Microscopic Theory of Wetting and Adhesion in Metal-Carbonitride Systems

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

SERGEY DUDIY

Department of Applied Physics CHALMERS UNIVERSITY OF TECHNOLOGY GOTEBORG¨ UNIVERSITY G¨oteborg, Sweden 2002 Microscopic Theory of Wetting and Adhesion in Metal-Carbonitride Systems SERGEY DUDIY ISBN 91-628-5296-5 Applied Physics Report 2002-45

c Sergey Dudiy, 2002

Chalmers University of Technology G¨oteborg University SE-412 96 G¨oteborg Sweden Telephone +46–(0)31–772 1000

Chalmersbibliotekets reproservice Göteborg, Sweden 2002 Microscopic Theory of Wetting and Adhesion in Metal-Carbonitride Systems SERGEY DUDIY Department of Applied Physics Chalmers University of Technology Göteborg University ABSTRACT Joints between metals and ceramics are increasingly important in the manufacturing of many high technology products, from microelectronic devices to cutting tools. Wetting of ceramics by metals is the driving force of metal-ceramic joining processes, such as brazing and sintering of WC-Co cemented carbides and TiC-Co cermets. Experimental studies suggest that wetting in metal-ceramic systems is most sensitive to microscopic factors, like local chemical composition at interfaces. This thesis is a theoretical study of the key microscopic mechanisms behind the wetting and adhesion, at the level of interatomic interactions. The ceramic materials considered are transition metal carbides and nitrides. The theoretical analysis is based on the results of first-principles density-functional calculations for a broad variety of model interface systems, using the plane-wave pseudopotential method. To deal with the problem of disordered interface structure, an approach based on comparative analysis of high-symmetry model systems is proposed. It is demonstrated that the dominating mechanism of the Co/Ti(C,N) interface adhesion is a strong Co-C(N) bond. The number of those bonds is determined by an interplay of the interface incoherence and the structure relaxation effects. The particular strength of the Co-C bond is explained in terms of interface-induced features 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 available data from wetting experiments with liquid Co on TiC surface. It is found that the Co ferromagnetism gives a significant change of the Co/TiC adhesion strength and interface energy, which is expected to be important during the solid-state sintering stage of the hardmetal manufacturing process. This effect can be adequately described within the Stoner model of itinerant ferromagnetism. The known fact of better wetting in WC-Co systems than in TiC-Co ones is confirmed and explained in terms of a larger contribution of the metal-metal Co-W bonding at Co/WC interfaces. The large scattering of the experimental wetting data for Cu and Ag on TiC and TiN is interpreted in terms of the different relative contributions of the elementary local atomic coordinations at the metal/Ti(C,N)(001) interfaces. Wetting is shown to be improved by C(N) vacancies and Ti segregation in the melt, in agreement with experimentally observed wettability improvements for hypostoichiometric carbides. The suggested simple microscopic picture of wetting in terms of different chemical bonds across the interface is also applied to the analysis wetting trends for Cu on HfC, ZrC, TaC, NbC, and VC. Keywords: metal-ceramic interfaces; structure; bonding; wetting; adhesion; interatomic interactions; total energy and electronic structure calculations; density functional theory; carbides and nitrides; cermets; composites; hardmetals; sintering; 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 TiC S.V. Dudiy, J. Hartford, and B.I. Lundqvist Phys.Rev.Lett.85, 1898 (2000)

II First-principles density-functional study of metal-carbonitride interface adhesion: Co/TiC(001) and Co/TiN(001) S.V. Dudiy and B.I. Lundqvist Physical Review B 64, 045403 (2001)

III Effects of Co magnetism on Co/TiC(001) interface adhesion: A ﬁrst-principles study S.V. Dudiy Surface Science 497, 171 (2002)

IV First-principles simulations of metal-ceramic interface adhesion: Co/WC versus Co/TiC Mikael Christensen, Sergey Dudiy, and G¨oran Wahnstr¨om Physical Review B 65, 045408 (2002)

V Wetting of TiC and TiN by Metals S.V. Dudiy and B.I. Lundqvist Applied 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 original drafts of the articles. In Paper IV, I have contributed to the setup of the calculations and article writing. Scientiﬁc publications not included in this thesis:

Density-functional bridge between surfaces and interfaces B.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. Schr¨oder, S.I. Simak, G. Wahnstr¨om and Y. Yourdshahyan Surface Science 493, 253-270 (2001)

Bridging between Micro- and Macroscales of Materials by Mesoscopic Models, Invited Paper B.I. Lundqvist, A. Bogicevic, S. Dudiy, P. Hyldgaard, S. Ovesson, C. Ruberto, E. Schr¨oder, and G. Wahnstr¨om Computational Materials Science, in print (2002)

Frequency dependence of the admittance of a quantum point contact I.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V.Dudiy Physical Review B 58, 9894-9906 (1998)

Modeling AC electronic transport through a two-dimensional quantum point contact I.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V.Dudiy Microelectronic Engineering 47, 357-359 (1999)

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

A.c. transport and collective excitations in a quantum point contact I.E. Aronov, N.N. Beletskii, G.P. Berman, D.K. Campbell, G.D. Doolen, S.V.Dudiy and R. Mainieri Semiconductor Science and Technology 13 , A104-A106 (1998)

Wigner function description of a.c. transport through a two-dimensional quantum point contact I.E. Aronov, G.P. Berman, D.K. Campbell, S.V. Dudiy Journal of Physics: Condensed Matter 9, 5089-5103 (1997) We always attract into our lives whatever we think about most, believe in most strongly, expect on the deepest level, and imagine most vividly.

– Shakti Gawain

Contents

1 Introduction 1

2 Motivation: Sintering and Brazing Technologies 5 2.1 Cemented Carbides and Cermets as High Performance Tool Materials 5 2.1.1 Hardmetal Sintering Process and Role of Wetting ...... 8 2.2 Brazing as Important Joining Technique ...... 9 2.2.1 Active-Metal Brazing of Ceramics in Overcoming Wettabil- ityChallenge...... 10 2.3 Wetting Experiments with Drops of Molten Metals on Transition Metal Carbide and Nitride Surfaces ...... 11 2.3.1 Wetting of TiC and WC by Co ...... 12 2.3.2 Wetting by Noble Metals ...... 13 2.3.3 ConcludingRemarks...... 14

3 Fundamental Framework for Materials Modeling: Density Functional Theory 15 3.1DensityFunctionalTheoryOverview...... 16 3.1.1 HohenbergandKohnTheorems...... 16 3.1.2 Kohn-Sham Equations ...... 17 3.1.3 AdiabaticConnectionFormula...... 18 3.2 Exchange and Correlation Approximations ...... 19 3.3NoteonApplicationstoSolids...... 21

4 Computational Method: Technical Aspects of Solving DFT Equations 23 4.1PlaneWavesasConvenientBasisSet...... 24 4.2 Pseudopotentials ...... 25 4.2.1 Key Steps in Pseudopotential Construction ...... 26 4.2.2 Essential Aspects of Pseudopotential Transferability . . . . 27 4.2.3 Ultrasoft Pseudopotentials for Efﬁcient Treatment of Tran- sition Metal and First Row Elements ...... 29

5 Transition Metal Carbides and Nitrides 31

ix Contents

5.1CrystalStructureandStoichiometry...... 31 5.2 Electronic Structure and Chemical Bonding ...... 34 5.2.1 Metal-C(N) Bonds ...... 34 5.2.2 Metal-Metal Bonds ...... 35 5.2.3 Bonding Trends and Population of Bonding and Antibond- ingStates...... 35 5.3 Free Surfaces ...... 36

6 Relevant Interface Thermodynamics Background 39 6.1 Deﬁnition of Interface Free Energy ...... 39 6.2 Thermodynamics of Wetting: Contact Angle and Work of Adhesion 42 6.3 Ideal Work Of Separation as Measure of Interface Adhesion Strength 43

7 Goals and Principles of Interface Geometry Modeling 45 7.1 Basic Deﬁnitions and Background: Geometrical Degrees of Freedom 46 7.1.1 Macroscopic Degrees of Freedom ...... 46 7.1.2 MicroscopicDegreesofFreedom...... 46 7.1.3 Interface Geometry Control in Experiment ...... 47 7.2ChoiceofInterfaceGeometryModels...... 48 7.2.1 Scenarios of Theory-Experiment Interaction ...... 48 7.2.2 Should We Search for Best Structure? ...... 50 7.2.3 Focus on Development of Theoretical Models of Wetting andAdhesion...... 51 7.2.4 Simpliﬁed Description of Bulk Phases ...... 53 7.2.5 ConcludingRemarks...... 54

8 Microscopic Interactions at Metal-Ceramic Interfaces 57 8.1 Dispersion Forces and Carbide Wetting Trends ...... 58 8.2ImageInteractionModel...... 60 8.3 Chemical Bonds across Interface ...... 61 8.4 Metal-C(N) Bonds across Interface as Opposed to those in Bulk Car- bidesandNitrides...... 62

9 Conclusions 65 9.1 Qualitative Microscopic Picture of Wetting and Adhesion ...... 65 9.2 Interpretations of Wetting Experiments ...... 66 9.3Outlook...... 69

Acknowledgements 71

Bibliography 73

x CHAPTER 1

Introduction

It is hard to find two classes of materials that are more dissimilar than metals and ceramics. Metals are typically tough. They are ductile rather than hard. They are quite unstable chemically and thermally. In particular, metals can relatively easily be affected by corrosion and chemical attacks, and they tend to expand or shrink noticeably with temperature changes. Metals are good conductors of heat and elec- tricity. Ceramics are just the opposite. Most of them, like many oxides, carbides, nitrides, and borides, are hard and wear resistant, though brittle. They can easily stand high 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, industrial cutting tools, or medical implants. The combination of properties of metals and ceramics within one device is crucial for such applications to work. To make metals and ceramics work together, what one needs before anything else is a way to join them. And this is where the dissimilarity of metals and ceramics shows itself again, but this time as a problem rather than a solution. Making reliable metal-ceramic joints is a well-known challenge, which often requires quite advanced metal-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 materials, or solving various performance problems with existing materials. It is broadly recognized that such development should be based on solid scientific foundations, on understanding of scientific principles involved in the joining processes. Without such an understanding, attempts to solve various performance and production problems tend to degenerate easily from empirical adjustment to random tinkering with process parameters. Motivated by those technology development needs, the present thesis deals with fundamental scientific principles behind metal-ceramic joining

1 1 Introduction processes.

METALS CERAMICS

Dissimilarity of properties problems

benefits

Applications Joining techniques cutting tools wetting, adhesion aerospace composites microelectronics medical implants .....

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 sintering and brazing. Most of those techniques are based on creating stable chemical interfaces between metal and ceramic components. Such joining processes are controlled by the conditions of wetting and adhesion. The first condition is that the metal should get in close contact with the ceramic surface. For this purpose the metal is melted and then allowed to flow freely over the ceramic surface. The liquid metal is expected to spread over the surface and fill all the narrow gaps between the ceramic components. Wetting is what makes this spreading happen. Non-wetting would mean that the liquid prefers to stay in drops, minimizing the area of its contact with the surface. In practice, for successful joining, e.g., via brazing or sintering, wetting should be good enough to fill the gaps completely, without any interface voids that degrade mechanical properties. To a large 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 still be strong bonding forces, that is, a strong adhesion that keeps the solid metal and ceramic parts together. The strength of the adhesion now determines the quality of the metal-ceramic joints, in particular, their mechanical properties. The present thesis aims to contribute to the scientific understanding of wetting and adhesion in metal-ceramic systems, in particular with ceramic carbides and nitrides, by exploring the microscopic origins of the adhesive forces at metal-carbide and metal-nitride interfaces.

2 Investigation of microscopic mechanisms of metal-ceramic wetting and adhesion has for long been a challenge for both experiment and theory. In particular, under the conditions of realistic wetting experiments it is very problematic to resolve the microscopic processes within a few atomic layers of the liquid-solid metal-ceramic interface. The lack of such experimental information is also a significant obstacle for developing theoretical models. The existing theoretical models of wetting in metal-ceramic systems are mostly phenomenological. At the same time, the recent wetting experiments point at in- adequacies of such phenomenological models. In order to understand experimental wetting behaviors one really needs to go to the microscopic level, they suggest. This is the level of interatomic chemical bonds across the interface. Such a microscopic theoretical understanding of wetting and adhesion is the goal of the research in this thesis. 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 sintering and brazing, which have been an important motivation for the research in this thesis. Chapter 3 summarizes the fundamental methodological framework for the theoretical simulations of materials systems, in particular, the density functional theory (DFT). Chapter 4 reviews important technical issues involved in solving the DFT equations, with main attention to the plane-wave pseudopotential method, which is used in first-principles calculations in the thesis. Chapter 5 collects relevant introductory information on transition metal carbides and nitrides, which is an important background for the simulation setup and the discussions of the results in the appended research papers. Chapter 6 defines the main quantities used in the thermodynamic description of interface systems and the computational experiments in the thesis. It aims to clarify the relations between the quantities measured in wetting experiments and those calculated in first-principles simulations. Chapter 7 discusses different aspects of making physically meaningful choices of the model interface structures in first-principles simulations of metal-ceramic systems. It also introduces a number of common definitions that are used in the interface geometry description. Chapter 8 describes various types of microscopic interactions across metal- ceramic interfaces, including a particular kind of metal-carbon covalent bond, an important finding of the research in the appended papers. Chapter 9 summarizes the progress in understanding of wetting and adhesion in metal-ceramic systems made in the cause of the present research.In addition, it contains examples of how experimental wetting behaviors can be understood within simple 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 Brazing Technologies

We are continuously faced by great opportunities brilliantly disguised as insoluble problems.

– 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 every industry, as metal-cutting inserts or sharp ends of the drills, cutting tools for coal or rocks, knives that slice paper or magnetic tape, etc. Such great success of the hardmetals is due to their outstanding properties, especially to their very high hardness and wear resistance, in combination with a reasonable price. Any further advance in the hardmetal technology can have a signiﬁcant impact on the efﬁciency of many industrial processes and open new areas of applications. The unique properties of the hardmetals are essentially due to the way they are made.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 of those components (Fig. 2.1). With such a combination of the constituting materials it appears to be possible to superimpose the positive properties of the carbonitrides and metals, while suppressing the negative ones. In particular, from the ceramic carbonitrides the hardmetals inherit the high hardness and wear resistance, while the metallic 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 material will combine the good properties of the components. On the contrary, it is quite easy to produce something that is worse than any of the components in its pure form. It is therefore no surprise that the performance of the hardmetals is very sensitive to the details 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 Schröter in the early twenties in Germany. They were produced by mixing together powders of tungsten carbide (WC) and cobalt, compacting that mixture, and then heating the system above the cobalt melting point. This was a major breakthrough in the hardmetal technology. After some further extensive developments, the WC-Co-based hardmetals, also called the cemented carbides, became the traditional cutting-tool materials. 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 put into the search of new solutions, which would allow higher speed and/or precision of cutting, more severe operating conditions, etc. Significant progress in the cutting performance was achieved by the introduction of coatings,3 i.e. by adding layers of alumina, titanium carbide and nitride and other materials onto the tool working surface. Such coatings improve the tool lifetimes by 5 to 10 times, but still they are more like technical improvements rather than real innovations, and there are many problems waiting to be solved. For instance, one of those problems is the plastic deformation of the tools at high temperatures, which is a serious limitation on the cutting 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 developments is 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 cermets are6 high cutting speed at moderate chip cross-sections, high surface quality of the machined workpiece, high wear resistance and reliability. The main drawback of the 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 adjusted by additions of other components (see Fig. 2.2), and they noticeably outperform the WC-Co-based hardmetals in many special applications, where the high performance of cutting is essential. The weight of the cermets on the tool market is expected to grow, 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 area requires a deeper and more systematic research. In particular, such issues as the metallurgical reactions during the cermet manufacturing, the microscopic processes in cermets, and the dependence of the cermet properties on their composition and microstructure have to be understood at a more fundamental level, which actually involves a very wide spectrum of materials science problems. The research in the present thesis is to a large extent an attempt to approach this spectrum of problems, at least a part of it. This implies that both the motivation and the experimental background of this work are to a large extent related to the cermets. To clarify this relation, the continuation of the present chapter provides a brief outline of the important aspects of the cermet manufacturing.

7 2 Motivation: Sintering and Brazing Technologies

2.1.1 Hardmetal Sintering Process and Role of Wetting

The hardmetal manufacturing is a many-stage process. The starting point is powders of the carbonitrides and of the binder metal. First the powders undergo the milling stage, during which the powder grain size is controlled, and a homogeneous mixture of the metal and carbonitride components is obtained. Then, during the pressing stage, such a mixture is compacted into the so-called green body. The compacted mixture 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 large amount of the empty space (pores) between them. The elimination of the pores is the task of the next stage, the sintering,7 which is the crucial step in the hardmetal technology. In the context of hard metals, sintering can be described as a thermally activated densification of the compacted powder mixtures. The densification during sintering is the result of the mass transport that rearranges the constituents in such a way that the pores are filled. The driving force for this mass transport is the excess of the surface 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 the free-surface area. On the other hand, instead of the free surfaces there are interphase boundaries, which is also more favorable since extra intergrain bonds are created. The main mechanisms of the mass transport during sintering depend on the sintering conditions. At high enough temperatures, but below the binder-metal melting point, one has a situation of the solid state sintering. During the solid state sintering the mass transport is mainly due to diffusion and plastic flow of the materials, and it is relatively slow. Although the solid state sintering should not be neglected, the major densification of the cemented carbides and cermets occurs during the liquid phase sintering, i.e. at temperatures above the metal melting point. The advantages of the liquid phase sintering are the enabled viscous flow of the metal phase and the significantly increased diffusion rates, which allow a faster and more complete penetration of the binder into the pores. In connection with the present work, the most interesting fact about the liquid phase sintering is that its result crucially depends on how good the wetting of the carbonitride grains by the binder phase is. If the wetting is poor then the liquid phase tends to minimize its surface area, pushing the hard grains apart rather than filling the pores between them. If the wetting is good then it becomes more favorable for the liquid phase to maximize its contact area with the grains, which means that the liquid penetrates into the pores, and pulls the grains together. Therefore, high wettability of the carbonitride grains by the metal binder is a necessary condition for successful sintering. The significance of good wetting also follows from the fact that pores left after sintering can act as internal sources of cracks, which noticeably affects the strength of the material. Although the amount of the residual porosity depends on many different factors, wettability is of primary importance, and to minimize the porosity it 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, wetting and ﬁlling of the pores are only one side of sintering. One more important aspect is the compositional rearrangements in the material. In particular, the carbonitride grains partially dissolve into the binder phase and then reprecipitate, which noticeably affects the size, shape and composition of the grains, the properties of the binder phase, and, as a result, the properties of the obtained material. In cermets such 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 surrounded by rims of carbonitrides of heavier metals, such as W, Mo, V, Nb or Ta. Inclusion of those processes in the analysis of this thesis, especially at the ﬁrst-principles level, would make the considered problems practically intractable. On the other hand, having a physical picture of metal-carbonitride interactions, some of such issues can be approached in future studies.

2.2 Brazing as Important Joining Technique

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

Brazing filler metal

Base material A Base material B Metallurgical bonding at interfaces

Figure 2.3: A schematic illustration of a brazed joint.

In brazing, the ﬁller metal should have the melting temperature below the melting points of the materials being joined, but above 450oC. If the same process is used below 450oC it is referred to as soldering. If such a process involves partial melting of the base components it is called welding. The ﬁller metals used in brazing are of higher 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 of overheating of the base components.

9 2 Motivation: Sintering and Brazing Technologies

There is a number of important advantages of brazing over many other joining techniques. Brazing is quite well suited for joining dissimilar materials. It is relatively 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 must adhere to the base component surfaces strongly enough so that to flow readily over those surfaces and fill the narrow gaps between the components. The wetting should be good enough for the filling of the gaps to be very complete. Any voids left at the interface would significantly degrade the mechanical properties of the joint. Good wetting is the first important criterion when choosing the material of the braze to use in a particular brazing applications. Another important requirement is the ductility of the braze metal, its ability to accommodate the stresses caused by temperature changes and the thermal expansion mismatch in the joined materials. The braze should be tough enough to create a strong joint. Depending on the specific application, the corrosion resistance of the braze metal may also be important. Most of the commercially available brazes are based on Al, Ag, Cu, Au, and Ni, which are 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 used in 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 types of 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 very problematic.9–14 The key difficulty in brazing of ceramics is that most of the ceramics are unwetted by the commercial alloys already developed for joining a wide variety of metallic component materials. There are two main approaches to overcome this wetting problem. The first approach, which is more traditional, uses pre-metallization, i.e. pre-coating of materials to be connected by conventional techniques, such as evaporation of metal in a vacuum furnace. The pre-metallized ceramics 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, and it is quite cumbersome. The strength of the joint is very sensitive to the quality of surface pre-treatments, and to the effectiveness of the fluxing agents used in the process. In brazing of ceramics, an important alternative to brazing through pre-metallization is 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 Cu and/or Ag with some small addition of Ti. The role of the active metal is to react with the ceramic, creating a layer of a reaction product that is more wettable by the

10 2.3 Wetting Experiments with Drops of Molten Metals on Transition Metal Carbide and Nitride 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 carbide and nitride ceramics, as well as on many oxides, borides, and so on. When using Ti-containing brazes, the reaction-product ceramic is typically TiC, TiN, or TiO on ceramic carbides, nitrides, or oxides, respectively. It is interesting to note that many chemically reactive molten metal alloys do not wet ceramic products because their reaction products are typically more stable than the original ceramics and hence, generally, less wettable. Fortunately, there are exceptions 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 wide

compositional 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 choice of the braze alloy composition one can create a well wettable interface layer of substoichiometric Ti compounds. The performance of the Cu- and Ag-based Ti-containing brazes depends on the relative concentrations of Cu and Ag in the alloy.9 While Cu-Ti alloys wet well a wide range of ceramics, they have found few applications as brazes, because they are stiff and, more importantly, result in reaction product layers that are thick and fragile. 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 Ag at a brazing temperature of 1000oC is only 7 atom%, as compared to 60 atom% in Cu. Even better characteristics can be achieved if Ag is alloyed with Cu. One of the most commonly used active braze alloys is the Ag-28Cu eutectic alloy, in which Ti solubility is less than 2 atom%. Ag-28Cu-2Ti have been found to wet a wide range of ceramics noticeably better than Cu-Ti alloys.

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

As discussed in the previous sections, the success of sintering and brazing is directly connected to the degree of control over wetting behaviors of liquid metals in contact with 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 the liquid metal on the surface of the corresponding ceramic material and analyzing the equilibrium shape (proﬁle) of the drop on the surface. This procedure is commonly referred to as a sessile drop wetting experiment (Fig. 2.4). What is actually measured in the sessile drop experiments is the contact angle, θ, which is the angle at the liquid front subtended by the liquid surface and the solid-liquid interface. Depending θ o on the value of the contact angle, there is a situation of wetting ( < 90 )ornon-

11 2 Motivation: Sintering and Brazing Technologies

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

θ B A

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

A number of sessile drop wetting experiments have been performed with molten metals on transition metal carbides.15–21 There are much fewer reports on wettability of transition metal nitrides.18 In the experiments the ceramic substrate is typically prepared by hot-pressing a carbide or nitride powder. The experiments are done in a vacuum furnace, with much attention given to the compositions of the materials and the surrounding atmosphere. It should be noted that those experiments do not provide any atomistic level information on the structure of the formed metal-ceramic interfaces. At best, they characterize the interface microstructure using the scanning electron 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 been performed by Ramqvist,15 who also gives a review of earlier work. In the context of hardmetal sintering, it is interesting to note that Ramqvist’s measurements show θ complete wetting for Co on WC, with ' 0, and a somewhat worse wetting for θ o Co on TiC, ' 25 . Those results are used as experimental support for theoretical results of Papers I, II, and IV. Though those experimental results were reported in sixties, there have not been any drastic changes in the sessile drop wetting experiment techniques since that time. Further, to our knowledge, there have not been any experimental measurements that would 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 angle θ o for Ni is ' 24 , which is the same, within experimental errors, as reported in the above-mentioned work of Ramqvist. Due to signiﬁcant similarities of Ni and Co properties, the reproducibility of the Ni/TiC results is a strong support for the reproducibility and reliability of the old Co/TiC, and likely Co/WC, wetting angle data.

12 2.3 Wetting Experiments with Drops of Molten Metals on Transition Metal Carbide and Nitride Surfaces

One more indication of the validity of the above-mentioned experimental results on wetting of TiC and WC by Co is the fact that they agree well with the wetting behaviors during liquid phase sintering of WC-Co cemented carbides and TiC-Co cermets. 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 TiC based 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 filled d-band, and they typically wet carbides worse than, e.g.,CoorNi.15,16 Wetting by the non-reactive noble metals is also more sensitive to the quality (contamination control) of the carbide substrate surface. Much attention in wetting experiments with metal-carbide systems has been given to wetting of TiC by Cu,15,16,18,21 partly due to its high relevance for brazing. Early works (see Ref. [15] and references therein) have reported that non-reactive o metals, like Cu and Ag, do not wet stoichiometric TiC and form high, > 120 , contact angles. At the same time, more recent wetting experiments of Li16 show that by giving special attention to contamination of the TiC surface by oxygen, using a Zr getter furnace tube, one can reach contact angles as low as 54o even for stoichiometric TiC. However, in later experiments of Xiao and Derby,18 the measured contact angle was 120o, even for the extremely low oxygen partial pressure. Those inconsistencies are addressed in a recent experimental work of Froumin and cowork- ers,21 who carefully analyze the presence and effects of the oxygen contamination of the TiC substrate surface. They find that the presence of oxygen on the TiC surface strongly inhibits the interaction between the ceramic and molten Cu, which is the most likely reason for non-wetting behaviors in the above-mentioned previous o studies. With an enhanced oxygen contamination control, a wetting angle of ' 89 is reached.21 The scattered wetting data for Cu and Ag on TiC is also analyzed in Paper V, based on comparisons with first-principles theoretical results. The above described situations with scattered data on the wetting of TiC by Cu is a strong reason to question earlier results15 on wetting of other carbides by Cu, in particular HfC, ZrC, TaC, NbC, and VC. It is interesting to note that those results were showing quite a pronounced trend, with the wetting angle decreasing noticeably along the series of HfC, ZrC, TiC, TaC, NbC, and VC. That trend has been given much attention in theoretical analysis, in particular in the search for a correlation between wettability and different carbide properties, like formation energies15,22 or plasmon frequencies.23,24 However, the range of contact angle variations covered by 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 of those interpretations. This situation raises the question of what is the real wetting trend for Cu on HfC, ZrC, TiC, TaC, NbC, VC, when oxygen contamination is sufficiently low. New experiments and theoretical predictions or interpretations are needed.

13 2 Motivation: Sintering and Brazing Technologies

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

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

14 CHAPTER 3

Fundamental Framework for Materials 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 how well we can describe the quantum mechanical behavior of the electrons in a given materials system. In principle, a complete description of the behavior of electrons in solid is given by the stationary Schr¨odinger equation for the many-electron wave

Ψ ; :::;

function r1 rNe :

! Ne Ne

1 2 1 1

5 + + Ψ = Ψ ;

∑ i Vion ri ∑ E (3.1)

2 2 jri r j

= =

i 1 j 6 i where Vion r is the potential of the ion cores, and E is the energy of the electron system. However, since the many-electron wave function is represented in the product space of the single-electron positions ri, the computational demands to solve equation (3.1) grow exponentially with the number of electrons Ne.Formorethan a few electrons a direct solution of the many-electron Schr¨odinger equation goes far beyond the capabilities of the present-day computers, and there is no way it can be solved for around 1023 electrons of solid. Thus, approximate methods are necessary. Among the approximate methods for the electron-structure calculations, one can distinguish 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 or from first-principles calculations. The calculations based on semiempirical methods are computationally very efficient, but they require great care in order not to go beyond the range of the applicability of the involved approximations and parameters. In contrast to the semiempirical methods, the first-principles or ab initio methods are parameter free, that is as input only the positions and atomic numbers of the ions are required. Although the first-principles calculations are computationally more demanding, they typically provide more reliable results and have much more predictive power. The most commonly used first-principles approach is the Density Functional 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 most successful first-principles electron-structure method in condensed matter physics, and it also gains more and more popularity among chemists. The DFT is based on the observation30 that ground-state properties of an electron system are functionals of the ground-state electron density alone, and for the ground-state total energy such a functional satisfies a variational principle. Moreover, as shown in Ref. [31], the problem of the minimization of the total-energy functional can be reduced to a set of effective single-particle Schrödinger-like equations, with all the many-body effects collected in the so-called exchange-correlation term. Since the DFT forms the basis for the methods used in this thesis, the present chapter 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 of N electrons in an external potential Vext r . According

to the theorems of Hohenberg and Kohn30 there exists such a universal functional

F fn r of the electron density n r that the ground-state density of the system

minimizes the following functional E fn r :

g = + f g; E fn r drVext r n r F n r (3.2)

under the constraint

= ;

drnr N (3.3)

and the minimum of E fn r gives the ground-state total energy,

f f gg :

Etot = min E n r (3.4)

The universality of the functional F fn r means that its form does not depend on N or Vext r .

16 3.1 Density Functional Theory Overview

The Hohenberg and Kohn theorems form the basis of the DFT. They mean that

g if we knew the form of F fn r then we would have a universal and exact method

to calculate the ground-state electron density and the total energy for any electron

g system in any external potential. However, as in reality the form of F fn r is un-

known, the Hohenberg and Kohn theorems by themselves are not enough for any

g practical calculations. Some reasonable approximations for F fn r have to be developed.

3.1.2 Kohn-Sham Equations

g The unknown functional F fn r in Eq. (3.2) includes the contributions from the kinetic energy and the electron-electron interactions, taking into account all possible many-body effects. All those contributions together are quite difﬁcult to analyze or make approximations for. In this context it appears to be very helpful to represent

the total-energy functional in the following form, suggested by Kohn and Sham:31 Z

1 Z

g = f g + Φ + + f g: E fn r T n r drn r r drn r V r E n r (3.5)

0 2 ext xc

g The idea of that representation is to extract from F fn r the meaningful contributions that can be evaluated exactly, separating all the rest into the term that requires

approximations. In particular, in Eq. (3.5) T0 is the kinetic energy functional that a system with density nr would have without electron-electron interactions; the next

Φ term is the classical electrostatic energy, with r being the classical Coulomb

potential for electrons,

0 nr

Φ 0 = ;

r dr (3.6)

j j

r r

g also called the Hartree potential. The last term in Eq. (3.5), Exc fn r , is actually deﬁned by Eq. (3.5) itself. This term is called the exchange-correlation energy functional, and it incorporates all the many-body effects. An important advantage of the representation (3.5) is also that the variational problem of the minimization of the total-energy functional (3.5) can be reformulated in 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

= jψ j ; nr ∑ i r (3.7)

i=1

ψ where i r are solutions of the Schr¨odinger equation for an effective system of N

noninteracting particles,

1 2

+ ψ =εψ : 5 V r r r (3.8) 2 eff

17 3 Fundamental Framework for Materials Modeling:

Density Functional Theory

Here Veff r is the effective single-electron potential deﬁned as

= +Φ + ;

Veff r Vext r r Vxc r (3.9)

where the exchange-correlation potential Vxcr is the variational derivative of the

exchange-correlation functional Exc fn r ,

δExc fn r

Vxc r : (3.10) δnr

In Eq. (3.7), the ﬁrst N solutions ψi of Eq. (3.8) with the lowest eigenvalues εi should

be taken. It is assumed that each ψi r is normalized, Z

ψ j = ; dr j i r 1 (3.11) which together with Eq. (3.7) provides the condition (3.3). If the ground-state density is found, the ground-state total energy of the system can be found using the total-energy functional (3.5) . The Kohn-Sham equations (3.8), together with expression (3.7), actually mean that 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 with each other but move in the effective potential (3.9) instead. A set of single-electron Kohn-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 functional Exc fn r ,

g it is not particularly convenient to use Eq. (3.5) as a deﬁnition of Exc fn r .The matter is that that equation does not reﬂect the structure and the physical nature of the exchange-correlation functional. A more explicit expression for the exchange-correlation functional can be obtained by considering an adiabatic switching-on of the electron-electron interactions,

starting from the noninteracting system. More speciﬁcally, let us consider an ef-

j j

fective system in which we replace the real Coulomb interaction, 1 = r r , with

j j λ λ = λ= r r , where the coupling constant can be varied between 0 an 1. When 0 λ we have a noninteracting system, which can be treated exactly. The case = 1cor- responds to the real system we want to describe. Now let us increase λ adiabatically from 0 to 1, and let us introduce an additional external potential Vλ in such a way

that the electron density is always the same as in the real system. By considering

g that procedure one can derive the following expression for Exc fn r , commonly referred to as the adiabatic connection formula:35–37

The total energy of the effective noninteracting system, ∑N ε , does not have any direct physical i =1 i meaning.

3.2 Exchange and Correlation Approximations Z

1 Z 1

0 0

f g = ; ;

Exc n r drn r dr nxc r r r (3.12)

j 2 jr r where

0 0 0

; : ; = λ ; λ nxc r r r n r d g r r 1 (3.13)

; λ Here g r r ; is the pair correlation function of the system with density n r and

λ 0 j j electron-electron interaction = r r . Expressions (3.12) and (3.13) introduce a simple physical interpretation of the exchange-correlation energy. The exchange-correlation energy originates from the energy of electrostatic interaction (see Eq. (3.12)) between the electron at r and the

exchange-correlation hole it creates, with the density of the exchange-correlation

; hole, nxc r r r , defined by Eq. (3.13). The exchange-correlation hole describes the effects that if there is an electron at r then the probability to find another electron at r0 close to r is reduced due to the Pauli principle (exchange) and the Coulomb electron-electron repulsion (correlation). An important property of the exchange- correlation hole is that its integration should give one electron, as reflected in the

following sum rule:35

0 0

; = : dr nxc r r r 1 (3.14)

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

3.2 Exchange and Correlation Approximations

The simplest approximation for the exchange-correlation functional is the local density 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 point r with the local electron density nr is the same as in the uniform electron gas with the corresponding electron density nr .Thatis,

the exchange-correlation functional takes the form Z

LDA unif

]= ε [ ] ; Exc [n drn r xc n r (3.15)

εunif where xc n is the exchange-correlation energy per particle in the homogeneous electron gas with density n. For the homogeneous electron gas, an accurate depen- 40 dence of εxc on the density n is extracted from quantum Monte Carlo calculations, and then, to simplify its applications, parametrized in one or another way.41–43

19 3 Fundamental Framework for Materials Modeling: Density Functional Theory

Although by construction the LDA approximation is expected to work only for systems 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-correlation hole. It is also important that the LDA exchange-correlation hole, taken from a real physical system, meets many important conditions, like Eq. (3.14), that should be satisfied 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 their compounds,46–49 and can give a wrong ground-state structure, like for iron.50 This fact reduces significantly the reliability of the LSDA description of bonding in the interface systems that are of interest in the present work. In addition, it should be mentioned that for most transition metals in the 3d-series LDA also underestimates the equilibrium volumes and overestimates the bulk moduli. A natural way to improve on the LDA (LSDA) is to include density gradients in the approximate exchange-correlation functional. The most straightforward procedure, suggested in Ref. [31], is to add to the LDA functional one more non-zero term in the gradient expansion of the exchange-correlation functional (gradient expansion approximation 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 important properties of the exact exchange correlation, like the sum rule (3.14), while the LDA functional does. To overcome these problems, the generalized gradient approximation (GGA) has been introduced.51 In GGA the gradient expansion is

replaced by generalized functionals of density gradients, Z

GGA GGA

]= ε [ ; 5 ] ; Exc [n drn r xc n n (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 of

εGGA ; 5 ] xc [n n , and many different functional forms have been suggested, as overviewed in, 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 components of the second-order expansion for the exchange-correlation hole allow to satisfy the sum rules on the exact hole. The GGA-PW91, or a similar version of Perdew, Burke and Ernzerhof53 (PBE), improve the accuracy of the description of the ground-state properties of many atomic, molecular and solid systems.44,54 And although in the general case some authors44 still recommend to consider both the GGA and LSDA results, the GGA-PW91 has been shown to correct, e.g., the serious deﬁciencies of the LSDA for transition metals, providing quite accurate descriptions of their structural55 and cohesive properties.56 In view of this fact the present work uses GGA-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 systems is the ability to separate the motion of the electrons from the motion of the atomic nuclei. 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 hence the characteristic velocities of the electronic motion are much higher than the ionic velocities. In the adiabatic approximation the electronic structure at any moment of time can be determined assuming that the ions are frozen in their instantaneous positions. That is, the ionic positions are just parameters of the external potential in Eq.(3.8). When the Kohn-Sham equation (3.8) is applied to periodic systems (solids), one can make use of the Bloch theorem,57 which states that the solutions of Eq. (3.8) can

be written in the form

= ;

ψr exp ikr uk r (3.17)

where uk r has the periodicity of the considered system. Substitution of (3.17) into

Eq. (3.8) leads to the equation for uk r ,

1 2

5 + + =ε ; ik V r u r u r (3.18) 2 eff k k k which needs to be solved only within one unit cell, with the periodic boundary con-

ditions. The electron density is then given by Z

= ε ε j j ; nr 2∑ dk f ik F uik r (3.19)

i : B :Z where i is the band index that enumerates different eigenstates, uik, of Eq. (3.18),

with the corresponding eigenvalues εik, the factor 2 accounts for the spin degeneracy, ε

f is the Fermi distribution function (zero temperature),

ε ε =θε ε ; f F F (3.20) and the integration is over the ﬁrst Brillouin zone in the k-space. The position of the Fermi level, εF, should be determined self-consistently from the electron-number

conservation condition,

= ;

drnr Nel (3.21) : U :C where the integration is over one unit cell and Nel is the number of electrons per unit cell.

21 3 Fundamental Framework for Materials Modeling: Density Functional Theory

22 CHAPTER 4

Computational Method: Technical Aspects 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 enormous progress in our ability to treat condensed matter systems quantum mechanically. The original many-body problem is overcome, and all what we need to do is to solve a set 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 computational effort, 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 extent this is due to the great progress in the computer technology, which has made a significant amount of the computational power available at a reasonable cost. However, as the computer power, although rapidly increasing, is always limited, a crucial role is also played by the high efficiency of the modern computational techniques and algorithms. These advances have moved the limit on the complexity of the system that can be treated by the density-functional method to as high as a few hundreds of symmetry-independent atoms. As without many well-developed computational techniques the problems addressed in this thesis would be unapproachable, the present chapter gives a brief overview of those techniques. This overview also explains the meaning of some of the 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 is to represent the Kohn-Sham wave functions as discrete sets of numbers. This is done by expanding the wave functions in some basis set, and truncating such an expansion at some finite number of terms. Then the desired discrete representation is given by the coefficients of the expansion. The choice of basis functions is crucial for the efficiency of the computational method. With the optimal choice, as few expansion terms as possible should be necessary to represent the wave functions with a reasonable accuracy. Not less significant 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 different density-functional computational methods are actually named after the used basis 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 plane waves eiqr. An important condition for an efficient use of the plane wave expansion is the periodicity of the functions that are expanded. Only for periodic functions the spectrum of the required q-values is discrete, being continuous otherwise. In this 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 functions in terms of the functions uk that retain the periodicity of the system. Expanding the functions uk in the plane wave basis set and substituting such an expansion into Eq. (3.17), one can represent the wave functions for each k-point and band index n as

ψ = [ + ] ; nk r ∑cnk G exp i k G r (4.1)

G where the summation is over all the reciprocal-space vectors G,andcnk G are the expansion coefﬁcients. Normally, the contributions of the plane waves with the

+ j = kinetic energies, jk G 2, higher than some finite value are sufficiently small to be neglected. Thus, instead of the full expansion (4.1) with an infinite number of terms, it is enough to use a finite basis set consisting only of the plane waves with the kinetic energy lower than some kinetic energy cutoff Ec,

+ j = < : jk G 2 Ec (4.2)

A plane wave basis set is very convenient in many respects, which explains its wide use in ﬁrst-principles density-functional calculations. Its mathematical simplicity makes it relatively easy to implement on the computer. With this basis the kinetic energy operator has a simple diagonal form, and the transformations between the real and reciprocal space representations can be done very efﬁciently with the modern fast Fourier transform algorithms67 (FFT). The accuracy of the plane wave 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 of the ionic positions, the plane waves provide an unbiased description of the whole unit 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 calculation of the ionic forces. The ionic forces are determined by the derivatives of the total energy with respect to the ionic positions. In principle, being able to compute the total energy, one can calculate such derivatives numerically. In practice, this is computationally very inefficient, since one has to perform a few extra total-energy calculations for displaced atomic configurations. Fortunately, there is a possibility to obtain the ionic forces from a single total-energy calculation, using an analytic expression for the ionic forces68 based on the Hellmann-Feynman theorem.69–71 The fact that the plane waves do not explicitly depend on the ionic positions allows us to use the Hellmann-Feynman theorem directly, without the need to include extra terms (the so-called Pulay forces72) to compensate for the contributions from the derivatives of the basis functions with respect to the ionic positions. Due to the fast calculation of the ionic forces the plane wave method is very efficient for the geometry optimization applications (atomic structure relaxation), which appears to be particularly relevant for the interface systems studied in the present work.

4.2 Pseudopotentials

In DFT implementations based on the plane wave basis set a major problem is the description of the core region of the atom. This is because the wave functions of the core electrons, as well as the wave functions of the valence electrons in the core region, are rapidly oscillating functions of the space coordinates. A reasonably accurate plane wave expansion of such functions would require a prohibitively large number 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 core approximation. This approximation assumes that the inner shell electronic orbitals are frozen in the sense that they do not change when the atom is transferred from one chemical environment to another. Such an assumption is based on the fact that most of the physical properties of solids are mainly determined by the behavior of the valence electrons, while the inner-shell orbitals are very close to those in the free atoms, being practically inactive. Thus, in the frozen core approximation, one has to describe only the changes in the valence-electron orbitals. The frozen core approximation still retains the problematic oscillatory behavior of the wave functions in the core region. To avoid this problem, the pseudopotential approximation for the valence-core interaction is introduced. In this approximation the action of the ion core on the valence electrons is replaced by an effective potential, the so-called pseudopotential. On the other hand, the wave functions of the

25 4 Computational Method: Technical Aspects of Solving DFT Equations valence electrons in the core region are replaced by pseudo wave functions, which are much smoother. In this way the necessary number of plane waves can be reduced to 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 optimally soft,88–90 and, ﬁnally, to ultrasoft91–94 pseudopotentials. In this thesis the focus is on the normconserving and ultrasoft pseudopotential schemes. For simplicity, we start from the normconserving pseudopotentials and then describe the new features introduced by the ultrasoft pseudopotentials, which are exploited in the calculations reported in the appended papers.

4.2.1 Key Steps in Pseudopotential Construction Pseudopotentials are generated by considering a one-atom problem. First the all

electron (AE) radial Kohn-Sham equation,

d l l 1 AE AE

+ φ =εφ ; + V r ε r ε r (4.3) dr2 r2 AE l l or its relativistic or scalar-relativistic generalization,95 is solved to ﬁnd the true

φAE

Kohn-Sham wave functions, lε r , for different atomic orbitals with different an-

; ; ; ;:: ε gular momenta l =0,1,2,3,.. (s p d f ) and energies .InEq.(4.3)VAE r is the screened potential in the all-electron atom, which is determined self-consistently for a given atomic conﬁguration. Then, from the wave functions of the valence orbitals,

φPS the pseudo wave functions, lε r , are constructed in such a way that they are equal to the corresponding true wave functions beyond some chosen cutoff radius, rcl,they are continuously differentiable at least twice at that cutoff radius, and they satisfy the norm-conservation constraint,

r r Z

Z cl cl

PS 2 AE 2

φ = φ : dr lε r dr lε r (4.4) 0 0 Since the pseudo wave functions should be orthogonal only to themselves, not to the core orbitals, they are taken to have no nodes inside rcl, which allows them to be much smoother than the true wave functions. At the same time, the cutoff radius

AE PS

φ φ rcl should be sufﬁciently large, so that lε r , and, hence, lε r , have no nodes outside rcl. After the pseudo wave functions are constructed, the pseudopotential can be determined by an inversion of the radial Kohn-Sham equation with the given

l and ε:

d l l 1 PS PS PS

+ + φ =εφ : V ε r ε r (4.5) dr2 r2 l l

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

26 4.2 Pseudopotentials

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

PS PS

j > < j; VSL = ∑ lm Vl r lm (4.6)

l ;m

> where jlm are the spherical harmonics, and Vl r are pseudopotentials obtained for each angular momentum channel l. Due to this semilocal form the computational PS effort of applying the VSL operator to the wave function in the basis of N plane waves grows as N2. This effort can be reduced considerably, to being proportional 2 PS to N instead of N , by transforming the semilocal operator VSL into a truly nonlocal

separable form, as suggested by Kleinman and Bylander:85

χ >< χ j

PS j lmε lmε

; +

V = Vloc r ∑ PS (4.7)

χ jφ > < ε l ;m lm lmε where

PS PS

χ >= jφ >; j lmε VSL Vloc lmε (4.8) and

PS 1φPS φ >= j >: j r lm (4.9)

lmε r lε Here Vloc r is the local potential, which, in principle, can be arbitrary. The choice of the 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 detailed discussion, and Refs. [99,100] for additional useful comments). Another important point is how to choose the pseudo wave functions and pseudopotential to optimize the smoothness of the pseudo potential, i.e. to minimize the necessary size of the plane wave basis set. The smoothness optimization techniques for normconserving 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 a pseudopotential. The transferability describes how accurately the pseudopotential can reproduce the quantum mechanical behavior of an all-electron atom when such an atom is placed in different chemical environments. Simple, although computationally costly, ways to improve the transferability are to decrease the cutoff radii rcl

27 4 Computational Method: Technical Aspects of Solving DFT Equations or to include the semicore orbitals into the valence states, i.e. to make the pseudopotential calculations closer to the all-electron ones. A brute-force approach to assessment of the transferability is to compare the results of the pseudopotential and all-electron calculations for a wide range of different systems and properties. Although useful to some extent, such an approach in full length is not very convenient, and it reduces the predictive power of the pseudopotential calculations. Thus it is important to be able to judge and control the quality of a given pseudopotential already at the pseudopotential generation stage. One commonly used characteristics of the pseudopotential transferability can be understood in terms of the following scattering problem. In some molecular or condensed 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 reproduce the valence wave functions outside the atomic spheres. The wave function outside the 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 on the surface of the spheres. Since the construction of the pseudopotential aims to provide 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 equation

φ = =φ in the considered problem can be given by the logarithmic derivatives, d r dr 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 energy region the logarithmic derivatives of the wave-functions on the chosen spheres around the all-electron atom can be reproduced with a given pseudopotential. In the modern pseudopotential schemes the behavior of the logarithmic derivatives (scattering properties) can be systematically improved by inclusion of more than one reference energy ε for the chosen angular momentum channels l in the pseudopotential construction.86,91 As pointed out in Refs. [101,102], the logarithmic derivatives do not give a complete measure of the transferability, especially when there is a signiﬁcant electron transfer between the components of a given chemical system. Indeed, the transferability criterion based on logarithmic derivatives works only under the condition that the changes in the chemical environment do not change the effective potential signiﬁcantly inside the core radius.101 As shown in Ref. [101], this condition can be met with some reasonably small core radius. In Ref. [102] this problem is addressed in terms of the so-called chemical hardness, which determines the changes in the atomic eigenvalues with the change of the valence state occupancies. With an improved description of the chemical hardness property, a major reduction of the errors of the pseudopotential approximation can be achieved.102 A further analysis and development of the pseudopotential transferability criteria related to the chemical hardness property are presented recently in Refs. [103,104] One more important factor in the pseudopotential transferability is related to the nonlinear character of the exchange-correlation potential. The nonlinear dependence of the exchange-correlation functional on density means that the exchange correla-

28 4.2 Pseudopotentials tion energy of an atom is not just a sum of the contributions of the core and valence electrons, and, hence, exclusion of the core electron density introduces a nonlinear error in the dependence of the exchange correlation on the valence density. This error can be noticeable when there is a signiﬁcant overlap between the densities of the core and valence states, like it is between the 3d valence and 3p semicore states of the 3d transition metals. To minimize such an error, the nonlinear core correction of Louie, Froyen, and Cohen105 can be used. The idea of that correction is that in the calculations of the exchange-correlation energy, instead of the valence density alone, the sum of the valence and the (frozen) core densities is used. Since the true core density has quite rapid oscillations, whose inclusion would require an increased number of plane waves, the core density is pseudized within some chosen radius (the so-called core correction radius). The nonlinear core correction is essential for a correct description of the magnetic systems, which is very relevant for Paper III of this thesis.

4.2.3 Ultrasoft Pseudopotentials for Efficient Treatment of Tran- sition Metal and First Row Elements With the above mentioned developments, the normconserving pseudopotentials are efficiently used for many systems, with quite a good control of the pseudopotential softness and transferability. However, there are problematic cases, like first-row elements and transition metals, for which the use of normconserving pseudopotentials requires significantly higher computational effort (plane wave cutoff) than for other elements. 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 transition metals and the 4 f -orbitals for the 4 f rare earths. To satisfy the norm-conservation constraint, the pseudo wave functions for such orbitals have to have quite a large amplitude in the core region, which does not allow to make them much smoother than the true wave functions. This problem is overcome in the Vanderbilt ultrasoft pseudopotential scheme. The main feature of the ultrasoft pseudopotentials is the released normconservation constraint, which gives much more freedom in softening the pseudo wave functions. The reasons why the normconservation constraint (4.4) is important in the 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 of the core region, and gives the right amplitude of the electron density outside the core. Moreover, from the point of view of the pseudopotential transferability, it is important that with the normconserving pseudopotentials not only the logarithmic derivatives (scattering properties) at the reference energies, but also their small variations around those reference energies, are equal to the corresponding all-electron values.

29 4 Computational Method: Technical Aspects of Solving DFT Equations

The ultrasoft pseudopotentials retain the useful properties of the normconserving pseudopotentials, and also add some extra improvements. To avoid the difﬁculties with a non-Hermitian Hamiltonian, the standard eigenvalue problem (4.5) is trans-

formed into a generalized eigenvalue problem

d l l 1 PS PS PS

+ + φ =ε φ ; V˜ r S r (4.10) dr2 r2 lε lε in which a non-Hermitian pseudopotential V PS is replaced by a Hermitian one V˜ PS at a price of introducing the so-called overlap operator S (see any of Refs. [91–94] for detailed definitions). To have an adequate description of the electron density, the expression for the charge density is also generalized by adding to a standard sum of squared absolute values of the wave functions an extra term correcting for the difference between the charge densities of the true and the pseudo orbitals, the so-called augmentation-charge term. For the reference states, the augmentation charge makes the density equal to the true valence density not only outside the core region, but also inside the core, down to the so-called inner core radius. With these generalizations, the ultrasoft pseudopotentials also give the correct logarithmic derivatives for the first order deviations from the reference energies, like in the normconserving case. In addition, the inclusion of extra reference energies, together with the improved description of the charge density, allow to increase the core radius without significant loss in the pseudopotential transferability. All these features make it possible to describe even the first row elements or 3d-transition metals with reasonable values of 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 pseudopotentials, in particular DACAPO106 and VASP65,66,107–109 codes. Brief outlines of further 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 and Nitrides

Microscopic pictures of metal-carbide and metal-nitride wetting and adhesion in appended Papers I - V are connected in many ways to microscopic understanding of transition-metal carbides and nitrides. In particular, the setup of computer simulations and the discussions of the obtained results for the studied interface systems often use what is known about the atomic and electronic structure of the transition metal carbides and nitrides (TMCN) and their surfaces. This Chapter reviews the known behaviors of the TMCN atomic and electronic structures,22,110–114 with the focus 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, atom types, 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,is

referred to as stoichiometric. A phase with a smaller concentration of Y atoms than :

of Me ones, MeYx,wherex < 1 0, is called substoichiometric or hypostoichiometric. : Phases with larger concentration of Y atoms, MeYx, x > 1 0, are very uncommon, if at all existing, and are not of interest here. Transition metal carbides and nitrides most often crystallize in the B1 structure (NaCl, see Fig. 5.1). For example, this structure describes TiC, ZrC, HfC, VC, NbC, TaC22,110 and TiN, ZrN, VN, NbN, TaN.111 Among the practically important carbides, the main deviations from the NaCl structure are present for WC, Mo2C,

31 5 Transition Metal Carbides and Nitrides

1 14 and Cr3C2, which have a hexagonal D3h, an orthorhombic D2h, and orthorhombic 12 22 D2h Pbnm structures, respectively. Here we discuss mainly MeY compounds with the cubic-symmetry NaCl crystal structure, which are often called cubic carbides and nitrides.

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

The NaCl-structured transition metal carbides and nitrides are often referred to as 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 behind this picture is that the effective radii of the Y atoms are around two times smaller than the Me ones. For example, the covalent radii of Ti and C are 1.32 and 0.77 A,˚ respectively.115 The insertion of the Y atoms requires only a relatively small expansion of the corresponding Y-free Me lattice. For example, the equilibrium lattice constant of pure fcc Ti calculated in Papers II and V is 4.10 – 4.11 A,˚ as compared to 4.33 – 4.34 A˚ for TiC (see Table I in Paper V.). That is, the expansion of 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 retained over 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 a change in the type of the crystal structure. Within that range of Y-to-Me ratios, the

substoichiometric MeYx compound can be viewed as stoichiometric MeY1 :0 one in which a certain number of Y atoms are replaced by vacancies. Those Y vacancies are typically randomly distributed over different Y sublattice positions, although there are examples of ordered MeYx phases at some speciﬁc values of x, e.g.,forVCand NbC.110 Under realistic experimental conditions, many of the cubic carbides and nitrides tend to be substoichiometric. For example, TiC most often contains at least a few percent 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.

5.1 Crystal Structure and Stoichiometry

: : x ' 0 96 0 97, and a pure C phase. The stoichiometric MeY compounds then should be considered a limiting case of substoichiometric MeYx ones, when x approaches unity, an approximation to the realistic experimental situations. A direct theoretical modeling of substoichiometric MeY compounds is a considerably more complex task than studying simple-structured stoichiometric compounds. This is the reason why theoretical simulations of TMCN materials use mainly stoichiometric compositions. This is also true for the investigation in Papers I - V, except a part of Paper V, where systems with C and N vacancies are also considered. As noted in Ref. [112], ﬁrst-principles theoretical studies of MeY surface electronic properties that assume stoichiometric compositions still explain quite well the majority of the behaviors observed in experiment. Finally, the effects of substoichiometry on bulk transition metal carbide electronic structure and phase stability are investigated in ﬁrst-principles theoretical 116–119 studies, including TiCx in Refs. [117, 119]. The situation with vacancies in interface systems is discussed in Paper V.

Figure 5.2: Phase diagram for Ti-C system.

33 5 Transition Metal Carbides and Nitrides

5.2 Electronic Structure and Chemical Bonding

Transition metal carbides and nitrides show quite an unusual combination of physical and chemical properties.22,110–113 They are extraordinarily hard. They have very high melting points and good corrosion resistance. At the same time they also have good electrical conductivity of metallic or semimetallic type. The origins of such combination of properties should ultimately be found in the type and behaviors of chemical bonds in those MeY compounds. This situation has motivated several theoretical and experimental studies of the nature of bonding in transition metal carbides and nitrides. There has been a signiﬁcant progress in understanding of the MeY cohesion during last two decades,113,114,120,121 mainly due to extensive electron structure calculations combined with analyses of experimental thermodynamical data and photoemission spectra. This section summarizes the important conclusions and insights from those studies, relevant for investigation in Papers I - V. Chemical bonding in transition metal carbides and nitrides is a complex mixture of covalent, metallic, and ionic components. All three types of bonding mainly involve 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 Y atoms. The covalent bonding is a result of hybridization of Y-2p and Me-d atomic orbitals. The essential conditions for a strong covalent bonding113 are strong over- laps between participating orbitals, as well as comparable size and energies of those orbitals. Those conditions are met quite well in TiC,113,120 andtoasimilardegree in other carbides and nitrides.113 A schematic illustration of the main participating Me-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, (ﬁlled circles) 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 the neighboring 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 antibonding states. The formation of Me-Y p-d bonding and antibonding states and their ﬁlling with electrons result in a redistribution of the electron charge density in space, which impliessomedegreeofchargetransferbetweentheMeandYatomicspheres.This charge transfer is responsible for the ionic contribution to the Me-Y bonding, which can 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. For example, 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 space between the Ti and C atomic spheres. As noted in Ref. [110], the electron transfer from metal to carbon found in electron structure calculations113,120 is consistent with experiments, in particular with electron 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 Y core electrons in MeY compared with those in the elements (core level shifts). This shift originates from a redistribution of electronic charge in the direction of the nonmetal atom.

5.2.2 Metal-Metal Bonds The metallic component of the MeY cohesion comes from the interaction of the d- orbitals of the neighbor Me atoms in the Me sublattice. Similarly to the covalent Me-Y bonding, Me-Me bonding is due to hybridization of the overlapping atomic orbitals (Fig. 5.3(b)). To get a simple insight into the Me-Me bonding one can start from a picture of metallic d-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 lattice to the size it has in the MeY compound. Such an expansion increases the distances between the nearest Me neighbors, decreasing the overlap between the atomic d- orbitals, and hence leading to weaker bonding. The energy cost of such an expansion is analyzed in Ref. [114]. An even more important modification of the Me-Me bonding is due to formation of the Me-Y bonding and antibonding states. This process significantly changes the Me-d component of the electronic local density of states (LDOS). The Me-Y antibonding states are localized mainly around the Me atoms (see, e.g.,Fig.9(c)in Paper 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 understanding of the MeY bonding trends when Me varies along the 3d ,4d,or5d transition metal series.114 Within the same transition metal row, the MeY compounds are characterized by a common LDOS pattern (Fig. 5.4) of the bonding and antibonding

35 5 Transition Metal Carbides and Nitrides states. Yet, the position of the Fermi level, which determines the ﬁlling of those states, changes from metal to metal, being ﬁxed by the number of valence electrons per Me-Y pair.

TiC TiN Antibonding C(N)−2p Metal−d Bonding Projected LDOS Energy CoC CoN

Figure 5.4: A schematic illustration of the characteristic pattern of bonding and antibonding states in the electronic local density of states (LDOS) of bulk transition metal carbides and nitrides. The LDOS projections on metal-d and C(N)-p orbitals are shown. The vertical lines 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 the trends in the strength of bonding, in particular, in the cohesive energy and enthalpy of formation.114 For example, TiC has the strongest bonding in the 3d-series, as the Fermi level lies between the energy intervals of the bonding and antibonding states (Fig. 5.4). The extra electrons in V, Cr, ..., Ni populate antibonding states, and the bonding becomes weaker. For CoC, CoN, NiC, or NiN, the antibonding states are practically totally ﬁlled, and those compounds are unstable in experiment (negative enthalpies of formation, see Ref. [114]). As noted in Ref. [114], there are no strict criteria for a clear separation of the energy 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 bonding energies and its correlation with the position of the Fermi level with respect to the LDOS pattern (Fig. 5.4).

5.3 Free Surfaces

There is a number of recent experimental and theoretical studies of structural and electronic properties of various carbide and nitride surfaces.112 However, the current data and understanding of the atomic and electronic structure of MeY surfaces is still very limited. Below there is an overview of the relevant ﬁndings in previous studies. The main attention is given to stoichiometric TiC, which is not only the most explored case in the literature112 but also of much importance for the present thesis. For MeY compounds in the NaCl structure one can consider three different low- index surfaces, in particular (001), (011), and (111).

36 5.3FreeSurfaces

Along the [001] and [011] directions the bulk NaCl structure is composed of equidistant layers. Each of those layers has the same structure and composition as the other ones. The layers are stacked in an ABAB... sequence. The difference between 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 concentration of Me and Y atoms. The (001) surface has the symmetry of a square lattice. There are 4-fold symmetry axes placed at each of the surface Me and Y atoms, aligned perpendicular to the surface. In the [111] direction, a MeY compound in the NaCl structure is a sequence of alternating Me and Y layers. Thus, in principle, one can consider two possible terminations of the MeY(111) surface. That is, there are Me- and Y- terminated MeY(111) surfaces. For TiC, the relative stability of the three low-index surfaces is investigated in Ref. [123]. The most stable TiC surface is TiC(001). This surface is the known cleavage plane of TiC.124 The relative stability of various MeY surfaces can be understood by comparing their surface energies. It is expected that the lower the surface energy, the more stable the surface. As one can see from Table IV in Paper V, the order of the TiC

and 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 explain 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 within a 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 determined by how many Ti-C bonds one has to break to cleave the TiC crystal to create a given TiC surface. For (001), (011), and (111) surfaces all the atoms of the topmost surface layers have one, two, and three broken bonds, respectively. The (001) surface has the smallest number of broken bonds, both per surface atom and unit area, and hence the lowest surface energy. The stability and abundance of the (001) surface of TiC, and likely so for many other NaCl-structure transition metal carbides and nitrides, makes this surface the most 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 compound surfaces in experiment and theory have to face the possibility that those structures 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 atoms are displaced in opposite directions, along the surface normal. The sum of the absolute values of those displacements, or the distance between the surface Me and Y

37 5 Transition Metal Carbides and Nitrides atoms along the surface normal, is called rippling. In experimental studies126 this rippling has been found very small, not larger than 0.1 A.˚ A recent, highly accurate measurements of work [127] have characterized the TiC(001) rippled relaxation as Ti 0.036 A˚ inwards and C 0.040 A˚ outwards. This rippling relaxation behavior is quite 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 0.07 A˚ inwards and C 0.04 A˚ outwards. In the same paper the calculated rippled relaxation of TiN(001) has the same direction as for TiC(001), and about twice as large magnitude. 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 calculations, as reviewed in Ref. [112]. The experimental and theoretical results suggest that rippled relaxation is likely to be a general phenomenon on the (001) surfaces of transition metal carbides and nitrides.112

38 CHAPTER 6

Relevant Interface Thermodynamics Background

The fundamental role in theoretical understanding of wetting and adhesion phenomena belongs to thermodynamic analysis. In particular, the difference between the wetting and non-wetting behavior, or the value of the wetting angle, is essentially determined by the relative values of different thermodynamic quantities, like interface energies, work of adhesion, and surface energies. Therefore, theoretical modeling of the interface thermodynamic quantities is the key part of the studies of wetting and adhesion in Papers I - V. The goal of this chapter is to introduce the main interface thermodynamic quantities and discuss the key assumptions and simpliﬁcations that are 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 its free energy per unit area, γ.128 This quantity is best deﬁned by considering a system that consists of two bulk phases, A and B, which are in contact along a planar internal interface. The system is growing in a container under some given equilibrium conditions by the accretion of atoms from suitable reservoirs. The growing interface system 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 the size of the system in all dimensions is much larger than the width of the interface regions. Then, in accordance with the ﬁrst and second laws of thermodynamics, the change in the internal energy of the system, E, due to the accretion can be expressed as128

39 6 Relevant Interface Thermodynamics Background

+ + γ ; dE = TdS PdV ∑ µidNi dA (6.1)

i=1 where S is the entropy,V the volume, Ni the amount of component i, C the number of components, and A the area of the planar interface. Compared to the corresponding expression for a bulk system, Eq. (6.1) contains an extra term, γdA, which describes the increase in the internal energy of the system associated with the increase in the

area of the interface. Equation (6.1) implies ∂

γ E : = (6.2)

∂A ; S ;V Ni Thus the interface free energy γ can be deﬁned as the increase in the internal energy of the entire system per unit increase in interface area at constant S and V of the system under closed conditions, i.e. at constant Ni. The state of a thermodynamic system under different thermodynamic constraints can be found by minimizing an appropriate thermodynamic variable. The most im-

portant examples of such thermodynamic variables are the Helmholtz free energy,

; F = E TS (6.3)

for a closed system at constant T and V, the Gibbs free energy,

+ ; G = E PV TS (6.4) for a closed system at constant T and P, and the grand potential,

Ω ; = E TS ∑ µiNi (6.5)

i=1 which is used to describe an open system under conditions of constant T, V,andµi.

With those deﬁnitions and Eq. (6.1) the interface free energy can also be deﬁned as ∂

γ F ; = (6.6)

∂A ;

T ;V Ni ∂

γ G ; = (6.7)

∂A ; T ;P Ni

6.1 Deﬁnition of Interface Free Energy

∂Ω ; γ = (6.8)

∂A ; T ;V µi There is one more interesting form of an interface free energy deﬁnition, which is also more relevant for the research work in the appended papers. Integrating Eq. (6.1),

+ + γ ; E = TS PV ∑ µiNi A (6.9)

i=1

and using Eq. (6.4), one can express γ as

" C

γ 1 : = G ∑ µ N (6.10) A i i

i=1 Here the quantity ∑C µ N can be identiﬁed as the total Gibbs free energy which i=1 i i the homogeneous A and B bulk phases would have together if they were made up of the same amounts of the components at the same chemical potentials. Thus, the interface free energy γ is the excess Gibbs free energy of the entire system per unit interface 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 relatively low compressibility of solids and liquids, in the context of the present work it is also reasonable to neglect the contribution of the PV term in Eq. (6.7). Thus,

Eq. (6.7) gives : G ' E (6.11) Under these assumptions, the interface free energy, or simply interface energy, can be viewed as the work per unit area required to form the two interfaces A/B and B/A

from the two bulk crystals A and B (see, e.g., Ref. [129]), as illustrated in Fig. 6.1,

γ bulk bulk = γ = = : j

AjB EA B EA EB A (6.12)

bulk bulk

Here EAjB is the total energy of the A/B-interface system, and EA , EB are the bulk total energies per each of the A and B half-crystals, i.e. the total energy per structural unit of the corresponding bulk crystal multiplied by the number of the structural units in the corresponding half-crystal. The interface energy shows how much weaker the bonding at the interface is than in the A and B bulk materials. AB + +2γ =AB + A B B A

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

41 6 Relevant Interface Thermodynamics Background

6.2 Thermodynamics of Wetting: Contact Angle and Work of Adhesion

The thermodynamic description of interface systems introduced in the previous section can be applied to the situation of wetting experiments. A typical situation of sessile drop wetting experiments, with a metal drop on a ceramic surface, is schemat- ically shown in Fig. 2.4. The contact angle θ characterizes the degree of wettability θ o of a given ceramic by a given metal: = 90 is considered as a boundary between

θ o o θ > the wetting ( < 90 ) and non-wetting ( 90 ) behaviors. The smaller the angle θ the better is the wetting. In addition to characterizing wettability itself, wetting experiments are practically the only feasible way to study interface thermodynamics (energetics) in metal-ceramic systems. Under thermodynamic equilibrium and steady state conditions, the contact angle

θ is determined by the Young equation

σ γ

θ A AjB : cos = (6.13) σB γ Here , σ ,andσ are the interface free energy values for the solid-liquid A jB, AjB A B solid-vapor, and liquid-vapor interfaces, respectively. The quantities σA and σB are at the same time the surface energies of solid A and liquid B, respectively. A decrease 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, this lowering of free energy is partially compensated for by the energy of the increasing area of the free liquid surface (liquid-vapor interface). The Young equation simply states the condition of a balance between those two opposite contributions, i.e. the condition of a free energy minimum. An interesting and important form of the contact angle expression (6.13) can be obtained by introducing one more important quantity, the work of adhesion, Wad. This quantity is deﬁned as the reversible free energy change for making free surfaces from interfaces,130 whereby the surfaces are in equilibrium with the solid and gaseous components. The work of adhesion Wad is connected to the corresponding

interface and surface energies via the Dupr´e equation

= σ + σ γ :

Wad A B AjB (6.14)

Combination of Eqs. (6.13) and (6.14) leads to the Young-Dupr´e equation

σ + θ: Wad = B 1 cos (6.15)

With this equation, the measured wetting angle directly gives the ratio of the interface adhesion work Wad and the liquid metal surface energy σB. The Young-Dupr´e equation (6.15) plays an important role in the analysis of wetting experiments. The contact angle θ 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 typically known from a separate experiment within the same study, or from the available data from other experimental works. The work of adhesion Wad is practically the only interface thermodynamics (energetics) quantity that can be directly extracted from the wetting experiments. A reliable measurement of the solid surface energy σA is a very problematic task, making it almost impossible to get any good estimates of a solid-liquid or solid-solid interface energies γ with Eq. (6.14). AjB

6.3 Ideal Work Of Separation as Measure of Interface Adhesion Strength

Another fundamental quantity in the interface thermodynamics is the ideal work 130 of separation Wsep, which is deﬁned as the reversible work needed to separate the interface into two free surfaces in a thought experiment, whereby plastic and diffusional degrees of freedom are suppressed. The ideal work of separation can be

expressed by a modiﬁed Dupr´e equation.

0 0

σ + σ γ : W = (6.16) sep A B AjB

The difference from the Dupr´e equation (6.14) is that the surface energies σA and σB are now replaced by the instantaneous values of surface energies before any plastic processes, like dislocation motion, or diffusional processes of chemical equilibra- tion, like surface segregation or surface contamination, take place. Due to such dissipative processes, the energy needed in a real cleavage experiment will always exceed Wsep. Yet, it is still a very useful quantity to characterize interface mechanical strength. AABB+W =

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 the contact angle or the work of adhesion Eq. (6.14), it is a much more manageable task to calculate the ideal work of separation Wsep, by comparing the total energy of the interface system with the total energy of the corresponding system in which the interface is cleaved, leaving two free surfaces (Fig. 6.2). Due to such calculational

difﬁculties, the theoretical analysis of wetting in the appended papers assumes ; Wad ' Wsep (6.17)

43 6 Relevant Interface Thermodynamics Background

where Wsep is calculated without any explicit inclusion of thermal motion, i.e. at

T = 0, as

= + = :

W = W E E E A (6.18) j j j sep AjB A B A B

Here E and E are the total energies of the separated half-crystals. j Aj B

44 CHAPTER 7

Goals and Principles of Interface Geometry 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 interface atomic structure. Interfaces that are of interest here can be viewed as two free 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 are placed relative to each other. This relative placement is affected by the fact that the interatomic interactions across the interface can significantly change both the interatomic interactions and the atomic structure within each of the contacting surface subsystems (interface relaxations). Besides, atoms can penetrate from one side of the interface into the other (interdiffusion). One more important factor is the difference in the structure and periodicity of each of the surfaces (lattice mismatch). This can result in significant irregularity of the interface structure, with an interface unit cell being much larger than the surface ones, with many defects (interface dislocations), or without any long-range order at all (amorphous). Moreover, if one of the contacting 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 correlations between dynamically changing atomic positions. Adequate choice of the interface model geometries (atomic structures) is a crucial component of the theoretical investigations in Papers I - V. This chapter introduces some common definitions related to the description of interface geometry. It also discusses different aspects of the experimental situation and theoretical understanding of metal-ceramic interfaces that help making physically meaningful choices of the interface structures in Papers I - V.

45 7 Goals and Principles of Interface Geometry Modeling

7.1 Basic Deﬁnitions and Background: Geometrical Degrees of Freedom

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

7.1.1 Macroscopic Degrees of Freedom The macroscopic degrees of freedom describe the relative orientation of the two crystals and the interface plane, i.e. the interface orientation relationship. The orientations of the interface plane and the two crystals can, in principle, be determined macroscopically, e.g., by the symmetry of the macroscopic properties. There are generally five macroscopic degrees of freedom. The relative orientation of the two crystals can be specified by their relative rotation in space. To describe such a relative rotation (orientation relationship) one needs three degrees of freedom: two for the unit vector along the rotation angle, and one for the rotation angle. Two more degrees of freedom are needed to specify the direction of the interface-plane unit normal (interface orientation). The macroscopic degrees of freedom constitute geometric thermodynamic variables which are required for a full thermodynamic description of the interface.128 One convenient way to specify macroscopic degrees of freedom is by writing the 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 imply that the Me/TiC interface plane is parallel to the (001) crystal planes of the both metal Me and TiC contacting crystals. Such a specification covers only four of the five microscopic degrees of freedom. What should also be included is the relative rotation of the parallel planes of the two crystals around the axis perpendicular to the interface plane, which is referred to as the interface rotation state.

7.1.2 Microscopic Degrees of Freedom The microscopic degrees of freedom provide a summary description of the atomic structure at the interface. The interface atomic structure is controlled by the microscopic relaxation processes, that is an adjustment of the relative positions of the interface crystal planes and particular interface atoms in order to lower the total energy of the systems. Such relaxation processes take place under the constraints introduced by the macroscopic degrees of freedom, which can be thought of as boundary conditions far from the interface. If the interface atomic structure is periodic, like for the interface model systems used in Papers I - V, then there are three main microscopic degrees of freedom, which are 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 component of t 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 of t perpendicular to the interface is described by the interface spacing, d, the interlayer distance at the interface (Fig. 7.1).

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

7.1.3 Interface Geometry Control in Experiment How much control does one have over those macroscopic degrees of freedom for metal-ceramic interfaces in experiment? The answer depends on the way the particular interface system is created. One common experimental situation of metal- ceramic interfaces encompasses particle-matrix interfaces in a nucleation and growth precipitation reaction, like for precipitate particles of Ti, V, Cr, and Nb carbides and nitrides in steels (see Refs. [132, 133] and references therein). In those systems the carbide and nitride particles often precipitate in a few known unique orientations with respect to the metal matrix, and a significant part of the formed metal-ceramic interface is planar and atomically flat. The crystollagraphy of such interfaces, i.e. their macroscopic orientation, is difficult if not impossible to influence in experiment. At the same time, the fact of a relatively well-defined interface structure for matrix-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 heteroepitaxial growth of thin films on single crystalline substrates.131 The heteroepitaxial growth allows us to choose and fix the interface orientation. There is much less control and flexibility in setting the orientation relationship, which is determined by the microscopic growth processes under the given growth conditions. Yet, often the film can be grown epitaxially, with a high-quality atomically flat and chemically pure interface, and with a well-defined and unique orientation relationship to the substrate. One of the most common and powerful heteroepitaxial growth techniques is molecular beam epitaxy (MBE). In the MBE growth process the atoms of the growing film impinge on the surface of the substrate as molecular beams. This technique has to be used in an ultra high vacuum to avoid contamination of the growing surface. The well ordered structure and macroscopic lateral dimensions of the MBE grown interfaces make such systems particularly suitable for characterization of their atomic and electronic structure with different surface science experimental techniques. For the same reasons, they are also nearly ideal systems for theoretical modeling. Probably most flexibility in choosing the interface macroscopic orientation re-

47 7 Goals and Principles of Interface Geometry Modeling lationship is provided by solid 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 via plastic deformation and closure of voids by diffusion. The plastic deformation and closure of voids by diffusion are also stimulated by applying a uniaxial pressure and elevating the temperature. While solid state bonding offers practically full control over 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 to uncontrollable microscopic changes in the near interface region, like faceting and formation of small angle grain boundaries. In such a situation the local interface orientation and orientation relationship in the near interface region can differ significantly from the orientation of the averaged interface plane and the relative orientation 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 atoms are 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 extent determine the value of the output of the computations, the validity and significance of 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 their results to experiment. Scenario I. One very attractive situation is when the system geometry, and probably other properties, are well characterized by experimental studies, and when the atomic structure is simple enough to be described by a hundred or so atoms put in a simulation supercell. First-principles simulations of such systems give good opportunities to test the reliability and accuracy of the approximations in the DFT exchange-correlation functionals and the first-principles computational methods. Such computational results can also significantly extend and complement the experimental data with helpful new predictions, interpretations, and insights. An example of such a situation in the field of metal-ceramic interfaces is the case of 134 Nb(111)/α-Al2O3 interface. Scenario II. A situation not too far from Scenario I is one when the experiments do not fully determine the system geometry but point at that the geometry realized in experiment should be found among a sufficiently small number of different known alternatives. Then, the first principles simulations can predict which of the alternative geometries is the real one, and again extend and complement the experimental data for that system. This kind of situation can be found in, e.g.,a

48 7.2 Choice of Interface Geometry Models

first-principles study of the Ag(001)/MgO(001) interface in Ref. [129], when the first-principles results point at which of a few alternative interface translation states should occur in the experimental MBE-grown interface system. Outside the field of metal-ceramic interfaces, one important example is also a first-principles determination of the κ-alumina atomic structure,135,136 when the first-principles calculations help to identify which structure, out of a considerable number of alternatives, should correspond to the experimental one. Scenario III. One more important situation is when the connection between first-principles calculations and experimental results is mediated by some additional physical model. That model relates the essential physics of a given physical property or process to a number of microscopic parameters, which can be provided by first- principles calculations. In that case the model atomic structures for first-principles calculations are chosen so that to effectively determine the parameters of the physical model. Often it may be relatively unimportant if those atomic structures are actually realized in experiment. Scenario III is typically used when the time or length scales of the physical problem of interest are beyond the capabilities of direct first-principles simulations. A characteristic example of Scenario III is given by the thermodynamic description of sizes and shapes of precipitates in metal alloys in Ref. [137]. The precipitates 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 the connection between experiment and first-principles simulations is mediated by the cluster-expansion model,138 together with Monte Carlo simulations. The parameters of the cluster expansion are determined from first-principles calculations for a set of simple structures that describe elementary geometry configurations, such as pairs, triangles, tetrahedra. Those atomic structures are not suggested by experiment. Instead, they are chosen with the main goal of determining the parameters of the cluster-expansion theoretical model. The existing first-principles studies do not necessarily belong to one and only one of the above described scenarios. They can mix various components of different scenarios, or emphasize other aspects, not mentioned here. Yet, it is interesting to analyze if we can identify any of the above described characteristic features in the research work of this thesis. First of all, the first-principles studies in the appended papers don’t have much in common with Scenario I. There is no experimental information about the interface atomic 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 possible alternatives for interface structures in the studied systems. The main structural clue we have from experiment is that the (001) surfaces of carbides and nitrides in the NaCl structure are more likely to be found than other surfaces (See Chapter 5 and discussions in Paper II and V). That allows us to focus mainly on the interfaces formed by the (001) carbide and nitride surfaces. Yet, that assumption still leaves too many different possible interface structures. The number of alternatives is beyond the capabilities of direct first-principles treatment.

49 7 Goals and Principles of Interface Geometry Modeling

7.2.2 Should We Search for Best Structure?

In the above discussion it was mentioned that one of the reasons why Scenarios I and II can not be followed in the present work is the lack of structural information from experiment. However, there is one more reason, which is more important and more interesting than the first one. That reason is in that there seems to be no meaning in the search for one preferred structure, as suggested by the following arguments. The interface systems of interest in this thesis are mainly found in brazing and cermet sintering, or in related wetting experiments. In cermets, the grains of the binder metal, e.g., Co, have to adjust to many surrounding carbide or nitride grains of various macroscopic orientations. Even if there is a special orientation relationship between the metal and one of the carbonitride grains, there will be a different macroscopic orientation relationship to other grains. That is, the interface macroscopic degrees of freedom should cover a relatively wide range of values. The interface atomic structure that should adjust to those various grain orientations, is then also 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 crystals forming the interface differ in their translation symmetry elements parallel to the plane of the interface. For example, there is around 25 percent difference in the lattice constants of TiC and fcc Co (see Paper I and II). When such metals and ceramics are in atomic contact, different situations are possible, depending on the relative strength of different chemical bonds inside the contacting materials and across the interface. If the interface bonds are too weak to cause any noticeable distortion in the materials, then the surface layers of the two materials will have different periodicity, forming an incoherent interface. Thus if one type of local atomic structure, like metal-near-non-metal configuration, applies to some point of the interface, then this type 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 specific lateral positions of the interface atoms of the metal phase relative to those in the ceramic one. This can be due to relatively strong bonds across the interface. For example at metal(001)/carbide(001) and metal(001)/nitride(001) interfaces studied in this thesis, it is energetically more favorable for the interface metal atoms to be placed 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 nearest favorable position. That is, the interface bonding tends to match the lattices of the surface layers of the two materials in order to increase the number of favored atomic configurations. Such matching introduces strain into the contacting crystals, mainly in the metal phase, which is generally less stiff than the ceramic one. For a reasonably small lattice misfit, the resulting interface structure is often described as consisting of well-matched (coherent) regions, with the misfit strain localized 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 7.2 Choice of Interface Geometry Models of misfit, so that at some point it is difficult to distinguish separate misfit dislocation. At that point those structure irregularities should merge, resulting in a totally irregular (amorphous) interface structure. Such situation is likely to occur at cermet Co/Ti(C,N) interfaces in cermets, where large lattice misfit can be caused by both the large difference in the lattice constant of Co and Ti(C,N) and a misorientation of the ceramic and metal grains in cermets.

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

The ﬁnal important argument against the existence of one best interface structure is 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 static structure, we should consider an ensemble of various instantaneous structures that are spanned by the moving atoms. There is no long range order in the structure, just some correlations between the positions of the neighboring atoms.

7.2.3 Focus on Development of Theoretical Models of Wetting and Adhesion The discussions above suggest that the present study does not fit in Scenarios I or II (see Section 7.2.1), but that there are similarities to Scenario III. The first noticeable similarity is that the physical problem of interest, i.e. wetting and adhesion in brazed and sintered metal-carbide and metal-nitride systems, is beyond a direct first-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 relation between first-principles simulations and reality should be mediated by some physical model. The physical model needed here should be able to treat the wetting and adhesion in metal-carbide and metal-nitride systems in microscopic terms. In the context of the discussion in Chapter 6, our main expectation from such a model is to describe the strength of the metal-ceramic interface bonding in the materials systems

51 7 Goals and Principles of Interface Geometry Modeling 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 we could take from other theoretical or experimental studies and adjust to our needs (see Chapter 8). The lack of adequate theoretical models for the problem of interest suggests two important directions for theoretical work in this thesis, which will also determine the choice of geometry models. The first direction is to analyze more general and qualitative relationships between different macroscopic and microscopic aspects of the considered problem, and then use those relationship in a role of a missing physical model. Those relationships will not be enough to formulate a specific model in a traditional sense, i.e. as mathematically expressed relations between a set of model parameters. Yet, they can still serve as an adequate base for a meaningful choice of geometric models for the system. The second direction, closely related to the first one, is to concentrate our research efforts on developing more specific physical models. That is, the first-principles simulations, including choice of geometry models, should be focused on exploring the key connections between different elements of the problem, so that we could gradually progress from the initial general and qualitative relationships to more specific and quantitative ones. As a starting point for building a theoretical model of wetting and adhesion we can take a general assumption that the strength of the interface bonding is determined by 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 and metal-nitride interfaces. Goal II: to explore and clarify the relationships between those microscopic interactions and the interface adhesion strength (the ideal work of separation). Goal III: to use first-principles simulations and general knowledge about the given materials systems to understand the relative role, and the interplay, of different microscopic interactions in the real metal-carbide and metal nitride interface systems. Those three goals determine the choice of the atomic structure models in the first-principles studies in Papers I - V. In connection to Goals I and II much attention in 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 distinguish more clearly the contributions of different kinds of chemical bonds and then analyze their nature (Goals I and II), as well as to compare the behaviors of those interactions, which is helpful in the context of Goal III. On the other hand, those simple structures represent the most distinct types of relative atomic arrangements at the interface, like metal-over-Ti(W) versus metal-over-C(N) structural configurations. First-principles calculations with such distinct structures help to identify the range in which the adhesion strength (the ideal work of separation) changes as a function of the interface structure, and to clarify the connections between the strength of interface bonding and the microscopic interactions (Goal II). More complex structures,

52 7.2 Choice of Interface Geometry Models 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 in Papers I and II.

M C(N) Ti

(a) (b)

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

(a) (c) (e) C Ti Co (b) (d) (f)

Figure 7.4: Various interface structure models for Co/Ti(C,N) interfaces studied in Papers I 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 typically identify parameters that describe the bulk structure of the contacting materials, as

53 7 Goals and Principles of Interface Geometry Modeling opposed to interface-specific parameters, like the interface interlayer spacing d, illustrated in Fig. 7.1. For a more clear identification of the key behaviors of the interface microscopic interactions, when choosing interface model structures, it is very advantageous to keep the number of bulk parameters within some reasonable minimum, focusing on variations of the interface-specific parameters instead. In particular, to decrease the necessary number of the bulk structural parameters, one can consider bulk structures with a relatively high symmetry. In the appended papers all the interface model structures are constructed with the assumption that the metal bulk phase is face centered cubic (fcc), rather than some disordered atomic structure representing an instantaneous state of a liquid metal. A description 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 metal bulk phases in solid-solid metal-carbide and metal-nitride interfaces for most of the metals 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-temperature phase of Co, above 418oC for pure Co. Yet, the fcc Co face is more common than the hcp one in the binder phase of cemented carbides and cermets,139 where the residual stresses and the presence carbon and tungsten stabilize the fcc phase even at room temperature. For Ti, the fcc phase is only slightly higher in energy, by less than 0.1 eV, than the hcp one. So that one could build commensurate (periodic) interface structures, the bulk fcc phase in the model interface systems is allowed to be distorted, as described by two parameters. The first parameter, the in-plane lattice constant, is the lattice parameter of the bulk crystal structure along the plane that is parallel to the interface plane. With given interface structural parameters, like the rotation state, the in-plane lattice constant is fully determined by the requirement of periodicity (commensu- rability) of the interface structure. The second parameter, the out-of-plane lattice parameter, aims to describe the expansion or contraction of the metal phase along the 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 in the 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 system is constructed, and they are not varied in interface calculations. Another important situation of the simplified description of the bulk phases is represented by the Co/WC study in Paper IV, where the WC phase, which is hexagonal in reality, is taken in the NaCl structure. This is done for a more direct comparison of the interface interactions in Co/TiC and Co/WC systems, as discussed in Paper IV.

7.2.5 Concluding Remarks It should be noted that in the appended papers the connection to experiment is mainly restricted to a comparison of the work of adhesion values measured in wetting ex-

54 7.2 Choice of Interface Geometry Models periments with the calculated data for the work of separation. The approximations involved in that comparison, like in the description of wetting, in the choice of structural models, and in the neglect of the thermal motion and diffusion, are expected to 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 close to the results of wetting experiments (Papers I, II, and VI), or help to understand the discrepancies in the experimental wetting data (Paper V). All this is an important indication of that the structural models used in the present thesis, together with the key principles discussed in this chapter, represent a very promising approach of ﬁrst-principles treatment of complex metal-ceramic interface systems.

55 7 Goals and Principles of Interface Geometry Modeling

56 CHAPTER 8

Microscopic Interactions at Metal-Ceramic Interfaces

In the thermodynamical, structural, mechanical and other properties of metal-ceramic interfaces 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 to study 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 shape of the drop is analyzed to obtain information on the interface energetics. For a long time, the wetting experiments were the main context of the theoretical work, which was mainly restricted to organizing the wetting data and to extracting the trends in the wetting behavior with respect to variations of different properties of the metal and the ceramic. Based on those trends various hypotheses on the nature of interface interactions were suggested. A considerable progress in the methods to study interface interactions has been made during the last decades, as reviewed, e.g., in Refs. [130, 131]. On the experimental side, there have been significant developments in the interface fabrication techniques, as well as in the ability to probe the details of the interface atomic and electronic structure. In particular, a detailed control of the interface structure and composition on the microscopic level can be achieved with the modern heteroepitaxial growth techniques, like molecular beam epitaxy (MBE). An atomic scale information on the interface structure is provided (see e.g. Refs. [130,131,140] and references therein) by the high-resolution transmission electron microscopy (HRTEM), Z-contrast scanning transmission electron microscopy, and atom-probe field ion microscopy. A similar level of resolution is obtained for the interface electronic structure, using the spatially resolved electron-energy-loss spectroscopy. On the theoretical side, some understanding of the metal-ceramic interface bonding has been gained through extensive studies of a few model interfaces, using different levels of description, from simplified scattered-wave,141 atomic-orbital,142 tight-

57 8 Microscopic Interactions at Metal-Ceramic Interfaces binding,143–146 and image-interaction147–152 models to ab initio density-functional calculations.129,134,140,153–160 However, this understanding still remains quite limited. As noted in, e.g., Refs. [131,140], the major difﬁculty in the theoretical description of the metal-ceramic interfaces is the difference in the nature of bonding on the two sides from the interface. The consequence of this difference is that, while there are reasonably good atomistic models of each of the contacting bulk materials, metal and ceramic with their bonding types, there are no acceptably reliable interatomic potential schemes that could describe the interatomic interactions at a metal-ceramic interface. Thus, ab initio calculations, of the type presented in this work, are practically the only reliable way to model metal-ceramic interfaces. This section discusses different types of interface interactions, together with the simpliﬁed 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 ﬂuctuating dipoles of the two interacting media. That is, in each medium there are quantum ﬂuctuations of the dipole moment, and the dipole moment of one medium interacts with the induced dipole moment in the other medium. The simplest way to take into account the dispersion forces is just to use the model with the interatomic potential in the form of the inverse sixth power of the distance. However, this sixth power law is an acceptable approximation only at large distances, where the van der Waals interaction is almost negligible. An alternative approach, which is more accurate and at the same time rather simple, is the dielectric continuum model of Barrera and Duke.161 In the dielectric continuum model, each of the media is described by a complex frequency-dependent dielectric function in the form

ε 2 2 2 ω= ω =[ω + ω=τ ∆ ]; 1 p i (8.1) where ωp is the plasma frequency, τ is the plasmon damping time, and ∆ is the band gap of the corresponding media. The dielectric continuum approximation leads to closed expressions for the surface and interfacial energies, and hence for the wetting contact angles, in terms of only three parameters for each medium (ωp,τ, ∆ ). The speciﬁc values of the parameters can be extracted directly from the bulk optical- absorption and electron energy-loss spectra. Thus the considered model makes it possible to estimate the contribution of the dispersion forces by just analyzing the accessible bulk experimental data. For ceramic oxides, the systematics of wetting by liquid metals does not correlate with the predictions of the dielectric continuum model, and it has been con- cluded24,147,162,163 that although the dispersion forces are not negligible they are not the major effect in the adhesion here. Instead, the trends in the wettability with changes in the substrate oxide has led to the view that the interfacial energetics is controlled 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 contribution of the dispersion forces to the interface interactions can be expected to be of more importance than for oxides, approaching the situation of metal-metal interfaces. For τ ∞ metal-metal interfaces, neglecting the plasmon damping ( = ), the dielectric continuum approximation gives the interface energy as161

∞ Z

h Z dq dω

γ = [ ω];

2 ln f i (8.2)

π π 4π 2 2 0 with

ω=[ε ω+ε ω] = ε ωε ω; f 1 2 4 1 2 (8.3)

ε ω where 1 ;2 are the dielectric functions of the two metals from Eq. (8.1) and q is the plasmon two-dimensional wave vector along the interface plane. To avoid

the divergence of the q-integration, this integration is performed within a cutoff qc,

j < assuming the existence of dispersionless plasmons only for jq qc. To calculate

ε ω surface energies, one of 1 ;2 should be taken to be unity. From the Young equation (6.13) and expression (8.2) and (8.3) for surface and interface energies, the

following expression for the wetting angle θ can be derived:23 p

p 2

1 + 2Z 2 1 Z

p ; cosθ = (8.4)

2 1 where Z is the ratio of the plasmon frequencies of the substrate, ωps, and the metal,

ω ω =ω θ

pm, Z = ps pm. Since cos should be between -1 and 1, the equation (8.4) is only valid for 0 Z 1, i.e. when

ω ω : ps pm (8.5)

In Ref. [23] expression (8.4) is used to analyze the wetting trends for Cu on carbides, and it is concluded that the above dielectric continuum model gives an adequate description of those trends. This leads to the view that adhesion of Cu to carbides is controlled by the van der Waals dispersion forces. However, a more recent work [18] suggests that such a view needs to be reconsidered. On the one hand, in Ref. [23], with reference to the lack of experimental data on the plasmon frequencies for carbides, the analysis is based on indirect arguments for those plasmon frequencies. At the same time, the experimentally measured plasmon frequencies for TiC in Refs. [164,165] are noticeably larger than the Cu plasmon frequency, which does not allow 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, for Cu on carbides, the description of the interface adhesion in terms of the dispersion interaction 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 Microscopic Interactions at Metal-Ceramic Interfaces

8.2 Image Interaction Model

A fruitful approach to understanding of the interface adhesion between metals and ceramic oxides is the image interaction model, suggested in Ref. [147] and then further developed in Refs. [148–152]. The idea of the image interaction comes from the basic electrostatics, from a picture describing the interaction of a charged object with a conductor surface. In particular, if there is a point charge q at a distance z0 from a conductor (e.g., metal) surface, then this charge induces a screening charge density on the surface given by

ρ z0q = ; r (8.6)

2 2 3 =2

π + 2 z0 r where r is the distance from the charge in the plane along the surface. The electrostatic potential from this induced charge density,

; = : V r z (8.7)

2 2 1 =2

+ + ] [r z0 z has the same form as if instead of the metal surface there were an effective point

charge q placed at the same distance z0 from the surface plane as the original charge, but on the other side from the surface, as illustrated in Fig. 8.1. That is, the interaction of a charge with a metal surface can be described in terms of electrostatic attraction between this charge and the effective image charge. The energy of such an image interaction is

1 q2

: E = (8.8) 2 2z0 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 the electrostatic interaction of a charged object with a metal surface. z q z0

-z 0 -q metal

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

60 8.3 Chemical Bonds across Interface

In the simplest variant of the image interaction model of the metal-ceramic adhesion, the metal is taken as a conducting continuum (a dielectric continuum with an inﬁnitely large dielectric constant). The ceramic oxide is considered as a lattice of ions, cations and anions, which interact with the metal through the attractive classical 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 per square meter, which is in the range of typical metal-oxide adhesion energies. The image 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 approximation to the real picture of metal-oxide adhesion, at the spatial scale down to a few Angstr¨˚ oms. At closer distances, quantum mechanical effects become important. In particular, in a real metal, in contrast to the dielectric continuum, the ﬁnite size of the variations of charge distribution has to be considered. Only variations with wave- lengths larger than the Fermi wave-length are allowed. One consequence of such a behavior is the so-called Friedel oscillations. A possible way to correct for this effect is to explicitly introduce the wave vector cutoff in the expression for the induced charge density, as it is done in Ref. [150]. A more consistent quantum mechanical theory of the image interactions is developed in Ref. [149], where a general linear response expression for the image interaction energy is derived in the framework of the density functional theory. This quantum mechanical analysis does not only give the correction to the classic expression for the image interaction energy, but also deepens its interpretation, showing that the simple form of the image interaction energy is actually a result of a delicate balance between more complex-structured contributions 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 interactions, are of a relatively long-range type, which makes them quite insensitive to the details of the atomic structure at the interface. Dominance of those interactions in the metal-ceramic interface bonding would lead to universal features in the adhesion trends. Besides the van der Waals and image interactions there are also possibilities for short-range ionic and covalent bonds. The short-range interactions generally complicate the picture of the interface bonding, not allowing us to construct simple and universal models, like the ones mentioned in the previous subsections. Compared to electrostatic image interactions, the localized ionic bonds involve some noticeable charge transfer between the interacting interface atoms. Such a mechanism of bonding has been demonstrated to dominate, e.g., at low metal cover-

156 134 ages on an Al2O3 substrate and for the bulk Ni(111)/α-Al2O3 0001 interface. One important aspect worth mentioning here is given by the trends in the nature

61 8 Microscopic Interactions at Metal-Ceramic Interfaces of the metal-ceramic interface bonding depending on whether the metallic phase is represented by a free-electron-like, transition or noble metal.129 The bonding between a simple metal and an oxide ceramic is controlled essentially by the Coulomb interaction between the ions and their screening charges in the metal, that is by the image interaction. The situation with transition metals is more complicated because they also have partially filled d-orbitals, and can form strong pd-covalent bonds across the interface with the oxygen atoms of the ceramic. Since the filling of the transition metal d-orbitals moves them closer in energy to the oxygen orbitals, the covalency of the interface pd-covalent bonds tends to increase towards the end of the transition metal series.143 These interfacial covalent bonds are also the main factor in determining the relative positions of the atoms at the interface: the transition metal atoms are usually placed above the oxygen atoms of the ceramic. For noble metals, there are still d-orbitals, and although they are filled and cannot participate in the covalent bonds, they are polarizable and can contribute to the interface bonding. In this case the bonding is expected to have an intermediate nature between the cases of simple and transition metals. The problem of the relative positions of the atoms at the interface is more complex and unclear, and probably there can be several metastable states of a noble metal-ceramic interface.

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

One efficient way of understanding something new is through exploring the similarities to and differences from something that is already known and understood. When we use this strategy to understand the chemical bonds across interfaces between metals and transition metal carbides and nitrides, one of the closest situations we 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 and nitrides 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 antibonding states (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 the antibonding are empty. Cohesion in CoC is significantly weaker, due to an almost complete filling of the antibonding states. As a first step, in the consideration of metal-C bonds across Co/TiC interfaces studied in Papers I and II, we can assume that the metal-C bonds at the interfaces are similar to those in the corresponding bulk carbides. Such an assumption leads to the expectation that the interface Co-C bonds are relatively weak, about as weak as those in bulk CoC. In a further step, we can still assume that the character of the Co-C bonding and antibonding states is similar to those states in bulk carbides, but allow the possibility that the interface environment changes the strength of the Co-C bonds by changing the 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 can make the interface Co-C bond stronger than in bulk CoC but still not much stronger than in TiC, where there is an optimal population of the bonding and antibonding states. As found in Papers I and II, the difference in the strength of the interface and bulk Co-C bonds is much more drastic than what could be expected with the above assumptions. In particular, there is a noticeable number of indications that point at that the interface Co-C bonds are considerably stronger than even the strong Ti-C bonds in bulk TiC. This is indicated by, e.g., the magnitude and structural dependence of the adhesion strength, the effects of structural relaxation, and the spatial distribution of the electron density (Fig. 8.2). A similar behavior is found for the Co-N and Co-C bonds at Co/TiN (Paper II) and Co/WC (Paper IV) interfaces, respectively. Such a situation suggests that the strength of the Co-C(N) interface bonds is beyond what could be explained within the theory of bonding in bulk transition metal carbides and nitrides. Based on the electronic structure analysis of the Co/Ti(C,N) interfaces, the unusual 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 interfaces is not significantly affected by the Co ferromagnetism, as discussed in Paper III. Noticeably weaker bonds are found when Co is replaced by noble metals, like Cu, Ag, and Au (see Paper V), though the characteristic electron structure features similar to those at Co/Ti(C,N) can still be identified.

C Ti Co C Co Ti Co C

Figure 8.2: Electron-density picture of bonding at the Co/TiC interface: constant-density surface at the level of 0.5 electrons/A˚ 3 (see Papers I-III).

63 8 Microscopic Interactions at Metal-Ceramic Interfaces

64 CHAPTER 9

Conclusions

Joints between metals and ceramics are becoming increasingly important in the manufacturing of many high technology products, from microelectronic devices to cutting 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 adhesion between metal and ceramics. Experimental studies of wetting suggest that wetting in metal-ceramic systems is highly sensitive to factors of microscopic nature, like local chemical composition at the interfaces. The main goal of the research in this thesis is to identify and analyze the key microscopic mechanisms behind the wetting and adhesion, at the level of interatomic interactions. The ceramic materials considered in the thesis are transition 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 structure for a variety of model interface systems. The first-principles calculations are quite accurate 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 qualitative microscopic picture of interface adhesion between metals and ceramic carbides and nitrides can be suggested. The adhesion between the metal and ceramic phases is due to two distinct kinds of chemical bonds across the interface: metal-C(N) and metal-metal ones. Its strength is determined by the number of bonds of each kind per 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 interface, e.g. at a solid-liquid interface in wetting experiments, can be estimated by analyzing the most distinct possible types of local structural arrangements at the interface. Such extreme cases, like the metal-over-C(N) and metal-over-metal configurations (Fig.1(a,b) in Paper V), determine the likely range of variations in the strength and number of different bonds and in the resulting adhesion strength. The experimental 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) and metal-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 bonding studied in Papers I and II. Effect II is a noticeable difference in the optimal bond-lengths of the interface metal-metal and metal-C(N) bonds. In particular, the metal-metal bonds tend to be about 30 per cent longer than the metal-C(N) ones. As a consequence of that difference, there is typically a much larger number of the interface metal-metal bonds than of the metal-C(N) ones. This effect increases significantly the relative weight of the metal-metal bonds in the total adhesion strength. Effect II is also important in the understanding of the differences between Co/TiC and 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 and adhesion trends from Co/TiC to Co/WC. Effect III is that the strength of the interface metal-metal bonds is quite sensitive 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. This third effect makes the competition between the contributions of the metal-metal and metal-C(N) bonds in Effects I and II even more dramatic. For example, if some of the 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 by the 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 interface adhesion. Effect III also plays an important role in the interface-orientation dependence 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 Co 15

on TiC (see Section 2.3.1). The estimate of the work of adhesion Wad (see Sec- : tions 6.2 and 6.3) from the ﬁrst-principles simulations in Papers I and II, Wad ' 3 3

2 15 2

= : = J=m , is quite close to the experimentally measured value, Wad 3 64 J m (at

66 9.2 Interpretations of Wetting Experiments

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

Another important experimental fact is that wetting by Co is noticeably better for WC than for TiC15 (see Section 2.3.1). This fact is of great signiﬁcance in the hardmetal industry, distinguishing the sintering of WC-Co cemented carbides from that of TiC-Co cermets (see Section 2.1). It can be described quantitatively by the

15 2

: = fact that the work of adhesion for Co/WC, Wad = 3 82 J m , is somewhat larger

2 o

: = than for Co/TiC, Wad = 3 64 J m (at 1420 C). This trend from Co/WC to Co/TiC is adequately well reproduced in the ﬁrst-principles simulations of Paper IV. Though the strong Co-C bonds should still be the main contribution to bonding, the further theoretical analysis shows that the difference between Co/WC and Co/TiC is due to the different strength of the metal-metal Co-W(Ti) bonds (see also Effect II in Section 9.1).

A more complex experimental situation is provided by wetting of carbides and nitrides by Cu, Ag, and Au (see Section 2.3.2 and Paper V), where there is a large scattering in the wetting data. That situation can also be given a simple microscopic interpretation, based on the qualitative picture of adhesion in Section 9.1 and specific first-principles results (Fig. 9.1). Most of the experimental Wad values are in the range between the theoretical Wsep values for the two most distinct types of interface structural configurations, in particular metal-over-C(N) and metal-over-Ti ones (see Fig. 1 in Paper V). Moreover, many of the experimental data points are very close to 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 the same 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 data and theoretical results fits well the qualitative microscopic picture of the interface adhesion in Section 9.1.

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

67 9 Conclusions

4.0 M−over−C(N) M−over−Ti 3.0 M−over−bridge )

2 Experimental Wad

2.0 (J/m sep W 1.0

0.0 Cu/TiC Ag/TiC Au/TiC Cu/TiN Ag/TiN Au/TiN

15–18,21 Figure 9.1: Values of the work of adhesion Wad extracted from wetting experiments for wetting of TiC and TiN by Cu, Ag, and Au, in comparison with the theoretical values

of the ideal work of separation Wsep (Wsep ' Wad, see Section 6.3) from the ﬁrst-principles calculations 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, respectively.

3.0

2.5

2.0 Cu−over−C ) 2 Cu−over−Me 1.5 Old exp

(J/m Recent exp sep 1.0 W

0.5

0.0 HfC ZrC TiC TaC NbC VC

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 structures in Fig. 1(a) and (b) of Paper V, respectively. “Old exp” and “Recent exp” refer to the work of Ramqvist15 and the more recent studies,16,18,21 respectively.

The same simple analysis as in Fig. 9.1 can give an insight into one more interesting issue in wetting of carbides by metals, in particular the experimental wetting trend 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 experimental Wad values can be made (Fig. 9.2). All the experimental points, including the

68 9.3 Outlook

16,18,21 more recent data for Cu on TiC, are covered by the Wsep region between the calculated values for the two extreme structural conﬁgurations, i.e. Cu-over-C and Cu-over-metal ones. Yet, the behavior of the theoretical curves does not show any ground 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 the two theoretical curves. A shift towards that middle region is seen in the data points from the more recent experiments with Cu/TiC.16,21 New experiments for Cu on other carbides are needed, with the same careful treatment of oxygen contamination as in Ref. [21].

9.3 Outlook

There are many interesting possibilities for the continuation of this work. One natural direction would be to apply the methodology used in this thesis to other similar interface systems. This could help to further develop and test the above described qualitative picture of metal-carbide and metal-nitride adhesion. For example, the effect 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 considering vacancies in Co/TiC or Co/WC systems. This direction could also contribute to the development 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 statistical nature. More of simple and valuable qualitative insights are likely to be found there. A simple example of statistical arguments in this thesis is the assumption that at disordered metal/Ti(C,N)(001) interfaces, e.g., when the metal is a liquid, a metal atom 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 between the metal-metal and metal-C(N) short-range interactions (chemical bonds) explored in this thesis affect the statistical correlations between the motion, and average positions, of the atoms at the interface. Probably the simplest way to approach this issue is to express those qualitative differences in some simple model of interatomic interactions, like pair potentials, and then perform molecular dynamics simulations with that model. A very promising possibility for a more systematic and reliable description of statistical effects in the metal-carbonitride adhesion is provided by the recently developed extended tight-binding method.166–168

69 9 Conclusions

70 Acknowledgements First 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 and encouraging me, and for creating the best environment for my scientific and personal growth during these years. Special thanks to Jan Hartford, Yashar Yourdshayan, and Lennart Bengtsson for introducing me into first-principles calculations, and to Ulf Rolander and Hans- Olof Andrén for very valuable discussions on the experimental and technological background 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 with various issues. Thanks to Gustaf Källen, Svetla Chakarova, Eleni Zimbaras, Anders Hellman, Niclas Jacobson, Martin Hassel, Per Sundell, and Fredrik Olsson for sharing their good mood and positive attitude. In various computer problems I got much of valuable help and support from Andy 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 this PhD opportunity, with the application process, and with various administrative issues afterwards. In connection to other administrative questions I would also like to thank Ing-Britt Bengtson and Camilla Eriksson. Many thanks to Sergei Simak, Mikael Christensen, and Behrooz Razaznejad for being very good friends, for support and sincerity, for being there, as well as for many 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-sacrifice through all these years. An additional thank you to Evgenia, who is now twenty months old, for teaching me many important life lessons.

Sergey Dudiy G¨oteborg, May 2002

Bibliography

[1] H. E. Exner, Physical and chemical nature of cemented carbides, Interna- tional Metals Reviews 24, 149 (1979).

[2] P. Ettmayer, Hardmetals and Cermets, Annu. Rev. Mater. Sci. 19, 145 (1989).

[3] J. Gurland, New scientiﬁc approaches to development of tool materials,Inter- national Materials Reviews 33, 151 (1988).

[4] H. Doi, Advanced TiC and TiC-TiN base cermets,inProc. 2nd Int. Conf. Science Hard Mater., Rhodes, edited by E. A. Almond, C. A. Brookes, and R. Warren (Adam Hilger Ltd., Bristol, 1986).

[5] E. B. Clark and B. Roebuck, Extending the Application Areas for Titanium Carbonitride Cermets, Int. J. of Refractory Metals and Hard Materials 11,23 (1992).

[6] P. Ettmayer, H. Kolaska, W. Lengauer, and K. Dreyer, Ti(C,N) Cermets – Metallurgy and Properties, Int. J. of Refractory Metals and Hard Materials 13, 343 (1995).

[7] R. M. German, Sintering theory and practice (John Wiley and Sons, Inc., New York, 1996).

[8] J. Zackrisson, Development of cermet microstructures during sintering and heat-treatment (PhD Thesis, Chalmers University of Technnology and G¨oteborg University, G¨oteborg, 1999).

[9] M. G. Nicholas, Joining processes introduction to brazing and diffusion bonding (Kluwer, Dordrecht, 1998).

[10] M. G. Nicholas and D. Mortimer, Ceramic/metal joining for structural applications, Materials Science and Technology 1, 657 (1985).

[11] J. Intrater, The Challenge of Bonding Metals to Ceramics, Machine Design 61, 95 (1989).

73 Bibliography

[12] M. G. Nicholas, Reactive metal brazing of ceramics, Scandinavian Journal of Metallurgy 20, 157 (1991).

[13] O. M. Akselsen, Advances in brazing of ceramics, Journal of Materials Sci- ence 27, 1989 (1992).

[14] M. G. Nicholas and S. D. Peteves, Reactive Joining; Chemical Effects on the Formation and Properties of Brazed and Diffusion Bonded Interfaces, Scripta Metallurgica et Materialia 31, 1091 (1994).

[15] L. Ramqvist, Wetting of Metallic Carbides by Liquid Copper, Nickel, Cobalt and Iron,Int.J.PowderMetall.1, 2 (1965).

[16] J. G. Li, Wetting of Titanium Carbide by Molten Metals, Ceramic Transactions 35, 103 (1993).

[17] J. G. Li, Wetting, Adhesion and Electronic Structure of Metal/Ceramic Inter- faces: A Comparison with Metal-Semiconductor Schottky Contacts, Ceramic Transactions 35, 81 (1993).

[18] P. Xiao and B. Derby, Wetting of Titanium Nitride and Titanium Carbide by Liquid Metals, Acta Mater. 44, 307 (1996).

[19] G. Levi, M. Bamberger, and W. D. Kaplan, Wetting of Porous Titanium Car- bonitride by Al-Mg-Si Alloys, Acta Mater. 47, 3927 (1999).

[20] N. Frumin, N. Frage, M. Polak, and M. P. Dariel, Wettability and Phase For- mationintheTiCx/Al System, Scripta Mater. 37, 1263 (1997).

[21] N. Froumin, N. Frage, M. Polak, and M. P. Dariel, Wetting Phenomena in the TiC/(Cu-Al) System, Acta Mater. 48, 1435 (2000).

[22] L. Ramqvist, Preparation, properties and electronic structure of refractory carbides and related compounds, Jernkont. Ann. 153, 159 (1969).

[23] A. M. Stoneham, D. M. Duffy, J. H. Harding, and P. W. Tasker, Wetting and Interfacial Energies for Carbides and Oxides,inDesigning Ceramic Inter- faces II: Understanding and Tailoring Interfaces for Coating, Composite and Joining Applications, edited by S. D. Peteves (CEC DGZIII, Luxembourgh, 1993).

[24] A. M. Stoneham, M. M. D. Ramos, and A. P. Sutton, Feature article: How do they stick together? The statics and dynamics of interfaces, Phil. Mag. A 67, 797 (1993).

[25] P. G. de Gennes, Wetting: statics and dynamics, Rev. Mod. Phys. 57, 827 (1985).

74 Bibliography

[26] W. Zisman, in Contact Angle, Wettability and Adhesion,editedbyF.M. Fowkes (American Chemical Society, Washington, D.C., 1964). [27] J. G. Li, Wetting of Ceramic Materials by Liquid Silicon, Aluminium and Metallic Melts Containing Titanium and Other Reactive Elements: A Review, Ceramic International 20, 391 (1994). [28] X. B. Zhou and J. T. M. D. Hosson, Reactive Wetting of Liquid metals on ceramic substrates, Acta Mater. 44, 421 (1996). [29] H. Nakae, R. Inui, Y. Hirata, and H. Saito, Effects of Surface Roughness on Wettability, Acta Mater. 46, 2313 (1998). [30] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964). [31] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects,Phys.Rev.140, A1133 (1965). [32] R. O. Jones and O. Gunnarsson, The density functional formalism, its applications and prospects, Rev. Mod. Phys. 61, 689 (1989). [33] R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989). [34] R. M. Dreizler and E. K. U. Gross, Density Functional Theory (Springer, Berlin, 1990). [35] O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, mole- qules, and solids by the spin-density-functional formalism,Phys.Rev.B13, 4274 (1976). [36] D. C. Langreth and J. P. Perdew, The exchange-correlation energy of a metallic surface, Solid State Commun. 17, 1425 (1975). [37] D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface: Wave-vector analysis, Phys. Rev. B 15, 2884 (1977). [38] J. von Barth and L. Hedin, A local exchange-correlation potential for the spin polarized case. i,J.Phys.C5, 1629 (1972). [39] O. Gunnarsson, B. I. Lundqvist, and S. Lundqvist, Screening in a spin- polarized electron liquid, Solid State Commun. 11, 149 (1972). [40] D. M. Ceperley and B. J. Alder, Ground state of the electron gas by a stochas- tic method,Phys.Rev.Lett.45, 566 (1980). [41] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis, Can. J. Phys. 58, 1200 (1980).

75 Bibliography

[42] J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23, 5048 (1981).

[43] J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Phys. Rev. B 45, 13244 (1992).

[44] S. Kurth, J. P. Perdew, and P. Blaha, Molecular and Solid-State Tests of Den- sity Functional Approximations: LSD, GGAs, and Meta-GGAs, Int. J. Quant. Chem. 75, 889 (1999).

[45] V. L. Moruzzi, J. F. Janak, and A. R. Williams, Calculated Electronic Prop- erties of Metals (Pergamon, New York, 1978). [46] J. Häglund, G. Grimvall, T. Jarlborg, and A. Fernández Guillermet, Band structure and cohesive properties of 3d-transition-metal carbides and nitrides with the NaCl-type structure, Phys. Rev. B 43, 14400 (1991). [47] J. Häglund, G. Grimvall, and T. Jarlborg, Electronic structure, x-ray photoemission spectra, and transport properties of Fe3C (cementite),Phys.Rev.B 44, 2914 (1991). [48] A. Fernández Guillermet, J. Häglund, and G. Grimvall, Cohesive properties of 4d-transition-metal carbides and nitrides in the NaCl-type structure, Phys. Rev. B 45, 11557 (1992). [49] A. Fernández Guillermet, J. Häglund, and G. Grimvall, Cohesive properties and electronic structure of 5d-transition-metal carbides and nitrides in the NaCl structure, Phys. Rev. B 48, 11673 (1993). [50] C. S. Wang, B. M. Klein, and H. Krakauer, Theory of magnetic and structural ordering in iron,Phys.Rev.Lett.54, 1852 (1985). [51] D. C. Langreth and M. J. Mehl, Easily implementable nonlocal exchange- correlation energy functional,Phys.Rev.Lett.47, 446 (1981).

[52] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. A. Pederson, D. J. Singh, and C. Fiolhais, Atoms, molecules, solids, and surfaces: Applica- tions of the generalized gradient approximation for exchange and correlation, Phys.Rev.B46, 6671 (1992). [53] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approxima- tion Made Simple,Phys.Rev.Lett.77, 3865 (1996).

[54] Y. M. Juan and E. Kaxiras, Application of gradient corrections to density- functional theory for atoms and solids,Phys.Rev.B48, 14944 (1993).

[55] V. Ozolin¸ˇsandM.K¨orling, Full-potential calculations using the generalized gradient approximation: Structural properties of transitional metals, Phys. Rev. B 48, 18304 (1993).

76 Bibliography

[56] M. K¨orling and J. H¨aglund, Cohesive and electronic properties of transition metals: The generalized gradient approximation,Phys.Rev.B45, 13293 (1992).

[57] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Sounders College Publishing, Fort Worth, 1976).

[58] O. K. Andersen, Linear methods in band theory, Phys. Rev. B 12, 3060 (1975).

[59] H. L. Skriver, The LMTO Method (Springer, Berlin, 1984).

[60] M. Methfessel, Elastic constants and phonon frequencies of Si calculated by a fast full-potential linear-mufﬁn-tin-orbital method,Phys.Rev.B38, 1537 (1988).

[61] H. Krakauer, M. Posternak, and A. J. Freeman, Linearized augmented plane- wave method for the electronic band structure of thin ﬁlms,Phys.Rev.B19, 1706 (1979).

[62] E. Wimmer, H. Krakauer, M. Weinert, and A. J. Freeman, Full-potential self- consistent linearized-augmented-plane-wave method for calculating the electronic structure of molecules and surfaces: O2 molecule, Phys. Rev. B 24, 864 (1981).

[63] M. Weinert, E. Wimmer, and A. J. Freeman, Total-energy all-electron density functional method for bulk solids and surfaces,Phys.Rev.B26, 4571 (1982).

[64] M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, It- erative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients, Rev. Mod. Phys. 64, 1045 (1992).

[65] G. Kresse and J. Furthm¨uller, Efﬁcient iterative schemes for ab initio total- energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).

[66] G. Kresse and J. Furthm¨uller, Efﬁciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis, Comput. Mater. Sci. 6, 15 (1996).

[67] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in FORTRAN (Cambridge University Press, Cambridge, 1992).

[68] M. T. Yin and M. L. Cohen, Theory of lattice-dynamical properties of solids: Applications to Si and Ge,Phys.Rev.B26, 3259 (1982).

[69] H. Hellmann, Einf¨uhrung in die Quanten Theorie (Deuticke, Leipzig, 1937).

[70] R. P. Feynman, Phys. Rev. 56, 340 (1939).

77 Bibliography

[71] J. Ihm, A. Zunger, and M. L. Cohen, Momentum-space formalism for the total energy of solids,J.Phys.C12, 4409 (1979).

[72] P. Pulay, Mol. Phys. 17, 197 (1969).

[73] V. Heine, The Pseudopotential Concept, Solid State Physics 24, 1 (1970).

[74] W. E. Pickett, Pseudopotential methods in condensed matter applications, Computer Physics Reports 9, 115 (1989). [75] J. C. Phillips and L. Kleinman, New Method for Calculating Wave Functions in Crystals and Molecules, Phys. Rev. 116, 287 (1959). [76] M. L. Cohen and V. Heine, The Fitting of Pseudopotentials to Experimental Data and Their Subsequent Application, Solid State Physics 24, 37 (1970). [77] J. A. Appelbaum and D. R. Hamann, Self-Consistent Pseudopotentials for Si, Phys.Rev.B8, 1777 (1973).

[78] S. G. Louie, M. Schlüter,J.R.Chelikowsky,andM.L.Cohen,Self-consistent electronic states for reconstructed Si vacancy models,Phys.Rev.B13, 1654 (1976). [79] A. Zunger and M. L. Cohen, First-principles nonlocal-pseudopotential approach in the density-functional formalism: Development and application to atoms,Phys.Rev.B18, 5449 (1978). [80] D. R. Hamann, M. Schlüter, and C. Chiang, Norm-Conserving Pseudopoten- tials,Phys.Rev.Lett.43, 1494 (1979). [81] M. T. Yin and M. L. Cohen, Theory of ab initio pseudopotential calculations, Phys.Rev.B25, 7403 (1982). [82] G. B. Bachelet, D. R. Hamann, and M. Schlüter, Pseudopotentials that work: From H to Pu,Phys.Rev.B26, 4199 (1982).

[83] D. R. Hamann, Generalized norm-conserving pseudopotentials,Phys.Rev.B 40, 2980 (1989).

[84] E. L. Shirley, D. C. Allan, R. M. Martin, and J. Joannopoulos, Extended normconserving pseudopotentials,Phys.Rev.B40, 3652 (1989).

[85] L. Kleinman and D. M. Bylander, Efﬁcacious Form for Model Pseudopoten- tials,Phys.Rev.Lett.48, 1425 (1982).

[86] P. E. Bl¨ochl, Generalized separable potentials for electronic-structure calculations,Phys.Rev.B41, 5414 (1990).

[87] M. Y. Chou, Reformulation of generalized separable pseudopotentials, Phys. Rev. B 45, 11465 (1992).

78 Bibliography

[88] D. Vanderbilt, Optimally smooth norm-conserving pseudopotentials, Phys. Rev. B 32, 8412 (1985). [89] A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joannopoulos, Optimized pseudopotentials,Phys.Rev.B41, 1227 (1990). [90] N. Troullier and J. L. Martins, Efficient pseudopotentials for plane-wave calculations,Phys.Rev.B43, 1993 (1991). [91] D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,Phys.Rev.B41, 7892 (1990). [92] K. Laasonen, R. Car, C. Lee, and D. Vanderbilt, Implementation of ultrasoft pseudopotentials in ab initio molecular dynamics,Phys.Rev.B43, 6796 (1991). [93] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, Car-Parinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials,Phys.Rev.B 47, 10142 (1993). [94] G. Kresse and J. Hafner, Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements, J. Phys.: Condens. Matter 6, 8245 (1994). [95] D. D. Koelling and B. N. Harmon, A technique for relativistic spin-polarised calculations,J.Phys.C10, 3107 (1977). [96] X. Gonze, P. Käckell, and M. Scheffler, Ghost states for separable, normconserving, ab initio pseudopotentials,Phys.Rev.B41, 12264 (1990). [97] X. Gonze, R. Stumpf, and M. Scheffler, Analysis of separable potentials, Phys. Rev. B 44, 8503 (1991). [98] A. Khein, Analysis of separable nonlocal pseudopotentials,Phys.Rev.B51, 16608 (1995). [99] K. Stokbro, Mixed ultrasoft/norm-conserved pseudopotential scheme, Phys. Rev. B 53, 6869 (1996). [100] E. G. Moroni, G. Kresse, J. Hafner, and J. Furthmüller, Ultrasoft pseudopotentials applied to magnetic Fe, Co, and Ni: From atoms to solids, Phys. Rev. B 56, 15629 (1997). [101] S. Goedecker and K. Maschke, Transferability of pseudopotentials, Phys. Rev. A 45, 88 (1992). [102] M. Teter, Additional condition for transferability in pseudopotentials, Phys. Rev. B 48, 5031 (1993). [103] N. J. Ramer and A. M. Rappe, Designed nonlocal pseudopotentials for enhanced transferability, Phys. Rev. B 59, 12471 (1999).

79 Bibliography

[104] I. Grinberg, N. J. Ramer, and A. M. Rappe, Quantitative criteria for transfer- able pseudopotentials in density functional theory,Phys.Rev.B(in press), (2002).

[105] S. G. Louie, S. Froyen, and M. L. Cohen, Nonlinear ionic pseudopotentials in spin-density-functional calculations,Phys.Rev.B26, 1738 (1982).

[106] L. Hansen et al, Dacapo-1.30, Center for Atomic Scale Materials Physics (CAMP), Denmark Technical University.

[107] G. Kresse and J. Hafner, Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium,Phys.Rev. B 49, 14251 (1994).

[108] G. Kresse and J. Hafner, Ab initio molecular dynamics for open-shell transition metals,Phys.Rev.B48, 13115 (1993).

[109] G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).

[110] W. S. Williams, Physics of Transition Metal Carbides, Materials Science and Engineering A105/106, 1 (1988).

[111] W. Lengauer and P. Ettmayer, Recent advances in the ﬁeld of transition-metal refractory nitrides, High Temperatures – High Pressures 22, 13 (1990).

[112] L. Johansson, Electronic and structural properties of transition-metal carbide and nitride surfaces, Surface Science Reports 21, 177 (1995).

[113] K. Schwarz, Band structure and chemical bonding in transition metal carbides and nitrides, CRC Critical Reviews in Solid State and Materials Sci- ences 13, 211 (1987).

[114] J. H¨aglund, A. F. Guillermet, G. Grimvall, and M. K¨orling, Theory of bonding in transition-metal carbides and nitrides,Phys.Rev.B48, 11 685 (1993).

[115] The Massachusetts Institute of Technology online datababase of covalent radii of elements at http://web.mit.edu.

[116] H. W. Hugosson, A Theoretical Treatise on the Electronic Structure of De- signer Hard Materials (PhD Thesis, Uppsala University, Uppsala, Sweden, 2001).

[117] H. W. Hugosson, O. Eriksson, U. Jansson, and B. Johansson, Phase stabilities and homogeneity ranges in 4d-transition-metal carbides: A theoretical study, Phys.Rev.B63, 134108 (2001).

80 Bibliography

[118] H. W. Hugosson, P. Korzhavyi, U. Jansson, B. Johansson, and O. Eriks-

son, Phase stabilities and structural relaxations in substoichiometric TiC1 x, Phys. Rev. B 63, 165116 (2001).

[119] P. Korzhavyi, L. V. Pourovskii, H. W. Hugosson, A. V. Ruban, and B. Johans- son, Ab Initio Study of Phase Equilibria in TiCx,Phys.Rev.Lett.88, 015505 (2002).

[120] P. Blaha, J. Redinger, and K. Schwarz, Bonding study of TiC and TiN. II. Theory,Phys.Rev.B31, 2316 (1985).

[121] V. A. Gubanov, V. P. Zhukov, and A. L. Ivanovsky, New achievements in theoretical calculations of electronic structure and properties of transition metal refractory compounds, Reviews of Solid State Science 5, 315 (1991).

[122] L. Ramqvist, K. Hamrin, G. Johansson, A. Fahlman, and C. Nording, Charge transfer in transition metal carbides and related compounds studied by ESCA, J. Phys. Chem. Solids 30, 1835 (1969).

[123] S. Zaima, Y. Shibata, H. Adachi, C. Oshima, S. Otani, M. Aono, and Y. Ishizawa, Atomic chemical composition and reactivity of the TiC(111) surface, Surf. Sci. 157, 380 (1985).

[124] G. E. Hollox, , Mater. Sci. Eng. 3, 121 (1968/69).

[125] C. Herring, , in Structure and Properties of Solid Surfaces, edited by R. Gomer (University of Chicago Press, Chicago, 1953).

[126] M. Aono, Y. Hou, R. Souda, C. Oshima, S. Otani, and Y. Ishizawa, Direct Analysis of the Structure, Concentration, and Chemical Activity of Surface Atomic Vacancies by Specialized Low-Energy Ion-Scattering Spectroscopy: TiC(001),Phys.Rev.Lett.50, 1293 (1983).

[127] Y. Kido, T. Nishimura, Y. Hoshino, S. Otani, and R. Souda, Rumpled relaxation of TiC(001) and TaC(001) determined by high-resolution medium- energy ion scattering spectroscopy, Phys. Rev. B 61, 1748 (2000).

[128] A. P. Sutton and R. W. Ballufﬁ, Interfaces in Crystalline Materials (Clarendon Press, Oxford, 1995).

[129] U. Sch¨onberger, O. K. Andersen, and M. Methfessel, Bonding at metal- ceramic interfaces; ab initio density-functional calculations for Ti and Ag on MgO, Acta Metall. Mater. 40, S1 (1992).

[130] M. W. Finnis, The theory of metal-ceramic interfaces, J. Phys.: Condens. Matter 8, 5811 (1996).

[131] F. Ernst, Metal-oxide interfaces, Mater. Sci. Eng. R Rep. 14, 97 (1995).

81 Bibliography

[132] J. Hartford, Total Energy Calculations for Predictive Modeling of Crystalline Materials Strength (PhD Thesis, Chalmers University of Technnology and G¨oteborg University, G¨oteborg, 1998).

[133] J. Hartford, Interface energy and electron structure for Fe/VN, Phys. Rev. B 61, 2221 (2000).

[134] I. G. Batirev, A. Alavi, M. W. Finnis, and T. Deutsch, First-Principles Calcu- lations of the Ideal Cleavage Energy of Bulk Niobium(111)/α-Alumina(0001) Interfaces, Phys. Rev. Lett. 82, 1510 (1999).

[135] Y. Yourdshahyan, C. Ruberto, M. Halvarsson, L. Bengtsson, V. Langer, B. I. Lundqvist, S. Ruppi, and U. Rolander, Theoretical Structure Determination of a Complex Material: κ-Al2O3,Phys.Rev.B61, 2221 (2000).

[136] Y. Yourdshahyan, Alumina (Al2O3) and Oxidation of Aluminum: A First- Principles Study (PhD Thesis, Chalmers University of Technnology and G¨oteborg University, G¨oteborg, 1999).

[137] S. M¨uller, C. Wolverton, L.-W. Wang, and A. Zunger, Prediction of alloy precipitate shapes from ﬁrst principles, Europhysics Letters 55, 33 (2001).

[138] A. Zunger, First Principles Statistical Mechanics of Semiconductor Alloys and Intermetallic Compounds,inNATOASIonStatics and Dynamics of Al- loy Phase Transformations, edited by P. Turchi and A. Gonis (Plenum Press, New York, 1994).

[139] M. Rettenmayr, H. E. Exner, and W. Mader, Electron microscopy of binder phase deformation in WC-Co alloys, Materials Science and Technology 4, 984 (1988).

[140] R. Benedek, A. Alavi, D. N. Seidman, L. H. Yang, D. A. Muller, and C. Woodward, First Principles Simulation of a Ceramic/Metal Interface with Misﬁt, Phys. Rev. Lett. 84, 3362 (2000).

[141] K. H. Johnson and S. V. Pepper, Molecular-orbital model for metal-sapphire interfacial strength, J. Appl. Phys. 53, 6634 (1982).

[142] K. Nath and A. B. Anderson, Oxidative bonding of (0001) α-Al2O3 to close- packed surfaces on the ﬁrst transition-metal series, Sc through Cu,Phys.Rev. B 39, 1013 (1989).

[143] M. Kohyama, S. Kose, M. Kinoshita, and R. Yamamoto, Electronic Structure calculations of Transition metal-Alumina Interfaces, J. Phys. Chem. Solids 53, 345 (1992).

[144] C. Noguera and G. Bordier, Theoretical approach to interfacial metal-oxide bonding, J. Physique III 4, 1851 (1994).

82 Bibliography

[145] P. Alemany, R. S. Boorse, J. M. Burlitch, and R. Hoffmann, J. Chem. Phys. 97, 8464 (1993).

[146] P. Alemany, Metal-ceramic adhesion: band structure calculations on transition-metal–AlN interfaces, Surf. Sci. 314, 114 (1994).

[147] A. M. Stoneham and P. W. Tasker, Metal-non-metal and other interfaces: the role of image interactions,J.Phys.C18, L543 (1985).

[148] M. W. Finnis, The interaction of a point charge with an aluminium (111) surface, Surf. Sci. 241, 61 (1991).

[149] M. W. Finnis, Metal-ceramic cohesion and the image interaction, Acta Met- all. Mater. 40, S25 (1992).

[150] D. M. Duffy, J. H. Harding, and A. M. Stoneham, Atomistic Modelling with Image Interactions, Acta Metall. Mater. 40, S11 (1992).

[151] D. M. Duffy, J. H. Harding, and A. M. Stoneham, Atomistic modelling of metal-oxide interfaces with image interactions, Phil. Mag. A 67, 865 (1993).

[152] M. W. Finnis, R. Kaschner, C. Kruse, J. Furthm¨uller, and M. Schefﬂer, The interaction of a point charge with a metal surface: theory and calculations for (111), (100) and (110) aluminium surfaces, J. Phys.: Condens. Matter 7, 2001 (1995).

[153] C. Li, R. Wu, and A. J. Freeman, Energetics, bonding mechanism, and electronic structure of metal-ceramic interfaces: Ag/MgO(001),Phys.Rev.B48, 8317 (1993).

[154] J. Goniakowski, Adsorption of palladium on the MgO(100) surface: Depen- dence on the metal coverage,Phys.Rev.B58, 1189 (1998).

[155] C. Verdozzi, D. R. Jennison, P. A. Schultz, and M. P. Sears, Sapphire (0001) Surface, Clean and with d-Metal Overlayers, Phys. Rev. Lett. 82, 799 (1999).

[156] A. Bogicevic and D. R. Jennison, Variations in the Nature of Metal Adsorp- tion on Ultrathin Al2O3 Films,Phys.Rev.Lett.82, 4050 (1999).

[157] D. J. Siegel, L. G. Hector, Jr., and J. B. Adams, Adhesion, atomic structure, and bonding at the Al(111)/α-Al2O3(0001) interface: A ﬁrst principles study, Phys. Rev. B 65, 085415 (2002).

[158] D. J. Siegel, L. G. Hector, Jr., and J. B. Adams, Adhesion, stability, and bonding at metal/metal-carbide interfaces: Al/WC, Surf. Sci. 498, 321 (2002).

[159] D. J. Siegel, L. G. Hector, Jr., and J. B. Adams, First-principles study of metal¯carbide/nitride adhesion: Al/VC vs. Al/VN, Acta Mater. 50, 605 (2002).

83 Bibliography

[160] J. B. Adams, L. G. Hector, Jr., D. J. Siegel, H. Yu, J. Zhong, and Y. T. Cheng, Adhesion, Lubrication, and Wear on the Atomic Scale, Surf. Interface Anal. 31, 619 (2001).

[161] R. G. Barrera and C. B. Duke, Dielectric continuum theory of the electronic structure of interfaces,Phys.Rev.B13, 4477 (1976).

[162] A. M. Stoneham, Systematics of metal-insulator interfacial energies: a new rule for wetting and strong catalyst-support interaction,Appl.Surf.Sci.14, 249 (1983).

[163] A. M. Stoneham, D. M. Duffy, and J. H. Harding, Understanding Wetting and Spreading: Non-Reactive Metals on Oxides and Similar Systems,inProc. Int. Conf. High Temperature Capillarity, edited by N. Eustathopoulos (Re- proprint, Bratislava, 1995).

[164] D. W. Lynch, C. G. Olson, D. J. Peterman, and J. H. Weaver, Optical prop-

: erties of TiCx (0:64 x 0 90)from0.1to30eV,Phys.Rev.B22, 3391 (1980).

[165] J. Rivory, J. M. Behaghel, S. Berthier, and J. Lafait, Optical properties of substoichiometric TiNx, Thin Solid Films 78, 161 (1981). [166] R. E. Cohen, M. J. Mehl, and D. A. Papaconstantopoulos, Tight-binding total-energy method for transition and noble metals,Phys.Rev.B50, 14694 (1994).

[167] M. J. Mehl and D. A. Papaconstantopoulos, Applications of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monoatomic metals,Phys.Rev.B54, 5419 (1996).

[168] DoD-TBMD code at http://cst-www.nrl.navy.mil/bind/dodtb/.