Electron-phonon interactions and high-temperature thermodynamics of vanadium and its alloys

Inelastic neutron scattering was used to measure the phonon densities of states (DOSs) for pure V and solid solutions of V with 6 to 7at% of Co, Nb, and Pt, at temperatures from 10 K to 1323 K. Ancillary measurements of heat capacity and thermal expansion are reported on V and V-7at%Co and used to help identify the different sources of entropy. Pure V exhibits an anomalous anharmonic stiffening of phonons with increasing temperature. This anharmonicity is suppressed by Co and Pt, but not by isoelectronic Nb solutes. The changes in phonon frequency with alloying and with temperature both correlate to the decrease in electron density of states (DOS) at the Fermi level as calculated using density functional theory. The effects of both temperature and alloying can be understood in terms of an adiabatic electron-phonon interaction (EPI), which broadens sharp features in the electron DOS. These results show that the adiabatic EPI can influence the phonon thermodynamics at temperatures exceeding 1000 K, and that thermal trends of phonons may help assess the strength of the EPI.

Figure 1

(Color online) Phonon DOS of V alloys measured with inelastic neutron spectrometry.

Figure 2

(Color online) Linear thermal expansion and coefficient of linear thermal expansion for pure V and V-7%Co measured by dilatometry (markers) and literature values of Ref. 18 for pure V (lines). The differential expansion, $\Delta l/l$, is referenced to 300 K.

Figure 3

(Color online) Anharmonic entropy for pure V and V-7%Co. The squares are the nonharmonic entropy $S_{\text{ph},\text{NH}}$ calculated from INS data (HFIR); triangles are the entropy $S_{\text{D}}$ calculated as in the text. Filled symbols: pure V, open symbols: V-7%Co. Dashed line: same as filled triangles, evaluated from the literature.

Figure 4

(Color online) Mean phonon energy of V and alloys. The markers are results from INS measurements. The dashed-dotted lines correspond to QH behavior for V-7%Co and pure V, with experimental Grüneisen parameters (see text).

Figure 5

(Color online) (a) Heat capacity of pure V and its different components. (b) Heat capacity of V-7%Co and its different components (see text). In both panels, the total constant pressure heat capacity, $C_{\text{P},\text{tot}}$, measured with DSC is given by the dots. Experimental error bars are comparable to the size of the markers. The sum of components are labeled $C_{\text{tot},\text{bare}}=C_{\text{ph},\text{H}}+C_{\text{D}}+C_{\text{el},\text{bare}}$, and $C_{\text{tot},\text{e}-\text{p}}=C_{\text{ph},\text{H}}+C_{\text{D}}+C_{\text{el},\text{e}-\text{p}}$. In panel $b$, $\Delta C_P=C_P\left(\text{V}-7\%\text{Co}\right)-C_P\left(\text{V}\right)$, measured with DSC. The value of $C_P$ for pure V recommended by Maglic (Ref. 20) is shown in panel a.

Figure 6

(Color online) Electronic density of states for V-6.25%X alloys, computed from first principles with the DFT computer code WIEN2K (see text). The electron DOS for V is the dashed red line (repeated and offset for comparison) and the DOS for the V-6.25%X alloys are given by the solid blue lines. (Green crosses: electron DOS for V computed with VASP.)

Figure 7

(Color online) (a) Electronic entropy and heat capacity for pure V obtained from the electron DOS calculated from first principles. (b) Electron entropy and heat capacity for V-6.25%Co obtained from the electron DOS calculated from first principles.

Figure 8

(Color online) (a) Change in $N\left(E_{\text{F}}\right)$ for 6.25% TM solutes, from DFT calculations (open markers) and heat capacity in the literature (solid markers). (b) Estimated superconducting critical temperature for solid solutions V-6.25%TM, as function of the number $N_d$ of $d$ electrons of the TM solute (in the $3d$, $4d$ or $5d$-series). (c) Calculated ${\lambda }_{\text{el}-\text{ph}}$ from $T_c$ measurements in literature (see text). The number of $d$ electrons of V is taken equal to 3.

Figure 9

(Color online) (a) Electronic DOS for V and ${\text{V}}_{15}{\text{X}}_1$ computed from first principles with VASP. (b) Electronic DOS for V and ${\text{V}}_{15}{\text{Co}}_1$ at 0 K and with broadening effect induced by the EPI at 1000 K and shift in $\mu \left(T\right)$ ($\lambda \left(\text{V}\right)=0.7$, $\Gamma \left(\text{V}\right)=190\text{meV}$; $\lambda \left(\text{V}-6.25\%\text{Co}\right)=0.5$, $\Gamma \left(\text{V}-6.25\%\text{Co}\right)=135\text{meV}$. Also shown are the derivatives of the Fermi-Dirac function $-\frac{\partial f}{\partial E}\left(E,T=1000\text{K}\right)$ and the Lorentzian representing the EPI broadening for V at 1000 K $\left(\Gamma =190\text{meV}\right)$.