I am confused as to which parameter I should start off with in order to optimizing the best performing solar cell. There are parameters like Thickness, Bandgap, Electron Affinity, relative Dielectric Permittivity, Effective conduction and valence band density, Electron and hole thermal velocity, Electron and hole mobility, Effective mass of Electron and Hole, Acceptor and donor concentration.

  • $\begingroup$ This is an extremely broad question, and moves quickly into engineering (as implied by 'optimizing' since you need to optimize against engineering criteria). $\endgroup$
    – Jon Custer
    Sep 17 '20 at 16:00
  • $\begingroup$ You need to specify an objective function to optimize against. What is your goal? Can your goal be specified mathematically? Can you write a system of equations that adequately describe operating solar cells? Do you know that the mathematical function specifying your goal is unimodal (e.g., has only one global optimum)? Are you optimizing against any constraints? Note - this is the "guts" of multi-variable optimization, and these questions can be VERY difficult to answer. $\endgroup$ Sep 30 '20 at 20:25

Simplest possible and most fundamental model of a solar cell comes from detailed balance of absorbed and emitted light.

This is not really a device model, which I think you are more interested in. But the electrical current that you get out of an ideal solar cell is the difference between the photon current the solar cell absorbes the photon current it emits,

$$ J(V) = q\left( N_a - N_e(V) \right) $$

Normally these are areal quantities: photons per square meter and current per square meter.

If you assume the appropriate blackbody spectrums, this is equivalent to the Shockley–Queisser model in the radiative limit (no non-radiative recombination).

Idealised model

We are going to calculate the current-voltage characteristic of an ideal Silicon solar cell.

Assuming the sun is a blackbody at temperature $T_s=5760K$, the photons per metre squares per second per Joule arriving at the Earth's surface is given by the Planck equation in terms of photon energy $\hbar\omega$,

$$ n_s(\hbar\omega) = f_s\left(\hbar\omega\right)^2\frac{2\pi}{c^2h^3}\frac{1}{\exp{\frac{\hbar\omega}{kT_s}}-1} $$

where the angular factor $f_s=6.8\times 10^{-5}$ considers the $1/r^2$ losses due to the Earth-Sun distance.

We are going to assume that the solar cell absorbs all sunlight above its bandgap energy $E_g$ and is completely transparent otherwise.

$$ a(\hbar\omega) = \begin{cases} 1,& \text{if } \hbar\omega \ge E_g\\ 0, & \text{otherwise} \end{cases} $$

The number of photons absorbed from the sun is,

$$ N_a = \int_{0}^{\infty} a(\hbar\omega) n_s(\hbar\omega) d(\hbar\omega) = \int_{E_g}^{\infty} n_s(\hbar\omega) d(\hbar\omega) $$

Dropping the minus one denominator we can write an approximate analytical solution,

$$ N_a \approx f_s\frac{2\pi}{c^2h^3 \beta_s^3}\left(2 + E_g \beta_s \left(2 + E_g \beta_s\right)\right) \exp(-E_g\beta_s) $$

where $\beta_s = 1/\left(kT_s\right)$.

Let's assume our solar cell is Silicon with bandgap $E_g = 1.1\text{eV}$.

Then $N_a \approx 1\times10^{22}$ photons m$^{-2}$s$^{-1}$.

We can use the generalised Planck expression to describe how a solar cell emits light as a function of the applied bias voltage $V$,

$$ n_e(\hbar\omega, V) = \left(\hbar\omega\right)^2\frac{2\pi}{c^2h^3}\frac{1}{\exp{\left( \frac{\hbar\omega - qV}{kT_e} \right)}-1} $$

We want to integrate this to know the total number of photons emitted,

$$ N_e = \int_{0}^{\infty} a(\hbar\omega) n_e(\hbar\omega, V) d(\hbar\omega) = \int_{E_g}^{\infty} n_e(\hbar\omega, V) d(\hbar\omega) $$

again dropping the minus one we can write,

$$ N_e(V) \approx \frac{2\pi}{c^2h^3 \beta_e^3}\left(2 + E_g \beta_e \left(2 + E_g \beta_e\right)\right) \exp(-\left(E_g - qV\right)\beta_e) $$

where the temperature of the solar cell is $T_e=300\text{K}$ and $\beta_e = 1/\left(kT_e\right)$.

We are now in the position to calculate the current voltage characteristics of an ideal solar cell.

$$ J(V) = q\left( N_a - N_e(V) \right) $$

With a little help from Mathematica,

Ideal JV curve Silicon solar cell

  • $\begingroup$ Thank you @boyfarrell for this illustration. $\endgroup$
    – Abhijeet
    Sep 29 '20 at 6:36
  • $\begingroup$ @Abhijeet you are welcome! And welcome to stack Physics. You can accept the answer and upvote if you like it. $\endgroup$
    – boyfarrell
    Sep 29 '20 at 6:38
  • $\begingroup$ Thank you @boyfarrell for this illustration. My question was regarding simulating a theorized solar cell device in particular software and it was intended to get information on the course of action regarding optimization of various device parameters. There are parameters like thickness of layers, electronic properties/parameters like mobility, dielectric constant, capture cross section, etc. I am confused as to which parametrs I should optimize first and whether optimization must be carried out one by one or simultaneously. $\endgroup$
    – Abhijeet
    Sep 29 '20 at 6:44

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