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Abstract
In recent years, researchers utilized Tamm plasmon polaritons (TPPs) in conventional heterostructures composed of a metal layer, a dielectric spacer layer and an all-dielectric one-dimensional (1-D) photonic crystal (PhC) to achieve high-efficiency absorption of graphene. According to the Bragg scattering theory, photonic bandgaps (PBGs) in all-dielectric 1-D PhC strongly shift toward shorter wavelengths (i.e., blueshift) as the incident angle increases. Therefore, TPPs in conventional heterostructures also show strongly blueshift property. Such strongly blueshift property of TPPs greatly limits the operating angle range of the high-efficiency absorption of graphene. Herein, we realize an angle-insensitive TPP in a heterostructure composed of a metal layer, a dielectric spacer layer and a 1-D PhC containing hyperbolic metamaterial layers. Empowered by the angle-insensitive property of the TPP, we achieve wide-angle high-efficiency absorption of graphene. The operating angle range (A > 80%) reaches 41.8 degrees, which is much larger than those in the reported works based on TPPs and defect modes. Our work provides a viable route to designing cloaking devices and photodetectors.
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1. Introduction
Over the past three decades, a kind of periodic nanostructures called photonic crystals (PhCs) have played an important role in both optical physics and devices [1–3]. PhCs are able to create ranges of frequencies called photonic bandgaps (PBGs) where light is strongly reflected [4–8]. In the early stage after the concept of PhCs was proposed, researchers mainly focused on two-dimensional (2-D) [9–12] and three-dimensional (3-D) PhCs [13–16]. In 1998, researchers found that omnidirectional PBGs can exist in all-dielectric one-dimensional (1-D) PhCs [17]. Since then, all-dielectric 1-D PhCs have attracted rich attention of the researchers working in nano-optics due to their simplicity [18–23]. In 2007, researchers found a kind of polaritons called Tamm plasmon polaritons (TPPs) in heterostructures composed of a metal layer and an all-dielectric 1-D PhC [24]. Assisted by the resonant property of TPPs, the electromagnetic fields are strongly localized at the interfaces between a metal layer and an all-dielectric 1-D PhC [25–28]. Therefore, TPPs have been widely utilized to design absorbers [29–32], solar cells [33], sensors [34,35], lasers [36], and filters [37].
As a kind of 2-D materials, graphene has been extensively explored since its unique optical property [38–40]. However, the absorptance of graphene is only 2.3% at both visible and near-infrared wavelengths, resulting in a weak interaction between the light and the graphene [41]. To enhance the absorptance of graphene, researchers proposed various resonant nanostructures [42–52]. Particularly, assisted by the field enhancement effect of TPPs, researchers achieved high-efficiency absorption of graphene [53,54]. In Ref. [53], Lu et al. achieved high-efficiency absorption of graphene at near-infrared wavelengths by introducing a graphene layer into a heterostructure composed of a metal layer, a dielectric spacer layer and an all-dielectric 1-D PhC. In Ref. [54], Wang et al. achieved high-efficiency absorption of graphene at terahertz frequencies in a heterostructure composed of a graphene layer, a dielectric spacer layer and an all-dielectric 1-D PhC. At terahertz frequencies, the graphene layer acts like a metal layer. However, according to the Bragg scattering theory, PBGs in all-dielectric 1-D PhCs strongly shift toward shorter wavelengths as the incident angle increases [17,55]. Therefore, TPPs in heterostructures composed of a metal layer, a dielectric spacer layer and an all-dielectric 1-D PhC also strongly shift toward shorter wavelengths as the incident angle increases [53,54]. Such strongly blueshift property of TPPs greatly limits the operating angle range of the high-efficiency absorption of graphene. In Refs. [53] and [54], the absorptance rapidly decreases from near 100% to lower than 80% as the incident angle slightly increases from 0 to 10 degrees.
Over the past two decades, hyperbolic metamaterials (HMMs) have attracted immense interest due to their strong anisotropy [56–58]. By virtue of their strongly anisotropy, HMMs have been utilized to design reflectors [59], absorbers [60–62], polarization selectors [63], and lasers [64,65]. By introducing HMMs into 1-D PhCs, researchers realized a kind of special PBGs called redshift PBGs [66,67]. Different from the strongly blueshift PBGs in all-dielectric 1-D PhCs, such PBGs shift toward longer wavelengths as the incident angle increases. The emergence of redshift PBGs provides us a possibility to expand the operating angle range of the high-efficiency absorption of graphene. In this paper, we propose a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers to realize an angle-insensitive TPP. At near-infrared wavelengths, the noble metal layer acts like a perfect electric conductor [68]. Hence, the reflection phase of the noble metal layer φM is near ‒π for all the incident angles. As the incident angle increases, the round-trip propagating phase within the dielectric spacer layer 2φS decreases. The PBG in the 1-D PhC containing HMM layers is designed to be redshifted. Hence, as the incident angle increases, the reflection phase of the reflection phase of the 1-D PhC containing HMM layers φPhC increases. The total phase of the heterostructure φTotal = φM + 2φS + φPhC can be designed to be angle-insensitive. According to the Fabry-Perot resonance condition, the TPP is also angle-insensitive. Then, we introduce a graphene layer into the heterostructure to achieve wide-angle high-efficiency absorption of graphene. The absorptance keeps higher than 80% in a wide angle range from 0 to 41.8 degrees. Compared with the reported works on high-efficiency absorption of graphene assisted by TPPs [46,53,54], the operating angle range is effectively widened. To the best of our acknowledge, this is the first demonstration to achieve wide-angle high-efficiency absorption of graphene assisted by TPP. Our work provides a viable route to designing cloaking devices and photodetectors.
Table 1. Comparison of the operating angle range of high-efficiency absorption between this work and the reported works based on TPPs and defect modes.
This paper is organized as follows. In Sec. 2, we propose a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers to realize an angle-insensitive TPP. In Sec. 3, we achieve wide-angle high-efficiency absorption of graphene assisted by the angle-insensitive TPP. Finally, the conclusion is also given in Sec. 4.
2. Angle-insensitive TPP in a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers
In this section, we realize an angle-insensitive TPP in a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers. The functionality of the dielectric spacer layer is as follows. At the wavelength of the TPP, the electric field distribution within the dielectric spacer layer shows like a standing wave (see Fig. 5). Then, we can put the graphene layer at the position where the electric field is strongest to achieve high-efficiency absorption of graphene. To illustrate the mechanism of the angle-insensitive property of the TPP, we start from the Fabry-Perot resonance condition in a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC, as shown in Fig. 1. Above the plasma wavelength, the noble metal layer can be viewed as an optical mirror. Within the wavelength range of the PBG, the 1-D PhC can also be viewed as an optical mirror. Then, the Fabry-Perot resonance condition in the heterostructure can be expressed as [24]
Fig. 1. Fabry-Perot resonance condition in a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC.
At near-infrared wavelengths, the noble metal layer acts like a perfect electric conductor [68]. Hence, the reflection phase of the noble metal layer φM is near ‒π for all the incident angles. In other words, the reflection phase of the noble metal layer φM is insensitive to the incident angle, i.e., ∂φM/∂θ ≈ 0. The round-trip propagating phase within the dielectric spacer layer can be given by
Now, we realize a redshift PBG in a 1-D PhC containing HMM layers according to the theory in Ref. [66]. The 1-D PhC is composed of alternating dielectric layers (A layers) and HMM layers (B layers), as schematically shown in Fig. 2(a). The material of the dielectric layer is chosen to be Si (silicon). By fitting the measured data from Ref. [70], the refractive index of Si nA as a function of the wavelength in the wavelength range from 800 to 2000 nm can be expressed as
where λ represents the wavelength in units of micrometer. As the wavelength increases from 800 to 2000 nm, the refractive index of Si decreases from 3.88 to 3.45. The HMM layer is mimicked by a subwavelength ITO (indium tin oxide)/Si multilayer (CD)4. The whole 1-D PhC can be denoted by [A(CD)4]10. As a kind of low-loss plasmonic materials at near-infrared wavelengths, the relative permittivity of ITO can be modelled by the Drude model [71]Fig. 2. (a) Schematic of the 1-D PhC composed of alternating dielectric layers and HMM layers. The HMM layer is mimicked by a subwavelength ITO/Si multilayer. (b) x and z components of the effective relative permittivity tensor of the subwavelength ITO/Si multilayer (CD)4 as a function of the wavelength. (c) Reflectance spectrum of the 1-D PhC [A(CD)4]10 as a function of the incident angle under transverse magnetic (TM) polarization.
According to the effective medium theory (EMT), the effective relative permittivity tensor of the subwavelength ITO/Si multilayer (CD)4 can be given by [57]
Here f = dC/(dC + dD) represents the filling ratio of ITO. In our design, the filling ratio of ITO is selected to be f = 0.65. According to Eqs. (6) and (7), we calculate the x and z components of the effective permittivity tensor of the subwavelength ITO/Si multilayer (CD)4 as a function of the wavelength, as shown in Fig. 2(b). As demonstrated, Re(εBx) > 0 and Re(εBz) < 0 are satisfied in the wavelength range from 989.4 to 1615.3 nm (shown by the purple shadow region). Hence, the subwavelength ITO/Si multilayer (CD)4 can be viewed as a type-I HMM layer in this wavelength range.
To realize a redshift PBG, the thicknesses of the dielectric layer and the HMM layer should satisfy [66]
Here, λBrg represents the designed Bragg wavelength. Notice that Re(εBx) and Re(εBz) are the real parts of the x and z components of the effective relative permittivity tensor at the Bragg wavelength. In our design, the Bragg wavelength is selected to be λBrg = 1310 nm. According to Eq. (9), the minimum thickness of the HMM layer for a redshift PBG can be calculated as dB,min = 126.1 nm. To realize a redshift PBG, the thickness of the HMM layer should be larger than dB,min. Here, the thickness of the HMM layer is chosen to be dB = 168.0 nm. Then, the thickness of the dielectric layer can be obtained as dA = 113.0 nm according to Eq. (8). Since f = 0.65, the thicknesses of the ITO and the Si layers within the HMM layer can be calculated as dC = fdB/4 = 27.3 nm and dD = (1‒f)dB/4 = 14.7 nm, respectively. Notice that the thickness of the unit cell within the HMM layer is only dC + dD = 42.0 nm (0.032 times of the Bragg wavelength), which ensures the accuracy of the EMT [72].
According to the transfer matrix method [73], we calculate the reflectance spectrum of the 1-D PhC [A(CD)4]10 as a function of the incident angle under transverse magnetic (TM) polarization, as shown in Fig. 2(c). The incident and the exit media (substrate) are selected to be air and BK7 with a refractive index 1.515 [74]. Clearly, a PBG is opened from 1166.3 to 1692.4 nm at normal incidence. Two band edges are estimated from the reflectance dips nearest the PBG. As the incident angle increases from 0 to 60 degrees, the PBG exhibits redshift property. Specifically, the short-wavelength band edge shifts from 1166.3 to 1168.3 nm an the long-wavelength one shifts from 1692.4 to 1707.0 nm. As we discussed above, the redshift property of the PBG provides us a possibility to realize an angle-insensitive TPP.
To realize an angle-insensitive TPP, we put a noble metal layer and a dielectric spacer layer in front of the 1-D PhC containing HMM layers to construct a special heterostructure, as schematically shown in Fig. 3(a). The heterostructure can be denoted by MS[A(CD)4]10. The materials of the noble metal layer and the dielectric spacer layer are chosen to be Ag (silver) and Si, respectively. At near-infrared wavelengths, the relative permittivity of Ag can be modelled by the Drude model [75]
Fig. 3. (a) Schematic of the heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers. (b) Reflectance spectrum and (c) total phase spectrum (in units of π) of the heterostructure MS[A(CD)4]10 at normal incidence.
According to the transfer matrix method [73], we calculate the reflectance spectrum of the heterostructure MS[A(CD)4]10 at normal incidence, as shown in Fig. 3(b). One can see that a reflectance dip (at λ = 1468.1 nm) occurs within the PBG. To confirm that the reflectance dip originates from the TPP, we also calculate the total phase spectrum (in units of π) of the heterostructure MS[A(CD)4]10 at normal incidence in Fig. 3(c). The purple dashed line represents φTotal = 2π. At the crossing point between the black solid line and the purple dashed line (λTPP = 1467.2 nm), the Fabry-Perot resonance condition [Eq. (1)] is satisfied. The wavelength of the reflectance dip agrees well with the wavelength satisfying the Fabry-Perot resonance condition. The slight deviation (Δλ = 0.9 nm) originates from the absorptance of the noble metal layer.
Next, we show the angle dependence of the TPP. Figure 4 gives the reflectance spectrum of the heterostructure MS[A(CD)4]10 as a function of the incident angle under TM polarization. As the incident angle increases from 0 to 60 degrees, the reflectance dip slightly shifts from 1468.1 to 1447.7 nm. The relative shift of the reflectance dip is only Δλ/λ = 20.4/1468.1 = 1.4%. Compared with the reported TPPs in conventional heterostructures [24,53,54], the achieved TPP shows a superior angle-insensitive property.
Fig. 4. Reflectance spectrum of the heterostructure MS[A(CD)4]10 as a function of the incident angle under TM polarization.
Finally, we calculate the electric field distribution |E| of the heterostructure MS[A(CD)4]10 at the reflectance dip λ = 1468.1 nm at normal incidence, as shown in Fig. 5. The incident electric field is set to be |Ein| = 1 V/m. The electric field distribution of the heterostructure is similar to those of the reported TPPs [24,53,54]. Specifically, the electric field within the noble metal layer decays exponentially. The electric field within the 1-D PhC decays exponentially with periodic oscillation, which agrees well with the characteristic of a Bloch wave. The electric field distribution within the dielectric spacer layer shows like a standing wave. The maximum electric field reaches 3.77 V/m. The position of the maximum electric field within the dielectric spacer layer is 72 nm away from the left interface of the dielectric spacer layer.
Fig. 5. Electric field distribution |E| of the heterostructure MS[A(CD)4]10 at the reflectance dip λ = 1468.1 nm at normal incidence.
3. Wide-angle high-efficiency absorption of graphene empowered by angle-insensitive TPP
In this section, we utilize the angle-insensitive TPP to achieve wide-angle high-efficiency absorption of graphene. To obtain high absorptance, we put the graphene layer (G layer) at the position where the electric field is strongest. Hence, the dielectric spacer layer (S layer) is divided into two sublayers (S1 and S2 layers). The thicknesses of S1 and S2 layers are 72.0 and 128.0 nm, respectively. To obtain a high absorptance ratio of the graphene layer, the graphene layer is chosen to contain five monolayers. The reason we do not choose a larger number of the monolayer graphene is that the interacting effect of the adjacent monolayer graphene should be considered when the number of the monolayer graphene is larger than 5 [76]. The whole heterostructure can be denoted by MS1GS2[A(CD)4]10, as schematically shown in Fig. 6(a). At near-infrared wavelengths, the refractive index of graphene can be expressed as [77]
where the constant is C = 5.446 µm-1 [77]. Notice that the wavelength λ should be in units of micrometers.Fig. 6. (a) Schematic of the heterostructure MS1GS2[A(CD)4]10. (b) Absorptance spectrum of the heterostructure MS1GS2[A(CD)4]10 at normal incidence. (c) Absorptance spectrum of the heterostructure MS1GS2[A(CD)4]10 as a function of the incident angle under TM polarization.
According to the transfer matrix method [73], we calculate the absorptance spectrum of the heterostructure MS1GS2[A(CD)4]10 at normal incidence, as shown in Fig. 6(b). Empowered by the TPP, an absorptance peak (at λ = 1468.7 nm) occurs. The maximum absorptance reaches 99.6%. To show the angle dependence of the absorptance peak, we also calculate the absorptance spectrum of the heterostructure MS1GS2[A(CD)4]10 as a function of the incident angle under TM polarization in Fig. 6(c). As the incident angle increases from 0 to 60 degrees, the absorptance peak slightly shifts from 1468.7 to 1447.6 nm. The relative shift of the absorptance peak is only Δλ/λ = 21.1/1468.7 = 1.4%. Such angle-insensitive property of the absorptance peak provides us a possibility to achieve wide-angle high-efficiency absorption of graphene.
Then, we calculate the electric field distribution |E| of the heterostructure MS1GS2[A(CD)4]10 at the absorptance peak λ = 1468.7 nm at normal incidence, as shown in Fig. 7. The incident electric field is set to be |Ein| = 1 V/m. Owing to the absorption of the graphene layer, the maximum electric field reduces to 2.41 V/m compared with Fig. 5. Besides, the graphene layer is indeed located at the position where the electric field is strongest. It is the reason why high-efficiency absorption of graphene can be achieved.
Fig. 7. Electric field distribution |E| of the heterostructure MS1GS2[A(CD)4]10 at the absorptance peak λ = 1468.7 nm at normal incidence.
Next, we calculate the absorptance of the heterostructure MS1GS2[A(CD)4]10 at λ = 1468.7 nm as a function of the incident angle, as shown in Fig. 8(a). The purple dashed line represents A = 80%. As the incident angle increases from 0 to 20 degrees, the absorptance slightly decreases from 99.6% to 98.0%. As the incident angle continues to increase to 60 degrees, the absorptance gradually decreases to 60.7%. The absorptance keeps higher than 80% in a wide angle range from 0 to 41.8 degrees. Notice that except for the graphene layer, the noble metal layer and the ITO layers within the heterostructure also absorb the incident light since they possess optical losses. It is necessary to figure out the absorptance ratio of the graphene layer. It is known that the power dissipation density of the incident light at the position z can be calculated by [78]
where |E(z)| denotes the electric field at the position z. The absorbed power of the noble metal layer, the graphene layer and the ITO layers can be calculated byFig. 8. (a) Absorptance of the heterostructure MS1GS2[A(CD)4]10 at λ = 1468.7 nm as a function of the incident angle under TM polarization. (b) Absorptance ratio of graphene of the heterostructure MS1GS2[A(CD)4]10 at λ = 1468.7 nm as a function of the incident angle under TM polarization.
According to Eq. (16), we calculate the absorptance ratio of graphene of the heterostructure MS1GS2[A(CD)4]10 at λ = 1468.7 nm as a function of the incident angle, as shown in Fig. 8(b). The absorptance ratio of graphene is 71.0% at normal incidence, which is slightly lower than that in conventional heterostructures supporting TPPs (about 80%) [53]. The reason is that the HMM layers within the 1-D PhC absorb part of the incident light. As the incident angle increases from 0 to 60 degrees, the absorptance ratio of graphene slightly decreases from 71.0% to 66.8%. Hence, most of the incident light is absorbed by the graphene layer. Wide-angle high-efficiency absorption of graphene is achieved. Notice that the heterostructure proposed in this paper is more complicated than conventional heterostructures composed of a graphene layer, a metal layer, a spacer layer, and an all-dielectric 1-D PhC. Fortunately, in 2022, researchers successfully fabricated a 1-D PhC containing HMM layers (mimicked by Nb2O5/ITO multilayers) by the ion beam sputtering deposition [59]. Hence, the heterostructure proposed in this paper is possible to be fabricated in the future.
Finally, we compare the operating angle range of high-efficiency absorption in our work with those in the reported works based on TPPs and defect modes, as given in Table 1. The operating angle range is defined as the angle range where A > 80%. The operating angle range of high-efficiency absorption in our work is much wider than those in the reported works.
4. Conclusions
In summary, we realize an angle-insensitive TPP in a heterostructure composed of a noble metal layer, a dielectric spacer layer and a 1-D PhC containing HMM layers. Then, we put a graphene layer at the position where the electric field is strongest into the heterostructure to achieve high-efficiency absorption of graphene. Empowered by the angle-insensitive property of the TPP, we achieve wide-angle high-efficiency absorption of graphene. The absorptance keeps higher than 80% in a wide angle range from 0 to 41.8 degrees. The operating angle range is much wider than those in the reported works based on TPPs and defect modes. Our work provides a viable route to designing cloaking devices and photodetectors.
Funding
National Natural Science Foundation of China (12104105, 11947065, 12264028); Science and Technology Program of Guangzhou (202201011176); Natural Science Foundation of Jiangxi Province (20202BAB211007); Start-up Funding of Guangdong Polytechnic Normal University (2021SDKYA033); Interdisciplinary Innovation Fund of Nanchang University (2019-9166-27060003).
Acknowledgment
Dr. Feng Wu would like to thank a mathematician engaged in the Landau-Siegel zeros conjecture, Yitang Zhang, for his tenacious spirit of exploration.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
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