Deconstructing the chemical potential of the cuprate superconductors

I have been reading through the nice review Finite temperature properties of doped antiferromagnets by Jaklic and Prelovsek from 2000. They summarise their studies of the t-J model by the Finite temperature Lanczos method.

At first sight the graph below of the temperature and doping dependence of the chemical potential does not look particularly interesting [at least to me]. However, they highlight its significance.

Here are a few points.

• In a simple Fermi liquid the chemical potential has a positive, quadratic and weak temperature dependence. This is only seen for doping c_h=x=0.3
• For a wide doping range [0.05 < c_h < 0.3] the temperature dependence is approximately linear. The slope changes sign for approximately optimal doping (c_h ~ 0.15).
• The weak temperature dependence for c_h ~ 0.15 means that optimal doping corresponds to maximum entropy!  [This can be deduced via the Maxwell relation below. Don't you love thermodynamics!]

• This relation is also related [approximately] to the thermopower via a relationship [equation 8.6], which is essentially a restatement of the Kelvin formula [discussed  by Peterson and Shastry].
• The latter means the thermopower should change sign around optimal doping, as is indeed observed [more on that later].
• The large entropy near optimal doping emerges from the interplay of the localised spins [from the remnants of the Mott insulator] and frustration of the antiferromagnetic spin interactions via doping.

I would be interested to see a similar calculation for the Hubbard model on the anisotropic triangular lattice at half filling to see how the chemical potential varies as a function of U/t as the Mott insulator is approached from within the metallic phase.

When the data is "too good to be true"

Remember Hendrik Schon! A decade ago he published a string of very impressive Nature and Science papers that eventually turned out to be "too good to be true". It seems a similar thing has been happening in the field of social psychology. The AP reports

 three graduate students grew suspicious of the data Stapel had supplied them without allowing them to participate in the actual research. When they ran statistical tests on it themselves they found it too perfect to be true and went to the university's dean with their suspicions.
In the future, the university plans to require raw data from studies to be preserved and made available to other researchers on request - a practice already common in most disciplines.

Nature News reports

The commission found that co-authors of Stapel's papers seem to have been unaware of the fraud, naively trusting in Stapel's reputation and fooled by elaborate preparations for tests that were never actually carried out..... Stapel and a colleague or student came up with a hypothesis, and then designed an experiment to test it. Stapel took responsibility for collecting data through what he said was a network of contacts at other institutions, and several weeks later produced a fictitious data file for his colleague to write up into a paper. On other occasions, Stapel received co-authorship after producing data he claimed to have collected previously that exactly matched the needs of a colleague working on a particular study.....
The data were also suspicious, the report says: effects were large; missing data and outliers were rare; and hypotheses were rarely refuted. Journals publishing Stapel's papers did not question the omission of details about where the data came from. 

This is part of a Nature News piece which has the misleading title "Report finds massive fraud at Dutch universities". A more responsible and accurate title would be "Report finds massive fraud by one Dutch professor of social psychology". In  the comments section several Dutch researchers rightly object to the title.

Chemistry driven by conical intersections

Nonadiabatic Quantum Chemistry is a nice Chemical Reviews article by David Yarkony.

It is quite succinct but covers a significant number of specific chemical systems where non-adiabatic effects [including conical intersections] are important and have been treated theoretically.

Here I just mention one example for which theory has failed so far, the vibrationally mediated photodissociation of NH3 (ammonia) to NH2 + H.
Experiments find that if the excited state contains a symmetric (asymmetric) N-H  stretch the dominant decay channel is to the NH2 ground state (excited state). Yarkony says that calculations [e.g. this one from Truhlar's group, which contains the figure below] have not yet captured this vibrational selectivity.

I thank Seth Olsen for bringing the article to my attention.

Should university be fun, fun, fun!

There is an interesting (and somewhat depressing) article in the New York Times Why Science Majors Change Their Minds (It's just so darn hard). It discusses how in the US there is a big push to have more STEM (Science, Technology, Engineering, and Mathematics) graduates but even if many start these degrees they do not finish.
One contributing factor is that these courses are graded harder than humanities courses.
The article also discusses initiatives, particularly in engineering courses, to make the courses more "fun" and "relevant", especially via projects.
I think this is all commendable and valuable. However, I have a sneaking discomfort that people [students, faculty, and administrators] just don't want to face the painful reality that engineering and science education does involve a certain amount of tedious hard work and that ultimately a lot of jobs (in any field) just aren't that exciting or satisfying.
Or am I just a grumpy old man?

I thank my wife for bringing the article to my attention.

A sign of something important

The Hall coefficient is a fundamental property of metals. In simple Fermi liquid metals it is temperature independent and inverse proportional to the charge carrier density. It has the same sign as the charge carriers (electrons or holes). A major triumph of the Bloch model of metals is that it could explain the sign of the Hall coefficient for simple metals in terms of their Fermi surface.

In contrast, the Hall coefficient of cuprate superconductors has a complex temperature and doping dependence which defies a simple description. Basic questions about the Hall coefficient are:

• What determines its sign?
• What is the origin of its temperature dependence?
• What is the relationship between it and the structure (or absence) of the Fermi surface?  

A 2006 PRB by Tsukada and Ono describes measurements of the Hall coefficient in the cuprate LSCO. The graph below shows the temperature dependence of the Hall coefficient for a range of dopings x of La2-xSrxCuO4 in the overdoped region. For reference, optimal doping is around x ~ 0.2, and for x larger than 0.3 there is no superconductivity. Note the sign change with increasing x.

The authors emphasize how this is a tricky measurement because one has to be careful that the current paths that are measured [to get both sigma_xx and sigma_xy needed for the Hall coefficient] really do lie in the plane of the layers and do not contain spurious contributions (see this earlier post about the challenge of electronic transport measurements in highly anisotropic materials).

The sign change may be an important signature of strong electronic correlations. I find it interesting (and surprising) that the observed sign change at x=0.3 is obtained in a high temperature series expansion of the high frequency Hall coefficient for the t-J model [in this 1994 PRL by Shastry, Shraiman, and Singh (SSS!)]. [An earlier post discusses Shastry's approach]. [Note: this calculation does not have a t' hopping term, which may be relevant. For example, it has a significant effect on the shape and  curvature of the Fermi surface and the proximity to van-Hove singularities. See below].

An alternative explanation of the sign change in terms of Mott physics was given by Stanescu and Phillips.

There may be a more mundane explanation in terms of changes in the Fermi surface associated with the proximity of the van Hove singularity in LSCO. Indeed ARPES experiments do find an electron-like Fermi surface for x~0.3. Furthermore, experiments on Tl2201 [which does not have a close van Hove singularity] do not see any hint of a decreasing Hall coefficient [or sign change] as one increases the doping on the overdoped side towards samples with Tc=0. [Higher dopings seem problematic for Tl2201].
Furthermore, one can quantitatively describe the temperature dependence of data for x=0.3  [including the sign change with temperature] if one uses a realistic Fermi surface and assumes that the impurity scattering rate is anisotropic over the Fermi surface. See this PRB; I thank Nigel Hussey for bringing it to my attention.

I thank Jure Kokalj for some helpful discussions.

Converging to the "right" answer

This week I had an interesting experience. I was doing a calculation and comparing my result to experiment. The comparison was poor, with a discrepancy of a factor of about two. This was disappointing, but then I decided that the theory was just too simple and one should not experiment anything better than qualitative agreement... I just had to accept this.
But then I found a mistake in my Mathematica code. I realised I had to check everything more carefully. .. One of my variables I had defined incorrectly... I redid the plot. The agreement of theory and experiment was excellent.

But, now there is a real danger. I could stop checking for errors. Afterall, given I already found a couple there may be another one which will lead to new discrepancies.
I will let you know if I find any. But, I have to confess the motivation to find errors is less than it was..

I wonder how often this happens in science.  I think I recall that there are some famous historical examples, e.g. that over years the value of the speed of light and the charge on the electron have drifted, but at any particular time peoples values have always been within a standard deviation of the latest measurements.

Just remember Feynman's warning: "The easiest person to fool is yourself."

The challenge of a simple measurement

Just because an experimentalist claims to have measured a specific physical quantity does not mean they actually have measured the desired quantity. Theorists need to be particularly wary at uncritically accepting data.

To most people, especially theorists, measuring the electrical resistivity of a metal sounds like an almost trivial measurement! Surely, you just stick a sample of the metal between the leads of an ohm-meter and read off the resistance!
The temperature dependence of the resistance can provide significant information about scattering of quasi-particles in the metal and any decent theory should be able to describe it. A famous case it the "linear in T" resistivity of optimally doped cuprate superconductors, a signature of non- Fermi liquid behaviour.

Most of the interesting strongly correlated metals (cuprates, organic charge transfer salts, iron pnictides, ....) have layered crystal structures leading to anisotropic electronic properties. These are sometimes referred to as quasi-two-dimensional metals.
Accurately, measuring the resistivity (and its temperature dependence) in the three different directions though is a highly non-trivial exercise. Basically, this is because you have to be sure that the current is going through the sample in the direction you think it is.

This is highlighted in a recent Nature Communications article from Nigel Hussey's group. They state:

 In a quasi-1D conductor, it is especially problematic to measure the smallest of the resistivity tensor components, because even a small admixture of either of the two larger orthogonal components can give rise to erroneous values and distort the intrinsic temperature dependence of the in-chain resistivity. In Li_0.9Mo₆O₁₇, reported room-temperature values for the in-chain (b axis) resistivity range from 400 μΩ cm²³ to more than 10 mΩ cm^34, 35.

Reported values for the ratio of the a to b axis resistivity vary from about 2 to 100!
This is a very large discrepancy!

I wrote this post because I thought I had come up with a fancy theoretical explanation of why in one paper the resistivity anisotropy ratio was only ~4, whereas band structure predicts a much larger value. However, when I surveyed the literature I discovered the result I was so proud of explaining is probably an artefact!