Wavefront depinning in semiconductor superlattices due to discrete-mapping failure

We investigate the wavefronts depinning in current biased, infinitely long semiconductor superlattice systems by the method of discrete mapping and show that the wavefront depinning corresponds to the discrete mapping failure. For parameter values near the lower critical current in both discrete drift model (DD model) and discrete drift--diffusion model (DDD model), the mapping failure is determined by the important mapping step from the bottom of branch $\gamma$ to branch $\alpha$. For the upper critical parameters in DDD model, the key mapping step is from branch $\gamma$ to the top of the corresponding branch $\alpha$, and we may need several active wells to describe the wavefronts.