Spin Effects in the Local Density of States of GaAs

We present spin-resolved measurements of the local density of states in Si doped GaAs. Both spin components exhibit strong mesoscopic fluctuations. In the magnetic quantum limit, the main features of the spin-up and spin-down components of the local density of states are found to be identical apart from Zeeman splitting. Based on this observation, we introduce a mesoscopic method to measure the $g$-factor in a material where macroscopic methods are severely restricted by disorder. Differences between the spin-up and spin-down components are discussed in terms of spin relaxation due to spin-orbit coupling.