Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum

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Cite This: J. Phys. Chem. C 2018, 122, 26584−26591

Anomalous Dipole−Dipole Interaction in an Ensemble of Quantum Emitters and Metallic Nanoparticle Hybrids Mahi R. Singh* and Kevin Black

J. Phys. Chem. C 2018.122:26584-26591. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 11/22/18. For personal use only.

Department of Physics and Astronomy, The University of Western Ontario, London N6A 3K7, Canada ABSTRACT: We have developed a theory for the photoluminescence (PL) and scattering cross section of a core− shell hybrid, where the core is the metallic nanoparticle and the shell is made of an ensemble of quantum emitters. A probe field is applied to calculate the scattering cross section of the core−shell hybrid. The surface plasmon polariton field in the metallic nanoparticle is calculated by solving the Maxwell equations in the quasi-static approximation. Dipoles are induced in the ensemble of quantum emitters because of the probe field and surface plasmon polariton field. Therefore, the dipole of one quantum emitter interacts with dipoles of other quantum emitters in the ensemble, and hence, there is the dipole−dipole interaction (DDI) between quantum emitters. We discovered an anomalous DDI, which is induced by the surface plasmon polaritons. It is shown that the strength of the DDI can be controlled by the surface plasmon polariton frequency, and it plays a dominant role in the phenomenon of the PL and scattering cross section. The surface plasmon polariton field can also interact with excitons of the quantum emitters via the exciton-surface plasmon polariton interaction. Using the density matrix method, the PL and scattering cross section are evaluated. It is found that the spectrum of the PL and the scattering cross section splits from one peak into two peaks mainly because of the strong coupling between the excitons and anomalous DDIs. It means that the PL and scattering spectrums can be switched ON (one peak) and OFF (two-peaks). This finding is consistent with the experimental data of the PL and scattering cross section of the J-aggregate and silver core−shell hybrid. We have found that the splitting and height of the two peaks can be increased or decreased by controlling mainly the strength of the anomalous DDI. The anomalous DDIs can be controlled by applying an external pulse pressure and pulse control laser. Hence, the present findings can be used for fabricating nanosensors and nanoswitches for applications in nanotechnology and nanomedicines.

1. INTRODUCTION Recent hybrid nanostructures have been fabricated by combining quantum emitters (QEs) and metallic nanoparticles (MNPs).1−16 When light falls on QEs, electron−hole pairs (excitons) are created in QEs. On the other hand, when light falls on an MNP, surface plasmon polaritons (SPPs) are created at the interface of the MNP. SPPs are spatially confined and have significantly enhanced fields at the nanoparticle surface relative to the incident light. The SPP resonances can be manipulated in these structures by changing the size and shape of MNPs. Similarly, exciton energies can be modified by size and shape of QEs. Because of their desirable SPP and exciton resonances, these nanostructures can be important building blocks for the study of light−matter interactions and their applications in nanomedicines and nanotechnology. These structures also have important technological applications in areas of chemical and biological sensors. The exciton−SPP interaction between QEs and MNPs plays a key role in the application of these materials. The enhanced SPP field can strongly modify the optical properties of QEs. When the exciton−SPP interaction is strong, the energies of both the SPP and the exciton are modified by their mutual interaction.1−6 This strong interaction can generate a new © 2018 American Chemical Society

hybrid state called the plexcitonic nanoparticle under the strong coupling regime. For example, Wersäll et al.1 synthesized QE−MNP hybrid nanoparticles by self-assembly of an ensemble of J-aggregate dye molecules over a Ag nanoprism surface. They measured the photoluminescence (PL) and scattering spectrum of these hybrids. They observed splitting not only in the scattering spectrum but also in the PL spectrum on the core−shell nanoparticle. Fofang et al.2 have fabricated hybrids from J-aggregate molecules and Au-nanoshells to study the coherent coupling between excitons from Jaggregate molecules and SPPs from metallic nanoshells. The metallic nanoshells are made of a spherical silica core coated with a thin, uniform Au layer. They measured the extinction coefficient by varying the core size/shell thickness ratio. They observed splitting of the extinction spectrum peak because of the strong exciton−SPP coupling. The strong exciton−SPP interaction has also been observed in a Ag nanosphere and Jaggregate molecule hybrid.3 In this paper, we have developed a theory for the PL and scattering cross section for a core−shell hybrid where the core Received: July 3, 2018 Revised: October 22, 2018 Published: October 24, 2018 26584

DOI: 10.1021/acs.jpcc.8b06352 J. Phys. Chem. C 2018, 122, 26584−26591

Article

The Journal of Physical Chemistry C

because of the probe field. The dipole of one QE interacts with dipoles of other QEs. This is called the DDI. Following ref 17, the DDI Hamiltonian is written as

is the MNP and the shell is made of an ensemble of QEs. A probe field is applied to calculate the scattering cross section of the core−shell hybrid. The SPP electric field in the MNP is calculated by solving the Maxwell equation in the quasi-static approximation. Dipoles are induced in the ensemble of QEs because of the probe field and SPP field. Therefore, there is a dipole−dipole interaction (DDI) between QEs. The DDI induced by the SPP field is called anomalous DDI because its strength can be controlled by the SPP frequency. The SPP field also interacts with the excitons in the QEs via the exciton−SPP interaction. Using the density matrix method in the presence of the anomalous DDI and the exciton−SPP interaction, the PL and scattering cross section are evaluated. It is found that the PL and scattering spectra with one peak splits into two peaks due mainly to the strong anomalous DDI. These findings are consistent with the PL and scattering cross section experiments of the J-aggregate molecules and MNP core−shell hybrids.1 We have also found that the splitting and height of the two peaks can be controlled by the strength of the anomalous DDI. These findings can be used for fabricating nanosensors and nanoswitches.

N

HDDI =

1 ∑ J p ·p , 2 i > j ij i j

Jij =

1 ϵbϵ0rij 3

(1)

where pi and pj are the induced dipole moments in the ith-QE and jth-QE, respectively. Here, Jij is the DDI coupling constant and rij is the distance between the ith and the jth QEs. In the mean field approximation,17−22 the DDI Hamiltonian can be rewritten as HDDI =

, ∑ pE i DDI i

E DDI =

1 ∑Jp 2 j ≠ i ij j

(2)

where EDDI is the average dipole electric field created by all QEs on the ith-QE. The average in the right side of the equation of EDDI has been evaluated in refs17,22 by using the method of Lorentz.23 The expression of EDDI has been evaluated as

2. ANOMALOUS DDIS We consider a hybrid that is fabricated by embedding an MNP in an ensemble of QEs. This hybrid can be called core−shell, where the core is the MNP and the shell is the ensemble of QEs. These types of hybrids have been fabricated by Wersäll et al.,1 where the core is made from a Ag nanoprism and the shell is made from an ensemble of J-aggregate molecules. A schematic diagram of the core−shell hybrid is depicted in Figure 1. We consider that this hybrid is surrounded by the

E DDI =

λQE⟨pi ⟩ (3)

3

where ⟨pi ⟩ =

PQE 4π ϵbR q 3

(4)

where λQE is the DDI constant and the typical value λQE is taken as unity.17 Here, ⟨pi⟩ is the average polarization of the ithQE and PQE is the polarization of a QE. The polarization of the QE and an electric field produced by PQE are calculated as follows. 2.1. Polarization and Electric Field in the Quasi-Static Approximation. When the probe field falls on a QE, an induce polarization occurs in the QE. It is denoted as produce PQE which in turn produces a dipole electric field denoted as EQE. The electric field produced by the QE can be calculated using the quasi-static approximation.24,25 The size of the QE is much smaller than the wavelength of the electromagnetic field in the optical region. For example, the wavelength of light in the visible region is of the order of 600 nm, whereas the size of the QE is of the order of 10 nm. In this case, one can consider a situation that the amplitude of electric field is constant over the nanoparticle. This condition is known as quasi-static approximation.24,25 When the probe field EP falls on a QE, it induces a polarization PQE, which in turn produces a dipole electric field EQE. Solving Maxwell’s equations in the quasi-static approximation, one can derive the following expression for the electric field produced by the QE as follows24,25

Figure 1. (Left Side) Schematic diagram of a hybrid, which consists of an MNP (blue), embedded in an ensemble of QEs (orange). (Right Side) A schematic diagram of a three-level QE. Energy levels are denoted as |a⟩, |b⟩, |c⟩. We consider that the probe field EP acts only on the transition |a⟩ ↔ |b⟩. The SPP field ESPP and the DDI field are acting on the |a⟩ ↔ |c⟩ transition only. The radius of the core is taken as Rc and the radius of the shell is denoted as Rs. The thickness of the shell is represented as ts and this gives Rs = Rc + ts.

biological cell with a dielectric constant ϵb. The dielectric constants of the core and shell are denoted as ϵc and ϵs, respectively. The radius of the core is taken as Rc, and the radius of the shell is denoted as Rs. The thickness of the shell is represented as ts and this gives Rs = Rc + ts. Let the radius of the QE be Rq and its dielectric constant be ϵq. A probe field with frequency/wavelength ω/λ and amplitude EP is applied in the hybrid. The frequency and wavelength are related as ω = 2πc/λ, where c is the speed of light. When the probe falls on the QE, induced dipoles are created in the QEs

EQE =

PQE 4π ϵbr 3

(5)

where the expression for PQE is found as ϵq − ϵb PQE = 4π ϵbR q 3gl ζQEE P , ζQE = ϵq + 2ϵb

(6)

The constant gl is called the polarization parameter and it has values gl = 2 and gl = −1 for PQE∥EP and PQE⊥EP, 26585


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The Journal of Physical Chemistry C respectively.24 For the calculations in this paper, we consider gl = −1. Similarly, when electric field falls on the MNP, an induced polarization occurs in the MNP. It is denoted as PSPP. The induced polarization produces a SPP field denoted as ESPP. There are two electric fields, namely, EP and EQE which are falling on the MNP. Solving Maxwell’s equations in the quasistatic approximation, one can derive the following expression for the SPP electric field produced by the MNP. ESPP =

Now we calculate ESPP produced by the MNS. We substitute the expression for EQE from eqs 10 into 11 and we get ESPP =

4π ϵbr

r3

βQEβSPP(R c + ts)3 R q 3

EP +

r6

EP (12)

Now we substitute the values of βQE and βSPP from eqs 10 and 11, respectively, into eq 12 which reduces to

PSPP 3

βSPP(R c + ts)3

ESPP =

(7)

gl ζSPP(R c + ts)3 r3

EP +

gl gl ζQEζSPP(R c + ts)3 R q 3 r6

(13)

where the expression for PSPP is found as PSPP = 4π ϵb(R c + ts)3 gl ζSPP(E P + EQE)

The above expression can further be rewritten in terms of physical parameters ΠPSPP of ΠQE SPP as follows

(8)

We want to make a comment on distance r being used in both eq 5 for EQE and eq 7 for ESPP. Here, the distance r is the same in both the equations because it is the distance from the center of the QE to the center of the MNP. The parameter ζSPP appearing in eq 8 is called the SPP polarization. It is found as ζSPP =

P ESPP = (ΠSPP + ΠQE SPP)E P

P ΠSPP =

ϵsc = ϵs

( − 2(R ) (

(R c + ts) + 2(R c)

ϵc − ϵs ϵc + 2ϵs

(R c + ts)3

3

ϵc − ϵs ϵc + 2ϵs

c

) )

EQE =

4π ϵbR q 3gl ζQE

EP =

4π ϵbr 3 βQER q 3 r

3

(9)

(E P),

R q 3gl ζQE r3

E DDI =

E DDI =

EP

E DDI =

βQE = gl ζQE

4π ϵ0ϵb(R c + ts)3 gl ζSPP =

ESPP =

4π ϵ0ϵbr 3 (R c + ts)3 gl ζSPP r3

βSPP(R c + ts)3 r3

(10)

ζQEζSPP

(15)

PQE

λQE

3 4π ϵbR q 3

(16)

λQE 4π ϵbR q 3gl ζQE(E P + ESPP) 4π ϵbR q 3

3 λQEgl ζQE 3

(E P + ESPP)

(17)

Putting the expression of ESPP from eq 14 into the above eq 17 and we get the expression of the DDI field as E DDI =

(E P + EQE)

λQEgl ζQE 3

EP +

λQEgl ζQE 3

P (ΠSPP + ΠQE SPP)E P

(18)

The final expression for the DDI electric field is written in terms of DDI parameters ΦP and ΦSPP as follows

(E P + EQE)

(E P + EQE),

r6

Note that two electric fields, namely, EP and ESPP are falling on the QEs. Putting the expression of PQE from eq 6 into the above eq 16 and we get

Similarly, to get a simple expression for the SPP field, we substitute eqs 8 into 7 and we get ESPP =

ζSPP , r3 gl gl (R c + ts)3 R q 3

It is worth mentioning that the SPP field depends on two P terms ΠPSPP and ΠQE SPP. The first ΠSPP term is the SPP field and it is induced by the probe field. Similarly, the second ΠQE SPP term is the SPP field and it is induced by the QE dipole field. Note that the first term depends on r−3 and the second term depends on r−6. Therefore, the SPP field induced by the probe field is stronger than the SPP field induced by the QE field. 2.2. DDI Electric Field. Now we calculate the DDI electric field found in eq 3. We substitute eqs 4 into 3 and we get

In the above expression, ϵc is the dielectric constant of metal and we have considered ϵc = ϵ∞ − ωp2/ω(ω + i/τ),26,27 where ωp (λp) is the plasmon frequency/wavelength and ϵ∞ is the dielectric constant of metal when light frequency is very large. Note that the real part of ϵc has a negative value when ω < ωp. It is interesting to find that when ϵc has a negative value, the denominator of the polarization factor ζSPP becomes zero at a certain value of the frequency. Let us call this value ω = ωsp. This means that the polarization and SPP field have a singularity when ω = ωsp. The frequency and wavelength are related as ωsp = 2πc/λsp, where c is the speed of light. Let us express eqs 5 and 7 in simple forms. To get a simple expression for EQE, we substitute eqs 6 into 5 and we get EQE =

gl (R c + ts)3

ΠQE SPP = 3

(14)

where

ϵsc − ϵb ϵsc + 2ϵb 3

EP

E DDI = (ΦP + ΦSPP)E P ,

βSPP = gl ζSPP

P ΦSPP = ΦSPP + ΦQE SPP

(19)

where

(11) 26586


The Journal of Physical Chemistry C ΦP =

λQEgl ζQE

ΦQE SPP =

,

3 λQEgl ζQE 3

P ΦSPP =

(ΠQE SPP)

λQEgl ζQE 3

ij μ ωabρ yz ab z zz|E |2 IPL = Q QE Imjjjj ab j 2Ep zz QE k {

P (ΠSPP ),

(20)

σscatt =

Note that the DDI found in eq 19 is made of two terms ΦP and ΦSPP. The first term ΦP is the DDI term induced by the probe field. We call it the classical DDI field induced by the external probe field. The second term ΦSPP is the DDI term induced by the SPP field. We call this term the anomalous DDI. P This term is further made of two terms ΦPSPP and ΦQE SPP. ΦSPP P depends on ΠSPP which is given by eq 15. This term depends QE on r−3. On the other hand, ΦQE SPP depends on ΠSPP which is −6 given by eq 15. This term depends on r . In other words, the −6 ΦQE SPP term depends on r . This means that the anomalous P DDI term due to the ΦSPP term is dominant compared to the P ΦQE SPP term. This is because the ΦSPP term is due to the SPP field induced by the probe field and it depends on the SPP polarization factor, ζSPP, which has a large value at ω = ωsp, whereas the ΦQE SPP term is induced by the QE electric field which is a very weak field. Further, we want to make a comment on the anomalous DDI. In the condensed matter physics literature, the classical DDI is calculated as follows. When light falls on the ensemble of particles, induced dipoles are created in the particles. These induced dipoles interact with each other and this is called the DDI. In this paper, we call it the classical DDI. It is important to note that this classical DDI is induced by the external (probe) light. On the other hand, in the present paper, we have invented a new type of DDI and we call it the anomalous DDI to distinguish it from the classical DDI. In this paper, the external (probe) field induces the SPP field in the MNP. This SPP field falls on the ensemble of QEs and induces dipoles in the QEs. Therefore, the induced dipoles in the ensemble of QEs interact with each other which we call the anomalous DDI. One can see from eq 19 that the DDI field is made of two terms: (i) the classical DDI induced by the external (probe) field denoted by ΦP and (ii) the anomalous DDI induced by the SPP field denoted by ΦSPP and it is a very strong field. One of the interesting points of our theory is that it provides a method to calculate the emission rate of a strongly coupled system between J-aggregates and plasmonic nanoparticles.

ω4 6πc 4 ϵ0 2

μab ρab

Article

2

Ep

(21)

Parameter QQE is called the PL efficiency factor and is taken as unity for simplicity. Here, ρab and μab represent the density matrix element and dipole moment of the QE, respectively, for the transition |a⟩ ↔ |b⟩. The density matrix element ρab for two levels |a⟩ and |b⟩ is calculated as follows. We consider that the SPP frequency ωsp lies closely to the exciton energy ωac. Hence, the SPP electric field (ESPP) and the DDI electric field (EDDI) are acting only between the transition |a⟩ ↔ |c⟩. Note that three fields are falling on the QE and they are written as EQT = EP + EDDI + ESPP, where ESPP and EDDI are given by eqs 14 and 19, respectively. These three fields induce dipoles in the QE and the induced dipoles interact with these three fields. In the dipole and the rotating wave approximation,29,30 the interaction Hamiltonian of the QE in the hybrid is expressed as P Hint = ℏΩ bσba + ℏ(ΦP)Ω bσba + ℏΩc(ΠSPP + ΠQE SPP)σca P + ℏΩc(ΦSPP + ΦQE SPP)σca + h. c.

(22)

where h.c. stands for the Hermitian conjugate. Parameter Ωn = μnaEP/ℏ (n = b, c) is called the Rabi frequency associated with the transitions |n⟩ ↔ |a⟩. Here, σna = |n⟩⟨a| is the exciton creation operator. The physical meaning of each term appearing in eq 22 is as follows. The first term is the interaction between excitons and the probe field. The second term is the exciton interaction with the classical DDI field induced by the probe field. The third term is the exciton interaction with the SPP field produced by the MNP. The last term is the exciton interaction with the anomalous DDI field. This term is called the anomalous DDI term because the dipoles in the QE are induced by the SPP field. This term is large near the SPP frequency ωsp. Equations of motion for different density matrix elements are found with the help of the density matrix equation method29,30 and Hamiltonian eq 22 as follows. dρcb dt

3. PL, SCATTERING CROSS SECTION, AND DENSITY MATRIX METHOD

dρba dt

Next, we calculate the interaction between excitons in a QE with the SPPs in the MNP and the DDI field in the ensemble of QEs. We consider a three-level QE whose energy levels are denoted as |a⟩, |b⟩, and |c⟩. Two excitons are induced in the QE with frequency/wavelength (ωab/λab) and (ωac/λac) because of transitions |a⟩ ↔ |b⟩ and |a⟩ ↔ |c⟩, respectively. A schematic diagram of the QE is shown in Figure 1. It is important to note that we have used two levels |a⟩ and | b⟩ to calculate the PL and SCS spectra. We applied the probe field Ep between the transition |a⟩ ↔ |b⟩ to monitor the PL and scattering cross section. Following the method of refs 25 and 28, we have evaluated the PL and scattering cross section for the QE in the hybrid system as

dρca dt

P P QE = −Ξcbρcb + iρab Ωc(ΠSPP + ΠQE SPP + ΦSPP + ΦSPP) P P QE − iΩc(ΠSPP + ΠQE SPP + ΦSPP + ΦSPP)*ρca

= −Ξ baρba − i(ρbb − ρaa )(Ω b + Ω b(ΦP)) P P QE + iρcb Ωc(ΠSPP + ΠQE SPP + ΦSPP + ΦSPP) P P = −Ξcaρca − i(ρcc − ρaa )Ωc(ΠSPP + ΠQE SPP + ΦSPP

+ ΦQE SPP) − iρcb (Ω b + Ω b(ΦP)) (23)

where Ξca = γc + iδc , Ξ ba = γb + iδ b , Ξcb = (γc + γb)/2 + i(δc − δ b) 26587

(24)


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The Journal of Physical Chemistry C Here, δb = ωab − ω and δc = ωac − ω are called the probe field detunings. The physical quantities γb and γc appearing in eq 24 represent the exciton decay rates from |a⟩ to levels |b⟩ and |c⟩, respectively. An analytical expression for PL and SCS can be obtained in the steady state if we consider that ρaa > ρbb and ρaa > ρcc. For simplicity, we also consider that μba = μca, which gives us Ωb = Ωc. An analytical expression for ρba is obtained from eq 23 as ρba = i[Ω b + Ω b(ΦP)](γac + iδab)

Figure 2. (a) Scattering cross section is plotted as a function of wavelength for the Ag-MNP alone. The dashed line and open circles represent the theoretical simulations and experimental data, respectively. Parameters used are γsp/Ω = 0.3, Ω = 0.3 eV, and λsp = 588 nm (b) Scattering cross section is plotted as a function of wavelength for the core−shell hybrid made from J-aggregate dye molecules and the Ag-MNP. The solid line is the theoretical curve and the open circles represent experimental data. Parameters used are γab/ Ω = 0.3, γac/Ω = 0.1, Ω = 0.3 eV, and λab = λsp = 588 nm. Notice that the peak not only splits from one to two peaks.

P P QE 2 (γac + iδab)(γac + iδab) + [Ωc(ΠSPP + ΠQE SPP + ΦSPP + ΦSPP)/2]

(25)

Substituting eqs 25 into 21, with the simplification Ωb = Ωc, we get the equations of the PL and scattering cross section as IPL = Q QEℏωabΩ b

yz i[Ω b + Ω b(ΦP)](γac + iδab)|(1 + ΦP)|2 ji zz Imjjjj zz P QE P QE 2z j (γ + iδab)(γ + iδab) + [Ωc(ΠSPP + Π + Φ + Φ )/2 ] SPP SPP SPP ac { k ab

(26) σscatt =

μab 4 ωsp2 6πc 4ℏ2Ω b2ϵ0 2 2

[Ω b + Ω b(ΦP)](γac + iδab) (γab + iδab)(γac + iδab) +

P [Ωc(ΠSPP

+

ΠQE SPP

+

P ΦSPP

+

2 ΦQE SPP)/2]

(27)

Note that the PL and scattering cross section depend on the anomalous DDI terms ΦPSPP + ΦQE SPP. It also depends on the exciton−SPP interaction (ΠPSPP term and ΠQE SPP term). Figure 3. (a) PL is plotted as a function of wavelength for the Jaggregate dye molecule alone. The dashed line and open circles represent the theoretical simulations and experimental data, respectively. Parameters used are γab/Ω = 0.126, γac/Ω = 0.15, Ω = 0.14 eV, and λab = 588 nm. (b) The solid line is the theoretical curve and the open circles represent experimental data for the core−shell hybrid. The parameters used are γab/Ω = 0.22, γac/Ω = 0.04, Ω = 0.6 eV, and λsp = λab = 588 nm. Notice that the peak splits from one to two peaks.

4. RESULTS AND DISCUSSION We compare our theory with the experimental data of the scattering cross section and the PL for the core-sell hybrid which is fabricated by embedding the Ag MNP in an ensemble of J-aggregate dye molecules.1 Here, dye molecules form a nanoshell and the Ag metallic nanoprism acts as the core. The numerical simulations are performed using wavelength λ rather than frequency ω. Almost all of the parameters used in our theory can be taken from the plasmonic and quantum optics literature.1,6,26,27,31−33 The references are provided beside each physical parameter. The plasmon frequency and the relaxation time for Ag are taken as ωp = 8.7 eV and τ = 1.45 × 10−14 s, respectively, which are taken from references Nash et al.27 and Jiang et al.31 The value of the dielectric constant ϵ∞ is taken as ϵ∞ = 6 from Yang et al.26 We have taken Rc = 25 nm,1,6 ts = 3 nm,1,6 and Rq = 0.5 nm.1,6,32 For a core of radius Rc = 25 nm, the volume is given by Vc = 4πRc3/3 = 6.5 × 104 nm3, which approximately agrees with the volume of the nanoprism ∼1.3 × 104 nm3.1 We have increased the volume of the sphere slightly to get SPP wavelength (frequency) approximately equal to the experimental value of ref 1. The size of the QEs Rq ≈ 0.5 nm is approximated as a Wannier−Mott exciton as suggested in Baranov et al.32 The shell thickness is taken as ts = 3 nm.1 The dielectric constant of the surrounding medium is taken as ϵb = 1.14.1 The values of γab/Ω and γac/Ω are shown in the captions of Figures 2−5 and are known from ref 33. Finally, the surface plasmon wavelength λsp and two exciton wavelengths λab and λac are taken from ref 1 and are written as λab = λac = λsp = 588 nm.

Figure 4. (a) Scattering coefficient is plotted as a function of the probe wavelength and the DDI constant (ΦSPP). Parameters used are γab/Ω = 0.16, γac/Ω = 0.02, Ω = 0.6 eV, and λsp = λab = 588 nm. (b) PL is plotted as a function of the probe wavelength and the DDI constant (ΦSPP). Parameters used are γab/Ω = 0.22, γac/Ω = 0.04, Ω = 0.6 eV, and λsp = λab = 588 nm. In both figures, we have neglected the effect of the EPI. Notice that as the DDI increases, one peak splits into two peaks. However, the height of the peaks decreases, and the splitting of peaks increases.

The relative dielectric constants of J-aggregates at high energies and the shell made of J-aggregate are taken as ϵs = ϵq 26588


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follows. In the case of the free J-aggregate dye molecule, there is no coupling between the exciton and the anomalous DDI. Hence, the scattering cross section and PL spectrum have one peak because of the transition |a⟩ ↔ |b⟩. In the presence of the strong anomalous DDI couplings, the dressed states are created in the system. The state |a⟩ splits into two dressed states called |a−⟩ and |a+⟩ because of the anomalous DDI coupling. Therefore, two transitions occur, |a−⟩ ↔ |b⟩ and |a+⟩ ↔ |b⟩, because of the presence of the dressed states. That is why one peak due to the transition |a⟩ ↔ |b⟩ in the PL and scattering cross section spectrum splits into two peaks because of two transitions |a−⟩ ↔ |b⟩ and |a+⟩ ↔ |b⟩. To observe the effect of the anomalous DDI, we have plotted the three-dimensional Figure 4a,b for the scattering cross section and PL spectrum for the core−shell hybrid, respectively. The results are plotted in Figure 4a,b for the scattering cross section and the PL as a function of wavelength and the anomalous DDI coupling parameter, respectively. Note that as the DDI increases, one peak splits into two peaks. The strength of the splitting increases as the DDI parameter increases. On the other hand, the height of the peaks decreases as the anomalous DDI parameter increases. Note that the splitting in PL is significantly smaller than the splitting in scattering for the same nanostructure. Finally, in Figure 5a,b, we have plotted the effect of the anomalous DDI on PL and SCS spectra, respectively. The dotted curves, plotted in both figures, represent when the anomalous DDI is absent (i.e., ΦSPP = 0). The dashed curves are plotted when λac ≠ λsp, where the experimental value of λsp ≈ 588 nm. This means that the SPP wavelength λsp is NOT in resonance with the wavelength λac of the exciton transition |a⟩ ↔ |c⟩. The dashed curve corresponds to λac ≠ λsp (off resonance). Next, we have plotted the solid curve when the SPP wavelength λsp is in resonance with the wavelength of the transition |a⟩ ↔ |c⟩, that is, λac ≈ λsp. Note that when both wavelengths are NOT in resonance with each other, the PL and SCS spectra splitting from one peak to two peaks are very small. This is because when both wavelengths are NOT resonant (i.e., λac ≠ λsp), the anomalous DDI has a very small value. However, when both wavelengths are in resonance with each other, the splitting in PL and SCS spectra is large from one peak to two peaks. This is because the anomalous DDI has the largest value when both wavelengths are in resonance (i.e., λac ≈ λsp). Note that the peaks are shifted in the presence of the DDI. The shift is due to the DDI coupling. In summary, one can say that the anomalous DDI is responsible for splitting the PL and SCS spectra. We have not included the effect of temperature in our theory because its effect is negligible as experimentally found.1 It is found as temperature is raised from T = 4 K to T = 293 K, the Rabi splitting decreases very little in the SCS spectra. See Figure 2d in ref 1. The inclusion of the temperature effect makes the theory very complicated and one cannot get an analytical expression for the PL and SCS. However, there is a simple way to include the effect of the temperature dependence in the density matrix formulation by following the method of ref 34. According to this reference, the effect of temperature can be included by multiplying the decay rate term γb by (1 + nB), where nB is the boson distribution function written as nB = 1/[exp(ℏω/kBT) − 1].34 The temperature-dependent part changes the height, the width, and locations of the peaks.1

Figure 5. (a) PL is plotted for different values of λsp. (b) The scattering cross section is plotted for different values of λsp. There are three curves. In both figures, the dotted curve is for the anomalous DDI = 0 [the free J-aggregate (a) and free MNP (b)], the solid line is for λsp = λab = 588 nm (on resonance), and the dashed line is for λsp ≠ λac(off resonance).

= 2.1.1,6 Note that there should be some difference between ϵs and ϵq. This is because the shell is made of the ensemble of QEs, and there is extra space between the close-packed QEs. It will be impossible to calculate the dielectric constant of the shell. The difference will be very small, and we found that it will not change the physics and calculations of the paper. That is why, in our numerical calculation, we have taken the dielectric constant of the shell (ϵs) and the dielectric constant of the QE (ϵq) to be the same. It is a good approximation. The results are plotted in Figure 2a,b for scattering cross section. In Figure 2a, the open diamonds and dashed curve represent the scattering data and theoretical calculations for the free MNP. Similarly, in Figure 2b, the open circles and solid curve represent the scattering data and theoretical simulations for the J-aggregate dye molecule in the core− shell hybrid. Note that a good agreement between our theory and experiment is found on the qualitative basis but not on the quantitative basis. The reason for the disagreement on the quantitative basis is because we used a nanosphere MNP whereas a nanoprism was used in the experiment. The nanosphere MNP is used so that we can get an analytical expression for the scattering cross section. The analytical expression for the scattering cross section will be useful to fabricate new core−shell nanohybrids and develop new experiments. These expressions can be also used in the fabrication of new nanoswitches and nanosensors. We have also compared our PL theory with the experimental data of the J-aggregate dye molecules and core−shell hybrid.1 The results are plotted in Figure 3a,b for the PL for free Jaggregate and the core−shell hybrid made from J-aggregate dye molecules and the Ag-MNP, respectively. In Figure 3a, the open diamonds and dashed curve represent the PL data and theoretical simulations for the free J-aggregate. Similarly, in Figure 3b, the open circles and solid curve represent the scattering data and theoretical calculations for the core−shell hybrid. Note that a good agreement between our theory and experiment is found on the qualitative basis but not on the quantitative basis because we used the spherical MNP instead of the prism MNP. It is an important note that our theory for the PL and scattering cross section shows that the peak of the free Jaggregate dye molecule splits into two peaks when the Jaggregate dye molecules are part of the core−shell hybrid. The splitting is mainly due to the strong coupling between excitons and the anomalous DDI. Physics of the splitting can be explained in the strong coupling limit and can be explained as 26589


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In Figure 3a, we studied the PL because of the QE (Jaggregates) only. In Figure 3b, we studied the PL because of the core−shell hybrid made from the Ag-MNP and the ensemble of QEs (J-aggregates). In the core−shell hybrid, the excitons in QEs and SPPs in the Ag-MNP interact with each other via the DDI. The DDI interaction is absent in Figure 3a because the experiment was performed on the QE (Jaggregates) alone. On the other hand, the DDI is present in Figure 3b because the experiment was performed on the core− shell hybrid. That is why decay rates are different in both Figure 3a,b. We can conclude from the above explanations about Figures 2 and 3 that the experimental environment and conditions in Figures 2 and 3 are different that is why the parameters used in these figures are also different. We found that the splitting in PL is significantly smaller than the splitting in scattering for the core−shell hybrid. In principle, these two splittings must be the same if the sample environment and experimental conditions were the same but they are not. The splitting is due to the strong DDI. This means that the DDI coupling in the SCS experiment in Figure 2b is stronger than that of the PL experiment in Figure 3b. In other words, in ref 1 the SCS experiment was performed in the stronger coupling limit than that of the PL experiments. The main aim of the paper is to develop a theory for core− shell structures and to obtain analytical expressions of the PL and SCS so that experimentalists in the plasmonic field can use to explain their experiments and plan new experiments. It will be impossible to get an analytical expression for the PL and SCS for a prism shape MNP. Because we are considering three level QEs and make use of the Maxwell and density matrix equations for the calculation of PL and SCS, it is recommended to use a numerical simulation method such as the Maxwell−Liouville−von Neumann FDTD method35,36 to calculate the PL and SCS for the prism MNP. The most important achievement of our theory is that with minor modifications, our theory can be applied for MNPs with different shapes such as single metallic nanoshells, double metallic nanoshells and metallic rods, metallic spheroidal, metallic ellipsoidals and so forth. Our theory predicts that there will be splitting in the PL and SCS spectra because of the strong DDI from one peak to two peaks for all shapes of MNP. The only thing which changes for different MNP shapes is the position of the peaks, the height of peaks, and energy splitting between two peaks. We have developed theories for differently shaped MNPs such as rods,37 ellipsoidal,10 single nanoshell,12 double nanoshells,38 and graphene flake,39 except prism. The most fascinating and interesting part of our theory is that the physics of our finding will NOT change and it can be applied to any shape of the MNP. For different shapes, the location of SPP frequency will vary. Therefore, in our theory one can substitute the SPP frequency for different shapes. For example, in the present paper, we mimic the SPP frequency of the prism by changing the spheroidal shape.

We have shown that the splitting and height of the scattering spectrum can be controlled by the anomalous DDI. Hence, core−shell hybrids can be used to develop nanosensors which can measure the splitting and height of the PL and scattering spectra. These hybrids can also be used to develop nanoswitches by switching the spectrum from one peak to two peaks. Here, one peak can be thought of as OFF position and two peaks can be thought of as ON position. We showed that the anomalous DDI depends on the thickness and dielectric constant of the shell. The thickness of the shell can be tuned by applying ultrafast pulse pressure or stress, whereas applying an external intense control laser can modify the dielectric constant of the shell. These very interesting findings and these nanosensors and nanoswitches can be used for applications in nanotechnology and nanomedicines. We want to comment on the three-level system used in the present theory. In the absence of the MNP, the PL and SCS have one peak. This means that there is one transition |a⟩ ↔ | b⟩, where |b⟩ is the ground state. In the presence of the MNP, PL and SCS spectra have two peaks. This is because of the strong coupling between the MNP and QEs due to the strong DDI coupling |a⟩ ↔ |c⟩. Because of this strong coupling, dressed states are created in the QEs. The excited state |a⟩ splits into two dressed states called |a−⟩ and |a+⟩ because of the strong DDI coupling. According to the quantum mechanics, the ground state does not split. Therefore, two transitions occur, |a−⟩ ↔ |b⟩ and |a+⟩ ↔ |b⟩. One can see clearly that in the calculation of the PL and SCS, we calculate ρab (eq 20) because of the two levels |a⟩ and |b⟩. This means basically we are using the two levels in the calculation of the PL and SCS. In other words, two levels play the main role in the present theory. In summary, the density matrix method creates the dressed states along with transition matrix elements to calculate the PL and SCS. This theory is very elegant, simple, and analytical expressions for the PL and SCS can be obtained. On the other hand, one can also use two levels |a⟩ and |b⟩ in the calculation of the PL and SCS. In this case, we have to solve the Schrodinger equation with the strong DDI coupling Hamiltonian and find the dressed states |a−⟩ and |a+⟩. In this case, we have three states |b⟩, |a−⟩, and |a+⟩ (three-level system). After that we have to use the density matrix method to calculate the transition matrix elements ρa−b (|a−⟩ ↔ |b⟩) and ρa+b (|a+⟩ ↔ |b⟩) to calculate the PL and SCS. This approach will be very messy and complicated mathematically. We tried this method for other systems, and we found it very tedious and complicated to get an analytical expression. Therefore, we abandoned this method. We want to make comments on the parameters used in figures Figures 2 and 3. In Figure 2a, we studied the scattering cross section because of the Ag-MNP only. In Figure 2b, we studied the SCS because of the core−shell hybrid made of the Ag-MNP and the ensemble of QEs (J-aggregates). In the core−shell hybrid, the excitons in QEs and SPPs in the AgMNP interact with each other via the exciton−SPP interaction and also the DDI. The DDI plays a very important role in decay rates of both excitons and SPPs. The DDI also plays a critical role in the splitting of the SCS spectrum. The DDI is absent in Figure 2a because the experiment was performed on the Ag-MNP alone. On the other hand, the DDI is present in Figure 2b because the experiment was performed on the core− shell hybrid. That is why parameters used in both figures are different.

5. CONCLUSIONS In conclusion, we have developed a theory for the PL and scattering cross section for core−shell nanohybrids using the density matrix method. We have compared our theory with the scattering cross section experiment of the J-aggregate-Ag MNP core−shell nanohybrid.1 A good agreement between our theory and experiments is found. It is found that the PL and scattering spectrum with one peak splits into two peaks due 26590


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The Journal of Physical Chemistry C mainly to the anomalous strong DDI. These findings can be used for fabricating nanosensors and nanoswitches.

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(14) Fainberg, B. D.; Rosanov, N. N.; Veretenov, N. A. LightInduced “Plasmonic” Properties of Organic Materials: Surface Polaritons and Switching Waves in Bistable Organic Thin Films. J. Appl. Phys. 2017, 110, 203301. (15) Yannopapas, V.; Paspalakis, E. Giant Enhancement of DipoleForbidden Transitions via Lattices of Plasmonic Nanoparticles. J. Mod. Opt. 2015, 62, 1435−1441. (16) Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P. Plasmons in Strongly Coupled Metallic Nanostructures. Chem. Rev. 2011, 111, 3913−3961. (17) Singh, M. R. Dipole-Dipole Interaction in Photonic-Band-Gap Materials Doped with Nanoparticles. Phys. Rev. A 2007, 75, 043809. (18) Mazenko, G. F. Quantum Statisitcal Mechanics; John Wiley & Sons Inc.: New York, 2000; Section 5.8. (19) Kittel, C. Introduction to Solid State Physics, 6th ed.; John Wiley & Sons Inc.: New York, 1996; Chapter 13. (20) Ali Omar, M. Elementary Solid State Physis; Addison-Wesley: New York, 1993; Section 8.11. (21) Gerstein, J. L.; Smith, F. W. The Physics and Chemistry of Materials; John Wiley & Sons Inc.: New York, 2001; Chapter 15. (22) Eyring, H.; et al. Statistical Mechanics and Dynamics; Wiley: New York, 1982. (23) Lorentz, H. The Theory of Electrons; Dover: New York, 1952. (24) Ohtsu, M.; Kobayashi, K. Optical Near Fields: Introduction to Classical and Quantum Theories of Electromagnetic Phenomena at the Nanoscale; Springer: Berlin, 2013. (25) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press: Cambridge, 2006; p 266. (26) Yang, U.; D’Archangel, J.; Sundheimer, M. L.; Tucker, E.; Boreman, G. D.; Raschke, M. B. Optical dielectric function of silver. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 235137. (27) Nash, D. J.; Sambles, J. R. Surface plasmon-polariton study of the optical dieletric function of silver. J. Mod. Optic. 1996, 43, 81−91. (28) Singh, M. R.; Cox, J. D.; Brzozowski, M. Photoluminescence and Spontaneous Emission Enhancement in Metamaterial Nanostructures. J. Phys. D: Appl. Phys. 2014, 47, 085101. (29) Singh, M. R. Electronic, Photonic, Polaritonic and Plasmonic Materials; Wiley Custom: Toronto, 2014. (30) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997. (31) Jiang, Y.; Pillai, S.; Green, M. A. Realistic Silver Optical Constants for Plasmonics. Sci. Rep. 2016, 6, 30605. (32) Baranov, D. G.; Wersäll, M.; Cuadra, J.; Antosiewicz, T. J.; Shegai, T. Novel Nanostructures and Materials for Strong LightMatter Interactions. ACS Photonics 2018, 5, 24−42. (33) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997; Sections 7.3 and 7.5. (34) Scully, M. O.; Zubairy, M. S. Quantum Optics; Cambridge University Press: London, 1997; Section 8.2. (35) Riesch, M.; Tchipev, N.; Senninger, S.; Bungartz, H.-J.; Jirauschek, C. Performance evaluation of numerical methods for the Maxwell-Liouville-von Neumann equations. Opt. Quantum Electron. 2018, 50, 112. (36) Sukharev, M.; Nitzan, A. Optics of exciton-plasmon nanomaterials. J. Phys.: Condens. Matter 2017, 29, 443003. (37) Singh, M. R.; Schindel, D. G.; Hatef, A. Dipole-dipole interaction in a quantum dot and metallic nanorod hybrid system. Appl. Phys. Lett. 2011, 99, 181106. (38) Guo, J.; Black, K.; Hu, J.; Singh, M. Study of Plasmonics in hybrids made from a quantum emitter and double metallic nanoshell dimer. J. Phys.: Condens. Matter 2018, 30, 185301. (39) Singh, M. R.; Brzozowski, M. J.; Apter, B. Effect of phononplasmon and surface plasmon polaritons on the energy absorption in quantum dot and graphene nanocomposite deposited on polar crystals. J. Appl. Phys. 2016, 120, 124308.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Mahi R. Singh: 0000-0003-0930-4003 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS One of the authors (M.R.S.) is thankful to the Natural Sciences and Engineering Research Council of Canada for the research grant. We also thank Timur Shegai, Department of Physics, Chalmers University of Technology, Goteborg, Sweden. Without his help, the paper would have not been completed.

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REFERENCES

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Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum

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