This Presentation "Energy band theory of solids" will help you to Clarify your doubts and Enrich your Knowledge. Kindly use this presentation as a Reference and utilize this presentation
SEMICONDUCTORS,BAND THEORY OF SOLIDS,FERMI-DIRAC PROBABILITY,DISTRIBUTION FUN...A K Mishra
This PPT contains valence band,conduction band& forbidden energy gap,Free carrier charge density,intrinsic and extrinsic semiconductors,Conductivity in semiconductors
Classification of magnetic materials on the basis of magnetic momentVikshit Ganjoo
I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
SEMICONDUCTORS,BAND THEORY OF SOLIDS,FERMI-DIRAC PROBABILITY,DISTRIBUTION FUN...A K Mishra
This PPT contains valence band,conduction band& forbidden energy gap,Free carrier charge density,intrinsic and extrinsic semiconductors,Conductivity in semiconductors
Classification of magnetic materials on the basis of magnetic momentVikshit Ganjoo
I made this presentation for my own college assignment and i had referred contents from websites and other presentations and made it presentable and reasonable hope you will like it!!!
This slide set corresponds to the MaterialsConcepts YouTube video "Muddiest Point- Electronic Properties I. Here's the link:
https://www.youtube.com/watch?v=BY8ZPobU8B0
To study the vocab used in this video, visit this site:
http://quizlet.com/24383440/71-electronic-properties-i-conductors-insulators-semiconductors-flash-cards/
This work was supported by NSF Grants #0836041 and #1226325.
Introduction
Formation Of Bond.
Formation Of Band.
Role Of Pauli Exclusion Principle.
Fermi Dirac Distribution Equation
Classification Of Material In Term Of Energy Band Diagram.
Intrinsic Semiconductor.
a)Drive Density Of State
b)Drive Density Of Free Carrier.
c)Determination Of Fermi Level Position
Extrinsic Semiconductor.
a) N Type Extrinsic Semiconductor
b) P Type Extrinsic Semiconductor
Compensated semiconductor.
E Vs. Diagram.
Direct and Indirect Band Gap.
Degenerated and Non-degenerated.
PN Junction.
Describe the Schroedinger wavefunctions and energies of electrons in an atom leading to the 3 quantum numbers. These can be also observed in the line spectra of atoms.
Hello all, This is the presentation of Graph Colouring in Graph theory and application. Use this presentation as a reference if you have any doubt you can comment here.
This Presentation Elliptical Curve Cryptography give a brief explain about this topic, it will use to enrich your knowledge on this topic. Use this ppt for your reference purpose and if you have any queries you'll ask questions.
This presentation about Conjestion control will enrich your knowledge about this topic.and use this presentation for your reference this presentation with the Leaky bucket algorithm.
This Presentation will useful to Enrich your knowledge on Cloud Computing Regarding to Networking. Use this presentation for Your reference purpose. Thankyou
This E Book Mangement system is Implemented in Case Tools Method like Unified Modeling Language models with more Diagrams. It will help you to an idea to develop your project.
This Presentation "Course Registration System" is Implemented in Case Tools. It will Help you to develop Your Project in Technical Manner. Kindly use this presentation for your Reference. If you have any doubts in this presentation mail me baranitharan@gmail.com
Clipping is used in the Computer Graphics sector and My presentation Will Help you to Enrich your idea's on Clipping. Utilize my Presentation as a Reference...
Water indicator Circuit to measure the level of any liquidBarani Tharan
This Presentation Will help you to an idea about this Water indicator circuit to measure the level of any liquid Topic. Use my Presentation for your Reference. It will enrich your Technical knowledge if any doubts mail me baranitharan2020@gmail.com
This Presentation will Use to develop your knowledge and doubts in Knapsack problem. This Slide also include Memory function part. Use this Slides to Develop your knowledge on Knapsack and Memory function
Cloud Computing is not only for Data Storage.. In this presentation I show it work to reduce the Death rate Of Heart Attack Patient. It is Totally Different in Domain. Use this Slide to Enrich your Knowledge in Cloud Domain
This Presentation will use to develop your knowledge in Fourier Transform mostly in Application side. So Kindly Use this presentation to enrich your knowledge in Fourier transform Domain and if any queries mail me baranitharan2020@gmail.com I'll solve your Doubts
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
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.
2. Nomenclature
For most purposes, it is sufficient to know the En(k) curves - the
dispersion relations - along the major directions of the reciprocal
lattice.
This is exactly what is done when real band diagrams of
crystals are shown. Directions are chosen that lead from the
center of the Wigner-Seitz unit cell - or the Brillouin zones - to
special symmetry points. These points are labeled according to
the following rules:
• Points (and lines) inside the Brillouin zone are denoted with
Greek letters.
• Points on the surface of the Brillouin zone with Roman
letters.
• The center of the Wigner-Seitz cell is always denoted by a Γ
3. For cubic reciprocal lattices, the points with a high symmetry on the
Wigner-Seitz cell are the intersections of the Wigner Seitz cell with
the low-indexed directions in the cubic elementary cell.
Nomenclature
simple
cubic
4. Nomenclature
We use the following nomenclature: (red for fcc, blue for bcc):
The intersection point with the [100] direction is
called X (H)
The line Γ—X is called ∆.
The intersection point with the [110] direction is
called K (N)
The line Γ—K is called Σ.
The intersection point with the [111] direction is
called L (P)
The line Γ—L is called Λ.
Brillouin Zone for fcc is bcc
and vice versa.
5. We use the following nomenclature: (red for fcc, blue for bcc):
The intersection point with the [100] direction is
called X (H)
The line Γ—X is called ∆.
The intersection point with the [110] direction is
called K (N)
The line Γ—K is called Σ.
The intersection point with the [111] direction is
called L (P)
The line Γ—L is called Λ.
Nomenclature
6. Real crystals are three-dimensional and we must consider
their band structure in three dimensions, too.
Of course, we must consider the reciprocal lattice, and, as
always if we look at electronic properties, use the Wigner-
Seitz cell (identical to the 1st
Brillouin zone) as the unit cell.
There is no way to express quantities that change as a
function of three coordinates graphically, so we look at a
two dimensional crystal first (which do exist in
semiconductor and nanoscale physics).
Electron Energy Bands in 3D
The qualitative recipe for obtaining the band structure
of a two-dimensional lattice using the slightly adjusted
parabolas of the free electron gas model is simple:
7. LCAO: Linear Combination of Atomic Orbitals
AKA: Tight Binding Approximation
• Free atoms brought together and the Coulomb interaction
between the atom cores and electrons splits the energy levels
and forms bands.
• The width of the band is proportional to the strength of the
overlap (bonding) between atomic orbitals.
• Bands are also formed from p, d, ... states of the free atoms.
• Bands can coincide for certain k values within the Brillouin
zone.
• Approximation is good for inner electrons, but it doesn’t
work as well for the conduction electrons themselves. It can
approximate the d bands of transition metals and the valence
bands of diamond and inert gases.
8.
9. The lower part (the "cup") is
contained in the 1st Brillouin zone,
the upper part (the "top") comes
from the second BZ, but it is
folded back into the first one. It
thus would carry a different band
index. This could be continued ad
infinitum; but Brillouin zones with
energies well above the Fermi
energy are of no real interest.
These are tracings along major
directions. Evidently, they contain
most of the relevant information
in condensed form. It is clear that
this structure has no band gap.
Electron Energy Bands in 3D
10. Electronic structure calculations such as our tight-binding method
determine the energy eigenvalues εn at some point k in the first
Brillouin zone. If we know the eigenvalues at all points k, then the
band structure energy (the total energy in our tight-binding method)
is just
LCAO: Linear Combination of Atomic Orbitals
where the integral is over the occupied states of below the Fermi
level.
11. The full Hamiltonian of the
system is approximated by
using the Hamiltonians of isolated atoms, each one centered at
a lattice point.
The eigenfunctions are assumed to have amplitudes that go to
zero as distances approach the lattice constant.
The assumption is that any necessary corrections to the
atomic potential will be small.
The solution to the Schrodinger equation for this type of single
electron system, which is time-independent, is assumed to be a
linear combination of atomic orbitals.
12. Band Structure: KCl
We first depict the band structure of an ionic crystal, KCl. The bands are very
narrow, almost like atomic ones. The band gap is large around 9 eV. For alkali
halides they are generally in the range 7-14 eV.
16. Electron Density of States: Free Electron
Model
+ + + + +
+ + + + +
+ + + + +
+ + + + +
+ + + + +
Schematic model of metallic
crystal, such as Na, Li, K, etc.
The equilibrium positions of
the atomic cores are
positioned on the crystal lattice
and surrounded by a sea of
conduction electrons.
For Na, the conduction
electrons are from the 3s
valence electrons of the free
atoms. The atomic cores
contain 10 electrons in the
configuration: 1s2
2s2
p6
.
17. Electron Density of States: Free Electron
Model
• Assume N electrons (1 for
each ion) in a cubic solid with
sides of length L – particle in a
box problem.
• These electrons are free to
move about without any
influence of the ion cores,
except when a collision
occurs.
• These electrons do not
interact with one another.
• What would the possible
energies of these electrons
be?
∞
0 L
18. How do we know there are free electrons?How do we know there are free electrons?
You apply an electric field across a metal pieceYou apply an electric field across a metal piece
and you can measure a current – a number ofand you can measure a current – a number of
electrons passing through a unit area in unitelectrons passing through a unit area in unit
time.time.
But not all metals have the same current for aBut not all metals have the same current for a
given electric potential. Why not?given electric potential. Why not?
20. The electron density of states is a key parameter in the
determination of the physical phenomena of solids.
Electron Density of States
Knowing the energy levels, we can count how many energy
levels are contained in an interval ∆E at the energy E. This is
best done in k - space.
In phase space, a surface of constant
energy is a sphere as schematically
shown in the picture.
Any "state", i.e. solution of the
Schrodinger equation with a specific k,
occupies the volume given by one of
the little cubes in phase space.
The number of cubes fitting inside the
sphere at energy E thus is the number
of all energy levels up to E.
21. Electron Density of States: Free Electrons
Counting the number of cells (each containing one possible state
of ψ) in an energy interval E, E + ∆E thus correspond to taking
the difference of the numbers of cubes contained in a sphere with
"radius" E + ∆E and of “radius” E. We thus obtain the density of
states D(E) as
1 ( , ) ( )
( )
1
N E E E N E
D E
V E
dN
V dE
+ ∆ −
=
∆
=
where N(E) is the number of states between E = 0 and E per
volume unit; and V is the volume of the crystal.
22. Electron Density of States: Free Electrons
The volume of the sphere in k-space is
3
3
4
kV π=
The volume Vk of one unit cell,
containing two electron states is
3
2
=
L
Vk
π
The total number of states is then
( )
3 3 3 3
23
4
2 2
33 8k
V k L k L
N
V
π
ππ
÷= = =
÷
23. Electron Density of States: Free Electrons
2
22
2
2
mE
k
m
k
E ±=⇒=
3/2
3 2 2
1 1 2
( )
2
dN m
D E E
L dE π
= = ÷
3 23 3 3
2 2 2
2
3 3
/
k L L mE
N
π π
= = ÷
27. Electron Density of States: Free Electron
Model
If an electron is added, it
goes into the next available
energy level, which is at the
Fermi energy. It has little
temperature dependence.
( )/
( )/
1
( )
1
1
1
B
F B
k T
k T
f
e
e
ε µ
ε ε
ε −
−
=
+
=
+
Fermi-Dirac Distribution
For lower energies,
f goes to 1.
For higher energies,
f goes to 0.
29. Free Electron Model: QM Treatment
and similarly for y and z, as well
2 4
0, , , ...
2 4
0, , , ...
2 4
0, , , ...
x
y
z
k
L L
k
L L
k
L L
π π
π π
π π
= ± ±
= ± ±
= ± ±
( )i k r
k eψ ×
=
31. Free Electron Model: QM Treatment
B
F
F
k
T
ε
=
1/32
F F
3 N
v k
m m V
π
= = ÷
h h
32. ( )
3/2
F2 2
3
2
2
3
ln ln constant
then
3
2
V m
N
N
dN N
D
d
ε
π
ε
ε
ε ε
= ÷
⇒ = +
= =
h
Free Electron Model: QM Treatment
The number of orbitals per
unit energy range at the Fermi
energy is approximately the
total number of conduction
electrons divided by the Fermi
energy.
33. Free Electron Model: QM Treatment
As the temperature
increases above T = 0
K, electrons from region
1 are excited into region
2.
This represents how
many energies are
occupied as a function of
energy in the 3D
k-sphere.
34.
35. Electron Density of States: LCAO
If we know the band structure at every point in the Brillouin zone, then the
DOS is given by the formula
( )
1
3
( ) k
4
n
n
dS
D ε ε
π
−
= ∇∑∫
where the integral is over the surface Sn(ε) is the surface in k space at
which the nth eigenvalue has the value εn.
Obviously we can not evaluate this integral directly, since we don't
know εn(k) at all points; and we can only guess at the properties of its
gradient. One common approximation is to use the tetrahedron method,
which divides the Brillouin zone into (surprise) tetrahedra, and then
linearly interpolate within the tetrahedra to determine the gradient. This
method is an approximation, but its accuracy obviously improves as we
increase the number of k-points.
36. When the denominator in the integral is zero, peaks due to van Hove
singularities occur. Flat bands give rise to a high density of states. It is also
higher close to the zone boundaries as illustrated for a two dimensional
lattice below.
Electron Density of States: LCAO
( )
1
3
( ) k
4
n
n
dS
D ε ε
π
−
= ∇∑∫
Leon van Hove
37. • For the case of metals, the
bands are very free electron-
like (remember we compared
with the empty lattice) and the
conduction bands are partly
filled.
• The figure shows the DOS for
the cases of a metal , Cu, and a
semiconductor Ge. Copper has
a free electron-like s-band,
upon which d-bands are
superimposed. The peaks are
due to the d-bands. For Ge the
valence and conduction bands
are clearly seen.
Electron Density of States: LCAO
38. Electron Density of States: LCAO
fcc
The basic shape of
the density of states
versus energy is
determined by an
overlap of orbitals. In
this case s and d
orbitals…
Calculated electron wave functions for atomic sodium, plotted around two nuclei separated by the nearest-neighbor distance in metallic sodium of 3.7 angstroms.
Overlap of the 1s wave functions is negligible – no alteration in the solid.
Overlap of the 2s and 2p orbitals is small – so you would find bands associated with these. The overlap is strongest for the 3s – where the valence electrons are. In the tight binding approximation we do not expect cases of strong overlap to make significant contribution to the band structure.
Every solid contains electrons. How those electrons respond to an electric field – whether they are conducting, insulating, or semi-conducting – is determined by the filling of available energy bands – separated by forbidden regions where no wavelike orbitals exist – these are the band gaps.
When an electron is confined to a cube-shaped box, the wavefunction is a standing wave.
Since we’ll also need this wavefunction to satisfy periodic boundary conditions, like we did for phonons, the wavefunctions will also be periodic in x, y, and z with period L.
Of course, we want to deal with free-particles, so Mr. Schrodinger with periodic conditions gives us wavefunctions that are traveling plane waves...
The traveling plane wave solution is valid as long as the components of the wavevector satisfy these relations...any component of k is of the form 2n/L where n is our positive or negative quantum number. This arises because when you satisfy the periodic boundary conditions, you find
Just as in the case of phonons...only certain wavelengths are possible for these electrons confined to the box.
The momentum i quantum mechanics is an operator
In the ground state of a system of N particles,
the occupied orbitals may be represented as points inside a sphere in k space...
The energy at the surface of the sphere is the Fermi energy and the wavevectors have a maximum value kF.
We can define a velocity at the Fermi surface...
This is for an electron at the highest occupied energy level, which can have a k-vector pointing in any direction.
We can also define what is called a Fermi temperature, but this is not a temperature of the electron gas.
It is a measure of where the Fermi energy is at (typically on the order of ~ 10000 K)
So, for most metals say at room temperature, not many electrons are excited above the Fermi energy.
The red curve is the square root of the energy.
The black curve is the red curve multiplied by the Fermi-Dirac distribution.
The blue line represents the Fermi energy, filled levels at 0 K.