Adiabatic Electron-Phonon Interaction and High-Temperature Thermodynamics of $A15$ Compounds

Inelastic neutron scattering was used to measure the phonon densities of states of the $A15$ compounds ${\mathrm{V}}_3\mathrm{Si}$, ${\mathrm{V}}_3\mathrm{Ge}$, and ${\mathrm{V}}_3\mathrm{Co}$ at temperatures from 10 to 1273 K. It was found that phonons in ${\mathrm{V}}_3\mathrm{Si}$ and ${\mathrm{V}}_3\mathrm{Ge}$, which are superconducting at low temperatures, exhibit an anomalous stiffening with increasing temperature, whereas phonons in ${\mathrm{V}}_3\mathrm{Co}$ have a normal softening behavior. First-principles calculations show that this anomalous increase in phonon frequencies at high temperatures originates with an adiabatic electron-phonon coupling mechanism. The anomaly is caused by the thermally induced broadening of sharp peaks in the electronic density of states of ${\mathrm{V}}_3\mathrm{Si}$ and ${\mathrm{V}}_3\mathrm{Ge}$, which tends to decrease the electronic density at the Fermi level. These results show that the adiabatic electron-phonon coupling can influence the phonon thermodynamics at temperatures exceeding 1000 K.

Figure 1

Phonon DOS of ${\mathrm{V}}_{3}X$ compounds ($X=\mathrm{Si}$, Ge, and Co), measured with inelastic neutron scattering, at different temperatures. Curves are extrapolated below 6 meV according to the long-wavelength limit.

Figure 2

Average phonon energy of ${\mathrm{V}}_{3}X$ compounds as a function of temperature. Markers are results from INS measurements, and dashed-dotted lines correspond to QH behavior, for the values of the Grüneisen parameters listed in the text. The QH curves were offset vertically to match the INS data at the highest $T$ measured.

Figure 3

Heat capacity at constant pressure $C_P$ for ${\mathrm{V}}_3\mathrm{Si}$, measured by DSC (several data sets), and contributions calculated from harmonic phonons ($C_{\mathrm{ph},H}$), bare electrons ($C_{\mathrm{el},\mathrm{bare}}$), electrons with EPI effect ($C_{\mathrm{el},e\mathrm{\text{−}}p}$, see text), and thermal dilation ($C_D$). Open squares are data from Knapp et al. [13].

Figure 4

(a) Electronic DOS for ${\mathrm{V}}_3\mathrm{Si}$, ${\mathrm{V}}_3\mathrm{Ge}$, and ${\mathrm{V}}_3\mathrm{Co}$, computed from first principles. (b) Electronic DOS for ${\mathrm{V}}_3\mathrm{Si}$ broadened by EPI at 25, 300, and 1000 K, using $\lambda =1.0$ at all $T$. Energies are with respect to $\mu (T)$. Inset: Broadening function at 300 and 1000 K.

Figure 5

Top: Phonon DOS for ${\mathrm{V}}_3\mathrm{Si}$ calculated from first principles for two different volumes ($V_{\mathrm{high}}=1.03\times V_{\mathrm{low}}$). Middle: Calculated DOS for different electronic broadenings $\sigma$. Bottom: The same as above for two different ($V$ and $\sigma$) combinations (${\sigma }_{\mathrm{low}}=50\mathrm{meV}$ and ${\sigma }_{\mathrm{high}}=200\mathrm{meV}$). Computations used a $2\times 2\times 2$ supercell (64 atoms) and a $4\times 4\times 4$ $k$-point grid. The calculated DOS was obtained from the neutron-weighted contributions of V and Si species and convolved with the experimental instrument resolution. The markers are experimental results for ${\mathrm{V}}_3\mathrm{Si}$ at 25 and 1023 K.