Ab Initio Study of the Structural Phase Transition in Cubic Pb₃GeTe₄

In the substitutionally disordered narrow-gap semiconductor Pb_{1-x}Ge_xTe, a finite-temperature cubic-rhombohedral transition appears above a critical concentration $x \approx 0.005$. As a first step towards a first-principles investigation of this transition in the disordered system, a (hypothetical) ordered cubic Pb_3GeTe_4 supercell is studied. First principles density-functional calculations of total energies and linear response functions are performed using the conjugate-gradients method with ab initio pseudopotentials and a plane-wave basis set. Unstable modes in Pb_3GeTe_4 are found, dominated by off-centering of the Ge ions coupled with displacements of their neighboring Te ions. A model Hamiltonian for this system is constructed using the lattice Wannier function formalism. The parameters for this Hamiltonian are determined from first principles. The equilibrium thermodynamics of the model system is studied via Metropolis Monte Carlo simulations. The calculated transition temperature, T_c, is approximately 620K for the cubic Pb_3GeTe_4 model, compared to the experimental value of T_c \approx 350K for disordered Pb_{0.75}Ge_{0.25}Te. Generalization of this analysis to the disordered Pb_{1-x}Ge_xTe system is discussed.