Emergence of an upper bound to the electric field controlled Rashba spin splitting in InAs nanowires

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.