Effect of front pyramid texture and/or rear grating on absorption in Si
Effect of front pyramid texture and/or rear grating on absorption in Si¶
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).
First, importing relevant packages:
[19]:
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
from solcore.material_system import create_new_material
import matplotlib.pyplot as plt
import matplotlib as mpl
import seaborn as sns
from sparse import load_npz
To make sure we are using the same optical constants for Si, load the same Si n/k data used in the paper linked above:
[20]:
create_new_material('Si_OPTOS', 'Si_OPTOS_n.txt', 'Si_OPTOS_k.txt')
You only need to do this one time, then the material will be stored in Solcore’s material database.
Setting options (taking the default options for everything not specified explicitly):
[21]:
angle_degrees_in = 8
wavelengths = np.linspace(900, 1200, 30)*1e-9
Si = material('Si_OPTOS')()
Air = material('Air')()
options = default_options()
options.wavelength = wavelengths
options.theta_in = angle_degrees_in*np.pi/180
options.n_theta_bins = 100
options.c_azimuth = 0.25
options.n_rays = 25*25*1300
options.project_name = 'OPTOS_comparison'
options.orders = 60 # decrease number of orders to speed up calculation
options.pol = 'u'
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.
[22]:
x = 1000
d_vectors = ((x, 0),(0,x))
area_fill_factor = 0.36
hw = np.sqrt(area_fill_factor)*500
back_materials = [Layer(si('120nm'), 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.
[23]:
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=60)
back_surf_planar = Interface('TMM', layers=[], name = 'planar_back')
bulk_Si = BulkLayer(201.8e-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.
[24]:
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)
process_structure(SC_fig7, options)
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 = 962.0689655172415 nm
INFO: RCWA calculation for wavelength = 982.7586206896552 nm
INFO: RCWA calculation for wavelength = 910.3448275862069 nm
INFO: RCWA calculation for wavelength = 941.3793103448277 nm
INFO: RCWA calculation for wavelength = 993.1034482758622 nm
INFO: RCWA calculation for wavelength = 900.0000000000001 nm
INFO: RCWA calculation for wavelength = 920.6896551724138 nm
INFO: RCWA calculation for wavelength = 972.4137931034483 nm
INFO: RCWA calculation for wavelength = 931.0344827586207 nm
INFO: RCWA calculation for wavelength = 951.7241379310346 nm
INFO: RCWA calculation for wavelength = 1003.448275862069 nm
INFO: RCWA calculation for wavelength = 1013.7931034482759 nm
INFO: RCWA calculation for wavelength = 1024.1379310344828 nm
INFO: RCWA calculation for wavelength = 1034.4827586206898 nm
INFO: RCWA calculation for wavelength = 1044.8275862068965 nm
INFO: RCWA calculation for wavelength = 1055.1724137931035 nm
INFO: RCWA calculation for wavelength = 1065.5172413793102 nm
INFO: RCWA calculation for wavelength = 1075.8620689655174 nm
INFO: RCWA calculation for wavelength = 1086.2068965517244 nm
INFO: RCWA calculation for wavelength = 1096.5517241379314 nm
INFO: RCWA calculation for wavelength = 1106.8965517241381 nm
INFO: RCWA calculation for wavelength = 1117.2413793103449 nm
INFO: RCWA calculation for wavelength = 1127.5862068965516 nm
INFO: RCWA calculation for wavelength = 1137.9310344827586 nm
INFO: RCWA calculation for wavelength = 1148.2758620689656 nm
INFO: RCWA calculation for wavelength = 1158.6206896551726 nm
INFO: RCWA calculation for wavelength = 1168.9655172413793 nm
INFO: RCWA calculation for wavelength = 1179.3103448275863 nm
INFO: RCWA calculation for wavelength = 1189.655172413793 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 = 910.3448275862069 nm
INFO: RT calculation for wavelength = 920.6896551724138 nm
INFO: RT calculation for wavelength = 931.0344827586207 nm
INFO: RT calculation for wavelength = 941.3793103448277 nm
INFO: RT calculation for wavelength = 951.7241379310346 nm
INFO: RT calculation for wavelength = 962.0689655172415 nm
INFO: RT calculation for wavelength = 972.4137931034483 nm
INFO: RT calculation for wavelength = 982.7586206896552 nm
INFO: RT calculation for wavelength = 993.1034482758622 nm
INFO: RT calculation for wavelength = 1003.448275862069 nm
INFO: RT calculation for wavelength = 1013.7931034482759 nm
INFO: RT calculation for wavelength = 1024.1379310344828 nm
INFO: RT calculation for wavelength = 1034.4827586206898 nm
INFO: RT calculation for wavelength = 1044.8275862068965 nm
INFO: RT calculation for wavelength = 1055.1724137931035 nm
INFO: RT calculation for wavelength = 1065.5172413793102 nm
INFO: RT calculation for wavelength = 1075.8620689655174 nm
INFO: RT calculation for wavelength = 1086.2068965517244 nm
INFO: RT calculation for wavelength = 1096.5517241379314 nm
INFO: RT calculation for wavelength = 1106.8965517241381 nm
INFO: RT calculation for wavelength = 1117.2413793103449 nm
INFO: RT calculation for wavelength = 1127.5862068965516 nm
INFO: RT calculation for wavelength = 1137.9310344827586 nm
INFO: RT calculation for wavelength = 1148.2758620689656 nm
INFO: RT calculation for wavelength = 1158.6206896551726 nm
INFO: RT calculation for wavelength = 1168.9655172413793 nm
INFO: RT calculation for wavelength = 1179.3103448275863 nm
INFO: RT calculation for wavelength = 1189.655172413793 nm
INFO: RT calculation for wavelength = 1200.0000000000002 nm
INFO: RT calculation for wavelength = 900.0000000000001 nm
INFO: RT calculation for wavelength = 910.3448275862069 nm
INFO: RT calculation for wavelength = 920.6896551724138 nm
INFO: RT calculation for wavelength = 931.0344827586207 nm
INFO: RT calculation for wavelength = 941.3793103448277 nm
INFO: RT calculation for wavelength = 951.7241379310346 nm
INFO: RT calculation for wavelength = 962.0689655172415 nm
INFO: RT calculation for wavelength = 972.4137931034483 nm
INFO: RT calculation for wavelength = 982.7586206896552 nm
INFO: RT calculation for wavelength = 993.1034482758622 nm
INFO: RT calculation for wavelength = 1003.448275862069 nm
INFO: RT calculation for wavelength = 1013.7931034482759 nm
INFO: RT calculation for wavelength = 1024.1379310344828 nm
INFO: RT calculation for wavelength = 1034.4827586206898 nm
INFO: RT calculation for wavelength = 1044.8275862068965 nm
INFO: RT calculation for wavelength = 1055.1724137931035 nm
INFO: RT calculation for wavelength = 1065.5172413793102 nm
INFO: RT calculation for wavelength = 1075.8620689655174 nm
INFO: RT calculation for wavelength = 1086.2068965517244 nm
INFO: RT calculation for wavelength = 1096.5517241379314 nm
INFO: RT calculation for wavelength = 1106.8965517241381 nm
INFO: RT calculation for wavelength = 1117.2413793103449 nm
/Users/z3533914/.pyenv/versions/3.11.5/envs/main_develop/lib/python3.11/site-packages/joblib/externals/loky/process_executor.py:752: UserWarning: A worker stopped while some jobs were given to the executor. This can be caused by a too short worker timeout or by a memory leak.
warnings.warn(
INFO: RT calculation for wavelength = 1127.5862068965516 nm
INFO: RT calculation for wavelength = 1137.9310344827586 nm
INFO: RT calculation for wavelength = 1148.2758620689656 nm
INFO: RT calculation for wavelength = 1158.6206896551726 nm
INFO: RT calculation for wavelength = 1168.9655172413793 nm
INFO: RT calculation for wavelength = 1179.3103448275863 nm
INFO: RT calculation for wavelength = 1189.655172413793 nm
INFO: RT calculation for wavelength = 1200.0000000000002 nm
INFO: Making matrix for planar surface using TMM for element 2 in structure
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.
[25]:
results_fig6= calculate_RAT(SC_fig6, options)
results_fig7 = calculate_RAT(SC_fig7, options)
results_fig8 = calculate_RAT(SC_fig8, options)
RAT_fig6 = results_fig6[0]
RAT_fig7 = results_fig7[0]
RAT_fig8 = results_fig8[0]
sim_fig6 = np.loadtxt('optos_fig6_sim.csv', delimiter=',')
sim_fig7 = np.loadtxt('optos_fig7_sim.csv', delimiter=',')
sim_fig8 = np.loadtxt('optos_fig8_sim.csv', delimiter=',')
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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.
[26]:
struc = tmm_structure([Layer(si('200um'), Si)], incidence=Air, transmission=Air)
options.coherent = False
options.coherency_list = ['i']
RAT = tmm_structure.calculate(struc, options)
Plot everything together:
[27]:
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 agreement between the reference values and our calculated values. 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:
[28]:
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()
Plot the redistribution matrix for the rear grating (summed over azimuthal angles) at 1100 nm:
[29]:
theta_intv, phi_intv, angle_vector = make_angle_vector(options['n_theta_bins'],
options['phi_symmetry'],
options['c_azimuth'])
path = get_savepath('default', 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=.5, rot=-.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()
