ORIGINAL PAPER

Is a Superconductor a Perfect Insulator?

Refael Gatt1

Received: 8 July 2019 /Accepted: 29 November 2019# Springer Science+Business Media, LLC, part of Springer Nature 2020

AbstractIt is suggested that the difference between the normal and the superconducting state is that the individual electronic states in theproximity of the Fermi level are standing wave states. Superconductivity may then be understood as a condensation of standingwave states. All other properties of superconductors and specifically their electrodynamic behavior are direct consequences ofthis single property. The London relation may be understood as a microscopic relation of single particles. The understandinggained provides a path for design of new superconductors.

Keywords Superconductor . StandingWave . BCS . London Equation . Fermi Volume . Superfluid . Residual Hamiltonian

1 Introduction

The quantum theory of superconductivity begins with the for-mulation of the London equation by means of the vector po-tential A [1],

J ¼ −nse2

mcA; ð1Þ

where J is the current density, A is the vector potential, ns is thesuper-electron density, e is the electronic charge, m is thesuper-electron mass, and c is the velocity of light. Londonshowed that this relation gives rise to the Meissner effectand zero resistivity. The Meissner effect is the ability of asuperconductor to expel a magnetic field. We therefore call asuperconductor “a perfect diamagnet.” Thus, for example,when we apply the London equation, combined withMaxwell’s equations, to a superconducting homogenous slabin a magnetic field, we get an exponential decay of the mag-netic field from the exterior into the bulk of the superconduc-tor (see below). Using the same equations, we get the distri-bution of the current in a superconducting slab, and we get thesame result: the current decays exponentially from the surfaceinto the bulk of the superconductor. In as much as a supercon-ductor is a perfect diamagnet, it is therefore also a perfect

insulator. In thick enough superconductors, there is zero cur-rent in the bulk in the same way as there is zero magnetic fieldin the bulk of thick enough superconductors. That simple factleads us to look for superconductors away from perfect metals(with simple parabolic band structure) and try to describe thephenomenon of superconductivity as a limit between a metaland an insulator.

London proposed such a direction, although not in theseterms, as a many body effect. He showed that Eq. (1) could bederived as a result of a “rigidity” of the many-body wavefunction:

ψ r1;…rn; tð Þ ð2Þ

In 1957, Bardeen, Cooper, and Schrieffer (BCS; hereafterreferring both to the original paper and to the theory developedin it) developed a microscopic theory of superconductivity thatcast this rigidity in the form of an energy gap [2, 3]. Severalpreceding studies contributed to the success of BCS theory:

(1) Fröhlich showed that the electron-phonon interactioncontained a term that could be approximately formulatedas a two-electron attraction [4].

(2) Cooper showed that a Fermi gas is unstable against anysmall effectively attractive interaction between two elec-trons [5].

(3) Landau had shown the importance of an energy gap inderiving the thermodynamic properties of superfluids [31].

BCS were able then to derive a gap equation thatreproduced remarkably the thermodynamic properties of

* Refael Gattrafi@qdm-tech.com

1 Quantum Designed Materials Ltd., P. O. Box 4146, 7414002 NessZiona, Israel

Journal of Superconductivity and Novel Magnetismhttps://doi.org/10.1007/s10948-019-05387-z

superconductors. The great success of BCS lay not only in itsability to explain previously observed phenomena but also tomake accurate predictions, confirmed by tunneling spectros-copy and other techniques [6–8]. Among the triumphs of BCSwere the Josephson effects, derived from the theory and sub-sequently observed, a physical phenomenon which gave riseto an entire field of sensing and fast electronics [9].

BCS made the following seven assumptions:

(1) The ground state is a paired state.

– Superconductivity is the result of coherent scattering ofCooper pairs by the attractive potential.

(2) The pairing is of opposite crystal momentum states (ormore generally of time reversed states [10]).

(3) The ground state is a singlet state (other pairing stateswere later described).

(4) The interaction is attractive, gaged by a matrix elementVkk .

(5) The ground state represents a “broadening” of the Fermisurface, namely, a correlated state with partial occupationof paired states:

ψBCS ¼ ∏k uk þ vkcþk↑cþ−k↓

� �0j i; ð3Þ

where |0⟩ is the vacuum state for creating Bloch wave func-tions in a crystal or, more generally, electronic structure statesin a metal that may be amorphous [10].

(6) The electronic system can be treated as a grand canonicalensemble.

Given these independent assumptions, the gap equationwas derived as:

Δk ¼ −1

2∑k ;

Δk ;

Δ2k ; þ ξ2k ;

� �Vkk ; ð4Þ

where

ξk ¼ εk−μN ; ð5Þand μN is the chemical potential in the normal state.

While the thermodynamic properties of superconductorswere reproduced with outstanding accuracy by BCS, it wasshown later that the Meissner effect could not be properlyderived without an additional assumption, namely, the gaugesymmetry breaking of the order parameter:

(7) The coefficients uk and vk in Eq. (3) are complex num-bers which have a definite phase difference that is aproperty of the condensate as a whole.

The work of Gorkov, Bogoliubov, and others showed arelation between the energy gap and the Ginzburg-Landauorder parameter [19]. The gap function may then be expandedto correspond to the “anomalous averages” [11]:

Δ rð Þ ¼ V Ψ rð Þ↑Ψ rð Þ↓D E

; ð6Þ

where Ψ(r) is the particle field

Ψ rð Þ ¼ ∑kφk rð Þck : ð7Þand ϕk(r) are the Bloch functions (or any other appropriatebasis set).

The picture of the BCS ground state emerging from theabove considerations is quite elaborate and detailed: Cooperpairs are “communicating” via virtual phonons. There may bea million other pairs separating the members of a given pair,yet this cooperative coherence is maintained “over miles ofdirty lead wires” [12].

The electrodynamic properties are derived from an orderparameter description. In that respect, they are not really mi-croscopic. In fact, the derivation of London rigidity from theBCS gap was questioned by Anderson [13].

It would be advantageous if we could derive these sevenassumptions from a single Ansatz that would serve as a mi-croscopic definition of a superconductor.

Superconductor development in practice suffers from aneven bigger problem. As is generally appreciated, whereaswe understand the phenomenology of superconductivity verywell, there is no known way from the current theory to designnew superconductors in chemical and physical detail, nor topredict qualitatively their transition temperatures. The prob-lem became apparent with the discovery of the cuprate-basedhigh-temperature superconductors (HTS) [20]. This failing isin a way inherent in the theory of superconductivity – BCSsolved the reduced Hamiltonian of any superconducting sys-tem, but not at all the full one. But chemistry, the specificatomistic state of matter which hosts superconductivity, is de-rived from solutions of the complete Schrödinger equation.

It is possible to calculate the density of states of a material,its phonon dispersion, and the electron-phonon coupling andfrom these to deduce Tc via a number of formalisms, such assolving the Eliashberg equations. The results are reliable insome cases but hardly all. Qualitative understanding – whatchange in the chemistry or the crystal structure will improveTc or impair it - is needed.

In the following, we offer a partial, yet detailed, remedy ofthese fundamental problems. While superconductivity mani-fests itself through a variety of many-body effects, we look atthe possibility of deriving the London rigidity and the attrac-tive interaction between electrons from the properties of asingle super-electron. We present a solution of the fullFröhlich Hamiltonian based on a new definition of a

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superconductor. We also show how the electrodynamics ofsuperconductors arise naturally through London rigidity of asingle super-electron. It will appear from the derivation thatthe BCS theory is accurate (with some modifications) but notcomplete. In fact, the reduced Hamiltonian of BCSwill appearas the residual Hamiltonian of the standing wave theory.

2 Superconductors Are Standing WaveCondensates

In the following discussion, we show that the seven assump-tions of BCS can be derived from a single Ansatz:

If ϕk is the normal state basis set, then the superconductingstate is characterized by the following condition on the disper-sion of the super-electron at the chemical potential, μ (thechemical potential here is not the normal state chemical po-tential but the superfluid chemical potential; see below):

∇kε ¼ 0jε¼μ; ð8Þ

where ε is the super-electron energy and k is its crystal mo-mentum index (k can be replaced by a more general index αwith the appropriate partial derivatives for amorphous metals).This is a single-electron relation, so it corresponds to theLondon description of super-electrons. Pairing correlation,as we shall see, emerges as a many-body result of that singlesuper-electron property. The assumption (8) does not implyany long-range coherence; such coherence may be derivedfrom this assumption. Condition (8) may be seen as a newdefinition of the superconducting state.

Assumption (8) has the quasi-classical meaning:

vg ¼ 1

ℏ∇kε ¼ 0; ð9Þ

so that the group velocity vg of the single super-electron prob-ability wave function is zero. That means that the single super-electron state is a standing wave state. We emphasize againthat this is an assumption, from which many consequencesensue.

Instead of applying periodic boundary conditions and usinga traveling wave representation, we consider standing wavesstates that represent interference between two Bloch states:

φα r; tð Þ ¼ 1ffiffiffi2

p ϕ rð Þ eiα rð Þ þ e−iα rð Þ� �

eiℏεt ð10Þ

In a crystalline material,

α rð Þ ¼ k � r

In a disorderedmaterial,α(r) can be any real function of r [10].

We could also choose the difference function:

φα r; tð Þ ¼ 1ffiffiffi2

p ϕ rð Þ eiα rð Þ−e−iα rð Þ� �

eiℏεt: ð11Þ

However, we will see that this state is of higher energy.Note that the standing wave probability function now cre-

ates a static distribution of electron density. This static distri-bution triggers some relaxation of the surroundings (phononsin low Tc materials, possibly other bosons, such as spin-waves, in other materials) to be explored below.

A detailed treatment of this relaxation for phonons nowfollows.

3 Thermodynamics and Quantum Dynamics

We start with the full Fröhlich Hamiltonian (with minor mod-ification) that includes the kinetic energy, the phonon part, theelectron-phonon interaction, and the electron-electron interac-tion [4].

H ¼ μ∑kcþk ck þ ∑k εk−μð Þcþk ck þ ∑qωq aþq aq þ

1

2

� �

þ ∑q;kMqcþkþqck aq þ aþ−q� �

þ ∑q;k;k ;v qð Þcþk−qcþk ;þqck ;ck :

ð12Þ

Here, μ is the anticipated chemical potential, εk is the nor-mal quasiparticle energy, cþk and aþq are electron and phonon

creation operators, respectively, ωq is the phonon frequency,Mq is an electron-phonon matrix element, and v(q) is ascreened Coulombic potential. Screening is maintained, evenin a narrow band if an additional dispersive band is crossingthe Fermi level [21]. We apply our assumption (Eq. 8), that is,all the states are degenerate with respect to k. We assume thatthe second term in Eq. (12) is a small perturbation so it can beneglected for now, and we can perform the following trans-formation:

Aq ¼ aq þM*

q

ωq∑kc

þk−qck ; ð13Þ

where Aq is a renormalized phonon annihilation operator. Wewould like to write down H in terms of the new operators Aq(see Eq. 17 below).

We confine ourselves here to the operators cþk creating thestates (10). We will deal with this constraint later.

We again apply our assumption (8),

ρ1 q; tð Þ ¼ ∑kcþk−q tð Þck tð Þ ¼ ∑kc

þk−qe

iℏμtcke−

iℏμt ¼ ρ1 qð Þ: ð14Þ

We assume here that the standing waves are eigenstates ofthe Hamiltonian H0 (H without the kinetic perturbation). Itremains to be seen if these functions can diagonalize theHamiltonian.

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It follows then that ρ1(q) is a constant of the motion, and Aq

retains the canonical relations:

Aq;Aþq0

h i¼ δq;q0; Aþ

q ;Aþq0

h i¼ 0 ¼ Aq;Aq0

: ð15Þ

Note that ⌊H0, ρ1(q)⌋ = 0 . Therefore, they can be diagonal-ized simultaneously. That means that in the standing wave

base, ρ1 qð Þ ¼ diag ρk11 qð Þ; ρk21 qð Þ;…; ρkn1 qð Þ� �, but ρ1(q) is

not a function of k, therefore,ρ1(q) = ρ1(q)I and (15) follows,if indeed H0 is diagonal in the standing wave basis.

The second term in the expression (13) represents the bo-son shift with respect to the standing wave electronic distribu-tion. It should be noted that this is not a charge density wavebut a detailed correlation of electrons and phonons [14].

Consider now the expression:

∑qωqAþq Aq ¼ ∑qωq a

þq aq þ ∑qMq aq þ aþ−q

� �∑kc

þkþqck

þ ∑qMq�� ��2ωq

∑kcþk−qck

� �∑k 0c

þk ;þqck ;

� � ð16Þ

where we have used the identityM*-q =Mq and rearranged thesummation order. Neglecting terms of order ns with respect toterms of order ns

2 (note, ns is of the order of Avogadro’snumber):

H ¼ μN þ ∑k εk−μð Þcþk ck þ ∑qωq Aþq Aq þ 1

2

� �

þ ∑q v qð Þ− Mq�� ��2ωq

!∑k;k ;c

þk−qc

þk ;þqck ;ck ð17Þ

where, again, the second term is considered a perturbation.Similarly,

ρ2 q; tð Þ ¼ ∑k;k 0cþk−q tð Þcþk 0þq tð Þck 0 tð Þck tð Þ ¼ ρ2 qð Þ; ð18Þ

ρ0 ¼ ∑k∈Ωcþk ck :

The Hamiltonian (17) is therefore exactly diagonalized un-der the standing wave condition (8), if we neglect the kineticperturbation,

H0 ¼ μN þ ∑qωq Aþq Aq þ 1

2

� �þ ∑q v qð Þ− Mq

�� ��2ωq

!ρ2 qð Þ:

ð19Þ

Using the arguments detailed after Eq. (15), we obtain:ρ2(q) = ρ2(q)I, and (19) follows self-consistently. Therefore, thestanding wave basis (10) diagonalizes the HamiltonianH0, if thedispersion is so low that it can be neglected to first order.

The definition (8) of a super-electron and the neglect(for the moment) of the kinetic energy of the electrons,made it possible to solve for H0 in a way similar to theproblem of a localized electron [14]. The difference is that

the super-electrons are itinerant, as in the normal state. It isonly the single-electron wave function that transforms to astanding wave function to allow for an electron-phononcorrelation in the form of Eq. (13). In contrast to theparticle-in-a box problem, these states are all degenerate.However, they are not required to be coherent. The elec-tronic eigenstates are the states represented by the wavefunctions (10) and (11). State (10) is symmetric relativeto the phonon shift (13). State (11) is antisymmetric rela-tive to the phonon shift. State (10) is therefore of lowerenergy.

We can now identify the observableρ2(q) as the order pa-rameter in the following way (other functions of ρ2(q) arepossible too),

ψj j2 ¼ ∑qρ2 qð Þ�� ��2; ð20Þ

with proper normalization.The order parameter can also be defined as the full pairing

potential:

∑q v qð Þ− Mq�� ��2ωq

!ρ2 qð Þ:

The Ginzburg-Landau (G-L) equation [11] can now be de-rived from the full Hamiltonian, and the G-L theory will bevalid in that case also at low temperature.

The pairing potential in the standing wave basis becomes

∑q v qð Þ− Mq�� ��2ωq

!ρ2 qð Þ•Ι :

We therefore obtained pairing correlation, not by imposingit but as a result of our assumption (8), and the canonicaltransformation (13). Note that the single-electron terms disap-peared from the Hamiltonian. They are included in a constantshift of the energy and as part of the renormalized phonons.Only pairing correlation amplitudes appear in the diagonal-ized standing wave Hamiltonian.

We are now ready to treat the kinetic perturbation,

H1 ¼ ∑k εk−μð Þcþk ck ; ð21Þ

with the unperturbed Hamiltonian:

H0 ¼ μN þ ∑qωq Aþq Aq þ 1

2

� �þ ∑q v qð Þ− Mq

�� ��2ωq

!ρ2 qð Þ:

ð22Þ

The unperturbed Hamiltonian H0 represents a fully degen-erate subspace of k-space,

Ω ¼ k ε kð Þ ¼ ηjf g

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Instead of the Fermi surface, this space should be called aFermi volume, as its dimension is one higher [see Fig. 1]. Asall of the k states in the Fermi volume subspace are degener-ate, we have to use degenerate perturbation expansion.Following the footsteps of BCS and the two-fluid model ofGorter and Casimir [18], we envisage a ground state of astanding waves condensate and a normal fluid of quasi-particle dispersive excitations.

In addition, we are facing other issues at the level of H0.In a normal metal, there is only one configuration at T = 0.All of the states are filled up to the Fermi level. Therefore,the entropy S = 0 at T = 0. This is not the case in a standingwave superconductor (see Fig. 1). While the subspace Ω iscompletely degenerate, minimizing the energy requiressome of the degenerate states to be empty (note that ρ2(q)is a sum over occupied and unoccupied states). Therefore,there are many different ways to arrange the ground state.While this is not in complete violation of the third law ofthermodynamics, as several ground states are possible insome systems, here we are dealing with an O (nln(n)) num-ber of degenerate standing wave states where n is of theorder of Avogadro number. As we will see immediately,the kinetic energy term H1 lifts that degeneracy, while con-serving the all-important condition (8). Here again we willuse the insight of BCS and apply partial occupation. (BCSassumption number 5). It is important to note that the par-tial occupation is a direct result of the standing wave as-sumption and the third law of thermodynamics.

Another issue is related to the flatness of the dispersion.The Fermi volume Ω is connected to a particle reservoir,which is the dispersive Fermi Sea. Therefore, the appropriatetreatment of the problem is in the grand canonical ensemble(BCS assumption number (6)).

The standing wave Bloch states form atomic-like or-bitals, where the local distribution amplitude (Eq. 10) isfixed in space. The low velocity bosons can thus cope with

the high velocity electrons in a way described by Eq. 13. Itis therefore energetically favored for another electron tooccupy the same standing wave state (atomic-like orbital)with opposite spin. This is the maximum occupationallowed by the Pauli exclusion principle. It is thereforeclear that the energy of a “relaxed” standing wave statecontaining two electrons is lower than a singly occupiedstanding wave state. In fact, a singly occupied standingwave state is an excited electron-hole state due to the localatomic-like nature of a standing wave state. This is theorigin of Cooper pairing. There is no “long range coher-ence” except for the Pauli principle, which is nonlocal, asis the case in all quantum systems.

A major difference between BCS pairing and the stand-ing wave model is the non-locality mentioned here. Whilein BCS pairing long-range order is assumed to maintainthe correlated pairing of two moving electrons, communi-cating at large distances, in the standing wave pairing, it isonly the Pauli principle that maintains this non-locality, asin any electron pair in atomic orbitals. Therefore, there isno additional assumption here, except for the usual non-locality related to the Pauli principle (as occurs in basicquantum mechanics.)

It is now clear that the pairing state is a singlet state.The two super-electrons retain their fermionic identity,they are not a boson, and therefore their total wave func-tion should remain antisymmetric. Therefore, the twoelectrons are in a singlet state as long as the atomic-likeorbital is symmetric (Eq. 10). Other antisymmetric config-urations are, of course, possible.

So far, we treated only the symmetric states (10). ThePhonon shift (13) actually splits the degenerate subspaceΩ into two subspaces where ΩS corresponds to the sym-metric states (10) and ΩA corresponds to the antisymmet-ric states (11); the symmetry is with respect to the phononshift (13). The split between ΩS and ΩA is determined by

Simple 2D metal 2D superconductor

Fermi

surface

Fermi

volume

Ω

Fig. 1 Left: isotropic 2D metal, the Fermi surface is a 1D circle. Right: isotropic 2D superconductor. The Fermi volume is a 2D ring

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the electron-boson couplingMqj j2ωq

and by the somewhat

reduced electron-electron repulsive potential v(q)A due tothe antisymmetric space correlation of the electrons.

Looking at the full Hamiltonian in the low-temperaturelimit (low density of quasi-particles) H =H0 + H1, we alreadysee the resemblance to the reduced BCS Hamiltonian. Afterwe have diagonalized the standing waves HamiltonianH0, weare left with the electronic residue:

He ¼ μNþ ∑k εk−μð Þcþk ck þ ∑q v qð Þ− Mq�� ��2ωq

!ρ2 qð Þ:ð23Þ

With |S⟩ designating the many body wave function of thesymmetric states (10) and |A⟩the many body wave functionof the antisymmetric states (11),

H0 Sj ¼ ES Sj iandH0 Aj i ¼ EA Aj i;

so that we can write:

H0 ¼ ES 00 EA

� �;

where H0 is block-diagonalized.Both states |S⟩ and |A⟩ are many body eigenstates of

the Hamiltonian with different energies ES and EA. Weknow that EA–ES > 0 because the antisymmetric statesare out of phase with respect to the phonon shift inEq. (13). Therefore, the states |A⟩ are orthogonal to thestates |S⟩. For the time being, we do not know theirdifference in energy except that it is greater than zero.Therefore, we can use simple perturbation theory evenin this highly degenerate case, by choosing the manybody eigenstates |S⟩ and |A⟩ as the basis for theHamiltonian, H0 and H1 as the perturbation.

By introducing the perturbationH1(ε(k)), we introduce dis-persive states as linear combinations of states (10) and (11).

ck ¼ ukjSk > þvkjAk > :

The operators cþk are now the original electron creationoperators of Eq. (12). We can use them now because thereare no single-standing wave states in H0. They always mixone |Sk > state with one |Ak > state, but never mix states ofthe same degenerate subspace.

We can define 2Δ = EA − ESas an energy gap yet to bedetermined.

A scenario where v(q) is larger thanMqj j2ωq

could happen, in

which case the order of the bands could reverse.Using two-level simple perturbation theory, we get for every k:

Ek1;2 ¼ ESþEA

2 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2 þ ξ2k

q, very reminiscent of the BCS

solution, assuming of course that the perturbation ξk is verysmall with respect to EA − ES.

The energy gap Δ should be determined self-consistently.We know that the ground state |S⟩with paired standing wavestates of energy EScannot contain a definite number of pairs inviolation of the third law of thermodynamics. Here comes

finally the BCS solution of partial occupation, and Δ ¼EA−ES

2 is determined self-consistently.We are left with the residual Hamiltonian:

He ¼ μN þ ∑k εk−μð Þcþk ck þ ∑q v qð Þ− Mq�� ��2ωq

!ρ2 qð Þ:ð23Þ

We have neglected here the anticipated quasi-particle inter-actions with the phonons and among themselves. FollowingBCS, we assume that these interactions are the same as in thenormal state.

It is important to note that we obtained a formally BCS-likeHamiltonian without assuming any pairing. Of course, onecan argue that the pairing assumption is equivalent to theassumption (8). As we have shown, however, other BCS as-sumptions are derived from (8). The advantage of the standingwave definition (8) will become clear in examining the elec-trodynamics and the chemistry of the materials.

Following BCS, we now minimize:

Ψh jHe−μ N̂−∑q λqρ2 qð Þ Ψj i ¼ ∑k2ξk vkj j2

þ ∑q v qð Þ− Mq�� ��2ωq

−λq

!ρ2 q; vk ; uk ; vk 0 ; uk 0ð Þh i;

ð24Þ

where Ψ ¼ ∏k uk þ vkcþk↑cþk↓

� �0j i and |0⟩ is the Fermi sea,

under the constraints:

∂∂t

ρ2 qð Þ ¼ 0; ð25Þ

and

Ψh jN Ψj i ¼ Nh iaverageλq here is a Lagrange multiplier, related to constraint (25).Since we have established pairing, we can now use the

reduced form of the interaction:

ρ2; q; vk ; uk ; vk 0 ; uk 0ð Þh i ¼ ∑k;k 0 ; cþk 0↓; tð Þ; cþk 0↑; tð Þ; ck↓; tð Þ; ck↑; tð Þ

D E¼ ∑k;k 0u

*kv

*k 0uk 0vk � k 0 ¼ k þ qð Þ

The pairing is, of course, of standing wave states, where allthe atomic-like orbitals are doubly occupied with fractionaloccupation of pairs.

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Similar to BCS, we obtain the solution:

Δk ¼ −1

2∑k 0Vkk0

Δk 0

Ek 0; ð26Þ

Ek ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiξ2k þΔ2

k

q; ð27Þ

with

Vkk ; ¼ Vk−k ; ¼ Vq ¼ v qð Þ− Mq�� ��2ωq

−λq: ð28Þ

The lowest energy excitations will create a standing waveparticle hole pair with zero dispersion. Any higher excitation en-ergy will create a moving particle combining states (10) and (11).

λq can therefore be called the rigidity potential.The interpretation of the BCS solution within the standing

wave model is therefore as follows: The ground state corre-sponds to partial occupation of dispersion-less standing wavestates. The dispersion is carried by the quasi-particle excita-tions evaporating from the condensate. The standing wavedispersion-less states are essential in order to benefit fromthe phonon (or any other boson) relaxation (Eq. 19). Withincreasing temperature, the evaporated dispersing quasi-particles deplete the condensate, and the energy gain relatedto the boson relaxation is diminished.

Since λq �→Ωj j→0

0, low Tc materials can generally be de-

scribed as:

Δk S:Wð Þ �→Ωj j→0

Δk BCSð Þ;

where S.W stands for Standing Waves.The Josephson effect can now be understood as two Fermi

seas (particle reservoirs) connected by their flat dispersionregions Ω (“Fermi ponds”). As in a semiconductor junction,the weak link between the two particle reservoirs brings themto equilibrium. In a semiconductor junction, it regulates thejunction voltage. In a Josephson junction, it regulates thephase difference by controlling the chemical potential andtherefore the occupation of the Fermi “ponds” at each sideof the junction. Similarly, in a d-wave superconductor, whenthe material undergoes a transition to the superconductingstate, the Fermi ponds “poison” the Fermi surface by partialoccupation, and the condensation includes the nodes as well.A similar thing happens in the proximity effect: The Fermisurface of the normal metal is connected with the Fermi pondsof the superconductor and therefore is “poisoned” by partialoccupation.

An important difference between the SW model and BCSis the relevant energy scale. While BCS consider the Debyefrequency ωD to be the relevant energy scale, in SW it is Δ,the superconducting gap. Therefore, in contrast to BCS, thefine structure of the energy bands at the Fermi level is all

important in SW. In SW, the isotope effect is strong for lowTc (more dispersive) materials and negligible for high Tc ma-terials, where the dispersion at the Fermi level is very low. Inthat respect, a phonon mechanism is not ruled out for high Tcmaterials.

The doping in high Tc materials is exactly the gradualfilling of the “Fermi volume” Ω. The entropy of the par-tially filled “Fermi volume” goes as x(1 − x) if x is thefractional filling of that volume. Therefore, the dopingdependence of Tc is parabolic, with a maximum at halffilling of Ω. We claim that this is a general property ofsuperconductors, not observed in low Tc materials due tothe very limited size of the “Fermi volume” and an inef-ficient doping mechanism.

It is important to note that BCS, while not providing asolution of the full Hamiltonian, captures the essence ofthe thermodynamics of superconductors. Therefore, thedrive for high density of states at the Fermi level is al-ready present in the BCS gap equation and the approxi-mation for the critical temperature. This led to a search fornew materials and understanding of HTS within the “VanHove Scenario” with limited success [22–28]. The VanHove Scenario got experimental support with the discov-ery of the “extended Van Hove singularity” (EVHS) in allmeasured HTS materials [29, 30]. This includes Bi2201,Bi2212, Bi2223, Y123, Y124, LSCO, and electron-dopedHTS. In the standing wave theory concept, the normalquasi-particle states are perturbed standing wave states.This is a cornerstone of the theory. Therefore, thesuperconducting transition is a transformation of the bandstructure from shallow dispersion in the normal state to aflat dispersion in the superconducting state. So far, allHTS compounds that were measured by photoemissionshow this property of an extended region of shallow dis-persion in the normal state in the proximity of the Fermilevel [29, 30].

The high Tc of these materials can be traced back to theextension of this shallow dispersion region as discussed below.

Recently, an EVHSwas discovered in twisted graphene bilay-ers [36–38] The topology of the Fermi surface and its proximitywas found byARPESmeasurements to be similar toBi2201 [36].The critical temperature of the best intercalated graphene super-conductor (12 K) and Bi2201 (12 K–24 K) are similar.

EVHS was also discovered in Sr2RuO4 [39]. It remains achallenge to explain the low Tc of this material (below 1 K).Perhaps the complex topology of the Fermi surface of thismaterial [40] figures in the explanation.

4 Electrodynamics

The most important difference between the standing wavemodel and the BCS treatment is the derivation of the

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electrodynamic properties. The original BCS treatment [2, 3]calculates the conductivity of the superconductor from whichthe London rigidity should be derived. P.W. Anderson andothers [13] criticized that treatment. Gorkov and others [11]made of the connection between the BCS energy gap and theGinzburg-Landau order parameter. This is an elegant solution,but it is phenomenological. In that respect, we still lack amicroscopic theory of zero resistance and the Meissner effect.

It is clear that the standing wave theory produces a newoperator:

ρ2 q; tð Þ ¼ ∑k;k 0cþk−q tð Þcþk 0þq tð Þck 0 tð Þck tð Þ ¼ ρ2 qð Þ;

that is a constant of the motion. Therefore[H, ρ2(q, t)] = 0, and

the operator ρ̂2 qð Þ can be diagonalized simultaneously with

the Hamiltonian. It is also clear that ρ̂2 qð Þ is not dependent onk. Therefore ρ̂2 qð Þ ¼ ρ2 qð ÞI , where I is the identity matrix,

and we have used the same notation for the operator ρ̂2 qð Þ andthe eigenvalue ρ2(q) as in x̂Ψ ¼ xΨ for the position operator.The set of eigenvalues ρ2(q) therefore generate a wave func-tion through the relation:

|ψ|2 = |∑qρ2(q)|2, with proper normalization.

This macroscopic wave function can be defined as the or-der parameter of the system, in analogy with the Ginzburg-Landau order parameter.

Thus, we can treat the macroscopic spatial variation of thesuper-electron density using the G-L theory-derived micro-scopically. We will not proceed along this route but rathershow how the London relation is by itself a microscopic rela-tion between standing wave electrons and the vector potential,without requiring the rigidity of the many-body wave function[15].

Let us look again at the single standing wave electron func-tion (10). The pairing is not a necessary condition to derive theLondon relation, but, as we have seen, pairs are favored ener-getically, so we will look at the single pair wave function. Thiswill preserve the 2e charge observed experimentally.

To be precise, we are looking at a single superconductingstanding wave state at T = 0. The pair wave function is given as,

ψ r1; r2; tð Þ ¼ Cφ r1; tð Þφ r2; tð Þ ↑↓j i; ð30Þwhere C is any complex constant, |↑↓⟩ denotes a singlet state, andϕ(r, t) is a standing wave function given by Eq. (10). The spatialpart of ϕ(r, t) can be set to be a real function with respect to avector potential in the London gauge:

∇ � A ¼ 0; A⊥ ¼ 0 at the surface of an isolated body(31).

Then the single-pair probability current will be:

J r; tð Þ ¼ Re ψ* ℏim

∇−qmc

A� �

ψ

� : ð32Þ

But

ψ* ℏim

∇ψ ¼ CC* ℏim

φ r1; tð Þφ r2; tð Þ ∇ 1φ r1; tð Þφ r2; tð Þ þ φ r1; tð Þ∇ 2φ r2; tð Þ½ �;

ð33Þso that

Re ψ* ℏim

∇ψ

� ¼ 0;

and

J r; tð Þ ¼ −qmc

A ψj j2: ð34Þ

Thus, the London relation appears as a single-particle mi-croscopic property. It is a linear relation between the probabil-ity current of each superconducting pair and the vector poten-tial. Since all the pairs individually obey this relation, the totalcurrent is proportional to the vector potential, giving the mac-roscopic London relation:

JTOTAL r; tð Þ ¼ −qmc

∫drK r−r0ð ÞA r0ð Þ; ð35Þ

where the kernel K(r-r’) expresses the collection of these mi-croscopic currents into the macroscopic current JTOTAL,

J total rð Þ ¼ −e2

mc∫B r;ξ0ð Þ dr0A r0ð ÞK r−r0ð Þ ð36Þ

Where B(r, ξ0) is a sphere around r with radius ξ0 and

K r−r0ð Þ ¼ ∑k∈Ω

cos2 k � r−r0ð Þð Þ ð37Þ

Again, note that the derivation of the London relationdid not require any assumption about macroscopic coher-ence of any kind. The Pippard integral now appears as asummation over standing wave states, a summation car-ried out over the single-electron probability currents togive the total current. It is emphasized again that thepairing is not necessary to derive the London relation.The same treatment over a single super-electron state ofthe type (10) would yield the London relation.

Any other potential should be related to the London poten-tial by a gauge transformation,

A; ¼ Aþ ∇χ; ð38Þwhere χ(r) is any scalar function. The corresponding transfor-mation of the wave function is then

ψ; ¼ ψeiqℏcχ rð Þ; ð39Þ

and

J r; tð Þ ¼ Re ψ;* r; tð Þ ℏim

∇−qmc

A; r; tð Þ� �

ψ; r; tð Þ�

¼ qmc

∇χ rð Þ ψ;j j2− qmc

∇χ rð Þ ψ;j j2− qmc

A r; tð Þ ψ;j j2ð40Þ

so that again

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J r; tð Þ ¼ − qmcA r; tð Þ ψj j2 [1].

It is suggested here that the rigidity of the many-body wavefunction, which is maintained by the energy gap in the BCSdescription, can be replaced by a single-body relation that isthe property of any real wave function in the London gauge.Thus, the long-range coherence, explained as the phase rigid-ity of the order parameter in BCS, can now be understood asmerely reflecting this single-electron behavior.

What is then the symmetry that is broken in a system ofstanding wave functions? The periodic boundary conditions,which allow for arbitrary phase of the wave function, are bro-ken, driven from the bulk, by the relaxation of the phononcloud (or other bosons, for that matter) against a standingelectronic wave function. This is the proposed physical under-standing of gauge symmetry breaking in the case of asuperconductor.

Alternatively, it can be said that the temperature of the solidis low enough to “feel” the boundaries. The grain boundariesof the material, impurities, and defects are supplying the “sup-port” for maintaining the standing wave function. The bound-ary conditions determine the shape of the wave function. For asuperconductor to be formed, the London relation has to existon the surface. That excludes the antisymmetric wave func-tions (11).

An important observation, due to H. Thomas [17], is thatthe single electronic current in the standing wave model is notdivergence-free:

∇ � J r; tð Þ ¼ −qmc

A r; tð Þ � ∇ ψ r; tð Þj j2 ð41Þ

This is in accord with a BCS-like ground state where stand-ing wave pair states are constantly being created anddestroyed. Again, J(r, t) is the probability current of a singlepair.

For the total current, we would have

∇ � J total þ dρdt

¼ 0: ð42Þ

We now apply again our single assumption of zero disper-sion,

dρdt

¼ ddt

∑qρq ¼ddt

∑k;qcþkþq tð Þck tð Þ

¼ ddt

∑k;qcþkþqe

−iℏμtcke

iℏμt ¼ 0; ð43Þ

so that the total current is divergence-free.Consider again Eq. (41). For a superconductor, we would

like the surface current to be divergence-free. Therefore, wewould require ∇|ψ(r, t)| = 0 at the surface of the superconduc-tor (in the London gauge). That means that we must choosethe standing wave sum function (10) and not the differencefunction (11) for a superconductor.

It is helpful to look at the problem quasi-classically. In thequasi-classical picture, a standing wave is a wave-packet withgroup velocity vg= 0. The magnetic force on such a particle isF = vg ×B = 0. Therefore, the semi-classical standing wave par-ticle is unaffected by magnetic fields as long as it is not evapo-rated from the condensate. This is the rigidity that was aimed atby BCS [2, 3]. It is only the vector potential that affects the phaseof the single-electron wave function, as in the Aharonov-Bohmeffect. The super current is therefore a current of all the super-electrons, not a collective effect due to coherence of the many-body wave function but simply a phase current. It therefore doesnot interfere with the relaxation (19), being a uniform current forall of the super-electrons. The Pippard integral comes from therelation between the vector potential and the macroscopic cur-rent. In order to obtain the macroscopic current, one must inte-grate over all the standing wave k-states. This affects real spaceintegration over a region on the order of the coherence length asexpressed in Eq. (36) above.

The question arises of what is the difference between oursuperconductor and a charge density (spin density) insulator.The answer is simple: Inasmuch as a superconductor is a per-fect diamagnet, it is also a “perfect insulator.” The magneticfield in the superconductor appears only on the surface anddecays exponentially. Thus, for a plain boundary located atx = 0,

B xð Þ ¼ B 0ð Þe− xλ ; ð42Þ

where λ is the London penetration depth.So is the electrical current,

J xð Þ ¼ J 0ð Þe− xλ : ð43Þ

Here is the justification for the title of our paper – a super-conductor is therefore a “perfect insulator,” but not a dielectricinsulator. It is a conductor condensed into standing wavestates in order to benefit from the relaxation of low-level ex-citations. It allows for current to flow without resistance alongits boundary while keeping the interior of the material free ofcurrents and fields. A superconductor and a charge densitywave insulator (or a spin density wave one for that matter)are therefore different phases that compete for the same inter-action with different relaxation modes.

5 Chemistry

Now, that we have the elements of a solution to the total many-body Hamiltonian, we can begin to understand the chemistryrequired for superconductivity. AWilson-type rule for SWwouldstate the following: An insulator obtains where the Fermi level isin the gap and a conductor where the Fermi level cuts the energyband. A superconductor occurs where the Fermi level is “tan-gent” to the energy band. This is not a mathematical tangent in

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the normal state. However, the dispersion should be so low,below the superconducting gap, in order to be considered as aperturbation. The transition to the superconducting state is thecomplete flattening of the band at the superconducting chemicalpotential, creating the Fermi volume Ω.

The key to increasing Tc is therefore to increase the Fermivo lume Ω = {k |ε (k ) = η} , and wi th i t the fac to rρ2 q; tð Þ ¼ ∑k;k 0c

þk−q tð Þcþk 0þq tð Þck 0 tð Þck tð Þ ¼ ρ2 qð Þ.

This can be done by controlling the band dispersion near theFermi level. Using quantum chemistry, we have identified in thecuprates, in addition to the dispersive Cu-Oσ bond (the referenceis to the band arising from this bond), a low dispersion Oπ bond[16]. If this bond is responsible for the low dispersion bandobserved in the cuprates near the M point, we could control itand the resulting Fermi volume by controlling its chemistry.Materials so designed can show superconductivity at tempera-tures several times higher than observed so far. Experimentalsteps in this direction will soon be published.

6 Summary

The standing wave model provides a new direction for theunderstanding of superconductivity and for the design ofnew superconductors.

The electrodynamics of superconductors, their zero DCresistivity, and the Meissner effect can be understood as aresult of a microscopic London relation between the vectorpotential and the standing wave super-electrons.

The analysis can only be correct if the normal state could bedescribed as a perturbed standing wave state. The BCS solu-tion appears as a perturbation on the standing wave solution.Therefore the dispersion of at least one band in the normalstate should be extremely low. The transition to thesuperconducting state is the transition from a nearly flat bandin the normal state to a completely flat band in thesuperconducting state. Other bands in the proximity of theFermi surface will follow through by means of poisoning theFermi surface with partial occupation. Their contribution to Tcwill be negligible if their dispersion is high.

The extension of this low dispersion region over theBrillion zone determines the strength of the interactionthrough the term (18)

That is, the number of k-states in the sum (18) determinesTc if the other parameters are varying slowly.

This number is expressed through the term Ω.

Therefore it is the extension of the shallow regions inthe normal state that determines Tc. A detailed calculationfor LSCO with comparison to experimental data will bepublished soon.

Another condition is high dispersion of at least one anotherband crossing the Fermi surface to allow for the screening ofthe electron-electron repulsion expressed in v(q). [21]

The BCS theory indicates high density of states at theFermi level as one of the parameters enabling high criticaltemperature. This understanding triggered a search for flatdispersion at the Fermi energy, what is known as the “VanHove Scenario”. [25–28, 32–35].

An important difference between the van Hove Scenarioand the standing wave model is the following: While super-conductivity is in general a many-body property, the standingwave model brings about a single particle property that is thebasis for the whole phenomenon. This single particle propertyis the one allowing for design of new superconductors.

Acknowledgements Avi Schiller is remembered warmly for intensivediscussions and critical review of the formalism. Roger Kornberg is great-ly appreciated for a critical review of the manuscript and very instructiveremarks. Roald Hoffmann is greatly appreciated for a critical review ofthe manuscript. Robert Loughlin is acknowledged for stimulatingdiscussions.

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