## Abstract

Crystalline silicon is an attractive photovoltaic material because of its natural abundance, accumulated materials and process knowledge, and its appropriate band gap. To reduce cost, thin films of crystalline silicon can be used. This reduces the amount of material needed and allows material with shorter carrier diffusion lengths to be used. However, the indirect band gap of silicon requires that a light trapping approach be used to maximize optical absorption. Here, a photonic crystal (PC) based approach is used to maximize solar light harvesting in a 400 nm-thick silicon layer by tuning the coupling strength of incident radiation to quasiguided modes over a broad spectral range. The structure consists of a double layer PC with the upper layer having holes which have a smaller radius compared to the holes in the lower layer. We show that the spectrally averaged fraction of photons absorbed is increased 8-fold compared to a planar cell with equivalent volume of active material. This results in an enhancement of maximum achievable photocurrent density from 7.1 mA/cm^{2} for an unstructured film to 21.8 mA/cm^{2} for a film structured as the double layer photonic crystal. This photocurrent density value approaches the limit of 26.5 mA/cm^{2}, obtained using the Yablonovitch light trapping limit for the same volume of active material.

©2010 Optical Society of America

## 1. Introduction

Crystalline Si (c-Si) is an attractive material for photovoltaic cells due to its natural abundance, nearly ideal band gap, and leverage of existing process and materials knowledge. A potential approach to significantly lower the cost of crystalline Si (c-Si) solar cells is to use thin films of Si on low-cost substrates, that are formed by the deposition of a thin film using chemical vapor deposition [1] or epitaxy followed by release [2,3], or, alternatively, by a mechanical step such as that used in the SLIM-cut process developed at IMEC. Using thin layers of active material also allows poorer quality Si with shorter carrier diffusion lengths to be used and minimizes Auger recombination, leading to larger open circuit voltages and fill-factors [4–6]. Thin-film cells also have the advantage of lower energy consumption for device fabrication and offer potential for light-weight and flexible photovoltaics.

Most of the work done on thin-film Si cells has focused on cells which are microns to tens of microns thick. Relatively little work has focused on the design and fabrication of c-Si solar cells whose film thickness is in the sub-micron range. Such ultrathin c-Si cells are of interest because they further limit the amount of material needed, and allow for adequate electrical performance despite reduced carrier diffusion lengths. Danos, et al. [7] designed a 200 nm-thick cell with a short-circuit current density of only 6.5 mA/cm^{2}. Yoon, et al. [8] have proposed a process to make submicron cells by KOH etching of bulk wafers and transfer printing on substrates but the cells they fabricated were 15-20 μm in thickness. One of the main challenges in realizing the efficiency potential of ultrathin c-Si cells is the weak optical absorption in the near-infrared spectral range with an absorption length of just over 10 μm at λ = 800 nm to >1 mm at λ = 1108 nm.

Conventional c-Si solar cells use a texturing approach whose effects can be understood using geometric optics [9,10]. Strongly scattering surfaces that bound the absorbing layer randomize the direction of light internally within the absorber into a uniform distribution of forward angles, thus randomizing the occupation of the photon density-of-states (PDOS). This increases the effective absorption in a weak absorber by a factor of 4n^{2} (where n is the index of refraction) over that of a planar slab with the same volume [9,10]. This geometrical optics limit is also known as the Yablonovitch limit and corresponds to an enhancement in absorption of ~50 for c-Si for near the band edge. This is the highest enhancement that can be obtained in the geometric optics regime when angles of incidence spanning the full hemisphere are considered.

The geometric optics approach is much less effective in thin-film solar cells where the wavelength of the incident light is comparable or larger than the film thickness. The local PDOS averaged over the thickness of a thin film is significantly less than the bulk PDOS such that even for an ideal Lambertian scattering surface, the enhancement in optical path length cannot approach the Yablonovitch limit [11]. In addition, subwavelength Lambertian scattering surfaces that work across a wide spectral range are challenging to realize.

Because of the shortcomings of geometric light trapping schemes, light harvesting approaches based on wave optics in thin-film cells have attracted significant interest. It has been proposed that coherent light trapping schemes can achieve absorption enhancements superior to geometric approaches because the wave-optics approach can target a specific spectral range [12]. Several wave optics light-trapping approaches were explored such as surface plasmon based light-trapping [13,14], scattering into guided modes by metal nanoparticles [15] and grating couplers [16–19], and photonic crystals (PCs) (in 1D [20–24], 2D [21,22] and 3D [25–27]). Zeng, et al. have explored c-Si cells with photonic crystals (dielectric distributed Bragg reflectors (DBR) and 1D gratings) as back reflectors [23] but the observed enhancement in efficiencies was small for the thick cells they studied. The optimization of such structures for application in thin-film cells was explored by Feng, et al. [28]. O'Brien, et al. studied the effect of a PC back reflector, demonstrating up to a 1.65 and 2.5-fold enhancement in optical absorption over a narrow spectral range near the band edge of a 10 μm thick μc-Si cell and a 250nm thick a-Si cell, respectively [29,30]. Bermel, et al. showed that 2 μm-thick c-Si cells with a back reflector consisting of a 2D PC in addition to a DBR as back reflector had superior light absorption compared to a 3D PC back reflector with a maximum enhancement in optical absorption given by a factor of nearly 5 as compared to a planar slab [22]. Another interesting theoretical result is that by Chutinan et al. [31] where they showed that structuring the active region as a 2D PC increased the efficiency of 2 μm thick c-Si cells by 11.15% and 10 μm thick c-Si cells by 3.87%.

Here, we focus on ultrathin c-Si films and analyze the achievable enhancement in optical absorption across the solar spectrum by structuring the c-Si film as a 2D PC (see Fig. 1
). By exploring the design space using numerical simulations, we show that such structures can provide a spectrally-averaged absorption enhancement that approaches the limit derived in the geometric optics regime [9]. This is consistent with our theoretical analysis of light trapping in the wave-optics regime that yields an upper limit for the achievable enhancement of optical absorption of 4n^{2}, irrespective of the photonic structure that is built around the volume of interest and irrespective of the spectral range considered [11], when one integrates over all angles of incidence.

## 2. Simulation methods

The light trapping structures discussed here are modeled using Rigorous Coupled Mode Analysis (RCWA) [32,33] also known as the Fourier Modal Method (FMM) [34]. Initially developed to study the diffraction efficiencies of optical gratings, RCWA has been used to study more complicated near-field optical effects such as the dispersion curves of quasi-guided modes in waveguides, resonant scattering, diffraction of surface plasmon polaritons, etc. In the most basic RCWA formulation, the simulated structure is surrounded by unbounded media on both sides, corresponding to the cover and substrate. The structure, which may have one or two-dimensional periodicity, can be decomposed into several layers with the constraint that the each layer has the same periodicity. The fields inside each layer are expanded in terms of in-plane spatial harmonics, the amplitudes of which are allowed to vary in the out-of-plane direction. The fields in the cover and substrate region are described by Rayleigh expansion. The wave vectors of these diffracted orders outside of the periodic structure are phase-matched to the in-plane spatial harmonics inside the structure. The electromagnetic problem now consists of solving for the variations of the amplitudes of the spatial harmonics along the out-of-plane direction. The fields are then boundary matched to calculate the amplitudes of the Rayleigh orders in the cover and substrate. The accuracy of RCWA depends on the number of diffraction modes retained in the structure. It should be noted that the method has now been generalized for aperiodic structures [35] and for structures in which different layers have different periodicity [36].

We developed an efficient implementation of the RCWA method to allow for rigorous optimization of PC structures as solar absorbers. The eigenvalue problem in each layer of the 2D periodic structure was set up in terms of two second-order differential equations rather than four first order equations which improved computational efficiency by a factor of eight [37]. Lifeng Li’s Fourier factorization rules [34] for factorization of a continuous space function into two stepwise continuous functions were applied. Mirror symmetries were exploited for faster convergence with fewer diffraction orders. A 2D generalization of the enhanced transmittance matrix method was used for enforcing boundary conditions at the interface of two layers. The enhanced transmittance matrix method is unconditionally stable for arbitrary thickness of gratings or homogeneous layers in the geometry or with inclusion of arbitrarily high number of harmonics in the simulation [33]. A similar enhanced transmittance matrix formalism was used for reverse propagating the diffraction amplitudes to evaluate 3D field profiles inside the structure. For homogeneous layers amongst the stack of periodic layers, the eigenvalue problem was more efficiently solved following [38]. To further speed up calculations, an adaptive version of the code was used. The adaptive method assumes that the most important spatial harmonics in the field profile are directly related to those Fourier coefficients of the 2D permittivity profile which have the largest amplitudes. In the non-adaptive scheme, a fixed number of lowest spatial frequency harmonics (including positive and negative harmonics) are retained in the field expansion which means that certain higher order Fourier coefficients, and consequently spatial harmonics, that are more important might potentially be overlooked. In the adaptive version, the same number of spatial harmonics is retained as in the non-adaptive version, but the harmonics retained are not the lowest spatial harmonics but those with the largest absolute value of the corresponding Fourier coefficients. Since the total number of spatial harmonics retained need not be larger, greater accuracy is obtained without increasing computational requirements. It should be noted that the adaptive method uses the form of the Fourier coefficient matrices proposed by Moharam, et al. [32,33].

## 3. Results

The structures investigated consist of a 400nm thick c-Si film. The thickness was chosen as 400 nm because the goal was to simulate an ultrathin c-Si solar cell and see how far the performance could be improved with the light trapping structures proposed. Having a submicron thickness also means that poorer quality silicon with diffusion lengths of the order of tens of microns or less can be used to make the solar cell.

A schematic of an unit cell of the structure is shown in Fig. 1 with an antireflection coating (ARC) (shown in green) on top of the active layer (shown in gray) and a 400 nm-thick SiO_{2} layer (shown in yellow) below the active layer. The entire structure is coated on the back with a metal layer (shown in blue) to ensure double-pass reflection. The ARC and active layers are perforated by a square lattice of cylindrical holes fashioned into a double-layer 2D PC. We considered double-layer PCs in which the hole diameter varies in two steps throughout the film thickness because this offers additional degrees of freedom in the design while still being amenable to fabrication. The fabrication involves starting with a silicon layer with the appropriate doping, etching the top layer for the smaller hole, passivating and protecting the sidewalls and then isotropically etching the larger hole in the bottom. Additional carrier recombination due to the etched surfaces in the double-layer PC structure is a potential concern. With proper passivation, the surface recombination can however be greatly reduced [39]. Also, for these thin structures, the benefits of light trapping are far likely to outweigh the loss of efficiency due to increased surface recombination. Investigation of surface recombination and carrier collection is an important aspect that should be investigated in measurements with such patterned devices. Since this is a paper on modeling light trapping in ultra-thin silicon layers, we ignore surface recombination effects.

The geometry is fully parametrized by the period (p) and ratio of the hole diameter (d) over period (p) in the upper and lower c-Si layers. Light is incident at normal incidence from the top. Electromagnetic fields as a function of wavelength, geometry and angle of incidence are calculated using RCWA, as described above. The structure is symmetrical in the two in-plane directions to maintain polarization-independence for normal incidence. Figure 1 also shows a vertical cross-section of the same structure. It is seen that the optimized structure is a double layer PC which looks like an inverted pyramid of silicon, if two unit cells are seen side by side. It should be noted that even though Fig. 1 shows the final structure with the ARC and back reflector, all the initial simulations were done on structures without ARCs or back reflectors. The purpose was to focus on light-trapping effects due to the PC structures alone.

#### 3.1 Convergence

Using the above method, the fraction of incident photons absorbed by the patterned structure was calculated over the spectral range 400-1200 nm with a step size of 3 nm. The accuracy of our model was tested by verifying the convergence as a function of the number of spatial harmonics. The convergence for a structure in which the upper layer has a d/p ratio of 0.65 and the lower layer a d/p ratio of 0.9 (cf. Fig. 2 inset) was tested over the spectral range considered by evaluating the fraction of photons absorbed for normal incidence as a function of the number of harmonics used in the optical simulation, as shown in Fig. 2. When less than 200 harmonics are used the absorption values change for a given wavelength and the resonant peaks shift when the number of harmonics is increased. The absorption spectra converge when >200 harmonics are used with <1% variation in absorption. The simulations reported here were carried out using 700 harmonics. The graph is plotted on a logarithmic scale so that the difference in convergence between simulations is magnified.

#### 3.2 Optimization with respect to period

Initially, the geometric parameters of the upper and lower PC layers were optimized to get the maximum achievable photocurrent density (MAPD) for a structure with a 100 nm-thick SiO_{2} ARC and without backside metal coating. As mentioned above, this was done in order to focus on light-trapping effects due to the PC structures alone. Each layer of the 2D PC was taken to span half the c-Si film thickness for the initial unoptimized structure. To calculate the MAPD, it was assumed that each absorbed photon generates an electron-hole pair and that both carriers are collected by their respective electrodes. This is a useful assumption for a thin device when the minority carrier diffusion length is much greater than the distance traveled by each of the carriers

The MAPD is a useful metric since it allows one to evaluate the effect of structuring on the optical performance. The MAPD is calculated in Eq. (1) as follows

*dI*/

*d*λ is the light intensity incident on the solar cell per unit area per unit wavelength (the AM1.5 global tilt spectrum was used), α(λ) is the fraction of photons absorbed, e is the electron charge, h is Planck’s constant and c is the speed of light. Fig. 3 shows the MAPD as a function of period for the inverted pyramidal structures (see Fig. 1). For each period, the d/p ratios of the upper and lower c-Si layers were optimized. Based on this data, it was seen that the MAPD was the highest for a period of 600 nm even though the other values were comparable.

#### 3.3 Optimization with respect to ratio of diameter of hole over periodicity of structure

Figure 4
shows the variation in MAPD as a function of the d/p ratios of the upper and lower PC layer for a 600 nm period. We note that optimum values of MAPD are obtained both for pyramid (d/p of lower layer < d/p of upper layer) as well as inverted pyramid (d/p of lower layer > d/p of upper layer) structures. In the remainder of this work, we focus on the inverted pyramid structures. In terms of optimal MAPD, the structure with a d/p of 0.9 for the upper layer and 0.65 for the lower layer is optimal with a MAPD of 16.9 mA/cm^{2}.

Since the thickness of the active layer is kept constant at 400 nm, the volume of active material is different for each design in Fig. 4. The enhancement in MAPD relative to a solid slab (i.e. unstructured slab) cell with the same active layer volume as the structured layer, and a 70nm thick SiNx ARC, is shown in Fig. 5 . The enhancement is more pronounced for higher d/p ratios since the equivalent solid slab is much thinner.

#### 3.4 Optimization with respect to relative thicknesses of the two layers

In Fig. 6
, we vary the thicknesses of the two layers while keeping the total thickness and d/p ratios constant. A thickness of 250 nm for the upper layer and 150 nm for the lower layer results in a maximum in MAPD and corresponds to an average fraction of photons absorbed of 34% over the wavelength range 400-1200 nm, resulting in a MAPD of 17.1 mA/cm^{2}.

#### 3.5 Back reflector and antireflection coating

The average absorption probability can be further increased by adding a reflective Ag coating on the backside of the cell to avoid loss of light by transmission. This results in an increase in average absorbance from 33.9% to 40.6%, and an increase in MAPD from 17.1 mA/cm^{2} to 21 mA/cm^{2}. Figure 7
shows the resulting change in the absorption spectrum.

A final optimization is the replacement of the 100 nm-thick SiO_{2} ARC by a 70 nm-thick SiN_{x} ARC, resulting in a MAPD of 21.8mA/cm^{2} and an average absorption of 43.2%.

#### 3.6 Comparison with solid slab and Yablonovitch limit

Figure 8
shows the fraction of photons absorbed as a function of wavelength for the optimized structure (blue solid line) and for a c-Si slab of equivalent volume which is calculated to have a thickness of 221.6 nm (green solid line). Since the PC structures would likely be fabricated by etching holes into a 400 nm-thick slab of silicon, the performance of the structure is also compared to a 400 nm thick slab (cyan solid line). This calculation is done for normal incidence. The Yablonovitch limit for the same active layer volume is also shown for reference (red dotted line).This limit was calculated by considering multiple reflections of a light ray inside the absorber. The enhanced percentage of photons absorbed *F _{y}* is calculated in Eq. (2) as follows

*1*is the mean thickness of the sample,

*a*is the optical absorption coefficient,

*n*

_{1}and

*n*

_{2}are the refractive indices of the textured medium and the surrounding dielectric, respectively, and

*T*is the Fresnel transmission coefficient which is taken to be 1 assuming a perfect ARC. In the case of a perfect Lambertian surface, the averaged path length of a randomized ray is twice the mean thickness

*1*of the film, which accounts for the factor of 2 in the exponent in the above equation. The average absorption of the solid slab of thickness 221.6 nm with a silver back reflector (double pass absorption) is 15.3% while the average absorption of the solid slab of thickness 400 nm is 21.7%. The average absorption for the optimized PC cell with back reflector is 43.2%, The Yablonovitch limit for the c-Si slab of equivalent volume (which has a thickness of 221.6 nm) is 53.2%. The corresponding MAPDs are 21.8 mA/cm

^{2}for the PC cell, 7.1 mA/cm

^{2}for the solid slab with equivalent volume, 10.3 mA/cm

^{2}for the 400 nm-thick solid slab and 26.5 mA/cm

^{2}when the Yablonovitch limit is used for the equivalent volume.

The enhancement in optical absorption of the PC cell over a solid slab of equivalent volume is shown in Fig. 9
(green solid line). The maximum enhancement in absorption is as high as 169 in narrow peaks and on average, the enhancement is a factor of 8 over the spectral range considered. This enhancement in optical absorption (Fig. 9, green solid line) is compared to the maximum enhancement calculated using the geometric optics limit for a structure with the same volume (Fig. 9, red dotted line). In the case of the geometric optics limit, the average enhancement in optical absorption is 15.9. This is smaller than the optical path length enhancement of 4n^{2} because the spectral range includes wavelengths where absorption is already strong (<700 nm). The enhancement in optical absorption of the PC cell over a solid slab of the same thickness (400 nm) is also shown in Fig. 9 (cyan solid line). In this case, the maximum enhancement in absorption is a factor of 84.6 and the average enhancement is a factor of 4.4 over the spectral range considered .The enhancement is plotted on a logarithmic scale to emphasize the degree of enhancement in various parts of the spectrum.

#### 3.7 Angular simulations

The enhancements in optical absorption that exceed the Yablonovitch limit shown in Fig. 9 do not violate the limit since it only applies when averaged over all angles of incidence. In Fig. 10a , the optical absorption averaged over all incident angles is shown. The hemispherically averaged absorption is plotted as a function of wavelength for both the s (blue solid line) and p (cyan solid line) polarization and compared to the Yablonovitch limit for a 221.6 nm thick solid slab (equivalent total volume). The average absorption for the s and p polarizations is 36.7% and 34.2%, respectively, as compared to 14.5% and 13.9% for a solid slab. The enhancement in optical absorption for the s and p polarizations is shown in Fig. 10b. The average hemispherically integrated enhancement is 2.53 for the s-polarization (blue solid line) and 2.47 for the p-polarization (green solid line).

In Fig. 11, the absorption is plotted as a function of wavelength and polar angle. Here, the absorption is averaged over all azimuthal angles. It is seen that under oblique incidence, new peaks appear and some of the peaks shift. This behavior is consistent with coupling of incident light to quasiguided modes [41].

#### 3.8 Analysis of simulated spectra

The location of the absorption peaks in the optimized PC structure can be explained by a simple model that approximates the PC as a waveguide with effective dielectric constants. Each layer of the PC is approximated as a homogeneous layer of volumetric averaged refractive index and the band structure of the corresponding structure is calculated by picking the reflectivity peaks in the complex k_{xy} plane [41]. The band structure of the s-polarized (green) and p-polarized (red) modes supported by the PC waveguide is shown in the lower panel of Fig. 12
. We note that these modes are in reality quasi-guided modes since the PC couples these modes to free space modes. The horizontal black lines correspond to the reciprocal lattice vectors of the periodic lattice of the PC, given by Eq. (3) as follows

*m*and

*n*are integers and

*p*refers to the period. Coupling of normally incident light into quasi-guided modes occurs when the wavevector of a guided mode is equal to a reciprocal lattice vector. The location of most peaks in the absorption spectrum can be attributed to such a match and are predicted within +/ 10 nm, despite the approximations used. The unexplained peaks in the spectrum may be due to the mode dispersion that is not modeled accurately by the simple effective medium model.

A number of quasiguided modes do not show up as peaks in the absorption spectrum when their in-plane wavevector matches a reciprocal lattice vector. This is attributed to weak coupling. The presence of these modes can be detected as shifts in the phase of reflection from the structure for a very weak absorber. For modes that couple very weakly to incident free-space modes, the corresponding resonances are very sharp and hard to detect unless the spectral response is scanned finely.

Figure 13a shows the phase of reflected waves (red line) over the spectral range 488-497 nm. There are two instances of sudden jumps in the otherwise monotonically increasing phase. The first corresponds to a wavelength of 489 nm, indicating that there is a resonance at this wavelength which is not apparent in the corresponding absorption plot (black solid line). From around 492 nm to 495 nm there is an approximately 4π jump in the phase indicating two resonances in close proximity. In Fig. 13b, corresponding detail of the band structure is shown, showing that each match of a quasi-guided mode wavevector to a reciprocal lattice vector is detected as a phase shift in the reflection. No absorption peaks are observed because the coupling is too weak compared to the strong absorption. Figure 13c shows the phase (red dashed line) and absorption (black solid line) over the spectral range 620-637 nm. Sudden jumps in phase are observed at 623 nm, 630 nm and 635 nm. There is an absorption peak at 625 nm which corresponds to a match between a quasi-guided mode wavevector and a reciprocal lattice vector (Fig. 13d). No absorption peaks are observed corresponding to the phase jumps at 630 nm and 635 nm. However, for these wavelengths of 630 nm and 635 nm there is a match between the wavevectors and the reciprocal lattice vectors in Fig. 13d at 631.5 nm and 635 nm respectively indicating weakly coupled modes.

#### 3.9 Electric field distribution

Figures 14a and 14b show the electric field intensity in the xz-plane for 918 nm and 826 nm, respectively. Since both these wavelengths correspond to resonances, most of the intensity is concentrated in the active layer with electric field intensity values exceeding the incident electric field intensity. One concern could be that in Fig. 14a, a lot of the electric field intensity is located at the edges of the structure which could lead to losses due to surface recombination. However, with proper passivation it should be possible to reduce the effects of surface recombination and harvest the carriers generated. Figure 15 shows the electric field intensity in xy-planes located in the center of each of the three patterned layers at the same wavelengths of Fig. 14. Hot spots in the field intensity profile as high as 60 times the incident field intensity are observed.

## 7. Conclusion

In conclusion, we have designed an effective sub-wavelength light trapping scheme through nano-scale restructuring of the active layer in ultrathin c-Si solar cells. The structure reported consists of a 400 nm-thick c-Si layer patterned into a double-layer PC, supported by a 400 nm-thick SiO_{2} layer back-coated with Ag. The structure achieves an enhancement of approximately 2.5 in average absorption over a flat cell with equivalent volume of active material when averaged over all angles. Compared to an equivalent volume of active material, the optimized structure gives an enhancement in MAPD of 3.1 for normal incidence. Compared to an equivalent thickness of active material, the optimized structure gives an enhancement in MAPD of 2.1 for normal incidence. Thus, we demonstrate that the enhancement of optical absorption that can be realized using 2D PCs approaches the Yablonovitch limit. When considering a limited range of incident angles, larger enhancements can be obtained which may provide a route for the further optimization of PCs for ultrathin solar cells used in applications where the cells are placed on a solar tracker.

## Acknowledgements

This work was supported by the Department of Energy award DE-FG02-07ER46426. We would like to thank Nicholas Sergeant, Zongfu Yu and Shanhui Fan for useful discussions. We would also like to acknowledge computational help provided by Dr. Changsoon Kim.

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