Neutron scattering study of the charge-density wave transitions in $2H-\mathrm{Ta}{\mathrm{Se}}_2$ and $2H-\mathrm{Nb}{\mathrm{Se}}_2$

We have used the triple-axis neutron-scattering technique to study $2H-\mathrm{Ta}{\mathrm{Se}}_2$ and $2H-\mathrm{Nb}{\mathrm{Se}}_2$ which undergo charge-density wave transitions at $T_0=122.3\mathrm{}$ K and $T_0=33.5\mathrm{}$ K, respectively. The transitions in both compounds appear to be second-order and involve atomic displacements of ${\Sigma }_1$ symmetry. At inception, the superlattices in both compounds have nearly identical incommensurate wave vectors with magnitude $q_{\delta }=\frac{1}{3(1-\delta )a^{*}}$, with $\delta \sim 0.02$. The Nb${\mathrm{Se}}_2$ superlattice remains incommensurate to 5 K but Ta${\mathrm{Se}}_2$ undergoes a first-order lock-in transition where $\delta \to 0$ at 90 K. The temperature dependence of the superlattice wave vector $\overrightarrow{\mathrm{q}}$ in the incommensurate phase and the lock-in transition are discussed using a free energy involving third-order "umklapp" terms and a secondary order parameter. The secondary lattice distortion which is predicted in this model is observed experimentally. Most phonon branches having energies less than 10 meV with propagation vectors in the [$\zeta 00$] and [$00\zeta$] directions have been measured at 300 K. Strong anomalies are found in the ${\Sigma }_1\left[\zeta 00\right]$ phonon branches in both materials near the wave vector ${\overrightarrow{\mathrm{q}}}_c=(\frac{1}{3},0,0)$ characteristic of the low-temperature superlattices. Substantial softening of this phonon is observed as the transition is approached. In addition, the spectral profile exhibits a central peak which is not measurably inelastic.