Ab initio phonon coupling and optical response of hot electrons in plasmonic metals

Ultrafast laser measurements probe the nonequilibrium dynamics of excited electrons in metals with increasing temporal resolution. Electronic structure calculations can provide a detailed microscopic understanding of hot electron dynamics, but a parameter-free description of pump-probe measurements has not yet been possible, despite intensive research, because of the phenomenological treatment of electron-phonon interactions. We present ab initio predictions of the electron-temperature dependent heat capacities and electron-phonon coupling coefficients of plasmonic metals. We find substantial differences from free-electron and semiempirical estimates, especially in noble metals above transient electron temperatures of 2000 K, because of the previously neglected strong dependence of electron-phonon matrix elements on electron energy. We also present first-principles calculations of the electron-temperature dependent dielectric response of hot electrons in plasmonic metals, including direct interband and phonon-assisted intraband transitions, facilitating complete theoretical predictions of the time-resolved optical probe signatures in ultrafast laser experiments.

Figure 1

Schematic electron and lattice temperature evolution with time following laser pulse illumination of a plasmonic metal like gold, along with the relevant material properties that determine this evolution. The vertical position of the gold atoms on the plot corresponds to electron temperature, and the vibration marks around the atoms schematically indicate lattice temperature. We show that both the electron heat-capacity $C_e\left(T_e\right)$ [from electronic density of states (DOS)] that sets the peak electron temperature $T_e$, and the electron-phonon coupling strength $G\left(T_e\right)$ (from electron-phonon matrix element $M_{\mathrm{e}-\mathrm{ph}}$) that affects the relaxation time of $T_e$, vary with $T_e$ in a manner sensitive to details of $d$ electrons in noble metals. Only the lattice heat capacity $C_l\left(T_l\right)$, that determines the lattice temperature rise, does not vary substantially between the detailed phonon DOS and simpler models.

Figure 2

Comparison of electronic density of states for (a) Al, (b) Ag, (c) Au, and (d) Cu from our relativistic PBEsol+$U$ calculations, previous semilocal PBE DFT calculations [20] (less accurate band structure), and a free electron model.

Figure 3

Comparison of the electronic heat capacity as a function of electron temperature, $C_e\left(T_e\right)$, for (a) Al, (b) Ag, (c) Au, and (d) Cu, corresponding to the three electronic density-of-states predictions shown in Fig. 2. The free electron Sommerfeld model underestimates $C_e$ for noble metals at high $T_e$ because it neglects $d$-band contributions, whereas previous DFT calculations [20] overestimate it because their $d$-bands are too close to the Fermi level.

Figure 4

Comparison of DFT-calculated phonon density of states and the Debye model for (a) Al, (b) Ag, (c) Au, and (d) Cu.

Figure 5

Comparison of DFT and Debye model predictions of the lattice heat capacity as a function of lattice temperature, $C_l\left(T_l\right)$, for (a) Al, (b) Ag, (c) Au, and (d) Cu. Despite large differences in the density of states (Fig. 4), the predicted lattice heat capacities of the two models agree within 10%.

Figure 6

Energy-resolved electron-phonon coupling strength $h\left(\varepsilon \right)$, defined by (11), for (a) Al, (b) Ag, (c) Au, and (d) Cu. For the noble metals, $h\left({\epsilon }_F\right)$ is substantially larger than its value in the $d$-bands, which causes previous semiempirical estimates [20] using a constant $h\left(\varepsilon \right)$ to overestimate the electron-phonon coupling $\left[G\left(T_e\right)\right]$ at $T_e\gtrsim 3000$ K, as shown in Fig. 7.

Figure 7

Comparison of predictions of the electron-phonon coupling strength as a function of electron temperature, $G\left(T_e\right)$, for (a) Al, (b) Ag, (c) Au, and (d) Cu, with experimental measurements where available [3, 14, 15, 32, 48]. The DFT-based semiempirical predictions of Lin et al. [20] overestimate the coupling for noble metals at high temperatures because they assume an energy-independent electron-phonon coupling strength (Fig. 6) and neglect the weaker phonon coupling of $d$-bands compared to the conduction band. The experimental results (and hence the semiempirical predictions) for aluminum underestimate electron-phonon coupling because they include the effect of competing electron-electron thermalization which happens on the same time scale.

Figure 8

DFT predictions of the complex dielectric functions for (a) Al, (b) Ag, (c) Au, and (d) Cu at room temperature (300 K) compared with ellipsometry measurements [52]. The $y$ axis is scaled by ${\omega }^2/{\omega }_p^2$ in order to represent features at different frequencies such as the Drude pole and the interband response on the same scale.

Figure 9

Change in the DFT-predicted complex dielectric function (solid lines) for (a) Al, (b) Ag, (c) Au, and (d) Cu from room temperature (300 K) to electron temperature $T_e=400$ K (with the lattice remaining at room temperature). In comparison, the analytical $d\to s$ model (16) (dashed lines) captures essential features of the DFT results for noble metals at lower temperatures, but misses the contributions of broadening due to electron-electron scattering at higher temperatures. Note that the $y$ axis is scaled as in Fig. 8 for clarity.

Figure 10

Change in the DFT-predicted complex dielectric function (solid lines) for (a) Al, (b) Ag, (c) Au, and (d) Cu from room temperature (300 K) to electron temperature $T_e=1000$ K (with the lattice remaining at room temperature), compared to the analytical $d\to s$ model (16) (dashed lines).

Figure 11

Change in the DFT-predicted complex dielectric function (solid lines) for (a) Al, (b) Ag, (c) Au, and (d) Cu from room temperature (300 K) to electron temperature $T_e=5000$ K (with the lattice remaining at room temperature), compared to the analytical $d\to s$ model (16) (dashed lines).

Figure 12

Critical interband transitions determining the “sharp” features in the dielectric function change for (a) noble metals (gold shown; similar shapes for silver and copper) and (b) aluminum. A parabolic band model around the L point (parameters in Table 2) approximates the critical transition in noble metals. This is difficult in aluminum because of four such transitions in a narrow energy range $\approx 1.3–1.6$ eV.