Talk:Electronic band structure

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Proposed merger[edit]

Proposed merger from Energy band because that article is a stub with nothing to indicate why its topic is different from this one. Further, the history indicates that over several months, nobody has editted the stub significantly. --The Photon 02:32, 5 September 2006 (UTC)


I'm sure the diagram I've just added could use some improvement. Please comment here with your suggestions. -- The Photon 04:19, 6 September 2006 (UTC)i don't understand the band diagram .so plze explain it ??????

I believe the definition of the fermi level used here in the diagram to be less useful in terms of visualisation than the alternative definition where it is designated as the energy of the highest occupied energy level of the solid at Absolute zero. By this definition the fermi level lies within or at the top of the valence band. In conductors, overlap of valence and conduction bands are not essential as long as its fermi level is sufficiently below the top of the valence band. In semiconductors and insulators the fermi level would be very near the top of the valence band. Hence the band gap is important in these cases. (talk) 11:35, 19 September 2009 (UTC)


I think it would be a good ideea to replace the diagram with an example (Si for example) and also show on the diagram how the line in single atom beacoms a band in atom in crystal with high number of atoms (so that the splitting of orbitals is also shown) --Fractografie 20:26, 10 October 2006 (UTC)

What is a band?[edit]

The article talks about bands without stating explicitly what they are.

The answer is in the first sentence of the article: "ranges of energy that an electron is 'forbidden' or 'allowed' to have". However I would definitely agree the article is not totally clear that this is the definition of a band. -- The Photon 00:49, 26 November 2006 (UTC)
I edited the lead to include a definition for energy band, which is in boldface because a search on that term redirects here. Hopefully I got it right (I'm just learning this stuff by reading Wikipedia, and my understanding of the details doesn't go very deep into the article). Wbm1058 (talk) 02:26, 20 December 2011 (UTC)

What's a quantum mechanical electron wave?[edit]

doesn't seem to be explained anywhere...

First, you've forgotten to sign properly. Second, it is related to the quantum-mechanical wave associated with every particle. I suppose you know that waves show particle-like behavior, when they interact with matter. Well, this implies that the opposite is also true, namely, particles show wave-like properties when they ancounter conditions within which they reveal their wave-like behavior, such as openings whose order of magnitude is compared to their associated wavelength. E.g., they can undergo diffraction. See also matter wave or de Broglie. BentzyCo (talk) 18:12, 13 July 2008 (UTC)


Quote article: In recent physics literature, a large majority of the electronic structures and band plots are calculated using density-functional theory (DFT), which is not a model but rather a theory, i.e., a microscopic first-principles theory of condensed matter physics that tries to cope with the electron-electron many-body problem via the introduction of an exchange-correlation term in the functional of the electronic density. DFT-calculated bands are in many cases found to be in agreement with experimentally measured bands, for example by angle-resolved photoemission spectroscopy (ARPES). In particular, the band shape is typically well reproduced by DFT. But there are also systematic errors in DFT bands when compared to experiment results. In particular, DFT seems to systematically underestimate by about 30-40% the band gap in insulators and semiconductors.

It must be said that DFT is, in principle an exact theory to reproduce and predict ground state properties (e.g., the total energy, the atomic structure, etc.). However, DFT is not a theory to address excited state properties, such as the band plot of a solid that represents the excitation energies of electrons injected or removed from the system. What in literature is quoted as a DFT band plot is a representation of the DFT Kohn-Sham energies, i.e., the energies of a fictive non-interacting system, the Kohn-Sham system, which has no physical interpretation at all. The Kohn-Sham electronic structure must not be confused with the real, quasiparticle electronic structure of a system, and there is no Koopman's theorem holding for Kohn-Sham energies, as there is for Hartree-Fock energies, which can be truly considered as an approximation for quasiparticle energies. Hence, in principle, DFT is not a band theory, i.e., not a theory suitable for calculating bands and band-plots. End quote

To begin: DFT is a theory as far as the derivation of the Kohn-Sham Equations is concerned. However, once we start with (obviously necessary) approximations to the exchange energy, it becomes a model or method. Most people use the term DFT to refer to the method/model isntead to the theory, therefore, the first sentence is misleading. Especially as DFT is relly just a rather narrow branch of many-body quantum mechanics, and not really what most people undestrand under a theory (a wide-reaching concept). The exchange-correlaton term is not relatied to the functional of the electron density, instead, the Hamiltonian is written as a function of the density and incorporates a xc-Term. The DFT Bands are (in principle) exact for all the occupied states in the ground state. What is correct is, that the eigenstates above the Fermi energy are not necessarily equivalent to exited states - however, at least in theory: If we do excite an "electron" in the real world, we do change the band structure anyway - but that is a problem of the concept of a band structure, not necessarily of DFT. Also, the Kohn-Sham system is as much "physical" as any other represantation - it's properties (which can be calculated through the density matrix) are - assuming exact exchange energy - the same as those of the "real" system, and all observables are identical with the real ones. Of course, other opinions are heartily invited, but if no one protests, I'd modify the article accordingly... —Preceding unsigned comment added by (talk) 03:32, 13 February 2009 (UTC)

band gap as definition of semiconductor or insulator?[edit]

To my mind that is not correct. There are many materials we use as semiconductors and refer to as semiconductors that have a far higher band gap than many insulators. It is the ability of a material to be doped that makes it a useful semiconductor or just another insulator. --Adarah85 (talk) 11:33, 13 August 2010 (UTC)

John Clarke Slater[edit]

Prior omission of Slater is very worrisome. I have put in a place holder pending expansion by an expert. Also, OPW methods require mention. Michael P. Barnett (talk) 03:09, 22 February 2011 (UTC)

Conductivity and Probability flow[edit]

Apologies for my ignorance. I understand in classical electrodynamics conductivity means flow of electrons, now since here conductivity is understood in term of solution of Schrodinger's equation, what is conductivity then, which differs conductor from insulator. Now if one looks at the Bloch wave, as one approximation mentioned in this article, I will intuitively assume conductivity, the flow of electrons, is connected with the probability flow of electrons, that is (from the conservation of probability implied by the Schrodinger's equation)

\vec J=\frac{\hbar}{2im}(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*)

substituting the expression of Bloch wave, I get \vec J=\frac{\hbar \vec k}{m}. So no matter where the electron is, it is moving like as a flow. Now I kinda guess, the summation of all the reciprocal vector \vec k inside the Brillouin zone cancels out when the valence band is filled up. Is it true? Electrons are actually move collectively in the valance band but do not present macroscopic flow due to cancellation?

Then another question is, how to understand this in the case superconductivity? Gamebm (talk) 17:13, 27 November 2012 (UTC)

I don't think your calculation is right. With \Psi = ue^{ikr}, I get
\Psi^* \nabla \Psi - \Psi \nabla \Psi^* = 2ik|u|^2 + u^* \nabla u - u \nabla u^*
Can you please double-check? --Steve (talk) 23:29, 27 November 2012 (UTC)
Also as a side note, this won't help you understand superconductivity at all, since it doesn't arise from the band structure...a13ean (talk) 23:47, 27 November 2012 (UTC)
Thanks for pointing that out. I was study the topic and for some reason incorrectly assumed that u must be real. Now it seems the first term (|u|^2) will not change the conclusion since if one sums up all the k at a given spatial point, it is just a constant. However if the second term does remain, it will give a non-zero contribution to the flow (locally). Then I may understand that a filled band does present some flow locally since the charge flow calculated from QM does not vanish(? - is it measurable?). Now globally, since the second term is surface like (or due to its periodic behavior), it still may not lead to macroscopic flow, (since the length scale in our question is of lattice size.) and for the very same argument, the first term reduces to a irrelevant normalization constant.
I have no idea what superconductivity is. I was just trying to think of it following this line of thought, if all those electron pairs are in the same state (no idea what the wave function shall look like, will study it), and one is again legitimate to calculate its probability flow, but if everything is in the same state, and the resulting flow does not vanish (since it conducts not by excited to another state, but simply staying in this state?), then does this imply that there is some preferred direction in space? Gamebm (talk) 18:04, 28 November 2012 (UTC)
Electron motion is properly calculated using the formula for the group velocity of the electron wavepacket ... not the formula for probability current. They must be related ... I would not be surprised if they are identical ... but I'm not sure of the details off the top of my head.
As pointed out by A13ean, this line of thought will not lead you towards understanding superconductivity. Band structures are part of the "single-particle picture", where you by-and-large ignore the fact that electron quasiparticles interact with each other. But interactions between electron quasiparticles are the entire basis for superconductivity. --Steve (talk) 19:14, 28 November 2012 (UTC)
If you're interested in that bit, the last chapter of Ashcroft and Mermin gives an overview of how superconductivity arises from electron-electron interactions. a13ean (talk) 19:25, 28 November 2012 (UTC)

Wording issue[edit]

In the section "why bands and band gaps occur", the article states: "If multiple atoms are brought together into a molecule, their atomic orbitals split into separate molecular orbitals...". According to the article on Linear combinations of atomic orbitals, when atoms join to form molecules, the resulting molecular orbital is a superposition of the existing atomic orbitals. The term "split" caused confusion for me when reading, "...their atomic orbitals combine to form molecular orbitals..." seems more clear. Ducksandwich (talk) 17:24, 29 October 2013 (UTC)

Also slightly re-worded the following few sentences to make it clear how the molecular orbitals are formed. Ducksandwich (talk) 16:50, 6 November 2013 (UTC)

File:Metals and insulators, quantum difference from band structure.ogv[edit]

"Animation" not accessable without explanation. — Preceding unsigned comment added by Jangirke (talkcontribs) 20:39, 23 February 2014 (UTC)