Electron-Phonon Coupling and Energy Flow in a Simple Metal beyond the Two-Temperature Approximation

The electron-phonon coupling and the corresponding energy exchange are investigated experimentally and by ab initio theory in nonequilibrium states of the free-electron metal aluminium. The temporal evolution of the atomic mean-squared displacement in laser-excited thin freestanding films is monitored by femtosecond electron diffraction. The electron-phonon coupling strength is obtained for a range of electronic and lattice temperatures from density functional theory molecular dynamics simulations. The electron-phonon coupling parameter extracted from the experimental data in the framework of a two-temperature model (TTM) deviates significantly from the ab initio values. We introduce a nonthermal lattice model (NLM) for describing nonthermal phonon distributions as a sum of thermal distributions of the three phonon branches. The contributions of individual phonon branches to the electron-phonon coupling are considered independently and found to be dominated by longitudinal acoustic phonons. Using all material parameters from first-principles calculations except the phonon-phonon coupling strength, the prediction of the energy transfer from electrons to phonons by the NLM is in excellent agreement with time-resolved diffraction data. Our results suggest that the TTM is insufficient for describing the microscopic energy flow even for simple metals like aluminium and that the determination of the electron-phonon coupling constant from time-resolved experiments by means of the TTM leads to incorrect values. In contrast, the NLM describing transient phonon populations by three parameters appears to be a sufficient model for quantitatively describing electron-lattice equilibration in aluminium. We discuss the general applicability of the NLM and provide a criterion for the suitability of the two-temperature approximation for other metals.

Popular Summary

All condensed matter is composed of atomic ions and valence electrons moving swiftly through the material, which act as glue for the otherwise-repellent ions. The ions and electrons are never at rest but continuously affect one another. These interactions between the electrons and vibrations of the atomic ions are a central subject in solid-state physics; they dictate fundamental effects such as the electron and thermal conductivity of materials, energy dissipation in electronic devices, and the emergence of quantum phenomena like superconductivity. Theoretical descriptions of these phenomena rely on knowledge of the coupling strength between electrons and the lattice, which cannot be directly measured. Here, we present an experimental and theoretical investigation of electron-lattice interactions in nanometer-thick films of aluminum, a prototypical metal.

We study how energy is transferred from electrons to phonons by visualizing the aluminum’s internal response to a sudden external disturbance. The disturbance is imposed by a very short (50-femtosecond) infrared laser pulse, which instantly heats up the electrons. The excited electrons then—because of their mutual interactions—start to equilibrate with the atomic vibrations. By taking snapshots of the atomic vibrations at various times after excitation, we obtain a “movie” of the relaxation pathway. Combining our experimental findings with state-of-the-art numerical calculations of atomic mean-squared displacements, we are able to revise the existing model of these interactions.

Our work highlights that the standard model for describing the energy exchange between the moving electrons and the vibrating ions needs to be refined. We expect that our findings will improve our understanding of the inner workings of solids and motivate future studies of phonon distributions.

Figure 1

Normalized phonon density of states (PDOS) and Eliashberg function of aluminium, calculated with DFT and the abinit package. The solid green lines show Bose-Einstein statistics for three lattice temperatures.

Figure 2

(a) Diffraction image of 30-nm-thick polycrystalline aluminium (inset) and radial average, retrieved by angular integration. Peaks with multiple labels are too close together to separate experimentally. (b) Evolution of the relative intensity of three selected Bragg peaks and the background (BG) of diffusely scattered electrons with respect to negative time delays for an absorbed energy density of $440\mathrm{J}/{\mathrm{cm}}^3$. The solid lines represent a global fit to the data (see text for details).

Figure 3

(a) Measured lattice temperature (circles) and simulated lattice temperature with a TTM with optimized parameters for three different absorbed energy densities $u_E$. In the bottom, the residuals for the highest excitation density are shown. (b) Electron-phonon coupling parameter $G_{ep}$ as a function of absorbed fluence. (c) Electron-phonon coupling dependence on electronic temperature obtained by our measurements. The green line shows $G_{ep}(T_e)$ obtained from DFT calculations by evaluating Eq. (9). The dashed light green line is the result from Lin et al. [6].

Figure 4

(a) DFT calculation of the Eliashberg function and its contributions from three phonon branches ${\alpha }^2{F}_i(\omega )$, with $i={TA}_1,{TA}_2,LA$. (b) Lattice heat capacities of the three branches and of the entire lattice in equilibrium as a function of temperature. (c) Electron-phonon coupling calculated for the entire lattice and the individual phonon branches as function of $T_e$.

Figure 5

(a) Measured mean-squared displacement (MSD) as a function of delay (circles) and calculated evolution predicted by the nonthermal lattice model with a phonon-phonon coupling of $3.5\times {10}^{17}\mathrm{W}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$ (lines). All other material parameters are taken from DFT calculations. In the bottom, the residuals for the highest excitation density are shown. (b) Evolution of the temperatures of electrons (blue) and the three phonon branches according to the NLM for an excitation density of $680\mathrm{J}/{\mathrm{cm}}^3$. The inset shows the full range of $T_e(t)$.

Figure 6

(a),(b) Schematic representation of the TTM and the NLM. In the latter, the phononic system is subdivided into its branches and the couplings are considered separately. The widths of the arrows are proportional to the coupling strengths retrieved from theory and experiment. (c) Temporal evolution of the energy content in the electronic (dash-dotted lines) and lattice (solid lines) subsystems for an excitation density of $680\mathrm{J}/{\mathrm{cm}}^3$ according to the NLM and the TTM. The energy content of the individual phonon branches is shown as dotted lines. The black dashed line indicates 95% of the energy which is eventually transferred from electrons to the lattice.

Figure 7

DFT calculation of the electronic density of states in aluminium. The dashed line illustrates the $\sqrt{E}$ behavior of a free $e$ gas.

Figure 8

Comparison of the spectral shapes of the partial Eliashberg functions ${\alpha }^2{F}_i$ and partial phonon density of states $F_i$. Each function is normalized to its integral.