1. Introduction

Optical modes at the dielectric-metal interface have gained increasing attention in recent years as people seek modes that can be confined into a near- or sub-diffraction length scale [1]. The planar geometry, with permittivity varying in only one direction, has been widely used as a model system to understand how different optical modes propagate within the multi-layered structures [2]. It is straight forward to solve for the transverse magnetic (TM) mode at a single dielectric-metal interface. The calculation gets more complicated when solving for modes that exist at metal-insulator-metal or insulator-metal-insulator three layered structure. The analytical solutions for the three layered structure can still be found in text books such as Ref. [2]. However, once another dielectric layer is added into the configuration, such as dielectric-semiconductor-insulator-metal 4-layered structure, the geometry known to support ‘hybrid plasmonic’ mode [3], the analytical solution is yet to be solved.

Researchers tend to analyze the fundamental TM₀ mode in this 4-layer structure using a mode hybridization picture, which was originally proposed by Oulton et al. [3]. However, due to the lack of analytical solution to this TM₀ mode, it is not fully clear to what extend the mode hybridization picture is valid to explain this fundamental TM₀ mode in this 4-layered structure. Researchers have been using parameters such as insulator layer (between metal and higher refractive indexed layer) thickness to distinguish the ‘hybrid-plasmonic’ and ‘plasmonic-only’ aspects of this TM₀ mode [4]. Nevertheless, the transition also depends on the refractive index difference between the semiconductor and insulator layers and semiconductor/insulator thickness ratio, which add more complexity into the analysis [5].

In this paper, the author first presents the analytical solution to the fundamental TM₀ mode in the dielectric-semiconductor-insulator-metal 4-layered planar structure. A change from sinusoidal function to exponential function in the electric (magnetic) field in the higher refractive indexed semiconductor layer clearly marks the transition from a ‘hybrid plasmonic’ to a ‘plasmonic-only’ behavior in the fundamental TM₀ mode. The effective mode index, alternatively the propagation constant β, naturally appears as the parameter that separates the ‘hybrid-plasmonic’ solution from its ‘plasmonic-only’ counterpart. If the effective mode index exceeds the refractive index of semiconductor, only exponential functions of electric (magnetic) field are allowed in the solution for the semiconductor region and the original mode hybridization picture needs to be modified.

To further illustrate this transition, an approximation method based on mode hybridization [3,6] in the context of variational technique [7,8] is developed to calculate the dispersion relation ω (β) of the TM₀ mode. It is demonstrated that using trial functions based on mode hybridization the dispersion curve of the TM₀ mode in the ‘hybrid plasmonic’ region can be well reproduced but similar trial function fails to reproduce the dispersion curve in the ‘plasmonic only’ region, revealing the limitation of mode hybridization approximation. In addition, the functional developed for the variational method can be used to approximate the dispersion relationship of other modes coexisting in this 4 layered structure as long as a suitably approximated wave function is applied. As an example, it is shown that dispersion curve of the hybrid photonic mode of transverse electric (TE₀) characteristics supported on the same 4 layered structure is accurately produced using the wave functions constructed from the TE₀ mode in the top three dielectric layers and its image mode due to the metal response. This approximation method provides a reasonable estimation to the dispersion relations of fundamental TM₀ and TE₀ modes with an easily understood physical picture. The method proposed here can in general be applied to other structures of complex shapes.

In the following analyses, the x axis is defined as the direction of light propagation and z the direction perpendicular to the interface of different media, as depicted in Fig. 1(a). If the electromagnetic wave is assumed of a simple harmonic time dependence on a single frequency ω and the relative permittivity of the materials $\varepsilon (z )$ is a piecewise function that only depends on the frequency ω and z, the electromagnetic wave can then take a simple form (E, H) (r, t) = (E(z), H(z))e^iβx-iωt. Here E, H are vectors. Without any external charges or currents, the Maxwell equations can be written as [9]:

 

Fig. 1. (a) Diagram of the dielectric-semiconductor-insulator-metal 4 layered planar structure. (b) Dispersion curves of TM₀ mode obtained from Eq. (4) (black) and Eq. (6) (blue) and from COMSOL simulation package (dashed green) respectively with permittivity ɛ_d = ɛ_i = 2.9, ɛ_m = 1-ω_p²/ω² with ω_p= 1.4 × 10¹⁶ rad·Hz, 2a = 110 nm and h = 6 nm. The intersection point between the light line (dashed red) and the dispersion curve corresponds to the transition frequency ω_T. (c) The ratio of propagation constant β of TM₀ mode with respect to the propagation constant of light in bulk semiconductor as a function of β. Here ${n_c} = \sqrt {{\varepsilon _c}} $ and ${k_0} = \frac{\omega }{c}$.

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In this paper, the author only considers non-magnetic materials and the relative permeability $\mu $ is set to 1. It is worth noting that Eq. (1) does not require $\varepsilon (z )$ to be real numbers. In order to demonstrate a clear physical picture, it is first assumed in section 2 that all permittivities are real numbers. The results obtained from section 2 can be equally derived for $\varepsilon (z )$ of complex numbers.

2. Analytical solution to the fundamental TM₀ mode in the dielectric-semiconductor-insulator-metal planar structure

As discussed in Refs. [10] and [11], both TM and TE modes are supported by this 4 layered structure. Since the analytical solution to the fundamental TE₀ mode has been solved in Ref. [11], the author will only discuss the analytical solution to the fundamental TM₀ mode in this section. The TM₀ mode in ‘hybrid plasmonic’ region can be obtained by solving Eq. (1) in each of the four layers. H_y and E_x are written as:

For simplicity, we further assume ${\varepsilon _d} = {\varepsilon _i} \;<\;{\varepsilon _c}.\; $The continuity of H_y and E_x at three interfaces leads to the dispersion relationship of the TM₀ mode in the ‘hybrid plasmonic’ region:

In the simplest cases, the dielectric material can be assumed nondispersive with the permittivity$\; {\varepsilon _d}$ and the middle semiconductor layer a nondispersive material with a higher permittivity ${\varepsilon _c}$ and the bottom metallic layer a lossless Drude metal of ${\varepsilon _m} = 1 - \frac{{\omega _p^2}}{{{\omega ^2}}}$, where ω_p is the plasma frequency of the metal. Figure 1(b) illustrates the ‘hybrid plasmonic’ to ‘plasmonic only’ transition with a given set of geometric values (2a = 110 nm, h = 6 nm, similar to what were used in Ref. [11]) with ${\varepsilon _c} = 13$ and ${\varepsilon _d} = 2.9$. The dispersion curves obtained from Eq. (4) and Eq. (6) are plotted along with that of TM₀ mode directly obtained from COMSOL simulation package. The ‘hybrid plasmonic’ to ‘plasmonic only’ transition frequency ${\omega _T}$ can be obtained graphically by plotting the light line $\beta = \frac{{\omega \sqrt {{\varepsilon _c}} }}{c}\; $in Fig. 1(b), where the intersection point between the light line and the dispersion curve indicates the transition point. The ratio of $\beta /\frac{{\omega \sqrt {{\varepsilon _c}} }}{c}$ as a function β is plotted in Fig. 1(c) to further illustrate the ‘hybrid plasmonic’ to ‘plasmonic only’ transition.

The transition frequency ${\omega _T}$ depends closely on the values of permittivities ${\varepsilon _d}$, ${\varepsilon _c}$ and the thicknesses 2a and h. The value of ${\omega _T}$ can be solved by letting $\beta - \frac{{\omega \sqrt {{\varepsilon _c}} }}{c} \to 0$ and calculating Eq. (4) or (6) at this extreme. The transition frequency can then be written as:

Figure 2 shows ${\omega _T}$ as a function of ɛ_c in the range of 3 to 13 [Fig. 2(a)), of ɛ_d in the range of 1 to 12.8 (Fig. 2(b)), of core thickness 2a in the range of 20 nm to 420 nm (Fig. 2(c)) and thin gap thickness h in the range of 4 nm to 300 nm (Fig. 2(d)). These values are chosen to reflect various semiconductor and thin oxide materials and geometries used in literature for the 4-layered structures [1]. Though the plots are mostly straightforward to understand, it is worth keeping in mind that the transition frequency$\; {\omega _T}$ occurs at the interception of the light line $\omega = \frac{{\beta c}}{{\sqrt {{\varepsilon _c}} }}$ with the dispersion curve of TM₀ (Fig. 1(b)). For the same ɛ_c value, the smaller the transition$\; {\omega _T}$ (also β), the more closely the dispersion curve resembles that of a single semiconductor (ɛ_c)-metal interface. For example, in both Fig. 2(a) and 2(b), the smallest $\beta c/{\omega _p}$ occurs around 0.4 when ɛ_d approaches the value of ɛ_c. When the gap layer is maintained very thin (Fig. 2(c)), the thicker the core layer, the smaller the transition β value, which again indicates the dispersion curve behaves similar to that of a single semiconductor (ɛ_c)-metal interface. On the other hand, when the thickness of the core layer is fixed, increasing the gap thickness h yields the transition happens at larger β value, consistent with the situation that the dispersion relation closely resembles that of a single dielectric (ɛ_d) - metal interface. This is why the transition frequency plateaus at larger h values (Fig. 2(d)). From the transition frequency ${\omega _T}$, we can easily obtain the transition propagation constant β_T = $\frac{{{\omega _T}\sqrt {{\varepsilon _c}} }}{c}$ and vice versa.

 

Fig. 2. Hybrid plasmonic to plasmonic only transition frequency ${\omega _T}$ (solid curves) and corresponding propagation constant β (dashed curves) as a function of (a) relative permittivity of core semiconductor ${\varepsilon _c}$ with three different$\; {\varepsilon _d}$ values at 2a = 110 nm and h = 6 nm, (b) relative permittivity of thin gap material ${\varepsilon _d}$ with three different ${\varepsilon _c}$ values at 2a = 110 nm and h = 6 nm, (c) core semiconductor thickness 2a with three different $\; {\varepsilon _d}$ values at ${\varepsilon _c} = 13$ and h = 6 nm, and (d) gap thickness h with three different ${\varepsilon _c}$ values at ${\varepsilon _d}$ = 2.9 and 2a = 110 nm. In this figure, ɛ_m = 1-ω_p²/ω² with ω_p= 1.4 × 10¹⁶ rad·Hz.

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It is clear from above analyses that at a given set of geometric parameters, the fundamental TM₀ mode in this 4-layered structure can be either hybrid plasmonic with a photonic contribution or pure plasmonic depending on the operating frequency of the mode. In addition, the dispersion relation does not require β and/or ${\varepsilon _n}$ (n = d, c, i, m) to be real numbers and therefore can be used for materials with losses.

For the more complex situation of the top cladding and thin gap with different refractive indices (permittivities) ɛ_d < ɛ_i < ɛ_c, the dispersion relation and transition frequency ${\omega _T}\; $can be derived using similar approaches detailed above from Eq. (2) and Eq. (3). One important point that requires further attention is that for $\beta\;<\;\frac{{\omega \sqrt {{\varepsilon _i}} }}{c}$, in the low propagation constant (and low frequency) region, k_i in Eq. (2) and Eq. (3) becomes imaginary number.

3. Variational method to calculate the dispersion relationship of the fundamental TM₀ mode in the ‘hybrid plasmonic’ region

Alternatively, the dispersion curve of ‘hybrid plasmonic’ mode can be obtained using variational techniques. Following the variational methods described in Ref. [8], we can define ω (E, H) as a functional:

 

Fig. 3. (a) Diagrams showing the E_x and E_z components of fundamental TM₀ mode at the metal-insulator interface (red), in the top three layers (blue) and the superposition of the two modes (pink) in ‘hybrid plasmonic’ region. (b) Dispersion curves of TM₀ mode at the metal-dielectric interface (red), that in the dielectric-semiconductor-dielectric 3 layers (blue), that obtained through the variational method assuming a hybrid plasmonic mode with lossless materials (pink) and those obtained from Eq. (4) (dashed black) and Eq. (6) (dashed green). The geometric and material parameters used here are the same as used in Fig. 1. The grey dashed line indicates the transition point obtained from Fig. 1. (c) Change of photonic to plasmonic component ratio in the hybrid plasmonic mode as a function of β with dispersion curve shown in (b). A = sinφ, B = cosφ.

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and

Here ω_t, ɛ_t(z) and ω_b, ɛ_b(z) are the corresponding frequency and permittivity of the the fundamental TM₀ mode in the dielectric-semiconductor-insulator planar structure without the metal and that for the fundamental TM₀ mode at the insulator-metal interface, at a given $\beta. \,\, \omega$ is then obtained by Eq. (9) divided by Eq. (10). It is now obvious to see that depending on the overlap of the wave function (E_t, H_t) with (E_b, H_b) and the difference between ɛ and ɛ_t (ɛ_b), the frequency ω of the hybridized TM₀ mode can vary with the change of the materials and geometric factors. In addition, ω also depends on the A/B ratio of E_t to E_b, which describes the character of the mode being more photonic like or plasmonic like.

It is essentical to identify a method to obtain an appropriate A/B ratio in this calculation, in other words to look for a stationary point of functional ω with A/B ratio as the variable. If all permitivities are lossless real numbers (in the simplest case), both ω and β are real numbers. It is reasonable to assume A and B are real numbers and A/B is allowed in the range of –∞ to +∞. It is therefore convenient to re-define a single variable φ, so that A = sinφ, B = cosφ and A/B = tanφ. Because the fundamental TM₀ mode supported by this structure would have the lowest frequency at a given propagation constant β, using the ground state assumption, the appropriate value φ can be expected to occur at the lowest frequency ω where $\frac{{\partial \omega }}{{\partial \varphi }} = 0,$ at the given material and geometric configuration and at a given β. In a more general case, A/B could be written as a complex number where A/B = tanφe^iϕ. The stationary point occurs at $\frac{{\partial \omega }}{{\partial \varphi }} = 0$ and $\frac{{\partial \omega }}{\partial \phi } = 0$.

3.1. Fundamental TM₀ mode with lossless materials

For the simplest case, it is again assumed that the top dielectric and bottom insulator are composed of the same nondispersive material with a real permittivity ${\varepsilon _d}$, the middle semiconductor layer a nondispersive material of a higher real permittivity ${\varepsilon _c}$ and the bottom metallic layer a lossless Drude metal of ${\varepsilon _m} = 1 - \frac{{\omega _p^2}}{{{\omega ^2}}}$. The expressions of the electric and magnetic fields of the fundamental photonic TM₀ mode in a symmetric dielectric-semiconductor-dielectric planar structure and those of the fundamental plasmonic TM₀ mode in a planar insulator-metal interface can be found in Refs. [2,12]. The dispersion relation for the insulator-metal interface is given by ${\omega _{plasmon}} = \beta c\sqrt {\frac{{{\varepsilon _m} + {\varepsilon _d}}}{{{\varepsilon _m}{\varepsilon _d}}}} $, where β is the propagation constant and c the speed of light in vacuum. The dispersion relation for the guided TM₀ mode supported by the insulator-dielectric-insulator 3 layers is $\textrm{tan}{k_c}a = \frac{{{k_d}{\varepsilon _c}}}{{{k_c}{\varepsilon _d}}}$, where $k_d^2 = {\beta ^2} - \frac{{{\omega ^2}{\varepsilon _d}}}{{{c^2}}},{\; }k_c^2 = \frac{{{\omega ^2}{\varepsilon _c}}}{{{c^2}}} - {\beta ^2}$ and a is half of the thickness of the central semiconductor layer. Figure 3(b) shows the calculated hybrid plasmonc TM₀ mode dispersion curve using the same set of material and geometric parameters as Fig. 1 that satisfies $\frac{{\partial \omega }}{{\partial \varphi }} = 0$ as a function of β. Simulated dispersion curves of the fundamental TM₀ modes in a symmetric dielectric-semiconductor-dielectric planar structure, that of the TM₀ mode at the dielectric-metal interface and that of TM₀ modes in the 4 layered structure obtained directly from Eq. (4) and Eq. (6) are also given in the same plot for comparision. (See Appendix A for detail) The corresponding angle φ as a function of the propagation constant β is given in Fig. 3(c). It is clear from the plots that the appoximation method yields very good agreement with the analytical solution in the ‘hybrid plasmoninc’ region (β <$\frac{{{\omega _T}\sqrt {{\varepsilon _c}} }}{c}$, indicated by the grey dashed line). However, in the ‘plasmonic only’ region (β >$\frac{{{\omega _T}\sqrt {{\varepsilon _c}} }}{c}$,) in particular β >$\; 0.9{\omega _p}/c$ for current case, the approximation method deviates from the analytical result. The deviation of approximation method at higher frequency is attributed to the deviation of trial wave function, based on the hybridization picture, from the true analytical solution, which is of pure plasmonic characteristics. Using the variational method, we can again confirm that the hybridization picture is only valid in the region of $\beta \le \frac{{\omega \sqrt {{\varepsilon _c}} }}{c}$, where the trial wave function based on mode hybridization is a good approximation to the true wave function.

For the more complex situation of the top cladding and thin gap with different permittivities (ɛ_d < ɛ_i < ɛ_c), solution to the fundamental photonic TM₀ mode in an asymmetric dielectric-semiconductor-insulator 3-layered planar structure can be found in most textbooks [12]. Using modes hybridization, the dispersion relation in the 4-layered structure that satisfies $\beta \le \frac{{\omega \sqrt {{\varepsilon _c}} }}{c}\; $can be obtained following a similar method outlined above. One thing worth noting is that in the asymmetric 3-layer cut-off region ($\beta \;<\;\frac{{\omega \sqrt {{\varepsilon _i}} }}{c}$), no waveguided photonic TM₀ mode exists in the top 3 layers. To use mode hybridization method in $\beta \;< \;\frac{{\omega \sqrt {{\varepsilon _i}} }}{c}$ region, a leaky photonic TM₀ mode can be assumed in the top 3 layers [13]. A method dealing with lossy modes is detailed in 3.2.

3.2. Fundamental TM₀ mode with lossy materials

For realistic materials with losses, the relative permittivities are complex numbers, resulting in propagation constant β a complex number as well, with the imaginary part Imag(β) indicating the loss of the mode. In this case, both (E_t, H_t) and (E_b, H_b) have an exponential decay term along x direction that is determined by Imag(β_t) and Imag(β_b). Since Imag(β_t) is usually not equal to Imag(β_b) when Real(β_t) = Real(β_b), we need to modify the trial fuction of hybrid plasmonic mode (E, H). Different from the lossless case, with complex β, ${\boldsymbol E}\; ({\boldsymbol H} ){\; } = A{e^{ - \Delta \beta x}}{{\boldsymbol E}_t}({{{\boldsymbol H}_t}} )+ B{e^{\Delta \beta x}}{{\boldsymbol E}_b}({{{\boldsymbol H}_b}} ),{\; }$where $\Delta \beta = \frac{{Imag({{\beta_b}} )- Imag({{\beta_t}} )}}{2}$ and A and B are still constants. The modified trial fuction (E, H) can be plugged back into Eq. (8) to calculate the dispersion relation as a function of A/B = tanφe^iϕ. Similar to the lossless case, the A/B ratio corresponding to the lowest frequency fundamental mode can be identified when $\frac{{\partial \omega }}{{\partial \varphi }} = 0\; $and $\frac{{\partial \omega }}{{\partial \phi }} = 0\; $conditions are satisfied. If we use ${\varepsilon _m} = {\varepsilon _{m,\; \infty }} - \frac{{\omega _p^2}}{{{\omega ^2} + i\gamma \omega }}$ for Ag [10], where γ is associated with the loss of the metal, and$\; {\varepsilon _c} = {\varepsilon _{c,\infty }} - \frac{{\omega _{c,p}^2}}{{\omega _c^2 - {\omega ^2} - i{\gamma _c}\omega }}$ for the semiconductor GaInP layer [14], where γ_c is related to the absorption loss of GaInP, Fig. 4(a) and Fig. 4(b) describe the dsipersion curves of TM₀ mode at an dielectric-Ag interface (${\varepsilon _d} = 2.9$), TM₀ mode at the dielectric-GaInP-dielectric, TM₀ mode in the 4 layers obtained from the variational method and simulated TM₀ mode in the 4 layers by COMSOL and their corresponding losses. The trial function assumes the imaginary part of the β as the average of (E_t, H_t) and (E_b, H_b) modes ($Imag(\beta )= \frac{{Imag({{\beta_b}} )+ Imag({{\beta_t}} )}}{2}$), which is overestimated compared to the simulation results of COMSOL. Nevertheless, the dispersion curve from the variational method well reproduces the one obtained from COMSOL. Only in the region Real($\beta )\;>\;\frac{{\omega {n_c}}}{c}$, where n_c is the real part of the refractive index of GaInP, the dispersion curve from the vationational method deviates more from that obtained from COSMOL simulation. Consistent with the lossless case, it is suggested again that the mode hybridization picture is a good description for the TM₀ mode supported by the 4 layered structure, as long as the the real part of the mode propagation constant, Real(β), does not exceed the real part of the propagation constant in bulk semiconductor material.

 

Fig. 4. (a) Dispersion curves of TM₀ mode at the Ag-insulator interface (red), that in the insulator-GaInP-insulator 3 layers (blue), that obtained through the variational method assuming a hybrid plasmonic mode with lossy materials (pink) and that obtained from COMSOL simulation package (dashed green). The geometric parameters used here are the same as used in Fig. 1 and Fig. 3. The permittivities of Ag and GaInP are obtained from Ref. [17] and [14] respectively. The right axis indicates the ratio of Real(β) of TM₀ mode obtained from the variational method with respect to Real(β) of light in bulk GaInP. (b) Imag(β) of the four modes described in (a). The inset shows the angle φ associated with the change of photonic to plasmonic component ratio in the hybrid plasmonic mode. Angle $\phi \; $for the optimized trial wave function occurs at $\phi = 0.$

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4. Approximation method to calculate the dispersion relationship of the fundamental TE₀ mode

Equation (8) can be used to obtain the dispersion curve of other higher frequency modes, if a plausible wave function can be assumed. Though it is difficult to guess a wave function of an arbitrary higher order mode, the physical meaning of the TE₀ mode that coexists in this 4-layer structure has been elaborated in Ref. [11]. As detailed in Ref. [11], the TE₀ mode is photonic-like and has a higher frequency compared to that of the hybrid plasmonic TM₀ mode at a given propagation constant β. Its dispersion curve is very close to that of the waveguided TE₀ mode in the top three layers. The effect of the metal to this TE mode is like a mirror, reflecting the TE mode off the metal. In a typical insulator-dielectric-insulator 3 layered geometry, the waveguided mode is regarded as the light propagating in the dielectric core, bouncing off the top and bottom cladding insulator layers. The bouncing angle is determined by the thickness of the dielectric core and the permittivities of the core and cladding materials [15]. Once a metallic substrate is added as an extra bottom layer, the bouncing angle from the bottom layers becomes larger, resulting a slightly decreased mode refractive index compared to the case without the metal, especially in the low frequency region. In ray optics, the light reflected off a mirror is considered as if the light source were placed inside the mirror (at the mirror-image position). The electromagnetic wave outside the metal is considered as the sum of the field generated from the light source itself and that from its mirror image. In the 4 layered case, the TE₀ mode can be constructed by adding the waveguided TE₀ mode supported by the top 3 layers with its image mode (Fig. 5(a)), E (z) = E_t (z) – E_t (z + 2a + 2 h). If it is again assumed that the top dielectric use the same material as the insulator, the approximated solution of the electric field in the 4 layers is then written as:

 

Fig. 5. (a) Diagrams showing the E_y field of fundamental TE₀ mode in the top three layers (green solid), its image mode (green dash) and the superposition of the two modes (cerulean blue). (b) Dispersion curves of TE₀ mode calculated from Eq. (8) using wave functions (Eq. 11) detailed in section 4 (cerulean blue), that obtained from COMSOL simulation package (black dash). The dispersion curve of TE₀ mode in insulator-semiconductor-insulator 3 layers is given for comparison (green). The geometric and material parameters used here are the same as used in Fig. 1.

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where $k_d^2 = {\beta ^2} - \frac{{{\omega ^2}{\varepsilon _d}}}{{{c^2}}},{\; }k_c^2 = \frac{{{\omega ^2}{\varepsilon _c}}}{{{c^2}}} - {\beta ^2}$ and A a constant. Here the electric field of the image mode has an opposite sign to the original mode, so that the total electric field equals to zero at the interface, satisfying the continuity condition where no electric field penetrates into the metal in the idea metal case.

The dispersion curve of the TE₀ mode using the approximated wave function is plotted in Fig. 5(b), agreeing very well with the COMSOL simulation results. This approximation solution for the TE₀ mode can also be compared with the analytical solution given as [11]:

5. Conclusion

In summary, the author demonstrates that at a given configuration of the dielectric-semiconductor-insulator-metal planar geometry the nature of the fundamental TM₀ mode is either ‘hybrid plasmonic’ or ‘plasmonic only’. The transition occurs when the real part of the propagation constant β of TM₀ mode reaches the same value as the real part of the wave number in the bulk semiconductor material. At larger β values, the wave function in the semiconductor layer assumes the form of exponential functions rather than sinusoidal functions and the mode hybridization picture is no longer valid. Nevertheless, this result confirms that when operating above the transition frequency ${\omega _T}$, the fundamental TM₀ mode can be confined into a deep subwavelength scale. Even though this conclusion is obtained through the simplest 4-layered planar geometry, the same idea can be extended to semicondutor (or higher refractive index dielectric) waveguides of various cross sections shapes [16–19] near metal. The same conclusion is also supported by using a variational method where the trial wave functions are constructed based on mode hybridization. It is noted that the functional developed to obtain the dispersion relation of TM₀ mode can also be exploited for other modes as long as the wave function is a good approximation to its analytical solution.