Shortwave infrared surface plasmons in multilayered two-dimensional Ti₃C₂T_x MXenes

Two-dimensional (2D) MXenes, a family of transition metal carbides or nitrides, have recently garnered significant interest due to their unique properties, such as outstanding electrical conductivity [1, 2], high mechanical strength [3], and electrochemical performance [4, 5], even with their ultra-thin structures. These attributes have spurred research across diverse applications, such as energy storage [5, 6], electromagnetic shielding [7, 8], gas sensors [9, 10], saturable absorbers [11], and transparent electrodes [12, 13]. Among the various types of MXenes, Ti₃C₂T_x MXene exhibits unique optical properties, revealing metallic characteristics in the shortwave infrared (SWIR) range that are challenging to observe in other traditional 2D materials, and these properties facilitate the formation of surface plasmons in the SWIR region. Certainly, traditional conductive 2D materials like graphene and black phosphorus support surface plasmons as well [14–19]; however, their relatively low electron density has limited them to the terahertz to mid-infrared frequency ranges [14, 15]. This spectral limitation has impeded the direct implementation of 2D material-based plasmonic applications, including biomolecular sensors [14, 17], optical modulators [20], and metasurfaces [21], in communication wavelength regions such as SWIR. In contrast, Ti₃C₂T_x MXene, possessing a high electron density, has been reported to exhibit plasma frequencies within the 280–300 THz range, allowing for the excitation of surface plasmons across the full SWIR spectrum. Nevertheless, Ti₃C₂T_x MXene plasmons show considerably lower field confinement in a structure with a single interface compared to other plasmons, owing to substantial intrinsic absorptions stemming from the high imaginary permittivity of Ti₃C₂T_x in the SWIR region. For this reason, MXene plasmons have been primarily employed as broadband SWIR absorbers [22]. To effectively employ 2D MXene-based SWIR plasmons in diverse applications, it is necessary to find a method that enhances the field confinement of these plasmons.

In this study, we investigate various multilayer structures that enhance the field confinement of Ti₃C₂T_x MXene plasmons in the SWIR region. We began by conducting experimental measurements of the optical permittivity of Ti₃C₂T_x and used these values to analytically calculate the characteristics of MXene plasmons in different multilayer configurations, including insulator–MXene–insulator (IMI), MXene–insulator–MXene (MIM), and insulator–MXene–insulator–MXene (IMIM) structures. Among these structures, we discovered that the IMIM structure exhibited the strongest confinement of MXene plasmon fields, which intensifies as the thickness of MXene or insulating layer decreased. For an MXene plasmon with a frequency of 150 THz (λ₀ = 2.0 μm), the wavelength and effective field size in an IMIM structure with a 1.3 nm-thick MXene monolayer and a 1 nm-thick SiO₂ insulator layer were calculated to be 24.61 nm and 1.50 nm, respectively. These values exhibit a reduction by factors of 55 and 596, respectively, compared to those obtained in a single SiO₂–MXene interface structure.

We experimentally obtained the optical permittivity of Ti₃C₂T_x MXene and carried out a theoretical analysis of MXene plasmons based on the acquired data. To measure the permittivity, we prepared a 10 nm-thick Ti₃C₂T_x MXene film on a quartz substrate using a two-step process [see Methods]: (1) creating a Ti₃C₂T_x MXene solution using the minimum intensity layer delamination (MILD) method, which selectively etches the 'A' layer from the MAX phase (Ti₃AlC₂), and (2) spin-coating the MXene solution onto the quartz substrate, resulting in a thin Ti₃C₂T_x MXene layer. As illustrated in the cross-sectional transmission electron microscopy (TEM) image in figure 1(a), multiple uniform MXene monolayers, each 1.3 nm thick, are stacked atomically thin with air gaps in between, ultimately composing a 10 nm-thick film [23]. Then, we measured the permittivity ( = ' + ''i) of the prepared Ti₃C₂T_x MXene film for the frequency (f = ω/2π) range of 50–300 THz using two different ellipsometers: the IR-VASE Mark II (J.A. Woollam) for the 50–176 THz range, and the M-2000 Ellipsometer (J.A. Woollam) for the 176–300 THz range. As shown in figure 1(b), the plasma frequency (f_p) where ' = 0 was measured to be 288 THz, and Ti₃C₂T_x MXene exhibited metallic properties (' < 0 and '' > 0) in the entire shortwave infrared (SWIR) region below f_p. The measured permittivity was fitted using a Drude model ( = _∞–ω₀ ² / ω(ω+ γi)) for analytical calculations, where the parameters _∞, ω₀, and γ were determined to be 11.35 +5.11i, 4.08 eV, and 0.219 eV, respectively. We additionally measured for Ti₃C₂T_x MXene films with various thicknesses ranging from 2 nm to 1 μm. The results revealed that both ' and '' exhibited similar values with a variation of approximately ±5% and ±1%, respectively, compared to the values measured for the films with a thickness of 10 nm.

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Figure 1(c) illustrates the calculated dispersion relation of surface plasmons generated at the interface between air and MXene, assuming that the MXene layer is sufficiently thick. The calculation was conducted using the relation k_x ² = k₀ ²(_MXene)/(1 + _MXene) [24], where k_x represents the wavenumber in the propagation direction x, k₀ denotes f / c, and _MXene refers to the measured permittivity of MXene. The surface plasmon frequency (f_sp = f_p/√2) is calculated to be 202 THz, and k_x displays a slight increase compared to k₀ in the SWIR range below f_sp, indicating that the plasmons are confined to the air-MXene interface. The effective refractive index (n_eff = k_x /k₀) was calculated to be 1.013 for f = 150 THz (λ₀ = 2.0 μm). Figure 1(d) illustrates the field distribution of the surface plasmon propagating along the air–MXene interface at a frequency of 150 THz. The measured effective wavelength of 1974 nm agrees well with the calculated value of λ_sp (= λ₀/n_eff), where n_eff is 1.013 as shown in figure 1(c). Despite the high |'| value of Ti₃C₂T_x MXene, the surface plasmon exhibits a low effective refractive index due to the relatively high '' value. A low n_eff implies that the energy of MXene plasmons is predominantly concentrated in the air region, resulting in weak field confinement. This is further confirmed by the notably weakened field confinement observed in figure 1(d), indicating that the majority of the fields exist within the air region with a noticeable lack of such fields within the MXene. To effectively utilize MXene plasmons in plasmonic applications requiring high optical density, it is crucial to enhance their field confinement. Consequently, we explored various multilayer MXene structures with the goal of enhancing the field confinement of MXene plasmons.

To investigate the behavior and properties of plasmons in multilayer structures, we employed a theoretical formalism [25, see Methods] and conducted calculations. We applied boundary conditions at each interface to calculate the dispersion relations and field distributions of plasmons in the multilayer structures. Using the calculated solutions, we derived and analyzed four factors that pertain to the field confinement and absorption losses—plasmon wavelength (λ_sp), propagation length (D), and effective modal and field sizes (L_e and L_f) [24–26, see Methods].

The thin MXene layer synthesized on a quartz substrate inherently exhibits a multilayer configuration consisting of insulator (air)–MXen–insulator (SiO₂) (IMI). This multilayer structure can enhance the field confinement through the interaction between plasmons generated at the two interfaces (air–MXene and MXene–SiO₂ interfaces), depending on the thickness of the MXene layer. When the distance between the two interfaces exceeds the decay length into the MXene layer, the surface plasmons at both interfaces weakly couple, causing the fields to behave independently. This leads to a field distribution similar to that of a single interface, as depicted in figure 1(d). However, when the MXene layer is sufficiently thin, the surface plasmons at both interfaces can strongly couple, resulting in enhanced field confinement [27]. Figure 2(a) illustrates the |E_z | field distribution of 150 THz plasmons for MXene layer thicknesses (h_m) of 100, 50, 20, and 10 nm. In structures with MXene layers of 100 and 50 nm thicknesses, the plasmon fields at the two interfaces weakly couple with each other and predominantly expand into the air and SiO₂ directions. However, in structures with thin MXene layers of 20 and 10 nm thicknesses, the plasmons at the two interfaces couple more effectively, leading to a stronger confinement of fields near the MXene layer and a more symmetric field distribution with respect to the MXene layer.

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Figure 2(b) illustrates the calculated dispersion relations of surface plasmons in air-MXene-SiO₂ multilayer structures with different values of h_m, covering a frequency range of 50–250 THz. In all structures, the effective refractive index (n_eff) of plasmon mode is greater than the refractive index of SiO₂ at frequencies below f_sp, indicating vertical field confinement. Furthermore, as the MXene layer becomes thinner, the plasmon mode has a higher k_x value, corresponding to a larger n_eff. For plasmons at a frequency of 150 THz (λ₀ = 2.0 μm), the n_eff values are calculated to be 1.47, 1.48, 1.74, and 2.78 for h_m values of 100, 50, 20, and 10 nm, respectively. The increase in n_eff for structures with thin MXene layers is attributed to the strong interaction between plasmons at the two MXene-insulator interfaces. This enhanced interaction increases k_x of the MXene plasmon mode, which directly influences the degree of field confinement and localization. In other words, a larger k_x indicates a stronger confinement of the electromagnetic field and a more localized plasmon mode within the MXene layer. At frequencies above f_sp, quasibound leak modes that exhibit significant propagation losses are observed [24].

Figure 2(c) displays the E_z field distribution of MXene plasmons propagating in the x-direction at 150 THz for structures with h_m values of 100, 50, 20, and 10 nm. The figure reveals that as h_m decreases, both the plasmon wavelength (λ_sp) and the effective modal and field sizes (L_e and L_f) are reduced. Additionally, the propagation length (D) also decreases as more energy is concentrated in the MXene absorption layer with decreasing h_m. For the structure with h_m = 10 nm, λ_sp, L_e, L_f and D are calculated to be 719.27 nm, 79.40 nm, 257.05 nm, and 1215.07 nm, respectively, at a frequency of 150 THz (λ₀ = 2.0 μm). To quantify the dependence of plasmon properties in the IMI multilayer structure on h_m, we calculated λ_sp, L_e, L_f, and D for h_m ranging from 1 to 1000 nm at a frequency of 150 THz and plotted the results in figure 2(d). As shown in this figure, the plasmonic properties undergo a rapid change as h_m decreases below 100 nm. As h_m decreases from 100 to 1 nm, λ_sp decreases by a factor of 17.22 from 1353.59 nm to 78.64 nm. In addition, L_e decreases by a factor of 55.63 from 473.45 nm to 8.48 nm. Likewise, L_f also decreases by a factor of 34.36 from 896.52 nm to 26.09 nm, reflecting an increase in field confinement by more than 30 times in terms of the square of field amplitude. It is worth noting that in the IMI structure, L_f tends to be larger than L_e. Additionally, D also significantly decreases to 114.6 nm when h_m decreases to 1 nm, indicating limitations in utilizing MXene plasmons as a waveguide. These findings clearly demonstrate that the multilayer structure can have a significant impact on plasmonic properties, particularly on the field confinement characteristics.

The metal–insulator–metal plasmonic structure has been a subject of extensive research for decades due to its ability to create a gap plasmon within the insulator spacer, resulting from the interaction between surface plasmons at the two metal surfaces. Decreasing the thickness of the insulating spacer can confine the plasmon field, which in turn enhances its intensity, and this compression can theoretically be infinite [28]. Similarly, in an MXene–insulator (SiO₂)–MXene (MIM) multilayer structure composed of two thick MXene layers separated by a nanometer-thick insulating layer, MXene plasmons can be confined within a very thin SiO₂ spacer, which is expected to enhance the field confinement. Figure 3(a) illustrates the variation of |E_z | distribution in the MIM structure as the SiO₂ thickness is decreased. This figure clearly displays the strong compression and confinement of the electric field within the SiO₂ spacer. Figure 3(b) presents the calculated dispersion relations of MXene plasmons in the MIM structure for SiO₂ spacer thicknesses (h_s) of 100, 50, 20, and 10 nm in the frequency range of 50–250 THz. It is observed that the plasmon mode in the MIM structure has a higher k_x than that in the IMI structure shown in figure 2(b), and the k_x of the plasmon increases as the thickness of the SiO₂ spacer decreases. At a frequency of 150 THz, the n_eff of the plasmon is calculated to be 2.11, 2.63, 3.85, and 5.45 for h_s of 100, 50, 20, and 10 nm, respectively.

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Figure 3(c) displays the E_z distribution of a 150 THz frequency plasmon that occurs in the MIM structure at h_s values of 100, 50, 20, and 10 nm. In all structures, the plasmon field is compressed into the SiO₂ spacer, and the plasmon wavelength decreases with h_s. Figure 3(d) shows the values of λ_sp, L_e, L_f, and D of the plasmon mode at a frequency of 150 THz (λ₀ = 2.0 μm) as a function of h_s in the range of 1–1000 nm. Here, we restricted the minimum value of h_s to 1 nm to exclude quantum effects occurring in the MIM structure [29]. As h_s decreases, the plasmon field confinement in the MIM structure becomes stronger than that of the IMI structure shown in figure 3(d), and the field confinement in MIM structure is continuously enhanced from 1000 to 1 nm, unlike in the IMI structure. When h_s is 10 nm, λ_sp, L_e, and L_f are calculated as 367.19 nm, 53.67 nm, and 10.99 nm, respectively, which are reduced by 1.96, 1.45, and 23.39 times, respectively, compared to the values of the IMI structure with h_m of 10 nm. As h_s decreases, it is worth noting that L_f becomes progressively smaller than L_e, which indicates that the electric fields of plasmon modes are mostly concentrated in the SiO₂ spacer. When h_s decreases to 1 nm, λ_sp, L_e, and L_f decrease to 58.08 nm, 10.16 nm, and 1.17 nm, respectively. However, due to the strong absorption of MXene, D also decreases to 1246.36 nm and 83.87 nm for h_s of 10 nm and 1 nm, respectively, and these values are comparable to those observed in the IMI structure.

Finally, we investigated the plasmonic properties in an insulator (air)—MXene—insulator (SiO₂)—MXene (IMIM) structure, which contains both IMI and MIM structures. Figure 4(a) shows the |E_z | distribution in the IMIM structure with a 5 nm thick MXene layer (h_m = 5 nm) and a 5 nm thick SiO₂ spacer (h_s = 5 nm) on the sufficiently thick MXene substrate. In this structure, we can expect that the field confinement occurs through the plasmonic interactions generated at the three interfaces (air–MXene, MXene–SiO₂, and SiO₂–MXene interfaces). The plasmon mode in the IMIM structure is confined within the SiO₂ spacer between the thin MXene film and the thick MXene substrate, and it also interacts with the counter-phase charge oscillation at the top air–MXene interface, resulting in even stronger field confinement [17, 18]. To observe the plasmonic properties of the IMIM structure, both the thickness of the insulating spacer (h_s) and the thickness of the thin MXene film (h_m) need to be sufficiently small. When h_s is too large, the plasmons exhibit characteristics similar to those of the IMI structure discussed in figure 2, and when h_m is too large, they exhibit characteristics similar to those of the MIM structure discussed in figure 3.

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Figure 4(b) shows the dispersion relation of plasmon modes in the IMIM structure for four cases of h_m and h_s values in the frequency range of 50–250 THz: (Case 1) h_m = 20 nm, h_s = 20 nm, (Case 2) h_m = 20 nm, h_s = 5 nm, (Case 3) h_m = 5 nm, h_s = 20 nm, and (Case 4) h_m = 5 nm, h_s = 5 nm. In Case 2 and Case 3, the plasmon mode has a larger wavevector compared to Case 1, while Case 4 exhibits a significantly larger wavevector than the other cases. At a frequency of 150 THz, the effective refractive index (n_eff) of the plasmon mode is calculated to be 5.12, 9.47, 10.65, and 18.48 for Cases 1, 2, 3, and 4, respectively. These results demonstrate that the IMIM structure can provide higher effective refractive index and stronger field confinement than the IMI or MIM structures alone, where both h_m and h_s should be thin for maximum effectiveness. Figure 4(c) shows the E_z field distributions of plasmons at a frequency of 150 THz (λ₀ = 2 μm) in the IMIM structure for the four cases discussed in figure 4(b). It can be observed in Case 4 that there is a significant reduction in λ_sp and that the field is strongly concentrated within the SiO₂ spacer. In Case 4, L_e and L_f are calculated to be 47.38 nm and 28.77 nm, respectively. Furthermore, it is observed that Case 2 dominantly exhibits the plasmonic properties of the MIM structure, while Case 3 dominantly exhibits those of the IMI structure.

Figures 4(d)–(g) display the λ_sp, L_e, L_f, and D of the plasmon modes as a function of h_m and h_s, ranging from 1–100 nm, in the IMIM structure at a frequency of 150 THz (λ₀ = 2.0 μm). When h_s is fixed at a large value of 100 nm, the IMIM structure becomes similar to the IMI structure. In this case, as h_m decreases from 100 to 1 nm, λ_sp, L_e, and L_f decrease by factors of 12.01, 17.21, and 4.08, respectively, and D decreases by a factor of 77.33. Similarly, when h_m is fixed at a large value of 100 nm, the IMIM structure becomes similar to the MIM structure. In this case, as h_s decreases from 100 to 1 nm, λ_sp, L_e, and L_f decrease by factors of 16.26, 14.53, and 99.73, respectively, and D decreases by a factor of 102.78. When both h_m and h_s simultaneously decrease from 100 to 1 nm, λ_sp and L_e experience more significant reductions by factors of 43.36 and 45.85, respectively. L_f also significantly decreases by a factor of 67.99, but its value is slightly larger compared to that of the structure with h_m = 100 nm and h_s = 1 nm, due to the penetration of a portion of the electric field through the thin MXene layer. Additionally, D decreases by a factor of 152.21. When h_m is 1.3 nm, corresponding to the thickness of the MXene monolayer, and h_s is 1 nm, the λ_sp, L_e, and L_f are calculated to be 24.61 nm, 3.27 nm, and 1.50 nm, respectively. These results clearly demonstrate that the IMIM multilayer structure is the most optimized configuration for confining the field of MXene plasmons.

We have demonstrated the strong field confinement of MXene plasmons through the investigation of various multilayer structures. Among the diverse configurations, the IMIM structure proved to be the most effective in confining the plasmon mode, as it combines the plasmonic confinement capabilities of both IMI and MIM structures. Our observations indicate that field confinement is enhanced when the thickness of both MXene and insulator layers are reduced in multilayer structures. With a monolayer MXene thickness of 1.3 nm and a 1 nm thick SiO₂ layer between the two MXene layers, the wavelength of the plasmon at a frequency of 150 THz (λ₀ = 2.0 μm) in the IMIM structure is reduced to 24.61 nm, and the effective modal and field sizes are reduced to be 3.27 nm and 1.50 nm, respectively. The strong field confinement of MXene plasmons in the IMIM structure offers a solution that transcends the limitations of MXene plasmons, significantly expanding the potential for implementing various 2D MXene-based applications such as subwavelength nonlinear optics, photodetectors, biosensors, metasurfaces, light-emitting devices, and optical modulators in the SWIR region.

7.1. Synthesis of Ti₃C₂T_x

7.2. Theoretical formalism

For theoretical calculations of plasmon wavevector k_x in multilayered structure, we solved the boundary condition of Maxwell's equations of each interface deriving the continuity of electromagnetic field [24, 25]. Assuming propagation of the wave in the positive x-direction at each interface, the surface plasmons are represented as transverse-magnetic (TM) waves:

$$\begin{align}E = ({E_x}\hat x + {E_z}\hat z){{\text{e}}^{i{k_x}x}}{{\text{e}}^{ - i\omega t}},\end{align} \tag{ 1 }$$

$$\begin{align}H = ({H_y}\hat y){{\text{e}}^{i{k_x}x}}{{\text{e}}^{ - i\omega t}}.\end{align} \tag{ 2 }$$

The tangential electric field E_x and normal displacement field D_z (E_z ) components of the nth interface can be expressed of parallel to x and perpendicular to z:

$$\begin{align}{E_{x,n}}(z = {d_n}) = {a_n}{{\text{e}}^{{k_z}z}} + {b_n}{{\text{e}}^{ - {k_z}z}},\end{align} \tag{ 3 }$$

$$\begin{align}{{\text{D}}_{\text{z},\text{n}}}(z = {d_n}) = \left( {\frac{{i{k_x}{\varepsilon _n}}}{{{k_{z,n}}}}} \right)\left( { - {a_n}{e^{{k_{z,n}}z}} + {b_n}{e^{ - {k_{z,n}}z}}} \right),\end{align} \tag{ 4 }$$

where d_n is the distance of nth interface from the 1_st interface, a_n and b_n are the field coefficients, k_z,n is the perpendicular wavevector at the nth interface, and the k_x is defined by the momentum conservation k_x ² = k_z,n ² _—n (ω/c)² [25], and _n is the permittivity of nth layer. At the nth interface between n_th and (n + 1)th layers, the continuity of the tangential electric field and normal displacement field must be satisfied according to the boundary conditions:

$$\begin{align}{E_{x,n}}(z = {d_n}) = {E_{x,n + 1}}(z = {d_n}),\end{align} \tag{ 5 }$$

$$\begin{align}{D_{z,n}}(z = {d_n}) = {D_{z,n + 1}}(z = {d_n}).\end{align} \tag{ 6 }$$

In multilayered condition, solving equations (10) and (11) as simultaneous equations can yield the multilayered structure plasmon wavevector k_x . Especially, the plasmon wavevector k_x in the four layers structure is determined by adopting the determinant of the field coefficient a_n and b_n yielding the plasmon dispersion relation:

$$\begin{align}\det \left[ {\begin{array}{*{20}{c}} 0&1&0&0&0&0&0&0 \\ {{e^{{k_{z1}}{d_1}}}}&{{e^{ - {k_{z1}}{d_1}}}}&{ - {e^{{k_{z2}}{d_1}}}}&{ - {e^{ - {k_{z2}}{d_1}}}}&0&0&0&0 \\ { - \tfrac{{{\varepsilon _1}}}{{{k_{z1}}}}{{\text{e}}^{{k_{z1}}{d_1}}}}&{\tfrac{{{\varepsilon _1}}}{{{k_{z1}}}}{{\text{e}}^{ - {k_{z1}}{d_1}}}}&{\tfrac{{{\varepsilon _2}}}{{{k_{z2}}}}{{\text{e}}^{{k_{z2}}{d_1}}}}&{ - \tfrac{{{\varepsilon _2}}}{{{k_{z2}}}}{{\text{e}}^{ - {k_{z2}}{d_1}}}}&0&0&0&0 \\ [5pt] 0&0&{{{\text{e}}^{{k_{z2}}{d_2}}}}&{{{\text{e}}^{ - {k_{z2}}{d_2}}}}&{ - {{\text{e}}^{{k_{z3}}{d_2}}}}&{ - {{\text{e}}^{ - {k_{z3}}{d_2}}}}&0&0 \\ 0&0&{ - \tfrac{{{\varepsilon _2}}}{{{k_{z2}}}}{{\text{e}}^{{k_{z2}}{d_2}}}}&{\tfrac{{{\varepsilon _2}}}{{{k_{z2}}}}{{\text{e}}^{ - {k_{z2}}{d_2}}}}&{\tfrac{{{\varepsilon _3}}}{{{k_{z3}}}}{{\text{e}}^{{k_{z3}}{d_2}}}}&{ - \tfrac{{{\varepsilon _3}}}{{{k_{z3}}}}{{\text{e}}^{ - {k_{z3}}{d_2}}}}&0&0 \\ [5pt] 0&0&0&0&{{{\text{e}}^{{k_{z3}}{d_3}}}}&{{{\text{e}}^{ - {k_{z3}}{d_3}}}}&{ - {{\text{e}}^{{k_{z4}}{d_3}}}}&{ - {{\text{e}}^{ - {k_{z4}}{d_3}}}} \\ 0&0&0&0&{ - \tfrac{{{\varepsilon _3}}}{{{k_{z3}}}}{{\text{e}}^{{k_{z3}}{d_3}}}}&{\tfrac{{{\varepsilon _3}}}{{{k_{z3}}}}{{\text{e}}^{ - {k_{z3}}{d_3}}}}&{\tfrac{{{\varepsilon _4}}}{{{k_{z4}}}}{{\text{e}}^{{k_{z4}}{d_3}}}}&{ - \tfrac{{{\varepsilon _4}}}{{{k_{z4}}}}{{\text{e}}^{ - {k_{z4}}{d_3}}}} \\ 0&0&0&0&0&0&1&0 \end{array}} \right] = 0.\end{align} \tag{ 7 }$$

By applying this determinant equation, the plasmon wavevector k_x of more complex multilayered structures can be obtained by adding boundary conditions in the matrix that correspond to additional layers.

7.3. Definition of analysis parameters

The plasmon wavelength, λ_sp, which represents the distance between two adjacent intensity maxima of the plasmon mode, can be directly calculated from the real part of k_x using equation (8). This parameter serves as a significant metric for characterizing the spatial extent of plasmon field confinement and determining the overall performance of plasmonic devices

$$\begin{align}{\lambda _{sp}} = \frac{{2\pi }}{{\operatorname{Re} [{k_x}]}}.\end{align} \tag{ 8 }$$

In addition, the propagation length, D, which quantifies the distance over which a plasmon mode can propagate before being significantly attenuated due to absorption losses, can be directly calculated from the imaginary part of k_x using equation (9). This parameter is critical for determining the feasibility and efficiency of plasmonic devices based on multilayer structures, as well as for designing strategies to mitigate absorption losses and enhance the performance of the devices

$$\begin{align}D = \frac{{2\pi }}{{\text{Im} [{k_x}]}}.\end{align} \tag{ 9 }$$

The effective modal size, L_e, representing the effective size of the energy distribution of the plasmon mode, can be obtained by calculating the effective size of the energy density (W) using equations (10) and (11) [24]. These parameters reflect the energy confinement of plasmon modes in multilayer structures and are a crucial for optimizing the performance of plasmonic devices, such as sensors, waveguides, and filters

$$\begin{align}{L_{\text{e}}} = \frac{1}{{\max [W(z)]}}\mathop {\mathop \int \nolimits }\limits_{}^{} W{(z)^{}}{\text{d}}z,\end{align} \tag{ 10 }$$

$$\begin{align}W(z) = \frac{1}{2}\text{Re} \left[ {\frac{{{\text{d}}\left\{ {{\omega ^{}}{{\varepsilon (z)}}} \right\}}}{{{\text{d}}\omega }}} \right]{\left| {E(z)} \right|^2} + \frac{1}{2}{\mu _{\text{0}}}{\left| {H(z)} \right|^2},\end{align} \tag{ 11 }$$

where E and H represent the electric and magnetic fields, respectively, (z) denotes the real part of the permittivity at the specific location z, and μ₀ represents the vacuum permeability.

Finally, the effective field size, L_f, can be obtained by calculating the effective size of the square of electric field amplitude (|E|²) using equation (12) [26]. These parameters directly reflect the electric field confinement of plasmon modes in multilayer structures

$$\begin{align}{L_{\text{f}}} = \frac{{{{\left( {\int {|E{|^2}{\text{d}}z} } \right)}^2}}}{{\int {|E{|^4}{\text{d}}z} }}.\end{align} \tag{ 12 }$$

7.4. Numerical simulation

This study was supported by the National Research Foundation of Korea (2020R1A2C2010967, 2022R1A2C3006227, 2021M3H4A1A03047327), Institute for Information & Communications Technology Planning & Evaluation (IITP) Grants (2020-0-00947, 2022-0-00198 and 2022-RS-2022-00164799), National Research Council of Science & Technology (NST) Grants (CRC22031-000 and CAP22051-102), Fundamental R&D Program for Core Technology of Materials (20020855) funded by the Ministry of Trade, Industry and Energy, KIST Institutional Program (2E32801-23-P027), and KU-KIST School Project. Part of this study has been performed using facilities at IBS Center for Correlated Electron Systems, Seoul National University.

The datasets associated with this research paper are available from the authors upon request.

J K, C M K, and M K K conceived the research project. J K developed the physical concept, conducted theoretical analysis, and measured MXene permittivity. C P contributed to useful discussions and also measured MXene permittivity. H K executed the fabrication and characterization of MXene, while P N R captured the TEM image. All authors participated in discussing the results and writing the manuscript.