Solid state electronics

Jamia Millia Islamia
M.Tech Nanotechnology
2020-2022
Presentation on
Solid State Electronics
Submitted To: Submitted By:
PROF. S.S ISLAM SULTAN SAIFUDDIN

Content
• Introduction
• Formation Of Bond.
• Formation Of Band.
• Role Of Pauli Exclusion Principle.
• Fermi Dirac Distribution Equation
• Classification Of Material In Term Of Energy Band Diagram.
• Intrinsic Semiconductor.
a)Drive Density Of State
b)Drive Density Of Free Carrier.
c)Determination Of Fermi Level Position
o Extrinsic Semiconductor.
a) N Type Extrinsic Semiconductor
b) P Type Extrinsic Semiconductor
• Compensated semiconductor.
• E Vs. Diagram.
• Direct and Indirect Band Gap.
• Degenerated and Non-degenerated.
• PN Junction.

Solid-State Electronic Materials
• Electronic materials fall into three categories (WRT
resistivity):
– Insulators  > 105 -cm (diamond  = 1016 )
– Semiconductors 10-3 <  < 105 -cm
– Conductors  < 10-3 -cm (copper  = 10-6 )
• Elemental semiconductors are formed from a single type of
atom of column IV, typically Silicon.
• Compound semiconductors are formed from combinations
of elements of column III and V or columns II and VI.
• Germanium was used in many early devices.
• Silicon quickly replaced germanium due to its higher band
gap energy, lower cost, and ability to be easily oxidized to
form silicon-dioxide insulating layers.
NJIT ECE-271 Dr. S. Levkov Chap 2 - 3

Solid-State Electronic Materials (cont.)
• Band gap is an energy range in a solid where no electron
states can exist. It refers to the energy difference between
the top of the valence band and the bottom of the
conduction band in insulators and semiconductors.

NJIT ECE-271 Dr. S. Levkov Chap 2 - 5
Semiconductor Materials (cont.)
Semiconductor
Bandgap
Energy EG (eV)
Carbon (diamond) 5.47
Silicon 1.12
Germanium 0.66
Tin 0.082
Gallium arsenide 1.42
Gallium nitride 3.49
Indium phosphide 1.35
Boron nitride 7.50
Silicon carbide 3.26
Cadmium selenide 1.70

Covalent Bond Model
Silicon diamond lattice unit
cell.
Corner of diamond lattice
showing four nearest
neighbor bonding.
View of crystal
lattice along a crystallographic
axis.
• Silicon has four electrons in the outer shell.
• Single crystal material is formed by the covalent bonding of each
silicon atom with its four nearest neighbors.

Silicon Covalent Bond Model (cont.)
Silicon atom Silicon atom
Covalent bond

Silicon atom Covalent bonds in silicon

• Near absolute zero, all bonds are complete
• Each Si atom contributes one electron to each of
the four bond pairs
• The outer shell is full, no free electrons, silicon
crystal is an insulator
• What happens as the temperature
increases?

Energy Bands in Solids:
• According to Quantum Mechanical Laws, the energies of electrons in a
free atom can not have arbitrary values but only some definite
(quantized) values.
• However, if an atom belongs to a crystal, then the energy levels are
modified.
• This modification is not appreciable in the case of energy levels of
electrons in the inner shells (completely filled).
• But in the outermost shells, modification is appreciable because the
electrons are shared by many neighboring atoms.
• Due to influence of high electric field between the core of the atoms and
the shared electrons, energy levels are split-up or spread out forming
energy bands.
• Consider a single crystal of silicon having N atoms. Each atom can be
associated with a lattice site.
• Electronic configuration of Si is 1s2 , 2s2 , 2p6 ,3s2 , 3p2 . (Atomic No. is
14)

Role Of Pauli Exclusion Principle
The electronic system should obey Paulie’s exclusion principal , which states
that no 2 e- are in the system can have same amount of energy.
To obey this 3s and 3p level spilt into multiple levels so that each e- can
occupy a district energy level.
When inter atomic distance is further reduced. The 2N ‘s’ levels and 6N ‘p’
levels combined into a single band.
At lattice spacing, single band is split into 2 bands with upper band containing
4N and lower band with 4N levels.
At 00 k e- possess lowest energy , they fill up valence band and conduction
band remaining empty.

Formation of Energy Bands in Solids:

(i) r = Od (>> Oa):
Each of N atoms has its own energy levels. The energy levels are identical,
sharp, discrete and distinct.
The outer two sub-shells (3s and 3p of M shell or n = 3 shell) of silicon
atom contain two s electrons and two p electrons. So, there are 2N
electrons completely filling 2N possible s levels, all of which are at the
same energy.
Of the 6N possible p levels, only 2N are filled and all the filled p levels
have the same energy.
(ii) Oc < r < Od:
There is no visible splitting of energy levels but there develops a tendency
for the splitting of energy levels.
(iii) r = Oc:
The interaction between the outermost shell electrons of neighboring
silicon atoms becomes appreciable and the splitting of the energy levels
commences.
(iv) Ob < r < Oc:
The energy corresponding to the s and p levels of each atom gets slightly
changed. Corresponding to a single s level of an isolated atom, we get 2N
levels. Similarly, there are 6N levels for a single p level of an isolated
atom.

Since N is a very large number (≈ 1029 atoms / m3 ) and the energy of each
level is of a few eV, therefore, the levels due to the spreading are very
closely spaced. The spacing is ≈ 10-23 eV for a 1 cm3 crystal.
The collection of very closely spaced energy levels is called an energy band.
(v) r = Ob:
The energy gap disappears completely. 8N levels are distributed
continuously. We can only say that 4N levels are filled and 4N levels are
empty.
(vi) r = Oa:
The band of 4N filled energy levels is separated from the band of 4N
unfilled energy levels by an energy gap called forbidden gap or energy gap
or band gap.
The lower completely filled band (with valence electrons) is called the
valence band and the upper unfilled band is called the conduction band.
Note:
1. The exact energy band picture depends on the relative orientation of
atoms in a crystal.
2. If the bands in a solid are completely filled, the electrons are not
permitted to move about, because there are no vacant energy levels
available.

15
Fermi-Dirac distribution and the Fermi-level
Density of states tells us how many states exist at a given energy E. The Fermi
function f(E) specifies how many of the existing states at the energy E will be
filled with electrons. The function f(E) specifies, under equilibrium conditions,
the probability that an available state at an energy E will be occupied by an
electron. It is a probability distribution function.
EF = Fermi energy or Fermi level
k = Boltzmann constant = 1.38 1023 J/K
= 8.6  105 eV/K
T = absolute temperature in K

16
Fermi-Dirac distribution: Consider T  0 K
For E > EF :
For E < EF :
0
)
(
exp
1
1
)
( F 



 E
E
f
1
)
(
exp
1
1
)
( F 



 E
E
f
E
EF
0 1 f(E)

17
If E = EF then f(EF) = ½
If then
Thus the following approximation is valid:
i.e., most states at energies 3kT above EF are empty.
If then
Thus the following approximation is valid:
So, 1f(E) = Probability that a state is empty, decays to zero.
So, most states will be filled.
kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small in comparison.
kT
E
E 3
F 
 1
exp F 





 
kT
E
E





 


kT
E
E
E
f
)
(
exp
)
( F
kT
E
E 3
F 
 1
exp F






 
kT
E
E





 


kT
E
E
E
f F
exp
1
)
(
Fermi-Dirac distribution: Consider T > 0 K

18
Temperature dependence of Fermi-Dirac distribution

Classification of material in term of energy band picture

Metals:
• The highest energy level in the conduction
band occupied by electrons in a crystal, at
absolute 0 temperature, is called Fermi Level.
• The energy corresponding to this energy level
is called Fermi energy.
• If the electrons get enough energy to go
beyond this level, then conduction takes place.
The first possible energy band
diagram shows that the conduction
band is only partially filled with
electrons.
With a little extra energy the
electrons can easily reach the empty
energy levels above the filled ones and
the conduction is possible.
The second possible energy band
diagram shows that the conduction
band is overlapping with the valence
band.
This is because the lowest levels in
the conduction band needs less energy
than the highest levels in the valence
band.
The electrons in valence band
overflow into conduction band and
are free to move about in the crystal
for conduction.

(a) Energy levels in a Li atom are discrete.
(b) The energy levels corresponding to outer shells of isolated Li atoms form an energy
band inside the crystal, for example the 2s level forms a 2s band. Energy levels form a
quasi continuum of energy within the energy band. Various energy bands overlap to give
a single band of energies that is only partially full of electrons. There are states with
energies up to the vacuum level where the electron is free.
(c) A simplified energy band diagram and the photoelectric effect.
Energy Bands in Metals

(a) Above 0 K, due to thermal excitation, some of the electrons are at energies above EF.
(b) The density of states, g(E) vs. E in the band.
(c) The probability of occupancy of a state at an energy E is f(E). The product g(E)f(E)
is the number of electrons per unit energy per unit volume or electron concentration per
unit energy. The area under the curve with the energy axis is the concentration of
electrons in the band, n.








 


T
k
E
E
E
f
B
F
exp
1
1
)
(
2
/
1
2
/
1
3
2
/
3
)
2
(
4
)
( AE
E
h
m
E e 
 

g
Density of states Fermi-Dirac function
dE
E
f
E
n
F
E




0
)
(
)
(
g
3
/
2
2
3
8
















n
m
h
E
e
FO

Semiconductors:
As an electron leaves the valence band, it leaves
some energy level in band as unfilled.
Such unfilled regions are termed as ‘holes’ in
the valence band. They are mathematically
taken as positive charge carriers.
Any movement of this region is referred to a
positive hole moving from one position to
another.
At absolute zero temperature, no
electron has energy to jump from
valence band to conduction band and
hence the crystal is an insulator.
At room temperature, some valence
electrons gain energy more than the
energy gap and move to conduction
band to conduct even under the
influence of a weak electric field.
Its conductivity can be tuned or
tailored by addition of impurities.

(a) A simplified two dimensional view of a region of the Si crystal showing covalent
bonds.
(b) The energy band diagram of electrons in the Si crystal at absolute zero of
temperature. The bottom of the VB has been assigned a zero of energy.
Energy Bands in Semiconductors

(a) A photon with an energy hu greater than Eg can excite an electron from the VB to
the CB.
(b) Each line between Si-Si atoms is a valence electron in a bond. When a photon
breaks a Si-Si bond, a free electron and a hole in the Si-Si bond is created. The result is
the photogeneration of an electron and a hole pair (EHP)
Energy Bands in Semiconductors

Insulators:
Electrons, however heated,
can not practically jump to
conduction band from valence
band due to a large energy
gap. Therefore, conduction is
not possible in insulators.

Intrinsic or Pure Semiconductor:

Intrinsic Semiconductor is a pure semiconductor.
The energy gap in Si is 1.1 eV and in Ge is 0.74 eV.
Ge: 1s2 , 2s2 , 2p6 ,3s2 , 3p6 , 3d10, 4s2 , 4p2 . (Atomic No. is
32)
Si: 1s2 , 2s2 , 2p6 ,3s2 , 3p2 . (Atomic No. is 14)
In intrinsic semiconductor, the number of thermally generated
electrons always equals the number of holes.
So, if ni and pi are the concentration of electrons and holes
respectively, then
ni = pi .
The quantity ni or pi is referred to as the ‘intrinsic carrier
concentration’.

Electrons and Holes:
On receiving an additional energy, one of the electrons from a covalent band
breaks and is free to move in the crystal lattice.
While coming out of the covalent bond, it leaves behind a vacancy named
‘hole’.
An electron from the neighboring atom can break away and can come to the
place of the missing electron (or hole) completing the covalent bond and
creating a hole at another place.
The holes move randomly in a crystal lattice.
The completion of a bond may not be necessarily due to an electron from a
bond of a neighboring atom. The bond may be completed by a conduction
band electron. i.e., free electron and this is referred to as ‘electron – hole
recombination’.

A pictorial illustration of a hole in the valence band (VB) wandering around the crystal due to
the tunneling of electrons from neighboring bonds; and its eventual recombination with a
wandering electron in the conduction band. A missing electron in a bond represents a hole as in
(a). An electron in a neighboring bond can tunnel into this empty state and thereby cause the
hole to be displaced as in (a) to (d). The hole is able to wander around in the crystal as if it were
free but with a different effective mass than the electron. A wandering electron in the CB meets a
hole in the VB in (e), which results in the recombination and the filling of the empty VB state as
in (f)
Hole Motion in a Semiconductor

Fermi-Dirac Distribution for an Intrinsic Semiconductor
Intrinsic Semiconductor with a
Valence and Conduction Band
EV
EC
g
Band Gap Energy (E )
Eg ~2 eV
Here, there is a high likelihood of the
electrons remaining in the valence
band due to the large gap in energy
between the valence and conduction
bands. This can be written as:
E  EF  kT
f (E) 
f (E) 
This Allows for the Fermi-Dirac
Distribution to be Simplified
1
 
 
 
kT
1
exp(E  EF ) 1
kT
f (E)  exp(EF  E) 1
kT
exp(E  EF )
f(E) is the probability that a state at
energy E IS populated. 1 – f(E) is the
probability that a state at energy E IS
NOT populated.

Calculation of the Electron Density
Recall the Density of States Equation
EV
EC
Band Gap Energy (Eg)
Eg ~2 eV
Any electron in the conduction band
must have an energy:
2me
h2
k2
Ek  so
E  Ec  Ek and
Ek  E  Ec
E  Ec   Ec  Ek or
2me
h2
k2
Intrinsic Semiconductor with a Rearrangement of this equation yields:
If we assume that we are working over
a unit volume, substitution of these
two equations leaves an expression
density of states of the free electron
k
E1/2
 2m 
3/ 2
k
k
D(E ) 
dN V

 

dE 2 2
 h
2

Calculation of the Electron Density (Part II)
c
(E  E )1/2

3/ 2
e

1 2m
2 2
 h
2
e
D (E) 

dE
kT
n  F
c
e
c

EE


(E  E)
exp





 
E 


 
(E  E )
1  2m
The Density of Electrons in the Conduction Band Must Just be Equal to the
Density of States in the Conduction Band Multiplied by the Probability that an
Electron Can Occupy One of Those States Integrated Over All Energies
E
n  De (E) fe (E) dE
EEc
1/2

3/ 2
2
2
h
2
The Integral Can be Simplified for Terms That Are Not Functions of Energy

Calculation of the Electron Density (Part III)
dE
E
c
F
kT c
E

EE



 


 




    
  
n  
(E)
 kT 
exp
exp
1 2m
(E E )1/ 2

3/2
e
2
2
h
2
Integrating Over the Definite Integral Yields The Following



 

 

 
F c
kT
n  2e  exp
 m kT 
3/ 2
 (E  E ) 
2
2 h


 
Often, The Effective Density of States for the Conduction Band (Nc) is Defined:



 
2
2 h
 m kT 
3/ 2

Nc  2e  
 
kT
c
So, The Following Holds
 Ec )
n  N exp(EF

Absence of an Electron is Termed a Hole
Intrinsic Semiconductor with a
EV
EC
g
Band Gap Energy (E )
g
E ~2 eV
If there is an electron present in the
conduction band, that electron must
have been promoted from the valence
band and across the band gap.
This “missing” electron in the valence
band is termed a hole.
 1
1
fh (E) 1
fh (E) 1 fe (E)
 1
1
fe (E) 

exp

(E  E )

exp

(E  E )
kT
kT
F
F
The Probability of Not Finding an
Electron is the Same as the Probability
of Finding a Hole
If EF – E >> kTThen:



 kT
F
h
(E  E )
f (E) exp

Calculation of the Hole Density
E)1/ 2
(Ev

3/ 2
h

1 2m
2 2
 h
2
h
D (E)

dE
kT
p  F
v
h
E
E
EEv
 

(E  E )
(E  E)1/ 2
exp



3/ 2

1  2m
The Density of Holes in the Valence Band Must Just be Equal to the Density of
States in the Valence Band Multiplied by the Probability that a Hole Can
Occupy One of Those States Integrated Over All Energies
EEv
p  Dh (E) fh (E) dE
2 2
 2
Again, We Simplify the Integral for Terms That Are Not Functions of Energy

Calculation of the Hole Density (Part II)
dE
EEv

E
v
 E 
kT 
(E  E)1/ 2
exp
F
h

 

 
 E


   kT 
p   exp 
1  2m 
3/ 2
2
2
h
2
Integrating Over the Definite Integral Yields The Following



 m kT 
3/ 2


 

 
 
v F
kT
(E  E )
exp
p  2h 
2
2 h


 
Often, The Effective Density of States for the Conduction Band (Nc) is Defined:



 
2
2 h
 m kT 
3/ 2

Nv  2h  
 
kT
v
So, The Following Holds
 EF )
p  N exp(Ev

Solving in Terms of the Bandgap Energy
If We Take the Product of n and p, We Obtain the Following
v
c
e h
e h
g
kT
v c
kT
 g
v F
kT
F c
kT
kT



   
 kT





; E  E  E
(E )
 m m  exp
np  4
(E  E ) 
 m m  exp
np  4










 


 







 

(E  E )
 2h  exp
(E  E ) 
np  2e  exp

3/2

3
2
3/2

3
2 h
 2 h2

2
 m kT 
3/ 2
  m kT 
3/ 2

2
2 h
2 h




 kT g
e h
 2kT 
 
(E )
 m m  exp
i i
n  p  2 3/4

3/ 2
This Expression is Only a Function of Masses, Temperature, and the Bandgap.
Furthermore, Because in an Intrinsic Semiconductor, n = p
np  n2
 p2
and
i i
2
2 h

Fermi Energy of an Intrinsic Semiconductor
If We Set the Equation We Found for n Equal to the Equation Found Earlier for p,
We Obtain the Following
Rearrangement of This Lower Equation Shows That:






 
 

  

 

  



 










 



kT
kT
kT
kT
v F
F c (E  E )
 2h  exp
(E  E ) 
2e  exp
 m kT 
3/ 2

2 
 m kT 
3/ 2

v F
(E  E )
exp

p  2h 
 m kT 
3/ 2

2 
F c
n  2e  exp
 m kT 
3/ 2
 (E  E ) 
2 h
2
2 h
2 h
2
2 h







kT
 kT 
 (E  E ) 
2E
 exp
F
exp
c v

kT
 EF )
exp(Ev
kT
h 
 me 
(EF  Ec ) 
 m 
3/ 2 exp

Fermi Energy of an Intrinsic Semiconductor (Part II)
Simplification of the Previous Equation Yields:
kT 
E 
exp
g

 

 kT  kT
 me   me 
(E  E ) 
exp
c v
   h 
2E
exp
F
   h 
 m 
3/ 2
 m 
3/ 2
Rearrangement of This Equation Demonstrates That:
 e 
 mh 
EF 
2
Eg 
4
kTln
m

1 3
If mh = me, We Find That the Fermi
Energy is in the Middle of the
Bandgap for an Intrinsic
Semiconductor
2
1
EF  Eg EV
EC
Band Gap
Energy (Eg)
Eg ~2 eV
EF

Plots of D(E), f(E), n, and p as a Function of Energy
Density of States Fermi-Dirac
Distribution
Electron and Hole
Densities

Doping a Semiconductor:
• Doping is the process of
deliberate addition of a very
small amount of impurity into
an intrinsic semiconductor.
• The impurity atoms are called
‘dopants’.
• The semiconductor containing
impurity is known as ‘impure or
extrinsic semiconductor’.
Methods of doping:
• i) Heating the crystal in the
presence of dopant atoms.
• ii) Adding impurity atoms in the
molten state of semiconductor.
• iii) Bombarding semiconductor
by ions of impurity atoms.

Extrinsic or Impure Semiconductor:
N - Type Semiconductors:
When a semiconductor of Group IV
(tetra valent) such as Si or Ge is
doped with a penta valent impurity
(Group V elements such as P, As or
Sb), N – type semiconductor is
formed.
When germanium (Ge) is doped
with arsenic (As), the four valence
electrons of As form covalent bonds
with four Ge atoms and the fifth
electron of As atom is loosely bound.

The energy required to detach the fifth loosely bound electron is
only of the order of 0.045 eV for germanium.
A small amount of energy provided due to thermal agitation is
sufficient to detach this electron and it is ready to conduct current.
The force of attraction between this mobile electron and the
positively charged (+ 5) impurity ion is weakened by the dielectric
constant of the medium.
So, such electrons from impurity atoms will have energies slightly
less than the energies of the electrons in the conduction band.
Therefore, the energy state corresponding to the fifth electron is in
the forbidden gap and slightly below the lower level of the
conduction band.
This energy level is called ‘donor level’.
The impurity atom is called ‘donor’.
N – type semiconductor is called ‘donor – type semiconductor’.

Extrinsic Semiconductors: n-Type
a)The four valence electrons of
As allow it to bond just like Si
but the fifth electron is left
orbiting the As site. The
energy required to release to
free fifth-electron into the CB
is very small.
b) Energy band diagram for an
n-type Si doped with 1 ppm
As. There are donor energy
levels just below Ec around
As+ sites.

Carrier Concentration in N - Type Semiconductors:
When intrinsic semiconductor is doped with donor impurities, not only does the
number of electrons increase, but also the number of holes decreases below that
which would be available in the intrinsic semiconductor.
The number of holes decreases because the larger number of electrons present
causes the rate of recombination of electrons with holes to increase. Consequently,
in an N-type semiconductor, free electrons are the majority charge carriers and
holes are the minority charge carriers.
Carrier Concentration in N - Type Semiconductors: If n and p represent the electron
and hole concentrations respectively in N-type semiconductor, then
where ni and pi are the intrinsic carrier concentrations.
The rate of recombination of electrons and holes is proportional to n and p. Or, the
rate of recombination is proportional to the product np. Since the rate of
recombination is fixed at a given temperature, therefore, the product np must be a
constant.
When the concentration of electrons is increased above the intrinsic value by the
addition of donor impurities, the concentration of holes falls below its intrinsic
value, making the product np a constant, equal to ni2 .

Extrinsic Semiconductors: n-Type
e
d
h
d
i
e
d eN
N
n
e
eN 


 










2
Nd >> ni, then at room temperature, the
electron concentration in the CB will
nearly be equal to Nd, i.e. n ≈ Nd
A small fraction of the large number of
electrons in the CB recombine with holes
in the VB so as to maintain np = ni
2
n = Nd and p = ni
2/Nd
np = ni
2

P - Type Semiconductors:
• When a semiconductor of
Group IV (tetra valent) such as
Si or Ge is doped with a tri
valent impurity (Group III
elements such as In, B or Ga),
P – type semiconductor is
formed.
• When germanium (Ge) is
doped with indium (In), the
three valence electrons of In
form three covalent bonds with
three Ge atoms. The vacancy
that exists with the fourth
covalent bond with fourth Ge
atom constitutes a hole.

Extrinsic Semiconductors: p-Type
(a) Boron doped Si crystal. B
has only three valence
electrons. When it substitutes
for a Si atom one of its bonds
has an electron missing and
therefore a hole.
(b) Energy band diagram for a
p-type Si doped with 1 ppm
B. There are acceptor energy
levels just above Ev around B
sites. These acceptor levels
accept electrons from the VB
and therefore create holes in
the VB.

The hole which is deliberately created may be filled with an electron from
neighboring atom, creating a hole in that position from where the electron
jumped.
Therefore, the tri valent impurity atom is called ‘acceptor’. Since the hole is
associated with a positive charge moving from one position to another,
therefore, this type of semiconductor is called P – type semiconductor.
The acceptor impurity produces an energy level just above the valence band.
This energy level is called ‘acceptor level’.
The energy difference between the acceptor energy level and the top of the
valence band is much smaller than the band gap.
Electrons from the valence band can, therefore, easily move into the
acceptor level by being thermally agitated.
P – type semiconductor is called ‘acceptor – type semiconductor’.
In a P – type semiconductor, holes are the majority charge carriers and the
electrons are the minority charge carriers.
It can be shown that,

Extrinsic Semiconductors: P-Type
• Na >> ni, then at room
temperature, the hole
concentration in the VB
will nearly be equal to
Na, i.e. p ≈ Nd
• A small fraction of the
large number of holes in
the VB recombine with
electrons in the CB so
as to maintain np = ni
2
h
a
e
a
i
h
a eN
N
n
e
eN 


 










2
p = Na and n = ni
2/Na
np = ni
2

Intrinsic, i-Si
n = p = ni
Semiconductor energy band diagrams
n-type
n = Nd
p = ni
2/Nd
np = ni
2
p-type
p = Na
n = ni
2/Na
np = ni
2

Semiconductor energy band diagrams
Energy band diagrams for
• (a) intrinsic
• (b) n-type
• (c) p-type semiconductors.
• In all cases, np = ni
2. Note
that donor and acceptor
energy levels are not shown.
CB = Conduction band, VB =
Valence band, Ec = CB edge,
Ev = VB edge, EFi = Fermi
level in intrinsic
semiconductor, EFn = Fermi
level in n-type
semiconductor, EFp = Fermi
level in p-type
semiconductor. c is the
electron affinity. , n and
p are the work functions for
the intrinsic, n-type and p-
type semiconductors

Compensated
Semiconductor:
A compensated
semiconductor is
one that contains
both donor and
acceptor impurity
atoms in the same
region
Energy-band diagram of a
compensated semiconductor
showing ionized and un-
ionized donors and acceptors

Compensation Doping
Compensation doping describes the doping of a semiconductor with both donors and
acceptors to control the properties.
Example: A p-type semiconductor doped with Na acceptors can be converted to an n-
type semiconductor by simply adding donors until the concentration Nd exceeds Na.
The effect of donors compensates for the effect of acceptors.
The electron concentration n = Nd  Na > ni
When both acceptors and donors are present, electrons from donors recombine with the
holes from the acceptors so that the mass action law np = ni
2 is obeyed.
We cannot simultaneously increase the electron and hole concentrations because that
leads to an increase in the recombination rate which returns the electron and hole
concentrations to values that satisfy np = ni
2.
n = Nd  Na
p = ni
2/(Nd  Na)
Nd > Na

Summary of Compensation Doping
a
d N
N
n 

a
d
i
i
N
N
n
n
n
p



2
2
i
a
d n
N
N 

More donors than acceptors
More acceptors than donors i
d
a n
N
N 

d
a N
N
p 

d
a
i
i
N
N
n
p
n
n



2
2

Direct and Indirect band Gap Semiconductor

(a) In GaAs the minimum of the CB is directly above the maximum of the VB. GaAs is
therefore a direct band gap semiconductor.
(b) In Si, the minimum of the CB is displaced from the maximum of the VB and Si is an
indirect band gap semiconductor.
(c) Recombination of an electron and a hole in Si involves a recombination center .
E vs. k Diagrams and Direct and Indirect Bandgap
Semiconductors

Direct Band Gap Semiconductor
• For these Semiconductor ,
Conduction band minima
and valence band maxima
occurs at same value of
momentum.
• An e- from CB directly
return to VB without
changing It’s momentum.
And releases energy in the
form of light (photon hv).
• Ex: GaAS, Gap.GaAsP,

Indirect Band Gap Semiconductor
• Conduction band minima
and valance band maxima
occurs at different value
of momentum.
• When e- from CB returns
VB after changing its
momentum is called
indirect band gap sc.
• E- changes its momentum
by releasing phonon
which is a heat particle.
• Ex: Si, Ge

Degenerate and Non-degenerate
Semiconductor

Degenerate and Non-degenerate Semiconductor

Degenerate and Non-degenerate Semiconductor
• As the donor concentration further increases, the band of
donor states widens and may overlap the bottom of the
conduction band.
• This overlap occurs when the donor concentration becomes
comparable with the effective density of states.
• When the concentration of electrons in the conduction band
exceeds the density of states Nc , the Fermi energy lies
within the conduction band. This type of semiconductor is
called a degenerate n-type semiconductor.
• In the degenerate n-type semiconductor, the states between
Ef and Ec are mostly filled with electrons; thus, the electron
concentration in the conduction band is very large.

Degenerate Semiconductors
(a) Degenerate n-type semiconductor. Large number of donors form a band
that overlaps the CB. Ec is pushed down and EFn is within the CB.
(b) Degenerate p-type semiconductor
.

What is a PN Junction?
• A PN junction is a device formed by combining p-type ( doped with
B,Al) and n-type (doped with P,As,Sb) semiconductors together in
close contact.
• PN junction can basically work in two modes,
– forward bias mode (as shown below: positive terminal connected to p-
region and negative terminal connected to n region)
– reverse bias mode ( negative terminal connected to p-region and positive
terminal connected to n region)
PN junction device

Law of the Junction
Apply Boltzmann Statistics (can only be
used with non-degenerate
semiconductors)
N1 = ppo N2 = pno
E1 = PE2 = 0 E2 = PE1 = eVo





 


T
k
E
E
N
N
B
)
(
exp 1
2
1
2





 


T
k
eV
p
p
B
o
po
no )
0
(
exp










T
k
eV
p
p
B
o
po
no
exp

Law of the Junction
Apply Boltzmann Statistics










T
k
eV
p
p
B
o
po
no
exp










T
k
eV
n
n
B
o
no
po
exp

















 2
ln
ln
i
d
a
B
no
po
B
o
n
N
N
e
T
k
p
p
e
T
k
V

Oxford University Publishing
Microelectronic Circuits by
Adel S. Sedra and Kenneth C.
Figure: The pn junction with no applied voltage (open-circuited
terminals).
n-type semiconductor
filled with free electrons
p-type semiconductor
filled with holes
junction
Step #1: The p-type and n-type semiconductors are
joined at the junction.

terminals).
Step #2: Diffusion begins. Those free electrons and holes which are
closest to the junction will recombine and, essentially, eliminate one
another.

The depletion region is filled with “uncovered” bound charges – who
have lost the majority carriers to which they were linked.
Step #3: The depletion region begins to form – as diffusion occurs and
free electrons recombine with holes.
terminals).

Step #4: The “uncovered” bound charges effect a voltage differential
across the depletion region. The magnitude of this barrier voltage (V0)
differential grows, as diffusion continues.
voltage
potential
location (x)
barrier voltage
(Vo)
No voltage differential exists across regions of the pn-junction
outside of the depletion region because of the neutralizing effect of
positive and negative bound charges.

terminals).
Step #5: The barrier voltage (V0) is an electric field whose polarity
opposes the direction of diffusion current (ID). As the magnitude of V0
increases, the magnitude of ID decreases.
diffusion current
(ID)
drift current
(IS)

Step #6: Equilibrium is reached, and diffusion ceases, once the
magnitudes of diffusion and drift currents equal one another – resulting
in no net flow.
diffusion current
(ID)
drift current
(IS)
Once equilibrium is achieved, no net current flow exists (Inet = ID – IS)
within the pn-junction while under open-circuit condition.
p-type n-type
depletion
region

Ideal pn Junction
Depletion Widths
n
d
p
a W
N
W
N 

 )
(
net x
dx
d

E
Field (E) and net space charge density
Field in depletion region


x
Wp
dx
x
x )
(
1
)
( net


E
Acceptor concentration Donor concentration
Net space charge density
Permittivity of the medium
Electric Field

Ideal pn Junction
Built-in field

n
d
o
W
eN


E








 2
ln
i
d
a
o
n
N
N
e
kT
V
Built-in voltage
Depletion region width
 
2
/
1
2





 

d
a
o
d
a
o
N
eN
V
N
N
W

 = o r
where Wo = Wn+ Wp is the total width of the depletion region under a zero applied
voltage
ni is the intrinsic concentration

Working of a PN junction
.
Forward Bias
Reverse Bias
Zener or
Avalanche
Breakdown
Voltage
Current
I-V characteristic of
a PN junction diode.
•PN junction diode acts as a rectifier as seen in the IV characteristic.
•Certain current flows in forward bias mode.
•Negligible current flows in reverse bias mode until zener or
avalanche breakdown happens.

• Figure to right shows pn-
junction under three
conditions:
– (a) open-circuit – where
a barrier voltage V0
exists.
– (b) reverse bias – where
a dc voltage VR is
applied.
– (c) forward bias – where
a dc voltage VF is
applied.
Figure 3.11: The pn junction in:
(a) equilibrium; (b) reverse bias;
(c) forward bias.
1) no voltage
applied
2) voltage differential
across depletion zone
is V0
3) ID = IS
1) negative voltage
applied
across depletion zone
is V0 + VR
3) ID < IS
1) positive voltage
applied
across depletion zone is
V0 - VF
3) ID > IS

Forward biased pn junction and the injection of minority carriers (a) Carrier concentration
profiles across the device under forward bias. (b). The hole potential energy with and
without an applied bias. W is the width of the SCL with forward bias
Forward Biased pn Junction

Forward Bias: Apply Boltzmann Statistics
Note: pn(0) is the hole concentration just outside the depletion region on the n-side
N1 = ppo
N2 = pn(0)





 


T
k
E
E
N
N
B
)
(
exp 1
2
1
2























 


T
k
eV
T
k
eV
T
k
V
V
e
p
p
B
B
o
B
o
po
n
exp
exp
)
(
exp
)
0
(
pno/ppo









T
k
eV
p
p
B
no
n exp
)
0
(

Microelectronic Circuits by Adel S. Sedra and Kenneth C. Smith
(0195323033)
Forward-Bias Case
• Observe that decreased
barrier voltage will be
accompanied by…
– (1) decrease in stored
uncovered charge on
both sides of junction
– (2) smaller depletion
region
• Width of depletion
region shown to right.
0
0
p
p
p
p
p
p
0
width of depletion region
electrical permiability of silicon (11.7 1.04 12 )
magnitude of electron charge
con
replac
P
P
P
/
e
with
0
2 1 1
( )
A
F
S F c
V
W
q
m
S
n p F
A D
N
V V
W x x V V
q N N
 


  



 
    
 
  action:
E
p
p
p
p
p
p
0
p
p
centration of acceptor atoms
concentration of donor atoms
barrier / junction built-in voltage
externally applied forward-bias voltage
P
P
P
0
P
2 (
D
F
A D
J S F
A D
N
V
V
N N
Q A q V V
N N




 
 
 

 
0
p
p
0
magnitude of charge stored on either side of
rep
dep
lace
wit
letion regionP
h
)
J
F
V V
Q
V


action:

Microelectronic Circuits by Adel S. Sedra and Kenneth C. Smith
(0195323033)
Reverse-Bias Case
• Observe that
increased barrier
voltage will be
accompanied by…
– (1) increase in stored
uncovered charge on
both sides of junction
– (2) wider depletion
region
• Width of depletion
region shown to right.
p
p
p
0
p
0
0
width of depletion region
electrical permiability of silicon (11.7 1.04 12 )
magn
replace
with
itude of electron ch
/
0
arge
P
P
(eq3.31)
2 1 1
( )
S
R
F cm
S
n p R
V
V V
W
q
A D
W x x V V
q N N
 


  


 
    
 
  action:
E
p
p
p
p
p
p
p
0 p
p
p
concentration of acceptor atoms
concentration of donor atoms
barrier / junction built-in voltage
externally applied reverse-bias volta
P
P
P
g P
e
P
(eq3.3 2
2)
A
D
R
N
N
V
J
V
Q A





0
p
p
0
magnitude of charge store
0
d on either side of depletion re
replace
with
gi P
on
( )
J
R
V
V V
A D
S R
A
Q
D
N N
q V V
N N



 

 

  action:

Built in charge, electric field and potential at equilibrium.
Built in charge
Built in electric field
N-region
P-region
P-region N-region
Built in
potential,
Vbi= 0.834V

Built in charge, electric field and potential at forward bias
Decreasing charge with applied bias due to
thinning of depletion width.
P-region N-region
N-region
P-region
Decreasing electric field with applied bias due to
thinning of depletion width.
Potential
difference
Vbi-Va= 0.234V
Positive bias at P side reduces the
barrier leading to increase in diode
current.
Increased diffusion of
electrons across the
barrier lowered by Va.

Doping= 1e16 cm3
Doping= 1e18 cm3
• Increasing doping leads to increasing
built in potential, Vbi [1],[2].
Na : P region doping level (cm-3).
Nd : N region doping level (cm-3).
ni : Intrinsic carrier density (cm-3).
KbT : Thermal voltage (= 0.0259 V).
= V
bi
On changing doping for both n-type and p-type regions from 1e16 cm3
to 1e18 cm3.



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Solid state electronics

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