Semiconductor Diode Detectors
Electron energy diagram – (electrons fall downward)
carrier density
conduction band
ED, donors → n-type (e’s conduct)
EA, acceptors → p-type (holes conduct)
valence band
“Fermi level” EF (analogous to sea level)
Vopen circuit
Egap ≅ 1.2 V (Si)
p-type
n-type
pn junction
z
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C1
Conductivity of biased diodes
Forward-biased diode
If T = 300K, kT ≅ 26 mv
-
+
p-type
n-type
E
Reversed-biased diode
electrons
i
hf photoexcitation
Egap
trapped electrons
cannot cross junction
trapped
holes
0
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v
-
holes
+
C2
Photodiodes
-
+
Reverse current
i
Increases with
hf (power = Ps)
~ eav − 1 for v > 0
i∝
photo-excitation
photodiode
and T (therefore cooled
circuit
photodetectors have
+
less “dark current”)
+
v
vo
i
V
R
0
Forward
Quantum efficiency
biased
η≅ 0.8 – 0.95
behavior
conduction electrons
per photon
ηPs ⎞
⎛
v o = ⎜ idark +
e ⎟R
if hf > Egap
⎝
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DHS 12/27/00
hf
⎠
C3
Avalanche Photo Diodes, “APD”
i
APDs have
current gain
per photon
E
L
v
cascaded
ionizations
produce
“avalanche”
avalanche
region
breakdown
photoexcitation hf
i(t)
collisionally
excited electron
100
10
zj z 0
1
t
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C4
Avalanche Photodiodes, “APD”
i(t)
100
10
1
t
Simple model for avalanche current and noise:
ith photon has gain gi ≅ e
go zi
L
1 ⎛ goL ⎞ ∆
1 g z
g z
− 1⎟[L junction thickness ]
G = E ⎡e o i ⎤ ≅ ∫ e o dz =
⎜e
⎢⎣
⎥⎦ L
⎠
goL ⎝
[ ]
E g2 = E ⎡e
⎢⎣
0
2go zi ⎤
⎥⎦
≅
1 ⎡ 2goL ⎤
e
−1
⎥⎦
2goL ⎢⎣
In practice E ⎡⎢ g2 ⎤⎥ G2 ≅ Gx ; where x ≅ 0.25; ( x = 0.2-0.5 is typical )
⎣ ⎦
G ≅ 200 ± ~6 dB, for typical applications
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C5
Carrier-to-Noise Ratio, “CNR”
for photon detectors
signal
+ shot
dark
+ shot
thermal
Rd
i(t)
T°K
Photodetector circuit model
RL = Rd
0°K
0 B
CNR ∆ is (t ) E[i - i] where is (t ) is the signal
current through RL and i is the total current
2
2
≅ constant for PMT
signal power (W)
is (t ) = ηPseG hf
in2 shot = 2B(eG ) is +D
Only in/2 flows in RL from Rd noise (assume TL << Tdiode),
and (in 2)2 RL = kTB since RL matched to diode R d
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DHS 12/27/00
Therefore in2 thermal R = 4kTB RL
L
E1
Carrier-to-Noise Ratio, “CNR”
CNR ∆ is (t ) E[i - i] where is (t ) is the signal
current through RL and i is the total current
2
2
≅ constant for PMT
signal power (W)
is (t ) = ηPseG hf
in2 shot = 2B(eG ) is +D
in2 thermal R = 4kTB RL
(ηPseG hf )2
CNR =
(2BeGη(Ps + PD ))eG hf + (4kTB RL )
in2
shot
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DHS 12/27/00
in2
L
for photo diodes
or PMT with
constant G
thermal
E2
CNR for constant-G photodiodes, photomultipliers
(ηPseG hf )2
CNR =
(2BeGη(Ps + PD ))eG hf + (4kTB RL )
in2
shot
CNR =
in2
for photo diodes
or PMT with
constant G
thermal
ηPs hf 2B
1 + PD Ps + 2kThf RL ηPs (eG )2
For PD = T =0, or Ps → ∞; ideal quantum limit (denominator equals unity)
In the quantum limit, we want large η and Ps, and small B
Why not let RL → ∞?
Because RLC = τ sec; C = detector capacitance ≅ 10-12 (say)
Then RL ≅ 1000 Ω for τ = 10-9 (f ≅ 150 MHz)
Also, generally:
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set T so PD < Ps
set RLG2 large to contain thermal noise
E3
CNR for variable-G avalanche photodiodes
i(t)
gie
η(Ps + PD )
n=
events s­ 1
hf
gie/δ
δ
0
Assume rectangular pulses
to simplify analysis
t
P +P
g2 e2η s D
hf
φi
AC
-δ
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DHS 12/27/00
(τ )
Ps + PD
η
δ
hf
2 e
g
0
δ
2
τ
Φi
0
shot noise
AC
B
(f )
f
~1/δ
E4
CNR for variable-G avalanche photodiodes
2
g
φi
AC
-δ
(τ )
0
2
σi2 = ⎡(i − i) =
⎢⎣
⎥⎦
2⎤
Φi
τ
δ
0
AC
(f )
f
B
~1/δ
B
2 2
(
)
Φ
f
df
=
2
B
g
e η(Ps + PD ) hf
∫ i
−B
Therefore CNR (APD) =
2B g2
(ηPseG hf )2
e2η(Ps + PD ) hf + 4kTB RL
σi2 shot
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DHS 12/27/00
shot noise
hf
P +P
η s D
δ
hf
2 e
g
2 Ps + PD
e η
in2 thermal
E5
CNR for variable-G avalanche photodiodes
Therefore CNR (APD) =
2B g2
(ηPseG hf )2
e2η(Ps + PD ) hf + 4kTB RL
σi2 shot
CNR (APD) =
in2 thermal
ηPs hf 2B
g2 ⎛ P ⎞
2kThf
⎜1 + D ⎟ +
⎜
⎟
G2 ⎝ Ps ⎠ RL ηPs (eG )2
Only change is g2
G 2 ≅ G x , x ≅ 0 .2 − 0 .5
In general, to maximize APD CNR we want G2 large to make
thermal noise negligible, but small so Gx is still modest;
e.g. Gx ≅ 4 is typical
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E6
Infrared Detection
Types of electromagnetic detectors
hf << kT
hf >> kT
hf = kT
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“radio”
“optical,” “visible,” etc.
“infrared”
G1
Infrared Detection
Bolometers (measure heat)
I = bias
Rbias
+
vout
-
R(T)
Ps(W)
+
-
Signal power Ps raises T,
changing R(T) and vo.
Heat flows to heat sink and
to bath; Tbath ~
< 4 – 250K.
heat sink
Tbath
Thermal conductance
(
Gt WK −1
R ≅ Ro e
) ∆ ∆Powert ∆T
Td T
E
conduction band
where
T increases with Ps.
kTd
donors
valence band
z
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G2
Responsivity S of a bolometer
I = bias
Rbias
+
vout
-
R(T)
Ps(W)
+
-
heat sink
Tbath
∂v o
∂R ∂P
∆ S " responsivi ty" = I
•
where P = I2R + Ps
∂Ps
∂P ∂Ps
∂P
∂R ∂P
∂R ⎞
⎛
Thus
= I2
+ 1 = ⎜1 − I2
⎟
∂Ps
∂P ∂Ps
∂
P
⎝
⎠
Thus S = I
S=
Lec07- 14
DHS 12/27/00
−1
(Rbias >> R)
∂R ∂R ∂T
∂R ⎛
2 ∂R ⎞
=
•
⎜1 − I
⎟ where
∂P ⎠
∂P ⎝
∂P ∂T ∂P
− ITdR
Gt T 2
⎛ I T R⎞
⎜1 + d ⎟
⎜ G T2 ⎟
⎝
⎠
t
2
=
= 1 Gt
⎞
∂ ⎛
Td T ⎞ − RoTd ⎛ Td
R
e
≅
1
<<
⎟
⎜
⎟
⎜ o
2
⎠
∂T ⎝
⎝T
⎠
T
Td << T may not apply
G3
Responsivity S of a bolometer
S=
− ITdR
Gt T 2
⎛ I2T R ⎞
⎜1 + d ⎟ If T << T
d
⎜ G T2 ⎟
⎝
⎠
t
0
Thus S → 0 and thermal noise in Rbias dominates as Ibias →
→∞
[
]
Thus there is optimum Ibias = f TdR Gt T 2 (near maximum S)
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DHS 12/27/00
G4
Bolometer noise sources
1)
2)
3)
4)
Can be recombination noise
Johnson noise in R, Rbias (thermal)
Phonon noise via Gt
Photon noise (“radiation noise”)
Recombination Noise
traditional shot noise
•
t
carrier creations
t
carrier recombinations (deaths)
~
< × 2 shot noise
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DHS 12/27/00
H1
Johnson noise
(
VJ rms = 4kTR v Hz ­ 1 2
) [Recall v 2 = 4kTRB]
(
" Noise - Equivalent Power" ≡ " NEP" W Hz ­ 1 2
)
⎛
∂V ⎞
⎟
NEPJ = VJ S ; ⎜⎜ S ∆
⎟
∂
P
⎝
s⎠
vo
rms
NEPJ is analogous to : ∆Trms =
∂v o ∂TA
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DHS 12/27/00
H2
Johnson noise minimization
Bolometer equivalent circuit for VJ , VJ
B
R
from bias and detector resistors:
RB
R
VJ
+
VJ
-
R
VJ
Bout
+
B
vo
-
RB
R
→∞
=
VJ → 0 as
R + RB B
R
∝ RB
Therefore set RB >> R or Tbias ≅ 0°K and then VJ
mostly from R
out
VJ
Rout
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RB
≅
• VJ ≅ VJ = 4kTbathBR volts rms
R
R
R + RB
H3
Phonon noise
Approximate intuitive analysis:
P+ watts →
For thermal phonons hf ≅ kT
Say nR phonons s-1
moving right, nL moving left
(
nR →
← nL
chip
Gt
bath
phonons carry heat
)
Heat flux P+ ≅ k nR Tchip − nL Tbath ≅ kn∆T(n = n ≅ nL ) = Gt ∆T
Gt
Phonon shot noise: NEPG ≅ σP ≅ kT 2n ≅ kT 2Gt k = 2kG t T 2
t
+
Actually,
NEPG ≅ 4kG t T 2 WHz ­ 1 2
t
If Gt → 0, NEPG → 0 but then T rises, so there is
t
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DHS 12/27/00
an optimum Gt which can be computed
H4
Photon shot noise
Thermal radiation contributes photon shot noise
Assume radiation comes from cavity attached to the detector
Cavity has ni photons in mode i at energy Ei = hfi
We seek σn2 for a single mode, i.e. the variance of n
p(n ) = D e­ n[hf kT ] where p(n) = probability of
n photons in each mode with frequency f;
D is to be determined
This is “Maxwell-Boltzmann” distribution for thermal equilibrium
Lec07- 20
DHS 12/27/00
J1
Photon shot noise
p(n ) = D e­ n[hf kT ] where p(n) = probability of
n photons in each mode with frequency f, is to be determined
This is “Maxwell-Boltzmann” distribution for thermal equilibrium
∞
To solve for D let x ∆ hf kT and define Sk (x ) ≡ ∑ nk e −nx
n=0
−d
Sk ( x )
then Sk +1(x ) =
dx
so So ( x ) = 1 1 − e− x
(
∞
Since
)
∑ p(n) = 1 =
n=0
∞
­ nx
D
e
= D • So
∑
n=0
If follows that D = 1 So = 1 − e −hf kT
Lec07- 21
DHS 12/27/00
J2
Photon shot noise
If follows that D = 1 So = 1 − e −hf kT
∞
−d
k −nx
(
)
(
)
≡
⇒
=
then Sk x
Sk +1 x
Sk (x )
∑n e
dx
n=0
Therefore S1(x ) = e
−x
(1− e )
−x 2
(1− e )
2
2
Here we seek σn2 ≡ E ⎡(n − n) ⎤ = E[n2 ]− (E[n])2 ≡ n2 − n
⎥⎦
⎣⎢
S2 (x ) = S1(x ) + 2e
∞
−2x
−x 3
(
∞
1 −nx
n = ∑ n p(n ) = ∑ n
e
= S1 So = e − x 1 − e − x
n=0
n = 0 So
)
−2x
S
2
e
2
2
2
1
+
= n + 2n
n = ∑ n p(n ) = S2 So =
2
So
n=0
1 − e− x
∞
Lec07- 22
DHS 12/27/00
(
)
J3
Photon shot noise
n=
∞
∑ n p(n) =
n=0
(
∞
1 −nx
∑ S e = S1 So = e − x 1 − e − x
o
n=0
n
)
S1
2e − 2 x
2
+
= n + 2n
n = ∑ n p(n ) = S2 So =
2
S
x
−
o
n=0
1− e
2
∞
2
(
σn2
2
2
)
2
= n − n = n + n = σn2
(
) (
)
where n = 1 e x − 1 = 1 ehf kT − 1 photons mode
Optical limit: hf >> kT ⇒ n << 1, σn ≅ n
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DHS 12/27/00
Radio limit:
hf << kT ⇒ n >> 1, σn ≅ n
IR regime:
hf ≅ kT ⇒ n ≅ 1, σn ≅ n + n
2
J4