Page 247 - Organic Electronics in Sensors and Biotechnology

224 Cha pte r S i x

2
2
and the current variance σ is related to σ by
I N
σ e ⎛ ⎞ 2 e 2 σ 2
σ = N = N
2
I ⎜ ⎝ Δ t ⎠ ⎟ t Δ 2 (6.36)
In situations where the motion of the individual electrons is com-
pletely uncorrelated, the number of electrons detected in the mea-
surement time Δt will be described by a Poisson distribution, for
50
2
which the mean μ equals the variance σ . Therefore the current vari-
ance can be written
e 2 σ 2 e 2 μ e 2 It Δ eI
σ = N = N = = (6.37)
2
I t Δ 2 t Δ 2 t Δ 2 e t Δ
Since each measurement takes a time Δt to complete, the bandwidth B,
i.e., the highest frequency that can be measured (the Nyquist Fre-
quency) is given by

1
B = (6.38)
2 t Δ
The mean-squared variation in the current per unit frequency is
therefore given by
2
σ = σ 2 I = 2 eI
I B (6.39)
which, from Eq. (6.11), can be written in the expanded form

2
V +
σ = 2 eI () 2 eI () (6.40)
V
I dark ph
The first term in Eq. (6.40) is zero when V = 0, meaning the shot noise
is minimized (and hence the signal-to-noise ratio is maximized) for
measurements obtained under short-circuit conditions. This mode
(known as the photovoltaic mode) is the preferred mode of operation
for high-precision measurements. Under an applied bias (the photo-
conductive mode), charges are extracted more quickly, allowing for
improved speed and linearity, but this comes at the expense of
increased shot noise. The signal-to-noise ratio (SNR) for a signal
dominated by shot noise is given by

σ 2eI
SNR = 1 = ∝ I (6.41)
I I

Shot noise is therefore most important at low photocurrents where
the statistical fluctuations in the flowing electrons are most evident.
In passing, we point out that Eq. (6.39) was obtained using Poisson