  The experimental assessment of the strength ($\alpha_R$) of the Rashba
spin-orbit coupling is rather indirect and involves the measurement of the spin
relaxation length from magnetotransport, together with a model of weak
antilocalization. The analysis of the spin relaxation length in nanowires,
however, clouds the experimental assessment of the $\alpha_R$ and leads to the
prevailing belief that it can be tuned freely with electric field--a central
tenant of spintronics. Here, we report direct theory of $\alpha_R$ leading to
atomistic calculations of the spin band structure of InAs nanowires upon
application of electric field-- a direct method that does not require a theory
of spin relaxation. Surprisingly, we find an {\it upper bound} to the electric
field tunable Rashba spin splitting and the ensuing $\alpha_R$; for InAs
nanowires, $\alpha_R$ is pinned at about 170 meV{\AA} irrespective of the
applied field strength. We find that this pinning is due to the quantum
confined stark effect, that reduces continuously the nanowire band gap with
applied electric field, leading eventually to band gap closure and a
considerable increase in the density of free carriers. This results in turn in
a strong screening that prevents the applied electric field inside the nanowire
from increasing further beyond around 200 kV/cm for InAs nanowires. Therefore,
further increase in the gate voltage will not increase $\alpha_R$. This finding
clarifies the physical trends to be expected in nanowire Rashba SOC and the
roles played by the nano size and electric field.
