Solid State Sciences 5 (2003) 47–57 www.elsevier.com/locate/ssscie A general structure model for Bi–Se phases using a superspace formalism Hanna Lind ∗ , Sven Lidin Inorganic Chemistry, Arrhenius Laboratory, Stockholm University, 106 91 Stockholm, Sweden Received 3 June 2002; accepted 3 September 2002 Dedicated to Sten Andersson for his scientific contribution to Solid State and Structural Chemistry Abstract Solid-state synthesis in the Bi–Se system produced both commensurate and incommensurate phases of compositions ranging from Bi 2 Se 3 to Bi 4 Se 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 ¯ 11, with approximate cell parameters of a ≈ 4.2 Å and c ≈ 5.7Å 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.  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. 1. Introduction The binary systems Bi–Se, Bi–Te and Sb–Se have similar phase diagrams with extreme phase richness in a narrow region of composition. With the only exception of cubic (Fm ¯ 3m) BiSe (BiTe) found in a thin film [1], all phases in the systems crystallise at normal pressure and room temperature in rhombohedral or trigonal layer structures. The a parameters for these phases are all highly similar (around 4 Å) but the reported c parameters vary from 10 Å to over 100 Å. Table 1 presents the known phases of the systems. The structures can be described as superstructures of NaCl with stacking of close-packed layers along the z-axis. Conventional illustrations of the structures are combinations of stacking sequences, such as the Bi 2 Se 3 (Bi 2 Te 3 , Sb 2 Se 3 ) structure with a stacking repeat of five layers (Se–Bi– Se–Bi–Se), with van der Waals gaps between the stacks. Another example is the Bi 4 Se 3 structure with a double Bi layer sandwiched between all five-layer stacks of Bi 2 Se 3 . * Corresponding author. E-mail address: hanna@inorg.su.se (H. Lind). Intermediate compositions are generated by the less frequent insertion of double Bi layers. The oxidation states of Bi and Se (+III and -II, respectively) give rise to the five-layer stack of Bi 2 Se 3 in order to preserve the overall charge zero. Gaudin et al. [2] have shown by band structure calculations that metallic Bi is formed at adjacent Bi–Bi layer contact, while Se–Se anionic bonding does not occur, producing a van der Waals-gap instead. Fig. 1 illustrates four compounds in the Bi–Se system. As the composition is changed from the most chalcogen- rich compound Bi 2 Se 3 (Bi 2 Te 3 , Sb 2 Se 3 ) to more metal- rich phases, the c axes and the different space groups vary in an irregular manner, despite the fact that the corresponding diffraction patterns are highly similar. The diverse stacking sequences as a function of composition generate a wide range of lattice repeat distances but the characteristic feature of the diffraction pattern remains, cf. Fig. 2. The preparation of single-phase specimen at specific compositions is combined with long annealing treatments for many phases in the systems [3–5]. This can be explained by the predominant formation of some phases at moderate temperatures but above all by the compositionally very close phases. The Bi–Se system might even be better described as a solid solution of Se in the Bi lattice. The continuous, and clearly composition-dependent, vari- ation of stacking sequences is an indication that all the com- 1293-2558/03/$ – see front matter  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1293-2558(02)00080-8