Category Archives: Review

The photoelectric effect does not imply photons

When I first learned quantum mechanics, I was told that we knew that the photon existed because of Einstein’s explanation of the photoelectric effect in 1905. As the frequency of light impinging on the cathode material was increased, electrons came out with higher kinetic energies. This led to Einstein’s famous formula:

K.E. = \hbar\omega - W.F.

where K.E. is the kinetic energy of the outgoing electron, \hbar\omega is the photon energy and W.F. is the material-dependent work function.

Since the mid-1960s, however, we have known that the photoelectric effect does not definitively imply the existence of photons. From the photoelectric effect alone, it is actually ambiguous whether it is the electronic levels or the impinging radiation that should be quantized!

So, why do we still give the photon explanation to undergraduates? To be perfectly honest, I’m not sure whether we do this because of some sort of intellectual inertia or because many physicists don’t actually know that the photoelectric effect can be explained without invoking photons. It is worth noting that Willis E. Lamb, who played a large role in the development of quantum electrodynamics, implored other physicists to be more cautious when using the word photon (see for instance his 1995 article entitled Anti-Photon, which gives an interesting history of the photon nomenclature and his thoughts as to why we should be wary of its usage).

Let’s return to 1905, when Einstein came up with his explanation of the photoelectric effect. Just five years prior, Planck had heuristically explained the blackbody radiation spectrum and, in the process, evaded the ultraviolet catastrophe that plagued explanations based on the classical equipartition theorem. Planck’s distribution consequently provided the first evidence of “packets of light”, quantized in units of \hbar. At the time, Bohr had yet to come up with his atomic model that suggested that electron levels were quantized, which had to wait until 1913. Thus, from Einstein’s vantage point in 1905, he made the most reasonable assumption at the time — that it was the radiation that was quantized and not the electronic levels.

Today, however, we have the benefit of hindsight.

According to Lamb’s aforementioned Anti-Photon article, in 1926, G. Wentzel and G. Beck showed that one could use a semi-classical theory (i.e. electronic energy levels are quantized, but light is treated classically) to reproduce Einstein’s result. In the mid- to late 1960’s, Lamb and Scully extended the original treatment and made a point of emphasizing that one could get out the photoelectric effect without invoking photons. The main idea can be sketched if you’re familiar with the Fermi golden rule treatment to a harmonic electric field perturbation of the form \textbf{E}(t) = \textbf{E}_0 e^{-i \omega t}, where \omega is the frequency of the incoming photon. In the dipole approximation, we can write the potential as V(t) = -e\textbf{x}(t)\cdot\textbf{E}(t) and we get that the transition rate is:

R_{i \rightarrow f} = \frac{1}{t} \frac{1}{\hbar^2}|\langle{f}|e\textbf{x}(t)\cdot\textbf{E}_0|i \rangle|^2 [\frac{\textrm{sin}((\omega_{fi}-\omega)t/2)}{(\omega_{fi}-\omega)/2}]^2

Here, \hbar\omega_{fi} = (E_f - E_i) is the difference in energies between the initial and final states. Now, there are a couple things to note about the above expression. Firstly, the term in brackets (containing the sinusoidal function) peaks up when \omega_{fi} \approx \omega. This means that when the incoming light is resonant between the ground state and a higher energy level, the transition rate sharply increases.

Let us now interpret this expression with regard to the photoelectric effect. In this case, there exists a continuum of final states which are of the form \langle x|f\rangle \sim e^{i\textbf{k}\cdot\textbf{r}}, and as long as \hbar\omega > W.F., where W.F. is the work function of the material, we recover \hbar\omega = W.F. + K.E., where K.E. represents the energy given to the electron in excess of the work function. Thus, we recover Einstein’s formula from above!

In addition to this, however, we also see from the above expression that the current on the photodetector is proportional to \textbf{E}^2_0, i.e. the intensity of light impinging on the cathode. Therefore, this semi-classical treatment improves upon Einstein’s treatment in the sense that the relation between the intensity and current also naturally falls out.

From this reasoning, we see that the photoelectric effect does not logically imply the existence of photons.

We do have many examples that non-classical light does exist and that quantum fluctuations of light play a significant role in experimental observations. Some examples are photon anti-bunching, spontaneous emission, the Lamb shift, etc. However, I do agree with Lamb and Scully that the notion of a photon is indeed a challenging one and that caution is needed!

A couple further interesting reads on this subject at a non-technical level can be found here: The Concept of the Photon in Physics Today by Scully and Sargent and The Concept of the Photon – Revisited in OPN Trends by Muthukrishnan, Scully and Zubairy (pdf!)

Critical Slowing Down

I realize that it’s been a long while since I’ve written a post, so the topic of this one, while unintentionally so, is quite apt.

Among the more universal themes in studying phase transitions is the notion of critical slowing down. Most students are introduced to this idea in the context of second order phase transitions, but it has turned out to be a useful concept in a wide range of systems beyond this narrow framework and into subjects well outside the purview of the average condensed matter physicist.

Stated simply, critical slowing down refers to the phenomenon observed near phase transitions where a slight perturbation or disturbance away from equilibrium takes a really long time to decay back to equilibrium. Why is this the case?

The main idea can be explained within the Landau theory of phase transitions, and I’ll take that approach here since it’s quite intuitive.  As you can see in the images below, when the Landau potential is far from T_c, the potential well can be approximated by a parabolic form. However, this is not possible for the potential near T_c.

LandauPotentials

Mathematically, this can be explained by considering a simple form of the Landau potential:

V(\phi) = \alpha (T-T_c) x^2 + \beta x^4

Near T_c, the parabolic term vanishes, and we are left with only the quartic one. Although it’s clear from the images why the dynamics slow down near T_c, it helps to spell out the math a little.

Firstly, imagine that the potential is filled with some sort of viscous fluid, something akin to honey, and that the dynamics of the ball represents that of the order parameter. This puts us in the “overdamped” limit, where the order parameter reaches the equilibrium point without executing any sort of oscillatory motion. Far from T_c, as aforementioned, we can approximate the dynamics with a parabolic form of the potential (using the equation for the overdamped limit, \dot{x} = -dV/dx):

\dot{x} = -\gamma(T) x

The solution to this differential equation is of exponential form, i.e. x(t) = x(0)e^{-\gamma(T) t}, and the relaxation back to equilibrium is therefore characterized by a temperature-dependent timescale \tau =1/\gamma(T).

However, near T_c, the parabolic approximation breaks down, as the parabolic term gets very small, and we have to take into consideration the quartic term. The order parameter dynamics then get described by:

\dot{x} = -\beta x^3,

which has a solution of the form x(t) \sim 1/\sqrt{\beta t}. Noticeably, the dynamics of the order parameter obey a much slower power law decay near T_c, as illustrated below:

ExpVsPowerLaw_Decay

Now, naively, at this point, one would think, “okay, so this is some weird thing that happens near a critical point at a phase transition…so what?”

Well, it turns out that critical slowing down can actually serve as a precursor of an oncoming phase transition in all sorts of contexts, and can even be predictive! Here are a pair of illuminating papers which show that critical slowing down occurs near a population collapse in microbial communities (from the Scheffer group and from the Gore group). As an aside, the Gore group used the budding yeast Saccharomyces cerevisiae in their experiments, which is the yeast used in most beers (I wonder if their lab has tasting parties, and if so, can I get an invitation?).

Here is another recent paper showing critical slowing down in a snap-through instability of an elastic rod. I could go on and on listing the different contexts where critical slowing down has been observed, but I think it’s better that I cite this review article.

Surprisingly, critical slowing down has been observed at continuous, first-order and far-from-equilibrium phase transitions! As a consequence of this generality, the observation of critical slowing down can therefore be predictive. If the appropriate measurements could be made, one may be able to see how close the earth’s climate is to a “tipping point” from which it will be very difficult to return (due to hysteresic effects) (see this paper which shows some form of critical slowing down in previous climatic changes in the earth’s history). But for now, it’s just interesting to look for critical slowing down in other contexts that are a little easier to predict and where perhaps the consequences aren’t as dire.

*Thanks to Alfred Zong who introduced me to many of the above papers

**Also, a shout out to Brian Skinner who caught repeated noise patterns in a recent preprint on room temperature superconductivity. Great courage and good job!

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:

image117

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:

PeriodicInfiniteSquareWell

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:

PeriodicFiniteSquareWell

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:

PeriodicFiniteSquareWellBloch

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:

 

PeriodicFiniteSquareWellBlochLevels2

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):

PeriodicFiniteSquareWellBloch2ndLevel

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:

 

PeriodicFiniteSquareWellReducedBarrier

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):

InfiniteWell

1870-3542-rmfe-62-02-00096-gf3

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.

ArgandPlaneResonance

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)

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.

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.).