Dr Jo Shien Ng (黄若瑄)
Royal Society Research Fellow/Senior
Lecturer Elect
Dept. of Electronic & Electrical Engineering
1.  Simulations of Avalanche Breakdown
Statistics: Probability & Timing
2.  InAs Avalanche Photodiodes for X-ray
Detection
Simulations of Avalanche
Breakdown Statistics
Probability & Timing
Important parameters
•  Single Photon Avalanche photo-Diodes (SPADs)
•  Quantum efficiency: absorption efficiency &
breakdown probability (Pb)
•  Dark count rate (DCR)
•  After-pulsing
•  Time-to-breakdown (tb)
Avalanche process
- Ionization coefs (α & β)
- Carriers’ dead spaces
- Carrier transit time
Breakdown probability
•  Pb increases with overbias (rV) or overbias ratio
(rV/Vb)
•  Ideal: a Step function
•  Actual: a gradual increase with reverse bias (V)
Ideal
•  Abrupt Pb with small ii
1
coefficients ratio, k = β/α
Actual
(β < α & injecting e)1, 2 Pb
•  Carriers’ dead spaces
•  Avalanche width: due to
changing k
1 R.J.
McIntyre, IEEE TED, ED-20, p.637 (1973).
2 S. Wang et al, APL, 82, p.1971 (2003).
0
Vb
Reverse bias
Time to breakdown
•  Avalanche current builds up with time differently,
reaching Ith at different time.
•  Mean = <tb>, standard deviation = σ
•  Carriers’ dead spaces, Avalanche width: transit time
scaling & k
10-1
10-2
pdf of tb
current (mA)
100
10-3
10-4
10-5
0
150
300
450
time (ps)
600
14
12
10
8
6
4
2
0
0
10 20 30 40 50 60
number of carrier trans it time
70
Simulation model
•  RPL (Random Path Length) model
•  Inputs
ionization
•  Ionization coefficients
•  Carriers’ dead spaces
•  Device structure
•  Outputs (as functions of reverse bias)
•  Distribution in gain: mean gain & excess noise (APDs)
•  Pb (SPADs)
•  Avalanche current: bandwidth (APDs) & tb (SPADs)
•  Avalanche duration (APDs) & duration of selfsustaining avalanche (SPADs)
RPL model
•  Example outputs: Carrier concentration profile in
the avalanche region at threshold condition (10
and 100 µA)
100uA
9e+5
•  k = 1
8e+5
7e+5
10uA
Num Carriers
6e+5
9e+4
8e+4
7e+4
4e+5
3e+5
6e+4
Num Carriers
5e+5
Electron
Hole
Total
2e+5
5e+4
1e+5
4e+4
0
0.0
3e+4
0.2
0.4
0.6
0.8
Position (micron)
2e+4
1e+4
0
0.0
0.2
0.4
0.6
0.8
Position (micron)
1.0
1.2
1.4
1.0
1.2
1.4
RPL model
•  Monte Carlo numerical method: useful when the
process of interest is random
•  Avalanche gain: how far does an electron drift
along the field before it initiates an II event? Not
a single value: a distribution of II path length
•  Throwing the dice many times: Build up statistics
(outputs) by repeating many trials with the same
simulation conditions
RPL model
•  Each trial (for SPAD simulations)
•  Inject an electron-hole-pair at the same
position
•  Allow avalanche current to increase upto the
threshold before ending the trial
•  Record time taken to reach threshold (tb)
•  Keep track of number of trials that did reach
threshold (Pb)
Breakdown probability:
dead spaces & avalanche width
Pb: dead space
•  p-i-n diode with w = 1µm (constant electric field
in avalanche region)
•  Carriers’ velocity = 105 m/s
1.0
•  Fixed k = 1 (same
trend in other k
values)
0.8
0.6
Pb
•  Abrupt Pb vs
overbias ratio when
dead spaces are
significant
d/w =0
d/w =0.1
d/w = 0.2
d/w = 0.3
0.4
0.2
0.0
0.00
0.05
0.10
overbias ratio
0.15
0.20
Pb: avalanche width
•  Including dead spaces
•  InP’s ionization coefficients & dead spaces
(injecting holes)
•  Excluding dead spaces:
as expected from prior
work
1.0
0.8
Pbh
0.6
w = 2µm
w = 1µm
w = 0.5µm
w = 0.2µm
L ocal model
0.4
0.2
0.0
0.00
0.03
0.06
0.09
overbias ratio
0.12
0.15
•  Including dead spaces:
wider avalanche region
still gives more abrupt
Pb vs. overbias ratio
•  Much smaller difference
though
Timing:
dead spaces & avalanche width
Timing: k
•  p-i-n diode with w = 1µm & no dead spaces
<tb> (ps)
103
more similar
α and β
•  β/α = 0.1, 0.5 & 1.0
(top to bottom)
•  Similar α & β for a
faster breakdown
with less variation
102
102
σ (ps)
more similar
α and β
101
0.2
0.4
0.6
Pbe
0.8
•  More carrier
feedbacks (essential
for breakdowns)
Timing: dead space
•  Fixed k = 1.0 and 0.1: d/w = 0 or 0.1
d/w = 0.1
d/w = 0
103
<tb> (ps)
k=1
k = 0.1
•  Fixed k: slower
breakdown & more
variation with dead
space
102
σ (ps)
102
101
0.2
0.4
0.6
Pbe
0.8
1.0
•  Regardless of dead
space, similar α & β for
a faster breakdown with
less variation
Timing: carrier transit time
•  Si (ionization coefficients and dead spaces)
•  Different w: increasing k and d/w with w
1.0µm
0.5µm
0.1µm
<tb> (ps)
103
102
•  As w reduces, effect
of increased k cancels
effect of increased d/
w (almost exactly)
101
102
σ (ps)
•  <tb> and σ roughly
scale with carrier
transit time
101
100
0.2
0.4
0.6
Pbe
0.8
Minimum avalanche width:
dark count rate & photon count
probability
Minimum w
•  Dark count rate: bulk leakage current & Pb
•  Photon count probability: Pb
•  Minimum w expected due to increased
tunneling current (as in high-speed APDs)
•  Ramirez et al3: Band-to-band tunneling current
considered (in DCR) but lacked accurate
ionization parameters for Pb then.
•  This work: compare DRC vs. photon count
prob. for different w
3 Ramirez
et al, IEEE JQE, 42(2), pp. 137–145, 2006.
Minimum w
•  DCR = 1 - exp (-NdPd)
•  Nd = Itun × tgate / q ( tgate assumed 2ns)
•  Photon count probability = 1-exp(-ηN0Pb)
•  η = QE (assumed 0.5)
•  N0 = number of photon during the on-time
(assumed 1)
1 Ramirez
et al, IEEE JQE, 42(2), pp. 137–145, 2006.
Minimum w
•  InP
(open
symbols)
Al
As AND InP SPADS
•  InAlAs (closed symbols)
EHAVIOR OF In
569
0.20µm (□, ■)
0.30µm (◊, ♦)
0.40µm (∇,▼)
0.50µm (∆,▲).
Dark counts
more likely than
photon counts
Photon counts
more likely than
dark counts
Fig. 7. Dark count probability calculated as a function of photon count proba-
Minimum w
•  InP (red symbols)
•  InAlAs (black symbols): a slight advantage
24x10-12
1.0
20x10-12
0.8
16x10-12
0.6
0.4
12x10-12
0.2
0.0
0.15
8x10-12
0.20
0.25
0.30
0.35
0.40
0.45
Avalanche widths (µm)
0.50
0.55
Timing jitter (s)
Dark count probability
1.2
With quenching resistor
•  More realistic simulation for <tb>
•  Use the Simple Monte Carlo model (for InAlAs)
and add a quenching (series) resistor
<tb>
1e-10
•  Much longer <tb>
(more similar to
experimental results)
1e-11
•  Observations from
RPL model work
expected to hold true
No resistor
R = 50k ohm
R = 30k ohm
R = 20k ohm
R = 10k ohm
1e-12
0.5
0.6
0.7
0.8
0.9
Breakdown probability
1.0
Which material then?
•  Abrupt Pb vs overbias ratio: k→ 0
•  Fast timing with small jitter: k → 1
•  If using quenching resistor, effects of k on
timing become relatively unimportant.
•  k→ 0, but from where other than Si (with large
w only so slow breakdown and high voltage)?
•  New candidate: InAs*
•  k = 0 & bias < 15V: sub-Geiger mode
InAs Avalanche Photodiodes
for X-ray Detection
Introduction: InAs & InSb
•  Si & Ge X-ray detectors
•  Smaller bandgap (Eg)
è smaller pair creation energy (ε)
è larger number of electron-hole pairs (N) created per
absorbed photon
è better fano-limited energy resolution
•  InAs (0.36 eV) & InSb (0.17eV)
•  Also large atomic numbers (In: 49, As: 33 & Sb: 51) è
high absorption coefficients than Si and Ge
•  Material properties suggest potential for direct detection of
X-ray
•  But leakage currents in actual devices???
25
Introduction: InAs & InSb
26
•  InSb for X-ray detection
•  Diodes made in Kyoto University
•  Detections of gamma ray [1] & alpha particle [2]
•  Cooled to 4-5 K due to high leakage currents
•  InAs has larger bandgap. Maybe lower leakage currents?
•  Zn-diffused pixel matrix detector reported in 2006 [3]
•  When cooled to 77 K, detected alpha particles but not X-ray &
gamma ray
•  Sheffield’s InAs APDs for vis-NIR APDs [4, 5]
•  Excellent (low) avalanche noise
•  Appreciable gain at low reverse bias
[1] S Hishiki et al, Nucl. Instr. & Meth. A 559 (2006) 558.
[2] Y Sato et al, Radiat. Meas., In Press, Corrected Proof.
[3] A Säynätjoki et al., Nucl. Instr. & Meth. A, 563 (2006) 24.
[4] A R J Marshall et al, IEEE Photon. Tech. Lett., 21 (2009) 866.
[5] P J Ker et al, IEEE J. Quan. Electron. 47 (2011) 1123.
Device structure
27
•  InAs n-i-p diode on commercial p-InAs substrate
•  Wafer growth (MBE) & device fabrication at Sheffield
•  Circular mesa diodes by photolithography & chemical
etching. Passivated with SU-8 photoresist
InAs, n-type, 2µm
InAs, undoped, 6µm
InAs, p-type, 2µm
InAs, p-type, substrate
mesa
diodes
bond
pads
Leakage current, Jd
Leakage currents at 77K
diodes with diameter = 100µm
•  100 µm diameter diodes
•  Packaged into TO headers
•  At low reverse bias, RT Jd ~
150 mAcm-2 due to small Eg
•  At 77 K, Jd ~150 nAcm-2
•  Increased leakage current
after packaging
!
X-ray detection: 55Fe Spectra
•  T = 77 K, 55Fe radioactive X-ray source of 185 MBq activity
•  Devices in TO header
•  Collection time ~ 2.5 minutes, shaping time constant = 0.5µs
Bias
InAs APD
X-ray
source
Liquid
nitrogen
Preamplifier
MCA
Post Amplifier (Ortec 570)
Metal Dewar
!
X-ray detection: 55Fe Spectra
•  Spectra at different reverse biases
•  Main peak at 5.9 keV
•  As reverse bias increases, main peak moves away from
noise (towards higher channel numbers) --- Expected
750
Counts per channel
unity gain
500
1V
3V
5V
6.9V
8.7V
9.5V
250
0
50
100
150
200
Channel number
250
300
X-ray detection: FWHM for 5.9 keV
•  2.02 keV at M = 1.58
•  950 eV at M = 5.3
10
laser measurements
D1
D2
8
Multiplication, M
•  FWHM improves (drops)
with increasing gain
6
4
2
0
0
!
2
4
6
8
10
Reverse Bias (V)
12
14
!
X-ray detection: Energy resolution
ΔE2 = (2.36)2 f Eε + Δ 2noise
Fano limit
Noise of detection
system & detector
•  For APDs, additional noise from avalanche gain
statistics
•  Random Path Length (RPL) model to provide
the avalanche gain statistics
http://www.shef.ac.uk/eee/research/smd/research/apd_simulator
X-ray detection: Data cf. simulation
Counts (normalised to peak)
•  Simulation vs data for electron M = 5.3 (at 10.8 V)
1.4
simulated
experimental
gain distribution
1.2
1.0
0.8
5.9 keV
unmultiplied peak
0.6
6.49 keV
0.4
0.2
0.0
50
100
150
200
250
300
Channel number
Lots of holes injected from ncladding (they don’t experience
avalanche gain)
Electron-hole pairs created
within avalanche region,
giving 1 < M < 5.3
X-ray detection: Gain Statistics
Fano limit
measured
gain statistics
FWHM (eV)
1000
100
1
2
3
4
5
Mean gain, M
6
7
8
•  Single carrier ionization (k=0 characteristics) produce low excess
noise
•
Dead space effect can further improve the energy resolution in
APDs
•  InAs shows gain statistics spread independent of the gain value.
Conclusions
•  Direct detection of X-ray (5.9 keV) using InAs APD
demonstrated
•  FWHM of 5.9 keV peak improves with gain (950 eV at M =
5.3)
•  Simulation: FWHM due to avalanche gain spread is
independent of M for M > 3
•  Improvements required:
•  Reduction of electronic noise in measurement setup
•  Reduction of APD leakage current through better device packaging
•  More optimised APD design (thick InAs absorber)