| الوصف: |
In the present work, we have performed the phonon dispersion calculations of body-centered cubic Vanadium (V) and Niobium (Nb) with the supercell approach using different supercell size. Using DFT method, the calculated phonon spectra of V and Nb are found to be in a good agreement with the available experimental data. Our calculated results show a \enquote{dip}-like feature in the longitudinal acoustic phonon mode along the $\Gamma$-H high symmetric path for both transition metals in the case of supercell size 4$\times$4$\times$4. However, in supercell size 2$\times$2$\times$2 and 3$\times$3$\times$3, the \enquote{dip}-like feature is not clearly visible. In addition to this, thermodynamical properties are also computed, which compare well with the experimental data. Apart from this, the phonon lifetime due to electron-phonon interactions ($\tau_{\rm{eph}}^{\rm{ph}}$) and phonon-phonon interactions ($\tau_{\rm{ph}}^{\rm{ph}}$) are calculated. The effect of phonon-phonon interactions (PPI) is studied by computing the average phonon lifetime for all acoustic branches. The value of $\tau_{\rm{eph}}^{\rm{ph}}$ of V (Nb) is found to be 23.16 (24.70)$\times 10^{-15}$ s at $100$ K, which gets decreased to 1.51 (1.85)$\times 10^{-15}$ s at 1000 K. The $\tau_{\rm{ph}}^{\rm{ph}}$ of V (Nb) is found to be 8.59 (18.09)$\times 10^{-12}$ and 0.83 (1.76)$\times 10^{-12}$ s at 100 and 1000 K, respectively. Nextly, the lattice thermal conductivity is computed using linearized phonon Boltzmann equation. The present work suggests that studying the variation of phonon dispersion with supercell size is crucial for understanding the phonon properties of solids accurately. |
