Quantitatively Connecting the Thermodynamic and Electronic ...
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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
Transcript of Quantitatively Connecting the Thermodynamic and Electronic ...
Quantitatively Connecting the Thermodynamic andElectronic Properties of Molten Systems
by
Charles Cooper Rinzler
Submitted to the Department of Materials Science and Engineeringin 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 ProfessorThesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Donald Sadoway
Chairman, Department Committee for Graduate Students
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Quantitatively Connecting the Thermodynamic and
Electronic Properties of Molten Systems
by
Charles Cooper Rinzler
Submitted to the Department of Materials Science and Engineeringon April 9, 2017, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
AbstractThe electronic and thermodynamic properties of noncrystalline systems are inves-tigated and quantitatively connected through the application of theory presentedherein. The electronic entropy is confirmed to control the thermodynamics of moltensemiconductors. The presented theory is applied to predict the thermodynamic prop-erties of the prototypical Te-Tl molten semiconductor from empirical electronic prop-erty data and the electronic properties from empirical thermodynamic data. Thetheory is able to answer a question posed in the literature regarding a correlationbetween features of phase diagrams and molten semiconductivity. The quantitativeconnection is extended to predict thermodynamic properties of fusion, and a stabil-ity criterion to predict whether a system will behave as a molten semiconductor isdeveloped and verified.
The investigation and prediction of electronic transitions, such as metallization ofhigh temperature systems, is enabled by the theory provided herein. The thermo-dynamic bases for key features of phase diagrams in the molten state are explainedand quantified. Methods to rapidly collect electronic and entropy data in the moltenphase are provided and enable access to key thermodynamic data for high temper-ature systems. The connection of electronic entropy to short-range order allows thedetection and prediction of solid-phase compounds through the collection of electronicproperty data in the molten phase and the prediction of thermodynamic quantitiesof fusion. An absolute reference for entropy at temperatures substantially above 0�
K is proposed.
Thesis Supervisor: Antoine AllanoreTitle: Associate Professor
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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.
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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...
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Contents
1 Introduction 17
1.1 Structure of the Present Work . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 Connecting Electronic and Thermodynamic Properties in the
Molten Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Background on Molten Semiconductors . . . . . . . . . . . . . . . . . 22
1.2.1 Electronic Properties of Noncrystalline Systems . . . . . . . . 22
1.2.2 Molten Semiconductors . . . . . . . . . . . . . . . . . . . . . . 22
1.2.3 Theory of Molten Semiconductors . . . . . . . . . . . . . . . . 24
1.2.4 Previous Approaches . . . . . . . . . . . . . . . . . . . . . . . 26
1.2.5 Solid vs. Molten Semiconductors . . . . . . . . . . . . . . . . 32
1.3 Thermodynamics of Molten Semiconductors . . . . . . . . . . . . . . 33
1.3.1 Prediction of Phase Diagrams . . . . . . . . . . . . . . . . . . 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 Transport Entropy . . . . . . . . . . . . . . . . . . . . . . . . 38
7
1.4.2 Previous Attempts at Connection . . . . . . . . . . . . . . . . 39
1.5 Electronic Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.5.1 Forms of Electronic Entropy . . . . . . . . . . . . . . . . . . . 40
1.5.2 Contribution of Electronic Entropy to Total Entropy . . . . . 41
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Hypothesis 53
2.1 Features of Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Scientific Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4 Consequences for Materials Modeling . . . . . . . . . . . . . . . . . . 57
2.5 Framework for Validation of Hypothesis . . . . . . . . . . . . . . . . 59
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Theory Relating Electronic Entropy to Electronic Properties 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 Discussion of Theoretical Basis . . . . . . . . . . . . . . . . . . . . . 66
4 Prediction of Properties of Te-Tl 71
4.1 Applied Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Extension of Framework to Predicting Thermodynamic Quantities
of Fusion 79
5.1 Calculation of the Entropy of Fusion . . . . . . . . . . . . . . . . . . 79
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8
6 A Criterion for Molten Semiconductivity 85
6.1 Stability Analysis of Molten State . . . . . . . . . . . . . . . . . . . . 85
6.2 Application to the Te-Tl System . . . . . . . . . . . . . . . . . . . . . 87
6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 Prediction of Metallization Temperature of Molten Semiconductor
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
8 Prediction of Features of Phase Diagrams 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
9 Experimental Methods and Results 111
9.1 Review of Apparatuses from Previous Researchers . . . . . . . . . . . 111
9.1.1 Quartz Test Cell . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.1.2 Boron Nitride Test Cell . . . . . . . . . . . . . . . . . . . . . . 112
9.2 Dynamic Induction Test Cell . . . . . . . . . . . . . . . . . . . . . . . 113
9.2.1 Apparatus Design . . . . . . . . . . . . . . . . . . . . . . . . . 113
9.2.2 Apparatus Performance . . . . . . . . . . . . . . . . . . . . . 116
9.2.3 Results for Pb-S . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.3 Static Test Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.3.1 Apparatus Design . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.3.2 Apparatus Performance . . . . . . . . . . . . . . . . . . . . . 120
9.3.3 Results for Sn-S . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.4 Discussion of the Experimental Methods . . . . . . . . . . . . . . . . 122
9
10 Extension to Metallic and Ionic Systems 127
10.1 Extension of Theory to Metallic Systems . . . . . . . . . . . . . . . . 127
10.2 Extension of Theory to Ionic Systems . . . . . . . . . . . . . . . . . . 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 Molten Semiconductor Systems . . . . . . . . . . . . . . . . . 134
11.1.2 Metallic Systems Exhibiting Congruent Melting Compounds . 134
11.1.3 Multicomponent Systems . . . . . . . . . . . . . . . . . . . . . 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 Demonstrated Consequences of Theory . . . . . . . . . . . . . . . . . 139
12.1.1 Modeling of Molten Semiconductors . . . . . . . . . . . . . . . 139
12.1.2 Beyond Molten Semiconductors . . . . . . . . . . . . . . . . . 140
12.2 Potential Impact of Work . . . . . . . . . . . . . . . . . . . . . . . . 140
12.2.1 Absolute Reference for Entropy . . . . . . . . . . . . . . . . . 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
Connection to Thermodynamics . . . . . . . . . . . . . . . . . 142
12.3 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A Overview of Solution Theory 147
B Thermoelectrics Overview 149
C Heuristic Arguments for Theory 153
D Modified Richard’s Rule 159
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E Relationship of Enthalpy of Mixing to Enthalpy of Fusion 165
11
12
List of Figures
1-1 Electronegativity and electronic behavior . . . . . . . . . . . . . . . . 25
1-2 Conductivity vs. temperature . . . . . . . . . . . . . . . . . . . . . . 27
1-3 DOS vs. temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2-1 Notional phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 54
2-2 Notional thermopower vs. temperature . . . . . . . . . . . . . . . . . 56
4-1 Entropy of mixing vs. at. % Tl . . . . . . . . . . . . . . . . . . . . . 73
4-2 Thermopower vs. at. % Tl . . . . . . . . . . . . . . . . . . . . . . . . 74
5-1 Electronic entropy of fusion of compounds . . . . . . . . . . . . . . . 81
6-1 Phase diagram of the Te-Tl system . . . . . . . . . . . . . . . . . . . 88
6-2 �Se
vs. �Sideal
for the Te-Tl system . . . . . . . . . . . . . . . . . . 89
6-3 �Se
vs. ⇠ for the Te-Tl system . . . . . . . . . . . . . . . . . . . . . 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 Thermopower vs. temperature for PbS . . . . . . . . . . . . . . . . . 117
9-4 CV of PbS at 1120� Celsius . . . . . . . . . . . . . . . . . . . . . . . 118
9-5 Static test cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
9-6 Sn-S phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
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9-7 Entropy of mixing of Sn-S . . . . . . . . . . . . . . . . . . . . . . . . 122
10-1 Entropy of mixing of the Mg-Bi system . . . . . . . . . . . . . . . . . 128
D-1 Modified Richard’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . 160
14
List of Tables
1.1 Molten Semiconductor Classification . . . . . . . . . . . . . . . . . . 23
1.2 Thermodynamic Models of Free Energy . . . . . . . . . . . . . . . . . 35
1.3 Contributions to Entropy of Mixing . . . . . . . . . . . . . . . . . . . 42
15
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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 ex-
traction, heat transfer materials, batteries, photovoltaics, and more. Noncrystalline
systems can offer unique benefits such as high temperature operation, tunable elec-
tronic 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 chal-
lenges 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” demanded by the metallurgical community for a practical theory to
“provide readily accessible models for the appraisal of the thermodynamics of multi-
component [systems]” [1]. It has been put forth by Fultz, and other members of the
thermodynamics community, that one explanation for the challenges in the develop-
ment 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 sys-
tems 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 interac-
tion” 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 com-
munity 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 con-
densed 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 be-
tween 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 en-
thalpic 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 non-
crystalline 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 quanti-
tative, 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 elec-
tronic transport properties (e.g. thermopower). The theory could enable empirical
access to entropy in high temperature, noncrystalline systems and provide a theoreti-
cal 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 in-
vestigation 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 provid-
ing 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 semicon-
ductors are equilibrium systems that exhibit a lack of long-range order, but significant
short-range order.
A quantitative theory that predicts the properties of molten semiconductors is
potentially broadly useful to the study of noncrystalline systems. The electronic
properties of systems evolve as a function of temperature, pressure, and other ther-
modynamic 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 rela-
tion 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 fea-
ture 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 elec-tronic conductivity (�) and thermopower (↵) in the molten phase near the meltingtemperature [11]
Transition �(⌦�1cm�1)
d�dT
↵(µV K�1)
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, ex-
plained below in section 1.2.4). Consequently, within this document the two cate-
gories 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 semicon-
ducting 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
0.5 1.5
Semiconductor Metallic Insulator
FeTe FeS FeO
Pauling Electronegativity
Difference
Figure 1-1: Semiconductor behavior as a function of Pauling electronegativity differ-ence. The classification of metallic, semiconductor, and insulator systems is based ontypical solid-state properties. The boxed region indicates the range of electronegativ-ity 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 semicon-
ductor systems and an overview of the role of solid-state crystal structure on molten
short-range order and electronic properties. Many systems that retain semiconduc-
tivity 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 semicon-
ducting 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 liq-
uid 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 temper-
atures above the melting point, molten semiconductor systems metallize and expe-
rience a loss of semiconductivity [18, 24]. It has been shown empirically that the
electronic conductivity of semiconducting systems increases monotonically as a func-
tion 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 semiconduc-
tors are most frequently compound systems (with tellurium and selenium excepted)
and typically show a minimum conductivity and thermopower at the stoichiometry of
a congruent melting compound (a compound that melts homogeneously such that the
composition of the liquid phase is identical to that of the solid phase). The conduc-
tivity 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 ametallization event occurs at the minimum metallic conductivity. SC-M systems ex-hibit typical metallic conductivity behavior, showing a decrease in conductivity withtemperature.
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 descrip-
tion 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 con-
ductivity 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 conduc-
tivity and thermopower of molten semiconductor systems that have been empirically
validated [14].
� = �min
e�T�E0
kT (1.1)
↵ = �E0 + (k � �)T
eµT(1.2)
�min
is the minimum metallic conductivity, defined in Section 1.2.4. � is a parame-
ter 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 functionof energy (E) vs. temperature. T
m
and Tc
are respectively the melting and criticaltemperature. Shaded regions identify pseudogaps in the density of states, and theupper curve represents a fully metallized system. E
F
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 de-
scriptions 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 evo-
lution 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 sys-
tems 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 sys-
tems. 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 semi-
conducting 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 mod-
els 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 quali-
tative 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 investiga-
tors, 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 substan-
tial 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 evo-
lution 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 atom-
istic 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 pre-
diction 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 re-
search. 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 noncrys-
talline 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 counter-
parts 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 ad-
dressable 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 proper-
ties 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 de-
velop 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 reg-
ular 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
Quasichemical Model [50–52]
Adds temperature-dependententropy term to regular solution
model (non-random mixing).This model relies on the pair
approximation for calculation ofthe free energy.
Effective at predicting ternaryphase diagrams from binarydata. Hundreds of oxide and
sulfide systems have beenmodeled [53].
Pair approximation fails toaccount for all relevant SRO forcomplex slag systems such as
sulfides.
Cluster Variation Method(CVM) [54]
Extension of quasichemicalphilosophy not limited to pair
approximation, allows for largerbase units for calculation of free
energy.
Effective at modeling systemsthat tend to exhibit clustering /
ordering in the liquid state.
Requires substantially more fitparameters to develop
self-consistent free energy modelreflecting SRO.
Central Atom Model [55, 56]
Models the free energy byconsidering permutations of
central atoms (anions orcations), the nearest neighborcation shell, and the nearest
neighbor anion shell.
Used to great success inmodeling sulfides for the steelindustry. Requires fewer fit
parameters than CVM.
Not currently incorporated intoenergy minimization softwareoutside of industry-specific
packages.
Associate Model [57–61]
Assumes formation of molecular‘associates’ replicating thestoichiometry of solid-state
compounds in the liquid. Modelsthe solution as a mixture of thepure elements and associates.
Simple interaction parameter fitto data. Very successful in
modeling binary sulfide systemsexhibiting retention of SRO.
Poor at predicting ternary phasediagrams from binary data. Does
not easily extend tomulticomponent systems.
35
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 geologi-
cal 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 con-
tinued, 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 accu-
rate 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
a thermodynamic description and the electronic properties of molten semiconductor
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 dia-
grams 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 ) is on the order of 5-10
kJ mol�1 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 com-
pound 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 or-
dering 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 lever-
ages 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 prop-
erties 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
dne
⌘
T,P,N
= �F↵ (1.3)
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 descrip-
tion). 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 con-
necting transport properties (irreversible thermodynamic quantities) and equilibrium
thermodynamic properties must provide empirically verifiable predictions to be vali-
dated.
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].
Se
= �kB
Z 1
0
N(E) (p(E) ln p(E) + (1� p(E)) ln(1� p(E))) dE (1.4)
p(E) is defined by the Fermi function, where Ef
is the Fermi level.
p(E) =1
eE�E
f
kT + 1(1.5)
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 con-
tributions (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
) is dependent on the degree of short-
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
) is the difference
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 contribu-
tions 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 semicon-ductor systems [20, 73–75]
Component Value for Molten Semiconductors (Jmol
�1K
�1)
�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 ther-
modynamic 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.
A theory that connects the thermodynamic and electronic properties of noncrys-
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 pre-
diction 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
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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 par-
ticular, the entropy and electronic properties of molten systems exhibiting short-range
order (SRO). A quantitative framework that connects the thermodynamic and elec-
tronic properties of noncrystalline systems would enable the use of existing thermo-
dynamic data to predict electronic properties and the use of experimental techniques
to generate electronic property data to facilitate the generation of thermodynamic
databases.
A novel means to access entropy in disordered phases would be most impactful for
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 semi-
conducting 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 di-
agram 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 asemiconductor-to-semiconductor (SC-SC) transition and systems that undergo asemiconductor-to-metal (SC-M) transition at melting. SC-SC systems exhibit at leastone miscibility gap in the molten phase. Solid shaded regions correspond to metal-lic molten systems. Thatched shaded regions correspond to semiconducting moltensystems.
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 pres-
ence 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 proper-
ties 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 mis-
cibility. 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 semiconduct-
ing properties of liquid systems has thus been thoroughly demonstrated by correlation
studies and relates to the chemical ordering of molten systems. However, the predic-
tive utility of such qualitative correlations of binary systems remains limited.
55
A
Tem
pera
ture
B AB Thermopower
Tc
Figure 2-2: Notional representation of the thermopower of a binary (A-B) moltensemiconductor system as a function of temperature. There is a transition of ther-mopower from semiconductor to metallic behavior at the temperature correspondingto the critical point of the miscibility gap T
c
.
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.
A quantitative connection between an equilibrium thermodynamic variable (entropy)
and transport properties (e.g. thermopower) is proposed.
It is posited that electronic entropy substantially contributes to the thermody-
namics 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 semi-
conductors 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 espe-
cially the liquid state (see Chapter 1). However, quantifying absolute entropy is not
empirically possible today, and the modeling community is afforded a singular abso-
lute 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� K reference state. For example, the absolute entropy of a system in the
liquid state at temperature T at the composition of a congruent melting compound
that is stable from 0� K to the melting temperature Tm
would be:
S =
ZT
m
0
Csolid
p
TdT +
�Hf
Tm
+
ZT
T
m
C liquid
p
TdT (2.1)
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:
S = S(Tx
) +
ZT
T
x
C liquid
p
TdT (2.2)
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 hypoth-
esis 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, ar-
senides, 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 mod-
eling 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 mix-
ing 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 mea-
sured 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
A previously noted correlation between certain features of phase diagrams (i.e. mis-
cibility 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
• A connection between electronic properties and electronic entropy extends to
broader classes of high-temperature noncrystalline materials
Validating these hypotheses would enable empirical access to a new absolute ref-
erence state for the entropy of a material in a disordered state (noncrystalline ma-
terials). 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
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4 (2002), pp. 38–42.
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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 con-
nected to the electronic properties of molten semiconductor systems. A discussion of
the key findings of a new theory connecting electronic entropy and electronic proper-
ties 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
Equation 3.1 connects the integral electronic entropy, Se
(J mol�1 K�1), to the electron
density, n (mol�1), the fundamental charge, e, and the thermopower, ↵ (V K�1) 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
= np
e↵p
� nn
e↵n
(3.3)
np
and nn
are, respectively, the density of positively and negatively charged car-
riers. ↵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
shown in Equation 3.4 where � (⌦�1 m�1) is the conductivity, µ (m2 V�1 s�1) is the
electron mobility, m (m�3) is the electron density, and e 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 en-
tropy Se
, on one side and empirically measurable electronic properties on the other.
This relationship will be explored and validated in Chapter 4. The conversion factor
w (m3 mol�1) is used to convert from volumetric to molar quantities.
Se
= �w�↵
µ(3.5)
For systems with positively charged carriers the equation follows.
64
Se
= w�↵
µ(3.6)
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.
�Se
= Se
� xA
SA
e
� xB
SB
e
= �w�↵
µ+ x
A
nA
e↵A
+ xB
nB
e↵B
(3.7)
�Se
is the electronic entropy of mixing. SA
e
and SB
e
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 equi-
librium electronic entropy of that charge. Consequently, the total electronic entropy
is, in the simplest assumption, the number of charged carriers (n) multiplied by the
entropy per charge (e↵).
There are several assumptions associated with this interpretation. First, only mo-
bile electrons (those electrons contributing to the conductivity) participate in ther-
mopower, 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 ‘ac-
cessible’ 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 addi-
tional 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 contribut-
ing 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)) could be
used to formulate an equation equivalent to Equation 3.1 for which assumption 2
would not be necessary.
Se
=
Z
E
n(E) e ↵(E) (3.8)
n(E) represents the number of occupied states at energy E. However, the energy-
dependent thermopower in Equation 3.8 is not, to the author’s knowledge, a measur-
able entity and thus is of limited utility.
67
68
References
[1] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically ac-
cessible 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 elec-
tronic 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�↵
µ� x
T l
nT l
e↵T l
� xTe
nTe
e↵Te
(4.1)
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
w�(�S
mix
+ xTe
nTe
e↵Te
+ xT l
nT l
e↵T l
) (4.2)
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� K as measured by Nakamura and
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 maxi-
mum 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 capac-
ity. 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
at. % Tl
ΔS
mix /
J m
ol-1
K-1
Nakamura
Terpilowski
Te
Oh
Prediction
Figure 4-1: Entropy of mixing vs. atomic % Tl as measured by Nakamura (bluecircles), Terpilowski (red circles), and as modeled by Oh (squares) and as predictedby Equation 4.1 (solid line) [1, 5, 6].
73
at. % Tl
α /
μV
K-1
CutlerPrediction
Te
Figure 4-2: Thermopower vs. atomic % Tl as measured by Cutler (circles) and aspredicted 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
[1] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically ac-
cessible electronic properties in high temperature systems”. In: Philosophical
Magazine 96.29 (2016), pp. 3041–3053.
[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 compo-
sition 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 semi-
conducting 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
) and the number of
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
) are more sparse than data on the
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
, and �s
) and the liquid (↵l
, µl
, and �l
).
�Sf
= ws
�s
↵s
µs
� wl
�l
↵l
µl
(5.2)
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
Ge
InSb
GaSb
GeTe
Sb2Te
3
Te
GaAs AlSb
GePbTe
Bi2Se
3
In2Te
3
Ga2Te
3
PbSe
Cu2S
GaSbAlSbAlSb
15 2065
70P
redic
ted Δ
Sf /
J m
ol-1
K-1
Reported ΔSf / J mol-1 K-1
Figure 5-1: Predicted vs. reported electronic entropy of fusion for congruent melt-ing semiconductor compounds [1]. Solid symbols show systems where Equation 5.2provides accurate prediction. Hollow symbols show systems where Equation 5.2 failsto provide accurate prediction. Blue symbols show the application of Equation 5.1for systems where carrier concentration data are available [3]. The red symbol showsthe Ge system corrected to include configurational electronic entropy [4]. The insetshows the out-of-range data for the AlSb and GaSb compounds. Reproduced fromReference [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
) from electronic prop-
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 ac-
cessible 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
A Criterion for Molten
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 anal-
ysis 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.
�GSC
mix
< �GM
mix
(6.1)
85
�GSC
mix
is the Gibbs free energy of mixing of the hypothetical molten semiconductor
(superscript SC) and �GM
mix
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
mix
� T�SSC
mix
< �HM
mix
� T�SM
mix
(6.2)
As discussed in Chapters 1 and 3, and Reference [2], the entropy of mixing of
a molten semiconductor, �SSC
mix
, is approximated by the electronic component of
the entropy of mixing �Se
. Further, systems that metallize exhibit minimal short-
range order and the entropy is dominated by the configurational component, and
consequently the entropy of mixing of the metallized state (�SM
mix
) is maximally the
ideal entropy of mixing �Sideal
.
Thus, Equation 6.2 can be simplified and rearranged to provide a stability condi-
tion for the electronic entropy of mixing of the molten semiconductor state.
�Se
>�HSC
mix
��HM
mix
T+�S
ideal
(6.3)
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
mix
��HM
mix
⇡ �HSC
f
��HM
f
(6.4)
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 temper-
ature 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.
�Se
>�HSC
f
��HM
f
T+�S
ideal
(6.5)
For simplicity we define the term ⇠:
⇠ =�HSC
f
��HM
f
T+�S
ideal
(6.6)
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 com-
positions 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 mix-
ing 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
) and the ideal entropy of mixing (�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
Te Tl
T /
K
at. frac. Tl
1 2 3
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-phaseregion. Region 3 indicates a metallized molten phase. [9]
compositions, without appeal to enthalpic terms. However, it is not possible to de-
termine 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
Te
ΔS
mix
/ J
mol-1
K-1
Tl
∆Se (Rinzler 2016)
∆Sideal
1 2 32222
at. frac. Tl
Figure 6-2: �Se
(solid) as calculated by Reference [1] and the ideal entropy of mixing(�S
ideal
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
Te Tl
J m
ol-1
K-1
∆Se
ξ
1 2 3222222
at. frac. Tl
Figure 6-3: �Se
(solid) as calculated by Reference [2] and the ⇠ parameter fromEquation 6.6 (dashed) for Te-Tl at 770� K. The ⇠ parameter is not interpreted tohave 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 prop-
erties 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 crite-
rion 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 proper-
ties of molten semiconductors: the role of electronic entropy”. In: Philosophical
Magazine 6435.January (2016), pp. 1–11.
[2] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically ac-
cessible electronic properties in high temperature systems”. In: Philosophical
Magazine 96.29 (2016), pp. 3041–3053.
[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 semicon-
ductor systems.
7.1 Method
The key result of Chapter 6 is restated here:
�HSC
mix
� T�SSC
mix
< �HM
mix
� T�SM
mix
(7.1)
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 mix-
ing 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:
�HSC
mix
��HM
mix
= �HSC
f
��HM
f
+
ZT
T
m
cSCP
� cMP
dT (7.2)
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 cal-
culated 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 of-
ten 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 tem-
perature, 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 semi-
conductor 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:
Tc
=�HSC
f
��HM
f
+ Tm
(cMP
� cSCP
)
�Se
��Sideal
+ cMP
� cSCP
(7.3)
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
f
� �HM
f
. The electronic entropy
of mixing (�Se
) was taken as the �Sex
term reported in Reference [1]. The heat
97
capacities of the semiconductor and metallic states are 62.55 J mol�1 K�1 and 42.02
J mol�1 K�1 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
metallic state, �SM
mix
, is assumed to be the ideal mixing term (see Reference [2] for a
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 temperatureprediction 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
f
� �HM
f
. The electronic entropy of mixing
(�Se
) was taken from the results of Chapter 4. The heat capacities of the semicon-
ductor and metallic states are 38.16 and 29.29 J mol�1 K�1 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
mixing of the metallic state, �SM
mix
, is assumed to be the ideal mixing term (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,
a full analysis of the Fe-S system is provided that does not depend on an assumption
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.
[2] C. Rinzler and A. Allanore. “Connecting electronic entropy to empirically ac-
cessible electronic properties in high temperature systems”. In: Philosophical
Magazine 96.29 (2016), pp. 3041–3053.
[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 sys-
tems 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 thermo-
dynamic 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 qua-
sichemical, 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
a function of composition or temperature, but is rather, typically, a polynomial fit to
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. elec-
tronic conductivity and thermopower) of molten semiconductor systems as a function
of temperature have been developed. The theory presented herein (see Chapter 3) pro-
vides 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.
Se
= �w�min
e�T�E0
kT
E0 + (k � �)T
eµT(8.1)
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 frame-
work. The improved model for excess entropy is shown to correct an erroneous pre-
diction 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
A describes the method to translate electronic entropy into excess entropy of mixing.
The key result (shown for a two-component A-B molten semiconductor system) is
reproduced here.
�Sex
= Se
� xA
SA
e
� xB
SB (8.3)
xA
and xB
are the concentration of species A and B respectively. SA
e
and SB
e
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:
�Sex
= �Se
= �w�min
e�T�E0
kT
E0 + (k � �)T
eµT� x
A
SA
e
� xB
SB
e
(8.4)
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 elec-
tronic 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
used is 300 ⌦�1 cm�1. � is 6.35 x 10�23.
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-
S system as empirically determined and provided by Reference [5], as predicted by
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
0 0.2 0.4 0.6 0.8 1
S / at. frac.
500
1000
1500
2000T
/ C
Figure 8-1: The Fe-S phase diagram. The dashed black line is the empirically de-termined miscibility gap [5]. The blue dashed line is the miscibility gap predictedby Pelton et al. through use of the quasichemical method per Reference [1]. Thered dashed line is the miscibility gap predicted by the use of the modeled entropy ofmixing per Equation 8.4.
Reference [1] is 0.000067 J mol�1 K�2, that of the modified excess entropy of mixing
incorporating the temperature dependency of electronic properties is 0.006 J mol�1
K�2, a factor of over 90 greater.
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 improve-
ment 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 ac-
cessible 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 equi-
librium 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 elec-
tronic 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 con-
ductivity 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 pro-
vided 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% rhodium thermocouples were used, and the pure platinum leads were used as
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 proper-
ties 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 prop-
erty 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
304 SS Shield
Molybdenum SusceptorAlumina Crucible
Probe
Copper Coil
Figure 9-1: The dynamic induction test cell comprises a copper induction coil induc-tively coupled to a molybdenum susceptor and contained within an argon-purged 304stainless steel shield. A sample is contained in an alumina crucible and a probe isvertically positioned by a linear stage external to the shield.
A schematic of a typical probe is provided in Figure 9-2.
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 ter-
minating 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 proxi-
mally 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
Alumina 4-bore Tube
Graphite Electrode
Thermocouple
Figure 9-2: The probe is comprised of a 4-bore alumina tube containing 3 molyb-denum wires terminating in graphite electrodes. The 4th bore contains a B-Typethermocouple 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
a temperature gradient. However, the presence of large oscillating electromagnetic
fields that may be operated discontinuously (i.e. in an on-off fashion) presents some
challenges for certain electrochemical measurements that depend on potential stabil-
ity (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
μV
K-1
Reported Value
Measured Data
Figure 9-3: The thermopower as a function of temperature for the compound PbSin the molten phase is provided. The black circles represent data gathered in thedynamic induction test cell. The red circle represents a literature-reported value [4].
9.3.1 Apparatus Design
A Lindberg Blue TF-55035A split tube furnace is used as the primary heating element.
A sealed quartz tube that is purged with argon hosts the test-cell. Figure 9-5 shows
a schematic of the test cell.
The cell is comprised of a quartz tube sealed with graphite electrodes (bonded to
the quartz by carbon paste (Ted Pella Pelco)). The sample of desired composition is
loaded into the cell before sealing in an argon purged environment.
117
Projected Reported ValueMeasured Data
Figure 9-4: The Cyclic Voltammogram at 1120� Celsius for the compound PbS in themolten phase is provided. The black circles represent data gathered in the dynamicinduction 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).
Thermocouple
Molybdenum Wires
Quartz Tube
Ni-Chrome
Graphite Electrode
Figure 9-5: The static test cell comprises a quartz tube sealed with graphite elec-trodes. Molybdenum wires are threaded into the graphite to establish a 4-probe mea-surement. Thermocouples are attached to the graphite electrodes for measurementof the thermal gradient. A Ni-Chrome wire coil is used to establish a temperaturegradient.
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 mea-
suring 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 ex-
periment 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 composi-
tion.
9.3.3 Results for Sn-S
A study of the entropy of mixing of molten tin-sulfur system as a function of com-
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 stud-
ied. 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
0.46 0.48 0.5 0.52
at. frac. S
850
900
950
1000
1050
1100
Tem
pera
ture
/ C
Figure 9-6: The partial phase diagram of the Sn-S system [6]. Red dashed linesindicate the selected compositions and temperature ranges for measurement with theStatic Test Cell.
compositions. This is expected due to the activation of additional carriers as a func-
tion 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
940 C
960 C
980 C
1000 C
1020 C
1040 C
1060 C
1080 C
1100 CΔ
Sm
ix /
J m
ol-1
K-1
Figure 9-7: The entropy of mixing (�Smix
) of the Sn-S system as a function oftemperature. The electronic conductivity and thermopower of the Sn-S system weremeasured with the Static Test Cell. The method of Chapter 4 was used to generateentropy of mixing data.
9.4 Discussion of the Experimental Methods
Experimental methods have been developed to investigate the electronic conductiv-
ity 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 macro-
scopic 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.
[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. 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 mixingof the Mg-Bi system at 827� Celsius as empirically provided by reference [3] and aspredicted by the theory provided in Chapter 3 from electronic property data providedby 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
entropy of mixing with a peak of 6 J mol�1 K�1. The maximum contribution of
electronic entropy of mixing is 2 J mol�1 K�1. 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) is indeed reflected in the phase diagram of
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 elec-
tronic 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 elec-
tronic 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 con-
tributions to the thermopower may contribute or dominate. Whereas the electronic
thermopower describes the electronic response of a system to a perturbation in tem-
perature, 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 contribu-
tion 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 ther-
modynamic (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 equi-
librium 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.
[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] 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 elec-
tronic 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
a function of composition and temperature. It is suggested that the experimental
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 com-
pound. 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 avail-
ability 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 measure-
ments 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
a CALPHAD Framework
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 semi-
conductivity (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
[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] 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 proper-
ties 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 illus-
trated 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 pre-
dict 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 pre-
dict 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 dom-
inated by electronic entropy (Chapter 10). By providing empirical access to a mea-
surement 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.
A new method to detect and predict new compounds is potentially valuable. It is
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 appli-
cation 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. tem-
perature 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 equi-
librium 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 thermody-
namics. Thus, thermodynamic models of the electronic entropy of systems that span
multiple-phases can be potentially translated into descriptions of the electronic prop-
erties of phases.
It is proposed that an investigation into the applicability of the theory of Chapter
3 to the low temperature solid phase, molten phase, gas phase, and plasma phase
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 sys-
tems [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 non-
crystalline 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
[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] 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) of the
actual solution and the mechanical Gibbs free energy (Gm) of the end-members A
and B, defined as:
Gm = xA
GA
+ xB
GB
(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
.
�Gmix
= G�Gm = �Hmix
� T�Smix
(A.2)
In general, the mixing term of a thermodynamic variable � (�mix
) can be described
as the difference between the total value and the mechanical mixing term.
��mix
= �� �m (A.3)
147
148
Appendix B
Thermoelectrics Overview
Onsager’s application of the assumption of local equilibrium to irreversible thermo-
dynamic 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:
�J = L111
Trµ+ L12r
1
T(B.1)
Q = L121
Trµ+ L22r
1
T(B.2)
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
rT(B.3)
This can be expressed in terms of the coefficients as
149
=1
T 2L11(L11L22 � L122) (B.4)
The electronic conductivity can similarly be defined in an isothermal system by
� =�eJ1e
rµ(B.5)
This can be expressed in terms of the coefficients as
� =e2L11
T(B.6)
The entropy current density S can be expressed in terms of the coefficients as well.
S = � L12
TL11J+
L11L22 � L212
TL11r 1
T(B.7)
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 � L12TL11
. The ther-
mopower is defined as
↵ = �1
e
L12
TL11(B.8)
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 em-
pirically accessible, we first propose a reminder of the relevant thermodynamic terms.
Entropy describes the change in Gibbs free energy of a system with temperature:
S = �⇣dGdT
⌘
N,P,etc.
(C.1)
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].
Se
= �⇣dG
e
dT
⌘
N,P,etc.
(C.2)
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)e
�G(b)e
(C.3)
m is the number of electrons exchanged, F is Faraday’s constant, and � is the
electrical potential.
From the definition of the thermopower (↵, V K�1):
d�
dT= �↵ (C.4)
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:
mF⇣d�(a)
dT� d�(b)
dT
⌘=
dG(a)e
dT� dG
(b)e
dT(C.5)
mF⇣d�(a)
dT� d�(b)
dT
⌘= �S(a)
e
+ S(b)e
(C.6)
Collecting terms:
mFd�(a)
dT+ S(a)
e
= mFd�(b)
dT+ S(b)
e
(C.7)
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� K be 0, and, under the assumption that entropy is a positive quantity, this
constant is equal to 0.
Thus, rearranging and eliminating indices, we can relate
mFd�
dT= �S
e
(C.8)
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 equi-
librium [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
contribute to nonlocal transport phenomena (i.e. mobile electron density n (mol�1)
of charge e) provide a quantification of the absolute electronic entropy of a system in
units of J mol�1 K�1:
Se
= ne↵ (C.10)
155
This is identical to Equation C.9 where Se
is in units of J mol�1 K�1.
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-
mate 9.6 J mol�1 K�1 [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
mol�1 K�1, 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 metal-
lic 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
reflected in the slope of the line of fit for the SC-M data of 41.2 J mol�1 K�1. 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
Tm / K
ΔH
f / J
mol-1
SC-SCSC-M
TeTl2
Cu2S
Cu2Se
Ag2S
Ag2Se
GaSe
In2Se
3
Tl5Se
3
AlSb
GaAs
GaSb
InSb
ΔSf ~ 10.0
ΔSf ~ 41.2
Figure D-1: Enthalpy of fusion (�Hf
) vs. melting temperature (Tm
) for systemsthat 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 fitare 10.0 J mol�1 K�1 and 41.2 J mol�1 K�1 respectively. The slope of best fit for theSC-SC systems is similar to the slope predicted by Richard’s Rule for metallic alloysystems (a slope of 9.6 J mol�1 K�1) [2].
160
tems. This rule can be leveraged in the stability analysis of molten semiconductor
thermodynamics.
161
162
References
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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
f
= HSC
L
�HSC
S
(E.1)
HSC
L
and HSC
S
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
f
= HM
L
�HM
S
(E.2)
In this analysis, the solid state remains unchanged. Hence:
HSC
S
= HM
S
(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
f
��HM
f
= HSC
L
�HM
L
(E.4)
165
The enthalpies are defined in terms of mixing and mechanical terms:
HSC
L
= �HSC
mix
+HSC
mech
(E.5)
HM
L
= �HM
mix
+HM
mech
(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:
HSC
mech
= HM
mech
= xA
HA
+ xB
HB
(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
L
�HM
L
= �HSC
mix
��HM
mix
(E.8)
Therefore the difference in enthalpies of fusion can be equated to the difference in
enthalpies of mixing at the melting temperature:
�HSC
mix
��HM
mix
= �HSC
f
��HM
f
(E.9)
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.
�HSC
mix
��HM
mix
= �HSC
f
��HM
f
+
ZT
T
m
cSCP
� cMP
dT (E.10)
Tm
is the melting temperature and cSCP
and cMP
are the specific heats of the semi-
conducting and metallic phases.
166
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