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