The basics of conductivity in metals

Currently my experiment involves determining the limits to a classical model for conductivity in metals.  So how do we model the conductivity of metals?

First, by conductivity, we are refering to the ability for current to pass through a material.  Current is moving charges, and the charges that are moving are either electrons (negative charges, which move in a direction opposite to the current – this is just the convention) or holes (positive charges, basically a place in the material where an electron no longer is).  Electrons and holes are known as charge carriers.

In metals the holes are basically stationary, and it is free electrons in the material that move around.  Now the ions (the atoms that are missing electrons) in the metal are very heavy (relative to the electrons), so they make up a background sludge while the electrons move essentially freely, like a gas through the material.

So if we treat the electrons as a free gas of particles against a background drag (caused by the electric attraction between the electrons and the ions), we have the Drude model of conductivity (invented in 1900, all equations will be ommited here but are available on the Wikipedia page).  Applying the classical laws of physics (Newton’s), we can find that the velocity of the electrons in a metal should be proportional to the mobility times the electric field magnitude.

The mobility is a measure of the charge over mass times the mean free time of the charge carrier.  This free time represents the time each electron is free before getting trapped by another ion – but charge will continue to flow, as long as an electric field is applied, since more electrons are liberated.  The free time is equivalent to the mass of the electron divided by the drag coefficient that is felt by the electrons.  This means that the mobility is just the charge of the electron over the drag coefficient.

The physical result of this model is that when you turn a light switch on, it takes at least that mean free time before the current begins flowing between the switch and lightbulb in your room.  However, average times for this are well under a picosecond (for gold, its on the order of femtoseconds).

The Drude model can be extended to AC signals, and fairly accurately represents what happens when pulsed laser light is incident on a sample (either metal or semiconductor).

So where doesn’t it work?  Our lab has found that for thin films of gold (under 30nms), the model fails to account for the conductivity found under terahertz spectroscopy probes.¹ What accounts for the discrepancy then?

An extrapolation of the Drude model, called the Drude-Smith model, attempts to add the effect of the free electrons scattering off of various sources, be they other electrons, atoms, ions, or defects.²  This results in an extra “fudge” factor and is not completely backed theoretically, but fits the data seen from experiments.

The idea with my current experiment, is that by limiting the path length that the electrons are able to travel we can force the demonstration of non-Drude behaviour and can either back evidence behind the Drude-Smith approach, or attempt to discover other processes that are involved.

---

1. M. Walther, D. G. Cooke, C. Sherstan, M. Hajar, M. R. Freeman, and F. A. Hegmann, “Terahertz conductivity of thin gold films at the metal-insulator percolation transition”, Phys. Rev. B 76, 125408 (2007). [PDF]
2. N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity”, Phys. Rev. B 64, 155106 (2001). [citation]