This document discusses the physics of semiconductor devices and band structure theory. It introduces band structure, the density of states, Fermi-Dirac statistics, and the Fermi energy. The Fermi-Dirac distribution describes the probability of an electron occupying an energy state based on the Fermi energy and temperature. Near the Fermi energy, the Maxwell-Boltzmann approximation can be used. The position of the Fermi energy determines whether a material is a metal, semiconductor, or insulator by defining the number of charge carriers available to conduct electricity.
Describes electrostatic principles and concepts.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as electric fields, magnetic fields, and light, and is one of the four fundamental interactions (commonly called forces) in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation.[1] At high energy the weak force and electromagnetic force are unified as a single electroweak force.
Describes electrostatic principles and concepts.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as electric fields, magnetic fields, and light, and is one of the four fundamental interactions (commonly called forces) in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation.[1] At high energy the weak force and electromagnetic force are unified as a single electroweak force.
Basics of Band Structure and semiconductors.pdfDr Biplab Bag
Basics of Band Structure and semiconductors: How the energy bands and energy gaps are formed, Classification of metals/insulators/semiconductors, Fermi level, conduction & valance bands have been discussed
Solid State Electronics.
this slide is made from taking help of
TextBook
Ben.G.StreetmanandSanjayBanerjee:SolidStateElectronicDevices,Prentice-HallofIndiaPrivateLimited.
A dimensionless quantity described as a fundamental physical constant characterizing the coupling strength of the electromagnetic interaction. Introduced by Sommerfeld in 1916 to describe the spacing of splitting of spectral lines in multi-electron atoms, it is formed from four physical constants: electric charge, speed of light in vacuo, Planck's constant and electric permittivity of free space.
The inverse fine structure constant (=137.035999...) represents the spin precession whirl no. of the electron. The electron exhibits a slight precession due to an imbalance of electrostatic and magnetostatic energy levels. Electric charge is a result of this spin precession and represents a loop closure failure (torsion defect) similar to topological charge.
Rest mass results from quantum wave interference due to precession. Hence, electric charge, rest mass and the fine structure constant are interrelated and directly calculable.
A free electron model is the simplest way to represent the
electronic structure and properties of metals.
According to this model, the valence electrons of the constituent
atoms of the crystal become conduction electrons and travel
freely throughout the crystal.
The classical theory fails to explain the heat capacity and the
magnetic susceptibility of the conduction electrons. (These are
not failures of the free electron model, but failures of the classical
Maxwell distribution function.)
Condensed matter is so transparent to conduction electrons
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
1. Physics of Semiconductor Devices
an Introduction
Dr. Wolfgang Ploss
Texas Instruments Germany
Dr. Wolfgang Ploss, Studien Material, 2022 1
wolfgang.ploss@extern.oth-regensburg.de
My Email OTH
mastermem@oth-regensburg.de
Master Mail Box
For questions, please use the OTH email addresses
3. Dr. Wolfgang Ploss, Studien Material, 2022 3
Band Structure
• Electron Statistics
• Fermi Function, Fermi Energy
• Electrons and Holes in Bands
4. Dr. Wolfgang Ploss, Studien Material, 2022 4
r
Potential energy
Potential Energy ~ -
1
𝑟
+
5. Dr. Wolfgang Ploss, Studien Material, 2022 5
+ + +
The nuclei in the lattice are positive charged and we have an attraction force between
the positive nuclei and negative charged electrons. If we move the electrons along the
Electrical field with an external force, which is opposite to the internal field,
we increase the potential energy of the electrons in the field.
E – Field
of nucleus
Direction
In crystal
+
+
+
6. Dr. Wolfgang Ploss, Studien Material, 2022 6
At Edges of Conduction Band and Valence Band:
Electrons or Holes have no kinetic energy, no velocity
Negative electrons increase energy if we move them up in the conductive band.
Positive charged holes increase the potential energy if we move them down in the Valence band
8. Some Statistics
Questions:
What is the probability of an event ?
o Expected event divided by all possibilities of the event
o e.g. dice: what is probability to get number 5 to play a dice ? Dice has 6 areas. On
every area is written a number 1,2,.. 6. All areas are same. The expected event is one
area. We have 6 areas on a dice. Probability = 1 / 6 ( to get a special number, e.g. 5 or 1
or 4 etc.)
What is the probability to find an electron between E …E +dE
Dr. Wolfgang Ploss, Studien Material, 2022 8
x
probability
x x + dx
dp (x) = p(x) dx = area
Probability to find an electron between x and x + dx.
Probability to find the particle between a and b
should be 1.
p(x)
area
a b 𝑎
𝑏
𝑝 𝑥 𝑑𝑥 = 1 p = probability function
If we have a function f(x) which shows us opportunities , but not all opportunities
are possible or allowed between a and b , which is defined by a probability function p(x),
we need to calculate: 𝑎
𝑏
𝑝 𝑥 𝑓 𝑥 𝑑𝑥
9. BOLTZMANN – Statistic: Particle are capable of being differentiated
BOSE – Statistic: Particle are NOT capable of being differentiated
Fermi – Statistic: Particle are NOT capable of being differentiated and
only one particle is allowed to take a certain condition
Example (next page) :
2 particles should be distributed to 4 possible states with the same energy.
N= 2 (particle) , g = 4 (states with same energy)
How many possibilities do we get for the 3 different Statistics ?
Dr. Wolfgang Ploss, Studien Material, 2022 9
Nature of particles requires different statistics
10. Example: 2 particle distributes on 4 possible states with the same energy
𝒈𝑵
N + g -1
N
g
N
( )
( )
N = 2 particle with same energy
g = 4 states (options to be with
same energy)
Dr. Wolfgang Ploss, Studien Material, 2022 10
11. Energy Ei : gi states , Ni particle are distributes on gi states
Assumptions:
𝑁𝑖 𝐸𝑖 = 𝑐𝑜𝑛𝑠𝑡= E ; 𝑁𝑖 = 𝑐𝑜𝑛𝑠𝑡
Thermal Equilibrium State:
Particle distribution according to the highest probability, P
P maximum of probability
S = K ln (P ) , Entropy S get maximum because of P
Calculation of Entropy for different particle types gives statistic distribution
Bose
Boltzmann
Fermi
Probability P (E)
Dr. Wolfgang Ploss, Studien Material, 2022 11
12. Dr. Wolfgang Ploss, Studien Material, 2022 12
Probability to find an electron within Energy E + dE
Fermi Function
13. Dr. Wolfgang Ploss, Studien Material, 2022 13
Probability to find an electron within Energy E + dE
Fermi Function
Important about the Fermi Energy : Definition of Fermi Energy
At T= 0 , Fermi Energy is max Energy. All States below Fermi Energy are occupied, f (E<=EF)=1
At T>0, at Fermi Energy the probability to find a Fermion (electron). f(E=EF ) = ½ , 50%
Note, to be exact:
in Physics and Theory of Heat, the Fermi Energy for T > 0 K is called “Chemical Potential”.
In our course we call also for T > 0 the Energy with 50% probability the Fermi Energy.
14. Dr. Wolfgang Ploss, Studien Material, 2022 14
For E > EF :
For E < EF :
0
)
(
exp
1
1
)
( F
E
E
f
1
)
(
exp
1
1
)
( F
E
E
f
Fermi-Dirac distribution: Consider T 0 K
0
∞
For Energy E < = Fermi Energy
the probability to find a electron
with this energy is 1.
For Energy > Fermi Energy we
will find no electron.
Probability = 0
15. 15
If E = EF then f(EF) =
𝟏
𝟐
for all temperatures
If then
Thus the following approximation is valid:
i.e., most states at energies 3kT above EF are empty.
If then
Thus the following approximation is valid:
kT
E
E 3
F
1
exp F
kT
E
E
kT
E
E
E
f
)
(
exp
)
( F
kT
E
E 3
F
1
exp F
kT
E
E
kT
E
E
E
f F
exp
1
)
(
Fermi-Dirac distribution: Consider T > 0 K
Maxwell - Boltzmann
Mathematics:
1
1+𝑥
= 1 – x, if x << 1, x =
Dr. Wolfgang Ploss, Studien Material, 2022
17. Dr. Wolfgang Ploss, Studien Material, 2022 17
• kT (at 300 K) = 0.025eV (25 meV) , 3KT = 0.075eV (75 meV)
• In comparison to Eg(Si) = 1.1eV,
• 3kT is very small in comparison to band gap
Fermi Dirac Distribution and Maxwell – Boltzmann Distribution
For E > EF +3KT
Maxwell – Boltzmann
is good enough to get
the probability
kT
E
E
E
f
)
(
exp
)
( F
18. Dr. Wolfgang Ploss, Studien Material, 2022 18
T >0
EF – 3KT
EF + 3KT
For Energies +- 3KT around EF Fermi Dirac Function can be described
with Maxwell Boltzmann Probability Function
3KT = 75meV at 300K
19. Dr. Wolfgang Ploss, Studien Material, 2022 19
Which statistics must be used depends on where the Fermi Energy level is
Fermi Level in this Section
Fermi Level in this Section
Fermi Level in this Section
20. 20
Fermi-Dirac distribution and the Fermi-level
Density of states tells us how many states exist at a given energy E. The Fermi
function f(E) specifies how many of the existing states at the energy E will be
filled with electrons. The function f(E) specifies, under equilibrium conditions,
the probability that an available state at an energy E will be occupied by an
electron. It is a probability distribution function.
EF = Fermi energy or Fermi level
k = Boltzmann constant = 1.38 1023 J/K =
8.6 105 eV/K
T = absolute temperature in K
Dr. Wolfgang Ploss, Studien Material, 2022
21. 21
Equilibrium distribution of carriers
Distribution of carriers = DOS probability of occupancy
= g(E) f(E)
(where DOS = Density of states)
Total number of electrons in CB (conduction band) =
top
C
d
)
(
)
(
C
0
E
E
E
E
f
E
g
n
Total number of holes in VB (valence band) =
V
Bottom
d
)
(
1
)
(
V
0
E
E
E
E
f
E
g
p
Ec
Ev
CB
VB
E top
E bottom
Dr. Wolfgang Ploss, Studien Material, 2022
22. Dr. Wolfgang Ploss, Studien Material, 2022 22
K = Boltzmann Constant Charge of electron:
K = 8.617 10 -5 eV / K e = q = 1.6 10 -19 As
KT for T = 300K, Energy at 300K
KT = 25.8 meV , KT /e = 25.8mV
What is 1eV ? electron with charge q passes potential difference of 1V and get
energy of 1eV.
Energy = q U, unit in eV
Energy per electron at 300K = KT / e = 25.8mV
KT at 300K
K T = 25.8 meV = 25.8 1.6 10 -19 10 -3 As V ~ 4.1 10 -21 Ws
K T NA = R T, in J / mole ; R / NA = K ,
NA = 6 10 +23 particle per mole, Avogadro Const
information
23. Dr. Wolfgang Ploss, Studien Material, 2022 23
Free Electrons in MET: Fermi Energy of ‘Free Electron Gas’
Fermi Energy
Depends only of
electron number per volume
in MET
Typical Value for Fermi
Energy in Metal ~ 4eV
Fermi Temperature Typical Value ~ 50,000K
Fermi Wave Lengths Typical Value ~ 1 A
Fermi Velocity Typical Value ~ 10 +8 cm / s
information
24. Dr. Wolfgang Ploss, Studien Material, 2022 24
Fermi Energy moves to lower Values with T > 0
T0 = 0
All states E > Ef are empty
All states E<= Ef occupied
T1 > T0 > 0
2 electrons have higher
Energy. probability
f(E F ) = ½ moves
to lower values.
T2 > T1
3 electrons have higher energy
Fermi Energy gets lower again.
Density of states
25. Dr. Wolfgang Ploss, Studien Material, 2022 25
Energy state density g(E) in Silicon
Lines (delta function)
from electrons in
Shelves close to nuclei
Energy state density
In valence – and conduction
bands
26. Dr. Wolfgang Ploss, Studien Material, 2022 26
Density of states g (E) for Silicon – simple Model
Close to Ev and Ec : g(E) ~ 𝑬
Question: Where is Fermi Energy ?
g ( E) ~ 𝑬
27. Dr. Wolfgang Ploss, Studien Material, 2022 27
Thermal Equilibrium and Fermi Energy
The definition of the Fermi Energy has a useful and important consequence:
In an electronic system in thermal equilibrium, there is an unique Fermi Energy
that is constant throughout the entire system.
In other words, if an electronic system is characterized by Fermi Energy that
varies with location, the system is not in thermal equilibrium. This makes sense:
We have regions with high energy and the high - energy states are occupied.
There are also regions where the Fermi Energy is low and low- energy states are empty.
The hole system can lower its energy by allowing the electrons in high-energy states to
move to low- energy states that are empty. This movement of electrons stops when
the Fermi Energy becomes constant.
28. Dr. Wolfgang Ploss, Studien Material, 2022 28
Position of Fermi Energy and Consequences
Metal
Above Fermi Energy are
enough free states for
electrons. They can
gain energy and move.
We have current in Metals.
Semiconductor
Fermi Energy is within
The band gap. For T>0 K,
some electrons are in
conduction band and “free”.
Holes are “free”
+ +
Isolator
The band gap in isolator
is big. No ‘free’ electron can be
In the condition band .
No current is possible
valence band
conduction band
29. 29
Equilibrium distribution of carriers
Distribution of carriers = DOS probability of occupancy
= g(E) f(E)
(where DOS = Density of states)
Total number of electrons in CB (conduction band) =
top
C
d
)
(
)
(
C
0
E
E
E
E
f
E
g
n
Total number of holes in VB (valence band) =
V
Bottom
d
)
(
1
)
(
V
0
E
E
E
E
f
E
g
p
Ec
Ev
CB
VB
E top
E bottom
Dr. Wolfgang Ploss, Studien Material, 2022
30. Dr. Wolfgang Ploss, Studien Material, 2022 30
How to calculate the number of electrons in the conduction band ?
top
C
d
)
(
)
(
C
0
E
E
E
E
f
E
g
n ; gc ( E) ~ 𝐸
Non- degenerate semiconductor: EF is in the gap.
If Ef is > 3KT away from Ec we can use Maxwell – Boltzmann.
Degenerated semiconductor: EF is in conduction band. You must use Fermi Dirac.
37. Dr. Wolfgang Ploss, Studien Material, 2022 37
What does it mean ?
Nc and Nv describes all available states in the conduction band for electrons
or holes in the valence band - the effective density of states
Assumptions:
• Maxwell Boltzmann (Fermi Energy within Conduction and Valence band)
• Non - degenerated Silicon
Nc ~ 3 x 10+19 cm-3
Nv ~ 1 x 10+19 cm-3
Silicon at 25C
38. Dr. Wolfgang Ploss, Studien Material, 2022 38
Intrinsic Semiconductor, no doping
We have electrons and holes in intrinsic semiconductor and call:
Electrons per cm-3 = ni i stands for ‘intrinsic’
Holes per cm-3 = p i
The Fermi Energy (Reference) for this System is EF = E i
Maxwell Boltzmann
for n i
for p i = n i
𝐍𝐜
𝐍𝐯
> 0
Ec – Ev = Egap
Ei =
𝐄𝐜+𝐄𝐯
𝟐
- K T ln (
𝐍𝐜
𝐍𝐯
)
𝟏
𝟐
39. Dr. Wolfgang Ploss, Studien Material, 2022 39
Intrinsic Semiconductor, no doping
EF = Ei =
𝐄𝐜+𝐄𝐯
𝟐
- K T ln (
𝐍𝐜
𝐍𝐯
)
𝟏
𝟐 ~
𝐄𝐜+𝐄𝐯
𝟐
= Ei
If mass of electron and holes would be equal, EF or Ei would be in the middle of Gap.
Ei is slightly below the middle of the gap, because Nc > Nv,
~
𝐄𝐜+𝐄𝐯
𝟐
40. Dr. Wolfgang Ploss, Studien Material, 2022 40
Semiconductor, ni and pn Product
ni = 𝐍𝐯 𝐍𝐜 𝒆−
𝟏
𝟐
(𝑬𝒈/𝑲𝑻)