Page 246 - Organic Electronics in Sensors and Biotechnology

An Intr oduction to Or ganic Photodetectors 223

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Unlike a resistor, a capacitor generates no noise. The voltage across
the capacitor is fixed the instant the two electrodes are disconnected
and thereafter undergoes no further thermal fluctuations, since the iso-
lated plates cannot spontaneously acquire additional charge through
thermal motion. It is only when the plates of the capacitor are con-
nected by a resistor that fluctuations can arise. It follows that the only
circuit element that contributes directly to the intrinsic noise character-
istics of the photodiode is the shunt resistance R . It is possible to
sh
derive the following expression for the voltage variance per unit fre-
50
quency due to a resistor R (see Appendix and Usher ):
2
σ = 4 Rk T (6.33)
V B
The thermal noise can be alternatively expressed in terms of
2
2
σ = σ / R , the mean-squared current per unit frequency flowing
2
V
I
through the resistor
2
σ = 4 kT (6.34)
B
I
R
The current noise is therefore smallest for large values of the shunt
resistance. Importantly, biasing the resistor has virtually no effect on
the voltage and current fluctuations since it has the minor effect of
superimposing a very small constant drift velocity on the (much larger)
instantaneous velocities of the charge carriers. The thermal noise of a
discrete resistor is therefore largely independent of the applied bias. In
the specific case of a photodiode, the shunt resistor is in parallel with a
diode, and both “components” contribute thermal noise (whose vari-
ances combine in a sum of squares manner). In short circuit, the diode
resistance is much higher than the shunt resistance, and the shunt resis-
tance is therefore the dominant source of thermal current noise; at suf-
ficiently high applied biases, the diode resistance drops substantially,
causing it to become the dominant source of thermal noise.
Shot Noise
The second major source of noise is shot noise, which is due to statis-
tical fluctuations in the flowing current caused by the discrete nature
of the electrons. Consider an experiment in which we determine the
number N of particles that flow through the external circuit in a meas-
urement time Δt. N will fluctuate randomly from one measurement to
2
the next, but we can assign a mean μ and a variance σ for the
N N
number of electrons detected per measurement. The mean current I is
related to μ by
N
eμ
I = N (6.35)
t Δ