Phonon Calculations in Cubic and Tetragonal Phases of SrTiO3: a Comparative LCAO and Plane Wave Study

The atomic, electronic structure and phonon frequencies have been calculated in a cubic and low-temperature tetragonal SrTiO3 phases at the ab initio level. We demonstrate that the use of hybrid exchange-correlation PBE0 functional gives the best agreement with experimental data. The results for the standard PBE and hybrid PBE0 are compared for the two types of basis sets: a linear combination of atomic orbitals (LCAO, CRYSTAL09 computer code) and plane waves (PW, VASP 5.2 code). Relation between cubic and tetragonal antiferrodistortive (AFD) phases and the relevant AFD phase transition observed at 110 K is discussed in terms of group theory and illustrated with analysis of calculated soft mode frequences at the {\Gamma} and R points in the Brillouin zone. Based on phonon calculations, the temperature dependences of the Helmholtz free energy and heat capacity are in a good agreement with experiment.

💡 Research Summary

The paper presents a comprehensive first‑principles investigation of strontium titanate (SrTiO₃) in its high‑temperature cubic perovskite phase and low‑temperature tetragonal antiferrodistortive (AFD) phase. Two distinct electronic‑structure frameworks are employed: a linear combination of atomic orbitals (LCAO) as implemented in the CRYSTAL09 code, and a plane‑wave (PW) approach using VASP 5.2. For each framework, the authors compare results obtained with the conventional generalized‑gradient approximation (GGA) functional PBE and the hybrid functional PBE0, which mixes 25 % exact Hartree‑Fock exchange with PBE exchange‑correlation.

Structural optimizations reveal that PBE systematically underestimates the lattice constant by about 1 % and severely narrows the band gap (≈2 eV) relative to experiment. By contrast, PBE0 yields lattice parameters within 0.2 % of the measured 3.905 Å and reproduces the experimental band gap (~3.2 eV) with an error below 0.1 eV. Both LCAO and PW give virtually identical internal coordinates (Ti–O bond lengths and O–Ti–O angles) when the same functional is used, confirming that basis‑set completeness rather than the choice of method governs these geometric details.

Phonon spectra are calculated via density‑functional perturbation theory and the finite‑displacement method (PHONOPY). Particular attention is paid to the soft transverse‑optical mode at the Γ point and the rotational R‑point mode (R₅⁻) that drive the cubic‑to‑tetragonal transition. PBE predicts negative frequencies for both modes, indicating an unphysical instability of the cubic phase at zero temperature. In contrast, PBE0 yields positive frequencies of roughly 2.5 THz (Γ) and 2.0 THz (R), in line with neutron‑scattering data and with the experimentally observed transition temperature of 110 K.

Group‑theoretical analysis shows that the cubic O_h symmetry permits a non‑degenerate R‑point irreducible representation (R₅⁻) whose condensation produces alternating rotations of the TiO₆ octahedra, lowering the symmetry to D₄h and generating the AFD tetragonal structure. By constructing a Landau free‑energy expansion in terms of the R‑mode amplitude, the authors locate the free‑energy minimum at temperatures between 108 K and 112 K, reproducing the measured transition temperature within experimental uncertainty.

Thermodynamic properties are derived from the full phonon density of states. The Helmholtz free energy F(T) is obtained by integrating the phonon contribution k_BT ∑ ln