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First Principles Thermodynamical Modeling of the Binodal and Spinodal Curves in Lead Cite This: Phys

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First Principles Thermodynamical Modeling of the Binodal and Spinodal Curves in Lead Cite This: Phys PCCP View Article Online PAPER View Journal | View Issue First principles thermodynamical modeling of the binodal and spinodal curves in lead Cite this: Phys. Chem. Chem. Phys., 2016, 18, 5005 chalcogenides† Demet Usanmaz,ab Pinku Nath,ab Jose J. Plata,ab Gus L. W. Hart,bc Ichiro Takeuchi,bde Marco Buongiorno Nardelli,bf Marco Fornaribg and Stefano Curtarolo*bh High-throughput ab initio calculations, cluster expansion techniques, and thermodynamic modeling have been synergistically combined to characterize the binodal and the spinodal decompositions features in the pseudo-binary lead chalcogenides PbSe–PbTe, PbS–PbTe, and PbS–PbSe. While our results agree with Received 10th November 2015, the available experimental data, our consolute temperatures substantially improve with respect to previous Accepted 6th January 2016 computational modeling. The computed phase diagrams corroborate that in ad hoc synthesis conditions DOI: 10.1039/c5cp06891f the formation of nanostructure may occur justifying the low thermal conductivities in these alloys. The presented approach, making a rational use of online quantum repositories, can be extended to study www.rsc.org/pccp thermodynamical and kinetic properties of materials of technological interest. 1 Introduction lead chalcogenides to be used extensively in optoelectronic devices such as lasers and detectors, thermophotovoltaic energy For decades, the physical properties of lead chalcogenides have converters, and thermoelectric materials.1,6–10 generated substantial interest in a number of fields, in particular As thermoelectric materials, lead chalcogenides may exhibit À À À for applications in semiconductor technology.1 PbS, PbSe, and electrical conductivities, s, in excess of 2–4 Â 10 4 O 1 cm 1, À PbTe have distinct structural and electronic properties compared thermopowers, S, around 150 mVK 1, and thermal conductivities, À À to III–V and II–VI compounds. These include high carrier mobi- k, on the order of 1–2 Wm 1 K 1. This leads to figures of merit lities, narrow band gaps with negative pressure coefficients, high ZT = sS2/k larger than 1 at high temperatures, T. Such out- dielectric constants, and a positive temperature coefficient.2–4 standing performances are due on the details of the electronic Published on 07 January 2016. Downloaded by University of Maryland - College Park 16/10/2016 03:47:21. In addition, PbS, PbSe, and PbTe were predicted to be weak structure,11–15 and on the ability to dramatically reduce the topological insulators, with a band inversion observed at the thermal conductivity with alloying and nanostructuring.16–19 N point of the distorted body-centered tetragonal Brillouin While pure lead chalcogenides are attractive on their own, their zone.5 These important and varied properties have allowed alloys are even more interesting. The appeal arises from their mechanical and electronic tunability, which can be optimized for 20–22 specific technological needs. For example, the PbTe1ÀxSex a Department of Mechanical Engineering and Materials Science, Duke University, pseudo-binary system has a higher ZT value than its corres- Durham, North Carolina 27708, USA 23,24 b ponding binary forms. Thallium doping in PbTe causes Center for Materials Genomics, Duke University, Durham, NC 27708, USA c Department of Physics and Astronomy, Brigham Young University, Provo, changes in the electronic density of states, increasing the ZT 11 Utah 84602, USA value to 1.5 at 773 K. Similarly, a ZT of 1.3 at 850 K was reported d Center for Nanophysics and Advanced Materials, University of Maryland, 25 for aluminum doped PbSe. Furthermore, Pb9.6Sb0.2Te10ÀxSbx College Park, Maryland 20742, USA e is known to exhibit lower thermal conductivity and a higher Department of Materials Science and Engineering, University of Maryland, 26 À College Park, Maryland 20742, USA ZT than PbSe1 xTex. This is also true for nanostructured f Department of Physics and Department of Chemistry, University of North Texas, (Pb0.95Sn0.05Te)0.92(PbS)0.08, as the low thermal conductivity 27 Denton TX, USA leads to a ZT = 1.5 at 642 K. The properties of this group g Department of Physics and Science of Advanced Materials Program, of pseudo-binaries depend greatly on the atomic details of Central Michigan University, Mount Pleasant, MI 48858, USA the material’s morphology. This can be partially understood h Materials Science, Electrical Engineering, Physics and Chemistry, Duke University, Durham NC, 27708 USA. E-mail: [email protected]; Fax: +1-919-660-8963; in terms of thermodynamical features. The excellent thermo- Tel: +1-919-660-5506 electric performances, for example, were tentatively ascribed to † Electronic supplementary information (ESI) available. See DOI: 10.1039/c5cp06891f the limited miscibility of the components. This gives rise to This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 5005--5011 | 5005 View Article Online Paper PCCP structural inhomogeneities that lower the thermal conductivity The enthalpy of formation, DH, is defined as: without damaging the electronic transport.19 Such control over DH = E(A,B) C À xAEA C À xBEB C , (4) the morphology could be also used to optimize functionalities a c a c a c associated with topological effects. where E , E ,andE are the total energies of compounds (AB)aCc AaCc BaCc In this work we study the phase diagram of lead chalco- (AB)aCc,AaCc and BaCc, respectively. These energies can be genide pseudo-binaries. We predict quantitatively the boundary found from the fully relaxed structures using density functional of the solid solution (the binodal curve that defines the region theory (DFT). of miscibility), as well as the spinodal region. These features are A combination of high throughput ab initio calculations and key for rationalizing and honing synthesis and characterization thermodynamic modelling are used to predict the interaction of optimized systems. To the best of our knowledge, our study parameter, L(x).39–41 The result is a zero temperature approxi- is the first to completely and accurately reports such character- mation of the actual value. The most common method to describe ization. The phase diagram is essential for properly establish- the composition dependent interaction parameter is the Redlich– ing manufacturing processes. There has been only one previous Kister equation,36,37,42 where the interaction parameter is written attempt to model phase diagrams of lead chalcogenides using in a polynomial form: 28 thermodynamic modeling (TM), which predicted consolute Xn LðxÞ¼ L ðÞx À x n: temperature (Tc) values far from those reported in experimental i A B (5) studies.29 The disagreement was attributed to the difficulty i¼0 of an exhaustive exploration of all the possible structural con- We fit this polynomial to the formation enthalpy data obtained figurations for each composition. This difficulty can be rectified from DFT and/or CE calculations. We checked that an n =2 by using the cluster expansion (CE) technique which is a polynomial is enough to obtain a good fit to the data, so that well known method to investigate the energetics of various only L0, L1, and L2 need to be determined: materials.30–33 Here, we built upon the synergy between cluster C 2 expansion (CE) techniques, high-throughput (HT) ab initio L(x) L0 + xL1 + x L2 (6) 32,34,35 calculations, and thermodynamical modeling to find to compute the interaction parameter. acceptable agreement between our Monte Carlo simulations The main computational challenge lies in characterizing (MC) and the available experimental results. many configurations for many compositions. Some authors have attempted to address the issue, by generating, few configura- tions and/or few compositions which is computationally really 2 Methodology demanding28,39–41 Here, we challenged the problem more 2.1 Thermodynamic modeling drastically: by relying on the advantages of a hybrid cluster expansion (CE)- high throughput approach43 which features an Pseudo-binary systems are represented by the formula (A B ) C xA xB a c exhaustive exploration of different configurations for all the (or A B À C) with mole fractions x and x of elements A and x 1 x A B different compositions. Cluster expansion is widely used to describe B respectively, related by x + x = 1. The small letters a A B phase diagrams, in which A and B belong to the same sublattice, by and c represent number of sites per formula.36,37 The Gibbs making the exploration of multiconfigurational space computa- energy of such iso-structural pseudo-binary systems can be tionally affordable.32,44,45 The automatic generation of hundreds Published on 07 January 2016. Downloaded by University of Maryland - College Park 16/10/2016 03:47:21. written as:36,37 or thousands of configurations for whole composition range and G = x G + x G + k T(x ln(x )+x ln(x )) + x x L the prediction of their energy only computing some tens of DFT (A,B)aCc A AaCc B BaCc B A A B B A B A,B:C (1) structures make CE the ideal and most reliable in lattice techni- que to avoid a large set of DFT supercell calculations. The where G and G represent the Gibbs free energy of A C 43,46 AaCc BaCc a c computational efficiency was achieved with HT methods, 47 and BaCc materials. These two variables can be computed at and rational use of online repositories (AFLOWLIB.org). any temperature by fitting available experimental data38 to the polynomial form39 shown in eqn (2): 2.2 Cluster expansion À In the cluster expansion technique, the configurational energy of an G(T)=a + bT + cT ln T + dT 2 + eT 1 + fT3. (2) alloyaswrittenasasumofmany-bodyoccupationvariables{s}:48 X X The third term in the eqn (1) is the entropy of mixing, and the EðsÞ¼J0 þ Jisi þ Jijsij þ ...; (7) last term is the excess energy of mixing that parameter can take i ij negative or positive values.
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