Anharmonic suppression of charge density waves in 2$H$-NbS${}_2$

The temperature dependence of the phonon spectrum in the superconducting transition-metal dichalcogenide 2$H$-NbS${}_2$ is measured by diffuse and inelastic x-ray scattering. A deep, wide, and strongly temperature-dependent softening of the two lowest-energy longitudinal phonon bands appears along the $\Gamma \mathrm{M}$ symmetry line in reciprocal space. In sharp contrast to the isoelectronic compound 2$H$-NbSe${}_2$, the soft phonons energies are finite, even at very low temperature, and no charge density wave instability occurs, in disagreement with harmonic ab initio calculations. We show that 2$H$-NbS${}_2$ is at the verge of the charge density wave transition and its occurrence is only suppressed by the large anharmonic effects. Moreover, the anharmonicity and the electron phonon coupling both show a strong in-plane anisotropy.

Figure 1

Reconstruction of the x-ray diffraction in the (H,K,0) plane of 2$H$-NbS${}_2$ at 300 and 77 K, from a 3D mapping of thermal diffuse scattering. A symmetrization has been applied. Diffuse intensity is present only along $\Gamma \mathrm{M}$ and, more precisely, only where scattering geometry selects longitudinal phonons (white ellipses). No diffuse features are visible for transverse geometry (black ellipse). At 77 K, small extra spots are visible.

Figure 2

Experimental phonon spectrum of 2$H$-NbS${}_2$ at room temperature (full red circles) and 2 K (open blue diamond), compared to ab initio, zero-temperature, and harmonic phonons (dashed lines) calculations. The two soft phonon branches along $\Gamma \mathrm{M}$ are nearly degenerate at room temperature, and slightly split away as temperature decreases. At the exception of the scans along $\Gamma \mathrm{M}$ at room temperature, we limited ourselves to energies below 30 meV.

Figure 3

(a) IXS scans at (H,K,L) $=$ (3.184,$-$0.072,0) fitted by the convolution of a DHO (with Bose factor) and the Lorentzian experimental resolution. Ab initio phonon calculations, convoluted by the experimental resolution, are in good agreement with experimental results. (b) IXS scans at (H,K,L) $=$ (3.375,0,0) for different temperatures. For clarity, scans are shifted up. In the superimposed fits, the linewidths and amplitude ratio of the two phonons are fixed according to the ab initio calculations (see text). Ab initio phonon calculations, convoluted by the experimental resolution, are also in good agreement with experimental results when using higher $T_{\mathrm{elec}}$.

Figure 4

Experimental phonon energies, squared, from our fitting procedure with two phonons (see text), as a function of temperature at ($h$,$k$,$l$) = (3.375,0,0). Energies are compatible with the mean-field theory that predicts a power law $T^{\frac{1}{2}}$. Lines are guides to the eyes.

Figure 5

Calculated phonon energy squared for the two lowest branches at 0.8$\Gamma \mathrm{M}$ as a function of the smearing of the Fermi surface with $T_{\mathrm{elec}}$.