Multiple bosonic mode coupling in the electron self-energy of (La2-xSrx)CuO4.
ABSTRACT High resolution angle-resolved photoemission spectroscopy data along the (0,0)-(pi,pi) nodal direction with significantly improved statistics reveal fine structure in the electron self-energy of the underdoped (La2-xSrx)CuO4 samples in the normal state. Fine structure at energies of (40-46) meV and (58-63) meV, and possible fine structure at energies of (23-29) meV and (75-85) meV, have been identified. These observations indicate that, in (La2-xSrx)CuO4, more than one bosonic modes are involved in the coupling with electrons.
arXiv:cond-mat/0405130v3 [cond-mat.str-el] 29 Jul 2005
Multiple Bosonic Mode Coupling in Electron Self-Energy of (La2−xSrx)CuO4
X. J. Zhou1,2, Junren Shi3, T. Yoshida1,4, T. Cuk1, W. L. Yang1,2, V. Brouet1,2, J. Nakamura1, N. Mannella1,2, Seiki
Komiya5, Yoichi Ando5, F. Zhou6, W. X. Ti6, J. W. Xiong6, Z. X. Zhao6, T. Sasagawa1,7, T. Kakeshita8, H. Eisaki1,8,
S. Uchida8, A. Fujimori4, Zhenyu Zhang3,9, E. W. Plummer3,9, R. B. Laughlin1, Z. Hussain2, and Z.-X. Shen1
1Dept. of Physics, Applied Physics and Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, CA 94305
2Advanced Light Source, Lawrence Berkeley National Lab, Berkeley, CA 94720
3Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
4Dept. of Complexity Science and Engineering, University of Tokyo, Kashiwa, Chiba 277-856, Japan
5Central Research Institute of Electric Power Industry, Komae, Tokyo 201-8511, Japan
6National Lab for Superconductivity, Institute of Physics,
Chinese Academy of Sciences, Beijing 100080, China
7Department of Advanced Materials Science, University of Tokyo, Japan
8Dept. of Superconductivity, University of Tokyo, Bunkyo−ku, Tokyo 113, Japan
9Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996
(Dated: February 2, 2008)
High resolution angle-resolved photoemission spectroscopy data along the (0,0)-(π,π) nodal di-
rection with significantly improved statistics reveal fine structure in the electron self-energy of the
underdoped (La2−xSrx)CuO4 samples in the normal state. Fine structure at energies of (40∼46)
meV and (58∼63)meV, and possible fine structure at energies of (23∼29)meV and (75∼85)meV,
have been identified. These observations indicate that, in LSCO, more than one bosonic modes are
involved in the coupling with electrons.
PACS numbers: 74.25.Jb,71.18.+y,74.72.Dn,79.60.-i
The recent observation of the electron self-energy
renormalization effect in the form of a “kink” in the
dispersion has generated considerable interest because it
reveals a coupling of the electrons with a collective bo-
son mode of the cuprate superconductors. However,
the nature of the bosons involved remains controversial
mainly because the previous experiments can only be
used to determine an approximate energy of the mode
and this energy is close to both the optical phonon[2, 3]
and the spin resonance. Determining the nature of the
mode(s) that couple to the electrons is likely important
in understanding the pairing mechanism of superconduc-
In conventional superconductors, identification of the
fine structure for the phonon anomalies in the tunnelling
spectra has played a decisive role in reaching a con-
sensus on the nature of the bosons involved.
fine structure provides fingerprints for much more strin-
gent comparison with known boson spectra.
such fine structure has not been detected in the angle-
resolved photoemission spectroscopy (ARPES) data. In
this Letter we present significantly improved high resolu-
tion ARPES data of (La2−xSrx)CuO4(LSCO) that, for
the first time, reveal fine structure in the electron self-
energy, demonstrating the involvement of multiple boson
modes in the coupling with electrons.
The photoemission measurements were carried out on
beamline 10.0.1 at the ALS, using Scienta 2002 and
R4000 electron energy analyzers. As high energy res-
olution and high data statistics are crucial to identify
fine structure in the electron self-energy, the experimen-
tal conditions were set to compromise between these two
conflicting requirements. The measurement is particu-
larly challenging for LSCO system because of the ne-
cessity to use a relatively high photon energy (55eV).
Different energy resolution between 12 and 20 meV was
used for various measurements on different samples, and
the angular resolution is 0.3 degree. An example of the
high quality of the raw data is shown in Figs. 1a and
1b. Due to space charge problem, the Fermi level cali-
bration has a ±5 meV uncertainty. We mainly present
our data on the heavily underdoped LSCO x=0.03 (non-
superconducting), LSCO x=0.063 (Tc=12 K) and LSCO
x=0.07 (Tc=14 K) samples. These heavily underdoped
LSCO samples are best candidates because they exhibit
a stronger band renormalization effect above Tc; a rel-
atively large magnitude of the real self-energy makes the
identification of the fine structure easier. The LSCO sin-
gle crystals are grown by the travelling solvent floating
zone method. The samples were cleaved in situ in vac-
uum with a base pressure better than 4×10−11Torr. The
measurement temperature was ∼20K so all samples were
measured in the normal state.
Fig. 1c shows the energy-momentum dispersion rela-
tion along the (0,0)-(π,π) nodal direction extracted by
the MDC (momentum distribution curves) method. Be-
cause of the larger band-width along the nodal direction,
the MDC method can be reliably used to extract high
quality data of dispersion in searching for fine structures.
It has also been shown theoretically that this approach
is reasonable in spite of the momentum-dependent cou-
pling if we are only interested in identifying the mode
energies. As seen in Fig. 1c, there is an abrupt slope
change (“kink”) in the dispersions for LSCO samples
k - kF (A-1)
E - EF (eV)
Momentum k (A
EDC Intensity (Arb. Unit)
-0.6 -0.4 -0.2
E - EF (eV)
E - EF (eV)
FIG. 1: (a) Raw data of a two-dimensional image showing
the photoelectron intensity as a function of momentum and
energy for LSCO x=0.063 sample.
sented by the false-color. The measurement was taken along
the (0,0)-(π,π) nodal direction at a temperature of ∼20 K.
(b) The photoemission spectra (energy distribution curves,
EDCs) for the LSCO x=0.063 sample corresponding to Fig.
1a. The spectrum at the Fermi momentum kF=(0.44π/a,
0.44π/a) (red curve) shows a sharp peak. (c) The energy-
momentum relation determined from MDC method for LSCO
x=0.03(c1), x=0.063 (c2) and x=0.07 (c3) samples. The green
dashed lines connecting the two points at the Fermi energy
and -0.2 eV are examples of a simple selection of the bare
The intensity is repre-
with different dopings, similar to that reported before.
However, the new data with improved statistics indicates
that the “kink” has fine structure and subtle curvatures
in it, as seen for example in the LSCO x=0.03 sample
Since the bare dispersion is expected to be smooth in
such a small energy window, the “kink” and its fine struc-
ture represent effects associated with the electron self-
energy. The real part of the electron self-energy, ReΣ,
can be extracted from the measured dispersion (as in
Fig. 1c) by subtracting the “bare dispersion”. Following
the convention[8, 10], within a small energy range near
the Fermi level, the “bare dispersion” can be assumed as
ǫ0(k)=α1(k-kF)+α2(k-kF)2. As we will describe later,
the values of α1 and α2 are determined so as to yield
the best fit of the measured dispersion from the maxi-
mum entropy method (MEM); the choice of smooth bare
band has little effect on the fine structure that originate
from the abrupt change in the dispersion. The “effective”
ReΣ for LSCO at various doping levels is shown in Fig.
2a. Also note the data were taken at different energy
resolutions, using different analyzers and under different
measurement conditions, as described in the caption of
As seen from Fig. 2a, even with the most optimal ex-
perimental conditions we can achieve and in these sam-
ples that exhibit the strongest self-energy effects, there
remains considerable noise in the data statistics. How-
ever, by searching for peaks or curvature changes, we
can clearly identify some fine structure in the data. Two
clear features are around 40∼46 meV and 58∼63 meV,
as indicated by arrows in Fig. 2a, which show up clearly
in x=0.03 (Fig.2a1) and ∼0.06 samples (Fig.
although less clearly in x=0.063 and x=0.07 samples.
There may be another structure near 23∼29 meV, which
shows up mainly as a shoulder that is visible in x=0.03
(Fig. 2a1), 0.063 (Fig. 2a2), 0.07 (Fig. 2a3) and pos-
sibly in x=∼0.06 sample (Fig. 2a4). While these fine
features are subtle and one may argue about individual
curves, the fact that we have invariably observed them
near similar energies in many different samples and un-
der different measurement conditions make the presence
of the fine structure convincing.
A natural question is whether the fine structure can
be due to instrumental artifact which may be related to
detector inhomogeneity and/or system noise. The detec-
tor problem can be ruled out because: (1). The data
were taken using the “swept mode” of the electron ana-
lyzer, i.e., each energy point was averaged over the en-
tire detector range along the energy direction. Therefore,
all the energy points for the same angle were taken un-
der the same condition. (2). The inhomogeneity along
the angle direction is also minimized by normalizing the
photoemission spectra with the spectral weight above the
Fermi level which comes from the high harmonics of syn-
chrotron light and is angle-insensitive. (3). Among differ-
ent measurements, the corresponding detector angle with
respect to the dispersion bands vary due to slight changes
in measuring geometry. The fact that we observed con-
sistent results suggest that the results are intrinsic; (4).
We have carried out our measurements using different
analyzers, from SES2002 to the latest R4000, yielding
qualitatively similar results. One can also rule out the
possibility of noise because they are supposed to be ran-
dom. But for many different measurements on different
samples and under different experimental conditions, the
multiple structures are similar in energy within the un-
certainty caused by data statistics.
The direct inspection in the raw data (Fig. 2a) has
clearly established the existence of fine structure in the
electron self-energy. This is independent of any mod-
els that are used in the data analysis, including the
MEM method we use below. In order to better quan-
tify the characteristic energies of these fine features, we
take an approach to fit a smooth curve through the mea-
sured ReΣ data and then perform second-order deriva-
tive to the fitted curve.Given the statistics level in
the data, a spline through the data is difficult because
it is somewhat subjective. The MEM procedure, which
has been exploited to extract the spectral features of the
electron-phonon coupling from ARPES data for the two-
dimensional surface state of Be, is well suited for this
purpose. By incorporating prior knowledge, including
positiveness of the bosonic function, and zero value of
the bosonic function at zero frequency and above a max-
imum frequency of 100meV, the MEM is robust against
the random noise in the data, insensitive to the fitting de-
tails, and is therefore more objective. Since it remains
unclear whether the method developed for conventional
metal can be extended to cuprate superconductors, as
the first step, we use it only as a procedure for curve
The fitted curves are shown in Fig.
with the measured ReΣ; the corresponding second-order
derivative of the fitted curves are shown in Fig 2b. As
expected, two dominant features near 40∼46 meV and
58∼63 meV, and one possible feature near 23∼29, have
been resolved in Fig. 2b. This is consistent with the
fine structure identified to a naked eye in the raw data of
the electron self-energy (Fig. 2a). In addition, this data
analysis process also allows us to identify one more pos-
sible feature that may exist at high energy near 75∼85
meV, which seems to get stronger with increasing dop-
ing. We note that, while the agreement is not perfect
at this stage, there is sufficient similarity to suggest the
detection of multiple modes.
The fine structure in the electron self-energy originates
from the underlying bosonic spectral function. The mul-
tiple features in Fig. 2b show marked difference from the
magnetic excitation spectra measured in LSCO which is
mostly featureless and doping dependent. In compar-
ison, the features in Fig. 2b show more resemblance to
the phonon density-of-states (DOS), measured from neu-
tron scattering on LSCO (Fig. 2c), in the sense of
the number of modes and their positions. This similarity
between the extracted fine structure and the measured
phonon features favors phonons as the nature of bosons
involved in the coupling with electrons. In this case, in
addition to the half-breathing mode at 70∼80 meV that
we previously considered strongly coupled to electrons,
the present results suggest that several lower energy op-
tical phonons of oxygens are also actively involved.
For conventional metals, the MEM procedure can be
used to extract the Eliashberg function which gives spec-
tral features of the electron-phonon coupling.
strongly correlated cuprate superconductors, a priori it
is unclear whether the Eliashberg formalism is applica-
ble or not. However, we note that the nodal excitation of
LSCO in the normal state may provide a closest case for
the procedure to be applicable. Recent transport mea-
surements reveal that a quasiparticle picture may be still
reasonable for electrons near the nodal direction, even for
very low doping. This is consistent with the ARPES
data which show a well-defined peak in the nodal spec-
tra in the lightly doped samples. The nodal dispersion
does not have pronounced curvature in its bare band,
unlike the strong curvature near the saddle point of the
antinodal region. The selection of the nodal direction in
the normal state also minimizes complications due to the
existence of a superconducting gap or pseudogap in the
E - EF (eV)
Effective ReΣ (meV)
Second Derivative of Fitted ReΣ (Arb.Unit)
E - EF (eV)
Phonon Energy (meV)
Phonon DOS (meV-1)
FIG. 2: (a).The effective real part of the electron self-energy
for LSCO x=0.03 (a1), 0.063 (a2), 0.07 (a3)and ∼0.06 (a4)
samples. Data (a1-a3) were taken using Scienta 2002 ana-
lyzer, 10eV pass energy at an overall energy resolution (con-
voluted beamline and analyzer resolution) of ∼18meV. Data
(a4) were taken using Scienta R4000 analyzer, 5eV pass en-
ergy at an overall energy resolution of ∼12meV. For clarity,
the error bar is only shown for data (a4) which becomes larger
with increasing binding energy. The arrows in the figure mark
possible fine structures in the self-energy. The data are fitted
using the maximum entropy method (solid red lines). The
values of (α1, α2) (the unit of α1 and α2 are eV·˚ A and eV·˚ A2,
respectively) for bare band are (-4.25,0) for (a1), (-4.25, 13)
for (a2), (-3.7,7) for (a3) and (-4.3, 0) for (a4). (b). The
second-order derivative of the calculated ReΣ. The rugged-
ness in the curves is due to limited discrete data points. The
four shaded areas correspond to energies of (23∼29), (40∼46),
(58∼63) and (75∼85) meV where the fine features fall in. (c)
The phonon density of state F(ω) for LSCO x=0 (red) and
x=0.08 (blue) measured from neutron scattering.
Given these considerations and the fact that there is
no better alternative available, we made an attempt by
applying the Eliashberg formalism and the MEM proce-
dure to extract the effective bosonic function from the
real part of the electron self-energy (Fig. 3a). It is clear
that the multiple features are rather robust against the
choice of the bare band by varying α1and α2. All other
tests as detailed in  have been carried out. The fine
structure as obtained from LSCO x=0.03 is in agree-
ment with that in the second-order derivative shown in
Fig. 2b. The calculated real part of the electron self-
energy is plotted in Fig. 3a together with the measured
data. Fig. 3b shows the MDC width which is directly
related to ImΣ=(Γ/2)v0, with Γ being the MDC width
(Full-Width-at-Half-Maximum, FWHM) and v0the bare
Effective Bosonic Function
MDC Width (A-1)
E - EF (eV)
E - EF (eV)
Effective ReΣ (meV)
FIG. 3: (a)Real part of the electron self-energy for LSCO
x=0.03 as obtained from the dispersion shown in Fig. 1c1
(solid square) and calculated from the extracted effective
bosonic spectral using the MEM procedure (red solid curve)
with α1=-4.25 and α2=0. Also plotted are the effective
bosonic functions obtained by using different bare bands as
represented by the different sets of α1 and α2 values. (b)
The MDC width of LSCO x=0.03 (open circles). The contri-
bution from the electron-phonon coupling (blue line) is cal-
culated from the effective bosonic function in Fig. 3a with
α1=-4.25 and α2=0. The “impurity” contribution is assumed
to be a constant, 0.053˚ A−1(dotted black line). The momen-
tum resolution here is 0.019˚ A−1. After subtracting all the
electron-phonon and “impurity” contributions, the residual
part is fitted by Cωαwith C∼0.7 and α∼1.5 (green line).
is an overall increase of the MDC width with increas-
ing binding energy (Fig.3b), which is different from
simple electron-phonon coupling systems such as Be.
The MEM analysis allows us to calculate the contribu-
tion of the electron-phonon coupling which gives rise to
the abrupt drop in ImΣ. We note that this calculation
has some uncertainty related to the bare band selection
(Fig. 3a). After subtracting the contributions from the
electron-phonon coupling, “impurity” scattering, and an-
gular resolution, the residual part is found to be propor-
tional to ωα(Fig. 3b). This term most likely represents
the contribution of the electron-electron interaction. The
corresponding electron-electron contribution in the real
part of the self-energy is a smooth function and may
be absorbed into the “bare dispersion” because we fo-
cus only on abrupt structure in ReΣ in extracting the
bosonic spectral function. Here we also note that while
the imaginary part of the electron self-energy is consis-
tent with the existence of electron-phonon coupling, it is
difficult to identify the fine structure as has been done
for the real part because of the larger experimental un-
certainty in determining the peak width over the peak
position. This analysis also shows that there is an inter-
nal consistency in the MEM procedure that connects the
real and imaginary parts of the self-energy.
In summary, by taking high resolution data on heavily
underdoped LSCO samples with high statistics, we have
While there is a drop near 75 meV, there
detected fine structure in the electron self-energy. This
indicates multiple bosonic modes are involved in the cou-
pling to electrons in the LSCO system.
The work at the ALS and SSRL is supported by
the DOE’s Office of BES, Division of Material Sci-
ence, with contract DE-FG03-01ER45929-A001 and DE-
AC03-765F00515. The work at Stanford was also sup-
ported by NSF grant DMR-0304981 and ONR grant
N00014-04-1-0048-P00002. EWP is supported by DOE
DMS and NSF-DMR-0451163. The work at Oak Ridge
National Laboratory was partially supported through
DOE under Contract DE-AC05-00OR22725. The work
in Japan is supported by a Grant-in-Aid from the Min-
istry of Education, Culture, Sports, Science and Tech-
nology of Japan and the NEDO. The work in China
is supported by NSF of China and Ministry of Science
and Technology of China through Project 10174090 and
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