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(PACS: 74.20.Mn 74.25.F-) (Keywords: mechanism of electron pairing between E(k) faces from different Brillouin zones with definite binding energy in crystals)

A mechanism of electron pairing relating to superconductivity

Author: Q. LI Affiliation: JHLF

Date: 29 March 2010

Abstract

Mechanism of superconductivity is based on a mechanism of electron pairing near EF with definite binding energy. A candidate mechanism of such electron pairing is described in this paper. Electron pairs are induced by EM wave modes generated by corresponding lattice wave modes. The pairs are formed between E(k) faces from different Brillouin zones, with definite binding energies. The binding energy of an electron pair is characterized by the frequency of the EM mode that induces the pairing. Introduction

Mechanism of superconductivity is based on a mechanism of electron pairing near EF with a definite binding energy. A candidate mechanism of such electron pairing is described in this paper. A new mechanism of electron pairing

Lattice wave modes exist in crystals [1]. These lattice wave modes generate oscillating potentials, which in turn generate electromagnetic (EM) wave modes [2]. The EM wave modes would drive an electron in a crystal to perform stimulated transition if an unoccupied target energy state is available, in which the electron emits/absorbs a photon to transit to the target state.

If, however, the target state has already been occupied (such as in the situation shown in Fig. 1), a virtual stimulated transition “would” occur, in which the electron at the initial state exchanges its state with the electron at the target state, with the electron originally at the higher energy state emitting a photon of the threshold frequency, which photon would be directly absorbed by the other electron. The photon emission/absorption is virtual in that they do not result in photon exchange with the EM wave mode that induces them, and in that they are confined between the two electrons concerned. Such an exchange of states between two electrons by virtual photon emission/absorption is an “electron pairing by virtual stimulated transitions”.

Now we consider the binding energy of such an electron pair. None of the two electrons under such pairing is in a stationary state. So what will be the measured energy of anyone of them?

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The author would suggest that both electrons in the pair be bound by the photon to the ground energy state of the pair (that is, the two electrons might be said to “condense” on the ground state), so when one of the electrons is knocked out of pair the photon do not go with the exiting electron. This understanding could be in line with some of the basic experimental results [3] [6], as will be further explained later. With this, when no energy state lower than the upper energy level in the pair is available for transition by electrons in the pair, the pair will have a binding energy no smaller than the energy of the photon.

For such a virtual emission/absorption of photon happening in a crystal, it would be reasonable to anticipate that they follow the same wavevector selection rule as real emission/absorption of photon in crystal [4] [5], so that only nearly vertical transitions between E(k) faces from different Brillouin zones be allowed.

FIG. 1: Vertical pairing of electrons under stimulated transitions. Two electrons, at energy levels of E1 and E2 respectively, perform stimulated transitions under the stimulation of electromagnetic wave mode of frequency hν=E2-E1 by exchanging their states as well as a “binding photon” of the same frequency ν, thus resulting in pairing of the two electrons. Photon emissions/absorptions associated with the stimulated transitions are virtual (not real). The binding energy of the pair, in this particular system of levels E1, E2 and E3 as shown, is hν+(E3-E2).

electron E2

E1

EM wave mode hν=E2- E1

Electron 1

Electron pairing by virtual stimulated transitions

E3

Electron 2

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In so far as high-Tc superconductivity is concerned, the “upper band” and

“lower band”, as shown above and below the band gap respectively in Fig. 2, may not necessarily be a conducting band and a valence band respectively. The lower-upper band combination could also be one of:

- the lower band being a valence band, the upper band being an acceptor band, with a donor band being formed above the acceptor band, where the donor band has fewer energy levels than the acceptor band, as shown in Fig. 3;

- the lower band being combined acceptor band/valence band, the upper band being a donor band, where the donor band has fewer energy levels than the acceptor band;

- the lower band being an acceptor band, the upper band being a donor band formed separately or in connection to the conduction band, where the donor band has more energy levels than the acceptor band;

- the lower band being the valence band, the upper band being combined acceptor band/donor band, with the donor band having fewer energy levels than the acceptor band;

Lower band E-(k)

Pairing with ΔE=hνm≤hνM

Upper band E+(k) E= E(k)

k

Band gap

Gap width

Peak width

π/a

FIG. 2: Schematic illustration of (nearly) vertical pairing between electrons on E(k) faces from different Brillouin zones. Each of the dashed lines with double arrows indicates the two electrons in a pair exchanging their states as well as their binding photon of an energy ΔE=hνm, which is the lower limit of the binding energy of the pair. The virtual stimulated transitions, and the associated electron pairings, occur between all electrons on the two bands for which ΔE≤hνM can hold and wavevectors and energy can be conserved (with νM being the highest frequency of lattice wave modes in the crystal). The upper and lower bands need not be conduction and valence bands, respectively. “Band gap” could be small or even zero while the reduced E+(k) and E-(k) curves might intercept, as the scenario of superconductivity in metals might be.

EF

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- and etc. The donor band and/or acceptor band may include, or be formed (partly) of, a

deep energy level system.

: :

: :

Discussion and conclusions

The argument that both electrons in the pair are bound by the photon in the ground state of the pair could be in line with the results of some basic experiments [6] [3] on superconducting gap and pseudogap, where, with the present pairing mechanism, the peak just below EF could be understood as double-accounting of both upper and lower electrons in the pairs at the energy positions of the lower electrons.

More specifically, in the current mechanism “pseudogap” or “SC gap” are understood as “gap width” as illustrated in Fig. 2, while “peak width” is understood as is shown in Fig. 2. The width of a pseudogap does not characterize the binding energy of an electron pair, which is characterized by the frequency of the EM mode that induces the formation of the pair. Therefore, the distance between the gap and peak generally characterizes the binding energy of the pairs involved, and a peak just below EF may be a more reliable evidence of electron pairing.

Notably, it might not be a coincidence that the dip and the SC/pseudo gap as reported in some experiments [3] [7] are positioned substantially symmetrical with

Pairing for donor type superconductivity

Acceptor band

Electron

ΔE≤hνM

Donor band

Valence band

Conduction band

Pairing for acceptor type superconductivity

ΔE≤hνM

FIG. 3: Schematic illustration of a possible exemplary model of band structure in a high-Tc superconductor. The system may start with certain acceptors and donors, which neutralize with each other. With further acceptor doping, the acceptor band becomes partly full and can be conductive, so superconducting pairs could be formed across the acceptor and valence bands. On the other hand, if further donors are added, superconducting pairs could be formed across the donor and acceptor bands.

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respect to the peak. According to the present mechanism, when an electron pair near EF is destroyed by such as an incident photon in ARPES, the binding photon would not go with the exiting electron; rather, it would be emitted by the remaining electron under the stimulation of the EM wave mode that induced the original pairing and then be absorbed by the EM wave mode through resonant absorption; as the EM wave mode is directly coupled to its lattice wave mode, the photon absorbed is directly converted to an added phonon of the lattice wave mode. This phonon could then be absorbed by electrons on energy levels at the dip for transition to the energy levels at the peak, which are left by the knocked out electrons. It is to be noted that this phonon could hardly interact with electrons already in pairs because of the binding energy as described in the present mechanism. Thus, the origin of the dip could be interpreted as depletion of electrons due to their interaction with phonons produced from the released binding electrons, which would not go with the electrons knocked out of their original pairs; and it may have been evidenced that such depletion further enhanced the height of the peak [8].

“Band gap” as shown in Fig. 2 could be small or even zero while the reduced E+(k) and E-(k) curves might intercept, and the “Gap width” could also be small, as might be in the scenarios of metal superconductivity.

[1] Kittel Charles Introduction To Solid State Physics 8Th Edition, pages 95-99 and

Figs, 7, 8(a) and 8(b). [2] “Solid State Physics”, by Prof. HUANG Kun, published (in Chinese) by People’s

Education Publication House, with a Unified Book Number of 13012.0220, a publication date of June 1966, and a date of first print of January 1979, page 201-205.

[3] Phys. Rev. Lett. 82, 2179 (1999): Fedorov et al. Temperature Dependent Photoemission Studies of Optimally Doped Bi2Sr2CaCu2O8.

[4] page 205, Equ. 7-93 of [2]. [5] page 189, Figs. 4 and 5 of [1]. [6] T. Timusk, B.W. Statt, http://arxiv.org/abs/cond-mat/9905219v1. [7] Figs. 6, 16, 17, 18 of [6]. [8] Fig. 1(b) of [3].