Thermodynamic modeling of the PbX (X=S,Te) phase diagram using a five sub-lattice and two sub-lattice model
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 and ,, , and 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 , 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
References (50)
Ab initio study of intrinsic point defects in PbTe: an insight into phase stability
Acta Mater.
(2015)- et al.
The compound energy model for compound semiconductors
J. Alloy. Compd.
(1996) - et al.
Relations between the concentration of imperfections in solids
J. Phys. Chem. Solids
(1958) - et al.
Optimization and calculation of the Pb-Te System
Calphad
(1986) - et al.
Thermodynamic description of the Pb-Te system using ionic liquid model
J. Alloy. Compd.
(2009) - et al.
Thermodynamic modelling of solid gallium arsenide
J. Alloy. Compd.
(1995) - et al.
Thermodynamic modeling of native point defects and dopants of GaN semiconductors
J. Electron. Mater.
(2002) - et al.
Thermodynamic modeling of native defects in ZnO
Opt. Mater.
(2013) CALPHAD
(1991)Thermoelectric cooling and power generation
Science
(1999)
Thermopower enhancement in lead telluride nanostructures
Phys. Rev. B
High thermoelectric efficiency of n-type PbS
Adv. Energy Mater.
Spinodal decomposition and nucleation and growth as a means to bulk nanostructured thermoelectrics: enhanced performance in Pb1-xSnxTe-PbS
J. Am. Chem. Soc.
Computational Thermodynamics: the CALPHAD Method
Integrated Computational Materials Engineering: a transformational Discipline for Improved Competitiveness and National Security
First-principles calculations and CALPHAD Modeling of thermodynamics
J. Phase Equilibria Diffus.
Thermodynamic descriptions and phase diagrams for Pb-S and Bi-S binary systems
J. Electron. Mater.
Composition limits of stability of PbTe
J. Chem. Phys.
Composition stability limits of PbTe. II
J. Chem. Phys.
Precipitation of Te and Pb in PbTe Crystals
Phys. Rev.
Control of imperfections in crystales of Pb1-xSnxTe, Pb1-xSnxSe, and PbS1-xSex
J. Nonmetals
Annealing studies of PbTe and Pb1-xSnxTe
J. Appl. Phys.
Philos. Res. Rep.
TEM precipitation studies in Te-Rich As-Grown PbTe single crystals
Phys. Status Solidi
Nonstoichiometry and point defects in PbTe
Cryst. Res. Technol.
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