What is known about the minimal absolute value in $\mathbb{Z}[\zeta_n]\setminus\{0\}$?

Question

Here $\mathbb{Z}[\zeta_n]$ is the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the $n$th root of unity. In my mind $\mathbb{Z}[\zeta_n]$ looks like a grid in the complex plane (is that clear/known?). In particular an element $\alpha \in \mathbb{Z}[\zeta_n]\setminus \{0\}$ with minimal absolute value $|\alpha| > 0$ should exists. Are there results about this minimal absolute value $|\alpha|$? In particular are there lower bounds on it with respect to $n$?