FERMI ENERGY & FERMI LEVEL |1
FERMI ENERGY & FERMI LEVEL |2
Question: What is Fermi level and Fermi energy? Write down the differences
between Fermi energy and Fermi level.
ii) Contact potential: The potential generated by the contact of two
dissimilar materials in air or in a vacuum is called contact potential.
iii) Ohmic contact: An ohmic contact is a non-rectifying electrical junction.
A junction between two conductors that has a linear current voltage (I-V)
curve as with Ohm’s law is called Ohmic contact.
Answer:
Fermi level: The Fermi level is the total chemical potential for electrons and
usually denoted by μ or EF. The Fermi level of a body is a thermodynamic
quantity and its significance is the thermodynamic work required to add one
electron to the body.
Fermi energy: The Fermi energy is a concept in quantum mechanics usually
referring to the energy difference between the highest and lowest occupied
single particle states in a quantum system of non-interacting fermions at
absolute zero temperature.
Difference between Fermi energy and Fermi level:
1. The Fermi energy is only defined at absolute zero temperature while the
Fermi level is defined for any temperature.
2. The Fermi energy is an energy difference usually corresponding to a
kinetic energy, whereas the Fermi level is a total energy level including
kinetic energy and potential energy.
3. The Fermi energy can only be defined for non-interacting fermions,
whereas the Fermi level remains well defined even in complex interacting
systems at thermodynamic equilibrium.
Question: Discuss the limitation of classical free electron theory/statistics.
Answer: The energy possessed by an electron at highest energy level at 0°K is
called Fermi energy. The classical free electron theory is used to calculate the
transport properties of gas including thermal conductivity which agrees with
general features observations. But this theory when used to calculate electronic
specific heat, mean free path, diamagnetic, paramagnetic, susceptibility the
result have no resemblance to observed values. These drawbacks are explained
by Summerfield. He replaced Bose-Einstein statistics by Fermi-Dirac statistics.
Question: Define (i) Work function (ii) Contact potential (iii) Ohmic contact.
Question: Derive the expression of density of states and prove that density of
states is proportional to the square root of Fermi energy.
Answer: According to free energy theory of electron, the solids behave as free
electron gas. Solid is a non-interacting gas of electron.
Let us consider free electrons are confined in a square potential well of size ‘L’.
Dimension of the potential well is thought as crystal well.
The energy level of electrons is found by solving Schrödinger equation,
En =
Here, h = Plank’s constant
m = Mass
In three dimensional case Schrödinger equation is,
=0
E=
where E=EF and EF =
Let us consider a rectangular parallelepiped volume V. The shape of Fermi
surface in momentum space for free electron model is the Fermi radius kF.
In momentum space, the volume occupied by each
solid,
=
=
Answer:
i) Work function: The minimum amount of energy required to remove an
electron from the surface of a metal is called work function.
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
=
[∵ V = LxLyLz]
Figure: Fermi sphere
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
FERMI ENERGY & FERMI LEVEL |3
FERMI ENERGY & FERMI LEVEL |4
Let, no. of states inside Fermi sphere be nF and N be the total number of
electrons in it. Each level occupies two electrons with spin up and another spin
down.
∴ No. of Fermi electrons, nF =
i.e. D(EF) ∝ √
+
where, kF = Fermi radius
[
=
Fermi energy, EF =
En =
In three dimensional case Schrödinger equation is,
We know,
EF =
(
=
*
+
(
) (
) =
N=
=
Here, h = Plank’s constant
m = Mass
∴ D(EF) =
∴ D(EF) =
Let us consider free electrons are confined in a square potential well of size ‘L’.
Dimension of the potential well is thought as crystal well.
The energy level of electrons is found by solving Schrödinger equation,
]
Density of states: The density of states of Fermi level is defined as the number
of states per unit volume per unit energy range.
⇒
) √
Answer: According to free energy theory of electron, the solids behave as free
electron gas. Solid is a non-interacting gas of electron.
=
⇒
(
) √
Question: From the electron theory and Schrödinger equation, find out the
expression for Fermi energy, Fermi momentum, Fermi velocity and Fermi
temperature.
=
⇒ kF = *
(
Hence density of states is proportional to the square root of Fermi energy when
all the roots are constant.
= nF
⇒
∴ D(EF) =
=
Again,
⇒
=
[
(
(
)
=0
E=
) (
(
)
)
where E=EF and EF =
Let us consider a rectangular parallelepiped
volume V. The shape of Fermi surface in
momentum space for free electron model is the
Fermi radius kF.
)]
In momentum space, the volume occupied by each
solid,
Figure: Fermi sphere
=
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
FERMI ENERGY & FERMI LEVEL |5
=
=
⇒
[∵ V = LxLyLz]
=
*
+
Answer:
Distribution function: The distribution function gives the number of electrons
or other fermions per single particle state for a microscopic system at non-zero
temperature.
1.
2.
3.
4.
= nF
⇒
=
Classification of distribution function:
Again,
⇒
TF =
Question: What is distribution function? Classify distribution function.
Let, no. of states inside Fermi sphere be nF and N be the total number of
electrons in it. Each level occupies two electrons with spin up and another spin
down.
∴ No. of Fermi electrons, nF =
FERMI ENERGY & FERMI LEVEL |6
=
Fermi-Dirac distribution function
Bose-Einstein distribution function
Maxwell-Boltzmann distribution function
Probability distribution function
Question: Write down the properties of Fermi-Dirac function.
=
Answer:
⇒ kF = *
+
where, kF = Fermi radius
i) Fermi energy, EF =
[
=
]
ii) Fermi momentum, ρF = h.kF = h. *
iii) Fermi velocity =
= EF
+
-------------- (1)
But, ρF = mvF
Now, putting the value of mvF and EF in equation (1),
ρFvF =
⇒ h. *
⇒ vF =
*
*
+ vF =
+
+
Properties of Fermi-Dirac function:
1. Fermi level is the energy which correspond to the centre of gravity of
conduction electrons and holes weighted according to their energies.
2. Since bandwidths are small, all levels on conduction band have same
energy.
3. Energies of all level in the conduction band are equal to gap energy Eg.
4. In a semiconductor, Fermi level lies midway between valence band and
conduction band.
Question: Discuss Fermi-Dirac distribution.
*
Answer: The Fermi-Dirac distribution gives the probability that an energy state
with energy E is occupied by an electron.
+
The Fermi-Dirac distribution,
F(E) =
iv) Fermi temperature:
We know,
EF = KBTF
where,
(
)⁄
------------ (i)
E = Total energy
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
FERMI ENERGY & FERMI LEVEL |7
EF = Fermi energy
KB = Boltzmann’s constant =
J/K
FERMI ENERGY & FERMI LEVEL |8
⇒
⇒
0.01 =
⁄
0.01 =
[Let, x =
]
⇒
⇒
⇒
⇒
The significance of the Fermi energy is most clearly seen by setting T=0.
At T=0 for E < EF,
F(E) =
At T = 0 for E > EF,
F(E) =
At t > 0 for E = EF,
F(E) =
⇒
⇒
x = 2.303
⇒
T=
⇒
The above equations indicate that at absolute zero temperature, the probability
of finding the electrons having energy less than the Fermi energy is 1 and the
probability becomes zero for the electrons having energies greater than the
Fermi energy. On the other hand, with the increase of the temperature, the
probability of finding the electrons becomes 0.5 within the energy given by the
Fermi energy.
x = ln 99
⇒
= 2.303
T=
Problem: Calculate the temperature at which there is one percent probability
that a state with an energy of 0.5eV above the Fermi energy will be occupied by
an electron.
Solution: We know, Fermi-Dirac function,
F(E) =
(
)⁄
(
)⁄
Given that, E = EF+0.5
Now, F(E) =
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
DIFFUSION CURRENT & DRIFT CURRENT |1
DIFFUSION CURRENT & DRIFT CURRENT |2
Question: Define- (i) Diffusion current (ii) Drift current (iii) Drift velocity.
e
n
Answer:
i) Diffusion current: The current in a semiconductor caused by the
diffusion of charge carriers (holes or electrons) is called diffusion
current.
ii) Drift current: The current due to the motion of charge carriers caused
by the force exerted on them by an electric field is called drift current.
iii) Drift velocity: When potential difference (or voltage) is applied across
the ends of a metallic wire, the free electrons start drifting towards the
positive terminal of the source. The average velocity with which free
electrons get drifted in a metallic conductor under the influence of
potential difference is called drift velocity (vd) of the free electron. The
drift velocity of free electrons is very small, of the order of
ms-1.
Question: What are the differences between diffusion current and drift current?
Answer: The differences between diffusion current and drift current are given
below:
Diffusion current
Drift current
1. It may occur even if there isn’t any 1. It requires an electric field to be
electric field in the semiconductor.
present.
2. The magnitude of diffusion current 2. The magnitude of drift current
depends on the slope of the carriers
depends
on
the
carrier
concentration
and
not
the
concentration.
concentration itself.
3. It does not obey Ohm’s law.
3. It obeys Ohm’s law.
= Charge of electron
= Particle density of electron
Now let us select a certain point
in a semiconductor and add up
the particle currents at that point
which are generated at a rate ‘g’.
The particle can disappear
because of recombination of
holes with electrons at a rate ‘r’.
They may flow away depending
upon
.
∴ Rate of change of particle density,
Here, p = Space charge density
en = Electron concentration
∴J=
∴
∴
=
=
=
=
This is known as diffusion equation.
Question: Derive the equation of diffusion current./Establish diffusion equation.
Question: Derive the equation of drift velocity/drift motion of particles.
Answer: Let us consider one dimensional semiconductor model where electrons
and holes diffusion currents are given by,
Answer: Drift motion of particles occurs due to the external field or force. Here
the electrons or holes in semiconductor are driven by external electric field E of
battery voltage.
Jn =
and
Jp =
where,
Jn
Jp
Dn
= Current density of electron
= Current density of holes
= Diffusion constant for electron
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
∴ Drift velocity, vd = μE
where,
μ = Mobility of electron
E = Battery field
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Solid State (EEE 701)
DIFFUSION CURRENT & DRIFT CURRENT |3
When a field is applied, the free flight of the electron is bent by the acceleration
due to that field. This acceleration is given by,
a=
If the average free time between collision is t, then the speed due to that field at
the end of the flight is,
DIFFUSION CURRENT & DRIFT CURRENT |4
where n and μ vary with temperature.
Question: Establish the Einstein’s equation for both n type and p type
semiconductors./Deduce Einstein’s equation/relation for both electrons and
holes.
Answer: For special case where the current density is zero, the Maxwell
Boltzmann equation becomes,
v = t.a = t.
μnnE
On collision, the electron losses the entire velocity gained in the direction of the
field. Thus the average velocity is,
= no
which is the equation of drift velocity.
Now, we know,
⇒
Drift current, J = nevd = neμE
×
=
×
=
( E)
×
---------------- (ii)
Now, from equation (i),
Here, n = No. of particles
=
= neμ
Question: Show that, the conductivity of a solid is σ = neμ, where the symbols
have their usual meanings.
Answer: The current due to the motion of charge carriers caused by the force
exerted on them by an electric field is called drift current.
∴ Drift current, J = nevd = neμE
Here, vd = Drift velocity = μE
n = No. of particles to carry current
e = Charge of electron
μ = Mobility of electrons
E = Electric field
We know,
Conductivity, σ = =
---------------- (i)
where, n = no
vavg =
Current conductivity, σ = =
=0
Putting the value of
=
⇒
⇒
in equation (ii), we have,
( E)
=
Dn =
μn
which is the Einstein equation.
∴ Dn ∝ μn
i.e. for electron diffusion is proportional to mobility because KB, T, e are
constant.
For holes that relation is,
= neμ
Dp =
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
μp
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)
DIFFUSION CURRENT & DRIFT CURRENT |5
Now,
=
⇒
=
Thus diffusion constant is directly proportional to mobility.
Problem: Find the diffusion co-efficient of electrons and holes of Si (Silicon)
single crystal at 27°C if the mobilities of electrons and holes are 0.17 and 0.025
m2v-1sec-1 respectively at 27°C.
Solution:
We know,
μ
D=
Here,
KB = Boltzmann constant = 1.38×
J/K
e = Charge of electron = 1.609×
C
μe = Mobility of electron = 0.17 m2v-1sec-1
μp = Mobility of holes = 0.025 m2v-1sec-1
T = Temperature = 273+27 = 300K
Einstein equation for electrons,
Dn =
μn
=
×0.17
m2sec-1
= 44×
Einstein equation for holes,
Dp =
μp
=
= 6.47×
×0.025
m2sec-1
Prepared By: Hasan Bin Firoz (533)
Solid State (EEE 701)