  Let $F$ be a free group of positive, finite rank and let $\Phi\in Aut(F)$ be
a polynomial-growth automorphism. Then $F\rtimes_\Phi\mathbb Z$ is strongly
thick of order $\eta$, where $\eta$ is the rate of polynomial growth of $\phi$.
  This fact is implicit in work of Macura, but her work predates the notion of
thickness. Therefore, in this note, we make the relationship between polynomial
growth and thickness explicit. Our result combines with a result independently
due to Dahmani-Li, Gautero-Lustig, and Ghosh to show that free-by-cyclic groups
admit relatively hyperbolic structures with thick peripheral subgroups.
