# Non-equilibrium variation in electron density in a metal

Consider two atoms of platinum, A and B, in a sample of platinum. Atom A lives deep within the sample, and atom B lives at the tip of a sharp protuberance at the surface. My understanding is that electrons in the sample will accumulate within a surface defect such as the tip of a sharp needle. The following questions come up for me (not necessary to address here, but you can if you like):
• In the equilibrium case, what is the range of variation in electron density in the volumes on the atomic scale immediately surrounding A and B?
• In the non-equilibrium case, where there is a dynamic perturbation such as the application of a strong time varying electric potential or magnetic field, what is the range in variation in electron density? What are the average differences and what are the minima and maxima over time?
• Are these questions that have been explored experimentally, or are they ones that are accessible only through modeling?
In this context my question for this forum is: is there a textbook or a set of papers that can help me to become acquainted with what is known about the dynamic, non-equilibrium case, where there is a perturbation of some kind?

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To make my question a little clearer, here is a schematic diagram that depicts the intuition behind my question, which ignores everything but lattice sites A and B in the platinum sample:

The thought is that the varying circumstances of A and B, which are situated in very different areas of the platinum sample, might be such that if you compared the electron charge density distributions ##\rho(\mathbf{r}, t)## you would see a non-trivial difference under non-equilibrium conditions. If so, what kind of differences are we talking about?

Last edited:
Henryk
Gold Member
Eric,
Very interesting questions. I don't have answers but here are a few thoughts.
Solid state theory, deals with atoms arranged in a regular crystal structure. Given the properties of atoms and their arrangement one calculates energy bands and all the physical properties of a bulk solid. As you probably know, if you have a local defect, the energy levels near the vicinity of that defect differ from that of a perfect crystal.
Now, a surface of a solid is one huge defect !! and all bulk properties calculations are off.
Here are a few things:
Well inside a solid, each atom is surrounded by neighbours from all sides. At the surface, one or more of the neighbour is missing. The spacing between the atom at the surface and his neighbours may not be the same as it is in the bulk.
The surface, a big defect, creates localized energy states different from that of the bulk materials. In semiconductors, for example, you find surface energy states lying within the energy gap of a bulk material. I actually studied surface states for my master thesis.
I cannot comment about what exactly happens for metals but I would bet that the electron densities around atoms A and B are different at equilibrium on atomic distance scale.
In a non-equilibrium case, the penetration of external fields is give by the skin depth. Here is a link to the Wikipedia page
https://en.wikipedia.org/wiki/Skin_effect
The skin depth is much larger than atomic spacing by a few orders of magnitude and effects of just one or two atomic layer on the penetration of external fields can be usually ignored, at least for metals. In semiconductors, the density of surface states can be high enough that they can effectively screen external field even if the fields are static. This is why GaAs electronics never took off. No one figured out how to passivite GaAs surface to eliminate surface states and therefore how to make MOS transistors using GaAs.

As for studying the properties of solid surface, yes, there is a branch of solid state physics called surface science. There is theoretical and experimental work. The experimental work is actually very challenging. The main reason is that is very difficult to control what happens at the surface. At ambient condition, you can't have a clean surface of a given material. Water, oxygen, nitrogen becomes absorbed and even a monolayer of absorbed molecules can alter the electronic states at the surface.
By the way, platinum surface is most likely covered with platinum oxide.

Thanks for bringing the subject up. I gave you no answer your questions but yes, the questions are good.

Henryk

HARSHARAJ and Eric Walker