Spin-orbit interaction in tubular prismatic nanowires

TL;DR

This work analyzes spin-orbit interaction in the outer, tubular regions of core-shell nanowires with polygonal cross sections, where corner localization creates a low-energy, quasi-1D subspace and large gaps to side states. Using a folded-down eight-band Kane model with a conduction-band offset and extended band-offset potential, plus possible external fields, the authors show that SOI strength and state degeneracies depend sensitively on geometry (hexagonal, triangular, and Tri-Hex) and on core-shell intermixing. Corner states in triangular and Tri-Hex geometries can behave as three independent wires at low energy, while higher-energy states exhibit inter-state interactions; SOI can be significantly enhanced by deeper core penetration into the shell (larger intermixing). These results provide a pathway to engineer strong SOI and robust corner-state subspaces, with implications for Majorana-state platforms and geometry-driven spintronics; data are available in Zenodo, and the findings highlight the importance of cross-sectional geometry in prismatic nanowires.

Abstract

We theoretically study the spin-orbit interaction in the outer regions of core-shell nanowires that can act as tubular, prismatic conductors. The polygonal cross section of these wires induces non-uniform electron localization along the wire perimeter. In particular, low-energy electrons accumulate in the corner regions, and in the case of narrow shells, conductive channels form along the sharp edges. In contrast, higher-energy electrons are shifted toward the facets. These two groups of states may be separated by large energy gaps, which can exceed the room-temperature energy in the case of triangular geometries. We compare the impact of spin-orbit interaction on the corner and side states of hexagonal and triangular shells grown on hexagonal cores as well as on triangular shells grown on triangular cores. We find that the spin-orbit splitting, and thus the degeneracy of energy states at finite wave vectors, strongly depend on the tube's geometry. We demonstrate that the weak spin-orbit coupling observed in clean wires can be significantly enhanced if the intermixing of core and shell materials takes place. Moreover, we show that the energy spectrum in the presence of spin-orbit interaction allows for estimating the interaction between states and shows that triangular shells can act as three independent wires in the low-energy regime, while they behave as interacting systems at higher-energy ranges.

Figures (10)

• Figure 1: Cross sections of (a) hexagonal, (b) triangular, and (c) Tri-Hex wires. $R_{\mathrm{c}}$ and $R_{\mathrm{s}}$ represent the external radii of the core and shell (wire), respectively. The shell thickness is denoted by $d$ and the intermixing length is indicated by $r$. $R_{\mathrm{c}}=40$ nm, $d=8$ nm, and $r=2$ nm.
• Figure 2: Probability distributions corresponding to the lowest energy level (lower row) and the first energy level above the gap $\Delta$ (upper row) for (a) hexagonal, (b) triangular, (c) and Tri-Hex cross sections ($d=8$ nm, $R_{\mathrm{c}}=40$ nm).
• Figure 4: Low-energy states for the cross sections shown in Fig. \ref{['fig:SOI_Samples']} without SOI. The energies are shifted with respect to the ground state energy at $k_z=0$. (a) States $l=1$, $l=4$, $l=5$, and $l=8$ are twofold degenerate, while states $l=2$, $l=3$, $l=6$, and $l=7$ are fourfold degenerate. (b) and (c) States $l=1$ and $l=4$ are twofold degenerate, while states $l=2$ and $l=3$ are fourfold degenerate.
• Figure 5: Energy dispersions for the corner ($l=1, \dots, 6$) and side ($l=7, \dots, 12$) states of hexagonal shells with varying side thicknesses ($d$), where $R_{\mathrm{c}}=40$ nm and $r=2$ nm. The energies are shifted with respect to the ground state energy at $k_z=0$. The line description defined on the right-hand side of the figure applies for all panels. The label $l$ refers to a pair of degenerate states. In (b) and (c) one of the overlapping curves in each pair (representing levels: 2, 4, 8, and 10) is shown as dashed, with the dashed curve color corresponding to the same energy level as the solid line in (a).
• Figure 7: Energy dispersions for the corner and side (gray lines) states of the 8-nm-wide hexagonal shell in the presence of an external electric field. Here, $R_{\mathrm{c}}=40$ nm, $r=8$ nm, $E=0.1$ mV/nm, and $\varphi=\pi/12$. The energies are shifted with respect to the ground state energy at $k_z=0$.