Phonons in Vanadium Alloys at High Temperatures
6.4 High-Temperature Properties of BCC Transition Metals
6.4.1 Phonon DOS
The phonons in elements of group 5 present an intermediate behavior between those of groups 4 and 6. In the high-temperature BCC phase of group 4 metals, the phonon DOS exhibits a stiffening with temperature. This is particularly pronounced for the transverse modes, while longitudinal modes do not vary much with temperature [91, 92], see figure 6.7. On the other hand, in the BCC metals of group 6, the phonons soften significantly with increasing temperature. This was observed by direct measurements of the phonon dispersions at high temperature in Cr (see [142]; see also Fig. 6.8) and was inferred from a thermodynamic analysis of the heat capacity for Mo and W [141].
Measurements of phonon dispersions in BCC Nb up to 2223K have been performed by G¨uthoff and coworkers [143]. Their results are shown in Fig. 6.9. From room temperature to 773 K, the phonon DOS of Nb exhibits a stiffening of the transverse modes with little change in the position of the longitudinal peak and cutoff, very much like the behavior observed in the BCC group 4 metals. At temperatures between 773 K and 1773 K, all vibrational modes start to soften, with similarity to the behavior in vanadium. The magnitude of the high-temperature softening in Nb is rather smaller than that expected based on thermal expansion alone, similarly as for V [56, 77].
It thus appears that the softening of phonons expected from thermal expansion is over-
0.08 0.06 0.04 0.02 0.00
DOS (1/meV)
40 30
20 10
0
E (meV) 1073K 293K
1473K 1773K
Figure 6.8: Phonon DOS of BCC Cr at elevated temperatures. Adapted from [142].
0.3
0.2
0.1
0.0
Phonon DOS (1/meV)
25 20
15 10
5 0
E (meV) Nb 293K
Nb 773K
Nb 1773K Nb 2223K
Figure 6.9: Phonon DOS of BCC Nb at elevated temperatures determined by G¨uthoff et al. [143].
come by an intrinsic temperature stiffening (that is, a stiffening with increasing temperature at constant volume) in BCC metals of groups 4 and 5. This intrinsic stiffening appears to dominate over the full temperature range of stability of the BCC phase in group 4 met- als, but it is suppressed at very high temperatures in the BCC group 5 metals V and Nb.
Between groups 5 and 6, the mechanism responsible for this intrinsic stiffening becomes ineffective and the phonons soften even at moderate temperatures.
6.4.2 Elastic Constants
It is instructive to compare the temperature dependence of phonons in BCC transition metals to the temperature variation of their elastic constants. Room-temperature ultra- sonic measurements of elastic constants for BCC transition metals and alloys with varying electron-to-atom ratios, e/a, have been reported [59, 60, 61, 62]. These were discussed in chapter 4. Several studies have been reported of such measurements in single crystals of V and Nb, as well as single-crystal alloys of V-Cr, Zr-Nb and Nb-Mo at varying compositions and over wide ranges of temperature [65, 66, 64, 67]. These studies have shown that theC44 shear elastic constant exhibits an anomalous temperature variation in V, Nb and Ta and in BCC alloys of composition close to V and Nb. Published results for V and Nb-Mo alloys are shown in figures 6.10 and 6.11, respectively. As seen on these figures,C44for V and Nb follows a non-monotonic temperature evolution, with first a decrease at low temperatures leading to a local minimum located between 300 K and 800 K, followed by a stiffening that proceeds up to melting in Nb, but reverses around 1500 K in V.
This behavior can be tuned by alloying V and Nb with neighboring elements Cr and Mo, respectively, which form solid-solutions over the whole range of compositions. As seen in Fig. 6.11, the addition of Mo in Nb shifts the position of the local minimum inC44to lower temperatures. For 33%Mo, the minimum is at 0 K and for concentrations above 56% Mo, the local minimum has disappeared, leaving a normal, monotonic temperature decrease ofC44. The same effect is observed in V-Cr alloys [67]. Through electronic structure calculations, this behavior has been interpreted as a Fermi surface effect [65, 66, 63]. In pure V, the distortion of the unit cell in a trigonal strain corresponding to the C44 elastic constant leads to the opening of a gap around the Γ25 point of the band structure, about 25 mRy
20 10 0
2000 1500
1000 500
0
Temperature (K) 200
150
100 50
0
GPa
2000 1500
1000 500
0
Temperature (K) C11
B
C12
Figure 6.10: Elastic moduli of vanadium single crystal as function of temperature, measured by Walker [64].
60
40
20
0 C44 (GPa)
2000 1500
1000 500
0
Temperature (K) Nb44Mo56
Nb67Mo33
Nb
Figure 6.11: Shear elastic constant C44 of Nb and Nb-Mo single crystals as function of temperature, measured by Bujard and coworkers [66].
1.0 0.8 0.6 0.4 0.2 0.0
Fermi distribution
-3 -2 -1 0 1 2 3
eV
60
40
20
0
states / supercell / eV
295K 1273K
Γ25'
Figure 6.12: Electronic DOS for BCC V calculated with Wien2k and position of Γ25 point.
above the Fermi level, associated with a decrease in the symmetry of the lattice [63]. This has only a small energy cost when the upper band above this gap is unoccupied as is the case in pure V and Nb at low T. However, it is expected to become more energetically unfavorable when electrons are thermally excited into these states. The position of the Γ25
point in the electronic DOS of BCC V and Fermi distribution for different temperatures are illustrated in Fig. 6.12. Alternatively, in the rigid-band model, the states above Γ25
become filled around 5.4e/a and, at this composition, the stiffening of C44 is shifted to lower temperatures, as band-filling and temperature play similar roles. At higher band- filling and higher temperature, the effect is not as prominent, as Fermi integrals involve the derivative (−∂f /∂E), which is either not centered on these states or not as sharply peaked [65, 66]. Besides, for band-fillings crossing the Γ25 point, the Fermi surface undergoes an electronic topological transition, as discussed in chapter 5. The significant modifications of the geometry of the Fermi surface during this transition have also been invoked to explain the behavior in theC44 elastic constant [112].
The temperature and composition variations of C44 presents some striking similarities with the variations observed in the phonon DOS, as described above. This is particularly true for the transverse phonon modes, as expected since those have a close connection to shear elastic constants. In particular, the stiffening (or absence of softening) of these
5.3e/a, which is not in bad agreement with the composition at which the anomaly inC44is suppressed above 300 K. However, we expect that Co and Pt have additional effects, related to the electronic redistribution described in chapter 5. Nevertheless, this similarity between C44 and the phonons strongly suggests the importance of the coupling of the phonons with the Fermi surface in theT-dependence of the phonons in vanadium and niobium.