Time-dependent simulations of electron transport through a quantum ring: Effect of the Lorentz force

The time-dependent Schrödinger equation for an electron passing through a semiconductor quantum ring of nonzero width is solved in the presence of a perpendicular homogeneous magnetic field. We study the effects of the Lorentz force on the Aharonov-Bohm oscillations. Within the range of incident momentum for which the ring is transparent at zero magnetic field, the Lorentz force leads to a decrease of the oscillation amplitude, due to the asymmetry in the electron injection in the two arms of the ring. For structures in which the fast electrons are predominantly backscattered, the Lorentz force assists in the transport, producing an initial increase of the corresponding oscillation amplitude. Furthermore, we discuss the effect of elastic scattering on a potential cavity within one of the arms of the ring. For the cavity tuned to shift maximally the phase of the maximum of the wave packet we observe a $\pi$ shift of the Aharonov-Bohm oscillations. For other cavity depths oscillations with a period of half of the flux quantum are observed.

Figure 1

Geometry and dimensions of the circular (a) and diamond (b) quantum rings studied in the present paper. In the calculations the positions of the centers of the Gaussians (2) are chosen along the drawn lines with a spacing of $20\mathrm{nm}$.

Figure 2

(Color online) Charge (contours) and current (vectors) densities for a Gaussian wave packet with kinetic energy ${\hslash }^2{q}^2∕2m=1.42\mathrm{meV}$ being transferred through a quantum ring of radius $132\mathrm{nm}$ for zero magnetic field. (a)–(d) correspond to $t=2.17$, 4.35, 6.35, and $8.8\mathrm{ps}$. Scale for the charge and current density is the same in all the plots, with the exception of the current density vectors in (a) which were shortened by a factor of $1∕2$ with respect to the other plots.

Figure 3

(Color online) Same as Fig. 2 but for $B=0.0378\mathrm{T}$, which corresponds to the flux of the magnetic field through the ring $\Phi =h∕2e={\Phi }_0∕2$. (d) corresponds to $t=13.06\mathrm{ps}$ (the others to $t$’s as in Fig. 2).

Figure 4

(Color online) Same as Fig. 2 but for $B=0.34\mathrm{T}$, i.e., $\Phi =4.5{\Phi }_0$.

Figure 5

The transmission probability of the wave packet through the circular (solid line), and diamond (dotted line) quantum rings as function of the flux passing through the ring in units of the flux quantum. The dashed line shows the transmission probability through a wire of semicircular shape obtained from the circular quantum ring after removal of one of its arms.

Figure 6

Probability of finding the electron inside the semicircular wire above and below it as function of time for $\Phi =10{\Phi }_0$. The solid (dashed) lines show the results for $B<0$ $(B>0)$. The reflected probability density for the $B<0$ is marked by the dots.

Figure 7

(Color online) Probability density of the transferred wave packet through the circular ring in wave vector space. Lines corresponding to 0, 1, 2.5, 7, and 7.5 flux quanta passing through the ring are labeled. The dashed line shows the shape of the initial wave packet. The arrows show the wave vector values equal to $n∕R$.

Figure 8

(Color online) Transfer probability for fixed values of the incident wave vector as a function of the magnetic field flux for the circular ring. The plots for $k=0.072$, 0.06, and $0.053\text{per}\mathrm{nm}$ were shifted by 0.5, 1, and 1.5, respectively.

Figure 9

Probability density of the packet transferred through the diamond ring in wave vector space for $B=0$. The dashed line shows the shape of the initial wave packet.

Figure 10

(Color online) Transmission probability (solid lines) and the phase shift (dotted lines) as functions of the wave vector for a strictly one-dimensional Gaussian potential cavity of width $28\mathrm{nm}$ and different depths $V_0$.

Figure 11

(Color online) (a) Transmission probability of the wave packet through a circular ring with a Gaussian quantum well [Eq. (8)] in the left arm as a function of the depth of the well $V_0$. Values for fluxes equal to integer flux quanta are plotted with solid lines and for half flux quanta with dashed lines. (b) The transferred momentum distribution for $V_0=0$ and $5.5\mathrm{meV}$ at $\Phi =0$. (c) Same as (b) but now for $\Phi =0.5{\Phi }_0$.

Figure 12

(Color online) Probability of transmission of the wave packet through a circular ring with a Gaussian quantum well [Eq. (8)] in the left arm as a function of the magnetic field flux for well depths: $V_0=5.5$, 4, 2.75, 1 and $0\mathrm{meV}$. Lines for $V_0=4$, 2.75, 1, and $0\mathrm{meV}$ have been shifted for clarity by $+0.12$, $+0.24$, $+0.36$, and $+0.48$, respectively.