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A monolayer of the high-$T_c$ superconductor FeTe$_{1-x}$Se$_x$ has beenpredicted to realize a topologically non-trivial state with helical edge modesat its boundary, providing a novel intrinsic system to search for topologicalsuperconductivity and Majorana zero modes. Evidence in favor of a topologicalphase transition and helical edge modes has been identified in recentexperiments \cite{Peng2019}. We propose to create Majorana zero modes byapplying an in-plane magnetic field to the FeTe$_{1-x}$Se$_x$ monolayer and bytuning the local chemical potential via electric gating. Owing to theanisotropic magnetic couplings on edges from a topological band inversion, anin-plane magnetic field drives the system into an intrinsic high-ordertopological superconductor phase with Majorana corner modes, withoutfabricating heterostructures. Furthermore, we demonstrate that Majorana zeromodes can occur at other different locations, including the domain wall ofchemical potentials at one edge and certain type of tri-junction in the 2Dbulk. Our study not only demonstrates FeTe$_{1-x}$Se$_x$ monolayer as apromising Majorana platform with scalability and electrical tunability andwithin reach of contemporary experimental capability, but also provides ageneral principle to search for realistic realization of high-order topologicalsuperconductivity.

High-$T_c$ Superconductor Fe(Se,Te) Monolayer: an Intrinsic, Scalable and Electrically-tunable Majorana Platform