BAND BENDING CHALLENGE!

The challenge

Instytut Fotonowy will award a prize of ONE THOUSAND USD to the first person that will provide a sound argument that band bending really exists.

Many explanation attempts in solid state physics or in electrochemistry involving semiconductors relay on the, so called, band bending. It is supposedly a consequence of electric carrier density imbalance near surface or an interface that would bend upwards or downwards a conduction or valence electronic bands in energy diagrams for semiconductors.

Our claim is that no band bending exists.

The argument against band bending

The reason for the claim is the following.
Energy diagrams describe single electron energy levels that originate from quantum mechanics. Indeed, the existence of band-like structures and forbidden single electron energies is a direct consequence of solving a time-independent Schrödinger equation with periodic external potential that corresponds to the periodic lattice structure of crystals.

The Schrödinger equation in 1D has the following form:
$$\hat{H}(x){\psi }_n(x)=E_n{\psi }_n(x),$$
where $\hat{H}(x)\equiv -\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+\hat{V}(x)$ is the energy operator for the external potential $\hat{V}(x)$, ${\psi }_n(x)$ is the n-th eigenfunction of the energy operator to the eigenvalue $E_n$.
It is $E_n$ that form bands, like conduction or valance bands, for periodic potentials $\hat{V}(x)$. And those bands are depicted in energy diagrams.
In particular, the lowest energy level of the conduction band is interpreted as one of the eigenvalues of the Schrödinger equation.
However, the eigenvalues of Schrödinger equation are just numbers. For example $E_{193}=1.3\text{eV}$ and they are independent of $x$. All the space dependence of the Schrödinger equation parts sits firmly in the potential $\hat{V}(x)$ and wavefucntions ${\psi }_n(x)$. Never in eigenvalues $E_n$.

Since $E_n$ is just position independent number it cannot bend!
Thus, band bending has no roots in quantum mechanics.

Before one shouts out “electric potential bends energy levels!”, please notice, that any such potential should be included in $\hat{V}(x)$, the Schrödinger equation should be solved anew and one will arrive at new set of eigenvalues $E_n^{\prime }$ that remain just numbers with no position dependence whatsoever.

Who can participate

Anybody with a bank account where we can transfer the funds when successful answer is provided.

The deadline for the challenge

The challenge will last until November the 18th, 2022 and can be terminated earlier if a valid argument is delivered before that date.

How to provide a response to the challenge

Just post a replay to that post on this forum. Anybody to be able to write a replay needs to sign up to the forum.
You don’t have to reveal your true identity when signing up. Only when you win, you will need to provide the wire transfer details.

beranekr

Dear Radim,
Thank you for taking up on the challenge!

The concept of band bending is a direct consequence of the formalism of solid state physics that requires that the Fermi level across an (electronically communicating) system in equllibrium is constant across the whole system (device). In other words, if you take a Schottky diode as a device, you will typically observe formation of a depletion layer because the Fermi levels of the semiconductor and metal will equillibriate. Importantly, this equillibriation that results in band bending is associated also with bending of the vacuum level (indeed, this contains information on any electrostatic potential changes across the system)

If I understand your argument, it would boil down to the following:
Energy bands bend because the vacuum level bends.
Am I correct?
If I am, that would be prooving the thesis by assuming it in the proof. That does not, typically, constitute a valid explanation of anything.
The vacuum level is a quantum level as well, so it cannot bend.

Based on this, the formalism is completely consistent with, for example, results of photoelectron spectroscopy measurements, which would be otherwise difficult to understand (if we had no concept of “band bending”).

If you think you can understand something with the help of a demonstrably incorrect notion, you might be in for a surprise. Photoelectron spectroscopy can be well understood in terms of quantum energy levels and electrostatic potentials. No need for “band bending”.

Obviously, while the concept proved useful for generation of solid state physicists, it is a completely valid request to show its consistency with basics of quantum mechanics, as the initial post calls for. Such a quantum mechanical treatment of band bending has been presented, for example, here: https://www.tandfonline.com/doi/pdf/10.1080/10408437308245840 (in short, it explains that the complete solution for a depletion layer requires that the Schrödinger equation and the Poisson equation are solved self-consistently).

Could you be, please, more specific with the reference? The book has over 500 pages, can you point me to the “complete solution”?

In my view, this is perhaps analogous to the claim, for example, that a “hole” does not really exist (there is just collective movements of electrons in a band where an electron is missing.

I think that you are right. The analogy is there. If you argument depends on the existence of holes then you do not understand the problem. Let alone the solution.
That is the case in textbooks trying to “explain” Seebeck effect for type-n and type-p semiconductors.
The current flows in opposite direction in those two cases: from cold to hot in type-n and from hot to cold in type-p.
The common attempt at an explanation is that the current is carried by electrons in the first case and by holes in the second.
This is plain wrong.

beranekr

Well, I think that, when trying to identify the source of disagreement, it all boils down to your claims such as “Vacuum level is a quantum level as well, so it cannot bend.” This might be true from the purely quantum mechanical point of view, but, as I wrote above, the conventional picture of devices in solid state physics presupposes that this bending happens, and the reason is that one wants to have two things in the same picture: the energy levels calculated using Schrödinger equation AND the spatial distribution of these levels as dictated by the Poisson equation; therefore the complete solution for a depletion layer requires that the Schrödinger equation and the Poisson equation are solved self-consistently, as shown, for example, in that article I cited above (it is a 15-page article, you can download it here: https://www.theoechem.org/owncloud/index.php/s/MrwpfA0oY4ryHCk)

In the article you have referred to, there is no band bending at all. The energy levels found from the Schrödinger equation (SE) solved self-consistently with the Poisson’s equation (PE) are given in Eq. 3 of the article.
The levels DO NOT depend on position as they shouldn’t.
The PE is solved to find the potential $V(z)$ that enters the SE. It needs to be solved in a self-consistent manner with SE because the charge density $\rho (z)$ from PE depends on the energy levels found from SE as stated in Eq. 4b and Eq.4d.
So, the authors did exactly what I proposed in the first post:

zbyszek Before one shouts out “electric potential bends energy levels!”, please notice, that any such potential should be included in $\hat{V}(x)$ , the Schrödinger equation should be solved anew and one will arrive at new set of eigenvalues $E_n^{\prime }$ that remain just numbers with no position dependence whatsoever.

beranekr I need to go sentence by sentence because we are starting running in circles here.

I agree.

You just gave this article as a proof for the band bending. The article shows the opposite: that no band bending is needed and that my approach to the problem is valid. So, what do you agree with?

The concept of band bending is a direct consequence of the convention of solid state physics that requires that the Fermi level across an (electronically communicating) system in equllibrium is constant across the whole system (device).

The Fermi level is constant in a conductor in equilibrium. But I don’t see that the “band bending” results from that observation directly or otherwise.

In other words, the band bending is the consequence of the fact that - in a conventional energy diagram - the vacuum level is not a “straight line” but follows the spatial changes of electric potential (more accurately, of the so-called outer/Volta potential), i.e., the vacuum level will show the same bending like the bands.

And here you use the circular argument again: The bands bend because the vacuum level bends . Perhaps you can break the circle by explaining why does the vacuum level bend? And just stating that it “follows the spatial changes of electric potential” is no explanation because the vacuum level also originates from Schrödinger equation and can bend as much as any other quantum eigenvalue or a number $\pi$, for example.

The energy diagrams constructed in this way have proven very useful for generations of solid state physicists in order to explain the behavior of various devices.

By “very useful” you mean “commonly used”. Unfortunately, either way, it does not make them correct.

Footnote 1: When I speak about the vacuum level, I mean the so-called “local vacuum level”, that is the energy of an electron at rest (zero kinetic energy) “just outside” of the material. With respect to this “local” vacuum level all spectroscopy is measured, hence only this vacuum level is relevant; for details see this wonderful reference: https://www.theoechem.org/owncloud/index.php/s/ph6ueyJ6dClp3Eh

This time you fall into a trap of the Cahen and Kahn article and their “local vacuum level” notion.
First of all it is not a quantum level. It is some energy of an electron at rest inserted “by hand” by the authors.
Second, for finite-size samples, as authors describe it at the beginning of section 2, their proposition is impossible.
Namely, position for the resting electron does not exist.
The place is supposed to be far enough from the surface of the conductor, so that the image charge attraction has no effect. Still close enough so that the electron is affected by the electronic surface dipole. Those two conditions are impossible to be fulfilled simultaneously for any finite-size sample in 3-dimensional space. This is because the image charge potential decays like $\propto 1/r$ (Coulomb potential) while the surface dipole potential decays like $\propto 1/r^2$, where $r$ is the distance from the surface.
So, if the electron is far enough due to the image charge, there is no surface dipole potential anymore.
The reference is full of wonders indeed .

beranekr Said as a true believer.
If the circular reasoning and the impossible definition, each being pretty serious “no go” in logic, doesn’t shake your confidence in the “band bending” picture, I don’t think any scientific reasoning will.

Yes, I have an alternative picture, that I have shown to you a couple of years ago. But this is beyond the scope of the challenge. Yet, it seems, it changed nothing in your firm believes .
Besides, you have delivered an alternative yourself in the reference in the post beranekr.

beranekr In the challenge at hand a winner was supposed to provide a sound argument for the band bending. It does not matter what someone thinks. What matters is if there is a series of logical steps that would lead from known laws of physics to the band bending picture. So far, no such reasoning has been proposed.

These days, asking for a basic scientific rigor may appear disturbing to some, as it hinders their publication potency.
If you should choose to relay on a superstition instead, you will not be alone.
Anyway, in the old school of reasoning, the burden of the proof is on proponents of a new notion not on the rest of the community.

… inaccuracies …

You are kidding, right? An impossible proposition is not an “inaccuracy”. Volta potential does not remove the impossible part of the definition.