# Topological Phase Diagram of BiTeX-Graphene Hybrid Structures
## 1 Introduction  
After the first successful isolation of graphene in 2004, graphene has proved to be an excellent template material with outstanding mechanical properties for revolutionary applications [1]. Due to the small atomic weight of the carbon atoms building up the graphene lattice, the effect of spin-orbit coupling (SOC) on the electronic band structure is negligible [2]. This makes graphene a semimetal. There has recently been a strong push to find ways to enhance spin-orbit coupling in graphene [3] in order to enable spintronics applications [4]. Theoretical investigations showed that one can overcome this limitation in several ways. Introducing curvature in the graphene sheet enhances the effect of the spin-orbit interaction up to \(\approx\)17 meV in realistic situations [5]. It has been also shown that impurities influence the SOC in a positive way and can amplify its strength up to \(\approx\)7 meV [2]. A more practical approach, from an engineering point of view, is to use 2D heterostructures in order to introduce spin-orbit interaction into graphene, since the already existing and understood fabricating procedures [6] and van der Waals heterostructures provide more robust control towards technological reproducibility of devices. SOC can be introduced to graphene by van der Waals constituents, which itself is characterized by strong instrinsic spin-oribt interaction. Prospective candidates are bismuth tellurohalides (e.g., BiTeX with X = I, Br, and Cl). The key element of these compounds is bismuth with strong atomic SOC. The structure element of BiTeX is a trilayer with the X-Bi-Te stacking. The already large intrinsic SOC in Bi and the structural asymmetry combined with a large in-plane gradient of the crystal field in this lattice results in giant Rashba spin-splitting semiconductors [7; 8; 9; 10; 11]. A singletrilayer of BiTeI, that shows the Rashba spin-split band structure can be exfoliated from the bulk BiTeI as it was recently shown by [PERSON] et al., using a novel exfoliation technique [12]. It was also shown theoretically that centrosymmetric thin films composed from topologically trivial BiTeI trilayers are quantum spin Hall insulators and properly stacked bulk compounds of BiTeX results in topological insulating phase [13; 14]. In monolayer BiTeI the band gap is increased, compared to the bulk material, still the basic characteristics are preserved, hence it is a good candidate as a SOC inducing component in graphene-based heterostructures, as it was previously proposed by earlier works [15; 16].  
The fact that fabricated devices are usually subject to mechanical stress requires the investigation of the impact of such effects on the electronic properties of the systems. In experimental setups a considerable effort has been made to control and manipulate the electronic and optical properties of novel heterostructures by strain fields [17; 18; 19; 20].  
To realize mechanically driven 2D \(\mathbb{Z}_{2}\) topological insulators, several theoretical proposals have been made. For example, in two-dimensional dumbbell stanene a topological phase can be tuned by compression and in hydrogenated ultra-thin tin films a topological gap can be opened by mechanical distortions [21].  
As we claimed in our previous work, the single sided heterostructures of BiTeX and graphene under appropriate conditions are topological insulators, however we showed evidence for only the BiTeBr-graphene structure [22]. In this manuscript we investigate in details the phase diagram of the BiTeCl-graphene and BiTeI-graphene systems in the level of density functional theory calculations, as well as the BiTeBr-graphene structure. We analyze and discuss the differences of the electronic properties of the heterostructures.