The magnitude of an apparent energy gap is recognized as a measure of relative instability of electron pairing at the gap location, for it indicates that stabilized pairing can only be realized at a greater binding energy. At a low temperature, the chemical potential of a system like Bi2212 is determined by the most stable pairing, and will drop by about the energy of the mediating mode when pairing is stable. Bogoliubov quasiparticles are explained as excitations by lattice modes that win a mode competition among mediating modes and therefore have a large number of phonons depleted from losing modes. Thus, the energy of BQP peak is that of upper states of pairs mediated by the winning modes (~70 meV), while the energy of the nodal kink is that of the base states of these pairs, which shifts upward due to band topology on leaving the node. The “superconducting gap” corresponds to a kinked band section, a kink at the lower edge of which varnishes as nodal gap transform into the antinodal one.

1. Energy gap as a measure of pairing instability, Bogoliubov quasiparticles as
excitations by depleted phonons, and transformation between nodal and
antinodal pairing features in Bi2212
(posted on Slideshare on 13 March 2012)
Qiang Li
Jinheng Law Firm, Beijing, China
Abstract: The magnitude of an apparent energy gap is recognized as a measure of
relative instability of electron pairing at the gap location, for it indicates that stabilized
pairing can only be realized at a greater binding energy. At a low temperature, the
chemical potential of a system like Bi2212 is determined by the most stable pairing,
and will drop by about the energy of the mediating mode when pairing is stable.
Bogoliubov quasiparticles are explained as excitations by lattice modes that win a
mode competition among mediating modes and therefore have a large number of
phonons depleted from losing modes. Thus, the energy of BQP peak is that of upper
states of pairs mediated by the winning modes (~70 meV), while the energy of the
nodal kink is that of the base states of these pairs, which shifts upward due to band
topology on leaving the node. The “superconducting gap” corresponds to a kinked
band section, a kink at the lower edge of which varnishes as nodal gap transform into
the antinodal one.
Contents
1 Introduction
2 Gap magnitude as a measure of pairing instability
3 Determination of antinodal kink energy
4 Significance of pairing model based on nonstationary steady (NSS) state
5 Stronger pairing vs. more stable pairing, and binding energy
6 Transformation from antinodal to nodal pairing features
7 Bogoliubov quasiparticles (BQP) as excitations by depleted phonons
8 A dip as new energy scale corresponding to BQP peak in ARPES spectra
9 Origin of BQP peak energy: kinked band at Fermi level
10 The nodal kinked band as embryo of antinodal gap
11 Conclusion
1. Introduction
In a previous paper[1], discussions were made to antinodal gap features of Bi2212 with
respect to the angle–resolved photoemission spectroscopy (ARPES) results of
Gromko et al[2], with the application of a model of electron pairing based on non-
stationary steady state (NSS state) [3]. According to the model, if two occupied
stationary electron states ( E1 ,k1 ) and ( E 2 ,k2 ) matches a lattice mode (hν , q) with
1

2. hν = E 2 −E1 and q = k2 − k1 , electrons on the two states are tuned by the lattice mode
and are set into NSS state, in which the distribution of probability of the measured
energy of each of the two electrons depends on the average phonon number n of the
lattice mode, and when n→0 the probability that any of the electrons is measured at
E2 effectively goes to zero.
Thus, as shown in Fig. 1 from Gromko et al (we add reference numbers 101-105 and
the associated lines), states on the suppositional antibonding band (AB) part from 104
to 105 and states on the bonding band (BB) part from 101 to 102 match their
corresponding lattice modes respectively, so at sufficiently low temperature (when
n→0 for these lattice modes) the measured energy (and wavevector) of each of the
electrons associated with states on AB part from 104 to 105 is basically that of the
respective matched state on BB part from 101 to 102. The same mechanism holds true
for states on BB part from 102 to 103 with respect to the states on BB part from 101
to 102. In other words, electrons on states on AB part from 104 to 105 and BB part
from 102 to 103 are measured as if they “sink” to their respective matching states on
BB part from 101 to 102. (It is to be noted that BB part from 102 to 103 needs not to
be linear.)
However, to existing nodal ARPES results for bilayer Bi2Sr2CaCu2O8+δ (Bi2212)[4]-[7],
explanation according to the above model of electron pairing meet difficulty. ARPES
results for Bi2212 are typically featured by a kink and bilayer splitting band structure
including substantially parallel antibonding band (AB) and bonding band (BB)
extending from the kink to the Fermi level FL’. In such a nodal bilayer splitting band
structure, as schematically shown in Fig. 2, electrons on states 203 and 204 near FL’
should tend to sink to their matching state 201, so below superconducting transition
temperature Tc a remarkable gap should exist at or immediately below FL’, which is
not in conformity with the existing nodal ARPES results. In this paper, explanation is
to be made to the “missing” nodal gap, with respect to some existing nodal ARPES
results of Bi2212, by application of the above-mentioned model of electron pairing
based on NSS state.
2. Gap magnitude as a measure of pairing instability
Conventionally[8], determination of Fermi level by ARPES was realized by comparing
the photoelectron energies from a reference metal to those of the sample under study,
where the reference metal and the sample are electrically connected.
We would argue that the applicability of this method is model-dependent. To a model
of electron pairing in which a superconducting gap (SG) or a psuedogap (PG) opens
symmetrically with respect to Fermi level, application of the method can be justified.
However, to the above-mentioned model of electron pairing based on NSS state, the
2

3. method would not be applicable. As schematically shown in Fig. 2, if electrons on
states 203 and 204 at the “original” Fermi level FL’ sink to state 201, and electrons on
states between 201 and 203 sink to corresponding states below 201, state 201 would
become the lower edge of gap (for simplicity we omit the effect of thermal excitation
here), and the chemical potential (CP) of electrons in a reference metal electrically
connected to the system of Fig. 2 would sink to the level of state 201 and be measured
at lowered level FL. More specifically, for a pair on two states E2>E1, when the
phonon number n of the mediating mode goes to zero, the pair will become
increasingly stable, and the chemical potential FL→E1, that is, the measured chemical
potential would tend to be aligned with the lower state of the pairing; by contrast,
when n gets greater, the pair becomes increasingly instable, and there would be
FL→E2. Insofar that the system shown in Fig.2 resembles nodal bilayer splitting band
structure typically shown in existing ARPES data of Bi2212, the above analysis may
explain why no remarkable nodal gap is apparent in ARPES measurements of
superconducting Bi2212 sample.
3. Determination of antinodal kink energy
Then, why antinodal gap is apparent in existing ARPES results of Bi2212? Band
topological features seem to decide the lower edge of the antinodal gap. As shown in
Fig. 1, electrons on the AB part between states 104 and 105 and on BB part between
states 102 and 103 sink to respective states on BB part between states 101 and 102, so
none of the states on AB part from state 104 to the Fermi level (zero energy) and
states on BB part from 102 to the Fermi level can be the “base” state of electron
pairing of our model, for all these states are upper states pairing with respective lower
states on BB part from 101 to 102. Thus, it could be said that the magnitude ∆A of the
antinodal gap is determined by the energy EkA of the antinodal kink (that is, the energy
of state 101 in Fig. 1) as:
∆ A = E kA −hν , (1)
with ν being the frequency of the lattice mode mediating pairing between states 101
and 102.
Then, what decides the EkA the energy antinodal kink? We would propose that it is the
competition between electron pairing at the node and that at the anti-node, which
competition could be characterized by the difference in stabilities of electron pairing
at these two sites. In the anti-nodal scenario of Fig. 1, state 102 is associated with the
double pairing of between 103 and 102 and between 105 and 102; similarly, in the
nodal scenario of Fig. 2, state 201 is associated with the double pairing of between
203 and 201 and between 204 and 201. As the pairing at the node and that at the anti-
node compete with each other, the less stable pairing can only be realized at an energy
level further below the chemical potential of the system. This interpretation agrees
with the results of Gromko et al that the antinodal gaps of OD58, OD75, and OP91
3

4. Bi2212 samples at T=10K increase in this order, as can be seen clearly in Fig. 3.
Although that the spacing between states 101 and 102 in Fig. 1 corresponds to the
energy (hν) of the lattice mode mediating the intraband pairing in BB is a very strong
limitation, such confinement of the length of the apparent BB section seems to agree
very well with existing experimental results. As shown in Fig. 4 from Gromko et al,
even at (0.7π,0) the length of the apparent BB section above the kink is still confined
very well. While data for momentum cut further closer to the node is not available in
Reference 2, one may roughly identify corresponding antinodal AB and BB band
features in Fig. 5 taken from Lee et al[9], to further examine how anti-nodal band
pattern like that of Figs. 1 transforms to two parallel bands above the nodal kink (like
for example those shown in References 6 and 7). In data up to cut C4 (for T=10K) of
Fig. 5, the energy range of apparent BB above the kink still seems basically confined,
while the range of apparent AB band increases dramatically. Turning back to Fig. 4,
we see that the range of apparent AB band indeed increases remarkably at (0.7π,0).
Such widespread confinement of apparent bonding band above the kink supports our
model of electron pairing based on NSS state. Another associated feature in
conformity with our pairing model is ∆A< hν, or EkA< 2hν. The validity of these
relations is evidenced by the results of Figs. 3, 4 and 5. Further and detailed
interpretations relating to details relating to transformation between the antinodal and
nodal gap/pairing features will be provided later, particularly in Sections 6 and 10.
Thus, with the confinement of the apparent BB band above anti-nodal kink, if anti-
nodal electron pairing becomes weaker, the kink, and thus the entire BB structure as
that shown in Fig. 1, would tend to sink deeper downward with respect to the
measured chemical potential (Fermi level) so that antinodal pairing can be realized at
lower energy, and surplus electrons left on states above sunken states 105 and 103
would go to nodal region to build up the lower edge of nodal gap, which is basically
the measured chemical potential, leading to the overall effect of enlarging the anti-
nodal apparent gap.
That weight transfer from the dip has no contribution to the pileup of the peak
signifies the significance of the antinodal and nodal kinks: to isolate pairings above
the kinks from interacting with processes below the kinks. Due to the antinodal kink,
the lattice mode mediating intraband pairing in antinodal BB above the kink is kept
different from the modes mediating pairings below the kink, and this is also true for
nodal intraband and interband pairings above and below the nodal kink. It could be
anticipated that such isolation helps to stabilize the band structure above the kinks.
4. Significance of pairing model based on NSS state
As explained in References 1 and 3, our model of electron pairing is entirely based on
traditional time-dependent perturbation equation for electron-lattice scattering. But a
scattering interpretation emphasizes transitions among stationary states. In our model,
however, time-dependent perturbation for electron-lattice interaction is interpreted as
4

5. representation of a non-stationary electron state in crystal by stationary states ( E n ,kn )
, as lattice modes leading to time-dependent term in the Hamiltonian for the
perturbation equation are intrinsic to crystal; each such non-stationary electron state is
typically associated with two stationary states ( E1 ,k1 ) and ( E 2 ,k2 ) matching a
mediating lattice mode (hν , q) with hν = E 2 −E1 and q = k2 − k1 , and whether such a
non-stationary electron state can be “realized” depends on its competition with other
available non-stationary electron state(s). The non-stationary electron state is a steady
state. An electron in such a non-stationary electron state can be measured either at
( E1 ,k1 ) or ( E 2 ,k2 ) . When the number of (real) phonon of the mediating lattice mode
goes to zero, the probability that the electron is measured at ( E 2 ,k2 ) also goes to
zero. As one such non-stationary electron state is associated with two stationary states,
it is 2-fold “degenerate”. The 2-fold degeneracy results in “pairing”, and the vanishing
probability that the electron is measured at ( E 2 ,k2 ) leads to “binding energy”.
5. Stronger pairing vs. more stable pairing, and binding energy
When two candidate pairs, such as those shown between states 201 and 204 and
between 201 and 202 in Fig. 2, compete with each other, the stronger one has greater
probability to be realized. According to time-dependent perturbation equation, the
magnitude of transition matrix element is proportional to the magnitude of mediating
lattice wave, so the probability of transition is proportional to (n+1/2), with n being
the average number of phonons of the mediating lattice mode. Thus, the greater n is,
the stronger the pairing is. But as discussed above, when n gets greater, the pair
becomes increasingly instable, and can only be realized at greater binding energy.
When n is large, the upper stage E2 of the pair would be just above the chemical
potential, or FL→E2. The most stable pairing is also the weakest one. For a lattice
mode with phonon energy of ~30-70 meV and temperature of ~100K or lower, the
average phonon number of the mode is much smaller than one for an electron gas, so
variation of n would be of little effect on pairing competition. However, when
interaction or correlation, like lattice mode competition discussed below, is involve, n
can become great and have vital effect on pairing competition, as explained in Ref. 3
and in Sec. 7 and 9 of this paper.
At a non-zero temperature, there would be vacant states even below FL, to which an
electron from a broken pair could transit. So at a limited temperature, binding energy
of a pair is somewhat indefinite even in a traditional pairing model. In the framework
of the present pairing model, states in the energy range of 0<E<70 meV are partly
occupied, so binding energy would be even more indefinite. Thus, binding energy
5

6. becomes less reliable as a factor determining the outcome of pairing competition.
However, for states immediately below FL, the pairing-upward option is favored in
the scenario of bilayer band structure. As far as the ~30 meV and ~70 meV modes are
concerned, a state at the range of 0<E<30 meV may have as many as four candidate
upward pairings: one interband 70 meV, one interband 30 meV, one intraband 70 meV,
and one intraband 30 meV; the four pairing can be realized simultaneously. But only
one downward competing pairing can be realized at a time. This may explain why
pairs in Bi2212 based on nodal BB states in the range 0<E<30 meV are popular.
It is seen that pairing competition discussed above may function in two major ways:
1) if at a location (such as an antinode in Bi2212) pairing cannot be realized on a base
state sufficiently close to FL, electrons will tend to transit to another location (such as
a node in Bi2212) where pairing can be realized on a base state at or immediately
below FL; and 2) it decides whether a particular electron (as one at state 201) pairs
upward or downward.
6. Transformation from antinodal to nodal pairing features
Some details of transition from antinodal pairing to nodal pairing could be identified
in the data of Fig. 5, particularly in the data of cuts C2-C5 for 10K. The confined
apparent antinodal BB part (corresponding to the part from 101 to 102 in Fig. 1) can
still be traced even in cut C2, but increasing weight adds as its lower extension, and
finally the added weight becomes indistinguishable from the antinodal BB part, and
transforms into nodal BB part above the nodal kink together with the antinodal BB
part; the confinement never collapses, it merely thins out. Moreover, this
transformation suggests a switch of the pair-mediating lattice mode from the antinodal
mediating mode(s) to the nodal mediating mode (Details concerning the
transformation and the onset of the gap at the nodal region are to be discussed later in
this paper.) Scenarios in which apparent AB and BB parts stay in juxtaposition are
already shown in the existing results for OP91 of Fig. 3 and for cut at (0.7π,0) of Fig.
4, where pairing is considered as being “based on” the apparent BB part, although
these parts are not “parallel”. Another experimental evidence is a nodal peak of about
the same width as its antinodal counterpart (about 30-35 meV as identified in Fig. 1)
[10]
, which indicates that a mode of this energy is involved in mediating nodal pairs (it
is noted, however, that this peak is a result of integration over a nodal area).
With these, we would anticipate a scenario of nodal pairing as schematically shown in
Fig. 6, in which some “main” interband pairs are mediated between BB states below
measured chemical potential FL and AB states above FL by modes of relatively great
energy (~70 meV[11][12]) so that all AB states in interband pairs “based on” BB states
are above FL, intraband pairs like that between 601 and 602 are mediated by modes of
30-35 meV or so and lead to a nodal peak[10], and some interband pairs “based on” AB
states might be realized between AB states 607 and BB states 608. Such a picture
could be in conformity with the results of Figs. 4 and 5 and References 5-7 and 10.
6

7. In the scenario shown in Fig. 6, not all the states below 603 are occupied. A pair like
that between states 601 and 603 or states 604 and 605 has to be stable for the
corresponding upper state 603 or 605 to be occupied. Thus, a pair on a “base state”
near FL like state 601 may not be guaranteed of a binding energy corresponding to the
energy (hν) of its mediating lattice mode; but at sufficiently low temperature, a pair
with a base state below FL like state 604 should have a binding energy basically no
smaller than the binding energy of the base state and no greater than the binding
energy of the base state plus hν.
7. Bogoliubov quasiparticles (BQP) as excitations by depleted phonons
Of special interest are Bogoliubov quasiparticles (BQP) detected by ARPES,
interpreted as thermally excited electrons in an upper branch of a superconducting
binding-back band. [9][13][14] We propose, however, that BQPs are paired electrons
measured at the upper state ( E 2 ,k2 ) of their pairs due to excitations by concentrated
phonons in its mediating lattice mode. In an earlier paper by this author, a mechanism
of phonon depletion from lattice modes mediating electrons at the nodal region in
superconducting cuprates was proposed.[15] It is to be noted that phonon transfer as
proposed in Ref. 15 can be along various directions, and the electron states involved
are not necessary in spatially communication in k-space. Generally, electron pairs
based on NSS states need not to be confined in one plane as shown in Fig. 6, but due
to symmetry restriction, stabilized pairing at the node could only be realized along Γ-
Χ. Thus, for a lattice mode mediating pairing at the node to be depleted, the
destination mode(s) of the phonon depletion/transfer should not deviate too much
from Γ-Χ direction.
BQPs in Bi2212 were reported in a region near the node. [9][14] We propose here that
these BQPs in Bi2212 are due to excitation by phonons of destination modes of
phonon depletion suffered by lattice modes mediating interband pairings “based on”
BB states at the node; such interband pairings are schematically shown in Fig. 6 as
pairing between 604 and 605 and between 601 and 603. A consideration is that while
the slope of the band decreases upon moving from the node to the antinode, the
separation of the band bilayer splitting tends to increase. [6] So interband pairing
between bilayer band structures of various slopes can be mediated by the same mode,
so long as the bilayer separation varies accordingly. It is to be noted that, according to
time-dependent perturbation, the more pairs a mode tunes, the more competitive the
mode tends to be, for more tuned pairs tends to result in more real phonons and thus
the greater magnitude of the corresponding matrix element. This mode competition is
essentially the same as that in a laser, and the winning mode(s) corresponds to laser
modes that generate radiation output. While phonons concentrated to the winning
lattice modes are not for output from the crystal of superconducting cuprates, they do
lead to excitation of paired electrons from their “base” states as schematically shown
in Fig. 6 at 604 to their “upper” states as schematically shown at 605, where they
might be detected by ARPES as BQPs.
7

8. As explained above, some of the states from FL to the upper limit of BQPs’
distribution are not occupied, allowing phonon to be dissipated by transitions of BQPs
to these non-occupied states. At a limited temperature, as processes like anharmonic
interactions or so may lead to an inflow of phonons to the depleted modes, so phonon
dissipation or “phonon sink” like this would allow a corresponding flow of depleted
phonons to balance the inflow. The phonon depletion suffered by lattice modes
mediating BB-based interband pairs at the node leads to greatly enhanced stability of
these pairs, since it will be far less probable for electrons in these pairs to be
“measured” at their upper states (E2, k2). These losing modes should be responsible to
high-temperature superconductivity.
8. A dip as new energy scale corresponding to BQP peak in ARPES spectra of
Bi2223
In such an interpretation, BQPs would be associated with a corresponding dip
immediately above the nodal kink; the dip is separated from the BQP peak by the
phonon energy (~70meV). As shown in Fig. 5, in cut C1 a seemingly slight dip could
be identified at 50-60 meV but no dip is apparent in cuts C2-C4. But due to
dispersion, the weight of the main part of BB at around 50-70 meV might have not
been included in the data of Fig. 5, as can be seen from Figs. 4 and 5 of Ref. 14,
where the main part of BB at ~50-70meV is clearly excluded from the data-collecting
(shade) area. However, in Fig. 7(b) of Ref. 14, a clear dip is seen at the energy of
about 50-70 meV while a BQP peak centers at about 20 meV; the energy of this dip is
not easy to be determined as it is partly at the shoulder of the steep peak.
Results of Matsui et al are of triple-layered Bi2223[13], with a remarkable feature that
BQPs are measured at locations closer to antinode than to node, as shown in Fig. 7
taken from Ref. 13. But triple-layered system is more flexible in pairing match
because switch between different pairs of layers is possible. For example, a mode that
mediates pairing between the upper and middle bands at the nodal region may switch
to mediate pairing between the middle and the lower bands at the antinodal region.
Such a switch would allow a phonon sink located much farther away from the node
and possible multiplicity of phonon-dissipating areas. Details of pairing mechanism in
triple-layered Bi2223 are still to be investigated.
In Fig. 8 from Matsui et al, a dip occurs in the spectra at points B and C at energy
slightly smaller than 50 meV, together with a BQP peak at about -18 meV. Moreover,
we can even identify a slight dip and a corresponding BQP peak in the spectrum at
point A. And while the BQP peak slightly shifts toward higher energy from point C to
point A, the dip exhibits substantially the same shift, effectively keeping the energy
separation between the dip and the BQP peak at about 68 meV at each of points A, B
and C. The strength of the dip at point C, however, seems much weaker than that of its
BQP peak. But this mismatch can be explained in the framework of our interpretation.
As shown in Fig. 9, which is drawn in view of the inset of Fig. 7 taken from Matsui et
8

9. al, line 901 represents the central line of the Fermi arc in the detected area, the (0,0)-
(π,π) direction is shown by the arrow, and the detected area can be represented by a
horizontal stripe schematically shown at 902; thus, points A, B, and C are
schematically shown here as corresponding to points 1, 2, and 3 on line 901
respectively. According our interpretation, BQPs at point C are “based on” states in an
area schematically shown at 903C, located directly upward with respect to point 3,
and so are BQPs at points B and A with respect to areas 903B and 903A respectively.
While points 1, 2 and 3 along the central line 901 of the Fermi arc are well covered by
stripe 902 representing the detected area, part of area 903C is left out of stripe 902,
leading to a weakened dip in the spectrum at point C.
9. Origin of BQP peak energy: kinked band at Fermi level
An important question is why the BQP peak appears at certain energy above FL near
the node. First, in the antinodal scenario as shown in Fig. 1, the energy EkA of the
antinodal kink is determined by the energy ∆A of the lower edge of the antinodal gap
and the energy hν of mediating mode, as indicated in Equ. (1), due to that BB
intraband pairing (as between states 101 and 102 in Fig. 1) is decisive in determining
the kink energy. The mechanism for determining nodal kink energy is basically the
same, so with zero gap at the node we would have E kN = hν , where EkN is the energy
of the nodal kink and hν is the energy of the nodal mediating mode. (It is to be noted
that, in the framework of the present pairing model, any further downward shifting of
the kink from EkN =-hν does not make any sense, for such a shift would destroy the
pairing immediately below FL to lead to a corresponding downward shift of FL.)
When phonon depletion occurs, the phonon transfer must be downward, as from
modes mediating states at 601 and 603 to modes mediating states at 604 and 605 in
Fig. 6, because the mode(s) winning the mode competition will have a large number
of phonons so, as explained above with respect to determination of chemical potential
(FL), pairs mediated by the winning mode will be highly instable and can only be
realized with their upper state just above FL, that is, they can only be “based on”
states at or slightly above the kink at the node.
It is to be noted that, among all electron pairs mediated by modes involved in a mode
competition, as a general rule, the electron pairs mediated by the winning modes in
the competition are the least stable ones and therefore always appear as having the
greatest binding energy in ARPES spectra, particularly at the bottom of a nodal kink.
Conversely, pairs “based on” states at or immediately below FL are the most stable
ones; the closer the base state of a pair is to FL, the more stable the pair is. For
example, in Fig. 1 a pair based on state 102 is slightly more stable than a pair based
on state 104, indicating that at the antinode pairs based on BB states above the
antinodal kink generally tend to be more stable than pairs based on AB states.
It is known in the art that “strong electron coupling with other excitations ……
concomitantly makes the electron appear as a heavier and slower quasiparticle”,[7] and
9

10. that these interactions or correlation effects give electrons “an enhanced mass or
flatter E vs. k. dispersion”. [2] It was evidenced that “the quasiparticle peak
dramatically sharpens on crossing |ω| ~ 70 meV (kink) towards the Fermi level”,
while at T>Tc such sharpening would not be present even though a nodal kink still
exists. [6] Thus, insofar that the “strong electron coupling with other excitations” is
essentially the electron-electron interaction mediated by lattice modes, which sets
electrons into NSS states, then electrons below the kink are not subject to the same
interactions or correlation effects as electrons above the kink, particularly in
consideration of the effect of the large number of phonons of the winning modes
acting at the kink; in other words, the nodal kink is formed at the base states of
electron pairs mediated by the winning modes of the phonon depletion. On leaving the
node, due to band topology, the states of pairs matching the winning modes, and thus
the kink (on BB), gradually shift upward, so the upper states of these pairs also shift
in the same way from FL to an energy Δ’>0 (it is to be noted that the samples reported
in both Refs. 9 and 13 are of very high Tc and are thus “optimal systems”;) at the
same time, due to the above-explained pairing competition, the upper limit of
interband pairs based on BB states falls from FL to -|Δ|, as schematically shown in
Fig. 10. Again, electrons between the two levels Δ’>0 and -|Δ| are not subject to the
same interactions or correlation effects as those above Δ’ or below -|Δ|, for they can
only pair with electrons below the nodal kink. Thus, the band between -|Δ| and Δ’ is
an additional kinked band section, delineated by an upper FL kink at Δ’>0 and a lower
FL kink at -|Δ|, which correspond to the upper and lower edges of the traditionally
called “superconducting gap” respectively.
In this way, we have not only explained the energy variation of the BQP peak upon
moving away from the node but also explained the origin and structure of the
superconducting gap, as a kinked band section at the FL. The lower edge –Δ of the
kinked band section is determined by the instability of pairing at the location with
respect to that at the node, while the upper edge Δ’ of the kinked band section is
determined by the energy of the upper states of pairs mediated by the winning modes,
so there is ∆ ≠ Δ’; that is, the upper and lower edges of the kinked band section are
not symmetrical with respect to FL.
As illustrated in Fig. 10, with the presence of the upper and lower FL kinks, pairing
relations similar to those at the node can be realized, by essentially the same
mediating modes (~70meV), in which BB state 1001 at Δ pairs with AB state 1005,
and BB state 1004 at the lower kink pairs with AB state 1006 to give rise to BQPs.
Thus, the band above the gap should have approximately the same slope as that of the
band below the gap (the kinked band section), as the AB states above the gap are in
interband pairing with respective BB states below the gap. Intraband pairing mediated
by ~70meV lattice mode might exist between states 1001 and 1003, which mode
might also mediate pairing between 1004 and 1002.
10

11. Thus, upon leaving the node, while the upper limit of pairing (i.e. the lower edge of
traditional superconducting gap) is lowered due to pairing competition with respect to
pairing at the node, the lower limit of pairing (i.e. the nodal kink) is raised as the base
states on BB in pairs matching to the winning modes in mode competition among the
~70 meV mediating lattice modes. As shown in Fig. 5(b), at cut C4 the BQP peak is
slightly less than 20 meV above FL, accordingly the BB side of the kink is expected
to be at about 50 meV below FL, basically in agreement with the result shown in Fig.
5(a) at Cut 4 (T=10K), in which the BB side of the kink indeed seems to be at 50 meV
(the weights extending beyond 50 meV should mostly be those from AB; extension of
AB beyond the BB side of the kink could also be seen in the Fig. 4 at the cut of
(0.7π,0).)
The presence of the upper and lower FL kinks could also lead to the seemingly
dispersion of measured BQPs.[13][14] In fact, the BQP distribution shown in Fig. 11
taken from Ref. 14 looks very much like a kink. It is to be further noted that the slope
of the BQP band part should be more likely to duplicate that of the BB part
immediately above the traditional nodal kink (the BB part at state 1004 as shown in
Fig. 10), rather than that of the band part immediately below the “superconducting
gap”. Another factor would be that, since BQPs are on AB in Bi2212, they would be
measured as if they have a slight shift to the AB side when photon energy like 22.7 eV
is used to allow simultaneous measurement of AB and BB.
10. The nodal kinked band section as embryo of antinodal gap
The gap associated with kinked band at FL is the embryo of the antinodal gap, which
grows in magnitude upon moving toward the antinode as evidence in Fig. 5(a). As
explained with Fig. 1, the antinodal gap is not symmetry with respect to FL, which is
in line with Δ≠Δ’. It is to be noted that “normal” density of state is present in the gap,
but some of the electrons on the states within the gap engage in downward pairing and
are measured as being on their respective base states. As the gap grows, pairings
mediated by the ~70 meV modes begin to lose their dominance over those mediated
by the ~30 meV modes. Pairs mediated by the winning modes quickly varnish as Δ’
increases on leaving the node, thus the nodal kink (on BB) becomes very indefinite
even at locations not far from the node, as could be seen at cuts C2 and C3 in Fig.
5(a), and the antinodal features relating to the ~30 meV modes seem to sit in even at
the nodal location of cut C3. The slope of the kinked band section in the gap becomes
gentle until the lower FL kink varnishes somewhere on the way to the antinode, while
the pairing and gap patterns transform to those at antinode as shown in Fig. 1. It is not
determined whether such transformation is a critical one. The fate of the upper FL
kink and the pairing mediated by the ~70 meV modes needs further investigation. It is
seen that although both antinodal and nodal pairings are dominated by the more stable
pairing at the node, the antinodal kink is lowered by such dominance while the nodal
kink is raised due to pairing match to the winning modes with respect to band
topology.
11

12. 11. Conclusion
In a Bi2212 (and Bi2223) systems, the magnitude of an apparent energy gap is a
measure of relative instability of electron pairing at the gap location, for a greater gap
indicates that a stabilized pairing (NSS state) of electrons can only be realized at a
higher binding energy. At a low temperature, the chemical potential of a system like
Bi2212 is determined by the most stable pairing in it, and when pairing is sufficiently
stable the chemical potential would be lowered by about the energy of the mediating
mode with respect to a free electron scenario. Bogoliubov quasiparticles could be
explained as excitations by lattice modes functioning to dissipate the depleted
phonons in competition of lattice modes mediating pairings, the superconducting gap
is a kinked band section, and the energy of BQP peak is that of upper states of pairs
mediated by the winning modes of the mode competition. The depleted modes should
be responsible to high-temperature superconductivity. The kinked band section is an
embryo of the antinodal gap, and on its way of transformation the kink at the bottom
of the kinked band section varnishes.
20 103
105
0
104
102
-20
-40
101
-60
-80
-100
-120
Fig. 1. From Gromko et al. (The reference numbers 101-105 and dashed or double-arrowed lines
are added by this author for explaining relevant features.) BB part from 101 to 102 is the visible
12

13. kink section, the line from 104 to 105 roughly indicates the suppositional AB part engaging in
interband pairing with BB part from 101 to 102. AB part from 104 to 105 is basically invisible for
nearly all electrons on this section are measured as “sinking” to their pairing counterparts on the
BB part from 101 to 102. Intraband pairing may exist between some of the states from 102 to 103
and their counterparts on the BB part from 101 to 102.
E
AB
BB
204 203
FL’
FL
201
202
q
Fig. 2. Schmetical illustration of pairing relations in a bilayer band. Pairing between states 201
and 204 competes with pairing between states 201 and 202. Pairings between states 201 and 204
and between states 201 and 203, however, do not compete with each other; they tend to enhance
each other.
13

14. Fig. 3. From Gromko et al. The magnitude of antinodal gaps of OD58, OD75, and OP91 Bi2212
samples at T=10K increases in this order.
Fig. 4. From Gromko et al. Data along four momentum slices centered around the k values of
(π,0), (0.9π,0), (0.8π,0), and (0.7π,0). Obviously, even at (0.7π,0) the length of the apparent BB
section above the kink is still confined very well to the phonon energy of ~30-35 meV.
14

15. (b) (c)
Fig. 5. Taken from W.S. Lee et. Al. (a)Transformation from antinodal features to nodal features.
(b) Bogoliubov quasiparticles. (c) Locations of cuts C1-C8
15

16. E
AB
BB
603
608
605 602
601 FL
607
606
604
Nodal kink
q
Fig. 6. Pairing relations in a cut at the node. Lattice mode of ~70 meV mediates states
605 at FL and states 604 at the kink, thus determining the energy of kink at the node.
Fig.7. Taken from Matsui el. Al. BQPs’ spectra of Bi2223 at the location shown in
inset.
16

17. Fig. 8. (a) same spectra as in Fig. 7. (b) EDCs at the three points A, B, and C locations
are shown. Particular attention should be paid to the small dip at 50 meV with their
positions shift matches that of BPQ peaks at about -20 meV. The seemingly mismatch
between intensity of dip at C and the corresponding BQP can be explained by the
experimental setup in view of pairing model we proposed.
901
(π, π) 903C
903B
(0, 0)
3 902
903A
2
1
A B C
Fig. 9. Schematic illustration of experimental setup relating to results shown in Figs. 7 and 8.
BQPs at position C come from the area generally represented by square 903C, which might have
been partly left out, as schematically indicated by partial coverage of square 903C by stripe
between lines 902.
1005 1003
17

18. 1005 1003
1006 1002
Upper FL kink
Δ’
FL
Δ
1001 Lower FL kink
1004 Kink
Fig. 10. Proposed bilayer band splitting in Bi2212 with a kinked band section at FL,
at a location not too far away from the node. Pairings are represented by dashed lines
with double-arrows. Mode mediating interband pairing between 1004 and 1006 and
states 1001 and 1005 may be in competition with the mode(s) mediating interband
pairs at the node. The FL kink allows the depleted mode(s) at the node to be adapted
to a shortened lower kink. BQPs are generated at 1006 by mode mediating 1004 and
1006.
18

19. Fig. 11. Taken from Balatsky et al. The gap below BQPs looks like a kinked band section.
According to our model, BQPs are excitations in the antibonding band above a kinked band
section corresponding to the gap, there is “normal” state density within the gap. The slope of the
BQP band should be more likely to duplicate that of the BB part immediately above the nodal
kink.
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