Theory of drift diffusion modelling

Outline

OghmaNano’s electrical model is a 1D/2D drift-diffusion model (like many others) however the special thing about OghmaNano which makes it very good for disordered materials (Think organics, perovskites and a-Si) is that it goes to the trouble of explicitly solving the Shockley-Read-Hall equations as a function of energy and position space. This enables one to model effects such as mobility/recombination rates changing as a function of carrier population and enables one to correctly model transients as one does not have to assume all the carriers in the trap states have reached equilibrium. Things such as ToF transients, CELIV transients etc.. can be modelled with ease. Of course can be used for more ordered materials as well, you then just need to turn the traps off.

Electrostatic potential

The conduction band/valance band (or LUMO/HOMO in organic semiconductor speak) are defined as

\[E_{LUMO}=-\chi-q\phi\]

\[E_{HOMO}=-\chi-E_g-q\phi\]

To obtain the internal potential distribution within the device Poisson’s equation is solved,

\[\label{eq:pos} \nabla \cdot \epsilon_0 \epsilon_r \nabla = q (n_{f}+n_{t}-p_{f}-p_{t}-N_{ad}+-N_{ion}+a),\]

where \(n_{f}\), \(n_{t}\) are the carrier densities of free and trapped electrons; \(p_{f}\) and \(p_{t}\) are the carrier densities of the free and trapped holes; and \(N_{ad}\) is the doping density. \(N_{ion}\) is the background density of perovskite ions and a is the density of mobile ions.

Free charge carrier statistics

For free carriers the model can either use Maxwell-Boltzmann statistics i.e.

\[n_{l}=N_c exp \left (\frac{F_n-E_{c}}{kT} \right)\]

\[p_{l}=N_v exp \left(\frac{E_{v}-F_p}{kT} \right)\]

or full Fermi-dirac statistics i.e.

\[n_{free}(E_{f},T)=\int^{\infty}_{E_{min}} \rho(E) f(E,E_{f},T) dE\]

\[p_{free}(E_{f},T)=\int^{\infty}_{E_{min}} \rho(E) f(E,E_{f},T) dE\]

where

\[f(E)=\frac{1}{1+e^{{E-E_f}/kT}}\]

When using FD statistics free carriers are assumed to move in a parabolic band:

\[\rho(E)_{3D}=\frac{\sqrt{E}}{4\pi^2} \left ( \frac{2m^{*}}{\hbar^2}\right )^{3/2}\]

The average energy of the carriers is defined as

\[\label{eq:energy} \bar{W}(E_{f},T)=\frac{\int^{\infty}_{E_{min}} E \rho(E) f(E,E_{f},T) dE}{\int^{\infty}_{E_{min}} \rho(E) f(E,E_{f},T) dE}\]

Carrier trapping and Shockley-Read-Hall recombination

The model provides two methods to account for carrier trapping and recombination via trap states. The first by equation [eq:ss_srh], this assumes that the trapped carrier distribution has reached equilibrium. It also assumes there are relatively few trapped charge carriers compared the the number of free carriers, and thus the trapped charges do not significantly change the electrostatic potential. These assumptions are valid when the material is very ordered (i.e. GaAs) or at a push in steady state for some moderately disordered material systems. However if you wish to simulate transient or frequency domain experiments, then you can no longer use [eq:ss_srh]. Instead, one must use a non-equilibrium SRH approach which does not assume trapped carriers have reached equilibrium. Unlike many other models, OghmaNano has such a non-equilibrium SRH model built in this is described in section 9.4.2. In fact, it is turned on by default so when using OghmaNano you have to go out of your way to turn on equation [eq:ss_srh].

To understand the importance of such a dynamic solver, consider the following example: You are performing a transient photocurrent experiment (TPC). You photo-excite your device with a laser, carriers very quickly become trapped during the first 1-2\(\mu s\) after photoexcitation, as time passes, the carriers gradually de-trap from deeper and deeper trap states and produce the long photocurrent transient . These transients can often extend out to over 1 second after photo-excitation. Current at the start of the transient originates from shallow traps while current at the end of the transient originates from carriers from very deep trap levels. To simulate this one has to be able to account for the gradual emptying of trap states firstly starting at the shallow traps, then progressing to deeper and deeper trap states. Were one to assume all trap states were in equilibrium one would not be able to simulate this process.

So in summary, although many others have used [eq:ss_srh] to model disordered devices in time DON’T you results won’t make sense. If you want to simulate anything but steady state in an ordered device turn ON the non-equilibrium solver.

Equilibrium Shockley-Read-Hall recombination

For some very ordered material systems where there are not many trap states it is enough to describe SRH trap states using the equation:

\[\label{eq:ss_srh} R^{SRH}=\frac{np-n_{0}*p_{0}}{\tau_{p} (n+n_{1})+\tau_{n} (p+p_{1})}\]

where \(R_{SRH}\) is the rate of SRH recombination, \(n,p\) are the density of free charge carriers \(n_0, p_0\), are the equilibrium density of charge carriers, \(\tau_{n,p}\) are the SRH life times and \(n_{1}\) and \(p_{1}\) are the trapped electron and hole densities when the Fermi-level matches the trap state energy. This can be turned on in the electrical parameter editor.

Charge carrier transport

To describe charge carrier transport, the bi-polar drift-diffusion equations are solved in position space for electrons, \[\label{eq:ndrive} \boldsymbol{J_n} = q \mu_e n_{f} {\nabla E_{c}} + q D_n {\nabla n_{f}},\] and holes, \[\label{eq:pdrive} \boldsymbol{J_p} = q \mu_h p_{f} {\nabla E_{v}} - q D_p {\nabla p_{f}}.\]

Conservation of charge carriers is forced by solving the charge carrier continuity equations for both electrons, \[\label{eq:contn} \nabla \boldsymbol{J_n} = q (R-G+\frac{\partial n}{\partial t}),\] and holes \[\label{eq:contp} \nabla \boldsymbol{J_p} = - q (R-G+\frac{\partial p}{\partial t}).\]

where \(R\) and \(G\) are the net recombination and generation rates per unit volume respectively.