Elsevier

Calphad

Volume 58, September 2017, Pages 17-24
Calphad

Thermodynamic modeling of the PbX (X=S,Te) phase diagram using a five sub-lattice and two sub-lattice model

https://doi.org/10.1016/j.calphad.2017.05.001Get rights and content

Abstract

Five-sublattice and two-sublattice models have been built to describe the PbX (X=S,Te) semiconductors using the CALPHAD method. The five-sublattice model has three additional sublattices to explicitly model interstitials, electrons, and holes. The experimental literature has been critically reviewed and many high temperature carrier concentration results used in previous assessments have been shown to be unreliable. First-principle calculations for the formation energies of neutral defects and their respective ionization energies have also been incorporated into the models. The models show an improvement over previous assessments of PbTe and are the first time PbS has not been treated as a stoichiometric compound in a CALPHAD assessment.

Introduction

Due to the high demand for energy in today's world, alternative energy resources need to be developed in order to sustain our needs. Thermoelectric (TE) devices provide a viable option, due to their ability to convert waste heat into usable electricity [1]. The measurement of their efficiency is given as a dimensionless figure of merit, ZT, where ZT=S2σT/κ and S,σ, T, and κ are the Seebeck coefficient, electrical conductivity, temperature, and thermal conductivity, respectively. PbTe has been identified as a high ZT material [2] as well as PbS [3], [51]. The Pb(S,Te) system has also been recognized to further increase the ZT due to introduction of precipitates that help scatter heat-carrying phonons [4]. It is clear that this alloy system could lead to new high ZT materials.

Each of the previously mentioned alloys requires optimization of composition and processing parameters for them to be of use. The parameters used to calculate ZT are often intrinsically linked and separate manipulation of each variable is difficult. However, much work has gone into nanostructuring as a method for reducing the thermal conductivity without lowering the electrical conductivity or the Seebeck coefficient. Nucleation and growth as well as spinodal decomposition as a method for nanostructuring was recently investigated in the Pb(S,Te) system [4]. The experimenters were able to vary the concentration of S in order to induce nucleation or spinodal decomposition. Their study determined that nucleation of a second phase was a much more effective method for scattering heat carrying phonons, thereby reducing the thermal conductivity without adversely affecting the electrical conductivity. The barrier between spinodal decomposition and nucleation and growth at a given temperature is determined where 2G/x2=0, which can be calculated if the Gibbs free energy of the material is known. To achieve this, free energies of the two constituent semiconductors is necessary. The CALPHAD method is a well-established technique for building such databases and is a particularly powerful tool for multicomponent systems [5]. It is also a fundamental step in the Integrated Computational Materials Engineering (ICME) process for the optimization of the properties in novel materials [6]. The CALPHAD method is a semi-empirical method that models Gibbs free energy curves to experimental and theoretical information. Recently, the field has seen great success in supplementing experimental information with those obtained through first-principle calculations [7] thereby giving greater physical significance to the models.

The Pb-Te and Pb-S systems have recently been assessed by Bajaj et al. [8] and Huang et al. [9], respectively, however the aim of this paper is to improve the accuracy of the PbX (X=S,Te) semiconductor through a five-sublattice model specifically developed for binary semiconductors [10]. This model can reproduce both phase stability as well as carrier concentrations, which are of the utmost importance for describing electronic materials. This model utilizes both experimental and first-principle calculations and shows a large improvement over previous models. In addition to the 5SL model, a two-sublattice model is also developed for compatibility in multicomponent databases.

Section snippets

Pb-Te

The thermodynamics and off-stoichiometric nature of PbTe has been studied extensively. Due to the extremely narrow homogeneity range of this compound, typical metallographic techniques for measuring the solubility cannot be used. Instead, measurements of the carrier concentration through Hall coefficients, coupled with assumptions of the defects, determine the solubility limits. There are a number of such studies done in this way for PbTe. One of the first such studies was conducted by Brebrick

5-Sublattice model

The CALPHAD method has been used several times to describe the Pb-Te system, notably by Kattner et al. [29] and even more recently by Gierlotka et al. [30], and Bajaj et al. [8]. The crystal structure of PbS and PbTe are both NaCl B1 [31], which is generally modeled using two sublattices, one for each atomic site. Gierlotka modeled PbTe using a two sublattice anti-site model. First-principle calculations indicate this to likely not be the most accurate model, as formation energies for antisite

Results and discussion

The PARROT [40] module in ThermoCalc [41] was used to optimize the parameters V1 to V4 for PbTe and PbS. The temperature independent terms V1 and V3 represent the formation energy of the neutral vacancies on the X and Pb sites, respectively. These values were provided from the authors of [20] as they were calculated in their recent study, but are not explicitly given in the paper due to their high formation energy compared to the other defects. Values for these calculations are found in Table 2

Conclusion

This study has developed new model descriptions for the binary PbTe and PbS semiconductors. The model is different than previous ones used as it explicitly has sublattices dedicated to electrons and holes. An excellent fit to the carrier and phase boundaries has been found for both systems. The PbTe free energies show a marked improvement from previous models and the PbS is new and consistent with the experimental and first-principles data. The carrier concentration has been converted into

Acknowledgements

The authors would like to acknowledge S. Bajaj and J. Snyder for sharing their POP files from their recent PbTe assessment. Ursula Kattner for her mentorship. The authors gratefully acknowledge thermoelectrics research at Northwestern University through the Center for Hierarchical Materials Design (CHiMaD) and financial support from the DARPA SIMPLEX program through SPAWAR (Contract #N66001-15-C-4036). M. Peters was supported by the Department of Defense (DoD) through the National Defense

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