Transcript CHAPTER 1 Wave Nature of Light

CHAPTER 3
Semiconductor Science
and Light Emitting Diodes
1
The first transistor was demonstrated on Dec.
23, 1947, at Bell Labs by William Shockley.
This new invention consisting of P type and N
type semiconductive materials (in this case
germanium) has completely revolutionized
electronics.
William Shockley (seated), John Bardeen (left),
and Walter Brattain (right) invented the transistor
at Bell Labs and thereby ushered in a new era of
semiconductor devices. The three inventors
shared the Nobel prize in 1956
2
3.1
SEMICONDUCTOR CONCEPTS
AND ENERGY BANDS
3
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
A. Energy Band Diagrams
4
Energy of electron in an atom or a
molecule
• The energy of the electron in an atom is
quantized and can have only certain
discrete values.
• The same concept also applies to the
electron energy in a molecule with several
atoms.
5
Energy levels of the Li atom
• Li: 1s22s1
– Two electrons in the 1s shell
– One electron in the 2s subshell
6
Energy bands of Li metal crystal
• When we bring together 1023 Li atoms to form a metal
crystal, the interatomic interaction result in the formation
of electron energy band.
• 2s energy level splits into some 1023 closely spaced
energy levels  2s energy band
• Other higher energy levels also form bands.
7
Energy band structure of a metal
• The energy bands overlap to form one continuous
energy band that represents the energy bands structure
of a metal.
• The 2s energy level in the Li atom is half full
 The 2s band in the crystal will also be half full.
• Metals characteristically have partially filled energy
bands.
8
Electron energies in
semiconductor crystal
• Si: 1s22s22p63s23p2  4 valence electrons
• Si crystal:
Each Si atom is bonding to four neighbors.
The interactions between Si atoms and valence electrons results in
the electron energy falling into 2 bands  valence band (VB) and
conduction band (CB) that are separated by bandgap (Eg)
9
Bandgap
• There are no allowed electron
energies in the bandgap.
• It represents the forbidden
electron energies in the crystal.
10
Valence band (VB)
• Electron wavefunctions correspond to
bonds between the atoms.
• Electrons that occupy these
wavefunctions are the valence
electrons.
• At 0 K, all the energy levels in the VB
are normally filled with these
electrons.
11
Conduction band (CB)
• The CB represents electron
wavefunctions in the crystal that
have higher energies than those
in the VB.
• CB are normally empty at 0 K.
• An electron placed in the CB is
free to move around the crystal.
12
Band diagram at 0 K
the bottom of CB
= Ec  Ev
the top of VB
• Electron affinity : The energy difference between the vacuum level
and the bottom of the CB, i.e. the width of the CB.
13
Effective mass
• We can treat an electron in the CB as if it
were free with the crystal by simply

assigning an effective mass mc to it.
• Effective mass:
– A quantum mechanical quantity that take into
account the electron in the CB interacts with a
periodic potential energy as it moves through
the crystal.

 mc  me The mass of electron in vacuum
14
Creation of an electron-hole pair
• When an incident photon (h > Eg)
interacts with an electron in the VB
The electron absorbs the photon and
reach the CB.

charge :  e
 
free electron in the CB  "e " 

effective
mass
:
m
e



free hole in the VB  "h" charge :  e



effective mass : mh

The empty electronic state, or the missing electron, in the bond.
15
Creation of electron-hole pair
16
Thermal generation of electron-hole pair
• Due to thermal energy, the atom in the
crystal are constantly vibrating.
• Energetic vibrations can rupture bonds
and thereby create electron and hole pairs
(EHPs) by exciting electrons from the VB
to the CB.
17
Recombination
• When an electron in the CB meets a hole
in the VB, the electron falls from the CB to
the VB to fill the hole.
Recombination of EHP.
Annihilation of an electron from the CB
and a hole in the VB.
photon, e.g. in GaAs and InP
 Excess energy  
lattice vibration (heat), e.g. in Si and Ge
18
In steady state
• Thermal generation rate = Recombination
rate
Electron concentration n in CB = constant
Hole concentration p in VB = constant
• Both n and p depend on temperature:
n = n(T)
p = p(T)
19
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
B. Semiconductor Statistics
20
Two important concepts
• Density of states (DOS)
• Fermi-Dirac function
21
Density of states (DOS) g(E)
• The number of electronic
states (electron
wavefunctions) in a band per
unit energy per unit volume
of the crystal.
• For an electron confined
within a 3-D potential well
g(E)  (E  Ec)1/2
• DOS gives information only
available states and not on
their actual occupation.
22
Fermi-Dirac function f (E)
• The probability of finding an
electron in a quantum state
with energy E.
• It is a fundamental property
of a collection of interaction
electron in thermal
equilibrium.
•
1
 E  EF 
1  exp 

k
T
 B

k B : Boltzmann constant, T : temperature (K)
f (E) 
EF : Fermi energy (chemical potential)
23
Fermi energy
• If V is the potential difference between two points
 EF = eV
 A
VA
EFA
B
VB
EFB
 EF = e(VA VB)= eV
• For a semiconductor in the dark, in equilibrium,
and with no applied voltage or no emf generated
EF = 0
 EF must uniform across the system
24
f (E) and EF
1-f(E): the probability of
finding a hole
EF is located in the bandgap
f(E = EF) = 1/2
f (E) 
1
 E  EF 
1  exp 

k
T
 B 
f(E): the probability of
finding an electron
25
Electron concentration n in CB
• nE(E) = gCB(E) f (E) : the actual number of
electrons per unit energy per unit volume
in the CB.
• nE(E)dE = gCB(E) f (E)dE is the number of
electron in the energy range E to E+dE per
unit volume in the CB.
• Electron concentration in the CB
n
Ec  
Ec
nE ( E)dE  
Ec  
Ec
gCB ( E) f ( E)dE
26
Nondegenerate semiconductor
• Whenever (Ec – EF) >> kBT, i.e. EF is at least a
few kBT below Ec, then
f (E)  exp (E  EF ) / kBT 
Boltzmann probability function
• Such semiconductors are called nondegenerate.
• It implies that the number of electrons in the CB
is far less than the number states in this band.
27
Electron concentration in CB
• For nondegenerate semiconductor
 (Ec – EF) >> kBT
n
Ec  
Ec
 ( Ec  EF ) 
gCB ( E ) f ( E )dE  Nc exp  

k
T
B


 2 m k T 
in which Nc  2 

h



e B
2
3/2
Nc : Effective density of states at the CB edge
Nc is a temperature-dependent constant
28
Effective density of states at the CB edge
 ( Ec  EF ) 
n  Nc exp  

k
T
B


 2 m k T 
Nc  2 

h



e B
2
3/2
: Effective density of states at the CB edge
• If we take all the sates in the CB and replace
them with Nc (number of states per unit volume)
at Ec and then multiply this by the Boltzmann
probability function, we obtain the concentration
n in the CB.
29
Hole concentration in VB
• If (EF – EV) >> kBT
p
Ev
0
 ( EF  Ev ) 
gVB ( E )(1  f ( E ))dE  N v exp  

k
T
B


 2 m k T 
in which Nv  2 

h



h B
2
3/2
Nv : Effective density of states at the VB edge
Nv is a temperature-dependent constant
30

nE ( E)  gCB ( E) f ( E)
pE ( E)  gVB ( E)(1  f ( E))
31
Electron and hole concentrations
 ( Ec  EF ) 
 2 m k T 
• n  N c exp  
 , Nc  2 

k BT 
h




e B
2
3/2
 ( EF  Ev ) 
 2 m k T 
• p  N v exp  
 , Nv  2 

k BT 
h




h B
2
3/2
The location of EF determined the electron
and hole concentrations.
32
Fermi level in intrinsic crystal
• In an intrinsic semiconductor (a pure crystal)
n p
 ( Ec  EFi ) 
 ( EFi  Ev ) 
n  N c exp  
  p  N v exp  

k
T
k
T
B
B




 1

 1

Nc

 exp  
( EFi  Ev  Ec  EFi )   exp  
(2 EFi  2 Ev  Eg ) 
Nv
 k BT

 k BT

 Nc 
1
1
 EFi  Ev  Eg  k BT ln 
  Fermi level in
2
2
N
 v
intrinsic crystal
 Nc 
Typically, N c  N v  ln 
0
 Nv 
 EFi  Ev 
1
Eg  EFi is very approximately in the middle of the bandgap
2
33
Mass action law
 ( Ec  EF ) 
 ( EF  Ev ) 
np  N c exp  
  N v exp  

k BT 
k BT 


 Ec  Ev 
 Eg 
2
 N c N v exp  

N
N
exp


n

c v


i
k BT 

 k BT 
 temperature T
2
ni is a constant that depends on 
 energygap Eg
does not depend on Fermi level EF
• Intrinsic concentration in an undoped (pure)
crystal  ni = n = p
34
Average energy of electrons in CB
• The electron in the CB is “free” in the

crystal with an effective mass me
Average kinetic energy
1  2
3
me v  kBT
2
2
• Thermal velocity (root mean square
velocity)
v
2
5
10 m/s
35
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
C. Extrinsic Semiconductors
36
Extrinsic semiconductors
• Extrinsic semiconductor  Pure crystal +
small amount of impurities
• e.g.
As + Si  n-type semiconductor  n >> p
Arsenic: pentavalent impurities
B + Si  p-type semiconductor  p >> n
Boron: trivalent impurities
37
Arsenic doped Si crystal
• As+ ion + e-  hydrogenic
impurity
ionization energy ~ 0.05eV
• At room temp.
~ kBT = 0.025 eV
The fifth valence electron
can be readily freed by
thermal vibrations of Si
lattice
38
Energy diagram for n-type Si doped As
Ed : Donor level
Ec Ed ~ 0.05 eV
39
Donor impurity
• As atom : donates an electron into CB
donor impurity
• Nd : donor atom concentration
• If Nd >> ni
 n = Nd at room temp.
 p =ni2/n = ni2/Nd
 p << ni
40
Conductivity 
  ene  ep  h
e : drift mobility of electrons
h : drift mobility of holes
For n - type semiconductors
n  Nd , p  n / Nd
2
i
   eN d e  e(n / N d )  h
2
i
 eN d e (
n-type conductivity
N d  p  n / N d )
2
i
41
Arsenic doped Si crystal
• When B substitutes for a Si
atom one of its bonds has
an electron missing and
therefore a hole.
• Bonding energy of hole ~
0.05eV
• At room temperature, the
thermal vibrations of the
lattice can free the hole
away from the B site.
A free hole exits in the VB.
42
Energy diagram for p-type Si doped B
Ea : Acceptor level
Ea Ev ~ 0.05 eV
• There are acceptor energy levels just above Ev around B site.
• These acceptor levels accept electrons from the VB and therefore
create holes in the VB.
43
Acceptor impurity
• The B atom introduced into the Si crystals
acts as an electron acceptor impurity.
• Na : acceptor atom concentration
• If Na >> ni
 p  Na at room temp.
 n = ni2/p  ni2/Na << p
44
Conductivity 
  ene  ep  h
e : drift mobility of electrons
 h : drift mobility of holes
For p - type semiconductors
p  Na , n  n / Na
2
i
   e(n / N a ) e  eN a  h
2
i
 eN a  h (
p-type conductivity
N a  n  n / N a )
2
i
45
Energy band diagrams
• EFi: intrinsic, EFn:n-type, EFp: p-type
• The energy distance of EF from Ec and Ev determines the
electron and hole concentrations by
 ( Ec  EF ) 
 2 me k BT 
n  N c exp  
 , Nc  2 

2
k
T
h
B




3/2
 ( E  Ev ) 
 2 m k T 
p  N v exp   F
,
N

2

v


k
T
h
B





h B
2
3/2
46
n-type semiconductors
•
•
•
•
n >> p
Electrons: majority carriers
Holes: minority carriers
n  nn0
in equilibrium
n type
 nn0 = Nd
• p  pn0
• Mass action law nn0  pn0 = ni2
47
p-type semiconductors
•
•
•
•
p >> n
Electrons: minority carriers
Holes: majority carriers
n  np0 in equilibrium
p type
• p  pp0
 pp0 = Na
• Mass action law np0  pp0 = ni2
48
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
D. Compensation Doping
49
Compensation doping
• Compensation doping describes the doping of
a semiconductor with both donors and acceptors
to control the properties.
• A p-type semiconductor can be converted to an
n-type semiconductor by simply adding donors
until Nd > Na
• Electron concentration n  Nd  Na
• The effect of donors compensates for the effect
of acceptors and vice versa.
50
When both acceptors and donors
are present
• Electrons from donors recombine with the holes
from the acceptors so that the mass action law
np = ni2 is obeyed.
We can not simultaneously increase the electron
and hole concentrations because that leads to
an increase in the recombination rate.
• If we have more donors than acceptors
 n = Nd  Na
• If we have more acceptors than donors
 p = Na  Nd
51
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
E. Degenerate and Non-degenerate
Semiconductors
52
Non-degenerate semiconductors
• Number of states in CB >> Number of
electrons
Probability of 2 electrons occupying the
same state ~ 0
Pauli exclusion principle can be neglected
and electron statistics can be described by
the Boltzmann statistics
f (E)  exp (E  EF ) / kBT 
53
Non-degenerate semiconductors
• The Boltzmann express
 ( Ec  EF ) 
 2 m k T 
n  N c exp  
 , Nc  2 

k BT 
h




e B
2
3/2
for n is valid only when n << Nc
• Those semiconductors for which n << Nc
and p << Nv are termed non-degenerate
semiconductors.
54
Degenerate semiconductors
• When semiconductor has been excessively doped
with donors
n ~ 1019-1020 cm-3  n is comparable to Nc
• The Pauli exclusion principle becomes important
and we have to use the Fermi-Dirac statistics.
• Such a semiconductor are more metal-like, e.g.
resistivity  as T .
• Semiconductors that have n > Nc or p > Nv are
called degenerate semiconductors.
55
Degenerate semiconductors
• Heavy doping  Donor atoms become so close to each
other  Their orbitals overlap to form a narrow band,
which overlaps and becomes part of the conduction
band.
• Degenerate n-type semiconductor  EFn > Ec
• Degenerate p-type semiconductor  EFp < Ev
56
Degenerate semiconductors
• The dopant concentration is so large that
they interact with each other.
Not all dopants are ionized
Carrier concentration saturates ~ 1020 cm-3
 n  Nd or p  Na
• The mass action law np = ni2 is not valid
for degenerate semiconductors.
57
3.1 SEMICONDUCTOR CONCEPTS AND
ENERGY BANDS
F. Energy Band Diagram in an Applied
Field
58
Band diagram in an applied field
• A n-type semiconductor is
connected to a voltage
supply of V.
• The whole band structure
tilts because the electron
now has an electrostatic
potential energy as well.
• EF is not uniform throughout
the whole system.
• EF = EF(A) – EF(B) = eV
• CB, VB and EF all bend by
the same amount.
-
59
EXAMPLE 3.1.1 Fermi level in
semiconductors
 n-type Si, doped with 1016 cm-3
antimony (Sb) (donor) Nd = 1016cm-3
Calculate EFn – EFi @ 300 K
 When the wafer is further doped with
boron (B), Na = 21017 cm-3 > Nd =
1016cm-3  p-type
Calculate EFp – EFi @ 300 K
60
EXAMPLE 3.1.1 Fermi level in
semiconductors
Solution
 Nd = 1016cm-3  Nd >> ni (= 1.451010 cm-3)
 n = Nd = 1016cm-3
• For intrinsic semiconductor
ni  Nc exp (Ec  EFi ) / kBT 
•
For doped Si
n  Nc exp (Ec  EFn ) / kBT   Nd
61
EXAMPLE 3.1.1 Fermi level in
semiconductors
Solution
ni  N c exp  ( Ec  EFi ) / k BT 
n  N c exp  ( Ec  EFn ) / k BT   N d
Nd
 exp  ( EFn  EFi ) / k BT 
ni
Nd
1016
EFn  EFi  k BT ln( )  (0.259 eV) ln(
)
10
ni
1.45 10
 0.348 eV
62
EXAMPLE 3.1.1 Fermi level in
semiconductors
Solution
 Si doped with boron (B)
Na = 2  1017cm-3 > Nd = 1016 cm-3
 p = Na – Nd = 2  1017cm-3  1016 cm-3
= 1.9017 cm-3
• For intrinsic semiconductor
p  ni  Nc exp  ( EFi  Ev ) / kBT 
•
 1.45 1010 cm3
For doped Si
p  N v exp  ( EFp  Ev ) / k BT 
63
EXAMPLE 3.1.1 Fermi level in
semiconductors
Solution
p  ni  N v exp  ( EFi  Ev ) / k BT 
p  N v exp  ( EFp  Ev ) / k BT 
p
 exp  ( EFp  EFi ) / k BT 
ni
p
1.9 1017
EFp  EFi  k BT ln( )  (0.259 eV) ln(
)
10
ni
1.45 10
 0.424 eV
64
EXAMPLE 3.1.2 Conductivity
•
•
•
n-type Si, Nd = 1016cm-3 phosphorus (P)
atoms (donors)
e: drift mobility e = 1350 cm2V-1s-1
What is the conductivity?
65
EXAMPLE 3.1.2 Conductivity
Solution
Since N d  1016 cm 3  ni  1.45 1010 cm 3
 n  Nd
Conductivity   ene  ep  p  ene
2
ni
 N d )
(p 
Nd
  (1.6 1019 C)(11016 cm 3 )(1350 cm 2 V 1s 1 )
=2.161cm 1
66
3.2
DIRECT AND INDIRECT
BANDGAP SEMICONDUCTOR:
E-k DIAGRAMS
67
Infinite potential well
• When the electron is within an infinite potential energy
well, the wavevector kn is a quantum number
n
kn 
, n  1, 2,3,
L
• The electron momentum is kn
• Its energy is quantized
( kn ) 2
En 
2me
V(x)=
V(x)=
V(x)
Electron
 e
0
L
 En increases parabolically with kn
• This description can be used to represent the behavior
of electrons in a metal.
68
Nearly free electron model
• When electrons are in a metal, their average
potential energy can be taken very roughly zero.
• We take
within the metal crystal
0
V ( x)  
V0 (several eV)   outsid the metal crystal
so that the electron is contained within the metal.
• This is the nearly free electron model of a metal.
• This model is too simple since it does not take
into the actual variation of the electron potential
energy in the crystal.
69
Periodic potential energy in a crystal
• The E-k relationship will no longer be simply En = (kn)2/2me
70
Bloch wavefunctions
V ( x) : Periodic potential in crystal
V ( x)  V ( x  a )  V ( x  2a )   V ( x  ma), m  1, 2,
One has to solve the Schrodinger equation
d 2 2me
 2 [ E  V ( x)]  0
2
dx
Solutions are Bloch wavefunctions
 k ( x)  U k ( x) exp( jkx)
Bloch wave in the crystal
U k ( x  ma)  U k ( x) traveling wave
periodic function
The electron wavefunction in the crystal is a
traveling wave that is modulated by Uk(x).
71
E-k diagram
•  k ( x)  Uk ( x)exp( jkx)
• There are many such Bloch
wavefunction solutions to the
crystal.
• kn acts as a kind of quantum
number.
• Each k(x) corresponds a
particular kn and represents
a state with an energy Ek
The dependence of Ek on k
 E-k diagram
/a
/a
72
Crystal momentum
• k :crystal momentum
k is the momentum invloved in its
interaction with external fields, such as
those involved in photon absorption
processes.
d( k)
•
 Fext
dt
Fext: the externally applied force , such as
due to an electric field E  Fext = eE
73
E-k diagram
• E-k diagram Energy
vs. crystal momentum
plot.
• Lower E-k curve 
Valence band (VB).
• Upper E-k curve 
Conduction band (CB).
• All the valence electrons
at 0K fill the states in the
lower E-k diagram.
74
E-k diagram
• The E-k curve consists of
many discrete points with
each point corresponding
to a possible state k(x).
• In the energy range from Ev
to Ec, there are no
solutions to the
Schrödinger equation.
• E-k behavior is not a simple
parabolic relationship
except near the bottom of
the CB and the top of the
VB.
75
Direct bandgap semiconductor
• e.g. GaAs
• The minimum of the CB is
directly above the maximum of
the VB.
• Electron-hole pairs can
recombine directly and emit a
photon
• The majority of light-emitting
devices use direct bandgap
semiconductors to make use
of direct recombination.
76
Indirect bandgap semiconductor
• e.g. Si, Ge
• The minimum of the CB is
not directly above the
maximum of the VB, but it is
displaced on the k-axis.
• An electron at the bottom of
the CB cannot recombine
directly with a hole at the top
of the VB because it is not
allowed by the law of
conservation of momentum.
77
Indirect bandgap semiconductor
with a recombination center
• Recombination center
– Crystal defects
– Impurities
• The electron is first captured by the
defect at Er.
• The change in the energy and
momentum of the electron by this
capture process is transferred to
lattice vibrations, i.e., to phonons.
• The captured electron at Er can
readily fall down into an empty state
at the top of the VB and thereby
recombine with a hole.
• The electron transition from Ec to Ev
involves the emission of further
lattice vibration.
78
GaP doped nitrogen impurities
• GaP: indirect bandgap
semiconductor
• Nitrogen (N) impurities
 Recombination center
• The recombination of the
electron with a hole at the
recombination centers results
in photon emission.
• The electron transition from Ec
to Ev involves photon emission.
79
3.3
pn JUNCTION PRINCIPLES
80
3.3 pn JUNCTION PRINCIPLES
A. Open Circuit
81
pn junction
• One side is n-type and the other is p-type.
• There is an abrupt discontinuity between the p and n
region.  Metallurgical junction, M
• n region:
– fixed (immobile) ionized donors
– free electrons (in CB)
• p region :
– fixed ionized acceptors
– holes (in VB)
82
Hole concentration gradient
• p = pp0 (p-side) > p = pn0 (n-side)
Holes diffuse towards the right and enter the n-region
and recombine with the electrons (majority carriers) in
this region.
The n-side near the junction becomes depleted of
majority carriers and has exposed positive donors of
concentration Nd.
83
Electron concentration gradient
• n = nn0 (n-side) > n = np0 (p-side)
Electrons diffuse towards the left and enter the p-region
and recombine with the holes (majority carriers)
The p-side near the junction becomes depleted of
majority carriers and has exposed negative acceptors of
concentration Na.
84
Space charge layer
• The regions on both sides of M becomes
depleted of free carriers in comparison with the
bulk regions far away from the junction.
• There is a space charge layer (SCL) around M.
• Also known as the depletion region around M.
85
Electron and hole concentration profiles
logarithmic scale
• Note: under equilibrium condition (e.g., no applied bias ,
or photoexcitation) pn = ni2 everywhere.
86
Internal electric field E0
• There is an internal electric field E0 in the SCL.
• E0 tries to drift the holes back into the p-region and
electrons back into the n-region.
• This field drives the carriers in the opposite direction to
their diffusion.
• In equilibrium
– Hole diffusion flux  =  Hole drift flux
– Electron drift flux  =  Electron diffusion flux
87
Net space charge density net(x)
• For uniformly doped p and
n regions
 eN a , Wp  x  0
net ( x)  
eN d , 0  x  Wn
• For overall charge neutrality
NaWp  NdWn
88
Depletion widths
• NaWp = NdWn
• If Na > Nd  Wp < Wn
Depletion width of heavily
doped side < Depletion width
of lightly doped side
• If Na >> Nd  Wp << Wn
 The depletion region is
almost entirely on the lightly
doped side.
• We generally indicate heavy
doped regions with the
superscript plus sign as p+.
89
Built-in field
Gauss's law
 net (r )  net (r )
dE  net ( x)
  E (r ) 




 0 r
dx

 ( x ')
 E ( x)   net
dx '

 eN a , W p  x  0
 net ( x)  
eN d , 0  x  Wn
eN a x
 x (eN a )

 Wp  dx ',
E0   , W p  x  0
 E ( x)  

0
x
(eN a )
(eN d )
eN d x


dx '  
dx ',
E0 
,
0  x  Wn
0
 Wp 




eN aW p
eN dWn  Bulit-in field
where E 0  E (0)  



90
Built-in potential
x
dV
E 
 V ( x)    E ( x ')dx '
O
dx
eN a x

E0   , W p  x  0
E ( x)  
 E  eN d x ,
0  x  Wn
0


Wn
 V0  V (Wn )    E ( x ')dx '
O
eN a N dWo 2
1
V0   E0W0 
2
2 ( N a  N d )
in which W0  Wn  W p
Bulit-in potential
If we know V0, we can calculate W0.
91
Boltzmann statistics
• If carrier concentration in the CB << Nc
we can use Boltzmann statistics
Effective density of states
n(E)  exp(E / kBT )
at the CB edge
• The concentrations n1 and n2 of carriers at
potential energies E1 and E2 are related by
 ( E2  E1 ) 
n2
 exp 

n1
k
T
B


in which E = qV is the potential energy (PE).
92
Potential energy of carriers
• Electrons
– q = -e
– PE = 0 on the p-side far
away from M where n = np0
– PE = -eV0 on the n-side far
away from M where n = nn0.
• Holes
– q=e
– PE = 0 on the p-side far
away from M where p = pp0
– PE = eV0 on the n-side far
away from M where p = pn0.
93
Built-in potential
nn 0
k BT
n p 0 / nn 0  exp(eV0 / k BT )  V0 
ln( )
e
np0
pn 0 / p p 0
p p0
k BT
 exp(eV0 / k BT )  V0 
ln(
)
e
pn 0
p p 0  N a , pn 0  ni2 / nn 0  ni2 / N d
k BT  N a N d 
 V0 
ln  2 
e
 ni 
Built-in potential
V0 is related to the dopant and
host material properties via Na,
Nd, and ni2 [= NcNv exp(-Eg/kBT)]
94
Built-in potential
•
kBT  N a N d 
V0 
ln  2 
e
 ni 
 Eg 
where n  Nc Nv exp  

k
T
 B 
2
i
• V0 is the potential across the pn
junction, going from p to n-type
semiconductor, in an open circuit.
• Once we know V0 from above Eq.,
we can then calculate the width
W0 of the depletion region form
eNa NdWo 2
V0 
2 ( Na  Nd )
95
3.3 pn JUNCTION PRINCIPLES
B. Forward Bias
96
Forward Bias
• A battery with a voltage V is connected across a pn
junction:
– Positive terminal  p-side.
– Negative terminal  n-side.
The applied voltage drops mostly across the depletion
width W.
V directly opposes V0 and the potential barrier against
diffusion is reduced to (V0-V).
97
Injection of excess minority carriers
• The applied voltage effectively reduces the guilt-in
potential and the built-in field that acts against diffusion.
• Excess holes can diffuse across SCL and enter the nside.
• Excess electrons can diffuse across SCL and enter the
p-side.
 Injection of excess minority carriers
98
Small increase in the majority carriers
• When holes are injected into the neutral n-side, they draw
some electrons from the bulk of n-side (and hence from the
battery).
• There is a small increase in the electron concentration in the
n-side.
• Similarly, there is a small increase in the hole concentration
in the p-side.
99
Hole concentration just outside SCL
• pn(0) = pn (x = 0): the hole concentration just outside the
depletion region.
• pn(0) is determined by the probability of surmounting the
new potential energy barrier e(V0V),
pn (0)  pp0 exp[e(V0 V ) / kBT ]
100
Law of the junction
• Holes are injected from the p-side to the n-side
pn (0)  p p 0 exp[e(V0  V ) / k BT ]
pn 0 / p p 0  exp(eV0 / k BT )
 pn (0)  pn 0 exp(eV / k BT )
• Electrons are injected from the n-side to the p-side
np (0)  np0 exp(eV / kBT )
Law of the junction
where np(0) is the electron concentration just outside the
depletion region at x = Wp
101
Current due to diffusion of minority carriers
• The current due to holes (electrons)
diffusing in the n-region (p-region) can be
sustained because more holes (electrons)
can be supplied by the p-region (n-region),
which can be replenished by the positive
(negative) terminal of battery.
An electric current can be maintained
through a pn junction under forward bias,
and that the current flow seems to be due
to the diffusion of minority carriers.
102
Excess minority carrier concentration
• The hole concentration pn(x) profile on the nside falls exponentially toward to the thermal
equilibrium value.
pn ( x ')  pn ( x ')  pn 0  pn (0) exp( x '/ Lh )
pn ( x ') : excess minority carrier concentration
Lh : hole diffusion length
Lh : Dh h
Dh : diffusion coefficient of holes
 h : mean hole recombination lifetime
(minority carrier lifetime) in the n - region
The average distance diffused by a minority carrier before it disappears by recombination.
103
Hole diffusion current density
Hole diffusion flux
pn ( x ')  pn ( x ')  pn0  pn (0)exp( x '/ Lh )
dpn ( x ')
d pn ( x ')
J D ,hole ( x ')   Dh
 (e)  eDh
dx '
dx '
 eDh 
 x' 
Hole diffusion current

 pn (0) exp   
depends on location
L
L
 h 
 h
104
Total current
• In n-region
– Jhole  Minority carrier diffusion current
– Jelec  Majority carrier drift current
• In p-region
– Jhole  Majority carrier drift current
– Jelec  Minority carrier diffusion current
The field in the neutral
region is not totally zero
but a small value, just
sufficient to drift the huge
number of majority carriers.
• J = Jelec + Jhole= constant
105
Hole diffusion current density
 eDh 
 x' 
J D ,hole ( x ')  
 pn (0) exp   
 Lh 
 Lh 
 eV 
pn (0)  pn 0 exp 
  Law of the junction
 k BT 
pn 0  ni2 / nn 0  ni2 / N d
 ni2  

 eV  
 exp 
  pno  
  1

 k BT  
 Nd  
 eDh 
 eDh ni2  
 eV   Hole diffusion

 pn (0)  
 exp 
  1 current
 k BT  
 Lh 
 Lh N d  
 eV
pn (0)  pn (0)  pno  pn 0 exp 
 k BT
 J D ,hole
x ' 0
Just outside the depletion region
106
Total current density
• We assume that the electron and hole currents do not
change across the depletion region because, in general,
the width of this region is narrow.
 eDh ni2  
 eV  
J D ,hole x W or W  J D ,hole x '0  
exp 
 1



n
p
 kBT  
 Lh N d  
 eDe ni2  
 eV  
J D ,elec x W or W  
 exp 
  1
n
p
 kBT  
 Le N a  
• The total current density is
 eDh
 eV
eDe  2 
J  J D ,hole  J D,elec  

 ni exp 
 k BT
 Lh N d Le N a  

 eV
 J  J so exp 
 kBT

 
  1
 
 
  1
 
Schockley diode equation
107
J D ,hole
 eDh ni2  
 eV

 exp 
 k BT
 Lh N d  
 
  1
 
J D ,elec
 eDe ni2  
 eV

 exp 
 k BT
 Le N a  
 eDh
eDe
J 

 Lh N d Le N a
 2
 eV
 ni exp 
 k BT
 
 
  1
 
 
  1
 
108
Shockley equation

 eV
• J  J so exp 
 k BT

 eDh
 
eDe

  1 , where J so  
 
 Lh N d Le N a
 2
 ni

• The constant Jso depends Na, Nd, ni, Dh, De, Lh and
Le.
• If we apply a reverse bias V = Vr > kBT/e (= 25
mV)

 eVr
J  J so exp 
 k BT

 
  1   J SO
 
 JSO is known as the reverse saturation current
density.
109
Recombination in SCL
• Under forward bias, the minority carriers
diffusing and recombining in the neutral
regions are supplied by external current.
• However, some of the minority carriers will
recombine in the depletion region.
The external current must therefore also
supply the carriers lost in the
recombination process in the SCL.
110
Recombination current
•
•
•
•
•
•
pM = hole concentration at M
nM = electron concentration at M
A symmetrical pn junction
pM = nM (due to symmetry)
area ABC  electrons recombining in p-side
area BCD  holes recombining in n-side
Recombination current
J recom 
eABC
e

eBCD
h
1
1
e W p nM e Wn pM
 2
 2
e
h
e : mean electron recombination time in Wp
h : mean hole recombination time in Wn
111
Recombination current
• PE = 0 at A, PE = e(V0V)/2 at M
 ( PE M  PE A ) 
 e(V0  V ) 
pM

 exp  

exp



p p0
k
T
2
k
T
B
B




k BT  N a N d 
V0 
ln  2  , p p 0  N a
e
 ni 
 e(V0  V ) 
 eV
 pM  p p 0 exp  
  ni exp 
2 k BT 

 2 k BT
J recom

  nM

1
1
e W p nM e Wn pM
 eV 
eni  W p Wn 
2
2






 exp 
e
h
2  e h 
2
k
T
 B 
• From a better quantitative analysis
J recom  J r 0 exp(eV / 2kBT ) 1
112
Total current
• The total current into the diode will supply
carriers for minority carrier diffusion in the
neutral regions and recombination in SCL.

 eV
J  J so exp 
 k BT

J recom
 
  1
 

 eV
 J r 0 exp 
 2 k BT


 eV
 I  I 0 exp 
  k BT

 
  1
 
 
  1
 
• : the diode ideality factor
•  is 1 for diffusion controlled and 2 for SCL
recombination controlled characterisitics.
113
3.3 pn JUNCTION PRINCIPLES
C. Reverse Bias
114
Reverse Bias
• Positive terminal  n-side
 Electrons in n-side move away from SCL
• Negative terminal  p-side
 Holes in p-side move away from SCL
Wider SCL
• The movement of electrons
in the n-region towards the
positive terminal cannot be
sustained because there is
no electron supply to this nside.
115
Reverse Bias
• Built-in potential  V0+Vr
• Electric field in SCL  E0+E
• Small number of holes on the n-side near the SCL become
extracted and swept by the field across the SCL over to the
p-side.
• This small current can be
maintained by the diffusion
of holes from the n-side
bulk to the SCL boundary.
116
Minority carrier diffusion current
• pn(0)  pn0 exp(eVr/kBT)  law of the junction
• If Vr > 25 mV = kBT/e  pn(0)  0 < pn0
Small concentration gradient
Small hole diffusion current toward SCL
• Similarly, there is a small
electron diffusion current
from bulk p-side to SCL.
• Within the SCL, these
carriers are drifted by the
field.
117
Reverse current

 eVr
• J  J so exp 
 k BT

 
  1  Shockley equation
 
• If Vr >> KBT/e = 25 mV
 J  Jso  Reverse saturation current density
 eDh
eDe

• J so  
 Lh N d Le N a
 2
 ni

Dh = hkBT/e, De = ekBT/e
Lh = (Dhh)1/2, Le = (Dee)1/2
Jso depends only on the material via ni, h,e, the
dopant concentrations, etc., but not on the voltage.
 Jso depends ni2, and hence is strongly temperature
dependent.
118
EHP thermal generation in SCL
• The thermal generation of EHPS in the SCL can
also contribute to the observed current since the
internal field will separate the electron and hole
and drift them toward the neutral region.
119
EHP thermal generation in SCL
• g: the mean time to generate an EHP by thermal
vibrations of lattice (mean thermal generation time)
• ni/g :the rate of thermal generation per unit volume
• A : the cross-sectional area of the depletion region
• WA: the volume of the depletion region
The reverse current density component due to thermal
generation of EHPs within SCL:
J gen
ni
1 eWni
 e AW 
g
A
g
120
The total reverse current density
• The total reverse current density is the sum of
the diffusion and generation component:
J rev  J diffusion  J gen
 eDh
eDe


 Lh N d Le N a
 2 eWni
 ni 
g

• Jgen increases with Vr because the SCL width W
increase with Vr.
121
The total reverse current density
• J rev  J diffusion  J gen
 eDh
eDe


 Lh N d Le N a
 2 eWni
 ni 
g

The reverse current are predominantly
controlled by ni2 and ni.
• Since ni ~ exp(Eg /2kBT), the reverse
current depends on the temperature.
122
Reverse current
J rev
ni
 eDh
eDe
 J diffusion  J gen  

 Lh N d Le N a
exp( Eg / 2k BT )
 2 eWni
 ni 
g

 I rev  Aexp( Eg / k BT )  Bexp( Eg / 2k BT )
Eg 1
 ln  I rev   ln( A) 
if I rev is controlled by ni2
kB T
Eg 1
 ln  I rev   ln( B) 
if I rev is controlled by ni
2k B T
123
Reverse current Irev in a Ge pn junction
• T > 238 K
slope of ln(Irev) vs. 1/T 
0.63 eV  0.66 eV (Eg of Ge)
Irev is controlled by ni2
• T < 238 K
slope of ln(Irev) vs. 1/T 
0.33 eV (Eg/2 of Ge)
 Irev is controlled by ni
124
3.3 pn JUNCTION PRINCIPLES
D. Depletion Layer Capacitance
125
Stored charge in depletion region
• The depletion region of a pn junction has
positive and negative charges separated over a
distance W similar to a parallel plate capacitor.
• Storged charge:
+Q  +eNdWnA on n-side
Q  eNaWpA on p-side
• Unlike in the case of a
parallel plate capacitor, Q
does not depend linearly on
the voltage V across the
device.
126
Depletion layer capacitance
• We can define an incremental capacitance:
When V changes by dV to V+dV, then W
also changes and the Q becomes Q+dQ.
The depletion layer capacitance Cdep is
defined by
Cdep
dQ

dV
127
Depletion layer width
• If the applied voltage is V, the voltage
across the depletion layer W is V0 V
eN a N dW 2
V0  V 
2 ( N a  N d )
Built-in potential
 2 ( N a  N d ) V0  V  
W  

eN a N d


1/2
128
Depletion layer capacitance
Q  eN dWn A  eN aW p A
Wn  Q / (eN d A), W p  Q / (eN a A)
 2 ( N a  N d ) V0  V  
W 

eN
N
a d


1/2
Q 1
1
Q Na  Nd
 Wn  W p 
(

)
eA N a N d
eA N a N d
eAN a N d  2 ( N a  N d ) V0  V  
Q


( Na  Nd ) 
eN a N d

1/2
Cdep
eAN a N d 1  2 ( N a  N d ) V0  V  
dQ




dV ( N a  N d ) 2 
eN a N d

1/2
Cdep


eN a N d
  A

2

(
N

N
)
V

V
 

a
d  0

1/2
2 ( N a  N d )
eN a N d
A
W
129
Depletion layer capacitance

1/2

eN a N d

 2 ( N a  N d ) V0  V  
• Cdep   A 

A
W
 Cdep is given by the same expression as that for
the parallel plate capacitor, A/W, but with W
being voltage-dependent.
• For reverse bias, V = Vr
1/2
Cdep


eN a N d
  A

 2 ( N a  N d ) V0  Vr  
Cdep decreases with increasing Vr
– Typically, Cdep~ pF
130
3.3 pn JUNCTION PRINCIPLES
E. Recombination Lifetime
131
Excess carrier injection
• Consider recombination in a direct
bandgap semiconductor.
• Suppose that excess electrons and hole
have been injected:
pp : the excess hole concentration in the
neutral p-side.
np : the excess electrons concentration in
the neutral p-side.
pp = np for charge neutrality
132
Net rate of change of np
• At any instant:
n p  n po  n p  instantaneous minority carrier concentration
p p  p po  n p  instantaneous majority carrier concentration
• Instantaneous recombination rate  nppp
• Thermal generation rate of EHPs is Gthermal.
• The net rate of change of np is
n p
t
  Bn p p p  Gthermal
in which B is called the direct recombination
capture coefficient.

133
Net rate of change of np
• In equilibrium
n p
t
0
• Using np = np0, pp = pp0, we find
n p
  Bn p p p  Gthermal   Bn p 0 p p 0  Gthermal  0
t
 Gthermal  Bn p 0 p p 0
The rate of change of np is
n p
t
 B  np p p  np0 p p0 
Rate of change
due to recombination
134
Recombination time e
• In many instance
n p
 n p
t
• Excess minority carrier recombination
time (lifetime) e is defined by
n p
t

n p
e
135
Weak injection
• In practical cases: np >> np0
Actual equilibrium minority
carrier concentration
• Weak injection: np << pp0
np  np , pp  pp0 + pp  pp0  Na

n p
t
  B  n p p p  n p 0 p p 0    B  n p N a  n p 0 N a    BN a n p
Comparing with
n p
t

n p
e
  e  1/ BN a  constant
Weak injection
recombination lifetime

136
Strong injection
• Strong injection: np >> pp0
n p
t
  B  n p p p  n p 0 p p 0    B  n p p p  n p 0 p p 0 
  Bn p ( p p 0  p p )   Bn p p p   B  n p 
2
Comparing with
n p
n p

e
t
1
1

 e 
Bn p n p
 Under high level injection conditions the lifetime e is
inversely proportional to the injected carrier concentration.
137
EXAMPLE 3.3.1 A direct bandgap
pn junction
• A symmetrical GaAs pn junction:
– A = 1 mm2 (cross sectional area)
– Na (p-side doping) = Nd (n-side doping) = 1023 m-3
– B = 7.2110-16 m3s-1 (direct recombination capture
coefficient)
– ni = 1.8 1012 m-3
– r = 13.2
– h (in the n-side) = 250 cm2V-1s-1 (drift mobility)
– e (in the p-side) = 5000 cm2V-1s-1 (drift mobility)
– Forward voltage across the diode = 1V

138
EXAMPLE 3.3.1 A direct bandgap
pn junction
• What is the diode current due to the minority
carrier diffusion at 300 K assuming direct
recombination?
• If the mean minority carrier recombination
time in the depletion region is of the order of
~ 10 ns, estimate the recombination
component of the current.
139
EXAMPLE 3.3.1 A direct bandgap
pn junction
Solution
• Assuming weak injection

1
1
h  e 

BN a (7.211016 m3s 1 )(11023 m3 )
 1.39 108 s
140
EXAMPLE 3.3.1 A direct bandgap
pn junction
Solution
• Einstein relation:
Diffusion coefficients are
Dh  h k BT / e, De  e k BT / e
 Dh  h k BT / e  (0.2585)(250 10 )
4
 6.46 104 m 2s -1
 De  e k BT / e =(0.2585)(5000 104 )
 1.29 10
2
2 -1
ms
141
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
• The diffusion length
Lh  ( Dh h )1/2  [(6.46 104 m 2s -1 )(1.39 108 s)]1/2
 3.00 106 m
Le  ( De e )1/2  [(1.29 102 m 2s-1 )(1.39 108 s)]1/2
 1.34 105 m
142
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
• The diffusion length
Lh  ( Dh h )1/2  [(6.46 104 m 2s -1 )(1.39 108 s)]1/2
 3.00 106 m
Le  ( De e )1/2  [(1.29 102 m 2s-1 )(1.39 108 s)]1/2
 1.34 105 m
143
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
• The reverse saturation current due to diffusion
in the neutral regions is
IS 0
 Dh
De  2
 A

 eni
 Lh N d Le N a 
4
2


6.46

10
1.29

10
6
19
12 2
 (10 ) 

(1.6

10
)(1.8

10
)
6
23
5
23 
 (3.00 10 )(10 ) (1.34 10 )(10 ) 
 6.13 1021 A
144
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
• The forward diffusion current
I diff  I so exp(eV / K BT )  (6.13 10
27
 1.0 V 
A) exp 

0.0258
V


 3.9 104 A
•
The built-in voltage V0
 10231023 
Na Nd
K BT
V0 
ln( 2 )  (0.0258) ln 
 1.28 eV
12 2 
e
ni
 (1.8 10 ) 
145
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
•
The depletion layer width
 2 ( N a  N d ) V0  V  
W 

eN
N
a d


1/2
1/2
 2(13.2)(8.85 10 Fm )(10  10 )m (1.28  1)V 


19
23
3
23
3
(1.6

10
C
)(10
m

10
m
)


 9.0 108 m or 0.090 m
12
-1
23
23
3
146
EXAMPLE 3.3.1 A direct
bandgap pn junction
Solution
•
For a symmetric diode, Wp = Wn = W/2, and taking e = h = r  10
ns
Aeni  Wp Wn  Aeni  W 
I r0 

 


2  e h 
2  e 
•
(106 )(1.6 1019 )(1.8 1012 )  9.0 108 
12


1.3

10
A


9
2
 10 10 
So that
I recom
 eV 


1.0 V
12
 I r 0 exp 
  (1.3 10 A)exp 

2
k
T
2(0.02585)
V


 B 
 3.3 104 A
The diffusion and recombination components are about the same
order.
147
3.4
THE pn JUNCTIONN BAND
DIAGRAM
148
3.4 THE pn JUNCTION BAND DIAGRAM
A. Open Circuit
149
Energy diagram of a pn junction under
open circuit
• EFp : Fermi level in the p-side
• EFn : Fermi level in the n-side
• In equilibrium and in the dark  EFp = EFn
150
Energy diagram of a pn junction under
open circuit
• Ec on the n-side is close to EFn whereas on the p-side it is far away
from EFp.
• EFp = EFn through the whole system.
 We have to bend the bands Ec and Ev near the junction at M.
151
Energy diagram of a pn junction under
open circuit
• PE of the electron: 0 (p-region)  eV0 (n-region)
• The electron in the n-side at Ec must overcome a PE barrier to go to
Ec at the p-region
• PE barrier = eV0 (V0 : built-in potential)
152
3.4 THE pn JUNCTION BAND DIAGRAM
B. Forward and Reverse Bias
153
Energy diagram of a pn junction under
forward bias
• Applied voltage V   Built in potential V0.
• PE barrier: eV0  e(V0  V)
154
Energy diagram of a pn junction under
forward bias
• The electrons in the nside can overcome the
PE barrier and diffuse to
the p-side.
• The diffusing electrons
can be replenished by the
negative terminal.
• Similarly, holes can
diffuse from p- to n-side.
The positive terminal can
replenish those holes.
 A current flow through the junction and around the
circuit.
155
Energy diagram of a pn junction under
forward bias
• p  exp [e(V0V)/kBT]
 the probability that electron in the n-side overcomes
the PE potential and diffuses to the p-side.
 Forward current  exp [e(V0V)/kBT]  exp (V/kBT)
156
Energy diagram of a pn junction under
reverse bias
• When reverse bias:
V= Vr .
• PE barrier:
eV0  e(V0 + Vr)
• Field in SCL:
E0  E0 + E
There is hardly any reverse current because if
an electron were to leave the n-side to travel
to the positive terminal, it can not be
replenished from the p-side.
157
Thermal generations
• Thermal generation of EHPs in the SCL
– The field separates the pairs
– Electrons  n-side
– Hole  p-side
• Thermal generation of minority carriers within a
diffusion length to the SCL
– A thermally generated hole in the n-side can diffuse to
the SCL and then drift cross the SCL
– A thermally generated electron in the p-side can
diffuse to the SCL and then drift cross the SCL
Small reverse current
158
Thermal generation of EHPs in SCL
• When EHPs are thermally generated in the SCL, the field
separates the pairs
Electrons fall down the PE hill  n-side
Holes fall down the PE hill  p-side
Small reverse current
159
3.5
LIGHT EMITTING DIODES
160
3.5 LIGHT EMITTING DIODES
A. Principles
161
Light Emitting Diodes
• A light emitting diode (LED) is a pn
junction diode typically made from a direct
bandgap semiconductor in which the
electron hole pair recombination results in
the emission of a photon.
• Emitted photon energy
h  Eg
Bandgap energy
162
Band diagram of a pn+ junction device
• pn+ junction device:
– The n side is more
heavily doped than the
p-side.
– The depletion region
in a pn+ device
extends mainly into
the p-side.
(a) Without any bias:
• Uniform Fermi level
• PE barrier = eV0
 Built-in potential V0 prevents electrons from diffusing
from n+ to p side.
163
Band diagram of a pn+ junction device
(b) Forward bias
• Built in potential
V0 V0 – V
It allows the
electrons from the
n+ side to diffuse or
become injected
into the p-side.
=
• The recombination of injected electrons in the
depletion region as well as in the neutral p-side
results in the spontaneous emission of photons.
164
Active region
• Recombination primary occurs within:
The depletion region
A volume extending over the diffusion
length Le of the electrons in the p-side
• The recombination zone is called the
active region.
165
Injection electroluminescence
• The phenomenon of light emission from
EHP recombination as a result of minority
carrier injection.
• Spontaneous emission process
The emitted photons are in random
direction.
166
3.5 LIGHT EMITTING DIODES
B. Device Structure
167
Planar surface emitting LED devices

(a) Epitaxial layers: first n+-layer and then the p-layer.
(b) p-side is formed by diffusing dopants into the epitaxial
n+-layer.
• Narrow p-side (~ few m)  Photons can escape without
being absorbed.
• Heavily doped n-side (n+)  Most of recombination take
place in the p-side.
• Segmented back electrode  Encourages reflection
from the semiconductor-air interface.
168
Lattice mismatch
• Lattice mismatch:
– The epitaxial layer and the substrate crystal
have different lattice parameter
Lattice strain in the LED layer
Crystal defect
Recombination centers
Radiationless EHP recombinations
• It is important to lattice-match the LED
layer to the substrate crystal.
169
Lattice match system
• AlGaAs alloys
– direct bandgap
– bandgap  red-emission region
• It can be grown on GaAs substrates with
excellent lattice match.
High efficiency LED devices.
170
TIR in LED
• Not all light rays reaching the semiconductor-air interface
can escape because of total internal reflection (TIR)
• E.g. GaAs-air interface, c  16
 Much of the light suffers TIR
171
Domed surface
• Domed or hemisphere surface of semiconductor
 i < c  No TIR
• Main drawback:
– additional difficult process
– Increase in expense
172
Plastic dome for LED
• The encapsulation of the pn junction within a transparent
plastic medium (an epoxy) that has a higher refractive
index.
 More light escape from the LED
• Many individual LEDs are sold in similar types of plastic
bodies.
173
3.6
LED MATERIALS
174
Photon energy, wavelength and color
E  h 

hc


(4.14 1015 eV  s)  (2.9979  1017 nm/s)

1240 eV  nm

1240 eV  nm
  (nm) 
E (eV)
175
LED materials from VIS to IR
y = 0.45
y=0
y=1
x=0
176
GaAs1y Py
•
•
•
•
Ga : group III, As, P: group V
GaAs1y Py  III-V ternary alloy
Eg (GaAs1y Py )  as y 
y < 0.45
– direct bandgap
– EHP recombination is direct
– 630 nm <  < 870 nm  y = 0, GaAs
 y = 0.45, GaAs0.55P0.45
• y > 0.45
– indirect bandgap
– EHP recombination occur through
recombination centers and involve lattice
vibrations rather than photon emission.
177
N doped indirect GaAs1-yPy (y > 0.45)
• N and P atoms
– same group V  same valency
– different electric cores
 The positive nucleus of N is less shielded by
electrons compared with that of the P atom.
• If some N atoms are added into GaAsP crystal
– N atoms are isoelectric impurities
– N atoms substitute for P atoms
• same number of bonds
• do not act as donors or acceptors
A conduction electron near an N atom will be attracted
and may become trapped at this site.
N atoms introduce localized energy levels, or
electron trap EN near the conduction band edge.
178
N doped indirect GaAs1-yPy (y > 0.45)
• N atom  electron trap
• Trapped electron at EN can
attract a hole by Coulomb
attraction.
• Direct recombination between
a trapped electron and a hole
emits a photon.
• h  EN– Ec < Eg
 Inexpensive green, yellow, and orange LEDs
• Efficiency < Efficiency (direct bandgap
semiconductors)
179
UV and blue LED materials
• InxGa1-xN alloy
–
–
–
–
Direct bandgap
Eg  as x
GaN: Eg = 3.4 eV  UV emission
InGaN: Eg = 2.7 eV  blue emission
• Al doped silicon carbide (SiC)
– Indirect bandgap
– EHP recombination is through an acceptor level Ea
• ZnSe (II-VI compound)
– Direct bandgap
– Drawback: difficulty in appropriately doping to
fabricaye efficient pn junction
180
AlxGa1xAs
• Al, Ga : group III
• As: group V
• AlxGa1xAs III-V ternary alloy
– Direct bandgap for x < 0.45
– Eg  as x 
– GaAs: Eg = 1.43 eV
  = 870 nm  IR emission
– AlxGa1-xAs (x < 0.45) : Eg = 1.941.43 eV
  = 640  870 nm  deep red to IR emission
181
In0.49Ga0.51-xAlxP
• Al, Ga, In : group III
• P: group V
• InGaAlP III-V quarternary alloy
– Direct bandgap
– Lattice-matched to GaAs substrates when in the
composition range In0.49Ga0.34Al0.17P to
In0.49Ga0.452Al0.058P
–  = 590  630 nm
 Amber, orange, red LEDs
– Material for high-intensity visible LEDs
182
In1-xGaxAs1-yPy
• Ga, In : group III
• P, As: group V
• In1-xGaxAs1-yPy III-V quarternary alloy
– Direct bandgap
–  = 870nm (GaAs)  3.5 m (InAs)
includes the optical communication
wavelength of 1.3 and 1.55 m
183
External quantum efficiency external
• ext  The efficiency of conversion of electrical
energy into an emitted external optical energy.
• It incorporates the “internal” efficiency of
radiative recombination process and the
subsequent efficiency of photon extration from
the device.
• external
Pout (Optical)

100%
IV
The input of electrical power into an LED
• Indirect bandgap LEDs: external < 1%
• Direct bandgap LEDs: higher external
184
LED materials
orange,
185
3.7
HETEROJUNCTION HIGH
INTENSITY LEDs
186
Homo and heterojunctions
• Homojunction:
A junction (such as a pn junction) between two
differently doped semiconductors that are of the
same material (same bandgap Eg)
• Heterojunction:
A junction between two different bandgap
semiconductors
• Heterojunction device (HD):
A semiconductor device structure that has
junctions between different bandgap materials
187
Refractive index and bandgap
• The refractive index n of semiconductor
material depends on its bandgap Eg:
n  as Eg 
• By constructing LEDs from heterostructures,
we can engineer a dielectric waveguide
within the device and thereby channel
photons out from the recombination region.
188
Drawbacks of homojunction LEDs
The p-region must be narrow to allow the
photons to escape without much
reabsorption.
 When the p-side is narrow, some of the
injected electrons in the p-side reach the
surface by diffusion and recombine
through crystal defects near the surface.
 Radiationless recombination process
decreases the light output.


189
Drawbacks of homojunction LEDs
• If the recombination occurs over a relative
large volume (or distance), due to long
electron diffusion lengths, then the
chances of reabsorption of emitted
photons becomes higher.
The amount of reabsorption increases with
the material volume.
190
Double heterostructure (DH) device
• DH device based on two junctions between
different semiconductor materials with different
bandgaps.
• Different materials
– Eg (AlGaAs)  2 eV
– Eg (GaAs)  1.4 eV
• Two heterjunctions
– n+ p heterojunction between n+-AlGaAS and p-GaAs
– p p heterojunction between p-GaAS and p-AlGaAs
• p-GaAS thin layer
– a fraction of m
– lightly doped
191
Band diagram with zero bias
• No bias  Fermi level EF = constant
• eV0 : PE barrier between n+-AlGaAS and p-GaAs
• Ec = effective PE barrier between p-GaAS and p-AlGaAs
Bandgap change between p-GaAs and p-AlGaAs
192
Forward biased band diagram
• PE barrier eV0 eV0-V
• Electrons in the CB of n+AlGaAs are injected (by
diffusion) into p-GaAs
• These electrons are
confined to the CB of pGaAs by the barrier Ec
• Wide gap AlGaAs layers
act as confining layers.
193
Double heterojunction LED
• EHP recombination presents in the
p-GaAs layer
• Eg (AlGaAs) > Eg (GaAs)
 The emitted photons do not get
reabsorbed as they escape the
active region and can reach the
surface of the device.
• Since light is also not absorbed in
p-AlGaAs, it can be reflected to
increase the light output.
• DH LED is much more efficient
than homojunction LED
194
3.7
LED CHARACTERISTIC
195
Energy of an emitted photon
• The energy of an emitted
photon from an LED is
not simply equal to Eg
because electrons in the
CB are distributed in
energy and so are the
holes in the VB.
196
Energy distribution of carriers
• Electron concentration
in CB as a function of
energy:
nE(E) = gCB(E)f(E)
A peak at (1/2) kBT
above Ec
Energy spread ~ 2KBT
• The hole concentration
is similarly spread from
Ev in VB
nE ( E)  gCB ( E) f ( E)
pE ( E)  gVB ( E)(1  f ( E))
197
• Transition 2 (E = h2)
– has the maximum probability as both electron and hole
concentrations are largest at these energy.
Relative intensity is maximum, or close maximum
• Idealized spectrum
– Energy for the peak emission ~ Eg+KBT
– Linewidth (h) ~ 2.5 KBT to 3 KBT
198
Output spectrum from an LED
• The output spectrum from an LED
depends not only on the semiconductor
material but also on the structure of the on
junction diode, including the dopant
concentration level.
199
Effect of Heavy doping
• Electron wavefunctions at the donors
overlap to generate a narrow impurity
band overlaps the CB and effectively
lowers Ec
Minimum emitted photon energy < Eg
200
Typical characteristics of a red
GaAsP LED
• Linewidth  = 24 nm  2.7 kBT
• LED current   injected minority carrier concentration 
 rate of recombination   output light intensity 
• Turn-on voltage ~ 1.5 V from which point the current
increases sharply with voltage.
201
Turn-on voltage
• The turn-on voltage depends on the
semiconductor and generally increases
with the energy bandgap Eg
• Blue LED  Vturn-on~3.5-4.5 V
• Yellow LED  Vturn-on~ 2 V
• GaAs IR LED  Vturn-on~ 1 V
202
EXAMPLE 3.8.1 LED Output Spectrum
• Typical width of the relative intensity vs.
photon energy spectrum ~ 3kBT
• Linewidth 1/2 = ?
203
EXAMPLE 3.8.1 LED Output Spectrum
Solution
•   c / v  hc / E ph  d    hc2
dE ph
E ph

d
hc
hc

   2 E ph 
E ph
2
E ph dE ph
E ph
(hc /  )
   
2
E ph
hc
We are given E ph  ( h )  3k BT
  
2
E ph
3k BT

hc
hc
2
204
EXAMPLE 3.8.1 LED Output Spectrum
Solution
•   
2
E ph
3k BT

hc
hc
2
23
J/K)(300 K)
10

3(1.38
2
  870 nm    (870 nm)
(6.626 1034 J  s)(3 1017 nm/s)
 (870 nm) 2 (6.24 105 nm 1 )  47.2 nm
  1300 nm    (1300 nm) 2 (6.24 10 5 nm 1 )  105.4 nm
  1550 nm    (1550 nm) 2 (6.24 10 5 nm 1 )  149.9 nm
•
These linewidths are typical values and the exact
values depend on the LED structure.
205
EXAMPLE 3.8.2 LED Output
wavelength variations
•
•
•
•
Consider a GaAs LED.
Eg = 1.42 eV @ 300 KT
dEg/dT = 4.5  104 eVK-1
What is the change in the emitted
wavelength if the temperature change is
10C ?
206
EXAMPLE 3.8.2 LED Output
wavelength variations
Solution
•   c / v  hc / Eg
d
hc  dEg 

 2

dT
Eg  dT 
(6.626 1034 )(3 108 )
4
19

(

4.5

10

1.6

10
)
19 2
(1.42 1.6 10 )
 2.77 1010 m K 1  0.277 nm K 1
  (d  / dT )T  (0.277 nm K 1 )(10 K )  2.8 nm
•
Eg decrease with T,  increases with T.
207
EXAMPLE 3.8.3 InGaAsP on InP
substrate
• Quarternary alloy In1-xGaxAsyP1-y grown on InP
substrate
Commercial semiconductor material for IR LED
and laser diode.
• In1-xGaxAsyP1-y is lattice matched to InP if y  2.2 x
• Eg (In1-xGaxAsyP1-y) = 1.350.72y+0.12y2, 0 x  0.47
• Calculate the compositions of InGaAsP
quarternary alloy for peak emission at   1.3m.
208
EXAMPLE 3.8.3 InGaAsP on InP substrate
Solution
• E ph  hc  Eg  k BT

hc k BT
 Eg 

(in eV)
e
e
  1.3 106  m, T  300 K
(6.626 1034 )(3 108 )
Eg 
 0.0259 eV  0.982 eV
19
6
(1.6 10 )(1.3 10 )
0.982 eV  1.35  0.72 y  0.12 y 2  y  0.66
y  2.2 x  x  y / 2.2  0.66 / 2.2  0.3
The quarternary alloy is In0.7Ga0.3As0.66P0.34
209
3.9
LEDS FOR OPTICAL FIBER
COMMUNICATIONS
210
Light sources for optical
communications
• LEDs
–
–
–
–
simpler to drive
more economic
have a longer lifetime
provide the necessary output power
Short haul application (e.g. local networks)
• Laser diodes
– narrow linewidth
– high output power
– higher signal bandwidth capability
Long-haul and wide bandwidth communications
211
Two types of LED devices
• Surface emitting LED (SLED)
– The emitted radiation emerges from an area in the plane of the
recombination layer
• Edge emitting LED (ELED)
– The emitted radiation emerges from an area on an edge of the
crystal, i.e. from an area on a crystal perpendicular to the active
layer.
• ELEDs provide a greater intensity light and also a beam
that is more collimated than the SLEDs.
212
Light coupling for SLEDs
213
Burrus type device
• The method is to etch a well in the planar LED structure
and lower the fiber into the well as close as possible to
the active region.
• An epoxy resin is used to bond the fiber and provide
refractive index matching between the fiber and the LED
material.
214
A microlens focuses light into fiber
• The method is to use a truncated spherical lens
(a microlens) with a high refractive index (n = 1.9
-2) to focus the light into the fiber.
215
Structure of a edge emitting LED
• The light is guided to the edge of the crystal by a
dielectric waveguide formed by wider bandgap
semiconductors surrounding a double heterostructure.
216
Light coupling for ELEDs
• Light from an edge emitting LED is coupled into a fiber
typically by using a hemispherical lens or a GRIN rod
lens.
• A GRIN rod lens (typical diameter ~ 0.5-2 mm) can be
used to focus the light into a fiber. This coupling is
particular useful for single mode fibers because their
core diameters are typically ~ 10 m.
217
• GRIN rod lenses and a spherical lens (a ball lens) used
in coupling light into fibers.
218