Friday, November 13, 2015

Comparing theory and experiment for metals: look at the frequency dependence of the reflectivity not the conductivity

The frequency dependence of the real part of the conductivity of a metal gives a lot of information, both qualitative and quantitatively. For example, one can extract a scattering rate and see if a Drude model is relevant. Hence, it is natural that experimentalists present “measurements” of this quantity.

However, it is important to acknowledge that the conductivity is not directly measured; rather, the reflectivity or absorption of a thin film or single crystal.
The real and imaginary parts of the conductivity are then extracted from a Kramers-Kronig analysis. This procedure is only stable and reliable if there is experimental data out to sufficiently high frequencies.
Several experimentalists have privately told me this can be a can of worms. It is not clear how high a frequency cutoff you need and interband transitions can complicate things…
Hence, one should be particularly nervous about people claiming exotica such as quantum criticality and non-Fermi liquid behaviour such as anomalous power laws.

There is a simple way to avoid these complications and ambiguities when comparing theory and experiment. The reflectivity can be written in terms of the conductivity as follows

From theory one can calculate the full complex conductivity and thus the reflectivity and
compare this to experiment.

This is the procedure followed by Jure Kokalj, Nigel Hussey, and I in this paper about overdoped cuprates.

I thank Swagata Acharya for motivating this post.


  1. Hi Ross. Not all optical techniques require Kramers-Kronig. Its quite standard now for low frequency reflectivity to be supplemented in the analysis by the complex dielectric function that can be measured directly by spectroscopic ellipsometry in the visible and UV range. And now there are procedures for incorporating a number of different experimental measures of the same complex response function. One might measure the DC resistance, the IR reflectance, and the visible and UV complex dielectric functions and use them as global constraints to output a complex conductivity for all measured frequencies. I believe that the errors introduced in such procedures are very minor, and may only be important in correlated systems when discussing very very small spectral weight shifts that (for instance) could be interpreted as changes in KE above and below superconducting transitions.

    I believe the errors are far far greater in the interpretation of spectra e.g. should some shoulder on some low frequency Drude-like peak be interpreted as a coherent part of the same intraband conduction processes that give the Drude or should it be considered a different excitation altogether. These debates have raged for 25 years in the transition metal oxide community and will continue to rage.

    I should also mention that there is the technique of time-domain THz spectroscopy in which the complex transmission function is measured directly. It can be used to directly generate complex response functions in the frequency range of interest to correlated systems.

    1. Dear Peter,

      Would you have an introductory reference to suggest to read for this time-domain THz spectroscopy technique? I would be interested to know more about it.

    2. I can recommend an excellent thesis from my group, from my first student L.S. Bilbro.

      I first learned these techniques from reading the thesis of J. Corson, who was one of Orenstein's students.

    3. Dear Peter,

      Thanks for the helpful comment.
      I am glad that there are now methods that avoid Kramers-Kronig.
      However, I still see papers that do use it and do not have the multiple constraints and careful checks that you are doing.
      I fear what is "standard" for you is not for others.