Perfect-absorption, High-sensitivity, and Low-thickness THz Gas Sensor Based on Tamm Plasmon Polaritons

In this paper, an ultra-high sensitivity and low-thickness gas sensor based on the excitation of Tamm plasmon polaritons (TPPs) and Fabry-Perot resonances in the frequency range of 0.1 to 1THz is presented. The sensor consists of a graphene sheet, a spacer layer, a gas tank, and a one-dimensional dielectric-metallic photonic crystal (DM-PhC). The transfer matrix method is used to evaluate the effects of various parameters, including the presence of the graphene sheet, the number of periods, the chemical potential of graphene, and the thickness of the layers, on the absorption spectrum of the sensor. The simulation results reveal that the presence of the graphene sheet is mandatory for exciting TPPs. Additionally, the use of a DM-PhC enhances the absorption and reduces the thickness of the sensor. The absorption and sensitivity of the sensor can reach high values of 99.98% and 0.887THz/RIU for adjusting the geometrical parameters. The sensor benefits from high absorption and sensitivity, fabrication-friendly structure, tunability by the chemical potential of graphene, and low thickness. The proposed sensor can find many applications in industrial fields.


INTRODUCTION
Electromagnetic surface waves (ESWs) are electromagnetic waves that propagate along the interface of two materials.However, they are severely attenuated inside the materials [1,2] .The strong confinement of light and the narrow spectrum pave the way for the development of photonic integrated circuits.ESWs, known as surface plasmon polaritons (SPPs), are excited at the interface between a metal and a dielectric [3] .SPPs are very sensitive to changes in the refractive index of the materials.Therefore, SPPs are a good candidate for sensing applications.SPP-based sensors can be used to detect biomolecules [4,5] , chemical substances [6,7] , temperature [8,9] and pressure [10] .However, SPP-based devices face several drawbacks.For the excitation of SPPs, the wave vector of the incident light must match that of the surface plasmon wave.Therefore, the excitation of SPPs is not straightforward, and phasematching mechanisms like diffraction grating or prism must be employed.Additionally, SPPs are polarizationdependent waves and are stimulated by incident light with transverse-magnetic (TM) polarization [11,12] .These obstacles prevent the design of integrated SPP-based sensors.
Another ESW is Tamm plasmon polaritons excited at the interface between a metal film and a distributed bragg reflector (DBR) [13,14] .Initially, Kaliteevski et al.  2007 theoretically proposed the existence of TPPs [15] .Since the dispersion curve of the Tamm state lies within the light cone, no optical component is required for excitation, unlike SPPs.Moreover, TPPs are excited by any incident polarization and at any angle of incidence.The spectral response of both SPPs and TPPs depends on the geometrical parameters and materials of the structure, as well as the operating frequency [13][14][15][16] .It has been reported that a one-dimensional photonic crystal (PhC) can be used instead of a DBR.In addition, a graphene sheet can replace the metal film to excite TPPs in the terahertz (THz) range [17] .THz waves with frequencies of about 0.1 to 10THz have many applications in information and communication technology, imaging, security, medicine, and industry.The bandwidth of TPPs is narrower than that of SPPs.Therefore, TPPs show a larger resonant mode when the local field increases [18] .The first TPP-based refractive index sensor was proposed by Zhang et al [19] .After that, various TPP-based refractive index sensors were presented for the detection of fluids [20] , gases [21] , and blood components [22] , as well as for temperature [23] sensing.Das et al. [16] described a refractive index sensor based on a Tamm Fabry-Perot hybrid resonance.It has a sensitivity of 87nm/RIU, and a figure of merit (FOM) of 7.5RIU -1 for refractive index ranges from 1.25 to 1.38.The sensor consists of a Bragg mirror and a silver (Ag) grating, and the hybrid resonance occurs at the interface between the Bragg mirror and the metal grating.Hu et al. reported a multi-channel refractive index sensor using graphene and a hybrid Tamm plasmonic structure [18] .It consists of two identical Ag-DBR structural units joined together to form a symmetric structure.The highest values for sensitivity and FOM were 950nm/RIU and 161RIU -1 , respectively.A THz gas sensor based on TPPs was presented by Zaky et al [24] .The sensor consists of a gas cavity located between a one-dimensional porous silicon PhC and an Ag layer deposited on a prism.The sensitivity of the sensor is 1.9×10nm/RIU.All of the above sensors are composed of all-dielectric PhCs (AD-PhC) and are very thick.It is worth mentioning that PhCs can be divided into all-dielectric and dielectricmetallic PhCs (DM-PhC), depending on the materials forming them.AD-PhCs consist of two dielectric materials arranged alternately next to each other.DM-PhCs, on the other hand, are formed by stacking dielectric and metallic layers [25,26] .To date, much research has been done in the field of AD-PhCs, while DM-PhCs have received much less attention.In this paper, we will show that DM-PhCbased structures benefit from the advantages of stronger resonances and thinner thickness compared to AD-PhCbased structures.In this context, the performance of a high-sensitivity gas sensor based on the excitation of TPPs in the THz range from 0.1 to 1THz is evaluated using the modified transfer matrix method (TMM).

Proposed Sensor
The proposed sensor is schematically shown in Figure 1.It consists of a graphene sheet, a thin dielectric spacer layer, a gas tank, and a DM-PhC, all placed on a dielectric substrate.The PhC consists of N periods of alternating dielectric (D) and metallic (M) layers.The period of the PhC is a=T D +T M , where T D and T M represent the thickness of the dielectric and metallic layers, respectively.The total thickness of the PhC is equal to N×a.The thickness of the graphene sheet is 0.34nm.Additionally, the thickness of the spacer and the gas tank is denoted by T S and T G , respectively.The refractive indices of the gas tank, the spacer layer, and the dielectric layers are specified by n Gas , n S , and n D , respectively.The dielectric constant of the metallic layers is shown by ε M .The incident light with a wavenumber of k 0 and an incident angle of θ 0 irradiates the surface of the sensor.The gas entry vent is seen in the figure.The PhC part of the sensor can be easily fabricated using traditional deposition approaches, such as chemical vapor deposition (CVD), atomic layer deposition (ALD), and sputtering methods [27][28][29] .Also, it is possible to grow CVD graphene on copper and then transfer it on the spacer layer by utilizing the wet transfer method [30,31] .

Modified TMM
The modified TMM is used to achieve the spectra response of the proposed sensor.The proposed sensor's structure can be considered as a system with a graphene sheet on the left side and a DM-PhC on the other side.For T G >T S and n Gas ≈n S , two amplitude reflection coefficients (r Graphene and r PhC ) arise when light hits the graphene sheet, as shown in Figure 2. The reflection coefficient for light propagating in the -z direction is represented by the parameter r Graphene and the reflection coefficient for light propagating in the +z direction is represented by the parameter r PhC .The amplitude matching requirement must be met to stimulate TPPs.The transfer matrix approach yields the following formula for light propagation in the top layer cavity [32] .
where ϕ is the phase change caused by light traveling through the gas tank, and A is a constant.Following a series of mathematical simplifications and after eliminating the coefficient A, the excitation condition of Tamm plasmons can be obtained as follows [32] : Equation ( 2) can be further simplified in the form of And For a P-polarized incident wave, the transmission characteristics of light through the interface between the mth layer and the nth layer in the existence of the graphene sheet is calculated by the D mn matrix as follows [32] : where Here, ε 0 and σ are the dielectric constant of free space and the surface conductivity of graphene, respectively.In addition, ε m and ε n are the permittivity of the mth and nth layers, respectively.It is important to note that a P-polarized light, also known as a TM mode, is a light wave with an electric field vector in the plane of incidence.In contrast, an S-polarized light, also known as a transverse electric (TE) mode, has an electric field vector that is perpendicular to the plane of incidence [26] .For an S-polarized wave, the D mn matrix is as follows [32] : where Here, µ 0 is the permeability of the free space.With the application of the D mn matrices, the transmission and reflection spectra for P and S polarizations are obtained.Therefore, the absorption spectrum A(f) is obtained as follows [33] : where R(f) and T(f) denote the reflection and transmission spectra, respectively.

Modeling the Structure
The performance of the proposed sensor is evaluated using the analytical TMM approach.We employed the commercial Ansys Lumerical software [34] to simulate the suggested structure.The graphene sheet can be modeled by its surface conductivity (σ) using the Kubo formula as follows [35] : The graphene surface conductivity depends on several variables, including angular frequency (ω), chemical potential (μ c ), scattering rate (Γ), and temperature (T).
The parameters e, ħ, and k B are constant coefficients, including the electron charge, reduced Planck's constant, and Boltzmann's constant, respectively.The charge carrier density of graphene is controlled by applying chemical doping or an external voltage [36] .A change in charge carrier density affects the chemical potential and, consequently, the surface conductivity of graphene.
Assuming T=300°K and Γ=0.11meV, the real and imaginary parts of the surface conductivity of graphene in the frequency range of 0.1 to 1THz are plotted in Figure 3, where μ c values vary from 0.1 to 0.9eV in steps of 0.2eV.
It is assumed that the gas, spacer layer, and dielectric layers are of non-dispersive materials and can be described by a simple refractive index.The dielectric constant of the metallic layers is modeled using the Drude-Lorentz model as follows [37] : where ε r,∞ is the dielectric constant at infinite frequencies and ω p is the plasma frequency.In addition, the parameters f k , ω k , and Γ k are the strength, resonant frequency, and damping frequency of the kth oscillator, respectively.The Drude-Lorentz model incorporates the effects of bound electrons into the original Drude model.The validity range of the Drude model can be extended by including the Lorentz term.The Lorentz-Drude model uses K-damped harmonic oscillators to characterize small resonances in the spectral response of metals [37] .As a high-loss metal, tungsten (W) is selected for metallic parts of the PhC.In addition to high losses of W metal, this metal shows a better performance at high operating temperatures than other metals, such as gold (Au), Ag, chromium (Cr), copper (Cu), and Titanium Nitride (TiN), due to its high melting point of 3,422℃ [38,39] .The coefficients of the Drude-Lorentz model for W metal are taken from Ref [40] .Figure 4 shows the real and imaginary parts of the dielectric constant of W in the frequency range of 0.1-1THz.

RESULTS AND DISCUSSION
It is assumed that the initial values of the sensor's parameters are T S =0.5µm, T G =1mm, T D =1µm, T M =10nm, n Gas =1, n S =n D =3.53, M=W, µ c =0.5eV, θ 0 =0°, and N=4.transmission, and absorption spectra in the presence and absence of the graphene sheet.In the presence of the graphene sheet, six resonances are seen in the reflection spectrum in the frequency range of 0.1-1THz (Figure 5A).The transmission is zero due to the use of DM-PhC.In other words, the ultra-thin metal layers of the PhC prevent light propagation toward the outside of the sensor.As a result, six absorption peaks are observed exactly at the reflection resonance frequencies.As seen in Figure 5B, there is no resonance in the reflection spectrum when the graphene sheet is removed.Almost all the incident light is reflected, and a negligible amount (less than 10%) is absorbed.In this case, TPPs are not stimulated; therefore, no resonance occurs, and the small amount of absorption is due to materials' losses.
It was expressed that the thin metallic layers in the PhC make the transmission equal to zero and cause strong resonances in the reflection spectrum.For this purpose, dielectric layers with a refractive index of n D2 =1.9 replace the metallic layers of the PhC, and the spectral response of the sensor is plotted.The thickness of both dielectric layers (T D1 and T D2 ) is equal to 1µm.The refractive index of the first dielectric layer (n D1 ) is 3.53.It should be noted that changing the order of the dielectric layers (with a thickness of 1µm) does not affect the absorption spectrum of the sensor.In this case, several resonances are observed in the reflection and transmission spectra, especially at the higher frequencies (Figure 6).Unlike the DM-PhC-based structure, the transmission is not zero and has a significant value at resonance frequencies in the AD-PhC-based structure.In this structure, a small amount of incident light is absorbed, and the rest is either reflected or passes through the structure.This structure has a thicker thickness compared to the DM-PhC-based structure, and its spectral response is not appropriate for sensing applications.Now, we find that utilizing DM-PhC instead of AD-PhC reduces the thickness of the sensor and causes higher absorption peaks.Knowing this, the effect of the number of periods of the DM-PhC is discussed.Figure 7 shows the absorption spectrum of the sensor for different number of periods.When N=0, there is no PhC, and the gas tank is connected to the substrate.Therefore, no absorption occurs within the sensor's structure.When N=1, strong resonance peaks appear in the absorption spectrum.Increasing the number of periods leads to a slight change in the absorption of the structure.The two zoomed-in figures show this matter.
As a gas sensor, the proposed sensor must recognize small changes in the refractive index in the gas tank.The absorption spectrum of the sensor in the frequency range of 0.5-0.6THz(the fourth resonance in the frequency range of 0.1-1THz) for n Gas changes is illustrated in Figure 8, where n Gas changes from 1 to 1.1 with steps 0.02.The resonant frequencies and the corresponding absorption values are summarized in Table 1.
The sensitivity (S) of the sensor is evaluated using the following equation [41] :

A B
where Δf=f 2 -f 1 is the resonant frequency displacement due to the refractive index change of Δn Gas =n Gas,2 -n Gas,1 .
The more the resonant frequency displacement, the more the sensor sensitivity.Considering n Gas,1 =1 and n Gas,2 =1.1, the resonant frequency displacement of 0.0537THz is achieved, which results in the sensitivity of S=0.537THz/ RIU.
The effect of the graphene sheet on the excitation of TPPs was investigated earlier.Actually, the graphene sheet acts as a metal in the proposed structure.Additionally, it was shown that the surface conductivity and, consequently, the refractive index of graphene depends on the chemical potential.Hence, the absorption spectrum of the sensor for different values of µ c in the frequency range of 0.1 to 1THz is plotted in Figure 9A.The change of the absorption peak of the resonances can be determined by a Gaussian-like distribution function so that its average shifts towards higher frequencies with the increase of the chemical potential of graphene.For µ c =0.5eV, the average of the Gaussian distribution coincides with the fourth resonance.So, the maximum absorption in the fourth resonance corresponds to the chemical potential equal to 0.5eV.A zoomed-in figure of the fourth resonance in the range of 0.52-0.62THz is given in Figure 9B. Figure 9C shows the absorption and sensitivity of the sensor for the fourth resonance as a function of µ c .The maximum absorption and sensitivity values are 99.95% and 0.5406THz/RIU, which are obtained for chemical potentials of 0.5eV and 0.9eV, respectively.Figure 9D indicates the resonant frequency displacement of the six resonances as a function of µ c .It is seen that the resonant frequency shift is more for the higher-order resonances.Furthermore, the resonant frequency displacement rate decreases as the chemical potential increases.
Figure 10 reveals the absorption spectrum and sensor sensitivity dependency on the geometrical parameters of the structure.First of all, the effect of the dielectric thickness changes is investigated so that T D changes from 1 to 30µm, and the absorption spectrum is calculated for each thickness, as seen in Figure 10A.For T D >20µm, another resonance (seventh resonance) is apparent in the absorption spectrum.Figure 10B shows the absorption and sensitivity for the fourth resonance as a function of T D changes in the frequency range of 0.45 to 0.65THz, indicating that T D =1µm gives the best result.As much as T D increases, both absorption and sensor   sensitivity decrease.Figure 10C shows the absorption spectrum for changes in the thickness of the spacer layer from 0.1 to 30µm.The seventh resonance is seen for T S values higher than 10µm.According to Figure 10D, the absorption of the fourth resonance has a negligible dependency on T S changes.On the other hand, the sensor sensitivity decreases as the spacer layer gets thicker.The absorption spectrum of the sensor as a function of the metal thickness, varying from 5 to 100nm, is observed in Figure 10E.The number of resonances appears to be almost independent of T M changes.Additionally, no extra resonance is excited by changing the thickness of the metallic layers.Figure 10F gives a better view of this issue.The absorption has a nonlinear relationship with T M for the fourth resonance (in the frequency range of 0.45-0.65THz),and the highest absorption of 99.98% is achieved for T M =12nm.However, T M variation has the least effect on the sensor sensitivity, among other geometrical parameters, among other geometrical parameter (Figure 10F).For this case, the sensor sensitivity is equal to a fixed value of 0.537THz/RIU.The thickness of the gas tank layer significantly affects the number of resonances in the absorption spectrum (Figure 10G).For example, for T G =100µm, no resonance is observed, while for T G =500µm, there are three resonances in the absorption spectrum, and the fourth resonance is seen for about T G >600µm. Figure 10H demonstrates the absorption and sensitivity of the fourth resonance for 600µm<T G <1000µm.For the lower values of T G , the fourth resonance is not excited.The absorption and sensitivity have different behaviors when T G increases.The higher the T G value, the higher the absorption.At the same time, the sensor sensitivity decreases as T G increases.It is clear that in exchange for reducing the absorption peak to 84.44%, it is possible to reach a sensitivity of 0.887THz/ RIU for T G =600µm.Furthermore, the two absorption and sensitivity curves intersect for T G =710µm.Therefore, values of 92.74% and 0.752THz/RIU (equivalent to 36.32µm/RIU) can be achieved simultaneously for absorption and sensitivity, respectively.

CONCLUSION
In this paper, a highly efficient optical gas sensor based on the excitation of TPPs and utilizing a DM-PhC was presented.The sensor indicates several narrowband high absorption peaks in the frequency range of 0.1-1THz.The number and frequency of resonances depend on the geometrical parameters of the structure and the chemical potential of the graphene sheet.The use of a DM-PhC enhances the absorption properties and reduces the thickness of the whole structure compared to sensors employing AD-PhCs.Adjusting the structure parameters makes it possible to achieve absorption of 92.74% and a sensitivity of 0.752THz/RIU (36.32µm/RIU) at a frequency of 0.620THz.Due to the remarkable characteristics of the suggested sensor, it can be used in different parts of industrial applications.

Figure 1 .
Figure 1.Schematic of the proposed gas sensor.

Figure 2 .
Figure 2. Reflection waves in the sensing region.

Figure 5 Figure 3 .
Figure 3.The graphene surface conductivity for different values of chemical potentials.A: Real parts; B: Imaginary parts.

Figure 4 .
Figure 4. Real and imaginary parts of the dielectric constant of tungsten (W) metal in the frequency range of 0.1-1THz.

Figure 5 .
Figure 5. Spectral response of the proposed sensor in the (A) presence and (B) absence of the graphene sheet.

Figure 6 .
Figure 6.Spectral response of the sensor with an AD-PhC.

Figure 8 .
Figure 8. Absorption spectrum of the suggested sensor for different gas refractive.

Figure 7 .
Figure 7. Effect of the number of periods on absorption spectrum of the sensor.

Figure 9 .Figure 10 .
Figure 9. Absorption spectra of the sensor for different values of µ c .A: In the frequency range of 0.1-1THz; B: for the fourth resonance in the frequency range of 0.52-0.62THz;C: Absorption and sensitivity; D: Resonant frequency displacement of the sensor as a function of µ c .