Superconducting energy gap of $2H\text{−}\mathrm{NbS}{\mathrm{e}}_2$ in phonon spectroscopy

We present a high-energy-resolution inelastic x-ray scattering data investigation of the charge-density-wave (CDW) soft phonon mode upon entering the superconducting state in $2H\text{−}\mathrm{NbS}{\mathrm{e}}_2$. Measurements were done close to the CDW ordering wave vector ${\mathit{q}}_{\mathrm{CDW}}$ at $\mathit{q}={\mathit{q}}_{\mathrm{CDW}}+\left(0,0,l\right),0.15\le l\le 0.5$, for $T=10\mathrm{K}$ (CDW order) and $3.8\mathrm{K}$ (CDW order+superconductivity). We observe changes of the phonon line shape that are characteristic for systems with strong electron-phonon coupling in the presence of a superconducting energy gap $2{\mathrm{\Delta }}_c$ and from which we can demonstrate an $l$ dependence of the superconducting gap. Reversely, our data imply that the CDW energy gap is strongly localized along the $c$* direction. The confinement of the CDW gap to a very small momentum region explains the rather low competition and easy coexistence of CDW order and superconductivity in $2H\text{−}\mathrm{NbS}{\mathrm{e}}_2$. However, the energy gained by opening ${\mathrm{\Delta }}_{\mathrm{CDW}}$ seems to be too small to be the driving force of the phase transition at $T_{\mathrm{CDW}}=33\mathrm{K}$, which is better described as an electron-phonon coupling driven structural phase transition.

Figure 1

(a) Raw data of inelastic x-ray scattering in $2H\text{−}\mathrm{NbS}{\mathrm{e}}_2$ at $T=10\mathrm{K}$ (dots) and $3.8\mathrm{K}$ (circles), i.e., above and below the superconducting transition temperature $T_c=7.2\mathrm{K}$. Data were taken at $Q=\left(2.671,0,l\right)$ and $0.15\le l\le 0.5$. (b) Observed phonon energies at $T=10\mathrm{K}$ (dots) in comparison to the reported dispersion of the soft mode at $T=T_{\mathrm{CDW}}=33\mathrm{K}$ (line) [16].

Figure 2

(a) Exemplarily analysis of IXS data (dots) taken in the nonsuperconducting phase of $2H\text{−}\mathrm{NbS}{\mathrm{e}}_2$. The solid line is a fit using damped harmonic oscillator functions for the phonons (dashed blue lines), a resolution limited pseudo-voigt function for the elastic line (dotted line), and a constant background (dashed-dotted line). The vertical bar represents the experimental resolution of $1.5\mathrm{meV}$ (FWHM). (b) Calculation of the spectral weight distribution in the superconducting phase with an energy gap of $\mathrm{\Delta }=0.9\mathrm{meV}$. The solid black line corresponds to the DHO fit of the lower-energy phonon in (a). The dashed red line is the calculation based on the theory of Allen et al. [25] and the solid red line includes the effect of the finite experimental resolution. (c) Difference between the nonsuperconducting and calculated superconducting phonon line shape shown in (b). The position of the superconducting energy gap $2\mathrm{\Delta }=1.8\mathrm{meV}$ is indicated by the vertical dashed line.

Figure 3

(a) Difference curves of the observed intensities at $Q=\left(2.671,0,l\right),0.15\le l\le 0.5$, at $T=10$ and $3.8\mathrm{K}$. Data for $l<0.5$ are offset for clarity (offsets indicated by the thin horizontal lines). Red solid lines are calculated differences (see text and Fig. 2) employing $\mathrm{\Delta }=0.6\mathrm{meV}$ $\left(l=0.15\right),0.9\mathrm{meV}$ $\left(l=0.3\right)$, and $1.2\mathrm{meV}$ $(l=0.4$ and $0.5)$. Dashed lines are results calculated with $\mathrm{\Delta }=1.2\mathrm{meV}$ $\left(l=0.15\right)$ and $0.6\mathrm{meV}$ $(l=0.4$ and $0.5)$. (b) $\mathrm{\Delta }$ values as a function of $l$ (dots) yielding the best agreement between experiment and theory. The solid line is a guide to the eye assuming $\mathrm{\Delta }=0$ at the CDW superlattice wave vector, i.e., at $l=0$. The dashed horizontal lines indicate the largest and smallest $\mathrm{\Delta }$ values used, for which calculated differences are also plotted in (a) for comparison.