Vibrational contributions to the phase stability of PbS-PbTe alloys

Abstract

The thermoelectric figure of merit $(ZT)$ of semiconductors such as PbTe can be improved by forming nanostructures within the bulk of these materials. Alloying PbTe with PbS causes PbS-rich nanostructures to precipitate from the solid solution, scattering phonons and increasing $ZT$. Understanding the thermodynamics of this process is crucial to optimizing the efficiency gains of this technique. Previous calculations of the thermodynamics of PbS-PbTe alloys [(J. W. Doak and C. Wolverton, Phys. Rev. B 86, 144202 (2012)] found that mixing energetics alone were not sufficient to quantitatively explain the thermodynamic driving force for phase separation in these materials: first-principles calculations of the thermodynamics of phase separation overestimate the thermodynamic driving force for precipitation of PbS-rich nanostructures from PbS-PbTe alloys. In this work, we re-examine the thermodynamics of PbS-PbTe, including the effects of vibrational entropy in the free energy through frozen-phonon calculations of special quasirandom structures (SQS) to explain this discrepancy between first-principles and experimental phase stability. We find that vibrational entropy of mixing reduces the calculated maximum miscibility gap temperature ${T}_{G}$ of PbS-PbTe by 470 K, bringing the error between calculated and experimental ${T}_{G}$ down from 700 to 230 K. Our calculated vibrational spectra of PbS-PbTe SQS exhibit dynamic instabilities of S ions that corroborate reports of low-$T$ ferroelectriclike phase transitions in solid solutions of PbS and PbTe, which are not present in either of the constituent compounds. We use our calculated vibrational spectra to obtain phase transition temperatures, which are in qualitative agreement with experimental results for PbTe-rich alloys, as well as to predict the existence of a low-$T$ displacive phase transition in PbS-rich PbS-PbTe, which has not yet been experimentally investigated.