A statistical theory of the photoluminescence determination of the band gap energy in nano-crystals and layered materials

Semiconductor nano-crystals as well as two-dimensional nanostructures manifest an abnormal dependence of the optical band gap energy on the temperature. In contrast to bulk systems, in the present case the optical band gap energy increases with temperature or may show a non-monotonic blueshift-redshift behavior. It was suggested that this abnormal temperature behavior is associated with the recombination of excitons and free electron-hole pairs under conditions of quantum confinement. Here, we show that the presence of nanometer lengths and anharmonic interactions makes it necessary to rescale the energy and thermal energy of the material according to the invariance of statistical mechanics on the energy/thermal-energy ratio. In addition, considering the effects of the thermal expansion of the material, we managed to derive a formula for the optical band gap energy. Using this formula, the photoluminescence spectra are accounted for by emphasizing that the exciton and free electron-hole pair recombination is a non-thermal statistical process following a Poisson distribution in which the average value of the energy and the full width at half maximum are both directly related to the optical band gap energy. Our results account remarkably well for several experimental data reported in recent literature. Statistical mechanics explains the abnormal bandgap energy temperature dependence in nanocrystals and 2D materials via energy/thermal-energy ratio invariance.