Semiconductor Detectors -...

Calorimetry and Tracking with High PrecisionSemiconductor 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

ATLAS Pixel Detector


ATLAS Pixel Detector[Details]

Pixel Sensor

ATLAS Pixel Module


Solid State Detectors

Lecture for Summer Students at DESY

Georg Steinbrück

Hamburg UniversityAugust 15, 2008

CMS Inner Barrel

Event Display


Front EndElectronics

Structure 4.2

Structure 2.8

Structure 1.4

Active area[Pads: 10x10 mm2]

ECAL ‘Physics’ Prototype[CALICE]

CALICESiW ECAL


pnCCD CameraX-ray astronomy satellite XMM-Newton

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


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

➙


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):


N(E) = E32

�2me

�2

� 32 V

3π2

∝√

E

g(E) =dN

dE= E

12 ·

�2me

�2

� 32 V

2π2


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)


f(E)

EF

0.5

1 T = 0

T > 0

f(E, T ) =1

e(E−µ)/kBT + 1


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


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


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:


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


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


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


Conductionband

Valenceband

f(E,T>0)

n(E)

g(E)

Chemical Potential μ

...

Band structure and electron density

EC

EVμ

calculation continued ...


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


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.


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

�


Intrinsic semiconductors; no impurities ➛ number of electrons in conduction band is equal to number of holes in valence band.


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

✝


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



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 ...


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


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


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:


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:


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]

Semiconductor Detectors -...

Documents

Transcript of Semiconductor Detectors -...