Structural Stability and Elastic and Electronic Properties of W2X (X = B, C, N):First-Principles Investigations

Gui Yang1+ and Bao Feng Chen2

1College of Physics and Electrical Engineering, Anyang Normal University, Anyang 455000, P. R. China2Anyang Institute of Technology, Anyang, Henan 455000, P. R. China

(Received August 19, 2013; accepted October 22, 2013; published online December 12, 2013)

The structural stability, elastic property, and mechanical stability of W2X (X = B, C, N) are investigated using first-principles calculations with density functional theory. The calculations show that Re2N-type W2N is boththermodynamically and elastically stable. The calculated elastic properties indicate that W2X compounds are potentiallow-compressibility materials. The large elastic modulus, low Poisson’s ratios, small B/G ratios, and high Debyetemperatures of thecompounds indicate that they are potential hard materials. The analysis of the density of states andelectronic localization function provides further understanding of the elastic and chemical properties of these compounds.

1. Introduction

The introduction of light and covalent-bond-formingelements into transition-metal (TM) lattices is expected tohave profound effects on the chemical, mechanical, andelectronic properties of these lattices. These investigations aremotivated by the search for new ultra-incompressible andsuperhard materials for various technological applications.1)

In synthesizing and designing new superhard materials,besides the traditional B–C–N–O systems, transition metalsare also very attractive owing to their relatively high bulkmodulus such as the parent metal Re (380GPa),2) but theirshear strength is low owing to the nondirectional metallicbonding, and they have low hardness. Therefore, it isexpected that through the insertion of B, C, N, and O atomsinto the transition metals, superhard materials may be formedby inducing the nondirectional metallic bonding in puretransition metals to highly directional covalent bonding in thecorresponding carbides, borides, nitrides or oxides. Recentdesign of new intrinisically hard materials has concentratedon light-element TM compounds with high elastic moduli,such as OsB2,3) ReB2,4) IrN2,5) Re2N,6) Re3N,7) Re2C,8) andB2O.9) To the best of our knowledge, W2B10) and W2C11)

have been synthesized for decades. However, their basicproperties (such as structure stability, chemical bonding,elastic properties, and electronic structure) are still unclear.As is well known, chemically related compounds mayhave similar structures. For W2N, we choose the recentlysynthesized Re2N structure as its basic structure. Our maingoal is to determine the trends of the elastic and electronicproperties in the series of B, C, and N. Therefore, a detailedstudy of the structural, elastic, and electronic properties ofW2X (X = B, C, N) is undoubtedly necessary and helpful forunderstanding their unique properties.

In this work, we present a detailed ab initio study of thestructural, elastic, and electronic properties and chemicalbonding of W2X (X = B, C, N). Our results show that thesethree phases exhibit excellent mechanical properties, whichcan be grouped into incompressible materials. Re2N-typeW2N is both mechanically and dynamically stable, asverified by calculations of its elastic constant and phonondispersion. The chemical bonding was also studied by ananalysis of the electronic localization function and density ofstates.

2. Computational Details

The calculations in this study were performed by theprojector-augmented wave (PAW) method12) implemented inthe Vienna ab initio simulation package (VASP).13,14) Thegeneralized gradient approximation (GGA)15) was used todescribe the exchange–correlation function. Geometry opti-mization was performed using the conjugate gradientalgorithm method with a plane-wave cutoff energy of500 eV. For hexagonal structures, the ¥-centered k meshwas used, and for other structures, the Monkhorst–Pack kmesh was used to ensure that the energy change of each atomwas less than 0.005 eV. The structures were relaxed withrespect to both lattice parameters and atomic positions. Thestrain-stress method was used to obtain the elastic constants.From the calculated elastic constants Cij, the polycrystallinebulk modulus B, the shear modulus G, Young’s modulus E,and Poisson’s ratio ¯ were further estimated using theVoigt–Reuss–Hill approximation.16) In addition, the Young’smodulus Y and Poisson’s ratio ¯ were obtained using theequations Y ¼ ð9GBÞ=ð3Bþ GÞ, and � ¼ ð3B� 2GÞ=ð6Bþ2GÞ, respectively. Formation enthalpies were calculated from�H ¼ EðW2XÞ � 2Eðsolid WÞ � Eðsolid XÞ.3. Results and Discussions

3.1 Structural determination and featureAfter full geometry optimization, all the structures

maintain the same symmetry as the initial symmetries. Thecalculated lattice parameters, cell volume, and formationenthalpies of W2X are listed in Table I. As is well known, thenegative values of the formation enthalpies indicate thatthey are thermodynamically stable. From Table I, we can seethat all the considered compounds have negative formationenthalpies (�H), indicating that the formation of thesephases is exothermic and that they are all thermodynamicallystable. From Table I, we can see that the calculated latticeparameters of W2B, a and c, are in good agreement with theexperimental data,10) with differences of about 0.003 and0.027Å. The calculated lattice constants of W2C (a ¼4:7392, b ¼ 6:0877, and c ¼ 5:2255Å) are very close tothe experimental values (a ¼ 4:721, b ¼ 6:03, and c ¼5:18Å),11) confirming the reliability of the present calculationmethod. The optimized crystal structures of W2B, W2C, andW2N are shown in Figs. 1(a)–1(c), respectively.

Journal of the Physical Society of Japan 83, 014708 (2014)

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014708-1 ©2014 The Physical Society of Japan

At zero temperature, a stable crystalline structure requiresall phonon frequencies to be positive. The calculated phonondispersion curves at high symmetry points of W2N are shownin Fig. 2. No imaginary phonon frequency is observed in theBrillouin zone, indicating that Re2N-type W2N is dynami-cally stable.

On the basis of the correlation between bulk modulus andvalence electron density,3,17) it is expected that the bulkelastic properties of these compounds will have possiblecorrelations with their valence-electron density. Thus, it isnecessary to calculate the valence electron density of W2Xfirstly. As shown in Table I, our calculated valence electrondensities for W and W2X (X = B, C, N) are 0.376, 0.405,0.425, and 0.464 electrons/Å3, respectively. These valuesshow that incorporating X (X = B, C, N) into W increaseselectron density. The bulk moduli and their pressure deriva-tives (shown in Table I) are obtained by fitting pressures andcell volume with the third-order BirchMurnaghan equation ofstate. It can be seen that the general trend of the bulk moduliof these compounds is the same as that of the valenceelectron densities. It thus can be concluded that valenceelectron density is highly related to bulk modulus.

3.2 Elastic and physical propertiesMechanical stability is a necessary condition for a crystal

to exist. Accurate elastic constants can help us to understand

the mechanical properties and also provide very usefulinformation to estimate the hardness of a material. Thecalculated elastic constants of the W2X are listed in Table II.It is clearly seen that all the studied compounds satisfy themechanical stability criteria,18) indicating that they areelastically stable. On the other hand, the positive eigenvaluesof the elastic constant matrix for each considered compoundfurther prove that they are elastically stable. We observe highincompressibilities along the a-, b-, and c-axes, as demon-strated by the large C11, C22, and C33 values, respectively.

Bulk and shear moduli are still the most importantparameters for identifying the material hardness. Superhardmaterials have high bulk modulus to represent theirincompressibility and high shear modulus to restrict defor-mation. The bulk moduli, shear moduli, Young’s moduli, andPoisson’s ratios of W2X have been estimated from thecalculated elastic constants. The calculated results show thatthe bulk moduli (B) are in good agreement with the measuredB0 values (shown in Table I), which were employed from thethird order Birch–Murnaghan equation of state, demonstrat-ing the reliability of the present theoretical method. FromTable II, we can see that all the compounds have large bulkmoduli, which indicates that these materials are difficultto compress. Moreover, with the increases in the electrondensity, the bulk modulus increases. Among the compounds,W2N has the largest bulk modulus (363GPa), indicating thatit is the most difficult to compress among these compounds,followed by W2C (347GPa) and W2B (340GPa). The largemoduli of W2X show that they can be grouped into low-compressibility materials. As we know, shear modulus showsa much better correlation with hardness than with bulkmodulus. From Table II, it can be seen that W2C has thelargest shear modulus (194GPa), suggesting that it can

(a)

(b)

(c)

Fig. 1. Crystal structures of W2B (a), W2C (b), and W2N (c). The large andsmall spheres represent W and X (X = B, C, N) atoms, respectively.

ΓA HK Γ ML H0

5

10

15

20

Freq

uenc

y/T

Hz

Fig. 2. Phonon dispersion curves at 0GPa of Re2N-type W2N structure.

Table II. Calculated elastic constants (in GPa), bulk moduli B (in GPa),shear moduli G (in GPa), Young’s moduli Y (in GPa), and Poisson’s ratios (¯)of W2X (X = B, C, N).

Structure C11 C12 C13 C22 C33 C44 C66 B G B=G Y ¯

W2B 565 206 249 565 519 157 180 340 180 1.09 459 0.28W2C 530 266 278 559 557 163 213 347 194 1.79 491 0.26W2N 543 311 218 543 689 143 166 363 169 2.15 439 0.30

Table I. Calculated formation enthalpies (�H), valence electron densities(�0 in electrons/Å3) equilibrium lattice parameters a (Å), b (Å), and c (Å),cell volumes (V in Å3), bulk moduli (B0 in GPa), and their pressurederivatives at zero pressure B0 of W2X (X = B, C, N).

Structure �H �0 a b c V B0 B0

W2B ¹0.80 0.405 5.571 4.771 37.02 341 4.12W2B10) 5.568 4.744 36.77W2C ¹0.17 0.425 4.739 6.087 5.225 37.69 348 4.35W2C11) 4.721 6.030 5.180 36.87W2N ¹0.63 0.464 2.862 10.326 36.63 364 4.36

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withstand shear strain to the largest extent. W2N has thesmallest shear modulus (169GPa). The shear modulus ofW2X is larger than that of superhard WB4 (104GPa).19)

Poisson defined the ratio ¯ between transverse strain (et)and longitudinal strain (el) in the elastic loading direction as� ¼ �et=el. The relatively low values of ¯ for hard materialsin general indicate the high degree of directional covalentbonding. As seen from Table II, the small Poisson’s ratios ofW2B (0.28), W2C (0.26), and W2N (0.30) indicate their highdegree of directional covalent bonding, which are smallerthan that of WB4 (0.3420)). The directional covalent bondingis helpful for their shear moduli. This may be the reason whyW2C has the largest shear modulus. Young’s modulus (Y) isused to provide a measure of the stiffness of a solid, which isdefined as the ratio of stress and strain. A large Y indicates astiff material. As a result, W2C is stiffer than the other twocompounds because of its largest Y (491GPa). A high (low)B=G ratio of a material indicates that it is ductile (brittle) andthe critical B=G is about 1.75.21) The calculated B=G ratio forW2B is below the critical value (1.75), indicating that it isbrittle. W2C and W2N are ductile. In other words, the largeelastic modulus, low B=G ratio, and small Poisson’s ratioshow that W2X compounds are potential hard materials.

Mechanical properties (particularly, hardness of solids) canbe related to thermodynamical parameters such as Debyetemperature, specific heat, thermal expansion, and meltingpoint.22) In this concept, Debye temperature (�D) is oneof the fundamental parameters for the characterization ofmaterials and the microhardness of a solid.23,24) We canobtain some information on the thermal properties fromthe zero-temperature elastic constants. The transverse andlongitudinal acoustic velocities can be calculated as

Vt ¼ffiffiffiffiG

�

s; ð1Þ

Vl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Bþ 4G

3�

s: ð2Þ

Here, µ is the mass density of the material. We can thendefine the mean sound velocity as

1

V3m

¼ 1

3

1

V3tþ 1

V3l

� �: ð3Þ

For a solid with N atoms in the volume V, the Debyetemperature �D may be estimated from the mean soundvelocity as

�D ¼ hVm

kB

� �3N

4�V

� �1=3

; ð4Þ

where h is Plank’s constant and kB is the Boltzmann constant.The computed longitudinal, transverse, and average soundvelocities and the Debye temperatures obtained from theaverage sound velocity are shown in Table III. The Debyetemperature of W2C (575K) is larger than those of W2B(554K) and W2N (535K), but is smaller than that of ReB2

(756K,25) 858K26)). It is known that the higher the �D of thematerials, the larger their microhardness. Therefore, W2Cmay have the largest hardness among these compounds.

3.3 Electronic structure and chemical bondingThe electronic structure is crucial to understanding the

origin of the physical properties of these compounds. Thetotal and partial densities of states (DOSs) of W2X (X = B,C, N) were calculated at zero pressure as shown in Fig. 3. Allthree phases show similar metal bonding features because ofthe finite DOS at the Fermi level (EF), which also shows seenthat the W-5d states of these three compounds contributemost to their total DOS and dominate the DOS at the Fermilevel; the role of X-2p states is relatively small. The W-p,W-s, and X-s states are localized in the low-energy range, andnaturally their effect on bonding is negligible.

The electronic localization function (ELF)27) offers areliable measure of electron pairing and localization.According to its original definition, the ELF values arescaled between 0 and 1, where ELF = 1 corresponds to theperfect localization characteristic of covalent bonds or lonepairs. To analyze the chemical bonding of W2X (X = B,C, N), we plot their ELF in Fig. 4 with the isosurface atELF = 0.70. A partially covalent bonding interaction be-tween the W and X atoms in these three compounds isindicated by the ELF maxima biased toward the X atoms. ForW2B, its ELF is localized in the interstitial triangular refions,which is made up of two W atoms and one B atom shown inFig. 4(a). There is a small ELF between single W and Batoms. For W2C, the ELF between W and C atoms attains

Table III. Density (g cm¹3), longitudinal, and transverse, average soundvelocities (Vt, Vl, and Vm in km s¹1), and the Debye temperatures �D of W2X(X = B, C, N) obtained from the average sound velocity.

Structure µ Vt Vl Vm �D

W2B 17.01 3.253 5.840 3.623 554W2C 16.75 3.403 6.014 3.785 575W2N 17.32 3.123 5.828 3.488 535

-12 -9 -6 -30

5

10

15

20 totalW-dW-pW-sX_sX_p

-15 -12 -9 -6 -3

0 3

0 30

5

10

15

-15 -10 -5 0 5 100

2

4

6

8

10

(a)

(b)

(c)

Den

sity

of

Stat

ess

(sta

tes/

eV)

Energy (eV)

Fig. 3. (Color online) Total and partial DOS for W2B (a), W2C (b), andW2N (c), respectively. The Fermi level is at zero.

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local maximum values close to the C sites, indicatingpartially covalent and ionic interactions. For W2N, the ELFbetween W and N atoms is very close to the N sites,indicating the weak covalent interactions. Moreover, W–Wmetallic bonds also exist in W2N shown in Fig. 4(c), whichcan reduce the hardness of a material. Our analysis revealsthat bonding in these species is complex and may bedescribed as a mixture of metallic, ionic, and covalentcontributions. Such a conclusion was also drawn in othertransition metal compounds.28,29) Clearly, the covalentbonding between W and X (X = B, C, N) atoms will surelyincrease their structural stabilities and elastic moduli. Therelative strength order of covalent bonding for thesecompounds is W2C > W2B > W2N. This is why W2C hasthe largest hardness. This is consistent with the results ofanalyses of their elastic properties and Debye temperature.

4. Conclusion

In conclusion, we have extensively analyzed the latticeparameters, structural stability, elastic moduli, Poisson’sratio, valence electron density, Debye stiffness, and electroniclocalization function of W2X (X = B, C, N) using first-principles calculations. The calculations show that the Re2N-type W2N is both thermodynamically and elastically stable.The calculations of the bulk modulis of these stable phasesreveal that they are potential low-compressibility and hardmaterials. In addition, the results of the studies indicate thatthe bulk elastic properties of these stable phases correlatewith their valence electron density. The results of analyses

of the density of states and electronic localization functionindicates that the covalent W–X bonds are the driving forcefor their large bulk moduli.

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China Nos. 11047108, 11005002, and11005003, the Research Project of Basic and Cutting-EdgeTechnology of Henan Province, China (No. 112300410183),and the Education Department of Henan Province, China(Nos. 2011B140002 and 14A140016).

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(a)

(b)

(c)

Fig. 4. (Color online) Contours of the electronic localization function(ELF) of (a) W2B, (b) W2C, and (c) W2C with an isosurface value of 0.70.The red and blue spheres represent W and X (X = B, C, N) atoms,respectively.

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