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 $\mathbf{S}$ matrix. This matrix is expressed in terms of two parameters, one of which $\left(\lambda \right)$ 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.

Figure 1

A ring of radius $R$ connected to two leads, indicated by triangular arrows, with two elastic scatterers indicated by the black squares. The transmission and reflection amplitudes of the scatterers are denoted by $t_i$ and $r_i$, $i=1$,2 respectively. A flux $\Phi$ pierces the ring through its center.

Figure 2

Transmission vs $\varphi$ (a) for two values of the flux and vs flux (b) for two values of $\varphi$. The solid and dashed curves are for $\lambda =1$ and the dotted and dash-dotted curves for $\lambda =3$. The value of $a$ is zero, i.e., there is no reflection at the left lead-ring junction.

Figure 3

Transmission vs $\varphi$ (a) and vs flux (b) when one scatterer is present in one of the arms with strength $T_s=0.25$. The curves are marked as in Fig. 2.

Figure 4

Transmission vs $\varphi$ (a) for $\Phi ∕{\Phi }_0=0.4$ and vs flux (b) for $\lambda =3$ and $\varphi =\pi ∕2$ with $a$ as indicated.

Figure 5

Top panel: Transmission vs flux with $\varphi$ such that the solid, dashed, and dotted curve correspond to the wave vectors $k=0.091∕\mathrm{nm}$, $k=0.006∕\mathrm{nm}$, and $k=0.053∕\mathrm{nm}$, in Fig. 8 of Ref. 6. The values of $a$, $s$, and $c$ used are $a=0.25$, $s=1.25$, and $c=1$. No scatterer is present in the ring’s arms. Bottom panel: As in top panel with one scatterer present in one of the arms. The dashed-dotted, dotted, dashed, and solid curves are for $T_s=0.25$, 0.5, 0.75, and 1, respectively. The other parameters are $\varphi =6.95\pi$, $a=0.4$, $s=1.5$, and $c=1$.