2.1. Surface phonon-plasmon polaritons

Phonon-plasmon polaritons are quasiparticles that result from strong interactions between photons and phonons, which can occur as surface and volume confined waves. The dispersion relation for phonon polaritons is highly dependent on the material. While volume polaritons tend to approach asymptotically to the longitudinal optical phonon frequency of the material, the surface polaritons tend to approach the surface phonon frequency.^[9]

One reason for interest in recent research on bulk and surface polaritons is to obtain optoelectronic devices that operate with electromagnetic waves in the THz and mid-infrared (MIR) ranges.^[10–12] We will comment on the results of studies on surface plasmon-phonon polaritons (SPPPs) in doped GaAs, which resulted in the emission of radiation in the THz range.^[13]

It is noteworthy that SPPPs occur at the interface of two media with dielectric permittivities of different signs. The dispersion relation for SPPPs is given by^[14]

where k_SPPP is the surface plasmon-phonon polariton wave vector, w is the angular frequency, c is the speed of light, ε₁(w) is the dielectric permittivity of the first medium (n-doped GaAs in this case), ε₂ is the dielectric permittivity of the second medium (vacuum in this case). For the same frequency (w), the SPPP wave vector is larger than the photon wave vector, and the SPPPs located on the interface can be considered as non-radiative modes. However, SPPPs can be transformed into photons (and vice versa). For this to occur, a regular surface grating should be used in order to reach the phase-matching condition for SPPP and photon^[1] (and references therein), i.e.,

with k_Ph = w/c being the wave vector of photon in vacuum, θ the incidence angle, k_G = 2π/a the reciprocal vector of the grating, and a the grating period.

Other research has revealed that the electrical permittivity for electron-doped GaAs (n-GaAs), considering the contributions of the lattice, as well as the free electrons, is given by^[15]

Here, ε_∞ is the high frequency permittivity, ε₀ the low frequency permittivity, w_TO the transverse optical phonon frequency, γ the damping constant for the transverse optical phonon, (where N_e is the free electron concentration, q is the elementary charge, and m_e is the electron effective mass) the plasma frequency, and τ = μm_e/q (where μ is the mobility) the electron relaxation time.

For the existence of surface plasmon-photon polaritons at the GaAs/vacuum interface, it is necessary that Re(ε₁) < 0, and |Re(ε₁)| > 1, the curves referring to the real and imaginary parts of ε₁ are plotted as a function of the energy of the modes (in meV) present in a GaAs/vacuum plane interface (Fig. 1).^[13]

It is noteworthy that the following values were considered: the free electron concentration N_e = 2 × 10¹⁸ cm⁻³, and mobility μ = 2500 cm²/V·s. Moreover, as the GaAs conduction band is nonparabolic, the dependencies of μ and γ on the doping level were taken into account,^[16,17] as well as the electron concentration dependence on m_e.^[17]

As we can see in Fig 1, for Re(ε₁(ω)) < −1, the n-GaAs SPPPs can occur in the ranges between 0 meV and 31.7 meV, and 33.2 meV and 56.9 meV, i.e., for these two energy ranges, there may be surface plasmon-phonon polariton at the n-GaAs/vacuum interface.^[13]

From Eqs. (1) and (3) we find the dispersion ratio (SPPP mode energy versus K_SPPP, or versus k_Ph sin θ + mk_G, as shown in Fig. 2(a)^[13]) for θ = 11° (solid thick line) and for normal incidence (θ = 0°, Fig. 2(b) solid thick line, adapted from Ref. [13]).

Notice in Fig. 2(a) that the SPPP dispersion relation is made up of two branches, which are separated by a narrow gap. It occurs because the real part of the GaAs permittivity exceeds the value −1, close to the transverse optical phonon energy so that for this energy range, there can be no plasmon-phonon polaritons. The dashed lines shown in this figure indicate the coordinates of the points where there is resonance for different values of m, considering the incident angle θ = 11°.

Figure 2(b) shows the same results for the normal incidence angle (θ = 0°).

The reflectivity that occurs on a flat GaAs/vacuum interface was also investigated, as shown in Fig. 3(a) (adapted from Ref. [13]). As we can see, there are two distinct dips (due to bulk plasmon-phonon polariton excitations).^[18]

The insertion of a regular grating at the semiconductor surface (for instance, using photolithography technique) may cause changes in the reflectivity spectrum, which may provide the conversion of photons into surface plasmon-phonon polaritons, as well as the conversion of surface plasmon-phonon polaritons into photons. It is noteworthy that the occurrence of such a conversion may lead to the emergence of resonant dips in the reflectivity spectrum.

The reflectivity spectrum points were determined (where these dips exist, i.e., SPPP modes excitation occurs), considering the surface grating period a = 60 μm, as shown in Fig. 3(a) (arrows), using Eqs. (1) and (2). In these surveys, the absorptivity spectrum of a GaAs semi-infinite surface (plane surface) was also determined, considering the normal incidence of radiation, as shown in Fig. 3(b).

Inserting a surface grating can also alter the absorptivity spectrum of GaAs significantly. It is noteworthy that in this case, as the non-equilibrium SPPPs modes can scatter on the surface grating, there may be the occurrence of a transformation of surface plasmon-phonon polaritons into photons. Consequently, this transformation can cause resonant peaks in the absorptivity spectrum, which are shown in Fig. 3(b), by arrows (obtained via Eqs. (1) and (2), for a = 60 μm). Therefore, according to Kirchhoff’s law, photon emission in the THz and MIR frequency ranges is expected to occur at the resonant frequencies (where transformations of the surface plasmon-phonon polaritons to photons in n-GaAs occur).

2.2. Excitons-plasmons polaritons

The incidence of photons on the surface of a solid may lead to the occurrence of electron (in the conduction band)–hole (in the valence band) pairs, i.e., excitons, due to the absorption of photons.^[19,20] There are two types of excitons: Wanner^[21] and Frenkel.^[22] Wanner excitons are generally present in semiconductor materials, have weak coupling interaction and wide distributions of electrons and holes. On the other hand, the Frenkel excitons are usually located in insulating materials, have strong Coulomb interaction, and the electrons and holes are bound in somatic cells. Indeed, the incidence of photons in a two-level system (for example, an atom or ion characterized by a ground state and an excited state) can cause the periodic exchange of energy between the light field and this two-level system. The oscillations arising from the above mentioned process, called Rabi oscillations (named after Nobel Prize laureate Isidor Isaac Rabi), are associated with oscillations of electrons and photons. In other words, these oscillations can be defined as a periodic change between absorption and stimulated emission of photons and can be modeled by the Bloch vector formalism. The angular frequency of the Rabbi oscillations is called the Rabi frequency.

The coupling between photon and exciton (exciton-polariton) can be detailed from Maxwell’s equations so that the equation that determines the dispersion relationship for excitons-polaritons is given by^[23]

where k is the wave vector, w_L the resonance frequency of the longitudinal mode, and w_T the resonance frequency of the transverse mode.

The dispersion relationship for bulk exciton-plasmon polaritons, related to the GaP, is shown in Fig. 4(a) (solid lines) (adapted from Ref. [23]).

The horizontal dashed line represents the longitudinal resonance frequency, and the dashed-dotted line represents the transverse resonance frequency. Note that there are two branches, upper and lower branches. The longitudinal-transverse splitting, i.e., w_LT = w_L − w_T, represents the possibility of coupling strength between the exciton and photon.

The surface exciton-plasmon polariton is made up of strong coupling between photons and excitons, and these modes have the wave vector component in the surface plane. From Maxwell’s equations, considering the continuity of the electric field at the surface, it is possible to obtain the dispersion relation of the surface exciton-plasmon polaritons according to the equations shown below:

with being the dielectric function of the material.^[23]

Equations (5) and (6) determine that ε < −1, so that for the existence of surface exciton-plasmon polariton, it is necessary that w_T < w < w_L, as shown in Fig. 4(b) (the solid line between w_T and w_L).

It is noteworthy that both phonon-polariton and exciton-polariton have longitudinal-transverse splitting, i.e., longitudinal-transverse symmetry breaking. However, the LT-splitting for exciton-polariton is small. Note that w_LT = 008 meV for GaAs,^[24] and w_LT = 5.00 meV for ZnO so that the dispersion relation for the surface exciton-plasmon polariton is approximately linear. Therefore, the effective mass of exciton-polariton approaches infinity, which means that it can not move, and the states in this band can be neglected.

The interaction between localized surface plasmons (LSPs) and photons may provide some advantages regarding optical properties, such as, a longer lifetime of excitons and a large increase in the quantum yield.^[25,26] Taking into account that in metal nanostructures can have an energy of localized EM field, the intensity of the electric field can be increased,^[27] and there may be nonlinear effects.^[28] To achieve a composite structure with enhanced optical properties, hybrid metal/semiconductor structures are being much researched, because of the interaction between SPs and excitons.^[29–31]

The combination of metal and semiconductor constituting a hybrid structure provides optical properties which can be used for the nanometer scale of electromagnetic energy flow control. For example, interactions in hybrid structures can be obtained between quantum-confined electronic states located in the semiconductor and confined electromagnetic modes located in the metal. In other words, exciton-plasmon interactions obtained in this way can provide various effects, such as absorption and emission, control of nanoscale energy-transfer processes, creation of new excitations in the strong coupling regime, and increase of optical nonlinearities.^[32]

2.3. Surface plasmon polaritons

Surface plasmon polaritons (SPPs) can occur due to the coupling between the oscillations of the electron plasma with photons, which propagate at the interface between a dielectric and a conductor.

To enable the detailing of SSPs in graphene (GSPPs) as well as in other 2D materials, we start from the theory that involves SPPs in metallic interfaces.

From the Maxwell and Helmholtz equations, we can determine the equations that represent the behavior of the metal/dielectric interface SPPs, referring to both sides of this interface, as shown in Fig. 5.

The following equations can be used for TM modes:^[14]

Here, k_{i, z} ≡ k_i (i = 1, 2) is the component of the wave vector in the direction perpendicular to the interface between the two media, and β(k_x) is often referred to as the propagation constant. It is noteworthy that the inverse of the wave vector determines the decay length of the fields in the direction perpendicular to the interface, and consequently, the wave confinement is given by E_z = 1/e^|k_z||z|.

Recall that a metal (in this case located at z < 0) has a negative real part dielectric function (Re[ε₁(w)] < 0) for frequencies below the plasmon frequency (w_p), and the dielectric (in this case located at z > 0) has positive dielectric constant ε₂.

The continuity condition of H_y ad ε_iE_z at the interface requires that A₁ = A₂ so that

From Eqs. (7)–(12), we can state that due to the exponent signals adopted in these equations, the confinement of the interface modes requires that the real part of the metal-dielectric constant is negative (Re(ε₁) < 0), and the dielectric constant of the dielectric is positive (ε₂ > 0). Therefore, surface waves at the metal/dielectric interface can only exist when the materials that make up the interface have the real parts of their dielectric constants with opposite signals, i.e., the interface materials are a conductor and a dielectric. We must state that for the H_y equations related to TM modes (Eqs. (9) and (12)) to comply with the conditions of the wave equation, it is necessary that

On the other hand, from Eqs. (13)–(15), we can arrive at the important expression for the dispersion relation of TM SPP modes that propagate at the interface, given by

Note that equation (16) can be applied to real and complex values of ε₁, that is, for propagating signals in conductors with or without attenuation. On the other hand, for TE modes, we can state that

Due to the continuity condition of E_y and H_x in the interface, we have

Therefore, for the existence of TE modes in a metal/dielectric interface, it is necessary that Re[k₁] > 0 and Re[k₂] > 0. Hence, we can conclude that to have TE SPP modes on this interface, it is needed that A₁ = A₂ = 0, i.e., there are no TE modes at the metal/dielectric interfaces. However, as will be seen later, in graphene/dielectric interfaces, it is possible to have TE modes (although TE modes suffer greater attenuation than the TM modes).

Getting back to the SPPs TM on metal/dielectric interfaces, we can state that the energy of electric and magnetic fields of SPPs at these interfaces has exponential evanescent decay, as shown in Fig. 6.

We can apply the Drude model by adding the effective parameters w_p (plasmon frequency in eV), γ_bulk (proportional to the reciprocal of the mean free time between electron collisions), and ε₀ (contribution of interband transitions to the polarizability)^[33] to determine the dielectric function of the metal, as given by

However, the parameters w_p, γ_bulk, and ε₀ found in the international scientific literature present different values. Taking into account the values that presented the results closest to those obtained experimentally,^[34] we adopted the following values for gold:^[35] w_p = 9.01 eV, γ_bulk = 0.072 eV, and ε₀ = 9.84.

From Eq. (16), we got Fig. 7, by which we can observe the dependence of the real and imaginary parts of the propagation constant as a function of the incident photons energy.

We used Eq. (24), adopting the parameters mentioned above, for the determination of the metal dielectric function ε₁, and the dielectric constant of the dielectric (in this case the silica) ε₂ = 2.25.

In Fig. 7, the solid line represents the real part, and the dashed line represents the imaginary part of the SPP modes propagation constant.

Note that we used the energy range between ∼ 0.8 eV (λ = 1.55 μm) and ∼ 4.133 eV (λ = 0.3 μm) for the incident photons, i.e., in the range from infrared to ultraviolet. Interestingly, in this case, the maximum value of the real part for SPP modes is β_m ≈ 4.332 × 107 + 3.469 × 107i, the energy of this mode is E_{β m} ≈ 2.561 eV, and λ_{β m} ≈ 2π/Re(β_m) ≈ 145.05 nm. It is noteworthy that for values greater than E_{β m}, the real part of SPPs modes drops dramatically, but from certain energy, the real part of SPPs modes starts to grow again, but less intensely. This means that from this maximum wavenumber value, the value of the wavenumber in the z direction () decreases, and consequently, the amount of mode confinement perpendicular to the interface (E_z = 1/e^|k_z||z|) decreases too. Thus, for energy higher than E_{β m}, the modes on the interface are no longer SPPs modes and are called quasibound modes.

Besides, for the occurrence of SPPs at the metal/dielectric interfaces, it is necessary to use an insertion technique to achieve the match for the wave vectors of the SPPs and the incident light. Recall that the imaginary part of the propagation constant represents the energy loss of the SPPs modes, that is, the attenuation of the SPPs modes so that the propagation length and wavelength of SPPs modes are given by

We can still state that SSP modes can be considered as consisting of an electromagnetic wave located in the dielectric, with (E_z2 = E_z = 1/e^|k_z2||z|) and in metal (E_z1 = E_z = 1/e^|k_z1||z|).^[14] Hence, as |k₁| ≫ k₂|, we can conclude that the amount of mode confinement in metal is much smaller than in the dielectric.

6.1. Single graphene layer embedded in a dielectric medium

The dispersion relationship obtained from Maxwell’s equations for TM modes, considering the dielectric layer embedded in two distinct dielectrics, is given by^[45]

where ε_d1 and ε_d2 are the dielectric constants of the dielectric medium, β_TM is the SSP wavevector (TM mode), k₀ and w are the wavenumber and angular frequency of the incident photon, and σ is the graphene conductivity. Therefore, from Eq. (36), we can obtain the dispersion relation for TM modes, assuming the graphene sheet embedded in a single dielectric, which is given by^[45,46]

with being the intrinsic impedance of free space.

We can state that for TE modes using the same reasoning as above, the dispersion relation for graphene embedded in a dielectric medium is given by

We saw in Section 4 that considering the real and imaginary parts of conductivity (σ = σ′ + σ″), the effective relative permittivity of graphene is given by . Hence, we can get both the real and imaginary parts of the graphene permittivity, which are given by and , respectively. Therefore, for convenience, repeating what we have shown in Section 3, the real part of the graphene permittivity may be positive or negative, depending on the value of the imaginary part of the graphene conductivity. It is noteworthy that the real part of this permittivity being negative may provide the occurrence of TM modes. On the other hand, being positive can provide the occurrence of TE modes. Consequently, graphene may have TM (, i.e., σ″ > 0) or TE modes ( that is σ″ < 0).^[54]

The plot of the imaginary part of graphene conductivity in air calculated by Eqs. (30) and (31) (dashed line), as well as the graph of the real part of the relative permittivity of graphene (solid line), according to Eqs. (30) and (31), is shown in Fig. 10.

In Fig. 11, we can see in more detail that for TM GSPP mode and TE GSPP mode to be supported on graphene, it is necessary that 0 < ħw/μ_g < 1.667 (σ″ > 0) and 1.667 < ħw/μ_g < 2 (σ″ < 0), respectively.^[55]

Note that for ħw/μ_g > 2, the real part of the graphene conductivity is greater than zero so that the TE modes are attenuated, and there are no TE modes.

It is noteworthy that the increase in temperature causes a slight change in the real part of graphene conductivity around the bifurcation point (where σ′ = 0) so that the real part of conductivity becomes slightly positive for some values of ħw/μ_g ≈< 2, which results in small attenuations of GSPPs modes. However, TE modes suffer greater attenuation than TM modes.^[45]

6.2. Two graphene sheets embedded in a dielectric medium

The schematic representation of a double-layer of graphene, embedded between 3 dielectric media, is shown in Fig. 12(a). We can see the schematic representation of the electric field profile of the hybrid GSPP mode in Fig. 12(b), which is constituted by the overlap of the GSPP modes located in each of the graphene layers.

From the equations that determine the electric fields, applying the boundary conditions on both interfaces, and assuming exponential decay of the fields, we can get the dispersion relation for this device^[56]

In the case that medium 1 is equal to medium 3, equation (39) becomes

Indeed, the first term represents the even mode and the second term represents the odd mode.

Assuming all dielectrics are equal, that is, ε₁ = ε₂ = ε₃ = ε_d and (non-delayed approximation or plasmon approximation), equation (40) reduces to

It is noteworthy that while the + sign in Eq. (41) represents the dispersion relation for the even mode, the − sign represents the dispersion relation for the odd mode.

On the other hand, according to the conventional waveguide theory, the electric fields in the different regions of the double-layer graphene device (Fig. 13, considering ε₁ = ε₂ = ε₃ = ε_d) can be determined by the following equations:^[46]

where , , β is the propagation constant (wave vector in the propagation direction) of the GSPPs modes, and coefficients A,…,H are the amplitudes of the modes in the different regions.

Taking into account the boundary conditions (the transverse electric and magnetic fields are continuous at the interfaces), as well as using the limit for the smallest possible value of the distance d, when d → 0, the dispersion relation of this device is given by

There are two solutions concerning Eq. (43), i.e, the coupled, symmetrical (even, β_even), and anti-symmetrical (odd, β_odd) modes, which can be roughly determined adopting β_even = β + Δβ₊ and β_odd = β + Δβ₋, where Δβ₊ and Δβ₋ represent small amounts with respect to the propagation constant in an isolated graphene sheet (β)^[46]

where + and − signs represent the symmetric and anti-symmetric modes, respectively, and .

We plot the real parts of the even and odd mode propagation constants using Eq. (44) (left y-axis for Fig. 14), considering the same parameters adopted in Ref. [46], i.e., λ₀ = 10 μm (wavelength of incident light), μ = 0. 15 eV (applied to both graphene layers), τ = 0. 5 ps, 0.02 μm < d < 100 μm, and ε_d = 1, in order to compare the results obtained.

The results are identical, and the slight difference between the values may have been caused by the fact that we used the general graphene conductivity equation (σ = σ_intra + σ_inter) according to Eqs. (30) and (31). Note that the real part values related to the wavenumbers in the propagation direction of the even modes are greater than the real part values of the odd modes. We can also see that for values from 100 nm onwards, the real parts of the propagation constants of the even and odd modes approximate the value of the propagation constant on a single graphene layer.

It is noteworthy that this device can operate as a directional coupler whose coupling coefficient (C) is given by^[46]

as we can see in the right-hand axis of Fig. 14.

For comparison, in Fig. 15, we also plot the same parameters determined in Fig. 14, but using λ₀ = 1. 55 μm, μ = 0. 5 eV (applied to both graphene layers), and 2 nm < d < 2.5 nm using hexagonal boron nitride (h-BN) as the dielectric. We considered the mobility of graphene μ_m = 60000 cm²/V·s and the dielectric constant between 3 and 4^[57,58]). In Section 8, we will detail the most commonly used substrates to support graphene.

10.1. Back gate voltage

Gate voltage means applying a voltage between an external medium and a single graphene sheet, between two parallel graphene tapes, or between an external medium and one of the parallel sheets, depending on the geometry of the nanophotonic device which is analyzed. Through the gate voltage, it is possible to control the chemical potential of graphene, which can operate in the conduction or valence band, depending on the applied voltage, due to the diffusion of negative (positive) charges in the graphene plasmonic system (electric field effect).^[94]

In Fig. 23, we show an example of a scheme for obtaining a gate voltage between silicon and a graphene sheet. In this case, the graphene sheet has been deposited on the silicon oxide substrate with a well-defined thickness, which is supported by highly doped silicon. As the highly doped silicon functions similarly to a metallic contact, silicon dioxide is an insulator, and graphene is a zero-gap semimetal, an electric field effect occurs, similar to that occurring within a capacitor of flat plates so that it is possible to control the charge density in graphene.

This type of gate voltage, called back gate voltage, can be used in graphene-based nanophotonic devices, such as a transistor, whose schematic representation is shown in Fig. 23. As we can see in this figure, from the transistor, it is possible to measure the dependence of conductivity and resistivity of graphene as a function of the voltage between silicon and graphene (in this case, back gate voltage). Note that the electrode located on the left side of Fig. 23 acts as the source, while the other electrode acts as the drain. Then, by applying an alternating current between the source and the drain, and making the measurements of the voltage between the source and drain, for the value of the applied back gate voltage (V_bg), we obtain the graphs related to the conductivity (V_bg, σ) and resistivity (V_bg, ρ) of graphene,^[95] as shown in Fig. 24.

The conductivity of graphene increases (while obviously, the resistivity decreases) as the absolute value of the back gate voltage increases, regardless of whether the voltage applied to the highly doped silicon is positive (and negative in graphene, causing electron transfer to graphene) or negative (and positive in graphene, causing the transfer of holes to graphene).

The position of the Fermi level, whose value is practically the same as the chemical potential of graphene, is controlled in this case by the back gate voltage V_bg. It is noteworthy that considering graphene as ideal, for V_bg = 0 eV graphene would have the minimum conductivity (σ_min). However, considering real graphene, the minimum conductivity does not occur under this condition, given the unwanted doping of the graphene from the environment and the device manufacturing process, among other factors.

The Dirac voltage (V_D), i.e., the gate voltage where the minimum conductivity (or maximum resistivity), for example, in graphene on SiO₂ occurs, has been studied in several experiments, where the influence of the manufacturing and cleaning process of graphene was verified. Additionally, at the interface, there are energy states that behave as charge acceptors or donors. Therefore, there are different values for the graphene Dirac voltage.^[96–100] It is noteworthy that the residual charge density in graphene n₀ has typical values between ∼ 5 × 10¹⁰ cm⁻² to ∼ 30 × 10¹⁰ cm⁻² (p-type doping), for graphene on SiO₂/Si depending on the charge impurity density in SiO₂.^[101] So, even without gate voltage, graphene has a residual charge density and for this reason, it is necessary to exclude this charge density from the value of the charge density provided by the gate voltage as shown in Eq. (58).

In the case of back gate voltage, considering graphene and highly doped silicon surface, such as the plates of a flat capacitor and SiO₂ as the dielectric between the plates, the density of graphene charge carriers due to the application of V_bg is given by

where C_ox = ε_dε₀/d is the capacitance per unit area of the graphene/SiO₂/Si (oxide capacitance) structure, d and ε_d = 3 9 are the thickness and the dielectric constant of the SiO₂ layer, respectively. It is noteworthy that in Eq. (58) the quantum capacitance of graphene was not taken into account, as will be detailed below, since for dielectric thickness greater than 200 nm, for example, this quantum capacitance can be neglected.^[102]

On the other hand, the graphene conductivity can be, theoretically, determined from a semiclassic diffusive transport model according to the following equation:

with n_t being the density of graphene charge carriers (n_t = n₀ + n_Vbg) and μ the mobility of these graphene charge carriers.

10.2. Top gate voltage

Considering that practically all potential difference occurs in the dielectric medium, which in a back gate structure has a large thickness, a large potential difference (ddp) is needed to obtain moderate increments of charge density. However, the application of high values (ddp > 100 V) causes the damage of SiO₂. Therefore, as we have mentioned above, we can state that the back gate voltage process is limited. To overcome this problem, it is necessary to decrease the thickness of the SiO₂ layer, but obtaining thinner layers of this dielectric to support the graphene layer requires more advanced techniques. Another solution is to use a medium with high dielectric constant, such as HfO₂. Thus, the most used solution is the application of a top gate voltage on graphene, so that the dielectric has a small thickness.

One option for improving the top gate voltage efficiency is to replace SiO₂ with h-BN, the thickness of the h-BN layer being thin enough (usually between 10 nm and 20 nm), to increase the efficiency of charge generation in graphene. Figure 25 shows an example scheme for obtaining a top gate voltage between silicon and a graphene sheet.

In the top gate voltage structure, a graphene sheet shall be deposited on the silicon, and the insulator shall be deposited above the graphene so that the insulator is in contact with the gate electrode. Since in this case, the dielectric thickness is small (generally 5 nm to 15 nm), the quantum capacitance per unit area cannot be neglected. As the quantum capacitance (C_q) is inherent to the graphene layer, we should consider the total capacitance per unit area of the structure (C_t) shown in Fig. 25 as a series association of C_ox and C_q. C_q is given by^[103]

considering eV_ch ≫ K_BT, where n_Vtg is the charge density provided by the top gate voltage. Therefore,^[104]

with

Here, V_ch is the voltage on the channel (over graphene), which depends on the potential difference between the source port and the drain port (V_ds), as well as on the potential difference between the gate electrode and the source port (V_gs).

10.3. Top gate voltage + back gate voltage

An even more efficient system for applying gate voltage is the simultaneous use of a back gate voltage and a top gate voltage, according to the schematic representation shown in Fig. 26.

Top gate + back gate experiments were performed using a MOSFET where a graphene film was used as the channel of this transistor (length = 5 μm, width = 1 μm). The graphene film has been grown in metal and transferred to a SiO₂ layer, which was located on a silicon substrate. The dielectric used in the top gate was HfO₂ (ε_dt = 16) and in the back gate was SiO₂ (ε_db = 3.9).^[105]

The charge density on the graphene channel (Q_sh) has been determined by the hole and electron density difference, multiplied by the elementary charge value.^[106] Hence, the capacitance has been obtained by the Q_sh derivative concerning the graphene potential, i.e., C_q = − dQ_sh/dV_ch, where V_ch = E_F/e.

In these studies, it has been found that the minimum quantum capacity at room temperature had a value C_qmin ≈ 0. 8 μ F·cm⁻², without taking into account the intrinsic charge density of graphene. However, considering, for example, n₀ = 4 × 10¹¹ cm⁻² and V_tg = 0 V, it has been obtained that C_qmin = 1732 μF·cm⁻².^[105]

Note that the total capacity (C_t2) per unit area for the double gate voltage device, shown in Fig. 26, is given by the serial association of capacitance per unit area (quantum, top-gate oxide (C_ox-top) and back-gate oxide (C_ox-back)), i.e.,

Here, t_g, t_b, ε_t, and ε_b represent the top-gate dielectric thickness, back-gate dielectric thickness, top-gate dielectric constant, and back-gate dielectric constant, respectively.

Another research used a transistor operating with top gate and back gate together consisting of graphene over Si/insulator, in which the source and drain electrodes (left part of Fig. 27) were made of a gold film with 50 nm thickness and titanium (thickness 5 nm), which presented ohmic contact with graphene less than few kΩ.^[107]

Between these two electrodes, a 30 nm thick aluminum film was inserted, which was directly deposited on the graphene layer and was used as the top gate electrode. It is noteworthy that the insulation between the aluminum film (top electrode) and the graphene layer was not achieved by the introduction of a conventional insulating film. This isolation was achieved simply after this device remained exposed to the air for a few hours.^[108] It was shown that the resistance between the top gate electrode and the source and drain electrodes exceeded 100 MΩ, but the interior of the aluminum electrode remained intact and conductive.

The greater possibility of the occurrence of this insulation film (obtained between the aluminum film and the graphene layer) with an estimated thickness between 5 nm and 9 nm is due to the oxygen diffusion present in the atmosphere, and to the very weak physical contact between the aluminum electrode and graphene layer.^[109–111]

In Fig. 27(b), we can observe the physical representation regarding the electric fields and potentials through this device. The electric fields and potentials on the graphene channel (V_gr) are related as follows: V_bg − V_gr = E_b d_b V_gr − V_tg = E_t d_t, with V_bg and V_tg being the applied voltages at the top gate electrode and bottom electrode (highly doped silicon), respectively, E_t and E_b the electric fields between the top electrode and the graphene and between the graphene and the bottom electrode, respectively. By Gaussian law, considering that the graphene layer has charge density q, we can say that ε_td_t − ε_bd_b = q, where ε_t is the dielectric constant of the top gate, ε_b is the dielectric constant of the back gate, and d_t and d_b are the dielectric thicknesses for the top and bottom capacitors, respectively. From the above equations, we can arrive at the following equation:^[107]

where . Note that the value of α corresponds to the value of the ratio between the back gate capacitance and the top gate capacitance per unit area. After some mathematical manipulations, the value of α has been got (α = 0.013) by measurements, using the device shownin Fig. 27(a). Hence, taking into account that the SiO₂ dielectric constant is ε_b = 3.9 and the silica layer thickness is d_b = 300 nm, the top gate capacitance for this device was estimated to be 8.9 × 10⁻³ F/m². Additionally, assuming that the aluminum oxide dielectric constant has a value between 5 and 9, the top gate dielectric thickness should be between 5 nm and 9 nm. Note that obtaining a dielectric film with this small thickness is very difficult to achieve by using conventional methods.^[107]

10.4. Fermi level control via gate voltage

The graphene’s Fermi level (approximately the same value as graphene’s chemical potential) is related to the voltage applied to graphene (back voltage) according to the following equation:^[112]

where ε₀ε_d is the electrical permittivity of the dielectric and V_D is the Dirac voltage (gate voltage for which the maximum graphene resistivity occurs, or in other words, for which the minimum conductivity of graphene occurs). In Eq. (65), the + sign is used for V ≤ V_D and the − sign for V > V_D.

Note that for ideal graphene, the gate voltage at which graphene’s maximum resistivity occurs is V_g = V_D = 0. In an experiment, it was shown that via the dry transfer of graphene films to the standardized substrate (SiO₂/Si), as well as subsequently modified RCA cleaning, devices with low graphene Dirac voltages were obtained, i.e., V_D < 4.5 V, with a mean value around 2.5 eV.^[113] However, during the investigation of a low flicker frequency in a graphene layer on h-BN/SiO₂/Si as well as on SiO₂/Si, it was found that V_D ≈ 5 V for graphene on h-BN/SiO₂/Si and V_D ≈ 25 V for graphene over SiO₂/Si.^[114]

Given the importance of this subject, we will detail the research conducted on gate voltage in a graphene structure over h-BN, which is supported by Cu(111).^[115] In this research, the electrostatic doping of graphene, as a function of the applied voltage and the thickness of the h-BN layer, was detailed, using first-principles density-functional theory calculations (DST). It has been observed that even without a gate voltage there is a spontaneous charge transfer between Cu (111) and graphene through h-BN so that graphene maintains intrinsic doping. This charge transfer occurs due to the difference between the work functions of Cu(111) and graphene, but it is strongly altered by displacement of the charges due to the poor chemical interactions at the metal/h-BN interfaces, and h-BN/graphene. However, the application of a variable gate voltage can be used to provide the controlled level of the graphene doping, as it has been shown through other research, where it has been showed that the graphene Fermi level is dependent on the square root of the voltage.^{[64,116–119]}

The structure used in this research has been modeled in a periodic supercell in the normal direction to the plane of the graphene layer (z), with six atomic layers of copper atoms in the surface orientation (111), a thin film with 1 to 6 layers of h-BN, a single graphene layer, and a vacuum region of 15 Å. It has been shown that in this structure (Cu (111)/h-BN/graphene), graphene has intrinsic n-type (electrons) doping. However, the difference between the Fermi level of this structure and that of the graphene decreases with an increasing number of h-BN layers. This intrinsic doping of graphene originates from the transfer of electrons from copper to graphene via h-BN. The working functions related to copper and graphene, without applying a gate voltage, presented the values W_M = 5.25 eV and W_G = 4.48 eV, respectively. Therefore, in a structure consisting only of copper and graphene, there is an electron transfer from graphene to copper (average electron energy level in graphene is higher than that in copper), so that in this case, graphene is intrinsically positively doped, i.e., the difference in the graphene Fermi level is positive (Δ_EF > 0). However, a great influence of the h-BN layer has been proven, because, in the aforementioned structure, this does not occur. Incidentally, in the Cu(111)/h-BN/graphene structure, the difference between the copper and graphene Fermi levels is Δ_EFCG = −0.49 eV (average electron energy level in copper is higher than that in graphene), which means that in this case, in order to achieve equilibrium, it is necessary to transfer electrons from copper to graphene so that graphene receives an intrinsic n-type doping.

The explanation for the above can be seen in Fig. 28. Note that a potential drop occurs at the copper/h-BN interface (Δ_EpMH = 1.12 eV), as well as at the h-BN/graphene interface (Δ_EpHG = 0.14 eV).

The energy difference between the two interface media is explained by the formation of dipoles between the two materials (at the interface), which occurs due to the Pauli exchange repulsion principle. In other words, this means that the adsorbed charges on the surface cause repulsions concerning substrate charges so that, in this case, electrons transfer to copper occur.^[71,120] It is noteworthy that the intensity of these dipoles does not depend on the number of layers of h-BN, for the number of layers greater than 1.^[115] In Fig. 28, d₀ is the average distance between the charges constituting the dipoles.

Note that the difference between copper and graphene Fermi levels is given by ΔE_FMG = W_M − ΔE_pMH − ΔE_pHG − W_G. Then, due to charge transfer, a potential difference occurs between copper and graphene so that the h-BN layer is crossed by an electric field, which provides the polarization of the h-BN layer.

On the other hand, by applying a positive gate voltage between copper and graphene, so that an electric field is generated, which opposes the intrinsic electric field mentioned above, it is possible to alter the doping of graphene and consequently its Fermi level. Incidentally, the application of a gate voltage that provides an electric field E_g = 0, 1 V/Å (contrary to the intrinsic electric field) causes a null electric field in the h-BN layers, so that h-BN behaves as if it were isolated. Obviously, the application of an electric field due to a negative gate voltage causes an increase in the polarization of the h-BN layers. Therefore, the application of gate voltage provides the control of graphene doping, and consequently, its Fermi level.

Through further research, an analytical model has been developed for current tunneling and contact resistivity in a metal/insulating/graphene (MIG) structure using gate voltage, where the metal/graphene (MG) junction can be considered as a particular case of the MIG structure.^[121] In this model, a voltage drop has been considered between the insulation edges, i.e., between metal and graphene (ΔV). The reason for this voltage drop across the insulator is the difference between the work functions of graphene (W_g) and metal (W_m), so that a charge transfer between these two materials occurs. Moreover, there is also the influence of chemical interactions between graphene and metal.^[122]

The displacement of the graphene Fermi level (E_Fg) concerning the energy level at the Dirac point (E_D) as a function of the graphene charge doping is defined as ΔE_Fg = E_D − E_Fg. In these researches, the relationship between the applied voltage (V) and ΔE_Fg is presented as

The potential drop at the interface is represented by ΔV = Δ_tr + Δ_ch, where Δ_tr is a parameter that determines the influence of charge transfer between metal and graphene, and Δ_ch represents the influence of short-range interaction due to the overlap of wave functions in metal and graphene. However, according to Khomyakov’s model,^[111] the value of Δ_ch depends greatly on the separation distance (d) between metal and graphene (insulation thickness), so that it becomes negligible for d ≈> 4 nm.

The energy level at the Dirac point (E_D = eV_D) can be determined by the following equation:

After some mathematical manipulations, the variation regarding the Fermi level due to a gate voltage application is given by^[121]

where , with x = d–d₀ being the effective distance between graphene and semiconductor charges (which form the dipole) and V_F the Fermi velocity. The d₀ represents the offset of the charges that form the dipole at the interfaces M/I, and G/I (in this case, d₀ = 0, 24 nm). The + and − values in Eq. (68) have the same meanings given in Eq. (65), i.e., the + sign is used for V ≤ V_D, and the − sign for V > V_D.

Although equation (68) has been determined for temperature T → 0, it can be used at room temperature since the values found are approximately the same. In addition, this equation can also be applied to MG structures, which is considered a particular case of the MIG structure. However, while for MIG structures the value of Δ_ch = 0, for MG structures (without the intermediate insulator) the value of Δ_ch must be taken into account. Additionally, in the MG structure, the equivalent distance (d_e) between interfaces must replace the d value. We will adopt silver as the metal, just to show the detail of this analytical model we are showing. It is noteworthy that for silver, W_ag = 4, 92 eV and Δ_ch = 0.88 eV,^[111] so that, in this case, according to Eq. (67), for the MIG structure, V_D = 0, 42 V, and for the MG structure, V_D = −0,46 V. This means that in a MIG structure, graphene has intrinsic p-type doping, but in an MG structure, graphene has n-type doping. In Fig. 29(a), we plot the graphene Fermi level displacement as a function of the gate voltage (−2 V to 2 V) relative to the MG structure.

In Fig. 29(b), the graph referring to the MIG structure is shown, where the adopted insulation thickness was 10 nm and the dielectric constant 3.9. For comparison, this plot was elaborated from Eqs. (65) and (68), and as we can see, the values referring to the graphene Fermi level displacement were almost the same.

We also plot this same graph for d = 50 nm (not shown in Fig. 29) and found that in this case, the Fermi level displacement ranges from approximately −0.08 eV to 0.09 eV. Therefore for this voltage range and insulator thickness greater than 50 nm, we can neglect the displacement of the Fermi level of graphene. However, for higher voltage ranges, this referred Fermi level shift needs to be taken into account.

The possibility of applying a gate voltage to graphene and other 2D materials is perhaps the most important property of these materials. According to we have detailed in this section, it is possible to change the charge density in graphene by a gate voltage, so that a change occurs in its Fermi level^[123,124] and consequently in its work function and conductivity. So, managing graphene’s work function has many benefits that are not possible to achieve using conventional 3D materials. For example, changing the Fermi level makes it possible to use graphene in increasingly efficient optoelectronic devices such as photodetectors and solar cells. On the other hand, changing the controlled conductivity of the graphene via gate voltage causes the change of its effective dielectric constant so that much more efficient optoelectronic devices, such as waveguides, modulators, and full optical switching cells, among many others, can be obtained. We close this review paper, showing the progress and challenges of tuning concerning the electrical and optical properties of graphene via gate voltage.

For some time now, graphene-based transistors have been researched using back, top, or back and top gate voltage, from low frequencies to infrared.^{[101,125,126]} The gate voltage applied to graphene, operating as the channel in a transistor, changes its Fermi level to the conduction (or valence) band so that its conductivity increases with the addition of electrons (or holes) caused by the application of the gate voltage. Hence, it is possible to control the current flowing between the source and the drain electrodes. It is noteworthy that the gate port, i.e., where we apply the gate voltage, the source and the drain ports are located over graphene.

More recent research has presented a logic inverter based on a transistor consisting of p-doped graphene associated with a transistor made up n-doped graphene. Graphene doping was obtained through lanthanide dopants, and the operation of this device is based on C-H-π interactions between graphene and lanthanide dopants.^[127] This research may lead to the development of new technologies based on interactions of graphene and other 2D materials with aggregates of molecules, or ions, (supermolecules) joined by noncovalent interactions.

Advances in technology regarding fully optical networks depend on obtaining optoelectronic devices which can be integrated into photonic integrated circuits (PICs). A graphene-based nanophotonic Mach–Zehnder interferometer (MZI) has been introduced to work as a signal follower, switch, splitter, and as a multiplexer/demultiplexer, with dimensions that allow integration into PICs. This MZI consists of an input coupler, two waveguides in the middle (one waveguide longer than the other, so that the modes propagating within the MZI undergo a relative phase shift), and an output coupler. As in graphene-based nanoribbon waveguides, it is possible to insert 90° bends we can reach the difference between the lengths of the two waveguides by inserting a bend in one of these waveguides. However, as the bend is very small and made up of two bends (to allow the same direction), changes were made in its shape to avoid attenuations in the propagating optical signal. Apart from the difference in waveguide lengths, the operation of this device is based on the application of a gate voltage, in order to shift the conductivity and the effective dielectric constant of graphene so that it enables the operation of this device as any of the options shown above. It is noteworthy that the input and output couplers have lengths of 80 nm, and the distance between the two graphene nanoribbons is 2 nm. The difference between the lengths of the two arms has values between 3 nm and 6 nm, which are controlled by the number of carbon atoms located in the propagation direction.^[128]