Magnetoconductance of rectangular arrays of quantum rings

Electron transport through multiterminal rectangular arrays of quantum rings is studied in the presence of Rashba-type spin-orbit interaction (SOI) and of a perpendicular magnetic field. Using the analytic expressions for the transmission and reflection coefficients for single rings we obtain the conductance through such arrays as a function of the SOI strength, of the magnetic flux, and of the wave vector $k$ of the incident electron. Due to destructive or constructive spin interferences caused by the SOI, the array can be totally opaque for certain ranges of $k$, while there are parameter values where it is completely transparent. Spin resolved transmission probabilities show nontrivial spin transformations at the outputs of the arrays. When pointlike random scattering centers are placed between the rings, the Aharonov-Bohm peaks split, and an oscillatory behavior of the conductance emerges as a function of the SOI strength.

Figure 1

The notations used for the spinor part of the wave functions in the case of (a) two-, (b) three-, and (c) four-terminal rings.

Figure 2

The geometry of the device in the simplest case of a $3\times 3$ array with three or one (without leads displayed with dashed lines) input terminals. The notations can easily be generalized to larger arrays.

Figure 3

(Color online) The conductance $G/G_0$ $\left(G_0=e^2/h\right)$ of (a) $3\times 3$, (b) $4\times 4$, and (c) $5\times 5$ rectangular arrays with 3, 4, and 5 input terminals, respectively, for zero magnetic flux as a function of the SOI strength and $ka$.

Figure 4

(Color online) The conductance $G/G_0$ $\left(G_0=e^2/h\right)$ of (a) $3\times 3$, (b) $4\times 4$, and (c) $5\times 5$ rectangular arrays with 3, 4, and 5 input terminals, respectively, for $ka=19.6$ as a function of the SOI strength and the magnetic flux $\Phi$ (in units of ${\Phi }_0=h/e$).

Figure 5

(Color online) The conductance $G/G_0$ $\left(G_0=e^2/h\right)$ of a $5\times 5$ rectangular array with a single input lead attached to ring {31} for zero magnetic flux as a function of the SOI strength and $ka$.

Figure 6

(Color online) The conductance $G/G_0$ $\left(G_0=e^2/h\right)$ of a $5\times 5$ rectangular array with a single input lead attached to ring {31} for $ka=19.57$ as a function of the SOI strength and the magnetic flux $\Phi$ (in units of ${\Phi }_0=h/e$).

Figure 7

(Color online) The probabilities of the (a) $|{\uparrow }_x⟩$, (b) $|{\uparrow }_y⟩$, and (c) $|{\uparrow }_z⟩$ outputs at ring {55} of a $5\times 5$ rectangular array with one input lead (attached to ring {31}) for $ka=19.6$ as a function of the SOI strength and the magnetic flux $\Phi$ (in units of ${\Phi }_0=h/e$). The incoming spin state is chosen to be $|{\uparrow }_z⟩$.

Figure 8

The spin transformation properties of a $5\times 5$ array with input leads attached to all rings and only to ring {31} (black and gray curves, respectively). The $z$ component of the normalized spin states transmitted via the output terminals attached to ring {25} (solid line) and ring {45} (dashed line). The incoming spin state is chosen to be $|{\uparrow }_z⟩$.

Figure 9

(Color online) The conductance $G$ (in units of $G_0=e^2/h$) of a $5\times 5$ rectangular array with and without pointlike random scatterers between the rings as a function of the magnetic flux $\Phi$ (in units of ${\Phi }_0=h/e$) for $ka=20.2$ and ${\omega }_{\text{SO}}/\Omega =13.0$.

Figure 10

Fourier spectra of the data shown in Fig. 9. Notice that the relative weight of peaks corresponding to $2\Phi /{\Phi }_0$ oscillations increases when scattering effects are introduced.

Figure 11

The conductance $G$ (in units of $G_0=e^2/h$) of a $5\times 5$ rectangular array with pointlike random scatterers between the rings for different root-mean-square deviations $D$ as a function of the SOI strength for $ka=19.6$ and $\Phi =0.3{\Phi }_0$.