Chirality-induced selectivity of angular momentum by orbital Edelstein effect in carbon nanotubes

Abstract

Carbon nanotubes (CNTs) are promising materials exhibiting exceptional strength, electrical conductivity, and thermal properties, making them promising for various technologies. Besides achiral configurations with a zigzag or armchair edge, there exist chiral CNTs with a broken inversion symmetry. Here, we demonstrate that chiral CNTs exhibit chirality-induced orbital selectivity (CIOS), which is caused by the orbital Edelstein effect and could be detected as chirality-induced spin selectivity (CISS). We find that the orbital Edelstein susceptibility is an odd function of the chirality angle of the nanotube and is proportional to its radius. For metallic CNTs close to the Fermi level, the orbital Edelstein susceptibility increases quadratically with energy. This makes the CISS and CIOS of metallic chiral nanotubes conveniently tunable by doping or applying a gate voltage, which allows for the generation of spin- and orbital-polarized currents. The possibility of generating large torques makes chiral CNTs interesting candidates for technological applications in spin-orbitronics and quantum computing.

Figures (5)

• Figure 1: Chiral carbon nanotube.a Geometrical construction of a $(n,m)=(7,1)$ nanotube from a two-dimensional graphene lattice. The color indicates the $x$ coordinate of each atom. The circumferential vector $\mathbf{C}$ and the translational vector $\mathbf{T}$ span a rectangle containing the atoms of the unit cells of the corresponding chiral carbon nanotube in b. The chirality angle $\alpha$ is indicated. c Top view of the nanotube.
• Figure 2: Chirality-induced orbital selectivity of a chiral carbon nanotube.a Band structure of a metallic $(n,m)=(7,1)$ nanotube for which the color indicates the value of the orbital angular momentum $L_{\nu,z}(k)$ normalized to a unit tube length (see legend). b Orbital Edelstein susceptibility $\chi_z^{L_z}\cdot a/T$ as a function of energy for a $(n,m)=(7,1)$ nanotube (blue) and for the corresponding nanotube with opposite chirality $(n,m)=(8,-1)$ (red). The dashed line indicates an energy of $E=0.49\,\mathrm{eV}$ in the range where the orbital Edelstein susceptibility increases quadratically with energy. c Band structure and d orbital Edelstein susceptibility (blue) of an insulating $(n,m)=(4,2)$ nanotube and for the corresponding nanotube with opposite chirality $(n,m)=(6,-2)$ (red).
• Figure 3: Band structure of a metallic carbon nanotube by backfolding the graphene band structure.a Brillouin zone of two-dimensional graphene (hexagon) and smaller Brillouin zone (gray) corresponding to the unit cell spanned by $\mathbf{C}$ and $\mathbf{T}$ (gray rectangle in Fig. \ref{['fig:nanotubes']}a) for a $(n,m)=(7,1)$ tube. Black lines represent multiples of the reciprocal lattice vector $\mathbf{b}_T$ shifted by multiples of the reciprocal lattice vector $\mathbf{b}_C$. b Band structure of two-dimensional graphene. The black lines from panel a give rise to the band structure of the corresponding nanotube in c once they are shifted back to the small Brillouin zone (gray). The color resembles $L_{\nu,z}^\mathrm{approx}$ that approximates $L_{\nu,z}(k)$ of the corresponding carbon nanotube. The quantity has been normalized by $a/T$ for comparability. Colored dots indicate the Dirac points originally located at the points $K$ and $K'$ of the Brillouin zone of graphene.
• Figure 4: Comparison of metallic carbon nanotubes. The diagram shows all metallic nanotubes that are characterized by a circumference $C\leq14a$ (plotted along the radial direction) and a chirality angle $|\alpha|\leq30^\circ$ (plotted along the polar direction). The color of the hexagon indicates the value of the orbital Edelstein susceptibility at an energy of $E=0.49\,\mathrm{eV}$ (dashed line in Fig. \ref{['fig:bands_edelstein']}). This value has been normalized by the length of the tube $T$, for comparability: $\chi_z^{L_z}\cdot a/T$ (see legend). The color in the background is the function proportional to $C\sin(6\alpha)$.
• Figure 5: Chirality dependence of the orbital Edelstein susceptibility per atom.a The blue points are the calculated values $\chi_{z,\mathrm{atom}}^{L_z}$ at an energy of $E=0.49\,\mathrm{eV}$ for all metallic nanotubes shown in Fig. \ref{['fig:classification']}. Here the band structure exhibits four nearly linear bands and the effect is governed by the chirality of the tube as can be seen by comparison with the gray line that is a function proportional to $\sin(6\alpha)$. b shows the equivalent plot at the energy $E=7.91\,\mathrm{eV}$ where the effect is mostly governed by the sub-band structure leading to pronounced differences of the orbital Edelstein susceptibility between the different nanotubes.