Power devices are key components in power electronic systems and are widely used in aerospace, electric vehicles, high-voltage direct current (HVDC)/flexible alternating current transmission systems (FACTs), AC/DC motor drives, and household appliances. As silicon is limited by its narrow bandgap and low critical electric field, the performance of silicon-based power devices is approaching the theoretical material limit. Owing to its wide bandgap, high critical electric field, excellent thermal stability, and high carrier saturation velocity, gallium nitride (GaN) has emerged as a leading candidate material for next-generation power devices. Enabled by advances in free-standing n-type GaN substrates, fully vertical GaN devices have achieved rapid progress, featuring high current capability, high breakdown voltage, a compact chip footprint, and superior 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 based on the drift-diffusion equations is developed for a vertical GaN SBD. The effects of the drift-layer doping concentration on the forward and reverse characteristics are quantitatively analyzed, and the forward conduction behavior at various ambient temperatures is investigated to elucidate the temperature-dependence behavior of the J-V characteristics. Furthermore, a merged pn-Schottky (MPS) structure is proposed, 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 insights and practical design guidelines for optimizing high-performance vertical GaN power diodes.