Power devices are key components in power electronic systems and are widely used in aerospace, electric vehicles, high-voltage DC/ flexible alternating current transmission systems, AC/DC motor drives, and household appliances. Limited by the narrow bandgap and low critical electric field of silicon, the performance of silicon-based power devices is approaching the material limit. Owing to its wide bandgap, high critical electric field, excellent thermal stability, and high carrier saturation velocity, gallium nitride (GaN) has become a promising material for next-generation power devices. With the development of free-standing n-type GaN substrates, fully vertical GaN devices have achieved rapid progress, featuring high current capability, high breakdown voltage, compact chip area, and improved thermal management. Among them, vertical GaN Schottky barrier diodes (SBDs) have attracted considerable attention because of their low forward voltage drop and fast switching characteristics. In this work, an electrothermal physical model for a vertical GaN SBD is established based on the drift-diffusion equations. The effects of drift-layer doping concentration on the forward and reverse characteristics are quantitatively analyzed, and the forward conduction behavior under different ambient temperatures is investigated to clarify the temperature dependence of the J-V characteristics. Furthermore, a merged pn-Schottky (MPS) structure is introduced, and the influences of p-region geometry and doping parameters on the electric-field distribution, forward conduction, and reverse blocking performance are systematically studied. The results provide theoretical support and design guidelines for the optimization of high-performance vertical GaN power diodes.