Visualizing Critical Spatial Correlations for Electronic States near the Metal-Insulator Transition in Ga$_{1-x}$Mn$_{x}$As

Semiconductors have long been used to study critical phenomena near the disorder-induced (Anderson) metal-insulator transition (MIT). We studied the dilute magnetic semiconductor Ga$_{1-x}$Mn$_{x}$As with dopings near the MIT using low temperature cross sectional scanning tunneling microscopy (STM). This allows us to visualize the electronic states near the Fermi level (E$_{F})$ which display unique critical properties. Suppression of the density of states (DOS) around E$_{F}$ due to electron-electron interactions is observed. In this energy range, the electronic states show a diverging correlation length approaching E$_{F}$, where the suppression of the DOS is strongest. The distance dependence of the correlations at E$_{F}$ is consistent with a power law decay, expected for states near criticality, while away from E$_{F}$ the correlations fall off exponentially. These results highlight the importance of electron-electron interactions and represent some of the first experimental observations of states near the Mott-Anderson MIT, where both disorder and interactions are equally important for the localization of electronic states.