EEE465 Mini Project 2 Solution

The goal of the series of mini-projects is to build a model to allow us to calculate and optimize a PV system. In the first project, we re-acquainted everyone with programming, and calculated solar power density and energy on a surface with arbitrary tilt and azimuth. In this project, we will calculate and optimize the efficiency of a solar cell. Mini Project 3 will calculate the power from a module; Project 4 calculates the PV system parameters.
Overview
A solar cell is characterized by the equation:
(𝑉−𝐽𝑅𝑠𝑒𝑟𝑖𝑒𝑠)
𝐽 = 𝐽0 (𝑒 𝑘𝑇 − 1) − 𝐽𝐿
𝑉𝑜𝑐𝐽𝑠𝑐𝐹𝐹
efficiency =
𝑃𝑖𝑛
We divide the assignment into five parts:
Part 1: Read in material and solar cell device parameters.
Part 2: Calculate the IV curve parameters: J0, JL and Rseries. As part of this we also calculate the quantum efficiency.
Part 3: Calculate the efficiency parameters VOC, JSC, FF and Pin and calculate efficiency.
Part 4: Vary parameters to optimize efficiency.
Part 5: Use your Solar Cell Simulator to Analyze a solar cell.









Set up your program to read in or acquire the material and device parameters. “Hardwiring” them in the code is OK. The default values for Silicon (Si) are in the table below.
Name in code Emitter Value Base Value Units
MATERIAL PARAMETERS ASSUMED INDEPENDENT OF DOPING
N-p or p-n junction n-type
Intrinsic carrier concentration (ni) ni 8.6  109 cm-³ or 1  1010 (first is more accurate, second is more commonly used.) cm-³
Minority carrier lifetime 0 0 = 1; calculate eff based off of this value and parameters given below in table msec
Absorption coefficient () ab Absorption coefficient and AM1.5G spectrum as a function of  in “10_nm_AM15G_absorption.txt” cm-1
EMITTER AND BASE PARANETERS
Diffusivity (D) DE DB 15 30 cm²/s
Doping (N) NE NB 1  1018 5  1016 cm-3
Surface
Recombination (S) SE, SB 500 1000 cm/sec
Thickness (xj, H) WE , WB 1 300 Microns or cm
CALCULATED PARAMETERS
Diffusion Length (L) LE or LB 𝐿 = √𝐷𝜏𝑒𝑓𝑓 Microns or cm
Mobility () 𝐷 𝑘𝑇
=
𝜇 𝑞 cm²/V.s
Minority carrier lifetime 1 = 𝐶𝐴𝑢𝑔𝑒𝑟𝑛2 + 1
𝜏𝑒𝑓𝑓 𝜏0 𝐶𝐴𝑢𝑔𝑒𝑟 = 1.5 × 10−30 cm6/s

****** WATCH OUT FOR THE UNITS ******
The biggest mistake (by far) I see in debugging is unit errors, which admittedly is not helped by the semiconductor convention of specifying thickness in microns, and all the other parameters in units that use “cm”. If you get strange numbers, first check unit consistency. Output:
(1) Plot the minority carrier lifetime as a function of doping for the n-type emitter – use a log plot for the x-axis doping and y-axis lifetime. LABEL UNITS!!
(2) Plot the absorption coefficient as a function of wavelength, labelling units and using log plot for .
(3) Print the emitter, base and material parameters in a readable format and compare to tabulated values.

1. Calculate J0 for a device with the test solar cell parameters and compare with the values in Appendix A: Output of the Python script for the test case. The variation generally comes from two sources: accuracy (significant figures) in physical constants; and variation in material parameters. For example, the most commonly used value of ni in Si is 1  1010 cm-³, but an updated value is 8.6  109 cm-³. You can use either.
For reference, the J0 equation for electrons in p-type material is repeated from the notes below; ntype is the same, expect all hole carrier parameters are used and NA is replaced by ND.
𝑆𝑛𝐿𝑛 𝐻 𝐻
𝐷
𝐽0𝑛 = 𝑞 𝐿𝑛𝑛 𝑁𝑛𝐴𝑖2 [𝑆𝐷𝑛𝐿𝑛𝑛𝑛 sinhcosh((𝐿𝐿𝐻𝑛𝑛))++coshsinh((𝐿𝐿𝐻𝑛𝑛))]
𝐷
Output: Print the emitter, base and total J0 for the test solar cell and compare to the given value in Appendix A.
2. Calculate JL.
Calculating JL is conceptually straight forward (although computationally long). It is done by:
a. Calculating the number of photons/m² in each wavelength band in the solar spectrum – this is called photon flux.
b. Calculating the quantum efficiency at each wavelength, which physically corresponds to the fraction of incident photons converted to current at each wavelength.
c. Multiplying to two above and adding them all up over all the wavelengths.
More details on each of the a,b,c parts from above:
a. Calculate the photon flux at each wavelength.
For this, we need to read in the standard solar spectrum (which is the spectral irradiance or the POWER DENSITY per wavelength range). Then we convert it to the number of photons at each wavelength (photon flux) by
𝜆𝑖
Φ𝑖 = 𝑆𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝐼𝑟𝑟𝑎𝑑𝑖𝑎𝑛𝑐𝑒𝑖 × [Δ𝜆] ×
ℎ𝑐
Where “i” denotes the wavelength at which the spectral irradiance is given. Please, make sure that you calculate the photon energy ℎ 𝑐 in Joules. If you use the expression 1240 ,
𝜆𝑖 𝜆𝑖 (𝑛𝑚)
you need to multiply the energy in eV by the electron charge q. [Δ𝜆] is the wavelength interval the spectral irradiance corresponds to. However, you need to be careful if the spectral irradiance has already been normalized to 1 nm bin width or already been multiplied by [Δ𝜆]. You need to look at the file header or other information on the units of the spectral irradiance. If you are uncertain, sum the spectral irradiance and compare it to the value of approximately 840 W/m2, which is the integral of the spectral irradiance over the wavelength range.
In Mini Project 1, you read in a file, did a calculation and printed it out, so this is the same except the file you read is the AM1.5G solar spectrum and the calculation is the equation above.
b. Calculate and plot the quantum efficiency as a function of wavelength in both the emitter and the base of the solar cell as a function of wavelength. The quantum efficiency equations are calculated as functions in the python library called IQE_emitter(ab, We, Le, De, Se)
The parameter “α” is the absorption coefficient and is dependent on wavelength (we omitted the (λ) after the α to keep the equations to one line. In the function call in the code it is an array called “ab”. α(λ) is in a file called “10_nm_AM15G_absorption.txt”, which you read into your program in Part 1.
*** We do not use Wd, the width of the depletion region, so use Wd = 0.
Output: The two plots of QE_emitter and QE_base are typically combined into one plot with two lines, so please plot it this way.

(𝑆𝑒𝑚𝐿𝑒𝑚 +𝛼𝐿
𝑄𝐸_𝑒𝑚𝑖𝑡(λ) = (𝛼2𝐿𝛼2𝑒𝑚𝐿𝑒𝑚−1) [ 𝐷𝑒𝑚 𝑒𝑚𝐷)𝑒𝑚−𝑒−sinh𝛼𝑥𝑗 ((𝑆𝐿𝑒𝑚𝐷𝑒𝑚𝑒𝑚𝐿𝑒𝑚 cosh(𝐿𝑥𝑒𝑚𝑗 )+sinh(𝐿𝑥𝑒𝑚𝑗 )) −α𝑥𝑗
𝑆𝑒𝑚𝐿𝑒𝑚 𝑥𝑗 )+cosh( 𝑥𝑗 ) −α𝐿𝑒𝑚𝑒
𝐿𝑒𝑚 ]

𝑄𝐸_𝑏𝑎𝑠𝑒(λ) = ( 𝛼𝛼𝐿2𝑏𝐿𝑒2𝑏−−α𝑥1𝑗′) [𝛼𝐿𝑏 − 𝑆 𝐷𝑏𝐿𝑏𝑏 ( cosh𝑆(𝐷𝑏𝐿𝐻𝐿𝑏𝑏𝑏)sinh− 𝑒−(α𝐿𝐻𝐻𝑏))++sinhcosh((𝐿𝐻𝐿𝐻𝑏𝑏))+ 𝛼𝐿𝑏𝑒−α𝐻]

For Matlab users, copy the equation for these in the definitions into Matlab and define a function around it. One thing you have to change in the equation for Matlab (or other programming languages) is how you call the hyperbolic functions; in python it is np.cosh, so change this for another programming language.
c. Calculate JL. To find the JL, we multiply the photon flux (part a) by the quantum efficiency (part b), and then sum over all the wavelengths.

𝜆𝑖
𝐽𝐿𝑖 = 𝑆𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝐼𝑟𝑟𝑎𝑑𝑖𝑎𝑛𝑐𝑒𝑖 × [Δ𝜆] × × 𝑞 × 𝑄𝐸𝑖 = Φ𝑖 × 𝑞 × 𝑄𝐸𝑖 ℎ𝑐

Output: Plot JLi as a function of wavelength and in the graph title give the total JL in mA/cm².
𝜆𝑖
𝑆𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒𝑖 = 𝑞 × 𝑄𝐸𝑖 ℎ𝑐 To get:
𝐽𝑠𝑐𝑖 = 𝐽𝐿𝑖 = 𝑆𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝐼𝑟𝑟𝑎𝑑𝑖𝑎𝑛𝑐𝑒𝑖 × 𝑆𝑝𝑒𝑐𝑡𝑟𝑎𝑙 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒𝑖
Note 2: if you want to use other absorption coefficients (e.g, you choose to do a different solar cell material), the wavelength at which the absorption coefficient is defined must be the same wavelength at which the spectral irradiance is defined. Since the absorption coefficient and spectral irradiance don’t in general have the same wavelength spacing, we have to convert either the absorption or the spectral irradiance to the same wavelength interval. We did this already for the files we handed out.
Output: Print the emitter, base and total JL for the test case.
3. Calculate the emitter series resistance. In the notes, we calculate multiple terms of the series resistance; for simplicity, here we only calculate the emitter series resistance. The area-normalized series resistance is given by the equation:
𝑆²
𝑅𝑠𝑒𝑟 ≈ 𝑅𝑒𝑚𝑖𝑡𝑡𝑒𝑟 = 𝜌
12
where S is the finger spacing (NOT the surface recombination velocity – semiconductor notation is terrible!) and 𝜌 is the sheet resistance of the emitter calculated by the equation:
𝜌 1 1
𝜌 = =
𝑊𝑒 𝑞𝜇𝑁𝑒 𝑊𝑒
where We is the emitter thickness, N is the emitter doping, and  is the mobility in the emitter.
We have set S = 0.2 cm corresponding to a typical value for a commercial silicon grid. If you want to see the optimization of the top metallization from which we get this value, it is at: http://www.pveducation.org/pvcdrom/5-design-of-silicon-cells/optimization-of-finger-spacing.
The python library has two function definitions which calculate the sheet resistivity and emitter series resistance for you. They are: def emitter_resistance(Sf, Rsheet): and def sheet_resistivity(doping, thickness):
Output:
(1) Print out the series resistance (including units) and compare to test case.
(2) Plot the IV curve, labelling the graph.

(1) Calculate JSC using JSC = JL. Unless the series resistance is very high (in which case we aren’t using the solar cell anyway), the current at V=0 (which is how JSC is defined) is the same as JL.
Note: If you are bored or writing the code in a more general fashion, you can find JSC from:
(𝐽𝑆𝐶𝑅𝑠𝑒𝑟𝑖𝑒𝑠)
𝐽𝑆𝐶 = 𝐽0 (𝑒 𝑘𝑇 − 1) − 𝐽𝐿
This equation is solved recursively (you can’t solve exactly in closed form), which is the reason we typically ignore it.
(2) Calculate Voc using:
𝑘𝑇 𝐽𝑆𝐶
𝑉𝑜𝑐 = 𝑙𝑛 ( )
𝑞 𝐽0
(3) Calculate FF of the IV curve.
The FF is defined as the voltage and current where the power (P=V x I) is the largest. We can find this value one of two ways:
• Trace out the IV curve and find at what values of V and I the power is the largest. Advantage: you need the full IV curve in another mini-project, so might as well do it now. Disadvantage: Longer to compute (not important now but depending on what you are doing in your final project, it might be an issue).
• Use the closed form for FF from the two equations below:
𝑣𝑜𝑐−ln(𝑣𝑜𝑐+0.72) 𝑞
𝐹𝐹0 = 𝑣𝑜𝑐+1 where 𝑣𝑜𝑐 = 𝑛𝑘𝑇 𝑉𝑜𝑐 , n being the ideality factor and
𝑅𝑠𝑒𝑟 𝑉 𝑉𝑂𝐶
𝐹𝐹 = 𝐹𝐹0 (1 − 𝑅𝐶𝐻) where 𝑅𝐶𝐻 𝐼𝑀𝑃 𝐼𝑆𝐶
(4) Calculate the efficiency. and the impact of the maximum power point & FF.
𝐽𝑆𝐶𝑉𝑂𝐶𝐹𝐹
𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 = 𝜂 =
𝑃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡
Where P is the incident power density. Since we are using the AM1.5G spectrum, it is 1000 W/m² (or equivalently 0.1 W/cm²), but if you are generalizing the program, you should calculate it from the input spectrum.
Output: Print Voc, Fill Factor and Efficiency.

The goal of this part of the problem is to optimize the solar cell, changing only the emitter doping and emitter thickness.
IMPORATANT: The maximum emitter doping you should use is Nemitter = 1  1020 cm-³. Values above this will give higher efficiency in the calculations, but the equations for minority carrier lifetime do not adequately work for such high doping – in practice the minority carrier lifetime is so low at higher doping that the material is not particularly electrically active. The minimum thickness of the emitter should be 100 nm – below this in practice the solar cell is subject to shunting.
Output:
(1) Emitter doping and thickness, JSC, VOC, FF, and efficiency for the maximum efficiency.
(2) Justify that your answer is the optimum. You can do a plot of efficiency vs doping and thickness; you can describe how you optimized it and what changed and why it makes sense; or you can prove it’s the optimum in other ways. You cannot say “this is what my program spit out”. Basically, any time you give an optimum result for an engineering problem, you have to also show that your answer is reasonable and correct.

The goal of this part of the problem is to analyze what the losses are in the solar cell, and how we might improve the solar cell. Parameters you can look at are the quantum efficiency, compare J0 from the emitter and from the base. You can plot different parameters (JSC, VOC) as function of doping and thickness. You can change a material parameter and see what its effect is.
Output:
Give the material and device parameters which are key in limiting the device efficiency. From this, suggest some approaches to increasing cell efficiency.


This is just for you to check if your Python installation works or if your code in Excel or Matlab produces realistic values for the test case.
emitter diffusion length (cm) 0.00387104830569 base diffusion length (cm) 0.0925820099773 emitter resistivity (ohm/sqr) 208.0503042999924 series resistance (ohm.cm²) 0.6935010143333081
light generated current density in emitter (A/cm²) 0.012054716461 light generated current density in base (A/cm²) 0.0268994750224 total light generated current density (A/cm²) 0.0389541914834
J0 emitter (A/cm²) 7.08712167027e-15
J0 base (A/cm²) 1.32750778369e-13
J0 total (A/cm²) 1.39837900039e-13
Voc 0.677074345174
FF without series resistance 0.842761558955
FF with series resistance 0.806066612651
Efficiency (%) 21.2599131551