# Double Diode Model

The single diode equation assumes a constant value for the ideality factor n. In reality the ideality factor is a function of voltage across the device. At high voltage, When the recombination in the device is dominated by the surfaces and the bulk regions the ideality factor is close to one. However at lower voltages, recombination in the junction dominates and the ideality factor approaches two. The junction recombination is modeled by adding a second diode in parallel with the first and setting the ideality factor typically to two.

Circuit diagram of the double diode model including the parasitic series and shunt resistances

The equation of the double diode model under illumination is:

$J={J}_{L}-{J}_{01}\left\{exp\left[\frac{q\left(V+J{R}_{s}\right)}{kT}\right]-1\right\}-{J}_{02}\left\{exp\left[\frac{q\left(V+J{R}_{s}\right)}{2kT}\right]-1\right\}-\frac{V+J{R}_{s}}{{R}_{shunt}}$

Practical measurements of the illuminated equation are difficult as small fluctuations in the light intensity overwhelm the effects of the second diode. Since the double diode equation is used to characterise the diode it is more common to look at the double diode equation in the dark.

$J={J}_{01}\left\{exp\left[\frac{q\left(V-J{R}_{s}\right)}{kT}\right]-1\right\}+{J}_{02}\left\{exp\left[\frac{q\left(V-J{R}_{s}\right)}{2kT}\right]-1\right\}+\frac{V-J{R}_{s}}{{R}_{shunt}}$

In both the light and dark cases the -1 terms in the exponential are typically ignored as it makes the analysis far easier.

Under illumination:

$J={J}_{L}-{J}_{01}exp\left[\frac{q\left(V+J{R}_{s}\right)}{kT}\right]-{J}_{02}exp\left[\frac{q\left(V+J{R}_{s}\right)}{2kT}\right]-\frac{V+J{R}_{s}}{{R}_{shunt}}$

In the dark:

$J={J}_{01}exp\left[\frac{q\left(V-J{R}_{s}\right)}{kT}\right]+{J}_{02}exp\left[\frac{q\left(V-J{R}_{s}\right)}{2kT}\right]+\frac{V-J{R}_{s}}{{R}_{shunt}}$

The double diode equation in the dark is graphed below:

I01

I02

Rs

Rshunt

Graph showing the double diode model. The device in gray has no parasitic resistance losses. The device in red has the loss of series and shunt resistance included.

### Limitations of the Double Diode Model

In actual silicon devices the recombination components are a complex function of the carrier concentration. For example, in high efficiency PERL solar cells as the number of carriers increase with the applied voltage, the recombination at the rear surface changes dramatically with voltage. In such cases the analysis is best performed by a single diode, but allowing both the ideality factor and the saturation current to vary with voltage. In such cases, which are quite common in silicon devices, a double diode fit yields erroneous values.