Phase-field-crystal modelling of the (2 × 1)–(1 × 1) phase transitions of Si(0 0 1) and Ge(0 0 1) surfaces

Abstract

We propose a two-dimensional phase-field-crystal model for the (2 × 1)–(1 × 1) phase transitions of Si(0 0 1) and Ge(0 0 1) surfaces. The dimerization in the 2 × 1 phase is described with a phase-field-crystal variable which is determined by solving an evolution equation derived from the free energy. Simulated periodic arrays of the dimerization variable are consistent with scanning tunnelling microscopy images of the two dimerized surfaces. The calculated temperature dependence of the dimerization parameter indicates that normal dimers and broken ones coexist between the temperatures describing the characteristic temperature width of the phase transition, TL and TH, and a first-order phase transition takes place at a temperature between them. The dimerization over the whole temperature is determined. These results are in agreement with experiment. This phase-field-crystal approach is applicable to phase transitions of other reconstructed surface phases, especially semiconductor n × 1 reconstructed surface phases.