Ab initiostudy of the ferroelectric transition in cubicPb3GeTe4

In the substitutionally disordered narrow-gap semiconductor ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te},$ a finite-temperature cubic-to-rhombohedral transition appears above a critical concentration $x\approx \mathrm{0.005.}$ As a first step towards a first-principles investigation of this transition in the disordered system, a (hypothetical) ordered cubic ${\mathrm{Pb}}_3{\mathrm{GeTe}}_4$ 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 ${\mathrm{Pb}}_3{\mathrm{GeTe}}_4$ 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 620 K for the cubic ${\mathrm{Pb}}_3{\mathrm{GeTe}}_4$ model, compared to the experimental value of $T_c\approx 350\mathrm{K}$ for disordered ${\mathrm{Pb}}_{0.75}{\mathrm{Ge}}_{0.25}\mathrm{Te}.$ Generalization of this analysis to the disordered ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ system is discussed.