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.).