1.

Introduction

Commercial silicon solar cells have a thickness of about 50 to 100 μm to absorb as much light as possible, especially in the near-infrared regime.¹ Excessive thickness of the solar cell does not only increase its manufacturing cost but also its carbon footprint during its manufacture. To reduce both, thin-film photovoltaic (PV) solar cells made of amorphous and polycrystalline silicon are being actively investigated.

The absorption efficiency of these thin-film solar cells can be improved by incorporating antireflection coatings^2–4 and texturing of the front surface.^5–7 The metallic back-reflector of a solar cell can also be textured at transverse length scales greatly larger than a thousand nanometers, but planar backing appears to perform better.⁸ In contrast, periodic texturing of the metallic back-reflector was indicated even in the early 1980s to help absorb light better, if the period were a few hundred nanometers.⁹ Hence, the use of a metallic surface-relief grating as the back-reflector may result in higher efficiency.^10,11

A metal/semiconductor interface can guide surface-plasmon-polariton (SPP) waves. This was appreciated by Anderson in the 1980s,^12,13 but the prism-coupled technique suggested to exploit the broad solar spectrum seems not to have gained traction in the research community. A few years ago,¹⁴ research interest returned when it was realized that an electric field of large magnitude, existing close to the back-reflector inside the semiconductor when an SPP wave is excited, is favorable to the generation of electron–hole pairs.

In single-junction amorphous-silicon (a-Si) solar cells backed by a periodically corrugated metallic back-reflector, the semiconductor is almost homogeneous. Therefore, only one SPP wave, that too of the $p$-polarization state, can be launched at a given free-space wavelength of light,^15,16 leading to modest gains in light absorption efficiency.^14,17 But multiple SPP waves of both linear polarization states can be guided by a periodicially corrugated metal/semiconductor interface, if the semiconductor is periodically nonhomogeneous along the direction normal to the mean plane of the metal/semiconductor interface; this results in enhanced absorption of light of both linear polarization states, according to a predecessor study.¹⁸ However, the semiconductor in that study¹⁸ was taken to have a continuously varying relative permittivity, which is not easily implementable.

Therefore, we set out to theoretically investigate the usefulness of exciting multiple SPP waves in a realistic tandem solar cell with a piecewise homogeneous semiconductor, as shown schematically in Fig. 1. The solar cell is supposed to be made of a-Si alloys, and the metallic back-reflector is periodically corrugated. The a-Si alloys can be fabricated using plasma-enhanced chemical vapor deposition over planar and patterned substrates.¹⁹ The composition of the alloys can be controlled by controlling the ratio of precursor gases (e.g., silane, hydrogen, methane, and germane) and the deposition temperature.²⁰ A top layer of aluminum-doped zinc oxide (AZO) allows electrical contact. Also, an AZO layer is sandwiched between the metal and the semiconductor to avoid the deterioration of the electrical properties of the a-Si alloy closest to the metal.¹

Fig. 1

Schematic of the tandem solar cell with a periodically corrugated metallic back-reflector. Specular components of the reflected and transmitted light are identified as of order 0, whereas nonspecular components are identified as of nonzero orders.

The plan of this paper is as follows. The methodology for computation is provided in brief in Sec. 2, and the numerical results are presented in Sec. 3. Concluding remarks are presented in Sec. 4. An $\exp (-i\omega t)$ dependence on time $t$ is implicit, with $\omega$ denoting the angular frequency and $i=\sqrt{-1}$. The free-space wavenumber, the free-space wavelength, and the intrinsic impedance of free space are denoted by $k_0=\omega \sqrt{{ϵ}_0{\mu }_0}$, ${\lambda }_0=2\pi /k_0$, and ${\eta }_0=\sqrt{{\mu }_0/{ϵ}_0}$, respectively, with ${\mu }_0$ being the permeability and ${ϵ}_0$ the permittivity of free space. Vectors are in boldface, and the Cartesian unit vectors are identified as ${\widehat{\mathbf{u}}}_x$, ${\widehat{\mathbf{u}}}_y$, and ${\widehat{\mathbf{u}}}_z$.

2.

Methodology

3.

Numerical Results and Discussion

For the illustrative numerical results presented in this section, the various layers of the tandem solar cell in Fig. 1 were assumed to be made of a-Si alloys as follows:

Thus the tandem solar cell would function as a spectrum-splitting solar cell.^1,20 All the $n$- and $p$-type layers were taken to be 20-nm thick, and the thickness of all intrinsic layers was taken to be 200 nm. The relative permittivities of the a-Si alloys as functions of ${\lambda }_0$ are presented in Fig. 2. They were computed using a model provided by Ferlauto et al.²⁰ The thicknesses of the two AZO layers were set as $d_T=100\mathrm{nm}$ and $d_a=140\mathrm{nm}$. The refractive index of AZO was obtained as a function of ${\lambda }_0$ by using a single-oscillator model developed for AZO films deposited by reactive magnetron sputtering at a substrate temperature of 170°C.³⁰ The imaginary part of the refractive index of AZO is negligible in the visible and the near-infrared regimes.³⁰ The information provided in this paragraph is sufficient to specify ${ϵ}_d(z)$.

Fig. 2

(a) Real and (b) imaginary parts of the relative permittivities of a-Si alloys as functions of ${\lambda }_0$. These materials were used for the tandem solar cell, as detailed at the beginning of Sec. 3.

The metal was taken to be silver, and its relative permittivity was taken from a standard source.^31,32 Moreover, we set $L_m=30\mathrm{nm}$.

3.1.

Planar Back-Reflector

Before we present the results for the periodically corrugated metallic back-reflector, let us present the results when the back-reflector is planar (i.e., $L_g=0$). The total reflectances $R_p$ and $R_s$ are presented in Fig. 3 for $\theta \in [0,70]\mathrm{deg}$ and ${\lambda }_0\in [\mathrm{400,900}]\mathrm{nm}$. The corresponding absorptances are $A_p=1-R_p$ and $A_s=1-R_s$, because transmittances are negligible as the thickness $L_m$ exceeds the penetration depth of silver. Since the absorption of light by AZO is negligible, light is absorbed only by the $p-i-n$ cells and the metal. Furthermore, the nonspecular components of the reflected field are absent because the metallic back-reflector is planar and $N_t=0$ then suffices.

Fig. 3

Total reflectances (a) $R_p$ and (b) $R_s$ as functions of $\theta$ and ${\lambda }_0$ when $L_g=0$. No nonspecular Floquet harmonics and no SPP waves are excited due to the absence of the periodic corrugations in the metallic back-reflector.

Figure 3 shows that both total reflectances are very low when ${\lambda }_0<700\mathrm{nm}$, but both are very high when ${\lambda }_0>750\mathrm{nm}$. This dimorphism is due to the higher values of the imaginary parts of the relative permittivities of the a-Si alloys at shorter wavelengths. Therefore, considerable room for enhancement of the absorption of light by the semiconductor layers is available when ${\lambda }_0>700\mathrm{nm}$.

The absence of sharply defined bands with low reflectance in Fig. 3 indicates that no SPP waves are excited. Parenthetically, we note that high-phase-speed SPP waves can indeed be excited with a planar interface if the partnering dielectric material is several-periods thick³³; but, as these SPP waves will not play a significant role in enhancing the absorption of light in solar cells, their excitation was not investigated.

3.2.

Simple Surface-Relief Grating as the Back-Reflector

Let us next consider that the tandem solar cell is backed by a simple surface-relief grating with

Eq. (17)

$$g(x)=L_d+L_g[1-\sin (\frac{2\pi x}{L_x}\left)\right]$$

as the grating-shape function.

For the computation of the reflectances, the wavelength range was restricted to ${\lambda }_0\in [\mathrm{450,900}]\mathrm{nm}$ because of convergence problems with the RCWA when ${\lambda }_0\in [\mathrm{400,450}]\mathrm{nm}$. This is due to the fact that a-Si alloys are highly dissipative for ${\lambda }_0\in [\mathrm{400,450}]\mathrm{nm}$, as can be seen from the plots of the real and the imaginary parts of the relative permittivities in Fig. 2. However, the high dissipation in turn implies that the light absorption efficiency in this narrow spectral regime is very high and does not offer much room for improvement.

3.2.1.

L_x = 400 nm

Suppose next that the period $L_x=400\mathrm{nm}$ and the trough-to-crest height $L_g=80\mathrm{nm}$. We set $N_t=9$ after ascertaining that the RCWA provided converged solutions. The total reflectances $R_p$ and $R_s$ as functions of $\theta$ and ${\lambda }_0$ are presented in Figs. 4(a) and 4(b), respectively. Also in the same figure, the angle

Eq. (18)

$${\theta }_{\mathrm{SPP}}={\sin }^{-1}[\frac{\mathrm{Re}(q)}{k_0}-n\frac{{\lambda }_0}{L_x}]$$

is plotted in relation to ${\lambda }_0$ when an SPP wave is predicted to be excited as a Floquet harmonic of order $n$,²³ $q$ being the wavenumber of either a $p$- or an $s$-polarized SPP wave delivered by the solution of the canonical boundary-value problem described in Sec. 2.2. As the transmittances were found to be negligible, the higher that the total reflectance is, the lower is the corresponding absorptance.

Fig. 4

Total reflectances (a) $R_p$ and (b) $R_s$ as functions of $\theta$ and ${\lambda }_0$ when the metallic back-reflector is sinusoidally corrugated with period $L_x=400\mathrm{nm}$ and depth $L_g=80\mathrm{nm}$, per Eq. (17). Values of ${\theta }_{\mathrm{SPP}}$ defined in Eq. (18) in relation to ${\lambda }_0$ when wavenumber $k_x^{(n)}$ of a Floquet harmonic of order $n$ is the same as wavenumber $q$ of (c) $p$- and (d) $s$-polarized SPP waves delivered by the solution of the underlying canonical boundary-value problem.

The two upper panels in Fig. 4 show the presence of low-reflectance bands when ${\lambda }_0>750\mathrm{nm}$. These bands could be due to the excitation of either SPP waves or waveguide modes³⁴ that propagate in the bulk of the tandem solar cell. The two lower panels in Fig. 4 show the locations in ${\lambda }_0\times \theta$ space where SPP waves are predicted to be excited for the chosen solar cell. Comparisons of Figs. 4(a) and 4(c), and of Figs. 4(b) and 4(d), suffice to identify the low-reflectance bands that represent the excitation of SPP waves. Clearly then, multiple $p$- and $s$-polarized SPP waves are excited due to the periodic corrugation of the metallic back-reflector. The low-reflectance bands that do not correspond to the branches in Figs. 4(c) and 4(d) most likely represent the excitation of waveguide modes that are guided by the tandem solar cell.

Comparisons of Figs. 3(a) and 4(a), and of Figs. 3(b) and 4(b), show that $R_p$ and $R_s$ are generally smaller for ${\lambda }_0>700\mathrm{nm}$, when the metallic back-reflector is periodically corrugated than when it is planar. The decreases in $R_p$ and $R_s$ are surely due to the excitation of SPP waves and waveguide modes, thereby resulting in enhanced absorption of light by the tandem solar cell. The decrease in $R_p$ is more than the decrease in $R_s$, which is in line with the theoretical prediction from the canonical boundary-value problem of a smaller number of $s$-polarized SPP waves than of $p$-polarized SPP waves.

The absence of the low-reflectance bands (representing the excitation of SPP waves) for both linear polarization states when ${\lambda }_0<700\mathrm{nm}$, even though the canonical boundary-value problem predicts the excitation of SPP waves, is inconsequential. This is because the absorption of light by the tandem solar cell is very high in this spectral regime. Most likely, those bands are not easily evident.

3.2.2.

L_x = 500 nm

Let us now increase the period of corrugations to $L_x=500\mathrm{nm}$, but $L_g=80\mathrm{nm}$ remains fixed. The RCWA algorithm delivered converged reflectances for $N_t=13$.

The total reflectances $R_p$ and $R_s$ are presented as functions of $\theta$ and ${\lambda }_0$ in Figs. 5(a) and 5(b), respectively. The points in ${\lambda }_0\times \theta$ space when either a $p$- or an $s$-polarized SPP wave is predicted to be excited are identified in Figs. 5(c) and 5(d), respectively. Comparisons of Figs. 4(c) and 5(c), and of Figs. 4(d) and 5(d), show that the points in ${\lambda }_0\times \theta$ space when $L_x=500\mathrm{nm}$ are at different positions and are different in number for $L_x=500\mathrm{nm}$ than for $L_x=400\mathrm{nm}$ for both the linear polarization states. Parenthetically, we note that the wavenumbers $q$ of SPP waves (for any given ${\lambda }_0$) obtained by the solution of the underlying canonical boundary-value problem are independent of the period $L_x$ of the corrugations; however, the condition $k_0\sin {\theta }_{\mathrm{SPP}}+n\frac{2\pi }{L_x}\approx \mathrm{Re}(q)$ that needs to be satisfied for those SPP waves to be excited involves the period $L_x$.

Fig. 5

Same as Fig. 4 except that $L_x=500\mathrm{nm}$.

Comparisons of Figs. 5(a) and 5(c), and of Figs. 5(b) and 5(d), allow us to conlcude that several low-reflectance bands (for ${\lambda }_0>700\mathrm{nm}$) are due to the excitation of SPP waves. This excitation of multiple SPP waves is accompanied by overall low reflectances in the near-infrared spectral regime (${\lambda }_0>700\mathrm{nm}$) as compared to Fig. 3 for the planar metallic back-reflector.

3.3.

Compound Surface-Relief Grating as the Back-Reflector

An SPP wave is excited in the grating-coupled configuration when the condition $k_0\sin {\theta }_{\mathrm{SPP}}+n\frac{2\pi }{L_x}\approx \mathrm{Re}(q)$ is met.^16,23 To increase the number of SPP waves, a compound surface-relief grating with each period having several periods of more than one simple surface-relief grating can be used.^35,36 For this purpose, we chose a compound grating with

Eq. (19)

$$g(x)=\{\begin{array}{cc}{L}_d+L_g[1-\sin (2\pi \frac{x}{L_{x1}}\left)\right],& x\in (0,N_1{L}_{x1})\\ L_d+L_g[1-\sin (2\pi \frac{x-N_1{L}_{x1}}{L_{x2}}\left)\right],& x\in (N_1{L}_{x1},N_1{L}_{x1}+N_2{L}_{x2}=L_x)\end{array},$$

specified in the reference unit cell of the region $L_d<z<L_d+L_g$. A schematic representation of the compound grating is shown in Fig. 6. The density plots of the reflectances $R_p$ and $R_s$ as functions of $\theta$ and ${\lambda }_0$ for a compound grating with $N_1=N_2=2$, $L_{x1}=401\mathrm{nm}$, $L_{x2}=501\mathrm{nm}$, and $L_g=80\mathrm{nm}$ are given in Fig. 7. We chose each period of the compound grating to comprise two periods each of the simple sinusoidal gratings with periods $L_{x1}=401\mathrm{nm}$ and $L_{x2}=501\mathrm{nm}$—instead of $L_{x1}=400\mathrm{nm}$ and $L_{x2}=500\mathrm{nm}$—to avoid difficulties in computing the Fourier coefficients used in Eq. (10). Moreover, $N_t=16$ was chosen after ensuring the convergence of the reflectances.

Fig. 6

Schematic of the compound surface-relief grating when $g(x)$, $x\in (0,L_x)$ is given by Eq. (19) with $N_1=N_2=2$.

Fig. 7

Total reflectances (a) $R_p$ and (b) $R_s$ as functions of $\theta$ and ${\lambda }_0$ when the solar cell is backed by a compound grating defined by Eq. (19) with $L_{x1}=401\mathrm{nm}$, $L_{x2}=501\mathrm{nm}$, and $L_g=80\mathrm{nm}$.

The low-reflectance bands in Fig. 7 representing the excitation of SPP waves could either be attributed to an individual simple surface-relief grating in the compound grating or the compound grating itself, though the efficiency of the excitation is generally smaller with the compound grating than with either of the simple grating alone.³⁶ Furthermore, not all SPP waves excited by individual simple gratings may be excited by the compound grating. A comparison of Fig. 7 with Fig. 3 shows that the reflectance in the near-infrared regime is generally lower when a compound grating is used than when the metallic back-reflector is planar.

3.4.

Solar-Spectrum-Integrated Absorption Efficiency

The solar-spectrum-integrated (SSI) absorption efficiency is defined by³⁷

Eq. (20)

$$\eta (\theta )=\frac{1}{{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}{\lambda }_{0}S({\lambda }_0)\mathrm{d}{\lambda }_0}{\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}\frac{2-R_p({\lambda }_0,\theta )-R_s({\lambda }_0,\theta )}{2}{\lambda }_{0}S({\lambda }_0)\mathrm{d}{\lambda }_0,$$

for unpolarized light coming from the sun, where $S({\lambda }_0)$ is the solar irradiance spectrum for AM1.5.³⁸ The SSI absorption efficiency is plotted in Fig. 8 as a function of $\theta$ when the metallic back-reflector is (1) planar, (2) the simple grating of Sec. 3.2.1, (3) the simple grating of Sec. 3.2.2, or (4) a compound grating of Sec. 3.3, with ${\lambda }_{\min }=450\mathrm{nm}$ and ${\lambda }_{\max }=900\mathrm{nm}$. The figure shows that $\eta$ increased by up to 20% when the solar cell is backed by a periodically corrugated reflector than a planar reflector. For $\theta \in (2\text{,}23)\mathrm{deg}$, the simple grating with period $L_x=400\mathrm{nm}$ gives the highest SSI absorption efficiency. When either $\theta \approx 0\mathrm{deg}$ or $\theta \in (23\text{,}70)\mathrm{deg}$, the compound grating yields the highest SSI absorption efficiency.

Fig. 8

SSI absorption efficiency $\eta$ as a function of the incidence angle $\theta$ when ${\lambda }_{\min }=450\mathrm{nm}$ and ${\lambda }_{\max }=900\mathrm{nm}$. The dashed and dotted lines represent simple gratings ($L_g=80\mathrm{nm}$), the chain-dashed line represents the compound grating defined by Eq. (19) with $L_{x1}=401\mathrm{nm}$ and $L_{x2}=501\mathrm{nm}$ ($L_g=80\mathrm{nm}$), and the solid line represents the planar back-reflector.

To delineate the enhancement of absorption efficiency in the near-infrared spectral regime, the SSI absorption efficiencies are presented in Figs. 9(a) and 9(b) when ${\lambda }_{\min }=450\mathrm{nm}$ and ${\lambda }_{\max }=700\mathrm{nm}$, and ${\lambda }_{\min }=700\mathrm{nm}$ and ${\lambda }_{\max }=900\mathrm{nm}$, respectively. Figure 9(a) shows that $\eta$ is enhanced up to 5% only when ${\lambda }_0\in [\mathrm{450,}700]\mathrm{nm}$. This is due to the fact that semiconductor materials used in the solar cell are highly dissipative in this spectral regime. Furthermore, in this spectral regime, both the simple gratings and the compound grating have very similar effects on $\eta$. This is also evident from Fig. 9(a) since $\eta$ is about 80% over a wide range of the incidence angle $\theta$ even when the metallic back-reflector is planar.

Fig. 9

Same as Fig. 8 except that (a) ${\lambda }_{\min }=450\mathrm{nm}$ and ${\lambda }_{\max }=700\mathrm{nm}$, and (b) ${\lambda }_{\min }=700\mathrm{nm}$ and ${\lambda }_{\max }=900\mathrm{nm}$.

The plots of $\eta$ in Fig. 9(b), however, show an enhancement of up to 100% when ${\lambda }_0\in [\mathrm{700,}900]\mathrm{nm}$. Let us repeat that multiple SPP waves excited with either the simple or the compound grating are all dominant in this spectral regime. Therefore, the enhancement in $\eta$ can be attributed to the excitation of multiple SPP waves.

4.

Concluding Remarks

The effect of multiple SPP waves on the SSI absorption efficiency of a thin-film tandem solar cell made of a-Si alloys with a periodically corrugated metallic back-reflector was theoretically investigated. The boundary-value problem to find the total reflectances as functions of the free-space wavelength ${\lambda }_0\in [450\text{,}900]\mathrm{nm}$ and the angle of incidence $\theta \in [0\text{,}70]\mathrm{deg}$ for incident linearly polarized plane waves was set up and solved for a typical tandem solar cell. The low-reflectance bands representing the excitation of SPP waves by simple gratings were identified by comparing the total-reflectance spectrums with the solution of the underlying canonical boundary-value problem. A compound grating with each period occupying several periods of two simple gratings was also investigated.

The total reflectance in the near-infrared regime was found to decrease for both linear polarization states when the metallic back-reflector is periodically corrugated than when it was planar. The identification of multiple SPP waves showed that most of the decrease in total reflectance results from the excitation of SPP waves. Furthermore, the use of different periods for the corrugations of the metallic back-reflector showed that the position of low-reflectance bands in the ${\lambda }_0\times \theta$ space can be changed by changing the period of the corrugations. A compound grating was found to deliver a larger number of SPP waves than a simple grating, but the excitation efficiency was generally lower than when a simple grating was used.

The SSI absorption efficiency as a function of the incidence angle using an AM1.5 solar spectrum showed that an enhancement of the absorption efficiency of as much as 20% is achievable when the back reflector is periodically corrugated. A further analysis of the efficiency by dividing the spectral regime into ${\lambda }_0\in [\mathrm{450,}700]\mathrm{nm}$ and ${\lambda }_0\in [\mathrm{700,}900]\mathrm{nm}$ showed that most of the enhancement in absorption comes from the enhancement in the near-infrared spectral regime.

Let us note that the absorption efficiency of the tandem solar cell depends not only on the period $L_x$ of the corrugations of the metallic back-reflector, but also on the other parameters such as the depth, shape, and the duty cycle of the corrugations.³⁷ Furthermore, the choice of periodically corrugated back-reflector depends on the thicknesses and the materials used for making a solar cell. Since multiple SPP waves are guided due to the periodic nonhomogeneity of the partnering semiconductor material, an enhancement in the absorption efficiency should be expected in any thin-film solar cell. Therefore, we hope that our numerical studies have provided experimentalists with some useful guidelines to exploit surface multiplasmonics¹⁶ for harvesting solar energy not only in those cells that are made of a-Si alloys but also of compound semiconductor materials.