Harder than diamond: dreams and reality

PHILOSOPHICAL MAGAZINE A, 2002, VOL. 82, N O. 2, 231±253

Harder than diamond: dreams and reality Vadim V. Brazhkin, Alexander G. Lyapi ny Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow Region, 142190, Russia

and Russell J. Hemley Geophysical Laboratory and Center for High Pressure Research, Carnegie Institution of Washington, Washington, DC 20015, USA [Received 12 July 2000 and accepted in revised form 11 May 2001]

Abstract Analysis of correlations between various physical properties of solids provides a formulation of simple criteria applicable to the search for new superhard materials. The prospects for the synthesis of new substances with the key elastic moduli, whose values approach or even exceed those of diamond are also discussed. We introduce the concept of ideal hardness and strength, which relates the elastic properties of materials to the corresponding mechanical characteristics, the concepts that are less unambiguously determined and that depend on the conditions of measurements. The control of material nanostructure makes it possible to approach the ideal hardness. The formulation of trends interrelating physical properties is expected to allow an essential guidance in the synthesis of new classes of superhard materials.

} 1. Introduction It is generally agreed at present that diamond is the hardest and most incompressible substance; polycrystalline diamond has the highest shear modulus and Young’s modulus of all materials. Nevertheless, the search for new superhard materials that might surpass diamond in elastic properties and hardness has a long history and has witnessed a renewed activity in recent years. Synthesis of superhard materials is important both for understanding the fundamental correlations between microscopic characteristics of interatomic interactions and macroscopic properties and for purely technological applications. Two lines of investigations in this ®eld can be distinguished: the experimental preparation of new phases or ceramics, often with the use of high pressures, and the theoretical calculations of elasticity for hypothetical phases. The empirical correlation between compressibility and hardness has received considerable attention in recent years (LeÂger et al. 1994, 1996, Sung and Sung, 1996). The predictive power of this correlation, comparing theoretically calculated bulk moduli for diŒerent structures and transferring this information to hardness, requires careful examination. For practical applications, both elastic properties and macroscopic mechanical characteristics, such as the hardness and strength, are of y Author for correspondence. Email: [email protected] Philosophica l Magazin e A ISSN 0141±8610 print/ISSN 1460-699 2 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080 /0141861011006774 3

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interest. Elastic properties (the bulk modulus, shear modulus and Young’s modulus and elastic stiŒness coe cients) are governed by microscopic interatomic interactions. Of the mechanical characteristics, such properties as the hardness, strength and yield stress are also directly connected to the cohesive forces between atoms of the substance. The strength and yield stress are de®ned as the limiting stresses corresponding to failure or plastic ¯ow respectively of the substance. At the same time, the hardness, or the capability of a material to withstand imprinting or scratching with another material, does not have such an unambiguous de®nition. The hardness can depend on the elastic properties and plasticity of a substance, on the radius of the indenting or scratching device point, on the load applied to the indenter and on the testing method. Therefore the hardness, like other mechanical characteristics, must in the general case depend not only on microscopic properties (i.e., interatomic forces) but also on macroscopic properties of the material (defects, morphology, admixtures, stress ®elds, and possible inhomogeneity or superstructure). Due regard for the fact that the de®nition of hardness is ambiguous has become particularly important for the problem of superhard materials and, in particular, in comparing substances with similar hardness values. We review the synthesis of new superhard materials and methods for measuring their hardness. First we identify the classes of superhard substances. The factors governing the experimentally measured value of hardness, as well as the correlation between the elastic characteristics, hardness and other properties are then discussed. In addition, we consider criteria that can be used for the prediction of superhard properties of substances, and e cient directions for searching for such materials that may approach (and perhaps surpass) diamond in useful characteristics, such as the hardness and elastic moduli.

} 2. Islands of hardness in the ocean of materials Three classes of superhard substances may be distinguished, including both already synthesized and hypothetical phases: (i) covalent and ionic±covalent compounds formed by light elements from periods 2 and 3 of the periodic table; (ii) speci®c covalent substances, including various crystalline and disordered carbon modi®cations; and (iii) partially covalent compounds of transition metals with light elements, such as borides, carbides, nitrides and oxides. Often these superhard phases, such as diamond, are metastable at normal conditions. From the chemical standpoint, the majority of superhard materials are both covalent and ionic in nature, although superhard compounds of transition metals possess both covalent bonding and metallic bonding. A common feature of superhard substances is the fact that they consist of elements positioned in the middle groups of the periodic table. These are precisely the elements having the smallest ionic, covalent or metallic radii and the greatest cohesive forces between atoms in the solid state. The periodic ®lling of electron shells of the elements leads to regularities in the variations in the atomic radii or molar volume of their solid phases. In this case, the solid phases of elements with the least molar volume in the period have the highest bulk modulus, cohesive energy and melting temperature (®gure 1). There is a clear correlation between the molar

Harder than diamond: dreams and reality

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Figure 1. Physical properties of elemental substances, namely the molar volume Vm , bulk modulus K, and cohesive energy Ec , which depend periodically on the atomic number. In the search for superhard materials, the interesting elements are those having small molar volumes and strong interatomic interactions characterized by high elastic constants and a high cohesive energy. For carbon, the data on diamond are presented. Nitrogen and oxygen have relatively large volumes per atom in elemental molecular forms, while in solid compounds their atomic volumes are (as a rule) much smaller.

volume Vm , bulk modulus K and cohesive energy Ec (see the data from Young (1991)). In particular, the relationship K/

Ec Vm

…1†

was proposed by Aleksandrov et al. (1987). For elemental solids, the ratio KVm =Ec c varies approximatel y from 0.5 to 16 (®gure 2), although for the overwhelming majority of elements it lies within the much narrower interval of 1±4. If the normalized energy and volume for a certain group of substances are related by the universal scaling function, E ˆ E0 f …V=V0 †, where E0 and V0 are the

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Figure 2. The KVm =Ec ratio for elemental solids presented as a function of the atomic number. The input data are taken from Young (1991). The data for crystals of group VII and VIII elements are designated by open symbols.

energy and volume respectively at zero pressure and zero temperature, then the similarity relationship K0 V0 =E0 ˆ f 000 ˆ constant will hold for these substances (Aleksandrov et al. 1987). For solids, whose energy can be described by a potential of the form Ec ˆ A=Vmm B=Vmn , c ˆ mn (Aleksandrov et al. 1987). In particular, for rare-gas solids, well described by the Lennard-Jones potential (m ˆ 4, n ˆ 2), the condition c ˆ 8 is ful®lled. Starting from argon, the experimental value of c is close to 8. The values of c are also high for molecular crystals formed from ordinary binary gases (®gure 2). One can suggest that equation (1) is applicable to various categories of solid compounds. 2.1. Low-Z compounds The ®rst group of superhard materials consists of compounds of elements from the middle of periods 2 and 3, such as beryllium, boron, carbon, nitrogen, oxygen, aluminium, silicon and phosphorus. These elements are capable of forming threedimensional rigid lattices with shortened covalent bonds. Typical examples of ionic± covalent compounds are oxides, such as corundum (Al2 O3 ) and stishovite (the highpressure phase of SiO2 ). The high moduli and hardness values of stishovite (table 1) are connected with the fact that silicon atoms here are in the sixfold (octahedral ) coordination, in contrast with the fourfold coordination in the common (low-pressure) phases of SiO 2 (quartz, cristobalite, tridymite and glass), whereas oxygen atoms exhibit the threefold coordination. The oxides BeO and B6 O possess a high hardness. In early publications devoted to boron oxide (B22 O), Badzian (1988) reported that this substance scratched diamond. In this connection it should be noted that many properties of boron oxides with a high boron content have not been studied as yet, and these materials show great promise.


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Table 1. Mechanical properties of substances with large elastic moduli, where K is the bulk modulus, G is the shear modulus, E is Young’s modulus and H is the hardness measured by the Vickers or microindentation techniques (room temperature). We have chosen the most reliable results from the widely varied experimental data, which can be attributed to natural di culties in the study of superhard materials. The intervals shown for elastic moduli indicate the extreme values of those from diŒerent sources, although individual data sets are more accurate (we approximate all data to an uncertainty of 1 GPa). In contrast, the data on hardness display not only variations in the measured values from diŒerent experiments but also a rather ambiguous nature of the hardness measurements in general (see § 3). It should be noted that, for the homogeneous medium approximation, elastic moduli are related by the equation E ˆ 9KG=…3K ‡ G†. Substance

K (GPa)

Diamond

442±433 a±f

Londsdaleite (hexagonal C) Amorphous carbon ®lms Three-dimensional polymers from C60 Amorphous carbon from C60 B ( phase) BN (cubic) Wurtzite (BN) BP B13C2 B4C CN0.2 ®lms ¬-SiC (hexagonal) -SiC (cubic) Si3N4 Al2O3 B6O Stishovite (SiO2) BeO Cr2O3 TiB2 ZrB2 HfB2 VB2 TiC ZrC HfC WC TiN ZrN Re Os

170a 369±382 d,n,o,p 401e 390p 169 4f 152±175 e,p 200q 247f 221±234 a 210e , 227n 249 3f

E (GPa)

G (GPa)

1142 a 1164 h

534±535a, 544 g

942 a 200±300 j

382 a

400 k 700 k 390 m 973 a, 840b 790 b 480 a 474 a 493 g,i 900 r 457±466 a 401±410 a,i 280 a

246 11e, f 403±141 a,i 200±208 f,q 305 11 f 467 a 316 8t 250±254 a, 237 e 394±400 a 397 a f 244 446±540 a 218a 420±430 a,g a 222 480±510 a,g 286a 340±347 a,g 200e , 241 1 f 383±437 a 195±223 a,e, f 353±386 a,g 505 a 421 f 700±720 a,g 280±292 a,e 431±440 a e f 270 , 217 350 g 265±267 a 380±400 a a,e 363±365 462±520 a 373a 515±559 a

H (GPa) e, f

409 6e, f 330 b 174 f 136 e 203 a 201±205a,g 171 11f 198±200a 170±173a,e,i 123 f 160±166a,e, f,i 204 f 187 a 220 12 f,t 159±162e,a 162 a 263 f 221 a 228 a 130±137a,g 182±196a,e, f,g 162±168a,e, f 221 a 269±280a,g 160 a, 210e 118 f 156±160a 179±206a 224 a

60±150 a,b,e,f,g 60±70i 30±65j 25±60k 60±130 k,l 30±34g,h,j 46±80b,e, f,g,j 50±60i 33 3e, f,g,i 57g 30 2f,q 42±49g,h 60 r 21±29s 26±37e,h 33h,i 21 3a 20±27e±i 35 5 f,q 33 2 f,q 17±23u 10±15e,g,h 27±29h 33±34e, f,g 23±36g,i 23±29g,h,i 28g 18±32e±i 25±30e±h 20±29g,h,i 28±32 f,h 18 2 f,g,h 21e 17±19 f,g 2.5±6m 3.5±3.9m (continued)

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V. V. Brazhkin et al. Table 1.

Substance Ru Mo W Be

K (GPa) 285 a 268±273a 299±310a,e 111±115a,e

(Continued) E (GPa) 422±463 a,m 317±330 a,g 389±395 a 287±320 a

G (GPa) 160±173a,m 119±127a,g 149±160a,e,g 149±155a,e

H (GPa) 2.6±4.9m 1.5±2.3g,h 2.2±3.8e,g,h 0.9±1.34 e,h

a

From Frantsevich et al. (1982). From Novikov (1987) and Kurdumov et al. (1994). c From Aleksandrov et al. (1987). d From Aleksandrov et al. (1989a). e From Clerc and Ledbetter (1998). f From Teter (1998). g From Kislyi (1985). h From Saifulin (1983). i From Frantsevich et al. (1980). j From Weiler et al. (1996). k From Brazhkin et al. (1998). l From Brazhkin et al. (1999). m From Drits (1985). n From Aleksandrov et al. (1989b). o From Yakovenko et al. (1989). p From RuoŒand Li (1995). q From LeÂger et al. (1996). r From SjoÈstroÈm et al. (1995). s From LeÂger et al. (1994). t From Weidner et al. (1982). u From Stishov and Popova (1961). b

These observations suggest that the search for new superhard oxides should be carried out by considering the range of dense high-pressure phases; such phases may also be thermodynamically metastable under ambient pressure conditions. The hypothetical phases of the oxides B1 x Ox , P2 O5 , and phases in the Al±B±O system are possible candidates. The recent discovery of the high-pressure phase of Al2 O3 with the Rh2 O3 -II structure type is of interest (Funamori and Jeanloz 1997). The problem of the existence of the B2 O high-pressure phase with the diamond-like structure has been discussed at length by Endo et al. (1987) . The recent prediction and discovery of quartz-like CO2 (Iota et al. 1999, Serra et al. 1999) add another candidate to the group of potentially superhard oxides. Quartz-like CO2 can be preserved down to at least 1 GPa at room temperature (Iota et al. 1999). The calculated value of the bulk modulus for this phase, K ˆ 183 GPa (Serra et al. 1999), implies that the new phase of solid CO2 can possess a high hardness. However, it is likely that these oxides correspond to the lower threshold of superhard materials. This appears to be connected with a low coordination of oxygen atoms (in most cases), which reduces the topological stiŒness of the relevant structures. Interest in carbon nitrides has grown considerably in recent years. Comprehensive experimental and theoretical studies of the search for hypothetical superhard modi®cations of C3 N4 have been reported (see Teter and Hemley (1996) and Teter (1998), and references therein). Experimental eŒorts to synthesize superhard carbon nitrides are most probably still at the initial stages. High hardness values (60 GPa) have been obtained in distorted graphite-like ®lms CNx (x ˆ 0:2)


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(SjoÈstroÈm et al. 1995). For a wide range of nitrogen content, the structural transformation from primarily sp3 -bonded to sp2 -bonded carbon and a density decrease from 3.3 to 2.1 g cm 3 were observed as the nitrogen concentration increases from 11 to 17% (Hu et al. 1998). Nevertheless, one can hope to overcome this tendency and to synthesize hard CNx networks with a high nitrogen content using high pressure. In fact, the prediction of various low-compressibility C3 N4 (K 425±496 GPa) (Teter and Hemley 1996, Teter 1998) and CN (K 286 375 GPa) (CoÃteÂ and Cohen 1997, CoÃteÂ et al. 1998) modi®cations, including three-dimensional all-sp2 structures, display a potentially vast variety of metastable carbon±nitrogen phases. Sphalerite and wurtzite boron nitrides, the binary isoelectronic analogues of cubic and hexagonal diamond, are well known as having among the highest hardness values and also key elastic moduli. Other covalent materials, such as the carbides of silicon (SiC) and beryllium (Be2 C), and compounds of boron with carbon, phosphorus, and silicon (B13 C2 , B4 C, BP and B4 Si), as well as silicon nitride (Si3 N 4 ), are of particular interest. The hardness values for these substances are in the range 20± 50 GPa. The recent synthesis by Zerr et al. (1999) of cubic Si3 N 4 (the cubic spinel structure) with its high calculated bulk and shear elastic moduli has demonstrated that a high pressure is a powerful tool for the preparation of new dense and hard modi®cations of covalent compounds. Ternary compounds in the B±C±N, B±C±O, and B±C±Be systems are possible candidates (Zhogolev et al. 1981, Sasaki et al. 1993, Nakano et al. 1996, Hubert et al. 1998, Teter 1998). The search for new high-pressure phases of elemental boron should also be considered (see, for example, the results of Ma et al. (1997)). Even polymorphic boron modi®cations that exist at standard conditions and have complicated structures based on icosahedral clusters display su ciently high elastic moduli and hardness. In its compounds, boron tends to adopt the volume per atom, that is about 20% less than the atomic volume of solid boron at standard pressure (S. V. Popova 1998, private communication). This suggests that it may become possible to synthesize denser boron phases by highpressure techniques. 2.2. Carbon materials Carbon materials may be regarded as a special group. Because of the existence of diŒerent types of chemical bond between carbon atoms, there is a great variety of carbon allotropes and disordered phases. The cubic sp3 carbon modi®cation, diamond, is the hardest substance known to date. The Vickers hardness for diamond single crystals, depending on the crystal type and chosen crystal face, varies from 70 to 140 GPa with the load on the indenter equal to 2±10 N (Novikov 1987, Kurdumov et al. 1994). Variations in the load and testing method lead to a still wider scatter of the reported values. Diamond single crystals have record values of the elastic constants c11 and c44 , as well as an anomalously low Poisson’s ratio equal to 0.07 (Frantsevich et al. 1982, Novikov 1987, Kurdumov et al. 1994). The polycrystalline shear modulus of diamond exceeds by more than twofold the shear moduli of other known superhard substances, except for BN. Another sp3 modi®cation of carbon, lonsdaleite, possesses mechanical characteristics similar to those of diamond (table 1). In recent years, the technology has been developed for producing amorphous carbon ®lms with a high degree (about 80%) of sp3 bonding (see, for example, the work of Weiler et al. (1996)). The microhardness values of 70 GPa (Weiler et al. 1996), close to that of diamond, were reported for such ®lms. The estimated elastic

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constants for the tetrahedral amorphous network from tetrahedrally coordinated carbon atoms also prove to be close to but somewhat less than those of diamond (Kelires 1994). A number of hypothetical structures formed from trivalent sp2 atoms of carbon have been investigated theoretically (Liu et al. 1991, Townsend et al. 1992, CoÃteÂ et al. 1998, Jungnickel 1998). These phases exhibit a wide range of densities (0.5±3.14 g cm 3 ) and bulk moduli (50±370 GPa); these are obviously less than those for diamond (fully sp3 phase of carbon). Nevertheless, for some of the proposed sp2 phases of carbon, the elastic constants and bulk moduli are by only 20±50% less than those of diamond. These phases diŒer substantially from graphite, whose weak mechanical characteristics arise from the weak bonding between the covalent hexagonal layers in the structure (along the c axis). The discovery of fullerites, a condensed carbon phase having carbon in the sp2 state, has opened up a new avenue in the search for superhard carbon materials. The formal evaluation of the bulk modulus for an individual C60 molecule yields an extremely high value of 800±900 GPa (RuoŒ and RuoŒ, 1991a,b, Wang et al. 1991), a factor of about two higher than that of diamond. Although fullerite C60 is a soft molecular crystal with a low bulk modulus, empirical estimations have shown that under a pressure of 50±70 GPa, when C60 molecules almost `touch’ each other, it becomes less compressible than diamond and has the bulk modulus K ˆ 600±700 GPa (RuoŒand RuoŒ1991a,b, Wang et al. 1991). The heating of C60 under a high pressure has permitted the synthesis of a number of polymeric sp2 sp3 amorphous and nanocrystalline phases (Brazhkin et al. 1997, 1998), having hardness values similar to that of diamond (see table 1) and exceeding those of a large number of other materials. Reports of the synthesis of phases from C60 that may be harder than diamond, however, have been based on indirect observations of scratching (Blank et al. 1995, 1997); no quantitative hardness measurements have been reported. The amorphous and nanocrystalline diamond prepared using the shockwave loading of C60 is reported to scratch sapphire (Hirai et al. 1994, 1995, Hirai and Kondo 1998). There has been considerable interest in other carbon phases that could be denser than diamond; the elastic moduli and hardness values of such phases may be expected to exceed those of diamond. According to the theoretical calculations of Yin and Cohen (1983), Biswas et al. (1984, 1987), Yin (1984), and Clark et al. (1995), the hypothetical high-pressure BC8 and R8 phases (having the distorted tetrahedral coordination) and the sc structure are predicted to be denser than diamond; however, these phases have not (yet) been observed experimentally (i.e. either at high pressures or as a metastable form at ambient pressure). Nevertheless, it should be noted that the estimated zero-pressure bulk modulus of 410 GPa for the BC8 carbon phase at standard conditions is less than that of diamond (Fahy and Louie, 1987), and only under a higher pressure might it exceed that of the latter. 2.3. Transition-metal compounds Compounds of transition metals from groups IVa to VIa (titanium, vanadium, chromium, zirconium, niobium, molybdenum, hafnium, tantalum and tungsten) with boron, carbon, nitrogen and oxygen belong to the third group (some compounds with the highest in hardnesses are listed in table 1). Compounds containing metals from the neighbouring higher groups of the rhenium boride type can also exhibit a signi®cant hardness. The leaders in this class of materials are the tungsten borides (WB 4 , WB2 and WB with approximate hardnesses of 36±40 GPa), but the


239

existence of denser high-pressure phases of other transitional metals cannot be ruled out. The borides of transition metals form the largest set of superhard materials with the hardness values exceeding 20 GPa (Frantsevich et al. 1980, Saifulin 1983, Kislyi 1985). Carbides and nitrides of these (although also numerous) are inferior to the borides in hardness (see table 1). New modi®cations of TaN and ReC (Boiko and Popova 1970, Popova and Boiko 1971, Popova et al. 1972) serve as examples of successfully synthesized high-pressure phases (HV ˆ 15±25 and 30±35 GPa respectively). Elemental metals from group VIa to Xa have the lowest molar volume and the highest bulk moduli (with the extreme values for elements of group VIII). Evidently, metals with a smaller number of electrons in the outermost shell (from four to six) turn out to be more preferable from the viewpoint of forming hard, partially covalent compounds with boron, carbon and nitrogen. Elemental metals of the platinum group (rhenium, osmium and iridium) have very high bulk moduli (about 360 GPa); however, their hardness is low (less than 7 GPa). It is also appropriate to mention oxides and silicides of transition metals, whose hardnesses vary chie¯y within the 5±20 GPa range. Recently, the high-pressure phase of RuO2 with the ¯uorite structure has been synthesized by Haines and LeÂger (1993). A very high bulk modulus of 399 GPa was reported for RuO2 on the basis of X-ray diŒraction under static compression (LeÂger et al. 1994, Lundin et al. 1998), which may be connected with a strong covalent bonding between the Ru d and O p states (Lundin et al. 1998). High values of K were also predicted by Lundin et al. (1998) for the isoelectronic analogues OsO 2 (411 GPa) and RuNF (315 GPa). An important feature of the structures speci®ed is the fact that the transition-meta l atom has eight covalently bonded nearest neighbours. The recent quenching (and the equation-ofstate measurement) of the high-density ZrO2 and HfO2 phases with the nine-coordinated PbCl2 -type structure is also very promising because both phases have very high reported bulk moduli, 444 15 and 340 10 GPa, respectively (Desgreniers and Lagarec 1999). } 3. Superhard materials: less compressible or less deformable? Because of the di culty inherent in obtaining accurate ®rst-principle calculations of hardness (as well as plasticity and rheology), the establishment of possible correlations between hardness and other elastic parameters has become an extremely useful guide for predicting these properties. In fact, the ab initio calculations of elastic constants for both crystalline lattices and fragments of amorphous networks have recently become almost routine. The existence of a correlation between the elasticity and mechanical properties (in particular, hardness) is beyond question, since all these characteristics are in one way or another connected with interatomic forces. In order to clarify these relationships, we now examine in more detail the methods used for hardness testing. In so doing, the existence of correlations between hardness and other mechanical characteristics should not be ruled out. For example, superhard substances have high values of the yield stress ¼y and high ¼y =E ratios (Kislyi 1985), whereas metals have low values of ¼y =E. (E is Young’s modulus.) 3.1. Methods of measurement In mineralogy, hardness is typically measured on the Moh scale, which is based on the scratch tests with the use of certain standard substances from talc to diamond. Unfortunately, a single step on the Moh scale turns out to be substantially nonlinear in terms of hardness measurements of standards based on more quantitative physical

240


methods (see the quantitative data presented by Kislyi (1985)). In addition, experiments based on the scratching of one substance by another are much less reliable from the viewpoint of hardness; substances with comparable but not equal hardnesses are able to scratch each other. The possibility of scratching depends not only on the hardness but also on the speci®c details of the contact between the given two materials. In physics and technology the hardness is measured by the Brinell, the Rockwell and the Vickers methods, according to which the material is indented by a hardened steel ball, diamond cone or diamond pyramid respectively. The hardness value is determined from the ratio of the load to the indentation area, P=S. For diŒerent indentation techniques, the measured hardness value will substantiall y depend on the load in the framework of the method (see ®gure 3 and the hardness versus load curves for diamond and BN in the paper by Taniguchi et al. (1996)). Only those measurements for which the similarity law (P=S ˆ constant) is ful®lled in the regime

Figure 3. The indentation process shown diagrammatically in (a) occurs according to the loading curve given (b), demonstrating the crossover between the elastic and plastic modes.


241

of plastic deformation can be regarded as correct. The similarity law is only satis®ed for loads exceeding the critical value Pcr , when the major part of the body deformation energy is spent on the plastic deformation around the indenter. At small loads, the relative contribution of the energy consumed by the elastic deformation and formation of new surfaces under the indenter increases substantially (Kislyi 1985), and the indentation techniques yield hardness values that are overestimated by a factor of about two to ten, depending on the load. For softer substances, this dependence becomes noticeable only at very small loads P < 0:1 N. For superhard materials, the region of the critical load value lies in the interval 10±100 N (Novikov 1987, Kurdumov et al. 1994, Taniguchi et al. 1996), that is, is higher than the standard loads for ordinary materials. An insu cient load value can lead to deviations in the measurements. Naturally, diŒerent methods will yield diŒerent hardness values, since the pattern of both elastic and plastic deformations around indenters of diŒerent shapes varies with the shape itself, as well as with the method applied. In the case of nanoindentation , the loading curve can be nonlinear in all its segments, thereby preventing an unambiguous determination of hardness (see the example presented by SjoÈstroÈm et al. (1995)). Elastic stiŒness coe cients do not suŒer these problems and oŒer a more unequivocal and meaningful means for describing microscopic properties of a substance. However, care must be exercised with respect to the frequency dependence, and in the case of polycrystalline aggregates the continuity of stress across grain boundaries (Reuss or Voigt) must be taken into account. The leading elastic constants are represented by the bulk modulus and second-order elastic tensor for single crystals and by the bulk modulus, shear modulus and Young’s modulus for isotropic polycrystals. 3.2. Connections with elasticity Although there is a correlation between the bulk moduli and the hardness values for particular classes of substances (LeÂger et al. 1994, 1996, Sung and Sung, 1996), there are limitations to strictly using the bulk modulus for predicting hardness. For example, the bulk modulus of corundum exceeds that of B6 O, and yet its hardness is signi®cantly less. A better correlation is observed between the hardness and Young’s modulus or the shear modulus (Gilman 1968, 1996, Gerk 1977, Clerc and Ledbetter 1998, Teter 1998), although, in this case too, the dependence is not unequivocal and monotonic. The correlation between the shear modulus and the hardness or strength for diŒerent substances is, of course, not accidental: ®rstly, the majority of mechanical tests are connected with deformations consisting of a substantial shear component and, secondly, one may presume the existence of a relationship between the interatomic forces and the material resistance to elastic and plastic deformations. The recent studies of the interrelation between the hardness, the structure geometry, and the intrinsic atomic-level properties (Gilman, 1996, Clerc and Ledbetter 1998), as well as the report of the correlation between the shear modulus c44 and microhardness for transition-meta l carbonitrides (Jhi et al. 1999) clearly supports this idea. From the correlation between the hardness and the shear modulus, it is readily seen that the hypothetical cubic phase C3 N4 with the bulk modulus exceeding that of diamond must have a lower hardness; the calculated elastic constants for this phase yield the shear modulus G equal to 320 GPa (Teter and Hemley, 1996, Teter 1998), which is appreciably less than that of diamond. Qualitatively, this can be explained

242


by the fact that the threefold atoms of nitrogen reduce the topological stiŒness of the lattice compared with the purely tetrahedral lattice of diamond. However, such a structure remains stiŒwith respect to the uniform compression: its stiŒness is mainly determined by the radial compressibility of the bonds, leading to the lower value of G=K for C3 N4 than for diamond. The correlation between Young’s modulus and the hardness is well known for carbon ®lms (Weiler et al. 1996). The magnitudes of both of these characteristics are in turn determined unambiguously by the ®lm density and the proportion of atoms in the sp2 and sp3 states (Weiler et al. 1996). It may be supposed that the correlations between density and hardness (Young’s modulus) are common for various carbon modi®cations, whose structures correspond to the three-dimensional covalent network. In fact, the experimental data on the hardness and density of carbon phases prepared from C60 (Brazhkin et al. 1998, 1999) are in good quantitative agreement with the data already cited for amorphous ®lms (®gure 4). At the same time, for the low-dimensional carbon phase the interrelation between the hardness and the density becomes more complicated (Lyapin et al. 2000). The correlations between the density and the hardness allow us to understand why it has been impossible to prepare from fullerite C60 any phases signi®cantly harder than diamond. The origin of the extremely high incompressibility value for the C60 molecule (800±900 GPa (RuoŒand RuoŒ1991a,b, Wang et al. 1991)) may be found in the fact that, since the molecule represents a separate carbon cluster, the formal molecular density of an individual molecule (6.3 g cm 3 ) exceeds that of diamond by a factor of almost two (Brazhkin and Lyapin 1996). The phase consisting of closely packed C60 molecules will be less compressible than diamond only at densities comparable with or exceeding that of

Figure 4. Comparison of the Vickers hardness versus density for diŒerent carbon phases prepared from C60 at 8 GPa (*, !), 9 GPa (!), and 12.5 GPa (!), with the experimental data for diamond (^) and for amorphous carbon ®lms (*, *); (- - -), linear least squares interpolation for experimental points from the work of Weiler et al. (1996).


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diamond. In this case, it is straightforward to show that the molecules must be arranged very close to each other, almost in direct contact. However, such a situation is not readily obtainable because of large repulsive forces between atoms and the formation of rigid covalent bonds between molecules at certain critical intermolecular distances (Adams et al. 1994, Brazhkin et al. 1997). For this reason, at pressures of about 20 GPa, C60 molecules appear to collapse into an amorphous network (Regueiro et al. 1992a,b, Snoke et al. 1992, Hirai et al. 1994), the density and hardness of which are close to those of diamond (Hirai et al. 1994, Brazhkin and Lyapin 1996). As regards the relationship between the hardness and elasticity, another important mechanical characteristic, plasticity, should be taken into account (Kislyi, 1985). Transition metals have elastic constants comparable with those of their compounds with boron and carbon, whereas the hardness of pure metals is less by one order of magnitude (table 1); this eŒect is usually associated with the metallic nature and high plasticity of these materials. Temperature variations can drastically change the material plasticity and, consequently, its hardness. For example, the hardness of diamond decreases by one order of magnitude at T 1600 K (Novikov 1987, Kurdumov et al. 1994), whereas the decrease in the elastic moduli at the same temperatures does not exceed 10% (Zouboulis 1998). In the plastic regime, the hardness is mostly determined by the kinetics (strain rate), or by the frequency of generation and mobility of dislocations (Frantsewich 1980, Jhi et al. 1999), whereas the relation between the hardness and elasticity for brittle materials is more straightforward. 3.3. Ideal hardness: the bridge to elasticity Although the hardness de®nition and the process of its measurement are ambiguous, the nonlinear dependence of the indentation area on the load contains information relating the elastic characteristics of a substance to the conventional hardness measured in the regime of plastic deformation (®gure 3). In the elastic limit of very small loads on the identer, it is possible to introduce the notion of the ideal hardness Hid as the ratio of the load to the contact surface of the indenter. During loading of the absolutely hard cone or pyramid with the point angle 2¿, Hid ˆ E

cot ¿ ; 2…1 ¸ 2 †

…2†

where E is Young’s modulus and ¸ is Poisson’s ratio (Sneddon 1948, 1965). For the majority of non-metals, ¸ ˆ 0.1±0.25 (Frantsevich et al. 1982); that is ¸2 1. For the standard pyramids cot ¿ 0:5, and we obtain Hid ˆ E=4. For most materials, the shear modulus is noticeably less than the bulk modulus; therefore, E ˆ 9KG=…3K ‡ G† 3G and Hid 3G=4. For the ideal theoretical shear strength, the analogous relationship ¼ G=2º has been proposed (Kittel 1971). Therefore, the ideal hardness and strength are directly related to elastic characteristics and, in particular, to Young’s modulus and the shear modulus. Experimental measurements of Young’s modulus from the initial loading curve are used in nanoindentatio n techniques. In this connection the correlation between the hardness and Young’s modulus or the shear modulus is not surprising. The real hardness H measured at high loads is usually (0.01±0.2)Hid . The diŒerence between the real and ideal hardnesses is associated with the fact that, in the former, the plastic deformation connected with the presence of dislocations and other defects is involved, and structural transformation s under conditions of extreme

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stress are possible. Moreover, the load increase usually leads to the generation of additional defects of the dislocation type. In general, variations in the number and mobility of dislocations, caused, for example, by temperature changes can signi®cantly aŒect the material plasticity and its H=Hid ratio. It should be noted that at the initial moment of the indentation experiment (½ 1 ms), the contact stress caused by the indentation can attain the value of about one third of the Young’s modulus and is comparable with Hid (Golovin and Tyurin 1994, 1995). The ideal hardness is obviously the upper limit of the true hardness, which can be attained only within the elastic limit of small loads. We discuss below the question as to how the real hardness can approach the ideal value.

} 4. Correlation of physical properties In carbon phases, the density and mechanical properties are governed by the topological parameter, that is by the ratio of atoms in the threefold and fourfold positions (Weiler et al. 1996, Brazhkin et al. 1998, 1999). The bonds in the sp2 network are in themselves stiŒer (e.g. graphite) than those in the sp3 network. As shown below, sp2 carbon reduces the topological stiŒness of the system, and sp2 bonds in the network are aŒected by weaker eŒective `bond-bending’ forces, which results in the lowering of both the bulk modulus and the shear modulus with the population of sp2 bonds. The carbon phases are not the only example illustrating the relationship between the density and elastic properties. The bulk modulus of ionic compounds is represented by the general relation K / Za Zc =V, where Za and Zc are the formal charges of the anion and the cation, respectively and V is the speci®c volume per ionic pair (Anderson and Nafe 1965). For crystals with the structure of diamond or zincblende, Cohen (1985) proposed the empirical relation K ˆ …1971 220¶†d 3:5 , where d is the distance between the nearest neighbours, and ¶ is the empirical parameter that takes into account the ionicity eŒects (¶ ˆ 0, 1, 2 for semiconductors of groups IV, III±V and II±VI respectively). The analogous relation has also been proposed by Kelires (1997) on the basis of Monte Carlo simulations for crystalline alloys Si1 x Cx . For the above-mentioned cases, the relationship K / »n holds, where » is the valence electron density, and the exponent n is equal or close to unity. The relation K / »n , n 1, is directly connected with the empirical equation » / 1=Vm . Note that for the degenerate electron gas, when there is no compensating charge of positive ions, the » dependence of the bulk modulus is K / »5=3 . As far as the shear modulus and Young’s modulus are concerned, there are no obvious empirical correlations but, in this case too, the electron valence density aŒects the values of the moduli. Moreover, the shear modulus must be more sensitive to the directionally non-uniform distribution of electron density, since the lattice shear must lead to a concomitant distortion in the directional distribution of electron density. At the same time, the bulk modulus is more likely to be governed by the spatially averaged electron density. The shear moduli of many ionic±covalent and covalent compounds (e.g., SiO2 , SiC and B4 C) exceed the corresponding values for transition metals (e.g. tungsten and rhenium), whereas the bulk moduli of the latter are substantially higher. A dramatic illustration is provided by the phases of tin. The metallic modi®cation -Sn (white tin) exceeds by approximatel y 20% the semiconductor phase with the diamond structure (grey tin), in both the density and the bulk modulus. At the same time, the shear modulus of grey tin estimated from the single-crystal stiŒ-


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ness coe cients (Ravelo and Baskes, 1997), is one and a half times of that of white tin. A relatively high value for the shear modulus in comparison with the bulk modulus in the case of diamond is connected with a high degree of covalency and large bond-bending forces that appear during the lattice distortion. Formally, the high shear modulus in diamond is associated with a very low Poisson’s ratio (0.07). Hereinafter we shall use the homogeneous medium approximation, in which G=K depends monotonically on Poisson’s ratio according to the relationship ¸ ˆ …3 2G=K †=…6 ‡ 2G=K †. The increasing metallicity correlates with increasing Poisson’s ratio and with the decreasing relative values of the shear modulus and hardness, as observed for transition metals and metallic compounds (Frantsevich et al. 1982). In metals, the number of nearest neighbours is typically larger than in covalent compounds; geometrically, the lattice becomes topologically stiŒer. However, the bond length increase and, ®rst and foremost, the bond type change (an increase in the metallicity degree) lead to a general reduction in the shear stiŒness in metals. The topological stiŒness of the lattice seems to play an important role in covalent structures, when the coordination number decreases from four towards lower values. The presence of trivalent, and especially divalent atoms reduces signi®cantly the topological stiŒness of the covalent or ionic±covalent lattice and, consequently, the shear modulus. On the other hand, with the increase in coordination number above four, the system moves toward a spherical disposition of nearest neighbours which gives rise to spherical symmetry in the distribution of valence electrons, that is the loss of directional covalent bonds and hence, the decrease in bond-bending forces. It may be predicted from these arguments that lattices with the tetrahedral ordering of nearest neighbours (e.g., diamond or lonsdaleite) are ideal for attaining the highest shear modulus and hardness. However, the possibility of obtaining a carbon phase with a shear modulus higher than that of diamond still remains an open question. The last issue is in many aspects related to another fundamental problem: whether substances with a high ®rst coordination number (z 5 6) and with a small Poisson’s ratio (i.e., with a high G=K ratio) exist. Poisson’s ratio may formally take values between 1 and 0.5, where the lower limit corresponds to the material that does not change its shape, and the upper limit corresponds to the unchanged volume. There are few systems with ¸ < 0 (Rothenburg et al. 1991). For substances with predominantly central interatomic forces (ionic and Van der Waals crystals), Poisson’s ratio is close to 0.25, which corresponds to G=K ˆ 0:6. For metals, it lies between 0.3 and 0.4, corresponding to 0:2 < G=K < 0:5. For many covalent compounds, ¸ < 0:25, that is G=K > 0:6. Only a few substances have a shear modulus exceeding the bulk modulus, G > K (¸ < 0:125). Among them are diamond, cubic-BN, quartz (¬-SiO2 , pyrites, beryllium, and, probably, lonsdaleite, BP, B6 O, HfB2 , ZrB2 and TiB2 (Frantsevich et al. 1982, Teter 1998). The theoretical calculation by Kelires (1994) for amorphous sp2 ± sp3 networks shows that the elastic constants …c11 c12 †=2 and c44 (and, consequently, the shear modulus) exceed the bulk modulus for materials with a wide range of sp2 -to-sp 3 ratios. For some substances, for example LiF (Frantsevich et al. 1982), G exceeds K at high temperatures. The reasons for high values of the G=K ratio in covalent crystals are associated with a high angular stiŒness of directional covalent bonds. This circumstance can be easily analysed using diamond as an example within the framework of the simple

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valence force ®eld model. This model consists of bond-stretching kr r2 =2 and bondbending k¿ ¿2 =2 interactions, where r and ¿ are the deviations in the nearestneighbour distance and the angle between two bonds respectively from their equilibrium values, while kr and k¿ are the bond-stretching and bond-bending force constants. In this case, the eigenvalues of the elastic constant tensor, corresponding to the tetragonal and rhombohedral shear deformations, are known (Musgrave and Pople, 1962) to be …c11 c12 †=2 ˆ 3k¿ =a and c44 ˆ 6kr k¿ =a…kr ‡ 8k¿ † respectively (here a is the lattice constant). Consequently, G / k¿ is approximately correct. Likewise, the contribution of bond-bending forces to the stiŒness moduli of diamond is close to that of the bond-stretching interaction. In fact, diamond has an extremely low relative value of c12 (c11 ˆ 1076 GPa, c12 ˆ 125 GPa, and c44 ˆ 577 GPa). In the framework of the force ®eld model discussed above, c11 ˆ …kr ‡ 12k¿ †=3a and c12 ˆ …kr 6k¿ †=3a, that is the contribution of bond-bending forces to c12 is negative and to c11 positive. Beryllium, however, is a metal for which G > K (¸ ˆ 0:05). Despite the facts that beryllium has a high coordination number (z ˆ 6 ‡ 6, corresponding to the hcp structure with c=a ˆ 1:567) and is of the metallic nature, a high G=K ratio and small Poisson’s ratio are observed for it. The large G=K value for beryllium is due to considerable directional anisotropy in the electron density, which is governed by the characteristics of the beryllium electronic structure with its two valence electrons (Cracnell and Wong 1973). At the same time, a fairly important role is played by the electronic core of beryllium that contains only two s electrons and has a very small radius. This results in a very non-uniform radial distribution of the electron density, the degree of non-uniformity exceeding signi®cantly that in the phases of heavier elements from the next periods. Another striking example is the sixfold-coordinate d ionic crystals of light-element LiF, that also have G > K at high temperatures (Frantsevich et al. 1982). The very non-uniform angular distribution of the electron density in diamond (Novikov 1987, Kurdumov et al. 1994), where ¸ ˆ 0:07, is again connected with a very small radius of the carbon core. For the other group IV elements that have the diamond structure, namely silicon and germanium, ¸ 0:22 and 0.21 respectively (Frantsevich et al. 1982). It may therefore be expected that hypothetical carbon phases having the -Sn-type structure or the simple cubic structure (both with a coordination number of six), not only will be less compressible than diamond but also will possess a higher shear modulus, and this behaviour will diŒer substantially from the properties of heavier elements of group IV (cf. the relationships between K and G in ¬- and -Sn). The situation for the shear modulus in denser hypothetical carbon phases having the BC8 and R8 structures (with coordination numbers of four but distorted angles) also remains to be understood. The above arguments lead to the prediction that the entire class of high-pressure phases containing elements with small core radii (beryllium, boron, carbon, nitrogen and oxygen) and high coordination numbers (z > 6) may have very small Poisson’s ratios (G > K). It will, of course, be equally important to determine whether or not such hypothetical substances, if formed, can be preserved in the metastable form under standard conditions (Mailhiot and McMahan 1991). } 5. Control of nanostructure: towards the ideal hardness The above discussion leads to an examination of the possible limitations on the increase in shear modulus and ideal hardness that were introduced earlier. For


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structures with a low coordination number, the limitations arise from a low topological stiŒness, whereas with the increasing coordination number the limitation is caused by the decrease in shear elasticity, connected to a more uniform distribution of the electron density and enhancement of the metallicity. However, it is possible to reduce the diŒerence between the real and ideal hardnesses, and the key to this is the material plasticity decrease, through the in¯uence on its morphology or inhomogeneity. The real hardness of a homogeneous material can increase in the two limiting cases: in the ideal defect-free crystal and in the amorphous or nanocrystalline states. In the ®rst case, dislocations are absent whereas, in the second case, the formation and motion of dislocations are kinetically hindered. It is obvious that the defect-free crystal, as a rule, cannot be obtained (e.g. owing to the thermal activation of point defects). In the case of amorphous solids, the hardness increase is often limited by the presence of defects in the amorphous network (e.g. dangling bonds) connected to local deviations of the structure from the `ideal’ ordering of the amorphous network (see discussions given by Davis (1976) and Mott and Davis (1979)). From the practical standpoint, the preparation of nanocrystalline composites is of particular interest. The natural composites of diamond, known as `carbonado’ (Orlov, 1973), represent the well-known example of a hard polycrystal with ultra®ne grains. The hardness of such carbon composites can exceed that of diamond single crystals (Orlov 1973). It would be appropriate to note that the reported hardness of single-crystal stishovite (17±23 GPa (Stishov and Popova, 1961)) appears to be smaller than that obtained for polycrystalline specimens (33 GPa (LeÂger et al. 1996)). Together with the use of nanocomposites (nanoceramics), the development of superhard coatings based on superlattices is promising (see Barnett and Madan, (1998), and the references therein). The stressed boundaries between the layers in superlattices or between the grains in nanoceramics serve as barriers for the motion of dislocations and favour the approach to the ideal values of hardness and strength. The hardness of TiN/NbN and TiN/AlN superlattices can exceed that of the bulk crystalline components by a factor of two to three (Barnett and Madan 1998). The maximum hardness for superlattices has been observed with a periodicity of 6±8 nm (Barnett and Madan 1998). For the preparation of hard nanoceramics, the optimum grain size is about 10 nm (Yip 1998). The existence of the optimum size in either case (®gure 5) is quite understandable: the size increase results in a mixture of bulk materials, while the decrease in the characteristic distance between the boundaries causes them to cease to serve as barriers for the motion of dislocations (Barnett and Madan 1998, Yip 1998). When the grain size decreases, the mechanism of sliding grain-boundar y motions becomes important, whereas the superlattice period decrease causes the material to become approximately homogeneous, similar to a solid solution. } 6. Harder than diamond: trends in the search The main groups of hard compounds and the ways for creating new hard materials discussed above are summarized in ®gure 6. It is possible to indicate the lines in the search for new superhard materials. The ®rst line involves the search for substances with an average electron valence density (and for pure carbon phases, the ordinary density as well) higher than that of diamond. This case can probably be likely correlated with elastic and mechanical characteristics exceeding those of diamond. However, it should be remembered that the electron density enhancement leads, as

248


Figure 5. Nanostructured materials, including (a) superlattices, (b) heterogeneous and (c) homogeneous nanocrystalline composites, may demonstrate (d) the maximum hardness for a de®nite size of crystallites or superlattice period.

a rule, to an increasing bulk modulus, but not necessarily to an increasing shear modulus. One way to implement this important criterion would be represented by the synthesis of new compounds having the covalent bonding and high ®rst coordination numbers (the average number of nearest neighbours is greater than four). Superhard phases are expected to have a small Poisson’s ratio (about 0±0.2). Using beryllium and carbon as examples, it is proposed that the combination of strong anisotropy in the electron density along the key directions (e.g. along the bonds, as in carbon) and high radial inhomogeneity for ions with small cores provides the crucial microscopic constituent of the superhardness. The question of whether the recovery of any such phases is possible at standard conditions remains open at present. The second line involves the synthesis of new borides, carbides, nitrides and possibly oxides of transition metals, as well as the synthesis of high-pressure phases of known compounds. It is possible that interesting superhard materials will be discov-


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Figure 6. Diagram illustrating the relations between the periodic table, the main groups of hard substances, and the basic technological approaches for the creation of new hard materials.

ered among three-component and more complicated covalent compounds, consisting, for example, of two light elements and a transition metal. Another line in the search for new superhard materials is technological, and includes the synthesis of nanocomposite materials with diŒerent morphologies and distributions of crystalline grain sizes. It is not improbable that this aspect of the search should be carried out among multicomponent systems. Recently obtained nanocomposites in the systems B±C±N (Hubert et al. 1998), diamond+Al2 O3 (Liu and Ownby 1991), SiC+Al2 O3 (Lei et al. 1991), TiC+Al2 O3 (Tamari et al. 1995), WC+TiC (Fahrmann et al. 1991), TiN+TiB2 (Zhang et al. 1995), and Si3 N4 +TiC (Wayne et al. 1991) are intriguing candidates. One can expect that this list of ceramics combined from superhard materials will be expanded in the near future. It is unlikely that the hardness of such ceramic materials will exceed the record hardness of polycrystalline diamond; however, the combination of a high hardness with the possibility of controlling other physical properties is extremely advantageous for technical applications. The synthesis of nanocrystalline carbon phases from various fullerites also appears promising. Preliminary experiments have shown that all-carbon nanocrystalline composites of diamond with graphite that possess a high degree of homogeneity and bonding between grains can be produced from C60 under pressure (Brazhkin et al. 1998, 1999). The hardness values for such composites

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(80±130 GPa) (Brazhkin et al. 1998, 1999) are substantially higher than the values obtained by Kondo et al. (1996) for composites compacted from nanocrystalline diamond powders (20±30 GPa). Unfortunatel y for practical applications, many superhard phases often have a potential disadvantag e that stems from their metastable nature. For example, stishovite is virtually of no technological value, as a superhard material, since it transforms to an amorphous low-density phase at 500o C. The possible existence of denser metastable carbon phases at standard conditions is not clear as yet. Nevertheless, the su cient high-temperatur e stability of diamond that is of course metastable at zero pressure inspires us with a certain optimism. For diverse technological applications, the development of still harder materials should not constitute the ®nal purpose in itself, and the combination of other useful properties with a high hardness is frequently much more important. For example, very light-weight and hard substances may be possibly produced using sp2 carbon crystalline nanotubes and ropes of nanotubes (Thess et al. 1996) that can possess extremely high Young’s moduli (Treacy et al. 1996). Substantial eŒorts to synthesize new phases in the B± C±N system have been spurred on by the search for a material that is harder than cubic BN and chemically resistant to iron (as is BN (Teter 1998)). Such a material is urgently needed for technological applications connected with the processing and polishing of the iron-based alloys. A unique combination of high hardness (30±40 GPa) and plasticity at standard conditions has been found recently for threedimensional polymers of fullerite C60 (Brazhkin et al. 1997, 1998). The problem of comparing superhard substances with each other is of fundamental importance for these investigations. From the physical standpoint, only the elastic characteristics of various phases can be compared directly and quantitatively . Such a comparison requires not only the bulk moduli but also the shear moduli and the single-crystal elastic tensors. A comparison of hardness values obtained by different experimental procedures cannot be performed correctly in full. The ideal hardness and strength corresponding to the possible upper limits of these mechanical characteristics can serve as a bridge between the unambiguously determined elastic properties of the material and its macroscopic mechanical characteristics. Diamond still remains the hardest substance, with record values of the shear modulus and Young’s modulus, but the promising search for new superhard phases continues both theoretically and experimentally. ACKNOWLEDGEMENTS The authors are grateful to S. M. Stishov, S.V. Popova, N. W. Ashcroft and D. M. Teter for useful discussions, and to S. A. Gramsch for very helpful comments on the manuscript. This work was supported by the Russian Foundation for Basic Research (grants 99-02-1740 8 and 00-15-99308 ) and the US National Science Foundation (NSF). The Center for High-Pressure Research is a NSF Science and Technology Center. References Adams, G. B., Page, J. B., Sankey, O. F., and O’ Keeffe, M., 1994, Phys. Rev. B, 50, 17 471. Aleksandrov, I. V., Goncharov, A. F., Makarenko, I. N., Zisman, A. N., Yakovenko, E. V., and Stishov, S. M., 1989a, High Pressure Research, 1, 333. Aleksandrov, I. V., Goncharov, A. F., Stishov, S. M., and Yakovenko, E. V., 1989b, Pis’ma Zh. ekp. teor. Fiz., 50, 116 (Engl. transl., 1989, JETP Lett., 50, 127).


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