The authors have calculated the energy differences between the four members of the 2 x 1 family (b(2 x 1), c(4 x 2), p(2 x 2) and p(4 x 1)) of the Ge (001) surface by taking only electrostatic interactions between the dimers into account. They have found that p(2 x 2) was the lowest-energy reconstruction at zero temperature. The phase transitions from ordered p(2 x 2) to disordered (2 x 1) and ordered c(4 x 2) to disordered (2 x 1) were discussed using the authors' energy calculations, the mean-field approximation, the Ising model and Onsager’s exact solution. A second-order phase transition was found at 200 to 250K from an ordered p(2 x 2) (c(4 x 2)) dimer reconstruction to a disordered (2 x 1) dimer reconstruction assuming a charge transfer of 0.08e between the up atom and the down atom of the asymmetric dimer.

Reconstructions and Phase Transitions of the Ge(001) Surface. H.J.W.Zandvliet, D.Terpstra, A.van Silfhout: Journal of Physics - Condensed Matter, 1991, 3[4], 409-15