  We discuss operators of the type $H = -\Delta + V(x) - \alpha
\delta(x-\Sigma)$ with an attractive interaction, $\alpha>0$, in
$L^2(\mathbb{R}^3)$, where $\Sigma$ is an infinite surface, asymptotically
planar and smooth outside a compact, dividing the space into two regions, of
which one is supposed to be convex, and $V$ is a potential bias being a
positive constant $V_0$ in one of the regions and zero in the other. We find
the essential spectrum and ask about the existence of the discrete one with a
particular attention to the critical case, $V_0=\alpha^2$. We show that
$\sigma_\mathrm{disc}(H)$ is then empty if the bias is supported in the
`exterior' region, while in the opposite case isolated eigenvalues may exist.
