The surface also plays an important role in recombination. Typically the surfaces complicate the measurement of the bulk lifetime. Recombination at the surfaces is typically described by a *surface lifetime* τ_{s}, which includes the fundamental decay mode but ignores higher decay modes. τ_{s} is a function of the surface recombination velocities S_{1} and S_{2}, 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 3in transient measurements that works in all cases is:

$$ S = \sqrt{D \left(\frac{1}{\tau_{eff}}- \frac{1}{\tau_b}\right)} \tan \left[\frac{W}{2} \sqrt{\frac{1}{D}\left(\frac{1}{\tau_{eff}}- \frac{1}{\tau_b}\right)}\right]$$

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 = S_{1} = S_{2}.

One surface is perfectly passivated so S_{2} = 0.

Both surfaces are perfectly passivated so S_{1} = S_{2} = 0.

Both surface have high recombination so S_{1} and S_{2} are large.

One surface has a high recombination and the other has a low recombination so S_{1} = 0 and S_{2} = ∞

### Effective Lifetime

The combination of the effects of surface and bulk recombination give the effective lifetime of carriers τ_{eff} within a sample.

For many measurements, it is possible to reduce surface recombination of the surfaces to a low enough value so that τ_{b} ≫ τ _{eff.} Alternatively, if τ_{b} is sufficiently high the lifetime measurement is completely dominated by the surfaces and τ_{s} ≫ τ_{eff}. 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, т. 76, с. 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, т. 61, с. 2282-2293, 1987.
- 3. «Analytical approximation of effective surface recombination velocity of dielectric-passivated p-type silicon», Solid-State Electronics, т. 45, № 9, с. 1549 - 1557, 2001.

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