Families of two dimensional modular $(\varphi,\Gamma)$-modules

Let $F/{\mathbb Q}_p$ 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 $(\varphi,{\mathcal O}_F^{\times})$-modules over $k$. We construct algebraic families of such integral structures and show that these comprehensively reflect the degeneration behaviour of $(\varphi,{\mathcal O}_F^{\times})$-modules. These results reveal new combinatorial structures of the moduli stack of $(\varphi,{\mathcal O}_F^{\times})$-modules, and allow us, in particular, to rederive the fact that the Serre weights assigned to a two dimensional ${\rm Gal}(\overline{F}/F)$-representation over $k$ can be read off from the geometry of the stack.