Figure 1

(Color online) Hubbard-model SCF-theory phase diagram as a function of doping per length $\delta n$ (defined in the text) and ribbon width $W$ for nearest-neighbor hopping ${\gamma }_0=2.6\text{eV}$. The on-site repulsion strength was chosen to have a value $U=2\text{eV}$ which reproduce the ribbon band gaps obtained in the LDA-DFT calculations in Ref. 17. We used 1200 $k$ points for Brillouin-zone sampling. The energetic preference for opposite spins (AF) on opposite edges is replaced by a preference for parallel F spins at larger doping. Above a critical doping $\delta n\sim 0.7$ the SCF calculation does not find magnetic states. Solutions at finite doping sometimes (AFb and Fb) break the inversion symmetry of the ribbon. When NC is allowed canted spin solutions midway between AF and F configuration become energetically favored at low doping.Reuse & Permissions

Figure 2

(Color online) Total energy differences per edge atom between the AF (or AFb) and F, Fb, or P states as a function of doping $\delta n$ for a relatively narrow ribbon with $N=8$ atom pairs per unit cell. For low doping AFb-type solutions with broken charge symmetry are energetically favored over AF solutions although the energy difference is very small. The NC spin solutions are lowest in energy in the weakly doped regime.Reuse & Permissions

Figure 3

(Color online) Net spin polarization obtained from the total electron spin densities $\zeta =\left(n_{\uparrow }-n_{\downarrow }\right)/\left(n_{\uparrow }+n_{\downarrow }\right)$ for AFb, F, and Fb solutions as a function of doping. AFb solutions collapse into AF solutions with zero net spin polarization for high enough doping. F and Fb configurations also progressively lose net spin polarization as they approach the nonmagnetic $P$ limit. The shaded region represents the doping regime where noncollinear solutions are favored energetically.Reuse & Permissions

Figure 4

(Color online) Upper row. Band structures corresponding to AFb, AF, F, Fb, and P spin collinear solutions of the Hubbard-model SCF equations for a zigzag nanoribbon with $N=8$ atom pairs in the unit cell. At finite doping the energy gain due to the gap present in the AFb and AF solutions is reduced favoring the F solution which does not have a gap. Lower row. Up- and down-spin electron occupation per lattice site in the unit cell across the ribbon. The AFb configuration has broken charge-distribution symmetry relative to the ribbon center due to an unequal occupation of up- and down-spin bands. All mean-field bands are invariant under $k\to -k$. We show only the portion of the 1D BZ with states close to the Fermi level.Reuse & Permissions

Figure 5

(Color online) In the weakly doped region canted spin orientations develop in order to minimize the total energy when noncollinear solutions are allowed. Upper row. Band structure and spin-resolved electron occupation per lattice for a zigzag ribbon with $N=8$ and $\delta n=0.05$. The occupation and spin polarization at each lattice site are represented in a local frame where the spin is diagonal. Lower row. Spin polarization and relative orientation of the spin direction between different lattice sites represented with the arrow heads.Reuse & Permissions