Current Control by Resonance Decoupling and Magnetic Focusing in Soft-Wall Billiards

The isolation of energetically persistent scattering pathways from the resonant manifold of an open electron billiard in the deep quantum regime is demonstrated. This enables efficient conductance switching at varying temperature and Fermi velocity, using a weak magnetic field. The effect relies on the interplay between magnetic focusing and soft-wall confinement, which rescale the scattering pathways and decouple quasibound states from the attached leads, the field-free motion being forwardly collimated. The mechanism proves robust against billiard shape variations and qualifies as a nanoelectronic current control element.

Figure 1

System setup and sketch of pathways. (a) Billiard defined by hard-wall confinement (solid line) and soft-wall potential $V(x,y)$ decreasing along elliptic contours to zero (dotted contour), opening up along $y=0$ to attached leads of width $w$. ($\mathrm{a}{}^{\prime }$) Cross section at $x=0$ for linear wall potential, with central contour at the threshold of the first propagating channel. (b) Without the soft wall, a magnetically backscattered pathway (cyclotron radius $R_1$, red arrows) turns into a transmitted pathway ($R_2<R_1$, orange arrows) for sufficiently lower energy, while (c) backscattering would be retained for a correspondingly smaller billiard. For appropriate $V$ in (a), similar backscattered paths can persist in varying energy $E$ for common field $B_s$, with forward propagation favored for $B=0$ (blue arrow).

Figure 2

Transmission $T$ ($\text{black}=0$ to $\text{white}=1$) for varying magnetic field $B$ (or flux $\phi$) and scaled incoming momentum $\kappa =\sqrt{2E}w/\pi$ within the first open channel, for (a) the soft-wall potential of Fig. 1(a′) with $(a,b,d,w)=(84,128,96,32)a_0$, and (b) the same billiard without the soft wall. $B$ is in units of ${10}^{-3}{B}_0$. For $a_0=2\mathrm{nm}$, $B_0=\hslash /e{a}_0^2=164.55\mathrm{T}$. Right panels: cuts through $T(B,\kappa )$-maps at $B=0$ (○) and $B_s=0.63\times {10}^{-3}{B}_0$ (⊙) or ${\phi }_s=7.32{\phi }_0$ (flux quantum ${\phi }_0=h/e$).

Figure 3

(a) Dimensionless conductance $g=G/G_0$ (with quantum $G_0=2e^2/h$) around the first open channel for $B=0$, for the same billiard as in Fig. 2, at different temperatures $\mathrm{\Theta }$. For $a_0=2\mathrm{nm}$ and $m=0.069m_e$ ($\mathrm{GaAs}/\mathrm{AlGaAs}$ interface): $E_0=276\mathrm{meV}$ and $\mathrm{\Theta }=0,0.5,1.0,3.0\mathrm{K}$. Scaled (b) LDOS $\sqrt{\rho }$ and (c) current density $\sqrt{|\mathit{j}|}$ shown at momenta indicated by vertical lines in (a), for electrons incident in the left lead. ($\mathrm{a}{}^{\prime }$, $\mathrm{b}{}^{\prime }$, $\mathrm{c}{}^{\prime }$) Same as above, but for $B=B_s$.

Figure 4

(a), ($\mathrm{a}{}^{\prime }$): Channel-averaged transmission for varying midwall semiaxes $a$ and $b$ of the billiard in Fig. 1 with $d=96a_0$, at (a) $B=0$ and ($\mathrm{a}{}^{\prime }$) $B=B_s$. Dotted lines indicate the geometry $(a,b)=(128,84)a_0$ chosen in Figs. 2, 3, and below. (b), (c), (d): Cross section $V(x=0,y)$ for different (b), (c) linear and (d) parabolic (with Wood-Saxon-type [40] boundary, thin dashed line) soft wall profiles. ($\mathrm{b}{}^{\prime }$), ($\mathrm{c}{}^{\prime }$), ($\mathrm{d}{}^{\prime }$): Corresponding switching contrast $\mathrm{\Delta }g$ at $k_B\mathrm{\Theta }/E_0=0.312\times {10}^{-3}$ ($\mathrm{\Theta }=1.0\mathrm{K}$ for $a_0=2\mathrm{nm}$, $m=0.069m_e$) for optimal fields $B_s$ within the first open channel (threshold at vertical lines). Dashed-dotted line in ($\mathrm{d}{}^{\prime }$) corresponds to dotted profile in (d) for alternative $B_s$. Arrows indicate each curve’s maximum. Lengths, energies, and fields are in units of $a_0$, ${10}^{-3}{E}_0$, and ${10}^{-3}{B}_0$, respectively.