Let us assume n-type material with electron traps at the grain boundaries. After capturing free electrons from the conduction band, the traps are charged negatively. The figure below represents the variation of the charge density across three adjacent grains. Inside the grains the free carrier density is depleted, leaving behind the positive cores of the ionized dopants (red), whereas a high electron density of the trapped charges is located at the grain boundaries (blue).
We can distinguish two cases; in a material with high doping concentration or large grains the number of free carriers within the grain may exceed the number of traps at the grain surface. Thus, all traps will be filled and the grain is only partially depleted. If we apply the depletion approximation similar to the case of the pn-junction theory, the depletion will be limited to the region close to the grain boundary, whereas the centre of the grain will be neutral. This situation is depicted in the left panel of the figure. If we lower the free carrier density in the grains, the depletion region will grow and eventually extend all the way towards the middle of the grain. This situation is called total depletion. If we lower the doping concetration further, some traps may even stay uncharged. This situatin is shown in the right panel.
Obviously in our simple one dimensional model the condition for transition between the two cases is met when the trap density Qt equals the product of donor concentration ND and grain size d.
We can calulate the variation of the potential across the grain boundary by solving the Poission equation. Assuming constant charge density throughout the depletion regions, integration is easy and results in a parabolic band bending, just as in the case of the pn-junction.
The height of the transport barrier Φb is just the difference between the values of the potential in the centre and the grain boundary. For the two situations above we get different dependence on the doping concentration; in case of total depletion the potential barrier increases linearly with the doping concentration:
In the case where we have high doping concentration the grain boundary barrier is inversely proportional N which means it decreases for higher doping. Note that only in this case the trapping state density Qt enters the formula for the barrier.
For the calculation of the mobility Seto [1] assumed ballistic transport of the charge carriers over this barrier. He arrived at the following formula:
For very high doping Tarng suggested to use Schottky thermionic emission rather than ballistic transport [2]. However, this regime is difficult to test experimentally because the barrier becomes very small and ionized impurity scattering within the grain may dominate.
A very important assumption of this simple theory is not always met; Seto assumed that there is only one trapping level, and that it is deep in the gap, so that it is always filled. Under certain circumstances this trap coud accumulate a vast number of charges which would result in barrier heights in the order of 0.1 eV and more. This will get it close to the Fermi-Level, and there is a probability that trapped charges escape by thermal excitation. It becomes necessary to calculate the occupancy in thermal equilibrium. This extension of the theory was suggested by Baccarani [3]. His considerations limit the barrier height to more reasonable levels.