Particle production at a finite potential step. Transition from Euler-Heisenberg to Klein paradox

Spontaneous pair production for spin-1/2 and spin-0 particles is explored in a quantitative manner for a static tanh-Sauter potential step (SS), evaluating the imaginary part of the effective action. We provide finite-valued per unit-surface results, including the exact sharp-edge Klein paradox (KP) limit, which is the upper bound to pair production. At the vacuum instability threshold the spin-0 particle production can surpass that for the spin-1/2 rate. Presenting the effect of two opposite sign Sauter potential steps creating a well we show that spin-0 pair production, contrary to the case of spin-1/2, requires a smoothly sloped wall.