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The Physics of Infrared Detectors
(with a special emphasis on
MID-IR detectors)
Massimo Robberto
(ESA – STScI)

Outline
• Semiconductors as crystals
• Energy bands
• Intrinsic semiconductors
• Light on intrinsic semiconductors
• Extrinsic Semiconductors
• Impurity band conductors (IBC)

Periodic Table
Atomic structure: (four 2s2p) covalent bonds per atom

Crystal Structure: 2-d
simple cubic
Bravais lattice R ma nb pc
  

  
In two dimensions, there are
five distinct Bravais lattices

Crystal Structure: 3-d
simple cubic body-centered cubic face centered cubic
In three dimensions, there are fourteen distinct Bravais lattices
A large number of semiconductors are cubic
(a1=a2=a3= a “lattice constant”)

Diamond Structure
Two interpenetrating face-centered cubic lattices
Diamond lattice:
All same atoms
(e.g. Si)
Zincblende lattice:
different atoms
in each sublattice
(e.g.CdTe, GaAs)

Lattice planes in crystals:
Miller indexes
Silicon and Germanium break on {1,1,1} planes
never confuse the spacing between lattice planes
with the spacing between crystal planes

Example
The surface density of Silicon is
14 2
14 2
14 2
11.8 10 atoms/cm on {111}
9.6 10 atoms/cm on {110}
6.8 10 atoms/cm on {100}




Wigner-Seitz cell
The smallest (“primitive”) cell which displays the full symmetry
of the lattice is the Wigner-Seitz cell.
Construction method: surfaces passing through the middle points
to the nearest lattice points
In 3-d think to a polyhedron

Reciprocal lattice
g ha kb lc
  
  

  
The Bravais lattice after Fourier transform
real space reciprocal lattice
normals to the planes(vectors) points
spacing between planes 1/distance between points
(actually, 2/distance)
 (distance, wavelength) 2k (momentum, wave
 number)
Bravais cell Wigner-Seitz cell

Bragg conditions
When a wave impinges on a crystal - and it doesn't matter if it is an electromagnetic wave,
e.g. X-rays, or an electron, or neutron "wave" - it will be reflected at a particular set
of lattice planes {hkl} characterized by its reciprocal lattice vector g only if the so-called
Bragg condition is met
k k g

 
  
If the Bragg condition is not met, the incoming wave just moves through the lattice
and emerges on the other side of the crystal (neglecting absorption)

Elastic scattering
k k

 
For a given k, The Bragg conditions is met on surfaces
normal to particular g.
These surfaces define cells in the k-space, called
Brillouin zones

Brillouin Zone construction
All wave vectors that end on a BZ, will fulfill the Bragg condition and thus are diffracted.
Wave vectors completely in the interior of the 1. BZ, or in between any two BZs, will never get
diffracted; they move pretty much as if the potential would be constant, i.e. they behave very close
to the solutions of the free electron gas.

First BZ
Second BZ
Brillouin zone
The Brillouin zone is defined in the reciprocal lattice.
The first BZ is the volume enclosed within a Wigner-Seitz cell in
the k-space.
In 3-d think to a nested set of polyhedra
Third BZ

WS zone and BZ
Lattice Real Space Lattice K-space
bcc WS cell Bcc BZ (fcc lattice in K-space)
fcc WS cell fcc BZ (bcc lattice in K-space)
The WS cell of bcc lattice in real space transforms to a Brillouin zone in a fcc lattice in
reciprocal space while the WS cell of a fcc lattice transforms to a Brillouin zone of a bcc
lattice in reciprocal space.
Bcc: body-centered cubic; fcc: face-centered cubic

Brillouin zone of Silicon
Points of high-symmetry on the Brillouin zone have specific
importance. The most important point for optoelectronic devices
is the center at k = 0, known as the gamma point Γ.
Note the points Γ, X, W, K,…

Wave vectors near or at a BZ - let's call
them kBZ
electrons - feel the periodic
potential of the crystal while the others do
not. E.g., they are diffracted.

ENERGY GAP in CRYSTALS
• On the BZ it is k=-k: these are two standing waves
described by  ~eikr
and  e-ikr
• Their combination can be symmetric
eikr
+ e-ikr
~ cos(kr)
or antisymmetric
 eikr
- e-ikr
~ sin(kr)
• The probability density  have different values at
each point
• 2 different values of the energy, varying with k on the
BZ: ENERGY GAP.

 
2
BZ
E(k ) = ± ( )
2
k
U g
m


Energy bands
2
2
( ) ( ) ( )
2
k k k
V r r E r
m
 
 
   
 
 

If V(r) is periodic with periodicity of the lattice, then the wave
Function is a plane wave (free electron) with periodic modulation
( ) ( , ) Block function
ik r
k n
r e U k r
 
 
k is a wave vector in the reciprocal lattice, Un(k,r) is periodic in r,
i.e. U(r+R)=U(r), and n is the band index.
-For a given n, it is sufficient to use k’s in the primitive cell of the
reciprocal lattice (Brillouin zone). The rest is redundant!
F. Block solved the Shroedinger equation for an electron in the
lattice:

Band Structure and Bloch’s
Theorem
( ) ( , ) Block function
ik r
k n
r e U k r
 
 
From:
It is also:
( )
( ) ( , ) ( )
i K g r
k g n k
r e U k g r r
 
 
   
Then:
( ) ( )
E k g E k
 
for dispersion curves
that have a different origin
There are many energy values for
one given k. In particular, all
possible energy values are contained
within the first Brillouin zone
(between -1/2g1
and +1/2g1
in the
picture).

Band Structure and Bloch’s
Theorem
reduced representation of the band diagram
Every energy branch in principle should carry
an index denoting the band (often omitted)
Energy functions of a periodic potential
The electron at k1can go to the upper
band if someone gives him
1. E > bandgap, AND
2. k2=kl+g
THIS IS THE BRAGGS LAW FOR
INELASTIC SCATTERING

Band diagram of Silicon
• Si has a band gap of about 1 eV.
• Si is an indirect semiconductor because the maximum of the valence band (at )
does not coincide with the minimum of the conduction band (to the left of X).

Direct and Indirect
Semiconductors

Simplified band diagram
Conduction band
Valence band
Eg bandgap
1.24
( )
co
g
E eV
 

Fermi Energy and Carrier Concentration
Density of electrons in the energy interval E, E + E =
density of states  probability for occupancy  energy interval
The number (or density) of something is given by the density of
available places times the probability of occupation.
( ) ( , )
dN D E f E T dE
  

Density of states
2 2
2
k
k
E
m


A free (or under constant potential)
particle in a rectangular box has:
1) only kinetic energy
2) k is discrete (stationary waves)
, , , ,
2
x y z x y z
k n
L


Therefore, the energy is quantized
 
2
2
2 2 2
2
2
k x y z
E n n n
m L

 
  
 
 

kx
ky
kz

Density of states Ds
• In phase space a surface of constant energy is a sphere.
The volume is
•Any "state", i.e. solution of the Schroedinger equation
with a specific k, occupies with 2 electrons the volume
• The number of cubes fitting inside the sphere at energy
E thus is the number of all energy levels up to E:
3
4
3
V k


3
2
k
V
L

 
 
 
3 3
2
2
3
k
s
V k L
N
V 
 
3/ 2
1/ 2
3 2 2
1 1 2
2
s
s
dN m
D E
L dE 
 
   
 

Finally:

Density of states Ds
At 0 K one can place electrons up to the Fermi level EF
3/ 2
1/ 2
2 2
1 2
2
s
m
D E

 
  
 


Probability of occupancy
F
E-E
kT
1
f(E,T)=
e 1

Fermi
distribution
E – EF
>> kT
Fermi level

Intrinsic Semiconductors
Si
HgCdTe
InSb

The concentration of electrons in the conduction band is
3/ 2 2
( ) ( , )
c
F
E
e
E
E E
e KT
eff
Eg
KT
n D E f E T dE
N e
AT e








effective density of states
at the band edge
3/ 2
2
2
4
e e
eff
m kT
N
h
 
 
  
 
With A=4.831021
electrons m-3
K-3/2
.
The nr. of electrons in the conduction band depends on Eg and T.
In Si ne
doubles for ~8 degree rise in temperature

Same for the holes in the valence band:
ne
=nh
=ni
“INTRINSIC CONCENTRATION”
And the mass action law is
 
3
3/2
2
2
2
4
g
E
i e p kT
e h
kT
n n n e m m
h
 
 
 
   
 
Eg
(300 K) = 1.1242 eV
Eg
(0 K) = 1.700 eV
The bandgap energy and the effective mass depend on T
for Silicon:

The CONDUCTIVITY
 
e e h h
e n n
  
 
The mobility µ is the proportionality constant between the average drift velocity vD
of carriers in the presence of an electrical field E:
D
v E


Mobility depends on the average time between scattering processes
s
e
m

 
decreases with Eg. Leakage currents are lower in large Eg
materials.

Radiation on intrinsic
semiconductors
• at low T the conduction band is “empty”. Low intrinsic
conductivity
• radiation with h>Eg creates electron-hole pairs: ne
=nh
• both electrons and holes contribute to the photocurrent,
depending on mobilities
• The conductivity changes: PHOTOCONDUCTOR
 
e e h h e e h h
e n n n n
    
     
Where
ne,h
= quantum efficiency)  photon flux)
• all atoms count: absorption occurs in thin layer (~10micron)

Extrinsic semicondictors
Semiconductor
donors
acceptors
Doped semiconductors

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( , , )
e e
eff c F
n N f E E T

( , , )
A A A F
N N f E E T


 
1 ( , , )
h h
eff V F
n N f E E T
 
 
1 ( , , )
D D A F
N N f E E T

 
e h
A D
n N n N
 
  
Charge neutrality
gives EF and therefore the concentrations

Extrinsic semiconductors
Normally :
“Dopant or majority” vs. “ residual impurities”
Ex: Si:Ga (III group) is p-type
Si:As (V group) is n-type
• There are shallow- and deep-level impurities
• shallow-level impurities increase the  response
• Atoms of dopant and residual impurities do not interact
with each other.
• p-type: at low T the conduction band is “empty”; donor
impurities have lost their electrons and are ionized, acceptor
dopant has electrons from valence band and impurities; valence
band has free holes.
D A
N N

D A
N N

D A
N N


Opposite requirements
To move away and collect charges we need an electric field:
To have a good drop of potential, no current must flow:
Detector material must have high impedence
(I.e. low conductivity)
On the other hand, we want high concentration of absorbers:
this goes in the direction of high conductivity
x
dV
E
dx


Impurity band conduction
The donor band (As in Si:As ) can be heavily doped
Photons are absorbed
• in thin layer (smaller volume)
• providing higher QE
• higher radiation immunity
• lower applied bias
• better uniformity
• faster response
High dopant concentration creates a band ~1meV wide,
thus the cutoff increasees from 24 to 28m.
However:
The donor band becomes a conduction band:
“Impurity Band Conduction”

Impurity band conduction
The donor band (As in Si:As ) is heavily doped
Electric field is applied (bias) and the free charge carriers
are driven out from the IR active region (depletion):
High resistance and high electric field
Transparent contact (V+)
Blocking layer (intrinsic Si)
IR active region (heavily As doped)
Substrate (heavily doped n-type) at ground
Problem: charges “hop” in the impurity conduction band;
If they are sensed there is an extra dark current: need for a blocking layer (BIB)
h+ e-

Photo-electrons and relative holes are collected
hole

Si:As Engeneering
• Dopant As is n-type. P-type impurities (e.g.
B) are a potential problem
• They are “neutralized” by As, leaving them
as negative charge centers in the “depletion
region”
• These NA charges create an electric field
that limits the extention of the depletion
region w

Poisson equation for the ionized impurities:
0 0 0 0
x A
dE eN
dx k k

 

 
1/ 2
2
0 0
2
b B B
A
k
w V t t
eN

 
  
 
 
-tB
The width of the depletion region depends on the bias and
on the impurity concentration
Assuming NA = 1012
cm-3
and Vb = 1V it is w=32m

An acceptable arsenic concentration is ND=3 x 1017
cm-3
.
For arsenic in silicon, the absorption cross section is Si:As= 2.2 x 10-15
cm2
the absorption length is l=1/ ND Si:As = 15m.
Since l<w, a high quantum efficiency detector can be built.
Most of the failures in detector processing have to do with unwanted impurities,
and the improvement of the performance of this detector type is closely linked to
driving down NA.
Spitzer Si:As (BIB)
128x128 array

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