Ask me something.

 I realized today that I had not had an open "Ask me something" post since December, 2018.  Seems like it's time - please have at it.

What is disorder, to condensed matter physicists?

Condensed matter physicists throw around the term "disorder" quite a bit - what does this mean, and how is it quantified?  This is particularly important when worrying about comparatively delicate, exotic quantum states, as in the recent discussions of the challenge of experimentally observing emergent Majorana fermions at the interfaces between semiconductor nanowires and superconductors.  

Latent in the use of the word "disorder" is a contrast with "order".  One of the most powerful ideas in condensed matter is Bloch's theorem:  In (infinite) crystalline solids, the spatial periodicity of the arrangement of atoms in a lattice leads to the conservation of a quantity \(\hbar \mathbf{k}\), the crystal momentum, for the electrons.  The allowed energies of single-electron states in that lattice (neglecting electron-electron interaction effects) is then a function \(E(\mathbf{k})\), and it is possible to think about a wavepacket (blob) of electrons with some dominant \(\hbar \mathbf{k}\) propagating along, as discussed extensively here for example.   "Disorder" in this context is some break with perfect spatial periodicity, which breaks \(\mathbf{k}\) conservation - in the Drude picture, this is what causes electron trajectories to scatter and do a random, diffusive walk.  

Now, not all disorder is created equal.  In a metal like gold, there is a quantitative difference between having a dilute concentration of silver atoms substituted on gold sites, and alternately having the same concentration of vacancies on gold sites.  Surely the latter is somehow more disordered.  In quantum classes, we learn to think about scattering lengths, and in conductors one can ask the physically motivated question, how far would a wavepacket propagate between scattering events (a "mean free path", \(\ell\), compared to its dominant wavelength \(\lambda\)?  For a metal we can think of the product  \(k_{\mathrm{F}} \ell\), where \(k_{\mathrm{F}}\) is the Fermi wavevector, \(2 \pi/ \lambda_{\mathrm{F}}\).  A "good metal" has \( k_{\mathrm{F}} \ell >> 1 \).  When \(k_{\mathrm{F}} \ell\  < 1\), it doesn't make sense to think of propagating wavepackets anymore.  

In other contexts, it's more helpful to think of disorder explicitly as associated with an energy scale that I'll call \(\delta\).  Some sort of structural change in a material away from ordered perfection leads, on some length scale, to a shift in electronic energies by an amount of typical magnitude \(\delta\).  The question then becomes, how does \(\delta\) compare with other energy scales in the material?  The case above where \(k_{\mathrm{F}} \ell < 1\) roughly corresponds to \(\delta\) being comparable to the electronic bandwidth (the energetic extent of \(E(\mathbf{k})\).  When one wants to think about the effects of disorder on superconductors, an important ratio is \(\delta/\Delta\), where \(\Delta\) is the superconducting gap energy scale of the ordered case.   When one wants to think about the effects of disorder on some fragile emergent phase like a fractional quantum Hall state, then a relevant comparison is between \(\delta\) and the relevant energy scale associated with that state.  

TL/DR version:  "Disorder" is a catch-all term, and it is quantified by how strongly the system is perturbed away from some target ordered condition.  

It's worth remembering that some of the progenitors of modern physics thought that it would be impossible to learn much about the underlying physics of real materials because disorder would be too severe and too idiosyncratic (that is, that each kind of defect would have its own peculiar impacts).  That's why Pauli derisively said "Festkörperphysik ist eine Schmutzphysik" (solid-state physics is the physics of dirt).   Fortunately, we have been able to learn quite a bit, and disorder has its own beautiful results, even if it continues to be the bane of some problems.

Catching up

As may be obvious from my pace of posting, the last couple of weeks have been very busy and intense for multiple reasons.  I hope that once the academic year really ends I can get back into more of a routine.

Two notable stories this week:

• Two papers were published back-to-back in Science (here and here, with commentary here) that demonstrate (a) that comparatively macroscopic mechanical oscillators - drumheads - can be operated as true quantum objects (cooled down to the point where the thermal energy scale \(k_{\mathrm{B}}T\) is small compared to the quantum energy level spacing \(\hbar \omega\) (this has been done before); and that these resonators can be quantum mechanically entangled, so that the two have to be treated as a single quantum system when understanding measurements performed on each individually.   This can be used, in the case of the second paper, to allow clever measurement schemes that shift measurement back-action (see here for a nice tutorial) away from a target system, enabling precision measurements of the target better than standard quantum limits.  
• IBM has demonstrated 300 mm wafer fabrication of integrated circuits with features and techniques for the upcoming "2 nm node".  As I've mentioned before, we have fully transitioned to the point where labeling new semiconductor manufacturing targets with a length scale is basically a marketing ploy - the transistors on this wafer do not have 2 nm channel lengths, and the wiring does not have 2 nm lines and spaces.  However, this is a very impressive technical demonstration of wafer-scale success in a number of new approaches, including triple-stacked nanosheet gate-all-around transistors.