Kinetic roughening of the (001) singular surface during steady crystal growth is studied on the basis of a lattice model using the Monte Carlo method. At a sufficiently low temperature, there are known to be two kinetic roughening points as the driving force for crystal growth \(\Delta \mu \) increases. At a low driving force \(\Delta \mu _\text{KPZ}^{(001)}\) , there is the Karder–Parisi–Zhang (KPZ) roughening transition point. On the KPZ rough surface, elementary steps around islands are well defined though the surface is thermodynamically rough, with a roughness exponent \(\alpha \) consistent with the KPZ universal value of 0.3869. Island-on-island structures were found to be crucial in forming the KPZ rough surface. To understand the effects of the atomical roughness of the (001) surface and the interplay of steps on long-period undulations on this surface, the dependence on the temperature T and driving force for crystal growth \(\Delta \mu \) of surface quantities is investigated. At higher temperatures, additional Berezinskii–Kosterlitz–Thouless (BKT) rough and re-entrant KPZ regions are found for large \(\Delta \mu \) , where the crystal surface grows adhesively. A T– \(\Delta \mu \) kinetic roughening diagram is also presented.
