Berry Curvature of Low-Energy Excitons in Rhombohedral Graphene

Abstract

We investigate low energy excitons in rhombohedral pentalayer graphene encapsulated by hexagonal boron nitride (hBN/R5G/hBN), focusing on the regime at the experimental twist angle $θ= 0.77^\circ$ and with an applied electric field. We introduce a new low-energy two-band model of rhombohedral graphene that captures the band structure more accurately than previous models while keeping the number of parameters low. Using this model, we show that the centres of the exciton Wannier functions are displaced from the moiré unit cell origin by a quantised amount - they are instead localised at $C_3$-symmetric points on the boundary. We also find that the exciton shift is electrically tunable: by varying the electric field strength, the exciton Wannier centre can be exchanged between inequivalent corners of the moiré unit cell. Our results suggest the possibility of detecting excitonic corner or edge modes, as well as novel excitonic crystal defect responses in hBN/R5G/hBN. Lastly, we find that the excitons in hBN/R5G/hBN inherit excitonic Berry curvature from the underlying electronic bands, enriching their semiclassical transport properties. Our results position rhombohedral graphene as a compelling tunable platform for probing exciton topology in moiré materials.

Figures (9)

• Figure 1: Rhombohedral graphene encapsulated between two hBN layers. The hopping parameters of our tight-binding model are labelled $t_{0\dots 4}$ in $(a)$. The $A$ sublattice of graphene is shown in red and the $B$ sublattice in blue. $(b)$ Dispersions of the low energy model and the full model of the free-standing rhombohedral graphene (with displacement field $u_D = 20\mathrm{meV}$). $(c)$ The Berry curvature [$B(k_x, k_y)$] distribution of the low energy model.
• Figure 2: The mini Brillouin zone $(a)$ and unit cell $(b)$ for hBN/R5G/hBN with twist angle $\theta = 0.77^\circ$. The moiré reciprocal lattice vectors are labelled $\boldsymbol{g_1}, \:\boldsymbol{g_2}$, and the high symmetry points are $\gamma$, $\kappa_+$, $\kappa_-$ and $\mu$. The real space unit cell [panel $(b)$] shows the lattice vectors $\boldsymbol{L_1}, \: \boldsymbol{L_2}$ and maximal Wyckoff positions $1a,\: 1b, \: 1c$.
• Figure 3: Electronic (non-interacting) dispersion at $\theta = 0.77^\circ$ for displacement field $u_D = 20 \:\mathrm{meV}$ and $u_D = -20 \:\mathrm{meV}$. The $C_3$ symmetry eigenvalues are shown at the high symmetry points in terms of $\omega = \exp \left(\mathrm{i} 2\pi/{3}\right)$.
• Figure 4: Electronic Berry curvature for the occupied ($v$) and empty band ($c$) for parameters $\theta = 0.77^\circ$ and $u_D = \pm 20\:\mathrm{meV}$.
• Figure 5: Exciton dispersion (in red) with particle-hole continuum above (in blue) for $\theta = 0.77^\circ$$(a)$$u_D = 20 \:\mathrm{meV}$ and $(b)$$u_D = -20 \:\mathrm{meV}$. In both cases the $C_3$ symmetry eigenvalues of the excitons are marked. The dispersion of the excitons throughout the whole BZ for both parameter choices is shown in $(c)$.