Spring term 2019 | LITH-IFM-A-EX—19/3736--SE
Phase stability and mechanical properties
of M
4
AlB
4
(M = Cr, Hf, Mo, Nb, Ta, Ti, V, W,
Zr) from first principles
Adam Carlsson
Examiner, Johanna Rosén Supervisor, Martin Dahlqvist Co-supervisor, Hans Lind
Department of Physics, Chemistry and Biology
Linköping University
URL för elektronisk version
ISBN
ISRN: LITH-IFM-A-EX--19/3736--SE
_________________________________________________________________
Serietitel och serienummer ISSN
Title of series, numbering 1____________________________
Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________ Titel
Phase stability and mechanical properties of M4AlB4 (M = Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr) from first principles
Författare Adam Carlsson
Sammanfattning
The recent discovery of Cr4AlB4, a laminated ternary metal boride belonging to the family of layered MAB-phases,
where the transition metal boride layers are interleaved by an A layer, has spurred theoretical investigation for novel M4AlB4 phases. In this study, first-principles calculations were applied in order to investigate the thermodynamical
stability and mechanical properties of M4AlB4 where M = Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr and the A layer was kept
fixed as Al.
The thermodynamical stability calculations validate the recently discovered Cr4AlB4 phase's stability and suggest the
neighboring phase Mo4AlB4 to be stable. Additionally, the phases Mo3AlB4, Mo2AlB2, Ti4AlB4 and Ti2AlB2 indicates
phases close to stable with a formation enthalpy within the range of 0 < ΔH < 25 meV per atom compared to competing phases. Hence dynamical stability investigations were carried out, which indicates Mo4AlB4 to be
dynamically stable. The stability of Mo4AlB4 does encourage synthesizing attempts to be kept in mind as a future
project. Phase stability trends of the 111, 212, 314 and 414 compositions were discovered, where a 212, 314 and 414 composition is seen to be more stable for an M-element with lower electron configuration.
Furthermore, the mechanical properties of the 414 compositions were investigated by systematically applying different strains to the unit cell. The bulk-, shear- and Young's-modulus were derived and are presented, where Ti4AlB4
demonstrates values similar to the commended Ti2AlC MAX-phase. Finally, ductility plots are presented which
purposes a linear trend between the elements of group IV, V and VI. Based on the results, further studies with a focus on the temperature and magnetization's impact on the stability and mechanical properties are suggested.
The recent discovery of Cr4AlB4, a laminated ternary metal boride belonging to the
family of layered MAB-phases, where the transition metal boride layers are interleaved by an A layer, has spurred theoretical investigation for novel M4AlB4 phases. In this study,
first-principles calculations were applied in order to investigate the thermodynamical stability and mechanical properties of M4AB4 where M = Cr, Hf, Mo, Nb, Ta, Ti, V, W,
Zr while the A layer was kept fixed as Al.
The thermodynamical stability calculations validate the recently discovered Cr4AlB4
phase’s stability and suggest the neighboring phase Mo4AlB4 to be stable. Additionally,
the phases Mo3AlB4, Mo2AlB2, Ti4AlB4 and Ti2AlB2 indicates phases close to stable
with a formation enthalpy within the range of 0< ∆H < 25 meV per atom compared to competing phases. Hence dynamical stability investigations were carried out, which indicates Mo4AlB4 to be dynamically stable. The stability of Mo4AlB4 does encourage
synthesizing attempts to be kept in mind as a future project. Phase stability trends of the 111, 212, 314 and 414 compositions were discovered, where a 212, 314 and 414 composition is seen to be more stable for an M-element with lower electron configuration. Furthermore, the mechanical properties of the 414 compositions were investigated by systematically straining the unit cell in different directions. The bulk-, shear- and Young’s-modulus were derived and are presented, where Ti4AlB4 demonstrates values
similar to the commended Ti2AlC MAX-phase. Finally, ductility plots are presented
which purposes a linear trend between the elements of group IV, V and VI. Based on the results, further studies with a focus on the temperature and magnetization’s impact on the stability and mechanical properties are suggested.
First and foremost, I want to express my gratitude towards my supervisor Martin Dahlqvist for introducing me to the field of material science, which led me to apply for a masters project at the thin film division at IFM. My co-supervisor Hans Lind who guided me through the field of mechanical computations with his invaluable expertise. Thank you both for the incredible amount of patience in regards to my many silly questions. I want to thank Johanna Ros´en for giving me the opportunity to evolve and the (mostly) fantastic experience. And finally, my warmest thanks to my loved ones and my classmates of physics class 2014, LIU, for enlightening my spirit on a daily basis.
1 Introduction 2
1.1 Motivation and Goals . . . 5
1.2 Outline of the thesis . . . 6
2 Theory and Methods 7 2.1 solid-state theory . . . 7
2.1.1 The many-body problem . . . 7
2.1.2 Density functional theory (DFT) . . . 8
2.1.3 Bloch’s theorem . . . 10 2.2 Stability . . . 11 2.2.1 Thermodynamical stability . . . 11 2.2.2 Dynamical stability . . . 14 2.3 Mechanical properties . . . 17 2.4 Theoretical methods . . . 28
2.4.1 Vienna Ab-Initio package . . . 28
2.4.2 Evaluating the formation enthalpy . . . 30
2.4.3 Phonon calculations . . . 31
2.4.4 Mechanical properties . . . 31
3 Results 35 3.1 Thermodynamical stability . . . 35
3.1.1 The formation enthalpy . . . 35
3.2 Dynamical stability . . . 36
3.2.1 Phonon dispersion plots of Mo-Al-B . . . 37
3.3 Mechanical results . . . 38
3.3.1 The elastic constants . . . 38
3.3.2 Mechanical properties . . . 38
4 Discussion 41
5 Conclusion 45
6 Appendix 52
6.1 Supercell convergence of Mo4AlB4 . . . 52
6.2 Phonon dispersion plots of Cr-systems and Ti-systems . . . 53
6.2.1 Cr-systems . . . 53
6.2.2 Ti-system . . . 54
6.3 competing phase tables . . . 55
Materials based on nitrides and carbides of transition metals are now well accepted materials used as wear- and corrosion resistant coatings to increase the lifetime of tools and engineering components. Materials such as TiN, TiAlN2 and TiCN are members
of the mentioned group and are materials commonly applied as hard coatings [1–4]. Furthermore, boron is one of the hardest materials known and is by no mean entirely explored. The uniqueness of B12and the superhard phase of γ-B28 are fields of interest
as of today [5–7]. The mentioned characteristics of borides may hence be seen as a promising pathway to achieve strong materials with unique properties. The structure and boding diversity of borides may contribute to the broad range of properties. Yelon demonstrated the ternary boride Nd2Fe14B to be the strongest permanent magnet as
of this date [8]. Yeung et al. proved the binary borides ReB2 and WB4 to be members
of the superhard materials group [9]. An additional feature of borides are the ceramic characteristics. ZrB2 and HfB2 are two binary transition metal borides which belong to
the ultra-high temperature ceramic (UHTC) group. Materials of the UHTC group are of interest as they generally exhibit both ceramic and metallic properties. Properties of this matter may be utilized in areas such as wear resistant coatings, battery electrodes and high-temperature structural materials [10, 11]. However, both ZrB2 and HfB2 suffer
from low fracture toughness and damage tolerance [12, 13]. Additional compounds of the UHTC group are the layered MAB-phases. A MAB-phase consists of a transition metal boride sublattice intervened by an A layer, which commonly consists of Al atoms. One may notice the layered composition of the crystal structures for the MAB, M2AlB2,
Figure 1.1: The crystal structure of the a) MAlB composition, b) M2AlB2 composition,
c) M3AlB4 composition and d) M4AlB4 composition
As of today, the compositions of layered MAB-phases consists of MAB, M2AB2, M3AB4,
M4AB4 and M4AB6 for different combinations of M = {Cr, Mo, W, Mn, Fe, Ru} and A =
{Al, Zn} with Ru2ZnB2 being the only case for A= Zn [10, 14]. Figure 1.2 demonstrates
the periodic table with denoted elements of experimentally known MAB-phases with layered planes, which includes MoAlB, WAlB, Fe2AlB2, W2AlB2 Cr2AB2, Cr3AlB4,
Cr4AlB4 and Ru2ZnB2.
Ade and Hillebrecht experimentally synthesized the ternary MAB-phases MoAlB, WAlB, Cr2AlB2, Cr3AlB4 and Cr4AlB6, which in this study serves as guidelines [15].
Most recently Zhang et al., discovered the layered orthorhombic phase Cr4AlB4 of
space group Immm, 71, which serves as the main reference throughout this study [16]. A great number of both theoretical and experimental studies based on layered MAB-phases have been carried out. Zhou et al. conducted DFT computations in order to evaluate electrical conductive and damage-tolerant nanolaminated MAB-phases, including Cr2AlB2, Cr3AlB4 and Cr4AlB6. Zhou’s research concluded MAB-phases (CrB2)nCrAl
Figure 1.2: The periodic table with denoted elements of MAB-phases
et al. predicted quaternary MAB-phases of composition M2M′AlB4 for M = {Mn, Fe,
Co} and M′ = {Cr, Mo, W} based on Cr3AlB4, to exhibit a strong tendency of ordering.
The ordering is suggested to be caused by the different bonding strengths between the Al-and M-elements [18]. The layered compositions of MAB-phases may additionally serve as a pathway to 2D materials. By etching the A layer, 2D MBenes may be obtained [19, 20].
A similar material family to the MAB-phases are MAX-phases, which belong to the class of ternary transition carbides or nitrides known as Mn+1AXn. In which M is an early
transition metal, A is a group IIIA or IVA element, X is either C or N and commonly n= 1, 2 or 3 [21, 22]. MAX-phases have as of late obtained a great amount of attention [23–26], where MXenes have been obtained by selectively etching the A layer [27, 28]. The stoichiometry of MAX-phases does resemble the stoichiometry of MAB-phases with similar M and A elements. However, the lattices of MAX-phases are hexagonal while MAB-phases are orthorhombic or tetragonal. In figure 1.3, the unit cells of M2AX, M3AX2
and M4AX3 are illustrated.
The list of studies utilizing first principle stability calculations in order to predict or confirm materials can be made long [31–33]. Historically, the development of new materials has been performed in a trial and error manner, which is usually a high cost and low efficient procedure. Due to the recent rapid development of computational algorithms and hardware, the theoretical opportunities for exploring material properties have grown over the last two decades. The aim of these theoretical frameworks is usually to predict material properties, but also to guide experimentalists in the right direction.
Figure 1.3: The crystal structures of M2AX, M3AX2 and M4AX3 phases seen from left
to right. From Eklund et al. [22], adapted from H¨ogberg et al [29] and an earlier version by Barsoum [30]. © Elsevier, reproduced with permission
1.1 Motivation and Goals
Recently, Zhang et al. published the discovery of an additional stable MAB-phase of the M4AlB4 composition in the M = Cr family, induced by mixing CrB and Al powders at
1000°C [16]. The stoichiometry of Cr4AlB4 is demonstrated in figure 1.1 and resembles
the symmetry of the neighboring MAlB, M2AlB2 and M3AlB4 phases. The discovery of
Cr4AlB4 introduces the issue of additional families of the M4AlB4 composition. Hence
theoretical stability computations based on the formation enthalpy and phonon dispersion were investigated for hypothetical M4AlB4 phases where M = {Cr, Hf, Mo, Nb, Ta, Ti,
V, W, Zr}. Trends such as the electron configuration and element size were investigated as a contribution to future studies. In addition to stability, mechanical properties of the M4AlB4 systems were investigated in order to characterize materialistic properties, such
as the bulk-, the shear- and Young’s-modulus. Properties of this matter which may be compared to neighboring MAB-phases and other well-known materials in the field of hard coatings.
1.2 Outline of the thesis
The thesis builds on a theoretical foundation which is initially presented, followed by the applied methods. Thereafter, results are presented followed by discussion and conclusions. Each chapter is generally divided into three subsections, being the thermodynamical stability, the dynamical stability and mechanical properties. The thermodynamical section is introduced with general concepts, followed by the theoretical framework of evaluating the phase stability based on the formation enthalpy. The dynamical section contains the essential theory behind a phonon computation and the correlation between phonons and the stability of a material. The mechanical sections introduce the most common mechanical properties of a material, such as the bulk-, shear- and Young’s-modulus. Initially, the essential theory of solid-state physics and computational material science will be introduced in the section solid-state theory.
The current chapter presents the fundamental theory applied throughout the thesis, along with the approached methods. Initially, the solid-state theory will be introduced with a focus on the many-body problem, the density functional theory and Bloch’s theorem. The section to follow aims to introduce the theoretical framework of which the stability of a material is defined, and is divided into the subsections thermodynamical stability and dynamical stability. This section is followed by Mechanical properties which strive to introduce the fundamentals of mechanical materialistic properties. Finally, the used methods are presented in the same manner as the theoretical framework, based on the previous three sections.
2.1 solid-state theory
Any solid system may completely be described by the time-dependent Shr¨odinger equation [34],
i̵hd
dt∣Ψ(t)⟩ = ˆH∣Ψ(t)⟩ , (2.1)
in which i is the imaginary unit and ̵h being the reduced Plack’s constant.∣Ψ(t)⟩ describes the time dependent state vector of a quantum system and ˆH being the Hamiltonian operator.
2.1.1 The many-body problem
The many-body problem is self-described if one considers a non-relativistic system consisting of Neinteracting electrons and Nnnuclei. The many-body Hamiltonian operator
may be used to describe the system as the following = ˆT + ˆT + ˆU + ˆU + ˆU
In which ˆTe denotes the electrons kinetic energy contribution while ˆTn being the kinetic
energy contributed by the nuclei. The remaining terms denote the potentials, or rather ˆ
Uee the electron-electron (the Coulomb interaction) interaction, ˆUnn the nuclear-nuclear
interactions and finally ˆUen correlates to the potential energy between the electron-nuclei
interactions. Each term of eq.(2.2) may be further expanded resulting in the following expression, ˆ H= − Ne ∑ i=1 ̵h2 2me∇ 2 i − Nn ∑ k=1 ̵h2 2mk∇ 2 k+ 1 2 Ne ∑ i≠j e2 ∣ri− rj∣+ 1 2 Nn ∑ k≠l ZkZle2 ∣rk− rl∣− Ne ∑ i Nn ∑ k eZk ∣ri− rk∣ . (2.3)
The meand mkterms being the electron mass and the nucleus mass, respectively. e denotes
the negative charge of the electron, Zk,l the atomic number and ri and rk represents the
position of the electron and nuclei, respectively. Further notice, the expression has been simplified by implicitly including the spin coordinates σi of the electrons. The indices i
and j runs through the electron’s degrees of freedom while k and l through the nuclei’s degrees of freedom. Eq.(2.3) becomes practically impossible to solve for any system of interest as the number of degrees of freedom rapidly expands for larger systems, with the only analytically solvable system being the hydrogen atom. Hence several approximations must be applied. With the Born-Oppenheimer approximation, one may approximate a solution to the many-body problem with theories such as the density functional theory (DFT) [35] and Hartree-Fock method [36]. DFT was the applied method throughout this
study.
2.1.2 Density functional theory (DFT)
The theoretical framework of this study is under the influence of the Born-Oppenheimer (BO) approximation, i.e., the mass of an atomic nucleus is much larger than the mass of an electron [37]. This mass difference leads to the static interpretation of the nucleus with respect to the electrons. A stationary state for a N-electron system is described by the wave function Ψ( ⃗r1, ...,r⃗N) which must satisfy the many-body time-independent
Shr¨odinger equation
ˆ
With ˆH being the Hamiltonian, ˆT the kinetic energy, vext the potential energy of
the external field and ˆU the electron-electron interaction energy. DFT builds on the two Hohenberg-Kohn theorems which relates to any system which consists of electrons moving under the influence of an external potential, vext [35]. The theorems states that
the total energy is a functional of the ground state electron density E[ρ(⃗r)], due to the external potential being a unique functional of the electron density in the ground state. Secondly, the total energy may be minimized with respect to the electron density, ρ(⃗r), for a correct ground state energy, EGS.
EGS= E[ρ(⃗r)] = min
ρ(⃗r){∫ vext(⃗r)ρ(⃗r)d⃗r+ F[ρ(⃗r)]} (2.5)
where F[ρ(⃗r)] is the sum of ˆT[ρ(⃗r)] and ˆU[ρ(⃗r)] which result in yet another problematic functional. Where a functional is a function which builds on a function and is denoted as [ ]. Kohn and Sham simplified procedures by approximating the system as a fictitious system consisting of N non-interacting electrons of the same density ρ(⃗r) under the influence of the effective potential vef f. The single-particle Shr¨odinger equation takes the
following form,
{− ̵h2 2m∇
2+ v
ef f} ψn(⃗r) = εnψn(⃗r). (2.6)
Where ψn(⃗r) is a N-body time-independent wave function and the effective potential
vef f may be written as
vef f(ρ(⃗r)) = vext(ρ(⃗r)) + ∫
ρ(⃗r′)
∣⃗r− ⃗r′∣d ⃗r′+ vxc (2.7)
and for a system consisting of N electrons, the correlation between the single-particle wave function and the electron density is expressed as
ρ(⃗r) =
n
∑
i=1∣ψi(⃗r)∣
2. (2.8)
The eigenvalue equation, eq.(2.6) is commonly referred to as the Kohn-Sham equations and depends on the electron density. The electron density depends on the single-particle wave function, as seen in (2.8). Hence the solution to (2.7) may be found self-consistently
2.1.3 Bloch’s theorem
The approximated solution given by DFT to the many-body problem is however not sufficient and additional theories are needed. A solid body may consist of an infinite number of electrons, which is an obstacle even for DFT. Bloch’s theorem state the wave function of an electron within a perfect periodic potential may be expressed as,
ψ(r) = eik⋅ru(r) (2.9)
where r is the position, k denotes the crystal wave vector confined in the first Brillouin zone and u(r) is a periodic function of the same periodicity as the potential, or rather the crystal with lattice constant a. The periodicity may be expressed as
u(r) = u(r+ R) (2.10)
Where R is an primitive translation vector. The first Brillouin zone is defined as the space −π
a ≤ k < π
a. The number of points covered in this space are denoted as the k-point mesh,
which is an essential input parameter in computation physics which will be brought up in sections to come. Due to the periodicity of u(r), u(r) may be Fourier transformed into,
u(r) = ∑
G
eiG⋅r (2.11)
where G is the reciprocal space lattice vector which is defined as the following
G⋅ R = 2πm. (2.12)
R denotes the real space lattice vector and being the primitive translation vector while m is an integer. One may combine eq.(2.11) and eq.(2.9) into
ψ(r) = ∑
G
Cei(k+G)⋅r. (2.13)
Due to the wave function being almost identical for sufficient close values of k, one may represent the wave function by considering a single k-point. Therefore the electronic state at a finite number of k-points is sufficient to consider. Meaning, the Bloch theorem allows one to only consider the number of electrons in the unit cell at a finite number of k-points [39].
2.2 Stability
The stability section is divided into two subsections, thermodynamical stability and dynamical stability. Initially, the general theoretical background of thermodynamical stability will be introduced, followed by the formation enthalpy and convex hull diagrams. The dynamical stability is presented by initially focus on the general aspects of phonons, which is followed by the phonon stability.
2.2.1 Thermodynamical stability
The fundamentals of theoretical phase stability computations are centered at the laws of thermodynamics, with the goals of predicting future systems which may be of experimental interest. The phase equilibrium of a system is based around the Gibbs free energy being a local minimum with respect to minor deformations. Gibbs free energy is defined as
G= H − TS, (2.14)
where H is the enthalpy, T the temperature and S the entropy. Expression (2.14) refers to the thermodynamic potential that is minimized when a system reaches chemical equilibrium. At zero temperature, eq.(2.14) depends only on the enthalpy H, which is defined as
H= U + pV. (2.15)
Hence Gibbs free energy, eq.(2.14), may be expressed as a sum of the internal energy, U , and the work term pV which represents the amount of work required to make room for the system by displacing its environment and establishing its pressure p, and volume V . Eq.(2.14) is expressed as the following at zero temperature,
G= U + pV. (2.16)
Additionally, the structural optimization calculations strive to minimize the external pressure of the unit cell which allows the correct symmetry properties to be obtained. Hence the phase equilibrium may be obtained by minimizing Gibbs free energy [40].
Formation enthalpy
The formation enthalpy determines the stability of a phase in this study. Even with an intrinsic stability obtained for a system of interest, one is recommended to evaluate and compare the local energy minima with the energy of competing equilibrium phases which may be more thermodynamically favorable. The set of most competing phases is determined using the simplex linear optimization procedure for a given elemental composition of bM, bA and bB and the optimization problem may be framed as
min{Ecomp(bM, bA, bB)} = n
∑
i=1xiEi. (2.17)
The Ecomp term represents the energy that shall be minimized while xi denotes the
amount and Ei the energy of compound i. The minimization is subject to the following
constraints xi ≥ 0, bM = n ∑ i=1x M i , bA= n ∑ i=1x A i , bB= n ∑ i=1x B i (2.18)
with bM={1, 2, 3, 4}, bA={1} and bB={1, 2, 4} for the compounds of interest in this study. The formation enthalpy per atom is thus calculated using
∆H= E(i) − min{Ecomp(b
M, bA, bB)}
Ni
, (2.19)
where E(i) is the energy of the compound of interest and Ni denotes the number of
atoms per formula unit of phase i. The sign of ∆H determines the stability, with ∆H> 0 indicating a non-stable phase or metastable phase and ∆H< 0 denotes a stable phase. The described approach is considered reliable as single elements, binary and ternary phases are taken into account instead of only analyzing the minimal energy of the compounds. Additionally, the approach has been utilized and validated in several previously carried out stability studies [31, 41–43].
Convex hull diagrams
Convex hull diagrams were used as a procedure to validate the energies of the competing phases. The convex hull is defined as the smallest convex set which contains subset X, where subset X is a set of points in the Euclidean space. A 2D phase diagram of the
binary and singularly competing phases were illustrated with a convex hull plot. 2D phase diagrams for several phases are provided by databases such as materials project [44], and open quantum materials database, (OQMD) [45, 46]. In this study, the subset X consists of the formation enthalpy of each competing phase. Figure (2.1) demonstrates the convex hull of the MoxB1−x binary system, containing data points provided by Materials project,
OQMD and the obtained data of the relaxation computations [44–46].
Figure 2.1: 2D The convex hull plot of competing phases in the Mo-B system. The graph contains data from Material project (green) [44] and OQMD (red) [45, 46], (blue) represents the evaluated data while (black) is the corresponding convex hull
The data sets of figure 2.1 are overlapping, hence the difficulty in observing the different sets. 3D phase diagrams are obtained by combining the binary phase diagrams of which a ternary phase consists of. This also allows the diagram to include ternary phases. Instead of analyzing 27 graphs, this allows one to evaluate nine, one for each composition. Figure 2.2 demonstrates the calculated ternary phase diagram of the Mo-Al-B system with focus on Mo4AlB4. Where red data points represent a phase included in the Mo-study.
Figure 2.2: Ternary phase diagram of the Mo-Al-B system where the red squares (∎) indicates a phase included in the Mo-study
2.2.2 Dynamical stability
Phonon computations date back to Born and Huang and have greatly expanded the field of condensed matter physics since it became a routine during the last decade [47]. The ab-initio force constant approach was applied, which builds on the displacement of a single atom in a supercell which induces forces acting on the neighboring atoms [48, 49]. Depending on the symmetry of the phase, several atoms may be displaced. Hence a large enough supercell is required to make sure that the force field does not act on the displaced atom in its periodic repetition. The supercell may be obtained with the software phonopy [50]. The force acting between a pair of atoms may be seen as a spring according to Hooke’s law which is commonly defined as
F= −k(r − r0) (2.20)
where k denotes the spring constant and (r− r0) the dislocation of the initial placement.
Figure 2.3 demonstrates the well-known quantum harmonic oscillator which describes the fundamental behavior of an oscillating particle around an energy minimum.
Figure 2.3: The quantum harmonic oscillator with distinct energy levels En around
particle position r
En= (n +
1
2)̵hω where n = 0, 1, 2... (2.21)
The angular frequency, denoted as ω in eq.(2.21), of the particle is further described by the following equation
ω= √
k
µ, (2.22)
where µ is the reduced mass. Once the frequencies are known, thermal properties may be derived. Properties such as the entropy, the heat capacity and Helmholtz free energy may be evaluated.
Phonon stability
Essentially, the particles in a system may take two cases. Either located at an energy hill, seen in figure 2.4 a) or at an energy minima demonstrated in figure 2.4 b). The cases are demonstrated in figure 2.4 below.
Figure 2.4: Two cases of an particle in a lattice. Where a) denotes a particle located on a energy hill and b) a particle located at an energy minima
The equilibrium of a crystal is found if the potential energy of the lattice always increases against any combination of atomic displacement, which very much resembles the behaviour case b) in figure 2.4 and the quantum harmonic oscillator demonstrated in figure 2.3. This procedure is equivalent to all phonons vibrating with real and positive frequencies. However, imaginary frequencies may occur which indicates a dynamical instability of the given crystal structure. An imaginary frequency indicates a corrective atomic displacement which would reduce the potential energy in the vicinity of the equilibrium of the crystal. However, this would also, according to eq.(2.22), indicate an imaginary spring constant or an imaginary mass which is not a physical solution. Meaning, in order to validate the dynamical stability of a phase, an analysis of the respectively phonon dispersion plot is necessary [51, 52]. Phonopy is a tool which allows one to design phonon dispersion plots of the frequencies over the Brillouin zone and evaluate the density of states (DOS). Phonopy was the tool utilized for the phonon computations and builds on translating a Hessian matrix into vibrational energy. A Hessian matrix H of f(r1, r2, ..., rn) is a n × n squared matrix defined as
H= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ ∂2f ∂r2 1 ∂2f ∂r1∂r2 . . . ∂2f ∂r1∂rn ∂2f ∂r2∂r1 ∂2f ∂r2 2 . . . ∂r2∂2∂rnf ⋮ ⋮ ⋱ ⋮ ∂2f ∂rn∂r1 ∂2f ∂rn∂r2 . . . ∂2f ∂r2 n ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ .
2.3 Mechanical properties
The following section aims to introduce the theoretical framework of the mechanical computations. The fundamental material related properties such as the bulk-, the shear-, Young’s-modulus, Poisson’s ratio and binding properties are introduced. Initially, the fundamental framework together with Voigt notation are presented, as the notation is applied throughout the remaining sections. Finally, the essential theoretical framework behind mechanical computations based on the orthorhombic crystal system is introduced. Theory of elasticity
Throughout the thesis, Voigt notations will be used. Voigt notation is a method of representing a symmetric tensor by reduced order. The foundation of the mechanical computations builds on solid-state theory which generally utilizes Voigt notation. Com-monly, during small linear deformations of a crystal, the change in energy of the strain is given by
dE= σijdεij (2.23)
which may be rewritten as
dE dεij = σ
ij. (2.24)
In which σij is the natural stress tensor and εij are the elements of the strain tensor.
Indices i, j run from values 1-3 representing the spatial coordinates. Furthermore, the elastic constants of a material describe the resistance of elastic deformations under strain. The formal definition is
cijkl= 1 V ∂E ∂εij∂εkl . (2.25)
E denotes the internal energy while cijklrepresents the elements of the fourth-rank elastic
constant tensor C. The stress tensor, σij, of eq.(2.24) is generally written as,
σ=⎡⎢⎢⎢ ⎢⎢ σxx σxy σxz σyx σyy σyz ⎤⎥ ⎥⎥ ⎥⎥. (2.26)
By applying Voigt notation to the stress tensor, eq.(2.26), one obtains the vector, σ= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ σxx σyy σzz σyz σxz σxy ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ σ1 σ2 σ3 σ4 σ5 σ6 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ . (2.27)
In Voigt notation, the stress, strain and elastic stress constants tensor may be denoted as the following [53], σ= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ σ1 σ2 σ3 σ4 σ5 σ6, ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ , = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ 1 2 3 4 5 6 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ and C= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ c11 c12 c13 c14 c15 c16 c12 c22 c23 c24 c25 c26 c13 c23 c33 c34 c35 c36 c14 c24 c34 c44 c45 c46 c15 c25 c35 c45 c55 c56 c16 c26 c36 c46 c56 c66 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ .
The symmetry of the unit cell generally simplifies C as many elements become zero or equal to each other, which simplifies the framework of computing mechanical properties. Additional to the elastic constant, one may define the elastic compliance constant tensor S, which may be defined as CS= I, where I is the unit matrix. The relationship between the stress, strain and elastic constant tensors is, [54],
σ= C. (2.28)
Orthorhombic crystal system
The orthorhombic lattices are a result of stretching two base vectors of a regular cubic lattice. The stretching is done along with two of the orthogonal lattices with different stretching factors. As a consequence, the lattice parameters (a, b, c) will construct a rectangular prism where a≠ b ≠ c while the lattices all intersect at 90°[55].
Figure 2.5: The orthorhombic crystal symmetry where (a≠b≠c) and (a, b, c) all intersect at 90°
The elastic constant tensor representing an orthorhombic lattice builds on nine inde-pendent elastic constants. The tensor takes the following form
C = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ c11 c12 c13 0 0 0 c12 c22 c23 0 0 0 c13 c23 c33 0 0 0 0 0 0 c44 0 0 0 0 0 0 c55 0 0 0 0 0 0 c66 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ . (2.29)
Hence, in order to evaluate the nine elastic constants of an orthorhombic unit cell, nine independent linear equations are needed. And each application of eq.(2.28) may potentially give six equations. A set of strain tensors may be defined specifically for elasticity computations. The applied strain tensors of this study were defined to depend only on one deformation parameter δ. Hence the same amplitude of strain is applied but in different directions. The tensors are independent of the lattice, however, which tensor to be used depends on the symmetry of the unit cell. In order to evaluate the nine independent elastic constants presented in the matrix (2.29), the following three strain matrices were applied.
Matrix 1 = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ δ δ 0 0 0 0 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦
which applied gives( + I)a =⎡⎢⎢⎢ ⎢⎢ ⎣ 1+ δ 0 0 0 1+ δ 0 0 0 1 ⎤⎥ ⎥⎥ ⎥⎥ ⎦ a (2.30)
Where a is an arbitrary vector, I is the unit matrix and δ is the applied strain. Giving rise to the following stresses
σ1= (c11+ c12)δ σ2= (c12+ c22)δ (2.31) σ3 = (c13+ c23)δ. Matrix 2 = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ δ −δ 0 0 0 0 ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦
which applied gives( + I)a =⎡⎢⎢⎢ ⎢⎢ ⎣ 1+ δ 0 0 0 1− δ 0 0 0 1 ⎤⎥ ⎥⎥ ⎥⎥ ⎦ a (2.32)
Giving rise to the following stresses
σ1= (c11− c12)δ
σ2= (c12− c22)δ (2.33)
Matrix 3 = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ 0 0 δ δ δ δ ⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦
which applied gives( + I)a =⎡⎢⎢⎢ ⎢⎢ ⎣ 1 δ2 δ2 δ 2 1 δ 2 δ 2 δ 2 1+ δ ⎤⎥ ⎥⎥ ⎥⎥ ⎦ a (2.34)
The1/2 terms of the applied matrix originates from the symmetry of the strain tensor,
where 12= 21 etc. Giving rise to the following stresses
σ1= c13δ σ2= c23δ σ3= c33δ σ4= c44δ (2.35) σ5= c55δ σ6= c66δ.
Eq.(2.31) and eq.(2.33) combined is enough to evaluate c11, c22, c12, c13 and c23 while
eq.(2.35) provides the additional constants c33, c44, c55 and c66 if the stress of the applied
strain is known. The Bulk Modulus
The bulk modulus of a polycrystalline material is defined as the volume derivative of the pressure
B(V ) = −V ∂P ∂V = V
∂2E(V )
∂V2 (2.36)
where the pressure is given by the negative derivation of the energy P(V ) = −∂E(V )
∂V . (2.37)
B= ∆P
∆V/V =
volume stress
volume strain (2.38)
where ∆P denotes the applied pressure, ∆V being the volume change and V the initial volume. This is further illustrated in figure 2.6.
Figure 2.6: A schematic view of the bulk modulus of a system, where P is the applied pressure.
One may also evaluate the bulk modulus through the elastic constants as seen in eq.(2.39) and eq.(2.40),
BV = ( c11+ c22+ c33) + 2(c12+ c13+ c23) 9 (2.39) and BR= 1 (s11+ s22+ s33) + 2(s12+ s13+ s23) . (2.40)
Where BV and BR denotes the Voigt and Reuss, which are methods of evaluating the
bulk modulus of a polycrystalline material through the elastic constants cij and the
elastic stiffness constants sij. The bulk modulus for polycrystalline materials are generally
represented as the arithmetic average of BRand BV, denoted as the Hill bulk modulus
BH = (
BV + BR)
2 . (2.41)
The shear modulus
The shear modulus is described by the response to transverse internal forces acting on a body. The formal definition is
G= F/A
∆l/l =
shear strain
shear stress. (2.42)
Where F is the applied force, A the area of which the force is applied, ∆l the extended length induced by the applied force while l is the uncontracted length of the body. A simplified explanation would be to consider cutting a body with a dull pair of scissors. Two forces acting on a body with parallel direction, commonly a transverse force and friction which is seen in figure 2.7.
Figure 2.7: A schematic view of the shear modulus of a body, where ∆l denotes the change in length caused by the force F applied on area A, and l denoting the initial height.
Just as the bulk modulus, the shear modulus may be defined as the arithmetic average of the Voigt and Reuss shear moduli denoted as GV and GR respectively, which are
GV = ( c11+ c22+ c33) − (c12+ c13+ c23) + 3(c44+ c55+ c66) 15 (2.43) and GR= 15 4(s11+ s22+ s33) − 4(s12+ s13+ s23) + 3(s44+ s55+ s66) . (2.44)
The arithmetic average, or rather the Hill average is henceforth expressed as GH =
GV + GR
2 . (2.45)
Young’s modulus
The stiffness of a body is distinguished by measuring the load of an induced deforma-tion. The characteristics of Young’s modulus is commonly referred to as the material’s strength to resist deformations. Hooke concluded the load to be approximately linearly for small values of the applied deformation. The Young’s modulus is defined as the elastic deformation a body undergoes as a load is applied
Y = F/A
∆l/l =
normal stress
strain . (2.46)
Where F is the applied force, A is the cross-sectional area, ∆l the extended length caused by the applied force and l being the original length of the body. Figure 2.8 demonstrates a schematic view of Young’s modulus of a body.
Figure 2.8: A schematic view of the Young’s modulus of a body, where l denotes the initial length, ∆l denoting the change in length caused by the applied force F on the cross-section area A.
Young’s modulus may be, for a polycrystalline material, expressed explicitly by the bulk modulus and the shear modulus as the following
Y = 9BG
3B+ G. (2.47)
Poisson’s ratio
The Poisson’s ration may be described by a positive tensile strain in one direction which induces a negative compressive strain in the orthogonal direction. Just as a rubber band stretched in one direction will become thinner in the other. This induced lateral contraction along with a longitudinal expansion is referred to the Poisson’s effect. The Poisson’s ratio is the ratio between the lateral and the longitudinal strain.
ν= −∆r/r
∆l/l = (−1)
transverse strain
longitudional strain. (2.48)
Where ∆r is the change in lateral direction, r being the lateral the length, ∆l the extended longitudinal length and l being the original length of the body. The Poisson’s ratio may for a polycrystalline material be expressed to explicitly depend on the bulk- and the shear-modulus
ν= 3B− 2G
2(3B + G). (2.49)
Figure 2.9: A schematic view of the Poisson’s ratio of a system, where r denotes the initial lateral length, ∆r the change in lateral length, l the initial longitudinal length while ∆l the change in the longitudinal direction.
Ductility
Plastic deformation is the process where a material has undergone an irreversible elastic deformation which is generally followed by the materialistic characteristics referred to as the strain hardening. During the plastic deformation, new equilibrium positions of the microscopic structure are generally taken. As more strain is applied, maximum stress may be achieved where a fracture is next to imminent. In between the maximum stress and fracture, a phase referred to the strain hardening is achieved. During the strain hardening, as more strain is applied, the stress is decreased. Materials which do not possess the characteristics of a plastic deformation are said to be brittle and tend to rupture immediately as enough strain is applied. Materials which do possess these characteristics are said to be ductile. Figure 2.10 demonstrates the fundamental characteristics of a ductile and a brittle material as strain is applied.
Figure 2.10: A schematic strain stress plot illustrating a ductile and a brittle material. With ductile being the left and brittle the right figure.
One may, based on different criterion, assume the bonding characteristics of a material. Based on these properties, one may further assume properties such as the ductility or the brittleness. The assumption is based on metallic materials are commonly strong and ductile while covalent and ionic materials are generally hard and brittle.
Cauchy criteria
The Cauchy pressure (CP) is defined as C12−C44 for all bravais lattices [58]. The CP
describes the angular characteristic of interatomic bonds of a compound. CP> 0 is a criterion for a metallic material hence a ductile material is suggested. [59].
Pugh criteria
The Pugh criteria suggest a material with G/B ratio< 0.57 to be ductile. Alternatively,
considering the inverse B/G greater than 1.75. Moreover, the bonding characteristics of a
material may be assumed [60]. Fransevich criteria
Based on the Poisson’s ratio, eq.(2.49), a material with ν > 0.26 is said to be ductile. This criteria further allows assumptions regarding the bonding properties of the material
2.4 Theoretical methods
The applied methods throughout the project may be divided into four sections: Vienna Ab-initio package, Evaluation of the formation enthalpy, Phonon calculations and Mechanical properties. The Vienna ab-initio package covers the essential settings and input parameters of the thermodynamical computations followed by the atomic simulation package (ASE). Thereafter, the method behind the evaluation of the formation enthalpy will be discussed which builds on structural optimization calculations. Phonon calculations cover the general phonon framework and the carried out convergence tests of the supercell and k-points. Finally, the strain induced method of systematically deforming the unit cell in order to evaluate mechanical properties is explained.
2.4.1 Vienna Ab-Initio package
The Vienna Ab-Initio Package (VASP) [63], is a computational software written originally by Mike Payne which also was the basis of one of the leading codes for calculating material properties from first principles, CASTEP [64]. VASP allows an approximate solution to the many-body Schr¨odinger equation to be evaluated either by the Hartree-Fock approximation and solving the Roothaan equations or by solving the Kohn-Sham equations within the density functional theory. VASP was utilized through this project. ASE
The atomic simulation environment (ASE) is a toolbox specified for material computations built in python. Tools which simplifies visualization, running and analyzing atomistic simulations. The main usage of ASE for this project was the visualization of 2D and 3D phase diagrams [65].
Potentials and energy cutoff
Throughout the project, ab-initio calculations were carried out using the Generalized Gra-dient Approximation (GGA) Perdew-Burke-Ernzerhof (PBE) approximation as exchange-correlational functional [66, 67].
k-point convergences
Based on the suggested Cr4AlB4 crystal structure by Zhang et al. [16], convergence test
of the k-grid was performed. The initial simulations were aimed towards broadening the knowledge of the constructed unit cell. Hence the k-grid was converged in the steps of 6n×n×6n where n = 1, 2, . . . , 8 and relaxed. The irreducible k-points was plotted versus the ground state energy per atom of the relaxed Cr4AlB4 system with k-grids in sets of
6n×n×6n where n = 1, 2, . . . , 8 which resulted in figure 2.11.
Figure 2.11: The energy convergence of the Cr4AlB4 system for k-grids 6n× n × 6n where
n= 1, 2, . . . , 8
The k-mesh of 36×6×36 denotes an energy convergence as the energy per atom deviates with less than 10 µeV afterward. Hence the k-grid was set to 36×6×36 for the following relaxation simulations for each M4AlB4 system. Figure 2.11 was further utilized as a
guideline when the structural optimization calculations were performed on the competing phases as the number of k-points per reciprocal ˚A may be approximated from figure2.11. Relaxation of phases
The structural optimization computations was executed on the M4AlB4 systems with
transition metals M = {Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr}. The relaxation computations were carried out in steps of eight, with the settings gradually increased. The k-grid was
The same procedure was also used for relaxation of the considered competing phases. The plain-wave cut-off energy was set to 400 eV and the energy break condition was gradually increased from 1E-2 to 1E-6 eV in order to gradually increase the accuracy. The cell shape and the atomic positions were set as degrees of freedoms during the two initial calculations, allowing a semi-stable structure to be found before initiating heavier demanding computations. During the following five simulations, the unit cell volume was added as an additional degree of freedom. For the final computational step, volume, unit cell shape, and atomic positions were kept fixed. All calculations were treated as non-magnetic using non-spin polarized functionals.
2.4.2 Evaluating the formation enthalpy
The procedure for evaluating the formation enthalpy consisted of two parts. The initial step included the selection of competing phases while the second part involved the relaxation procedure for equilibrium structures, which resulted in a calculated ground state energy. To ensure the calculated internal energies are valid, 2D phase diagrams were plotted and compared to databases such as Materials Project [44], and OQMD [45, 46]. The results are illustrated in figure 2.1, which demonstrates the evaluated data along with the obtained data sets of Materials Project and OQMD. The data is seen to overlap, hence the carried out computations were validated. With the competing phases validated, eq.(2.19) may be evaluated. The crystal structures of the competing phases were obtained through the databases Springer Material [68], which includes experimentally phases of a great set of databases, and Materials Project [44], which is a database on its own. Experimentally known ternary, binary and single atomic phases were extracted and relaxed with k-grids chosen explicitly for each structure based on the previously k-grid convergence. The experimentally known ternary phases Cr4AlB4, Cr3AlB4, Cr2AlB2 and
MoAlB were used as prototypical structures. The tables demonstrating the complete set of competing phases included in the analysis may be found in the appendix under section 6.3. With the competing phases systematically confirmed, the formation enthalpy was derived using eq.(2.19) for the MAlB, M2AlB2, M3AlB4 and M4AlB4 phases where
2.4.3 Phonon calculations
To ensure reliable phonon calculations, it is necessary to use proper settings in terms of both the number of k-points and supercell size. Hence, convergence tests with respect to k-grid and supercell size were carried out. The procedure was initiated by running convergence tests of the zero point energy versus the number of irreducible k-points and the symmetry of the supercell. The threshold of the phonon simulations were set to accurate and all phonon calculations were performed using a cut off energy of 500 eV. Through density functional perturbation theory, the Hessian matrix was determined, which is a requirement of phonopy in order to determine the vibrational energy. The partial occupancies were set as a Gaussian smearing with a width of 60 meV.
Convergence tests
Supercell and k-grid convergence tests were initially carried out. The supercells were set to 2×1×2, 3×1×3, 4×1×4, 5×1×5, 2×2×2 and 4×2×4 times the unit cell of Mo4AlB4. The
size differences of the supercells allowed a complete investigation of the most optimized structure which captured the complete range of incident plane waves to be carried out. Figure 2.12 demonstrates the two convergence tests of the zero point energy versus a) the supercell size and b) the number of irreducible k-points. A supercell of the size 4×1×4 times the original unitcell of Mo4AlB4 was used throughout the dynamical stability
computations. The observed energy differences are within a 1 meV interval which sufficed for this study. The supercell convergence may be observed in section 6.1 as the phonon dispersion plots of the 2×1×2, 3×1×3, 4×1×4 and 2×2×2 supercell sizes are demonstrated. With the mentioned supercell applied, the k-grid convergence was carried out in the sets of k-grid = {2×1×1, 2×1×2, 2×2×2, 3×2×3, 6×4×5}. Plot b) in figure 2.12 demonstrates an small change in the zero-point energy, within the range of 1 meV, as the k-grid was increased. Hence the k-grid 2×1×1 was used throughout the remaining dynamical computations.
2.4.4 Mechanical properties
The relaxed structures of the M4AlB4 systems, where M={Cr, Hf, Mo, Nb, Ta, Ti, V,
W, Zr}, was used as the initial atomic structures for the strain induced systems with a cut of energy of 650 meV. The strain matrices 1, 2 and 3 were applied with the strain
divided into two sections. Initially, the cell volume and cell shape were fixed for each strain while the atomic position was relaxed. During the second computation, the single relaxed property was the electron state.
(a) (b)
Figure 2.12: The convergence of the zero point energy versus a) the supercell size sym-metry b) the irreducible k-points
Energy - Strain relation
The internal energy per atom of each system was extracted and illustrated as a function versus the applied strain. This procedure was carried out for all systems for the applied strain matrices 1, 2 and 3. The energy per atom versus strain graphs, based on the M = V system, is demonstrated in figure 2.13.
Figure 2.13: A demonstration of the correlation between the internal energy and the applied strain seen as matrix 1, 2 and 3 was applied to the system V4AlB4.
Elastic stiffness and compliance constants
Eq.(2.28) denotes the general relation between the strain, stress and elastic constants. For an orthorhombic system, eq.(2.28) requires one to apply matrices 1, 2 and 3 which results in the linear eq.(2.31), eq.(2.33) and eq.(2.35). A linear interpolation may be applied, where the regression may be extracted as the elastic constants of eq.(2.31), eq.(2.33) and eq.(2.35). The elastic constants C13, C23, C33, C44, C55 and C66 may be extracted
directly as the regression of eq.(2.35) while eq.(2.31) and eq.(2.33) may be written as linear combinations in order to evaluate the remaining elasticity constants C11, C22,
C12, C13 and C23. Figure 2.14 demonstrates the mentioned characteristics of the system
V4AlB4 where matrix 1 and a linear interpolation have been applied.
Figure 2.14: Strain-Stress relation of V4AlB4 as matrix 1 have been applied. The
Remaining mechanical properties
The remaining mechanical properties: the bulk-, the shear-, Young’s-modulus and Pois-son’s ratio may be evaluated as the elastic constants have been determined. However, the binding properties are an exception. TheG/Bratio was plotted versus the Poisson’s ratio
and versus the Cauchy pressure while the Pugh criteria and the Fransevich criteria were taken into account. The resulting graphs may be utilized to predict mechanical properties of the compound. Properties such as the binding which determines the ductility or brittle characteristics of a material. The characteristics are denoted in the obtained graphs as sections based on the Cauchy pressure, Pugh and Fransevich criterion.
The results are divided into three sections, namely: thermodynamical stability, dynamical stability and mechanical results. The formation enthalpy is presented as the main result of the thermodynamical stability followed by the band structure graphs, which refers to the dynamical stability. The last section covers the mechanical results where the elastic constants, additional mechanical properties and ductility graphs are presented.
3.1 Thermodynamical stability
The complete set of competing phases included throughout the study and the evaluated internal energies may be found in the appendix under section 6.3, competing phase tables. Additionally, the 3D phase diagrams are demonstrated in the appendix section 6.4.
3.1.1 The formation enthalpy
The formation enthalpy per atom was evaluated using eq.(2.19). The result is given in figure 3.1 in meV per atom for the MAlB, M2AlB2, M3AlB4 and M4AlB4 phases where
M ={Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr}. The color scheme goes from blue towards red, where blue color indicates a negative formation enthalpy (stable phase) while a bright red indicates a positive formation enthalpy (metastable or not stable phase). Additionally, a black square illustrates an experimentally known stable phase.
Figure 3.1: Formation enthalpy per atom for the MAlB, M2AlB2, M3AlB4 and M4AlB4
phases where M = {Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr}.
3.2 Dynamical stability
With the thermodynamical stability evaluated, the result of figure (3.1) reveals Mo4AlB4
to be thermodynamically stable which have as of yet not been reported as synthesized. Hence dynamical stability computations were mainly carried out on the M = {Mo} systems. Systems where M = {Cr, Ti} may be found in appendix but are not included in the analysis to come.
3.2.1 Phonon dispersion plots of Mo-Al-B
Figure 3.2 illustrates the resulting frequencies over the Brillouin zone for the MoAlB, Mo2AlB2, Mo3AlB4 and Mo4AlB4 compounds. The dynamical stability of the phases is
evaluated based on the sign of the frequency. Only real frequencies indicate a dynamical stable phase while negative components indicate a dynamically unstable phase with imaginary frequencies.
Figure 3.2: Phonon dispersion plots of the M=Mo family where a) denotes the MoAlB phase, b) denotes the Mo2AlB2 phase, c) denotes the Mo3AlB4 phase and d)
3.3 Mechanical results
The mechanical properties of the M4AlB4 compounds, where M = {Cr, Hf, Mo, Nb, Ta,
Ti, V, W, Zr}, are demonstrated in the tables and figures to follow. Primarily, the elastic constants of the systems are presented, followed by the general mechanical properties. Finally, binding plots are illustrated by two separate graphs which build on the different criterion.
3.3.1 The elastic constants
Eq.(2.29) demonstrates the significant elastic constant components required of an or-thorhombic unit cell. The obtained values may be seen in table 3.1 below.
Table 3.1: The elastic constants Cij where all units are in GPa.
3.3.2 Mechanical properties
The different shear and bulk moduli are demonstrated in table 3.2 together with Young’s modulus. The ratio Gh/Bh, the Poisson’s ratio ν and the Cauchy pressure are stated but
may be better represented in the binding plots. The maximum value is denoted with a red square (<=) and the lowest with a blue (<=).
Table 3.2: The complete set of mechanical properties for the M4AlB4 systems. All units
are in GPa except for Gh/Bh and ν which are dimensionless. The maximum is
denoted with a red rectangle (<=) and minimum with a blue rectangle (<=)
Additionally, the trends of the bulk- shear- and Young’s-modulus with respect to elements of group numbers 4,5 and 6 are demonstrated in figure 3.3.
Figure 3.3: Trends of the investigated modules for different group numbers. The figures demonstrates the a) Bulk-, b) Shear- and c) Young’s-modulus
3.3.3 Ductility plots
Figure 3.4 demonstrates two different ductility plots of the investigated M4AlB4 phases
which can be used to indicate their bonding characteristics/properties. The Poisson’s ratio and the Cauchy pressure are illustrated versus the Gh/Bh ratio, in which Gh/Bh is
denoted as G/B, with implemented Cauchy pressure, Pugh and Fransevich criterion.
Figure 3.4: Binding plots based on criterion where a) includes the Pugh criteria and Fransevich criteria and b) includes the Pugh criteria and the Cauchy pressure.
The calculated formation enthalpy in figure 3.1 validates the experimentally synthesized MAB-phases as theoretically stable and also indicate additional hypothetical phases, as Mo4AlB4, to be stable and a candidate for synthesis. Hence the dynamical stability
was mainly investigated for the Mo-based MAB-phases, where figure 3.2 demonstrates only real frequencies for the Mo4AlB4 system, which further validates its stability. The
remaining Mo-systems also demonstrates dynamically stable properties, as only non-imaginary frequencies may be observed in figure 3.2. The dynamical stability and the negative formation enthalpy of Mo4AlB4 is an indication that it is a likely candidate for
experimental verification. A black square in figure 3.1 indicates an experimentally stable phase. Cr3AlB4have already been synthesized [15], however, figure 3.1 indicates a positive
formation enthalpy but rather low. Hence a metastable phase might be considered as more correct wording. Additionally, one may observe the formation enthalpy in figure 3.1 to increase, as an M-element with higher electron configuration is included for the 212, 314 and the 414 compositions, whereas the group 4 of the 414 compositions may be seen as an exception. The 111 compositions demonstrates a similar trend with an increased formation enthalpy as the group number of the M-element is increased. These trends may be due to symmetry differences demonstrated in figure 1.1 and investigations are recommended as a future study.
Previous studies evaluated the standard deviation of 24 meV per atom when the formation enthalpy of ternary and binary oxides was predicted [69]. OQMD conducted a statistical research based on their own database and several more, assessing the accuracy of DFT stability computations. One may, based on OQMD’s research, concluded a mean average error of 96 meV per atom between experimental and calculated formation enthalpies [46].
A similar estimation for metal borides is lacking. However, one may consider using the same standard deviation for this study. With the standard deviation taken into account, several more phases of figure 3.1 should be considered metastable. Including Mo3AlB4,
6.3, of Mo4AlB4 does however, include the experimental known stable phase MoAlB.
MoAlB demonstrates the lowest presented formation enthalpy among the included phases. The very negative formation enthalpy of MoAlB may contradict Mo4AlB4’s anticipated
stability. However, the Mo4AlB4 system does look promising based on the negative
formation enthalpy and being dynamically stable.
The systematic applied strain method resulted in mechanical properties demonstrated in tables 3.1 and 3.2 while the ductility plots of figure 3.4 suggests the bonding characteristics of the investigated compounds. The compounds in figure 3.4 are arranged in a linear pattern which is as intended. Hence the combination of ductile and covalent/ionic or brittle and metallic materials are impossible. With the electron configuration (going from group IV to group VI) kept in mind, one may distinguish a trend of group VI and V. However, group IV demonstrates a distinction as it is not following the trend of νTi < νZr < νHf. Figure 3.3 demonstrates the behaviour of the investigated moduli
for m-elements of group numbers 4, 5 and 6. The trends resembles previous studies and are therefor argued to be correct [70, 71]. The majority of compounds may be classed as brittle and possess a covalent/ionic bonding. The brittle characteristics of a hard coat may be preferred as plastic deformation is not always coveted. The ceramic characteristics of MAX-phases is a sought property in the field of hard coatings. Studies regarding mechanical properties temperature dependence of MAB-phases have previously been carried out with a focus on specific ternary phases, but also the 212 compositions based on Cr2AlB2. Fe2AlB2 demonstrated a decrease in compressive strength at room
temperature compared to 600℃[72], while the borides tend to become more ductile for heavier compounds going from Cr2AlB2 to Ni2AlB2 [73]. Studies of this matter have not
yet been carried out for MAB-phases in general, which is recommended as a future study based on the obtained results. One is further recommended to compare the materialistic properties of the 414 compositions to the neighboring 111, 212 and 314 compositions as to investigate ongoing trends. The experimental success of synthesizing Cr4AlB4 may for
certain expand the field of mechanical properties of MAB-phases as with new materials comes the possibility of expanding the field of material science which allows one to tailor materials based on their specific properties. One may not draw any direct conclusions based on the results of this study in regards to the investigated compounds as potential hard coating materials. Instead, further investigations of the temperature dependence for various properties of M4AlB4 are needed to find their usability. The following ductility
plots are figures 3.4 including results of previous studies which includes a few MAB-phases and some MAX-phases [14, 17].
Figure 4.1: Binding plots where computed data is demonstrated as (x), neighbouring (both experimentally and theoretically) MAB phases as (x) [14, 17], theoretical hard coatings as (x), and experimentally hard coatings as (O) [74, 75]. Where plot a) includes the Pugh criteria and Fransevich criteria and b) includes the Pugh criteria and the Cauchy pressure.
The previously obtained results, denoted with (x) was obtained from [17, 76, 77]. The graph includes the well-known MAX-phases Ti2AlC, Ti3AlC4 and Ti3SiC4 as well TiAlN
and TiCN, which are common thin films applied as hard coatings. The Ti-Al-C family have attracted a great amount of interest due to the mechanical merits of being both ceramic and metallic [78, 79]. One may observe Ti4AlB4 and Hf4AlB4 as close neighbors
to Ti2AlC while the discussion regarding the stability of the M = Ti family have already
been held in the previous paragraph. One may based on the excellent properties of Ti2AlC
conclude future investigations of Ti4AlB4 to be conducted as Ti4AlB4 may be considered
close to stable with a formation enthalpy of +10 meV per atom.
The study did not consider any contribution from spin-polarization. The magnetic ordering of Hf, Mo, Nb, Ta, Ti, V, W and Zr are all non-magnetic while Cr being a magnetic element [80]. One might argue that the magnetic contrition may affect the stability computations. However, Dahlqvist et al. demonstrated the contrast between magnetic and nonmagnetic stability computations of MAX-phases, M={Ta, V, Cr, Mn}, where
between the magnetic and nonmagnetic computations [31, 81]. One may assume the same deviation holds for borides hence stability-wise, the magnetic contribution may not be significant. Furthermore, the mechanical computations should, however, be observed with a more critical perspective. Farzan et al.[82] demonstrated the mechanical properties dependence of TiN of the applied exchange correlation solution. The solutions being the local density approximation (LDA) [83] and GGA: Perdew–Burke–Ernzerhof (PBE) [67], Wu–Cohen (WC) [84], Perdew–Burke–Ernzerhof solid (PBEsol) [85] approximations. The mechanical properties, such as the Young modulus, demonstrated by M. Farzan was 141 GPa in contrast to 176 GPa utilizing the GGA(PBE) versus LDA, respectively, for zinc-blended TiN. An additional approximation is the LDA+U. Lind suggests LDA+U, which is considered to be a method for treating systems of strongly correlated electrons, for which one may perhaps include systems with M = Cr [86]. Perhaps the result of table 3.2 may deviate from experimental values. However, the trends of figure 3.4 should still be valid. One should, based on the previous topics, consider to investigate the magnetic contribution and temperature dependence on the phase stability of MAB-phases in general. The observed trends of figure 3.1 should be further investigated. One suggestion would be to apply a special quasirandom structure (SQS) in order to observe formation enthalpy changes as the structures are randomized. Additionally, one may refine the obtained mechanical results by considering the mentioned characteristics as a future study.
Theoretical computations at 0 K based on the stability and mechanical properties of M4AB4 phases, where M = {Cr, Hf, Mo, Nb, Ta, Ti, V, W, Zr} has been performed. The
stability computations included thermodynamic- and dynamical-stability computations. Figure 3.1 revealed a negative formation enthalpy for Mo4AlB4 which suggest a stable
phase. Neighboring compositions of the 414 phases where M = Mo and M = Ti, such as Mo3AlB4, Mo2AlB2, Ti4AlB4 and Ti2AlB2 indicated phases close to stable which are
suggested to be further investigated. Stability trends based on the electron configurations was observed in the 212, 314 and 414 compositions. The 212, 314 and 414 compositions revealed a lower formation enthalpy as the M-element was of a lower order of electron configuration. The 111 composition revealed a similar trend but based on the group number of the M-element. A higher group number of the M-element resulted in a lower formation enthalpy. The Mo-based compositions, the 111, 212, 314 and 414 phases were carried onto the dynamical computations where the phonon dispersion plots revealed no complex frequencies which indicates dynamically stable phases. The phonon dispersion plots also validates figure 3.1, where the phase stability of the MoAlB, Mo2AlB2 and
Mo3AlB4 phases may be seen as metastable. The stability computations do suggest the
Mo4AlB4 phase to be further investigated, both theoretically and experimentally. Based
on the obtained results, one is recommended to further investigate stability trends in regards to temperature and magnetic contributions on MAB-phases in general. Addition-ally, the phase stability trends of figure 3.1 should be investigated.
The ductility plots of group IV, V and VI revealed a linear trend based on the electron configuration, which is demonstrated in figure 3.4.The majority of phases are seen to be brittle and demonstrates properties similar to materials which are commonly utilized as hard coatings. The phase Cr4AlB4 demonstrated the highest Young’s modulus while
Zr4AlB4 demonstrates the lowest. The obtained mechanical properties may not be
1. Barshilia, H. C., Prakash, M. S., Jain, A. & Rajam, K. Structure, hardness and thermal stability of TiAlN and nanolayered TiAlN/CrN multilayer films. Vacuum 77, 169–179. issn: 0042-207X (2005). 2. Ichimura, H. & Kawana, A. High-temperature oxidation of ion-plated TiN and TiAlN films. Journal
of Materials Research 8, 1093–1100 (1993).
3. Chim, Y., Ding, X., Zeng, X. & Zhang, S. Oxidation resistance of TiN, CrN, TiAlN and CrAlN coatings deposited by lateral rotating cathode arc. Thin Solid Films 517. 4th International Conference on Technological Advances of Thin Films and Surface Coatings, 4845–4849. issn: 0040-6090 (2009).
4. Jindal, P., Santhanam, A., Schleinkofer, U. & Shuster, A. Performance of PVD TiN, TiCN, and TiAlN coated cemented carbide tools in turning. International Journal of Refractory Metals and Hard Materials 17, 163–170. issn: 0263-4368 (1999).
5. Oganov, A. R. et al. Ionic high-pressure form of elemental boron. Nature 457, 863 (Jan. 2009). 6. Solozhenko, V. L., Kurakevych, O. O. & Oganov, A. R. On the hardness of a new boron phase,
orthorhombic γ-B28. Journal of Superhard Materials 30, 428–429. issn: 1934-9408 (Dec. 2008). 7. Ogitsu, T. et al. Imperfect Crystal and Unusual Semiconductor: Boron, a Frustrated Element.
Journal of the American Chemical Society 131, 1903–1909 (2009).
8. HERBST, J., CROAT, J., PINKERTON, F. & YELON, W. Relationships between crystal-structure and magnetic-properties in Nd2Fe14B. English. PHYSICAL REVIEW B 29, 4176–4178. issn: 0163-1829 (1984).
9. Yeung, M. T., Mohammadi, R. & Kaner, R. B. Ultraincompressible, Superhard Materials. Annual Review of Materials Research 46, 465–485 (2016).
10. Fahrenholtz, W. G., Hilmas, G. E., Talmy, I. G. & Zaykoski, J. A. Refractory Diborides of Zirconium and Hafnium. Journal of the American Ceramic Society 90, 1347–1364 (2007).
11. Fahrenholtz, W. & Hilmas, G. Ultra-high temperature ceramics: Materials for extreme environments. Scripta Materialia 129. doi:10.1016/j.scriptamat.2016.10.018 (Oct. 2016).
12. Zapata-Solvas, E., Jayaseelan, D., Lin, H., Brown, P. & Lee, W. Mechanical properties of ZrB2 -and HfB2-based ultra-high temperature ceramics fabricated by spark plasma sintering. Journal of the European Ceramic Society 33, 1373–1386. issn: 0955-2219 (2013).
13. Zhang, G.-J., Guo, W.-M., Ni, D.-W. & Kan, Y.-M. Ultrahigh temperature ceramics (UHTCs) based on ZrB2 and HfB2 systems: Powder synthesis, densification and mechanical properties. Journal of Physics: Conference Series 176, 012041 (June 2009).
