# Mdpj _sciences_
## 2 Model Creation  
For ADI, it is assumed that the morphology of the top grating is symmetrical (as shown in Figure 1a), which consists of a uniform photoresist or anti-reflection coating, due to undergoing various planar process steps such as spin-coating and resist coating [24; 25]. In contrast, the asymmetric deformation of bottom grating frequently occurs [13]. The tilt angle is represented by the symbol \(\phi\), the grating groove depth is represented by the symbol \(h\), the symbol \(p\) refers to the pitch, and the duty cycle is defined as \(a/p\). The \(h\) value was chosen to be 80 nm [26]. The duty cycle was 0.5. The pitch was chosen as 6000 nm, 3000 nm, 1600 nm, 1200 nm, 1000 nm, 900 nm, 800 nm, 700 nm, and 600 nm, respectively. The wavelength ranged from 160 to 200 nm or 360 to 600 nm, depending on the pitch. The light was linearly polarized and the grating axis was aligned parallel to the laser polarization direction. The diffraction values of the light intensity of positive (\(I_{+\text{1 top}}\)) and negative (\(I_{-\text{1 top}}\)) diffraction order, generated by the top grating with symmetric morphology, were equal. The use of multi-wavelength measurements to measure overlay in advanced DBO can ensure that the interference between the top light and the bottom light (i.e., \(I_{\text{top}}\) and \(I_{\text{bottom}}\)) of the composite light intensity source is enhanced at a certain wavelength. The error signal only comes from the impact of the asymmetric bottom grating. Thus, we used an asymmetric grating as a model to investigate how to mitigate the impact of asymmetric deformation of the bottom grating (see Figure 1b) in the overlay metrology. The overlay error was set to 0. At normal incidence, the asymmetric grating shows asymmetric diffraction into the positive and negative diffraction orders. An error signal, caused by the asymmetric grating, is expressed by the difference in the diffraction efficiency of positive and negative diffracted light (\(\pm\)1 st through \(\pm\)4 th diffraction order). To compare, we also provided the error signal for the symmetric grating. Rigorous coupled-wave analysis (RCWA) is one of the widely used methods for the modeling of diffraction gratings [27]. The diffraction efficiency was analyzed using RCWA. In this study, GSolver (GSolver V5.2) was employed for the simulation [28]. One of the trends in overlay metrology is to use new materials for fabricating overlay targets, aiming to increase the signal-to-noise ratio (S/N) and sensitivity of overlay measurement [15]. In this simulation, aluminum (Al) was applied for Layer A, and silicon nitride (Si\({}_{3}\)N\({}_{4}\)) or silicon dioxide (SiO\({}_{2}\)) was applied for Layer B (see Figure 1), respectively. The index of refractive (n) and extinction coefficient (k) versus wavelength of Si\({}_{3}\)N\({}_{4}\) and SiO\({}_{2}\) were shown in Figure 2.  
Figure 1: Schematic diagram of the cross section of (**a**) an overlay target and (**b**) diffraction from an asymmetric bottom grating. \(\phi\): tilt angle, \(h\): grating groove depth, \(p\): pitch, and \(a/p\): duty cycle, normal incidence. Figure 2: The index of refractive (n) and extinction coefficient (k) versus wavelength of (**a**) Si\({}_{3}\)N\({}_{4}\) and (**b**) SiO\({}_{2}\). Note that the data are from GSolver V5.2 [28]. It was assumed that the morphology of the bottom grating was linearly tilted. The slope was set as 0.01 (or \(\phi\) was set as 0.57\({}^{\circ}\)). Layer A was divided into two parts--the tiled top part (see the red triangle in Figure 3a) and the bottom rectangle. The tiled top part was layered (with the number n-2 in Figure 3a). To obtain the optimal number of layers, taking the material system of SiO\({}_{2}\) for example, as shown in Figure 3b, we presented the relationship between the normalized ratio and the number of layers (i.e., n-2) in various diffraction orders (0 th through \(\pm\)4 th diffraction order).