qEELS: Momentum-Resolved Energy-Loss Spectroscopy

• qEELS is an electron-scattering technique that simultaneously measures energy loss and momentum transfer, producing detailed energy–momentum maps of collective excitations.
• It extends conventional EELS by accessing finite wave vectors to capture excitations such as phonons, plasmons, excitons, and magnons beyond the light cone.
• Instrumental implementations in TEM and STEM balance energy resolution, spatial selectivity, and momentum precision to reveal material-specific dispersion and interference effects.

Searching arXiv for recent and foundational qEELS papers on phonons, plasmons, excitons, and unified theory. arXiv search query: "momentum-resolved electron energy-loss spectroscopy qEELS phonon plasmon exciton" Momentum-resolved electron energy-loss spectroscopy, commonly abbreviated qEELS, is an electron-scattering technique in which the energy loss $\hbar\omega$ and the momentum transfer $\mathbf{q}$ are resolved simultaneously, so that the measured signal is an energy–momentum map rather than an angle-integrated loss spectrum. In transmission electron microscopes and scanning transmission electron microscopes, qEELS extends conventional EELS from the optical or near-forward limit into finite wave vectors, allowing direct access to phonons, plasmons, excitons, magnons, phonon–polaritons, and related collective modes, including excitations outside the light cone and beyond the first Brillouin zone. Depending on geometry, it can emphasize either high angular precision in parallel-beam TEM or nanometer-scale spatial selectivity in STEM (Nicholls et al., 2018, Elgvin et al., 10 Oct 2025, Shekhar et al., 2017).

1. Kinematics and measured quantity

The basic kinematics are those of inelastic electron scattering. An incident electron with wavevector $\mathbf{k}_i$ and energy $E_i$ scatters to $\mathbf{k}_f$ and $E_f$, transferring momentum and energy

$$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$

For small scattering angles $\boldsymbol{\theta}$, the transverse momentum transfer obeys

$\mathbf{q}\approx k_0\,\boldsymbol{\theta},$

with $k_0=2\pi/\lambda$ the incident electron wavevector magnitude. In paraxial relativistic descriptions, the longitudinal component is commonly approximated by $\mathbf{q}$0, where $\mathbf{q}$1 is the electron velocity (Nicholls et al., 2018, Lourenço-Martins et al., 2020).

What qEELS records experimentally is the double differential scattering cross section, or an intensity proportional to it, as a function of both $\mathbf{q}$2 and detector angle. In diffraction-plane implementations, detector position maps directly onto momentum transfer, so that a single acquisition produces $\mathbf{q}$3 or a one-dimensional slice $\mathbf{q}$4 along a chosen reciprocal-space direction. This distinguishes qEELS from conventional STEM-EELS, which integrates over scattering angles at the spectrometer entrance and therefore washes out momentum dependence, and from optical spectroscopies, which are largely restricted to $\mathbf{q}$5 and cannot reach modes outside the light cone (Senga et al., 2018, Elgvin et al., 10 Oct 2025).

The accessible $\mathbf{q}$6 range and the momentum resolution are set by beam energy and angular optics. In STEM notation, the probe convergence semi-angle $\mathbf{q}$7 and collection semi-angle $\mathbf{q}$8 control the accepted momentum space, with a common estimate

$\mathbf{q}$9

Concrete implementations span markedly different operating points: phonon qEELS in STEM at $\mathbf{k}_i$0 kV with $\mathbf{k}_i$1 mrad achieved $\mathbf{k}_i$2 and $\mathbf{k}_i$3–$\mathbf{k}_i$4 meV; parallel-beam graphene experiments reported $\mathbf{k}_i$5–$\mathbf{k}_i$6 at $\mathbf{k}_i$7 meV resolution; free-standing monolayer WSe$\mathbf{k}_i$8 experiments used $\mathbf{k}_i$9 and $E_i$0–$E_i$1 meV (Nicholls et al., 2018, Senga et al., 2018, Hong et al., 2019).

2. Response functions and theoretical formalisms

In first-Born treatments, qEELS is naturally expressed in terms of the target dynamic structure factor $E_i$2 or, in dielectric language, the loss function $E_i$3. For phonons in the electron microscope, one widely used form is

$E_i$4

whereas for low-loss electronic excitations it is often written as

$E_i$5

These are not competing definitions so much as different representations of the same inelastic response in different approximations and material regimes (Nicholls et al., 2018, Senga et al., 2018, Leon et al., 2024).

For crystalline phonons, the one-phonon structure factor can be written as

$E_i$6

with a mode-resolved matrix element containing both a polarization projection $E_i$7 and basis-phase factors $E_i$8. The former enforces polarization selectivity, while the latter produces coherent interference across the basis and generates strong branch-dependent visibility variations (Nicholls et al., 2018).

At a more general level, qEELS can be cast in terms of a mixed dynamic form factor (MDFF) or, in a fully retarded treatment, in terms of the dyadic photon propagator. A scalar relativistic QED formulation showed that the imaginary part of the retarded photon propagator and the relativistic MDFF are related exactly, thereby connecting the nano-optical Green-dyadic language used for photonic excitations with the condensed-matter MDFF language used for core and valence excitations (Lourenço-Martins et al., 2020). In homogeneous bulk media this reduction recovers the familiar dielectric-loss form, but finite structures do not in general preserve a simple one-to-one intensity equivalence with the momentum-resolved photonic density of states. For thin films, qEELS and q-PDOS can track the same dispersion while exhibiting different high-$E_i$9 intensity laws and different sensitivity to bulk-plasmon channels (Shekhar et al., 2017).

A further extension appears in core-loss qEELS, where the relevant operator is the reciprocal-space electron density $\mathbf{k}_f$0 and the dynamic structure factor becomes a many-electron density–density correlator. In that regime the dipole approximation yields a quadratic form in the Cartesian dipole–dipole spectral functions $\mathbf{k}_f$1, while beyond-dipole terms become relevant only at larger $\mathbf{k}_f$2 (Kunitsa et al., 21 Aug 2025).

3. Instrumental realizations and resolution trade-offs

qEELS has developed along several instrumental lines. In parallel-beam TEM, the specimen is illuminated broadly, the diffraction plane is imaged with high angular precision, and a slit or aperture selects a narrow momentum interval while the spectrometer disperses energy. This geometry underlies early plasmonic qEELS work and the high-$\mathbf{k}_f$3 graphene and WSe$\mathbf{k}_f$4 measurements. In STEM, by contrast, the probe is scanned in real space while the spectrometer entrance aperture is positioned in the diffraction plane; this preserves angle selectivity and enables spatial mapping at fixed $\mathbf{k}_f$5 or along selected reciprocal-space paths (Shekhar et al., 2017, Senga et al., 2018, Hong et al., 2019).

A distinct surface-sensitive lineage is provided by high-resolution EELS with hemispherical analyzers. There the analyzer disperses energy along one detector axis and emission angle along the other, so that a full $\mathbf{k}_f$6 stripe is acquired in parallel. In the demonstrated Cu(111) implementation, a full surface-phonon dispersion was obtained in about seven minutes with $\mathbf{k}_f$7 meV energy resolution (Ibach et al., 2016).

The instrumental compromise is always between energy resolution, momentum resolution, signal level, and spatial resolution. In graphene qEELS, $\mathbf{k}_f$8 corresponded to a probe size of $\mathbf{k}_f$9 nm, while integrating a wider $E_f$0 allowed $E_f$1 nm probe size, explicitly reflecting the Heisenberg trade-off between spatial and momentum resolution (Senga et al., 2018). Atomic-resolution vibrational STEM-EELS sharpens that trade-off further: large convergence semi-angles are needed for Å-scale probes, but fine $E_f$2 resolution prefers small angular spreads. A proposed multi-rotation acquisition strategy therefore reconstructs $E_f$3 from several one-momentum-axis acquisitions, rather than attempting fine two-dimensional momentum resolution in a single atomic-resolution frame (Haas et al., 2024).

Recent practice has also diversified into serial qEELS, slit qEELS, and 4D qEELS. Serial qEELS steps an aperture through the diffraction plane; slit qEELS aligns a rectangular entrance slit along a high-symmetry direction to capture an $E_f$4–$E_f$5 map in one shot; 4D qEELS records a full $E_f$6–$E_f$7 dataset at each probe position. Direct detectors with high dynamic range have become important because they permit simultaneous recording of a strong zero-loss peak and much weaker low-loss features (Elgvin et al., 10 Oct 2025).

4. Vibrational qEELS and phonon selection

Vibrational qEELS is the branch of the technique in which the low-energy losses are phonons. Its central result is that finite-$E_f$8 EELS does not merely measure phonon energies; it measures phonon dispersion, polarization, and coherence across the crystal basis. In cubic and hexagonal boron nitride, a first-principles DFPT-based one-phonon theory reproduced the measured momentum-dependent spectra and showed that only selected branches appear along a given direction because the matrix element carries both the projection factor $E_f$9 and the unit-cell interference factor $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$0 (Nicholls et al., 2018).

That same framework clarifies why qEELS differs from conventional near-$$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$1 vibrational EELS. Conventional EELS predominantly probes dipole-active excitations at vanishing momentum transfer, whereas qEELS resolves phonon branches across finite wavevectors. In non-polar materials, the cross section is suppressed near $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$2 and becomes useful only at larger $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$3, where impact scattering dominates and valence screening weakens. This was essential for graphene and graphite: near $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$4 the phonon signal vanishes, but in the large-$$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$5 impact regime, modes such as LA, LO, TO, ZA, and ZO become visible and can be mapped directly in monolayer graphene and graphene nanoribbons (Senga et al., 2018).

The interpretation of vibrational qEELS has acquired a more refined language since 2025. Simulations based on frequency-resolved multislice, spectral energy density, and lattice dynamics identified an “interferometric Brillouin zone” controlled by the smallest interatomic spacing rather than simply by the primitive cell. In this picture, “missing branches” are not necessarily instrumental failures; they can result from destructive basis-phase interference. Similarly, apparent “phantom branches” near $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$6 can arise from depth-dependent, nonuniform sampling that relaxes perfect destructive interference and couples through-plane modes into the measured dispersion. The same work showed that finite apertures reduce polarization selectivity because they incoherently mix local $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$7 directions (Pfeifer et al., 12 Mar 2025).

A related development concerns localization. Long-range dipole scattering is intrinsically delocalized, while impact scattering is atomically localized. Atomic-resolution vibrational STEM-EELS therefore relies on suppressing the forward dipole cone and emphasizing dark-field impact scattering. This is straightforward in non-polar materials such as Si, and more subtle in polar materials where dipole and phonon–polariton channels coexist. A plausible implication is that momentum-resolved dark-field acquisition is not merely a convenience but a central design principle for atomic-scale vibrational spectroscopy (Haas et al., 2024).

5. Plasmons, excitons, and photonic modes

In the electronic and photonic regimes, qEELS has become a principal probe of finite-momentum collective response. On plasmonic thin films, it mapped surface plasmon polaritons to wavevectors far beyond the optical light line and showed close agreement between measured dispersion, fast-electron simulations, and the momentum-resolved photonic density of states. At the same time, it established an important caveat: in finite films the qEELS intensity is not identical to the local external-emitter q-PDOS, even when the dispersions coincide, because the moving electron and an external dipole couple differently to longitudinal and surface channels (Shekhar et al., 2017).

For graphene, high-$$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$8 qEELS resolved a long-standing controversy over the $$\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i,\qquad \omega=\frac{E_i-E_f}{\hbar}.$$9 plasmon. With $\boldsymbol{\theta}$0 across the full in-plane Brillouin zone, the $\boldsymbol{\theta}$1 plasmon was shown to follow the two-dimensional form

$\boldsymbol{\theta}$2

with $\boldsymbol{\theta}$3 eV and $\boldsymbol{\theta}$4 for $\boldsymbol{\theta}$5, rather than the linear dispersion reported in earlier lower-resolution work. The same data revealed an in-plane anisotropy at larger $\boldsymbol{\theta}$6, attributed to nonvertical $\boldsymbol{\theta}$7 interband transitions along $\boldsymbol{\theta}$8–M (Liou et al., 2014).

In semiconductors and insulators, qEELS has become a many-body spectroscopy of finite-$\boldsymbol{\theta}$9 dielectric response. A detailed ZnO study combined low-loss qEELS with $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$0-sum-rule normalization and Kramers–Kronig analysis to reconstruct $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$1 and $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$2, identifying an excitonic onset near $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$3 eV, a double-peaked bulk plasmon at $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$4 and $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$5 eV, and strong anisotropy along and perpendicular to the $\mathbf{q}\approx k_0\,\boldsymbol{\theta},$6 axis. The same study used IPA, RPA, and BSE calculations to distinguish interband features from genuine many-body effects (Leon et al., 2024).

Finite-$\mathbf{q}\approx k_0\,\boldsymbol{\theta},$7 excitons in two-dimensional semiconductors are another major application. In free-standing monolayer WSe$\mathbf{q}\approx k_0\,\boldsymbol{\theta},$8, qEELS measured a parabolic A-exciton dispersion,

$\mathbf{q}\approx k_0\,\boldsymbol{\theta},$9

with $k_0=2\pi/\lambda$0–$k_0=2\pi/\lambda$1 eV and $k_0=2\pi/\lambda$2, and inferred a nearly $k_0=2\pi/\lambda$3-independent binding energy $k_0=2\pi/\lambda$4 eV. The same experiment also observed a sub-gap exciton at $k_0=2\pi/\lambda$5–$k_0=2\pi/\lambda$6 eV with approximately linear dispersion over the measured range and attributed it to prolific Se vacancies identified by STEM-ADF imaging (Hong et al., 2019).

More broadly, qEELS on 2D materials and heterostructures now covers phonon–polaritons in hBN, excitonic spectral-weight redistribution in hBN and TMDCs, thickness-dependent plasmon response in MoS$k_0=2\pi/\lambda$7, and twist-dependent excitonic suppression in WSe$k_0=2\pi/\lambda$8/MoS$k_0=2\pi/\lambda$9 heterobilayers. The common thread is that qEELS resolves excitations at finite momentum, including dark or high-$\mathbf{q}$00 states inaccessible to optics, while retaining nanoscale spatial selectivity (Elgvin et al., 10 Oct 2025).

6. Magnetic channels, interpretation challenges, and emerging directions

Momentum-resolved EELS has also been extended to spin excitations. A theory for YIG separated two magnon channels: a purely spin interaction formally analogous to magnetic inelastic neutron scattering, and a charge-dependent interaction arising from the minimal-coupling substitution $\mathbf{q}$01. The spin channel carries the familiar transverse projector $\mathbf{q}$02, whereas the charge channel introduces an additional orientation matrix and an explicit $\mathbf{q}$03 factor in the derived cross section. Both channels probe the same dynamical spin structure factor, but with different momentum and orientation weightings (Nascimento et al., 2024).

A fully dynamical treatment of coupled low-energy excitations was demonstrated for bcc Fe by extending the TACAW method to atomistic spin-lattice dynamics. In that study, a $\mathbf{q}$04 kV parallel beam propagated through a $\mathbf{q}$05 nm specimen, an annular dark-field detector with inner and outer angles $\mathbf{q}$06 and $\mathbf{q}$07 mrad was identified as optimal for magnon collection, acoustic phonons appeared below $\mathbf{q}$08 meV, and a broad magnon band extended to $\mathbf{q}$09 meV. The full coupled signal was explicitly non-additive, containing interference and spectral redistribution that were absent from separate phonon-only or magnon-only simulations (Castellanos-Reyes et al., 9 Aug 2025).

Interpretation at finite momentum is further complicated by geometry. In tilted planar samples under optical-mode excitation, the momentum delivered to the sample modifies the apparent dispersion measured by qEELS. For a tilt angle $\mathbf{q}$10, one reported correction was

$\mathbf{q}$11

with $\mathbf{q}$12. Under specific conditions, the sample can even receive momentum opposite to the electron-beam direction. This establishes that finite-momentum interpretation is not purely a matter of detector geometry; recoil and sample tilt can enter directly into the recovered dispersion (Yasuhara et al., 17 Nov 2025).

A final development concerns computation. In the core-loss regime, a 2025 study reformulated qEELS simulation in terms of dynamic-structure-factor evaluation from off-diagonal dipole–dipole time-domain correlators and proposed a quantum algorithm for that task. Applied to an oxygen-centered Li$\mathbf{q}$13MnO$\mathbf{q}$14 cluster with an active space of 18 orbitals, the workflow required a circuit depth of $\mathbf{q}$15 T gates, about 100 logical qubits, and roughly $\mathbf{q}$16 shots for the representative calculation. This suggests that, as qEELS expands from vibrational and low-loss spectroscopy into strongly correlated core-loss problems, its theoretical infrastructure is beginning to move beyond conventional electronic-structure pipelines (Kunitsa et al., 21 Aug 2025).