Tag Archives: C. Zener

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.).

Wannier-Stark Ladder, Wavefunction Localization and Bloch Oscillations

Most people who study solid state physics are told at some point that in a totally pure sample where there is no scattering, one should observe an AC response to a DC electric field, with oscillations at the Bloch frequency (\omega_B). These are the so-called Bloch oscillations, which were predicted by C. Zener in this paper.

However, the actual observation of Bloch oscillations is not as simple as the textbooks would make it seem. There is an excellent Physics Today article by E. Mendez and G. Bastard that outline some of the challenges associated with observing Bloch oscillations (which was written while this paper was being published!). Since the textbook treatments often use semi-classical equations of motion to demonstrate the existence of Bloch oscillations in a periodic potential, they implicitly assume transport of an electron wave-packet. To generate this wave-packet is non-trivial in a solid.

In fact, if one undertakes a full quantum mechanical treatment of electrons in a periodic potential under the influence of an electric field, one arrives at the Wannier-Stark ladder, which shows that an electric field can localize electrons! It is this ladder and the corresponding localization which was key to observing Bloch oscillations in semiconductor superlattices.

Let me use the two-well potential to give you a picture of how this localization might occur. Imagine symmetric potential wells, where the lowest energy eigenstates look like so (where S and A label the symmetric and anti-symmetric states):

Now, imagine that I start to make the wells a little asymmetric. What happens in this case? Well, it turns out that that the electrons start to localize in the following way (for the formerly symmetric and anti-symmetric states):

G. Wannier was able to solve the Schrodinger equation with an applied electric field in a periodic potential in full and showed that the eigenstates of the problem form a Stark ladder. This means that the eigenstates are of identical functional form from quantum well to quantum well (unlike in the double-well shown above) and the energies of the eigenstates are spaced apart by \Delta E=\hbar \omega_B! The potential is shown schematically below. It is also shown that as the potential wells slant more and more (i.e. with larger electric fields), the wavefunctions become more localized (the image is taken from here (pdf!)):

screenshot-from-2016-12-01-222719

A nice numerical solution from the same document shows the wavefunctions for a periodic potential well profile with a strong electric field, exhibiting a strong wavefunction localization. Notice that the wavefunctions are of identical form from well to well.

StarkLadder.png

What can be seen in this solution is that the stationary states are split by \hbar \omega_B, but much like the quantum harmonic oscillator (where the levels are split by \hbar \omega), nothing is actually oscillating until one has a wavepacket (or a linear superposition of eigenstates). Therefore, the Bloch oscillations cannot be observed in the ground state (which includes the the applied electric field) in a semiconducting superlattice since it is an insulator! One must first generate a wavepacket in the solid.

In the landmark paper that finally announced the existence of Bloch oscillations, Waschke et. al. generated a wavepacket in a GaAs-GaAlAs superlattice using a laser pulse. The pulse was incident on a sample with an applied electric field along the superlattice direction, and they were able to observe radiation emitted from the sample due to the Bloch oscillations. I should mention that superlattices must be used to observe the Wannier-Stark ladder and Bloch oscillations because \omega_B, which scales with the width of the quantum well, needs to be fast enough that the electrons don’t scatter from impurities and phonons. Here is the famous plot from the aforementioned paper showing that the frequency of the emitted radiation from the Bloch oscillations can be tuned using an electric field:

PRLBlochOscillations.png

This is a pretty remarkable experiment, one of those which took 60 years from its first proposal to finally be observed.