# Topological Phase Diagram of BiTeX-Graphene Hybrid Structures
## 2 Results  
To set up the investigated systems, the experimental characteristics of the BiTeX crystals were used. The in-plane lattice constants are \(a_{\text{BiTeX}}\) = 4.34 A, 4.24 A, 4.27 A for X = I, Br, Cl respectively [23]. Inasmuch as these values are close to a \(\sqrt{3}\times a_{\text{Gr}}\), a \(30^{\circ}\) rotated, \(\sqrt{3}\times\sqrt{3}\) unit cell of graphene was placed on top of the BiTeX layer as a starting point for the calculations. This choice of configuration carries a stress between the two systems due to the lattice mismatch. The scale of discrepancy is \(a_{\text{BiTeX}}/a_{\sqrt{3}\times\sqrt{3}\text{Cr}}=1.86\%,-0.49\%,0.21\%\) for X = I, Br, Cl respectively. Performing a full geometry optimization on the hybrid systems it shows that the crystal structure of graphene is preserved and the lattice constant of BiTeX layer was adjusted. This phenomenon is comprehensible considering the fact that the bonds in the BiTeX crystals have more degrees of freedom to accommodate. Hence, this reasonable strain only effects the BiTeX in the heterostructures, as it may alter the band structure of the BiTeX layers, but it does not influence our main conclusions as we focus on the Dirac cones of the graphene sheet. The supercell is depicted in Figure 1, consisting of six carbon atoms, one bismuth, one tellurium, and one halogen atom. As it was already shown previously, the most stable horizontal configuration is the so called hollow, when the adjacent atom of the BiTeX layer (i.e., Te or X) is positioned above the center of a hexagon of carbon atoms in the graphene layer [12; 16]. The structures in I-Bi-Te-C configurations are characterized by lower total energy than those in the structural model with the Te-Bi-I-C sequence [16]. In order to simulate the effect of tensile uniaxial and uniform strain we stretched the in-plane unit cell vectors parallel to a carbon-carbon bond and let the atomic positions in the constrained unit cell relax. The compressive strain was taken in consideration by reducing the distance between the BiTeX and the graphene layers. In these calculations we did not let the atomic positions relax. As we applied both types of strain to the system we calculated the size of the band gap and the topological index in every case. In Figure 2 we show the corresponding phase diagrams. In relevant cases we include the band structures as well. The sign of the band gap indicates the topological invariant, it is negative if the system is topological and positive otherwise. The first subfigure (a) presents the calculations on the BiTeCl-graphene heterostructure. At the optimal geometry the system is semi-metallic due to the touching altered Dirac cones of graphene as it can be seen in subfigure (I.). As the tensile strain increases, a topologically non-trivial band gap opens in graphene (subfigure (II.)). Turning on the compressive out-of-plane strain, it widens the band gap in the beginning, but later closes it (see the corresponding band structure in subfigure (III.)) and changes its topological favor to trivial. In this scenario, a 50 meV topological and a 120 meV trivial gap can be reached. The second subfigure (b) corresponds to the BiTeBr-graphene system. The same statements can be made with a slightly different energy scale. The behavior of the band gap is similar to the case of the BiTeCl-graphene structure. At fixed tensile strain, applying pressure widens the topological gap in the beginning up to 70 meV, but later closes the gap and the reopened gap is trivial, which can be as high as 140 meV. The third subfigure (c) shows the BiTeI-graphene structure. The effect of in-plane strain is the same as above: it promotes the topological phase. The consequence of the compressive strain is somewhat different. This system turns metallic instead of a trivial insulator due to some non graphene bands reaching the Fermi level as pressure is increased (see subfigure (IV.)). The deviating behavior of the BiTeI-graphene structure can be explained by the different work function of the BiTeX materials.