Category Archives: Research

Wannier-Stark Ladder, Wavefunction Localization and Bloch Oscillations

Most people who study solid state physics are told at some point that in a totally pure sample where there is no scattering, one should observe an AC response to a DC electric field, with oscillations at the Bloch frequency (\omega_B). These are the so-called Bloch oscillations, which were predicted by C. Zener in this paper.

However, the actual observation of Bloch oscillations is not as simple as the textbooks would make it seem. There is an excellent Physics Today article by E. Mendez and G. Bastard that outline some of the challenges associated with observing Bloch oscillations (which was written while this paper was being published!). Since the textbook treatments often use semi-classical equations of motion to demonstrate the existence of Bloch oscillations in a periodic potential, they implicitly assume transport of an electron wave-packet. To generate this wave-packet is non-trivial in a solid.

In fact, if one undertakes a full quantum mechanical treatment of electrons in a periodic potential under the influence of an electric field, one arrives at the Wannier-Stark ladder, which shows that an electric field can localize electrons! It is this ladder and the corresponding localization which was key to observing Bloch oscillations in semiconductor superlattices.

Let me use the two-well potential to give you a picture of how this localization might occur. Imagine symmetric potential wells, where the lowest energy eigenstates look like so (where S and A label the symmetric and anti-symmetric states):

Now, imagine that I start to make the wells a little asymmetric. What happens in this case? Well, it turns out that that the electrons start to localize in the following way (for the formerly symmetric and anti-symmetric states):

G. Wannier was able to solve the Schrodinger equation with an applied electric field in a periodic potential in full and showed that the eigenstates of the problem form a Stark ladder. This means that the eigenstates are of identical functional form from quantum well to quantum well (unlike in the double-well shown above) and the energies of the eigenstates are spaced apart by \Delta E=\hbar \omega_B! The potential is shown schematically below. It is also shown that as the potential wells slant more and more (i.e. with larger electric fields), the wavefunctions become more localized (the image is taken from here (pdf!)):


A nice numerical solution from the same document shows the wavefunctions for a periodic potential well profile with a strong electric field, exhibiting a strong wavefunction localization. Notice that the wavefunctions are of identical form from well to well.


What can be seen in this solution is that the stationary states are split by \hbar \omega_B, but much like the quantum harmonic oscillator (where the levels are split by \hbar \omega), nothing is actually oscillating until one has a wavepacket (or a linear superposition of eigenstates). Therefore, the Bloch oscillations cannot be observed in the ground state (which includes the the applied electric field) in a semiconducting superlattice since it is an insulator! One must first generate a wavepacket in the solid.

In the landmark paper that finally announced the existence of Bloch oscillations, Waschke et. al. generated a wavepacket in a GaAs-GaAlAs superlattice using a laser pulse. The pulse was incident on a sample with an applied electric field along the superlattice direction, and they were able to observe radiation emitted from the sample due to the Bloch oscillations. I should mention that superlattices must be used to observe the Wannier-Stark ladder and Bloch oscillations because \omega_B, which scales with the width of the quantum well, needs to be fast enough that the electrons don’t scatter from impurities and phonons. Here is the famous plot from the aforementioned paper showing that the frequency of the emitted radiation from the Bloch oscillations can be tuned using an electric field:


This is a pretty remarkable experiment, one of those which took 60 years from its first proposal to finally be observed.

Kapitza-Dirac Effect

We are all familiar with the fact that light can diffract from two (or multiple) slits in a Young-type experiment. After the advent of quantum mechanics and de Broglie’s wave description of matter, it was shown by Davisson and Germer that electrons could be diffracted by a crystal. In 1927, P. Kapitza and P. Dirac proposed that it should in principle be possible for electrons to be diffracted by standing waves of light, in effect using light as a diffraction grating.

In this scheme, the electrons would interact with light through the ponderomotive potential. If you’re not familiar with the ponderomotive potential, you wouldn’t be the only one — this is something I was totally ignorant of until reading about the Kapitza-Dirac effect. In 1995, Anton Zeilinger and co-workers were able to demonstrate the Kapitza-Dirac effect with atoms, obtaining a beautiful diffraction pattern in the process which you can take a look at in this paper. It probably took so long for this effect to be observed because it required the use of high-powered lasers.

Later, in 2001, this experiment was pushed a little further and an electron-beam was used to demonstrate the effect (as opposed to atoms), as Dirac and Kapitza originally proposed. Indeed, again a diffraction pattern was observed. The article is linked here and I reproduce the main result below:


(Top) The interference pattern observed in the presence of a standing light wave. (Bottom) The profile of the electron beam in the absence of the light wave.

Even though this experiment is conceptually quite simple, these basic quantum phenomena still manage to elicit awe (at least from me!).

Diversity in and of Physics

When someone refers to a physicist from the early twentieth century, what kind of person do you imagine? Most people will think of an Einstein-like figure, but most likely, one will think of a white male from western Europe or the US.

Today, however, things have changed considerably; physics, both as a discipline and in the people that represent it, has become more diverse. This correlation is probably not an accident. In my mind, the increased diversity is an excellent development, but as with everything, it can be further improved. There are a couple excellent podcasts I listened to recently that have championed diversity in different contexts.

The first podcast was an episode of Reply All entitled Raising the Bar (which you should really start listening to at 11:52 after the rather cringe-worthy Yes-Yes-No segment!). The episode focuses on the lack of diversity in many companies in Silicon Valley. In doing so, they interview an African-American man named Leslie Miley who was a security guard at Apple and went on to work as a software developer and manager at Twitter, Apple, and Google among other companies (i.e. he possessed a completely unorthodox background by Silicon Valley standards). He makes an interesting statement about companies in general (while referring specifically to Twitter) saying:

If you don’t have people of diverse backgrounds building your product, you’re going to get a very narrowly focused product.

He also goes onto say that including people from different backgrounds is not just appropriate from a moral standpoint, but also that:

Diverse teams have better outcomes.

There is plenty of research to support this viewpoint. In particular, Scott Page from the Santa Fe institute and University of Michigan – Ann Arbor is interviewed in the episode and suggests that when teams of people are selected and asked to perform a task, teams of “good people” from diverse backgrounds generally outperform many “excellent people”/experts from similar backgrounds (i.e. the same Ivy League schools, socio-economic status, age etc.).

There is a caveat that is presented in this episode, however. They suggest that it may take longer for a diverse team to gel and to communicate and understand each other. But again, the outcomes in the long-term are generally better.

There is an excellent episode of Hidden Brain that also covers similar topics, but focuses on building a better workplace. The host of the podcast, Shankar Vendantam, interviews the (then) head of human resources at Google, Laszlo Bock, to gain some insight into how Google has been able to build their talent pool. Of specific interest to physicists was how much Google borrows from places like Bell Labs to build a creative workplace environment. Again, Bock stresses the importance of diversity among the employees at Google in order for the company to be successful.

In physics departments across the country, I think it is necessary to take a similar approach. Departments should strive to be diverse and hire people from different backgrounds, schools, genders, and countries. Not only that, graduate students with unorthodox backgrounds should also be welcomed. This again, is not just important for the health of the department, but for the health of the discipline in general.

I strongly suspect that Michael Faraday was one of the greatest experimental physicists in the past few hundred years not in spite of his lack of mathematical acuity, but probably because of it. His mathematical ability famously did not extend much beyond basic algebra and not even as far as trigonometry.

An Interesting Research Avenue, an Update, and a Joke

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.

A Joke: 

Q: What did Mrs. Cow say to Mr. Cow?

A: Hubby, could you please mooo the lawn?

Q: What did Mr. Cow say back to Mrs. Cow?

A: But, sweetheart, then what am I going to eat?