Enhancement of the Electron–Phonon Scattering Induced By

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Cite This: ACS Photonics XXXX, XXX, XXX−XXX pubs.acs.org/journal/apchd5

Enhancement of the Electron−Phonon Scattering Induced by Intrinsic Surface Plasmon−Phonon Polaritons † † ‡ † † David Hagenmüller,*, 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 metallic crystals where plasmons coexist with phonons exhibiting large oscillator strength. We demonstrate theoretically that this coexistence can lead to strong light−matter interactions without external resonators. When the frequencies of plasmons and phonons are comparable, hybridization of these collective matter modes occurs in the crystal. We show that the coupling of these modes to photonic degrees of freedom gives rise to intrinsic surface plasmon−phonon polaritons, which oﬀer the unique possibility to control the phonon properties by tuning the electron density and the crystal thickness. In particular, dressed phonons with reduced frequency and large wave vectors arise in the case of quasi-2D crystals, which could lead to large enhancements of the electron−phonon scattering in the vibrational ultrastrong coupling regime. This suggests that photons can play a key role in determining the quantum properties of certain materials. A nonperturbative self-consistent Hamiltonian method is presented that is valid for arbitrarily large coupling strengths. KEYWORDS: plasmonics, polaritons, ultrastrong coupling, electron−phonon coupling, 2D materials, superconductivity

sing light to shape the properties of quantum materials is plasmonic resonators. The latter allow the propagation of a long-standing goal in physics, which is still attracting tightly confined surface plasmon polaritons (SPPs), which are U − much attention.1 3 Since the 1960s, it is known that evanescent waves originating from the coupling between light superconductivity can be stimulated via an amplification of and collective electronic modes in metals, called plasmons.30 4,5 the gap induced by an external electromagnetic radiation. Similarly, the collective coupling between light and vibrations Ultrafast pump−probe techniques have been recently used to of ions in solids gives rise to surface phonon polaritons. While control the phases of materials such as magnetoresistive suitable quantized theories to describe the ultrastrong coupling Downloaded via UNIV STRASBOURG on March 28, 2019 at 13:56:45 (UTC). 6 7 manganites, layered high-temperature superconductors, and between SPPs or surface phonon polaritons and other 8 fi alkali-doped fullerenes, by tuning to a speci c mid-infrared quantumemitterssuchasexcitonshavebeenalready molecular vibration. It is an interesting question whether the reported,31,32 most quantized models used in the liter- See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. phases of quantum materials may also be passively modified by 24,33,34 − ature are not valid in this regime. strong light matter interactions in the steady state, without Strong coupling between SPPs in a 2D material and surface external radiation. Recently, the possibility of modifying phonons of the substrate has stimulated considerable superconductivity by coupling a vibrational mode to the 35−40 fi fi 9 interest. While signi cant couplings cannot generally vacuum eld of a cavity-type structure has been suggested. occur in the same crystal due to the screening of the ion This idea was then investigated theoretically in the case of an charges by free electrons, exceptions have been recently FeSe/SrTiO superconducting heterostructure embedded in a 3 reported considering surface phonons in InP or SiC nano- Fabry−Perot cavity.10 In this work, enhanced superconductiv- crystals coupled to photoexcited carriers.41 Other exceptions ity relies, however, on unrealistically large values of the phonon−photon coupling strength, in a regime where light exist as a result of a transfer of oscillator strength from 11 interband electronic transitions to the lattice. These include and matter degrees of freedom totally decouple. This occurs 42 43,44 beyond the ultrastrong coupling regime, which is defined when bilayer graphene, alkali fullerides A6C60, organic − 45−47 the light−matter coupling strength reaches a few tens of conductors (e.g., K TCNQ ), and some transition metal − 48 percents of the relevant transition frequency.12 14 compounds. Interestingly, some of these materials or similar Thanks to the confinement of light below the diffraction − limit,15 an alternative approach to engineer strong16 26 and Received: February 14, 2019 − even ultrastrong light−matter coupling27 29 is the use of Published: March 17, 2019

© XXXX American Chemical Society A DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article compounds feature superconductivity and charge density photons could lead to large enhancements of the electron− waves under certain conditions that are believed to be driven, phonon scattering in the crystal. − at least partly, by electron−phonon scattering.49 53 Since the coexistence of collective electronic and ionic modes with large ■ QUANTUM HAMILTONIAN oscillator strength in the same crystal opens up the possibility We consider a metallic crystal of surface S and thickness l in to tune bulk phonon resonances,42,54 it is interesting to ask air, which contains a free electron gas and a transverse optical − ω whether strong light matter interactions between phonons, phonon mode with frequency 0. The excitation spectrum of plasmons, and photons could occur in these materials in the the electron gas features a collective, long-wavelength plasmon ff ω absence of an external resonator, and if so, can this a ect the mode with plasma frequency pl, as well as a continuum of electron−phonon scattering. individual excitations called electronic dark modes that are In this work, we propose and investigate the possibility to orthogonal to the plasmon mode. Both plasmons and phonons tune material properties of certain crystals via a hybridization are polarized in the directions uz and u∥ (Figure 1). The of phonons, plasmons, and photons, which is intrinsic to the material, and without the use of an external resonator. The coexistence of plasmons and phonons within the same crystal gives rise to hybrid plasmon−phonon modes, which have been extensively 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 at the level of the random phase approximation, strong interactions between these modes and photons offer a unique possibility to control the energy and momentum of the Figure 1. Intrinsic surface plasmon−phonon polaritons: Metal of resulting surface plasmon−phonon polaritons by tuning the ω thickness l in air featuring a plasmon mode with frequency pl (gray “ ” ω electron density and the crystal thickness: Dressed phonons dashed line) and an optical phonon mode with frequency 0 (green with reduced frequency arise in the ultrastrong coupling dashed line). Both plasmons and phonons are polarized in the ν ≠ ω ≈ ω regime, where the electronic and ionic plasma frequencies directions uz and u∥. When pn 0 and pl 0, the matter − become comparable to the phonon frequency. These dressed Hamiltonian Hmat provides two hybrid plasmon phonon modes with ω̃ ω̃ ω ω phonons are shown to exhibit unusually large momenta frequencies 1 and 2 [eq 1] represented as a function of pl/ 0 for ν ω comparable to the Fermi wave vector in the case of quasi-2D pn = 0.5 0. The surface polaritons generated by the coupling of these modes to light propagate along the two metal−dielectric interfaces crystals, which could lead to large enhancements of the with in-plane wave vector q. electron−phonon scattering. Simply put, such enhancements fi are signi cant only for thin enough crystals, as they rely on the Hamiltonian is derived in the Power−Zienau−Woolley dressing of 3D phonon modes by 2D surface polaritons. These representation,61 which ensures proper inclusion of all results suggest that photons can play a key role in determining photon-mediated dipole−dipole interactions and can be the quantum properties of certain materials. decomposed as H = Hpol + Hel−pn with Hpol = Hpt + Hmat−pt We utilize a self-consistent Hamiltonian method based on a fi + H mat.The rst term in H pol reads generalization of that introduced in ref 60, which is valid for all 1 2 1 2 Hpt =+∫∫ddRD() R2 RH () R and corresponds to regimes of interactions including the ultrastrong coupling 2ϵ0 2ϵ0c regime, and in the absence of dissipation. This quantum the photon Hamiltonian. c denotes the speed of light in ≡ ϵ description of surface plasmon−phonon polaritons provides an vacuum, R (r, z) the 3D position, 0 is the vacuum ideal framework to investigate how the latter can affect the permittivity, and D and H are the displacement and magnetic quantum properties of the crystal. Furthermore, our method fields, respectively. The light−matter coupling term reads can be generalized to various surrounding media and =−1 RP R · D R P P P Hmat− pt ϵ ∫ d () (). Here, = pl + pn denotes geometries of interest such as layered metallo-dielectric 0 the matter polarization field associated with the dipole moment metamaterials and simple nanostructures and differs from the density, where P and P correspond respectively to the usual effective quantum description of strong coupling between pl pn 24,33,34 plasmon and phonon contributions. [Our method can be easily excitons and quantized SPPs. We show in the generalized when phonons are replaced by excitons, by Supporting Information that the latter model leads to considering the appropriate polarization field.] The matter unphysical behaviors in the ultrastrong coupling regime. 2 fi Hamiltonian is decomposed as Hmat = Hpn + Hpl + HP , where The paper is organized as follows: (i) We rst introduce the H and H denote the contributions of free phonons and total Hamiltonian of the system under consideration and (ii) pn pl − plasmons, which are provided in the Supporting Information, diagonalize the matter part leading to plasmon phonon 1 2 ∝ 2 2 and HP2 = ∫ dRP() R contains terms Ppl, Pph, and a hybridization in the crystal. (iii) We derive the coupling 2ϵ0 − ∝ · Hamiltonian of these hybrid modes to photonic degrees of direct plasmon phonon interaction Ppl Pph. Finally, Hel−pn freedom, which gives rise to intrinsic surface plasmon−phonon includes the contribution of the free, individual electrons polaritons. (iv) We determine the Hamiltonian parameter and (electronic dark modes), as well as the coupling Hamiltonian its eigenvalues self-consistently using the Helmoltz equation between electronic dark modes and phonons. and characterize the properties of the resulting surface plasmon−phonon polaritons. (v) We show that while ■ DIAGONALIZATION OF THE MATTER plasmon−phonon hybridization cannot solely modify the HAMILTONIAN electron−phonon scattering at the level of the random phase Hybrid plasmon−phonon modes have been extensively studied − approximation (RPA), the coupling of these hybrid modes to in doped semiconductors and semimetals.55 59 In this section,

B DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article ω we propose a derivation of these hybrid modes using a 0 (Figure 1). In the following, we focus on the most ω ≈ ω nonperturbative quantum method. In order to diagonalize interesting case occurring close to the resonance pl 0, − Hmat, the phonon polarization Ppn is written in terms of the where plasmon phonon hybridization occurs. bosonic phonon annihilation and creation operators BQ,α and † ≡ α ∥ BQ,α, with the 3D wave vector Q (q, qz) and = , z the ■ COUPLING TO PHOTONS fi polarization index. The lattice polarization eld Ppn is Due to the breaking of translational invariance in the z 2 proportional to the ion plasma frequency ν = A N , direction, the 3D photon wave vector can be split into an in- pn ϵ 3 Ma0 plane and a transverse component in each media as Q = qu∥ + which plays the role of the phonon−photon coupling iγ u , where n = d, cr refers to the dielectric medium and the 62 n z strength. Here, N denotes the number of vibrating ions crystal, respectively. For a given interface lying at the height z0, with effective mass M and charge A in a unit cell of volume a3. the electromagnetic field associated with surface waves decays ±γ (z − z ) The plasmon polarization Ppl is provided by the dipolar exponentially on both sides as e n 0 with the (real) γ fi description of the free electron gas corresponding to that of the penetration depth n. The displacement and magnetic elds RPA.60 In the long-wavelength regime, P is written in terms fi pl † can be written as superpositions of the elds generated by each of the bosonic plasmon operators PKQ and PKQ (K denotes the interfaces m = 1, 2, namely, 3D electron wave vector), superpositions of electron−hole excitations with wave vector Q across the Fermi surface. As 4ϵℏ0 c iqr· D()Ru= ∑ e()qqm zDm explained in the Supporting Information, Hmat can be put in S ∑ ℏω̃Π† Π q,m the diagonal form Hmat = Q,α,j=1,2 j Qjα Qjα, where the − Π hybrid plasmon phonon operators Qjα are superpositions of and BQ,α, PKQ, and their Hermitian conjugates, and satisfy the Π Π† δ δ δ bosonic commutation relations [ Qjα, Q′j′α′]= Q,Q′ j,j′ α,α′. 4ϵℏc · ω̃ HR()= ∑ w 0 e()iqr v zH The hybrid mode frequencies j are represented in Figure 1 q S qqm m and are given by q,m

2 2 2 2 22 2 2 with wq the frequency of the surface waves (still undetermined) 2ωωω̃j = ̃0 +±pl ()4 ωω̃0 −+pl νωpn pl (1) ≡ | | fi and q q . The mode pro le functions uqm(z) and vqm(z) γ Here, j = 1 and j = 2 refer respectively to the signs − and +, depending on n are provided in the Supporting Information. and the longitudinal phonon mode with frequency Using these expressions, the photon Hamiltonian takes the 2 2 form ωων̃00=+pn is determined by the combination of the mm′ mm′ short-range restoring force related to the transverse phonon Hpt =ℏcDDHH∑ ()()q qqmm−′ + q qqmm−′ resonance, and the long-range Coulomb force associated with q,,mm′ the ion plasma frequency. The diagonalization procedure of where the overlap matrix elements (mm′ and )mm′ depend on Hmat further provides the well-known transverse dielectric q q 55 γ γ fi function of the crystal: the parameters cr and d. The next step corresponds to nding fi 2 2 the electromagnetic eld eigenmodes, which consist of a ωpl νpn symmetric and an antisymmetric mode, such that the photon ϵ=−−()ω 1 cr 2 2 2 Hamiltonian can be put in the diagonal form: ω ωω− 0 (2)

One can then use the eigenmodes basis of the matter Hpt =ℏcDDHH∑ ()αβqσσqq−− σ + qσ qqσσ Hamiltonian to express the total polarization ﬁeld as q,σ=±

2 11 12 11 12 ℏϵ ω with αqqq± =±((, βq± =±))qq, = 0pl Π+Π†−·iQR PR() ∑ gj ()e−QQjjαα uα 2ω̃ V DDD± =±()/2, and a similar expression for Hq±. Q,,α j j (3) qqq21 fi The new eld operators Dqσ and Hqσ satisfy the commutation † − δ δ with relations [Dqσ, Hq′σ′]= iC σ q q′ σ σ′, together with the † † q , , 2 2 2 2 4 2 2 properties Dqσ = D−qσ and Hqσ = H−qσ. The constant Cqσ is ωνωω̃j (1+−pn /pl ) 0 ωpl νωpn pl determined using the Amperè ’s circuital law, which provides g = + j 22 4 2 22 wq 60 ωω̃jj− ̃ ′ ω̃j ()ωω̃j − 0 Cqσ = . 2cβqσ − j′ ≠ j, and VS= l. In the absence of phonon−photon coupling The light matter coupling Hamiltonian Hmat−pt is derived ν using the expression of D(R) in the new basis together with eq ( pn = 0), the two modes j = 1, 2 reduce to the bare phonon ν ≠ 3. While q is a good quantum number due to the in-plane and plasmon. A similar situation occurs for pn 0, in the case − ω ≫ translational invariance, the perpendicular wave vector of the of large plasmon phonon detunings. Both in the high pl ω ω ≪ ω 3D matter modes q is not. In the light−matter coupling 0 and low pl 0 electron density regimes, the hybrid z modes reduce to the bare plasmon and phonon excitations, Hamiltonian Hmat−pt, photon modes with a given q interact and the plasmon contribution prevails in the polarization field with linear superpositions of the 3D matter modes exhibiting ff “ ” eq 3. As explained in the following, when the latter interacts di erent qz. The latter are denoted as quasi-2D bright modes fi fi π σ ∑ α ασ Π α ασ with the photonic displacement eld D, this simply results in and are de ned as q j = qz, f (Q) Qj , where f (Q) stem the formation of SPPs coexisting with bare phonons. The latter from the overlap between the displacement and the polar- ω ω ≫ ω fi are either transverse phonons with frequency 0 for pl 0 ization elds and is determined by imposing the commutation ω̃ ω ≪ π π† δ δ δ or longitudinal phonons with frequency 0 in the case pl relations [ qσj, q′σ′j′]= q,q′ σ,σ′ j,j′.

C DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article

ω ω ω Figure 2. Normalized frequency dispersion wqσζ/ 0 and mode admixtures of the surface polaritons as a function of q/q0 for pl = 1.5 0. (a) = = ν ql0 10. (b) ql0 0.01. Top panels: The SPPs obtained for pn = 0 are depicted as black lines, while the surface lower and upper polaritons ν ω obtained for pn = 0.5 0 are represented as thick red and blue lines, respectively. Resonance between SPPs and phonons (horizontal dotted lines) is indicated by thin vertical lines. The antisymmetric (σ = −) and symmetric (σ = +) modes correspond to the solid and dashed lines, and the light ω ω cone with boundary = qc is represented as a gray-shaded region. Inset: Frequency wqσζ/ 0 of the lower (red line) and upper (blue line) surface ν ω = ω ω polaritons versus pn/ 0 for ql0 10 and pl = 1.5 0. The phonon frequency is depicted as a horizontal dotted line. Bottom panels: Plasmon (gray LP dashed lines), photon (magenta solid lines), and phonon (green dotted lines) admixtures of the antisymmetric lower polariton Wi,qσ = − (i = pl, pt, ω ω ν ν ω pn) as a function of q/q0 for pl = 1.5 0. The top and bottom subpanels correspond to pn = 0 and pn = 0.5 0, respectively.

Using a unitary transformation,62 the matter Hamiltonian equation can be combined with the Helmholtz equation ∑ ℏω̃π† π ϵ ω ω2 2 2 − γ2 60 can be decomposed as Hmat = q,σ,j j qσj qσj + Hdark, where n( ) /c = q n in order to determine these parameters. the second term is the contribution of the “dark” modes that We use a self-consistent algorithm which starts with a given γ are orthogonal to the bright ones and which do not interact with frequency wqσLP, then determine n from the Helmholtz ϵ ϵ photons. Without this contribution, the polariton Hamiltonian equation with d = 1 and cr(wqσLP) given by eq 2 and use γ reads Hpol = ∑ σ /qσ with these n to compute the parameters entering the Hamiltonian q, 62 Hpol. The latter is diagonalized numerically, which allows / =ℏαβ + + ℏ ωππ̃ † qqσσσσcDD()qq−−qσ HHqqσσ∑ jjj qq σσ determining the new wqσLP. The algorithm is applied j independently for the symmetric (σ = +) and antisymmetric σ − −ℏΩ+ππ† ( = ) modes until convergence, which is ensured by the ∑ qjσσ()−qqq j σσ jD fi 60 j discontinuity of both light and matter elds at each interface. We now use this method to study the surface polaritons in ω ω and the vacuum Rabi frequency our system. As an example, we consider the case pl = 1.5 0 with different crystal thickness ql= 10 (Figure 2a) and 2 0 cωpl ql= 0.01 (Figure 2b) and compute the surface polariton Ω= 0 qjσ gj ωj̃ frequencies wqσζ (top panels), as well as their phonon (i = pn), LP ν plasmon (i = pl), and photon (i = pt) admixtures Wi,qσ for pn 2 + γ2 ν ω fi q cr −−22llγ 2 = 0 and pn = 0.5 0 (bottom panels). Precise de nitions of × (1−+ eγcr ) 2lqσγ ecr ( − ) γ cr these quantities are provided in the Supporting Information, cr ω and l and q are both normalized to q0 = 0/c. Considering ℏω typical mid-infrared phonons with 0 = 0.2 eV, the two ■ SURFACE PLASMON−PHONON POLARITONS dimensionless parameters ql= 10 and ql= 0.01 correspond 0 0 fi The Hamiltonian Hpol exhibits three eigenvalues in each to l = 10 μm ( rst case) and l ≈ 10 nm (second case), subspace (q, σ), and only the lowest two (referred to as lower respectively. and upper polaritons) correspond to surface modes (below the In the first case (Figure 2a), the two surface modes at each light cone). At this point, we have built a Hamiltonian theory interface have negligible overlap, and the modes σ± therefore fi γ ν − providing a relation between the eld penetration depths n coincide. For pn = 0, the plasmon photon coupling is ζ and the surface wave frequencies wqσζ, where = LP, UP refers responsible for the appearance of an SPP mode (black line) 0 to the lower (LP) and upper (UP) polaritons. This eigenvalue with frequency wq, which enters in resonance with the phonon

D DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article ≈ ≪ mode at qr 2.2q0. While for q q0 this SPP is mainly the coupling constant 4 does not depend on Q as shown 0 ∼ − 63 composed of light (wq qc), it features a hybrid plasmon theoretically for intramolecular phonons in certain crystals. ∼ − photon character for q q0 and becomes mostly plasmon-like Electron phonon interactions are usually characterized by the 0 λ fi as wq approaches the surface plasmon frequency ωpl/2for q dimensionless coupling parameter , which quanti es the ≫ ν ≠ electron mass renormalization due to the coupling to q0. For pn 0, a splitting between the two polaritons 55 λ fi branches (thick red and blue lines) is clearly visible, and the phonons. At zero temperature, is de ned as latter consist of a mix between phonons, plasmons, and 1 photons in the vicinity of q = q . In the regime q > q , since the λ =ℜ−∂Σ∑ δξ()(KKωω̅ () ω| =0 ) 0 0 N mostly phonon-like LP exhibits a ∼25% plasmon admixture, 3D K this polariton mode can be seen as a “dressed” phonon with ω “ ” where ℜ stands for real part, N = ∑ δξ()= VmKF is the frequency red-shifted from 0.Similarly,the dressed 3D K K 2π 2ℏ plasmon mode (UP) is blue-shifted from the surface mode 3D electron density of states at the Fermi level, and ∼ ∂ Σ̅ω | frequency ωpl/2due its 25% phonon weight. ω K( ) ω=0 denotes the frequency derivative of the retarded Σ̅ ω ω The frequencies of the LP (red line) and UP (blue line) at electron self-energy K( ) evaluated at = 0. An equation of ν ω motion analysis64 of the electron Green’s function (GF) resonance are represented as a function of pn/ 0 in the inset. fi 0 ω † Here, the resonance is de ned by the condition wq = 0, which .KKK()ττ=−⟨ic () c (0)⟩ allows to write the electron self- ≈ provides qr 2.2q0. We observe that the polariton splitting energy as − ω wqσUP wqσLP is symmetric with respect to wqσζ = 0 (black ν ≪ ω dω′ dotted line) for pn 0 and becomes slightly asymmetric in Σ ()ω =||i 4.2 ()()ωω+′B ω ′ 12 ν K ∑ ∫ KQ− Qα the ultrastrong coupling regime, when pn is a non-negligible 2π ω Q,α (5) fraction of 0. Furthermore, the splitting is found to decrease ω rapidly as the electronic plasma frequency pl is increased. where B ()ωττ=−id∫ eiωτ ⟨)) () (0) ⟩is the phonon In the second case ql= 0.01 (Figure 2b), the crystal is Qα QQαα− 0 ) =+† thinner than the penetration depth γ of the surface waves at GF written in the frequency domain, and QQααBB− Qα. cr − ν the two interfaces, and the latter overlap. This results in two In the absence of phonon photon coupling ( pn = 0), there is σ ± ff ν no hybridization between phonons and plasmons, nor is there sets of modes = with di erent frequencies. For pn = 0, the 0 symmetric and antisymmetric SPPs with frequency w σ ± are coupling to photons. Phonons therefore enter the Hamiltonian q = ∑ ℏω † represented as black dashed and solid lines, respectively. The Hpol only in the free contribution Hpn = Q,α 0BQαBQα.In resonance between the symmetric SPP and the phonon mode this case, the equation of motion analysis simply provides ≈ 2 2 occurs for q q0, in the regime where the symmetric SPP is BQα()ωωωω=− 20 /(0 ). Using this expression together 0 ∼ mostly photon-like with wqσ=+ qc. Interestingly, the 0 with the noninteracting electron GF .K()ωωξ=− 1/(K )in resonance between the antisymmetric SPP and the phonon − ≈ eq 5, one can compute the electron phonon coupling mode is now shifted to a large wave vector qr 220q0, where parameter for ν =0as62 the antisymmetric SPP exhibits a pure plasmonic character. For pn → ∞ σ ± q/q0 , the two SPPs = converge to the surface 2 2 2 ||4 2 ||4 N3D plasmon frequency ω /2. For ν ≠ 0, the antisymmetric LP λ0 = ∑∑δξ()KQ δξ (K− )= pl pn N ω ω (thick red solid line) and UP (thick blue solid line) are split in 3D 0 KQ 0 ≈ the vicinity of the resonance qr 220q0, while the symmetric In the presence of phonon−photon coupling (ν ≠ 0), the polaritons (thick colored dashed lines) do not feature any pn ∼ phonon dynamics is governed by the Hamiltonian Hpol, which anticrossing behavior for q q0. Similarly to that in (a), the σ ± includes the coupling of phonons to plasmons and photons. It LPs = can be seen as dressed phonon modes with is therefore convenient to express the 3D phonon operators ω † † frequencies red-shifted from 0 for large q due to their Π Π BQα and BQα in terms of the 3D hybrid modes Qαj and Qαj plasmon admixtures. Here, however, these dressed phonons and then project the latter onto the quasi-2D bright and dark can propagate with very large wave vectors comparable to the modes such that the electron−phonon Hamiltonian eq 4 takes electronic Fermi wave vector. ∑ ℏξ † (B) (D) the form Hel−pn = K KcKcK + Hel−pn + Hel−pn.The contribution of the bright modes reads ■ ELECTRON−PHONON SCATTERING (B) * † † We now show that the ability to tune the energy and wave Hel− pn =ℏ∑∑ 4ηππjfQccασ ()KK−− Q ( qσσjj+ q ) vector of the dressed phonons can affect the electron−phonon KQ,,,ασj scattering in the crystal. The coupling Hamiltonian between (D) individual electrons (electronic dark modes) and phonons while the contribution Hel−pn of the dark modes that do not reads62 interact with photons is given in the Supporting Information. ωω0 / j̃ νωpn pl ††† Here, η = χ where χ = is associated jjωω̃ 42+ χ j ωω̃2 − 2 Hel− pn =ℏ∑∑ξKccKK + ℏ4 ccKK−− Q() B Qαα + B Q (/)pl j j j 0 K KQ,,α with the hybrid mode weights of the phonons. The electron (4) self-energy due to the interaction with bright modes reads † The Fermionic operators cK and cK annihilate and create an dω′ electron with wave vector K and energy ℏξ (K ≡ | K|) relative Σ=(B)()ωη∑ ifQ22 |4. ( ) | ()ωω+′ K K j ασ ∫ π KQ− to the Fermi energy. For simplicity, we assume a 3D spherical Q,,,ασj 2 ξ ℏ 2 − 2 Fermi surface providing K = (K KF)/(2m), with m the ff Bqjσ ()ω′ (6) electron e ective mass and KF the Fermi wave vector, and that

E DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article

φ σ ν ω = ω ω σ ω Figure 3. (a) Function q versus q/q0 for pn/ 0 = 0.5, ql0 0.01, and pl = 1.5 0. Inset: Normalized frequency dispersion of the LPs wq LP/ 0 corresponding to the red solid and dashed lines in Figure 2b. The contributions of symmetric (σ = +) and antisymmetric (σ = −) modes are depicted as dashed and solid lines, respectively. (b) Relative enhancement of the electron−phonon coupling parameter Δλ given by eq 7 versus ql λ0 0 ν ω ω ω 3 and pn/ 0, for pl = 1.5 0 and KF =10q0.

iωτ ϒ π for q ≲ q . Note that for ν /ω → 0, the LPs for q > q are where Bqjσ ()ωττ=−id∫ e ⟨ϒqjσσ () ϒ− qj (0)⟩ and qσj = qσj 0 pn 0 r † fully phonon-like with w σ → ω , and one finds that φ σ → 1 + π− σ . As detailed in the Supporting Information, we now use q LP 0 q q j Δλ φ the equation of motion theory to calculate the GF B ()ω . For does not contribute to in this region. For q < qr, qσ does qjσ λ0 ν 22 Δλ pn = 0, one simply obtains Bqjσ ()ωωωω= 2 j̃ /(− ̃ j ), which not contribute to because of the vanishing phonon weight λ0 provides B ()ωω= B ()since ω̃= ω (and ω̃= ω ). In ν ω → qQσα1 1 0 2 pl of the LP for pn/ 0 0. η η this case, 1 =1, 2 = 0, and only the pure phonon term j =1 The relative enhancement of the electron−phonon coupling Σ(B) Δλ therefore contributes to K . We then calculate Bqjσ ()ω and is represented in Figure 3b as a function of q l and ν /ω . λ 0 pn 0 ν ≠ − 0 the self-energy eq 6 for pn 0 and decompose the electron fi ≳ ν ≳ ω phonon coupling parameter into its bright and dark mode We nd that it can reach large values 10% for pn 0.4 0 λ λ(B) λ(D) ν ≠ λ λ(B) λ(D) and l sufficiently small (ql≈ 0.01). The enhancement of Δλ contributions: = + for pn 0, and 0 = 0 + 0 0 λ ν λ(B) λ(B) ff 0 for pn = 0. The contributions and 0 are di erent when decreasing the crystal thickness l can be understood by because bright modes interact with photons for ν ≠ 0, while 2 pn realizing that the function |fασ(Q)| entering eq 7 scales as they do not for ν = 0. Similarly, λ(D) and λ(D) are a priori pn 0 ∼1/(γ l), as shown in the Supporting Information. This stems different since dark modes are hybrid plasmon−phonon modes cr ν ≠ from the fact that the dressing of 3D phonons by 2D for pn 0, while they reduce to bare phonons and plasmons ν for pn = 0. However, we show in the Supporting Information evanescent waves decaying exponentially at each interface λ(D) λ(D) that = 0 , which means that, at the level of the RPA, the increases when decreasing the crystal thickness. However, hybridization between plasmons and phonons cannot solely when the crystal thickness becomes too small (e.g., when fi − fi lead to a modi cation of the electron phonon scattering. KlF ≲ 1), the description of the matter excitations by 3D elds Finally, the relative enhancement of the electron−phonon breaks down, and quantum confinement effects have to be coupling parameter reads62 taken into account in the model. The parameters of Figure 3b are l = 10 nm for ℏω = 0.2 eV, and K =1nm−1.Wefind that Δλ λλ− 0 F ≡ 0 the enhancement of the electron−phonon coupling strongly λ λ 0 0 increases with the ratio q0/KF, i.e., when the mismatch between 1 the typical photonic and electronic wave vectors is reduced. =−||∑∑δξ() ( φ 1)()fQ2 δξ ( ) 2 K qσ ασ KQ− This can occur in materials with larger phonon frequency and/ N3D KQ,,ασ or lower electron density. In the latter case, however, (7) decreasing KF requires considering larger l for the 3D φ where the function qσ is given in the Supporting Information. description of the matter fields to be valid, which in turn The latter describes the renormalization of the phonon energy leads to a reduction of Δλ . due to the coupling to plasmons and photons and depends on λ0 65 the polariton frequencies wqσζ. It is represented in Figure 3a For instance, bilayer graphene with l ≈ 0.4nm exhibits an ω ℏω 42 together with the frequency dispersion of the LPs wqσLP/ 0 infrared-active phonon mode at 0 = 0.2 eV and has been corresponding to the red solid and dashed lines in Figure 2b. recently shown to feature superconductivity at low carrier fi φ ≈ × 12 −2 First, we nd that the contributions of the LPs to qσ largely densities 2 10 cm when the two sheets of graphene are dominate over the contributions of the other polaritons. twisted relative to each other by a small angle.52 In this regime, φ −4 Second, we observe that qσ is anticorrelated with the we find that ω ≈ ω , ql≈×710, K ≈ 1.6 × 103q , and fi pl 0 0 F 0 dispersion of the LPs, whenever the latter exhibit a nite − ν ≈ ω 42 φ the large phonon photon coupling strengths pn 0.25 0 phonon weight. For instance, qσ=− reaches large values in the ≲ allow reaching the ultrastrong coupling regime. Using these region q q0 where the frequency of the antisymmetric LP ≈ ν ω parameters, we find that the contribution of surface plasmon− ( 35% phonon for pn = 0.5 0) is far away from the bare ≪ ω phonon polaritons to the electron−phonon scattering is phonon frequency. In contrast, while wqσ=+LP 0 in this φ ≈ Δλ region, qσ=+ 1 since the symmetric LP is fully photon-like expected to reach a very large value, ≈ 2.7. However, the λ0 F DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX ACS Photonics Article small thickness of bilayer graphene such that KlF ∼ 1 in this Thomas W. Ebbesen: 0000-0002-3999-1636 fi ff weakly doped regime implies that quantum con nement e ects Notes become important. Furthermore, the failure of the 3D model fi ω → → The authors declare no competing nancial interest. assuming pl const as q 0 can be seen in the dispersion of the optical plasmon mode in bilayer graphene,58 which is → ■ ACKNOWLEDGMENTS known to be ωpl ∼ q as q 0. We gratefully acknowledge discussions with M. Antezza, T. ■ CONCLUSION Chervy, V. Galitski, Y. Laplace, Stefan Schütz, A. Thomas, and “ In summary, we have carried out a nonperturbative quantum Y. Todorov. This work is supported by the ANR - ERA-NET ” “ ” theory of intrinsic, lossless plasmon−phonon polaritons valid QuantERA - Projet RouTe (ANR-18-QUAN-0005-01) and for arbitrary large coupling strengths and discussed the IdEx Unistra - Projet “STEMQuS”. different regimes of interest obtained by tuning the crystal thickness and the phonon−photon coupling strength. This ■ REFERENCES theory allows characterizing the hybridization between (1) Schlawin, F.; Cavalleri, A.; Jaksch, D. Cavity-mediated electron- plasmons and phonons in the crystal, which cannot lead solely fi − photon superconductivity. ArXiv e-prints, 2018. to a modi cation of the electron phonon scattering at the (2) Curtis, J. B.; Raines, Z. M.; Allocca, A. A.; Hafezi, M.; Galitski, V. level of the RPA. 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I DOI: 10.1021/acsphotonics.9b00268 ACS Photonics XXXX, XXX, XXX−XXX