Families of two dimensional modular (φ,Γ)-modules

Let F/ Qₚ be a finite unramified extension, let k be a finite extension of the residue field of F. We provide explicit constructions of integral structures for all rank two \'{e}tale Lubin-Tate (φ, OF×)-modules over k. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of (φ, OF×)-modules. These results reveal new combinatorial structures of the moduli stack of (φ, OF×)-modules, and allow us, in particular, to rederive the fact that the Serre weights assigned to a two dimensional Gal(F̄/F)-representation over k can be read off from the geometry of the stack.