Layman's explanation, please. Let's consider an Esaki diode.
How do physicists measure/obtain the velocity of a would-be-tunneling electron in order to justify the claim that the potential barrier is higher in energy in comparison with the tunneling electron?
And, a follow-up question: how can one be certain that the electron (on the other side of the barrier) is "the same" electron whose velocity one have measured before the quantum tunneling?
This is a long comment.
At the level of diodes one is in the calculational range of quantum mechanics, the measuring the velocity of a single electron within the diode has no meaning. In quantum mechanics one deals with probabilities, which means the measurable quantities come from the accumulation of many events with the exact boundary conditions .
Now at the level of the diode, the velocities of the electrons are calculated from knowing the interactions they come from
Tunnel diodes have a heavily doped positive-to-negative (P-N) junction that is about 10 nm (100 Å) wide. The heavy doping results in a broken band gap, where conduction band electron states on the N-side are more or less aligned with valence band hole states on the P-side. They are usually made from germanium, but can also be made from gallium arsenide and silicon materials.
So there is no meaning to the "the same" electron whose velocity one have measured. It is all a matter of accumulating statistics biased with the quantum mechanical states.
See this for the tunneling effect.
