Unusual lattice dynamics of vanadium under high pressure
The electronic structures and lattice dynamics of pressure-induced complex phase transitions [bcc → hR1(110.5°) → distorted-hR1(108.2°) → bcc] in vanadium as a function of pressure up to 400 GPa have been investigated with an ab initio method using density functional perturbation theory (DFPT). At ambient pressure, the soft transverse acoustic phonon mode corresponding to Kohn anomaly appears at a wave vector q = 2k F along [ξ00] Γ → H high symmetry direction. The nondegenerate transverse acoustic branches TA1 on 〈110〉 and TA2 on 〈001〉 show an exceptionally large split at high symmetry point N (0.5 0.5 0.0). The lattice dynamical instability starts at a pressure of 62 GPa (V/V 0 = 0.78, where V 0 is experimental volume of bcc-V at ambient conditions), derived by phonon softening that results in phase transition of bcc → hR1 (alpha = 110.5°). At compression around 130 GPa (V/V 0 = 0.67), the rhombohedral angle of hR1 phase changed to 108.2°, and the electronic structure changed drastically. At even higher pressure, ≈250 GPa (V/V 0 = 0.57), lattice dynamic calculations show that the bcc structure becomes stable again.
A vast number of extremely interesting and fundamental physical and chemical changes in solids have been reported in high-pressure experiments. The Kohn anomaly in bcc transition metals is considered to be one of the unsolved problems in the field of high-pressure structural phase transition. Kohn anomaly is characterized by electron–phonon features. Vanadium is an interesting case because this metal is one of the so-called hard superconductors (1) and it has been shown that the application of high pressure increases the superconducting temperature Tc (2, 3). The experimental phonon dispersion curves at ambient conditions in bcc phase using inelastic neutron scattering has been measured and for the longitudinal (111) branch it exhibits a dip near the (2/3 2/3 2/3) position (4). In another experiment by Page (5), a peak in the frequency distribution for vanadium is observed at 2 × 1012 c/s, and this peak is attributed to Kohn anomalies (5). As far as calculations are concerned, the lattice dynamics study of bcc vanadium at high pressure using a full-potential linear muffin-tin orbital (LMTO) method has been performed by Suzuki and Otani (3), and these calculations show a strong change of phonon density of states with volume at 130 GPa. It has been found that the transverse acoustic phonon mode TA [ξ00] around ξ = 1/4 shows a dramatic softening under pressure and becomes imaginary at pressures >130 GPa, indicating the possibility of a structural phase transition. Landa et al. (6) have performed the elastic constants calculations for vanadium using exact muffin-tin orbital (EMTO) and shown the trigonal shear elastic constant (C44) softening at ≈200 GPa due to the intraband nesting of the Fermi surface (FS).
Very recently, Ding et al. (7) performed the high-pressure experiment for bcc vanadium using diamond-anvil cell and synchrotron x-ray diffraction and, for the first time, observed a transition from bcc to a rhombohedral phase (hR1) at 69 GPa; this is a new type of high-pressure structural phase transition that was not found in earlier experiments for any element or compound. Therefore, it will be very interesting to perform first-principles calculation using this new high-pressure phase to look for the origin of such structural phase transitions. Here, we present calculations for the phonon dispersion, phonon density of states, and electronic band structures for different phases as a function of pressure. Details about the performed calculations can be found in Methods. We observed the structural sequence bcc → hR1(110.5°) → distortion-hR1(108.2°) → bcc with increasing pressure.
Very recently, Ding et al. (7) performed the high-pressure experiment for bcc vanadium using diamond-anvil cell and synchrotron x-ray diffraction and, for the first time, observed a transition from bcc to a rhombohedral phase (hR1) at 69 GPa; this is a new type of high-pressure structural phase transition that was not found in earlier experiments for any element or compound. Therefore, it will be very interesting to perform first-principles calculation using this new high-pressure phase to look for the origin of such structural phase transitions. Here, we present calculations for the phonon dispersion, phonon density of states, and electronic band structures for different phases as a function of pressure. Details about the performed calculations can be found in Methods. We observed the structural sequence bcc → hR1(110.5°) → distortion-hR1(108.2°) → bcc with increasing pressure.
Results and Discussion
We performed the calculations for the phonon dispersion curves along the high symmetry direction Γ–H–P–Γ–N for bcc-V at ambient conditions (Fig. 1). The solid lines are obtained by calculation, and the open symbols are the experimental data taken from the experimental work that were observed with x-ray thermal scattering (4). There is a steep slope observed at 1/3(Γ–H), caused by Kohn anomaly, on computed transverse acoustic mode (TA) along [ξ00] direction. Our calculations are in overall good agreement with experimental results, including anomalous frequency distribution near high-symmetry point N along Γ–N. There are two branches of one LA mode and other degenerated TA1 (〈1̄10〉) and TA2 (〈001〉) modes along the [ξξ0] direction. There exists a crossover of LA mode with TA2 mode at zone boundary point N. The transverse dispersion curve for [ξξ0] TA2 is higher than those of the longitudinal dispersion curve near the BZ of [ξξ0] symmetry direction. The present computed result gives better ordering of the branches than previous calculations (8, 9).
Fig. 2 shows the phonon density of state (PDOS) for bcc-vanadium (bcc-V) as a function of pressure. The lower frequency phonon mode caused by the Kohn anomaly is found to be grown and softened, but all other phonon frequencies move to higher values with increasing pressure. At V/V 0 ≈ 0.78 (62 GPa), negative frequency appears in the PDOS spectrum, but the PDOS still shows the typical bcc structural spectrum. After V/V 0 = 0.78 (62 GPa), the PDOS shows a
different profile of frequency distribution, indicating the dynamic instability of bcc-V at V/V 0 ≈ 0.78.
After comparing our calculated phonon spectrum with experiments performed under ambient conditions, we examined the effect of pressure on the phonon spectrum. Here, we chose three volumes: V/V 0 = 0.78, 0.75, and 0.56. Fig. 3 a shows that, at values up to V/V 0 = 0.78, bcc phase with the phonon mode linked to a Kohn anomaly has softened to negative values, and this can induce a high-symmetry loss and breaking in the H direction. This also drives the distortion in the bcc phase along (100) direction. The phonon spectra near Γ point also show softening. At V/V 0 ≈ 0.75 (112 GPa), the hR1 structure with α = 110.5° is more stable in our calculations and the corresponding phonon dispersion at this pressure is presented along high-symmetry direction Γ–FA–T–Γ–L (Fig. 3 b). As pressure increased to 130 GPa, i.e., volume compression V/V 0 = 0.67, the rhombohedral angle of hR1 phase changes to 108.2°. At V/V 0 = 0.56 (260 GPa), the rhombohedral angle changes back to 109.47° (i.e., the bcc phase), indicating that the bcc phase is dynamic stable or hR1 with an ideal angle of 109.47°. The phonon dispersion at this pressure is calculated and shown in Fig. 3 c. Compared to the ambient condition bcc phonon dispersion spectrum, there appears a deep dip of LA mode in (ξξξ) direction. This phonon softening in high-pressure bcc phase might make bcc phase unstable under extreme conditions (e.g., >500 GPa). Although the phonon dispersion of high-pressure bcc phase indicates dynamic stability at this pressure, the Kohn anomaly is still there. The anomaly in frequency at symmetry point N is absent in the high-pressure bcc phase.
The hR1 structure of V can be easily characterized starting from a parent bcc structure with primitive unit cell and considering the atomic displacements along the cubic diagonal, which transform as a transverse zone boundary instability at point N of the bcc Brillouin zone (BZ). At ambient pressure, vanadium crystallizes in a body-centered cubic phase Im3̄m, which has anomaly at symmetry point N mainly due to the larger elastic anisotropy (10). This is why, in our studies, the frequencies of transverse branch at N point are also found to be higher than those of longitudinal branch. We have shown that the frequencies of longitudinal mode LA and the transverse mode TA2 [polarized along (001)] mode as a function of pressure (Fig. 4). The frequency dispersion of LA and TA2 at V/V 0 ≈ 0.80 change back to the normal order (i.e., TA2 mode frequency is lower than LA mode frequency).
To understand high-pressure structural phase transition of vanadium by dynamical stability, we plotted the frequencies at Γ center along high-symmetry Γ–H and Γ–N directions of bcc BZ (Fig. 5, solid lines). The frequencies at Kohn anomaly [H(1/3 0 0)] soften mode are shown by dashed lines as a function of pressure. As mentioned above, the soft modes caused by Kohn anomalies are in the lower-frequency part along Γ–H direction. Under compression, the frequencies at (1/3 0 0) soften, and the phonon modes smoothly and slowly go to negative values up to V/V 0 = 0.79. It is reasonable to suggest that the phonon mode softening related to Kohn anomaly drives the structural phase transition from bcc to hR1. Above this pressure (up to V/V 0 = 0.59), the phonon dispersion of bcc phase shows a complete dynamical instability. At Γ point, all of the phonon branches along high-symmetry directions are negative, suggesting that no active phonon modes are present in bcc phase. This finding also indicates that bcc symmetry is broken under high pressure. It is noteworthy that the angle of rhombohedral hR1 phase changes from 110.5° to 108.2° at V/V 0 ≈ 0.67. A further compression changes this angle back to 109.47° at V/V 0 = 0.57. The origin of these two changes in rhombohedral angle α cannot be explained by the lattice dynamic calculation alone. Unlike the phase transition of bcc to hR1 at V/V 0 = 0.78, the transition of hR1 with angle 108.2° to bcc at V/V 0 = 0.57 shows an abrupt change in phonon frequency.
At ambient pressure, vanadium crystallizes in a body-centered cubic phase Im3m, and bcc structure can also be shown in rhombohedral structure. If the rhombohedral angle is 109.47°, then this structure will be equal to the bcc structure. To avoid the effect of symmetry, we have done all of the electron band structural calculations for vanadium using rhombohedral structural setting. The electronic band structure of bcc and hR1 phases as function of volume (or pressure) are shown in Fig. 6 a–d. For comparison, we have also plotted the band structure of bcc (Im3m) phase at ambient condition (Fig. 6 e).
From ambient volume to V/V 0 = 0.80, no pronounced change can be seen from the electronic band structure. In Fig. 6 a, we show the electronic band structure at volume compressed to V/V 0 = 0.80. There are two conduction bands: band 1 has symbols Γ1–FA1–T1–L1 and band 2 has the symbols Γ2–FA2–T2–L2 at high symmetry points, respectively. Both bands are partly filled by conduction electrons and above Fermi level at Γ point. In the rhombohedral hR1 structure with a 110.5° angle, shown in Fig. 6 b, there are still two conduction bands. The noticeable difference in hR1 band structure compared to band structure in Fig. 6 a is that, at Γ point, band 1 moves below the Fermi level and separates from the other two bands. Under pressure (i.e., at V/V 0 = 0.70), the Fermi level moves up and conduction band 1 is almost full filled by conduction electrons (data not shown). There is only one band that crosses the Fermi level, just like the electronic band structure of fcc metals.
At V/V 0 = 0.65, the rhombohedral angle α changes from 110.5° to 108.2°; the corresponding band structure is shown in Fig. 6 c. For α = 108.2° rhombohedral phase, band 2 moves from above to below the Fermi level and links with band 1 at Γ point, as shown in Fig. 6 a. At this point, rhombohedral angle α transforms from 110.5° to 108.2°; it is reasonable to suggest that this transition is driven by the electronic phase transition shown in Fig. 6 c. At even higher pressure, in high-pressure bcc phase, two conduction bands connected below the Fermi level at Γ center as shown in Fig. 6 d. One can see a huge change in electronic band structure going from Fig. 6 c to d. An empty band (≈1 eV above the Fermi level at point Γ) at V/V 0 = 0.60, as shown in Fig. 6 c, moves down as pressure increases and merges with two occupied bands at point Γ (Fig. 6 d). This change in electronic structure is responsible for an abrupt change in phonon frequencies at the transition from hR1 (108.2°) to high-pressure bcc phase.
We have also calculated the bulk modulus for vanadium using the Birch–Murnaghan equation of state for ambient bcc and hR1 phases. Our calculation gives the bulk modulus for bcc phase, B = 177.7 GPa with B′ = 4.3, and hR1 phase, B = 182.5 GPa with B′ = 3.6. These data compare very well with our experimental results (7).
In conclusion, we have performed ab initio calculations of lattice dynamics and electronic band structure as a function of pressure for vanadium. Based on our lattice dynamics calculations, we show the following structural phase transitions in vanadium bcc → hR1(110.5°) → distorted-hR1(108.2°) → bcc as a function of pressure. Our phonon calculations confirm the phase transition from BCC to hR1, which takes place at ≈62 GPa and compares very well with our experimental value of 69 GPa (7). At high pressure, based on our phonon calculations, we predict two new phase transitions: one from hR1 (110.5°) to distorted-hR1 (108.2°) at ≈120 GPa, and finally the reentrance of bcc phase at ≈250 GPa. We have explained our lattice dynamics results in terms of electronic band structure. A huge change in the electronic structure is driving force behind the structural phase transition. We welcome more experiments to look into the unusual lattice dynamics of vanadium.
MethodsAt V/V 0 = 0.65, the rhombohedral angle α changes from 110.5° to 108.2°; the corresponding band structure is shown in Fig. 6 c. For α = 108.2° rhombohedral phase, band 2 moves from above to below the Fermi level and links with band 1 at Γ point, as shown in Fig. 6 a. At this point, rhombohedral angle α transforms from 110.5° to 108.2°; it is reasonable to suggest that this transition is driven by the electronic phase transition shown in Fig. 6 c. At even higher pressure, in high-pressure bcc phase, two conduction bands connected below the Fermi level at Γ center as shown in Fig. 6 d. One can see a huge change in electronic band structure going from Fig. 6 c to d. An empty band (≈1 eV above the Fermi level at point Γ) at V/V 0 = 0.60, as shown in Fig. 6 c, moves down as pressure increases and merges with two occupied bands at point Γ (Fig. 6 d). This change in electronic structure is responsible for an abrupt change in phonon frequencies at the transition from hR1 (108.2°) to high-pressure bcc phase.
We have also calculated the bulk modulus for vanadium using the Birch–Murnaghan equation of state for ambient bcc and hR1 phases. Our calculation gives the bulk modulus for bcc phase, B = 177.7 GPa with B′ = 4.3, and hR1 phase, B = 182.5 GPa with B′ = 3.6. These data compare very well with our experimental results (7).
In conclusion, we have performed ab initio calculations of lattice dynamics and electronic band structure as a function of pressure for vanadium. Based on our lattice dynamics calculations, we show the following structural phase transitions in vanadium bcc → hR1(110.5°) → distorted-hR1(108.2°) → bcc as a function of pressure. Our phonon calculations confirm the phase transition from BCC to hR1, which takes place at ≈62 GPa and compares very well with our experimental value of 69 GPa (7). At high pressure, based on our phonon calculations, we predict two new phase transitions: one from hR1 (110.5°) to distorted-hR1 (108.2°) at ≈120 GPa, and finally the reentrance of bcc phase at ≈250 GPa. We have explained our lattice dynamics results in terms of electronic band structure. A huge change in the electronic structure is driving force behind the structural phase transition. We welcome more experiments to look into the unusual lattice dynamics of vanadium.
Electronic structural properties and phonon frequencies in vanadium are analyzed by using the density-functional perturbation theory (DFPT) (11). The vibrational properties have been computed by using a recent implementation of ultrasoft pseudopotentials (12) into DFPT formalism in the PWSCF (Plane-Wave Self-Consistent Field) code (www.pwscf.org). We have used Vanderbilt ultrasoft pseudopotential with 3s, 3p, 3d, and 4s states as valence states (13). The calculations are carried out by using generalized gradient approximation for the exchange and correlation potential as described by Perdew et al. (14).
The electronic wave functions are represented in a plane-wave basis set with a kinetic energy cutoff of 50 Ry. The Brillouin zone (BZ) integrations are carried out by the Gaussian smearing technique using a 12 × 12 × 12 k point mesh with shift from origin. Total energies are converged to within 10−4 Ry/atom with respect to energy cutoff changed from 30 to 90 Ry. The dynamical matrices were calculated on a 6 × 6 × 6 grid. All of the structural parameters are fully relaxed.
Cesar Hernandez
19.502.806
Electronica De Estados Solidos
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