) simulated using WTD of form (2) with $\mu =0.5$ (double log scale). The dashed line is a power law $Q\propto t^{\mu }$. (b) The same plot in the linear scale. The large charge noise in a single channel is due to the lack of self-averaging for a wide-tail WTD. (c) Current in a single channel with a wide distribution of waiting times (schematic). Short packets of current pulses are separated by very long waiting times $\tau ⪢{\tau }_0$.

Figure 2

(Color online) Measured mean current and noise spectrum, with averaging performed over 50 current transients on the same sample. The current Fourier harmonic mean $⟨I_{\omega }⟩$ [Eq. (5)] and variance $⟪I_{-\omega }{I}_{\omega }⟫$ [Eq. (6)] are shown. Both the current and the noise are described by power law ${\omega }^{-\mu }$ with the same $\mu \approx 0.72$. The $1∕f$ dependence is shown for comparison (see text).