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A kinetic Monte Carlo (KMC) model coupled with the vapor diffusion above the Al-polar (0001) surface of AlN is constructed for the physical vapor transport (PVT) growth of AlN crystals. Most of the important surface events and the vapor diffusion of Al atoms are taken into account. Based on the numerical simulations, an analytical model of the step-flow growth on the (0001) surface is attained and the time evolution of random terrace widths under homogeneous and linearly inhomogeneous vapor fluxes of Al atoms is explored. Using the KMC model and the analytical model it is found that under the growing conditions in this work the rate limiting step for the PVT growth of AlN is the supply of Al atoms due to the tiny flow of Al atoms in the vapor phase (Alg) at the steady state. The energy barriers for adsorbed AlN (AlNad) incorporated at different configurations of neighboring AlN dimers can influence the growth morphology significantly. If the adsorption rate of Alg is much slower than the rates of the surface events, the step-bunching caused by the randomness of the terrace widths can be avoided under either the homogeneous or linearly inhomogeneous flux of Alg.