Computational Materials Science 47 (2009) 491–500

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Computational Materials Science

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Theoretical study of mechanical, electronic, chemical bonding and opticalproperties of Ti2SnC, Zr2SnC, Hf2SnC and Nb2SnC

M.B. Kanoun a,1, S. Goumri-Said b,*,1, A.H. Reshak c

a Unité de Chimie-Physique Théorique et Structurale (UCPTS), Laboratoire de Chimie Théorique (LCT), Facultés Universitaires Notre-Dame de la paix, 61 Rue deBruxelles, B-5000 Namur, Belgiumb Centre de recherche en Physique de la Matitére et du Rayonnement (PMR), Laboratoire de Physique du Solide, Facultés Universitaires Notre-Dame de la paix, 61 Rue de Bruxelles,B-5000 Namur, Belgiumc Institute of Physical Biology, South Bohemia University, Institute of System Biology and Ecology, Academy of Sciences, Nove Hrady 37333, Czech Republic

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 August 2009Received in revised form 17 September2009Accepted 20 September 2009Available online 21 October 2009

PACS:61.66.Fn62.20.Dc71.15.Mb

Keywords:Ceramic maxAb-initioElastic constantsOptical propertiesMechanical stabilityElectronic description

0927-0256/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.commatsci.2009.09.015

* Corresponding author.E-mail addresses: mohammed-benali.kanoun@f

Souraya.Goumri-Said@fundp.ac.be (S. Goumri-Said).1 These authors contributed equally to this work.

In this work, we investigate structural parameters, elastic stiffness, electronic, bonding and optical prop-erties of four 211 MAX phases compounds, Ti2SnC, Zr2SnC, Hf2SnC and Nb2SnC. These systems exhibitnanolaminated structure where MC layers are interleaved with Sn. We employ first-principles calcula-tions based on density functional theory (DFT) by means of two methods, the full-potential linearizedaugmented plane-wave and plane-wave pseudopotential. Geometrical optimization of the unit cell isfound in good agreement with the available experimental data. Electronic and bonding properties arestudied throughout the calculation of densities of states, Fermi surfaces and charge densities. Further-more, the optical properties are investigated via the calculation of the dielectric tensor component inorder to characterize the linear optical properties. The optical spectra are analyzed by means of the elec-tronic structure, which provides theoretical understanding of the conduction mechanism of these ceram-ics. We gave more attention to the elastic constants, where we have derived bulk and shear modulususing Voigt–Reuss–Hill approach. Elastic parameters and mechanical properties are analyzed fromanisotropy and ductility calculations. The Debye temperature is derived using Debye–Grüneisen model.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction without lubrication. They demonstrate good thermal and electrical

Mnþ1AXn (n = 1–3), which is the generic chemical formula forthe so called MAX phases in literature, represent a class of techno-logically important solids exhibiting unusual properties associatedwith metals and ceramics [1]. Here, M is an early transition metal,A is a group IIIA or IVA element, X is either carbon (C) and/or nitro-gen (N). These materials gained renewed interest since Barsoumand co-workers [2] succeed to synthesis single phase, bulkdense samples of Ti3SiC2. Consequently, many of the roughly60 Mnþ1AXn phases known to date have been studied both experi-mentally and theoretically [3–9]. Generally speaking, the MAXphases are good thermal shock and oxidation resistance [10,11].They exhibit high strength at high temperatures, but are relativelysoft and can be machined using conventional high-speed tools

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undp.ac.be (M.B. Kanoun),

conductors [12–15]. Experimental investigations stated that Ti2SnC[16] exhibits high electrical conductivity than that of other layeredternaries like Ti3SnC2 [17], Ti2AlC and Ti2AlN [18–20], but negligi-ble thermoelectric power. The negligible thermoelectric power isattributed to the fact that both electrons and holes contribute tothe conductivity [17].

Recently, we have studied the steric effect on the M site of nan-olaminate compounds M2SnC (M = Ti, Zr, Hf and Nb) [21], where ithas been shown that the structural parameters obtained afterrelaxation are in good agreement with the experimental ones. Fur-thermore, from this work we have evidenced the existence of a ste-ric effect on the M site in the M2SnC 211 MAX phases. For Melements which atomic radius (Zr, Hf) is larger than the tin one,the distortions of the polyhedra (octahedron and trigonal prism)constituting the hexagonal unit cell are minimum. These resultswhere obtained using an hybrid all-electron full-potential linear-ized augmented plane-wave method [22] within the generalizedgradient approximation. More recently, it has been reported adetailed study of the effect of the pressure on the elastic constants

492 M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500

[23]. To our knowledge, no systematic study of the correlationbetween mechanical and electronic properties of M2SnC phaseswas discussed before. In the present work, we will calculate theelastic constants in order to deduce mechanical properties suchas bulk, shear and Young’s modulus, Poisson’s factor, anisotropyand Debye temperature. Moreover, we will give special attentionto the correlation between the electronic and elastic propertiesas function of the metal nature (M). We have to point out thatmany theoretical calculations have been carried out on the elec-tronic structures of MAX phases, but the optical properties of theMAX phases have not received adequate attention from neitherexperimentalists nor theorists.

In the present paper, we would like to deepen our understand-ing of these systems by the determination of their structural, elas-tic, electronic, bonding and optical properties, using conjointly anall electron approach and a plane-waves pseudopotential (appliedmainly for elastic properties calculations). The paper is organizedas follows: in the next section, we will give the technical detailsof the employed methods. Section 3 is devoted to the structuralproperties. In Section 4, we will calculate elastic constants, the bulkand shear moduli, derived from Voigt–Reuss–Hill approach usingboth methods. Mechanical properties of the polycrystalline statewill be derived also. Section 5 concerns the electronic and thebonding nature of these compounds. The optical properties willbe investigated in Section 6. We summarize our main finding inthe last section.

2. Methods of calculation

Our calculations were carried out using the full-potential (line-arized) augmented plane-waves plus local orbitals (FP-L/APW + lo)method [22], based on the density functional theory (DFT) [24].The influence of the exchange and correlation potential has beentested within the generalized gradient approximation using Wuand Cohen scheme (GGA-WC) [25]. The muffin-tin radii ðRMT ) areconsidered to be equal to 2.0, 2.21, 2.1, 2.12, 2.5 and 1.9 a.u. (Bohrunits) for Ti, Zr, Hf, Nb, Sn and C atoms, respectively. The maximumvalue lmax for the wave function expansion inside the atomicspheres is limited to 10. To achieve the energy eigenvalues conver-gence, the wave functions in the interstitial region are expanded inplane-waves with a cutoff RMT � Kmax ¼ 8, where Kmax is the maxi-mum modulus for the reciprocal lattice vector. Moreover, local

Table 1The equilibrium lattice parameters a and c (in Å) are compared to experimental ones. The renergy ðDEf Þ are also reported.

a c c=a

Ti2SnCThis work 3.136 13.641 4.350Exp.a 3.163 13.679 4.324Calc. 3.171b, 3.096c 13.857b, 13.404c 4.387c

Zr2SnCThis work 3.352 14.681 4.170Exp.a 3.357 14.57 4.34Calc. 3.295c 14.458c 4.384c

Hf2SnCThis work 3.308 14.450 4.368Exp.a 3.32 14.388 4.333Calc. 3.2c 13.534c 4.228c

Nb2SnCThis work 3.244 13.754 4.240Exp.a 3.241 13.802 4.258Calc. 3.292c 14.608c 4.346c

a Ref. [33].b Ref. [7].c Ref. [23].

orbitals have been added for all atoms and valence states. Theself-consistency cycle was achieved taking 1500 points in the firstBrillouin zone. The convergence has been followed with respect tothe energy and the density.

Furthermore, we have conjointly used the ABINIT package [26],a plane-waves pseudopotential DFT code with the generalized gra-dient approximation [27], which in addition to usual ground-statecalculations, allows linear-response computations of the phononfrequencies, Born effective charges, dielectric tensors, and elasticconstants. In this work, we are using ABINIT package in order tocalculate the elastic constants.

These two DFT-based codes are therefore complementary andallows us to perform a complete investigation of M2SnC (M = Ti,Zr, Hf and Nb) properties. The all-electron potentials were replacedin ABINIT by norm-conserving pseudopotentials generated accord-ing to the Troullier–Martins scheme [28,29]. The self-consistent cy-cles were converged within a tolerance of 10�14 Ha on the potentialresidual. The electronic wave functions were expanded in plane-waves up to a kinetic energy cutoff of 60 Ha in ABINIT. For elasticconstants calculations, the Brillouin zone integration was sampledusing a 6 � 6 � 6 special k-points grid according to the Monkhorst–Pack scheme [30].

3. Geometry and structure optimization

Like all MAX phases, M2SnC belongs to P63/mmc space group.This hexagonal structure can be considered as a nearly close-packed stacking of M2SnC layers interleaved with a pure tin layerevery fourth layer. The atoms are located at the following Wyckoffpositions: M atoms at 4f ðZMÞ, Sn at 2d and C at 2a. The coordinateZM is the only internal free parameters in the crystal structure. Forthe atoms in 4f positions, the ideal values of the internal dimen-sionless parameter Z is 1/12 for metal atom M. A set of differentpredefined volumes are chosen and for each of these volumes dif-ferent calculations are made for several c/a ratio values. Duringeach step, the free parameter ZM is allowed to vary for optimiza-tion. The energy curves as function of the c/a ratio are then fittedwith a polynomial function. This allows us to find a minimum inenergy as a function of the c/a ratio for each selected volume. Theseminima give a set of energies as a function of volume in which allother parameters are optimized. In the last step, this energy versus

elaxed atomic coordinates ZM are also given. The bulk modulus (B), and the formation

Zm B NðEf Þ DEf

0.0817 170 3.30 �1.032– – – –0.0807b, 0.0826c 167b, 159.29c – �2.32b

0.0850 156 2.71 �1.269– – – –0.0864c 149c – –

0.0850 165 2.97 �1.189– – – –0.0846c 188c

0.0830 206 3.71 �1.071– – – –0.0876c 167c

Table 2Calculated elastic constants, Cij (in GPa) and anisotropy factors ðA ¼ C33=C44Þ for Ti2SnC, Zr2SnC, Nb2SnC, and Hf2SnC compounds.

C11 C33 C44 C12 C13 C66 A

Ti2SnCOur work 337.43a 329.40a 168.80a 86.01a 101.67a 125.70a 1.018a

323b 329b 125b 82b 87b 120b 0.976b

Calc.c 303 308 121 84 88 109 –

Zr2SnCOur work 269.02a 290.44a 148.48a 80.50a 107.30a 94.26a 1.080a

283b 276b 121b 76b 93b 103b 1.079b

Calc.c 279 272 111 70 89 104 –

Hf2SnCOur work 330.04a 291.63a 167.34a 53.70a 125.50a 138.17a 0.886a

318b 301b 123b 96b 99b lllb 0.978b

Calc.c 315 309 124 99 141 108 –

Nb2SnCOur work 340.94a 320.64a 183.35a 105.65a 169.00a 117.64a 0.940a

321b 311b 132b 101b 99b lllb 0.968b

Calc.c 311 306 119 96 97 109 –

a Using FP-L/APW + lo.b Using PP-PW.c Ref. [23] CASTEP (GGA).

M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500 493

volume curve is fitted using the third-order Birch–Murnaghanequation of state [31,32] to find the optimized parameters.

Our calculated values for a; c and ZM are summarized in Table 1as well as the experimental data [33] for comparison. Compared tothe experiment, the calculated a and c lattice parameters wereunderestimated about 0.85% and 0.28% for Ti2SnC, respectively.For Zr2SnC, the changes from experimental values are 0.15% and0.76%, while for Hf2SnC they are 0.36% and 0.43%, respectively.The deviations are estimated to be about 0.09% and 0.34% forNb2SnC. It should be observed that our results for Ti2SnC are closerto the experimental data than those reported in Ref. [33]. Thisslight difference originates likely from the fact we have used forthe exchange–correlation the new scheme of Wu and Cohen [25]instead of the most popular one of Perdew et al. [27] as in Ref.[7]. The formation energies DEf of these MAX phases, calculatedfrom the difference between their total energies and the sum ofthe isolated atomic energies of the pure constituents, are also re-ported in Table 1. The values for DEf are always negative, indicatingthat the formation of these MAX phases is exothermic and thatthey are intrinsically stable in agreement with the experiment[33]. The predicted bulk modulus of Ti2SnC, Zr2SnC, Hf2SnC andNb2SnC are found to be 170, 155.5, 178.2 and 206.16 GPa, respec-tively. From Table 1, it is clear that the bulk modulus of Nb2SnC arehigher than those of IVB based M2SnC compounds. Indeed the cal-culated bulk modulus of M2SnC is significantly lower than the re-lated binary carbide (for TiC, ZrC, HfC and NbN respectively).

4. Elastic stiffness tensor calculation

Let us discuss the elastic constants and mechanical parametersas obtained within the framework of the FP-L/APW + lo andpseudopotential calculations for M2SnC compounds. The valuesof the five independent elastic constants ðCij, namely C11; C12;

C13; C33; C44; while C66 ¼ ðC11 � C12Þ=2Þ summarized in Table 2were found by imposing different deformations [22,34] to theequilibrium lattice of the hexagonal unit cell and by determiningthe dependence of the resulting energy changes on the deforma-tion. In this work, we report only the independent elements re-laxed-ion elastic tensors at constant electric field frompseudopontential calculation and full potential method.

The mechanically stable phases or macroscopic stability isdependent on the positive definiteness of stiffness matrix [35].

The conditions following are known as the Born criteria [36]. Forour M2SnC compounds, we get the following conditions:

C11 � jC12j > 0

ðC11 þ C12ÞC33 � 2C213 > 0

C44 > 0

8><>: ð1Þ

At present, there are some general conclusions we would like topoint out from the obtained results. It can to be see that ourM2SnC compounds are mechanically stable because all these elasticconstants are positive and satisfy the Born mechanical stabilityrestrictions. Results obtained are in same order with available the-oretical values [23]. We also observe that the elastic constants ob-tained from FP-LAPW calculation are larger compared with valuesobtained from pseudopotential calculation.

The calculated elastic constants allow us to obtain the macro-scopic mechanical parameters of M2SnC compounds, namely theirbulk B and shear G moduli. The bulk modulus BV and the shearmoduli GV in the Voigt (V) [37] approximation are expressed as:

BV ¼19½2ðC11 þ C12Þ þ C33 þ 4C13� ð2Þ

GV ¼1

30ðC11 þ C12 þ 2C33 � 4C13 þ 12C44 þ 12C66Þ ð3Þ

The above elastic parameters are obtained from first-principle cal-culations of M2SnC monocrystals. Meanwhile, the majority of syn-thesized and experimentally examined M2SnC materials areprepared and investigated as polycrystalline ceramics [38–41], i.e.in the form of aggregated mixtures of microcrystallites with a ran-dom orientation. Thus, it is useful to estimate the elastic parametersfor the polycrystalline M2SnC materials. For this purpose we usedthe Voigt–Reuss–Hill (VRH) approximation. In this approach,according to Hill [42], two main approximations, Voigt [37] and Re-uss (R) [43], are used. In fact, in the Reuss model the bulk and shearmoduli BR and GR are given by:

BR ¼ðC11 þ C12ÞC33 � 2C2

13

C11 þ C12 þ 2C33 � 4C13ð4Þ

GR ¼52

½ðC11 þ C12ÞC33 � 2C213�C44C66

3BV C44C66 þ ðC11 þ C12ÞC33 � 2C213

� �ðC44 þ C66Þ

h i ð5Þ

Then, the Voigt and Reuss averages are limits and the actualeffective moduli for polycrystals could be estimated in the VRH

494 M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500

approach by the arithmetic mean of these two limits. In this way,when the bulk modulus BVRH and the shear modulus GVRH are ob-tained from BV ;R and GV ;R by the VRH approach in simple forms asBVRH ¼ ðBV þ BRÞ=2 and GVRH ¼ ðGV þ GRÞ=2, the averaged compress-ibility bVRH ¼ 1=BVRH and Young’s modulus YVRH may be calculatedby the expression:

YVRH ¼9BVRHGVRH

3BVRH þ GVRHð6Þ

Then, Poisson’s m was obtained for polycrystalline M2SnC speciesfrom BVRH; GVRH and YVRH as:

m ¼ 3BVRH � 2GVRH

2ð3BVRH þ GVRHÞð7Þ

The computed bulk, shear, Young’s modulus, averaged compress-ibility and Poisson’s are summarized in Table 2. From our resultswe see that the difference between bulk moduli as well as shearmoduli is comparatively small. It was found for all compounds thatBVRH > GVRH; this implies that the parameter limiting the stability ofthese materials is the shear modulus GVRH . Moreover, according tocriterion [44,45], a material is brittle (ductile) if the B/G ratio is lessthan 1.75. In our case, for all compounds, B/G is much less than 1.75,i.e. these materials will behave in a brittle manner.

Furthermore, it can be seen that the bulk, shear and Young’smodulus for Zr2SnC are rather small compared to the other nanola-minates systems, whereas the compressibility of Zr2SnCðbVRH � 0:00636 1=GPaÞ adopts the maximal value. Thus, it is obvi-ous to conclude that the Zr2SnC is a relatively soft material.

Values of the Poisson ratio m for covalent materials are smallðm ¼ 0:1Þ, whereas for ionic materials 0.25 is a typical value of m[46]. In our case, the values of m for M2SnC vary in the interval[0.19–0.25], indicating a considerable ionic contribution in intra-atomic bonding should be assumed for these materials. Besides,for covalent and ionic materials, the typical relations between bulkand shear modulus are G � 1:1B and G � 0:6B, respectively [44].Moreover, the calculated value of GVRH=BVRH vary between 0.6 and0.79, which also indicates that the mixed ionic–covalent bondingis suitable for M2SnC. Estimations of the anisotropy factorA ¼ C33=C11 are given in Table 2. For an isotropic crystal, A is equalto 1, while any value smaller or larger than 1 indicates an elasticanisotropy. The magnitude of the deviation from 1 is a measureof the degree of elastic anisotropy possessed by the crystal. Wehave calculated other estimations of elastic anisotropy frompercentage anisotropy in compression and shear, respectively, byusing the following equations [47]:

Table 3Calculated bulk moduli ðBR; BV ; BVRH ; in GPaÞ, compressibility ðb; in GPa�1Þ, shear modpercentage anisotropy in compression and shear.

Methods

Ti2SnC Zr2SnC

FP-LAPW PP-PW FP-LAPW PP-PW

BV 175.90 165.22 157.63 151.77BR 175.87 165.18 156.60 151.73BVRH 175.87 165.20 157.12 151.75GV 140.20 121.83 113.80 107.56GR 136.31 121.77 106.98 106.20GVRH 138.25 121.80 110.40 106.88BVRH=GVRH 1.272 1.356 1.423 1.411bVRH 0.00569 0.00605 0.00636 0.006YVRH 328.65 293.32 268.33 259.68m 0.188 0.204 0.215 0.214Acomp 0.0054 0.0113 0.3280 0.013Ashear 1.404 0.025 3.090 0.639

Acomp ¼BV � BR

BV þ BR� 100% ð8Þ

Ashear ¼GV � GR

GV þ GR� 100% ð9Þ

A value of zero means that the crystal shows isotropic propertiesand a value of 100% represents the maximum anisotropy. From Ta-ble 3, our calculated values show that Ti2SnC are largely isotropic incompression and anisotropic slightly in shear. While Zr2SnC, Hf2SnCand Nb2SnC are slightly anisotropic in compression and shear.Moreover, we see that these compounds will exhibit small elasticanisotropy.

One of the most important parameter that determine the ther-mal characteristics of materials is the Debye temperature ðhDÞ. As arule of thumb, a higher hD implies a higher associated thermal con-ductivity. The knowledge of such a numerical figure is essential fordeveloping and manufacturing electronic devices. At low tempera-tures, the vibrational excitations arise solely from acoustic vibra-tions. Therefore, the Debye temperatures calculated from elasticconstants at low temperatures is the same as that determined fromspecific heat measurements. In order to estimate the magnitude ofDebye temperature, we used the simple Debye–Grüneisen model.The Debye temperature can be defined in terms of the mean soundvelocity and gives explicit information about lattice vibrations[48]. It can be determined from the following equation [49]:

hD ¼hkB

3n4p

NAqM

� �� �1=3

vm ð10Þ

where h is the Planck’s constant, kB is the Boltzmann’s constant, NA

is the Avogadro’s number, q is the density, M is the molecularweight, n is the number of atoms in the unit cell, and vm is the meansound velocity given by the following relations:

v l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3BVRH þ 4GVRH

3q

� �sð11Þ

v t ¼

ffiffiffiffiffiffiffiffiffiffiGVRH

q

sð12Þ

vm ¼13

2v3

tþ 1

v3l

� �� ��1=3

ð13Þ

where v l and v t are longitudinal and transverse elastic wave veloc-ity of the polycrystalline materials and can be obtained using thepolycrystalline shear modulus and bulk modulus from Naviersequation [50]. The calculated values of Debye temperatures arelisted in Table 4. Our results predict that theses ceramics have a

uli ðGR ; GV ; GVRH ; in GPaÞ, Young’s moduli ðYVRH ; in GPaÞ, Poisson’s ratio ðmÞ, and

Hf2SnC Nb2SnC

FP-LAPW PP-PW FP-LAPW PP-PW

173.45 170.44 209.98 192.33172.92 170.43 208.98 191.81173.20 170.44 209.48 192.07137.70 114.86 134.12 112.40127.00 114.46 117.52 108.25132.35 114.66 125.82 110.32

1.300 1.486 1.665 1.74159 0.00577 0.00587 0.00477 0.00521

316.45 281.00 314.50 277.800.195 0.225 0.250 0.260

5 0.1538 0.0025 0.2386 0.13464.043 0.177 6.600 1.880

Table 4Calculated density ðq in g=cm3Þ, longitudinal, transverse and average sound velocityðv l; v t ; vm in km=sÞ, and Debye temperatures ðHD in KÞ of nanolaminate ternaryTi2SnC, Zr2SnC, Nb2SnC, and Hf2SnC.

q Vt Vl Vm HD

Ti2SnC6.473 4.337a 7.113a 4.790a 584.83a

4.621b 7.459b 5.095b 622.05b

Zr2SnC7.280 3.831a 6.357a 4.236a 482.75a

3.893b 6.464b 4.305b 490.64b

Hf2SnC11.828 3.113a 5.228a 3.446a 398.32a

3.345b 5.437b 3.691b 426.56b

Nb2SnC8.388 3.626a 6.358a 4.030a 479.70a

3.872a 6.706b 4.299b 511.74b

a Using FP-L/APW + lo.b Using PP-PW.

M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500 495

relative high hD value as to indicate that they possess a rather stifflattice and therefore good thermal conductivity. The progressivedecreasing of mean sound velocities in the Ti ! Zr ! Nb ! Hfseries further explains the proneness of lowering Debye tempera-tures along the same sequential order.

5. Electronic and bonding properties

To obtain a deeper insight into the electronic structure, we plotin Fig. 1 total density of states (TDOS). At the Fermi level ðEFÞ, thevalues of TDOS for Ti2SnC, Zr2SnC, Hf2SnC and Nb2SnC, were 3.30,

Fig. 1. Total and local densities of states (TDOS and LDOS) as a function of energy (E

2.71, 2.97 and 3.71 states per unit cell per eV, respectively. Henceone may to conclude that Nb2SnC is more conducting than Ti2SnC,Zr2SnC and Hf2SnC. More understanding can be gained by exami-nation of the local l-projected LDOS which are also displayed forselected compounds in Fig. 1. These LDOS shows several interest-ing features. In the energy range from �6 to �1.8 eV, the C p LDOShas the same shape as the d electronic states of M (M = Ti, Zr, Hf,Nb) atoms located between C layers. It is indicative of a hybridiza-tion between M d and C p states and thus of a covalent interaction.In the energy range from �2.5 to �0.6 eV, the Sn 5p states interactmainly with the M d states. But these Sn p–M d hybrids are higherin energy than the C p–M d ones (i.e. closer to the Fermi level) sug-gesting that M–C bonds are stronger than M–Sn bonds. It is thisstrong C p–M d hybridization which stabilizes the structure ofM2SnC, a general trend in MAX phases [51–53].

The calculated Fermi surfaces are shown in Fig. 2. All the Fermisurfaces (FS) contain tubes with ‘‘hexagonal” cross sections cen-tered along the C–A direction. For Ti2SnC, Hf2SnC and Zr2SnC, thecentre of Fermi surface consists of several sheets. The first sheetis cylindrical-like, and the last sheet adopts a prismatic-like. Whilefor Nb2SnC, the centre of FS contain filled prismatic-like form.Additional sheets appear in all compounds at the corners of theBrillouin zone, around the H–K direction except Nb2SnC that it con-tain the other sheets in L–M direction. They arise from the doublydegenerate bands along the A–H direction.

In order to understand the nature of chemical bonding, we dis-play, in Fig. 3, the contours of charge densities in ð11 �20Þ plane forthe considered systems. It is known that MAX phases are usuallystacks of ‘hard’ M–X bond and ‘soft’ M–A bond along c directions.From Table 5, the M–C bond strength is much stronger than M–Sn. In addition to the Ti–C hybridization, there is also a stronginteraction between M and Sn atoms. This means that the ‘soft’M–C bond is strengthened in M2SnC. Following previous works

) for M2SnC (M = Ti, Zr, Hf, Nb). The vertical dotted line denotes the Fermi level.

Fig. 2. Fermi surfaces for: (a) Ti2SnC, (b) Zr2SnC, (c) Hf2SnC and (d) Nb2SnC.

496 M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500

[7], it can seen that the strengthening effect contributed consider-ably to the increase in bulk modulus and shear modulus of MAXphase. According to the correlation between the shear modulus,bulk modulus and the hardness, it can be expected that Nb2SnCwill possess high hardness compared others compounds. It is clearthat a covalent interaction occurs between the constituting ele-ments due to the fact that states are degenerate with both angularmomentum and lattice site, and also due to the difference in elec-tronegativity between the comprising elements, some ionic char-acter can be expected. In fact, the bonding character may be

described as a mixture of covalent–ionic and, due to the d-reso-nance in the vicinity of the Fermi level, be partly metallic.

6. Optical properties

The investigated compounds Ti2SnC, Zr2SnC, Hf2SnC andNb2SnC have a hexagonal symmetry. For compounds having hex-agonal or tetragonal symmetry, the experiments are performedwith electric vector E

!parallel or perpendicular to the c axis. The

corresponding dielectric functions are ezzðxÞ and exxðxÞ. The calcu-lations of these dielectric functions involve the energy eigenvaluesand electron wave functions. These are natural outputs of bandstructure calculations. We have performed calculations of theimaginary part of the interband frequency dependent dielectricfunction using the expressions [54]:

ezz2 ðxÞ ¼

12mx2

ZBZ

Xnn0

jPZnm0 ðkÞj

2dSk

5xnm0 ðkÞð14Þ

exx2 ðxÞ ¼

xmx2

ZBZ

Xnn0

jPXnm0 ðkÞj

2 þ PYnm0 ðkÞj

2h i

dSk

5xnm0 ðkÞð15Þ

The above expressions are written in atomic units with e2 ¼ 1=m ¼2 and �h ¼ 1, where x is the photon energy. PX

nm0 ðkÞ and PZnm0 ðkÞ are

the X and Z component of the dipolar matrix elements betweeninitial jnkiand final jn0ki states with their eigenvalues EnðkÞ andEn0 ðkÞ, respectively. xn0nðkÞ is the energy difference, wherexn0mðkÞ ¼ EnðkÞ � En0 ðkÞ and Sk is a constant energy surface withSk ¼ fk;xnn0 ðkÞ ¼ xg.

For the metallic materials there are two contribution to eðxÞ,namely intra-band and inter-band transitions. As the investigatedcompounds are metallic we must include the Drude term (intra-band transitions) [55]:

e2ðxÞ ¼ e2interðxÞ þ e2intraðxÞ ð16Þ

where

exx2intraðxÞ ¼

xxxp s

xð1þx2s2Þ ð17Þ

where xp is the plasma frequency [56] and s is the mean free timebetween collisions.

x2p ¼

8p3

Xkn

#2kndðeknÞ ð18Þ

where enkðxÞ is EnðkÞ � EF and #kn is the electron velocity (in basalplane) squared.

Fig. 4a, shows the calculated eðxÞ and exxðxÞ and ezzðxÞ for theinvestigated compounds using GGA. We have performed calcula-tions of eðxÞ, with and without the Drude term. The effect of theDrude term is significant for energies less than 1 eV. The sharp riseat low energies is due to the Drude term. The effect of the Drudeterm is significant for energies less than 1 eV, the sharp rise atlow energies.

As can be seen these compounds have a different optical spec-trum and the optical absorption varies from one compound to an-other. That is attributed to the fact that the band structures forthese compounds are different with minor similarities, and thesymmetries of the wave functions dictate that the selection rulesare fully reflected in the matrix moment elements (MME’s). Thesedifferences with minor similarities cause some changes in the opti-cal transitions resulting in changes the peaks positions and thepeaks height. We notice that in the Ti2SnC compound, at low ener-gies less 0.2 eV, the ezzðxÞ component show the major contribution

M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500 497

with considerable anisotropy with exxðxÞ component. SubtractingTi by Hf causing to shift the entire structures towards higherenergies and reducing all the peaks height with respect to Ti2SnCcompound and making both exxðxÞ and ezzðxÞ components are iso-tropic at the low energies up to 1.8 eV and 2.8 eV and high energiesbetween 5.0 and 10.0 eV except at 7.0 eV. Whereas subtracting Tiby Nb causing to push all the structures further towards higherenergies with reducing the peaks height. When we subtract Ti byZr cause to shift whole the structure towards lower energies withincreasing the peaks height and show considerable anisotropy be-tween exxðxÞ and ezzðxÞ at whole the energy range and again push-ing the exxðxÞ component to be the dominant in the lower energies

Fig. 3. Charge-density distributions in ð11 �20Þ p

less than 0.2 eV. The shifting of the optical structures is consistentwith our calculated band structure and density of states.

Generally the peaks in the optical response are determined bythe electric–dipole transitions between the occupied and unoccu-pied bands. These peaks can be identified from the band struc-ture. In order to identify these peaks we need to look at theoptical transition dipole matrix elements. When we analyze theoptical spectra of e2ðxÞ, we found that the transitions which areresponsible for the structures below 5 eV in e2ðxÞ, are dominatedby transitions from bands just below Fermi energy ðEFÞ to bandsjust above it. Whereas the transitions which are responsible forthe structures above 5 eV in e2ðxÞ, are dominated by transitions

lane of Ti2SnC, Zr2SnC, Hf2SnC and Nb2SnC.

Fig. 3 (continued)

498 M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500

from top (and bottom) of the occupied states to the unoccupiedstates.

The optical properties of the Ti2SnC and Zr2SnC compoundsshow highly anisotropic in the energy range between 0.0 and

Table 5Calculated bond lengths for M2SnC with (M = Ti, Zr, Hf and Nb).

M–C Sn–C M–Sn

Ti2SnC 2.126 3.861 2.923Zr2SnC 2.302 4.15 3.100Hf2SnC 2.270 4.086 3.055Nb2SnC 2.193 3.915 2.963

7.0 eV for Ti2SnC and between 0.0 and 7.0 eV for Zr2SnC while itis isotropic in the higher energy range. While the Hf2SnC andNb2SnC compounds are isotopic at lower and higher energies.

It would be worthwhile to attempt to identify the band transi-tions that are responsible for the spectral structures exxðxÞ andezzðxÞ using our calculated band structure, for more details see[57]. As a prototype we have shown the band structure for Ti2SnC(Fig. 4b). Fig. 3b shows the transitions which are responsible forthe spectral structures exxðxÞ and ezzðxÞ. For simplicity we have la-beled the transitions in Fig. 3b, as A, B, and C. The transitions A areresponsible for the structures in exxðxÞ and ezzðxÞ for energies up to5.0 eV, transitions B are responsible for the structures in the energy

Fig. 4a. Calculated exx2 ðxÞ (dashed curve) and ezz

2 ðxÞ (solid curve) for the investigated compounds using GGA.

Fig. 4b. Calculated band structure of Ti2SnC along with the optical transitions. Thezero of the energy is taken at the Fermi energy EF shown by a line.

M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500 499

range 5.0–10.0 eV, and the transitions C are responsible for thestructures between 10.0 eV and 14.0 eV.

7. Conclusion

In summary, we have carried out ab-initio calculations of struc-tural, elastic, electronic and optical properties of M2SnC, withM = Ti, Zr, Hf and Nb using a hybrid all-electron full-potential lin-earized augmented plane-wave method and a pseudopotential ap-proach within GGA-WC. It has been shown that the structuralparameters obtained after relaxation are in good agreement withthe experimental ones. These materials have an interesting bulkmodulus among the MAX phases experimentally or theoreticallystudied so far. From the plane-wave pseudopotential method wehave calculated the elastic constants and derived the bulk modulusand the shear modulus. All the investigated compounds aremechanically stable and the obtained values of the bulk modulusand the shear modulus show that these materials with high B orG are likely to be hard materials. Among these systems, Ti2SnC isductile. Moreover, the electronic structure showed that theseMAX compounds have a metallic character which allowed us tocalculate Fermi surfaces. Hence it is tempting to conclude thatNb2SnC is more conducting compared to the other compounds.The analysis of charge densities contours, leads us to conclude thatthe bonding character in these compounds may result from as amixture between covalent–ionic and metallic behavior. When we

500 M.B. Kanoun et al. / Computational Materials Science 47 (2009) 491–500

observe more carefully, the calculated optical properties via thecalculated optical spectrum and the optical absorption, we mayconclude that the slight differences observed in these curves aremainly attributed to the fact that the band structures for thesecompounds are different and have consequences in optical transi-tions resulting in changes the peaks positions.

Acknowledgments

Calculations using Wien2k and ABINIT have been performed atNamur Interuniversity Scientific Computing Facilities (Namur-ISCF, Belgium). Namur-ISCF is a common project between FNRS,SUN Microsystems, and Les Facultés Universitaires Notre-Dame dela Paix (FUNDP). We acknowledge F. Wautelet (ISCF-FUNDP) forpermanent computer assistance. Alberto Otero de la Roza fromUniversidad de Oviedo (spain) is acknowledged for providing usRunwien code.

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