It is important to increase light absorption and quantum efficiency in image sensor pixels, especially for wavelength ranges in which material absorption is weak. Surface textures, including nanostructure lattices, have been invented for significant improvement in light absorption. Those nanostructures typically support a number of physical processes for which the optimal geometries are different. We design a class of nanostructure superlattices to enable the co-optimization of different physical processes for further enhancement in light absorption.

## I. INTRODUCTION

In past few decades, the CMOS image sensor (CIS) design for the near infrared (NIR) range has attracted great attention among researchers. The majority of smartphones, laptops, digital cameras, and night vision goggles are now equipped with one or more crystalline silicon (c-Si) based CIS modules. However, due to its indirect bandgap, c-Si absorbs weakly in the NIR range. Therefore, for certain situations that require NIR illuminations, for example, distance measurement, three-dimensional reconstruction, and facial recognition, it is crucial to improve light absorption efficiency in CISs.^{1–3}

One way to enhance the NIR sensitivity of CISs is to use thicker light absorption layers for a longer effective optical path. Nevertheless, this results in the reduction of imaging quality, for example, because of more lateral color crosstalk, and the increase in manufacturing cost, for example, because of the need for high energy ion implanters.^{4} In view of these issues, a typical Si photo-absorption layer in back-illuminated CISs (BI-CISs) has an optimal thickness of 3 *µ*m.^{5} Another way is to apply needle-shaped black silicon to the surface, which, however, is not applicable to consumer-grade image sensors where the pixels are relatively small.^{6–8} It has also been shown that a BI-CIS architecture, with uniform inverted pyramid array (IPA) structures on the c-Si surface, could improve significantly in IR sensitivity, when compared to samples with a flat surface.^{4}

The working principle of light absorption for CIS pixels in Ref. 4 has been described as the easy fulfillment of the total reflection condition because of the large gap in the refractive index of silicon and dielectric material of IPAs. Nevertheless, efficient light absorption requires both effective antireflection and effective light trapping techniques, whose geometrical requirements are often different.^{9–13} To provide a graded refractive index layer, recent designs have looked at the possibility of using inverted pyramid array (IPA) structures^{14–16} or double-sided nanocone gratings,^{13} which are inspired by the anti-reflective qualities of moth eyes.^{17} These methods are effective because a smooth transition of the refractive index from one medium to another is provided, instead of having a rapid jump from the minimum value of the refractive index to the maximum. In this paper, we denote the wavelength of light by *λ* and denote the base size and the height of an inverted pyramid structure in Ref. 4 by *a* and *h*, respectively. The geometrical requirements demanded by Ref. 4 will be as follows: On the one hand, it requires *a* < *λ* for incoming light to see a sufficiently effective spatially averaged refractive index, i.e., to have only a small number of channels through which the radiation can leak.^{18} On the other hand, it requires *a* > *λ* to support a sufficiently large number of guided resonances for absorption to occur, since light trapping relies on the excitation of guided resonances, based on the statistical temporal coupled-mode theory.^{18} Therefore, there is a distinct mismatch between the two sets of geometrical requirements, and only one kind of IPA in each period can meet both of them at the same time. This suggests that a better choice is to use a superlattice that can contain more than one type of IPA in each period, thus jointly satisfying the two sets of requirements.

## II. DESIGN CONSIDERATIONS

We design a CIS structure with two types of IPAs in each of the two lateral directions in a period of the superlattice on the surface of the silicon substrate. The periodic structure contains the silicon portion, which is covered with a dielectric material, and the non-silicon portion (IPAs), which is filled with some specific dielectric material. A diagram of the CIS pixels using the proposed approach is shown in Fig. 1. In this paper, we only consider two adjacent IPA structures in each lateral direction in a period. Since instead of global optimization, we aim to show that the light absorption of CIS pixels could be significantly improved by employing more than one type of IPA in each period compared to uniform ones, results of two types of IPAs could already strongly support this claim. The increase in the number of types of IPAs employed in each period on the surface can be taken into consideration in further optimization work.

For comparison, we now start by considering three structures of the absorption layer in 2D in air, as shown in Fig. 2, wherein the non-silicon portions (if applicable) are filled with air as well. Structure A, shown in Fig. 2(a), is the reference structure, consisting of a uniform bulk silicon substrate in air without any arrays on the surface. Structure B, shown in Fig. 2(b), is the one proposed in Ref. 4, which has one kind of IPA in each period on the silicon surface, with a base of *a* and a height of *h*. Structure C, shown in Fig. 2(c), is the improved one put forward in this paper, which has two different types of IPAs in each period on the silicon surface. One of the two IPAs has a base denoted by *a*_{1} and a height denoted by *h*_{1}, while the other has a base denoted by *a*_{2} and a height denoted by *h*_{2}. Then, the lattice constant of structure C denoted by *L* is equal to the sum of *a*_{1} and *a*_{2}. We require *a*_{1} ≠ *a*_{2} or *h*_{1} ≠ *h*_{2} for structure C; otherwise, if *a*_{1} = *a*_{2} and *h*_{1} = *h*_{2}, then there is only one type of IPA in each period, which is the configuration for structure B. We denote the width and the equivalent thickness of the bulk silicon structure by *w* and *t*, respectively.

## III. RESULTS AND DISCUSSION

Next, we use the rigorous coupled wave analysis (RCWA)^{19} method to simulate the behavior of the three structures in Fig. 2 and calculate their absorption of light for wavelengths between 800 and 1000 nm. For all three structures, *w* = 6 *µ*m and *t* = 3 *µ*m because we assume each pixel to be a 6 × 6 *μ*m^{2} square and the photo-absorption layer to have a typical 3 *µ*m thick c-Si substrate. For structure B, to be consistent with the optimal IPA size stated in the conclusion of Ref. 4, h is equal to 400 nm and a is equal to 566 nm (=2*h*/tan 54.7°). As for structure C, we select some reasonable parameters without optimization. *L*, *a*_{1}, *a*_{2}, *h*_{1}, and *h*_{2} are chosen to be equal to 932, 445, 487, 330, and 296 nm, respectively. Since this set of parameters is not optimized, the performance of structure C could be further improved. Assuming normal incidence, the corresponding results are shown in Fig. 3, where the dashed line refers to structure A, the dotted line refers to structure B, and the solid line refers to structure C. It could be calculated from the plot that structure B achieves a significant improvement in averaged absorption compared to structure A, while structure C achieves more than 18% absorption improvement compared to structure B, from *λ* = 800 nm to *λ* = 1000 nm. This indicates that in 2D, absorption efficiency in c-Si could be significantly enhanced by applying two types of IPAs in each period to the surface compared to using uniform IPAs on the surface.

In Fig. 4, we demonstrate some heatmaps concerning the relation between configurations of *a*_{1}, *a*_{2}, *h*_{1}, and *h*_{2} for structure C and corresponding light absorption behaviors. Since light trapping depends on the excitation of guided resonances,^{18} it is essential to select a proper period. There are two principal factors to be considered. First, a larger period allows more guided resonances for light trapping. Second, however, a smaller period would reduce the number of coupled resonances or those leaking to external channels, i.e., the diffraction directions.^{20–22} Therefore, the optimal period is typically chosen close to the wavelength of interest. In this paper, we mainly focus on the NIR range from *λ* = 800 nm to *λ* = 1000 nm. We expect to see obvious differences in the comparison of the three structures. A lattice constant that is slightly smaller than the wavelength of interest could allow bulk structures to have best performance in absorption.^{21} Therefore, the lattice constant *L* in Fig. 4 is assumed to be fixed to 900 nm, which is approximately close to the optimal number. It is worth noting that *L* chosen for Fig. 4 is different from that for structure C in Fig. 3 because *L* in Fig. 4 is assigned without optimization work. The color in Fig. 4 represents the Figure of Merit (FOM) that is consistent with earlier work.^{4} The FOM is defined as an average of the normalized absorption enhancement for structure C from a wavelength of 800–1000 nm, when the absorption enhancement is defined as the absorption coefficient of structure C divided by the absorption coefficient of structure A. In order to control the variables, we assume *h*_{1}/*a*, *h*_{1}/*h*_{2}, and *a*_{1}/*a*_{2} to be fixed numbers and plot the relation of other parameters accordingly. In Fig. 4, we present some representative plots for the relations. In Fig. 4(a), the ratio of *h*_{1} to *L* is fixed to 1/3. In Fig. 4(b), the ratio of *h*_{1} to *h*_{2} is fixed to 1. In Fig. 4(c), the ratio of *a*_{1} to *a*_{2} is fixed to 3. Note in Fig. 4(b) that the FOM is relatively small when the ratio of *a*_{1} to *a*_{2} is equal to 1 (indicated by the solid vertical line). At this point, we have *a*_{1} = *a*_{2} and *h*_{1} = *h*_{2}, which means only one type of IPA is applied, and this is the configuration for structure B. In this way, the lattice constant *L* of the periodic structure is decreased from 1100 to 550 nm and the efficiency of light trapping is undermined. In Figs. 4(a) and 4(c), some similar features could also be found. For instance, the FOM is relatively small at *a*_{1}/*a*_{2} = 1 or *h*_{1}/*h*_{2} = 1. Moreover, it is shown from the plot that in most cases, as the difference between *a*_{1} and *a*_{2} or *h*_{1} and *h*_{2} increases, the FOM gradually increases and the light absorption is strengthened again. This indicates that it is easier to achieve efficient light trapping by taking advantage of the superlattice. To further understand the physics of it, we compare structure C (*a*_{1} ≠ *a*_{2} or *h*_{1} ≠ *h*_{2}) and structure B (*a*_{1} = *a*_{2} and *h*_{1} = *h*_{2}) with fixed *h*_{1}/*L*. We calculate the electric fields for structure A, point B (*a*_{1} = *a*_{2}, *h*_{1} = *h*_{2}, *h*_{1}/*L* = 1.4) in Fig. 4(b), and point C (*a*_{1}/*a*_{2} = 4, *h*_{1} = *h*_{2}, *h*_{1}/*L* = 1.4) in Fig. 4(b) under normal incidence at *λ* = 800 nm. Figures 5(a)–5(c), respectively, correspond to the electric field distribution in structure A, point B, and point C. As shown in Fig. 5, the reflection of light at point B is almost the same as in structure A, but the reflection of light at point C is much weaker than at point B. Additionally, it could be seen that some guided resonances are excited in the absorbed layer of the structure in Fig. 5(b), compared to structure A in Fig. 5(a) with mere Fabry–Pérot resonances. Compared to the structure at point B in Fig. 5(b), for the structure at point C in Fig. 5(c), there are more guided resonances excited in the absorbed layer. In view of the light trapping theory, the enhancement of light absorption depends on the number of optical resonances supported by the structure that could be excited.^{9} This may explain why the absorption is improved in the structure at point C compared to the structure at point B. This result shows that using more IPAs in a period to enlarge the lattice constant could enhance the light absorption. Therefore, designing a class of nanostructure superlattices to co-optimize different physical processes is an effective way to further enhance the light absorption for image sensor pixels.

Moreover, it could be seen from Fig. 6 that the performance enhancement of the superlattice is robust. Every point in Fig. 6 corresponds to the absorption enhancement of the structure with a superlattice in contrast to the structure with only one type of IPA. We use the CMA-ES method^{23} to optimize each of the two structures with the chosen equivalent thickness in the selected wavelength range and calculate the absorption enhancement at every point. In Fig. 6, the center wavelength varies from 500 to 1100 nm, while the equivalent thickness varies from 0.5 to 5 *µ*m. The datapoint with an equivalent thickness of *d* *μ*m and a center wavelength of *p* nm refer to the structures with the equivalent thickness of *d* *μ*m in the wavelength range from (*p* − 100) to (*p* + 100) nm. It is shown that in most areas of Fig. 6, the absorption enhancement is statistically significant.

Although we have only used 2D examples in air so far, the findings are also generally applicable in 3D. We conduct research on those three structures in air in 3D when *w* = 6 *µ*m and *t* = 3 *µ*m. Structure A is the flat c-Si substrate. For structure B, the base length *a* and the height *h* of the kind of IPA are equal to 566 and 400 nm, respectively. Similar to 2D, some proper parameters are selected without optimization for structure C. The configuration for structure C in 3D is set as *L* = 1100 nm, *a*_{1} = 977 nm, *a*_{2} = 123 nm, *h*_{1} = 1200 nm, and *h*_{2} = 400 nm. The shapes and positions of the IPAs in a superlattice in 3D could be arranged in more than one way with these parameters. Therefore, we only choose one of them as an example to discuss. It is assumed that there are four types of IPAs in structure C, each of which, respectively, has a base *a*_{1} × *a*_{1} and a height *h*_{1}, a base *a*_{1} × *a*_{2} and a height *h*_{2}, a base *a*_{2} × *a*_{2} and a height *h*_{1}, and a base *a*_{2} × *a*_{1} and a height *h*_{1}. It is also assumed that they are arranged in a clockwise direction to form a square superlattice with a (*a*_{1} + *a*_{2}) lattice constant. We simulate the three structures using the RCWA method^{19} and calculate their absorption of light in the wavelength range of 800–1000 nm. Assuming normal incidence, the corresponding results are shown in Fig. 7, where the dashed line refers to structure A, the dotted line refers to structure B, and the solid line refers to structure C. It could be calculated from the plot that from *λ* = 800 nm to *λ* = 1000 nm, structure B improved a lot in averaged light absorption compared to structure A, while structure C achieves a significant further improvement of 28.7% compared to structure B. This indicates that in 3D, absorption efficiency in c-Si could also be greatly enhanced by applying the superlattice to the surface compared to using uniform IPAs on the surface.

## IV. CONCLUSION

We employ wave physics and devise an improved structure that can significantly enhance the absorption of light in CISs. We show that a seemingly small modification with a superlattice leads to a significant enhancement in absorption. Such a superlattice leverages both antireflection and light trapping. As for prototyping, the IPA structures on the c-Si surface could be fabricated with the method of lithography and wet etching process. Subsequently, the passivation process would be done over the pyramid arrays.^{4,24} Furthermore, we could use more comprehensive photon management techniques to optimize the symmetry, shape, and Bravais lattice for further enhancement up to theoretical limits.^{15,25–29} We could also improve overall performance, including considerations on the reduction of crosstalk, on using structures such as deep trench isolation (DTI), and on the application of anti-reflection coatings on the interface between c-Si and dielectric superstates. The method is applicable to other wave absorbers, in addition to CISs, such as solar cells, and acoustic absorbers.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

^{4}: A free electromagnetic solver for layered periodic structures