FERMI LEVEL AND EFFECT OF
TEMPERATURE ON SC
FERMI LEVEL AND EFFECT OF
TEMPERATURE ON INTRINSIC SC
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At absolute zero all the electronic states of the
valence band are full and those of conduction band
are empty
Classically all electrons have zero energy at 00K (i.e.,
practically insulator. When temp is increased then
electrons jump from VB to CB) But
Quantum Mechanically all electrons are not having
zero energy at 00K
The maximum energy that electrons may posses at
00k is the Fermi energy EF
Quantum mechanically electrons actually have
energies extending from 0 to EF at 00K
Electron Energy
Conduction Band
EF
00K
Valence Band
For intrinsic semiconductors like silicon and germanium,
the Fermi level is essentially halfway between the valence
and conduction bands
FERMI LEVEL
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Throughout nature, particles seek to occupy the lowest
energy state possible. Therefore electrons in a solid will
tend to fill the lowest energy states first. Electrons fill up
the available states like water filling a bucket, from the
bottom up. At T=0 , every low-energy state is occupied,
right up to the Fermi level, but no states are filled at
energies greater than EF .
"Fermi level" is the term used to describe the top of the
collection of electron energy levels at absolute zero
temperature
At absolute zero electrons pack into the lowest available
energy states and build up a "Fermi sea" of electron
energy states. The Fermi level is the surface of that sea
at absolute zero where no electrons will have enough
energy to rise above the surface.
Illustration of the Fermi function for zero temperature. All electrons are
stacked neatly below the Fermi level.
FERMI ENERGY
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The Fermi energy is a concept in quantum
mechanics usually referring to the energy of the
highest occupied quantum state in a system of
fermions at absolute zero temperature.
The Fermi level is an energy that pertains to
electrons in a semiconductor. It is the chemical
potential μ that appears in the electrons' FermiDirac distribution function
The Fermi-Dirac distribution, also called the
"Fermi function," is a fundamental equation
expressing the behaviour of mobile charges in
solid materials
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FERMI FACTOR OR FERMI FUNCTION
Although no conduction occurs at 0 K, at higher
temperatures a finite number of electrons can
reach the conduction band and provide some
current
The increase in conductivity with temperature
can be modelled in terms of the Fermi function,
which allows one to calculate the population of
the conduction band
Fermi factor tells us how many of the energy
states in the VB and CB will be occupied at
different temperatures OR we can say that
The Fermi function tells us the probability that a
state is occupied
FERMI FACTOR or Fermi function
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FERMI FACTOR or Fermi function is the
number that expresses the probability
that the state of a given energy (E) is
occupied by an electron under the
condition of thermal equilibrium. This
number has a value between zero and
unity and is a function of temperature and
energy
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The probability that the particle will have an
energy E is
F (E ) 
1
E EF
1 e

KT
At absolute zero, the probability is equal to 1 for
energies less than the Fermi energy and zero for
energies greater than the Fermi energy. We
picture all the levels up to the Fermi energy as
filled, but no particle has a greater energy
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Fermi factor is independent of the energy
density of states, it is the probability that the
states occupied at that level, irrespective of the
number of states actually present i.e., the
occupancy of possible states
Case-I When T=00K , then for E  E F
F (E ) 
and for
F (E ) 
1
1 e

0
E  EF
1
1 e

1
Case-II When t=T0K , then at E=EF
1
1
F (E ) 

0
1 e
2
This means that when the temperature is not
00K but some higher value say T=10000K, then
some covalent bonds will be broken and some
electrons will be available in CB
The Fermi energy level , EF , is the energy at
which the probability of occupancy is exactly 1/2
for temperatures greater than zero
 This is similar to a bucket of hot water. Most of
the water molecules stick around the bottom of
the bucket. The Fermi level is like the water line.
A fraction of water molecules are excited and
drift above the water line as vapour, just as
electrons can sometimes drift above the Fermi
level.
Illustration of the Fermi function for temperatures
above zero. Some electrons drift above the Fermi
level. Their density at higher energies is
proportional to the Fermi function.
In a semiconductor, not every energy level is
allowed. For example, there are no allowed
states within the forbidden gap
The density of electrons in a semiconductor, showing
how the Fermi function is modulated by the density of
allowed states (which is zero inside the forbidden gap).
• In a solid with numerous atoms, a large number of states
appear at energy levels very close to each other. A crystal
weighing 1mg contains 1019 atoms. If valence band is sband then there are 2x1019 levels. Suppose the width of
energy band is 2 eV then 2x1019 levels are spread over an
energy band width of 2 eV. Hence spacing between
different levels = 2/ 2x1019 = 10-19 eV
• We approximate these states as a continuous "band" and
imagine that an "energy level" is a vanishingly small
energy interval of width dE
• An energy level may contain several sublevels, all with
the same energy. Each sublevel is called a "state," and can
be occupied by exactly one electron.
• In continuous-band theory we represent S(E) as a density
of available states.
•The density of states complements the Fermi
function by telling us how many states actually
exist in a particular material
• The density of available states , S(E) , is the
fraction of all allowed states that lie within E
and E+dE . This is a density function
• We can multiply S(E) and F(E) together,
resulting in units of electrons per energy level
per unit volume
•Suppose there are N(E) occupied states at
energy E . Then the probability of finding an
occupied state at energy E is
S(E)×F(E)
POSITION OF THE FERMI LEVEL
IN AN INTRINSIC SC
Let
The available number of states = S(E)
Probability of their occupancy = F(E)
Then the total number of occupied states N(E) by
electrons in conduction band with energy
between E and E+dE
N ( E ) dE  S ( E ) F ( E ) dE
In a solid semiconductor at thermal equilibrium, every
mobile electron leaves behind a hole in the valence band.
Since holes are also mobile, we need to account for the
density of "hole states" that remain in the valence band.
Because a hole is an unoccupied state, the probability of
a mobile hole existing at energy E is 1−F(E)
P ( E ) dE  S ( E ) 1  F ( E ) dE
ENERGY LEVEL IN C.B = E1
ENERGY LEVEL IN V.B = E2
THEN
N ( E 1 ) dE  S ( E 1 ) F ( E 1 ) dE
AND
P ( E 2 ) dE  S ( E 2 ) 1  F ( E 2 ) dE
IN CASE OF INTRINSIC SC: S ( E 1 )  S ( E 2 )
N ( E 1 ) dE
P ( E 2 ) dE

S ( E 1 ) F ( E 1 ) dE
S ( E 1 ) 1  F ( E 2 ) dE
N ( E1 )

P(E2 )
F ( E1 )
1  F ( E 2 ) 
AT E1 = 3000K
F  E1   e
  E1  E F
KT
AND
1  F E 2   e
E2  EF 
KT
ALSO
N ( E1 )
P(E2 )
THEREFORE
EF 


ni
1
pi
E1  E 2
2
•This equation shows that the Fermi level lies at the centre
of the forbidden gap for intrinsic semiconductor and it is
independent of the temperature
The density of mobile electrons is shown in the conduction band. The
corresponding density of mobile holes is shown in the valence band
FERMI LEVEL IN EXTRINSIC SC
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In intrinsic SC the number of electrons is equal
to the number of holes (ni=pi)
Fermi level is a measure of the probability of
occupancy of the allowed energy states by the
electrons, so when ni=pi Fermi level is at the
centre of the forbidden gap
Now in n-type SC, number of electrons ne>ni
and number of holes pe<pi
This means ne>pe , hence the Fermi level must
move upward closer to the conduction band
For p-type SC, pe>ne so Fermi level must move
downward from the center of the forbidden gap
closer to the valence band
VARIATION OF FERMI LEVEL WITH TEMP
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For intrinsic SC (ni=pi) and as the temperature increases
both ni and pi will increase
Fermi level will remain approximately at the center of
the forbidden gap
This means Fermi level is independent of the
temperature
But in extrinsic SC it is different
In n-type SC electrons come from two source
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From donor state- which are easily separated from parent atom
and do not vary much as the temperature is increased
Intrinsically produced electrons- which increases with
increase in temperature
This shows that as the temperature rises the
material becomes more and more intrinsic and
Fermi level moves down closer to intrinsic position
(at the center of the forbidden gap)
VARIATION OF FERMI LEVEL WITH TEMP
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Similarly for p-type SC as the temperature rises
the material becomes more and more intrinsic
and Fermi level moves up closer to intrinsic
position (at the center of the forbidden gap)
Thus both n- and p-type SC become more and
more intrinsic at high temperature
This puts a limit on the operating temperature
of an extrinsic SC
P-N JUNCTION
P-N JUNCTION
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Single piece of SC material with half n-tpye and
half p-type
The plane dividing the two zones is called
junction (plane lies where density of donors and
acceptors is equal)
P
+ + +
+ + +
+ + +
Junction
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N
-
-
Three phenomenas take place at the junction
 Depletion layer
 Barrier potential
 Diffusion capacitance
P-N JUNCTION
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Formation of depletion layer
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Also called Transition region
Both sides of the junction
Depleted of free charge carriers
Density gradient across junction (due to greater
difference in number of electrons and holes)-Results
in carrier diffusion-diffusion of holes and electrons
Diffusion current is established
Devoid of free and mobile charge carriers (depletion
region)
It
seems that all holes and
electrons would diffuse!!!
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But there is formation of ions on both sides of the
junction
Formation of fixed +ve and –ve ions- parallel rows of
ions
Any free charge carrier is either
Diffused by fixed ions on own side
 Repelled by fixed ions of opposite side
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Ultimately depletion layer widens and equilibrium
condition reached
P
+ +
+ +
+ +
-
N
+ + + -
-
BARRIER VOLTAGE
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Inspite of the fact that depletion region is cleared of
charge carriers, there is establishment of electric
potential difference or Barrier potential (VB) due to
immobile ions
P
N
+
+
-
+
+
+
-
+
+
+
-
+
-
-
-
-
-
-
VB
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VB for Ge is 0.3eV and 0.7eV for Si
Barrier voltage depends on temperature
VB for both Ge and Si decreases by about 2 mV/0C
Therefore VB= -0.002 t
where t is the rise in temperature
VB causes drift of carriers through depletion layer
Barrier potential causes the drift current which is equal
and opposite to diffusion current when final equilibrium
is reached- Net current through the crystal is zero
PROBLEM
Calculate the barrier potential for Si junction at 1000C
and 00C if its value at 250C is 0.7 V
Explanation of P-N junction on the basis
of Energy band theory
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Operation of P-N junction in terms of energy bands
Energy bands of trivalent impurity atoms in the P-region
is at higher level than penta-valent impurity atoms in Nregion (why???)
However, some overlap between respective bands
Process of diffusion and formation of depletion region
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High energy electrons near the top of N-region conduction band
diffuse into the lower part of the P-region conduction band
Then recombine with the holes in the valence band
Depletion layer begins to form
Energy bands in N-region shifts downward due to loss of high
energy electrons
Equilibrium condition- When top of conduction band reaches at
same level as bottom of conduction band in P-region- formation
of steep energy hill
CB
CB
CB
VB
VB
VB
P-Region
N-Region
P-Region
CB
VB
N-Region
Biasing of P-N junction
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Forward Biasing
 Positive
terminal of Battery is connected with
P-region and negative terminal with N-region
 Can be explained by two ways. One way is
Holes in P-region are repelled by +ve terminal of the battery
and electrons in N-region by –ve terminal
 Recombination of electrons and holes at the junction
 Injection of new free electrons from negative terminal
 Movement of holes continue due to breaking of more
covalent bonds- keep continuous supply of holes
 But electron are attracted to +ve terminal of battery
 Only electrons will flow in external circuit
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 Another
 Forward
way to explain conduction
bias of V volts lowers the barrier potential
(V-VB)
 Thickness of depletion layer is reduced
 Energy hill in energy band diagram is reduced
 V-I Graph for Ge and Si
 Threshold or knee voltage (practically same as
barrier voltage)
 Static (straight forward calculation) and dynamic
resistance (reciprocal of the slope of the forward
characteristics)
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Reverse Biasing
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Battery connections opposite
Electrons and holes move towards negative and
positive terminals of the battery, respectively
So there is no electron-hole combination
Another way to explain this process is
The applied voltage increases the barrier potential (V+VB)blocks the flow of majority carriers
 Therefore width of depletion layer is increased
 Practically no current , but small amount of current due to
minority carriers (generated thermally)
 Also called as leakage current
 V-I curve and saturation
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PROBLEMS
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Compute the intrinsic conductivity of a specimen of pure silicon at
6
3
2
room temperature given that n i  1 . 4  10 m , e  0 . 145 m / V  s
2
 h  0 . 05 m / V  s and e  1 . 6  10  19 C . Also calculate the
individual contributions from electrons and holes.
Find conductivity and resistance of a bar of pure silicon of length 1
cm and cross sectional area 1mm 2 at 3000k. Given
n i  1 . 5  10

16
m
3
 e  0 . 13 m / V  s
 h  0 . 05 m / V  s
2
e  1 . 6  10
 19
C
A specimen of silicon is doped with acceptor impurity to a density
2
16
3
22


0
.
145
m
/V  s
n

1
.
4

10
m
of 10 per cubic cm. Given that i
e
 h  0 . 05 m / V  s
2

2
e  1 . 6  10
 19
C
All impurity atoms may be assumed
to be ionized
Calculate the conductivity of a specimen of pure Si at room
16
3
2
temperature of 3000k for which n i  1 . 5  10 m  e  0 . 13 m / V  s
2
 h  0 . 05 m / V  s e  1 . 6  10  19 C The Si specimen is now doped
2 parts per 108 of a donor impurity. If there are 5x1028 Si atoms/m3,
calculate its conductivity.
• In particle physics, fermions are particles which
obey Fermi–Dirac statistics; they are named after
Enrico Fermi. In contrast to bosons, which have
Bose–Einstein statistics, only one fermion can
occupy a quantum state at a given time; this is the
Pauli Exclusion Principle.
• Thus, if more than one fermion occupies the same
place in space, the properties of each fermion (e.g.
its spin) must be different from the rest.
• In the Standard Model there are two types of
elementary fermions: quarks and leptons. In total,
there are 24 different fermions: being 6 quarks and
6 leptons, each with a corresponding antiparticle
FERMI DIRAC DISTRIBUTION FUNCTION
• This concept comes from Fermi-Dirac statistics.
Electrons are fermions and by the Pauli exclusion
principle cannot exist in identical energy states
• F–D statistics applies to identical particles with half-integer
spin in a system in thermal equilibrium. Additionally, the
particles in this system are assumed to have negligible mutual
interaction. This allows the many-particle system to be described
in terms of single-particle energy states. The result is the Fermi–
Dirac distribution of particles over these states and includes the
condition that no two particles can occupy the same state, which
has a considerable effect on the properties of the system.
• Since Fermi–Dirac statistics applies to particles with halfinteger spin, they have come to be called fermions. It is most
commonly applied to electrons, which are fermions with spin 1/2.
• 12 quarks - 6 particles (u · d · s · c · b · t) with 6
corresponding antiparticles (u · d · s · c · b · t);
• 12 leptons - 6 particles (e− · μ− · τ− · νe · νμ · ντ)
with 6 corresponding antiparticles (e+ · μ+ · τ+ · νe
· νμ · ντ).