Completing the parameter map of the De Vos–Szymańska–Badescu down-conversion solar-cell model: non-radiative recombination, concentration, band-gap ordering, and non-ideal anti-reflection coatings

Down-conversion (quantum cutting) in a luminescent front converter is a proposed route to reduce thermalization losses in single-junction solar cells: one absorbed high-energy photon is converted into two lower-energy photons absorbed by the cell. De Vos, Szymańska, and Badescu (Energy Conversion and Management, 2009) built a ray-optical model of a solar cell covered by a front down-converter, an anti-reflection coating, and light trapping, but reported results only for pure radiative recombination, non-concentrated sunlight, the band-gap ordering $E_g \le E_d/2$ (ideal cutting at $E_d = 2E_g$), and idealized broad-band anti-reflection coatings of nearly zero reflectance, even though the underlying model could treat more general cases. Completing this parameter map — quantifying how the reported efficiency and optical-efficiency conclusions change under non-radiative recombination in the converter or cell, concentrated illumination up to the thermodynamic ceiling, the other converter/cell band-gap orderings, and realistic non-ideal anti-reflection coatings — remains unreported for this optical framework.

Background

A single-junction solar cell wastes a large fraction of the solar spectrum: photons with energy well above the cell band gap $E_g$ dump their excess energy as heat (thermalization), and the Shockley–Queisser detailed-balance analysis (the standard thermodynamic method that equates absorbed and emitted photon fluxes in equilibrium) bounds the resulting efficiency of an ideal single-junction cell at about 31% for non-concentrated 6000 K blackbody radiation. Down-conversion, also called quantum cutting, attacks the thermalization loss: a luminescent converter layer with transition energy $E_d$ absorbs one photon of energy $h\nu > E_d$ and emits two photons each just above the cell band gap, so one incident photon can generate two electron–hole pairs. Trupke, Green, and Würfel (2002) showed by detailed balance that an ideal front or rear down-converter with $E_d = 2E_g$ and purely radiative recombination raises the one-sun limiting efficiency of the coupled system to 39.63% for a 6000 K blackbody spectrum, versus 30.9% for the bare cell. A front converter sits on the sun-facing side of the cell; a rear converter sits behind it. In practice the converter layer is combined with an anti-reflection (AR) coating, a thin dielectric stack that suppresses Fresnel reflection at the air/glass interface, and with light trapping, in which a textured surface with an optically thick cell randomizes internal ray directions so that weakly absorbed photons make many passes (the Yablonovitch intensity-enhancement effect). Realistic modeling must also track the refractive-index mismatch between air, converter, and cell, which traps luminescence by total internal reflection (étendue conservation), as treated in the classic analysis of fluorescent collectors by Markvart and in the planar-converter model studies of van Sark (2005). Within this program, De Vos, Szymańska, and Badescu (2009) published an improved ray-optical computer model of a solar cell with a front down-converter, AR coating, and light trapping, refining two earlier companion models (Badescu, De Vos, Badescu, and Szymańska 2007, which introduced realistic refractive indices, and Badescu and De Vos 2007, which included non-radiative recombination and concentration). The 2009 paper, however, reports results only for converters with pure radiative recombination, non-concentrated sunlight, and the band-gap ordering $E_g \le E_d/2$ emphasizing the ideal cutting regime $E_d = 2E_g$, and it treats the AR coating as an ideal broad-band, all-angle element of approximately zero reflectance; the code itself could in principle treat non-radiative recombination, concentration factors up to $C \approx 4.6 \times 10^4$ (the thermodynamic concentration ceiling for the solar disk), and all three band-gap orderings $E_g \le E_d/2$, $E_d/2 \le E_g \le E_d$, and $E_d \le E_g$. The paper's own acknowledged but unpublished parameter space therefore spans four axes: non-radiative recombination in the converter and/or cell, concentrated illumination, general converter/cell band-gap ordering, and realistic coating reflectance. Two of these axes, non-radiative recombination and concentration, were already analyzed in the authors' companion model without the anti-reflection-coating and light-trapping optics (Badescu and De Vos 2007), and lossy front-converter limits were obtained independently by Abrams, Niv, and Zhang (2011).

Problem Statement

Within the ray-optical model of De Vos, Szymańska, and Badescu (2009) — a solar cell covered by a luminescent front down-converter, an anti-reflection coating, and light trapping, with full refractive-index/étendue treatment of luminescence extraction — determine how the reported efficiency and optical-efficiency conclusions change when the model's previously unreported parameter space is covered: (i) non-radiative recombination in the converter and/or the cell; (ii) concentrated sunlight up to the thermodynamic ceiling $C \approx 4.6 \times 10^4$; (iii) converter/cell band-gap orderings beyond the emphasized $E_d = 2E_g$ regime, i.e., the regimes $E_d/2 \le E_g \le E_d$ and $E_d \le E_g$ in addition to $E_g \le E_d/2$; and (iv) realistic anti-reflection coatings with non-zero, spectrally and angularly dependent reflectance in place of the idealized broad-band, all-angle, near-zero-reflectance coating. The single scientific objective is a self-consistent, complete four-axis map of converter-coupled cell performance in this framework, together with the identification of which of the 2009 ideal-case conclusions survive. All four axes are mandatory and must be treated within the 2009 optical framework; axes (i) and (ii), although already analyzed in the authors' companion model without anti-reflection-coating and light-trapping optics (Badescu and De Vos 2007), are to be re-derived under the 2009 optics rather than merely cited. Because axes (i)–(iv) each span a continuous range — the non-radiative recombination rate, the concentration factor $C$ up to the $4.6 \times 10^4$ ceiling, the converter/cell band-gap ratio $E_g/E_d$, and a spectrally and angularly dependent coating reflectance — the four-axis map must cover each axis across its full admissible range rather than through exhaustive sampling, and it must include at least one explicitly specified realistic non-ideal anti-reflection coating design. Locating which of the 2009 ideal-case conclusions survive is a required deliverable of the same objective, not an optional remark.

References

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