Role of the upper branch of the hour-glass magnetic spectrum in the formation of the main kink in the electronic dispersion of high-T$_\text{c}$ cuprate superconductors

2 MB·

by  Dominique Geffroy

Role of the upper branch of the hour-glass magnetic spectrum in the formation of the
main kink in the electronic dispersion of high-T cuprate superconductors
c
Dominique Geffroy,1,∗ Jiˇr´ı Chaloupka,1,2 Thomas Dahm,3 and Dominik Munzar1,2,†
1Department of Condensed Matter Physics, Faculty of Science,
Masaryk University, Kotla´ˇrska´ 2, 611 37 Brno, Czech Republic
2Central European Institute of Technology, Masaryk University, Kamenice 753/5, 62500 Brno, Czech Republic
3Universita¨t Bielefeld, Fakulta¨t fu¨r Physik, Postfach 100131, D-33501 Bielefeld, Germany
(Dated: January 26, 2016)
We investigate the electronic dispersion of the high-T cuprate superconductors using the fully
c
6 self-consistent version of the phenomenological model, where charge planar quasiparticles are cou-
1 pled to spin fluctuations. The inputs we use —the underlying (bare) band structure and the spin
0
susceptibilityχ—areextractedfromfitsofangleresolvedphotoemissionandinelasticneutronscat-
2
teringdataofunderdopedYBa Cu O byT.Dahmandcoworkers(T.Dahmet al.,Nat.Phys.5,
2 3 6.6
n 217 (2009)). Our main results are: (i) We have confirmed the finding by T. Dahm and coworkers
a thatthemainnodalkinkis,forthepresentvaluesoftheinputparameters,determinedbytheupper
J branchofthehour-glassofχ. Wedemonstratethatthepropertiesofthekinkdependqualitatively

4 on the strength of the charge-spin coupling. (ii) The effect of the resonance mode of χ on the
2 electronic dispersion strongly depends on its kurtosis in the quasimomentum space. A low (high)
kurtosis implies a negligible (considerable) effect of the mode on the dispersion in the near-nodal

] region. (iii) The energy of the kink decreases as a function of the angle θ between the Fermi sur-
n
face cut and the nodal direction, in qualitative agreement with recent experimental observations.
o
We clarify the trend and make a specific prediction concerning the angular dependence of the kink
c
energy in underdoped YBa Cu O .
- 2 3 6.6
r
p
u
I. INTRODUCTION between the Fermi surface cut and the Brillouin zone di-
s
agonal,fromabout65meVatthenode(i.e.,atthediago-
.
t
a The kink at 50–80 meV in the electronic dispersion nal),toabout55meVroughlyone-thirdofthewaytothe
m antinode. In addition, when going from the node to the
along the Brillouin zone diagonal (i.e., from (0,0) to
- (π,π)) of high-T cuprate superconductors1–8 has been antinode, the kink and also the underlying structures of
d c thequasiparticleself-energysharpendramatically. These
the object of intense scrutiny by the scientific com-
n trends of the kink energy and sharpness have been com-
munity since it was first reported. Understanding of
o
pared with simple estimates for several phonon modes
c the kink may be of importance in the context of the
and for the upper branch of the hourglass of spin fluctu-
[ quest for the mechanism of high temperature supercon-
ations,andthegreatestsimilarityhasbeenfoundforthe
ductivity. Unfortunately, a satisfactory understanding
1 latter.
has not yet been achieved. While there is a broad
v
9 (yet not unanimous9–13) consensus that the kink is Theaimsofthepresentstudyare(a)toaddressthean-
0 due to an interaction with bosonic excitations, the na- gulardependenceofthekinkusingthefullyselfconsistent
4 ture of the latter excitations remains controversial. It versionoftheEliashbergequationsemployedinprevious
6 is debated whether they are of lattice4,14–21 (phonon), studies by some of the authors50,51, and the same inputs
0 magnetic3,22–36 (spin fluctuation), or more complex37–42 (band structure and spin susceptibility) as in Ref. 48,
.
1 origin43. andtofindoutwhetherthemodeliscapableofaccount-
0 Regarding the magnetic scenario, it has been claimed ing for—in addition to the nodal dispersion—the trends
6 for some time that the kink reflects the coupling of the reported recently by Plumb et al. (b) To clarify the in-
1
chargedquasiparticlestotheresonancemodeobservedby terplay between the roles of the resonance mode and of
:
v neutronscattering44–47. InamorerecentstudybyDahm theupperbranchofthehourglassintheformationofthe
Xi andcoworkers48,however,itwasstronglysuggestedthat kink.
r in underdoped YBa2Cu3O6.6 (YBCO), the kink is due Therestofthepaperisorganizedasfollows. InSec.II
a to the upper branch of the hourglass dispersion of spin we summarize the equations employed in the calcula-
fluctuations, rather than to the resonance mode. This tions, present important computational details and dis-
has opened the question of how the influence of the res- cuss our choice of the values of the input parameters.
onance mode and that of the upper branch cooperate, Our results are presented in Secs. III and IV. In Sub-
under which conditions the former is the dominant one, section IIIA, we address qualitative aspects of the nodal
and under which the latter. kink, among others the role played by the kurtosis of
A relevant piece of information was recently reported the resonance mode of the spin susceptibility. In Sub-
by Plumb et al.49. These authors have shown that in section IIIB, we provide a detailed account of the re-
nearly optimally doped Bi Sr CaCu O (Bi2212), the lation between the energy and the shape of the nodal
2 2 2 8+δ
energy of the kink decreases as a function of the angle kink, and the structures of the quasiparticle self-energy.
2
In particular, we highlight the effect of the magnitude of
the coupling constant on the properties of the kink. In χ00(k,ω) [µB2/(eV f.u.)]
Sec. IV we address the evolution of the kink when going
from the node to the antinode. First (in Subsec. IVA),
6 60 600
we use the effective self-energy approach of Ref. 49 and
then (in Subsec. IVB) our own approach based on an 0.7
approximate relation between the properties of the kink
and those of the quantity S(k,E)≡Σ (k,E)+φ(k,E).
0 0.6
Here Σ (k,E) and φ(k,E) are the τ component of the
0 0
self-energy and the anomalous self-energy, respectively.
In Sec. V we compare our results with the experimental qx 0.5
dataofRefs.48and49. Itisshownthataminormodifi- a2π
cationoftheinputparametervaluesbringstherenormal-
0.4
ized (nodal) Fermi velocity and the energy of the nodal
kink close to the experimental values for YBCO48. The
calculated magnitude of the slope of the angular depen- 0.3
dence of the kink energy is only slightly larger than that 0 20 40 60 80 100
of Bi221249. We make a prediction concerning the angu- ω [meV]
lar dependence of the kink energy in underdoped YBCO
and provide a possible qualitative interpretation of the
FIG. 1. Cut of the spin excitation spectrum χ(cid:48)(cid:48)(q,ω) along
difference between the kink in underdoped YBCO and
the nodal axis, calculated using the set of parameter values
that in Bi2212.
S . Thesolidredlinecorrespondstothepositionofthevector
1
Q shown in Fig. 2.
0
II. SPIN-FERMION MODEL BASED
CALCULATIONS
the spin susceptibility, the dispersion relations (cid:15)A/B, the
k
Within the spin-fermion model28,43,52–55, the self- chemical potential µ, and the coupling constant g. For
all of them except for g, and except otherwise stated, we
energies Σ(cid:98)A(k,iEn) and Σ(cid:98)B(k,iEn) of the antibonding have used the parametrization published in Ref. 48, that
and bonding bands of a bilayer cuprate superconductor,
is based on fits of the neutron57 and photoemission data
such as Bi2212 or YBCO, are given by24:
ofunderdopedYBa Cu O . Thespinsusceptibilityex-
2 3 6.6
Σ(cid:98)A/B =g2(cid:2)χoSF∗G(cid:98)B/A+χeSF∗G(cid:98)A/B(cid:3). (1) hibits the hourglass shape with the resonance mode at
q = (π/a,π/a), illustrated in Figure 1 by a cut of the
Here g is the coupling constant, whose dependence on spectrum of χ(cid:48)(cid:48)(q,ω) along the nodal axis. The Fermi
kanidsenveegnleccotmedp,oχnoSeFn(tqs,oifωtnh)easnpdinχseSuFsc(qep,itωibni)litayr4e7,threespoedcd- s(cid:15)uBrfaarceesshcoowrrnesipnonFdigin.g2.toTthheeddiisstpaenrcseiosnfrroemlattiohnesΓ(cid:15)Akpoainndt
k
tively, and the symbol χSF∗G(cid:98)stands for to the Fermi surfaces, alonπg√the Brillouin zone diagonal
β1N (cid:88) χSF(k−k(cid:48),iEn−iEn(cid:48))×G(cid:98)(k(cid:48),iEn(cid:48)). (2) aknBd e=xp0re.3ss9e3d. Tinheunciatlscuolfataions2,aareredkonFA,eNfo=r T0.=34220,Kan.d
k(cid:48),iE(cid:48) F,N
n Finally,weaddressthecouplingconstantg. InRef.48,
Further,G(cid:98)A/B(k,iEn)aretheNambupropagatorsofthe themagnitudeofthesuperconductinggap∆SC wasfixed
renormalized electronic quasiparticles: (∆SC = 30meV), so that the value of the coupling con-
stant g could be obtained by imposing that the value of
1
G(cid:98)A/B(k,iEn)= iEnτ(cid:98)0−((cid:15)Ak/B −µ)τ(cid:98)3−Σ(cid:98)A/B(k,iEn), twhiethcatlhceulaatnegdlererneosormlveadlizpedhoFteoremmiisvseilooncit(yARbePEcoSn)sidstaetnat.
(3) This choice leads to a high value of the superconducting
where τ and τ are the Pauli matrices, (cid:15)A and (cid:15)B are transition temperature T of 174K. In the present work,
(cid:98)0 (cid:98)3 k k c
the bare dispersion relations of the two bands, and µ is the iterative solution of Eqs. (1) and (3) has been per-
thechemicalpotential. Wehaveconsideredonlytheodd formed in a fully self-consistent manner, along the lines
channel (i.e., only the term with χo in Eq. (1)). This of Refs. 50 and 51. The renormalized dispersions are
SF
channelhasbeendemonstrated24tobethedominantone, adjusted at each iteration, following the approach devel-
in particular because χe does not exhibit a pronounced oped in Refs. 48 and 58, in such a way that the renor-
SF
resonance mode56. A broadening factor δ is used in the malized Fermi surfaces are fixed and match the ARPES
analytic continuation of the propagators to the real axis profiles used as inputs. Within this framework, ∆ is
SC
(iE →E+iδ), δ =1meV. not constrained, so that its dependence on g has allowed
n
The input parameters of the model are the imaginary us to fix the value of g by requiring that ∆ =30meV.
SC
component χ(cid:48)(cid:48) (the indices are omitted for simplicity) of The resulting value of g of 1.0eV is considerably smaller
3
III. THE KINK IN THE DISPERSION
χ00(k,ω) [µB2/(eV f.u.)] RELATION ALONG THE NODAL AXIS
A. Role of the upper branch of χ(cid:48)(cid:48)
6 60 600
π
The solid blue line in Fig. 3 represents the electronic
dispersionalongthenodalaxisforthebondingband. For
θ a given energy, the associated value of k is obtained as
therootoftherealpartofthedenominatorofEq.(3). It
A2 B1 coincides with the value of k corresponding to the max-
Q imum of the spectral function for the given energy. The
0
dashed line connects the quasiparticle peak at k and
F
ω=38 meV
ky 0 ω=80 meV tthhee mhiagxhimenuemrgyofctuhteoffspoefct2r5a0lmfuenVc.tioTnhecokrrineskpiosnsdminogottho
and broad, with a relatively small amplitude. The dis-
A crepancy between this profile and the result of Ref. 48 is
1 B
2 mainly due to the lower value of g used in the present
study, as discussed in detail in Subsec. IIIB.
The position and the profile of the kink can be under-
stood in terms of a combination of the geometrical fea-
tures of the Fermi surfaces and those of the spin suscep-
π
− π 0 π tibility spectrum. Consider a scattering process whereby
−
k an electron from the bonding band, of quasimomentum
x
k and energy E, is scattered to the antibonding band,
quasimomentum k−q and energy E −ω, while a spin
FIG. 2. The Fermi surfaces for the antibonding (dashed
excitation of quasimomentum q and energy ω is emitted
line), and bonding (solid line) bands, obtained using the set
(an example with k = k and q = Q ≡ k −k is
of parametervalues S . The solid green arrowrepresentsthe B1 0 B1 A1
1 shown in Fig. 2). The process can occur with a con-
interband scattering vector Q . The red dashed-dotted line
0 siderable probability only if the momentum q is such
(the nearby dashed line) indicates an example of the Fermi
that χ(cid:48)(cid:48)(q,ω) is significant. Let us consider scattering
surfacecutusedinSubsec.IVA(IVB).Alsoshown aretwo
(suitablyshifted)constantenergycutsofthespinsusceptibil- processes along the direction of the Brillouin zone diag-
ity. The one shown in the upper right quadrant corresponds onal, from the region around kB1 to the region around
to χ(cid:48)(cid:48)(k−k ,ω = 38meV), the one shown in the bottom k = k −Q . Figure 2 shows that such processes
right quadraAnt1to χ(cid:48)(cid:48)(k−kA2,ω=80meV). haAv1e a neBg1ligible0 probability for ω (cid:39) 40meV (see the
constantenergycutshownintheupperrightquadrantof
Fig. 2). The contribution of the resonance mode to the
quasiparticle self-energy Σ(cid:98)B can thus be expected
|k=kB1
to be negligible, and the nodal dispersion to be almost
unaffected by the presence of the resonance mode. For
ω (cid:39)80meV – the energy of the crossing point of the red
line and the upper branch of the hourglass in Fig. 1 –,
than that of Ref. 48 (the coupling constant of the latter
(cid:113) however,theprobabilityisconsiderable(seetheconstant
reference U¯ is connected to our g by U¯ =g 2, and the
3 energy cut in the lower right quadrant of Fig. 2). The
valueofU¯ usedthereincorrespondstog =1.95eV). The nodal dispersion can thus be expected to be strongly in-
renormalizationofthenodalFermivelocityisweakerand fluenced by the coupling to spin excitations of the upper
tsheteovfalpuaeraomf eTtcerlovwaelruewsitjuhstthiinstrsomdaulcleerdviaslutheeofmga.inTsheet bshraowncnh.inInFdige.ed5,,tdhoeescanlocutleaxthedibsitpeacntyrusmignoifficImanΣt(cid:98)B|fke=atkuBr1e,
used throughout the paper, and is referred to as set S1. around 40meV due to the resonance mode. Instead, it
displays a steep onset around 80meV due to the upper
branch.
The calculations have been performed using the fast The kink itself (defined as the minimum of the sec-
Fouriertransformalgorithm,takingfulladvantageofthe ond derivative of the dispersion) is located at a higher
symmetries of the system. We have used a grid of 256× energy of about 130meV. The difference is due to two
256 points in the Brillouin zone and a cutoff of 4eV to facts. (a) The kink energy corresponds to the energy
limit the number of Matsubara frequencies. We have of the maximum of the real part of the self-energy (con-
checked,byvaryingthedensityofthegridandthecutoff, nectedtoitsimaginarypartthroughtheKramers-Kronig
that these values are sufficient. relation). This maximum is located at an energy higher
4
0 3.0
100
50 2.0
] )
5
meV] 100 1.0 [eV.k χ(0.00int10-1
d / Eschrig-Norman
[E 150 0.0 d∆/disp χ(q)00int10-2 MDaBhCm et al.
200 -1.0
10-3
250
0.3 0.4 0.5
0.39 0.41 0.43 0.45 a q
k [ −1] 2π x
FIG. 3. Dispersion relation along the Brillouin zone di- FIG. 4. The quantity χ(cid:48)(cid:48) , defined in the text, as a func-
int
agonal for the bonding band. The solid line represents the tion of q along the Brillouin zone diagonal. The three lines
x
renormalized dispersion. The dashed line represents a linear correspond to the three profiles of χ(cid:48)(cid:48)(q,ω) discussed in the
approximation to the bare dispersion. The dotted line is the text. The vertical red dashed line indicates the position of
derivative of the difference ∆ between the renormalized the interband vector Q .
disp 0
dispersion and the bare dispersion. The vertical dash-dotted
line is a guide to the eye. The calculations have been per-
formed using the set of parameter values S .
1 cluding the resonance mode. The solid green line, corre-
spondingtothespectrumofχ(cid:48)(cid:48)usedinthepresentstudy,
exhibits a broad peak and thin tails, both characteristic
than that of the onset of the imaginary part. This issue of a distribution with low kurtosis. The dashed blue line
isdiscussedindetailinSubsec.IIIB.(b)Theself-energy corresponds to the form of the spin susceptibility used
is k-dependent and in the region of k-space around the by two of the present authors in previous studies50,61,62
kink (where |k| < kB ), its imaginary part sets on at (the MBC form in the following). It possesses a higher
F,N
a higher energy than for k close to kB . This can be kurtosis, with both a narrower peak and fatter tails. Fi-
F,N
inferred from Figure 1: the energy of the crossing point nally, the black dash-dotted line represents the suscepti-
of the upper branch of χ(cid:48)(cid:48) with a fixed q horizontal line bility profile used by Eschrig and Norman in their thor-
increaseswhenthemagnitudeofqdecreases. Theimpact oughanalysisofthedispersionanomalieswithinthespin-
of the k-dependence of the self-energy on the energy of fermion model26 (see also Ref. 28). It also displays a rel-
the kink is quantitatively assessed in Subsec. IIIB. The atively high kurtosis. The vertical red dashed line sits
validity of the simple relation between the kink energy at the position of the interband vector Q . It can be
0
and the boson energy has been examined, in a different seen that both for the MBC profile and for the Eschrig-
context, by Schachinger and Carbotte59. Norman one, χ(cid:48)(cid:48) (|Q |) is significant, approximately an
int 0
order of magnitude larger than the corresponding value
The above analysis confirms the conclusions of Ref. 48
for the present spectrum of χ(cid:48)(cid:48). This has a direct im-
regardingtheoriginofthekink. However,itadditionally
pact on the magnitude of the contribution of the reso-
revealsthatthepresenceoftheupperbranchperseisnot
nance mode to the quasiparticle self-energy. Note, that
a sufficient condition for it to play the prominent role in
the spectrum of χ(cid:48)(cid:48) used here was obtained from a fit to
the formation of the nodal kink. Another necessary con-
dition is the simultaneous occurrence of a low kurtosis60 experimental inelastic neutron scattering data, while the
of χ(cid:48)(cid:48)(q,ω ) (where ω is the frequency of the reso- othertwospectra(MBCandEschrig-Norman)arebased
res res
on assumptions about the q-dependence. The considera-
nancemode)andofarelativelysmallvalueof|Q |. Only
0
tionsherearecomplementarytothoseofapreviouswork
under these conditions is the contribution of the reso-
nance mode negligible. A higher kurtosis of χ(cid:48)(cid:48)(q,ω ) byChubukovandNorman25,wheretheweakeningofthe
res
effect of the resonance on the near nodal dispersion has
or a larger value of |Q | would allow the contribution
0
been addressed using an analytical approach.
of the resonance to be large enough and dominate that
of the high-energy branch. This effect was confirmed by
separate calculations of the respective contributions of
the resonance mode and of the upper branch/continuum B. Impact of the magnitude of the coupling
for various shapes of the spectrum of χ(cid:48)(cid:48). constant
The low kurtosis exhibited by χ(cid:48)(cid:48)(q,ω ) is illustrated
res
in Fig. 4, which displays χ(cid:48)(cid:48) (q)=(cid:82)40meVχ(cid:48)(cid:48)(q,ω)dω as In this subsection, we examine the link between the
int 0
a function of q for q along the Brillouin zone diagonal. kink in the nodal dispersion and the features of the
The figure allows us to assess the q-space distribution of fermionic self-energy. Using Eq. (3), we find that the
the spectral weight of low energy spin fluctuations in- renormalized velocity v for a quasimomentum k along
5
the nodal axis is given by:
V] 100 25
v ((cid:15)¯ )+∂ Σ(cid:48)(k,(cid:15)¯ ) e
v((cid:15)¯ )= 0 k k k , (4) m
k 1−∂EΣ(cid:48)(k,(cid:15)¯k) [E) 50 20 1]−
where v0 is the bare velocity and (cid:15)¯k the renormalized Σ(00k 0 15 [eV
dispersion. The known form of the bare velocity allows E), 10 E)
one to approximate v0((cid:15)¯k) by its value at the Fermi sur- Σ(0k -50 A(k
face, v . Moreover, it is usually assumed that the mo- 5
mentumF0dependence of the self-energy is weak28, so that †,k
−
the term ∂ Σ(cid:48)(k,(cid:15)¯ ) in Eq. (4) can be neglected, and E -100 0
k k -300 -200 -100 0
the term ∂ Σ(cid:48)(k,(cid:15)¯ ) replaced with ∂ Σ(cid:48)(k = k ,(cid:15)¯ ).
E k E F k E [meV]
With these approximations, the energy dependence of
v is determined by the renormalization factor Z((cid:15)¯ ) =
k
1−∂ Σ(cid:48)(k = k ,(cid:15)¯ ), and the energy of the kink coin-
E F k FIG.5. Graphicalsolutionoftheequationforthequasipar-
cides with the energy of the extremum of Σ(cid:48)(k =kF,(cid:15)¯k). ticle energy (cid:15)¯k, for two different values of k along the nodal
In the following, we quantitatively assess the impact of axis: k = kB and k (i.e., the value of quasimomentum
F,N kink
themomentumdependenceoftheself-energyonthekink forwhichthenodalkinkoccurs),andthecorrespondingspec-
energy and shape, and identify two qualitatively distinct tra of the real and imaginary parts of the self-energy, and of
regimes. thespectralfunctionAk(E). Thecalculationshavebeenper-
Figure 5 illustrates the relationship between the en- formed using the set of parameter values S1. The solid lines
correspond to k = kB , the dashed lines to k = k . The
ergy of the kink and the energies of the features of the F,N kink
black lines represent the linear functions E−(cid:15) −µ, the red
self-energy, for the set of parameter values S . It shows k
1 lines the imaginary parts of the self-energy, whose real parts
the graphical solution of the equation for the quasipar-
are shown in blue. The green line represents the spectral
ticle energy (cid:15)¯ , for two values of k along the nodal axis:
k function for k .
kB and k (the value of quasimomentum at which kink
F,N kink
thekinkoccurs). Alsoshownarethecorrespondingspec-
tra of the real and imaginary components of the normal ] 300
V
self-energy,andfork ,inaddition,thenormalspectral e
kink m
afusnhcatiropnqAuka(sEip)a.rtTichleesppeeacktraatlEfun=ct0io.nFfoorrekaFBc,hNopfothsseestsweos [E) 150 4 1]−
values of k, (cid:15)¯k is determined as the energy of the cross- Σ(00k 0 [eV
ing between the corresponding black line (representing E), E)
ERe−Σ(cid:15)(kk+,Eµ)))a.nTdhteheenceorrgrieesspoofntdhiengcrbolsuseinlginpeo(irnetpsrecsoeinntciindge Σ(0k -150 2 A(k
,
with those of the quasiparticle peaks of Ak(E), as ex- †k−
pected. It can be seen that Σ(cid:48)(cid:48) sets on at around E -300 0
k=kFB,N -300 -200 -100 0
80meV as discussed in Sec. IIIA, and that the maxi-
E [meV]
mum of its Kramers-Kronig transform Σ(cid:48) occurs
k=kB
F,N
at a higher energy (approximately 110meV) due to the
finite width of the step in Σ(cid:48)(cid:48) . Finally, the afore- FIG. 6. The same quantities as in Fig. 5, calculated with
k=kFB,N the same input parameter values, except for g = 1.95eV,
mentioned assumption of weak momentum dependence
consistentwithRef.48. Noticethechangeinthescaleofthe
of the self-energy can be seen to be valid: even though
left axis, compared with Fig. 5.
the energy of the maximum of Σ(cid:48) is higher than
that of the maximum of Σ(cid:48) k=bkykin∆k (cid:39) 20meV,
k=kB kink
the shapes of the profiles are qFu,aNlitatively very similar. Thus, the maximum value of Σ(cid:48) is much larger,
k=kB
In particular, a sharp maximum is present in both pro- andthedistancebetweenkB andkF,N aswell. Figure6
files. This explains why the energy of the kink is only F,N kink
showsthatoversuchabroadk-interval,thequasimomen-
slightly (by ∆kink) higher than that of the maximum of tumdependenceofΣ(cid:48)(k,E)maynolongerbeconsidered
Σ(cid:48)k=kB , and why the kink is relatively sharp. to be weak. The flattening of Σ(cid:48) as k moves away from
F,N
It is worth contrasting these findings with the results the Fermi surface (expected irrespective of the chosen
ofthefullyself-consistentapproachwiththevalueofthe setofparametervalues)islargeenoughfortheprofileto
coupling constant g of 1.95eV (as in Ref. 48) in place of changequalitatively. Inparticular,thepronouncedmax-
g =1.0eV. Figure 6 illustrates the properties of the sys- imumofΣ(cid:48)disappearsbeforetheE−(cid:15) +µlinereachesit.
k
teminthiscase. Thelargevalueofthecouplingconstant Therefore,thepositionandtheshapeofthisextremumat
induces much larger magnitudes of the real and imag- kB are not the critical factors determining the energy
F,N
inary parts of the self-energy than in the former case. and the shape of the kink anymore. Instead, the depen-
6
denceoftheself-energyonk hasasubstantialimpacton
ReΣ (θ,E) [meV]
the profile of the kink. In terms related to Eq. (4), this eff
means that the weak momentum approximation breaks
down. 10 20 40 60
The interpretation of the formation of the kink there-
fore differs qualitatively between the former and the lat-
ter case. In the low-g regime, the energy of the kink -60
is approximately given by the energy of the maximum of
Σ(cid:48)(kB ,(cid:15)¯ ),andthekinkissharp. Inthehigh-g regime, -80
F,N k V]
the kink is made smoother by the influence of the mo- e
m -100
mentum dependence of Σ(cid:48). [
E
-120
-140
IV. THE KINK IN THE DISPERSION
RELATION AWAY FROM THE NODAL AXIS
-160
0 5 10 15
Havinganalyzedthebehaviorofthekinkinthedisper- θ [arc degree]
sion relation along the Brillouin zone diagonal, we now
proceed to examine how the situation evolves away from
FIG. 7. Heat map of the real part of the effective self-energy
the nodal axis, as a function of the angle θ between the Σ (θ,E) defined in the text, calculated using the set of pa-
eff
direction of the Fermi surface cut and the diagonal (for rametervaluesS . Foreachoftheselectedvaluesofθ,thered
1
a definition of θ, see Fig. 2). circle represents the energy of themaximum of ReΣ (θ,E),
eff
which coincides with the energy of the kink.
A. Effective self-energy approach
First, we follow the approach introduced by Plumb
et al.49. Figure 7 shows a heat map of ReΣ (θ,E), the the maximum decreases as q increases towards 0.5 from
eff
real part of the effective self-energy defined by Eq. (1) below. The profile of the self-energy can be expected to
of Ref. 49, and used in order to track the angular de- follow the same trend, which indeed occurs in Fig. 7.
pendence of the kink49. For the convenience of the
reader, the definition of Σ (θ,E) will be restated here. Following this analysis, we are in a position to conjec-
eff
Denote the inverse of the renormalized dispersion rela- turethatforlargevaluesofθ,thecontributionoftheres-
tion for a given value of θ by k¯(θ,E). Then we define onancemodetothescatteringbecomeslarge,andeventu-
ReΣeff(θ,E)≡(cid:15)¯k=k¯(θ,E)−(cid:15)k=k¯(θ,E). Inthepresentwork, ally dominates the profile. This should be accompanied
we have followed the approach of Ref. 49, and approxi- by a change of sign of the slope of Ωkink(θ) at a crit-
mated the bare dispersion by a straight line connecting ical angle θc. Simple geometrical considerations based
the quasiparticle peak at kF and the maximum of the on Fig. 2 provide θc (cid:39) 28◦. The coupling to the reso-
spectralfunctioncorrespondingtothehighenergycutoff nance mode has been put forward as the source of the
of200meV. Theheatmaphasbeenobtainedbyaninter- dispersion anomalies in earlier spin-fermion model based
polation of the results for a discrete set of θ-values. For studies26,28. Withintheframeworkofthesestudies,how-
each of these values, the red circle indicates the energy ever, the scattering mechanism does not exhibit a very
of the maximum of ReΣ , coinciding with the energy strongangulardependence,giventhehighkurtosisofthe
eff
Ω (θ) of the kink in the fermionic dispersion. resonancemode. Amorepreciseanalysisofthesituation,
kink
The most striking aspect of the result is the strong presented in Sec. IVB, shows that θc is larger than 20◦,
angular dependence of Ωkink. With increasing θ, Ωkink and that for θ > θc, the effective self-energy approach
decreasesandtheintensityandthesharpnessofthekink introduced above does not provide reliable estimates of
increase. Both observations are in qualitative agreement the kink energy.
with the experimental findings of Ref. 49. These trends
can be understood in terms of the interplay between the Note finally that the scenario outlined above is – from
fermionicdispersionandthebosonicspectrum,discussed the qualitative point of view – analogous to the one pro-
for the case of θ = 0◦ in Sec. IIIA. As the Fermi sur- posed by Hong and Choi39. These authors have also
face cut moves away from the nodal axis, the modulus arguedthattheobservedcomplexstructureofthequasi-
of the interband scattering vector along the (π/a,π/a) particle self-energy and its evolution when going from
direction increases. As a consequence, the section of χ(cid:48)(cid:48) the nodal cut to the antinodal one is determined by the
which contributes most to the scattering, changes. As presenceoftwoindependentcontributions: thatofares-
Fig. 1 shows, the spectral weight of the constant-q cut onancemodeandtheoneofaseparatebranchofbosonic
of the upper branch of χ(cid:48)(cid:48) increases, and the energy of excitations.
7
B. Relation between the kink and the features of
the quasiparticle self-energy 200 12
10
Here we present a different approach to determine the ] 150 ]
angular dependence of the kink energy, based on a nu- meV 8 1eV−
merical procedure for estimating the roots of the real [ 100 6 [
partofthedenominatoroftheGreen’sfunction(3). This Ta E)
methodisparticularlywellsuitedtothestudyofthekink , Te 50 4 A(k
for larger values of θ. For numerical reasons we use here 2
slightly different Fermi surfacecuts thanin Subsec.IVA.
0 0
The present ones are parallel to the Brillouin zone di- -100 -75 -50 -25 0
agonals. For an example of the two types of cuts, see E [meV]
Fig. 2.
The2×2self-energymatrixcanbeexpressedinterms
of the Pauli matrices: FIG. 8. Comparison of the expression from
Eq. (6), T ≡ (cid:0)Re[(E−S(k,E))(E−D(k,E))](cid:1)1/2
Σ(cid:98)(k,E)≡Σ0(k,E)τˆ0+ξ(k,E)τˆ3+φ(k,E)τˆ1, (black solide line) with its approximation T ≡
a
and the Nambu propagator as (cid:0)Re[E−S(k,E)]Re[E−D(k,E)](cid:1)1/2 used in Eq. (7)
(dashed blue line), for k = k corresponding to the cut
kink
(cid:20) (cid:21)−1 defined by θ = 26.9◦. The dashed-dotted line represents
G(cid:98)(k,E)= G(cid:98)−01(k,E)−Σ(cid:98)(k,E) the spectral function Ak(E). The calculations have been
performed using the set of parameter values S .
(cid:2) (cid:3) 1
E−Σ (k,E) τˆ +(cid:15)(k,E)τˆ +φ(k,E)τˆ
0 0 (cid:101) 3 1
= .
(cid:2) (cid:3)2
E−Σ (k,E) −(cid:15)(k,E)2−φ(k,E)2
0 (cid:101)
For large values of θ, where the gap is fully developed,
We have dropped the band index for simplicity, and
Σ (k,E)andφ(k,E)havecomparablemagnitudes. Asa
(cid:15)(k,E) stands for (cid:15)(k,E) − µ + ξ(k,E). The normal 0
(cid:101) consequence, Re[E−S(k,E)] and Re[E−D(k,E)] ex-
component of the propagator is given by
hibit very different profiles, while both remain weakly
E−Σ (k,E)+(cid:15)(k,E) k-dependent along a fixed cut. This is illustrated by
G(k,E)= 0 (cid:101) . (5)
(cid:2) (cid:3)2 Fig. 9, which shows the approximately linear profile
E−Σ (k,E) −(cid:15)(k,E)2−φ(k,E)2
0 (cid:101) of Re[E−D(k,E)] , contrasting with the peaked
|θ=26.9◦
Theapproachweintroducehereismosteasilypictured shape of Re[E−S(k,E)]|θ=26.9◦. The former profile,
asanextensionofSec.IIIBandFig.5tothecasewhere close to linear, emerges as the difference between two
φ(k,E) is finite. Provided the quasiparticle is well de- similarlypeakedfunctionsΣ (k,E)andφ(k,E)(plusthe
0
fined, its energy E is equal to the root of the real part linear function E). The similarity is due to the fact that
of the denominator, i.e., to the solution of the following both functions result from the convolution in Eq. (2).
equation in E, parametrized by k: The latter profile represents the sum of the two func-
tions (plus the linear function E), and therefore exhibits
Re(cid:2)(E−S(k,E))(E−D(k,E))−(cid:101)(cid:15)(k,E)2(cid:3)=0, (6) a peaked shape reminiscent of the similar shape of both
functions.
where S(k,E) ≡ Σ (k,E) ± φ(k,E) and D(k,E) ≡
0
The expressions entering Eq. (7) can be interpreted in
Σ (k,E) ∓ φ(k,E). The upper (lower) sign is used if
0
simple terms. The one on the left hand side displays
ReΣ (k,E) and Reφ(k,E) have the same (opposite)
0
a peak whose magnitude increases with increasing θ as
signs (recall that Reφ(k,E) possesses d-wave symmetry,
a consequence of the lengthening of the interband scat-
while ReΣ (k,E) is positive in the momentum-energy
0
tering vector, and of the corresponding increase of the
section we are considering). Assuming that the imagi-
spectral weight of the section of χ(cid:48)(cid:48) which contributes
nary parts of E − S(k,E) and E − D(k,E) are small
to the scattering processes. The term on the right-hand
comparedtotheirrealparts,wemayapproximateEq.(6)
side of Eq. (7) involves the inverse of an approximately
by:
linear expression. For fixed values of θ and k, the value
Re(cid:2)(cid:15)(k,E)2(cid:3) of this expression at the origin equals |Reφ(k,E =0)|.
(cid:101)
Re[(E−S(k,E))](cid:39) . (7) These observations allow us to interpret the profile of
Re[(E−D(k,E))]
the right-hand side of Eq. (7) as that of a hyperbola-
The validity of this assumption is related to that of the like function, with the origin of the E-axis displaced
quasiparticle picture, for an illustration, see Fig. 8. by −|Reφ(k,E =0)| (cid:39) −|Reφ(k=k (θ),E =0)| =
F
For θ = 0◦, S(k,E) = D(k,E) = Σ (k,E) ∆ (θ), as illustrated in Fig. 10. As k moves away from
0 SC
and Eq. (7) reduces to the simple equation deter- the Fermi surface, a family of hyperbola-like functions
mining the quasiparticle energy employed in Sec. III, (“hyperbolas”inthefollowing)isgenerated, withamul-
Re[E−Σ (k,E)−(cid:15)(k,E)]=0. tiplicativefactorRe(cid:2)(cid:15)(k,E)2(cid:3)appliedtothey-axis. The
0 (cid:101) (cid:101)
8
ReS(θ,E) [meV]
200
]
V Re[E D(k,E)]
me 100 Re[E−S(k,E)]
− 0 50 100 150 200
[
S]− 0
E
Re[ -100 -40
D],
−-200 -60
E
e[ V]
R e
-300 m -80
-100 -75 -50 -25 0 [E
E [meV] -100
-120
FIG. 9. Profiles of the terms Re[E−D(k,E)] and
-140
Re[E−S(k,E)] entering Eq. (7), for θ=26.9◦ and for a set 0 5 10 15 20 25
of values of the quasimomentum k, calculated using the set θ [arc degree]
of parameter values S . The lowest curves correspond to the
1
Fermisurface. Thequasimomentumk differsby∆k=π/128
from one curve to the next. For readability, each curve is FIG. 11. Heat map of the real part of the quantity S(k,E)
shiftedby20meVwithrespecttothepreviousoneaskmoves definedinthetext,calculatedusingthesetofparameterval-
away from the Fermi surface. ues S1. Foreach ofthe selected values ofθ, the pink triangle
represents the energy of the extremum of ReS(k,E), which
coincideswithΩ ,asdiscussedinthetext. Thesolidwhite
kink
linerepresentstheexpressionω +∆ (θ). Thesolidredcir-
res SC
300 40 cles, displayed for comparison, are taken from Fig. 7.
Re[†(k,E)2]
]
V Re[E D(k,E)]
me 150 Re[Ee−−S(k,E)] 20 ]
[E) 1V− Re(cid:101)(cid:15)(k,E)norD(k,E)exhibitapronouncedkink,weare
e
T(k,l 0 0 [E) ifnerampioonsiictiodnisptoercsoionnclulideseitnhatthethkeinokrigeixnhoibfitthedekbiynkthine ltehfet
k,E), -150 -20 A(k hand side of Eq. 7, Re[(E−S(k,E))]. The position of
( the kink can now be reliably evaluated by exploring the
Tr
-300 smooth quantity Re[(E−S(k,E))] defined on the fine
-40
-60 -50 -40 -30 -20 energy mesh.
E [meV] The approach detailed below has been used to obtain
the profile of Ω (θ) displayed in Fig. 11: For each se-
kink
lected value of θ, the momentum dependence of the self-
FIG. 10. Profiles of both sides of Eq. (7) and of the quasi- energy is examined. We then define k (θ) as the value
particle spectral function A (E) for θ = 26.9◦ and for a 0
k of k on the computational k-mesh, along the considered
set of values of the quasimomentum k, calculated using the
θ-cut (recall that the k-space cuts we use in this subsec-
set of parameter values S . As in Fig. 9, the quasimomen-
1 tionhavetheadvantageofmatchingthegeometryofthe
tum k differs by ∆k = π/128 from one curve to the next.
computationalk-mesh),whichisclosesttok (θ). This
The set of dashed blue (solid black) lines represents the kink
term Tr(k,E) ≡ Re(cid:2)(cid:101)(cid:15)(k,E)2(cid:3)/Re[(E−D(k,E))] (the term processisillustratedinFig.10. Givenavalueofθ,k0(θ)
T(k,E)≡Re[(E−S(k,E))]). Note that the energies of the is the value of k, such that the dashed line representing
pleaks of the spectral function (dotted red line) coincide with Re(cid:2)(cid:101)(cid:15)(k,E)2(cid:3)/Re[(E−D(k,E))] crosses the solid line
those of the crossing points of the corresponding blue and representing Re[(E−S(k,E))] close to its extremum.
black lines. Once k is fixed, we obtain the energy of the kink as
0
that of the extremum of ReS(k ,E) (we have checked
0
that in the present context the two energies coincide).
right-hand side of Eq. (7) thus evolves from a very sharp As discussed above, in the θ → 0 limit, this method for
hyperbola, for k → k (θ) , to a smooth hyperbola, for estimatingtheenergyofthekinkisequivalenttotheone
F
large values of |k−k (θ)|. used in Sec. IIIB , but there is one caveat: for small val-
F
This analysis shows that the left-hand (right-hand) uesofθ,thegapissmall,sothatthekinkinE−S(k,E)
side term of Eq. (7), indexed by (k,θ), is strongly isweakandmaynotalwaysdominatetheveryweakkink
(weakly) dependent on θ, but weakly (strongly) depen- in E−D(k,E). As a consequence, for small values of θ,
dent on k. In other words, Eq. (7) allows us to disen- the former method may be more accurate in estimating
tangle the sensitivities of the quantities of interest with the energy of the kink.
respect to k and θ. At this point, noticing that neither It can be seen in Fig. 11 that the present Ω (θ) is
kink
9
closetotheresultshowninSec.IVA.Themaindiscrep- order to reconcile the results of the calculations with the
anciesappearintheθ →0region(discussedabove), and experimental data. It can be expected that Ω (θ =0)
kink
for large values of θ. The latter arise because the kink decreases with increasing interband distance |Q | (see
0
becomes so intense, and sharp in momentum space, that Fig. 2 for a definition), but that it is not very sensi-
the former method, based on interpolations of the renor- tivetothedopinglevelorthebonding-antibondingsplit-
malized dispersion in k-space, does not provide a precise ting (provided that |Q | and the Fermi velocity are kept
0
estimate of the kink energy. fixed). Ouranalysisalsoindicatesthatawideningofthe
Theincreasedextentoftheaccessibleθ-domainallows upper branch of the hourglass should lead to a shift of
foraconfirmationoftheconjectureexposedinSec.IVA, Ω (θ =0)towardslowerenergiesandtoareductionof
kink
related to the role of the resonance mode. Figure 11 theslopeofΩ (θ). Finally, reducingthebandwidthof
kink
clearly shows that the slope of Ω (θ) changes sign at thebaredispersionshouldinducealoweringoftherenor-
kink
θ (cid:39) 23◦. We argued in Sec. IVA that if the kink is malizedFermivelocity. Wehavecheckedthesetrendsby
c
duetotheupperbranchofχ(cid:48)(cid:48),thentheslopeofΩ (θ) performing calculations of the same type as described in
kink
must be negative. This is the trend observed for θ <θ . sections III and IV for many different sets of values of
c
Conversely,iftheresonancemodeisthedominantsource the input parameters.
ofscattering,theθ-dependenceofΩ (θ)isdetermined As an example, and an illustration of the sensitivity
kink
mainly by that of ∆ (θ) and Ω (θ) must therefore of the results of the calculations to the input parame-
SC kink
display a positive slope close to that of ∆ (θ). This is ter values, we present below results of our calculations
SC
what we observe in the θ > θ region of Fig. 11, where obtained using a set of parameter values (S in the fol-
c 2
theprofileofΩ (θ)followsthatofω +∆ (θ),repre- lowing), where some of the values have been modified
kink res SC
sentedbythesolidwhiteline. ThefactthattheΩ (θ) along the lines of the previous paragraph. The values
kink
line is located somewhat above the ω +∆ (θ) line is of kA and kB are increased to 36.0% and 40.7% of
res SC √F,N F,N
likelyduetotheinfluenceofthelowerbranchofχ(cid:48)(cid:48). The π 2/a, respectively63. This shift applied to the band
discontinuityofΩkink(θ)atθ =θcisanartifactrelatedto structure leaves the system well within the limits given
the method for the numerical determination of Ωkink(θ). bypublishedexperimentalvalues: thevaluesofkA and
F,N
Finally, wenotetheremarkablesimilaritybetweenthe kB remainsmallerthan41%, thecommonvalueofthe
background of the heat map shown in Fig. 11 and the F,N
two parameters reported in Ref. 64. Furthermore, the
profile of the upper branch of χ(cid:48)(cid:48) displayed in Fig. 1,
correspondingincreaseinthemagnitudeof|Q |issmall,
arising from the selfenergy-χ(cid:48)(cid:48) relation (1). It illustrates 0
so that the resonance mode does not participate in the
the major role played by the upper branch of χ(cid:48)(cid:48) in the
scattering along the nodal cut, and the qualitative fea-
formation of the angular dependence of Ω (θ) in the
kink turesofFig.2areconserved. Thebandwidthofthebare
near nodal region.
dispersion is reduced by 40%, so that the value of the
renormalized Fermi velocity is close to the experimental
one,andwesetg =0.8eV,sothatthemaximumvalueof
V. COMPARISON WITH EXPERIMENTAL the gap remains unchanged at 30meV. Finally, the up-
DATA per branch of χ(cid:48)(cid:48) is made wider, so as to further reduce
the value of Ω (θ = 0) and the slope of the profile of
kink
The main trend of Subsection IVA, i.e. the decrease Ω 65.
kink
of |Ω | when going from the nodal cut to the antin- Figure 12 displays the renormalized dispersion calcu-
kink
odal one, is consistent with the experimental findings of latedusingthesetofparametervaluesS . Itcanbeseen
2
Ref. 49. Our results provide support for the conjecture that the kink is much more pronounced. As expected,
that the decrease is associated with the dispersion of the the energy of the kink (ca 90meV) and the renormalized
upper branch of the hourglass. The calculated value of Fermi velocity (ca 1.5eV˚A) are considerably lower than
the energy of the nodal kink ((cid:39) 130meV), however, is inFig.3,andclosetotheexperimentalvaluesofRef.48.
much higher than that of underdoped YBCO reported The corresponding angular dependence of Ω is
kink
in Ref. 48 (80meV). In addition, the calculated magni- shown in Fig. 13. It can be seen that the magnitude
tude of the slope of Ω (θ) (3.5meV per arc degree) is of the slope of Ω is reduced to only 1.1meV per arc
kink kink
much higher than the experimental value of Bi2212 re- degree, reasonably close to the experimental value for
ported in Ref. 49 (0.8meV per arc degree). Finally, the Bi221249. The value of θ of Fig. 13 (ca 26◦) is higher
c
renormalizedFermivelocityof2.8eV˚Aonthenodalaxis thanthatofFig.11. Thedifferenceismainlyduetothat
(see Fig. 3), is much larger than the experimental value between the bare dispersion relations of S and those of
1
of underdoped YBCO of 1.8eV˚A. This discrepancy is S . TheinterpretationexposedattheendofSec.IVstill
2
connected with the fact that the value of g used in the applies. Based on this interpretation and the above dis-
set S is much smaller than that of Ref. 48. cussionwecanmakeapredictionconcerningtheangular
1
Based on our interpretation of the origin of the kink, dependence of Ω in underdoped YBCO. We predict
kink
it is possible to understand the influence of the model that there exists a critical value θ , such that for θ < θ
c c
parameters on the profile of Ω (θ). We are also well (θ > θ ), Ω (θ) is a decreasing (weakly increasing)
kink c kink
equipped to find out which adjustments are necessary in function. The minimum Ω (θ ) of Ω is determined
kink c kink
10
a fairly high kurtosis of χ(cid:48)(cid:48)(q,E) [see Fig. 2 (c)
|E=42meV
0 3.0 of Ref. 66], and that the higher energy cuts of χ(cid:48)(cid:48)(q,E)
shown in Figs. 2(a) and 2(b) of Ref. 66 are considerably
40 2.0
] wider than those of underdoped YBCO. In particular,
V] 80 1.0 eV. the values of χ(cid:48)(cid:48) for q = 0.19 r.l.u. (corresponding to
me [dk |Q0| of Fig. 1) and ω = 42meV, 54meV and 66meV in
[E 120 0.0 ∆/disp Fofigas.c2om(cp)a,r(abb)leanmda(gan)itoufdRe.ef.M6o6t,ivaareteadllbsyigtnhiifiscaonbts,eravnad-
160 -1.0 d tion and by the large width of the nodal kink in Bi2212
(seeFig. 1(d)ofRef.49),weproposethefollowingqual-
200
itative interpretation of the angular dependence of Ω
0.38 0.40 0.42 0.44 0.46 kink
1 in Bi2212: we suggest that the nodal kink is not deter-
k [ − ]
mined by a single narrow cut through the upper branch
ofthehourglass, asinthecaseofunderdopedY-123(see
FIG. 12. The same quantities as in Fig. 3. The calculations Fig. 1), but rather by a broad band of χ(cid:48)(cid:48) ranging from
have been performed using the set of parameter values S . ca 40meV to ca 100meV. Even the 42meV cut con-
2
tributes because of the high kurtosis. With increasing
θ, lower energy segments of χ(cid:48)(cid:48) become more influential,
ReS(θ,E) [meV] for the same reasons as discussed in Sec. IVA, and as a
consequence, the energy of the kink slighly decreases.
0 50 100 150 200 250 300 350
VI. SUMMARY AND CONCLUSIONS
We have investigated the effect of the upper branch
-60
] of the hour-glass magnetic spectrum on the electronic
V
e dispersion of high-T cuprate superconductors using
m c
[ the fully self-consistent version of the phenomenological
E
-80 model, where charged planar quasiparticles are coupled
to spin fluctuations. The same input band structure and
thesameinputspinsusceptibilityasinthepreviousstudy
by T. Dahm and coworkers have been used.
-100
0 5 10 15 20 25 First, we have confirmed the finding by Dahm et al.,
θ [arc degree] that the nodal kink is determined, for the present val-
ues of the input parameters, by the upper branch of χ(cid:48)(cid:48).
We have further demonstrated that the position and the
FIG.13. ThesamequantitiesasinFig.11,calculatedusing
shape of the kink depend strongly on the strength of the
the set of input parameter values S . The apparent steps in
2
the pink triangle profile are due to the reduced energy range charge-spin coupling. For low (but still realistic) values
of the E-axis, and the discretization of the energy mesh. of the coupling constant, the position of the kink can
be estimated using the common approximation, where
the quasimomentum dependence of the self-energy along
by ∆ (θ ) and by the lower branch of χ(cid:48)(cid:48). A value in the Fermi surface cut is neglected. The kink is weak but
SC c
the range from 40meV to 60meV can be expected. This sharp. Forhighvaluesofthecouplingconstant,however,
prediction could be tested in ARPES experiments. thedependenceoftheself-energyonthequasimomentum
Finallyweaddress,inlightofourfindings,theΩ (θ) plays an important role. The kink is less sharp, but has
kink
line for nearly optimally doped Bi2212 reported in a larger amplitude.
Ref. 49, which was one of our starting points. The Second, we have shown that the kurtosis of the res-
energy of the nodal kink in Bi2212 of ca 65meV is onance mode of the spin susceptibility in the quasi-
roughly 15meV lower than that of underdoped YBCO momentum space has a major influence on the mecha-
and 25meV lower than our result shown in Fig. 13. The nism of the fermionic scattering. If the kurtosis is low
magnitudeoftheslopeofΩ inBi2212isonlyslightly (high), as in the present study (as in several previous
kink
smallerthanthatofourcalculations. Thedifferencemay studies26,50,61,62),theeffectoftheresonancemodeinthe
becausedbyadifferenceintheFermisurfacesand/orby near-nodal region of the Brillouin zone is weak (large),
adifferenceinχ(cid:48)(cid:48). Sincethemagnitudeoftheinternodal and the upper branch of the hour-glass (the resonance
distance, |Q |, of optimally doped Bi2212 is almost the mode)playsthemajorroleintheformationofthenodal
0
sameasthatofunderdopedYBCO,itappearsthatsome kink.
differenceinχ(cid:48)(cid:48) playsthecrucialrole. Notethattheneu- Third, thecalculatedenergyofthekinkdecreasesasa
tron scattering data of optimally doped Bi221266 reveal functionoftheangleθbetweentheFermisurfacecutand