Importance of Coulomb correlation on the quantum anomalous Hall effect in V-doped topological insulators

The presence of the quantum anomalous Hall effect in a V-doped topological insulator (TI) has not yet been understood from band-structure studies. Here, we demonstrate the importance of including the correlation effect in density-functional-theory (DFT) calculations, in the format as simple as the Hubbard $U$, for the determination of the topological properties of these materials. Our results show that the correlation effect turns a V-doped TI thin film into a Mott insulator and facilitates it entering the quantum anomalous Hall phase. Even the ferromagnetic ordering is also strongly affected by the inclusion of the $U$ term. This work satisfactorily explains recent experimental observations and highlights the essentialness of having the Coulomb correlation effect in DFT studies of magnetic TIs.

Figure 1

(a) Calculated band structures of V-doped (1.0%) $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ with different $U$ values. (b) The spin density of V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ ($U=0$) in the Sb layer with V dopants (left panel) and its line profile corresponding to the shaded regions of the Sb layer (right panel), respectively. (c) The spin density of V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ ($U=0$) in the Te layer with V dopants (left panel) and its line profile corresponding to the shaded regions of the Te layer (right panel), respectively. The color scale covers the range from ($-4.0\times {10}^{-4}$) to $(4.0\times {10}^{-4})e/{\AA }^3$. For the $\mathrm{SOC}+U$ case, the $U$ value is set to 1 eV. (d) Exchange energies as a function of V-V distance for $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$. Positive (negative) values imply ferromagnetic and antiferromagnetic alignments, respectively.

Figure 2

(a) Calculated band structures of V-doped (2.8%) ${\left(\mathrm{Bi},\mathrm{Sb}\right)}_2\mathrm{T}{\mathrm{e}}_3$ and $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ in a 4-QL-slab geometry without the $U$ effect. (b) Calculated band structures of 4-QL V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ without the $U$ effect for varying SOC of V with respect to its native strength. V dopants are placed in the outermost QL. Spin-up (-down) states are marked by red (blue) dots.

Figure 3

Calculated band structure of 4-QL (2.8%) V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ for varying $U$ values. V dopants are placed in the outermost QL. Spin-up (-down) states are marked by red (blue).

Figure 4

Density of states of V $d$ orbitals for varying the $U$ value from 1 to 3 eV in 4-QL (2.8%) V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$.

Figure 5

Calculated band structure of 4-QL (6.3%) V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ for varying the $U$ value. V dopants are located at the outermost QL. The $U$ value of the V atom is indicated at the top of the panels. Spin-up (-down) states are marked by red (blue) dots. Dashed lines are set to the Dirac point of V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ thin films.

Figure 6

(a) The variation of the anomalous Hall conductivity (${\sigma }_{xy}$) upon different $U$ values of the V atom in 4-QL V-doped (6.3%) $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$. (b) Calculated anomalous Hall conductivity (${\sigma }_{xy}$) of 4-QL (6.3%) V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ with the $U$ effect (2 eV). The horizontal dashed line marks the Hall conductivity with a Chern number of 1. (c) Calculated edge state for the semi-infinite boundary of 4-QL V-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$. The vertical dashed line in (b) and the horizontal dashed line in (c) are set to the Dirac point.