## 2 Design of Unit Cell Structure  
The designed unit cell of the all-dielectric nanostructure consists of a silicon nanodisk and a silicon substrate separated by a Si\({}_{3}\)N\({}_{4}\) dielectric layer, as shown in Figure 1. \(D\) denotes the diameter of the nanodisk with height \(h\). \(H_{\text{d}}\) is the height of the Si\({}_{3}\)N\({}_{4}\) dielectric layer, and \(H_{\text{s}}\) is the height of the silicon substrate. \(p\) denotes the period of the unit cell. The central axis of the nanodisk coincides exactly with the central axis of the dielectric layer.  
We use the finite-difference time-domain (FDTD) solutions software to analyze the reflection characteristics of the nanostructure illuminated by a vertical incident light, whose electric field is polarized along the x direction, and set the working wavelength range as 400-700 nm. The boundary conditions in x and y directions are periodic, whereas the boundary conditions in the z directions are perfect matching layers (PML). According to previous studies [5], we chose \(h\) = 130 nm and \(H_{\text{d}}\) = 70 nm. We found the optimal height of the substrate \(H_{\text{s}}\) to be about 140 nm by running Parameter Sweep and observing the reflection spectra at different \(H_{\text{s}}\), which was also verified by the distance between the chromaticity coordinate of reflection spectra and that of the standard color.  
Simulation results are shown in Figure 2. When \(D\) is 190 nm and both of the period \(p\) in the x and y directions are 290 nm, the maximal reflection peak of the spectra is lying in the red band, as shown in Figure 2a. When \(D\) is 115 nm and the period \(p\) is 290 nm, the reflection spectra with maximal peak in green band can be obtained, as shown in Figure 2b. When \(D\) is 100 nm and the period \(p\) is 190 nm, the maximal reflection peak of spectra is in the blue band, as shown in Figure 2c.  
To understand the resonant modes inside the nanostructures, we take the structure with \(D\) = 190 nm and \(p\) = 290 nm as an example. The corresponding electric dipole (ED) and magnetic dipole (MD) resonances are shown in Figure 2d,e, respectively. It can be seen clearly from Figure 2 that magnetic resonance and electric resonance coexist, but the electric resonance shows higher intensity. The electric dipole and magnetic dipole resonances play an important role in increasing the absorption of light and enhancing the transmission efficiencies [21; 22; 23].  
Figure 1: Schematic diagram of the nanostructrue unit cell.  
Using above nanostructures, we can obtain a variety of reflection spectra corresponding to different colors by changing the diameter \(D\) or the period \(p\) of nanostructures. Assuming the obtained reflection spectra are \(R(\lambda)\), the \(X\), \(Y\), \(Z\) components can be obtained from the following formulae:  
\[\left\{\begin{array}{l}X=\int_{400}^{700}R(\lambda)\overline{x}d\lambda\\ Y=\int_{400}^{700}R(\lambda)\overline{y}d\lambda\\ Z=\int_{400}^{700}R(\lambda)\overline{z}d\lambda\end{array}\right., \tag{1}\]  
where \(\overline{x}\), \(\overline{y}\), \(\overline{z}\) are the matching functions. Then, we can obtain the chromaticity coordinates (x, y) by \(x=X/(X+Y+Z)\) and \(y=Y/(X+Y+Z)\). The corresponding chromaticity coordinates in the International Commission on Illumination's (CIE) 1931 chromaticity diagram based on the measured reflectance spectra from simulations are shown in Figure 2f. On the one hand, the resonances are not equally excited under the different parameters of the nanostructure. On the other hand, the absorption of silicon to blue light is more intense from the trend of the permittivity of silicon with wavelength. Hence, we can find that the blue is weaker than red and green from the chromaticity diagram. Even so, we think that rich colors can be obtained by improving the parameters of nanostructures, and this can be potentially used for colorful display.  
Figure 2: _Cont._