Moir{\'e}-like superlattice generated van Hove singularities in strained CuO$_2$ double layer

While it is known that the double-layer Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$ (BSCCO) cuprate superconductor exhibits a one-dimensional (1D) incommensurate superlattice (IS), the effect of IS on the electronic structure remains elusive. Following the recent shift of the interest from underdoped to optimum and overdoped phase in BSCCO by increasing the hole doping $x$, controlled by the oxygen interstitials concentration $y$, here we focus on the multiple splitting of the density of states (DOS) peaks and emergence of higher order van Hove singularities (VHS) due to the 1D incommensurate superlattice. It is known that 1D incommensurate wave vector $\textbf{q} = \epsilon \textbf{b}$ (where $\textbf{b}$ is the reciprocal lattice vector of the orthorhombic lattice) is controlled by the misfit strain between different atomic layers in the range $0.209$ - $0.215$ in BSCCO and in the range $0.209$ - $0.25$ in Bi$_2$Sr$_2$Ca$_{1-x}$Y$_x$Cu$_2$O$_{8+y}$ (BSCYCO). This work reports the theoretical calculation of a complex pattern of VHS due to the 1D incommensurate superlattice with large 1D quasi-commensurate supercells with the wave vector $\epsilon =9/\eta$ in the range $36 > \eta > 43$. The similarity of the complex VHS splitting and appearing of higher order VHS in a mismatched CuO$_2$ bilayer with VHS due to the moir{\'e} lattice in strained twisted bilayer graphene is discussed. This makes mismatched CuO$_2$ bilayer quite promising for constructing quantum devices with tuned physical characteristics.