## Monday, November 27, 2023

### Noise in a strange metal - pushing techniques into new systems

Over the holiday weekend, we had a paper come out in which we report the results of measuring charge shot noise (see here also) in a strange metal.   Other write-ups of the work (here and especially this nice article in Quanta here) do a good job of explaining what we saw, but I wanted to highlight a couple of specific points that I think deserve emphasis.

In thermal equilibrium at some temperature $T$, there are current and voltage fluctuations in a conductor - this is called Johnson-Nyquist noise - and it is unavoidable.  Shot noise in electrical current results from the granularity of charge and, as shown in its original incarnation (pdf is in German), from the statistical variation in the arrival times of electrons.   Shot noise is an "excess" noise that appears in addition to this, only when a conductor is driven out of equilibrium by an applied voltage and carries a net current.

While the idea of shot noise is tunnel junctions and vacuum tubes had been worked out a long time ago (see the above 1918 paper by Schottky), it was in the 1990s when people really turned to the question of what one should see in noise measurements in small metal or semiconductor wires.  Why don't we see shot noise in macroscopic conductors like your house wiring?  Well, shot noise requires some deviation of the electrons from their thermal equilibrium response - otherwise you would just have Johnson-Nyquist noise.  The electrons in a metal or semiconductor are coupled to the vibrations of the atoms (phonons) - the clearest evidence for this is that the decrease in scattering of the electrons by the phonons explains why metals become more conductive as temperature is decreased.  In conductors large compared to the (temperature-dependent) electron-phonon scattering length, the electrons should basically be in good thermal equilibrium with the lattice at temperature $T$, so all that should be detected is Johnson-Nyquist noise.  To see shot noise in a wire, you'd need the wire to be small compared to that e-ph length, typically on the order of a micron at low temperatures.  In the 1980s and 1990s, it was now possible to make structures on that scale.

 Fig. 4 from the paper
The theory of what should be seen was worked out in a couple of different ways, initially assuming that it is safe to describe the conductor as a Fermi gas (ignoring electron-electron interactions).  One approach started from the conduction-as-wave-transmission picture of Landauer (see here and here for two examples).  A complementary approach (see here) calculated noise from the electronic distribution functions and got the same answer for non-interacting electrons, that the current noise should be 1/3 of the classic Schottky result.  That factor of 1/3 is called the Fano factor, $F$.   If electron-electron interactions are "turned on", allowing the electrons to exchange energy amongst themselves but not lose energy to the lattice, the noise is actually a bit larger, $F \rightarrow \sqrt{3}/4$.   It turns out that these values were verified in experiments in gold wires (see here and here, though one has to be careful in experimental design to see  $F \rightarrow \sqrt{3}/4$).  This confirmation is a great triumph of our understanding of physics at these mesoscopic scales.  (Interestingly, similar results are expected even with a non-degenerate electron gas - see here and here.)

We applied these same experimental approaches to nanowires we made from exquisite films of a strange metal, YbRh2Si2, and we found that the noise is much reduced from the usual result seen in Au wires (which we also confirmed).  We tested whether phonons could be responsible for the noise suppression, applying the same approach as had been done in the '90s (measurements on wires tens of microns long, where e-ph scattering should be important), and found (in addition to further confirming the e-ph energy loss results in Au from the '90s) that energy loss to phonons can't explain what we see in YbRhsSi2.

Some further points of interest:

• Until recently there really has not been much attempt to push the theoretical analysis of these kinds of measurements beyond the 1990s/early 2000s results.  My colleague Qimiao Si and his group have looked at whether strong Fermi liquid corrections affect the expected noise, and the answer is "no".  Of course, there are all kinds of additional complications that one could imagine.
• This work was only possible because of the existence of high quality thin films of the material, and the ability to fabricate nanostructures from this stuff without introducing so much disorder or chemical change as to ruin the material.  My collaborator Silke Bühler-Paschen and her group have spent years learning how to grow this and related materials, and long-term support for materials growth is critically important.  My student, the lead author on the study, did great work figuring out the fabrication.  It's really not trivial.
• I think it's worthwhile to consider pushing older techniques into new regimes and applying them to new materials systems.  The heyday of mesoscopics in the 1990s doesn't need to be viewed as a closed, somewhat completed subfield, but rather as a base from which to consider new ways to characterize the rich variety of materials and phases that we have to play with in condensed matter.

Stefan Bringuier said...

This is really interesting, although I'm not familiar with the topic.

If I understand correctly the Fano factor quantifies the relative magnitude of shot noise compared to with Johnson-Nyquist noise, correct? Does the temperature dependence of the Fano factor indicate any specific feature of the system? I guess I'm trying to understand the meaning of the larger variation in the Fano factor for YbRh2Si2 over the temperature range. Is it some kind of e-e to e-ph scattering transition, but I think you are saying it's not strictly due to phonons? The other thing I'm trying to follow is as you "turn on" e-e interactions in gold the Fano factor increases, but as the temperature increases the e-e weak scattering (i.e., thermal screening) and Fano factor decreases? If this is all explained in the paper, I apologize in advance.

Douglas Natelson said...

Hi Stefan, Fano factor here is really the ratio between the shot noise you measure and the Poissonian value (2eI A^2/Hz, in current noise, where I is the avg current and e is the electronic charge), when the system is well out of equilibrium (eV>>kT, where V is the voltage drop across the device, and kT is the usual thermal scale). There are analytical expressions for how the noise should cross over from the J-N noise at zero current/equilibrium to the shot noise at high bias. When F = 1, the simple expectation is that the current noise should look like 2eI coth(eV/2kT), which goes to 4kT/R at zero bias (the J-N result) and 2eI at high bias.

The temperature variation that we see is interesting and we do discuss possibilities in the paper. For the Au device, F stays pretty close to 1/3 all the way up to 20 K (as seen in the SM of the paper), and I actually want to do the measurement up to even higher temperatures to really see how it drops off.

Anonymous said...

Nice work! Are there other strange metals that are amenable to nanowire growth that this is likely to be applied to? Also a bit funny that there is a typesetting glitch in the first word of the abstract in the PDF "S trange-metal".

Anonymous said...

Is it at all feasible to measure shot noise like this with two STM probes on bulk materials rather than nano wires? Or are there a lot of complications there that ruin things

Douglas Natelson said...

Anon@8:21, good question. I think there are more possibilities starting with thin films and patterning into nanowires than there are trying to grow nanowires directly. If people have suggestions, I'm always interested in learning more.

Anon@9:42, the big issue that comes to mind is that schematically you want all dimensions of the system to be small compared to the e-ph scattering length. If carriers can diffuse sideways and lose energy to the lattice, that makes it hard to interpret any suppression of noise.

Don Monroe said...

Fantastic work, Doug! This provides a much-needed new window into strange metals. In spite of the widespread temperature-linear resistance, it has always worried me to hang so much on a particular functional dependence.

I remember in the 1980s trying to understand why ordinary resistors don't have shot noise. After some discussions with Bob Dynes I came to some comfort with that fact, but the rationalization was quite different from the electron-phonon scattering length that seems to be the modern framing.

Instead, the idea concerned how the motion of an individual charge turned up in the leads. For a tunneling electron, this is fast, but for a mm-scale resistor, there are something like a Coulomb of conduction electrons. A 10uA current only needs about 10^-5 of this charge to cross each second, so it takes 100,000 s for each one to move across. So you would expect the shot-noise only to be visible below 10^-4 Hz or so, but lower than that the true dc results might still be Johnson.

Thinking about the screening of each electron by a vast number of others ended up in a similar place.

Anonymous said...

This explanation is what came to mind for bulk objects. I guess the point is that e-phonon scattering is what matters in more microscopic dimensions?

Doug, could you clarify? Maybe missing something fundamental here

Anonymous said...

I don't think Doug reads old posts, better luck posting on a newer post to get an answer. Or just email

Douglas Natelson said...

I do read old posts sometimes - just very busy. Don, I think your intuition leads you astray a bit. Yes, bulk solids contain many many electrons at very high densities, but at 10 microamps, that still means 6e13 electrons per second going through. The classical noise doesn't care about whether a particular electron makes it all the way across, just how many electrons per unit time are transported and what the statistical fluctuations are in their arrival times. If you compare, say, a vacuum diode passing a 10 microamp average current and a copper wire passing a 10 microamp average current, it's true that the voltage fluctuations in the copper wire will be much less than in the vacuum diode, because the Cu resistance will be low and screening will be very good. If you're looking to describe the noise of diffusing electrons within the solid, though, it comes down to what the electronic distribution function looks like. You can compute noise from a Boltzmann equation type approach as the current-current correlation. That's where the importance of e-e and e-ph interactions really come in. If the electronic distribution in the bulk object is always basically the same as a thermal equilibrium distribution at the lattice temperature, all you will see is JN noise at that temperature. My understanding of the modern interpretation is, it's deviations away from thermal distributions or deviations of the electrons away from what the phonons are doing that can give "excess" noise.

Anonymous said...

Doug, the shot noise In Fig. 2B is negative for some frequencies. The paper do not say anything about. Why? My intuition is that you substract thermal (JN) noise from noise to obtain shot noise, hence a negative shot noise occurs when it is smaller than JN noise, is it the reason?