Plasmonic Polarons: Electron–Plasmon Satellites

• Plasmonic polarons are many-body satellite features in the spectral function, arising from electron–plasmon coupling and shifted by roughly the plasmon energy.
• The GW+cumulant methodology accurately captures their momentum-dependent band structure and broadening, overcoming limitations of plain GW approximations.
• They serve as experimental fingerprints in semiconductors and 2D materials, aiding the analysis of carrier density, dielectric screening, and electron dynamics.

Plasmonic polarons are many-body spectral structures generated by electron–plasmon coupling, most commonly observed as broadened replicas of quasiparticle bands at larger binding energy by roughly a plasmon energy in angle-resolved photoemission spectra. In the modern formulation, they are not sharp “plasmaron” poles in the older $GW$ sense, but plasmon-induced satellite structures captured more faithfully by $GW$ plus cumulant theory; in momentum-resolved form, these satellites can themselves exhibit a recognizable band structure (Caruso et al., 2015, Caruso et al., 2016). The term has also acquired broader usages, including a topological collective mode in which a surface plasmon is strongly renormalized by bulk phonons (Shvonski et al., 2018). This breadth makes terminological precision essential, especially because several nearby fields study plasmonic polaritons, plexcitons, or plasmon-assisted readout of other polarons rather than plasmonic polarons in the strict many-body spectral-function sense.

1. Definition and conceptual scope

In the spectral-function literature, a plasmonic polaron is the signature of a photoemission process in which an electron is removed and a plasmon is excited simultaneously. The resulting state is not a new sharp elementary particle in the same sense as an ordinary electron or hole quasiparticle. Rather, it appears as a broader, weaker satellite that tracks the parent dispersion and is shifted to higher binding energy by approximately the plasmon energy. This is the sense in which “plasmonic polaron bands” were formulated for semiconductors and then clarified in the homogeneous electron gas (HEG) (Caruso et al., 2015, Caruso et al., 2016).

This definition is deliberately narrower than older plasmaron language. In the HEG, plain $GW$ produces an additional sharp peak at about $1.5\,\omega_{\rm pl}$ below the quasiparticle, but the cumulant treatment replaces this with a broader and weaker satellite centered near $\omega_{\rm pl}$, identifying the physically relevant structure as a plasmonic polaron band rather than a distinct plasmaron pole (Caruso et al., 2016). In this usage, the relevant observable is the one-electron spectral function $A(\mathbf{k},\omega)$, not a cavity spectrum, not a thresholded optical emission line, and not a generic hybridization between plasmons and some other resonance.

A broader but distinct usage appears in topological surface-state physics, where a “plasmon-polaron” denotes a low-frequency acoustic collective branch that splits off from the conventional spin-plasmon because surface electrons are screened by a polarizable bulk lattice. That object is a collective mode, not a one-particle spectral satellite, although the same term emphasizes that the mode has strong polaronic character through phonon-modified dielectric screening (Shvonski et al., 2018). The coexistence of these meanings is a recurrent source of confusion.

2. Many-body formulation

The theoretical backbone of the subject is the spectral function and its connection to the frequency-dependent self-energy. In the $GW$ approximation, the spectral function is written as

$$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$

with the screened interaction

$$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$

Because plasmon poles appear in $\epsilon^{-1}({\bf q},\omega)$, they enter $GW$0, then $GW$1, and finally generate satellites in $GW$2 (Caruso et al., 2016).

The central methodological result is that plain $GW$3 is inadequate for satellite physics. In the HEG, it produces the spurious plasmaron peak around $GW$4 below the quasiparticle. The cumulant expansion corrects this by treating plasmon shake-up processes in a form appropriate to electron–boson coupling. In the $GW$5cumulant framework,

$GW$6

so the quasiparticle contribution is retained while the cumulant term generates the plasmon satellite (Caruso et al., 2016).

For semiconductors, the same logic was cast in band-resolved form: $GW$7 There, $GW$8 was explicitly identified with plasmonic polarons, and the first cumulant was retained to describe one-plasmon excitations, while higher-order $GW$9-plasmon processes were noted to be suppressed by the Lang-Firsov factor $GW$0 (Caruso et al., 2015).

The theoretical picture that emerges is specific. Plasmonic polarons are intrinsic many-body satellites generated by dynamical screening. They inherit the momentum dependence of the parent quasiparticle band because the underlying hole is still created in a definite Bloch state, but the additional plasmon cost shifts the spectral weight downward in an ARPES plot. Their broadening reflects the finite lifetime and phase space of plasmon-assisted removal processes.

3. Band-structure manifestations and observability criteria

A major advance of the field was the recognition that plasmon satellites possess an internal band structure. In silicon, the calculated quasiparticle valence manifold occupies roughly $GW$1–$GW$2 eV binding energy, while plasmonic structures appear at roughly $GW$3–$GW$4 eV and disperse similarly to the normal valence bands. The integrated plasmon satellite in photoemission can therefore be understood as the Brillouin-zone average of underlying plasmonic-polaron bands, with Van Hove-related features at $GW$5 eV, $GW$6 eV, and a shoulder at $GW$7 eV (Caruso et al., 2015).

The same calculations established two practical criteria for observing plasmonic polarons in ARPES: low-energy plasmon excitations in the loss spectrum, and a gap in the valence-band manifold. Monolayer group-IV transition-metal dichalcogenides satisfy these conditions unusually well because the $GW$8 plasmon is strongly red-shifted relative to the bulk. For monolayer MoS$GW$9 and WS$1.5\,\omega_{\rm pl}$0, the $1.5\,\omega_{\rm pl}$1 plasmon is around $1.5\,\omega_{\rm pl}$2 eV, which places the $1.5\,\omega_{\rm pl}$3-plasmonic-polaron window in the $1.5\,\omega_{\rm pl}$4–$1.5\,\omega_{\rm pl}$5 eV range, inside the gap between the upper TM-$1.5\,\omega_{\rm pl}$6/Ch-$1.5\,\omega_{\rm pl}$7 bands and the lower chalcogen-$1.5\,\omega_{\rm pl}$8 bands. Monolayer MoSe$1.5\,\omega_{\rm pl}$9 was identified as the best testbench because its plasmonic structures have essentially no overlap with ordinary valence bands (Caruso et al., 2015).

The HEG analysis generalized this interpretation. There the plasmonic polaron band is a broadened replica of the ordinary quasiparticle dispersion, shifted by approximately $\omega_{\rm pl}$0, and its spectral weight increases with increasing $\omega_{\rm pl}$1, that is, with decreasing electron density. This connected plasmonic polarons in semiconductors to a generic consequence of electron–plasmon coupling in solids rather than a peculiarity of specific band structures (Caruso et al., 2016).

A more recent consequence of dimensional reduction is the emergence of low-energy carrier plasmons in doped two-dimensional semiconductors. In $\omega_{\rm pl}$2-doped monolayer MoS$\omega_{\rm pl}$3, the relevant plasmons span $\omega_{\rm pl}$4 to $\omega_{\rm pl}$5 eV for the representative doping analyzed in theory, with a Landau-damping cutoff at $\omega_{\rm pl}$6. The observed ARPES satellite energy is then controlled not by a single sharply defined boson, but by the average energy of the two-dimensional plasmon manifold (Caruso et al., 2021).

4. Experimental realizations in photoemission and loss spectroscopy

Several material platforms now exhibit spectroscopic signatures that have been interpreted as plasmonic polarons. The common pattern is a weak satellite or replica near a conduction- or valence-band quasiparticle, with an energy scale that tracks a carrier plasmon rather than a phonon.

System	Reported signature	Characteristic scale
Epitaxial SrIrO$\omega_{\rm pl}$7 films	Replica bands $\omega_{\rm pl}$8, $\omega_{\rm pl}$9, and weaker $A(\mathbf{k},\omega)$0	$A(\mathbf{k},\omega)$1 meV, second order $A(\mathbf{k},\omega)$2 meV
$A(\mathbf{k},\omega)$3-doped monolayer MoS$A(\mathbf{k},\omega)$4	Conduction-band satellites near K	$A(\mathbf{k},\omega)$5, $A(\mathbf{k},\omega)$6, $A(\mathbf{k},\omega)$7 meV; second satellite $A(\mathbf{k},\omega)$8 meV
Alkali-doped $A(\mathbf{k},\omega)$9-HfS$GW$0	Shoulder below the conduction-band quasiparticle	$GW$1 meV
Self-intercalated $GW$2-TiS$GW$3	ARPES plasmon-loss satellite plus HR-EELS plasmon	$GW$4 meV; $GW$5 to $GW$6 meV with doping

In semimetallic epitaxial SrIrO$GW$7, the favorable ingredients are a low carrier density, $GW$8, and narrow bands of about $GW$9 meV for $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$0, $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$1 meV for $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$2, and $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$3 meV for $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$4. ARPES resolves weak replica bands $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$5 and $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$6 separated from the main bands by about $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$7 meV, with a still weaker $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$8 at about $$A({\bf k},\omega) = \frac{1}{\pi} \frac{|\Sigma^{\prime\prime}_{\bf k}(\omega)|} {\left[\omega-\epsilon_{\bf k}-\Sigma'_{\bf k}(\omega)\right]^2 +\left[\Sigma^{\prime\prime}_{\bf k}(\omega)\right]^2},$$9 meV. The separation is thickness-independent from 10 to 35 unit cells, but increases with carrier density up to about $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$0 meV under La substitution, consistent with a plasmon energy rather than a quantum-well or phonon scale. A Poisson-type analysis yielded $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$1, a Fröhlich-like coupling constant $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$2, and a plasmonic-polaron radius of approximately $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$3 Å (Liu et al., 2021).

In $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$4-doped monolayer MoS$$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$5, ARPES near K shows conduction-band satellites with separations of $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$6, $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$7, and $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$8 meV for carrier concentrations $$W({\bf q},\omega)=\frac{v({\bf q})}{\epsilon({\bf q},\omega)}.$$9, $\epsilon^{-1}({\bf q},\omega)$0, and $\epsilon^{-1}({\bf q},\omega)$1, respectively, as well as a second satellite at $\epsilon^{-1}({\bf q},\omega)$2 meV for the intermediate doping. Optical phonons in monolayer MoS$\epsilon^{-1}({\bf q},\omega)$3 lie only in the $\epsilon^{-1}({\bf q},\omega)$4–$\epsilon^{-1}({\bf q},\omega)$5 meV range, so the observed scale and its doping dependence point to carrier plasmons. The cumulant calculations reproduced the first and second satellites and showed that the measured satellite energy corresponds closely to the average energy of the two-dimensional plasmon density of states (Caruso et al., 2021).

In heavily electron-doped $\epsilon^{-1}({\bf q},\omega)$6-HfS$\epsilon^{-1}({\bf q},\omega)$7, produced by in situ potassium deposition, ARPES resolves a sharp conduction-band quasiparticle peak and an additional shoulder about $\epsilon^{-1}({\bf q},\omega)$8 meV below the Fermi level, with a broader tail extending to about $\epsilon^{-1}({\bf q},\omega)$9 eV below $GW$00. The carrier density extracted from the Fermi pockets is $GW$01. Theoretical analysis attributed the feature to coupling to a doping-induced carrier plasmon, and argued that reduced screening at the surface, modeled by $GW$02, is primarily responsible for making the plasmonic-polaron signature observable (Emeis et al., 2023).

The most direct combined spectroscopic evidence comes from self-intercalated $GW$03-TiS$GW$04, where ARPES reveals a satellite band around $GW$05 eV below $GW$06 at the $GW$07 point and HR-EELS independently measures a high-energy mode near $GW$08 meV, while all phonons remain below $GW$09 meV. The quasiparticle–satellite separation increases from $GW$10 meV at $GW$11 to $GW$12 meV at $GW$13, and the satellite is about 8 times weaker than the quasiparticle peak. Temperature increases broaden the satellite far more strongly than the main quasiparticle, with the satellite/plasmon FWHM increasing by about $GW$14 meV versus about $GW$15 meV for the quasiparticle, linking plasmon damping directly to plasmonic-polaron stability (Choi et al., 4 Mar 2026).

5. Collective-mode variants and topological plasmon-polarons

Not all plasmon-polaron literature concerns one-electron satellites. In topological metallic surface states, a distinct usage denotes a low-frequency acoustic branch of the surface charge dynamics. There the starting point is a helical two-dimensional electron gas with Hamiltonian

$GW$16

whose conventional collective mode is the spin-plasmon. When bulk acoustic phonons are incorporated through a phonon-modified dielectric function,

$GW$17

the plasmon equation acquires a second solution. This new branch follows the acoustic scale $GW$18 and was termed a plasmon-polaron because it is an electronic collective mode with strong polaronic character inherited from lattice screening (Shvonski et al., 2018).

The same work argued that this collective mode inherits the helical spinor-overlap suppression of the underlying surface electrons, so scattering at the Brillouin-zone boundary is strongly suppressed and the dispersion can continue without an Umklapp gap. In this context, a plasmon-polaron is therefore neither a plasmaron nor a cumulant satellite. It is a topologically protected, phonon-dressed surface plasmon branch (Shvonski et al., 2018).

This broader usage is conceptually related but not identical to the spectral-function definition. Both invoke bosonic dressing of an electronic excitation by a plasmonic degree of freedom, but the first concerns the one-particle Green’s function and ARPES satellites, while the second concerns a collective mode in the charge response.

6. Distinction from polaritons, plexcitons, and plasmon-assisted readout

The term should not be applied indiscriminately to all plasmon-containing hybrids. A plasmon–exciton polariton condensate in a silver nanorod lattice is a condensate of plasmon–exciton polaritons formed by strong coupling between a surface lattice resonance and Frenkel excitons. The associated work explicitly notes that this system should not be classified as involving “plasmonic polarons” in the many-body condensed-matter sense; it is a polariton condensate supported by lattice-assisted plasmonic modes, not a polaronic many-body state (Giorgi et al., 2017).

The same distinction applies to cavity-free plasmon–interband hybridization in nickel nanostructures. There the relevant objects are plasmon–interband polaritons inside metals, observed by EELS as split upper and lower hybrid modes near the $GW$19 eV interband transition. The conceptual overlap is that an electronic excitation is dressed by a plasmonic mode, but the paper is explicit that this is not a plasmonic polaron in the usual many-body spectral-function sense; it is a polaritonic coupled-oscillator problem in dielectric response (Assadillayev et al., 2022).

Strong coupling between localized surface plasmons and molecules, as treated by coupled-cluster theory in plasmonic nanocavities, is likewise framed in terms of polaritons or plexcitons. That work is highly relevant to plasmon dressing because it includes mutual polarization, mixed plasmon–molecule correlation, and ground-state density renormalization, but it does not explicitly study plasmonic polarons in the charged-quasiparticle or ARPES-satellite sense (Fregoni et al., 2021).

A different but equally important distinction concerns Fermi polarons in doped WS$GW$20 on a plasmonic metasurface. There the many-body quasiparticles remain ordinary Fermi polarons of the exciton–Fermi-sea problem. The plasmonic metasurface provides strong optical coupling and hot-electron-assisted control, yielding branch-resolved scattering spectra and Fermi polaron polaritons with lower, middle, and upper branches, but it does not create an intrinsic new plasmon-polaron many-body quasiparticle (Wu et al., 15 Jun 2026).

7. Control parameters, limitations, and outlook

Across the literature, three control parameters recur: carrier density, dimensionality, and dielectric screening. Carrier density shifts the plasmon energy and therefore the satellite separation, as seen in SrIrO$GW$21 and self-intercalated $GW$22-TiS$GW$23 (Liu et al., 2021, Choi et al., 4 Mar 2026). Dimensionality can lower plasmon energies into an experimentally accessible sub-eV range, as in monolayer MoS$GW$24, where the coupling is to two-dimensional carrier plasmons rather than the high-energy plasmons of ordinary metals (Caruso et al., 2021). Screening can enhance or suppress the effect; in alkali-doped $GW$25-HfS$GW$26, reduced surface screening was argued to be primarily responsible for the emergence of the plasmonic-polaron signature, while in $GW$27-TiS$GW$28 the temperature dependence implied that increasing dielectric screening weakens the plasmonic polaron even as $GW$29 rises (Emeis et al., 2023, Choi et al., 4 Mar 2026).

Methodologically, the field has converged on cumulant-based treatments as the appropriate framework for satellites, since simpler quasiparticle approaches can misplace or overemphasize them. At the same time, several experimental papers rely on physically motivated but simplified analyses rather than a complete material-specific $GW$30cumulant extraction of $GW$31, and intrinsic versus extrinsic plasmon losses can remain a nontrivial issue in photoemission (Caruso et al., 2016, Liu et al., 2021).

A plausible implication is that layered semiconductors and self-doped van der Waals materials will remain especially productive platforms. They permit simultaneous control of carrier density, surface localization, and dielectric environment, while keeping plasmon energies in the $GW$32–$GW$33 eV range where ARPES and loss spectroscopies can be compared directly. In that regime, plasmonic polarons are no longer only a formal consequence of dynamical screening; they become experimentally tractable quasiparticle fingerprints of electron–plasmon coupling.