Conductance measurements remain our primary window into quantum behavior in mesoscopic systems. My work on single-body electron transport algorithms helps expand our ability to simulate these experiments to novel geometries and larger systems.
The crux of the Outward Wave algorithm is to permute a matrix called the Hamiltonian, which describes the behavior of an electron in a mesoscopic system. Normally, the Hamiltonian is extremely sparse, much like an adjacency matrix. When a small device is connected by wires, they perturb the Hamiltonian by contribute dense blocks. In the picture above, the dense blocks appear in blue.
My work shows how to permute the Hamiltonian in a fashion that makes it easy to invert it in waves that emanate from the center of the system towards the wires that connect to it. This allows to us easily calculate properties like the conductance (red block in lower-right corner), but also the density of states (along the diagonal) and the density matrix (vertical block).
Extensions to this algorithm then allow us to calculate scattering matrices off of unusual geometries. One manipulation shows how to compute the reflection off of arbitrary boundaries in mesoscopic systems. By automating the process and rendering it numerically tractable, my work opens up the ability to know precisely how electrons reflect off each boundary in a realistic system, a key ingredient to predicting future experiments in graphene.