Enhanced Thermoelectricity in Nanowires with inhomogeneous Helical states

TL;DR

This work studies thermoelectric transport in semiconductor nanowires with strong Rashba spin–orbit coupling under a magnetic field, focusing on inhomogeneous RSOC profiles to control helical states. Using a tight-binding model and a scattering-matrix approach (via KWANT), the authors show that misaligning RSOC directions between two NW segments (characterized by the angle $\phi$) can drastically modify transmission and energy filtering, leading to strong Seebeck-coefficient enhancement and significant WF-law violations in the antiparallel (Dirac-paradox) case. The key finding is that optimal thermoelectric performance, including $ZT$ up to about 0.9, is achieved in the antiparallel configuration near the magnetic gap boundaries and at intermediate temperatures set by $E_Z$, with further improvements possible by increasing NW length up to a few magnetic lengths. The results suggest a versatile, electrically tunable route to high-performance thermoelectric devices and temperature sensors in quantum-coherent nanowires.

Abstract

Semiconductor nanowires (NWs) with strong Rashba spin-orbit coupling (RSOC), when exposed to a suitably applied Zeeman field, exhibit one-dimensional helical channels with a spin orientation locked to the propagation direction within the magnetic energy gap. Here, by adopting a scattering-matrix approach applied to a tight-binding model of the NW, we demonstrate that the thermoelectric (TE) properties can be widely controlled by tuning the misalignment angle $φ$ between the spin-orbit directions of two NW segments. In particular, when the RSOC vectors are antiparallel (Dirac paradox configuration) we predict a significant violation of the Wiedemann-Franz law, and a strong enhancement of the Seebeck coefficient and the $ZT$ figure of merit. We also show that the Zeeman gap determines the optimal energy window for doping and temperatures. These results suggest that controlling the spin-orbit field direction, which can be achieved with suitably applied wrap gates, is a promising alternative for tuning and optimizing the TE response in quantum-coherent semiconducting NW devices.

Figures (8)

• Figure 1: Sketch of the setup: (a) Rashba NW oriented along the $x$ direction and deposited on an insulating substrate (green). An external magnetic field is applied along the direction of $x$. The Rashba spin-orbit interaction is oriented along a unit vector $\hat{\mathbf{n}}_\alpha^{L(R)}$, which differs between the left and right regions of the NW. In the left region, $\hat{\mathbf{n}}_\alpha^{L} = (0,1,0)$, pointing along the $y$-direction. In the right region, the vector lies in the $yz$-plane, i.e., $\hat{\mathbf{n}}_\alpha^{R} = (0, \cos\phi, \sin\phi)$, forming an angle $\phi$ with the $\hat{\mathbf{n}}_\alpha^{L}$. (b) 1D chain with TB parameters as described by Eqs. \ref{['H0-def']}--\ref{['HZ-def']} and the number of sites $N$ in the chain sets the NW length $2L = (N-1)a$.
• Figure 2: The electronic band structure of a 1D NW with a constant RSOC profile, for different values of the Zeeman field. (a) The Zeeman field is absent, and the RSOC lifts the spin degeneracy of the parabolic band (dashed gray line). This results in two spin-split bands, each shifted by $\pm k_{\mathrm{SO}}$ in momentum and lowered in energy by $E_{\mathrm{SO}}$. (b) A finite Zeeman field along the $x$-axis opens a gap at $k = 0$, creating a single helical state when the chemical potential lies within the gap. Spins align with the Zeeman field at low $k$ and tilt toward the RSOC direction at higher $k$. (c) Zeeman-dominated regime resulting in almost complete spin polarization of the bands along the field direction.
• Figure 3: The electrical zero-temperature conductance $G$ of an inhomogeneous RSOC NW, shown in units of the conductance quantum $G_0=e^2/h$, is plotted as a function of the normalized chemical potential $\mu/E_\mathrm{Z}$ for different misalignment angles $\phi$, and for various lengths: (a) $L= 0.05L_\mathrm{Z}$, (b) $L= 1.5L_\mathrm{Z}$, and (c) $L= 6L_\mathrm{Z}$. We set $E_{\mathrm{SO}}/E_\mathrm{Z} = 10$, i.e. we are in the deep Rashba-dominated regime.
• Figure 4: (a) Wiedemann–Franz ratio $\mathcal{L}/\mathcal{L}_0={K}/{(G\,T\mathcal{L}_0)},$ as a function of the normalized chemical potential $\mu/E_\mathrm{Z}$ for three different RSOC misalignment angles (see label) where RSOC NW segments have a length $L=6\,L_\mathrm{Z}$. (b) Thermoelectric Onsager coefficient $L_{eh}$ for the same $\phi$ values as in panel (b). The temperature is set such that $k_B T=0.05 E_\mathrm{Z}$. Other parameters as in Fig. \ref{['fig:G_0']}.
• Figure 5: Thermoelectric transport properties of a RSOC NW with segments of length $L=6\,L_\mathrm{Z}$ as functions of the normalized chemical potential $\mu/E_\mathrm{Z}$ for different RSOC misalignement angle $\phi$ (see label). (a) Seebeck coefficient $S$, (b) thermoelectric power factor $GS^2$, and (c) figure of merit $ZT$ in logarithmic scale. Temperature is fixed such that $k_B T=0.05\,E_\mathrm{Z}$. Other parameters as in Fig. \ref{['fig:G_0']}.