Page 233 - Organic Electronics in Sensors and Biotechnology

210 Cha pte r S i x

2
100 mW/cm , so the linear range can extend over many orders of
magnitude.
Negligible recombination does not imply a quantum efficiency
of 100 percent. The most obvious reason is that only a fraction γ of
the incident photons lead to the successful generation of free carri-
ers (due to incomplete absorption and non-dissociative exciton
decay channels). A second, more subtle, reason is that some of the
photogenerated carriers reach the “wrong” electrodes. To generate
the maximum (negative) photocurrent, the photogenerated charges
should be extracted only by their “parent” electrodes––electrons by
the cathode and holes by the anode. In reality, the electrons and
holes move by a combination of diffusion and drift. The diffusive
trajectories of the electrons and holes resemble random walks onto
which the electric field superimposes ordered drift: diffusion drives
charges indiscriminately to both electrodes whereas drift drives the
electrons and holes systematically to their parent electrodes. The
randomizing effects of diffusion mean that only a fraction α of elec-
e
trons reach the cathode and only a fraction α of holes reach the
h
anode. The remaining charges migrate to the “wrong” electrode
where they are extracted into the external circuit, giving rise to a
positive photocurrent that partially cancels the negative one. Hence,
if photons strike the photodiode at a rate ℜ, the short-circuit photo-
current will equal

I ⎛α + α ⎞ ⎡ 1( − α ) + 1( − α ) ⎤
γ
SC =−γ ⎜ e h ⎟ ℜ+ ⎢ e h ⎥ ℜ (6.3)
e ⎝ 2 ⎠ ⎣ 2 ⎦
where the first term on the right-hand side corresponds to the nega-
tive photocurrent generated by charges that reach the “correct” elec-
trode and the second term corresponds to the positive photocurrent
generated by charges that reach the wrong electrode. (The factor 2 in
the denominator is required to avoid double counting since for every
photogenerated electron-hole pair, at most one electron can be
extracted into the external circuit.) Equation (6.3) can be rewritten in
the simpler form

I
SC =−γα + α − ) ℜ (6.4)
(
1
e e h
which, dividing through by eℜ, yields the following expression for
the short-circuit quantum efficiency η:
η(E ) =− γ(E )[ α (E ) + α (E ) − 1 ] (6.5)
BI BI e BI h BI

In writing this last equation, we have made explicit the dependence
of γ, α , and α on the built-in field strength E . The larger the built-in
e h BI