Angle-resolved spin noise spectroscopy in semiconductors: Two-state theory and experiments

Traditional spin noise spectroscopy generally does not address spatial spin correlations, which are in fact crucial for an in-depth understanding of spin relaxation. In semiconductors, one requires spin noise measurements at substantial angles from the incident beam to study spatial correlations on the relevant submicron scale. Recording electron spin noise over a wide range of angles via goniometric homodyne detection enables us to recover the spatiotemporal spectra of electron spin fluctuations with virtually no losses in spin noise power in reciprocal space. Applied to a series of insulating <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>n</a:mi></a:math>-doped CdTe samples, the technique permits the separation of various spin motion mechanisms in reciprocal space. We show that cross correlations between localized and delocalized spins are crucial in explaining the observed long-distance spin jumps between neutral donors, and they have a profound impact on the spin relaxation of localized electrons even at temperatures that are low compared to the activation energy. We develop a theoretical two-state model based on Bloch equations that take these cross correlations into account via a release-and-trapping spin transport mechanism. The model is in quantitative agreement with the entire body of our experimental data, where spatiotemporal electron spin noise spectra are measured as functions of temperature, donor density, and magnetic field.