# Help: How to interpret energy band?

1. Feb 21, 2005

### leoant

As we all known, energy band is very important to everyone who want to go longer in physics, especially in condensed matter physics. However, as a student of physics, I am shamed to say that I cannot interpret the picture of energy bands well, thus would someone be kind to tell me the secret or refer me to some books or articles.
As an example of my problems, I would take the picture attached(electronic structure of magnesium dibrode)as an example: how can one assign some curves belong to a band(in the picture, one could assign one of the curves to sigma bands which belongs to B-B)?
And further more, if the curve stands for the energy of electrons in solids, what can we learn for the band structure? Since one coordinate of the picture is K vector, the change of energy wiht respect to K means what exactly?
Thanks very much!

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2. Feb 21, 2005

### ZapperZ

Staff Emeritus
First of all, what you are asking isn't easy to explain online since it requires me sketch stuff.

To be able to decipher any band structure, you first need to know AND have in front of you the crystal structure of the material, or more specifically, the reciprocal structure of the material, preferably the 1st Brillouin zone. This will allow you to know the designation (those letters) of the different momentum or k directions. Different crystals and structures sometime use different symbols and letters. However, I think $$\Gamma$$ is universally used to indicate the center of the zone.

Once you have the picture of the 1st Brillouin zone, then you can follow the "path" indicated in the band structure. For example, if you look at the horizontal axis going from Gamma to M, let's say, in your band structure, then you are doing the same thing along that direction in the Brillouin zone. So the horizontal axis simply tells you the momentum or k values, and what bands are there as you go along all the high-symmetry directions of that crystal.

Next, look at where E=0 is. This is your Fermi energy level. For metals, there will be at least ONE band that crosses this level. For insulators, you will usually have no bands crossing this (there are exceptions). If you can find the gap between the lowest energy minima above the Fermi level and the highest energy maxima below the Fermi level, then you have found the band gap of this insulator. If they occur at different different k (different horizontal axis value), then you have an indirect band gap.

The "spaghetti-like" picture that you normally get out of one of these graphs is due to the calculation of several different bands coming from all the constituents of the material. In many cases, you are only interested in the valence band, which is normally the band closest to the Fermi level. Take note that these band-structure calculations typically use the Fermi-liquid model, meaning they don't usually consider strong electron-electron interactions. So for many "exotic" compounds, such as high-Tc superconductors, these things can be spectacularly wrong. This, however, may not be the case for MgB2.

A good reference for band structures would be Walter Harrison's book "Electronic Structure and the Properties of Solids : The Physics of the Chemical Bond". It's a Dover book, so it isn't that expensive to get.

Zz.

3. Feb 22, 2005

### leoant

Thank you very much again!~~~~~~~
As to the symmetry of crystal and the first Briillouin zones, I think I have got some knowledge of them. And what have tortured me for so long a time is how to assign one of the curves to, for example, sigma of B-B. And from the energy band what exactly we can learn about properties of materials.
I will get the book and try to find the answer, but it seems that a book is too much for me and is there any paper or review which can help me?

4. Feb 22, 2005

### ZapperZ

Staff Emeritus
I will have to say that you are being baptized by fire by using the MgB2 band structure here to learn these things. MgB2 has a very complex structure because superconductivity in these materials appear to come out of two different bands - the pi band and the sigma band. You cannot tell from just looking at the figure to know which is which. So I suggest you start off by using a simpler material with a less complicated band structure.

Unless, of course, for some reason, you have to analyze MgB2.....

Zz.

5. Feb 22, 2005

### Gokul43201

Staff Emeritus
I second this suggestion. Any solid state/electronics text will show you the E-k curves for simpler materials like Si, Ge, GaAs. Start here. Identify the band gap, and determine whether or not the material is photonic. Then look at dopant levels, curvatures (effective mass), etc.

After you are comfortable with this, you may look at superconductors. And it would be instructive to look at curves for the normal state as well as the superconducting state. Do you notice any change in the periodicity ? Can you identify the appearance of a gap ? Are there any directions where you see a "gap" in the normal state ? In the case of MgB2, I recall something about two different gaps (from the sigma and pi bands?) but never really followed up on it. I still find it strange that you can have two fairly differing gaps, yet only one Tc, and no other funniness at some other characteristic temperature.

PS : I think any paper/review reporting ARPES or other photoemission measurements will give you enough curves to look at, but go with Zz's word on this.

Last edited: Feb 22, 2005
6. Feb 22, 2005

### ZapperZ

Staff Emeritus
I can make an explicit recommendation on what paper to read. Try Valla et al, Phys. Rev. Lett. 83, 2085-2088 (1999). I know this paper very well. I joined this particular group right after they published this paper (damn it!). I also used this paper as a strong comparison with the overdoped Bi-2212 high-Tc superconductor in my own paper. So I know it inside out.

But what is more amazing is that the Valla et al. paper shows the band that's crossing the Fermi surface directly from the ARPES measurement. The image that you get in Fig. 1 was obtained directly from measurement and clearly shows "text book" band dispersion that most of us only read about. One literally gets the E vs k disperson using this technique. What is even more amazing is that one can actually see the many-body interactions from this disperson from the deviation or "kink" in the disperson near the Fermi energy. You can literally obtain the self-energy terms from the experimental data directly with minimal data processing.

Zz.

7. Feb 23, 2005

### leoant

Thanks very much :!!)

Last edited: Feb 23, 2005
8. Feb 23, 2005

### leoant

HELP: how to build unit cell?

Shame that there are so many things I don't understand~~~~Here is my another question.

Since dear ZapperZ and Gokul43201 have told me what basically a energy band mean, and I have read some. However, when I want to calculate band structure of some simple material, it seems too hard and I am frustrated at the first step: how to build a unit cell of a kind of concrete material which can be used in a program, for example, Gaussian.

Someone said that reduced coordination should be used, for example, in diamond, the unit cell is tetrahedron, and there two atoms in the unit cell, thus we can get C(0,0,0) and C(1/4, 1/4, 1/4), assuming lattice constant to be 1. However, when I use this to write an input file for Gaussian, it doesn't work and in the result there are only 2 atoms. The space group is not right and it seems that I have to find some method to find the point group. Or should I have to have my unit cell rebuild?

May someone tell me how to build unit cell?

9. Feb 23, 2005

### ZapperZ

Staff Emeritus
You should NOT double post! That's against PF rules. I will not respond to this on here but rather stick to the original thread. A PF moderator should delete this thread.

Zz.

10. Feb 23, 2005

### Tom Mattson

Staff Emeritus
Why delete, when it's so easy to merge?

leoant, if you have a question that continues on with the material from an already existing thread, you should just go ahead and continue that first thread. It makes discussions much easier to follow, and you won't shake loose the good people who are helping you!

Last edited: Feb 23, 2005
11. Feb 23, 2005

### kanato

I don't use gaussian, I use abinit.. is there a way in gaussian to define your primitive vectors? abinit has a tutorial (#3) on doing electronic structure calculations of elemental silicone, which is a diamond structure. The primitive vectors given in there are
$$\vec{a_1} = \left( 0, \frac12, \frac12 \right)$$
$$\vec{a_2} = \left( \frac12, 0, \frac12 \right)$$
$$\vec{a_3} = \left( \frac12, \frac12, 0 \right)$$
with the correct primitive vectors and correct atomic positions (which you have) you should get the correct space group and point group. At least, that's how abinit works.

12. Feb 23, 2005

### leoant

Sorry for bringing inconvenience to everyone but I really want it to be a new post... Anyway, hope your help sincerely and I will follow the rule next time.

13. Feb 25, 2005

### Gokul43201

Staff Emeritus
I just remembered there's a nice review article by Timusk and Statt on experimental probes used on High Tc Superconductors. The PDF can be downloaded from Timusk's website.

14. Feb 26, 2005

### leoant

I have fixed it. Thanks very much

So lazy to ask a so easy question, and now I have fixed it.
I use PWSCF to do some calculation and now I have two powerful softs: XCrySDen and PWgui. With them,I find it easy to try again and again until you get a correct data.
Thanks everyone here.

15. Mar 2, 2005

### Pieter Kuiper

The absolute value of $$k$$ is $$2 \pi/\lambda$$. For free electrons this is easy: you take the de-Broglie-wavelenth $$\lambda = h/p$$ and you will find that kinetic energy $$E=p^2/2m= \hbar^2 k^2$$, which is the dispersion relation for free electrons.

Reciprocal space is a bit abstract, but there was a paper (in Science?) about STM on surface states on copper. Inside a quantum corral these were standing waves. The wavelength was dependent on the STM voltage, a clearly electronic effect. And a plot of the voltage against reciprocal wavelength showed the free-electron parabola with a relative effective mass close to unity.

For angle-resolved photoemission on MgB2, this is a good paper: http://arxiv.org/abs/cond-mat/0111152
(I have used it in exam questions.)