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REQUIREMENTS OF SILICON FOR CRYSTALLINE SOLAR CELLS 189
10 −4
10 −6
Co* Ni
Mn 8
Cr 18
Cu
Diffusivity
[cm 2 /s]
10 −8
10 −10
v
Mn*
Ti 7
10 −12 Ti 17
Cr 17
Fe 17
Fe 3
10 −14
Mn 17
10 20
30
Reciprocal temperature
[10 4 /K]
Figure 5.8 Diffusivities of 3d transition elements. Reproduced from Metal Impurities in Silicon-Device
Fabrication, Graff K, 29, 2000, © Springer-Verlag GmbH & Co. KG
junction). The low-injection-level minority-carriers lifetime, τ 0 , is inversely proportional
to the impurity concentration, N(1/cm 3 ) [28]:
τ 0 = (σ vN) −1
where v is the thermal velocity, which is the average speed of the electrons as they
randomly collide with atoms, impurities or other defects, and σ , having the units of cm 2 ,
represents the impurity atoms effective cross-section for the capture of a minority carrier.
Here the carrier capture cross-section for electrons, σ e (cm 2 ), must be inserted for
p-type silicon and σ h for holes in n-type silicon. The thermal diffusion velocity v of
electrons at room temperature is 2 × 10 7 cm/s.
The capture cross-sections for different transition metals can differ by several orders
of magnitude. As a consequence, the carrier lifetime of a silicon sample can even be
determined by an impurity of minor concentration if this is a “lifetime killer” with a high