Spatial Distribution of Electron-Hole Pairs
Created by Photons in Detector Materials
Fei Gao, Luke Campbell, Yulong Xie, Ram Devanathan,
Anthony Peurrung, William Weber
Pacific Northwest National Laboratory
Supported by the Radiation Detection Materials Discovery
Initiative at PNNL
Outline
•
Introduction
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™
™
Spatial distribution of electron-hole pairs (e-h) – previous simulations
Calculation of e-h pair yield
Several issues to be addressed
•
Monte Carlo Model
•
Production of e-h pairs
™
™
•
Electron cascade and number distribution of e-h pairs
W value and Fano Factor
Spatial Distribution of Electron-Hole Pairs
™
™
Conventional detailed MC method
Spatial distributions of electron-hole pairs (in Ge)
2
Introduction
‰ Most previous simulations of e-h pairs – only consider track
structure of secondary electrons created by photons to evaluate
dE/dx [NIMA 563 (2006) 116]
106 photons
Total deposited energy: 100 keV
electron in CdZnTe – linear electron
cascade
¾ e-h pair density (or number of e-h pairs) - ρ =
1 dE
W dx
W – mean energy required to create an electron-hole pair
3
Introduction
‰ An attempt to carry out the full e-h – only considers inner shell
ionizations [J. Phys. D: Appl. Phys. 33 (2000) 1417]
e-h pair distribution by a 140
keV photon in amorphous-Si
‰ Several issues – detailed distribution of e-h pairs
¾ e-h pair recombination
¾ Nonlinear electron cascade- more accurate calculations of intrinsic
properties
¾ Nonlinearity observed in scientilators
4
Monte Carlo Model
‰ Model – interaction of photon with materials:
¾ Photoelectric absorption
¾ Compton scattering
¾ Electron-positron pair production
‰ Model – detailed energy-loss mechanisms for electrons (inelastic)
¾ Electron-Phonon interaction
¾ Valence Band Ionization
¾ Excitation of Plasmons and Decay
¾ Core shell ionization (K and L-shell…..)
¾ Shake-off electrons
¾ Bremsstrahlung emission – energy loss
¾ Band structure effects (for semiconductors)
¾ Stopping criteria for electrons – 2.5 eV
‰ Model – elastic interaction
‰ Mean free path between collisions;
λ−1 = N ( σ el + σ in )
5
Electron Cascades in Ge
¾ Photon energy – 50 eV ~ 2 MeV
¾ Number of simulations for each energy – 105 (statistics)
¾ Band gap – 0.74 for Ge
‰ Electron distribution in Ge
4
No. Events (x10 )
3.0
50 eV
150 eV
600 eV
2.5
¾
Asymmetric
¾
Not perfect Gaussian
distribution
¾
No. electrons/event =
16.8, 54.3 and 224.6
for 50, 150 and 600
eV, respectively
2.0
1.5
1.0
0.5
0.0
0
50
100
150
200
250
No. Electrons (n)
6
Electron Cascades in Ge
1200
40 keV
662 keV
No. Events
1000
662 keV gamma
ray from 137Cs
800
600
400
200
0
1.500e+4
1.530e+4 2.496e+5
2.508e+5
No. Electrons (n)
¾ Symmetric distribution
¾ Approximate Gaussian distribution
¾ No. electrons ~ 557.4 and 2893.2 for 40 and 662 keV, respectively
¾ Height of the distribution decreases with increasing energy
¾ Width of the distribution increases with increasing energy
7
Electron Cascades in Ge
‰ W value and Fano Factor
MII/MIII LII/LIII
3.0
LI
0.25
K
Ge
MI
0.20
0.15
F
W (eV)
2.9
2.8
0.10
2.7
0.05
2.6
0.01
0.1
1
10
100
1000
0.00
10000
E (keV)
¾W:
For E (>2 keV), ~ 2.64 eV (exp: 2.6 – 2.9 eV)
¾F:
For E (>2 keV) 0.11 (exp: 0.09 ~ 0.195)
8
Electron Cascades in Several Materials
ΔEin
‰ Intrinsic resolution:
E
5.0
4.0
0.40
CZT
Simulations
Experiments
4.5
= 8 ln( 2 )WF / E
0.35
0.30
Si
SiGe
0.20
Ge
3.0
0.15
2.5
2.0
0.6
F
W
0.25
3.5
0.10
0.8
1.0
1.2
1.4
0.05
1.8
1.6
Band gap (eV)
⎡ 0.12 E g2 + 0.22 E g + 0.057 ⎤
ΔEin
= 2.355 ⎢
⎥
E
E
⎢⎣
⎥⎦
1/ 2
9
Spatial Distribution of e-h Pairs
‰ Distribution of electron-hole pairs in Ge (2 keV)
Photon
Electron (interband)
Electron (plasmon)
Electron (ionization)
Electron (relaxation)
Hole
¾Electrons created by
plasmon – along the track
¾Electrons created by
interband transition – at
periphery of cascade
volume
¾There are some single
electrons and holes.
¾Most electrons and
holes are close to each
other.
¾Thermalized electrons
can move away from
cascade volume.
40 nm
10
Spatial Distribution of e-h Pairs
‰ Distribution of electron-hole pairs in Ge (2 keV)
¾Distribution of electron-hole pairs is different for different cascade (same energy)
¾Density of electron-hole pairs along main path is high
¾Density of electron-hole pairs at the periphery of cascade volume is low.
11
Spatial Distribution of e-h Pairs
‰ Distribution of electron-hole pairs in Ge (10 keV)
Photon
Electron (interband)
Electron (plasmon)
Electron (ionization)
Electron (relaxation)
Electron (shake off)
Hole
¾Nonlinear electron cascade
12
Spatial Distribution of e-h Pairs
‰ Dynamic process of electron-hole pair creation (1 keV)
1 keV photon
13
Conclusions
‰
A Monte Carlo code has been developed to simulate electron
cascade, production of e-h pairs and their number distribution in
detector materials.
‰
Intrinsic resolution in semiconductors follows;
ΔEin
⎡ 0.12 E g2
= 2.355 ⎢
E
⎢⎣
+ 0.22 E g + 0.057 ⎤
⎥
E
⎥⎦
1/ 2
‰
Plasmon and interband transition are major mechanisms to
generate e-h pairs in semiconductors.
‰
Spatial distribution of e-h pairs indicates that electrons created
by plasmon are along the track, while electrons created by
interband transition distribute at the periphery of cascade
volume - nonlinear electron cascade
14
Thank You
15
Monte Carlo Model
‰ Cross sections – based on the generalized oscillator strength
model, a new model has been developed.
1e-15
Total
1e-16
Interband transition
σ (cm2)
1e-17
LII/LIII
1e-18
Plasmon excitation
1e-19
1e-20
LI
1e-21
1e-22
K
Si
1e-23
1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9
Ee (eV)
¾
Total cross section – within 2%-8% of experimental measurements
¾
Band structure effects on cross sections at low energy (<20 eV) must be
calculated
16
First
-Principles Approach
First-Principles
‰ Differential electronic scattering cross section determined by dielectric
function
[
]
∂σ (q, ω )
8π ℑ ε −1 (q, ω )
=− 2
vn
∂q ∂ω
q
ab initio data model
‰ Dielectric function is a function of band energies and orbitals
ε ( q ,ω ) = f ( En ,k ,< r | n ,k > )
‰ Need band energies En, k, and orbitals
r | n, k
• Use ABINIT code to get band structure - Plane wave basis set,
pseudopotentials
• Correct DFT band energies using GW method.
• Use DFT Kohn-Sham eigenfunctions for orbitals
17
First
-Principles Approach
First-Principles
‰ Our calculated dielectric function for silicon is similar to experiments
Our calculations
Philipp & Ehrenreich, Phys. Rev., 129, 1550, (1963)
Experiment
¾With differential cross section known, we can
integrate over q , ω to find total cross section &
electron mean free paths.
¾Determine secondary particle (electron, hole)
spectrum from plasmon decay
Walter & Cohen, Phys. Rev. B, 5, 3101, (1972)
Theory
18
First
-Principles Approach
First-Principles
‰ Cross section – ab initio data model
1e-15
Si
σ (cm2)
1e-16
Total
1e-17
1e-18
Plasmon excitation
1e-19
Ab Initio data model
Interband transition
1e-20
1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9
Ee (eV)
¾
Total cross section is in reasonable agreement with that from ab initio data
model.
¾
We will apply the ab initio data model to known and unknown materials.
19
Spatial Distribution
‰ Elastic DCS – Walker, Adv. Phys. 20, 257 (1971)
¾
Solving the partial-wave expanded Dirac equation for the motion of the
projectile in the field of the target atom
dσ el
2
2
= f ( θ ) + g( θ )
dθ
f (θ ) =
1 ∞
∑ {(l + 1)[exp(2iδ l − ) − 1] + l [exp(2iδ l − ) − 1]}Pl (cosθ )
2ik l =0
g( θ ) =
1 ∞
1
∑ {exp( 2iδ l − + exp( 2iδ l + }Pl (cos θ )
2ik l =0
¾
DCS
δl – phase shifts
Pl(cosθ) – Legendre polynomials
1e-14
Ge
σ el
dσ
= ∫ el
dθ
σ(cm2)
1e-15
1e-16
1e-17
1e-18
1e-19
Inelastic
Elastic
1e-20
1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 1e+9
E (eV)
20
Spatial Distribution
‰ Approach – conventional detailed MC method
¾
Each simulated path of a particle – characterized by a series of states;
{rn , En ,cos θ n }
λ−1 = N ( σ el + σ in )
¾
Mean free path between collisions;
¾
Length s of the free path to the next collision is obtained by random
sampling from the distribution;
p( s ) = λ−1 exp( − s / λ )
rn+1 = rn + s cos θ n
1 dσ el
p( θ ) = el
σ dθ
¾
If elastic collision, θ is sampled from the distribution;
¾
If inelastic collision, θ is fixed by energy and momentum conservation
Q( Q + 2 mc 2 ) = c 2 ( p 2 + p' 2 −2 pp' cosθ )
Depends on type of
inelastic collisions
Momentum transfer – q=p-p’
Recoil energy, Q – Q=[(cq)2+mc2]1/2-mc2
¾
Azimuthal scattering angle, φ – φ=2πrand()
21