Magneto-orbital effect without spin-orbit interactions in a noncentrosymmetric zeolite-templated carbon structure

A peculiar manifestation of orbital angular momentum is proposed for a zeolite-templated carbon system, C${}_{36}$H${}_9$. The structure, being a network of nanoflakes in the shape of a “pinwheel,” lacks inversion symmetry. While the unit cell is large, the electronic structure obtained with a first-principles density-functional theory and captured as an effective tight-binding model in terms of maximally localized Wannier functions, exhibits an unusual feature that the valence band top comes from two chiral states having orbital magnetic momenta of $\pm 1$. The noncentrosymmetric lattice structure then makes the band dispersion asymmetric, as reminiscent of, but totally different from, spin-orbit systems. The unusual feature is predicted to imply a current-induced orbital magnetism when holes are doped.

Figure 1

(a) Crystal structure, theoretically optimized here, of the zeolite-templated carbon. Solid line represents a unit cell, which contains eight hydrocarbon C${}_{36}$H${}_9$ patches [with 288 carbon (darker spheres) and 72 hydrogen atoms (lighter)] with different orientations. (b) A C${}_{36}$H${}_9$ patch with its three nearest-neighbor patches, which form a three-winged “pinwheel” with a threefold axis.

Figure 2

LDA band structure of C${}_{36}$H${}_9$. The origin of energy is set to be the top of the valence band. Inset shows the first Brillouin zone.

Figure 3

Maximally localized Wannier functions, ${\psi }_1$, ${\psi }_2$, and ${\psi }_3$, for the 24 bands at the top of the valence band. The surfaces with different hues represent positive (negative) isosurfaces.

Figure 4

Structure of the 24 topmost valence bands with the linewidth representing the weights of (a) the ${\psi }_{\mathrm{A}}$ state or (b) ${\psi }_{\mathrm{E}}^{\pm }$.

Figure 5

The C${}_{36}$H${}_9$ represented by the obtained tight-binding model. Each black sphere represents a C${}_{36}$H${}_9$ patch, while wings of a pinwheel illustrate Wannier orbitals. The transfer integrals are $t_1=-0.255$ eV, $t_2=0.116$ eV, $t_3=0.022$ eV, and $t_4=-0.092$ eV.

Figure 6

(a) A blow-up of Fig. 4, where bands originating from ${\psi }_{\mathrm{A}}$ (black dotted line), ${\psi }_{\mathrm{E}}^{+}$ (red solid), and ${\psi }_{\mathrm{E}}^{-}$ (blue dashed) on a patch normal to (1,1,1) are displayed, respectively, along the $\Gamma$ - R line. (b) Two topmost valence bands plotted on a two-dimensional $k$ space. The two bands anticross with each other along the (1,1,1) direction, and arrows represent the orbital moment, $l_z$. (c) The total orbital magnetic moment per unit cell as a function of the electric field with the number of holes, $\delta =0.01,0.02,0.03,0.04,$ and $0.05$ (from bottom to top). The applied electric field is here represented by $\Delta k$, which is the crystal momentum acquired by an electron due to the field in the (1,1,1) direction.