Noncollinear Spin-Orbit Magnetic Fields in a Carbon Nanotube Double Quantum Dot

We demonstrate experimentally that noncollinear intrinsic spin-orbit magnetic fields can be realized in a curved carbon nanotube two-segment device. Each segment, analyzed in the quantum dot regime, shows near fourfold degenerate shell structure allowing for identification of the spin-orbit coupling and the angle between the two segments. Furthermore, we determine the four unique spin directions of the quantum states for specific shells and magnetic fields. This class of quantum dot systems is particularly interesting when combined with induced superconducting correlations as it may facilitate unconventional superconductivity and detection of Cooper pair entanglement. Our device comprises the necessary elements.

Figure 1

(a) SEM image of nanotube with Au normal leads (50 nm), a $\mathrm{Ti}/\mathrm{Al}$ superconducting lead ($5/15\mathrm{nm}$), and two side gates. Note that the lead geometries are overlaid on the image digitally. (b) Schematic of the device. The angle $\phi$ reflects the angle between the external magnetic field ${\mathbf{B}}_{\text{ext}}$ and the tangent of the midpoint of the right nanotube segment. A rigorous definition of $\phi$ is given below. Note that ${\mathbf{B}}_{\text{ext}}$ is constricted to the $x\text{−}z$ plane. The expected valley-dependent directions of the spin-orbit magnetic fields $\pm {\mathbf{B}}_{\mathrm{SO},(\mathrm{L}/\mathrm{R})}$ are indicated by the two-headed arrows.

Figure 2

(a),(c) Bias spectroscopy of the left and right quantum dots showing charging energies of $U_{\mathrm{L}}\approx 6\mathrm{meV}$ and $U_{\mathrm{R}}\approx 9\mathrm{meV}$, respectively. Both dots show fourfold shell filling characteristic of carbon nanotubes. Kondo resonances caused by strong lead-dot coupling are visible at zero bias for multiple fillings. They are most pronounced in shell $e$ in the left dot. The blue arrow shows the onset of inelastic cotunneling close to zero bias in shell $M$. The dot filling is $n_e=53–56$ electrons for shell $N$ and $n_h=20$ holes for shell $f$. (b),(d) A superconducting gap is visible in both dots with a magnitude of about $60\mu \mathrm{eV}$. Cuts in (c) and (d) show the superconducting gap.

Figure 3

Excitation spectroscopy of shell $N$ in the right dot. (a)–(c) In the top row the magnetic field magnitude is swept with the field aligned $-11°$ away from parallel orientation (denoted by ${\parallel }^{*}$). The misalignment does not change the spectrum qualitatively compared to parallel orientation. The zero-field splitting is denoted by $\mathrm{\Lambda }=\sqrt{{\mathrm{\Delta }}_{K{K}^{\prime }}^2+{\mathrm{\Delta }}_{\mathrm{SO}}^2}$. (d)–(f) Bottom row are magnetic field angle sweeps at 2 T. The electron filling $n_e=53–56$. For all plots fitting parameters are ${\mathrm{\Delta }}_{\mathrm{SO}}=150\mu \mathrm{eV}$, ${\mathrm{\Delta }}_{K{K}^{\prime }}=70\mu \mathrm{eV}$, $J=120\mu \mathrm{eV}$, $|g_{\text{orb}}|=2.6$. (g) Plot of the nanotube spectrum with the parameters for the shell shown. (h) Schematic of an inelastic cotunneling process involving level $\alpha$ and $\beta$ at the red line in (g). This process is allowed when $|V_{\mathrm{SD}}|$ is equal to or above the energy difference between levels $\alpha$ and $\beta$.

Figure 4

Excitation spectroscopy of (a) shell $h$ in the left segment and (b) shell $N$ in the right segment. Annotations show the angle where ${\mathbf{B}}_{\text{ext}}$ is oriented perpendicular to the spin-orbit magnetic field. This occurs for different angles in the two segments. Inset: SEM image of the device. The average angle difference between the left and right dot of 21° is illustrated by tangents drawn on the nanotube segments. Drawing the tangents at the segment midpoints would instead give $\mathrm{\Delta }{\phi }_{\mathrm{SO}}^{\mathrm{av}}\approx 32°$. (c) Spin-orbit magnetic field angles plotted for the shells measured. Red (blue) denotes left (right) segment. Dashed lines are averages of angles in a segment. Error bars are obtained as the minimum and maximum value for ${\phi }_{\mathrm{SO}}^{\nu }$ that make the model fit the data.

Figure 5

(a),(c) Spectrum and spin expectation value $⟨\mathbf{S}⟩$ as a function of external magnetic field angle $\phi$. (b),(d) Energy levels and $⟨\mathbf{S}⟩$ at the position of the solid black lines in (a),(c). The model parameters are for shell $h$ and shell $N$ in (a),(b) and (c),(d), respectively. The spin vectors are shown in the $x\text{−}z$ plane since the effective magnetic field is constricted to this plane. The dotted vertical lines show the angle at which the magnetic field is parallel to the tube segment. All spins are shown in a global basis relative to shell $N$.