The devil's stair case transformation in incommensurate lattices

Abstract

Free energy models describing defects in crystal structures contain necessarily strong anharmonic terms allowing metastable configurations (defectible models). If such a model involves conflicting forces (frustrated model), its response to the relative variation of these forces is qualitatively different from a linear or quasi-linear response, and the structure evolves by discontinuous processes of defects creation or annihilation. These transformations turn out to be described by pathological functions at the macroscopic scale.

A model used for epitaxy and incommensurate structures illustrates these concepts. An elastic chain of atoms is submitted to a periodic modulating potential with a period different from the atomic spacing. This model allows to study the many defects structures (epitaxy dislocations) at fixed concentration or at fixed pressure (or chemical potential) which corresponds to different physical situations.

An incommensurate structure with a given wave-vector is represented by a fixed defect concentration. When the amplitude of a modulating potential increases, the defects structure exhibits a transition from a “fluid” regime (called analytic regime) with a zero frequency phason mode to a locked regime (called non-analytic regime) with a finite gap in the phonon excitation spectrum. Frustration variation in the model corresponds to pressure variation. The wave-vector of the modulated structure (i.e. the defect concentration) varies continuously with piece wise constant parts at each commensurate value (the resulting curve is called a devil's stair case).

When the modulating potential is small enough, the devil's stair case is rather smooth. The locking forces which oppose to any structure change are either null or small. True incommensurate structures (with phasons) are possible with finite probability (this variation curve is called an incomplete devil's stair case). When the modulating potential increases, the devil's stair case becomes steeper. Locking forces appear and oppose to any structure change which results physically into hysteresis associated with this continuous transformation. Commensurate configurations only are obtained (but possibly with high order which appears experimentally as incommensurate). Phasons at zero frequency do not exist (the devil's stair case is then called complete).

Recent experiments on thiourea (S=C−(NH₂)₂) have been found in satisfactory qualitative agreement with the predictions of this model.