The thick-base estimate fails in a wafer that is not thick compared with

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Energy from Physics, Ch The illuminated junction: generation, collection and the cell's characteristic

Problem

The thick-base estimate

f_c(x)\approx e^{-x/L}

fails in a wafer that is not thick compared with

L

, where the rear surface decides. In a base of thickness

W

with the junction at

x = 0

, the collection probability of a pair born at depth

x

obeys the same equation as an excess of minority carriers in the dark,

D\,\mathrm{d}^2f_c/\mathrm{d}x^2 - f_c/\tau = 0

, with

f_c(0) = 1

at the junction. A perfectly passivated rear loses no carriers, so

\mathrm{d}f_c/\mathrm{d}x = 0

x = W

; an ohmic rear contact recombines every carrier that reaches it, so

f_c(W) = 0

. Take

W = 180\,\mu\mathrm{m}

L = 187\,\mu\mathrm{m}

and

1000\,\mathrm{nm}

light with absorption length

1/\alpha = 156\,\mu\mathrm{m}

, entering at the front (reflection already excluded) and making a single pass through the wafer. ((a)) Solve for

f_c(x)

with each rear, and check both against the far-side values

0

and

0.67

quoted for this wafer in the text. ((b)) Find the fraction of entering photons whose pairs are collected,

\int_0^W\alpha e^{-\alpha x}f_c(x)\,\mathrm{d}x

, with each rear and with the thick-base estimate cut off at

x = W

. ((c)) Find the fraction of entering photons absorbed at all, and hence the share of absorbed photons collected with each rear. Once the rear is passivated, which is the larger loss: light passing straight through, or pairs lost to recombination?

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