Enhancement of the Electron−Phonon Scattering Induced byIntrinsic Surface Plasmon−Phonon PolaritonsDavid Hagenmuller,*,† Johannes Schachenmayer,†,‡ Cyriaque Genet,† Thomas W. Ebbesen,†

and Guido Pupillo*,†,‡

†ISIS (UMR 7006) and icFRC, University of Strasbourg and CNRS, Strasbourg 67081, France‡IPCMS (UMR 7504), University of Strasbourg and CNRS, Strasbourg 67081, France

*S Supporting Information

ABSTRACT: We investigate light−matter coupling in metalliccrystals where plasmons coexist with phonons exhibiting largeoscillator strength. We demonstrate theoretically that this coexistencecan lead to strong light−matter interactions without externalresonators. When the frequencies of plasmons and phonons arecomparable, hybridization of these collective matter modes occurs inthe crystal. We show that the coupling of these modes to photonicdegrees of freedom gives rise to intrinsic surface plasmon−phononpolaritons, which offer the unique possibility to control the phononproperties by tuning the electron density and the crystal thickness. Inparticular, dressed phonons with reduced frequency and large wavevectors arise in the case of quasi-2D crystals, which could lead to largeenhancements of the electron−phonon scattering in the vibrational ultrastrong coupling regime. This suggests that photons canplay a key role in determining the quantum properties of certain materials. A nonperturbative self-consistent Hamiltonianmethod is presented that is valid for arbitrarily large coupling strengths.

KEYWORDS: plasmonics, polaritons, ultrastrong coupling, electron−phonon coupling, 2D materials, superconductivity

Using light to shape the properties of quantum materials isa long-standing goal in physics, which is still attracting

much attention.1−3 Since the 1960s, it is known thatsuperconductivity can be stimulated via an amplification ofthe gap induced by an external electromagnetic radiation.4,5

Ultrafast pump−probe techniques have been recently used tocontrol the phases of materials such as magnetoresistivemanganites,6 layered high-temperature superconductors,7 andalkali-doped fullerenes,8 by tuning to a specific mid-infraredmolecular vibration. It is an interesting question whether thephases of quantum materials may also be passively modified bystrong light−matter interactions in the steady state, withoutexternal radiation. Recently, the possibility of modifyingsuperconductivity by coupling a vibrational mode to thevacuum field of a cavity-type structure has been suggested.9

This idea was then investigated theoretically in the case of anFeSe/SrTiO3 superconducting heterostructure embedded in aFabry−Perot cavity.10 In this work, enhanced superconductiv-ity relies, however, on unrealistically large values of thephonon−photon coupling strength, in a regime where lightand matter degrees of freedom totally decouple.11 This occursbeyond the ultrastrong coupling regime, which is defined whenthe light−matter coupling strength reaches a few tens ofpercents of the relevant transition frequency.12−14

Thanks to the confinement of light below the diffractionlimit,15 an alternative approach to engineer strong16−26 andeven ultrastrong light−matter coupling27−29 is the use of

plasmonic resonators. The latter allow the propagation oftightly confined surface plasmon polaritons (SPPs), which areevanescent waves originating from the coupling between lightand collective electronic modes in metals, called plasmons.30

Similarly, the collective coupling between light and vibrationsof ions in solids gives rise to surface phonon polaritons. Whilesuitable quantized theories to describe the ultrastrong couplingbetween SPPs or surface phonon polaritons and otherquantum emitters such as excitons have been alreadyreported,31,32 most quantized models used in the liter-ature24,33,34 are not valid in this regime.Strong coupling between SPPs in a 2D material and surface

phonons of the substrate has stimulated considerableinterest.35−40 While significant couplings cannot generallyoccur in the same crystal due to the screening of the ioncharges by free electrons, exceptions have been recentlyreported considering surface phonons in InP or SiC nano-crystals coupled to photoexcited carriers.41 Other exceptionsexist as a result of a transfer of oscillator strength frominterband electronic transitions to the lattice. These includebilayer graphene,42 alkali fullerides A6C60,

43,44 organicconductors (e.g., K−TCNQ45−47), and some transition metalcompounds.48 Interestingly, some of these materials or similar

Received: February 14, 2019Published: March 17, 2019

Article

pubs.acs.org/journal/apchd5Cite This: ACS Photonics XXXX, XXX, XXX−XXX

© XXXX American Chemical Society A DOI: 10.1021/acsphotonics.9b00268ACS Photonics XXXX, XXX, XXX−XXX

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compounds feature superconductivity and charge densitywaves under certain conditions that are believed to be driven,at least partly, by electron−phonon scattering.49−53 Since thecoexistence of collective electronic and ionic modes with largeoscillator strength in the same crystal opens up the possibilityto tune bulk phonon resonances,42,54 it is interesting to askwhether strong light−matter interactions between phonons,plasmons, and photons could occur in these materials in theabsence of an external resonator, and if so, can this affect theelectron−phonon scattering.In this work, we propose and investigate the possibility to

tune material properties of certain crystals via a hybridizationof phonons, plasmons, and photons, which is intrinsic to thematerial, and without the use of an external resonator. Thecoexistence of plasmons and phonons within the same crystalgives rise to hybrid plasmon−phonon modes, which have beenextensively studied in doped semiconductors and semi-metals.55−59 Here, we show that while hybrid plasmon−phonon modes do not affect the electron−phonon scattering atthe level of the random phase approximation, stronginteractions between these modes and photons offer a uniquepossibility to control the energy and momentum of theresulting surface plasmon−phonon polaritons by tuning theelectron density and the crystal thickness: “Dressed” phononswith reduced frequency arise in the ultrastrong couplingregime, where the electronic and ionic plasma frequenciesbecome comparable to the phonon frequency. These dressedphonons are shown to exhibit unusually large momentacomparable to the Fermi wave vector in the case of quasi-2Dcrystals, which could lead to large enhancements of theelectron−phonon scattering. Simply put, such enhancementsare significant only for thin enough crystals, as they rely on thedressing of 3D phonon modes by 2D surface polaritons. Theseresults suggest that photons can play a key role in determiningthe quantum properties of certain materials.We utilize a self-consistent Hamiltonian method based on a

generalization of that introduced in ref 60, which is valid for allregimes of interactions including the ultrastrong couplingregime, and in the absence of dissipation. This quantumdescription of surface plasmon−phonon polaritons provides anideal framework to investigate how the latter can affect thequantum properties of the crystal. Furthermore, our methodcan be generalized to various surrounding media andgeometries of interest such as layered metallo-dielectricmetamaterials and simple nanostructures and differs from theusual effective quantum description of strong coupling betweenexcitons and quantized SPPs.24,33,34 We show in theSupporting Information that the latter model leads tounphysical behaviors in the ultrastrong coupling regime.The paper is organized as follows: (i) We first introduce the

total Hamiltonian of the system under consideration and (ii)diagonalize the matter part leading to plasmon−phononhybridization in the crystal. (iii) We derive the couplingHamiltonian of these hybrid modes to photonic degrees offreedom, which gives rise to intrinsic surface plasmon−phononpolaritons. (iv) We determine the Hamiltonian parameter andits eigenvalues self-consistently using the Helmoltz equationand characterize the properties of the resulting surfaceplasmon−phonon polaritons. (v) We show that whileplasmon−phonon hybridization cannot solely modify theelectron−phonon scattering at the level of the random phaseapproximation (RPA), the coupling of these hybrid modes to

photons could lead to large enhancements of the electron−phonon scattering in the crystal.

■ QUANTUM HAMILTONIANWe consider a metallic crystal of surface S and thickness l inair, which contains a free electron gas and a transverse opticalphonon mode with frequency ω0. The excitation spectrum ofthe electron gas features a collective, long-wavelength plasmonmode with plasma frequency ωpl, as well as a continuum ofindividual excitations called electronic dark modes that areorthogonal to the plasmon mode. Both plasmons and phononsare polarized in the directions uz and u∥ (Figure 1). The

Hamiltonian is derived in the Power−Zienau−Woolleyrepresentation,61 which ensures proper inclusion of allphoton-mediated dipole−dipole interactions and can bedecomposed as H = Hpol + Hel−pn with Hpol = Hpt + Hmat−pt+ H m a t . T h e fi r s t t e r m i n H p o l r e a d sH d dRD R RH R( ) ( )pt c

12

2 12

20 0

2∫ ∫= +ϵ ϵand corresponds to

the photon Hamiltonian. c denotes the speed of light invacuum, R ≡ (r, z) the 3D position, ϵ0 is the vacuumpermittivity, and D and H are the displacement and magneticfields, respectively. The light−matter coupling term readsH dRP R D R( ) ( )mat pt

1

0∫= − ·− ϵ . Here, P = Ppl + Ppn denotes

the matter polarization field associated with the dipole momentdensity, where Ppl and Ppn correspond respectively to theplasmon and phonon contributions. [Our method can be easilygeneralized when phonons are replaced by excitons, byconsidering the appropriate polarization field.] The matterHamiltonian is decomposed as Hmat = Hpn + Hpl + HP

2, whereHpn and Hpl denote the contributions of free phonons andplasmons, which are provided in the Supporting Information,and H dRP R( )P

12

22

0∫= ϵ

contains terms ∝ Ppl2 , Pph

2 , and a

direct plasmon−phonon interaction ∝Ppl·Pph. Finally, Hel−pnincludes the contribution of the free, individual electrons(electronic dark modes), as well as the coupling Hamiltonianbetween electronic dark modes and phonons.

■ DIAGONALIZATION OF THE MATTERHAMILTONIAN

Hybrid plasmon−phonon modes have been extensively studiedin doped semiconductors and semimetals.55−59 In this section,

Figure 1. Intrinsic surface plasmon−phonon polaritons: Metal ofthickness l in air featuring a plasmon mode with frequency ωpl (graydashed line) and an optical phonon mode with frequency ω0 (greendashed line). Both plasmons and phonons are polarized in thedirections uz and u∥. When νpn ≠ 0 and ωpl ≈ ω0, the matterHamiltonian Hmat provides two hybrid plasmon−phonon modes withfrequencies ω1 and ω2 [eq 1] represented as a function of ωpl/ω0 forνpn = 0.5ω0. The surface polaritons generated by the coupling of thesemodes to light propagate along the two metal−dielectric interfaceswith in-plane wave vector q.

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we propose a derivation of these hybrid modes using anonperturbative quantum method. In order to diagonalizeHmat, the phonon polarization Ppn is written in terms of thebosonic phonon annihilation and creation operators BQ,α andBQ,α† , with the 3D wave vector Q ≡ (q, qz) and α = ∥, z the

polarization index. The lattice polarization field Ppn is

proportional to the ion plasma frequency NM apn

2

03ν =

ϵ,

which plays the role of the phonon−photon couplingstrength.62 Here, N denotes the number of vibrating ionswith effective mass M and charge in a unit cell of volume a3.The plasmon polarization Ppl is provided by the dipolardescription of the free electron gas corresponding to that of theRPA.60 In the long-wavelength regime, Ppl is written in termsof the bosonic plasmon operators PKQ and PKQ

† (K denotes the3D electron wave vector), superpositions of electron−holeexcitations with wave vector Q across the Fermi surface. Asexplained in the Supporting Information, Hmat can be put inthe diagonal form Hmat = ∑Q,α,j=1,2ℏωjΠQjα

† ΠQjα, where thehybrid plasmon−phonon operators ΠQjα are superpositions ofBQ,α, PKQ, and their Hermitian conjugates, and satisfy thebosonic commutation relations [ΠQjα, ΠQ′j′α′

† ] = δQ,Q′δj,j′δα,α′.The hybrid mode frequencies ωj are represented in Figure 1and are given by

2 ( ) 4j2

02

pl2

02

pl2 2

pn2

pl2ω ω ω ω ω ν ω = + ± − + (1)

Here, j = 1 and j = 2 refer respectively to the signs − and +,and the longitudinal phonon mode with frequency

0 02

pn2ω ω ν = + is determined by the combination of the

short-range restoring force related to the transverse phononresonance, and the long-range Coulomb force associated withthe ion plasma frequency. The diagonalization procedure ofHmat further provides the well-known transverse dielectricfunction of the crystal:55

( ) 1crpl2

2pn2

202ω

ω

ω

ν

ω ωϵ = − −

− (2)

One can then use the eigenmodes basis of the matterHamiltonian to express the total polarization field as

gV

P R u( )2

( )ej

jj

j ji

QQ Q

Q R

, ,

0 pl2

∑ω

ω=

ℏϵ

Π + Πα

α α α−† − ·

(3)

with

g(1 / )

( )jj

j j j j

2pn2

pl2

02

2 2pl4

4pn2

pl2

202 2

ω ν ω ω

ω ω

ω

ω

ν ω

ω ω=

+ −

− +

−′

j′ ≠ j, and V Sl= . In the absence of phonon−photon coupling(νpn = 0), the two modes j = 1, 2 reduce to the bare phononand plasmon. A similar situation occurs for νpn ≠ 0, in the caseof large plasmon−phonon detunings. Both in the high ωpl ≫ω0 and low ωpl ≪ ω0 electron density regimes, the hybridmodes reduce to the bare plasmon and phonon excitations,and the plasmon contribution prevails in the polarization fieldeq 3. As explained in the following, when the latter interactswith the photonic displacement field D, this simply results inthe formation of SPPs coexisting with bare phonons. The latterare either transverse phonons with frequency ω0 for ωpl ≫ ω0or longitudinal phonons with frequency ω0 in the case ωpl ≪

ω0 (Figure 1). In the following, we focus on the mostinteresting case occurring close to the resonance ωpl ≈ ω0,where plasmon−phonon hybridization occurs.

■ COUPLING TO PHOTONSDue to the breaking of translational invariance in the zdirection, the 3D photon wave vector can be split into an in-plane and a transverse component in each media as Q = qu∥ +iγnuz, where n = d, cr refers to the dielectric medium and thecrystal, respectively. For a given interface lying at the height z0,the electromagnetic field associated with surface waves decaysexponentially on both sides as e±γn(z − z0) with the (real)penetration depth γn. The displacement and magnetic fieldscan be written as superpositions of the fields generated by eachinterfaces m = 1, 2, namely,

cS

z DD R u( )4

e ( )m

im m

q

q rq q

,

0∑=ϵ ℏ ·

and

wc

Sz HH R v( )

4e ( )

mq

im m

q

q rq q

,

0∑=ϵ ℏ ·

with wq the frequency of the surface waves (still undetermined)and q ≡ |q|. The mode profile functions uqm(z) and vqm(z)depending on γn are provided in the Supporting Information.Using these expressions, the photon Hamiltonian takes theform

H c D D H H( )m m

qmm

m m qmm

m mq

q q q qpt, ,

∑= ℏ +′

′− ′

′− ′

where the overlap matrix elements qmm′ and q

mm′ depend onthe parameters γcr and γd. The next step corresponds to findingthe electromagnetic field eigenmodes, which consist of asymmetric and an antisymmetric mode, such that the photonHamiltonian can be put in the diagonal form:

H c D D H H( )q qq

q q q qpt,

∑ α β= ℏ +σ

σ σ σ σ σ σ=±

− −

w i t h q q q11 12α = ±± , q q q

11 12β = ±± ,

D D D( )/ 2q q q2 1= ±± , and a similar expression for Hq±.The new field operators Dqσ and Hqσ satisfy the commutationrelations [Dqσ, Hq′σ′

† ] = −iCqσδq,q′δσ,σ′, together with theproperties Dqσ

† = D−qσ and Hqσ† = H−qσ. The constant Cqσ is

determined using the Ampere’s circuital law, which provides

Cqw

c2q

q=σ β σ

.60

The light−matter coupling Hamiltonian Hmat−pt is derivedusing the expression of D(R) in the new basis together with eq3. While q is a good quantum number due to the in-planetranslational invariance, the perpendicular wave vector of the3D matter modes qz is not. In the light−matter couplingHamiltonian Hmat−pt, photon modes with a given q interactwith linear superpositions of the 3D matter modes exhibitingdifferent qz. The latter are denoted as quasi-2D “bright” modesand are defined as πqσj = ∑qz,α fασ(Q)ΠQjα, where fασ(Q) stemfrom the overlap between the displacement and the polar-ization fields and is determined by imposing the commutationrelations [πqσj, πq′σ′j′

† ] = δq,q′δσ,σ′δj,j′.

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Using a unitary transformation,62 the matter Hamiltoniancan be decomposed as Hmat = ∑q,σ,jℏωjπqσj

† πqσj + Hdark, wherethe second term is the contribution of the “dark” modes thatare orthogonal to the bright ones and which do not interact withphotons. Without this contribution, the polariton Hamiltonianreads H q qpol ,= ∑ σ σ with

c D D H H

D

( )

( )

q qj

j j j

jq j j j

q q q q q q q

q q q

∑

∑

α β ωπ π

π π

= ℏ + + ℏ

− ℏΩ +

σ σ σ σ σ σ σ σ σ

σ σ σ σ

− −†

−†

and the vacuum Rabi frequency

gc

ql q(1 e ) 2 e ( )

q j jj

l l

pl2

2cr2

cr

2 2cr2

cr cr

ωω

γγ

σ γ

Ω =

×+

− + −

σ

γ− −γ

■ SURFACE PLASMON−PHONON POLARITONSThe Hamiltonian Hpol exhibits three eigenvalues in eachsubspace (q, σ), and only the lowest two (referred to as lowerand upper polaritons) correspond to surface modes (below thelight cone). At this point, we have built a Hamiltonian theoryproviding a relation between the field penetration depths γnand the surface wave frequencies wqσζ, where ζ = LP, UP refersto the lower (LP) and upper (UP) polaritons. This eigenvalue

equation can be combined with the Helmholtz equationϵn(ω)ω

2/c2 = q2 − γn2 in order to determine these parameters.60

We use a self-consistent algorithm which starts with a givenfrequency wqσLP, then determine γn from the Helmholtzequation with ϵd = 1 and ϵcr(wqσLP) given by eq 2 and usethese γn to compute the parameters entering the HamiltonianHpol. The latter is diagonalized numerically,62 which allowsdetermining the new wqσLP. The algorithm is appliedindependently for the symmetric (σ = +) and antisymmetric(σ = −) modes until convergence, which is ensured by thediscontinuity of both light and matter fields at each interface.60

We now use this method to study the surface polaritons inour system. As an example, we consider the case ωpl = 1.5ω0

with different crystal thickness q l 100 = (Figure 2a) andq l 0.010 = (Figure 2b) and compute the surface polaritonfrequencies wqσζ (top panels), as well as their phonon (i = pn),plasmon (i = pl), and photon (i = pt) admixtures Wi,qσ

LP for νpn= 0 and νpn = 0.5ω0 (bottom panels). Precise definitions ofthese quantities are provided in the Supporting Information,and l and q are both normalized to q0 = ω0/c. Consideringtypical mid-infrared phonons with ℏω0 = 0.2 eV, the twodimensionless parameters q l 100 = and q l 0.010 = correspondto l 10 mμ= (first case) and l 10 nm≈ (second case),respectively.In the first case (Figure 2a), the two surface modes at each

interface have negligible overlap, and the modes σ± thereforecoincide. For νpn = 0, the plasmon−photon coupling isresponsible for the appearance of an SPP mode (black line)with frequency wq

0, which enters in resonance with the phonon

Figure 2. Normalized frequency dispersion wqσζ/ω0 and mode admixtures of the surface polaritons as a function of q/q0 for ωpl = 1.5ω0. (a)q l 100 = . (b) q l 0.010 = . Top panels: The SPPs obtained for νpn = 0 are depicted as black lines, while the surface lower and upper polaritonsobtained for νpn = 0.5ω0 are represented as thick red and blue lines, respectively. Resonance between SPPs and phonons (horizontal dotted lines) isindicated by thin vertical lines. The antisymmetric (σ = −) and symmetric (σ = +) modes correspond to the solid and dashed lines, and the lightcone with boundary ω = qc is represented as a gray-shaded region. Inset: Frequency wqσζ/ω0 of the lower (red line) and upper (blue line) surfacepolaritons versus νpn/ω0 for q l 100 = and ωpl = 1.5ω0. The phonon frequency is depicted as a horizontal dotted line. Bottom panels: Plasmon (graydashed lines), photon (magenta solid lines), and phonon (green dotted lines) admixtures of the antisymmetric lower polariton Wi,qσ = −

LP (i = pl, pt,pn) as a function of q/q0 for ωpl = 1.5ω0. The top and bottom subpanels correspond to νpn = 0 and νpn = 0.5ω0, respectively.

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mode at qr ≈ 2.2q0. While for q ≪ q0 this SPP is mainlycomposed of light (wq

0 ∼ qc), it features a hybrid plasmon−photon character for q ∼ q0 and becomes mostly plasmon-likeas wq

0 approaches the surface plasmon frequency / 2plω for q≫ q0. For νpn ≠ 0, a splitting between the two polaritonsbranches (thick red and blue lines) is clearly visible, and thelatter consist of a mix between phonons, plasmons, andphotons in the vicinity of q = q0. In the regime q > q0, since themostly phonon-like LP exhibits a ∼25% plasmon admixture,this polariton mode can be seen as a “dressed” phonon withfrequency red-shifted from ω0. Similarly, the “dressed”plasmon mode (UP) is blue-shifted from the surface modefrequency / 2plω due its ∼25% phonon weight.The frequencies of the LP (red line) and UP (blue line) at

resonance are represented as a function of νpn/ω0 in the inset.Here, the resonance is defined by the condition wq

0 = ω0, whichprovides qr ≈ 2.2q0. We observe that the polariton splittingwqσUP − wqσLP is symmetric with respect to wqσζ = ω0 (blackdotted line) for νpn ≪ ω0 and becomes slightly asymmetric inthe ultrastrong coupling regime,12 when νpn is a non-negligiblefraction of ω0. Furthermore, the splitting is found to decreaserapidly as the electronic plasma frequency ωpl is increased.In the second case q l 0.010 = (Figure 2b), the crystal is

thinner than the penetration depth γcr of the surface waves atthe two interfaces, and the latter overlap. This results in twosets of modes σ = ± with different frequencies. For νpn = 0, thesymmetric and antisymmetric SPPs with frequency wqσ=±

0 arerepresented as black dashed and solid lines, respectively. Theresonance between the symmetric SPP and the phonon modeoccurs for q ≈ q0, in the regime where the symmetric SPP ismostly photon-like with wqσ=+

0 ∼ qc. Interestingly, theresonance between the antisymmetric SPP and the phononmode is now shifted to a large wave vector qr ≈ 220q0, wherethe antisymmetric SPP exhibits a pure plasmonic character. Forq/q0 → ∞, the two SPPs σ = ± converge to the surfaceplasmon frequency / 2plω . For νpn ≠ 0, the antisymmetric LP(thick red solid line) and UP (thick blue solid line) are split inthe vicinity of the resonance qr ≈ 220q0, while the symmetricpolaritons (thick colored dashed lines) do not feature anyanticrossing behavior for q ∼ q0. Similarly to that in (a), theLPs σ = ± can be seen as dressed phonon modes withfrequencies red-shifted from ω0 for large q due to theirplasmon admixtures. Here, however, these dressed phononscan propagate with very large wave vectors comparable to theelectronic Fermi wave vector.

■ ELECTRON−PHONON SCATTERINGWe now show that the ability to tune the energy and wavevector of the dressed phonons can affect the electron−phononscattering in the crystal. The coupling Hamiltonian betweenindividual electrons (electronic dark modes) and phononsreads62

H c c c c B B( )KK

K KK Q

K K Q Q Qel pn, ,

∑ ∑ξ= ℏ + ℏ +α

α α−† †

− −†

(4)

The Fermionic operators cK and cK† annihilate and create an

electron with wave vector K and energy ℏξK (K ≡ | K|) relativeto the Fermi energy. For simplicity, we assume a 3D sphericalFermi surface providing ξK = ℏ(K2 − KF

2)/(2m), with m theelectron effective mass and KF the Fermi wave vector, and that

the coupling constant does not depend on Q as showntheoretically for intramolecular phonons in certain crystals.63

Electron−phonon interactions are usually characterized by thedimensionless coupling parameter λ, which quantifies theelectron mass renormalization due to the coupling tophonons.55 At zero temperature, λ is defined as

N1

( ) ( ( ) )K KK3D

0∑λ δ ξ ω= ℜ −∂ Σ |ω ω=

where ℜ stands for real part, N ( )KVmK

K3D 2F

2δ ξ= ∑ =π ℏ

is the

3D electron density of states at the Fermi level, and∂ωΣK(ω)|ω=0 denotes the frequency derivative of the retardedelectron self-energy ΣK(ω) evaluated at ω = 0. An equation ofmotion analysis64 of the electron Green’s function (GF)

i c c( ) ( ) (0)K K Kτ τ= − ⟨ ⟩† allows to write the electron self-energy as

Bid

( )2

( ) ( )QKQ

K Q,

2 ∫∑ω ωπ

ω ω ωΣ = | | ′ + ′ ′α

α−(5)

where B i d( ) e ( ) (0)Qi

Q Q∫ω τ τ= − ⟨ ⟩αωτ

α α− is the phonon

GF written in the frequency domain, and B BQ Q Q= +α α α−† .

In the absence of phonon−photon coupling (νpn = 0), there isno hybridization between phonons and plasmons, nor is therecoupling to photons. Phonons therefore enter the HamiltonianHpol only in the free contribution Hpn = ∑Q,αℏω0BQα

† BQα. Inthis case, the equation of motion analysis simply providesB ( ) 2 /( )Q 0

202ω ω ω ω= −α . Using this expression together

with the noninteracting electron GF ( ) 1/( )KK0 ω ω ξ= − in

eq 5, one can compute the electron−phonon couplingparameter for νpn = 0 as62

NN2

( ) ( )2

K QK Q

K0

2

3D 0

23D

0∑ ∑λ

ωδ ξ δ ξ

ω= | | =

| |−

In the presence of phonon−photon coupling (νpn ≠ 0), thephonon dynamics is governed by the Hamiltonian Hpol, whichincludes the coupling of phonons to plasmons and photons. Itis therefore convenient to express the 3D phonon operatorsBQα and BQα

† in terms of the 3D hybrid modes ΠQαj and ΠQαj†

and then project the latter onto the quasi-2D bright and darkmodes such that the electron−phonon Hamiltonian eq 4 takesthe form Hel−pn = ∑KℏξKcK

† cK + Hel−pn(B) + Hel−pn

(D) . Thecontribution of the bright modes reads

H f Q c c( ) ( )j

j j jK Q

K K Q q qel pn(B)

, , ,

∑ ∑ η π π= ℏ * +α σ

ασ σ σ−†

− −†

while the contribution Hel−pn(D) of the dark modes that do not

interact with photons is given in the Supporting Information.

Here, j j/

( / )j

j j

0

pl4 2η χ= ω ω

ω ω χ

+where j

j

pn pl2

02χ =

ν ω

ω ω −is associated

with the hybrid mode weights of the phonons. The electronself-energy due to the interaction with bright modes reads

B

i f Qd

( ) ( )2

( )

( )

jj

q j

KQ

K Q(B)

, , ,

2 2 ∫∑ω η ωπ

ω ω

ω

Σ = | | ′ + ′

′

α σασ

σ

−

(6)

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where B i d( ) e ( ) (0)q ji

q j q j∫ω τ τ= − ⟨ϒ ϒ ⟩σωτ

σ σ− and ϒqσj = πqσj+ π−qσj

† . As detailed in the Supporting Information, we now usethe equation of motion theory to calculate the GF B ( )q j ωσ . For

νpn = 0, one simply obtains B ( ) 2 /( )q j j j2 2ω ω ω ω= − σ , which

provides B B( ) ( )q Q1 ω ω=σ α since ω1 = ω0 (and ω2 = ωpl). Inthis case, η1 = 1, η2 = 0, and only the pure phonon term j = 1therefore contributes to ΣK

(B). We then calculate B ( )q j ωσ andthe self-energy eq 6 for νpn ≠ 0 and decompose the electron−phonon coupling parameter into its bright and dark modecontributions: λ = λ(B) + λ(D) for νpn ≠ 0, and λ0 = λ0

(B) + λ0(D)

for νpn = 0. The contributions λ(B) and λ0(B) are different

because bright modes interact with photons for νpn ≠ 0, whilethey do not for νpn = 0. Similarly, λ(D) and λ0

(D) are a prioridifferent since dark modes are hybrid plasmon−phonon modesfor νpn ≠ 0, while they reduce to bare phonons and plasmonsfor νpn = 0. However, we show in the Supporting Informationthat λ(D) = λ0

(D), which means that, at the level of the RPA, thehybridization between plasmons and phonons cannot solelylead to a modification of the electron−phonon scattering.Finally, the relative enhancement of the electron−phononcoupling parameter reads62

Nf Q

1( ) ( 1) ( ) ( )K q

K QK Q

0

0

0

3D2

, ,

2∑ ∑λ

λ λλ

δ ξ φ δ ξ

Δλ ≡−

= − | |α σ

σ ασ −

(7)

where the function φqσ is given in the Supporting Information.The latter describes the renormalization of the phonon energydue to the coupling to plasmons and photons and depends onthe polariton frequencies wqσζ. It is represented in Figure 3atogether with the frequency dispersion of the LPs wqσLP/ω0corresponding to the red solid and dashed lines in Figure 2b.First, we find that the contributions of the LPs to φqσ largelydominate over the contributions of the other polaritons.Second, we observe that φqσ is anticorrelated with thedispersion of the LPs, whenever the latter exhibit a finitephonon weight. For instance, φqσ=− reaches large values in theregion q ≲ q0 where the frequency of the antisymmetric LP(≈35% phonon for νpn = 0.5ω0) is far away from the barephonon frequency. In contrast, while wqσ=+LP ≪ ω0 in thisregion, φqσ=+ ≈ 1 since the symmetric LP is fully photon-like

for q ≲ q0. Note that for νpn/ω0 → 0, the LPs for q > qr arefully phonon-like with wqσLP → ω0, and one finds that φqσ → 1does not contribute to

0λΔλ in this region. For q < qr, φqσ does

not contribute to0λ

Δλ because of the vanishing phonon weight

of the LP for νpn/ω0 → 0.The relative enhancement of the electron−phonon coupling

0λΔλ is represented in Figure 3b as a function of q l0 and νpn/ω0.

We find that it can reach large values ≳ 10% for νpn ≳ 0.4ω0

and l sufficiently small (q l 0.010 ≈ ). The enhancement of0λ

Δλ

when decreasing the crystal thickness l can be understood byrealizing that the function |fασ(Q)|

2 entering eq 7 scales asl1/( crγ∼ ), as shown in the Supporting Information. This stems

from the fact that the dressing of 3D phonons by 2Devanescent waves decaying exponentially at each interfaceincreases when decreasing the crystal thickness. However,when the crystal thickness becomes too small (e.g., whenK l 1F ≲ ), the description of the matter excitations by 3D fieldsbreaks down, and quantum confinement effects have to betaken into account in the model. The parameters of Figure 3bare l 10 nm= for ℏω0 = 0.2 eV, and KF = 1 nm−1. We find thatthe enhancement of the electron−phonon coupling stronglyincreases with the ratio q0/KF, i.e., when the mismatch betweenthe typical photonic and electronic wave vectors is reduced.This can occur in materials with larger phonon frequency and/or lower electron density. In the latter case, however,decreasing KF requires considering larger l for the 3Ddescription of the matter fields to be valid, which in turnleads to a reduction of

0λΔλ .

For instance, bilayer graphene with l 0.4nm≈ 65 exhibits aninfrared-active phonon mode at ℏω0 = 0.2 eV42 and has beenrecently shown to feature superconductivity at low carrierdensities ≈2 × 1012 cm−2 when the two sheets of graphene aretwisted relative to each other by a small angle.52 In this regime,we find that ωpl ≈ ω0, q l 7 100

4≈ × − , KF ≈ 1.6 × 103q0, andthe large phonon−photon coupling strengths νpn ≈ 0.25ω0

42

allow reaching the ultrastrong coupling regime. Using theseparameters, we find that the contribution of surface plasmon−phonon polaritons to the electron−phonon scattering is

expected to reach a very large value, 2.70

≈λΔλ . However, the

Figure 3. (a) Function φqσ versus q/q0 for νpn/ω0 = 0.5, q l 0.010 = , and ωpl = 1.5ω0. Inset: Normalized frequency dispersion of the LPs wqσLP/ω0

corresponding to the red solid and dashed lines in Figure 2b. The contributions of symmetric (σ = +) and antisymmetric (σ = −) modes aredepicted as dashed and solid lines, respectively. (b) Relative enhancement of the electron−phonon coupling parameter

0λΔλ given by eq 7 versus q l0

and νpn/ω0, for ωpl = 1.5ω0 and KF = 103q0.

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small thickness of bilayer graphene such that K l 1F ∼ in thisweakly doped regime implies that quantum confinement effectsbecome important. Furthermore, the failure of the 3D modelassuming ωpl → const as q → 0 can be seen in the dispersionof the optical plasmon mode in bilayer graphene,58 which isknown to be qplω ∼ as q → 0.

■ CONCLUSIONIn summary, we have carried out a nonperturbative quantumtheory of intrinsic, lossless plasmon−phonon polaritons validfor arbitrary large coupling strengths and discussed thedifferent regimes of interest obtained by tuning the crystalthickness and the phonon−photon coupling strength. Thistheory allows characterizing the hybridization betweenplasmons and phonons in the crystal, which cannot lead solelyto a modification of the electron−phonon scattering at thelevel of the RPA. Considering the coupling of these hybridplasmon−phonon modes to photons, we have found that inthe regime where both the electronic and ionic plasmafrequencies become comparable to the phonon frequency, adressed low-energy phonon mode with finite plasmon weightarises. In the case of quasi-2D crystals, the in-plane wave vectorof this dressed phonon can reach very large values comparableto the Fermi wave vector, which is shown to lead to anenhancement of the electron−phonon scattering. It is aninteresting prospect to investigate whether this effect couldmodify superconductivity in certain molecular crystals. Whileour model specifically addresses the case of metallic crystalswith large ionic charges supporting far/mid-infrared pho-nons,42−45,47 it can be directly generalized to describe othertypes of excitations such as excitons and extended to othergeometries. Further useful extensions include the presence ofquantum confinement in 2D materials and dissipation. Inaddition to the intrinsic phonon linewidth, the main source ofdissipation is the damping of plasmons with large wave vectorsdue to the electron−hole continuum (electronic darkmodes).55 However, since the dressed phonons feature onlya ∼25% plasmon admixture at large wave vectors, this effect isnot expected to dramatically affect our results.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acsphoto-nics.9b00268.

Detailed diagonalization of the matter Hamiltonian,derivation of the total Hamiltonian and diagonalization,derivation of the electron−phonon coupling Hamilto-nian, calculation of the relative change of the electron−phonon coupling parameter, derivation of the effectivequantum model, figure showing the failure of theeffective quantum model in the ultrastrong couplingregime (PDF)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: dhagenmuller@unistra.fr.*E-mail: pupillo@unistra.fr.ORCIDDavid Hagenmuller: 0000-0003-4129-9312Cyriaque Genet: 0000-0003-0672-7406

Thomas W. Ebbesen: 0000-0002-3999-1636NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

We gratefully acknowledge discussions with M. Antezza, T.Chervy, V. Galitski, Y. Laplace, Stefan Schutz, A. Thomas, andY. Todorov. This work is supported by the ANR - “ERA-NETQuantERA” - Projet “RouTe” (ANR-18-QUAN-0005-01) andIdEx Unistra - Projet “STEMQuS”.

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