Signatures of spin-orbit interaction in transport properties of finite carbon nanotubes in a parallel magnetic field

The transport properties of finite nanotubes placed in a magnetic field parallel to their axes are investigated. Upon including spin-orbit coupling and curvature effects, two main phenomena are analyzed that crucially depend on the tube’s chirality: (i) Finite carbon nanotubes in a parallel magnetic field may present a suppression of current due to the localization at the edges of otherwise conducting states. This phenomenon occurs due to the magnetic-field-dependent open boundary conditions obeyed by the carbon nanotube’s wave functions. The transport is fully suppressed above threshold values of the magnetic field, which depend on the nanotube chirality, length, and on the spin-orbit coupling. (ii) Reversible spin-polarized currents can be obtained upon tuning the magnetic field, exploiting the curvature-induced spin-orbit splitting.

Figure 1

System and coordinates. (a) Schematic illustration of the carbon nanotube with the local and global coordinate systems used for its description. The direction of the magnetic field is parallel to the tube axis. (b) Projection of the nanotube onto the $X-Z$ plane. $i$ and $j$ schematically show the projection of two nearest-neighbor atoms (at an exaggerated distance in order to visualize the different variables). The angles are measured from the positive $Z$ axis and $|{\theta }_j-{\theta }_i|$ is very small for large radii, so that in fact both angles can be seen as $\tilde{\theta }$. $R$ is the radius of the nanotube, whereas ${\mathbf{R}}_{ij}$ is the vector connecting the two neighboring atoms (here we plot actually the projection of this vector). (c) Real-space representation of a fragment of a graphene lattice with translational vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. The chiral vector is $\mathbf{C}$ and the unit vectors ${\widehat{x}}_{\perp }$ and ${\widehat{x}}_{\parallel }$ reflect the parallel and perpendicular directions with respect to the nanotube axis. The three $B$ nearest-neighbor atoms of an atom $A$ are located at positions given by ${\mathbf{d}}_l$ with $l=1,2,3$. (d) Reciprocal space for the graphene, spanned by the vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$. The central hexagon is the symmetric Brillouin zone. Its corners are the so-called $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points or Dirac points.

Figure 2

Schematic representation of the effect of the curvature of a nanotube on the transfer integrals between the orbitals of nearest-neighbor atoms. The $p_z$ orbitals are not parallel anymore, so that a mixing with the orbitals building the $\sigma$ bonds takes place.

Figure 3

Allowed ${\kappa }_{\perp }^{\prime }$ and ${\kappa }_{\parallel }^{\prime }$ values for a zigzag-like CNT fulfilling the boundary condition given by Eq. (18). At ${\kappa }_{\perp }^{\prime }=0$, as expected, ${\kappa }_{\parallel }^{\prime }\frac{L}{\pi }=n+\frac{1}{2}$ with $n\in \mathbb{Z}$, and when including the spin, the Kramers degeneracy holds.

Figure 4

Linear dispersion relation of finite-size armchair carbon nanotubes with open boundary conditions. (a) The band structure of an armchair CNT with periodic boundary conditions around the two inequivalent Fermi points, $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$. For finite-length tubes, only certain values of ${\kappa }_{\parallel }^{\prime }$ are allowed, which are indicated here by the sketched vertical cuts. Branches of left movers are indicated by $l$, whereas $r$ indicate those of right movers. (b) These four branches can be combined into the band structure shown on the right, which represents the solutions for the open boundary problem. We have chosen explicitly the case in which ${\kappa }_{\perp }=0$ and the allowed ${\kappa }_{\parallel }$ do not lie symmetrically around the Fermi points in order to easily observe the bands of equal energy. Including SOI and the magnetic field in ${\kappa }_{\perp }^{\prime }$ leads to shifted cuts of the Dirac cones, and we would have the same picture as here but instead with parabolas.

Figure 5

Schematic drawing of the setup of our system, where a suspended CNT connects the source and drain leads, between which a bias voltage $V_b$ is applied.

Figure 6

Density of states of a (9,0) CNT of a length of about 42 nm (100 unit cells) (a) within a $\pi$-orbital tight-binding approximation, neglecting all other effects, (b) with the inclusion of curvature effects as described in Sec. 2b, and (c) taking into account the Zeeman effect. Finally, in (d) spin-orbit coupling is additionally considered. The value of the SOI parameter $\delta$ is set to $0.013$ and ${\Sigma }_{\text{WB}}\left(E\right)\approx -\mathrm{i}0.85$ eV.

Figure 7

$dI/dV$ characteristics in units of the conductance quantum $G_0=e^2/h$, for a (9,0) CNT of a length of about 42 nm as in Fig. 6 with the respective effects being successively included.

Figure 8

Spatially resolved wave function (obtained numerically) of a (9,0) CNT with 100 unit cells and with SOI, corresponding to the valence eigenstate getting localized in the magnetic field as a spin-up state (see Fig. 6). In the top figures, dots represent the atomic positions and their color corresponds to the amplitude of the wave function at the site (going from zero to its maximal value). The two sublattices $A$ and $B$ are plotted separately. The corresponding magnetic fluxes are (a) $\phi =0.02{\phi }_0$ and (b) $\phi =0.05{\phi }_0$. The decay of the wave-function amplitude for these two flux values is shown in (c) and (d), respectively, where sublattice $A$ is plotted with a dashed line and a solid line is used for the results corresponding to sublattice $B$. Red marks the spin-up component of these states while blue is used for spin-down, which is negligible here.

Figure 9

(a) Density of states of a chiral (12,9) CNT ($\eta ={25}^{\circ }$) with a length of 16 unit cells (about 42 nm) taking into account curvature effects, the spin-orbit coupling, and the Zeeman effect. (b) Density of states of an armchair CNT (10,10) of 172 unit cells (about the same length of 42 nm) and with the same approximations as in (a). (c) and (d) are the $\mathrm{d}I/\mathrm{d}V$ characteristics in units of $G_0$ corresponding, respectively, to (a) and (b). The value of the SOI $\delta$ is set to $0.0028$. Notice the absence of localization in the armchair case.

Figure 10

Comparison of numerical and analytical results for the projected density of states in different CNT’s of approximately the same length (about 42 nm). The dashed lines in the analytical calculation correspond to states in the $\mathbf{K}$ cone and the solid lines correspond to ${\mathbf{K}}^{\prime }$ states. Spin up is represented in red, and spin down in blue. All the lines shown correspond to the band with ${\kappa }_{\perp }=0$, except for the states appearing in the zigzag CNT at the Fermi energy at $\phi =0$, which are the edge states corresponding to neighboring bands (the next allowed ${\kappa }_{\perp }$ values, ${\kappa }_{\perp }=\pm \frac{1}{R}$) and for which dotted lines are used. (a) A (9,0) CNT as in Fig. 6d. (b) The same (9,0) CNT but with a more realistic value of $\delta$, as in the calculations shown in Fig. 9. (c) A (12,9) CNT as in Fig. 9a. (d) A (10,10) CNT as in Fig. 9b.

Figure 11

(a) Spin-resolved energy spectrum for the CNT (12,9) with a length of 100 unit cells and a SOI parameter $\delta$ of 0.013. The colors correspond to those in Fig. 10. (b) Corresponding differential conductance, where the spectrum lines have been symmetrized due to the considered symmetrical voltage drop.

Figure 12

Spin-resolved energy spectrum for (12,9) CNT’s of several lengths with a SOI characterized by the parameter $\delta =0.0028$. (a) A 100-unit-cell-long CNT, that is, about 259 nm long. (b) A CNT with 200 unit cells, and thus a length of about 518 nm. (c) A CNT with a length of about 1295 nm (500 unit cells). (d) Evolution with the magnetic field of the band edges for an infinite CNT. The colors correspond to those in Fig. 10.

Figure 13

(a) Evolution of the spectrum in the magnetic field and (b) differential conductance as well as (c) integrated conductance for a (12,9) CNT of 200 unit cells. The points around $V_b=0$ have been suppressed to avoid a division by zero. (d) Growth with the CNT length of the separation in magnetic flux of the conductance peaks around the Fermi energy.

Figure 14

(a) Spectrum of a zigzag CNT, (9,0), as in Fig. 10b. The dashed lines correspond to states in the $\mathbf{K}$ cone and the solid lines correspond to ${\mathbf{K}}^{\prime }$ states. Up and down spins are represented, respectively, in red and blue or in magenta and cyan when the valley- and spin-dependent energy shift is included. In the inset, a zoom in of the first Kramers doublet in the conduction band allows us to see the widening of the splitting. (b) Spectrum for a (12,9) CNT of 200 unit cells with the same color code as in (a). A dotted line marks the Fermi energy, cut by the energy levels at the magnetic fluxes where conduction peaks are observed. The inset shows the change in the separation in magnetic flux of the conductance peaks for the chiral CNT (12,9) in Fig. 13d when including the valley-dependent Zeeman-like shift as a function of the CNT length (red dashed line). The vertical dotted line marks a length of about 500 nm.