Toughness Enhancement in Transition Metal Nitrides

Linköping Studies in Science and Technology

Licentiate Thesis No. 1462

Toughness Enhancement

in Transition Metal Nitrides

Davide G. Sangiovanni

LIU-TEK-LIC-2011:2

Thin Film Physics Division

Department of Physics, Chemistry and Biology (IFM)

Linköping University

© Davide G. Sangiovanni, 2011

ISBN: 978-91-7393-257-8

ISSN: 0280-7971

Printed by LiU-tryck

Linköping, Sweden, 2011

Abstract

Toughness enhancements can be induced in cubic-B1 transition metal nitride alloys by an increased occupation of the d-t2g metallic states. In this

Licentiate Thesis I use density functional theory to investigate the mechanical properties of TiN and VN and of the ternaries obtained by replacing 50% of Ti and V atoms with M (M = V, Nb, Ta, Mo, and W) to form ordered structures with minimum number of inter-metallic bonds. The calculated values of elastic constants and moduli show that ternary alloys with high valence electron concentrations (M = Mo and W), have large reductions in shear moduli and C44 elastic constants, while retaining the

typically high stiffness and incompressibility of ceramic materials. These results point to significantly improved ductility in the ternary compounds. This important combination of strength and ductility, which equates to material toughness, stems from alloying with valence electron richer d-metals. The increased valence electron concentration strengthens metal– metal bonds by filling metallic d-t2g states, and leads to the formation of a

layered electronic configuration upon shearing. Comprehensive electronic structure calculations demonstrate that in these crystals, stronger Ti/V – N and weaker M – N bonds are formed as the valence electron concentration is increased. This phenomenon ultimately enhances ductility by promoting dislocation glide through the activation of an easy slip system.

Preface

This Thesis is part of my PhD studies in the Thin Film Physics Group at Linköping University. The aim of the Thesis is to provide deeper insights on the origin and effects of electronic structure processes on the mechanical properties of cubic transition metal nitride alloys. In particular, I look for materials which can be used as hard-ductile coatings.

The work was supported by the Swedish Research Council (VR) and the Swedish Strategic Research Foundation (SSF) Program on Materials Science and Advanced Surface Engineering. All calculations were performed on the Neolith and the Kappa clusters located at the National Supercomputer Centre (NSC) in Linköping.

Included Papers

Paper I Electronic mechanism for toughness enhancement in

TixM1-xN (M = Mo and W)

D.G. Sangiovanni, V. Chirita, and L. Hultman Physical Review B 81, 104107 (2010)

Paper II Supertoughening in B1 transition metal nitride alloys by

increased valence electron concentration

D.G. Sangiovanni, L. Hultman, and V. Chirita

Accepted for publication in Acta Materialia

Paper III Structure and mechanical properties of TiAlN-WNx thin

films

T. Reeswinkel, D.G. Sangiovanni, V. Chirita, L. Hultman, and J.M. Schneider

Acknowledgements

Thanks to Vio, my main supervisor, for guiding and supporting me in this project. You are an unlimited source of ideas and the best motivator ever! Thanks to Lars, my supervisor, for all the useful feedback and advice; Mattias, Arkady, and Peter Steneteg for all the times you helped me out; Hanna Kindlund for all the interesting discussions and our nice collaboration; Björn, Gueorgui, Peter Münger, Igor, Sergey, and Ferenc for many useful discussions and suggestions; Finn, Javier, Carina, and Jens for your time dedicated on growth and analysis of TiWN films. I thank all colleagues and friends at IFM, and all friends in Linköping, for the great time spent together.

Ringrazio i miei genitori Roberto e Palma, per il continuo supporto psicologico e le attenzioni, ma soprattutto per credere sempre in me. Grazie a tutta la mia famiglia e i miei amici italiani. Tack till Erik, Helga, Malte och Gustaf, min familj här i Sverige!

Un ringraziamento speciale va ad Elinor. Sei la principale fonte di ispirazione ed energie, vicina ogni giorno con affetto e cure. Vivere con te è la vera ragione per cui mi trovo in Svezia.

Table of contents

1. INTRODUCTION 1

2.

METHODOLOGY 5

2.1. Density functional theory 5

2.2. Exchange and correlation potential 8

2.3. Crystal orbital overlap population (COOP) analysis 9

3.

RESULTS 15

3.1. Paper I 15

3.2. Paper II 20

3.3. Paper III 20

4.

CONCLUSIONS AND FUTURE WORK 23

References 25

PAPER I 27

PAPER II 37

1. INTRODUCTION

Toughness, in materials science, is one of the most desired mechanical properties, as it reflects the ability of materials to withstand stresses before fracture.

In a typical stress vs. strain graph, toughness corresponds to the area under the curve, so in other words, it is the energy the material can absorb up to fracture (grey colored area in Fig. 1).

Figure 1. Stress-strain curve. (1) - elastic region (Hooke’s law). (2) - plastic

region. (3) - necking. (4) - rupture. (A) - yield strength (yield point). (B) - ultimate strength. Grey area - toughness.

Toughness is generally defined as a suitable combination of hardness/strength and ductility, which yields better response to elastic and plastic deformations, or compressive and tensile stresses. There are numerous applications in which this combination of properties is obviously extremely useful, but this is particularly the case in the cutting tools industry. To endure the extreme temperatures and pressures involved in the cutting process, tools are typically coated with protective films, consisting primarily of hard and/or wear resistant materials.

For several decades now, transition metal nitrides (TMN) have widely been used as protective coatings, as they possess remarkable physical and mechanical properties such as high melting temperature, oxidation and corrosion resistance, high hardness and wear resistance.1, 2 _{TMN often}

crystallize in the cubic-B1 rocksalt (NaCl) structure; a configuration in which each metal or N atom is located at the centre of an octahedron of six N, respectively metal atoms. In such cubic close-packed structure, metal and N atoms are nearest neighbors and form strong ionic – covalent bonds, which confer high hardness to the material. Due to this mechanical behavior, these compounds are usually labeled as ceramics. Their electrical properties, however, are similar to those of parent metals.3

The first, and most studied, TMN employed as a protective coating material is TiN.4 _{Later, partial replacements of Ti atoms by a second}

metallic element have been tested to improve mechanical and physical features. In particular, many researchers have focused on TiAlN coatings.5

Additions of Al are proved to enhance hardness, wear and oxidation resistance. Other well known TiN-based hard ternary alloys are, for instance, TiZrN6 _{and TiCrN.}7, 8 _{In most cases, the method of alloying is}

adopted to increase the hardness of the films. This technique has been successfullyusedtoenhancethehardnessofTMN,morepreciselyNbZrN,9, 10

increase in hardness, however, typically results in an increase in brittleness, which leads to crack initiation and propagation in the coating.11

During the time I spent in the Thin Film Physics Group I performed a theoretical study on the mechanical properties of binary, ternary and quarternary TMN. I explored the possibility of developing hard, but less brittle alloys. The three papers included in this thesis analyze the electronic mechanisms which, upon alloying TMN, are shown to promote ductility, and hence enhance toughness, in these hard ceramic compounds. However, since even the most advanced computational techniques are based on a number or form of approximations, the predictions presented herein need experimental confirmations.

2. METHODOLOGY

The theoretical technique employed for my calculations is called

density functional theory, which is nowadays widely applied in materials

science. This technique can predict with high accuracy and reliability many physical and electronic material properties. The shortcoming, however, is to be limited to small simulation cells (hundreds of atoms), since in terms of computational time the method is expensive.

2.1. Density functional theory

For solid systems (~1023 _{interacting particles) it is not possible to find}

exact solutions to the Schrödinger equation. Different approximations, however, are adopted to simplify this many-body problem.

First of all, as shown by Born and Oppenheimer,12 _{given the large}

electron-nucleus difference in mass, and hence frequency of vibration, one can solve the electronic equation by assuming fixed nuclei positions throughout the calculation.

Within density functional theory (DFT),13 the many-electrons Schrödinger equation can be reduced to a single-particle equation. The DFT is based upon Hohenberg and Kohn theorems, which demonstrate that the total energy E is: (i) a unique functional

!

E

_{[ ]}

" of the charge density

!

"(r )! , and (ii) minimized for:

! "(r ) = # ! " (r )! , (1) where ! "

# (r )! is the ground state charge density.

!

"

# (r )! represents a system of non-interacting electrons immersed in an effective potential field

!

V_eff(r ). !

Kohn and Sham showed14 _{that the} V_eff(r ) is formed by an attractive part !

divided in a classical Hartree potential ( ) and non-classical exchange and correlation potential ( ):

! V_eff(r ) = V! ext( ! r ) + VHartree( ! r ) + VXC( ! r ). (2) All these quantities are computed from

!

"(r )! . The resulting single-particle equation, namely, Kohn-Sham equation is:

! "1 2# 2 + Veff(r )! $ % & ' ₍ ) *j(r ) = e! j*j(r ). (3) ! The solutions to eq. (3) are found self-consistently as showed

schematically in Fig. 2. At the beginning of any iteration, excepting the first, the initial charge density is computed by a combination of output charge

densities obtained from an m number of previous steps. Such

procedure, as for example showed in Ref. [15], reduces the number of iterations needed to achieve the convergence.

A DFT calculation stops when the difference between input and output charge densities is within a fixed tolerance !. The so found ground state charge density

!

"

# (r )! is a fundamental property of the system. From

!

"

# (r )! one can in principle determine all physical information related to a particular system.

False

Mixing

Figure 2. DFT self consistent loop

START from a trial charge density

! "in( ! r ) Compute !

V_eff(r ) from eq. (2) !

Compute the wavefunctions

!

"j(r ) from eq. (3) !

Compute the new charge density

! "out(r ) =! #j(r )! 2 j=1 N occ

$

?

T ru e END

2.2. Exchange and correlation potential

Although an exact mathematical formulation for the exchange and correlation potential

!

V_XC(r )! is not known, different forms of approximation

are available.

In the local-density approximation (LDA), the exchange and correlation energy EXC is a functional of the electron density:

! E_XC

_{[ ]}

" = dr #! XC "( ! r )

[

]

"(r )!

$

. (4)

The exchange and correlation energy density is equivalent to that of a

homogeneous electron gas of density corresponding to the electron density at any point in space. While in solid state calculations the LDA generally yields accurate results for various materials properties,16 it fails in describing some features of finite systems (molecules and clusters), and low dimensional systems (such as surfaces).17

In the generalized gradient approximation (GGA), electron density gradient corrections are applied to the local value of , so that the exchange and correlation energy EXC takes the following form:18

! E_XC

_{[ ]}

" = dr #! XC GGA "(r ), $"(! r )!

[

]

"(r )!

%

. (5) The functional is available in different forms.18-22 Despite the fact that

GGA potentials are designed to improve beyond the accuracy of DFT+LDA calculations, results from solid state calculations performed with GGA are often less satisfactory than those obtained with LDA itself. Nevertheless, GGA describe the equilibrium volume of 3d metals better than LDA,16 _and

the GGA potential, (PW91),20 was selected and used in the calculations discussed herein.

2.3. Crystal orbital overlap population (COOP) analysis

To understand the mechanical properties and trends in materials, it is important to investigate their binding characteristics. The generally adopted method for quantifying contributions to bonding is the analysis of density of states (DOS) or band structure. This procedure, however, does not easily resolves the bonding and anti-bonding states, particularly for complex compounds characterized by intricate bonding.

To circumvent this problem, crystal orbital overlap population23

(COOP) analysis can be used to investigate binding contributions, of primarily covalent character, in the chemical bonds of a crystal. The fundamental COOP idea is to partition the wavefunctions into molecular orbitals (MO) derived from linear combinations of atomic states.

2.3.1. Mathematical formalism of COOP

In the original two-center MO formulation:23

!

" = c1 #1 + c2 #2 , (6)

the electron distribution in orbital

!

" is described by the square of the wavefunction module ! "

_{( )}

r ! 2. Here, ! "1 and !

"2 denote the normalized

states, but not mutually orthogonal, of different atoms. By imposing the normalization of ! " : 1 = " " =

_#

dr ! "(r )! 2 = c1 2 + c2 2 + 2 Re c

(

1$c2 %1 %2

)

= = c1 2 + c2 2 + 2 Re c

(

1"c2S12

)

, (7)

Figure 3. Linear combinations of atomic orbitals. In (a), (b), and (c) a

constructive superposition gives rise to a bonding state. Figures (d), (e), and (f) show a destructive wave interference with the formation of an anti-bonding state.

in which the numbers

!

c₁2, and

!

c₂2, quantify the localization of the electron

around each atomic center, one can obtain the all important overlap population (OP): ! OP₁₂= Re c1 "_c 2S12

(

)

. (8) Typically, OP estimates the molecular orbital bonding or anti-bonding covalent character in the following manner: positive, respectively negative values are assigned to bonding, respectively anti-bonding contributions, while its absolute value returns the strength of the bond.

The OP value is connected to the probability of finding the electron in between the atoms. Figure 3 illustrates the meaning of OP. For simplicity, the space considered here is one-dimensional. In Fig. 3(a), (b), and (c), the constructive interference between two atomic wavefunctions is shown. As it can be seen, in between the atoms, in this case the two atomic orbitals have the same sign, and hence the inner product

!

c₁"_c

2S12 =

%

dr c! 1"#1"(r $! r ! 1)& c2#2(r $! r ! 2) must necessarily be positive. The

opposite situation occurs in the formation of an anti-bonding state (Fig. 3(d), (e), and (f)). The complex numbers c1, c2, and c3 are chosen so as to

normalize MO. If all c numbers are assumed positive and real, one can easily verify that OP is positive or negative, when a bonding or anti-bonding MO is formed. It is also interesting to note that, at negative OP values, the MO normalization constraint equates to concentrating more electron density onto atomic sites (see the red color in Fig. 3f). In the figure, arrows show that the effect of negative OP is to “remove” electrons from a covalent bond and “place” them in the cores.

By generalizing this formalism to the solid state case, one can partition the wavefunctions into “two-center-molecular-orbitals” and

states. Thereafter, the COOP is defined as the overlap population weighted density of states, in which bonding, anti-bonding orbitals are resolved in energy and identified by positive, respectively, negative values.

COOP calculations typically use tight-binding parameterizations, based on localized atomic-like orbital basis sets, which simplify the chemical bonding analysis. Nevertheless, a more correct description of chemical bonds in solids can be obtained by combining accurate electronic structure and COOP calculations. In Paper II we report results of such an improved COOP analysis, based on VASP calculations using PAW potentials, known to compare very well in terms of accuracy with full-potential methods. No previous COOP evaluations using VASP calculations have been reported in the literature, hence the procedure is detailed below.

VASP computes the projections P of the wavefunctions !

"_n!

k (

!

r ) onto

non-zero spherical harmonics

!

Y_lm(r )! within spheres of a radius R, which depend on the atomic type i:

! P_lm,n! k Ri _{= Y} lmRi "_n! _k , (9)

It then follows that for a

!

Y_lmRi " Y_{l # #} m

R j _{orbital, COOP can be calculated using:}

! COOP_{lm, "} i, j_{l " m} (#) = $(# % #_n! k ) & Re Plm, nk ! Ri * _{& S} lm, " l " m i, j & P_{l " " m , n}! k R j

(

)

nk !

'

, (10)

where S is the overlap matrix given by:

S_lm,l'm'i, j = YlmRi Yl'm'

performed for all interacting pairs of atoms in the unit/calculation cell, and for all occupied states, to obtain the strength of a particular covalent bond in the crystal. The resulting integrated COOP (ICOOP) is thus the equivalent for solids of the bond order in molecules and measures the

!

Y_lmRi " Y_{l # #} m R j

chemical bond strength:

! ICOOP_{lm, " l " m} = d# COOP_{lm, "} i, j_{l " m} (#) i, j i$ j

%

&' #Fermi

(

. (12)

Upon undertaking this COOP analysis, I should mention the following limitations of this approach: (i) the radii and of atomic spheres are arbitrarily chosen; (ii) the radius is unique for the atom i, i.e. all orbitals have the same spatial extension; (iii) COOP and ICOOP estimate the covalent character of a bond but provide no information about the ionic character. Consequently, COOP and ICOOP only qualitatively estimate bond strength and one should be cautious when comparing bonding of different compounds. In Paper II, COOP and ICOOP are primarily used to assess the effects of strain/shear on the strength of a particular chemical bond, namely between pairs of first and second neighbors in the crystal.

3. RESULTS

3.1. Paper I

This paper reports the findings related to ductility and toughness enhancements in B1-Ti0.5Mo0.5N and Ti0.5W0.5N. The main result is that in

response to shear deformations, a layered electronic structure is formed in these alloys. This layered electron density distribution consists in alternating layers of high/low charge density, aligned to metal stacking and normal to the direction of the applied strain (see Fig. 4c and 4d).

Figure 4. Charge densities on the (001) plane of 10% shear strained (a) TiN,

(b) Ti0.5Al0.5N, (c) Ti0.5Mo0.5N, and (d) Ti0.5W0.5N (from Ref. [24]). Color

As shown in the paper, the electronic mechanism responsible for these charge distribution effects is rooted in the additional valence electrons of the alloying species, which ultimately enhance the occupancy of d-t2g metallic

states in ternary compounds. This phenomenon is the key to understanding the significant decrease in C44 and shear moduli G values observed upon

alloying TiN or TiAlN with W and Mo, and the resulting ductility trends. A more intuitive picture of the entire process emerges if one applies ligand field theory arguments, which can be used to understand the properties of transition metal complexes. Particularly for cubic lattices, the method allows for straightforward calculations of transition metal d-orbital energies, and can be used to interpret the changes induced in the density of states of nitrides by external fields.

TiN is a typical example in this sense, as shown in Fig. 5, where one can see that the five “initial” d-states have the same energy.

Figure 5. The d-orbitals of a Ti atom not interacting with its N neighbors

To approach the situation in the actual TiN lattice, one can assume the six N atoms as approaching Ti from both verses of each Cartesian direction. This octahedral electronic field splits the 5 orbitals in the two well known, eg and

t2g, d-orbital sets (Fig. 6). Note that the eg orbitals, and , which

point in the direction of neighboring atoms, have stronger interaction with N electrons, and consequently these states have higher energy.

Figure 6. Energy splitting of d-orbitals.

In the actual TiN lattice, two d-eg, one s and three p orbitals hybridize

to form six sp3_d2 _{orbitals, which overlap with the 2p N orbitals to form}

bonding (!) and anti-bonding (!*) states (Fig. 7). In addition to this 1st neighbors Ti-N orbitals interaction, the Ti d-t2g orbitals form bonding and

anti-bonding states with the corresponding d-t2g orbitals of 2nd neighbors Ti

atoms. Nevertheless, given the greater distance between 2nd _{neighbors, the}

superposition of Ti d-t2g orbitals yields a considerably smaller energy

Figure 8. Formation of Ti-Ti bonding and anti-bonding states.

Figure 9. Schematic analogy/comparison of discrete energy states in the

“local” molecular orbitals view with the energy spectrum of a DOS plot. Dashed line indicates Fermi energy.

To qualitatively assess the role played by the d-d bonding states in the compounds discussed in this thesis, it is useful to compare the molecular orbital diagram obtained for TiN with the typical continuous energy spectrum of a TiN crystal (Fig. 9). As it can be seen, d-t2g states are

positioned very close to the Fermi energy. It then becomes evident that by increasing or decreasing the valence electron concentration, which equates to increasing/decreasing, or tuning, the electron population of these d-d bonding states, one can significantly influence the mechanical properties of transition metal nitrides. The significant enhancement in ductility and toughness reported in the first paper for TiMoN and TiWN clearly demonstrates the validity and usefulness of this approach to materials properties design.

3.2. Paper II

The promising results published in the previous article led to further investigations of mechanical properties of cubic transition metal nitride alloys. The main idea pursued in the research for Paper II was the possibility of controlling ductility by tuning the population of d-t2g metallic states.

Therein, a systematic study, of the effects of valence electron concentration (VEC) on the properties of ternary alloys, is reported. VEC spans the range between 9 (TiN) and 10.5 (V0.5W0.5N) electrons per unit cell. The results

validate and extend the findings reported in Paper I to a significant number of Ti and V ternary alloys.

3.3 Paper III

This paper is dedicated to the study of a recently synthesized quarternary in Aachen, namely TiAlWN. The compound was grown as a thin film by DC sputtering and High Power Pulsed Magnetron Sputtering

properties, primarily lattice constants and Young modulus, for several concentrations of W in TiAlN and to probe the role of vacancies on the properties of the compound. The theoretical results are confirmed and support the composition, structure and mechanical properties correlation found experimentally.

4. CONCLUSIONS AND FUTURE WORK

In this Thesis I discuss the findings related to toughness enhancements in cubic TMN alloys induced via increased occupation of the metallic d-d states. Theoretical hardness estimations (as reported in Paper II), corroborated with indentation measurements suggest that all these alloys are inherently hard, and can therefore be classified as hard-ductile materials. Nevertheless, the final confirmation of predictions made herein can only be obtained through experimental validation and certification.

In addition to this project, completely based on ab-initio calculations, I will continue my current study of TiN surface-related phenomena and properties, using classical molecular dynamics (MD) simulations. This technique complements DFT calculations, as it allows for truly large scale simulations, of systems containing millions of atoms and nanosecond time scales, at a reasonable computational cost. The expectation here is to obtain detailed information, not available experimentally, of the key mechanisms governing the nucleation, thin film growth, and microstructure evolution, of this most important model system for nitrides.

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