Optical spectra of plasmon–exciton core–shell nanoparticles: an anisotropic classical model eliminates discrepancies in experiments

Alexey D. Kondorskiy

doi:10.3788/COL202422.093602

1. Introduction

Studies of hybrid metal–organic nanoparticles have attracted considerable interest due to the wide range of potential applications of such structures in nanophotonics^[1-6]. Because of their unique properties, core–shell nanoparticles are being exploited in a wide range of applications, including light harvesting^[7-9] and light emitting devices^[9,10], sensing^[11,12], biotechnology^[12,13], and neuromorphic optical computing^[14,15]. Nanoparticles composed of a metallic core and an organic shell of ordered J- and H-aggregates of molecules are of particular interest^[6,16-21]. The optical properties of such structures are determined by the electromagnetic near-field interaction between Frenkel excitons in the outer organic shell and plasmons localized at the surface of the metallic core. As the strength of this plasmon–excitonic (plexcitonic) coupling increases, the plasmonic band in the metallic core and the excitonic band in the organic shell form hybrid polariton states of the entire nanosystem with qualitatively new properties compared to those of the individual components^[17]. The spectral properties of these polariton states can be controlled by tuning the parameters of the plasmonic and excitonic bands. Advances in the synthesis of plasmonic nanostructures allow the fabrication of metallic cores of different sizes and shapes with tunable plasmonic band parameters^[22]. The use of molecular J-aggregates of a cyanine dye in the organic shell significantly enhances the plexcitonic coupling in the hybrid system due to the narrow and intense absorption features of these aggregates^[23,24]. The size, shape, and optical constants of the metallic and organic compounds of the nanoparticle determine different possible regimes of plexcitonic coupling. These regimes include weak, strong, and ultra-strong coupling, which have been extensively discussed in several review articles^[1,6,17,25].

Plexcitonic systems are typically simulated using classical electromagnetic tools such as Mie theory and its extensions^[26-28] or numerical Maxwell equation solvers^[29]. The dielectric functions of the materials are described by the Drude–Lorentz model for the metallic core and the scalar (isotropic) classical Lorentz oscillator model for the organic shell. When the plexcitonic coupling is sufficiently strong, predictions obtained with the described classical approach are found to disagree with experimental results^[30]. The discrepancy is particularly noticeable in the studies of metal–organic structures with a shell made of the J-aggregate of the TDBC dye since the value of the reduced oscillator strength of its excitonic band is large ( $f = 0.36$ )^[31,32]. The plexcitonic coupling energy (Rabi splitting) in the system reaches hundreds of meV. In this case, the above isotropic classical model predicts a third resonance between the two polariton bands^[16,33], which is associated with an uncoupled shell mode. However, this resonance has never been observed experimentally in spectra of pure metal–organic particles with a shell of J-aggregates (see discussion in Ref. [30]), except for samples containing residual uncoupled emitters^[34].

Regarding this problem, it is worthwhile to mention here some recent discussions about the validity of the classical description of the nanoparticle excitonic subsystem. A remarkable number of works is devoted to the discussion of the importance of the consideration of quantum effects induced by the enhancement of the electromagnetic field in the vicinity of the metallic surface^{[6,17,30,31,33]}. In Ref. [30], it was suggested that by considering the quantum effects, one can avoid the discrepancies between the predictions of the classical model and the experimental data in the cases of strong and ultra-strong plexcitonic coupling. The heuristic quantum model introduced therein provides an expression for the frequency-dependent dielectric function of the shell material that includes the collective vacuum Rabi frequency, which depends on the size and shape of the structure. Using this model, the authors obtained excellent agreement with a series of experimental extinction spectra of metal–organic particles with different coupling strengths due to systematic size variation. To the best of our knowledge, Ref. [30] is the only work that explicitly addresses the problem of discrepancies between the predictions of the classical model and the experimental results.

In this Letter, we give a completely different theoretical explanation for the discrepancies between the predictions of the isotropic classical model and the experimental results on the spectral properties of metal–organic particles with a shell made of J-aggregates. We believe that the appearance of the uncoupled shell mode resonance in the theoretical spectra, which is not observed experimentally, is not a result of the limitation of the classical description of the optical properties of the J-aggregate as compared to the quantum description. This peak appears in the theoretical results under the assumption that the optical response of molecular J-aggregates is isotropic, so that the frequency-dependent dielectric function of the organic shell is taken in conventional scalar form, $ε_{J} (ω) = ε_{J}^{\infty} + \frac{f_{J} ω_{J}^{2}}{ω_{J}^{2} - ω^{2} - i ω γ_{J}},$ (1)where $ω_{J}$ is the resonance frequency of the J-band, $γ_{J}$ is its full width, $f_{J}$ is the reduced oscillator strength, and $ε_{J}^{\infty}$ is the permittivity outside the J-band. Recently, it has been demonstrated that several experimental results for optical spectra of metal–organic nanostructures^[35-37] cannot be explained in terms of the isotropic response model. The experimental data can be well described if the anisotropic and orientational effects of the outer J-aggregate shell are taken into account in the theoretical treatment of the optical properties of hybrid metal/J-aggregate and metal/spacer/J-aggregate nanoparticles^[21].

2. Theoretical Approach

To confirm our statement, we perform numerical simulations of the extinction spectra of bare and J-aggregate coated gold spherical nanoparticles and nanorods considered in Ref. [30] with the dielectric function of an anisotropic molecular J-aggregate represented in tensor form with frequency dependence described by the classical Lorentz oscillator model. The theoretical approach used is described in Ref. [21].

To calculate the spectra of core–shell spherical nanoparticles, we use the formulas of the anisotropic version of Mie theory^[26]. These formulas are similar to those of the original Mie theory with appropriately modified indices and arguments of spherical Bessel functions. This similarity could be used to extend the anisotropic single core–shell theory to treat the aggregate of nanoparticles using various approaches^[27,28].

Numerical simulations of the extinction spectra of TDBC-coated gold nanorods are performed using the finite-difference time-domain (FDTD) method implemented in the freely available software package MIT Electromagnetic Equation Propagation (MEEP)^[29]. The details of the calculation method can be found in Ref. [38]. Using the numerical method, it is possible to evaluate the cross sections for a given polarization of the incident radiation. We denote by $σ_{α}^{(ext)}$ the extinction cross section obtained for the case when the electric field is directed along the $α$ axis, with $α = X$ , $Y$ , $Z$ [see Figs. 1(a) and 3(d)]. Due to the axial symmetry of the nanorod around the Y-axis, one has $σ_{X}^{(ext)} = σ_{Z}^{(ext)}$ . The final results are presented for orientationally averaged cross sections corresponding to naturally polarized light incident on the randomly oriented nanorods in colloidal solutions, $⟨ σ^{(ext)} ⟩ = [σ_{Y}^{(ext)} + 2 σ_{Z}^{(ext)}] / 3$ .

We assume axial symmetry of the optical properties of the molecular J-aggregate and describe the components of the dielectric function tensor of the organic shell in terms of $ε_{∥}$ and $ε_{⊥}$ , which are the tensor components associated with the direction parallel to the aggregate orientation axis (longitudinal component) and perpendicular to it (transverse component), respectively. We assume that the TDBC J-aggregate could be described by the longitudinal Lorentzian only (a general expression is discussed in Ref. [21]), ${\begin{cases} ε_{∥} (ω) = ε_{J}^{\infty} + \frac{f_{J}^{∥} ω_{J}^{2}}{ω_{J}^{2} - ω^{2} - i ω γ_{J}} \\ ε_{⊥} (ω) = ε_{J}^{\infty} . \end{cases}$ (2)

This assumption is confirmed by experimental observations of the orientation-dependent optical spectra of TDBC-like aggregates, which exhibit anisotropic absorption with a strong sharp band with the dipole oriented along a single axis^[39,40].

Following Ref. [21], we consider three cases of molecular arrangements in the J-aggregate shell: (I) normal aggregate orientation, when the direction of the $ε_{∥}$ component is parallel to the normal direction to a nanoparticle surface; (II) tangential aggregate orientation, in which the direction of the $ε_{∥}$ component is perpendicular to the normal direction, and we assume the equiprobable orientation of the aggregate axis along the tangential plane of a nanoparticle; (III) equiprobable orientation of the aggregate axis in space. We describe these cases with a tensor oriented along the $e_{X}$ , $e_{Y}$ , and $e_{Z}$ axes [see Figs. 1(a) and 3(d)]. For the case of Eq. (2), the general tensor expressions are simplified and take the form of $\hat{ε} (ω) = ε_{J}^{\infty} \hat{I} + \frac{ω_{J}^{2}}{ω_{J}^{2} - ω^{2} - i ω γ_{J}} \hat{f},$ (3)where $\hat{I}$ is the unit tensor, and the reduced oscillator strength tensor, $\hat{f}$ , can be written for normal, tangential, and equiprobable orientations, respectively, as ${\hat{f}}_{norm} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & f_{J}^{∥} \end{matrix}),$ (4) ${\hat{f}}_{tang} = (\begin{matrix} \frac{1}{2} f_{J}^{∥} & 0 & 0 \\ 0 & \frac{1}{2} f_{J}^{∥} & 0 \\ 0 & 0 & 0 \end{matrix}),$ (5) ${\hat{f}}_{epo} = (\begin{matrix} \frac{1}{3} f_{J}^{∥} & 0 & 0 \\ 0 & \frac{1}{3} f_{J}^{∥} & 0 \\ 0 & 0 & \frac{1}{3} f_{J}^{∥} \end{matrix}) .$ (6)

Figure 1.(a) Schematic view of a Drude sphere with a dye shell under study. The vectors e_X, e_Y, and e_Z show the basis for the permittivity tensor of the shell material: e_Z is directed normally to the surface, while e_X and e_Y lie in the tangential plane. (b)–(j) Light extinction cross section of a Drude sphere with a dye shell with varying reduced oscillator strength, f_J, and different molecular arrangements in the organic shell. (b)–(d) Results are obtained for f_J = 0.15 and (b) isotropic (equiprobable) J-aggregate orientation, (c) tangential J-aggregate orientation, and (d) normal J-aggregate orientation in the dye shell. In (e)–(g) and (h)–(j), the same results are calculated for f_J = 0.08 and f_J = 0.03, respectively.

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The case of the equiprobable orientation of Eq. (6) is equivalent to the isotropic model of Eq. (1) with the oscillator strength of the single longitudinal band being three times larger than the oscillator strength of the isotropic model. To compare the results obtained with isotropic and anisotropic models, we take a certain value of $f_{J}$ and assume that it can be estimated from the situation of the equiprobable orientations that take place in solution of molecular aggregates. The results for normal and tangential orientations are obtained with Eqs. (4) and (5) with $f_{J}^{∥} = 3 f_{J}$ . Because of this relation, we have a single value of $f_{J}$ to refer to parameters of both the isotropic and anisotropic models used.

3. Results and Discussion

First, we perform calculations of the light extinction cross section for a model Drude sphere with a dye shell in water. Our results for this system are shown in Figs. 1 and 2. The permittivity of the core is described by the Drude model for gold, and the parameters for the permittivity of the J-aggregate shell are $ε_{J}^{\infty} = 1.7$ , $ω_{J} = 2.5 eV$ (wavelength $λ = 495 nm$ ), and $γ_{J} = 0.05 eV$ . The geometrical parameters are $R = 12.5 nm$ and $s = 3 nm$ [see Fig. 1(a)]. The results of calculations performed with the isotropic classical description of the dye permittivity clearly show the growth of the uncoupled shell band around $λ = 490 nm$ with increasing reduced oscillator strength, $f_{J}$ . However, there is no such band in the results obtained with the anisotropic shell model for both the tangential and normal J-aggregate orientations in the organic shell. The electromagnetic energy density distributions around a sphere shown in Fig. 2 demonstrate a significant effect of the J-aggregate orientations on the shape of the electromagnetic field spatial profile.

Figure 2.(a) Light extinction cross section of a bare Drude sphere and a Drude sphere with a dye shell of isotropic J-aggregate orientation with f_J = 0.15 [see Fig. 1(b)]. (b)–(i) Electromagnetic energy density distributions in the plane lying along the light polarization and passing through the particle center. Results are presented for light wavelengths, λ, corresponding to different resonances in the light extinction spectra (see Fig. 1). (b) Bare Drude sphere, λ = 485 nm. (c)–(e) Drude sphere with a dye shell with isotropic J-aggregate orientation: (c) λ = 458 nm, (d) λ = 490 nm, and (e) λ = 517 nm. (f)–(g) Normal J-aggregate orientation: (f) λ = 433 nm and (g) λ = 512 nm. (h)–(i) Tangential J-aggregate orientation: (h) λ = 470 nm and (i) λ = 510 nm. The reduced oscillator strength of a dye shell, f_J = 0.15 for all cases. Light polarization is parallel to the horizontal axis. The color maps represent the electromagnetic energy density in the logarithmic scale.

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Second, we perform calculations of the extinction spectra of bare and TDBC-coated gold nanorods of different sizes as measured and simulated in Ref. [30]. Our results are shown in Fig. 3. The notation of the sizes of the nanorods studied corresponds to the transverse diameters of 11, 18, 34, and 37 nm as measured in Ref. [30]. We use the following actual sizes of the nanorods: for a transverse diameter of 11 nm, $2 R = 11.5 nm$ , $L = 19.5 nm$ , $r = 5 nm$ ; for diameter of 18 nm, $2 R = 19.5 nm$ , $L = 32 nm$ , $r = 7 nm$ ; for diameter of 34 nm, $2 R = 36 nm$ , $L = 58 nm$ , $r = 14 nm$ ; for diameter of 37 nm, $2 R = 41 nm$ , $L = 66 nm$ , $r = 15 nm$ . The permittivity of gold was taken from Ref. [41]. For bare nanorods, a 1 nm thick citrate capping layer with a permittivity of 2.37 is included (see Ref. [42]). The TDBC shell thickness is $s = 3 nm$ . For the TDBC J-aggregate, we use the parameters of the Lorentz oscillator model, which are in agreement with those previously reported^[30,32,34]: $ε_{J}^{\infty} = 2.3$ , $ω_{J} = 2.12 eV$ ( $λ = 585 nm$ ), $γ_{J} = 0.048 eV$ , and $f_{J} = 0.22$ . We use the same parameters of the dielectric function of the TDBC J-aggregate for all nanorod sizes in our calculations. The permittivity of the environment was set to 1.96.

Figure 3.(a)–(c) Light extinction spectra of bare nanorods of different sizes: (a) 11 nm, (b) 18 nm, and (c) 34 nm. For the sake of brevity, the results for 37 nm are not shown because they are similar to those for 34 nm. In these figures, the solid orange lines represent the experimental results^[30], and the solid black lines represent our calculations. (d) Schematic view of a TDBC-coated gold nanorod. (e)–(h) and (i)–(l) Light extinction spectra of TDBC-coated gold nanorods of different diameters: (e) and (i) 11 nm, (f) and (j) 18 nm, (g) and (k) 34 nm, and (h) and (l) 37 nm. (e)–(h) show a comparison of experimental results^[30] (solid orange lines) and numerical simulations performed with the heuristic quantum model^[30] (dashed–dotted blue lines) with our results obtained with the classical anisotropic model for tangential J-aggregate orientation in the TDBC shell (solid black lines). (i)–(l) show the effect of J-aggregate orientation: the solid orange lines represent the experimental results^[30], the solid black lines represent tangential orientation, the thin red lines with dashes represent isotropic (equiprobable) orientation, and the dotted green lines represent normal orientation.

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Figures 3 and 4 show our results for the extinction spectra of bare and TDBC-coated gold nanorods. It is seen that the results obtained for the tangential J-aggregate orientation in the organic shell of the TDBC-coated gold nanorods are in good agreement with the experimental data^[30]. Our calculations for the cases of isotropic (equiprobable) and normal J-aggregate orientations do not show a sufficient quality of agreement with the experimental data for all nanorod sizes. Therefore, we conclude that J-aggregates of the TDBC dye are assembled on the metallic surface so that their axis is directed along the surface. Note that the uncoupled shell resonance appears in the spectra calculated for isotropic J-aggregate orientation around $λ = 585 nm$ [see Figs. 3(i) and 3(j)].

Figure 4.(a)–(c) Light extinction cross sections of a TDBC-coated gold nanorod of size 34 nm for (a) isotropic, (b) normal, and (c) tangential J-aggregate orientations in a shell. The black curves (A) are orientation-averaged cross sections, ⟨σ^(ext)⟩. The red curves (B) and blue curves (C) are cross section contributions from cases of light polarization parallel and perpendicular to the nanorod axis, respectively (i.e., σ_Y^(ext)/3 and 2σ_Z^(ext)/3). (d)–(o) Electromagnetic energy density distributions in the Y–Z plane passing through the nanorod center [see Fig. 3(d)]. Results are presented for light wavelengths, λ, corresponding to different resonances in the light extinction spectra. (d)–(i) Isotropic J-aggregate orientation: (d) λ = 551 nm, (e) λ = 580 nm, (f) λ = 635 nm, (g) λ = 520 nm, (h) λ = 570 nm, and (i) λ = 590 nm. (j)–(l) Normal J-aggregate orientation: (j) λ = 635 nm, (k) λ = 546 nm, and (l) λ = 566 nm. (m)–(o) Tangential J-aggregate orientation: (m) λ = 577 nm, (n) λ = 625 nm, and (o) λ = 520 nm. The direction of the light polarization is shown in the upper right corner of each figure. The nanorod size is 34 nm in all cases. The color maps represent the electromagnetic energy density in the logarithmic scale.

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The predictions of the anisotropic classical model have a similar quality of agreement with experimental data as those of the heuristic quantum model^[30] (see Fig. 3). However, our model is much simpler because it does not use quantum effects. Also, this classical model does not require ad hoc modification of the spectroscopic parameters of the organic material due to its assembly on the metal surface, while the heuristic quantum model does. We emphasize that our theoretical results are obtained with the same set of optical parameters of the excitonic subsystem for all nanostructure sizes.

4. Conclusion

In conclusion, we report the results of theoretical studies of the optical properties of hybrid metal–organic nanoparticles with a metallic core and an organic shell made of molecular J-aggregates. We address the problem of drastic discrepancies between the theoretical predictions of the traditional isotropic classical oscillator model and some of the available experimental spectra, which appear in such systems in cases of strong and ultra-strong plasmon–exciton coupling. These discrepancies are manifested in the difference in the number of spectral bands predicted by theory and observed experimentally. The traditional classical model predicts a third resonance between the two polariton bands, which is not observed experimentally. We show that the observed discrepancies are not due to the limitations of the classical oscillator model, but to the treatment of the organic shell as the optically isotropic medium.

Since optical anisotropy is an essential feature of molecular J-aggregates, we have studied the effects of the orientation of molecular J-aggregates in the organic shell on the optical spectra of the hybrid nanostructure. We describe anisotropic molecular J-aggregates with a dielectric function represented in tensor form with a frequency dependence described by the classical Lorentz oscillator model. We show that, in the case of strong and ultra-strong plasmon–exciton coupling, the extinction spectra of the hybrid nanoparticle differ significantly for isotropic, tangential, and normal orientations of the molecular J-aggregate in the organic shell. While the third peak is observed in the case of isotropic orientation, it is not observed in either the case of normal orientation or tangential orientation. Furthermore, we show that the experimental extinction spectra of TDBC-coated gold nanorods^[30] can be excellently reproduced if one assumes a tangential orientation of molecular J-aggregates in shell when describing the permittivity of the J-aggregate with the use of the classical oscillator model. However, this system cannot be treated with the conventional isotropic shell model.

Our results clearly demonstrate the importance of incorporating the anisotropic optical properties of J-aggregates into the theoretical description of the optical properties of hybrid metal–organic core–shell nanostructures.

Category: Nanophotonics, Metamaterials, and Plasmonics

Received: Dec. 8, 2023

Accepted: May. 8, 2024

Published Online: Sep. 12, 2024

The Author Email: Alexey D. Kondorskiy (kondorskiy@lebedev.ru)

DOI:10.3788/COL202422.093602

CSTR:32184.14.COL202422.093602