Momentum-Resolved Electron-Phonon Interaction in Lead Determined by Neutron Resonance Spin-Echo Spectroscopy

Neutron resonance spin-echo spectroscopy was used to monitor the temperature evolution of the linewidths of transverse acoustic phonons in lead across the superconducting transition temperature $T_c$ over an extended range of the Brillouin zone. For phonons with energies below the superconducting energy gap, a linewidth reduction of maximum amplitude $\sim 6\mu \mathrm{eV}$ was observed below $T_c$. The electron-phonon contribution to the phonon lifetime extracted from these data is in satisfactory overall agreement with ab initio lattice-dynamical calculations, but significant deviations are found.

Figure 1

Neutron resonance spin-echo profiles for the $T_1$ phonon at $(0.3,0.3,0)$ at selected temperatures above and below the superconducting transition temperature. The inset shows a triple-axis scan in focusing mode for the same phonon. From these data, $\Gamma$ was extracted as $25.5\pm 0.6$, $31.4\pm 0.8$, and $30.9\pm 0.7\mu \mathrm{eV}$ for $T=3.7$, 5.5, and 8.0 K, respectively.

Figure 2

(a)–(e) Temperature dependence of the observed NRSE relaxation rate $\Gamma$ for $T_1$ phonons along $(\xi \xi 0)$. (f) Superconducting energy gap from tunneling data [17]. (g),(h) $\Gamma$ versus temperature for $T_2$ phonons along $(\xi \xi 0)$ and (i) for a transverse phonon along $(\xi \xi \xi )$. The dashed lines in panels (a)–(e),(g),(i) indicate the temperatures at which the superconducting gap opens, according to the data in panel (f).

Figure 3

Experimentally determined and numerically calculated electron-phonon linewidth of transverse phonons along $(\xi \xi 0)$. The experimental points were obtained by subtracting the linewidths at 10 K (see Fig. 2) from those at the lowest temperature (0.5 K).