# Ubiquitous Coexisting Electron-Mode Couplings in High Temperature Cuprate
Superconductors

Hongtao Yan1,2, Jin Mo Bok3, Junfeng He1†, Wentao Zhang1‡, Qiang Gao1, Xiangyu
Luo1,2, Yongqing Cai1, Yingying Peng1§, Jianqiao Meng1¶, Cong Li1, Hao
Chen1,2, Chunyao Song1,2, Chaohui Yin1,2, Taimin Miao1,2, Genda Gu4, Chengtian
Lin5, Fengfeng Zhang6, Feng Yang6, Shenjin Zhang6, Qinjun Peng6, Guodong
Liu1,2,7, Lin Zhao1,2,7, Han-Yong Choi8, Zuyan Xu6 and X. J. Zhou1,2,7,9,∗
1National Lab for Superconductivity, Beijing National laboratory for Condensed
Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing
100190, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3Department of Physics, Pohang University of Science and Technology (POSTECH),
Pohang 37673, Korea
4Condensed Matter Physics, Materials Science Division of Brookhaven National
Laboratory, Upton, NY 11973-5000, USA
5Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569
Stuttgart, Germany
6Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China
7Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
8Department of Physics, Sungkyunkwan University, Suwon 16419, Korea
9Beijing Academy of Quantum Information Sciences, Beijing 100193, China
†Present address: Department of Physics and CAS Key Laboratory of Strongly-
Coupled Quantum Matter Physics, University of Science and Technology of China,
Hefei, Anhui 230026, China
‡Present address: Key Laboratory of Artificial Structures and Quantum Control
(Ministry of Education), Shenyang National Laboratory for Materials Science,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai,
China
§Present address: International Center for Quantum Materials, School of
Physics, Peking University, Beijing 100871, China
¶Present address: School of Physics and Electronics, Central South University,
Changsha, Hunan 410083, China
∗Corresponding author<EMAIL_ADDRESS>
In conventional superconductors, the electron-phonon coupling plays a dominant
role in pairing the electrons and generating superconductivity. In high
temperature cuprate superconductors, the existence of the electron coupling
with phonons and other boson modes and its role in producing high temperature
superconductivity remain unclear. The evidence of the electron-boson coupling
mainly comes from the angle-resolved photoemission (ARPES) observations of the
$\sim$70 meV nodal dispersion kink and the $\sim$40 meV antinodal kink.
However, the reported results are sporadic and the nature of the involved
bosons are still under debate. Here we report new findings of ubiquitous two
coexisting electron-mode couplings in cuprate superconductors. By taking
ultra-high resolution laser-based ARPES measurements, combined with the
improved second derivative analysis method, we discovered that the electrons
are coupled simultaneously with two sharp phonon modes with energies of
$\sim$70 meV and $\sim$40 meV in different superconductors with different
doping levels, over the entire momentum space and at different temperatures
above and below the superconducting transition temperature. The observed
electron-phonon couplings are unusual because the associated energy scales do
not exhibit an obvious change across the superconducting transition. We
further find that the well-known “peak-dip-hump” structure, which has long
been considered as a hallmark of superconductivity, is also omnipresent and
consists of finer structures that originates from electron coupling with two
sharp phonon modes. These comprehensive results provide a unified picture to
reconcile all the reported observations and pinpoint the origin of the
electron-mode couplings in cuprate superconductors. They provide key
information to understand the role of the electron-phonon coupling in
generating high temperature superconductivity.

High temperature cuprate superconductors are derived from doping the parent
Mott insulatorsYTokura1998MImada ; XGWen2006PALee . They exhibit anomalous
normal state properties and unconventional superconductivity which are
attributed to strong electron correlation and electron interactions with other
collective excitations (boson modes) like phonons, magnetic fluctuations and
so onBStatt1999TTimusk ; JZaanen2015BKeimer . Revealing such many-body effects
is crucial to understanding the unusual properties and superconductivity
mechanism in cuprate superconductorsPWAnderson2007PWAnderson . With the
dramatic improvement of the instrumental resolutions, angle-resolved
photoemission spectroscopy (ARPES) has emerged as a powerful technique to
probe the many body effects in cuprate superconductorsZXShen2003ADamascelli ;
JRSchrieffer2006JRSchrieffer ; WTZhang2018XJZhou ; ZXShen2021JASobota . It has
been found that the band dispersion along the nodal direction exhibits a kink
at $\sim$70 meV in various cuprate superconductors although its origin remains
under debateZXShen2000PVBogdanov ; MRNorman2000MEschrig ; DGHinks2001PDJohnson
; KKadowaki2001AKaminski ; ZXShen2001ALanzara ; ZXShen2003XJZhou ;
YAndo2006AAKordyuk ; XJZhou2008WTZhang2 ; XJZhou2013JFHe . Another dispersion
kink at $\sim$40 meV is also revealed near the antinodal region which is
attributed to electron coupling with either phonons or magnetic resonance
modeDSDessau2003ADGromko ; JFink2003TKKim ; KKadowaki2003TSato ;
ZXShen2004TCuk ; NNagaosa2004TPDevereaux ; XJZhou2013JFHe . The well-known
peak-dip-hump structure observed in the superconducting state near the
antinodal region is considered as a hallmark of the electron-mode coupling in
cuprate superconductorsHRaffy1999JCCampuzano ; KKadowaki2001AKaminski ;
ZXShen2018YHe . It has been shown that superconductivity is closely related to
the strength of the electron-mode couplingsZXShen2018YHe ; GDGu2020TValla .
However, the observations of the electron-boson couplings in cuprate
superconductors have been mostly sporadic and sometimes controversial (Fig.
S1), lacking the consistency to pin down their nature and their role in
generating high temperature superconductivity.

In this paper, we report new findings and new understanding of the electron-
mode couplings from our comprehensive high resolution laser-based ARPES
measurements of the Bi-based cuprate superconductors. We discovered that the
$\sim$40 meV mode-coupling, which was observed only in Bi2Sr2CaCu2O8+δ
(Bi2212) near the antinodal region, is also present in Bi2Sr2CuO6+δ (Bi2201);
it is also observed off the antinodal region and even along the nodal
direction. On the other hand, the $\sim$70 meV mode-coupling, which was
observed mainly near the nodal area, is clearly revealed near the antinodal
region. We have found the ubiquitous coexistence of the $\sim$40 meV and
$\sim$70 meV mode-couplings in different cuprate superconductors over the
entire momentum space, both in the normal and superconducting states and at
different doping levels. The well-known “peak-dip-hump” structure, which is
usually observed in the photoemission spectra of Bi2212 near the antinodal
region in the superconducting state, is actually composed of a finer “peak-
double dip-double hump” structure that originates from electron coupling with
two sharp modes. These observations provide a unified picture to understand
all the previous results and the nature of the electron-boson couplings in
high temperature cuprate superconductors.

ARPES measurements were performed using our lab-based laser ARPES systems
equipped with the 6.994 eV vacuum-ultra-violet (VUV) laser and a hemispherical
electron energy analyzer or a time-of-flight electron energy analyzer (ARToF)
with the capability of simultaneous two-dimensional momentum space
detectionXJZhou2008GDLiu ; WTZhang2018XJZhou . The energy resolution was set
at $\sim$1 meV and the angular resolution is $\sim$0.3 ∘, corresponding to a
momentum resolution of $\sim$0.004 Å-1 at the photon energy of 6.994 eV. High
quality single crystals of Bi2201, Bi2212 and Bi2Sr2Ca2Cu3O10+δ (Bi2223) were
grown by the traveling solvent floating zone method and post annealed in
different atmospheres to get different doping levelsXJZhou2009JQMeng2 ;
XJZhou2010LZhao ; XJZhou2012SYLiu ; XJZhou2016YXZhang2 ; CTLin2016AMaljuk ;
XJZhou2019YDing  (Fig. S2 and Fig. S3). For convenience, we define hereafter
the name of the samples according to their doping level and superconducting
transition temperature ($T_{c}$) by using UD for the underdoped, OP for the
optimally-doped and OD for the overdoped. All the samples were cleaved
$\mathit{in\,situ}$ at a low temperature and measured in vacuum with a base
pressure better than 5$\times$10-11 Torr. The Fermi level is referenced by
measuring on a clean polycrystalline gold that is electrically connected to
the sample.

Figure 1 shows the observation of two coexisting mode-couplings in the band
structures measured in Bi2201 and Bi2212 along the nodal direction. Fig. 1a1
shows the band structure of an overdoped Bi2212 OD73K sample measured along
the nodal direction at 17 K. In order to reveal the fine electronic
structures, we took second derivative of the original data with respect to
energy to get images in Fig. 1b1 and Fig. 1c1. The detailed analysis of the
band is presented in Fig. S4 in Supplementary Materials. The comparison
between the original data (Fig. S4a) and the second derivative images (Fig.
S4e and Fig. S4f), together with the comparison between the original
photoemission spectra (energy distribution curves, EDCs, Fig. S4g) and their
corresponding second derivative EDCs (Fig. S4h), indicates that the
utilization of the second derivative method emphasizes on the curvature
change, thus suppressing the dramatic intensity variation in the original data
and sharpening the subtle features. This makes it possible to reveal fine
structures in the second derivative images (Fig. S4e and Fig. S4f, Fig. 1b1
and Fig. 1c1) that are hard to see in the original data (Fig. S4a and Fig.
1a1). For this reason, such an improved analysis method will be used
throughout the paper. The second derivative images are presented in two
different ways with one to highlight the peak and hump features (Fig. 1b1) and
the other to highlight the dip features (Fig. 1c1). As we will see below, the
identification of both the hump structure and the dip structure are necessary
to understand the electron-boson couplings in cuprate superconductors.

As seen in the second derivative image in Fig. 1b1, in addition to the main
band that is clear in the original data (Fig. 1a1), four new features can be
observed below the Fermi level. They show up as two flat bands on the left
side of the main band, LLH (left-low energy-hump) and LHH (left-high energy-
hump), and the other two flat bands on the right side of the main band, RLH
(right-low energy-hump) and RHH (right-high energy-hump). In the other second
derivative image in Fig. 1c1, four new features can also be observed below the
Fermi level marked as LLD (left-low energy-dip), LHD (left-high energy-dip),
RLD (right-low energy-dip) and RHD (right-high energy-dip). As illustrated
from a typical second derivative EDC in Fig. 1d1, the new features in Fig. 1b1
represent the hump structures in EDCs (marked as LH and HH in Fig. 1d1) while
in Fig. 1c1 they correspond to the dip structures in EDCs (marked as LD and HD
in Fig. 1d1). These results indicate that, in addition to the main peak,
coexisting double hump and double dip structures are observed in EDCs over a
wide range of momentum space. These can also be directly observed from the
original EDCs (Fig. S4g) and the corresponding second derivative EDCs (Fig.
S4h). The energy position of the two hump and two dip structures in EDCs does
not change when the momentum moves away from the main band and they have the
same energy position on both the left and right sides of the main band (Fig.
S4h). The energy position of the low energy hump (LH) and the high energy hump
(HH) in Fig. 1(b1,d1) is $\sim$60 meV and $\sim$91 meV, respectively, while
the energy position of the low energy dip (LD) and the high energy dip (HD) in
Fig. 1(c1,d1) is $\sim$38 meV and $\sim$75 meV, respectively.

We emphasize that the high resolution and high statistics data from our laser
ARPES measurements are critical in revealing the fine electronic structures in
Bi2212 (Fig. 1(a1-d1)) that were not observed before. We also took high
resolution laser ARPES measurements on the nodal band structure of the heavily
overdoped non-superconducting Bi2201 (ODNSC) sample (Fig. 1a2). The observed
results (Fig. 1a2-1d2) are rather similar to those in Bi2212 (Fig. 1a1-1d1).
Four new features can be clearly observed below the Fermi level in the second
derivative images (Fig. 1b2 and Fig. 1c2). In addition to the main peak,
coexisting double hump (LH: $\sim$52 meV and HH: $\sim$85 meV in Fig.
1(b2,d2)) and double dip (LD: $\sim$34 meV and HD: $\sim$65 meV in Fig.
1(c2,d2)) structures are observed in Bi2201 in EDCs over a wide range of
momentum space.

The peculiar fine structures in the measured nodal band structures of Bi2212
(Fig. 1a1-1d1) and Bi2201 (Fig. 1a2-1d2) can be understood by considering
electron coupling with two sharp boson modes. Fig. 1a3 shows the simulated
single-particle spectral function by involving electron-boson coupling with
two modes at 70 meV and 40 meV in the normal state (simulation details are
described in Supplementary Materials). The corresponding second derivative
images are shown in Fig. 1b3 and Fig. 1c3. It is clear that the simulated
results (Fig. 1a3-1d3) show remarkable resemblance to the measured results in
Bi2212 (Fig. 1a1-1d1) and Bi2201 (Fig. 1a2-1d2). These demonstrate
unambiguously that in the nodal electron structures of Bi2212 and Bi2201
electrons are coupled with two sharp boson modes. In order to understand the
measured data, several points need to be noted. First, as shown in Fig. S5 in
Supplementary Materials, two sharp boson modes must be included to produce two
hump structures (Fig. S5g) and two dip structures (Fig. S5h); one mode can
only produce one hump structure (Fig. S5b) and one dip structure (Fig. S5c)
that are not consistent with the measured results. Second, as shown in Fig. S5
and Fig. S6, the associated boson mode must have a finite linewidth which
causes an energy separation between the hump and dip structures in EDCs. The
boson mode with zero linewidth will result in the same energy position of the
hump and dip structures in EDCs that are apparently not consistent with the
measured results. Third, as shown in Fig. S6, when the involved boson modes
have finite linewidths, the energy position of the dips in EDCs does not
represent the mode energy. Instead, the mode energy is determined by the
energy positions of both the hump and dip structures as a group, close to the
middle position between them (Fig. S6d and Fig. S6h). In this way, the two
energy scales in Bi2212 can be determined to be $\sim$83 meV and $\sim$49 meV
from Fig. 1a1-1d1, and they are $\sim$75 meV and $\sim$43 meV in Bi2201 as
determined from Fig. 1a2-1d2.

The electron-boson coupling along the nodal direction has been studied
extensively in Bi2212 and Bi2201 through the electron self-energy analysis and
the $\sim$70 meV dispersion kink is ubiquitously observed ZXShen2000PVBogdanov
; MRNorman2000MEschrig ; DGHinks2001PDJohnson ; KKadowaki2001AKaminski ;
ZXShen2001ALanzara ; ZXShen2003XJZhou ; YAndo2006AAKordyuk ;
XJZhou2008WTZhang2 ; XJZhou2013JFHe . We also carried out such an analysis on
the high quality laser ARPES data of Bi2212 (Fig. 1a1) and Bi2201 (Fig. 1a2).
Fig. 1e1 shows the nodal dispersion of Bi2212 obtained from Fig. 1a1 by
fitting the momentum distribution curves (MDCs) at different binding energies.
To reveal the kink structures in the dispersion, several straight lines are
assumed as bare bands and the resultant “effective real part of electron self-
energy” is presented in Fig. 1f1. Two kinks are observed at $\sim$73 meV and
$\sim$39 meV although the energy position slightly depends on the bare band
selection. The observation of these two kinks in dispersion is consistent with
the finding of the electron coupling with two modes in Bi2212 as shown in Fig.
1a1-1d1. From a similar self-energy analysis of the Bi2201 data in Fig. 1a2,
two kinks can be identified at $\sim$73 meV and $\sim$41 meV (Fig. 1f2). It is
consistent with the observation of the electron coupling with two modes in
Bi2201 as shown in Fig. 1a2-1d2. We also carried out self-energy analysis on
the simulated data in Fig. 1a3. The two dispersion kinks at $\sim$70 meV and
$\sim$40 meV are in a good agreement with the electron coupling with two modes
shown in Fig. 1a3-1d3. Overall, we have found clear evidence of electron
coupling with two coexisting sharp modes in the nodal electronic structure of
Bi2212 and Bi2201.

Now we come to check on the electron coupling near the antinodal region using
our second derivative analysis method. Fig. 2a1 shows a high resolution laser
ARPES measured band of Bi2212 (OD73K) near the antinodal region; the
corresponding second derivative images displayed in two different ways are
shown in Fig. 2b1 and Fig. 2c1. Fig. 2d1 shows two EDCs at two typical
momentum points; their corresponding second derivative EDCs are shown in Fig.
2e1. As seen in Fig. 2b1, in addition to the main band, two flat features can
be observed below the bottom of the main band. These two flat features in Fig.
2b1 correspond to the two hump structures in EDCs at different momentum
points, marked as LH and HH in Fig. 2d1 and Fig. 2e1 (detailed analysis can be
found in Fig. S7 in Supplementary Materials). Likewise, the two flat features
in Fig. 2c1 represent the two dip structures in EDCs, marked as LD and HD in
Fig. 2d1 and Fig. 2e1. Fig. 2a2-2e2 show the measured band structure of Bi2201
(ODNSC) near the antinodal region (Fig. 2a2), its corresponding second
derivative images (Fig. 2b2 and Fig. 2c2), typical EDCs (Fig. 2d2) and the
corresponding second derivative EDCs (Fig. 2e2). Overall, the observed fine
structures in Bi2201 are similar to those observed in Bi2212. Two flat
features can also be observed in the second derivative images (Fig. 2b2 and
Fig. 2c2) although they are relatively weaker than those in Bi2212 (Fig. 2b1
and Fig. 2c1). They also correspond to the double hump and double dip
structures observed in EDCs (Fig. 2d2) and second derivative EDCs (Fig. 2e2).

The unusual fine structures in the measured band structures of Bi2212 (Fig.
2a1-2d1) and Bi2201 (Fig. 2a2-2d2) near the antinodal region can also be
understood by considering electron coupling with two sharp boson modes. Fig.
2a3 shows the simulated band structure by considering electron coupling with
two boson modes in the normal state. In the simulation, the energy position of
the original main band bottom (35 meV) is smaller than the energies of the two
modes (70 meV and 40 meV). As seen in Fig. 2a3-2e3, the simulated results can
well reproduce the observed band structures in Bi2212 and Bi2201, including
the observation of two flat features in the second derivative images (Fig. 2b3
and Fig. 2c3) and the double hump and double dip structures in EDCs (Fig. 2d3)
and second derivative EDCs (Fig. 2e3). These demonstrate unambiguously that in
Bi2212 and Bi2201 electrons near the antinodal region also couple to two
coexisting sharp boson modes. The two energy scales, determined from both the
hump and dip positions, are $\sim$50 meV and $\sim$90 meV for Bi2212 (Fig.
2b1-2e1) and $\sim$40 meV and $\sim$75 meV for Bi2201 (Fig. 2b2-2e2).

We note that, when the bandwidth is significantly larger than the mode energy,
the electron-mode coupling produces kink in the dispersion near the mode
energy as demonstrated in the simulated data of Fig. 1(a3,e3,f3) and observed
in Bi2212 (Fig. 1(a1,e1,f1)) and Bi2201 (Fig. 1(a2,e2,f2)). On the other hand,
when the bandwidth is comparable or smaller than the mode energy, the
electron-mode coupling can no longer produce kink in the dispersion. Instead,
the electron-boson coupling causes band renormalization; the bandwidth shrinks
from the original 35 meV to 23 meV as shown in the simulated data of Fig. 2a3.
The above results in Fig. 1 and Fig. 2 have demonstrated the advantages of the
second derivative method in identifying electron-mode coupling. When the
bandwidth is larger than the mode energy, although both the band dispersion
analysis and the second derivative analysis can be applied to identify the
mode-coupling and the results from these two methods are consistent, the
second derivative analysis is more reliable than the self-energy analysis
because the flat features in the second derivative images are more
characteristic and robust than the subtle slope change in the electron self-
energy. In particular, when the bandwidth is comparable or smaller than the
mode energy, the traditional self-energy analysis is no longer useful and the
second derivative analysis provides a more general and decisive way to pin
down the electron-mode coupling.

The electron-mode coupling has been observed in Bi2212 near the antinodal
region, famously known as the $\sim$40 meV antinodal mode-coupling that is
identified through the band dispersion and electron self-energy
analysisDSDessau2003ADGromko ; JFink2003TKKim ; KKadowaki2003TSato ;
ZXShen2004TCuk ; NNagaosa2004TPDevereaux ; XJZhou2013JFHe . Through the band
dispersion analysis, we observed a similar $\sim$40 meV energy scale in Bi2212
near the antinodal region on both the bonding band and the antibonding band,
as shown in Fig. S8a. The measured results and related analyses in Fig. 2
provide significant new insights on the electron-mode coupling in cuprates
superconductors. First, the $\sim$40 meV antinodal mode-coupling has been
observed only in Bi2212 so far; we have observed, for the first time, the
$\sim$40 meV antinodal mode-coupling in Bi2201 even in a heavily overdoped
non-superconducting sample (Fig. 2a2-2e2). Second, we have clearly observed
the coexisting $\sim$40 meV and $\sim$70 meV energy scales near the antinodal
region in Bi2212 and, for the first time, also in Bi2201. The dispersion kink
at $\sim$70 meV was once observed in optimally-doped Bi2212 near the antinodal
region in the superconducting state and was interpreted as due to the $\sim$40
meV mode-coupling with its energy position shifted by the superconducting
gapDSDessau2003ADGromko ; ZXShen2004TCuk . Our simultaneous observation of the
$\sim$40 meV and $\sim$70 meV energy scales near the antinodal region in
heavily overdoped non-superconducting Bi2201 indicates that these two mode-
couplings are separate and coexisting.

Figure 3 shows the momentum evolution of the two mode-couplings between the
nodal and the antinodal regions in Bi2212 and Bi2201. Fig. 3a shows the band
structures of Bi2212 OD73K sample measured along different momentum cuts; here
the second derivative images are presented to highlight the hump structures
caused by the electron-boson couplings. The two mode-couplings, observed along
the nodal direction (Fig. 1a1-1f1 and Fig. 3a1) and near the antinodal region
(Fig. 2a1-2e1 and Fig. 3a6), are present in the entire measured momentum space
as evidenced by the observation of two coexisting hump structures marked as LH
and HH in Fig. 3a1-3a6. Fig. 3b shows the second derivative EDCs from the
corresponding band structures in Fig. 3a1-3a6. Here the EDCs on the right side
of the Fermi momentum are chosen because as shown in the simulated data in
Fig. S5, when the momentum point moves away from the dispersion region, the
hump and dip structures become more pronounced. Two hump structures (LH and
HH) and two dip structures (LD and HD) can be clearly observed in each EDC in
Fig. 3b. From the energy positions of the hump and dip structures in Fig. 3b,
the energy scales of the two mode-couplings are determined and plotted in Fig.
3c. When the momentum cuts change from the nodal direction to the near
antinodal region, the two energy scales show little change with momentum
within the uncertainty of our data measurements and analysis. The coexistence
of the two mode-couplings is also observed in Bi2201 over the entire measured
momentum space although the signal of the low energy $\sim$40 meV energy
scales is relatively weak (Fig. 3e1-3e6 and Fig. 3f). These two energy scales
also exhibit little change with momentum variation from the nodal direction to
the antinodal region (Fig. 3g). We also determined the energy scales of the
two mode-couplings by using the traditional self-energy method in the momentum
space where such a method is applicable. Fig. 3i shows the measured electron
self-energy of Bi2212 along six momentum cuts (the details of the self-energy
extraction can be found in Supplementary Materials and Fig. S9). Two energy
scales are observed and the low energy scale (LScale) becomes pronounced with
the momentum cuts changing from the nodal direction to the near antinodal
region. The two energy scales observed (marked as LScale and HScale in Fig.
3i) are plotted in Fig. 3c; they show a good agreement with the results
obtained from the second derivative method. Fig. 3j shows the electron self-
energy of Bi2201 measured along different momentum cuts from the nodal
direction to the antinodal regions (the details of the self-energy extraction
can be found in Supplementary Materials). Two main energy scales can be
observed with the low energy scale (LScale) getting relatively pronounced and
becoming dominant when the momentum cuts change from the nodal direction to
the near antinodal region. The two energy scales determined from the self-
energy in Fig. 3j are plotted in Fig. 3g; they are also consistent with those
results obtained from the second derivative analysis.

We have observed the coexisting two mode-couplings in different
superconductors with various doping levels. As seen from Fig. S10 in
Supplementary Materials, the two mode-couplings can be observed from the band
structures along the nodal direction in Bi2201 over a wide doping range (Fig.
S10a), in underdoped, optimally-doped and overdoped Bi2212 (Fig. S10b) and in
Bi2223 (Fig. S10c). Note that these two mode-couplings are clearly observable
even in heavily overdoped non-superconducting Bi2201 and heavily overdoped
Bi2212 with a $T_{c}$ of 35 K ($p$$\sim$0.25). Our observation of the two
mode-couplings over the entire doping range and even in the heavily overdoped
non-superconducting Bi2201 sample is distinct from the previous reports that
the antinodal low energy mode-coupling becomes significantly weakened upon
crossing the critical doping of $\sim$0.19ZXShen2018YHe  or completely
disappears when the Bi2212 sample is heavily-overdoped to be non-
superconductingGDGu2020TValla . The energy scales of the two mode-couplings,
within the experimental uncertainty, fall into two categories: the high energy
scale lies in the energy range of (79$\pm$10) meV and the low energy scale
lies in the energy range of (46$\pm$10) meV (Fig. S10f). The two mode-
couplings are ubiquitous; they can be observed in Bi2201, Bi2212 and Bi2223
superconductors, at different doping levels and in the entire momentum space
(Fig. 3, Fig. S10 and Fig. S11). These results indicate that the coexisting
two mode-couplings is a common phenomenon in cuprate superconductors.

In order to understand the nature of the two coexisting mode-couplings, it is
crucial to investigate their temperature dependence, particularly the change
across the superconducting transition. To this end, we have taken high
resolution laser ARPES data on Bi2212 and Bi2201 at different temperatures, as
shown in Fig. 4. Fig. 4a shows the band structures of the Bi2212 OD73 K sample
measured along the nodal direction at different temperatures. The
corresponding second derivative EDCs are plotted in Fig. 4d. The high energy
mode-coupling can be clearly observed at 100 K and the low energy one is
visible at 90 K; both are above the $T_{c}$ of 73 K of the measured sample.
Fig. 4b shows the band structures of the Bi2212 OD73 K sample measured near
the antinodal region at different temperatures. The corresponding second
derivative EDCs are plotted in Fig. 4e. Both the high and the low energy mode-
couplings can be observed up to 100 K that is above the $T_{c}$ of 73 K of the
measured sample. Fig. 4c shows the band structures of the Bi2201 OD17 K sample
measured at different temperatures. The corresponding second derivative EDCs
are plotted in Fig. 4f. The high energy mode-coupling can be clearly observed
at 100 K and the low energy one is visible at 80 K; both are well above the
$T_{c}$ of 17 K of the measured sample. These results indicate that both mode-
couplings exist above $T_{c}$. The pseudogap temperature $T^{*}$ of the Bi2212
OD73 K sample is $\sim$89 KSShin2015TKondo  and it is $\sim$30 K for the
Bi2201 OD17K sampleCTLin2005GQZheng . These further indicate that the two
mode-couplings are present above $T^{*}$. In fact, the observation of the two
energy scales even in the heavily overdoped non-superconducting Bi2201 (Fig.
1a2-1f2, Fig. 2a2-2e2 and Fig. 3(e1-e6,f,g)) already demonstrates their
presence in the normal state that has nothing to do with the superconducting
transition or the pseudogap transition. Our observation on the persistence of
the low energy $\sim$40 meV energy scale in both Bi2212 and Bi2201 above the
temperatures of $T_{c}$ and $T^{*}$ is distinct from the previous reports that
the $\sim$40 meV energy scale observed in the superconducting state of Bi2212
near the antinodal region disappears above the superconducting transition in
terms of the self-energy analysisDSDessau2003ADGromko ; KKadowaki2003TSato ;
KKadowaki2006KTerashima2 ; XJZhou2013JFHe ; GDGu2020TValla . We note that the
temperature dependence of the two mode-couplings observed in the Bi2212 OD73 K
sample along the nodal direction (Fig. 4a and Fig. 4d) is quite similar to
that observed in the Bi2201 OD17 K sample (Fig. 4c and Fig. 4f). This strongly
indicates that the weakening of the mode-coupling signatures with the
increasing temperature in Bi2212 (Fig. 4d) and Bi2201 (Fig. 4f) is mainly due
to thermal broadening rather than superconducting transition.

The energy scales of the two mode-couplings do not exhibit an obvious change
across the superconducting transition or pseudogap temperature. Fig. 4g shows
the temperature dependence of the two energy scales in the band structures
along the nodal direction for the Bi2212 OD73K sample. The two energy scales
are obtained from both the second derivative EDCs in Fig. 4d (solid circles in
Fig. 4g) and from the self-energy analyses in Fig. 4j for the high energy
scale and in Fig. 4k for the low energy scale (empty circles in Fig. 4g). The
results obtained from these two methods are consistent with each other. Both
the energy scales show little change with temperature; there is no obvious
change at either the $T_{c}$ of 73 K or the pseudogap temperature $T^{*}$ of
$\sim$89 K. Fig. 4h shows the temperature dependence of the two energy scales
in the band structures near the antinodal region for the Bi2212 OD73K sample
obtained from the second derivative EDCs in Fig. 4e (solid circles in Fig. 4h)
and from the self-energy analysis in Fig. 4l for the low energy scale (empty
circles in Fig. 4h). Again, both the energy scales keep nearly constant with
temperature and no obvious change at the $T_{c}$ or the pseudogap temperature
$T^{*}$ is observed. Fig. 4i shows the temperature dependence of the two
energy scales for the Bi2201 OD17K sample obtained from the second derivative
EDCs in Fig. 4f (solid circles in Fig. 4i). Both the energy scales also show
little change with temperature. There is no obvious change at either the
$T_{c}$ of 17 K or the pseudogap temperature $T^{*}$ of $\sim$30 K. We note
that, even in two different superconductors Bi2212 and Bi2201 with quite
different $T_{c}$s of 73 K and 17 K and different superconducting gap sizes of
$\sim$20 meV and $\sim$8 meV, the observed two coexisting energy scales
($\sim$80 meV and $\sim$40 meV) and their temperature dependence are quite
similar (Fig. 4g-4i). These results rule out the possibility that, in the
previous reports, the observed $\sim$70 meV kink near nodalZXShen2008WSLee  or
antinodal regionsZXShen2004TCuk ; ZXShen2018YHe  and the $\sim$40 meV kink
near the antinodal regionZXShen2004TCuk  in the superconducting state of
Bi2212 are considered to originate from a low energy mode shifted by the
superconducting gap.

Our new findings of the ubiquitous two coexisting sharp mode-couplings in
different materials, in the entire momentum space, at different doping levels
and at different temperatures have put strong constraints on the nature of the
associated electron-boson coupling. The similar energy scales and the similar
momentum-, doping- and temperature-dependences observed in Bi2201, Bi2212 and
Bi2223 indicate that the two mode-couplings have a common origin. As the
energy scales exhibit a weak temperature dependence and do not show abrupt
change across $T_{c}$ and $T^{*}$, the energy of the two modes can be
determined with one being $\sim$70 meV and the other being $\sim$40 meV
although the exact values may vary slightly with temperature, doping or in
different materials. The previous observations of the nodal $\sim$70 meV mode-
couplingZXShen2000PVBogdanov ; MRNorman2000MEschrig ; DGHinks2001PDJohnson ;
KKadowaki2001AKaminski ; ZXShen2001ALanzara ; ZXShen2003XJZhou ;
YAndo2006AAKordyuk ; XJZhou2008WTZhang2 ; XJZhou2013JFHe  and the antinodal
$\sim$40 meV mode-coupling in the superconducting state
Bi2212DSDessau2003ADGromko ; JFink2003TKKim ; KKadowaki2003TSato ;
ZXShen2004TCuk ; NNagaosa2004TPDevereaux ; XJZhou2013JFHe  are consistent with
our present results but cover only a portion of the global electron-boson
coupling landscape (Fig. S1). The origin of the $\sim$70 meV nodal kink
remains unsettled with possible mechanisms like electron-phonon
couplingZXShen2001ALanzara ; MKrauss2003KPBohnen ; SGLouie2008FGiustino ;
DManske2008RHeid ; TPDevereaux2008DReznik ; NNagaosa2004TPDevereaux ;
SGLouie2021ZLLi , electron coupling with magnetic resonance mode or spin
fluctuationDGHinks2001PDJohnson ; BKeimer2009TDahm , or purely electron-
electron correlationDVollhardt2007KByczuk . The nature of the $\sim$40 meV
mode in the antinodal kink observed in the superconducting state of Bi2212 is
also under debate between possible candidates of phononsZXShen2004TCuk ;
NNagaosa2004TPDevereaux  or magnetic resonance modeDSDessau2003ADGromko ;
GDGu2020TValla . Our observations of the $\sim$70 meV mode-coupling in the
broader material, momentum, doping and temperature spaces lend further support
that the involved boson most likely corresponds to the phonon of the oxygen
breathing modeYEndoh1999RJMcQueeney . We have found that the $\sim$40 meV
mode-coupling is no longer limited to the antinodal region of Bi2212 in the
superconducting state; it is also present in other momentum space, above the
superconducting transition temperature and even in heavily overdoped non-
superconducting Bi2201 sample. These results rule out the possibility of the
magnetic resonance mode as the origin of the $\sim$40 meV mode are strongly in
favor of the phonon of the oxygen buckling modeIAAksay1995DReznik ;
AZawadowski1999TPDevereaux .

The momentum and temperature dependences of the identified $\sim$70 meV and
$\sim$40 meV mode-couplings are quite unusual in that their energy scales do
not change with momentum and temperature in the $d$-wave superconductors. In a
conventional picture where the electron-boson coupling vertex is momentum
independent, an Einstein phonon mode with an energy $\mathrm{\Omega}$ in the
normal state is expected to be shifted to
$\mathrm{\Omega}$+$\mathrm{\Delta_{0}}$ ($\mathrm{\Delta_{0}}$ is the maximum
of the $d$-wave superconducting gap) in the superconducting state over the
entire Fermi surfaceNEBickers2004AWSandvik . We have carried out similar
simulations by considering phonon modes with a finite linewidth (Fig. S12 and
Fig. S13 in Supplementary Materials) and found the same conclusion. Therefore,
such a conventional electron-phonon coupling picture can not explain the
temperature independence of the two mode-couplings we observed. In an
alternative electron-phonon coupling picture where the dominant forward
scattering is considered, a phonon mode with an energy $\mathrm{\Omega}$ in
the normal state is expected to be shifted to
$\mathrm{\Omega}$+$\mathrm{\Delta_{\mathbf{k}}}$
($\mathrm{\Delta_{\mathbf{k}}}$ is the size of the local $d$-wave
superconducting gap) in the superconducting state along the Fermi
surfaceOVDolgov2005MLKulic ; TPDevereaux2012SJohnston . This picture can
explain the zero energy shift of the modes across $T_{c}$ along the nodal
direction where the local gap is zero. But it can not explain the absence of
energy shift for the two modes crossing $T_{c}$ over the entire momentum space
when the local gap increases from zero at nodal to the maximum
$\mathrm{\Delta_{0}}$ at the antinodal region. Further theoretical efforts are
needed to understand the unusual momentum and temperature dependences of the
two electron-phonon couplings we have observed.

Our discovery of the two coexisting sharp mode-couplings that is omnipresent
in cuprate superconductors has significant implications on the origin of the
well-known peak-dip-hump structure observed in the superconducting state of
Bi2212 near the ($\mathit{\pi}$,0) antinodal regionAKapitulnik1991DSDessau ;
NKoshizuka1999AVFedorov ; HRaffy1999JCCampuzano . Since it was observed only
near the antinodal region, it was proposed that the peak-dip-hump structure
originates from electron coupling with collective excitations with a
$\mathbf{Q}$=($\mathit{\pi}$,$\mathit{\pi}$) like spin fluctuations or
magnetic resonance modeJRSchrieffer1997ZXShen ; KKadowaki1997MRNorman ;
DKMorr1998AVChubukov ; AVChubukov1999AAbanov ; MRNorman2000MEschrig . This
peak-dip-hump structure is considered as a hallmark of superconductivity and
is used to study the relationship between the electron-boson coupling and
superconductivityAJMillis2000JOrenstein ; ZXShen2018YHe . Our present work has
provided fundamentally new insights on the nature of the peak-dip-hump
structure in cuprate superconductors. First, as shown in Fig. 5, the peak-dip-
hump structure actually consists of one peak, two humps (LH and HH) and two
dips (LD and HD), forming a peak-double dip-double hump structure. This comes
directly from the two coexisting mode-couplings we have observed. Second,
different from the previous observations of the peak-dip-hump structure only
near the antinodal region, the peak-double dip-double hump structure is
observed over the entire momentum space, as shown in Fig. S4, Fig. S7 and Fig.
5. It can even be observed near the nodal region (Fig. 5(a1,c1) and Fig. S14).
Third, unlike the previous observation of the peak-dip-hump structure only in
Bi2212, the peak-double dip-double hump structure can also be observed in
Bi2201 as shown in Fig. 5(c1-c2,d1-d3) and Fig. S15. Fourth, different from
the previous observation of the peak-dip-hump structure only in the
superconducting state, the peak-double dip-double hump structure can also be
observed in the normal state as shown in Fig. S15. It can even be clearly
observed in the heavily overdoped non-superconducting Bi2201 (Fig.
5(c1-c2,d1-d3)). These observations strongly indicate that the peak-double
dip-double hump structure is also ubiquitous. It originates from electron
coupling with two sharp phonon modes other than the spin fluctuation or
magnetic resonance mode as proposed beforeJRSchrieffer1997ZXShen ;
KKadowaki1997MRNorman ; DKMorr1998AVChubukov ; AVChubukov1999AAbanov ;
MRNorman2000MEschrig .

Our present work has provided new and comprehensive information on the
electron-phonon coupling in cuprate superconductors. The observation of the
two coexisting sharp mode-couplings provides a unified picture to reconcile
the previous results of the nodal kink and antinodal kink in cuprate
superconductors. In conventional superconductors, the electron-phonon coupling
is always present at different temperatures both above and below $T_{c}$. This
does not mean that the electron-phonon coupling is not relevant to
superconductivity. Instead, it has been well established that it is the
electron-phonon coupling that induces the electron pairing and the
superconductivityJRSchrieffer1957JBardeen . Now we have found the ubiquitous
existence of the electron-coupling simultaneously with two sharp phonon modes
at $\sim$70 meV and $\sim$40 meV in different cuprate superconductors, at
different doping levels and at different temperatures above and below $T_{c}$.
We have also shown that the observed electron-phonon couplings are unusual
because their temperature and momentum dependences can not be understood in
the conventional electron-phonon coupling picture. This may be related to the
fact that the cuprate superconductors are doped Mott insulators with strong
electron-electron correlationXGWen2006PALee . Our results ask for further
theoretical efforts to understand the role of the observed electron-phonon
couplings in generating high temperature superconductivity in cuprate
superconductors.

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Acknowledgement This work is supported by the National Natural Science
Foundation of China (Grant Nos. 11888101, 11922414 and 11974404), the National
Key Research and Development Program of China (Grant Nos. 2016YFA0300300,
2016YFA0300602, 2017YFA0302900, 2018YFA0305602 and 2018YFA0704200), the
Strategic Priority Research Program (B) of the Chinese Academy of Sciences
(Grant No. XDB25000000 and XDB33000000), the Youth Innovation Promotion
Association of CAS (Grant No. 2017013), the Research Program of Beijing
Academy of Quantum Information Sciences (Grant No. Y18G06). J.M.B. was
supported by National Research Foundation (NRF) of Korea through Grants No.
NRF-2019R1I1A1A01057393. The work at BNL was supported by the US Department of
Energy, oﬃce of Basic Energy Sciences, contract No. DOE-sc0012704. H.Y.C. was
supported by the National Research Foundation of Korea under
NRF-2021R1F1A1063697.

Author Contributions
X.J.Z. and H.T.Y. proposed and designed the research. H.T.Y., J.F.H., W.T.Z.,
Q.G., X.Y.L., Y.Y.P., J.Q.M. and L.Z. carried out the ARPES experiments.
H.T.Y., Y.Y.P., J.Q.M., G.D.G., C.T.L. and L.Z. grew the single crystals.
H.T.Y., J.F.H., W.T.Z., Q.G., X.Y.L., Y.Q.C., Y.Y.P., J.Q.M., C.L., H.C.,
C.Y.S., C.H.Y., T.M.M., F.F.Z., F.Y., S.J.Z., Q.J.P., G.D.L., L.Z., Z.Y.X. and
X.J.Z. contributed to the development and maintenance of Laser-ARPES systems.
J.M.B. and H.Y.C. contributed to theoretical analysis. H.T.Y and X.J.Z.
analyzed the data and wrote the paper. All authors participated in discussions
and comments on the paper.

Additional information
Competing financial interests: The authors declare no competing financial
interests.

Figure 1: Observation of two coexisting mode-couplings in the band structures
along the nodal direction. (a1) Band structure of the Bi2212 OD73K sample
taken at 17 K along the nodal direction. The location of the momentum cut is
shown in the bottom-right inset by the red line. (b1) Second derivative image
of (a1) with respect to energy showing the absolute magnitude of the negative
values. (c1) Second derivative image of (a1) with respect to energy showing
the absolute magnitude of the positive values. (d1) Second derivative EDC at
the momentum point marked by the arrow in (b1). For convenience, we show the
negative of the second derivative values throughout the paper. In this case,
the peaks correspond to the observed band structure in (a1). (e1) Band
dispersion obtained by fitting MDCs at different energies from (a1). The pink,
purple and blue dashed straight lines connecting two energy positions in the
dispersion at the Fermi level and -0.3 eV, -0.15 eV and -0.06 eV,
respectively, represent empirical bare bands to extract effective real parts
of electron self-energy in (f1). The inset shows the corresponding MDC width
(Full width at half maximum, FWHM). (f1) Effective real parts of electron
self-energy from the dispersion in (e1) by subtracting different empirical
bare bands. (a2-f2) Same as (a1-f1) but for the Bi2201 ODNSC sample measured
at 15 K. (a3-f3) Same as (a1-f1) but for the simulated band structure. Two
modes, 40 meV and 70 meV, are assumed in the simulation, as shown in the inset
of (b3) (the details of the simulation are described in Supplementary
Materials). In (b1,b2,b3), in addition to the main band, four features appear
marked as LLH, RLH, LHH and RHH. In (c1,c2,c3), four features are observed
marked as LLD, RLD, LHD and RHD. In (d1,d2,d3), the observed features are
marked as HH, LH, HD and LD. As shown in Fig. S4 in Supplementary Materials,
the hump and dip features in (b1,b2,b3) and (c1,c2,c3) correspond to the hump
and dip structures in the corresponding EDCs and second derivative EDCs.
Figure 2: Observation of two coexisting mode-couplings in the band structures
near the antinodal region. (a1) Band structure of the Bi2212 OD73K sample
taken at 17 K near the antinodal region. The location of the momentum cut is
shown in the bottom-right inset by the red line. (b1) Second derivative image
of (a1) with respect to energy showing the absolute magnitude of the negative
values. (c1) Second derivative image of (a1) with respect to energy showing
the absolute magnitude of the positive values. (d1) EDCs at two momentum
points, k1 and k2, as marked in (a1). For clarity, the data are normalized by
the peak intensity near the Fermi level and are offset along the vertical
axis. (e1) Second derivative EDCs obtained from (d1). For clarity, the data
are normalized by the intensity difference between the HH and LD features.
(a2-e2) Same as (a1-e1) but for the Bi2201 ODNSC sample measured at 15 K.
(a3-e3) Same as (a1-e1) but for the simulated band structure. Two modes, 40
meV and 70 meV, are assumed in the simulation, as shown in the inset of (b3)
(the details of the simulation are described in Supplementary Materials). In
(b1,b2,b3), in addition to the main band, two features appear marked as LH and
HH. In (c1,c2,c3), two features are observed marked as LD and HD. In
(d1,d2,d3), the observed features are marked as HH, LH, HD and LD.
Correspondingly, in (e1,e2,e3), the observed features are marked as HH, LH, HD
and LD. As shown in Fig. S7 in Supplementary Materials, the hump and dip
features in (b1,b2,b3) and (c1,c2,c3) correspond to the hump and dip features
in the measured EDCs. Figure 3: Momentum dependence of the two mode-couplings
in Bi2212 and Bi2201. Details of the electron self-energy extraction are
described in Supplementary Materials. (a) Momentum-dependent band structures
of the Bi2212 OD73K sample taken at 17 K. These are the second derivative
images with respect to energy showing the absolute magnitude of the negative
values. The location of the six momentum cuts are shown in (d) by blue lines.
Two features are marked by LH (pink dashed line) and HH (blue dashed line).
(b) Second derivative EDCs obtained from (a). The corresponding momentum
position is marked by arrow in each panel of (a). For clarity, the data are
normalized by the intensity difference between the HH and LD features, and are
offset along the vertical axis. The two dip and two hump features in these
second derivative EDCs are marked by LD, HD, LH and HH. (c) The two energy
scales obtained from the second derivative EDCs in (b) (solid circles) and the
real part of electron self-energy in (i) (empty circles). From the second
derivative EDCs in (b), the high energy scale (HScale) is determined from the
middle point between HH and HD features while the low energy scale (LScale) is
from the middle point between LH and LD features. (d) The location of the
momentum cuts used in (a). (e-h) Same as (a-d) but for the Bi2201 ODNSC sample
measured at 15 K. (i) Real parts of the electron self-energy of the Bi2212
OD73K sample taken at 17 K along different momentum cuts. Here the momentum
cut positions are defined by the angle $\varphi$ as marked in (d). For
clarity, the data are offset along the vertical axis with the offset values
marked on the right side of each curve. Two energy scales can be observed as
marked by HScale and LScale. Their positions are plotted also in (c) as empty
circles. (j) Real parts of the electron self-energy of the Bi2201 ODNSC sample
taken at 15 K along different momentum cuts. The momentum cut positions are
defined by the angle $\varphi$ marked in (h). They are taken every two degrees
between $\varphi$=45 ∘ and $\varphi$=19 ∘. For clarity, the data are offset
along the vertical axis with the offset values marked on the right side of
each curve. Two energy scales can be seen as marked by HScale and LScale.
Their positions are also plotted in (g) as empty circles. Figure 4:
Temperature dependence of the two mode-couplings in Bi2212 and Bi2201. The
locations of the momentum cuts are shown in (m) by blue lines. (a) Band
structures of the Bi2212 OD73K sample taken along the nodal direction at
different temperatures. These are the second derivative images with respect to
energy showing the absolute magnitude of the negative values. The Fermi-Dirac
distribution is removed from the images. Two features are marked by LH (pink
dashed line) and HH (blue dashed line). (b) Same as (a) but for the momentum
Cut 2 near the antinodal region. (c) Same as (a) but taken from Bi2201 OD17K
sample along the momentum Cut 3. (d) Second derivative EDCs obtained from (a).
The corresponding momentum position is marked by an arrow in the left-most
panel of (a). For clarity, the data are offset along the vertical axis. The
two dip and two hump features are marked by LD, HD, LH and HH, respectively.
(e) Same as (d) but from (b). (f) Same as (d) but from (c). (g) The energy
positions of the two modes obtained from the second derivative EDCs in (d)
(solid circles) and the real part of electron self-energy in (j) (empty
circles). (h) Same as (g) but from (e) and (l). (i) Same as (g) but from (f).
(j) Real parts of the electron self-energy of the band structure measured
along the nodal directional Cut 1 at different temperatures. Details of the
electron self-energy extraction are described in Supplementary Materials. For
clarity, the data are offset along the vertical axis with the offset values
marked on the right side of each curve. The energy scale, marked as HScale,
can be clearly observed and its energy position is plotted in (g) as empty
circles. (k) Effective real parts of the electron self-energy from dispersions
taken along the nodal direction at different temperatures. They are obtained
by subtracting the empirical bare bands that are straight lines connecting two
energy positions in the dispersion at the Fermi level and -0.06 eV. The energy
scale, marked as LScale, can be observed and its energy position is also
plotted in (h) as empty circles. (l) Same as (j) but for the momentum Cut 2
near the antinodal region. The energy scale, marked as LScale, can be observed
and its energy position is included in (h) as empty circles. Figure 5:
Observation of the “peak-double dips-double humps” structure in Bi2212 and
Bi2201. (a1-a2) EDCs of the Bi2212 OD73K sample measured along the nodal
direction at two momentum points k1 (Fermi momentum, a1) and k2 (a2). The
measured band structure is shown in the inset of (a2) where the location of
the two momentum points k1 and k2 are marked by the arrows. The location of
the momentum cut is shown in the inset of (b3) by a blue line marked as Cut 1.
Two EDCs are shown in (a1) and (a2) that are measured at 17 K (black curve)
and 100 K (gray dashed curve). The inset purple line in (a1) is the difference
between the EDC at 17 K and a fitted Lorentzian; for clarity, it is multiplied
by five times ($\times$5) (the details to extract the purple line are
described in Supplementary Materials Fig. S14c). From the purple line in (a1)
and the EDC at 17 K in (a2), two dip and two hump features can be observed
marked as LD, HD, LH and HH, respectively. (b1-b3) EDCs of the Bi2212 OD73K
sample taken along the momentum cut near the antinodal region at three
momentum points k3 (Fermi momentum, b1), k4 (b2) and k5 (point at the
(0,0)-(0,$\mathit{\pi}$) line, b3). The measured band structure is shown in
the inset of (b2) where the location of the three momentum points k3, k4 and
k5 are marked by the arrows. The location of the momentum cut is shown in the
inset of (b3) by a blue line marked as Cut 2. Two EDCs are shown in (b1), (b2)
and (b3) that are measured at 17 K (black curve) and 100 K (gray dashed
curve). In (b1-b3), from the EDCs at 17 K, two dip (LD and HD) and two hump
(LH and HH) features can be observed and marked. (c1-c2) Same as (a1-a2) but
for the EDCs measured in the Bi2201 ODNSC sample at 15 K. (d1-d3) Same as
(b1-b3) but for the EDCs measured in the Bi2201 ODNSC sample at 15 K.