Heteroepitaxy provides an effective path for the synthesis of diamond wafers. After more than 20 years of development, the diamond nucleation and growth technology on iridium substrates has enabled to prepare crystals with a maximum diameter of 3.5 inches, which opens a door to application diamond as ultimate semiconductor in the future chip industry. However, a series of problems that occur on heterogeneous substrates, such as surface nucleation, bias process window, and diamond epitaxial growth, need to overcome from the perspective of growth thermodynamics. In this study, aiming at the key issue how diamond can achieve epitaxial nucleation and growth in chemical vapor deposition atmosphere, a simulation study was carried out on the nucleation and growth process of diamond at the atomic scale based on the first-principle calculation. The results show that the adsorption of C atoms on the surface of the Ir substrate is more stable than that on the bulk phase, which indicates that diamond nucleation can only occur on the substrate surface. The number of C atoms of sp³ hybridization in the amorphous hydrogenated carbon layer increases firstly and then decreases with the increase of ion kinetic energy under ion bombardment, confirming the existence of the ion kinetic energy or bias voltage window in the high-density nucleation of diamond. The interfacial binding energy is the lowest (about -0.58 eV/C) when diamond is epitaxially grown along the Ir substrate, meaning that the interface binding energy is the decisive thermodynamic factor for the epitaxial growth. In conclusion, this study clarifies the thermodynamic mechanism of single crystal diamond epitaxial growth under the bias-assisted ion bombardment, and points out a great significant guidance for the growth of diamond and other carbon based semiconductors.