  We consider (AlAs)_n/(GaAs)_n superlattices with random thickness
fluctuations Delta-n around the nominal period n. Using three-dimensional
pseudopotential plane-wave band theory, we show that (i) any amount Delta-n/n
of thickness fluctuations leads to band-edge wavefunction localization, (ii)
for small Delta-n/n the SL band gap is pinned at the gap level produced by a
single layer with ``wrong'' thickness n + Delta-n, (iii) the bound states due
to monolayer thickness fluctuations lead to significant band-gap reductions,
(iv) <001> AlAs/GaAs SL's with monolayer thickness fluctuations have a direct
band gap, while the ideal <001> SL's are indirect for n<4.
