Graphene-Based Optical Waveguides for Surface Plasmon Polariton Transmission

Abstract

In this chapter, various ideas and designs related to plasmonic waveguides are reviewed. As plasmonic structures can confine light in small sizes, they can be considered a suitable option in the design of optical devices. In recent years, graphene has been known as an attractive material in optical applications and can be used to guide surface plasmons. By photo-stimulating surface plasmons at the semiconductor-graphene interface, plasmon polaritons can be confined near graphene and guided to several micrometers. Using this feature, various optical waveguides can be designed. The real and imaginary parts of the effective refractive index of the waveguide, the figure of merit, coupling length, crosstalk, and loss in decibels per micrometer are the most important parameters in the design of plasmonic waveguides. In recent years, various ideas such as the use of ridges, graphene nano-ribbons or strips, and graphene sheets have been proposed, which provide waveguides with different characteristics. In this chapter, the theory and basic relationships in light confinement in plasmonic waveguides are investigated first. In the following, different structures designed in recent years are reviewed.

1. Introduction

With the advancement of technology toward the realization of integrated photonic circuits, plasmonic structures and plasmonic waves have been the focus of attention and utilization. Most of these nanostructures are made up of metals, graphene, and dielectrics, with sizes below the excitation wavelength, which is the wavelength of the incident radiation that excites the plasmonic waves. Plasmons are described based on the interaction between electromagnetic waves and conduction electrons in metals with nanosizes or graphene. Metals exhibit reflective behavior in the vicinity of the visible frequency spectrum and do not allow the propagation of electromagnetic waves inside the metal. Therefore, metals are used for waveguide structures and oscillators. In higher frequencies such as visible light frequency, the penetration of the field into the metal increases significantly, leading to severe losses and consequently limiting the fabrication of optical device structures. In ultraviolet frequencies, metals behave like insulators, and electromagnetic waves can penetrate inside them. All the scattering properties mentioned above are described by the permittivity function as a function of frequency [1].

2. Surface plasmon polariton

Surface plasmon polaritons (SPPs) are excitations or electromagnetic waves that couple to the collective and oscillating free electrons on the surface of metals to propagate longitudinally on the interface between the metal and dielectric [2]. However, the incident electromagnetic radiation should meet certain conditions, including being incident in a way that creates total internal reflection and having longitudinal polarization. Figure 1 illustrates the excitation and propagation of plasmonic waves at the interface of two media.

Figure 1.

Propagation of surface plasmon polariton waves at the interface of dielectric and metal and penetration of electric field into the two materials.

The reason for the importance of SPPs is their ability to concentrate the electromagnetic field. As a result, the scattering of plasmons decreases, and the propagation of localized fields improves by several orders of magnitude [3]. Therefore, it causes transverse electric or transverse magnetic fields to be confined to the surfaces perpendicular to their propagation direction.

3. Graphene

Researchers have been able to improve certain desired features in devices using new materials with special electro-optic properties. One of the most important materials that can be used for sensing and providing unique characteristics is graphene. Graphene has become a unique material due to its exceptional properties in electrical and thermal conductivity, high density, carrier mobility, optical conductivity, and mechanical properties, which have been employed in optical and electronic devices alongside three-dimensional materials such as silicon [4]. In recent years, graphene plasmons have garnered significant attention from researchers. Graphene, in the plasmonic phase, has led to the proposal of new optical devices in the terahertz to visible frequency range with very high speeds, low excitation voltages, low power consumption, and reduced size [5, 6, 7, 8].

Surface plasmon waves are not radiative and appear exponentially at the boundary between two regions. This wave enables efficient coupling with input signals across a wide range of wavelengths while minimizing losses. Previously, noble metals like gold and silver were primarily used as plasmonic materials, allowing surface plasmon propagation from visible to infrared frequencies. However, these structures, particularly in the visible light range, exhibited significant losses and limited adjustability. Additionally, there are limitations on the ratio of the SPP propagation length at the metal-dielectric interface. One of the main reasons for using graphene is the possibility of tuning due to energy bandgap control, increased limitation due to graphene’s thickness being comparable to an atom, and higher speed due to reduced electron scattering and fewer collisions [9]. Furthermore, in the optical frequency range, the incident light polarization has no significant effect on the graphene refractive index due to its high refractive index. For these reasons, graphene has been proposed as a suitable option for plasmonic structures in the terahertz frequency range. The reason for using graphene is that photons can easily couple with graphene surface plasmons in the terahertz frequency range, leading to remarkable features such as tunability with external voltage application and low losses.

Graphene is a two-dimensional crystal composed of carbon atoms arranged in a hexagonal lattice with sp2 bonding orbitals. In this structure, there is no energy gap between the valence and conduction bands, which have a symmetric Dirac-like cone shape and touch at the k = 0 point in the Brillouin zone. The possibility of conducting in this material is provided by engineering the energy gap through chemical doping or using a biased electric field or voltage. Using the voltage connection to graphene, the Fermi energy level can be changed. If a positive voltage is applied, the Fermi level increases, resulting in an increase in electron density and graphene acting as a p-type semiconductor [9].

The absence of the forbidden energies in the energy bands of graphene has led to unique optical, electronic, and mechanical properties, making it suitable for a wide range of optical applications. As a result, extensive research on graphene has been conducted by research centers and industries. A group of researchers successfully extracted graphene using a micro-mechanical approach from graphite material [10]. When we write with a pencil on paper, the remaining mark on the paper is graphite. If the applied pressure is adjusted so that only a one-atom-thick layer of graphite remains on the paper, graphene is produced. The graphene layer is then placed on silicon dioxide and compressed, transferring the graphene layer to the substrate. Although this method provides better quality compared to other methods, it is not proper for mass production [10]. Chemical synthesis is another common method for producing graphene. The first step is graphite oxidation using acids such as nitric acid and sulfuric acid to produce graphene oxide. Then, graphene oxide is converted to graphene using a thermal shock or ultrasonic waves.

Graphene is classified into three categories based on layer thickness or atomic layers: single-layer graphene, few-layer graphene (2–10 layers), and multilayer graphene (more than 10 layers). Graphene with more than 100 layers, or thick graphene, possesses graphite-like properties [11]. Atomic force microscopy is one of the methods used to determine the number of graphene layers [12]. Single-layer graphene absorbs about 2–3% of white light regardless of the size of the light’s wavelength, which makes it suitable for optical devices such as photodetectors, solar cells, switches, and polarizers [13]. Due to its high mobility, controllable conductivity without carrier scattering, and increased penetration depth by a few micrometers, graphene is suitable for plasmonic structures.

The integrated structure of graphene allows for faster phonon transfer and increased heat transfer. The tunability of graphene is due to its mixed dynamic conductivity, which is determined by the Kubo formula. This formula indicates that dynamic conductivity depends on parameters such as chemical potential, scattering coefficient, and temperature. The concept of dynamic conductivity in graphene may vary in different frequency ranges depending on the positive or negative chemical potential. In intrinsic graphene or graphene with low chemical potential, the concept of dynamic conductivity is negative. In this state, graphene can transmit TE surface plasmon modes, acting as a semiconductor. By applying an external voltage, graphene has a large chemical potential; therefore, it can pass surface plasmon modes TM and act as a metal. Increasing temperature also increases the real part of dynamic conductivity, and when TM modes are conducting, the propagation constant in this state also increases [14, 15].

The conductivity of graphene is generally a complex quantity resulting from intraband and interband transitions [16]. In the visible frequency range, if the energy of photons is lower than the Fermi energy, the Fermi energy is centered around the Dirac point, and all the valence bands are filled and the conduction band is empty. There always exists a pair of excitable electron-holes with a corresponding photon energy, which results in effective photon absorption across all frequency ranges up to the visible range. When the Fermi level increases, lower energy photons below the Fermi level have a lower probability of absorption because of the Pauli exclusion. Therefore, in the visible frequency range, photons cause the transition of electrons from the valence band to the conduction band by creating electron-hole pairs. If excitation occurs, the photon energy is transferred to the electron. This type of transfer is called intraband transfer.

For calculating the conductivity factor of graphene in the visible frequency range, the intraband transition term is important, while the interband transition term is disregarded. In the terahertz frequency range, as there are impurities present in the medium, the Fermi energy is much higher than the photon energy. Consequently, according to the Pauli exclusion principle, intraband excitation in the terahertz frequency spectrum is negligible. Therefore, in terahertz frequencies, only interband transitions are significant [16].

The optical properties of graphene can be examined using the surface conductivity. The surface conductivity of graphene depends on various parameters, including frequency, temperature, chemical potential, and carrier mobility. By changing the surface conductivity of graphene, the optical properties of this two-dimensional material can be altered, which can be obtained from the Kubo formula and includes both intraband and interband contributions [17].

$$\mathrm{\sigma }\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)={\mathrm{\sigma }}_{int\mathrm{ra}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)+{\mathrm{\sigma }}_{int\mathrm{er}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)$$E1

$${\mathrm{\sigma }}_{int\mathrm{ra}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)=\frac{-\mathrm{i}{\mathrm{e}}^2}{\mathrm{\pi }{\mathrm{\hslash }}^2\left(\mathrm{\omega }+\mathrm{i}2\mathrm{\Gamma }\right)}{\int }_0^{\infty }\mathrm{\xi }\left(\frac{\partial {\mathrm{f}}_{\mathrm{d}}\left(\mathrm{\xi }\right)}{\partial \left(\mathrm{\xi }\right)}-\frac{\partial {\mathrm{f}}_{\mathrm{d}}\left(-\mathrm{\xi }\right)}{\partial \left(\mathrm{\xi }\right)}\right)\mathrm{d\xi }$$E2

$${\mathrm{\sigma }}_{int\mathrm{er}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)=\frac{\mathrm{i}{\mathrm{e}}^2\left(\mathrm{\omega }+\mathrm{i}2\mathrm{\Gamma }\right)}{\mathrm{\pi }{\mathrm{\hslash }}^2}{\int }_0^{\infty }\mathrm{\xi }\left(\frac{{\mathrm{f}}_{\mathrm{d}}\left(-\mathrm{\xi }\right)}{{\left(\mathrm{\omega }+\mathrm{i}2\mathrm{\Gamma }\right)}^2}-\frac{{\mathrm{f}}_{\mathrm{d}}\left(\mathrm{\xi }\right)}{4{\left(\mathrm{\xi }/\mathrm{\hslash }\right)}^2}\right)\mathrm{d\xi }$$E3

where σ_intra is the intraband conductivity which represents absorption due to photon-electron scattering within the band. σ_inter refers to the interband conductivity and is related to electron transitions between two bands. The intraband conductivity is affected by temperature (T), carrier scattering rate (Γ), chemical potential (μ_c), angular frequency (ω), and reduced Planck constant (ћ). It is described by the Fermi-Dirac distribution function (f_d) and can be obtained from Eq. (4) [18].

$${\mathrm{f}}_{\mathrm{d}}={\left(1+exp\left(\left(\mathrm{\epsilon }-{\mathrm{\mu }}_{\mathrm{c}}\right)/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\right)\right)}^{-1}$$E4

By utilizing the above equations, the intraband conductivity of graphene can be rewritten as Eq. (5) [18].

$${\mathrm{\sigma }}_{int\mathrm{ra}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)=\frac{{\mathrm{e}}^2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{\pi }{\mathrm{\hslash }}^2\left(2\mathrm{\Gamma }+\mathrm{i\omega }\right)}\left[\frac{{\mathrm{\mu }}_{\mathrm{c}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}+2ln\left({\mathrm{e}}^{-{\mathrm{\mu }}_{\mathrm{c}}/{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}+1\right)\right]$$E5

At room temperature, the interband conductivity is obtained from Eq. (6) [19].

$${\mathrm{\sigma }}_{int\mathrm{er}}\left(\mathrm{\omega },\mathrm{T},\mathrm{\Gamma },{\mathrm{\mu }}_{\mathrm{c}}\right)=-\mathrm{j}\frac{{\mathrm{e}}^2}{4\mathrm{\pi }{\mathrm{h}}^2}ln\left(\frac{2\left|{\mathrm{\mu }}_{\mathrm{c}}\right|-\left(\mathrm{\omega }-\mathrm{j}2\mathrm{\Gamma }\right)\mathrm{h}}{2\left|{\mathrm{\mu }}_{\mathrm{c}}\right|+\left(\mathrm{\omega }-\mathrm{j}2\mathrm{\Gamma }\right)\mathrm{h}}\right)$$E6

For ℏω ≺ 2μ_c, graphene conductivity is purely intraband, and the imaginary part of graphene’s surface conductivity is positive, behaving like a metal and sustaining TM polarization. For ℏω ≻ 2μ_c, graphene conductivity is interband, and the imaginary part of conductivity is negative, behaving like an insulator and supporting propagating TE polarization [20]. Therefore, in the average infrared wavelengths, graphene conductivity is defined as a semi-metallic conductivity model and only includes the intraband conductivity term, which is obtained from Eq. (7) at room temperature [19].

$${\mathrm{\sigma }}_{\mathrm{g}}=\frac{\mathrm{i}{\mathrm{e}}^2{\mathrm{\mu }}_{\mathrm{c}}}{\mathrm{\pi }{\mathrm{e\hslash }}^2\left(\mathrm{\omega }+\mathrm{i}{\tau }^{-1}\right)}$$E7

where $\tau$ is the carrier relaxation time and is obtained from Eq. (8) [19].

$$\mathrm{\tau }=\frac{\mathrm{\upsilon }{\mathrm{\mu }}_{\mathrm{c}}}{\mathrm{e}{\mathrm{V}}_{\mathrm{F}}^2}$$E8

where υ is the carriers’ mobility and its value is 10⁴ cm²/Vs. V_F represents the speed and equals 10⁶ m/s. The chemical potential of graphene is adjustable by applying an external electric field or doping. The chemical potential is related to the bias voltage according to Eq. (9) [21].

$${\mathrm{\mu }}_{\mathrm{c}}=\mathrm{\hslash }{\mathrm{V}}_{\mathrm{F}}\sqrt{\frac{\mathrm{\pi }{\mathrm{\epsilon }}_0{\mathrm{\epsilon }}_{\mathrm{r}}{\mathrm{V}}_{\text{bias}}}{\mathrm{e}{\mathrm{D}}_1}}$$E9

where ɛ_r, D₁, and ɛ₀ are the relative permittivity, substrate layer thickness, and vacuum permittivity, respectively. The sheet conductivity of graphene is obtained from Eq. (10) [22].

$${\mathrm{\epsilon }}_{\mathrm{q}}=1+\frac{\mathrm{i}{\mathrm{\sigma }}_{\mathrm{g}}}{{\mathrm{\epsilon }}_0\mathrm{\omega t}}$$E10

4. Theory of SPP excitation

A wave with TM polarization incident on a surface with an angle θ₁ is shown in Figure 2. The momentum of the photon in the dielectric equals hk_d/2π, where n_d is the refractive index of the dielectric and k_d = 2πn_d/λ.

Figure 2.

The interface between a metal and a semiconductor, considering optical illumination with TM polarization.

When a wave illuminates the metal-dielectric interface, a portion of it at a similar angle is reflected, and its momentum is preserved in the dielectric. Additionally, a portion of the incident optical wave is transmitted through the metal at an angle of θ₂, carrying a momentum of ħk_m in the metal. Wavevector k_m represents the metal’s index of the metal through k_m = 2πn_m/λ. The component at the x-axis is conserved, meaning k_mx = k_dx, where k_dx = k_dsinθ₁ and k_mx = k_m sinθ₂. Thus, it can be written as [3]:

$${\mathrm{n}}_{\mathrm{d}}{\text{sinθ}}_1={\mathrm{n}}_{\mathrm{m}}{\text{sinθ}}_2$$E11

The above relation is known as Snell’s law. Generally, in the visible wavelength range, n_d is greater than n_m. As n_d > n_m, θ₂ is maximized at 90 degrees, and θ₁ is limited. For angles greater than the limiting value, the incident wave is not transmitted into the metal. The limiting value of the incidence angle is known as the critical value θ_c, and it can be concluded that:

$${\text{sinθ}}_{\mathrm{c}}={\mathrm{n}}_{\mathrm{m}}/{\mathrm{n}}_{\mathrm{d}}$$E12

A wave with θ₁ > θ_c has a greater momentum along the interface compared to the metal. Incident-polarized P at the interface of the metal and dielectric possesses electric and magnetic fields that create oscillating surface charges on the common surface between them. Although the wave with an angle greater than the critical angle is completely reflected from the common surface, it causes the surface charges to undergo oscillatory motion and generate a field. This field penetrates into the metal, and these induced fields are perpendicular to the common surface.

When a wave illuminates the surface at the critical angle, there is an infinite extent of the surface where surface charges can oscillate. However, as the incident angle increases further, the oscillation of surface charges decreases and is emitted at a shorter distance from the surface. In these cases, the induced plasmon fields created due to the wave incident at an angle greater than the critical angle are considered as factors in coupling photons to surface plasmons.

There are no boundaries for E_x, so it is preserved at the overall surface. This is not true for E_z, which is the vertical component of the electric field. The vertical component D_z is continuous, and E_z changes with the displacement and the dielectric coefficient because D_z = ε_dε₀E_z and D_z = ε_mε₀E_zm. Discontinuity in component E_z leads to polarization changes at the surface.

A normal incident wave with s polarization does not create surface charges at the interface, but a radiating wave with p polarization makes polarization charges. Due to the need for electric field components perpendicular to create surface charges, only electromagnetic waves with p polarization should be considered. Additionally, any surface wave should satisfy the electromagnetic wave equation in both dielectric and metal materials. Assuming the x-y plane for the metal-dielectric, one has the following formulas in the x direction for z > 0:

$${\mathrm{E}}_{\mathrm{d}}=\left({\mathrm{E}}_{\mathrm{xd}},0,{\mathrm{E}}_{\mathrm{zd}}\right)exp\left(-{\mathrm{k}}_{\mathrm{zd}}\mathrm{z}\right)exp\left[\mathrm{i}\left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}-\mathrm{\omega t}\right)\right]$$E13

$${\mathrm{H}}_{\mathrm{d}}=\left(0,{\mathrm{H}}_{\mathrm{yd}},0\right)exp\left(-{\mathrm{k}}_{\mathrm{zd}}\mathrm{z}\right)exp\left[\mathrm{i}\left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}-\mathrm{\omega t}\right)\right]$$E14

For z < 0;

$${\mathrm{E}}_{\mathrm{m}}=\left({\mathrm{E}}_{\mathrm{xm}},0,{\mathrm{E}}_{\mathrm{zm}}\right)exp\left(-{\mathrm{k}}_{\mathrm{zm}}\mathrm{z}\right)exp\left[\mathrm{i}\left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}-\mathrm{\omega t}\right)\right]$$E15

$${\mathrm{H}}_{\mathrm{m}}=\left(0,{\mathrm{H}}_{\mathrm{ym}},0\right)exp\left(-{\mathrm{k}}_{\mathrm{zm}}\mathrm{z}\right)exp\left[\mathrm{i}\left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}-\mathrm{\omega t}\right)\right]$$E16

Taking Eqs. (13) and (15) into account in Maxwell’s equation ∇.E = 0, the components of the electric field can be written as follows:

$${\mathrm{E}}_{\mathrm{zd}}=\mathrm{i}\frac{{\mathrm{k}}_{\mathrm{x}}}{{\mathrm{k}}_{\mathrm{zd}}}{\mathrm{E}}_{\mathrm{xd}}$$E17

$${\mathrm{E}}_{\mathrm{zm}}=-\mathrm{i}\frac{{\mathrm{k}}_{\mathrm{x}}}{{\mathrm{k}}_{\mathrm{zm}}}{\mathrm{E}}_{\mathrm{xm}}$$E18

Taking Eqs. (13) and (14) into account, Maxwell’s equation can be applied as Eq. (19)

$$\text{∇}\times \mathrm{E}=-\frac{1}{\mathrm{c}}\frac{\mathrm{\partial H}}{\mathrm{\partial t}}$$E19

Furthermore, we have:

$$-{\mathrm{k}}_{\mathrm{zd}}{\mathrm{E}}_{\mathrm{xd}}-{\mathrm{ik}}_{\mathrm{x}}{\mathrm{E}}_{\mathrm{zd}}={\mathrm{ikH}}_{\mathrm{yd}}$$E20

where k refers to the wave vector in the free space and equals ω/c. By considering Eqs. (15), (16), and (19), we have:

$${\mathrm{k}}_{\mathrm{zm}}{\mathrm{E}}_{\mathrm{xm}}-\mathrm{i}{\mathrm{k}}_{\mathrm{x}}{\mathrm{E}}_{\mathrm{zm}}=\mathrm{i}\mathrm{k}{\mathrm{H}}_{\mathrm{ym}}$$E21

With regard to Eqs. (17), (18), (20) and (21) can be expressed as follows:

$${\mathrm{\varepsilon }}_{\mathrm{d}}{\mathrm{kE}}_{\mathrm{xd}}={\mathrm{ik}}_{\mathrm{zd}}{\mathrm{H}}_{\mathrm{yd}}$$E22

$${\mathrm{\varepsilon }}_{\mathrm{m}}{\mathrm{kE}}_{\mathrm{xm}}=-{\mathrm{ik}}_{\mathrm{zm}}{\mathrm{H}}_{\mathrm{ym}}$$E23

In a way that:

$${\mathrm{k}}_{\mathrm{zd}}^2={\mathrm{k}}_{\mathrm{x}}^2-{\mathrm{\varepsilon }}_{\mathrm{d}}{\mathrm{k}}^2$$E24

$${\mathrm{k}}_{\mathrm{zm}}^2={\mathrm{k}}_{\mathrm{x}}^2-{\mathrm{\varepsilon }}_{\mathrm{m}}{\mathrm{k}}^2$$E25

Furthermore, the tangential components E and H are continuously related at the z = 0 boundary condition, meaning E_xd = E_xm and H_yd = H_ym. The relationship between ɛ and the perpendicular k components in the metal and dielectric is expressed in Eqs. (4)–(16).

$$\frac{{\mathrm{k}}_{\mathrm{zd}}}{{\mathrm{k}}_{\mathrm{zm}}}=-\frac{{\mathrm{\varepsilon }}_{\mathrm{d}}}{{\mathrm{\varepsilon }}_{\mathrm{m}}}$$E26

Based on Eqs. (24), (25), and (26), one can conclude:

$${\mathrm{k}}_{\mathrm{x}}=\mathrm{k}\sqrt{\frac{{\mathrm{\varepsilon }}_{\mathrm{d}}{\mathrm{\varepsilon }}_{\mathrm{m}}}{{\mathrm{\varepsilon }}_{\mathrm{d}}+{\mathrm{\varepsilon }}_{\mathrm{m}}}}$$E27

If we replace k_x with k_spp, the scattering relationship SPP can be expressed as follows:

$${\mathrm{k}}_{\mathrm{spp}}=\mathrm{k}\sqrt{\frac{{\mathrm{\varepsilon }}_{\mathrm{d}}{\mathrm{\varepsilon }}_{\mathrm{m}}}{{\mathrm{\varepsilon }}_{\mathrm{d}}+{\mathrm{\varepsilon }}_{\mathrm{m}}}}$$E28

The momentum of SPP (ħk_spp) is greater than the momentum of a photon in free space (ħk) for the same frequencies, which causes the mismatch between light and SPP. It is necessary to overcome the challenge of the mismatch between light and SPP at the common interface. Therefore, it can be written as follows:

$${\mathrm{\varepsilon }}_{\mathrm{d}}+{\mathrm{\varepsilon }}_{\mathrm{m}}=0$$E29

The metal permittivity, ε_m, is similar to that of free electrons.

$${\mathrm{\varepsilon }}_{\mathrm{m}}=1-\frac{{\mathrm{\omega }}_{\mathrm{p}}^2}{{\mathrm{\omega }}^2}$$E30

where ω_p is the plasma frequency, and for many metals, it falls in the ultraviolet wavelength region. Based on Eqs. (29) and (30), the SPP frequency can be written as:

$${\mathrm{\omega }}_{\mathrm{spp}}=\frac{{\mathrm{\omega }}_{\mathrm{p}}}{\sqrt{1+{\mathrm{\varepsilon }}_{\mathrm{d}}}}$$E31

5. Graphene-based plasmonic waveguides

This section represents a brief literature review of graphene-based plasmonic waveguides. Electrical conductivity in graphene is tuned by applying a bias voltage or doping, so different optical analog and digital devices can be designed using the graphene layer. Plasmonic waveguides due to their applications in communications have great attention.

5.1 Graphene on silver

Bagheri et al. proposed a plasmonic waveguide that features a graphene monolayer on silver, positioned between two SiO₂ layers [22]. The SiO₂ and silver layers have thicknesses of 200 and 30 nm, respectively. Figure 3 depicts the transmission loss in the plasmonic waveguides across varying waveguide widths at chemical potentials of 0.4, 0.6, 0.8, and 1 eV. Increasing the chemical potential results in an elevation of the Fermi level and a reduction in the probability of occupation of energy levels exceeding the Fermi level. This phenomenon leads to a minor absorption in graphene, with loss diminishing as the chemical potential increases. Consequently, the modulation of transmittance is achievable by adjusting the chemical potential value. The high absorption exhibited at μ_c = 0.4 eV classifies graphene as a light absorber, which is unsuitable for this investigation. Enhancing transmittance is crucial for guiding terahertz waves in both the waveguides and the resonant ring. Therefore, a chemical potential of 1 eV is adopted in this analysis. As illustrated in Figure 3, an increase in waveguide width from 10 to 100 nm corresponds to a monotonic reduction in loss. The simulations indicate that field concentration at the waveguides’ edges is significant for widths narrower than 30 nm. Considering the fundamental needs of optical circuits, a small area is recognized as a pivotal factor in devising optical devices. Hence, a waveguide width of 50 nm is chosen.

Figure 3.

Transmission loss in terms of waveguide width for different values of the graphene chemical potential.

5.2 Parallel graphene with spacing

The first study in the field of parallel graphene plasmonic waveguides was conducted by Hanson [23]. In this research, the Maxwell equation for the propagation of quasi-transverse electromagnetic (TEM) mode in a parallel graphene-layer-based waveguide was solved. He considered a fundamental structure including three layers with the permittivity of ɛ₁, ɛ₂, and ɛ₃ where the graphene layer was at ɛ₁-ɛ₂ and ɛ₂-ɛ₃ interfaces. He calculated the mixed wavevector diagram β/k₀ and the surface conductivity of graphene σ/σ_min as a function of frequency for both the monolayer and 10-layer cases. The increase in β/k₀ frequency for both the monolayer and 10 layers converges to an approximately constant value. The imaginary part of β/k₀ tends toward zero due to the Pauli blocking principle. The behavior of the real part σ/σ_min for monolayer graphene shows a decreasing trend with frequency, while the imaginary part σ/σ_min exhibits an initial decrease followed by an increase.

Table 1 depicts the attenuation or loss coefficient (α) for monolayer graphene at a frequency of 100 GHz and distances of 100 nm for various values of chemical potential. The attenuation decreases with increasing chemical potential of graphene and, likewise, with the spacing among layers.

μc (eV)	α (dB/μm)
	d = 0.1 μm	d = 1 μm
0	1.22	0.394
0.4	0.362	0.115
0.8	0.256	0.081
1.2	0.209	0.066
1.6	0.198	0.061
2	0.162	0.05

Table 1.

Attenuation of monolayer graphene wave for various values of chemical potential.

5.3 Graphene tape on SiO₂

Mohammadi et al. introduced a graphene monolayer 20 nm in width situated on the top of a silicon dioxide layer that is 50 nm thick, for the purpose of confining and guiding terahertz waves [24]. The configuration is subject to illumination in the transverse magnetic mode. As illustrated in Figure 4, the waves can pass through and be filtered using an air gap at a graphene chemical potential of 0.9 eV. The transmitted wave spectrum is concentrated around a central wavelength of 10 μm. The real component of the effective refractive index is approximately 80, indicating the effective confinement of surface plasmon polaritons. The imaginary part of the effective refractive index is 0.038, a suitable value for SPP transmission over short distances of a few micrometers.

Figure 4.

The transmission spectrum for graphene tapes with a width of 20 nm and air gaps [24].

5.4 Graphene-SiO₂-Si

He and Li conducted a comparison between surface plasmon modes in the transverse electric field (TE) and transverse magnetic field (TM) configurations for a graphene-SiO₂-Si structure [25]. A plasmonic channel was formed below the graphene in the silicon dioxide layer.

In this study, dispersion relations for surface plasmon modes in TE and TM configurations were extracted using propagation equations and boundary conditions. The effective refractive index and transmission loss for TM and TE modes in the graphene-SiO₂-Si structure with various SiO₂ layer thicknesses (50, 100, 200, and 300 nm) were calculated. The chemical potential of graphene was set to 0.2 eV. Calculations indicate that the TM mode is confined, and the refractive index of graphene in the TM mode increases with frequency. Consequently, the TM mode exhibits better confinement compared to metal-based structures. The maximum loss occurs at a frequency range of 2.5–3 THz. For frequencies higher than 4 THz, the effect of silicon dioxide thickness on the loss factor is negligible.

5.5 Graphene on gold

Ono et al. reported the use of graphene plasmonic waveguides with sizes smaller than 230 × 20 nm [26]. In this thin-layered waveguide, a gold layer is deposited on a SiO₂ substrate, and on top of it, a single layer of graphene is placed. The waveguide particularly confines light for a chemical potential of 0.2 eV. In this structure, graphene nonlinear absorption has increased, resulting in fast switching. The switching energy is 35 fJ, and the response time is 260 fs. The energy of switching is lower than conventional devices, and the switching time is the smallest reported to date. This device can effectively integrate with common silicon waveguides used in photonic integrated circuits. They calculated the field enhancement factor for changes in the thickness of gold from 20 to 100 nm. The results demonstrated the thinner gold layer makes the higher enhancement.

5.6 Multilayers of graphene and dielectric

A graphene-based plasmonic waveguide was introduced [27]. This waveguide utilized a multilayer graphene and dielectric structure positioned on the top of a silicon waveguide core, resulting in reduced loss and significantly increased propagation length.

According to the reported results, an increase in the number of dielectric layers leads to a reduction in the effective refractive index of the waveguide, diminishing light confinement. Simultaneously, the rise in the number of dielectric layers results in an extension of the propagation length, reaching approximately 30 μm. As a result, losses have been mitigated with the increase of graphene layers.

5.7 Ridge-based waveguide

Haddadan and Soroosh proposed a plasmonic waveguide [28]. This waveguide consists of a silicon ridge with a height (h) of 50 nm and a width (W) of 450 nm placed on silicon substrates with a thickness of 200 nm. On this ridge, a single-layer graphene is sandwiched between layers of silicon dioxide, with thicknesses of 20 and 12 nm. The space between silicon and graphene serves as the plasmonic waveguide or channel, aiding in trapping the incoming light in the waveguide and propagating it along the length of the waveguide (as shown in Figure 5).

Figure 5.

(a) A view of the structure and (b) a representation in the x-y plane for a better understanding of the ridge-based channel [28].

The graphene chemical potential makes a difference in the refractive index of the silicon stack and its surrounding area. Therefore, due to the significant increase in the electric field, light is focused in the plasmonic channel. This phenomenon is attributed to the slot effect. The slot confines terahertz waves in a region with a lower index due to the overall reflection trapping of light.

The analysis indicates that the height and width of the channel have a direct impact on FOM in the plasmonic waveguide. Increasing the width brings the behavior of the waveguide closer to a slab behavior, while increasing the height leads to an increase in FOM due to reduced loss. Additionally, raising the chemical potential and confinement results in an increased FOM, while the corresponding graphs for effective refractive index N and loss α show a decrease. These findings highlight the enhancement of light guidance performance in plasmonic channels through increased height and reduced loss (as depicted in Figure 6).

Figure 6.

FOM factor in terms of (a) chemical potential and (b) waveguide width for various channel heights [28].

5.8 SiO₂-graphene-ridge

In another research, a plasmonic waveguide was designed by Maleki et al. [5]. As shown in Figure 7a, with the help of a ridge and adjustment of chemical potential, cross-talk reduces in comparison with the waveguide of Ref. [28].

Figure 7.

(a) A view of the device in the cross-section and (b) FOM and loss in terms of the channel width in the x direction [5].

In Figure 7b, the graph illustrates the relationship between the channel width and both figures of merit and loss. The figure of merit ranges between 944.79 and 1036.63 as the width increases from 75 to 310 nm, while the loss decreases from 1.14 to 0.92 dB/μm within this width range. The superior figure of merit and minimal loss validate the exceptional performance of the channel under consideration. Hence, a width of 80 nm is selected for designing a compact waveguide, resulting in a figure of merit of 950.92 and a loss of 1.14 dB/μm.

5.9 SiO₂-graphene-SiO₂

A plasmonic structure utilizing graphene was suggested to guide and manipulate surface plasmon polaritons (as shown in Figure 8) [8]. The design involved positioning a graphene nanoribbon between two silicon dioxide layers. By inducing a 1 eV chemical potential to the graphene, a plasmonic waveguide formed close to the graphene monolayer. The effective confinement of the waveguide led to a decrease in channel loss to 0.029 dB/μm. Additionally, by reducing field penetration into the surrounding areas, the coupling length was extended to 187.9 μm.

Figure 8.

A graphene monolayer between two silicon dioxide layers for the confinement of surface plasmon polaritons [8].

5.10 Graphene nano-ribbon on SiO₂

Maleki et al. have designed a plasmonic channel with significant confinement to guide surface plasmon polaritons, utilizing 50 nm wide graphene nano-ribbons on a 230 nm thick silicon dioxide layer (as illustrated in Figure 9) [6]. A gold layer, with a thickness of 180 nm, was placed beneath the silicon dioxide to create a potential difference with respect to the graphene. The simulation results show that, by applying voltages of 1.5 and 8.3 V to graphene nano-ribbons, chemical potentials of 0.1 and 0.5 eV can be obtained, and the channel loss can be changed from 0.91 to 88.23 dB/μm. Based on this, two logical zero and one states and switching operations can be realized.

Figure 9.

Graphene nano-ribbon on silicon dioxide with a width of 50 nm for SPP confinement below the graphene [6].

6. Conclusion

This chapter considers the theory of exciting surface plasmon polaritons and introduces the excellent features of graphene in the fields of optics and electricity. It also explores various concepts and structures designed to confine surface plasmon polaritons using graphene. The ridge-base structure, graphene nano-ribbon, and graphene sheet are highlighted as important methods for designing plasmonic devices. The adjustable chemical potential of graphene offers a promising avenue for manipulating terahertz wave transmission. The dependence of transmission loss on the chemical potential is extensively utilized in designing optical fast switches for both digital and analog applications. Through the investigation of relevant articles, it is demonstrated that a loss of 0.21 dB/μm and a figure of merit of up to 845 can be achieved for wavelengths ranging from 25 THz to 40 THz. These studies suggest that the graphene-dielectric interface holds considerable potential for guiding surface plasmons in plasmonic devices.

Acknowledgments

This work was supported by Shahid Chamran University of Ahvaz, grant number SCU.EE1402.672.

Conflict of interest

The authors declare no conflict of interest.