Hardness of T-carbon: Density functional theory calculations
Xing-Qiu Chen, Haiyang Niu, Cesare Franchini, Dianzhong Li, Yiyi Li
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug , Hardness of T-carbon: Density functional theory calculations
Xing-Qiu Chen , ∗ Haiyang Niu , Cesare Franchini , , Dianzhong Li , and Yiyi Li Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Shenyang 110016, P. R. China and University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, Austria (Dated: January 24, 2014)We revisit and interpret the mechanical properties of the recently proposed allotrope of carbon,T-carbon [Sheng et al. , Phys. Rev. Lett., , 155703 (2011)], using density functional theory incombination with different empirical hardness models. In contrast with the early estimation basedon the Gao’s model, which attributes to T-carbon an high Vickers hardness of 61 GPa comparableto that of superhard cubic boron nitride ( c -BN), we find that T-carbon is not a superhard mate-rial, since its Vickers hardenss does not exceed 10 GPa. Besides providing clear evidence for theabsence of superhardenss in T-carbon, we discuss the physical reasons behind the failure of Gao’sand ˇSim˚unek and Vack´aˇr’s (SV) models in predicting the hardness of T-carbon, residing on theirimproper treatment of the highly anisotropic distribution of quasi- sp -like C-C hybrids. A possibleremedy to the Gao and SV models based on the concept of superatom is suggest, which indeedyields a Vickers hardness of about 8 GPa. PACS numbers: 64.60.My, 64.70.K-, 62.25.-g, 62.20.Qp
Recently, on the basis of first-principles calculationsSheng et al. proposed a carbon allotrope which theynamed T-carbon . Strictly speaking, its actual stabilityneeds a highly large negative pressure which is far be-yond currently available technologies. Structurally, thisphase can be obtained by substituting each carbon atomin diamond with a carbon tetrahedron (Fig. 1), and thuscrystallizes in the same cubic structure of diamond (spacegroup Fd3m)) with the carbon atoms at the Wyckoff site32 e (0.0706, 0.0706, 0.0706). It has been noted that T-carbon has a large lattice constant of 7.52 ˚A and a lowbulk modulus of B =169 GPa, only 36.4% of the bulkmodulus of diamond . In particular, its equilibrium den-sity, 1.50 g/cm , is the smallest among diamond (cubicand hexagonal diamond) , graphite , M -carbon , bct-C , W -carbon , chiral-carbon as well as the newly pro-posed dense hp tI
12- and tP phases. Thisresults in an highly porous structural pattern, which canbe viewed as a diamond-like array of superatoms (tetra-hedral C4 clusters), as depicted in Fig. 1. Given this pe-culiar clusterized arrangement of atoms exhibiting a quitelow shear modulus of G = 70 GPa , it is very surprisingthat T-carbon was predicted to be superhard, with anexceptionally high Vickers hardness (H v ) of 61.1 GPa ,comparable to that of superhard cubic boron nitride ( c -BN).The aim of our present study is to elucidate the ori-gin of this anomalous hardness. We do this by exploringin details the mechanical properties of T-carbon throughthe application of several different empirical approaches:the Gao’s formula , the SV model and our recently pro-posed empirical treatment based on the Pugh’s modu-lus ratio . Our systematic analysis provides an unam-biguous and physically sound results: T-carbon is nothard. We will show that the conventional application ofGao and SV models leads to a much too high Vickers hardness, H Gao v =61.1 GPa and H SV v =40.5 GPa, substan-tially overestimated with respect to the value obtainedusing our formalism, H Chen v =5.6 GPa. The predictionof a low Vickers hardness in T-carbon is consistent withthe estimation of a low shear strength (7.3 GPa alongthe (100) < > slip system), which represents the up-per bound of the mechanical strength. FIG. 1: Lattice structure of T-carbon (space group Fd3m).By considering each carbon tetrahedron (C4 unit) as an ar-tificial superatom, the corresponding structure is isotypic tothat of diamond. The local environment of each superatom isillustrated in the right panel.
The improper assignments derived by a conventionalapplication of the Gao and SV models can be attributedto the fact that these two models assume that the chem-ical bonds, which are significant for hardness, are dis-tributed uniformly in the lattice. But in T-carbon, asalready pointed out be Sheng et al. , though the car-bons atoms are tetrahedrally coordinated and apparentlyresembling a three-dimensional quasi- sp -like hybrid ,their bonds are ordered in an extremely anisotropic andporous framework, highly different from the bonding dis-tribution in ideal sp -hybrid. We propose a remedy tocure the limitations of Gao and SV models in dealing withanisotropic and porous systems by assuming each carbontetrahedron cluster as an artificial superatom. Indeed,this cluster-like approach leads to low Vickers hardnessin the range of 7-8 GPa, in agreement with the estimatedvalue of 5.6 GPa using our proposed model .All calculations were performed using the Vienna abinitio Simulation Package (VASP) in the frameworkof density functional theory (DFT), and we adopted thePerdew, Burke and Ernzerhof approximation to treatthe exchange-correlation kernel. Well converged resultswere obtained using an energy cut-off of 500 eV and ak-point grid 11 × × . The DFT results were thenemployed as input for the three different hardness empir-ical models, with which we have computed the Vickershardness H v :(a) Gao’s model : H Gao v = 350[( N / e ) e − . f i / ( d . )] (1)where N e is the electron density of valence electrons per˚A , d is the bond length and f i is the ionicity of thechemical bond in a crystal scaled by Phillips. As alreadymentioned, this model gives H Gaov =61.1 GPa (Ref. 1).(b) SV model : H SV v = C Ω √ e i e i / ( d ii n ii ) (2)where C is the constant of 1550 and Ω is the equilib-rium volume of T-carbon. e i = Z i / R i represents the ref-erence energy, with Z i indicating the valence number ofelement i . For carbon e i = 4.121 (taken from Ref. 8). n ii and d ii are the number of bonds and bonding lengths be-tween atom i . In T-carbon, each carbon has four nearest-neighbors with two different bonding lengths: three in-tratetrahedron carbon-carbon bonding length of 1.502 ˚Aand one intertetrahedron bonding length of 1.417 ˚A .By using the average bonding length of 1.48075 ˚A we ob-tained H v = (1550/26.5785) × ×
4) = 40.5GPa, which is 33.5% smaller than the correspondingGao’s value. (c) Chen’s model . This is the empirical formula whichwe have recently proposed, based on the Pugh’s modulusratio k = G / B : H Chen v = 2( k G ) . − . (3)This model not only reproduced well the experimentalvalues of Vickers hardness of a series of hard materialsincluding all experimentally verified superhard materials(see Fig. 2 and Table I), but also provides a theoret-ical foundation of Teter’s empirical correlation in itssimplified form .Before discussing the results for T-carbon we start bypresenting some general considerations regarding the cal-culation of the Vickers hardness and the trustability of our proposed model . Hardness is a highly complex prop-erty, which depends on the loading force and on the qual-ity of samples (i.e., presence of defects such as vacanciesand dislocations). Because Vickers hardness is exper-imentally measured as a function of the applied load-ing forces, the saturated hardness value (or experimen-tal load-invariant indentation hardness) is usually consid-ered to be the hardness value of a given material. There-fore, the theoretically estimated Vickers hardness withinGao’s, SV’s and Chen’s models should be directly com-pared to the experimentally saturated hardness value ofpolycrystalline materials. An overview on the experimen-tal and theoretical values of H v for the experimentallyverified superhard materials (diamond, c -BC N, c -BN, c -BC , γ -B ) is summarized in Fig. 2 and Tab. I. Theexperimental results are highly scattered, reflecting theinherent difficulties in achieving a trustable and preciseestimation of hardness. For instance, the reported valuesfor the hardness of diamond, the archetype superhardmaterials, range from 60 GPa to 120 GPa . Similartrends have been observed for the other two well-knownsuperhard materials c -BC N and c -BN. The most typ-ical case is probably ReB , whose actual hardness hasbeen extensively debated after the first value of itsVickers hardness (48 ± . Depending on different samples, syn-thetic methods and measurement technique, the obtainedvalues range from 18 to 48 GPa (Table I). In contrast toexperiment, theoretical estimations of the Vickers hard-ness given by different models agree within few GPa,including the data obtained by our proposed model (Eq.3). Overall, the comparative trend displayed in Fig. 2provides robust evidence for the reliability of our pro-posed formalism .Now, let’s turn the attention to T-carbon. By usingthe values of the shear and bulk moduli from Ref.1 as in-put ( B = 169 GPa and G = 70 GPa) for Eq. 3 we obtaina Vickers hardness of 5.6 GPa, dramatically smaller thanthe corresponding Gao (61.1 GPa) and SV (40.5 GPa) es-timations. Furthermore, we noted that Sneddon definedthe concept of ideal elastic hardness by H id = Ecotφ − v ) where E is Young’s modulus, v is Poisson’s ratio andcot φ ≈ andsuggested that the real hardness would be (0.01 ∼ id at high loads . Utilizing this definition and the derived E = 185 GPa, the real hardness for T-carbon should bein the range from 0.5 GPa to 10 GPa, in agreement withour obtained value. In particular, it still needs to notethat the occurrence of this serious discrepancy among thethree different methods (Gao’s, SV’s and Chen’s models),which is not observed for the other test cases of Fig. 2and Table I, urges for a clarification aiming to discernwhich method provides the more reliable description ofthe hardness of T-carbon and, consequently, to help usto answer a naturally arising question: is T-carbon a realsuperhard material?An useful concept for understanding strong mechani-cal strength – but still relying on elastic properties – is k G (GPa)020406080100120 V i ck e r s ha r dne ss ( G P a ) Ref. 17Ref. 18Ref. 19Ref. 17Ref. 18Ref. 21Ref. 18Ref. 17Ref. 21Ref. 22Ref. 14Ref. 27Ref. 28Ref. 32Ref. 31Ref. 31Ref. 40Ref. 33Ref. 33Ref. 34Ref. 36 k G (GPa)020406080100120 V i ck e r s ha r dne ss ( G P a ) details refer to Ref. [9]This workGao et al.’s modelSimunek and Vackar’s modelT-carbonReB γ -B c -BN c -BC c -BC N Diamond Expt. Ref.
FIG. 2: Vickers hardness H v as a function of a product ( k G )of the squared Pugh’s modulus ratio ( k = G / B ) and shearmodulus ( G ). The curve corresponds to the empirical relationof Eq. 3 (For other data and more details, see Ref. 9). Elas-tic moduli and experimental Vickers hardness are collected inTable I. Note the huge discrepancies among the three theo-retical estimations for T-carbon. based on ideal shear and tensile strengths , at whicha material is getting unstable under direction-dependentdeformation strains . To shed some light on the na-ture of T-carbon we have thus investigated ideal tensilestrength along the < > direction and shear strengthalong the (100) < > slip system. We found that a ten-sile strength of 40.1 GPa along the < > direction and ashear strength of 7.3 GPa in the (100) < > slip system(see Fig. 3). Therefore, we can conclude that the failuremode in T-carbon is dominated by the shear deforma-tion type in the (100) < > slip system. The calculatedshear stress of 7.3 GPa basically sets the upper bound onits mechanical strength at zero pressure , because theideal strength is the stress where a defect-free crystal be-comes unstable and undergoes spontaneous plastic defor-mation. It is well-known that the measurement of hard-ness has to first encounter the elastic deformation andthen experience permanent plastic deformation. There-fore, it can be conjectured that the hardness of T-carbonshould not exceed 7.3 GPa. These arguments provide astrong support for our estimated Vickers hardness of 5.6GPa on the basis of Eq. 3.In order to gain further insights on this intricate sub-ject and to reach a consistent and satisfactory conclusionon the hardness of T-carbon we consider now the rela-tion between hardness and brittleness on the basis of thePugh’s modulus ratio . There is no doubt that all exper-imentally verified superhard materials, such as diamond, c -BN, c -BC N, γ -B and c -BC are intrinsically brittle.As shown in Table I the Pugh’s modulus ratio of thesesuperhard materials ( k = 1.211-1.178 (diamond), 0.999- TABLE I: Comparison between measured ( H Expv ) and the-oretically computed values of the Vickers’ values (in GPa),along with available bulk modulus (B, GPa), shear modulus(G, GPa) and Pugh’s modulus ratio k = G / B . G B k H Chen v H Exp v H Gao v H SV v Diamond 536 a a d a a e b b ± f c c c -BC N 446 g g d ,75 e
78 71.9445 c c ± e,h c -BC i i e ,73 e c -BN 405 j j d a a h a a h k k l c c ± c γ -B m m n ,58 ± o ReB p p ± s r r ± r r r ± r x x t y z t y z u v w ± A ± B T-carbon 70 C C a Ref. , b Ref. , c Ref. , d Ref. , e Ref. , f Ref. , g Ref. , h Ref. , i Ref. , j Ref. , k Ref. , l Ref. , m Ref. , n Ref. , o Ref. , p Ref. , q Ref. , r Ref. , s Ref. , t Ref. , u Ref. , v Ref. , w Ref. , x Ref. , y Ref. , z Ref. , A Ref. , B Ref. , C Ref. . I dea l s t r eng t h ( G P a ) (100)<001> shear strength<001> tensile strength FIG. 3: DFT calculated ideal tensile and shear strengths ofT-carbon. c -BN), 1.107-1.091 ( c -BC N), 1.054 ( γ -B ) and1.048 ( c -BC )) are larger than 1.0. They clearly obey tothe empirical relation that considers the Pugh’s modulusratio as an indicator of the brittleness or ductility of ma-terials. The higher k the more brittle (and less ductile)the material is. Pugh still proposed, when k is largerthan 0.571 the materials are brittle and with k being lessthan 0.571 the materials are ductile . This relation hasbeen extensively applied not only to metals and alloysbut also to high-strength materials. In the case of T-carbon, the calculated Pugh’s modulus ratio k = 0.414 issmaller than 0.571, clearly in the range of ductility. Theductile behavior of T-carbon is a further indication of itsnon-superhardness.On the basis of the above consideration we can nowunderstand why T-carbon is not an superhard material.One common feature of superhard materials is that theynot only need a three-dimensional network composed ofshort, strong, and covalent bonds but also have a uni-form distribution of strong covalent bonds. The proto-typical example is diamond, which is characterized byan isotropic array of tetrahedrally bonded sp carbonatoms. Conversely, in soft graphite the sp -type covalentbonds, though strong, are localized in two-dimensionalsheets. At first glance, T-carbon seems to be a goodcandidate for superhardness since each carbon atom hasfour nearest-neighboring carbon tetrahedrally bonded byshort and strong carbon-carbon covalent bonds. How-ever, due to the extreme-anisotropic arrangement of thesecarbon-carbon bonds and the associated formation of alarge proportion of porosity in lattice space as well as thelow density of bonds, the framework of T-carbon be moreeasily bendable in comparison with that of diamond, asmanifested by its low shear strength.Having this in mind, we can look back Gao’s and SVmodels. Although these two models perform very wellfor many hard materials, they deliver questionable num-bers for T-carbon in sharp contrast with our findings,as we have documented above. The reason for this ap-parent failure is that in these two models all bonds aretreated as uniformally distributed in the lattice space.Clearly, this constrain will not affect the predictions forisotropic material but it will be inadequate to describethe hardness of extremely-anisotropic compounds suchas T-carbon. However, if we give a closer look to eachindividual C4 tetrahedron unit (see Fig. 1), the distribu-tion of six strong carbon-carbon covalent bonds withineach C4 unit is highly dense. It is therefore trustfullyexpected, that the Vickers hardness of each individualC4 unit can be comparable (or even harder) to that ofdiamond because its bonds density and strengths withineach C4 unit are higher than those of diamond. Thestrength and rigidity of each individual C4 unit appearto be such strong that it cannot be broken easily. Basedon this fact, in order to render Gao’s and SV’s meth- ods applicable to T-carbon, each carbon tetrahedron (C4unit) is considered to be an artificial superatom (See rightpanel of Fig. 1). The cubic unit cell of T-carbon consistsof eight superatoms and each superatom has four nearestneighbors with the bonding length of d = 3.257 ˚A . Interms of Gao’s and SV’s methods, this distance d shouldbe the bonding length between exact atomic positionswith positively charge cores, representing the real forcecenter of each atom. Based on our assumption, the d distance is defined as the spatial separation between twonearest neighbor superatom positions, d =3.257 ˚A. Al-though it remains disputable whether the center of massof the C4 superatom could be assigned its real force cen-ter (thus allowing the applicability of Gao’s and SV’smodels) the high strength and rigidity of each individ-ual C4 unit manifested by the dense and strong carbon-carbon bonds seem to validate this assumption of d dis-tance. Obviously, each superatom contains 16 valenceelectrons, and N e = 8/26.61 = 0.3. By inserting thesevalues of d and N e in Gao’s formula (Eq. 1), we derivea Vickers hardness of 8.2 GPa, in agreement with ourvalue of 5.6 GPa. To apply the same adjustment to theSV model one needs to define the crucial parameter R i forthe superatom. From our first-principles calculations, itcan be inferred that R i = 2.32 ˚A represents the optimumradius containing all 16 valence electrons for each super-atom. By inserting e i = 16/2.32 = 6.896 in Eq. 2 a Vick-ers hardness of 7.7 GPa is obtained, again in agreementwith our analysis. Within this superatom approach, allthree methods discussed in the present paper convey thesame answer: T-carbon is not superhard. The anomalousbehavior of Gao and SV models observed in Fig.2 for T-carbon is cured and the general agreement among thethree Gao, SV and Chen models is re-established. Thisprovides clear evidence that the hardness of T-carbonshould not exceed 10 GPa. Acknowledgements
We greatly appreciate thethought of the treatment of “superatom” to successfullyapply Gao et al.’s model to hardness of T-carbon fromProf. Faming Gao and useful discussions with Prof.Gang Su and Dr. D.-E. Jiang for his critical reading.X. -Q. C. acknowledges the support from the “Hun-dred Talents Project” of CAS and the NSFC (Grant No.51074151). The authors also acknowledge the compu-tational resources from the Supercomputing Center (in-cluding its Shenyang Branch in the IMR) of CAS. ∗ Corresponding author: [email protected] X.-L. Sheng, et al., Phys. Rev. Lett., , 155703 (2011). Q. Li, et al., Phys. Rev. Lett., , 175506 (2009). K. Umemoto, et al., Phys. Rev. Lett., , 125504 (2010);R.H. Baughman, et al., Chem. Phys. Lett., , 110(1993); Y. Omato, et al., Physica (Amsterdam) , 454(2005); P. Y. Wei, et al., Appl. Phys. Lett., , 061910(2010). J. -T. Wang, et al., Phys. Rev. Lett., , 075501 (2011).H. Y. Niu, et al., Appl. Phys. Lett., , 031901 (2011). C. J. Pickard et al., Phys. Rev. B, , 014106 (2010). Q. Zhu, et al., Phys. Rev. B, , 193410 (2011). F. M. Gao, et al., Phys. Rev. Lett., , 015502 (2003). A. ˇSim˚ u nek et al., Phys. Rev. Lett., , 085501 (2006). X.-Q. Chen, et al., Intermetallics, , 1275 (2011). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994); G. Kresse et al., Comput. Mater. Sci. , 15 (1996); G. Kresse et al.,Phys. Rev. B, , 1758 (1999). J. P. Perdew, et al., Phys. Rev. Lett., , 3865 (1996); J.P. Perdew, et al., J. Chem. Phys. , 9982 (1996). X.Q. Chen, et al., J. Phys.: Condens. Matter S. F. Pugh, Philos. Mag. Ser. 7, ,823 (1954). D. M. Teter, MRS Bulletin , 22 (1998). H. Z. Yao, et al., J. Am. Ceram. Soc., , 3194 (2007). H.J.MoShimin et al., J. Appl. Phys., , 2944 (1972). Y. Zhao, et al., J. Mater. Res., , 3139 (2002). V. L. Solozhenko, et al.,, Phys. Rev. Lett., , 015506(2009). R. A. Andrievski, Int. J. Refract. Met. Hard. Mater., ,447 (2001). J Chang, et al., Physica B,
5, 3751 (2010). V. L. Solozhenko, et al., Appl. Phys. Lett., , 1385 (2001). Y. J. Wang et al., J. Appl. Phys., , 043513 (2009). M. Grimsditch, et al., J. Appl. Phys., , 832-834 (1994). I. G. Steinle-Neumann et al., Euro. Phys. J. B , 127(2007). J. H. Westbrook et al., The Science of Hardness Testingand its Research Applications (ASM, Metals Park, Ohio,1973). C. Jiang, et al., Appl. Phys. Lett., , 191906 (2009). V.L. Solozhenko, et al., J. Superhard Mater., , 428(2008). A. R. Oganov, et al., Nature, , 863 (2009) E. Y. Zarechnaya, et al., Phys. Rev. Lett., , 185501(2009). J. B. Levine, et al., Adv. Func. Mater., , 3519 (2009). J. B. Levine, et al., Acta Mater, , 1530 (2010). H. Y. Chung, et al., Science , 436 (2007). J. B. Levine, et al., J. Am. Chem. Soc., , 16953 (2008). Q. Gu, et al., Adv. Mater., , 3620 (2008). S. Otani, et al., J. Alloys Compd., , L28 (2009). J. Q. Qin, et al., Adv. Mater., , 4780 (2008). X.Hao, et al., Phys. Rev. B, , 224112 (2006). W. Zhou, et al., Phys. Rev. B, , 184113 (2007). X. Hao, et al., J. Phys.: Condens. Matter., , 196212(2007). M. R. Koehler, et al., J. Phys. D: Appl. Phys., , 095414(2009). N. Dubrovinskaia, et al., Science, , 1550c (2007). H.-Y. Chung, et al., Appl. Phys. Lett., , 261904 (2008). X.-Q. Chen, et al., Phys. Rev. Lett., , 196403 (2008). V. V. Brazhkin, et al. , Phil. Mag. A, , 231 (2002); I. N.Sneddon, Int. J. Engng. Sci., , 47 (1965). D. Roundy, et al., Philos. Mag. A, , 1725 (2001); D.Roundy, et al., Phys. Rev. Lett. , 2713, (1999). J. S. Tse, J. Superhard Mater., , 177 (2010); A. R.Oganov et al., J. Superhard Mater., , 3 (2010). R. B. Kaner, et al., Science,308