  We present explicit examples of semi-realistic heterotic models with
spontaneously broken supersymmetry, which dynamically lead to breaking scales
much smaller than $M_{\rm Planck}$ and exponentially small positive values for
the cosmological constant. Contrary to field theoretic intuition, we find that
the global structure of the effective potential is significantly affected by
contributions of massive and non-level matched string states and we investigate
the conditions that dynamically ensure a number of desired properties.
