What is weak localization?

A few days ago I wrote about localization, where waves in a medium can become trapped due to interference by scattering off disorder.  This is an extremely general phenomenon that applies to light, sound, and electronic waves in solids.  

Now I want to write about a phenomenon that is specific to electrons (or at least wavepackets that carry electronic charge, if we want to be very general).  Rather than the completely general arguments about conductivity scaling, now we are going to consider particular sets of trajectories in the weak scattering limit.  

We can define "weak" scattering here in terms of the ratio of the mean free path \(\ell\), the typical distance a wavepacket of electrons travels between being redirected by elastic scattering off disorder (vacancies, impurities, surfaces, grain boundaries), and the Fermi wavelength of the electrons, \(\lambda_{\mathrm{F}}\).  If \(\ell/\lambda_{\mathrm{F}} \gg 1\), then the scattering is weak.  (If you have some measurement that allows you to calculate that ratio for a given system and you find instead that you get \(\ell/\lambda_{\mathrm{F}} \ll 1\), then the disorder is so strong that the model of propagating electronic waves really fails and you have to worry about conduction by something like thermally assisted hopping between localized states.)

Electron wavepackets scattering around
a loop trajectory clockwise (red) or 
counterclockwise (blue).  Gray circles
are scattering sites.  Magnetic field \(B\)
is shown pointing out of the page.

In weak localization, as initially explored here, we consider electronic wavepackets bopping through a disordered environment, as shown.  There are many possible trajectories for the electrons, and bouncing off disorder (symbolized here as gray circles) leads to a shift in the phase of the waves as well as a direction change, but it's all deterministic and reversible.  An electron can bounce around a particular loop trajectory from defect to defect in two ways, clockwise or counterclockwise.  The reversibility means that whatever phase the wavepacket racks up going clockwise, it would accumulate the same phase if it went counterclockwise.  This means that there is constructive interference from the loop trajectories for the electron to end up back where it started - that tends to localize the electrons.  Each particular loop trajectory has its own amount of accumulated phase, but all of them have this "constructive interference for back-scattering" issue.

How can we tell this is really going on?  We can turn on a magnetic field \(\mathbf{B} = \nabla \times \mathbf{A}\) that threads flux through the loops.  As I described here, the propagating electrons then pick up an additional phase \(\delta \varphi = (q/\hbar)\int \mathbf{A}\cdot d\mathbf{r}\) as they go along a trajectory.  This means that the clockwise and counterclockwise versions of the loop trajectories are now offset in phase by an amount proportional to the magnetic flux through the loop and in general no longer interfere constructively for back-scattering.   

How large of loops do we need to consider?  Because of inelastic interactions with other electrons, lattice vibrations, etc., the phase of the electronic waves gets scrambled on a characteristic coherence timescale \(\tau_{\phi}\), and a corresponding coherence length scale \(L_{\phi} = \sqrt{D \tau_{\phi}}\), where \(D\) is the diffusion constant for the electrons.  (See here.)

The result of all this is a positive magnetoconductance (equivalently a negative magnetoresistance), since applying the magnetic field suppresses the back-scattering.  The magnetic field scale over which the zero-field conductance dip gets suppressed is on the order of \(B_{c} \sim (h/e)/L_{\phi}^{2}\), though the detailed functional form of \(\delta \sigma (B)\) depends on the relative size of \(L_{\phi}\) and the sample dimensions.  (See here for a key reference if you want details.) Weak localization is one of the main techniques used to infer coherence properties of metals and semiconductors.  A classic review by Gerd Bergmann is here.  Note that this is also closely related to the physics of universal conductance fluctuations.

(One additional point for experts.  I hadn't mentioned spin or spin-orbit coupling.  It turns out that in the strong spin-orbit coupling limit (\(\tau_{\mathrm{so}} \ll \tau_{\phi}\)), the accumulated phases for the time-reversed loop trajectories are no longer of the same sign, but instead are of opposite signs.  The result is destructive interference for back-scattering, and therefore a negative magnetoconductance and "weak antilocalization" (WAL), where the analytic expressions for WAL differ from the WL forms by a factor of -1/2.)

What is localization?

Physicists love simplifying idealizations, and this is especially true in the physics of materials.  The simplest decent model for metals, for example, is the ideal Fermi gas, where we neglect the existence of atoms entirely and just model the electrons as noninteracting particles in some box.  One step up from there, the Sommerfeld model, assumes that the electrons are in a perfectly periodic crystal lattice.  In both cases, the standard semiclassical approach treats the electrons as waves but basically ignores quantum interference.  

Real conductors have defects that break the lattice periodicity, like vacancies, interstitials, impurities, grain boundaries, surfaces and interfaces, etc.  It's natural to wonder, are there major consequences to this "disorder"?  Common sense suggests that sufficiently minor or dilute disorder can't be too important.  Sure, once you break the lattice symmetry, the electronic wavefunctions can't be exactly Bloch waves anymore, but if only one atom out of 10 billion is out of place, how big a deal can it be?

In the late 1970s, a number of theorists were thinking about this problem, and they came up with some impressive insights about the role of disorder, leading to the concept of localization.  The key point to consider is whether the wavefunctions in the presence of disorder are delocalized (extending "to infinity", like plane waves or Bloch waves), or whether they are localized (decaying exponentially away from some origin region where their magnitude is large).  This idea can apply to wavefunctions for electrons, but it can also apply to other kinds of waves, including electromagnetic waves in inhomogeneous dielectric media (think light bouncing around in a cloud).  

Update:  As Andrew Millis pointed out to me, the genesis of this key idea came earlier, from Phil Anderson in this 1957 paper, "Absence of Diffusion in Certain Random Lattices".  Into the 1960s, Sir Nevill Mott introduced the idea of the "mobility edge" - that in a disordered system, the electronic states in the middle of a band are delocalized, but there is an energy threshold at the band edge beyond which the electronic states are localized.  

A major result that came out of the resurgence of this thinking in the 1970s was the scaling theory of localization.  That link points to some excellent lecture notes and a couple of youtube videos by Piet Brouwer for people interested in a more technical explanation.  Intuitively, if the electronic states are exponentially localized, then making a block of material bigger should lead to the conductance of that material dropping exponentially.  Alternately, if the electronic states are delocalized, making a hunk of material larger should generally increase its conductance.  (Think about a piece of copper wire.  Now double both the length and the diameter of the wire.  The conductance \(= \sigma (\pi d^2)/(4L)\) has doubled.)  

Let's call \(g(L) = G(L)/(e^2/h)\) the (dimensionless) conductance of some hunk of material of size \(L\).  The question is, if you increase \(L\), what happens to \(g\)?  There is a scaling function \(\beta(g) \equiv d \ln g/d \ln L\) that describes this.  If \(\beta(g)\) is positive, then the system is metallic.  If \(\beta(g)\) is negative, then the system is insulating in the large size limit, a situation called strong localization.  The technical bit is figuring out what \(\beta(g)\) looks like.   (This scaling idea had many contributors, including most famously people like Anderson and Thouless)  

Remarkably, in this famous paper, the conclusion is that in 2D and 1D, any disorder at all makes \(\beta(g)\) negative.  Thus the surprising conclusion is that, for this model (with no interactions), in principle there are no 2D or 1D metals.  (The distance scale over which the conductance decays with increasing size is the "localization length", \(\xi\), and it could be very long.  That's why seeing metal-like conduction in cm-scale gated graphene or 2D electron gas samples isn't surprising or necessarily inconsistent with this.  There are many subtleties here.)  In 3D, the situation depends on the actual magnitude of \(g\), where if \(g\) starts too small, the system runs away toward localization as system size is increased.

This idea, that interference of scattered waves from disorder can lead to exponentially confined waves, is called Anderson Localization.  This is generic to waves in disordered media, as in this famous paper where it was demonstrated for light.  By the way, you can think of localization of light as an effective cavity that confines the radiation via disorder scattering, an idea which in turn led to the random laser.  Just earlier this year, people successfully demonstrated 3D Anderson localization of ultrasound.

I used google gemini to code up a toy model of Anderson localization (of light) in HTML5, where the disorder is in the form of a spatially varying index of refraction. (I used periodic boundary conditions.) If the disorder is weak (5 in toy units), all the energy dumped into the middle of the space spreads out roughly equally to fill the whole region.  However, if the disorder is strong (50 in toy units), the energy of the waves is localized near the origin for long simulation times.  Here is the model.  (No deep claims of strict accuracy here; this was quick and dirty.  To really see localization in this small play area, we'd need to \(\xi\) to be small compared to the size of the region because of the periodic boundary conditions.)

The ideas here have had a very long reach, and I'll likely write more about related physics soon.

Thermometry at the mK scale, revisited

It's been almost a decade since I last wrote about this topic, and a preprint on the arXiv this week is a good jumping off point for more discussion.

Thermometers are devices that allow us to take some physical observable and infer temperature.  I wrote about the nature of temperature 17 years ago (!!!) in a way that did not completely satisfy me or most of my readers, so maybe I should take another crack at it.  Temperature is a statistically emergent quantity (it doesn't make sense to talk about the temperature of a single particle in isolation) that tells us whether there will be a net flow of energy when a system we care about is brought into contact (able to exchange energy via microscopic degrees of freedom that we aren't tracking, like jiggling of atoms bumping into each other or emission/absorption of radiated photons) with some other system.  Temperature is closely related to the energy stored in the microscopic degrees of freedom of a system.  Our definition of \(T\) is such that there will be a spontaneous, net, averaged flow of energy from hot (a high \(T\) system) to cold (a low \(T\) system).  Two systems in contact at the same \(T\) will still exchange energy microscopically, but on average there will be no net flow, and in the absence of other complications, these systems are said to be in thermal equilibrium.

Measuring temperature is serious business with a fascinating history.  The kelvin is, as of 2019 (see, told you it was time to revisit this), defined by using the fundamental definitions of the kilogram, the meter, and the second, and by declaring that Boltzmann's constant \(k_{\mathrm{B}}\) is exactly 1.380 649 ×10⁻²³ J/K or equivalently kg m²/s²K.   In practice, there are fixed, measurable reference points that help make sure temperatures are calibrated.  For example, the triple point of water is a standard reference point at 273.16 K.  In total, there are two internationally agreed temperature scales, ITS-90 (pdf) and PLTS-2000 (pdf), that include a total of 21 reference points spanning from 0.9 mK to 1357.77 K.  

It's extremely helpful to have primary thermometers, where the physics involved in some measurable quantity are so well known that it is possible to analyze a measurement and directly pull out \(T\) based only on the data and known fundamental and numerical constants.  The preprint linked at the top of this post does an extremely careful comparison of two nanostructure-based approaches.  

Adapted from Fig. 1 from here.

A Coulomb blockade thermometer consists of a series of tiny metal/insulator/metal junctions.  The energy required to move a single electron across one such junction is proportional to \(e^2/C\), where \(C\) is the capacitance of the junction structure.  When the temperature is low, that charging energy scale can exceed the thermal energy scale, \(k_{\mathrm{B}}T\), so that the conductance \(dI/dV\) of the junction near zero applied voltage is suppressed compared to its high voltage and high temperature value.   If temperature is very low, conductance is suppressed all the way to zero, and that is Coulomb blockade.  If instead you have an array of identical junctions in series, and the temperature isn't too low, there is a perturbative suppression of conductance at zero bias. Remarkably, in this regime the shape of \(dI/dV\) as a function of \(V\) is universal, independent of details, and for an array of \(N\) junctions in series, its width is \(5.44 N k_{\mathrm{B}}T/e\). (top panel of figure) 

In a single tunnel junction, it is possible to measure Johnson-Nyquist noise, the current (voltage) fluctuations that take place across the device due to thermally driven motion of the electrons in equilibrium, and the charge shot noise, the fluctuations due to the statistical variations in the arrival times of the electrons.  The theoretical expression for the noise as a function of bias voltage is known (Eq. (2) in the paper).  (bottom panel of figure).

The authors find that the two thermometric approaches are quantitatively consistent to better than 2.5% between 20 mK and 235 mK, and the biggest uncertainty comes from knowing the effective bandwidth of the noise measurement.  This is a characteristically careful, clean work from this Finnish group, who are world experts in the field.  

NAS "State of Science" 2026 address

I watched the webcast of the NAS State of Science address by outgoing NAS president Dr. Marcia McNutt.  (I did not watch the panel discussion afterward, so sorry if I missed critical pieces.)  A few thoughts on this:

• The intro music was a very classy baroque string quartet.  Hard not to think of this scene from Titanic.
• 
  The main theme was about ways to revitalize US science, and there were six main points that she wanted to emphasize, each with examples of relevant projects underway, ways to measure success, and the consequences of failure.  That's fine, and I'll relay them below with some comments, but first an overall impression:  This was largely an exercise in avoiding talking about the elephant in the room, the overt hostility toward and the attempted wanton dismantling of much of the publicly funded US research ecosystem by the executive branch.  I'm unfortunately not surprised that this was largely brushed over, given the position of the Academies (see here).  As the saying goes, I'm not mad, I'm just disappointed.  The realization that the National Academies leadership do not feel empowered to have a frank discussion about this publicly has been depressing.
• Dr. McNutt mentioned that in her previous address, she had pointed out the US vulnerability in STEM by being so reliant on international talent, and that now that other countries are heavily investing in research, the US STEM research world needs to do a better job getting US citizens in the workforce.  That's all true, but leaving out how the government leadership is explicitly trying to curtain international scholars and international collaboration seems like quite an omission.
• She mentioned in passing that industrial research in the US in the 1950s was tiny, nothing compared to the fraction of R&D it is today.  Is that actually correct?  I mean, that was the heyday of Bell Labs, IBM, GE, Westinghouse, and big research labs at companies like Ford and GM.  Much has been written about this.  
• The first big point was the need for improved relationships between universities and industry, and some examples of ways to encourage this, including relatively simple policy changes like making it easier for faculty and others to take leaves in industry.  Certainly it would be broadly good for the US research ecosystem to have more diverse forms of support, and as I've written before, major industrial sectors with lots of capital rely in the long term on trained people. 
• The second point was the need to realign the academic reward system, so that industrial/entrepreneurial/coalition-building activities are incentivized, rather than rewarding on lone-wolf PIs. That's fine, and honestly I think it's already happening to some large degree at major research universities. 
• The third point was meeting the needs of the STEM workforce, though increased interactions with industry (including, e.g., prospective industrial employers helping to define dissertation topics), co-op efforts, some training in businessy aspects (note:  the Sloan Foundation was pushing this 25 years ago.).  This is all laudable to try, but I don't see how any of this actually addresses the issue of fewer STEM workforce participation from US citizens, which is quite complicated.
• The fourth point was the need to reduce regulatory burden.  Sure, we all want to reduce bureaucratic BS.  I have to say, though, that it was genuinely baffling to me that the most Dr. McNutt had to say about the threatened OMB rule changes (apart from a passing mention early on) is that they would increase bureaucracy.  That isn't even in the top 15 problems raised by those changes.  Remember, the default position of those pushing those rules is that academics are fundamentally untrustworthy and poor stewards of public resources.  
• Fifth was the need for automated/self-driving labs.  I agree completely that advanced degree training should not be driven by the need for cheap labor to do tedious lab tasks (e.g. a zillion cell cultures or chemical syntheses).  Overall this was pretty innocuous.
• Sixth, Dr. McNutt emphasized the need to take on big challenges - researchers need to be bold and not play it safe, and peer review can be inherently biased toward incrementalism.  She gave examples of large privately endowed institutes as enabling such work (MBARI, the Allen Institute).  Apparently STAC will be proposing new multi-agency science and technology "breakthrough funds".  The argument in favor of public investment in science in this section sounded rote rather than heartfelt.  If anything, I thought knocking peer review right now at a time when OMB wants to ignore it at their pleasure was a weird position to take.

To be clear:  I don't think any of the ideas highlighted in the speech are actually bad (necessarily).  It just avoided emphasizing that publicly funded research has been incredibly beneficial, and that irreversible harm is being done.  The statement that science agencies "have seen a loss of key personnel" is the worst kind of passive voice garbage.  A hundred thousand technical personnel leaving agencies is not something that just "happened" like the weather.  Being quiet, avoiding confrontation, and only trying to work behind the scenes is not the leadership that is needed now.  (See, I can do passive voice, too.)

I will try to get back to more science posting....

Info gathering: Excellent intro undergrad lab courses and facilities?

Introductory undergraduate labs are a recurring challenge at nearly every university.  Is the purpose to teach students something about how experimental science works (formulating hypotheses, defining measurement needs, setting up equipment, acquiring and analyzing data)?  Is the purpose to emphasize and reinforce specific scientific points from the curriculum?  How structured should they be?  Where are there opportunities for interdisciplinary labs rather than traditional physics/chemistry/biology/earth sciences stovepipes?  

I'm interested in learning about US examples of outstanding introductory physics labs - both in their content/execution, and in the intro lab facilities that my readers consider to be particularly well done.  Please respond in the comments or via email.   I'd really appreciate your thoughts on this, even knowing that my blog readership is a highly biased sample.

(I tried launching a survey about undergrad physics lab instruction five years ago.  I got zero responses.  Hopefully this will be a little more successful.)

The Manhattan Project and public communication

The Manhattan Project was the largest government sponsored research and development project of its time.  Some things worth noting, in light of the present US government attitude toward science:

• It's hard to overstate the role played by immigrant scientists in this story.  Szilard, Einstein, Fermi, Wigner, Teller, von Neumann, and many more.  
• I was trying to remember when the Manhattan Project became publicly known in any detail.  It turns out, within three days of the US bombing of Nagasaki, the US released a tidily written report headlined by Henry DeWolf Smyth on all the essentials, including the administrative story of how the project came to be and was managed.  That report is available in many forms, including this cute version on the internet archive and simple pdf files at DOE and Princeton.  It's an outstanding piece of clear, spare writing.  It almost boggles the mind: Here was a technical topic that the national leadership considered important for the public to understand (!), so a highly readable report was prepared and released basically immediately following public knowledge of the bombs. (!!)
• The National Academies played a pivotal role in this story.  On page 51:  "In the spring of 1941, Briggs, feeling that an impartial review of the problem was desirable, requested [presidential science adviser Vannevar] Bush to appoint a reviewing committee. Bush then formally requested F. B. Jewett, president of the National Academy of Sciences, to appoint such a committee. Jewett complied, appointing A. H. Compton, chairman; W. D. Coolidge, E. O. Lawrence, J. C. Slater, J. H. Van Vleck, and B. Gherardi."  Once upon a time, the national leadership respected the National Academies and trusted them to provide impartial, accurate scientific advice to inform policy.  Somehow I doubt that Frank Baldwin Jewett, president of the NAS at the time, was worried that the government would cut off funding to the Academy if they didn't toe the line.  (As far as I know, no one from the Roosevelt administration was taking “donations” for lucrative government contracts on the bomb, and no one from the cabinet or the Department of War were personally betting for profit on whether it would work, either, but I digress.)

Just some food for thought.

Brief items - news roundup, AI, international issues, good reading

Several items worth reading about as we head into a long weekend in the US.  Starting with news related to funding and other aspects of US government policy:

• US government taking equity stakes in some quantum information sciences companies while investing around $2B (seemingly from the Department of Commerce and the CHIPs Act resources.  (Non-paywall news story here).  This raises a number of thorny issues. 
• Some US funding agencies (NIH, NASA) are enacting restrictions (Science article here, Inside Higher Ed article here) on publishing scientific papers with non-US coauthors.  It's understandable that US funding agencies are concerned about the possibility US funds directly or effectively supporting researchers in foreign countries.  This is not that, though.  Some people making policy seem to be moving toward wanting to ban any co-authorship, but even the agencies seem confused about what they want.
• In a move that will stress out many non-US-citizens in the country, the administration is floating making people leave the US to apply for green cards (PBS article here).  This just was sort of announced yesterday, so I don't know anything about this other than on its face it sounds to me like a terrible idea for multiple reasons.  
• The AAAS is pushing for a Senate hearing on the nominee for NSF director, on the theory that this issue and the nominee at least need to be discussed in a public forum rather than coasting along without a NSB and no end in sight to interim leadership.
• It would seem that some Republican congresspeople are pushing the idea of de-funding the National Academies.  This is directly related to the issues mentioned here.  I think the National Academies should be endowed and thus not so reliant on federal funding; this would be a way to make sure that they always feel secure in delivering reports even if the customer is a part of the government and the conclusions might be something the customer doesn't want to hear.

There was a lot of AI-related news this week:

• 
  There were three papers published in Nature about using AI agents to do science (here, here, and here, with a news and views).  The first two papers are both about drug discovery research, and the third is about using AI to help write scientific software models (also medically related).  It'll be interesting to see how this progresses.  
• One of OpenAI's tools solved an Erdos problem (that's the OpenAI release) by finding a counterexample to a conjecture long thought to be true.  Here is the accompanying paper, which includes commentary by several esteemed mathematicians.  The commentary parts of the paper are very much for non-mathematicians and fascinating to read.  It seems like the AI tools are genuinely good at pulling together complex arguments, and that so far a key advantage they have is an exhaustive familiarity with the full breadth of the literature.
• Unsurprisingly, university graduates are not fans of AI.  This cartoon from this week's New Yorker is topical.  
  

 Additional suggestions that look cool but I haven't had time to actually read:

• Jeremy Levy at Pitt has posted his draft graduate quantum mechanics textbook to the arXiv - looks like an interesting take on the pedagogy of the subject for sure. 
• Dimitrii Makarov at UT Austin has posted a list of critical papers in chemical physics.  
• Samuel Braunstein at York has put up this preprint regarding the Clay Millennium Prize problem about the Navier-Stokes equations.  His main points seem at a quick read to be (1) at finite temperature, fluctuation-dissipation requires some additional noise term be present in the N-S equations; and (2) the N-S equations assume a continuum fluid, so that infinitely-short wavelength excitations are permitted, while for all real fluids wavelengths get cut off at the molecule size.  Basically the issues w/ the N-S equations come from unphysical aspects of the formulation.