interatomic bonding, elastic properties, and ideal strength of

Interatomic bonding, elastic properties, and ideal strength of transition metal aluminides:A case study for Al3„V,Ti …

M. Jahnátek,1 M. Krajčí,1,2 and J. Hafner21Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK 84511 Bratislava, Slovakia

2Institut für Materialphysik and Center for Computational Materials Science, Universität Wien, Sensengasse 8/12,A-1090 Wien, Austria

(Received 9 March 2004; revised manuscript received 1 June 2004; published 5 January 2005)

On the basis ofab initio density-functional calculations we have analyzed the character of the interatomicbonding in the intermetallic compounds Al3sV,Tid with theD022 andL12 structures. In all structures we foundan enhanced charge density along the Al-sV,Tid bonds, a characteristic feature of covalent bonding. The bondsin Al3V with the D022 structure are more saturated and stronger than the corresponding bonds in Al3Ti. Highsymmetry of the transition metal sites in theL12 structure leads to higher metalicity of alloys assuming thisstructure. The bond strength is quantitatively examined by tensile deformations. The ideal strength of Al3V andAl3Ti under uniaxial tensile deformation was found to be significantly higher than that of both fcc Al and bccV. We investigated also the changes of the interatomic bonding in Al3V during tensile deformation. We foundthat the covalent interplanar Al-V bonds disappear before reaching the maximal stress. The weakening of thebonding between the atomic planes during the deformation is accompanied by a strengthening of in-planebonding and an enhanced covalent character of the intraplanar bonds. Interplanar bonding becomes moremetallic under tensile deformation.

DOI: 10.1103/PhysRevB.71.024101 PACS number(s): 61.50.Lt, 62.20.Dc, 71.20.Be, 71.20.Lp

I. INTRODUCTION

The mechanical properties of materials have been for along time studied extensively by various experimental meth-ods. The progress in the development ofab initio methodsduring the past decade allows to approach this interestingtopic also from the theoretical side. The elastic constantsdescribe the mechanical properties of materials in the regionof small deformations where the stress-strain relations arestill linear. An important quantity, which describes the me-chanical properties of the material beyond the linear region isthe ideal(or theoretical) strength. It represents the stress nec-essary for de-cohesion or fracture of the crystal. The idealstrength represents upper strength limit for a solid under agiven load. Morris and Krenn1 have investigated the condi-tions of elastic stability that set the upper limits of mechani-cal strength. The ideal strength is one of few mechanicalproperties that can be calculated from first principles. It is aninherent property of the crystal lattice and thus offers insightinto the correlation between the intrinsic chemical bondingand the symmetry of a crystal and its mechanical properties.2

Recently several theoretical studies of the ideal strengthof elemental metals have appeared.3–15 Černý et al.6 investi-gated elastic properties of Fe, Co, Ni, and Cr crystals underisotropic tri-axial deformation. The ideal shear strength ofthe bcc metals Mo, Nb, and W were studied by Luo, Krenn,and co-workers.7–9 Roundy, Krenn, and co-workers10,11 cal-culated ideal shear strength of fcc Al and Cu.

The most straightforward way to calculate the elastic lim-its from first principles is to stretch or shear the crystal in thedesired direction and to calculate the changes of the totalenergy and of the induced stress. The stress associated with aparticular strain is proportional to the derivative of the en-ergy with respect of the strain. One has to calculate the stress

as a function of strain, and to look for the maximum of thestress. The elastic limits can be strongly anisotropic.16 For acalculation of the ideal strength one has to deform the crystalalong a path of the weakest deformation. The maximumstress is the ideal strength, and is associated with the inflec-tion point in the curve of energy as a function of strain alongthe path toward the saddle point. The paths of weakest de-formation for common crystals are known.9,10 For an fcccrystal it is a shear deformation in ak112l direction in a{111} plane. For a bcc crystal it is a slip of{110} planesalong thek111l direction. This procedure yields an upperbound on the theoretical strength. Instabilities along eigen-vectors perpendicular to the direction of the stretch may ap-pear at lower values of the stress. For instance, bcc crystalsbecome unstable after a relatively small stretch along[100].This instability is responsible for their tendency to cleavealong{100} planes. Very recently Clatterbucket al.17 calcu-lated phonon spectra of aluminum as a function of strain andfound that phonon instabilities may occur before a materialbecomes elastically unstable.

The calculations of elastic limits of a solid are computa-tionally expensive, particularly when the distorted solid has alow symmetry. The crystal must be reconfigured at everystep to relax the lateral stresses. Such calculations are com-putationally very difficult even for simple monoatomic cubicsystems. Extension of these studies to more complex crystal-line structures requires an extreme computational effort closeto and often beyond what is now computationally feasible.

Transition metal aluminides are known to be of greattechnological importance. Al based compounds of transitionmetals(TM) are among the most promising candidates forhigh-performance structural materials.18,19 The reported ten-sile strength of, e.g., nanocrystalline Al94V4Fe2 is above1300 MPa, which exceeds the strength of usual technical

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steels.18 The enhanced strength of these materials is mostlyattributed to their nano-crystalline and nano-quasicrystallinestructure. However, in transition metal aluminides inter-atomic bonding between Al and transition metals may ex-hibit enhanced covalency, with a significant influence on themechanical properties. Covalency can increase the strengthof interatomic bonding20,21 and consequently increase themechanical strength and hardness, but on the other hand itleads also to an undesired higher brittleness.

Ab initio calculations of the elastic limits of transitionmetal aluminides are a great challenge. The crystals struc-tures of transition metal aluminides can be quite complex:For instance, Al10V has 176 atoms per unit cell. A search forthe path of easiest deformation can be a difficult task. In thepresent paper we attempt to achieve a deeper understandingof the relation between interatomic bonding and the me-chanical properties of the transition metal aluminides. Forthis study we have chosen Al3V and some related com-pounds as for these systems the enhanced covalency in thebonding is known. The trialuminide intermetallic Al3Tihas many attractive characteristics such as low densitys<3.3 g/cm3d, high hardness, good oxidation resistance, andmoderately high melting temperatures<1700 Kd.22 There-fore, this intermetallic may be useful for elevated tempera-ture applications.22 Both stable Al3sV,Tid compounds crys-tallize in the D022 sAl3Tid crystal structure. Nuclearmagnetic resonance(NMR) studies suggest a strong direc-tional bonding23,24 in Al3V. This conjecture is further cor-roborated by the structural analysis: The short Al-Vdistances25 indicate that the bonding may have a covalentcharacter. Recently, we have performed a detailed study ofthe electronic structure and bonding in Al3V.20 The elec-tronic density of states of this compound exhibits an unusu-ally deep pseudogap very close to the Fermi level. We havedemonstrated that a strong hybridization between the Alss,pdand Vsdd orbitals which is responsible for forming of a deeppseudogap near the Fermi level is also associated with theformation of covalent bonds. We analyzed the charge distri-bution in the elementary cell and found an enhanced chargedensity along the Al-V bonds and certain Al-Al bonds thatis characteristic for covalent bonding. The character of thebonds was investigated by the analysis of the density ofstates projected on bonding and antibonding configurationsof atomic orbitals. This analysis reveals that the Al-V bondsin theD022 structure are almost saturated. Because of a lowerd-band filling a similar result is not expected for Al3Ti withthe same crystal structure.

In addition to the stable Al3V phase in theD022 structurethere exists a metastable Al3V phase with theL12 sCu3Audstructure. It is also known that Al3Ti can be transformed tothe high-symmetryL12 cubic structure by alloying with tran-sition metals with mored-electrons in the valence band.22

Both theD022 and theL12 structures can be considered as aspecial decoration of a face-centered cubic lattice. In theL12structure the TM site has full octahedral symmetry, while inthe D022 structure the symmetry of this site is lower. It ap-pears that just this high symmetry of the TM site in theL12structure prevents a saturation of all Al-TM bonds. Highermetalicity of bonding is generally considered to be reason fora better ductility of these materials.

To investigate the relation between interatomic bondingand the mechanical properties we first calculated the elasticproperties of Al3sV,Tid compounds. Then we investigatedthe stress-strain relation under triaxial(homogenous) anduniaxial deformations and studied in detail the change ofinteratomic bonding under these tensile strains.

For the study of the uniaxial tensile deformation we havechosen the[001] direction. This choice is determined by thetetragonal symmetry of the crystals and the orientation of theobserved covalent bonds. The orientation of the bonds in theD022 structure indicates that the[001] direction is the direc-tion of the most difficult deformation. It would be interestingto know also a direction of the weakest deformation. It isknown that for fcc-based phases the weak direction for atensile deformation isk110l. Although the D022 and L12phases are fcc-like, the weakest direction is not determinedby the symmetry of the crystal alone but also by the aniso-tropy of the bonding which is very pronounced in tetragonalphases, and particularly in the case of Al3V in D022. Thedetermination of the weakest direction would be an interest-ing but rather extensive study which exceeds the scope ofour work.

In the next section computational tools are briefly de-scribed. In Sec. III we present a picture of bonding in theD022 and L12 structures. In Sec. IV we study the elasticproperties and the tensile deformation of Al3sTi,V d com-pounds and compare them with the properties of the constitu-ent metals: fcc Al, bcc V, and hcp Ti. In Sec. IV D we dem-onstrate how the interatomic bonding in Al3V changesduring the tensile deformations of the elementary cell.

II. METHODOLOGY

The electronic structure calculations have been performedusing two different techniques: The plane-wave based Vi-ennaab initio simulation packageVASP26,27has been used forcalculations of the electronic ground-state and for the opti-mization of the atomic volume and unit-cell geometry. TheVASP-program has also been used to calculate charge-distributions. In its projector-augmented-wave(PAW)version27 VASP calculates the exact all-electron eigenstates,hence it can produce realistic electron-densities. The plane-wave basis allows to calculate Hellmann-Feynman forcesacting on the atoms and stresses on the unit cell. The totalenergy may by optimized with respect to the volume and theshape of the unit cell and to the positions of the atoms withinthe cell. We usedVASP for highly accurate calculations of thetotal energy changes necessary for the determination of theelastic constants.VASP was successfully used also for thecalculation of stresses during our tensile computer experi-ments. All our calculation were performed with PAWpseudopotentials in generalized gradient approximation(GGA) according to Perdew and Wang.28

For Al and Ti we used the standard PAW pseudopotentials(PP) distributed by VASP with reference configurations

13Al: fNeg3s23p1 and 22Ti: fArg3d24s2. In the case of vana-dium we obtained better results with PP which treat the 3-pcore electrons as valence electrons, in the reference configu-ration 23V: fArg3p64d44s1. It is well known that the 3-p en-

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ergy levels of the early transition metals are close to thebottom of the valence band and exhibit non-negligible broad-ening. This seems to be the reason why the standard vana-dium PP does not reproduce the V-V interaction correctly.However, as in the studied Al-V compounds there are notany direct V-V interactions we could use the standard PP forthese compounds.

A plane-wave-based approach such as used inVASP pro-duces only the Bloch-states and the total density of states(DOS), a decomposition into local orbitals and local orbital-projected DOS’s requiring additional assumptions. Toachieve this decomposition, self-consistent electronic struc-ture calculations have been performed using the tight-binding linear muffin-tin orbital(TB-LMTO) method29,30 inan atomic-sphere approximation(ASA). The minimal LMTObasis includess, p, andd-orbitals for each Al and TM atom.The two-center TB-LMTO Hamiltonian has been constructedand diagonalized using the standard diagonalization tech-niques. The TB-LMTO basis is used to construct the symme-trized hybrid orbitals from which the DOS projected onbonding and antibonding configurations are calculated. Thedifference between bonding and antibonding configurationsof two orbitalsf1 andf2 located on two neighboring atomsis proportional tokf1udsE−Hduf2l. This information is es-sentially equivalent to the crystal-orbital overlap populations(COOP) introduced by Hoffmann.31 We note that also a moresophisticated approach to the characterization of the inter-atomic bonding has been proposed.32,33

III. STRUCTURE AND INTERATOMIC BONDING

A. Al 3„V,Ti … in the D022 crystal structure

Both stable Al3V and Al3Ti phases crystallize in theD022sAl3Tid crystal structure. The space group is I4/mmm(No.139). The primitive cell contains 4 atoms. The tetragonalelementary cell consists of two primitive cells, see Fig. 1(a).All atoms are located on the sites of a slightly tetragonallydeformed face-centered-cubic(fcc) lattice. The elementarycell consists of two fcc units stacked along thez-direction. Inthe z=0 plane the vertices of the cubic lattice are occupiedby transition metal(TM) atoms and the face-centers by alu-minum atoms. In thez=0.5 plane the arrangement is just theopposite. The planesz=0.25 andz=0.75 are occupied byaluminum atoms only. In the elementary cell there are oneTM crystallographic site(Wyckoff notation 2a) and two alu-minum sites Al1 (2b) and Al2 (4d). Each TM atom has 4 Al1neighbors and 8 Al2 neighbors. Al1 neighbors are located inthe sx,yd plane atz=0 andz=0.5 on vertices of a squarecentered by the TM atom. Considering only nearest neigh-bors the point-group symmetry of the TM-site would beOh,but nonequivalent neighbors from the second and highershells reduce the point-group symmetry toD4h. An Al1 atomsees 4 TM neighbors on the vertices of a square and 8 Al2neighbors forming a tetragonal prism centered by the Al1atom. Al2 has four Al2 neighbors in thesx,yd plane formingtogether a square grid of aluminum atoms. Al2 has also 4 Al1and 4 TM neighbors, both with tetrahedral coordination. Insummary, the TM atom has 12 aluminum nearest neighbors,

both aluminum atoms have 4 TM neighbors and 8 aluminumneighbors in the first coordination shell.

TheD022 structure may be considered as a superstructureof the simple cubicL12 cell, described in the next subsection.The D022 unit cell consists of twoL12 cubes stacked along,say, thez direction, with af1/2,1/2,0g antiphase shift be-tween the cubes.34 This structure can be also viewed as astacking of two types of tetragonal planes. The planeA con-sists of Al1 and TM atoms occupying vertices of a squarenetwork, the planeB consists of Al2 atoms only. The stack-ing of the planes isABA8BA. . ., whereA8 is a theA planeshifted horizontally with respect toA by a half of a squarediagonal.

The lattice parameters of the elementary cells are given inTable I. We note that during the tensile deformations dis-cussed later in this paper the distance between planes in-creases with increasingc parameter while thea parametershrinks.

B. Al3„V,Ti … in the L12 crystal structure

The L12 sCu3Aud crystal structure is similar to theD022

structure, but its symmetry is higher. The space group isPm3m (No. 221), all atoms are located on the sites of a

FIG. 1. An elementary cell of Al3V in the D022 (a) andL12 (b)structures. Aluminum atoms are represented by smaller spheres, va-nadium atoms are represented by bigger balls; Al1 atoms are black,Al2 atoms are gray.

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face-centered-cubic lattice, see Fig. 1(b). TM atoms are lo-cated on the vertices of the cubic lattice, aluminum atomsoccupy the face centers. In contrast to theD022 structurethere is only one Al site(3c) and one TM site(1a). Analuminum atom has 12 nearest neighbors, 8 aluminum, and 4TM atoms. A TM atom has 12 equivalent aluminum nearestneighbors. The total number of nearest neighbors around alu-minum and TM atoms is thus the same as in theD022 struc-ture. A notable difference between both structures is that allnearest-neighbor Al-TM and AluAl distances are the sameand that the point-group symmetry of the TM site is exactlyOh. TheL12 structure can be viewed as anABAB. . . stackingof the sameA andB planes as in theD022 structure.

C. Electronic structure

The electronic structure of Al-TM systems typically ex-hibits a narrow TMd-band superposed on a parabolic-like Alband. The total density of states is more or less modulated byvan Hove singularities, bonding-antibonding d-band splittingand possible hybridization effects. For transition metals theFermi level is located in a region of high density of states,which results in the characteristic metallic properties of thesystem.

The electronic structure of both Al3V and Al3Ti is wellknown.35,36,20 Figure 2 compares the total density of states(DOS) of Al3Ti and Al3V, both in theD022 structure andAl3V in the L12 crystal structure. Around the Fermi level theTM contribution to the DOS is almost entirely formed by thed-electrons. The most interesting feature is the deep mini-mum at the Fermi level in the DOS of Al3V in the D022structure. Although the V site in theL12 structure has fullOhsymmetry, the splitting of the vanadiumd-band is not soclearly resolved as in theD022 structure where the point-group symmetry of the vanadium site is onlyD4h. The split-ting of thed-band in theD022 structure, therefore, cannot be

attributed to theeg− t2g splitting of d-orbitals in a local fieldwith Oh point-group symmetry. Our recent study20 revealedthat the origin of the deep minimum is in the bonding-antibonding splitting of molecular orbitals formed predomi-nantly bysp3 hybrids on the Al2 sites andsdzx,dyzd orbitalson the vanadium sites. The bonding of the Al2 atoms with Vexhibits enhanced covalency as discussed in next Sec. III D.The similarity of the shape of the DOS of Al3Ti with that ofAl3V is not surprising as both are of the same crystal struc-ture. The minimum of the DOS of Al3Ti falls <0.85 eVabove the Fermi level. The minimum has apparently thesame origin as in Al3V, but the position of the Fermi levelindicates that the Al2-Ti bonds are not fully saturated.

D. Charge densities and bonding

Using the VASP program we calculated the charge-density distribution in the elementary cells of Al3V andAl3Ti. Figure 3 compares a contour plot of the valence-charge distribution in the elementary cells of theD022 andL12 structures for thesx,zd plane aty=0.5. If the character ofthe bonding is purely metallic, the charge distribution amongatoms should be homogenous. A possible covalency is indi-cated by an enhanced charge distribution along connectionsbetween atoms. In Fig. 3 we see regions of enhanced charge-density between the TM atom and neighboring Al atoms. ForAl3V and Al3Ti in the D022 structure we observe an en-hanced charge-density approximately halfway between thecentral TM atom and the neighboring Al2 atoms. Thesemaxima of charge density correspond tosp3 hybrids on theAl2 atoms are bonded withdzx orbitals on the TM atoms(seebelow). A similar bonding is observed also for Al3V andAl3Ti in the L12 structure. Here the enhanced charge distri-bution between Al and TM atoms corresponds to thesp2dhybrid orbitals on Al atoms bonded withdzx orbitals on thevanadium atoms. The maximum of the charge density in

TABLE I. Computed and experimental(Ref. 45) structural parameters for investigated materials.

System Structure a (Å) c (Å) c/a V0 sÅ3/atomd

Al fcc 4.046 ¯ ¯ 16.56

Exp. 4.045 ¯ ¯ 16.56

V bcc 2.995 ¯ ¯ 13.43

Exp. 3.022 ¯ ¯ 13.80

Ti hcp 2.919 4.621 1.583 13.43

Exp. 2.950 4.684 1.587 13.80

Al3V D022 3.766 8.312 2.207 14.73

Exp. 3.840 8.579 2.234 15.81

Al3V L12 3.897 ¯ ¯ 14.79

Al3Ti D022 3.851 8.578 2.227 15.93

Exp. 3.854 8.584 2.227 15.94

Exp. (Ref. 46) 3.851 8.611 2.236 15.96

Al3Ti L12 3.977 ¯ ¯ 15.74

Exp. (Ref. 46) 3.967 ¯ ¯ 15.73

Al10V cF176 14.460 ¯ ¯ 17.18

Exp. 14.492 ¯ ¯ 17.29


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halfway between the central V atom and the Al2 atoms(marked in Fig. 3 by a dot) has in the case of theD022structure the value of 0.24 electrons/Å3. In the case of theL12 structure this value of 0.23 electrons/Å3 is little lower.In the case of the Al3Ti compounds for both crystal struc-tures we found the same value of 0.22 electrons/Å3.

A similar analysis was performed for other bonding con-figurations in the elementary cells. A detailed discussion ofthe results for Al3V can be found in our previous work.20 Insummary, we identified three types of bonds, as sketched inFig. 4: Each vanadium atom forms two types of bonds:V-Al 2 bonds are marked as dashed lines, V-Al1 bonds aredisplayed as dash-dotted line segments. The Al2 atoms formalso Al2-Al2 bonds marked in Fig. 4 by dotted lines. Allbonds are characterized by an enhanced degree of covalency.Al1 and Al2 atoms are also nearest neighbors, but from theanalysis of the charge density distribution it is obvious that

FIG. 2. Comparison of the density of states of Al3V and Al3Tiin theD022 andL12 structures. Al3Ti in D022 (a), Al3V in D022 (b),Al3Ti in L12 (c), and Al3V in L12 (d). Thin lines represent thecontribution fromd-electrons.

FIG. 3. Comparison of the contour plots of valence electrondensity in thesx,zd plane of theD022 and L12 structures fory=0.5. High charge density—black regions corresponds to the TMatoms. Positions of the aluminum atoms are marked by small blacksAl1d and graysAl2d circles. The islands of enhanced charge be-tween the central TM atom and neighboring Al atoms marked byblack dots. Al3V in D022 (a), Al3Ti in D022 (b), Al3V in L12 (c),and Al3Ti in L12 (d).

FIG. 4. An elementary cell of Al3V in the D022 structure. Posi-tions of aluminum atoms are represented by smaller blacksAl1d andgray sAl2d circles, positions of vanadium atoms are represented bylarger circles. Three types of covalent bonds are sketched: V-Al2

bonds are marked as dashed lines, V-Al1 bonds are displayed asdash-dotted lines, Al2-Al2 bonds are marked in by dotted linesegments.


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there is no significant bonding between them. The picture ofbonding in Al3Ti is very similar to that in Al3V. The maindifference are less saturated and hence weaker Ti-Al bonds.A comprehensive analysis of the stability of Al3Ti inter-preted in a chemical language of interatomic bonding wasperformed recently by Bester and Fähnle33 and leads to thesame conclusion.

To gain a deeper understanding of the bonding, we con-structed sets of symmetrized hybrid orbitals oriented alongthe bonds and calculated the density of states projected ontobonding and antibonding combinations of these symmetrizedorbitals. The difference of these projected DOS’s is essen-tially equivalent to the differential crystal orbital overlappopulation(COOP) defined by Hoffmann.31 Figure 5 com-pares the projected DOS’s for Al3V and Al3Ti in the D022and in theL12 structures. In addition to the bonding and

antibonding DOS the difference between bonding and anti-bonding densities is presented.

The character of bonding in theD022 structure can bedescribed as follows:(i) VsTid -Al1 bonds. The V(Ti) atominteracts with four Al1 atoms in the A plane(see dot-dashedline segments in the Fig. 4). The corresponding set of sym-metrized hybrid orbitals centered at Al1 is sp2d with Al1, s,px, py, anddxy orbitals. From the point of view of the V(Ti)atom the VsTid -Al1 bonds are oriented from the centralV(Ti) atom to four Al1 atoms located at the vertices of asquare. The corresponding set of symmetrized orbitals isagainsp2d. However, as the contribution ofp-states to theTM DOS is negligible and the contribution ofs-states issmall, thedxy orbital is dominant. The calculated projecteddensity of states shows that for Al3V the states below theFermi level have almost completely bonding character whileabove the Fermi level the states are antibonding. For Al3Tithe bonding orbitals are not completely saturated.

(ii ) VsTid -Al2 bonds. The V(Ti) atom is bonded with fourAl2 atoms located in vertices of a tetragonal prism, seedashed lines in the Fig. 4. These bonds mediate inter-planarbonding between atomic planesA, B and A8, B. The Al2atoms have four V(Ti) neighbors located at the vertices of atetrahedron. The symmetrized orbitals on the Al2 atoms arethussp3 hybrids. We found that bonding from V(Ti) side ismediated byd4 hybrid-orbitals formed bydz2, dx2−y2, dzx, dyz

states(or sd3 hybridization if we considers orbital instead ofdz2). Four d4 hybridized orbitals possessing inversion sym-metry point towards the vertices of a tetragonal prism. Amore detailed analysis of the VsTidsd4d -Al2ssp3d bondinghas shown that the main bonding contribution out of the fourd4 hybridized orbitals comes from the componentsdzx anddyz both belonging to theEg irreducible representation. Fig-ures 5(a) and 5(b) represent the bonding and antibondingdensities corresponding to bonding of thedzx orbital on theV(Ti) atoms with sp3 hybrids on Al2 atoms. While forV-Al 2 the bonding-antibonding projected DOS changes signalmost exactly at the Fermi level, for Ti-Al2 this change ofthe sign is<0.85 eV above the Fermi level.

(iii ) Al2-Al2 bonds. Each Al2 atom has four Al2 neighborslocated at the vertices of a square located in planeB. Thesymmetrized orbitals correspond to ansp2d hybridization.Although the bond exhibits enhanced covalency, it is onlypartially populated by electrons. The calculated bonding-antibonding projected DOS does not change its sign at theFermi level, these bonds are not fully saturated.

The picture of bonding in Al3sV,Tid in the L12 structureis more simple as here only one type of bonding exists,namely VsTid -Al bonding. Each TM atom has twelve Alneighbors located in three orthogonal planes. The bonding ineach plane is the same. From the TM side the bonding ismediated predominantly byd-orbitals with eg symmetry,from the Al side we havesp2d hybrids. Figure 5(c) shows thebonding and antibonding densities of states in Al3V. Thesymmetry of the local neighborhood of the Al site favorsparticipation of Al d-states in the bonding. The bond is notsaturated as the bonding-antibonding projected DOS changessign <1 eV above the Fermi level. In Al3Ti [Fig. 5(d)] thesaturation of the Ti-Al1 bonds is even smaller.

FIG. 5. The density of states projected on bonding(B) and an-tibonding(A) combinations of symmetrized hybrid orbitals locatedon atoms(dotted lines). In addition the difference(B-A) betweenbonding and antibonding DOS is presented by a thick line. Thisfigure compares TMsdzxd−Al2ssp3d bonding for Al3Ti in D022 (a),Al3V in D022 (b), and TMsdxzd−Al1ssp2dd bonding for Al3Ti in L12

(c), and Al3V in L12 (d).


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Comparing the total DOS’s(Fig. 2) and the projectedDOS’s (Fig. 5) of both Al3V crystal structures we see thatwhile the high vanadiumd-peak at<0.8 eV above the Fermilevel in the DOS ofD022 has antibonding character, the simi-lar high vanadiumd-peak just above the Fermi level in caseof L12 structure has still bonding character. This will be con-firmed also in Sec. IV C by examination of the variation ofthe strength of bonding under tensile deformation.

IV. INTERATOMIC BONDING AND MECHANICALPROPERTIES

A. Elastic properties

For the calculation of the elastic constants we used thesymmetry-general least-squares extraction method proposedby Le Page and Saxe.37 This novel method allows automatedapproach to the calculation of a set of elastic constants for acrystal of arbitrary crystalline symmetry. The input consistsof a set of values of the total energy for special strainedstates of the crystal. We implemented the method in connec-tion to VASP. The combination of this two programs creates avery powerful tool for calculation of elastic constants.

All elastic constants and related quantities are in Table II.In this work we are dealing with crystals of three differentsymmetry classes: cubic(bcc, fcc,L12, and cF176 in Pearsonnotation), hexagonal(hcp) and tetragonalsD022d. For eachsymmetry there exists a set of independent elastic constants.For cubic symmetry there are three independent elastic con-stants, C11, C12, and C44, for a tetragonal crystal with4/mmm symmetry there are six constants,C11=C22, C33,C44=C55, C66, C12 and C13=C23. Hexagonal crystals have

five independent elastic constants, the same set of elasticconstants as for the tetragonal crystals exceptC66; C66= 1

2sC11−C12d, see Ref. 38. To obtain the set of elastic con-stants from the changes of the total energy one has to applya set of distortions(strains) on the crystal. For tetragonalsymmetry we applied the strains: ±e1, e1±e2, e1±e3, ±e3,±e4, and ±e6, for hexagonal symmetry the same set without±e6 and for crystals with the cubic symmetry: ±e1, e1±e2,and ±e4. For each set of strains ±ei three different magni-tudes 0.25%, 0.5%, and 0.75% were used.

The small total energy differences between the strainedstates must be calculated with a very high precision. It is alsoimportant that the internal atomic positions not fixed by sym-metry are fully relaxed. All total energy calculation weredone with VASP with the accuracy parameter set to HIGH,which results in a cutoff of the plane-wave basis increased by25%. For all cubic crystalline elements we used a very finek-point meshs25325325d. This gives almost 1000k-pointsin the irreducible part of the Brillouin zone(IBZ) for theunstrained state and about 2000 for strained states. For thetetragonal binary compounds, because of the higher numberof atoms in the elementary cell we could use a somewhatsmaller number ofk-points s20320310d. The total energywas calculated using the tetrahedron method with Blöchlcorrections,39 converged to 10−7 eV/atom.

A good check for the calculated elastic constants is a con-sistency test. The bulk modulus can be computed from theelastic constants and independently also from the energy vsvolume curve. A similar consistency test is possible for theYoung modulus. It can be calculated from the elastic con-stants or extracted from the stress-strain curve as the slope ofthis curve at the origin.

TABLE II. Computed and experimental values of elastic constants and related material properties of Al, V, Ti, and their compounds. Theline under each set correspond to experimental value for that element or structure,B represents bulk modulus in GPa,Cij is set of elasticconstants typical for each crystal, all in GPa,C8—shear modulus in GPa, relevant only for cubic structures,G/B—ratio between shearmodulusG and bulk modulus,Y—Young’s modulus in GPa andn is Poisson’s ratio(two values correspond to tension along thex andz axis).

System B C11 C12 C13 C33 C44 C66 C8 G/B Y n

Al (fcc) 75 111 57 ¯ ¯ 28 ¯ 28 0.37 73 0.34

Exp. (Ref. 47) 79 114 62 ¯ ¯ 32 ¯ 26 0.37 70 0.35

V ( bcc) 185 272 142 ¯ ¯ 20 ¯ 65 0.20 175 0.34

Exp. (Ref. 47 and 48) 161 238 122 ¯ ¯ 47 ¯ 58 0.32 155 0.33

Exp. (Ref. 49) 157 231 120 ¯ ¯ 43 ¯ 55 0.31 149 0.36

Ti(hcp) 118 193 72 85 188 40 ¯ ¯ 0.43 148/133 0.27/0.32

Exp. (Ref. 47) 107 162 92 69 181 47 ¯ ¯ 0.46 103/147 0.27

Al3V sD022d 118 233 77 47 258 104 129 ¯ 0.88 203/244 0.25/0.15

Al3V sL12d 118 178 88 ¯ ¯ 87 ¯ 45 0.59 120 0.33

Al3Ti sD022d 107 192 84 49 216 94 122 ¯ 0.84 152/199 0.30/0.18

other calc. 118,a,c,d 120b

Al3Ti sL12d 107 192 65 ¯ ¯ 74 ¯ 63 0.65 159 0.25

other calc. 110,a 150,b 118c,d

Al10V (cF176) 85 116 69 ¯ ¯ 62 ¯ 24 0.55 65 0.37

aFLAPW results of Ref. 50.bLMTO-ASA results of Ref. 51(LDA ).cFLAPW results of Ref. 52(LDA ).dFP-LMTO results of Ref. 53(LDA ).


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In Table II available experimental values are also pre-sented. The overall agreement of theory and experiment issatisfactory. One notable exception is the value of theC44parameter of bcc vanadium. The experimental value of47 GPa here differs substantially from the calculated one of20 GPa. Vanadium exhibits the highest value of the bulkmodulus, B=185 GPa. This value is somewhat overesti-mated by the calculations. The reason for the discrepancynoted for V (too high bulk modulus, too small resistance totetragonal shear) are not clear at the moment. Calculationswith full-potential linearized augmented plane wave methods(FP-LAPW) have led to similar conclusions. On the otherhand the bulk modulus of Al is the lowest,B=75 GPa. Thebulk moduli of transition metal aluminides lie between theseextremal values. The calculated values ofB approximatelyfollow the relationB~h2, whereh is the electron density.40

The comparison of the values of bulk moduli shows thatthe bulk modulus does not depend significantly on the struc-ture, but depends dominantly on the chemical composition.This is clearly seen from the results for Al3V and Al3Ti inboth crystal structures. Both calculated values ofB are thesame,B=118 GPa andB=107 GPa, respectively.

For engineering applications the most important quantityis the Young modulus. The Young modulus is highest forAl3V in the D022 structure. Its value is particularly high inzdirection,Y=244 GPa. This high value can be attributed toenhanced covalency of the Al-V bonds. Relative values ofYfor other systems well agree with the picture of bondingresulting from the analysis of the charge distribution. In thecase of vanadium the calculated Young modulusYk100l

=175 GPa is somewhat higher than the value ofYk100l

=155 GPa calculated from the experimental values of theelastic constants. This discrepancy is obviously related to theproblem with theC44 constant mentioned above.

B. Triaxial deformations

The ideal strength represents the upper strength limit for asolid under a given load. Triaxial(hydrostatic) tension isconsidered to lead to the highest value of the ideal strength.There exist some strong arguments suggesting that for cubicstructures no other types of instability(phase transformation,phonon resonance, shear collapse) occur before reaching thestrength limit during deformation under hydrostatic tension.3

The ideal strength is relatively easy to calculate. However,from the experimental point of view a verification of thesevalues is extremely difficult. Triaxial tension occurs in thesolid in the vicinity of some type of defects in the micro-structure of crystalline solids, e.g., cracks, voids, or grainboundaries.6 The ideal strength of solids under triaxial iso-tropic deformation was studied by several authors.4,6,17

Figure 6 shows the stress-strain relation of elemental met-als and intermetallic compounds under triaxial deformation.In addition to the systems considered so far the figure in-cludes also the curve for Al10V. All significant results fromthe triaxial deformations are collected in the Table III. Theideal strength corresponds to the maximum of the inducedstress.

The stress-strain curves of the elements demonstratehigher strength of bonding in TM elements in comparison

with Al. It is remarkable that for large strains the stress-straincurves of all transition metal aluminides converge to the ap-proximately same value as for pure fcc Al. This behavior isapparently the consequence of a shorter range of the Al-TMbond exhibiting enhanced covalency in comparison with thepredominantly metallic AluAl bonding. In this sense thestudy of triaxial strength can be considered as a probe of theinteratomic bonding.

The comparison of the values in Table III reveals someinteresting relations. It seems that all fcc-like structures(Al,Al3V and Al3Ti in L12, Al10V) have the same value of thetriaxial tensile strain,etis<14.4% at which the stress reachesits maximum. Similarly, allD022 structures reach their stressmaximum at aboutetis<15.2%. Although the difference ofthese values is small, it apparently reflects a difference in thebonding.

C. Uniaxial tensile deformations

A better probe for the characterization of interatomicbonding in crystals than the isotropic triaxial deformation is

FIG. 6. The stress-strain curves for triaxial tensile deformationof the elementary cells of the monoatomic crystals fcc Al, bcc V,and hcp Ti(a) and their compounds Al3V and Al3Ti in D022 andL12 structures(b). For the comparison a curve for Al10V is alsopresented.


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an uniaxial tensile deformation. We imposed a tensile strainon the Al3V and Al3Ti crystals in both theD022 and L12structures. The strain was imposed in the high-symmetry di-rection k001l common for both crystal structures. We con-sider a quasireversible deformation process at zero absolutetemperature. The total energy and induced stresses of thestrained crystals are calculated. For a higher accuracy of thestress tensor we increased the plane wave cutoff by 30% oreven more, which results in 350 eV for the binary com-pounds and 320, 245, 250 eV for Al, Ti, and V, respectively.Thek-point sampling was the same as used at the calculationof the elastic constant. The calculation proceeds by evaluat-ing the energy and stress as a function of the tetragonal lat-tice constantsa, and c. As the parameterc increases, thecrystal is reconfigured at every step to relax the lateralstresses. For comparison we have performed the same studyalso for all constituent elements: fcc Al, bcc V, and hcp Ti.As bothD022 andL12 structures are certain variants of an fccstructure it is possible to compare the results directly withfcc Al. The comparison with bcc V, and hcp Ti crystals is lessstraightforward.

Figure 7 collects the results of the tensile deformationcalculations. In the upper panel[Fig. 7(a)] the strain-stresscurves of constituent elements Al, V, and Ti are presented, inthe lower panel[Fig. 7(b)] the results for Al3V and Al3Ticrystals are shown. Full curves represent stresses induced bytensile strain applied on the elementary cells. Near the originone observes a short region of a linear behavior. The slope ofthe linear part of the stress-strain curve near the origin isrelated to the Young modulus. This important elastic propertyhas been calculated also from the elastic constants and isrepresented by the dotted straight lines. The agreement of theslopes of the stress-strain curves obtained from our tensileexperiments with the dotted straight lines representing theYoung moduli demonstrates the consistency of the both com-putational approaches. The first maximum in the stress-straincurve is the ideal strength for this deformation path, provided

no other instabilities occur before reaching it.The stress-strain curve for an fcc Al crystal deformed

along the [100] direction increases monotonously andreaches its maximum ofsuis=11.4 GPa at the straineuis=34.3%. These values can be compared with the resultsof other authors. Clatterbucket al.17 report a value ofsuis=12.92 GPa ateuis=34.0%. In addition to the[100] di-rection we have calculated for fcc Al the stress-strain curvefor tensile strain along the[111] direction. For this case amaximal stress ofsuisf111g=9.8 GPa is obtained for thestrain of euisf111g=34.7%. These values can again be com-pared with the those of Clatterbucket al. For [111] tensionthey report values ofsuisf111g=11.30 GPa ateuisf111g=33.0%. Our values of the maximal stresses are by about10% lower, but it is necessary to emphasize that Clatterbucket al. performed their calculations within the LDA while ourvalues were obtained using the GGA exchange-correlationfunctional. It is known that the LDA consistently gives

TABLE III. Computed values of the uniaxial and triaxial idealstrengths for Al, V, Ti, and their compounds,«tis represents engi-neering strain at the stress maximumstis during triaxial (hydro-static) tension,«uis—engineering strain at the stress maximumsuis

during uniaxial tension.

System Structure «tis (%) stis (GPa) «uis (%) suis (GPa)

Al k001l fcc 14.5 11.1 34.3 11.4

Al k111l fcc ¯ ¯ 34.7 9.8

Va bcc 16.5 32.3 18.2 18.9

Ti hcp 18.0 21.5 44.3 23.6

Al3V D022 15.3 20.2 24.7 28.2

Al3V L12 14.3 19.3 38.4 23.0

Al3Ti D022 15.2 17.7 29.4 25.2

Al3Ti L12 14.4 17.3 38.0 20.0

Al10V cF176 14.4 13.4 ¯ ¯

aVanadium undergoes two structural phase transitions duringuniaxial tensile deformation, at 27.4% it transforms from bcc→ fcc and at 51.5% from fcc→bct in which reaches stress maxi-mum 36.6 GPa at 98.0%.

FIG. 7. The results for the uniaxial tensile deformations of theelementary cells of the monoatomic crystals fcc Al, bcc V, and hcpTi (a) and their compounds Al3V and Al3Ti in D022 andL12 struc-tures(b). The full curves represent stresses induced by tensile strainapplied on the elementary cells. Near the origin one observes ashort region of a linear behavior. The slope of the linear part of thestress-strain curve near the origin is related to the Young modulusthat can be calculated also from the elastic constants—dottedstraight lines, cf. text.


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smaller lattice constants and a stress that is 10%–20% higherthan that obtained in the GGA.41 The ideal strength in atensile deformation is significantly higher than the value ofthe ideal shear strength. Roundyet al.10 report an ideal shearstrength for fcc Al of 1.85 GPa. We did not calculate theideal shear strengths as it would go beyond the scope of thiswork.

The value of the Young modulus ofY=73 GPa extractedform the linear(elastic) region agrees well with the experi-mental value, see Table II. Aluminum is known to haverather isotropic Young moduli calculated for different crystaldirections. A similar isotropy is observed also under tensiledeformations. The triaxial strength of Al isstis=11.1 GPa, avalue comparable with maximum stresses of 11.4 and9.8 GPa under uniaxial deformation along the[100] and[111] directions, respectively.

Figure 7(a) shows also the stress-strain curve of bcc va-nadium. The shape of this curve differs substantially fromthe other materials. A bcc crystal deformed tetragonally inthek001l direction transforms to a stress-free fcc crystal. Thestress-strain curve intersects here the zero value. This trans-formation path is known as the Bain path.42–44 Along thispath the crystal undergoes a bcc→ fcc phase transition. Thestrain at which the bcc cell assumes the fcc structure(Bainstrain) at constant volume ise=sÎ32−1d3100=25.99%. Atthis strain the c/a ratio of the tetragonally deformed bcc lat-tice is just equal to 1/Î2 and the lattice becomes fcc. Fortetragonally deformed bcc vanadium we got a value of Bainstrain eB=27.4% at an excess volume of 4.9%. The maxi-mum value of the stress issuis=18.9 GPa at a strain ofeuis=18.2%. The ideal strength of the related bcc metals Nb andMo was studied recently by Luoet al.7 For tetragonally de-formed bcc Nb homologous to V they report a similar valueof the peak stress ofsuis=18.8 GPa. However, they alsofound that between the bcc and the stress-free fcc structurethe body-centered tetragonal(bct) structure is not stableagainst orthorhombic deformations. The ideal strength alonga path allowing for orthorhombic deformations(the ortho-rhombic path) is suiso=13.1 GPa, i.e., it is significantly lowerthan that for the tetragonal path. A similar instability alongthe tetragonal Bain path is expected also for vanadium.

It is obvious that the existence of a stress-free structure onthe deformation path significantly reduces the peak value ofthe stress. This is seems to be the reason why the maximalstress of hcp Ti is higher than that of bcc V.(We note that ifone continues with the tensile deformations beyond 50%,vanadium undergoes another structural phase transitions at astrain of 51.5% from fcc to bct and then at a strain of 98.0%the stress reaches a maximum of 36.6 GPa.) For hcp Ti wefound a maximal stress ofsuis=23.6 GPa at a relatively highvalue of the strain ofeuis=44.3%. A small inflection on thestress-strain curve between strains of 10%–15% requires acomment. The hcp structure can be viewed as aAA8AA8. . .packing of hexagonal planes. We assume that the inflection isrelated to the disappearance of the covalent bonding compo-nent between the hexagonal planes. A detailed analysis ofbonding shows that three neighboring atoms from theAplane together with two other atoms from the oppositeA8planes form a strongly bonded cluster. The cluster has theshape of a trigonal bipyramid with two atoms from theA8

plane in the tips. We observed a strong bond charge insidethe bipyramid. This charge represents a covalent contributionto the inter-planar bonding. When the distance between theplanes increases the inter-planar bond with enhanced cova-lency breaks and the bond charge is distributed homoge-neously in the planes.

Figure 7(b) presents the stress-strain curves for Al3V andAl3Ti in the D022 andL12 crystal structures. The curves forboth D022 compounds have a simple bow shape. The peakvalue of the stress issuis=28.2 GPa at a straineuis=24.7%for Al3V, and suis=25.2 GPa ateuis=29.4% for Al3Ti. Theideal strengths of the same systems in theL12 structure arelower; suis=23.0 GPa ateuis=38.4% for Al3V and suis=20.0 GPa ateuis=38.0% for Al3Ti. All these results areconsistent with the picture of bonding that we have deducedfrom the charge distribution analysis. Al-V bonds are moresaturated and hence stronger than Al-Ti bonds, and this isreflected in higher values of the ideal strength of Al3V incomparison with those of Al3Ti in both structures. Thehigher symmetry of theL12 structure leads to a higher met-alicity manifest in higher values of the strains correspondingto the peak values of stresses.

Although at first sight this picture seems to be satisfac-tory, there is one unanswered question. All the values of themaximal stresses in the transition metal aluminides are sur-prisingly high. The maximal value of the induced stress ofAl3V in the D022 structure is 2.5 times higher than that of fccAl and 1.5 times higher than that of V. There is no doubt thatother deformation paths can lead to lower values of themaximal stresses, but this point is not so important. BothD022 andL12 structures are fcc-like structures and, therefore,so significant an increase of their strength along the samedeformation path asks for an explanation.

It is tempting to attribute this increased strength to theenhanced covalency of the interplanar bonding. However, adetailed analysis shows that it is not the case. The covalentcontribution to the interatomic bonding disappears at strainsbetween 10% and 15%, but the maximal stress is observed at24% or even at 38.4% in case of theL12 structure. Whatkeeps the crystal together at such high strains? A possibleanswer to this question is suggested by the results presentedin Fig. 8. Figure 8(a) shows the development of the latticeparametera in the plane normal to the direction of the tensilestrain during deformations. It is striking that for both Al3Vand Al3Ti in the D022 structure the change of this parameteris minimal, at 50% strain the lattice parametera is reducedonly by <3%. The change of thea parameter for both sys-tems in theL12 structure is similar to that of fcc Al. A smallchange of the lattice parameter indicates stiffness of bondingin the atomic planes perpendicular to the direction of thetensile deformation. Below we shall demonstrate that thisstiffness comes from an increased covalency of the in-planebonding. Figure 8(b) shows development of the cell volumeduring tensile deformation. The almost linear variation ob-served for Al3V and Al3Ti confirms this argument. Thechange of the volume is the highest for both systems in theD022 structure. Interesting variations of volume are observedunder tensile strain for V. The volume change increases up toa strain of 20%, at the stress-free fcc structure ateuis=27.4% the excess volume is about 5%, but betweeneuis


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=32% and 48% the excess volume is negative, i.e., the vol-ume of the cell even shrinks. While for vanadium one couldconsider the volume of the elementary cell as being approxi-mately constant(within 5% accuracy), other systems andparticularly theD022 structures are more close to the modelwhere volume increases linearly with the strain.

It is also remarkable that changes of the lattice parametera and volume changes for both systems Al3V and Al3Ti inthe D022 crystal structure are almost identical. For theL12structures a larger difference is seen. Such behavior can bededuced also from the elastic constants, see Table II. Theelastic constant that relates the changes of the crystal in thexandz direction is Poisson’s ration. The calculated values ofthis constant for Al3V and Al3Ti in D022 are n=0.15, and0.18, respectively, i.e., they are almost identical and rathersmall. On the other hand, for the cubicL12 structure thevalues of this constant for Al3V and Al3Ti are n=0.33 and

0.25, respectively. These values are larger and do not coin-cide so well as in case ofD022. The elastic constants describethe behavior of the crystal only for small deformations, butthe trends of this behavior in this linear region are continuedalso for larger deformations.

The D022 and L12 structures can be considered as twodifferent decorations of the fcc lattice. While the equilibriumlattice parametera in fcc Al is 4.05 Å in theD022 structurethe lattice parametera is 3.77 Å, i.e., 7% smaller. Thismeans that already in the un-strained system the distancesbetween the aluminum atoms in theB-plane are shrinked.

Figure 9 present changes of the total energy per atomduring the tensile deformations. The increase of the totalenergy is largest for Al3V in the D022 structure. This strongincrease reflects again the highest strength of bonding ofAl3V in the D022 structure. The figure also confirms that theobserved Al-TM bonding in Al3V and Al3Ti is stronger thanTM-TM bonding in bcc V and hcp Ti.

D. Changes of bonding in Al3V during tensile deformation

An uniaxial stretch of the crystal along the direction per-pendicular to the planes can provide an insight to the char-acter of bonding between the atomic planes. The structure ofAl3V can be considered as aABA8BA. . . stacking of theatomic planes, see Sec. III. With increasing strain the dis-tance betweenA andB or A8 andB increases equally.

Figures 10 and 11 display the difference electron-density,i.e., a superposition of the atomic charge-densities is sub-tracted from the total charge density. The contour plots rep-resent the regions of positive difference electron-density inselected planes; in the blank space the difference density isnegative. We remind the reader that vanadium forms twotypes of bonds: V-Al2 bonds are marked in Fig. 4 as dashedlines, V-Al1 bonds as dash-dotted lines. The Al2 atoms havein addition to the bonds with the V atom Al2-Al2 bondsmarked in Fig. 4 by dotted lines. Figures 10 and 11 show the

FIG. 8. The development of the lattice parametera (a) and ex-cess volume(b) during the tensile deformation. It is striking that forboth Al3V and Al3Ti in the D022 structure the change of this pa-rameter is minimal, at 50% strain the lattice parametera is reducedonly by <3%. Small change of the lattice parameter indicate thestiffness of bonding in the atomic planes perpendicular to the direc-tion of tensile deformation. The increase of the excess volume is thehighest for both systems in theD022 structures. Interesting varia-tions of volume are observed in case of V. The excess volumebetween 32% and 48% is negative, i.e., the volume of the cellshrinks, cf. text.

FIG. 9. The changes of the total energy during tensile deforma-tion of the monoatomic crystals Al, V, and Ti and their compoundsAl3V and Al3Ti. The increase of the total energy per atom in thecase of Al3V in the D022 structure is the highest.


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development of the difference electron-density for the threetypes of covalent bonds sketched in Fig. 4.

Figure 10 shows the difference charge density in thesx,0.5,zd plane. The tension is imposed along thez axis(horizontal). In the undeformed state one can clearly identifythe “bond-charges” between the V and Al2 atoms (graycircles). The bonds are directed from the central vanadiumatom to the eight vertices of a tetragonal prism formed by theAl2 atoms. The bond charges correspond to the tetrahedrallyorientedsp3 hybrids on the Al2 atoms. The overlap of thesp3

hybrids with the vanadiumdzx orbitals promotes the inter-planar bonding between the planesA andB or A8 andB.

In Fig. 11(a) bonding between the V and Al1 atoms(smallblack circles) is presented. The figure shows the difference

electron density in thesx,yd plane atz=0.5 (A8 plane) and atz=0.25 (B plane). The bonds are directed from the centralvanadium atom to the four vertices of a square formed by theAl1 atoms. The picture of bonding in theA plane is the sameas in theA8 plane. An enhanced charge density can be ob-served also between Al2 atoms in theB planes, Fig. 11(b).

When the tensile strain is imposed, the tetrahedrally ori-entedsp3 hybrids on the Al2 atoms tilt to a in-plane positionand gradually transform to nonbonding states. The covalentcomponent of the interplanar bonding disappears at strainsbetween 10% and 15%. Although the distance between va-nadium atoms in theA plane changes only little(less than3%) one observes the development of bond charges betweenneighboring V atoms. This is the consequence of a chargeflow from the interplanar region towards the atomic planes.This charge transfer has a significant effect on the in-planebonding. The covalent bonds in the atomic planes becomestronger. The increased strengths of the V-Al1 bonds in theatomic planeA8 and of Al2-Al2 bonds in the planeB aredistinctly observable in Figs. 11(a) and 11(b).

The right hand panels of Fig. 10 presents the developmentof the density of states in Al3V in the D022 structure duringtensile deformation. It is interesting that the deep pseudogapclose to the Fermi level disappears with the increasing strainsimultaneously with the disappearance of the covalency inthe interplanar bonding.

FIG. 10. The development of the bonding in Al3V in the D022

structure. The contours represent the difference charge density insx,0.5,zd plane in dependence on the tensile deformation, cf. text.The tension is imposed alongz axis (horizontal). In undeformedstates one can identify the bond-charges between the V(big blackcircle) and Al2 atoms(gray circles). The bonds are directed fromthe central vanadium atom to the vertices of a tetragonal prismformed by the Al2 atoms. These bonds mediate inter-planar bondingbetween atomic planesA, B, and A8. When the tensile strain isimposed the the inter-planar bonds are tilting to in-plane positionand gradually transform to nonbonding states. This covalent com-ponent of this bonding disappears at the strains between 10% and15%. On the other hand one observes development of bond chargebetween the neighboring vanadium atoms. The right hand panelsshow the the development of the density of states structure duringtensile deformation. The deep pseudogap close to the Fermi leveldisappears with increasing strain simultaneously with the disappear-ance of the covalent interplanar bonding.

FIG. 11. The development of the bonding in Al3V in the D022

structure. The contours represent the difference charge density inA8;sx,y,0.5d andB;sx,y,0.25d planes in dependence on the ten-sile deformation. In the planeA8 the bonds of enhanced are directedfrom the central vanadium atom to the four vertices of a squareformed by the Al1 atoms(small circles). An enhanced charge den-sity was found also in the planes at theB planes where Al2 atoms(small gray circles) are located. In the strained states the covalencyof bonds in both atomic planes becomes stronger.


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The increased covalency of the intraplanar bonding ex-plains its higher stiffness preventing the shrinking of the lat-eral parametera under uniaxial deformation, cf. Fig. 8. Onthe other hand, as the covalent component of inter-planarbonding parallel to the direction of the strain disappears at astrain of<15%, the question what keeps the crystal togetherat larger strains still arises. We assume that the origin of theinter-planar bonding forces under large tensile strains comesfrom an increased kinetic energy of electrons associated withan increased spatial variation of the charge density in thestrained crystal. When charge flows from the inter-planarspace into atomic planes, the variation of the spatial chargedensity distribution increases.

A flow of charge from the inter-planar region to theatomic planes can be expected to occur in all systems loadedby tensile strains. We quantitatively analyzed these flows. Aprerequisite for such an analysis is to characterize the chargedistribution in the unstrained state. Figure 12 shows the dis-tribution of charge density in and between the atomic planesat zero strain. The charge distributions were averaged overthe planes perpendicular to direction of the tension(z-direction). For sake of an easier comparison the total chargedensity in the elementary cell of each system is normalizedto unity. For theL12 structure plane A correspond toz=0 and

z=1, the planeB to z=0.5. There is no difference in theaveraged charge density between theA andA8 planes in theD022 structure. For fcc and bcc crystals there is also no dif-ference betweenA andA8 planes. Figure 12 shows that thecharge is most uniformly distributed in fcc Al. Going throughthe crystal along thez axis the planar average varies between0.9 and 1.06. A larger variation is seen in bcc vanadium, theaveraged charge density varies between 0.87 and 1.19. Thevariation of the charge density in hcp Ti is even higher. Thehigher variation of the charge density in hcp Ti in compari-son with that in bcc V has apparently origin in the higherseparation of the atomic planes. The distance between thehexagonal planes in hcp Ti is 2.31 Å while in bcc V thedistance between the considered atomic planes is 1.50 Å. FortheD022 andL12 structures the variation of charge density isdistinctly higher for theD022 structures. The Al3V com-pounds also exhibit a stronger variation than the Al3Ti al-loys. The variation of the charge density for Al3V in theD022structure is thus the highest; the charge varies between 0.78and 1.38, and for Al3Ti in the L12 structure is the lowest; thecharge varies between 0.85 and 1.14.

Figure 13 shows the relative differences of the averagedcharge densities in atomic planes between the strained and

FIG. 12. Normalized distribution of the charge in the elementarycells across the direction of the tensile deformation. The figureshows the variation of the charge distribution in unstrained state, cf.text.

FIG. 13. The relative change of the charge distribution in theelementary cells across the direction of the tensile deformation. Thefigure shows relative changes of the the charge distribution instrained state fore=50%, cf. with the previous figure.


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unstrained state. The difference increases with increasingstrain. In the figure the differences of the charge density forthe maximal imposed strain of 50% are presented. The figureprovides insight how the charge flows from the inter-planarregion into the atomic planes. In theA planes of bothD022and L12 structures the charge surplus is<60%. In theBplane the charge increase differs; for bothD022 structures itis <50% while for the bothL12 structures it reaches<42%.A similar difference between theD022 andL12 structures isseen for the charge deficit between the atomic planes. Thedevelopment of the charge deficit on the imposed strain ispresented in Fig. 14. Again we observe distinctly differentbehavior for theD022 and L12 structures, but on the otherhand differences between the Al3V and Al3Ti alloys areminimal.

V. DISCUSSION AND CONCLUSION

We have presentedab initio calculations of the tensilestrengths of the intermetallic compounds Al3V and Al3Ti,both in their stableD022 and in their metastableL12 struc-tures, and of the constituent elements: Face-centered cubicAl, body-centered cubic V, and hexagonal closepacked Ti.Our computer simulations of triaxial and uniaxial tensile de-formations(with complete relaxation of lateral stresses havebeen supplemented by the calculations of the elastic con-stants and by a detailed analysis of the changing character ofthe interatomic bonding during deformation.

For the cubic metals Al and V our results are in goodagreement with earlier theoretical estimates of the stress-strain relation and the ideal tensile strength,17,41 the investi-gation of the hexagonal close-packed metal Ti and of theintermetallic trialuminide compounds leads to interesting andnovel conclusions. The stress-strain relation of hcp Ti underuniaxial strain shows an inflection at strains in the rangebetween 10% and 15%, related to a change in the bondingproperties. The axial ratio of Ti is withc/a=1.583 distinctlysmaller than the valuec/a=1.633 for optimal dense packing,reflecting a certain degree of covalency of the inter-planarbonding. Under sufficiently strong uniaxial deformation thecovalent character of the bonding disappears and the bondingassumes a more metallic character. Mixed metallic and co-valent bonding also determines the physical properties of theintermetallic compounds Al3V and Al3Ti. In a previousstudy we had shown that the unique electronic and mechani-cal properties of Al3V with the tetragonalD022 structure(deep pseudogap at the Fermi level, high hardness) are re-lated to a substantial covalent contribution to both theV-Al and to some of the Al-Al bonds. The covalent V-Albonding arises from the overlap ofsp2d-type hybrid orbitalson both the V and Al1 sites, the covalent contribution to theV-Al 2 bonds from the overlap ofsp3 hybrids on the Al siteswith d4 hybrid orbitals on the V sites. The bonding Al-Alorbitals havesp2 character. In Al3V all occupied states havepredominantly bonding character, all empty states are anti-bonding. Due due a lower valence-electron concentration,the saturation of the covalent bonds is lower in tetragonalAl3Ti than in Al3V. In the metastable compounds with thecubicL12 structure, the higher site-symmetry at the TM sites

leads to aeg- t2g splitting, the bonding is mediated mainly bylinear combinations ofeg states andsp2d hybrids, but thesaturation of these bonds is lower than in the tetragonalphase. The increased covalency of Al3V compared to Al3Tiis reflected in higher values of the bulk modulus, the shearmodulus and of Young’s modulus. The tetragonal and to thecubic phases of a given compound have approximately thesame bulk modulus, but the more covalent cubic phase hashigher shear and Young’s moduli.

Under triaxial(homogeneous) deformation we calculate asomewhat higher tensile strength for Al3V than Al3Ti, butthere is no significant difference between the cubic and te-tragonal phases. This demonstrates that under homogeneousdeformation the relative mechanical properties deduced fromthe elastic limit may be extrapolated up to the maximuminduced stress. Under uniaxial tensile deformation things arequite different: (i) The D022 phases have a significantlyhigher tensile strength at a smaller strain than theL12 phases.(ii ) Al3V has a higher maximum stress than Al3Ti in bothstructures.(iii ) TheD022 phases show a very small reductionof the lateral lattice parameter in a plane perpendicular to thedirection of the strain, reflecting the strength of the covalentintraplanar bonding. TheL12 phases shrink in the lateral di-rections in an extent comparable to that observed for fcc Al.As a result we find a volume reduction which is almost equalto the linear strain for the tetragonal intermetallic compoundsand a much smaller volume contraction at the same strain inthe cubic compounds at in fcc Al. These differences in thestress-strain relations reflect the changing character of theinteratomic bonding forces under uniaxial tensile strain. Adetailed analysis of the charge-redistributions for tetragonalAl3V shows that with increasing tensile deformations thestrong covalent V-Al bonds are gradually destroyed whileintraplanar V-V and Al-Al bonds build up. The interplanarbonding acquires a much more metallic character. We notethat this is expected to lead to a lower resistance to shearingalong a{001} plane, but this has not been explored in thepresent study.

The D022 andL12 structures are certain occupation vari-ants of the fcc structure. It would be interesting to know how

FIG. 14. The development of the charge deficit between theatomic planes in the dependence on the imposed tensile strain.


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the anisotropy of bonding observed in theD022 and L12structures affects the the anisotropy of the mechanical prop-erties. Already in the fcc metals with a pure metallic and,therefore, isotropic interatomic bonding the mechanicalproperties exhibit strong anisotropy. For example, a tensilestrength of Al of 4.9 GPa for a pull along[110] is signifi-cantly lower than the values for[001] s12.92 GPad or [111]s11.30 GPad directions.17 In fcc metals there is also an inter-esting relation between the tensile and shear deformations.The tension ink110l direction is closely related to a shear onk112l {111}. This leads to an important result that an fccmetal does not cleave. It would be interesting to investigatewhether this behavior pertains to theD022 andL12 structureswhere the strong anisotropy of bonding competes with theanisotropy originating from the geometry. We believe that innear future we shall be able to extend our calculations toinvestigate these competing anisotropies and to study theirinfluence on the mechanical properties of the transition metalaluminides.

In summary: Our work represents one of the first attemptsto extend atomistic studies of the ideal strength of metallicsystems beyond the elemental metals. At the example of the

trialuminides Al3V and Al3Ti we have explored the influenceof a partially covalent bonding on the mechanical properties.In the elastic limit, covalent intra- and inter-planar bondingforces lead to a significant enhancement of the bulk, shear,and Young’s modulus, as well as to an increased idealstrength under triaxial loading. Under uniaxial stress the co-valent bonding between planes perpendicular to the appliedstrain is gradually reduced, while the covalent intraplanarbonding is even enhanced.

ACKNOWLEDGMENTS

We acknowledge support from the Grant Agency for Sci-ence in Slovakia(Grant No. 2/2038/22) and from the Agencyfor Support of Science and Technology(Grants No. APVT-51021102, APVT-51052702, SO-51/03R80603). Work at theUniversity of Wien has been supported by the Austrian Min-isters for Education and Research through the Center forComputational Materials Science. One of the authors(M.J.)acknowledges support from a Marie Curie Training SiteGrant (HPIYT-CT-2001-00255).

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