Friday, January 29, 2010

Beware of curve fitting

In a comment on a previous post about charge transport in organic materials Doug Natelson brought to my attention a recent preprint. It is a really nice paper and is a cautionary tale about drawing conclusions from curve fitting.

In a Nature Materials paper last year, Alan Heeger's group at UCSB considered the electric field and temperature dependence of the current in an organic field effect transistor. They fitted the observed dependence to that for a theory which describes slightly dirty long one dimensional conducting wires with strong electronic correlations (Tomonaga-Luttinger liquid theory). This theory gives a good description of charge transport in single carbon nanotubes. However, it is not clear if there is a physical reason to expect the TLL theory to be relevant to "dirty" crystals of small organic molecules.

However, Worne, Anthony, and Natelson show that for their data on similar devices the curve fitting to the TLL theory is problematic. In particular, the apparent "scaling collapse" is fortuitous. They state:

decreasing T moves subsequent temperature data sets up and to the right on the graph, even if the data themselves do not change with temperature at all. In fact, any weakly temperature dependent dataset that resembles a power-law can be made to fit onto a single line if plotted in this way with an appropriate choice of α. Data collapse with this plotting procedure is not sufficient to demonstrate TLL physics.

There is another reason why I am skeptical about any claim of "metalllic" behavior in these systems. Their mobility is much less than than the "minimum mobility"of 1 cm^2/Vsec that is necessary for the coherent transport associated with delocalised electrons and band structure.


  1. I think the problem is that the importance of mixed states in quantum mechanics is not taught to beginning students. If unitary equivalence is a requirement, then any material can be discussed in a basis of momentum eigenstates (or position eigenstates, whatever).

    The points that you are making rely heavily on the fact that in REAL LIFE, there is not always invariance. Reality does choose a representation. Isn't this right? Band structure is equivalent to saying that the system acts like momentum is always being measured? It will look like a pure quantum state in this representation, but a mixed state with respect to position observables?

    I think that it is a big problem that quantum mechanics and statistical mechanics are taught as separate subjects. They should not be - and statistical mechanics should be taught first.

  2. The same problem exists in the controversial 2D metal-insulator transition. Initially it was claimed that all rho(T) data on both sides of the transition could be collapsed onto a single curve by scaling the effective temperature.
    In the insulating regime this is not surprising, since VRH predicts we have rho(T)=rho_0 exp(T_0/T)^m. So of course everything scales as T/T_0.
    In the metallic regime, the scaling was alwways more problematic - but plotting data on a log-log scale is an excellent way of hiding the fact that there are deviations, or that there is almost no overlap between traces.
    I wonder if similar problems exist in other claims of quantum phase transitions?
    Alex Hamilton.