  In III-V semiconductors it was shown theoretically that under optical cooling
the nuclear spin polaron bound to neutral donors would form below some critical
nuclear spin temperature TC [I. A. Merkulov, Phys. Solid State 40, 930 (1998)].
The predicted critical behavior is a direct consequence of the use of the
mean-field approximation. It is known however that in any finite size system a
critical behavior must be absent. Here we develop a model of the optically
cooled nuclear polaron, which goes beyond the mean-field approximation. An
expression of the generalized free energy of the optically cooled nuclear
polaron, valid for a finite, albeit large, number of spins, is derived. This
model permits to describe the continuous transition from the fluctuation
dominated regime to the saturation regime, as the nuclear spin temperature
decreases. It is shown that due to the finite number of nuclear spins involved
in the polaron, the critical effects close to TC are smoothed by the spin
fluctuations. Particularly, instead of a divergence, the nuclear spin
fluctuations exhibit a sharp peak at TC, before being depressed well below TC.
Interestingly, the formation of the nuclear polaron can, in certain conditions,
boost the nuclear polarization beyond the value obtained solely by optical
pumping. Finally, we suggest that the nuclear polaron could be detected by spin
noise spectroscopy or via its superparamagnetic behavior.
