Lorentz force effects for graphene Aharonov-Bohm interferometers

We investigate magnetic deflection of currents that flow across the Aharonov-Bohm interferometers defined in graphene. We consider devices induced by closed $n\text{−}p$ junctions in nanoribbons as well as etched quantum rings. The deflection effects on conductance are strictly correlated with the properties of the ring-localized quasibound states. The energy of these states, their lifetime, and the periodicity of the conductance oscillations are determined by orientation of the current circulating within the interferometer. The formation of high harmonics of conductance at high magnetic field and the role of intervalley scattering are discussed.

Figure 1

AB interferometers: a ring etched out of graphene (a) and a circular $n\text{−}p$ junction induced within a graphene nanoribbon (b).

Figure 2

(a) Conductance of a narrow quantum ring ($R_1=41.05$ nm, $R_2=48.95$ nm) connected to a semiconducting armchair ribbon of width $W=17.23$ nm. (b) Counter of the localized states as determined by the stabilization method. Dashed lines separate the regions of varied filling factor $\nu$ in the ribbon. The solid vertical lines indicate the Fermi energies studied in detail.

Figure 3

Conductance (blue line): cross section of Fig. 2 for $E=0.0586$ eV, the current flux through the upper and lower arms of the ring (see Fig. 4), and the sign of the flux (upper panel).

Figure 4

Schematics of the electron flow at the maxima (a) and minima (b) of $G$ in Fig. 3 and (c) the orientation of the Lorentz force for the electron moving right in the out-of-plane magnetic field.

Figure 5

(a) Conductance (line of varied colors): same as in Fig. 3, and the resonance counter for $E=0.0586$ eV. (b) Fourier transform calculated for intervals of the magnetic field which are marked in the corresponding colors. Each plot in (b) is normalized so that the first peak has the same amplitude as in the blue curve.

Figure 6

Fourier transform of Fig. 2 for $B\in (0,10\mathrm{T})$ (a), and for $B\in (10\mathrm{T},30\mathrm{T})$ (b).

Figure 7

Same as Fig. 2 only for a wider ring to $R_1=23.4$ nm and $R_2=48.95$ nm.

Figure 8

Same as Fig. 3 only for a wider ring to $R_1=23.4$ nm and $R_2=48.95$ nm. The conductance plotted in blue is a cross section of Fig. 7 for $E=0.0542$ eV.

Figure 9

The current distribution for the results of Fig. 8 for (a) $B=6.2$ T (a peak of conductance) and (b) for $B=5.78$ T (a dip of conductance.)

Figure 10

Schematics of the electron flow for the AB interferometer induced within an $n$-type nanoribbon by external potential forming a circular $p$-type region in the center of the ribbon. The black arrows indicate the electron currents and the red arrows the direction of the Lorentz force for the out-of-plane magnetic field. In the simulation the electron is incident from the left near the upper edge.

Figure 11

Conductance of the $n\text{−}p$ inteferometer (a), its Fourier transform (b), and the resonance counter $F$ (c).

Figure 12

The electron density obtained from the diagonalization of a closed system (as in the stabilization algorithm) and the current distribution for (a) one of the peaks of conductance ($E=0.0756$ eV, $B=7.38\mathrm{T}$) with the circulation of the current along the $n\text{−}p$ junction and (b) one of the resonances formed inside the $p$ region ($E=0.077$ meV, $B=7.38$ T).

Figure 13

Same as Fig. 7 only for the ring connected to a zigzag ribbon.

Figure 14

Conductance for the circular $n\text{−}p$ junction defined within a zigzag ribbon without (blue line) and with (red line) edge defects (see text).

Figure 15

The conductance for the system of Fig. 14 for the Lorentzian tip potential ($V_2$) and for the exponent $n$ in Eq. (2) replaced by 20 for $V_{20}$ and by 40 for $V_{40}$. The top inset shows the profile of the tip potential and the bottom one a zoom of the main figure with $V_2$ and $V_{20}$ only.

Figure 16

Scheme of the extending of the leads length for the calculation of the density of ring-localized resonances for an armchair (a) and zigzag (b) nanoribbons serving as leads.

Figure 17

Energy spectrum calculated as a function of the lengths of the leads of a zigzag nanoribbon with a Lorentzian potential discussed in the paper, for $B=6.6$ T. Every second energy level is marked with a cross or a circle. The energy levels that are nearly independent of the leads length—corresponding to a localized resonance—are twofold degenerate.

Figure 18

Same as Fig. 5 only for $T=4.2\mathrm{K}$.

Figure 19

Same as Fig. 5 only for $T=20\mathrm{K}$.

Figure 20

Same as Fig. 5 only for current at source-drain bias of 10 meV. The inset to (a) shows the current-bias characteristics.