### 2.11:  The Shockley equation for a diode

So far, you have developed an understanding of solar cells that is mainly intuitive. Generally, it is very useful to connect intuition with a quantitative treatment. Therefore, let us use the gained intuition to understand the famous Shockley equation of the diode.

Let us summarize in the following how, at a given applied bias V, a current I can be driven through a diode in the dark.

On the selective contacts page, you learned that the contact on n-type (or p-type) allows mainly free electrons (or free holes) to pass through. Hence, to drive a current through the diode in the dark, the electrons need to recombine somewhere in the diode, see the currents in the dark page. On the recombination page, you learned that the recombination rate is limited by the minority carrier density. This density is proportional to the Boltzmann factor e–E/kT, where E is the barrier across the p-n junction. On the role of the p-n junction page, you learned that this barrier decreases by the amount of the applied bias.

For all these reasons, the minority carrier density increases exponentially with applied bias, and so does the recombination rate, and so does the current through the diode.

Hence, we can state purely methematically: To derive the constants A, B, and C, you may put physics into the above equation by having a look at certain situations:

• At V = 0 there is J = 0. Enter this into the above equation and you will receive a relation between A and C.
• It was measured, and it is indicated in the figure, that at negative V, J saturates to the so-called saturation current J0. This is a very small current, often between 100 and 1000 fA/cm2 (femto is 10–15). So, take V towards minus infinite, and you receive a physical value for c (and with the first step also for A).
• The assignment of B to a physical entity is done with the Boltzmann factor, where E is replaced by the voltage B: B = qV/kT, where k = 8.61758 x 105 eV/K. Then, kT/q is called the thermal voltage Vth and is 25.8 mV at 300K.

With all this you get the Shockley diode equation: Under illumination, the current-direction is reversed but, otherwise, the shape of the IV curve is the same as in the dark (see page "Cell under illumination"). This implies that we can simply subtract JSC form the above equation to get the Shockley diode equation under illumination: This is an important equation for understanding solar cells, and we will use it for deeper investigations in subsequent lessons. Figure: Current–voltage curves of a solar cell in the dark (blue) and light (red).