EE453 - Fundamentals of
Nanoelectronics
EE453
Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
Lecture 2: Two Key Concepts
EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
Learning Objectives Today:
• Review from EE305
• The idea of Density of State.
• The Fermi function
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Energy Band Diagram for semiconductor
 𝐸𝐶 : It is the lowest energy level in the conduction band, in Joule or eV
 𝐸𝑉 : It is the highest energy level in the valence band, in Joule or eV
 𝐸𝑔 : It is different from material to another and it depends on
temperature. It is given in the formula sheet in a form of graph.

It is the minimum required energy to move electron from the
valence band to the conduction band.

Another definition: It is the minimum required energy to break a
covalent bond between atoms.
𝐸𝐺 = 𝐸𝐶 − 𝐸𝑉
Type of Semiconductor using Fermi Level
Intrinsic Semiconductor (n=p)
 𝐸𝐹 is located in the middle of the band gap
 Note that we use 𝐸𝐹𝑖 or 𝐸𝑖 instead of 𝐸𝐹
n-type Semiconductor (n>p)
 Semiconductor has been doped with donors more than acceptors or has been
doped with donors only
𝐸𝑖
 𝐸𝐹 is located in the upper half of the band gap, closer to Ec
p-type Semiconductor (n<p)
 Semiconductor has been doped with acceptors more than donors or has been
doped with acceptors only
 𝐸𝐹 is located in the lower half of the band gap, closer to Ev
𝐸𝑖
EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
• When we apply voltage across the
source and drain, Electrons move and
current flow!
• But, let’s first start with the equilibrium
scenario!
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
At equilibrium, the applied voltage is zero volts.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
Before starting the investigation of the current flow through the channel, let us start
with equilibrium, where the applied voltage is equal to zero.
So, what are the energy levels that are available in the channel?
Rather than draw lots of lines to define the energy level or state, we define
a simple technique called the density of state D(E).
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
The density of state here tells us how
many states we have per unit energy in
the channel.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
For the current flow, what really matters is
the top part, near the Fermi level.
At equilibrium, all states are filled with
electrons in both the source and the drain
contacts.
The Fermi level separates all the filled
states from the empty states at 0 degrees
Kelvin.
At 0°K temperature, all states below the
fermi level will be filled, and all states
above the fermi level will be empty.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
What if we raise the temperature?
The answer is: some of the electrons will
receive enough thermal energy to actually
jump up and occupy these higher states.
Therefore, the electron distribution at
equilibrium will be given by what is called
the Fermi Function!
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
The Fermi Function:
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
The Fermi Function: 𝑓(𝐸)
The x axis is
𝐸−𝜇
𝑘𝑇
; where:
𝐸 is the energy,
𝜇 is the fermi level at equilibrium,
𝑘 is the Boltzmann constant,
𝑇 is the absolute temperature.
The y axis is the fermi function 𝑓(𝐸) in
term of energy 𝐸.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
The Fermi function is the
probability of having an electron
occupying a given state. It tells
us what fraction of the states
are filled.
For example, for energy levels
below 𝜇, we expect all states to
be filled; therefore the Fermi
function is equal to 1.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
On the other hand, for energy above 𝜇,
we expect the states are completely
empty, therefore is equal to 0.
The mathematical form of the Fermi
function is given as:
1
𝑓 𝐸 =
1
𝐸−𝜇
+ 𝑒 𝑘𝑇
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
Now when we apply a positive voltage at
the drain, we are not at the equilibrium
scenario anymore,
At the drain contact, the Fermi level will
go down due to the applied positive
potential.
The Fermi level is lover by 𝑞𝑉.
We will use this diagram to discuss the
formula for the current flow.
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EE 453 Fundamentals of Nanoelectronics
Unit 1 – The New Perspective
End of Unit 1, Lecture 2
Some materials/figures in this presentation are adapted/taken from (with permission):
Lessons from Nanoelectronics, S. Datta, World Scientific Publication, 2012, ISBN No.
978-981-4335-29-4.
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