Ferromagnetism is a state of matter whose existence has been known for millennia (Fe3O4 supposedly was the first known permanent magnet, but is actually a ferrimagnet). Coincidentally, it was also the material that took center stage in Expt 8 in this series of posts. Antiferromagnetism is, in some sense, “hidden order”. Although predicted in 1932 by Louis Neel, it was not apparent how to convincingly demonstrate that spins could align antiparallel to one another when the magnetic moments perfectly cancel. There is no net magnetization to speak of or measure!
Suggestions of a such a state were hinted at through the observation of a phase transition in magnetic susceptibility measurements, but the origin of such a phase transition was unclear. In 1949, Clifford Shull and J. Stuart Smart definitively demonstrated that antiferromagnetism was present in MnO. In their experiment, which used the new technique of neutron diffraction, magnetic peaks suddenly showed up below , which indicated a doubling of the unit cell. Thus, the question of “hidden order” was finally settled — the evidence for antiferromagnetism was almost indisputable! The original paper, surprisingly and explicitly, states that it was Smart’s idea to use neutron diffraction to detect antiferromagnetism. Below is the original plot from MnO, taken from here, showing at least three peaks corresponding to the antiferromagnetism:
As you can probably tell from my previous post, I have found the recent activism inspiring and genuinely hopeful of it translating into some form of justice and meaningful action. At the end of this post I share a few videos that I found particularly poignant.
It’s hard to imagine the history of condensed matter physics without both the discovery and theory of superconductivity. Superconductivity has played and continues to play an outsized role in our field, and it is quite easy to understand why this is the case. All one has to do is to imagine what our world would look like with room temperature superconductivity. Besides the potential technological implications, it has also garnered attention because of the wealth of stunning effects associated with it. A few examples include the Josephson effect, flux quantization, persistent superconducting currents, vortex lattices and the Meissner effect.
Now, these effects occur for various reasons, but there are a couple of them that can be viewed to some extent as a microscopic effect on a macroscopic scale. To show what I mean by that, I am going to focus on the Meissner effect and talk about how we can view it as an amplification of atomic diamagnetism. One could also extend the this microscopic to macroscopic amplification picture to the relationship between a Josephson junction in a superconducting ring and the Aharonov-Bohm effect, but I’ll leave that discussion to another day.
To understand what I mean by amplification, let’s first look at atomic diamagnetism. Here we can use a similar logic that led to the Bohr model of the atom. Two conditions are important here — (i) the de Broglie relation and (ii) the Bohr quantization condition which states that only integer wavelengths are allowed in a closed loop (such as an atomic orbit). See the image below for a simple picture (click the image for the source).
We can use the classical relation for the momentum in addition to equations (i) and (ii) above to get , which is what Bohr got in his atomic model. It’s worth noting here that when the atom is in its ground state (i.e. ), there is no “atomic current”, meaning that . Without this current, however, it is not possible to have a diamagnetic response.
So how do we understand atomic diamagnetism? To do so, we need to incorporate the applied field into the deBroglie relation by using the canonical momentum. By making the “Peierls substitution”, we can write that . Using the same logic as above, our quantization condition is now . Now, however, something has changed; we do get a non-zero current in the ground state (i.e. for ). Qualitatively, this current circulates to screen out the field that is trying to “mess up” the integer-number-of-wavelengths-around-the-loop condition. Note also that we have a response that is strictly quantum mechanical in nature; the current is responding to the vector potential. (I realize that the relation is not strictly gauge invariant, but it makes sense in the “Coulomb gauge”, i.e. when or when the vector potential is strictly transverse). In some sense, we already knew that our answer must look obviously quantum mechanical because of the Bohr-van Leeuwen theorem.
If we examine the equation for the electromagnetic response to a superconductor, i.e. the London equation, we obtain a similar equation , where is the superfluid density. The resemblance between the two equations is far from superficial. It is this London equation which allows us to understand the origin of the Meissner effect and the associated spectacular diamagnetism. Roughly speaking then, we can understand the Meissner effect as an amplification of an atomic effect that results in a non-zero ground state “screening” current.
I would also like to add that the Meissner effect is also visible in a multiply connected geometry (see below). This time, the magnetic field (for sufficiently small magnetic fields) is forbidden from going through the center of the ring.
What is particularly illuminating about this ring geometry is that you don’t have to have a magnetic field like in the image above. In fact, it is totally possible to have a superconducting ring under so-called Aharonov-Bohm conditions, where a solenoid passes through the center but the ring never sees the magnetic field. Instead, the superconducting ring “feels the vector potential”. In some sense, this latter experiment emphasizes the equation above where the current really responds (in a gauge-invariant way) to a vector potential and not just the magnetic field.
Understanding the Meissner effect in this way helps us divorce the Meissner effect from the at-first-sight similar effect of persistent currents in a superconducting ring. In the Meissner effect, as soon as the magnetic field is turned off, the current dies and goes back to zero. This is because through this entire process, the superconductor remains in its ground state. Only when the superconductor is excited to higher states (i.e. …) does the current persist in a metastable fashion for a quasi-infinitely long time.
To me, understanding the Meissner effect in this way, which exposes the connection of the microscopic to the macroscopic, harks back to an old post I made about Frank Wilczek’s concept of upward inheritence. The Meissner effect somehow seems clearer through his lens.
Now as promised, here are the couple videos (if the videos don’t play, click on the panel to take you to the twitter website because these videos are worth watching!):
Beautiful to see people of all colors coming together, but we have a system unable to respond. Looting is wrong and murder is wrong, but their legalized forms are wrong too. Look at the wickedness in high places first, then at the least of these.#watch: https://t.co/g9GcLOjG4R
(1/2) Trevor on George Floyd, the Minneapolis protests, Ahmaud Arbery & Amy Cooper:
“While everyone is facing the battle against coronavirus, black people in America are still facing the battle against racism… and coronavirus.” pic.twitter.com/eaVUdq6NzJ
An Interesting Research Avenue: A couple months ago, Stephane Mangin of the Insitut Jean Lamour gave a talk on all-optical helicity-dependent magnetic switching (what a mouthful!) at Argonne, which was fascinating. I was reminded of the talk yesterday when a review article on the topic appeared on the arXiv. The basic phenomenon is that in certain materials, one is able to send in a femtosecond laser pulse onto a magnetic material and switch the direction of magnetization using circularly polarized light. This effect is reversible (in the sense that circularly polarized light in the opposite direction will result in a magnetization in the opposite direction) and is reproducible. During the talk, Mangin was able to show us some remarkable videos of the phenomenon, which unfortunately, I wasn’t able to find online.
The initial study that sparked a lot of this work was this paper by Beaurepaire et al., which showed ultrafast demagnetization in nickel films in 1996, a whole 20 years ago! The more recent study that triggered most of the current work was this paper by Stanciu et al. in which it was shown that the magnetization direction could be switched with a circularly polarized 40-femtosecond laser pulse on ferromagnetic film alloys of GdFeCo. For a while, it was thought that this effect was specific to the GdFeCo material class, but it has since been shown that all-optical helicity-dependent magnetic switching is actually a more general phenomenon and has been observed now in many materials (see this paper by Mangin and co-workers for example). It will be interesting to see how this research plays out with respect to the magnetic storage industry. The ability to read and write on the femtosecond to picosecond timescale is definitely something to watch out for.
Update: After my post on the Gibbs paradox last week, a few readers pointed out that there exists some controversy over the textbook explanation that I presented. I am grateful that they provided links to some articles discussing the subtleties involved in the paradox. Although one commenter suggested Appendix D of E. Atlee Jackson’s textbook, I was not able to get a hold of this. It looks like a promising textbook, so I may end up just buying it, however!
The links that I found helpful about the Gibbs paradox were Jaynes’ article (pdf!) and this article by R. Swendsen. In particular, I found Jaynes’ discussion of Whifnium and Whoofnium interesting in the role that ignorance and knowledge plays our ability to extract work from a partitioned gases. Swendsen’s tries to redefine entropy classically (what he calls Boltzmann’s definition of entropy), which I have to think about a little more. But at the moment, I don’t think I buy his argument that this resolves the Gibbs paradox completely.
In a similar vein, there was an elegant experiment published in PRL back in 1977 by P. Grunberg and F. Metawe that shows a rather peculiar spectral signature observed with Brillouin scattering in thin film EuO. The data is presented below. For those who don’t know, Brillouin scattering is basically identical to Raman scattering, but the energy scale observed is much lower, typically a fraction of a ~ 5 (1 30GHz). Brillouin scattering is often used to observe acoustic phonons.
Anyway, in usual light scattering experiments, depending on the temperature, modes are observed on both sides (Stokes and anti-Stokes) with an intensity difference determined by Bose-Einstein statistics. In this case, two ingredients, the slab geometry of the thin film and the broken time-reversal symmetry give rise to the propagation of a surface spin wave that travels in only one direction, known as the Damon-Eshbach mode. The DE mode propagates on the surface of the sample in a direction perpendicular to the magnetization, B, of the thin film, obeying a right-hand rule.
When one thinks about this, it is truly bizarre, as the dispersion relation would for the DE mode on the surface would look something like the image below for the different magnetic field directions:
One-way propagation of Damon Eshbach Mode
The dispersion branch only exists for one propagation direction! Because of this fact (and the conservation of momentum and energy laws), the mode is either observed solely on the Stokes or anti-Stokes side. This can be understood in the following way. Suppose the experimental geometry is such that the momentum transferred to the sample, q, is positive. One would then be able to excite the DE mode with the incident photon, giving rise to a peak on the Stokes side. However, the incident photon in the experiment cannot absorb the DE mode of momentum -q, because it doesn’t exist! Similar reasoning applies for the magnetization in the other direction, where one would observe a peak in only the anti-Stokes channel.
There is one more regard in which this experiment relied on a serendipitous occurrence. The thin film was thick enough that the light, which penetrates about 100 Angstroms, did not reach the back side of the film. If the film had been thin enough, a peak would have shown up in both the Stokes and anti-Stokes channels, as the photon would have been able to interact with both surfaces.
So with a little fortune and a lot of ingenuity, this experiment set Peter Grunberg on the path to his Nobel prize winning work on magnetic multilayers. As far as simple spectroscopy experiments go, one is unlikely to find results that are as remarkable and dramatic.
and (ii) the Bohr quantization condition 
which states that only integer wavelengths are allowed in a closed loop (such as an atomic orbit). See the image below for a simple picture (click the image for the source). 
in addition to equations (i) and (ii) above to get 
, which is what Bohr got in his atomic model. It’s worth noting here that when the atom is in its ground state (i.e. 
), there is no “atomic current”, meaning that 
. Without this current, however, it is not possible to have a diamagnetic response.
. Using the same logic as above, our quantization condition is now 
. Now, however, something has changed; we do get a non-zero current in the ground state (i.e. 
for 
or when the vector potential is strictly transverse). In some sense, we already knew that our answer must look obviously quantum mechanical because of the 
, where 
is the superfluid density. The resemblance between the two equations is far from superficial. It is this London equation which allows us to understand the origin of the Meissner effect and the associated spectacular diamagnetism. Roughly speaking then, we can understand the Meissner effect as an amplification of an atomic effect that results in a non-zero ground state “screening” current.
…) does the current persist in a metastable fashion for a quasi-infinitely long time.
~ 5 
30GHz). Brillouin scattering is often used to observe acoustic phonons.
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