The objective of this investigation is to study the relaxation behavior of the space charge near injecting electrical contacts on high resistivity semiconductors. The transport of charge carriers in the regime dominated by dielectric relaxation of space charge is not well understood. The materials being investigated are high purity silicon and germanium at low temperatures and gallium arsenide at room temperature, where they are semi-insulating. The transition from low resistivity to high resistivity will be studied using current-voltage and capacitance-voltage techniques to observe the transition in the material from lifetime-dominated to relaxation=dominated conduction. High-resistivity semiconductors become more important as electronic circuits become more highly integrated (increasing the density of devices on a single chip or conversely decreasing the distance between neighboring electronic elements), which increases the necessity to insulate neighboring electronic elements from each other. Use of insulating or semi-insulating materials to store charge also increases the need to understand the dielectric relaxation effects in these materials. The transport physics of these materials is poorly understood, partially ignored, and controversial at the present time. This research aims to provide accurate and realistic data on the transport in semi-insulating materials.