  It has been recently suggested that the coexistence of ferroelectricity and
Rashba-like spin-splitting effects due to spin-orbit coupling in a single
material may allow for a non-volatile electric control of spin degrees of
freedom. In the present work, we compared the structural and ferroelectric
properties of tetragonal and rhombohedral phases of ferroelectric BiAlO$_3$ by
means of density-functional calculations. In both phases, we carefully
investigated Rashba and Dresselhaus effects, giving rise to spin-splitting in
their bulk electronic structure, particularly near the conduction band minimum,
supplementing our first-principles results with an effective $ k\cdot p$ model
analysis. The full reversal of the spin texture with ferroelectric polarization
switching was also predicted. BiAlO$_3$ can therefore be considered as the
first known oxide to exhibit a coexistence of ferroelectricity and
Rashba-Dresselhaus effects.
