Problem 6.1

Fundamentals of Solar Cells and Photovoltaic Systems Engineering

Solutions Manual - Chapter 6

Problem 6.1

Using the tabulated data for the Quantum Efficiency (QE) of a III-V triple-junction solar cell provided in the online repository of this book:

(a) Obtain the short-circuit current density \(J_{SC}\) under the AM1.5D spectrum for each subcell.

(b) What is the current flowing throughout the device?

(c) Calculate the current balance of the top and middle subcells (\(J_{SC,top}\)/\(J_{SC,middle}\)).

We will use the package pandas to handle the data and matplotlib.pyplot to plot the results.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

We start by importing the data for the solar spectra.

datafile = pd.read_csv('data/Reference_spectrum_ASTM-G173-03.csv', index_col=0, header=0) 
datafile

	AM0	AM1.5G	AM1.5D
Wvlgth nm	Etr W*m-2*nm-1	Global tilt W*m-2*nm-1	Direct+circumsolar W*m-2*nm-1
280	8.20E-02	4.73E-23	2.54E-26
280.5	9.90E-02	1.23E-21	1.09E-24
281	1.50E-01	5.69E-21	6.13E-24
281.5	2.12E-01	1.57E-19	2.75E-22
...	...	...	...
3980	8.84E-03	7.39E-03	7.40E-03
3985	8.80E-03	7.43E-03	7.45E-03
3990	8.78E-03	7.37E-03	7.39E-03
3995	8.70E-03	7.21E-03	7.23E-03
4000	8.68E-03	7.10E-03	7.12E-03

2003 rows × 3 columns

datafile.drop(datafile.index[0], inplace=True)   #remove row including information on units
datafile=datafile.astype(float)                  #convert values to float for easy operation
datafile.index=datafile.index.astype(float)      #convert indexes to float for easy operation

We can also plot the three spectra

plt.plot(datafile, 
         linewidth=2, label=datafile.columns)
plt.ylabel('Spectral distribution (Etr W*m-2*nm-1)')
plt.xlabel('Wavelength (nm)')
plt.legend()

<matplotlib.legend.Legend at 0x7f8343bd1310>

We define the relevant constants and import the QE of the triple junction solar cell.

h=6.63*10**(-34)   # [J·s] Planck constant
e=1.60*10**(-19)   # [C] electron charge
c =299792458       # [m/s] Light speed

QE_data = pd.read_csv('data/QE_triple_junction_solar_cell.csv', index_col=0, header=0) 
QE_top=QE_data['QE top'].dropna()
QE_data.set_index('nm middle', inplace=True)
QE_mid=QE_data['QE middle'].dropna()
QE_data.set_index('nm bottom', inplace=True)
QE_bot=QE_data['QE bottom'].dropna()

We can plot the Quantum Efficiency.

plt.plot(QE_top, linewidth=2, label='top')
plt.plot(QE_mid, linewidth=2, label='middle')
plt.plot(QE_bot, linewidth=2, label='bottom')
plt.ylabel('Quantum Efficiency (QE)')
plt.xlabel('Wavelength, $\lambda$ (nm)');
plt.legend()

<matplotlib.legend.Legend at 0x7f8343af7d90>

For the top subcell, we calculate the spectral response, interpolate the spectrum, and integrate to obtain the short-circuit current density using Eq. 3.5.

\(J=\int SR(\lambda) \cdot G(\lambda) \ d\lambda\)

QE=QE_top
SR=pd.Series(index=QE.index,
            data=[QE.loc[i]*e*i*0.000000001/(h*c) for i in QE.index])

spectrum='AM1.5G'
spectra=datafile[spectrum]
spectra_interpolated=np.interp(SR.index, spectra.index, spectra.values)

J_top = np.trapz([x*y for x,y in zip(SR, spectra_interpolated)], x=SR.index)*1000/10000 # A-> mA ; m2 -> cm2
print('Photocurrent density top= ' + str(J_top.round(1)) + ' mA/cm2')

Photocurrent density top= 13.9 mA/cm2

We repeat the analysis for the middle subcell.

QE=QE_mid
SR=pd.Series(index=QE.index,
            data=[QE.loc[i]*e*i*0.000000001/(h*c) for i in QE.index])

spectra=datafile[spectrum]
spectra_interpolated=np.interp(SR.index, spectra.index, spectra.values)

J_mid = np.trapz([x*y for x,y in zip(SR, spectra_interpolated)], x=SR.index)*1000/10000 # A-> mA ; m2 -> cm2
print('Photocurrent density middle= ' + str(J_mid.round(1)) + ' mA/cm2')

Photocurrent density middle= 13.7 mA/cm2

We repeat the analysis for the bottom subcell.

QE=QE_bot
SR=pd.Series(index=QE.index,
            data=[QE.loc[i]*e*i*0.000000001/(h*c) for i in QE.index])

spectra=datafile[spectrum]
spectra_interpolated=np.interp(SR.index, spectra.index, spectra.values)

J_bot = np.trapz([x*y for x,y in zip(SR, spectra_interpolated)], x=SR.index)*1000/10000 # A-> mA ; m2 -> cm2
print('Photocurrent density bottom= ' + str(J_bot.round(1)) + ' mA/cm2')

Photocurrent density bottom= 19.3 mA/cm2

The middle subcell determines the current flowing throughout the device since it is the subcell that generates the lowest current. The current balance of the top and middle subcells (\(J_{SC,top}\)/\(J_{SC,middle}\)) can be calculated as follows:

J_top/J_mid

1.0091728232834571