Hans J. Scheel and Peter Capper · Hans J. Scheel and Peter Capper Crystal Growth Technology From Fundamentals and Simulation to Large-scale Production. 9783527623457.jpg

Hans J. Scheel and Peter Capper

Crystal Growth Technology

From Fundamentals and Simulationto Large-scale Production

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Hans J. Scheel and Peter CapperCrystal Growth Technology

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Hans J. Scheel and Peter Capper


From Fundamentals and Simulationto Large-scale Production

The Editors

Dr.-Ing. Hans J. ScheelScheel ConsultingGr

..onstrasse Haus Anatas

3803 BeatenbergSwitzerland

Dr. Peter CapperSELEX Sensors and AirborneSystems Infrared Ltd.P.O. Box 217, MillbrookSouthampton SO1 5EGUnited Kingdom

CoverPolar bismuth zinc borate, Bi2ZnB2O7, pointsymmetry group mm2, grown from hightemperature solution using Top Seeded SolutionGrowth (TSSG) method.(Source: M. Burianek and M. M

..uhlberg;

Institute of Crystallography,University of Cologne)

� All books published by Wiley-VCH arecarefully produced. Nevertheless, authors,editors, and publisher do not warrant theinformation contained in these books,including this book, to be free of errors.Readers are advised to keep in mind thatstatements, data, illustrations, proceduraldetails or other items may inadvertently beinaccurate.

Library of Congress Card No.: applied for

British Library Cataloguing-in-Publication DataA catalogue record for this book is availablefrom the British Library.

Bibliographic information published bythe Deutsche NationalbibliothekDie Deutsche Nationalbibliothek lists thispublication in the Deutsche National-bibliografie; detailed bibliographic data areavailable in the Internet at.

© 2008 WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

All rights reserved (including those oftranslation into other languages). No part ofthis book may be reproduced in any form – byphotoprinting, microfilm, or any othermeans – nor transmitted or translated into amachine language without written permissionfrom the publishers. Registered names,trademarks, etc. used in this book, even whennot specifically marked as such, are not to beconsidered unprotected by law.

Printed in the Federal Republic of GermanyPrinted on acid-free paper

Composition Laserwords Private Ltd,Chennai, IndiaPrinting betz-druck GmbH, DarmstadtBookbinding Litges & Dopf GmbH,HeppenheimCover Design WMX-Design, Heidelberg

ISBN: 978-3-527-31762-2

V

Contents

Preface XI

List of Contributors XIII

Part I General Aspects of Crystal Growth Technology 1

1 Phase Diagrams for Crystal Growth 3Manfred M

..uhlberg

1.1 Introduction 31.2 Thermodynamics and Phase Diagrams 41.3 Phase Diagrams vs. Crystal Growth from Liquid Phases 131.4 Conclusions 23

References 25

2 Fundamentals of Equilibrium Thermodynamicsof Crystal Growth 27Klaus Jacobs

2.1 Introduction 272.2 Recapitulation of Some Basic Concepts 302.3 Relationships Between Thermodynamics and Kinetics 362.4 Thermodynamics of Melt Growth 382.5 Thermodynamics of Solution Growth 432.6 Thermodynamics of Crystal Growth from the Vapor 542.7 Solid–Solid Equilibria 652.8 Thermodynamics of Nucleation and Interfaces 672.9 Summary 71

References 71

3 Thermodynamics, Origin, and Control of Defects 73Peter Rudolph

3.1 Introduction 733.2 Native Point Defects 74

VI Contents

3.3 Dislocations 813.4 Dislocation Cells and Grain Boundaries 853.5 Second-Phase Particles 923.6 Summary and Outlook 95

References 96

4 Thermophysical Properties of Molten Silicon 103Taketoshi Hibiya, Hiroyuki Fukuyama, Takao Tsukadaand Masahito Watanabe

4.1 Introduction 1034.2 Density and Volumetric Thermal Expansion Coefficient 1064.3 Isobaric Molar Heat Capacity 1104.4 Emissivity 1134.5 Thermal Conductivity 1174.6 Surface Tension 1174.7 Diffusion Constant 1214.8 Viscosity 1234.9 Electrical Conductivity 1254.10 Sensitivity Analysis 1274.11 Recommended Thermophysical Property Data for Silicon

System 1274.12 Summary 129

References 131

Part II Simulation of Industrial Growth Processes 137

5 Yield Improvement and Defect Control in Bridgman-Type CrystalGrowth with the Aid of Thermal Modeling 139Jochen Friedrich

5.1 Introduction 1395.2 Principles of Thermal Modeling 1415.3 Verification of Numerical Models 1545.4 Yield Enhancement by Defect Control 1585.5 Conclusions 169

References 170

6 Modeling of Czochralski Growth of Large Silicon Crystals 173Vladimir Kalaev, Yuri Makarov and Alexander Zhmakin

6.1 Introduction 1736.2 Numerical Model 1746.3 Model Validation 1816.4 Conclusions 191

References 192

Contents VII

7 Global Analysis of Effects of Magnetic Field Configurationon Melt/Crystal Interface Shape and Melt Flow in a Cz-SiCrystal Growth 195Koichi Kakimoto and Lijun Liu

7.1 Introduction 1957.2 Model Description and Governing Equations Under a

Transverse Magnetic Field 1967.3 Computation Results for Model Validation 1977.4 Numerical Analysis of a TMCZ Growth 1997.5 Conclusions 202

References 203

8 Modeling of Semitransparent Bulk Crystal Growth 205Vladimir Kalaev, Yuri Makarov, Valentin Yuferev andAlexander Zhmakin

8.1 Introduction 2058.2 Numerical Model 2088.3 An Example: Growth of Bismuth Germanate Crystals 2138.4 Conclusions 226

References 227

Part III Compound Semiconductors 229

9 Recent Progress in GaAs Growth Technologiesat FREIBERGER 231Stefan Eichler, Frank B

..orner, Thomas B

..unger, Manfred Jurisch,

Andreas K..ohler, Ullrich Kretzer, Max Scheffer-Czygan, Berndt Weinert

and Tilo Flade9.1 Introduction 2319.2 Properties of GaAs 2329.3 Growth of Large-Diameter GaAs Single Crystals 2359.4 LEC versus VB/VGF GaAs Wafers 2469.5 Doping 2519.6 Summary 263

References 263

10 Interface Stability and Its Impact on Control Dynamics 267Frank J. Bruni

10.1 Introduction 26710.2 Diameter Control 26810.3 Interface Transitions 27110.4 Factors Influencing the Shape of the Solid/Liquid Interface 27610.5 Conclusions and Discussion 280

References 281

VIII Contents

11 Use of Forced Mixing via the Accelerated Crucible RotationTechnique (ACRT) in Bridgman Growth of Cadmium MercuryTelluride (CMT) 283Peter Capper

11.1 Introduction 28311.2 Elemental Purification (Mercury and Tellurium) 28411.3 Bridgman Growth of CMT 28511.4 Accelerated Crucible Rotation Technique (ACRT) 28911.5 Uses in IR Devices 30111.6 Summary 304

References 306

12 Crystal-Growth Technology for Ternary III-V SemiconductorProduction by Vertical Bridgman and Vertical Gradient FreezingMethods with Accelerated Crucible Rotation Technique 307Partha S. Dutta

12.1 Introduction 30712.2 Fundamental Crystal Growth Challenges for Ternary

Compounds 30812.3 Key Requirements for Ternary Substrates and Crystal-Growth

Process 31212.4 Optimization of Growth Parameters for Radially Homogeneous

Crystals 31412.5 Controlled Solute-Feeding Process for Axially Homogeneous

Crystals 31912.6 Steps in Ternary Crystal Production 32712.7 Current Status of Ternary Substrates 33212.8 Conclusion 332

References 333

13 X-Ray Diffraction Imaging of Industrial Crystals 337Keith Bowen, David Jacques, Petra Feichtingerand Matthew Wormington

13.1 Introduction 33713.2 Digital X-Ray Diffraction Imaging 33813.3 Applications Examples 34113.4 Summary 349

References 349

Part IV Scintillator Crystals 351

14 Continuous Growth of Large Halide Scintillation Crystals 353Alexander Gektin, Valentine Goriletskiy and Borys Zaslavskiy

14.1 Introduction 35314.2 Physical Principles Underlying Continuous

Single-Crystal Growth 355

Contents IX

14.3 Technological Platform for Family of ContinuousCrystal Growth Hardware 360

14.4 State-of-the-Art Crystal Performance for Continuous-GrowthTechniques 367

14.5 Crystal Press Forging for Large Scintillator Development 376References 378

Part V Oxides 379

15 Phase Equilibria and Growth of Langasite-Type Crystals 381Satoshi Uda, Shou-Qi Wang, Hiromitsu Kimura and Xinming Huang

15.1 Introduction – What is Langasite? 38115.2 Structure of Langasite-Type Crystals 38215.3 Study of Equilibrium Phase Diagram Around Langasite 38415.4 Study of Equilibrium Phase Diagram Around Langatate 39115.5 Conversion of Melting State of Langasite from

Incongruent to Congruent 39515.6 Direct Growth of Langasite from the Melt 39815.7 Optimal Composition for the Growth of Langasite via the

Czochralski Method 40115.8 Growth Technology of Four-Inch Langasite Along [0111] 40315.9 Growth of Langasite by the Bridgman Technique 406

References 413

16 Flame-Fusion (Verneuil) Growth of Oxides 415Hans J. Scheel and Leonid Lytvynov

16.1 Introduction 41516.2 Historical Background 41616.3 Impact of Verneuil’s Principles 41916.4 Apparatus 42016.5 Powder Preparation and Feeding Control 42316.6 Thermal Conditions 42716.7 Growth of Compounds with Volatile Constituents 43116.8 Conclusions 432

References 433

Part VI Crystal Growth for Sustaining Energy 435

17 Saving Energy and Renewable Energy Through CrystalTechnology 437Hans J. Scheel

17.1 Introduction 43717.2 Storage, Transport and Saving of Energy 44017.3 World Energy Consumption and Conventional Energy

Sources 441

X Contents

17.4 Future Energy Sources 44417.5 Costs and Risks of Conventional and of Future Energy

Sources 44717.6 Crystal Technology and its Role for Energy 44917.7 Future Technologies for Mankind 451

References 452

Part VII Crystal Machining 455

18 Crystal Sawing Technology 457Hans J. M

..oller

18.1 Introduction 45718.2 Multiwire Wafering Technique 45718.3 Basic Sawing Mechanism 45918.4 Experimental Results 47018.5 Summary 472

References 474

19 Plasma Chemical Vaporization Machining and Elastic EmissionMachining 475Yasuhisa Sano, Kazuya Yamamura, Hidekazu Mimura, KazutoYamauchi and Yuzo Mori

19.1 Introduction 47519.2 Plasma Chemical Vaporization Machining (PCVM) 47519.3 Elastic Emission Machining 48219.4 Catalyst-Referred Etching 488

References 493

Index 497

XI

Preface

This volume deals with the technologies of crystal fabrication, crystalcharacterization and crystal machining. As such it will be of interest toall scientists, engineers and students who are engaged in this wide field oftechnology. High-quality crystals form the basis of many industries, includingtelecommunications, information technology, energy technology (both energysaving and renewable energy), lasers and a wide variety of detectors of variousparts of the electromagnetic spectrum.

Of the approximately 20,000 tons of crystals produced annually the largestfraction consists of semiconductors such as silicon, gallium arsenide, indiumphosphides, germanium, group-III nitrides, cadmium telluride and cadmiummercury telluride. Other large fractions include optical and scintillator crystalsand crystals for the watch and jewellery industries.

For most applications these crystals must be machined, i.e. sliced, lapped,polished, etched or surface treated in various ways. These processes are criticalto the economic use of these crystals as they are a strong driver of yields ofusable material. Improvements are always sought in these various areas for aparticular crystal.

This book contains 19 selected reviews from the ‘Third InternationalWorkshop on Crystal Growth Technology’ held in Beatenberg, Switzerlandbetween 10–18 September, 2005. The first in the series ‘First InternationalSchool on Crystal growth Technology’ was also in Beatenberg between 5–14September 1998, while the second in the series was held between 24–29August 2000 in Mount Zao Resort, Japan. The latter generated a book of 29selected reviews that was published in 2003 by Wiley, UK entitled ‘CrystalGrowth Technology’ edited by Hans J. Scheel and Tsuguo Fukuda.

Part 1 covers general aspects of crystal growth technology, includingthermodynamics, phase diagrams, defects and thermophysical properties ofmelts. In Part 2 the emphasis is on theoretical modeling of the thermaland liquid/gaseous flows in the growth of both elemental and compoundsemiconductors, e.g. silicon and gallium arsenide, respectively. Part 3discusses the growth of several compound semiconductors in more detail.These include gallium arsenide, cadmium mercury telluride and galliumindium antimonide. There is also a chapter on the importance of X-ray

XII Preface

characterization in these materials. Part 4 covers the growth and applicationsof a range of halide and oxide crystals, including those grown by the Verneuiltechnique, which is recognized as the key technology from which most othertypes of bulk crystal growth are derived. The subject of crystal growth in theenergy industries, including energy saving and energy sources is detailed inPart 5. The final section of the book, Part 6, describes the current situation inboth crystal slicing and slice machining.

The editors would like to thank all the contributors for their valuable reviewsand the sponsors of IWCGT-3. Furthermore, the editors gratefully acknowledgethe patience and hard work of the following at Wiley-VCH: Martin Ottmar,Andreas Sendtko, Waltraud Wüst, Nele Denzau and Maike Petersen.

The editors hope that the book will contribute to the scientific developmentof crystal growth technologies and to the education of future generations ofcrystal growth engineers and scientists.

Beatenberg, Switzerland and Southampton, UK Hans J. ScheelOctober 2007 Peter Capper

XIII

List of Contributors

Frank B..orner

Freiberger Compund MaterialsGmbHAm Junger L

..owe Schacht 5

09599 FreibergGermany

Keith BowenBEDE X-ray MetrologyBelmont Business ParkDurham DH1 1TWUK

Frank J. BruniP.O. Box 2413Santa Rosa, CA 95405-0413USA

Thomas B..unger

Norddeutsche Raffinerie AGHovestrasse 5020539 HamburgGermany

Peter CapperSelex Sensors and AirborneSystems Infrared Ltd.P.O.Box 217MillbrookSouthampton SO1 5EGUK

Partha S. DuttaRensselaer Polytechnic InstituteCII-6015110 8th StreetTroy, NY 12180USA

Stefan EichlerFreiberger Compound MaterialsGmbHAm Junger L

..owe Schacht 5


Petra FeichtingerBEDE X-ray MetrologyBelmont Business ParkDurham DH1 1TWUK

Tilo FladeFCM GmbHAm Junger L

..owe Schacht 5


Jochen FriedrichFraunhofer IISBCrystal Growth LaboratorySchottkystr. 1091058 ErlangenGermany

XIV List of Contributors

Hiroyuki FukuyamaInstitute of MultidisciplinaryResearch for Advanced MaterialsTohoku University2-1-1 Katahira, Aoba-kuSendai 980-8577Japan

Alexander GektinInstitute for Scintillation Materials60 Lenin Ave.61001 KharkovUkraine

V. GoriletskiyInstitute for Scintillation Materials60 Lenin Ave.61001 KharkovUkraine

Taketoshi HibiyaGraduate School of System Designand ManagementKeio University2-15-45, MitaMinato-ku, Tokyo 108-8345Japan

Xinming HuangIMR, Tohoku University2-1-1 Katahira, Aoba-ku, SendaiMiyagi 980-8577Japan

Klaus JacobsInstitut f

..ur Kristallz

..uchtung

Max-Born-Strasse 212489 BerlinGermany

David JacquesBEDE X-ray MetrologyBelmont Business ParkDurham DH1 1TWUK

Manfred JurishFreiberger Compund Materials GmbHAm Junger L

..owe Schacht 5


Koichi KakimotoKyushu University6-1, Kasuga-KoenKasuga 816-8580Japan

Vladimir KalaevSoft-Impact Ltd.P.O. Box 83St.-Petersburg 194156Russia

Hiromitsu KimuraIMR, Tohoku University2-1-1 Katahira, Aoba-ku, SendaiMiyagi 980-8577Japan

Andreas K..ohler

Freiberger Compund Materials GmbHAm Junger L

..owe Schacht 5


Ullrich KretzerFreiberger Compund Materials GmbHAm Junger L

..owe Schacht 5


Lijun LiuKyushu University6-1, Kasuga-KoenKasuga 816-8580Japan

Leonid LytvynovInstitute for Single CrystalsLenin Ave. 60310001 KharkovUkraine

List of Contributors XV

Yuri MakarovSemiconductor Technology Res. Inc.10404 Patterson Ave., Suite 108Richmond, VA 23238USA

Hidekazu MimuraOsaka UniversityResearch Center for Ultra-Precision Science and Technology2-1 Yakadaoka, SuitaOsaka 565-0871Japan

Hans Joachim M..oller

Institut f..ur Experimentelle Physik

TU Bergakademie FreibergLeipziger Strasse 2509599 FreibergGermany

Yuzo MoriOsaka UniversityResearch Center for Ultra-Precision Science and Technology2-1 Yakadaoka, SuitaOsaka 565-0871Japan

Manfred M..uhlberg

Universit..at zu K

..oln

Z..ulpicher Strasse 49b

50674 K..oln

Germany

Peter RudolphInstitute for Crystal GrowthMax-Born-Strasse 212489 BerlinGermany

Yasuhisa SanoOsaka University2-1 Yamada-Oka, SuitaOsaka 565-0871Japan

Max Scheffer-CzyganFreiberger Compund Materials GmbHAm Junger L

..owe Schacht 5


Hans J. ScheelScheel ConsultingGr

..onstrasse Haus Anatas

3803 BeatenbergSwitzerland

Takao TsukadaDepartment of Chemical EngineeringOsaka Prefecture University1-1 Gakuen-choNaka-ku, SakaiOsaka 599-8531Japan

Satoshi UdaIMR, Tohoku University2-1-1 Katahira, Aoba-ku, SendaiMiyagi 980-8577Japan

Masahito WatanabeDepartment of PhysicsGakushuin University1-5-1 Mejiro, Tokyo 171-8588Japan

Shou-Qi WangElectronics Device Research andDevelopment CenterMitsubishi Materials Corp.2270 Yokoze, ChichibuSaitama 368-8502Japan

Berndt WeinertFreiberger Compund Materials GmbHAm Junger L

..owe Schacht 5


XVI List of Contributors

Matthew WormingtonBEDE X-ray MetrologyBelmont Business ParkDurham DH1 1TWUK

Kazuya YamamuraOsaka UniversityResearch Center for Ultra-Precision Science and Technology2-1 Yakadaoka, SuitaOsaka 565-0871Japan

Kazuto YamauchiOsaka University2-1 Yakadaoka, SuitaOsaka 565-0871Japan

Valentin YuferevIoffe Physical Technical InstituteRussian Academy of SciencesPolytechnicheskaya 26St.-Petersburg 194021Russia

Borys ZaslavskiyInstitute for Scintillation Materials60 Lenin Ave.61001 KharkovUkraine

Alexander ZhmakinIoffe Physical Technical InstituteRussian Academy of SciencesPolytechnicheskaya 26St.-Petersburg 194021Russia

Part IGeneral Aspects of Crystal Growth Technology

3

1

Phase Diagrams for Crystal Growth

Manfred Mühlberg

1.1Introduction

The operating abilities of a large part of modern technological hardware (electronicand optic devices, control and operating systems, watches, etc.) is based on activeand/or passive crystalline core pieces. For various applications the crystalline stateis required to be polycrystalline (ceramics), multigrained (solar cells), crystallinedesigned (thin film sequences) or single crystalline (semiconductor and NLOdevices, CaF2 lenses, prisms, etc.). The dimension of the crystallites and crystalscovers a wide range from (nanocrystallites) and (ceramics, thin-filmarrangements) up to and scales (electronics, optics), in special casesup to scale (silicon single crystals, natural ice and quartz crystals).

This chapter is only focused on the growth conditions of so-called single crystalsin the dimension of and scale. The fabrication of such singlecrystals is normally connected with the well-established growth methods namedafter Bridgman, Czochralski, Verneuil or zone melting, top seeded solution growth(TSSG), recrystallization techniques, etc. All these methods can be described bythe following definition of crystal growth:

1. In growing single crystals, one is primarily concerned with obtaining acrystal of predetermined size with a high degree of structural perfection anda well-determined chemical composition.

2. Growth of a single crystal requires the nucleation, subsequent growth,eventual termination of the process and, finally, removal of the crystal fromthe apparatus.

3. The transition into the solid/crystalline state can be realized from the vaporphase, liquids or a polycrystalline solid phase. Liquid phases are melts orhigh- or low-temperature solutions. The growth from liquid phases plays themost important role.

4. Each step of the growth process is affected by controlling the experimentalparameters pressure p, temperature T , and concentration (of components) xi.

4 1 Phase Diagrams for Crystal Growth

Paragraphs (1) and (2) are primarily determined by the growth method andoptimized technological parameters. Paragraphs (3) and (4) are correlated withsome thermodynamic terms: phases, pressure, temperature, and concentration.In a pictorial representation crystal growth means to start in a p − T − xi phasespace at any point po, To, xi(o). By default, the final point of the growth process isfixed at the normal atmospheric pressure, room temperature and a desired crystalcomposition. One (i), in some cases two (ii) challenges must be overcome betweenthe starting and final point in the phase space.

i. A phase transition (of first order) is necessary for the transfer into thesolid/crystalline state. They are denoted as sublimation, solidification, precip-itation, recrystallization, etc.

ii. Additionally, one or more phase transitions may exist in the solidified materialbetween the starting and final point. The kinds of solid/solid phase transitionsare very varied (Rao and Rao [1]), and the structural quality of the grown crystalis strongly influenced by the type of these phase transitions. Ferroelectriccompounds play an important role in several technical applications. Forthis reason, ferroelectric phase transitions, classified as phase transition ofsecond order, are of special interest in crystal growth. The most importantmaterials undergoing ferroelectric phase transitions are members of theperovskite group (LiNbO3, BaTiO3, KNbO3) and the tetragonal tungstenbronzes (strontium barium niobate (SBN), calcium barium niobate (CBN),potassium lithium niobate (KLN), and potassium titanyl phosphate (KTP =KTiOPO4).

Phase diagrams represent all these transitions. Consequently, the determination,knowledge and understanding of phase diagrams are one of the essential precon-ditions for selection and basic application of the growth method and the growthprocess.

1.2Thermodynamics and Phase Diagrams

Phase diagrams are the reflection of thermodynamic laws and rules betweendifferent phases in the p − T − xi phase space. The general thermodynamicbackground is given in textbooks (e.g. [2, 3]). There are also some distinguishedoverviews (e.g. [4]) and collections of selected phase diagrams (e.g. [5]). The aimof this chapter is to give an overview and understanding of phase diagramswith the dedicated focus to crystal growth. The basic thermodynamic functionsand variables are seen as prerequisite and are not included in this chapter (seeChap. 2).

As mentioned above, the crystallization process is a phase transition of first ordercharacterized by a jump of the latent (transition) heat, volume and several physicaland chemical properties like heat conductivities, densities etc. The latent heat ΔHtrmust be considered as the first important parameter.

1.2 Thermodynamics and Phase Diagrams 5

Table 1.1 Typical values for the heat and entropy of transformation.

Kind of transformation Heat of transformation Entropy of transformationΔ Htr [kJ/mole] ΔStr[

JK·mole ]

solid/solidfirst order 1–(5) 100 (comp.) ≈ 80

In particular, some growth processes from the melt have relatively high growthrates, being influenced by release of the heat of fusion. There are the followingrelationships between the different kinds of latent heat (legend: s/s – solid/solid;fus – fusion; vap – vapor; sub – sublimation):

ΔHs/s < ΔHfus < ΔHvap;ΔHsub = ΔHfus +ΔHvap;ΔHfus


Pre

ssur

e

0 1000 2000Temperature (°C)

Vapor

573 870 1470 1723

a-Qtz.

a

a b

bb′

b-Q

tz.

b-T

ridym

ite

Liquid

b-C

rist

(c)

p

Solid

ab

Vapour

Liquid

(a) (b)

T

0.10

0.00

−0.10

−0.20

−0.30

520 530 540 550 560 580 590 600 610570Temperature °C

Quartz, #1

DT

A r

espo

nse

a.u.

573 °C

Fig. 1.1 Ideal (top left) and real (SiO2 [2], top right)one-component system. DTA plot of the α–β quartz phasetransition.

at a simple one-component phase diagram, which can be articulated completely bythe CC equation.

For a solid/liquid phase transition the steepness of dp/dT can be greater orless than zero caused by a positive or negative volume difference between the twophases. In most cases, these differences are positive, i.e. the volume of the liquidphase is greater than the volume of the solid state (see top left image in Fig. 1.1).Furthermore, ΔV is very small for solid/liquid and solid/solid transitions, andthe pressure dependence on the melting point is also very small, typically in therange of 10−3 K/bar. Additionally, a differential thermal analysis (DTA) plot of thewell-known α ↔ β quartz transition is given in the top right picture of Fig. 1.1.The plot displays a heat effect for this transition being typical for phase transitionsof first order. On the other side, the α↔ β quartz transition can be specified usingthe Landau theory by a typical phase transition of second order. The rotation δ of theSiO2 tetrahedrons between 16

◦C (at room temperature) and 0 ◦C (at 573 ◦C) isthe order parameter and satisfies the classical rule δ∼ (T − T tr)1/2. Table 1.2 showssome examples for a positive and/or negative slope of the solid/liquid transition.


Table 1.2 Slope of dT/dp for the types of phase transitions:solid/solid, solid/liquid and liquid/vapor.

dTdp (K/bar) s ↔ s s ↔ l l ↔ v

Ag +4 × 10−3H2O −8 × 10−3 28.01CdSe −0.2 × 10−3HgTe +4.5 × 10−3α↔ β Quartz 0.021α = Quartz.↔ Tridymite 0.620

For transitions from a condensed phase into the vapor phase (the vapor phase isassumed to be perfect: Vv − Vcond ≈ Vv = R·T/p) the solution of the CC equationresults in

p = p0 · exp(

ΔHtrR

(1

T0− 1

T

))(1.6)

The one-component system can be easily expanded by Raoult’s and van’t Hoff’slaws if it is diluted. These laws describe that a low solute composition xB reducesthe freezing point of a solid phase and the partial pressure over a liquid phase(see Fig. 1.2).

Raoult’s law: ps = (1− xB) · p0(A) (1.7)van’t Hoff equations:

boiling point elevation :ΔT

Tv= xB · RTv

ΔHv(1.8)

freezing point depression :ΔT

Tm= −xB · RTm

ΔHf(1.9)

Equation (1.9) is useful to derive solubility curves from limited solubility data.

1.2.2Multicomponent Systems

For a multicomponent system Eqs. (1.2) and (1.3) can be primary extended by aterm describing the composition influence of the participated components xA,B,C,....The thermodynamic activity of any component (e. g. A) is expressed by the chemicalpotential μAi = μAi(0) + R·T ln xA(i); i corresponds to solid or liquid or vapor. Thechemical potential can be understood in terms of the Gibbs free energy per moleof substance, and it demonstrates the decreasing influence of a pure element or acompound in a diluted system. If any pure component is diluted then the term R·Tln xA(i) will always take values lower than zero (note, that only an ideal solutionbehavior is considered by the mole fraction xA. For real cases the so-called activity


Vap.

Sol.

PPure one-

compon. system

(III) (II)

(I)

Liqu.

T

Impure one-compon. system

Fig. 1.2 Extension of aone-component system byadding a solute; ps – va-por pressure over a dilutesolution; po(A) – vapor pres-sure of the pure solvent A;xB Δ mole fraction of animpurity; Tv, Tm –

boiling point, meltingpoint; ΔT = T − Tv;ΔT = T − Tm – absoluteboiling point elevation orfreezing point depression;ΔHv, ΔHf – heat of vapor-ization, heat of fusion.

1

0

−1

−2

−3

−4A 0.9 0.8 0.7

1200 K

800 K

400 K

μ A−μ

O A [k

J/m

ole]

XA

Fig. 1.3 The chemical potential of a pure componentis reduced in a binary system; it is also a function oftemperature; μA0 is the chemical potential of the purecompound A.

aA = f A·xA must be used. The activity coefficient f A collects all deviations froman ideal solution behavior). Figure 1.3 shows the reducing influence in a dilutedsolution as a function of the temperature.


Accepting that all processes have to be discussed in a p − T − xi phase space,Eq. (1.10) describes the complete change dG of the free energy of a multicomponentsystem.

dG = Vdp− SdT +n∑

i=1μidxi

︸︷︷︸T−xi phase diagrams

+ dγsurf + (..) dεelast (1.10)

For many processes the vapor pressure can be neglected and also the last twoterms must only be considered for small particles (surface influence) or nucleationinside of a solid phase (elastic strain). They can be neglected for any bulk growthprocesses from the liquid or vapor phase. These assumptions are the basis for thepresentation of the technical important T − x phase diagrams. Equation (1.10) isreduced for the case of a two-component system A–B to

dG = Vdp− SdT + xAdμA + xBdμB (1.11)Using the chemical potential as the ‘‘partial molar Gibbs free energy’’ in

Eq. (1.12) and accepting that many processes are running at a nearly constantpressure (p = const.; dp = 0) one can rewrite Eq. (1.12) to Eq. (1.13) for a twocomponent system A–B in solid(s)/liquid(l) equilibrium

dGsA = dG

1A dGA =

dG

dxA(1.12)

−(S sA − S1A)dT + RT · d ln

xsAx1A= 0 (1.13)

Replacing −(S sA − S1A) by

ΔHfusT and integrating Eq. (1.13) gives the final

expression for an ideal binary phase diagram of a solid solution system A–B. Thisequation is also indicated as the van Laar equation for a two-component systemA–B [3].

lnxsAx1A− ln x

sB

xlB= ΔHA

RT

(1− T

TA

)− ΔHB

RT

(1− T

TB

)(1.14)

The van Laar equation is only determined by the two melting points TA, TB andthe heats of fusion ΔHA, ΔHB of the end members A and B. Their influence onthe shape of a solid solution system can easily be shown on a PC if the equationis converted in parametric functions [Eqs. (1.15) and (1.16)] and calculated by anydata and function plotting utility (e.g. Gnuplot [6], see Fig. 1.4).

Equations (1.15) and (1.16) illustrate the parametric function for the solidus andliquidus curve

xsB =exp

{ΔHA

R

(1

T− 1

TA

)}− 1

exp

{ΔHA

R

(1

T− 1

TA

)− ΔHB

R

(1

T− 1

TB

)}− 1

(1.15)


1400

800

1000

1200

600

400

Tem

pera

ture

(K

)

Liquid

Solid (b–phase)

Solid (a–phase)

A B0.2 0.4 0.6 0.8

Mole fraction xB

Fig. 1.4 Application of the van Laar equationto a solid/liquid and a solid(α)/solid(β) with random,but typical values.

xlB =exp

{ΔHA

R

(1

T− 1

TA

)}− 1

exp

{ΔHA

R

(1

T− 1

TA

)}− exp

{ΔHB

R

(1

T− 1

TB

)}(1.16)

Examples for a binary complete solid solution system for thea) solid↔ liquid transition:(with ΔHf A = ΔHf B = 50 kJ/mole, and formelting points: TA = 1000 K; TB = 1400 K) and forb) α↔ β phase transition:(with ΔHtrA = ΔHtrB = 1.5 kJ/moletransition temperatures: TA = 500 K; TB = 900 K)are given in Fig. 1.4.

It can be seen in Fig. 1.5 that the higher the heats of fusion the broader thewidth between the liquidus and solidus lines of an ideal system. Furthermore,the difference between the heats of fusion determines the asymmetric shape of thephase diagram. In Section 1.3.1 the consequences of the shape of the solid solutionphase diagrams on the segregation behavior in normal freezing growth processeswill be discussed.

The extension to real cases and eutectic systems can be carried out in ananalogous manner and is described by Kubaschewski and Alcock [7].


1300

1200

1100

1000

900

800

7000.20.40.60.8A B

Mole fraction xB

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

1300

1200

1100

1000

900

800

7000.20.40.60.8A B

Mole fraction xB

Tem

pera

ture

(K

)T

empe

ratu

re (

K)

ΔHB = 10kJ/mole

ΔHA = 30kJ/Mol

ΔHB = 30kJ/Mol

ΔHB = 40kJ/Mol

ΔHA = 20kJ/mole

ΔHB = 20kJ/mole

ΔHA = 40kJ/mole

ΔHA = 10kJ/mole

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

Tem

pera

ture

(K

)T

empe

ratu

re (

K)

1300

1200

1100

1000

900

800

7000.2 0.4 0.6 0.8A B

Mole fraction xB

ΔHA = 25kJ/mole

ΔHA = 25kJ/mole

ΔHB = 75kJ/mole

ΔHA = 100kJ/mole

ΔHB = 50kJ/mole

ΔHA = 75kJ/Mol

ΔHB = 50kJ/Mol

ΔHB = 50kJ/mole

(b)(a)

Fig. 1.5 Influence of the heat of fusion on the design ofsolid solution phase diagrams; calculated by Eqs. (1.15) and(1.16).

1.2.3Gibbs Phase Rule and Phase Diagrams

The main key for the understanding of phase diagrams is the phase rule of WilliamGibbs (1876)

P + F = C + 2 (1.17)

where P is the number of phases, C is the number of components in thesystem, and F is the number of freedom, or variance. The definition forthe combined terms are: P – any part of a system that is physically ho-mogeneous within itself and bounded by a surface; component C – small-est number of independently variable chemical constituents and degree offreedom – smallest number of intensive variables (e.g. p, T, xi of componentsin each phase) that must be specified to completely describe the state of the system.

Phase diagrams are the graphical representations of the phase rule, and theyare classified by the number of components as follows: one-, two-, three-, . . .component systems. On the other side, the phase rule is the most important toolfor verifying phase diagrams. If pressure is omitted as a variable, the number ofvariables in a system is two: temperature and composition. The phase rule reducesto F=C− P + 1 and in this form is referred to as the condensed phase rule or phaserule for condensed systems. As an example, let us discuss the application of thephase rule on a simple three-component system A–B–C with one compound BC.


Bb bc

E

a

Dc

X

BC C

C + L

BC + C

B + BC

B + L BC + L

BC CB

A

Fig. 1.6 Application of the Gibbs phase rule on three spe-cial points in a simple three-component system. The phasediagram was taken from [2].

If at least one phase exists then there is a four-dimensional phase spacedetermined by the variables: p, T, xA, xB. As an illustration, we have to re-duce the dimension of the phase space. In the first step a constant pres-sure is assumed and the phase space is reduced to a trihedral prism withthe coordinates T, xA, xB. Normally, the projection onto this trihedral prismis used for printing processes. Figure 1.6 and the legend give the explana-tion for the relationship between the number of phases and the number offreedom.

P = 1 → F = 4 p, T , xA, xB phase spaceP = 1 → F = 3 T , xA, xB phase space, if p = const.P = 2 → F = 2 BCsol. +melt

(⊕ )P = 3 → F = 1 (A+ BC)sol. +melt(O)P = 4 → F = 1 (A+ B+ BC)sol. +melt(Δ)

There are several violations of the phase rule resulting in incorrect description ofphase relationships. Instructive examples of such thermodynamically impossible

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