Tag Archives: Review

Zener’s Electrical Breakdown Model

In my previous post about electric field induced metal-insulator transitions, I mentioned the notion of Zener breakdown. Since the idea is not likely to be familiar to everyone, I thought I’d use this post to explain the concept a little further.

Simply stated, Zener breakdown occurs when a DC electric field applied to an insulator is large enough such that the insulator becomes conducting due to interband tunneling. Usually, when we imagine electrical conduction in a solid, we think of the mobile electrons moving only within one or more partially filled bands. Modelling electrical transport within a single band can already get quite complicated, so it was a major accomplishment that C. Zener was able to come up with a relatively simple and solvable model for interband tunneling.

To make the problem tractable, Zener came up with a hybrid real-space / reciprocal-space model where he could use the formalism of a 1D quantum mechanical barrier:

Tunneling.png

In Zener’s model, the barrier height is set by the band gap energy, E_{g}, between the valence and conduction bands in the insulator, while the barrier width is set by the length scale relevant to the problem. In this case, we can say that the particle can gain enough kinetic energy to surpass the barrier if e\mathcal{E}d = E_{g}, in which case our barrier width would be:

d = E_{g} / e\mathcal{E},

where \mathcal{E} is the applied electric field and e is the electron charge.

Now, how do we solve this tunneling problem? If we were to use the WKB formalism, like Zener, we get that the transmission probability is:

P_T = e^{-2\gamma}             where           \gamma = \int_0^d{k(x) dx}.

Here, k(x) is the wavenumber. So, really, all that needs to be done is to obtain the correct funtional form for the wavenumber and (hopefully) solve the integral. This turns out not to be too difficult — we just have to make sure that we include both bands in the calculation. This can be done in similar way to the nearly free electron problem.

Quickly, the nearly-free electron problem considers the following E-k relations in the extended zone scheme:

E-kRelation

Near the zone boundary, one needs to apply degenerate perturbation theory due to Bragg diffraction of the electrons (or degeneracy of the bands from the next zone, or however you want to think about it). So if one now zooms into the hatched area in the figure above, one gets that a gap opens up by solving the following determinant and obtaining \epsilon(k):

\left( \begin{array}{cc} \lambda_k - \epsilon & E_g/2 \\ E_g/2 & \lambda_{k-G} - \epsilon \end{array} \right),

where \lambda_k is \hbar^2k^2/2m in this problem, and the hatched area becomes gapped like so:

ZoneBoundaryGap

In the Zener model problem, we take a similar approach. Instead of solving for \epsilon(k), we solve for k(\epsilon). To focus on the zone boundary, we first let k \rightarrow k_0 + \kappa and \epsilon \rightarrow \epsilon_0 + \epsilon_1, where k_0 = \pi/a (the zone boundary) and \epsilon_0 = \hbar^2k_0^2/2m, under the assumption that \kappa and \epsilon_1 are small. All this does is shift our reference point to the hatched region in previous figure above.

The trick now is to solve for  k(\epsilon) to see if imaginary solutions are possible. Indeed, they are! I get that:

\kappa^2 = \frac{2m}{\hbar^2} (\frac{\epsilon_1^2 - E_g^2/4}{4\epsilon_0}),

so as long as \epsilon_1^2 - E_g^2/4 < 0, we get imaginary solutions for \kappa.

Although we have a function \kappa(\epsilon_1), we still need to do a little work to obtain \kappa(x), which is required for the WKB exponent. Here, Zener just assumed the simplest thing that he could, that \epsilon_1 is related to the tunneling distance, x, linearly. The image I’ve drawn above (that shows the potential profile) and the fact that work done by the electric field is e\mathcal{E}x demonstrates that this assumption is very reasonable.

Plugging all the numbers in and doing the integral, one gets that:

P_T = \exp-\left(\pi^2 E_g^2/(4 \epsilon_0 e \mathcal{E} a)\right).

If you’re like me in any way, you’ll find the final answer to the problem pretty intuitive, but Zener’s methodology towards obtaining it pretty strange. To me, the solution is quite bizarre in how it moves between momentum space and real space, and I don’t have a good physical picture of how this happens in the problem. In particular, there is seemingly a contradiction between the assumption of the lattice periodicity and the application of the electric field, which tilts the lattice, that pervades the problem. I am apparently not the only one that is uncomfortable with this solution, seeing that it was controversial for a long time.

Nonetheless, it is a great achievement that with a couple simple physical pictures (albeit that, taken at face value, seem inconsistent), Zener was able to qualitatively explain one mechanism of electrical breakdown in insulators (there are a few others such as avalanche breakdown, etc.).

Mott Switches and Resistive RAMs

Over the past few years, there have been some interesting developments concerning narrow gap correlated insulators. In particular, it has been found that it is particularly easy to induce an insulator to metal transition (in the very least, one can say that the resistivity changes by a few orders of magnitude!) in materials such as VO2, GaTa4Se8 and NiS2-xSx with an electric field. There appears to be a threshold electric field above which the material turns into a metal. Here is a plot demonstrating this rather interesting phenomenon in Ca2RuO4, taken from this paper:

Ca2RuO4_Switch.PNG

It can be seen that the transition is hysteretic, thereby indicating that the insulator-metal transition as a function of field is first-order. It turns out that in most of the materials in which this kind of behavior is observed, there usually exists an insulator-metal transition as a function of temperature and pressure as well. Therefore, in cases such as in (V1-xCrx)2O3, it is likely that the electric field induced insulator-metal transition is caused by Joule heating. However, there are several other cases where it seems like Joule heating is likely not the culprit causing the transition.

While Zener breakdown has been put forth as a possible mechanism causing this transition when Joule heating has been ruled out, back-of-the-envelope calculations suggest that the electric field required to cause a Zener-type breakdown would be several orders of magnitude larger than that observed in these correlated insulators.

On the experimental side, things get even more interesting when applying pulsed electric fields. While the insulator-metal transition observed is usually hysteretic, as shown in the plot above, in some of these correlated insulators, electrical pulses can maintain the metallic state. What I mean is that when certain pulse profiles are applied to the material, it gets stuck in a metastable metallic state. This means that even when the applied voltage is turned off, the material remains a metal! This is shown here for instance for a 30 microsecond / 120V 7-pulse train with each pulse applied every 200 microseconds to GaV4S8 (taken from this paper):

GaVa4S8.PNG

Electric field pulses applied to GaV4S8. A single pulse induces a insulator-metal transition, but reverts back to the insulating state after the pulse disappears. A pulse train induces a transition to a metastable metallic state.

Now, if your thought process is similar to mine, you would be wondering if applying another voltage pulse would switch the material back to an insulator. The answer is that with a specific pulse profile this is possible. In the same paper as the one above, the authors apply a series of 500 microsecond pulses (up to 20V) to the same sample, and they don’t see any change. However, the application of a 12V/2ms pulse does indeed reset the sample back to (almost) its original state. In the paper, the authors attribute the need for a longer pulse to Joule heating, enabling the sample to revert back to the insulating state. Here is the image showing the data for the metastable-metal/insulator transition (taken from the same paper):

gava4s8_reset

So, at the moment, it seems like the mechanism causing this transition is not very well understood (at least I don’t understand it very well!). It is thought that there are filamentary channels between the contacts causing the insulator-metal transition. However, STM has revealed the existence of granular metallic islands in GaTa4S8. The STM results, of course, should be taken with a grain of salt since STM is surface sensitive and something different might be happening in the bulk. Anyway, some hypotheses have been put forth to figure out what is going on microscopically in these materials. Here is a recent theoretical paper putting forth a plausible explanation for some of the observed phenomena.

Before concluding, I would just like to point out that the relatively recent (and remarkable) results on the hidden metallic state in TaS2 (see here as well), which again is a Mott-like insulator in the low temperature state, is likely related to the phenomena in the other materials. The relationship between the “hidden state” in TaS2 and the switching in the other insulators discussed here seems to not have been recognized in the literature.

Anyway, I heartily recommend reading this review article to gain more insight into these topics for those who are interested.

An Excellent Intro To Physical Science

On a recent plane ride, I was able to catch an episode of the new PBS series Genius by Stephen Hawking. I was surprised by the quality of the show and in particular, its emphasis on experiment. Usually, shows like this fall into the trap of giving one the facts (or speculations) without an adequate explanation of how scientists come to such conclusions. However, this one is a little different and there is a large emphasis on experiment, which, at least to me, is much more inspirational.

Here is the episode I watched on the plane:

Nonlinear Response and Harmonics

Because we are so often solving problems in quantum mechanics, it is sometimes easy to forget that certain effects also show up in classical physics and are not “mysterious quantum phenomena”. One of these is the problem of avoided crossings or level repulsion, which can be much more easily understood in the classical realm. I would argue that the Fano resonance also represents a case where a classical model is more helpful in grasping the main idea. Perhaps not too surprisingly, a variant of the classical harmonic oscillator problem is used to illustrate the respective concepts in both cases.

There is also another cute example that illustrates why overtones of the natural harmonic frequency components result when subject to slightly nonlinear oscillations. The solution to this problem therefore shows why harmonic distortions often affect speakers; sometimes speakers emit frequencies not present in the original electrical signal. Furthermore, it shows why second harmonic generation can result when intense light is incident on a material.

First, imagine a perfectly harmonic oscillator with a potential of the form V(x) = \frac{1}{2} kx^2. We know that such an oscillator, if displaced from its original position, will result in oscillations at the natural frequency of the oscillator \omega_o = \sqrt{k/m} with the position varying as x(t) = A \textrm{cos}(\omega_o t + \phi). The potential and the position of the oscillator as a function of time are shown below:

harmpotentialrepsonse

(Left) Harmonic potential as a function of position. (Right) Variation of the position of the oscillator with time

Now imagine that in addition to the harmonic part of the potential, we also have a small additional component such that V(x) = \frac{1}{2} kx^2 + \frac{1}{3}\epsilon x^3, so that the potential now looks like so:

nonlinearharm

The equation of motion is now nonlinear:

\ddot{x} = -c_1x - c_2x^2

where c_1 and c_2 are constants. It is easy to see that if the amplitude of oscillations is small enough, there will be very little difference between this case and the case of the perfectly harmonic potential.

However, if the amplitude of the oscillations gets a little larger, there will clearly be deviations from the pure sinusoid. So then what does the position of the oscillator look like as a function of time? Perhaps not too surprisingly, considering the title, is that not only are there oscillations at \omega_0, but there is also an introduction of a harmonic component with 2\omega_o.

While the differential equation can’t be solved exactly without resorting to numerical methods, that the harmonic component is introduced can be seen within the framework of perturbation theory. In this context, all we need to do is plug the solution to the simple harmonic oscillator, x(t) = A\textrm{cos}(\omega_0t +\phi) into the nonlinear equation above. If we do this, the last term becomes:

-c_2A^2\textrm{cos}^2(\omega_0t+\phi) = -c_2 \frac{A^2}{2}(1 + \textrm{cos}(2\omega_0t+2\phi)),

showing that we get oscillatory components at twice the natural frequency. Although this explanation is a little crude — one can already start to see why nonlinearity often leads to higher frequency harmonics.

With respect to optical second harmonic generation, there is also one important ingredient that should not be overlooked in this simplified model. This is the fact that frequency doubling is possible only when there is an x^3 component in the potential. This means that the potential needs to be inversion asymmetric. Indeed, second harmonic generation is only possible in inversion asymmetric materials (which is why ferroelectric materials are often used to produce second harmonic optical signals).

Because of its conceptual simplicity, it is often helpful to think about physical problems in terms of the classical harmonic oscillator. It would be interesting to count how many Nobel Prizes have been given out for problems that have been reduced to some variant of the harmonic oscillator!

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.

Landau Theory and the Ginzburg Criterion

The Landau theory of second order phase transitions has probably been one of the most influential theories in all of condensed matter. It classifies phases by defining an order parameter — something that shows up only below the transition temperature, such as the magnetization in a paramagnetic to ferromagnetic phase transition. Landau theory has framed the way physicists think about equilibrium phases of matter, i.e. in terms of broken symmetries. Much current research is focused on transitions to phases of matter that possess a topological index, and a major research question is how to think about these phases which exist outside the Landau paradigm.

Despite its far-reaching influence, Landau theory actually doesn’t work quantitatively in most cases near a continuous phase transition. By this, I mean that it fails to predict the correct critical exponents. This is because Landau theory implicitly assumes that all the particles interact in some kind of average way and does not adequately take into account the fluctuations near a phase transition. Quite amazingly, Landau theory itself predicts that it is going to fail near a phase transition in many situations!

Let me give an example of its failure before discussing how it predicts its own demise. Landau theory predicts that the specific heat should exhibit a discontinuity like so at a phase transition:

specificheatlandau

However, if one examines the specific heat anomaly in liquid helium-4, for example, it looks more like a divergence as seen below:

lambda_transition

So it clearly doesn’t predict the right critical exponent in that case. The Ginzburg criterion tells us how close to the transition temperature Landau theory will fail. The Ginzburg argument essentially goes like so: since Landau theory neglects fluctuations, we can see how accurate Landau theory is going to be by calculating the ratio of the fluctuations to the order parameter:

E_R = |G(R)|/\eta^2

where E_R is the error in Landau theory, |G(R)| quantifies the fluctuations and \eta is the order parameter. Basically, if the error is small, i.e. E_R << 1, then Landau theory will work. However, if it approaches \sim 1, Landau theory begins to fail. One can actually calculate both the order parameter and the fluctuation region (quantified by the two-point correlation function) within Landau theory itself and therefore use Landau theory to calculate whether or not it will fail.

If one does carry out the calculation, one gets that Landau theory will work when:

t^{(4-d)/2} >> k_B/\Delta C \xi(1)^d  \equiv t_{L}^{(4-d)/2}

where t is the reduced temperature, d is the dimension, \xi(1) is the dimensionless mean-field correlation length at T = 2T_C (extrapolated from Landau theory) and \Delta C/k_B is the change in specific heat in units of k_B, which is usually one per degree of freedom. In words, the formula essentially counts the number of degrees of freedom in a volume defined by  \xi(1)^d. If the number of degrees of freedom is large, then Landau theory, which averages the interactions from many particles, works well.

So that was a little bit of a mouthful, but the important thing is that these quantities can be estimated quite well for many phases of matter. For instance, in liquid helium-4, the particle interactions are very short-ranged because the helium atom is closed-shell (this is what enables helium to remain a liquid all the way down to zero temperatures at ambient conditions in the first place). Therefore, we can assume that \xi(1) \sim 1\textrm{\AA}, and hence t_L \sim 1 and deviations from Landau theory can be easily observed in experiment close to the transition temperature.

Despite the qualitative similarities between superfluid helium-4 and superconductivity, a topic I have addressed in the past, Landau theory works much better for superconductors. We can also use the Ginzburg criterion in this case to calculate how close to the transition temperature one has to be in order to observe deviations from Landau theory. In fact, the question as to why Ginzburg-Landau theory works so well for BCS superconductors is what awakened me to these issues in the first place. Anyway, we assume that \xi(1) is on the order of the Cooper pair size, which for BCS superconductors is on the order of 1000 \textrm{\AA}. There are about 10^8 particles in this volume and correspondingly, t_L \sim 10^{-16} and Landau theory fails so close to the transition temperature that this region is inaccessible to experiment. Landau theory is therefore considered to work well in this case.

For high-Tc superconductors, the Cooper pair size is of order 10\textrm{\AA} and therefore deviations from Landau theory can be observed in experiment. The last thing to note about these formulas and approximations is that two parameters determine whether Landau theory works in practice: the number of dimensions and the range of interactions.

*Much of this post has been unabashedly pilfered from N. Goldenfeld’s book Lectures on Phase Transitions and the Renormalization Group, which I heartily recommend for further discussion of these topics.

Strontium Titanate – A Historical Tour

Like most ugly haircuts, materials tend to go in and out of style over time. Strontium titanate (SrTiO3), commonly referred to as STO, has, since its discovery, been somewhat timeless. And this is not just because it is often used as a substitute for diamonds. What I mean is that studying STO rarely seems to go out of style and the material always appears to have some surprises in store.

STO was first synthesized in the 1950s, before it was discovered naturally in Siberia. It didn’t take long for research on this material to take off. One of the first surprising results that STO had in store was that it became superconducting when reduced (electron-doped). This is not remarkable in and of itself, but this study and other follow-up ones showed that superconductivity can occur with a carrier density of only ~5\times 10^{17} cm^{-3}.

This is surprising in light of BCS theory, where the Fermi energy is assumed to be much greater than the Debye frequency — which is clearly not the case here. There have been claims in the literature suggesting that the superconductivity may be plasmon-induced, since the plasma frequency is in the phonon energy regime. L. Gorkov recently put a paper up on the arXiv discussing the mechanism problem in STO.

Soon after the initial work on superconductivity in doped STO, Shirane, Yamada and others began studying pure STO in light of the predicted “soft mode” theory of structural phase transitions put forth by W. Cochran and others. Because of an anti-ferroelectric structural phase transition at ~110K (depicted below), they we able to observe a corresponding soft phonon associated with this transition at the Brillouin zone boundary (shown below, taken from this paper). These results had vast implications for how we understand structural phase transitions today, when it is almost always assumed that a phonon softens at the transition temperature through a continuous structural phase transition.

Many materials similar to STO, such as BaTiO3 and PbTiO3, which also have a perovskite crystal structure motif, undergo a phase transition to a ferroelectric state at low (or not so low) temperatures. The transition to the ferroelectric state is accompanied by a diverging dielectric constant (and dielectric susceptibility) much in the way that the magnetic susceptibility diverges in the transition from a paramagnetic to a ferromagnetic state. In 1978, Muller (of Bednorz and Muller fame) and Burkard reported that at low temperature, the dielectric constant begins its ascent towards divergence, but then saturates at around 4K (the data is shown in the top panel below). Ferroelectricity is associated with a zone-center softening of a transverse phonon, and in the case of STO, this process begins, but doesn’t quite get there, as shown schematically in the image below (and you can see this in the data by Shirane and Yamada above as well).

quantumparaelectricity_signatures

Taken from Wikipedia

The saturation of the large dielectric constant and the not-quite-softening of the zone center phonon has led authors to refer to STO as a quantum paraelectric (i.e. because of the zero-point motion of the transverse optical zone-center phonon, the material doesn’t gain enough energy to undergo the ferroelectric transition). As recently as 2004, however, it was reported that one can induce ferroelectricity in STO films at room temperature by straining the film.

In recent times, STO has found itself as a common substrate material due to processes that can make it atomically flat. While this may not sound so exciting, this has had vast implications for the physics of thin films and interfaces. Firstly, this property has enabled researchers to grow high-quality thin films of cuprate superconductors using molecular beam epitaxy, which was a big challenge in the 1990’s. And even more recently, this has led to the discovery of a two-dimensional electron gas, superconductivity and ferromagnetism at the LAO/STO interface, a startling finding due to the fact that both materials are electrically insulating. Also alarmingly, when FeSe (a superconductor at around 7K) is grown as a monolayer film on STO, its transition temperature is boosted to around 100K (though the precise transition temperature in subsequent experiments is disputed but still high!). This has led to the idea that the FeSe somehow “borrows the pairing glue” from the underlying substrate.

STO is a gem of a material in many ways. I doubt that we are done with its surprises.