Electron Energy-Loss Function in Materials

• Electron Energy-Loss Function is a quantitative descriptor that measures the probability of fast electrons losing energy due to inelastic scattering in materials.
• The ELF is defined as the negative imaginary part of the inverse dielectric function, with peaks indicating collective excitations like plasmons and interband transitions.
• Advances in ultrafast EELS enable real-time tracking of transient dynamics, such as plasmon shifts and lattice heating, with high spatial and temporal resolution.

The electron energy-loss function (ELF) is a quantitative descriptor of the probability that a fast electron traversing a material will lose a specific energy ω (and, in general, a momentum q) due to inelastic interactions with the material's excitations. Mathematically, in many theoretical treatments, it is defined as the negative imaginary part of the inverse dielectric function:

$$L(q, \omega) = -\mathrm{Im} \left[ \frac{1}{\epsilon(q,\omega)} \right]$$

where $\epsilon(q,\omega)$ is the complex dielectric function. The ELF is a cornerstone of electron energy-loss spectroscopy (EELS), as it directly encodes the energy and momentum-resolved interaction between the electron beam and collective as well as single-particle excitations—such as plasmons, interband transitions, phonons, and core-level ionizations. Modern advances in ultrafast and time-resolved EELS have extended the reach of this approach from static characterization of materials to real-time tracking of dynamic processes in quantum materials, semiconductors, and energy-conversion systems (Lee et al., 6 Oct 2025).

1. Physical Basis and Mathematical Definition

The ELF emerges from the inelastic scattering cross-section in EELS experiments. When a fast electron interacts with a material, the differential scattering cross-section for energy loss is dominated by the ELF:

$$\frac{d^2\sigma}{d\Omega d\omega} \propto L(q, \omega) = -\mathrm{Im} \left[ \frac{1}{\epsilon(q,\omega)} \right]$$

Peaks in $L(q, \omega)$ correspond to high probability for the electron to excite collective modes (e.g., plasmons) or interband transitions in the target. The real part of $\epsilon$ determines the material's polarization response to external perturbations, while the imaginary part accounts for energy absorption. Consequently, maxima in $L(q,\omega)$ occur near energies where $\mathrm{Re}[\epsilon(q, \omega)] \approx 0$ (with a small $\mathrm{Im}[\epsilon]$), signaling collective excitations.

In static cases, ab initio calculations—using density functional theory (DFT) and time-dependent extensions (TDDFT)—yield $\epsilon(q, \omega)$ by summing over electronic transitions, and the ELF immediately follows. In non-equilibrium (time-resolved) experiments, the ELF becomes time-dependent and reflects transient modifications of the electronic and lattice structure.

2. Role in Time-Resolved and Ultrafast EELS Experiments

In ultrafast EELS, the ELF serves to probe both equilibrium and transient excitations on time scales ranging from femtoseconds to microseconds. In a pump–probe scheme, a laser pulse perturbs the material, creating non-equilibrium states; a subsequent electron pulse probes the dynamically evolving ELF at controlled time delays. The measured changes in $L(q,\omega,t)$ reveal:

• Shifts and amplitude changes in plasmon peaks, indicating variations in free-carrier concentrations, screening behavior, and transient electronic structure.
• Evolution of phonon features, broadenings, or shifts due to lattice heating and anharmonicity.
• Time-dependent changes in core-loss edges (e.g., K and L edges), which can reflect oxidation state changes, electronic redistribution, and bond-length relaxations during ultrafast processes.

Specific demonstrations include transient blueshifts/redshifts of the bulk and surface plasmon peaks in Au nanotriangles, and real-time K-edge shifts in graphite correlated with lattice heating and bond expansion (Lee et al., 6 Oct 2025). The ability to track $L(q,\omega,t)$ as the system evolves provides a direct route to quantifying ultrafast carrier, thermal, and structural dynamics with nanometer spatial and sub-picosecond temporal resolution.

3. Calculation and Interpretation: Ground-State and Transient Regimes

In the ground state, ELF computation is based on ab initio approaches such as TDDFT, where the full energy and momentum dependence is captured by evaluating the dielectric response of the electronic system. Features in $L(q,\omega)$ correspond to:

• Plasmon resonances (collective oscillations of free carriers)
• Interband transitions (single-particle electronic excitations)
• Phonon modes (vibrational excitations, resolved at meV energy resolution in monochromated systems)
• Core-level excitations (analogous to X-ray absorption)

Under photoexcitation or bias (transient regimes), differential forms such as $$\Delta L(q,\omega,t) = L(q,\omega,t) - L_0(q,\omega)$$ provide insight into the ultrafast processes of interest. For example, changes in the energy and width of the plasmon peak can quantify carrier multiplication or screening dynamics, while core-level edge shifts probe local chemical environment changes.

In time- and momentum-resolved EELS (tr-q-EELS), the dispersion and evolution of the ELF across reciprocal space are mapped, allowing disentanglement of valley-selective carrier relaxation and lattice rearrangements.

4. Technological and Theoretical Advances Enabling ELF Imaging

Recent developments have been central to enabling precise measurement and simulation of ELF in both steady-state and ultrafast EELS:

• Advanced monochromators reducing the energy spread of electron beams to the meV level, permitting the observation of fine phonon and excitonic features.
• High-speed direct electron detectors and time-to-digital converters (TDCs) improving both energy and temporal resolution, enabling experiments on femtosecond-to-picosecond timescales.
• Ab initio methods such as TDDFT with Liouville-Lanczos solvers for efficient calculation of the full $\epsilon(q, \omega)$ spectrum, even in large and complex systems. Such tools facilitate dynamic simulations of electronic response following photoexcitation and provide quantitative modeling for comparison with experiment (Lee et al., 6 Oct 2025).

These advances have allowed the ELF to become a genuinely dynamic probe for monitoring the evolution of nanoscale excitations, including real-time imaging of heat flow, valence electronic structure, and collective mode behavior.

5. Applications and Case Studies

The dynamical ELF measured via time-resolved and ultrafast EELS has provided new insights in several material systems:

System/Excitation Type	Observed ELF Dynamics	Reference
Au nanotriangles (plasmon modes)	Transient shifts in bulk and surface plasmon energies	(Lee et al., 6 Oct 2025)
Graphite (core-loss, K edge)	Redshift upon photoexcitation (lattice heating, bond expansion)	(Lee et al., 6 Oct 2025)
Manganites (Mn M₂,₃ edges)	Correlation of transient ELF shifts with structural changes	(Lee et al., 6 Oct 2025)
tr–q–EELS in graphite	q-dependent ELF evolution revealing valley physics	(Lee et al., 6 Oct 2025)

These studies underscore the versatility of the ELF as a fingerprint for transient carrier redistribution, lattice heating, phase transitions, and chemical reactions at the nanoscale.

6. Outlook and Future Directions

The evolution of ELF measurements and calculations is expanding the operable parameter space for nanoscale and ultrafast spectroscopies. Key frontiers include:

• Extension of ab initio and TDDFT-based ELF models to explicitly time-dependent and non-equilibrium regimes, including full-pump–probe simulation frameworks.
• Development of next-generation hardware for sub-10 meV energy and sub-100 fs temporal resolution.
• Real-time spectral imaging of ELF across low-loss (plasmons, phonons), valence (interband), and core-excitation regimes.
• In operando measurements tracking dynamic changes in solar energy conversion, phase-change, or quantum materials during operation or external stimulus.

These ongoing efforts position ELF-based EELS as a powerful tool for direct, time-resolved imaging of electron, phonon, and coupled field dynamics in quantum materials and devices (Lee et al., 6 Oct 2025).

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In summary, the electron energy-loss function encapsulates the probability and spectral character of electronic and vibrational excitations induced by fast electrons in materials. In the context of both static and ultrafast EELS, the ELF provides a direct, quantitative window into the fundamental processes governing carrier, thermal, and structural dynamics at unprecedented spatial and temporal resolution—a capability that is being continually advanced by theoretical and instrumental innovation in the field (Lee et al., 6 Oct 2025).