OK Ben; your staying power has forced my arm to go into aspects which I tried to avoid for reasons that will become apparent. The energy difference that you have been raising consistently I have already noticed about 5 years ago: I, however, decided that since people call me a "crackpot" after I have found that electrons extracted from an n-type diamond into an anode form part of a dipole layer within which the applied electric-field is totally cancelled, while charge is still transferred from the diamond to the anode, the most obvious implication of the energy-balance situation would have had me burning at the stake.
I have searched all these years to avoid mentioning this possibility in the hope that there is a way around it. Since you persistently pointed out that there is no way around it, I will take the plunge and brace myself to be called the crackpot of all crackpots. This means that this post will be rather lengthy; but please bear with me since objective criticism will be very useful indeed.
But before I give further background, first a few remarks on your energy diagram. (i) A trivial remark is that both you and Ziggurat claim that the charge-carriers within a superconductor are being accelerated, the voltage between the power supply and the junction should then have a slope. Since I, however, know that they move without being accelerated, I am happy with the flat voltage as you have drawn it. (ii) A more serious remark is that you ignored the Fermi-level and the matching of the Fermi-level at the contacts to the superconductor. By doing this, your diagram is not applicable to the BCS model, and also not to my model. The latter aspect is what I want to address; but it first requires a comparison between BCS and my model.
The BCS model and my model do have some similarities. The major similarity is that for both models there is an energy gap and a Fermi-level within the gap. And both models are based on quantum fluctuations: In the BCS model it is a "virtual phonon" in my model it is a "quantum-hop".
In the BCS model it is claimed that the relevant energy-gap represents the binding-energy of the two electrons forming a Cooper pair; and that both electrons have an energy (delta) below Fermi-level so that there is an energy gap 2(delta) between the Cooper pairs and the free electrons at higher energy: The postulate is that the Fermi-level is situated in the exact middle of this energy gap and stays centered at this position for all temperatures at an energy (delta) below the free electrons and an energy (delta) above the electrons forming the Cooper pairs . This energy gap of 2(delta) is supposedly "mediated" by a "virtual phonon" which causes the gap to be zero at the critical temperature and to increase to its maximum value at absolute zero. In other words when superconduction initiates, the electron-energy spectrum is exactly the same as it is above the critical temperature.
Before getting to the Fermi-levels at the contacts to the superconductor there are aspects which I find strange in this model; and I will now mention the two most obvious ones:
(i)The most obvious one is that if there is no drastic change in the electronic energy spectrum at the critical temperature, there should not be a sudden jump in the heat capacity of the electrons as is observed in the low temperature metal superconductors.
(ii) Phonons are positively charged and if the bonding of the two electrons is mediated by phonons the electron-pair will be accompanied by a positive charge (while it moves along) of at least +e; more probably +2e. Thus a Cooper pair cannot be a doubly-charged charge-carrier and is most probably a neutral entity which sure as hell can then not be a charge-carrier.
At both the injection contact (where single electrons are being injected by the power supply) and the ejecting contact where single electrons are being ejected from the superconductor, the Fermi-level within the contact must be in alignment with the Fermi-level within the superconductor, which according to the BCS model is situated in the middle of the energy-gap between the free electron energy levels and the electrons forming Cooper pairs.
Thus when electrons are being injected they enter into an insulating energy-gap: Each injected electron is required to lose energy (delta) in order to form one of the constituants of a Cooper pair.
Andreev tried to model how a Cooper pair is formed at the injection contact and his model has become known as "Andreev reflection": However, his model requires that an electron must be injected at an energy (delta) below the Fermi-level. Obviously this is a nonsense assumption. Furthermore, the back-reflected hole derived by him always occurs when you eject an electron from a metal into an insulator; and has nothing to do with Cooper pairs being formed.
A more serious situation occurs at the ejection contact: Both electrons forming the Cooper pair have an energy (delta) below the Fermi-level and must thus get this energy from somewhere to move into the contact. Where does this extra energy (delta) per electron comes from? Out of the "vacuum"?
At a Josephson junction there are even more complications which I have already mentioned on this thread: The major one is that the Cooper pair has a much larger size than the largest width of the junction through which it can "tunnel" as if it is a "point particle". Why is it not possible to tunnel through a junction that at least has a width equal to the size of the Cooper pair?
2. In my model the energy gap (delta)ES already exists at full strength at the critical temperature: In the low temperature metals the critical temperature coincides with the metal-insulator transition: i.e. when the Fermi-level moves into the gap between the free-electron energy states and the localized orbitals which are responsible for superconduction by means of "quantum-hopping". At this point the energy difference (delta)EA between the Fermi-level and the free electron-levels is zero, but the energy required to absorb heat must suddenly jump, since electrons must now primarily be excited from the stationary orbitals over the gap energy (delta)ES. Thus the jump in heat capacity must occur for this reason. As the temperature decreases the Fermi-level moves down within the gap (like Fermi-levels always do within an electron-energy spectrum) until at absolute zero temperature (delta)EA=(delta)ES. It is this increase in (delta)EA, which is interpreted in the BCS model as being the binding energy of a Cooper pair.
It is also the latter Fermi-level which must align with the Fermi-levels within the contacts. At temperatures above absolute zero it is at an energy (delta)EQ=(delta)ES-(delta)EA above the superconducting orbitals. Thus when an electron is injected in order to replace the nearest orbital to the injection contact, which then moves on further to replace the next orbitals etc., the injected electron must lose energy (delta)EQ; which can be easily done by radiating it away.
When it reaches the ejection contact, it needs this energy to enter the contact; but since the temperature is higher than absolute zero this energy is probably supplied by a temperature fluctuation; so that the charge is easily transferred from the superconductor to the external circuit. But how does it enter the contact at absolute zero? Fortunately at absolute zero the Fermi-level is at the energy-position of the orbital, so that an injected electron need not lose any energy and an ejected electron does not need extra energy.
But now comes the Josephson junction with a voltage VJ across it: To simplify the discussion let us assume that without the junction voltage the Fermi-level is situated at the energy position of the orbitals. Before switching on the voltage VJ the Fermi-level is the same on both sides of the junction. After switching on the voltage VJ the Fermi-level shifts by an energy qVJ over the junction where q is the charge on the charge-carriers. But as already explained on this thread, the charge-carriers on the low energy side shifts by an additional amount of energy qVJ owing to a decrease in their polarization-energy. Thus when an orbital with charge q moves through the junction it radiates energy 2qVJ.
But now when reaching the ejection contact, it has an energy qVJ lower than the position of the Fermi-level within the contact. Thus to move on it must gain this additional energy. Where does it come from? Well as Ben thought he correctly pointed out all along, it cannot come from the battery; even though I have tried for years to argue that it must be so since, if it does not, I have to conclude that it is supplied by a quantum fluctuation. It is all good and well to claim that the orbitals are borrowing energy (delta)ES+1/2m*v^2 to move from one point to another after which this energy is returned; but to argue that at the ejection contact part of this borrowed energy remains, and is not given back, sounds bizarre. It would mean that "free energy" is extracted from the "vacuum". I thus decided that there must be another mechanism for the appearance of this energy; and have been valiantly searching for this mechanism in order not to cement my position as a crackpot.
And of course there must be another mechanism since for th BCS model the situation is even more dire. In the latter case, even without the presence of a Josephson junction, each electron forming part of the Cooper pair must gain an energy (delta) and this energy is much much larger than the energy qVJ that must be gained by the orbital in my model after it has gone through a Josephson junction with a voltage V(sub)J across it. So if, after 50 years, each Cooper pair electron can find the energy (delta) to increase its energy at the ejection contcat why can my orbital not find the energy qVJ<<<(delta) to do the same? Furthermore, in the case of the BCS model there really is no other energy available to supply the energy (delta) required, whereas in my model there is polarisation energy available which must stay constant within the superconductor. Since this energy cancels the applied electric field, the voltage which initially caused the applied electric-field can top up this energy without requiring an increase in the measured voltage over the superconductor. As soon as one realizes that a superconductor is not just a conductor and also not just an insulator, but a conducting-dielectric, the energy conservation falls into place
The following paragraph was meant to point out the absurdity in Ben's arguments:Ben's persistence that this energy cannot come from the battery or any capacitor in parallel, now forces me to conclude that, if my model is correct, one might be able to use Josephson junctions to extract "free energy" from our surroundings. There: Ben has forced me to state this in the open! I now really feel like a crackpot!
