## 2 Morphology and Interface Structure of Epitaxial Graphene on SiC (0001)  
Graphene growth by thermal decomposition of hexagonal SiC relies on the interplay of three mechanisms: (i) the preferential Si sublimation from the topmost SiC layers, leaving an excess of carbon, (ii) the diffusion of these C atoms on the SiC surface and (iii) their reorganization in the 2D hexagonal graphene lattice. These mechanisms depend both on the annealing conditions (sample temperature and gas partial pressures in the chamber) and on the SiC surface termination. To date most of the studies on graphene growth have been carried out on the Si terminated (0001) face [40; 41; 42; 43] and on the C terminated (000-1) face [40; 41; 42; 43], due to the availability of large area SiC substrates with these orientations. Some experiments on the non-polar faces (1-100) and (11-20) have been recently reported as well [44]. Graphene films with very different structural and electronic properties (defectivity, thickness homogeneity, doping and mobility) have been obtained on these different crystal orientations, as a result of the different surface reconstructions during thermal decomposition. Multilayers of graphene rotationally disordered to each other and with respect to the substrate are typically obtained on the (000-1) face. On the contrary, monolayer or Bernal stacked few layers of graphene showing single crystalline epitaxial alignment with the SiC substrate are normally achieved on the (0001) face. This is due to the specific growth mechanism, mediated by the formation of an interfacial carbon layer, the so-called buffer layer (BL), with partial sp\({}^{3}\) hybridization with the Si face [45]. Figure 1 schematically depicts the structure of the BL, which is covalently bonded to Si atoms of the substrate with a large density of dangling bonds, and to the topmost monolayer of graphene with Van der Walls forces. This peculiar interface structure makes EG compressively strained, and the electrostatic interaction with the dangling bonds at the BL/SiC interface is responsible for a high n-type doping (approx. 10\({}^{13}\) cm\({}^{-2}\)) of the overlying EG [46; 47]. Although thermal decomposition of the (0001) face allows a good control of the number of layers, achieving uniform 1 L graphene coverage on the entire SiC surface remains one of the main challenges in this research field. Besides having an impact on the EG electronic properties, thickness inhomogeneities of EG have also a critical impact on the nucleation of insulating films grown by ALD, as discussed in the Section 3. The EG thickness uniformity on the (0001) face depends on the growth conditions (temperature and pressure) and on the substrate morphology, in particular the miscut angle, with better uniformity achieved for low miscut angle (nominally \"on-axis\") SiC. Under the ultra-high-vacuum (UHV) condition, EG formation has been achieved at temperatures as low as 1280 \({}^{\circ}\)C [3], but the topography and thickness distribution of the EG film was typically very inhomogeneous (composed of submicrometer 0 L, 1 L and 2 L patches) as shown by the atomic force microscopy (AFM) and low energy electron microscopy (LEEM) maps in Figure 2a,b. On the other hand, growth in inert gas (Ar) ambient at atmospheric pressure (approx. 900 mbar) allows it to greatly reduce the Si sublimation rate, raising the EG formation temperature at values as high as T = 1650 \({}^{\circ}\)C [3]. As shown in Figure 2c,d, the EG grown on \"nominally\" on-axis SiC(0001) is commonly composed of 1 L domains on the planar (0001) SiC terraces, separated by long and narrow 2 L or 3 L EG stripes at SiC step edges. Such steps are inherent of SiC crystal and their spacing is related to the miscut angle [48]. The preferential formation of 2 L and 3 L EG at their edges is related to the enhanced Si-desorption from these locations due to the weaker bonding in the SiC matrix.