Conductance behavior in nanowires with spin-orbit interaction: A numerical study

We consider electronic transport through semiconducting nanowires (W) with spin-orbit interaction (SOI) in a hybrid N-W-N setup where the wire is contacted by normal-metal leads (N). We investigate the conductance behavior of the system as a function of gate and bias voltage, magnetic field, wire length, temperature, and disorder. The transport calculations are performed numerically and are based on standard recursive Green's function techniques. In particular, we are interested in understanding if and how it is possible to deduce the strength of the SOI from the transport behavior. This is a very relevant question since so far no clear experimental observation in that direction has been produced. We find that the smoothness of the electrostatic potential profile between the contacts and the wire plays a crucial role, and we show that in realistic regimes the N-W-N setup may mask the effects of SOI, and a trivial behavior with apparent vanishing SOI is observed. We identify an optimal parameter regime, with neither too smooth nor too abrupt potentials, where the signature of SOI is best visible, with and without Fabry-Perot oscillations, and is most resilient to disorder and temperature effects.

Figure 1

Expected SOI-induced reentrant behavior in the W conductance. (left) N dispersion, with two right movers per energy, labeled by $1,2$ and $3,4$ within and outside of the gap, respectively. (middle) W-section dispersion, in the presence of Rashba SOI with energy ${\epsilon }_{\mathrm{so}}$ and $B$ field, giving rise to the Zeeman gap $2{\Delta }_z$ at zero momentum $k$, highlighted in yellow, with a single right mover per energy ($A$). Below the Zeeman gap there are two right movers per energy ($B,C$). (right) Perfect conversion of $3,4$ into $B,C$ yields the ideal conductance behavior $G^{\left(0\right)}\left(V_{\mathrm{g}}\right)$, where $G$ drops from $2e^2/h$ to $e^2/h$ within the Zeeman gap.

Figure 2

Typical N-W-N transport setup in which a nanowire (W) is placed along the $x$ direction on a substrate lying in the $xy$ plane and the two-terminal conductance $G$ of the wire is measured via normal electrodes (N, blue), while the chemical potential in the wire, or, better, in a portion of the wire, is tuned via an underlying gate (G, dark gray). The actual electrostatic gate potential $V_{\mathrm{g}}\left(x\right)$ (red curve) is smooth and varies along $x$ on some typical length scale $\lambda$.

Figure 3

Zero-temperature conductance behavior in the simple N-W measurement setup (semi-infinite wire with only one contact) as a function of gate voltage $V_{\mathrm{g}}$, with the profile given in Eq. (3) and for typical parameter values ${\epsilon }_{\mathrm{so}}=50$ $\mu \mathrm{eV}$ and ${\Delta }_z=10$ $\mu \mathrm{eV}$ [see text after Eq. (4)]. (a) Abrupt case $\lambda =0$. Different curves correspond to different total variations of $V_{\mathrm{g}}$ across the junction. In the case of large gate voltage variations ($\Delta V_{\mathrm{g}}=200{\epsilon }_{\mathrm{so}}\simeq 10$ meV), the conductance $G$ is largely reduced. and almost no signature of SOI appears compared to the ideal case shown in Fig. 1. By decreasing the induced Fermi energy variations one gets a $G\left(V_{\mathrm{g}}\right)$ curve that exhibits the desired reentrant behavior. The vertical dotted lines indicate the extension of the Zeeman gap. (b) Dependence of $G$ on the smoothness $\lambda$ of the gate potential profile at fixed $\Delta V_{\mathrm{g}}=10{\epsilon }_{\mathrm{so}}\simeq 0.5$ meV. At some optimal value $\lambda \simeq {\lambda }^{☆}$ the conductance shows the characteristic SOI-induced step at $2e^2/h$, but for increasing values of $\lambda$ the step is flattened down to a single $e^2/h$ plateau, like in the case of SOI = 0 (and finite Zeeman gap).

Figure 4

Real-space Landau-Zener-Stückelberger-like mechanism that determines the visibility of the SOI step in the adiabatic limit. As the gate potential $V_{\mathrm{g}}\left(x\right)$ varies along the wire (blue solid curve), the local dispersion relation is vertically shifted, and different states are involved at the given energy (dotted line). There exists a finite region where the chemical potential lies within the Zeeman gap (third picture from the left), and the tunneling through this forbidden region gives rise to Landau-Zener-Stückelberger physics.

Figure 5

Zero-temperature conductance behavior ($G$ in units of $e^2/h$) in the case of two (lower curves) and three (upper curves, shifted by $e^2/h$) occupied subbands. $G$ shows the desired reentrant behavior in an intermediate regime $\lambda \simeq {\lambda }^{☆}$, while for either too steep (black diamonds and blue circles) or too smooth (green squares and white triangles) gate potential profiles the SOI-related step becomes less and less visible.

Figure 6

Zero-temperature conductance behavior in a N-W-N setup with wire length $L=2$ $\mu \mathrm{m}$. (a) Dependence of $G\left(V_{\mathrm{g}}\right)$ on the smoothness $\lambda$ at fixed $\Delta V_{\mathrm{g}}=5{\epsilon }_{\mathrm{so}}$. (b) same as in (a), but with different $\Delta V_{\mathrm{g}}=50{\epsilon }_{\mathrm{so}}\sim$ meV.

Figure 7

Same as in Fig. 6, but for a wire twice as long, $L=4$ $\mu \mathrm{m}$ and intermediate $V_{\mathrm{g}}=10{\epsilon }_{\mathrm{so}}$. (a) $T=0$ conductance behavior, with Fabry-Pérot resonances exhibiting halved energy spacing in comparison with those of Fig. 6, as expected. (b) $T=50$ mK. The oscillations show reduced amplitude, and for the lowest values of $\lambda$ we get $G\simeq e^2/h$ within the SOI step. (c) $T=100$ mK. The lowest FP oscillations have disappeared and for $\lambda \ll {\lambda }^{☆}$ (in particular, the black curve, referring to $\lambda \simeq 100$ nm) give rise to a single almost flat $e^2/h$ plateau, masking any signature of SOI. Indeed the SOI = 0 curve, plotted in red circles, shows a very similar behavior.

Figure 8

Conductance for a finite wire of length $L=4$ $\mu \mathrm{m}$ in the presence of disorder, with optimal condition $\lambda ={\lambda }^{☆}$. The clean-case result is shown as green circles; all other curves refer to different disorder configurations. (a) Realistic mean free path values (${\ell }_{\mathrm{mfp}}\simeq 300$ nm) destroy the SOI-induced features, and no indication of SOI can be read off. (b) With milder disorder ($L/{\ell }_{\mathrm{mfp}}\simeq 1$–2) and $T=50$ mK it is still possible to observe the SOI step, depending on the disorder configuration. (c) For $T=100$ mK the SOI peak already gets smeared out and is no longer uniquely identifiable.

Figure 9

Zero-temperature Fabry-Pérot resonances observed in $G\left(B\right)$. When ${\Delta }_z>{\epsilon }_{\mathrm{F}}$, only one state is involved in transport, and regular single-period oscillations show up in $G\left(B\right)$; see the portion of the curves on the right of the vertical dotted line identifying ${\Delta }_z={\epsilon }_{\mathrm{F}}$. When the chemical potential lies outside the Zeeman gap, two states are involved, and more complicated oscillations are observed (left part of the curves). The plots refer to the case $L=10$ $\mu \mathrm{m}$ . (a) ${\epsilon }_{\mathrm{F}}=2{\epsilon }_{\mathrm{so}}$, (b) ${\epsilon }_{\mathrm{F}}=5{\epsilon }_{\mathrm{so}}$, (c) ${\epsilon }_{\mathrm{F}}=8{\epsilon }_{\mathrm{so}}$, and (d) ${\epsilon }_{\mathrm{F}}=5{\epsilon }_{\mathrm{so}}$ and $L=5$ $\mu \mathrm{m}$ . Note that this curve presents half-period oscillations, as expected [compare to (b)]. For all these plots we kept $\lambda =0.1{\lambda }^{☆}$.

Figure 10

Fabry-Pérot resonances observed in $G\left(B\right)$ in the presence of disorder in a 4-$\mu \mathrm{m}$ -long wire for ${\epsilon }_{\mathrm{F}}=5{\epsilon }_{\mathrm{so}}$, $\lambda =0.1{\lambda }^{☆}$, and finite temperature $T=50$ mK. (a) Realistic disorder corresponding to a mean free path ${\ell }_{\mathrm{mfp}}\simeq 300$ nm. Different curves refer to different disorder configurations. The green curve corresponds to the clean-case result. $G\left(B\right)$ for different configurations shows few variations, especially in the single-channel regime ${\Delta }_z>{\epsilon }_{\mathrm{F}}$. (b) Strong disorder ${\ell }_{\mathrm{mfp}}\simeq 150$ nm. The difference between the two regimes is still clearly discernible for each configuration, even if the different curves substantially deviate from each other.