Simulation of electron transport through a quantum dot with soft walls

We numerically investigate classical and quantum transport through a soft-wall cavity with mixed dynamics. Remarkable differences to hard-wall quantum dots are found which are, in part, related to the influence of the hierarchical structure of classical phase space on features of quantum scattering through the device. We find narrow isolated transmission resonances which display asymmetric Fano line shapes. The dependence of the resonance parameters on the lead mode numbers and on the properties of scattering eigenstates are analyzed. Their interpretation is aided by a remarkably close classical-quantum correspondence. We also searched for fractal conductance fluctuations. For the range of wave numbers $k_F$ accessible by our simulation we can rule out their existence.

Figure 1

(a) Bunimovich stadium with leads attached at the straight segments. (b) Reduced scattering geometry consisting of a semicircle of radius $R=1$, which is attached to a rectangle of length $L=2R$ and width $d=0.3$. The solid lines represent hard-wall boundary conditions and the dashed curves are contour lines of the soft-wall potential [for details see Eq. (2.3)]. (c) Illustration of the potential surface corresponding to contour plot (b).

Figure 2

(Color online) Log-log plot of the classical dwell probability $P\left(l\right)$ for the cavity with soft walls (red solid line) and hard walls (green dashed line).

Figure 3

(Color online) Poincaré surface of section for the half stadium with soft walls [see Figs. 1b, 1c]. A large number of chaotic trajectories (black crosses); six regular closed orbits (b)–(e),(g),(h); two unstable periodic orbits (f); and one trapped chaotic trajectory (i) are shown. For orientation, the arrows denote one point of the phase-space representation of each trajectory. The green dashed curve shows an approximation for the outermost partial transport barrier. For details see text.

Figure 4

(Color online) Comparison between total transmission $T$ through the soft-wall (left inset) and hard-wall geometry (right inset) as a function of the Fermi wave number $k_F$ (14 lead modes are open). Narrow isolated resonances on a smoothly varying background are clearly visible for the soft-wall billiard (red solid line), but are absent for the case of a hard-wall boundary (green dashed line).

Figure 5

(Color online) (a), (b) Classical periodic orbits in the soft-wall billiards which are closely mirrored in the density ${\mid \psi (x,y)\mid }^2$ of the quantum scattering wave functions at the resonant energies: (c) wave number $k_F=14.0592\pi ∕d$ for incoming lead mode number $m=14$ and (d) $k_F=14.0021\pi ∕d$, with $m=12$. (e) and (f) give the corresponding phase-space portraits: Husimi distribution (density plot, red) and classical PSS (islands, black). For reference, we also plot the outermost partial barrier (green dashed line) and the most prominent regular island (blue solid lines). The blue boxes in the lower-left corner of (e) and (f) indicate the size of $h$, which characterizes the quantum mechanical resolution of the classical phase space.

Figure 6

(Color online) Top row: Classical stable periodic orbits (a) and (b), a transiently trapped trajectory (c), and an unstable periodic orbit (d). Second row: Resonant wave function densities ${\mid \psi (x,y)\mid }^2$ closely mimicking these trajectories with wave numbers (a) $k_F=14.9203\pi ∕d$, (b) $k_F=14.0061\pi ∕d$, (c) $k_F=14.0260\pi ∕d$, and (d) $k_F=8.186\pi ∕d$. Third row: The Husimi distributions (density plot, red) and the PSS of the corresponding trajectories (island and crosses, black). Bottom row: Transmission resonances, numerical results (green dots), and fits to the Fano resonance formula Eq. (4.1) (red line). Fitted values for the width $\Gamma$ and Fano asymmetry parameter $q$ are indicated.

Figure 7

The ratio ${\xi }_{A_1}∕({\xi }_{A_1}+{\xi }_{A_2})$ of the HDs integrated over the chaotic area $A_1$ outside the outermost partial barrier and over the remaining phase-space area $A_2$ [see Eq. (3.2)] as a function of the resonance width $\Gamma$.

Figure 8

(Color online) (a) Wave function densities for selected incoming mode numbers $m$ of a typical Fano resonance $[k_F=14.1111\pi ∕d]$. For increasing $m$, the directly transmitted (nonresonant) part of the wave function, being dominant for $m=1$, becomes less pronounced, while the relative intensity of the ∩-shaped resonant part drastically increases. (b) The Fano parameter $q$ shows no correlation to the geometric mean value of the ingoing and outgoing mode number, $\sqrt{m\times n}$. (c) Amplitudes $A_{mn}$ are approximately proportional to $\sqrt{m\times n}$.

Figure 9

(Color online) Offset of transmission eigenvalues ${\overline{T}}^{\mathrm{offs}}$ (green dashed line, open circles) and amplitudes ${\overline{A}}_i$ (red solid line, squares) of the Fano resonances in the eigenmode $i$. All Fano resonances in the ensemble considered were taken from the energy interval where 14 lead modes are open, resulting in $i_{\max }=14$. The resonance parameters associated with the eigenmode $i$ (for ordering see text) are ensemble averaged within one specific eigenmode $i$. Note that the amplitudes ${\overline{A}}_i$ feature pronounced values only for intermediate modes $i$ where the resonance offset ${\overline{T}}^{\mathrm{offs}}$ does not take on the “classical” values close to 0 or 1.

Figure 10

Distribution $P\left(\mid q\mid \right)$ of the absolute value of the $q$ parameter in Fano resonances found in the transmission eigenvalues ${\lambda }_i$ (the same ensemble of resonances was considered as in Fig. 9). Fitting the data (histograms) with a Gaussian distribution (solid line) yields good agreement.