Energy gap as a measure of pairing instability, Bogoliubov quasiparticles as    excitations by depleted phonons, and trans...
hν = E 2 −E1 and q = k2 − k1 , electrons on the two states are tuned by the lattice modeand are set into NSS state, in whi...
method would not be applicable. As schematically shown in Fig. 2, if electrons onstates 203 and 204 at the “original” Ferm...
Bi2212 samples at T=10K increase in this order, as can be seen clearly in Fig. 3.Although that the spacing between states ...
representation of a non-stationary electron state in crystal by stationary states ( E n ,kn ), as lattice modes leading to...
becomes less reliable as a factor determining the outcome of pairing competition.However, for states immediately below FL,...
In the scenario shown in Fig. 6, not all the states below 603 are occupied. A pair likethat between states 601 and 603 or ...
As explained above, some of the states from FL to the upper limit of BQPs’distribution are not occupied, allowing phonon t...
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 ...
that these interactions or correlation effects give electrons “an enhanced mass orflatter E vs. k. dispersion”. [2] It was...
Thus, upon leaving the node, while the upper limit of pairing (i.e. the lower edge oftraditional superconducting gap) is l...
11. ConclusionIn a Bi2212 (and Bi2223) systems, the magnitude of an apparent energy gap is ameasure of relative instabilit...
kink section, the line from 104 to 105 roughly indicates the suppositional AB part engaging ininterband pairing with BB pa...
Fig. 3. From Gromko et al. The magnitude of antinodal gaps of OD58, OD75, and OP91 Bi2212samples at T=10K increases in thi...
(b)                               (c)Fig. 5. Taken from W.S. Lee et. Al. (a)Transformation from antinodal features to noda...
E                                   AB                                                     BB                            6...
Fig. 8. (a) same spectra as in Fig. 7. (b) EDCs at the three points A, B, and C locationsare shown. Particular attention s...
1005                   1003                           1006                   1002                                         ...
Fig. 11. Taken from Balatsky et al. The gap below BQPs looks like a kinked band section.According to our model, BQPs are e...
Shen: arXiv:0801.2819v2, DOI: 10.1038/nature06219.[10] D. S. Dessau et al., Phys. Rev. Lett. 66, 2160 (1991).[11] H. Iwasa...
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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

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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.

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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

  1. 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, ChinaAbstract: The magnitude of an apparent energy gap is recognized as a measure ofrelative instability of electron pairing at the gap location, for it indicates that stabilizedpairing can only be realized at a greater binding energy. At a low temperature, thechemical 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 amode competition among mediating modes and therefore have a large number ofphonons depleted from losing modes. Thus, the energy of BQP peak is that of upperstates of pairs mediated by the winning modes (~70 meV), while the energy of thenodal kink is that of the base states of these pairs, which shifts upward due to bandtopology on leaving the node. The “superconducting gap” corresponds to a kinkedband section, a kink at the lower edge of which varnishes as nodal gap transform intothe antinodal one. Contents1 Introduction2 Gap magnitude as a measure of pairing instability3 Determination of antinodal kink energy4 Significance of pairing model based on nonstationary steady (NSS) state5 Stronger pairing vs. more stable pairing, and binding energy6 Transformation from antinodal to nodal pairing features7 Bogoliubov quasiparticles (BQP) as excitations by depleted phonons8 A dip as new energy scale corresponding to BQP peak in ARPES spectra9 Origin of BQP peak energy: kinked band at Fermi level10 The nodal kinked band as embryo of antinodal gap11 Conclusion1. IntroductionIn a previous paper[1], discussions were made to antinodal gap features of Bi2212 withrespect to the angle–resolved photoemission spectroscopy (ARPES) results ofGromko 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 occupiedstationary electron states ( E1 ,k1 ) and ( E 2 ,k2 ) matches a lattice mode (hν , q) with 1
  2. 2. hν = E 2 −E1 and q = k2 − k1 , electrons on the two states are tuned by the lattice modeand are set into NSS state, in which the distribution of probability of the measuredenergy of each of the two electrons depends on the average phonon number n of thelattice mode, and when n→0 the probability that any of the electrons is measured atE2 effectively goes to zero.Thus, as shown in Fig. 1 from Gromko et al (we add reference numbers 101-105 andthe associated lines), states on the suppositional antibonding band (AB) part from 104to 105 and states on the bonding band (BB) part from 101 to 102 match theircorresponding lattice modes respectively, so at sufficiently low temperature (whenn→0 for these lattice modes) the measured energy (and wavevector) of each of theelectrons associated with states on AB part from 104 to 105 is basically that of therespective matched state on BB part from 101 to 102. The same mechanism holds truefor states on BB part from 102 to 103 with respect to the states on BB part from 101to 102. In other words, electrons on states on AB part from 104 to 105 and BB partfrom 102 to 103 are measured as if they “sink” to their respective matching states onBB part from 101 to 102. (It is to be noted that BB part from 102 to 103 needs not tobe 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. ARPESresults for Bi2212 are typically featured by a kink and bilayer splitting band structureincluding substantially parallel antibonding band (AB) and bonding band (BB)extending from the kink to the Fermi level FL’. In such a nodal bilayer splitting bandstructure, 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 transitiontemperature Tc a remarkable gap should exist at or immediately below FL’, which isnot in conformity with the existing nodal ARPES results. In this paper, explanation isto be made to the “missing” nodal gap, with respect to some existing nodal ARPESresults of Bi2212, by application of the above-mentioned model of electron pairingbased on NSS state.2. Gap magnitude as a measure of pairing instabilityConventionally[8], determination of Fermi level by ARPES was realized by comparingthe 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 modelof electron pairing in which a superconducting gap (SG) or a psuedogap (PG) openssymmetrically 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. 3. method would not be applicable. As schematically shown in Fig. 2, if electrons onstates 203 and 204 at the “original” Fermi level FL’ sink to state 201, and electrons onstates between 201 and 203 sink to corresponding states below 201, state 201 wouldbecome the lower edge of gap (for simplicity we omit the effect of thermal excitationhere), and the chemical potential (CP) of electrons in a reference metal electricallyconnected to the system of Fig. 2 would sink to the level of state 201 and be measuredat lowered level FL. More specifically, for a pair on two states E2>E1, when thephonon number n of the mediating mode goes to zero, the pair will becomeincreasingly stable, and the chemical potential FL→E1, that is, the measured chemicalpotential 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 beFL→E2. Insofar that the system shown in Fig.2 resembles nodal bilayer splitting bandstructure typically shown in existing ARPES data of Bi2212, the above analysis mayexplain why no remarkable nodal gap is apparent in ARPES measurements ofsuperconducting Bi2212 sample.3. Determination of antinodal kink energyThen, why antinodal gap is apparent in existing ARPES results of Bi2212? Bandtopological features seem to decide the lower edge of the antinodal gap. As shown inFig. 1, electrons on the AB part between states 104 and 105 and on BB part betweenstates 102 and 103 sink to respective states on BB part between states 101 and 102, sonone of the states on AB part from state 104 to the Fermi level (zero energy) andstates on BB part from 102 to the Fermi level can be the “base” state of electronpairing of our model, for all these states are upper states pairing with respective lowerstates on BB part from 101 to 102. Thus, it could be said that the magnitude ∆A of theantinodal gap is determined by the energy EkA of the antinodal kink (that is, the energyof state 101 in Fig. 1) as: ∆ A = E kA −hν , (1)with ν being the frequency of the lattice mode mediating pairing between states 101and 102.Then, what decides the EkA the energy antinodal kink? We would propose that it is thecompetition between electron pairing at the node and that at the anti-node, whichcompetition could be characterized by the difference in stabilities of electron pairingat these two sites. In the anti-nodal scenario of Fig. 1, state 102 is associated with thedouble pairing of between 103 and 102 and between 105 and 102; similarly, in thenodal scenario of Fig. 2, state 201 is associated with the double pairing of between203 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 energylevel further below the chemical potential of the system. This interpretation agreeswith the results of Gromko et al that the antinodal gaps of OD58, OD75, and OP91 3
  4. 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 theenergy (hν) of the lattice mode mediating the intraband pairing in BB is a very stronglimitation, such confinement of the length of the apparent BB section seems to agreevery 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 confinedvery well. While data for momentum cut further closer to the node is not available inReference 2, one may roughly identify corresponding antinodal AB and BB bandfeatures in Fig. 5 taken from Lee et al[9], to further examine how anti-nodal bandpattern like that of Figs. 1 transforms to two parallel bands above the nodal kink (likefor example those shown in References 6 and 7). In data up to cut C4 (for T=10K) ofFig. 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 ourmodel of electron pairing based on NSS state. Another associated feature inconformity with our pairing model is ∆A< hν, or EkA< 2hν. The validity of theserelations is evidenced by the results of Figs. 3, 4 and 5. Further and detailedinterpretations relating to details relating to transformation between the antinodal andnodal 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 asthat shown in Fig. 1, would tend to sink deeper downward with respect to themeasured chemical potential (Fermi level) so that antinodal pairing can be realized atlower energy, and surplus electrons left on states above sunken states 105 and 103would go to nodal region to build up the lower edge of nodal gap, which is basicallythe 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 peaksignifies the significance of the antinodal and nodal kinks: to isolate pairings abovethe 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 keptdifferent from the modes mediating pairings below the kink, and this is also true fornodal intraband and interband pairings above and below the nodal kink. It could beanticipated that such isolation helps to stabilize the band structure above the kinks.4. Significance of pairing model based on NSS stateAs explained in References 1 and 3, our model of electron pairing is entirely based ontraditional time-dependent perturbation equation for electron-lattice scattering. But ascattering interpretation emphasizes transitions among stationary states. In our model,however, time-dependent perturbation for electron-lattice interaction is interpreted as 4
  5. 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 theperturbation equation are intrinsic to crystal; each such non-stationary electron state istypically associated with two stationary states ( E1 ,k1 ) and ( E 2 ,k2 ) matching amediating lattice mode (hν , q) with hν = E 2 −E1 and q = k2 − k1 , and whether such anon-stationary electron state can be “realized” depends on its competition with otheravailable non-stationary electron state(s). The non-stationary electron state is a steadystate. 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 modegoes to zero, the probability that the electron is measured at ( E 2 ,k2 ) also goes tozero. 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 vanishingprobability that the electron is measured at ( E 2 ,k2 ) leads to “binding energy”.5. Stronger pairing vs. more stable pairing, and binding energyWhen two candidate pairs, such as those shown between states 201 and 204 andbetween 201 and 202 in Fig. 2, compete with each other, the stronger one has greaterprobability to be realized. According to time-dependent perturbation equation, themagnitude of transition matrix element is proportional to the magnitude of mediatinglattice wave, so the probability of transition is proportional to (n+1/2), with n beingthe 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 pairbecomes 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 chemicalpotential, or FL→E2. The most stable pairing is also the weakest one. For a latticemode with phonon energy of ~30-70 meV and temperature of ~100K or lower, theaverage phonon number of the mode is much smaller than one for an electron gas, sovariation of n would be of little effect on pairing competition. However, wheninteraction or correlation, like lattice mode competition discussed below, is involve, ncan become great and have vital effect on pairing competition, as explained in Ref. 3and in Sec. 7 and 9 of this paper.At a non-zero temperature, there would be vacant states even below FL, to which anelectron from a broken pair could transit. So at a limited temperature, binding energyof a pair is somewhat indefinite even in a traditional pairing model. In the frameworkof the present pairing model, states in the energy range of 0<E<70 meV are partlyoccupied, so binding energy would be even more indefinite. Thus, binding energy 5
  6. 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 inthe scenario of bilayer band structure. As far as the ~30 meV and ~70 meV modes areconcerned, a state at the range of 0<E<30 meV may have as many as four candidateupward 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 onlyone downward competing pairing can be realized at a time. This may explain whypairs 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 basestate sufficiently close to FL, electrons will tend to transit to another location (such asa node in Bi2212) where pairing can be realized on a base state at or immediatelybelow FL; and 2) it decides whether a particular electron (as one at state 201) pairsupward or downward.6. Transformation from antinodal to nodal pairing featuresSome details of transition from antinodal pairing to nodal pairing could be identifiedin the data of Fig. 5, particularly in the data of cuts C2-C5 for 10K. The confinedapparent antinodal BB part (corresponding to the part from 101 to 102 in Fig. 1) canstill be traced even in cut C2, but increasing weight adds as its lower extension, andfinally the added weight becomes indistinguishable from the antinodal BB part, andtransforms into nodal BB part above the nodal kink together with the antinodal BBpart; the confinement never collapses, it merely thins out. Moreover, thistransformation suggests a switch of the pair-mediating lattice mode from the antinodalmediating mode(s) to the nodal mediating mode (Details concerning thetransformation and the onset of the gap at the nodal region are to be discussed later inthis paper.) Scenarios in which apparent AB and BB parts stay in juxtaposition arealready 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, althoughthese parts are not “parallel”. Another experimental evidence is a nodal peak of aboutthe 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 (itis 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 inFig. 6, in which some “main” interband pairs are mediated between BB states belowmeasured chemical potential FL and AB states above FL by modes of relatively greatenergy (~70 meV[11][12]) so that all AB states in interband pairs “based on” BB statesare above FL, intraband pairs like that between 601 and 602 are mediated by modes of30-35 meV or so and lead to a nodal peak[10], and some interband pairs “based on” ABstates might be realized between AB states 607 and BB states 608. Such a picturecould be in conformity with the results of Figs. 4 and 5 and References 5-7 and 10. 6
  7. 7. In the scenario shown in Fig. 6, not all the states below 603 are occupied. A pair likethat between states 601 and 603 or states 604 and 605 has to be stable for thecorresponding 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 theenergy (hν) of its mediating lattice mode; but at sufficiently low temperature, a pairwith a base state below FL like state 604 should have a binding energy basically nosmaller than the binding energy of the base state and no greater than the bindingenergy of the base state plus hν.7. Bogoliubov quasiparticles (BQP) as excitations by depleted phononsOf special interest are Bogoliubov quasiparticles (BQP) detected by ARPES,interpreted as thermally excited electrons in an upper branch of a superconductingbinding-back band. [9][13][14] We propose, however, that BQPs are paired electronsmeasured at the upper state ( E 2 ,k2 ) of their pairs due to excitations by concentratedphonons in its mediating lattice mode. In an earlier paper by this author, a mechanismof phonon depletion from lattice modes mediating electrons at the nodal region insuperconducting cuprates was proposed.[15] It is to be noted that phonon transfer asproposed in Ref. 15 can be along various directions, and the electron states involvedare not necessary in spatially communication in k-space. Generally, electron pairsbased on NSS states need not to be confined in one plane as shown in Fig. 6, but dueto 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, thedestination mode(s) of the phonon depletion/transfer should not deviate too muchfrom Γ-Χ direction.BQPs in Bi2212 were reported in a region near the node. [9][14] We propose here thatthese BQPs in Bi2212 are due to excitation by phonons of destination modes ofphonon depletion suffered by lattice modes mediating interband pairings “based on”BB states at the node; such interband pairings are schematically shown in Fig. 6 aspairing between 604 and 605 and between 601 and 603. A consideration is that whilethe slope of the band decreases upon moving from the node to the antinode, theseparation of the band bilayer splitting tends to increase. [6] So interband pairingbetween 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 totime-dependent perturbation, the more pairs a mode tunes, the more competitive themode tends to be, for more tuned pairs tends to result in more real phonons and thusthe greater magnitude of the corresponding matrix element. This mode competition isessentially the same as that in a laser, and the winning mode(s) corresponds to lasermodes that generate radiation output. While phonons concentrated to the winninglattice modes are not for output from the crystal of superconducting cuprates, they dolead to excitation of paired electrons from their “base” states as schematically shownin Fig. 6 at 604 to their “upper” states as schematically shown at 605, where theymight be detected by ARPES as BQPs. 7
  8. 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 BQPsto these non-occupied states. At a limited temperature, as processes like anharmonicinteractions or so may lead to an inflow of phonons to the depleted modes, so phonondissipation or “phonon sink” like this would allow a corresponding flow of depletedphonons to balance the inflow. The phonon depletion suffered by lattice modesmediating BB-based interband pairs at the node leads to greatly enhanced stability ofthese 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 tohigh-temperature superconductivity.8. A dip as new energy scale corresponding to BQP peak in ARPES spectra ofBi2223In such an interpretation, BQPs would be associated with a corresponding dipimmediately above the nodal kink; the dip is separated from the BQP peak by thephonon energy (~70meV). As shown in Fig. 5, in cut C1 a seemingly slight dip couldbe identified at 50-60 meV but no dip is apparent in cuts C2-C4. But due todispersion, the weight of the main part of BB at around 50-70 meV might have notbeen 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 ofabout 50-70 meV while a BQP peak centers at about 20 meV; the energy of this dip isnot 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 thatBQPs are measured at locations closer to antinode than to node, as shown in Fig. 7taken from Ref. 13. But triple-layered system is more flexible in pairing matchbecause switch between different pairs of layers is possible. For example, a mode thatmediates pairing between the upper and middle bands at the nodal region may switchto 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 nodeand possible multiplicity of phonon-dissipating areas. Details of pairing mechanism intriple-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 energyslightly 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 atpoint A. And while the BQP peak slightly shifts toward higher energy from point C topoint A, the dip exhibits substantially the same shift, effectively keeping the energyseparation between the dip and the BQP peak at about 68 meV at each of points A, Band C. The strength of the dip at point C, however, seems much weaker than that of itsBQP 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. 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 ahorizontal stripe schematically shown at 902; thus, points A, B, and C areschematically shown here as corresponding to points 1, 2, and 3 on line 901respectively. According our interpretation, BQPs at point C are “based on” states in anarea 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 bystripe 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 levelAn important question is why the BQP peak appears at certain energy above FL nearthe node. First, in the antinodal scenario as shown in Fig. 1, the energy EkA of theantinodal kink is determined by the energy ∆A of the lower edge of the antinodal gapand the energy hν of mediating mode, as indicated in Equ. (1), due to that BBintraband pairing (as between states 101 and 102 in Fig. 1) is decisive in determiningthe kink energy. The mechanism for determining nodal kink energy is basically thesame, so with zero gap at the node we would have E kN = hν , where EkN is the energyof the nodal kink and hν is the energy of the nodal mediating mode. (It is to be notedthat, in the framework of the present pairing model, any further downward shifting ofthe kink from EkN =-hν does not make any sense, for such a shift would destroy thepairing immediately below FL to lead to a corresponding downward shift of FL.)When phonon depletion occurs, the phonon transfer must be downward, as frommodes mediating states at 601 and 603 to modes mediating states at 604 and 605 inFig. 6, because the mode(s) winning the mode competition will have a large numberof 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 berealized 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 modecompetition, as a general rule, the electron pairs mediated by the winning modes inthe competition are the least stable ones and therefore always appear as having thegreatest 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 stableones; the closer the base state of a pair is to FL, the more stable the pair is. Forexample, in Fig. 1 a pair based on state 102 is slightly more stable than a pair basedon state 104, indicating that at the antinode pairs based on BB states above theantinodal 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. 10. that these interactions or correlation effects give electrons “an enhanced mass orflatter E vs. k. dispersion”. [2] It was evidenced that “the quasiparticle peakdramatically 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 stillexists. [6] Thus, insofar that the “strong electron coupling with other excitations” isessentially the electron-electron interaction mediated by lattice modes, which setselectrons into NSS states, then electrons below the kink are not subject to the sameinteractions or correlation effects as electrons above the kink, particularly inconsideration of the effect of the large number of phonons of the winning modesacting at the kink; in other words, the nodal kink is formed at the base states ofelectron pairs mediated by the winning modes of the phonon depletion. On leaving thenode, due to band topology, the states of pairs matching the winning modes, and thusthe kink (on BB), gradually shift upward, so the upper states of these pairs also shiftin the same way from FL to an energy Δ’>0 (it is to be noted that the samples reportedin both Refs. 9 and 13 are of very high Tc and are thus “optimal systems”;) at thesame time, due to the above-explained pairing competition, the upper limit ofinterband pairs based on BB states falls from FL to -|Δ|, as schematically shown inFig. 10. Again, electrons between the two levels Δ’>0 and -|Δ| are not subject to thesame interactions or correlation effects as those above Δ’ or below -|Δ|, for they canonly pair with electrons below the nodal kink. Thus, the band between -|Δ| and Δ’ isan additional kinked band section, delineated by an upper FL kink at Δ’>0 and a lowerFL kink at -|Δ|, which correspond to the upper and lower edges of the traditionallycalled “superconducting gap” respectively.In this way, we have not only explained the energy variation of the BQP peak uponmoving away from the node but also explained the origin and structure of thesuperconducting gap, as a kinked band section at the FL. The lower edge –Δ of thekinked band section is determined by the instability of pairing at the location withrespect to that at the node, while the upper edge Δ’ of the kinked band section isdetermined 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 arenot symmetrical with respect to FL.As illustrated in Fig. 10, with the presence of the upper and lower FL kinks, pairingrelations similar to those at the node can be realized, by essentially the samemediating 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 theband below the gap (the kinked band section), as the AB states above the gap are ininterband pairing with respective BB states below the gap. Intraband pairing mediatedby ~70meV lattice mode might exist between states 1001 and 1003, which modemight also mediate pairing between 1004 and 1002. 10
  11. 11. Thus, upon leaving the node, while the upper limit of pairing (i.e. the lower edge oftraditional superconducting gap) is lowered due to pairing competition with respect topairing at the node, the lower limit of pairing (i.e. the nodal kink) is raised as the basestates 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 isslightly less than 20 meV above FL, accordingly the BB side of the kink is expectedto 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 ofAB 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 seeminglydispersion of measured BQPs.[13][14] In fact, the BQP distribution shown in Fig. 11taken from Ref. 14 looks very much like a kink. It is to be further noted that the slopeof the BQP band part should be more likely to duplicate that of the BB partimmediately above the traditional nodal kink (the BB part at state 1004 as shown inFig. 10), rather than that of the band part immediately below the “superconductinggap”. Another factor would be that, since BQPs are on AB in Bi2212, they would bemeasured as if they have a slight shift to the AB side when photon energy like 22.7 eVis used to allow simultaneous measurement of AB and BB.10. The nodal kinked band section as embryo of antinodal gapThe gap associated with kinked band at FL is the embryo of the antinodal gap, whichgrows in magnitude upon moving toward the antinode as evidence in Fig. 5(a). Asexplained with Fig. 1, the antinodal gap is not symmetry with respect to FL, which isin 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 andare measured as being on their respective base states. As the gap grows, pairingsmediated by the ~70 meV modes begin to lose their dominance over those mediatedby 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 indefiniteeven 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 atthe nodal location of cut C3. The slope of the kinked band section in the gap becomesgentle until the lower FL kink varnishes somewhere on the way to the antinode, whilethe pairing and gap patterns transform to those at antinode as shown in Fig. 1. It is notdetermined whether such transformation is a critical one. The fate of the upper FLkink and the pairing mediated by the ~70 meV modes needs further investigation. It isseen that although both antinodal and nodal pairings are dominated by the more stablepairing at the node, the antinodal kink is lowered by such dominance while the nodalkink is raised due to pairing match to the winning modes with respect to bandtopology. 11
  12. 12. 11. ConclusionIn a Bi2212 (and Bi2223) systems, the magnitude of an apparent energy gap is ameasure of relative instability of electron pairing at the gap location, for a greater gapindicates that a stabilized pairing (NSS state) of electrons can only be realized at ahigher binding energy. At a low temperature, the chemical potential of a system likeBi2212 is determined by the most stable pairing in it, and when pairing is sufficientlystable the chemical potential would be lowered by about the energy of the mediatingmode with respect to a free electron scenario. Bogoliubov quasiparticles could beexplained as excitations by lattice modes functioning to dissipate the depletedphonons in competition of lattice modes mediating pairings, the superconducting gapis a kinked band section, and the energy of BQP peak is that of upper states of pairsmediated by the winning modes of the mode competition. The depleted modes shouldbe responsible to high-temperature superconductivity. The kinked band section is anembryo of the antinodal gap, and on its way of transformation the kink at the bottomof the kinked band section varnishes. 20 103 105 0 104 102 -20 -40 101 -60 -80 -100 -120Fig. 1. From Gromko et al. (The reference numbers 101-105 and dashed or double-arrowed linesare added by this author for explaining relevant features.) BB part from 101 to 102 is the visible 12
  13. 13. kink section, the line from 104 to 105 roughly indicates the suppositional AB part engaging ininterband pairing with BB part from 101 to 102. AB part from 104 to 105 is basically invisible fornearly all electrons on this section are measured as “sinking” to their pairing counterparts on theBB part from 101 to 102. Intraband pairing may exist between some of the states from 102 to 103and their counterparts on the BB part from 101 to 102. E AB BB 204 203 FL’ FL 201 202 qFig. 2. Schmetical illustration of pairing relations in a bilayer band. Pairing between states 201and 204 competes with pairing between states 201 and 202. Pairings between states 201 and 204and between states 201 and 203, however, do not compete with each other; they tend to enhanceeach other. 13
  14. 14. Fig. 3. From Gromko et al. The magnitude of antinodal gaps of OD58, OD75, and OP91 Bi2212samples 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 BBsection above the kink is still confined very well to the phonon energy of ~30-35 meV. 14
  15. 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. 16. E AB BB 603 608 605 602 601 FL 607 606 604 Nodal kink qFig. 6. Pairing relations in a cut at the node. Lattice mode of ~70 meV mediates states605 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 ininset. 16
  17. 17. Fig. 8. (a) same spectra as in Fig. 7. (b) EDCs at the three points A, B, and C locationsare shown. Particular attention should be paid to the small dip at 50 meV with theirpositions shift matches that of BPQ peaks at about -20 meV. The seemingly mismatchbetween intensity of dip at C and the corresponding BQP can be explained by theexperimental setup in view of pairing model we proposed. 901 (π, π) 903C 903B (0, 0) 3 902 903A 2 1 A B CFig. 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 havebeen partly left out, as schematically indicated by partial coverage of square 903C by stripebetween lines 902. 1005 1003 17
  18. 18. 1005 1003 1006 1002 Upper FL kink Δ’ FL Δ 1001 Lower FL kink 1004 KinkFig. 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 lineswith double-arrows. Mode mediating interband pairing between 1004 and 1006 andstates 1001 and 1005 may be in competition with the mode(s) mediating interbandpairs at the node. The FL kink allows the depleted mode(s) at the node to be adaptedto a shortened lower kink. BQPs are generated at 1006 by mode mediating 1004 and1006. 18
  19. 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 bandsection corresponding to the gap, there is “normal” state density within the gap. The slope of theBQP band should be more likely to duplicate that of the BB part immediately above the nodalkink.[1] Qiang Li: “Explaining cuprates anti-nodal kink features on the basis of a model of electron pairing”, http://www.paper.edu.cn/index.php/default/en_releasepaper/content/4449070[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] TIAN, Duoxian: PCT/CN2010/075071.[4] Y.-D. Chuang, A. D. Gromko, A. Fedorov, Y. Aiura, K. Oka, Yoichi Ando, H. Eisaki, S. I. Uchida, and D. S. Dessau: Phys. Rev. Lett. 87, 117002 (2001). DOI: 10.1103/PhysRevLett.87.117002.[5] S. V. Borisenko, A. A. Kordyuk, V. Zabolotnyy, J. Geck, D. Inosov, A. Koitzsch, J. Fink, M. Knupfer, B. Buechner, V. Hinkov, C. T. Lin, B. Keimer, T. Wolf, S. G. Chiuzbăian, L. Patthey, and R. Follath: Phys. Rev. Lett. 96, 117004 (2006).[6] T. Yamasaki, K. Yamazaki, A. Ino, M. Arita, H. Namatame, M. Taniguchi, A. Fujimori, Z.-X. Shen, M. Ishikado, and S. Uchida: arXiv:cond-mat/0603006v2, Phys. Rev. B 75, 140513(R) (2007), DOI: 10.1103/PhysRevB.75.140513.[7] 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).[8] Tom Timusk and Bryan Statt: 1999 Rep. Prog. Phys. 62 61 doi:10.1088/0034-4885/62/1/002.[9] W.S. Lee, I. M. Vishik, K. Tanaka, D. H. Lu, T. Sasagawa, N. Nagaosa, T. P. Devereaux, Z. Hussain, Z. -X. 19
  20. 20. Shen: arXiv:0801.2819v2, DOI: 10.1038/nature06219.[10] D. S. Dessau et al., Phys. Rev. Lett. 66, 2160 (1991).[11] H. Iwasawa, J. F. Douglas, K. Sato, T. Masui, Y. Yoshida, Z. Sun, H. Eisaki, H. Bando, A. Ino, M. Arita, K. Shimada, H. Namatame, M.Taniguchi, S. Tajima, S. Uchida, T. Saitoh, D. S. Dessau, and Y. Aiura: arXiv:0808.1323v1, DOI: 10.1103/PhysRevLett.101.157005.[12] T. Cuk, D. H. Lu, X. J. Zhou, Z.-X. Shen, T. P. Devereaux, and N. Nagaosa: phys. stat. sol. (b) 242, No. 1, 11– 29 (2005) / DOI 10.1002/pssb.200404959.[13] H. Matsui, T. Sato, T. Takahashi, S.-C. Wang, H.-B. Yang, H. Ding, T. Fujii, T. Watanabe, A. Matsuda: arXiv:cond-mat/0304505v2, Phys. Rev. Lett. 90, 217002 (2003), DOI: 10.1103/PhysRevLett.90.217002.[14] Alexander V. Balatsky, W. S. Lee, Z. X. Shen: arXiv:0807.1893v1, DOI: 10.1103/PhysRevB.79.020505.[15] Qiang Li: “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/en_releasepaper/content/4390614 Or http://www.docin.com/p-123665085.html 20

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