Download A full Monte Carlo simulation code for silicon strip detectors

9th Topical Seminar on Innovative Particle and Radiation Detectors
May 23-26, 2004- Siena, Italy
A full Monte Carlo
simulation code for silicon
strip detectors
M. Brigida, C. Favuzzi, P. Fusco, F. Gargano,
N. Giglietto, F. Giordano, F. Loparco,
B. Marangelli, M. N. Mazziotta, N. Mirizzi,
S. Rainò, P. Spinelli
Bari University & INFN
Welcome Luciafrancesca !
The simulation chain
Charged particles
Interaction with silicon
Ionization energy
loss
Primary e-h pairs
Electronics simulation
Voltage signals
on the readout
strips
Photons
Photoelectric
absorption
Secondary e-h
pairs
Drift of charge
carriers
Electronic noise
Induced current
signals
Electronics chain
Propagation of carriers
Energy loss of charged particles in silicon
The energy loss in Si is
evaluated from the
collision cross section σ(E)
(H. Bichsel, Rev. Mod.
Phys. 60, 663)
The number of collisions
per unit path length is
evaluated as:
λ
1
E m ax
 N σ(E)dE
a
0
M-shell
(~17 eV)
L-shells
(~150 eV)
K-shell
(~1850 eV)
Generation of e-h pairs in silicon
Ionizing particle
Virtual γ
Si atom
Primary
e-h pairs
Phonon
scattering
Secondary
e-h pairs
Silicon energy levels
Conduction band
Energy gap
Energy
Eg = 1.12 eV @T=300K
Valence band
EV= [-12, 0] eV
L-shells
EL2-3= -99.2 eV
EL1= -148.7 eV
K-shell
Ek= -1839 eV
Generation of e-h pairs
• Primary carriers: are produced in the primary collisions of
the incident particle with the silicon absorber, with the
absorption of virtual photons by the medium.
• Secondary carriers: are produced by the subsequent energy
losses of primary (and secondary) carriers.
The relative absorption
probabilities depend on the photon
energy. For energies above the Kshell there is a 92% probability of
absorption by the K-shell and an
8% probability of absorption by the
L1-shell
Primary e-h pairs
Absorption by an inner shell (x=K, L1, L23):
• A hole is left in the shell with energy Eh=Ex
• A photoelectron is ejected with energy Epe=E-Ex-Egap
Absorption by the valence band (M shell):
• A hole is left with an energy Eh random distributed in the
range [0,EV] (EV=12eV)
• A photoelectron is ejected with energy Epe=E-Eh-Egap
The relaxation process following photon absorption yields
electrons and vacancies in the K, L1 and L23 shells.
Silicon shells relaxation trees
K-shell vacancy and photoelectron:
Eh=EK Epe=E-EK-Egap
Auger emissions (95.6%)
K-shell fluorescence (4.4%)
Transition
Chain
Proba
bility
Vacancies
Proba
bility
Photon
energy
KL1L1
19.2%
L1L1
59.3% 1740 eV
L3
KL1L23
38.9%
L1L23
29.6% 1740 eV
L2
KL23L23
23.3%
L23L23
11.1% 1836 eV
M
KL1M
7.5%
L1M
KL23M
10.4%
L23M
KMM
0.8%
MM
Vacancy
Electron and hole
energies are assigned
according to Sholze et
al, J. Appl. Phys. 84
(1998), 2926
L1-shell vacancy and photoelectron:
Eh=EL1 Epe=E-EL1-Egap
L23-shell vacancy and photoelectron:
Eh=EL23 Epe=E-EL23-Egap
Transition
Chain
Probabi
lity
Emission
Vacancies
L23MM
100%
Auger
MM
Transition
Chain
Probability
Emission
Vacancies
L1MM
2.5%
Auger
MM
L1L23M
97.5%
CosterKroning
L23M
Production of secondary e-h pairs
A primary electron (hole) with E > Ethr (Ethr=3/2 Egap) can
interact with the Si absorber by ionization or by phonon
scattering. The ratio between the ionization rate and the
phonon scattering rate is:
rPHON
105 ( E  E0 )1 / 2
 A

rION
2 ( E  E gap )7 / 2
where A=5.2 eV3 and E0 is the phonon energy (E0=63 meV @
T=300 K)
The generation of secondary pairs is a cascade process, that is
simulated with a MC method. At the end of each step, a
carrier can emit a phonon or can cause ionization. In this
case a new e-h pair is created.
Pair creation energy & Fano factor
Pairs generated by
photons
Pairs generated by
electrons (holes)
The pair creation energy approaches the value
W∞=3.645 eV for large primary energies
The Fano Factor approaches the limit
F∞=0.117 for large primary energies
Pair distribution along the track
βγ=5 electron tracks
in 400 m silicon
SSD geometry
p+ strips
p
h
w
d
n bulk
The p strips are grounded, the back is kept at a positive voltage V0
"Small pitch" geometry:
"Large pitch" geometry:
• d=325 m, p=25 m
• d=400 m, p=228 m
• w=12 m, h=5m
• w=60 m, h=5m
• V0=100 V
• V0=100 V
The electric field
"Large pitch"
configuration
The electric field has been
calculated by solving the
Maxwell equation:
 
D  
in an elementary detector
cell with the following
boundary conditions for
the potential:
V ( x  0 )  V0
V( x  d, y  w / 2 ) 0
V( y   p / 2 )V( y  p / 2 )
The calculation has been
performed using the ANSOFT
MAXWELL 2D field calculator.
Motion of charge carriers
After being produced, electrons and holes will drift under the action of the
electric field towards the n back and the p strips, according to the equation:


v  E
where the mobility is related to the E field by the parameterization:

v m / Ec
1  E / E  
 1/ 
c
The parameters vm, β and Ec are different for electron and holes and depend
on the temperature.
During their drift, carriers are diffused by multiple collisions according to
a gaussian law:
dN

N

1
r2 
dr
exp 
4DT
 4 DT 
Induced current signals
The current signals induced by the moving carriers on the
readout electrodes (p strips) are calculated using the ShockleyRamo's theorem:
 

ik ( t )    qv ( t )  E k r ( t )
carriers
The weighting field Ek describes the geometrical coupling
between the moving carrier and the k-th electrode. It has been
evaluated by solving the same Maxwell's equation as for the
electric field with ρ=0 and with the boundary conditions:
Vk  1 V
Vj  0
if j  k
Weighting potential
"Large pitch"
configuration
Readout strip
Back electrode
Adjacent
strips
Simulation of the electronics
Input current
signal i(t)
Front-end
electronics
H(s)
Output voltage
signal V(t)
The transfer function can be expressed as a ratio of polynomials
H( s ) 
m
a
s
m m
n
b
s
n n
V ( s )  H ( s )i ( s )

(n)
(m)
n bnV ( t )  m am i ( t )
V ( s )  H ( s )i ( s )
The output signals are evaluated in the time domain by solving the inverse
Laplace transform with the finite difference approximation for the time
derivatives
Noise contributions are added by taking into account the proper noise
transfer functions
Front-end electronics
Detector
Preamplifier
Shaper
Noise simulation
The electronic noise is due to the detector and to the electronic front-end.
Thermal noise due to the
feedback resistor:
i2nf=4KT/Rf
Shot noise due
to the leakage
current:
i2nd=2eIL
Thermal noise
due to the bias
resistor:
i2nb=4KT/Rb
Electronic noise due
to the amplifier:
i2na= 0
v2na = 2.7KT/gm
Charge sharing analysis (1)
To study the charge sharing a sample of MIPs has been
simulated, crossing the detector with null zenith angle, in the
region between two strips
The charge sharing has been studied with the η function:

Vleft
Vleft  Vright
Charge sharing analysis (2)
• Both the η distribution are
symmetric around the value
η=0.5
• In the large pitch geometry
the peaks are located at η≈0
and η≈1 → weak coupling
between adjacent strips
• In the small pitch geometry
the peaks are located at
η≈0.2 and η≈0.8 → strong
coupling between adjacent
strips
Comparison with experimental data
A beam test has been
carried out exposing a
400m thick SSD with
228m strip pitch to a 3
GeV/c π beam @ CERN-PS
T9 beam facility
Experimental data are in
good agreement with the
MC prediction
Conclusions
 We have developed a new MC full simulation code that includes all
the physical processes taking place in a SSD
 The MC code can be used with different detector geometries and
front-end electronics
 The temperature dependence of the physical processes is taken into
account, thus allowing a study of the SSD performance with the
temperature (an example will be given in S. Rainò's talk)
 A charge sharing analysis has been performed, showing that the
MC predictions are in good agreement with experimental data
 Our MC code allows to study the efficiency and the space
resolution of SSDs (an example will be shown in M. Brigida's talk)
 For further details: http://www.ba.infn.it/~mazziot/article.pdf