2/20/2025
Semiconductor Fundamentals
Instructor: Girish Pahwa
Emerging Device Technologies Lab
International College of Semiconductor Technology
National Yang Ming Chiao Tung University
Semiconductor Industry
• Heart of technological progress and innovation.
• Occupies virtually every domain of people's lives.
source: https://www.semiconductors.org/industry-impact/
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What is a Semiconductor?
•
•
A material with electrical conductivity falling between a metal and an insulator.
The unique properties of semiconductors have led to the development of several solidstate devices and circuits forming the basis of modern electronics.
D. Neamen, Semiconductor Physics and Devices, 4th edition (2012).
https://chemed.chem.purdue.edu/genchem/topicreview/bp/materi
als/defects3.html
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Crystal Potential
Coulomb potential due to a
one-electron atom
Net potential is periodic in
the bulk of the crystal
Coulomb potential due to
two one-electron atoms
Image: Adv. Semi. Fund. R.F.Pierret book
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Energy Band Formation
To understand the concept of energy bands, let’s first approximate the atomic potential with a square well
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Diatomic Molecule Model
Assume that there are two identical atoms far away from each other. They show identical values of
discrete energies.
Since they are far apart their wavefunctions do not interact. Particle in well A is confined to well A, and
particle in well B remains confined in well B.
B
A
𝐸
𝑈0
𝐸
𝑈0
𝑏→∞
Image: Modified from Wikipedia
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𝑎
𝑏
𝑎
𝑥
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Diatomic Molecule Model
Let’s bring them closer.
As 𝑏 becomes smaller, the wavefunctions in one well start to encroach in the other well. The particles that were
localized to their respective atoms now start to delocalize.
Further, each energy level of the isolated well will split into two levels in a double well, one below the given
energy and the other above it.
𝐸6
𝐸5
𝐸4
𝐸3
𝐸2
𝐸1
Approximately
𝜓1 ≈ 𝜓𝐴 + 𝜓𝐵
𝜓2 ≈ 𝜓𝐴 − 𝜓𝐵
promotes bonding
promotes anti-bonding
This is called the linear combination of atomic orbitals (LCAO)
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More Potential Wells
As more atoms interact, a given energy level
splits into more levels. N atoms will split a
given energy level into N levels.
https://aimath.org/~farmer/tmp/phys212supplement/sec_energy_bands.html
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Energy Band
For a very large number of atoms in a crystal, a given energy level will split into a large number of
extremely closely spaced levels. Corresponding to each given level, we will get an energy band of
almost continuous energies.
Different bands are separated by forbidden gaps, or band-gaps.
At T=0K, the highest filled band is called the Valance Band, the lowest empty band is called the
conduction band.
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Intrinsic semiconductor
A pure semiconductor free from any impurity or with negligible impurity.
Semiconductor at T ≠ 0𝐾
Semiconductor at T = 0𝐾
At any finite temperature in equilibrium, there are electrons in the conduction band and holes in the
valance band.
Since electrons and holes are created in pairs, their concentration is equal. Let us call that common
concentration as 𝑛𝑖 .
𝑛 = 𝑝 = 𝑛𝑖
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Extrinsic Semiconductor
A semiconductor with intentionally added impurities to increase the electron or hole concentration.
The process of adding the impurity atoms is called doping and the impurity atoms are called dopants.
There can be two types of dopants.
If the dopant atoms increase the electron concentration in the conduction band, they are n-type dopants,
and the resulting material is called n-type semiconductor.
If the dopant atoms increase the hole concentration in the valance band, they are p-type dopants an the
resulting material is called p-type semiconductor.
N-type dopant for Si
Addition of pentavalent impurity,
such as P, As from group V
Each P contributes one free electron
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Extrinsic Semiconductor
𝑇=0
𝑇>0
Donors contribute to energy levels slightly below
the conduction band edge (≈ 30meV to 60meV).
At 0K, all the donors are fully occupied. Around
50K-100K, almost all of them get sufficient
energy to ionize (become positively charged) and
contribute electrons to the conduction band.
Approximate calculation of energy required for a donor electron to excite to the conduction band
Assume that P atoms have four electrons that are forming covalent bonds and the fifth electron is
loosely bound. It is as if the loose electron is revolving around a positively charged ionic core with +e
charge. This situation can be approximated by Bohr’s model of hydrogen like orbit. The ground state
energy of in Bohr’s model is given by
𝑚∗ 𝑞 4
13.6 𝑚∗
𝐸=
=
−
eV
2 4𝜋𝜖𝑟 𝜖0 ℏ 2
𝜖𝑟2 𝑚0
∗
∗
For Si, 𝜖𝑟 = 11.7, 𝑚 = 0.26𝑚0 (𝑚 is conductivity effective mass) resulting in 𝐸 = −26meV.
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Extrinsic Semiconductor
P-type dopant for Si
Addition of trivalent impurity,
such as B, Al, etc. group III
Each B atom contributes one hole.
Acceptors contribute to energy levels slightly above the
valance band edge (≈ 30meV to 60meV).
At 0K, all the acceptor states are empty. Around 50K100K, the electrons in the valance band get sufficient
energy to occupy the acceptor states and leave behind
holes in the valance band. After accepting the
electrons, the acceptor atoms become negatively
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Law of Mass Action
In equilibrium condition
𝑛𝑝 = 𝑛𝑖2
for both intrinsic and extrinsic semiconductors.
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Fermi Function
Electrons are Fermions that follow the Fermi-Dirac statistics.
The Fermi function is a distribution function that determines the ratio
of the number of filled states to the number of allowed states at a
given energy. In other words, it is the probability of occupancy of
allowed states at a given energy.
𝑓 𝐸 =
1
𝐸−𝐸𝐹
1 + 𝑒 𝑘𝑇
where 𝐸𝐹 is the Fermi energy or Fermi level, 𝑘 = 8.617 × 10−5 eV/K
is Boltzmann constant and 𝑇 is the temperature.
For absolute zero (𝑇 = 0),
𝑓 𝐸 =𝑓 𝑥 =ቊ
1,
0,
𝐸 < 𝐸𝐹
𝐸 ≥ 𝐸𝐹
i.e. all the electrons occupy an energy level below 𝐸𝐹 .
Image: Adv. Semi. Fund. R.F.Pierret book
As the temperature increase, some electrons may acquire sufficient
energy to reach higher energy states and the probability of
occupancy for 𝐸 > 𝐸𝐹 becomes non-zero.
As 𝑇 keeps increasing the Fermi tail encroaches higher and higher
energy levels.
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Density of States
DOS: Number of states per unit energy per unit volume
For a bulk 3D system
𝑔 𝐸 ∝ 𝐸𝑛𝑒𝑟𝑔𝑦
For conduction band
3/2
2 𝑚𝑒∗
𝑔𝑐 (𝐸) = 2
𝐸 − 𝐸𝑐
2
𝜋 ℏ
∗
where 𝑚𝑒 is called electron effective mass.
For valance band
𝑔𝑣 (𝐸) =
2 𝑚𝑝∗
𝜋 2 ℏ2
𝐸𝑐
𝐸𝑣
3/2
𝐸𝑣 − 𝐸
where 𝑚𝑝∗ is called hole effective mass.
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Equilibrium Carrier Concentration
Number of electrons per unit volume of the semiconductor sample occupying energy levels
between 𝐸 and 𝐸 + 𝑑𝐸 are 𝑔𝑐 𝐸 𝑓 𝐸 𝑑𝐸.
The electron concentration (total number of electrons per unit volume) in the conduction
band can be obtained by adding the number of electrons in different energy levels ranging
from 𝐸 = 𝐸𝑐 to 𝐸 = ∞ as
∞
𝑛 = න 𝑔𝑐 𝐸 𝑓 𝐸 𝑑𝐸
𝐸𝑐
For 3D
2 𝑚𝑛∗
𝑔𝑐 𝐸 = 2
𝜋 ℏ2
2 𝑚𝑛∗
𝑛= 2
𝜋 ℏ2
3/2 ∞
න
3/2
𝐸 − 𝐸𝑐
𝐸 − 𝐸𝑐
𝐸−𝐸𝐹 𝑑𝐸
𝐸𝑐 1 + 𝑒 𝑘𝑇
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Maxwell-Boltzmann Approximation
𝑓 𝐸 =
1
𝐸−𝐸𝐹
1 + 𝑒 𝑘𝑇
For 𝐸𝑐 − 𝐸𝐹 > 3𝑘𝑇 (nondegenerate semiconductor), the Fermi-Dirac distribution function can be
approximated as the Maxwell-Boltzmann statistics
𝐸−𝐸𝐹
𝑓 𝐸 ≈ 𝑒 − 𝑘𝑇
Degenerate SC
The electron concentration can
then be calculated in an analytical
form as
∞
3/2 ∞
𝐸−𝐸𝐹
2 𝑚𝑛∗
𝑛 = න 𝑔𝑐 𝐸 𝑓 𝐸 𝑑𝐸 = 2
න 𝐸 − 𝐸𝑐 ⋅ 𝑒 − 𝑘𝑇 𝑑𝐸
𝜋 ℏ2
𝐸𝑐
Nondegenerate SC
𝐸𝑐
Assuming 𝐸 − 𝐸𝑐 = 𝑥, we have
∞
3/2
3/2
𝐸 −𝐸
𝑥
𝐸𝑐 −𝐸𝐹 𝑘𝑇 𝜋𝑘𝑇
2 𝑚𝑛∗
2 𝑚𝑛∗
− 𝑐 𝐹
−
𝑘𝑇 න 𝑥 ⋅ 𝑒 𝑘𝑇 𝑑𝐸 =
𝑛= 2
𝑒
𝑒 − 𝑘𝑇 ⋅
𝜋 ℏ2
𝜋 2 ℏ2
2
Degenerate SC
0
𝐸𝑐 −𝐸𝐹
𝑛 = 𝑁𝑐 𝑒 − 𝑘𝑇
𝑁𝐶 = 2
𝑚𝑛∗ 𝑘𝑇
2𝜋ℏ2
3/2
=2
2𝜋𝑚𝑛∗ 𝑘𝑇
ℎ2
3/2
For Si at 300K, 𝑁𝑐 = 2.8 × 1019 𝑐𝑚−3
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Maxwell-Boltzmann Approximation
1−𝑓 𝐸 =1−
1
1
𝐸−𝐸𝐹 =
𝐸−𝐸𝐹
1 + 𝑒 𝑘𝑇
1 + 𝑒 − 𝑘𝑇
Similarly, for 𝐸𝐹 − 𝐸𝑣 > 3𝑘𝑇, 1 − 𝑓 𝐸 can be approximated as
𝐸−𝐸𝐹
1 − 𝑓 𝐸 ≈ 𝑒 𝑘𝑇
The hole concentration in the valance band can then be calculated in an analytical form as
𝐸𝑣
3/2 𝐸𝑣
𝐸−𝐸𝐹
2 𝑚𝑝∗
𝑝 = න 𝑔𝑣 𝐸 (1 − 𝑓 𝐸 )𝑑𝐸 = 2
න 𝐸𝑣 − 𝐸 ⋅ 𝑒 𝑘𝑇 𝑑𝐸
𝜋 ℏ2
−∞
−∞
After some mathematical manipulations, we get
𝑝 = 𝑁𝑣 𝑒
𝑁𝑣 = 2
𝑚𝑝∗ 𝑘𝑇
2𝜋ℏ2
3/2
=2
2𝜋𝑚𝑝∗ 𝑘𝑇
ℎ2
𝐸𝑣 −𝐸𝐹
𝑘𝑇
3/2
For Si at 300K, 𝑁𝑣 = 1.04 × 1019 𝑐𝑚−3
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