# Quantum mechanical antireflection coating; what quantity would be analogous to index of refraction?

### Background

Quasiparticle interference (QPI) is a technique that can be used to study 2D surface state band structure; carrier reflections at boundaries or impurity sites will interfere producing standing waves which can be imaged by STM. Plotting the periods of the standing waves vary with STM tip bias (by Fourier transform analysis of the dI/dV STM images) produces $$E(\mathbf{k})$$-like plots that can be directly compared to band structures measured via photoelectron emission spectroscopy (ARPES) and predicted via DFT.

We have a system where grain boundaries are not showing QPI in a situation where one would expect to see it. There can be (more than) several explanations for this.

### Question

In optics it's easy to make antireflection coatings. The simplest and most well known is the quarter-wave index-matching layer with $$n_{\lambda / 4} = \sqrt{n_1 n_2}$$ but this has a limited range of wavelengths for which it is effective. Another transmissive technique is a slow gradient in index, and if absorption (complex $$n$$) is allowed one can impedance match to free space. Broadband AR coatings are optimized designs with multiple layers of different indices and thicknesses.

I have some history/background/familiarity with optics but not in QM. Nevertheless I must give this a try.

If I'm solving the 1D Schrodinger's equation for a wave incident on a surface defined by a change in potential, what quantity is most analogous to index of refraction from the perspective of antireflection optimization?

Is it energy $$E$$ or $$\sqrt{E}$$ or does it turn out to be something I shouldn't actually vary like effective mass? Or something else? Or none-of-the-above?

Potentially helpful but I'm not really sure as it focuses on enhanced tunneling rather than reflection of a freely propagating wave: Antireflection coating of barriers to enhance electron tunnelling: exploring the matter wave analogy of superluminal optical phase velocity

Famous IBM image of QPI standing waves for background only

The corral is an artificial structure created from 48 iron atoms (the sharp peaks) on a copper surface. The wave patterns in this scanning tunneling microscope image are formed by copper electrons confined by the iron atoms. Don Eigler and colleagues created this structure in 1993 by using the tip of a low-temperature scanning tunneling microscope (STM) to position iron atoms on a copper surface, creating an electron-trapping barrier. This was the first successful attempt at manipulating individual atoms and led to the development of new techniques for nanoscale construction.

IMAGING TOOL: Scanning tunneling microscope

In terms of the 1D Schrodinger equation formalism, this question has a relatively straightforward answer. I'm not sure, however, that the answer we get from the theory has a clear correlation with the an easy-to-measure property of the coating.

The simplest version of this would be a 1D particle scattering off a step potential.

$$V(x<0) = 0$$ $$V(0 $$V(a

Here $$V=0$$ is free space, $$V_b$$ represents your "barrier" or your antireflective coating) and $$V_0$$ is the material you're putting the coating on.

You now have a schodinger equation in three distinct regions, which you can solve by generalizing the step function solution. The algebra is difficult, but manageable. In the end, you will want to solve for the conditions that minimize $$R$$ in terms of the thickness of the coating, $$a$$.

You can make some simplifying assumptions: $$E>V_b, V_0$$ (since you want your particle to propagate. You will therefore have complex expotentials in all three sections.

• Thanks for your answer! You're confident that for a given $E$ and $V_0 < E$ there will be some $V_b, a$ that results in 100% transmission? It's that "easy"? I know I should just buckle down and do the math (it's been several decades but I can always use a numerical solver) but I'm still wondering if there's a closed form solution out there. In the optical analogy $n_{\lambda / 4} = \sqrt{n_1 n_2}$ works because the reflection coeffs. at each interface have the same magnitude so backwards reflected waves exactly cancel for the right thickness. I suppose that should be the case here as well?
– uhoh
Oct 26, 2021 at 20:56
• There is probably a 100% transmission solution, but I am not very confident there is. Before you get too far into the weeds on the math, it might be easier to check by just setting the reflection coefficient to zero and seeing if you can solve it that way. Oct 27, 2021 at 14:52
• We've got synchrotron beam time right now, but as soon as I can I'll just adapt the math of this really cool wave-packet demo jakevdp.github.io/blog/2012/09/05/quantum-python The standing waves show up quite nicely! The potential function can be user-defined, the animation turned off and the transmitted and reflected probabilities can be calculated. Right now the wave packet has a very large spread of energies $\Delta E$ so even if the layer is $\lambda / 4$ for the central wavelength there's still leakage. So it will need some parameters adjusted.
– uhoh
Oct 28, 2021 at 11:17
• I suppose a T-matrix approach is really what I should do. I used it for multilayer optical coatings, I think it will work here as well; one matrix for each interface, at least for the two-step problem.
– uhoh
Oct 28, 2021 at 16:50
• – uhoh
Oct 28, 2021 at 21:01