Transverse profile and three-dimensional spin canting of a Majorana state in carbon nanotubes

The full spatial 3D profile of Majorana bound states (MBS) in a nanowirelike setup featuring a semiconducting carbon nanotube (CNT) as the central element is discussed. By atomic tight-binding calculations, we show that the chiral nature of the CNT lattice is imprinted in the MBS wave function which has a helical structure, anisotropic in the transverse direction. The local spin-canting angle displays a similar spiral pattern, varying around the CNT circumference. We reconstruct the intricate 3D profile of the MBS wave function analytically, using an effective low-energy Hamiltonian accounting both for the electronic spin and valley degrees of freedom of the CNT. In our model, the four components of the Majorana spinor are related by the three symmetries of our Bogoliubov-de Gennes Hamiltonian, reducing the number of independent components to one. A Fourier transform analysis uncovers the presence of three contributions to the MBS, one from the $\mathrm{\Gamma }$-point and one from each of the Fermi points, with further complexity added by the presence of two valley states in each contribution.

Figure 1

Setup and bulk properties of a (12, 4) carbon nanotube with proximity-induced superconductivity. (a) Schematic of the system including the CNT which lies on top of an $s$-wave superconductor (SC) with a magnetic field applied perpendicular to the nanotube axis. The nearest-neighbor hopping $t_{ij,s{s}^{\prime }}$ is spin-dependent due to curvature. The superconducting substrate breaks the rotational symmetry of the nanotube with respect to the CNT axis, which induces a valley-mixing term in the Hamiltonian. Moreover, it generates an on-site superconducting pairing term ${\mathrm{\Delta }}_0$. The numerical values of the various parameters of the model can be found in Appendix pp1-s1. (b) The low energy spectrum of the CNT consists of 1D cuts across the Dirac cones, with two valleys and two spin directions at each energy. (c) The single-particle energy spectrum of a (12,4) nanotube in the vicinity of the $\mathrm{\Gamma }$-point for a magnetic field of $B_{\perp }=14\text{T}$. Color scale shows the expectation value of $\langle s_z\rangle$ for the corresponding energy state. A finite ${\mathrm{\Delta }}_0$ induces in the $k$-space two superconducting pairing terms ${\tilde{\mathrm{\Delta }}}_s\left(k\right)$ and ${\tilde{\mathrm{\Delta }}}_p\left(k\right)$ whose action is indicated by the magenta and green lines, respectively. (d) The two superconducting pairing terms ${\tilde{\mathrm{\Delta }}}_s\left(k\right)$ (interband), and ${\tilde{\mathrm{\Delta }}}_p\left(k\right)$ (intraband), as functions of $k$. (e) The action of the particle-hole $\mathcal{P}$, pseudo-time-reversal $\tilde{\mathcal{T}}$ and chiral $\mathcal{C}$ operations on the components of a Nambu spinor in the real space. (f) The counterpart of these relations in the reciprocal space. The fact that $\mathcal{P}$ relates $u_{\tau s}\left(k\right)$ and $v_{\tau s}^{*}\left(k\right)$ follows from $\mathcal{P}{\gamma }_k={\gamma }_{-k}^{†}$.

Figure 2

Spin-canting angle ${\theta }_{xy}\left(\vec{r}\right)$ and the amplitude $|u_{M\uparrow }\left(\mathbf{r}\right)|$ of the electronic component of the Majorana state, obtained in a real-space tight-binding calculation of a finite (12,4) CNT with 4000 unit cells ($L=6.14\mu \text{m}$) for a magnetic field $B_{\perp }=14\text{T}$. In all panels, the color corresponds to the local value of $\langle {\theta }_{xy}\rangle$. (a) The full Majorana state and its leftmost $0.5\mu \mathrm{m}$, with distance from the CNT surface encoding $|u_{\uparrow }\left(\mathbf{r}\right)|$. (b) 2D projection of the region with the first maximum of the Majorana wave function, with point size corresponding to $|u_{M\uparrow }\left(\mathbf{r}\right)|$. (c) The left termination (i.e., the first 1.8 nm) of the CNT lattice. Vector length corresponds to $|u_{M\uparrow }\left(\mathbf{r}\right)|$, its orientation to the spin canting angle. In both (b) and (c), note the variation of $\langle {\theta }_{xy}\rangle$ with the polar coordinate.

Figure 3

$k$-space properties of a proximitized CNT in magnetic field at low energies. (a) Quasiparticle energy ${\tilde{\xi }}_{+}\left(k\right)$ and superconducting order parameter ${\tilde{\mathrm{\Delta }}}_p\left(k\right)$ in the effective one-band model. The superconducting order paramater is an odd function of the momentum $k$. Three $k$ values generate the dominant contributions to zero energy modes: one comes from the $\mathrm{\Gamma }$ point and one from each of the Fermi points, $\pm k_F$. (b) The Fourier transform of the numerical Majorana wave function for different azimuthal cuts $\phi$ confirms that the zero mode contains only three dominant $k$ contributions.

Figure 4

Azimuthal cuts of the electron component $u_{A\uparrow }$ of the Majorana spinor for (a) $\phi =0^{\circ }$, (b) $\phi =24.{23}^{\circ }$, and (c) $\phi =114.{23}^{\circ }$. The position of the cut in the full MBS wave function is indicated in each inset. The analytical form of $u_{\uparrow }$ is given by Eq. (13), its parameters are obtained from fits to the modulus of the numerical solution.

Figure 5

The absolute value of the fitted amplitude $|A_R|$ of 28 different $\phi$ cuts. The colors correspond to different groups of atoms related by the $C_4$ symmetry (i.e., atoms at the same $z$ position). From the inset, we see the approximate $\pi$ periodicity of $A_R$ and thus the $C_2$ symmetry of the MBS wave function.

Figure 6

(a), (b) Numerical and analytical $\text{Re}\left(u_{A\uparrow }\left(z\right)\right)$ and $\text{Im}\left(u_{A\uparrow }\left(z\right)\right)$ for the polar angle $\phi =0^{\circ }$ with parameters from the $\left|u_{A\uparrow }\left(z\right)\right|$ fit. (c) The spin-canting angle ${\theta }_{xy}$, defined in Eq. (1), for the cut $\phi =0^{\circ }$.

Figure 7

The breaking of $\tilde{\mathcal{T}}$ as a function of $B_{\perp }$ and of $V_0$. (a) The substrate potential has the amplitude $V_0=0.4\mathrm{eV}$, the value taken in the simulations shown in the main text. (b) The magnetic field is $B_{\perp }=14\mathrm{T}$ as in the main text and the strength $V_0$ of the substrate potential is varied.

Figure 8

The influence of a localized superconducting pairing on the bulk spectrum and the wave function of the Majorana states. (a) For different values of the parameter ${\lambda }_{\mathrm{\Delta }}$, the band gap still closes, but at different values of $B_{\perp }$. For ${\lambda }_{\mathrm{\Delta }}=a_{\mathrm{CC}}$ (the carbon-carbon bond length), we set ${\mathrm{\Delta }}_0=0.8$ meV to keep a clear band gap. (b) The polar angle range with strongest pairing, i.e., $Rcos(\phi -{\phi }_0)<{\lambda }_{\mathrm{\Delta }}$; the grey circle represents constant amplitude over the CNT circumference, the arc fragments are color coded as in (a). Cuts across $|u_{i\uparrow }|$ (c) and the spin canting angle (d) along several values of the polar angle $\phi$. The corresponding values of $B_{\perp }$ are the same in (c) and (d). (e) The local spin direction of the particle component of the Majorana spinor; the arrows representing electron spin for various values of ${\lambda }_{\mathrm{\Delta }}$ are superposed on each atom to show that the spin direction is not much influenced by ${\lambda }_{\mathrm{\Delta }}$. The color code is the same as in (a)–(d), $B_{\perp }$ for each ${\lambda }_{\mathrm{\Delta }}$ is the same as in (c), (d).