# Interactions: from galaxies to the nanoscale

For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by $\vec{F}_{k}=-\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1}}^{N_{\rm donor}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1\\l\ne k}}^{N_{\rm elec}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+e \dot{\vec{r}}_k\times\vec{B}$

Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.

Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.

Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol. 10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact.