Longer-ranged interactions in quantum point contacts

Lukas Weidinger

04.12.2014 at 12:00 

Quantum point contacts (QPCs) show quantized conductance steps in terms of the conductance quantum GQ=2 e2/h when the voltage of the applied gates and thus the barrier height of the QPC is varied. This behavior can be understood in a non-interacting particle picture, assuming that the transport stems from single electrons traveling through the QPC. However, even the earliest
experiments on conductance quantization observed deviations from this ideal behavior. The most striking one, the so called ''0.7-anomaly'', is a shoulder in the conductance step between the pinchoff and the first conductance plateau at roughly G≈0.7⋅GQ . Ever since, that anomaly has been the subject of controversial discussion which is still going on. Recently, Bauer et al. managed to give a consistent explanation for the 0.7-anomaly, identifying it's origin in a smeared van Hove singularity in the center of the QPC. Using a short-ranged interaction model for the QPC, the conductance was explicitly calculated, employing the functional renormalization group (fRG) method. As an approximation within this method they applied a so called ''coupled ladder approximation'' (CLA), which reduces efficiently the degrees of freedom by exploiting the particular structure of the fRG flow  equations.
In this thesis, we will develop an fRG scheme suitable to take longer-ranged interactions into account. Explicitly, we will set up a coupled ladder approximation similar to the one of Bauer et al., but allowing also longer ranged contributions for the bare interaction. Using various symmetries of our system, we end up at a system of ODEs, which we solve numerically. We then study the results of this new algorithm using the previous model with short ranged interactions, as well as models with longer-ranged interactions. For the latter, we observe that in a certain regime, increased longer-ranged interactions are capable of actually increasing the conductance. We study this -at first sight- counterintuitive result by examining conductance, density, and magnetic susceptibility for various parameters. As cause of this physical behavior we suspect a Wigner like crystallization process of the QPC in the presence of long ranged interactions. To conclude this thesis, we further examine two more cases modeled by short-ranged interactions, namely the transition between a QPC and a quantum dot (QD) and a QPC with non-parabolic potential. This was motivated by the improved convergence of our new algorithm compared to the previous method, since the last two cases are known to suffer from convergence issues due to the relative flat barrier top.