Franz Kaiser
Current and Noise in Driven Heterostructures
Supervisor: PD Dr. Sigmund Kohler [Theoretical physics I]
Date of oral examination: 02/18/2009
91 pages, english
In this thesis we consider the electron transport in nanoscale systems driven by an external energy source. We introduce a tight-binding Hamiltonian containing an interaction term that describes a very strong Coulomb repulsion between electrons in the system. Since we deal with time-dependent situations, we employ a Floquet theory to take into account the time periodicity induced by different external oscillating fields. For the two-level system, we even provide an analytical solution for the eigenenergies with arbitrary phase shift between the levels for a cosine-shaped driving. To describe time-dependent driven transport, we derive a master equation by tracing out the influence of the surrounding leads in order to obtain the reduced density operator of the system. We generalise the common master equation for the reduced density operator to perform an analysis of the noise characteristics. The concept of Full Counting Statistics in electron transport gained much attention in recent years proven its value as a powerful theoretical technique. By combining its advantages with the master equation approach, we find a hierarchy in the moments of the electron number in one lead that allows us to calculate the first two cumulants. The first cumulant can be identified as the current passing through the system, while the noise of this transmission process is reflected by the second cumulant. Moreover, in combination with our Floquet approach, the formalism is not limited to static situations, which we prove by calculating the current and noise characteristics for the non-adiabatic electron pump. We study the influence of a static energy disorder on the maximal possible current for different realisations. The probability distribution of the currents in an open transport channel demonstrates that with increasing system length the effect of rather tiny fluctuations increases drastically. We conclude that the reason for this behaviour lies in the increasing probability to find one level in the system misaligned and so inhibiting effectively the transport. Further, we explore the possibility of non-adiabatically pumping electrons in an initially symmetric system if random fluctuations break this symmetry. We find that fluctuations may alter the distribution of the current, but already rather small bias voltages suppress pumping. Our analysis reveals that for longer systems, that is working with a large number of levels, even very small fluctuations might lead to an effectively isolating behaviour. Since this energy disorder is inevitable in the real setups, e.g., in a possible quantum computer that reads out the qubit by measuring a current, one should take care to (i) minimise the fluctuations or (ii) reduce the number of incorporated levels, such that the fluctuations cannot inhibit transport. Otherwise, the reproduceability of the results might be strongly decreased. Motivated by recent and upcoming experiments, we use our extended Floquet model to properly describe systems driven by propagating waves that induce a phase lag between neighbouring sites. For a qualitative analysis of surface acoustic wave driven quantum dot systems, we adopted our model to the static situation and extracted the numerical values. The study of the same system in a driven configuration is hindered by the very small resulting driving frequency and the consequent huge numerical effort. We see that in the theoretical model the sign of the current depends sensitively on the phase difference between the sites. The direct relation between phase lag and spacing of the dots as well as the well-defined wavelength of the SAW, may work as a ``ruler'' to measure the distance between the quantum dots. Furthermore, there are current attempts to emulate a propagating wave in a fully controllable triple quantum dot. Then, the study of the current for arbitrary phase lags becomes possible; opening the way to an experimental test for our model. Recent measurements of the photoconductive gain and more detailed studies on photo induced ballistic transport indicated an influence of geometrical constraints on the electron path. Within this thesis we numerically evaluate the current for latter in two distinct geometries. Unlike in the other systems discussed in this thesis, we simulate the electrons as free non-interacting particles. The comparison of our simplified model for the charge carriers with the measured current reproduce well the experimental results in both geometries. Moreover, the absolute values agree well with the experiment, taking our reduced model into account. Our results prove that a description of the electrons as free moving particles in the two dimensional electron gas is justified and already suitable to understand the experimental results. The deviations we observe near the edges might be reduced by improving the model, for example by taking charge accumulation due to holes at the boundaries into account or by employing a more sophisticated model for the scattering with the lattice.