In this paper, physics-based analytical models using two-dimensional (2D) Poisson equations for surface potential, channel potential, electric field, and drain current in AlN/$beta$-Ga$_2$O$_3$ high electron mobility transistor (HEMT) is presented. The analytical expression of different quantities is achieved based on full depletion approximation of the AlN barrier layer and polarization charge induced unified two-dimensional electron gas (2DEG) charge density model. For the validation of the developed model, results are compared with 2D numerical simulation results, and a good consistency is found between the two. The drain current model is also validated with experimental results of a similar dimension device. The developed model can be a good reference for different $beta$-Ga$_2$O$_3$-based HEMTs.
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