## Abstract

Splitting of a Tamm plasmon polariton at the interface between a metal and a nanocomposite layer conjugated with a photonic crystal (Bragg mirror) has been theoretically investigated. The splitting can be implemented when the Tamm plasmon polariton frequency approaches the resonance frequency of the nanocomposite, which consists of metallic nanoballs dispersed in a transparent matrix. The reflectance spectrum of the $p$-polarized waves at the normal incidence of light has been calculated. It has been established that the Tamm plasmon polariton splitting significantly depends on the concentration of nanoballs in the nanocomposite layer and its thickness. The solution of the dispersion equation derived for a sample with semi-infinite mirrors has been compared with the spectra obtained for a finite structure by the transfer matrix method.

© 2021 Optical Society of America

## 1. INTRODUCTION

The optical Tamm state (OTS) is a surface state implemented at the interface between two media where multiple rereflections occur and light gets trapped between two mirrors [1,2]. The OTS is an optical analog of the electronic Tamm state, in which the electron density is localized at the boundary of the periodic potential of a crystal. This state can be excited at the interface between a photonic crystal (PC) and a metallic mirror. In this case, a light wave appears associated with the surface plasmon, i.e., oscillations of free electrons at the surface of a metal, and is called the Tamm plasmon polariton (TPP) [3–7]. In contrast to the surface plasmon, the TPP can be excited for both the $p$ and $s$ polarizations of light, even at its normal incidence onto the interface; the energy does not propagate along the interface, but exponentially decreases with the distance from the interface in both directions. The TPP manifests itself in experiments as a narrow resonance in the optical transmittance or reflectance spectrum of a sample at wavelengths within the photonic bandgap.

The TPPs have found application in fundamentally new devices, including absorbers [8], switches [9], lasers [10], white organic light-emitting diodes [11], heat emitters [12], sensors [13,14], and spontaneous emission amplifiers [15]. A high degree of field localization at the TPP frequency makes it possible to reduce the threshold of generation of nonlinear effects [16–18] and implement the extremely high light transmission through a nanohole [19].

The TPPs and devices based on them are formed, as a rule, using planar metallic films conjugated with a PC. In this case, the optimization of the optical properties of such structures via changing the parameters of the films is exhausted by choosing their material and thickness. New opportunities for optimizing the characteristics of the TPPs are offered by the use of resonant materials as structural elements. Such materials can be metal–dielectric nanocomposites (NCs), which consist of a dielectric matrix and metallic nanoballs uniformly distributed over its volume. The position and width of the NC resonance region, which lies in the visible spectral range, depend on the permittivity of initial materials and the concentration, size, and shape of nanoparticles. The effective characteristics of an NC consisting of metallic nanoparticles suspended in a dielectric matrix are formed by their plasmon resonance and, in the optical range, can take unique values atypical of natural materials [20–24]. The features of the spectral properties of the TPPs implemented at the interface between a PC and a metal–dielectric NC layer characterized by an effective resonance permittivity of ${\varepsilon _{\rm{mix}}} \lt 0$ were explored in [25,26].

The aim of this study was to investigate the TPP localized at the interface between a metallic mirror and a resonant NC layer conjugated with a Bragg mirror. The TPP splitting effect is demonstrated and analyzed for the case when the NC resonance frequency coincides with the TPP frequency or is similar to it. The possibility of controlling the position and number of Tamm modes by changing the parameters of the metal–dielectric NC layer is shown. It is established that the numerical solution of the dispersion equation for an unbounded sample is consistent with the reflectance spectrum obtained by the transfer matrix method for a bounded structure.

The application of the TPP splitting effect in the creation of a two-mode laser [10] and an organic solar cell [27] may be an incentive for experimental research of the new results obtained in this work. In [10], the generation of a single-mode laser with Tamm plasmon modes at the boundary of a multilayer dielectric structure and a metal was obtained. Due to the features of these modes, Tamma lasers open up prospects for the implementation of integrated microlasers. In [27], a structure comprising metal film-NC-PC is suggested as an organic solar cell. In such a structure, a TPP can be excited. An additional radiation absorption line appears in the active layer (NC layer) at the TPP wavelength. This mechanism of increasing the efficiency of organic solar cells was demonstrated.

## 2. DESCRIPTION OF THE MODEL

We consider a structure schematically shown in Fig. 1. The structure consists of a metallic layer, an NC layer, and a one-dimensional (1D) PC. The PC unit cell includes layers $a$ and $b$ with respective permittivities ${\varepsilon _a}$ and ${\varepsilon _b}$ and thicknesses ${d_a}$ and ${d_b}$; the lattice period is $L = {d_a} + {d_b}$. The metallic layer has a permittivity ${\varepsilon _m}$. The NC with a thickness $d$ is formed by metallic nanoballs dispersed in a transparent matrix and has an effective permittivity ${\varepsilon _{\rm{mix}}}$. We assume that the medium surrounding the layered structure is vacuum.

To describe the optical properties of the NC layer, we use the effective medium approximation in the Maxwell–Garnett model, which is widely used in studying matrix media with a small volume fraction of isolated inclusions dispersed in the matrix material [28–30]:

The ${\varepsilon _{\rm{mix}}}(\omega)$ function is complex:

Ignoring the small factor ${\gamma ^2}$, we find the resonance frequency position, which is determined by the characteristics of the initial materials and the dispersed phase concentration $f$:

At the point $\omega = {\omega _0}$, the function ${\varepsilon ^\prime _{\rm{mix}}}(\omega)$ turns to zero, and the function ${\varepsilon ^{\prime \prime} _{\rm{mix}}}(\omega)$ takes the maximum value. In addition, the function ${\varepsilon ^\prime _{\rm{mix}}}(\omega)$ turns to zero at the following point:

In the interval $[{\omega _0},{\omega _1}]$, we have ${\varepsilon ^\prime _{\rm{mix}}}(\omega) \lt 0$, i.e., in this frequency range, the NC is similar to the metal.

Figure 2(a) shows the real and imaginary parts of the dielectric constant of silver as functions of the wavelength. Figure 2(b) shows dispersion dependence of the NC permittivity at filling factors, $f = 0.01$.

## 3. DISPERSION EQUATIONS

The dispersion equation for a mode localized in an anisotropic layer embedded between a semi-infinite 1D PC and a metal was obtained in the general case in [31]. In the case of an embedded isotropic layer, the dispersion equation describing the TPP is reduced to the following form:

The solution of Eq. (6) is the complex frequency $\omega = {\omega _r} + i\gamma$, whose real part determines the TPP spectral position.

## 4. TRANSFER MATRIX METHODS

The transmittance of a plane light wave polarized along the $x$ axis and propagating in the $z$ direction was calculated using the transfer matrix method [32]. The light field variation during passage through each structural layer is determined by the second-order transfer matrix. And the transfer matrix of the entire structure, which relates the amplitudes of the incident and outgoing waves, is calculated as the product of $2 \times 2$ matrices:

where the transfer matrix isHere, $h = \sqrt {{\varepsilon _n}/{\varepsilon _{n - 1}}}$, $\varepsilon (n)$ is the permittivity of the $n$th layer, ${\alpha _n} = (\omega /c)\sqrt {\varepsilon (n)}$, $\omega$ is the wave frequency, $c$ is the speed of light, ${\gamma _n} = {z_n} - {z_{n - 1}}$ (${{n}} = {{1}},{{2}},\cdots, {{N}}$) represents the layer thicknesses, and ${z_n}$ is the coordinate of the interface between the $n$th layer and the ${(n + 1)}$th layer adjacent from the right, ${\gamma _{N + 1}} = 0$. The transfer matrix for the orthogonally polarized wave is obtained from Eq. (11) via the replacement of $h$ by $\sqrt {{\varepsilon _{n - 1}}/{\varepsilon _n}}$.

The energy transmittance, reflectance, and absorptance are expressed as

## 5. RESULTS AND DISCUSSIONS

In the absence of the metallic mirror and the NC layer, the bandgap of an unbounded PC lies in the wavelength range from 377.6 nm to 469.3 nm; the bandgap boundaries are shown by black horizontal lines in Figs. 3(a), 4(a), and 5. Red dotted lines superimposed over the reflectance spectrum represent the TPP dispersion curves calculated by numerical solution of dispersion Eq. (6). Let us compare the obtained solution of the dispersion equation for the TPP localized in the sample with semi-infinite mirrors with the spectra calculated for a finite structure by the transfer matrix method.

In Fig. 3(a), it can be seen that the red dotted curve completely coincides with the gray solid curve, thereby demonstrating considerable agreement between the two calculation methods, even at a small number of PC periods. It should be noted that an increase in the number of PC layers improves the agreement of the solution of the dispersion equation at the PC bandgap edges. Upon increasing the layer thickness $d$, the phase of the reflection from the PC changes; the TPP frequency in the bandgap shifts to the long-wave region, which makes it possible to adjust the TPP position. In particular, at $d = 50\;{\rm{nm}}$, the TPP occurs at a wavelength of 437.7 nm, and the $Q$ factor of the mode is 292. At $d = 80\;{\rm{nm}}$, the TPP frequency attains the upper limit of the PC bandgap and occurs at a wavelength of 468.8 nm.

Figure 3(b) shows the spatial distribution of the local field intensity for the TPP frequency at $d = 50\;{\rm{nm}}$. The structure distribution of the electric field intensity normalized to the intensity of the incident light is colored in red. The field is localized in a region comparable with the wavelength.

When the NC layer is filled with metallic silver nanoballs with ${\varepsilon _0} = 5$, $\hbar \omega p = 9\;{\rm{eV}}$, $\hbar \gamma = 0.02\;{\rm{eV}}$ [33], the permittivity of the NC matrix is set by the parameter ${\varepsilon _d} = 2.56$; this permittivity is typical for some dense glasses, such as crown glass. The reflectance spectrum reveals the TPP frequency splitting effect [see Fig. 4(a)] for the case with $f = 0.01$, and the plasmon resonance frequency ${\omega _0}$ is similar to the TPP frequency or coincides with it. Then, in contrast to the case $f = 0$ [Fig. 3(a)], there are two dips in the reflectance spectrum, which are caused by the ${\varepsilon _{\rm{mix}}}$ dispersion in the NC layer.

In addition, Fig. 4(a) shows excellent agreement between the curves obtained by solving the dispersion equation with the reflectance spectrum calculated by the transfer matrix method in the NC resonance region. Hence, it can be seen that the solution of the dispersion equation for the upper curve is limited by the edge of the bandgap of an infinite seed PC.

Figure 4(b) presents the spatial distribution of the local field intensity for the Tamm modes at wavelengths of ${\lambda _1} = 422.7\;{\rm{nm}}$ and ${\lambda _2} = 455,4\;{\rm{nm}}$ with $Q$ factors of 201 and 207, respectively. Such wavelengths in the reflectance spectrum in Fig. 4(a) correspond to $d = 50\;{\rm{nm}}$. Consequently, the TPP splitting value is $\Delta \lambda = 32.4\;{\rm{nm}}$. Comparison of Figs. 3(b) and 4(b) shows that the maximum intensity decreased by an order of magnitude. It can be seen that the light is localized between the silver film and PC mirror and that the maximum of the local field strength is located in the NC layer and decreases exponentially in both directions.

Let us consider the effect of the concentration of nanoballs dispersed in the NC matrix on the reflectance spectrum. Figure 5 shows the reflectance spectra of the investigated structure at different volume fractions of nanoparticles in the NC layer at a concentration from 1% to 10%. It can be seen that, with a change in the $f$ value, the high- and low-frequency modes are symmetrically repulsed and the splitting value increases with the concentration of silver nanoballs in the NC layer. In particular, at $f = 0.05$, the TPP with a wavelength of 409 nm splits into two Tamm modes: a high-$Q$ mode with ${\lambda _1} = 397\;{\rm{nm}}$ and a low-$Q$ mode with ${\lambda _2} = 463\;{\rm{nm}}$; in this case, the TPP splitting value is $\Delta \lambda = 66\;{\rm{nm}}$, which is twice as large as the splitting value at a concentration of 1%. Figure 5 shows good agreement between the curves obtained by solving numerically dispersion Eq. (7) with the reflectance spectrum of the structure calculated using the numerical method.

Equation (1) was obtained in the quasielectrostatic approximation, which is applicable when nanoparticles are much smaller than the wavelength in the medium. For the optical range, the thickness of the layers of a 1D PC is about a quarter of the radiation wavelength, which also imposes a restriction on the maximal size of nanoparticles. The size effect takes place when the average free path of the conduction electrons exceeds the size of the nanoparticles and the processes of scattering of electrons by the surface of a particle start to contribute considerably to relaxation. The collision of electrons with the surface of the nanoparticle is phenomenologically accounted for by introducing an additive to the relaxation rate inversely proportional to the radius $r$ of the particle [34,35]:

where ${V_f} = 1.4 \cdot {10^8} \;{\rm sm/s}$ is the velocity of electrons at the Fermi energy, $r$ represents the nanoparticle radius, and $A$ is the proportionality coefficient determined by the electron scattering at the nanoparticle surface.The dependences in Fig. 6(a) show that the effective permittivity of the NC in the plasmon resonance region differs significantly from the values obtained without taking into account the size of nanoparticles. We note that the decrease in the nanoparticle size leads to the decrease in the amplitude values of the real and imaginary parts of the effective permittivity. At the same time, it can be seen from Fig. 6(b) that when taking into account the size adjustment, the TPP splitting effect is preserved.

## 6. CONCLUSION

Thus, the effect of TPP splitting at the interface between a metallic film and a resonant NC layer conjugated with a PC was demonstrated. It was shown that the TPP splitting is sensitive to the nanoparticle concentration and other parameters of an NC. The results obtained for a finite structure using the transfer matrix method were found to be in good agreement with the data determined by the independent method for solving the dispersion equation for an unbounded structure. The advantage of the proposed model is determined by the significantly extended possibility of effectively controlling the position of Tamm modes in the reflectance spectra of the PC structures via changing the parameters of the resonant NC layer. The significant dependence of the position of reflectance peaks on the NC layer thickness opens up the possibility of creating a tunable filter based on such a structure. To do this, it is necessary to fabricate an NC layer with a variable thickness, for example, wedge-shaped [36].

## Funding

Russian Foundation for Basic Research (19-42-240004, 19-52-52006).

## Acknowledgment

The reported study was funded by Russian Foundation for Basic Research, Government of Krasnoyarsk Territory, Krasnoyarsk Region Science and Technology Support Fund to the research project No. 19-42-240004 and by Russian Foundation for Basic Research, project No. 19-52-52006. The authors are thankful to Pavel S. Pankin for valuable discussions and comments.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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