Beyond the Yablonovitch limit: Trapping light by frequency shift

Tom Markvart
2011 Applied Physics Letters  
It is shown that randomizing the photon distribution over the frequency as well as orientation variables dramatically improves the efficiency of optical confinement in a weakly absorbing material such as crystalline silicon. The enhancement in average optical path length over the Yablonovitch limit ͓E. Yablonovitch, J. Opt. Soc. Am. 72, 899 ͑1982͔͒ is given by an inverse Boltzmann factor of the frequency shift, making it possible to manufacture, for example, efficient crystalline silicon solar
more » ... ells of thickness barely 1 m. Increasing the optical path lengths by surface texturing is a convenient way to increase light absorption and reduce the thickness of solar cells ͑see, for example, Refs. 1 and 2͒. Yablonovitch 3 showed that the maximum path length 4n 2 d ͑where n and d are the refractive index and thickness of the layer͒ can be achieved by a random ͑Lambertian͒ texture. In this paper, we show that a further substantial increase in the optical path length can be achieved by photonic structure which randomizes photon momentum as well as its direction. Such scheme is based on combining a weakly absorbing semiconductor layer with a highly absorbing luminescent film and a photonic band stop mirror to trap the thermalized light. In practical terms, this allows effective light absorption close to the silicon band edge and makes it feasible to contemplate silicon-based optoelectronic devices such as solar cells which are merely 1 m thick. Figure 1 shows a typical geometric light-trapping scheme with a textured rear surface. 4 The textured surface can be pictured as scattering photons between radiation ͑externally coupled͒ and trapped ͑internal͒ photon modes, producing a uniform distribution of photons over all directions. When incident from within onto the top surface, photons in modes with directions outside the "escape cone" are reflected while photons within the cone are emitted from the structure. Neglecting reflection of rays inside the escape cone, one obtains an average reflection coefficient of 1 / n 2 . Taking into account the average photon path length 4d per double passage through the layer yields the Yablonovitch result. The light-trapping scheme described in this letter extends this philosophy by adding a frequency dimension to the geometrical coordinates. Such scheme is obtained by replacing the textured reflecting rear surface of Fig. 1 with a highly absorbing fluorescent layer of a similar refractive index as silicon which is again capped by a perfect reflector ͑Fig. 2͒. We assume that all photons incident on this layer are absorbed and all absorbed photons reemitted as fluorescence. The isotropic emission process ensures random orientations as in the case reflection by the Lambertian surface ͑in other words, an equilibrium distribution of directions͒. The detailed balance between absorption and emission, expressed by the Kennard-Stepanov relations, 5 now adds a similar element to the frequency ͑or energy͒ distribution of the photon gas within the silicon slab. Interaction between radiation and matter through repeated absorption and emission events brings the radiation into thermal equilibrium at the temperature T o of the absorbing/fluorescent layer ͑or, more precisely, its Kennard-Stepanov temperature 5 ͒ as demonstrated, for example, in Ref. 6. Thus, the specific emission spectrum of the fluorescent layer evolves into the Bose-Einstein distribution, where is the ͑generally nonzero͒ chemical potential. The light-trapping scheme is completed by a photonic filter/mirror placed on the front face of the semiconductor slab. This mirror reflects all photons with frequency below some frequency o but transmits all incident light with frequency above this threshold. This restriction on which photon energy is allowed to enter or escape the silicon slab parallels the angular escape cone represented by total internal reflection. We shall now show that adding the fluorescent component dramatically extends the path length ᐉ of photons in the structure. Let us suppose that light absorption in the semiconductor is weak ͑␣ Ӷ 1 / d, where ␣ is the absorption coef-ficient͒. The probability of photon absorption rather than escape through the front face is then equal to where Ṅ abs is the rate of absorption and Ṅ em is the rate of photon emission by the structure. Clearly, a͒ Electronic mail: FIG. 1. A schematic depiction of a light-trapping scheme with back surface texturing and reflector ͑sin c =1/ n͒. APPLIED PHYSICS LETTERS 98, 071107 ͑2011͒
doi:10.1063/1.3554436 fatcat:335ucz4hgbe3jlwjpccsbiptei