Multifractal Behavior in Ga$_{1-x}$Mn$_{x}$As Close to the Metal-Insulator Transition*

We have used a low temperature scanning tunneling microscope (STM) to study the metal-insulator transition in Ga$_{1-x}$Mn$_{x}$As. STM topographies reveal a variety of electronic modulations indicating the presence of a high level of disorder. Furthermore, spectroscopic mapping reveals that the local density of states (LDOS) near E$_{F}$ is strongly influenced by e-e correlations. Therefore, Ga$_{1-x}$Mn$_{x}$As provides an opportunity to examine critical behavior of electronic states close to the MIT in a correlated system. We have studied the probability distribution and the multifractal character of the spatial distribution of the LDOS. Near the MIT, LDOS shows a transition from a Gaussian to a lognormal distribution. The multifractal character which was studied by calculating the singularity spectrum shows a trend towards strong multifractality with decreasing Mn doping. Our work demonstrates the fractal nature of wavefunctions in a correlated system, and constitutes the first experimental results to map the critical behavior of states near this quantum phase transition