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FERMI LEVEL AND EFFECT OF TEMPERATURE ON SC
FERMI LEVEL AND EFFECT OF TEMPERATURE ON INTRINSIC SC
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
Valence Band
Ele
ctro
n E
nerg
y
Conduction Band
EF
00K
For intrinsic semiconductors like silicon and germanium, the Fermi level is essentially halfway between the valence
and conduction bands
FERMI LEVEL 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 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' Fermi-Dirac distribution function
The Fermi-Dirac distribution, also called the "Fermi function," is a fundamental equation expressing the behaviour of mobile charges in solid materials
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
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
FERMI FACTOR or Fermi function
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
The probability that the particle will have an energy E is
KT
EE F
e
EF
1
1)(
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
FEE Case-I When T=00K , then for
01
1)(
e
EF
11
1)(
e
EF
FEE and for
Case-II When t=T0K , then at E=EF
2
1
1
1)(
0
eEF
This means that when the temperature is not 00K but some higher value say T=10000K, then some covalent bonds will be broken and someelectrons will be available in CBThe 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 s-band 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
dEEFESdEEN )()()(
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
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)
dEEFESdEEP )(1)()(
ENERGY LEVEL IN C.B = E1 ENERGY LEVEL IN V.B = E2
THEN
dEEFESdEEN )()()( 111
dEEFESdEEP )(1)()( 222 AND
IN CASE OF INTRINSIC SC: )()( 21 ESES
dEEFES
dEEFES
dEEP
dEEN
)(1)(
)()(
)(
)(
21
11
2
1
)(1
)(
)(
)(
2
1
2
1
EF
EF
EP
EN
AT E1 = 3000K
KT
EE F
eEF
1
1
AND
KT
EE F
eEF
2
21ALSO
1)(
)(
2
1 i
i
p
n
EP
EN
THEREFORE
221 EE
EF
The density of mobile electrons is shown in the conduction band. The corresponding density of mobile holes is shown in the valence band
•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
FERMI LEVEL IN EXTRINSIC SC
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
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
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
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
VARIATION OF FERMI LEVEL WITH TEMP
P-N JUNCTION
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 N
Junction Three phenomenas take place at the junction
Depletion layer Barrier potential Diffusion capacitance
P-N JUNCTION
Formation of depletion layer 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!!!
P-N JUNCTION
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
Ultimately depletion layer widens and equilibrium condition reached
+
+ + - -
+ ++ +
- -- -
P N
++
---
BARRIER VOLTAGE 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
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
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 N-region (why???)
However, some overlap between respective bands Process of diffusion and formation of depletion region
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
Explanation of P-N junction on the basis of Energy band theory
VBVB
CBCB
VBVB
CBCB
P-Region
N-Region
N-Region
P-Region
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
Biasing of P-N junction
Another way to explain conduction Forward 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)
Reverse Biasing 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
Compute the intrinsic conductivity of a specimen of pure silicon at room temperature given that ,
and . Also calculate the individual contributions from electrons and holes.
PROBLEMS
36104.1 mni sVme /145.0 2sVmh /05.0 2 Ce 19106.1
Find conductivity and resistance of a bar of pure silicon of length 1 cm and cross sectional area at 3000k. Given 21mm
316105.1 mni sVme /13.0 2 sVmh /05.0 2 Ce 19106.1 A specimen of silicon is doped with acceptor impurity to a density
of 1022 per cubic cm. Given that 316104.1 mni sVme /145.0 2
sVmh /05.0 2 Ce 19106.1 to be ionized
All impurity atoms may be assumed
Calculate the conductivity of a specimen of pure Si at room temperature of 3000k for which
316105.1 mni sVme /13.0 2sVmh /05.0 2 Ce 19106.1 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
• 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 half-integer spin, they have come to be called fermions. It is most commonly applied to electrons, which are fermions with spin 1/2.
FERMI DIRAC DISTRIBUTION FUNCTION
• 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).