On an infinite family of cubic fields with explicit fundamental units

For an integer $b\neq0,1$ let $θ$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\mathbb{Q}(θ)$. We exhibit an explicit set of $b$ of positive density for which $η_b=-1/(θ-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $η_b$ is the square of a unit: