1.

Introduction

Three items are needed to numerically optimize the design of an optical absorber such as a thin-film solar cell. The first item is a fast solver that can predict the performance of the device in a variety of configurations. The second item is an optimization code that can mitigate the effect of local minima without excessive computational time. The third item is a figure of merit that adequately captures the desired performance characteristics of the device so that a good design will emerge. Some of these choices are explored in this paper.

The rigorous coupled-wave approach (RCWA)^1,2 can be used to model the optical performance of thin-film optical absorbers, as has been shown for solar cells with metallic backreflectors that are periodically corrugated along one direction.^3–5 Indeed, the RCWA provides accurate results with high computational speed for boundary-value problems involving structures that are invariant only along, say, the $y$-axis and therefore are quasi-two-dimensional.^6,7 Furthermore, the RCWA can be coupled with the differential evolution algorithm (DEA)^8,9 for optimization.⁵ However, the computational requirements of RCWA increase significantly when the backreflector is periodically corrugated in two directions, i.e., the boundary-value problem is fully three-dimensional (3-D) in nature.^10,11 Additional design parameters enter the optimization process thereby to increase the computational burden further.

The dimensions of the unit cell of an optical absorber with a PCBR directly affect optical absorption.⁵ Optimization of thin-film solar cells with two-dimensionally corrugated backreflectors for maximum absorption has not been reported heretofore, to our knowledge. As a preliminary study showed that it is becoming a practicable proposition with commonly available computational resources, we decided to implement the RCWA + DEA approach to optimize a fully 3-D absorbing structure.¹² To demonstrate this approach, we report here the maximization of optical absorption in an idealized thin-film tandem solar cell fabricated over a periodically corrugated backreflector (PCBR) with hillock-shaped corrugations arranged on a hexagonal lattice. The active region of the chosen solar cell comprises three electrically isolated $p–i–n$ junctions. The semiconductor layers were taken to have the bandgap-dependent optical properties of amorphous silicon.¹³ Silver ¹⁴ is a good choice for the PCBR because its plasmonic nature can be harnessed to launch surface-plasmon-polariton (SPP) waves inside the device and thereby enhance the optical electric field and optical absorption.^15–17 With the foregoing choices, our results indicate that maximization of the optical short-circuit current density, the standard figure of merit,^18–20 does not result in a desirable design. Instead, we found that the maximum power density is a better figure of merit.

The plan of this paper is as follows: the optical boundary-value problem that is solved to determine the spectrally integrated number of absorbed photons per unit volume per unit time N_ph is presented in Sec. 2. The numerical techniques adopted for this work are presented in Sec. 3: the 3-D implementation of the RCWA is briefly described in Sec. 3.1, Sec. 3.2 discusses the diagonalization of a matrix that emerges in the RCWA implementation, and Sec. 3.3 briefly describes the DEA. Numerical results are provided in Sec. 4, Sec. 4.3 discusses the convergence of the numerical methods, whereas Sec. 4.4 briefly compares these numerical results to a thin-film tandem solar cell with a bisinusoidal PCBR. Closing remarks are presented in Sec. 5.

The free-space wavenumber, angular frequency, and intrinsic impedance of free space are denoted by $k_0=2\pi /{\lambda }_0$, $\omega =k_0{c}_0$, and ${\eta }_0=\sqrt{{\mu }_0/{ϵ}_0}$, respectively, where ${\lambda }_0$ is the free-space wave length, ${\mu }_0$ is the permeability of free space, ${ϵ}_0$ is the permittivity of free space, and $c_0=1/\sqrt{{ϵ}_0{\mu }_0}$ is the speed of light in free space, respectively. Vectors are underlined, column vectors and matrices associated with the RCWA are in boldface with breve notation, and the Cartesian unit vectors are identified as $\underset{\_}{\widehat{x}}$, $\underset{\_}{\widehat{y}}$, and $\underset{\_}{\widehat{z}}$. The imaginary unit is denoted by $i=\sqrt{-1}$.

2.

Model Boundary-Value Problem

We considered the boundary-value problem shown schematically in Fig. 1, which also defines the thicknesses $L_d$, $L_g$, and $L_m$. The device occupies the region

Fig. 1

(a) Schematic of the model boundary-value problem in the plane $y=0$. The $n$-type semiconductor layers are blue, the $p$-type semiconductor layers are red, and the $i$-type are gray. White regions are occupied by a material with real relative permittivity ${ϵ}_w({\lambda }_0)$. (b) Schematic of the grating region in the plane $z=z_g\in [L_d,L_d+L_g]$.

The region $0<z<L_d$ comprises an antireflection window and three $p–i–n$ junctions that are electrically isolated from each other by two windows, as shown in Fig. 1(a). The relative permittivity ${ϵ}_d(x,y,z,{\lambda }_0)$ of this multilayered material depends on ${\lambda }_0$. The layers are identified in the figure. All windows are made of a material of relative permittivity ${ϵ}_w({\lambda }_0)$. The ${\lambda }_0$-dependent relative permittivity of each semiconductor layer depends on the bandgap chosen for that layer.

The region $L_d+L_g<z<L_d+L_g+L_m$ is occupied by a metal with relative permittivity ${ϵ}_m({\lambda }_0)$. The region $L_d<z<L_d+L_g$, henceforth termed the grating region, contains a periodically undulating surface with period $L_x$ along the $x$-axis and period $L_y$ along the $y$-axis. The unit cell in the $xy$ plane was chosen to form a two-dimensional (2-D) rectangular lattice that is equivalent to a hexagonal lattice. If the side of the regular hexagons in this lattice is denoted by $L_h$, then $L_x=L_h$ and $L_y=\sqrt{3}{L}_h$ for the rectangular lattice.

The grating region is defined by hillocks arranged as in Fig. 1(b). Each hillock is a frustum of a sphere of radius $R_{sph}$. The base of the hillock is a circle of radius $R_g$, and the height of the hillock equals $L_g$, as shown in Fig. 1(a); accordingly

3.

Numerical Techniques Used

4.

Numerical Results and Discussion

In this paper, we have chosen to maximize optical absorption. The figures of merit defined later in this section take into account all optical effects such as the excitation of SPP waves and waveguide modes.²⁶ This allows a tradeoff between the various optical phenomena without prejudicing one mechanism over another. However, the parameter space is chosen so that the excitation of SPP waves and waveguide modes can be supported.

For all numerical results in this paper, the window layers were chosen to be made of aluminum-doped zinc oxide (AZO). For the two windows between $p–i–n$ junctions, the thicknesses were fixed so that $d_{2d}=d_{1d}=20\mathrm{nm}$. The relative permittivity ${ϵ}_w({\lambda }_0)$ of AZO was taken from a standard source.²⁷ The minimum thickness of the PCBR was fixed at $L_m=150\mathrm{nm}$. The metal was chosen to be silver, whose relative permittivity ${ϵ}_m$ also depends on ${\lambda }_0$.¹⁴

The bandgaps ${\mathsf{E}}_{\ell i}$, $\ell \in \{\mathrm{1,2},3\}$, of the $i$-layers in the triple-junction solar cell were kept variable in the range [1.3, 1.95] eV, but their thicknesses $d_{1i}=d_{2i}=d_{3i}=200\mathrm{nm}$ were kept fixed. The thicknesses of all three $n$-layers and all three $p$-layers were also kept fixed $d_{\ell n}=d_{\ell p}=20\mathrm{nm}$. The bandgaps of all three $n$-layers were fixed as ${\mathsf{E}}_{\ell n}=1.8\mathrm{eV}$, $\ell \in \{\mathrm{1,2},3\}$. The bandgaps of the $p$-layers were fixed as follows: ${\mathsf{E}}_{1p}=1.8\mathrm{eV}$ and ${\mathsf{E}}_{2p}={\mathsf{E}}_{3p}=1.95\mathrm{eV}$.

By introducing C or Ge into the lattice, a material is formed, but the $i$-layers are still of the a-Si:H_GeC family. This process changes the bandgap, where the ${\lambda }_0$-dependent relative permittivity of the material is obtained by an analytical model.^13,17 The electrical properties of the material also change, and can be found by applying Vegard’s law to known values.²⁸ As the electrical properties have no effect on our optical model, they have no role in this study.

Furthermore, the lattice parameter $L_h\in [\mathrm{200,800}]\mathrm{nm}$, the antireflection-window’s thickness $d_w\in [\mathrm{10,130}]\mathrm{nm}$, the base radius $R_g\in [\mathrm{10,400}]\mathrm{nm}$, and the corrugation height $L_g\in [\mathrm{0,300}]\mathrm{nm}$ were allowed to vary. Thus, the dimension of the search space $\mathcal{S}$ was $\overline{N}=7$, and we sought an optimal design over a candidate space of $9\times {10}^{11}$ possible configurations. We used parameter values $C_R=0.7$, $\alpha =0.8$, and $N_P=70$ for optimization.

4.1.

Optimization for Optical Short-Circuit Current Density

The figure of merit $C$ for the DEA optimization was initially chosen to be standard figure of merit for optical modeling of solar cells:²⁹ the optical short-circuit current density

Eq. (16)

$$J_{SC}^{\mathrm{Opt}}=\frac{q_e}{L_x{L}_y}{\iiint }_{{\mathcal{R}}_{sc}}{N}_{ph}(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$

where ${\mathcal{R}}_{sc}$ is the portion of the reference unit cell $\mathcal{R}$ occupied by the nine semiconductor layers in the solar cell and $q_e=1.6\times {10}^{-19}\mathrm{C}$ is the elementary charge. This figure of merit will maximize the number of photons absorbed in the solar cell, but disregards all electrical properties. In this section, we proceed to show that this results in a poor design.

Plots of $J_{SC}^{\mathrm{Opt}}$ against the bandgaps ${\mathsf{E}}_{\ell i}$, $\ell \in \{\mathrm{1,2},3\}$, for points sampled by the optimization exercise are shown in Fig. 2. In this figure as well as in Fig. 3, the data points from DEA are projected onto the plane formed by the variable being investigated and the figure of merit. The bandgap of the topmost $i$-layer is the most influential parameter that controls $J_{SC}^{\mathrm{Opt}}$. While attempting to maximize $J_{SC}^{\mathrm{Opt}}$, the DEA minimized the bandgap in this layer, and we see that many parameter sets with ${\mathsf{E}}_{3i}^{\mathrm{opt}}=1.3\mathrm{eV}$ at the boundary of the constraint set were evaluated. As ${\mathsf{E}}_{3i}$ increases from 1.3 to 1.95 eV, $J_{SC}^{\mathrm{Opt}}$ decreases throughout most of this interval as shown in Fig. 2(a).

Fig. 2

$J_{SC}^{\mathrm{Opt}}$ in relation to (a) ${\mathrm{E}}_{3i}$, (b) ${\mathrm{E}}_{2i}$, and (c) ${\mathrm{E}}_{1i}$. Each marker $(·)$ represents a choice of parameters by DEA as the algorithm progresses. Larger values of $J_{SC}^{\mathrm{Opt}}$ are desirable.

Fig. 3

$P_{\sup }$ in relation to (a) $L_h$, (b) $d_w$, (c) $\zeta =R_g/L_h$, and (d) $L_g$. Each marker $(·)$ represents a choice of parameters by DEA as the algorithm progresses. The large marker indicates the maximum value of $P_{\sup }$ achieved.

Maximization of $J_{SC}^{\mathrm{Opt}}$ showed that minimizing ${\mathsf{E}}_{3i}$ was the most important factor compared with the remaining $\overline{N}-1$ variable parameters (i.e., ${\mathsf{E}}_{1i}$, ${\mathsf{E}}_{2i}$, $L_h$, $d_w$, $R_g$, and $L_g$). This phenomenon is clearly exemplified in Fig. 2(a), where, for any fixed value of ${\mathsf{E}}_{3i}$, the remaining $\overline{N}-1$ variable parameters contribute only to a $\pm 2\mathrm{mA}{\mathrm{cm}}^{-2}$ variation in $J_{SC}^{\mathrm{Opt}}$. Unfortunately, such a configuration is likely to be electrically inefficient: with a narrow bandgap, more charge carriers are excited, but the operating voltage of the solar cell will be reduced.³⁰ Contrast to Figs. 2(b) and 2(c) wherein the variations of $J_{SC}^{\mathrm{Opt}}$ with the bandgaps ${\mathsf{E}}_{2i}$ and ${\mathsf{E}}_{1i}$, respectively, are shown. We see that while $J_{SC}^{\mathrm{Opt}}$ is maximum when ${\mathsf{E}}_{1i}={\mathsf{E}}_{2i}=1.35\mathrm{e}\mathrm{V}$, there are values of $J_{SC}^{\mathrm{Opt}}$ ranging from 18 to $27\mathrm{mA}{\mathrm{cm}}^{-2}$.

Parenthetically, when the thicknesses of the $i$-layers were included as variables in an optimization exercise, the DEA simply focused on the maximization of those thicknesses. The resulting configuration would also have poor electrical performance. Although this type of solar cell would absorb more light, an excited charge carrier would have to travel further to reach an electrode, thereby increasing recombination and decreasing efficiency. We note that Fig. 2 demonstrates the choice of figure of merit is very important. We only included these results to contrast them with numerical results in Fig. 5, since all the chosen parameters in the optimization exercise should affect the solar-cell performance.

4.2.

Optimization for Maximum Power Density

To improve the optimal design of the chosen solar cell without including a full electrical model, we devised a figure of merit for the DEA that penalizes the effect of minimizing the bandgap of any of the $i$-layers. We defined the power density

Eq. (17)

$$P_{\sup }=\frac{1}{L_x{L}_y}\sum _{\ell =1}^3{\mathsf{E}}_{\ell i}{\iiint }_{{\mathcal{R}}_{\ell }}{N}_{ph}(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z,$$

as the figure of merit, with ${\mathcal{R}}_{\ell }\subset \mathcal{R}$ being the region occupied by the $\ell$’th $p–i–n$ junction. Let us note that $P_{\sup }$ is a theoretical upper bound on the maximum extractable power density of the solar cell. Furthermore, the summation over the index $\ell$ indicates that the tandem solar cell is configured in the multiterminal format. Note that this power density is to be computed solely from the absorption of photons and the material bandgap. This estimates the maximum electrical power density but does not involve any electrical modeling (e.g., recombination and mobility of electrons and holes). A useful extension of our approach would be to include an electrical model, but that extension lies beyond the scope of this paper.

Figures 3 and 4 show the results of the DEA optimization for the variable geometric and bandgap parameters of the solar cell—namely, the lattice parameter $L_h$, the antireflection-window thickness $d_w$, the duty cycle $\zeta =R_g/L_h$, the corrugation height $L_g$, and the bandgaps ${\mathsf{E}}_{1i}$, ${\mathsf{E}}_{2i}$, and ${\mathsf{E}}_{3i}$. All seven of these parameters influence the figure of merit $P_{\sup }$ defined by Eq. (17). Indeed, a steady increase in $P_{\sup }$ is seen in Fig. 3(a) on the interval $200\le L_h\le 600\mathrm{nm}$; a very sharp increase in $P_{\sup }$ is evident in Fig. 3(b) as $d_w$ increases from 10 to 80 nm; $P_{\sup }$ increases steeply in Fig. 3(c) as $\zeta$ approaches 0.35 with a drop off thereafter; and $P_{\sup }$ peaks in the neighborhood of $L_g=200\mathrm{nm}$ in Fig. 3(d). Also shown is the hillock base radius $R_g$ in Fig. 4(d). We note the contrast between the behavior seen in Figs. 2(a) and 4(c). In the latter optimization exercise, the effect of the other parameters amounts to a variation of $\pm 5\mathrm{mW}{\mathrm{cm}}^{-2}$.

Fig. 4

$P_{\sup }$ in relation to (a) ${\mathrm{E}}_{1i}$, (b) ${\mathrm{E}}_{2i}$, (c) ${\mathrm{E}}_{3i}$, and (d) $R_g$. Each marker $(·)$ represents a choice of parameters by DEA as the algorithm progresses. The large marker indicates the maximum value of $P_{\sup }$ achieved.

The optimal values found are as follows: ${\mathsf{E}}_{1i}=1.35\mathrm{eV}$, ${\mathsf{E}}_{2i}=1.95\mathrm{eV}$, ${\mathsf{E}}_{3i}=1.65\mathrm{eV}$, $L_h=642\mathrm{nm}$, $d_w=89\mathrm{nm}$, $\zeta =0.36$, and $L_g=231\mathrm{nm}$. These values yielded a maximum $P_{\sup }=43.66\mathrm{mW}{\mathrm{cm}}^{-2}$. These optimal parameters, found by maximizing $P_{\sup }$, contrast sharply with our findings in Sec. 4.1, wherein maximization of $J_{SC}^{\mathrm{Opt}}$ was dominated by the minimization of ${\mathsf{E}}_{3i}$ with the remaining $\overline{N}-1$ parameters having very little effect.

The optimization exercise yielded two distinct categories of unit cells with relatively high $P_{\sup }$ values. The first comprises configurations for which $L_h$ lies in the interval [200, 300] nm, and the second comprises configurations for which $L_h\in [\mathrm{550,650}]\mathrm{nm}$. For the first category, the optimal value of $L_g$ is $\sim 40\mathrm{nm}$ with a base radius $R_g$ of 50 nm. The second type of configuration has much deeper corrugations, with $P_{\sup }$ maximized when $L_g$ takes values near 200 nm and $R_g$ around 230 nm. This phenomenon is evidenced by two distinct peaks in Figs. 3(a)–3(d) and 4(d).

4.3.

Convergence of RCWA and DEA

To ensure convergence of the optical short-circuit density, a representative configuration for the unit cell was used to determine an appropriate choice of $N_t$ and $M_t$. We let $N_t$ vary in the set {2,3,4,5,6}, and defined $M_t=\lceil \sqrt{3}{N}_t\rceil$, where $\lceil ·\rceil$ is the ceiling function. After determining that $P_{\sup }$ changed by $\le 1\%$ for two successive values of $N_t$ and $M_t$, the number of Fourier modes was fixed for all numerical results reported in this paper, and taken to be $N_t=2$, $M_t=4$. Since Eq. (16) has the charge-carrier generation rate integrated over the nine semiconductor layers, the susceptibility of $J_{SC}^{\mathrm{Opt}}$ to the effect of Gibbs’ phenomena⁶ on the electric field in the region $z\in [0,L_d]$ is negligible. Hence, the electric field converges everywhere in the semiconductor layers, even for relatively small values of $N_t$ and $M_t$. We performed 50 DEA iterations in the optimization exercise. Figure 5 shows the convergence of $P_{\text{sup}}$ with the number of DEA iterations. We see that the best value of $P_{\sup }$ does not change after the first 32 DEA iterations. As with any stochastic optimization method, it is always possible that further iteration would result in improvement of the computed maximum power density.

Fig. 5

The best value of $P_{\sup }$ versus number of DEA iterations when optimizing for maximum power density as in Sec. 4.2.

5.

Conclusions

As a model problem to demonstrate the practicability of the RCWA + DEA approach to design efficient optically absorbing 3-D structures, optimal values of four geometric parameters and the bandgaps of three $i$-layers were found for an idealized, multiterminal, thin-film tandem solar cell comprising three $p–i–n$ junctions with a silver PCBR with hillock-shaped corrugations arranged on a hexagonal lattice. The figure of merit for the DEA was either (i) the optical short-circuit current density in Sec. 4.1 or (ii) the power density in Sec. 4.2. Thus, two different optimization exercises using the RCWA + DEA approach were performed.

As the optical short-circuit current density takes into account only the optical constitutive properties of the solar cell, maximization of that quantity resulted in a poor design. In particular, we determined that only the bandgap of the topmost $i$-layer (i.e., ${\mathsf{E}}_{3i}$) was significant to the optimization of the optical short-circuit current density, and the remaining six parameters played minor roles. Although photon absorption in the topmost $p–i–n$ junction was maximized thereby, this configuration would have poor electrical performance.

This is because, when the thicknesses of the $i$-layers were included in the optimization, the DEA simply focused on the maximization of those thicknesses. Although increasing those thicknesses can enhance light absorption, the electrical performance may be sacrificed, thereby reducing efficiency. The design of thin-film solar cells must balance optical and electrical performances.^31,32

To avoid configurations with potentially poor electrical performance, we next used the power density—thereby weighting the optical short-circuit current density—as a figure of merit. The resulting configuration was optimal with respect to all seven design parameters. Another optimization exercise was then performed on a similar tandem solar cell but with a bisinusoidally corrugated PCBR. In this case, the configurationally averaged power density tested by the DEA was about 10% higher than with the hexagonally corrugated PCBR. In the future, we plan to supplement the optical model by an electrical drift-diffusion model³² and then optimize the overall electrical performance of the solar cell.

In closing, let us emphasize that the triple-junction tandem solar cell was chosen as a model problem to show here the capabilities of the RCWA + DEA approach developed for 3-D optically absorbing structures. Our approach can be extended not only to other types of photovoltaic solar cells^33,34 but also to optical absorbers^35,36 with 3-D morphology.