A general structure model for Bi–Se phases using a superspace formalism

Abstract

Solid-state synthesis in the Bi–Se system produced both commensurate and incommensurate phases of compositions ranging from Bi 2Se 3 to Bi 4Se 3, all crystallising in rhombohedral or trigonal layer structures. The a parameters are very similar for all phases but the c parameters vary irregularly between 10 and 100 Å. A general model for all these phases was developed, using single crystal X-ray diffraction and a four-dimensional superspace formalism. The 4D superspace group is P : R 3 ̄ : m 1 ̄ 1 , with approximate cell parameters of a≈4.2 Å and c≈5.7 Å valid for all phases. The q vector value in this model, ranging from 1.70[001] ∗ to 1.80[001] ∗ is the only parameter that varies with the composition. Two different modelling options were developed for the structures. One of them was a single-atom refinement model consisting of harmonic modulation waves and the other was a two-atom model based on a displacive sawtooth modulation of the Se atom. Both models converged to low R-values ( R all<0.07) and small residual electron density values for all phases. The q vector may be physically interpreted as arising from the average distance between homoatomic layers in the structures.