Curvature-enhanced spin-orbit coupling in a carbon nanotube

Structure of the spin-orbit coupling varies from material to material and thus finding the correct spin-orbit coupling structure is an important step toward advanced spintronic applications. We show theoretically that the curvature in a carbon nanotube generates two types of the spin-orbit coupling, one of which was not recognized before. In addition to the topological phase-related contribution of the spin-orbit coupling, which appears in the off-diagonal part of the effective Dirac Hamiltonian of carbon nanotubes, there is another contribution that appears in the diagonal part. The existence of the diagonal term can modify spin-orbit coupling effects qualitatively, an example of which is the electron-hole asymmetric spin splitting observed recently, and generate four qualitatively different behavior of energy-level dependence on parallel magnetic field. It is demonstrated that the diagonal term applies to a curved graphene as well. This result should be valuable for spintronic applications of graphitic materials.

Figure 1

(Color online) Schematic of the lowest conduction-band (red, $E>0$) and highest valence-band (blue, $E<0$) positions of a semiconducting CNT predicted by ${\mathcal{H}}_{\text{Dirac}}^{\text{K}}+{\mathcal{H}}_{\text{soc}}^{\text{K}}$ for (a) ${\delta }_{\text{K}}={\delta }_{\text{K}}^{\prime }=0$, (b) ${\delta }_{\text{K}}\ne 0$, ${\delta }_{\text{K}}^{\prime }=0$, and (c) ${\delta }_{\text{K}}\ne 0$, ${\delta }_{\text{K}}^{\prime }\ne 0$. In (c), the conduction or valence band has larger spin splitting depending on the sign of $\nu$. Arrows (green) show the spin direction along the CNT. The expressions for the energy-level spacing are also provided. When they are negative, the positions of the two spin-split bands should be swapped.

Figure 2

(Color online) Two-dimensional honeycomb lattice structure. $\mathbf{x}\left(\mathbf{y}\right)$ is the coordinate around (along) the CNT with chiral vector $n{\mathbf{a}}_1+m{\mathbf{a}}_2\equiv \left(n,m\right)$ and chiral angle $\theta$. ${\omega }_j\left(j=1,2,3\right)$, the length between $y$ axis passing $A$ atom and its parallel (red dashed) line is related with ${\xi }_j$ by ${\xi }_j\approx {\omega }_j/\left(2R\right)$ [Eq. (10)]. The coordinates for the CNT is illustrated on the right. Here, $x=\phi R$.

Figure 3

(Color online) Schematic of the second-order transition process generated by $H^{\text{K},\left(2\right)}$ [Eq. (6)]. Pseudospin transitions (between the sublattices $A$ and $B$) and interband transitions (between $\pi$ and $\sigma$ bands) are illustrated.

Figure 4

(Color online) Calculated energy spectrum of the conduction-band bottom (red, $E>0$) and valence-band top (blue, $E<0$) near K (solid lines) and ${\text{K}}^{\prime }$ (dashed lines) points in semiconducting CNTs with $R\approx 2.5\text{nm}$ as a function of magnetic field $B$ parallel to the CNT axis. The chiral vectors for each CNT are (a) (38,34), (b) (39,34), (c) (61,0), and (d) (62,0), respectively. Arrows (green) show spin direction along the CNT axis and ${\Delta }_{\text{so}}$ denotes the zero-field splitting. Assuming $k_y=0$, the energy $E$ including the SOC, the Aharonov-Bohm flux (Ref. 15) ${\varphi }_{\text{AB}}=B\pi R^2$, and the Zeeman coupling effects is, $E=\pm \hslash v_F\sqrt{{\left[k_x+\left(1/R\right)\left({\varphi }_{\text{AB}}/{\varphi }_0\right)\right]}^2}+E_{\uparrow (\downarrow )}^{\text{K}\left({\text{K}}^{\prime }\right),\left(2\right)}+\left(g/2\right){\mu }_{\text{B}}{\tau }_{\parallel }B$, with upper (lower) sign applying to the conduction (valence) band. ${\varphi }_0=hc/\left|e\right|$, ${\tau }_{\parallel }=+1\left(-1\right)$ for ${\chi }_{\uparrow (\downarrow )}$, $v_F=-a{V}_{pp}^{\pi }\sqrt{3}/2$, and $g=2$ (Ref. 12). For estimation of $E_{\uparrow (\downarrow )}^{\text{K}\left({\text{K}}^{\prime }\right),\left(2\right)}$, we use tight-binding parameters in Ref. 21; $V_{ss}^{\sigma }=-4.76\text{eV}$, $V_{sp}^{\sigma }=4.33\text{eV}$, $V_{pp}^{\sigma }=4.37\text{eV}$, $V_{pp}^{\pi }=-2.77\text{eV}$, ${\epsilon }_s=-6.0\text{eV}$, and ${\epsilon }_p=0$ (Ref. 22).