Triple-band metamaterial perfect absorber for refractive index sensing in THz frequency

A triple-band metamaterial perfect absorber (MPA) is presented in monolayer graphene in the THz region, which is adjustable and polarization-independent. The first layer of the structure from the top is patterned graphene which contains a graphene ring at the center of the frame with four graphene wheel-shaped around it and four graphene triangles in the corner. This proposed structure has caused the absorber to achieve 98.64%, 99.97%, and 99.98% perfect absorption peaks at 8.17 THz, 9.74 THz, and 11.95 THz, respectively. We can vary the absorption peak frequencies to our desirable frequencies by changing graphene's Fermi level. This absorber has good tolerance for up to 60-degree change in the angle of incident waves. Moreover, a significant matter about the suggested absorber structure design is that it is polarization independent, so by tuning the polarization angle, the frequency peaks and absorption value remain unchanged. These aspects make the suggested absorber proper for imaging, detecting, filtering, and sensing applications. We have investigated the application of the MPA in refractive index sensing. The refractive index of unknown material can be measured by measuring the shift of the frequency peaks. The perfect metamaterial absorber is a good candidate for biosensor application based on the obtained results.


Introduction
Metamaterials are man-made materials with unique electromagnetic properties that have attracted the attention of many researchers in recent years. In this section, we introduced metamaterial absorbers. Then, in the second part, considering the unique aspects of graphene, we have specifically investigated the features of this widely used metamaterial. Metamaterials (MTMs) are artificial structures made up of unit cell arrays that are periodic and very smaller than the operating wavelength in size. These present unique and different properties from ordinary materials, such as inverse Doppler effects, zero index of refraction, negative refractions, and so on (Moitra et al. 2013;Shelby et al. 2001). In recent decades a lot of research has been done for different frequency ranges such as GHz (Cao et al. 2019;Karaaslan et al. 2018), optical (Kumar et al. 2022;Mulla & Sabah 2016) and terahertz (Norouzi-Razani and Rezaei 2022a; Razani et al. 2022). Engineered designs using metamaterials in the terahertz region have fascinated much consideration due to their potential application in imaging, sensing, and absorption (Norouzi- Razani & Rezaei 2022b;Shen et al. 2022;Zamza et al. 2021). A perfect metamaterial absorber (MPA) is one of the important applications of MTMs. By precisely designing the absorber structure and optimizing its geometric parameters and engineered layout, researchers can create multiband or broadband absorbers with different characteristics such as polarization independent, adjustable, thin thickness, high absorption ratio, and small dimensions (Kantamaneni et al. 2022;. Metamaterial absorbers can be used in the terahertz range for chemical and biological sensing (Khani & Hayati 2021;Khodadadi et al. 2022;Nickpay et al. 2022a). When an unknown substance (analyte) is added to the absorber, the unknown substance can be analyzed by examining the response at the resonance frequency. Due to the added analyte, the interaction between this unknown substance and the absorber occurs, which results in a frequency shift. By examining and analyzing these displacements, the properties of unknown materials can be obtained .

Metamaterial perfect absorber
In this work, a new three-band MPA is designed and simulated in the terahertz field. This structure is a single-layer patterned graphene absorber with tunable narrow-band absorption suitable for sensing. Therefore, this work has two main parts. In the first part, a perfect metamaterial absorber based on graphene with narrow tunable bandwidth was designed, and these features were investigated in the absorber. Next, in the second part, the application of this MPA as a sensor for measuring the dielectric refractive index (RI) and important parameters of the sensor, such as sensitivity, FOM, etc., are also investigated.

Graphene
Graphene is an allotrope of carbon made up of single layers of atoms set them in a two-dimensional honeycomb nanostructure mesh. Graphene is displayed great adjustability by doping with specific chemicals and also with electrical/magnetic biases (Heydari and Vadjed Samiei 2020;Nagatsuma et al. 2016). The unique structure of graphene straightly influences the graphene conductivity as well as other items such as Fermi level, relaxation time, chemical doping, temperature, electron mobility, and operating frequency. Moreover, these features are associated together (Novoselov et al. 2012;Svintsov et al. 2016).
In the articles, Generally, effective surface conductivity is used to express the aspects and properties of graphene. The surface conductivity of graphene includes intra-band and interband electron transition, which is shown by using the following formula (Xiao-Peng et al. 2012): where N g is a number of graphene layers, e = 1.6 × 10 −19 illustrates the electron charge, angular frequency is defined by = 2 f , in Γ = 1∕2 is relaxation time, k B is the Boltzmann constant, ℏ is plank constant which is reduced, c is the Fermi level of graphene,and T = 300 is the absolute temperature in Kelvin.
When the Fermi level | | c | | ≫ ℏ ∕2 , the in-band conductivity intra has an important role, and when the frequency is under 30 THz,and the Fermi level is over 0.1 eV, the inter can be omitted,as stated by the Pauli principle of incompatibility. When the graphene has a Fermi level of | | c | | ≫ ℏ ∕2 , the conductivity of inter-band inter has a prominent impress, and the conductivity of in-band intra can be omitted (Teperik et al. 2008). The conductivity of the graphene is merely illustrated by using the conductivity of in-band. So, the surface conductivity of the graphene can be modified to a Drude-like model (Wang et al. 2020a, b) Although, the graphene conductivity behaves in a different manner at visible frequency, infrared and THz frequency due to negative real permittivity. In-plane permittivity can be defined by surface conductivity and the effective thickness of graphene sheet by Eq. (3) as: where h and g denote the graphene thickness and surface conductivity, respectively. Moreover, the out-plane permittivity might be initialized as dielectric permittivity with value of 2.5 .
The relation of relaxation time ( ) and graphene Fermi level ( c ) is shown in Eq. (4).
where v and V F = 9.5 × 10 5 m∕s represent electron mobility and Fermi velocity. Generally, by changing the voltage between the graphene and gold layer, the gating of the graphene layer adjusted the Fermi level of the graphene. By the gate voltage, the Fermi level is changed and the conductivity of graphene is adjusted for the desired application. Moreover, by varying the graphene's Fermi level because of carrier injection, the conductivity of graphene increases. Therefore, the Fermi level is adjusted by varying the bias voltage in the graphene layer. Equation (5) is shown the relationship between chemical potential and the bias voltage.
Here V g is the gate voltage, d is the electric permittivity, 0 is free space permittivity and h is the height of the dielectric layer. So, the c could be varied by putting on an F 0 r external DC bias voltage. As depicted, it can affect the graphene conductivity and influence the design structure for devices such as absorbers or sensors.

Perfect metamaterial absorber (MPA)
In this section, we have investigated the suggested perfect terahertz absorber in detail. For this purpose, we expressed the details of the designed three-layer structure in the first part. Then, in the second part, the simulation results of the proposed structure are given. We have also examined the structure in terms of the important parameters of the absorber, such as the tolerance of different values of the incident angles and polarization angles, etc.

Structure design and simulation
The goal of the absorber is that in addition to high absorption rates and narrow independent bands, the structure has no complexity as well as asymmetry. Symmetry has an important role in absorber structure design because it causes insensitivity to the polarization angle (Feng et al. 2021). For this purpose, the absorber structure was designed according to Fig. 1a. This classic absorber has only three layers of gold, dielectric, and graphene from bottom to top, respectively. The bottom layer of the structure is covered by gold,and its conductivity is gold = 4.56 × 10 7 . The thickness of the last layer is h g = 0.2 μm for eliminating the transmission of the absorber (Fu et al. 2018). The middle layer consists of dielectric SiO 2 with electrical conductivity r = 2.25 and thickness of h s = 4 μm . The top layer comprises patterned graphene with the thickness of h 1 = 0.01 μm that presented in Fig. 1b.
The unit cell consists of four equilateral triangles with dimensions P 4 at 4 corners and four circles with distance P 4 from the center with outer radius r 3 = 0.4 μm , inner radius r 4 = 0.3 μm and x = 0.1 μm . The outer radius of the center circle is equal to r 1 = 0.5 μm and the inner radius of this circle is equal to r 2 = 0.3 μm . The geometric dimensions of the unit cell are denoted in Fig. 1c. The repetition period is P = P x = P y = 3 μm.

Result and discussion
We simulated this absorber under normal incidence in TE and TM waves using the frequency domain solver in CST Microwave Studio. The unit cell boundary condition was employed along the x − y axis. The incident waves are moved in the z-direction, and the Fig. 1 a Schematic diagram of the absorber,P x = P y = P = 3 μm , h s = 4 μm , h g = 0.2 μm , b Top view, c Side view incident wave polarization is along the x-direction according to the coordinate system represented in Fig. 1.
The absorption can be obtained as here R is the reflectance,and T is the transmittance coefficients (B.-X. Wang et al. 2015). As depicted, because the bottom gold sheet transmission becomes zero (T = 0) and the absorption can be computed byA = 1 − R.
As denoted in Fig. 2a, the structure has three absorption peaks at 8.17THz , 9.74THz,and 11.95THz with absorption values of 0.9864, 0.9997, and 0.9998, respectively. Figure 2b shows that the curves of the polarization absorption are entirely the same as the curves of TM polarization absorption, which is a consequence of the symmetric structure of the absorber.
Then we examined the causes of the absorption peaks in the structure. As we can clearly see in the electric field distribution in Fig. 2c, in three resonance peaks, the resonance areas in the designed structure are different. At the first resonance frequency at f = 8.17 THz, the contribution caused by a central circle is the largest here, but the assistance of others is small. In the second peak at f = 9.74 THz, the distribution of electric fields in the four wheels is dominant, and other elements have a minor contribution here. In the third resonance frequency at f = 11.95THz, the largest distribution of electric fields is in one corner of the side triangles in the structure, and the contribution of the others is almost negligible.

Fig. 2
Absorption spectra of structure when fermi level is 0.9 eV and relaxation time is 0.4 ps a absorption peak frequencies with absorption values, b In the normal incidence of TE/TM waves, c electric field distribution of three peaks So, this method helps us analyze the peaks and also provides an idea to adjust the perfect absorption without changing the symmetry of the structure, as well as separate debugging for different resonance areas.
One of the privileges of the suggested absorber is frequency adjustment for the planned application without changing the dimensions or geometry of the designed structure, just by changing the Fermi level of the graphene ( c ) . The graphene surface conductivity is increased by increasing c , as seen by Eq.
(2). As demonstrated in Fig. 3a, the peak frequencies are red-shifted when the fermi level is increased. When c is grown, the carrier concentration increase, which is caused more powerful excitation ). In conclusion, the absorption efficiency of graphene becomes better. Another parameter of graphene that could be influenced by the absorption peak frequencies is relaxation time. As mentioned before, the factors affecting the graphene relaxation time are in Eq. (4). As represented in Fig. 3b, when the relaxation time is changed, the absorber structure spectrum is changed, too.
For an ideal absorber, the values of absorption and absorption peaks remain unchanged for any polarization state and the incidence angle of an EM wave (Aydin et al. 2011). To this end, controlling small changes in absorber performance is a challenging task. Fluctuations observed in absorption at a given radiation angle may be due to the occurrence of impedance mismatch, which increases the reflection of the incident wave (He et al. 2011). Figure 4(a) displays the absorption when the incident wave angle increases from = 0° up to = 60°. The denotes the incident angle of the TE waves and TM waves. It is interesting to notice that a slight shift in the absorption spectra appears as the incident angle varies. As you can see, with increasing , the rate of absorption is decreasedbecause more diagonal incidence increases reflection. So up to = 60 °, we have a wide-angle absorber. We have investigated the dependence of the absorber on the polarization of the incident waves for this structure. Fig. 4(b), p denotes the angle between the direction of the incident wave polarization and the y direction. It is obvious from Fig. 4(b) that the suggested structure has supreme absorption stability at various angles of polarization. The structure gives the same outcome in response to the various polarization of the waves from p = 0° up p = 90°. This is attained by using a symmetric structure. When the incident waves strike a symmetric resonator at various angles, the resonator treats it similarly and achieves the same response. This basic mechanism is accounted for independent polarization for symmetric structures. Also, the change in the relative electric permittivity dielectric parameter ( r ) was investigated in Fig. 5(a). We kept other parameters fixed and only change the relative electric permittivity of dielectric ( r ). This investigation shows that by reducing the values of r , the absorption frequency decrease and vice versa.
According to relation where h is the thickness of the dielectric layer and p is the phase path of the incident wave. Plane-wave is transferred on the homogeneous medium layer at the normal incident, so and p is constant. Therefore the shift of the frequency relies on r . Then we investigate the influences of the dielectric thickness (h s ) on the absorption characteristic. Fig. 5(b) presents the effect h s on the absorber characteristics, as the dielectric thickness increases, the absorption value, and bandwidth change. For a compromise consideration of the absorptivity and bandwidth, we choose the thickness of the dielectric h s = 4 μm in this work.

The sensor based on metamaterial perfect absorber in THz
In this part, we have investigated the application of the desired absorber in sensing. In the first part, by adding the analyte layer to the proposed perfect absorber, we examined the important parameters of the sensor, such as sensitivity, FOM, and quality factor by the change of refractive index. Then we compared the obtained results with some articles presented in recent years. In the second part, the application of the proposed sensor in malaria diagnosis is given.

Fig. 5
Absorption spectra of structure a for different relative permittivity of dielectric, b for different thickness of dielectric h s

The results and discussion
One of the applications of metamaterial perfect absorbers is sensing. As depictured in Fig. 2, the proposed absorber has three narrow band absorption in the THz. This result helps us to have a metamaterial sensor in the THz field. Figure 6a denotes this sensor with a test medium thickness (d) over the MPA structure. This method is operated by analyzing the frequency shift of the absorber peaks based on changing the RI of the surrounding medium. The two factors which are more important for measuring and ensuring the quality of the sensor are sensitivity(S) and figure of merit(FOM) Ma et al. 2019). Equation (7) shows sensitivity(S), which computes the absorption frequency shift when the refractive index (RI) changes from n 1 to n 2 (Δn = n 2 − n 1 ) (Yan et al. 2019). The FOM is calculated as Eq. (8), where FWHM denotes the full width at half-maximum.
The absorption spectra for various RI of the test medium are denoted in Fig. 6b. It is evident from Fig that as RI increases, the three resonant frequencies of MPA have redshift. The sensor resonance peak can be calculated as follows: where, L eq and C eq are the equivalent induction and capacitance of the absorber, respectively and C Analyte is the capacitance of the test medium. Since C eq has a tiny amount versus C Analyte , the sensor capacitance is more influenced by the test medium factors (Nickpay et al. 2022a).
As an outcome, a slight variation in the parameters of the test medium (analyte), such as thickness or RI, can cause notable variation in the frequency of MPA and make it appropriate for utilization in sensor applications. Besides, these parameters are correlated to the √ L eq C eq + C Analyte Fig. 6 a The proposed sensor based on absorber of Fig. 1a, b The absorption spectra for different refractive indexes of surrounding medium when d = 0.6 μm analyte dielectric constant r , which is related to RI (n) by r = n 2 and for many materials = 1 (Wu et al. 2013).
At the junction of metal and dielectric, plasmons are formed called surface plasmons. If these plasmons are excited by visible or ultraviolet photons, we call them "surface plasmon resonant" (SPR). Surface plasmons strongly react with light polaritons. For more information about surface plasmons, one should study the behavior of metals against a light field. The optical behavior of metals is related to their dielectric function, and plasmon plays an essential role in the optical properties of metals. According to the intended application, the reflected light frequency is coordinated with the plasma frequency for resonance. Nanoparticles have many surface atoms compared to the atoms inside their volume. This property is the reason for the greater importance of surface effects. Therefore, in response to external fields and forces, nanoparticles show effects that depend on the size and shape of the particle, as well as the dielectric constant of the medium and metal.
The particle dimensions in relation to electromagnetic radiation can control the dependence of the optical spectrum of large nanoparticles on their size. Minor changes in the dielectric surrounding the nanoparticle affect the resonance of surface plasmons in such a way that these changes are observed in the amount of reflected radiation, absorbed radiation, or wavelength change. We observe the phenomenon of ''local surface plasmon resonance'' (LSPR) when the frequency of the produced plasmon oscillation equals the frequency of the incident waves. In this case, the field is concentrated in a tiny space of about 100 cubic nanometers, called a nano-focus, and any object that enters this area will affect the LSPR (Razani et al. 2022).
SPR performs another function in chemical sensors and biosensors: it transmits the signal in a colorimetric sensor in response to external stimuli by changing spectral position and intensity. Also, in cases of ultra-sensitive detection, SPR modulates fluorescence emission by concentrating the incident field in a nanostructure. The enhanced fluorescence enables plasmon to be used for ultrasensitive detection (Li et al. 2015). We have examined the LSPRs of the designed structure by depicting electric fields at absorption peak in Fig. 2c.
Tables 1, 2 and 3 denotes the achieved values of S, f (peak) , FWHM, FOM, and quality factor Q = f ∕ FWHM of the sensor for three modes resonance peak frequencies with the tuning of the refractive index. The values obtained from Fig. 6b and formulas 7 and 8 have been used to calculate the information in these tables.
In Fig. 7a-e, the frequency shift, FWHM, sensitivity, FOM, and the quality factor of the proposed sensor under different refractive indexes from 1 to 1.8 are plotted with steps of 0.1, respectively. As shown in Fig. 7a, as n changes, the absorbers' peak frequencies obviously change. According to Eq. (7), these differences affect the sensor sensitivity, depicted in Fig. 7c. The highest sensitivity of the sensor is when n = 1.3, which makes it suitable for use in biomedical sensors [8]. FOM in Fig. 7d, according to Eq. (8), has a trend similar to the sensor's sensitivity. As depicted in Fig. 7e, the third peak has the highest quality factor. This sensor has a maximum sensitivity of 1867 GHz/RIU at 3rd resonance frequency with f peak = 11.41THz and Q = 13.11 ; this is because of the narrow bandwidth (1.87) of the absorption spectrum.
Finally, we have collected and compared the result achieved in this article with some of the articles published in recent years in Table 4. As observed, the suggested sensor has the highest sensitivity and a wide operational range. Liu et al. presented an article in 2016 about the sense of the refractive index for cell biology and disease diagnosis (Liu et al. 2016). They are stated that 250 million people get malaria every year. Consequently, rapid diagnosis of malaria is critical to prevent mortality from malaria progression. The healthy red blood cell (RBC) has a refractive index of 1.399, and an infected RBC refractive index, in the trophozoite and schizont stages, is 1.383 and 1.373, respectively, according to the progression of malaria (Liu et al. 2016). The proposed sensor can discern the steps of malaria, as given in Tables 5, 6 and 7, for three absorption peaks when the analyte thickness is 0.6 μm.

Conclusion
In total, we have provided a THz adjustable MPA sensor, which has three perfect independent narrow-bandwidth absorptions. The simulation outcomes demonstrate the absorber's tunability without changing structure geometry, which helps adjust the THz absorption frequency for the intended application. Also, this THz absorber is polarization independent and insensible to the incident angle up to 60. The designed absorber has a high sensitivity of 18.67 GHz/RIU at f = 11.41 THz. Uncomplex design, ultra-thin thickness, and good reactions are some of the advantages of the proposed sensor, which make it suitable for sensing in the THz region, such as diagnosis of malaria. Funding No funding was received for this research.
Data availability My submission has no associated data. All data generated or analyzed during this study are included in this published article.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Ethical approval This material is the authors' own original work, which has not been previously published else where and All authors will take public responsibility for its content.