Quantitatively Connecting the Thermodynamic and Electronic

Quantitatively Connecting the Thermodynamic and Electronic Properties of Molten Systems by Charles Cooper Rinzler Submitted to the Department of Materials Science and Engineering in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 c Massachusetts Institute of Technology 2017. All rights reserved.

Author...... Department of Materials Science and Engineering April 9, 2017

Certiﬁed by...... Antoine Allanore Associate Professor Thesis Supervisor

Accepted by...... Donald Sadoway Chairman, Department Committee for Graduate Students 2 Quantitatively Connecting the Thermodynamic and Electronic Properties of Molten Systems by Charles Cooper Rinzler

Submitted to the Department of Materials Science and Engineering on April 9, 2017, in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Abstract The electronic and thermodynamic properties of noncrystalline systems are investigated and quantitatively connected through the application of theory presented herein. The electronic entropy is confirmed to control the thermodynamics of molten semiconductors. The presented theory is applied to predict the thermodynamic properties of the prototypical Te-Tl molten semiconductor from empirical electronic property data and the electronic properties from empirical thermodynamic data. The theory is able to answer a question posed in the literature regarding a correlation between features of phase diagrams and molten semiconductivity. The quantitative connection is extended to predict thermodynamic properties of fusion, and a stability criterion to predict whether a system will behave as a molten semiconductor is developed and verified. The investigation and prediction of electronic transitions, such as metallization of high temperature systems, is enabled by the theory provided herein. The thermodynamic bases for key features of phase diagrams in the molten state are explained and quantified. Methods to rapidly collect electronic and entropy data in the molten phase are provided and enable access to key thermodynamic data for high temperature systems. The connection of electronic entropy to short-range order allows the detection and prediction of solid-phase compounds through the collection of electronic property data in the molten phase and the prediction of thermodynamic quantities of fusion. An absolute reference for entropy at temperatures substantially above 0 Kisproposed.

Thesis Supervisor: Antoine Allanore Title: Associate Professor

3 4 Acknowledgments

For my parents Denise Denton-Rinzler and Richard Rinzler. You gave everything so that I could have a chance at a life of purpose, fulﬁllment, and happiness. You have supported and encouraged me in all things. This PhD, and all success I may have in life, is only possible because of your dedication to my education, your commitment to giving me every opportunity by demolishing every obstacle to my wellbeing, and, most importantly, your support in my development as a human being. I love you and am so thankful to have you in my life.

To my sister, Marina (Mimi) Rinzler, who has believed in me despite my best eﬀorts to dissuade her and has been the best friend through all times of life to an extremely lucky brother. You have taught me more about how to live my best life than you could ever know.

To my mentor, advisor, and friend Professor Antoine Allanore, who has been my thought partner, an intellectual, academic, and moral guide, and who has, along with his family, supported me far beyond any reasonable expectation. Thank you for the opportunity to work, and work with you, on matters that are meaningful, challenging, and rewarding. This thesis is every bit as much yours as it is mine.

The Fannie and John Hertz Foundation has supported my work through a Hertz Fellowship. This generous grant enabled me to engage in research on a high-risk, high- impact subject and to work and collaborate with the ideal advisor. More critically, the fellowship has provided support on all axes (intellectual, academic, professional, and personal) in abundance. The Hertz community has become my family and it has been an absolute honor to be a part of such an incredible group of individuals. Our work together is just beginning.

I would like to thank and acknowledge Professors Eugene Fitzgerald and Jeﬀrey Grossman for actively participating on my committee. Professor Grossman has been a continual source of enthusiasm and perspective for this work. Professor Fitzgerald has kept my eye on the prize while enabling me to make the most out of my time at MIT.

5 Thank you to my colleagues in the Allanore Lab for your friendship, support, and for making the day-to-day and month-to-month of this PhD meaningful and engaging. You will be happy to know that I will no longer have a forum to talk at you about how exciting entropy can be in your lives. A special shout-out to Angelita Mireles, Elissa Haverty, and the whole DMSE administration for being complete rockstars, keeping me sane and on task, and al- ways taking the opportunity to make my life better and my PhD smoother. This department does not exist without you - thank you for all that you do for all of us every day. Finally, thank you to all of my friends for keeping me aﬂoat with copious amounts of love, humor, and (liquid) support. You know who you are. You make my life worth living. And to the ones that encouraged me to get this PhD - this is all your fault...

6 Contents

1Introduction 17 1.1 StructureofthePresentWork ...... 18 1.1.1 Connecting Electronic and Thermodynamic Properties in the MoltenPhase ...... 18 1.1.2 The Role of Entropy at High Temperature ...... 19 1.1.3 Connecting Transport Properties and Entropy: a Quantitative Theory...... 20 1.1.4 Molten Semiconductors as Materials of Focus ...... 20 1.1.5 Extensibility of the Theory to Other Systems ...... 21 1.2 BackgroundonMoltenSemiconductors ...... 22 1.2.1 Electronic Properties of Noncrystalline Systems ...... 22 1.2.2 MoltenSemiconductors...... 22 1.2.3 TheoryofMoltenSemiconductors ...... 24 1.2.4 PreviousApproaches ...... 26 1.2.5 Solidvs. MoltenSemiconductors ...... 32 1.3 Thermodynamics of Molten Semiconductors ...... 33 1.3.1 PredictionofPhaseDiagrams ...... 34 1.3.2 Interpretation of Phase Diagrams of Molten Semiconductor Sys- tems ...... 37 1.4 Connection of Transport Properties to Equilibrium Thermodynamic Variables...... 38 1.4.1 TransportEntropy ...... 38

7 1.4.2 Previous Attempts at Connection ...... 39 1.5 ElectronicEntropy ...... 39 1.5.1 FormsofElectronicEntropy ...... 40 1.5.2 Contribution of Electronic Entropy to Total Entropy . . . . . 41 1.6 Summary ...... 43

2Hypothesis 53 2.1 FeaturesofPhaseDiagrams ...... 54 2.2 Scientiﬁc Gap ...... 56 2.3 Hypothesis ...... 56 2.4 ConsequencesforMaterialsModeling ...... 57 2.5 Framework for Validation of Hypothesis ...... 59 2.6 Summary ...... 59

3TheoryRelatingElectronicEntropytoElectronicProperties 63 3.1 Theory...... 63 3.1.1 Electronic Entropy and Thermopower ...... 63 3.1.2 Formulation for Use of Empirical Data ...... 65 3.1.3 Assumptions Used in Application of Theory ...... 65 3.2 DiscussionofTheoreticalBasis ...... 66

4PredictionofPropertiesofTe-Tl 71 4.1 AppliedModel ...... 71 4.2 Results ...... 72 4.3 Discussion ...... 72

5ExtensionofFrameworktoPredictingThermodynamicQuantities of Fusion 79 5.1 CalculationoftheEntropyofFusion ...... 79 5.2 Results ...... 80 5.3 Discussion ...... 80

8 6ACriterionforMoltenSemiconductivity 85 6.1 Stability Analysis of Molten State ...... 85 6.2 ApplicationtotheTe-TlSystem...... 87 6.3 Discussion ...... 90

7PredictionofMetallizationTemperatureofMoltenSemiconductor Systems 95 7.1 Method ...... 96 7.2 Calculation of the Metallization Temperature of FeS ...... 97 7.3 Calculation of the Metallization Temperature of the Te-Tl system . . 99 7.4 Discussion ...... 99

8PredictionofFeaturesofPhaseDiagrams 103 8.1 Method ...... 105 8.2 Calculation of the Excess Entropy of the Fe-S System ...... 105 8.3 Calculation of the Miscibility Gap of the Fe-S System ...... 106 8.4 Discussion ...... 106

9ExperimentalMethodsandResults 111 9.1 Review of Apparatuses from Previous Researchers ...... 111 9.1.1 QuartzTestCell ...... 112 9.1.2 BoronNitrideTestCell...... 112 9.2 DynamicInductionTestCell...... 113 9.2.1 Apparatus Design ...... 113 9.2.2 Apparatus Performance ...... 116 9.2.3 ResultsforPb-S...... 116 9.3 StaticTestCell ...... 116 9.3.1 Apparatus Design ...... 117 9.3.2 Apparatus Performance ...... 120 9.3.3 Results for Sn-S ...... 120 9.4 DiscussionoftheExperimentalMethods ...... 122

9 10 Extension to Metallic and Ionic Systems 127 10.1 ExtensionofTheorytoMetallicSystems ...... 127 10.2 ExtensionofTheorytoIonicSystems ...... 129

11 Future Research 133 11.1 Extension of Experimental Methods for Measuring the Entropy of Mix- ing to New Systems ...... 133 11.1.1 MoltenSemiconductorSystems ...... 134 11.1.2 Metallic Systems Exhibiting Congruent Melting Compounds . 134 11.1.3 MulticomponentSystems...... 134 11.1.4 Ionic Systems ...... 134 11.2 Integration of Physical Models of Entropy into a CALPHAD Framework135 11.3 Atomistic Modeling of Molten Semiconductors ...... 135

12 Conclusion 139 12.1 DemonstratedConsequencesofTheory ...... 139 12.1.1 Modeling of Molten Semiconductors ...... 139 12.1.2 BeyondMoltenSemiconductors ...... 140 12.2 Potential Impact of Work ...... 140 12.2.1 AbsoluteReferenceforEntropy ...... 140 12.2.2 Predicting Solid Phase Compounds from Liquid Phase Property Data...... 141 12.2.3 Unifying Physics of Electronic Properties Across Phases Through ConnectiontoThermodynamics ...... 142 12.3FinalThoughts ...... 143

AOverviewofSolutionTheory 147

BThermoelectricsOverview 149

CHeuristicArgumentsforTheory 153

DModiﬁedRichard’sRule 159

10 ERelationshipofEnthalpyofMixingtoEnthalpyofFusion 165

11 12 List of Figures

1-1 Electronegativity and electronic behavior ...... 25 1-2 Conductivityvs.temperature ...... 27 1-3 DOS vs. temperature ...... 29

2-1 Notional phase diagrams ...... 54 2-2 Notional thermopower vs. temperature ...... 56

4-1 Entropyofmixingvs.at.%Tl ...... 73 4-2 Thermopowervs.at.%Tl...... 74

5-1 Electronicentropyoffusionofcompounds ...... 81

6-1 PhasediagramoftheTe-Tlsystem ...... 88

6-2 Se vs. Sideal fortheTe-Tlsystem ...... 89

6-3 Se vs. ⇠ fortheTe-Tlsystem ...... 90

7-1 Fe-S phase diagram with metalliazation prediction ...... 98

8-1 Fe-S phase diagram ...... 107

9-1 Dynamic induction test cell ...... 114 9-2 Dynamic induction test cell probe ...... 115 9-3 Thermopowervs. temperatureforPbS ...... 117

9-4 CV of PbS at 1120 Celsius ...... 118 9-5 Static test cell ...... 119 9-6 Sn-S phase diagram ...... 121

13 9-7 EntropyofmixingofSn-S ...... 122

10-1 EntropyofmixingoftheMg-Bisystem ...... 128

D-1 ModiﬁedRichard’sRule ...... 160

14 List of Tables

1.1 MoltenSemiconductorClassiﬁcation ...... 23 1.2 ThermodynamicModelsofFreeEnergy...... 35 1.3 Contributions to Entropy of Mixing ...... 42

15 16 Chapter 1

Introduction

The study of noncrystalline systems is a critical frontier of materials science. Noncrys- talline systems are systems that do not exhibit long-range order, such as amorphous and liquid systems. These systems have applications in materials processing and extraction, heat transfer materials, batteries, photovoltaics, and more. Noncrystalline systems can offer unique benefits such as high temperature operation, tunable electronic and optical properties, and a wide range of mechanical properties. However, the value and benefit of this broad class of materials is limited due to fundamental challenges in modeling and predicting, in particular, the thermodynamic and electronic properties of these systems without appeal to direct empirical evidence. Specifically, quantitative prediction of basic features of the phase diagram (e.g. the liquidus) and qualitative prediction of the electronic nature of a material in the molten phase (i.e. conductor vs. insulator) have been historically intractable. Alcock has described a“revolution”demandedbythemetallurgicalcommunityforapracticaltheoryto “provide readily accessible models for the appraisal of the thermodynamics of multicomponent [systems]” [1]. It has been put forth by Fultz, and other members of the thermodynamics community, that one explanation for the challenges in the development of such models is the inability to accurately model and predict the entropy of high temperature, noncrystalline phases of materials [2].

17 1.1 Structure of the Present Work

1.1.1 Connecting Electronic and Thermodynamic Properties in the Molten Phase

Hensel, in his 1999 monograph "Fluid Metals", discusses the connection of electronic structure to thermodynamic properties, as mediated through atomic structure. Bridg- ing this relationship is a key challenge for materials science. Indeed, he states that “it represents one of the basic problems of modern condensed matter physics” [3].

Hensel investigates metal-to-nonmetal (MNM) transitions in molten metallic systems by linking electronic and thermodynamic properties. The stated goal of his work is the ability to predict the electronic properties of materials from an understanding of the thermodynamics. To achieve this, Hensel introduces “state-dependent interaction” which connects thermodynamic state variables to atomic structure, from which calculations of the electronic structure can be made.

The challenge for the materials science community is to define a formalism that connects states of matter from the solid phase through the plasma phase. The community has built models for the plasma phase, the gas phase, the solid phase, and certain liquid phases (such as weakly-interacting liquid metals). However, inherent in the approach (where each phase has a different state-dependent interaction model) are “trade-offs” due to the inapplicability of, for example, plasma phase models to condensed state models. As stated by Hensel: “A complete solution of the ‘real problem,’ calculation of structure, electronic, and phase behavior over wide ranges of pressures and temperatures starting from realistic atomic properties, lies beyond the present capacity of theory” [3].

Much progress has been made for systems where theory enables the connection of atomic structure to both thermodynamic quantities and electronic properties [4]. For certain metallic and ionic systems, structure-property relations have been developed which enable simultaneous thermodynamic and electronic property prediction (for a given phase). For example, the free energy of alkali metal systems can be calculated

18 from a model of the atomic structure which is informed by analytical relationships between the electronic properties and the atomic structure [5]. There exist broad classes of material systems for which these models do not exist, most especially systems that exhibit strong short-range order (SRO) but no long-range order (i.e. noncrystalline systems). Hensel speciﬁcally identiﬁes molten semiconductors as systems exhibiting complex interactions that are not addressable with today’s theories. Unifying theory across phases and across material systems is still lacking.

It is the explicit goal of this document to describe a theory that quantitatively connects thermodynamic properties with electronic properties without depending on phase- or system-speciﬁc approximations of atomic interaction, thus enabling progress on the “basic problem of modern condensed matter physics” described above.

1.1.2 The Role of Entropy at High Temperature

As the temperature of a system increases, the relative importance that entropy plays in the free energy increases. While for low temperature crystalline systems the enthalpic contributions to the free energy control the thermodynamics of the system, at high temperatures and in long-range disordered systems, entropy can no longer be treated as a perturbation. There is a significant scientific gap in the ability of the thermodynamic community to model and accurately predict the entropy of noncrystalline phases and there is a corresponding gap in the community’s predictive capacity of the thermodynamics of these systems. Further, empirical quantification of entropy is non-trivial, and it has historically been viewed as untenable to measure the absolute entropy of a high temperature system. Thus, thermodynamic models of entropy are neither theoretically rigorous nor directly empirically determinable, presenting a substantial barrier to the application of high temperature noncrystalline systems, such as the molten state.

19 1.1.3 Connecting Transport Properties and Entropy: a Quan- titative Theory

Empirical access to entropy would have dramatic consequences on the ﬁeld. It has been previously suggested that certain transport properties of equilibrium material systems are related to the entropy [6–9]. However, there has not yet been a quantitative, empirically validated theory put forth to enable the use of transport property measurements to inform thermodynamic models, or thermodynamic property data to predict transport properties (see section 1.4.2). Presented herein (Chapter 3) is a quantitative, veriﬁable theory connecting empirically accessible transport properties to thermodynamic properties. Specif- ically, the electronic entropy is hypothesized to be quantitatively connected to electronic transport properties (e.g. thermopower). The theory could enable empirical access to entropy in high temperature, noncrystalline systems and provide a theoretical basis for developing models of entropy for high temperature systems. Further, the theory could enable transport property prediction from thermodynamic datasets. To- gether, the ability to model and predict the thermodynamic and transport properties can allow the faster investigation and broader application of noncrystalline systems.

1.1.4 Molten Semiconductors as Materials of Focus

A particular class of noncrystalline systems, known as molten semiconductors, has proven challenging for the thermodynamic modeling community to accurately model. These systems behave as semiconductors in the molten phase, but do not exhibit the long-range order associated with crystalline phases (see section 1.2.2 for a more detailed description). Their unique electronic properties have been the focus of investigation for 50 years, which has resulted in a detailed set of both thermodynamic and electronic property data for certain representative systems. However, there has been to-date no quantitative theory that allows for the practical prediction of molten semiconducting behavior. It is herein hypothesized (Chapter 2) and validated (Chapter 4) that electronic

20 entropy is a critical thermodynamic function for molten semiconductor systems. The theory presented herein (Chapter 3) enables the prediction of electronic entropy. Molten semiconductors are selected as a focus of investigation for validation of the theory due to 1) availability of relevant property data, 2) the scientiﬁc gap in providing a thermodynamic basis for the high temperature properties of these systems, and 3) the dominant role that electronic entropy plays in the thermodynamics of these systems which enables the more direct assessment of the validity of the quantitative theory. The tellurium-thallium (Te-Tl) system is the most studied molten semiconductor system and property data are available over a broad range of composition. It has been veriﬁed that the Te-Tl system is representative of the broader class of molten semiconductors, and the same physics that determine the properties of Te-Tl control the properties of molten semiconductors as a class [10–12]. Consequently, herein Te-Tl is selected as the system of focus as the archetypal molten semiconductor.

1.1.5 Extensibility of the Theory to Other Systems

Both the thermodynamic and electronic properties of molten semiconductor systems are demonstrably determined by short-range order (SRO) (see section 1.2.3). SRO has been shown to control the electronic properties of a wide range of material systems via structure-property relations [13, 14]. The term ‘molten semiconductor’ refers to a classiﬁcation of systems based on electronic properties. However, the chemical nature of these systems is incredibly broad, including oxides, sulﬁdes, tellurides, selenides, arsenides, antimonides, and more (indeed, nearly all metallic systems experience a metal-to-nonmetal transition at a critical temperature, exhibiting semiconducting and then insulating behavior above the critical temperature [3]). The variety of chemical ordering expressed by molten semiconductor systems is vast, and thus a theory that is applicable to this electronic class of materials, and is based on short-range order (i.e. chemical ordering), may be extensible to systems of distinct electronic class (e.g. insulators and conductors). The applicability of the theory to systems beyond molten semiconductors is ex-

21 plored in Chapter 10, which extends the theory of Chapter 3 to a system that behaves metallically in the molten state and discusses the potential value of extension to ionic systems.

1.2 Background on Molten Semiconductors

1.2.1 Electronic Properties of Noncrystalline Systems

Noncrystalline systems may exhibit a wide variety of electronic properties. Borosil- icate glass, for example, behaves as an insulator while molten silver is an electronic conductor. The breadth of electronic behavior expressed by noncrystalline materials has met with challenge in ﬁnding unifying predictive theories for electronic properties. Particularly problematic are systems that exhibit neither fully insulating nor fully conducting electronic behavior: the noncrystalline semiconductors. Molten semiconductors are equilibrium systems that exhibit a lack of long-range order, but signiﬁcant short-range order. Aquantitativetheorythatpredictsthepropertiesofmoltensemiconductorsis potentially broadly useful to the study of noncrystalline systems. The electronic properties of systems evolve as a function of temperature, pressure, and other thermodynamic variables. Iron oxide, for example, is an insulator at STP, but exhibits semiconductivity above its melting temperature. Silver is an electronic conductor over a wide range of temperatures, but experiences a metal-to-nonmetal transition at high temperatures in the molten phase. Thus, for most material systems of interest, there are regions of the phase diagram where the study of molten semiconductors is relevant.

1.2.2 Molten Semiconductors

The investigation of the semiconducting properties of liquids has a rich history. Molten semiconductors exhibit many similar properties to their solid counterparts including the eﬀect of temperature on electronic conductivity, thermoelectric behav-

22 ior, and optical band gaps [10, 11, 15]. However, not all systems that behave as semiconductors in the solid state retain their semiconducting properties once molten, and the initial eﬀorts to describe these liquid systems sought to understand the relation of the properties of the liquid state to those of the solid semiconductor.

Early studies of these systems resulted in a phenomenological classification of molten semiconductors into three categories: those that experience a semiconductor- to-metal (SC-M) transition upon melting, those that experience a semiconductor-to- semiconductor (SC-SC) transition upon melting, and those that experience a semiconductor- to-semimetal transition upon melting (SC-SM) [14]. The primary differentiating feature of these systems is the impact of temperature on the electronic properties - specifically, the electronic conductivity and thermopower (See Table 1.1).

Table 1.1: Classiﬁcation of molten semiconductor systems according to their electronic conductivity ()andthermopower(↵)inthemoltenphasenearthemelting temperature [11]

1 1 d 1 Transition (⌦ cm ) dT ↵(µV K ) SC-M >5000 - ↵<90 SC-SM 5000 >>500 + 90 <↵<120 SC-SC <500 + ↵>120

While the above classification may seem arbitrary or tautological, empirical evi- dences support this effort and demonstrate that most systems do indeed fall squarely into one of the categories. Physically, the distinction between a molten semimetal and a molten semiconductor is not critical. The different property ranges can be explained by the magnitude of the effective gap in the density of states (DOS, explained below in section 1.2.4). Consequently, within this document the two categories semiconductor-to-semiconductor (SC-SC) and semiconductor-to-metal (SC-M) are used, and semiconductor-to-semimetal (SC-SM) is considered a subset of SC-SC.

23 1.2.3 Theory of Molten Semiconductors

The Role of Short-Range Order (SRO)

The pioneers in the field sought a theoretical description that would support the empirical classification and in 1960, with the publication in the West of a Russian review article by Ioffe and Regel [13], researchers began working on a theoretical description of molten semiconducting behavior. In the article, Ioffe and Regel describe the quintessential connection of short-range order (SRO) to the electronic properties of disordered materials. Theories of solid-state electronic behavior typically had relied upon the existence of long-range order (i.e. crystallinity) to predict features such as band gaps. However, the prescriptive and paradigm-shifting realization of Ioffe and Regel laid the groundwork for a new field of physics: the study of disordered systems.

Sir Nevill Mott built upon the Russian work and created a new framework and theory for the electronic properties of disordered systems, as rigorously described in his 1967 article [16] and 1971 monograph Electronic Processes in Non-Crystalline Ma- terials [14]. Empirical studies of elemental and binary molten semiconductor systems laid the groundwork for a chemical description of the foundation of SRO in semiconducting melts; studies by Bosch [17], Regel [18], Belotskii [19], and others describe qualitatively how the nature of chemical bonding in a system relates to its SRO and hence electronic properties.

Binary systems that exhibit semiconducting behavior tend to be composed of elements of specific electronegativity differences. While difference in electronegativity does not contain sufficient physics to fully describe whether a system will behave as a metal, semiconductor, or insulator, a general trend exists [24]. Figure 1-1 qualitatively outlines the difference in Pauling electronegativity associated with semiconductivity in the liquid state. It should be made clear that this categorization does not accurately capture all systems. Systems with too extreme a difference in electronegativities between constituents tend to behave ionically and act as true insulators, whereas systems with too minimal a difference in electronegativity have a strongly metallic character and fail to exhibit semiconducting properties.

24 FeTe FeS FeO

Pauling Electronegativity Metallic Semiconductor Insulator Difference 0.5 1.5

Figure 1-1: Semiconductor behavior as a function of Pauling electronegativity difference. The classification of metallic, semiconductor, and insulator systems is based on typical solid-state properties. The boxed region indicates the range of electronegativity difference typically associated with molten semiconductivity. Based on data from [10].

Molten semiconductors exhibit a wide variety of chemical ordering and the solid- state compounds correspondingly exhibit a wide range of crystal structures. Belotskii [19] and Glazov [20] both provide descriptions of the chemistry of molten semiconductor systems and an overview of the role of solid-state crystal structure on molten short-range order and electronic properties. Many systems that retain semiconductivity in the molten state exhibit Van der Waals interaction between linear or two- dimensional molecular structures. Upon melting, the Van der Waals interactions are insufficient to maintain long-range order, but the short-range order associated with the formation of molecular structures are retained [19]. However, this is only one mechanism of retention of molten semiconductivity, and systems with a wide variety of solid-state crystal structure may exhibit semiconducting properties in the melt. Rigorous quantitative support for the role of short-range order in molten semiconducting behavior came in the form of neutron scattering data in the 1970s. Bhatia, Thornton, and Hargrove describe a series of three structure factors that describe the SRO of the system. These structure factors can be transformed into the radial pair distribution functions and are measurable via high energy diffraction experiments [21, 22]. Armed with a useful formalism, experimentalists tackled the problem of investigating the evolution of short-range order upon melting and as a function of temperature in liquid state via high energy diffraction [10]. Numerous studies show the degradation of long-range order upon melting in terms of structure factors and/or radial distribution functions, and confirm that systems that exhibit metallization (SC- M) correspondingly exhibit a reduction of short-range order. However, systems that

25 experience a SC-SC transition in fact retain many of the structural features of the solid state [4, 10, 11, 22, 23]. There is general consensus in the ﬁeld on the prescriptive connection of SRO to the semiconducting properties of the liquid state.

1.2.4 Previous Approaches

With strong foundations for the nature of the transition from the solid to the liquid state, efforts to develop an understanding of the electronic behavior of molten semiconductor systems above the liquidus continued in the 1970s and 1980s. It has been found experimentally that systems that experience a SC-SC transition across the liquidus do not retain semiconducting properties indefinitely [11]. At temperatures above the melting point, molten semiconductor systems metallize and experience a loss of semiconductivity [18, 24]. It has been shown empirically that the electronic conductivity of semiconducting systems increases monotonically as a function of temperature until such a point as it reaches what is referred to in the field as the “minimum metallic conductivity," which is the typical electronic conductivity in a metallic system when the mean free path of an electron is of the same order as the interatomic spacing [16]. Further, at sufficiently high temperatures, both SC-SC and SC-M systems experience a metal-to-insulator transition [5, 25–27]. Figure 1-2 shows the regions of behavior for typical SC-SC and SC-M systems as a function of temperature. Molten semiconductors differ from their solid-state counterparts in the impact of defects, dopants, and off-stoichiometry. While part-per-million level defects can induce meaningful electronic property modification in crystalline semiconductors, molten semiconductors show a reduced sensitivity to impurities. Molten semiconductors are most frequently compound systems (with tellurium and selenium excepted) and typically show a minimum conductivity and thermopower at the stoichiometry of acongruentmeltingcompound(acompoundthatmeltshomogeneouslysuchthatthe composition of the liquid phase is identical to that of the solid phase). The conductivity increases as a function of off-stoichiometry. The thermopower typically changes from exhibiting n to exhibiting p type behavior at the composition of the compound.

26 (SC-M)

(SC-SC)

Figure 1-2: Evolution of conductivity of semiconducting and metallizing melts, im- age recreated from [11]. SC-SC systems exhibit an increase in conductivity until a metallization event occurs at the minimum metallic conductivity. SC-M systems exhibit typical metallic conductivity behavior, showing a decrease in conductivity with temperature.

Two primary frameworks were proposed to account for the observed behavior of these systems. The ﬁrst, pioneered by Mott and leveraging the work of Anderson [27, 28], relies upon a description of the band structure of disordered systems [14, 16]. The second, led by Hodgkinson, relies upon a heterogeneous description of the liquid state and leverages Percolation Theory to account for electronic properties [29]. There has been debate about which description reﬂects physical reality, but both frameworks have led to moderate successes in describing the semiconducting properties of the liquid state and both will be described accordingly.

Mott/Anderson - Mobility Edge

The Mott/Anderson model of molten semiconductivity relies on a qualitative description of the evolution of the density of states (DOS) of the system as a function of

27 temperature. Replacing the complete band gap in crystalline solid state devices, Mott suggests the formation of a ‘pseudogap’, or a dip in the electronic density of states for disordered systems exhibiting semiconducting behavior [14, 16]. The lack of long- range order makes the possibility of a true band gap unlikely. However, the notion of localization of electrons within the pseudogap provides an alternative mechanism to create a critical phenomenological feature of semiconducting behavior: the thermal excitation of electrons across a mobility gap. The localization is hypothesized to be Anderson Localization, caused by the mean free path of the electrons being of the same order as the distance between atoms [4, 10, 11]. Thus, while electronic states do exist in the pseudogap, the mobility of electrons in the gap is substantially inhib- ited due to localization eﬀects. A ‘mobility edge’ takes the place of a band edge for disordered systems. As temperature is raised, short-range order is presumed to degrade resulting in a ‘ﬁlling-in’ of the pseudogap such that the semiconducting properties gradually dimin- ish (see Figure 1-3). At the point at which the mobility edges overlap (the critical temperature), thermal activation of electrons to the conduction band ceases to be the dominant mechanism of transport and metallization occurs. The electrical conductivity and thermopower can be modeled by application of the Kubo-Greenwood equations if the DOS of a system is fully characterized [10, 11, 14]. Mott developed temperature-dependent relationships for the electronic conductivity and thermopower of molten semiconductor systems that have been empirically validated [14].

T E 0 = mine kT (1.1)

E0 +(k )T ↵ = (1.2) eµT

min is the minimum metallic conductivity, defined in Section 1.2.4. is a parameter reflecting the temperature dependence of the effective pseudogap of the system. E0 is the magnitude of the difference between the valence edge and Fermi level (typically

28 Figure 1-3: Schematic variation of the electronic density of states (N(E)) as a function of energy (E) vs. temperature. Tm and Tc are respectively the melting and critical temperature. Shaded regions identify pseudogaps in the density of states, and the upper curve represents a fully metallized system. EF represents the Fermi level.

EF /2). µ is the electronic mobility.

While providing qualitative agreement with the data, the Mott formalism has met with substantial challenges that severely limit its utility in providing quantitative descriptions of the liquid state. Most critically, the framework provides no means to predict whether a system will behave as a semiconductor without appeal to direct measurement of the electronic properties. Further, without a continuous definition of the energy-dependent conductivity or density of states, the description of the evolution of a system to metallization, while qualitatively accurate, does not accurately quantify the transition point. This has been repeatedly demonstrated by efforts to apply the formalism to specific material systems [30–34]. Molten semiconducting systems express complex electronic behavior that varies as a function of temperature, and thus simplifying approximations to the full rigor of applying the Kubo-Greenwood equations regularly fail to provide even qualitative agreement with experiment.

29 Hodgkinson - Cluster

An alternative to the Mott/Anderson description of molten semiconductivity relies upon a presupposition of microscopic inhomogeneity of molten semiconductor systems. It is hypothesized that the strong tendency for short-range order in molten semiconductor systems is manifested by the retention of molecular entities which in- homogeneously cluster together in the molten state and reﬂect the stoichiometry of a solid state compound [29]. Thus, microscopic clusters of semiconducting species are present in a dominantly metallic matrix [35]. When the volume fraction of clusters is suﬃciently high (greater than approximately 70%), no continuous path through the metallic matrix is present in the system and conductivity is dominated by the semiconducting element of the heterogeneous system, as described by Percolation Theory [36, 37]. As the temperature is raised, the tendency of molecular entities to cluster degrades [38].

This theory can qualitatively describe the semiconductor-to-metal transition, the change in character of semiconducting behavior from n to p type at the stoichiometry of the solid state compound, and the thermoelectric properties of the system as a function of temperature [33, 39]. Further, as described below, thermodynamic models of molten semiconductor systems may support a description of the liquid state wherein molecular entities reﬂecting the stoichiometry of the solid state compound are present in large concentrations in the liquid state [40]. However, despite its qualitative success, as yet the theory has no means to predict whether a system will behave a semiconductor in the liquid state without direct empirical comparison. Further, the description of the semiconductor-to-metal transition is not quantitative, and does not provide a means to predict the temperature of the transition. This framework has been met with much skepticism, and high energy diﬀraction experiments attempting to resolve the presence of microscopic inhomogeneities have proven inconclusive. It is most likely that this description is useful for a certain subset systems, but not for the general category of molten semiconductors [41, 42].

30 Atomistic Modeling

The advent and rise to prominence of atomistic simulation provided a new tool with which to explore the structure and properties of the liquid state. Several investigators, starting in the 1990’s, began using these tools to examine the behavior of molten semiconductors and the transitions between metallic, semiconductor, and insulator regimes [43]. The complex nature of semiconducting liquid systems confers substantial challenges to the atomistic modeler. Specifically, the absence of long-range order and strong influence of short-range order make the use of traditional, classical poten- tials difficult. Thus, the majority of explorations of the use of computer simulation to describe molten semiconductors have leveraged first-principles, or ab initio, po- tentials, typically within a Density Functional Theory (DFT) framework, coupled with Molecular Dynamics (MD) (Car-Parrinello approach [44]) or Monte Carlo (MC) simulations.

For example, in 1999 Godlevskey performed the ﬁrst ab initio MD simulation of a II-VI system in the liquid state: CdTe [43]. He was able to reproduce the experimental structure factors, transition from semiconductor to metal, and transition from metal to insulator. Shimojo has studied several selenide and telluride systems with ab initio MD [45–47]. He too was able to evaluate the evolution of structure through metallization of molten semiconductor systems. More recently, Akola used ab initio MD to model liquid tellurium and, leveraging high energy x-ray diﬀraction, was able to provide detailed information about the evolution of structural and electronic behavior of the material as a function of temperature that qualitatively agreed with experiment [48].

These and other eﬀorts have validated much of the phenomenological description of the previous decades and have provided direct access to probing the structural evolution associated with molten semiconductor systems and their transitions. Further, the ab initio approach gives a quantum-mechanical description of the electrons that allows for the direct probing of the density of states and band gap of the system. Molecular Dynamics simultaneously provides transport property information, such

31 as diffusivity, which can be related to physical properties of the system of practical interest. However, while atomistic simulation has proven a capstone achievement in con- firming much of the theory and phenomenology of molten semiconductors, there are substantial practical challenges with this approach. The simulations are computation- ally intensive due to the need to perform quantum mechanical calculations at multiple steps in the MD simulation. Consequently, the presence of additional species, and the simulation of multiple concentrations, requires additional simulations, each requiring substantial input and tuning from the modeler. Thus, the ability to leverage atomistic simulation as a tool for screening complex systems for semiconducting potential is reduced by the time and effort intensiveness of the process. Still further, atomistic modeling has proven quantitatively inadequate in the prediction of temperature of critical features such as the liquidus and semiconductor-to- metal transition. Simulations are considered successful when errors are in the 100’s of degrees Celsius range [47, 48]. Thus, while incredibly useful in probing the structural foundation for electronic properties in the liquid state, atomistic modeling has yet to demonstrate itself as a practical tool for quantifying critical thermodynamic and electronic properties of noncrystalline systems.

1.2.5 Solid vs. Molten Semiconductors

It is of interest to outline key connections and differences between the study of liquid systems, and more specifically molten semiconductor systems, and other fields of research. The study of solid-state semiconducting disordered systems, i.e. amorphous semiconductors, has achieved significant results of practical interest in the decades since Mott. Kolomiets, in a well-regarded 1964 review article, discusses the role of short-range order and covalent bonding character in defining the semiconducting properties of solid-state amorphous systems [49]. Indeed, the description of noncrystalline solid-state systems and liquid-state systems are highly complimentary and in fact do not exhibit substantial differences in the source and behavior of electronic properties. This is reflected by Bhatia and Thornton in an article describing the role

32 of short-range order in deﬁning the properties of disordered systems [22], as well as Cutler in his 1977 review of molten semiconductors [12].

However, several practical differences do present themselves when considering the distinction between noncrystalline liquid and solid-state systems. The temperature ranges of liquid systems exceed those of amorphous systems. Molten semiconducting systems are true equilibrium systems, whereas their solid state amorphous counterparts are metastable systems. This has several consequences. First, it is a significant challenge to the experimentalist wishing to study amorphous systems to achieve re- peatable samples due to the influence of thermal history on the structure of system. Second, the presence of thermodynamic equilibrium for liquid state systems allows the use of the full range of thermodynamic modeling to describe the system. While the thermodynamic modeling of equilibrium liquid systems is more immediately addressable by the current methods than that of amorphous systems, the same physics control the electronic properties of amorphous and liquid systems [14]. Consequently, it is proposed that a study of the semiconducting properties of liquid-state systems will shed substantial light on the physics of amorphous systems.

1.3 Thermodynamics of Molten Semiconductors

While physicists and material scientists pursued a description of the electronic properties of molten semiconductors, metallurgists and geologists, in parallel, began thermo- physically and thermochemically characterizing complex slag systems for the metals extraction industry and geological studies. Many natural minerals, such as chalcopy- rite, galena, cinnabar, molybdenite, and sphalerite, exhibit molten semiconductivity and in order to improve the eﬃciency of extraction processes, and to further develop an understanding of rock formation processes, many researchers sought a more complete description of the high temperature, molten behavior of these and related systems. More speciﬁcally, the practical engineer and geologist sought phase diagrams for these systems.

33 1.3.1 Prediction of Phase Diagrams

The field of predicting phase diagrams has achieved great practical success in the past 50 years. Thermodynamic descriptions of material systems amenable to phase diagram interpretation typically seek a description and functional form of the Gibbs free energy of species for each phase. Molten semiconductor systems in particular, as detailed above, tend to exhibit strong short-range order and complex interactions. Consequently, simplistic thermodynamic models for the free energy, such as the regular solution model, are rendered ineffective for accurate prediction of key features of the phase diagram. More complex models of the free energy, reflecting a more physically realistic description of the entropy of complex liquids, have been a focus of metallurgists and thermodynamicists for decades. Many models of Gibbs free energy relevant for complex liquid systems (e.g. systems rich in sulfur) have been proposed over the years, and each framework has relative advantages for the thermodynamic modeler. Table 1.2 outlines some of the key methods to model the Gibbs free energy used by practicing thermodynamicists.

34 Table 1.2: Thermodynamic models for the free energy used to model liquid systems

Model Description Benefits Challenges Adds temperature-dependent Effective at predicting ternary entropy term to regular solution Pair approximation fails to phase diagrams from binary model (non-random mixing). account for all relevant SRO for Quasichemical Model [50–52] data. Hundreds of oxide and This model relies on the pair complex slag systems such as sulfide systems have been approximation for calculation of sulfides. modeled [53]. the free energy. Extension of quasichemical Requires substantially more fit philosophy not limited to pair Effective at modeling systems Cluster Variation Method parameters to develop approximation, allows for larger that tend to exhibit clustering /

35 (CVM) [54] self-consistent free energy model base units for calculation of free ordering in the liquid state. reflecting SRO. energy. Models the free energy by considering permutations of Used to great success in Not currently incorporated into central atoms (anions or modeling sulfides for the steel energy minimization software Central Atom Model [55, 56] cations), the nearest neighbor industry. Requires fewer fit outside of industry-specific cation shell, and the nearest parameters than CVM. packages. neighbor anion shell. Assumes formation of molecular ‘associates’ replicating the Simple interaction parameter fit Poor at predicting ternary phase stoichiometry of solid-state to data. Very successful in diagrams from binary data. Does Associate Model [57–61] compounds in the liquid. Models modeling binary sulfide systems not easily extend to the solution as a mixture of the exhibiting retention of SRO. multicomponent systems. pure elements and associates. On their own, the utility of the above-described Gibbs free energy models are limited, but when coupled with computer-automated energy minimization software, their potency is multiplied and each can be used to generate self-consistent phase diagrams and perform thermodynamic calculations. Several primary thermodynamic software packages for the generation of phase diagrams (the CALPHAD approach) have been developed, including FactSage and Thermo-Calc. Of critical importance to these software packages, and the free energy models, is the availability of empirical data with which to optimize the thermodynamic description, and it is indeed in this aspect that the tools are differentiated.

The availability and utility of thermodynamic data for molten systems has been dramatically improved by the thorough investigations of metallurgical and geological professionals and academics. In 1970, Kullerud produced a review article titled “Sulﬁde Phase Relations” summarizing the available thermodynamic data for sulﬁde systems [62]. The compendium included a plethora of binary, ternary, and quaternary systems. Generation of phase information for molten semiconductor species has continued, and many investigators have continued to populate thermodynamic databases and produce phase diagrams relevant for a practical study of molten semiconductor behavior [63, 64]. Critically, the generated databases have been used successfully to predict ternary and higher order multicomponent system properties from binary data with modern software packages, demonstrating the power that a thermodynamic modeling approach can have for the practicing engineer [60, 61].

Thus, whereas atomistic simulation has struggled to achieve quantitatively accurate predictions of melting points and semiconductor to metal transitions, modern calculation of phase diagrams has provided a consistent framework with which to accurately predict critical elements of phase diagrams. However, thermodynamic models do not as yet explicitly engage with the electronic nature of the systems, and are highly dependent on experimentally-intensive empirical datasets. Further, they often fail to predict the high-temperature features of phase diagrams, such as closure of miscibility gaps, due to a temperature-independent description of entropy (see Chapter 8). Consequently, it has been challenging to date to bridge the gap between

36 athermodynamicdescriptionandtheelectronicpropertiesofmoltensemiconductor systems, especially as constrained by the conceptual framework of Mott et al. which requires a knowledge of the evolution of the electronic density of states of a system to accurately predict semiconducting behavior. However, the free energy of a species is fundamentally dependent upon structure. As mentioned above, free energy models that accurately account for short-range order have better predictive ability. Thus, it should not be surprising that the elements of phase diagrams connected to the SRO of the system may correlate to, if not explicitly be reﬂective of, elements of electronic properties.

1.3.2 Interpretation of Phase Diagrams of Molten Semicon- ductor Systems

Glazov has written extensively about the information contained within phase diagrams in his 1989 book “Semiconductor and Metal Binary Systems” [65]. The use of geometrical analysis of an empirically validated phase diagram can provide insight to the structure, properties, and energetics of the phases and components. In particular, the presence of congruent melting compounds and the shape of the liquidus in the vicinity of said compounds has been shown to reﬂect the degree of ordering in the melt. The presence of a molecular entity that resembles a compound in the molten phase is typically associated with a congruent melting compound [65]. This can be understood by comparing the typical heats of mixing of liquids to the typical thermal energy at melting (i.e. RT where R is the gas constant). For the majority of binary metallic systems, the magnitude of the thermal energy (RT)isontheorderof5-10 1 kJ mol while the energies of mixing are an order of magnitude lower. This leads to the dissociation of molecular entities. However, for some systems, the energies of mixing are of the same order as the thermal energy. This allows for the formation of molecular entities due to the energetic favorability of bonding. Put another way, for certain systems, the free energy is minimized by a retention of molecular bonding

37 in the molten phase, despite the reduction in the configurational entropy of mixing of the system associated with short-range order. Large negative enthalpies of mixing are associated with compound formation. When the liquid phase accommodates a molecular entity with stoichiometry similar to the solid-state compound, the compound melts congruently. The entropic benefit of off-stoichiometry is magnified as a function of the degree of ordering of the molecular entity and consequently the slope of the liquidus is greater for systems that do not dissociate vs. systems that do exhibit dissociation of bonding in the molten phase. Thus, a qualitative connection between features of phase diagrams and the ordering of the molten state has been previously identified. Researchers have used geometric analysis of phase diagrams to interpret the relationship between structure and properties. However, a gap in connecting the these features of phase diagrams to electronic properties still exists.

1.4 Connection of Transport Properties to Equilib- rium Thermodynamic Variables

1.4.1 Transport Entropy

The derivation of the thermopower in the ﬁeld of irreversible thermodynamics leverages the concept of a transport entropy (for a more complete description, see Appendix B). Onsager and Callen describe the transport of entropy by electrons and ions in systems exposed to a thermal gradient [66]. This transport entropy term is not ana- lyzed in the framework of equilibrium thermodynamics, and has been leveraged as a useful construct by which to calculate the transport property of thermopower [66]. The utility of this relationship of transport entropy to measurable transport properties has been long recognized; for example Wagner describes experimental methods to separate the individual contributions of ions and electrons to the total thermopower which could enable the diﬀerentiation of transport entropy associated with ions and electrons [67].

38 1.4.2 Previous Attempts at Connection

In the years following Onsager, several researchers have sought to investigate whether the irreversible thermodynamic property of transport entropy described above might have a physical interpretation in the context of equilibrium thermodynamics. Hensel describes the relationship between a reduction of entropy of a system due to electron self-trapping (polaron eﬀects) and the thermopower [3]. Alcock describes a method to measure the partial molar entropy of oxygen ions through use of thermopower [68]. Rockwood and Tykodi describe a theoretical connection between the partial molar entropy of electrons and the thermopower in systems with mobile electrons [6–9].

dS = F↵ (1.3) dne T,P,N ⇣ ⌘

ne is the number of electrons, F is the Faraday constant, and ↵ is the thermopower.

These investigations have proven theoretically intriguing, but are not empirically validatable, and do not provide any predictive utility. This can be understood by contemplating the physical interpretation of the partial molar entropy of an electron, which describes the incremental entropy associated with the addition of an electron to a system at equilibrium (see References [8] and [9] for a more thorough description). The utility of this property to the thermodynamicist seeking a description of entropy to incorporate into a model of the free energy is minimal. A theory connecting transport properties (irreversible thermodynamic quantities) and equilibrium thermodynamic properties must provide empirically veriﬁable predictions to be validated.

1.5 Electronic Entropy

The theory presented in Chapter 3 connects measurable electronic properties of melts to the electronic entropy. A brief background on the topic of electronic entropy follows.

39 1.5.1 Forms of Electronic Entropy

Electronic entropy, broadly speaking, describes the size of the available state space of electrons. However, there are multiple contributions to the electronic entropy. For the majority of systems, electrons that contribute to the overall electronic entropy of the system adhere to the physics of indistinguishable particles (i.e. they are not localized). For these systems, the density of states (DOS) describes the available states and can be directly related to the electronic entropy via Equation 1.4, where p(E) is the probability of occupation of an electronic state [69].

1 Se = kB N(E)(p(E)lnp(E)+(1 p(E)) ln(1 p(E))) dE (1.4) Z0

p(E) is deﬁned by the Fermi function, where Ef is the Fermi level.

1 p(E)= E E (1.5) f e kT +1

This form of electronic entropy is herein named the electronic state entropy. The theory contained within this document relates to electronic state entropy, and mov- ing forward reference to the ‘electronic entropy’ should be understood to mean the electronic state entropy.

However, for a certain small subset of systems, there is an additional contribution to the total entropy of electrons. The configurational electronic entropy refers to the entropy associated with electrons that exhibit the statistics of distinguishable particles (i.e. are localized on the lattice). This entropy can be thought of as analogous to configurational entropy of atoms on a lattice. Configurational electronic entropy has been shown to be relevant for mixed valence oxides [70] or certain semiconductor systems at melting [71].

40 1.5.2 Contribution of Electronic Entropy to Total Entropy

Size of Accessible State Space

Electronic state entropy is maximized for systems that exhibit a large accessible density of states. Only states near the Fermi level are accessible at ﬁnite temperatures, and consequently electronic entropy is maximized for systems with large DOS near the Fermi level. For metallic systems such as alkali metals, the DOS is frequently small near the Fermi level and consequently the contribution of the electronic entropy to the total entropy is small. Molten semiconductor systems, however, often exhibit large DOS near the Fermi level; indeed, thermoelectric materials (typically a subset of semiconductors) are designed for maximum DOS near the Fermi level. Thus, for these systems, electronic entropy plays a larger role in the thermodynamics of the system at high temperature as it substantially contributes to the total entropy.

Components of Entropy for Molten Semiconductors

The free energy of mixing of a system is comprised of enthalpic and entropic contributions (see Appendix A for a background on the thermodynamics of mixing). A more complete description can be found in Reference [72].

Gmix =Hmix T Smix (1.6)

The entropy of mixing can approximated as a linear combination of contributions from conﬁgurational (Sc), vibrational (Sv), electronic (Se), and other terms

(Sr)[73].

Smix =Sc +Sv +Se +Sr (1.7)

Se is found to dominate for molten semiconductor systems (see Chapter 4).

The conﬁgurational entropy of mixing (Sc)isdependentonthedegreeofshort- range order in the system. More ordered systems exhibit less conﬁgurational entropy. Molten semiconductors exhibit short-range order and consequently do not exhibit

41 substantial configurational entropy of mixing (i.e. there are not substantially more available configurational states available in the mixed state than in the unmixed state because mixing induces ordering). Further, molten semiconductors, as described above, exhibit liquid phase immiscibility. This further restricts the configurational entropy of mixing, because mixing at an atomic level does not occur.

Sc Se (1.8) | |⌧| |

Vibrational entropy of mixing is also aﬀected by the loss of long-range order in the liquid state [2]. The vibrational entropy of mixing (Sv)isthediﬀerence between the vibrational entropy of the mixed system and that of the unmixed system. Both states are long-range disordered and have similar magnitudes of vibrational entropy. Consequently, the contribution of the vibrational entropy of mixing to the total entropy of mixing is small.

Sv Se (1.9) | |⌧| |

Sr comprises magnetic entropy and other system-dependent entropic contributions which are not considered herein. The typical contributions of the various components of entropy of mixing for molten semiconductor systems is summarized in Table 1.3.

Table 1.3: Typical values of components of entropies of mixing for molten semiconductor systems [20, 73–75]

1 1 Component Value for Molten Semiconductors (Jmol K )

Se 3

Sc <1

Sv 1 ⌧ Sr 1 ⌧

42 1.6 Summary

The description and prediction of the properties of noncrystalline systems present a critical frontier in materials science. Success in providing a rigorous physical and thermodynamic basis for the properties of molten metal systems begs extension to systems exhibiting short-range order. Of particular interest are theories that quantitatively predict the thermodynamic and electronic properties of the molten phase of systems that prove challenging for atomistic modeling, such as molten semiconductors. Atheorythatconnectsthethermodynamicandelectronicpropertiesofnoncrys- talline systems is needed. In particular, a method of predicting and measuring the entropy of high temperature phases is necessary to enable the investigation and prediction of noncrystalline material properties. Historical work attempting to connect transport properties to the entropy of electrons hints that a quantitative relationship between electronic entropy and electronic transport properties may exist. This thesis will explore the following question and its consequences: can the electronic entropy be quantitatively connected to measurable electronic properties without the use of phase- and system-speciﬁc models of atomic interaction? Molten semiconductors have been chosen as materials of focus to investigate this question due to the lack of existing predictive theory, availability of molten-phase data, and the relative importance of electronic entropy to the thermodynamics of these systems.

43 44 References

[1] C. Alcock. “Measurements, Models, and Mathematics of Alloy Systems”. In: Journal of Phase Equilibria 15.3 (1994), pp. 295–302.

[2] B. Fultz. “Vibrational Thermodynamics of Materials”. In: Progress in Materials Science 55.4 (2010), pp. 247–352.

[3] F. Hensel and W. Warren. Fluid Metals. Princeton University Press, 1999.

[4] M. Saboungi, W. Geertsma, and D. Price. “Ordering in liquid alloys”. In: Annu. Rev. Phys. Cher 41 (1990), pp. 207–244.

[5] F. Hensel. “Transition to Nonmetallic Behavior in Expanded Liquid and Gaseous Metals”. In: Angew. Chem. Internat. 13.7 (1974), pp. 446–454.

[6] A. Rockwood. “Comments on "The Seebeck coeﬃcient and the Peltier eﬀect in a polymer electrolyte membrane cell with two hydrogen electrodes"”. In: Electrochimica Acta 107 (2013), pp. 686–690.

[7] A. Rockwood. “Partial molar entropy of electrons in a jellium model: Impli- cations for thermodynamics of ions in solution and electrons in metals”. In: Electrochimica Acta 112 (Dec. 2013), pp. 706–711.

[8] A. Rockwood. “Relationship of thermoelectricity to electronic entropy”. In: Physical Review A 30.5 (1984), pp. 2843–2844.

[9] R. Tykodi. Thermodynamics of Systems in Nonequilibrium States.Davenport: Thinkers’ Press, 2002.

[10] J. Enderby and A. Barnes. “Liquid Semiconductors”. In: Rep. Prog. Phys. 53 (1990), pp. 85–179.

45 [11] V. Alekseev, A. Andreev, and M. Sadovskil. “Semiconductor-metal transition in liquid semiconductors”. In: Sov. Phys. Usp. 23.9 (1980), pp. 551–575.

[12] M. Cutler. Liquid Semiconductors. New York, NY: Academic Press, 1977.

[13] A.Ioﬀe and A. Regel. Progress in Semiconductors.Vol.4.NewYork,NY:Wiley, 1960.

[14] N. Mott and E. Davis. Electronic Processes in Non-Crystalline Materials.Ox- ford, UK: Clarendon Press, 1971.

[15] A. Allanore. “Features and Challenges of Molten Oxide Electrolytes for Metal Extraction”. In: Journal of the Electrochemical Society 162.1 (2015), E13–E22.

[16] N. Mott. “Electrons in disordered structures”. In: Advances in Physics 16.61 (Jan. 1967), pp. 49–144.

[17] A. Bosch, J. Moran-Lopez, and K. Bennemann. “Electronic theory for the metal- nonmetal transition in simple metal alloys”. In: J. Phys. C: Solid State Phys. 11 (1978), pp. 2959–2966.

[18] A. Regel, V. Glazov, and V. Koltsov. “Magnetic Susceptibility of Semiconduc- tors in the Liquid State”. In: Sov. Phys. Semicond. 23.7 (1989), pp. 707–714.

[19] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the diagrams of state - 3. Short-range order structure and chemical bond nature”. In: Journal of Thermo- electricity 3(2001),pp.3–23.

[20] V. Glazov. Liquid Semiconductors. New York, NY: Plenum Press, 1969.

[21] A. Bhatia and W. Hargrove. “Concentration ﬂuctuations and thermodynamic properties of some compound forming binary molten systems”. In: Physical Re- view B 10.8 (1974), pp. 3186–3196.

[22] A. Bhatia and D. Thornton. “Structural Aspects of the Electrical Resistivity of Binary Alloys”. In: Physical Review B 2.8 (1970), pp. 3004–3012.

[23] Y. Poltavtsev. “Structure of semiconductors in noncrystalline states”. In: Sov. Phys. Usp. 19.12 (1976), pp. 969–987.

46 [24] I. Bandrovskaya et al. “Optical and thermoelectric properties of arsenic selenide in glassy and liquid states”. In: phsyica status solidi b 107 (1981), pp. 95–98.

[25] F. Hensel. “Liquid metals and liquid semiconductors”. In: Angewandte Chemie (International ed. in English) 99.1976 (1980), pp. 593–606.

[26] V. Prokhorenko et al. “The Semiconductor-Metal Transition in Liquid Tel- lurium”. In: Physica Status Solidi (B) 113 (Oct. 1982), pp. 453–458.

[27] P. Lee and T. Ramakrishnan. “Disordered electronic systems”. In: Reviews of Modern Physics 57.2 (Apr. 1985), pp. 287–337.

[28] D. Thouless. “Electrons in disordered systems and the theory of localization”. In: Physics Reports 13.3 (Oct. 1974), pp. 93–142.

[29] R. Hodgkinson. “Miscibility gaps, ordered liquids and liquid semiconductors”. In: Philosophical Magazine 22.180 (Aug. 1970), pp. 1187–1199.

[30] S. Ohno, A. Barnes, and J. Enderby. “The electronic properties of liquid Ag1 xSex”. In: J. Phys.: Condens. Matter 6(1994),pp.5335–5350.

[31] S. Ohno et al. “The electrical conductivity and thermopower of liquid Ag1 cSec alloys”. In: Journal of Non-Crystalline Solids 207 (1996), pp. 98–101.

[32] V. Sklyarchuk and Y. Plevachuk. “Thermophysical properties of liquid ternary chalcogenides”. In: High Temperatures-High Pressures 34 (2002), pp. 29–34.

[33] V. Sklyarchuk et al. “Semiconductor-metal transition in semiconductor melts with 3d metal admixtures”. In: Journal of Physics: Conference Series 98 (Feb. 2008), pp. 1–4.

[34] A. Barnes. “Liquid metal-semiconductor transitions”. In: Journal of Non-Crystalline Solids 158 (1993), pp. 675–682.

[35] J. Jortner. Electrons in Fluids.Berlin:Springer,1973,pp.257–285.

[36] M. Cohen and J. Jortner. “Electronic structure and transport in liquid Te”. In: Physical Review B 13.12 (1976), pp. 5255–5260.

47 [37] M. Cohen and J. Sak. “Electronic structure of liquid and amorphous alloys with clusters”. In: Journal of Non-Crystalline Solids 10 (1972), pp. 696–701.

[38] V. Glazov. “Investigation of Postmelting in Semiconductor Melts”. In: Inorganic Materials 32.11 (1996), pp. 1287–1305.

[39] B. Sklyarchuk and Y. Plevachuk. “Nonmetal-Metal Transition in Liquid Cu- Based Alloys”. In: Zeitschrift fur Physikalische Chemie 215.1 (2001), pp. 103– 109.

[40] A. Bhatia and V. Ratti. “Structure factors of compound forming liquid mixtures”. In: Physics Letters 51A.7 (1975), pp. 386–388.

[41] M. Cutler. “Thermodynamic evidence against large-scale ﬂuctuations in liquid semiconductors”. In: Physical Review B 14.12 (1976), pp. 5344–5350.

[42] R. Hodgkinson. “Interpretation of the electrical properties of liquid semiconductors”. In: J. Phys. C: Solid State Phys. 9(1976),pp.1467–1482.

[43] V. Godlevsky et al. “First-principles calculations of liquid CdTe at temperatures above and below the melting point”. In: Physical Review B 60.12 (Sept. 1999), pp. 8640–8649.

[44] R. Car and M. Parrinello. “Uniﬁed approach for molecular dynamics and density- functional theory”. In: Physical Review Letters 55.22 (1985), pp. 2471–2474.

[45] F. Shimojo et al. “Temperature dependence of the atomic structure of liquid

As2Se3:abinitiomoleculardynamicssimulations”.In:J. Phys.: Condens. Mat- ter 12 (2000), pp. 6161–6172.

[46] F. Shimojo and M. Aniya. “Eﬀects of Cu alloying on the structural and elec-

tronic properties in amorphous Cux(As0.4S0.6)1 x:Abinitiomolecular-dynamics simulations”. In: Physical Review B 73.23 (June 2006), pp. 235212–1–235212– 11.

[47] F. Shimojo et al. “The microscopic mechanism of the semiconductor-metal transition in liquid arsenic triselenide”. In: J. Phys.: Condens. Matter 11 (1999), pp. 153–158.

48 [48] J. Akola et al. “Density variations in liquid tellurium: Roles of rings, chains, and cavities”. In: Physical Review B 81 (Mar. 2010), pp. 094202–1–094202–7.

[49] B. Kolomiets. “Vitreous Semiconductors (II)”. In: phys. stat. sol 7(1964),pp.713– 731.

[50] E. Guggenheim. Mixtures. Oxford, UK: Oxford University Press, 1952.

[51] A. Pelton and M. Blander. “Thermodynamic Analysis of Ordered Liquid Solu- tions by a Modiﬁed Quasichemical Approach - Application to Silicate Slags”. In: Metallurgical Transactions B 17B (1986), pp. 805–815.

[52] A. Pelton and M. Blander. “A least squares optimization technique for the analysis of thermodynamic data in ordered liquids”. In: CALPHAD 12.1 (1988), pp. 97–108.

[53] A. Pelton et al. “The Modiﬁed Quasichemical Model I - Binary Solutions”. In: Metallurgical and Materials Transactions B 31B (2000), pp. 651–659.

[54] R. Kikuchi. “The cluster variation method”. In: Journal de Physique Colloques 38 (1977), pp. 307–313.

[55] E. Foo and C. Lupis. “The "Central Atoms" model of multicomponent inter- stitial solutions and its applications to carbon and nitrogen in iron alloys”. In: Acta Metallurgica 21 (1973), pp. 1409–1430.

[56] J. Lehmann and H. Gaye. “Modeling of thermodynamic properties of sulphur bearing metallurgical slags”. In: Rev. int. Hautes Temper. Refract. 28 (1993), pp. 81–90.

[57] R. Sharma and Y. Chang. “Thermodynamics and phase relationships of transition metal-sulfur systems: Part III. Thermodynamic properties of the Fe-S liquid phase and the calculation of the Fe-S phase diagram”. In: Metallurgical Transactions B 10B (1979), pp. 103–108.

[58] R. Sharma and Y. Chang. “A thermodynamic analysis of the copper-sulfur system”. In: Metallurgical Transactions B 11B (1980), pp. 575–583.

49 [59] R. Sharma. “Thermodynamics and Phase Relationships of Transition Metal- Sulfur Systems : IV. Thermodynamic Properties of the Ni-S Liquid Phase and the Calculation of the Ni-S Phase Diagram”. In: Metallurgical Transactions B 11B (1980), pp. 139–146.

[60] F. Kongoli, Y. Dessureault, and A. Pelton. “Thermodynamic Modeling of Liquid Fe-Ni-Cu-Co-S Mattes”. In: Metallurgical and Materials Transactions B 29B (1998), pp. 591–601.

[61] F. Kongoli and A. Pelton. “Model Prediction of Thermodynamic Properties of Co-Fe-Ni-S Mattes”. In: Metallurgical and Materials Transactions B 30B (1999), pp. 443–450.

[62] G. Kullerud. “Sulﬁde phase relations”. In: Mineral Soc. Amer. Spec. Pap. 3 (1970), pp. 199–210.

[63] G. Kullerud. “The Lead-Sulfur System”. In: American Journal of Science 267.A (1969), pp. 233–256.

[64] D. Chakrabarti and D. Laughlin. “The Cu-S (Copper-Sulfur) System”. In: Bul- letin of Alloy Phase Diagrams 4.3 (1983), pp. 254–270.

[65] V. Glazov and L. Pavlova. Semiconductor and Metal Binary Systems. Ed. by E A D White. New York: Plenum Press, 1989.

[66] H. Callen. “The Application of Onsager’s Reciprocal Relations to Thermoelec- tric, Thermomagnetic, and Galvanomagnetic Eﬀects”. In: Physical Review 73.11 (1948), pp. 1349–1358.

[67] C. Wagner. “The Thermoelectric Power of Cells wIth Ionic Compounds Involv- ing Ionic and Electronic Conduction”. In: Progress in Solid State Chemistry 7 (1972), pp. 1–37.

[68] C. Alcock, K. Fitzner, and K. Jacob. “An entropy meter based on the thermoelectric potential of a nonisothermal solid-electrolyte cell”. In: Journal of Chem- ical Thermodynamics 9(1977),pp.1011–1020.

[69] C. Kittel. Introduction to Solid State Physics.7th.Hoboken,NJ:Wiley,1996.

50 [70] F. Zhou, T. Maxisch, and G. Ceder. “Conﬁgurational Electronic Entropy and

the Phase Diagram of Mixed-Valence Oxides: The Case of LixFePO4”. In: Phys- ical Review Letters 97.15 (Oct. 2006), pp. 155704–1–155704–4.

[71] B. Chakraverty. “Conﬁgurational entropy of electrons in semiconductors”. In: Radiation Eﬀects 4.1 (1970), pp. 39–43.

[72] D. Gaskel. Introduction to the Thermodynamics of Materials.5thed.BocaRa- ton, FL: CRC Press, 1995.

[73] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically accessible electronic properties in high temperature systems”. In: Philosophical Magazine 96.29 (2016), pp. 3041–3053.

[74] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the state diagrams - 5. Clas- siﬁcation of electronic semiconductor melts”. In: Journal of Thermoelectricity 1 (2004), pp. 32–47.

[75] A. Regel, V. Glazov, and A. Aivazov. “Calculation of components of fusion entropy of some semiconducting compounds”. In: Sov. Phys. Semicond. 8.11 (1975), pp. 1398–1401.

51 52 Chapter 2

Hypothesis

The study of high temperature and noncrystalline phases of materials has been plagued by challenging experimental conditions for thermodynamic measurements and diﬃculty in applying ab initio modeling techniques to the prediction of, in particular, the entropy and electronic properties of molten systems exhibiting short-range order (SRO). A quantitative framework that connects the thermodynamic and electronic properties of noncrystalline systems would enable the use of existing thermodynamic data to predict electronic properties and the use of experimental techniques to generate electronic property data to facilitate the generation of thermodynamic databases.

Anovelmeanstoaccessentropyindisorderedphaseswouldbemostimpactfulfor systems where ab initio modeling ﬁnds challenge in providing accurate quantitative prediction. Molten semiconductors are thus chosen as a system through which to investigate a novel theory to quantify the role of entropy in high temperature phases, precisely because there is no other framework available. The electronic properties (e.g. electronic conductivity, mobility, and thermopower) of molten semiconductors have proven challenging to model with suﬃcient accuracy to support practical investigation and only qualitative predictions are supported by the current literature (see Chapter 1).

53 2.1 Correlation Between Features of Phase Diagrams and Molten Semiconductivity

The notion that features of phase diagrams are correlated to the electronic properties of molten semiconductors is not new. Cutler, in his comprehensive 1977 monograph, reﬂects on the correlation of liquid phase immiscibility with systems displaying semiconducting properties in the liquid state [1]. This idea was recently furthered by a rigorous correlation study of hundreds of binary systems exhibiting semiconductor-to- semiconductor (SC-SC) and semiconductor-to-metal (SC-M) transitions by Belotskii et al. in a series of articles [2–5]. Belotskii goes further to describe particular phase diagram features that correlate to the diﬀerent transitions that may occur upon melting. Binary systems that metallize (SC-M) do not exhibit liquid-liquid miscibility gaps. Binary systems that remain semiconductors (SC-SC) have at least one liquid-liquid miscibility gap as illustrated in Figure 2-1.

Figure 2-1: Notional phase diagrams of binary A-B systems that undergo a semiconductor-to-semiconductor (SC-SC) transition and systems that undergo a semiconductor-to-metal (SC-M) transition at melting. SC-SC systems exhibit at least one miscibility gap in the molten phase. Solid shaded regions correspond to metallic molten systems. Thatched shaded regions correspond to semiconducting molten systems.

54 Belotskii and Cutler agree upon the source of liquid phase immiscibility: the liquid state accommodates both non-metallic and metallic phases. The inhomogeneity of these two phases leads to immiscibility [5]. Thus, the presence of the characteristic phase diagram features reﬂects the chemical bonding of the melt, which is similarly coupled to the short-range order (SRO), as described in Chapter 1.

It is therefore suggested that for semiconductor systems, prediction of the presence of miscibility gaps in the liquid state may be used as a proxy for prediction of molten semiconducting behavior for binary systems. Still further, recent studies on the behavior of molten semiconductors beyond the liquidus have revealed additional connections of features of phase diagrams to the evolution of semiconducting properties of liquid systems. Sokolovskii et al., followed by Didoukh et al., performed a series of experiments on selenide and telluride systems whereby they measured the electrical conductivity and Seebeck coeﬃcient as a function of temperature in the vicinity of a critical point of a liquid phase miscibility gap. The results are clear: metallization of the system is correlated with a critical point of a miscibility gap in the phase diagram [6–9]. See Figure 2-2 for a schematic representation of the behavior.

The critical point of the miscibility gap represents a second order transition, which reveals a deeper connection between the metallization and the onset of complete miscibility. The critical point reflects a continuous order-disorder transformation. The connection between the degradation of semiconducting properties and the reduction of SRO was demonstrated in Chapter 1: the semiconducting properties of liquids depend on SRO, which in turn is connected to electronic ordering. The continuous semiconductor-to-metal transition at the critical point of a miscibility gap in the phase diagram of a molten semiconductor (the metallization) thus reflects a ‘filling in’ of the pseudogap (see Figure 1-3), which, as described in Chapter 1, is connected with metallization. Per Belotskii and Cutler, immiscibility is no longer possible without a non-metallic phase. The connection between the phase diagram and the semiconducting properties of liquid systems has thus been thoroughly demonstrated by correlation studies and relates to the chemical ordering of molten systems. However, the predictive utility of such qualitative correlations of binary systems remains limited.

55 Tc Temperature Temperature

A AB B Thermopower

Figure 2-2: Notional representation of the thermopower of a binary (A-B) molten semiconductor system as a function of temperature. There is a transition of thermopower from semiconductor to metallic behavior at the temperature corresponding to the critical point of the miscibility gap Tc.

2.2 Scientiﬁc Gap

The lack of a quantitative explanation, or thermodynamic basis, of the correlation between features of phase diagrams and molten semiconductivity constitutes a sci- entiﬁc gap. Further, the current theory is incapable of predicting a priori whether a system will behave as a semiconductor in the molten phase. Finally, the existing theory is unable to predict over what range of temperature and composition a system will exhibit semiconducting properties in the molten phase.

2.3 Hypothesis

The role that electrons play in the thermodynamics of molten semiconductor systems, as manifested in features of phase diagrams, is related to structure and bonding properties, as described above. It is here hypothesized that it is possible to deﬁne and measure a form of electronic entropy, the electronic state entropy (see Chapter

56 1), that reﬂects the ordering of electrons, which in turn determines the electronic properties of molten systems. It is further hypothesized that the electronic entropy and certain electronic properties of molten materials are quantitatively connected. Aquantitativeconnectionbetweenanequilibriumthermodynamicvariable(entropy) and transport properties (e.g. thermopower) is proposed. It is posited that electronic entropy substantially contributes to the thermodynamics of mixing for molten semiconductor systems. The electronic entropy is thus responsible for driving the key thermodynamic behavior of the molten state for molten semiconductor systems.

2.4 Consequences for Materials Modeling

The role of electronic entropy in the macroscopic thermodynamics of molten semiconductors will be investigated herein. However, electrons play a key role in the energetics and structure of material systems beyond molten semiconductors. There- fore it would be surprising if additional useful information about structure could not be provided thanks to a better description of the electronic entropy of a system. More generally, entropy is a state function and plays a significant role in defining the macroscopic thermodynamic behavior of high temperature systems, and especially the liquid state (see Chapter 1). However, quantifying absolute entropy is not empirically possible today, and the modeling community is afforded a singular absolute reference state at 0 K. It is critical to the study of high temperature systems, and to the study of solids near the liquidus, to have a means to access entropy. Today, it is most common to calculate entropy by integration of the heat capacity from the 0 Kreferencestate.Forexample,theabsoluteentropyofasysteminthe liquid state at temperature T at the composition of a congruent melting compound that is stable from 0 KtothemeltingtemperatureTm would be:

Tm solid T liquid Cp Hf Cp S = dT + + dT (2.1) T T T Z0 m ZTm This method requires accurate models for the temperature dependent heat capac-

57 ities of solid and liquid states which can be challenging to generate.

An absolute measurement of entropy in the melt is unprecedented, and would give the thermodynamic and atomistic modeling communities a new reference state beyond

0 K by which to describe the thermodynamics of high temperature systems. This could in principle enable modeling of liquids in their own right, without reference to the solid state, and provide a foundation for the modeling of entropy for noncrystalline systems at high temperature. For example, if the absolute entropy of a liquid system at temperature Tx is known, then the entropy at temperature T in the liquid phase at temperature T can be determined by:

T liquid Cp S = S(Tx)+ dT (2.2) T ZTx

If the hypothesis that electronic entropy dominates the thermodynamics of mixing for a certain class of systems (i.e. molten semiconductors) is valid, and if the hypothesis that electronic entropy can be quantiﬁably connected to empirically accessible electronic properties is correct, then an absolute reference state for entropy can be achieved and accessed empirically.

As discussed in Chapter 1, demonstrating this connection in the prototypical tellurium-thallium system has been shown in the literature to validate extension to the broader class of molten semiconductor systems. Molten semiconductor systems encompass a wide variety of chemistries (oxides, sulﬁdes, selenides, tellurides, arsenides, and more) and thus there is reason to believe that a demonstration of the quantitative viability of the theory for molten semiconductors will extend to a broader class of noncrystalline systems at high temperature.

Access to absolute entropy can thus provide a critical bridge between the modeling and empirical communities investigating the structure and properties of high temperature material systems.

58 2.5 Framework for Validation of Hypothesis

To validate the hypothesis, a quantitative connection between electronic entropy and electronic properties of molten semiconductor systems is required. This theory must be validated by appeal to empirical data of the properties of the melt. The electronic entropy must be quantitatively demonstrated to impact the thermodynamics of mixing of molten semiconductor systems and to materially impact key features of the phase diagram. This should ideally be achieved by reference to 3rd party data that have been generated by the community. Further, the extension of the predictive connection to systems not previously measured is required to validate a predictive capacity (as opposed to simply the ability to reproduce existent data). Finally, it is necessary to determine the classes of material systems for which the predictions herein are valid. Speciﬁcally, investigating the extension of the model to systems that do not exhibit molten semiconductivity can bound the utility of the proposed theory.

2.6 Summary

Apreviouslynotedcorrelationbetweencertainfeaturesofphasediagrams(i.e.miscibility gaps) and the electronic properties (e.g. electronic conductivity, mobility, and thermopower) of molten systems indicates a connection between macroscopic thermodynamic quantities and electronic properties. It is herein hypothesized that:

Electronic entropy is quantitatively connected to the electronic properties of • molten semiconductor systems

Electronic entropy substantially contributes to the thermodynamics of mixing • for molten semiconductor systems

Electronic entropy drives the macroscopic thermodynamic behavior of the molten • state for molten semiconductor systems

59 Aconnectionbetweenelectronicpropertiesandelectronicentropyextendsto • broader classes of high-temperature noncrystalline materials

Validating these hypotheses would enable empirical access to a new absolute reference state for the entropy of a material in a disordered state (noncrystalline materials). This may enable a new method to connect and evaluate the predictions of the atomistic and macroscopic thermodynamic modeling communities and provide a foundation for modeling entropy in high temperature noncrystalline systems.

60 References

[1] M. Cutler. Liquid Semiconductors. New York, NY: Academic Press, 1977.

[2] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the diagrams of state - 3. Short-range order structure and chemical bond nature”. In: Journal of Thermo- electricity 3(2001),pp.3–23.

[3] D. Belotskii and O. Manik. “On the interrelation between parameters of certain thermoelectric materials and state diagrams”. In: Journal of Thermoelectricity 4(2002),pp.38–42.

[4] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the diagrams of state - 4. Thermal conductivity of semiconductors with delamination in the diagrams of state”. In: Journal of Thermoelectricity 4(2003),pp.32–38.

[5] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the state diagrams - 5. Clas- siﬁcation of electronic semiconductor melts”. In: Journal of Thermoelectricity 1 (2004), pp. 32–47.

[6] B. Sokolovskii et al. “Critical Phenomena and Metal-Nonmetal Transition in

the Miscibility Gap Region of Liquid Tl1 xSex Alloys”. In: phys. stat. sol. (a) 139 (1993), pp. 153–159.

[7] V. Didoukh, Y. Plevachuk, and B. Sokolovskii. “Liquid-liquid equilibrium in immiscible In-Se alloys suﬀering metal-nonmetal transition”. In: Journal of Phase Equilibria 17.5 (Oct. 1996), pp. 414–417.

61 [8] V. Didoukh, B. Sokolovskii, and Y. Plevachuk. “The miscibility gap region and properties of liquid ternary alloys”. In: J. Phys.: Condens. Matter 9(1997), pp. 3343–3347.

[9] V. Didoukh, Y. Plevachuk, and B. Sokolovskii. “Experimental Investigations of Phase Equilibria in Binary Liquid Immiscible Alloys”. In: International Journal of Thermophysics 20.1 (1999), pp. 343–351.

62 Chapter 3

Theory Relating Electronic Entropy to Electronic Properties

There has been an ongoing discussion in the literature since the 1980s regarding a potential connection between electronic entropy and thermopower. However, to date no relationship between integral electronic entropy and the thermopower of a system has been proposed.

3.1 Theory

It is herein hypothesized (see Chapter 2) that electronic entropy is quantitatively connected to the electronic properties of molten semiconductor systems. A discussion of the key ﬁndings of a new theory connecting electronic entropy and electronic properties follows. For a more complete derivation, see Reference [1]. For background on the relevant thermodynamic and electronic properties, see Chapter 1. For a derivation and heuristic argument of the following relations, see Appendix C.

3.1.1 Electronic Entropy and Thermopower

1 1 Equation 3.1 connects the integral electronic entropy, Se (J mol K ), to the electron 1 1 density, n (mol ), the fundamental charge, e,andthethermopower,↵ (V K )for

63 negatively charged carriers as per Reference [1].

Se = ne↵ (3.1)

Equation 3.2 shows the relation formulated for positively charged carriers.

Se = ne↵ (3.2)

For systems exhibiting ambipolar conductivity, the relation follows.

Se = npe↵p nne↵n (3.3)

np and nn are, respectively, the density of positively and negatively charged carriers. ↵p and ↵n are, respectively, the thermopowers of positively and negatively charged carriers. For simplicity, we will, unless explicitly stated otherwise, assume systems exhibit dominantly unipolar conductivity. The Drude model of conductivity has been shown to provide accurate predictions of the conductivity of many molten semiconductor systems [2]. The Drude model is 1 1 2 1 1 shown in Equation 3.4 where (⌦ m )istheconductivity,µ (m V s )isthe 3 electron mobility, m (m )istheelectrondensity,ande is the fundamental charge.

= meµ (3.4)

Combining Equations 3.1 and 3.4 results in an equation for systems of negatively charged carriers that has a macroscopic thermodynamic variable, the electronic entropy Se,ononesideandempiricallymeasurableelectronicpropertiesontheother. This relationship will be explored and validated in Chapter 4. The conversion factor 3 1 w (m mol )isusedtoconvertfromvolumetrictomolarquantities.

↵ Se = w (3.5) µ

For systems with positively charged carriers the equation follows.

64 ↵ Se = w (3.6) µ

These relations are valid for systems with deﬁned transport properties, and are thus applicable to mixtures as well as single component systems.

3.1.2 Formulation for Use of Empirical Data

Available thermodynamic datasets typically provide for mixing quantities rather than integral quantities (i.e. entropy of mixing rather than total entropy). Consequently, it is practical to reformulate the theory to relate thermodynamic quantities of mixing to empirically measurable electronic properties. See Appendix A for a discussion on the relevant thermodynamics of solutions that allow reformulation of total quantities into mixing quantities. Quantities of mixing are typically expressed per mole of system. The formula for a binary A-B system is shown in Equation 3.7.

A B ↵ Se = Se xAS xBS = w + xAnAe↵A + xBnBe↵B (3.7) e e µ

A B Se is the electronic entropy of mixing. Se and Se are, respectively, the absolute entropies of the end members A and B. xA and xB are, respectively, the concentration of A and B. µ, ,and↵ still refer to the measurable bulk material properties (i.e. not values of mixing). All components are assumed to have dominantly negatively charged carriers in Equation 3.7. The absolute entropies of the end members are predicted without use of the Drude model because the concentration of carriers is typically available for single elements. Equation 3.7 assumes that the end members are approximately modeled by Equa- tion 3.2. The validity of this assumption is discussed in Chapter 5.

3.1.3 Assumptions Used in Application of Theory

A framework is provided to apply Equation 3.7 to empirical datasets. The following assumptions underlie the framework [1]:

65 1. The entropy of mixing is accounted for by the electronic entropy of mixing

2. The Drude model of conductivity describes the conductivity of the system

3. The Hall mobility is used for the carrier mobility

4. The thermopower is dominated by the electronic contribution

Support for Assumptions 1 and 4 is provided in Chapter 1. For a more thorough discussion on Assumption 4 see Reference [1].

3.2 Discussion of Theoretical Basis

Equations 3.1-3.3 relate the electronic component of thermopower of a material to the integral electronic entropy. As discussed in Chapter 1 and detailed in Appendix B, the thermopower can be understood to represent the entropy transported per charge from thermal gradient induced electron migration in a material. As an electron migrates from, for example, high temperature to low temperature in a piece of material in a thermal gradient, it transports entropy. One interpretation of Equation 3.1 is to identify this transported entropy as the contribution to the equilibrium electronic entropy of that charge. Consequently, the total electronic entropy is, in the simplest assumption, the number of charged carriers (n)multipliedbythe entropy per charge (e↵). There are several assumptions associated with this interpretation. First, only mobile electrons (those electrons contributing to the conductivity) participate in thermopower, and consequently only the contribution of mobile electrons to the electronic entropy is considered in Equations 3.1-3.3. Only electronic states near the Fermi level contribute to the conductivity of the system. However, these are precisely the electronic states that are considered ‘accessible’ and contribute to the electronic entropy (see Chapter 1). Consequently, for most systems, the electrons that do not participate in transport behavior occupy fully occupied, or fully unoccupied, states and thus do not contribute to electronic entropy. Therefore, the ﬁrst assumption appears reasonable.

66 The second assumption is that the thermopower represents the average entropy per charge for all charges participating in transport behavior. Previous researchers have discussed the physical interpretation of the thermopower as the partial molar entropy of electrons (i.e. the contribution to electronic entropy of the addition of one electron to a system) [3–6]. For systems with a gap in the density of states (e.g. molten semiconductors), transport is thermally activated and the electronic states that contribute to transport behavior (for unipolar systems) are approximately equivalent in energy. Consequently, the value of the entropy of transport for an additional charge (i.e. the thermopower) well-represents the average entropy of transport for an electron in the system. However, for systems such as metallic systems, where electronic states contributing to transport behavior span a broader range of energies, the bulk thermopower measured may reﬂect a net measurement inclusive of electrons migrating from higher to lower temperatures as well as from lower to higher temperatures due to the energy- dependence of the impact of temperature on the chemical potential of electrons. Thus, the assumption that the contribution of electronic entropy of one additional electron represents the average contribution to electronic entropy of all electrons participating in transport processes may be suspect. It could be hypothesized that an energy-dependent thermopower (↵(E))couldbe used to formulate an equation equivalent to Equation 3.1 for which assumption 2 would not be necessary.

Se = n(E) e↵(E) (3.8) ZE n(E) represents the number of occupied states at energy E.However,theenergy- dependent thermopower in Equation 3.8 is not, to the author’s knowledge, a measurable entity and thus is of limited utility.

67 68 References

[1] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically accessible electronic properties in high temperature systems”. In: Philosophical Magazine 96.29 (2016), pp. 3041–3053.

[2] N. Mott and E. Davis. Electronic Processes in Non-Crystalline Materials.Ox- ford, UK: Clarendon Press, 1971.

[3] A. Rockwood. “Comments on "The Seebeck coeﬃcient and the Peltier eﬀect in a polymer electrolyte membrane cell with two hydrogen electrodes"”. In: Electrochimica Acta 107 (2013), pp. 686–690.

[4] A. Rockwood. “Partial molar entropy of electrons in a jellium model: Impli- cations for thermodynamics of ions in solution and electrons in metals”. In: Electrochimica Acta 112 (Dec. 2013), pp. 706–711.

[5] A. Rockwood. “Relationship of thermoelectricity to electronic entropy”. In: Physical Review A 30.5 (1984), pp. 2843–2844.

[6] R. Tykodi. Thermodynamics of Systems in Nonequilibrium States.Davenport: Thinkers’ Press, 2002.

69 70 Chapter 4

Prediction of Properties of Te-Tl

The application of Equation 3.7 and the assumptions laid out in subsection 3.1.3 to a validated electronic property dataset for a model molten semiconductor system follows. As the most studied molten semiconductor system, both thermodynamic and electronic property data are available for the Te-Tl system. Pure thallium is a metal in the molten state while pure tellurium behaves as a semimetal (see Chapter 1). Inter- mediate compositions exhibit molten semiconductivity.

4.1 Applied Model

Equation 3.7 can be speciﬁed for the prediction of the electronic entropy of mixing of the Te-Tl system (formulated for p-type conduction).

↵ Se = w xTlnTle↵Tl xTenTee↵Te (4.1) µ

The Drude model is assumed to hold for the Te-Tl system. Equation 3.7 can be reformulated for the prediction of the thermopower of the Te-Tl system.

µ ↵ = (Smix + xTe nTe e↵Te + xTl nTl e↵Tl) (4.2) w

71 Equation 4.2 is reproduced from Reference [1]. The assumption that the entropy of mixing is approximated by the electronic component of the entropy of mixing is reﬂected by the inclusion of Smix in lieu of Se.

4.2 Results

Property data for the mobility, electronic conductivity, and thermopower is provided by Donally and Cutler [2–4]. Data for the entropy of mixing of the Te-Tl system was compiled and assessed by Oh [5]. Figure 4-1 shows the total entropy of mixing of the

Te-Tl system as a function of composition at 800 KasmeasuredbyNakamuraand Terpilowski and as compiled by Oh, and the predicted electronic entropy of mixing of the Te-Tl system per Equation 4.1 [5, 6]. Figure 4-2 shows the thermopower of the Te-Tl system as a function of composition at 800 K as measured by Cutler and the predicted thermopower of the Te-Tl system per Equation 4.2 [3].

4.3 Discussion

Agreement between prediction and reported data is excellent (within 10% at maximum divergence). There are no ﬁtting parameters in this analysis. See Reference [1] for a complete discussion on the results presented herein. The predictive capacity of the theory presented in Chapter 3 demonstrates the validity of the assumption that the entropy of mixing is dominated by the electronic entropy of mixing (Smix Se). Thus, electronic entropy is indeed critical to ⇡ the thermodynamics of mixing of molten semiconductor systems and a quantitative connection between electronic entropy and electronic properties has predictive capacity. A bidirectional predictive relationship between electronic and thermodynamic properties is demonstrated. Entropy is a thermodynamic state function with a value at equilibrium. Conse- quently, if Equation 4.2 is valid, and thermopower is quantitatively related to the

72 Nakamura Oh Terpilowski Prediction -1 K -1 / J / mol mix Δ S

Te at. % Tl

Figure 4-1: Entropy of mixing vs. atomic % Tl as measured by Nakamura (blue circles), Terpilowski (red circles), and as modeled by Oh (squares) and as predicted by Equation 4.1 (solid line) [1, 5, 6].

73 Cutler Prediction -1 V K μ V / α

Te at. % Tl

Figure 4-2: Thermopower vs. atomic % Tl as measured by Cutler (circles) and as predicted by Equation 4.2 (solid line) [1, 3].

74 entropy of mixing of a system at equilibrium, the thermopower can be interpreted to be a material property with a physical interpretation at equilibrium.

75 76 References

[2] J. Donally and M. Cutler. “Hall Measurements in Liquid Thallium-Tellurium”. In: Physical Review 176.3 (1968), pp. 1003–1004.

[3] M. Cutler. Liquid Semiconductors. New York, NY: Academic Press, 1977.

[4] M. Cutler and C. Mallon. “Thermoelectric Study of Liquid Semiconductor So- lutions of Tellurium and Selenium”. In: The Journal of Chemical Physics 37.11 (1962), pp. 2677–2683.

[5] C. Oh. “Assessment of the Te-Tl (Tellurium-Thallium) System”. In: Journal of Phase Equilibria 14.2 (1993), pp. 197–204.

[6] Y. Nakamura and M. Shimoji. “Thermodynamic Properties of the Molten Thal- lium + Tellurium System”. In: Trans. Faraday Soc. 67 (1971), pp. 1270–1277.

77 78 Chapter 5

Extension of Framework to Predicting Thermodynamic Quantities of Fusion

There are few molten semiconductor systems for which validated empirical data exist for both the electronic and thermodynamic properties of the melt over broad composition and temperature ranges. Chapter 4 shows the application of the theory to the Te-Tl system. However, there exist substantial data in the literature on the properties of compounds at melting. To validate the extension of the framework to additional systems beyond Te-Tl, the electronic entropy of fusion of congruent melting semiconducting compounds is predicted leveraging the theory presented in Chapter 3. Implicit in this application is the assumption that Equations 3.3 and 3.4 apply to solid semiconductor compounds near melting.

5.1 Calculation of the Entropy of Fusion

Equations 3.3 and 3.4 are applied to the solid and molten phases of compounds. The following discussion assumes negatively charged carriers.

Equation 5.1 shows the electronic entropy of fusion, Sf as a function of the number of carriers and thermopower of the solid (ns and ↵s)andthenumberof carriers and thermopower of the liquid (nl and ↵l).

79 Sf = ns e↵s nl e↵l (5.1)

Data on the number of carriers (ns and nl)aremoresparsethandataonthe conductivity. Consequently, the Drude model is assumed below and Equations 3.5 and 3.6 are applied.

Equation 5.2 shows the electronic entropy of fusion, Sf as a function of the electronic properties of the solid (↵s, µs,ands)andtheliquid(↵l, µl,andl).

s↵s l↵l Sf = ws wl (5.2) µs µl

5.2 Results

Figure 5-1 shows the predicted electronic entropy of fusion vs. the reported electronic entropy of fusion (as provided by Belotskii) [1]. Reported values of electronic entropy of fusion are typically derived by the subtraction of conﬁgurational and vibrational components of the entropy of fusion from the total entropy of fusion. For a complete description of the results reported in Figure 5-1 see Reference [2].

5.3 Discussion

For a subset of evaluated systems (shown as solid symbols) the predicted value of the electronic entropy from Equation 5.2 approximates the reported value from Belotskii

[1]. However, Sf of certain evaluated systems (shown as hollow symbols) does not agree well with the reported value. This disagreement is explained by two factors. First, for some systems, especially systems that metallize and thus exhibit a larger electronic entropy of fusion, the Drude model of conductivity may not be accurate. Certain systems (such as GaSb) have reported carrier concentration data (ns and nl), and thus it is possible to leverage Equation 5.1 in lieu of the approximate Equation 5.2. These results are shown as blue symbols in Figure 5-1 and are in better agreement with reported values. Second,

80 70 Ge GaSb AlSb

-1 65 15 20 InSb K -1

GaSb / J / mol f

Bi Se GeTe 2 3 Sb Te In Te 2 3

Δ S Predicted 2 3 Ga Te Te PbSe 2 3 GaAs AlSb Cu S PbTe 2 Ge

Reported S / J mol-1 K-1 Δ f

Figure 5-1: Predicted vs. reported electronic entropy of fusion for congruent melting semiconductor compounds [1]. Solid symbols show systems where Equation 5.2 provides accurate prediction. Hollow symbols show systems where Equation 5.2 fails to provide accurate prediction. Blue symbols show the application of Equation 5.1 for systems where carrier concentration data are available [3]. The red symbol shows the Ge system corrected to include conﬁgurational electronic entropy [4]. The inset shows the out-of-range data for the AlSb and GaSb compounds. Reproduced from Reference [2].

conﬁgurational electronic entropy, as described in Chapter 1, may play a role in certain systems such as Ge. The reported value of the conﬁgurational electronic entropy of fusion of Ge is included in the red symbol shown in Figure 5-1.

81 The ability to predict the electronic entropy of fusion (Sf )fromelectronicprop- erties of the solid and liquid phases of a molten semiconductor material demonstrates the validity of the framework presented in Chapter 3. The electronic properties and the macroscopic thermodynamic variable electronic entropy have been shown to be quantitatively connected through the use of published datasets from literature.

82 References

[1] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the state diagrams - 5. Clas- siﬁcation of electronic semiconductor melts”. In: Journal of Thermoelectricity 1 (2004), pp. 32–47.

[2] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically accessible electronic properties in high temperature systems”. In: Philosophical Magazine 96.29 (2016), pp. 3041–3053.

[3] A. Regel, V. Glazov, and A. Aivazov. “Calculation of components of fusion entropy of some semiconducting compounds”. In: Sov. Phys. Semicond. 8.11 (1975), pp. 1398–1401.

[4] B. Chakraverty. “Conﬁgurational entropy of electrons in semiconductors”. In: Radiation Eﬀects 4.1 (1970), pp. 39–43.

83 84 Chapter 6

ACriterionforMolten Semiconductivity

The model provided herein (Chapter 3) and validated by appeal to existing datasets (Chapters 4 and 5) can be used to predict whether a solid-state semiconductor compound will, upon melting, retain its semiconducting properties by undergoing a semicondutor-to-semiconductor transition (SC-SC) or metallize by undergoing a semiconductor-to-metal transition (SC-M). A thermodynamic stability condition analysis is provided herein. For a more complete derivation and discussion see Reference [1].

6.1 Stability Analysis of Molten State

A system with a congruent melting solid-state semiconductor compound is considered. The question of whether the liquid phase of this system will metallize (SC-M) or retain its semiconducting properties (SC-SC) is equivalent to the question of which of these hypothetical material states has the lowest free energy. A molten semiconductor is a system for which the inequality expressed in Equation 6.1 holds.

SC M Gmix < Gmix (6.1)

85 SC Gmix is the Gibbs free energy of mixing of the hypothetical molten semiconductor M (superscript SC) and Gmix is the Gibbs free energy of mixing of the hypothetical metallized molten state (superscript M). Breaking Equation 6.1 into enthalpic and entropic terms yields Equation 6.2.

HSC T SSC < HM T SM (6.2) mix mix mix mix As discussed in Chapters 1 and 3, and Reference [2], the entropy of mixing of SC amoltensemiconductor,Smix,isapproximatedbytheelectroniccomponentof the entropy of mixing Se.Further,systemsthatmetallizeexhibitminimalshort- range order and the entropy is dominated by the conﬁgurational component, and M consequently the entropy of mixing of the metallized state (Smix)ismaximallythe ideal entropy of mixing Sideal. Thus, Equation 6.2 can be simpliﬁed and rearranged to provide a stability condition for the electronic entropy of mixing of the molten semiconductor state.

SC M Hmix Hmix Se > +Sideal (6.3) T

The diﬀerence in enthalpies of mixing of the molten semiconductor and metallized molten states can be approximated by a diﬀerence in the enthalpies of fusion of these two hypothetical molten states for temperatures in the vicinity of the liquidus. A derivation is provided in Appendix E. A more thorough analysis and explanation is provided in Reference [1].

HSC HM HSC HM (6.4) mix mix ⇡ f f Equation 6.4 allows the use of the Richard’s Rule equivalent provided in Appendix D which quantiﬁes the relationship between enthalpy of fusion and melting temperature for molten semiconductor and metallized molten systems. The investigation of the thermodynamics of molten semiconductor systems has revealed that enthalpies of fusion of solid semiconductor systems follow predictable trends - an analogy to the Richard’s Rule for metallic systems [3]. We herein use this novel relationship to

86 enable the stability analysis. Thus Equation 6.3 is approximately equivalent to Equation 6.5.

SC M Hf Hf Se > +Sideal (6.5) T

For simplicity we deﬁne the term ⇠:

SC M Hf Hf ⇠ = +Sideal (6.6) T

Thus, the stability condition for a system to behave as a molten semiconductor is simply:

Se >⇠ (6.7)

6.2 Application to the Te-Tl System

Electronic property data (Donally and Cutler) and thermodynamic mixing properties (Nakamura and Terpilowski) are available for the Te-Tl system over a range of compositions at 770 K [4–8]. The electronic entropy of mixing (Se) of the Te-Tl system was calculated by in Reference [2]. The phase diagram of the Te-Tl is provided in Figure 6-1. Equation 6.5 relates the electronic entropy of mixing to the ideal entropy of mixing and an enthalpy-dependent term. Reference to the Richard’s Rule equivalent presented in Appendix D demonstrates that this enthalpic term is negative. It can thus be stated that if the electronic entropy of mixing exceeds the ideal entropy of mixing, the system will behave as a molten semiconductor. The electronic entropy of mixing (Se)andtheidealentropyofmixing(Sideal) of the Te-Tl system are presented in Figure 6-2. The electronic entropy of mixing exceeds the ideal entropy of mixing for certain compositions of the Te-Tl system. As described above, from this alone it can be concluded that the system must behave as a molten semiconductor for a subset of

87 1 2 3 T / K / T

Te Tl at. frac. Tl

Figure 6-1: Phase diagram of the tellurium-thallium system as reported by Okamoto. Region 1 indicates the semiconducting molten phase. Region 2 indicates a two-phase region. Region 3 indicates a metallized molten phase. [9] compositions, without appeal to enthalpic terms. However, it is not possible to determine the speciﬁc range of compositions over which a system behaves as a molten semiconductor without the addition of the enthalpic terms.

The enthalpies of fusion of the hypothetical molten semiconductor and metallized molten systems are predicted by the method described in Appendix E leveraging the above-described Richard’s Rule equivalent. These values are used in conjunction with Equation 6.6 to predict the value of ⇠ as a function of composition. Figure 6-3

88 S (Rinzler 2016) ∆ e S ∆ ideal 1 2 3 -1 K -1 J mol / /

mix Δ S

Te Tl at. frac. Tl

Figure 6-2: Se (solid) as calculated by Reference [1] and the ideal entropy of mixing (Sideal dashed) for Te-Tl at 770 K [2]. The miscibility gap (region 2) is shaded.

provides ⇠ and Se as a function of composition.

From Figure 6-3 it can be seen that for the entire range of composition except for Region 3, the electronic entropy of fusion exceeds the ⇠ parameter. From Figure 6-1, Regions 1 and 2 correspond to the molten semiconductor phase and the miscibility gap respectively. Region 3 corresponds to a metallic phase. Consequently Equation 6.7 has accurately predicted the electronic nature of the molten phases of the Te-Tl system.

89 ∆S ξ e

1 2 3 -1 K -1 J mol

Te Tl at. frac. Tl

Figure 6-3: Se (solid) as calculated by Reference [2] and the ⇠ parameter from Equation 6.6 (dashed) for Te-Tl at 770 K. The ⇠ parameter is not interpreted to have a physical meaning in the presence of a miscibility gap (the shaded region) [1, 2].

6.3 Discussion

The stability analysis provided herein can provide a prediction of the electronic properties of the molten state through the use of electronic properties of the solid state and the Richard’s Rule equivalent provided in Appendix D. It is proposed that this method could be coupled with a method to predict the enthalpy of fusion (i.e. via

90 atomistic modeling) to provide a more broadly applicable method for prediction of molten semiconductivity. The success in predicting the electronic behavior of the Te-Tl system as a function of composition provides additional support for the validity of the theory presented in Chapter 3. The method of prediction of electronic behavior by stability criterion presented herein overcomes the previous inability of the theory of noncrystalline high temperature materials to provide empirically veriﬁable predictions of use for the material scientist and practicing metallurgist. Further, the ability to predict molten semiconductivity without appeal to direct empirical measurement based on thermodynamic analysis addresses the scientiﬁc gap discussed in Chapter 1.

91 92 References

[1] C. Rinzler and A. Allanore. “A thermodynamic basis for the electronic properties of molten semiconductors: the role of electronic entropy”. In: Philosophical Magazine 6435.January (2016), pp. 1–11.

[3] T. Iida and R. Guthrie. The Thermophysical Properties of Metallic Liquids. Oxford, UK: Oxford University Press, 2015.

[4] Y. Nakamura and M. Shimoji. “Thermodynamic Properties of the Molten Thal- lium + Tellurium System”. In: Trans. Faraday Soc. 67 (1971), pp. 1270–1277.

[5] C. Oh. “Assessment of the Te-Tl (Tellurium-Thallium) System”. In: Journal of Phase Equilibria 14.2 (1993), pp. 197–204.

[6] J. Donally and M. Cutler. “Hall Measurements in Liquid Thallium-Tellurium”. In: Physical Review 176.3 (1968), pp. 1003–1004.

[7] M. Cutler. Liquid Semiconductors. New York, NY: Academic Press, 1977.

[8] M. Cutler and C. Mallon. “Thermoelectric Study of Liquid Semiconductor So- lutions of Tellurium and Selenium”. In: The Journal of Chemical Physics 37.11 (1962), pp. 2677–2683.

[9] H. Okamoto. “Te-Tl (Tellurium-Thallium)”. In: Journal of Phase Equilibria 12.4 (1991), pp. 507–508.

93 94 Chapter 7

Prediction of Metallization Temperature of Molten Semiconductor Systems

Chapter 6 presents a method of performing a thermodynamic stability analysis to assess the molten behavior of electronic systems and, specifically, to predict whether a system that is a semiconductor in the solid phase will retain semiconducting properties in the molten phase. As described in Chapter 1, systems that undergo a semiconductor-to-semiconductor (SC-SC) transition at melting are known as molten semiconductors. These systems exhibit evolving electronic properties as a function of temperature. At some critical temperature Tc molten semiconductors experience a semiconductor-to-metal (SC-M) transition (if not preempted by vaporization). It is desirable to predict this critical temperature of metallization for several reasons. First, the study of electronic order- disorder transitions is of interest to the fields of high temperature superconductivity, geology, astrophysics, and materials processing. Secondly, as described in Chapter 1, the SC-M transition is correlated with the critical point of a miscibility gap in binary molten semiconductor systems. This correlation exists because immiscibility is driven by a difference in short-range order between two phases, one of which is a molten semiconductor. The short-range order is also responsible for the electronic

95 properties of molten semiconductors. Consequently, an evolution of the short-range order to allow full miscibility is correlated with a reduction of the short-range order that leads to molten semiconductivity. Presented herein is an extension of the stability analysis of Chapter 6 to predict the critical temperature of the semiconductor-to-metal transition of molten semiconductor systems.

7.1 Method

The key result of Chapter 6 is restated here:

HSC T SSC < HM T SM (7.1) mix mix mix mix

Equation 7.1 indicates the stability condition for the molten semiconductor phase to be energetically preferred over the molten metallic phase. The temperature at which the inequality becomes an equality is the critical temperature Tc. Appendix E derives the relationship between the diﬀerences in enthalpies of mixing of a system and the diﬀerences in the enthalpies of fusion of a system between the hypothetical molten semiconductor and metallized molten states. The result is restated here:

T HSC HM =HSC HM + cSC cM dT (7.2) mix mix f f P P ZTm The diﬀerence in enthalpy of fusion between the molten semiconductor and molten metal states has been shown to follow a regular pattern and can be predicted by appealing to the Richard’s Rule analog presented in Appendix D.

At a speciﬁc critical temperature (Tc) the inequality in Equation 7.1 will no longer hold, and the system will exhibit a SC-M transition. This temperature can be calculated by combining Equations 7.1 and 7.2 and solving for T. However, knowledge of the heat capacity as a function of temperature for hypothetical states is not often readily available, nor necessarily are the terms associated with the temperature

96 dependence of the electronic entropy of mixing.

In lieu of performing the full calculation, a first approximation to Tc can be made by making some simplifying assumptions. There is an excess stability to the molten semiconductor state above the molten metal state at melting. The difference in heat capacities between the molten semiconductor and metal states, integrated over temperature, decreases this excess. At a critical temperature, there is no excess stability to the molten semiconductor state and a SC-M transition occurs. The question then becomes at what temperature above melting is the excess stability of the molten semiconductor state eliminated by a change to the differences in free energies of mixing of the molten semiconductor and molten metal states as accounted for by integrated heat capacities. The most basic assumptions one could make are to linearize the temperature-dependent components of Equation 7.2. To facilitate a calculation of the critical temperature, several assumptions are thus employed:

1. The Richard’s Rules of Appendix D hold for enthalpies of mixing

2. Heat capacities are constant in temperature

3. The heat capacity accounts for the temperature dependence of Se

4. The Drude model predicts the conductivity of the molten semiconductor state

Under these assumptions, the critical temperature can be solved for:

SC M M SC Hf Hf + Tm(cP cP ) Tc = (7.3) S S + cM cSC e ideal P P

7.2 Calculation of the Metallization Temperature of FeS

The melting temperature of FeS (1194 Celsius) was used in conjunction with the Richard’s Rule of Appendix D to calculate HSC HM . The electronic entropy f f of mixing (Se)wastakenastheSex term reported in Reference [1]. The heat

97 1 1 capacities of the semiconductor and metallic states are 62.55 J mol K and 42.02 1 1 Jmol K respectively. The heat capacities of the semiconductor and metallic states of FeS are taken from the NIST database at melting temperature. All values are calculated for the compound composition FeS. The entropy of mixing of the M metallic state, Smix,isassumedtobetheidealmixingterm(seeReference[2]fora discussion on the validity of this assumption).

Solving Equation 7.3 predicts a metallization temperature of 1305 Celsius. Fig- ure 7-1 shows the empirically determined phase diagram of the Fe-S system with a measured critical point of the phase diagram at 1320 Celsius. Thus, the stability analysis prediction reasonably predicts the critical point of the miscibility gap of the Fe-S system.

1305

Figure 7-1: The Fe-S phase diagram per Reference [3]. The metallization temperature prediction of FeS of 1305 Celsius according to Equation 7.3 is shown in blue.

98 7.3 Calculation of the Metallization Temperature of the Te-Tl system

The melting temperature of Tl2Te (415 C) was used in conjunction with the Richard’s Rule of Appendix D to calculate HSC HM . The electronic entropy of mixing f f (Se) was taken from the results of Chapter 4. The heat capacities of the semicon- 1 1 ductor and metallic states are 38.16 and 29.29 J mol K respectively. The heat capacities of the semiconductor and metallic states of Tl2Te are taken from Reference

[4]. All values are calculated for the compound composition Tl2Te. The entropy of M mixing of the metallic state, Smix,isassumedtobetheidealmixingterm(see Reference [2] for a discussion on the validity of this assumption).

Solving Equation 7.3 for Tl2Te predicts a metallization temperature of 746 Cel- sius. Reference [5] reports the measured metallization temperature of Tl2Te as 757.

7.4 Discussion

The predictions provided herein predict metallization temperatures that are lower than the empirically measured values. This can be explained by the approximate nature of our assumption of constant heat capacity. Having a temperature-dependent heat capacity would enable a more accurate deployment of this model. In Chapter 8, afullanalysisoftheFe-Ssystemisprovidedthatdoesnotdependonanassumption of heat capacity data and the resulting prediction of metallization temperature is shown to more closely agree with measurement. The stability analysis provided in Chapter 6 has been extended to predict the metallization temperature of systems that behave as molten semiconductors. The key enabling factor in this stability analysis (i.e. why it was not possible previously) is the quantitative prediction of the entropy of mixing of the system. The agreement with literature-reported values for the critical point and metallization of the Fe-S and Te-Tl systems, respectively, indicates the validity of assigning the electronic entropy provided by the theory in Chapter 3 to the equilibrium thermodynamic entropy.

99 100 References

[1] F. Kongoli, Y. Dessureault, and A. Pelton. “Thermodynamic Modeling of Liquid Fe-Ni-Cu-Co-S Mattes”. In: Metallurgical and Materials Transactions B 29B (1998), pp. 591–601.

[3] E. Ehlers. The interpretation of geological phase diagrams.W.H.Freeman,San Francisco, CA, 1972.

[4] F. Kakinuma, S. Ohno, and K. Suzuki. “Heat Capacity of Liquid Tl-Te Alloys”. In: Journal of the Physical Society of Japan 60.4 (1991), pp. 1257–1262.

[5] J. Enderby and A. Barnes. “Liquid Semiconductors”. In: Rep. Prog. Phys. 53 (1990), pp. 85–179.

101 102 Chapter 8

Prediction of Features of Phase Diagrams

The prediction of thermodynamic properties of high temperature molten slag systems in a CALPHAD framework relies on the optimization of free energy models of solutions by fitting to empirical data (see Chapter 1 for an overview of thermodynamic modeling of phase diagrams). For systems exhibiting subregularity, excess entropy terms are often required to accommodate the complexity of the energetics of the molten phase. The different methods for generating free energy models (i.e. quasichemical, CVM, etc.) typically employ configurational entropy models that at- tempt to approximate the physical reality of the system. For multicomponent systems where configurational entropy dominates the total entropy of mixing in the molten phase, the quasichemical approach proves accurate in modeling key features of phase diagrams, including miscibility gaps. However, certain systems exhibiting strong short-range order in the melt have substantial contributions to the entropy of mixing that are not calculated by the quasichemical, or other, methods. Instead, these contributions are lumped into an excess term. The form of the excess term is not based on a physical model of the entropy as afunctionofcompositionortemperature,butisrather,typically,apolynomialfitto empirical data. It has been shown that for certain important systems (i.e. sulfides, tellurides, selenides, antimonides, arsenides, and certain oxides), known as molten

103 semiconductors, the electronic entropy plays a critical role in the thermodynamics of mixing of the molten phase. These systems have been correspondingly challenging to model through existing CALPHAD methodologies. In particular, the accurate prediction of miscibility gaps, and the critical points of miscibility gaps, has proven elusive for systems as well-studied and industrially relevant as Fe-S.

As described in Chapter 1, physical models for the electronic properties (i.e. electronic conductivity and thermopower) of molten semiconductor systems as a function of temperature have been developed. The theory presented herein (see Chapter 3) provides a quantitative connection between these properties and the electronic entropy of a system. Thus, for molten semiconductors, a temperature-dependent determination of the electronic entropy of a system from electronic properties is possible.

T E 0 E0 +(k )T Se = wmine kT (8.1) eµT

The terms in Equation 8.1 are deﬁned in Chapter 1.

For systems where the electronic entropy of mixing approximates the total entropy of mixing (i.e. molten semiconductors), the excess entropy term can be approximated by the electronic entropy of mixing term. Consequently, a functional form of the excess entropy vs. temperature with a physical basis in the electronic properties of high temperature systems is determinable based on the relationship in Equation 8.1.

Sex Se (8.2) ⇡

Se is the electronic entropy of mixing as described in Chapter 3.

Herein, these methods are applied to the Fe-S system in a quasichemical framework. The improved model for excess entropy is shown to correct an erroneous prediction of the miscibility gap critical point for the Fe-S system.

104 8.1 Method

Equation 8.1 provides the functional form of the temperature dependence of the electronic entropy. It is desirable to translate this into a description of the excess entropy of mixing as a function of temperature to enable incorporation into existing quasichemical methods deployed in, for example, the FactSage framework. Appendix Adescribesthemethodtotranslateelectronicentropyintoexcessentropyofmixing. The key result (shown for a two-component A-B molten semiconductor system) is reproduced here.

A B Sex = Se xAS xBS (8.3) e

A B xA and xB are the concentration of species A and B respectively. Se and Se are the electronic entropies of the end-members A and B respectively. This method is equally applicable to multicomponent systems. Equations 8.1 and 8.3 can be combined to give:

T E 0 E0 +(k )T A B Sex =Se = wmine kT xAS xBS (8.4) eµT e e The following assumptions are employed in the present method:

1. The electronic entropy of mixing approximates the excess entropy of mixing

2. The Drude model predicts the electronic conductivity of the melt

3. The electronic entropies of the end-members are constant in temperature

4. The electronic mobility is constant in temperature

8.2 Calculation of the Excess Entropy of the Fe-S System

Fe-S has been modeled in the quasichemical framework by Pelton et al. in Reference [1] and incorporated into FactSage. For an overview of the quasichemical framework,

105 see Reference [2]. This model was recreated in software of the author’s design to enable direct modiﬁcation of excess parameters. The excess entropy of mixing as calculated by Pelton et al. was optimized to reproduce the empirically-determined liquidus, and thus well-reﬂects the excess term at the melting point. To incorporate the temperature dependence associated with the physics of electronic properties in molten semiconductor systems, Equation 8.4 was solved. The properties of the compound FeS at melting were used to calculate the temperature- dependent excess entropy of mixing. The E0 of FeS used is 0.5 eV [3, 4]. The min 1 1 23 used is 300 ⌦ cm . is 6.35 x 10 .

The calculated Sex replaced the optimized Sex of Pelton and the quasichemical equilibrium was reestablished and free energy minimized in the authors’ software.

8.3 Calculation of the Miscibility Gap of the Fe-S System

The minimized free energy of the Fe-S system was used to generate predictions of miscibility gaps in the software. Figure 8-1 shows the phase diagram of the Fe- SsystemasempiricallydeterminedandprovidedbyReference[5],aspredictedby Pelton et al. by the quasichemical method implemented in FactSage, and as predicted by the method of this chapter. The excess entropy of mixing predicted by Equation 8.4 has corrected the error in the prior calculation. The critical point of the miscibility gap is predicted to be 1312 Celsius.

8.4 Discussion

The miscibility gap modeled with the modiﬁed excess entropy of mixing, informed by Equation 8.4, exhibits superior agreement with the empirically determined miscibility gap than the prior predictions within the FactSage framework. The chief reason for this improvement is the temperature dependency of the excess term. Whereas the slope of the excess entropy of mixing according to the optimized parameters of

106 2000

1500 T / C / T

1000

500

0 0.2 0.4 0.6 0.8 1 S / at. frac.

Figure 8-1: The Fe-S phase diagram. The dashed black line is the empirically determined miscibility gap [5]. The blue dashed line is the miscibility gap predicted by Pelton et al. through use of the quasichemical method per Reference [1]. The red dashed line is the miscibility gap predicted by the use of the modeled entropy of mixing per Equation 8.4.

1 2 Reference [1] is 0.000067 J mol K ,thatofthemodiﬁedexcessentropyofmixing 1 incorporating the temperature dependency of electronic properties is 0.006 J mol 2 K ,afactorofover90greater.

The electronic properties, which have been shown to quantitatively determine the electronic entropy, evolve as a function of the short-range order of the system. This in turn is a function of the temperature - as the system increases in temperature, the entropic beneﬁt of reduction of short-range order exceeds the enthalpic beneﬁt of bonds in the melt. The molten semiconductor transitions to a metallic state, typically within a few hundred degrees. The impact of this transition on the entropy of mixing

107 is substantial, and determines the closure of the miscibility gap (see Reference [6]). By bringing a physical description of entropy into a CALPHAD framework, the need for challenging-to-gather empirical data was reduced, and substantial improvement of the predictive capacity of the model was demonstrated.

108 References

[1] F. Kongoli, Y. Dessureault, and A. Pelton. “Thermodynamic Modeling of Liquid Fe-Ni-Cu-Co-S Mattes”. In: Metallurgical and Materials Transactions B 29B (1998), pp. 591–601.

[2] A. Pelton et al. “The Modiﬁed Quasichemical Model I - Binary Solutions”. In: Metallurgical and Materials Transactions B 31B (2000), pp. 651–659.

[3] A. Rohrbach, J. Hafner, and G. Kresse. “Electronic correlation eﬀects in transition- metal sulﬁdes”. In: Journal of Physics: Condensed Matter 15.6 (2003), pp. 979– 996.

[4] F. Ricci and E. Bousquet. “Unveiling the Room-Temperature Magnetoelectric- ity of Troilite FeS”. In: Physical Review Letters 116.22 (2016), pp. 1–6.

[5] E. Ehlers. The interpretation of geological phase diagrams.W.H.Freeman,San Francisco, CA, 1972.

[6] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically accessible electronic properties in high temperature systems”. In: Philosophical Magazine 96.29 (2016), pp. 3041–3053.

109 110 Chapter 9

Experimental Methods and Results

Theory has been presented which connects measurable electronic properties to equilibrium thermodynamic properties (Chapter 3). The theory has been validated by appeal to existing empirical data for, in particular, molten semiconductor systems (Chapters 4 and 5). It has been proposed that the new connection identiﬁed can enable the empirical generation of thermodynamic data through the collection of electronic property data. Herein are provided methods and results for empirical access to electronic entropy in the molten phases of systems. Under assumption of the validity of the Drude model of conductivity (see Chapter 1), the key electronic measurements required to permit a prediction of the electronic entropy of a system per the theory presented in Chapter 3 are the electronic conductivity and the thermopower. A review of a selection of the previous apparatuses developed for measurement of these properties in the molten phase of systems is provided and is followed by a brief description of two new methods developed for the investigation of molten semiconductor systems.

9.1 Review of Apparatuses from Previous Researchers

Typical apparatuses for investigating the electronic properties of the molten state comprise a sealed crucible material of quartz, alumina, or boron nitride and one or more electrode materials including graphite, molybdenum, and platinum. Follow-

111 ing are descriptions of two apparatuses that are indicative of those used by past researchers.

9.1.1 Quartz Test Cell

Some of the earliest measurements on molten semiconductor systems were performed by Cutler and Mallon [1]. A sealed quartz ampoule was used with two platinum leads to measure the electronic conductivity and thermopower of tellurium. Two platinum /10%rhodiumthermocoupleswereused,andthepureplatinumleadswereusedas additional electrodes for establishing a 4-point measurement. The cell was placed in a resistive heater, and Joule heating of the system was provided by transiently providing current between the platinum leads. Thermopower measurements were then recorded by measuring a diﬀerence in potential between the pure platinum leads and the thermal gradient.

This cell design works well for systems with melting temperatures below 1000 Celsius, but the quartz limits the maximum temperature of operation. Many molten semiconductors of interest have substantially higher melting temperatures.

9.1.2 Boron Nitride Test Cell

Sklyarchuk et al. designed and operated a sealed boron nitride test cell for use with, for example, telluride and selenide molten systems [2]. The test cell comprised a vertical boron nitride crucible with 2 internal diameters and 4 radially impinging graphite electrodes (2 each per diameter) each with a thermocouple embedded in the graphite. 4-point electronic measurements were made by establishing a 10 to 20 K temperature gradient provided by a resistive heating furnace containing the crucible. Argon pressures of up to 50 MPa were provided to stabilize the vapor phase. This cell designed was used to measure the metallization temperature of molten semiconductor systems. The robust design relied on the machinability and toughness of boron nitride. Unfortunately, this crucible material is not compatible with all molten semiconductor systems [3].

112 9.2 Dynamic Induction Test Cell

The above-described apparatuses provided empirical access to the electronic properties of the molten state for a subset of material systems at a subset of temperatures of interest. Additionally, the systems typically required long cycles to achieve steady- state thermal conditions due to the method of heating (resistive). There is an interest in providing an experimental tool that enables the rapid collection of electronic property data of molten systems as a function of temperature.

9.2.1 Apparatus Design

For this purpose, an induction-heated dynamic test cell was developed. A schematic of the apparatus is provided in Figure 9-1. An UltraFlex M25/150 induction heater provided with a 5-loop internally water- cooled copper coil, capable of delivering 25 kW of power at a frequency of up to 150 kHz is inductively coupled to a molybdenum susceptor. This is enclosed in a 304 stainless steel container that is actively purged with laboratory grade argon (to protect the molybdenum susceptor from oxidation). The enclosure additionally provides electromagnetic shielding from the high frequency induction system. An alumina crucible is placed within the molybdenum susceptor to contain the material sample. The sample is sealed by Swagelok vacuum fittings outside of the hot-zone. The vacuum fittings pass the probe through a radial o-ring seal. The probe is attached to a Zaber T-LSR linear stage that is digitally controlled such that the probe can be moved vertically through the hot zone and into and through the sample. A crash-detector is provided that ceases the motion of the linear stage should a sufficient force (0.5 lbs) be placed upon the probe. The entire system is contained within a transparent sealed acrylic container that is purged with argon. Pass-throughs are provided for the electronic connections to the probe and linear stage. The furnace is controlled by a temperature controller (Omega CNI3233) connected to a B-Type thermocouple probe in contact with the molybdenum susceptor.

113 Probe

304 SS Shield

Molybdenum Susceptor Alumina Crucible

Copper Coil

Figure 9-1: The dynamic induction test cell comprises a copper induction coil inductively coupled to a molybdenum susceptor and contained within an argon-purged 304 stainless steel shield. A sample is contained in an alumina crucible and a probe is vertically positioned by a linear stage external to the shield.

AschematicofatypicalprobeisprovidedinFigure9-2.

The probe is comprised of a 4-bore alumina tube (Coorstek 99.8% alumina). A B-Type thermocouple probe is sealed with alumina paste into one of the bores terminating distally. Three molybdenum rods are threaded into sharpened graphite electrodes (EDM grade) which are alumina pasted into place, one into a distally terminating bore and two into proximally terminating bores.

Electronic conductivity measurements are performed using a Gamry Reference 3000 using Electrochemical Impedance Spectroscopy (EIS) between the two proximally terminating graphite electrodes. Amplitudes of less than 20 mA and a frequency range of 5-100 Hz are used to perform EIS measurements. Potential measurements are made using a Gamry Reference 3000 measuring the Open Circuit Potential (OCP) between one proximally terminating graphite electrode and one distally terminating graphite electrode.

114 Thermocouple

Alumina 4-bore Tube

Graphite Electrode

Figure 9-2: The probe is comprised of a 4-bore alumina tube containing 3 molybdenum wires terminating in graphite electrodes. The 4th bore contains a B-Type thermocouple probe.

The molybdenum susceptor establishes a vertical temperature gradient in the sample. Thus, the proximally and distally terminating bores of the probe are located in regions of distinct temperature. The probe is scanned at a slow rate (typically 0.1 mm per minute) and a temperature proﬁle of the system is measured while electronic conductivity and potential data are gathered. From these data the thermopower and temperature dependent electronic conductivity are determined.

Probes are typically calibrated in a material of known electronic property (such as gallium or tin).

By this method, electronic conductivity and thermopower vs. temperature data can be generated in a single experiment. Further, the time to achieve temperatures in excess of 1000 Celsius is less than one hour due to the high power coupling of the induction furnace to the sample.

115 9.2.2 Apparatus Performance

The induction furnace test cell allows for rapid testing and dynamic scanning through atemperaturegradient.However,thepresenceoflargeoscillatingelectromagnetic fields that may be operated discontinuously (i.e. in an on-offfashion) presents some challenges for certain electrochemical measurements that depend on potential stability (such as Alternating Current Voltammetry). For measurements of thermopower and electronic conductivity that require a stable temperature gradient, however, the apparatus provides sufficient thermal stability.

9.2.3 Results for Pb-S

The lead-sulfur system is a p-type molten semiconductor with a melting temperature of 1114 Celsius.

Figure 9-3 shows the thermopower vs. temperature for molten PbS. The literature reported value at melting is shown for reference. Sigma Aldrich 99.9% PbS was used (item number 372595).

Figure 9-4 shows a Cyclic Voltammetry (CV) plot of PbS at 1120 Celsius. The literature reported expected relationship is shown for reference.

The experimental results for PbS agree with literature reported values. These can be used to generate entropy of mixing predictions for the PbS system in accordance with the methods of Chapter 3.

9.3 Static Test Cell

Certain molten semiconductor systems (i.e. sulﬁdes) exhibit large vapor pressures of certain species in the molten phase. It is thus desirable to control the gas phase environment above the melt to suppress the vaporization of volatile species. A sealed- cell apparatus was built to satisfy this experimental requirement.

116 oddit h elbfr eln na ro ugdenvironment. purged argon is an composition in desired sealing of before sample cell The the into Pelco)). Pella loaded (Ted paste carbon by quartz the shows 9-5 Figure test-cell. the hosts argon with purged aschematicofthetestcell. is that element. tube heating primary quartz the sealed as used A is furnace tube split TF-55035A Blue Lindberg A Design Apparatus the in gathered [4]. value 9.3.1 data literature-reported represent a PbS circles represents compound circle black red the The The for cell. provided. temperature test of is induction function phase dynamic a molten as the thermopower in The 9-3: Figure h eli opie faqat uesae ihgaht lcrds(oddto (bonded electrodes graphite with sealed tube quartz a of comprised is cell The μV K-1 Measured Data Reported Value 117 Measured Data Projected Reported Value

Figure 9-4: The Cyclic Voltammogram at 1120 Celsius for the compound PbS in the molten phase is provided. The black circles represent data gathered in the dynamic induction test cell. The red line represents a literature-reported value [5].

Threaded molybdenum wires are attached to the graphite electrodes forming a 4-probe circuit. K-type thermocouple probes are contacted to predetermined insets into the graphite electrodes. Both the molybdenum wires and the thermocouples are contained within a 4-bore alumina tube (Coorstek 99.8% Alumina). A Ni-Chrome wire heating element is provided on the upstream side of the sample cell (with respect

118 to the direction of the argon ﬂow). The Ni-Chrome heating element is controlled independently from the furnace with a DC power supply (BK Precision 1735).

Quartz Tube Ni-Chrome Molybdenum Wires

Graphite Electrode Thermocouple

Figure 9-5: The static test cell comprises a quartz tube sealed with graphite electrodes. Molybdenum wires are threaded into the graphite to establish a 4-probe measurement. Thermocouples are attached to the graphite electrodes for measurement of the thermal gradient. A Ni-Chrome wire coil is used to establish a temperature gradient.

Electronic conductivity measurements are performed using a Gamry Reference 3000 using Electrochemical Impedance Spectroscopy (EIS) between the two graphite electrodes. Potential measurements are made using a Gamry Reference 3000 measuring the Open Circuit Potential (OCP) between the graphite electrodes. Both measurements are made with a 4-probe conﬁguration.

Experiments are conducted by heating the cell in the split-tube furnace at a rate of 5 Celsius per minute to 10 Celsius below the melting temperature of the material. The furnace is then set to a speciﬁed temperature and the sample is allowed to reach steady-state (typically around 15 minutes). An EIS spectrum is taken in isothermal conditions. The secondary Ni-Chrome heating element is then activated and the system is allowed to reach a steady state temperature. OCP is then recorded along with the temperature gradient across the cell. This is repeated at each temperature interval desired by the operator.

119 9.3.2 Apparatus Performance

The thermal stability of this apparatus is superior to the induction cell described above. Further, there are no sources of electromagnetic interference. A typical experiment takes on the scale of 10 hours of preheating and 4 hours of operation to determine the electronic properties vs. temperature for a speciﬁc sample composition.

9.3.3 Results for Sn-S

Astudyoftheentropyofmixingofmoltentin-sulfursystemasafunctionofcom- position and temperature was performed with the static test cell. Sn-S is a molten semiconductor. 4 compositions on the tin-rich side of the phase diagram were studied. These are indicated on the phase diagram of the Sn-S system shown in Figure

9-6. The temperature study was performed at least approximately 60 Celsius above the melting temperature of the compound to ensure that no resolidiﬁcation occurred during the experiment. Samples were prepared per the method of Zhao [7]. 99.5% sulfur from Sigma- Aldrich (item number 84683) and 99.85% tin (mesh 100 powder) from Alfa Aesar (item number 00941) were used. Electronic conductivity and thermopower measurements were performed using the static test cell. The method presented in Chapter 4 was used to translate these data to predictions of the entropy of mixing of the Sn-S system vs. temperature. The results are shown in Figure 9-7. There are several features of the entropy of mixing that are worth noting. At the compound composition (SnS) the entropy of mixing is negative near the melting temperature. This is anticipated due to the strong short-range order expected for molten semiconductor systems. The entropy of mixing achieves a minimum at the stoichiometric compound SnS. This is anticipated as is discussed in Chapter 1. The entropy of mixing increases as a function of temperature for all measured

120 1100

1050

1000

950 Temperature / C / Temperature

900

850 0.46 0.48 0.5 0.52 at. frac. S

Figure 9-6: The partial phase diagram of the Sn-S system [6]. Red dashed lines indicate the selected compositions and temperature ranges for measurement with the Static Test Cell. compositions. This is expected due to the activation of additional carriers as a function of temperature, as described by the temperature-dependent electronic property relationships discussed in Chapter 1.

To the author’s knowledge, this is the ﬁrst reported data for entropy of mixing for the molten tin-sulfur system. It is hoped that an experimental investigation into the thermodynamic properties of Sn-S will be pursued to validate the predictions of the entropy of mixing of molten Sn-S as provided herein.

121 1100 C 1080 C 1060 C 1040 C 1020 C 1000 C -1

K 980 C -1 960 C

/ J / mol 940 C mix Δ S

Figure 9-7: The entropy of mixing (Smix)oftheSn-Ssystemasafunctionof temperature. The electronic conductivity and thermopower of the Sn-S system were measured with the Static Test Cell. The method of Chapter 4 was used to generate entropy of mixing data.

9.4 Discussion of the Experimental Methods

Experimental methods have been developed to investigate the electronic conductivity and thermopower of molten systems. These methods, when combined with the theory presented herein, enable the measurement and prediction of the entropy of mixing of systems in the molten phase. Where available, literature data support predictions based on experimental data generated by these methods. The ability to rapidly generate entropy data through the collection of electronic property data in the

122 melt may enable the rapid thermodynamic investigation of molten phase systems and provide access to previously challenging-to-measure properties that drive the macroscopic thermodynamic behavior of systems, including key features of phase diagrams such as the liquidus and miscibility gaps.

123 124 References

[1] M. Cutler and C. Mallon. “Thermoelectric Study of Liquid Semiconductor So- lutions of Tellurium and Selenium”. In: The Journal of Chemical Physics 37.11 (1962), pp. 2677–2683.

[2] B. Sklyarchuk and Y. Plevachuk. “Nonmetal-Metal Transition in Liquid Cu- Based Alloys”. In: Zeitschrift fur Physikalische Chemie 215.1 (2001), pp. 103– 109.

[3] V. Sklyarchuk and Y. Plevachuk. “Thermophysical properties of liquid ternary chalcogenides”. In: High Temperatures-High Pressures 34 (2002), pp. 29–34.

[4] V. Glazov. Liquid Semiconductors. New York, NY: Plenum Press, 1969.

[6] B. Predel. S-Sn (Sulfur-Tin). Springer Berlin Heidelberg, 1998. Chap. 5 J.

[7] Y. Zhao, C. Rinzler, and A. Allanore. “Molten Semiconductors for High Temper- ature Thermoelectricity”. In: ECS Journal of Solid State Science and Technology 6.3 (2017), N3010–N3016.

125 126 Chapter 10

Extension to Metallic and Ionic Systems

Chapter 1 discusses the dependency of electronic properties on short-range order (SRO). For certain systems (i.e. molten semiconductors) the electronic properties, which determine the electronic entropy, dominate the thermodynamics of mixing (as shown in Chapter 4). However, even for systems where other contributions to the thermodynamics mixing cannot be ignored, the structure-property relationship between SRO and electronic properties still holds. Consequently, information on the ordering properties of materials is accessible through an investigation of the electronic properties of materials. A discussion on the consequences of the theory presented in Chatper 3 on the investigation of the electronic and ordering properties of metallic and ionic systems follows.

10.1 Extension of Theory to Metallic Systems

The Mg-Bi system is homogeneously metallic across composition in the liquid phase. Electronic property data and thermodynamic mixing property data are both available in the liquid phase at 827 Celsius. Figure 10-1 shows the electronic entropy of mixing, as predicted by the method of Chapter 3 from the empirically provided electronic property data gathered by Ratti

127 and Enderby [1, 2]. The empirically-determined total entropy of mixing is provided by Hultgren [3].

at. frac. Bi

Figure 10-1: The total (solid line) and electronic (black circles) entropies of mixing of the Mg-Bi system at 827 Celsius as empirically provided by reference [3] and as predicted by the theory provided in Chapter 3 from electronic property data provided by References [1] and [2].

Because of the metallic nature of the melt, the approximation that the electronic entropy of mixing comprises the total entropy of mixing of the system is not expected to hold. Indeed, the total entropy of mixing is seen to roughly approximate the ideal

128 1 1 entropy of mixing with a peak of 6 J mol K . The maximum contribution of 1 1 electronic entropy of mixing is 2 J mol K . Consequently, the thermodynamics of mixing cannot be explained by the electronic properties alone. However, the electronic entropy of mixing shows a sharp minimum (and negative entropy of mixing) at the composition of 40% Bi. This reﬂects the present of short- range order in the molten phase and, as described in Chapter 2 suggests a compound.

A congruent melting compound (Mg3Bi2)isindeedreflectedinthephasediagramof the Mg-Bi system [4]. Therefore, the connection between electronic entropy, electronic properties, and short-range order enables the qualitative assessment of ordering in metallic systems. The ordering of solid phases is reflected in the short-range order of the molten phase near the liquidus, and electronic property measurements, as translated into the electronic entropy of mixing, offer access to the SRO of the system. Use of experimental methods to measure molten properties (as presented in Chapter 9) to predict solid- phase ordering is thus suggested.

10.2 Extension of Theory to Ionic Systems

Chapter 3 presents a connection between the thermopower of a system and the electronic entropy. However, this implicitly assumes that the electronic properties of a system dominate (i.e. the measured thermopower is due to electronic, rather than ionic, effects). For some systems, and in particular ionic systems, additional contributions to the thermopower may contribute or dominate. Whereas the electronic thermopower describes the electronic response of a system to a perturbation in temperature, the Soret effect describes the chemical potential response of charged atomic species (i.e. ions) to thermal gradients. Wagner has demonstrated an experimental method to isolate the electronic and ionic contributions to the thermopower [5]. The electronic thermopower has been shown to reflect a change in the contribution of electronic entropy to the free energy as a function of temperature. It is thus proposed by analogy that the Soret effect reflects a change in the contribution of the

129 partial entropy of a species to the free energy as a function of temperature. The method of Chapter 3 can be interpreted as measuring a response of the equilibrium property of free energy to a perturbation in temperature. The electronic response provides the electronic entropy. The chemical response provides the partial entropy of species. More broadly, it is proposed that the determination of reversible thermodynamic (equilibrium) properties of systems is possible through the measurement of responses of properties described in an irreversible thermodynamic framework to small perturbations in thermodynamic variables such as temperature. The impact of the measurement of electronic entropy on the quantiﬁcation of equilibrium thermodynamic quantities of ionic systems is further discussed by Rockwood and Tykodi [6–8].

130 References

[1] V. Ratti and A. Bhatia. “Electrical properties of compound forming molten systems : Mg-Bi and Tl-Te”. In: J. Phys. F 5(1975),pp.893–902.

[2] J. Enderby and A. Barnes. “Liquid Semiconductors”. In: Rep. Prog. Phys. 53 (1990), pp. 85–179.

[3] R. Hultgren et al. Selected Values of Thermodynamic Properties of Metals and Alloys.Hoboken,NJ:Wiley,1963.

[4] A. Nayeb-Hashemi and J. Clark. “The Bi-Mg ( Bismuth-Magnesium ) System”. In: Bulletin of Alloy Phase Diagrams 6.6 (1985), pp. 528–533.

[5] C. Wagner. “The Thermoelectric Power of Cells wIth Ionic Compounds Involv- ing Ionic and Electronic Conduction”. In: Progress in Solid State Chemistry 7 (1972), pp. 1–37.

[8] R. Tykodi. Thermodynamics of Systems in Nonequilibrium States.Davenport: Thinkers’ Press, 2002.

131 132 Chapter 11

Future Research

The quantitative connection of electronic and thermodynamic properties of materials has been validated for molten semiconductor and certain metallic systems. The utility of the connection has been discussed for the generation of thermodynamic data of high temperature systems, the prediction of solid-phase compounds from liquid-state data, the prediction of the electronic properties of molten systems, and the analysis of electronic transitions such as metallization. However, there are substantial eﬀorts outside of the scope of the current investigation with are suggested following the proposal of the theory presented in Chapter 3.

11.1 Extension of Experimental Methods for Mea- suring the Entropy of Mixing to New Systems

The experimental methods provided in Chapter 9 provide quick access to the electronic entropy of mixing (and in certain cases total entropy of mixing) of molten systems. To demonstrate the utility of these methods, a more thorough investigation of systems with a wide range of electronic properties is proposed to bound the realm of applicability of the experimental tools.

Four main areas of focus are proposed to extend the current work.

133 11.1.1 Molten Semiconductor Systems

A wide range of molten semiconductor systems have been identiﬁed [1–3]. However, only a small fraction have published thermodynamic and electronic property data as afunctionofcompositionandtemperature.Itissuggestedthattheexperimental methods of Chapter 9 be leveraged to generate these data for additional systems.

11.1.2 Metallic Systems Exhibiting Congruent Melting Com- pounds

Mg-Bi, a system exhibiting a congruent melting compound and metallic conduction in the molten phase, was investigated in Chapter 10. It was shown that the electronic entropy of mixing of the molten phase indicated the presence of a solid-phase compound. It is suggested that the predictive capacity of molten phase measurements on solid-state ordering be further investigated.

11.1.3 Multicomponent Systems

The investigations herein focus on binary (two-component) systems due to the availability of thermodynamic and electronic property data in the molten phase. However, the theory presented in Chapter 3 applies to multicomponent systems as well. It is suggested that experimental investigations into the electronic and thermodynamic properties of ternary and higher-order multicomponent systems are performed to conﬁrm the extension of the present theory.

11.1.4 Ionic Systems

As described in Chapter 10, the quantitative connection between electronic entropy and electronic properties can ﬁnd analogy in ionic systems through the connection of the partial entropy of chemical species to the Soret eﬀect. Electrochemical measurements of ionic systems may thus provide access to entropy data that are otherwise challenging to measure. It is suggested that an analogous investigation as described

134 herein is performed on liquid ionic systems exhibiting the Soret eﬀect.

11.2 Integration of Physical Models of Entropy into aCALPHADFramework

Chapter 8 illustrates the power of implementing a physical model of entropy within a CALPHAD framework to improve the prediction of critical features of phase diagram such as the liquidus and miscibility gaps. For slag systems that exhibit molten semiconductivity (a broad class of chemistries as described in Chapter 1), entropy models based on electronic property relations are directly implementable. Not all systems exhibit the molten semiconductor property of the electronic contribution dominating the total entropy of mixing. However, for these systems the electronic contribution to the total entropy of mixing is still predictable through the use of models of electronic behavior. The implementation of physical models for entropy reduces the burden of obtaining accurate empirical data for the entropy of systems at high temperature (which are often challenging to measure or not investigated). It is proposed that an integration of the physical model for electronic entropy provided in Chapter 3 be implemented in a CALPHAD framework and the consequences on the prediction of key features of phase diagrams be investigated.

11.3 Atomistic Modeling of Molten Semiconductors

The theory provided enables the prediction and measurement of the entropy of mixing of molten semiconductor systems. The entropy of disordered systems is challenging to model with today’s methods. However, the applicability of atomistic modeling to the prediction of the enthalpy of systems, when combined with the theory presented herein, may enable the prediction of the total free energy of molten phases that were previously challenging to model.

135 136 References

[2] V. Glazov. Liquid Semiconductors. New York, NY: Plenum Press, 1969.

[3] M. Cutler. Liquid Semiconductors. New York, NY: Academic Press, 1977.

137 138 Chapter 12

Conclusion

Entropy is critical to the thermodynamics of high temperature phases. Electronic entropy has been shown to substantially contribute to the thermodynamic properties of certain classes of systems, such as molten semiconductors. Theory has been presented (Chapter 3) that quantitatively connects electronic entropy to electronic transport properties (e.g. thermopower). The consequences of this theory, as illustrated herein, and the potential broader impact of this theory on the ﬁeld of materials science are discussed below.

12.1 Demonstrated Consequences of Theory

12.1.1 Modeling of Molten Semiconductors

The electronic entropy has been conﬁrmed to control the thermodynamics of molten semiconductors (Chapter 4). The presented theory (Chapter 3) was applied to predict the thermodynamic properties of the prototypical Te-Tl molten semiconductor from empirical electronic property data and the electronic properties from empirical thermodynamic data (Chapter 4). The theory was able to answer a question posed in the literature regarding a correlation between features of phase diagrams and molten semiconductivity [1]. In Chapter 5 the quantitative connection was extended to predict thermodynamic properties of fusion, and a stability criterion to predict whether

139 a system will behave as a molten semiconductor was developed and veriﬁed with the Te-Tl system (Chapter 6).

12.1.2 Beyond Molten Semiconductors

It has been shown that the investigation and prediction of electronic transitions, such as metallization (the transition of a system from a nonmetallic to a metallic state) of high temperature systems, is enabled by the theory provided herein (see Chapter 7). The thermodynamic basis for key features of phase diagrams in the molten state were explained and quantiﬁed (see Chapter 8). Methods to rapidly collect electronic and entropy data in the molten phase enable access to key thermodynamic data for high temperature systems (see Chapter 9). The connection of electronic entropy to short- range order allows the detection and prediction of solid-phase compounds through the collection of electronic property data in the molten phase (see Chapter 10) and the prediction of thermodynamic quantities of fusion (see Chapter 5). An absolute reference for entropy at temperatures substantially above 0 K is enabled (see Chapter 2). The connection of electronic entropy to short-range order enables the presented model to predict the ordering of systems for which the thermodynamics are not dominated by electronic entropy (Chapter 10). By providing empirical access to a measurement of entropy, predictive thermodynamic models of noncrystalline systems are enabled and a thermodynamic basis for the electronic properties of noncrystalline systems is achieved.

12.2 Potential Impact of Work

12.2.1 Absolute Reference for Entropy

The third law of thermodynamics establishes that the entropy of a system at 0 K is 0. Today, all absolute quantiﬁcations of the free energy, whether empirically determined or modeled atomistically, depend on this reference as an initial point of

140 integration. This requires knowledge of the low-temperature properties of materials (i.e. heat capacity), which are often challenging to measure or model. For the study of high-temperature, and in particular noncrystalline, phases of matter, it would be useful to establish a new absolute reference for entropy. Entropy is a state function. Thus, determination of the absolute entropy of a system is path-independent. This enables the broad use of absolute reference points for the entropy.

It has been suggested in this document that the absolute electronic entropy of certain systems is determinable through the measurement of electronic properties. For certain systems, at certain points in the phase diagram, it has been shown that the entropy of mixing is fully determined by the electronic entropy of mixing. Thus, if the thermodynamic properties (i.e. absolute entropy) of end-members of a system are known (as they are for most pure materials), an absolute measurement of the total entropy of a system is possible.

It is proposed that future research should investigate the consequences of providing these reference states for the atomistic and thermodynamic modeling communities.

12.2.2 Predicting Solid Phase Compounds from Liquid Phase Property Data

Chapter 10 demonstrated the potential utility of using electronic measurements in the liquid phase, as quantitatively transformed into electronic entropy of mixing by the theory of Chapter 3, to detect short-range order that can indicate solid-phase ordering. Anewmethodtodetectandpredictnewcompoundsispotentiallyvaluable.Itis proposed that an experimental investigation of connection between electronic entropy of mixing and the presence of compounds be performed for material systems of interest to determine the extent of utility of the present method for predicting and detecting solid-phase compounds.

141 12.2.3 Unifying Physics of Electronic Properties Across Phases Through Connection to Thermodynamics

Physical models of the electronic properties of systems are typically phase-specific. Solid-state physics can predict the electronic properties of crystalline materials. Liq- uid metals have descriptive relations that predict the electronic properties by application of, for example, mean field theories. The field of plasma physics has developed relations that enable the modeling and prediction of the electronic properties of plas- mas.

As a single system evolves as a function of a thermodynamic variable (e.g. temperature or pressure), ﬁrst (e.g. melting) and second (e.g. metallization) order phase transitions occur. Physical theories of electronic properties are typically phase-speciﬁc and rarely bridge transitions between or within phases. Ab initio quantum mechanical calculations are fundamentally capable of performing this function (i.e. the physics of quantum mechanics holds across the phases of interest to the materials scientist), however these methods have proven quantitatively intractable for high-throughput calculations of certain critical phases of matter (i.e. high temperature noncrystalline systems exhibiting short-range order) and for the property prediction over large ranges of temperature and composition.

The quantitative connection between electronic transport properties and the equilibrium thermodynamic property of entropy may oﬀer a means to provide phase- independent models of the electronic properties of materials. The evolution of entropy during ﬁrst and second order phase transitions follows from the laws of thermodynamics. Thus, thermodynamic models of the electronic entropy of systems that span multiple-phases can be potentially translated into descriptions of the electronic properties of phases.

It is proposed that an investigation into the applicability of the theory of Chapter 3tothelowtemperaturesolidphase,moltenphase,gasphase,andplasmaphase be attempted for a system that exhibits an evolution of electronic properties as a function of phase transition (such as caesium).

142 12.3 Final Thoughts

The study of noncrystalline systems is fundamentally interdisciplinary. Researchers have attempted to describe the thermodynamic and electronic properties of this di- verse group of materials through the extension of theories developed in the context of distinct phases of matter. The extension of the physics of the gas phase to the molten phase of matter provides some insight to the thermodynamics of molten systems [2]. The extension of solid-state physics to the study of noncrystalline phases provides insight to the electronic properties of noncrystalline systems [3]. However, no comprehensive theory has been able to capture and describe the physics of noncrystalline systems or successfully bridge them to the existing pillars of theory (such as the kinetic theory of gases or solid-state physics). The universality of the thermodynamic framework has proven a productive lens through which to view the challenge of modeling noncrystalline phases exhibiting short-range order. It has been found that bringing physical descriptions of material properties into a thermodynamic framework enables the material scientist to make tangible progress on the investigation of the properties of materials, and for this progress to broadly and accretively contribute to the ﬁeld.

143 144 References

[2] J. Frenkel. Kinetic Theory of Liquids. London: Oxford University Press, 1946.

[3] N. Mott and E. Davis. Electronic Processes in Non-Crystalline Materials.Ox- ford, UK: Clarendon Press, 1971.

145 146 Appendix A

Overview of Solution Theory

For simplicity, a binary solution of A and B is considered, though mixing formalism is not restricted to binary systems. For a solution in the A-B system, the Gibbs free energy of mixing (Gmix) is the diﬀerence between the Gibbs free energy (G)ofthe actual solution and the mechanical Gibbs free energy (Gm)oftheend-membersA and B, deﬁned as:

m G = xAGA + xBGB (A.1)

xA and xB are the concentrations of components A and B respectively.

Gmix is conventionally expressed in terms of the enthalpy and entropy of mixing, respectively Hmix and Smix.

m Gmix = G G =Hmix T Smix (A.2)

In general, the mixing term of a thermodynamic variable (mix)canbedescribed as the diﬀerence between the total value and the mechanical mixing term.

m mix = (A.3)

147 148 Appendix B

Thermoelectrics Overview

Onsager’s application of the assumption of local equilibrium to irreversible thermodynamic systems (such as materials in a temperature gradient experience ﬂux) forms the theoretical basis for the derivation of the key features of thermoelectricity. See Callen for a detailed derivation of the thermoelectric eﬀect from Onsager’s reciprocal relations [1]. The key results as they apply to the current discussion are reproduced here.

The ﬂuxes of current and heat are coupled:

1 1 J = L11 µ + L12 (B.1) T r rT

1 1 Q = L12 µ + L22 (B.2) T r rT

In a system with no electronic ﬂux (i.e. J =0), the system of equations can be solved and the physical interpretation of the coeﬃcients can be solved for.

The thermal conductivity is deﬁned by

Q  = (B.3) T r This can be expressed in terms of the coeﬃcients as

149 1 2  = 2 (L11L22 L12 ) (B.4) T L11 The electronic conductivity can similarly be deﬁned in an isothermal system by

eJ = (B.5) 1 µ e r This can be expressed in terms of the coeﬃcients as

2 e L11 = (B.6) T

The entropy current density S can be expressed in terms of the coeﬃcients as well.

2 L12 L11L22 L 1 S = J + 12 (B.7) TL11 TL11 rT Thus there are two sources of entropy current density. The second term of Equa- tion B.7 describes the entropy transported by the flux of heat through the system. The first term implies that associated with the current produced by the motion of electrons in a temperature gradient is an entropy per particle of L12 . The ther- TL11 mopower is defined as

1 L12 ↵ = (B.8) e TL11 Thus, the entropy per charged particle transported in a thermal gradient is

Sparticle = e↵ (B.9)

150 References

[1] H. Callen. “The Application of Onsager’s Reciprocal Relations to Thermoelec- tric, Thermomagnetic, and Galvanomagnetic Eﬀects”. In: Physical Review 73.11 (1948), pp. 1349–1358.

151 152 Appendix C

Heuristic Arguments for Theory

Chapter 3 presents a theory connecting the electronic and thermodynamic properties of systems. The validity of this theory has been supported by empirical evidence proving its predictive power. Provided herein is a heuristic argument supporting a physical explanation of the theory. The microscopic basis for the connection between thermopower and entropy has previously been discussed by Peterson and Chaikin [2,3], and the speciﬁc statistical mechanical basis for the electronic entropy has been presented in Wallace [4]. Seeking to identify the macroscopic consequences of this connection on quantities that are empirically accessible, we ﬁrst propose a reminder of the relevant thermodynamic terms. Entropy describes the change in Gibbs free energy of a system with temperature:

dG S = (C.1) dT N,P,etc. ⇣ ⌘ Consequently, the electronic entropy is proportional to the change in the electronic contribution to the free energy of the system (Ge) with temperature. For a discussion on the validity of decomposing the free energy function into components see Smith [1].

dGe Se = (C.2) dT N,P,etc. ⇣ ⌘ The electronic entropy is related to the accessible density of states (DOS) of

153 electrons; only states vicinal to the chemical potential of electrons (or the Fermi level) are accessible and contribute to the electronic entropy of the system. For derivation we describe a system comprising two materials (a) and (b). The materials are at a uniform temperature and the electrochemical potential difference between (a) and (b) can be measured. We can relate the measured electrochemical potential difference between (a) and (b) to a difference in the electronic component of the free energy of a system, by analogy to the classic relation of electrochemical cells:

mF ((a) (b))=G(a) G(b) (C.3) e e

m is the number of electrons exchanged, F is Faraday’s constant, and is the electrical potential. 1 From the deﬁnition of the thermopower (↵,VK ):

d = ↵ (C.4) dT

We can thus relate the variation of free energy with temperature, and hence the entropy, to the change in thermopower. From Eqs. C.2 and C.3:

(a) (b) (a) (b) d d dGe dGe mF = (C.5) dT dT dT dT ⇣ ⌘

d(a) d(b) mF = S(a) + S(b) (C.6) dT dT e e ⇣ ⌘ Collecting terms:

d(a) d(b) mF + S(a) = mF + S(b) (C.7) dT e dT e

At this point the derivation is general and does not specify the composition of system (a) and system (b). Thus, thermodynamic functions of system (a) and system (b) are not required to co-vary and for Equation C.7 to hold generally each side must be equal to a constant. The third law of thermodynamics requires that the entropy

154 at 0 Kbe0,and,undertheassumptionthatentropyisapositivequantity,this constant is equal to 0. Thus, rearranging and eliminating indices, we can relate

d mF = Se (C.8) dT

Combining with Equation C.4:

mF ↵ = Se (C.9)

Measuring the thermopower of a system consists of measuring the response of the electronic component of the free energy to a perturbation in temperature. This measurement is not without analogy to electromotive force (e.m.f.) measurement, one of the few experimental methods to determine the (total) entropy of a system by monitoring a relative chemical potential difference as a function of temperature [2]. However, because the absolute chemical potential of electrons is directly accessible via the proposed connection, the absolute electronic entropy of the system can be calculated by measurement of the thermopower. Thus, the thermopower of a system is quantifiably related to the electronic entropy and gives access to the absolute electronic entropy of a system. The thermopower can then be considered a material property with physical meaning for a material at equilibrium [2, 3]. The same conclusion can be drawn using irreversible thermodynamics. From this perspective, the thermopower reflects the entropy transported during the thermally driven motion of charged particles. The thermopower is then defined as the entropy per unit charge associated with mobile electrons in the system [4]. Therefore, the thermopower of a material and the number of charged particles that 1 contribute to nonlocal transport phenomena (i.e. mobile electron density n (mol ) of charge e) provide a quantification of the absolute electronic entropy of a system in 1 1 units of J mol K :

Se = ne↵ (C.10)

155 1 1 This is identical to Equation C.9 where Se is in units of J mol K . Thus, it is possible to macroscopically probe the electronic properties of a system and relate them to an essential thermodynamic quantity: entropy.

156 References

[1] P. Smith and W. Gunsteren. “When Are Free Energy Components Meaningful?” In: J. Phys. Chem. 98.51 (1994), pp. 13735–13740.

[2] C. Wagner. “The Thermoelectric Power of Cells wIth Ionic Compounds Involv- ing Ionic and Electronic Conduction”. In: Progress in Solid State Chemistry 7 (1972), pp. 1–37.

[3] D. Adler. Physics of Disordered Materials. New York, NY: Plenum Press, 1985, pp. 275–285.

[4] H. Callen. “The Application of Onsager’s Reciprocal Relations to Thermoelec- tric, Thermomagnetic, and Galvanomagnetic Eﬀects”. In: Physical Review 73.11 (1948), pp. 1349–1358.

157 158 Appendix D

Modiﬁed Richard’s Rule

Reference [1] presents a modified Richard’s Rule that relates the enthalpies and entropies of fusion of systems that undergo semiconductor-to-metal (SC-M) and semiconductor-to-semiconductor (SC-SC) transitions at melting. The key results are reproduced herein for convenience. Richard’s Rule provides that the entropies of fusion of metallic systems approxi- 1 1 mate 9.6 J mol K [2]. Figure D-1 plots the enthalpies of fusion vs. the melting temperature for several congruent melting compounds of systems that exhibit SC-SC and SC-M transitions at melting. There is a clear distinction between the slope of the line of best fit for SC-SC systems and SC-M systems. The slope of the line fit to the SC-SC data is 10.0 J 1 1 mol K , which approximates Richard’s Rule. This congruence can be explained by the retention of similar degrees of short-range order across the melt for both metallic systems and systems that undergo a semiconductor-to-semiconductor transition. Metallic systems exhibit strong disorder in both the molten and solid phase. SC-SC systems exhibit strong ordering in both the molten and solid phases. SC-M systems, however, experience a dramatic evolution in short-range order upon melting. This is 1 1 reflected in the slope of the line of fit for the SC-M data of 41.2 J mol K .Large configurational and electronic entropy of fusion contribute [3–7]. The regular and predictive pattern of entropies of fusion for SC-SC and SC-M systems describes a rule analogous to Richard’s Rule for molten semiconductor sys-

159 SC-SC GaAs SC-M Δ Sf ~ 41.2

-1 AlSb GaSb / J / mol f

H InSb Δ

GaSe Cu Se Δ 2 Sf ~ 10.0

Ag2Se

Tl5Se3 Cu S In Se 2 TeTl2 2 3 Ag2S

Tm / K

Figure D-1: Enthalpy of fusion (Hf )vs.meltingtemperature(Tm)forsystems that exhibit semiconductor-to-semiconductor (SC-SC) transitions at melting (circles) and systems that exhibit semiconductor-to-metal (SC-M) transitions at melting (dia- monds). Calculated from measurements reported in [3]. The slopes of best line of ﬁt 1 1 1 1 are 10.0 J mol K and 41.2 J mol K respectively. The slope of best ﬁt for the SC-SC systems is similar to the slope predicted by Richard’s Rule for metallic alloy 1 1 systems (a slope of 9.6 J mol K )[2].

160 tems. This rule can be leveraged in the stability analysis of molten semiconductor thermodynamics.

161 162 References

[2] D. Gaskel. Introduction to the Thermodynamics of Materials.5thed.BocaRa- ton, FL: CRC Press, 1995.

[3] D. Belotskii and O. Manik. “On the interrelation between electronic properties and structure of thermoelectric material melts and the state diagrams - 5. Clas- siﬁcation of electronic semiconductor melts”. In: Journal of Thermoelectricity 1 (2004), pp. 32–47.

[4] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically accessible electronic properties in high temperature systems”. In: Philosophical Magazine 96.29 (2016), pp. 3041–3053.

[5] V. Glazov. Liquid Semiconductors. New York, NY: Plenum Press, 1969.

[6] A. Regel, V. Glazov, and A. Aivazov. “Calculation of components of fusion entropy of some semiconducting compounds”. In: Sov. Phys. Semicond. 8.11 (1975), pp. 1398–1401.

[7] D. Adler. Physics of Disordered Materials. New York, NY: Plenum Press, 1985, pp. 275–285.

163 164 Appendix E

Relationship of Enthalpy of Mixing to Enthalpy of Fusion

The enthalpy of fusion of a system undergoing a semiconductor-to-semiconductor (SC-SC) transition at melting is:

HSC = HSC HSC (E.1) f L S

SC SC HL and HS are the enthalpies of the liquid and solid state at the melting temperature respectively. Similarly, we can deﬁne the enthalpy of fusion of a system undergoing a semiconductor-to-metal transition:

HM = HM HM (E.2) f L S

In this analysis, the solid state remains unchanged. Hence:

SC M HS = HS (E.3)

Thus, the diﬀerence in magnitude of the enthalpies of fusion of the semiconductor (SC) and metallic (M) molten phase at melting is:

HSC HM = HSC HM (E.4) f f L L 165 The enthalpies are deﬁned in terms of mixing and mechanical terms:

SC SC SC HL =Hmix + Hmech (E.5)

M M M HL =Hmix + Hmech (E.6)

However, because, in this analysis, the end members are the same in both the semiconductor and metallic molten states, the mechanical terms are equivalent:

SC M Hmech = Hmech = xAHA + xBHB (E.7)

xA and xB are the concentrations of the end-members A and B and HA and HB are the absolute enthalpies of the end-members A and B (for a two component system A-B). Thus, the diﬀerences in enthalpies of the semiconducting and metallic phases is:

HSC HM =HSC HM (E.8) L L mix mix Therefore the diﬀerence in enthalpies of fusion can be equated to the diﬀerence in enthalpies of mixing at the melting temperature:

HSC HM =HSC HM (E.9) mix mix f f At temperatures above the melting temperature, the difference in enthalpies of mixing between the semiconductor and metallic phases includes a temperature-dependent term due to a difference in the heat capacities of the phases. This is reflected in Equa- tion E.10.

T HSC HM =HSC HM + cSC cM dT (E.10) mix mix f f P P ZTm SC M Tm is the melting temperature and cP and cP are the speciﬁc heats of the semiconducting and metallic phases.

166