Monday, January 05, 2009

What does it mean for a material to be a "metal"?

Continuing on from my earlier posts about insulators, it's worth thinking about what we mean by a "metallic" state. Colloquially, people have an image of what they think is a metal: a material that is shiny, electrically conducting, and probably relatively ductile and malleable. Let's not discuss the elastic properties at the moment, since their origin is rather subtle. The electrical conduction is what really stands in contrast to insulators, and the shiny surface is a consequence of the electrical conduction at high frequencies (optical, ~ 1015 Hz). (By the way, for those interested in why some metals have color to them, this site has a pretty nice explanation. The short answer: interband transitions alter the absorption at short wavelengths.)

It's important to understand that, from the condensed matter physicist's perspective, there's a big difference between a substance that is merely electrically conductive and one that is a "real" metal. In a real metal, the electrical resistivity decreases as temperature is decreased. There are conduction mechanisms (e.g., ionic conduction in glasses; hopping conduction in doped organic semiconductors) that become much less effective at lower temperatures - those systems are not metals, just moderately conducting at room temperature. Similarly, lightly doped semiconductors aren't metals either; as T approaches 0 they have no mobile charge carriers. It would be nice to be able to find a ground state property that lets us decide whether something is a metal or an insulator rather than worrying about temperature dependences. Fortunately, there is. As discussed here (a nice pdf that I found while learning more about what Peter had written in the comments to the previous post), when placed between capacitor plates at T = 0, a metal develops only a surface charge, while an insulator develops a bulk dielectric polarization (dipole moment per unit volume) throughout itself.

There are different types of metals. Conventional metals are Landau Fermi liquids. The low energy electronic excitations of Fermi liquids are "quasiparticles" that act very much like non-interacting electrons - they have spin-1/2, charge -e, and have a lifetime much longer than h/kBT. In bulk Fermi liquids, electronic excitations can have arbitrarily low energies. The spectrum of these excitations is said to be gapless. The hallmark of Fermi liquids is that they have properties that look much like those we find in undergrad statistical mechanics treatments of noninteracting Fermi gases. For example, their heat capacities vary at low temperatures as T, and their resistivities vary at low temperatures as T2.

There are other metallic states known variously as bad metals or strange metals. The classic example of a bad metal is the normal state of optimally doped high temperature superconductors. These systems have a metallic ground state, but near T = 0, their resistivities vary linearly in T rather than quadratically. This may not seem like a big deal, but it has major implications. It implies that the low energy electronic excitations of these materials are not well described as quasiparticles; they must somehow involve collective excitations of many correlated electrons, and may not have easily intuitive quantum numbers. That is, they're non-Fermi liquids. Trying to understand these systems and their excitations is a major outstanding challenge in condensed matter physics today. It's hard because it involves understanding excitations of a system of many strongly interacting quantum particles, and also because our intuition has been shaped by our classical ideas about simple quasiparticles. By the way, this idea of excitations that are complicated and lack particle-like quantum numbers has come into vogue in high energy physics in the form of "unparticles".


Anonymous said...

It's actually harder than you might think to distinguish between a surface charge and a polarization. The net charge distribution, coarse-grained over many unit cells, looks like a surface charge in both cases. Within a unit cell, it's not always easy to decide whether a particular bit of excess charge came from the the right or the left, which is what you need to know to define the polarization.

I encountered this issue in writing a story on the polarization of ice. It turns out to be quite tricky to determine how much of ice's high dc dielectric constant should be assigned to polarization of individual water molecules and how much to collective motions of many molecules. The reason is that, in this highly hydrogen-bonded solid, there is not clear way to isolate a particular water molecule.

By the way, Raffaele Resta, whose 2000 paper you cite, was a co-author of the PRL my story was about. He and David Vanderbilt have also written a comprehensive book chapter on this subject.

Douglas Natelson said...

Great comment, Don. I understand your point, and thanks for the links.

After reading Resta's paper, I was wondering if this bulk polarization vs. surface charge business was analogous to some of the definitions of solid vs. liquid. Take a volume of material and apply a shear stress to the top surface. A solid will develop a uniform shear strain throughout itself (analogous in my feeble brain to the uniform volume polarization). A liquid can't support uniform shear strain, so the surface acquires a velocity. This analogy is rather poor, though, since shear here is in the plane of the surface, while polarization is basically normal to the surface in the dielectric problem.

Anonymous said...

are you going to have a discussion of semi-metals too?

Anonymous said...

1. I was wondering how would one distinguish metal from a superconductor within the approach mentioned in Resta's paper??

I am aware of a slightly different approach (also related to Kohn's idea) which does that (again) using only ground state wave-fn. This involves looking at current-current correlations with different limits of freq./wave-vector going is zero. For example, see
the paper by Scalapino et al, Phys. Rev. B 47, 7995 - 8007 (1993). So is there a simple extension of Resta's paper to encompass superconductors too?

2. Regd. 'non-Fermi' metals you wrote that "they must somehow involve collective excitations of many correlated electrons, and may not have easily intuitive quantum numbers". Come to think of it, aren't there theoretical predictions of non-Fermi metals which could involve strongly correlated *bosons* instead ('Bose metals')?

Anonymous said...
