Explaining Cuprates’ Anti-Nodal Kink and Pseudogap

Features on the Basis of a Model of Electron Pairing

Posted on 04 December 2011

Author: Qiang Li

Jinheng Law Firm

Abstract:

We provide an explanation of anti-nodal pseudogap features of cuprates on the

basis of a proposed electron pairing model, and recognize the bosonic modes

responsible for electron pairings leading to the anti-nodal pseudogap features, as

having and energy range estimated at about 25-30 meV.

Introduction

In a previous article by the author,[1] a model of electron pairing is described.

According to the model, in the context of high temperature superconductivity in

cuprates, a pair of electrons on two suitable states are tuned by a matching bosonic

(phonon) mode, setting the two electrons into a nonstationary steady state (NSS state),

in which the threshold boson becomes redundant and releasable to establish a binding

energy of the pair. Phonon is the quanta of interaction between a lattice mode and an

object concerned, where the interaction is in a general form , which can be

electromagnetic, gravitational, or so on. A phonon is associated with a lattice mode,

but it is not the lattice mode itself; the threshold phonon of a tuned electron pair may

leave the pair, but the pair is still tuned, mediated, and maintained by the lattice mode

of the threshold phonon, even if the lattice mode is depleted of (real) phonon. There

may be boson transfer among phonon modes in the nodal kink of superconducting

cuprates, particularly in the presence of a bilayer/multi-layer band structure, resulting

in boson depletion of the bosonic mode mediating electrons at/near Fermi Surface

(FS). The boson depletion should be responsible for at lease some of the exotic

features of the nodal kink.

In this report, we will show that a clear explanation of anti-nodal pseudogap

features can be obtained on the basis of the proposed electron pairing model, in view

of the results by Gromko et al[2] on a cuprate sample, with bosonic modes responsible

for anti-nodal pairings being identified and their energy ranges being reliably

estimated to be at about 25-30 meV.

Electron Pairing and Band Dispersion Effects

It is known in the art that “strong electron coupling with other excitations ……

concomitantly makes the electron appear as a heavier and slower quasiparticle”, [3] and

that these interactions or correlation effects give electrons “an enhanced mass or

flatter E vs. k. dispersion”. [2] In Reference [1], we have shown that the outcome of

pairing competitions is related to the band dispersion with respect to specific phonon

dispersion and band structure features such as bilayer splitting. In a single linear band,

pairing cannot be stable. As schematically shown in Fig. 1, electron at state 101 may

pair up with electron at matching state 103 by releasing their threshold phonon, but

electron at state 102, which is symmetrical to state 103 with respect to state 101, can

also pair up with electron at state 101 by releasing a threshold phonon of basically the

same energy; so pairing competition between states 102 and 103 with respect to state

101 is even. Moreover, on a linear band section, every pairing is associated with the

same threshold phonon, so a pair (such as that between states 101 and 103) can be

destroyed by any of these threshold phonons, that is, the pair can be freely transferred

over all matching couples of electron states on a linear band section. Thus, if a linear

band section stays by its own, there could hardly be any gap feature.

In the situation of bilayer splitting band structure, where the spacing between the

two bands is within the range of lattice mode frequencies, there is competition

between intraband pairing and interband pairing. As shown in Fig. 2, pairing between

states 201 and 204 competes with pairing between states 201 and 202. Pairing

between states 201 and 204 and pairing between states 201 and 203, however, do not

compete with each other; rather, they tend to enhance each other. But in a parallel

two-band structure, such enhancement would not be sufficient to establish any gap

feature because the interband pairing can also be transferred along the bands as long

as the linear bands are parallel, unless all electrons on one of the band have paired up.

Explaining Anti-nodal Pseudogap Features and Energy Scales of Mediating Boson Modes

Gromko et al resolved the band and pseudogap features in over-doped 58 K

Bi2212 sample[2], as reproduced in Fig. 3. We now explain these features with our

model of electron pairing. As shown in Fig. 3, the band section from 301 to 302 is the

visible kink section of the bonding band (BB), the line segment from 304 to 305

schematically indicates the roughly estimated position of the antibonding band (AB)

section engaging in interband pairing with the BB section from 301 to 302. According

to our electron pairing model, that the AB section from 304 to 305 is not visible

indicates nearly all electrons on this section are in pairing with their counterparts on

the BB section from 301 to 302 so that they sink to their counterpart states on the BB

section from 301 to 302. Moreover, it is determined that state 301 must also be the

lower state of an intraband pairing, for otherwise the interband pairing between states

301 and 304 would be subjected to destruction by an intraband pairing between state

301 and a matching state below it. Since no gap feature is visible in the BB section

from 301 to 302, it is further determined that state 301 pairs up with a BB state

immediately above state 302. Thus, we have identified that the mode mediating the

interband pairing between states 301 and 304 has a frequency of about 25meV (which

equals to the difference between the binding energies of states 301 and 304,) and the

mode mediating the intraband pairing between states 301 and 302 has a frequency of

about 30meV.

Although the exact position of AB section above state 304 is not visible, the AB

state (as indicated by 305) pairing with state 302 should very likely be above FS, as

state 302 has a binding energy of only about 10 meV, which is less than half of the

phonon energy of the mode mediating states 301 and 304. That state 302 pairs up with

state 305 above FS is understandable in view of virtual transition, in which an

electron virtually transits to state 305 and sinks to state 302 by pairing to enjoy the

binding energy of state 302. The electron virtually transiting to state 305 can be from

a state at or near FS outside the cut as shown. Moreover, the electron at state 302

should also engage in intraband pairing with an electron virtually transiting to state

303, which is generally determined by the phonon mode mediating states 301 and 302

as we expect the straight band section from state 301 to state 302 would extend to

state 303.

In this way, it has been shown that the anti-nodal kink features as shown in Fig.

3 are caused by or associated with the interband and intraband double pairings, which

are so complete that the entire AB section from 304 to FS and its virtual extension

from FS to state 305 sink to BB section from 301 to 302, and BB section from 302 to

FS with its extension from FS to state 303 also sinks to BB section from 301 to 302.

Thus, with respect to this result by Gromko et al., we have explained the origin of

cuprates’ pseudogap features, the pseudogap scale, and the energy scales of the

bosonic (phonon) modes that mediating the electron pairing leading to the pseudogap

features.

Conclusion

In this paper, using our pairing model, we have explained anti-nodal pseudogap

features in connection to results by Gromko et al with respect to bilayer splitting

bands. We have also discussed some factors concerning the origin of the nodal kink

features and the associated pairing mechanism there. In our model, the nodal kink and

anti-nodal pseudogap have the same origin of NSS state-based electron pairing

mediated by bosonic modes, with the difference (possibly) in the presence/absence of

boson depletion due to local line shapes, which difference should be responsible for

the apparent absence/presence of gap features. The bosonic modes responsible for

anti-nodal pairing are identified as at about 30 meV. While this is in agreement with

the energy scales of phonon modes, our model is not specific to phonon modes as the

pairing mediators; rather, it is applicable to any bosonic modes capable of tuning

electrons on matching states, particularly across/along a splitting band structure.

Fig. 1 Electron at state 101 may pair up with electron at matching state 103 by releasing their

threshold phonon, but electron at state 102, which is symmetrical to state 103 with respect to state

101, can also pair up with electron at state 101 by releasing a threshold phonon of basically the

same energy; so pairing competition between states 102 and 103 with respect to state 101 is even.

EF

103

101

102

E

q

104

Fig. 2 Pairing between states 201 and 204 competes with pairing between states 201 and 202. Pairing between states 201 and 204 and pairing between states 201 and 203, however, do not compete with each other; rather, they

tend to enhance each other.

E

203

201

BB

202

204

AB

q

Fig. 3 Band section from 301 to 302 is the visible kink section of the bonding (BB) band, the line

segment from 304 to 305 schematically indicates the roughly estimated position of the

antibonding (AB) band section engaging in interband pairing with the BB section from 301 to 302.

According to our electron pairing model, that the AB section from 304 to 305 is not visible

indicates nearly all electrons on this section are in pairing with their counterparts on the BB band

section from 301 to 302 so that they sink to their counterpart states on the BB band section from

301 to 302.

References

[1] LI, Qiang: Golden Rule Characteristics of Electron-Lattice Interaction, Electron-pairing, and Phonon Depletion at Fermi Surface in Cuprates http://www.paper.edu.cn/index.php/default/releasepaper/content/201011-261[2] A. D. Gromko, A. V. Fedorov, Y.-D. Chuang, J. D. Koralek, Y. Aiura, Y. Yamaguchi, K. Oka, Yoichi Ando, and D. S. Dessau: PHYSICAL REVIEW B 68, 174520 ~2003

[3] H. Anzai, A. Ino, T. Kamo, T. Fujita, M. Arita, H. Namatame, M. Taniguchi, A. Fujimori, Z.-X. Shen, M.

Ishikado, and S. Uchida: Phys. Rev. Lett. 105, 227002 (2010)

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