# The nuclear polaron beyond the mean-field approximation

## TL;DR

This paper develops a finite-size model of the nuclear polaron in III-V semiconductors, showing how spin fluctuations smooth out the mean-field predicted critical behavior and suggesting experimental detection methods.

## Contribution

It introduces a generalized free energy model for the nuclear polaron beyond mean-field, accounting for finite spin numbers and fluctuation effects.

## Key findings

- Critical behavior is smoothed by spin fluctuations in finite systems.
- Nuclear polarization can be enhanced beyond optical pumping limits.
- Potential detection via spin noise spectroscopy or superparamagnetic behavior.

## Abstract

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.