Two-dimensional mixed crystals

We show that the long-range positional order in a two-dimensional crystal A1-xBx is destroyed by size-mismatch disorder. The size-mismatch disorder is characterized by an effective temperature kBTD=Kx(1-x)(LB0-LA0)2 associated with the strain energy, where K is a force constant and LB0-LA0 is the length mismatch. We study a model that has a fixed triangular-net topology. By comparison with computer simulations, we show that a linearized small displacement theory is adequate for small size mismatches. The long-range orientational order remains. The positional correlation function decays algebraically, which leads to power-law peaks that replace the Bragg peaks in the diffraction pattern. We argue that this model should provide a reasonable qualitative description of real two-dimensional mixed crystals, in the limit of a small size mismatch.