Effect of intervalley interaction on band topology of commensurate graphene/EuO heterostructures

Recent experiments demonstrating proximity induced ferromagnetism in graphene motivate this study of commensurate graphene/EuO heterostructures. Due to the commensurability of graphene with the (111)-EuO layer, graphene's Dirac points are mapped to the $\mathbf{\Gamma }$ point of the commensurate Brillouin zone. The Eu atoms not only induce proximity exchange on the graphene layer, but they also introduce intervalley interactions resulting in a nonlinear dispersion at $\mathbf{\Gamma }$. We develop a model Hamiltonian, consistent with the lattice symmetries, that includes proximity induced exchange splitting, spin-orbit coupling, and intervalley interactions with parameters fitted to ab initio calculations. The intervalley interaction opens up a trivial gap preventing the system from crossing into a nontrivial state. The model Hamiltonian is analyzed to determine the conditions under which the heterostructures can exhibit topologically nontrivial bands.

Figure 1

(a) Schematic view of a heterostructure with graphene between two EuO layers. The O layers are terminated with H atoms, and the surfaces abutting the graphene are the Eu (111) planes. (b) The reciprocal lattice corresponding to the unit cells shown in (c) and (d) maps the $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points of the hexagonal graphene Brillouin zone (BZ) indicated by the outer red hexagon to the $\mathbf{\Gamma }$ point of the commensurate BZ of the graphene/EuO unit cell indicated by the central green hexagon. The ${\mathbf{g}}_i$'s are the reciprocal lattice vectors of the heterostructure unit cells in (c) and (d). Elevation and plan views of the unit cells corresponding to the two graphene/EuO geometries are shown in (c) for the Eu-misaligned structure and (d) for the Eu-aligned structure.

Figure 2

Band dispersions in the absence of spin-orbit coupling for different values of m, ${\mathrm{\Delta }}_{\mathrm{ex}}$, and ${\mathrm{\Delta }}_v$. (a) $m>{\mathrm{\Delta }}_v$ and ${\mathrm{\Delta }}_{\mathrm{ex}}<|{\mathrm{\Delta }}_v-m|$, (b) $m>{\mathrm{\Delta }}_v$ and ${\mathrm{\Delta }}_{\mathrm{ex}}>|{\mathrm{\Delta }}_v-m|$, (c) $m<{\mathrm{\Delta }}_v$ and ${\mathrm{\Delta }}_{\mathrm{ex}}<|{\mathrm{\Delta }}_v-m|$, and (d) $m<{\mathrm{\Delta }}_v$ and ${\mathrm{\Delta }}_{\mathrm{ex}}>|{\mathrm{\Delta }}_v-m|$.

Figure 3

Band dispersion with spin-orbit coupling of Eu-misaligned structure calculated from model Hamiltonian $H$ along the path $\mathbf{\Gamma }$ to $\mathbf{K}$ using parameters obtained from ab initio calculations. $\hslash v_F=3.5\mathrm{eV}$Å, ${\mathrm{\Delta }}_{\mathrm{ex}}=80$ meV, $m=48$ meV, ${\mathrm{\Delta }}_v=17$ meV, ${\lambda }_R=5$ meV, and ${\lambda }_I=1$ meV.

Figure 4

The Chern number calculated as a function of ${\mathrm{\Delta }}_{\mathrm{ex}}$ for four different cases of ${\mathrm{\Delta }}_v$ and $m$. The red open circles show the Chern number of the occupied bands, the blue ‘x’ symbols show the Chern number of the unoccupied bands, and the black triangles show the Chern number of the summation of all bands. The Chern number of the occupied bands is 2 for all values satisfying (a) ${\mathrm{\Delta }}_v\ne 0,m=0$ and (b) ${\mathrm{\Delta }}_v>m,m\ne 0$. For condition (c), ${\mathrm{\Delta }}_v=0,m\ne 0$, the Chern number of the occupied bands becomes 2 for ${\mathrm{\Delta }}_{\mathrm{ex}}\ge m$. For this example, $m$ is chosen to be 0.04 eV. (d) For ${\mathrm{\Delta }}_v\ne 0$ and $m>{\mathrm{\Delta }}_v$ (in this example ${\mathrm{\Delta }}_v=0.01$ eV and $m=0.04$ eV), the topological transition is pushed to a higher value of ${\mathrm{\Delta }}_{\mathrm{ex}}=0.165$ eV.

Figure 5

Phase diagram as a function of $m$ and ${\mathrm{\Delta }}_v$ for fixed values of ${\lambda }_R=5$ meV, ${\lambda }_I=1$ meV, and ${\mathrm{\Delta }}_{\mathrm{ex}}=80$ meV. The red triangle in the figure represents the fitted band structure shown in Fig. 3. The curve indicates the phase boundary between a Chern number of 2 on the left side of the curve and 0 on the right side of the curve.

Figure 6

Band structure of the Eu-misaligned structure without spin-orbit coupling. Left inset: close-up of the low-energy band structure of the misaligned structure near $\mathbf{\Gamma }$. Right inset: Band structure of the Eu-aligned structure.

Figure 7

Band structure with spin-orbit coupling of Eu-misaligned structure calculated along the path $\mathbf{\Gamma }$ to $\mathbf{K}$ where $\mathbf{K}$ is 0.57 Å ${}^{-1}$ away from $\mathbf{\Gamma }$.