What are Friedel oscillations and how are they caused?

[AlanKirby]

Hi, and thank you to anyone who replies. I was hoping that someone could please elucidate as to what Friedel oscillations are and what causes them. All of the material that I can find on it is either too simple ("it's an oscillation in the charge density..." or "quantum mechanical analog of charge screening...") or way too advanced that I just can't even start to understand it.

Thanks again for any response.

 

[Simon Bridge]

What do you need the description for?
Unless we know this, we cannot hope to fulfil your requirements that the description provided be neither "too simple" nor "way too advanced" for you.
It may help if you gave us some idea of your education level.

 

[DrDu]

Basically it is a consequence of Fourier transforming the fermi dirac distribution. A step function in momentum space becomes am oscillation in real space.

 

[AlanKirby]

What do you need the description for?
Unless we know this, we cannot hope to fulfil your requirements that the description provided be neither "too simple" nor "way too advanced" for you.
It may help if you gave us some idea of your education level.

Thanks for the response. I'm currently doing a masters project and an STM image appears to be showing these Friedel oscillations, which have been reported in the literature as appearing on other surfaces of the same complex metallic alloy. Hence I'm reading into them to try and write some background theory for the observation, however most literature doesn't explain what they are, it just talks about their particular observations, which is fair enough but doesn't help someone trying to initially wrap their head around them. One of the reasons that I'm seeking help is that I've read that the confinement of the electron wavefunction between the two surfaces of a thin film can cause oscillations in the density of states at the Fermi energy(this is relavent for the project). So the first issue is to clarify whether these are also Friedel oscillations, or completely separate oscillations of a different nature.

Secondly, I'm looking to clarify whether my current interpretation of a Friedel oscillation is correct (so that I can progress with reading more of the theory behind it in detail), which i'll lay out below.

A metal can be roughly described by a Jellium model whereby positive ions occupy lattice sites and the associated electrons are mobile and form an electron gas which is evenly distributed (in a perfect metal with no impurities) such that the total charge cancles everywhere. Now if we introduce an impurity into the metal, depending on its charge, the electrons will be either attracted or repelled by it through the coulomb interaction in such a way as to effectively restore the charge neutrality. However the probability of finding an electron in a given state is given by the Fermi-Dirac distribution, and given the quantum nature of the electrons, the energy spectrum is not continuous, thus the electrons can only occupy certain distances/radius from the impurity. This influneces the charge screening in such a way that the lowest energy electrons overcompensate and effectively cause another impurity at a given radius from the actual impurity (at their location), and the process repeats. The charge density away from the impurity falls with a power law (r^-3) and oscillates between positive and negative. This oscillating charge density around an impurity is a Friedel oscillation.

 

[AlanKirby]

Basically it is a consequence of Fourier transforming the fermi dirac distribution. A step function in momentum space becomes am oscillation in real space.

Thanks for the reply. Your first sentence causes an odd confusion for me. You're saying that it's a consequence of Fourier transforming the Fermi-Dirac distribution. Fourier transforming as far as I'm aware is simply a tool for taking a description in terms of position and obtaining an equivilant description in terms of wavelector (or the reverse), which in solid state physics, may be easier to work with. So your first sentence confuses me because the description is of nature (the friedel oscillation and behaviour of the charge density) however the cause of the nature is a change in the description? I would think that nature has a reason for the phenomenon, and that our choice of description for it, is irrelevant to the phenomenon itself having ocurred or not ocurred. So with that, could you please reword your initial sentence. I realize that I may just be being daft when reading your post and not seeing the obvious. If so, then whoops, and sorry. Thanks again.

 

[DrDu]

I don't remember the details too well, myself. The point is that the spatial response of charge density to a point charge at the origin will involve in lowest order perturbation theory some integral over the Fermi Dirac distribution which has the form of a Fourier transform. As the Fermi Dirac distribution in metals has a sharp step at the Fermi energy, this will induce the oscillations in real space. I found a source in internet. See eq. 1.2.15:
http://www.lassp.cornell.edu/clh/Book-sample/1.2.pdf