1. Ideality Factor

2. I am the lumped element’s function
General comments
Most of the simple and basic relations we develop are based on considering the device under study as a lumped element. H IN
OUT
I am the lumped element’s function
The illustration to the left is the engineering presentation where J(V) would be J=H(V)
Few ways to illustrate the simplifications we make:
LUMO
EFn
EFn
EFn
EFp
J(n,p) V H H V J J V J
EFi
EFp
EFp
HOMO

3. The two routes should give the same result (hmmm)
General comments
Few ways to illustrate the simplifications we make:
LUMO
EFn
EFn
EFn
EFp
J(n,p) V H H V J J
EFi J V
EFp
EFp
HOMO
The two routes should give the same result (hmmm)
To simplify the expression for J(n,p) we take one of two routes
J(n,p) =Drift diffusion current
J(n,p)
J(n,p )=Recombination current

4. What does ideality factor mean
Theories describing the characteristics (J-V) of diodes predict that in the ideal case such characteristics should have the form of the expression on the left.
Practical diodes rarely follow exactly the ideal relation and it was found that in many cases a good agreement can be found if one introduces an ideality factor (n) to produce the expression on the right.
[* We use n instead of n to not confuse with charge density]

5. Justifying the use of n - #1 – drift diffusion
Lets consider a P-N diode where it is known that the currents across the junction can be described as diffusion currents:
If we assume that D is independent of the charge density (n) than one arrives at:
Y. Vaynzof, Y. Preezant, and N. Tessler, "Current voltage relation of amorphous materials based pn diodes-the effect of degeneracy in organic polymers/molecules," Journal of Applied Physics, Article vol. 106, no. 8, p. 6, Oct 2009.

6. Justifying the use of n - #1 – drift diffusion
If the charge density is such that the material is degenerate two effects [1] take place:
Accounting for the GER (D/m) and m(n) [2]:
for Long diode:
Since both b and h are associated with the same rise in the carrier energy [1] - In most cases ndegen is close to 1.[3]
[1] As degeneracy is associated with increase in the energy of the charge population. It affects both the GER and the mobility: D. Mendels and N. Tessler, "Drift and Diffusion in Disordered Organic Semiconductors: The Role of Charge Density and Charge Energy Transport," JPC C, vol. 117, no. 7, pp , Feb 2013.
[2] Y. Vaynzof, Y. Preezant, and N. Tessler, "Current voltage relation of amorphous materials based pn diodes-the effect of degeneracy in organic polymers/molecules," JAP, vol. 106, no. 8, p. 6, 2009.
[3] N. Tessler, "Experimental techniques and the underlying device physics," J. of Polymer Science Part B: Polymer Physics, vol. 52, no. 17, pp , 2014.

7. Justifying the use of n - #2 – generation recombination
Let’s assume that all the current that flows into the diode recombines inside. Let’s also assume that the recombination is bimolecular:
In the ideal case the material is non degenerate and the Boltzmann statistics can be used:
And if the current can be described as recombination current:
If the material is degenerate one can use the Boltzmann factor with a correction however, I am not aware of attempt to use this to deduce ideality factor (based on the drift-diff analysis and the need for both to be the same - the “C” would be density dependent too).

8. Justifying the use of n - #2 – generation recombination
Let’s now assume that the recombination is governed by SRH process (trap assisted recombination):
Traps as recombination centers
EFi
Traps Et
LUMO
HOMO
Intrinsic material and very low charge density (traps not full)
Intrinsic material and high charge density (n=p)
n=2
* Note that if the traps are not full the SRH results in n=1. In BHJ they are often full only close to 1 Sun or above [1].
[1] L. Tzabari, J. Wang, Y.-J. Lee, J. W. P. Hsu, and N. Tessler, "Role of Contact Injection, Exciton Dissociation, and Recombination, Revealed through Voltage and Intensity Mapping of the Quantum Efficiency of Polymer:Fullerene Solar Cells," The Journal of Physical Chemistry C, vol. 120, no. 19, pp , 2016/05/

9. Justifying the use of n - #2 – generation recombination
SRH process (trap assisted recombination) in the presence of dark carriers N.
If N doesn’t fill the traps:
Intrinsic material and very low charge density (n,p< P & traps not full)
Intrinsic material and very low charge density (n,p> P & traps not full)
Intrinsic material and high charge density (n=p)

10. Other scenarios n ≤ 2 EFi Et Et1 Et1 EFi EFi Et2 Et2
Single level, low density traps
Multi level, low density traps
Multi level, high density traps
EFi
Traps Et
LUMO
HOMO
LUMO
LUMO
Traps
Et1
Traps
Et1
EFi
EFi
Traps
Et2
Traps
Et2
HOMO
HOMO
Inter-level recombination/hopping can lead to n > 2
n ≤ 2

11. Effect of trap assisted recombination on the J-V
The extra recombination paths make the recombination current much stronger already at V~0 but then it grows slower.

12. Currents slopes Exponential Power law
Note that using the numerical formula it is always possible to extract n(V), even from a current that follows a power law.
Parallel (leakage) resistance would modify n at low currents and serial resistance or space charge would modify it in the high current regime.
Although the ideality factor is a powerful tool you need to make sure first that it is relevant.
There are too many papers applying it where it is not relevant and dwelling on why it is voltage dependent