Why graphene growth is very different on the C face than on the Si face of SiC: Insights from surface equilibria and the (3$times$3)-3C-SiC($bar{text{1}}bar{text{1}}bar{text{1}}$) reconstruction

We address the stability of the surface phases that occur on the C-side of 3C-SiC($bar{1} bar{1} bar{1}$) at the onset of graphene formation. In this growth range, experimental reports reveal a coexistence of several surface phases. This coexistence can be explained by a Si-rich model for the unknown (3$times$3) reconstruction, the known (2$times$2)$_{C}$ adatom phase, and the graphene covered (2$times$2)$_{C}$ phase. By constructing an $ab$ $initio$ surface phase diagram using a van der Waals corrected density functional, we show that the formation of a well defined interface structure like the buffer-layer on the Si side is blocked by Si-rich surface reconstructions.