Ideality Factor

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

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

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]

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.

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. 3287-3293, 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. 1119-1152, 2014.

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).

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. 10146-10155, 2016/05/19 2016

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)

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

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.

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