Calorimetry and Tracking with High Precision Semiconductor Detectors
Applications
1. Photon spectroscopy with high energy resolution2. Vertex detection with high spatial resolution 3. Energy measurement of charged particles [few MeV]
Main advantages:
(i) Possibility to produce small structures using micro-chip technology; 10 μm precision; relatively low costs ...
(ii) Comparably low energy deposition per detectable electron-hole pair required ...
Silicon : 3.6 eV per electron-hole pairIonization (LAr): O(30 eV) for a single ion pair; see laterScintillators : O(100 eV) depending on light yield [typical 1-10%]
e.g.
Applications – Examples
Solid State Detectors
Lecture for Summer Students at DESY
Georg Steinbrück
Hamburg UniversityAugust 15, 2008
CMS Inner Barrel
Event Display
Applications – Examples
Front EndElectronics
Structure 4.2
Structure 2.8
Structure 1.4
Active area[Pads: 10x10 mm2]
ECAL ‘Physics’ Prototype[CALICE]
CALICESiW ECAL
Basic Semiconductor PropertiesC
onduction band
Vale
nce
band
Conduction band
Valenceband
Valenceband
Conduction band
Eg ≈ 1 eVEg ≈ 6 eV
Ene
rgy
gap
Gap
Insulator Semiconductor Metal
Electrons
Holes
Basic Semiconductor Properties
Intrinsic semiconductor:
Very pure material; charge carriers are created by thermal, optical or other excitations of electron-hole pairs; Nelectrons = Nholes holds ...
Commonly used: Silicon (Si) or Germanium (Ge); four valence electrons ...
Doped or extrinsic semiconductor:
Majority of charge carriers provided by donors (impurities; doping)
n-type: majority carriers are electrons (pentavalent dopants)p-type: majority carriers are positive holes (trivalent dopants)
Pentavalent dopants (electron donors): P, As, Sb, ... [5th electron only weakly bound; easily excited into conduction band]
Trivalent dopants (electron acceptors): Al, B, Ga, In, ...[One unsaturated binding; easily excepts valence electron leaving hole]
E(�k) =�2k2
2me=
�2
2me(k2
x + k2y + k2
z)
g(E) ∝�
2me
�2
� 32 √
E ∝√
E
Basic Semiconductor Properties
Conduction band
Valenceband
EnergyGap
Ec
Ev
Energy bands : Regions of many discrete energy levels with very close spacing
Arise from interaction of electrons with the very many atoms of the crystalline/solid material ...
Energies treated like particles in a box ...[Fermi gas model]
Yields:
[Dispersion relation]
[Density of states]
IntrinsicSemiconductors
ki =ni2π
L[i = x, y, z]
=�2
2me
�2π
L
�2
n2 n2 = n2x + n2
y + n2z
N
2=
43πn3
max
nmax =�
3N
8π
� 13
V = L3
Emax =�2
2me
�2π
L
�2
n2max
Emax =�2
2me
�3π2N
V
� 23
➙
Basic Semiconductor Properties
E(�k) =�2k2
2me=
�2
2me(k2
x + k2y + k2
z) with
[... quantized due to boundary conditions]
with
We have to fill all states within a sphere of radius nmax or, alternatively, kmax:
Number of electrons
Each stateoccupied twice
[Volume of solid]Derivation of g(E):
IntrinsicSemiconductors
N(E) = E32
�2me
�2
� 32 V
3π2
∝√
E
g(E) =dN
dE= E
12 ·
�2me
�2
� 32 V
2π2
Basic Semiconductor Properties
Emax =�2
2me
�3π2N
V
� 23
Derivation of g(E):
➙
Density of states follows from:
Emax is also called Fermi Energy EF; can be associated with highest kinetic energy of electrons in a solid at T = 0 K ...
Remark: Characteristics of solids determined by location of Fermi Energy
Metal: EF below top of an energy band Insulator: EF at top of valence band; large gap Semiconductor: EF at top of valence band; smaller gap
EEF
Density of states g(E)
IntrinsicSemiconductors
f(E)
EF
0.5
1 T = 0
T > 0
f(E, T ) =1
e(E−µ)/kBT + 1
Basic Semiconductor Properties
At a temperature T the occupation probability of the available states is given by the Fermi-Dirac distribution ...
with chemical potential μ[metals: EF = μ; often identified with EF]
Fermi-Dirac distribution
μ (= EF)
T = 0:
Step function; only states below μ are occupied ...
T > 0:
Fermi-Dirac distributions develops a 'soft zone' ...
Notice: EF ~ several eV soft zone: 100 meV @ 300 K
IntrinsicSemiconductors
E
0.5
1
1
0
f(E)
f(E)
n(E)
g(E)
EF
T 0=
T 0>
n(E)
N =� Emax
0g(E)f(E)dE
n =1V
� Emax
0g(E)f(E)dE
Basic Semiconductor Properties
μ (= EF)
μ (= EF)
Electron density isgiven by:
Sometimes extra factor 2if g(E) does not account for spins ...
Total number of electronsup to an energy Emax:
Electron density:
IntrinsicSemiconductors
n(E) = g(E)f(E)½
f(E, T ) ≈ e−(E−µ)/kBT
f(E, T ) =1
e(E−µ)/kBT + 1n =
1V
� ∞
Eg
V
2π2
�2me
�2
� 32 �
E − Eg e−(E−µ)/kBT dE
Basic Semiconductor Properties
Conductionband
Valenceband
Carrier concentration in conduction and valence band:
Eg is generally large compared to 'soft zone', i.e. (E – μ) » kBT, such that ...
f(E,T>0)
n(E)
g(E)
Chemical Potential
... for calculation of the electron density.
Using above conventions (see Fig.):
...
Band structure and electron density
IntrinsicSemiconductors
n =1V
� ∞
Ec
gc(E)f(E, T )dE
p =1V
� Ev
−∞gv(E)[1− f(E, T )]dE
n =(2me)
32
2π2�3eµ/kBT
� ∞
Eg
�E − Eg e−E/kBT dE
[using symmetry of f(E,T)]
with Xg = (E − Eg)/kBT[ substitution ]
using� ∞
0X
12g e−Xg dXg =
√π
2...
n =(2me)
32
2π2�3eµ/kBT
� ∞
Eg
�E − Eg e−E/kBT dE
n =√
π
2(2mekBT ) 3
2
2π2�3e−(EC−µ)/kBT = NC · e−(EC−µ)/kBT
p =√
π
2(2mhkBT ) 3
2
2π2�3e−(µ−EV )/kBT = NV · e−(µ−EV )/kBT
Basic Semiconductor Properties
Conductionband
Valenceband
f(E,T>0)
n(E)
g(E)
Chemical Potential μ
...
Band structure and electron density
EC
EVμ
calculation continued ...
IntrinsicSemiconductors
m*: effective mass[electrons in crystal]
n =(2me)
32
2π2�3(kBT )
32 e−(Eg−µ)/kBT
� ∞
0X
12g e−Xg dXg
n =√
π
2(2mekBT ) 3
2
2π2�3e−(Eg−µ)/kBT
*
*
p = NV · e−(µ−EV )/kBT
n = NC · e−(EC−µ)/kBT
np = NCNV e(EV −EC)/kBT ∝ (m∗em
∗h)
32
Basic Semiconductor Properties
Carrier concentration in conduction and valence band:
NC: effective density of electrons at edge of conduction band
NV: effective density of holes at edge of valence band
Pure semiconductors: carrier concentration depends on separation of conduction/valence band from chemical potential or Fermi level ...
NC,V ~ (m*T)3/2
T dependent
Location of Fermi level determines n and p ...But, product is independent of location of Fermi level ...
At given temperature characterized by effective mass and band gap.
IntrinsicSemiconductors
Law ofmass action
n = p ni = pi
µ =EC + EV
2− kBT
2ln
�NC
NV
�=
EC + EV
2− 3
4kBT ln
�m∗
e
m∗h
�
Basic Semiconductor Properties
Intrinsic semiconductors; no impurities ➛ number of electrons in conduction band is equal to number of holes in valence band.
IntrinsicSemiconductors
At T = 0: Fermi-level (EF = μ) lies in the middle between valence and conduction band ...
At T > 0: In case the effective masses of electrons and holes are non-equal, i.e. NC ≠ NV the Fermi-level changes with temperature ...
or to characterize that this holds for intrinsicsemiconductors only
The expressions for n,p then yield:
EC
EV
μ = EF
✝
Basic Semiconductor Properties
Si Ge GaAs[III-V Semiconductor]
Egap [eV] 1.11 0.67 1.43
ni @ 150 K [m-3] 4.1⋅106 — 1.8⋅100
ni @ 300 K [m-3] 1.5⋅1016 2.4⋅1019 5.0⋅1013
me/me 0.43 0.60 0.065
mh/me 0.54 0.28 0.50
Energy/e+e–-pair [eV] 3.7 3.0 —
*
*
✝
✝ at 77 K
Some properties of intrinsic semiconductors
IntrinsicSemiconductors
Basic Semiconductor Properties
Introducing impurities (doping) ➛ balance between holes and electrons in conduction band can be changed; yields higher carrierconcentrations.
DopedSemiconductors
n-doping
p-doping
n-doping: extra electron resides in discreteenergy level close to conduction band ...
p-doping: additional state close to the valence band can accept electrons ...
n-doping: majority carriers = electrons [holes don't contribute much; minority carriers]
p-doping: majority carriers = holes [electrons are minority carriers]
n-doping: Sb, P, As ...
p-doping: B, Al, Ga ...
Basic Semiconductor Properties
B Al GaEV
Sb P As Ec
0.045 eV 0.067 eV 0.085 eV
EF (reines Si)
0.039 eV 0.044 eV 0.049 eV
Abbildung 6.2: Lage der Fermi–Niveaus verschiedener Elemente relativ zur Kante des Valenz–(EV ) und des Leitungsbandes (Ec) [5]
Folgende allgemeine Trends konnen beobachtet werden :
• Die Lage der Fermi–Energie EF hangt von der Art des dotierten Sto!es und vom Grund-material ab.
• Die Zahl der Elektronen im Leitungsband hangt von der Dotierung und der Temperaturab (Abb.6.3).
ne. 1
0−16
[cm−3
]
2
1
00 400200
[ ]600
T K
Abbildung 6.3: Elektronendichte als Funktion der Temperatur fur reines Si (- - - -) und Sidotiert mit 1016As–Atome/cm3 ( ) (in Anlehnung an [5])
Zwei Großen sind fur die Beschreibung des elektrischen Verhaltens eines Halbleiters wichtig :
Mobilitat µ [m2 V !1 s!1]
spez. Widerstand ! [" m]
Es gilt naherungsweise (siehe Kap.7)"vD = µ "E
fur die Driftgeschwindigkeit und fur den Widerstand (l Lange, A Flache des Leiters ! "E )
R = !l
A.
89
DopedSemiconductors
μ (= EF)[pure Si]
Energy levels for silicon with different dopants
Basic Semiconductor Properties
Position of chemical potential for n-doped semiconductor:
DopedSemiconductors
High temperature (intrinsic) Intermediate temp. (extrinsic) Low temp. (freeze-out)
All donors and someintrinsic carriers ionized
Position of chemical potential for n-doped semiconductor:
Almost all donors; very few intrinsic carriers
ionized
Only few donors are ionized
Basic Semiconductor Properties
B Al GaEV
Sb P As Ec
0.045 eV 0.067 eV 0.085 eV
EF (reines Si)
0.039 eV 0.044 eV 0.049 eV
Abbildung 6.2: Lage der Fermi–Niveaus verschiedener Elemente relativ zur Kante des Valenz–(EV ) und des Leitungsbandes (Ec) [5]
Folgende allgemeine Trends konnen beobachtet werden :
• Die Lage der Fermi–Energie EF hangt von der Art des dotierten Sto!es und vom Grund-material ab.
• Die Zahl der Elektronen im Leitungsband hangt von der Dotierung und der Temperaturab (Abb.6.3).
ne. 1
0−16
[cm−3
]
2
1
00 400200
[ ]600
T K
Abbildung 6.3: Elektronendichte als Funktion der Temperatur fur reines Si (- - - -) und Sidotiert mit 1016As–Atome/cm3 ( ) (in Anlehnung an [5])
Zwei Großen sind fur die Beschreibung des elektrischen Verhaltens eines Halbleiters wichtig :
Mobilitat µ [m2 V !1 s!1]
spez. Widerstand ! [" m]
Es gilt naherungsweise (siehe Kap.7)"vD = µ "E
fur die Driftgeschwindigkeit und fur den Widerstand (l Lange, A Flache des Leiters ! "E )
R = !l
A.
89
Carrier density dependson doping and temperature ...
Electrons in conduction band
Extrinsic
Intrinsic
pure Sin-doped Si
Neutrality condition:
ND + p = NA + n
ND: donor concentrationNA: acceptor concentration
Extrinsic region:
n-type: n ≈ ND [NA = 0; n » p]p-type: p ≈ NA [ND = 0; p » n]
Typical concentrations:
Dopants: ≥ 1013 atoms/cm3
[Strong doping: 1020 atoms/cm3; n+ or p+]
Compare to Si-density: 5⋅1023/cm3
DopedSemiconductors
The np-Junction
Abbildung 6.5: Schematische Ortsabhangigkeit der Energiebander (in Anlehnung an [5])
Was passiert wahrend der Einstellung des Gleichgewichts?e! und + di!undieren aufgrund der stark inhomogenen Dichteverteilung solange, bis dasentstehende !E–Feld zwischen Donatoren und Akzeptoren dem Fluß entgegenwirkt und ihnschließlich beendet. Im Grenzbereich verschwinden die beweglichen Ladungstrager.Wir wollen fur die quantitative Beschreibung die folgenden Naherungsannahmen machen, dienach der oben beschriebenen Eigenschaft epitaktischer Schichten sinnvoll sind:
Abrupte Anderung von n > 0 ! n = p = 0 ! p > 0.
In Realitat tritt dies in einem Bereich von etwa 0.1µ"1µ auf (Debye–Lange) (Abb.6.4).
Qualitatives Modell:
• Ec, EV , Ei haben gleiche x–Abhangigkeit
• Setze fur das Potential an (EF = frei gewahlter Bezugspunkt)
" = "1
e(Ei " EF ) ,
d.h. reines Si " = 0p Si " < 0n Si " > 0
Die Ortsabhangigkeit der interessierenden Großen ist in Abb.6.5 gezeigt.
Aus nD = n = ni e (EF!Ei)/kT ! "n = kTe #n nD
ni
nA = p = ni e (Ei!EF )/kT ! "p = "kTe #n nA
ni
Damit erhalt man als Potentialbarriere
"" = "i = "n " "p =kT
e#n
nD · nA
n2i
92
Function of present-day semiconductor detectors depends on formation of a junction between n- and p-type semiconductors ...
μ (= EF)
μ (= EF) EV
EC
E0
μ (= Ei )
p-type n-type
Thermodynamic equilibrium ➛ Fermi energies should become equal ...
moves up whenforming junction
moves down whenforming junction
Abbildung 6.5: Schematische Ortsabhangigkeit der Energiebander (in Anlehnung an [5])
Was passiert wahrend der Einstellung des Gleichgewichts?e! und + di!undieren aufgrund der stark inhomogenen Dichteverteilung solange, bis dasentstehende !E–Feld zwischen Donatoren und Akzeptoren dem Fluß entgegenwirkt und ihnschließlich beendet. Im Grenzbereich verschwinden die beweglichen Ladungstrager.Wir wollen fur die quantitative Beschreibung die folgenden Naherungsannahmen machen, dienach der oben beschriebenen Eigenschaft epitaktischer Schichten sinnvoll sind:
Abrupte Anderung von n > 0 ! n = p = 0 ! p > 0.
In Realitat tritt dies in einem Bereich von etwa 0.1µ"1µ auf (Debye–Lange) (Abb.6.4).
Qualitatives Modell:
• Ec, EV , Ei haben gleiche x–Abhangigkeit
• Setze fur das Potential an (EF = frei gewahlter Bezugspunkt)
" = "1
e(Ei " EF ) ,
d.h. reines Si " = 0p Si " < 0n Si " > 0
Die Ortsabhangigkeit der interessierenden Großen ist in Abb.6.5 gezeigt.
Aus nD = n = ni e (EF!Ei)/kT ! "n = kTe #n nD
ni
nA = p = ni e (Ei!EF )/kT ! "p = "kTe #n nA
ni
Damit erhalt man als Potentialbarriere
"" = "i = "n " "p =kT
e#n
nD · nA
n2i
92
μ (= EF)
EV
EC
E0
μ (= Ei )
p-type
n-type
The np-Junction
EC
E0
EV
– – – –– – – –
+ + + + + + + +acceptors donors
ni
ND
NA
n
p
N acceptorsA
N donorsD
p-type n-type
!( )x
+
"x
e UD#
Löcher
+" " " "
""
+++++
EV
EC
x
n p( )
n n( )
p n( )
p p( )
ln , , ,n p N ND A
eUD = ∆Epot = E(p)C − E(n)
C
eUD = kBT · lnnn−type
np−type= kBT · ln
NDNA
n2i
[ using n = NC · e−(EC−µ)/kBT , p = ... ]
The np-Junction
Equilibration process:
Electrons diffuse from n to p-typesemiconductor and recombine ...
Holes diffuse from p to n-typesemiconductor and recombine ...
Resulting electric field counteractsand stops diffusion process ...
At the boundary concentrationof mobile carriers is depleted ...[depletion layer]
ρ(x) =−eNA − xp < x < 0
eND 0 < x < xn{
NAxp = NDxn
dV
dx=
d2V
dx2= −ρ(x)
�
−eND/� · x + Cn 0 < x < xn
eNA/� · x + Cp − xp < x < 0
eNA/� · (x + xp) − xp < x < 0
−eND/� · (x− xn) 0 < x < xn
The np-Junction
Depletion depth:
Model for calculatingdepletion zone
{dV
dx= {
as electric field E = -dV/dx must vanish for x=xn and x= -xp
Depletion depth:
NAxp = NDxn
V (x) =eNA/� · (x2/2 + xpx) + C� − xp < x < 0
−eND/� · (x2/2− xnx) + C 0 < x < xn
V0 =eND
2�x2
n +eNA
2�x2
pC =eNA
2�x2
p
xn =
����2�V0
eND
�1 + ND
NA
� xp =
����2�V0
eNA
�1 + NA
ND
�
d = xn + xp =
�2�V0 (NA + ND)
eNAND
The np-Junction
Depletion depth:
Model for calculatingdepletion zone
Depletion depth:
{Solution must be continuous at x=0; thus C = C';Also V(-xp) = 0 and V(xn) = V0 with V0 contact potential ...Thus:
Using NAxp = NDxn yields:
p-type n-type
With:
and
Remark: If one side more heavily doped, depletion zone will extend to lighter doped side; e.g. NA » ND, xn » xp ...
NAxp = NDxn
d = xn + xp =
�2�V0 (NA + ND)
eNAND
d ≈ xn ≈�
2�V0
eND
≈�
2�ρnµeV0
The np-Junction
Depletion depth:
Model for calculatingdepletion zone
Depletion depth:
p-type n-type
If e.g. NA » ND [as in figure] ...
using conductivity σ = 1/ρ = e(n μe + p μh),with n = ND and mobility μ = v/E.
Depletion depth determined by mobility of charge carriers ...
Typical values: Silicon: Germanium:
0.53 (ρnV0)0.5 μm (n-type); 0.32 (ρpV0)0.5 μm (p-type) 1.00 (ρnV0)0.5 μm (n-type); 0.65 (ρpV0)0.5 μm (p-type)
[Typical ρ ≈ 20000 Ωcm and V0 = 1V ➛ d = 75 μm]
For large depth chooseasymmetric doping!
The np-Junction
Application of an external voltage:
+
+
–
–
No voltage Forward bias Reverse bias
Equilibrium: drift of minority electrons from p-side compensates diffusion current from n-side which
have to move against E-field
Here: consider only electrons[similar for holes]
Voltage drop over depletion zone; diffusion current higher due to shift
of chemical potential; current increases exponentially with bias
Voltage drop over depletion zone; diffusion current smaller due to shift of chemical potential; widening of
the depletion zone
I = I0 (eeV/kT -1) I = I0 (e-eV/kT -1)
The np-Junction
Leakage currentU
I
! I / 100
UB
U
pn
Characteristic I(V) curve of a diode
Forward bias
Reversebias
E =U
d=
100 V300 · 10−6 m
≈ 3 · 105 Vm
= 100 V
U =e
2�NAd2
d ≈ xp ≈�
2�U
eNA
Basic Semiconductor Detector
Requirement:Large sensitive region ...
We know:
Typical: NA = 1015/cm3
n+ region highly doped: ND » NA
p[ρ = 10 kΩcm; NA]
1 μm
1 μm
300 μm
n+ and p+ needed to allow metallic contacts ...[High doping = small depletion zone]
p+ dead layer
Metal contact
Sensitive volume
Bias
n+
Bias voltage supplied through series resistor ...
Signal
Electric field:
[Safe. Breakdown limit at 107 V/m]