A Model for Understanding the Formation Energies of Nanolamellar Phases in Transition Metal Carbides and Nitrides

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A model for understanding the formation energies of nanolamellar phases in transition metal carbides and nitrides

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Modelling and Simulation in Materials Science and Engineering

Modelling Simul. Mater. Sci. Eng. Modelling and Simulation in Materials Science and Engineering

Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 (22pp) doi:10.1088/0965-0393/24/5/055004 24

2016

055004 Hang Yu1, Matthew Guziewski1, Gregory B Thompson2 1 H Yu et al and Christopher R Weinberger 1 Mechanical Engineering and Mechanics Department, Drexel University, A model for understanding the formation energies of nanolamellar phases 3141 Chestnut, Philadelphia PA 19104, USA 2 Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa AL 35487, USA Printed in the UK E-mail: [email protected]

MSMS Received 21 December 2015, revised 12 April 2016 Accepted for publication 14 April 2016 Published 6 May 2016 10.1088/0965-0393/24/5/055004 Abstract In this paper we introduce a stacking-fault based model to understand the Paper energetics of formation of the nanolamellar-based metal carbide and nitride structures. The model is able to reproduce the cohesive energies of the stacking fault phases from density functional theory calculations by fitting the energy 0965-0393 of different stacking sequences of metal layers. The model demonstrates that the first and second nearest metal-metal neighbor interactions and the nearest 1 metal-carbon/nitrogen interaction are the dominant terms in determining the cohesive energy of these structures. The model further demonstrates that above a metal to non-metal ratio of 75%, there is no energetic favorability 22 for the stacking faults to form a long-range ordered structure. The model’s applicability is demonstrated using the Ta-C system as its case study from which we report that the interfacial energy between ζ-Ta C and TaC or Ta C 5 4 3 2 is negligible. Our results suggest that the closed packed planes of these phases should be aligned and that precipitated phases should be thin, which is in agreement with experiments.

Keywords: nanolamellar, phase stability, stacking faults, interfacial energies, microstructure, DFT (Some figures may appear in colour only in the online journal)

1. Introduction

The group IVB and VB transition metal carbides and nitrides are refractory ceramics that are known for their ultra-refractory nature and corresponding high hardness [1–4]. These properties manifest themselves in the complex bonding in these materials, which is often described as a mixture of covalent, metallic and ionic bonds [5, 6]. Covalent bonds form between the metal and non-metal atoms from the overlap of their respective d and p electrons. The metallic bonds form from overlap in the metal-metal d electron orbitals and some charge trans- fer occurs from the metal atoms to the non-metal atoms. The unique nature of these bonds give rise to a range of properties of these materials [3], their complex phase diagrams [7, 8], as well as the potential to form nanolemellar structures [9]. However, there is still a lack of understanding of how and why certain phases form within these systems, despite their great potential to tailor and enhance the properties of the material. As an example, the ζ-M4C3 phase forms as a complex set of thin laths or stacking faults in the B1 matrix structure in both the carbides and nitrides [10, 11]. Between the carbides and nitrides, this phase has been most heavily studied in the tantalum carbides because high volume fractions of ζ-M4C3 has been correlated to a fracture toughness that is approximately twice in either monolithic TaC or Ta2C [12–14]. Furthermore, the phase stability of this zeta phase is debated [7, 15, 16]. While no other materials with similar volume fractions of the ζ-M4C3 have been evaluated for similar high fracture toughness or stability arguments, it stands to reason that its properties and phase stability could also be present in other carbides and nitrides. To initiate our discussion of the ζ-M4C3 and other related stacking fault phases, we will provide a brief primer on the major crystal structures within in these transition metal IVB and VB carbides and nitrides with varied non-metal atom concentrations. At equal parts metal and non-metal concentrations, most group IVB and VB carbides and nitrides nominally form the B1 or rocksalt structure with the exception of vanadium carbide which cannot form this stoichiometric phase, but reverts to a carbon depleted B1 structure [17], as well as TaN and NbN, both of which have additional polymorphs [18–20]. With the depletion of the non- metal atoms, intermediate phases (between the metal atom and the rocksalt structure) are able to form. These include vacancy ordered and faulted phases [21]. At low non-metal vacancy concentrations (relative to the B1 structure), a vacancy ordered phase can be observed at the chemical composition of M6X5, where M denotes the metal atoms and X denotes the non- metal atoms [22]. In particular to the carbides, this phase is known to exist in the group VB carbides and is predicted to exist in the IVB carbides, although the exact crystal structure of this phase is still debated [23]. The current literature has not identified any vacancy ordered structures at this composition in the nitrides, however phase diagrams do show the potential to create both metal and nitrogen vacancies in the B1 lattice. At the composition M2X, most of the group IV carbides and nitrides either form a vacancy ordered form of the rock salt structure or the metal atoms reorganize into an hcp array and the non-metal atoms order in the octahedral interstices. The Ti2N structure, the prototypical structure that is predicted to exist in the group IVB carbides as well, is related to the M6X5 structure in that it is a vacancy ordered form of the B1 structure [24]. Alternatively, the M2X structures of the group VB carbides (e.g. Ta2C, Nb2C and V2C) is based on an HCP array of the metal atoms but differ in the ordering of the non-metal sub-lattice at low temperatures [23, 25–28]. However, at high temperatures the non-metal sub-lattice disorders forming the L’3 structure. Between the M2X composition and the M6X5 compositions, number of vacancy ordered phases have the ability to form, which have been proposed theoretically by a number of studies [22, 23, 29, 30] and vary with the system as well as the concentration of non-metal atoms.

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Another interesting set of the carbon/nitrogen depleted phases that form between the MX and M2X compositions have been collectively described by Demyashev as the nanolamellar stacking fault phases (SFPs) [9]. These phases are also related to the B1 structure, just as the vacancy ordered phases are, however they involve the formation of layers of non-metal vacancies accompanied by the formation of a stacking fault. The two prevalent SFPs in these materials are epsilon (ε) and zeta (ζ) phases as described by Yvon and Parthé [17]. The epsilon phase is formed by removing every third layer of carbon/nitrogen atoms in the B1 structure and subsequently shearing the depleted layer by a Shockley partial dislocation: a 112 . Thus, 6 ⟨⟩ we can denote the epsilon phase with its chemical composition, i.e. ε-M3X2. Similarly, the zeta phase is formed by the removal of every fourth layer of non-metal atoms and the associated shearing via a Shockley partial and can be written as the ζ-M4X3 phase. The epsilon phase is known to be stable at low temperatures in the Hf-N system and stable at high temperatures in the Ti-N system, its stability is not known in the Zr-N system [19, 31–33]. The zeta phase has similar stability in the group IV nitrides as the epsilon phase. However, the zeta phase is thought to form in all the group VB carbides, but, as mentioned previously for ζ-Ta4C3, the nature of the stability of this phase is debated [15, 16]. These phases are not known to be stable in other compounds to the author’s knowledge. It is worth pointing out that the C6 structure, as observed in Ta2C, is also related by the same non-metal atom removal and subsequent shearing, and thus can be classified as a SFP. While the epsilon, zeta and C6 structures are all SFPs known to exist, it is easy to postulate the existence of related SFPs. These phases would all be related back to the parent B1 structure and can be formed from it by a coupled non-metal atom depletion and associated shear. For example, Demyashev has postulated a M6X5 SFP whose existence has yet to be verified [9]. It is also possible to postulate the existence of other related SFPs such as a M7X6 phase, M8X7 phase and so on. In the limit of the largest index, these SFP converge on a single isolated plane that is depleted of non-metal atoms and sheared by a Shockely partial dislocation, a process quite important for moderate temperature slip in some of these materials [34–36]. While the formation energies of vacancy ordered phases in the rocksalt structures have been studied extensively [22, 23, 29, 30], there has been less attention to developing models of the nanolamellar phases. In this work, we start to develop a model for the formation energies of the SFP based on the interaction energies between the layers themselves. Since many of these phases either do not form or because they cannot be isolated in experiments to concretely determine their formation enthalpy, we utilize electronic structure density functional theory to compute the enthalpies of formation of these structures using the tantalum-carbon system as our case study. The Ta-C system is an ideal case study system since the stability of the zeta phase is debated [15, 16], though current phase diagrams from computational thermodynam- ics list it as a stable phase [7], the low temperature microstructure of this phase has been studied extensively [10, 37], and its presence can dramatically alter mechanical properties [12, 13]. The general framework, using Ta-C, will provide insight into the nature of stability and microstructures that form from the existence of the zeta phase and other SFPs in the tantalum carbon system as well as other carbide and nitride systems.

2. Cohesive energy calculations using density functional theory

In order to test the model developed below, we will use electronic structure density functional theory (DFT) to evaluate the cohesive energies of the structures at zero Kelvin. We choose the approach of using DFT because it allows us to evaluate the energies not only of well-known structures, i.e. TaC and Ta2C, but also those structures that either cannot be

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isolated in experiments (ζ-Ta4C3) or hypothetical phases, such as ε-Ta3C2, which have not been experimentally reported to form. This will allow us to fully populate any model as well as provide insight into why certain phases are energetically favorable to form and others are not. The thermodynamically stable structures computed from DFT lie on a convex hull of the enthalpies of formation and we will use the convex hull construction throughout this paper to compare the energetics of SFPs. However, the model developed here will focus on representation of the cohesive energy for the structures at zero Kelvin. In this case, the convex hull for the negative of the cohesive energy also represents the collection of stable structures, just as the convex hull of enthalpies of formation. Thus, throughout this manuscript we will focus on plotting the negative of the cohesive energy. The cohesive energy of all of the structures in this paper were computed using electronic structure density functional theory as implemented in the Vienna ab initio Simulation Package (VASP) [43–46] using a plane-wave basis. The projector augmented wave (PAW) [47, 48] psuedopotentials were used in all calculations and the exchange correlation energies were evaluated using the formulations of Perdew–Burke–Ernzerhoff (PBE) [49, 50] within the generalized gradient approximation (GGA). The plane-wave cutoff energy of 400 eV was found to be sufficient to converge the cohesive energies. Integration in k-space was performed using a Monkhorst–Pack scheme where the number of integration points was kept roughly constant −1 in reciprocal space at density of 2π ⋅ 0.03 A˚ amongst the phases. The initial atomic positions were assigned based on the arrangement in the rocksalt structures and the lattice constants and atomic positions were relaxed using a conjugate gradient method. The structure was described by a trigonal base a, b, c which are along [11¯ 0], [101¯] and [111] direction relative to the rocksalt structure. The electrons involved in the calculation were the 5ds326 in Ta and 2sp222 in C. To demonstrate the robustness of this approach and the suitability of the PBE parameterization within GGA applied to the tantalum carbon system, we first compute the cohesive energy and lattice constants of bulk TaC and Ta2C and compare our predictions with experimental results. We note that our results are in general agreement with our previous use of PBE to model the tantalum carbon system [30, 51, 52]. Table 1 lists the cohesive energies and lattice constants computed from DFT along with experimental measures. In general, we note good agreement between the predictions and experiments and the error in the DFT calculations demonstrate an over prediction of the lattice constants that are typical for GGA. The prediction of the bulk modulus of TaC is also in good agreement with experiments. The authors are unaware of experimental reports for the elastic constants of Ta2C or the surface energies of either compound that would enable additional comparison with DFT predictions.

3. Elastic and dynamic stability

As mentioned in the introduction, the stability of the SFPs in the transition metal carbides and nitrides is debated. While the energetics of formation, which is the basis for most of this paper, are key in determining the stability of such phases an equally important consideration is the elastic and dynamic stability of these compounds. If any of the nanolamellar phases are either elastically or dynamically unstable, the phases will never form regardless of the energetics. Thus, prior to developing a model for the cohesive energy of the SFPs, it is important to estab- lish elastic and dynamic stability. To accomplish this, we utilized density functional theory to compute the elastic constants, which include the effects of ion relaxation, of five different SFPs: TaC, Ta6C5, Ta4C3, Ta3C2, and Ta2C. From a direct analysis of the elastic constants of all of these phases we found that the elastic stiffness matrixes are all positive definite, demonstrating elastic stability. Figure 1

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Table 1. A comparison between DFT calculations and experimental data for known structure in the tantalum carbon system.

Ecoh 1 a (Å) c (Å) (eV f.u.− ) Bulk modulus (GPa)

TaC(exp) 4.457 [38] — 17.0 [39] 329 [40] 344 [41] TaC(DFT) 4.476 — 17.2 340 Ta2C(exp) 3.103 [28] 4.938 25.6 [42] — Ta2C(DFT) 3.118 4.959 26.1 270

Note: The cohesive energy per formula unit of TaC and Ta2C are computed from the enthalpy of TaC −1 Ta2C formation reported from experiments, where ∆H f =−144.1 kJ mol and ∆H f =−197.6 kJ mol−1.

shows the variation of the elastic constants of the SFPs as a function of carbon content, which is approximately linear. This includes TaC whose elastic constants have been rotated to a hexagonal setting to facilitate a direct comparison with the other phases. Given that all of the studied SFPs are elastically stable, the higher order SFPs to lie between Ta6C5 and TaC, and the linear trend in the elastic constants, it stands to reason that higher order SFPs will be elastically stable as well. However, elastic stability is not an absolute guarantee of material stability and dynamic stability should be confirmed using phonon dispersion curves. The phonon dispersion curves for TaC, Ta4C3, Ta3C2, and Ta2C are shown in figure 2 as computed using density functional pertabation theory. The dispersion curves were computed in high symmetry directions in the trigonal class for Ta2C through Ta4C3, while cubic directions were used for TaC. The phonon dispersion curves show no negative frequencies, confirming the conclusions from elastic constant calculations that these materials are stable. It is reasonable to assume that, given the elastic constant trends, the trends in phonon dispersion curves will hold for higher order stacking fault phases as well. Thus, we can conclude that all of the SFPs are elastically and dynamically stable and, as such, this type of stability does not dictate phase formation in the tantalum carbon system.

4. Nanolamellar model

With elastic and dynamic stability established, we proceed with our model. Our construction will be developed from the transition metal carbides and nitrides structure descriptions of Demyashev [9]. Our contribution will be extending this model as well as quantifying the energetics for predicting stability, which has not been extensively done in these material classes. In this model, two types of close-packed metal layers have been defined: c and h, where c denotes a layer in a cubic close-packed (ccp) metal stacking sequence and h denotes a layer in a hexagonal close-packed (hcp) metal stacking sequence. Thus, the pure B1-MX phase can be written as simply as c, the pure C6-M2X phase as h, and the secondary ζ-M4X3 phase can be represented as hhcc. In this notation, the location of the non-metal atoms has to be assumed to lie in the octahedral interstices between the metal layers and that an h layer only has 1 non-metal nearest neighbor while a c layer has two. Note that the implied number of adjacent non-metal metals can and will be relaxed later. In this paper, we will refer to this model as ⟨ch, ⟩ model, or the 1mm model since it only considers the first nearest neighbor metal-metal layer interactions. This conceptual model is a good starting point because all of the SFPs of the transition metal carbides can be obtained by the sequential removal of non-metal layers followed by a shear associated with a Shockley partial relative to the B1 structure and thus can

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800 C11 C33 C44 C12 C13 C14

700

600

500

400

300

200 Elastic Moduli (GPa)

100

-100 0.3 0.35 0.4 0.45 0.5 0.55 C/(Ta+C)

Figure 1. Calculated elastic constant for different nanolamellar phases. From left to right they are: Ta2C, Ta3C2, Ta4C3, Ta6C5 and TaC.

TaC Ta2C (a) 25 (b) 25

20 20

15 15

10 10

5 5 Frequency (THz) Frequency (THz)

0 0

-5 -5 Γ XW Γ L Γ MK Γ ALH A|L M|KH

20 20

15 15

10 10

5 5 Frequency (THz ) Frequency (THz )

0 0

-5 -5 Γ MK ΓALH Γ L B1|B Z Γ X|Q FP1 Z|L P

Figure 2. Phonon dispersion relations calculated using density functional perturbation theory (DFPT) as implemented in VASP [53] [54] for (a) TaC, (b) Ta2C, (c) Ta3C2 and (d) Ta4C3. The lack of negative frequencies in the phonon dispersion curves confirms dynamic stability of these materials.

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A A A B B B Shear C C A Remove A A B B B C C C A

(a) (b) (c)

Figure 3. The process of creating a carbon depleted stacking fault within the TaC stacking sequence. (a) The initial stacking sequence of TaC illustrating two complete repeats of the stacking sequence along a ⟨⟩111 direction, i.e. ……ABγαCAβγBCαβ . (b) The removal of a carbon layer at the α position followed by (c) a shear in the ⟨⟩112 direction creating a carbon depleted stacking fault which is denoted as …↑ABγγABαβCAγ …

be written as sequences of h and c layers. For example, if a stacking fault forms between the first B and C layer in the B1 structure:…… ABγαCAβγBCαβ , the faulted stacking sequence will be …↑ABγγABαβCAγ … where ↑ represents a shear and non-metal vacancy pair as illustrated in figure 3. This takes the ……cccccc structure to the ……chhccc using the ⟨ch, ⟩ model. Note that one advantage of the ⟨ch, ⟩ model is that it already includes the information of the first nearest neighbors. The other feature of SFPs is that they should have long range ordered stacking sequences, which results in a periodic structure. Thus, it is straight forward to create a number of SFPs of the form MnXn−1, where n is an integer greater than 1. The full stacking sequence for the common SFPs are listed in the table 2. In this paper, the M6X5 phase will be treated as a SFP as proposed by Demyashev and should not be confused with the vacancy ordered M6X5 phase that is observed to form in the VB carbides [55–58]. Here, we extend the simple stacking sequence idea to create a stacking-fault-based model for the cohesive energies of these phases. First, we assume that the cohesive energy of a nanolamellar structure can be written solely as a sum over the metal layers, and each metal layer i has λi metal atoms,

Ecoh =Γ∑ λii, (1) i where Γi is the energy of this stacking sequence per metal atom on that layer. This assumption implies that the energy contributions from the non-metal atoms as well as the metal atoms are included in the determination energies Γi. The chemical potential of the non-metal atoms can be extracted from this information, as is done later in this manuscript. Furthermore, for these nanolamellar structures, it is possible to categorize the metal layers via their stacking sequence such that each metal layer is either a c type or an h type. If we note the number of h layers as ηh and the number of c layers as ηc, the cohesive energy of any nanolamellar structure can be written as Ecoh =Γηλh hh+Γηλc cc. The cohesive energy per-atom εcoh would then be:

⟨⟩ch, ∑i ηλi iiΓ ε coh = (2) ⟨⟩ch, nc + ∑i ηλi i

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Table 2. The stacking sequences of the 5 SFPs introduced in the literature. SFPs Stacking sequence 1mm representation

TaC ()Aγ 3 c Ta6C5 ()ABγγ↑ ABαβCAγ 3 chhccc Ta4C3 ()ABγγ↑ ABα 3 chhc Ta3C2 ()ABγγ↑ A 3 chh Ta2C AγB ↑ hh

Note: The stacking sequences in parentheses are the shortest repeat unit and the rest of them follow the pattern of this unit with permutation of the layers. For example, (AAγγ)3 ≡ BCαβ.

where nc is the number of carbon/nitrogen atoms in the structure. In our DFT calculations, which is typically used to parameterize the models described herein, the nanolamellar structures were created in a trigonal basis where x lies along ⟨112⟩ and y lies along ⟨110⟩ and z lies along ⟨111⟩ such that each layer contains only one atom, either M or X. Thus, in the parameterization of the model, the values of the λs are usually taken as one. For example, in a perfect B1-TaC crystal the Γs of each layer will have a single value, Γc, because each layer is stacked in a cubic closed packed arrangement and the structure is completely periodic. Similarly, Ta2C can be completely described with Γh. Using the 1mm model, the cohesive energies per-atom in TaC and Ta C can be expressed as 1 Γ and 2 Γ and thus the 2 2 c 3 h model can be fully parameterized with the cohesive energies of TaC and Ta2C. This allows the 1mm model to predict the cohesive energies of other SFPs, such as those listed in table 2. Figure 5 plots the cohesive energies per-atom of the SFPs computed using this model with those found directly using DFT. Note that in this figure, the negative of the cohesive energy per-atom is plotted so that it is consistent with plots of the formation enthalpy, ∆Hf . Obviously the model fails to capture the trends observed except at the end points, i.e. the 1mm model cannot predict the energy of the other SFPs. In addition, the model suggests that all nanolamellar phases are equally favorable since they all lie along a line on the convex hull and thus they could decompose into each other without any change in energy. This prediction is a consequence of the model only accounting for the first nearest metal atom interactions. In order to improve the model to capture more information than these interactions, the 1mm model will now be extended to consider the adjacent carbon/nitrogen layers to differentiate the stoichiometry of each metal layer as shown in figures 4(a) and (b). For example, c1 can be used to denote the cubic-closed-packed layer with only one adjacent carbon/nitrogen layer, figure 4, distinguishing it from c2, which has two adjacent non-metal layers figure 4(b). In this model, we will not consider metal layers that have no adjacent non-metal layers, which means the model will not include the c0 and h0 metal layers, since these depleted metal layers do not appear in our SFPs or other carbides/nitrides of interest. This improved model has four types of metal layers ⟨cc12,,hh12, ⟩ and we will henceforth be referred to as the 11mm − mc model because it takes the influence from the nearest metal-non-metal interactions into account. The 1mm − 1mc model, when applied to predict the energetics of the SFPs, only uses c2 and h1 in representing their cohesive energies. This is because all of the SFPs are related through a coupled carbon/nitrogen depletion and shearing process that always changes c2 to h1, a result of the model only considering first nearest neighbor interactions. Therefore, the prediction of a straight line in figure 5 does not change with the 11mm − mc model. The values of h2 and c1 can be obtained by fitting a series of additional phases, which we will call the extended phases, that have the same metal stacking sequences as the SFPs but with differing non-metal content. Thus, the 11mm − mc model can distinguish between the SFPs and the

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(a) (b) c A h2 2

B c2 C c h1 1

(d) (c) C A

B c20 c10 h22

Figure 4. The 1mm − mc and 21mm − mc models can be understood through local stacking environments. (a) The rocksalt stacking squence can be analyzed in terms of a single layer and its first nearest neighbor metal and carbon atoms. The local environment is described as c2 which indicates the middle layer is in a cubic closed packed metal arrangement with two carbon layers. (b) The four different stacking sequences of the 11mm − mc model. (c) The 21mm − mc model is obtained by further included the second nearest neighbor metal atoms. In this model, the rocksalt stacking sequence can be written as c20, where the additional (second) index indicates the number of second nearest neighbors that have the same latin index in the stacking sequence. For example, the atom in the ‘B’ stacking sequence of the rocksalt structure has no second nearest neighbors with a ‘B’ label, thus c2 becomes c20. (d) The c10 and h22 sequences are among the many possible in the 21mm − mc model. It is worth noting that even though the 2nd nearest neighbor carbon atoms are included in the diagram, they are not included in the model.

extended phases. However, to be able to accurately capture trends in the cohesive energy of SFPs, a model that describes higher order metal-metal interactions must be developed. The second nearest neighbor model (2mm − 1mc model) is obtained by expanding the 11mm − mc model by including second nearest neighbor layer interactions as shown in figures 4(c) and (d). In this model, we will denote metal layers as ⟨xij⟩ where x is the stacking sequence of the metal layer, i is the number of adjacent non-metal layers and j is the number of second nearest neighbor metal layers that have the same stacking position. For example, the repeating stacking sequence in TaC can be written as: AγαBCβ in terms of its stacking sequence and can be represented in the 2mm − 1mc model as c20cc20 20 as illustrated in figure 4. This stacking sequence can then be reduced to a single formula unit of c20. The resulting 2mm − 1mc model has 12 distinct layers: ⟨cc10,,11 cc12,,20 cc21,,22 hh10,,11 hh12,,20 hh21, 22⟩, three of which are illustrated in figures 4(c) and (d). While this representation is sufficiently general for our purposes in modeling the SFPs and related phases under consideration here, we have chosen to ignore two distinct stacking types. First, we omit all possibilities of metal

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1mm model prediction for SFP DFT calculation model prediction −8.62

−8.64 (eV) co h

- −8.66

−8.68

0.35 0.4 0.45 0.5 C/(C+Ta) ratio

Figure 5. The cohesive energy per-atom for the SFPs predicted using the basic ⟨⟩ch, model (Γc = 17.231 eV and Γh = 13.037 eV) and compared with that computed directly from DFT as a function of carbon concentration. The model predicts a straight line which implies that all of the SFPs are equally favorable.

layers with no adjacent non-metal layers, i.e. c0x and h0x, as mentioned prevoiusly. Hence, the current model does not capture the stacking sequence: ...ββABC .... Second, we omit the possibility that two metal layers will stack up on one another: e.g. ...AAγ .... This type of stacking occurs in I1 faults in stochiometric B1 structures [59] but is not useful in describing SFPs since they simply do not exist in these phases. The model also assumes that non-metal atoms stay in their octahedral interstices of the metal atom lattice. It is possible to extend the model easily to include c0x and h0x layers, or the AγA stacking sequences. However, adding these effects will not affect the results presented in this paper.

5. Model parameter estimation

In order to estimate the energy of each layer in the 2mm − 1mc model, we must have at least as many structures as there are variables in the model in order to determine all of the parameters. In order to achieve this, we have created a number of phases (see appendix) that are related to the SFPs which, as mentioned previously, have the same metal stacking sequences as the SFPs but have had carbon layers either removed or added to cover a wide range of carbon concentrations. The cohesive energies of the supercells listed in table A1 were computed using DFT under full periodic boundary conditions and are given in table 3. The cohesive energies of the SFPs as well as the extended phases can be written as a sum of the energies of the layers in our model and thus a linear mapping can be established between the cohesive energies of the phases and the energies of individual metal layers. This results in a linear system of equations: Ax = b, where A is a coefficient matrix, x is the energy per-atom for each type of layer and b is the cohesive energy. A least squares method was used to estimate the value of the layer energy corresponding to a best-fit solution.

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Table 3. The cohesive energies of the extended phases in (eV) listed in appendix computed using DFT.

Structure MX M6X5 M4X3 M3X2 M2X 0 206.78 309.09 205.76 154.11 204.18 1 198.25 301.20 197.89 146.17 196.31 2 189.78 293.32 190.03 138.17 188.39 3 181.41 285.44 182.12 130.14 180.47 4 172.21 — 173.19 — 172.46 5 163.01 — 164.16 — 166.44 6 153.81 — 155.11 — 156.44

Note: The structure number, i.e. 0–6, indicates the number of non-metal layers have been removed following the sequence shown in table A1.

However, it can be proven that the coefficient matrix A does not have full column rank for any model that considers interactions beyond the first nearest neighbor layers, which means the best fit solution is not unique. First, we note that the layers c12, h10, and h20 do not exist in any of the extended phases listed in table A1 used in our fitting procedure and results in rank deficiency of the coefficient matrix. Although it is possible to include these 3 variables by creating twin-like structures such as ... ABγαCBαγA ..., we ignore these stacking sequences to further simplify the model. Removing these three variables from our coefficient matrix, however, does not eliminate the rank deficiency. This additional rank deficiency arises from the fact that some of the layers (based on their second nearest neighbor interactions) cannot exist independently. This dependence arises as a non-empty null space for our coefficient matrix A. The null space of this revised coefficient matrix A, in which c12, h10 and h20 are absent, is:

Γ cc11 +Γ21 +Γ20ch22 −Γ11 −Γh21 = . (3) It is possible to prove that the rank deficiency that is still present cannot be eliminated by creating additional nanolamellar structures, and thus it is a property of the 2 mm interactions included in the model. Since one can add or remove non-metal layers independently of the metal stacking sequence, metal layers with a different number of neighbors (but the same stacking sequence) can be collapsed into a single type when analyzing the null space. For example, C0 can be used to represent the total number of c10 and c20 layers in the structure and thus it is possible define Xxjj=+12x j for any metal stacking sequence. This 2 mm model can be used to analyze the null space of 2mm − 1mc model without loss of generality, which has

6 variables ⟨ΓΓCC01,,ΓΓCH20,,ΓΓHH12, ⟩. First, note that it is impossible to form a pure C1 stacking sequence, which indicates that the rank deficiency of this model must be equal to or greater than 1, and thus a null space must exist. Next consider the 6 nanolamellar structures that can be represented using the 2 mm model tabulated in table 4. The coefficient matrix for these structures is a 66× coefficient matrix:

⎛000002⎞ ⎜1 00000⎟ ⎜020020⎟ A = ⎜ ⎟, ⎜001020⎟ ⎜002200⎟ ⎝021121⎠

11 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

Table 4. The structures selected structures used to analyze the null space of the coefficient matrix. Structures Model

(AB)ABAB H22H (A)BCABC C0 (ABAB)CACABCBC C11HHC11 (ABA)BCBCAC C21HH1 (ABCB)ABCB H02CH02C (ABABCBA)CACABAC H12HH12CH01CC1

Note: These select structures are sufficient to create a coefficient matrix of rank 5 and it is possible to demonstrate that the rank of the matrix cannot be increased through the inclusion of additional structures.

The rank of this coefficient matrix is 5, indicating that the null space has a dimension no greater than 1. Considering the requirement that the dimension of the null space is at least 1, the dimension of the null space must be 1 and the null space can be directly determined from the reduced matrix A. Without the loss of generality, the null space can be represented as

(Γ−CH11Γ+)(20Γ−CH20Γ=) . This argument can be finally extended to the full2 mm − 1mc model that includes our previously excluded metal layers to the completely general null space as:

( Γ+cc11 Γ−21 Γ−hh11 Γ+21)(20Γ+cc12 Γ−22 Γ−hh10 Γ=20) , (4) Considering our omitted variables, the null space collapses to equation (3). This proof shows that it is impossible to remove the null space from the 2mm − 1mc model except by degenerating the model back to 11mm − mc. Thus, the values of some of the layers cannot be fully determined if they are not perpendicular to the null space. Although this makes the parameter estimation a rank deficient least squares problem, it is still solvable using the singular value decomposition (SVD). In addition, since we want to reproduce the cohesive energies of the SFPs more than the extended phases, a weighting matrix W is applied to adjust the fitting procedure. By adjusting the value in the weighting matrix, we can fix the values that are perpendicular to the null spaces by themselves, i.e. the cx0 and hx2 layers. The other parameters in the weighting matrix were adjusted to provide good fits toε and ζ phases. The values from this weighted least squares fit are listed in table 5. Figure 6 shows a plot of the 2mm − 1mc model predictions for the cohesive energy per- atom compared with DFT values. Overall, the fit is quite good. However, the dashed line shows a clear deviation between DFT and the 2mm − 1mc model when the metal atoms are in a ccp arrangement. We believe this is a result of a higher order interactions between metal and non-metal atoms, i.e. 2mc interactions, which are not considered in our model and potentially occur at low non-metal concentrations in the ccp structure. However, these 2mc interactions do not appear to affect the model’s ability to predict the energetics of the other SFPs or extended phases. Since the focus of this work is in describing the SFPs, expanding the 2mm − 1mc model to count for this type of interaction will unnecessarily increase the complexity of the model and thus is not pursued in this paper. However, we have corrected our model by specifying an additional interaction term for the MX extended phases with the energy of the corrected c being ∗ 12.818 eV for higher carbon concentrations in the ccp 10 Γc10 = structure and ∆cc10 20 = 0.179 eV as a correction factor for lower carbon concentrations in the case study of the tantalum carbide system.

12 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

Table 5. The cohesive energy per-atom in each type of metal layer described by 21mm − mc model.

ccp layer Γi (eV) hcp layer Γi (eV)

c10 12.871 h10 — c11 8.483 h11 17.375 c12 — h12 13.036 c20 17.231 h20 — c21 12.980 h21 21.346 c22 8.629 h22 17.024

Note: After applying the correction mentioned in the text to the FCC metal atom stacking sequence, ∗ 12.818 eV. The s for the c , h and h stacking sequences are undetermined Γc10 = Γ 12 10 20 because the twin-like structures, which are required to determine these values, were not considered in the construction of the extended phases and thus are not included in the fitting of the layer energies.

2mm-1mc model prediction for extended phases

−8.55

(eV) −8.6 co h -

M3X2 −8.65 MX M6X5

M4X3

M2X MX orig. −8.7 0.35 0.4 0.45 0.5 C/(C+Ta) ratio

Figure 6. The predictions of the extended phases from the 21mm − mc model for the tantalum carbides system plotted with the values determine directly from DFT. The short dashed line represent the original model without using the correction factor to the cc20 10 case. 6. Results and discussion

The 2mm − 1mc model, in conjunction with DFT calculations, can now be used to understand the SFPs within the tantalum carbon system as shown in figure 7. The model is able to accurately capture all of the cohesive energy values of the stacking fault phases that were included in the fitting procedure. This includes the higher value associated with Ta3C2, which breaks the trends of the other phases. One key outcome of the model is the prediction of a straight line between ζ-Ta4C3 and B1-TaC. This straight line implies that all of the SFPs between TaC and ζ-Ta4C3 are equally favorable. To test this prediction, we computed the cohesive energies of nanolamellar Ta7C6 and Ta8C7 phases, which appear to follow the trends in figure 7 further

13 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

validating the model’s prediction. Keep in mind the line is a guide for the reader and true stacking fault phases would occupy discrete points, becoming infinitesimally closer together as they approach TaC. This prediction, which will be elaborated upon further in the discussion that follows, is important because it suggests that there is minimal driving force to create well formed, thickened ζ-Ta4C3 grains. The DFT calculations of the extended phases and our nanolamellar model can now be used to compute the chemical potential of carbon atoms in different faulted structures. The identification of this chemical potential is important because it allows us to separate, in part, contributions from carbon depletion and the faulting process. The chemical potential of the carbon layers is: ∂G dε µεC ≡− =+()1 − x (5) ∂nc dx

where xn=+cm/(nnc) is the carbon concentration. Since the carbon chemical potential comes from the slope of the cohesive energy shown in figure 6 via equation (5), the nanolamellar model creates a simplified representation of the different carbon chemical potentials which are listed in table 6. As stated above, the chemical potentials will be useful in separating the energetics of faulting from the addition/removal of carbon. Returning to the discussion on the energetics of the faulting process, the 2mm − 1mc model predicts a straight line between the cohesive energy per-atom of the zeta phase and the B1 structure. For SFP compositions near the B1 structure, conceptually there should be little difference between the long range ordered SFPs and random carbon depleted faults in the B1 structure. To demonstrate this connection, the analytical cohesive energy per-atom of the SFPs between the zeta phase and TaC will be derived and compared with those of a collection of isolated faults. The cohesive energy per-atom of a SFP that lies between TaC and the zeta phase can be constructed by removing every y ⩾ 4 carbon layers and shearing via a Shockley partial dislocation. This transforms B1-TaC to TayCy−1 and changes the stacking sequence from, ...cccccc20 20 20 20 20 20... to ...cc20 21hhc11 11 21c20... . Using the nanolamellar model for the energies, the cohesive energy per-atom can be written as:

()y −Γ42cc20 +Γ21 +Γ2 h11 ε = . (6) 21y − In order to connect the cohesive energy per-atom directly with the carbon concentration x, the number of carbon depleted faults can be directly related to the carbon concentration x. This can be substituted into equation (6) to derive the cohesive energy per-atom as a function of carbon concentration:

ε =−()73xxΓ−cc20 ()42−Γ21 −−()42x Γh11, (7) This equation represents the energy of the SFPs predicted by the 2mm − 1mc model for x ⩾ 0.43 and creates a straight line between the ζ phase and B1 phases in figure 7. The slope of this line is: ∂ε =Γ74cc−Γ −Γ4.h (8) ∂x 20 21 11 This is a formal derivation of the SFPs from the 2mm − 1mc model and assumes periodicity of the faulted structure. Now, as mentioned previously, it is possible to connect this energy to a group of random faults, which are often observed in experiments. This derivation starts with the change in energy associated with the formation of a set of non-interacting carbon depleted faults within the B1 structure. Thus, the energy associated with the fault is

14 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

2mm-1mc model prediction for SFP and other phases

model prediction −8.62 DFT calculation

Ta7C6 and Ta8C7 from DFT

−8.64 (eV) coh - −8.66

−8.68

0.35 0.4 0.45 0.5 C/(C+Ta) ratio

Figure 7. A plot of the cohesive energies per-atom for the SFPs computed from DFT as well as those predicted by the second nearest neighbor model (solid line). The stacking fault phases, Ta7C6 and Ta8C7 are shown in the figure are not used in the fitting procedure for the model.

Table 6. The chemical potential for carbon atoms in different bonding environments within the tantalum carbon system computed from the 21mm − mc nanolamellar model.

−1 Structure Ratio range Model µC (eV atom )

c (0.5,0.43) cc20 20 ∧ cc20 20 8.441 c (0.43,0.33) cc10 20 ∧ cc20 10 9.196 ccchhc (0.5,0.45) ch21 21 ∧ hc21 21 7.945 chhc (0.5,0.43) ch21 21 ∧ hc21 21 7.945 chhc (0.43,0.33) hc11 21 ∧ ch21 11 9.008 hch (0.5,0.29) ch22 21 ∧ hc21 22 7.945 h (0.5,0.33) hhxx2222∧ hh22 7.982

Note: The ∧ symbol indicates the removal of a carbon layer without the associated shear.

U =+UU0 ∆ , (9) where U is the internal energy of the faulted structure, U0 is the energy for perfect B1 structure and the energy caused by faults is:

∆ UA= γSF SF, (10) with ASF the total area of the faults and γSF the stacking fault energy per unit area. The total area of the faults can be computed from the number of atoms that are involved in the faults times the area per-atom. We note that the area is approximately constant with respect to carbon content and thus the total faulted area is ASF =−()nnmcA0 where A0 is the area per metal atom and nmi= ∑ λ is the total number of metal atoms.

15 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

The cohesive energy per-atom can be obtained from ε =−Un/( mc+ n ). Since U0 is the internal energy of the B1 structure, it can be computed directly from our model as: U02=−ncm 0. The model also provides an estimate of the energy required to create an iso-

lated fault: γSF A0 =−22Γ−ch21 Γ+11 48Γ=c20 .217 eV. Hence the cohesive energy per-atom is given as:

nm nnmc− ε = Γ−c20 ()−Γ22ch21 −Γ11 +Γ4,c20 (11) nnmc+ nnmc+

ε =−()11xxΓ−cc20 ()−−22()Γ−21 24Γ+hc11 Γ 20 (12) Simplifying equation (12) reveals that it is identical to equation (7) which was computed previously for the periodic SFPs. These results indicate that the energy to create the SFPs or to create a set of random faults are the same as long as the random faults do not interact strongly with each other. The strong interaction associated with faults impinging upon one another, as may occur in the precipitation of the zeta phase on multiple {111} planes in TaC, has the potential to increase the energetic cost of random faults and is neglected here. We note that the stacking fault energies used in the preceding discussions include the chemical potential of the carbon atoms and thus the computed value in Ta-C system is ˚ 22˚ − −2 γSF ==8.217 /eV 8.675 A 0.95 eV A = 15.2 J m . The large value for this term is associated with the loss of carbon. There are two possible ways to treat the loss of carbon and the related shear process. We first consider the case when shear occurs prior to carbon removal, and thus the carbon chemical potential is associated with the stacking sequence of ...ch21 21 ∧ hc21 21... (the ∧ symbol denotes the removal of carbon without shear) with µC = 7.945 1 −2 eV atom− . This results in a SFE that considers only the shearing process of 0.031 eV A˚ or 500 mJ m−2. If the shearing process occurs after carbon removal, the carbon chemical potential would be associated with a stacking sequence of ...cc20 20 ∧ cc20 20... with µC = 8.441 eV atom−1. In this case SFE is −410 mJ m−2. This demonstrates that when the carbon layer is depleted first, shearing is an energetically favorable process and should happen spontaneously if there is no barrier to the shearing process. In contrast, it is energetically unfavorable to shear prior to carbon loss, as demonstrated by the positive fault energy. This result might suggest that the nucleation of carbon depleted faults may occur first through a critical concentration of coalesced vacancies on the close packed planes, similar to the formation of Frank loops in closed packed metals, followed by an energetically favorable shear. Such circular formation of ζ-Ta4C3 nuclei has been noted to occur in TaC-Ta reaction diffusion couples [10]. Alternatively, these faults could nucleate through the pref- erential binding of carbon vacancies to dislocation cores, creating wide stacking faults also observed in experiments [34, 35] and eventually creating complete carbon depleted stacking faults. In this proposed scenario, the carbon likely diffuses through the dislocation cores leaving a carbon depleted stacking fault in its wake ultimately extending the carbon depleted fault. It is important to note that the growth of the faults, regardless of the nucleation process, is likely to occur through a coupled diffusion-shear process. Never-the-less, the results of this model demonstrate that there should be a chemical driving force for carbon vacancies to diffuse to the dislocation cores that bound these carbon depleted faults, ultimately extending them. Similarly, the SFE associated with transforming from ζ-Ta4C3 to Ta2C can be also calculated through this model. For each stacking fault, the energy difference can be obtained from the difference in their stacking sequence: ...h11cch21 21 11... and ...hhhh12 12 12 12.... This results in an energy difference of γSF A0 =Γ24()hc11 +Γ21 −Γh12 = 8.563 eV including the

16 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

Table 7. The structures of and interfacial energies of the interfaces between TaC, ζ-Ta4C3, and α-Ta2C. Interface Stacking sequence Energy difference

TaC // ζ-Ta4C3 AγαBCβγAB↑↑ABγαCAβ ... 0 Ta2C // ζ-Ta4C3 AγγγBA↑↑BABCαβA ↑ ... 0

TaC // Ta2C AγαBCββββAC↑↑↑ACAC... Γch21 +Γ11 −Γch20 −Γ12

depletion of carbon. If carbon depletion occurs prior to shear, the carbon chemical potential, −1 µC = 9.007 eV atom , is associated with removal of carbon from the stacking sequence −2 ...hc11 21 ∧ ch21 11... and the stacking fault energy is −820 mJ m . If shear happens prior to −1 carbon depletion, the chemical potential of carbon is µC = 7.982 eV atom and is associated −2 with the stacking sequence of hhxx2222∧ hh22 resulting in a SFE that is about 1070 mJ m , which is quite large. It is interesting to note the values are roughly twice that of the faults forming in compositions between TaC and the ζ phase. As noted before, this demonstrates that the transformation from ζ-Ta4C3 to Ta2C is energetically favorable to shear after carbon removal but not before. Finally, this model can be used to predict interfacial energies between the closed packed planes of the TaC, Ta2C, and zeta phases. This corresponds to the alignment of the {111} planes in the B1 structure and the {0001} planes in Ta2C and zeta phases. The stacking sequences associated with the interfaces are listed in table 7 with the energy difference obtained from the energy of those stacking sequences and bulk structures listed in table 7. Our results show zero interfacial energy between both the B1 and zeta phase as well as the zeta phase and Ta2C structure. This is attributed to the fact that this alignment does not alter the local stacking sequence. However, if the B1 and C6 structures are placed together, the interface changes the stacking

sequence locally and the interfacial energy becomes γint =Γ()ch21 +Γ11 −Γch20 −Γ12 /A0. In the tantalum carbon system, this amounts to −166 mJ m−2 suggesting that the material would prefer to form an interface between TaC and Ta2C with parallel {111} and {0001} planes. While such an occurrence seems strange, i.e. a negative interfacial energy, it actually does make physical sense. The presence of a negative interfacial energy should drive the system to create a lot of interfaces, which in turn changes small portion of TaC and Ta2C to the zeta phase, which is energetically favored over the coexistance of TaC and Ta2C. It is worth pointing out that this explains the experimental observations of the coexistence of the three phases [11], with a high density of the faulted phases within the microstructure itself [10, 60, 61]. If TaC and Ta2C grains are placed next to each other, the interface will form a section of the zeta phase in-between and the negative interfacial energy will further drive the local formation of the zeta phase while kinetics (carbon diffusion) likely limits the full conversion resulting in a three phase microstructure.

7. Conclusion

In this work, we developed a nanolamellar model representation of the cohesive energies of the stacking fault phases of transition metal carbides and nitrides and applied it to the tantalum carbon system as a case study. In this model, the cohesive energy of the structure is represented as a summation of the energy associated with each close-packed metal layer and categorized the layers using a model that accounts for the second nearest neighbor metal- metal interactions and first nearest neighbor metal-carbon interactions. This model is able to reproduce the cohesive energy for most of the SFPs and extended phases in the tantalum

17 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

carbide system. This suggests that the second order metal-metal interactions are very important for reproducing the energetic hierarchy of these structures. However, the model failed to predict the pure depletion of the carbon layer in B1 structure, which is thought to be caused by second order metal-carbon interactions. Further expansion of the model would further compli- cate predictions without improving its ability to represent the energetics of the stacking fault phases. Despite this limitation, the model does show promise in understanding the formation of the nanolamellar phases in all the group VB carbides and the IVB nitrides as well as the microstructures that they form. The model, as applied to the tantalum carbon system, as well as DFT calculations demonstrate that all the stacking fault phases between the zeta phase and B1 phase lie on a straight line. This means that none of the phases are energetically favorable over the others. Furthermore, we have shown that the energetics associated with random isolated stacking faults also lie along this line on the convex hull. This is important because it suggests that there is little driving force for the formation of ordered faulted structures in the tantalum carbon system. Thus, one would expect the formation of random faults as carbon is depleted from the B1 structure, which is what is readily observed in experiments [10, 11, 61, 27]. This result does not preclude the formation of large grains of the zeta phase in tantalum carbides but provides understanding of why they are not readily observed in experiments. Finally, the model is also able to determine the carbon chemical potential between different stacking sequences in the extended phases, which allows us to extract the stacking fault energies associated with transformations between TaC to Ta2C while removing the effects of the carbon. It is found that the stacking fault energy depends on whether carbon is removed before or after faulting. If removed first, the stacking fault energy is negative if carbon is depleted and if the shear occurs with carbon is place, the energy is positive. This demonstrates that faulting is an energetically favorable process as carbon is removed from the B1 structure leading to the zeta phase. Moreover the fault energies going from TaC to the zeta phase are about half those of going from the zeta phase to the Ta2C phase. Using this model we were also able to show that the interfacial energies between TaC and the zeta phase as well as the zeta phase and the Ta2C structure are zero, which agrees directly with DFT calculations. The interfacial energy between the Ta2C structure and B1 structure is negative and thus energetically favorable to form, a consequence of the interface being a section of the zeta phase, which is indeed more energetically stable. This creates a microstructure that can have a high density of stacking faults, as observed in experiments.

Acknowledgments

H Yu and C R Weinberger recognize Air Force Office of Scientific Research grant FA9550-15- 1-0217, Dr A Sayir program manager. GB Thompson recognize Air Force Office of Scientific Research grant FA9550-15-1-0095, Dr A Sayir program manager.

Appendix. Extended phases

The stacking sequences and the order of carbon/nitrogen removal for extended phases are given in table A1. Each column represents a metal stacking sequence that corresponds to a SFP and the non-metal atoms associated with the SFP are also listed in the columns. The extended phases can be formed from these stacking sequences through the removal (or addition) of non-metal atoms. The extended phases can be formed by filling all of the

18 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al

Table A1. The stacking sequences of the extended phases.

MX M6X5 M4X3 M3X2 M2X A A A A A γ γ γ γ γ B B B B B α()1 ↑ ()1 ↑ ()1 ↑ ()1 ↑ ()1 C A A A A β γ γ γ γ A B B B B γ()4 α α()4 α ↑ ()4 B C C C A α β β ↑ ()2 γ C A A B B β()2 γ ↑ ()2 α ↑ ()2 A B C C A γ α β β γ B C A A B α()5 ↑ ()2 γ()5 ↑ ()3 ↑ ()5 C B B C A β α α β γ A C C B γ()3 β ↑ ()3 ↑ ()3 B A B A α γ α γ C B C B β()6 α β()6 ↑ ()6 C β A ↑ ()3 C β A γ B α C β

Note: The column headers denote the chemical formulate of the SFP which shares the metal stacking sequence of the extended phase. The number in bracket indicates the sequence of carbon removal.

non-metal depleted layers in table below (if any) followed by the ordered removal of the non-metal layers. The choice of removal was chosen such that the largest distance between depleted non-metal atoms was maintained during the removal process and the progres- sion of removal is given in parenthesis in the table. For example, the extended phases that form the M2X metal stacking sequence are given in the 5th column of table A1. The first extended phase is formed by first filling every other layer with γa non-metal atom creating a NiAs prototype stacking sequence (A γ B γ). The next extended phase is formed by

19 Modelling Simul. Mater. Sci. Eng. 24 (2016) 055004 H Yu et al removing the non-metal atom with the (1) label and the process is repeated until a true M2X phase is formed.

References

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