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

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Int. Journal of Refractory Metals and Hard Materials

journal homepage: www.elsevier.com/locate/IJRMHM

Microscopic theory of 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 article info abstract

Article history: Hardness can be defined microscopically as the combined resistance of chemical bonds in a material to inden- Received 19 October 2011 tation. The current review presents three most popular microscopic models based on distinct scaling schemes Accepted 18 February 2012 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 Keywords: known crystals. More importantly, hardness prediction based on the designed crystal structures becomes Hardness feasible with these models. Consequently, a straightforward and powerful criterion for novel superhard Superhard materials Modeling materials is provided. The current focuses of research on potential superhard materials are also discussed. Chemical bond © 2012 Elsevier Ltd. All rights reserved. Crystal design

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 indenter, several scales such as the Vickers, Knoop, Brinell, and Rock- well scales have been developed. The most common are the Vickers In 1722, the French scientist R.A.F. de Réaumur coined the term and Knoop scales, whose indenters are a pyramidal-shaped hardness [1]. Since then, hardness has been used as one of the funda- with a square base and an elongated lozenge base, respectively. Hard- mental mechanical properties of materials. Hardness can be defined ness is measured from the ratio of the indenter force to the associated macroscopically as the ability of a material to resist being scratched indentation area. The deduced hardness usually depends on the or dented by another. Although hardness governs the technological shape of the indenter, loading force and rate, indentation size and applications of numerous materials, it is not as well defined as other time, sample orientation, as well as surface condition. For brittle ma- physical properties [2], especially at the atomic scale. Experimentally, terials or material whose hardness approaches that of , the hardness is accurately characterized by the indentation of a material indentation process is not controlled by plastic deformation alone. using a hard indenter. According to the nature and shape of the 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 ⁎ Corresponding author. Tel.: +86 335 8057047; fax: +86 335 8074545. increase with decreasing indentation size. Large strain gradients E-mail address: [email protected] (Y. Tian). inherent in small indentations produce geometrically necessary

0263-4368/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmhm.2012.02.021 94 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 dislocations, causing enhanced hardening [5]. A reliable hardness hardness characterizes a permanent plastic deformation, it is now could be determined from the asymptotic-hardness region of a well- accepted that hardness does not depend monotonously on bulk mod- controlled indentation process. Hardness values widely vary for differ- ulus or according to the simple linear correlation, ent materials, i.e., from tenths of a gigapascal or less for ionic crystals, considering that these elastic moduli correspond to reversible elastic to several gigapascals or less for pure metals, and to tens of gigapascals deformation [30]. Consequently, Chen et al. have proposed a macro- for covalent crystals. As generally accepted by the materials science scopic model of hardness by considering the Pugh's modulus ratio, community, materials with Vickers hardness larger than 40 GPa k=G/B[30]. The parameter k is closely correlated to the brittleness/ are classified as superhard materials [2,6]. Readers can also refer to ductility of materials, as well as highlights a relationship between Ref. [7] for a discussion about definition of superhard/ultrahard and the elastic and plastic properties of pure polycrystalline metals: Brit- how they evolve with times. All as-known superhard compounds are tle materials have high k values, and ductile materials have low ones covalent and polar covalent crystals. [31]. In principle, covalent materials with high hardness are obviously Over the past several decades, a large number of studies have brittle with a larger Pugh modulus ratio. Chen correlated k with hard- been devoted to novel superhard materials for both practical and ness, and a better reliability is reached because k responds to both scientific purposes. One is to synthesize robust materials with desir- material elasticity and plasticity. The Vickers hardness can be calcu- 2 0.585 able properties for modern technologies, and the other is to reveal lated as HV =2(k G) −3, which can be used to predict the the controlling factors that determine the hardness of materials at hardness of a variety of materials [30]. This formula fairly well agrees the microscopic level. Superhard materials are of great importance with experimental data. However, both bulk and shear moduli are in various industry areas, such as wear-resistant coating, , macroscopic concepts, and the origin of hardness is still not cutting and polishing tools [2,8–10]. Diamond is thus far the hardest completely understood. On the other hand, a direct quantification of known substance (HV =95 GPa) with the highest shear modulus and hardness with microscopic parameters may reveal the fundamental Young's modulus. However, the applicability of diamond is limited factors controlling materials hardness, and provide valuable bases because of its chemical reactivity with ferrous materials and nonre- for pursuing new superhard materials. sistance to oxidation. Cubic , the second hardest mate- Some models with different physical considerations have been rial with a diamond structure, can be used to cut ferrous metals. recently proposed to evaluate the intrinsic hardness of ideal crystals However, its hardness is only 66 GPa, or 30% lower than that of dia- with microscopic parameters [32–37]. These models can provide rea- mond. Hence, the syntheses of novel inertial superhard materials sonable results based on crystal structures or parameters from with hardness comparable to or even harder than diamond are high- first principles calculations. Considering that the input parameters ly anticipated. Recent searches for new superhard materials mainly for hardness evaluation are either directly obtained from the crystal focus on two classes of materials. The first class includes light- structure or deduced from the constituent elements, these models element compounds in a B-C-N-O system with short and strong are called “microscopic” models. These microscopic models enable three dimensional (3D) covalent bonds, which are crucial for super- the hardness prediction for covalent, polar covalent, and even ionic hard materials. The experimental syntheses of BCxN, BCx, γB28,B6O crystals based on crystal structures, thus greatly aid the design of [9,11–15], etc., have significantly progressed. The second class con- new superhard materials. However, a satisfying and general descrip- sists of materials formed by light elements (B, C, and N) and heavy tion of hardness for covalent crystals, ionic crystals, and metals still transition metals (TMs) that could introduce a high valence electron eludes materials scientists due to inherent complexities [8,38,39]. density into the corresponding compounds, such as ReB2, OsB2,WB4, The outline of the current review is as follow. First, microscopic PtC, IrN2, OsN2, and PtN2 [16–22]. The high valence electron density models for hardness quantification are presented with a brief discus- enables resistance to elastic and plastic deformations. Although the sion of the key factors governing the hardness of materials. Sub- assignment of superhard materials to ReB2 is under debate [21,23], sequently, current developments in superhard materials research this class of materials is still a great search pool for semiconducting are discussed. The paper is concluded with a brief perspective. superhard materials. The properties of materials depend on electrical structures. In 2. Microscopic models for hardness prediction principle, new materials with expected properties can be designed. However, this aspiration is currently very unrealistic. The quantitative Hardness quantifies the crystal resistance to deformation. This re- connections between electronic structures and macroscopic “engi- sistance is related to the bonding types of chemical bonds in crystals. neering” properties remain as one of the foremost challenges in mod- In simple metals, the bonding is delocalized. The deformation resis- ern computational materials science [24]. Alongside the difficulty of tance depends not only on the dislocation density created by a rigid synthesizing new superhard materials are two theoretical problems indentation, but also on the previously stored dislocation density that have mystified scientists for more than a century. One is the [5]. Usually, the stored dislocation density in metals is sufficiently design of a hard material based on atomic arrangements in crystal high to dominate hardness value. In this case, the measured hardness structure, which urgently need reliable models for hardness quantifi- is extrinsic for metals. In covalent and polar covalent crystals, the cations in the field of superhard materials [25]. The other is the defi- bonding is localized in electron pairs; consequently, the hardness is nition of hardness at the microscopic level, which is very fundamental intrinsic and entirely depends on the resistance of the chemical for understanding the physical origin of hardness. bonds in the crystal within indentation area. For simplicity, the pre- Empirical models originally correlate hardness with the elastic sent discussion is limited to the intrinsic hardness of single crystals. properties of crystals. Historically, Gilman and Cohen have estab- Several strategies for establishing the microscopic theory of hardness lished a linear correlation between hardness and are presented, and the main points are analyzed. For hardness predic- [26,27]. Later in 1998, an improved correlation between hardness tion, these microscopic hardness models can be applied to covalent and shear modulus was proposed by Teter [28]. However, these em- and polar covalent crystals, and in some cases, to ionic crystals. pirical correlations between hardness and bulk (or shear) modulus turn out to be physically questionable. The bulk modulus character- 2.1. Bond resistance model izes the incompressibility of a material, and has a direct relationship with valence electron density; more electrons correspond to greater When an indenter is forced into the surface of a single crystal, as repulsions within a structure [29]. The shear modulus characterizes shown in Fig. 1a, the chemical bonds below the indenter withstand the resistance to shape change at a constant volume. A larger shear compression, and the bonds around the indenter withstand bending modulus results in a greater ability to resist shearing forces. While or even stretching. Based on this simple physical picture, Tian et al. Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 95

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) the distribution of valence electrons shifting away from the center from pure to polar covalent bond.

have proposed an intuitive and important assumption that the hard- An equation relating the Vickers hardness to fi, Ne, and d is then ness for covalent and polar covalent crystals is equivalent to the sum obtained: of the resistance of each bond to the indenter per unit area [34]. The − : = − : key hypothesis is to correlate the plastic deformation (associated N e 1 191f i N2 3e 1 191f i H ðÞ¼GPa 556 a ¼ 350 e : ð4Þ with the creation and motion of dislocations) with the breaking of V d2:5 d2:5 electron-pair bonds in crystals. Hardness then measures the com- bined resistance of chemical bonds to indentation. More bonds in a The average hardness for a multicomponent system is assumed to region of the surface correspond to harder crystals. A scheme to link be the geometrical mean of the hardness of different types of covalent the Vickers hardness for a broad class of covalent and polar covalent bonds in the system via: crystals to their microscopic properties has been suggested. In cova- μ μ 1=∑n lent crystals, energetically breaking an electron-pair bond means μ μ H ¼ Π H n ; ð5Þ two electrons excited from the valence band to the conduction V V band. The activation energy required for a plastic glide is twice the μ μ − μ μ − 2/3 1.191fi 2.5 band gap Eg [40], as schematically shown in Fig. 1b. The resistance where HV =350(NV) e (d ) is the hardness of a binary μ force of a bond can be evaluated with the corresponding Eg. The hard- compound composed of μ bond, and n is the number of μ-type ness of pure covalent crystals should have the following form: bonds in the unit cell. Readers can refer to the original paper for other details [34]. HðÞ¼Gpa AN E ; ð1Þ a g Except for the Phillips ionicity fi, the calculation parameters can all be deduced from a first-principles calculation, which makes this where A is the proportional constant, and Na is the covalent bond microscopic model a powerful tool for hardness estimation from number per unit area which can be evaluated from the valence elec- designed crystal structures, and save a great deal of experimental tron density Ne as: efforts. Authors from the same group have sequentially developed this model by defining a new ionicity scale based on the first- 2=3 ¼ ∑ = ¼ ðÞ= 2=3; ð Þ principles calculation and generalizing to metallic systems [32,33]. Na niZi 2V Ne 2 2 i For a specific crystal structure or cluster containing the same type of

coordinates, an ionicity scale, fh,isdefined for a bond based on the where ni is the number of the ith atom in the unit cell, Zi is the valence Mulliken's bond overlap population as: electron number of the ith atom attributing to the covalent bond, and −jj− = V is the volume of the unit cell. ¼ − Pc P P; ð Þ f h 1 e 6 For polar covalent crystals, the valence electrons are preferentially distributed to the anion side, which weakens the binding of two where P is the overlap population of a bond in a calculated crystal, atoms, as demonstrated in Fig. 1c. An ionic component needs to be and Pc is the overlap population of a bond in a pure covalent crystal counted for hardness calculation in addition to the covalent compo- with an identical structure as the calculated one. A power-law fitof nent. Eg for a binary polar covalent ABm crystal can be separated m fi as a function of fh, fi =fh , yields m=0.735. All the inputs for hard- into a covalent homopolar gap Eh and an ionic heteropolar gap C sug- ness can then be obtained through the first-principles calculations. gested by Phillips [41]: This model was further developed by including a small metallic component of chemical bond and considering the orbital form of s-p 2 ¼ 2 þ 2: ð Þ Eg Eh C 3 or s-p-d [32]. To account for the metallicity effect on hardness, a factor

of metallicity fm =nm/ne has been introduced, where nm =kBTDF is the The homopolar component Eh determines the activation energies numbers of electrons that can be excited at ambient temperature, and of a dislocation glide in polar covalent crystals [42], and can be ne is the total number of valence electrons in the unit cell. At ambient estimated in electronvolts with the empirical expression Eh = temperature, kBT≈0.026eV and DF is the density of electronic states 39.74d−2.5, where d is the bond length in angstroms [41]. The ionic at the Fermi level which can be acquired via electronic structure component results in a loss of covalent bond charge and is accounted calculations. Similar with the ionicity contribution to hardness [34], for by introducing a correction factor, exp(−afi), to Eq. (1). This cor- the screening effect of the metallic component may be phenome- rection factor describes the screening effect for each bond, where α is nologically described by introducing another correction factor of 2 2 n a constant and fi =1−Eh/Eg is the ionicity of the chemical bond in a exp(−βf m), where β and n are constants. The contributions of d crystal scaled by Phillips [41]. The constants A and α are determined valence electrons in TM compounds to hardness have also been by fitting the hardness expression to a standard set of materials counted. The bond strength of an s-p-d hybridized chemical bond is with known Vickers hardness to be 14 and 1.191, respectively [34]. greater 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- where Nij is the multiplicity of the binary system ij, and k is the num- portional coefficient A' for the s-p-d hybridized crystals different ber of different atoms in the system. from that for the s-p hybridized crystals in Eq. (1). As a result, the To address some issues rooted in the above model (e.g., different hardness equation for metallic and s-p-d hybridized crystals (bonds) coordination number for constituent atoms) [56], and “to estimate is written as: hardness of crystals on a pocket calculator”, a generalization has been proposed [36,57]. Bond strength is redefined as pffiffiffiffiffiffiffiffi = − : − : 0:55 ¼ = 2 3 1 191f i 32 2f m sij eiej ninjdij , where ni and nj are the coordination number of ðÞ¼ Ne e : ð Þ HV GPa 1051 : 7 atom i and j, respectively. To use s in calculations, a number b for d2 5 ij ij counting individual bonds of the ij-type in the unit cell is introduced. − − Subsequently, all that remain to be performed in Eq. (11) is the With f =f0.735 =(1−e |Pc P|/P)0.735, all of the parameters in i h replacement of S and N with s and b , respectively. The radius R , Eqs. (4) and (7) can now be determined from ab initio calculations. ij ij ij ij i which is determined from the first-principles calculation in the orig- As suggested by Eq. (7), the metallic component of the bond (f ) m inal formula, is also replaced by the atomic radius. Constants C and has a stronger negative effect on hardness than the ionic component σ are chosen to be 1450 and 2.8 respectively for the hardness calcula- (f ). The d valence electrons play an important role in increasing hard- i tion. Consequently, no constant determined by ab initio methods is ness by enhancing the directionality and orbital strength of the chem- required for the estimation of hardness [36]. Given that constants C ical bond. It should be noticed that, for IB and IIB metals with a fully and σ are determined from experimental data, this model is also a filled d orbital, the prefactor for hardness calculation is 350. The semi-empirical one. This bond strength model works well for the extension of the formula to multicomponent systems is similar with hardness estimation of covalent, polar covalent, and ionic crystals. Eq. (5), with attentions on the metallicity term and prefactor. In recent years, Eqs. (4) and (7) as well as its generalization to a 2.3. Electronegativity model multicomponent system, Eq. (5), have been extensively used for the prediction of hardness for a large class of proposed superhard struc- Based on our assumption [34], the third empirical model is pro- tures from first-principles calculations [44–55]. This method is appli- posed recently to predict the hardness of single and multiband mate- cable to polar covalent crystals, oxides with some contributions of rials in terms of electronegativity (EN) [35,58]. The EN of an element ionic bonds and ionic crystals, as well as for some multi-component is defined as: crystals with mixed types of interatomic bonds. The overall agree- ment with experiments is highly satisfactory. This consistency indi- χ ¼ : = ; ð Þ j 0 481nj Rj 12 cates that hardness can be defined microscopically as the combined resistance of chemical bonds in a crystal to indentation. 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- 2.2. Bond strength model less coefficient [59]. For any covalent bond a–b with coordination

numbers CNa and CNb of atoms a and b, respectively, it can be An alternate scheme for hardness prediction has been proposed by assumed that this bond is composed of (1/CNa) a atom and (1/CNb) Simunek and Vackar [37]. Instead of relating the resistance to the b atom. A bond EN can be defined as an average of the electron- bond energy gap, the resistance is assumed to be proportional to holding energy of two atoms distributed to the a–b bond as: the bond strength S between atoms i and j as: ij rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χ χ pffiffiffiffiffiffiffiffi  ¼ a b : ð Þ ¼ = ; ð Þ Xab 13 Sij eiej dijnij 8 CNa CNb

Subsequently, bond hardness can be expressed as H =X /(Ω/N), where ei =Zi/Ri is a reference energy, Zi is the valence electron num- ab ab where Ω is the volume of the unit cell, and N is the number of ber of atom i, and nij is the number of bonds between atom i and its covalent bonds per unit cell. After defining an ionicity indicator, neighboring atoms j at the nearest neighbor distance dij. The radius pffiffiffiffiffiffiffiffiffiffiffiffi f ¼ jjχ −χ =4 χ χ , Xue et al. proposed a hardness expression Ri for each atom in a crystal is determined such that a sphere around i a b a b for covalent and polar covalent crystals as [35]: an atom with radius Ri contains exactly Zi valence electrons. The hard- ness of the ideal single crystal is proportional to the bond strength Sij − : ðÞ¼ : 2 7f i − : : ð Þ and the bond number in the unit cell. For a simple crystal with one Hk GPa 423 8NvXabe 3 4 14 element, hardness is expressed as: pffiffiffiffiffiffiffiffi For a crystal with n types of bonds, hardness can be expressed as: ¼ ðÞ=Ω =ðÞ: ð Þ H C eiej diinii 9 ! 1=n 423:8 n − : ðÞ¼ ∏ 2 7f iabðÞ − : : ð Þ For a binary compound with two different atoms a and b, hardness Hk GPa Ω n NabXabe 3 4 15 a;b¼1 is expressed as:

pffiffiffiffiffiffiffiffiffi −σ 2.4. Some points about microscopic hardness models ¼ ðÞ=Ω =ðÞf e ; ð Þ H C eaeb dabnab e 10 a) The above three microscopic hardness models differ from one where the exponential factor phenomenologically describes the dif- another in physical treatment and mathematical formula. Neverthe- ference between ea and eb. For a multicomponent system, hardness less, they are all based on the assumption that hardness is equivalent can be calculated as: to the sum of the resistance of each chemical bond to the indenter ! per unit area. The differences are that the deformation resistance is 1=n n −σ expressed by the energy gap for the bond resistance model, by the H ¼ ðÞC=Ω n ∏ N S e f e ; ij ij bond strength consisting of the reference energy for the bond "#i;j¼1 ð Þ = 2 11 strength model, and by the bond EN consisting of the element EN k 1 k Xk ¼ − ∏ ; f e 1 k ei = ei for the EN model, respectively. A comparison of these models sheds ¼ i 1 i¼1 light on the factors that should be considered for pursuing superhard Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 97 materials. The factors include short and strong chemical bonds, high from Eq. (4) for MgO, NaCl and KCl are 4.5 GPa, 0.4 GPa and valence electron density or high bond density, and strongly direc- 0.18 GPa, respectively. The values well agree with the experimental tional bonds (as suggested by the larger prefactor for d orbitals). Ion- ones [60]. icity is adverse for hardness, as clearly demonstrated by the c) A development of the bond strength model has been proposed exponential factor, and metallicity is even worse. recently to describe the anisotropy of hardness with respect to crystal b) The bond resistance, bond strength, and EN models work well orientation [61]. The earlier model is updated by mathematically for pure covalent and polar covalent crystals. As demonstrated in weighing the contributions of each bond to the total hardness based Ref. [37], in addition, the bond strength model can also estimate hard- on its direction, with the bonds perpendicular to the applied force ness of ionic crystals. Still, the definition of ionicity indicator is some- being given the most weight. what arbitrary, more mathematical than physical in the latter two d) The EN model has been recently developed by Lyakhov and models. In the EN model, the intercept term (−3.4) aids in more ac- Oganov by considering the dependence of the EN on the environment curately reproducing the experimental values at the lower hardness and deviations of actual bond lengths from the sum of covalent radii end. However, the physics behind this term is unclear, and in some [62]. The atomic EN and effective coordination numbers are rede- cases, it gives an unrealistic negative value of hardness, i.e., for BaO. fined. Only the bond density Nv needs to be calculated in the EN Every parameter in the bond resistance model is well defined. model, which is convenient for programming. Therefore, the updated

With special attention paid to the density of valence electrons, Ne, model combined with the evolution crystal structure prediction algo- this model can be easily generalized to ionic crystals and TM com- rithm, USPEX [62,63], provides a way of systematic discovery of new pounds. In Fig. 2, the typical charge densities of the chemical bonds hard and superhard materials. in diamond(110), Al2O3(100), MgO(100), and NaCl(100) planes are e) The calculated hardness values for selected crystals from both demonstrated. A variation in valence electron distribution is clearly microscopic and macroscopic models are compared with the experi- observed from diamond to NaCl: From a symmetric accumulation in mental values in Table 1 and Fig. 3. The mutual agreement between the middle of two neighboring C atoms (Fig. 2a for diamond), to an the two different strategies, as well as between the experimental asymmetric gathering preferential to O side (Fig. 2b for Al2O3), and and calculated hardness values is satisfactory. The first-principles cal- to a spherical distribution around O and Cl atoms (Fig. 2c for MgO culations of elastic constants are now routine, and the conversions to and Fig. 2d for NaCl). In the last two cases, the valence electrons shear or bulk moduli for all crystal classes are well known [64,65]. from anions show little evidence of contributing to covalent bond Hence, these two strategies are complementary and can be used to formation. In the most ionic NaCl, Na atom may even lose part of its estimate the hardness for new crystals. In the inset of Fig. 3, the hard- valence electron to Cl, leading to a very small covalent component. ness values for Al2O3, MgO, some ionic crystals, as well as TM mono- A scheme for counting the valence electron density of ionic AxBy carbides and - are emphasized. Obviously, the bond resistance (B=O, F, and Cl) crystals is herein proposed. The valence electrons model reproduces experimental values more accurately by consider- from atom B are omitted in density counting as long as χB ing the contributions of the metallic component (weakening) and d −χA >1.7 in Pauling's scale, which indicates the formation of a strong orbitals (strengthening). The macroscopic model, however, fails to re- ionic bond. In addition, for the most ionic crystals (fluoride and chlo- produce correct hardness values for ionic crystals and MgO, suggest- ride with alkali metals), the charge transferred from the ing the deficiency of describing hardness with the elastic properties atom to F or Cl atom, which can be determined from the first princi- of materials. ples calculations, is also deducted from the counting of valence elec- On a closing note of this session, the intercept term (−3) in the tron density since the transferred electrons are now seized by the modulus strategy is observed as unnecessary due to the lack of phys- anions. With these considerations, the hardness values estimated ical basis. Similar with Xue's model, negative hardness values are predicted for some ionic crystals, such as KI and RbCl, using Chen's formula. This promotes a continuous exploration with the current modulus model. Chen's dataset is fitted and a revised formula is given without the intercept term,

¼ : 1:137 0:708: ð Þ HV 0 92k G 16

The comparison with the original formula is emphasized in Fig. 4 using 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 possibility of 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 are several 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. d valence electrons can effectively enhance the proportional coefficient AinEq. (1). However, in TM compounds, this enhancement effect from 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 and fully covalent bonds, satisfies all the criteria. Boron-rich materials Fig. 2. Typical charge densities of chemical bonds for a) diamond (110), b) Al2O3 (100), c) MgO (100), and d) NaCl (100). (BCx) as superhard materials are particularly appealing [70]. They 98 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

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

Crystal HExp (GPa) HTian (GPa) HSimunek (GPa) HXue (GPa) HChen (GPa) C96a 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 Fig. 3. Comparison of calculated hardness from different models with experimental InAs 3.8a 3.8 5.7b 4.5e 3.3f values. See text for the details of the inset. InSb 2.2a 4.3 3.6b 2.2e 2.4f ZnS 1.8b 6.8 2.7b 2.4e 2.4f b b e f ZnSe 1.4 5.5 2.6 1.8 2.7 Ta) have a high number of valence electrons and can form very strong ZnTe 1b 4.1 2.3b 0.9e 2.1f TiC 32c 34 18.8b 23.9e 27f bonds, although the bonds formed are not very directional. Their TiN 20.6c 21.6 18.7b 23.8h 23.3f compounds with light elements, namely B, C, and N, should be care- ZrC 25c 21 10.7g 15.7h 27.5f fully examined [16,17,22,29,38,73–75]. Examples of these potential c g h – ZrN 15.8 16.7 10.8 15.9 superhard materials and their recent progresses are now presented. 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 3.1. B-C crystals VN 15.2c 14.9 26.5g 16.5h – c b h f NbC 17.6 16.1 18.3 12.8 15.4 Most of B-C binary systems exhibit high resistance to oxidation c b h f NbN 13.7 13.6 19.5 12 14.7 and reaction with ferrous metals, compared with the carbon-based TaC 24.5c 26 19.9g 14.7h – – TaN 22c 20 21.2g 14.3h – materials [76 78]. Boron carbide B4C is a hard crystal that can be pro- CrN 11c 11 36.6g 19.2h – duced at ambient pressure [79,80], whereas B-doped diamond shows WC 30c 31 21.5b 20.6e 31.3f a superconducting transition temperature of 4 K [71]. It is of great in- j j g h i Re2C 17.5 19.7 11.5 16.2 26.4 c g h i terest in diamond-like BCx systems to pursue superior superhard Al2O3 20 18.8 13.5 18.4 20.3 MgO 3.9d 4.5 4.4g 5.4h 24.8i crystals that are not only thermally and chemically more stable than LiF 1d 0.8 2.2g – 8.5i diamond, but also possess interesting electrical properties [78,81,82]. d g – i NaF 0.6 0.85 1 5.7 Experimentally, turbostratic graphite-like g-BCx compounds are rou- d b i NaCl 0.2 0.4 0.4 – 2.4 tinely used as precursors for the synthesis of novel diamond-like d b – i KCl 0.13 0.18 0.2 2.3 phases of the B-C system. Stoichiometric diamond-like BC (d-BC ) KBr 0.1d 0.23 0.2g – 0.1i 5 5 and BC3 (d-BC3) have been recently synthesized under high pressure a Reference [34]. and high temperature [13,70,83]. b Reference [37]. c Reference [32]. At 24 GPa and 2200 K, d-BC5 is synthesized from g-BC5, exhibiting d Reference [60]. extremely high Vickers hardness (71 GPa), unusually high fracture e Reference [58]. toughness (9.5 MPam0.5) for superhard materials, and high thermal f . Reference [30] stability (1900 K, which is about 500 K more thermally stable g . Calculated by authors using method [36] than pure diamond) [13]. Theoretically, Calandra and Mauri have h Caculated using [35]. i Calculated with [30]. j Referenece [52]. show excellent physical and chemical properties, including low mass density, very high hardness, high mechanical strength, high , excellent wear-resistance, and high chemical inertness. Superconductivity has been recently found in boron-doped diamond synthesized at high pressure and temperature [71]. Experimental and theoretical studies have indicated an effective way to increase the superconducting Tc by increasing dopant concentrations [45,47]. Technically important superhard and superconductive materials can now be pursued from boron-rich systems. Another interesting family of carbon allotropes including 3D carbon nanotube polymers has recently attracted a lot of interest. Given the unique configurations of these 3D polymers, they have distinctive electronic properties, high Young's moduli, high tensile strength, ultrahigh hardness, good ductility, and low density. Hence, these polymers may be potentially Fig. 4. Comparison of Chen's hardness formula and the refitting formula with the applied to a variety of needs [72]. Some TMs (e.g., Os, Re, W, and experimental values. Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 99

demonstrated that BC5 is superconducting with a critical temperature values for different structures are from 65 to 80 GPa, indicating that of the same order as that of MgB2 [69]. These combined mechanical d-BC7 is a superhard crystal. d-BCx phases with higher boron content and electrical properties make d-BCx systems appealing materials may exhibit superior electrical conductivity and improved chemical for electronics application under extreme conditions. Theoretical stability. These appealing properties of d-BCx systems call for studies have been performed to address the crystal structures and follow-up experimental syntheses. Selected d-BCx systems with spe- physical properties of d-BC5 phase. Liang et al. studied the mechanical cific crystal structures are presented in Fig. 5. Predicted properties, and electronic properties of diamond-like BCx phases and pointed out such as symmetry, lattice parameter, hardness, and Tc, are summa- a higher energy barrier of d-BC5 (0.057 eV/atom higher than that of rized in Table 2. diamond), implying that d-BC5 is about 500 K more kinetically stable In our previous study [32], a small metallic component has been than diamond [84]. A decreasing trend in mechanical properties with found to have a strong negative effect on hardness. Electrons deloca- increasing boron content in d-BCx systems was revealed by analyzing lized to contribute to the conduction should be excluded from the the electronic structures [84]. Yao et al. predicted a thermodynami- hardness calculation, and the correction from conducting electrons cally stable metallic I4m2 phase with a superconducting critical tem- is necessary to account for the experimental hardness. In d-BCx perature of 47 K, and another possible candidate structure with P1 systems, the major carriers are holes and the valence electrons are symmetry whose X-ray diffraction (XRD) pattern fits well with the mainly localized to form covalent bonds. As a result, the metallic experimental one [85]. Jiang et al. argued that d-BC5 may be correction in the hardness calculation need not be considered. disordered with boron atoms randomly distributed in the diamond lattice using first-principles density functional calculations [86]. 3.2. Carbon allotropes

Li et al. proposed two Pmma structures where the synthesized BC5 adopts the diamond-[100] structure with an atomic packing of Due to its unique ability to form sp-, sp2-, and sp3-hybridized the form ABCABC… along the [100] crystallographic direction of bonds, carbon can adopt a wide range of structures, including dia- diamond. The simulated XRD patterns, Raman modes, and Vickers mond, , graphite, fullerenes, nanotubes, graphene, and hardness remarkable agree with the experimental data [47]. amorphous carbon. Fullerenes, nanotubes, and graphene are current-

An earlier attempt to synthesize d-BC3 phase was unsuccessful ly the focus of research in nanotechnology, electronics, optics, and because g-BC3 decomposed into a composite bulk of B4C and boron- many other fields of materials science and technology. Nevertheless, doped diamond, although with a very high hardness of 88 GPa [87]. there exists a long-term endeavor for searching superhard phases

Until recently, a d-BC3 phase produced from g-BC3 at 2000 K under among carbon allotropes in addition to diamond and lonsdaleite very high pressure (40–50 GPa) was confirmed with combined XRD, [72,94–101]. Raman, TEM,and EELS measurements [70,83]. The authors argued Recently, a carbon allotrope has been obtained by cold compres- that boron and carbon atoms are randomly distributed in the eight sing graphite with pressure over 17 GPa, where the original sp2 positions of the diamond-like structure given that four sharp peaks bonds in graphite are transformed into bonds of sp2 and sp3 mixture in XRD patterns can be indexed as the (111), (220), (311), and [101]. This high-pressure phase has hardness at least comparable to (400) diffractions, respectively, of the diamond structure with the diamond, and arouses a continuous debate about its structure. Mono- right respective intensities [83]. The properties of d-BC3 have been cline polymorph M-carbon [62,96], body centered tetragonal poly- studied with proposed structures, indicating conductive superhard morph bct-carbon [98,99], and orthorhombic polymorph W-carbon crystals [45,81,82]. Liu et al. explored the crystal structures with [102] structures are suggested, as demonstrated in Fig. 6. All these a particle swarm optimization (PSO) algorithm combined with first- phases are superhard with a hardness of about 90 GPa [100]. The principles structural optimizations [45]. Three metallic configura- actual high-pressure phase is likely a mixture of these metastable tions, namely Pmma−a, Pmma−b and P4m2 phases were uncov- carbon phases. Another quenchable superhard carbon phase was syn- ered. With the bond resistance model, the Vickers hardness for all thesized by cold compression of carbon nanotubes at 75 GPa with an 3 three phases is larger than 60 GPa, indicating the superhard nature sp -rich bonding configuration [94]. A superhard (HV =95 GPa) car- of these polymorphs, which should be tested experimentally. In addi- bon allotrope of C-centered orthorhombic C8 (Cco-C8), which can tion, all phases have a superconductive transition at low temperature. account for the experimental data for this superhard carbon phase, We simulated the XRD patterns for these structures and found the has been proposed [103]. This theoretical work sheds light on a new agreement with the experimental data is reasonably good. Hence, strategy to design and synthesize novel metastable carbon allotropes an unambiguous determination of the real crystal structure needs by directly compressing carbon nanotube bundles or other carbon more investigations. structures. Metastable phases of other materials (e.g. BN) with higher

Other d-BCx systems are either synthesized or predicted, such as energy and unique physical properties may also be produced using BC, BC2,BC4, and BC7 [44,68,88–91]. Most are predicted to be super- similar compression techniques [103]. hard and potential superconductive crystals, and a systematical in- Individual C60 molecules are estimated to have an extremely high vestigation of BCx systems with variable x is of fundamental interest elastic modulus of 800–900 GPa [104,105]. However, C60 crystal [84,92,93]. Xu et al. investigated a tetragonal BC2 (t-BC2) phase orig- shows a very soft lattice and a very small elastic modulus due to inating from the cubic diamond structure by first-principles calcula- very weak intermolecular interactions. By the formation of strong 3 tions [88]. The structural stability of BC2 has been confirmed by the sp -like bonds between fullerenes, the rigidity, stability, and hardness calculated elastic constants and phonon frequencies. The electron of the original phase would be greatly enhanced. Experimentally, deficiency introduced by the B atom is distributed to each atom in distinct carbon phases can be prepared by high-pressure high- the system, leading to a 3D conductivity. The calculated theoretical temperature treatment of C60 with subsequent quenching to the Vickers hardness of t-BC2 is 56.0 GPa, indicating that it is a potential ambient conditions [106]. A unique and promising combination of conductive superhard crystal. In addition, the calculated B/G ratio of sufficiently high hardness and high plasticity has been found for the t-BC2 is larger than that of diamond, suggesting that t-BC2 is more 3D-polymerized C60-based phases [106–108]. The hardness of disor- ductile than diamond. Xu et al. also predicted a d-BC7 with P4m2 dered and composite phases attains the values close to that of dia- symmetry possessing a Vickers hardness of 78 GPa and Tc of 11.4 K mond, whereas the fracture toughness coefficient may be even [68]. Most recently, Liu et al. considered a series of proposed structur- higher. In particular, the conductivity of these polymerized 3D C60 al configurations for d-BC7, and concluded that all the simulated crystals may vary considerably from metallic to semiconducting structures are metallic due to the introduction of one-electron defi- conductivity depending on the relative concentration of sp3/sp2 cient B atoms into the system [44]. The calculated Vickers hardness bonds, topology, system structure, etc. [95,108,109]. 100 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

Fig. 5. Crystal structures of a) I41/amd BC2,b)P4m2BC3,c)Pmma−a BC3,d)Pmma−b BC3,e)I4m2BC5,f)Pmma−1BC5,g)Pmma−2BC5,h)P43m BC7,i)P3m1BC7,j)Pmm2BC7, k) P4m2BC7, and l) R3m BC7. Boron atoms are shown in orange (lighter gray in print).

Many theoretical and experimental attempts have been made on nanotube polymers show excellent mechanical properties, including associates of carbon nanotubes (CNTs) to search for possible super- high Young's moduli, high tensile strength, ultrahigh hardness, hard materials, especially on the polymerized (covalently bonded) good ductility, and low density. They also show interesting electric carbon nanotubes [94,95]. A series of ordered 3D structures of parallel properties tuning from semiconducting to linear, planar, or 3D con- CNTs have been proposed [110–112]. As suggested by a recent theo- ducting depending on the specific crystal structures. The combination retical work about CNT polymers [72], some of the proposed struc- of these properties implies a great application prospect for CNT poly- tures, which are demonstrated in Fig. 7, have lower ground-state mers, such as optical or electronic nanodevices under extreme energies than the corresponding nanotube parents and may be syn- conditions. thesized through the treatment of high-energy CNT. These 3D

3.3. Transition metal compounds

Table 2 Structure symmetry, lattice parameter, calculated hardness, and superconducting Compounds of TMs with light elements, B, C, N, and O are cur- critical temperature for proposed BCx crystals. rently the subjects of the intensive research activities searching for novel superhard and ultra-incompressible materials. Several im- Crystal Symmetry Lattice parameter (Å) HV (GPa) Tc (K) portant design parameters for selecting possible superhard TM abc compounds have been pointed out, including a high electron con- a – BC2 I41/amd 2.520 2.52 11.919 56 centration (EC defined as electrons per atomic volume) and the b – BC3 P4m2 2.5015 2.5105 3.915 65.8 13.4 19.5 presence of directional covalent bonding [38,113]. The introduction Pmma−a 2.5132 2.5202 7.7878 61.9 16.6–23.4 Pmma−b 2.4834 2.5311 7.8914 64.8 4.9–8.8 of light and covalent-bond-forming elements into TM lattices can c fl BC5 I4m2 2.525 2.525 11.323 80 47 have profound in uences on their chemical, mechanical, and elec- Pmma−1d 2.5005 2.5238 11.4789 74 8.4–11.3 tronic properties. d Pmma−2 2.5172 2.5265 11.3352 70 18.1–22.6 Several TM borides have been synthesized and claimed to be BC P4m2e 2.5141 2.5141 7.4663 78 8.4–11.4 7 superhard, such as ReB [17], OsB [19,114], and WB [29,73]. The P4m2f 2.5158 2.5158 7.4497 75.2 – 2 2 4 P3m1f 2.5356 2.5356 8.5188 65.3 – structural models for these phases are shown in Fig. 8. The structure f P43m 3.6205 3.6205 3.6205 77.6 – of diborides of heavy TMs (ReB2 and OsB2) includes two dimensional Pmm2f 2.5132 5.124 3.687 80.7 – (2D) lattices of boron atoms. Enhanced mechanical properties are f – R3m 2.5876 2.5876 25.525 65.4 attributed to the high valence electron density as well as the B-B a Reference [88]. and TM-B covalent bonds. In tetraborides of heavy TMs (WB4), b Reference [45]. boron atoms form hexagonal 2D lattice with additional covalent B c Reference [85]. dimers located perpendicularly in between 2 boron layers, leading d Reference [47]. e Reference [68]. to a 3D network. The enhanced mechanical properties are attributed f Reference [44]. to this quasi-3D B lattice. Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 101

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

Gu et al. studied the properties of a series of TM borides [16]. Un- WB2, ReB2, and Os0.5W0.5B2, where a high EC governs a low compress- fortunately, none of the synthesized TM borides shows a comparable ibility, and covalent bonding generates low plasticity as well as (asymptotic) hardness larger than 40 GPa. Their results clearly sequentially high hardness. demonstrate that, while the bulk modulus (compressibility) can be Unlike TM borides, which can usually be acquired under ambient understood with EC of crystals, the difference in hardness can be un- pressure using arc melting, most of TM carbides and nitrides of cur- derstood by means of a chemical bonding analysis. The most incom- rent (superhard) interest have to be synthesized under high pressure pressible OsB, for example, has the smallest hardness, which can be and high temperature [18,20,22,52,115–121]. The extreme mechani- accounted for by the high EC and absence of B-B bonds in the crystal. cal properties of TM nitrides and carbides, such as high strength,

In contrast, WB4 possesses a relatively small bulk modulus due low compressibility, and high hardness have attracted tremendous to small EC, and the hardness is greatly enhanced because of the for- research interests, both experimentally [18,22,118,122–124] and the- mation of a covalently bonded 3D framework of boron atoms. For oretically [53,119–121,125–133]. Superhardness has been predicted compounds with intermediate B content, a good compromise of for some of the TM nitrides and carbides, such as PtN2 [32] and RuC ultra-incompressibility and high hardness may be reached, such as [121].InTable 3, structural information and calculated hardness

Fig. 7. Proposed 3D CNT polymer structures. a) (4, 0) carbon, b) (3, 3) carbon, c) (5, 0) carbon, d) (4, 4) carbon, e) (7, 0) carbon, f) (8, 0) carbon, g) (6, 0) carbon, h) (9, 0) carbon, and i) (6, 6) carbon. 102 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

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

from the band resistance models of some potential semiconducting suppressed with decreasing grain size due to the Hall–Petch effect superhard TM nitrides and carbides are summarized with the crystal [155], while the resistance to the plastic deformation gets weak structures presented in Fig. 9 [134]. Combining with their electronic due to increasing grain boundary shear. As a result, there exists a properties, these materials can be applied to electronics devices “strongest size” of 10–20 nm for grains of crystallite where the hard- under extreme conditions. A recent review of Friedrich et al. provides ness can be increased by a factor of two for many materials [156,157]. an excellent guide for TM borides, nitrides, and carbides [135]. Bulk nanocrystalline diamonds were synthesized from graphite under HPHT conditions with an enhanced hardness as high as 3.4. Other systems 140 GPa [158]. Another example of this extrinsic hardness enhance- ment is demonstrated for c-BN (diamond-like) and w-BN (lonsda- Boron allotropes, B-N [136], B-O [76], C-N [137–141], B-C-N leite-like) nanocomposites where the load-invariant hardness of [49,142–149], B-C-O [50], and Si-C-N [55,150] systems are appealing 85 GPa, approaching to the value of single-crystal diamond, has for searching novel and/or superhard materials. Recently, several been reached [159]. These nanocrystalline bulk materials usually superhard phases of these systems have been synthesized under possess very high fracture toughness, excellent wear resistance, and high pressure high temperature conditions, such as orthorhombic high thermal stability in addition to the superhardness [159], making

BC2N [12],BC4N [11], γ-B28 [14,151], rhombohedral B13N2 [152], them deal materials for cutting, grinding and drilling. and B6O [15,153]. Notably, in these systems, there are some phases Vepřek et al. have further suggested that, through the formation suggested to be superhard, but have not been experimentally con- of nc-TmN/a-Si3N4 nanocomposites with strong and shear resistant firmed, such as C3N4 [62,154],B2O [46], and B2CO [50]. interfaces, the limit (enhancement factor of two) imposed by the Up to now, the considered systems all belong to the intrinsic grain boundary shear in materials composed of very small nano- superhard materials where high hardness is achieved through their crystals can be avoided, and much higher hardness enhancement strong chemical bonding. Superhardness can also be extrinsically can be achieved [160]. Theoretical calculations using nonlinear finite achieved through structuring. For example, an increase of hardness element modeling have predicted hardness as high as 158 GPa in by 80% (from 77 GPa to 140 GPa) was observed for polycrystalline the nc-TiN/a-Si3N4 nanocomposites [161]. However, experimentally CVD diamond in a hybrid ultrahard polycrystalline composite materi- reported hardness in these nanocomposites is less than 115 GPa, al [7]. In polycrystals, the dislocation activities inside the grain is much lower than the predicted value. This hardness decrease is

Table 3 Calculated bond parameters and Vickers hardness of proposed semiconducting transition metal compounds.

μ μ μ Crystal Structure Bond type d (Å) N Ne PPc fi HV, theor (GPa) HV, theor (GPa)

Ti3N4 Th3P4 Ti–N 2.034 48 0.653 0.39 0.43 0.181 108.1 49.3 Ti–N 2.393 48 0.401 0.15 0.43 0.884 22.5

PdN2 Pyrite N–N 1.271 4 2.553 0 359.7 47.6 Pd–N 2.215 24 0.403 0.29 0.57 0.703 33.9

HfN2 Pyrite N–N 1.500 4 1.517 0 167.9 59.9 Hf–N 2.281 24 0.331 0.51 0.57 0.199 50.5

PtN2 Pyrite N–N 1.140 4 1.827 0 221.8 70.9 Pt–N 2.136 24 0.438 0.44 0.57 0.367 58.7

TiN2 Pyrite N–N 1.443 4 1.694 0 199.0 71.9 Ti–N 2.105 24 0.418 0.45 0.57 0.344 60.7

NiN2 Pyrite N–N 1.334 4 2.153 0 284.2 78.6 Ni–N 1.998 24 0.534 0.36 0.57 0.549 63.8

PdN2 Marcasite N–N 1.265 2 2.583 0 366.4 45.5 Pd–N 2.181 4 0.420 0.33 0.57 0.615 40.3 Pd–N 2.235 8 0.390 0.24 0.57 0.807 28.7

PtN2 Marcasite N–N 1.396 2 1.855 229.7 65.8 Pt–N 2.219 4 0.436 0.44 0.57 0.367 59 Pt–N 2.148 8 0.424 0.40 0.57 0.459 50.8 FeC FeSi Fe–C 1.894 4 0.438 0.48 0.75 0.538 64.7 60.3 Fe–C 1.908 12 0.428 0.45 0.75 0.589 58.9 OsC FeSi Os–C 2.028 4 0.329 0.60 0.75 0.329 57.7 45.3 Os–C 2.063 12 0.313 0.48 0.75 0.538 41.7 FeC Zinc blende Fe–C 1.836 16 0.367 0.50 0.75 0.504 64.7 RuC Zinc blende Ru–C 1.968 16 0.298 0.45 0.75 0.589 42.8

FeC2 Rutile Fe–C 1.888 8 0.405 0.47 0.57 0.297 82.4 85.9 Fe–C 1.890 4 0.403 0.64 0.57 0.189 93.3 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106 103

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. attributed to sample imperfectness, oxygen contamination, and more hardness and crystallite size in c-BN and w-BN nanocomposites fundamentally, Friedel oscillations of valence charge density that perfectly [159]. develop in TiN nanograins adjacent to the SiNx type interfacial layers caused by the negative charge at the strengthened SiNx type interfa- 4. Perspectives cial layer [162,163]. It is also noted that different nc-TiN/a-Si3N4 nanocomposite coatings have elastic moduli similar with those of Both Chen's macroscopic model and the microscopic models pre- TiN, whereas the hardness greatly differs [164]. Hardness estimated sented in Section 2 serve effectively for hardness estimation. The with the macroscopic hardness model yield a value of 22 GPa [30], macroscopic model relates hardness to materials' elastic moduli, similar with TiN, which is inconsistent with the experimental obser- while the microscopic models evaluate hardness based on the param- vations. A microscopic explanation based on the specific atomic struc- eters at atomic scale. Understanding of materials hardness at the tures is thus highly anticipated. Nevertheless, this extrinsic hardness atomic scale with the microscopic models can greatly promote the enhancement from nanocomposites points out a very important quest for novel superhard materials. Microscopically, hardness can be pathway in the quest for superhardness [165–169]. defined as the combined resistance of chemical bonds in a crystal to We provide some qualitative considerations of materials harden- indentation. 3D bond network structures with short and strong ing from the Hall–Petch and quantum confinement effects before bond, high bond density, and high valence electron density are the we end this session. The hardening effect of grain boundaries can be determining factors of superhardness. The novel superhard materials expressed through the Hall–Petch equation as [170]: must possess parts or all of these characteristics. The three micro- pffiffiffiffi scopic models discussed in this review can predict hardness reliably ¼ þ = ; ð Þ H H0 KHP D 17 for pure covalent and polar covalent crystals, with the bond resis- tance model and bond strength model applicable to ionic crystals. where H0 is the hardness of the bulk single crystal, KHP is the Hall– While the bond strength model and the EN models have some advan- Petch hardening coefficient which is sample dependent, and D is the tages as mentioned in c) and d) of Section 2.4, these two models grain size in nanometer. In addition, Tse et al. suggested another should be used with cautions to metallic systems where the hardness hardening effect for nanocrystals [171]. Based on the Kubo theory weakening effect from metallicity is not taken into account. and including the quantum confinement effect, the band gap Eg in Light element compounds, such as carbon allotropes and B-C-N Eq. (1) can be updated for nanocrystals as: systems, are still the most appealing materials family for novel super- hard materials. The fascinating variety of possible carbon allotropes ¼ þ δ ¼ þ = 1=3ðÞ: ð Þ – Eg;nano Eg;bulk p Eg;bulk 24 DNe eV 18 is one of the greatest topics of materials science. C C bond strength is known to be stronger in sp2 hybridization (graphite) than in sp3 As a result, the hardness of a nanocrystal can be obtained by hybridization (diamond) [172,173]. However, graphite is a fragile updating Eq. (4) as: semimetal with 2D conductive atomic planes while diamond is the hardest material as-known due to the formation of 3D network  − : = − = − : fi 2 5 2 3 1 1 3 1 191f i with high bond density. Atomic steric con guration, electronic prop- H ðÞ¼GPa AN E ; ¼ 350d N þ 211D N e : ð19Þ V a g nano e e erty, and hardness can be tuned by varying the sp2/sp3 bond ratio. Through high pressure (hydrostatic or non-hydrostatic) treatments By including the Hall–Petch and quantum confinement effects, we of glassy carbons, fullerenes, and nanotubes, superhard crystalline can estimate the hardness for a nanocrystalline bulk as: and amorphous carbon phases with interesting electronic properties pffiffiffiffi are highly anticipated by forming mixed sp2- and sp3-hybridized C– ¼ þ = þ = ; ð Þ H H0 KHP D Kqc D 20 C bonds in a 3D network. d-BCx systems, with the properties of super- hardness, enhanced thermal stability, and tunable metallicity and 1/3 − 1.191fi where Kqc =211Ne e is the quantum confinement hardening superconductivity with varying B/C ratio, are very attractive super- coefficient. This equation describes the observed relations between hard materials for multifunctional applications. 104 Y. Tian et al. / Int. Journal of Refractory Metals and Hard Materials 33 (2012) 93–106

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