Textured surfaces: pyramids and light trapping

Anisotropic etching of a (100) silicon wafer in hot alkali exposes the slowly etching (111) planes and leaves a field of upright pyramids whose facets make $\alpha = 54.74°$ (the angle between (100) and (111)) with the wafer plane. The pyramids are several micrometres across, large compared with the wavelength, so the texture does not change the Fresnel coefficients of each facet; it changes where a reflected ray goes next, and what happens to a ray once it is inside. Both effects are geometric, and both are why every crystalline silicon solar cell is textured.

Two chances to enter

A vertical ray meets a facet at the local angle of incidence $\theta_1 = \alpha$. The specularly reflected ray leaves rotated by $2\alpha$ from the vertical; for $\alpha > 45°$ it still points downward and strikes the opposite facet, at

Light escapes only if it is reflected both times, so the front-surface reflectance becomes the product

and bare silicon's 35 % at 55° times 32 % at 16° gives 11 %, against 35 % for a polished wafer. Each factor is the reflectance of the coated silicon surface at that angle, which is a film calculation: with a 75 nm silicon nitride layer, a quarter wave at 600 nm for the facet's angle, the factors fall to a few percent each and their product well below one percent. The coating must be designed for the facet's incidence angle, not for normal incidence, and for the mix of s and p that the two reflections produce (the first reflection partly polarizes the light, so the second sees a polarization-weighted average). Rays that hit near a pyramid's base can escape after one reflection, and the fraction that does depends on the exact geometry; a hemispherical reflectance measurement of a textured wafer sums all of this and is what the geometric model is checked against.

import numpy as np
from pystratify import planar
from pystratify.docs import material  # n + ik of a tabulated page (refractiveindex.info format)

n = lambda wl: [1.0, material('docs/data/tutorials/Si3N4_Luke.yml', wl), material('docs/data/tutorials/Si_Green-2008.yml', wl)]  # incident medium, layers, substrate
d = [np.inf, 0.075, np.inf]  # thicknesses, µm
wl = 600 * 1e-3  # wavelength, µm
angles = 0.25 + 0.5 * np.arange(180)  # degrees
R = {pol: np.array([planar.coh_tmm(pol, n(wl), d, np.radians(a), wl)['R'] for a in angles]) for pol in ('s', 'p')}

Trapping inside

Once in the silicon, refracted rays travel obliquely, by $\alpha - \theta_t$ from the vertical, about 41° for the first refraction. Weakly absorbed near-infrared light (silicon's absorption length is 100 µm at 1000 nm and a millimetre at 1100 nm, against a 150 µm wafer) reaches the back at an angle, reflects from the back contact or back mirror, and meets the front facets beyond their critical angle of 16°, so it is reflected back in and bounces many times before an escape cone lets it out. The path length is multiplied, and with it the absorption in the long-wavelength tail that sets a silicon cell's current.

The Lambertian limit: 4n²

How much can any texture achieve? Yablonovitch (1982) answered with a thermodynamic argument. A surface that randomises the direction of light completely (a Lambertian surface) fills the whole solid angle inside the slab with radiation of uniform brightness. The density of optical modes inside a medium of index $n$ is $n^2$ times that in air, and only the modes within the escape cone, a fraction $1/n^2$ of the hemisphere's solid angle (by Snell's law and $\sin^2\theta_c = 1/n^2$), can leave through the front. The mean path a ray travels before it escapes is therefore enhanced by $2n^2$ relative to a single vertical pass if the back is a mirror, and by $4n^2$ once the double pass between front and back is counted:

for silicon ($n \approx 3.5$ in the near infrared). The argument is one of étendue, or equivalently of detailed balance between the radiation inside and outside the slab, and the limit holds for any geometry that randomises without absorbing; it is the yardstick against which every light-trapping scheme, from pyramids to nanophotonic gratings, is measured, and in the weakly absorbing limit the absorption of a Lambertian slab is

(Yablonovitch and Cody 1982). Pyramids come close to it: Campbell and Green (1987) traced rays through regular and inverted pyramids and found enhancements within a factor of two of the limit, with inverted pyramids and random "tiler's" patterns doing best, because a regular array returns some rays to their entry facet.

Where the ray picture stops

The geometric treatment holds while the features are many wavelengths across and the surfaces between them are locally flat. A texture of a few hundred nanometres, such as the random "black silicon" of the moth-eye section or a periodic nanograting, is neither: it diffracts, and its light trapping is a question of coupling into the guided modes of the wafer, which can exceed $4n^2$ over a narrow band and falls short over a broad one. Between the two regimes, pyramids of a micrometre, nothing simple applies and a rigorous solver is needed. Following rays through the whole wafer, with its coatings and its back reflector, also needs ray tracing rather than a transfer matrix; the calculator's surface textures do it with RayFlare, a separate engine that applies the transfer-matrix reflectances of the coated facets to each ray and counts where it goes. The per-facet reflectance of the example above is the ingredient of that calculation, and the check that it is right.