Topology and zero energy edge states in carbon nanotubes with superconducting pairing

We investigate the spectrum of finite-length carbon nanotubes in the presence of onsite and nearest-neighbor superconducting pairing terms. A one-dimensional ladder-type lattice model is developed to explore the low-energy spectrum and the nature of the electronic states. We find that zero energy edge states can emerge in zigzag class carbon nanotubes as a combined effect of curvature-induced Dirac point shift and strong superconducting coupling between nearest-neighbor sites. The chiral symmetry of the system is exploited to define a winding number topological invariant. The associated topological phase diagram shows regions with nontrivial winding number in the plane of chemical potential and superconducting nearest-neighbor pair potential (relative to the onsite pair potential). A one-dimensional continuum model reveals the topological origin of the zero energy edge states: a bulk-edge correspondence is proven, which shows that the condition for nontrivial winding number and that for the emergence of edge states are identical. For armchair class nanotubes, the presence of edge states in the superconducting gap depends on the nanotube's boundary shape. For the minimal boundary condition, the emergence of the subgap states can also be deduced from the winding number.

Figure 1

(a) Schematic figure of a SWNT proximity coupled to a superconducting substrate. (b) Hexagonal lattice structure. Depicted are unit vectors ${\mathit{a}}_1$, ${\mathit{a}}_2$, alternative unit vectors ${\mathit{C}}_h/d$, $\mathit{H}$, and vectors to the nearest-neighbor and next-nearest-neighbor sites ${\mathbf{\delta }}_j^{\left(t\right)}$ for an unrolled $(n,m)=(6,3)$ SWNT, where $d=\gcd (n,m)=3$. A and B sublattices are denoted by gray and white circles, respectively. (c) An effective 1D lattice model, which is obtained by a partial Fourier transform in the circumferential direction, and is a projection of the 2D lattice structure onto the 1D nanotube axis $z$ (see the dashed lines). Solid lines denote nearest-neighbor bond connections in the original lattice structure.

Figure 2

Conduction and valence bands of (a) $(n,m)=(6,3)$ metal-1 SWNT, classified into the zigzag class, and (b) $(n,m)=(8,2)$ metal-2 SWNT, classified into the armchair class. For both cases (a) and (b), $p_s=1$ and $q_s=0$. The angular momentum for each band is indicated, the blue curves show bands with $\mu ={\mu }_K$, and the purple curves in (a) show bands with $\mu ={\mu }_{K^{\prime }}$. Curvature-induced energy gaps at zero energy and spin-orbit splitting are not seen on this energy scale.

Figure 3

(a) Energy band, and (b) BdG excitation spectrum near the $K$ point for an $(n,m)=(6,3)$ SWNT. The chemical potential is set to be ${\mu }_{\mathrm{c}}=500$ meV. The two arrows with $k_{\pm }^{(K,s)}$ in (a) indicate the two Fermi points. In (b) the superconducting coupling parameters are chosen to be ${\mathrm{\Delta }}_0=0.5$ meV and ${\mathrm{\Delta }}_1=2$ meV. Each inset in (b) shows the enlarged BdG spectrum near the two Fermi points. The blue and red curves show the BdG spectra for the spin-up and -down components, respectively.

Figure 4

BdG spectrum of a $(6,3)$ nanotube with a length of $16.1\mu \mathrm{m}$ in the $\mathbf{\mu }=({\mu }_K=2,1)$ subspace. (a) Spectrum as a function of the superconducting pairing ${\mathrm{\Delta }}_1$, and (b) as a function of the chemical potential ${\mu }_{\mathrm{c}}$. Blue circles show the calculated spectrum and the dashed lines show the superconducting gaps ${\varepsilon }_{\mathrm{g},r}^{(K,s=1)}$ of the bulk system given in Eq. (18). The inset in (b) shows the real part components ${\varphi }_{\chi \mathrm{A}}$ (blue) and ${\varphi }_{\xi \mathrm{A}}$ (red) in arbitrary units as a function of lattice site $\ell$ for the calculated eigenfunction at ${\varepsilon }_{\mathrm{BdG}}=0$ with ${\mathrm{\Delta }}_0=0.5$ meV, ${\mathrm{\Delta }}_1=2$ meV, and ${\mu }_{\mathrm{c}}=650$ meV [indicated by the red arrow in (b)]. The definition of ${\varphi }_{p\sigma }$ ($p=\chi ,\xi$) is given in Eq. (22).

Figure 5

Phase of $det{h}_{\mathbf{\mu }}$, $argdet{h}_{\mathbf{\mu }}$, appearing in the integrand of the winding number in Eq. (28), for an $(n,m)=(6,3)$ nanotube near the $K$ point for which the angular momentum is ${\mu }_K=2$. The parameters are the same as those in Fig. 3. The continuous change of the function in the interval $-\pi \le argdet{h}_{\mathbf{\mu }}\le 3\pi$ is clearly seen. The blue and red curves show the spin components $s=1$ and $-1$, respectively. Note that both of them are almost equal $\pi$ in the regions of $k{a}_z\lesssim -2.21$. For this case, the integrand gives contribution $+1$ ($-1$) to the winding number for $s=1$ ($s=-1$).

Figure 6

Schematics of the trajectories of the complex function $det{h}_{\mathbf{\mu }}$ in the complex plane when $k$ changes from $k\ll k_{-}$ to $k\gg k_{+}$. The solid curve shows an example for a nontrivial winding number $w_{\mathbf{\mu }}=1$, and the dashed curve shows a case for a trivial winding number $w_{\mathbf{\mu }}=0$.

Figure 7

Topological phase diagram for an $(n,m)=(6,3)$ nanotube estimated from Eq. (32) for $(\tau ,s)=(1,1)$ in the ${\mu }_{\mathrm{c}}$ and ${\mathrm{\Delta }}_1/{\mathrm{\Delta }}_0$ plane, where ${\mathrm{\Delta }}_0=0.5$ meV. The light blue areas show the region of nontrivial winding number, $|w_{\mathbf{\mu }}|=1$. The dashed curves show Eq. (36), the analytical expression for the border of the topological phases. The region between the dashed vertical lines is the band-gap region of the normal state. The red lines indicate the parameter region of Fig. 4. The inset shows the phase diagram for the value $w_{(\mu ,1)}+w_{(\mu ,-1)}$ near the region marked by the red point, which has a nontrivial value only near the border of the main figure.

Figure 8

BdG spectrum of an armchair class $(5,2)$ nanotube with length of $13.6\mu \mathrm{m}$ in the $\mathbf{\mu }=(\mu ,s)=({\mu }_K,+1)$ subspace. (a) Unrolled tube near the left end. The boundary is formed by a simple cut of the lattice in the plane orthogonal to the nanotube axis, represented by the solid line perpendicular to the $z$ axis. Removed lattice sites adjacent to the boundary sites are represented by dashed circles, and the dangling bonds are represented by the dashed lines. The orthogonal boundary is given by keeping the Klein site indicated by ${\mathrm{K}}_{\mathrm{L}}$, and the minimal boundary is given by removing the Klein site. (b) Minimal and (c) orthogonal boundaries, respectively, in the 1D model. (d) BdG spectrum as a function of the superconducting pairing ${\mathrm{\Delta }}_1$, and, (e) that as a function of the chemical potential ${\mu }_{\mathrm{c}}$, respectively, for the minimal boundary, and, (f) and (g) show those for the orthogonal boundary. Each inset in (e) and (g) shows the real part components ${\varphi }_{\chi \mathrm{A}}$ (blue) and ${\varphi }_{\xi \mathrm{A}}$ (red) in arbitrary units as a function of lattice site $\ell$ for the calculated eigenfunction at the state indicated by the red arrow.