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Simulation and modeling of BEGe detectors

Matteo Agostini, Calin A. Ur,

• E. Bellotti, D. Budj´

aˇ s, C. Cattadori, A. di Vacri, A. Garfagnini, L. Pandola, S. Sch¨

• nert

Max-Plank-Institute f¨ ur Kernphysik

MaGe meeting, January 18th 2010

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 1

Outline

1

The BEGe detectors

2

The simulation The structure of the simulation Design and implementation of the simulation

3

Validation of the simulation Validation of the MaGe simulation Validation of the PSS

4

Conclusion

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 2

The BEGe detectors

The BEGe geometry

71 mm p+ contact (read–out electrode) n+ contact p-type Ge 3500 V 0 V groove 31 mm Al endcap thick-window

PRE

BEGe

N2

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 3

The BEGe detectors

The (LNGS) BEGe features

Electrical Characteristics: Depletion voltage +3000 V Operational bias voltage +3500 V Integral nonlinearity < 0.05% Physical Characteristics: Active diameter 71 mm Active area 3800 mm2 Thickness 32 mm Distance from window 5 mm Efficiency > 34% Energy Resolution at 1332.5 keV: FWHM (nominal) 1.752 keV FWHM (measured) 1.607 ± 0.003 keV FWTM 3.259 keV

1332 1250 1173 1100 105 104 103 102 101 digital analogue Energy [keV] 1.607 keV counts 2500 2000 1500 1000 500 106 105 104 103 102 101 digital analogue fitting function: f (x) = √0.31 + 0.0018 x f (0) = √a ∼ 0.55 keV Energy [keV] FWHM [keV] 300 2500 2000 1500 1000 500 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 4

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

interaction points anode cathode

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

anode cathode electrons holes

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

anode cathode electrons holes

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 3e-07 3.5e-07 4e-07 4.5e-07 adc counts time [ns] total pulse h pulse e pulse

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

anode cathode electrons holes

• 0.01

0.01 0.02 0.03 0.04 0.05 0.06 5e-08 1e-07 1.5e-07 2e-07 2.5e-07 3e-07 3.5e-07 4e-07 4.5e-07 adc counts time [ns] total pulse + pre total pulse h pulse e pulse

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

anode cathode electrons holes 200 400 600 800 1000 1e-07 2e-07 3e-07 4e-07 5e-07 6e-07 7e-07 8e-07 9e-07 adc counts [a.u.] time [s] 0.5 MeV SSE (66,34,26) 0.4 MeV SSE (46,34,26) 1.1 MeV SSE (28,34,16) 2.0 MeV MSE

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The structure of the simulation

I. MC simulation

–> coordinates and energy of the hits

• II. Signal formation and development

<– coordinate of each hit –> electron and hole trajectories –> the signal induced on the point size electrode

• III. DAQ simulations

<– energy and signal for each hit in an event <– the Preamplifier Transfer Function (PTF) –> each pulse is convolved with the PTF –> all the pulses of an event are added up –> the noise is added to the total pulse

anode cathode electrons holes 200 400 600 800 1000 1e-07 2e-07 3e-07 4e-07 5e-07 6e-07 7e-07 8e-07 9e-07 adc counts [a.u.] time [s] 0.5 MeV SSE (66,34,26) 0.4 MeV SSE (46,34,26) 1.1 MeV SSE (28,34,16) 2.0 MeV MSE

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 5

The simulation

The simulation design

step 0. Create a library of pulses:

0.1 divide the detector in cubic cell (1 mm × 1 mm × 1 mm) and generate a pulse for each cell 0.2 convolve each pulse with the PTF 0.3 store all the pulses in a library

step 1. Run the MC simulation step 2. For each hit compute the pulse as weighted average of the pulses stored in the library step 3. For each event compute the total pulse by adding up the pulse of each hit step 4. Add the noise

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 6

The simulation

The simulation design

step 0. Create a library of pulses:

0.1 divide the detector in cubic cell (1 mm × 1 mm × 1 mm) and generate a pulse for each cell

<– MGS

0.2 convolve each pulse with the PTF 0.3 store all the pulses in a library

step 1. Run the MC simulation step 2. For each hit compute the pulse as weighted average of the pulses stored in the library step 3. For each event compute the total pulse by adding up the pulse of each hit step 4. Add the noise MGS v 5r02 : Multi Geometry Simulation is a MATLAB software developed for the AGATA project (http://www.iphc.cnrs.fr/-MGS-.html )

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 6

The simulation

The simulation design

step 0. Create a library of pulses:

0.1 divide the detector in cubic cell (1 mm × 1 mm × 1 mm) and generate a pulse for each cell

<– MGS

0.2 convolve each pulse with the PTF 0.3 store all the pulses in a library

step 1. Run the MC simulation

<– MaGe

step 2. For each hit compute the pulse as weighted average of the pulses stored in the library step 3. For each event compute the total pulse by adding up the pulse of each hit step 4. Add the noise MGS v 5r02 : Multi Geometry Simulation is a MATLAB software developed for the AGATA project (http://www.iphc.cnrs.fr/-MGS-.html ) MaGe: BEGe geometry used munichteststand/GELNGSBEGeDetector.hh

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 6

The simulation

MGS: simulation of the signal formation and development

Trajectory simulation Fourth–order Runge–Kutta method (∆t = 1 ns): r(t + ∆t) = r(t) + f (v(r(t)), ∆t) where the velocity is computed by using the mobility model of L. Mihailescu and B. Bruynell: vh = µh(r, E) · E ve = µe(r, E) · E Simulation of the Electric Field SOR and relaxation method to solve the Poisson’s eq: ∇2φ(r) = − ρ(r) ε → E(r) = −∇ (ϕ(r)) – cathode at 0 V, anode at 3500 V – detector completely depleted: ρ(r) = eNA(r) Signal computation Shockley-Ramo Theorem: Q(t) = −qφw (r(t)) where φw(r(t)) is the weighting potential

holes electrons cathode anode 40 35 30 25 20 15 10 80 70 60 50 40 30 20 10 80 70 60 50 40 30 20 10

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 7

The simulation

MGS: simulation of the signal formation and development

Trajectory simulation Fourth–order Runge–Kutta method (∆t = 1 ns): r(t + ∆t) = r(t) + f (v(r(t)), ∆t) where the velocity is computed by using the mobility model of L. Mihailescu and B. Bruynell: vh = µh(r, E) · E ve = µe(r, E) · E Simulation of the Electric Field SOR and relaxation method to solve the Poisson’s eq: ∇2φ(r) = − ρ(r) ε → E(r) = −∇ (ϕ(r)) – cathode at 0 V, anode at 3500 V – detector completely depleted: ρ(r) = eNA(r) Signal computation Shockley-Ramo Theorem: Q(t) = −qφw (r(t)) where φw(r(t)) is the weighting potential

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 7

The simulation

MGS: simulation of the signal formation and development

Trajectory simulation Fourth–order Runge–Kutta method (∆t = 1 ns): r(t + ∆t) = r(t) + f (v(r(t)), ∆t) where the velocity is computed by using the mobility model of L. Mihailescu and B. Bruynell: vh = µh(r, E) · E ve = µe(r, E) · E Simulation of the Electric Field SOR and relaxation method to solve the Poisson’s eq: ∇2φ(r) = − ρ(r) ε → E(r) = −∇ (ϕ(r)) – cathode at 0 V, anode at 3500 V – detector completely depleted: ρ(r) = eNA(r) Signal computation Shockley-Ramo Theorem: Q(t) = −qφw (r(t)) where φw(r(t)) is the weighting potential The weighting potential is defined as the electric potential calculated when the considered electrode is kept at a unit potential, all other electrodes are grounded and all charges inside the device are removed.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 7

Validation of the simulation

Validation of the MaGe simulation

–> Housing absorption scanning with a collimated source of Am along a diameter and the side:

position [mm] counts 60 40 20

• 20
• 40
• 60

0.001 0.0008 0.0006 0.0004 0.0002 experimental data MC position [mm] counts 20 15 10 5

• 5
• 10
• 15
• 20

0.3 0.25 0.2 0.15 0.1 0.05

–> Dead layer measurements (nominal dead layer 0.8 mm) ratio between the counts in the peaks at 81 keV and at 356 keV of 133Ba:

dead layer thickness [mm] Ratio 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 MC experimental value dead layer thickness [mm] 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0.39 0.36 0.33 0.3 0.27 0.24 0.21 0.18 0.15

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 8

Validation of the simulation

Validation of the PSS

The validation was carried out by comparing directly the simulated and the experimental signals:

241Am colimated source ⇒ well localized events close to the detector surface;

averaging up the experimental and simulated signals ⇒ reduction of noise

100 200 300 400 500 600 700 1600 1700 1800 1900 2000 2100 2200 2300 2400 charge [a.u.] time [sample] exp sim 100 200 300 400 500 600 700 1900 1920 1940 1960 1980 2000 2020 2040 2060 charge [a.u.] time [sample] exp sim

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 9

Validation of the simulation

Validation of the PSS

The validation was carried out by comparing directly the simulated and the experimental signals:

241Am colimated source ⇒ well localized events close to the detector surface;

averaging up the experimental and simulated signals ⇒ reduction of noise The averaging algorithm steps:

1

the experimental signal (sampled at 10 ns) is resampled at 1 ns interpolating the original points with the FADC transfer function;

2

the resampled signal is fitted with the average in order to obtain the best possible time alignment;

3

if the average rms is minor than the threshold value, the resampled and shifted signal is accepted in the average.

single pulse average exp pulse average sim pulse time [s] adc counts 1e-06 8e-07 6e-07 4e-07 2e-07

• 2e-07

500 450 400 350 300 250 200 150 100 50

• 50

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 9

Validation of the simulation

Radial scanning

–> 241Am source –> 2 mm collimator –> 600 s acquisitions for each position

cathode anode 40 35 30 25 20 15 10 80 70 60 50 40 30 20 10 80 70 60 50 40 30 20 10 100 200 300 400 500 600 700 1000 1010 1020 1030 1040 1050 1060 1070 1080 charge [a.u.] time [10 ns] exp data 0 mm 10 mm 15 mm 20 mm 25 mm 30 mm 35 mm

The holes are dragged to the center of the detector and then drift to the p+ contact with a common trajectory ⇒ pulse shape discrimination parameter A/E a depends on the final rising part only which is largely independent of the position of interaction inside crystal

aA → max amplitude of the current pulse; E → total energy of the event Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 10

Validation of the simulation

Radial scanning

–> 241Am source –> 2 mm collimator –> 600 s acquisitions for each position

cathode anode 40 35 30 25 20 15 10 80 70 60 50 40 30 20 10 80 70 60 50 40 30 20 10 10 20 30 40 50 60 1000 1010 1020 1030 1040 1050 1060 1070 1080 current [a.u.] time [10 ns] exp data 0 mm 10 mm 15 mm 20 mm 25 mm 30 mm 35 mm

The holes are dragged to the center of the detector and then drift to the p+ contact with a common trajectory ⇒ pulse shape discrimination parameter A/E a depends on the final rising part only which is largely independent of the position of interaction inside crystal

aA → max amplitude of the current pulse; E → total energy of the event Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 10

Validation of the simulation

Circular Scanning

–> 241Am source –> 1 mm collimator –> 500 s acquisitions for each position

We study the rise time as a function of the angle. –> To observe variations we used the rise time between 1% and 90%

experimental data simulated data angle [deg] rise time variations [ns] 360 315 270 225 180 135 90 45 30 20 10

• 10
• 20

experimental data simulated data simulated data + shift rise time [ns] rise time [ns] 800 600 400 200 200 400 600 800 1000 800 600 400 200 200 400 600 800

Although the experimental data show a behaviour coherent with the simulation, the agreement is only qualitative. ⇒ the result is remarkable taking into account the problems related to the identification of the time corresponding to the 1% of the maximum amplitude

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 11

Conclusion

Conclusion

Results:

the simulations performed with the nominal geometry is in reasonable quantitative agreement with the experimental data the impact of detector parameters (i.e. geometry description, grid step, impurity distribution, bias voltage, etc.) on the signal pulse shape has been studied and the simulation accuracy could be improved.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 12

Conclusion

Conclusion

Results:

the simulations performed with the nominal geometry is in reasonable quantitative agreement with the experimental data the impact of detector parameters (i.e. geometry description, grid step, impurity distribution, bias voltage, etc.) on the signal pulse shape has been studied and the simulation accuracy could be improved. Future works: investigate the pulse shape discrimination performances of BEGe detectors by using simulations: compare PS discrimination performance of experimental data with the simulation study the impact of the detector parameters on pulse shape discrimination performances and the robustness of A/E method determine the depletion voltage and the best operational voltage validate the simulation with a precise inner scanning of the detector generate library for the Phase I detectors and study PSA detector

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 12

Conclusion

Conclusion

Results:

the simulations performed with the nominal geometry is in reasonable quantitative agreement with the experimental data the impact of detector parameters (i.e. geometry description, grid step, impurity distribution, bias voltage, etc.) on the signal pulse shape has been studied and the simulation accuracy could be improved. Future works: investigate the pulse shape discrimination performances of BEGe detectors by using simulations: compare PS discrimination performance of experimental data with the simulation study the impact of the detector parameters on pulse shape discrimination performances and the robustness of A/E method determine the depletion voltage and the best operational voltage validate the simulation with a precise inner scanning of the detector generate library for the Phase I detectors and study PSA detector –> We are writing a paper containing these results (March-April) –> The beta version of the simulation software will be soon uploaded to the MaGe repository.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 12

Backup slides

backup slides

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 13

Backup slides

BEGe detector

71 mm p+ contact (read–out electrode) n+ contact p-type Ge 3500 V 0 V groove 31 mm Al endcap thick-window

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 14

Backup slides

DAQ systems

PRE

BEGe

N2 AMPLIFIER Ortec 672 ETHERNIM DIGITIZER N1728B PC GAMMAVISION PC TUC software processing

PULSES

Digital signal (Gast MWD)

SPECTRA ENERGY

93 Ω 50 Ω ADC

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 15

Backup slides

HV scanning

Peak Position [ch] 2000 1600 1200 800 400 Peak Area [counts] 60000 50000 40000 30000 20000 10000 HV [V] FWHM [ch] 3500 3000 2500 2000 1500 1000 500 80 60 40 20

region I [2045 V, 3500 V]: excellent performances, detector full depleted, rise time ∼ 0.5µ s, amplitude ∼ 0.3 V. region II [1860 V, 2045 V]: anomalous behaviour, pulses still fast but their amplitudes four times smaller. region III [ 100 V, 1860 V]: detector partially depleted, charge collection not complete, detector capacitance increment, slower rise time ∼ 5µs.

3500 V 2201 V 2040 V 2000 V 1842 V 1000 V sample [10 ns] adc counts 350 3000 2500 2000 1500 1000 9800 9600 9400 9200 9000 8800 8600 8400 8200 8000 7800 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 16

Backup slides

Characterization measurements - Linearity

digital electronics analogue electronics Energy [keV] channel 3000 2500 2000 1500 1000 500 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 digital electronics analogue electronics Energy [keV] 3000 2500 2000 1500 1000 500 Energy [keV] Energy [keV] 2500 2000 1500 1000 500 1 0.5

• 0.5
• 1

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 17

Backup slides

Characterization measurements - Resolution

1332 1250 1173 1100 105 104 103 102 101 digital analogue Energy [keV] counts 2500 2000 1500 1000 500 106 105 104 103 102 101

Energy [keV] Analogue DAQ system Digital DAQ system peak counts FWHM [keV] peak counts FWHM [keV] 1173 259899 (510) 1.529 (0.002) 224857 (506) 1.520 (0.002) 1332 225023 (474) 1.617 (0.002) 200137 (518) 1.607 (0.003)

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 18

Backup slides

Characterization measurements - Preamplifier

HV Detector Qin A Cf Rf V0 stage 1: integrator amplifier stage 2: pole zero network stage 3:

• utput buffer

T.P. 93 Ω 50 Ω Energy Timinig

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 19

Backup slides

Characterization measurements - Preamplifier noise

HV Detector Cf Rf stage 1: integrator amplifier in νn

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 20

Backup slides

Characterization measurements - Preamplifier noise

corner frequency Frequency [kHz] Power [arbitrary units] 10000 1000 100 10 9 8 7 6 5 4 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 21

Backup slides

Validation of the MaGe simulation - Absorption

position [mm] counts 60 40 20

• 20
• 40
• 60

0.001 0.0008 0.0006 0.0004 0.0002 experimental data MC position [mm] counts 20 15 10 5

• 5
• 10
• 15
• 20

0.3 0.25 0.2 0.15 0.1 0.05 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 22

Backup slides

Validation of the MaGe simulation - Barium spectrum

Energy [keV] counts 450 400 350 300 250 103 102 101 MC + background experimental data Energy [keV] counts 1000 400 200 100 103 102 101 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 23

Backup slides

Validation of the MaGe simulation - DL

dead layer thickness [mm] Ratio 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 MC experimental value dead layer thickness [mm] 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0.39 0.36 0.33 0.3 0.27 0.24 0.21 0.18 0.15 Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 24

Backup slides

Mobility model

vexp =

µ0E

(1+(E/E0)β)

1/β − µnE

vd = A (|E|, T) P

j nj n γj E0

(E0γj E0)

1/2

vd ≈ @ vr vθ vφ 1 A = v100(E) @ 1 − Λ(k0) sin4(θ0) sin2(2φ0) + sin2(2θ0) Ω(k0) ˆ 2 sin3(θ0) cos(θ0) sin2(2φ0) + sin(4θ0) ˜ Ω(k0) sin3(θ0) sin(4φ0) 1 A

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 25

Backup slides

Drift velocity

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 26

Backup slides

How to compute the electric field in a semiconductor detector

Since a semiconductor detector can be considered as an electrostatic system, the electric field can be computed by solving the following Maxwell’s equations

• r, equivalently, by solving the Poisson’s equation:

∇ · E = ρ ε ∇ × E = ⇒ E = −∇φ 9 > = > ; ∇ · ∇φ = −ρ ε ⇒ ∇2φ = −ρ ε To solve the Poisson’s equation ∇2φ = −ρ/ε and find the potential φ we need to know: the charge density distribution ρ the boundary conditions (the value of φ on some surfaces): φ0|Scathode = Vcathode and φ0|Sanode = Vanode

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 27

Backup slides

The semiconductor junction

The semiconductor detector functioning is based on the properties of a semiconductor junction:

Donor Ions electrons Acceptor Ions holes n-type p-type rho E Charge density Electric Field e h E e h

• I. Spontaneus diffusion
• II. Recombination
• III. Thermodinamic equilibrium

e h The junction formation: because of the difference in the concentration of electrons and holes between the two materials, there is an initial diffusion of the holes towards the n-region and a similar diffusion of electrons towards the p-region the diffusing electrons fill up holes in the p-region while the diffusing holes capture electrons in the n-side the recombination creates a net charge distribution inside the seminconducor. This creates an electric field gradient across the junction whitch halts the diffusion process.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 28

Backup slides

The charge distribution dependence of an external electric field

n-type p-type rho Charge density Eext Eext Eext = 0 rho rho

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 29

Backup slides

The charge distribution dependence of the impurity concentrations

n-type p-type rho Charge density rho rho

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 30

Backup slides

The charge distribution in a real detector

In a real semiconductor the junction is created between an heavily doped semiconductor and a high-purity semiconductor:

n+ contact p-type rho e Nd

• e Na

Eext

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 31

Backup slides

The assumptions

In all the potential computation we will assume that: the detector is fully depleted p-type detector ⇒ ρ = −eNd n-type detector ⇒ ρ = eNa the boundary conditions are: the voltage on the electrodes is defined by the HV supply ⇒ φ|cathode = 0 V ⇒ φ|anode = 3000 V the detector is enclosed in a vacuum chamber ⇒ φ|ext = 0 V

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 32

Backup slides

The linear superposition principle – the potential

From the linear superposition principle the potential can be separated into two contribution: φ(r) = φ0(r) + φρ(r) where: φ0 is the potential calculated considering only the electrode potentials (ρ(r) = 0 ∀r ) φρ is the potential obtained grounding all the electrodes The linearity of the Maxwell’s equation allows for computing the Poisson’s equation for each contribution and then add up all the contribution: ∇2φ0(r) = 0 with: φ0|Scathode = Vcathode φ0|Sanode = Vanode ∇2φρ(r) = −ρ(r)/ε with: φ0|Scathode = 0 φ0|Sanode = 0 where Sanode and Scathode are the boundary surface of the two electrodes.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 33

Backup slides

The linear superposition principle – the field

Similarly, since the electric field is determined by the linear relation E = −∇φ, it can be divided into two components: E(r) = E0(r) + Eρ(r) where: E0(r) = −∇φ0(r) Eρ(r) = −∇φρ(r)

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 34

Backup slides

How to solve the Poisson’s equation

The Poisson’s equation is solved analytically only in the simplest problem, usually it is solved by using numerical methods. In our simulation we use two algorithms which works on a grid: The Successive Over Relaxation (SOR) method converges to a solution replacing at each iteration the current approximated solution at a given grid point by a weighted average of its nearest neighbour on the grid the relaxation method converges by replacing at each iteration the current approximated solution with its Taylor expansion computed for each point on the grid.

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 35

Backup slides

Comparison

φρ Eρ φ0 E0 φtot Etot

Simulation and modeling of BEGe detectors Matteo Agostini (MPIK) 36