  We give a new construction of $(\varphi, \hat G)$-modules using the theory of
prisms developed by Bhatt and Scholze. As an application, we give a new proof
about the equivalence between the category of prismatic $F$-crystals in finite
locally free $\mathcal{O}_\Delta$-modules over $(\mathcal{O}_K)_\Delta$ and the
category of lattices in crystalline representations of $G_K$, where $K$ is a
complete discretely valued field of mixed characteristic with perfect residue
field. We also generalize this result to semi-stable representations using the
absolute logarithmic prismatic site defined by Koshikawa.
