## Abstract

Thin silicon solar cells suffer from low light absorption compared to their thick counterparts, especially in the near infra-red regime. In order to obtain high energy conversion efficiency in thin solar cells, an efficient light trapping scheme is required. In this paper, we theoretically demonstrate significant enhancement in efficiency of thin crystalline silicon solar cells by using photonic crystals as the light absorbing layer. In particular, a relative increase of 11.15% and 3.87% in the energy conversion efficiency compared to the optimized conventional design is achieved for 2μm and 10μm thicknesses, respectively.

©2009 Optical Society of America

## 1. Introduction

In order to reduce the amount of silicon required for the production of solar cells, recent silicon solar cell designs use thin silicon wafers or thin silicon films [1]. However, the absorption of sun light decreases with decreasing silicon thickness, especially in the long wavelength part of the solar spectrum. In order to increase light absorption in thin solar cells, various light trapping schemes have been proposed [2-6].

Traditionally, light trapping is based on geometrical optics. A variety of light scattering surface structures, ranging from random to periodic, have been used at the front and/or back side of the absorbing layer to refract, diffract, and reflect light with the aim of increasing the total path length of light inside the material. In addition, external mirrors have been used to reflect light, that was not absorbed in the initial pass through the material, to return and traverse the material again. It has been shown that on average the absorption is enhanced by a factor of 4*n*
^{2}, where *n* is the refractive index of the material, compared to the absorption for normal incident light in an untextured, free-standing slab (when the intrinsic absorption is small) [2]. This factor is primarily related to the photon density of state (PDOS) in the absorbing medium and applies to most surface texturization designs [2].

Recently, a wave optical approach, which includes dielectric structures with an optical-wavelength feature size, has begun to be studied [3-6]. It is well known that by employing an optical resonant cavity, absorption, in a weakly absorbing material, can be enhanced by a factor larger than 4*n*
^{2} at the resonant frequency of the cavity. Maximal enhancement (complete absorption) at the resonant frequency in an arbitrarily weakly absorbing material is obtained by matching the cavity decay time with the intrinsic absorption decay time of the material together with the use of back reflectors [7]. However, the spectral range of enhanced absorption is limited by the linewidth of resonance. The greater the required enhancement for complete absorption, the narrower the linewidth. Thus, over a wide spectral range, the average absorption enhancement by a single-mode cavity does not exceed 4*n*
^{2} [7]. One approach would be to use many cavity resonant modes with dense frequency spacing. To achieve maximal overall absorption, the cavity decay time must match the absorption decay time in the material for each resonant frequency. It is unclear what dielectric architecture would satisfy the above condition.

Photonic crystals (PCs) are periodic dielectric microstructures that exhibit photonic band gaps – frequency ranges in which no electromagnetic propagation is allowed. In principle, photonic crystals can enhance light absorption in a certain wavelength range by using high photon density of states near the photonic band edge [4,8,9]. However, until now, it has been a challenge to design a solar cell with improved efficiency using photonic crystals. While an enhancement greater than 4*n*
^{2} for certain bandwidths has been reported [3-6], it seems that such enhancement over a broader bandwidth is required to achieve improved solar cell efficiency. To the best of our knowledge, considering the absorption bandwidth of crystalline silicon (c-Si) (~300-1,100nm), there has been no clear demonstration of the advantage of photonic crystal solar cells over optimized conventional (non-photonic crystal) solar cells, in terms of energy conversion efficiency. In this paper, by investigating the use of photonic crystals as the light absorbing layer we theoretically demonstrate significant solar cell efficiency improvement over conventional design. Specifically, the question we address is “Given a certain amount of c-Si, which can be rearranged into any physical structure with unlimited addition of a non-absorbing material, what structure yields the highest energy conversion efficiency over a wavelength range of 300-1,100nm under AM1.5 solar radiation?” We also assume a perfect back mirror in all cases in order to eliminate differences in reflectivity at the back surfaces of different solar cell architectures. We show that a photonic crystal structure can be designed to yield a solar cell with greater energy conversion efficiency than an optimized non photonic crystal structure.

## 2. Calculation method

For simplicity, we consider 2D systems in this work. Figure 1 shows a schematic of conventional solar cells comprising an anti-reflective (AR) coating, a uniform c-Si layer, a back scatterer (grating), and a back reflector. We consider S-polarization, where the electric field vector is perpendicular to the page. The optical absorption in the solar cells is calculated using the scattering matrix method [10]. In the scattering matrix method, the structure is divided into a number of vertically uniform layers that can be periodic in the lateral direction. The electromagnetic field in each layer is represented by an infinite set of plane waves. The method rigorously solves the Maxwell equations by imposing matching conditions for the tangential field components at each layer boundary. The refractive index of the AR coating is assumed to be 1.91 (corresponding to silicon nitride) and wavelength-independent. The complex dielectric constant as a function of wavelength for c-Si is obtained from [11].

The procedure for obtaining the energy conversion efficiency is as follows. First, the optical absorption of normally incident light impinging onto the front surface of the solar cell is calculated for the wavelength range 280-1,107nm. In our calculations, the number of calculated sampling (wavelength) points is 1,000. The number of absorbed photons is then calculated using the ASTM AM1.5 (Global tilt) solar spectrum [12]. Assuming each absorbed photon with energy greater than the c-Si band gap energy (1.12eV) generates an electron-hole pair that contributes to the photocurrent, the photo-induced current density *J _{ph}* is given by

where *dI*/*dλ* is the power density per unit wavelength of the solar radiation (ASTM AM1.5), *A*(*λ*) is the optical absorption calculated by the scattering matrix method, and *λ _{Eg}* is the wavelength corresponding to the c-Si band gap energy (=1,107nm). Following [13], we assume this photo current exists in the solar cell’s active region, that is, into a region where a built-in electrostatic potential of

*V*exists, and get an electric current density,

*J*(

*V*), of

where *e* is the electron charge, *n* is the average refractive index of silicon (we assume 3.6), *E _{g}* is the c-Si band gap energy,

*k*is the Boltzmann constant,

*T*is the operating temperature (we assume 300 K),

*ħ*is the reduced Planck constant, and

*c*is the light speed in vacuum. Finally, the energy conversion efficiency is obtained by finding the ratio of the maximum power point from Eq. (2) and the incident solar radiation power (~1kW/m

^{2}). The open circuit voltage,

*V*, can be calculated from Eq. (2) by finding the voltage at which

_{oc}*J*(

*V*)=0. On the other hand, the short circuit current,

*J*, which is obtained when

_{sc}*V*=0, is almost identical to

*J*. The fill factor (

_{ph}*FF*) is then a ratio between the maximum power and the product of

*J*and

_{sc}*V*. While the Eq. (2) gives slightly different values of

_{oc}*V*and

_{oc}*FF*for different

*J*, the typical values for the optimized cell parameters are

_{sc}*V*=0.80V and

_{oc}*FF*=0.86. Again, as stated above, these calculations assume complete, loss-less charge collection.

## 3. Conventional design

A schematic of conventional solar cell architecture with a grating scatterer is shown in Fig. 1. It consists of an anti-reflective (AR) coating, a uniform c-Si layer, a back scatterer (grating), and a back reflector. As mentioned above, the back reflector is assumed to be a perfect mirror. We optimize the structure by iteratively scanning all the structural parameters: 1) the AR coating thickness (*t _{AR}*), 2) the grating periodicity (

*a*), 3) the grating depth (

*t*), and 4) the grating duty cycle (

_{g}*dc*). In the process, the total amount of c-Si is kept equal to that of an untextured, uniform cell with a certain thickness. The optimization is performed for two different cell thicknesses of 2μm and 10μm.

First, we use gratings with a rectangular shape as shown in Fig. 1(a). For the cell thickness of 2μm, the optimized cell efficiency of 16.76% is obtained when *t _{AR}*=70nm,

*a*=620nm,

*t*=180nm, and

_{g}*dc*=0.5. For the cell thickness of 10μm, the optimized cell efficiency of 22.24% is obtained when

*t*=76nm,

_{AR}*a*=720nm,

*t*=220nm, and

_{g}*dc*=0.5. Next, we use gratings with triangular shape as shown in Fig. 1(b). It is expected that light scattering is improved since the triangular shape represents a gradual change from the uniform layer (at its base) to the periodic structure (at the apex) (see Fig. 1(b)), compared to an abrupt change from the uniform layer to the periodic structure as in the rectangular grating (Fig. 1(a)). This allows for an adiabatic coupling between the normal-incident light to the laterally scattered light. The optimized cell parameters for the 2μm thickness are

*t*=72nm,

_{AR}*a*=750nm, and

*t*=260nm. The triangular grating yields an efficiency of 18.82%, which is plus 2 percentage points over the rectangular grating. For the 10μm thickness, the optimized cell efficiency of 23.24% is achieved when

_{g}*t*=77nm,

_{AR}*a*=860nm, and

*t*=310nm, an expectedly smaller gain of plus 1 percentage point.

_{g}## 4. Photonic crystal design

We explore a solar cell design using PCs as an absorbing layer. Figures 2(a) and 2(b) show a schematic for a 2D PC structure with a square lattice of square dielectric rods in an air background and the corresponding band diagram, respectively. In principle, one could benefit from the high PDOS of photonic crystals by positioning the photonic band gap in the non-absorbing region of the material and the photonic band edge at the weakly absorbing region. For c-Si, this corresponds to placing the photonic band gap in a wavelength range just beyond *λ*=1,107nm (e.g., placing the photonic band gap in the wavelength range 1,107-1,300nm), and accordingly the frequency range above the band gap (indicated by the black arrow in Fig. 2(b)) corresponds to *λ*<1,107nm. However, no improvement in the energy conversion efficiency is obtained using this approach. We attribute this to inefficiency of coupling light into the photonic crystals. In general, it is not a trivial issue to design an efficient coupler for PCs, especially when good coupling is required for a wide bandwidth spanning multiple photonic bands. Conversely, one may place the lower first band edge (indicated by the green arrow in Fig. 2(b)) around the weakly absorbing range of 800-1,100nm. However, this leads to positioning the band gap around 600-800nm and consequently the reflection loss of light in this wavelength range. We also could not find improvement in the energy conversion efficiency using this approach.

Finally, we choose to allocate the entire relevant wavelength range of ~300-1,100nm into the first photonic band and place the band gap at *λ*~200nm. This corresponds to PCs with a lattice constant of ~160nm. Since there is no diffraction due to photonic crystals in the first band, a grating with a larger periodicity (than the PC) is required to scatter light into off-normal directions. For simplicity, the periodicity of grating is chosen to be an integer multiple of the PC lattice constant. The larger the periodicity of the grating, the longer the wavelength corresponding to the diffraction threshold. A large increase in efficiency is observed when the grating periodicity is 6 times the PC lattice constant. We show in Fig. 3(a) a schematic of our photonic crystal solar cell design. Note that the shape of triangular grating at the back surface is inverted compared to Fig. 1 to avoid abrupt changes in the dielectric structure.

An AR coating and a coupler are required to efficiently couple light from free space into the photonic crystal absorbing layer. Figure 3(b) shows our design of the coupler. As in the conventional design, the topmost surface is covered with a uniform AR coating. To provide an adiabatic change from the uniform layer into the discrete structure of a PC, we use the following approach. For an adiabatic change in the lateral direction, a tapered 1D grating gradually connects the uniform AR layer to the square PC (lateral coupler). In the longitudinal direction, the initially connected square dielectric rods are gradually isolated to form the PC as shown in Fig. 3(b) (longitudinal coupler). For the longitudinal coupler with *n _{lc}* layers of square dielectric rods, the spacing between the

*i*

^{th}and (

*i*+1)

^{th}rods is given by

*i*/

*n*times the spacing in the regular PC region.

_{lc}Similar to the previous section, we optimize the structure by iteratively scanning all the structural parameters; 1) the AR coating thickness (*t _{AR}*), 2) the back grating periodicity (

*a*) (the ratio between the back grating periodicity and the PC lattice constant is maintained at 6), 3) the grating depth (

*t*), 4) the lateral coupler thickness (

_{g}*t*), and 5) the number of layers in the longitudinal coupler region (

_{lc}*n*). In the process, the total amount of c-Si is kept equal to that of an untextured, uniform cell with a certain thickness. The optimization is performed for two different cell thicknesses of 2μm and 10μm. For the cell thickness of 2μm, the optimized cell efficiency of 20.92% is obtained when

_{lc}*t*=74nm,

_{AR}*a*=708nm,

*t*=75nm,

_{g}*t*=90nm, and

_{lc}*n*=9. There is a significant enhancement of plus 2 percentage points over the conventional design. We note that a good coupler is of critical importance in achieving high energy conversion efficiency. In fact, the same design without the longitudinal coupler only yields an efficiency of 18.62%. For the cell thickness of 10μm, the optimized cell efficiency of 24.21% is obtained when

_{lc}*t*=81nm,

_{AR}*a*=780nm,

*t*=85nm,

_{g}*t*=290nm, and

_{lc}*n*=10, an expectedly smaller gain with increasing thickness.

_{lc}While we have discussed in the above that the in-coupling of light into the c-Si absorbing layer is critical to the cell efficiency, the reason for the PC design showing a significantly higher efficiency over the conventional design is not simply due to the better in-coupling. To demonstrate this, we calculate the efficiency of the conventional design with optimized graded index AR coating, which allows a near-perfect in-coupling of light. The energy conversion efficiency only reaches 19.75% in this case. The optimized PC solar cell (without graded index AR coatings) exceeds this value by plus 1 percentage points. Figure 4 shows the absorption spectra for the conventional and PC design for 2μm thickness. Clearly, the PC solar cell absorbs considerably more light in the 700-1,100nm wavelength range; the average absorption level is apparently higher. It is noted that dense interference fringes are observed in the absorption spectra for the PC solar cell as compared to the conventional design. While the origins of these dense fringes are not completely understood, one of the factors may be the fact that there is an extra interference mechanism apart from the Fabry-Perot, that is, the Bragg reflection in the photonic crystal design. The above results are summarized in Table 1.

## 5. Discussion and conclusion

We discuss the implication of the Yablonovitch and Cody limit in terms of energy conversion efficiency for c-Si solar cells with a certain thickness. In three dimensions (3D), it has been shown that the intensity enhancement factor due to light trapping is 2*n*
^{2}. The absorption enhancement factor is calculated by taking an average over different optical path lengths inside the absorbing film due to different refracted angles (the larger the refracted angle (measured relative to the surface normal), the longer the optical path length) [2]. This angular integral in 3D gives an average optical path length of twice the film thickness, yielding an absorption enhancement factor of 4*n*
^{2}. In 2D, we follow the same approach of analysis and find that the intensity enhancement factor using a back mirror is 2*n*. The angular average of the optical path length in 2D gives, however, a factor of *π*/2, resulting in the absorption enhancement factor of π*n*.

We consider the upper limit for the energy conversion efficiency of c-Si solar cells with a certain thickness, *t*, assuming the enhancement factor of π*n*. This corresponds to the case where the incident light completely enters the c-Si layer (no reflection at the front surface). The absorption in the Yablonovitch limit is then equal to the one-pass absorption in a c-Si layer with the thickness *t* equal to *πn*∙
*t* for each wavelength. Using this absorption coefficient, the upper limits for energy conversion efficiency for 2μm- and 10μm-thick c-Si solar cell are 21.70% and 26.05% respectively. These values are, however, exceeded by the PC solar cells. For PC solar cells with optimized graded index AR coatings, the efficiency reaches 22.26% and 26.43% for 2μm and 10μm thickness, respectively. While careful consideration is needed in interpreting the above results, they seem to suggest that our PC solar cell design provides, for the first time, an overall absorption enhancement exceeding the classical limit over a wavelength range as broad as 800nm, centered at 700nm. Certainly, the achieved high energy conversion efficiency for c-Si results from complicated balance between many factors, including 1) providing a good light coupler between air and photonic crystals for the broad solar spectrum, 2) retaining strong absorption (and good coupling) for the highly absorbing wavelength range (~500-700nm), 3) providing extra absorption (above the classical average) for the weakly absorbing wavelength range (800-1,100nm). It should be noted that the strong absorption enhancement using photonic crystals as an absorbing layer is found for the specific polarization described above (the electric field vector is perpendicular to the plane of photonic crystals). We have also investigated photonic crystal solar cell design for the other polarization (the magnetic field vector is perpendicular to the plane of photonic crystals) but have not yet been able to obtain improvement in energy conversion efficiency.

In conclusion, using photonic crystals as an absorbing layer we have theoretically shown a crystalline-silicon-based solar cell architecture with significant improvement in optical absorption and accordingly attainment of enhanced energy conversion efficiency. The relative increase of 11.15% and 3.87% compared to the conventional design is achieved for 2μm and 10μm thicknesses, respectively. With the trend toward continual wafer thinning, this implies potential for applications in very thin cells, whereas the significance of photonic crystal designs may be less pronounced for thicker cells (50-200μm). While the improved efficiency has been shown only for specific polarization in 2D systems, we expect that the results can be extended to the 3D cases as well as for materials other than crystalline silicon.

## Acknowledgments

This work was supported through funding from the Natural Sciences and Engineering Research Council of Canada, the Ontario Research Fund – Research Excellence, and Arise Technologies Corporation. The authors would also like to acknowledge the inspiration of colleagues G.A. Ozin and S. John to investigate photonic crystal photovoltaics.

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