# Tag Archives: Education

## 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.

## DIY Garage Work

Recently, I heard about a string of YouTube videos where Ben Krasnow of the Applied Sciences YouTube Channel makes a series of scientific instruments in his garage. One of the particularly impressive achievements is his homemade Scanning Electron Microscope, where he constructs a pretty decent instrument with approximately $1500. This is definitely outstanding from an educational viewpoint —$1500 will probably be affordable for many high schools and will enable students to see how to image objects with electrons.

Here are a couple videos showing this and another one of his projects where he uses a laser and a couple optical elements to construct a Raman spectroscopy setup:

Lastly, I’d like to point out that Christina Lee has put together an excellent set of Jupyter code (i.e. IPython Notebook code) to solve various condensed matter physics problems. It’s definitely worth having a look.

## 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:

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

and

$\theta_B + \theta_2 = 90^o$

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.

## Book Review – The Gene

Following the March Meeting, I took a vacation for a couple weeks, returning home to Bangkok, Thailand. During my holiday, I was able to get a hold of and read Siddhartha Mukherjee’s new book entitled The Gene: An Intimate History.

I have to preface any commentary by saying that prior to reading the book, my knowledge of biology embarrassingly languished at the middle-school level. With that confession aside, The Gene was probably one of the best (and for me, most enlightening) popular science books I have ever read. This is definitely aided by Mukherjee’s fluid and beautiful writing style from which scientists in all fields can learn a few lessons about scientific communication. The Gene is also touched with a humanity that is not usually associated with the popular science genre, which is usually rather dry in recounting scientific and intellectual endeavors. This humanity is the book’s most powerful feature.

Since there are many glowing reviews of the book published elsewhere, I will just list here a few nuggets I took away from The Gene, which hopefully will serve to entice rather than spoil the book for you:

• Mukherjee compares the gene to an atom or a bit, evolution’s “indivisible” particle. Obviously, the gene is physically divisible in the sense that it is made of atoms, but what he means here is that the lower levels can be abstracted away and the gene is the relevant level at which geneticists work.
• It is worth thinking of what the parallel carriers of information are in condensed matter problems — my hunch is that most condensed matter physicists would contend that these are the quasiparticles in the relevant phase of matter.
• Gregor Mendel, whose work nowadays is recognized as giving birth to the entire field of genetics, was not recognized for his work while he was alive. It took another 40-50 years for scientists to rediscover his experiments and to see that he had localized, in those pea plants, the indivisible gene. One gets the feeling that his work was not celebrated while he was alive because his work was far ahead of its time.
• The history of genetics is harrowing and ugly. While the second World War was probably the pinnacle of obscene crimes committed in the name of genetics, humans seem unable to shake off ideas associated with eugenics even into the modern day.
• Through a large part of its history, the field of genetics has had to deal with a range of ethical questions. There is no sign of this trend abating in light of the recent discovery of CRISPR/Cas-9 technology. If you’re interested in learning more about this, RadioLab has a pretty good podcast about it.
• Schrodinger’s book What is Life? has inspired so much follow-up work that it is hard to overestimate the influence it has had on a generation of physicists that transitioned to studying biology in the middle of the twentieth century, including both Watson and Crick.

While I could go on and on with this list, I’ll stop ruining the book for you. I would just like to say that at the end of the book I got the feeling that humans are still just starting to scratch the surface of understanding what’s going on in a cell. There is much more to learn, and that’s an exciting feeling in any field of science.

Aside: In case you missed March Meeting, the APS has posted the lectures from the Kavli Symposium on YouTube, which includes lectures from Duncan Haldane and Michael Kosterlitz among others.

## Disorganized Reflections

Recently, this blog has been concentrating on topics that have lacked a personal touch. A couple months ago, I started a postdoc position and it has gotten me thinking about a few questions related to my situation and some that are more general. I thought it would be a good time to share some of my thoughts and experiences. Here is just a list of some miscellaneous questions and introspections.

1. In a new role, doing new work, people often make mistakes while getting accustomed to their new surroundings. Since starting at my new position, I’ve been lucky enough to have patient colleagues who have forgiven my rather embarrassing blunders and guided me through uncharted territory. It’s sometimes deflating admitting your (usually) daft errors, but it’s a part of the learning process (at least it is for me).
2. There are a lot of reasons why people are drawn to doing science. One of them is perpetually doing something new, scary and challenging. I hope that, at least for me, science never gets monotonous and there is consistently some “fear” of the unknown at work.
3. In general, I am wary of working too much. It is important to take time to exercise and take care of one’s mental and emotional health. One of the things I have noticed is that sometimes the most driven and most intelligent graduate students suffered from burnout due to their intense work schedules at the beginning of graduate school.
4. Along with the previous point, I am also wary of spending too much time in the lab because it is important to have  time to reflect. It is necessary to think about what you’ve done, what can be done tomorrow and conjure up experiments that one can possibly try, even if they may be lofty. It’s not a bad idea to set aside a little time each day or week to think about these kinds of things.
5. It is necessary to be resilient, not take things personally and know your limits. I know that I am not going to be the greatest physicist of my generation or anything like that, but what keeps me going is the hope that I can make a small contribution to the literature that some physicists and other scientists will appreciate. Maybe they might even say “Huh, that’s pretty cool” with some raised eyebrows.
6. Is physics my “passion”? I would say that I really like it, but I could have just as easily studied a host of other topics (such as literature, philosophy, economics, etc.), and I’m sure I would have enjoyed them just as much. I’ve always been more of a generalist in contrast to being focused on physics since I was a kid or teenager. There are too many interesting things out there in the world to feel satiated just studying condensed matter physics. This is sometimes a drawback and sometimes an asset (i.e. I am sometimes less technically competent than my lab-mates, but I can probably write with less trouble).
7. For me, reading widely is valuable, but I need to be careful that it does not impede or become a substitute for active thought.
8. Overall, science can be intimidating and it can feel unrewarding. This can be particularly true if you measure your success using a publication rate or some so-called “objective” measure. I would venture to say that a much better measure of success is whether you have grown during graduate school or during a postdoc by being able to think more independently, by picking up some valuable skills (both hard and soft) and have brought a  multi-year project into fruition.

Please feel free to share thoughts from your own experiences! I am always eager to learn about people whose experiences and attitudes differ from mine.

A few nuggets on the internet this week:

1. For football/soccer fans:
http://www.espnfc.us/blog/the-toe-poke/65/post/3036987/bayern-munichs-thomas-muller-has-ingenious-way-of-dodging-journalists

2. Barack Obama’s piece in Science Magazine:
http://tinyurl.com/jmeoyz5

3. An interesting read on the history of physics education reform (Thanks to Rodrigo Soto-Garrido for sharing this with me):
http://aapt.scitation.org/doi/full/10.1119/1.4967888

4. I wonder if an experimentalist can get this to work:
http://www.bbc.com/news/uk-england-bristol-38573364

## Diffraction, Babinet and Optical Transforms

In an elementary wave mechanics course, the subject of Fraunhofer diffraction is usually addressed within the context of single-slit and double-slit interference. This is usually followed up with a discussion of diffraction from a grating. In these discussions, one usually has the picture that light is “coming through the slits” like in the image below:

Now, if you take a look at Ashcroft and Mermin or a book like Elements of Modern X-ray Physics by Als-Nielsen and McMorrow, one gets a somewhat different picture. These books make it seem like X-ray diffraction occurs when the “scattered radiation from the atoms add in phase”, as in the image below (from Ashcroft and Mermin):

So in one case it seems like the light is emanating from the free space between obstacles, whereas in the other case it seems like the obstacles are scattering the radiation. I remember being quite confused about this point when first learning X-ray diffraction in a solid-state physics class, because I had already learned Fraunhofer diffraction in a wave mechanics course. The two phenomena seemed different somehow. In their mathematical treatments, it almost seemed as if for optics, light “goes through the holes” but for X-rays “light bounces off the atoms”.

Of course, these two phenomena are indeed the same, so the question arises: which picture is correct? Well, they both give correct answers, so actually they are both correct. The answer as to why they are both correct has to do with Babinet’s principle. Wikipedia summarizes Babinet’s principle, colloquially, as so:

the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.

To get an idea of what this means, let’s look at an example. In the images below, consider the white space as openings (or slits) and the black space as obstacles in the following optical masks:

What would the diffraction pattern from these masks look like? Well, below are the results (taken from here):

Apart from minute differences close to the center, the two patterns are basically the same! If one looks closely enough at the two images, there are some other small differences, most of which are explained in this paper.

Hold on a second, you say. They can’t be the exact same thing! If I take the open space in the optical mask on the left and add it to the open space on the mask to the right, I just have “free space”. And in this case there is no diffraction! You don’t get the diffraction pattern with twice the intensity. This is of course correct. I have glossed over one small discrepancy. First, one needs to realize that intensity is related to amplitude as so:

$I \propto |A|^2$

This implies that the optical mask on the left and the one on the right give the same diffraction intensity, but that the amplitudes are 180 degrees out of phase. This phase doesn’t affect the intensity, though, as in the formula above intensity is only related to the magnitude of the amplitude. Therefore the masks, while giving the same intensity, are actually slightly different. The diffraction pattern will then cancel when the optically transparent parts of the two masks are added together. It’s strange to think that “free space” is just a bunch of diffraction patterns cancelling each other out!

With this all in mind, the main message is pretty clear though: optical diffraction through slits and the Ashcroft and Mermin picture of “bouncing off atoms” are complementary pictures of basically the same diffraction phenomenon. The diffraction pattern obtained will be the same in both cases because of Babinet’s principle.

This idea has been exploited to generate the excellent Atlas of Optical Transforms, where subtleties in crystal structures can be manipulated at the optical scale. Below is an example of such an exercise (taken from here). The two images in the first row are the optical masks, while the bottom row gives the respective diffraction patterns. In the first row, the white dots were obtained by poking holes in the optical masks.

Basically, what they are doing here is using Babinet’s principle to image the diffraction from a crystal with stacking faults along the vertical direction. The positions of the atoms are replaced with holes. One can clearly see that the effect of these stacking faults is to smear out and broaden some of the peaks in the diffraction pattern along the vertical direction. This actually turns out to gives one a good intuition of how stacking faults in a crystal can distort a diffraction pattern.

In summary, the Ashcroft and Mermin picture and the Fraunhofer diffraction picture are really two ways to describe the same phenomenon. The link between the two explanations is Babinet’s principle.