microscopic theory of hardness and design of novel superhard crystals

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Microscopic theory of hardness and design of novel superhard crystals Yongjun Tian , Bo Xu, Zhisheng Zhao State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao, Hebei 066004, China abstract article info Article history: Received 19 October 2011 Accepted 18 February 2012 Keywords: Hardness Superhard materials Modeling Chemical bond Crystal design Hardness can be dened microscopically as the combined resistance of chemical bonds in a material to inden- tation. The current review presents three most popular microscopic models based on distinct scaling schemes of this resistance, namely the bond resistance, bond strength, and electronegativity models, with key points during employing these microscopic models addressed. These models can be used to estimate the hardness of known crystals. More importantly, hardness prediction based on the designed crystal structures becomes feasible with these models. Consequently, a straightforward and powerful criterion for novel superhard materials is provided. The current focuses of research on potential superhard materials are also discussed. © 2012 Elsevier Ltd. All rights reserved. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2. Microscopic models for hardness prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.1. Bond resistance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.2. Bond strength model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.3. Electronegativity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.4. Some points about microscopic hardness models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3. Novel superhard crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.1. B-C crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2. Carbon allotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3. Transition metal compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4. Other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 1. Introduction In 1722, the French scientist R.A.F. de Réaumur coined the term hardness [1]. Since then, hardness has been used as one of the funda- mental mechanical properties of materials. Hardness can be dened macroscopically as the ability of a material to resist being scratched or dented by another. Although hardness governs the technological applications of numerous materials, it is not as well dened as other physical properties [2], especially at the atomic scale. Experimentally, hardness is accurately characterized by the indentation of a material using a hard indenter. According to the nature and shape of the indenter, several scales such as the Vickers, Knoop, Brinell, and Rock- well scales have been developed. The most common are the Vickers and Knoop scales, whose indenters are a pyramidal-shaped diamonds with a square base and an elongated lozenge base, respectively. Hard- ness is measured from the ratio of the indenter force to the associated indentation area. The deduced hardness usually depends on the shape of the indenter, loading force and rate, indentation size and time, sample orientation, as well as surface condition. For brittle ma- terials or material whose hardness approaches that of diamond, the indentation process is not controlled by plastic deformation alone. The brittle microcracking of the sample and the deformation of the indenter also play roles, leading to hardness changes with different loads [3,4]. For metals and their alloys, hardness is observed to increase with decreasing indentation size. Large strain gradients inherent in small indentations produce geometrically necessary Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93106 Corresponding author. Tel.: + 86 335 8057047; fax: + 86 335 8074545. E-mail address: [email protected] (Y. Tian). 0263-4368/$ see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2012.02.021 Contents lists available at SciVerse ScienceDirect Int. Journal of Refractory Metals and Hard Materials journal homepage: www.elsevier.com/locate/IJRMHM

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Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

Contents lists available at SciVerse ScienceDirect

Int. Journal of Refractory Metals and Hard Materials

j ourna l homepage: www.e lsev ie r .com/ locate / IJRMHM

Microscopic theory of hardness and design of novel superhard crystals

Yongjun Tian ⁎, Bo Xu, Zhisheng ZhaoState Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao, Hebei 066004, China

⁎ Corresponding author. Tel.: +86 335 8057047; fax:E-mail address: [email protected] (Y. Tian).

0263-4368/$ – see front matter © 2012 Elsevier Ltd. Alldoi:10.1016/j.ijrmhm.2012.02.021

a b s t r a c t

a r t i c l e i n f o

Article history:Received 19 October 2011Accepted 18 February 2012

Keywords:HardnessSuperhard materialsModelingChemical bondCrystal design

Hardness can be definedmicroscopically as the combined resistance of chemical bonds in a material to inden-tation. The current review presents three most popular microscopic models based on distinct scaling schemesof this resistance, namely the bond resistance, bond strength, and electronegativity models, with key pointsduring employing these microscopic models addressed. These models can be used to estimate the hardness ofknown crystals. More importantly, hardness prediction based on the designed crystal structures becomesfeasible with these models. Consequently, a straightforward and powerful criterion for novel superhardmaterials is provided. The current focuses of research on potential superhard materials are also discussed.

© 2012 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932. Microscopic models for hardness prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

2.1. Bond resistance model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942.2. Bond strength model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.3. Electronegativity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.4. Some points about microscopic hardness models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3. Novel superhard crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.1. B-C crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.2. Carbon allotropes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.3. Transition metal compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.4. Other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4. Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

1. Introduction

In 1722, the French scientist R.A.F. de Réaumur coined the termhardness [1]. Since then, hardness has been used as one of the funda-mental mechanical properties of materials. Hardness can be definedmacroscopically as the ability of a material to resist being scratchedor dented by another. Although hardness governs the technologicalapplications of numerous materials, it is not as well defined as otherphysical properties [2], especially at the atomic scale. Experimentally,hardness is accurately characterized by the indentation of a materialusing a hard indenter. According to the nature and shape of the

+86 335 8074545.

rights reserved.

indenter, several scales such as the Vickers, Knoop, Brinell, and Rock-well scales have been developed. The most common are the Vickersand Knoop scales, whose indenters are a pyramidal-shaped diamondswith a square base and an elongated lozenge base, respectively. Hard-ness is measured from the ratio of the indenter force to the associatedindentation area. The deduced hardness usually depends on theshape of the indenter, loading force and rate, indentation size andtime, sample orientation, as well as surface condition. For brittle ma-terials or material whose hardness approaches that of diamond, theindentation process is not controlled by plastic deformation alone.The brittle microcracking of the sample and the deformation of theindenter also play roles, leading to hardness changes with differentloads [3,4]. For metals and their alloys, hardness is observed toincrease with decreasing indentation size. Large strain gradientsinherent in small indentations produce geometrically necessary

94 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

dislocations, causing enhanced hardening [5]. A reliable hardnesscould be determined from the asymptotic-hardness region of a well-controlled indentation process. Hardness valueswidely vary for differ-ent materials, i.e., from tenths of a gigapascal or less for ionic crystals,to several gigapascals or less for puremetals, and to tens of gigapascalsfor covalent crystals. As generally accepted by the materials sciencecommunity, materials with Vickers hardness larger than 40 GPaare classified as superhard materials [2,6]. Readers can also refer toRef. [7] for a discussion about definition of superhard/ultrahard andhow they evolve with times. All as-known superhard compounds arecovalent and polar covalent crystals.

Over the past several decades, a large number of studies havebeen devoted to novel superhard materials for both practical andscientific purposes. One is to synthesize robust materials with desir-able properties for modern technologies, and the other is to revealthe controlling factors that determine the hardness of materials atthe microscopic level. Superhard materials are of great importancein various industry areas, such as wear-resistant coating, abrasives,cutting and polishing tools [2,8–10]. Diamond is thus far the hardestknown substance (HV=95 GPa) with the highest shear modulus andYoung's modulus. However, the applicability of diamond is limitedbecause of its chemical reactivity with ferrous materials and nonre-sistance to oxidation. Cubic boron nitride, the second hardest mate-rial with a diamond structure, can be used to cut ferrous metals.However, its hardness is only 66 GPa, or 30% lower than that of dia-mond. Hence, the syntheses of novel inertial superhard materialswith hardness comparable to or even harder than diamond are high-ly anticipated. Recent searches for new superhard materials mainlyfocus on two classes of materials. The first class includes light-element compounds in a B-C-N-O system with short and strongthree dimensional (3D) covalent bonds, which are crucial for super-hard materials. The experimental syntheses of BCxN, BCx, γB28, B6O[9,11–15], etc., have significantly progressed. The second class con-sists of materials formed by light elements (B, C, and N) and heavytransition metals (TMs) that could introduce a high valence electrondensity into the corresponding compounds, such as ReB2, OsB2, WB4,PtC, IrN2, OsN2, and PtN2 [16–22]. The high valence electron densityenables resistance to elastic and plastic deformations. Although theassignment of superhard materials to ReB2 is under debate [21,23],this class of materials is still a great search pool for semiconductingsuperhard materials.

The properties of materials depend on electrical structures. Inprinciple, new materials with expected properties can be designed.However, this aspiration is currently very unrealistic. The quantitativeconnections between electronic structures and macroscopic “engi-neering” properties remain as one of the foremost challenges in mod-ern computational materials science [24]. Alongside the difficulty ofsynthesizing new superhard materials are two theoretical problemsthat have mystified scientists for more than a century. One is thedesign of a hard material based on atomic arrangements in crystalstructure, which urgently need reliable models for hardness quantifi-cations in the field of superhard materials [25]. The other is the defi-nition of hardness at the microscopic level, which is very fundamentalfor understanding the physical origin of hardness.

Empirical models originally correlate hardness with the elasticproperties of crystals. Historically, Gilman and Cohen have estab-lished a linear correlation between hardness and bulk modulus[26,27]. Later in 1998, an improved correlation between hardnessand shear modulus was proposed by Teter [28]. However, these em-pirical correlations between hardness and bulk (or shear) modulusturn out to be physically questionable. The bulk modulus character-izes the incompressibility of a material, and has a direct relationshipwith valence electron density; more electrons correspond to greaterrepulsions within a structure [29]. The shear modulus characterizesthe resistance to shape change at a constant volume. A larger shearmodulus results in a greater ability to resist shearing forces. While

hardness characterizes a permanent plastic deformation, it is nowaccepted that hardness does not depend monotonously on bulk mod-ulus or shear modulus according to the simple linear correlation,considering that these elastic moduli correspond to reversible elasticdeformation [30]. Consequently, Chen et al. have proposed a macro-scopic model of hardness by considering the Pugh's modulus ratio,k=G/B[30]. The parameter k is closely correlated to the brittleness/ductility of materials, as well as highlights a relationship betweenthe elastic and plastic properties of pure polycrystalline metals: Brit-tle materials have high k values, and ductile materials have low ones[31]. In principle, covalent materials with high hardness are obviouslybrittle with a larger Pugh modulus ratio. Chen correlated kwith hard-ness, and a better reliability is reached because k responds to bothmaterial elasticity and plasticity. The Vickers hardness can be calcu-lated as HV=2(k2G)0.585−3, which can be used to predict thehardness of a variety of materials [30]. This formula fairly well agreeswith experimental data. However, both bulk and shear moduli aremacroscopic concepts, and the origin of hardness is still notcompletely understood. On the other hand, a direct quantification ofhardness with microscopic parameters may reveal the fundamentalfactors controlling materials hardness, and provide valuable basesfor pursuing new superhard materials.

Some models with different physical considerations have beenrecently proposed to evaluate the intrinsic hardness of ideal crystalswith microscopic parameters [32–37]. These models can provide rea-sonable results based on crystal structures or parameters fromfirst principles calculations. Considering that the input parametersfor hardness evaluation are either directly obtained from the crystalstructure or deduced from the constituent elements, these modelsare called “microscopic” models. These microscopic models enablethe hardness prediction for covalent, polar covalent, and even ioniccrystals based on crystal structures, thus greatly aid the design ofnew superhard materials. However, a satisfying and general descrip-tion of hardness for covalent crystals, ionic crystals, and metals stilleludes materials scientists due to inherent complexities [8,38,39].

The outline of the current review is as follow. First, microscopicmodels for hardness quantification are presented with a brief discus-sion of the key factors governing the hardness of materials. Sub-sequently, current developments in superhard materials researchare discussed. The paper is concluded with a brief perspective.

2. Microscopic models for hardness prediction

Hardness quantifies the crystal resistance to deformation. This re-sistance is related to the bonding types of chemical bonds in crystals.In simple metals, the bonding is delocalized. The deformation resis-tance depends not only on the dislocation density created by a rigidindentation, but also on the previously stored dislocation density[5]. Usually, the stored dislocation density in metals is sufficientlyhigh to dominate hardness value. In this case, the measured hardnessis extrinsic for metals. In covalent and polar covalent crystals, thebonding is localized in electron pairs; consequently, the hardness isintrinsic and entirely depends on the resistance of the chemicalbonds in the crystal within indentation area. For simplicity, the pre-sent discussion is limited to the intrinsic hardness of single crystals.Several strategies for establishing the microscopic theory of hardnessare presented, and the main points are analyzed. For hardness predic-tion, these microscopic hardness models can be applied to covalentand polar covalent crystals, and in some cases, to ionic crystals.

2.1. Bond resistance model

When an indenter is forced into the surface of a single crystal, asshown in Fig. 1a, the chemical bonds below the indenter withstandcompression, and the bonds around the indenter withstand bendingor even stretching. Based on this simple physical picture, Tian et al.

Fig. 1. Schematic diagrams of a) the chemical bond breaking with the indenter pushing into the surface, b) electrons excitation accompanying chemical bond breaking, and c) thedistribution of valence electrons shifting away from the center from pure to polar covalent bond.

95Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

have proposed an intuitive and important assumption that the hard-ness for covalent and polar covalent crystals is equivalent to the sumof the resistance of each bond to the indenter per unit area [34]. Thekey hypothesis is to correlate the plastic deformation (associatedwith the creation and motion of dislocations) with the breaking ofelectron-pair bonds in crystals. Hardness then measures the com-bined resistance of chemical bonds to indentation. More bonds in aregion of the surface correspond to harder crystals. A scheme to linkthe Vickers hardness for a broad class of covalent and polar covalentcrystals to their microscopic properties has been suggested. In cova-lent crystals, energetically breaking an electron-pair bond meanstwo electrons excited from the valence band to the conductionband. The activation energy required for a plastic glide is twice theband gap Eg [40], as schematically shown in Fig. 1b. The resistanceforce of a bond can be evaluated with the corresponding Eg. The hard-ness of pure covalent crystals should have the following form:

H Gpað Þ ¼ ANaEg ; ð1Þ

where A is the proportional constant, and Na is the covalent bondnumber per unit area which can be evaluated from the valence elec-tron density Ne as:

Na ¼ ∑iniZi=2V

� �2=3¼ Ne=2ð Þ2=3; ð2Þ

where ni is the number of the ith atom in the unit cell, Zi is the valenceelectron number of the ith atom attributing to the covalent bond, andV is the volume of the unit cell.

For polar covalent crystals, the valence electrons are preferentiallydistributed to the anion side, which weakens the binding of twoatoms, as demonstrated in Fig. 1c. An ionic component needs to becounted for hardness calculation in addition to the covalent compo-nent. Eg for a binary polar covalent ABm crystal can be separatedinto a covalent homopolar gap Eh and an ionic heteropolar gap C sug-gested by Phillips [41]:

E2g ¼ E2h þ C2: ð3Þ

The homopolar component Eh determines the activation energiesof a dislocation glide in polar covalent crystals [42], and can beestimated in electronvolts with the empirical expression Eh=39.74d−2.5, where d is the bond length in angstroms [41]. The ioniccomponent results in a loss of covalent bond charge and is accountedfor by introducing a correction factor, exp(−afi), to Eq. (1). This cor-rection factor describes the screening effect for each bond, where α isa constant and fi=1−Eh

2/Eg2 is the ionicity of the chemical bond in acrystal scaled by Phillips [41]. The constants A and α are determinedby fitting the hardness expression to a standard set of materialswith known Vickers hardness to be 14 and 1.191, respectively [34].

An equation relating the Vickers hardness to fi, Ne, and d is thenobtained:

HV GPað Þ ¼ 556Nae

−1:191f i

d2:5¼ 350

N2=3e e−1:191f i

d2:5: ð4Þ

The average hardness for a multicomponent system is assumed tobe the geometrical mean of the hardness of different types of covalentbonds in the system via:

HV ¼ Πμ

HμV

� �nμ� �1=∑nμ

; ð5Þ

where HVμ=350(NV

μ)2/3e−1.191fi

μ

(dμ)−2.5 is the hardness of a binarycompound composed of μ bond, and nμ is the number of μ-typebonds in the unit cell. Readers can refer to the original paper forother details [34].

Except for the Phillips ionicity fi, the calculation parameters canall be deduced from a first-principles calculation, which makes thismicroscopic model a powerful tool for hardness estimation fromdesigned crystal structures, and save a great deal of experimentalefforts. Authors from the same group have sequentially developedthis model by defining a new ionicity scale based on the first-principles calculation and generalizing to metallic systems [32,33].For a specific crystal structure or cluster containing the same type ofcoordinates, an ionicity scale, fh, is defined for a bond based on theMulliken's bond overlap population as:

f h ¼ 1−e− Pc−Pj j=P; ð6Þ

where P is the overlap population of a bond in a calculated crystal,and Pc is the overlap population of a bond in a pure covalent crystalwith an identical structure as the calculated one. A power-law fit offi as a function of fh, fi= fh

m, yields m=0.735. All the inputs for hard-ness can then be obtained through the first-principles calculations.

This model was further developed by including a small metalliccomponent of chemical bond and considering the orbital form of s-por s-p-d [32]. To account for the metallicity effect on hardness, a factorof metallicity fm=nm/ne has been introduced, where nm=kBTDF is thenumbers of electrons that can be excited at ambient temperature, andne is the total number of valence electrons in the unit cell. At ambienttemperature, kBT≈0.026eV and DF is the density of electronic statesat the Fermi level which can be acquired via electronic structurecalculations. Similar with the ionicity contribution to hardness [34],the screening effect of the metallic component may be phenome-nologically described by introducing another correction factor ofexp(−βfmn ), where β and n are constants. The contributions of dvalence electrons in TM compounds to hardness have also beencounted. The bond strength of an s-p-d hybridized chemical bond isgreater than that of an s-p hybridized chemical bond [43]. The

96 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

intrinsic influence of the d valence electrons should result in a pro-portional coefficient A' for the s-p-d hybridized crystals differentfrom that for the s-p hybridized crystals in Eq. (1). As a result, thehardness equation for metallic and s-p-d hybridized crystals (bonds)is written as:

HV GPað Þ ¼ 1051N2=3

e e−1:191f i−32:2f 0:55m

d2:5: ð7Þ

With fi= fh0.735=(1−e− |Pc−P|/P)0.735, all of the parameters in

Eqs. (4) and (7) can now be determined from ab initio calculations.As suggested by Eq. (7), the metallic component of the bond (fm)has a stronger negative effect on hardness than the ionic component(fi). The d valence electrons play an important role in increasing hard-ness by enhancing the directionality and orbital strength of the chem-ical bond. It should be noticed that, for IB and IIB metals with a fullyfilled d orbital, the prefactor for hardness calculation is 350. Theextension of the formula to multicomponent systems is similar withEq. (5), with attentions on the metallicity term and prefactor.

In recent years, Eqs. (4) and (7) as well as its generalization to amulticomponent system, Eq. (5), have been extensively used for theprediction of hardness for a large class of proposed superhard struc-tures from first-principles calculations [44–55]. This method is appli-cable to polar covalent crystals, oxides with some contributions ofionic bonds and ionic crystals, as well as for some multi-componentcrystals with mixed types of interatomic bonds. The overall agree-ment with experiments is highly satisfactory. This consistency indi-cates that hardness can be defined microscopically as the combinedresistance of chemical bonds in a crystal to indentation.

2.2. Bond strength model

An alternate scheme for hardness prediction has been proposed bySimunek and Vackar [37]. Instead of relating the resistance to thebond energy gap, the resistance is assumed to be proportional tothe bond strength Sij between atoms i and j as:

Sij ¼ffiffiffiffiffiffiffiffieiej

p= dijnij

; ð8Þ

where ei=Zi/Ri is a reference energy, Zi is the valence electron num-ber of atom i, and nij is the number of bonds between atom i and itsneighboring atoms j at the nearest neighbor distance dij. The radiusRi for each atom in a crystal is determined such that a sphere aroundan atomwith radius Ri contains exactly Zi valence electrons. The hard-ness of the ideal single crystal is proportional to the bond strength Sijand the bond number in the unit cell. For a simple crystal with oneelement, hardness is expressed as:

H ¼ C=Ωð Þ ffiffiffiffiffiffiffiffieiej

p= diiniið Þ: ð9Þ

For a binary compound with two different atoms a and b, hardnessis expressed as:

H ¼ C=Ωð Þ ffiffiffiffiffiffiffiffiffieaeb

p= dabnabð Þe−σ f e ; ð10Þ

where the exponential factor phenomenologically describes the dif-ference between ea and eb. For a multicomponent system, hardnesscan be calculated as:

H ¼ C=Ωð Þn ∏n

i;j¼1NijSij

!1=n

e−σ f e ;

f e ¼ 1− k ∏k

i¼1ei

� �1=k

=Xki¼1

ei

" #2;

ð11Þ

where Nij is the multiplicity of the binary system ij, and k is the num-ber of different atoms in the system.

To address some issues rooted in the above model (e.g., differentcoordination number for constituent atoms) [56], and “to estimatehardness of crystals on a pocket calculator”, a generalizationhas been proposed [36,57]. Bond strength is redefined assij ¼ ffiffiffiffiffiffiffiffi

eiejp

= ninjdij� �

, where ni and nj are the coordination number ofatom i and j, respectively. To use sij in calculations, a number bij forcounting individual bonds of the ij-type in the unit cell is introduced.Subsequently, all that remain to be performed in Eq. (11) is thereplacement of Sij and Nij with sij and bij, respectively. The radius Ri,which is determined from the first-principles calculation in the orig-inal formula, is also replaced by the atomic radius. Constants C andσ are chosen to be 1450 and 2.8 respectively for the hardness calcula-tion. Consequently, no constant determined by ab initio methods isrequired for the estimation of hardness [36]. Given that constants Cand σ are determined from experimental data, this model is also asemi-empirical one. This bond strength model works well for thehardness estimation of covalent, polar covalent, and ionic crystals.

2.3. Electronegativity model

Based on our assumption [34], the third empirical model is pro-posed recently to predict the hardness of single and multiband mate-rials in terms of electronegativity (EN) [35,58]. The EN of an elementis defined as:

χj ¼ 0:481nj=Rj; ð12Þ

where nj is the number of valence electrons of atom j, Rj is its crystal-line covalent radius expressed in angstrom, and 0.481 is a dimension-less coefficient [59]. For any covalent bond a–b with coordinationnumbers CNa and CNb of atoms a and b, respectively, it can beassumed that this bond is composed of (1/CNa) a atom and (1/CNb)b atom. A bond EN can be defined as an average of the electron-holding energy of two atoms distributed to the a–b bond as:

Xab ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiχa

CNa

χb

CNb

r: ð13Þ

Subsequently, bond hardness can be expressed as Hab=Xab/(Ω/N),where Ω is the volume of the unit cell, and N is the number ofcovalent bonds per unit cell. After defining an ionicity indicator,f i ¼ χa−χbj j=4 ffiffiffiffiffiffiffiffiffiffiffiffi

χaχbp

, Xue et al. proposed a hardness expressionfor covalent and polar covalent crystals as [35]:

Hk GPað Þ ¼ 423:8NvXabe−2:7f i−3:4: ð14Þ

For a crystal with n types of bonds, hardness can be expressed as:

Hk GPað Þ ¼ 423:8Ω

n ∏n

a;b¼1NabXabe

−2:7f i abð Þ

!1=n

−3:4: ð15Þ

2.4. Some points about microscopic hardness models

a) The above three microscopic hardness models differ from oneanother in physical treatment and mathematical formula. Neverthe-less, they are all based on the assumption that hardness is equivalentto the sum of the resistance of each chemical bond to the indenterper unit area. The differences are that the deformation resistance isexpressed by the energy gap for the bond resistance model, by thebond strength consisting of the reference energy for the bondstrength model, and by the bond EN consisting of the element ENfor the EN model, respectively. A comparison of these models shedslight on the factors that should be considered for pursuing superhard

97Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

materials. The factors include short and strong chemical bonds, highvalence electron density or high bond density, and strongly direc-tional bonds (as suggested by the larger prefactor for d orbitals). Ion-icity is adverse for hardness, as clearly demonstrated by theexponential factor, and metallicity is even worse.

b) The bond resistance, bond strength, and EN models work wellfor pure covalent and polar covalent crystals. As demonstrated inRef. [37], in addition, the bond strength model can also estimate hard-ness of ionic crystals. Still, the definition of ionicity indicator is some-what arbitrary, more mathematical than physical in the latter twomodels. In the EN model, the intercept term (−3.4) aids in more ac-curately reproducing the experimental values at the lower hardnessend. However, the physics behind this term is unclear, and in somecases, it gives an unrealistic negative value of hardness, i.e., for BaO.

Every parameter in the bond resistance model is well defined.With special attention paid to the density of valence electrons, Ne,this model can be easily generalized to ionic crystals and TM com-pounds. In Fig. 2, the typical charge densities of the chemical bondsin diamond(110), Al2O3(100), MgO(100), and NaCl(100) planes aredemonstrated. A variation in valence electron distribution is clearlyobserved from diamond to NaCl: From a symmetric accumulation inthe middle of two neighboring C atoms (Fig. 2a for diamond), to anasymmetric gathering preferential to O side (Fig. 2b for Al2O3), andto a spherical distribution around O and Cl atoms (Fig. 2c for MgOand Fig. 2d for NaCl). In the last two cases, the valence electronsfrom anions show little evidence of contributing to covalent bondformation. In the most ionic NaCl, Na atom may even lose part of itsvalence electron to Cl, leading to a very small covalent component.

A scheme for counting the valence electron density of ionic AxBy(B=O, F, and Cl) crystals is herein proposed. The valence electronsfrom atom B are omitted in density counting as long as χB

−χA>1.7 in Pauling's scale, which indicates the formation of a strongionic bond. In addition, for the most ionic crystals (fluoride and chlo-ride with alkali metals), the charge transferred from the alkali metalatom to F or Cl atom, which can be determined from the first princi-ples calculations, is also deducted from the counting of valence elec-tron density since the transferred electrons are now seized by theanions. With these considerations, the hardness values estimated

Fig. 2. Typical charge densities of chemical bonds for a) diamond (110), b) Al2O3 (100),c) MgO (100), and d) NaCl (100).

from Eq. (4) for MgO, NaCl and KCl are 4.5 GPa, 0.4 GPa and0.18 GPa, respectively. The values well agree with the experimentalones [60].

c) A development of the bond strength model has been proposedrecently to describe the anisotropy of hardness with respect to crystalorientation [61]. The earlier model is updated by mathematicallyweighing the contributions of each bond to the total hardness basedon its direction, with the bonds perpendicular to the applied forcebeing given the most weight.

d) The EN model has been recently developed by Lyakhov andOganov by considering the dependence of the EN on the environmentand deviations of actual bond lengths from the sum of covalent radii[62]. The atomic EN and effective coordination numbers are rede-fined. Only the bond density Nv needs to be calculated in the ENmodel, which is convenient for programming. Therefore, the updatedmodel combined with the evolution crystal structure prediction algo-rithm, USPEX [62,63], provides a way of systematic discovery of newhard and superhard materials.

e) The calculated hardness values for selected crystals from bothmicroscopic and macroscopic models are compared with the experi-mental values in Table 1 and Fig. 3. The mutual agreement betweenthe two different strategies, as well as between the experimentaland calculated hardness values is satisfactory. The first-principles cal-culations of elastic constants are now routine, and the conversions toshear or bulk moduli for all crystal classes are well known [64,65].Hence, these two strategies are complementary and can be used toestimate the hardness for new crystals. In the inset of Fig. 3, the hard-ness values for Al2O3, MgO, some ionic crystals, as well as TM mono-carbides and -nitrides are emphasized. Obviously, the bond resistancemodel reproduces experimental values more accurately by consider-ing the contributions of the metallic component (weakening) and dorbitals (strengthening). The macroscopic model, however, fails to re-produce correct hardness values for ionic crystals and MgO, suggest-ing the deficiency of describing hardness with the elastic propertiesof materials.

On a closing note of this session, the intercept term (−3) in themodulus strategy is observed as unnecessary due to the lack of phys-ical basis. Similar with Xue's model, negative hardness values arepredicted for some ionic crystals, such as KI and RbCl, using Chen'sformula. This promotes a continuous exploration with the currentmodulus model. Chen's dataset is fitted and a revised formula isgiven without the intercept term,

HV ¼ 0:92k1:137G0:708: ð16Þ

The comparison with the original formula is emphasized in Fig. 4using a logarithmic scale. Both formulas well agree with the experi-mental values when hardness is larger than 5 GPa, but yield overesti-mated hardness values at low hardness side. However, the possibilityof unrealistic negative hardness is eliminated with the new formula.

3. Novel superhard crystals

The above models of hardness enable the possibility of the sys-tematic prediction and design of new superhard crystals. There areseveral criteria that can guide the search for superhard crystals,namely three-dimensional network structure, strong chemical bond,short bond length, and high bond density or high charge density. dvalence electrons can effectively enhance the proportional coefficientA in Eq. (1). However, in TM compounds, this enhancement effectfrom d valence electrons is largely offset by the large bond length.

High hardness occurs in the compounds of light elements,where extremely short and strong bonds are formed[9,12–14,44,45,47,66–69]. Diamond, a dense solid with strong andfully covalent bonds, satisfies all the criteria. Boron-rich materials(BCx) as superhard materials are particularly appealing [70]. They

Table 1Hardness values from experimental measurements and different hardness models forselected crystals.

Crystal HExp (GPa) HTian (GPa) HSimunek (GPa) HXue (GPa) HChen (GPa)

C 96a 93.6 95.4b 90e 94.6f

Si 12a 13.6 11.3b 14e 11.2f

Ge 8.8b 11.7 9.7b 11.4e 10.4f

SiC 31b 30.3 31.1b 27.8e 33.8f

BN 63a 64.5 63.2b 47.7e 65.3f

BP 33a 31.2 26b 24.9e 29.3f

BAs 19b 26 19.9b 21.1e –

AlN 18a 21.7 17.6b 14.5e 16.8f

AlP 9.4a 9.6 7.9b 7.4e 7.2f

AlAs 5.0a 8.5 6.8b 6.3e 6.6f

AlSb 4.0a 4 4.9b 4.9e 4.4f

GaN 15.1a 18.1 18.5b 13.5e 13.9f

GaP 9.5a 8.9 8.7b 8e 9.9f

GaAs 7.5a 8 7.4b 7.1e 7.8f

GaSb 4.5a 6 5.6b 4.5e 5.8f

InN 9a 10.4 8.2b 7.4e 7.4f

InP 5.4a 6 5.1b 3.9e 3.7f

InAs 3.8a 3.8 5.7b 4.5e 3.3f

InSb 2.2a 4.3 3.6b 2.2e 2.4f

ZnS 1.8b 6.8 2.7b 2.4e 2.4f

ZnSe 1.4b 5.5 2.6b 1.8e 2.7f

ZnTe 1b 4.1 2.3b 0.9e 2.1f

TiC 32c 34 18.8b 23.9e 27f

TiN 20.6c 21.6 18.7b 23.8h 23.3f

ZrC 25c 21 10.7g 15.7h 27.5f

ZrN 15.8c 16.7 10.8g 15.9h –

HfC 26.1c 26.8 10.9g 15.6h –

HfN 16.3c 18 10.6g 15.2h 19.2f

VC 27.2c 23 25.2g 17.5h 26.2f

VN 15.2c 14.9 26.5g 16.5h –

NbC 17.6c 16.1 18.3b 12.8h 15.4f

NbN 13.7c 13.6 19.5b 12h 14.7f

TaC 24.5c 26 19.9g 14.7h –

TaN 22c 20 21.2g 14.3h –

CrN 11c 11 36.6g 19.2h –

WC 30c 31 21.5b 20.6e 31.3f

Re2C 17.5j 19.7j 11.5g 16.2h 26.4i

Al2O3 20c 18.8 13.5g 18.4h 20.3i

MgO 3.9d 4.5 4.4g 5.4h 24.8i

LiF 1d 0.8 2.2g – 8.5i

NaF 0.6d 0.85 1g – 5.7i

NaCl 0.2d 0.4 0.4b – 2.4i

KCl 0.13d 0.18 0.2b – 2.3i

KBr 0.1d 0.23 0.2g – 0.1i

a Reference [34].b Reference [37].c Reference [32].d Reference [60].e Reference [58].f Reference [30].g Calculated by authors using method [36].h Caculated using [35].i Calculated with [30].j Referenece [52].

Fig. 3. Comparison of calculated hardness from different models with experimentalvalues. See text for the details of the inset.

Fig. 4. Comparison of Chen's hardness formula and the refitting formula with theexperimental values.

98 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

show excellent physical and chemical properties, including low massdensity, very high hardness, high mechanical strength, high thermalconductivity, excellent wear-resistance, and high chemical inertness.Superconductivity has been recently found in boron-doped diamondsynthesized at high pressure and temperature [71]. Experimentaland theoretical studies have indicated an effective way to increasethe superconducting Tc by increasing dopant concentrations [45,47].Technically important superhard and superconductive materials cannow be pursued from boron-rich systems. Another interesting familyof carbon allotropes including 3D carbon nanotube polymers hasrecently attracted a lot of interest. Given the unique configurationsof these 3D polymers, they have distinctive electronic properties,high Young's moduli, high tensile strength, ultrahigh hardness, goodductility, and low density. Hence, these polymers may be potentiallyapplied to a variety of needs [72]. Some TMs (e.g., Os, Re, W, and

Ta) have a high number of valence electrons and can form very strongbonds, although the bonds formed are not very directional. Theircompounds with light elements, namely B, C, and N, should be care-fully examined [16,17,22,29,38,73–75]. Examples of these potentialsuperhard materials and their recent progresses are now presented.

3.1. B-C crystals

Most of B-C binary systems exhibit high resistance to oxidationand reaction with ferrous metals, compared with the carbon-basedmaterials [76–78]. Boron carbide B4C is a hard crystal that can be pro-duced at ambient pressure [79,80], whereas B-doped diamond showsa superconducting transition temperature of 4 K [71]. It is of great in-terest in diamond-like BCx systems to pursue superior superhardcrystals that are not only thermally and chemically more stable thandiamond, but also possess interesting electrical properties [78,81,82].Experimentally, turbostratic graphite-like g-BCx compounds are rou-tinely used as precursors for the synthesis of novel diamond-likephases of the B-C system. Stoichiometric diamond-like BC5 (d-BC5)and BC3 (d-BC3) have been recently synthesized under high pressureand high temperature [13,70,83].

At 24 GPa and 2200 K, d-BC5 is synthesized from g-BC5, exhibitingextremely high Vickers hardness (71 GPa), unusually high fracturetoughness (9.5 MPam0.5) for superhard materials, and high thermalstability (1900 K, which is about 500 K more thermally stablethan pure diamond) [13]. Theoretically, Calandra and Mauri have

99Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

demonstrated that BC5 is superconducting with a critical temperatureof the same order as that of MgB2 [69]. These combined mechanicaland electrical properties make d-BCx systems appealing materialsfor electronics application under extreme conditions. Theoreticalstudies have been performed to address the crystal structures andphysical properties of d-BC5 phase. Liang et al. studied the mechanicaland electronic properties of diamond-like BCx phases and pointed outa higher energy barrier of d-BC5 (0.057 eV/atom higher than that ofdiamond), implying that d-BC5 is about 500 K more kinetically stablethan diamond [84]. A decreasing trend in mechanical properties withincreasing boron content in d-BCx systems was revealed by analyzingthe electronic structures [84]. Yao et al. predicted a thermodynami-cally stable metallic I4m2 phase with a superconducting critical tem-perature of 47 K, and another possible candidate structure with P1symmetry whose X-ray diffraction (XRD) pattern fits well with theexperimental one [85]. Jiang et al. argued that d-BC5 may bedisordered with boron atoms randomly distributed in the diamondlattice using first-principles density functional calculations [86].Li et al. proposed two Pmma structures where the synthesized BC5

adopts the diamond-[100] structure with an atomic packing ofthe form ABCABC… along the [100] crystallographic direction ofdiamond. The simulated XRD patterns, Raman modes, and Vickershardness remarkable agree with the experimental data [47].

An earlier attempt to synthesize d-BC3 phase was unsuccessfulbecause g-BC3 decomposed into a composite bulk of B4C and boron-doped diamond, although with a very high hardness of 88 GPa [87].Until recently, a d-BC3 phase produced from g-BC3 at 2000 K undervery high pressure (40–50 GPa) was confirmed with combined XRD,Raman, TEM,and EELS measurements [70,83]. The authors arguedthat boron and carbon atoms are randomly distributed in the eightpositions of the diamond-like structure given that four sharp peaksin XRD patterns can be indexed as the (111), (220), (311), and(400) diffractions, respectively, of the diamond structure with theright respective intensities [83]. The properties of d-BC3 have beenstudied with proposed structures, indicating conductive superhardcrystals [45,81,82]. Liu et al. explored the crystal structures witha particle swarm optimization (PSO) algorithm combined with first-principles structural optimizations [45]. Three metallic configura-tions, namely Pmma−a, Pmma−b and P4m2 phases were uncov-ered. With the bond resistance model, the Vickers hardness for allthree phases is larger than 60 GPa, indicating the superhard natureof these polymorphs, which should be tested experimentally. In addi-tion, all phases have a superconductive transition at low temperature.We simulated the XRD patterns for these structures and found theagreement with the experimental data is reasonably good. Hence,an unambiguous determination of the real crystal structure needsmore investigations.

Other d-BCx systems are either synthesized or predicted, such asBC, BC2, BC4, and BC7 [44,68,88–91]. Most are predicted to be super-hard and potential superconductive crystals, and a systematical in-vestigation of BCx systems with variable x is of fundamental interest[84,92,93]. Xu et al. investigated a tetragonal BC2 (t-BC2) phase orig-inating from the cubic diamond structure by first-principles calcula-tions [88]. The structural stability of BC2 has been confirmed by thecalculated elastic constants and phonon frequencies. The electrondeficiency introduced by the B atom is distributed to each atom inthe system, leading to a 3D conductivity. The calculated theoreticalVickers hardness of t-BC2 is 56.0 GPa, indicating that it is a potentialconductive superhard crystal. In addition, the calculated B/G ratio oft-BC2 is larger than that of diamond, suggesting that t-BC2 is moreductile than diamond. Xu et al. also predicted a d-BC7 with P4m2symmetry possessing a Vickers hardness of 78 GPa and Tc of 11.4 K[68]. Most recently, Liu et al. considered a series of proposed structur-al configurations for d-BC7, and concluded that all the simulatedstructures are metallic due to the introduction of one-electron defi-cient B atoms into the system [44]. The calculated Vickers hardness

values for different structures are from 65 to 80 GPa, indicating thatd-BC7 is a superhard crystal. d-BCx phases with higher boron contentmay exhibit superior electrical conductivity and improved chemicalstability. These appealing properties of d-BCx systems call forfollow-up experimental syntheses. Selected d-BCx systems with spe-cific crystal structures are presented in Fig. 5. Predicted properties,such as symmetry, lattice parameter, hardness, and Tc, are summa-rized in Table 2.

In our previous study [32], a small metallic component has beenfound to have a strong negative effect on hardness. Electrons deloca-lized to contribute to the conduction should be excluded from thehardness calculation, and the correction from conducting electronsis necessary to account for the experimental hardness. In d-BCxsystems, the major carriers are holes and the valence electrons aremainly localized to form covalent bonds. As a result, the metalliccorrection in the hardness calculation need not be considered.

3.2. Carbon allotropes

Due to its unique ability to form sp-, sp2-, and sp3-hybridizedbonds, carbon can adopt a wide range of structures, including dia-mond, lonsdaleite, graphite, fullerenes, nanotubes, graphene, andamorphous carbon. Fullerenes, nanotubes, and graphene are current-ly the focus of research in nanotechnology, electronics, optics, andmany other fields of materials science and technology. Nevertheless,there exists a long-term endeavor for searching superhard phasesamong carbon allotropes in addition to diamond and lonsdaleite[72,94–101].

Recently, a carbon allotrope has been obtained by cold compres-sing graphite with pressure over 17 GPa, where the original sp2

bonds in graphite are transformed into bonds of sp2 and sp3 mixture[101]. This high-pressure phase has hardness at least comparable todiamond, and arouses a continuous debate about its structure. Mono-cline polymorph M-carbon [62,96], body centered tetragonal poly-morph bct-carbon [98,99], and orthorhombic polymorph W-carbon[102] structures are suggested, as demonstrated in Fig. 6. All thesephases are superhard with a hardness of about 90 GPa [100]. Theactual high-pressure phase is likely a mixture of these metastablecarbon phases. Another quenchable superhard carbon phase was syn-thesized by cold compression of carbon nanotubes at 75 GPa with ansp3-rich bonding configuration [94]. A superhard (HV=95 GPa) car-bon allotrope of C-centered orthorhombic C8 (Cco-C8), which canaccount for the experimental data for this superhard carbon phase,has been proposed [103]. This theoretical work sheds light on a newstrategy to design and synthesize novel metastable carbon allotropesby directly compressing carbon nanotube bundles or other carbonstructures. Metastable phases of other materials (e.g. BN) with higherenergy and unique physical properties may also be produced usingsimilar compression techniques [103].

Individual C60 molecules are estimated to have an extremely highelastic modulus of 800–900 GPa [104,105]. However, C60 crystalshows a very soft lattice and a very small elastic modulus due tovery weak intermolecular interactions. By the formation of strongsp3-like bonds between fullerenes, the rigidity, stability, and hardnessof the original phase would be greatly enhanced. Experimentally,distinct carbon phases can be prepared by high-pressure high-temperature treatment of C60 with subsequent quenching to theambient conditions [106]. A unique and promising combination ofsufficiently high hardness and high plasticity has been found for the3D-polymerized C60-based phases [106–108]. The hardness of disor-dered and composite phases attains the values close to that of dia-mond, whereas the fracture toughness coefficient may be evenhigher. In particular, the conductivity of these polymerized 3D C60crystals may vary considerably from metallic to semiconductingconductivity depending on the relative concentration of sp3/sp2

bonds, topology, system structure, etc. [95,108,109].

Fig. 5. Crystal structures of a) I41/amd BC2, b) P4m2 BC3, c) Pmma−a BC3, d)Pmma−b BC3, e) I4m2 BC5, f) Pmma−1 BC5, g) Pmma−2 BC5, h) P43m BC7, i) P3m1 BC7, j)Pmm2 BC7,k) P4m2 BC7, and l) R3m BC7. Boron atoms are shown in orange (lighter gray in print).

100 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

Many theoretical and experimental attempts have been made onassociates of carbon nanotubes (CNTs) to search for possible super-hard materials, especially on the polymerized (covalently bonded)carbon nanotubes [94,95]. A series of ordered 3D structures of parallelCNTs have been proposed [110–112]. As suggested by a recent theo-retical work about CNT polymers [72], some of the proposed struc-tures, which are demonstrated in Fig. 7, have lower ground-stateenergies than the corresponding nanotube parents and may be syn-thesized through the treatment of high-energy CNT. These 3D

Table 2Structure symmetry, lattice parameter, calculated hardness, and superconductingcritical temperature for proposed BCx crystals.

Crystal Symmetry Lattice parameter (Å) HV (GPa) Tc (K)

a b c

BC2a I41/amd 2.520 2.52 11.919 56 –

BC3b P4m2 2.5015 2.5105 3.915 65.8 13.4–19.5

Pmma−a 2.5132 2.5202 7.7878 61.9 16.6–23.4Pmma−b 2.4834 2.5311 7.8914 64.8 4.9–8.8

BC5 I4m2c 2.525 2.525 11.323 80 47Pmma−1d 2.5005 2.5238 11.4789 74 8.4–11.3Pmma−2d 2.5172 2.5265 11.3352 70 18.1–22.6

BC7 P4m2e 2.5141 2.5141 7.4663 78 8.4–11.4P4m2f 2.5158 2.5158 7.4497 75.2 –

P3m1f 2.5356 2.5356 8.5188 65.3 –

P43mf 3.6205 3.6205 3.6205 77.6 –

Pmm2f 2.5132 5.124 3.687 80.7 –

R3mf 2.5876 2.5876 25.525 65.4 –

a Reference [88].b Reference [45].c Reference [85].d Reference [47].e Reference [68].f Reference [44].

nanotube polymers show excellent mechanical properties, includinghigh Young's moduli, high tensile strength, ultrahigh hardness,good ductility, and low density. They also show interesting electricproperties tuning from semiconducting to linear, planar, or 3D con-ducting depending on the specific crystal structures. The combinationof these properties implies a great application prospect for CNT poly-mers, such as optical or electronic nanodevices under extremeconditions.

3.3. Transition metal compounds

Compounds of TMs with light elements, B, C, N, and O are cur-rently the subjects of the intensive research activities searchingfor novel superhard and ultra-incompressible materials. Several im-portant design parameters for selecting possible superhard TMcompounds have been pointed out, including a high electron con-centration (EC defined as electrons per atomic volume) and thepresence of directional covalent bonding [38,113]. The introductionof light and covalent-bond-forming elements into TM lattices canhave profound influences on their chemical, mechanical, and elec-tronic properties.

Several TM borides have been synthesized and claimed to besuperhard, such as ReB2 [17], OsB2 [19,114], and WB4 [29,73]. Thestructural models for these phases are shown in Fig. 8. The structureof diborides of heavy TMs (ReB2 and OsB2) includes two dimensional(2D) lattices of boron atoms. Enhanced mechanical properties areattributed to the high valence electron density as well as the B-Band TM-B covalent bonds. In tetraborides of heavy TMs (WB4),boron atoms form hexagonal 2D lattice with additional covalent Bdimers located perpendicularly in between 2 boron layers, leadingto a 3D network. The enhanced mechanical properties are attributedto this quasi-3D B lattice.

Fig. 6. Crystal structures of a) M-Carbon, b) bct-Carbon, and c) W-Carbon.

101Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

Gu et al. studied the properties of a series of TM borides [16]. Un-fortunately, none of the synthesized TM borides shows a comparable(asymptotic) hardness larger than 40 GPa. Their results clearlydemonstrate that, while the bulk modulus (compressibility) can beunderstood with EC of crystals, the difference in hardness can be un-derstood by means of a chemical bonding analysis. The most incom-pressible OsB, for example, has the smallest hardness, which can beaccounted for by the high EC and absence of B-B bonds in the crystal.In contrast, WB4 possesses a relatively small bulk modulus dueto small EC, and the hardness is greatly enhanced because of the for-mation of a covalently bonded 3D framework of boron atoms. Forcompounds with intermediate B content, a good compromise ofultra-incompressibility and high hardness may be reached, such as

Fig. 7. Proposed 3D CNT polymer structures. a) (4, 0) carbon, b) (3, 3) carbon, c) (5, 0) carband i) (6, 6) carbon.

WB2, ReB2, and Os0.5W0.5B2, where a high EC governs a low compress-ibility, and covalent bonding generates low plasticity as well assequentially high hardness.

Unlike TM borides, which can usually be acquired under ambientpressure using arc melting, most of TM carbides and nitrides of cur-rent (superhard) interest have to be synthesized under high pressureand high temperature [18,20,22,52,115–121]. The extreme mechani-cal properties of TM nitrides and carbides, such as high strength,low compressibility, and high hardness have attracted tremendousresearch interests, both experimentally [18,22,118,122–124] and the-oretically [53,119–121,125–133]. Superhardness has been predictedfor some of the TM nitrides and carbides, such as PtN2 [32] and RuC[121]. In Table 3, structural information and calculated hardness

on, d) (4, 4) carbon, e) (7, 0) carbon, f) (8, 0) carbon, g) (6, 0) carbon, h) (9, 0) carbon,

Fig. 8. Crystal structures of a) ReB2, b) OsB2, and 3) WB4. Boron atoms (smaller ones) are shown in orange (lighter gray in print).

102 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

from the band resistance models of some potential semiconductingsuperhard TM nitrides and carbides are summarized with the crystalstructures presented in Fig. 9 [134]. Combining with their electronicproperties, these materials can be applied to electronics devicesunder extreme conditions. A recent review of Friedrich et al. providesan excellent guide for TM borides, nitrides, and carbides [135].

3.4. Other systems

Boron allotropes, B-N [136], B-O [76], C-N [137–141], B-C-N[49,142–149], B-C-O [50], and Si-C-N [55,150] systems are appealingfor searching novel and/or superhard materials. Recently, severalsuperhard phases of these systems have been synthesized underhigh pressure high temperature conditions, such as orthorhombicBC2N [12], BC4N [11], γ-B28 [14,151], rhombohedral B13N2 [152],and B6O [15,153]. Notably, in these systems, there are some phasessuggested to be superhard, but have not been experimentally con-firmed, such as C3N4 [62,154], B2O [46], and B2CO [50].

Up to now, the considered systems all belong to the intrinsicsuperhard materials where high hardness is achieved through theirstrong chemical bonding. Superhardness can also be extrinsicallyachieved through structuring. For example, an increase of hardnessby 80% (from 77 GPa to 140 GPa) was observed for polycrystallineCVD diamond in a hybrid ultrahard polycrystalline composite materi-al [7]. In polycrystals, the dislocation activities inside the grain is

Table 3Calculated bond parameters and Vickers hardness of proposed semiconducting transition m

Crystal Structure Bond type dμ (Å) Nμ Ne

Ti3N4 Th3P4 Ti–N 2.034 48 0.653Ti–N 2.393 48 0.401

PdN2 Pyrite N–N 1.271 4 2.553Pd–N 2.215 24 0.403

HfN2 Pyrite N–N 1.500 4 1.517Hf–N 2.281 24 0.331

PtN2 Pyrite N–N 1.140 4 1.827Pt–N 2.136 24 0.438

TiN2 Pyrite N–N 1.443 4 1.694Ti–N 2.105 24 0.418

NiN2 Pyrite N–N 1.334 4 2.153Ni–N 1.998 24 0.534

PdN2 Marcasite N–N 1.265 2 2.583Pd–N 2.181 4 0.420Pd–N 2.235 8 0.390

PtN2 Marcasite N–N 1.396 2 1.855Pt–N 2.219 4 0.436Pt–N 2.148 8 0.424

FeC FeSi Fe–C 1.894 4 0.438Fe–C 1.908 12 0.428

OsC FeSi Os–C 2.028 4 0.329Os–C 2.063 12 0.313

FeC Zinc blende Fe–C 1.836 16 0.367RuC Zinc blende Ru–C 1.968 16 0.298FeC2 Rutile Fe–C 1.888 8 0.405

Fe–C 1.890 4 0.403

suppressed with decreasing grain size due to the Hall–Petch effect[155], while the resistance to the plastic deformation gets weakdue to increasing grain boundary shear. As a result, there exists a“strongest size” of 10–20 nm for grains of crystallite where the hard-ness can be increased by a factor of two for many materials [156,157].Bulk nanocrystalline diamonds were synthesized from graphiteunder HPHT conditions with an enhanced hardness as high as140 GPa [158]. Another example of this extrinsic hardness enhance-ment is demonstrated for c-BN (diamond-like) and w-BN (lonsda-leite-like) nanocomposites where the load-invariant hardness of85 GPa, approaching to the value of single-crystal diamond, hasbeen reached [159]. These nanocrystalline bulk materials usuallypossess very high fracture toughness, excellent wear resistance, andhigh thermal stability in addition to the superhardness [159], makingthem deal materials for cutting, grinding and drilling.

Vepřek et al. have further suggested that, through the formationof nc-TmN/a-Si3N4 nanocomposites with strong and shear resistantinterfaces, the limit (enhancement factor of two) imposed by thegrain boundary shear in materials composed of very small nano-crystals can be avoided, and much higher hardness enhancementcan be achieved [160]. Theoretical calculations using nonlinear finiteelement modeling have predicted hardness as high as 158 GPa inthe nc-TiN/a-Si3N4 nanocomposites [161]. However, experimentallyreported hardness in these nanocomposites is less than 115 GPa,much lower than the predicted value. This hardness decrease is

etal compounds.

P Pc fi HV, theorμ

(GPa) HV, theor (GPa)

0.39 0.43 0.181 108.1 49.30.15 0.43 0.884 22.5

0 359.7 47.60.29 0.57 0.703 33.9

0 167.9 59.90.51 0.57 0.199 50.5

0 221.8 70.90.44 0.57 0.367 58.7

0 199.0 71.90.45 0.57 0.344 60.7

0 284.2 78.60.36 0.57 0.549 63.8

0 366.4 45.50.33 0.57 0.615 40.30.24 0.57 0.807 28.7

229.7 65.80.44 0.57 0.367 590.40 0.57 0.459 50.80.48 0.75 0.538 64.7 60.30.45 0.75 0.589 58.90.60 0.75 0.329 57.7 45.30.48 0.75 0.538 41.70.50 0.75 0.504 64.70.45 0.75 0.589 42.80.47 0.57 0.297 82.4 85.90.64 0.57 0.189 93.3

Fig. 9. Typical crystal structures of transition metal nitrides and carbides. a) Th3P4-type, b) pyrite-type, c) marcasite-type, d) FeSi-type, e) zinc blende-type, and f) rutile-type.Nitrogen and carbon atoms are shown in blue and black (both black in print), respectively.

103Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

attributed to sample imperfectness, oxygen contamination, and morefundamentally, Friedel oscillations of valence charge density thatdevelop in TiN nanograins adjacent to the SiNx type interfacial layerscaused by the negative charge at the strengthened SiNx type interfa-cial layer [162,163]. It is also noted that different nc-TiN/a-Si3N4

nanocomposite coatings have elastic moduli similar with those ofTiN, whereas the hardness greatly differs [164]. Hardness estimatedwith the macroscopic hardness model yield a value of 22 GPa [30],similar with TiN, which is inconsistent with the experimental obser-vations. A microscopic explanation based on the specific atomic struc-tures is thus highly anticipated. Nevertheless, this extrinsic hardnessenhancement from nanocomposites points out a very importantpathway in the quest for superhardness [165–169].

We provide some qualitative considerations of materials harden-ing from the Hall–Petch and quantum confinement effects beforewe end this session. The hardening effect of grain boundaries can beexpressed through the Hall–Petch equation as [170]:

H ¼ H0 þ KHP=ffiffiffiffiD

p; ð17Þ

where H0 is the hardness of the bulk single crystal, KHP is the Hall–Petch hardening coefficient which is sample dependent, and D is thegrain size in nanometer. In addition, Tse et al. suggested anotherhardening effect for nanocrystals [171]. Based on the Kubo theoryand including the quantum confinement effect, the band gap Eg inEq. (1) can be updated for nanocrystals as:

Eg;nano ¼ Eg;bulk þ δp ¼ Eg;bulk þ 24=DN1=3e eVð Þ: ð18Þ

As a result, the hardness of a nanocrystal can be obtained byupdating Eq. (4) as:

HV GPað Þ ¼ ANaEg;nano ¼ 350d−2:5N2=3e þ 211D−1N1=3

e

e−1:191f i : ð19Þ

By including the Hall–Petch and quantum confinement effects, wecan estimate the hardness for a nanocrystalline bulk as:

H ¼ H0 þ KHP=ffiffiffiffiD

pþ Kqc=D; ð20Þ

where Kqc=211Ne1/3e−1.191fi is the quantum confinement hardening

coefficient. This equation describes the observed relations between

hardness and crystallite size in c-BN and w-BN nanocompositesperfectly [159].

4. Perspectives

Both Chen's macroscopic model and the microscopic models pre-sented in Section 2 serve effectively for hardness estimation. Themacroscopic model relates hardness to materials' elastic moduli,while the microscopic models evaluate hardness based on the param-eters at atomic scale. Understanding of materials hardness at theatomic scale with the microscopic models can greatly promote thequest for novel superhard materials. Microscopically, hardness can bedefined as the combined resistance of chemical bonds in a crystal toindentation. 3D bond network structures with short and strongbond, high bond density, and high valence electron density are thedetermining factors of superhardness. The novel superhard materialsmust possess parts or all of these characteristics. The three micro-scopic models discussed in this review can predict hardness reliablyfor pure covalent and polar covalent crystals, with the bond resis-tance model and bond strength model applicable to ionic crystals.While the bond strength model and the EN models have some advan-tages as mentioned in c) and d) of Section 2.4, these two modelsshould be used with cautions to metallic systems where the hardnessweakening effect from metallicity is not taken into account.

Light element compounds, such as carbon allotropes and B-C-Nsystems, are still the most appealing materials family for novel super-hard materials. The fascinating variety of possible carbon allotropesis one of the greatest topics of materials science. C–C bond strengthis known to be stronger in sp2 hybridization (graphite) than in sp3

hybridization (diamond) [172,173]. However, graphite is a fragilesemimetal with 2D conductive atomic planes while diamond is thehardest material as-known due to the formation of 3D networkwith high bond density. Atomic steric configuration, electronic prop-erty, and hardness can be tuned by varying the sp2/sp3 bond ratio.Through high pressure (hydrostatic or non-hydrostatic) treatmentsof glassy carbons, fullerenes, and nanotubes, superhard crystallineand amorphous carbon phases with interesting electronic propertiesare highly anticipated by forming mixed sp2- and sp3-hybridized C–C bonds in a 3D network. d-BCx systems, with the properties of super-hardness, enhanced thermal stability, and tunable metallicity andsuperconductivity with varying B/C ratio, are very attractive super-hard materials for multifunctional applications.

104 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

Another family of materials, TM compounds with light elements,also bears great expectation for superhardness. Although decisive ex-perimental proofs are still missing, theoretical works on this family(especially carbides and nitrides) have predicted many semiconduct-ing superhard materials, which need to be validated experimentally.Other promising strategies for fabricating superhard materialsinclude nanocomposites engineering and formation of complicatedcrystal structures with high packing index to increase bond density.

On a final note, in the quest for novel superhard materials, devel-opments in synthesis technology are equally important as the under-standing of hardness. The reproduction of some naturally formedsuperhard materials (e.g., superhard carbon polymorphs identifiedfrom the meteorites [174,175]) is not easy, more so the productionof the designed superhard materials. Combining the microscopichardness models with evolutionary searching algorithms of crystalstructures [62,176], numerous superhard materials have beendesigned theoretically. Experimental synthesis and hardness valida-tion of these designed materials are highly expected, and pose agreat challenge for materials science.

Acknowledgements

This work was supported by NSFC (Grant Nos. 51121061 and91022029) and NBRPC (Grant No. 2011CB808205).

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