1. Introduction

Absorption is an intrinsic characteristic of materials, which is important for the construction of optoelectronic devices. The absorptions of ordinary materials usually stay static with a fixed structure. As a two-dimensional material, graphene has a tunable absorption behavior, whose absorption can be attributed to the inter-band and intra-band transitions [1]. In the infrared and terahertz regions, the absorption of graphene is dominated by the intra-band transitions, and its conductivity can be described by the Drude model [2]. The patterning of graphene can enhance its absorption by exciting the graphene plasmons in a specific resonant wavelength region [3–6]. To further enhance the graphene-light interaction, Fabry–Pérot cavities are usually employed to form the one port coupling system [7–10]. In such way, the absorptions of graphene plasmons could reach 100% in principal.

To date, much work has focused on the absorption modulation based on graphene nanostructures and the Fabry–Pérot cavities. For instance, tunable infrared metasurface by making use of graphene plasmons [11–13], electronic modulation of infrared radiation based on graphene nanostructures [14–16], plasmonic absorbers utilizing multilayer graphene nanoribbons [17–19]. Generally, the resonant wavelength would undergo a blue shift with the increase of Fermi level [20], and the absorption curve becomes narrower as the mobility of graphene increases [21]. In this paper, we firstly show that the anomalous redshift of graphene absorption occurs as the mobility of graphene is low. Such phenomenon is caused by the competition between the cavity mode and the plasmon mode. These results give an insight into the mode interaction, offering an alternative method to modulate the absorption, and could benefit the design of tunable graphene devices.

2. Results and discussions

The proposed system is shown in Fig. 1(a), which is composed of graphene ribbons, Al₂O₃ spacer, and a gold reflector. Figure 1(b) illustrates the cross-section of the unit cell, where the ribbon width is w, the period is p, and the thickness of the resonant cavity is h. The structure is modeled by the commercial software COMSOL Multiphysics, in which double-layer graphene is utilized to enhance the absorption, and a surface current boundary condition is set. The surface conductivity of the double-layer graphene is set as σ = 2*i4πe²E_f/h²(ω+iτ⁻¹) [18]. The relaxation time is expressed as τ = μE_f/ev_f², where μ is the mobility and v_f ≈10⁶ m/s is the Fermi velocity. When a transverse electromagnetic wave impinges on the graphene ribbons, the cavity mode and the surface plasmon mode would be excited. On one hand, the electric field near the cavity surface would reach a maximum when the optical path length of the cavity is equal to the odd multiples of a quarter wavelength, due to the constructive interference between the incident and reflected lights [7]. Owing to the motion of free electrons caused by the enhanced electric field, the ohmic loss in graphene, namely the cavity-induced absorption, would produce a peak at the wavelength λ = 4nh. On the other hand, as the plasmon mode is excited, the accompanying electric field would also drive the free electrons to oscillate, which could induce an absorption peak in the spectrum, called as plasmon-induced absorption. The completion between these two kinds of absorpitons become obvious when the mobility of graphene is low (i.e., the ohmic loss of graphene is large). Thus, the total electric field that drives the motions of free electrons can be divided into two parts, that is, the cavity mode related field E₁, and the graphene plasmon related part E₂. Figure 1(c) describes the interference effect between E₁ and E₂. For the constructive interference, these two electric fields lead to a higher absorption; while for the destructive interference, the coupled absorption would be reduced. Consequently, the dynamic competition between the cavity mode and the plasmon mode may induce the anomalous shifting of absorption. Namely, with the increase of E_f, the absorption peak would undergo a redshift, and the diagrammatic sketch is shown in Fig. 1(d).

 

Fig. 1. (a) Schematic diagram of the studied structure. (b) Cross-section of the unit cell, where w = 150 nm, p = 240 nm, h = 1 μm, and the refractive index of Al₂O₃ is 1.5. (c) Diagrams of the constructive and destructive interference between electric field E₁ and E₂. (d) Diagrammatic sketch of the anomalous redshift.

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Firstly, we separately analyzed the absorption characteristics of the plasmon mode and the cavity mode. Figure 2(a) shows the variation of absorption spectra versus the mobility when the graphene ribbon is suspended in vacuum (i.e., without the Fabry–Pérot cavity). It can be seen that the resonant peak related to the plasmon mode decays when the mobility of graphene declines, and the linewidth of the spectra becomes wider due to the growing loss of graphene. The position of the resonant peak is independent of the mobility, which remains around 8 μm, indicating the mobility of graphene does not affect plasmon dispersion relations too much. Figure 2(b) presents the absorption spectra versus the Fermi level when the mobility is set as 100 cm²/(V*s). It’s revealed that the peak undergoes a blue shift with increasing E_f, which is the ordinary dispersion of graphene plasmon [22,23]. Next, we obtained the absorption spectra of the cavity mode versus the mobility of graphene by modeling the continuous graphene sheet on a resonant cavity. As shown in Fig. 2(c), the peak of the cavity mode appears at λ = 4nh. When mobility is as high as 1000 cm²/(V*s), the peak is around 5%. However, as the mobility decreases from 1000 to 100 cm²/(V*s), the absorption gradually increases to near 20%, getting comparable with the plasmon mode. Such a phenomenon is reasonable because the ohmic loss of low mobility graphene is quite large. Figure 2(d) displays the spectra of cavity mode versus the Fermi level as the mobility of graphene is 100 cm²/(V*s). The position of absorption peak is independent of the Fermi level, which is different from the case of plasmon mode. It’s worth noting that the linewidths of spectra are relatively wider at low mobility, implying the inevitable interaction between two modes. Furthermore, the absorption of cavity mode is larger than that of plasmon mode at low Fermi levels. As the Fermi level increases, the absorption of the plasmon mode gradually becomes comparable with the cavity mode. Therefore, the shifting trend of absorption peaks will be influenced by the competition between the two modes, which is different from the variation trend of graphene plasmon absorption.

 

Fig. 2. (a) Absorption as a function of the wavelength and the graphene mobility of the suspending graphene ribbons. (b) Absorption spectra under different Fermi levels of the suspended graphene ribbons with μ = 100 cm²/(V*s). (c) Absorption as a function of the wavelength and the graphene mobility for a graphene sheet on the optical cavity. The cavity height h is 1 μm. (d) Graphene absorption of cavity mode under different Fermi levels at the graphene mobility of 100 cm²/(V*s).

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To investigate the interaction between the cavity mode and plasmon mode when the graphene is with low mobility, we calculated the absorption spectra versus the Fermi level, as shown in Fig. 3(a), where the mobility of graphene is set to be 100 cm²/(V*s). The maximum absorption varies from 2% to 93% as the Fermi level increases from 0.1 to 1.2 eV, and the peaks show a blue shift when the Fermi level ranges from 0.6 to 1.2 eV. However, the redshift is not evident when the Fermi level range of 0.1 to 0.6 eV due to the low absorptions. Hence, we extracted and normalized the representative spectra in Fig. 3(a), and plotted them in Fig. 3(b). It can be seen that the absorption peak gradually undergoes a redshift when the Fermi level is less than 0.6 eV. As the Fermi level further increases to 1.2 eV, the curve of peak returns to the plasmon dispersion of graphene. Such variation trend is further added to Fig. 3(a), together with the variation trends of the absorption peaks caused by merely cavity mode or plasmon mode (i.e., the dispersion curve of cavity mode or the plasmon mode). By comparing the three curves in Fig. 3(a), we can see that the absorption is dominated by cavity mode at low Fermi level, but as the Fermi level is increased to larger than 0.6 eV, the contribution by plasmon mode becomes dominant.

 

Fig. 3. (a) Absorption as a function of the wavelength and the Fermi level with μ = 100 cm²/(V*s) and h = 1 μm. The solid line is the curve of the absorption peak of graphene at the low mobility, the dot-dash line marks the dispersion curve of the cavity mode, and the dotted line denotes the dispersion curve of graphene plasmon. (b) Normalized curves of absorption spectra extracted from (a). (c)–(d) Anomalous redshift of absorption peaks with different h (w = 150 nm and h = 0.75 μm, 1 μm, 1.25 μm, and 1.5 μm) and w (h = 1 μm and w = 75 nm, 100 nm, 125 nm, and 150 nm), respectively. The mobility of graphene is μ = 100 cm²/(V*s). The solid, dot, and dot-dash lines represent the curves of redshift, the normal dispersion curves of graphene plasmon, and the positions of cavity mode, respectively.

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To further confirm the variation trend of absorption peaks, the cavity length h or the width of graphene nanoribbon w is varied, so that the dispersion curve of the cavity mode or the plasmon mode can be changed. Then, the absorptions in these systems are calculated numerically, and the absorption peaks are extracted and plotted in Figs. 3(c) and 3(d). In Fig. 3(c), only the cavity height h is varied and the dispersion curve of the plasmon mode is fixed. In the cases with different cavity height h, as Fermi level increases from 0.1 to 1.2 eV, the absorption peaks firstly exhibit a redshift, and then have a blueshift close to plasmon dispersion curve. In Fig. 3(d), the cavity height is fixed but the dispersion curve of the plasmon mode is varied, similar variation trend of the absorption peaks can be observed as in Fig. 3(c). Namely, as the Fermi level increases, the absorption peaks have a redshift from the cavity mode related dispersion curve and then coincide with the dispersion curve of plasmon mode gradually. These results give an insight into the mode interaction and offer an alternative method to modulate the absorption, which could benefit the design of tunable graphene devices.

Next, the mechanism of such anomalous redshift phenomena is quantitatively explained. The absorption of graphene is caused by the Ohmic loss of free electrons in graphene, and these electrons are driven by two kind of electric field, i.e., the cavity mode related electric field ${\mathop{E}\limits^\rightharpoonup}_1$ and the plasmon mode related electric field ${{\mathop{E}\limits^\rightharpoonup} _2}$. On the other hand, the Ohmic loss power P_loss is given by ${P_{loss}} = {\textrm{Re}} (\sigma )\cdot {\mathop{E}\limits^\rightharpoonup} _{total}^\ast{\cdot} {{\mathop{E}\limits^\rightharpoonup} _{total}}/2$, where $\sigma$ is the optical conductivity of graphene, and ${{\mathop{E}\limits^\rightharpoonup} _{total}}$ is the total electric field. Since ${{\mathop{E}\limits^\rightharpoonup} _{total}}$ is the sum of ${{\mathop{E}\limits^\rightharpoonup} _1}$ and ${{\mathop{E}\limits^\rightharpoonup}_2}$, and the total absorption of graphene is ${A_{total}} = {P_{loss}}/{P_{in}}$ with ${P_{in}}$ being the incident power, hence the absorptions can be written as,

In the equation, ${A_{cavity}} = {\textrm{Re}} (\sigma ){|{{{\mathop{E}\limits^\rightharpoonup }_1}} |^2}/2{P_{in}}$ is the cavity mode-related absorption, ${A_{plasmon}} = {\textrm{Re}} (\sigma ){|{{{\mathop{E}\limits^\rightharpoonup }_2}} |^2}/2{P_{in}}$ is the absorption caused by the plasmon mode, and ${A_{interference}} = {\textrm{Re}} (\sigma )({{{\mathop{E}\limits^\rightharpoonup }_1}^\ast{\cdot} {{\mathop{E}\limits^\rightharpoonup }_2}}$ ${+ {{\mathop{E}\limits^\rightharpoonup }_1} \cdot {{\mathop{E}\limits^\rightharpoonup }_2}^\ast } )/2{P_{in}}$ is the interference term. This equation indicates that the total absorption of graphene is not simply the sum of the absorptions, and the interference also plays an important role.

The absorption terms in Eq. (1) were extracted from the simulation results, as illustrated in Fig. 4. Firstly, the electric field ${{\mathop{E}\limits^\rightharpoonup}_1}$ is calculated from the configuration shown in Fig. 4(a), where the graphene is a continuous sheet so that the graphene plasmon mode cannot be excited. Then, the graphene is patterned into nanoribbons as shown in Fig. 4(b), both the cavity mode and the plasmon mode can be activated, and the total electric field ${{\mathop{E}\limits^\rightharpoonup}_{total}}$ can be calculated and extracted. After that, the electric field related to the plasmon mode can be calculated out using ${{\mathop{E}\limits^\rightharpoonup}_2}\textrm{ = }{{\mathop{E}\limits^\rightharpoonup}_{total}} - {{\mathop{E}\limits^\rightharpoonup}_1}$. Thus, the extracted fields are presented in the upper panel of Fig. 4(d), it can be seen that ${{\mathop{E}\limits^\rightharpoonup}_1}$ is constant, while ${{\mathop{E}\limits^\rightharpoonup}_2}$ varies along the graphene nanoribbon direction. The phase difference between ${{\mathop{E}\limits^\rightharpoonup}_1}$ and ${{\mathop{E}\limits^\rightharpoonup}_2}$ is exhibited in the lower panel of Fig. 4(d), which shows the phase difference also varies along the graphene ribbon direction. The absorptions A_total, A_cavity, A_plasmon, and A_interference can be computed from Eq. (1). In Fig. 4(c), the absorptions at low (0.2 eV) and high (1 eV) Fermi levels are shown in the upper and lower panels, respectively. For the case of the low Fermi level (the upper panel), A_plasmon is much smaller than A_cavity, and hence, the total absorption A_total is mainly determined by A_cavity. For the case of the high Fermi level (the lower panel), A_plasmon becomes dominant, and the interference term A_interference in the range of 6 to 12 μm becomes negative, indicating the electric fields ${{\mathop{E}\limits^\rightharpoonup}_1}$ and ${{\mathop{E}\limits^\rightharpoonup}_2}$ interfere with each other destructively. In this case, the total absorption curve A_total is similar to the curve A_plasmon, and thus the absorption of graphene is dominated by the plasmon-induced absorption.

 

Fig. 4. (a)–(b) Schematic diagrams of models for calculating E₁ and E₂. w = 150 nm, h = 1 μm, μ = 100 cm²/(V*s). (c) Total absorptions, the absorptions of the cavity mode and the plasmon mode as well as the interference terms when the values of Fermi level are 0.2 eV (upper panel) and 1 eV (lower panel). (d) Amplitudes of E₁ and E₂ (upper panel) and phase difference between E₁ and E₂ (lower panel) along the X-axis at the surface of graphene at 8 μm. The Fermi level is 1 eV.

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In what follows, we experimentally demonstrate the anomalous redshift of the absorption peak of graphene. The schematic of the fabricated device is shown in Fig. 5(a). In order to reduce the mobility, we patterned the graphene film into the dolmen structures to create a large number of defects. The ion-gel is used to electrically tune the Fermi level by the top gate. The thicknesses of the resonant cavity and the gold reflective layer are 0.8 μm and 60 nm, respectively. Figure 5(b) shows the SEM image of the fabricated graphene nanostructures, where the polarization angles of incident light for the spectra measurement are also marked. Figures 5(c) and 5(d) show the simulation spectra of the dolmen structure under the incident polarization angle of 0° and 90°, respectively. It can be observed that three and two plasmon modes are excited, respectively, when the mobility of graphene is 1000 cm²/(V*s). However, these modes decay with the decrease of mobility. As a result, the modes cannot be distinguished from the spectra when the mobility is reduced to 100 cm²/(V*s). The absorption spectra of the dolmen structure only exhibit one peak, which is similar to the case of graphene nanoribbons.

 

Fig. 5. (a) Schematic diagram of the graphene dolmen nanostructures for experimental study of the redshift phenomenon; (b) SEM photograph of the fabricated graphene nanostructures. The size of the graphene holes is 300 nm*100 nm and 250 nm*100 nm, the two legs are separated by 150 nm, the gap between legs and the top hole is 100 nm, and the periods in both directions are 500 nm. The cavity height h is 0.8μm. (c) and (d) Simulated absorption spectra of the dolmen structure versus electron mobility for the polarization angle of (c) 0° and (d) 90°.

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Next, the absorption spectra of the dolmen structure are studied. Figures 6(a) and 6(b) show the simulated absorption spectra versus the Fermi level when the incident polarization angle is 0° and 90°, respectively. We can find that the peaks marked as the triangles take a redshift. For the 0° polarization, the peak shifts from 5.6 μm to 6.0 μm; while for the 90° polarization, the peak shifts from 5.8 μm to 6.0 μm. Figures 6(c) and 6(d) are the extinction spectra versus the bias voltage measured by Fourier transform infrared spectroscopy. The extinction spectra are extracted as 1-R/R_0V, where R_0V is the reflectance under zero bias voltage. Similar to the simulation, the redshift phenomenon is observed as the gate voltage is enlarged to increase the Fermi level. For the 0° polarization, the absorption peak moves from 5.5 μm to 5.95 μm. For the 90° polarization, the absorption peak moves from 5.4 μm to 5.72 μm. Generally, the modulation of the Fermi level is limited in the experiment, making it difficult to tune the Fermi level from 0.1 eV to 1.2 eV. Hence, the peaks only show the redshift trend because the Fermi level is not high enough.

 

Fig. 6. (a)–(b) Absorption spectra of the dolmen structure at different Fermi levels when the incident polarization is (a) 0° and (b) 90°, respectively. The mobility of graphene is 100 cm²/(V*s). (c) and (d) Measured extinction spectra of the graphene dolmen structures at different gate voltages measured by FTIR when the incident polarization angle is (c) 0° and (d) 90°.

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At last, although graphene with high quality is expected to pave a new way to optoelectronic applications, the electron mobility usually decays seriously due to the defects and impurities after the patterning and transferring processes of graphene. Different from the performance of high-quality graphene, the graphene plasmon with low mobility would show a broadband property. We find the anomalous redshift of graphene absorption, which can be explained by the competition between the plasmon mode and the cavity mode.

3. Conclusions

We have systematically studied the influence of electron mobility on the interaction between graphene plasmon resonance and Fabry–Pérot cavity, and have discovered the redshift of absorption peaks. The competition between the plasmon mode and cavity mode has been explained to cause the anomalous shift. As the Fermi level increases, the absorption peak would move from the dispersion curve of cavity mode to the ordinary plasmon mode. The longer the distance between the two modes, the more obvious the redshift would be. Finally, experiments have been carried out to verify the redshift phenomenon by patterning graphene into dolmen nanostructures. These results provide a new insight to understand the absorption behavior of graphene at low mobility and could benefit the design of graphene-based active devices.