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Abstract
The strong absorption and reflection from atomically thin graphene nanoribbons has been demonstrated over the past decade. However, due to the significant band dispersion of graphene nanoribbons, the angle of incident wave has remained limited to a very narrow range. Obtaining strong absorption and reflection with a wide range of incident angles from atomically thin graphene layers has remained an unsolvable problem. Here, we construct a tunable moiré superlattice composed of a pair of graphene nanoribbon arrays to achieve this goal. By designing the interlayer coupling between two graphene nanoribbon arrays with mismatched periods, the moiré flat bands and the localization of their eigen-fields realized. Based on the moiré flat bands of graphene nanoribbons, highly efficient reflection and nearly perfect absorption was achieved with a wide range of incident angles. Even more interesting, is how these novel phenomena can be tuned through the adjustment of the graphene's Fermi energy, either electrostatically or chemically. Our designed moiré graphene nanoribbons suggest a promising platform to engineer moiré physics with tunable behaviors, and may have potential applications in the field of wide-angle absorbers and reflectors in the mid-infrared region.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Moiré superlattices in the twisted bilayer graphene provide an unprecedented platform to investigate a wide range of exotic quantum phenomena [1–15], including the correlated insulation [1], unconventional superconductivity [2–4], ferromagnetism [5,6], and so on . These fascinating phenomena are induced by the interplay between low-energy moiré flat bands and the long-range Coulomb interaction. Recently, the twist degree of freedom is incorporated into classical wave systems that showcases various interesting features in manipulating wave dynamics [16–27]. For example, the localization-delocalization transition of light has been observed in the optical moiré superlattice, and the threshold-less optical soliton enabled by the extremely flat moiré band is proved [16,17]. The topological transition of iso-frequency curves is demonstrated in twisted bilayer α-MoO3 flakes [18]. There are also some exact analogues to twist bilayer graphene based on dielectric photonic crystals [21–23], where the fundamental properties of band dispersions are determined by the interaction between the twist angle and the interlayer electromagnetic coupling of photonic crystals. In this case, the photonic magic-angle flat band as an analogy to the graphene counterpart could also be fulfilled. Hence, the interlayer electromagnetic coupling and twist angles could introduce an additional degree of freedom in designing nanostructures with novel photonic phenomena. In particular, twisted bilayer graphene could also exhibit their unique properties in photonics, including the realization of atomic photonic crystals [24] and the novel plasmons [25–27]. These twist-enabled phenomena in photonics could inspire us to design next-generation optical devices with novel performances.
On the other hand, the two-dimensional graphene nanoribbon is a promising candidate for achieving the flexible wave-control in mid- and far-infrared regions. The localized plasmonic modes of graphene nanoribbons have been experimentally observed through prominent absorption peaks in transmission spectroscopy measurements [28,29]. Moreover, the graphene plasmon resonance could also be controlled by changing the carrier density electrostatically or chemically [30–33], making the graphene nanoribbon become a tunable platform to realize various novel applications [34–36].
Recently, there has been significant interest in achieving the total absorption [37–49] and high reflection [50–54] based on the patterned atomically thin materials. While owing to the significant band dispersion, the high-performance absorption and reflection could only be fulfilled in a narrow range of incident angles at a fixed frequency. In this case, it is meaningful to ask whether the high-efficient absorption and reflection with wide angles could be realized by an atomically thin layer. Due to the fact that the moiré superlattice could induce the photonic flat bands, in this case, combining the moiré physics with unique properties of graphene nanoribbons may show much more advantages in achieving the high-efficient absorption and reflection with wide angles.
In this work, we numerically demonstrate that the moiré flat bands could be realized based on moiré graphene nanoribbons, which comprise of two overlapped one-dimensional (1D) graphene nanoribbon arrays with mismatched periods. The formation of moiré flat bands is resulting from the interplay between the lattice dislocation of the bilayer graphene nanoribbon and the interlayer coupling, which can be finely tuned by changing the separation distance between bilayer graphene nanoribbons and the Fermi energy of graphene. Moreover, based on the moiré flat bands of graphene nanoribbons, the highly efficient reflection and the nearly perfect absorption have been achieved in a wide range of incident angles. Our finding may have potential applications in the field of wide-angle absorbers and reflectors in the mid-infrared region.
2. Tunable moiré flatband and eigen-field localization in moiré graphene nanoribbons
We consider the structure consisting of two overlapped graphene nanoribbon arrays with different periods, as shown in Fig. 1(a). The periods P1 and P2 of the bottom and top graphene nanoribbon arrays are slightly different but satisfy the commensurate condition N1P1 = N2P2 with N1 and N2 being different integers. In this case, the 1D graphene moiré superlattice is achieved with the corresponding moiré period being P = N1P1 = N2P2. Figure 1(b) presents the top view of the moiré graphene nanoribbon with N1 = 9 and N2 = 8 (the red dash block marks the moiré unit), where the spatially varying local misalignment between two layers of graphene nanoribbon is clearly illustrated. The ratio between the width and period of the top (bottom) graphene nanoribbon array is defined as R1 = W1/P1 (R2 = W2/P2), which is also called the filling fraction. Moreover, the separation distance between two layers is represented as d.
Fig. 1. The schematic diagram of moiré graphene nanoribbons. (a) The side view of the structure for a pair of 1D graphene nanoribbon arrays with different periods. (b) The top view of the unit cell for moiré graphene nanoribbons. The red dash block marks the moiré unit.
The optical conductivity of graphene can be expressed as ${\sigma _s} = {\sigma _i} + {\sigma _D}$ under the low temperature condition [55–58]. Here, ${\sigma _i} = \sigma _i^{\prime} + i\sigma _i^\prime $ describes the inter-band contribution, which could be expressed as:
In the following, we numerically study the influence of interlayer distance between two graphene nanoribbon arrays (d) and the Fermi energy of graphene (${E_f}$) on the width of moiré bands. Other geometric parameters are set as N1 = 9, N2 = 8, P = 10.165µm, R1 = 0.85 and R2 = 0.85, respectively. Figure 2(a) and 2(b) present the calculated band dispersions of the system with the Fermi energy and interlayer distance being set as (${E_{f1}}$=1.0 eV, ${E_{f2}}$=1.0 eV, d = 80 nm) and (${E_{f1}}$=1.0 eV, ${E_{f2}}$=0.5 eV, d = 29nm), respectively. In the system of bilayer moiré graphene nanoribbons, the band structure is produced by the folding of bands for a single-layer graphene nanoribbon, and the hybridization of bands between two layers. For both structures, we note that there are significant widths of the moiré bands marked by red dots. Interestingly, by suitably tuning the separation distance or the Fermi energy of graphene nanoribbons, that is to change the interlayer distance from d = 80nm to d = 29nm (in Fig. 2(a)) or to change the Fermi energy from ${E_{f2}}$=0.5 eV to ${E_{f2}}$=1.0eV (in Fig. 2(b)), the considered moiré band could become much flatter, as shown by red dots in Fig. 2(c). Such a fascinating effect is similar to the formation of low-energy moiré bands of twisted bilayer graphene at a magic angle. In our case, there are magic interlayer distance and Fermi energy of graphene nanoribbons, which could make a specific moiré band become extremely flat. As for other moiré bands with different eigen-frequencies (marked by black dots), the associated band dispersion could also be flattened by suitably tuning the interlayer distance and Fermi energy to different magic values. Moreover, different from moiré flat bands existing in other dielectric photonic systems, the graphene nanoribbon-based moiré flat bands could be tuned by the external voltage (as proved in Fig. 2(b) and 2(c)), which can adjust the carrier density and Fermi energy of graphene, with fixed structural parameters. And, it is worthy to note that the magic value of Fermi energy is depended on the interlayer distance of moiré graphene nanoribbons. Hence, we could also design moiré flat bands at different Fermi energies when the interlayer distance is changed.
Fig. 2. Numerical results of band dispersion for moiré graphene nanoribbons. (a), (b) and (c) is the band dispersions of the moiré graphene nanoribbon with the Fermi energy and the interlayer distance being (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 80nm), (${E_{f1}}$=1.0eV, ${E_{f2}}$=0.5eV, d = 29nm) and (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 29nm), respectively. The red dots mark the considered moiré band.
One exotic property of moiré flat bands is that their eigen-modes could exhibit the significant localization of the near field. Next, we will show that such a novel effect of the field localization also exists in the moiré graphene nanoribbon. Figure 3(a), 3(b), 3(c) and 3(d) plot distributions of four normalized eigen-fields (at kx = 0) belonging to different bands of the periodic moiré graphene nanoribbon with ${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV and d = 29nm, where the associated eigen-frequencies are f1 = 19.63THz, f2 = 19.37THz, f3 = 19.12THz and f4 = 18.35THz, respectively. It is shown that the wider bandwidth of the moiré band possesses, the weaker localization of the corresponding eigen-field becomes. In particular, the eigen-fields of flat band (marked by red dots around 18.35THz) are significantly concentrated around the AA-stacking regions of the moiré graphene nanoribbon, as shown in Fig. 3(d). As for the most dispersive band with f1 = 19.63THz at kx = 0, the delocalization of the eigen-field is clearly illustrated (Fig. 3(a)). Hence, it is found that the localization of eigen-field appears accomplished with the flatten of moiré bands. Such a localized eigen-modes in moiré graphene nanoribbons may possess useful applications for surface-enhanced spectrum detections.
Fig. 3. Numerical results for the localization of eigen-fields in moiré graphene nanoribbons. (a), (b), (c) and (d) plots distributions of four eigen-fields (at kx = 0) belonging to different bands of the periodic moiré graphene nanoribbon with ${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV and d = 29nm, where the associated eigen-frequencies are f1 = 19.63THz, f2 = 19.37THz, f3 = 19.12THz and f4 = 18.35THz, respectively.
3. Highly efficient reflections with wide angles based on moiré graphene nanoribbons
In this part, we will numerically demonstrate that our designed moiré graphene nanoribbons could exhibit highly efficient reflections with wide angles assisted by the moiré flat band. Figure 4(a) and 4(b) display the calculated reflection spectra with different incident angles, when the Fermi energy and interlayer distance of the moiré graphene nanoribbon are set as (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 80nm) and (${E_{f1}}$=1.0eV, ${E_{f2}}$=0.5eV, d = 29nm), respectively. Other parameters are the same to that used in Fig. 2. It is clearly shown that resonance peaks of different incident angles in both systems locate at different frequencies, manifesting the significant dispersion of the excited moiré bands. These results are consistent with the calculated band dispersion in Fig. 2(a) and 2(b). Then, by tuning the separation distance or Fermi energy of the moiré graphene nanoribbon to magic values, that is d = 29nm, ${E_{f1}}$=1.0eV and ${E_{f2}}$=1.0eV, the calculated reflection spectra with different incident angles are shown in Fig. 4(c). We note that reflection peaks for different incident angles nearly locate at the same frequency, indicating the excitation of moiré flat band. Such a moiré flat band could exhibit highly efficient reflection at a single frequency with wide angles. To qualify this effect, we calculate the amplitude of reflection as a function of incident angles at a fixed frequency, as shown in Fig. 4(d). The blue, green and red lines correspond to results belonging to moiré graphene nanoribbons with associated parameters being (${E_{f1}}$=1.0 eV, ${E_{f2}}$=1.0 eV, d = 80 nm), (${E_{f1}}$=1.0 eV, ${E_{f2}}$=0.5 eV, d = 29 nm) and (${E_{f1}}$=1.0 eV, ${E_{f2}}$=1.0 eV, d = 29 nm), respectively. The selected frequency corresponds to the resonance peak with the incident angle being 0deg. It is clearly shown that the moiré graphene nanoribbon sustaining flat bands could show highly efficient reflection in a much wider range of incident angles.
Fig. 4. Numerical results of reflection spectra in moiré graphene nanoribbons. (a), (b) and (c) presents the calculated reflection spectra with different incident angles when the Fermi energy and the interlayer distance of moiré graphene nanoribbons are setting as (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 80nm), (${E_{f1}}$=1.0eV, ${E_{f2}}$=0.5eV, d = 29nm) and (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 29nm), respectively. The inset in Fig. 4(b) shows the reflection spectrum of a single layer graphene nanoribbon with associated parameters being ${E_f}$=1.0eV, P = 4.5µm and R = 0.85. (d) The amplitude of reflection for moiré graphene nanoribbons with different parameters as a function of incident angles at a fixed frequency, which corresponds to the resonance peak with the incident angle being 0deg.
Further, for comparation purposes, we also calculate the reflection spectrum of the single-layer graphene (${E_f}$=1.0eV, P = 4.5µm and R = 0.85) with different incident angles, as shown in the inset of Fig. 4(b). The excited mode of the single-layer graphene nanoribbon corresponds to the original mode that unfolds and hybrids into the flat bands in moiré graphene nanoribbons (red dots in Fig. 2(c)). It is shown that due to the low radiation loss rate, the reflection of a monolayer graphene nanoribbon is relatively low. This can be clearly shown by the black line in Fig. 4(d), which presents the calculated reflection as a function of incident angles for the monolayer graphene nanoribbon with the incident frequency locating at the resonance peak of 0deg. In this case, we can see that the moiré graphene nanoribbon could make the low-reflective mode of the single layer graphene nanoribbon display much enhanced wide-angle reflection owing to the amplified radiation loss rate and moiré potential induced flat dispersions.
4. Nearly perfect absorption with wide angles based on moiré graphene nanoribbons
Except for the highly efficient reflections, in this part, we demonstrate that the moiré graphene nanoribbon on a mirror substrate could exhibit nearly perfect absorption in a wide range of incident angles. The schematic diagram is presented in Fig. 5(a), where the distance between the bottom layer of moiré graphene nanoribbon and the mirror substrate is set as 3.8 µm. Figure 5(b) and 5(c) display the calculated absorption spectra with different incident angles, when the Fermi energy and interlayer distance of the moiré graphene nanoribbon are set as (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 85 nm) and (${E_{f1}}$=0.41eV, ${E_{f2}}$=1.0 eV, d = 55 nm), respectively. Other parameters are set as N1 = 4, N2 = 5, P = 4.8 µm, R1 = 0.958 and R2 = 0.958. We can see that absorption peaks at different incident angles of both systems locate at different frequencies, meaning that no moiré flat band is excited. Also, we note that the maximum values of the absorption at different incident angles approach to one. Such nearly total absorption is resulting from the realization of the critical coupling condition, where the absorption loss rate equals to the radiation loss rate of the system.
Fig. 5. Numerical results for the absorption spectra of moiré graphene nanoribbons on the mirror substrate. (a) The side view of the moiré graphene nanoribbon arrays on the mirror substrate. (b), (c) and (d) presents the calculated absorption spectra with different incident angles when the Fermi energy and the interlayer distance of moiré graphene nanoribbons are setting as (${E_{f1}}$=1.0 eV, ${E_{f2}}$=1.0 eV, d = 85 nm), (${E_{f1}}$=0.41 eV, ${E_{f2}}$=1.0 eV, d = 55 nm) and (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 55nm), respectively. The inset in Fig. 5(d) shows the absorption spectrum of a single layer graphene nanoribbon with associated parameters being ${E_f}$=1.0 eV, P = 2.287 µm and R = 0.958. (e) The amplitude of absorption for moiré graphene nanoribbons with different parameters as a function of incident angles at a fixed frequency, which corresponds to the resonance peak with the incident angle being 0deg.
It is worthy to note that the critical coupling and moiré flat bands could be realized simultaneously in our designed moiré graphene nanoribbons by suitably tuning the interlayer distance or Fermi energy to the magic value, which is d = 55 nm and ${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV. The calculated absorption spectra with different incident angles for such a structure (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d= 55 nm) are shown in Fig. 5(d). It is clearly shown that absorption peaks for different incident angles nearly locate at the same frequency, indicating the excitation of moiré flat bands. Moreover, such a moiré flat band could exhibit total absorption at a suitable incident angle, indicating the realization of critical coupling. Then, we calculate the absorption as a function of incident angles under different values of separation distances and Fermi energies, as shown in Fig. 5(e). Blue, green and red lines correspond to the moiré graphene nanoribbon with associated parameters being (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 85 nm), (${E_{f1}}$=0.41 eV, ${E_{f2}}$=1.0eV, d = 55 nm) and (${E_{f1}}$=1.0eV, ${E_{f2}}$=1.0eV, d = 55 nm), respectively. Similarly, the selected frequency corresponds to the absorption peak with the incident angle being 0deg. We can see that the moiré graphene nanoribbon with flat bands could show the nearly perfect absorption (larger than 90%) in a wider range of incident angles.
Additionally, we also design a single-layer graphene nanoribbon to exhibit the total absorption at the normal incidence, where parameters are set as ${E_f}$=1.0 eV, P = 2.287 µm and R = 0.958. The calculate absorption spectra with different incident angles are plotted in the inset of Fig. 5(d). We can see that the frequencies of absorption peaks at different incident angles are deviated from each other, making the wide-angle absorption with high efficiency become difficult to achieve. This phenomenon can be clearly shown by the black line in Fig. 5(e), that presents the calculated absorption as a function of incident angles at the resonance frequency with 0deg. From the above result, we can see that that due to the existence of flat bands, only the moiré graphene nanoribbon could realize nearly perfect absorption in a wide range of the incident angle.
It is worthy to note that the precise control of the interlayer distance is very difficult in practice. With the help of tunable property of graphene, moiré flat bands can still be realized by suitably tuning the associated Fermi level even the interlayer distance is slightly deviated from the designed value. For example, when the interlayer distance in Fig. 5(d) is changed from 55 nm to 65 nm, moiré flat band can also be realized by changing Ef1 from 1.0 eV to 0.71 eV, as shown in Fig. 6. It is shown that the resonance frequencies of absorption peaks at different incident angles are nearly the same of this system (Ef1 = 0.71 eV, Ef2 = 1.0 eV, d = 65 nm), indicating the excitation of moiré flat bands.
Fig. 6. Numerical results for the absorption spectra of moiré graphene nanoribbons on the mirror substrate with different incident angles when the Fermi energy and the interlayer distance of moiré graphene nanoribbons are setting as ${E_{f1}}$=0.71 eV, ${E_{f2}}$=1.0 eV, d = 65 nm.
5. Conclusion
In conclusion, we have constructed a tunable moiré superlattice based on a pair of graphene nanoribbon arrays with mismatched periods. We numerically demonstrate that the tunable moiré flat bands and the associated eigen-field localization could be realized based on moiré graphene nanoribbons. Moreover, based on the flat band of moiré graphene nanoribbons, the highly efficient reflection and the nearly perfect absorption have been achieved in a wide range of incident angles. Although there are only numerical results in the current work, our proposed moiré graphene nanoribbons could also be fabricated in experiments, where the appropriate dielectric slab (like CaF2 and BaF2) without absorptions in the mid-infrared region could be embedded between two layers of graphene nanoribbon. Thanks to the tunable property of graphene, when the thickness of this dielectric slab is slightly different from the desired value in practice, moiré flat bands can also be realized by tuning Fermi energy of graphene nanoribbons. Such a device is compatible with standard nanofabrication processes, and can be fabricated with electron beam lithography and plasma-assisted etching. Our designed moiré graphene nanoribbons may have potential applications in the field of designing wide-angle absorbers and reflectors, and the flat bands enhanced nonlinear effects may also be fulfilled based on the moiré graphene nanoribbons.
Funding
National Key Research and Development Program of China (No. 2017YFA0303800); National Natural Science Foundation of China (11904022, 12004038).
Disclosures
The authors declare no conflicts of interest.
Author Contributions. The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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