The surface also play an important role in recombination. Typically the surfaces complicate the measurement of bulk lifetime. Recombination at the surfaces is typically described by a surface lifetime τs, which includes the fundamental decay mode but ignores higher decay modes. τs is function of the surface recombination velocities S1 and S2, the cell width W and the minority carrier diffusivity D. The precise solution is quite complicated but approximate solutions exist for special cases and are sufficiently accurate for most purposes 1.
The simplified formulas given below can be used for transient measurements when the effective lifetime is much greater than the transient surface lifetime (). The more exact expression 2 in transient measurements that works in all cases is:
For quasi-steady-state measurements the formulas are accurate if .
There is a list of approximate solutions for several important cases below:
The surfaces are identical so S = S1 = S2.
One surface is perfectly passivated so S2 = 0.
Both surfaces are perfectly passivated so S1 = S2 = 0.
Both surface have high recombination so S1 and S2 are large.
One surface has a high recombination and the other has a low recombination so S1 = 0 and S2 = ∞
The combination of the effects of surface and bulk recombination give the effective lifetime of carriers τeff within a sample.
For many measurements it is possible to reduce surface recombination of the surfaces to a low enough value so that τb ≫ τ eff. Alternatively if τb is sufficiently high the lifetime measurement is completely dominated by the surfaces and τs ≫ τeff. An accurate measurement of surface recombination is then possible.
- 1. , “Dimensionless solution of the equation describing the effect of surface recombination on carrier decay in semiconductors”, Journal of Applied Physics, vol. 76, pp. 2851-2854, 1994.
- 2. , “Analysis of the interaction of a laser pulse with a silicon wafer: Determination of bulk lifetime and surface recombination velocity”, Journal of Applied Physics, vol. 61, pp. 2282-2293, 1987.