Aharanov-Bohm oscillations and perfectly transmitted mode in amorphous topological insulator nanowires

TL;DR

This work investigates how Aharanov-Bohm (AB) oscillations and the perfectly transmitted mode, hallmark features of crystalline topological insulator nanowires, survive in amorphous nanowires. Using a Bi$_2$Se$_3$-like tight-binding model with Peierl's flux and Landauer transport, the authors analyze layered and fully amorphous geometries, employing local chiral markers to connect transport to topology. They find that at low energies and moderate amorphicity, a perfectly transmitted mode persists—protected by an emergent effective time-reversal symmetry in layered wires and by chiral or statistical TRS in fully amorphous wires—while strong amorphicity suppresses AB oscillations and yields sharp bound-state resonances indicating a transition to a trivial insulating phase. The results provide a framework for probing amorphous topological phases and suggest routes to realize robust topological transport in amorphous nanowire platforms, with implications for nanoelectronics and Majorana-based applications.

Abstract

Crystalline topological insulator nanowires with a magnetic flux threaded through their cross section display Aharanov-Bohm conductance oscillations. A characteristic of these oscillations is the perfectly transmitted mode present at certain values of the magnetic flux, due to the appearance of an effective time-reversal symmetry combined with the topological origin of the nanowire surface states. In contrast, amorphous nanowires display a varying cross section along the wire axis that breaks the effective time-reversal symmetry. In this work, we use transport calculations to study the stability of the Aharanov-Bohm oscillations and the perfectly transmitted mode in amorphous topological nanowires. We observe that at low energies and up to moderate amorphicity the transport is dominated, as in the crystalline case, by the presence of a perfectly transmitted mode. In an amorphous nanowire the perfectly transmitted mode is protected by chiral symmetry or, in its absence, by a statistical time-reversal symmetry. At high amorphicities the Aharanov-Bohm oscillations disappear and the conductance is dominated by nonquantized resonant peaks. We identify these resonances as bound states and relate their appearance to a topological phase transition that brings the nanowires into a trivial insulating phase.

Figures (8)

• Figure 1: Conductance as a function of Fermi energy for a single layered-amorphous realization with amorphicity $w=0.1$. The nanowire length is $L=200$, the dimensions of the cross section are $N_x=N_y=10$, and we set the chemical potential in the leads to $\mu=0$. The conductance of the crystalline case is indicated with dashed lines. The upper inset shows the amorphous cross section for a sample nanowire of dimensions $N_x=N_y=5$, with the crystalline cross section depicted in grey. The lower inset shows the full nanowire constructed from the cross section in the upper inset connected to the leads (depicted in pink).
• Figure 2: (a-c) Realization-averaged conductance of a layered-amorphous nanowire for three different amorphicities as a function of Fermi energy. The average is performed over $300$ layer amorphicity realizations of nanowires with $L=100$ and $N_x=N_y=10$. The chemical potential in the leads is $\mu=-1$. The shaded regions indicate the standard deviation of the distribution. (d) Single-realization conductance at $E_F=0$ as a function of magnetic flux and for different $w$. The parameters are $\mu=-1$, $L=200$ and $N_x=N_y=10$. (e) Band gap for the different nanowires in (d).
• Figure 3: Effect of chiral symmetry breaking on the conductance of a single realization of a fully amorphous nanowire for different cross section linear size $N$, such that $N_x \times N_y = N \times N$. We model onsite disorder by adding the term $K_{\boldsymbol{r}} \delta_{\boldsymbol{r}\boldsymbol{r}'}$, with $K_{\boldsymbol{r}}\in[-K, K]$ drawn from a random uniform distribution for each $\boldsymbol{r}$, to the Hamiltonian \ref{['eq: Amorphous hamiltonian']}. Chiral symmetry is broken by the disorder, resulting in the conductance given by the solid lines. The crosses indicate the conductance for the same nanowire without onsite disorder. We used $w=0.15$, $L=200$ and $\mu=-1$.
• Figure 4: (a) Single-realization conductance of a fully amorphous nanowire at $E_F=0$ as a function of flux and amorphicity. The parameters are $L=200$, $N_x=N_y=10$, and $\mu = -1$. (b) Total density of states per unit area as a function of the cutoff parameter $R$ for a perfectly transmitted mode (circle), localized mode (triangle) and resonant mode (star) in (a). The meaning of the cutoff $R$ is sketched below the central panel in (c), where $R=0$ corresponds to considering only the central point in the cross section of the nanowires. Each curve is normalized to its total density of states per area at $R=5$. (c) Local density of states for the perfectly transmitted mode, localized mode and the resonant mode shown in (b).
• Figure 5: Single-realization conductance of a fully amorphous nanowire at $E_F=0$ as a function of flux and nanowire length $L$, with $w=0.2$ in (a) and $w=0.4$ in (b). Nanowires with different length are taken as cuts of the same parent nanowire of cross section dimensions $N_x=N_y=10$ and $\mu=-1$. (c) Single-realization conductance as a function of length for the resonances indicated in (b) with a blue circle, green square and purple triangle. The dashed lines show a decaying exponential fit for the three families of resonances.