There is something that I would like to confirm regarding the energy when aligning the bands of heterostructures.

There seems to be a general consensus that there is no simple rule to align two material energy bands. The most simple rules for aligning bands are the Schottky Mott rule and Anderson rule. In both of these, the vacuum level is taken as a reference when aligning bands. However, both proved to give the wrong results.

According to a review published in 2015 by Professor Tung,

When a crystal is in isolation, the positions of its internal energy bands can be referenced to the vacuum level outside.

Furthermore, he states

One deduces that the Schottky-Mott rule describes the Schottky Barrier Height of the metaphysical, non-interacting MS interfaces, put together without charge or atomic relaxation


This is what I understand so far. The first statement says that the energy bands inside a crystal can only be referenced to its vacuum level if it is isolated. It is like a reference point that is used only how "far" the energy levels are, and it is valid only inside the crystal.

However, there is band bending theory http://pubs.acs.org/doi/pdfplus/10.1021/cr3000626

As you can see, here they align the vacuum levels. And they use this to justify the direction of charge transfer and rationalize depletion and accumulation layers at equilibrium.

This is what I am confused about. From what I understand the vacuum level (as well as fermi level, etc) are only internal references. This is why it is so hard to align bands between two materials. And this is also the reason as to why the location of the zero energy in bandstructure diagrams is trivial (as mentioned here) https://www.researchgate.net/post/How_does_shift_fermi_energy_to_zero_in_bandstructure_p

In band bending, however, it seems that the vacuum level is more than just an internal reference. It can be used to describe which electrons have more energy (i.e are in higher energy states). Thus, it can be used to show the direction to which electrons will flow. If the vacuum levels were only an internal reference, such description would not make sense in my opinion. After all, just because it takes less energy to "remove" an electron doesn't mean that the electron will be in a lower energy state once outside the system. Consider one has two wells. It takes less energy to take something out of well 1 than well 2. However, in this scenario, it wouldn't be correct to say that an object on top of well 1 has a lower potential than an object in top of well 2 if we removed the wells.

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Nevertheless, the second statement from Professor Tung seems to suggest that the reason why band alignments are not simple is because there is charge transfer and atomic relaxation. Not because of the trouble in referencing vacuum levels, Fermi levels, etc. But I don't think this is the case. This is because the whole review places a significant emphasis on constructing model solids which can be used as references to predict Schottky Barriers.

So my question is: What am I misunderstanding? Are the vacuum levels and fermi levels an abolute energy scale (i.e. they tell what is the position in the hill in the drawing) or are they relative (i.e. they tell the position inside a well)? If they are relative, why can we use them to rationalize charge transfer? If they are not, why is is it so difficult to align bands in heterostructures? (other reasons than the fact that band alignments have to take into consideration things like atomic structure, vacancies, and defects)


1 Answer 1


I have taught myself and several other people how to understand band diagrams properly and I find that universally (myself included) there is a certain need to "unlearn" things that you may have been mistaught in classes or misunderstood. Major problems occur if one has taken a solid-state physics course which only studied band structure in infinite homogeneous perfect crystals. The intuitions one gains about electron energies in these courses can break down completely once inhomogeneities and interfaces are introduced.

The most important things I had to learn to fully understand band diagrams and band bending are the following facts:

  • In the actual semiconductor device, when all materials are assembled and in equilibrium, the vacuum level is completely irrelevant. There is no vacuum inside the device, and so it should be possible to explain everything without having to refer to the vacuum. If you feel you must draw a vacuum level, remember it is not a flat line, and remember it has a material-dependent offset relative to the average internal electrostatic potential.
  • In equilibrium the Fermi level is flat, absolutely. In other words, the Fermi level does not bend. Only the bands bend. Moreover the Fermi level is an absolute, not relative, energy measure, and when people say they are "tuning the Fermi level" they are being imprecise---they are actually tuning the band energies!
  • Potential energies (band edge energies) are best defined absolutely, i.e., all potential energies measured with reference to the same point. Important: The vacuum is not a single point and is not a valid reference. In your picture, well 1 has higher potential than well 2.
  • Surprisingly, band alignments are not predictable based on the properties of the two bulk materials involved. People have tried and failed to find an easy way. But in the end, it can be extremely important how the two lattices join together, diffuse together, if there are impurities, etc.. In the end the band alignment is a well defined quantity in a real device, but we pretty much have to discover its value empirically. There is no bulk reference property that can exactly predict band alignment.

Band bending at equilibrium is basically a consequence of two phenomena, and Tung only addresses the first point (the latter is a separate concern):

  • Band jump (band alignment): The interfacial band alignment/schottky barrier between a pair of materials, which can be empirically measured. Vacuum rules such as Schottky-Mott or Anderson aim to predict this alignment but ultimately they are only estimates based on way-oversimplified physics. Essentially they equate to demanding that the vacuum level is continuous across an interface (continuous, not flat). This is what Tung mockingly refers to as "metaphysical", because as he points out the vacuum level has an atomically sharp step in real interfaces.
  • Band curve: Screening effects, which occur over long distances in semiconductors.
  • $\begingroup$ I was under the impression that the vacuum level at infinity is flat. Not only that, but it is the same "energy" for any system. I mean, an electron that is infinitely far away is pretty much unaffected by anything. As a result, I thought that it could be used as a reference. The vacuum level near the surface is not flat due to the surface. I found out that there is ambiguity in the term vacuum level that lends to the confusion. You mention that in a semiconductor device it is possible to explain everything without a vacuum, what would serve as a reference to align the bands then? $\endgroup$ Apr 21, 2016 at 14:43
  • $\begingroup$ Lastly, if the Fermi energy is absolute, why is it so difficult to determine a Schottky Barrier Height? I thought that the lack of an adequate reference was the main reason why Schottky Barrier Heights were hard to determine. If Fermi energy levels were absolute, wouldn't it be easy to determine band offset and Schottky Barrier Height? $\endgroup$ Apr 21, 2016 at 14:46
  • $\begingroup$ Basically, my point is that we always know the Fermi level, but the hard part is to find where the bands are. Deep inside a bulk material it's easy but near interfaces it gets messy. $\endgroup$
    – Nanite
    Apr 21, 2016 at 19:09
  • $\begingroup$ And yes, you can often say infinity is a point, in which case there is often a well defined vacuum level at the infinity point. But consider, the earth is at thousands of volts relative to outer space so all your band diagrams would have the vacuum level at thousands of eV. :-) $\endgroup$
    – Nanite
    Apr 21, 2016 at 19:09
  • $\begingroup$ But usually when people say vacuum level they mean the vacuum level just outside the material, and that is definitely varying from surface to surface and from material to material. $\endgroup$
    – Nanite
    Apr 21, 2016 at 19:11

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