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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: joelyuen@ucsd.edu
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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The Journal of Physical Chemistry Letters
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.
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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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The Journal of Physical Chemistry Letters
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
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|yk ; 0U HP i
U HP ~ωK =
~c|K| √ , d
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|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
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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׈ z |K׈ 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 ∈