Nanowires with surface disorder: Giant localization length and dynamical tunneling in the presence of directed chaos

We investigate electronic quantum transport through nanowires with one-sided surface roughness in the presence of a perpendicular magnetic field. Exponentially diverging localization lengths are found in the quantum-to-classical crossover regime, controlled by tunneling between regular and chaotic regions of the underlying mixed classical phase space. We show that each regular mode possesses a well-defined mode-specific localization length. We present analytic estimates of these mode localization lengths which agree well with the numerical data. The coupling between regular and chaotic regions can be determined by varying the length of the wire leading to intricate structures in the transmission probabilities. We explain these structures quantitatively by dynamical tunneling in the presence of directed chaos.

Figure 1

Wire with one-sided surface disorder in a magnetic field $B$, applied perpendicular to the scattering area. The solid line shows a regular skipping trajectory and the dashed line an irregular trajectory scattering at the disordered surface.

Figure 2

(Color online) Poincaré sections for (a) right-moving $\left(p_x>0\right)$ and (b) left-moving $\left(p_x<0\right)$ classical electrons in the wire shown in Fig. 1. In (a) a single regular island with invariant tori (solid green lines) can be identified, which is separated from the chaotic region (blue dots) by the outermost torus (dashed). In (b), no such island appears. The gray-shaded part $\left(y>W-\delta /2\right)$ indicates the $y$ range affected by disorder.

Figure 3

(Color online) (a) Nanowire with the regular transverse modes ${\chi }_m\left(y\right)$ $m=4,3,2,1$ (green) for $k_{\text{F}}W/\pi =14.6$, i.e., with a total of 14 open channels. The gray-shaded part indicates the $y$-range affected by disorder. (b) Poincaré-Husimi functions of these modes and their quantizing tori. The axes are as in Fig. 2a.

Figure 4

(Color online) (a) Averaged conductance $⟨g⟩$ vs length $L$ of the wire for $r_c=3W$ and $k_{\text{F}}W/\pi =14.6$. The step-wise decrease is accompanied by the disappearance of the regular modes (lower set of Husimi distributions) and the flooding of the island region by the chaotic states (upper set). The Poincaré-Husimi representations to the left (right) of the curve correspond to scattering from left to right (right to right) lead, respectively. (b) Transmission ${\mathcal{T}}_m=\text{exp}\left(⟨\text{ln}{T}_m⟩\right)$ of the incoming mode $m$ vs $L$. The gray vertical lines correspond to the predicted localization lengths ${\xi }_m$, Eq. (13).

Figure 5

(Color online) (a) Mode localization lengths ${\xi }_m$ for constant cyclotron radius $r_c=3W$. (b) Mode localization lengths ${\xi }_m$ for constant magnetic field $B=10.05/W^2$. For both (a) and (b), the Fermi wavenumber is given by $k_{\text{F}}=\pi /\left(W{h}_{\text{eff}}\right)$. The symbols connected by full lines display the numerical results from the full calculation. The dashed lines show the analytical predictions from (a) Eq. (13) and (b) Eq. (14), while the vertical gray lines indicate the positions where the size of the regular island is large enough to accommodate $m$ modes (i.e., where $A_{\text{reg}}=mh$).

Figure 6

(Color online) Three-step process for transmission from regular mode $m$ at module boundary $n$ to the left-transporting chaotic modes and finally to the regular mode $m^{\prime }$ at the module boundary $n^{\prime }$.

Figure 7

(Color online) (a) Transmission probabilities $T_m$ of the incoming mode $m=1,2,3,4$ vs $L$ showing the same data as in Fig. 4b, but on a logarithmic scale. (b) Transmission probabilities $T_{m^{\prime }4}$. (c) Transmission probabilities $T_{m^{\prime }1}$. The dashed lines show the predictions from Eq. (15), using the analytical results for the localization lengths ${\xi }_m$. For the numerical results, the arithmetic means $⟨T_m⟩$ and $⟨T_{m^{\prime }m}⟩$ are taken. In regions where the distribution of transmission probabilities is log normal, the geometric mean (i.e., arithmetic mean of the logarithms) might be more appropriate. For simplicity we use the arithmetic mean everywhere, for which a slightly better agreement between analytical and numerical results is found. The purple ellipse indicates the region where the direct transition between island modes dominates (see text). The results shown were obtained with $k_{\text{F}}W/\pi =14.6$, i.e., with 14 open modes.

Figure 8

(Color online) Transmission probabilities $T_{m^{\prime }4}$ as in Fig. 7b, but with the dashed lines showing the predictions from Eq. (15) with second-order processes (i.e., coupling from mode $m$ to $m^{\prime }$ and then from $m^{\prime }$ to $n$) also taken into account.

Figure 9

Connection of two scattering systems. Transmission proceeds by transmission through the first system $\left(t^{\left(1\right)}\right)$, an arbitrary number of reflections between the two systems (each of which gives a term $r^{\prime \left(1\right)}{r}^{\left(2\right)}$), and then transmission through the second system $\left(t^{\left(2\right)}\right)$, leading to Eq. (A1).

Figure 10

Two leads with widths $w_L$ and $w_R$.

Figure 11

(Color online) Effective potential $V\left(y\right)$ and lowest mode in infinitely wide lead, for $h_{\text{eff}}^{-1}=9.6$ and $r_c=3$. In addition, $w_L=h_{\text{min}}$ for the geometry chosen in this paper is shown.

Figure 12

(Color online) Sketch of the first mode $\left(m=1\right)$ transverse wave function ${\chi }_m^{L+}\left(y\right)$ in a lead of width $w_L$, ${\chi }_m^{\infty +}\left(y\right)$ in an infinite lead, and the difference ${\sigma }_m\left(y\right)$, for $1/h_{\text{eff}}=9.6$ and $r_c=3$. The inset shows an enlargement of the region around $y=w_L$.