Anisotropy of the spin-orbit coupling driven by a magnetic field in InAs nanowires

We use the $\mathbf{k}·\mathbf{p}$ theory and the envelope function approach to evaluate the Rashba spin-orbit coupling induced in a semiconductor nanowire by a magnetic field at different orientations, taking explicitly into account the prismatic symmetry of typical nanocrystals. We make the case for the strongly spin-orbit-coupled InAs semiconductor nanowires and investigate the anisotropy of the spin-orbit constant with respect to the field direction. At sufficiently high magnetic fields perpendicular to the nanowire, a sixfold anisotropy results from the interplay between the orbital effect of field and the prismatic symmetry of the nanowire. A backgate potential, breaking the native symmetry of the nanocrystal, couples to the magnetic field inducing a twofold anisotropy, with the spin-orbit coupling being maximized or minimized depending on the relative orientation of the two fields. We also investigate in-wire field configurations, which shows a trivial twofold symmetry when the field is rotated off the axis. However, isotropic spin-orbit coupling is restored if a sufficiently high gate potential is applied. Our calculations are shown to agree with recent experimental analysis of the vectorial character of the spin-orbit coupling for the same nanomaterial, providing a microscopic interpretation of the latter.

Figure 1

(a) Schematics of a NW with a bottom gate. In our simulations, anisotropy is evaluated with a magnetic field $\mathbf{B}$ either perpendicular to the NW axis ($\varphi =\pi /2$) and rotated with an azimuthal angle $\theta$, or with $\theta =\pi /2$ and rotated in the $y\text{−}z$ plane. (b) Occupation of the lowest subband as a function of the wave vector $k_z$ at each chemical potential $\mu$ at $B=4\mathrm{T}$. As $\mu$ increases, the occupation saturates to one at any $k_z$ below the Fermi energy. Of course, in general several subbands are occupied. The nonparabolic dispersion is clearly appreciated, with the field inducing a seemingly Landau level dispersion. Two vertical dashed lines mark values of $\mu$ selected for the further analysis.

Figure 2

(a)–(c) Intra- (${\alpha }_i^{nn}, n=1,2,3$) and (d), (e) intersubband (${\alpha }_i^{1m}$) selected Rashba SO coupling constants as a function of the magnetic field $B$ and the wave vector $k_z$. (f) ${\alpha }_i^{nn}(B,k_z)$ at $B=4\mathrm{T}$ for the three lowest states. Results are shown for $\mu =0.3\mathrm{eV}$ and a magnetic field perpendicular to the NW axis and along the corner-corner direction ($\varphi =\pi /2,\theta =0$).

Figure 3

Squared envelope functions of the three lowest magnetic subbands, at $k_z=0$ and $0.2{\mathrm{nm}}^{-1}$, with a transverse magnetic field (red arrow) at intensities (a) $B=1\mathrm{T}$ and (b) $B=4\mathrm{T}$. Right panels show the electron density $n_e$ and the self-consistent potential profile $V$ at the corresponding field intensities.

Figure 4

(a) The intrasubband SO constant ${\alpha }_x^{11}$ as a function of $k_z$ at $B=1\mathrm{T}$ and $B=4\mathrm{T}$ and at chemical potentials $\mu =0.30\mathrm{eV}$ and $\mu =0.35\mathrm{eV}$. (b) The intrasubband SO constant ${\alpha }_x^{nn}\left(k_{n,z}^F\right),n=1,2,3$ (left axis) calculated at $k_{n,z}^F$ and number of occupied subbands (black line, right axis) as a function of the magnetic field intensity $B$.

Figure 5

Maps of ${\alpha }_x^{nn}$ and ${\alpha }_y^{nn}$ as a function of $\theta$ and wave vector $k_z$. Results are shown for $\mu =0.30\mathrm{eV}$ and magnetic fields (a) $B=1\mathrm{T}$ and (b) $B=4\mathrm{T}$.

Figure 6

Maps of ${\alpha }^{11}$ as a function of $\theta$ and wave vector $k_z$. Results are shown for $\mu =0.30\mathrm{eV}$ and magnetic fields (a) $B=1\mathrm{T}$ and (b) $B=4\mathrm{T}$. Insets under the main panels zoom in the $k_z$ range marked by dashed black rectangle of the corresponding panel.

Figure 7

Squared envelope function of the ground state at selected angles $\theta$ at $B=4\mathrm{T}$. (a) $k_z=0.04{\mathrm{nm}}^{-1}$, (b) $k_z=0.4{\mathrm{nm}}^{-1}$.

Figure 8

(a)–(c) Angular dependence of the $x$ (blue) and $y$ (red) components of intrasubband SO coupling constant (in units of meV nm) at $k_{n,z}^F$ together with the modulus ${\alpha }^{nn}=\sqrt{{\left({\alpha }_x^{nn}\right)}^2+{\left({\alpha }_y^{nn}\right)}^2}$ (black) for the three occupied states. Panel (d) presents the total SO coupling constant ${\alpha }_{\text{tot}}$ averaged over all occupied states. Results for $B=4\mathrm{T}, \mu =0.3\mathrm{eV}$.

Figure 9

The angular dependence of the total SO coupling constant (in units of meV nm) ${\alpha }_{\text{tot}}$, averaged over all $N$ occupied subbands at $k_{n,z}^F$. (a) $B=0.1\mathrm{T}, \mu =0.3\mathrm{eV}$ ($N=8$) and (b) $B=1\mathrm{T}, \mu =0.3\mathrm{eV}$ ($N=8$). (c) ${\alpha }^{11}$ at $\mu =0.3\mathrm{eV}$ (dashed line) and $\mu =0.35\mathrm{eV}$ (solid line).

Figure 10

The total intrasubband SO ${\alpha }_{\text{tot}}$, as a function of $V_g$ at selected magnetic fields $B=0,1,4\mathrm{T}$ directed in the $\theta =0$ (corner-to-corner) direction. Results are shown for $\mu =0.30\mathrm{eV}$.

Figure 11

The angular dependence of the $x$ (blue) and $y$ component (red) of the intrasubband SO constant (in units of meV nm) ${\alpha }_i^{11}$ calculated at the Fermi wave vector $k_{1,z}^F$ for the lowest subband and the total SO constant averaged over all occupied states at $k_{n,z}^F$ (green line). Insets in panel (b) show the squared envelope functions of the lowest subband at $k_{1,z}^F$ for the magnetic field with $\theta =0$ and $\theta ={180}^{\circ }$. Calculations are performed with $\mu =0.30\mathrm{eV}$ and $V_g=0.1\mathrm{eV}$.

Figure 12

The total intrasubband SO constant ${\alpha }_{\text{tot}}$ as a function of the gate voltage $V_g$ for different axial magnetic fields. Inset: squared envelope functions of the lowest subband for different magnetic fields at $V_g=0$.

Figure 13

Angular dependence of the $x$ (blue) and $y$ (red) component of the intrasubband SO (in units of meV nm) of the ground state ${\alpha }_i^{11}$ calculated at $k_{1,z}^F$ together with the total SOC ${\alpha }_{\text{tot}}$ (green). The magnetic field is rotated in the $y\text{−}z$ plane. Results for $\mu =0.30\mathrm{V}, B=1\mathrm{T}$ and (a) $V_g=0$ and (b) $V_g=0.1\mathrm{V}$.

Figure 14

(a) Schematic illustration of the experimental setup. (b), (c) The $x$ and $y$ component of the intrasubband SO coupling ${\alpha }_i^{11}$ as a function of the angle $\theta$ and the wave vector $k_z$. The magnetic field is rotated in the $x\text{−}y$ plane. (d), (e) The angular dependence of the total SO constant ${\alpha }_{\text{tot}}$. Results for $B=0.1\mathrm{T}$ and $\varphi =\pi /2$.