In trying to solve the Poisson equation in heterojunctions involving undoped organic semiconductors(Nd=Na=0), can I use the depletion approximation in this case(n=p=0)? If not, how do I go about solving this equation to obtain the potential distribution as a function of position across the hetero interface?
If you assume, Nd=Na=n=p=0, then the voltage distribution will be liner drop form one electrode to the other, i.e., V(x) = kx. However, organic semiconductors are generally p-doped. and at the heterojunction interface there are generally pile up of carriers which cannot be neglected.
Hi Ananth,
as a good book, which contains all necessary information and equations to calculate the properties of a heterojunction, I can recommend: Physics of Semiconductor Devices by Sze and Ng
This calculation involves at least these steps:
1) Determine the Fermi energy and the carrier concentrations (n, p) based on the valence band (VB) and conduction band (CB) density of states (Ncb and Nvb) and the donor and acceptor density and energy (Nd and Na). The calculation is based on the Fermi-Dirac statistic and the charge neutrality argument. All necessary equations are in chapter 1.4.
2) calculate the space charge (e.g. by using the depletion approximation), and the electric field and the potential. The thermal equilibrium can be obtained by requiring, that the Fermi-Energy in both parts has the bulk value far away of the junction and is the same throughout the system. All necessary equations are in the beginning of chapter 2.
As Biswajit Ray already mentioned, organic semiconductors are rarely intrinsic and are doped by their intrinsic defects, or abundant impurities like O2, H2O, OH molecules. I have the same fact in mind, that most organic semiconductors are in fact p-type doped.
Good luck,
Deniz
Dr. Ray,
If charge pile up does occur at the interface, is it responsible for the electric field at the interface? In that case, how do I calculate/assume a distribution of those charges? I am trying to use Dr.Dragica Vasileska's method for bi-layer organic solar cell I-V simulation in Matlab. Thanks
Charge pile up at the interface takes place for several reasons: Low mobility, low built in field etc. Even if charge pile up does not take place, carrier concentration is exponentially distributed in the device. However, if the magnitude of the charge concentration is low enough to change the Poisson equation we can still assume constant electric field, or in other words n=p=0 in the Poisson equation. All these you can verify in OPV lab tool: https://nanohub.org/tools/opv
I am not aware of the Dr. Vasileska's work on Bi-layer OPV. Can you please post a link?
You can solve the drift diffusion equation by finite element method. You can see this article for details JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 25, NO. 9, SEPTEMBER 2007 page 2828 in the case of OLEDs. Same method you can apply for OSCs, just you have to play with generation current.
n=p=0 will cause a constant electric field in the material. However, space charge pile up near the contacts changes the electric field in that zone drastically, so that can not be ignored for the calculation of potential distribution. You may solve the potential distribution function with the help of drift-diffusion equation just like as in diode.
Experience of calculating crystalline semiconductor junctions suggests:
1) This is strange situation that you do not have Nd, Na at all, if there would not be a junction between 2 materials, it should be just piece of insulator without charge carriers.
2) At heterojunction you must know space charge Nd-Na+p-n for Poisson, if Nd and Na are zero then obviously you CAN NOT OMIT n and p;
3) Further - heterojunction you must construct from the flat Fermi level EF and continuous electrostatic potential; depending on affinity at both sides of heterojunction you will get jumps in the edges of band gap Ec and Ev. Since distances Ec-EF and EF-Ev are shifted away from equilibrium near the junction, you get also p and n values even if you do not have doping. YOU CAN NOT OMIT THOSE n,p as you need them in Poisson eq.
In conclusion, your Poisson task seems opposite to simple complete depletion task - your charge concentrations will depend nonlinearily from unknown potential in every spatial point and thereby your task is nonlinear and self-correlated (something like Shockley-Poisson eq. to find equilibrium potential in semiconductor devices).
Drift-diffusion comes later, if you have solved equilibrium field and potential.
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