Download Semiconductor Detectors (Solid State Detectors)

Semiconductor Detectors
(Solid State Detectors)
energy, position
particles & photons
energy loss
conversion
electron-hole pairs
energetic “cheap”
improved resolution
Semiconductor Detectors
(Solid State Detectors)
conduction band
electrons
6 eV
1 eV
holes
valence band
insulator
semi conductor
metal
 −E g 
 −E g 
3/2
ni = N c N v exp
 = AT exp

 2kT 
 2kT 
NC,v
Eg
A
number of states conduction,valence band
gap at 0 K
constant independent from T
for pure Si: ni=1.45x1010 cm-3
Semiconductor Detectors
how to get a signal ?
particle passing through
a thin layer of Si:
300 µm
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 !
Semiconductor Detectors
“doped” n-type
donor
level
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
acceptor
level
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 with
1022 atoms/cm-3 density Ge & Si
heavily doped semiconductors:
(“+” sign after the material)
1020 atoms/cm-3
Semiconductor Detectors
Si, 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 length
for M.I.P.’s.: 390 eV/mm, ~108 e-h/
high mobility
me=1450 cm2/Vs, mh = 450 cm2/Vs
small dimensions
fast charge collection (<10 ns)
rigidity of silicon
thin self supporting structures
typical thickness 300
µm
~3.2 x104 e-h (average)
µm (average)
Semiconductor Detectors
“np-junction”
n
p
n
p
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
Semiconductor Detectors
increasing depletion
depletion zone
-
+
n
reversed bias junction
(bias voltage about 100 V)
apply negative voltage to p-side
-> attract holes
apply positive voltage to n-side
-> attract electrons
-> increase depletion zone
p
Semiconductor Detectors
signal read out
depletion zone
n+
n
bias
p
metal contacts
to prevent depletion zone between metal and semi
conductor
-> layer of highly doped material
p+
Semiconductor Detectors
characteristics
depletion depth
V0
eND
-xp
xn
-eNA
Poissons' s equation :
-xp
xn
d 2V
ρ (x)
=−
2
dx
ε
charge density :
ρ(x) = eN D 0 < x < x n
= −eN A − x p < x < 0
charge conservation :
N A x p = N D xn
Semiconductor Detectors
characteristics
depletion depth d
integration yields V(x):
0 < x < xn

eN D  x 2
V (x) = −
 − x n x + C
ε 2

−x p < x < 0

eN A  x 2
V (x) =
 − x p x + C'
ε 2

matching at x = 0 =>
C = C'
eN
C = A x p2
2ε
Semiconductor Detectors
characteristics
depletion depth d
integration yields contact potential V(xn)=V0:
V0 =
e
N D x n2 + N A x p2 )
(
2ε
and
xn =
2εV0
eN D (1 + N D / N A )
xp =
2εV0
eN A (1 + N A / N D )
depletion zone extends farther into the
lighter-doped
side
€
Semiconductor Detectors
characteristics
depletion depth d
under assumption NA >> ND :
2εV0
d = xn + x p ≈
eN D
resistivity of n-type material:
€
1
ρn ≈
eN D µ e
µe
electron mobility
d ≈ 2ερ n µ eV0
[cm 2 /Vs]
Semiconductor Detectors
characteristics
“an application”
ρ=20 kΩcm for heavily doped n-type Si and
V0=100 V. The dielectric constant ε/ε0=12
Assume
and
µe (300K)=1350 cm2/Vs.
How thick is the depletion zone ?
Semiconductor Detectors
characteristics
junction capacitance
for planar geometry:
A
C =ε
d
for ND>>NA or NA>>ND, respectively
Si:
€
2.2
C/A =
pF / mm 2
ρ N V0
n - type
3.7
pF / mm 2
ρ N V0
p - type
C/A =
€
Semiconductor Detectors
characteristics
pulse shape, rise time
charge collection time
for planar geometry, heavily doped p-type
p
n+
+V
+
0
x0
d
E
Semiconductor Detectors
characteristics
pulse shape, rise time
charge collection time
for planar geometry, heavily doped p-type:
qdx
dQ =
d
integration of Poisson' s equation :
eN A
dV
x
=E =−
x=−
dx
ε
µhτ
τ = ρε
€
with
1/ ρ = eN A µ h
Semiconductor Detectors
characteristics
dx e
µe x
ve =
= −µ e E =
dt
µh τ
if mobility independent of E
 µet 
x e (t) = x 0 exp

µ h τ 
electron reaches eletrode at x = d :
µh d
t=τ
ln
µ e x0
€
Semiconductor Detectors
characteristics
µet
e dx
e 
Qe (t) = − ∫ dt = x 0 1 − exp 
d dt
d 
µh 
analogue for holes, with
x
vh = µ h E = −
τ
e 
−t 
Qh (t) = − x 0 1 − exp 
d 
τ
Semiconductor Detectors
characteristics
Q(t)
Qtot
Qe
Qh
t
Semiconductor Detectors
characteristics
“an application”
Estimate the rise time of the solid state detector discussed
above !