Guide for assessable presentations:

Just a useful guide for assessable presentations.
http://sydney.edu.au/health_sciences/pdfs_docs/assign_guide.pdf

Good luck all with your presentations.

Superconductivity

A pdf that includes all main points about superconductivity and their magnetic property :

http://www.imprs-am.mpg.de/summerschool2003/muramatsu_notes.pdf

Student presentations

I expect you all to attend all the presentations and to fill out the feedback and assessment forms I will provide.

I will bring the data projector but it is your responsibility to bring a laptop and to check beforehand that you can get it to work with the data projector. I will set up the data projector at least 15 minutes before class time so you can do that.

Time limits will be strictly enforced.

Pseudogap phase

At the pseudogap phase there are still conductors with different properties of the usual conductors such as, linear resistivity. Pseudogap state has some similar properties with the superconducting state, but at a very high doping and temperature they act like normal Fermi-liquid.

semiconductor and Quantum dots

It is predicted that one day it may be possible to use the array of quantum dots driven by ultra fast laser pulse to carry out quantum information proceesing. Some semiconductors with conduction band electrons with effective mass less than 1/10th of an free electron mass this maske the size of semiconductor 10nm in the quantum limit (which makes easier to adress and manipulate individual quantum dots). This size is supposed to very useful to invernt some quantum-computers with much smaller size and weight. A paradigmshift toward direct use of the quantumproperties of information processingdevices is now underway.
Useful information about the semicondutor as quantum dots can be found in;
http://www.ph.utexas.edu/classes/li/QuantumDotIntro.pdf

Archive of Semiconducting materials

Here is an archive of semiconducting elements and compounds as well as heterostructures that are based on them. It gives information such as energy gap, carrier concentration, binding energies of dopants, effective masses of charge carriers and so on.
Not much in the way of theory but I thought it looked interesting.

http://www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html

Semiconductors lecture for reference

htp://videolectures.net/mit3091f04_sadoway_lec13/
A video about Intrinsic and Extrinsic Semiconductors, Doping, Compound Semiconductors, Molten Semiconductors. It's from MIT, a famous uni in the US.

Direct gap and indirect gap semiconductors

On the basis of band structure, there are two types of semiconductor: direct gap and indirect gap. The difference is that in direct gap semiconductors, the top and bottom of the valance and conduction bands are directly above each other (hence the name). In an indirect gap semiconductor however, the bottom of the conduction band is not directly above the top of the valence band.
This leads to a difference in the process to bring an electron from the valence band to the conduction band. A direct gap semiconductor only requires the transfer of energy (from say a photon) to reach the conduction band while an indirect gap semiconductor requires a change in both energy (from a photon) and momentum (from a phonon) to reach the conduction band.

problem

I tried several times today, but i still have a problem to post comments.

Doping

Generally, doping is the process of adding impurities to pure semiconductors in order to change its electrical properties. Doping in organic conductor can be chemical or electrical. There is also magnetic doping that can effect some properties like specific heat by adding small amount of impurities.

post a comment

i have the same Lan problem does anyone know how we can post a comment??

wide band gap in semiconductor

From the lecture note the energy gap in semiconductor is less than 2eV.However, i found in the wikipedia there is semiconductor that has wide band gap(more than 2eV) i couldn't understand what are the differences between this and insulator??

Impurity in semiconductor

Impurity know as donors or acceptors. The differences between them is that the former impurities provide extra electrons to the conduction band. Whereas, the acceptor impurities provide extra holes to the valence band. Also, the chemical valence helps to determine the impurity type if it is donor or acceptor. The donor impurities have high chemical valence, while the acceptors have low chemical valence.(Ashcroft and Mermin,1976)

Something's wrong

I cannot comment on any posts. Does anyone have the same problem?

An interesting guide to semi-conductors

http://britneyspears.ac/lasers.htm

For those who were a bit confused about the discussion of semi-conductors in lectures, the above link is a useful guide to their behaviour. Don't be alarmed by the URL, it is in fact a quite serious guide to semi-conductor physics, purportedly *written* by *Britney Spears*.

It has sections on the basics of semi-conductors, junctions, recombination and the other topics we touched on. I found this part on p-n junctions particularly useful.

http://britneyspears.ac/physics/pn/pnjunct.htm

The p-n junction

one of the most important application in semiconductors is tje p-n junction. when p(holes)-type and n(electrons)-type are joined the current will move in only one direction and not revers.
to see a clear picture about the p-n junction see this link
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/pnjun.html

Semiconductor and Band-gap

Some of the semiconductors like silicon (Si) and germanium (Ge) have band gap energy of 1.1 and 0.7 eV respectively, they absorb visible radiation. from the statical law of thermal energy distribution in solid, at room temperature there are significant numbers of electrons in valance band which acquire sufficient energy to cross the energy gap, this is semiconductivity. Unlike in metal, semiconductivity increases with the temperature not with the increased lattice vibration. The properties of semiconductor depends on the number and types of charge carriers n-type (negative- electrons), p-type (holes, positive) and can be controlled by the apropriate dopant elements. For more information;
http://ocw.mit.edu/courses/materials-science-and-engineering/3-091sc-introduction-to-solid-state-chemistry-fall-2010/syllabus/MIT3_091SCF09_aln03.pdf

question

i am confused about the physical meaning of j(j+1),s(s+1),and l(l+1) in paramagnetism and diamagnetism in chapter 31 in Ashcroft and Mermin
please can any one explain it ?

Course summary

You can hand this to me on friday June 24 but I will not start marking them until monday June 27.

Read the course profile for more information for what the summary should involve.

The summary should sent to me via email as a pdf file.

Since there have already been several incidents of plagiarism in this course, all the summaries will be run through the program Turnitin.

Tutorial today at 2pm

this will be open office hours
I will be available in the interaction room to answer questions about lectures, past tutorial and exam questions, your papers for presentations, ...
you might also try answering the problems for chapter 33 and 34 of ashcroft and mermin.

Updated lecture notes

You can download the latest version of my notes here.

Semiconductor Notes

Today's lecture notes are available for download here.

Paper presentation schedule

Here is the proposed schedule. You are welcome to swap with one another.
Each presentation will be 12 minutes plus 3 minutes for questions.
Time limits will be rigidly enforced.

The material below is from John Wilkins one page guides and should be read and applied before giving your talk.

• Preparing a talk. [html ], [postscript ]
• Talk Grit [html], [postscript] (Jim Garland)
• How to give a Terrible Talk 
• Viewgraphs Preparation: NOT 

----------
Monday May 30 2pm

Robert

FLUCTUATING VALENCE IN A CORRELATED SOLID AND THE ANOMALOUS PROPERTIES OF -PLUTONIUM

Shishir

THE BIRTH OF TOPOLOGICAL INSULATORS

Negar

Quasiparticles at the Verge of Localization near the Mott Metal-Insulator Transition in a Two-Dimensional Material

---------

Tuesday May 31 11am

Sam

A UNIFIED EXPLANATION OF THE KADOWAKI–WOODS RATIO IN STRONGLY CORRELATED METALS

Thurayana

TUNABLE FRÖHLICH POLARONS IN ORGANIC SINGLE-CRYSTAL TRANSISTORS

Lan

EXPERIMENTAL OBSERVATION OF THE QUANTUM HALL EFFECT AND BERRY'S PHASE IN GRAPHENE
-------
Wednesday June 1 noon

Josh

COMPLEX THERMOELECTRIC MATERIALS

Saeed

QUANTUM CRITICALITY IN HEAVY-FERMION METALS

Shahd

FLUCTUATING SUPERCONDUCTIVITY IN ORGANIC MOLECULAR METALS CLOSE TO THE MOTT TRANSITION

Updated lecture slides on superconductivity

are available here.
Again, reading chapter 34 of Ashcroft and Mermin is highly recommended (essential).

Flying frogs and levitating magnets with your hand

With all the assignments/speeches we have to sort through at the moment I though this would be a pleasant diversion on a Monday morning:

http://www.physics.ucla.edu/marty/diamag/diajap00.pdf

The paper is based on Earnshaw's theorem, which proves that there exists no stable equilibriums for 1/r^2 forces. In the simplest context, it explains why no matter how hard you try you've never been able to balance two magnets with their opposing poles as a kid.

The paper discusses how pre-WW2 it was shown by Braunbeck that it was possible to achieve stable equilibria with diamagnetic materials (hence why superconductors can levitate). They apply some of his basic theory to some very interesting examples, i.e. finding the stable equilibrium point to levitate a frog in mid-air. This is accomplished due to the dia-magnetic nature of many molecules, such as water and proteins. They even show how the minute diamagnetic nature of your fingers (and apparently a book on the Feynmann lectures) can be used to stabilise the equilibrium point of a small magnet in a field.

High temperature superconductivity

A critical temperature of 30K is the theoretical limit of the BCS theory of superconductors. However, some materials have been found to have Tc as high as ~130K (with one claimed as high as 164K). These materials cannot be explained by BCS theory although paired electrons may still be involved in at least some of these cases.

High temperature superconductors is currently one major problem within theoretical physics that remains unsolved (for now) and theory surrounding them is an active area of research.

http://adsabs.harvard.edu/abs/1993Natur.365..323C
http://prb.aps.org/abstract/PRB/v50/i6/p4260_1
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/hitc.html#c1

Confusing about supeconductivity devices

Superconductor exists in an acociatate with a critical temperature Tc. In fact, temperature does not stay at the same scale forever unless it is in lab condition. It means for every devices having supeconductivity character, they have to include a temperature adjuster to keep them always in Tc. I wonder whether my assumption is true?

Meissner Effect and Superconductivity

Meissner Effect:

When any superconductor is cooled below its critical temperature Tc, the external magnectic field is expelled.

some more useful informations can be found in this link.

http://www.phi.kit.edu/ee2/abb//Superconductivity-03.pdf

Superconductor

"The best information online about
superconductivity."
- Energy Science News
http://www.superconductors.org/

Magnetic structure

There are three different kinds of magnetic which are:

1. Ferromagnetic: In ferromagnetic all the moments are in the same direction of the spontaneous magnetisation.

2. Antiferromagnetic: In antiferromagnetic all the moments are in opposite direction of the spontaneous magnetisation and give the total zero moment.

3. Ferrimagnetic: In ferrimagnetic there are a mixture of moment in opposite direction but give the total non-zero moment.

Limit of superconductivity

Destroying superconductivity in a material requires the breaking of the cooper pairs. There are at least two ways to do this: heat the system to the point that the thermal energy is equal to the binding energy of the cooper pairs,
or the absorbtion of electromagnetic radiation of frequency w such that hbarw ~binding energy.

Superconductivitey

Superconducting phenomenon was observed in 1911 by the German scientist Onnes through his experiments on electrical conductivity of metals at low temperatures.He observed that the resistivity of the pure mercury disappeared suddenly at low temperature (4.2 K) this temperature ,which has this property,called critical temperature and above this temperature the resistivity is small and metal is in normal condition .

Energy gap in superconductor

For a normal metals there is a Drude peak, but the important point about the energy gap in superconductor is, some say that the condition for superconductivity is having the energy gap which is wrong. the energy gap is not a necessary condition for superconductivity.

Magnetism due to unfilled atomic shells

Magnetism in isolated atoms and ions can be due to the electrons of unfilled atomic shells. Although nuclei have a magnetic moment, it is usually ignored because it is much smaller than the magnetic moments of the electrons in the unfilled shells. In an isolated atom, the total magnetic moment for the atom is the sum of the moments of each electron. That can be found by summing the magnetic moment of the orbital ML and the moment of the electron spin Ms. These two moments will vanish for filled shells, so only the unfilled shells moment is count in the total magnetic moment.

Draft slides on superconductivity

Here is the current draft version of my Slides for the lectures on superconductivity.
But, they are no substitute for reading chapter 34 of Ashcroft and Mermin.

Antiferromagnetism

An antiferromagnetic interaction acts to anti-align neighboring spins. Antiferromagnetic interaction exists where energy J < 0, where J is the sum over all pairs, i, j, of an interaction term J(i, j), times the spin of atom i times the spin of atom j. J > 0 indicates a ferromagnetic interaction. The combination of both can lead to spin glass behavior.

When no external field is applied, the antiferromagnetic structure corresponds to a vanishing total magnetization. In a field, a kind of ferrimagnetic behavior may be displayed in the antiferromagnetic phase, with the absolute value of one of the sublattice magnetizations differing from that of the other sublattice, resulting in a nonzero net magnetization.

The magnetic susceptibility of an antiferromagnetic material typically shows a maximum at the Néel temperature. In contrast, at the transition between the ferromagnetic to the paramagnetic phases the susceptibility will diverge. In the antiferromagnetic case, a divergence is observed in the staggered susceptibility.

Various microscopic (exchange) interactions between the magnetic moments or spins may lead to antiferromagnetic structures. In the simplest case, one may consider an Ising model on an bipartite lattice, e.g. the simple cubic lattice, with couplings between spins at nearest neighbor sites. Depending on the sign of that interaction, ferromagnetic or antiferromagnetic order will result. Geometrical frustration or competing ferro- and antiferromagnetic interactions may lead to different and, perhaps, more complicated magnetic structures.

Superparamagnetism

Doing some reading I stumbled onto an interesting magnetic behaviour. Beyond our discussions of ferromagnetism, paramagnetism etc there exist more complex states of magnetic order/disorder. In particular there is a state known as superparamagnetism. This occurs in small ferro- or ferrimagnets, and is defined by flips in the overall spin state of the system due to temperature. The ordering of the system is not lost, only the orientation of the ordering (i.e. all spins may flip from up to down but retain <|S|> = 1).

This flipping is characterized by the Neel relaxation time:

tau_N = tau_0*exp ((K*V)/(k_B*T))

Where K is the particles magnetic anisotropic energy. The relaxation time may range from nanoseconds to years for particles. However over a time scale much greater than tau_N the net magnetization will be measured to be zero.

The above was assumed to be in zero magnetic field. However if we now apply a field to the superparamagnetic particle it will order accordingly, much like a paramagnet. Due to this ordering, the particle can be said to have a magnetic susceptibility. This magnetic susceptibility will be far larger than a normal paramagnet hence the term superparamagnet.

This state of magnetism is of great importance to technology, as it is a limiting factor for storage efficiency on magnet-based hard drives. Superparamagnetism sets a lower limit on the particle size which may be used to store information, as the relaxation time is proportional to exp(V).

Assignment 5 - due Monday May 30

You can get the latest assignment here.

In tomorrow's tutorial I suggest we do more practice questions. I have uploaded past exam papers from 2005 and 2007 that we can work on, but feel free to bring any other questions that you need help with.

Updated lecture notes

The latest version of my notes are available to download here.

Course schedule until end

Week 11
Tuesday 11am - Paul - magnetism
Wed. noon - Paul - tute
Wed. 2pm - Ross - lecture - superconductivity - read ch. 34 in Ashcroft and Mermin
Thursday 11am - Ross - optional problem solving session

Week 12

Monday 2pm - Ross - lecture - superconductivity - read ch. 34 

Tuesday 11am - Paul - lecture - semiconductors - read ch. 28

Wed. noon - Paul - lecture - semiconductors - read ch. 29

Wed. 2pm - tutorial

Thursday 1 pm - Ross - optional problem solving session

Week 13

student presentations

Energies of Singlet and Triplet States

Triplet spatial wavefunction is zero when the two electrons are at the same position, wheras,
the singlet wavefunction is nonzero. Because the electrons repel each other more when they are close to one another, we therefore expect the singlet to have more electronelectron repulsion and a higher energy. This rule turns out to hold quite generally and is called Hund’s rule : for degenerate noninteracting states, the configuration with highest spin multiplicity lies lowest in energy. Hence, triplet are expected with lower energies. more can be found on this site which may be helpful to understand soem more topics discussed in class.
http://ocw.mit.edu/courses/chemistry/5-61-physical-chemistry-fall-2007/lecture-notes/lecture26.pdf

Quantum Heisenberg Model

http://en.wikipedia.org/wiki/Heisenberg_model_%28quantum%29

The above is a link to a full description of the Quantum Heisenberg Model as we derived in class. Note that in class we considered the case of zero external field and unlimited neighbor interactions. Usually the Heisenberg model is referred to when considering a lattice with nearest-neighbor interactions only. The model is generally solvable for the ground-state, however as most of us are finding in Phys4040, it is not simple.

In general the Heisenberg model is much simpler when solved computationally, and in fact is much more efficient computationally than the classical Ising model. Using a suitable eigenvalue technique, for instance the Lanczos algorithm, it is possible to calculate the ground-state and lower excited states for an NxN lattice (and higher dimensions, although memory can become a problem for any lattice size which is a reasonable approximation to the thermodynamic limit). An important consequence one finds upon solving, is that the ferromagnet and anti-ferromagnet arrangements seen in class only occur for zero field. When a magnetic field is apply, a superposition of different spin states is found.

Optional problem solving session on thursdays at noon

For the next five weeks at thursday I will be in the interaction room from noon-1pm to help you with any problems you want to work through.

Notes on magnetic ordering

You can find the notes for today's lecture here. I would also recommend reading Chapter 32 from Ashcroft & Mermin beforehand.

Ions with a partially filled shell

Ashcroft Mermin p.650. Assuming that we have a free ion with a partially filled shell. Why did not electrons interact with one another, the ionic ground state will be degenerate? Please help!

Bohr magneton calculation

It is known that Bohr magneton is defined as (e.h_bar)/(2.m.c) where e is electron charge, h_bar is Planck's constant, m is electron rest mass. I'm just not sure about c. Is it the speed of light in vacuum? When calculating it, c is assumed to be 1, the same as in general relativity. Can anyone help?

Question about assignment 4

Can anyone find out how to get the factor of 2 in q2 for m*/m = UG/2E0(G/2)? becuase like the tutorial again I got m*/m = 2UG/E0(G). In the tutorial we also got E0(G) instead of E0(G/2), and I got the same result as tutorial again. I think maybe we should follow some other way. can anyone help me with it?

Kadowaki–Woods ratio

The Kadowaki–Woods ratio is the ratio of A, the quadratic term of the resistivity and γ2, the linear term of the specific heat. This ratio is found to be a constant for transition metals, and for heavy-fermion compounds, although at different values.

In 2009, Anthony Jacko, John Fjaerestad, and Ben Powell showed that the different ratios could be understood on the basis of different materials specific properties, such as the density of states and the electron density, even before electron-electron interactions were taken into account.

Fermi liquid theory

Landau's Fermi liquid theory can explain that why the independent electron approximation works, but it fails when there is a strong interaction between electrons, in one and two dimensions (where there are strong interaction between electron for example in cuprate materials), for high Tc and for heavy fermions near a quantum critical point.

Hund's Rule

Developed by the German scientist, Friedrich Hund (1896-1997), Hund's rule allows scientists to predict the order in which electrons fill an atom's suborbital shells. Hund's rule is based on the Aufbau principle that electrons are added to the lowest available energylevel (shell) of an atom.

Around each atomicnucleus, electrons occupy energy levels termed shells. Each shell is identified with quantum number, n, that defines the mainenergy level. Each main level is made up of a number of sublevels. These sublevels are identified by their shapes: s sublevels have 1 orbital, p sublevels have 3 orbitals, d sublevels have 5 orbitals; and f. sublevels have 7 orbitals. Each orbital can contain only 3 electrons spinning in opposite directions .

Although each suborbital can hold two electrons, the electrons all carry negative charges and, because like charges repel, electrons repel each other. In accord with Hund's rule, electrons space themselves as far apart as possible by occupying all availablevacant suborbitals before pairing up with another electron. The unpaired electrons all have the same spin quantum number (represented in electron configuration diagrams with arrows all pointing either upward or downward).

The Pauli exclusion principle states that each electron must have its own unique set of quantum numbers that specify its energy. Accordingly, because all electrons have a spin of 1/2, each suborbital can hold up to two electrons only if their spins are paired +1/2 with -1/2. In electron configuration diagrams, paired electrons with opposite spins are represented by paired arrows pointing up and down.

For example, if there are three available p orbitals (p^x, p^y, p^z) the first three electons will fill these one at a time, each with the same spin. When the fourth electron is added, it will enter the (p^x orbital and will adopt the opposite spin since this is a lower energy configuration.

Although Hund's rule accurately predicts the electron configuration of most elements, exceptions exist, especially when atoms and ions have the opportunity to gain additional stability by having filled s orbitals or half- filled or filled d or f orbitals.

Diamagnetism and Paramagnetism notes

Here are some notes on diamagnetism and paramagnetism from the national university of taiwan.
They cover similar areas to our lectures but also mention and includes crystal field splitting and nuclear demagnetisnation.

http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/chap11.pdf

Nuclear Diamagnetism

While reading about Diamagnetism, I came across something called nuclear diamagnetism. Which is similar to diamagnetism except that the external field effects the nucleons of an atom. I couldn't seem to find much else on it but I expect that since atomic nuclei are usually screened by electrons, this is only relevant for ions that have been stripped of most or all of their electrons such as plasmas.

Diamagnetism

The most important keys for diamagnetism are that the materials are very weakly respond to magnetic fields. Also,the atoms or molecules of diamagnetic materials contain no unpaired spins and the electrons orbiting in atoms oppose the action of the external field as with Lenz’s law. lastly.diamagnetism is not affected by changes in temperature and the value of the susceptibility is always negative.

Magnetisation

we understood from the lecture that magnetisation is " a measure of how a material act when a magnetic field is applied to it" . Also, i found this website that includes many information about magnetisation and answered in more detail the question of what happen to an atom/ion in a uniform magnetic field?

http://physics.unl.edu/~tsymbal/teaching/SSP-927/Section%2015_Magnetic_Properties_1.pdf

Updated lecture notes

The latest version of my notes is available here.

Instabilities of Fermi liquid

I found some interesting informations related to instabilities of fermi liquid.
Generally, Fermi liquid theory is very successful for single particle excitation in solids, but also it can have some phase transition into another state with some degree of order. Susceptibility is responsible for the phase transition in Fermi liquid. Some perturbation h is present in Fermi liquid which give the variation in O (charge, spin, current etc.), then (q,E) = xi(q,E) h(q,E), q = charge, E = energy. For E = 0 we observe the static perturbation and static response, so if there is increase in susceptibility then the small amount perturbation h can also effect the process. At a critical temperature Tc = 0 the susceptibility diverges so in this case a small perturbation would induce a finite response in the system. Now if the calculation is made over the susceptibility, there is prediction over the Fermi liquid phase, with symmetry, where the system wants to go with the change is temperature. This means the various quantities can diverge with the lower temperature (Tc). Lower the critical temperature the ground state breaks the symmetry. Hence the calculation made over the susceptibility does not hold any accuracy.

Some papers for student presentations

Here is a preliminary list of possible papers you can use for your presentation.
First come, first served.
Claim yours with a comment below.
I welcome alternative suggestions.

You will have to give a 15 minute presentation where you
-summarise the key ideas and results of the paper
-relate the contents to what you have learnt in the course
-state things you did not understand
-any weaknesses you see in the paper

Marks will be based on
-quality of presentation
-level of understanding of the paper
-ability to relate the paper to what you have learned in the course
-ability to answer questions

Experimental observation of the quantum Hall effect and Berry's phase in graphene

Ideal diode equation for organic heterojunctions. I. Derivation and application

Complex thermoelectric materials

Tunable Fröhlich polarons in organic single-crystal transistors

Understanding ion motion in disordered solids from impedance spectroscopy scaling

Quasiparticles at the Verge of Localization near the Mott Metal-Insulator Transition in a Two-Dimensional Material

The birth of topological insulators

Fluctuating valence in a correlated solid and the anomalous properties of -plutonium

Fluctuating superconductivity in organic molecular metals close to the Mott transition

Quantum criticality in heavy-fermion metals

A unified explanation of the Kadowaki–Woods ratio in strongly correlated metals

Magnetism lecture notes

My current notes for tomorrow's lecture can be downloaded here.

Fermi Liquid Theory and the Independent electron Approximation

In the lecture, the argument was made that at T>0K, the scattering rate:

1/tau = A(E1 - Ef)^2 + B(Kb.T)^2, where A, B are T-independent constants.

So the scattering time tau is approximately proportional to T^2. Arguments are made in A&M that the scattering time tau depends on the interaction potential from Thomas Fermi screening: 4.pi.e^2/k_0^2.

Performing quite a rough dimensional analysis on this quantity, they get that:

1/tau ~ (Kb.T)^2 / h-bar.Ef

At room T, tau is of order 10^-10 seconds, which is 4 orders of magnitude longer than the typical scattering due to impurities. This suggests that the e-e interactions do not have a significant effect on the validity of the independent electron approximation.

Slides for lecture on Fermi liquid theory - hard copies in lecture

Getting the most out of lectures

The lectures closely follow Ashcroft and Mermin. I suggest that before each lecture you read the suggested part of the text.
The lecture is designed to highlight the key ideas, concepts, equations, and experimental results. There is insufficient time to explain everything in detail, particularly to go through every step of the algebra in every derivation. If you are so inclined you should do this by yourself.
There is also no point in me writing out lectures notes which just say the same thing as what the text says.
You should then re-read the relevant part of the text after the lecture.

Todays lecture will be on Landau's Fermi liquid theory of metals and quasi-particles. The relevant parts of the text are pages 345-351.

My final lectures will be on Superconductivity, covering all of chapter 34.

Question about Mid semester exam

Can anyone tell me why there is not any factor of 1/a^2 in question 2 part b for g(E)?

Quasiparticle

Quasi particles can be defined as a compensation of a particle and its effect on the surrounding. A good example is holes: which are the missing of electrons, and that cause holes to be carriers for positive charges. Quasi particles approach is important in simplifying the many particles Schrodinger equation.

Hartree Approximation

Schrodinger equation can become very complex if the electron-electron interaction was included. So, to simplify the equation and actually solve it, Hartree Approximation is used to introduce the molecular orbital approximation and make the particles more independent.

Jellium model

I found a note that's somehow interesting to me. In the note, the Screening of the Coulomb interaction, Friedel oscillations, plasmons (collective oscillations)
are explained very detailed.
people.web.psi.ch/mudry/FALL01/lecture03.pdf

The basic eq of nonlinear Thomas-Fermi theory

While reviewing chap 17 (Ashcroft/Mermin), I cannot get the basic eq of nonlinear Thomas-Fermi(17.46) by combining (17.44) and (17.45). Can anyone help?

Some links regarding Hartree-Fock

Here some links about the Hartree-Fock method.

http://www.physics.uc.edu/~pkent/thesis/pkthnode13.html

http://www.chm.davidson.edu/ronutt/che401/HartreeFock/HartreeFock.htm

SI Units

Ross has been reminding us recently about the need for us to use units in calculations. He has a good point, but I for one have not sought to remember the necessary conversions between various different units, like J = kg.m/s^2, as I've always calculated in straight SI units and put the units on at the end. In a discussion about it, another student pointed out this is in fact an incomplete approach, as it turns out there are only 7 fundamental units. They are:

metre for length
kilogram for mass
second for time
ampere for electric current
kelvin for temperature
candela for luminous intensity
mole for the amount of substance.

I thought this was quite interesting, not just the fact that a joule is in fact technically not an SI unit, but more for the choice of ampere for electric current. This means technically a charge can be measured properly as A.s and not C. I would argue that C is much more fundamental, as if you change reference frames a current may disappear, but a collection of charge will not. Anyone know why ampere's are chosen, or disagree with my choice of C?

Plasmons

Plasmons are about to be brought up in the lecture material, but I stumbled across them in some reading of the Jellium model. They are a quasi-particle brought about by oscillations of the free-electron density against the positive ions in a metal.

An interesting consequence of plasmons is the optical properties of a metal. The frequency of the density oscillation is known as the plasma frequency. When light above the plasma frequency is incident on a metal it is transmitted, whereas light below this frequency is reflected due to the collective screening behavior of the electrons.

Paper presentation

I'm wondering about the paper presentation.

In regards to the presentation, what guidelines are there besides a (close) connection to the course? Are recent papers expected as opposed to older ones?

Question

Does anyone answer correctly part four of question two in the four assignment that we have done?if yes please, present it here?

Pauli Principle and e-e interaction

1. Fermi sphere cannot hold two electrons due to energy conservation and Pauli principle. So one of them must be outside the shpere for scattering.
2. Electrons can scatter itself at finite temperature, dE1 = KT = 1/40 eV at room temperature. For Ef = 2.5 eV, 10exp-4 electrons have chance for scattering.
3. Scattering rate tou ^-1 is directly proportional to T^2, low temperature and pure sample eliminates the thermal and impurity scattering so relatively good e-e scattering can be observed.
4. Fermi surface is stable because of Pauli principle.
Fell free to add some more points related e-e scattering.

jellium model definition

I found the definition of the jellium model that is "A model of electron-electron interactions in a metal in which the positive charge associated with the ion cores immersed in the sea of conduction electrons is replaced by a uniform positive background charge terminating along a plane that represents the surface of the metal"
http://www.answers.com/topic/jellium-model.

Limitations of Hartree Approximation

1. The Pauli principle doesnot hold good for many body approximations.
2. The total energy calculation comes to be positive i.e. electron gas is unstable.
Are there some more limitations?

Thomas-Fermi Screening

A basic illustration of screening involves considering a positive ion in an electron gas, which will aggregate electrons around itself. The presence of this negative charge surrounding the ion reduces the electric field of the ion; this is called screening.

The Thomas-Fermi theory of screening considers a slowly varying (in r) potential. The modification in the electron number density due to the presence of this potential is then calculated, as in the lectures. From this, after some maths, the total potential as a function of r, which is found to decay exponentially in r. The distance at which the charge is effectively screened (becomes negligible) is of the same order as the inter-atomic spacing, meaning the effect of the charge is not long ranged at all.

Energy levels of Helium atom

I found this websit when i search for explaination of Helium energy level. It might help.


http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html

Optional extra tutorial

Some students have asked for extra feedback and advice on passing the course.
Tomorrow at 1pm I will be available in the interaction room. If you wish to work through tutorial problems and/or past exam questions, and get feedback I will be available then. If it is popular enough I will do this every week, at a mutually agreeable time.

Exam practice and feedback

Several students have asked for concrete suggestions on improving their performance on the final exam. I suggest the following exercise.

Write out a new formula sheet.
Take the mid-semester exam and try and do it again in one hour by yourself.
Give me your written solutions and I will mark it as I would normally.
Note this will not change your mark, but hopefully it will help you get a better grasp of the material, and get more feedback of what I expect.

Hartree Approximation

The Hartree approximation arises from a first attempt to account for the effects of electron-electron interactions, which are neglected when using the independent electron approximation (as in the bloch model etc). The single-electron wavefunction seems insufficient to properly describe how the electrons in an N-particle system interact. So the Schrodinger equation for an N-particle system is considered, i.e. we want a wavefunction for the N electron system. However, attempts to solve this equation prove to be futile.

Instead, the single electron wavefunction that best represents the interactions is analysed, specifically, a potential that includes the potential of the ions U-ion and the electric fields from the other electrons in the system U-el (the approximation is made that this is given by a smooth distribution of negative charge). The contributions from all the electrons to the potential energy of the electron considered can then be worked out (A&M pg. 330-331) and plugged into the Schrodinger equation for that particular electron. If such an equation is written for each electron, the set of equations is known as the Hartree Equations. The Hartree equations are solved computationally by proposing a form of the potential U-el, solving the equation to obtain the wavefunction, from which the next U-el is found, and so forth until the potential reaches a certain degree of accuracy (does not change very much between successive iterations).

Energy bands

Here is a useful Pdf for better understanding of energy bands and periodic potential.

http://www.teknik.uu.se/ftf/education/ftf1/forelasningar/overview/energybands.pdf

Lecture slides on electron-electron interactions

Here is are the draft slides for the next few lectures.
At the tutorial on wednesday I will work through the answers to the mid-semester exam.

NFE model

Here is a very useful pdf for a nearly free electron model. (NFE)

http://www.cmmp.ucl.ac.uk/~ahh/teaching/3C25/Lecture19s.pdf

Summary of tight binding model

As it appears from the model name, in this approximation electrons are tightly bind to their atoms with limited interaction with other atoms in their neighbor. this approximation minimise the overlap. but small correction in Hamiltonian is needed.