Application Note
Photodiodes
The voltage-current characteristic for an idealized diode
is given by:
The effect of the additional current term is shown in
Figure 1 as curve B.
Equation (1)
By examining equation (2), an equivalent circuit can be
constructed which will characterize the photodiode.
Because most photodiodes are operated near the bias
point where V = 0 this condition will be used to
characterize the photodiode. Photodiodes may be
represented by the Norton or the Thevenin equivalent
circuit model. However, the Norton (constant current
generator) equivalent circuit is the simplest model to
characterize and understand (see Figure 2).
Photodiodes are essentially photon counters. That is,
one photon with energy equal to or greater than the
band gap energy of the semiconductor will generate one
free charge carrier in the theoretical photodiode. This is
the basis of the term - ISC in equation (2), and is also the
current generator in the equivalent circuit. The
magnitude of the current generator is Nq, where N is the
number of incident "effective" photons per second and q
is the charge on the electron. The term effective photons
is used to describe photons with energy equal to or
greater than the band gap energy of the semiconductor.
The only current generators present in the equivalent
circuit are the constant current generator and the noise
current generator. This can be seen by expanding
equation (2) and allowing V = 0:
qv
I = IS(e kT -1)
Where:
I = Junction current
IS = Saturation current
q = Charge on the electron = 1.6 x 10-19 C
V = Voltage across junction
k = Boltzmann’s constant = 1.38 x 1023 joules/K
T = Absolute temperature
Equation (1) is shown graphically in Figure 1, as Curve A.
Figure 1
Equation (3)
A
qv
B
I = - ISC + ISe kT- IS = - ISC + IS - IS
Equation (3) can be reduced to:
When a photon’s energy is equal to or greater than the
band gap energy of the photodiode material it impinges
on the photodiode, an additional current term is present in
the above expression. It now becomes:
Equation (2)
qv
I = - ISC + IS(e kT - 1)
Equation (4)
I = - ISC
The small signal ac conductance of the photodiode can
be found by finding the slope of the diode curve at a
particular operating point by:
Equation (5)
Where:
ISC = The current which is caused by the impinging
photons’ energies which are equal to or greater
than Ei
Ei = The band gap energy in eV
350
© Honeywell Europe S.A.
qv
qI e kT
dI
= s
= Gd
dV
kT
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changes in order to improve design and
supply the best products possible.
Application Note
Photodiodes
At the operating point of V = 0, Gd becomes:
Equation (6)
Where:
λi
h
c
Ei
I q
Gd = S
kT
and
Equation (7)
Rd =
= The long wavelength cutoff in microns
= Planck’s Constant
= The speed of light
= The band gap energy in eV
The spectral response of the theoretical photodiode to a
constant number of photons is shown in Figure 3.
kT
qIS
This is the value of the Norton equivalent resistance in
Figure 2.
Figure 3
Figure 2
R
Isc
Gd
Norton
equivalent
circuit
i
The spectral response of the theoretical photodiode to a
constant quantity of energy is shown in Figure 4.
The band gap energy equation is:
Equation (8)
Figure 4
Ei = h = hc/
This equates the minimum amount of energy required
for a photon to cause a charge carrier to flow. The
wavelength associated with this energy is the maximum
wavelength of an effective photon, and is known as the
long wavelength cutoff of the photodiode, or:
Equation (9)
i
=
hc
1.24
=
Ei
Ei
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changes in order to improve design and
supply the best products possible.
R
i
The difference in the shapes of the two response curves
is due to the relationship between the energy and the
wavelength of a photon. Equation (8) shows that the
energy of a photon varies inversely with its wavelength.
The photodiode requires the photon to have energy
equal to or greater than its band gap energy, so any
energy in excess of the band gap energy is "wasted".
351
Application Note
Photodiodes
Therefore, the photodiode is most energy efficient at the
band gap energy point or long wavelength cutoff.
The equivalent circuit of the theoretical photodiode
shown in Figure 2 must be expanded to make it more
like an actual photodiode. Figure 5 shows the expanded
model of the photodiode. Three noise current generators
have been added in addition to a shunt capacitor, a
shunt conductance, and a series resistor. The shunt
capacitor represents the junction capacitance which
varies with diode area, doping level, and junction bias.
Shunt conductance is the conductance of the
photodiode material. The series resistance represents
the sheet resistance of the top layer of the
semiconductor photodiode material.
Figure 5 General Equivalent Circuit for Photodiode
Equation (10)
Pn = kT f
Where:
Pn = Noise power
k = Boltzmann’s constant
T = The absolute temperature in degrees Kelvin
∆f = The equivalent noise bandwidth
A resistor can be characterized by a noiseless resistor in
shunt with a noise current generator.
Figure 6
Rs
IL
IR
in
Isc
1
f
Gs
in Gd in Gs insh
GR
Gd
Three of the more important noise sources found in
photodiodes are semiconductor noise (1/f), thermal
noise, and shot noise. One of these sources usually
dominates the photodiode noise.
Little is know about 1/f noise, however it has been found
that surfaces and contacts affect these noises.
Semiconductor noise usually dominates the photodiode
noise at frequencies below 500 Hz. In wideband
operation the semiconductor noise contributed to the
total noise is usually unimportant. However, in narrow
band applications at 500 Hz or below, the effect of 1/f
noises must be taken into account.
Thermal noise is due to thermal motions of the charge
carriers. Every conductor above 0° Kelvin exhibits
thermal noise. Nyquist has shown that the average
noise power which is available from a conductor is:
352
GL
Cd
For maximum power transfer, the source and load
resistances or conductances must be equal. If the
source and load resistances are equal, the magnitude of
the noise current generator can be found. Power can be
expressed as:
Equation (11)
I 2n
Pn
=
f
G
Equation (10) may be written as:
Equation (12)
I 2n
Pn
=
= kT
f
G
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changes in order to improve design and
supply the best products possible.
Application Note
Photodiodes
From Figure 5 and Figure 6 Gd = GI, Id = IL and In = 2IL,
equation (12) can be written as:
Equation (13)
IL2 =
I n2
= kTGd f
4
The shunt conductance shows full thermal noise and is
given by equation (14):
Equation (14)
In2 =
4kT f
= 4kTGd f
Rd
Shot noise is associated with the random arrival of
charge carriers at the junction and is given by the
following equation:
Equation (15)
Insh2 = 2qldc f
Where
q = The charge on the electron
Idc = The current in amperes
∆f = The equivalent noise bandwidth
Insh = Shot noise
The noise is independent of frequency and is known as
white noise.
There is a shot noise current connected to each current
term in equation (2) I = -ISC + IS (e - 1). Although it was
shown that the sum of the forward current and the
reverse saturation current is zero at the bias point V = 0,
the shot noise due to these currents is not zero. Each of
these currents develop shot noise currents which add as
the square of these currents. The shot noise current of
the forward and reverse currents at the V = 0) bias point
is:
Equation (16)
Insh2
= 2q (Is = If) = 4qIs
f
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changes in order to improve design and
supply the best products possible.
It can be calculated as follows:
Equation (17)
Is =
kTGd
kT
=
q
qRd
If this value of IS in equation (17) is substituted for IS in
equation (16), a value of noise current will be obtained. It
is inversely proportionate to the photodiode resistance or
proportionate to the photodiode conductance.
Equation (18)
Insh2 4kT
=
= 4kTGd
f
Rd
This shows that the noise characteristics of the small
signal ac impedance of the photodiode is the same as
the thermal noise current of a resistor of the same value.
The squared values of the individual thermal noise
currents can be added together to give the squared
value of the total thermal noise currents.
Equation (19)
Insh2
= 4kT (Gs + Gd) = 4kTGeq
f
Where:
Geq = The equivalent parallel conductance of the
shunt and diode conductances.
There is also a shot noise current associated with the
photon induced current ISC in equation (2). It has been
shown that the ISC term is proportionate to the impinging
photons. Therefore the shot noise current associated
with photon current will be proportionate to the square
root of the number of photons. Equation (20) shows the
magnitude of this shot noise current generator.
Equation (20)
In2
= 2q2 N
f
Where:
N = Number of photons/sec
353
Application Note
Photodiodes
The magnitude of the current generator, ISC, must be
changed in the theoretical photodiode equivalent circuit.
This is necessary because every impinging photon with
sufficient energy will not cause an electron to flow in the
external circuit. Some of the photons are reflected at the
surface of the semiconductor material. If the angle of
incidence of the photons is 90° and the area surrounding
the photodiode is a vacuum of air, the fraction of
photons lost by reflection is:
Equation (21)
r=
(n-1) 2
(n+1) 2
Where:
n = The index of refraction of the semiconductor
material at the wavelength of interest.
capacitance Cd, and the semiconductor noise current
generator. This equivalent circuit is valid in the
mid-frequency range. The mid-frequency range is above
the frequency range where semiconductor noise
dominates and is below the frequency range where
photodiode junction capacitive reactance becomes
appreciable, relative to the photodiode equivalent
resistance Req. The validity of the circuit also requires
the load impedance to be large compared to the series
resistance Rs. The magnitude of the current generator
ISC and the magnitude of the shot noise current
generator are given in terms of quantum efficiency and
effective photons. This circuit will be used as the
equivalent circuit in the following discussions of
photodiode figures of merit and photodiode applications.
Figure 7 Photodiode Equivalent Circuit
By using a non-reflecting coating on the photodiode, the
reflected loss can be reduced.
In addition to this loss, some of the electrons that are
produced by the photons are lost by recombining in the
bulk material and at the surface of the photodiode. The
bulk recombination is affected by the quality of the
photodiode material. The surface recombination is
affected by fabrication processes. The ratio of electrons,
that are caused to flow in the external circuit, to the
number of impinging photons is termed quantum
efficiency, η. The magnitude of the current generator in
the equivalent circuit becomes:
Equation (22)
Isc = q N
Because the shot noise current generator depends on
the value of the current generator, ISC, the shot noise
magnitude will also be affected by the quantum
efficiency. Equation (20) can be changed to show the
dependency of shot noises by:
Equation (23)
In2
= 2q2 N
f
This shows that the magnitude of the shot current
generator is proportionate to the square root of the
quantum efficiency.
Figure 7 shows a photodiode equivalent circuit that has
omitted the series resistance RS, the parallel diode
354
N
InN
insh
inGeq
Geq
Although both noise current generators may contribute
equal amounts of noise current, one will usually
dominate the total photodiode noise. Thermal noise
limited condition is the term used to refer to operating
conditions when the thermal noise current generator
dominates. From equations (19) and (22), the signal to
noise ratio for a one Hz noise equivalent bandwidth for
photodiodes in this operating condition can be
expressed as:
Equation (24)
qNt
Is
qNt Req
=
=
In
4kTGeq
4kT
Where:
Nt = The number of target or source photons
This shows that the signal to noise ratio is proportionate
to the quantum efficiency η and the square root of the
equivalent diode resistance, Rq. The signal to noise
ratio may be increased by increasing the quantum
efficiency to one or by increasing the equivalent
photodiode resistance. However, a point may be
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changes in order to improve design and
supply the best products possible.
Application Note
Photodiodes
reached where the thermal noise generator is no longer
dominant. This operating point may be reached where
the thermal noise generator is no longer dominant. This
operating condition is known as the background limited
condition and yields the maximum signal to noise ratio
for any operating condition. By referring to equations
(22) and (23), the signal to noise ratio for a one Hz
bandwidth for photodiodes in the background limited
condition can be expressed as:
Equation (25)
nqNt
n
Is
=
=
Nt
In
2q2NB
2NB
Where:
NB = The number of background photons
The maximum signal to noise ratio will be obtained when
η = 1.
Equation (26)
Nt
Is
=
In
2NB
This is the signal to noise ratio for a theoretical
photodiode in the background limited condition.
Honeywell reserves the right to make
changes in order to improve design and
supply the best products possible.
355