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Journal of Physics and Chemistry of Solids 67 (2006) 2056–2064

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Ab initio calculations of elastic constants of the bcc V–Nb system athigh pressures

A. Landaa,�, J. Klepeisa, P. Soderlinda, I. Naumovb, O. Velikokhatnyic, L. Vitosd,e, A. Ruband

aPhysics and Advanced Technologies Directorate, Lawrence Livermore National Laboratory, University of California, P.O. Box 808,

Livermore, CA 94550, USAbDepartment of Physics, University of Arkansas, Fayetteville, AR 72701, USA

cDepartment of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USAdDepartment of Materials Science and Engineering, Royal Institute of Technology, SE-10044 Stockholm, Sweden

eResearch Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary

Abstract

First-principles total energy calculation based on the exact muffin-tin orbital and full-potential linear muffin-tin orbital methods were

used to calculate the equation of state and shear elastic constants of bcc V, Nb, and the V95Nb05 disordered alloy as a function of

pressure up to 6Mbar. We found a mechanical instability in C44 and a corresponding softening in C0 at pressures �2Mbar for V. Both

shear elastic constants show softening at pressures �0.5Mbar for Nb. Substitution of 5 at% of V with Nb removes the instability of V

with respect to trigonal distortions in the vicinity of 2Mbar pressure, but still leaves the softening of C44 in this pressure region. We argue

that the pressure-induced shear instability (softening) of V (Nb) originates from the electronic system and can be explained by a

combination of the Fermi surface nesting, electronic topological transition, and band Jahn–Teller effect.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: A. Metals; C. Ab initio calculations; D. Elastic properties; D. Electronic structure; D. Fermi surface

1. Introduction

The group-VB transition metals (V, Nb, and Ta) havebeen subject of numerous experimental and theoreticalstudies [1–15] in 60–80 s due to the high superconductingtransition temperatures of its components (for example, Nbhas the highest superconducting transition temperatureTc ¼ 9.25K among the elemental metals) and uniquemechanical properties at extremely high temperatures thatplace these metals in the position of basic building blocksof intermetallic compounds. However at that time, most ofthis research was restricted to ambient pressure conditionsor pressures below 50 kbar. During the past 15 years, thedevelopment of new techniques for electric and magneticmeasurements in a diamond-anvil cell (DAC) has allowedfor performing measurements at megabar pressures.Struzhkin et al. [16] measured the superconducting transi-

front matter r 2006 Elsevier Ltd. All rights reserved.

s.2006.05.027

ng author. Tel.: +1925 424 3523.

ss: landa1@llnl.gov (A. Landa).

tion temperature of Nb in the pressure range up to1.32Mbar. They observed anomalies in Tc(p) at 50–60 and600–700 kbar and suggested that these anomalies arisefrom electronic topological transitions (ETT). LaterIshizuka et al. [17] performed similar DAC experimentson V in the pressure range up to 1.2Mbar. They discoveredthat Tc ¼ 5.3K at ambient pressure and increases linearlywith pressure and reaches 17.2K (the highest Tc among theelemental metals reported so far) at 1.2Mbar. However, incontrast to Nb, these measurements did not find anyindications of anomalies in Tc for V within this pressurerange. Takemura [18] performed high-pressure DACpowder X-ray diffraction experiments on V and Nb andfound no anomalies, which could be connected with ETT,in the equations of state (EOS) up to the maximumpressure of 1.54Mbar. In order to explore the possibility ofa structural phase transition from bcc to another phase,Suzuki and Otani [19] performed the first-principlescalculations of lattice dynamics of V in the pressure rangeup to 1.5Mbar. They found that the transverse acoustic

ARTICLE IN PRESSA. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–2064 2057

phonon mode TA [x 0 0] around x ¼ 1/4 shows a dramaticsoftening under pressure and becomes imaginary atpressures higher than �1.3Mbar, indicating a possibilityof such structural phase transition. Since in the limit oflong-wave lengths (q-0) this mode is directly related tothe elastic constant C44, clear understanding of theanomaly in the TA curve is principally important in theproblem of the shear lattice stability of bcc V. The authors,however, did not discuss the physical reasons behind thedip in the TA.

This paper is devoted to the ab initio study of the EOSand shear elastic constants of V, Nb, and the V95Nb05disordered alloy in the pressure range up to 6Mbar. Weused selected density functional theory approaches tomodel the V–Nb system at ambient pressure as well asextreme conditions. We discuss the physical nature of theanomalies in the shear elastic constants of V in terms of thecombined Fermi surface (FS) nesting, ETT, and bandJahn–Teller (J–T) effect.

The paper is organized as follows. Pertinent details ofcomputation methods are described in Section 2, followedby results in Section 3 and discussion in Section 4. Wepresent our conclusions in Section 5.

2. Computational details

In the present paper, we employ two complimentarycomputational techniques. The calculations we havereferred to as exact muffin-tin orbitals (EMTO) areperformed using a scalar-relativistic Green’s functiontechnique based on an improved screened Korringa–Kohn–Rostoker method, where the one-electron potentialis represented by optimized overlapping muffin-tin(OOMT) potential spheres [20–24]. Inside the potentialspheres the potential is spherically symmetric and it isconstant between spheres. The radii of the potentialspheres, the spherical potentials inside the spheres, andthe constant value from the interstitial are determined byminimizing (i) the deviation between the exact and over-lapping potentials, and (ii) the errors coming from theoverlap between spheres. Thus, the OOMT potentialensures a more accurate description of the full potentialcompared to the conventional muffin-tin or non-over-lapping approach.

Within the EMTO formalism, the one-electron states arecalculated exactly for the OOMT potentials. As an outputof the EMTO calculations, one can determine the self-consistent Green’s function of the system and the complete,non-spherically symmetric charge density. Finally, the totalenergy is calculated using the full charge density technique[23,25]. For the exchange/correlation approximation weuse the generalized gradient approximation (GGA) [26].For the total energy of random substitutional alloys, theEMTO method has been recently combined with thecoherent potential approximation [24,27].

The calculations are performed for a basis set includingvalence s p d orbitals whereas the core states were

recalculated at each iteration. Integration over the irredu-cible wedge of the Brillouin zone (IBZ) is performed usingthe special k-point method [28] with 819 k-points for bcclattice. The Green’s function has been calculated for 60complex energy points distributed exponentially on asemicircle with a 3.5Ry (3.8Ry for V95Nb05 alloy)diameter enclosing the occupied states. The equilibriumdensity is obtained from a Murnaghan fit [29] to about 16total energies calculated as a function of the latticeconstant. For the calculation of C0 and C44 we use volumeconserving orthorhombic and monoclinic setups for tetra-gonal and trigonal distortions, respectively [30], performingintegration over the IBZ with 15625 (C0) and 17457 (C44) k-points.As EMTO method does not allow us to calculate the

energy bands, we applied a full-potential linear muffin-tinorbitals (FPLMTO) method [31–33] for this purpose. Thislatter method has no geometrical approximations to itsone-electron potential or charge density and is thereforewell suited for low symmetry or distorted crystals (we alsoused this method for shear elastic constants calculations forV and Nb). Within FPLMTO approach, the crystal isdivided into regions inside atomic spheres, where Schro-dinger’s equation is solved numerically, and an interstitialregion. In all LMTO methods, the wavefunctions in theinterstitial region are Hankel functions. An interpolationprocedure is used for evaluating interstitial integralsinvolving products of Hankel functions. The triple-k basisis composed of three sets of s, p, d, and f LMTOs per atomwith Hankel function kinetic energies of �k2

¼ �0.01,�1.0, and �2.3Ry (48 orbitals per atom). The Hankelfunctions decay exponentially as e�kr. The angularmomentum sums involved in the interpolation procedureare carried up to a maximum of l ¼ 6. For the exchange/correlation approximation we again use the GGA [26]. Theintegration over the IBZ was carried out using thetetrahedron method [34] with 48� 48� 48 shifted meshresulting in 14724 and 14700 k-points for C0 and C44 shearelastic constants, respectively.The generalized susceptibility of non-interacting elec-

trons was selected in order to detect the nesting at the FS.This function was calculated by using the standard linearmuffin-tin orbitals method in the atomic sphere approx-imation with so-called combined correction terms [35] andthe highly precise analytic tetrahedron method [36]: inorder to reach the high precision, the IBZ of the bcc latticewas divided into 3375 tetrahedra.

3. Results

Fig. 1 shows the EOS (EMTO) for V and Nb as well asfor the V95Nb05 disordered alloy. For V, calculatedequilibrium volume (V0) and bulk modulus (B0) are13.65 A3 and 1.832Mbar, respectively. FPLMTO calcula-tions give V0 ¼ 13.53 A3 and B0 ¼ 1.827Mbar. Theseresults are in good agreement with experimental densityof V, V0 ¼ 13.91 A3 [37], and recently measured bulk

ARTICLE IN PRESSA. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–20642058

modulus, B0 ¼ 1.880Mbar [18], as well as results ofFPLMTO calculations by Suzuki and Otani,B0 ¼ 1.946Mbar [19]. For Nb, EMTO calculations giveV0 ¼ 18.41 A3 and B0 ¼ 1.666Mbar, and FPLMTO calcu-lations give V0 ¼ 18.26 A3 and B0 ¼ 1.707Mbar. BothEMTO and FPLMTO results are in good agreement withexperimental density of Nb, V0 ¼ 17.97 A3 [37], andrecently measured bulk modulus, B0 ¼ 1.680Mbar [18].

0 2Pressur

0

1

2

3

4

5

6

Ela

stic

mod

uli (

Mba

r)

C44 - FPLMTO

C` - FPLMTO

Expt C44

Expt C`

C44 - EMTO

C` - EMTO

1

Fig. 2. Pressure dependence of the shear elastic moduli of V. Exp

6 7 8 10 11 12 13 14

Volu

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

6500

Pres

sure

(kb

ar)

9

Fig. 1. Equations of state of V, Nb, a

Figs. 2 and 3 show pressure dependence of the shearelastic moduli of V and Nb, respectively. EMTO calcula-tions reveal a mechanical instability in C44 and correspond-ing softening in C0 at pressures �2Mbar for V (similartendency is observed from FPLMTO calculations, but at alower pressure). Both shear elastic constants show soft-ening at pressures �0.5Mbar for Nb. One should alsomention that similar softening for C44 modulus, although

4 5 6e (Mbar)

V

3

erimental data at ambient pressure are according to Ref. [38].

15 16 17 18 19 20 21 22 23

me (A3)

V

V95Nb05

Nb

nd the disordered V95Nb05 alloy.

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0 0.25 0.5 0.75 1 1.25 1.5 1.75Pressure (Mbar)

0

0.5

1

1.5

2

2.5

3

Ela

stic

mod

uli (

Mba

r)

C44 - FPLMTO

C` - FPLMTO

Expt C44

Expt C`

C44 - EMTO

C` - EMTO

Nb

Fig. 3. Pressure dependence of the shear elastic moduli of Nb. Experimental data at ambient pressure are according to Ref. [38].

Fig. 4. Band energies of undistorted V at (a) experimental equilibrium

volume (�0.048Mbar); (b) 2.089Mbar; (c) 5.734Mbar.

Fig. 5. Band energies of distorted V at (a) experimental equilibrium

volume (�0.048Mbar); (b) 2.089Mbar; (c) 5.734Mbar.

A. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–2064 2059

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less pronounced, was predicted from FPLMTO calcula-tions for Ta [39], the remaining representative of the groupVB. In order to correlate these phenomena with theelectronic structure behavior, we constructed (FPLMTO)band energies for V (undistorted, Fig. 4, and under thetrigonal distortion, Fig. 5) at different pressures. As onecan see from Fig. 5a (experimental equilibrium volume),the trigonal distortion causes the levels with t2g symmetry(G250 and H250) to split and levels D2 and S3 to hybridizewith levels D5 and S1, respectively. Similar features of theband energies of V were also described by Ohta and

0 2Pressu

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

δ = 0.01

δ = 0.03

δ = 0.05

∆ N

(EF)

(1/R

y)

Press

0

1

2

3

4

5

6

7

Ele

ctro

stat

ic c

ontr

ibut

ion

to C

44 (

mR

y/at

)

δ = 0.00

δ = 0.03

δ = 0.05

1

0 21

(a)

(b)

Fig. 6. (a) The increment of the DOS at the Fermi level and (b) the electrosta

Shimizu [15]. The 3-fold degenerate G250 pure d-state isunoccupied at low pressure but shifts to lower energyand passes through the Fermi level as pressure increases(Fig. 5b,c).Consider now how the density of states (DOS) behaves

under pressure and distortion. Fig. 6a shows the incrementdue to the trigonal distortion of the DOS at the Fermi level,DN(EF), as a function of pressure. This plot shows that forV, DN(EF) increases under deformations at ambientpressure but begins to decrease as pressure increases andpasses a minimum in the vicinity of �3Mbar. Additional

4 5 6re (Mbar)

V

ure (Mbar)

V

3

4 5 63

tic contribution to the C44 elastic modulus of V as a function of pressure.

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(a)

(b)

(c)

Fig. 7. Central {1 0 0} and {1 1 0} cross-sections of the FS of V at (a)

ambient pressure; (b) 2.216Mbar; and (c) 3.302Mbar.

A. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–2064 2061

analysis indicates that as the value of the deformationdecreases (d-0), DN(EF) the minimum moves rightapproaching the region where the G250 point crosses theFermi level (�3.3Mbar according to EMTO calculations).Further compression causes DN(EF) to increase againmonotonically. On the other hand, the Madelung (electro-static) contribution to the C44 elastic constant is positivefor all pressures under consideration (Fig. 6b).

The pressure evolution of the FS of V can be understoodfrom its cross-sections in the central {1 0 0} (triangular(GHN)), and {1 1 0} (rectangular (GHNPN)) planes shownin Fig. 7a (the notations are borrowed from Mackintoshand Andersen [40]). Since V contains five valence electronsper atom, they fill entirely the first, most of the second anda significant part of the third conduction band. In thisfigure, the first filled band is not shown, the second band(2) has a closed hole-surface centered at the point G andhas the shape of a distorted octahedron. The third band (3)has closed distorted hole ellipsoids centered at points N

and multiple connected opened hole-tubes, so-called‘jungle-gym’ (J-G), which extend from G to H in the[1 0 0] (D) directions. At ambient pressure, the calculatedFS sections for V look similar to previously reportedtheoretical results [1,3,5,7,9,11–14], however, at elevatedpressures we did not find any evidences of the ETTpredicted by Papaconstantopoulos et al. [5] whose calcula-tions revealed that at the 5% reduced lattice spacing, whichcorresponds to �240 kbar pressure, the distorted holeellipsoids around points N merge with the J-G hole tubescreating such a kind of a hole neck along the G–N (S)direction. Instead we found that as pressure increases, thedistorted octahedron hole-pocket around the G pointshrinks, indicating a movement of this point towards theFermi level (Fig. 7b), and the J-G hole tube terminates atthe [1 0 0] direction (Fig. 7c).

Calculated at ambient pressure generalized susceptibil-ity, w(q), along the G-H ([x00]) direction (Fig. 8a) shows aslightly pronounced peak at xE0.24. Detailed analysis ofpartial contributions to the generalized electron suscept-ibility indicates that this peak is due to 3rd-3rd elec-tron transitions (Fig. 8b) or, in other words, due to thenesting properties of the FS in the 3rd band (intra-bandnesting). As can be seen from Fig. 8b, this peak shiftsswiftly towards the smaller x (lower wave vector q) aspressure is applied, for example, at 1.8Mbar the peak islocated at xE0.06. Half of the nesting vector (|qn|/2Ep/

a[0.24, 0, 0]) can be clearly seen in Fig. 7a. It corres-ponds to the position of the anomaly in the TA found bySuzuki and Otani [19]. As pressure increases, |qn| decreases(Fig. 7b) and finally becomes equal to zero at pressureof the above-mentioned termination of the J-G tube(Fig. 7c).

Figs. 9 and 10 show calculated (EMTO) pressuredependence of the C44 elastic modulus and DN(EF) of theV95Nb05 disordered alloy, respectively. Notice that anaddition of only 5 at% of Nb stabilizes V mechanically,whereas significant softening in the C44 shear modulus still

takes place at �2Mbar pressure. This feature correlateswith DN(EF) behavior—it decreases as pressure increasesup to �3Mbar, but for the V95Nb05 disordered alloy thiseffect is significantly smaller than for pure V.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ξ

10

15

20

25

χ (1

/Ry)

p = 0.000 Mbar

p = 1.839 Mbar

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ξ

0

0.5

1

1.5

2

2.5

χ (1

/Ry)

p = 0.000 Mbar

p = 1.839 Mbar

(a)

(b)

Fig. 8. (a) Generalized total and (b) partial (3rd-3rd intra-band transition) electron susceptibilities of V along the G–H ([1 0 0]) direction.

A. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–20642062

4. Discussion

From the data presented above, it is clear that softeningin the shear constants of V and Nb is entirely due to theband structure features and not to the electrostatic forces(the latter stabilize the lattice under pressure). Thissoftening can be attributed simultaneously to three differentelectronic structure peculiarities, namely, to the FS nesting,ETT, and the band J–T effect.

The nesting vector qn, spanning two flat pieces of the FSin the 3rd band, already exists at zero pressure and leads to

the Kohn anomaly in the transverse acoustic phonon modeTA [x 0 0] [19] for small |qn|/2E0.24p/a. This anomaly alsosoftens the elastic constant C44 because in the limit of longwaves (q-0) ro2(q)/q2-C44. It is remarkable that underpressure qn shrinks at a fast rate and the effect of the Kohnanomaly on C44 increases. As soon as qn turns to zero, theETT, when the neck between two electronic sheets of theFS appear (the J-G hole-tube terminates at the [1 0 0]direction), takes place. Qualitatively, it is analogous to theevent when the FS touches the BZ boundary and is alwaysaccompanied by a sharp minimum in the shear elastic

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0 2Pressure (Mbar)

0

0.5

1

1.5

C44

(M

bar)

V95Nb05

31

Fig. 9. Pressure dependence of the shear elastic modulus C44 of the disordered V95Nb05 alloy.

Fig. 10. The increment DOS at the Fermi level of V and the disordered V95Nb05 alloy as a function of pressure.

A. Landa et al. / Journal of Physics and Chemistry of Solids 67 (2006) 2056–2064 2063

constants [41]. So as qn-0, the softening in C44 developsup to the point when qn turns to zero. It is very importantthat the condition qn ¼ 0 is reached long before the Fermilevel passes the triple-degenerate term G250.

The pressure-induced shear instability in V can be also,partially, due to the ‘band J–T effect’. As was alreadymentioned, the trigonal distortion splits the terms G250

and H250 and, besides, leads to a hybridization between D2

and D5 from one side and between S3 and S1—fromthe other. All these splitting/shifting/hybridization effectstend to open the energy gaps on the Fermi level anddecrease the total energy. For example, the initially triple-degenerate G250 point splits in such a way that one of thesplit level goes up above and another two go down belowthe Fermi level causing the DOS at the Fermi level todecrease.

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5. Conclusions

First-principles total-energy calculations reveal a ten-dency for the group-VB transition metals V and Nb tosoften their shear elastic moduli at elevated pressures. Weconclude that this pressure-induced softening is due to theelectronic system, exclusively, and has a complicatedmechanism. For V, expansion of the pressure range tohigher values (above 1.5Mbar) is desirable to obtaininformation on possibility of a structural transition frombcc to another phase. The proposed study should be a realchallenge for experimentalists because these measurementsmust be performed at a maximum low temperature(softening of the shear elastic constants disappears astemperature increases) and the hydrogen should beremoved from the samples (there is a tendency of theelastic constant C44 of V and Nb to increase with increasinghydrogen concentration [42]).

Acknowledgements

A.L. thanks D.A. Papaconstantopoulos for helpfuldiscussion. This work was performed under the auspicesof the US Department of Energy by the University ofCalifornia Lawrence Livermore National Laboratoryunder contract W-7405-Eng-48. I.N. is grateful to theOffice of Naval Research, Grant No. 00014-03-1-0598.L.V. and A.R. are grateful to the Swedish ResearchCouncil, the Swedish Foundation for Strategic Research,and the Royal Swedish Academy of Sciences. L.V. alsoacknowledges the research projects OTKA T046773 andT048827 of the Hungarian Scientific Research Fund.

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