1. Introduction

Among many nanomaterials, two-dimensional (2D) materials have emerged as promising candidates for mechanical and optoelectronic applications [1–3]. Especially, graphene with an atomically thin structure (atom layers distance of $\sim$0.335 nm) exhibits a higher strength than steel (specific surface area of 2500 m$^{2}$/g), a stronger conductivity than copper [4–6], remarkable mechanical and photoelectric properties (young’s modulus of up to $\sim$1 TPa and charge-carrier mobility up to 200000 cm$^{2}$v$^{-1}$s$^{-1}$) [7–11], a special experimental ductility (20%) [12,13], and a fatigue life of more than $10^9$ cycles, higher than any material reported so far [14], thus making it an ideal candidate for micro- and nanoelectromechanical systems (MEMS and NEMS) [15–18]. In addition, the suspended graphene material eliminates the interaction with the substrate, and give them the freedom of movement, which makes it has been widely used in applications of MEMS devices [12,19–22], such as mechanical resonators [23,24], high performance sensors [25–30], electronic switches [31,32], microphones [33], and high resolution displays [34–36]. In particular, remarkable progress has been made in the application of graphene MEMS in high resolution displays.

In 2016, Dejan Davidovikj $et$ $al.$ visualized the motion of micrometer scale graphene nanodrums at ultrahigh frequencies with spatial resolution using a phase-sensitive interferometer [37]. Meanwhile, Samer Houri $et$ $al.$ found that the double-layer graphene (DLG) membranes recreated the interference effect of interferometer modulation displays [35]. In 2018, they further reported a Graphene Interferometric Modulator Display (GIMD) prototype with 5 $\mu$m-in-diameter pixels composed a high-resolution image (2500 pixels per inch), whose color can be changed at frequencies up to 400 Hz [34]. The results showed that the color reproducibility and speed of graphene membranes under a certain electro-optic modulation makes them suitable as pixels for high refresh rate displays. In recent years, display devices are developing diversely. With the increasing popularity of Virtual Reality, ultra-thin, fast-speed and high-resolution display technology is becoming the mainstream of current development [38–40]. Obviously, the displays based on graphene mechanical pixels have above advantages and are more durable, energy efficient, flexible and easy to control than LED screens [34–36,41]. However, previous studies on graphene MEMS display have obtained a limited color gamut, which greatly limits the high-resolution display effect.

In this work, we studied the dynamic color modulation in the composite structure of graphene MEMS-photonic crystal microcavity. First of all, we designed a symmetric photonic crystal microcavity with three resonant standing wave modes corresponding to the three primary colors of red (R), green (G) and blue (B). The localized light fields are formed in three modes at different positions of the microcavity. Then, when graphene MEMS is coupled with photonic crystals, we found that graphene has a strong absorption when it is located in the antinode of the standing wave, and therefore the transmittance of the structure is lower, whereas graphene has the opposite property when it is located in the node of the standing wave. In fact, the physical mechanism of our design is consistent with the Ref. [42], except that there is only one mode in the Ref. [42], whereas in our design there are three modes. Therefore, the graphene absorption of different colors of light can be regulated dynamically by applying voltages to tune the equilibrium position of the graphene MEMS in the microcavity, thus realizing the output of vivid monochromatic light or multiple mixed colors of light within a single pixel. Traditionally, a single pixel of a Liquid Crystal Display (LCD) screen is actually composed of RGB sub-pixels, the superposition of which comprise what is known as a single pixel. It means that an LCD screen with a horizontal resolution of $N$ whole pixels is actually composed of a linear array of 3$N$ monochromatic sub-pixels, setting obstacles to manufacture higher-resolution displays. Thus, multiple mixed colors of light within a single pixel can greatly improve the resolution and reduce the power consumption of the display. The single pixel design has been experimentally demonstrated in Ref. [34], but our design further extends the modulation gamut.

2. Theoretical model and method

The schematic diagram of photonic crystal microcavity is shown in Fig. 1. Since the slope of parabolic profile is very small, the structure can be regarded as a quasi-one-dimensional structure. The cavity is formed by two Bragg mirrors and an air gap. The distributed Bragg mirrors consists of the ZnS layer and the ${\textrm{MgF}_{2}}$ layer represented by green and blue, respectively. These two materials are alternately distributed on both sides of the microcavity with a period $N$ = 8. The refractive indices of the two medium layer in the calculation are taken as $n_1$ = 2.5 and $n_2$ = 1.38, respectively. All layers are non-magnetic ($\mu =1$). The thicknesses of the corresponding layers are $d_1$ = $\lambda _0/4n_1$ and $d_2$ = $\lambda _0/4n_2$, where $\lambda _0$ = 530 nm. The fabrication of similar air gap photonic crystals has been quite mature [43–46], which provides a reasonable guarantee for the experimental implementation of our scheme. The light is bounced back and forth between the Bragg mirrors repeatedly in this microcavity, forming a standing wave distribution and increasing the transmission. The resonant state of the microcavity is sensitive to the thickness of the intermediate air gap layer, and the resonant wavelength $\lambda _c$ satisfies the following condition $m_i\lambda _c/2\propto L_c$, where $L_c$ represents the optical path of the microcavity, and $m_i$ is a positive integer [47]. Multiple resonance modes appear in the visible band once the cavity is long. The distances between the graphene pixel and the Bragg mirrors are $D_1$ and $D_2$, respectively, as shown in Fig. 1.

 

Fig. 1. Structure of suspended graphene/optical microcavity ($D_1$ = 70 nm and $D_2$ = 1010 nm); (a) A perspective of the designed structure; (b) No voltage;(c) The membrane can be actuated electrostatically($V_{ext}>0$ V). The electrodes are connected to graphene and the conductive layer in the Bragg mirror below. (The orange columns are insulated supports, and the inset shows the structure of graphene).

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The deflection of graphene pixels can be used as a function of voltage, and the deflection can be used to correlate the electromechanical and optomechanical properties of the pixels [34]. It can be seen from Ref. [34] that the deflection position of graphene can be precisely controlled as long as the voltage is precisely controlled, and they have achieved this in the experiment. We describe the pixels’ deflection by an axisymmetric parabolic profile given by $\delta (r)=\delta _c[1-(r/a)^2]$ [34,48], where $\delta _c$ is the center deflection (equilibrium position), $r$ is the radial distance away from the pixel’s center, and $a$ is the radius of the pixel [34], the value of $a$ is 5 $\mu$m in this work. When the pixel radius $a$ is much larger than the wavelength of light, the transmission response is little affected by $a$. The refractive index of monolayer graphene can be expressed as $n_g=3.0+i(C_1/3)\lambda$, where $C_1=5.446$ $\mu m^{-1}$, $\lambda$ is the wavelength, and the thickness of the monolayer graphene is 0.34 nm [47,49]. Our simulations dependent on the absorption properties of graphene. The absorption of graphene is mainly related to the imaginary part of the refractive index, but not much related to the real part.

The graphene is approximately parallel to the Bragg mirror, and the size of the graphene is much larger than the distance between the upper and lower Bragg mirrors. When a voltage is applied between the graphene and the conductive layer, the graphene and the conductive layer attract each other, causing them to experience a downward force. Thus, the graphene deflects, but only slightly. The slope of the parabolic profile is directly proportional to $\delta _c/a$, and has little relationship with $a$. In this work, the maximum value of $\delta _c/a$ is approximately 0.06, so this approximation is reasonable. Thus, the graphene can be divided into many small regions, each of which is approximately parallel to the layers, and the average transmittance of the structure can be calculated using the transfer matrix method, which is reliable and is often used in wedge-shaped cavity [47,50], for a detailed derivation see the Appendix A.

Assuming standard illumination (CIE) D65 normally illuminate (white daylight) the proposed structure from the top, the generated colors can be characterized by the CIE XYZ color space with the tri-stimulus values given as [51–54].

The chromaticity coordinates can be computed from the tri-stimulus values by [53].

Each point (x,y) on the diagram represents a color described by its tristimulus value (X,Y,Z). In the CIE1931 chromaticity space, it is convenient to visualize the color trends [55,56].

For sRGB the relationship between XYZ and linear RGB is given by the following equation [53],

3. Numerical result analysis

3.1 Optical response of the microcavity

Light can be coupled into the optical microcavity from the normal direction. When the microcavity length is set as shown in Fig. 1(a), there are three resonance modes in the microcavity, the resonance peaks of which appear at the wavelengths of 473.1 nm, 540.5 nm and 630.8 nm, respectively. The transmission spectrum of the microcavity is shown in Fig. 2(a). The localized light fields are formed in three modes at different positions of the microcavity, so there are three resonant transmission peaks in the spectral region, and the transimittance of which are 99.9%, 99.1% and 99.8%, respectively. It can be seen that the transmission spectrum shows the R, G and B modes necessary to create white. The generated resonances are narrow with a full width at half-maximum ranging between 0.8 and 2.1 nm. Thus, the quality factors Q of these three transmission peaks are 394.3, 675.6 and 300.4, respectively. The higher Q values ensure the generation of a color with higher saturation. The electric field distribution diagram as shown in Fig. 2(b) is enhanced and and its electric energy ratio $|E/E_0|$ is greater than 7 in the three modes, here $E$ and $E_0$ represent the amplitude of local electric field and the amplitude of incident photoelectric field respectively. In calculation, we usually set the value of $E_0$ as 1. The reason is that the incident light interferes with the reflected light of the Bragg mirror generate three groups of light standing waves. As can be seen from the insets in Fig. 2(a), when the resonant wavelength is 473.1 nm, 540.5 nm and 630.8 nm, the electric field at the resonant wavelength of the three modes is effectively enhanced to produce a standing wave pattern.

 

Fig. 2. (a)Transmittance of the structure (The insets show the distribution of electric fields in the three modes, and the black dotted areas in the insets is air cavity). (b)Electric field intensity distribution at the resonant wavelength of the three modes (dashed lines 1-4 are the corresponding positions of 0, 70, 170 and 320 nm in the microcavity in Fig. 3).

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3.2 Color modulation using graphene with different deflections

According to the three modes of R, G and B presented by the photonic crystal microcavity transmission spectrum, if the relative intensities of the three modes are adjusted according to the human eye’s perception for these three colors, the colors of CIE gamut can be continuously covered [51]. When we coupled a 10 $\mu$m diameter graphene mechanical pixel into the microcavity shown in Fig. 1, the microcavity greatly enhanced the interaction between light and graphene, where the graphene mechanical pixel is made up of five layers of graphene stacked on top of each other. Meanwhile, the intensity of the transmission spectrum of the three modes in the microcavity can be governed by graphene deflection. The mechanism is the modulation of light absorption caused by graphene mechanical deflection. In other words, when graphene is located in the antinode of the standing wave (with a strong light field), it exhibits a stronger absorption and therefore the transmittance of the structure is lower. On the contrary, when graphene is located in the node of the standing wave (with a weak light field), it has a weaker absorption and therefore a higher transmittance of the structure. In this way, when graphene located between the standing wave node and the wave antinode, the transmitted light intensity follows the change of graphene’s equilibrium position in the cavity. Due to the different field distribution of the three resonance modes, the absorption of light with different colors can be adjusted by tuning the equilibrium position of graphene in the microcavity, allowing the single red, green and blue color or their mixed colors to pass through. Additionally, the equilibrium position of graphene membranes can be controlled by voltage [34], that is, the modulation of both the color and the intensity of the three-mode transmission can be tuned dynamically by voltage. The relationship between the equilibrium position of graphene and the voltage can be described by the non-dimensional formula, $\bar {\delta }+\bar {k_3}\bar {\delta }^3+\beta \frac {\bar {\delta }/2}{1-\bar {\delta }/2}=-\frac {\bar {V}^2}{2}(\frac {\bar {\delta }-(\bar {\delta }-1)\ln ({1-\bar {\delta }})}{(\bar {\delta }-1)\bar {\delta }^2})$, where the non-dimensional parameters are given as $\bar {k_3}=k_3D_2^2/k_1$, $\beta =AP_0/k_1D_2$, $\bar {V}^2=\epsilon _0\pi a^2V^2/k_1D_2^2$ and $\bar {\delta }=\delta _c/D_2$. $k_1$ is the linear elastic force, $k_3$ the cubic elastic force, $D_2$ is the distance between graphene and the lower surface of the microcavity, $P_0$ is the initial (ambient) pressure, $A$ is the area of the circular membrane. Typical values of $\bar {k_3}$ range between $10^{-1}-10^4$, of $\beta$ between $1-100$, and $\bar {V}$ between $(10^{-3}-1)V^2$ [34]. The constraint relationship between deflection and voltage is influenced by the distance between graphene and the lower surface of the microcavity. When $D_2$ is set as shown in Fig. 1, the relationship between the equilibrium position of graphene and the applied voltage is shown in the inset of Fig. 3(a).

 

Fig. 3. (a)-(d) Transmission spectra of the system when the equilibrium position of graphene is 0, 70, 170, 320 nm, respectively. (e) The color coordinates of 0, 70, 170, 320 nm and the corresponding positions in 1931 CIE chromaticity diagram. The white dashed lines with the arrow show the color trend as the deflection of graphene increases. (f) Simulation of the averaged transmission as a function of the equilibrium position of graphene. The inset in (a) is the relationship between the equilibrium position of graphene and applied voltage, and the inset in (d) is the simulation of the averaged transmission as a function of applied voltage.

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As a typical example, four extraction equilibrium positions of graphene are: 0 nm, 70 nm, 170 nm and 320 nm, which are used as the research objects. The corresponding dashed line positions 1-4 in the microcavity are present in Fig. 2(b), and the corresponding transmission spectra are shown in Fig. 3(a)-(d), respectively. When the equilibrium position of graphene is 0 nm, graphene is located at the wave node of B mode, and it displays a lower absorption to B mode and a weaker influence on the electric field distribution of B mode, while a higher absorption to G and R mode. Therefore, the graphene membrane allows B mode to pass through, where the transmission peak of B mode is the highest and the interference color appears in blue, as shown in Fig. 3(a). Similarly, when the equilibrium position is 70 nm (170 nm), graphene located at the wave node of G (R) mode exhibits a lower absorption to G (R) mode and a weaker influence on the electric field distribution of G (R) mode, the interference color is green (red), as shown in Fig. 3(b)-(c). When the transmittance of one mode reaches the maximum, the other two modes are lower and close to zero, and the color formed is of a non-monochromatic light, but the color at these points reveals a higher saturation. As described in Fig. 3(e), the color coordinates of the above three cases are located in (0.236, 0.187), (0.268, 0.481) and (0.513, 0.290), respectively, which fully reflects the wide gamut of color modulation. When the equilibrium position of graphene is 320 nm (this position just completes one primary color and white modulation cycle, requiring approximately 24.5 V of modulation voltage), the graphene membrane is located near the wave antinode of the three standing waves. Affected by the strong light fields of the three modes, the absorption of graphene to the three modes is high, thus the transmittance of the three modes is low, as shown in Fig. 3(d). Its interference color is close to white as depicted in the CIE chromaticity diagram, and its color coordinate is (0.268, 0.337), marked in Fig. 3(e). As shown in Fig. 3(f), the average transmittance of the microcavity changes with the equilibrium position of graphene, indicated that the graphene membranes with different deflections affect directly the light field distribution of the whole device when the three standing waves are propagated in the microcavity. The variation of average transmittance of microcavity with applied voltage and wavelength is shown in the inset in Fig. 3(d). In fact, there is more than one place where graphene can achieve the above effect. Through analysis, we can also find other placement positions that achieve the similar conclusions, such as the position close to $z$ = 1240 nm in the microcavity (see the Appendix B), which provides a basis for the flexibility of our structural design.

Furthermore, the electro-optical response of the proposed structure, when averaged over the graphene membranes, is radius independent. When $a$ changes, the height distribution curve of graphene changes, but the proportion of each height area to the total area remains unchanged, so the electro-optic response is little affected by $a$. Therefore, changing the radius of the suspended graphene does not result in any qualitative change in the behavior of the pixels [34]. This phenomenon lays the foundation for graphene’s mechanical pixels to form high-resolution displays. The RGB responses of graphene pixels are shown in Fig. 4. The highlighted pixels in the CIE1931 chromaticity space are marked with black dots showing the gamut pixel trajectory in the chromaticity diagram. The simulation results show that with the increase of graphene layers, the modulation amplitude is larger and the color gamut is wider. Here, for few layers, the absorption of graphene in the visible range increases linearly with the number of layers, with the same refractive index [57]. In order to obtain more abundant colorimetric responses, a thicker membrane is required, while adding layers does not change the setup of the experiment. Although the light absorption of both the antinode and the node is enhanced with the increase of graphene layers, this enhancement is the enhancement of relative amplitude, and the difference of light absorption amplitude between the antinode and the node will increase with the increase of graphene layers. The colorimetric response of five-layer graphene is better, with a maximum deflection of about 400 nm. Because this research mainly discusses the influence of ultra-thin graphene on interferometric modulator display technology, only the influence of five-layer of graphene on the structure is discussed. If more graphene layers are added, the colorimetric response should improve. However, this response is no longer captured by the analytical model described above, it may also require a larger voltage difference, and may distort the standing wave pattern because of the Fbray-Perot effect. In conclusion, by analyzing the RGB channels of different layers of graphene mechanical pixels, we obtained the average color range of 2-5 layers of graphene mechanical pixels and their variation with applied voltage. The white dashed line with arrows shows the color trend as the increase of the equilibrium position of graphene deflection, indicating that electro-optical modulation among the three primary colors can be achieved using a single pixel. In terms of the color gamut coverage, the calculated color gamut area is around 43% of the standard RGB (sRGB) and 32% of the Adobe RGB, from Fig. 4(d). Due to the consistency of the proposed graphene MEMS with that in Ref. [34], the modulated color can be changed at frame rate at least 400 Hz and even up to kHz. In addition, if the superlubricated micromechanical system of graphene is introduced, the modulation speed can be greatly increased and the modulation voltage can be further reduced [58–60].

 

Fig. 4. Transmission colors with the number of graphene layers (the white dashed lines with the arrow represent the evolution trend for the colors as the deflection of graphene increases), where (a), (b), (c) and (d) correspond to two-layer, three-layer, four-layer and five-layer of graphene, respectively. The black line (red line) in (d) is the Adobe RGB (sRGB) color gamut.

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The system realizes the rich dynamic modulation of visible light color without introducing complex structure. As shown in Table 1, the equilibrium position of graphene gradually changes from 0 nm to 380 nm (step length interval is 20 nm), and CIE chroma coordinates (x, y) with respect to the visible light transmission spectrum and intuitive color blocks of simulation are given accordingly. As the equilibrium position of graphene gradually increases, the interference color of the transmission spectra revolve periodically, and the color displayed gradually changes from blue to green and finally to red, covering the whole visible light range. While color saturation gradually decreases due to the influence of parabolic profile of the graphene drums. Since the ideal white color coordinate of the standard illumination D65 is (0.313,0.329) [51], the chromaticity coordinates of the interference color of different equilibrium positions of graphene presents the phenomenon of taking points around the light source in the CIE chromaticity diagram. According to the calculation of color block changes, the voltage can be used to manipulate the deflection of graphene, thus verifying the rationality and effectiveness of vivid visible light color realization. It should be pointed out that changing the Fermi level of graphene in visible light by doping or voltage requires very high doping concentration or voltage, so the Fermi level of graphene will not change under general doping or voltage. It is worth noting that the color rendering system compared with the metarsurface plasma effect can also realize dynamic color adjustment for display devices [52,55]. Although dynamic structural colors based on phase change materials have shown remarkable properties such as high saturation and high efficiency [61–63], our design enables multiple mixed colors of light within a single pixel, thus greatly improving resolution.

Table 1. Different equilibrium positions of graphene correspond to transmission spectral color coordinates and interference color

View Table

Finally, although it is beyond the scope of this study, the feasibility of the experiment is also discussed. It is noted that the results are obtained through simulation, but all the parameters used in the simulation are based on the experimental results (i.e., the deflection of graphene). Especially, the configuration shown in Fig. 1 can be experimentally achieved following the previous preparation method. On the one hand, the dielectric DBRs were deposited by plasma-enhanced chemical vapor deposition (PECVD) on the epitaxial structure at low temperatures [43,45]. The use of DBRs in combination with a polymer sacrificial layer enables the achievement of the air-cavity [45,64]. On the other hand, multilayer graphene films grown by chemical vapor deposition (CVD) can be transferred into a DBR cavity using a semi-dry transfer technique [34,35,65]. It is noted that the growth and production cost of one-dimensional film is much lower, and the technology is relatively simple, which is conducive to the actual production. The application of two-dimensional materials with large loss and two-dimensional heterojunction can improve the absorption of the film, thus improving the color gamut of the device.

When the pixels are small, the resolution of the display depends not only on the size of the pixels, but on the spatial coherence of the backlight. For example, when one wants to show two white lines that are very close to each other, we can not see two white lines but the coherent stripes if the distance between the white lines is less than the coherent length of the light. Similarly, a single slit will also show a diffraction pattern, which means that it can not improve the resolution of displays when the size of pixels is smaller than the coherent length of light. It is known that the coherence length of most light sources is much larger than the wavelength of light (e.g. the coherence length of sunlight is as high as 50 microns) [66]. Although the size of our study is much larger than operation wavelength, it is still far smaller than the pixel size of existing display technologies on the market.

4. Conclusions

In conclusion, dynamic color modulation in the composite structure of graphene microelectromechanical systems (MEMS)-photonic crystal microcavity is investigated. The designed photonic crystal microcavity has three resonant standing wave modes corresponding to the wavelength of red, green and blue, and the light of these three modes forms the localized light field at different positions of the microcavity. Once graphene is added, it can govern the transmittance of three modes. When graphene is located in the antinode of the standing wave, which has strong light absorption and therefore the structure’s transmittance is low, or shows opposite properties when graphene is located in the node of the standing wave. Thus, we can tune the absorption of light of different colors by voltage regulating the equilibrium position of the graphene MEMS in the microcavity. As a result, we use a single pixel to achieve the output of monochromatic light or multiple mixed colors, greatly expanding the gamut. This method can achieve a high resolution, fast and dynamic color modulation by comparing the color rendering system with the metasurface plasma effect. Our findings reveal the physics behind dynamic color modulation with graphene and may provide guidance for the design and manufacture of high resolution, fast modulation and wide color gamut interferometric modulator displays.

Appendices

Appendix A: details of the transfer matrix method

The transfer-matrix method can be used in the calculation to analyze the Bragg mirror micro-cavities [47,50,67,68], and the TM (the magnetic field of light paralell to the interface) mode light beam is launched onto the photonic structure. As shown in the Fig. 1, the designed structure is divided into $1,2,3\cdots 2N+5$ layers from top to bottom with the angle of incidence is $\theta =0^{\circ }$, only considered in the case of normal incidence. Derived from Maxwell’s equations and boundary conditions, the reflection (incident) electric field amplitudes $A_{l-1}$($B_{l-1}$) and $A'_{l}$($B'_{l}$) on the up and down sides of the interface can be expressed as [67,68].

(4)$$\begin{pmatrix}B_{l-1}\\A_{l-1}\end{pmatrix}=\begin{pmatrix}\dfrac{1}{t_l} & \dfrac{r_l}{t_l}\\\dfrac{r_l}{t_l} & \dfrac{1}{t_l}\end{pmatrix}\begin{pmatrix}B'_{l}\\A'_{l}\end{pmatrix}=M_l\begin{pmatrix}B'_{l}\\A'_{l}\end{pmatrix}\\,$$

here, $t_l$ and $r_l$ are the transmission and reflection coefficients, which can be achieved from the Fresnel equations. $t_l=2n_{l-1}cos\theta _{l-1}/(n_{l-1}cos\theta _l+n_lcos\theta _{l-1})$, and $r_l=(n_{l-1}cos\theta _l-n_lcos\theta _{l-1})/(n_{l-1}cos\theta _l+n_lcos\theta _{l-1})$. $n_l$ is the refractive index of the material in the $l\textrm {-}th$ layer. If the incident light is the TE (the electric field of light paralell to the interface) mode light, the transmission and reflection coefficients are expressed as $t_l=2n_{l-1}cos\theta _{l-1}/(n_lcos\theta _l+n_{l-1}cos\theta _{l-1})$, and $r_l=(n_lcos\theta _l-n_{l-1}cos\theta _{l-1})/(n_lcos\theta _l+n_{l-1}cos\theta _{l-1})$.

The angle of light propagation in different layers is governed by Snell’s Law $n_{l-1}sin\theta _{l-1}=n_lsin\theta _l$. The electric field at two interfaces of the same layer with a thickness of $d_l$ satisfies the transfer equation

(5)$$\begin{pmatrix}B'_l\\A'_l\end{pmatrix}=\begin{pmatrix}e^{{-}2\pi d_ln_lcos\theta_l/\lambda} & 0\\0 & e^{2\pi d_ln_lcos\theta_l/\lambda}\end{pmatrix}\begin{pmatrix}B_{l}\\A_{l}\end{pmatrix}=P_l\begin{pmatrix}B_{l}\\A_{l}\end{pmatrix}\\,$$

where $n_0=n_{2N+5}=1$. The field amplitudes of incoming and outgoing light in the structure can be characterized by the transfer equation

(6)$$\begin{pmatrix}B_0\\A_0\end{pmatrix}=\begin{pmatrix}\prod_{l=1}^{2N+4}M_lP_l\end{pmatrix}M_{2N+5}\begin{pmatrix}B'_{2N+5}\\A'_{2N+5}\end{pmatrix}=\begin{pmatrix}Q_{11} & Q_{12}\\Q_{21} & Q_{22}\end{pmatrix}\begin{pmatrix}B'_{2N+5}\\A'_{2N+5}\end{pmatrix}\\.$$

Since there is no light reflected back after the light passes through the photonic crystal, the light in the $N+1$ layer has no light wave propagating along the negative direction of the $z$ axis, that is, $A'_{2N+5}=0$. The transmittance and reflectance of the photonic crystal are $R(\lambda )=\left |A_0/B_0\right |=\left |Q_{21}/Q_{11}\right |^2$ and $T(\lambda )=\left |B'_{2N+1}/B_0\right |=\left |1/Q_{11}\right |^2$, respectively.

Although graphene is not exactly parallel to the Bragg mirror, due to the small slope of the graphene working region, we can divide the graphene into many small pieces, each of which can be considered parallel to the Bragg mirror, and the transfer matrix can be used. We can calculate the transmittance of each of the pieces, and then sum them up to get the average transmittance.

 

Fig. 5. (a)-(b) Transmission spectra of the system when the equilibrium position of graphene is 0, 40, 140, 240 nm, respectively. The inset in Fig. 5(a) is the electric field intensity distribution at the resonant wavelength of the three modes (dashed lines 1-4 are the corresponding positions of 0, 40, 140 and 240 nm in the microcavity in Fig. 5(a)-(b)).The inset in Fig. 5(d) is the transmission colors with five-layer graphene.

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Appendix B: simulation of different graphene placement positions

All our simulations are performed using the transfer-matrix method. In the simulations, we consider that the distances between the graphene pixel and the Bragg mirrors are $D_1$ and $D_2$, where $D_1=636$ nm and $D_2=444$ nm, as shown the black dashed lines in the inset in Fig. 5(a).

A description similar to the text, we take the equilibrium positions of the four cases of graphene extracted are: 0 nm, 40 nm, 140 nm and 240 nm as the research objects. The corresponding dashed line positions 1-4 present in Fig. 5(a), and the corresponding transmission spectra are shown in Fig. 5(a)-(d), respectively.

The RGB responses of five-layer graphene pixels are shown in the inset in Fig. 5(d). This fully demonstrates the flexibility of our structural design.

Funding

National Natural Science Foundation of China (11764008, 11964007, 61865002); Science and Technology Talent Support Project of the Department of Education in Guizhou Province (KY[2018]045); Science and Technology Program of Guizhou Province ([2020]1Y026); Construction project of characteristic key laboratory in Guizhou Colleges and Universities (KY[2021]003).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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