Tunable magnetic topological insulating phases in monolayer ${\mathrm{CrI}}_3$

We present results indicating that Chern insulator states can be achieved in the recently synthesized pristine chromium triiodide (${\mathrm{CrI}}_3$) by either electron or hole doping. Our first-principles density-functional-theory calculations confirmed that monolayer ${\mathrm{CrI}}_3$ show nontrivial Chern number $C$ in both the valence and conduction bands. By introducing on-site Coulomb interaction or epitaxial strain, the doped ${\mathrm{CrI}}_3$ exhibit a series of topological quantum phase transitions between multiple Chern insulator phases as well as semimetal-to-insulator transitions. We show that the covalency of Cr $d$-I $p$ bands controlled by the on-site Coulomb interaction $U$ and strain is one of the key ingredients determining the topological phase and semimetal-insulator phase boundary.

Figure 1

(a) The crystal structure of pristine monolayer chromium triiodide (${\mathrm{CrI}}_3$) without doping. (b) The partial density of states of chromium (Cr)-$d$ states and I-$p$ states. The energy manifolds of the Cr-$t_{2g}$ and $e_g$ states are marked in the figure. (c) The Perdew-Burke-Ernzerhof (PBE) band structure. (d) The PBE+spin orbit coupling (SOC) band structure with Chern number $C$ of each individual band within the energy manifold around the Fermi level is marked according to band color. The two dash-dotted lines labeled by $\mathrm{\Delta }{n}_e=+1$ and $\mathrm{\Delta }{n}_e=-1$ represent two Fermi levels for one electron and hole doping per (${\mathrm{CrI}}_3{)}_2$-unit cell, respectively.

Figure 2

Electronic structure and topological quantum phase transition (TQPT) of hole-doped single-layer ${\mathrm{CrI}}_3$. (a) The TQPT diagram for the Hubbard $U$ versus the epitaxial strain. (b) PBE+SOC band structure at $4\%$ epitaxial strain with total Chern number $C=-2$. (c) The band structure with quantum critical point $U_{c1}=0.48$ eV exhibiting gap closing at $M$. (d) The PBE+$U$+SOC band structure at $\mathit{U}>{\mathit{U}}_{c1}$ with total Chern number $C=1$.

Figure 3

The electronic structure of electron-doped single-layer ${\mathrm{CrI}}_3$. (a) The crystal structure of Na doped ${\mathrm{CrI}}_3$ in the $ab$ plane. (b) Fatband projection of Cr $d$ orbitals in local coordinates under the PBE approximation for Na doping. (c)–(g) The evolution of the HOB and LUB near the Fermi level with increasing Hubbard $U$ under the PBE+$U$+SOC approximation. The gap closing occurs at two $U$ values ${\mathit{U}}_{c1}=0.58$ eV and ${\mathit{U}}_{c2}=1.39$ eV. (h) The evolution of the band gap at the $K$ and $M$ points with the Hubbard $U$.

Figure 4

The topological phase evolution with strong correlations due to band inversion in Na-doped ${\mathrm{CrI}}_3$. (a)–(f). Schematic diagram of the evolution of the HOB and LUB of the Dirac crossings at all $K$ and $M$ points in the Brillouin zone (BZ) under the PBE+$U$+SOC approximation and the Real (Re) and imaginary (Im) parts of the corresponding MOs symmetry of the HOB and LUB are shown. The Fermi level ($E_F$) is indicated by the dashed cyan colored line between the HOMO and LUMO.

Figure 5

Topological properties in Na-doped ${\mathrm{CrI}}_3$. (a) The anomalous Hall conductivity ${\sigma }_z$ as a function of the energy for $\mathit{U}=0$ eV, 1 eV, and 2 eV. (b) The Berry curvature under the PBE+$U$+SOC model, at $\mathit{U}=0$ eV, 1 eV, and 2 eV in the BZ. (c) The edge Green's function for the left edge of the slab under PBE+$U$+SOC at $\mathit{U}=0$ eV, 1 eV, and 2 eV.