Fermi surface net

So, with lecture, we've mentioned Fermi surfaces.

In an attempt to visualise this, I thought it appropriate to print off a net and make one, as follows the the other shapes...This one didn't turn out as nice as it did in the picture.

There was a net found on (another) condensed matter physics blog here.

Difficulties with Definitions: Bravais/Reciprocal Lattices

Hello Everyone,
As I'm sure we have all noticed, the exact definitions of the relationships between Bravais Lattices and Reciprocal Lattices are not exactly firmly defined. Question 3 on the assessed assignment, for example, seems to ask us to prove a definition! In this case, I can share my collected thoughts: this question is asking us to prove that the crystallography definition of Miller indices (i.e. the picture of the direct lattice axes and a plane intersecting them, according to Ashcroft and Mermin) is equivalent to the Physics definition of Miller indices (the co-efficients of the normal vector to the (hkl) plane. Confusing it all is how the plane is called the hkl plane: according to Ash.&Merm. the reason the plane is called the hkl plane is because these are the co-efficients of the normal vector that defines the plane!

Another problem I've come across is Question 1 of the assessed problems, where you need to find the reciprocal lattice of a 2D Bravais lattice. Obviously, if you take the 3rd lattice vector to be zero, you get zeroes everywhere (also known as the Australian Zero Party :~) )! I found a site suggesting that the 3rd vector should be a unit vector in the 3rd dimension, which seems to make a bit of sense to me, and the problem seems to work out when you do this. I'm not sure of the proper mathematical basis for it though (no pun intended); perhaps I should revise the parallelepipeds that were considered 'unimportant' by my maths C teacher (along with eigenvalues...)? Does anyone else know? Specifically, anyone who has done lots of maths at uni?

In conclusion, I think that Ash.&Merm. again have shown their canny textbook writing skills, by writing the most unambiguous description of the relationship between Miller indices and direct lattices I have seen (CHEM3004 was clearer, but only about the diffraction pattern relationships to Miller indices, not the direct lattice relationships. Admittedly, you use software to do that bit for you using the structure factors)!

Hopefully, this is helpful.
From down the Hill,
Josh Harbort

Mid-semester exam wed april 18 1pm

I propose to hold the mid-semester exam on wednesday April 18 at 1pm.
It will be worth 20% of the summative assessment.
It may cover all material up to and including week 6 (i.e. before the mid-semester break).
Here is a copy of last years exam. This years exam will be of similar structure and of comparable difficulty.

Lecture Slides on the tight-binding method

You can download my lecture slides on the tight-binding method from here. We will probably start to cover this topic in this morning's lecture.

Latest Lecture Slides

Today we may start looking at a specific application of Bloch's theorem to the case of an electron in a weak periodic potential. You can get a copy of my latest lecture slides here.

Crystal defects

I was thinking about about x-ray crystallography and how it can be also used to determine crystal defects. 

After all, real crystals are not perfect- there is usually some regions where the arrangement of the ions differ. Examples include point defects-consisting of missing or additional ions or impurities and line defects-also known as dislocations whereby the some of the atoms in the lattice are misaligned. Crystal defects are interesting cause they can significantly change the properties of the crystal. One of the obvious effects is the electrical conductivity of the material. Misalignment of the atoms in the lattice will reduce the strength of the crystal to external forces. Reading up on this topic, in some cases,  crystal defects can also change the colour of the material. When an negative ion vacancy is filled by an electron, the localized electron can absorb light in the visible spectrum such that the otherwise transparent perfect crystal becomes coloured. 

X-Ray Sources

X-ray sources are obviously an important part of x-ray crystallography. In the modern age there are now several sources of x-rays that are used for crystallography. It seems that the most important features of the source are to produce definite x-ray frequencies, monochromatic x-rays and a spectrum of different wavelength x-rays.

• X-ray tubes. Work by accelerating electrons from a source (air or a metal) along a tube and into a dense metal. When the electrons strike the metal, they lose kinetic energy as heat or as high frequency photons (x-rays). With this method, there is a definite maximum frequency (the fastest electrons) but there is a spectrum below it because of the random nature of the final collision.
• Cyclotrons. Work by accelerating electrons in a spiral with an alternating electric field pointing along the plane of the spiral and a magnetic field pointing perpendicular to the spiral (this is what makes the electrons spiral...:) ). Apparently x-rays are often produced in the same way as in x-ray tubes, by firing electrons at a piece of metal, but also from accelerating a charge in a magnetic field. The radiation frequency is dependent on the frequency of oscillation, and you also get higher harmonics of this radiation, but there is some spread due to non-uniformity of the magnetic field. 
• Synchrotrons. What could be better than a cyclotron than a relativistic cyclotron! Basically, when you solve all the equations and include relativistic considerations, you get two regimes of x-ray production: monochrome and broadband. Synchrotrons produce the radiation via emissions from the electrons. So synchrotrons are the best source, but one of the most expensive!

Are Bravais lattice related to Platonic solids?

So just thinking.
Bravais lattices have symmetry, when translated, their symmetry is preserved at all lattice sites, i.e. each lattice is the same.
There is the unit cell (of some sort) and these packed together form our (infinite) lattice.

Now consider the Platonic solids (i.e. cube, tetrahedron, octahedron, dodecahedron, icosahedron).
We can see that our simple cubic structure is just cubes packed around each other.
Are the others related? Visually, I can't see so, e.g. hexagonal, but otherwise?

Also, when you find the reciprocal lattice of a Bravais lattice (at least in the examples we've seen thus far), you get another Bravais lattice. E.g. Cubic goes to cubic. Face-centred goes to body-centre and body-centred goes to face-centred.
With the Platonic solids, if you consider the dual of it (swaps vertices and faces), you get another Platonic solid,i.e. tetrahedron duals with tetrahedron, cube with octahedron and dodecahedron with icosahedron.
I'm finding this similarity a bit suspicious - is it that common or just a nice shared characteristic?

Maybe they're related in the way they're stacked around each other?
You're trying to fill a space (infinite space) with a repeated pattern with no gaps. This is in the Bravais lattice and with Platonic solids.

Can anyone else see a similarity?

I have been reading a paper about trap states in P3HT, which is an organic semiconductor. These materials aren't as well defined as inorganic semiconductors (i.e. silicon etc), because they are largely amorphous. This makes them a nightmare to model. The trap states are basically energy levels that exist inside the "band gap" of these materials. In the paper, two traps are found. One is attributed to oxygen exposure, and one is found to be an intrinsic property of the material itself. By this, we really mean that there are some energy levels with low density of states in between the highest occupied molecular orbital (HOMO, analogous to the valence band) and the lowest unoccupied molecular orbital (LUMO, analogous to the conduction band). This got me thinking: imagine trying to model the density of states of an entire polymer chain, that would be crazy!

It makes one really appreciate looking at highly ordered systems like crystals.

Assignment 3: due April 4

This assignment on crystal structures and diffraction is due on wednesday April 4 at 1 pm. You will need to do some exercises with the program bravais in solid state simulations. A scan of the relevant section of the book is here.

Bloch theorem lecture slides

Today will start looking at the Bloch theorem. You can get a copy of my lecture slides here.

sommer program

Just having a play around on the sommer program, checking my interpretation.

Opening the program, we have the two halves.

On the left, there  is the red circle representing our Fermi surface (So size proportional to the Fermi energy - can alter this in the middle bars). Inside the circle are all of the possible wavevectors of electrons at zero temperature. So spatially, we have the x- and y-components of our wavevector represented by the white dots. The green dot would be the average.

As we run a simulation, we see that some of the white dots jump out of the red circle. This would be equivalent to the energies above the Fermi energy.
Applying an electric field, we see a drift of the wavevectors, i.e. white dots (and mean) drift to a side and outside the red circle.
Applying a magnetic field, we have the wavevectors (white dots) rotating about the mean.

I'm not sure how this relates back to our energy levels.


Sidenote: if you have a large scattering time (~1e3ps) and some electric field (~ 2 * 1e6V/m), everything drifts out of the Fermi surface, then it looks like it's snowing...

Crystallography Lecture Slides

You can download a copy of my slides here for today's and tomorrow's lectures.

visualising Bravais lattices

So, visualising Bravais lattices.
I'm finding this challenging in that these lattices are infinite, but when you draw/make a model, it's finite and you can't really see the symmetry as the lattice has finished.

There are more infinite pictures of the cubic lattices about halfway down this webpage, which I find makes sense. You can see how the adjacent unit cells fit in with each other and that all vertices in the lattice are the same.

Sidenote: Mathematica recognises input of ==face centered cubic  :D

Reciprocal Lattice

The most important concepts we learnt this week are the Bravais lattice and the reciprocal lattice. The Bravais lattice can be imagined physically as coordinates or holes in space where you can place atoms/molecule etc(basis). The reciprocal lattice being in k-space a bit harder, at least for me to imagine physically what it is. It is defined as a set of all wave vectors that yield plane waves with periodicity of a given Bravais lattice so I think each dot in the reciprocal lattice as the coordinate of a vector from the origin. A family of lattice planes correspond to a single wave vector in the reciprocal lattice which is convenient. Defining W-Z cell in real space makes sense but its k-space equivalent the first Brillouin zone, what does it represent?

Tutorial on wednesday 1pm: bring a laptop

We will look at the sommer program in solid state simulations and some tutorial problems [some of these have been exam questions....]

Make sure you bring a laptop with a working version of solid state simulations.

Maximum exam and assignment marks

You must include details of your calculations (including keep track of units) if you don't want to lose marks.
Just writing down the answer you got from plugging numbers into a calculator is not good enough. How do I know you did not just copy the answer off a another student?

Crystal Structures lecture slides

For the next part of the course we will be looking at crystal structures. You can download my lecture slides, which are based on chapters 4-5 of Ashcroft & Mermin, from here. I would recommend reading these chapters from the textbook in advance.

Boundary Conditions: Interesting Physics the Drude Model Misses

Hello everyone,
Here is something interesting I was considering about the Drude model: most of today's 'interesting' use of materials in the realm of computer hardware couldn't be modelled with the Drude model because of its ignorance of the boundaries of the substances used. Of course, the Drude model doesn't really lend itself to semiconductors either.... But it is interesting that the leading model of its time would be unable to model some of the most important condensed matter problems of the last century.

I suppose a question could be posed about the historical development of condensed matter: if semiconductors had been discovered earlier, would another model besides the Sommerfeld model replaced the Drude model? Also, was semiconductance discovered before or after a theoretical model suggested it existed?

Josh H

Universality in Ultracold Fermi Gas

This was the topic of the Physics Colloquium today. It was interesting talk by Chris Vale. It was about how  two-component Fermi gas can display universal properties at the cross-over between  Bardeen-Cooper-Schrieffer state (a pair weakly correlated  spin up and spin down electron form a Cooper pair in superconductors)  and BEC state (a pair strongly correlated  spin up and spin down electron form a boson). I.e. as you slowly increase the strength of the correlation you move from a BCS to BEC and somewhere in the middle (it was called unitary) weird things happen. Only a single parameter was needed to characterize this behaviour, called "the contact" because it is measure on how likely we can find a pair of particles at small separations. 

Being a experimentalist, he bragged about how his team used Bragg's spectroscopy to probe the Fermi gas while varying the strength of interaction via Feshbach resonance (something I still don't really get, if someone is familiar with this tell me). Anyway, what other universal parameters do you guys know? I think we just learnt one in class. (p.s. No Josh. You posted to many comments already =P).

I recall that the first time I learnt about the concept of universality was in a Phys2020 lecture Ross was giving ,where he emphasized several times on  the universality near critical points. e.g. law of corresponding states-real gases at the same reduced volume and reduce temperature exert the same reduced pressure.

 

An Application of the Thermoelectric Effect

The Voyager spacecraft (among many others) were powered by a Radioisotope thermoelectric generator. This sounds pretty cool, although admittedly not as cool as a Triaxial Fluxgate Magnetometer. The RTG works using the Thermoelectric effect, creating a temperature difference using a suitable radioactive material. It has the supreme advantage of not having any moving parts, resulting in very good reliability. Unlike solar cells, they can be used anywhere in the solar system, or beyond. Unlike conventional batteries, they will produce power for decades.

It is important to remember that these kind of amazing devices would not be possible without suitable models of metals, radioactivity etc. I guess this is one of the many reasons we study Physics, and especially solid state physics...

On a complete side note, according to wikipedia (the source of all knowledge and truth), the copyright holders of the music and images on the golden record placed on the voyager spacecraft "signed agreements which only permitted the replay of their works outside of the solar system." I find it absurd that such a distinction need be made.

Does anyone else know of some other cool applications of the stuff we've learned so far?

Why are metals shiny?

I believe the question asked in one of the classes this week was that why are metals so shiny? The explanation given in class was:

Metals have high free/conduction electron density > which means they have high plasma frequency > higher than the  frequency of visible light > the dielectric constant in the wave equation is negative >the solutions to the wave equation decay exponentially> no radiation can propagate> the electromagnetic wave is reflected> metals appear shiny.

In a classical picture, we imagine all the free electrons as oscillators which upon incident em fields absorb the em field and then re emit it. Why it is not lost dissipated as heat due to collision? Its because the frequency of the incident field, w>1/tau where tau is the relaxation time. i.e. the frequency of the oscillation is faster than the frequency of collision. As pointed in class, above the plasma frequency the metals become transparent such as in the UV region for alkali metals. One way of thinking about it is that the oscillating electrons cannot respond fast enough so em radiation just travels through as if nothing is there.

Then someone asked why other non-metalic objects which also reflect light are not as shiny? It probably has something to do with insulators not having as many free electrons which can respond to the field so they cannot reflect as well.  Maybe someone can give a better explanation? It was also pointed out that is also the fact that reflection of a metal surface is specular due to the smoothness of the metallic surface so it looks shinier.

Ok, that probably was boring to some of you so read this http://www.coolsciencefacts.com/2006/metal.html and tell me what you think.

Assignment 2: due March 21

This assignment on the Sommerfeld model is due on wednesday March 21 at 5pm. You will need to do some exercises with the program sommer in solid state simulations. A scan of the relevant section of the book is here.

Lecture slides on Sommerfeld model

In the second lecture today we will start to discuss the Sommerfeld model. The lecture slides largely follow Chapter 2 in Ashcroft and Mermin and so reading it is an excellent idea.

Saving data and figure of SSS

Okay. 

To add a title or change axes, labels in the figure, click [configure] then type in the box, then press [Enter] on your keyboard.

To save a figure: 

1.    When the graph has popped up, click [copy graph] in the top right. This opens a new window with the plot.

2.    Click [save postscript...] then in the new window type the name you wish in the Selection box with '.ps' at the end, e.g. T100K.ps . Then click [OK] or press [Enter] on your keyboard.

This saves my plot in the current window as a postscript in the sss/bin folder where all the drude.exe files are. These images can be opened in GSview, Inkscape, GIMP, PowePoint  and handled in the usual manner.

To save the data in the plot: 

1.    When the graph has popped up, click [copy graph] in the top right. This opens a new window with the plot.

2.    Click  [save data...] then in the new window, type the name in the Selection box followed by '.csv' at the end, eg. T10K.csv . Then click [OK] or press [Enter] on your keyboard.

This saves two or four columns of data of current window as a postscript in the sss/bin folder where all the drude.exe files are. 

For 2-column data, guess is that second column is the deviation, first column is unknown. For 4-column data, guess is that last three columns is deviation (or maybe just column 2 and 4), no idea for what first column is.

For those who were at tute, it is just what we were doing (at one stage), but with .ps or .csv at the end of the name!

Constantly looking at constants...

So, with Assignment 1, with the question of the particle in a 3D box, the energy eigenvalues to be shown seem a bit odd.

If we work with ℏ in our derivation then we will have the 2 in the denominator as in the question, but there is ℏ² missing in the numerator. If we do our derivation with h, then we don't have the π as is absent in the question, but we end up with a denominator of 8, not 2.
I think Andy found this too. Anyone else?
Apologies I didn't ask this in tute.

Digressing slightly now, we discussed this today, but it seems interesting that we would bother to define a whole new symbol, say ℏ, for an existing constant h, just divided by 2π.
This also led to a rather lengthy discussion involving the upcoming Pi Day, should we use π or τ (so 2π).

So, if we redefined π to be τ, then our current ℏ would just be h divided by τ. However, the new ℏ, would we still keep it as h divided by 2 π (so then it would be h divided by 2τ, or 4π) or would we still keep the ℏ definition (so h divided by τ, or 2π)?
I acknowledge that it is just constants which are (arbitrarily) assigned, but just out of curiosity...

Assignment due times

Ross, a few of us have been wondering at what time is the assignment due on Wednesdays? Do we hand them in an assignment box or to you in class?

Fermi-Dirac Distribution constant

I was wondering about the constant in front of equation (2.2) in A and M and its origin. Shown below:

The rest of the equation is probability of finding an electron in a state with the velocity v. So then,

must be the amount of states per volume within the velocity range. To derive this consider
,
and then rearranging:

The lambda terms express the volume occupied by one electron with this state. So this is the amount of electrons that can fit into one unit volume with the given velocities.
And since electrons are spin half particles we can fit double the electrons into any volume. So the origins of C are demystified.

Andy Posting

Hello, this is a test for Andy.

Scaling

We discussed this quite a bit in BIPH3001 since it's quite an important problem for biological systems. I find the relationships between energy scales quite interesting phenomena. For example, is it possible for a living creature to function in space, drawing its energy only from the sun? How big would it be? (cf. Tyranids) The modelling of behaviours of populations in a system is also an interesting question.

Another, more condensed matter related example is the question of if we could model the galaxies in the universe with the kinetic theory model.

The scaling part of these problems is whether the to the internal composition of the parts of the system is related to its overall 'macro' behaviour. Using the examples before, one can model the living organism as having certain rates/parameters that scale with the relative sizes and volumes of the organism, without looking at the constituent proteins of the animal; likewise, how does the behaviour of a population relate to the neural networks of the constituent animals? And how does the precise arrangement of planets in the galaxies relate to their overall placement?

These are all tricky questions of importance!

Lattice Crystal Structures

Since Prof. McKenzie brought it up, I thought it would be sensible to post about Bravais lattices and space groups (the 230 different symmetry groups allowed with the 14 Bravais lattices). Essentially, a space group is a particular arrangement of the elements in a unit cell which has a certain number of symmetry elements.

This description does not give us any information about the stability of these space groups, so it turns out that some groups are overwhelmingly common in nature and others have only a few examples (I'll have to check my notes to remember which ones are rarest).

Another interesting (but very useful fact) is that proteins (which are chiral in nature, i.e. have a handedness) cannot only form crystals with space groups that have mirror plane symmetry unless the other conformation is present also. Since biological proteins generally only occur with one handedness, it's impossible for them to form these crystal symmetries. This cuts down on the possible numbers of space groups that are allowed and thus makes x-ray crystallography of proteins much easier (but still not easy).

Free electron densities

Having a look at the free electron densities presented in Table 1.1 of Ashcroft and Mermin. Decided to plot them --- it looks much neater one colour at a time!
Red, green, blue denote group 1, 2, 3 elements respectively. The crosses represent the free electron density, n, and the the circles represent the radius of the atom wrt the Bohr radius, rs/a0.


The free electron density is defined as the number of free electrons per cubic centimetres. One would suspect that for bigger atoms, the free electron density would decrease, i.e. inverse correlation between atom size and free electron density.
So consider the groups of elements, so 1, 2, 3 free electrons for red, green, blue in plot above.

If we look at the free electron densities (crosses), then we see a general trend of moving down the group (following ----), the free electron density decreases. This would be expected since we have the same number of free electrons, but our atoms would be getting bigger to hold the rest of the electrons and the protons.
We can confirm this by looking at the radius of the atom (circles). Moving down the groups (following ....), we see that the radius does indeed increase.

Now consider the periods of the elements, so points with close atomic number.
We see that the atom size decreases (red -> green -> blue), so the greatest radii (circles) are for the group 1 elements. In agreement, considering moving to the right with the period (red-> green->blue), the free electron density (crosses) does indeed increase.

Trend seems intuitive, but thought it'd be nice to see a plot of it, just to check :)

Mean free path length

After reading Chapter 1 of Ashcroft and Mermin, I started to wonder about classical models. The relaxation time, when calculated experimentally, turns out to be 10E-14 to 10E-15. This can be used to find the mean free path, as long as you know the average electronic speed. Drude estimated this speed using classical equipartition energy, i.e.

1/2mv^2=3/2kT

This leads to a mean free path length of 1 to 10 angstroms. This seems reasonable. However, the chapter then goes on to say that the mean free path length is orders of magnitude larger than this, debunking the theory that the only collisions are off ions.

I always find it interesting when I encounter situations where different, not entirely correct, assumptions cancel each other out. It really drives the point home that one must be rigorous with one's assumptions, even when the end result is reasonable. 

Relaxation Time

The idea in the Drude model that I found to be interesting while reading chapter 1 was the idea of introducing the relaxation time, t -the average time between collisions and the fact that we only need one new variable in this model. It turns out that the original interpretation that it is time between electrons simply bouncing off ions was wrong but the idea of relaxation is till used in the later models. Quoting Ashcroft and Mermin "for in many respects the precise quantitative treatment of the relaxation time remains the weakest link in modern treatments o f metallic conductivity" which is why relaxation time independent quantities are of much interest. Even so, I can't think of any other way to develop a solid state theory without using this concept. The later chapters of the book may prove me wrong.

Where are your blog posts?

Each student should make one post on the blog each week and write a comments on three other student posts, in order to gain formative assessment points.

This does not have to be profound. e.g. A simple summary of one key idea, equation, or experimental data graph is fine.

Here are a few ideas you might post about:

the integer quantum Hall effect

the fractional quantum Hall effect

the quantum of resistance

what are the main assumptions of the Drude model?

what are the differences between the Drude, Sommerfeld, and Bloch models?

who were Drude, Sommerfeld, and Bloch?

Also, here are a few good examples from last year

Peltier effect

Bravais lattice symmetries

Neutron scattering

Postscript files in SSS