Lecture slides on superconductivity

They are available here.
Reading chapter 34 of Ashcroft and Mermin is highly recommended (essential).

Updated lecture slides

The latest version of my lecture slides on magnetism are available here.

Confusion in today lecture

Hey, I was thinking about the question in class today about how the applied field does not change the Fermi energy or number of electrons in the bulk matter. There was some confusion about the approximation of integral,

(at zero K) as

 In particular I am writing about why g(E_f) appears in the second term. This should clearly be there purely from the definite integral being evaluated close to the Fermi energy. At this point we have not assumed the Fermi energy under the applied field is the same as the Fermi energy of the free metal (no applied field). The fact that this second term (i.e. the first order expansion of the integral) is zero regardless of the Fermi energy of the free metal, is the proof the Fermi energy is constant under the applied magnetic field. Recall the integral from E= 0 up to Ef of the density of states times the F-D distribution function at zero Kelvin must be equal to the number of particles (electrons in this case). Note this is exactly the first term in the above equation, where the upper bound is exactly the Fermi energy of the free metal. However the above equation is the expression for the number of particles under the applied field, hence the act of applying the field does not change (to first order) the Fermi energy.

Week 13 student presentation schedule

Monday 11am
Kiran, Josh, Dale

Tuesday 11am
Andy, Joseph

Wednesday 11am
Ann, Dave

Lecture slides

You can download a copy of my latest lecture slides here.

Superconductors: The Meissner Effect

This came up in Joseph's blog post, I figured I might expand on it here.

First thing is that the Meissner Effect is dependent on the incoming magnetic field strength.  If it exceeds some particular value then full magnetic penetration will occur and the effect will be lost.  Secondly the condition of critical temperature is required here too in order to set up the superconducting state.  It should be noted that there is a difference between a superconductor, and a material with an infinite conductivity.  A superconductivity requires the Meissner Effect to take place, if not it's the latter.

When a superconductor meets these requirements eddy currents are created on the surface of the material.  These happen to take on values that exactly cancel the magnetic field inside the material, resulting in zero magnetic field up to some depth.  Due to the mechanisms that creates these currents, they are in fact non-deteriorating.  This means that the magnetic fields generated are also non-deteriorating, ie the interior of the material has zero magnetic field always!

Whilst in general the interior of the material has this zero field, around the surface the fields don't exactly cancel.  The depth that this occurs at is known as the London penetration depth, and is dependent on both the geometry and composition of the material.

At a certain critical magnetic field it suddenly becomes more energetically favourable for the magnetic fields to penetrate through the material.  As a result the Meissner Effect breaks down, and the field jumps right back up in a discontinuous fashion.  Again, each material has their own individual critical magnetic field where this happens.

Anyone else looking forward to the presentations?

So, I know this is a little way away, but just been thinking about the presentations (for this course and Honours).

There's an interesting talk on symmetry (reminds me of the crystal structures!). Topic is interesting (and a little relevant to our course), but his style, I find curious.
He has a very tangible and personal opening.
His slides are pretty much just pictures. They're used quite effectively. We see the symmetries and they are draw/rotated so we don't have to try to hard to visualise it.
A Cayley table for the rotation operator is shown, but it's presented in a very subtle and gentle manner. He has shown quite technical methods, but has really made it uderstandable and usable.
(Sidenote: I like how he had a little competition/quiz in there, although I suppose not too appropriate to our course presentations)

I note that this talk is perhaps too general and not detailed enough for our course presentations. I, personally, would be interested in how each article relates to what we have learnt.

As a distraction, there's a great proposal for presentation methods here.

Magnetism

Magnetism is an interesting topic. There are many classifications of magnetism-ferromagnetism, ferrimagnetism, antiferromagnetism, diamagnetism, paramagnetism etc. It gets kind of confusing. The first kind of magnetism to be discovered and the one people are most familiar with is that found in permanent magnets-ferromagnetism since it is  the strongest type of magnetism.The usage of lodestones as a compass was probably the first application of magnetism.

Ferromagnetism, ferrimagnetism and antiferromagnetism are the permanent magnetic ordering in the material leading to spontaneous magnetisation in the material. Ferromagnetism is when all the magnetic moments are aligned in the same direction (If some of them are not aligned in the same direction but the net is not zero then its called ferrimagnetism) whereas in antiferromagnetism there is equal amount of magnetic moment aligned in both directions. Diamagnetism and paramagnetism are only temporary magnetic moments induced in the presence of external fields- due to interaction of the field with localized electrons in shells/ions (Larmor,Van Vleck and Curie)and conduction electrons (Landau and Pauli) .

We learnt in class that diamagnetism or floating frog magnetism is a universal magnetism exhibited by all materials although would be overshadowed if the material has other kinds of magnetism.  It has negative magnetic susceptibility-repels magnetic fields. It is general cause it due to the motion/response of the electrons to the external magnetic field. Ions with a filled shell (J=L=S=0) only exhibit (Larmor) diamagnetism.  For comparison, there is also Landau diamagnetism for free electron gas (metals). Superconductors exhibit fundamentally different, perfect diamagnetism-'Meissner Effect' and completely repels external magnetic fields.

In contrast to diamagnetism, paramagnetism has a positive magnetic susceptibility so the material is attracted towards the magnetic field. If the shell is one electron short from filled (J=0) it exhibits Van Vleck paramagnetism. For other unfilled shells (J not 0), it exhibits paramagnetism (no fancy name probably should call it Curie paramagnetism) which unlike diamagnetism is temperature dependent-Curie Law.  In contrast to paramagnetic effects due to ions, there is Pauli paramagnetism for the the nonlocalized conduction electons on the Fermi surface for metals which is weaker and independent of temperature.

Preferred Posting Topics

Out of curiosity, I was wondering if people preferred to post on things they can relate back to their projects/other things they have learnt. Obviously, this could make it easier to do a post (:) ). For me, it means I post stuff on NMR (which is interesting but hard physics!).

And if you do post on things from the course relating to other stuff, is it good because you are linking up different things or less good because you aren't spending the time to understand something new from the course material? :) Thoughts, anyway.

Another question: can this post count towards my posts for next week? :)

EPR, NMR and Condensed Matter

It is interesting that now we've looked at more of the electron interactions, there is actually quite a bit of overlap between PHYS4030 and my Honours project (a nice change!).

The first thing that comes to mind is the description of pulsed-experiment magnetisation using the Bloch equations: since one detects magnetisation in a resonance measurement, these are quite crucial equations to describe the problem! Of course, this is possible because an ensemble of electrons/nuclei in a sample can be looked at as an crystal and always acts as if it's in some kind of 'lattice' anyway with the nearby electrons/nuclei.

And, of course, the equations describing the energy of a system of electrons in a magnetic field is very similar to the Hartree-Fock method of generating solutions. For my project we are looking at biradicals, with each radical electron near a nitrogen atom. The Hamiltonian is a bit hard to write here but essentially looks like this:

H = 2*hbar.gamma.(S.B) + 2*hbar.gamma2.(I.B) + dipolar coupling terms between the nitrogens and the electrons + scalar coupling terms between the nitrogens and the electrons + an exchange term between the two electrons + an exchange term between the two nitrogens.

As you can see, even with words this equation gets pretty complex! The hbar.gamma.(S.B) and hbar.gamma2.(I.B) terms are the Zeeman interaction of the respective particle and the magnetic field B (S.B is a dot product) and you get two of them because there are two electrons and two nitrogens in my case; this bit is already simplified because we assume the electrons and nitrogens are completely symmetrical on either side of the molecule, which is not necessarily the case because the molecule changes shape in solution. The dipolar and scalar coupling terms between the electrons and the nuclei also have 4 terms each for the possible pairs (although some of these terms will be small compared with the other terms), and are known as hyperfine couplings in spectro-speak. For my project, we are essentially investigating how the changes in the exchange couplings of the electrons affect the energy.

After all of this, you get your experimental spectrum as a squiggly line showing the resonance frequency of all of these features! So you do see the energies directly, but they're all jumbled together with noise—this is why I spend so much time in the lab! It takes about 3'10" to do a 1 scan and you need about 100 for reasonable signal-to-noise ratio!

So, the reach of Felix Bloch was far beyond simple condensed matter! It's also pretty amazing that Albert Overhauser (I think his first name was Albert) and a few other guys you can look at on Wikipedia essentially derived all of this stuff in the 60s, and now almost every chemistry lab uses NMR or EPR every week (even every day if they're fast at synthesising stuff)!

Timetabling of student presentations

I was originally planning to do 2 per regular time slot.
However, Ian McCulloch has requested the student honours talks be 12 noon-2pm on wednesday.
This is fine with me, but may not be ideal for you.
Here are two possibilities:

1.
Monday 11am-12:10pm - 3 students give talks
Tues 11am-11:50am - 2 students give talks
Wed11am-11:50am - 2 students give talks
but n.b. those 2 students may have to also give their honours talk in the following hour!

2.

Monday 11am-11:50am - 2 students give talks

Tues 11am-11:50am - 2 students give talks

The three remaining students give talks at a mutually agreed

time on monday, tuesday, or thursday.

I will leave you all to decide what you want and negotiate with Ian. 

Student presentations - week 13

10% of the summative assessment is based on class presentation.

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

5 minutes will be allocated for questions and discussions.

Time limits will be strictly enforced.

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

Superconductivity gets an iron boost

Thermal and electrical transport across a magnetic quantum critical point

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

The birth of topological insulators

Quantum spin liquid emerging in two-dimensional correlated Dirac fermions

Wednesday's Tutorial - 1 pm

In tomorrow's tutorial we will have a look at questions 1 and 3 from Chapter 31 of Ashcroft & Mermin. To get the most out of this exercise I recommend attempting the questions in advance.

Lecture slides on paramagnetism and diamagnetism

You can download my lecture slides on paramagnetism and diamagnetism from here.

The Noble Metals

So following through the notes and the book simultaneously, it seems that chapter 15 isn't really touched on.  This is probably because it doesn't introduce too many new concepts, but gives multiple examples of band structure in different metals.  One thing that I decided to check up on was the noble metals.  I've heard it mentioned a few times, but at this stage had never known what exactly distinguishes a metal as being noble.

This wasn't entirely helped by the fact that there seem to be different definitions, the general definition given in the quick blurb on wiki states that a metal is noble as long as it is resistant to corrosion and oxidation in moist air.  However, the definition in physics for a noble metal is somewhat different, and it talks about them on pages 288-293.  The only three metals that meet these requirements to be noble are copper, gold and silver.

Essentially, these three elements inner electron shells are all filled, and  can be ignored in all further calculations.  The outer electrons are all found in six energy bands around the fermi energy.  However, at (essentially) all k values, 5 of these bands are located in a very small energy range, just beneath the fermi energy.  Meanwhile the sixth band wanders all over the place.  However, it's not a case of following the path of one band that goes across a wide range of energies, but all the bands seem to take turns moving about.  It just happens that whilst one band is wandering about, the other five bands are all located very close together (approximately 2-5eV beneath the fermi energy).  This isn't the case always, there are some moments where all six bands lie within this range, but for the most part this is how it works. 

Because of this structure, it's usual to classify the 5 close bands as the f-bands, and the wandering level as the 4s band.  This 4s band's energy closely resembles the lowest free electron band of a face centred cubic crystal, and it is this band that gives the metal its various properties.  The diagram on page 289 shows all of this in more detail.

Fermi surfaces in reduced layout

Something we mentioned in tutorial.

In the Drude and Sommerfeld our electrons were non-interacting, so we just had our sea of electrons and our Fermi surfaces were just circles (or spheres). We only had one circle as well as we did not have a condition of periodicity.

In the Bloch model, there is periodicity, so there is a Fermi surface in each unit cell (since each unit cell must be identical).

The Bloch model also introduces interaction between electrons, so our Fermi surfaces are no longer spheres, but can be oddly shaped things (as we've seen throughout the later part of the course).

Here, we also meet Fermi surfaces which can be open or closed. An open surface would continue forever in k-space.

Consider a closed Fermi surface (in 2D) that extends past the boundaries of a unit cell, arbitrary, take the First Brillouin Zone.

Now this Fermi surface also has to be repeated in the neighbouring cell.

Suppose we go from the repeated diagram (left) to the reduced diagram (right).

How are they equivalent?

What happens at the overlap?

There still has to be periodicity in each unit cell as we are in the Bloch model.

Ross suggested that the top would come down and take a bite out of the bottom.
Dave pointed out that the top position is equivalent to the bottom position (due to periodicity).

Somewhere in there, there was a mention of how they could be considered as from electrons or holes...

I'm probably just mixing bands and Fermi surfaces.

(Thanks Joseph and David).

Assignment 6: due May 16th

The next assignment is due Wednesday May 16 at 1 pm. You will need to do some exercises with the ising model from solid state simulations. You can get a copy of the relevant section of the book here.
In preparation for next week, where we will start learning about magnetism, please read chapters 31-33 from A&M.

Nuclear Overhauser Effect and Condensed Matter

The nuclear Overhauser effect (nOe) is a well-known phenomenon in NMR circles. Essentially, it is the transfer of spin polarisation from a species with a bigger gyromagnetic ratio (e.g. electrons) to a species with a smaller gyromagnetic ratio (e.g. protons, carbon-13 atoms). NOe occurs through space, and so this is quite useful in NMR because it allows you to work out which atoms are closest to which other atoms (so you can find out how many hydrogens a given carbon has, for example). The bit that's relevant to condensed matter is that it was first theorised by Overhauser (the guy who basically derived NMR physics) as a way to enhance the spin-resonance signal of metal atoms via electron spin-resonance. The NMR spectra of metal atoms is today used (among other things) to check the Korringa ratio—metals that do weird stuff don't follow the Korringa ratio! And we all know that weird stuff is an excellent starting point for condensed matter study!

See you at the tutorial,
Josh

(TMTSF)_2ClO_4

Hey guys, I decided to look up what this (TMTSF)_2ClO_4 thing is. Turns out it's an organic semiconductor. There is a picture here:

http://hoffman.physics.harvard.edu/materials/Cuprates.php

This was the first material to be superconducting at ambient pressure. Cool. These guys have some STM images of the material, which are quite pretty also:

http://avspublications.org/jvstb/resource/1/jvtbd9/v9/i2/p1013_s1

some of the other papers look like they have some pretty crazy maths:

http://prb.aps.org/pdf/PRB/v55/i3/p1299_1

but it looks like a really interesting material. Thoughts?

Azulene vs. anthracene

I posted about this on condensed concepts.