Giant Topological Insulator Gap in Graphene with $5d$ Adatoms

Two-dimensional topological insulators (2D TIs) have been proposed as platforms for many intriguing applications, ranging from spintronics to topological quantum information processing. Realizing this potential will likely be facilitated by the discovery of new, easily manufactured materials in this class. With this goal in mind, we introduce a new framework for engineering a 2D TI by hybridizing graphene with impurity bands arising from heavy adatoms possessing partially filled $d$ shells, in particular, osmium and iridium. First-principles calculations predict that the gaps generated by this means exceed 0.2 eV over a broad range of adatom coverage; moreover, tuning of the Fermi level is not required to enter the TI state. The mechanism at work is expected to be rather general and may open the door to designing new TI phases in many materials.

Figure 1

(a) $4\times 4$ supercell employed to simulate periodic H-site adatoms (cyan) at 6.25% coverage. (b)–(d) Corresponding tight-binding band structures calculated with graphene hopping strength $t=2.7\mathrm{eV}$ and adatom on-site energy $ϵ=-0.5\mathrm{eV}$. The adatom-graphene hopping $t_c$ and adatom SOC ${\Lambda }_{\mathrm{SO}}$ are given by (b) $t_c={\Lambda }_{\mathrm{SO}}=0$; (c) $t_c=1.5\mathrm{eV}$, ${\Lambda }_{\mathrm{SO}}=0$; and (d) $t_c=1.5\mathrm{eV}$, ${\Lambda }_{\mathrm{SO}}=0.2\mathrm{eV}$. When the Fermi level sits at the dashed green line, ${\Lambda }_{\mathrm{SO}}$ generates a giant TI gap.

Figure 2

(a) Density of states for periodic (solid curve) and random (dashed curve) adatoms at 6.25% coverage on a graphene strip with armchair edges along $x$ and periodic boundary conditions along $y$. The parameters are the same as for Fig. 1c. The finite density of states within the bulk gap reflects edge states. Examples of edge states for the periodic and random cases respectively appear in (b) and (c).

Figure 3

(a) First-principles band structure for Os on graphene at 6.25% coverage. The green dashed line indicates the Fermi level. (b) Corresponding PDOS for the Os $5d$ levels. The large gap ${\Delta }_{\mathrm{SO}}$ visible in (a) arises from hybridization between graphene and the spin-orbit-split $d_{xz/yz}$ orbitals, as in our tight-binding model. (c) Coverage dependence of the gap for graphene with Os adatoms (circles) and Cu–Os dimers (triangles).

Figure 4

(a) Magnetic moment $M_s$ for graphene with Os versus external electric field $\epsilon$ applied perpendicular to the graphene sheet (see the inset for the direction of positive and negative $\epsilon$). (b) Band structure of an Os–graphene system with $\epsilon =-0.5\mathrm{V}/\AA$ corresponding to a vanishing moment. The large gap at the Fermi level (green dashed line) thus reflects a true TI phase. (c) Possible diffusion path of a Cu atom or a Cu–Os dimer, beginning from position $E$. The Cu atom ends above an Os atom at position $A$; the Cu–Os dimer ends at position $B$, adjacent to another Cu–Os dimer at position $A$. (d) Energy profile for Cu (circles) and the Cu–Os dimer (triangles) along the diffusion trajectory in (c). The small diffusion barrier evident for the Cu atom indicates that Cu–Os dimers should readily form. In contrast, the $O(\mathrm{eV})$ diffusion barrier for the Cu–Os dimer suggests a suppression of clustering at low coverages, even at room temperature. (e) Band structure for Cu–Os dimers on graphene. Time-reversal symmetry is preserved here even at $\epsilon =0$, although the Fermi level now resides in the valence band. (f) Band structure for Cu–Ir dimers on graphene. This system preserves time-reversal symmetry, eliminates the shift in Fermi level, and also supports a large TI gap. Coverage in (b), (e), and (f) is 6.25%.