The authors have calculated the energy differences between the four members of the 2*1 family (b(2*1), c(4*2), p(2*2) and p(4*1)) of the Ge (001) surface by taking only electrostatic interactions between the dimers into account. They have found that p(2*2) is the lowest-energy reconstruction at zero temperature. The phase transitions from ordered p(2*2) to disordered (2*1) and ordered c(4*2) to disordered (2*1) are discussed using the authors' energy calculations, the mean-field approximation, the Ising model and Onsagers' exact solution. A second-order phase transition is found at about 200-250 K from an ordered p(2*2) (c(4*2)) dimer reconstruction to a disordered (2*1) dimer reconstruction assuming a charge transfer of 0.08e between the up atom and the down atom of the asymmetric dimer.
Zandvliet, H. J. W., Terpstra, D., & van Silfhout, A. (1991). Reconstructions and phase transitions of the Ge(100) surface. Journal of physics: Condensed matter, 3, 409-415. https://doi.org/10.1088/0953-8984/3/4/003