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 $\hat H(x) \equiv -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + \hat V(x)$ 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 $E_{193} = 1.3 \mbox{eV}$ 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 $E’_n$ 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.
