Microscopic Mechanisms of Superconducting Flux Quantum and Superconducting and Normal Persistent Currents

In view of energy, according to Drude model, and considerring the kinetic energy

of carrier electrons as: , where is number density of

electrons and m is electron’s mass, there would be ( Michael Tinkham: Introduction To Superconductivity, Second Edition, McGraw-Hill, Inc., 1996, sec. 2.5.1):

，or

.

Obviously, when , becomes London equation. (But here we do not divide into superconducting and normal currents.)

Next we discuss superconducting flux quantum. According to Maxwell equation, for magnetic flux threading a superconducting ring, there is:

By substituting London equation into and integrating with respect to

time, we will obtain:

Assuming that the system contains only one superconducting electron (we would later see that this assumption is not necessary,) the current density could be (see: Michael Tinkham: Introduction to Superconductivity, Second Edition, McGraw-Hill, Inc., 1996, sec. 1.5):

where the wave function is Bloch function:

As London equation can also be in the form of: , and as ,

becomes:

By substituting and into , with , we obtain

.

As a trial treatment to “number density of electrons” , letting (we would

modify this;) thus, the 2nd and 3rd terms on the right of become:

1

,

thus the superconducting current in is in the direction of wave vector .

Since we can always take the integral loop as along , that is, along , there would

be: , where and are the length of the loop and the x

component of the wave vector respectively. According to Born–von Karman boundary condition: , where s is an integer, there is:

,

which is a result of magnetic flux quantization. This result leads to some further conclusions. First, in an ideal crystal, it is impossible for a loop to be kept along ; but in the initial experiment verifying flux quantum, the ring is formed by deposition of superconducting material on a cylindrical substrate (B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43(1961).). Thus, we could understand that the ring should include a single crystal having a series of dislocations, and the direction of its

should have been gradually altered by the dislocations, thereby “loop always along

the direction of ” was realized. On the other hand, it seems that these dislocations

did not affect the superconductivity of the samples. Second, the result of is

twice of the experimentally measured value of flux quantum . So

modification has to be made.Due to non-zero flux, corresponding perturbation exists. According to existing

perturbation theory, the original degenerated electronic states would linearly combine to form new zero order quantum states. For example, if the original degenerated states

before disturbance is Block functions: , where , then the

newly combined electronic states after perturbation could be like:

.

Although other states of may also be included, in the absence of external field the weight of these other states can be negligible due to that the perturbation corresponding to magnetic field of the order of is very weak. According to the principle of superposition, the measured electronic state of each of the two on states

must be one of of . But according to , if one of the two electrons

is measured at , the other electron cannot be measured at or the current would

become zero according to and the flux would also become zero. Thus, a single

flux quantum has to be provided by a pair of carrier electrons, which are

originally on two degenerated states of .

Thus, we need to consider the wave function for two electrons. First, we cannot directly substitute into or the antisymmetrical wave function:

2

would become zero when and . The problem is due to that

although both of contain , the two states are not the same state in fact, because they are of different energy. That is, “approximate” single-fermion wave functions may not be used to construct the anti-symmetrical wave function of a multi-fermion system. Thus, we need to differentiate the two states; we do this by

marking one of them as , as:

In addition, can no longer represent “number density” of two electrons,

as (among other things) cannot be greater than one; we first try to modify it

by multiplying it with the number N of electrons (and as we will see, this is not correct.) Thus, with , we would have:

The lass two terms on the right of has the form of , so their loop

integrals are zero (for example, we can let one of and equals to and let the

other one be , with , thereby doing the loop integral.) Substituting

into , and replacing with , we would obtain:

where s is an integer, L is the perimeter of the loop, and there is (with in the present example.) But this result still differs from the experimental result of

. The problem is with the representation of by . First, cannot

be , for it miss the cross terms. The scenario of the present case is: two

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coupled electrons, which are at respectively, jointly forms a

constant current, and there are four combinations of electronic states: , ,

and , where only can provide a positive and constant current; this means that all measurements corresponding to any of the other three combination states are not allowed to be “expressed” (due to limitation of energy conservation law), and weights corresponding to the three combination states are lost in the

measured current value. Thus, when we represent with of , we

miss a factor of 4 (instead of the electron number 2). The physics in it is: during the time slots of the combination states not allowed to be expressed, physical effects relating to are still kept valid, so we need to add the ’s parts corresponding the

“lost slots” into the representation of , and the factor to be multiplied with

should be “the total number of micro states divided by the number of the

micro states allowed to be expressed”. Thus, in the present case, we should have

, and the result of becomes:

which is the same as the experimental results. As indicated by the above operations and results, the flux quantum of is generated by two electrons, which are

originally at respectively and degenerate, at very weak coupling. This understanding is consequential. But before further discussing it, we first look at the situation where superconducting current is in the direction of . Assuming that

the current is co-generated by the two electrons at respectively,

with . Now we would have:

And is still valid. becomes：

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The loop integral of the 2nd term on the right of is obviously zero. The result corresponding to that of is:

.

That the direction of loop is always along vector can be realized in the presence of dislocations, as long as the latter do not destroy the periodic potential

substantially; then, with , becomes:

.

where is the perimeter of the loop, and are the base vectors

along x and y directions respectively, , and , then:

when , and lead to:

;

that is, when the current in the ring is carried by the pair of electrons at and

, the flux quantum is , which is the same as was concluded elsewhere (“Flux-Periodicity Crossover from hc/e in Normal Metallic to hc/2e in Superconducting Loops” ， Loder, Florian; Kampf, Arno P.; Kopp, Thilo. arXiv:1206.1738.) For such “diagonal” current, an exemplary integral loop could be arranged as: the “ring” is in the form of a long cylinder, with y axis extending along the axial direction of the cylinder, z axis being perpendicular to the surface, and x axis being along the circumferential direction of the surface of the cylinder; the x axis is kept to be along the tangential by displacements, and the z axis is also kept perpendicular to the cylinder surface by displacements; the integral loop starts from the middle of the cylinder and diagonally extends in opposite directions, forming half of the loop, and joints the other half formed similarly on the opposite side of the cylinder to form a complete loop; the perimeter along x direction is , the length

along y direction is , and the completeness of such a loop, as well as a result

close to that of , could be ensured with being somewhat greater than

. Apparently, determination of flux of a superconducting ring depends on the electronic states and of the pair of carriers. The effect of is remarkable, which depends on “the total number of micro states divided by the number of the states allowed to be expressed”; when the number of pairs engaging in coupling increases, the quotient increases and the flux quantum decreases accordingly. For example, when the four

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electrons at are in “full coupling” to form a “carrier team” of four

electrons and carrying a current corresponding to in x direction, the quotient is

16, and the four electrons carry a current corresponding to a flux quantum of .

But whether is multiplied by the same quotient among electron pairs having very weak or substantially no coupling, the answer seems “no”. Perhaps some limitation based on intensity of coupling should be introduced; but the cases discussed so far, like , are in full coupling. The author still could not decide on this. But it could

be sure that needs to be at least doubled for two pairs of electrons.The understanding, discussion and results so far, while being in agreement with some existing experimental evidences, lead to new questions. First, electrons with

are mostly not near , indicating that these electrons are “deep electrons”, contrary

to the understanding that carrier electrons should be near , and that carrier electrons should be participants of phonon process leading to electrical resistance and energy dissipation. If at least some carriers are “deep electrons”, the dual functions of electrical conductance and electrical energy dissipation are respectively undertaken by deep electrons and surface electrons. Such understanding seems more reasonable, as screening to phonon interaction by binding energy of electron pairing, albeit remarkable, seems not sufficient to supports things like persistent current, whose mechanism will be discussed later.Another question relates to mechanism of electricity. Symmetrical distribution of electrons in space would eliminate current, so generation of current should relate to destruction of such symmetry. The discussion relating to flux quantum provides a mechanism of such destruction, as well as some explanation of construction of ,

and provides some explanations to flux quantum. The expression of was previously discussed by some people, but flux quantum was not derived from it, with the understanding that current was zero within the body of superconductor. is indeed consequential, and we have not fully addressed it so far. But our derivations relating to and are correct. The key lies in ; the loop integral of

is with respect to rather than ; thus, there is the possibility that even if the integral on still has a finite value. In fact, as is finite at the surface

and within the body of superconductor due to magnetic field (where is the penetration depth and z is the coordinate perpendicular to the surface,) the

current ought to have similar relation ; as such, to maintain at a

finite value for all integral loops within the body of the material, there must be:

.

Such an electronic state does exist, and it is the surface state(s) of crystal. We thus could conclude that superconducting carrier electrons are those of surface states.

The results that a pair of electrons at generate flux and a team

of four electrons on states generate flux is due to that the more microstates are involved, the smaller the ratio of allowed microstates in them, and the greater the ratio of “the lost slots”.

Clearly, the “lost slots” (corresponding to “measurement collapse of wave

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function from such as to ”) and associated effect have

certain inherent association and similarity to the “measurements of transitions in nonstationary state” discussed with respect to AC Josephson effect; but there is a difference, the collapse effect here happens in stationary effect, while in the AC Josephson case it happens in non-stationary states. This difference is also associated with the matrix elements, as the matrix for degenerate perturbation of two electrons is:

, and that for “non-stationary stimulated transitions between two energy

levels” would be ; the former has matrix elements while the latter has

matrix elements of 1, that is why “lost slots” appear in the former but not in the latter. And it is due to the “lost slots” that more electrons in coupling carry a smaller current than fewer electrons. It is also due to that no slot is lost in non-stationary measurements of transitions that the peak in the “peak-dip-hump” in ARPES results of superconducting cuprates (such as B2212 system) can be observed.Moreover, we need to distinguish the states of the two electrons in , in agreement with our proposal “two measured states of the same energy and wave vector in non-stationary context are not the same state ”. In particular, Single-fermion wave functions in their “approximate” representation may not be used to construct the anti-symmetrical wave function of a multi-fermion system.

The conclusion that carriers can be “deep electrons” is in agreement with the experimental evidences of normal persistent cuurent (Persistent Currents in Normal Metal Rings,Phys. Rev. Lett. 102, 136802 – Published 30 March 2009,Hendrik Bluhm, Nicholas C. Koshnick, Julie A. Bert, Martin E. Huber, and Kathryn A. Moler.).

We are now going to establish a model of normal and superconducting persistent currents. As indicated schematically in Fig. 3, assuming that two state of wave vectors

and respectively are indicated by and D, and that the two states couple to form states in the form of:

When the two electrons of are both measured at , current in the form of

is formed, and establishes corresponding magnetic field and energy and also stores

current energy , which corresponds to momentum:

,

where is the density of carrier number. Thus, when is determined, flux and could be estimated from:

and

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,

and kinetic energy T and momentum could be estimated from .

As such, however,it seems that when both the electron are measured at , it

would correspond to that the electron initially at now stays on a state C,

whose energy E is increased by corresponding to “energy of the

magnetic field plus T”, and whose is increased by corresponding to . Thus, three effects emerge. First, microscopically, the electron state at C is

unstable, as the electron might jump back to state by a single or multiple phonon process corresponding to , so the initial energy of electric field is turned to phonon energy, as in a resistance process. Second, would correspond to an increase of the system’s internal energy, and a corresponding increase of the system’s Gibbs function, with which the system might leave superconducting phase. Third, most importantly, the momentum of the electron cannot be determined as to whether it is or , and such a situation might not be allowed to exist (particularly

after field disappears), for the momentum of an electron in a periodic potential

could only be the value of an eigen states while are not eigen state generally.

As such, the only possible situation seems that when the two electrons are both measured as being at , the electron initially at does not acquire the

energy and momentum of , and that one or more phonons are produced in the process in which the electron is stimulated by electric field to form current, which phonon(s) has a (total) energy and momentum of , and the electron

initially at only has its momentum changed from to . Thus, when

the current of the system changes from 0 to , while the energy of the system increases by , this increment is not in the electron system; rather, it is in the phonon system, which also has a momentum change of ; the electron system

only has a momentum change of . Since corresponds to a deviation from equilibrium by the phonon system, it would be transferred to the environment by thermal equilibrium process, allowing the system to discharge energy and maintaining the internal energy of the system to be the same as that in zero current state. But in such a situation, either of the two electrons at may return to state

any time, nullifying the current, without any change of the total energy of the system. So persistent current could not be formed in such a context.( But when the electron’s transition to state C is virtual, there would be no phonon being produced.)

So we need to explain persistent current with respect to another scenario. As shown in Fig. 4, the horizontal axis is , and direction (that of the total current)

is perpendicular to the plane of the paper. Letting state has and

, and state D has and ,and

，where , that is, energy of is higher than that of state D. Assuming that the electron at is energized by electric field and

virtually transits to state C; when states C and (one of the two states at) D

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are matched by one phonon mode and states and D are matched by

another phonon mode , then as discussed previously with respect to “non-stationary stimulated transitions between two energy levels”, the electron virtually transiting to state C, by (virtually) emitting a phonon ,

“condenses” to (the state at) D. (By “condense” we mean that two

electrons in non-stationary states both have the measurement results corresponding to one stationary state.) However, as C is an intermediate state of a multiple-process, there could be neither emission of any phonon nor absorption of any

energy from field , instead only a real phonon is emitted in the whole process.

As such, there would be two electrons both having measurement results

corresponding to state , and accordingly no electron could be measured at

the state at . Thus, a current contribution of is formed, which has a

non-zero y component because state lacks one electron while

state has two electrons. Obviously, in order to form a net total current in

x direction, there must be an electron of undergoing the similar process

with respect to the state of , whereby the x component of current

increases by while its y component increases by , so the electron distribution in y direction resumes symmetrical and the current along y direction cancels; thus, a net current of in x direction is formed. Most importantly, however, now each of the two carrier electrons obtains a binding energy of . In the situation as above, where one phonon mode matches states C and D and

another phonon mode matches states and D, the process has the greatest probability of occurrence; states and C are coupled by interaction of electric field and does not required phonon matching. If states C and D and/or states D and cannot be matched by one phonon mode, the probability of occurrence of the process will be greatly reduced, and it would be likely that corresponding persistent current carrier cannot be obtained. Also, the state D, which is matched by the electron virtually transiting to state C by phonon mode ,

could have a random distribution, and particularly could have a negative

component , so as to form a persistent current in –x direction, which is in

agreement with the known results that the normal persistent current could be random. （ Persistent Currents in Normal Metal Rings,Phys. Rev. Lett. 102, 136802 – Published 30 March 2009,Hendrik Bluhm, Nicholas C. Koshnick, Julie A. Bert, Martin E. Huber, and Kathryn A. Moler.）

Whether an electron could “condense” should also depends on attributes of the initial state (C), the final state (D) and the mediating phonon mode ; the specific relationships of are unknown; but it seems that the more stable the final state (D) is, the more unstable the middle state (C) is, and the greater the energy of the mediating phonon mode , the more likely “condensation” would occur

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and more stable the latter would be. Persistent current carriers are those electrons which could condense and obtain binding energy ; electrons that could not condense or that could not obtain adequate binding energy become normal current carriers.

When both processes and are virtual, the overall process does not absorb any energy from the electric field , and no phonon is

emitted; instead, only one phonon is emitted. Phonon mode merely mediates the virtual transition from the middle state C to the final state D, and it also mediates the “non-stationary interactions” (condensation) between states C and D after the transition takes place, which at low temperature the mediation is basically carried out by the zero-point of the phonon mode . As such, the generation of persistent current carriers itself would not dissipate energy of the electric field, and instead a phonon , the energy of which would be released by thermal equilibrium to the environment, allowing the energy of the present system to be kept at a accordingly low level. On the other hand, the generation of persistent current carriers includes build-up of energy of the middle state C as well as build-up of the probability of virtual transition of electrons to the latter, so corresponding relaxation should exist.

Since many (N) electrons participate the process relating to electric field, the energy of the middle state C which each carrier electron can reach depends on N. The smaller N is, the greater the energy of C is, and the greater the energy of phonon

. As interaction of the electric field might lead to a random distribution of middle states and final states, the resulted normal electron pairs and superconducting electron pairs would also have their distribution, which should correspond to the shape and/or symmetry of the energy band and the phonon spectra. Clearly, smaller ring allows greater energy per electron transiting to the above middle states C, and the limitation as would tend to reduce the order of carrier number by 1/3 or 1/2, which all lead to increase of each carrier candidate’s energy.

With respect to reduction of carrier number, superconducting phase has a clear advantage in that superconducting carrier electrons must be in surface states, while normal carrier do not have such a limitation. The experiments like that of Bluhm et. al. used thin film samples, which is equivalent to limitation of surface states. Low temperature reduces the number of phonons, leading to sufficiently low probability that the binding energy is released by multiple phonon process.

Thus, we have provided a microscopic explanation to (superconducting and normal) persistent current. The charging of superconducting current is accompanied by resistance dissipation, which is a necessary outcome of “two fluid” model. According to our present model, the generation of persistent carrier electron does not dissipate energy; instead there would be emission of real phonons and release of corresponding energy into the environment; but the normal carrier electrons involved still dissipate energy. There should be a build-up of energy of the middle state C and a build-up of the probability of virtual transition of electrons to the middle state, and the corresponding relaxation should exist accordingly.

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Fig. 3

Fig. 4

A

C

D

Δ

E

xk

B

D’

( , ) q

C

D

E

yk

ED’

CD CD( , ) q

y1k

D ' D D ' D( , ) q

11

12