Thermalization of Optically Excited Fermi Systems: Electron-Electron Collisions in Solid Metals

TL;DR

The paper addresses ultrafast thermalization of optically excited electrons in metals by deriving the full electron-electron Boltzmann collision integral within the random-$\boldsymbol{k}$ framework and comparing it to relaxation-time approaches. It introduces a screened-Coulomb matrix element and averages over momentum to obtain an energy-dependent collision integral, then examines both constant and energy-dependent relaxation times from Fermi-liquid theory. The results show that the full collision integral captures detailed energy- and time-resolved dynamics, including reoccupation effects, while a constant $\tau$ fails away from $E_F$; an energy-dependent $\tau_E$ with a fitted prefactor best reproduces high-energy behavior and the approach to a hot Fermi distribution, though some ensemble features require the full kinetics. These findings quantify the limitations of simplified relaxation-time models for ultrafast electron dynamics and have implications for interpreting pump-probe measurements and energy-resolved transport in metals.

Abstract

Ultrafast optical excitation of metals induces a non-equilibrium energy distribution in the electronic system, with a characteristic step-structure determined by Pauli blocking. On a femtosecond timescale, electron-electron scattering drives the electrons towards a hot Fermi distribution. In this work, we present a derivation of the full electron-electron Boltzmann collision integral within the random-k approximation. Building on this approach, we trace the temporal evolution of the electron energy distribution towards equilibrium, for an excited but strongly degenerate Fermi system. Furthermore, we examine to which extent the resulting dynamics can be captured by the numerically simpler relaxation time approach, applying a constant and an energy-dependent relaxation time derived from Fermi-liquid theory. We find a better agreement with the latter, while specific features caused by the balance of scattering and reoccupation can only be captured with a full collision integral.

Figures (5)

• Figure 1: (a) Direct and exchange scattering of two electrons with momentum $\boldsymbol{p}$ and $\boldsymbol{k}$ and spin $\sigma$, respectively. (b) Spin conserving scattering between opposite spin electrons with spin $\sigma$ and $\overline{\sigma}$. (c) Scattering between opposite spin electrons with spin flip. Adapted from Ref. Penn1985.
• Figure 2: Step-shaped, non-thermal distribution of an excited, degenerate free-electron gas (blue solid line) together with its corresponding hot Fermi distribution (dashed line) as well as a cold Fermi distribution at 300 K (dotted line). The distribution is plotted in a symmetric logarithmic representation know as the $\Phi$-function introduced in Ref. Rethfeld2002 (right y-axis) or the mathematically equivalent logit-function (left y-axis). In this plot, Fermi functions are represented as a linear function with a slope inversely proportional to temperature, and deviations from a straight line visualize the non-equilibrium.
• Figure 3: Normalized MAD of the non-thermal distribution from equilibrium for the FCI with random-$\boldsymbol{k}$ (black, dotted line) compared to three types of the RTA: constant $\tau_{\text{const}}$ (red, dashed line), energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{calc}}$ (green, long dash-dotted line), and energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{fit}}$ (blue, short dash-dotted line). An exponential fit provides the same fitted thermalization time of $15.2\,$fs for all curves except for the green one which relaxes with a characteristic time of $39.8\,$fs.
• Figure 4: Thermalization of the distribution after laser excitation. a) Calculated with FCI with random-$\boldsymbol{k}$, b) to d) calculated with RTA with different relaxation times: constant $\tau_{\text{const}}$ (b), energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{calc}}$ (c), and energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{fit}}$ (d).
• Figure 5: Energy-dependent occupation times for the FCI with random-$\boldsymbol{k}$ (black) and the RTA, compared for different relaxation times: constant $\tau_{\text{const}}$ (red), energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{calc}}$ (green), and energy-dependent $\tau_\text{E}$ with $\tau_{0,\text{fit}}$ (blue).