## Abstract

In spite of rapidly increasing demand and various applications of infrared (IR) detectors, their design process for the performance improvement has been mostly dependent on researchers’ intuition and knowledge. We present two-dimensional unit structure design of the absorbing layer in IR detectors. A systematic approach is introduced to enhance the absorbing efficiency of incident beam in the near-infrared wavelength range. We derived a layered structure composed of a silicon nitride (Si_{3}N_{4}) layer and an amorphous silicon (a-Si) one in turn by the so called topology optimization in association with the time variant finite element analysis (FEA). It is confirmed that thickness at each layer is in associated with the IR wavelength so that detail dimensions of each layer are inferred. A prototype of the layered structure was fabricated and its performance has been verified through experimental measurement.

© 2013 Optical Society of America

## 1. Introduction

IR detectors are widely used in various fields such as an early diagnosis of cancer through measuring temperature of human body, non-destructive inspections of a building or a structure, reliability estimation of electronic products, night vision devices, and so forth [1–3]. Therefore, the research demands for smaller and lower priced IR detectors are growing. In spite of the fact that an appropriate structure of the absorbing layer in IR detectors is essential to enhance the detecting performance [4], researches about detail design focused on the absorbing layer of IR detectors have not been so vigorous compared with new concept suggestions of whole IR detector structures or developments of IR detecting materials [5]. On the contrary, increasing attention has been given for thin film solar cell design to obtain high sunlight transmittance [6, 7] or to enhance the light trapping ability for enhancing the light path [8–10]. Moreover, in researches regarding thin film solar cells, various types of absorbing layers, that is, periodic wavelength-scale structures or nano-wires patterned into the active layer have been also suggested [11–14]. Layered texturing design for the absorbing layer in thin film silicon solar cells has been proposed using the topology optimization scheme by Soh et al [15].

As in researches of thin film solar cell design, the study to obtain high transmittance of an IR detector through the modification of absorbing layer design is necessary because it is directly related to improve the detector performance as can be confirmed in numerous studies on thin film solar cells. The selection of the appropriate material for the absorbing layer is critical; therefore, several materials such as HgCdTe and InGaAs have been taken into account to improve the efficiency [5]. In addition to appropriate selection of the material, optimal structural design of the absorbing layer must be also suggested to maximize the performance of an IR detector.

Topology optimization is generally regarded as one of the most flexible structural optimization methods that may allow changing topology as well as shape of the target structure and offer effective conceptual or detail design. It can propose structural configurations of the target structure and it has been applied to various problems. Since the first proposal by Bendsøe and Kikuchi proposed in the name of the homogenization design method (HDM) [16], various topology optimization methods have been proposed [17]. In the HDM, an infinite number of micro-structure in an element is assumed in associated with the homogenization theory and the material property of each element should be determined to satisfy the design objective in macro scale level and optimal material distribution can be obtained as a result. However, using the HDM includes quite cumbersome calculating procedures to get material properties from the composition of predetermined database of a micro-structure in spite of its strong theoretical background. In contrast to the HDM, the basic concept of the solid isotropic material with penalization (SIMP) method [17] is to represent the material property of an element as a simple function of its density which expresses a fictitious isotropic material property using the characteristic function and the penalization factor. Whatever method is used, topology optimization can be referred to as an effective way to get the optimal material distribution so that a conceptual structure is introduced.

Topology optimization has been extended to magnetic field problems [18] and it has been also applied to electromagnetic wave propagation problems based on not only just stationary problems but also dynamic field or time dependent field [19–22]. The dynamic field problem considering electromagnetic wave propagation problem was introduced by Jensen and Sigmund [19] and they proposed the topology optimization of photonic crystal structure with T-junction waveguide. Helmholtz’s equation is generally adopted to solve the steady state wave analysis and it has been expanded to the time transient wave field problems for the reflection wave to be taken into account [15].

In this study, we focus on optimal shape design of an absorbing layer of an IR detector at 1064nm wavelength which is in the near IR range. We introduce a systematic design approach and its result is verified through the experimental measurement using fabricated prototypes. The topology optimization scheme based on the SIMP method is applied to obtain the optimal design of an absorbing layer. We set the main objective of the design process as increasing the transmittance of the 1064nm wave passing through the absorbing layer. The design objective function is defined to maximize the Poynting vector in a prescribed specific measuring domain. Structural design of the IR wave absorbing layer, which is composed of a-Si and Si_{3}N_{4}, is proposed. Figure 1(a) shows the constitution of the absorbing layer of a detector and the schematic of the model for the analysis and design process is displayed in Fig. 1(b). The Si_{3}N_{4} layer is represented as a black part and the a-Si portion is expressed as a white part. The Helmholtz’s equation as a governing equation is solved using the commercial FEA package COMSOL^{TM} and the time dependent analysis mode is adopted for taking the time varying field into account. As a result of the topology optimization process, the optimal material distribution of Si_{3}N_{4} and a-Si can be obtained as a form of layout design. To confirm the optimal design structure, the fabrication of the absorbing layer prototype and its experimental verification are followed. We have measured the reflectance of the prototype using an integrating sphere as well as its transmittance using a power sensor [23, 24]. As a result, theoretical values and experimental values are compared [6] by calculating absorption via reflectance and transmittance values.

## 2. The absorbing layer modeling

Infrared detectors are generally used to detect and measure patterns of the thermal heat radiation emitted from any objects. Early thermal detector types such as thermocouples and bolometers, which rely on the temperature change, are still widely used. In general, thermal detectors are sensitive in overall IR wavelength range and work at the room temperature. However, they are comparatively slow in response time and have relatively low sensitivity; therefore, photovoltaic IR detectors using HgCdTe materials have been developed to overcome such defects [25]. Fabrication techniques combined with semiconductor are applied to make such type detectors and allowed the custom tailored specific detecting range using the band gap of the semiconductor.

This study focuses on absorbing layer design of the photovoltaic type IR detector. As can be confirmed in Fig. 1(a), the absorbing layer is located above of the semiconductor part in the detector. Incident radiation is passing through the absorbing layer and reaches the semiconductor part. In the analysis model represented in Fig. 1(b), the design domain is composed of a rectangular area with 600nm width and the height of 935nm taking the incident beam wavelength into account. The initial model has been set as a wedge shaped Si_{3}N_{4} layer as displayed in the figure. The periodic boundary condition is applied both along the left and the right boundaries to realize a periodic pattern derived from unit structure design. The absorbing layer is composed of Si_{3}N_{4} material and a-Si in the other parts. Si_{3}N_{4} has almost same optical property with zinc oxide (ZnO) which is widely used for the TCO layer in solar cells [26]. Because the resolution limit is generally larger than 20nm in case of evaporating the ZnO layer, Si_{3}N_{4} is replaced as a substitute material for the prototype fabrication for the purpose of experimental verification.

The time-varying electromagnetic wave propagation problems are solved by Maxwell’s equations for a two-dimensional (2D) wave propagation problem. Especially, assuming time-harmonic wave propagation and transverse magnetic (TM) polarization, only *z*-directional component of the field vector is needed for the analysis as follows:

*c*represents the speed of light.

_{0}Considering that IR radiation is irradiated vertically into the IR detector, a normal incident beam is assumed. For the load condition definition of such an incident beam phenomenon, following Helmholtz’s equation is derived from the Sommerfeld radiation condition assuming a time harmonic case [20]:

**H**

_{o}expresses the field strength of the incident beam and

*ω*is the frequency of the incident IR radiation. In this study,

*ω*becomes 281.95THz because the incident IR wavelength is set to 1064nm.

Figure 2 displays the contour of the wave propagation plot for the initial model shown in Fig. 1(b) and the time history of the **H _{z}** plot calculated at the measuring domain for the incident beam with 1064nm wavelength. For defining the design objective value, the energy flux expressed by the Poynting vector formulation is employed in this study. On account of

*y*-directional incident beam, the energy flux can be formulated as follows:

**H**represents the complex conjugate of the field vector

_{z}***H**.

_{z}Objective Poynting vector values in the measuring domain and along the incident boundary are calculated by following Eqs. (4) and (5), respectively.

where*Ω*represents the measuring domain and

_{obj}*A*means its area. Also,

*Γ*and

_{inc}*L*are the incident boundary and its length, respectively. They are designated in Fig. 1(b).

Usually, a time averaged form of the Poynting vector is employed in the steady state analysis for time harmonic wave fields. However, this study is based on the time dependent analysis so that the reflected beam effect from bottom layers would not be taken into account in the calculation of Poynting vector along the incident boundary. Poynting vectors must be calculated by the time integration scheme:

where*ψ*and

_{obj}*ψ*are time integration values of the Poynting vector at the measuring domain and the incident boundary, respectively. The time history is displayed in Fig. 2 and the time integration period is marked. The starting point of the time integration is selected as the second peak point of the history plot to avoid the confusion by mixing the incident wave with the reflected wave and also to reduce the total analysis time. The light transmittance which represents the efficiency is defined as the following equation:

_{inc}## 3. Topology optimization process

#### 3.1 Problem formulation

The topology optimization process is objected to obtain an optimal material distribution in a design domain so that the optimal design offers the best performance for a given physical problem. It is achieved through the iterative process by changing the design variable values by solving the problem and calculating sensitivity values repeatedly. As mentioned previously, this study intends to apply the topology optimization scheme based on the SIMP method to accomplish optimal design of the absorbing layer in a IR sensor. Figure 3 gives a brief account of the topology optimization concept. In the design domain, the material is distributed as a solid, void, and gray scale according to the density value of each element.

Since the design process can be classified into two phase material design, the solid region means the Si_{3}N_{4} and its density is defined as 1. On the other hand, void region represents the a-Si and its density is determined to 0 in this problem. The material property in the design domain is determined according to the density of each element. Since the design layout of absorbing layer composed of Si_{3}N_{4} and a-Si, the dielectric constant *ε _{r}*, i.e., the material property in two phase material case, can be written as

*γ*means the density of each element in the design domain and

*ε*is the relative permittivity.

_{r}*ε*and

^{’}*ε*represent the real and the imaginary part of the permittivity, respectively. The penalization parameter

^{”}*p*widely used in SIMP method to avoid gray scale element, is selected as 3 in this problem.

The design objective of this problem is to improve the transmittance of the 1064nm incident wave passing through the absorbing layer; therefore, the optimization problem is formulated using Eqs. (1) and (8) as follows:

#### 3.2 Time dependent field analysis and sensitivity calculation

The electromagnetic wave problem is analyzed through the governing equation defined in Eq. (1). This study requires time dependent analysis so that the vector field of next step ${\left({H}_{z}\right)}_{n+1}$is obtained using a central difference time explicit scheme.

In the gradient based optimization process, it is necessary to calculate the sensitivity of the objective function or the design constraints to update the design variable. The governing equation of the problem in this study is simply described as

where**K**means the coefficient matrix and

**φ**is the state variable and

**f**is the load vector. The state variable

**φ**represents the

*z*-directional magnetic field strength vector

**H**

_{z}in this study. The adjoint variable method is employed for the sensitivity calculation and the final formulation of sensitivity becomes

**is the adjoint variable and**

*λ**γ*is the design variable, i.e., the element density. $\stackrel{\u2322}{F}$ is the time integration form of the design objective function as

**is computed by solving the following adjoint equation.**

*λ*#### 3.3 Topology optimization result

The design domain is illustrated in Fig. 1(b) and it is located between the upper air layer and the measuring domain in the glass layer. The height and the width of design domain are 935nm and 600nm, respectively. Mapped meshing is employed to discretize the design domain with the element number of 300x450.

Optimization has been started from a wedge shaped initial design as a first trial because wedge structure is generally known as the good design for transmittance of the light in some waveband range [7, 9, 11]. The objective function value is computed by Eq. (8) during the optimization process. The convergence history of the objective function and shape change of the design domain at various iteration are displayed in Fig. 4(a). The optimal result is obtained with the shape of stacked layer as described at iteration 50 in Fig. 4(a). In the figure, black region represents the Si_{3}N_{4} while white region is the a-Si part in the patterned layer. It is remarkable that the wave transmittance of the optimal result is much better than the initial wedge shaped case. Figure 4(b) shows the convergence history of the design objective and the shape change from another initial shape where the design domain is filled with Si_{3}N_{4} material. The optimal shape defined at 50th iteration is almost same to that from the wedge initial shape. However, the efficiency of the full Si_{3}N_{4} model is better than the final optimal model as confirmed in the graph. No gray scale portions are occurred because this process is focused on a specific wavelength.

Taking that topology optimization can only offer conceptual results into account, it is necessary to determine the each layer thickness in detail. The thickness value of the most upper layer in the optimal result is set as 135nm which comes from *λ/4n* where *λ* is 1064nm of the incident beam wavelength and *n* is the refractive index of Si_{3}N_{4} material. The a-Si layer and the Si_{3}N_{4} layer are stacked sequentially with the thickness value of *λ/2n.* Therefore, thicknesses of the a-Si layer and the Si_{3}N_{4} layer become 130nm and 270nm, respectively, due to the refractive index value difference. It is verified that the structure composed of layers with *λ/2n* thickness gives good transmittance of the light while the first layer thickness value of *λ/4n* is effective as an anti-reflection coating for a single layer [27, 28]. Therefore, the final model is suggested to have *λ/4n* thickness at the top layer and *λ/2n* thickness in following stacked layers. Figure 5 display shapes of two initial models and final optimal model suggested. Efficiencies at each case are computed as 0.0988, 0.3686 and 0.4480 for wedge initial shape, full Si_{3}N_{4} initial shape and suggested optimal shape, respectively. For the suggested model, it gives improved transmittance efficiencies as 353.4% and 21.54% compared with the wedge shaped model and full Si_{3}N_{4} model, respectively. Figure 6 compares the wave propagation plot for each cases and strong wave plot can be confirmed in the absorbing layer for the optimal case.

## 4. Experiment

#### 4.1 Experiment set-up

In order to confirm the proposed result from the optimization process, a prototype had been fabricated and experiments for verification were performed. SEM images of two prototype models are displayed in Figs. 7(a) and 7(b). The Si_{3}N_{4} mono-layer model was fabricated by the plasma-enhanced chemical vapor deposition (PECVD) process while the multi-layer model stacked of Si_{3}N_{4} and a-Si layers was fabricated using the low pressure chemical vapor deposition (LPCVD) process [29, 30]. The multi-layered model stems from the suggested optimal result. SEM image of two models are displayed with detailed thickness of each layer as expressed in Figs. 5(b) and 5(c). We calculate the absorption (A) from measurement of the reflectance (R) and the transmittance (T) for those two prototype models as *A = 1-R-T*.

Among various devices for measuring the absolute absorptance of a laser component such as laser calorimetry, gonioreflectometer, integrating sphere or integrating mirror reflectometry [31, 32], we adopt the method using an integrating sphere for the purpose of minimizing the diffraction effect. Figure 8(a) shows the experiment set-up for reflectance measurement using an integrating sphere. In case of measuring the transmittance, the transmitted light power is directly measured by a power sensor as described in Fig. 8(b). Nd:YVO4 laser is employed to generate the infrared beam having 1064nm wavelength and Thorlabs PM310D power sensor is used in both cases. We also select Avantes AvaSphere-30-REFL as an integrating sphere. The transmittance and the reflectance of fabricated models are measured through two experiment processes and the resultant absorptance is estimated.

#### 4.2 Experiment results

Measuring results are summarized in Table 1 where for the Si_{3}N_{4} mono-layer model, the transmittance is measured as 0.751 and the reflectance is measured as the value of 0.157. The absorptance of full Si_{3}N_{4} model becomes 0.092 according to the relation of *A = 1-R-T*. Si_{3}N_{4}/a-Si multi-layered model, that is, the optimized model derived from the suggested process shows the value of 0.685 in transmittance and 0.201 in the reflectance; therefore, the absorptance becomes 0.114 as a result.

The enhancement ratio 23.91% from the measurement of fabricated models is comparable with the improvement rate of 21.54% obtained through the simulation process in spite of the difference in measuring process. In general, comparison results between the initial and the optimal model confirm that multi-layered structure design of an IR detector by the suggested process can guarantee a better performance.

## 5. Conclusion

This work suggests an absorbing layer design process for an IR detector of 1064nm near IR wavelength coupled with the numerical simulation for the time dependent wave analysis and the topology optimization design scheme. A specific multi-layered structure deposited by Si_{3}N_{4} layer and a-Si layer in turn has been obtained and each layer shows *λ/4n* thickness in the top layer and *λ/2n* in other layers. It turns out that those thicknesses are effective for anti-reflection and light transmittance, respectively.

Two prototype models, that is, Si_{3}N_{4} mono-layered one and Si_{3}N_{4}/a-Si multi-layered model, were fabricated and the absorptance is measured by measuring the reflectance and the transmittance. Improvement factors up to 23.91% (by experiment) and 21.54% (by simulation) show similar absorption response. The enhancement of the absorptance is pointing out that the suggested design process is valid in absorbing layer design of IR detectors.

We found that the suggested topology optimization process is not enough to determine detail thickness of each layer; therefore, further process or theoretical approach is required. Also, taking that the given result is optimized at specific 1064nm wavelength into account, further study is necessary to make the suggested method apply into broad wave-band design.

## Acknowledgment

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0017512).

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