Lithium dendrites formation is an inherent drawback in the development of all solid-state lithium batteries technology. These issues remain unsolved due to the multiple and competing troubleshooting during lithium plating and stripping. The inhomogeneous mechanical and chemical properties of the Solid Electrolyte Interphase (SEI) [1] lead to uneven current distribution and unsmooth lithium plating and stripping [2,3,4]. A fine understanding of the relationship between the SEI composition and the dynamic of dendrite growth under polarization is not well established. All computational studies are based on homogeneous SEI on the top of the lithium electrode [5, 6]. The local deformation of the SEI under polarization may lead to several mechanical defect such as cracking and therefore a dendrite nucleation. To link the crack dynamic to the current density, we developed a novel multiphase-field approach in order to probe the first instants of the dendrite formation. Indicative variables are used in order to define dynamics phases of lithium electrode (in black on the figure), electrolyte (green) and two SEI domains (blue and red) and their interaction. This method allows to consider the SEI as a multi-domains object, where each region has its own (chemical, mechanical and ion transport) properties. Our results show that a smaller current density result in homogeneous deposition (top row on the figure), where a high current leads to a fracture of the SEI and a dendrite growth (bottom row). We highlighted the influence of SEI properties such as the thickness, the ratio between SEI domains or the surface tension on the deposition dynamic.On the left of the figure, the simulation box constituted of lithium (in black color) in contact with a solid electrolyte interphase constituted of a dense (blue) and porous (red) SEI, and electrolyte (green). The dynamic evolution of lithium and SEI, under low and high current density 0.12 and 12 mA/cm² are reported on top and bottom figures respectively.