Temperature dependence of the band gap of Si28 :P at very low temperatures measured via time-resolved optical spectroscopy

We measure the temperature dependence of the indirect band gap of isotopically purified $^{28}\mathrm{Si}$:P in the regime from 0.1 K to 3 K by high-resolution absorption spectroscopy of the donor bound exciton transition. The measurements increase the up-to-date precision of the temperature-dependent band gap change by more than one order of magnitude and reveal a ${\mathrm{T}}^{4}$ dependence which is about a factor of two less than observed in previous measurements. Such a ${\mathrm{T}}^{4}$ dependence is predicted by theory, but the absolute values differ between our experiment and the most up-to-date calculations by a factor of 30, corroborating that the electron-phonon interaction at low temperatures is still not correctly included into theory. What is more, the ability of such very high-precision band-gap measurements facilitates the use of time- and spatially resolved $^{28}\mathrm{Si}$:P absorption as a contactless, local thermometer and electric field sensor with a demonstrated time resolution of milliseconds.