Supertoughening in B1 transition metal nitride alloys by increased

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Acta Materialia xxx (2011) xxx–xxx

Supertoughening in B1 transition metal nitride alloysby increased valence electron concentration

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

Thin Film Physics, Department of Physics, Chemistry and Biology, Linkoping University, SE-581 83 Linkoping, Sweden

Received 12 November 2010; received in revised form 2 December 2010; accepted 4 December 2010

Abstract

We use density functional theory calculations to explore the effects of alloying cubic TiN and VN with transition metals M = Nb, Ta,Mo or W at 50% concentrations. The ternary systems obtained are predicted to be supertough, as they are shown to be harder and sig-nificantly more ductile compared with reference binary systems. The primary electronic mechanism of this supertoughening effect isshown in a comprehensive electronic structure analysis of these compounds to be the increased valence electron concentration intrinsicto these ternary systems. Our investigations reveal the complex nature of chemical bonding in these compounds, which ultimatelyexplains the observed selective response to stress. The findings presented in this paper thus offer a design route for the synthesis of super-tough transition metal nitride alloys via valence electron concentration tuning.� 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Nitrides; Mechanical properties; Ductility; Ab initio calculations

1. Introduction

Transition metal nitride alloys are well known for theirexcellent properties, such as high hardness and wear resis-tance, high melting temperatures, and good chemical inert-ness [1–3]. Among various technological applications theyare employed as protective coatings in the cutting toolindustry, to extend tool life and improve machining perfor-mance. To develop thin film nitride alloys with suitablemechanical and physical properties the material chemicalcomposition can be tuned and the growth and processingparameters optimized to control stoichiometry, microstruc-ture, and texture. For protective coatings high hardness isobviously a very sought after characteristic [4–8], as it pre-vents the surface from being scratched and worn at extremeworking pressures. Typically, hardness enhancement isachieved by hindering dislocation mobility and, hence,reducing plastic deformation in the material [9,10]. Never-theless, hardness improvements are often accompanied by

1359-6454/$36.00 � 2010 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2010.12.013

⇑ Corresponding author. Tel.: +46 13282623; fax: +46 13137568.E-mail address: [email protected] (D.G. Sangiovanni).

Please cite this article in press as: Sangiovanni DG et al. Supertougelectron concentration. Acta Mater (2011), doi:10.1016/j.actamat.201

embrittlement, which beyond a certain load results in filmcracking. To achieve tool durability under various operat-ing conditions it is, therefore, necessary to design hardcoating materials with enhanced ductility.

Previous studies have shown that it is possible to attainrelatively ductile, yet hard, materials. For example, innon-isostructural superlattices [11] the alternating layersof ceramic B1 transition metal nitrides and more ductilebody-centered cubic (bcc) metals yield varying rates of plas-ticity while retaining hardness. Similarly, the nanocompos-ite structures developed by Voevodin and Zabinski [12]were designed to be hard at stresses below the elasticstrength limit, while at extreme loading their mechanicalbehavior switches to ductile, thus preventing brittle failure.In transition metal nitride alloys the common approach totailor the mechanical properties to applications require-ments is to combine different metallic species and/or to varytheir ratio. Although brittleness in these hard ceramics is anobvious issue, the few theoretical studies [13,14] which haveinvestigated this problem were in fact confined to the anal-ysis of calculated elastic constants values and their trends.Clearly, more rigorous electronic structure investigations

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hening in B1 transition metal nitride alloys by increased valence0.12.013

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2 D.G. Sangiovanni et al. / Acta Materialia xxx (2011) xxx–xxx

are required to understand the mechanisms leading to anappropriate/improved hardness to ductility ratio, or tough-ness, in materials.

Recently [15] we reported on a supertoughening processin ordered B1 Ti0.5M0.5N, for M = Mo and W. In thatstudy we demonstrated that this effect stems from a pro-nounced layered electronic arrangement on the metalstacking, not observed in the binary (TiN), which allowsa selective response to strain and shear deformations ofthe crystal. Such a charge distribution is induced in thecrystal by the substitution of Ti with Mo or W atoms, asthe exceeding valence electrons in these elements enhancethe occupation of d-t2g metallic states. These resultsimply the possibility to control ductility trends in materialsby tuning the electron population in the d-t2g metallicstates, and broaden the research perspectives in the questfor new supertough B1 transition metal nitride alloys.

In the present paper we investigate the mechanical prop-erties of cubic B1 Ti0.5M0.5N and V0.5M0.5N (M = V, Nb,Ta, Mo, and W) by means of ab initio density functionaltheory (DFT) calculations to identify candidate materialsfor potentially hard coatings with enhanced ductility. Pri-marily, this choice is based on our previous results forTiMoN and TiWN alloys [15] and the fact that the thinfilms containing combinations of these elements grown inthe rocksalt structure have been reported to have compara-ble hardness to TiN and TiAlN [16–20]. In the alloys pro-posed herein, the metallic elements are selected so as tospan the valence electron concentration (VEC) per unit for-mula from a minimum of 9 (in the reference material TiN)up to a maximum of 10.5 (in V0.5Mo0.5N), as beyond thisvalue further filling of the d-t2g metallic states might leadto instability of the cubic phase [21]. Our elastic constantsestimations show that with increasing VEC Ti0.5M0.5N andV0.5M0.5N are progressively less resistive to shear deforma-tions, while retaining stiffness and very low compressibility.The electronic structure and crystal orbital overlap popula-tion (COOP) [22] calculations reported herein indicate thatincreasing VEC is a key factor in ductility enhancement inthe ternaries studied. In addition, we analyze the stress–strain relationship in these alloys and find that an increasein VEC activates the f110gh1�10i slip system and promotesdislocation motion. Furthermore, our hardness estimationsobtained using theoretical methods [23,24] are in goodagreement with indentation measurements and suggest thatall these crystalline phases are excellent candidates for thesynthesis of supertough B1 transition metal nitride alloys.

2. Computational details

The DFT calculations reported herein were performedwith the Vienna ab initio simulation package (VASP) [25]in the generalized gradient approximation of Perdew–Wang (GGA-PW91) [26], and the electron–ion interactionswere described by the projector augmented wave potentials(PAW) [27]. In all calculations we used a large energy cut-off of 500 eV for the plane wave basis to achieve total


energy convergence within 10�5 eV. Structure relaxationswere carried out with 4 � 4 � 4 k-point grids, while thedensity of states (DOS), charge density distribution, andCOOP were computed with 8 � 8 � 8 k-point grids in theMonkhorst–Pack scheme [28]. The supercells employed inour investigations contained 64 atoms with a minimumnumber of intermetallic bonds (C#3 structure) [15], closelymatching the CuPt-type atomic ordering observed experi-mentally in TixW1�xN films [29] and consistently foundmost stable energetically in our calculations. We computedthe lattice constants a, bulk moduli B, elastic constants C11,C12 and C44, Young’s moduli E, shear moduli G, Poisson’sratios m, and theoretical hardness H for all alloys. The ideallattice constant and bulk modulus were evaluated by fittingthe total energy-volume curve to the Birch–Murnaghanequation of state [30]. More details about elastic constantsand related moduli calculations can be found in our previ-ous report [15].

We assessed the theoretical Knoop hardness Hk of allcompounds using the semi-empirical method proposed bySimunek for covalent and ionic crystals [23]. For the struc-tures employed in our calculations, in which nitrogenatoms (N) and metal atoms (M) are nearest neighbors,Simunek’s formula becomes:

Hk ¼ ðC=XÞX

M

bN;MsN;M expð�rf2Þ; ð1Þ

in which bN,M accounts for the number of N–M inter-atomic bonds in a unit cell of volume X, while C (=1450)and r (=2.8) are parameters fitted so that Eq. (1) repro-duces the Knoop hardness values (in GPa) for selectedcovalent and ionic crystals. The quantity sN,M, termedbond strength, is expressed as:

sN;M ¼ffiffiffiffiffiffiffiffiffiffiffieNeM

p=ðnNnMdN;MÞ; ð2Þ

where n (=6) is the coordination number, dN,M is the inter-atomic distance between N and M, and e is the ratio of thevalence electrons number to the atomic radius value takenfrom Pearson’s book [31]. Finally, the expression for f2 is:

f2 ¼eN � eM

eN þ eM

� �2

: ð3Þ

Furthermore, we used a recently reported methoddesigned to predict the Vickers hardness in transition metalcarbides and nitrides [24]. This parameter-free approachinvolves the estimation of, in addition to bond length, den-sity and ionicity, the intrinsic metallicity in the chemicalbonds of these compounds from ab initio calculations.The technique quantitatively reproduces the negative effectof d valence electrons on the hardness of primarily covalen-cy dominant binary transition metal carbides and nitrides,and is based on using Mulliken atomic and bond popula-tions to estimate the different components in chemicalbonds. According to this formulation the Vickers hardnessHv in binary systems with a single type of M–N bond canbe estimated using:

H v ¼ 1051N 2=3e d�2:5 expð�1:191f i � 32:2f 0:55

m Þ; ð4Þ



mmsguo

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mmsguo

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D.G. Sangiovanni et al. / Acta Materialia xxx (2011) xxx–xxx 3

where Ne is the valence electron density defined as in theoriginal paper [32] and d is the bond length. In the aboveformula, fi and fm, are the Phillips ionicity and metallicity,respectively, of a chemical bond, and are used as correctionfactors to reflect the screening effects of ionic and metalliccomponents on covalent bonds. The Phillips ionicity is de-fined as:

fi ¼ ½1� expð�jP C � P j=PÞ�0:735; ð5Þ

with P as the overlap population of the bond and PC theoverlap population of a bond in a hypothetical pure cova-lent crystal with the same structure. Metallicity can be cal-culated as:

fm ¼ 0:026DF=ne; ð6Þwhere DF is the electron density of states at the Fermi leveland ne is the number of valence electrons per unit cell. Formulti-component compound systems the hardness can bepredicted as the average of hardness of all hypothetical bin-ary systems in the respective compound:

H v ¼Yl

Hlv

� �nl� �1

Pnl=

ð7Þ

where Hlv is the hardness of the hypothetical binary con-

taining l-type bonds and nl is the number of l-type bondsin the complex compound.

We calculated the charge density of crystal structures inreal space using 132 � 132 � 132 grid points. This tech-nique, which entails mapping the difference between self-consistent charge densities and charge densities derivedfrom the superposition of atomic wavefunctions, is thususeful in identifying the effects of shear deformations uponcharge distribution in crystals, by tracing the charge trans-fer from initially non-interacting atomic orbitals into thechemical bonds of the final atomic configurations.

COOP calculations are generally used to investigate thebinding character of chemical bonds in a crystal [22] byestimating the overlap population of molecular orbitals.The method resolves the bonding (positive values) andanti-bonding (negative values) contributions to covalentbonds, while absolute values are indicative of the bond

Table 1Present work DFT estimations of the elastic properties for binary and ternary

a (A) B (GPa) E (GPa) G (GPa) C44 (GP

TiNa 4.254 290 489 200 159Ti0.5V0.5N 4.188 312 432 170 144Ti0.5Nb0.5N 4.363 295 403 159 120Ti0.5Ta0.5N 4.348 304 366 141 107Ti0.5Mo0.5Na 4.300 321 382 147 77Ti0.5W0.5Na 4.298 336 394 151 60VN 4.121 320 478 191 139V0.5Nb0.5N 4.304 301 399 156 100V0.5Ta0.5N 4.291 313 379 146 73V0.5Mo0.5N 4.250 333 340 128 71V0.5W0.5N 4.246 340 372 141 61

a Ref. [15].


strength. Herein we report the results of a COOP analysisbased on VASP calculations using PAW potentials, knownto compare very well in terms of accuracy with full poten-tial methods. With appropriate manipulation of the VASPoutput data, COOP summations for all interacting atompairs in the calculation supercell can be performed overall occupied states, and the resulting integrated COOP(ICOOP) measures the strength of a particular covalentbond in the crystal. We emphasize the fact that any suchanalysis provides no information on the ionic characterof a bond. Consequently, COOP and ICOOP calculationsshould only be used to qualitatively estimate and/or com-pare bond strengths in different compounds. In the presentstudy we used COOP and ICOOP primarily to assess theeffects of strain/shear on the strength of a particular chem-ical bond, namely between pairs of first and second neigh-bors in the crystal.

Finally, to qualitatively assess the general trend in theresponse to deformations we estimated the stress–strainrelationship for a number of selected ternaries and com-pared it with the behavior of reference binaries. Thestress–strain curves were obtained by relaxing all atomicpositions and maintaining fixed supercell shapes for eachstrain in calculations using 8 � 8 � 8 k-point grids. Stressvalues in the direction of interest were then extracted fromthe stress tensor, directly from the VASP output. Severalcalculations with no constraints on shearing [33], i.e. allow-ing for cell shape relaxations in all directions except in thedirection of applied strain, were tested for TiN andTi0.5W0.5N at several strain values. Given the small differ-ences in stress values obtained with the two approaches,under a few per cent, the former method was chosen in thisstudy as it was less computationally expensive.

3. Results and discussion

Table 1 presents the results of our calculations for thebinaries and ternaries studied in this paper. As can be seen,the properties estimated herein for the two reference bina-ries, TiN and VN, are in very good agreement with previ-ous experimental [1,34–38] and ab initio [13,39–43]

nitrides.

a) C11 (GPa) C12 (GPa) C12–C44 (GPa) G/B m

640 115 �44 0.690 0.219592 172 28 0.545 0.269581 148 28 0.539 0.270559 176 69 0.464 0.299655 153 76 0.458 0.302720 145 85 0.449 0.305680 140 1 0.597 0.251621 141 41 0.518 0.279653 143 70 0.466 0.298617 191 120 0.384 0.330690 166 105 0.415 0.318



Table 3DFT calculated and experimental lattice parameters for ternary nitrides.

Lattice constant (A)

Present work Exp.

Ti0.5V0.5N 4.188 4.19b

Ti0.5Nb0.5N 4.363 4.32b, 4.41c

Ti0.5Ta0.5N 4.348 4.37c, 4.31d, 4.33e

Ti0.5Mo0.5N 4.300a 4.33c, 4.25f

Ti0.5W0.5N 4.298a 4.28c, 4.25g

a Ref. [15].b Ref. [46].c Ref. [44].d Ref. [47].e Ref. [48].f Ref. [45].g Ref. [29].


results, shown in Table 2. Not surprisingly, crystal proper-ties data for Ti0.5M0.5N and V0.5M0.5N is sparse, and adirect comparison with our predictions is thus not possible.Nonetheless, our estimations for the lattice constants of theTi0.5M0.5N alloys match within 1% the available experi-mental results [29,44–48], as shown in Table 3. In additionto the solid agreement with previous data for TiN and VN,the results in Table 1 are indicative of another interestingtrend. Namely, the alloying of both binaries with V (TiNonly), Nb, Ta, Mo and W, corresponding to an increasein valence electron concentration (VEC) from 9 to 10.5electrons per unit cell, has the following effects in the result-ing ternaries: a continuous increase in bulk modulus valuesaccompanied by a continuous decrease in G and C44 values.We also note that, on average, the alloys exhibit 20% lowerYoung’s modulus values compared with the referencebinaries.

As a first aspect of this trend in mechanical propertieswe discuss the hardness of Ti0.5M0.5N and V0.5M0.5Nalloys. Experimentally the hardness of thin films is mea-sured from indentation tests, in which different factorsaffect the result, such as texture, average grain size, stoichi-ometry, and lattice mismatch between the film and sub-strate materials. While the addition of a second transition

Table 2DFT data for TiN and VN, a comparison with experimental andtheoretical values.

Presentwork

Ab initio calculations Experimental

TiNa (A) 4.254a 4.221b, 4.246c, 4.270d, 4.275e 4.240h, 4.240i

B (GPa) 290a 270b, 287c, 292d, 264e, 295f 346i, 318j

E (GPa) 489a 487b, 456c, 470d, 514f 455i, 475j, 470k, 590l

G (GPa) 200a 203b, 189c, 191d, 213f 178i, 190j

C44 (GPa) 159a 168b, 165c, 162d, 166f 156i, 163j, 192m

C11 (GPa) 640a 610b, 585c, 604d, 671f 626i, 625j

C12 (GPa) 115a 100b, 137c, 136d, 106f 206i, 165j

m 0.219a 0.199b, 0.235c, 0.230d, 0.210f 0.281i, 0.251j

VNa (A) 4.121 4.128d, 4.110e, 4.132g, 4.127g 4.140h

B (GPa) 320 320d, 313e, 326f, 316g, 310g 268j

E (GPa) 478 434d, 441f 400j, 460l

G (GPa) 191 170d, 173f 160j

C44 (GPa) 139 126d, 137f 133j, 149m

C11 (GPa) 680 636d, 652f 533j

C12 (GPa) 140 162d, 163f 135j

m 0.251 0.270d, 0.270f 0.251j

a Ref. [15].b Ref. [39].c Ref. [40].d Ref. [41].e Ref. [42].f Ref. [13].g Ref. [43].h Ref. [1].i Ref. [34].j Ref. [35].

k Ref. [36].l Ref. [37].

m Ref. [38].


metal has often been connected to hardness enhancementsin B1 TiMN alloys [4–7], these findings may be due to filmmicrostructure features rather than to an inherent hardnessof perfect crystalline materials. To assess films hardnessdirectly from DFT calculations would require extremelylarge supercells and, consequently, massive computationaltime. Ab initio methods [32,49] may, however, be used indi-rectly to estimate the hardness of ideal crystals, by assum-ing that in the absence of defects the indented volume sizeis uniquely related to the resistance of the inner chemicalbonds and using first principles calculations to obtain thenecessary information on the respective bonds. This is themanner in which parameters are fitted to reproducethe Knoop’s hardness values of covalent and ionic crystals[23,49] and chemical bonds are characterized to predict theVickers hardness of transition metal nitrides and carbides[24,32]. Naturally, the growth and processing conditionsaffect the actual film microstructure, which ultimately issignificantly different from the defect-free crystal models.In spite of this aspect, and of the disputed merits and limitsof theoretical approaches [50,51], the methods for idealcrystals can be used to relatively assess the potential hard-ness of Ti0.5M0.5N and V0.5M0.5N alloys, since comparisonwith the same types of calculations for binaries and com-pounds well studied experimentally yields at least a qualita-tive trend of the hardness of these ternaries.

Table 4 gives the calculated hardness values of the com-pounds studied in this paper and compares them, wherepossible, with Knoop [16–20,52] and Vickers/Berkovich[4–7,36,48,53–59] indentation results. As expected, the cal-culations are generally in good agreement with the respec-tive experimental tests. It is also interesting to note that thetheoretical predictions systematically underestimate thehardness of most compounds. Significantly, the experimen-tal results obtained by both methods demonstrate that thehardness of all TiMN alloys is higher than that of TiN.Although for different compositions compared with theexperiments, these findings are confirmed by our calcula-tions for Ti0.5M0.5N. We find the same trend forV0.5M0.5N with respect to VN and, more importantly, V



mmsguo

Highlight

Table 4Predicted hardness values and experimental results obtained in Knoop and Vickers/Berkovich indentation tests.

Hardness (GPa)

Knoop Vickers/Berkovich

Theor. Exp. Theor. Exp.

TiN 16.6 17.7a, 21.2b (TiN0.91), 10.7c, 11.7d 21.7A 21.6B 19–21g, 17.2h, 21.6i, 26j (TiN0.9)Ti0.5Al0.5N 15.0 24.4c, 19.4d, 17.3e (Ti0.43Al0.57N) 29.4A 21.7B 27.8–30.4k

Ti0.5V0.5N 21.5 23.0e, 23.5f (Ti0.5V0.5Ny, 0.7 < y < 0.85) 25.2A 18.0B 20.0i, 23.1l (Ti0.43V0.57N)Ti0.5Nb0.5N 17.3 26.9A 17.1B 24.7m (Ti0.3Nb0.7N0.7), 40n, 50n

Ti0.5Ta0.5N 17.5 25.6e (Ti0.68Ta0.32N) 20.4A 20.8B 31o

Ti0.5Mo0.5N 21.1 23.7b (Ti0.54Mo0.46N0.84) 25.9A 15.9B 34.4p (Ti0.52Mo0.48N)Ti0.5W0.5N 21.1 24.7e (Ti0.66W0.34N), 32.6e (Ti0.36W0.64N) 27.3A 17.6B 33q (Ti0.6W0.4N), 40r (Ti0.3W0.7N)VN 27.2 24.0f (VNy, 0.62 < y < 0.66) 16.2A 14.9B 15.9i, 14.5–25.2s (VNy, 0.81 < y < 0.91)V0.5Nb0.5N 21.8 18.1A 14.2B 18.2h (V0.6Nb0.4N)V0.5Ta0.5N 22.1 17.5A 17.3B

V0.5Mo0.5N 25.8 24.9A 13.2B

V0.5W0.5N 25.9 24.1A 14.6B

A Values obtained using general definition in Eq. (4) and overlap populations calculated in this work, as used for the COOP analysis.B Values obtained using average formulation in Eq. (7) and overlap populations of hypothetical binaries published in original paper [24].a Ref. [16].b Ref. [17].c Ref. [20].d Ref. [52].e Ref. [18].f Ref. [19].g Ref. [54].h Ref. [53].i Ref. [59].j Ref. [36].

k Ref. [56].l Ref. [7].

m Ref. [58].n Ref. [4].o Ref. [48].p Ref. [6].q Ref. [5].r Ref. [55].s Ref. [57].


ternaries are generally predicted to have higher hardnessvalues compared with Ti ternaries. While the availableexperimental data on V ternaries does not allow direct ver-ification of these observations, the arguments presentedabove demonstrate that the Ti and V ternaries studiedherein have a hardness which is at least comparable with,if not significantly higher than, that of materials generallyaccepted as hard compounds.

The other notable feature of the results in Table 1, asalready discussed, is the continuous decrease in G andC44 values with increasing VEC in the ternaries obtainedby alloying the reference binaries. This tendency obviouslyaffects a number of properties of any material, amongwhich the G/B ratio, the Cauchy pressure (C12–C44) andthe Poisson ratio. In the present study the former twoquantities are most significant, as it is well known that theyare the main criteria for assessing the ductility of materials.As in a previous report [15], here we use the same Pugh [60]and Pettifor [61] criteria to map the ductility trend of theternary systems studied. These results are shown inFig. 1, and it can easily be observed that, almost without


exception, alloying has the effect of significantly enhancingthe ductility in all cases studied. The only compoundswhich are not within the ductile region of the map arethe Nb ternaries and Ti0.5V0.5N, yet one can clearly see thateven in these situations ductility is enhanced comparedwith the reference binaries (Ti0.5Al0.5N is included hereonly as a brittleness reference point). In conjunction withthe predicted hardness improvement discussed above, thedemonstrated ductility enhancement for these ternariesconfirms that the supertoughening effect reported forTiMoN and TiWN is operating in all ternaries analyzedherein. In addition, the data in Fig. 1 indicates that thiseffect is more pronounced with increasing VEC, and thebest results are obtained for M = Mo or W, i.e. for VECvalues of 10 and 10.5 electrons per unit cell.

To verify the above assertions, we investigated the elec-tronic structure of these ternary systems and compared itwith that of compounds for which the toughness enhance-ment mechanism has been explained [15]. For practical andclarity reasons the discussion in the present study concen-trates on the comparison between Ti0.5W0.5N and



0.3

0.4

0.5

0.6

0.7

0.8

-50 -25 0 25 50 75 100 125 150

G/B

C12 - C44 (GPa)

DUCTILE

TiN

Ti0.5Al0.5NVN

Ti0.5V0.5N

Ti0.5Nb0.5N

V0.5Nb0.5N

Ti0.5Ta0.5N

V0.5Ta0.5N

Ti0.5W0.5NTi0.5Mo0.5N

V0.5W0.5NV0.5Mo0.5N

BRITTLE

Fig. 1. Map of brittleness and ductility trends in compounds as estimatedin this work.


V0.5W0.5N, primarily by analyzing the effects of shearingupon chemical bonding in the two compounds. Neverthe-less, an analogous argument can be formulated for eachalloying case in this study, so the general conclusions arevalid for all ternary systems considered here.

The starting point in this comparison is an examinationof the overall charge density distributions in Ti0.5W0.5Nand V0.5W0.5N of the unstrained and shear strained struc-tures, respectively. The results are shown in Fig. 2, andas can be easily seen, there is an obvious similarity betweenthe charge density profiles of Ti0.5W0.5N (top panels) and

Fig. 2. Charge densities of Ti0.5W0.5N (upper panels) and V0.5W0.5N (lower pconfigurations. Color scale units are electrons A�3.


V0.5W0.5N (lower panels), in both the unstrained (Fig. 2aand c) and strained (Fig. 2b and d) situations. Evidently,the addition of W in VN has the same effect as in TiN,which is to delocalize the charge in the vicinity of W nuclei,leading to the formation of a layered electronic structureupon shearing, consisting in alternating layers of high/low charge concentration along the ½1�10� direction [15].As shown in the original report [15], this layered electronicstructure is the result of increased VEC in ternaries (onemore valence electron per unit cell compared with the ref-erence binaries), which in turn increases the occupancy ofd-t2g metallic states, due to increased overlapping of theseorbitals upon shearing.

In order to further probe the VEC ductility enhance-ment effects in these ternaries a deeper analysis of the rela-tionship between strain and bonding in these materials isrequired. We start by comparing the partial DOS ofunstrained and strained Ti0.5W0.5N configurations toobserve the d states response to 5% and 10% trigonal defor-mations. As can be seen in Fig. 3a, as shear strain increasesthe populated d-eg states are destabilized and progressivelyshift to higher binding energies. This is an expected ten-dency in ternary nitrides as the p(N)–d-eg(Ti, W) statesaccount primarily for first neighbor interactions and havea pronounced covalent/directional character. It has beenshown that at a VEC of 8.4 electrons per unit cell, thep(N, C)–d-eg(Ti) states are fully occupied and yield themaximum hardness and C44 values in B1 TiC1�xNx alloys[62]. Beyond the threshold of VEC = 8.4, which is the casehere, valence electrons start filling the d-t2g metallic states

anels) for: (a and c) unstrained structures; (b and d) shear strained (10%)



mmsguo

Highlight

-20-15

-10-5

0 50.10

0.05

0.00

20

40 d-egdx2-y2

E - EF (eV)δ (shear [110])

(a)D

OS

(sta

tes/

eV)

-20-15

-10-5

0 50.10

0.05

0.00

20

40

60

80 d-t2gdxy


(b)

DO

S (s

tate

s/eV

)

Fig. 3. Shearing induced effects on the Ti0.5W0.5N partial DOS at 0%, 5%and 10% strains: (a) shifting of d-eg states to higher energies; (b) increasingpopulation of d-t2g states.


and can reduce hardness and shear resistance. This trend isvisible in Fig. 3b, where one can see that with increasingstrain d states with t2g symmetry also shift to higher ener-gies, however, at maximum strain a clear secondary peakforms in the t2g DOS just below the Fermi level. The lowercurves in Fig. 3b (green1) also suggest that this peak ismainly induced by an enhanced occupancy of dxy states.As reported, [15,62], such an electronic response to [1 1 0]shear deformations may stem from the shortening ofmetal–metal second neighbor distance along the ½1�1 0�direction, favoring dxy–dxy orbitals overlapping. Similarchanges are expected in the electronic structures of theother ternaries considered here, as illustrated by the com-parison shown in Fig. 4, where one can see that the analo-gous d-t2g states response to shearing in V0.5W0.5N (lowerpanels) closely matches that in Ti0.5W0.5N (top panels).

In order to better illustrate the changes induced in chem-ical bonding upon shearing, in Fig. 5 we plot the results ofour COOP analysis, typically used to assess modificationsin covalent bonding. Fig. 5a shows the COOP results forTi0.5W0.5N obtained from M–N orbitals overlap. We notethat our Ti0.5W0.5N VASP-based COOP calculations forthe unstrained configuration are in good agreement with

1 For interpretation of color in Figs. 1–7, 9–11, 13 and 14, the reader isreferred to the web version of this article.


those obtained using the full potential linear muffin-tinorbital method for TiN/TiC [63]. The two bonding peaks,located close to �7 and �18 eV, correspond to the p(N)–d-eg(M) and s(N)–s(M) r orbitals, i.e. essentially firstneighbor interactions. As strain increases one can observea pronounced decrease in the peak close to �7 eV (greencurve), suggestive of the significant weakening in the M–N bonds during trigonal deformation. A totally differentsituation is observed in Fig. 5b, which depicts the COOPresults corresponding to second neighbor interactions,obtained from d-t2g(M)–d-t2g(M) orbitals overlap. Here,in correlation with the DOS results presented in Fig. 3b,d-t2g–d-t2g bonding states are clearly formed below theFermi energy level as strain increases.

Qualitatively, identical changes are induced withincreasing strain in the covalent character of bonding inV0.5W0.5N, as shown by the COOP results obtained fromthe overlap of the same orbitals as those for Ti0.5W0.5N.These are presented in Fig. 6, where one can clearly observethat first neighbor interactions, p–d bonds, are weakenedupon shearing (top panels), while second neighbor interac-tions, d–d bonds, become considerably stronger (lowerpanels) at higher strains. The overall trend of covalentbonding in the crystals can be assessed from our integratedCOOP (ICOOP) results, which are shown in Fig. 7 (actualvalues in Table 5). Once again, the similarity betweenTi0.5W0.5N (Fig. 7a–c) and V0.5W0.5N (Fig. 7d–f) withrespect to the dependence of first and second neighborinteractions on the applied strain is obvious.

These results also clarify the role played by theadditional electron per unit cell in orbitals overlap in theternaries. For illustrative purposes a schematic representa-tion of orbitals interaction in unstrained and strained con-figurations is given in Fig. 8, where we show the typical first(Fig. 8a and b) and second (Fig. 8c and d) neighbors orbitalarrangements, respectively. As can be seen, upon shearingthe overlap region decreases for first neighbor p–d orbitals,while for second neighbor d–d orbitals it increases. Natu-rally, this situation holds for both binaries and ternaries,but the additional electron per unit cell in ternaries playsthe decisive role, leading to significantly more d–d orbitalsoverlapping and filling of the d-t2g states. This observationis confirmed by the ICOOP results for binaries shown inFig. 9 (actual values in Table 5), in which the absolute val-ues at Fermi levels obtained for second neighbor interac-tions are considerably lower than for the ternaries, shownin Fig. 7.

Nevertheless, bonding in these compounds is notentirely covalent, and so one needs to go beyond a COOPanalysis in order to assess other significant changes inducedby strain in the bonding of these materials. To achieve thistask we mapped the difference between the self-consistentelectron density and the atomic charge density on the(0 0 1) plane. This technique allows tracing electronsmigration from an initially unperturbed atomic arrange-ment into the chemical bonds of the final crystal structure,i.e. changes of both ionic and covalent character induced



0

20

40

60

80

0

20

40

60

80

-20 -10 0 -20 -10 0 -20 -10 0

DO

S (s

tate

s/eV

)

E - EF (eV)

(a) (b) (c)

(d) (e) (f)

Ti0.5W0.5Nd-t2g

V0.5W0.5Nd-t2g

Fig. 4. d-t2g state responses to shearing in Ti0.5W0.5N (upper panels) and V0.5W0.5N (lower panels) at increasing strains. From left to right, each series ofpanels (a–c) and (d–f) corresponds to 0%, 5% and 10% strains.

-20-15

-10-5

0 50.10

0.05

0.00

-100

100

200

300

1st Neighb.p (N) - d-eg (M )


(a)

CO

OP

(arb

. uni

ts)

-20-15

-10-5

0 5

0.00

0.05

0.10

-300

-200

-100

100

200

2nd Neighb.


(b)

CO

OP

(arb

. uni

ts)

Fig. 5. COOP effects induced by shearing at 0%, 5% and 10% strains,resolved in first and second neighbor orbital interactions, in Ti0.5W0.5N:(a) progressive weakening of the covalent character of first neighbor N–Mbonds; (b) corresponding gradual increase in covalent bonding in secondneighbor M–M interactions.


by strain. These results are shown in Fig 10 for TiN (toppanels) and VN (lower panels), respectively in Fig. 11 forTi0.5W0.5N (top panels) and V0.5W0.5N (lower panels). Inall four compounds valence electrons partially transfer


from the metal atomic shells to neighboring N atoms toform p–d ionic–covalent bonds. In TiN and Ti0.5W0.5N(Figs. 10a and 11a) Ti–N bonds have a pronounced ioniccharacter, as implied by the spherical charge distributionsurrounding the atoms. In VN and V0.5W0.5N, on the otherhand, the clear squarish charge distribution surrounding Vatoms shows that V–N bonds are more directional(Figs. 10d and 11d). In both ternaries W–N bonds have adistinct directional character, as evidenced by the fourlobes pointing towards neighboring N atoms (Fig. 11aand d), in agreement with the plots in Fig. 2a and c. In thisinstance, however, it is important to note the significanttransfer of charge from W atoms towards N atoms (com-pare Figs. 10a with 11a and 10d with 11d). This effect is vis-ible in both ternaries, but especially in V0.5W0.5N, wherethe well-defined crescent shapes in the charge differencemap in the vicinity of N atoms clearly prove the existenceof this charge transfer (Fig. 11d). Thus, these results dem-onstrate that alloying affects the electronic structure ofbinaries, as it induces a charge migration process whichyields in ternaries stronger ionic, implicitly shorter, N–Ti/V bonds, and weaker/longer W–N bonds.

It is then interesting to note the changes induced byshearing upon both sets of compounds. In binaries onecan clearly observe that as the strain increases charge issmeared between constituent atoms along the [1 1 0] direc-tion of applied trigonal strain, as evidenced by the elon-gated charge shapes surrounding atoms in this direction(Fig. 10b and c for TiN respectively Fig. 10e and f forVN). Clearly, binaries resist shearing essentially in an ionicmanner, and as atoms are pulled apart there is also a ten-dency to somewhat increase directional/covalent bondingin the direction of the applied strain, as shown by thewidening (light blue) channels of charge transfer in this



-200

-100

0

100

200-100

0

100

200

0-10-200-10-200-10-20

CO

OP

(arb

itrar

y un

its)

E - EF (eV)

(a) (b) (c)

(d) (e) (f)2nd Neighb.

d-t2g (M ) - d-t2g (M )

1st Neighb.p (N) - d-eg (M )

Fig. 6. COOP analysis for V0.5W0.5N, resolved in first neighbor (upper panels) and second neighbor (lower panels) orbital interactions. From left to right,each series of panels (a–c) and (d–f) corresponds to 0%, 5% and 10% strains.

Fig. 7. Integrated COOP (ICOOP) analysis for Ti0.5W0.5N (upper panels) and V0.5W0.5N (lower panels). From left to right each series of panels (a–c) and(d–f) corresponds to 0%, 5% and 10% strains. ICOOP values at the Fermi level indicate the covalent bond strength.

Table 5Integrated COOP (ICOOP) values, in arbitrary units, showing bondstrength at the Fermi level and applied shear strains, for first and secondneighbor orbitals overlap.

1st neighb. 2nd neighb.

Strain (%) 0 5 10 0 5 10

TiN 352 191 168 35 48 82VN 334 268 233 33 74 132Ti0.5W0.5N 428 327 286 51 110 212V0.5W0.5N 434 318 276 47 112 209



direction. To some extent similar changes can be observedin the ternaries (Fig. 11b and c for Ti0.5W0.5N respectivelyFig. 11e and f for V0.5W0.5N). In this case, however, themost significant clearly visible change is the appearanceof well contoured, oriented lobes of positive charge transfer(darker blue) between the W atoms. The presence of theselobes demonstrates a significant increase in directional/covalent bonding along the W–W stacking planes, whichequates to a considerable strengthening of metal–metalbonds in the ½1�10� direction, perpendicular to that of the



Fig. 8. Schematic representation of first and second neighbor orbitals overlap in unstrained and shear strained B1 transition metal nitrides. Any existingp–d-eg first neighbor orbitals overlap in unstrained structures (a) decreases with applied strains (b). The opposite situation is observed for second neighbord-t2g–d-t2g orbitals, when overlapping is significantly enhanced during shearing deformations (c and d).

Fig. 9. Integrated COOP (ICOOP) analysis for TiN (upper panels) and VN (lower panels). From left to right each series of panels (a–c) and (d–f),corresponds to 0%, 5% and 10% strains.


applied strain. This result is consistent with the shearing-induced formation of a layered electronic structurereported for Ti nitrides [15], and fully supports our analysisof other ternaries considered in this study.

In addition to the arguments presented so far in thisstudy, the following rationale helps to further elucidatethe mechanism through which ductility is promoted and


enhanced in these ternary systems. Upon replacing Ti/Vatoms with W in TiN/VN, and forming the close to exper-imentally observed [29] C#3 configuration, the lattice pointsymmetry is reduced from Oh to C3v. In this configurationN is coordinated with Ti/V and W on opposite verses ofeach Cartesian direction. The electronic environmentanisotropy leads to a relaxation of N atomic positions,



Fig. 10. Charge density difference maps for TiN (upper panels) and VN (lower panels) and effects of shearing applied on the (0 0 1) plane. From left toright each series of panels (a–c) and (d–f), corresponds to 0%, 5% and 10% strains. Color scale units are electrons A�3.


and the displacement of N atoms from ideal B1 lattice sitesdepends on the relative bonding strengths with neighboringatoms. As shown in the preceding sections, the N–Ti/Vbonds are stronger than W–N bonds, so the B1 ? C#3 tran-sition will yield shortening of the Ti/V–N distances. Thistrend will be enhanced with increasing strain, as shown inFig. 12, where we plot the bond length ratios of the W–Nbonds with respect to Ti/V–N bonds in the two ternaries.

At the same time one should consider the well-knownfact that one of the main channels for dislocations glideat low temperatures in B1 nitrides is the f1 10gh1�10i slipsystem [1]. During dislocation motion bonds are brokenand reformed and, obviously, dislocation glide will occurmore easily in planes normal to those containing weakerbonds. In the nitrides considered here this is clearly thecase. In addition to the demonstrated weaker bondingbetween the W–N planes induced by the B1 ? C#3 transi-tion, the [1 1 0] trigonal deformation has been shown toyield stronger covalent bonding along the ½1�10� W–Wplanes. At a certain level of shearing strain this set of con-ditions will obviously favor the breakage of bonds betweenthe W–N atomic planes, allowing these planes to slideagainst each other and making the f11 0gh1�10i slip systema primary channel for dislocation glide. Equally importantduring dislocation glide, the strong covalent bondingwithin the W–W slip planes will further delay bondbreakage. These are the key mechanisms which promote


and enhance ductility in these ternary systems. This isnot, however, the situation in TiN and VN, where, aswas shown, the Ti/V–N bonds are equally robust in allplanes and resist any type of deformation, either tetragonalor trigonal strains, thus explaining their hardness as well astheir brittleness. These arguments are clearly supported byour analysis of the stress–strain relationship in TiN, VNand the corresponding W alloyed ternaries, as shown inFig. 13. As can be seen, the binaries are characterized bya definite linear-elastic response to shearing, while the ter-naries exhibit a visibly more plastic stress–strain responseto deformations.

As mentioned in the preceding sections of this study, thefindings reported for Ti0.5W0.5N and V0.5W0.5N areexpected to be valid for the other alloying combinationsconsidered herein, as these were shown to be a VEC-induced effect. The trend in ductility criteria is clearly illus-trated in Fig. 14, for both Ti- and V-based ternary nitrides.Therein we also include as a reference point the results forTi0.5Al0.5N, compound with a VEC of 8.5 electrons perunit cell, i.e. marginally higher than the threshold of 8.4for which maximum hardness is expected. The VEC depen-dence, consisting in decreasing G/B ratios (Fig. 14a) andincreasingly positive values for Cauchy pressures (C12–C44) (Fig. 14b), is obvious for all ternary systems consid-ered in this study. A similar effect is active to some extentin binary systems, as VN is found to have more ductile



Fig. 11. Charge density difference maps for Ti0.5W0.5N (upper panels) and V0.5W0.5N (lower panels) and effects of shearing applied on the (0 0 1) plane.From left to right each series of panels (a–c) and (d–f), corresponds to 0%, 5% and 10% strains. Color scale units are electrons A�3.

1.02

1.03

1.04

1.05

0 0.01 0.02 0.03 0.04 0.05

W-N

/ Ti

(V)-N

bon

d le

ngth

ratio

δ (shear strain [110])

Ti0.5W0.5NV0.5W0.5N

Fig. 12. Bond length ratio dependence on strain: W–N/Ti–N bonds inTi0.5W0.5N (black squares) and W–N/V–N bonds in V0.5W0.5N (blackcircles).

5

10

15

20

25

30

35

0.02 0.04 0.06 0.08 0.10

Stre

ss (G

Pa)

δ (shear strain [110])

TiNVN

Ti0.5W0.5NV0.5W0.5N

Fig. 13. Calculated stress–strain trends in TiN (green, open squares), VN(blue, open circles), Ti0.5W0.5N (red, solid squares) and V0.5W0.5N (blue,solid circles).


properties compared to TiN, although clearly not aspronounced as in the case of ternary systems.

4. Conclusions

Our DFT calculations predict the existence and explainthe origins of supertoughening in cubic B1 Ti and V


ternary nitrides obtained by alloying TiN and VN withNb, Ta, Mo, and W at 50% concentration. All ternary sys-tems considered in this study are predicted to have hard-ness values at least comparable with the reference binarysystems and significantly enhanced ductile characteristics.This phenomenon is shown to be primarily an effect ofincreased VEC per unit cell, equating to stronger reference



mmsguo

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mmsguo

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mmsguo

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mmsguo

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mmsguo

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0.4

0.5

0.6

0.7

8.5 9 9.5 10 10.5

G/B

VEC

TiN

VN

TiAlN

TiNbN

TiVN

TiTaN

VNbN

VTaN TiMoNTiWNVWN

VMoN

(a)

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8.5 9 9.5 10 10.5

C12

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44 (G

Pa)

VEC

TiN

VNTiAlN

TiNbNTiVN

TiTaN

VNbN

VTaN TiMoNTiWNVWN

VMoN

(b)

Fig. 14. VEC-induced trends in Ti- and V-based B1 transition metalnitrides. (a) G/B ratio dependence on VEC: Ti (red, open circles); V (blue,solid circles) nitrides. (b) VEC effect on Cauchy pressure in Ti (red, opensquares), and V (blue, solid squares) compounds.


metal Ti/V–N, respectively weaker alloying metal Nb/Ta/Mo/W–N bonds, which upon shearing yields an increasedoccupancy of d-t2g metallic states. This combination ofproperties leads to the formation of a layered electronicstructure and, ultimately, allows a selective response tostrain and shear deformations by assisting the activationof the f110gh1�10i slip system for dislocation glide.

Acknowledgements

The work was supported by the Swedish ResearchCouncil (VR) and the Swedish Strategic Research Founda-tion (SSF) Program on Materials Science and AdvancedSurface Engineering. All calculations were performed onthe Neolith and the Kappa clusters located at the NationalSupercomputer Centre (NSC) in Linkoping.

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Supertoughening in B1 transition metal nitride alloys by increased