Tag Archives: Excitons

Electron-Hole Droplets

While some condensed matter physicists have moved on from studying semiconductors and consider them “boring”, there are consistently surprises from the semiconductor community that suggest the opposite. Most notably, the integral and fractional quantum Hall effect were not only unexpected, but (especially the FQHE) have changed the way we think about matter. The development of semiconductor quantum wells and superlattices have played a large role furthering the physics of semiconductors and have been central to the efforts in observing Bloch oscillations, the quantum spin Hall effect and exciton condensation in quantum hall bilayers among many other discoveries.

However, there was one development that apparently did not need much of a technological advancement in semiconductor processing — it was simply just overlooked. This was the discovery of electron-hole droplets in the late 60s and early 70s in crystalline germanium and silicon. A lot of work on this topic was done in the Soviet Union on both the theoretical and experiment fronts, but because of this, finding the relevant papers online are quite difficult! An excellent review on the topic was written by L. Keldysh, who also did a lot of theoretical work on electron-hole droplets and was probably the first to recognize them for what they were.

Before continuing, let me just emphasize, that when I say electron-hole droplet, I literally mean something quite akin to water droplets in a fog, for instance. In a semiconductor, the exciton gas condenses into a mist-like substance with electron-hole droplets surrounded by a gas of free excitons. This is possible in a semiconductor because the time it takes for the electron-hole recombination is orders of magnitude longer than the time it takes to undergo the transition to the electron-hole droplet phase. Therefore, the droplet can be treated as if it is in thermal equilibrium, although it is clearly a non-equilibrium state of matter. Recombination takes longer in an indirect gap semiconductor, which is why silicon and germanium were used for these experiments.

A bit of history: The field got started in 1968 when Asnin, Rogachev and Ryvkin in the Soviet Union observed a jump in the photoconductivity in germanium at low temperature when excited above a certain threshold radiation (i.e. when the density of excitons exceeded \sim 10^{16}  \textrm{cm}^{-3}). The interpretation of this observation as an electron-hole droplet was put on firm footing when a broad luminescence peak was observed by Pokrovski and Svistunova below the exciton line (~714 meV) at ~709 meV. The intensity in this peak increased dramatically upon lowering the temperature, with a substantial increase within just a tenth of a degree, an observation suggestive of a phase transition. I reproduce the luminescence spectrum from this paper by T.K. Lo showing the free exciton and the electron-hole droplet peaks, because as mentioned, the Soviet papers are difficult to find online.

EHD-Lo.JPG

From my description so far, the most pressing questions remaining are: (1) why is there an increase in the photoconductivity due to the presence of droplets? and (2) is there better evidence for the droplet than just the luminescence peak? Because free excitons are also known to form biexcitons (i.e. excitonic molecules), the peak may easily interpreted as evidence of biexcitons instead of an electron-hole droplet, and this was a point of much contention in the early days of studying the electron-hole droplet (see the Aside below).

Let me answer the second question first, since the answer is a little simpler. The most conclusive evidence (besides the excellent agreement between theory and experiment) was literally pictures of the droplet! Because the electrons and holes within the droplet recombine, they emit the characteristic radiation shown in the luminescence spectrum above centered at ~709 meV. This is in the infrared region and J.P. Wolfe and collaborators were actually able to take pictures of the droplets in germanium (~ 4 microns in diameter) with an infrared-sensitive camera. Below is a picture of the droplet cloud — notice how the droplet cloud is actually anisotropic, which is due to the crystal symmetry and the fact that phonons can propel the electron-hole liquid!

Pic_EHD.JPG

The first question is a little tougher to answer, but it can be accomplished with a qualitative description. When the excitons condense into the liquid, the density of “excitons” is much higher in this region. In fact, the inter-exciton distance is smaller than the distance between the electron and hole in the exciton gas. Therefore, it is not appropriate to refer to a specific electron as bound to a hole at all in the droplet. The electrons and holes are free to move independently. Naively, one can rationalize this because at such high densities, the exchange interaction becomes strong so that electrons and holes can easily switch partners with other electrons and holes respectively. Hence, the electron-hole liquid is actually a multi-component degenerate plasma, similar to a Fermi liquid, and it even has a Fermi energy which is on the order of 6 meV. Hence, the electron-hole droplet is metallic!

So why do the excitons form droplets at all? This is a question of kinetics and has to do with a delicate balance between evaporation, surface tension, electron-hole recombination and the probability of an exciton in the surrounding gas being absorbed by the droplet. Keldysh’s article, linked above, and the references therein are excellent for the details on this point.

In light of the recent discovery that bismuth (also a compensated electron-hole liquid!) was recently found to be superconducting at ~530 microKelvin, one may ask whether it is possible that electron-hole droplets can also become superconducting at similar or lower temperatures. From my brief searches online it doesn’t seem like this question has been seriously asked in the theoretical literature, and it would be an interesting route towards non-equilibrium superconductivity.

Just a couple years ago, a group also reported the existence of small droplet quanta in GaAs, demonstrating that research on this topic is still alive. To my knowledge, electron-hole drops have thus far not been observed in single-layer transition metal dichalcogenide semiconductors, which may present an interesting route to studying dimensional effects on the electron-hole droplet. However, this may be challenging since most of these materials are direct-gap semiconductors.

Aside: Sadly, it seems like evidence for the electron-hole droplet was actually discovered at Bell Labs by J.R. Haynes in 1966 in this paper before the 1968 Soviet paper, unbeknownst to the author. Haynes attributed his observation to the excitonic molecule (or biexciton), which he, it turns out, didn’t have the statistics to observe. Later experiments confirmed that it indeed was the electron-hole droplet that he had observed. Strangely, Haynes’ paper is still cited in the present time relatively frequently in the context of biexcitons, since he provided quite a nice analysis of his results! Also, it so happened that Haynes died after his paper was submitted and never found out that he had actually discovered the electron-hole droplet.

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Excitons in Cuprous Oxide

There is an excellent pair of papers here and here demonstrating the existence of a Rydberg series of excitons in Cu_2O, a material that has long been the poster child for observing excitons. The data in both papers are pretty stunning — in the first one, they see a Rydberg series up to n=25! In the second, they see higher angular momentum (f-like and h-like) excitons apart from the usual p-like excitons.

Excitons can occur in semiconductors and insulators and are the result of the Coulomb interaction between electrons and holes. For those familiar with superconductivity, an exciton, in many ways, bears a qualitative resemblance to Cooper pairs, except here the quasi-particles are of opposite charge. Hence, the binding between the electron and hole arises from the usual attractive Coulomb interaction. However, it helps to think of the exciton (similar to Cooper pairs) as possessing a center of mass wavefunction and a relative wavefunction.

The relative wavefunction is then qualitatively and quantitatively similar to the quantum mechanical solution of the hydrogen atom. The main differences are that (1) the reduced mass of the electron-hole pair is usually much less than the reduced mass of the electron-proton system and also that (2) there exists a background dielectric constant, \epsilon, that further reduces the energy of the Rydberg series in an insulator. In the simple hydrogen-like picture of the exciton, the energy levels (with center of mass momentum taken to be zero) then have the following energies:

E_n =- \frac{\mu}{m\epsilon^2}\frac{R}{n^2}

where \mu is the reduced mass of the electron-hole system, m is the proton mass, and R is the hydrogenic Rydberg.

It turns out that this perfectly spherically symmetric model does not encapsulate completely the exciton in a solid, however, because the potential in a crystal is not perfectly rotation-symmetric. Because of the lattice, the “angular momentum” of the exciton cannot be labeled using an s, p, d, f, etc. kind of scheme and we must resort to some group theory.

For the case of Cu_2O discussed in the papers above, the symmetry of the crystal is cubic — therefore the symmetry is still relatively high. However, because continuous rotational symmetry is broken, the light used to detect the excitons in this experiment are no longer forbidden from exciting f-like and h-like excitons and it is remarkable that they are observable!

Even though the observations in these papers can be accounted for in a quite simple theoretical framework, the experimental results are nonetheless quite elegant and remarkable.

As an interesting aside, the reason that this experiment is possible is because of the extreme purity of the Cu_2O crystals used for these studies. It turns out the cleanest of these samples are naturally-occurring, rather than man-made.