Ashcroft and Mermin
At T = 0, a pure semimetal behaves as good conductor, as there are some partialy filled electrons and holes band.
At T = 0, a pure semiconductor behaves as insulator, as the carriers are either due to thermal excitation or presence of impurities.
4/26/11
Fermi Surface
The fermi surface of Lithium is not known. What is martensitic transformation ? Is it like not having a proper crystal structure at low temperature? Can anyone help me ?
Reading for next week
Beginning next tuesday I will lecture on "Beyond the independent electron approximation", closely following chapter 17 and pages 674-9 of Ashcroft and Mermin.
Key concepts include:
The many-body Schrodinger equation
Hartree approximation and the self-consistent field
Hartree-Fock approximation and the exchange interaction
Screening
Fermi liquid theory
Singlet-Triplet splitting
Key concepts include:
The many-body Schrodinger equation
Hartree approximation and the self-consistent field
Hartree-Fock approximation and the exchange interaction
Screening
Fermi liquid theory
Singlet-Triplet splitting
4/24/11
How are predictions derived from fermisurface ?
As we know that the is an boundary separated the states that are occupied and unoccupied. Moreover, its shape is derived from the periodicity and symmetry of the crystalline lattice. It's said that, for these characters, Fermi surface is useful for predicting the thermal, electrical, magnetic, and optical properties of metals, semimetals, and doped semiconductors. Although I have read carefully Bloch model, band energy, electron on periodic and weak potential, I cannot understand how such prediction can be derive from Fermi surface? Any one can help?
Fermi surfaces
Electronic Fermi surfaces have been measured through observation of the oscillation of transport properties in magnetic fields H, for example the de Haas-van Alphen effect (dHvA) and the Shubnikov–de Haas effect (SdH). The former is an oscillation in magnetic susceptibility and the latter in resistivity. The oscillations are periodic versus 1 / H and occur because of the quantization of energy levels in the plane perpendicular to a magnetic field, a phenomenon first predicted by Lev Landau. The new states are called Landau levels and are separated by an energy where ωc = eH / m * c is called the cyclotron frequency, e is the electronic charge, m * is the electron effective mass and c is the speed of light.
Landau Levels
I found this useful for better understanding of the Landau levels and how to get quantum Hall effects out of it.
Necessary condition for observing the quantum oscillation
For observing the quantum oscillation three factors are important:
1. High magnetic fields
2. Low temperature
3. Clean samples which means having pure crystals
4/22/11
use of superconductor in train
Ever since the discovery of superconductors, there has been great interest in their use in electronics. It turns out that the magnetic properties of superconductors has been more useful in a larger variety of applications than the lack of resistance.
 | Maglev trains use superconductors to levitate the train above magnetic rails. This enables them to operate without friction, and therefore acheive unheard of speeds. The maglev train below is being installed at the Old Dominion University in Hampton, Virginia. It is the first to be installed in the US. Unfortunately, due to the short track it is on, it can only reach speeds of 40 miles per hour. Maglevs, with sufficient track, can reach speeds over 300 mph. A new Maglev train in Shanghai recently broke the 500 Km/h barrier (310 mph). These trains are also more efficient because there less energy loss to friction between the train and the track. |
4/20/11
Visualising Landau level filling and the IQHE
4/19/11
de Haas van Alphen Effect
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SSG_note/grad_chap14.pdf
This is a nice little collection of info on the de Haas van Alphen effect I found (might be where Ross pinched some pictures from).
For an example of how you can actually apply the effect to construct the Fermi surface, have a look through this paper:
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-14075.pdf
If you don't want to get into the gritty details at least check out fig. 2 to look at the relations between measurement and theory.
This is a nice little collection of info on the de Haas van Alphen effect I found (might be where Ross pinched some pictures from).
For an example of how you can actually apply the effect to construct the Fermi surface, have a look through this paper:
http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-14075.pdf
If you don't want to get into the gritty details at least check out fig. 2 to look at the relations between measurement and theory.
Lecture on determination of the Fermi surface
Here are draft slides for some of todays lecture which is closely related to Chapter 14 and the beginning Chapter 15 in Ashcroft and Mermin.
4/18/11
Notes on semi-classical transport theory
The notes I will use in today's lecture follow chapter 12 of Ashcroft and Mermin closely.
4/17/11
New Organic Conductor for Electronic Devices
A new organic polymer has been developed. It can be laid down using simple printing techniques rather than the expensive and elaborate methods used to process silicon, 6 times faster than previous organic polymers.
Now just as fast as silicon plus much cheaper, this inexpensive organic conductor could be used in areas where silicon struggles to compete, eventually slashing the cost of transistors, PDA’s, flat panel screens and bringing electronic paper into common use.
Now just as fast as silicon plus much cheaper, this inexpensive organic conductor could be used in areas where silicon struggles to compete, eventually slashing the cost of transistors, PDA’s, flat panel screens and bringing electronic paper into common use.
The Development of Organic Conductors
I found a link that may be useful for someone need to know more about Organic Conductor. This has 5 luctures: Lecture 1. the Development of Organic Conductors
Metals, Superconductors and Semiconductors. Lecture 2A. Introduction and Synthesis of Important. Conjugated Polymers. Lecture 2B. Solid State Polymerization
Lecture 3. Fullerene Chemistry. Lecture 3B. Molecular Engineering. www.lios.at/VirtualAcademy/.../Organic_Conductors_Intro_Lecture.pdf - Similar
Metals, Superconductors and Semiconductors. Lecture 2A. Introduction and Synthesis of Important. Conjugated Polymers. Lecture 2B. Solid State Polymerization
Lecture 3. Fullerene Chemistry. Lecture 3B. Molecular Engineering. www.lios.at/VirtualAcademy/.../Organic_Conductors_Intro_Lecture.pdf - Similar
Filled bands are inert.
The consequences of this are:
1. having full bands in insulators.
2. only the partially filled band supply the current in a metal.
3. electron current adding to the hole current is equal to zero.
Huckel assumption
Huckle method is based on two important points:
1. we only consider the pi orbitals.
2. valence electrons move in the potential field of the inner shell electrons and nuclei.
Huckel Method
Here are some lecture note from another lecture on the Huckel method. Some examples are shown including the benzene ring.
http://ocw.kyushu-u.ac.jp/3063/0002/lecture/QC_AH3_LM2.pdf
http://ocw.kyushu-u.ac.jp/3063/0002/lecture/QC_AH3_LM2.pdf
4/16/11
Practice Mid-Semester Question
Thought I'd throw this out there to see if anyone can help.
Has anyone had success with question 2 iv) of the practice mid-semester? I know that:
h^2/2m* = d^2E/dk^2
and have tried a few other methods, but I can't quite seem to get the answer. Any suggestions? I'm figuring I'm probably not taking the approximation near the boundary correctly.
Has anyone had success with question 2 iv) of the practice mid-semester? I know that:
h^2/2m* = d^2E/dk^2
and have tried a few other methods, but I can't quite seem to get the answer. Any suggestions? I'm figuring I'm probably not taking the approximation near the boundary correctly.
4/15/11
Delocalisation of electrons
i read one article that said "Delocalisation is when electrons are not associated with one atom but are spread over several atoms.So the electrons are not directly bonded with any atoms but effectively 'float' above and below the molecule in electron clouds".This means In the case of hydrocarbons, delocalisation occurs in Benzene rings, where a hexagon of six carbon atoms has delocalised electrons spread over the whole ring.For metals, electrons are delocalised over the whole crystal structure and outer electrons of the metal atoms are shared in an electron sea, and are not confined to particular atoms.
4/14/11
Molecular Orbital Theory
In our discussion of organic conductors, the notion of molecular orbitals was introduced and used in the Huckel Method. It was interesting that the electrons were treated not as dependent on a single atom, but rather being shared over the whole molecule, that is to say, the electron's motion is influenced by all nuclei in the molecule.
Each such molecule has a set of molecular orbitals, which are simply linear combinations of the atomic orbitals, and the coefficients of these sums (in the linear combination) are found using the Huckel method.
Advantages of organic electronics
One of the main advantages of organic electronics is the possibility of manufacturing components and circuits over large areas, while silicon chips are restricted to the area of circular pads of limited sizes.
Most of these advantages, however, is certainly the fact that organic electronics can be fabricated on plastic substrates, thin and flexible.
Only the flash memory transistor, the silicon component found in the pen-drives in digital cameras and MP3 players, continued to resist the benefits of plastic.
Now, Physicists built the first organic transistor Flash memory plastic.
4/13/11
Huckle Method.
A simpler explanation of the Huckle Method, http://www.jce.divched.org/JCEDLib/LivTexts/pChem/JCE2005p1880_2LTXT/QuantumStates/Bookfolder/L21Huckel_MO_Theory.htm
Reading for next week
Ashcroft and Mermin, Chapter 13
The semi-classical theory of conduction in metals
Important equations are
13.11, 13.19
13.22-25
13.44-46
13.58, 13.61
The semi-classical theory of conduction in metals
Important equations are
13.11, 13.19
13.22-25
13.44-46
13.58, 13.61
Magnetoresistance for Open Orbits
A good feel for what saturation/non-saturation of magneto-resistance represents can be found around p235-239 in Ashcroft and Mermin.
For the open orbit case we discussed in the lecture the form of the current:
j = sigma(0)n(n.E) + sigma(1).E
Where n unit vector in the direction of the open orbit. It was argued that the component sigma(1) vanishes in the high-field limit, whereas the sigma(0) contribution does not. This means that unlike the closed-orbit case the magneto-resistance will not approach a constant.
To see this last assertion it is useful to examine a simple case where in the high-field limit the current flow is directed in a direction =/ n, say n'. From our statement above this means that the projection of the electric field, E.n = 0.
Following the working in Ashcroft and Mermin, this gives a particular form for the electric field which I will not write here. Suffice to say, defining the magneto-resistance rho as;
rho = E.j/|j|
One can arrive at the expression:
rho = (n'.j)^2/(n'.sigma(1).n')
Now as we have previously argued that sigma(1) vanishes in the high-field limit we clearly see that the magneto-resistance must diverge, i.e. it grows with increasing magnetic field.
For the open orbit case we discussed in the lecture the form of the current:
j = sigma(0)n(n.E) + sigma(1).E
Where n unit vector in the direction of the open orbit. It was argued that the component sigma(1) vanishes in the high-field limit, whereas the sigma(0) contribution does not. This means that unlike the closed-orbit case the magneto-resistance will not approach a constant.
To see this last assertion it is useful to examine a simple case where in the high-field limit the current flow is directed in a direction =/ n, say n'. From our statement above this means that the projection of the electric field, E.n = 0.
Following the working in Ashcroft and Mermin, this gives a particular form for the electric field which I will not write here. Suffice to say, defining the magneto-resistance rho as;
rho = E.j/|j|
One can arrive at the expression:
rho = (n'.j)^2/(n'.sigma(1).n')
Now as we have previously argued that sigma(1) vanishes in the high-field limit we clearly see that the magneto-resistance must diverge, i.e. it grows with increasing magnetic field.
Lecture notes
You can download a copy of my notes for today's lecture on the similarities between the tight-binding approximation and the Huckel model here.
4/12/11
Magnetoresistance in orbits
I tried to find out why magnetoresistance in open orbits is higher than it in closed orbits? If any one have the answer please, display it here
Semiclassical Electron Dynamics
The semiclassical model of electron dynamics sets out to describe the behaviour of Bloch electrons between collisions. In order to have energy En(k) that is approx constant over all Bloch levels in a wave-packet, the spreading of the wavevector must be small compared to the 1st Brillouin Zone: Δk << |G| ~ 1/a, where G is a reciprocal lattice vector and a is the lattice constant. This implies that Δk >> a, meaning the wave-packet is spread over many primitive cells in real space.
The semi-classical model looks at the effect on the electron dynamics of external electric or magnetic fields that are approximately uniform over the scale of one such wave-packet. The periodic potential considered in the Bloch model is responsible for the important distinction between the classical and semi-classical models (because the free electron approximation is no longer valid). This potential changes on a scale far smaller than that of the wave-packet, requiring a quantum mechanical treatment.
The semi-classical model looks at the effect on the electron dynamics of external electric or magnetic fields that are approximately uniform over the scale of one such wave-packet. The periodic potential considered in the Bloch model is responsible for the important distinction between the classical and semi-classical models (because the free electron approximation is no longer valid). This potential changes on a scale far smaller than that of the wave-packet, requiring a quantum mechanical treatment.
Hall Effect
If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor. This is most evident in a thin flat conductor as illustrated. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. The presence of this measurable transverse voltage is called the Hall effect after E. H. Hall who discovered it in 1879. Note that the direction of the current I in the diagram is that of conventional current, so that the motion of electrons is in the opposite direction. That further confuses all the "right-hand rule" manipulations you have to go through to get the direction of the forces. The Hall voltage is given by
VH = IB/ned
VH = IB/ned Show The Hall effect can be used to measure magnetic fields with a Hall probe.
More slides on "http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/hall.html"
Revised schedule for this week
Today 11am
Ross McKenzie lectures on transport in the Bloch model
Wednesday noon
Ross gives tutorial looking at old mid-semester exam questions
Wednesday 2pm
Paul Shaw lectures on the relationship between tight-binding and the Huckel model in chemistry.
Ross McKenzie lectures on transport in the Bloch model
Wednesday noon
Ross gives tutorial looking at old mid-semester exam questions
Wednesday 2pm
Paul Shaw lectures on the relationship between tight-binding and the Huckel model in chemistry.
4/11/11
Assignment 4: due Tuesday May 3
Slides for electron transport in the Bloch model
Here are the slides for todays lecture.
They need to be read in parallel with chapter 12 of Ashcroft and Mermin.
They need to be read in parallel with chapter 12 of Ashcroft and Mermin.
4/10/11
A good clip to watch for the introduction to quasicrystal
http://www.youtube.com/watch?v=k_VSpBI5EGM
Types of Quasicrystals
quasiperiodic in two dimensions (polygonal or dihedral quasicrystals)
There is one periodic direction perpendicular to the quasiperodic layers.
- octagonal quasicrystals with local 8-fold symmetry [primitive & body-centered lattices]
- decagonal quasicrystals with local 10-fold symmetry [primitive lattice]
- dodecagonal quasicrystals with local 12-fold symmetry [primitive lattice]
quasiperiodic in three dimensions, no periodic direction
- icosahedral quasicrystals (axes:12x5-fold, 20x3-fold, 30x2-fold) [primitive, body-centered & face-centered lattices]
new type (reported in Nature, Nov.2000)
- "icosahedral" quasicrystal with broken symmetry (stable binary Cd5.7Yb)
Types of Quasicrystal
- In two dimensions, there are 3 types:
* octagonal quasicrystals with local 8-fold symmetry [primitive & body-centered lattices]
* decagonal quasicrystals with local 10-fold symmetry [primitive lattice]
* dodecagonal quasicrystals with local 12-fold symmetry [primitive lattice]
- In three dimensions (no periodic direction), there is only one type:
* icosahedral quasicrystals (axes:12x5-fold, 20x3-fold, 30x2-fold) [primitive, body-centered & face-centered lattices]
- Interestingly, there is a New type that was reported in Nature, Nov.2000). it is "icosahedral" quasicrystal with broken symmetry (stable binary Cd5.7Yb).
* octagonal quasicrystals with local 8-fold symmetry [primitive & body-centered lattices]
* decagonal quasicrystals with local 10-fold symmetry [primitive lattice]
* dodecagonal quasicrystals with local 12-fold symmetry [primitive lattice]
- In three dimensions (no periodic direction), there is only one type:
* icosahedral quasicrystals (axes:12x5-fold, 20x3-fold, 30x2-fold) [primitive, body-centered & face-centered lattices]
- Interestingly, there is a New type that was reported in Nature, Nov.2000). it is "icosahedral" quasicrystal with broken symmetry (stable binary Cd5.7Yb).
The reciprocal lattice and first Brillouin zone
The link to the PDF helps me for better understanding of reciprocal lattice and first Brillouin zone in 2D and 3D.
http://www.chem.tamu.edu/rgroup/hughbanks/courses/673/handouts/translation_groups3.pdf
I hope it can be useful.
A site showing 2D reciprocal vectors
This page shows 2d reciprocal vectors.
http://www.matter.org.uk/diffraction/geometry/2D_reciprocal_lattices.htm
This should be helpful for anyone who hasn't figured this out yet.
It's not a perfect site though :P
http://www.matter.org.uk/diffraction/geometry/2D_reciprocal_lattices.htm
This should be helpful for anyone who hasn't figured this out yet.
It's not a perfect site though :P
4/9/11
General form of reciprocal basis vectors
A general form of a set of reciprocal basis vectors, bi, is
Where aj is the real-space basis.
bi ·aj = 2πδij
Where aj is the real-space basis.
Why 5 fold symmetry is a problem?
As we know from the lecture that the symmetry axis of 1,2,3,4,6 fold exist in a periodic lattice . However, when we try 5 or 7 fold symmetry to construct a periodic lattice,it is hard to do it. the reason for this ,is can not fill the area of plane with a connected array of pentagons.
The definition of quasicrysta
As i found in one website that the Quasicrystal is a form of solid matter whose atoms are arranged like those of a crystal but assume patterns that do not exactly repeat themselves.
Penrose and Aperiodic Tiling
The tiling shown in the lectures was in fact one of three forms of Penrose tiling, and was not in fact the original form. The original created by Penrose used 5 different tile styles, shown here.
http://en.wikipedia.org/wiki/File:Penrose_Tiling_(P1).svg
The symmetry and other properties are completely equivalent between the three styles, and is an example of of substitution. This is where one can break the original shapes into smaller variations, and use these to reconstruct the the tiling.
Due to this possible substitution the tiling shares some features with fractals, in that it may be inflated or deflated, yet present the same image and symmetry.
A list of other forms of aperiodic tiling in various geometries can be found here:
http://en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles
http://en.wikipedia.org/wiki/File:Penrose_Tiling_(P1).svg
The symmetry and other properties are completely equivalent between the three styles, and is an example of of substitution. This is where one can break the original shapes into smaller variations, and use these to reconstruct the the tiling.
Due to this possible substitution the tiling shares some features with fractals, in that it may be inflated or deflated, yet present the same image and symmetry.
A list of other forms of aperiodic tiling in various geometries can be found here:
http://en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles
4/8/11
Mid-semester exam
Details and some sample questions are here.
The exam is worth 20% of the summative assessment.
We will look at some of the sample questions at next wednesday's tutorial.
The exam is worth 20% of the summative assessment.
We will look at some of the sample questions at next wednesday's tutorial.
Two helpful websites
A tutorial website from the University of Cambridge helps to understand how to generate the reciprocal lattice and to construct Brillouin Zones.
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/aims.php
Self study material in Solid State Electronics using Multimedia: A great website that shows lots of plots for the Crystal structure, Brillouin zones, energy bands and energy surface...
http://people.seas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u7/sse_tut_1/solid1.html
http://www.doitpoms.ac.uk/tlplib/brillouin_zones/aims.php
Self study material in Solid State Electronics using Multimedia: A great website that shows lots of plots for the Crystal structure, Brillouin zones, energy bands and energy surface...
http://people.seas.harvard.edu/~jones/ap216/lectures/ls_2/ls2_u7/sse_tut_1/solid1.html
4/7/11
The advantage of Bloch model over Sommerfeld model
Firstly, the Bloch model can explain the existance of the insulators. Secondly, the sign of the Hall effect can be given by the Bloch model. The Sommerfeld model cannot explain these, however, many of the problems of Sommerfeld model can be solved by taking the interaction of the electrons with the periodic potential.
4/6/11
Reading and some notes for next week
Ashcroft and Mermin
Chapter 12
Take particular note of
Table 12.1
Equations 12.6
subsection: Filled bands are inert
subsection: Holes
Here are some preliminary notes.
Chapter 12
Take particular note of
Table 12.1
Equations 12.6
subsection: Filled bands are inert
subsection: Holes
Here are some preliminary notes.
Tight-Binding Model
http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture20s.pdf
For anyone interested this collection of slides is a useful discussion of the basics of tight-binding theory. It goes through the same mathematical derivation of the energy eigenvalues, but also gives a good graphical example in k-space.
For anyone interested this collection of slides is a useful discussion of the basics of tight-binding theory. It goes through the same mathematical derivation of the energy eigenvalues, but also gives a good graphical example in k-space.
4/5/11
Wednesday's Tutorial
This is just a reminder that during tomorrow's tutorial (now at noon) we will be looking at the "Bloch" model of "Solid State Simulations".
Please bring with you :
- A laptop with the software installed
- A copy of the "Bloch" chapter from "Simulations for Solid State Physics". You can download it here if you haven't already done so.
Schedule for wednesday April 6
Just for this week the tutorial and lecture time slots will be swapped.
noon - Paul will give the tutorial
2pm - Ross will give the lecture
noon - Paul will give the tutorial
2pm - Ross will give the lecture
Tomorrows lecture on quasi-crystals
Here is the current version of the slides that will go with my lecture on Quasi-crystals.
Nearly free Electron Metal
Here are some slides, easy to understand nearly free electrons metal. http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture19s.pdf
4/4/11
six of Brillioun zones step by step
I found in my papers how to find the Brillioun zones in a lattice . so , I want to share you that
4/3/11
Bragg plane
A Bragg plane for two points in a lattice is the plane which is perpendicular to the line between the two points and passes through the bisector of that line. The first Brillouin zone for a point in a lattice is the set of points that are closer to the point than the Bragg plane of any point. In other words one can reach any of the points in the first Brillouin zone of a lattice point without crossing the Bragg plane of any other point in the lattice.
The nth Brillouin Zone
I've found an interesting definition for Brillouin zones:
"The nth Brillouin zone is the set of k-space points reached from k=0 by crossing exactly n-1 zone boundaries in the outward direction."
"The nth Brillouin zone is the set of k-space points reached from k=0 by crossing exactly n-1 zone boundaries in the outward direction."
two ways to proof the Bragg's law
There are two ways to proof Bragg's law, one of them is in wikipedia on this link
http://en.wikipedia.org/wiki/Bragg's_law
while the other one which is much easier can be written like that
AB = 2BC = nλ
sin θ = BC/d
BC = d sin θ
2BC = 2d sin θ = nλ
Bragg and Van Laue diffraction
Bragg and Van Laue formulation are two approach for reaching to the same phenomenon. For a constructive interference K = k'-k must be reciprocal lattice vector with K = 2dsina = (2pi/d)n, where a is the angle between the incident ray and lattice plane. So we reach the Bragg condition which is 2dsina=n* wavelength of incident ray.
Ewald's sphere
Ewald's sphere can be used in electron, neutron and X-ray crystallography. The relation between the wavelength of the incident beams and diffracted X-ray beams, the diffraction angle for a given reflection and the reciprocal lattice can be understood by Ewald's sphere.
4/2/11
Fermi surface
In this website you can see Fermi surface for different elements http://www.phys.ufl.edu/fermisurface/
Cleavage (crystal)
Cleavage is an important mechanism in examining the structure of solids and also in some industries. Some crystals have the probability to split along certain crystal planes. And because of the repeated structure these distinct and week planes repeat and become visible even to the naked eye.
Pic. ref.: ( http://www.doitpoms.ac.uk/tlplib/atomic-scale-structure/single2.php?printable=1)
The Energy gap in superconductors
The energy gap in superconductors is the gap between the energy bands which are fully occupied by electrons, and the bands which are fully empty. That Eg is one of the superconductors' properties that don't show in any other materials. The size of the energy band is about 1 eV, which is the required energy to break the band between 2 electrons pairs. At zero temperature, the electrons jump over the energy gap and create holes.
Constructing Brillouin Zones
I read one paper about Brillouin Zones that said they are a significant feature of crystal structures and their constructing for a two dimensional lattice is easier than in a three dimensional lattices. with two dimensional square lattice, Brillouin Zones constructed from the perpendicular bisectors of the vectors joining a reciprocal lattice point to neighbouring lattice point. Below two pictures show the construction of first and second Brillouin Zones
4/1/11
The reduced zone scheme
The reduced zone scheme is the representation of all energy bands within the 1st Brillouin zone. I will here describe in detail how it is constructed. We all know that the graph of E vs k for free electron in 1-D is given by a parabola (E = h_bar^2 k^2 / 2m). One such parabola can be drawn at (centred on) each K. The intersection of two parabolae is the bragg plane (at K/2, ... etc). The energy band is distorted near this Bragg plane (in the presence of the weak periodic potential) such that there is in fact no intersection, the bands are cut off from one-another and reach a value Uk higher and lower than the value which they would have intersected at. This is the origin of the 'band gap' of thickness 2Uk. Each Bragg plane has this effect on the parabola, that is, there is a band gap (of certain thickness) at each bragg plane. The reduced zone scheme arises when we represent all of the bands and their gaps within the first brillouin zone by translating them in with the appropriate reciprocal lattice vector.
Basic concept of Energy Bands
In solid state ,we have three different material metal,semiconductor,and insulator . The band energy of these materials depends on the gap between valence band and conduction band . We clearly can see from the picture that the band gap reduces from insulator to metal (overlap) which means metal does not need heat to be conductor while semiconductor needs.
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