Magnetic proximity induced valley-contrasting quantum anomalous Hall effect in a graphene-${\mathrm{CrBr}}_3$ van der Waals heterostructure

The magnetic proximity effect is an imperative tool to comprehend the valley contrasting quantum anomalous Hall (QAH) effect in van der Waals (vdW) heterostructure. The introduction of magnetic exchange and spin orbit interaction together enables to realize topological phases, with a particular emphasis on an interface can be envisioned towards dissipationless electronics. Herein, the proximity coupled valley contrasting QAH effect is predicted in vdW heterostructure consisting of graphene and a ferromagnetic (FM) semiconductor ${\mathrm{CrBr}}_3$ with the implication of relativistic effect, from the ab initio density functional theory (DFT) simulation. The valley contrasting QAH effect is observed with a nonzero Chern number at a high-symmetry point stemming from Berry curvature and Wannier charge center (WCC), leading to a topologically nontrivial state. The occurrence of strong magnetic proximity coupling between graphene and ${\mathrm{CrBr}}_3$ monolayer is realized intrinsically, from shifting of Hall coefficient value near Fermi level. The opening of the global band gap (178 meV) is observed with the inclusion of spin orbit coupling (SOC). The anomalous Hall conductivity (AHC) demonstrates the presence of two maxima peaked at valley $K^{\prime }$ and $K$. The observation of AHC is mainly dominated by nonzero surface charge and localized potential at the heterointerface due to proximity interaction. The Fermi level is found to be located exactly inside the nontrivial global band gap, which can be tuned effectively by applying the external electric field or by introducing a staggered sublattice potential. This robustness makes experimental fabrication highly favorable for developing a valley contrasting QAH device prototype.

Figure 1

Crystal structure of Gr-${\mathrm{CrBr}}_3$ heterostructure. (a) Top view of atomic structure of Gr-${\mathrm{CrBr}}_3$ heterostructure in supercell $2\times 2\times 1$. (b) Top view of atomic structure of Gr-${\mathrm{CrBr}}_3$ heterostructure in unit cell. (c) Side view of Gr-${\mathrm{CrBr}}_3$ unit cell with interplanar spacing notation of $d=0.13$ Å. The grey, blue, and red balls describe carbon (C), chromium (Cr), and bromine (Br), respectively. (d) Electronic band structure of Gr-${\mathrm{CrBr}}_3$ van der Waals heterostructure in presence of SOC. Zoomed-in picture shows band topology near Fermi level. (e) Density of states (DOS) calculation for Gr-${\mathrm{CrBr}}_3$ heterostructure in presence of SOC. Zoomed-in inset shows the flat states near Fermi level.

Figure 2

(a) Spin-polarized electronic band structure calculation of Gr-${\mathrm{CrBr}}_3$ heterostructure in absence of spin-orbit coupling (SOC). The blue and cyan line depicts spin-up and spin-down states, respectively. Inset shows the zoomed-in image of the Dirac cone near the Fermi level. (b) Projected band structure of Gr-${\mathrm{CrBr}}_3$ heterostructure with the implementation of relativistic effects. Zoomed image shows the splitting of energy level and gapped state at $K$ point. The color bar represents the magnitude of projection (scale in atomic units). (c) Projected density of states (PDoS) calculation of the Gr-${\mathrm{CrBr}}_3$ heterostructure in the presence of SOC. (d) Modulation of the band gap with respect to the external applied electric field effect in the heterostructure. (e) DFT simulated local potential distribution of the Gr-${\mathrm{CrBr}}_3$ heterostructure with a contribution from both ${\mathrm{CrBr}}_3$ monolayer and graphene sheet along $z$ direction.

Figure 3

(a) Calculated Berry curvature of Gr-${\mathrm{CrBr}}_3$ heterostructure along high-symmetry points. The red dashed arrow shows $\mathrm{\Omega }\left(k\right)$ and $-\mathrm{\Omega }(-k)$ (in a logarithmic scale). The inset shows Berry curvature in 2D $k$ plane. (b) Density plots of Berry curvature for the occupied band for the first Brillouin zone of Gr-${\mathrm{CrBr}}_3$ heterostructure. (c) Evolution of Wannier charge center (WCC) for Gr-${\mathrm{CrBr}}_3$ bilayer system in the presence of SOC (d) in absence of SOC. The red solid line is a reference line. Odd number times the reference line is crossed by the evolution line indicating the system to be topologically nontrivial.

Figure 4

(a) Calculation of the Hall coefficient in the Gr-${\mathrm{CrBr}}_3$ heterostructure with respect to the chemical potential at different temperature. (b) The calculated anomalous Hall conductivity as a function of energy. The two dot-dashed lines depict the two valleys. Inset displays scheme of QVH effect in the vdW heterostructure of graphene on monolayer ${\mathrm{CrBr}}_3$.

Figure 5

Modulation of band topology with the application of perpendicular electric field in positive $Z$ direction from 0.1 to 0.5 V/Å in Gr-${\mathrm{CrBr}}_3$ bilayer systems.

Figure 6

Modulation of electronic band structure with the application of perpendicular electric field in negative $Z$ direction from $-0.1$ to $-0.5$ V/Å in the Gr-${\mathrm{CrBr}}_3$ bilayer system.

Figure 7

Spin configuration in the momentum space around $K$ and $K^{\prime }$ for Gr-${\mathrm{CrBr}}_3$ bilayer system.