Equation for diode saturation current in photovoltaic cell model

Question

I've been looking for mathematical models for photovoltaic cells. Most of the references I've found use the following circuit to the model the behavior:

Particularly, the diode current is modeled with the following equation:

$$ I_D = I_0\left( e^\frac{qV_{D}}{nkT} - 1 \right) $$

Regarding the reverse saturation current of the diode, almost all references I've found use the following equation:

$$ I_0 = I_{0,ref} \left( \frac{T}{T_{ref}} \right)^3 e^{\frac{qE_{g}}{nk}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)} $$

Analyzing the exponential in the equation, there seems to be a mistake because the resulting term inside is not dimensionless. Considering that q is measured in C, Eg in eV, k in J/K, T in K and n is dimensionless, the resulting term inside the exponent would have dimension C (Coulomb).

So far I've only found one reference that uses a different version of the previous equation, with a dimensionless term inside the exponent:

$$ I_0 = I_{0,ref} \left( \frac{T}{T_{ref}} \right)^3 e^{\frac{E_{g}}{k}\left(\frac{1}{T_{ref}}-\frac{1}{T}\right)} $$

What would be the correct equation for the diode saturation current in this case? Most references use the first one, but doing the dimensional analysis it seems to be incorrect. Am I missing or misunderstanding some important detail?

Answer 1

I thought I'd write a short set of interesting notes.

• The reason the temperature ratio is cubed is because of the number of quadratic degrees of freedom, which for a simple particle (same as a simple atom like argon) is 3. The reason why this should be parameterized and not necessarily always 3 is in part because of the temperature dependence of diffusivity in doped semiconductors, \$\frac{k\,T}{q} \mu_{_\text{T}}\$ and in part because in heavy doping narrows the bandgap. While other parts of the expression could be made more complex to account for such details, it turns out that altering the degrees of freedom slightly achieves a reasonable approximation of reality without making things insanely complicated.
• The remaining part of the expression (the exponential factor also known as the Boltzmann factor) is due to elementary probability theory being applied to working out the ratio of the difference in numbers of states between the same system in two different states of temperature. If you read Boltzmann's paper (I much prefer Shannon's rendition as far more readable, though) you know about the concept of the number of states in a system of particles and also about partition (equipartition) theory for those with 3 degrees of freedom to move about (where \$\frac12 k\,T\$ energy is, on balance, equally distributed.) Any book on statistical thermodynamics covers this very early.

I think it was Fleming (a year before Einstein's flurry of papers in 1905) who created the first vacuum diode and allowed the use of electrons to target the anode to provide an independent means (besides the use of photons) to calculate work function values for metals.

The wonderful thing about this vacuum tube diode is that it doesn't matter how far away the cathode and anode are from each other. Distance isn't relevant and therefore this removes one factor of experimental uncertainty. If you apply a voltage differential between them, then each electron will impact the plate with exactly \$1\:\text{eV}\$ of kinetic energy on impact, per volt of applied differential between the cathode and plate.

It's here where I imagine the one author got the \$q\cdot E_{g}\$ factor. If \$E_g\$ is expressed in the applied volts between cathode and anode, then the energy imparted to an electron will be \$1\:\text{eV}=q_e\cdot E_{g}\$, where in this case \$E_g\$ is in applied volts. I can easily see how this might get into the literature that way. Or it could just be a mistake by the author. It happens.