Weak (anti)localization in tubular semiconductor nanowires with spin-orbit coupling

We compute analytically the weak (anti)localization correction to the Drude conductivity for electrons in tubular semiconductor systems of zinc-blende type. We include linear Rashba and Dresselhaus spin-orbit coupling (SOC) and compare wires of standard growth directions $\langle 100\rangle ,\langle 111\rangle$, and $\langle 110\rangle$. The motion on the quasi-two-dimensional surface is considered diffusive in both directions: transversal as well as along the cylinder axis. It is shown that Dresselhaus and Rashba SOC similarly affect the spin relaxation rates. For the $\langle 110\rangle$ growth direction, the long-lived spin states are of helical nature. We detect a crossover from weak localization to weak antilocalization depending on spin-orbit coupling strength as well as dephasing and scattering rate. The theory is fitted to experimental data of an undoped $\langle 111\rangle$ InAs nanowire device which exhibits a top-gate-controlled crossover from positive to negative magnetoconductivity. Thereby, we extract transport parameters where we quantify the distinct types of SOC individually.

Figure 1

(a) Internal (blue) and external (red) electric fields that lead to Rashba SOC in a nanowire (here, $E_{\text{int}},E_{\text{ext}}<0$). The internal field can be a consequence of Fermi level pinning, the external due to a gate voltage. Figure (b) sketches the situation of a nanowire with radius $R_0=40\mathrm{nm}$. Here, the electron probabitity density ${\left|\psi \right|}^2$ (green) is focused at $R=35\mathrm{nm}$ below the surface and extends over an area of about 10 nm for a confinement parameter $\gamma =0.55{\mathrm{nm}}^{-1}$. The blue line illustrates the bending of the conduction band (CB) edge due to Fermi (F) level pinning. The resulting radial confinement is modeled by a harmonic potential in this work.

Figure 2

Spectrum of the Cooperon Hamiltonian for $\left[001\right]$ nanowires with parameter configurations ${\lambda }_1=0.4$ and ${\lambda }_2=-0.1$ for $\mathcal{B}=\kappa =0$ (green). Dashed lines correspond to vanishing Rashba SOC and black solid line to the singlet mode. The grid lines are plotted by use of the approximate formulas ${\mathrm{\Delta }}_{\chi }^{\left[001\right]}$ (gray) and ${\delta }_{\varphi }^{\left[001\right]}$ for ${\mathcal{Q}}_{\varphi }^{\text{min,[001]}}$ (red). The grid lines are plotted using the approximate formulas derived in Sec. 3b2.

Figure 3

Spectrum of the Cooperon Hamiltonian for $\left[111\right]$ nanowires with parameter configurations ${\lambda }_1=0.5$ and ${\lambda }_2=0.3$ for $\mathcal{B}=\kappa =0$ (green). Dashed lines correspond to vanishing Rashba SOC and black solid line to the singlet mode. The grid lines are plotted by use of the approximate formulas ${\mathrm{\Delta }}_{\chi }^{\left[111\right]}$ (gray) and ${\delta }_{\varphi }^{\left[111\right]}$ for ${\mathcal{Q}}_{\varphi }^{\text{min,[111]}}$ (red). The grid lines are plotted using the approximate formulas derived in Sec. 3b2.

Figure 4

Spectrum of the Cooperon Hamiltonian for $\left[110\right]$ nanowires with parameter configurations ${\lambda }_1=0.3$ and ${\lambda }_2=0.1$ for $\mathcal{B}=\kappa =0$ (green). Dashed lines correspond to vanishing Rashba SOC and black solid line to the singlet mode. The grid lines are plotted by use of the approximate formulas ${\mathrm{\Delta }}_{\chi }^{\left[110\right]}$ (gray), ${\delta }_{\varphi }^{\left[110\right]}$ for ${\mathcal{Q}}_{\varphi }^{\text{min,[110]}}$ (red), and ${\delta }_z^{\left[110\right]}$ for ${\mathcal{Q}}_z^{\text{min,[110]}}$ (blue). The grid lines are plotted using the approximate formulas derived in Sec. 3b2.

Figure 5

Spectrum of the Cooperon Hamiltonian for $\left[110\right]$ nanowires with optimal parameter configurations for a lowest possible gap along ${\mathcal{Q}}_z$ with ${\lambda }_1=-0.901$ and ${\lambda }_2=0.305$ for $\mathcal{B}=\kappa =0$ (green). Dashed lines correspond to vanishing Rashba SOC and black solid line to the singlet mode. The grid lines are plotted by use of the approximate formulas ${\mathrm{\Delta }}_{\chi }^{\left[110\right]}$ (gray) and ${\delta }_z^{\left[110\right]}$ for ${\mathcal{Q}}_z^{\text{min,[110]}}$ (blue). The grid lines are plotted using the approximate formulas derived in Sec. 3b2.

Figure 6

Crossover from WL to WAL in a $\langle 111\rangle$ nanowire for $\mathcal{B}=\kappa ={\lambda }_2=0,{\lambda }_1=0.3$, and $R_{so}=30$.

Figure 7

Relative magnetoconductivity $\mathrm{\Delta }{\sigma }_R\equiv \mathrm{\Delta }\sigma \left(B\right)-\mathrm{\Delta }\sigma (B=0)$ in a $\langle 111\rangle$ nanowire for $\kappa ={\lambda }_2=0,{\lambda }_1=0.3,R_{so}=30$, and $c_{\tau }=10$.

Figure 8

Gate-controlled crossover from positive to negative magnetoconductivity $\mathrm{\Delta }{\sigma }_R\equiv \mathrm{\Delta }\sigma \left(B\right)-\mathrm{\Delta }\sigma (B=0)$ in a $\langle 111\rangle$ InAs nanowire. The symbol-dotted lines correspond to experimental data for different top-gate voltages $V_g$ which is fitted by theory (solid lines) using Eq. (68) and varying the external Rashba SOC strength ${\alpha }_{\text{ext}}\propto {\lambda }_2$.

Figure 9

Extracted fitting parameters for a $\langle 111\rangle$ InAs nanowire. (a) External Rashba SOC strength $|{\alpha }_{\text{ext}}|$ as well as (b) spin relaxation and dephasing length $l_s$ and $l_{\varphi }$, respectively, in dependence of a top-gate voltage $V_g$.