Being a true two-dimensional crystal, graphene possesses a lot of exotic properties that would enable unique applications. Integration of graphene with inorganic semiconductors, e.g. silicon carbide (SiC) promotes the birth of a class of hybrid materials which are highly promising for development of novel operations, since they combine the best properties of two counterparts in the frame of one hybrid platform. As a specific heterostructure, graphene on SiC performs strongly, dependent on the synthesis method and the growth modes. In this article, a comprehensive review of the most relevant studies of graphene growth methods and mechanisms on SiC substrates has been carried out. The aim is to elucidate the basic physical processes that are responsible for the formation of graphene on SiC. First, an introduction is made covering some intriguing and not so often discussed properties of graphene. Then, we focus on integration of graphene with SiC, which is facilitated by the nature of SiC to assume graphitization. Concerning the synthesis methods, we discuss thermal decomposition of SiC, chemical vapor deposition and molecular beam epitaxy, stressing that the first technique is the most common one when SiC substrates are used. In addition, we briefly appraise graphene synthesis via metal mediated carbon segregation. We address in detail the main aspects of the substrate effect, such as substrate face polarity, off-cut, kind of polytype and nonpolar surfaces on the growth of graphene layers. A comparison of graphene grown on the polar faces is made. In particular, growth of graphene on Si-face SiC is critically analyzed concerning growth kinetics and growth mechanisms taking into account the specific characteristics of SiC (0001) surfaces, such as the step-terrace structure and the unavoidable surface reconstruction upon heating. In all subtopics obstacles and solutions are featured. We complete the review with a short summary and concluding remarks.
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