Optimal light trapping in ultra-thin photonic crystal ... - OSA Publishing

Optimal light trapping in ultra-thin photonic crystal crystalline silicon solar cells Shrestha Basu Mallick,1,2* Mukul Agrawal,3 and Peter Peumans2 1 Department of Applied Physics, Stanford University, Stanford, California 94305, USA Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA 3 Applied Materials Inc, Santa Clara, California 95054, USA *[email protected]

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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/cm2 for an unstructured film to 21.8 mA/cm2 for a film structured as the double layer photonic crystal. This photocurrent density value approaches the limit of 26.5 mA/cm2, obtained using the Yablonovitch light trapping limit for the same volume of active material. ©2010 Optical Society of America OCIS codes: (350.4238) Nanophotonics and photonic crystals; (350.6050) Solar energy; (6628) Subwavelength structures, nanostructures

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Received 3 Dec 2009; revised 16 Jan 2010; accepted 22 Feb 2010; published 5 Mar 2010

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41. S. G. Tikhodeev, A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, and T. Ishihara, “Quasiguided modes and optical properties of photonic crystal slabs,” Phys. Rev. B 66(4), 045102 (2002).

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 fillfactors [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 nmthick cell with a short-circuit current density of only 6.5 mA/cm2. 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 4n2 (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 #120919 - $15.00 USD

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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 4n2, 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 quasiguided 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 #120919 - $15.00 USD

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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 nonadaptive 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 SiO2 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 inplane 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.

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Fig. 1. Schematic of optimized structure

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 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/cm2. 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.

0.9 3.5 d/p of Upper PC

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Fig. 5. Enhancement in MAPD under normal incidence of a structured layer compared to a solid slab with the same active layer volume as the structured layer, as a function of d/p ratios of the upper and lower PC layers.

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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/cm2. 17.5

Photocurrent Density (mA/cm2)

17 16.5 16 15.5 15 14.5 14 13.5 50

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Fig. 6. MAPD as a function of relative thicknesses of the two layers

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/cm2 to 21 mA/cm2. Figure 7 shows the resulting change in the absorption spectrum. 100 No Ag on the backside Ag on the backside

Fraction of Photons Absorbed

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Fig. 7. Change in spectrum as the backside is coated with silver

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A final optimization is the replacement of the 100 nm-thick SiO2 ARC by a 70 nm-thick SiNx ARC, resulting in a MAPD of 21.8mA/cm2 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 Fy is calculated in Eq. (2) as follows Fy =

(1 − e−2 al ).T .100 n2 1 − e−2 al + 2 2 .T .e −2 al n1

(2)

The above equation is a modified version of an equation in [40]. Here 1 is the mean thickness of the sample, a is the optical absorption coefficient, n1 and n2 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/cm2 for the PC cell, 7.1 mA/cm2 for the solid slab with equivalent volume, 10.3 mA/cm2 for the 400 nm-thick solid slab and 26.5 mA/cm2 when the Yablonovitch limit is used for the equivalent volume.

Fraction of Photons Absorbed

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Fig. 8. Optical absorbance as a function of wavelength for an equivalent volume cell with flat interfaces (green), a 400 nm-thickness flat cell (cyan) and the optimized structure (blue). The extrapolated geometric light trapping limit (red) is also shown for reference.

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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 4n2 because the spectral range includes wavelengths where absorption is already strong (