Quantum current magnification in a multichannel mesoscopic ring

We have studied the current magnification effect in a multichannel open mesoscopic ring. We show that the current magnification effect is robust even in the presence of several propagating modes despite mode mixing and cancellation effects. The magnitude of circulating currents in the multichannel regime can be much larger than that in a single channel case. Impurities can enhance or degrade the current magnification effect depending sensitively on the system parameters. Circulating currents are mostly associated with Fano resonances in the total transport current. We further show that system-lead coupling qualitatively changes the current magnification effect.

Figure 1

Schematic diagram of an open multichannel mesoscopic ring of perimeter $L=l_1+l_2+l_3$ connected through the leads 1 and 2 to electron reservoirs (not shown in figure). Both the ring and the leads have the same width $W$. Several transverse modes are shown by horizontal lines in the leads. A delta function type static scatterer $V_d(x,y)=V\delta (x-l_1)\delta (y-y_i)$ of strength $V$ is shown in the upper arm at $\times$.

Figure 2

Plot of the circulating current density $d{j}_c∕d{j}_{\text{in}}$ in the ring as a function of $E∕E_0$ of the electron. In the whole energy range, we have 1 to 10 propagating modes (corresponding energy-ranges are indicated by vertical dashed lines). The different system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $2m_{e}V∕{\hslash }^2=1$, $y_i=0.21W$, $ϵ=0.2$.

Figure 3

Plots of different partial current densities $d{j}_i{\left(s\right)}_U∕d{j}_{\text{in}}$ and the total current in the upper arm of the ring as a function of $E∕E_0$. The dotted curve gives $d{j}_1{\left(s\right)}_U∕d{j}_{\text{in}}$, the dashed curve gives $d{j}_2{\left(s\right)}_U∕d{j}_{\text{in}}$, the dashed-dotted curve gives $d{j}_3{\left(s\right)}_U∕d{j}_{\text{in}}$, the long-dashed one is for $d{j}_4{\left(s\right)}_U∕d{j}_{\text{in}}$, and the solid one is for $d{j}_U∕d{j}_{\text{in}}$. In the above energy range we have 4 propagating modes. The different system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $2m_{e}V∕{\hslash }^2=1$, $y_i=0.21W$, $ϵ=0.2$.

Figure 4

Plot of the circulating current $d{j}_c∕d{j}_{\text{in}}$ (dotted lines) and the total current $d{j}_T∕d{j}_{\text{in}}$ (solid lines). Both the functions are plotted vs $E∕E_0$. The different system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $2m_{e}V∕{\hslash }^2=1$, $y_i=0.21W$, $ϵ=0.2$.

Figure 5

Plots of circulating current $d{j}_c∕d{j}_{\text{in}}$ vs incident energy $E∕E_0$ in the range of single mode propagation. Curves $1,2,3,4,\dots ,10$ are for potential strengths $2m_{e}V∕{\hslash }^2=0.5,1.0,1.5,2.0,\dots ,5.0$ respectively. The other system parameters are $l_1=3.125$, $l_2=3.125$, $l_3=3.75$, $W=1$, $y_i=0.21W$, $ϵ=4∕9$.

Figure 6

Plots of circulating current $d{j}_c∕d{j}_{\text{in}}$ vs incident energy $E∕E_0$ in the range of the two mode propagation. Curves $1,2,3,4,\dots ,10$ represent potential strengths $2m_{e}V∕{\hslash }^2=0.5,1.0,1.5,2.0,\dots ,5.0$, respectively. The other system parameters are $l_1=3.125$, $l_2=3.125$, $l_3=3.75$, $W=1$, $y_i=0.21W$, $ϵ=4∕9$.

Figure 7

Plots of circulating current $d{j}_c∕d{j}_{\text{in}}$ vs incident energy $E∕E_0$. In this regime six modes are propagating. The dotted curve is for $L∕W=20$, while the solid one is for $L∕W=10$. The other system parameters are $W=1$, $l_1:l_2:l_3=7:5:8$, $y_i=0.21W$, $2m_{e}V∕{\hslash }^2=1.0$ and $ϵ=0.2$.

Figure 8

The circulating current $d{j}_c∕d{j}_{\text{in}}$ vs $E∕E_0$ is plotted for the repulsive potential $2m_{e}V∕{\hslash }^2=1$ in the strong coupling regime $ϵ=0.48$. The different system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $y_i=0.21W$.

Figure 9

In (a) the partial current $d{j}_1{\left(s\right)}_U∕d{j}_{\text{in}}$ and in (b) the partial current $d{j}_2{\left(s\right)}_U∕d{j}_{\text{in}}$ in the lower arm of the ring are plotted as a function of $E∕E_0$. In both (a) and (b) dashed curves are for $ϵ=0.2$ and the solid curves are for $ϵ=0.48$. The other system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $2m_{e}V∕{\hslash }^2=1$, $y_i=0.21W$.

Figure 10

Both in (a) and (b), the dashed curve gives the partial current $d{j}_1{\left(s\right)}_L∕d{j}_{\text{in}}$, the dotted curve gives the partial current $d{j}_2{\left(s\right)}_L∕d{j}_{\text{in}}$ in the lower arm of the ring, and the solid curve gives the circulating current $d{j}_c∕d{j}_{\text{in}}$ in the ring. All three quantities are plotted as a function of incident energy $E∕E_0$ of the electron. (a) is for weak coupling $ϵ=0.2$ while (b) is for strong coupling $ϵ=0.48$. Other system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $2m_{e}V∕{\hslash }^2=1$, $y_i=0.21W$.

Figure 11

The circulating current $d{j}_c∕d{j}_{\text{in}}$ vs $E∕E_0$ is plotted for strong coupling $ϵ=0.48$ and attractive potential $2m_{e}V∕{\hslash }^2=-2.5$. The arrows on the graph denote the positions of different quasi-bound-states in the available energy range, $31.87E_0$, $49E_0$, $57.7E_0$, $76.53E_0$, $100E_0$, $120.12E_0$. The other system parameters are $l_1=3.5$, $l_2=2.5$, $l_3=4.0$, $W=1$, $y_i=0.21W$.

Figure 12

$d{j}_T∕d{j}_{\text{in}}$ (solid curve) and $d{j}_c∕d{j}_{\text{in}}$ (dotted curve) are plotted as a function of $E∕E_0$ for coupling strengths $\left(ϵ\right)$ 0.05 (a) and 0.2 (b) in the presence of two propagating modes. Other system parameters are $l_1=2.5$, $l_2=1.5$, $l_3=3.0$, $W=1$, $2m_{e}V∕{\hslash }^2=-2.4$, $y_i=0.21W$. The quasi-bound-state is at $5.1025E_0$.