${\mathrm{SmB}}_6$ electron-phonon coupling constant from time- and angle-resolved photoelectron spectroscopy

${\mathrm{SmB}}_6$ is a mixed valence Kondo system resulting from the hybridization between localized $f$ electrons and delocalized $d$ electrons. We have investigated its out-of-equilibrium electron dynamics by means of time- and angle-resolved photoelectron spectroscopy. The transient electronic population above the Fermi level can be described by a time-dependent Fermi-Dirac distribution. By solving a two-temperature model that well reproduces the relaxation dynamics of the effective electronic temperature, we estimate the electron-phonon coupling constant $\lambda$ to range from $0.13\pm 0.03$ to $0.04\pm 0.01$. These extremes are obtained assuming a coupling of the electrons with either a phonon mode at 10 or 19 meV. A realistic value of the average phonon energy will give an actual value of $\lambda$ within this range. Our results provide an experimental report on the material electron-phonon coupling, contributing to both the electronic transport and the macroscopic thermodynamic properties of ${\mathrm{SmB}}_6$.

Figure 1

(a) ${\mathrm{SmB}}_6$ unit cell of the CsCl type. Sm atoms (teal) and a ${\mathrm{B}}_6$ cage (black) occupy the corners and the body-centered position of the cubic cell. (b) Brillouin zone projected on the (001) cleavage plane. The measured momentum range along the $\overline{\mathrm{\Gamma }}\overline{X}$ high symmetry direction is indicated by a gray rectangle. (c) Electronic properties measured at 120 K, 500 fs before optical excitation. Three spectral features are present, two nondispersive Sm $f$ bands and a dispersive $d$-derived state, whose momentum distribution curve at $E_F$ is shown in the inset. (d) Difference between the out-of-equilibrium electronic properties 300 fs after and 500 fs before optical excitation. Depletion of intensity (blue) is reveled only in the $f$ state, and not in the $d$ band. (e) Temporal evolution of the change in intensity along the $d$-band dispersion [green dashed line in (d)].

Figure 2

(a) Nonequilibrium dynamics as obtained by integrating the recorded intensity within selected energy regions along the $d$-band dispersion. The color code is the same of the rectangles shown in Fig. 1. Regions 2 and 3, located symmetrically around $E_F$, display a similar characteristic time scale. Region 4, between the $f$ states, is characterized by a small positive and delayed dynamics, different with respect to the negative dynamics of the $f$ states in regions 3 and 5. (b) Energy distribution curves along the $d$-band dispersion at selected delay times before $(-400$ fs) and after (+300, +2000, +5300 fs) optical excitation. Black dashed lines indicate the best fit. The inset shows a zoom at $E_F$ of the EDCs at $-400$ and +300 fs.

Figure 3

(a), (b) Temporal evolution of the chemical potential shift $\mathrm{\Delta }\mu$ and of the electronic temperature $T_e$, as obtained from the fit of the EDCs extracted from Fig. 1. The results for two excitation energy densities are shown, equal to $\sim 30\pm 6$ J/${\mathrm{cm}}^3$ (black) and $\sim 19\pm 4$ J/${\mathrm{cm}}^3$ (green). Both $\mathrm{\Delta }\mu$ and $T_e$ relax after optical excitation with a single exponential behavior, with a characteristic time $\tau =800\pm 50$ fs. The evolution of $T_e$ is modeled within a 2TM, and the best fit is shown in (b). Red and blue lines indicate the dynamics of the electron and lattice temperature, respectively.