trying to understand Cooper Pairs ?

[Widdekind]

I understand, that a "Cooper pair" is a composite, bosonic, state, of two electrons. Vaguely, the electrons have "opposite spins", so that the CP is S=0; and, the electrons have "opposite momenta", so that the CP is p=0 (in an isolated, non-conducting, ground state, i.e. the CP BEC "just sitting there" at rest). Yet, the electrons are "entangled", so that they really represent only one single conjoined, co-mingled, composite, combined, quantum state.

 

Is the wave-function, of a CP, at rest, representable by the following wave function, indexed by the individual electron momenta k; and individual electron spin s=1/2:

 

[math]\Psi_{CP, k} \approx |+k +s\rangle_1 |-k -s \rangle_2 \; + \; |+k -s\rangle_1 |-k +s \rangle_2 \; + \; |-k +s\rangle_1 |+k -s \rangle_2 \; + \; |-k -s\rangle_1 |+k +s \rangle_2[/math]

i.e. a "mixed up mixture" of product-states, each of which individually represents a pair of electrons having opposite linear & angular momenta; but whose combination represents a bosonic CP in a "stationary standing wave" state ? In particular, if a CP were to exit a super-conductor, and enter into a conventional resistive-and-dissipative conductor, so that the wave-function of the CP were to "collapse" (after a propagating through a characteristic "coherence length"); then would the constituent electrons "emerge" into a product state, i.e. each electron defined by oppositely directed linear & angular momenta ?

 

 

ADDENDUM:

 

I understand, that the constituent electrons, still obey the FD statistics, of the "Fermi sea", of individual electron states, in the (SC) material. Does that mean, that the bosonic "basis states", of CPs, must be indexed by their individual electron momentum [math]\pm\vec{k}[/math] values; and that those momentum basis states range, in momenta (i.e. momentum phase space), from [math]0 \le \frac{\hbar^2 k^2}{2 m_e} \le E_F[/math] ?

 

And, if "entangled" states arise, from "intimate" interactions; then CPs emerge, from "oppositely moving & spinning" electrons, in the (SC) material, "meeting mingling & merging" ?? And, CPs, arising from momentum eigenstates, represent spatially de-localized electron pairs ??

 

CPs apparently form, from non-valence f,d orbital electrons:

 

The compounds they’ve studied are made out of elements in the actinide series, including neptunium and plutonium. In these materials, active electrons are in “f-orbitals.” In contrast, materials that make up today’s highest-temperature superconductors, including copper or iron, have active electrons in “d-orbitals.” The f-electron materials generally have lower superconducting temperatures than their d-electron counterparts; but they are easier to make and may be easier to understand

(SD 2008)

Is that because valence s orbital electrons, having energy states into which they can easily scatter, easily do so scatter ?? Could such "near-Fermi-energy" electrons be "suppressed", e.g. by forcing additional electrons into the material, to induce a net negative charge ??

 

 

SC "energy gap" emerges from sub-valence electron de-localization, into lower energy bonding orbitals ??

 

As potential super-conductors are cooled, to their threshold temperatures, an energy gap emerges, near the Fermi energy. Lower lying "sub-valence" momentum states, below the "surface" of the "Fermi sea", decrease in energy. Is that an indication, of the de-localization, of tightly-bound, sub-valence, f/d orbital electrons, from "highly localized" states, into spread-out "bonding orbitals", approximated by the in-phase superposition, of the isolated atomic orbitals, associated with each & every lattice site ?

 

One hallmark of a superconductor is a so-called "energy gap" that appears when the material transitions into its superconducting phase. The gap in electron energies arises when electrons pair off at a lower energy to do the actual job of superconducting electric current.

 

When most of these materials warm to the point that they can no longer superconduct, the electron pairs split up, the electrons start to regain their previous energies, and the gap closes. But in the cuprates, the gap persists even above superconducting temperatures. This is the pseudogap, and it doesn't fully disappear until a second critical temperature called T* (pronounced "T-star") is reached. T* can be 100 degrees higher than the temperature at which superconductivity begins...

 

The electrons in the pseudogap state aren't superconducting... electrons in the pseudogap phase are not pairing up. They reorganize into a distinct yet elusive order of their own. In fact, the new order is also present when the material is superconducting; it had been overlooked before, masked by the behavior of superconducting electron pairs (SD 2011)

If "interior", strongly localized electrons, "tunneled away" from their native nuclei, into de-localized bonding orbitals; then would they not "drop down" in energy, into a lower energy configuration, thereby "opening up" a gap / pseudogap ?

 

the pseudogap is thought to reflect the competition between superconductivity, and another condition of the material - some other "ground state" (SD 2008)

Indeed, the "pseudo-gap" phase is associated with tunneling:

 

the "pseudogap" phase...is non-superconducting, and is observed at a temperature above the superconducting phase

 

Across the entire copper-oxide crystal, the scientists found a remarkable difference in the electronic states associated with the mysterious pseudogap phase: The number of electrons able to "tunnel" to the microscope tip differed depending on the position of the oxygen atom relative to the copper atom. "Picture the copper atom at the center of the unit, with one oxygen to the 'north' and one to the 'east,' and this whole unit repeating itself over and over across the copper-oxide layer," Davis said. "In every single copper-oxide unit, the tunneling ability of electrons from the northern oxygen atom was different from that of the eastern oxygen."

 

The discovery of this asymmetrical behavior could be a breakthrough in understanding and controlling high-temperature superconductors... The scientists will pursue...how the directional asymmetry in electronic behavior affects the ability of electrons to flow through the system (SD 2010).

scientists had previously observed a second gap, or pseudogap, in some high-Tc materials, well above the transition temperature... electrons are forming pairs above the transition temperature, before the material becomes a superconductor... The pairing occurs only along certain directions in the crystalline lattice of atoms making up the material — only along the directions in which copper atoms are bonded with oxygen atoms. Together, the existence of preformed electron pairs and their directional dependence should help clarify the picture of high-Tc superconductivity (SD 2008).

Now, I understand, that the electrons, in CPs, are "entangled"; and, that for once-separate wave-functions to become "entangled", they must first interact "intimately". If so, then electron de-localization, from "deep interior" f/d orbitals, to "spread out" bonding orbitals, must precede interactions, with other similarly-spreading electrons ("meeting"); must precede intimate interactions ("mingling"); must precede CP formation ("marrying"). So, could the pseudo-gap phase represent de-localized individual electrons, not yet paired up, into CPs ?? And, the energy "benefit", from forming de-localized bonding orbitals, increases with the overlap integral, between localized orbitals, on neighboring nuclei. So, could the operating temperature regime, of SCs, be increased, by increasing the operating pressure regime, i.e. compress the material, decrease the lattice spacing, increase the overlap integral, and energetically "incentivize" delocalization into bonding orbitals, within which electrons would "meet their mates", and then form CPs ??

 

Indeed, the pseudo-gap phase competes ("cheaper singles apartments"), against the gap phase, for electrons ("cheaper doubles apartments"):

 

the pseudogap doesn't give way to superconductivity, but persists and co-exists – possibly even competes – with it... the pseudogap doesn't just precede superconductivity. It continues to co-exist once the material superconducts... (the pseudogap) in the superconductor state. It co-exists along with superconductivity and we think it is competing with superconductivity... The notion of competition implies that the pseudogap's mere presence draws electrons away from the superconductor gap (SD 2009).

And, CP formation involves lattice distortions, i.e. phonons (SD 2007); which lattice distortions dramatically influence tunneling (SD 2010).

Edited January 30, 2012 by Widdekind

[Widdekind]

And, CP formation involves lattice distortions, i.e. phonons (SD 2007); which lattice distortions dramatically influence tunneling (SD 2010). Altering lattice spacing influences SC, e.g. by doping (SD 2010); and by applied laser excitations (SD 2011). Applied pressure can be considered a "static phonon loading"; and, iron-based high-temperature SCs can be "pressured into" SC states:

 

Under moderate pressure, the volume of the material's crystal structure compressed an unusually high 5 percent. Intriguingly, it also became superconductive (SD 2008)

as can C-60 molecules:

 

At room pressure the electrons in the material were too far apart to super-conduct, and so we 'squeezed' them together using equipment that increases the pressure inside the structure. We found that the change in the material was instantaneous – altering from a non-conductor to a superconductor (SD 2009)

Indeed, pressure is commonly applied, to "tune" materials into SC states (SD 2011, cp. SD 2008). Germanium hydride becomes SC at 2Mbar (SD 2008). And, in manganite

 

Pressure has a unique ability to tune the electron interactions in a clean and theoretically transparent manner... It is a direct and effective means for manipulating the behavior of electrons (SD 2009).

 

 

Electron localization impedes SC (SD 2011); Mott localization prevents SC, and can be effected, by increasing the lattice spacing, affecting overlap integrals (SD 2010). Lattice spacing, influenced by atomic magnetic moments, influences SC (SD 2009).

The pseudogap phase exhibits properties similar to those of the gap-phase, suggesting similarity of electronic wave-functions, i.e. individual/paired de-localized bonding states ??

 

scattering by impurities occurs when a material is in the pseudogap state as well as the superconducting state. That finding challenges the theory that the pseudogap is only a precursor state to the superconductive state, and offers evidence that the two states may coexist (SD 2008)

Tunneling SC (SD 2011, SD 2009). CPs usually form in stationary state, i.e. k=0 (SD 2008). Confining electrons, into 2D planes, may enhance CP formation (SD 2008).

[Widdekind]

To observe quantum tunneling in any system, it must be very cold: the temperature

must be close to absolute zero. In fact, when the temperature is low enough, quantum tunneling alone causes random switching [e.g. between lattice sites] (Moss & Wiesenfeld, Benefits of Background Noise (Sci-Am Aug. 1995)).

Aromaticity is an example of electron charge delocalization, into lower-energy bonding orbitals.

Edited January 31, 2012 by Widdekind