Reaction Paths for the Pressure-Induced Second-Order Structural Phase Transition of SnO2

Rutile represents an important structural aristotype from which numerous structures can be derived. Experimental investigation of the high-pressure behavior of metal dioxides with the rutile-type of structure (TiO2, SnO2, MnO2, ZrO2, RuO2, PbO2) led to a large body of work in the past 20 years. The above mentioned oxides undergo several structural phase transitions (SPT's) under pressure. In particular, a second-order transition from the tetragonal rutile-type structure (P42/mnm, Z=2) to the metastable orthorhombic CaCl2-type structure (Pnnm, Z=2) has been observed for a series of dioxides: β-MnO2 [1], RuO2 [2], SnO2 [3], PbO2 [4] and SiO2 [5,6]. In general, the nature of this transition implies an accurate study in order to reproduce the progressive displacement of atoms from their positions within the cell of the higher simmetry phase. Important results have recently been obtained on SPT in stishovite SiO2, which are of great interest in the field of geology [7]. The present study reports a first attempt to study and compare the rutile-type (i.e. cassiterite) and CaCl2-type phases of SnO2 on the basis of an ab-initio restricted Hartree-Fock (RHF) approach. Unlike first order SPT's, the form of the energy surface calculated for displacive transitions was found to pose severe limits to the validity of the usual sequential univariate minimization procedure which is performed to find the energy minimum at each volume. In particular, due to the flat energy surface in the space defined by the structural parameters, the minimization of the energy in the pressure range where an SPT was expected to occur, was not achieved after several cycles, which makes a computational study of SPT's by ab-initio SCF methods unfeasible. To overcome this problem, the study of phase transition was carried out by setting a sequence of intermediate phases between the CaCl2type of structure and cassiterite and finding an "average" order parameter which allowed to interpolate the behavior of structural parameters between the two extremes. The computational results predict a behavior which well compares with recent experimental data [3]. The cassiterite phase of SnO2 has a tetragonal unit cell and two formula units per cell, while the CaCl2-type of structure crystallizes in the orthorhombic system. From a structural point of view, the CaCl2-type of structure represents a orthorhombic distortion of rutile, which occurs along with a rotation of rigid columns of edge-sharing MO6 (M=metal) octahedra aligned along the [001] axis [8,9]. First principle calculations were performed on a IBM Power PC workstation using the CRYSTAL code [10,11]. Pseudopotentials of the Durand type were used to simulate the electronic contribution from core levels of both tin and oxygen atoms, while GTOís were introduced to model the outer-shell electrons [12]. The sequential univariate search was followed for the optimization of equilibrium geometries [13]. This method changes one coordinate at a time and cycles over all the coordinates one or more times until convergence in energy is reached. With this procedure the total energy per unit cell as a function of volume for the two phases was studied. Convergence in energy for each SCF cycle was established by setting a convergence threshold on the total energy of 1x10-5 Hartree/cell. Threshold for convergence of a full set of parameters ( the volume parameters a, b, and c, and the internal coordinates xO and yO) was fixed at 5x10-4 Hartree/cell. Close to the transition pressure, the energy surface was very flat and differences in energy between successive cycles were close to the numerical error, which did not allow us to effectively trace the structural changes in the displacive phase transition. Different strategies can be envisioned to overcome this problem. One of them consists of pointwise sampling an energy functional depending on the unit cell volume and a parameter, k, which allows to linearly interpolate, for each volume, the structural parameter values between those of cassiterite and of CaCl2-type of structure, so defining a set of intermediate phases labelled by k. Since a second order phase transition is considered, the structural parameters must change continuously from the cassiterite to the CaCl2-type of structure. This strategy assumes that the structural parameters are independent on each other and change linearly with k. This parameter is related to the structural parameter qi by