We develop a periodically patterned conformal photonic-plasmonic crystal based solar architecture for a nano-crystalline silicon solar cell, through rigorous scattering matrix simulations. The solar cell architecture has a periodic array of tapered silver nano-pillars as the back-reflector coupled with a conformal periodic structure at the top of the cell. The absorption and maximal current, averaged over the entire range of wavelengths, for this solar cell architecture is at the semi-classical 4n2 limit over a range of common thicknesses (500-1500 nm) and slightly above the 4n2 limit for a 500 nm nc-Si cell. The absorption exceeds the 4n2 limit, corrected for reflection loss at the top surface. The photonic crystal cell current is enhanced over the flat Ag back-reflector by 60%, for a thick 1000 nm nc-Si layer, where predicted currents exceed 31 mA/cm2. The conformal structure at the top surface focuses light within the absorber layer. There is plasmonic concentration of light, with intensity enhancements exceeding 7, near the back reflector that substantially enhances absorption.
©2011 Optical Society of America
Micro-morph tandem solar cells are a very attractive low-cost thin film tandem solar cell architecture  consisting of a nano-crystalline silicon (nc-Si) bottom cell of lower band gap [1,2] coupled with a top high band gap cell of hydrogenated amorphous silicon (a-Si:H). Short wavelength light (blue-green) is absorbed by the top a-Si:H cell whereas the longer wavelengths (upto the band edge of 1100 nm) are absorbed in the bottom nc-Si cell. Advanced manufacturing technology can produce large area cells on both rigid and flexible substrates with a record stabilized cell efficiency of 12% reported [2–4] for micro-morph cells.
A serious drawback of thin film silicon solar cells is that red and near-infrared (IR) photons (with λ >650 nm) are very poorly absorbed in thin nc-Si absorber layers. Using experimental wavelength-dependent dielectric functions  for nc-Si, the absorption length of near-infrared (IR) photons  (la) exceeds 2 μm for λ>700 nm, exceeding the thickness of the nc-Si layer (typically less than 1.5 μm), in micro-morph solar cells. Similar light-harvesting problems exist for thicker c-Si solar cells . Nc-Si is a mixed phase material composed of Si crystallites, that nucleate within an amorphous matrix , resulting in nc-Si having higher absorption than c-Si [1,8]. Since long-wavelength photons are absorbed by nc-Si in micro-morph cells, it is critical to achieve light trapping in nc-Si rather than a-Si:H.
To enhance the absorption of solar photons upto wavelengths of the band edge (λ = 1100 nm i.e.1.1 eV) in nc-Si, a common solution  is to use a randomly roughened back-reflector of silver and zinc oxide (Ag/ZnO), formed by etching ZnO, or roughening Ag on a substrate. The randomly roughened Ag/ZnO back reflector, with feature sizes much less than the wavelength, scatters incoming light in nearly Lambertian manner in random directions. This increases the path length of near-IR and red photons, increasing absorption and photo-current. Yablonovitch  demonstrated that such Lambertian scattering increases the path length by 4n(λ)2 at each wavelength λ, for completely loss-less conditions, where n(λ) is the refractive index. This enhancement factor can approach ~50 in silicon . However there are significant losses [11,12] from excitation of surface plasmon modes in the randomly textured back reflector, and it is suggested  that experimental enhancements are considerably less than the 4n2 factor.
It is a long-standing goal to develop solar cell architectures that can exceed the semi-classical 4n2 limit averaged over all wavelengths in the solar spectrum [14–18]. There has been much activity with periodic photonic crystal based back-reflectors [19–24] where a periodically structured back reflector diffracts light, enhancing the photon path length and dwell time of long-λ photons within the absorber layer. The wavelength of light inside the absorber layer is λ’ = λ/n(λ). When λ’<a, the pitch of the photonic lattice in the back reflector, strong diffraction occurs , resulting in diffraction resonances within the absorber layer. It has been demonstrated that photonic crystal based back reflectors can enhance the absorption over the semi-classical 4n2 limit at selected wavelengths especially near the band edge [17,25], but generally not over the entire range of solar wavelengths. Recent simulations  predicted that the 4n2 limit can be reached or surpassed in thin low index organic absorber layers (ε = 2.5), with double-layer photonic crystal cavities in thin c-Si , and nanorod arrays in c-Si .
2. Conformal nc-Si solar cell
2.1 Simulation method
We propose a novel periodic plasmonic crystal (PC) back-reflector with a conformal nc-Si solar cell architecture (Fig. 1 ) that has approached the semi-classical 4n2 limit when averaged over the entire spectrum of absorbed wavelengths. This occurs at realistic nc-Si thicknesses comparable to those in micro-morph cells. Our simulations are based on the well-established scattering matrix approach, where Maxwell's equations are rigorously solved in Fourier space for both polarizations of the incident wave, with the electric and magnetic fields expanded in Bloch waves [17,26]. The structure is divided into layers in the z-direction. The dielectric function is periodic in two-dimensions within each layer. Scattering matrices for individual layers were found and convoluted to obtain the scattering matrix of the entire structure, from which we obtained the reflection and absorption at each incoming wavelength. We utilize the experimentally determined wavelength-dependent absorption coefficients (α) for nc-Si  to obtain n2 the imaginary component of the refractive index (n = n1 + in2). We take the real component n1 to the value for c-Si . There is significantly larger measured absorption in nc-Si than c-Si [1,8], resulting from the 2-phase morphology of the material.
2.2 The classical limit
We calculate the absorption <Aw> weighted by the solar spectrum (Fig. 2 ), as a function of the nc-Si thickness, whereFig. 2a).
The short-circuit current Jsc cannot be calculated from this optical model of carrier generation, since photo-excited carrier transport, collection and recombination must be included. However, we may obtain a theoretical limit for the maximal current Jsc max, based on the simplifying assumption that all absorbed photons yield a photo-excited carrier that can be collected, which yields,
As the nc-Si thickness increases from 500 nm to 1500 nm, Jsc max increases from 30.5 mA/cm2 to 34.3 mA/cm2 for the 4n2 limit and from 16.5 mA/cm2 to 22.7 mA/cm2 for a flat Ag back-reflector (Fig. 1b).
2.3 Results for enhancement
Our proposed structure (Fig. 1) consists of tapered pillars of silver on a silver-coated substrate, periodically arranged in a triangular lattice of pitch a, with a height h, and base radius R. The maximum pillar radius is a/2. The conformal solar cell growth results in the nano-pillar structure at the top surface of the nc-Si, that is coated with an anti-reflecting layer of ITO, with a thickness d0. Nano-cone back reflectors in thinner (~200nm) a-Si:H cells  have shown large absorption enhancement, with high photo-currents. In contrast there is a thick planar region of nc-Si in our structure. We rigorously simulated the absorption of light in nc-Si layers on this conformal periodically textured back reflector. By systematic structural optimization of the ITO thickness d0, nc-Si upper pillar thickness d1, Ag pillar height d3, pillar base radius R and the pitch a we found a remarkable set of solutions where the weighted absorption averaged over all wavelengths can be at the semi-classical 4n2 limit (Fig. 2), especially at small nc-Si thicknesses (d2~500-1000nm). The robust solution consisted of a radius R/a = 0.4, d3 = 240nm, d1 = 230 nm, and d0 = 65nm. The d0 = 65nm corresponds to a quarter wavelength thickness of the anti-reflecting ITO layer.
Our patterned nano-pillar back reflector has an absorption slightly exceeding the average classical 4n2 limit over the nc-Si thickness of 500-1000 nm (Fig. 2). This simulation does include the absorption in the Ag back-reflector, in addition to that of nc-Si. One convenient way to separate the metal absorption and include only the absorption in the nc-Si is to adopt a nearly loss-less model for Ag . This is achieved by substantially increasing n2 for Ag, preventing the fields from penetrating the metal, increasing the reflectance back into the nc-Si, thereby approaching the ideal loss-less metal limit. In the loss-less limit the computed Jsc max includes photons absorbed in nc-Si (Fig. 2b) and also show the Jsc max for the conformal PC reflector is at the classical 4n2 limit for d2~500-1500nm) and slightly above the 4n2 limit at d2 = 500nm. The weighted Aw for loss-less limit is only slightly lower than the 4n2 limit by 1-3% (Fig. 2b), indicating small losses in the back reflector.
A substantial fraction of photons do not enter the solar cell and are reflected away at the top of the solar cell. In addition to the Yablonovitch 4n2 limit, we define an alternative practical 4n2 limit corrected for reflection loss at the top surface. A flat infinitely thick nc-Si slab with the anti-reflecting coating used here, suffers reflection loss R(λ). The 4n2 limit corrected by the reflection loss i.e. 4n2(λ)(1-R(λ)) describes the amount of light entering the absorber layer, and is significantly below shown the full classical 4n2 limit (Fig. 2). The simulated weighted absorption and photo-current significantly exceed the 4n2 limit, corrected for reflection loss (Fig. 2), illustrating the benefit of the conformal PC.
We observed an enhancement of 45% in <Aw> and 60% in Jsc max over a flat Ag back-reflector for a thickness d2 = 1000 nm. As expected, enhancement factors rise with decreasing nc-Si thickness (Fig. 3 ). These enhancements are considerably larger than the previous predicted values [5,17] (~15%) and measurements  for a hole-array back-reflector.
The wavelength-dependent absorption A(λ) for the conformal PC back reflector (Fig. 4 ) considerably improves over the flat Ag back reflector at wavelengths above 600 nm, and below 450 nm, and is very successful in collecting photons throughout the red and near-IR range (600-1100 nm). The PC absorption exceeds the semi-classical 4n2 limit at many wavelengths in this near-IR region. The conformal PC absorption exceeds the 4n2 limit, corrected for reflection loss, over the entire long-λ region (λ>600nm) and for short wavelengths (λ<450nm). Simulations with the loss-less Ag, show slightly lower absorption (Fig. 5a ), but enhancements over the 4n2 limit as well for long-wavelengths.
The absorption does not vary sharply with the pitch of periodic structure (Fig. 5a). For loss-less limit the optimum pitch is a~600 nm, and is slightly lower (500 nm) for real Ag where it has approached the 4n2 limit. Over the entire range of pitch (400-1000 nm) <Aw> exceeds the 4n2 limit corrected for reflection loss (Fig. 5a), and substantially more than the flat Ag. The height and radius of the nano-pillars is critical. For a fully conformal structure with the same pillar height in the top nc-Si and back-reflector, the absorption is optimized at a height of d3~240 nm (Fig. 5b). The nano-pillars have an optimal base radius R/a~0.40 (Fig. 5c). This is expected since the first Fourier component of the dielectric function ε(G) = 2J1(GR)/GR has a maximum for R/a~0.38 , for the first reciprocal lattice vector G1 of the triangular lattice where J1 is the Bessel function of first order.
The simulated absorption (Fig. 4,5) is for normal incidence. It is very encouraging that the conformal solar cell exhibits very good photon harvesting as the incident angle is changed away from normal. The photo-current does increase (by >4%) for angles away from the normal reaching a maximum near 30°, which is the best angle for enhanced absorption. Jsc is significantly above the classical 4n2 limit over a wide range of angles upto 50° even for a thick 1500 nm nc-Si cell (Fig. 6 ), The angle averaged Jsc is 34.4 mA/cm2 for angles less than 60° for 1500 nm thick nc-Si, larger than the classical 4n2 limit (34.3 mA/cm2) .There is a however a decrease in absorption and Jsc for incidence angles larger than 60°–so highly diffuse light is not well collected. For 1000 nm nc-Si, the 0-60° averaged Jsc is 34.03 mA/cm2, significantly larger than the classical 4n2 limit of 33.09 mA/cm2.
3. Electric field profiles
The simulated electric field intensity inside the solar architecture provides insight into the enhanced absorption. For an incident wavelength of 600 nm, the conformal nano-pillar grating at the top surface focuses the incoming light into a high intensity region (where the intensity is enhanced by a factor of ~3) within the upper portion of the nc-Si (Fig. 7 ) similar to the effect of a micro-lens. The light is largely absorbed before reaching the back interface. We do not observe focusing effects for a flat top interface. The focusing of light near the front of the solar cell occurs over a broad range of wavelengths. The conformal top periodic structure of the cell, has a graded index from silicon to air, and reduces the reflection loss at the top surface. The conformal top structure acts like a micro-lens to focus light within the solar cell and simultaneously reduce the reflection loss. Thus the increased absorption in the solar cell is a result of the reduced reflection loss by the top conformal structure. This is particular evident in the wavelength band around 600 nm (Fig, 6) where the conformal solar cell shows very high absorption at the classical 4n2 limit.
At an incident wavelength of 825 nm, typical of the near-IR region, the photon absorption length exceeds ~4 μm and the light reaches the periodic back-reflector (Fig. 8 ). Light is a focused by the top periodic texture into a bright region near the upper nc-Si surface. The conformal structure with front and back diffraction gratings is necessary for this coupling as found in recent studies . The electric field intensity shows a standing wave pattern in the nc-Si expected for a guided mode. In addition, there is a plasmonic mode at the top of the Ag nano-pillar surface, characterized by dipolar intensity distribution, where the field intensity is enhanced by more than a factor of ~6 for both polarizations (TE, TM) of the incident wave.
The enhanced absorption at long wavelengths occurs from a combination of two mechanisms: i) plasmonic enhancement of light intensity at the periodic metallic back-reflector and ii) diffraction resonances. Diffraction resonances give rise to maxima in simulated absorption (Fig. 3) and occur when the phase difference of waves from the top and bottom of the absorber layer differ in phase by a multiple of 2π, or kz = mπ/d, where kz is the z-component of the wave-vector. The periodic corrugation has reciprocal lattice vectors G, where Gx = i(2π/a), Gy = (2j-i)(2π/a)/√3 in the triangular lattice (i, j are integers). The incident wave with wave-vector k|| is diffracted by the back-reflector to a wave-vector k ||’ = k || + G, where k||’ is the wave-vector of the mode within the structure. Since (kz = mπ/d) the diffraction resonance is specified by 3 integers (i, j, m) according to
These are densely spaced diffraction resonances or wave-guided modes in nc-Si where the absorption is a maximum for the PC back-reflector (Fig. 2). Since the wavelength inside the absorber layer λ/n(λ) is considerably smaller than the pitch a (600 nm) of the photonic crystal, high orders of diffraction modes (i, j, m)>1 generate multiple absorption peaks of absorption (Fig. 3) in the near IR region (600-1100 nm). It is necessary to have several diffraction resonances below the band edge to achieve enhanced photon absorption. As found previously [5,17], the optimum PC pitch a~600 nm is of the order of or slightly smaller than the wavelength of the incident photon to be absorbed. Increasing the pitch of the structure increases the depth of the valleys in absorption and lowers the absorption. Very small pitch structures a<100 nm, lead to absorption mostly in the Ag back-reflector. An insulating ZnO spacer layer between the Ag nano-pillars and Ag-coated substrate gives rise to isolated plasmonic particles which offer slightly lower enhancements.
4. Discussion and Conclusions
These results rely on the accuracy of the scattering matrix simulation. One very attractive feature of the scattering matrix method is that it does not rely on the discretization in real space (in the x,y directions) within each layer. The accuracy of the calculation within each layer is controlled by the number of Fourier components (plane waves) used to expand the electric/magnetic fields and describe the sharp variation of the fields, We have checked the simulation accuracy by increasing the number of Fourier components (plane waves). Increasing the number of Fourier components from 271 to 595, show good convergence of convergence of currents Jsc to ~0.4 mA/cm2 and <Aw> to 0.01 (1%). Both Jsc and Aw slightly increase by these amounts when larger Fourier components are used.
The real space discretization was utilized in the z direction. We utilize a division of the nano-pillar into 5 layers in these simulations. When we increase the division of the nano-pillars in the z-direction from 5 to 13 layers, at the back-reflector and in the top array, we find changes in Jsc of ~0.6%. This suggests that excessively fine division in the z-direction may not be necessary.
Below 500 nm, the absorption decreases due to reflection loss at the top ITO anti-reflection coating, where the photons do not enter the solar cell. The graded conformal top surface reduces the reflection loss for flat top surfaces (Fig. 3- flat Ag). We expect gradual density-graded nanoporous layers  that gradually grade from air to ITO, can further reduce the reflection loss, and improve the currents.
In conclusion we have developed a conformal photonic crystal based architecture for a nc-Si solar cell that has approached the classical 4n2 limit for absorption averaged over the entire wavelength range of interest. Enhancements are due to both plasmonic concentration of light at the back reflector and diffraction resonances. We expect that the classical 4n2 limit may be surpassed by improving the anti-reflection coating and coupling more light into the structure.
We thank V. Dalal, J. Jin and E. Schiff for discussions. This research was supported by the Ames Laboratory that is operated for the Department of Energy by Iowa State University under contract No. DE-AC0207CH11385. We also acknowledge support from the NSF under grant no. ECCS-06013177, and the Iowa Powerfund. We acknowledge use of computational resources at the National Energy Research Scientific Computing Center (NERSC).
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