A full Monte Carlo simulation code for silicon strip detectors
description
Transcript of A full Monte Carlo simulation code for silicon strip detectors
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
9th Topical Seminar on Innovative Particle and Radiation Detectors May 23-26, 2004- Siena, Italy
Welcome Luciafrancesca !
The simulation chainCharged particles Photons
Ionization energy loss
Photoelectric absorption
Primary e-h pairs
Secondary e-h pairs
Drift of charge carriers
Induced current signals
Electronics chain
Electronic noiseVoltage signals on the readout
strips
Interaction with silicon
Propagation of carriers
Electronics simulation
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)
M-shell (~17 eV)
L-shells (~150 eV)
K-shell (~1850 eV)
maxE
0
aσ(E)dEN
1λ
The number of collisions per unit path length is
evaluated as:
Generation of e-h pairs in silicon
Ionizing particle
Si atom
Virtual γ
Primary e-h pairs
Secondary e-h pairs
Phonon scattering
Silicon energy levelsE
ner
gy
K-shell
Ek= -1839 eV
L-shells
EL2-3= -99.2 eV
EL1= -148.7 eV
Valence band
EV= [-12, 0] eV
Conduction band
Energy gap
Eg = 1.12 eV @T=300K
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 K-shell there is a 92% probability of absorption by the K-shell and an
8% probability of absorption by the L1-shell
Primary e-h pairsAbsorption 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 treesK-shell vacancy and photoelectron:
Eh=EK Epe=E-EK-Egap
Auger emissions (95.6%) K-shell fluorescence (4.4%)
Transition Chain
Probability
VacanciesProbability
Photon energy
Vacancy
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
L1-shell vacancy and photoelectron:
Eh=EL1 Epe=E-EL1-Egap
Transition Chain
Probability Emission Vacancies
L1MM 2.5% Auger MM
L1L23M 97.5%Coster-Kroning
L23M
L23-shell vacancy and photoelectron:
Eh=EL23 Epe=E-EL23-Egap
Transition Chain
Probability
Emission Vacancies
L23MM 100% Auger MM
Electron and hole energies are assigned according to Sholze et al, J. Appl. Phys. 84
(1998), 2926
Production of secondary e-h pairsA 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:
2/7gap
2/10
ION
PHON
)EE(
)EE(
2
105A
r
r
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 electrons (holes)
Pairs generated by photons
The Fano Factor approaches the limit F∞=0.117 for large primary energies
The pair creation energy approaches the value W∞=3.645 eV for large primary energies
Pair distribution along the track
βγ=5 electron tracks in 400 m silicon
SSD geometry
ph w
d
n bulk
p+ strips
The p strips are grounded, the back is kept at a positive voltage V0
"Small pitch" geometry:
• d=325 m, p=25 m
• w=12 m, h=5m
• V0=100 V
"Large pitch" geometry:
• d=400 m, p=228 m
• w=60 m, h=5m
• 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:
)2/py(V)2/py(V
0)2/wy,dx(V
V)0x(V 0
The calculation has been performed using the ANSOFT MAXWELL 2D field calculator.
Motion of charge carriersAfter 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:
Ev
where the mobility is related to the E field by the parameterization:
/1
c
cm
E/E1
E/v
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:
drDT4
rexp
DT4
1
N
dN 2
Induced current signalsThe current signals induced by the moving carriers on the readout electrodes (p strips) are calculated using the Shockley-Ramo's theorem:
carriers
kk )t(rE)t(vq)t(i
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:
kj if 0V
V 1V
j
k
Weighting potential"Large pitch" configuration
Readout strip
Adjacent strips
Back electrode
Simulation of the electronics
Input current signal i(t)
Front-end electronics
H(s)
Output voltage signal V(t)
The output signals are evaluated in the time domain by solving the inverse Laplace transform with the finite difference approximation for the time derivatives
)s(i)s(H)s(V
n
nn
m
mm
sb
sa)s(H
n m
)m(m
)n(n )t(ia)t(Vb
)s(i)s(H)s(V
The transfer function can be expressed as a ratio of polynomials
Noise contributions are added by taking into account the proper noise transfer functions
Front-end electronicsDetector Preamplifier Shaper
Noise simulationThe electronic noise is due to the detector and to the electronic front-end.
Shot noise due to the leakage
current:
i2nd=2eIL
Thermal noise due to the bias
resistor:
i2nb=4KT/Rb
Thermal noise due to the feedback resistor:
i2nf=4KT/Rf
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
rightleft
left
VV
V
The charge sharing has been studied with the η function:
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 400m thick SSD with 228m 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