Wednesday, September 24, 2014

Temperature dependence of the resistivity is not a definitive signature of a metal

When it comes to elemental metals and insulators, a simple signature to distinguish them is the temperature dependence of the electrical resistivity.
Metals have a resistivity that increases monotonically with increasing temperature.
Insulators (and semiconductors) have a resistivity that decreases monotonically with increasing temperature.
Furthermore, for metals the temperature dependence is a power law and for insulators it is activated/exponential/Arrhenius.
This distinction reflects the presence or absence of a charge gap, i.e., energy gap in the charge excitation spectrum.
Metals are characterised by a non-zero value of the charge compressibility
and a non-zero value of the (one-particle) density of states at the chemical potential (Fermi energy for a Fermi liquid).

However, in strongly correlated electron materials the temperature dependence of the resistivity is not a definitive signature of a metal versus an insulator.

The figure below shows the measured temperature dependence of the resistivity (on a logarithmic scale)  of the organic charge transfer salt kappa-(ET)2Cu2(CN3) for several different pressures. At low pressures it is a Mott insulator and for high pressures a metal (and a superconductor at low temperatures).

The data is taken from this paper.
Note that at intermediate pressures the resistivity is a non-monotonic function of temperature, becoming a Fermi liquid and then a superconductor at low temperatures.
Some might say that the system undergoes an insulator-metal transition as the temperature is lowered.
I disagree.
The system undergoes a smooth crossover from a bad metal to a Fermi liquid as the temperature is lowered. It is always a metal, i.e. there is no charge gap.

This view is clearly supported by the theoretical results shown below (taken from this preprint) and earlier related work based on dynamical mean-field theory (DMFT).

The graph below is the calculated temperature dependence of the resistivity for a Hubbard model, for a range of interaction strengths U [in units of W, the half band width].
Note, the non-monotonic temperature dependence for values of U for which the system becomes a bad metal, i.e. the resistivity exceeds the Mott-Ioffe-Regel limit.

But, for the temperature and U range shown the system is always a metal.
This is clearly seen in the calculated non-zero value of the charge compressibility, shown below.

Aside: In the cuprates the question of whether the pseudogap phase is a metal, insulator, or semi-metal is a subtle question I do not consider here.

I thank Nandan Pakhira for emphasising this point to me and encouraging me to write this post.


  1. Nice physics!

    But, to add to the discussion and regardless of the interesting physics at play, I think this is a matter of definitions.

    The question is whether your definition of a metal is appropriate; you say the temperature coefficient for (some) correlated systems is not a good indicator of something being metallic or not. And you say that "there is no charge gap, i.e. it is (always) a metal".

    With similar strength one could argue that strongly correlated systems with increasing resistivity with decreasing T are not metallic - because they don't conform to the definition of a metal (rho(T)). And then there is an MIT.
    If one defines a metal according to some traits of the Fermi liquid (in this case rho(T)), then the conclusion is that most strongly correlated systems are not metals - as is conveniently born out by there rho(T) deviating from that of a Fermi liquid metal...

    So I find your post a bit of an open door: if you define a metal as you do, then there is no MIT.

    The fundamental discussion to be had first is what is the proper definition of a metal, going beyond the standard metal-semiconductor-insulator story.

    1. Thanks for the comment.
      I agree that the fundamental question is how you define a metal.
      My point is that a definition in terms of the temperature dependence of the resistivity is problematic.
      A key issue, that you allude to, is having a clear definition of a MIT (metal-insulator) transition. This should be a thermodynamic transition (i.e. singularities or discontinuities in the free energy or of its derivatives).

      I think defining a metal as a Fermi liquid is not helpful.

      If the metal is defined by a non-zero (non-activated) compressibility then there is clear MIT as a function of U and temperature.
      Such a transition is seen in DMFT.
      This transition does not necessarily correspond to changes in the temperature dependence of the resistivity.

  2. Interesting; I have to think more about your compressibility argument before I can say something smart enough to put on-line...

    (I don't disagree that "metal=FL" is not helpful, but it does go to the fundamental issue; what is a metal. Either one takes a complete set of characteristics (FL), or only one (e.g. rho or kappa).

    In the end I don't see you the problem you see when basing things on resistivity; setting a definition is not problematic, as long as one is open to re-assess other concepts, such as an MIT - if there is no metal, there is no MIT. That in itself is not problematic, and does not take away that the physics of what is going on is interesting. If one wants at all costs to keep calling a certain change an MIT, then there could be a problem if M is redefined...)

    But, how does the framework you advocate work in a Kosterlitz-Thouless metal-insulator transition?
    Does the compressibility of a typical (2D) KT-MIT change across the transition? Or would you run in trouble there...?

  3. Naively, I don't see how discussions of the compressibility get around the problem of defining the metal. If the Nernst-Einstein relation holds, then any state with a finite conductivity similarly will have a finite compressibility. This includes conventional insulators at finite temperature, where there is a well defined gap in the single particle spectrum not related to correlations, and conductivity is clearly activated. The only clear distinction is still in the T->0 limit where the compressibility of a gapped system will tend to zero, along with the conductivity. One has to make a strong distinction between classifications of metals that make sense at finite temperature, and those that make sense at zero temperature.

    The most rigorous definition of the metal is probably what side of the "separatrix" of one is on. In other words, if I increase U, does the curvature of the free energy with respect to deviations of the single particle Green function grow or shrink? Do I move closer to the Mott critical end point, or further?

    1. Hi Steve,

      Thanks for the comment.

      I agree the Serbian separatrix is a helpful and concrete definition. But, I think it only works for a theoretical calculation, not for experimental data.
      Importantly, by this definition all the curves shown in this host are metallic.

      The relation between compressibility and conductivity (a la Nernst-Einstein) is subtle. In an Anderson localised insulator the compressibility is non-zero but the conductivity and diffusion constant are both zero.