Families of two dimensional modular $(\phi ,\mathrm{\Gamma })$-modules

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