Direct Observation of the Superconducting Gap in Phonon Spectra

We show that the superconducting energy gap $2\Delta$ can be directly observed in phonon spectra, as predicted by recent theories. In addition, since each phonon probes the gap on only a small part of the Fermi surface, the gap anisotropy can be studied in detail. Our neutron scattering investigation of the anisotropic conventional superconductor ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ demonstrates this new application of phonon spectroscopy.

Figure 1

Top: Evolution of the neutron-scattering profile measured on ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ at $\mathbf{Q}=(0.5,0.5,7)$ at $T=T_c(15.2\mathrm{K})$ and below $T_c$. Solid lines are guides to the eye. The broken line denotes the background as determined from measurements at neighboring $\mathbf{q}$ points. Bottom: Calculated phonon line shapes based on the theory of Allen et al. [10], using parameters extracted from the experimentally observed line shape in the normal state (thin lines). Thick lines are obtained after convolution with the experimental energy resolution.

Figure 2

Left: Background subtracted energy scans taken at $\mathbf{Q}=(0.5,0.5,7)$ at three different temperatures including an offset of 300 counts for clarity. Lines were calculated from the theory of Allen et al. [10] after scaling to the experimentally observed step heights and folding with the experimental energy resolution. The vertical bars indicate the gap values used in the calculations, respectively. Right:  Temperature scans taken at $\mathbf{Q}=(0.5,0.5,7)$ for three different energy transfers including an offset of 35 counts for clarity. Lines are as in the left panel. Vertical bars indicate the temperature for which the energy transfer is equal to the gap value $2\Delta$ in the calculation.

Figure 3

Top: Evolution of the neutron-scattering profile measured on ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ at $\mathbf{Q}=(0.5,0,8)$ above and below $T_c$. Solid lines are guides to the eye. The broken line denotes the background as determined from measurements at neighboring $\mathbf{q}$ points. Bottom: Calculated phonon line shapes based on the theory of Allen et al. [10], using parameters extracted from the line shape observed in the normal state (thin lines). The thick lines are obtained after convolution with the experimental resolution.

Figure 4

Superconducting energy gap $2\Delta$ versus temperature as evaluated from phonon data at different momentum transfers, i.e., $\mathbf{q}=(0.5,0.5,0)$ and $(0.5,0,0)$, respectively. Broken lines are calculated from weak-coupling BCS theory scaled to the experimental values at $T=3\mathrm{K}$. Solid lines are guides to the eye.

Figure 5

Panels (a)–(c) show background subtracted energy scans taken at $\mathbf{Q}=(0.65–0.75,0,8)$ at $T=2\mathrm{K}$ (open symbols) and 17 K (filled symbols). Lines are guides to the eye. Panel (d) describes the evolution of the low temperature superconducting gap value extracted from phonon scans along the (100) direction.