Models of strong interaction in flat-band graphene nanoribbons: Magnetic quantum crystals

Graphene-based nanostructures exhibit flat electronic energy bands in their single-particle spectrum. We consider interacting electrons in flat bands of zigzag nanoribbons. We present a protocol for flat-band projection that yields interaction-only tight-binding models. We argue that, at low densities, flat bands can delocalize single-particle basis states to support ferromagnetic quantum crystal ground states.

Figure 1

Top: Schematic of a zigzag nanoribbon of carbon atoms. Bottom: Schematic of a ferromagnetic crystal with one electron for every three unit cells in one band. The shaded areas correspond to single unit cells and the arrows indicate aligned electron spins.

Figure 2

The dotted-dashed lines indicate the energy eigenvalues of Eq. (1) versus wave vector for a nanoribbon of width $L_y=4$ and a $q$-space mesh of $N=44$. The solid line shows the approximate expression for the energy [Eq. (3)]. Two flat bands form near $q{R}_0=\pi$. In the large-$L_y$ limit, the bands flatten for $2\pi /3\le q{R}_0\le 4\pi /3$.

Figure 3

Two-dimensional Wannier functions plotted as a function of position in the lattice for a ribbon of width $L_y=4$. The Wannier functions tend to localize near the ribbon edges.

Figure 4

Left panel: Schematic of the energy dispersion for a wide ribbon with the Fermi level between the degenerate energy bands $u$ and $d$. In this regime, low lattice filling allows us to accurately ignore the finite dispersion near the band edges. Right panel: The same as the left panel but at larger fillings of the upper $u$ band. Here, the flat-band approximation will only be a good approximation if the Coulomb interaction is much larger than the bandwidth.

Figure 5

The diagonal component of the interband Coulomb interaction ($V_{ij}^{{}^{\prime }}$) for a zigzag graphene nanoribbon with $L_y=4$ and $N=44$. Circles are from numerical evaluation of Eqs. (A1). The solid line is a fit with Eq. (15) and $D_w=1.02$.

Figure 6

The same as Fig. 5 but for $L_y=10$ with $D_w=2.12$.