Optimization of planar pixel detector

T. Habermann

Planar pixel detectorsPlanar pixel detectors

L

W

H

ground

U

T. Habermann

Distortion effectsDistortion effects

a) Surface leakage current(10 – 20 pA)

E.L. Hull, R.H.Pell, …NIMA 364(1995) 488-495

b) Effects of the enclosure

T. Habermann

poissons equation for the potential :

electric field :

boundary conditions :

Electric field calculation in germaniumElectric field calculation in germanium

grad

2

6.16

10854.8

:constant dielectric

120

0

r

r

VmC

316

19

10

106.1

:density charge

mN

Ce

eN

A

A

. )2(0 )1(

constEn

T. Habermann

applied potential :

U = 2000 - 3000V

20mm

80mm

d

Detector geometry Detector geometry 2-dimensional2-dimensional

pixel (φ = U)

ground (φ = 0)

pixel width :

d = 4 - 16mm

T. Habermann

Electric field calculationElectric field calculation2-dimensional2-dimensional

2

2

2

2

yx

0)0,( x

2-dimensional poisson equation :

x

y

H

L

),( UHx

0),0( yx 0),(

yLx

T. Habermann

Finite-Volume-MethodFinite-Volume-Method

The region is divided into N rectangular controlvolumes (CV). The potential isapproximated in the center of this CV’s

→

For every CV we get an equation of the form :aPΦP - awΦW - aNΦN - aEΦE - aSΦS = bP

→

linear system of equations with N variables : AΦ=b

2-d grid

3-d grid

(“Computational Methods for Fluid Dynamics”,J.H.Ferziger,M.Peric)

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Potential in germaniumPotential in germanium

12x12 grid pointsU0 = 3000VH = 0.02 mL = 0.02 m

T. Habermann

Potential in germaniumPotential in germanium

50x50 grid pointsU0 = 3000VH = 0.02 mL = 0.02 m

2 pixelwidth : 0.005 mdistance to the edge : 0.0025 m

T. Habermann

ConvergencyConvergency

L

dxHxy0

i ),( Flux

1-i

1-ii

FluxFluxFlux

ire

Flux in y-direction at the top surfacevs number of grid points in x-direction

(= number of grid points in y-direction)

relative errorvs # of grid points in x-direction

T. Habermann

outlookoutlook

• further examination of convergency behaviour(→ adjustment of the numerical method for better convergency)

• 3d model • examination of the electric field and reconstruction of the expected

effects :• distortion caused by the enclosure• distortion caused by surface leakage current

• How do the free parameters influence these effects ? (distance between pixels, pixelsize, size of the capsule, ...)

• GOAL : Minimize the electric field distortion by choosing the right parameter values

T. Habermann

Model of the coupled systemModel of the coupled system

0)( )1( Gv EEnContact conditions :

sGGVV

GV

EEnDDn

)( )( )2(

00

Parameters :distance from the left chamber wallLV= [5 10 15] mm

contact surface charge densityρS= [0.04 ... 10.0] μC/m^3

pixel sizeLP = [2 4 ... 18 20] mm

applied voltage U = [2500 3000 4000 5000] V

(→648 different parameter sets)

T. Habermann

Finite volumes for the coupled systemFinite volumes for the coupled system

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Results : Field distortion caused by Results : Field distortion caused by surface charge densitysurface charge density

0s 310 mC

s

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Results : depth of affected region in Results : depth of affected region in dependence of surface charge densitydependence of surface charge density

pixelsize = 16mmL1 = 15mm

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Results : different pixel sizesResults : different pixel sizes

LP = 2mm LP = 10mm

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Results : depth of affected region in Results : depth of affected region in dependence of pixel sizedependence of pixel size

T. Habermann

Results : depth of affected region in Results : depth of affected region in dependence of applied voltagedependence of applied voltage

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Results : depletion voltageResults : depletion voltage

electric field propagation for small (2mm) pixelsize and 3000V applied voltage :

T. Habermann

Results : depletion voltage in Results : depletion voltage in dependence of pixel sizedependence of pixel size

pixel size = 20mm → planar detector

according to Glenn KnollVd = 2177,22 VsimulatedVd = 2176,32 V

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Results : depletion voltage in Results : depletion voltage in dependence crystal thicknessdependence crystal thickness

T. Habermann

outlookoutlook

• investigation of the electric field strength inside the germanium in dependence of pixel size, ...

• investigation of the electric field in dependence of the crystal thickness

• 3d model• another solver could be used to decrease memory usage and

calculation time (essential for 3d calculations)• ...