2014-PPS-A genomic approach to the stability,elastic, and electronic properties of the MAX phases.pdf

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A genomic approach to the stability,

elastic, and electronic propertiesof the MAX phases

Sitaram Aryal1, Ridwan Sakidja1, Michel W. Barsoum2, and Wai-Yim Ching*,1

1 Department of Physics and Astronomy, University of Missouri-Kansas City, Kansas City, Missouri 64110, USA2 Department of Materials Science and Engineering College of Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA

Received 22 March 2014, revised 9 May 2014, accepted 20 May 2014Published online 24 June 2014

Keywordselastic properties, electronic structure, genomic approaches, MAX phases

* Corresponding author: e-mail [email protected], Phone: 816-235-2503, Fax: 816-235-5221

In this study, we report a comprehensive assessment on theelastic and electronic properties of 792 possible MAX(Mn1AXn) phases with n 14 using ab initio methods.These crystals are then screened based on their elastic andthermodynamic stability resulting in a large database of 665viable crystals. All the experimentally veried MAX phasespassed the screening. Various correlations among and betweenthem are fully explored. In particular, the key elements in theinterdependence between the elastic properties together withmechanical parameter derived from them and the electronicstructure are identied. Detailed analysis of various correlation

plots shows that there is a clear correspondence between bulkmodulus Kand total bond order density (TBOD). Calculationsshow a marked difference between the carbides and nitrides.This database is also used to test the efcacy of data miningalgorithms for materials genome. We further identied severalthermodynamically stable new MAX phases with unusualmechanical parameters that have never been synthesized in thelaboratory or theoretically investigated. The complete databaseon the elastic and electronic structure together with themechanical parameters for these 665 MAX phases compoundsare includedin the Supplementary Materials and fully accessible.

2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The anisotropic laminated transitionmetal compounds with hexagonal crystal structure calledMAX phases or Mn1AXnare a class of ternary compoundsthat possess an unusual combination of vastly differentmechanical and electronic properties [15]. They behaveboth like metal and ceramic, which roots from its highly

anisotropic crystal structure and their diverse chemicalcomposition. Depending on their unique composition,structural arrangement and bonding pattern, these ternaryalloys possess some of the most diverse and desirableproperties such as damage-tolerance, oxidation resistance,excellent thermal and electric conductivity, machinability,etc. suitable for applications as advanced materials in manydifferent technologies and under extreme conditions. So far,only about 60 such MAX phases have been conrmed orsynthesized [1]. MAX phases were rst reported in earlysixties [69], but there has been an explosive growth in itsinterest and applications since 2000 [1].

A systematic and comprehensive study of the properties

and trends in the MAX phases is of vital importance in

understanding the reason behind their diversied propertiesand in the quest for new phases. A fairly large number ofpublished works both experimental and theoretical alreadyexists, but they are somewhat scattered and concentratedmostly on the n 1 o r MTi phases, or those that have beensynthesized [1016]. Here we report an extensive study

using ab initiocalculations of the elastic, mechanical, andelectronic properties of 792 MAX phases with n 14.They are screened for elastic and thermodynamic stabilityresulting in a database of 665 viable compounds. This largedatabase is used to establish general trends, in seekingcorrelations on various elastic properties together withmechanical parameter derived from them and electronicproperties. It should be stated in the very beginning thatthe mechanical properties we refer to in this paper are thegeneral parameters derived from the linear elastic theory andshould not be construed to imply the mechanical propertiesmeasured in the laboratory such as fracture toughnessor failure behaviors even though they may be related in

some complicated way. We also used this database to test

Phys. Status Solidi B 251, No. 8, 14801497 (2014) / DOI 10.1002/pssb.201451226 p s sbasic solid state physics

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the efcacy of the data-mining, machine-learning algo-rithms [17] in a genomic approach [18] for this class ofmaterials.

The MAX phases are layered hexagonal crystals (space

group: P63/mmc NO 194). Figure 1 displays the MAXcrystals for n 14. (From now on, we will refer them as211, 312, 413, and 514 phases.) Also, in this paper, the termMAX phases we refer to are for these crystals even thoughphases with n 5 are known to exist. Most of the existingexperimental work on the MAX phases has been on the211 and 312 carbides. The reason for the scarcity ofMAX nitrides, especially with for higher n, has not beenelaborated. An important feature to be recognized is that inMAX compounds, layer A remains constant whereas layersof M and X increase with n in Mn1AXn.The X layers arealways in between the M layers and blocks of MX layers areconnected by single A layer, which can signicantly affect

the properties of a MAX phase. It is anticipated that theelastic properties of MAX phases will vary over a wide rangedepending on M, A, X, andn. The Poissons ratio h, which isclosely related to Pugh moduli ratio (G/K) [19], and the totalbond order density (TBOD) (to be described in the nextsection), which is an indicator of the strength of interatomicbonding in a crystal are designated as the key parameters,respectively, for their mechanical and electronic propertiesof the MAX phases. Of fundamental importance is to

understand the connection between the elastic properties andthe electronic structure of MAX phase compounds, which isstill lacking.

The main objective of this work is to esh out various

correlations among and between the elastic and electronicstructures based on the large data set obtained from carefuldensity functional calculations. What is not included aretopics related to more specic properties such as hardness,corrosion resistance, magnetic structures, optical, electricand thermo-conductivities, phonon spectra, superconductiv-ity, and thermoelectric properties, which will be dealt withelsewhere in future studies. We also want to emphasize thatwe are looking mostly for obvious trends and identiablecorrelations in MAX phase properties as a distinctive classof materials with the same structure, and not on individualMAX phases, whose properties can vary signicantly.

In Section 2, we rst describe the approach we adopt and

the computational methods used in this study. The completeresults on various elastic and electronic properties and theircorrelations are presented and discussed in Section 3 andin the Supplementary Materials section. In Section 4, wedemonstrate the application of the vast data we obtained tovalidate the efcacy of data mining algorithm and statisticalmachine learning approach for materials genome in supportof the conclusions reached so far. This is followed byfurther discussions of a few thermodynamic stable buthitherto unrecognized MAX phases with unusual correla-tions (outliers). Finally, a summary and some conclusionstogether with future directions for research are presented inthe last section. The complete database for the elastic and

electronic structure properties of the 665 MAX crystals isavailable in the Supplementary Materials (SM).

2 Approach and methods We start by assuming theexistence of all possible MAX phases based on the originalsuggestion by Barsoum [1] of the M, A, X elements in thePeriodic Table that exist in at least one known synthesizedcompound. The elastic constants and the elastic constantsderived for 792 crystals with M Sc, Ti, Zr, H, V, Nb, Ta,Cr, Mo; AAl, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, S and,XC or N, and n 14 are calculated based on fullyrelaxed structures using density functional theory (DFT)based ViennaAb initio Simulation Package (VASP) [20, 21].

The calculated results are then subjected to two levels ofscreening for their elastic and thermodynamic stability.The rst round of screening is based on the CauchyBornelastic stability criteria for hexagonal crystals [22], whicheliminated 71 crystals. Next, we calculated the heat offormation (HoF) for the same 792 crystals. The HoF wascalculated based on the relative stability of each MAX phaserelative to the formation energy of its elements in their moststable form at the ground state. As a result, 72 phases withpositive HoF are then eliminated of which 27 crystalsoverlap with the rst round elimination. We should reiteratethat since the main goal of this study is to detect theunderlying major trends that dene the elastic properties and

electronic structures of existing and potential MAX phases,

Figure 1 Ball and stick diagrams of M2AX, M3AX2, M4AX3, and

M5AX4.

Phys. Status Solidi B 251, No. 8 (2014) 1481

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we purposely avoided the use of the time-consumingthermodynamic assessment on all potential competingphases in the MAX ternary phase diagrams as a screeningcriterion. Rather, we only employed these two criteria

(Cauchy

Born and negative HoF criteria) such that a largeenough sampling can be carried out to enable us to appraiseany major trends. This two-level screening resulted in adatabase of 665 crystals for our focused study. In Section 5,we evaluate the stability of some outliers, at the ground statebased on the relative lattice stability of these MAX phasesagainst their neighboring and hence competing phases.

The crystal structures for the 792 hypothetical MAXphases used in the present study are rst optimized byunconstrained relaxation using VASP to calculate the elastictensor. We used projector augmented wave (PAW) [23, 24]method and GGA PBE [25, 26] for exchange correlationfunctional with an energy cutoff of 500eV as recommended

for a high precision calculation in VASP [27]. We usedstringent convergence criteria of 1.0 107 eV for electron-ic relaxation and 1.0 105 eV 1 for ionic forceconvergence. We also used a large G-centered, k-pointsmesh (15 15 3) along with MethfesselPaxton schemefor smearing since the MAX phases are metallic and asufcient number of k-points should be used to ensure highaccuracy.

Once the structures were fully relaxed with minimuminternal stresses, we used the strainstress analysis approachfor elastic properties calculation [28]. A small strain (1%for present case) for each independent strain element isapplied to the crystal and the structure is relaxed again while

keeping its volume and shape xed. The stress tensor sijunder a set of applied strain ejand the elastic stiffnessconstants (Cij) are obtained by solving the followingequation:

sijX

ij

Cijej: 1

From the calculated Cij, we obtain the polycrystallineelastic bulk properties using the well-tested VoightReussHill (VRH) approximation for polycrystalline aggre-gates [2931]. The Voight approximation assumes uniformstrain distribution in the structure, which results in an upper

limits of the polycrystalline bulk moduli. On the other hand,the Reuss approximation assumes a uniform stress distribu-tion resulting in the lower limits. The average of these twolimits gives the so-called Hill approximation, which is morerealistic and can be compared with measured data. The VRHapproximation has been validated in many metallic andinsulating crystals [14, 3237].

The electronic structure and bonding of these phases arecalculated using the Orthogonalized Linear Combination ofAtomic Orbitals (OLCAO) package [38, 39] with the VASPrelaxed structures. The OLCAO method is also a DFT basedmethod within local density approximation (LDA) devel-oped by us and is extremely efcient for large or complex

materials due to its economic use of the basis expansion. The

use of atomic orbitals in the basis expansion makes theinterpretation of results easier and more natural. In theOLCAO method, three types of basis expansions are usedfor different types of properties [38, 39]. The full basis (FB)

is used for the self-consistent potential and electronicstructure calculation; the minimal basis (MB) is used foreffective charge (Q) and bond order (BO)rabcalculationsbased on Mullikens scheme [40], which requires morelocalized orbitals. They are dened by the following twoequations:

Q

aX

i

X

n:occ

X

j;b

CniaC

njbSia;kb; 2

rabX

n:occ

X

i;j

CniaC

njbSia;jb; 3

where Cn

jbare the eigenvector coef

cients of the nth band,jthorbital and bth atom, and Sia,jb are the overlap integralsbetween theith orbital of the a atom andjth orbital of the bthatom. The extended basis (EB) is used to study spectroscopicproperties, which require higher accuracy for states highabove the Fermi level. The combination of VASP andOLCAO packages to study the structure and properties ofcomplex crystals and has been successfully demonstrated inmany recent publications [37, 4147].

In the present study, the investigation of the electronicstructure focuses on the BO using a MB. It is well known thatthe Mulliken scheme is basis-dependent and should usemore localized orbitals, which cannot be precisely dened

so there may be some uncertainty when applied to the MAXphases with different M elements. There are other methodsin calculating BO or similar quantities such as Baderanalysis [48]. However, to obtain quantitative results for alarge number of crystals, and for structures with lowsymmetry such as in MAX phases, we opted to employ theMulliken scheme due to its simplicity and easy to use. Thesummation of all pairs of BO values in the crystal givesthe total bond order (TBO). When divided by the crystalvolume, it gives the TBOD. The latter can be resolved intopartial bond order density (PBOD) components. For theMAX phases, we calculated the PBODs for the MX, MA,MM, and AA pairs, (see Section 3.3).

3 Results and discussion The results of the elasticproperties and the electronic structure for the 665 MAXphases are presented in the following three subsections.Section 3.1 is for elastic properties and correlations amongthem. Section 3.2 is on the electronic structures and theirspecic trends and features. Section 3.3 concentrates on thecorrelations between elastic properties and electronicstructure, the central theme of this paper. The MAX phases elastic properties have been calculated by many groups [34,4957] including us [58]. These calculations use differentmethods and targeted different MAX phases. They generallyagree well with each other for most of the MAX phases.

In particular, the comprehensive evaluation by Covers

1482 S. Aryal et al.: A genomic approach to stability and properties of the MAX phases

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et al. [59, 60], which evaluated the elastic properties of awide range of 211 MAX phases and Ti-based MAX phaseswith variousn. Their results are quite close to what we haveobtained for these segments of our database (see the

complete data set in the SM). We do emphasize that due tothe nature of the large database reported in this study, ourgoal is not to elaborate on specic comparisons between theelastic properties obtained by our calculations versus othersas the fundamental DFT approach employed is quite similar.Rather, our focus is to leverage the large database obtainedfrom our calculations to probe possible correlations amongstand between elastic and electronic structure properties. TheMAX phases electronic structures have also been studiedby many authors and they do not always agree with eachother due to mainly the different computational methodsemployed. Here, our results are exclusively calculated usingthe OLCAO method [32]. As for the correlations between the

elastic properties and electronic structures, there are far lesssuch studies and mostly using the electronic structure resultsto explain the experimentally measured elastic properties in asomewhat hand-waving style. The availability of a large dataset in the present study offers an opportunity to amelioratethe situation. For the sake of brevity, the MAX carbides andnitrides will henceforth referred to as MAC and MAN,respectively.

3.1 Elastic properties Detailed understanding of theelastic and mechanical properties is crucial for materialapplication especially for layered structures such as MAXphases where anisotropy in one of the most important

properties. It becomes even more critical when they are usedas composites since elastic anisotropy of one may offset theother under different stress components. Using the approachoutlined in the previous section, the elastic constants and themechanical parameters of the 665 MAX crystals that passedthe two level screening were calculated. In Fig. 2, we displayall the data forC11, C33, C44, and C66for both the carbides(solid symbols) and nitrides (open symbols). Data forother components of Cij can be found in SupplementaryMaterials. On the left column of Fig. 2, C11 are plottedagainstC33; on the right column; C44againstC66. Differentshapes of the symbols represent different M elements anddifferent colors designate different A elements. In this

display, each symbol along with associated color representstwoCijvalues for a particular MAX phase. Both axes in thegure have the same range and the diagonal lines in the plotsfor the 211, 312, 413, and 514 groups guide us to identify thedegree of anisotropy for each MAX phase. The farther thesymbol is away from the diagonal line, the more anisotropicit is.

Careful analysis of the data of the left column of Fig. 2shows some interesting trends:

(1) The elastic constants of the MAX phases vary widelydepending on M, A, and X elements.

(2) Despite the wide variations in bothC11andC33, theC11

values appear to be linearly related to C33. MAX

compounds have layered structures, the elastic responsealong the c-axis is expected to be different from thosealong the a- and b-axes. The matching increase in C11andC33is consistent and, to some extent, broadened the

critical

nding of Cover et al. [60] on the trends in C11orC33for a given M and A in the 211 phases. Ourndingsuggests that not only that both carbides and nitridesexhibit little differentiating effect as far as each of theC11andC33value are concerned (all types of carbide ornitride points with a given M and A are reasonablypaired up to some extent), but also that these two valuesincrease proportionately. Similar trends can be observedin the highernMAX phases.

(3) Some MAX phases are isotropic (C11ffiC33), some areanisotropic with C11> C33, in others C33> C11.Remarkably, and despite of the increasing c/a ratios,MAX phases with highernvalues, the degree ofC33/C11

ratio remains largely close to 1, which is indicative of thestrong correlation exhibited by these two elasticconstants.

(4) Despite of the generally linear correlation betweenC11and C33, most of the MAX phases show at least someelastic anisotropy and we can still distinguish thedifference in MAC and MAN. In general, the nitridestend to be slightly concentrated in the region where C33is higher thanC11, whereas carbides exhibit the oppositetrend, i.e., havingC33lower thanC11. This difference inanisotropic feature could be due to different bondingpatterns of the N or C layers with M layers in the axialand planar directions and will be elaborated later.

(5) For n 1 or the 211 phases, the data are distributedalmost equally above and below the diagonal line. Formost of the nitridesC33> C11; the opposite is true of thecarbides. In general, both C11and C33increase with n.However, the increases in C11 are larger than those inC33 resulting in an increased number of anisotropiccarbides compared to nitrides with the notable exceptionof the scandium containing MAX phases. This anomalyin the Sc-based carbides is quite perplexing since thenitrides show no such marked difference.

(6) As n increases, the distribution of the data points shiftsmore toward below the diagonal line with C11 beinglarger than C33. The MAX phases that exhibit higher

anisotropy depend on the specic M element in bothcarbides and nitrides, especially for MCr or Mo.(7) The A element in the MAX phases which connects

blocks of layered Mn1Xnshow discernable effects onC11versusC33resulting in clustered regions of the sameA elements (same colored symbols in Fig. 2). TheseA-based clusters become less obvious as n increasesindicating a gradual loss of inuence on elasticproperties when n increases as the structure progres-sively moves toward the binary Mn1Xncompounds.

We now shift our discussion to the right column ofFig. 2, which plots all shear elastic constants C44versusC66.

The difculty in describing the degree of shear anisotropy in



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a hexagonal crystal structure is well known [61]. Thus, forthis reason, we have employed the simplest representation ofthe shear anisotropy, namely the C66 to C44 ratio, whichcorresponds to a simple physical strain. In addition, the threealternative anisotropy factors involving Cij, which are alsoknown to represent the degree of anisotropy [61] can be

easily calculated using the data presented in SupplementaryTables. The distributions of the C66versus C44data are farmore scattered in comparison to those ofC11versusC33onthe left and the anisotropic behavior between C44andC66isfar more pronounced.

Similar to theC11versusC33, theC44versusC66for thecarbides and nitrides tends to be concentrated in differentregions. Most of the carbides have largerC66thanC44andare located above the diagonal line. In contrast, the nitrideshave largerC44than C66and are more concentrated belowthe diagonal line. With increasing n, the demarcation thathighlights this difference in shear anisotropy in carbidesversus nitrides becomes much more evident. With the 514

MAX phases, for example, there is barely an overlap

between those of carbides versus those of nitrides as shownin Fig. 2. So to some extent, this gives an added complexityto the notion that X exerts little inuence on some of theelastic constants of MAX phases as suggested by Cover [60].While this argument may be true forC11and C33, the samecase could not be made forC44 norC66. Furthermore, a wider

variation in terms of the degree of anisotropy in both casesbecome much more pronounced with increasing n, indicat-ing clearly that A still plays a major factor despite its lowercontent in say the 514 MAX phases. In contrast, this is notcase for theC11andC33values, where the variation is clearlyless. Thus, A exerts a critical inuence on the shearresistance in all MAX phases. This phenomenon subse-quently is also manifested in a similarly strong inuenceofAon shear modulus G.

As noted above, from the calculated Cij, the polycrys-talline bulk (K),shear(G), and Youngs (E)moduliaswellasPoissons ratio (h) can be obtained. They are presented intabular form in the SM together with the Cijvalues. These

bulk elastic properties clearly spread over a wide range

Figure 2 CalculatedC11versusC33(left) andC44versus C66(right) of MAX phases. Solidcircles for carbides and open circles fornitrides. Diagonal lines help indicate noelastic anisotropy for MAX phases.



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indicating the possibility of a broad range of potentialapplications in different areas. Bulk and shear moduli, whichrepresent resistance to change in volume and shape,respectively, are the most important parameters, whereas

the Poisson

s ratio or the Pugh moduli ratio G/Kgives abalanced overall assessment of the mechanical properties. InFig. 3a, we plot the Kversus G values for all screened 655MAX phases. In general, the nitrides have smallerGvaluesthan the carbides. LargerKand smallerGvalues result ineasier bond angle changes without changing bond length(BL) which, according to Pughs criteria based on ductilityanalyses of various polycrystalline metals [19], G/K is a

good indication of a possible increase in its relative ductilityin pure metals. The extension of such an analysis tointermetallic compounds like MAX phases, however, ismore far-fetched. Actually, our results show that there is

little correlation between the G/Kratio to say the hardnessvalues of the MAX phases.In Fig. 3b, we plot the GversusKresults for all screened

665 MAX phases and compare them to those for metalliccompounds and select binary MX compounds [62]. We notethat the MAX phases cover a wide region of bulk and shearmoduli, overlapping with those of the common metals.Clearly, if there was any correlation between the G/Kratio

Figure 3 (a) Calculated bulk modulus Kversus shear modulus G of MAX phases. Solidand open symbols represent carbides andnitrides, respectively. (b) Plot of bulk modulusversus shear modulus for all screened MAXphases. Solid circles and open circles are usedfor carbides and nitrides, respectively. Differ-ent color is used for differentn in Mn1AXn.The gure also shows the location of other

metals and binary MC and MN compounds.



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and mechanical properties then the MAX phases shouldexhibit similar ductility as the values for metals/alloys andMAX phases overlap. This is clearly not observed for thesimple reason that ductility requires at leastve independent

slip systems; the MAX phases that have been experimentallyfound so far [1] possess only two such systems. Furthermore,the Vickers hardness values of the MAX phases span arelatively narrow range of 28GPa,afactthatisnotreectedin Fig. 3b. Said otherwise, our results clearly show that thePugh ratio cannot be used as any indicator of ductility, atleast for the MAX phases.

With these caveats it is still instructive to understand thechanges in theG/Kratio as a function of MAX structure andchemistries. As shown in Fig. 3b, theG/Kratio ranges from0.12 to 0.8, with the MAX carbides appear to concentratemostly above G/K 0.42. The MAX nitrides, on the otherhand, especially those with a higher n values, possess

relatively lower G/Kvalues. In general, the MAX phaseswith highernvalues have higher bulk moduli (K) than thosewith lowernvalues. Some of the MAX phases with highernvalues, mostly carbides, have both Kand G values close tothose of the binary carbides. The highernMAX nitrides, onthe other hand, display a much wider range ofGresulting insome phases with low G/Kvalue. This implies that, unlikethe MAX carbides, the G values of the MAX nitridesare more sensitive to the nature of the A group elements. Weview this widening of theG/Krange as an additional meansto tune the elastic properties of the MAX nitrides, especiallythose of higher n values. It is worth noting here howeveralmost none of the high n MAX nitrides have been

synthesized to date. It is important to note that in the absenceof A, the shear moduli of binary MN compounds aregenerally large and of a limited range, resulting in a narrowrange ofG/K. This result highlights one more time that themechanical behavior of the MAX phases is quite distinctfrom their parentmono-carbides/nitrides.

The calculated Poissons ratio,h, which is closely relatedto theG/Kratio for the original 792 hypothetical MAX phasesare presented in Fig. 4 in the form of an innovative mapthat resembles a portion of the Periodic Table. Here, theM elements are plotted on they-axes and the A elements alongthe x-axes. The color of each small square represents the hvalue of the particular MAX phase along with other

information such as whether the phase has been synthesizedor not. The phases that have been eliminated by the CauchyBorn and HoF criteria are marked with and, respectively.The experimentally conrmed phases are marked by a whitestar. It is gratifying to see that none of the experimentallyconrmed phases are among the ones judged to be unstableand there are many boxes of different colors without the whitestar suggesting the possible existence of myriad of unexploredMAX phases. While the elastic properties of MAX phases canvary over a wide range as indicated by the variations in colorfor the different squares in Fig. 4, we are able to delineatethe boundaries of MAXs materials properties within whichoptimized functionalities can be further explored. Additional

information on the elastic properties of the MAX phases is

shown in SM. Supplementary Figure S1 displays the linearelastic anisotropy ratioC11/C33in all MAX phases in the formof innovative maps similar to Fig. 4. Supplementary FiguresS2 and S3 showsimilar maps but for the Pugh ratio G/Kand K,

respectively.3.2 Electronic structure and bonding Given the

large diversity in the elastic properties of MAX phasesdepending their chemistry andn, the natural question arises:how does their electronic properties vary and can acorrelation be found between the elastic and electronicproperties. We therefore now shift our focus to the electronicstructure of these potentially stable MAX phases. Typically,the electronic structure consist of the band structure, thedensity of states (DOS), atom-resolved partial DOS (PDOS),effective charges on each atom and the BO (also called bondoverlap population between all pairs of atoms in all MAX

phases (see method Section 3.2). Since we have calculatedall potentially stable MAX phases comprising of a large setof M, A, and X elements, it is not practical to include theband structures and DOS in our presentation even thoughthey are all available. On the other hand, the DOS at theFermi-levelN(EF) is a quantitative measure representing therelative metallic character of the crystals and are included inthe database and our discussion. We also report the effectivecharge (Q) for each element and the BO values betweeneach pair of atoms. Together with N(EF), these threeparameters exemplify the quintessence of electronicstructures. They are discussed below.

The BO measures the relative strength of a bond between

two atoms. The TBO of a crystal is the sum of all the BOvalues in that crystal. When the TBO is divided by thevolume of the crystal, the TBOD is obtained. The MAXphases have a layered structure and it is of great interest toascertain how the bonding pattern evolves between thelayers and their variations withn. There are four types of BOin the MAX phases or the PBOD, MX, MA, MM, andAA with the MX bonding dominating. The TBOD valuesfor the AA bonds are quite small and depend on theparticular MAX phase. Although detailed quantitativeinformation on the BO values for each MAX phases areall available, it is not practical or necessary to present them indetail. Instead, we show in Fig. 5 the percentages of each

bonding type to the TBO for 211, 312, 413, and 514 carbides(ad, respectively) and the nitrides (eh, respectively) inthe form of pie charts. There is considerable difference inthe contributions to the TBO in the carbides and nitrides. Thecarbides have noticeably larger contributions from the MXpairs than the nitrides. On the other hand, the nitrides havesignicantly larger contributions from the MA pairs.Contributions from the bonding in the M layer (MMbonding) appear to be slightly higher in the nitrides. TheAA bonding contributes the least and appears to be slightlylarger in the carbides.

As n increases the MX TBO percentage increases inboth the carbides and nitrides since the fraction of the A

layers systematically decreases asnincreases. It should also



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be noted that the TBO and TBOD are two different quantities

with different physical units. The latter accounts for theeffect of the crystal volume, which differs signicantly in theMAX phases due to their diverse chemical compositions.We found that a better correlation can be established whenthe TBO is normalized to the unit volume. This implies that itis the concentrationof the BO within the crystal structuresthat actually exerts the strongest inuence to the elasticproperties as far as electronic structure factors as concerned.This also reasserts the importance that all crystal structures inthe present study are fully optimized with high accuracy sothat the unit volume can be accurately obtained.

The effective chargesQ on each atom in the 665 MAXphases according to Eq. (3) using the MB in the OLCAO

method [39] were calculated. The deviation ofQ from its

charge in the neutral atom represents the charge transferDQ

for this atom. Positive values represent charge loss, whereasnegative values indicate charge gain. Note that these valuesshould not be confused with the popular notion of chemicalvalence. Figure 6 displays DQ in all 665 screened MAXphases. The top panel is for carbides; the lower for nitrides.Within each stacked panel of four segments (211, 312, 413,and 513) from left to right, the vertical axis represents thecharge transfer values DQM, DQA and DQX, respectively,and the horizontal axis stand for the columns of M series inthe sequence of Al-Ga-In-Tl-Si-Ge-Sn-Pb-P-As-S for the Aelements within each column. Admittedly, this is a busygure that warrants very careful analysis, but this is also thebest way to present all the data points in a concise gure

showing the overall trends in charge transfer.

VIA

VA

IVA

IIIA

VIB

VB

IVB

PbTl Sn sAnI GeGa SSi PAl

Ta

Hf

Mo

Nb

Cr

V

Ti

Sc

Zr

MAX-514-Nitrides

0.19

0.21

0.24

0.26

0.29

0.31

0.34

0.36

0.39

0.41

0.44

Poisson'sRatio

IIIB

VIA

VA

IVA

IIIA

VIB

VB

IVB


Ta

Hf

Mo

Nb

Cr

V

Ti

Sc

Zr

MAX-211-Carbides

0.19

0.21

0.24

0.26

0.29

0.31

0.34

0.36

0.39

0.42

0.44

Poi

sson'sRatio

IIIB

VIA

VA

IVA

IIIA

VIB

VB

IVB


Ta

Hf

Mo

Nb

Cr

V

Ti

Sc

Zr

MAX-211-Nitrides

0.19

0.21

0.24

0.26

0.29

0.31

0.34

0.36

0.39

0.42

0.44

Po

isson'sRatio

IIIB

VIA

VA

IVA

IIIA

VIB

VB

IVB


Ta

Hf

Mo

Nb

Cr

V

Ti

Sc

Zr

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Figure 4 Poissons ratio maps for all theMAX phases according to M (y-axis)and A (x-axis) elements. Left column for carbides andright column for nitrides. Color in the boxrepresents calculated G/Kvalues as indicatedin the color bar. Stars in the box indicate thisphase has been synthesized. The sign standsfor elastic instability and the sign indicatesthe phase is screened out for thermodynamicinstability due to positive heat of formation.



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Some researchers have tried to nd correlations betweenthe valence electron concentrations (VEC) and the Cijs.However, VEC is not the same as Nval, since it does nottake into account the unit cell volume. For example, Wangand Zhou [57] reported a linear positive correlation betweenC44 andVECintheM2AlC phases. Our expanded evaluationfor a similar correlation on all of the MAX carbide phasesdid not verify the existence of such a positive correlation.There are apparent scatters in the data precluding a linearrelationship especially for low n carbide phases (see

Supplementary Fig. S6 in SM).

Similar to the case of N(EF), we found that the MACphases with highern values seems to exhibit a better linearcorrelation, presumably due to the reduced effect of A.Unlike MAC, the MAN phases tend to deviate from thispositive correlation. In fact, we see no evidence that such apositive correlation betweenC44andNvalnorNval(

3) thatcan be rmly established. This nding is quite consistentwith the apparent lack of a trend between G and Kin theMAN phases. That is, there is a signicant portion of MANphases that possess much lower values ofG, despite of their

highKvalues. The only exception to this rule appears to be

Figure 6 Effective charge transferDQ for M, A, X elements in all MAX phases. Top: Carbides. Bottom: Nitrides. Positive valuesrepresent electron loss, whereas negative values denote electron gain (see text for details).



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the Sc-based phases where the linear correlation is quiteapparent irrespective of the types of n, A or X used. Wepostulate that this could be due to the absence of d-electronsin Sc, limiting the contribution of pd hybridization in theSc-containing MAX phases.

3.3 Correlations between mechanical propertiesand electronic structure In the two preceding subsec-tions, the elastic properties and the electronic structuresof 665 MAX phases were discussed separately. In thissubsection, we try to establish correlations between the two,

a nontrivial exercise rendered more difcult because: (i) ofthe layered nature of the MAX phases and (ii) vast amount ofdata available, and (iii) the diverse properties explored. Here,we try to meet this challenge by making use of the vastamount of high quality data and various correlation plots. Aspointed out above, TBO or TBOD is a good measure ofthe strengths of the crystal. In principle, the latter shouldcorrelate with elastic properties. As already pointed outabove, the TBO and TBOD can be further resolved intopartial components of different pairs of bonds (MM, MA,MX, and AA). The partial BO values are especiallyilluminating for crystal cohesion in relation to atomic scalegeometry of different atomic species in the crystal. In Fig. 5,

we showed that the MX BO dominate the bonding (48

81%) followed by the MA BO (1537%) and MM BO(5.911.4%) depending on n and X. Although the AAbonds have small contributions and the AX bondsapparently do not form, there are some MAX compoundswhere the BO values of the AA bonds are actually higherthan the corresponding MA bonds especially when A is Alor In.

In the MAX structure (Fig. 1), there are interlayer bondsand intra-layer bonds; the latter are neither aligned in theaxial direction or in the basal plane. It follows that a portionof the interlayer MX bonds are inclined in a direction

favoring stiffness along the basal plane. On the other hand,interlayer bonds between M and A are more inclined towardthe c-axis, enhancing the stiffness along the [0001] direction.The degree of inclination of the weak or strong bonds willhave an inuence upon the stiffness values along thec-axesorthe ab plane. Thus atomic arrangements together with thechemical diversity lead to complex correlations betweenbonding and elastic properties discussed below.

Based on the extensive data obtained, we plotted 160correlations plots of C11, C33, C44, C66, K, G, E, andPoissons ratio against the TBOD and its MX, MA, MM,and AA components forn 14. Such extensive correla-tion plots (not shown) enabled us to identify any existing

correlations (or lack thereof) between the elastic properties

Figure 7 Plot of DOS at Fermi levelN(EF) against total number of valence electrons per unit volume in the crystal. Note the outlier natureof MSc.



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with the electronic structures embodied in the TBOD or itspartial components. Figure 8 plotsK(left panel) andG(rightpanel) versus TBOD in the 211, 312, 413, 514 MAC andMAN compounds, respectively. It is evident thatKcorrelates

with TBOD better thanG. This is not too surprising since Kis a measure of the resistances to change in BL, which isdirectly related to BO values. We also note that the TBODsof the nitrides are larger than those of the carbides. Closerobservation reveals that for n > 2, there are roughly fourgroups of correlated bands, two for the carbides and two forthe nitrides based on TBOD. The MAX phases based on Sc,Zr, Hf, and Nb have smaller TBODs than those of Ti, V, Cr,Mo, Ta, and this trend becomes more evident as nincreases.This is mainly due to the nature of denition of the MBadopted for these two groups. The atoms in the second groupdo not contain the empty valence shell p-orbitals in the MBin order for the MB to be more localized. Nevertheless, the

generally positive trend that links the TBOD with Kremainsquite evident. Some of the trends on the elastic propertycorrelations in a more limited MAX 211 phases have alsoobserved by Cover et al. [60].

In Fig. 9, weplotKagainst the MA (left panel) and MX(right panel) portions of the TBODs. It is apparent that thecorrelation with MA BOD is far better than the correlationswith TBOD shown in the left panel of Fig. 8. Also Kis better

correlated with the M

A than the M

X TBOD, even thoughthe contribution fromMA bonds to the TBOD is smaller thanthat MX. The approximate slope of the linear correlationincreases with increasingn. This nding again points to theimportant role played by element A in all MAX phases.

There are also strong correlations between the elasticconstants and the TBODs, which are shown ingures in SM.Supplementary Figure S4 shows such correlation for C11(left column) and C33 (right column) with TBOD. Thedifference in the degree of correlations reects the differencein the bonding related to planer and axial components of theelastic tensor elements. In Supplementary Fig. S5, we showsimilar plots for C44 (left column) and C66 (right column)

showing a similarly positive correlation with TBOD with alln; the latter however have more scatters with some MAXphases actually exhibit a negative correlation, e.g., Cr-basedMAN phases (see highlighted dotted lines).

Figure 8 Correlation plots of: (left) bulk and(right) shear moduli versus TBOD for all

MAX phases.



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4 Data-mining approach The complete data set on665 MAX phase properties enables us to test the efcacy ofdata-mining statistical learning approach being applied tomaterials property predictions [6365] that has been quitepopular in recent years. The success of data mining methodis predicated upon the use of statistical tools to recognizepatterns among multivariate data sets. In particular, we

employed on one of the simplest data-mining approach, i.e.,the multiple linear regression y w0 w1x1 w2x2 wkxk whereby the weight (wi) of each linear factor (xi) isestimated from a designated training set. We applied thisapproach as implemented in a well-known data-mining codeWEKA [66] to obtain a linear pattern that links the complexmechanical properties as a function of a combination ofelectronic structure factors. From the previous sections, it isquite clear that while TBOD yields correlations with avariety of elastic properties, this correlation is often timesalso marked by data scatters that may indicate no singleelectronic factor alone can be used to predict these elasticproperties. The statistical methodologies employed in this

data mining approach allow us to construct an empirical, but

quantitative, formula that links properties of interest. Itfurther enables us to test the strengths of correlation betweenproperties. Of particular interest here is the link between bulkmodulus and Poissons ratio and electronic structures. Weshould note that the DFT calculations on these mechanicalparameters can be quite time-consuming whereas those forthe electronic structure, on the other hand, can be acquired

quite readily depending on the method of calculation. Thus,if a correlation can be established, then one may be able toutilize this new approach as an expedited route to obtaininformation related to mechanical properties solely based onelectronic structure properties. For this purpose, we used theentire database wherein, 50% of the data set was used as atraining set to predict Kand Poissons ratio for the other50%. We employed only the electronic structure factors asthe variables to predictKand Poissons ratio. Table 1 liststhe coefcients of the linear superposition of these factorsthat have been optimized to predict the two properties.

Figure 10 (left panel) shows the comparison betweenKobtained from ab initio calculations versus those obtained

from the formulas derived from the data-mining algorithm

Figure 9 Correlation plots of bulk modulus(K) versus (left) MA part of the TBOD and(right) MX part of the TBOD for all MAXphases.



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for the other 50% of the 211, 312, 413, and 514 MAXphases, respectively. An excellent correlation with over 90%correlations on each type of MAX phases for theKvalues isobtained. Figure 10 also shows the pie charts comparing the

relative contributions of each of the electronic structurefactors used to predictK. We further identied the four mostimportant factors, viz. TBOD, MA BOD, MX BOD, andthe charge transfer for the X elements consistent with those

Table 1 List of coefcients for K and G/Kestimates from data-mining algorithm using a linear superposition: YC(0)C(1)*TB_DensC(2)*MM_TBOC(3)*MA_TBOC(4)*MX_TBOC(5)*AA_TBOC(6)*AX_TBOC(7)*X_QC(8)*N_EF.

Y C(0) C(1) C(2) C(3) C(4) C(5) C(6) C(7) C(8) correlation

coefficientK(211) 206.84 5038.87 6.1583 3.2376 10.199 28.091 100.743 107.34 4.18E-01 0.915K(312) 187.74 3970.89 2.9706 11.89 7.2138 22.733 7.2138 82.193 1.31E-01 0.9323K(413) 230.35 4110.44 4.6844 11.738 5.487 21.675 262.329 126.4 0.00E 00 0.9516K(514) 237.71 4366.93 2.6619 12.386 4.663 20.655 186.82 116.11 4.27E-02 0.9565Poisson ratio (211) 0.389 3.0584 0.0091 0.0144 0 0.0332 0.28 0.174 1.40E-03 0.8009Poisson ratio (312) 0.443 2.7664 0.0149 0 0 0.0253 0.2349 0.191 1.10E-03 0.8504Poisson ratio (413) 0.392 1.9542 0.0203 0 0.023 0.011 0 0.156 1.10E-03 0.8919Poisson ratio (514) 0.479 1.7623 0.0091 0 0.0019 0.0133 0.3082 0.219 6.33E-04 0.7955

Figure 10 Left panel: Comparative plots ofbulk modulus fromab initioDFT calculationsversus that from data-mining algorithmusing half of randomly chosen MAX phases:(a) 211, (b) 312, (c) 413, and (d) 514. In eachpanel on the left, there is a pie-chart insetshowing the average relative contributionsof each electronic structure factors to thepredicted bulk modulus based on the data-mining algorithm. Right panel: Similar com-parative plots for the Poissons ratio: (e) 211,

(f) 312, (g) 413, and (h) 514 MAX phases.



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already discussed in Section 3.3. Based on the coefcientslisted in Table 1 and the pie charts (Fig. 10), TBOD clearlystands out as the most important factor in determiningKforall MAX phases. Figure 10 (right panel) shows the similar

prediction for Poisson

s ratio using the similar procedure.Although it is less impressive than that forK, the predictionfor the Poissons ratio obtained from data mining is stillreasonably good, with correlation coefcients around 80%or higher. In addition, as shown in Table 1, one can also seethat the Poissons ratio is also strongly affected by the TBOalthough it is a negative correlation unlike the positive oneexhibited by K. The linear correlation is less denitiveprobably due to the fact that Poissons ratio as well as theG/Kratio are more inuenced by the nature of the A element,an effect that apparently is not fully represented by the alinear combination of electronic structure parameters used inthis analysis. It is possible that there should be a nonlinearity

in the regression model better suited for the Poisson

s ratio.Nevertheless, a reasonably good estimate on these twoproperties can indeed be established solely from a linearcombination of electronic structure factors. Furthermore, theTBDO emerges as a signicant variable that controls theseelastic properties in agreement with the nding from theprevious sections. This data mining approach also demon-strates that a simple correlation can be quantied to linkelastic parameters such as Poissons ratio or the Pugh ratio toa series of electronic structure indicators. Further, the use ofonly 50% of the data as a training set gives credence to theparticular machine learning software and the philosophylying behind it. To some extent the success can be explained

by the nature of our database and the strategy we employedto establish the database. In general, the data miningmethod that is applied to materials property predictionsutilizes a heterogeneous set of databases extracted fromvarious different sources experimentally or computationally.In our case, we employed a well-tested DFT-based approachto generate the required large database with a systematiccompositional variation in M, A, X, and n.

5 Special analysis of outliers The completeness ofthe MAX database encompassing a wide combination of M,A, and X affords us the unique capability to identify certainphases that qualify as outliers. For some of these outliers, we

further evaluated their thermodynamic stability at the groundstate.One of our objectives in this study is to identify

compounds with relatively low G/Kratios. In addition, wealso focus on the Sc-based MAX phases due to their uniqueelastic and electronic properties as already detailed inthe previous sections. For these two objectives, we haveextended our evaluation on the relative stability of the MAXphases to include their neighboring binary and/or ternaryphases. The thermodynamic information on the competingphases were obtained from the ground-state thermodynamiccalculations and the lattice stability analysis follows theimplementation of the Calphad method [67]. Since no

temperature and its effect on the Gibbs free energy are

included in this thermodynamic consideration, we haveallowed a small positive enthalpy variation (SH 0.05eVatom1), to reect the contribution from the vibrationenergy, which is not included in the calculations. Using this

simpli

ed approach, we were able to identify examples ofpossibly stable MAX phase outliers showing relatively lowG/Kas well as a number of Sc-based MAX phases. They arelisted in Table 2. None of these phases has been synthesizedto the best of our knowledge. Certainly, there are othermethodologies by which outliers can be identied which arebeyond the scope of this paper. We do want to stress thatthe ability to identify outliers effectively is based on thecapability to generate a sufciently large database fromwhich an unusual deviation from the norm can be detected.Our approach here is unique in that regard since we used onlyone methodology to furnish all of the elastic properties of theMAX phases. While the accuracy of materials properties

obtained using the DFT calculations is crucial as demon-strated in our previous study [58], the real benetofusingthissingular methodology is to avoid complications arising fromvariations in data due to different methodologies employed.Such variations can potentially be large enough to mask theidentication of outliers. We should also point out that by nomeans these identied outliers are reective of an exhaustivesearch of the outliers. It only serves to demonstrate that withthe availability of a large database set for MAX phases withpossibly unique properties can be more readily identied andexamined further.

6 Summary and conclusions In this paper, we

conducted an extensive investigation on the Mn1AXnphases withnvarying from 1 to 4. The focus is on the elasticproperties and electronic structure and the correlationsbetween the two. Based on a carefully designed screeningstrategy, the data for 669 MAX compounds are obtained andmade available to all readers. In spite the anisotropicstructure, and related complications, a wide range of elasticproperties and electronic structures were calculated andtrends and correlations between them established. Mostimportantly, the elusive correlations between these twoclasses of properties are captured. From these correlations,possible new MAX phases with some unique properties thathave never been synthesized have been identied and their

thermodynamic stability assessed. This was only possiblebecause of the large and credible database we developed.Based on this database, we tested the efcacy of the genomicapproach of data mining and concluded that this approachis promising, especially if the integrity and quality of asufciently large data base is carefully scrutinized. Thisapproach enables us to use a faster screening route based oneasily calculated properties on much larger samples ofpotential candidates and predict other properties that areeither too onerous to evaluate or too expansive to measureexperimentally. The massive amount of data and thescattered nature of the properties calculated require aninnovative way in presenting these results without consum-

ing an unacceptably large amount of journal space. To that



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effect we present them in maps mimicking the PeriodicTable or in correlation plots with data representing individual

MAX compounds clearly identiable. Such visual presenta-tion of large data set is important in facilitating the rapiddissemination of main correlations and a deeper comprehen-sion of the properties of MAX phases.

In conclusion, we have established the following trendsand correlations that to date had not been well recognizedor sufciently elaborated in the literature due to the lack ofa comprehensive data set. We conclude that: (i) there aremarked differences between the elastic and electronicproperties of the carbides and nitrides; (ii) there are cleartrends in the properties and variations with increasingn.Thistrend is intimately related to the dependence on the singlelayer of element A; (iii) the Sc-based carbides stand out in

the nine metallic elements in its properties and correlations,which is related to the total valence electron density and theFermi level position which lies below the pseudo gap in theDOS; (iv) general trends in the charge transfer among M, A,and X elements, although quite complex, are established.All M atoms lose charge to the A and X elements; (v) theimportance and usefulness of using the concept of TBODand partial TBOD has been fully demonstrated; (vi) Withthe notable exception of the Sc-based carbides, a simplecorrelation exists betweenN(EF) and the density of valenceelectrons, rst pointed out by Barsoum [1] is afrmed;(vii) The unique role of the A layer in MAX compound isemphasized regardless ofn; (viii) We have also established

several major correlations between elastic properties andelectronic structures among themselves and between thetwo. The most important ones include: (i) correlations ofC11versus C33and C66versus C44and (ii) correlations or lackthereof between: K, G, E, h, G/K, C11, C33, C44, C66 andelectronic properties in the forms of TBOD and theircomponents for differentn values.

We have also tested and validated the data-mining,machine learning approach, and algorithms using our dataset as inputs. The most important parameter in the electronicstructure in predicting the elastic properties is the TBOD ofthe crystal. We further identied ten outliers with unusualproperties that are thermodynamically stable against their

competing phases.

The present work focuses on two properties, elasticproperties, and electronic structures. There are many other

properties such as vibrational, spectroscopic, transport,corrosion resistance, electric conductivity, superconductivi-ty, thermo-conductivity etc. just to mention a few, which areall intimately related to the electronic structures and some tothe elastic properties. It is possible to extend the genomicapproach used here to explore these properties. Some ofthese are currently in progress. Moreover, the currentdatabase covers only the Mn1AXn phases with n 14comprising 11 M elements and 9 A elements. It is possible toexpand this list to include the early transition metals suchas La, W and Sb, and Bi, Se and Te for the A elements, tofurther enlarge the data base. Very recently, with thediscovery of Mn2GaC [68], Mn can now be added to the list

of M elements. Another fruitful route is to consider solidsolutions between some of the more promising MAX phases,which can further optimize properties. Another direction is toinvestigate the materials beyond the MAX phases to thoseof its derivations such as MXenes [69, 70] or articialMAX-like layered structures by inserting multiple layers ofselective A elements into the MX layers of the MXenes.The time is also ripe to extend such studies to defects andmicrostructures in MAX phases using a similar approach.There are endless opportunities for future work that alloriginate from this fascinating class of ternary intermetalliccompounds, the MAX phases.

Finally, the strategy used in this work can be applied to a

wider range of crystal structure prototypes whereby theabundance in experimental and/or theoretical database isabsent and thus, correlations amongst their materialsproperties cannot be readily obtained. By evaluating theeffect of various constituents of the crystal structuresystematically, these trends can potentially be identiedand further rened. Furthermore, such a strategy, a genomicapproach, can also be used to assist the identication ofoutliers.

AcknowledgementsThis work was supported by NationalEnergy Technology Laboratory (NETL) of the U.S. Department ofEnergy (DOE) under Grant No. DE-FE0005865. This research

used the resources of the National Energy Research Scientic

Table 2 Examples of MAX phase outliers, either Sc-based or with a relatively low G/Kand potentially thermodynamically stable, i.e.,with a negative

PDHor with

PDH0.053eV.

MAX G/K chemical equilibria P

DH(eVatom1)

Mo2GeC 0.375 3Mo2C 2MoGe2C4Mo2GeC 0.05216Ti2AsC 0.430 TiCTiAsTi2AsC 0.02150Ta2GeC 0.671 Ta2CTaCTaGe2 2Ta2GeC 0.03134Ta3GeC2 0.700 Ta2C3TaCTaGe2 2Ta3GeC 0.00045Sc3TlN2 0.685 TlSc 2ScNSc3TlN2 0.20233Sc3InN2 0.701 8ScN 2Sc3InNScIn3 5Sc3InN2 0.01623Sc3SC2 0.610 8Sc4C3 2Sc15C19 31ScS 31Sc3SC2 0.00272Sc3SnN2 0.715 Sc5Sn3 2ScSn2 14ScN 7Sc3SnN2 0.05022Sc3AlN2 0.737 ScAl 2ScNSc3AlN2 0.01350Sc3SN2 0.739 ScS2ScNSc3SN2 0.00185



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Computing Center (NERSC) supported by the Ofce of BasicScience of DOE under Contract No. DE-AC03-76SF00098. M.W.B. was partially supported by DMR-1310245.

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