Phonon instability and structural phase transition in Vanadium under
  high pressure
First-principles studies on Vanadium
Phonon instability and structural phase transitions in Vanadium under 
High pressure 
Ashok K. Verma   and  P. Modak 
High Pressure Physics Division, Bhabha Atomic Research Centre, 
Trombay, Mumbai-400085, India 
Abstract 
Results of the first-principles calculations are presented for the group-VB metals V, Nb 
and Ta up to couple of  megabar pressure. An unique structural phase transition sequence 
~60 ~160hom1( 110.50 )GPa GPaBCC R α °⎯⎯⎯⎯→ = ⎯⎯⎯⎯ ~434hom 2( 108.50 ) GPa→ R BCCα °= ⎯⎯⎯⎯→  is 
predicted in V. We also find that BCC-V becomes mechanically and vibrationally 
unstable at around 112 GPa pressure. Similar transitions are absent in Nb and Ta. 
PACS No.:  71.15.Nc, 61.50.Ks, 61.66.Bi, 62.50.+p,63.20.-e 
Introduction: 
Group-VB transition metals V, Nb, and Ta crystallizes in body-centered cubic (BCC) 
structure at ambient pressure and temperature conditions. These metals are of great use 
due to their high thermal, mechanical and chemical stabilities. Recently these metals have 
been the subject of numerous experimental and theoretical studies [1-7] in Mbar pressure 
regions. The Nb is known to have the highest superconducting transition temperature (Tc) 
among elemental solids at ambient pressure [1]. The Ta is used in high-pressure 
experiments as a pressure standard. Experimentally it is known that V, Nb and Ta  remain 
stable in the BCC structure at least up to 150 GPa [2,4] pressure.  
                       However, the linear response phonon calculations using full potential 
linear muffin-in orbital (FP-LMTO) method of V by Suzuki and Otani had shown the 
softening of the transverse acoustics (TA) phonon mode at ~120 GPa pressure, which 
eventually becomes imaginary at pressures higher than 130 GPa [5]. The subsequent 
work of Landa et al. using exact muffin-tin orbital (EMTO) method had also shown the 
anomalous pressure behavior of the C44 elastic constant. They found that the C44 starts 
softening above 50 GPa pressure and drops to zero value at about 180 GPa pressure. In 
the pressure range of 180-270 GPa the value of C44 is negative. Above to this, the C44 
adopts the usual pressure  behavior. These are an indication of  the development of a 
structural instability in BCC lattice.  However, these authors corroborated  this behavior 
to the Fermi-surface nesting  and the possibility of any structural phase transition was not 
explored [6].   
In this paper we are reporting our ab-initio density functional theory based results for the 
V, Nb and Ta under high pressure. For V, BCC to rhombohedral structural phase 
transition has been predicted at around 60 GPa pressure and it remains in rhombohedral 
structure at least up to 332 GPa pressure. However, at 434 GPa pressure the BCC 
structure re-appears  in our calculations. This behavior is not found in the Nb and Ta.. 
But lattice expansion in Nb resulted a new minimum around 111° which is lower in 
energy relative to BCC. By analogy, we expect such features in Ta will appear even at  
higher lattice expansions. 
Computational method:  
The cubic structures namely simple cubic, body centered cubic (BCC) and face-centered 
cubic (FCC) crystal structures are related to a rhombohedral structure. The simple cube 
can be viewed as a rhombohedron with angle (αrhom) equal to 90°. The primitive cells of 
BCC and FCC are rhombohedron with angle αrhom equal to 109.47° and 60° respectively 
[8]. Thus it is possible  to generate all theses cubic structures from a rhombohedron with 
atom at (0,0,0) by varying the angle. Here, we have performed total energy calculations  
taking  a  rhombohedral unit cell for V, Nb and Ta as a function of αrhom in the range of 
54°-112° at several volumes. All the calculations  for V were done using the 
pseudopotential based PWSCF computer code [9]. The total energy calculations are 
based on density functional theory  and the phonon calculations on density functional 
perturbation theory [10]. An ultrasoft pseudopotential for V was used with 40 Ry plane 
wave energy cut off and a 400 Ry cut off in the expansion of the augmentation charges. 
Phonon dynamical matrices were computed on a uniform 6x6x6 grid of q-points in the 
BZ of BCC structure. This leads  to total 16 q-points in the irreducible BZ. Using Fourier 
interpolation, we then constructed and diagonalized the dynamical matrices on a denser 
grid (24x24x24). The calculations for the Ta and Nb were performed using the VASP 
computer program with PAW pseudopotentials taking  an energy cut off of 400 eV for 
plane wave expansions [11-13]. Results were cross checked with both programs at a few 
volumes for V and Nb. The s and p semicore levels of respective elements were included 
in the valence states. For the exchange-correlation term the generalized gradient 
correction approximation of Perdew-Buke-Ernzerof (GGA-PBE) [14] was used. A 
18x18x18 k-point mesh for the Brillouin zone (BZ) integration was used for all the total 
energy calculations. The energy convergence with respect to computational parameters 
was carefully examined.  
Results and discussion:  
The zero-pressure and zero-temperature results for lattice constant a0, bulk modulus BB0, 
and the first pressure derivative of the bulk modulus B’ for V, Ta, and Nb are obtained by 
fitting the total energies versus lattice constant data to a fourth order polynomial. 
Calculated equilibrium lattice constants are in good agreement with the experimental 
values (within the error of 1%). Bulk modulus and its first pressure derivates are also in 
good agreement with existing experimental data (see table for comparison). Calculated 
pressure-volume relations matches well with existing experimental data and shown in 
Fig.1. Hence justifies the use of respective presudopotentials and other computational 
parameters.  
Table-1: The experimental data shown in brackets by symbol (*) is of Takemura [2] and 
by symbol (+) is of Dewaele et. al.[7]. 
Element Lattice constant (a0) Bulk Modulus (BB0) B’ 
V 2.998 (3.00)* 182 (165,188)* 3.70 (3.5,4.04)* 
Nb 3.309 (3.30)* 172 (153,168)* 3.40 (3.9,2.2)* 
Ta 3.320 (3.30)+ 208 (194,)+ 3.13 (3.25)+ 
Fig.2 depicts our calculated total energy as a function of angle αrhom  for V at equilibrium 
volume.  The SC is on the absolute maximum and FCC is on the local maximum 
indicating the unstable nature of theses structures for a given rhombohedral distortion. 
The BCC is in the absolute minimum of the total energy curve as per expectation. 
However, there are two local minima around angles 56° and 68° (see the inset of Fig.2).  
To our best of knowledge no body has shown that for V,  FCC and SC structures are 
mechanically unstable. The energy variation in the range of angles 106° -111°  is about 
4.0 mRy per atom (see Fig.3a). Similar calculations were performed at  many volumes 
covering more than 400 GPa pressures. The Figs.3(a, b) show total energy variation as a 
function of angle αrhom  in the range of  106°-112° at several pressures. The reason for 
showing the energy variation only in this range of angles is two fold. Firstly, BCC is of 
lowest energy among all structures  and secondly no extra feature appear around the FCC 
except that local minima become more pronounced with few degree shift. Particularly at 
112 GPa pressure  the minimum around 68° is shifted to angle 70° and the minimum 
around 56° shifted to 57° . At this pressure 70° minimum is lower in energy by 7 mRy 
per atom relative to the  57° minimum and around 12 mRy per atom relative to FCC. 
However, SC remains on the absolute maximum of the total energy curve up to 400 GPa 
pressure, ruling out the possibility of BCC to SC transition as predicted earlier [15]. 
Below 60 GPa pressure the behavior of energy curves remain unchanged.  However at 60 
GPa a new minimum appear at angle 110.50° (say, Min1) leading to a structural phase 
transition. Now BCC energy is higher by amount of 0.05 mRy per atom relative to Min1. 
Further increase in pressure leads to development of another minimum at angle108.50° 
(say, Min2) and BCC lies on a local maximum. At 112 GPa pressure energy difference 
between Min1 and Min2 is about –0.057 mRy per atom and that between Min1 and BCC 
is -0.10 mRy per atom. Thus now BCC-V becomes mechanically unstable toward a 
rhombohedral distortion which can lead to softening of C44 elastic constant as it related to 
trigonal shear. In fact Land et al [6]  predicted C44 softening in the same pressure region. 
It is to be noted that in our work the BCC to rhombohedral structural phase transition has 
occurred at much lower pressure (~ 60 GPa).  These energy changes are  found to persist 
even with denser (e.g. 28x28x28 k-point) mesh for the BZ integration and with larger 
plane wave energy cut offs (e.g., 65 Ry). Thus it confirms that this feature is not an 
artifact of calculation but it is real. In fact, as these calculations were performed with 
same rhombohedral cell  for all values of angles; inaccuracies due to computational 
parameters will be same. At 112 GPa pressure the calculations were also repeated with 
VASP using PAW pseudopotential [11-13] but results were invariant. 
In our calculations the frequency of TA mode becomes imaginary at some q-values along 
the Γ-Η direction of the BZ (Fig. 3c) at 112 GPa pressure; lower compared to ~130 GPa 
pressure reported by Suzuki and Otani [5]. The reason of this mismatch of pressure can 
be due to the  difference in computational techniques and approximations used for 
exchange-correlation terms.  
The total energy behavior between angles 106°-112° at 160 GPa is similar to that at 112 
GPa except now the Min2 becomes lower in energy relative to Min1 by 0.20 mRy per 
atom leading to an iso-structural phase transition. Now Min2 energy  is lower by 0.14 
mRy per atom relative to BCC. Further increase in pressure lead to  disappearance of  
Min1 and the energy difference between Min2 and Min1 at  240 GPa is  around -0.55 
mRy per atom.  However, the energy difference between the BCC and  Min1  is 0.14 
mRy per atom. At 332 GPa, the Min2 is lower in energy by 0.019 mRy per atom relative 
to 109.47° (BCC) energy value. Further increase in pressure lead to disappearance of the 
Min2  also and at 434 GPa pressure there is only one minimum located at 109.47°. In our 
phonon calculations at 160 GPa pressure the TA mode frequency becomes positive and 
further increase in pressure lead to usual pressure behavior, i.e, hardening with pressure 
[Fig. 3(c)]. The calculations for pressures higher than 300 GPa were checked with VASP 
code using PAW pseudopotential including the  nonlinear core corrections which are 
supposed to be taken care the core-core overlap at higher pressures. Our results remain 
unchanged. 
We have also performed total energy calculations for hexagonal closed packed (HCP) 
lattice at optimized c/a axial ratio (1.829) at two pressures; one at ambient and other at 
332 GPa pressure and its energy is found always higher compared to BCC. The energy 
difference (19.6 mRy/atom at ambient pressure) between BCC and HCP increases with 
pressure (40 mRy/atom at 332 GPa ) thus rules out the BCC-HCP transition.  
It is to be noted that work reported by Ding et. al. had claim the observation of BCC to 
rhombohedral transition near 69 GPa pressure [17] which also supports our predictions. 
                           Similar calculations were repeated for the Nb and Ta and the results of 
total energy variations with αrhom at several pressures are shown in Figs. 4(a,b). No new 
features appear in total energy curves under pressure like those in V and thus no phonon 
calculations were attempted. These are in agreement with earlier elastic constant and 
phonon calculations [3,6,16]. The only known anomaly for these two elements in the C44 
is its slope change with pressure, which occurs at 40 GPa for Nb [6] and in 100 GPa 
region for Ta [16]. But lattice expansion in Nb resulted a new minimum around 111° 
which is lower in energy relative to BCC by 11 meV per atom at 6.7% lattice expansion. 
By analogy, we expect such features in Ta will appear even at  larger lattice expansions. 
Conclusion: 
       In conclusion, high pressure structural behavior of BCC metals of group-VB are 
investigated using density functional theory. Calculations for V show the onset of a 
phonon softening and a rhombohedral instability  at  60 GPa pressure which may lead to 
C44 elastic constant softening as predicted earlier [6] in the same pressure region. Thus,  
we predict a  BCC to rhombohedral structural phase transition (αrhom=110.5°) at around 
60 GPa pressure in V. It exists in rhombohedral structure at least up to 330 GPa pressure, 
however its angle is changed to 108.50° at 160 GPa pressure resulting to an iso-structural 
phase transition. Finally at  around 434 GPa pressure, it transforms to BCC structure 
again.  This behavior is not found in the Nb and Ta.  
References: 
[1] V. V. Struzhkin, Y. A. Timofeev, R. J. Hemley and H. K. Mao, Phys. Rev. Lett. 79, 
4262 (1197). 
 [2] K. Takemura,  ‘Porc. Int. Conf. On High Pressure Science and Technology, AIRAPT-
17’  (Honolulu, 1999), (Sci. Technol. High Pressure vol 1) ed. M. H. Maghnani, W. J. 
Nellis and M. F. Nicol (India: University Press) page 443 (2000). 
[3] J. S. Tse, Z. Li, K. Uehara, Y. Ma and R. Ahuja, Phys. Rev. B 69, 132101 (2004). 
[4] H. Cynn and C. S. Yoo, Phys. Rev. B 59, 8526 (1999). 
[5] N. Suzuki and M. Otani, J. Phys.:Condens. Matter 14, 10869 (2002). 
[6] A. Landa, J. Klepeis, P. Soderlind, I. Naumov, O. Velikokhatnyi, L. Vitos and A. 
Ruban, J. Phys.:Condens. Matter 18, 5079 (2006).  
[7] A. Dewaele, P. Loubeyre and M. Mezouar, Phys. Rev. B 70, 094112 (2004). 
[8] C. Kittel, ‘Introduction to Solid State Physics’  Fifth Edition  (1993). 
[9] Website, http://www.pwscf.org
[10] S.  Baroni, S. de Gironcoli, A. D. Corso and P. Giannozzi, Rev. Mod. Phys. 73, 515 
(2001). 
[11] G. Kresse and J. Furthmuller, Phys. Rev. B. 54, 11169  (1996). 
[12] G. Kresse   and J. Hafner, Phys. Rev. B 47, 558 (1993). 
[13] G. Kresse   and J. Joubert, Phys. Rev. B 59, 1758 (1999). 
[14]  J. P. Perdew, K Burke and, M Ernzerhof, Phys. Rev. Lett. 77, 3865  (1996). 
[15] C. Nirmala Louis and K. Iyakutti, Phys. Rev. B 67, 094509 (2003). 
[16] O. Gulseren and R. E. Cohen, Phys. Rev. B 65, 064103 (2002). 
[17] Ding et. al., Phys. Rev. Lett. 98, 085502  (2007). 
http://www.pwscf.org/
0 100 200 300 400
0 40 80 120 160 200
    Theory
 Experiment
    Theory
 Experiment
Pressure (GPa)
    Theory
 Experiment
 Experiment
Fig. 1: The calculated P-V relations are shown with solid lines. The experimental data for 
V and Nb are from Ref. [2]. The experimental data for Ta shown by symbol (Ο) is from 
Ref. [4] and other is form Ref. [7]. The V0’s are respective theoretical and experimental 
equilibrium volumes. 
40 50 60 70 80 90 100 110 120
-0.74
-0.73
-0.72
-0.71
-0.70
-0.69
-0.68
-0.67
50 55 60 65 70 75
-0.720
-0.716
-0.712 V
P ~ 0 GPa
BCCSC
 (Deg)
Fig. 2: Total energy variation with αrhom at volume 13.48 Å3 per atom. 
-0.739
-0.738
-0.737
-0.736
-0.735
-0.734
-0.7200
-0.7188
-0.7176
-0.7164
-0.694
-0.693
-0.692
-0.691
106 107 108 109 110 111 112
-0.663
-0.662
-0.661
-0.660
-0.659
P~0 GPaVol=13.48
P=42 GPaVol=11.36
P=60 GPa
Vol=10.37
P=112 GPaVol=9.64
Fig. 3(a) 
-0.617
-0.616
-0.615
-0.614
-0.613
-0.612
-0.611
-0.538
-0.536
-0.534
-0.532
-0.530
-0.462
-0.460
-0.458
-0.456
-0.454
106 107 108 109 110 111 112
-0.334
-0.332
-0.330
-0.328
-0.326
P=160 GPaVol=8.89
P=240 GPaVol=8.02
P=332 GPaVol=7.41
P=434 GPaVol=6.67
Fig. 3(b) 
Fig. 3(a,b): The total energy variation with angle αrhom  at different pressures for V. All 
the volumes are in unit of Å3 per atom. 
P=60 GPa
P=112 GPa
P=160 GPa
P=332 GPa
Fig. 3c: The pressure variation of the phonon modes frequency of V along Γ-Η direction 
of the BZ.  
-9.900
-9.895
-9.890
-9.885
-9.880
-9.875
-10.215
-10.200
-10.185
-10.170
-10.18
-10.16
-10.14
-10.12
106 107 108 109 110 111 112
-9.48
-9.44
-9.40
-9.36
Vol=22
P=-2 GPa Vol=18.5
P=13 GPa Vol=17
Vol=10
(Deg)
P=73 GPa Vol=14
Fig. 4a 
-11.84
-11.82
-11.80
-11.78
-11.76
-11.74
-11.72
-11.70
-11.68
-11.36
-11.32
-11.28
-11.24
-11.20
-11.16
-11.12
-10.92
-10.88
-10.84
-10.80
-10.76
-10.72
-10.68
-10.64
-10.60
-10.56
-10.52
-10.48
-10.44
-10.40
-10.36
-10.32
106 107 108 109 110 111 112
-7.60
-7.52
-7.44
-7.36
-7.28
-7.20
-7.12
P=14 GPa Vol= 17.5
P=63 GPa Vol=15
Vol=14
P=114 GPa
P=96 GPa
Vol=13.5
P=280 GPa
Vol=10.78
(Deg)
Fig. 4b 
Fig. 4 (a,b): The calculated total energy variation with αrhom  at several volumes for Nb 
and Ta. The volumes given in figures are in units of Å3 per atom. 
