Molecular Emission Near Metal Interfaces: The Polaritonic Regime

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Molecular Emission Near Metal Interfaces: The Polaritonic Regime Joel Yuen-Zhou, Semion K Saikin, and Vinod M. Menon J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02980 • Publication Date (Web): 29 Oct 2018 Downloaded from http://pubs.acs.org on November 1, 2018

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The Journal of Physical Chemistry Letters

Molecular emission near metal interfaces: the polaritonic regime 1

Joel Yuen-Zhou , Semion K. Saikin

†1 Department

2,3

, Vinod M. Menon

4∗,†

of Chemistry and Biochemistry, University of California San Diego, La Jolla, CA, USA.

‡2 Department

of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, USA.

¶3 Institute §4 Department

of Physics, Kazan Federal University, Kazan, Russian Federation.

of Physics, Graduate Center and City College of New York, City University of New York, New York, New York, USA.

E-mail: [email protected]

Abstract The strong coupling of a dense layer of molecular excitons with surface-plasmon modes in a metal gives rise to polaritons (hybrid light-matter states) called plexcitons. Surface plasmons cannot directly emit into (or be excited by) free-space photons due to the fact that energy and momentum conservation cannot be simultaneously satised in photoluminescence. Most plexcitons are also formally non-emissive, even though they can radiate via molecules upon localization due to disorder and decoherence. However, a fraction of them are bright even in the presence of such deleterious processes. In this letter, we theoretically discuss the superradiant emission properties of these bright plexcitons, which belong to the upper energy branch and reveal huge photoluminescence enhancements compared to bare excitons, due to near-divergences in the density of photonic modes available to them. Our study generalizes the well-known problem of molecular emission next to a metal interface to the polaritonic regime. ACS Paragon Plus Environment 1

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The study of molecular photoluminescence (PL) next to a metal-dielectric interface dates back to a classic experiment reported by Drexhage, Kuhn, and Schäfer (DKS) almost fty years ago

1,2

. By

controlling the distance between a molecular emitter and a metal-mirror lm, the aforementioned authors showed that the observed PL rate oscillated and then monotonically increased for short distances. These oscillations were attributed to interferences between the free-space and reected light from the mirror, and the monotonic decay was associated to irreversible energy transfer to surface plasmons (SPs) in the metal. These observations were later fully elucidated in a theory provided by Chance, Prock, and Silbey (CPS)

3,4

, by adapting the results of an even older problem

of antenna radiation next to the surface of the Earth, whose mathematical solution was oered

5

by Sommerfeld as early as 1909 . This problem has been revisited countless times to understand molecular energy transfer processes in condensed phases.

Figure 1: Plexciton setup. A molecular layer of thickness of thickness

z0 ,

Wz

sits on top of a thin dielectric spacer

which in turn, is placed on a metal lm. Strong coupling between excitons in the

molecular layer and surface-plasmons (SPs) in the metal lm imply energy exchange between the excitons and SPs is much faster than their respective decay processes, giving rise to polaritonic excitations called plexcitons. Reprinted (adapted) with permission from ACS Photonics

2018,

5

(1), 167176. Copyright 2018 American Chemical Society."

In this letter, we study a variation of the aforementioned problem which oers new phenomenology that, as far as we are aware, has not been reported before.

We study PL originating from

(delocalized) molecular excited states (excitons) when they are strongly coupled to a SP metal lm (see Fig. 1). We consider a layer of a thin dielectric spacer (of thickness

z0 ),

Nx Ny Nz

molecules (of thickness

which is on a thick metal lm.

Wz )

placed on top of

When the molecular

layer is suciently dense, the energy transfer between the exciton and SP modes is reversible and faster than each of the decay rates

6,7

. The resulting eigenmodes are no longer purely plasmonic or

excitonic, but rather, polaritonic, being coherent superpositions of such modes. Polaritons arising from SPs and excitons are termed plexcitons

813

. As with any polaritonic system, the transmission

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spectrum of the plexciton system at the wavevector giving rise to resonance between the uncoupled exciton and SP bands shows a Rabi splitting (anticrossing) between two new bands per and lower plexcitons (UPs, LPs)

9,14

7,8

, named up-

, according to their energy ordering. Molecular polaritons

have recently been the subject of intense investigation, as they oer new ways for coherent control of molecular processes

15

such as changes on reaction rates and thermodynamics by imprinting

electromagnetic coherence directly onto few

1621

or many-molecule

2231

ensembles (in this article,

we will be concerned with the latter). They also provide new platforms to induce remote energy transfer

3234

and to recreate exotic topological

ature in tabletop experiments

3639

.

12,35

and many-body phenomena at room temper-

In this study, we highlight superradiant properties

40,41

that

plexcitons exhibit which are not encountered in the weak coupling regime of a bare molecule next to a metal surface. These properties can potentially be harnessed for light harvesting and energy routing purposes in molecular materials and device applications. We rst briey lay out a quantum-mechanical formalism to describe the plexciton system (see details in Supporting Information, SI-I, II). Writing the Hamiltonian as sum runs over a discrete set of in-plane exciton wavevectors

k,

we have

H =

P

k

Hk ,

where the

6,7

Hk = ~ωkSP a†k ak + ~ωe σk† σk + (Jk σk† ak + h.c.) + Hdark,k + Humklapp,k .

Here, a

kth

~ωkSP

and

a†k (ak )

SP [exciton];

Jk

the bare exciton energy

[

~ωe

and

σk† (σk )]

are the energy and creation (annhilation) operator of

is the collective SP-exciton coupling.

~ωe ,

(1)

Hdark,k

which do not couple directly to SPs

27

describes dark excitons at

; we disregard

Humklapp,k

as a

negligible o-resonant contribution due to coupling with high-wavevector SP modes. Ignoring also

Hdark,k

for the time-being (we will discuss it at the end of the letter),

two-level system for every the form

|yk i =

h

(SP ) ζyk a†k

UP (LP), and |vaci

k,

+

Hk

in Eq. (1) becomes a

and can be diagonalized to yield two plexciton (polariton) states of

(exc) ζyk σk†

i |vaci

= |g; 0SP ; 0U HP i

with eigenenergies

~ωyk ,

where

y=±

are labels for the

is the tensor product of the ground state for the molecular

degrees of freedom (|gi) and the vacuum for the SP modes (|0SP i); see Fig. 2a.



Figure 2:

(a) Dispersion plot of excitations for an isotropic molecular layer.

Page 4 of 19

When the (at)

exciton (dashed blue) and the SP (dashed red) bands couple, they give rise to upper and lower plexciton (UP, LP). The UP

|k| → ∞;

|+k i

interpolates between exciton and SP between

|k| ≈ 0

and

the converse is true for the LP (|−k i). Strong hybridization occurs in the anticrossing

region. The yellow-shaded region contains states that emit photons, while the right region contains states that cannot due to mismatch in energy and in-plane wavevector with respect to free-space ∗ photons. The critical point at k = k separates bright and non-emissive UPs. (b) PL rates for

waves (i

hγ+k iiso =

P

i hγ+k ,i iiso normalized to the single-molecule rate γSM . Contributions to hγ+k iiso correspond to direct (i = 0), and interference terms between free-space and reected

the UP band

= ref ).

The divergences in

hγ+k∗ ,i iiso

are van Hove singularities.


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Fig. 2a shows a dispersion energy plot for exciton, SP, and resulting plexciton bands assuming

ωe = 3 eV, |µeg | = 10 Debye,

and relative permittivities

d = 1.3

the molecular (dielectric) layer and metal lm, respectively.

and

We used

m (ω) = 3.7 −

(8.6 eV)2 for ω2

ρ = 109 molecules/µm−3 ,

z0 = 1 nm, and Wz = 200 nm, to account for representative parameters in the literature 6 . isotropic averages of

Jk

We took

to describe a molecular layer with orientational disorder. Conservation of

energy and in-plane momentum constrains the allowed PL processes: an excitation of frequency

ω

and in-plane wavevector

k

must output a propagating photon with the same properties. This

means that, for PL to occur, there needs to be a real-valued

kzd

satisfying

c p 2 |k| + |kzd |2 . ω=√ d It is well known this is not a possibility for SP modes line,

, whose dispersion is to the right of the light-

√c |k|, and thus, must be probed by coupling to gratings 43 or nanoparticles 44 , d

ωkSP < ωkll ≡

for example.

42

(2)

The same holds formally true for LPs (ω−k

< ωkll )

and for a subset of UPs; to

Hdark,k ),

distinguish them from the dark exciton states (eigenstates of

we shall call them non-

emissive states. Importantly, there is another subset of UPs that are to the left of the light-line, thus being formally bright and featuring PL (see Fig.

2a).

We then ask:

what is the rate of

PL from these UPs, and how does the bright-to-non-emissive transition occur as a function of To answer these questions, we compute the PL rate of a state

|yk i

k?

into the free-space radiative

modes of the upper-half-plane (UHP) of the plexciton setup. The corresponding Wigner-Weiskopf expression (Fermi golden rule) is

γy k =

where

Hint = −

P

ns

45,46

2π X U HP |hvac; (K, χ)U HP |Hint |yk ; 0U HP i|2 δ(~ωyk − ~ωK ), ~ K,χ

(3)

ˆ U HP (r ns ) contains the interaction between each of the molecules in the ˆ ns · E µ

slab and the electric eld at the UHP. In Eq. (3),

|0U HP i denotes the UHP photonic vacuum state.

The matrix element couples an initial state with a plexciton and no UHP photons with a nal state featuring no plexcitons and an UHP photon with energy


|yk ; 0U HP i

U HP ~ωK =

~c|K| √ , d


|vac; (K, χ)U HP i = b†K,χ |vac; 0U HP i,

where

b†K,χ (bK,χ )

is the creation (annhilation) operator of

photons at the UHP mode with (three-dimensional) wavevector

χ = s, p,

ization index

Page 6 of 19

which satises the commutation relation

K = (Kx , Ky , Kzd )

and polar-

[bK,χ , b†K 0 ,χ0 ] = δK,K 0 δχ,χ0 .

The

electric eld at the UHP is given by a collection of radiative modes. Using the modal representation given by Arnoldus and George

47

,

 s  (b U HP ˜ ) ~ωK K,χ + bK,χ ˆ U HP (r) = ˆ) p E Θ(−K · z  1 + |rKχ |2 20 d V K,χ # ) " X

˜

+ h.c. . × eK,χ eiK·r + rK,χ eiK·r eK,χ ˜

The radiative modes can have

s

or

p

polarization,

√ ˆ +Ky y ˆ) −Kzd (Kx x |K|

√

Kx2 +Ky2

function, and

+

ˆ Kx2 +Ky2 z . |K|

V

eK,s =

K×ˆ z |K×ˆ z|

(4)

ˆ xy √y xˆ −K =K 2 2 Kx +Ky

is the quantization volume of the UHP,

Θ

and

eK,p =

eKs ×K |eKs ×K|

=

is the Heavyside step

˜ = K −2K · z ˆz ˆ is the reected wavevector for an incident wave into the metal with K

m Kzd −d Kzm 43 zd −Kzm and rK,p = , where Kzm (K, ω) = Kzd < 0 and Fresnel coecients rK,s = K Kzd +Kzm m Kzd +d Kzm q 2 − m (ω) ωc2 − |K ⊥ |2 . Note that although Kzd ∈

Molecular Emission Near Metal Interfaces: The Polaritonic Regime

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