Introduction

Coronavirus disease 2019 (COVID-19) is an easily transmissible and very contagious disease which is caused by severe respiratory syndrome coronavirus 2 (SARS-CoV-2). It was first reported in Wuhan, China in December 2019 and spread with an unbelievable speed worldwide. As people around the world were severely infected by this virus, it was identified as a severe and dangerous pandemic. Various symptoms were reported for this disease as fever, cough, headache, fatigue and breath difficulties. The respiratory symptoms are the most serious signs, which can put human’s health in jeopardy. As reported by the world health organization (WHO), more than 300 million (~390 million) people were infected by this virus and unfortunately more than 5 million (~5.7 million) lost their lives. If the infected cases can be diagnosed and monitored in early stages, they can undergo various treatment approaches. This can lead to the significant reduction of death cases. Therefore, developing new and strong technologies for detecting and monitoring this virus is of great importance and many researches have devotedly worked on them since the beginning of the pandemic [1]. Many approaches were proposed for detecting the virus in infected people including different tests and sensors. Various biosensors for detecting COVID-19 were reported which indicated interesting results [2, 3]. The optical biosensors can diagnose the COVID-19 very fast and with precise results. They mainly operate based on the changes of refractive index (RI) in the sensing medium [1]. They attracted excessive attentions due to their high sensitivities, low costs, small sizes (nano-meter dimensions) and high precisions [4].

The optical biosensors can be considered as the graphene or plasmonic nano-combinations in different configurations [5]. These structures have amazing behaviors such as very high optical confinement, low losses and tunability [4]. Plasmonic nano-structures are based on the combination of metal-insulator configurations which have very interesting specifications in nano-scaled dimensions [6]. Graphene structures are based on the combination of carbon atoms in organized configurations which have unbelievable electrical, optical and thermal characteristics [7]. They can be considered in different configurations like plate-shaped, cylindrical-shaped [8], ring-shaped [9] or 1-D (one-dimensional) photonic crystal (PC) -shaped [10] structures.

The most important aspect of the plasmonic-graphene structures is the surface plasmon resonances (SPR) [4, 6]. In fact, in the plasmonic/graphene nano structures, the interaction of free electrons (in plasmonic or graphene medium) and electromagnetic (EM) light-wave which is known as the SPR can help scale down the structure’s dimensions (about or less than the operating wavelength). In other words, SPR can be utilized to conquer the diffraction limits in nano-dimension optical structures. Therefore, the plasmonic-graphene nano structures can be considered as the biosensors for detecting different elements like hemoglobin [8], glucose [11] and also for detection of different viruses like COVID-19 [12], MERS [13] and etc.

Interesting applications of graphene were reported which were utilized to excite Tamm resonance in PCs [14–19]. In [14], by considering the resonant dip of the Tamm plasmon (in a structure based on layers of gyroidal graphene and porous silicon), a refractive index sensor was proposed. The sensitivity and figure of merit of 1347.7 THz/RIU and 554.405 /RIU were reported, respectively [14].

In another research [15], a gas sensor based on various layers of gyroidal graphene with the sensitivity of 2431 THz/RIU was suggested. Also in [16], a refractive index sensor based on gyroidal graphene (considering the Tamm plasmon polaritons) with the sensitivity and quality factor of 18.6 THz/RIU and 26300 was proposed. In [17], a refractive index gas sensor based on the graphene-metals layers considering the Tamm resonance was proposed. The sensor indicated acceptable sensitivity, figure of merit and quality factor of 5.5 THz/RIU, 341 RIU-1 and 210, respectively. Also in [18], a biosensor based on graphene-porous silicon photonic crystals considering the Tamm resonance was reported with the sensitivity and figure of merit of 4.75 THz/RIU and 475 RIU-1, respectively.

Many researchers have also been fascinated by the amazing specifications of the graphene-plasmonic nano-structures and reported their works in the bio-sensing technologies. In [20], a biosensor for detection of glucose concentration in blood sample was proposed based on graphene-gold ellipse shaped structure.

In [8], various graphene-plasmonic based structures were reported for detecting hemoglobin concentrations in the blood samples. They reported the sensitivity of 570 nm/RIU for their proposed sensor [8]. In another research [21], efficient miniaturized plasmonic sensor with narrow line shapes in THz frequencies was suggested for detection of SARS-CoV-2 virus protein at low level. In [22], graphene and graphene-related materials (GRMs) were considered as interesting candidates for conquering COVID-19. Another research indicated amazing results in detection of COVID-19 very fast and with low errors. In this research, graphene field-effect transistor decorated with gold nano particles was proposed for detection of COVID-19 virus [23]. Also in [24], detection of COVID-19 virus with graphene based field-effect transistor was considered. The proposed sensor indicated very good sensitivity leading to precise results. In another interesting report [2], by utilizing the laser spectroscopic techniques and fiber optic–nano photonic approaches, rapid detection of COVID-19 viruses and especially a single virus detection were feasible [2].

In this research, different configurations of graphene-plasmonic nano structures for detection of corona viruses and especially COVID-19 virus are considered. Eventually, a sensitive biosensor for diagnosing COVID-19 based on graphene-plasmonic combinations in the 1-d photonic crystal and half-sphere shaped nano structures is proposed.

Geometry and Theoretical Descriptions

The first designed structure is based on different layers of aluminum (Al), gold (Au), SiO2 and graphene in the plate and half-sphere shaped layers. It is shown in Fig. 1.

Fig. 1
figure 1

a) Schematic of the first proposed structure, b) View of the single array of the structure, c) Side view of a single array

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As can be seen, the proposed structure is based on the repetitive and periodic algorithm of half-sphere shaped arrays of Al on the plate-shaped Au, SiO2 and graphene layers. The geometric parameters which are shown in Figs. 1b, c are tabulated in Table 1.

Table 1 Geometric parameters of Fig 1.
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In this wok, simulations were considered based on two-dimensional finite element based software (COMSOL Multiphysics 5.5). The structure (half-sphere shaped arrays) is periodically expanded in the y-direction (the structure is periodic in the y-direction, in other words the length of the structure in the y-direction is much longer than x or z directions). Therefore, a 2-D model (with similar results) was conducted instead of the 3-D model. This approach was considered for decreasing the time and computer resources needed for the simulations [25, 26].

Considering the proposed structure of Fig. 1, the field distributions at resonant and non-resonant wavelengths along with the absorption, reflection and transmission spectra are presented in Fig. 2.

Fig. 2
figure 2

a) Schematic of the absorption, reflection and transmission spectrum, Field distribution of the proposed structure at b) λ=968 nm, c) Field distribution at λ=975 nm

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As can be seen in Fig. 2a, the maximum absorption peak of 0.44 is achieved for λ=968 nm.

Also as can be seen in Fig. 1a, three plasmonic layers (Al, Au and graphene) are considered in the structure which can lead to Fano resonance (considering Fig. 2a) [27, 28]

Considering Fig. 2a, the following relation can be considered between the transmission, absorption and reflection spectra [7, 8]:

$$A\left(\omega \right)=1-R\left(\omega \right)-T\left(\omega \right)$$
(1)

In order to improve the functionality of the structure (to be used as a sensor), the absorption should be increased, so the transmission and reflection should be decreased. Therefore, for having a proper sensor, the absorption parameter A(ω) should become as much as possible. This can be achieved by considering different combinations of graphene and plasmonic layers (in the plate and half-sphere shapes) [7, 8]. Graphene and plasmonic (metal) nano structures are considered for improving the absorption parameter. Kubo formula can be considered for realization of the surface conductivity of a graphene layer [7, 8]:

$${\sigma }_{g}\left(\omega ,{\mu }_{c},\Gamma , T\right)={\sigma }_{g-real}+{\sigma }_{g-imag}={\sigma }_{int ral}+{\sigma }_{int er}$$
(2)

in which \(\omega\), \({\mu }_{c}\), \(\Gamma\) and T stand for the operation frequency, the chemical potential, the phenomenological scattering rate and the absolute temperature, respectively.

In Eq. (2), the intra-band (σintra) and inter-band (σinter) electro-photon scattering parameters can also be described as [7, 8]:

$${\sigma }_{int ra}=-j\frac{{e}^{2}{k}_{B}T}{{\pi \eta }^{2}(\omega -j2\Gamma )}\times \left[\frac{{\mu }_{c}}{{k}_{B}T}+2 ln\left(exp\left(\frac{{\mu }_{c}}{{k}_{B}T}\right)+1\right)\right]$$
(3)
$${\sigma }_{int er}=-j\frac{{e}^{2}}{4\pi \eta }1n\left[\frac{2{|\mu }_{c}|-(\omega -j2\Gamma )\eta }{2{|\mu }_{c}|+(\omega -j2\Gamma )\eta }\right]$$
(4)

in which kB, T and e represent Boltzmann constant, temperature and electron charge, respectively.

In order to define the dielectric function of the metals, Drude model can be considered [29, 30]:

$$\varepsilon \left(\omega \right)={\varepsilon }_{\infty }-\frac{{\omega }_{p}^{2}}{{\omega }^{2}+i\omega \gamma }$$
(5)

in which \({\varepsilon }_{\infty }\) = 3.7 (dielectric constant at infinite frequency), \(\gamma\) =0.018 eV (collision plasma angular frequency) and \({\omega }_{p}\) = 9.1 eV (bulk plasma angular frequency) [29, 30].

By applying the incident field to the first proposed structure (Fig. 1a), field distribution spectrum can be considered. As can be seen by considering λ=968 nm and λ=975 nm, the resonant and non-resonant spectra can be achieved in Figs. 2b, c, respectively. In fact, the absorption’s peak value of 0.44 is achieved in λ=968 nm, which should be enhanced for improving the functionality of the structures. It should also be considered that, biosensors with the ability of detecting COVID-19, mainly operate in the wavelength range of (300 nm - 500 nm) [30–32], therefore, the structure should be engineered for functioning in this range. For enhancing the functionalities of the proposed structure, different combinations of graphene and plasmonic in the plate and half-sphere shapes are applied to the structures. Effects of different structural parameters (r, t, d) and chemical potentials are also investigated for increasing the resonance peak’s values. The results are presented in the following parts.

Results and Discussions

In this section, 4 different configurations based on the graphene-plasmonic nano structures in the plate and half-sphere shapes are considered. The first one was presented in Fig. 1a. As stated for improving its functionality (increasing the absorption’s peak value), effects of the structural parameters (r, t) and chemical potential (μc) should be investigated. By considering values from 0.2 ev to 1ev for the chemical potential, its effect on the wavelength and peak corresponding to the absorption peak can be shown in Fig. 3.

Fig. 3
figure 3

Schematic of the absorption spectrum versus wavelength for different values of chemical potential (µc)

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As can be seen in Fig. 3, by increasing µc from 0.2 ev to 1 ev, the wavelength and peak corresponding to the absorption peak experience a blue shift and very light and smooth enhancement, respectively.

The absorption’s peak wavelength variations can be explained by the circuit theory. In the circuit theory, graphene is considered as a shunt admittance which can be altered by the geometrical parameters (r, t) and chemical potential µc [8, 33]. In fact, graphene can be substituted by resistors, capacitors and inductors. Therefore, varying µc can change the absorption’s peak wavelength according to [8, 33]:

$$\lambda =2\pi c\sqrt{LC}$$
(6)

in which c, L and C stand for the speed of light in vacuum, capacitance and inductance of the circuit, respectively. It can be concluded that enhancing µc diminishes L [34], which would shift the absorption’s peak wavelength to lower values (blue shift). Also by changing µc from 0.2 ev to 1ev, the absorption peak’s value experience a smooth and slow increase. Therefore, by considering µc=1 ev, the peak and wavelength corresponding to the absorption peak of 0.455 and 963 nm are achieved, respectively. Effects of the geometrical parameters (r, t) are also considered and depicted in Fig. 4.

Fig. 4
figure 4

Absorption spectrum versus wavelength for different values of a) r, b) t

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As indicated in Fig. 4a, increasing half-sphere’s diameter (from 30 nm to 50 nm) causes both of the wavelength and peak corresponding to the absorption peak to experience higher amounts [35]. In fact, for r=50 nm, the absorption’s peak wavelength shifts to higher values (red-shifted to λ=966 nm) and the absorption’s peak value reaches 0.5. For the following structures r=50 nm should be considered.

In Fig. 4b, effects of the gold layer’s (Au) height (t) on the absorption spectrum is considered (r=50 nm). As shown, the wavelength and peak corresponding to the absorption peak lead to lower and higher amounts, respectively [36]. By changing t from 8 nm to 18 nm, the wavelengths and peaks corresponding to the absorption peak shift from 968.5 nm to 962 nm and from 0.57 to 0.645, respectively. For the first proposed structure, the best obtained wavelengths and peaks corresponding to the absorption peak are 962 nm and 0.645, respectively. As a result, r=50 nm and t=18 nm are considered as the geometrical parameters for the following structures. The second proposed structure with periodic 1-d photonic crystal (1-d PhC) combinations of graphene-SiO2 layers is presented in the next section.

Second Structure

The second proposed structure is depicted in Fig. 5. As shown, periodic 1-d PhC layers of graphene-SiO2 are added to the first structure (shown in Fig. 1a).

Fig. 5
figure 5

a) Schematic of the second proposed structure, b) Side view of a single array

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In Fig. 5(a), PhC layers were added gradually (not instantly), therefore effects of considering each part (for Fig. 1a) on the absorption spectrum were considered and depicted in Fig. 6.

Fig. 6
figure 6

Effects of different 1-d PhC layers on the absorption spectrum

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It is concluded from Fig. 6, that adding more layers of 1-d PhC (to Fig. 1a) for designing Fig. 5a) would shift the absorption’s peak wavelength to lower values [7] and also increase the absorption’s peak value smoothly. As stated, for designing a biosensor with the ability of detecting COVID-19 virus, the absorption spectrum should be placed in the wavelength range of 300 nm to 500 nm. For achieving this goal, a defect layer (with specific material) would be placed in the middle of 1-d PhC layers. Interesting materials which can replace materials like gold, silver, aluminum and etc. in plasmonic structures (they function almost the same as metals) with the ability of operating in visible ranges (~300 nm – 800 nm) with low losses were proposed in recent decade [37, 38]. Therefore, a defect array is considered in Fig. 5a (shown as Fig. 7b) which is filled with different materials (GZO (Gallium doped Zinc Oxide), TiN (Titanium Nitride), ZrN (Zirconium Nitride) and ITO (Indium Tin Oxide)), and the results are shown in Fig. 7b.

Fig. 7
figure 7

a) Side view of a single array containing defect, b) Absorption spectrum versus wavelength for different defects

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As shown in Fig. 7b, by considering a defect layer of GZO, TiN, ZrN and ITO, the absorption’s peak wavelength would be shifted to lower amounts (blue-shift) and would be placed in the visible range [38, 39]. GZO is considered as a defect layer as it experiences more shifting to lower wavelengths. For improving the functionality of Fig. 7a, effects of the defect layer’s properties should also be considered. By changing its height (d4) from 60nm to 120 nm, the absorption spectra versus wavelength would be depicted in Fig. 8.

Fig. 8
figure 8

Absorption spectrum versus wavelength for different values of d4

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As shown in Fig. 8, by increasing d4 from 60 nm to 120 nm, the absorption’s peak wavelength would experience a red shift [29, 30, 40]. In fact, by increasing d4, distance between graphene layers “g3 and g4” enhances and the coupling between layers become poorer. As stated for our proposed application (biosensor for detecting COVID-19 virus), d4=60 nm with the absorption’s peak wavelength of 445 nm is acceptable. In the second structure, different layers of 1-d PhC were considered which improve the absorption’s peak value to 0.7 and by considering a defect layer of (GZO) and investigating its height’s (d4) effects, the absorption’s peak wavelength of 445 nm was obtained. In the following structure, a hollow half-sphere shaped layer of graphene is positioned above the half-sphere Al layer.

Third Structure

The third proposed structure is shown in Fig. 9. As can be seen, a hollow half-sphere shaped graphene layer is added to the structure of Fig. 7a. The defect layer (gray layer) is filled with GZO.

Fig. 9
figure 9

Schematic of a side view of a single array of the third proposed structure

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In order to improve the functionality of the third proposed structure, effects of its chemical potential (μc of g6) on the absorption spectrum would be considered as Fig. 10.

Fig. 10
figure 10

Schematic of the absorption spectrum versus wavelength for the third structure for different values of chemical potential (µc)

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As can be concluded from Fig. 10, increasing µc from 0.2 ev to 1.4 ev, would shift the wavelength and peak corresponding to the absorption peak to lower and higher amounts, respectively. This phenomenon was explained by the circuit theory in the previous section [8, 33]. As stated in this theory, graphene which is dependent to the geometrical parameters and chemical potential µc is considered as a shunt admittance. Therefore, geometrical parameters and Fermi energy (chemical potential) can affect its specifications. Changing µc can affect the absorption’s peak wavelength according to Eq. 6. As known, increasing chemical potential would decrease the inductance of the equivalent circuit (related to graphene). As a result, increasing µc would decrease L (inductance) [8, 24], which would lead to the blue shift of the absorption’s peak wavelength. As a result, by considering µc=1.4 ev, the peak and wavelength corresponding to the absorption peak of 0.8 and 432 nm are achieved, respectively. For improving the functionality of the proposed structure (the wavelength and peak corresponding to the absorption peak), layers of graphene and Al in the half-sphere hollow shapes and 1-d PhC formats are considered in the structure of Fig. 9.

Fourth Structure

The fourth proposed structure is considered in Fig. 11. As shown, layers of graphene and Al in the half-sphere hollow shapes are added to Fig. 9 which form 1-d PhC combination. It should be considered that the defect layer (gray layer) is filled with GZO.

Fig. 11
figure 11

Schematic of a side view of a single array of the fourth proposed structure

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As different graphene layers are considered in Fig. 11, their chemical potential’s effects on the absorption spectrum are investigated Fig. 12, indicates effects of μc(g7), μc(g8) and μc(g9) on the absorption spectrum.

Fig. 12
figure 12

Schematic of the absorption spectrum versus wavelength for the fourth structure for different values of a) μc(g7), b) μc(g8) and c) μc(g9)

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As shown in Fig. 12 and stated in the previous sections, chemical potential can affect the wavelength and peak corresponding to the absorption peak. Dependency of the resonance wavelength with the chemical potential was stated in the previous sections (circuit theory). As can be seen in Fig. 12a, increasing μc(g7) from 0.2 ev to 1.4 ev, would shift the wavelength and peak corresponding to the absorption peak from 440 nm to 430 nm and from 0.83 to 0.89, respectively. Also by increasing μc(g8) from 0.2 ev to 1.4 ev, the wavelength and peak corresponding to the absorption peak would change from 432 nm to 421 nm and from 0.89 to 0.96, respectively. In another case by changing μc(g9) from 0.2 ev to 1.4 ev, the wavelength and peak corresponding to the absorption peak would alter from 422 nm to 413 nm and from 0.95 to 1, respectively. As a result, by considering μc(g7)=μc(g8)=μc(g9)=1.4 ev, the proposed structure operates in the wavelength range of 300 nm - 600 nm with almost unity absorption’s peak value. Therefore, this structure can be utilized in different medical applications for its proper operating wavelength range and high absorption peak [8]. In the following part, a biosensor based on Fig. 11 with the ability of corona viruses’ detection and especially COVID-19 is proposed. For this purpose, refractive indices of corona viruses were considered from [31], which was obtained through reverse calculation. Refractive indices are tabulated in Table. 2.

Table 2 Refractive indices of the corona viruses [31]
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Figure 13 is obtained for different corona viruses which were mentioned in [31]. As can be seen in Fig. 13, as the refractive index of the corona virus is increased, the absorption peak’s wavelength is shifted to higher values (red-shift) [8, 31, 41, 42]. By considering the following equation for the sensitivity of the proposed bio-sensor, the sensitivity value can be calculated.

Fig. 13
figure 13

Absorption spectrum versus wavelength for different corona viruses’ RIs

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$$S=\frac{\Delta \lambda }{\Delta n}(nm/RIU)nm$$
(7)

Considering Fig. 13 and Eq. (7), the sensitivity of ~664.8 nm/RIU can be achieved. As a result, the proposed biosensor can be an important and functional device for detection of corona viruses and especially COVID-19.

Conclusion

As COVID-19 pandemic is a vital concern to everyone in this era, biosensors for detection of corona viruses and especially COVID-19 are of great importance. For this purposed, four different configurations based on graphene-plasmonic nano structures were suggested. The structures were based on arrays in the shapes of half-sphere and 1-d photonic crystal plates of Al, Au, SiO2 and graphene. For improving the functionality of the proposed structures effects of structural parameters (r, t, d4) and chemical potentials (μc, μg7, μg8 and μg9) on the wavelength and peak corresponding to the absorption peak were also investigated. As for detecting corona viruses and specially COVID-19, the application wavelength range should be positioned in 300 nm – 600 nm range, a defect layer of GZO was positioned in the middle of the 1-d photonic crystal structure. In the last proposed structure (fourth structure), by considering the refractive indices of the corona viruses (H5N1, H5N2, H9N2, H4N6, FAdV, IBV) in the biomolecule layer, the acceptable sensitivity of ~664.8 nm/RIU was obtained. As a result, the proposed structures can function as biosensors for diagnosing corona viruses and especially COVID-19.