study of electric field distortion and charge careers collection

Samuel Lago

Study of electric field Study of electric field distortion and charge distortion and charge

careers collection careers collection

Samuel Lago

detector geometrydetector geometry

L

W

H

U

0

U

0

0

Samuel Lago

Electric field calculation in germaniumElectric field calculation in germanium

poisson equation for potential:

boundary conditions:

electric field:

const

E

0

),,(2 zyx

gradE

Samuel Lago

Electric field calculation in germaniumElectric field calculation in germanium

three dimension poisson equation:

),,(

2

2

2

2

2

2 zyx

zyx

6.16

10854.8

:constant dielectric

120

0

r

r

Vm

C

zh

zyx

mNNCe

m

CeNNe

DA

DA

)(),,(

10 106.1

6.1)( :density charge

010

3

1619

33

Samuel Lago

3D detector geometry3D detector geometry

0

U

0x

0

x

0y

0y

0z

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

2D grid

3D grid

• The detector is devided into rectangular controlvolumes.

• The potential is approximated through the grid points in the center of every controlvolume.

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

discretization of poisson equation:

),,(

2

2

2

2

2

2 zyx

zyx

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

discretization of poisson equation:

KK VVdV

zyxdV

zyx ),,(

2

2

2

2

2

2

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

discretization of poisson equation (source term):

2

2

2010 )

)((

),,(

zz

zz

KK

V

K

K

K

yxz

hz

dVzyx

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

KV

tne

bsw

dVzyx

yxz

zxy

zyx

yxz

zxy

zyx

),,(

discretization of poisson equation (left hand side):

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

))(( 2xOxxx PE

PE

e

discretization of poisson equation (left hand side):

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

:const

discretization of poisson equation (Dirichlet boundary):

0))(()())()(()(

)())()((

)())()(()(

))()((

)())()(()(

)(

222

2

2

xOxxxx

xxx

xxxx

xx

xxxx

x

x

eeww

wewW

eeww

ewP

eeww

wE

w

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

:constx

discretization of poisson equation (Neumann boundary):

constxOxxxx

xxx

xxxx

xx

xxxx

x

x

eeww

wewW

eeww

ewP

eeww

wE

w

))(()())()(()(

)())()((

)())()(()(

))()((

)())()(()(

)(

222

2

2

Samuel Lago

Finite-Volume-MethodFinite-Volume-Method

discretization of poisson equation:

with all coefficients we get

and a linear system of equations

bA

baaaaaaa PBBSSWWTTNNEEPP

Samuel Lago

PotentialPotential

p-type detector

blocking contact on unsegmented electrode

U = 4000 V

geometry :

0.06m x 0.06m x 0.02m

z

e-3, e-3

pixel-size: 10mm x 10mm

Samuel Lago

PotentialPotential

p-type detector

blocking contact on unsegmented electrode

U = 4000 V

geometry :

0.06m x 0.06m x 0.02m

z

e-3, e-3

ce-3

pixel-size: 10mm x 10mm

c

Samuel Lago

PotentialPotential

p-type detector

blocking contact on unsegmented electrode

U = 4250 V

geometry :

0.06m x 0.06m x 0.02m

x

e-3, e-3

pixel-size: 10mm x 10mm

Samuel Lago

PotentialPotential

p-type detector

blocking contact on unsegmented electrode

U = 4250 V

geometry :

0.06m x 0.06m x 0.02m

x

e-3, e-3

pixel-size: 10mm x 10mm

Samuel Lago

PotentialPotential

p-type detector

blocking contact on segmented electrode

U = 4000 V

geometry :

0.06m x 0.06m x 0.02m

z

e-3, e-3

pixel-size: 10mm x 10mm

Samuel Lago

PotentialPotential

n-type detector

blocking contact on segmented electrode

U = 4000 V

geometry :

0.06m x 0.06m x 0.02m

z

e-3, e-3

ce-3

pixel-size: 10mm x 10mm

c

c

Samuel Lago

Depletion chargeDepletion charge

blocking contact on segmented electrode

z

pixel 6x6 mm 10x10 mm 14x14 mm 18x18 mm

z up 5828 3178 2212 1836

z down 6219 3588 2629 2258

x, y 7478 4250 3064 2601

const:1.0E-3 4015 2256 1614 1365

const:2.0E-3 8030 4511 3227 2729

Samuel Lago

Depletion chargeDepletion charge

blocking contact on unsegmented electrode

pixel 6x6 mm 10x10 mm 14x14 mm 18x18 mm

z up 2265 2265 2262 2251

z down 1813 1813 1811 1806

x, y 2499 2542 2570 2575

const:1.0E-3 1360 1359 1358 1352

const:2.0E-3 2719 2718 2715 2704

Samuel Lago

Pulse shapesPulse shapes

dtrEvrEU

qdQ

dQdQtQttQ

W

he

))(()(

)()(

Total charge at time "t"

Samuel Lago

Pulse shapesPulse shapes

=const=1.6E-3 ,)(

),,( 010 z

hzyx

Samuel Lago

Calculations with the LOAX geometryCalculations with the LOAX geometry

Samuel Lago

Calculations with the LOAX geometryCalculations with the LOAX geometry

Samuel Lago

OutlookOutlook

• investigation of the position resolution with (x,y,z) in n- and p-type detector

• calculatulation of potential and electric Field with (x,y,z) in COMSOL

• export Data for investigation of the position resolution in Fortran-Code