Supplement Number 1

The content includes: the thermionic emission model formulas and the equations required for calculating the interface states of GaN (Eqs. (S1-S3)); The formula required for calculating the dislocation density of GaN single crystals ((Eqs. (S4-S5)); Decay time constant calculation formula ((Eqs. (S6-S7)); The circular-like ink marks left by the ink-stained indium ball and tungsten probe on the glass micron scale ruler (Fig. S1); The I-V curve of the metal (indium and tungsten)/GaN heterojunction, as well as the calculation results of the interface states of GaN (Fig. S2); The dislocation density of GaN single crystals with different lattice orientations was characterized by the cathodoluminescence microscopy method (Figs. S3-S7); The dark currents of graphene/h-BN/GaN heterojunctions fabricated on GaN with different lattice orientations (Table SI).