Textured Si: diffraction grating and pyramids

This example is based on Figures 6, 7 and 8 from this paper. This compares three different structures, all based on a 200 micron thick slab of silicon with different surface textures:

Planar front surface (TMM), crossed grating at rear (RCWA)
Inverted pyramids on the front surface (RT_Fresnel), planar rear surface (TMM)
Inverted pyramids on the front surface (RT_Fresnel), crossed grating at rear (RCWA)

The methods which will be used to calculate the redistribution matrices in each case are given in brackets. If case 1 and 2 are calculated first, then case 3 does not require the calculations of any additional matrices, since it will use the rear matrix from (1) and the front matrix from (2).

Setting up

First, importing relevant packages:

import numpy as np
import os

# solcore imports
from solcore.structure import Layer
from solcore import material
from solcore import si

from rayflare.structure import Interface, BulkLayer, Structure
from rayflare.matrix_formalism import process_structure, calculate_RAT
from rayflare.utilities import get_savepath
from rayflare.transfer_matrix_method import tmm_structure
from rayflare.angles import theta_summary, make_angle_vector
from rayflare.textures import regular_pyramids
from rayflare.options import default_options

import matplotlib.pyplot as plt
import matplotlib as mpl
import seaborn as sns
from sparse import load_npz

Setting options (taking the default options for everything not specified explicitly):

angle_degrees_in = 8 # same as in Fraunhofer paper

wavelengths = np.linspace(900, 1200, 20) * 1e-9

Si = material("Si")()
Air = material("Air")()

options = default_options()
options.wavelength = wavelengths
options.theta_in = angle_degrees_in * np.pi / 180 # incidence angle (polar angle) in radians
options.n_theta_bins = 30
options.c_azimuth = 0.25
options.n_rays = 5e4 # number of rays per wavelength in ray-tracing
options.project_name = "OPTOS_comparison"
# options.orders = 60 # number of RCWA orders to use (more = better convergence, but slower)
options.pol = "u" # unpolarized light
options.only_incidence_angle = True
options.RCWA_method = "Inkstone"
options.parallel = True

Defining the structures

Now, set up the grating basis vectors for the RCWA calculations and define the grating structure. These are squares, rotated by 45 degrees. The halfwidth is calculated based on the area fill factor of the etched pillars given in the paper.

x = 1000

d_vectors = ((x, 0), (0, x))
area_fill_factor = 0.36
hw = np.sqrt(area_fill_factor) * 500

back_materials = [
    Layer(width=si("120nm"), material=Si,
        geometry=[{"type": "rectangle", "mat": Air, "center": (x / 2, x / 2),
                "halfwidths": (hw, hw), "angle": 45}],
    )]

Now we define the pyramid texture for the front surface in case (2) and (3) and make the four possible different surfaces: planar front and rear, front with pyramids, rear with grating. We specify the method to use to calculate the redistribution matrices in each case and create the bulk layer.

surf = regular_pyramids(elevation_angle=55, upright=False)

front_surf_pyramids = Interface(
    "RT_Fresnel",
    texture=surf,
    layers=[],
    name="inv_pyramids_front_" + str(options["n_rays"]),
)

front_surf_planar = Interface("TMM", layers=[], name="planar_front")

back_surf_grating = Interface(
    "RCWA",
    layers=back_materials,
    name="crossed_grating_back",
    d_vectors=d_vectors,
    rcwa_orders=30,
)

back_surf_planar = Interface("TMM", layers=[], name="planar_back")

bulk_Si = BulkLayer(200e-6, Si, name="Si_bulk")

Now we create the different structures and ‘process’ them (this will calculate the relevant matrices if necessary, or do nothing if it finds the matrices have previously been calculated and the files already exist). We don’t need to process the final structure because it will use matrices calculated for SC_fig6 and SC_fig7.

%%capture

SC_fig6 = Structure([front_surf_planar, bulk_Si, back_surf_grating], incidence=Air, transmission=Air)

SC_fig7 = Structure([front_surf_pyramids, bulk_Si, back_surf_planar], incidence=Air, transmission=Air)

SC_fig8 = Structure([front_surf_pyramids, bulk_Si, back_surf_grating], incidence=Air, transmission=Air)

process_structure(SC_fig6, options, save_location='current') # if you want to overwrite previous results, add overwrite=Trues
process_structure(SC_fig7, options, save_location='current')

INFO: Making matrix for planar surface using TMM for element 0 in structure
INFO: RCWA calculation for element 2 in structure
INFO: RCWA calculation for wavelength = 900.0000000000001 nm
INFO: RCWA calculation for wavelength = 915.7894736842105 nm
INFO: RCWA calculation for wavelength = 931.578947368421 nm
INFO: RCWA calculation for wavelength = 947.3684210526316 nm
INFO: RCWA calculation for wavelength = 963.1578947368421 nm
INFO: RCWA calculation for wavelength = 978.9473684210526 nm
INFO: RCWA calculation for wavelength = 1010.5263157894736 nm
INFO: RCWA calculation for wavelength = 994.7368421052631 nm
INFO: RCWA calculation for wavelength = 1026.3157894736842 nm
INFO: RCWA calculation for wavelength = 1042.1052631578948 nm
INFO: RCWA calculation for wavelength = 1057.8947368421054 nm
INFO: RCWA calculation for wavelength = 1073.6842105263158 nm
INFO: RCWA calculation for wavelength = 1089.4736842105265 nm
INFO: RCWA calculation for wavelength = 1105.2631578947369 nm
INFO: RCWA calculation for wavelength = 1121.0526315789475 nm
INFO: RCWA calculation for wavelength = 1136.842105263158 nm
INFO: RCWA calculation for wavelength = 1152.6315789473683 nm
INFO: RCWA calculation for wavelength = 1168.421052631579 nm
INFO: RCWA calculation for wavelength = 1184.2105263157896 nm
INFO: RCWA calculation for wavelength = 1200.0000000000002 nm
INFO: Ray tracing with Fresnel equations for element 0 in structure
INFO: Calculating matrix only for incidence theta/phi
INFO: RT calculation for wavelength = 900.0000000000001 nm
INFO: RT calculation for wavelength = 915.7894736842105 nm
INFO: RT calculation for wavelength = 947.3684210526316 nm
INFO: RT calculation for wavelength = 963.1578947368421 nm
INFO: RT calculation for wavelength = 931.578947368421 nm
INFO: RT calculation for wavelength = 978.9473684210526 nm
INFO: RT calculation for wavelength = 994.7368421052631 nm
INFO: RT calculation for wavelength = 1010.5263157894736 nm
INFO: RT calculation for wavelength = 1026.3157894736842 nm
INFO: RT calculation for wavelength = 1042.1052631578948 nm
INFO: RT calculation for wavelength = 1057.8947368421054 nm
INFO: RT calculation for wavelength = 1073.6842105263158 nm
INFO: RT calculation for wavelength = 1089.4736842105265 nm
INFO: RT calculation for wavelength = 1105.2631578947369 nm
INFO: RT calculation for wavelength = 1121.0526315789475 nm
INFO: RT calculation for wavelength = 1136.842105263158 nm
INFO: RT calculation for wavelength = 1152.6315789473683 nm
INFO: RT calculation for wavelength = 1168.421052631579 nm
INFO: RT calculation for wavelength = 1184.2105263157896 nm
INFO: RT calculation for wavelength = 1200.0000000000002 nm
INFO: RT calculation for wavelength = 915.7894736842105 nm
INFO: RT calculation for wavelength = 900.0000000000001 nm
INFO: RT calculation for wavelength = 931.578947368421 nm
INFO: RT calculation for wavelength = 947.3684210526316 nm
INFO: RT calculation for wavelength = 963.1578947368421 nm
INFO: RT calculation for wavelength = 994.7368421052631 nm
INFO: RT calculation for wavelength = 978.9473684210526 nm
INFO: RT calculation for wavelength = 1010.5263157894736 nm
INFO: RT calculation for wavelength = 1026.3157894736842 nm
INFO: RT calculation for wavelength = 1042.1052631578948 nm
INFO: RT calculation for wavelength = 1057.8947368421054 nm
INFO: RT calculation for wavelength = 1073.6842105263158 nm
INFO: RT calculation for wavelength = 1089.4736842105265 nm
INFO: RT calculation for wavelength = 1105.2631578947369 nm
INFO: RT calculation for wavelength = 1121.0526315789475 nm
INFO: RT calculation for wavelength = 1136.842105263158 nm
INFO: RT calculation for wavelength = 1152.6315789473683 nm
INFO: RT calculation for wavelength = 1168.421052631579 nm
INFO: RT calculation for wavelength = 1184.2105263157896 nm
INFO: RT calculation for wavelength = 1200.0000000000002 nm
INFO: Making matrix for planar surface using TMM for element 2 in structure

Calculating R/A/T

Then we ask RayFlare to calculate the reflection, transmission and absorption through matrix multiplication, and get the required result out (absorption in the bulk) for each cell. We also load the results from the reference paper to compare them to the ones calculated with RayFlare.

%%capture

results_fig6 = calculate_RAT(SC_fig6, options, save_location='current')
results_fig7 = calculate_RAT(SC_fig7, options, save_location='current')
results_fig8 = calculate_RAT(SC_fig8, options, save_location='current')

RAT_fig6 = results_fig6[0]
RAT_fig7 = results_fig7[0]
RAT_fig8 = results_fig8[0]

sim_fig6 = np.loadtxt("data/optos_fig6_sim.csv", delimiter=",")
sim_fig7 = np.loadtxt("data/optos_fig7_sim.csv", delimiter=",")
sim_fig8 = np.loadtxt("data/optos_fig8_sim.csv", delimiter=",")

INFO: After iteration 1: maximum power fraction remaining = 0.3147547844901925
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Finally, we use TMM to calculate the absorption in a structure with a planar front and planar rear, as a reference.

struc = tmm_structure([Layer(si("200um"), Si)], incidence=Air, transmission=Air)
options.coherent = False
options.coherency_list = ["i"]
RAT = tmm_structure.calculate(struc, options)

Plotting

Plot everything together, including data from the reference paper for comparison:

palhf = sns.color_palette("hls", 4)

fig = plt.figure()
plt.plot(sim_fig6[:, 0], sim_fig6[:, 1],
    "--", color=palhf[0], label="OPTOS - rear grating (1)")
plt.plot(wavelengths * 1e9, RAT_fig6["A_bulk"][0],
    "-o", color=palhf[0], label="RayFlare - rear grating (1)", fillstyle="none")
plt.plot(sim_fig7[:, 0], sim_fig7[:, 1],
    "--", color=palhf[1], label="OPTOS - front pyramids (2)",)
plt.plot(wavelengths * 1e9, RAT_fig7["A_bulk"][0],
    "-o", color=palhf[1], label="RayFlare - front pyramids (2)", fillstyle="none")
plt.plot(sim_fig8[:, 0], sim_fig8[:, 1],
    "--", color=palhf[2], label="OPTOS - grating + pyramids (3)")
plt.plot(wavelengths * 1e9, RAT_fig8["A_bulk"][0],
    "-o", color=palhf[2],label="RayFlare - grating + pyramids (3)", fillstyle="none",)
plt.plot(wavelengths * 1e9, RAT["A_per_layer"][:, 0], "-k", label="Planar")
plt.legend(loc="lower left")
plt.xlabel("Wavelength (nm)")
plt.ylabel("Absorption in Si")
plt.xlim([900, 1200])
plt.ylim([0, 1])
plt.show()

We can see good (qualitative) agreement between the reference values and our calculated values. The default values for the number of rays, RCWA orders, and bins in the angular redistribution matrix in this example are set to be quite low – agreement between the literature values and the values calculated before improves if these are increased (but the calculation takes much longer). The structure with rear grating also behaves identically to the planar TMM reference case at the short wavelengths where front surface reflection dominates the result, as expected. Clearly, the pyramids perform much better overall, giving a large boost in the absorption at long wavelengths and also reducing the reflection significantly at shorter wavelengths. Plotting reflection and transmission emphasises this:

fig = plt.figure()
plt.plot(wavelengths * 1e9,RAT_fig6["R"][0],
    "-o", color=palhf[0], label="RayFlare - rear grating (1)", fillstyle="none")
plt.plot(wavelengths * 1e9, RAT_fig7["R"][0],
    "-o", color=palhf[1], label="RayFlare - front pyramids (2)", fillstyle="none")
plt.plot(wavelengths * 1e9, RAT_fig8["R"][0],
    "-o", color=palhf[2], label="RayFlare - grating + pyramids (3)", fillstyle="none")

plt.plot(wavelengths * 1e9, RAT_fig6["T"][0], "--o", color=palhf[0])
plt.plot(wavelengths * 1e9, RAT_fig7["T"][0], "--o", color=palhf[1])
plt.plot(wavelengths * 1e9, RAT_fig8["T"][0], "--o", color=palhf[2])

# these are just to create the legend:
plt.plot(-1, 0, "k-o", label="R", fillstyle="none")
plt.plot(-1, 0, "k--o", label="T")

plt.legend()
plt.xlabel("Wavelength (nm)")
plt.ylabel("Reflected/transmitted fraction")
plt.xlim([900, 1200])
plt.ylim([0, 0.6])
plt.show()

Redistribution matrices

Plot the redistribution matrix for the rear grating (summed over azimuthal angles) at 1100 nm:

theta_intv, phi_intv, angle_vector = make_angle_vector(
    options["n_theta_bins"], options["phi_symmetry"], options["c_azimuth"])

path = get_savepath(save_location='current', project_name=options.project_name)
sprs = load_npz(os.path.join(path, SC_fig6[2].name + "frontRT.npz"))

wl_to_plot = 1100e-9
wl_index = np.argmin(np.abs(wavelengths - wl_to_plot))

full = sprs[wl_index].todense()

summat = theta_summary(full, angle_vector, options["n_theta_bins"], "front")
summat_r = summat[: options["n_theta_bins"], :]
summat_r = summat_r.rename({ r"$\theta_{in}$": r"$\sin(\theta_{in})$",
        r"$\theta_{out}$": r"$\sin(\theta_{out})$"})

summat_r = summat_r.assign_coords({r"$\sin(\theta_{in})$": np.sin(summat_r.coords[r"$\sin(\theta_{in})$"]).data,
    r"$\sin(\theta_{out})$": np.sin(summat_r.coords[r"$\sin(\theta_{out})$"]).data})

palhf = sns.cubehelix_palette(256, start=0.5, rot=-0.9)
palhf.reverse()
seamap = mpl.colors.ListedColormap(palhf)

fig = plt.figure()
ax = plt.subplot(111)
ax = summat_r.plot.imshow(ax=ax, cmap=seamap, vmax=0.3)
plt.show()

Questions

If you can add only one of the textures (pyramids or a grating), which one is better? Why?
Why do the structures with a front-surface texture have high reflection at long wavelengths? The anti-reflection properties of pyramids (treated with ray optics) are mostly independent of the wavelength, so why does apparent reflection increase near the bandgap of Si?
Can you explain any of the features present in the angular redistribution matrix of the rear grating surface?