Magnetic field-induced control of transport in multiterminal focusing quantum billiards

By exploring the four-terminal transmission of a semielliptic open quantum billiard in dependence of its geometry and an applied magnetic field, it is shown that a controllable switching of currents between the four terminals can be obtained. Depending on the eccentricity of the semiellipse and the width and placement of the leads, high transmittivity at zero magnetic field is reached either through states guided along the curved boundary or focused onto the straight boundary of the billiard. For small eccentricity, attachment of leads at the ellipse foci can yield optimized corresponding transmission, while departures from this behavior demonstrate the inapplicability of solely classical considerations in the deep quantum regime. The geometrically determined transmission is altered by the phase-modulating and deflecting effect of the magnetic field, which switches the pairs of leads connected by high transmittivity. It is shown that the elliptic boundary is responsible for these very special transport properties. At higher field strengths edge states form and the multiterminal transmission coefficients are determined by the topology of the billiard. The combination of magnetotransport with geometrically optimized transmission behavior leads to an efficient control of the current through the multiterminal structure.

Figure 1

Geometry of the open billiard with indicated length parameters and lead labeling for eccentricity $\epsilon =f/a=\sqrt{1-b^2/a^2}=0.35$, lead width $w=a/5$, and inner lead positions $\pm l=\pm a/2$. The two dots at $(x,y)=(\pm f,0)$ are the foci of the (semi-)ellipse and the dashed lines correspond to alternative setups (see text).

Figure 2

(A) The four independent transmission coefficients $T_{\mathit{ij}}$ at zero magnetic field, as a function of the scaled energy $\kappa =kw/\pi =\sqrt{2E}w/\pi$, for the elliptic ($\epsilon ={\epsilon }_3=0.35$, black solid line) and circular(cyan dotted line) billiard border, both with $w=a/5$ and $l={\epsilon }_{3}a$. (B) LDOS for the semielliptic billiard at different energies (a), (b), (c), (d) indicated by the vertical lines in (A), with the particle incident in the (i) outer and (ii) inner left lead. The colormap scales with $\sqrt{\rho (x,y)}$ from white ($\rho =0$) to black ($\rho =\text{max}$) and is normalized to its maximal value in each plot.

Figure 3

Zero-field channel-averaged transmission coefficients ${\mathrm{T̄}}_{\mathit{ij}}$ between terminals $(i,j)=(2,1)$ (red dashed-dotted line), $(3,4)$ (blue solid line), $(1,3)$ (cyan dotted line), $(3,2)$ (green dashed line), with lead widths and positions as in Fig. 2 and eccentricities (from top to bottom row) ${\epsilon }_1=0.78$, ${\epsilon }_2=0.65$, ${\epsilon }_3=0.35$, for the semielliptic billiardwith a disk (a) centered at $(x,y)=(0,b)$ with radius $r$ varying from $0$ to $b$ and (b) centered at $(x,y)=(0,b/2)$ with $r$ varying from $0$ to $b/2$. The vertical lines denote the threshold radius $r_t$ for tunneling within the first channel (see text).

Figure 4

${\mathrm{T̄}}_{\mathit{ij}}$ for varying ratio $b/a$, with equidistantly attached leads of width (a) $w_0=a/3.5$ and (b) $w_2=a/7$. The light (orange) dashed line shows the eccentricity $\epsilon$. The lower horizontal edge of the billiard here lies at $y=-w$ (see Fig. 1), and the upper edge is varied from straight ($b=0$) to elliptic ($0<b<a$) to circular ($b=a$), as illustrated by the inset pictures in (b).

Figure 5

${\mathrm{T̄}}_{\mathit{ij}}$ in the unperturbed semielliptic billiard, for lead width (columns from left to right) $w_1=a/5$, $w_2=a/7$, $w_3=a/10.5$, and eccentricities(rows from top to bottom) ${\epsilon }_1=0.78$, ${\epsilon }_2=0.65$, ${\epsilon }_3=0.35$, as a function of the displacement $l$ of the inner leads from the origin. The vertical lines show the position of the focal points $f/a=\epsilon$ for each setup.

Figure 6

Transmission coefficients $T_{21}$ (red dashed-dotted line), $T_{34}$ (blue solid line), $T_{32}$ (green dashed line), $T_{13}$ (cyan dotted line), with offsets $3,2,1,0$, respectively, as a function of $\kappa$ within the first three transversal subbands $n=1,2,3$ of leads of width $w=w_0=a/3.5$, attached equidistantly with $l=(a-w/2)/3=0.84\hspace{0.5em}f$, (a) for the unperturbed semiellipse billiard with $\epsilon =0.35$ and (b) for the same geometry with a disk of radius $r=b/2-2w/3$ centered at $(0,b/2)$, at magnetic field (i) $B=0$, (ii) $B=0.002$, (iii) $B=0.005$, and (iv) $B=0.010$, with $\mathbf{B}=B\stackrel{\wedge }{\mathbf{z}}$ pointing outward from the billiard plane.

Figure 7

Mean transmission coefficients ${\mathrm{T̄}}_{\mathit{ij}}$ as a function of the applied magnetic field $B$ for the billiard of Fig. 6 with increasing disk radius $r$ in the $n=1$ subband(top to bottom panels in leftmost column); for zero and maximal $r$ (top and bottom row, respectively) also the $n=2$ and $n=3$ subbands are shown. Each plot is of height $max[{\mathrm{T̄}}_{\mathit{ij}}^{(n)}]=n$. The light (orange) dashed line in the bottom row panels shows the scaled Larmor radius $r_L/a$ at the $n$th channel’s center $\kappa =n+1/2$.

Figure 8

${\mathrm{T̄}}_{\mathit{ij}}(B)$ for two parallel wires at distance equal to their common width $w=a/3.5$, coupled by a smooth opening of width $d$ and circular edges of radius $w/2$, (a) with the wires bent by an angle $\pi$ across the coupling and (b) in straight configuration (parallel and equidistant to the $x$-axis). In the top plots there is no opening, and then the opening width is increased in steps of $w/4$ from $d=w/4$ to $d=3w$ (second from top to bottom, with transmission offsets decreasing by one), the latter yielding in (a) the semicircular billiard with smooth lead openings.

Figure 9

Multiterminal conductance coefficients $g_{\mathit{ij}}$ at scaled Fermi energy ${\kappa }_F=w/\pi ·\sqrt{2E_F}=1.425$ and temperature $\Theta =100\hspace{0.5em}\mathrm{mK}$, for $B=0$, $B=\pm 0.002$, and $B=\pm 0.010$. In each $4\times 4$ block, row $i$ labels the input and column $j$ the output current terminal.