Azulene vs. anthracene

I posted about this on condensed concepts.

Lecture slides on Fermi liquid theory

Tomorrow I will discuss screening.
I will then introduce Landau's Fermi liquid theory which explains why the independent electron approximation works so well. Here are the slides.
Reading pages 345-350 of A&M will enhance your total lecture experience.

Which physicist do you think made the greatest contribution to condensed matter physics?

U don't say? Felix Bloch. Cause without him there would be nothing to learn after chapter 2, is what I would like to say but I am not sure whether it was named after him or he derived it himself.  I nominate

Lawrence Bragg for "services in the analysis of crystal structure by means of x-rays (Nobel Prize page)." Apparently he was the youngest noble prize winner (he shared the nobel prize with his father) up to date at the age of 25. Bragg's Law  is probably one of the most famous laws in physics which according to wiki, he derived it during his first year as a research student. 

 

Semiclassical Model

So, the Semiclassical Model just uses classical equations of motion, of a particle hbar/k to calculate electron trajectories. We then make the assertion that these trajectories are just the trajectories of the wave packets, so we're not violating uncertainty. I was wondering; can anyone think of a way to describe the electrons probabilistically, without something horrendous?

Edit:
I just thought I'd clarify: I mean, rather than calculating wave packet trajectories, is there a more general approach? I can't think of any that wouldn't be horrible to work with. I guess what I'm saying is that the semiclassical approach is quite nice...

Dispersion relation

Something that's taken my attention recently has been the presence of the energy bands particularly of the form of the dispersion relation, i.e. expression of energy wrt wavevector.

Transport properties, such as the conductivity tensor, current density and the Onsager relation, have expressions in terms of the energy.

Something I fund a little unsettling is why the energy holds such significance? As in, what is the physical significance of the energy?

I can see that the wavevector could be taken as an independent variable as the wavevector is related to the momentum of electrons, hence could be controlled by input energy or voltage.
I suppose this energy is the response to electron behaviour?

Effect of Spin on Bloch Model

So in our model that predicts the various Landau energy levels for a crystal in a uniform magnetic field, so far no consideration of electron spin has been introduced.  As it turns out, there are two ways in which spin affects the system.  According to the book, the major effect is that the Landau levels will be either raised or lowered a uniform amount, corresponding to

g*e*hbar*H/(4*m*c)

Where, Ashcroft and Mermin have once again reverted to using the H field rather than the B field.  Whether the levels are raised or lowered depends on the orientation of the spin to field; raised for along the field, and lowered for opposite it.

The density of states is altered as a result, and if g0(energy) is the density of states when spin is ignored, the new expression is

g(energy) = 0.5*g0(energy + g*e*hbar*H/(4*m*c)) + 0.5*g0(energy - g*e*hbar*H/(4*m*c))

I know when I first read this in the book, I was a bit confused as it seemed that the two factors would cancel out.  That was until I realised that they were functions of g0, not values inside a bracket multiplied by g0.  Just found it a bit ambiguous at first when I read it.

The other consideration not taken care of in the model so far is that spin-orbit coupling hasn't been introduced.  Whilst it maintains that any effect is small in lighter elements (it doesn't give a rigorous definition of either of those unfortunately) it has the effect of splitting further energy levels, reducing the degeneracy, and ultimately altering the density of states as before.  From what I can gather, this can be ignored up until we deal with Diamagnetism and Paramagnetism, whereupon it is reintroduced, and added to the Hamiltonian in order to explain atomic susceptibilities.

Beyond independent electron approximation

We will start to look at how things change when you include the interactions between the electrons. We will follow Chapter 17 of Ashcroft and Mermin closely, so reading it would be an excellent idea. Here are the lecture slides for monday.

Visualising the quantum Hall effect

There is a nice simulation associated with the Wikipedia page.
It shows how the Landau level filling varies as the magnetic field increases.
For low fields this is also useful for understanding how the periodic oscillations associated with the SdH and dHvA effects arise.

Consequences of the Semiclassical Approach

Right, it's friday night, time to do some blogging!

So now that we've got our Bloch structure for metals, we're starting to study the dynamics of particles using this theory. But I just wanted to post some important requirements and restrictions of the theory here (most of what I'm saying comes from chapter 12 of Ashcroft and Mermin).

So by applying a semiclassical approach, we neglect various quantum features (such as position-momentum uncertainty principles) whilst maintaining the quantum structure of the Bloch model. Furthermore, three rules of motion are enforced in this description:

1. There are no interband transitions (n stays fixed)
2. The position and momentum of a particle are known and related via

dr/dt = (1/hbar)(de/dk)
(hbar)k = -e[E + (1/c)v x H]

(I apologise about the terrible notation on here! They're equations 12.6a and 12.6b respectively)

3. Two electrons described by the labels n, r, k and n, r, k + K where K is the reciprocal lattice vector are in fact not distinct electrons. This is of course a consequence of the lattice theory used to derive the Bloch model.

The consequences of the semiclassical theory are the many-carrier theory (due to rule 1), crystal momentum is not momentum (due to rule 2), and that filled bands are inert (due to all three rules).

Assignment 5 due may 2

This assignment involves the ziman program of solid state simulations. The relevant section of the
book is here.

How to gain (or lose) easy marks in this course

Some of you still struggling to plug numbers into simple equations, keep track of units, and get the correct answer. This is not good. I think honours students should be able to do this in their sleep. If you can't you are throwing away easy marks.

If you struggle in this area, there is a simple solution.
Do lots of problems until you have mastered this skill.
Maybe even do some problems from a first year text that has answers in the back.

The most important equation in the whole course

It is Bloch's theorem.

Why would I claim this?

p.s. Only 1 out of 7 students included it in their formula sheet.
    Is that because you all meditate on it and so feel no need to write it down?

Mid-semester exam marks

Marks are out of 20.
First three numbers are the last 3 numbers of your student number

712 - 18
653 - 14.5
375 - 19.5
235 - 10
762 - 8
613 -16
none of above - 7

You can pick up your books for perusal from my office

Holes in our three theories

One of the things with which I am most impressed is the consideration of holes in the Bloch model.
It seems intuitive as it's something that's still flowing, but it's not that something's flowing, it's that something's not flowing.
Considering holes is much like how we consider the movement of air bubbles in water.

If we have our valance band with only few electrons then it would be easier to consider only the movement of the few electrons instead of the many holes.
Similarly with holes.

Intuitive almost, but the holes are a big hole in the Drude and Sommerfeld models where only electrons are considered.



By considering the movement as being from electrons or holes, so negative or positive charges, the signs of constants come out correctly.
Example would be the Hall coefficient where the Drude and Sommerfeld have the wrong sign (since they only consider movement of negative carriers).
The Bloch model considers positive and negative carriers, so the sign of the Hall effect comes out as positive for movement of holes and negative for movement of electrons.

Completely non-serious post regarding midsem

Just something amusing from Joe Grotowski's course from 2007.
http://www.maths.uq.edu.au/courses/MATH1051/Semester2_2007/secretcowlevel.html

Although I would suspect the reverse is more accurate...for now.

Harmonic Oscillators and FTs

So I was thinking about how holes originate from an assumption of quadradically dependent dispersion relations. This is essentially the same as assuming a simple harmonic oscillator.

 I recall a article/forum post* that said all applied physics is simple harmonic oscillators and Fourier transforms. This is almost true for our case. We use Fourier transforms to move to reciproical space and this is the basis of the majority of our analysis. The Bloch model extends this even further as even energy band gaps are just the Fourier components!

 A simple harmonic oscillator allows us to understand holes, a very physically meaningful and intuitive quasi particle. In my earlier encounters with holes I just believed them to be real because they make sense.

 Its interesting that a couple of mathematical tools can bring so much physical insight into vastly different applications. Perhaps I should have paid more attention to Fourier transforms in previous years ;) Good thing simple harmonic oscillators are simple. Even in quantum mechanics they are really nice :)

I propose a new judge on the beauty of a theory. The simpler a simple harmonic oscillator is in this theory the more elegant the theory. That's why quantum is much more elegant than classical mechanics! But I digress...

 Can anyone else think of examples from this course of simple harmonic oscillator power??

 Andy PS Hugh David Politzer 2004 Noble Prize Laureate sung a song about the simple harmonic oscillator.

 *Citation Needed

WPP, TB and Hubbard model.

So Chapter 9 tells us that WPP works well for metals in group I,II,II and IV of the periodic table because the valence electrons are not tightly bonded to the ion core because of screening and pauli exclusion principle. And we were told in chapter 10 that  tight binding works well for transition metals. Is there a more concrete reason why it works well for transition metals which have partially filled d orbitals? If WPP and TB are the two extremes, what about those that are not either extreme case?

An extension to the tight binding model would be to include electron-electron interaction, the simplest case being the Hubbard model where we only consider on-site electron interaction, i.e. if there is an electron pair on the same lattice site then the Hamiltonian would include an extra energy term due to the repulsion between the two electron with opposite spins.

Lecture slides on SdH and dHvA

Tomorrow or next monday we will start to talk about quantum magnetic oscillations and determination of the Fermi surface. Much of this will follow chapter 14 of Ashcroft and Mermin.  Here is the current version of the slides.

Lecture notes on transport theory

Here are the lecture notes on Transport properties of metals in the Bloch model. They follow Ashcroft and Mermin closely. We will start on this tomorrow or wednesday.

Revision!

Honours Room.
3pm today...and again tomorrow!!

A&M on the Tight-Binding Model

The differences in approaches between the tight-binding and weak periodic potential approximations are interesting. The WPP approximation is an expansion of the plane-wave solutions to Schrödinger's equation  whereas the TB approximation comes about from making a correction to the free atom Hamiltonian. So, the TB approximation takes another step at the start and then one works out the appropriate solutions but the WPP approximation modifies the solutions to fit the weak potential case. I suppose that it must be easier to apply the 'strong' potential in the TB model with a change to the Hamiltonian. :)

PS: Another note: a suggestion by spell-checker for the apparently incorrectly spelt 'Schrödinger's' is 'Ladyfinger's'; also, while 'Ladyfinger's' is spelt correctly, 'Ladyfinger' is not!

A&M on the Electron in a Weak Potential

A&M go through this topic in a slightly different way to Dr Paul's lecture notes. The description of the approximation part of the text has two main sections, the second of which (Bragg plane example) is analogous to the lecture notes. The first section essentially justifies us looking only at the near-degenerate atomic orbitals (the two orbitals from different atoms that are similar in energy)—this is because the other energy levels are not shifted by much when compared to the shift in the near-degenerate orbitals—and continues to explain the near degenerate case (the result of this bit is partially in the lecture notes). The second section then uses this to describe the case where two near-degenerate free electron orbitals are distanced from the other energy levels, as in the notes. 

PS: An interesting question: why do no spell checkers recognise 'orbitals' as being a word, when it is very suggestive that it should have a plural from its grammatical structure? 

Hall coefficient and Lorenz number

So, starting to look back through the material from the beginning of semester,
We have the Hall effect, which is the potential difference that we get in a conductor perpendicular to a current.
The Hall coefficient is the ratio of the transverse electric field to the product of the current and magnetic field, i.e. E_y/j_xH.

We have the Lorenz number which is the ratio of κ/σT  essentially a ratio between the thermal and electrical conductivity.

Now, electrical conductivity would appear in the Hall coefficient in the current term. Can't see how the thermal term would enter the Hall coefficient.
Likewise, can't see how the magnetic field would enter the Lorenz number.

Besides being able to find an expression for both from the Drude model, are they (physically) related?
(Sorry if this sounds a bit random, my notes are quite squished together...)

Tutorial Wednesday 11/4/12

As requested there will be an additional tutorial tomorrow at 1 pm in the interaction room. The plan is to go through some practice questions relating to the the Bloch theorem. Please bring a copy of the tutorial problems with you.

As an additional tutorial attendance will not count towards your formative assessment mark.

Quasicrystals and logic

I like how the definition of the crystal (from lecture) had to be redefined to be that the crystal exhibits discrete x-ray diffraction peaks and no longer requires a periodic arrangement of atoms.
It reminds us that how we see and understand the world changes as we continue to undermore more.

The definition of periodic arrangement of atoms seems intuitive and if you were to crack a crystal small enough, that is what you would get.
X-rays were only discovered just before 1900.
Before then, I'd imagine that it'd be incomprehensible to even suggest that discrete x-ray diffraction peaks should be used to identify crystals.

The distinction between sufficient and necessary (or if and only if ☺) certainly makes me more careful of my interpretation of what I read. Particualarly in everyday language, I note that I have become a little sloppy with my language.
Few instances would be when we have a derivation and we say suffice when we mean suffices and necessitates, although that still holds. Main issue would be when we say necessitate when we actually mean suffices.
I'll make sure I am more careful now.

Sidenote: below is a simulation of five fold symmetry from an icosohedral quasicrystal from http://www.alienscientist.com/quasicrystals.html. Makes you think about what other patterns exhibit 5-fold symmetry...and it looks pretty ☺

Another sidenote: this diffraction stuff of lattices with recirocal space etc reminds me of fraunhofer diffraction. Anyone else see that?

Brief History of Superconductivity

I did a literature review on that superconductivity for my capstone course. Superconductivity celebrated its 100th anniversary last year. The first person to discover superconductivity was Onnes in 1911 and it was observed in mercury when cooled below its critical temperature of 4K using liquid helium. Later other superconducting materials were found but it took more than half a century after its first discovery before a successful theory was developed. The BCS theory (named after the physicists who discovered it- Bardeen, Cooper and Shrieffer) . The theory predicts that below the metals critical temperature, electrons (fermions) with opposite spins  form Cooper pairs(bosons) which flow without any resistance-Cooper pairs form a superfluid/BEC. I not sure how in detail how this works but we will learn about it later in the semester. 

But the fact is that everyone thought they understood superconductivity until high temperature superconductors (higher than the theoretical limit of  BCS theory~30K) were discovered by Bednorz and Muller in 1986. Bednorz and Muller found a copper oxide material superconducting at 35K. Just a few months later, another group found material in the same class to superconduct at 93K. Interestingly copper oxides are insulators at room temperature.  As you would expect, the physics community (and probably the public as well) would go crazy at that time. It was an exciting time to do condensed matter physics. BCS theory fails to explain high temperature superconductivity. 

Even now,  there no theory explains high temperature superconductivity. I say this cautiously because there are theories out there but not one that everyone can agree on to fully explain this phenomenon as good as BCS theory explains low temperature superconductivity. If it piques your interest, the two most prominent ones are the resonant valence bond (RVB) theory and spin fluctuation theory. Currently the highest critical temperature stands at 135 K, recorded in 1993.  Apart from high temperature and BCS superconductors, there are many other exotic superconductors like organic superconductors researched at UQ.  In 2008, a new family of high temperature superconductors were discovered - iron pnictides. 

It is interesting to mention that all the people I mentioned so far have a Nobel prize, and as every condensed matter physicists out there knows there is certainly one waiting for the person to explain high temperature superconductivity.  

Organic Conductors

Right, I've been a little behind with these, but better late than never I suppose!!!

So one thing that I found quite interesting was the synthesis of organic conductors, and especially how they relate to superconductivity. I found this paper http://iopscience.iop.org/1468-6996/10/2/020301/, which unfortunately only lets me view the abstract at this stage (anyone else have any better luck with it?). But it briefly describes the history and new physics that arise due to them.

In terms of temperature at which these compounds superconduct, the keeper of all knowledge wikipedia says that these guys have a critical temperature of up to 33K http://en.wikipedia.org/wiki/Organic_superconductor, which is much less than the ceramic ones that can be created. However, I found this article which suggests theoretically you could get organic superconductors that operate at hundreds of Kelvin http://prola.aps.org/abstract/PR/v134/i6A/pA1416_1. Interesting to see where things go with these guys

Assignment 4: due April 18

This assignment on the Bloch model is due Wednesday April 18 at 1 pm - note the time! You will also need to do some exercises with the Bloch model from solid state simulations. You can get a copy of the relevant section of the book here.

Lecture slides on semi-classical electron dynamics

Here is the current version of the slides.
They follow Ashcroft and Mermin closely so reading chapter 12 is an excellent idea.

Wednesday's tutorial at 1 pm

At this week's tutorial we will be looking at the problems in "Tutorial 4: The Bloch Model" so please bring a printout. To get the most out of tomorrow's tutorial I would also recommend having a go at answering some of these questions beforehand.

Lecture slides on metals and insulators

This week we will start to look at semi-classical transport theory within the Bloch model, following chapter 12 in Ashcroft and Mermin. First we will review the key differences in the band structures and band fillings between metals and insulators. Here are the relevant lecture slides.
Reading chapter 12 would be excellent preparation.

Lecture slides on quasicrystals

Tomorrows lecture will consider the fascinating subject of quasicrystals, the subject of last years Nobel Prize in Chemistry. Here are the lecture slides.

This is not covered in Ashcroft and Mermin, for profound reasons.
Instead you could read Section 5.8 of Marder, or the notes on the Nobel Prize site.
The lesson is don't believe everything the textbooks (and your lecturers) say!

Bloch models

The periodicity of the potential in crystal stuctures allows us to approximate the many body schrodinger equation  as a one electon schrodinger equation with an effective potential, U(r) that is periodic. One extreme was treating the electons to be bounded by a weakly periodic potential so they are nearly free and the other extreme was to treat them to be tightly bounded to the atom. Group I,II,II and IV  is best described by weakly bounded valence electrons whereas transition metals are better described by tight binding model. Which model works best is depends on how strongly bounded the valence electrons are?  In reality, would the electrons be somewhere in between this two extremes but I believe that there are other models as well. Paul just posted on the Huckel model which  is just the term chemist use for tight binding model (which physicist use for crystals)  for molecules.

Lecture slides for the Hückel method

Tomorrow we will learn about the Hückel method, which describes the molecular orbitals of conjugated organic molecules and bears strong similarities to the Tight-Binding method. You can download a copy of my lecture slides here.