Category Archives: Review

Pictures of Band Theory: A real space view of where bands and band gaps come from

In learning solid state physics, one of the most difficult conceptual hurdles to overcome is to understand band theory. This is partly due to the difficulty in thinking about reciprocal space, and is highlighted on Nanoscale Views blog in the post “The Tyranny of Reciprocal Space”. In this post, I will sacrifice accuracy in favor of an intuitive picture of band theory in real space. Hopefully, this post will help newcomers overcome those scary feelings when first exposed to solid state physics.

Firstly, it is necessary to recount the mathematical form of a Bloch wavefunction:

\psi_{k}(r) = e^{ikr}u(r)

Let’s pause for a second to take a look at what this means — the Bloch wave consists of a plane wave portion multiplied by a periodic function. In this post, for illustration purposes, I’ll simplify this by treating both parts of the Bloch wave as real.1 Take a look  at the image below to see what this implies:


Fig 1: (a) The periodic potential. (b) The Bloch wavefunction. (c) The periodic part of the Bloch wave function. (d) The sinusoidal envelope part of the Bloch wavefunction.

Within this seemingly simple picture, one can explain the origin of band structure and why band gaps appear.

Let’s see first how band structure arises. For ease, since most readers of this blog are likely familiar with the solution to the infinite square well problem, we shall start there. Pictured below is a periodic potential with infinitely high walls between each well and the first two wavefunctions for each well looks like so:


Fig. 2: n=1 and n=2 wavefunctions for the periodic infinite square well.

The wavefunctions from well to well don’t have to be in phase, but I’ve just drawn them that way for ease. Bands arise when we reduce the height between walls to let the wavefunctions bleed over into the neighboring wells. This most easily seen for the two-well potential case as seen below:



In the first row, I have just plotted the n=1 energy levels for each well. Once the barrier height has been reduced, the (formerly degenerate) energy levels split into a symmetric and anti-symmetric state. I have not plotted the n=2 levels — this is just what happens if the n=1 interact! How much the energy levels split will be determined by how much I reduce the barrier height: the more I reduce the barrier, the larger the splitting. In band language, as you’ll see below, this implies that the lower the barrier height, the greater the dispersion.

One important thing to take away from this picture is that both in the infinite and finite barrier cases, we can fit at most four electrons in these two levels (if we include spin). In the infinite barrier case, two electrons can fit in the n=1 level in each well, and in the finite barrier case, two electrons can go into the symmetric state and two in the anti-symmetric state.

Now, let’s return to the case where we have an infinite  number (okay, I only drew fifteen!) of finite potential wells. In analogy to the two-well problem, we can draw the states for the case where the heights of the potential wells have been reduced:


Fig. 3: n=1 and n=2 wavefunctions for the periodic finite square well. My lack of artistic skills is severely exposed for the n=2 level here, but imagine that the wavefunctions don’t look so discontinuous.


This is where things get interesting. How do we represent the n=1 states in analogy with the symmetric and anti-symmetric states in the two-well case? We can invoke Bloch’s theorem. It basically says that you just multiply this periodic part by a sinusoidal function!

The sinusoidal function ends up being an envelope function, just like in the very first figure above. Here is what the lowest energy level would look like for the periodic finite potential well:


Fig. 4: The lowest energy wavefunction for the n=1 level

This state is the analog of the symmetric state in the two-well case. To preserve the number of states in going from the infinite barrier case to the finite barrier case, I can only multiply the periodic part by N sinusoidal envelope functions, where N is the number of potential wells — in this case, fifteen!

Therefore the functions from the n=1 level end up looking like this:



Fig. 5: Wavefunctions that comprise the n=1 band


These are the wavefunctions that comprise a single band, that is, the band formed by the n=1 level. Interestingly, just from looking at the wavefunctions, you can see that the wavefunctions for the n=1 band increase in energy in going from the totally symmetric state to the totally antisymmetric state, as the number of nodes in the wavefunction increases. Notice here also how this connects to the reciprocal space picture — the totally antisymmetric wavefunction was multiplied with an envelope function that had wavelength 2a, which is the state at the Brillioun zone boundary!

Now, in this picture, why do band gaps exist? Understanding this point requires me to do the same envelope multiplication procedure to the n=2 levels. In particular, when one multiplies by the 2a envelope function, it essentially has the effect of flipping the wavefunction in each well so that we get something that looks something like this (again, imagine a continuous function here, my artistic skills fail me):


Fig. 6: The zone boundary (\pi/a) wavefunction for the n=2 level


Imagine for a second what this function would look like in the absence (or with a very small) barrier height. It turns out that it would end up looking very similar to the highest energy wavefunction for the n=1 band! This is pictured below:



Fig. 7: The zone boundary (\pi/a) wavefunctions for the n=1 and n=2 energy levels with a negligible barrier height

What you can see here is that at the zone boundary, the wavefunctions essentially look the same, and are essentially degenerate. This degeneracy is broken when the barriers are present.  The barriers “mess up” the wavefunction so that they no longer perfect sinusoids, changing the energies of both the zone boundary blue n=1 and the orange n=2 curves so that their energies are no longer the same. In other words, a gap has opened between the wavelength 2a n=1 and n=2 energy levels! You can sort of use your eyes to interpolate between Fig. 6 and Fig. 7 to see that the energy of the n=2 level must increase as it loses its pure sinusoidal nature and, by comparing Fig. 6 to the last image in Fig. 5, that the zone boundary wavefunction degeneracy has been lifted.

In this picture, you can also easily see that when the periodic part of the n=2 wavefunction is multiplied by the first sinusoidal function (i.e. the one with wavelength Na/2), it actually has the highest energy in the n=2 band. This can be seen by comparing the orange curves in Fig. 7 and Fig. 3. The curve in Fig. 3 has many more nodes. The lowest energy is actually obtained when the n=2 periodic function is multiplied by the sinusoidal function of wavelength 2a, i.e. at the zone boundary. This implies that in contrast to the first band, the second one disperses downward from the center of the Brillouin zone.

One more thing to note, which has been implicit in the discussion is that essentially the n=1 level has the symmetry of an s-like wavefunction whereas the n=2 level has the symmetry of a p-like wavefunction.  If one keeps going with this picture, you can essentially get d- and f-like bands as well.

I hope this post helps bring an end to the so-called “tyranny of reciprocal space”. It is not difficult to imagine the wavefunctions in real space and this framework shouldn’t be so intimidating to band theory newcomers!

I actually wonder what the limitations of this picture are — if anyone sees how to explain, for instance, the Berry phase within this picture, I’d be interested to hear it!


1 This of course is not strictly correct, but this helps in visualizing what is going on tremendously.


Bands Aren’t Only For Crystalline Solids

If one goes through most textbooks on solid state physics such as Ashcroft and Mermin, one can easily forget that most of the solids in this world are not crystalline. If I look around my living room, I see a ceramic tea mug nearby a plastic pepper dispenser sitting on a wooden coffee table. In fact, it is very difficult to find something that we would call “crystalline” in the sense of solid state physics.

Because of this, one could almost be forgiven in thinking that bands are a property only of crystalline solids. That they are not, can be seen within a picture-based framework. As is usual on this blog, let’s start with the wavefunctions of the infinite square well and the two-well potential. Take a look below at the wavefunctions for the infinite well and then at the first four pairs of wavefunctions for the double well (the images are taken from here and here):



What you can already see forming within this simple picture is the notion of a “band”. Each “band” here only contains two energy levels, each of which can take two electrons when taking into consideration spin. If we generalize this picture, one can see that when going from two wells here to N wells, one will get energy levels per band.

However, there has been no explicit, although used above,  requirement that the wells be the same depth. It is quite easy to imagine that the potential wells look like the ones below. The analogue of the symmetric and anti-symmetric states for the E1 level are shown below as well:

Again, this can be generalized to N potential wells that vary in height from site to site for one to get a “band”. The necessary requirement for band formation is that the electrons be allowed to tunnel from one site to the other, i.e. for them “feel” the presence of the neighboring potential wells. While the notion of a Brillouin zone won’t exist and nor will Bragg scattering of the electrons (which leads to the opening up of the gaps at the Brillouin zone boundaries), the notion of a band will persist within a non-crystalline framework.

Because solid state physics textbooks often don’t mention amorphous solids or glasses, one can easily forget which properties of solids are and are not limited to those that are crystalline. We may not know how to mathematically apply them to glasses with random potentials very well, but many ideas used in the framework to describe crystalline solids are applicable when looking at amorphous solids as well.

Response and Dissipation: Part 1 of the Fluctuation-Dissipation Theorem

I’ve referred to the fluctuation-dissipation theorem many times on this blog (see here and here for instance), but I feel like it has been somewhat of an injustice that I have yet to commit a post to this topic. A specialized form of the theorem was first formulated by Einstein in a paper about Brownian motion in 1905. It was then extended to electrical circuits by Nyquist and then generalized by several authors including Callen and Welten (pdf!) and R. Kubo (pdf!). The Callen and Welton paper is a particularly superlative paper not just for its content but also for its lucid scientific writing. The fluctuation-dissipation theorem relates the fluctuations of a system (an equilibrium property) to the energy dissipated by a perturbing external source (a manifestly non-equilibrium property).

In this post, which is the first part of two, I’ll deal mostly with the non-equilibrium part. In particular, I’ll show that the response function of a system is related to the energy dissipation using the harmonic oscillator as an example. I hope that this post will provide a justification as to why it is the imaginary part of a response function that quantifies energy dissipated. I will also avoid the use of Green’s functions in these posts, which for some reason often tend to get thrown in when teaching linear response theory, but are absolutely unnecessary to understand the basic concepts.

Consider first a damped driven harmonic oscillator with the following equation (for consistency, I’ll use the conventions from my previous post about the phase change after a resonance):

\underbrace{\ddot{x}}_{inertial} + \overbrace{b\dot{x}}^{damping} + \underbrace{\omega_0^2 x}_{restoring} = \overbrace{F(t)}^{driving}

One way to solve this equation is to assume that the displacement, x(t), responds linearly to the applied force, F(t) in the following way:

x(t) = \int_{-\infty}^{\infty} \chi(t-t')F(t') dt'

Just in case this equation doesn’t make sense to you, you may want to reference this post about linear response.  In the Fourier domain, this equation can be written as:

\hat{x}{}(\omega) = \hat{\chi}(\omega) \hat{F}(\omega)

and one can solve this equation (as done in a previous post) to give:

\hat{\chi}(\omega) = (-\omega^2 + i\omega b + \omega_0^2 )^{-1}

It is useful to think about the response function, \chi, as how the harmonic oscillator responds to an external source. This can best be seen by writing the following suggestive relation:

\chi(t-t') = \delta x(t)/\delta F(t')

Response functions tend to measure how systems evolve after being perturbed by a point-source (i.e. a delta-function source) and therefore quantify how a system relaxes back to equilibrium after being thrown slightly off balance.

Now, look at what happens when we examine the energy dissipated by the damped harmonic oscillator. In this system the energy dissipated can be expressed as the time integral of the force multiplied by the velocity and we can write this in the Fourier domain as so:

\Delta E \sim \int \dot{x}F(t) dt =  \int d\omega d\omega'dt (-i\omega) \hat{\chi}(\omega) \hat{F}(\omega)\hat{F}(\omega') e^{i(\omega+\omega')t}

One can write this more simply as:

\Delta E \sim \int d\omega (-i\omega) \hat{\chi}(\omega) |\hat{F}(\omega)|^2

Noticing that the energy dissipated has to be a real function, and that |\hat{F}(\omega)|^2 is also a real function, it turns out that only the imaginary part of the response function can contribute to the dissipated energy so that we can write:

\Delta E \sim  \int d \omega \omega\hat{\chi}''(\omega)|\hat{F}(\omega)|^2

Although I try to avoid heavy mathematics on this blog, I hope that this derivation was not too difficult to follow. It turns out that only the imaginary part of the response function is related to energy dissipation. 

Intuitively, one can see that the imaginary part of the response has to be related to dissipation, because it is the part of the response function that possesses a \pi/2 phase lag. The real part, on the other hand, is in phase with the driving force and does not possess a phase lag (i.e. \chi = \chi' +i \chi'' = \chi' +e^{i\pi/2}\chi''). One can see from the plot from below that damping (i.e. dissipation) is quantified by a \pi/2 phase lag.


Damping is usually associated with a 90 degree phase lag

Next up, I will show how the imaginary part of the response function is related to equilibrium fluctuations!


An Undergraduate Optics Problem – The Brewster Angle

Recently, a lab-mate of mine asked me if there was an intuitive way to understand Brewster’s angle. After trying to remember how Brewster’s angle was explained to me from Griffiths’ E&M book, I realized that I did not have a simple picture in my mind at all! Griffiths’ E&M book uses the rather opaque Fresnel equations to obtain the Brewster angle. So I did a little bit of thinking and came up with a picture I think is quite easy to grasp.

First, let me briefly remind you what Brewster’s angle is, since many of you have probably not thought of the concept for a long time! Suppose my incident light beam has both components, s– and p-polarization. (In case you don’t remember, p-polarization is parallel to the plane of incidence, while s-polarization is perpendicular to the plane of incidence, as shown below.) If unpolarized light is incident on a medium, say water or glass, there is an angle, the Brewster angle, at which the light comes out perfectly s-polarized.

An addendum to this statement is that if the incident beam was perfectly p-polarized to begin with, there is no reflection at the Brewster angle at all! A quick example of this is shown in this YouTube video:

So after that little introduction, let me give you the “intuitive explanation” as to why these weird polarization effects happen at the Brewster angle. First of all, it is important to note one important fact: at the Brewster angle, the refracted beam and the reflected beam are at 90 degrees with respect to each other. This is shown in the image below:

Why is this important? Well, you can think of the reflected beam as light arising from the electrons jiggling in the medium (i.e. the incident light comes in, strikes the electrons in the medium and these electrons re-radiate the light).

However, radiation from an oscillating charge only gets emitted in directions perpendicular to the axis of motion. Therefore, when the light is purely p-polarized, there is no light to reflect when the reflected and refracted rays are orthogonal — the reflected beam can’t have the polarization in the same direction as the light ray! This is shown in the right image above and is what gives rise to the reflectionless beam in the YouTube video.

This visual aid enables one to use Snell’s law to obtain the celebrated Brewster angle equation:

n_1 \textrm{sin}(\theta_B) = n_2 \textrm{sin}(\theta_2)


\theta_B + \theta_2 = 90^o

to obtain:

\textrm{tan}(\theta_B) = n_2/n_1.

The equations also suggest one more thing: when the incident light has an s-polarization component, the reflected beam must come out perfectly polarized at the Brewster angle. This is because only the s-polarized light jiggles the electrons in a way that they can re-radiate in the direction of the outgoing beam. The image below shows the effect a polarizing filter can therefore have when looking at water near the Brewster angle, which is around 53 degrees for water.

To me, this is a much simpler way to think about the Brewster angle than dealing with the Fresnel equations.


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:


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:


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:


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:


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):


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):


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: