Theory of the Dirac half metal and quantum anomalous Hall effect in Mn-intercalated epitaxial graphene

The prospect of a Dirac half metal, a material which is characterized by a band structure with a gap in one spin channel but a Dirac cone in the other, is of both fundamental interest and a natural candidate for use in spin-polarized current applications. However, while the possibility of such a material has been reported based on model calculations [H. Ishizuka and Y. Motome, Phys. Rev. Lett. 109, 237207 (2012)], it remains unclear what material system might realize such an exotic state. Using first-principles calculations, we show that the experimentally accessible Mn-intercalated epitaxial graphene on SiC(0001) transits to a Dirac half metal when the coverage is $>1/3$ monolayer. This transition results from an orbital-selective breaking of quasi-two-dimensional inversion symmetry, leading to symmetry breaking in a single spin channel which is robust against randomness in the distribution of Mn intercalates. Furthermore, the inclusion of spin-orbit interaction naturally drives the system into the quantum anomalous Hall (QAH) state. Our results thus not only demonstrate the practicality of realizing the Dirac half metal beyond a toy model, but also open up an avenue to the realization of the QAH effect.

Figure 1

Illustration of the quasi-2D inversion operation to characterize the symmetry physics for a transition metal (green ball) atom adsorption above the graphene hexagonal center. Under the quasi-2D inversion operation ($\vec{R}\to -\vec{R}$), the sign changes only for the $\mathbf{x}$ and $\mathbf{y}$ vectors and not for the $\mathbf{z}$ vector, as do their directional cosines $r_i$ ($i=x, y$, and $z$). Within the Slater-Koster approximation, the $p-d$ interaction energies are $\langle p_z\left|H\right|d_{xy}\rangle =r_x{r}_y{r}_z\left(\sqrt{3}{V}_{pd\sigma }-2V_{pd\pi }\right), \langle p_z\left|H\right|d_{x^2-y^2}\rangle =\frac{1}{2}{r}_z(r_x^2-r_y^2)\left(\sqrt{3}{V}_{pd\sigma }-2V_{pd\pi }\right), \langle p_z\left|H\right|d_{z^2}\rangle =r_z\{[r_z^2-(r_x^2+r_y^2)/2]V_{pd\sigma }+\sqrt{3}(r_x^2+r_y^2)V_{pd\pi }\}, \langle p_z\left|H\right|d_{xz}\rangle =r_x\left[\sqrt{3}{r}_z^2{V}_{pd\sigma }+(1-2r_z^2)V_{pd\pi }\right]$, and $\langle p_z\left|H\right|d_{yz}\rangle =r_y\left[\sqrt{3}{r}_z^2{V}_{pd\sigma }+(1-2r_z^2)V_{pd\pi }\right]$, respectively, where $V_{pd\sigma }$ and $V_{pd\pi }$ are two parameters. These clearly show that the sign change occurs under the quasi-2D inversion operation for $p_z$ hybridization with either $d_{xz}$ or $d_{yz}$ orbitals, meaning symmetry breaking. In contrast, they are unchanged for the other three $d$ orbitals ($d_{z^2}, d_{xy}$, and $d_{x^2-y^2}$), meaning the symmetry invariant $p-d$ interaction.

Figure 2

Geometric and electronic structures of Mn-intercalated epitaxial graphene on SiC(0001). (a) Top and (b) side views of the optimized geometry for Mn intercalation coverage of $\chi =5/12$ ML. The blue rhombus in (a) denotes the supercell. The numbers (1–5) mark the five different Mn atoms. Note that there is only one configuration for adding ${\mathrm{Mn}}_5$ into a $2\times 2$ supercell with $\chi =1/3\mathrm{ML}$. Here, ${\mathrm{Mn}}_3, {\mathrm{Mn}}_4$, and ${\mathrm{Mn}}_5$ atoms form a trimer. The corresponding spin-resolved band structure along the high-symmetry lines: (c) majority- and (d) minority-spin channels. (e) Total density of states as well as the partial contributions from graphene (G), intercalated Mn, and surface Si atoms. The Fermi level is set to zero (brown dot). Note that spin-orbit coupling is neglected in this calculation.

Figure 3

The $d$-electron occupations of two categories of Mn atoms and the schematic diagram of their electronic levels with respect to the graphene Dirac point. (a) ${\mathrm{Mn}}_1$ to ${\mathrm{Mn}}_4$, fully saturated by three Si DBs. (b) ${\mathrm{Mn}}_5$, atomiclike magnetic impurity. (c) Graphene Dirac cone. The relative energy alignments are deduced from the symmetry analysis in combination with our DFT calculations. The green box denotes those states important for $p-d$ coupling. (d) Schematic illustration of the disorder due to a possibly random distribution of ${\mathrm{Mn}}_5$. For clarity, only graphene and Mn (pink balls) are shown. The blue rhomboid denotes the $2\times 1$ supercell containing two Mn trimers used for test calculations on random disorder. (e) Geometry at $\chi =1/3$ ML, where there exist category (i) Mn atoms underneath the yellow hexagons. Because of this preestablished distribution of Mn, the symmetry breaking cannot be canceled on average and only the one on the sites (in triangle) bonding free to category (i) Mn is anticipated to be suppressed by the random effect.

Figure 4

Band structure including the SOC (top panel) and the Berry curvature (bottom panel) along the high-symmetry line of Mn-intercalated epitaxial graphene on SiC(0001) at $\chi =5/12$ ML. The red circles highlight the SOC gap. The Fermi level is set to zero.