microscopic theory of wetting and adhesion in metal...

THESIS FOR THE DEGREE OFDOCTOR OFPHILOSOPHY

Microscopic Theory of Wetting and Adhesion inMetal-Carbonitride Systems

SERGEY DUDIY

Department of Applied PhysicsCHALMERS UNIVERSITY OF TECHNOLOGY

GOTEBORG UNIVERSITYGoteborg, Sweden 2002

Microscopic Theory of Wetting and Adhesion in Metal-Carbonitride SystemsSERGEY DUDIYISBN 91-628-5296-5Applied Physics Report 2002-45c Sergey Dudiy, 2002

Chalmers University of TechnologyGoteborg UniversitySE-412 96 G¨oteborg SwedenTelephone +46–(0)31–772 1000

Chalmersbibliotekets reproserviceGoteborg, Sweden 2002

Microscopic Theory of Wetting and Adhesion in Metal-Carbonitride SystemsSERGEY DUDIYDepartment of Applied PhysicsChalmers University of TechnologyGoteborg UniversityABSTRACT

Joints between metals and ceramics are increasingly important in the manufac-turing of many high technology products, from microelectronic devices to cuttingtools. Wetting of ceramics by metals is the driving force of metal-ceramic joiningprocesses, such as brazing and sintering of WC-Co cemented carbides and TiC-Cocermets. Experimental studies suggest that wetting in metal-ceramic systems is mostsensitive to microscopic factors, like local chemical composition at interfaces.

This thesis is a theoretical study of the key microscopic mechanisms behind thewetting and adhesion, at the level of interatomic interactions. The ceramic materialsconsidered are transition metal carbides and nitrides. The theoretical analysis isbased on the results of first-principles density-functional calculations for a broadvariety of model interface systems, using the plane-wave pseudopotential method.To deal with the problem of disordered interface structure, an approach based oncomparative analysis of high-symmetry model systems is proposed.

It is demonstrated that the dominating mechanism of the Co/Ti(C,N) interfaceadhesion is a strong Co-C(N) bond. The number of those bonds is determined byan interplay of the interface incoherence and the structure relaxation effects. Theparticular strength of the Co-C bond is explained in terms of interface-induced fea-tures of the electronic states, in particular a novel metal-modified covalent bond.The obtained strength of the Co/TiC adhesion is in good agreement with availabledata from wetting experiments with liquid Co on TiC surface.

It is found that the Co ferromagnetism gives a significant change of the Co/TiCadhesion strength and interface energy, which is expected to be important during thesolid-state sintering stage of the hardmetal manufacturing process. This effect canbe adequately described within the Stoner model of itinerant ferromagnetism.

The known fact of better wetting in WC-Co systems than in TiC-Co ones isconfirmed and explained in terms of a larger contribution of the metal-metal Co-Wbonding at Co/WC interfaces.

The large scattering of the experimental wetting data for Cu and Ag on TiC andTiN is interpreted in terms of the different relative contributions of the elementarylocal atomic coordinations at the metal/Ti(C,N)(001) interfaces. Wetting is shownto be improved by C(N) vacancies and Ti segregation in the melt, in agreement withexperimentally observed wettability improvements for hypostoichiometric carbides.The suggested simple microscopic picture of wetting in terms of different chemicalbonds across the interface is also applied to the analysis wetting trends for Cu onHfC, ZrC, TaC, NbC, and VC.Keywords: metal-ceramic interfaces; structure; bonding; wetting; adhesion; in-teratomic interactions; total energy and electronic structure calculations; densityfunctional theory; carbides and nitrides; cermets; composites; hardmetals; sinter-ing; brazing;

LIST OF PUBLICATIONS

This thesis consists of an introductory text and the following papers:

I Nature of Metal-Ceramic Adhesion: Computational Experiments with Co on TiCS.V. Dudiy, J. Hartford, and B.I. LundqvistPhys. Rev. Lett.85, 1898 (2000)

II First-principles density-functional study of metal-car bonitride inter face ad-hesion: Co/TiC(001) and Co/TiN(001)S.V. Dudiy and B.I. LundqvistPhysical Review B64, 045403 (2001)

III Effects of Co magnetism on Co/TiC(001) interface adhesion:A first-principles studyS.V. DudiySurface Science497, 171 (2002)

IV First-principles simulations of metal-ceramic interface adhesion: Co/WCversus Co/TiCMikael Christensen, Sergey Dudiy, and G¨oran Wahnstr¨omPhysical Review B65, 045408 (2002)

V Wetting of TiC and TiN by MetalsS.V. Dudiy and B.I. LundqvistApplied Physics Report 2002-38,to be published

Comments on my contributions to the included papers:

In Papers I, II, III, and V, I have performed the calculations and written the originaldrafts of the articles. In Paper IV, I have contributed to the setup of the calculationsand article writing.

Scientific publications not included in this thesis:

Density-functional bridge between surfaces and interfacesB.I Lundqvist, A. Bogicevic, K. Carling, S.V. Dudiy, S. Gao, J. Hartford, P.Hyldgaard, N. Jacobson, D.C. Langreth, N. Lorente, S. Ovesson, B. Razazne-jad, C. Ruberto, H. Rydberg, E. Schröder, S.I. Simak, G. Wahnström and Y.YourdshahyanSurface Science493, 253-270 (2001)

Bridging between Micro- and Macroscales of Materials by MesoscopicModels, Invited PaperB.I. Lundqvist, A. Bogicevic, S. Dudiy, P. Hyldgaard, S. Ovesson, C. Ruberto,E. Schroder, and G. WahnströmComputational Materials Science, in print (2002)

Frequency dependence of the admittance of a quantum point contactI.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V. DudiyPhysical Review B58, 9894-9906 (1998)

Modeling AC electronic transport thr ough a two-dimensional quantumpoint contactI.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V. DudiyMicroelectronic Engineering47, 357-359 (1999)

On the Crossover of the Surface Plasmon Spectrum from Two-Dimensionalto Quasi One-Dimensional in a Quantum Point ContactI.E. Aronov, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V. DudiyPhysica B,253, 169-179 (1998)

A.c. transport and collective excitations in a quantum point contactI.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V. Dudiyand R. MainieriSemiconductor Science and Technology13 , A104-A106 (1998)

Wigner function description of a.c. transport thr ough a two-dimensionalquantum point contactI.E. Aronov, G.P. Berman, D.K. Campbell, S.V. DudiyJournal of Physics: Condensed Matter9, 5089-5103 (1997)

We always attract into our lives whatever we think about most, believe in moststrongly, expect on the deepest level, and imagine most vividly.

– Shakti Gawain

Contents

1 Introduction 1

2 Motivation: Sintering and Brazing Technologies 52.1 Cemented Carbides and Cermets as High Performance Tool Materials 5

2.1.1 Hardmetal Sintering Process and Role of Wetting . . . . . . 82.2 Brazing as Important Joining Technique . . . . . . . . . . . . . . . 9

2.2.1 Active-Metal Brazing of Ceramics in Overcoming Wettabil-ity Challenge . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Wetting Experiments with Drops of Molten Metals on TransitionMetal Carbide and Nitride Surfaces . . . . . . . . . . . . . . . . . 112.3.1 Wetting of TiC and WC by Co . . . . . . . . . . . . . . . . 122.3.2 Wetting by Noble Metals . . . . . . . . . . . . . . . . . . . 132.3.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 14

3 Fundamental Framework for Materials Modeling:Density Functional Theory 153.1 Density Functional Theory Overview . . . . . . . . . . . . . . . . . 16

3.1.1 Hohenberg and Kohn Theorems . . . . . . . . . . . . . . . 163.1.2 Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . 173.1.3 Adiabatic Connection Formula . . . . . . . . . . . . . . . . 18

3.2 Exchange and Correlation Approximations . . . . . . . . . . . . . . 193.3 Note on Applications to Solids . . . . . . . . . . . . . . . . . . . . 21

4 Computational Method: Technical Aspects of Solving DFT Equations 234.1 Plane Waves as Convenient Basis Set . . . . . . . . . . . . . . . . . 244.2 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1 Key Steps in Pseudopotential Construction . . . . . . . . . 264.2.2 Essential Aspects of Pseudopotential Transferability . . . . 274.2.3 Ultrasoft Pseudopotentials for Efficient Treatment of Tran-

sition Metal and First Row Elements . . . . . . . . . . . . . 29

5 Transition Metal Carbides and Nitrides 31

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Contents

5.1 Crystal Structure and Stoichiometry . . . . . . . . . . . . . . . . . 315.2 Electronic Structure and Chemical Bonding . . . . . . . . . . . . . 34

5.2.1 Metal-C(N) Bonds . . . . . . . . . . . . . . . . . . . . . . 345.2.2 Metal-Metal Bonds . . . . . . . . . . . . . . . . . . . . . . 355.2.3 Bonding Trends and Population of Bonding and Antibond-

ing States . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Free Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Relevant Interface Thermodynamics Background 396.1 Definition of Interface Free Energy . . . . . . . . . . . . . . . . . . 396.2 Thermodynamics of Wetting: Contact Angle and Work of Adhesion 426.3 Ideal Work Of Separation as Measure of Interface Adhesion Strength 43

7 Goals and Principles of Interface Geometry Modeling 457.1 Basic Definitions and Background: Geometrical Degrees of Freedom 46

7.1.1 Macroscopic Degrees of Freedom . . . . . . . . . . . . . . 467.1.2 Microscopic Degrees of Freedom . . . . . . . . . . . . . . 467.1.3 Interface Geometry Control in Experiment . . . . . . . . . 47

7.2 Choice of Interface Geometry Models . . . . . . . . . . . . . . . . 487.2.1 Scenarios of Theory-Experiment Interaction . . . . . . . . . 487.2.2 Should We Search for Best Structure? . . . . . . . . . . . . 507.2.3 Focus on Development of Theoretical Models of Wetting

and Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . 517.2.4 Simplified Description of Bulk Phases . . . . . . . . . . . . 537.2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 54

8 Microscopic Interactions at Metal-Ceramic Interfaces 578.1 Dispersion Forces and Carbide Wetting Trends . . . . . . . . . . . 588.2 Image Interaction Model . . . . . . . . . . . . . . . . . . . . . . . 608.3 Chemical Bonds across Interface . . . . . . . . . . . . . . . . . . . 618.4 Metal-C(N) Bonds across Interface as Opposed to those in Bulk Car-

bides and Nitrides . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

9 Conclusions 659.1 Qualitative Microscopic Picture of Wetting and Adhesion . . . . . . 659.2 Interpretations of Wetting Experiments . . . . . . . . . . . . . . . . 669.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Acknowledgements 71

Bibliography 73

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CHAPTER 1

Introduction

It is hard to find two classes of materials that are more dissimilar than metals andceramics. Metals are typically tough. They are ductile rather than hard. They arequite unstable chemically and thermally. In particular, metals can relatively easilybe affected by corrosion and chemical attacks, and they tend to expand or shrinknoticeably with temperature changes. Metals are good conductors of heat and elec-tricity.

Ceramics are just the opposite. Most of them, like many oxides, carbides, ni-trides, and borides, are hard and wear resistant, though brittle. They can easily standhigh temperatures and chemical attacks. Ceramics are typically good insulators.

Due to such a dissimilarity of properties, in many high-technology applications,and often in everyday situations, metals and ceramics work together (see Fig. 1.1).One can easily see such situations, for example, in microelectronic devices, indus-trial cutting tools, or medical implants. The combination of properties of metals andceramics within one device is crucial for such applications to work.

To make metals and ceramics work together, what one needs before anythingelse is a way to join them. And this is where the dissimilarity of metals and ceramicsshows itself again, but this time as a problem rather than a solution. Making reliablemetal-ceramic joints is a well-known challenge, which often requires quite advancedmetal-ceramic joining techniques.

There is a constant need and ongoing process of further development of metal-ceramic joining techniques, expanding their applicability to new types of materi-als, or solving various performance problems with existing materials. It is broadlyrecognized that such development should be based on solid scientific foundations,on understanding of scientific principles involved in the joining processes. Withoutsuch an understanding, attempts to solve various performance and production prob-lems tend to degenerate easily from empirical adjustment to random tinkering withprocess parameters. Motivated by those technology development needs, the presentthesis deals with fundamental scientific principles behind metal-ceramic joining

1

1 Introduction

processes.

bene

fits problems

of properties

Applications

METALS CERAMICS

Joining techniques

aerospacecomposites

microelectronicsmedical implants

.....

cutting tools

Dissimilarity

wetting, adhesion

Figure 1.1: A schematic diagram illustrating the situation with the practical use of metal-ceramic systems.

There is a variety of known techniques of joining metals and ceramics, like sin-tering and brazing. Most of those techniques are based on creating stable chemicalinterfaces between metal and ceramic components. Such joining processes are con-trolled by the conditions of wetting and adhesion.

The first condition is that the metal should get in close contact with the ceramicsurface. For this purpose the metal is melted and then allowed to flow freely overthe ceramic surface. The liquid metal is expected to spread over the surface andfill all the narrow gaps between the ceramic components. Wetting is what makesthis spreading happen. Non-wetting would mean that the liquid prefers to stay indrops, minimizing the area of its contact with the surface. In practice, for successfuljoining, e.g., via brazing or sintering, wetting should be good enough to fill the gapscompletely, without any interface voids that degrade mechanical properties. To alarge extent wetting is determined by how well the liquid adheres to the surface,i.e.how strong the adhesive forces are at the metal-ceramic interface.

The second important condition is that after metal solidification there should stillbe strong bonding forces, that is, a strong adhesion that keeps the solid metal andceramic parts together. The strength of the adhesion now determines the quality ofthe metal-ceramic joints, in particular, their mechanical properties.

The present thesis aims to contribute to the scientific understanding of wettingand adhesion in metal-ceramic systems, in particular with ceramic carbides and ni-trides, by exploring the microscopic origins of the adhesive forces at metal-carbideand metal-nitride interfaces.

2

Investigation of microscopic mechanisms of metal-ceramic wetting and adhesionhas for long been a challenge for both experiment and theory. In particular, underthe conditions of realistic wetting experiments it is very problematic to resolve themicroscopic processes within a few atomic layers of the liquid-solid metal-ceramicinterface. The lack of such experimental information is also a significant obstaclefor developing theoretical models.

The existing theoretical models of wetting in metal-ceramic systems are mostlyphenomenological. At the same time, the recent wetting experiments point at in-adequacies of such phenomenological models. In order to understand experimentalwetting behaviors one really needs to go to the microscopic level, they suggest. Thisis the level of interatomic chemical bonds across the interface. Such a microscopictheoretical understanding of wetting and adhesion is the goal of the research in thisthesis.

The thesis is organized as follows.Chapter 2 provides a brief introduction into the spectrum of physical problems

involved in technological processes of metal-ceramic joining, in particular sinteringand brazing, which have been an important motivation for the research in this thesis.

Chapter 3 summarizes the fundamental methodological framework for the the-oretical simulations of materials systems, in particular, the density functional theory(DFT).

Chapter 4 reviews important technical issues involved in solving the DFT equa-tions, with main attention to the plane-wave pseudopotential method, which is usedin first-principles calculations in the thesis.

Chapter 5 collects relevant introductory information on transition metal car-bides and nitrides, which is an important background for the simulation setup andthe discussions of the results in the appended research papers.

Chapter 6 defines the main quantities used in the thermodynamic descriptionof interface systems and the computational experiments in the thesis. It aims toclarify the relations between the quantities measured in wetting experiments andthose calculated in first-principles simulations.

Chapter 7 discusses different aspects of making physically meaningful choicesof the model interface structures in first-principles simulations of metal-ceramic sys-tems. It also introduces a number of common definitions that are used in the interfacegeometry description.

Chapter 8 describes various types of microscopic interactions across metal-ceramic interfaces, including a particular kind of metal-carbon covalent bond, animportant finding of the research in the appended papers.

Chapter 9 summarizes the progress in understanding of wetting and adhesionin metal-ceramic systems made in the cause of the present research.In addition, itcontains examples of how experimental wetting behaviors can be understood withinsimple microscopic pictures of wetting emerging from our theoretical modeling.Finally, it discusses possible directions of further investigations.

3

1 Introduction

4

CHAPTER 2

Motivation: Sintering and BrazingTechnologies

We are continuously faced by great opportunities brilliantly disguised as insolubleproblems.

– Lee Iacocca

2.1 Cemented Carbides and Cermets as High Perfor-mance Tool Materials

Sintered carbonitrides, also referred to as (refractory) hardmetals, are an indispens-able part of modern technologies. In one way or another, they enter almost everyindustry, as metal-cutting inserts or sharp ends of the drills, cutting tools for coal orrocks, knives that slice paper or magnetic tape,etc. Such great success of the hard-metals is due to their outstanding properties, especially to their very high hardnessand wear resistance, in combination with a reasonable price. Any further advancein the hardmetal technology can have a significant impact on the efficiency of manyindustrial processes and open new areas of applications.

The unique properties of the hardmetals are essentially due to the way they aremade.1,2 They consist of hard particles of carbonitrides, typically WC or Ti(C,N)bonded together with a metallic binder phase, usually Co, Ni, Fe or a mixture ofthose components (Fig. 2.1). With such a combination of the constituting materialsit appears to be possible to superimpose the positive properties of the carbonitridesand metals, while suppressing the negative ones. In particular, from the ceramiccarbonitrides the hardmetals inherit the high hardness and wear resistance, while themetallic binder phase provides ductility, toughness and thermal-shock resistance.

In general, if one makes a composite material by mixing two different compo-

5

2 Motivation: Sintering and Brazing Technologies

nents,e.g., metal and carbonitride, there is no guarantee that the resulting materialwill combine the good properties of the components. On the contrary, it is quite easyto produce something that is worse than any of the components in its pure form. It istherefore no surprise that the performance of the hardmetals is very sensitive to thedetails of the manufacturing process and to the structure and composition of the raw

Figure 2.1: An electron emission photograph showing the microstructure of a TiC-WC-TaC-Co cermet.1

materials. This makes the technology of the hardmetal production very complex,with each hardmetal company having a lot of recipes and secrets of its own.

Historically, the first successful sintered carbides were obtained by Schr¨oter inthe early twenties in Germany. They were produced by mixing together powders oftungsten carbide (WC) and cobalt, compacting that mixture, and then heating thesystem above the cobalt melting point. This was a major breakthrough in the hard-metal technology. After some further extensive developments, the WC-Co-basedhardmetals, also called thecemented carbides, became the traditional cutting-toolmaterials.

The WC-Co cemented carbides still dominate the tool market. At the same time,as the demands on the modern cutting operations increase rapidly, much effort is putinto the search of new solutions, which would allow higher speed and/or precisionof cutting, more severe operating conditions,etc.Significant progress in the cuttingperformance was achieved by the introduction of coatings,3 i.e. by adding layersof alumina, titanium carbide and nitride and other materials onto the tool workingsurface. Such coatings improve the tool lifetimes by 5 to 10 times, but still they aremore like technical improvements rather than real innovations, and there are manyproblems waiting to be solved. For instance, one of those problems is the plasticdeformation of the tools at high temperatures, which is a serious limitation on thecutting speed, and, hence, productivity.

6

2.1 Cemented Carbides and Cermets as High Performance Tool Materials

Currently, one of the most prospective directions of the hardmetal developmentsis cermets.2,4–6 The cermets are hardmetals that instead of WC use titanium carbide,nitride or carbonitride. Among the important advantages of cutting with cermetsare6 high cutting speed at moderate chip cross-sections, high surface quality of themachined workpiece, high wear resistance and reliability. The main drawback ofthe cermets is that they are more brittle than the WC-Co cemented carbides, and,hence, are less suitable for rough cutting. Yet, cermet’s properties can be adjustedby additions of other components (see Fig. 2.2), and they noticeably outperform theWC-Co-based hardmetals in many special applications, where the high performanceof cutting is essential. The weight of the cermets on the tool market is expected togrow, which is due to the increasing role of the high technologies, on the one hand,and to the continuing improvements in the cermet performance, on the other hand.

Figure 2.2: Properties of cermet cutting alloys as a function of composition.6

So far the development of the cermets has been based mainly on empiricism.However, it has been recognized that any further significant progress in this arearequires a deeper and more systematic research. In particular, such issues as themetallurgical reactions during the cermet manufacturing, the microscopic processesin cermets, and the dependence of the cermet properties on their composition andmicrostructure have to be understood at a more fundamental level, which actuallyinvolves a very wide spectrum of materials science problems.

The research in the present thesis is to a large extent an attempt to approach thisspectrum of problems, at least a part of it. This implies that both the motivationand the experimental background of this work are to a large extent related to thecermets. To clarify this relation, the continuation of the present chapter provides abrief outline of the important aspects of the cermet manufacturing.

7


2.1.1 Hardmetal Sintering Process and Role of Wetting

The hardmetal manufacturing is a many-stage process. The starting point is powdersof the carbonitrides and of the binder metal. First the powders undergo the millingstage, during which the powder grain size is controlled, and a homogeneous mixtureof the metal and carbonitride components is obtained. Then, during the pressingstage, such a mixture is compacted into the so-called green body. The compactedmixture of the powders does not fall apart, but it is still far from the final product.This is because the grains are not fully bound to each other due to relatively largeamount of the empty space (pores) between them. The elimination of the pores isthe task of the next stage, thesintering,7 which is the crucial step in the hardmetaltechnology.

In the context of hard metals, sintering can be described as a thermally activateddensification of the compacted powder mixtures. The densification during sinteringis the result of the mass transport that rearranges the constituents in such a way thatthe pores are filled. The driving force for this mass transport is the excess of thesurface energy in the porous system. The filling of the pores gives an energy gain,because, on the one hand, the grains grow in size, which reduces the amount of thefree-surface area. On the other hand, instead of the free surfaces there are interphaseboundaries, which is also more favorable since extra intergrain bonds are created.

The main mechanisms of the mass transport during sintering depend on the sin-tering conditions. At high enough temperatures, but below the binder-metal meltingpoint, one has a situation of the solid state sintering. During the solid state sinteringthe mass transport is mainly due to diffusion and plastic flow of the materials, andit is relatively slow. Although the solid state sintering should not be neglected, themajor densification of the cemented carbides and cermets occurs during the liquidphase sintering,i.e. at temperatures above the metal melting point. The advantagesof the liquid phase sintering are the enabled viscous flow of the metal phase andthe significantly increased diffusion rates, which allow a faster and more completepenetration of the binder into the pores.

In connection with the present work, the most interesting fact about the liquidphase sintering is that its result crucially depends on how good the wetting of thecarbonitride grains by the binder phase is. If the wetting is poor then the liquidphase tends to minimize its surface area, pushing the hard grains apart rather thanfilling the pores between them. If the wetting is good then it becomes more favorablefor the liquid phase to maximize its contact area with the grains, which means thatthe liquid penetrates into the pores, and pulls the grains together. Therefore, highwettability of the carbonitride grains by the metal binder is a necessary condition forsuccessful sintering.

The significance of good wetting also follows from the fact that pores left aftersintering can act as internal sources of cracks, which noticeably affects the strengthof the material. Although the amount of the residual porosity depends on manydifferent factors, wettability is of primary importance, and to minimize the porosityit is highly desirable that wetting is complete,i.e. that there is a total spreading of

8

2.2 Brazing as Important Joining Technique

liquid over the ceramic surface, like in the WC-Co system.As a concluding remark, it should be mentioned that, under real conditions, wet-

ting and filling of the pores are only one side of sintering. One more importantaspect is the compositional rearrangements in the material. In particular, the car-bonitride grains partially dissolve into the binder phase and then reprecipitate, whichnoticeably affects the size, shape and composition of the grains, the properties of thebinder phase, and, as a result, the properties of the obtained material. In cermetssuch dissolution-reprecipitation processes lead to formation of the so-called core-rim structure.6,8 That is, the carbonitride grains consist of Ti(C,N) cores surroundedby rims of carbonitrides of heavier metals, such as W, Mo, V, Nb or Ta. Inclusion ofthose processes in the analysis of this thesis, especially at the first-principles level,would make the considered problems practically intractable. On the other hand, hav-ing a physical picture of metal-carbonitride interactions, some of such issues can beapproached in future studies.

2.2 Brazing as Important Joining Technique

Brazing9 is the joining of metals with metals, metals with ceramics, or ceramicswith ceramics through the use of heat and a filler (braze) metal. The braze metal oralloy is heated so that it melts and flows over the surfaces of the components to bejoined. The heating can be done in a furnace or with a torch. The melt should fill anarrow gap between the components and then form a permanent bond by remainingadherent while solidifying (see Fig. 2.3).

Metallurgical bonding at interfaces

Brazing filler metal

Base material A

Base material B

Figure 2.3: A schematic illustration of a brazed joint.

In brazing, the filler metal should have the melting temperature below the melt-ing points of the materials being joined, but above 450oC. If the same process is usedbelow 450oC it is referred to assoldering. If such a process involves partial meltingof the base components it is calledwelding. The filler metals used in brazing are ofhigher melting points, and hence of higher strength, than the ones used in soldering.The lower temperatures of brazing than of welding mean lower risk of the effects ofoverheating of the base components.

9


There is a number of important advantages of brazing over many other joiningtechniques. Brazing is quite well suited for joining dissimilar materials. It is rela-tively fast, simple, flexible, and economical. Brazed joints are strong and ductile,able to withstand considerable stress, shock, and vibrations.

The key physical phenomenon that lets brazing work is wetting. The liquid mustadhere to the base component surfaces strongly enough so that to flow readily overthose surfaces and fill the narrow gaps between the components. The wetting shouldbe good enough for the filling of the gaps to be very complete. Any voids left at theinterface would significantly degrade the mechanical properties of the joint.

Good wetting is the first important criterion when choosing the material of thebraze to use in a particular brazing applications. Another important requirement isthe ductility of the braze metal, its ability to accommodate the stresses caused bytemperature changes and the thermal expansion mismatch in the joined materials.The braze should be tough enough to create a strong joint. Depending on the specificapplication, the corrosion resistance of the braze metal may also be important. Mostof the commercially available brazes are based on Al, Ag, Cu, Au, and Ni, whichare aimed for different ranges of flow and application temperatures.

Brazing has been used since ancient times to join metal parts, like, for example,steal and iron parts of swords. Currently metal-metal brazing is very heavily usedin various industries, as well as in households, like when brazing bicycle frames.And there is a variety of brazes available, which are well suited for different typesof metal-metal brazing situations.

2.2.1 Active-Metal Brazing of Ceramics in Overcoming Wettabil-ity Challenge

Brazing of ceramics has been used in more recent times and has always been veryproblematic.9–14 The key difficulty in brazing of ceramics is that most of the ce-ramics are unwetted by the commercial alloys already developed for joining a widevariety of metallic component materials. There are two main approaches to over-come this wetting problem. The first approach, which is more traditional, usespre-metallization,i.e. pre-coating of materials to be connected by conventionaltechniques, such as evaporation of metal in a vacuum furnace. The pre-metallizedceramics can then be brazed with standard filler alloys,i.e. the ones used in metal-metal brazing. The brazing through pre-metallization is a multi-step process, andit is quite cumbersome. The strength of the joint is very sensitive to the qualityof surface pre-treatments, and to the effectiveness of the fluxing agents used in theprocess.

In brazing of ceramics, an important alternative to brazing through pre-metallizationis active-metal brazing,9–14 the technique developed at General Electric. Active-metal brazing uses special braze alloys that contain an “active metal”, in particular,Ti, and sometimes Hf or Zr. The active braze alloys are typically based on Cuand/or Ag with some small addition of Ti. The role of the active metal is to reactwith the ceramic, creating a layer of a reaction product that is more wettable by the

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2.3 Wetting Experiments with Drops of Molten Metals on Transition Metal Carbide andNitride Surfaces

metal braze than the original ceramic surface. No pre-metallization is necessary,and the joint can be made in one step. Active metal brazing works well on carbideand nitride ceramics, as well as on many oxides, borides, and so on. When usingTi-containing brazes, the reaction-product ceramic is typically TiC, TiN, or TiO onceramic carbides, nitrides, or oxides, respectively.

It is interesting to note that many chemically reactive molten metal alloys donot wet ceramic products because their reaction products are typically more stablethan the original ceramics and hence, generally, less wettable. Fortunately, there areexceptions from this rule, and alloys containing Ti, Hf, Zr, or Cr are some of these.

An important advantage of Ti is its ability to form ceramic compounds with widecompositional ranges, as will be discussed in Chapter 5. For example, at 1000oC,TiC is stable at compositions between TiC0:60 and TiC0:94. The wettability of TiC,TiN, and TiO by metals tends to increase with decreasing C, N, or O concentrations,which might be connected to the fact that they become more metallic.9 Furthermore,metals are better wetted by metals than the ceramics are. With an adequate choiceof the braze alloy composition one can create a well wettable interface layer ofsubstoichiometric Ti compounds.

The performance of the Cu- and Ag-based Ti-containing brazes depends on therelative concentrations of Cu and Ag in the alloy.9 While Cu-Ti alloys wet well awide range of ceramics, they have found few applications as brazes, because theyare stiff and, more importantly, result in reaction product layers that are thick andfragile. Brazes with high activity but low concentration of Ti are more desirable.These requirements can be satisfied by using solvents in which Ti is relatively in-solvable. In this respect Ag is more attractive than Cu. The solubility of Ti in Agat a brazing temperature of 1000oC is only 7 atom%, as compared to 60 atom% inCu. Even better characteristics can be achieved if Ag is alloyed with Cu. One of themost commonly used active braze alloys is the Ag-28Cu eutectic alloy, in which Tisolubility is less than 2 atom%. Ag-28Cu-2Ti have been found to wet a wide rangeof ceramics noticeably better than Cu-Ti alloys.

2.3 Wetting Experiments with Drops of Molten Metalson Transition Metal Carbide and Nitride Surfaces

As discussed in the previous sections, the success of sintering and brazing is directlyconnected to the degree of control over wetting behaviors of liquid metals in contactwith ceramics. For those behaviors to be controlled, they should be measurable.The most widely used way to characterize wettability is by putting a drop of theliquid metal on the surface of the corresponding ceramic material and analyzing theequilibrium shape (profile) of the drop on the surface. This procedure is commonlyreferred to as asessile drop wetting experiment(Fig. 2.4). What is actually measuredin the sessile drop experiments is thecontact angle, θ, which is the angle at theliquid front subtended by the liquid surface and the solid-liquid interface. Dependingon the value of the contact angle, there is a situation ofwetting(θ < 90o) or non-

11


wetting(θ > 90o). The smaller the contact angle, the better is wetting. When thecontact angle is zero, the wetting is considered to be complete (total spreading). Thecomplete wetting is what is typically required in liquid phase sintering of hardmetals.

θA

B

Figure 2.4: A schematic picture of the sessile drop method.

A number of sessile drop wetting experiments have been performed with moltenmetals on transition metal carbides.15–21There are much fewer reports on wettabilityof transition metal nitrides.18 In the experiments the ceramic substrate is typicallyprepared by hot-pressing a carbide or nitride powder. The experiments are done ina vacuum furnace, with much attention given to the compositions of the materialsand the surrounding atmosphere. It should be noted that those experiments do notprovide any atomistic level information on the structure of the formed metal-ceramicinterfaces. At best, they characterize the interface microstructure using the scanningelectron microscopy (SEM) techniques, with a resolution in the micrometer scale,which is still a few orders of magnitude larger than the atomic scale.

2.3.1 Wetting of TiC and WC by Co

One of the most systematic and detailed studies on wetting of carbides has beenperformed by Ramqvist,15 who also gives a review of earlier work. In the contextof hardmetal sintering, it is interesting to note that Ramqvist’s measurements showcomplete wetting for Co on WC, withθ ' 0, and a somewhat worse wetting forCo on TiC,θ' 25o. Those results are used as experimental support for theoreticalresults of Papers I, II, and IV.

Though those experimental results were reported in sixties, there have not beenany drastic changes in the sessile drop wetting experiment techniques since that time.Further, to our knowledge, there have not been any experimental measurements thatwould attempt to question the above results for Co on carbides. At the same time,there is a recent work with Ni on TiC.16 According to that study, the wetting anglefor Ni is θ ' 24o, which is the same, within experimental errors, as reported inthe above-mentioned work of Ramqvist. Due to significant similarities of Ni andCo properties, the reproducibility of the Ni/TiC results is a strong support for thereproducibility and reliability of the old Co/TiC, and likely Co/WC, wetting angledata.

12

2.3 Wetting Experiments with Drops of Molten Metals on Transition Metal Carbide andNitride Surfaces

One more indication of the validity of the above-mentioned experimental resultson wetting of TiC and WC by Co is the fact that they agree well with the wettingbehaviors during liquid phase sintering of WC-Co cemented carbides and TiC-Cocermets. In particular, there is a fast and complete penetration of Co into WC pores,while some extra measures have to be taken for a complete filling of pores in TiCbased cermets.

2.3.2 Wetting by Noble Metals

Noble metals, like Cu, Ag, and Au, are chemically less active than transition metals,like Co or Ni. This is due to their filledd-band, and they typically wet carbidesworse than,e.g., Co or Ni.15,16 Wetting by the non-reactive noble metals is alsomore sensitive to the quality (contamination control) of the carbide substrate surface.

Much attention in wetting experiments with metal-carbide systems has beengiven to wetting of TiC by Cu,15,16,18,21partly due to its high relevance for brazing.Early works (see Ref. [15] and references therein) have reported that non-reactivemetals, like Cu and Ag, do not wet stoichiometric TiC and form high,> 120o, con-tact angles. At the same time, more recent wetting experiments of Li16 show thatby giving special attention to contamination of the TiC surface by oxygen, usinga Zr getter furnace tube, one can reach contact angles as low as 54o even for stoi-chiometric TiC. However, in later experiments of Xiao and Derby,18 the measuredcontact angle was 120o, even for the extremely low oxygen partial pressure. Thoseinconsistencies are addressed in a recent experimental work of Froumin and cowork-ers,21 who carefully analyze the presence and effects of the oxygen contaminationof the TiC substrate surface. They find that the presence of oxygen on the TiC sur-face strongly inhibits the interaction between the ceramic and molten Cu, which isthe most likely reason for non-wetting behaviors in the above-mentioned previousstudies. With an enhanced oxygen contamination control, a wetting angle of' 89o

is reached.21 The scattered wetting data for Cu and Ag on TiC is also analyzed inPaper V, based on comparisons with first-principles theoretical results.

The above described situations with scattered data on the wetting of TiC by Cuis a strong reason to question earlier results15 on wetting of other carbides by Cu, inparticular HfC, ZrC, TaC, NbC, and VC. It is interesting to note that those resultswere showing quite a pronounced trend, with the wetting angle decreasing notice-ably along the series of HfC, ZrC, TiC, TaC, NbC, and VC. That trend has been givenmuch attention in theoretical analysis, in particular in the search for a correlationbetween wettability and different carbide properties, like formation energies15,22orplasmon frequencies.23,24 However, the range of contact angle variations coveredby that trend is comparable to the range of scattering in the Cu/TiC data,15,16,18,21

which casts doubt on the presence of that particular trend, and on the relevance ofthose interpretations. This situation raises the question of what is the real wettingtrend for Cu on HfC, ZrC, TiC, TaC, NbC, VC, when oxygen contamination is suf-ficiently low. New experiments and theoretical predictions or interpretations areneeded.

13


In the context of brazing, it is important to note that the wetting of TiC and TiNby noble metals can be improved significantly by lowering the carbon or nitrogenconcentration in TiC or TiN ( so-called hypostoichiometry) or by adding Ti or Aladditives to the metal melt, as discussed in Refs. [15, 18, 21]. A number of simpleinsights into atomic scale mechanisms of such wetting improvements are given inPaper V, based on first-principles theoretical simulations with a variety of modelinterface systems.

2.3.3 Concluding Remarks

In general, wetting is a complex physical phenomenon, involving many delicateprocesses at different length and time scales.15–18,21,24–29In the first approximation,it is controlled by the strength of the adhesion between the liquid and the substrate, incomparison with the liquid surface tension. However, even at this simplest level, themicroscopic mechanisms behind the wetting of carbonitrides by metals for a longtime have lacked understanding, being a challenge to both experiment and theory(see the appended papers and references therein). This is one of the main reasonswhy the nature of metal-carbonitride interactions, an important step to understandingof the hardmetal sintering and metal-ceramic brazing, is the main subject of thepresent work.

14

CHAPTER 3

Fundamental Framework forMaterials Modeling:

Density Functional Theory

Return to the root and you will find the meaning.

– Sengstan

Our ability to predict properties of materials depends in an essential way on howwell we can describe the quantum mechanical behavior of the electrons in a givenmaterials system. In principle, a complete description of the behavior of electronsin solid is given by the stationary Schr¨odinger equation for the many-electron wavefunctionΨ(r1; :::; rNe):(

Ne

∑i=1

1

252

i +Vion(r i)+12

Ne

∑j 6=i

1jr i r j j

!)Ψ = EΨ; (3.1)

whereVion(r) is the potential of the ion cores, andE is the energy of the electronsystem. However, since the many-electron wave function is represented in the prod-uct space of the single-electron positionsr i , the computational demands to solveequation (3.1) grow exponentially with the number of electronsNe. For more thana few electrons a direct solution of the many-electron Schr¨odinger equation goes farbeyond the capabilities of the present-day computers, and there is no way it can besolved for around 1023 electrons of solid. Thus, approximate methods are necessary.

Among the approximate methods for the electron-structure calculations, one candistinguish semiempirical and first-principles approaches. In semiempirical ap-proaches,e.g., the tight-binding method, the description of the electron structure

All over this chapter the atomic units are used,~ = 1, me= 1, e= 1.

15

3 Fundamental Framework for Materials Modeling:Density Functional Theory

contains parameters that have to be determined by fitting to experimental data orfrom first-principles calculations. The calculations based on semiempirical meth-ods are computationally very efficient, but they require great care in order not to gobeyond the range of the applicability of the involved approximations and parameters.

In contrast to the semiempirical methods, the first-principles orab initio meth-ods are parameter free, that is as input only the positions and atomic numbers ofthe ions are required. Although the first-principles calculations are computationallymore demanding, they typically provide more reliable results and have much morepredictive power. The most commonly used first-principles approach is the DensityFunctional Theory.

The Density Functional Theory30–34(DFT) established by the works of Hohen-berg and Kohn30 and Kohn and Sham31 in the mid-sixties has become the mostsuccessful first-principles electron-structure method in condensed matter physics,and it also gains more and more popularity among chemists. The DFT is based onthe observation30 that ground-state properties of an electron system are functionalsof the ground-state electron density alone, and for the ground-state total energy sucha functional satisfies a variational principle. Moreover, as shown in Ref. [31], theproblem of the minimization of the total-energy functional can be reduced to a set ofeffective single-particle Schr¨odinger-like equations, with all the many-body effectscollected in the so-called exchange-correlation term.

Since the DFT forms the basis for the methods used in this thesis, the presentchapter gives a more detailed discussion of the essential points of that theory.

3.1 Density Functional Theory Overview

3.1.1 Hohenberg and Kohn Theorems

Let us consider a system ofN electrons in an external potentialVext(r ). Accordingto the theorems of Hohenberg and Kohn30 there exists such a universal functionalFfn(r)g of the electron densityn(r ) that the ground-state density of the systemminimizes the following functionalEfn(r)g:

Efn(r)g=Z

dr Vext(r )n(r )+Ffn(r)g; (3.2)

under the constraintZ

dr n(r) = N; (3.3)

and the minimum ofEfn(r)g gives the ground-state total energy,

Etot = minfEfn(r )gg : (3.4)

The universality of the functionalFfn(r)g means that its form does not depend onN or Vext(r ).

16

3.1 Density Functional Theory Overview

The Hohenberg and Kohn theorems form the basis of the DFT. They mean thatif we knew the form ofFfn(r)g then we would have a universal and exact methodto calculate the ground-state electron density and the total energy for any electronsystem in any external potential. However, as in reality the form ofFfn(r)g is un-known, the Hohenberg and Kohn theorems by themselves are not enough for anypractical calculations. Some reasonable approximations forFfn(r)g have to be de-veloped.

3.1.2 Kohn-Sham Equations

The unknown functionalFfn(r)g in Eq. (3.2) includes the contributions from thekinetic energy and the electron-electron interactions, taking into account all possiblemany-body effects. All those contributions together are quite difficult to analyze ormake approximations for. In this context it appears to be very helpful to representthe total-energy functional in the following form, suggested by Kohn and Sham:31

Efn(r)g= T0fn(r)g+ 12

Zdr n(r)Φ(r )+

Zdr n(r )Vext(r)+Excfn(r)g: (3.5)

The idea of that representation is to extract fromFfn(r )g the meaningful contribu-tions that can be evaluated exactly, separating all the rest into the term that requiresapproximations. In particular, in Eq. (3.5)T0 is the kinetic energy functional that asystem with densityn(r) would have without electron-electron interactions; the nextterm is the classical electrostatic energy, withΦ(r) being the classical Coulombpotential for electrons,

Φ(r ) =Z

dr 0n(r 0)jr 0 r j; (3.6)

also called the Hartree potential. The last term in Eq. (3.5),Excfn(r)g, is actuallydefined by Eq. (3.5) itself. This term is called the exchange-correlation energy func-tional, and it incorporates all the many-body effects.

An important advantage of the representation (3.5) is also that the variationalproblem of the minimization of the total-energy functional (3.5) can be reformulatedin a much more convenient way. In particular, as it is shown by Kohn and Sham,31

the density that minimizes the Kohn-Sham functional (3.5) can be represented as

n(r) =N

∑i=1jψi(r )j2; (3.7)

whereψi(r ) are solutions of the Schr¨odinger equation for an effective system ofNnoninteracting particles,

1252+Veff(r)

ψ(r ) = εψ(r ): (3.8)

17


HereVeff(r ) is the effective single-electron potential defined as

Veff(r) =Vext(r )+Φ(r)+Vxc(r); (3.9)

where the exchange-correlation potentialVxc(r ) is the variational derivative of theexchange-correlation functionalExcfn(r)g,

Vxc(r) =δExcfn(r)g

δn(r): (3.10)

In Eq. (3.7), the firstN solutionsψi of Eq. (3.8) with the lowest eigenvaluesεi shouldbe taken. It is assumed that eachψi(r) is normalized,

Zdr jψi(r )j2 = 1; (3.11)

which together with Eq. (3.7) provides the condition (3.3). If the ground-state den-sity is found, the ground-state total energy of the system can be found using thetotal-energy functional (3.5).

The Kohn-Sham equations (3.8), together with expression (3.7), actually meanthat the ground-state density of a many-electron system is the same as the ground-state density of some effective system in which the electrons do not interact witheach other but move in the effective potential (3.9) instead. A set of single-electronKohn-Sham equations (3.8) is incomparably easier to solve than the original many-body problem (3.1). However, one still needs approximations for the exchange-correlation functional.

3.1.3 Adiabatic Connection Formula

When constructing approximations for the exchange-correlation functionalExcfn(r)g,it is not particularly convenient to use Eq. (3.5) as a definition ofExcfn(r )g. Thematter is that that equation does not reflect the structure and the physical nature ofthe exchange-correlation functional.

A more explicit expression for the exchange-correlation functional can be ob-tained by considering an adiabatic switching-on of the electron-electron interactions,starting from the noninteracting system. More specifically, let us consider an ef-fective system in which we replace the real Coulomb interaction, 1=jr 0 r j, withλ=jr 0 r j, where the coupling constantλ can be varied between 0 an 1. Whenλ = 0we have a noninteracting system, which can be treated exactly. The caseλ = 1 cor-responds to the real system we want to describe. Now let us increaseλ adiabaticallyfrom 0 to 1, and let us introduce an additional external potentialVλ in such a waythat the electron density is always the same as in the real system. By consideringthat procedure one can derive the following expression forExcfn(r)g, commonlyreferred to as the adiabatic connection formula:35–37

The total energy of the effective noninteracting system,∑Ni=1εi , does not have any direct physical

meaning.

18

3.2 Exchange and Correlation Approximations

Excfn(r)g= 12

Zdr n(r )

Zdr 0

1jr 0 r jnxc(r ; r 0 r); (3.12)

where

nxc(r ; r 0 r) = n(r 0)

1Z

0

dλg(r ; r 0;λ)1

: (3.13)

Hereg(r ; r 0;λ) is the pair correlation function of the system with densityn(r ) andelectron-electron interactionλ=jr 0 r j.

Expressions (3.12) and (3.13) introduce a simple physical interpretation of theexchange-correlation energy. The exchange-correlation energy originates from theenergy of electrostatic interaction (see Eq. (3.12)) between the electron atr and theexchange-correlation hole it creates, with the density of the exchange-correlationhole,nxc(r ; r 0 r), defined by Eq. (3.13). The exchange-correlation hole describesthe effects that if there is an electron atr then the probability to find another electronat r 0 close tor is reduced due to the Pauli principle (exchange) and the Coulombelectron-electron repulsion (correlation). An important property of the exchange-correlation hole is that its integration should give one electron, as reflected in thefollowing sum rule:35

Zdr 0 nxc(r ; r 0 r) =1: (3.14)

The adiabatic connection formula (3.12), with (3.13) and (3.14), appears to be veryuseful for the development of the approximate exchange-correlation energy func-tionals.

3.2 Exchange and Correlation Approximations

The simplest approximation for the exchange-correlation functional is the local den-sity approximation (LDA), first suggested in the original work of Kohn and Sham [31],and then generalized to spin-polarized systems35,38,39(local spin density approxi-mation or LSDA). In the LDA it is assumed that the contribution to the exchange-correlation energy from each pointr with the local electron densityn(r) is the sameas in the uniform electron gas with the corresponding electron densityn(r ). That is,the exchange-correlation functional takes the form

ELDAxc [n] =

Zdr n(r )εunif

xc [n(r)] ; (3.15)

whereεunifxc (n) is the exchange-correlation energy per particle in the homogeneous

electron gas with densityn. For the homogeneous electron gas, an accurate depen-dence ofεxc on the densityn is extracted from quantum Monte Carlo calculations,40

and then, to simplify its applications, parametrized in one or another way.41–43

19


Although by construction the LDA approximation is expected to work only forsystems with slowly varying density, it appears to be quite accurate for many atomic,molecular and condensed matter systems even with significant density variations.To a large extent, this is due to the fact that, as noted in Ref. [35], the exchange-correlation energy depends only on the spherical average of the exchange-correlationhole. It is also important that the LDA exchange-correlation hole, taken from a realphysical system, meets many important conditions, like Eq. (3.14), that should besatisfied by the exact hole.

Not surprisingly, there are quite many situations where the LDA leads to unac-ceptably large errors and even qualitatively wrong results (for details see Refs. [32,44]). What is particularly relevant for this thesis is that the LDA (and the LSDA)noticeably overestimates the bonding strength in most transition metals45 and theircompounds,46–49 and can give a wrong ground-state structure, like for iron.50 Thisfact reduces significantly the reliability of the LSDA description of bonding in theinterface systems that are of interest in the present work. In addition, it should bementioned that for most transition metals in the 3d-series LDA also underestimatesthe equilibrium volumes and overestimates the bulk moduli.

A natural way to improve on the LDA (LSDA) is to include density gradients inthe approximate exchange-correlation functional. The most straightforward proce-dure, suggested in Ref. [31], is to add to the LDA functional one more non-zero termin the gradient expansion of the exchange-correlation functional (gradient expansionapproximation or GEA). In reality, the GEA appears to be much worse than the LDA.This is because the GEA exchange-correlation functional does not retain many im-portant properties of the exact exchange correlation, like the sum rule (3.14), whilethe LDA functional does. To overcome these problems, the generalized gradientapproximation (GGA) has been introduced.51 In GGA the gradient expansion isreplaced by generalized functionals of density gradients,

EGGAxc [n] =

Zdr n(r )εGGA

xc [n;5n]; (3.16)

which are designed to incorporate the important features of the exact exchange-correlation functional. In contrast to the LDA, there is no unique definition ofεGGA

xc [n;5n], and many different functional forms have been suggested, as overviewedin, e.g., Ref. [44].

The most commonly used version of GGA is that of Perdew and Wang 1991(PW91) [52], in which the real-space cutoffs of the spurious long-range componentsof the second-order expansion for the exchange-correlation hole allow to satisfy thesum rules on the exact hole. The GGA-PW91, or a similar version of Perdew, Burkeand Ernzerhof53 (PBE), improve the accuracy of the description of the ground-stateproperties of many atomic, molecular and solid systems.44,54 And although in thegeneral case some authors44 still recommend to consider both the GGA and LSDAresults, the GGA-PW91 has been shown to correct,e.g., the serious deficienciesof the LSDA for transition metals, providing quite accurate descriptions of theirstructural55 and cohesive properties.56 In view of this fact the present work usesGGA-PW91, exclusively.

20

3.3 Note on Applications to Solids

3.3 Note on Applications to Solids

An important condition for the applicability of DFT to condensed matter systemsis the ability to separate the motion of the electrons from the motion of the atomicnuclei. This can be done within the adiabatic (Born-Oppenheimer) approximation,57

which is based on the fact that electrons are much lighter than nuclei, and hencethe characteristic velocities of the electronic motion are much higher than the ionicvelocities. In the adiabatic approximation the electronic structure at any momentof time can be determined assuming that the ions are frozen in their instantaneouspositions. That is, the ionic positions are just parameters of the external potential inEq.(3.8).

When the Kohn-Sham equation (3.8) is applied to periodic systems (solids), onecan make use of the Bloch theorem,57 which states that the solutions of Eq. (3.8) canbe written in the form

ψ(r ) = exp(ikr )uk(r); (3.17)

whereuk(r) has the periodicity of the considered system. Substitution of (3.17) intoEq. (3.8) leads to the equation foruk(r),

12(5+ ik)2+Veff(r )

uk(r) = εkuk(r); (3.18)

which needs to be solved only within one unit cell, with the periodic boundary con-ditions. The electron density is then given by

n(r ) = 2∑i

Z

B:Z:

dk f (εik εF)juik(r)j2; (3.19)

wherei is the band index that enumerates different eigenstates,uik , of Eq. (3.18),with the corresponding eigenvaluesεik , the factor 2 accounts for the spin degeneracy,f (ε) is the Fermi distribution function (zero temperature),

f (ε εF) = θ(εF ε); (3.20)

and the integration is over the first Brillouin zone in thek-space. The position of theFermi level,εF, should be determined self-consistently from the electron-numberconservation condition,

Z

U:C:

dr n(r) = Nel; (3.21)

where the integration is over one unit cell andNel is the number of electrons per unitcell.

21


22

CHAPTER 4

Computational Method: TechnicalAspects of Solving DFT Equations

Great things are not done by impulse, but by a series of small things brought together.

– Vincent Van Gogh

The density functional theory discussed in the previous section is an enormousprogress in our ability to treat condensed matter systems quantum mechanically. Theoriginal many-body problem is overcome, and all what we need to do is to solve aset of single-particle Kohn-Sham equations (3.8) or (3.18). However, in practice,the solution of the Kohn-Sham equations still requires a noticeable computationaleffort, and for many years the practical applicability of the first-principles density-functional method was limited mainly to simple bulk systems and molecules.

During the last decade the situation has changed dramatically. To a large extentthis is due to the great progress in the computer technology, which has made a sig-nificant amount of the computational power available at a reasonable cost. However,as the computer power, although rapidly increasing, is always limited, a crucial roleis also played by the high efficiency of the modern computational techniques andalgorithms. These advances have moved the limit on the complexity of the systemthat can be treated by the density-functional method to as high as a few hundreds ofsymmetry-independent atoms.

As without many well-developed computational techniques the problems ad-dressed in this thesis would be unapproachable, the present chapter gives a briefoverview of those techniques. This overview also explains the meaning of some ofthe computational parameters referred to in the attached papers.

23

4 Computational Method: Technical Aspects of Solving DFT Equations

4.1 Plane Waves as Convenient Basis Set

The first step of any numerical implementation of the density-functional method isto represent the Kohn-Sham wave functions as discrete sets of numbers. This is doneby expanding the wave functions in some basis set, and truncating such an expansionat some finite number of terms. Then the desired discrete representation is given bythe coefficients of the expansion.

The choice of basis functions is crucial for the efficiency of the computationalmethod. With the optimal choice, as few expansion terms as possible should be nec-essary to represent the wave functions with a reasonable accuracy. Not less signifi-cant is how complex the form of the Kohn-Sham functional (3.5) and equation (3.8)becomes in that basis set. The choice of the basis is so important that the differ-ent density-functional computational methods are actually named after the used ba-sis set, like for example, the linearized-muffin-tin-orbital58–60 (LMTO), the linear-combination-of-atomic-orbitals (LCAO), the full-potential-linearized-augmented-plane-wave61–63(FLAPW) and the plane-wave pseudopotential (PWPP) methods.64–66

In the calculations of the present work the exploited basis set consists of planewaveseiqr . An important condition for an efficient use of the plane wave expansionis the periodicity of the functions that are expanded. Only for periodic functionsthe spectrum of the requiredq-values is discrete, being continuous otherwise. Inthis situation, an important role is played by the Bloch theorem (see Section 3.3,Eq. (3.17)), which, for periodic systems, allows us to represent the wave functionsin terms of the functionsuk that retain the periodicity of the system.

Expanding the functionsuk in the plane wave basis set and substituting such anexpansion into Eq. (3.17), one can represent the wave functions for eachk-point andband indexn as

ψnk(r) =∑G

cnk(G)exp[i(k +G)r ] ; (4.1)

where the summation is over all the reciprocal-space vectorsG, andcnk(G) arethe expansion coefficients. Normally, the contributions of the plane waves with thekinetic energies,jk +Gj2=2, higher than some finite value are sufficiently small tobe neglected. Thus, instead of the full expansion (4.1) with an infinite number ofterms, it is enough to use a finite basis set consisting only of the plane waves withthe kinetic energy lower than some kinetic energy cutoffEc,

jk +Gj2=2< Ec: (4.2)

A plane wave basis set is very convenient in many respects, which explains itswide use in first-principles density-functional calculations. Its mathematical sim-plicity makes it relatively easy to implement on the computer. With this basis thekinetic energy operator has a simple diagonal form, and the transformations be-tween the real and reciprocal space representations can be done very efficiently withthe modern fast Fourier transform algorithms67 (FFT). The accuracy of the planewave expansion is controlled quite easily, and it can be systematically improved to

24

4.2 Pseudopotentials

a desired level by increasing the plane wave cutoff energy. Being independent ofthe ionic positions, the plane waves provide an unbiased description of the wholeunit cell (supercell) and are equally accurate for atomic, molecular, solid-state bulk,surface, and interface systems.

One more important advantage of the plane wave basis set concerns the calcu-lation of the ionic forces. The ionic forces are determined by the derivatives of thetotal energy with respect to the ionic positions. In principle, being able to computethe total energy, one can calculate such derivatives numerically. In practice, this iscomputationally very inefficient, since one has to perform a few extra total-energycalculations for displaced atomic configurations. Fortunately, there is a possibility toobtain the ionic forces from a single total-energy calculation, using an analytic ex-pression for the ionic forces68 based on the Hellmann-Feynman theorem.69–71 Thefact that the plane waves do not explicitly depend on the ionic positions allows usto use the Hellmann-Feynman theorem directly, without the need to include extraterms (the so-called Pulay forces72) to compensate for the contributions from thederivatives of the basis functions with respect to the ionic positions. Due to thefast calculation of the ionic forces the plane wave method is very efficient for thegeometry optimization applications (atomic structure relaxation), which appears tobe particularly relevant for the interface systems studied in the present work.


In DFT implementations based on the plane wave basis set a major problem is thedescription of the core region of the atom. This is because the wave functions ofthe core electrons, as well as the wave functions of the valence electrons in thecore region, are rapidly oscillating functions of the space coordinates. A reasonablyaccurate plane wave expansion of such functions would require a prohibitively largenumber of plane waves, which would make the plane wave basis set very impractical.This problem is quite efficiently solved within the pseudopotential approximation,which will be discussed in this section.

The first important component of the pseudopotential approach is the frozen coreapproximation. This approximation assumes that the inner shell electronic orbitalsare frozen in the sense that they do not change when the atom is transferred fromone chemical environment to another. Such an assumption is based on the fact thatmost of the physical properties of solids are mainly determined by the behavior ofthe valence electrons, while the inner-shell orbitals are very close to those in the freeatoms, being practically inactive. Thus, in the frozen core approximation, one hasto describe only the changes in the valence-electron orbitals.

The frozen core approximation still retains the problematic oscillatory behaviorof the wave functions in the core region. To avoid this problem, the pseudopotentialapproximation for the valence-core interaction is introduced. In this approximationthe action of the ion core on the valence electrons is replaced by an effective po-tential, the so-called pseudopotential. On the other hand, the wave functions of the

25


valence electrons in the core region are replaced by pseudo wave functions, whichare much smoother. In this way the necessary number of plane waves can be reducedto a reasonable level.

The pseudopotential formalism has gone through many stages of development,73,74

from the original ideas of Phillips and Kleinman75 and the empirical pseudopoten-tials76–78 to ab initio79 ones, then to normconserving80–84 separable,85–87 to opti-mally soft,88–90 and, finally, to ultrasoft91–94 pseudopotentials. In this thesis thefocus is on the normconserving and ultrasoft pseudopotential schemes. For sim-plicity, we start from the normconserving pseudopotentials and then describe thenew features introduced by the ultrasoft pseudopotentials, which are exploited in thecalculations reported in the appended papers.

4.2.1 Key Steps in Pseudopotential Construction

Pseudopotentials are generated by considering a one-atom problem. First the allelectron (AE) radial Kohn-Sham equation,

d2

dr2 +l (l +1)

r2 +VAE(r)

φAE

lε (r) = εφAElε (r); (4.3)

or its relativistic or scalar-relativistic generalization,95 is solved to find the trueKohn-Sham wave functions,φAE

lε (r), for different atomic orbitals with different an-gular momental =0,1,2,3,.. (s; p;d; f ; ::) and energiesε. In Eq. (4.3)VAE(r) is thescreened potential in the all-electron atom, which is determined self-consistently fora given atomic configuration. Then, from the wave functions of the valence orbitals,the pseudo wave functions,φPS

lε (r), are constructed in such a way that they are equalto the corresponding true wave functions beyond some chosen cutoff radius,rcl, theyare continuously differentiable at least twice at that cutoff radius, and they satisfythe norm-conservation constraint,

rclZ

0

dr φPSlε (r)

2 =

rclZ

0

dr φAElε (r)2: (4.4)

Since the pseudo wave functions should be orthogonal only to themselves, not tothe core orbitals, they are taken to have no nodes insidercl, which allows them tobe much smoother than the true wave functions. At the same time, the cutoff radiusrcl should be sufficiently large, so thatφAE

lε (r), and, hence,φPSlε (r), have no nodes

outsidercl. After the pseudo wave functions are constructed, the pseudopotentialcan be determined by an inversion of the radial Kohn-Sham equation with the givenl andε:

d2

dr2 +l (l +1)

r2 +VPS

φPSlε (r) = εφPS

lε (r): (4.5)

i.e. including only the kinematic relativistic effects due to the high kinetic energy of electronsnear heavy nuclei, but neglecting the spin-orbit splitting of the electronic levels

26


The above procedure of the pseudopotential generation still leaves some free-dom in the specific choice of the pseudo wave functions and the pseudopotential.This freedom can be used to optimize the computational efficiency of the con-structed pseudopotential. In this context, an important issue is the separability ofthe pseudopotential representation. In general, the pseudopotential produced in theabove-described way has a semilocal (i.e. local in the radial coordinate but not inthe angular coordinates) operator form

VPSSL =∑

l ;m

jlm>VPSl (r) < lmj; (4.6)

wherejlm> are the spherical harmonics, andVPSl (r) are pseudopotentials obtained

for each angular momentum channell . Due to this semilocal form the computationaleffort of applying theVPS

SL operator to the wave function in the basis ofN planewaves grows asN2. This effort can be reduced considerably, to being proportionalto N instead ofN2, by transforming the semilocal operatorVPS

SL into a truly nonlocalseparable form, as suggested by Kleinman and Bylander:85

VPS=Vloc(r)+∑l ;m

jχlmε >< χlmεj< χlmεjφPS

lmε >; (4.7)

where

jχlmε >= (VPSSL Vloc)jφPS

lmε >; (4.8)

and

jφPSlmε >=

1r

φPSlε (r)jlm> : (4.9)

HereVloc(r) is the local potential, which, in principle, can be arbitrary. The choice ofthe local potential actually can be quite important,e.g., to avoid such artificial non-physical effects as the so-called ghost states (see Refs. [86, 96–98] for a detaileddiscussion, and Refs. [99,100] for additional useful comments).

Another important point is how to choose the pseudo wave functions and pseudopo-tential to optimize the smoothness of the pseudo potential,i.e. to minimize the nec-essary size of the plane wave basis set. The smoothness optimization techniques fornormconserving pseudopotentials are addressed quite in detail in Refs. [88–90].

4.2.2 Essential Aspects of Pseudopotential Transferability

In the pseudopotential formalism a fundamental issue is the transferability of apseudopotential. The transferability describes how accurately the pseudopotentialcan reproduce the quantum mechanical behavior of an all-electron atom when suchan atom is placed in different chemical environments. Simple, although computa-tionally costly, ways to improve the transferability are to decrease the cutoff radiircl

27


or to include the semicore orbitals into the valence states,i.e. to make the pseudopo-tential calculations closer to the all-electron ones.

A brute-force approach to assessment of the transferability is to compare the re-sults of the pseudopotential and all-electron calculations for a wide range of differ-ent systems and properties. Although useful to some extent, such an approach in fulllength is not very convenient, and it reduces the predictive power of the pseudopo-tential calculations. Thus it is important to be able to judge and control the qualityof a given pseudopotential already at the pseudopotential generation stage.

One commonly used characteristics of the pseudopotential transferability canbe understood in terms of the following scattering problem. In some molecular orcondensed matter system each atom is surrounded by a sphere, inside which an all-electron atom is replaced by a pseudopotential. The purpose is to correctly reproducethe valence wave functions outside the atomic spheres. The wave function outsidethe spheres is determined by the Schr¨odinger (Kohn-Sham) equation in that region,the norm of the wave function, and the boundary conditions for the wave function onthe surface of the spheres. Since the construction of the pseudopotential aims to pro-vide the first two factors like in the all-electron system, it is the boundary conditions(the scattering properties of the atomic spheres) that require extra attention.

A complete specification of the boundary conditions to the Schr¨odinger equationin the considered problem can be given by the logarithmic derivatives,(dφ(r)=dr)=φ(r),of the wave functions on the surface of the sphere. Therefore, an important indica-tor of the pseudopotential transferability is how accurately and in how wide an en-ergy region the logarithmic derivatives of the wave-functions on the chosen spheresaround the all-electron atom can be reproduced with a given pseudopotential. In themodern pseudopotential schemes the behavior of the logarithmic derivatives (scat-tering properties) can be systematically improved by inclusion of more than one ref-erence energyε for the chosen angular momentum channelsl in the pseudopotentialconstruction.86,91

As pointed out in Refs. [101,102], the logarithmic derivatives do not give a com-plete measure of the transferability, especially when there is a significant electrontransfer between the components of a given chemical system. Indeed, the trans-ferability criterion based on logarithmic derivatives works only under the conditionthat the changes in the chemical environment do not change the effective potentialsignificantly inside the core radius.101 As shown in Ref. [101], this condition canbe met with some reasonably small core radius. In Ref. [102] this problem is ad-dressed in terms of the so-called chemical hardness, which determines the changesin the atomic eigenvalues with the change of the valence state occupancies. Withan improved description of the chemical hardness property, a major reduction of theerrors of the pseudopotential approximation can be achieved.102 A further analysisand development of the pseudopotential transferability criteria related to the chemi-cal hardness property are presented recently in Refs. [103,104]

One more important factor in the pseudopotential transferability is related to thenonlinear character of the exchange-correlation potential. The nonlinear dependenceof the exchange-correlation functional on density means that the exchange correla-

28


tion energy of an atom is not just a sum of the contributions of the core and valenceelectrons, and, hence, exclusion of the core electron density introduces a nonlinearerror in the dependence of the exchange correlation on the valence density. Thiserror can be noticeable when there is a significant overlap between the densities ofthe core and valence states, like it is between the 3d valence and 3p semicore statesof the 3d transition metals. To minimize such an error, the nonlinear core correctionof Louie, Froyen, and Cohen105 can be used. The idea of that correction is that inthe calculations of the exchange-correlation energy, instead of the valence densityalone, the sum of the valence and the (frozen) core densities is used. Since the truecore density has quite rapid oscillations, whose inclusion would require an increasednumber of plane waves, the core density is pseudized within some chosen radius (theso-called core correction radius). The nonlinear core correction is essential for a cor-rect description of the magnetic systems, which is very relevant for Paper III of thisthesis.

4.2.3 Ultrasoft Pseudopotentials for Efficient Treatment of Tran-sition Metal and First Row Elements

With the above mentioned developments, the normconserving pseudopotentials areefficiently used for many systems, with quite a good control of the pseudopotentialsoftness and transferability. However, there are problematic cases, like first-row ele-ments and transition metals, for which the use of normconserving pseudopotentialsrequires significantly higher computational effort (plane wave cutoff) than for otherelements. This is because those elements have strongly localized valence orbitals,like the 2p-orbitals for the first-row elements, the 3d-orbitals for the 3d transitionmetals and the 4f -orbitals for the 4f rare earths. To satisfy the norm-conservationconstraint, the pseudo wave functions for such orbitals have to have quite a largeamplitude in the core region, which does not allow to make them much smootherthan the true wave functions. This problem is overcome in the Vanderbilt ultrasoftpseudopotential scheme.

The main feature of the ultrasoft pseudopotentials is the released normconser-vation constraint, which gives much more freedom in softening the pseudo wavefunctions. The reasons why the normconservation constraint (4.4) is important inthe previous (normconserving) pseudopotential schemes, and is not released before,is that, on the one hand, the norm conservation secures that the pseudized Hamil-tonian of the electron motion remains Hermitian, which guarantees that the Hamil-tonian can be diagonalized and that its eigenvalues are real. On the other hand,the norm conservation allows an adequate description of the electrostatic field ofthe core region, and gives the right amplitude of the electron density outside thecore. Moreover, from the point of view of the pseudopotential transferability, it isimportant that with the normconserving pseudopotentials not only the logarithmicderivatives (scattering properties) at the reference energies, but also their small vari-ations around those reference energies, are equal to the corresponding all-electronvalues.

29


The ultrasoft pseudopotentials retain the useful properties of the normconservingpseudopotentials, and also add some extra improvements. To avoid the difficultieswith a non-Hermitian Hamiltonian, the standard eigenvalue problem (4.5) is trans-formed into a generalized eigenvalue problem

d2

dr2 +l (l +1)

r2 + VPS

φPSlε (r) = εSφPS

lε (r); (4.10)

in which a non-Hermitian pseudopotentialVPSis replaced by a Hermitian oneVPSata price of introducing the so-called overlap operatorS(see any of Refs. [91–94] fordetailed definitions). To have an adequate description of the electron density, theexpression for the charge density is also generalized by adding to a standard sum ofsquared absolute values of the wave functions an extra term correcting for the differ-ence between the charge densities of the true and the pseudo orbitals, the so-calledaugmentation-charge term. For the reference states, the augmentation charge makesthe density equal to the true valence density not only outside the core region, but alsoinside the core, down to the so-called inner core radius. With these generalizations,the ultrasoft pseudopotentials also give the correct logarithmic derivatives for thefirst order deviations from the reference energies, like in the normconserving case.In addition, the inclusion of extra reference energies, together with the improveddescription of the charge density, allow to increase the core radius without signifi-cant loss in the pseudopotential transferability. All these features make it possible todescribe even the first row elements or 3d-transition metals with reasonable valuesof the plane wave cutoffs, within 25-30 Ry.

The first-principles dansity functional calculations in this thesis use existing im-plementatations of the plane wave pseudopotential method with ultrasoft pseudopo-tentials, in particular DACAPO106 and VASP65,66,107–109codes. Brief outlines offurther technical details related to the use of those codes can be found in Papers II,III (DACAPO), and V (VASP) and references therein.

30

CHAPTER 5

Transition Metal Carbides andNitrides

Microscopic pictures of metal-carbide and metal-nitride wetting and adhesion in ap-pended Papers I - V are connected in many ways to microscopic understanding oftransition-metal carbides and nitrides. In particular, the setup of computer simula-tions and the discussions of the obtained results for the studied interface systemsoften use what is known about the atomic and electronic structure of the transitionmetal carbides and nitrides (TMCN) and their surfaces. This Chapter reviews theknown behaviors of the TMCN atomic and electronic structures,22,110–114with thefocus on what is most relevant for the investigation in Papers I - V.

5.1 Crystal Structure and Stoichiometry

Transition metal carbides and nitrides are compounds containing two types of atoms.One type is a transition metal, like Ti, W,etc. The other one is carbon or nitrogen.For brevity, they are labeled as metal, Me=Ti, W, ... , and non-metal, Y=C, N, atomtypes, respectively.

For a system of Me and Y atoms there exists a variety of possible MeY com-pound phases. A phase with an equal concentration of Me and Y atoms, MeY1:0, isreferred to asstoichiometric. A phase with a smaller concentration of Y atoms thanof Me ones, MeYx, wherex<1:0, is calledsubstoichiometricor hypostoichiometric.Phases with larger concentration of Y atoms, MeYx, x> 1:0, are very uncommon, ifat all existing, and are not of interest here.

Transition metal carbides and nitrides most often crystallize in theB1 structure(NaCl, see Fig. 5.1). For example, this structure describes TiC, ZrC, HfC, VC,NbC, TaC22,110and TiN, ZrN, VN, NbN, TaN.111 Among the practically importantcarbides, the main deviations from the NaCl structure are present for WC, Mo2C,

31

5 Transition Metal Carbides and Nitrides

and Cr3C2, which have a hexagonalD13h, an orthorhombicD14

2h, and orthorhombicD12

2hPbnmstructures,22 respectively. Here we discuss mainly MeY compoundswith the cubic-symmetry NaCl crystal structure, which are often called cubic car-bides and nitrides.

Figure 5.1: Unit cell of the NaCl crystal structure of the transition metal carbides and ni-trides. The darker spheres represent nonmetal atoms and the lighter spheres metal atoms.

The NaCl-structured transition metal carbides and nitrides are often referred toas interstitial compounds. The structure can be viewed as an fcc lattice of Me atoms,in which the interstitial sites are occupied by the Y atoms. A simple argument behindthis picture is that the effective radii of the Y atoms are around two times smallerthan the Me ones. For example, the covalent radii of Ti and C are 1.32 and 0.77A, respectively.115 The insertion of the Y atoms requires only a relatively smallexpansion of the corresponding Y-free Me lattice. For example, the equilibriumlattice constant of pure fcc Ti calculated in Papers II and V is 4.10 – 4.11A, ascompared to 4.33 – 4.34A for TiC (see Table I in Paper V.). That is, the expansionof the Ti lattice due to insertion of C atoms is only about 6 per cent.

An interesting fact is that the fcc structure of the Me sublattice is often retainedover quite a wide range of Y-to-Me ratios,x, in substoichiometric compounds MeYx.For example, in TiCx, the C-to-Ti ratio can change from 0.97 to about 0.5 without achange in the type of the crystal structure. Within that range of Y-to-Me ratios, thesubstoichiometric MeYx compound can be viewed as stoichiometric MeY1:0 one inwhich a certain number of Y atoms are replaced by vacancies. Those Y vacancies aretypically randomly distributed over different Y sublattice positions, although thereare examples of ordered MeYx phases at some specific values ofx, e.g., for VC andNbC.110

Under realistic experimental conditions, many of the cubic carbides and nitridestend to be substoichiometric. For example, TiC most often contains at least a fewpercent of C vacancies. According to the Ti-C phase diagram in Fig. 5.2, the ide-ally stoichiometric TiC phase would prefer to split into substoichiometric TiCx,

Half the distance between two identical atoms bonded together by a single covalent bond.

32

5.1 Crystal Structure and Stoichiometry

x ' 0:96 0:97, and a pure C phase. The stoichiometric MeY compounds thenshould be considered a limiting case of substoichiometric MeYx ones, whenx ap-proaches unity, an approximation to the realistic experimental situations.

A direct theoretical modeling of substoichiometric MeY compounds is a con-siderably more complex task than studying simple-structured stoichiometric com-pounds. This is the reason why theoretical simulations of TMCN materials usemainly stoichiometric compositions. This is also true for the investigation in PapersI - V, except a part of Paper V, where systems with C and N vacancies are also con-sidered. As noted in Ref. [112], first-principles theoretical studies of MeY surfaceelectronic properties that assume stoichiometric compositions still explain quite wellthe majority of the behaviors observed in experiment.

Finally, the effects of substoichiometry on bulk transition metal carbide elec-tronic structure and phase stability are investigated in first-principles theoreticalstudies,116–119including TiCx in Refs. [117, 119]. The situation with vacancies ininterface systems is discussed in Paper V.

Figure 5.2: Phase diagram for Ti-C system.

33


5.2 Electronic Structure and Chemical Bonding

Transition metal carbides and nitrides show quite an unusual combination of phys-ical and chemical properties.22,110–113 They are extraordinarily hard. They havevery high melting points and good corrosion resistance. At the same time they alsohave good electrical conductivity of metallic or semimetallic type. The origins ofsuch combination of properties should ultimately be found in the type and behaviorsof chemical bonds in those MeY compounds. This situation has motivated severaltheoretical and experimental studies of the nature of bonding in transition metalcarbides and nitrides. There has been a significant progress in understanding ofthe MeY cohesion during last two decades,113,114,120,121mainly due to extensiveelectron structure calculations combined with analyses of experimental thermody-namical data and photoemission spectra. This section summarizes the importantconclusions and insights from those studies, relevant for investigation in Papers I -V.

Chemical bonding in transition metal carbides and nitrides is a complex mixtureof covalent, metallic, and ionic components. All three types of bonding mainlyinvolve Me-d and Y-2p atomic orbitals.

5.2.1 Metal-C(N) Bonds

The main contribution to bonding is given by covalent bonds between the Me and Yatoms. The covalent bonding is a result of hybridization of Y-2p and Me-d atomicorbitals. The essential conditions for a strong covalent bonding113 are strong over-laps between participating orbitals, as well as comparable size and energies of thoseorbitals. Those conditions are met quite well in TiC,113,120and to a similar degreein other carbides and nitrides.113 A schematic illustration of the main participatingMe-d and Y-2p orbitals and the areas of their overlap is given in Fig. 5.3.

(a)

−−

−

(b)

+ + ++

−

−

−

−+

+

+

+

+

+

−

−

−

Figure 5.3: Schematic illustration of the main overlaping orbitals of the metal, Me, (filledcircles) and nonmetal, Y=C,N, (unfilled circles) atoms in the (001) plane of a NaCl structure:(a) σ-bonding between Me-d and Y-p orbitals; (b)σ-bonding between Me-d orbitals of theneighboring Me atoms

As discussed in,e.g., Ref. [113] within a simple molecular orbital theory, the

34

5.2 Electronic Structure and Chemical Bonding

overlapping orbitals of appropriate symmetry (see Fig. 5.3) form bonding and anti-bonding states.

The formation of Me-Yp-d bonding and antibonding states and their filling withelectrons result in a redistribution of the electron charge density in space, whichimplies some degree of charge transfer between the Me and Y atomic spheres. Thischarge transfer is responsible for the ionic contribution to the Me-Y bonding, whichcan be viewed as electrostatic interaction between charged Me and Y atomic spheres.The direction of the electron transfer is typically from metal to nonmetal atoms. Forexample, in TiC there is a transfer of approximately 0.3 electrons from Ti to C,120

though this number can be affected by the ambiguity in dividing the crystal spacebetween the Ti and C atomic spheres.

As noted in Ref. [110], the electron transfer from metal to carbon found in elec-tron structure calculations113,120is consistent with experiments, in particular withelectron spectroscopy for chemical analysis (with X-ray photoelectron scattering)studies.122 This is seen from slight shift in the binding energies of the Me and Ycore electrons in MeY compared with those in the elements (core level shifts). Thisshift originates from a redistribution of electronic charge in the direction of the non-metal atom.

5.2.2 Metal-Metal Bonds

The metallic component of the MeY cohesion comes from the interaction of thed-orbitals of the neighbor Me atoms in the Me sublattice. Similarly to the covalentMe-Y bonding, Me-Me bonding is due to hybridization of the overlapping atomicorbitals (Fig. 5.3(b)). To get a simple insight into the Me-Me bonding one can startfrom a picture of metallicd-d bonding in the corresponding Y-free fcc Me metal.The first step to formation of the MeY compound is to expand the fcc Me latticeto the size it has in the MeY compound. Such an expansion increases the distancesbetween the nearest Me neighbors, decreasing the overlap between the atomicd-orbitals, and hence leading to weaker bonding. The energy cost of such an expansionis analyzed in Ref. [114].

An even more important modification of the Me-Me bonding is due to formationof the Me-Y bonding and antibonding states. This process significantly changesthe Me-d component of the electronic local density of states (LDOS). The Me-Yantibonding states are localized mainly around the Me atoms (see,e.g., Fig. 9(c) inPaper II), being a substantial part of the Me-d LDOS (Fig. 5.4).

5.2.3 Bonding Trends and Population of Bonding and Antibond-ing States

The picture of Me-Y bonding and antibonding states is important for understandingof the MeY bonding trends when Me varies along the 3d , 4d, or 5d transitionmetal series.114 Within the same transition metal row, the MeY compounds arecharacterized by a common LDOS pattern (Fig. 5.4) of the bonding and antibonding

35


states. Yet, the position of the Fermi level, which determines the filling of thosestates, changes from metal to metal, being fixed by the number of valence electronsper Me-Y pair.

Energy

Pro

ject

ed L

DO

S C(N)−2pMetal−d

TiC TiN

CoC CoN

Bonding

Antibonding

Figure 5.4: A schematic illustration of the characteristic pattern of bonding and antibondingstates in the electronic local density of states (LDOS) of bulk transition metal carbides andnitrides. The LDOS projections on metal-d and C(N)-p orbitals are shown. The verticallines indicate the positions of the Fermi level for TiC, TiN, CoC, and CoN.

The population of the bonding and antibonding states is shown to dominate thetrends in the strength of bonding, in particular, in the cohesive energy and enthalpyof formation.114 For example, TiC has the strongest bonding in the 3d-series, as theFermi level lies between the energy intervals of the bonding and antibonding states(Fig. 5.4). The extra electrons in V, Cr, ..., Nipopulate antibonding states, and thebonding becomes weaker. For CoC, CoN, NiC, or NiN, the antibonding states arepractically totally filled, and those compounds are unstable in experiment (negativeenthalpies of formation, see Ref. [114]).

As noted in Ref. [114], there are no strict criteria for a clear separation of the en-ergy spectrum into regions of bonding, antibonding, and other, nonbonding, states.Such a separation is to a large extent based on a clear trend in the variation of bond-ing energies and its correlation with the position of the Fermi level with respect tothe LDOS pattern (Fig. 5.4).

5.3 Free Surfaces

There is a number of recent experimental and theoretical studies of structural andelectronic properties of various carbide and nitride surfaces.112 However, the currentdata and understanding of the atomic and electronic structure of MeY surfaces isstill very limited. Below there is an overview of the relevant findings in previousstudies. The main attention is given to stoichiometric TiC, which is not only themost explored case in the literature112 but also of much importance for the presentthesis.

For MeY compounds in the NaCl structure one can consider three different low-index surfaces, in particular (001), (011), and (111).

36

5.3 Free Surfaces

Along the [001] and [011] directions the bulk NaCl structure is composed ofequidistant layers. Each of those layers has the same structure and composition asthe other ones. The layers are stacked in an ABAB... sequence. The differencebetween the neighboring layers is only a relative in-plane shift. The (001) and (011)surface layers of stoichiometric MeY, like the bulk ones, have an equal concentrationof Me and Y atoms. The (001) surface has the symmetry of a square lattice. Thereare 4-fold symmetry axes placed at each of the surface Me and Y atoms, alignedperpendicular to the surface.

In the [111] direction, a MeY compound in the NaCl structure is a sequenceof alternating Me and Y layers. Thus, in principle, one can consider two possibleterminations of the MeY(111) surface. That is, there are Me- and Y- terminatedMeY(111) surfaces.

For TiC, the relative stability of the three low-index surfaces is investigated inRef. [123]. The most stable TiC surface is TiC(001). This surface is the knowncleavage plane of TiC.124

The relative stability of various MeY surfaces can be understood by comparingtheir surface energies. It is expected that the lower the surface energy, the morestable the surface. As one can see from Table IV in Paper V, the order of the TiCand TiN surface energies is as follows:

(001)< (011)< (111) (5.1)

Thus, the (001) surface of TiC and TiN have the lowest surface energy. This can ex-plain why the TiC(001) surface is more stable and more common than the TiC(011)or TiC(111) ones.

The order of the surface energies described by Eq. (5.1) is understandable withina simple broken bond model.123,125 The model accounts for only the Ti-C bonds,which are expected to be the strongest ones in TiC. The surface energy is determinedby how many Ti-C bonds one has to break to cleave the TiC crystal to create a givenTiC surface. For (001), (011), and (111) surfaces all the atoms of the topmost surfacelayers have one, two, and three broken bonds, respectively. The (001) surface hasthe smallest number of broken bonds, both per surface atom and unit area, and hencethe lowest surface energy.

The stability and abundance of the (001) surface of TiC, and likely so for manyother NaCl-structure transition metal carbides and nitrides, makes this surface themost relevant case for the theoretical analysis of wetting experiments in Papers I -V.

Realistic considerations of the crystal structures of the TiC or other MeY com-pound surfaces in experiment and theory have to face the possibility that those struc-tures can differ from ideal bulk truncations.

The TiC(001) surface shows only small deviation from the ideal bulk truncation,which is described as rippled relaxation. In particular, the Me and Y surface atomsare displaced in opposite directions, along the surface normal. The sum of the ab-solute values of those displacements, or the distance between the surface Me and Y

37


atoms along the surface normal, is called rippling. In experimental studies126 thisrippling has been found very small, not larger than 0.1A. A recent, highly accuratemeasurements of work [127] have characterized the TiC(001) rippled relaxation asTi 0.036A inwards and C 0.040A outwards. This rippling relaxation behavior isquite well described with the computational methods used in the appended papers.In particular, in Paper II the calculated result for the rippled relaxation is Ti 0.07A inwards and C 0.04A outwards. In the same paper the calculated rippled relax-ation of TiN(001) has the same direction as for TiC(001), and about twice as largemagnitude. This type of rippled relaxation is also reported in TaC(001), HfC(001),and VN(001) experimental studies and in TaC(001) and TiC(001) theoretical calcu-lations, as reviewed in Ref. [112]. The experimental and theoretical results suggestthat rippled relaxation is likely to be a general phenomenon on the (001) surfaces oftransition metal carbides and nitrides.112

38

CHAPTER 6

Relevant Interface ThermodynamicsBackground

The fundamental role in theoretical understanding of wetting and adhesion phenom-ena belongs to thermodynamic analysis. In particular, the difference between thewetting and non-wetting behavior, or the value of the wetting angle, is essentially de-termined by the relative values of different thermodynamic quantities, like interfaceenergies, work of adhesion, and surface energies. Therefore, theoretical modeling ofthe interface thermodynamic quantities is the key part of the studies of wetting andadhesion in Papers I - V. The goal of this chapter is to introduce the main interfacethermodynamic quantities and discuss the key assumptions and simplifications thatare used in calculations of those quantities in Papers I - V.

6.1 Definition of Interface Free Energy

The most fundamental property in a thermodynamic description of an interface is itsfree energy per unit area,γ.128 This quantity is best defined by considering a systemthat consists of two bulk phases, A and B, which are in contact along a planar inter-nal interface. The system is growing in a container under some given equilibriumconditions by the accretion of atoms from suitable reservoirs. The growing interfacesystem and the reservoirs are maintained at constant temperature,T, pressure,P,and chemical potential,µi, of each of the components. It is also assumed that thesize of the system in all dimensions is much larger than the width of the interfaceregions. Then, in accordance with the first and second laws of thermodynamics, thechange in the internal energy of the system,E, due to the accretion can be expressedas128

39

6 Relevant Interface Thermodynamics Background

dE= TdSPdV +C

∑i=1

µidNi + γdA; (6.1)

whereSis the entropy,V the volume,Ni the amount of componenti, C the number ofcomponents, andA the area of the planar interface. Compared to the correspondingexpression for a bulk system, Eq. (6.1) contains an extra term,γdA, which describesthe increase in the internal energy of the system associated with the increase in thearea of the interface. Equation (6.1) implies

γ =

∂E∂A

S;V;Ni

: (6.2)

Thus the interface free energyγ can be defined as the increase in the internal energyof the entire system per unit increase in interface area at constantS andV of thesystem under closed conditions,i.e. at constantNi.

The state of a thermodynamic system under different thermodynamic constraintscan be found by minimizing an appropriate thermodynamic variable. The most im-portant examples of such thermodynamic variables are the Helmholtz free energy,

F = ETS; (6.3)

for a closed system at constantT andV, the Gibbs free energy,

G= E+PVTS; (6.4)

for a closed system at constantT andP, and the grand potential,

Ω = ETSC

∑i=1

µiNi ; (6.5)

which is used to describe an open system under conditions of constantT, V, andµi.With those definitions and Eq. (6.1) the interface free energy can also be defined as

γ =

∂F∂A

T;V;Ni

; (6.6)

γ =

∂G∂A

T;P;Ni

; (6.7)

40

6.1 Definition of Interface Free Energy

γ =

∂Ω∂A

T;V;µi

; (6.8)

There is one more interesting form of an interface free energy definition, whichis also more relevant for the research work in the appended papers. IntegratingEq. (6.1),

E = TSPV+C

∑i=1

µiNi + γA; (6.9)

and using Eq. (6.4), one can expressγ as

γ =1A

"G

C

∑i=1

µiNi

#: (6.10)

Here the quantity∑Ci=1µiNi can be identified as the total Gibbs free energy which

the homogeneous A and B bulk phases would have together if they were made upof the same amounts of the components at the same chemical potentials. Thus, theinterface free energyγ is the excess Gibbs free energy of the entire system per unitinterface area due to the presence of the interface.

In the theoretical simulations of the interface systems in this thesis, for simplic-ity, no thermal motion of atoms is included,T = 0. Moreover, in view of the rela-tively low compressibility of solids and liquids, in the context of the present workit is also reasonable to neglect the contribution of thePV term in Eq. (6.7). Thus,Eq. (6.7) gives

G' E: (6.11)

Under these assumptions, the interface free energy, or simplyinterface energy, canbe viewed as the work per unit area required to form the two interfaces A/B and B/Afrom the two bulk crystals A and B (see,e.g., Ref. [129]), as illustrated in Fig. 6.1,

γ = γAjB = (EAjBE(bulk)A E(bulk)

B )=A: (6.12)

HereEAjB is the total energy of the A/B-interface system, andE(bulk)A , E(bulk)

B arethe bulk total energies per each of the A and B half-crystals,i.e. the total energyper structural unit of the corresponding bulk crystal multiplied by the number of thestructural units in the corresponding half-crystal. The interface energy shows howmuch weaker the bonding at the interface is than in the A and B bulk materials.

+2γ = +B

A B

A+

A

A B

B

Figure 6.1: A schematic diagram illustrating the definition of the interface energy.

41


6.2 Thermodynamics of Wetting: Contact Angle andWork of Adhesion

The thermodynamic description of interface systems introduced in the previous sec-tion can be applied to the situation of wetting experiments. A typical situation ofsessile drop wetting experiments, with a metal drop on a ceramic surface, is schemat-ically shown in Fig. 2.4. The contact angleθ characterizes the degree of wettabilityof a given ceramic by a given metal:θ = 90o is considered as a boundary betweenthe wetting (θ < 90o) and non-wetting (θ > 90o) behaviors. The smaller the angleθ the better is the wetting. In addition to characterizing wettability itself, wettingexperiments are practically the only feasible way to study interface thermodynamics(energetics) in metal-ceramic systems.

Under thermodynamic equilibrium and steady state conditions, the contact angleθ is determined by the Young equation

cosθ =(σA γAjB)

σB: (6.13)

HereγAjB, σA, andσB are the interface free energy values for the solid-liquid AjB,solid-vapor, and liquid-vapor interfaces, respectively. The quantitiesσA and σB

are at the same time the surface energies of solid A and liquid B, respectively. Adecrease in the contact angle, on the one hand, lowers the free energy of the system.This is due to the fact that that part of the substrate free surface is replaced by liquid-solid interface area, which normally has a lower energy. On the other hand, thislowering of free energy is partially compensated for by the energy of the increasingarea of the free liquid surface (liquid-vapor interface). The Young equation simplystates the condition of a balance between those two opposite contributions,i.e. thecondition of a free energy minimum.

An interesting and important form of the contact angle expression (6.13) can beobtained by introducing one more important quantity, thework of adhesion, Wad.This quantity is defined as the reversible free energy change for making free sur-faces from interfaces,130 whereby the surfaces are in equilibrium with the solid andgaseous components. The work of adhesionWad is connected to the correspondinginterface and surface energies via the Dupré equation

Wad = σA +σB γAjB: (6.14)

Combination of Eqs. (6.13) and (6.14) leads to the Young-Dupré equation

Wad = σB(1+cosθ): (6.15)

With this equation, the measured wetting angle directly gives the ratio of the inter-face adhesion workWad and the liquid metal surface energyσB.

The Young-Dupré equation (6.15) plays an important role in the analysis of wet-ting experiments. The contact angleθ is what is actually measured in the sessile drop

42

6.3 Ideal Work Of Separation as Measure of Interface Adhesion Strength

wetting experiments. The liquid surface energy at a given temperature is typicallyknown from a separate experiment within the same study, or from the available datafrom other experimental works. The work of adhesionWad is practically the onlyinterface thermodynamics (energetics) quantity that can be directly extracted fromthe wetting experiments. A reliable measurement of the solid surface energyσA isa very problematic task, making it almost impossible to get any good estimates of asolid-liquid or solid-solid interface energiesγAjB with Eq. (6.14).

6.3 Ideal Work Of Separation as Measure of InterfaceAdhesion Strength

Another fundamental quantity in the interface thermodynamics is theideal workof separation130 Wsep, which is defined as the reversible work needed to separatethe interface into two free surfaces in a thought experiment, whereby plastic anddiffusional degrees of freedom are suppressed. The ideal work of separation can beexpressed by a modified Dupr´e equation.

Wsep= σ0A +σ0

B γAjB: (6.16)

The difference from the Dupr´e equation (6.14) is that the surface energiesσA andσBare now replaced by the instantaneous values of surface energies before any plasticprocesses, like dislocation motion, or diffusional processes of chemical equilibra-tion, like surface segregation or surface contamination, take place. Due to suchdissipative processes, the energy needed in a real cleavage experiment will alwaysexceedWsep. Yet, it is still a very useful quantity to characterize interface mechanicalstrength.

+W =BA A B

Figure 6.2: Illustration for the definition of the work of separation.

As discussed by Finnis,130 while it is very problematic to directly calculate thecontact angle or the work of adhesion Eq. (6.14), it is a much more manageabletask to calculate the ideal work of separationWsep, by comparing the total energy ofthe interface system with the total energy of the corresponding system in which theinterface is cleaved, leaving two free surfaces (Fig. 6.2). Due to such calculationaldifficulties, the theoretical analysis of wetting in the appended papers assumes

Wad'Wsep; (6.17)

43


whereWsep is calculated without any explicit inclusion of thermal motion,i.e. atT = 0, as

Wsep=WAjB = (EAj+EjBEAjB)=A: (6.18)

HereEAj andEjB are the total energies of the separated half-crystals.

44

CHAPTER 7

Goals and Principles of InterfaceGeometry Modeling

Heaven on Earth is a choice you must make, not a place we must find.

– Dr. Wayne Dyer

Practically all properties of metal-ceramic interfaces directly depend on the in-terface atomic structure. Interfaces that are of interest here can be viewed as twofree surfaces of different materials that are put atomically close to each other. In-terface atomic structure describes how the atoms of the two contacting surfaces areplaced relative to each other. This relative placement is affected by the fact that theinteratomic interactions across the interface can significantly change both the inter-atomic interactions and the atomic structure within each of the contacting surfacesubsystems (interface relaxations). Besides, atoms can penetrate from one side ofthe interface into the other (interdiffusion). One more important factor is the differ-ence in the structure and periodicity of each of the surfaces (lattice mismatch). Thiscan result in significant irregularity of the interface structure, with an interface unitcell being much larger than the surface ones, with many defects (interface disloca-tions), or without any long-range order at all (amorphous). Moreover, if one of thecontacting phases is in a liquid state, like a melted metal in the problem of wetting,there is no well-ordered interface structure, there are just short-range correlationsbetween dynamically changing atomic positions.

Adequate choice of the interface model geometries (atomic structures) is a cru-cial component of the theoretical investigations in Papers I - V. This chapter intro-duces some common definitions related to the description of interface geometry. Italso discusses different aspects of the experimental situation and theoretical under-standing of metal-ceramic interfaces that help making physically meaningful choicesof the interface structures in Papers I - V.

45

7 Goals and Principles of Interface Geometry Modeling

7.1 Basic Definitions and Background: GeometricalDegrees of Freedom

The geometry of an interface can be specified by a number of variables, which arereferred to as the geometrical degrees of freedom. It is common to distinguish be-tween macroscopic and microscopic degrees of freedom.

7.1.1 Macroscopic Degrees of Freedom

The macroscopic degrees of freedom describe the relative orientation of the twocrystals and the interface plane,i.e. the interfaceorientation relationship. The ori-entations of the interface plane and the two crystals can, in principle, be determinedmacroscopically,e.g., by the symmetry of the macroscopic properties. There aregenerally five macroscopic degrees of freedom. The relative orientation of the twocrystals can be specified by their relative rotation in space. To describe such a rel-ative rotation (orientation relationship) one needs three degrees of freedom: twofor the unit vector along the rotation angle, and one for the rotation angle. Twomore degrees of freedom are needed to specify the direction of the interface-planeunit normal (interface orientation). The macroscopic degrees of freedom constitutegeometric thermodynamic variables which are required for a full thermodynamicdescription of the interface.128

One convenient way to specify macroscopic degrees of freedom is by writingthe indices of the crystal planes that are parallel to the interface plane. For example,the Me(001)/TiC(001) notations that are often used in the appended papers implythat the Me/TiC interface plane is parallel to the (001) crystal planes of the bothmetal Me and TiC contacting crystals. Such a specification covers only four of thefive microscopic degrees of freedom. What should also be included is the relativerotation of the parallel planes of the two crystals around the axis perpendicular tothe interface plane, which is referred to as the interfacerotation state.

7.1.2 Microscopic Degrees of Freedom

The microscopic degrees of freedom provide a summary description of the atomicstructure at the interface. The interface atomic structure is controlled by the mi-croscopic relaxation processes, that is an adjustment of the relative positions of theinterface crystal planes and particular interface atoms in order to lower the total en-ergy of the systems. Such relaxation processes take place under the constraints intro-duced by the macroscopic degrees of freedom, which can be thought of as boundaryconditions far from the interface.

If the interface atomic structure is periodic, like for the interface model systemsused in Papers I - V, then there are three main microscopic degrees of freedom, whichare described by the rigid body displacement,t, of one crystal relative to the other.The type of interface atomic structure corresponding to a specific value of the com-ponent oft along the interface plane, the relative translation, is sometimes referred

46

7.1 Basic Definitions and Background: Geometrical Degrees of Freedom

to as the interface (lateral)translation state.131 In Papers I - V, the component oftperpendicular to the interface is described by theinterface spacing, d, the interlayerdistance at the interface (Fig. 7.1).

d

Figure 7.1: A schematic illustration of the definition of the interface interlayer spacingd.

7.1.3 Interface Geometry Control in Experiment

How much control does one have over those macroscopic degrees of freedom formetal-ceramic interfaces in experiment? The answer depends on the way the par-ticular interface system is created. One common experimental situation of metal-ceramic interfaces encompasses particle-matrix interfaces in a nucleation and growthprecipitation reaction, like for precipitate particles of Ti, V, Cr, and Nb carbides andnitrides in steels (see Refs. [132,133] and references therein). In those systems thecarbide and nitride particles often precipitate in a few known unique orientationswith respect to the metal matrix, and a significant part of the formed metal-ceramicinterface is planar and atomically flat. The crystollagraphy of such interfaces,i.e.their macroscopic orientation, is difficult if not impossible to influence in experi-ment. At the same time, the fact of a relatively well-defined interface structure formatrix-precipitate systems facilitates much computer simulations of such interfaces,like in a first-principles theoretical study of the Fe/VN interface.132,133

A better control over the interface structure can be reached in a heteroepitaxialgrowth of thin films on single crystalline substrates.131 The heteroepitaxial growthallows us to choose and fix the interface orientation. There is much less control andflexibility in setting the orientation relationship, which is determined by the micro-scopic growth processes under the given growth conditions. Yet, often the film canbe grown epitaxially, with a high-quality atomically flat and chemically pure inter-face, and with a well-defined and unique orientation relationship to the substrate.One of the most common and powerful heteroepitaxial growth techniques is molec-ular beam epitaxy (MBE). In the MBE growth process the atoms of the growing filmimpinge on the surface of the substrate as molecular beams. This technique has tobe used in an ultra high vacuum to avoid contamination of the growing surface. Thewell ordered structure and macroscopic lateral dimensions of the MBE grown in-terfaces make such systems particularly suitable for characterization of their atomicand electronic structure with different surface science experimental techniques. Forthe same reasons, they are also nearly ideal systems for theoretical modeling.

Probably most flexibility in choosing the interface macroscopic orientation re-

47


lationship is provided bysolid state bonding,131 also known as diffusion bonding,when planar surfaces of a metal and ceramic are joined in high or ultra high vacuum.An intimate microscopic contact between the two contacting surfaces is reached viaplastic deformation and closure of voids by diffusion. The plastic deformation andclosure of voids by diffusion are also stimulated by applying a uniaxial pressure andelevating the temperature. While solid state bonding offers practically full controlover the macroscopic degrees of freedom, the interface formation process is diffi-cult to control on the atomic scale. The material transport by diffusion can lead touncontrollable microscopic changes in the near interface region, like faceting andformation of small angle grain boundaries. In such a situation the local interfaceorientation and orientation relationship in the near interface region can differ signif-icantly from the orientation of the averaged interface plane and the relative orienta-tion of the macroscopic bulk parts of the contacting crystals.

7.2 Choice of Interface Geometry Models

The geometry of the system, the information on where different kinds of atomsare located in space, is the main input of first-principles simulations. The goals,principles, insights, and specific data incorporated in that input to a large extentdetermine the value of the output of the computations, the validity and significanceof the obtained theoretical results.

7.2.1 Scenarios of Theory-Experiment Interaction

There is a number of common scenarios that can be identified in many existing first-principles DFT studies in the way they choose the system geometry and relate theirresults to experiment.

Scenario I.One very attractive situation is when the system geometry, and prob-ably other properties, are well characterized by experimental studies, and whenthe atomic structure is simple enough to be described by a hundred or so atomsput in a simulation supercell. First-principles simulations of such systems givegood opportunities to test the reliability and accuracy of the approximations in theDFT exchange-correlation functionals and the first-principles computational meth-ods. Such computational results can also significantly extend and complement theexperimental data with helpful new predictions, interpretations, and insights. Anexample of such a situation in the field of metal-ceramic interfaces is the case ofNb(111)/α-Al2O3 interface.134

Scenario II. A situation not too far from Scenario I is one when the experi-ments do not fully determine the system geometry but point at that the geometryrealized in experiment should be found among a sufficiently small number of dif-ferent known alternatives. Then, the first principles simulations can predict whichof the alternative geometries is the real one, and again extend and complement theexperimental data for that system. This kind of situation can be found in,e.g., a

48


first-principles study of the Ag(001)/MgO(001) interface in Ref. [129], when thefirst-principles results point at which of a few alternative interface translation statesshould occur in the experimental MBE-grown interface system. Outside the field ofmetal-ceramic interfaces, one important example is also a first-principles determina-tion of theκ-alumina atomic structure,135,136when the first-principles calculationshelp to identify which structure, out of a considerable number of alternatives, shouldcorrespond to the experimental one.

Scenario III. One more important situation is when the connection betweenfirst-principles calculations and experimental results is mediated by some additionalphysical model. That model relates the essential physics of a given physical propertyor process to a number of microscopic parameters, which can be provided by first-principles calculations. In that case the model atomic structures for first-principlescalculations are chosen so that to effectively determine the parameters of the phys-ical model. Often it may be relatively unimportant if those atomic structures areactually realized in experiment. Scenario III is typically used when the time orlength scales of the physical problem of interest are beyond the capabilities of directfirst-principles simulations.

A characteristic example of Scenario III is given by the thermodynamic descrip-tion of sizes and shapes of precipitates in metal alloys in Ref. [137]. The precip-itates have to be treated at relatively large length scales and at finite temperatures,which puts the problem beyond direct first principles calculations. In this case theconnection between experiment and first-principles simulations is mediated by thecluster-expansion model,138 together with Monte Carlo simulations. The parame-ters of the cluster expansion are determined from first-principles calculations for aset of simple structures that describe elementary geometry configurations, such aspairs, triangles, tetrahedra. Those atomic structures are not suggested by experi-ment. Instead, they are chosen with the main goal of determining the parameters ofthe cluster-expansion theoretical model.

The existing first-principles studies do not necessarily belong to one and onlyone of the above described scenarios. They can mix various components of differentscenarios, or emphasize other aspects, not mentioned here. Yet, it is interesting toanalyze if we can identify any of the above described characteristic features in theresearch work of this thesis.

First of all, the first-principles studies in the appended papers don’t have much incommon with Scenario I. There is no experimental information about the interfaceatomic structure for any of the materials systems considered in Papers I - V.

In the spirit of Scenario II, we can try to narrow down the range of different pos-sible alternatives for interface structures in the studied systems. The main structuralclue we have from experiment is that the (001) surfaces of carbides and nitrides inthe NaCl structure are more likely to be found than other surfaces (See Chapter 5and discussions in Paper II and V). That allows us to focus mainly on the interfacesformed by the (001) carbide and nitride surfaces. Yet, that assumption still leaves toomany different possible interface structures. The number of alternatives is beyondthe capabilities of direct first-principles treatment.

49


7.2.2 Should We Search for Best Structure?

In the above discussion it was mentioned that one of the reasons why Scenarios I andII can not be followed in the present work is the lack of structural information fromexperiment. However, there is one more reason, which is more important and moreinteresting than the first one. That reason is in that there seems to be no meaning inthe search for one preferred structure, as suggested by the following arguments.

The interface systems of interest in this thesis are mainly found in brazing andcermet sintering, or in related wetting experiments. In cermets, the grains of thebinder metal,e.g., Co, have to adjust to many surrounding carbide or nitride grainsof various macroscopic orientations. Even if there is a special orientation relation-ship between the metal and one of the carbonitride grains, there will be a differentmacroscopic orientation relationship to other grains. That is, the interface macro-scopic degrees of freedom should cover a relatively wide range of values. The inter-face atomic structure that should adjust to those various grain orientations, is thenalso unlikely to realize a strong preference for only one particular interface structure.Moreover, the sizes and shapes of grains change during various stages of sintering.

The next argument is the lattice misfit,i.e., that the metal and ceramic crystalsforming the interface differ in their translation symmetry elements parallel to theplane of the interface. For example, there is around 25 percent difference in the lat-tice constants of TiC and fcc Co (see Paper I and II). When such metals and ceram-ics are in atomic contact, different situations are possible, depending on the relativestrength of different chemical bonds inside the contacting materials and across theinterface.

If the interface bonds are too weak to cause any noticeable distortion in thematerials, then the surface layers of the two materials will have different periodicity,forming an incoherent interface. Thus if one type of local atomic structure, likemetal-near-non-metal configuration, applies to some point of the interface, then thistype of interface structure will not be periodically repeated in the nearby areas.

A more common situation is when there is a clear preference for some specificlateral positions of the interface atoms of the metal phase relative to those in theceramic one. This can be due to relatively strong bonds across the interface. Forexample at metal(001)/carbide(001) and metal(001)/nitride(001) interfaces studiedin this thesis, it is energetically more favorable for the interface metal atoms to beplaced over the carbon or nitrogen sites than over the metal ones (Papers I - V).Those interactions result in forces that tend to displace each atom towards the nearestfavorable position. That is, the interface bonding tends to match the lattices of thesurface layers of the two materials in order to increase the number of favored atomicconfigurations. Such matching introduces strain into the contacting crystals, mainlyin the metal phase, which is generally less stiff than the ceramic one.

For a reasonably small lattice misfit, the resulting interface structure is oftendescribed as consisting of well-matched (coherent) regions, with the misfit strain lo-calized as line defects, referred to as the misfit dislocations131 (Fig. 7.2). However,the distance between the misfit dislocations should decrease with increasing degree

50


of misfit, so that at some point it is difficult to distinguish separate misfit disloca-tion. At that point those structure irregularities should merge, resulting in a totallyirregular (amorphous) interface structure. Such situation is likely to occur at cermetCo/Ti(C,N) interfaces in cermets, where large lattice misfit can be caused by boththe large difference in the lattice constant of Co and Ti(C,N) and a misorientation ofthe ceramic and metal grains in cermets.

Figure 7.2: A schematic illustration of localized lattice distortions that form an interfacemisfit dislocation.131 The main region of the misfit dislocation is shown by a dashed-linecircle.

The final important argument against the existence of one best interface structureis that, in the context of both brazing and sintering, our primary interest is wetting,when the metal is in the liquid state. In this case, instead of looking at one staticstructure, we should consider an ensemble of various instantaneous structures thatare spanned by the moving atoms. There is no long range order in the structure, justsome correlations between the positions of the neighboring atoms.

7.2.3 Focus on Development of Theoretical Models of Wettingand Adhesion

The discussions above suggest that the present study does not fit in Scenarios I orII (see Section 7.2.1), but that there are similarities to Scenario III. The first notice-able similarity is that the physical problem of interest,i.e. wetting and adhesionin brazed and sintered metal-carbide and metal-nitride systems, is beyond a directfirst-principles treatment. This is especially due to the variety, complexity, and un-certainty of the interface atomic structure, as discussed in Section 7.2.2. The relationbetween first-principles simulations and reality should be mediated by some physi-cal model.

The physical model needed here should be able to treat the wetting and adhe-sion in metal-carbide and metal-nitride systems in microscopic terms. In the contextof the discussion in Chapter 6, our main expectation from such a model is to de-scribe the strength of the metal-ceramic interface bonding in the materials systems

51


of interest, as can be characterized by the ideal work of separation. Unfortunately,there are no specific microscopic models of metal-ceramic interface systems that wecould take from other theoretical or experimental studies and adjust to our needs (seeChapter 8).

The lack of adequate theoretical models for the problem of interest suggests twoimportant directions for theoretical work in this thesis, which will also determinethe choice of geometry models. The first direction is to analyze more general andqualitative relationships between different macroscopic and microscopic aspects ofthe considered problem, and then use those relationship in a role of a missing phys-ical model. Those relationships will not be enough to formulate a specific modelin a traditional sense,i.e. as mathematically expressed relations between a set ofmodel parameters. Yet, they can still serve as an adequate base for a meaningfulchoice of geometric models for the system. The second direction, closely relatedto the first one, is to concentrate our research efforts on developing more specificphysical models. That is, the first-principles simulations, including choice of geom-etry models, should be focused on exploring the key connections between differentelements of the problem, so that we could gradually progress from the initial generaland qualitative relationships to more specific and quantitative ones.

As a starting point for building a theoretical model of wetting and adhesion wecan take a general assumption thatthe strength of the interface bonding is determinedby the microscopic interactions at the interface. In the context of the present study,that general statement leads to three important specific goals for our research.

Goal I: to identify the key microscopic interactions at the metal-carbide andmetal-nitride interfaces.

Goal II: to explore and clarify the relationships between those microscopic in-teractions and the interface adhesion strength (the ideal work of separation).

Goal III: to use first-principles simulations and general knowledge about thegiven materials systems to understand the relative role, and the interplay, of dif-ferent microscopic interactions in the real metal-carbide and metal nitride interfacesystems.

Those three goals determine the choice of the atomic structure models in thefirst-principles studies in Papers I - V. In connection to Goals I and II much attentionin the appended papers is given to various simple model structures, like metal-over-Ti(W) or metal-over-C(N) structures in Papers I-II, IV, and V (Figs. 7.3 and 7.4(a,b)).On the one hand, due to the simplicity of those structures it is easier to distinguishmore clearly the contributions of different kinds of chemical bonds and then analyzetheir nature (Goals I and II), as well as to compare the behaviors of those interac-tions, which is helpful in the context of Goal III. On the other hand, those simplestructures represent the most distinct types of relative atomic arrangements at theinterface, like metal-over-Ti(W) versus metal-over-C(N) structural configurations.First-principles calculations with such distinct structures help to identify the rangein which the adhesion strength (the ideal work of separation) changes as a functionof the interface structure, and to clarify the connections between the strength of inter-face bonding and the microscopic interactions (Goal II). More complex structures,

52


like in Figs. 7.4(c)-(f), are intended to mimic the situation of realistic interfaces(Goal III), where different local configurations are mixed together, as discussed inPapers I and II.

(b)(a)

Ti

C(N)

M

Figure 7.3: The two main types of simple model interface structures for metal/Ti(C,N) in-terfaces studied in Papers I,II, and V. Half the elevation of the simulation supercell is shownfor metal-over-C(N) (a) and metal-over-Ti (b) structures.

CTi

Co(f)

(e)(c)

(d)(b)

(a)

Figure 7.4: Various interface structure models for Co/Ti(C,N) interfaces studied in PapersI and II. The position of the first Co layer with respect to the outermost layer of TiC(001)surface is shown within one unit cell.

7.2.4 Simplified Description of Bulk Phases

In a specification of the atomic structure of an interface system one can typicallyidentify parameters that describe the bulk structure of the contacting materials, as

53


opposed to interface-specific parameters, like the interface interlayer spacingd, il-lustrated in Fig. 7.1. For a more clear identification of the key behaviors of theinterface microscopic interactions, when choosing interface model structures, it isvery advantageous to keep the number of bulk parameters within some reasonableminimum, focusing on variations of the interface-specific parameters instead. Inparticular, to decrease the necessary number of the bulk structural parameters, onecan consider bulk structures with a relatively high symmetry.

In the appended papers all the interface model structures are constructed with theassumption that the metal bulk phase is face centered cubic (fcc), rather than somedisordered atomic structure representing an instantaneous state of a liquid metal. Adescription in terms of interface misfit dislocations is not used either.

It should be noted that the fcc structure is a very realistic description of the metalbulk phases in solid-solid metal-carbide and metal-nitride interfaces for most of themetals considered in Papers I - V. In particular, fcc is the ground state structure of Cu,Ag, Au, and Al. The ground state structure of Co and Ti is hexagonal close packed(hcp). Fcc structure, which is also close packed, is realized in the high-temperaturephase of Co, above 418oC for pure Co. Yet, the fcc Co face is more common than thehcp one in the binder phase of cemented carbides and cermets,139 where the residualstresses and the presence carbon and tungsten stabilize the fcc phase even at roomtemperature. For Ti, the fcc phase is only slightly higher in energy, by less than 0.1eV, than the hcp one.

So that one could build commensurate (periodic) interface structures, the bulkfcc phase in the model interface systems is allowed to be distorted, as described bytwo parameters. The first parameter, the in-plane lattice constant, is the lattice pa-rameter of the bulk crystal structure along the plane that is parallel to the interfaceplane. With given interface structural parameters, like the rotation state, the in-planelattice constant is fully determined by the requirement of periodicity (commensu-rability) of the interface structure. The second parameter, the out-of-plane latticeparameter, aims to describe the expansion or contraction of the metal phase alongthe direction perpendicular to the interface. For a given in-plane lattice parameter,the out-of-plane lattice parameter is adjusted so that to minimize the bulk strain inthe metal phase. That adjustment is done in a separate set of bulk calculations. Thus,both bulk parameters of the metal phase are fully determined before an interface sys-tem is constructed, and they are not varied in interface calculations.

Another important situation of the simplified description of the bulk phases isrepresented by the Co/WC study in Paper IV, where the WC phase, which is hexag-onal in reality, is taken in the NaCl structure. This is done for a more direct com-parison of the interface interactions in Co/TiC and Co/WC systems, as discussed inPaper IV.

7.2.5 Concluding Remarks

It should be noted that in the appended papers the connection to experiment is mainlyrestricted to a comparison of the work of adhesion values measured in wetting ex-

54


periments with the calculated data for the work of separation. The approximationsinvolved in that comparison, like in the description of wetting, in the choice of struc-tural models, and in the neglect of the thermal motion and diffusion, are expectedto introduce a larger inaccuracy than the approximations in the used DFT exchange-correlation functional (i.e., GGA). Yet, in spite of many approximations involved,our first-principles estimates of the work of adhesion still appear to be very closeto the results of wetting experiments (Papers I, II, and VI), or help to understandthe discrepancies in the experimental wetting data (Paper V). All this is an impor-tant indication of that the structural models used in the present thesis, together withthe key principles discussed in this chapter, represent a very promising approach offirst-principles treatment of complex metal-ceramic interface systems.

55


56

CHAPTER 8

Microscopic Interactions atMetal-Ceramic Interfaces

In the thermodynamical, structural, mechanical and other properties of metal-ceramicinterfaces the crucial role is played by the interatomic interactions at those interfaces.

Since the early days of the metal-ceramic interface research, a common way tostudy metal-ceramic interface interactions is wetting experiments (See Section 2.3).A drop of liquid metal is placed onto some ceramic substrate, and then the shapeof the drop is analyzed to obtain information on the interface energetics. For a longtime, the wetting experiments were the main context of the theoretical work, whichwas mainly restricted to organizing the wetting data and to extracting the trends inthe wetting behavior with respect to variations of different properties of the metaland the ceramic. Based on those trends various hypotheses on the nature of interfaceinteractions were suggested.

A considerable progress in the methods to study interface interactions has beenmade during the last decades, as reviewed,e.g., in Refs. [130, 131]. On the exper-imental side, there have been significant developments in the interface fabricationtechniques, as well as in the ability to probe the details of the interface atomic andelectronic structure. In particular, a detailed control of the interface structure andcomposition on the microscopic level can be achieved with the modern heteroepitax-ial growth techniques, like molecular beam epitaxy (MBE). An atomic scale infor-mation on the interface structure is provided (seee.g.Refs. [130,131,140] and refer-ences therein) by the high-resolution transmission electron microscopy (HRTEM),Z-contrast scanning transmission electron microscopy, and atom-probe field ion mi-croscopy. A similar level of resolution is obtained for the interface electronic struc-ture, using the spatially resolved electron-energy-loss spectroscopy.

On the theoretical side, some understanding of the metal-ceramic interface bond-ing has been gained through extensive studies of a few model interfaces, using differ-ent levels of description, from simplified scattered-wave,141 atomic-orbital,142tight-

57

8 Microscopic Interactions at Metal-Ceramic Interfaces

binding,143–146and image-interaction147–152models toab initio density-functionalcalculations.129,134,140,153–160However, this understanding still remains quite lim-ited.

As noted in,e.g., Refs. [131,140], the major difficulty in the theoretical descrip-tion of the metal-ceramic interfaces is the difference in the nature of bonding onthe two sides from the interface. The consequence of this difference is that, whilethere are reasonably good atomistic models of each of the contacting bulk mate-rials, metal and ceramic with their bonding types, there are no acceptably reliableinteratomic potential schemes that could describe the interatomic interactions at ametal-ceramic interface. Thus,ab initio calculations, of the type presented in thiswork, are practically the only reliable way to model metal-ceramic interfaces.

This section discusses different types of interface interactions, together with thesimplified models that have earlier been used to describe those interactions.

8.1 Dispersion Forces and Carbide Wetting Trends

The van der Waals dispersion interaction is the interaction between the fluctuatingdipoles of the two interacting media. That is, in each medium there are quantumfluctuations of the dipole moment, and the dipole moment of one medium interactswith the induced dipole moment in the other medium. The simplest way to take intoaccount the dispersion forces is just to use the model with the interatomic potentialin the form of the inverse sixth power of the distance. However, this sixth powerlaw is an acceptable approximation only at large distances, where the van der Waalsinteraction is almost negligible. An alternative approach, which is more accurateand at the same time rather simple, is the dielectric continuum model of Barrera andDuke.161 In the dielectric continuum model, each of the media is described by acomplex frequency-dependent dielectric function in the form

ε(ω) = 1ω2p=[ω

2+ iω=τ∆2]; (8.1)

whereωp is the plasma frequency,τ is the plasmon damping time, and∆ is the bandgap of the corresponding media. The dielectric continuum approximation leads toclosed expressions for the surface and interfacial energies, and hence for the wettingcontact angles, in terms of only three parameters for each medium (ωp,τ, ∆ ). Thespecific values of the parameters can be extracted directly from the bulk optical-absorption and electron energy-loss spectra. Thus the considered model makes itpossible to estimate the contribution of the dispersion forces by just analyzing theaccessible bulk experimental data.

For ceramic oxides, the systematics of wetting by liquid metals does not cor-relate with the predictions of the dielectric continuum model, and it has been con-cluded24,147,162,163that although the dispersion forces are not negligible they arenot the major effect in the adhesion here. Instead, the trends in the wettability withchanges in the substrate oxide has led to the view that the interfacial energetics iscontrolled by the electrostatic image interaction, as discussed below.

58

8.1 Dispersion Forces and Carbide Wetting Trends

For carbides, which are less ionic but rather metallic (∆ = 0), the contributionof the dispersion forces to the interface interactions can be expected to be of moreimportance than for oxides, approaching the situation of metal-metal interfaces. Formetal-metal interfaces, neglecting the plasmon damping (τ = ∞), the dielectric con-tinuum approximation gives the interface energy as161

γ =h

4π

Zdq

(2π)2

∞Z

0

dω2π

ln[ f (iω)]; (8.2)

with

f (ω) = [ε1(ω)+ ε2(ω)]2=4ε1(ω)ε2(ω); (8.3)

whereε1;2(ω) are the dielectric functions of the two metals from Eq. (8.1) andqis the plasmon two-dimensional wave vector along the interface plane. To avoidthe divergence of theq-integration, this integration is performed within a cutoffqc,assuming the existence of dispersionless plasmons only forjqj < qc. To calculatesurface energies, one ofε1;2(ω) should be taken to be unity. From the Young equa-tion (6.13) and expression (8.2) and (8.3) for surface and interface energies, thefollowing expression for the wetting angleθ can be derived:23

cosθ =1+

p2Z

p2(1+Z2)p

21; (8.4)

whereZ is the ratio of the plasmon frequencies of the substrate,ωps, and the metal,ωpm, Z = ωps=ωpm. Since cosθ should be between -1 and 1, the equation (8.4) isonly valid for 0 Z 1, i.e. when

ωps ωpm: (8.5)

In Ref. [23] expression (8.4) is used to analyze the wetting trends for Cu on car-bides, and it is concluded that the above dielectric continuum model gives an ade-quate description of those trends. This leads to the view that adhesion of Cu to car-bides is controlled by the van der Waals dispersion forces. However, a more recentwork [18] suggests that such a view needs to be reconsidered. On the one hand, inRef. [23], with reference to the lack of experimental data on the plasmon frequenciesfor carbides, the analysis is based on indirect arguments for those plasmon frequen-cies. At the same time, the experimentally measured plasmon frequencies for TiC inRefs. [164,165] are noticeably larger than the Cu plasmon frequency, which does notallow to satisfy the condition (8.5) of applicability of Eq. (8.4). On the other hand,in Ref. [18] it is shown that for hypostoichiometric TiC or TiN the contact angle de-creases with the plasmon frequency, which is inconsistent with Eq. (8.4). Thus, forCu on carbides, the description of the interface adhesion in terms of the dispersioninteraction seems to be inadequate. For more chemically active transition metals,including Co, the role of the dispersion forces should be even less significant.

59


8.2 Image Interaction Model

A fruitful approach to understanding of the interface adhesion between metals andceramic oxides is the image interaction model, suggested in Ref. [147] and thenfurther developed in Refs. [148–152]. The idea of the image interaction comes fromthe basic electrostatics, from a picture describing the interaction of a charged objectwith a conductor surface. In particular, if there is a point chargeq at a distancez0

from a conductor (e.g., metal) surface, then this charge induces a screening chargedensity on the surface given by

ρ(r) =z0q

2π(z20+ r2)3=2

; (8.6)

wherer is the distance from the charge in the plane along the surface. The electro-static potential from this induced charge density,

V(r;z) = q

[r2+(z0+z)2]1=2: (8.7)

has the same form as if instead of the metal surface there were an effective pointchargeq placed at the same distancez0 from the surface plane as the originalcharge, but on the other side from the surface, as illustrated in Fig. 8.1. That is, theinteraction of a charge with a metal surface can be described in terms of electrostaticattraction between this charge and the effective image charge. The energy of suchan image interaction is

E =12

q2

2z0: (8.8)

Since an arbitrary charge distribution can be represented as a set of point charges,it is straightforward to move from the above equations to the general case of theelectrostatic interaction of a charged object with a metal surface.

-q

qz0

z

0-zmetal

Figure 8.1: The image interaction picture for the electrostatic interaction of a point chargewith a metal surface.

60

8.3 Chemical Bonds across Interface

In the simplest variant of the image interaction model of the metal-ceramic ad-hesion, the metal is taken as a conducting continuum (a dielectric continuum withan infinitely large dielectric constant). The ceramic oxide is considered as a latticeof ions, cations and anions, which interact with the metal through the attractive clas-sical image interactions, balanced by a short-range repulsion. A simple estimate147

shows that the energy of the image interaction can be of the order of Joules persquare meter, which is in the range of typical metal-oxide adhesion energies. Theimage interaction picture is also supported by the fact that the adhesion of a non-reactive metal to a ceramic oxide is mainly determined by the material of the oxide,not the metal.

So far we have been discussing the image interaction mainly in classical terms.As a matter of fact, even the classical image potential gives a fairly good approxi-mation to the real picture of metal-oxide adhesion, at the spatial scale down to a fewAngstroms. At closer distances, quantum mechanical effects become important. Inparticular, in a real metal, in contrast to the dielectric continuum, the finite size of thevariations of charge distribution has to be considered. Only variations with wave-lengths larger than the Fermi wave-length are allowed. One consequence of such abehavior is the so-called Friedel oscillations. A possible way to correct for this ef-fect is to explicitly introduce the wave vector cutoff in the expression for the inducedcharge density, as it is done in Ref. [150]. A more consistent quantum mechanicaltheory of the image interactions is developed in Ref. [149], where a general linearresponse expression for the image interaction energy is derived in the framework ofthe density functional theory. This quantum mechanical analysis does not only givethe correction to the classic expression for the image interaction energy, but alsodeepens its interpretation, showing that the simple form of the image interactionenergy is actually a result of a delicate balance between more complex-structuredcontributions from the kinetic and exchange-correlation energies.

8.3 Chemical Bonds across Interface

The interactions discussed above,i.e. the van der Waals and electrostatic image in-teractions, are of a relatively long-range type, which makes them quite insensitive tothe details of the atomic structure at the interface. Dominance of those interactions inthe metal-ceramic interface bonding would lead to universal features in the adhesiontrends. Besides the van der Waals and image interactions there are also possibili-ties for short-range ionic and covalent bonds. The short-range interactions generallycomplicate the picture of the interface bonding, not allowing us to construct simpleand universal models, like the ones mentioned in the previous subsections.

Compared to electrostatic image interactions, the localized ionic bonds involvesome noticeable charge transfer between the interacting interface atoms. Such amechanism of bonding has been demonstrated to dominate,e.g., at low metal cover-ages on an Al2O3 substrate156and for the bulk Ni(111)/α-Al2O3(0001) interface.134

One important aspect worth mentioning here is given by the trends in the nature

61


of the metal-ceramic interface bonding depending on whether the metallic phase isrepresented by a free-electron-like, transition or noble metal.129 The bonding be-tween a simple metal and an oxide ceramic is controlled essentially by the Coulombinteraction between the ions and their screening charges in the metal, that is by theimage interaction. The situation with transition metals is more complicated becausethey also have partially filledd-orbitals, and can form strongpd-covalent bondsacross the interface with the oxygen atoms of the ceramic. Since the filling of thetransition metald-orbitals moves them closer in energy to the oxygen orbitals, thecovalency of the interfacepd-covalent bonds tends to increase towards the end ofthe transition metal series.143 These interfacial covalent bonds are also the main fac-tor in determining the relative positions of the atoms at the interface: the transitionmetal atoms are usually placed above the oxygen atoms of the ceramic. For noblemetals, there are stilld-orbitals, and although they are filled and cannot participatein the covalent bonds, they are polarizable and can contribute to the interface bond-ing. In this case the bonding is expected to have an intermediate nature betweenthe cases of simple and transition metals. The problem of the relative positions ofthe atoms at the interface is more complex and unclear, and probably there can beseveral metastable states of a noble metal-ceramic interface.

8.4 Metal-C(N) Bonds across Interface as Opposed tothose in Bulk Carbides and Nitrides

One efficient way of understanding something new is through exploring the sim-ilarities to and differences from something that is already known and understood.When we use this strategy to understand the chemical bonds across interfaces be-tween metals and transition metal carbides and nitrides, one of the closest situationswe can compare to is the chemical bonds in bulk carbides and nitrides (Section 5.2).

As discussed in Section 5.2, the bonding in bulk transition metal carbides andnitrides is mainly due to covalent bonding between C(N)-2p and metal-d orbitals,and the bonding strength is controlled by the filling of bonding and antibondingstates (see Section 5.2.3). For example, among the 3d-transition-metal carbides,TiC has the largest cohesive energy, because the bonding states are filled, while theantibonding are empty. Cohesion in CoC is significantly weaker, due to an almostcomplete filling of the antibonding states.

As a first step, in the consideration of metal-C bonds across Co/TiC interfacesstudied in Papers I and II, we can assume that the metal-C bonds at the interfacesare similar to those in the corresponding bulk carbides. Such an assumption leads tothe expectation that the interface Co-C bonds are relatively weak, about as weak asthose in bulk CoC.

In a further step, we can still assume that the character of the Co-C bonding andantibonding states is similar to those states in bulk carbides, but allow the possibilitythat the interface environment changes the strength of the Co-C bonds by changingthe degree of population of the Co-C bonding and antibonding states. For example,

62

8.4 Metal-C(N) Bonds across Interface as Opposed to those in Bulk Carbides and Nitrides

there can be partial or complete emptying of the antibonding states. Such effects canmake the interface Co-C bond stronger than in bulk CoC but still not much strongerthan in TiC, where there is an optimal population of the bonding and antibondingstates.

As found in Papers I and II, the difference in the strength of the interface and bulkCo-C bonds is much more drastic than what could be expected with the above as-sumptions. In particular, there is a noticeable number of indications that point at thatthe interface Co-C bonds are considerably stronger than even the strong Ti-C bondsin bulk TiC. This is indicated by,e.g., the magnitude and structural dependence ofthe adhesion strength, the effects of structural relaxation, and the spatial distributionof the electron density (Fig. 8.2). A similar behavior is found for the Co-N and Co-Cbonds at Co/TiN (Paper II) and Co/WC (Paper IV) interfaces, respectively. Such asituation suggests that the strength of the Co-C(N) interface bonds is beyond whatcould be explained within the theory of bonding in bulk transition metal carbidesand nitrides.

Based on the electronic structure analysis of the Co/Ti(C,N) interfaces, the un-usual strength of the interface Co-C(N) bonds can be explained in terms of interface-induced modifications in the character of the Co-C bonding and antibonding states.This interface-modified covalent bond is discussed in more detail in Paper II.

It should be noted that the strength of the Co-C bonds across the Co/TiC inter-faces is not significantly affected by the Co ferromagnetism, as discussed in PaperIII. Noti ceably weaker bonds are found when Co is replaced by noble metals, likeCu, Ag, and Au (see Paper V), though the characteristic electron structure featuressimilar to those at Co/Ti(C,N) can still be identified.

CoTi

C

Co

Co

Ti

C

C

Figure 8.2: Electron-density picture of bonding at the Co/TiC interface: constant-densitysurface at the level of 0.5 electrons/A3 (see Papers I-III).

63


64

CHAPTER 9

Conclusions

Joints between metals and ceramics are becoming increasingly important in the man-ufacturing of many high technology products, from microelectronic devices to cut-ting tools. The wetting of ceramics by metals is the key driving force in metal-ceramic joining processes. To a large extent it is controlled by the strength of adhe-sion between metal and ceramics.

Experimental studies of wetting suggest that wetting in metal-ceramic systemsis highly sensitive to factors of microscopic nature, like local chemical compositionat the interfaces. The main goal of the research in this thesis is to identify andanalyze the key microscopic mechanisms behind the wetting and adhesion, at thelevel of interatomic interactions. The ceramic materials considered in the thesis aretransition metal carbides and nitrides.

Due to the lack of reliable simplified models of interatomic interactions in metal-ceramic systems, the theoretical analysis of the thesis is based on the results of first-principles density-functional calculations of the total energy and electron structurefor a variety of model interface systems. The first-principles calculations are quiteaccurate and reliable, though computationally very demanding.

9.1 Qualitative Microscopic Picture of Wetting and Ad-hesion

Based on the conclusions and insights of the appended Papers, the following qualita-tive microscopic picture of interface adhesion between metals and ceramic carbidesand nitrides can be suggested. The adhesion between the metal and ceramic phasesis due to two distinct kinds of chemical bonds across the interface: metal-C(N) andmetal-metal ones. Its strength is determined by the number of bonds of each kindper unit area of interface, as well as by the strength of each of those bonds.

65

9 Conclusions

The strength and number of bonds at an irregularly-structured experimental in-terface,e.g. at a solid-liquid interface in wetting experiments, can be estimated byanalyzing the most distinct possible types of local structural arrangements at theinterface. Such extreme cases, like the metal-over-C(N) and metal-over-metal con-figurations (Fig.1(a,b) in Paper V), determine the likely range of variations in thestrength and number of different bonds and in the resulting adhesion strength. Theexperimental situation is expected to be in the middle of that range.

The total adhesion strength results from an interplay of three important effects,which come from the distinctions in the character of the interface metal-C(N) andmetal-metal bonds.

Effect I is that a metal-C(N) bond is often significantly stronger than a metal-metal bond at a given interface. This effect increases the relative role of the metal-C(N) bonds in the metal-carbide or metal-nitride interface adhesion. In particular,the strong metal-C bonds are expected to dominate the Co/TiC(001) interface bond-ing studied in Papers I and II.

Effect II is a noticeable difference in the optimal bond-lengths of the interfacemetal-metal and metal-C(N) bonds. In particular, the metal-metal bonds tend to beabout 30 per cent longer than the metal-C(N) ones. As a consequence of that dif-ference, there is typically a much larger number of the interface metal-metal bondsthan of the metal-C(N) ones. This effect increases significantly the relative weightof the metal-metal bonds in the total adhesion strength.

Effect II is also important in the understanding of the differences between Co/TiCand Co/WC adhesion (Paper IV). In spite of the large contribution of the strong Co-C bonds, it is still the metal-metal Co-Ti(W) bonds that determine the wetting andadhesion trends from Co/TiC to Co/WC.

Effect III is that the strength of the interface metal-metal bonds is quite sen-sitive to the presence of C or N atoms near that bond (see Paper V). In particular,those C or N atoms can weaken the interface metal-metal bonds significantly. Thisthird effect makes the competition between the contributions of the metal-metal andmetal-C(N) bonds in Effects I and II even more dramatic. For example, if some ofthe C or N atoms in the interface layer are removed (creation of C- or N-vacancies),the corresponding metal-C(N) bonds are lost. Yet, that loss can be outweighed bythe gain in the strength of a larger number of metal-metal bonds. In case of,e.g.,Ag/Ti(C,N)(001) interface, vacancies increase the overall strength of the interfaceadhesion. Effect III also plays an important role in the interface-orientation depen-dence of the interface bonding (Paper V).

9.2 Interpretations of Wetting Experiments

The first experiment of interest in the present thesis is wetting experiments for Coon TiC15 (see Section 2.3.1). The estimate of the work of adhesionWad (see Sec-tions 6.2 and 6.3) from the first-principles simulations in Papers I and II,Wad' 3:3J=m2, is quite close to the experimentally measured value,15 Wad = 3:64 J=m2 (at

66

9.2 Interpretations of Wetting Experiments

1420oC). The theoretical analysis shows that bonding at Co/TiC interfaces can beunderstood in terms of Co-Ti and Co-C bonds, within a qualitative picture discussedin the previous section. As shown in Papers I and II, the Co-C bonds are particularlystrong, and they should give the main contribution to the adhesion (see Effect II inSection 9.1).

Another important experimental fact is that wetting by Co is noticeably betterfor WC than for TiC15 (see Section 2.3.1). This fact is of great significance in thehardmetal industry, distinguishing the sintering of WC-Co cemented carbides fromthat of TiC-Co cermets (see Section 2.1). It can be described quantitatively by thefact that the work of adhesion for Co/WC,15 Wad = 3:82 J=m2, is somewhat largerthan for Co/TiC,Wad = 3:64 J=m2 (at 1420oC). This trend from Co/WC to Co/TiCis adequately well reproduced in the first-principles simulations of Paper IV. Thoughthe strong Co-C bonds should still be the main contribution to bonding, the furthertheoretical analysis shows that the difference between Co/WC and Co/TiC is dueto the different strength of the metal-metal Co-W(Ti) bonds (see also Effect II inSection 9.1).

A more complex experimental situation is provided by wetting of carbides andnitrides by Cu, Ag, and Au (see Section 2.3.2 and Paper V), where there is a largescattering in the wetting data. That situation can also be given a simple microscopicinterpretation, based on the qualitative picture of adhesion in Section 9.1 and specificfirst-principles results (Fig. 9.1). Most of the experimentalWad values are in therange between the theoreticalWsepvalues for the two most distinct types of interfacestructural configurations, in particular metal-over-C(N) and metal-over-Ti ones (seeFig. 1 in Paper V). Moreover, many of the experimental data points are very closeto the theoretical results for the metal-over-bridge structure (Fig. 1(c) in Paper V),which is an intermediate case of the interface structure, half way between the metal-over-C(N) and metal-over-Ti structures. Those experimental data points are at thesame time quite close to a simple average of the results for the metal-over-C(N)and metal-over-Ti configurations. Such a relation between the experimental dataand theoretical results fits well the qualitative microscopic picture of the interfaceadhesion in Section 9.1.

However, there are also experimental data points in Fig. 9.1 that are quite farfrom the middle region of the theoreticalWseprange. Those points come from worksthat report poor wetting of TiC and TiN by Cu and Ag.15,18 What is interesting aboutthose points is that they correlate quite well with the theoretical results for only oneof the extreme structural configuration, the metal-over-Ti one. That implies that theadhesion in those system is controlled by a limited number of longer range metal-Tibonds (see Effect II in Section 9.1), while metal-C(N) bonds and part of the metal-Tibonds are disabled, probably due to contamination of the substrate surface. Otherrelated aspects of wetting of TiC and TiN by metals are discussed in Paper V.

67

9 Conclusions

Cu/TiC Ag/TiC Au/TiC Cu/TiN Ag/TiN Au/TiN0.0

1.0

2.0

3.0

4.0

Wse

p (J

/m2 )

M−over−C(N)M−over−TiM−over−bridgeExperimental W ad

Figure 9.1: Values of the work of adhesionWad extracted from wetting experiments15–18,21

for wetting of TiC and TiN by Cu, Ag, and Au, in comparison with the theoretical valuesof the ideal work of separationWsep (Wsep'Wad, see Section 6.3) from the first-principlescalculations in Paper V. Labels “metal-over-C(N)”, “metal-over-metal”, and “metal-over-bridge” refer to the model interface structures (a), (b), and (c) in Fig. 1 of Paper V, respec-tively.

HfC ZrC TiC TaC NbC VC0.0

0.5

1.0

1.5

2.0

2.5

3.0

Wse

p (J

/m2 ) Cu−over−C

Cu−over−MeOld expRecent exp

Figure 9.2: Trends for wetting of carbides by Cu. Curves “Cu-over-C” and “Cu-over-Me”correspond to the results of first-principles calculations that use the model interface struc-tures in Fig. 1(a) and (b) of Paper V, respectively. “Old exp” and “Recent exp” refer to thework of Ramqvist15 and the more recent studies,16,18,21respectively.

The same simple analysis as in Fig. 9.1 can give an insight into one more inter-esting issue in wetting of carbides by metals, in particular the experimental wettingtrend reported for Cu on HfC, ZrC, TiC, TaC, NbC, and VC15 (See discussion in Sec-tion 2.3.2). When the calculations in Paper V for Cu/TiC are repeated for Cu/MeC,Me=Hf, Zr, Ta, Nb, and VC, a simple comparison with the corresponding experi-mentalWad values can be made (Fig. 9.2). All the experimental points, including the

68

9.3 Outlook

more recent data for Cu on TiC,16,18,21are covered by theWsepregion between thecalculated values for the two extreme structural configurations,i.e. Cu-over-C andCu-over-metal ones. Yet, the behavior of the theoretical curves does not show anyground for such a rapid increase of the work of adhesion from Cu/HfC to Cu/VC.A more smooth trend is expected, with data points in the middle region between thetwo theoretical curves. A shift towards that middle region is seen in the data pointsfrom the more recent experiments with Cu/TiC.16,21 New experiments for Cu onother carbides are needed, with the same careful treatment of oxygen contaminationas in Ref. [21].

9.3 Outlook

There are many interesting possibilities for the continuation of this work. One nat-ural direction would be to apply the methodology used in this thesis to other similarinterface systems. This could help to further develop and test the above describedqualitative picture of metal-carbide and metal-nitride adhesion. For example, theeffect of C(N) neighbors on the metal-metal bonding across interfaces is worth in-vestigating for a number of other metal-ceramic combinations,e.g., by consideringvacancies in Co/TiC or Co/WC systems. This direction could also contribute to thedevelopment of new materials.

From the point of view of fundamental microscopic understanding of wetting,one attractive direction of future research is to explore microscopic effects of statis-tical nature. More of simple and valuable qualitative insights are likely to be foundthere. A simple example of statistical arguments in this thesis is the assumption thatat disordered metal/Ti(C,N)(001) interfaces,e.g., when the metal is a liquid, a metalatom at the interface is about as likely to appear over a C(N) site as over the Ti site.One interesting issue for future studies is how the qualitative differences betweenthe metal-metal and metal-C(N) short-range interactions (chemical bonds) exploredin this thesis affect the statistical correlations between the motion, and average po-sitions, of the atoms at the interface. Probably the simplest way to approach thisissue is to express those qualitative differences in some simple model of interatomicinteractions, like pair potentials, and then perform molecular dynamics simulationswith that model. A very promising possibility for a more systematic and reliable de-scription of statistical effects in the metal-carbonitride adhesion is provided by therecently developed extended tight-binding method.166–168

69

9 Conclusions

70

AcknowledgementsFirst of all, it is my great pleasure to thank Professor Bengt Lundqvist, my supervi-sor, for giving me the opportunity to do my PhD study in his group, for guiding andencouraging me, and for creating the best environment for my scientific and personalgrowth during these years.

Special thanks to Jan Hartford, Yashar Yourdshayan, and Lennart Bengtssonfor introducing me into first-principles calculations, and to Ulf Rolander and Hans-Olof Andren for very valuable discussions on the experimental and technologicalbackground of this work.

I would like to thank Alexander Bogicevic, Nicolas Lorente, and Yashar Yourd-shayan, whose help and support has been very important to me in many ways.

I also got a number of valuable opportunities to learn from Göran Wahnström,Michael Mehl, Anders Carlsson, Sergei Simak, Per Hyldgaard, and Mats Persson.

I thank Henrik Rydberg, Elsebeth Schröder, Carlo Ruberto, Mathias Hedouin,Staffan Ovesson, Karin Carling, Johan Carlsson, Mattias Slabanja, Javier Aizpu-rua, Shiwu Gao, and Natalia Skorodumova for interesting discussions and help withvarious issues.

Thanks to Gustaf Källen, Svetla Chakarova, Eleni Zimbaras, Anders Hellman,Niclas Jacobson, Martin Hassel, Per Sundell, and Fredrik Olsson for sharing theirgood mood and positive attitude.

In various computer problems I got much of valuable help and support fromAndy Polyakov, Anna Stedt, and other people from Physics Computer at Chalmers,as well as from Lars Hansen and Asbjorn Christensen at Denmark Technical Uni-versity.

I am indebted much to Margaretha Lövgren, who helped me with finding thisPhD opportunity, with the application process, and with various administrative is-sues afterwards. In connection to other administrative questions I would also like tothank Ing-Britt Bengtson and Camilla Eriksson.

Many thanks to Sergei Simak, Mikael Christensen, and Behrooz Razaznejad forbeing very good friends, for support and sincerity, for being there, as well as formany enjoyable moments of talking about life and science.

Finally, words are not enough to thank my family, my wife Yana and our daugh-ter Evgenia, for all their love, support, patience, understanding, and self-sacrificethrough all these years. An additional thank you to Evgenia, who is now twentymonths old, for teaching me many important life lessons.

Sergey DudiyGoteborg, May 2002

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