A surface of a single crystalline semiconductor-carbon alloy layer having a surface normal along or close to a major crystallographic direction is provided by mechanical means such as cutting and/or polishing. Such a surface has naturally formed irregular surface features. Small semiconductor islands are deposited on the surface of single crystalline semiconductor-carbon alloy layer. Another single crystalline semiconductor-carbon alloy structure may be placed on the small semiconductor islands, and the assembly of the two semiconductor-carbon alloy layers with the semiconductor islands therebetween is annealed. During the initial phase of the anneal, surface diffusion of the semiconductor material proceeds to form vicinal surfaces while graphitization is suppressed because the space between the two semiconductor-carbon alloy layers maintains a high vapor pressure of the semiconductor material. Once all semiconductor material is consumed, graphitization occurs in which graphene layers can be formed on the vicinal surfaces having atomic level surface flatness.