Semiconductor Detectors(Solid State Detectors)

energy, positionparticles & photons

energy loss

conversionelectron-hole pairs

energetic “cheap”improved resolution

Semiconductor Detectors(Solid State Detectors)

NC,v number of states conduction,valence bandEg gap at 0 KA constant independent from T

for pure Si: ni=1.45x1010 cm-3

insulator semi conductor metal

6 eV 1 eV

conduction band

valence band

electrons

holes

€

ni = NcN v exp−Eg

2kT

= AT 3 / 2 exp

−Eg

2kT

Semiconductor Detectorshow to get a signal ?

mimimum ionizing particle will produce:about 3.2x104 e-h pairs

but 4.5x108 free charge carriers in the same volume !

reduce number of free charge carriers,i.e. deplete material !

particle passing through a thin layer of Si:300 µm

Semiconductor Detectors“doped” n-type

donorlevel n-type

+ e-

add donor elements: Vth group elements like P, As, Sb

extra electrons resides at discrete energy level in the

energy gap, close to the CB (separated 0.01 - 0.05 eV)

conductivity enhanced by electrons

=> n-type semiconductors

Semiconductor Detectors“doped” p-type

acceptorlevel

p-type+ holes

add acceptor elements: IIIrd group elements like Ga, B, In

extra electron states close to the VB, electrons easily excited

into these states, creating hole states in the VB.

conductivity enhanced by holes states

=> p-type semiconductors

Semiconductor Detectors“doped”

Typical doping levels for detector silicon:

1012 atoms/cm-3

which has to be compared with1022 atoms/cm-3 density Ge & Si

heavily doped semiconductors:(“+” sign after the material)

1020 atoms/cm-3

Semiconductor DetectorsSi, characteristics

band gap: Eg =1.12 V.

E(e--hole pair) = 3.6 eV(~30 eV for gas detectors)

high specific density (2.33 g/cm3)

ΔE/track lengthfor M.I.P.’s.: 390 eV/mm, ~108 e-h/ µm (average)

high mobilityme=1450 cm2/Vs, mh = 450 cm2/Vs

small dimensionsfast charge collection (<10 ns)

rigidity of siliconthin self supporting structurestypical thickness 300 µm~3.2 x104 e-h (average)

Semiconductor Detectors“np-junction”

initial diffusion:holes -> n-region, electrons -> p-region

charge building up:n-region-> positive, p-region -> negative

emerging electric field (contact potential)stops diffusion

region of changing potential:depletion zone, space charge region

n p

n p

Semiconductor Detectorsincreasing depletion

reversed bias junction(bias voltage about 100 V)apply negative voltage to p-side-> attract holesapply positive voltage to n-side-> attract electrons

-> increase depletion zone

n p

+ -

depletion zone

Semiconductor Detectorssignal read out

to prevent depletion zone between metal and semiconductor-> layer of highly doped material

n p

depletion zonen+ p+

metal contactsbias

Semiconductor Detectorscharacteristics

depletion depth

xn

-xp

-eNA

eND

V0

-xp xn

€

Poissons' s equation : d 2Vdx 2

= −ρ(x)ε

charge density : ρ(x) = eND 0 < x < xn = −eNA − xp < x < 0

charge conservation : NA xp = NDxn

Semiconductor Detectorscharacteristics

depletion depth d

integration yields V(x):

€

0 < x < xn

V (x) = −eND

εx 2

2− xn x

+ C

−xp < x < 0

V (x) =eNA

εx 2

2− xp x

+ C '

matching at x = 0 =>

C = C'

C =eNA

2εxp2

Semiconductor Detectorscharacteristics

depletion depth d

integration yields contact potential V(xn)=V0:

€

V0 =e2ε

NDxn2 + NA xp

2( )

and

xn =2εV0

eND 1+ ND /NA( )

xp =2εV0

eNA 1+ NA /ND( )

depletion zone extends farther into thelighter-doped side

Semiconductor Detectorscharacteristics

depletion depth d

under assumption NA >> ND :

resistivity of n-type material:

€

d = xn + xp ≈2εV0eND

€

ρn ≈1

eNDµ e

µ e electron mobility [cm 2 /Vs]

d ≈ 2ερnµ eV0

Semiconductor Detectorscharacteristics

“an application”

Assume ρ=20 kΩcm for heavily doped n-type Si and

V0=100 V. The dielectric constant ε/ε0=12andµe (300K)=1350 cm2/Vs.

How thick is the depletion zone ?

Semiconductor Detectorscharacteristics

junction capacitance

for planar geometry:

for ND>>NA or NA>>ND, respectively

Si:€

C = εAd

€

C /A =2.2ρNV0

pF /mm 2 n - type

C /A =3.7ρNV0

pF /mm 2 p - type

Semiconductor Detectorscharacteristics

pulse shape, rise timecharge collection time

for planar geometry, heavily doped p-type

+V

n+p

+

-

0x0 d

E

Semiconductor Detectorscharacteristics

pulse shape, rise timecharge collection time

for planar geometry, heavily doped p-type:

€

dQ =qdxd

integration of Poisson' s equation :

dVdx

= E = −eNA

εx = −

xµ hτ

τ = ρε with 1/ρ = eNAµ h

Semiconductor Detectorscharacteristics

€

ve =dxedt

= −µ eE =µ e

µ h

xτ

if mobility independent of E

xe (t) = x0 expµ e tµ hτ

electron reaches eletrode at x = d :

t = τµ h

µ e

ln dx0

Semiconductor Detectorscharacteristics

€

Qe (t) = −ed

dxdtdt =

ed

∫ x0 1− expµ e tµ h

analogue for holes, with

vh = µ hE = −xτ

Qh (t) = −edx0 1− exp

−tτ

Semiconductor Detectorscharacteristics

Q(t)

t

Qtot

Qe

Qh

Semiconductor Detectorscharacteristics

“an application”

Estimate the rise time of the solid state detector discussedabove !