Aharonov-Bohm oscillations in a mesoscopic ring with asymmetric arm-dependent injection

Electron transport through mesoscopic, one-dimensional rings with asymmetric injection into the arms of the ring is studied, in the presence of a Aharonov-Bohm flux, by means of an appropriate $\boldsymbol{S}$ matrix. This matrix is expressed in terms of two parameters one of which ($\lambda$) accounts phenomenologically for this asymmetric injection into the arms of the ring. In addition, the effect of a scatterer placed in one arm of the ring is considered. Explicit expressions are obtained for the transmission as a function of the incident electron energy, the magnetic field, the asymmetry parameter $\lambda$, and the strength of the scatterer. Results of the literature for symmetric rings are described by $\lambda=1$ and readily recovered. We relate our results to rings of finite width.