Nanophotonics 2019; 8(4): 629–639

Research article

Jacob B. Khurgin* Pliable : Wannier - coupling in metal-semiconductor structures https://doi.org/10.1515/nanoph-2018-0166 the vacuum fluctuations of the electromagnetic field and Received October 3, 2018; revised October 25, 2018; accepted the matter, quantified by the Rabi energy ħΩ. When the October 26, 2018 Rabi energy exceeds the dissipation rates of both photons and , the exciton-photon system is said to enter Abstract: Plasmonic structures are known to support the so-called strong coupling regime [7] and polaritons the modes with sub-wavelength volumes in which the get formed, as manifested by characteristic Rabi split- field/matter interactions are greatly enhanced. Coupling ting. Furthermore, a number of fascinating phenomena between the molecular excitations and leading take place, most notably the Bose-Einstein condensa- to the formation of “plexcitons” has been investigated for tion and lasing. Satisfying the strong coupling a number of organic molecules. However, plasmon-exci- criterion is far from being easy, and it is only in the last ton coupling in metal/semiconductor structures has not 20 years that the progress in microcavity fabrication and experienced the same degree of attention. In this work, we growth of high-quality III–V semiconductors has led to show that the “very strong coupling” regime in which the the observation of large Rabi splitting, exciton condensa- Rabi energy exceeds the exciton binding energy is attain- tion [8], and polariton lasing in multiple quantum wells able in semiconductor-cladded plasmonic nanoparticles (MQW) GaAs [9]- or GaN [10]-based structures. The term and leads to the formation of Wannier exciton-plasmon ultrastrong coupling (USC) had also been introduced to polariton (WEPP), which is bound to the metal nanopar- describe the situation where the coupling (Rabi) energy ticle and characterized by dramatically smaller (by a fac- becomes comparable to the photon energy itself, meaning tor of a few) excitonic radius and correspondingly higher that the coupling changes the electronic structure of the ionization energy. This higher ionization energy, which material itself. This regime is difficult to achieve in the exceeding approaches 100 meV for the CdS/Ag structures, optical range of frequencies,­ but it can be feasible with may make room-temperature Bose-Einstein condensa- THz ­radiation [11, 12]. tion and polariton lasing in plasmonic/semiconductor At the same time, with the MQW exciton-polariton in ­structures possible. the semiconductor cavity, another intermediate regime of Keywords: exciton; plasmon; plexciton; polariton. the so-called very strong coupling (VSC) exists in which the Rabi energy ħΩ is much less than the photon energy ħω but it is comparable or larger than the exciton binding

energy EX. Then as shown in ref. [13], one can no longer 1 Introduction consider coupling as a weak perturbation to the “rigid” exciton – the exciton in cavity polariton becomes flexible Polaritons [1] are bosonic quasi-particles formed by or, better said, “pliable” as its radius decreases for the photons and matter excitations such as phonons [2], lower polariton and increases for the upper one, causing excitons [3, 4], intersubband transitions [5], or others. increase of the effective binding energy in the lower Polaritons can be formed in a relatively unconstrained polariton. The simple semi-analytical results of Khurgin geometry (“free space”), in the waveguides [6], or in the [13] were confirmed by a rigorous Green function analy- fully confining resonant structures, as in the case of the sis in ref. [14] and, after numerous attempts [15, 16], were widely studied exciton-photon polaritons in microcavi- ­experimentally verified in ref. [17]. ties [4]. Polaritons are enabled by the coupling between Microcavities used in all the above experiments typi- cally provide confinement in only one dimension, which

*Corresponding author: Jacob B. Khurgin, Johns Hopkins University, limits the strength of vacuum field and which is inversely Baltimore, MD 21218, USA, e-mail: [email protected]. proportional to the square root of the cavity volume. Even https://orcid.org/0000-0003-0725-8736 three-dimensional all-dielectric cavities have volumes

Open Access. © 2018 Jacob B. Khurgin, published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License. 630 J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling larger than (λ/n)3. Therefore, in recent years, the atten- that the VSC regime can be rather easily reached. In this tion of the polaritonic community had been turned in the VSC regime, WEPP becomes quite flexible or pliable, as direction of plasmonic structures, which enable localized the excitonic radius decreases at least threefold and the surface plasmon (LSP) modes whose volume is far below WEPP becomes bound to the Ag nanoparticle, forming a the diffraction limit (V < <(λ/n)3) [18] and in which elec- shell with the thickness of a few nanometers. tromagnetic field can be greatly enhanced in the vicinity of resonance. For this reason, LSPs are widely used (or, at least, on their way to being used) in a variety of sensing [19], spectroscopic [20], and other applications [21]. 2 Theory of WEPP Obviously, vacuum field strength in LSP is also greatly enhanced, which means that the coupling between LSP The Hamiltonian of the system consisting of a localized and the medium placed in close proximity to it also gets surface plasmon (LSP) and collective excitation of elec- stronger, which results in the strong Purcell enhance- tron-hole pairs with momenta ke and kh forming Wannier ment of spontaneous radiation rate. Once the Rabi energy exciton can be written as follows [36]: exceeds the damping (broadening) rate of LSP Γ, a new 1 ˆ =+ ω ˆˆ††ˆˆ− ˆˆ† type of quasiparticle – plasmon-exciton-polariton or H plaa ∑∑Ekk, bbkk, kk, Vbc,,kk− kk bkk, 2 eh eh eh eh eh eh kkeh, kkeh, “plexciton” – is formed [22, 23]. Plexcitons had been ˆˆ†† +−iHint,()abˆˆkk abkk, , observed with a variety of metals [24, 25] in the shape of ∑ eh eh kkeh, both individual nanoparticles/nanoantennas and in their arrays [26]. Strong coupling has led to such exotic behav- where aaˆˆ† , , and bbˆˆ† , are the creation-annihilation kkeh, kkeh, ior as plexcitonic lasing [27] and condensation [28, 29]. operators for the plasmons with energy ħωpl and - There had been reports of USC regime [30] with Rabi split- hole pairs with energies E, Ve=−22/ εε||kk is kkeh, cs, kkeh− 0 eh ting exceeding hundreds of meV [31, 32]. the Coulomb attraction energy, εs is the static dielectric

Most of the plexciton research had been performed permittivity, and Hint is the interaction energy between with various types of organic and other molecules in which the plasmons electric field and the electron hole pair. The the exciton is essentially a local excitation localized on a WEPP state can be described as follows: scale of a few atoms and capable of moving around in the P =+αβ1, 0 0, 1 , (1) medium. One can easily model these excitons as isolated αβ, pl ex pl ex two-level entities that get coupled via interaction with LSP cavity and thus form plexcitons. The spatial characteris- where α and β are the relative weight of plasmon and tics of excitons do not change – one can say that these exciton (Hopfield coefficients [37]), the ket vectors refer excitons, usually referred to as Frenkel excitons [33, 34] in to the states with 1(0) plasmons and 0(1) excitons, the condensed matter science, remain “rigid”. But in an inor- plasmon (or rather LSP) is described by its electric field ganic semiconductor, medium excitons are formed from E(R), and the Wannier exciton, by its wavefunction Фex free carriers in the conduction and valence band attracted (R, ρ), which can be approximated by the product of to each other; their wavefunctions are usually spaced normalized wavefunctions for the center of mass (COM) over many lattice sites, and they are usually referred to as motion Ф(R) and the relative motion of the electron and Wannier excitons [35]. The radius of Wannier excitons can hole Ψ(ρ). Here the relative electron-hole coordinate is vary depending on the environment, and as was shown ρ = re–rh and the COM coordinate is R = (mere–mhrh)/M, in ref. [13], it can change in the VSC regime. One can where M = me + mh. say that unlike rigid Frenkel excitons, Wannier excitons The total energy of the WEPP can then be written as are quite pliable. Furthermore, due to their lower mass, follows: Wannier excitons can travel through the medium easier EE==PHˆ P αω22 ++ββ22EE+ β than Frenkel excitons do, and they can also be localized in WEPP αβ, αβ, pl gap cm eh (2) −− βα2E2βΩ , the region where their potential energy is lower, forming C bound excitons.

In this paper, we consider the interaction between where Egap is the bandgap energy, 2*2 Wannier excitons in the semiconductor and LSPs in metal E/cm =− 2(M ΦΦRR)(∇ ) is the kinetic energy of the 2*2 nanoparticles to form a quasiparticle of a different flavor COM motion in exciton, E/eh =− 2(mr ΨΨρ)(∇ ρ) – Wannier exciton-plasmon polariton (WEPP). Using is the kinetic energy of the relative electron and −−11−1 the example of WEPP in an Ag/CdS structure, we show hole motion, mmre=+()mh is the reduced mass, J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling 631

2* −1 E(Cs=−e Ψρρ)(Ψπρ)/4 εε0 is the Coulomb energy of field and the creation/annihilation operators [41] for the electron-hole attraction, and ħΩ is the strength of exciton- modes in vacuum, with the only difference being the defi- plasmon coupling (Ω is a Rabi frequency). The strength of nition of the mode volume (5). the coupling can be found as follows: Substitute (5) into (3) and obtain the following:  1 1/2 1/2 22 ΩΨ=⋅(0)(Φ RE)(μcv vac RR),d (3)  ∫ ωpl e 2 ΩΨ=⋅(0)(Φ )(μˆ ˆ ),d (6)  ∫ Recv RR 22ε VmE 0gpl e ap where a factor of ½ accounts for the fact that only the pos- itive frequencies component of the “vacuum” electric field where μˆ is the unit vector collinear with the momentum E ~ cosω t is responsible for the coupling between the cv vac pl of the interband transition. Integration over the volume states and μ is the matrix element of the dipole moment cv involves averaging over the direction so the mean value between the valence and conduction bands. It can be of the projection of μˆ onto the direction of electric field related to the interband matrix element of the momentum cv is 1/ 3. Then we introduce the effective overlap between P as μ = eħP /m E , where E is the bandgap energy, cv cv cv 0 gap gap the plasmon electric field and the exciton wavefunction and then using the relation between P and the conduc- cv κΦ=Ve1/2 ()ˆ()d 22 as pl ∫ RRR and finally obtain (assuming tion band effective mass, mm0g/2ec= P v /E ap [38] to obtain 2 2 1/2 ħωpl ~ Egap) the following: μcv = (e ħ /2meEgap) . 1/2 22 Finding the “vacuum field” Evac (R) of the LSP mode 1 e responsible for vacuum Rabi oscillations and splitting are Ωκ= Ψ(0). (7) 23meε0 different from the case of dielectric microcavities, where the total energy is equally split between electric and mag- Finally, we can normalize the dimensions to netic energies. In subwavelength LSP modes with charac- the free exciton radius a = 4πε ε ħ2/m e2 and the teristic size s < < λ, the magnetic energy U is very small in 0 0 s r eff H energy to the binding (Rydberg) energy of 1S exciton comparison to the electric energy U , roughly U ~ (2πs /λ)2 E H eff E/= me4232πε 22ε2 , where ε is the static dielectric con- U , and can be neglected. Then the energy of one LSP Rr 0 s s E 1 stant of the semiconductor and obtain the following: can be found as ωε=∂((ωε )/∂ω)|Ed()RR|,2 pl 4 0v∫ r ac where ε is the co-ordinate-dependent relative per- 1/2 r 2π1/2 m = 3/2 r mittivity at optical frequencies. Furthermore, in the ΩεE(Rsa0 κΨ 0) (8) 3 me metal described by the Drude relative permittivity 22 22 εωmp()=−1/ωω, ∂∂()ωεmp//ωω≡ ω and as shown in refs. [39, 40], in the absence of the magnetic field, and also exactly one-half of the total energy of the plasmon is con- tained in the form of kinetic energy of the free , 2* 2 EEeh =− Ra0 ΨΨ()ρ ∇ ()ρ 2 11 22 ==εωω 2* 2 UUKE0v(/p )(Edac RR). It follows then EE=− amΦΦ()∇ () /.M (9) 24∫metal  cm Rr0 RR *1− 1 2 E2=− E(a Ψρρ)(Ψ ρ) that ωε=−(1 ε )(EdRR) . Let us normalize CR0 pl 2 0v∫metal m ac the field of the plasmonic mode to the field at some spe- The wavefunction for the relative electron-hole motion is cific point, for example, point R , where the electric field 0 that of the usual lowest hydrogen state Ψ(ρ) = exp(–ρ/a)/ is at maximum, eEˆ()RR= ()/(ER), and then we immedi- 0 π1/2 a3/2. Therefore, if all the dimensions are normalized to ately obtain the following: the Bohr radius of exciton and the energy, to that of the 1/2  1 ωpl binding energy, we obtain the following: (4) ERvac()= eRˆ(), 22ε V 22 0 pl ββ E(=−α∆22βΦ*2RR)(∇+Φ )/mM − 2 WEPP pl r a2 a where the “effective volume” of the plasmon is as follows: 1/2 (10) 4 m κ 2 − αβ ε r , Ve=−(1 ε )(ˆ RR).d (5) s 3/2 pl ∫metal m 3 me a

ˆ Note that the coefficient in front of eR() in ref. (4) is where ∆ωpl =−(E pl gap )/ER and the energy EWEPP is now precisely the coefficient relating operator of the electric measured relative to the bandgap. 632 J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling

3 WEPP bound to a metal Then the angular integral in the expression for к is 1 2(πθedˆ )(Φθ)sin()θθ= 2π and κε=+6/(1 2)Fκ , nanosphere ∫−1 θ s, opt where F = RR3/21Φ ()Rd− R. We now find the kinetic To continue the analysis, it is necessary at this point to κ 0 ∫ R energy of the angular motion as follows: choose a particular geometry. In order to facilitate carrying 2 on of analytical derivations for as long as possible before E(kin,θθR)(=∇Φθ)(θθΦθ) finally switching to numerical calculations, we consider the 11∂∂1 π (12) ==Φθ() Φθ() . most simple example of a spherical metal nanoparticle with θθR2 sinsθθ∂∂inθθ 2 23R radius R0, as shown in Figure 1A. These nanospheres are capable of supporting a strong dipole surface plasmon mode Next, the radial wavefunction must be chosen, and it whose electric field magnitude relative to the maximum must satisfy the boundary condition ФR(R0) = 0, hence, an field just outside the nanoparticle (R = R0) is as follows: appropriate radiation wavefunction can be as follows:

1  23 −− θ +> 2 ()RR0 /d 203cos 1(RR/) RR0 Φ =− eˆ()R =  . R()RR3/22 21/2 ()Re0 , (13) 1 (11) ++ RR< dR(300Rd 3)d  2 0

where d is the distance of the peak from the nanosphere The condition for the surface plasmon resonance surface, to which we can refer as polariton shell thickness. is ε (ω ) = –2ε , where ε is the relative dielectric m pl s,opt s,opt This wavefunction is plotted in the curve in Figure 2 ­permittivity of semiconductor cladding at optical frequen- next to the normalized electric field distribution, while cies whose dispersion (not counting the exciton) is insig- 2 the total COM probability density ΦΦ()R ()θ is shown nificant over the range of frequencies considered here. R θ 1 in Figure 1A and B. Now we calculate the all-important Therefore, from (5), VR=+πε3(1 2). To avoid the pl 3 0,s opt overlap integral in the following: surface recombination of excitons, the wide gap spacer 2 of a few Angstrom thickness must surround the metal, as Fκ(/dR0 ) = 1/2 ++23 (/dR00)3(/dR)3(/dR0 ) shown in Figure 1A. This will cause some distortions of the (14) electric field due to the difference between the permittivi- Rd/ R 1 − exp(0 0 Ei Rd/), ties, but this difference can be ignored in the present order d 10  of magnitude estimate. Next we consider the COM wavefunction of the ∞ where Eiz()=−exp( tz)/tdt is the exponential integral. bound exciton Ф(R) = Ф (R)Ф (θ). To start, we assume 1 ∫1 R θ The function is shown in Figure 3A, and it reaches the that the angular function of the exciton is the same maximum value of roughly 0.5 for d ~ 0.22R . 1 0 as the field, Φθ()=+3cos2 θπ1/ 2, where (2π)−1/2 θ 2 Now all that is left is to calculate the kinetic energy *2 2 stands for ­normalization in the angular domain. term ΦΦRR()RR−∇ ()= Fdkin0(/RR)/ 0 , where

A B 1 2

, θ )| 0.5 R

2 |Φ (R,θ)| | Φ ( 0 R e 2 d 0 a R 1 0 h 0 y/R0 –1 2 1 0 –2 –1 –2 x/R0

Figure 1: Explanation of Wannier exciton localization on the metal nanosphere of radius R0. (A) Plot of |Φ(R, θ) | 2 – the probability density of the exciton center of mass. Also shown is a the excitonic radius. (B) Three-dimensional representation of center of mass probability density around metal nanosphere. J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling 633

0.08 three terms in (16), where the positive kinetic energy −2 −1 0.07 eˆ(R) term ~a is counteracted by the Coulomb term ~a and Rabi splitting ~a−3/2. 0.06

0.05

0.04 d ΦR(R) d 0.03 4 Material choice and background 0.02 oscillation strength 0.01 Ag CdS We shall now consider a choice of materials. The metal 0 10 15 20 25 30 sphere can be made of silver, and among the semicon- Distance from the center (nm) ductors, cadmium sulfide (CdS) appears to be a good can-

Figure 2: Overlap between the electric field distribution of the didate because it supports robust free Wannier excitons [42, 43] with a radius of a ~ plasmon mode eRˆ() and center-of-mass exciton wavefunction ФR (R) 0 3 nm and binding energy of of the WEPP. ER = 30 meV observable at room temperature. Further- more, a surface plasmon supported by a small Ag nano-

sphere surrounded by CdS has an energy of ħωpl ~ 2.30 eV, 2 (/Rd)2++(/Rd)/π 23 which places it in the region where intrinsic losses in F (/dR) = 00 (15) kin0 3(dR/)2 ++3(dR/)1 silver are small and also very close to the bandgap exciton 00 energy 2.42 eV. It means that detuning Δpl can be always is shown in Figure 3B. Now, the equation for the total adjusted within a fairly broad range of positive and nega- lower WEPP energy follows from (10), tive values by either small variations in the Ag nano- ββ22m β2 particle shape (making it slightly oblong) or changing =+2 r +− E(Ppα∆l 22Fdkin0/)R 2 semiconductor composition by introducing small fraction Ra0 Ma (16) of selenium [44] to shift the bandgap energy of CdS1−xSex 4αβ m 2ε − rsFd( /)R . slightly downward. a3/2 m 12+ ε κ 0 es,opt Interestingly, given the relevant material characteris- 0 tics of CdS: me = 0.2m0, mv = 0.8m0, εs = 8.9, and εs,opt = 6.35,

Clearly, for large nanosphere radius R0, the exciton- the square root term in (16) is equal to 1.02, i.e. very close polariton will probably localize within the thin shell to unity. In fact, this term is not that far from unity for any d ~ 0.22R0, but for smaller radii, the COM kinetic energy polar semiconductor in which εs > εs,opt and me < < mv (it is 2 term, inversely proportional to R0 , may become too large 1.1 for GaN, for instance, and 1.01 for GaAs). and it will force the polariton shell thickness to increase If we compare (16) with the corresponding equa- and thus reduce the coupling strength ħΩ. For the exci- tion for VSC in the microresonators incorporating QWs tonic radius, it is determined by the interplay of the last [13], two major differences can be identified. First, one is

A B 0.6 3

kin 10 F

0.5 102 κ F 0.4 101

0.3 100 Overlap

0.2 10–1

–2

0.1 Center of mass kinetic energy 10 0 0.5 1 1.5 2 2.5 00.5 11.5 22.5 Relative polariton thickness d/R 0 Relative polariton thickness d/R0

Figure 3: Dependences of (A) the overlap between the exciton and plasmon Fκ and (B) center of mass kinetic energy Fkin on relative thickness of the polariton shell d/R0. 634 J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling

that the non-uniform electric field in the plasmon mode 103 causes exciton-polariton to become bound and intro- duces the third (in addition to Hopfield coefficients and 102 exciton radius) variational parameter – shell thickness d. The second difference is the three-dimensional character of exciton, which is manifested in the presence of a−3/2 in 101 place of a−1 in the last term in (16). This stronger depend- ence of oscillator strength and Rabi energy tends to reduce 100 the excitonic radius in CdS WEPP from an already small value of 2.8 nm by at least an order of magnitude, where –1 it practically becomes a Frenkel exciton with a very large Plasmon resonance frequency shift (meV) 10 0123456 oscillator strength [45]. But this giant oscillator strength Exciton radius (nm) of the exciton with a small radius does not appear from Figure 4: Upward shift of the “undressed LSP resonance” as a nowhere. As the exciton wavefunction is simply a super- function of the exciton radius a. Shaded region indicates the range position of the free carrier states in valence and conduc- of excitonic radii at which the upward shift of LSP precludes its tion bands and as the oscillator strength of exciton grows effective coupling to the exciton. with the reduction of its radius, the sum of the oscillator strengths of continuum interband transitions decreases. And it is these transitions that are primarily responsible plasmon. This is to be expected as (as will be discussed for the susceptibility εopt – 1 at optical frequencies. There- below) the formation of polariton is essentially a classical fore, when the radius of the exciton approaches the radius phenomenon. From this point of view, the “mobile” polar- of the unit cell, r0, essentially the entire oscillator strength izable medium is attracted towards the electromagnetic of the interband transitions gets transferred to the exciton mode (which is manifested by the phenomena taking and the susceptibility decreases. The reduction of “back- place in optical tweezers [46] and multitude of micro opto- ground” optical dielectric constant can be approximated mechanical devices [47]). Therefore, the easily polarizable 00 33 as εεss,opt ()ar=−,opt (1εs,opt − )/0 a . This ensures that the and very mobile exciton is naturally attracted towards the same electron-hole pair transitions are not counted plasmon mode. At the same, if the polarizable medium is twice – first, as the free carrier transition and then again not mobile, then it is the electromagnetic mode itself that as constituents of excitonic transition. This reduction will tends to gravitate towards the more polarizable medium cause a small change in the coupling strength term under – hence, all the waveguiding and other light-confining the square root in (16), but the main effect of it will be in phenomena in dielectrics. It is only natural then to expect the upward shift of the “undressed”, i.e. in absence of that the shape and energy of the plasmon mode will also exciton frequency of the plasmonic mode, which can be change in the presence of polarizable medium. Interest- found as follows: ingly, in most situations, one side of the light-polariza-

−−ble medium interaction easily dominates the other – in ∆∆()aa=+01 ωε[(12+−())(/2 12+ ε01)]/2 . (17) pl pl ps,opt s,opt optical tweezers, for instance, the micro-particles move while the optical mode stays largely undistorted. In the As shown in Figure 4, the reduction of exciton radius optical waveguides, the situation is exactly the opposite means that the oscillator strength of the free carrier inter- – the light “moves” while the medium stays put. What band transitions gets reduced, which causes the upward makes the case of WEPP special is the fact that both shift of the “undressed” plasmon, which prevents cou- optical mode and the exciton move and neither motion pling with the exciton into polariton mode. While the should be neglected. exact form of εopt (a) dependence is difficult to obtain, our approximation works really well in ensuring that the exciton radius can only approach the unit cell size but never get smaller than as that indicated by the shaded 5 Results region in Figure 4. Introduction of dependence of plasmon resonance on We can now perform a minimization of plasmon-exciton- excitonic radius means that VSC between the exciton and polariton energy (16) over three variational parameters, plasmon not only affects the exciton energy and shape α, d, and a, for different nanoparticle sizes R0 and detun- but also (albeit less strongly) the energy and shape of ing between the bandgap (in the absence of exciton) and J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling 635

0 plasmon mode ∆ pl . Note that unlike the typical dispersion ZnxCd1−xS [48] for positive detuning and CdS1−xSex [44] for curves for cavity polaritons [3, 4] where change in detun- negative ones, with the alloy fraction x not exceeding ing is achieved simply by changing the in-plane kinetic 30%. Of course, for experimental purposes, one should energy of COM motion (i.e. the angle of incidence) in real consider only a few compositions where the Rabi splitting time, for bound exciton-polaritons, each data point cor- is at a maximum and dressed exciton radius is small, as responds to the change of either semiconductor composi- shown in the results displayed in Figures 5–7. tion or the aspect ratio of Ag nanoparticle. The entire span First, in Figure 5, the plasmon and exciton weights of detunings of ±200 meV can be handled by growing |α|2 and |β|2 in the WEPP state are shown for three

ABC 1 1 1 R = 2.5 nm 2 R = 10 nm 2 R = 5 nm 0 0.8 α 0 0.8 α 0 0.8 α2 0.6 0.6 0.6

0.4 0.4 0.4 β2 0.2 0.2 0.2 Polariton composition Polariton composition Polariton composition 2 β2 β 0 0 0 –200 –150 –100 –500 50 100150 200 –200 –150 –100 –500 50 100 150 200 –200 –150 –100 –50 050 100 150 200 Detuning (meV) Detuning (meV) Detuning (meV)

Figure 5: Polariton composition (α2 is plasmonic and β2 is excitonic weight) versus the energy of the “undressed” surface plasmon relative to the bandgap energy for three different radii of the Ag nanosphere.

(A) R0 = 10 nm, (B) R0 = 5 nm and (C) R0 = 2.5 nm.

A B C 200 200 200

150 R0 = 10 nm 150 R0 = 5 nm 150 R0 = 2.5 nm 100 100 100 Upper rigid polariton Upper rigid polariton 50 50 50 Upper rigid polariton 0 0 0 –50 –50 –50 ∆hΩ Lower rigid polariton –100 Lower rigid polariton ∆hΩ Lower rigid polariton –100 –100 ∆hΩ –150 –150 Polariton energy (meV) Polariton energy (meV)

Polariton energy (meV) –150 –200 –200 –200 –200 –150 –100 –50 050 100 150 200 –200 –150 –100 –50 050 100 150 200 –200 –150 –100 –50 050 100 150 200 Detuning (meV) Detuning (meV) Detuning (meV)

Figure 6: Lower polariton dispersion versus the energy of the “undressed” surface plasmon relative to the bandgap energy for three different radii of the Ag nanosphere.

(A) R0 = 10 nm, (B) R0 = 5 nm and (C) R0 = 2.5 nm. Dashed lines show polariton dispersion under assumption of “rigid” exciton with unchanged radius. The energies are all relative to the bandgap energy.

A B 5 2.5 4.5 4 2 3.5 3 1.5 R = 10 nm R = 2.5 nm 2.5 0 0 R0 = 5 nm 2 1 R0 = 10 nm R0 = 5 nm Exciton radius (nm) 1.5 R0 = 2.5 nm Polariton layer thickness (nm) 1 0.5 –200 –150 –100 –50 050 100 150 200 –200 –150 –100 –50 050 100 150 200 Detuning (meV) Detuning (meV)

Figure 7: (A) Lower polariton shell thickness d and (B) excitonic radius a versus the energy of the “undressed” surface plasmon relative to the bandgap energy for three different radii of the Ag nanosphere. The free exciton radius is equal to 2.8 nm. 636 J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling

different values of the nanosphere radius R0. One can atomic physics [50] where the energy of atomic transition see a substantial difference between these curves and in hydrogen gets reduced due to absorption and reemis- what one would expect from the case where the plasmon sion of photons. At higher plasmon energies, as the weight mode interacts with the “rigid exciton”, which can be of plasmon in the lower polariton decreases and so is the described by the two-level system. The curves in Figure product αβ in (16), the lowering of potential energy of the 5 are not symmetrical around the resonance, especially electron-hole pairs becomes less pronounced and the for the larger nanospheres in Figure 5A and B. While electron-hole pairs spread out away from the nanoparti- plasmon resonance stays well below the bandgap, cle surface to lower their COM kinetic energy (the value of the WEPP composition varies slowly, as its character Fkin in (16) increases according to Figure 3B). The onset of gradually changes from 100% plasmonic to being par- this sharp increase in the shell thickness d occurs earlier tially excitonic. But once the 50%/50% composition is for the small radii of nanoparticles where the COM kinetic achieved, the character of lower WEPP quickly becomes energy is larger. mostly excitonic. Note that at positive detunings, “rigid” As one can see from Figure 7A, the ratio d/R0 at polariton is expected to have mostly an excitonic char- which maximum Rabi splitting is achieved decreases acter α < β; however, in the VSC regime, adding more of from roughly 0.56 for the 2.5-nm sphere to 0.27 for the plasmonic (increased α) character causes energy lower- 10-nm sphere, which causes commensurate increase ing and, as a result, 50%/50% split occurs at positive of the overlap factor Fк in Figure 3A from 0.33 to 0.48. detunings. Therefore, Rabi splitting increases with radius R0, as

The dispersion of WEPP for different radii is shown shown in Figure 6. When the radius R0 further increases,

Figure 6. Two dashed lines show dispersion of lower this behavior is expected to saturate; the ratio d/R0 is and upper polaritons in the “rigid” approximation, i.e. further reduced and reaches the optimum value of 0.22. disregarding the effect of VSC on the exciton radius In practice though, one should be careful about further and location. Once that coupling is introduced, the increasing R0 as the polariton dimensions may exceed lower polariton dispersion curve shifts downward. That the coherence length and that will reduce the oscillator shift, ΔħΩ, is the largest (nearly 50 meV) for the largest strength. nanosphere radius R0 = 10 nm, because for that radius, Finally, in Figure 7B, the dependence of exciton radius the entire exciton shell of thickness d is located in the a on the detuning is shown. For negative detunings, the high field region, while for the smaller radii, the electric composition of polariton includes relatively small exci- field of plasmon mode drops down inside the exciton tonic fraction β2 (as shown in Figure 5); hence, the impact shell, causing the decrease of the overlap factor Fк (see of kinetic energy of relative electron-hole motion is not sig- Figure 3). The ­lowering of the polariton energy makes nificant, while the small exciton radius favors the energy the effective binding energy as large as 5kBT, where T is lowering due to virtual plasmon emission and reabsorp- at room temperature, which means that it is expected tion (Rabi energy, the last term in (16)). The exciton radius to be very robust, which can be beneficial for observing a gets reduced by almost a factor of 3 relative to the “rigid” such phenomena as Bose-Einstein condensation at room exciton radius a0 = 2.8 nm. The behavior is similar to the temperature. Furthermore, the effective binding energy one in [13] but is more prominent due to the stronger (a−3/2 is larger than the linewidth of the LSP mode (~80 meV), rather than a−1) dependence of the Rabi frequency on exci- which ensures that the lower polariton is observable. tonic radius. As the polariton composition changes first The upper polariton is not shown in Figure 6 because it to become more equal and then predominantly excitonic, is pushed within the absorption band of CdS and thus the kinetic energy of relative electron hole motion starts cannot be observed. playing a more important role, while the last term in (16) In Figure 7A, the thickness of polariton shell d is decreases with the product αβ. Naturally, the uncoupled plotted versus detuning Δ0 – for lower plasmon energies, exciton radius then gradually increases towards free the polariton gets strongly bound around the nanosphere exciton radius a0. as electron-hole pairs get attracted into the high field Note that the decrease of the excitonic radius is the region where their potential energy gets lowered by emit- most important and, in fact, the only sign of VSC as it ting and re-absorbing virtual plasmons (the value of Fк in can be easily detected by the reduction of the diamag- (16) increases according to Figure 3A). This phenomenon netic shift, as it was done in ref. [17]. Simple change of the is akin to lowering the electron energy via emission and resonant frequency between “dressed” and “undressed” reabsorption of phonons when the polarons are formed exciton would be difficult to detect due to a variation in [49]. It is also conceptually similar to the Lamb shift in samples. J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling 637

6 Classical picture of WEPP difference between the kinetic energy of relative electron hole motion Eeh and the Coulomb attraction energy EC will formation increase, causing the excitonic energy to move upward to E ~ 2.41 eV, as indicated by the right-pointing arrow in Before finalizing the discussion, it may be worthwhile ex+ Figure 8. At the same time, reduced excitonic radius will to offer an alternative, fully classical explanation for increase the oscillator strength of exciton so that its dis- the reduction of excitonic radius in the presence of VSC persion εωex+ () will develop more prominent anomalous with a LSP mode. For that, all that one needs to do is to s, opt Lorentzian feature around E ~ 2.41 eV as shown by the recall the equation defining the localized surface plasmon ex+ solid curve, which now intersects the metal dispersion at ­frequency, ε (ω ) + 2ε (ω ) = 0, or m pl s,opt pl point C leading to the downward shift of the lower WEPP 1 energy to Elp+ ~ 2.32 eV, as indicated the left-pointing arrow −=εωmp()lsεω,opt ()pl (18) 2 in Figure 8. The actual radius a will correspond to the minimum value of E where the positive kinetic energy and consider its graphic solution in Figure 8. lp+ 1 is balanced by negative Coulomb energy and negative The left hand side of (18) (metal dispersion − εω() 2 m coupling with plasmon energies. Thus the whole process is plotted as the solid curve, while the dashed curve repre- of WEPP formation can be understood from a perfectly sents εω0 () dispersion of the semiconductor permittiv- s,opt classical point of view, although using quantum picture ity disregarding the exciton. The intersection at point A greatly facilitates the determination of exciton radius, as defines ħω ~ 2.385 eV – the “undressed” LSP frequency pl it was done here. disregarding the exciton effects. Once the “rigid” exciton with radius a0 is included in the dispersion (dotted curve), ex εωs, opt () becomes anomalous around excitonic energy

Eex ~ 2.38. As a consequence, the metal dispersion curve 1 7 Conclusions and perspective − εω() intersects εωex () at three different points, of 2 m s,opt In this work, we considered the VSC between the Wannier which only the first one, B, and the third one, B′, are exciton in semiconductor and localized surface plasmon stable solutions corresponding to lower and upper polari- (LSP). Using a simple variational technique, we showed tons. As the energy of the upper polariton B′ is above the that once the coupling (Rabi) energy approaches the value absorption edge of semiconductor, this polariton is not of exciton binding energy, two previously not considered observable while the lower polariton can be observed at phenomena ensued. First of all, the Wannier exciton gets energy E ~ 2.35 eV. Let us see what is going to happen if lp localized in the region of high field of LSP so that a quasi- the excitonic radius is reduced to a < a . First of all, the net 0 particle – WEPP – is formed. In addition, the radius of the excitonic component of WEPP gets reduced by a factor 12 of a few in comparison to free excitonic radius and the 1 effective Rabi splitting increases accordingly. Notably, the 11 – εm (ω) Continuum 2 ex+ εs,opt (ω) absorption increase in Rabi splitting beyond large broadening associ- 10 ated with LSP makes WEPP observable and robust even at C 9 room temperature and thus one can guardedly anticipate ε0 (ω) 8 B s,opt ex that Bose-Einstein condensation and polaritonic lasing εs,opt (ω) A

Permittivity 7 may be within reach in WEPPs. B′ Although we considered only the simplest and easily 6 C′ (almost analytically) solvable problem of WEPP bound to a 5 spherical metal nanoparticle embedded into the semicon- Elp+ Elp Eex Eex+ 4 ductor, one can consider far more interesting configura- 2.15 2.2 2.25 2.3 2.35 2.42.452.5 2.55 Energy (eV) tions that would allow real-time manipulation of WEPPs. For instance, one can contemplate a WEPP formed by a Figure 8: Classical interpretation of WEPP. two-dimensional exciton in a transition metal dichalcoge- Shown are the graphic solutions of Eq. (18) for the case of (A) nide material, such as MoS or WSe and a strong fringe “undressed LSP” in a semiconductor without exciton (dashed line), 2 2 (B) “rigid polariton” in a semiconductor with free exciton of radius (in-plane) field of a metal nano-tip, as shown in Figure 9. Then vertical motion of the tip will cause the change of a0 (dotted line), and (C) “pliable” WEPP in semiconductor with bound exciton of radius a < a0 (solid line). the excitonic radius and energy in WEPP, while the lateral 638 J.B. Khurgin: Pliable polaritons: Wannier exciton-plasmon coupling

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