Simulation of carrier profiles in thick samples
The measurement of thick samples as bricks leads to new questions and problems. One of these questions is how the carrier profiles that develops in a sample effect the lifetime measurements. To solve this problem, we developed a simulation tool for the modelling of carrier profiles. This tool consists of a partial differential equation system of the transport equations for electrons and holes and the Poisson equation.
\(\frac{\partial}{\partial t}n(x,t)=\frac{\partial}{\partial x} \left[ -\mu_nn(x,t)\frac{\partial}{\partial x} \Psi(x,t) + D_n\frac{\partial}{\partial x}n(x,t) \right]+ G^o(x,t) - U(x,t)\)
\(\frac{\partial}{\partial t}p(x,t)=\frac{\partial}{\partial x} \left[ \mu_pp(x,t)\frac{\partial}{\partial x} \Psi(x,t) + D_p\frac{\partial}{\partial x}p(x,t) \right]+ G^o(x,t) - U(x,t)\)
\(\frac {\partial^2}{\partial x^2} \Psi (x,t) = -\frac{q}{\in_0\in_y} \left[ -n(x,t) + p(x,t)_\text{dot} \right]\)
Transport equations for electrons and holes + Poisson equation
The simulations for measurements with a long (typical MDP condition) or a very short light pulse (typical µ-PCD conditions) at thick unpassivated samples are shown in figure 1 and 2. It becomes clear, that the carrier profile of a long light pulse expands through a large volume of the sample, where as the carrier profile of the only 200 ns long light pulse is very surface near. This has a direct effect on the measured lifetime, since the surface recombination has a much stronger effect on the µ-PCD measurements.
Figure 3 shows the quantitative effect of the surface recombination on the effective lifetime for both measurement conditions. The MDP measurements are less prone to the surface effect, so that MDP measurements are more suited for the investigation of bulk properties. Accordingly µ-PCD is an ideal method to investigate the surface properties of a sample.
Fig.3: evaluated effective lifetimes as a function of bulk lifetime for both methods
