Aspects of Electron-Phonon Self-Energy Revealed from Angle-Resolved
  Photoemission Spectroscopy
Aspects of Electron-Phonon Self-Energy Revealed from Angle-Resolved
Photoemission Spectroscopy
W.S. Lee,1 S. Johnston,2 T.P. Devereaux,2 and Z.-X. Shen1
Department of Physics, Applied Physics, and Stanford Synchrotron
Radiation Laboratory, Stanford University, Stanford, CA 94305
Department of Physics, University of Waterloo,Waterloo, Ontario, Canada N2L 3G1
(Dated: November 4, 2018)
Lattice contribution to the electronic self-energy in complex correlated oxides is a fascinating
subject that has lately stimulated lively discussions. Expectations of electron-phonon self-energy
effects for simpler materials, such as Pd and Al, have resulted in several misconceptions in strongly
correlated oxides. Here we analyze a number of arguments claiming that phonons cannot be the
origin of certain self-energy effects seen in high-Tc cuprate superconductors via angle resolved pho-
toemission experiments (ARPES), including the temperature dependence, doping dependence of the
renormalization effects, the inter-band scattering in the bilayer systems, and impurity substitution.
We show that in light of experimental evidences and detailed simulations, these arguments are not
well founded.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
The microscopic pairing mechanism of the high-Tc
superconductivity remains an unsolved question even
after twenty years of its discovery. Observations of
a kink at around 40-70 meV in the band dispersion
along the diagonal of the Brillouin zone (nodal diec-
tion) and a peak-dip-hump (PDH) structure at the zone
boundary by angle-resolved photoemission spectroscopy
(ARPES)1,2,3,4,5,6,7,8,9,10,11,12,13 have drawn a great deal
of recent attention as they may shed some light on this
problem. Although an agreement has been established
that the kink and PDH structure are signatures of the
electrons coupled to a sharp bosonic mode, it is still
strongly debated about the origin of this bosonic mode.
Influenced by the fact that the high-Tc cuprate is a doped
antiferromagnetic insulator, a common belief is that this
bosonic mode has a magnetic origin2,3,4,5,6,7,8,9. How-
ever, an alternative view is that the electron-phonon cou-
pling in such a doped-insulator can be very strong and
anomalous because of a number of unusual effects, such
as poor screening, complex structure as well as the in-
terplay with correlations. Indeed, oxygen related op-
tical phonons have been invoked to explain the tem-
perature and doping dependence of the renormalization
effects10,11,12,13,14. This idea of phonons being mainly
responsible for this low energy band renormalization ef-
fect observed by ARPES has stimulated intense debate.
There is currently no consensus whether a phonon, a set
of phonons, or a magnetic mode is the primary cause of
the band renormalization.
As mentioned, some important reasons to invoke
phonon interpretation of the ARPES data are: the
presence of an universal energy scale in all materials
at all doping10,15, particularly in the normal state of
very low Tc materials
16; the strong inferred momen-
tum dependence11,14; the existence of multiple bosonic
mode couplings12 and the decrease in the overall cou-
pling strength with increased doping, interpreted as a
screening effect, especially for phonons with eigenvectors
along the c-axis13. Yet, there is still a widespread belief
that phonons are not responsible for the kink features. In
the following sections, with a comprehensive look at all
experimental data as well as some recent simulations, we
address some of the criticisms that have been commonly
used to argue against the phonon interpretation. These
include the temperature and doping dependence of the
renormalization effects, inter-band scattering for bilayer
system, and the ARPES experiments on impurity substi-
tuted Bi2212 crystal, Bi2Sr2Ca(Cu2−xMx)O8+δ with M
= Zn or Ni. Our goal is to clarify these misconceptions
as being due to oversimplifying the effects of electron-
phonon coupling in cuprates as well as other strongly
correlated transition metal oxides.
II. ASPECTS OF THE ELECTRON-PHONON
SELF-ENERGY
A. Temperature Dependence
In the standard treatment of electron-phonon coupling
effects, the Debye temperature sets a characteristic tem-
perature scale, which is well above Tc in conventional
superconducting materials. However in the cuprates and
other low Fermi energy systems, these two energy scales
can be comparable. As a result, the temperature depen-
dence of phonon induced self-energies can be very differ-
ent from that of conventional superconductors. Accord-
ing to the ARPES measurements on Bi2212 system, the
band renormalization in the antinodal region (peak-dip-
hump structure) shows a dramatic superconductivity-
induced enhancement when the system goes through a
phase transition from the normal state to the supercon-
ducting state. It has been argued that only a mode which
emerges in the superconducting state and vanishes in
http://arxiv.org/abs/0704.0093v1
the normal state can explain this temperature-dependent
renormalization effect2,3,4,5. Phonons are thereby ex-
cluded.
The sharpness of the renormalization effects due to
electron-phonon coupling is strongly temperature depen-
dent, given by the fact that Tc of optimally-doped Bi2212
is close to 100K. To demonstrate this temperature de-
pendence, we consider the normal state (120 K) and su-
perconducting state (10 K) of a d-wave superconductor
coupled to a 36 meV B1g, 55meV oxygen A1g, and 70
meV breathing phonons14,17. The electron-phonon cou-
pling for the B1g and breathing phonons are those used
in Ref. 14. The A1g modes involve c-axis motions of
both planar and apical oxygens, and have two branches
around 55 and 80 meV. The apical electron-phonon cou-
pling, derived in Ref. 17, involves a charge-transfer from
the apex oxygen into the CuO2 plane via the Cu 4s or-
bital, the same pathway that gives rise to bi-layer split-
ting. However, for simplicity, the apical electron-phonon
coupling is treated as a constant in the calculations pre-
sented in this paper. The reason to include three modes
in the calculation was inspired by the earlier success of
the two-mode calculation11 as well as the recent discov-
ery of multiple mode coupling12,13. For this calculation,
the tight-binding band structure described in Ref. 19
has been used. The real part and imaginary part of the
self-energy Σ(k, ω) and the spectral functions A(k, ω) at
k = (0, π) are then obtained within weak coupling Eliash-
berg formalism14 and plotted in Fig. 1. Details of the
calculations are presented in the Appendix.
At high temperature, both ReΣ(k, ω) and ImΣ(k, ω)
do not exhibit a sharp renormalization feature as shown
by the dashed curves in Fig. 1 (a) and Fig. 1(b), respec-
tively. This demonstrates that the thermal broadening
effect smears out the sharp renormalization features; in
addition, broadening effects due to additional many body
effects would smear out the renormalization features fur-
ther. Thus, one should not expect to observe any sharp
renormalization features at k = (0, π) in the normal state
(∼ 100K) from the electron-phonon coupling. In the su-
perconducting state, the renormalization features of the
self-energy become much sharper, due to smaller ther-
mal broadening effect as well as the opening of a su-
perconducting gap. Consistent with the optimally-doped
Bi2212 and Bi2223 measurements2,4,11,18, the PDH struc-
ture of the spectral function at k = (0, π) emerges at low
temperature and disappears at high temperature (nor-
mal state), as illustrated in Fig. 1(c) and Fig. 1(d),
respectively. While this behavior is generally expected
for any phonon, we note that in addition, the self-energy
from electron-phonon couplings which involve momen-
tum transfers within and between anti-nodal regions of
the Fermi surface, such as the apex A1g and B1g phonons,
are greatly enhanced for all k-points due to the large
density of state enhancements in these regions via the
opening of a d-wave gap. A detailed momentum depen-
dence of the renormalization effects in the normal state
and superconducting state due to the coupling to the B1g
FIG. 1: The calculated (a) real part, ReΣ, (b) imaginary
part of the self-energy, ImΣ , and the corresponding spectral
functions, A(k,ω) in (c) normal state and (d) superconducting
state. An extra 5 meV is added to the imaginary part of the
self energy in 120K simulation to account for the thermal
broadening of the quasi-particle life time. The location for
this calculation is indicated in inset of (a) by a red dot with
a red curve representing the FS. Insets of (c) and (d) are the
data of optimally-doped Bi2223 system (Tc=110K) taken at
the normal state (120K) and superconducting state(25K)18,
respectively.
phonon has been discussed in Ref. 11 and Ref. 14. Fur-
thermore, both the dispersion kink and the PDH struc-
ture in the nodal region have been clearly observed in
the normal state when measured at a low temperature
on samples with a lower Tc
16. This lends further support
to the strongly temperature dependent renormalization
features due to electron-phonon coupling.
B. Doping Dependence
Another problematic statement against the phonon
scenario stems from the apparent strong doping depen-
dence of the kink position and strength. Based on the
wisdom for conventional good metals, phonons should
not have a strong doping dependence, either in frequency
of the mode or in strength of the coupling. This is
not necessary valid for layered, doped insulators with
strong correlation effects, such as cuprates where dop-
ing dramatically changes the metallicity and the abil-
ity of electrons to screen charge fluctuations. We first
note that many experiments on various cuprates have re-
ported strongly doping dependent anomalies for several
phonons, which implies a strongly doping dependent e-p
coupling for these phonons. For example, from inelastic
neutron scattering measurements, the breathing mode,
half-breathing mode, and the bond-stretching modes ex-
hibit prominent doping dependence of dispersion and en-
ergy renormalizations20,21. In Raman and infrared spec-
FIG. 2: The intensity plots of the (a) spectral functions with-
out resolution convolution and (b) resolution convoluted spec-
tral functions in the superconducting state (10K) along the
nodal direction, as indicated in the inset of (b) by the blue
line. Black curves are the band dispersion extracted from
the maximum positions of the momentum distribution curves,
which cut the spectral functions at a fixed energy. The MDC-
derived dispersions in (a) exhibit three sharp ”subkinks” due
to the coupling to the three phonon modes used in the model;
while in (b) the subkinks are washed out by the finite instru-
ment resolution effect leaving an apparent single kink in the
band dispersion. The white dashed line illustrates the bare
band for extracting ReΣ shown in Fig. 3 (a).
troscopy, the Fano lineshapes of phonon modes in B1g
and B1u symmetry show strong doping dependences
Furthermore, the strength of the phonon energy shift and
linewidth variation across Tc also changes strongly with
doping23.
Second, the doping dependence of the renormalization
effects to the electronic self-energy is sophisticated as in-
ferred by two recent ARPES studies. One is the observa-
tion of multiple bosonic mode couplings along the nodal
direction12. The other is the doping dependence of the
c-axis screening effect to the coupling between the elec-
tron and c−axis phonons. As proposed by Meevasana et
13,24 and Devereaux et al.17, for electron-phonon cou-
pling at long wavelengths, the screening becomes more
effective at reducing the coupling strength when the c-
axis conductivity becomes more metallic. Given these
two results plus the variation of the superconducting gap
magnitude with doping, the doping dependence of the
kink energy is highly convoluted in Bi2212 whose super-
conducting gap has an energy comparable to some of the
phonons.
In Fig. 2, we present the intensity plot of calculated
spectral functions demonstrating a doping dependence
of the dispersion kink in the superconducting state. The
superconducting gap size was set to be 40, 20, and 10
meV for the optimally-doped and more overdoped sys-
tems. In addition, the coupling strength of the breathing
mode, whose appreciable coupling is only for short wave-
lengths and large momentum transfers20,21, remains un-
FIG. 3: (a) The ReΣ extracted from Fig. 2(a) (dashed lines)
and Fig. 2(b) (solid lines) by subtracting a linear bare band
(dashed line in Fig. 2(a)) from the band dispersion. The ar-
rows indicate the maximum positions of the ReΣ where the
“single” apparent kink in the band dispersion is usually de-
fined. (b) Summary of the doping dependence of the apparent
kink energy and the apparent mode energy extracted by as-
suming a single mode scenario.
changed for our doping dependence simulation; while, a
filter function, ω2/(ω2+ω2sc) with different value of c-axis
screening frequency ωsc is applied to the c-axis phonons
(B1g and A1g), to simulate the doping-dependent cou-
pling strength due to the change of the c-axis screening
effect13,24. We note that although this is a simplifica-
tion, it represents the general behavior of screening con-
siderations for phonons involving small in-plane momen-
tum transfers. Full consideration of screening has been
given in Ref. 17 and Ref. 24. In addition, a component
0.005+ ω2 eV is added in the imaginary part of the self-
energy to mimic the quasiparticle life time broadening
due to electron-electron interaction.
As shown in Fig. 2(a), the coupling to multiple phonon
modes induces several “subkinks” in the dispersion. The
positions of these subkinks mostly correspond to the en-
ergies of phonons plus the maximum d-wave SC gap,
∆0, even through there is no gap along the nodal di-
rection. This is because when calculating the self-energy,
one needs to integrate the contributions from the entire
zone, which makes the electrons in the nodal region ”feel”
the presence of the gap. Furthermore, revealed by the ex-
tracted real-part of the self-energy, ReΣ (dashed curves
in Fig. 3 (a)), the dominant feature in ReΣ for the OP
case is induced by 36 meV B1g mode, while for the OD1
and OD2 case, the features of the 55 meV A1g mode and
70 meV breathing mode gradually out-weight the con-
tribution from the B1g mode. This demonstrates that
the change of the SC gap magnitude and the increasing
screening effect to these phonons because of increased
doping alters the relative strength of the subkinks caused
by different modes.
To simulate the experimental data, we convoluted the
spectral functions shown in Fig. 2(a) with a typical
ARPES instrumental resolution: 12.5 meV in energy res-
olution and 0.012 π/a in momentum resolution. As illus-
trated in Fig. 2(b) and the extracted ReΣ (solid curves
in Fig. 3(a)), the subkinks are less obvious and become
a broadened “single” kink in the dispersion which is lo-
cated at roughly the energy of the dominant phonon fea-
ture determined by the maximum position of the ReΣ
(the arrows in Fig. 3(a)). The doping dependence of
the kink position is summarized as the solid symbols in
Fig. 3(b). Assuming a single mode scenario, one can
obtain the “doping dependence” of the mode energy by
subtracting out the SC gap size. However, we note that
this extracted “apparent” mode energy does not match
any of the modes used in the model; instead, it is a av-
erage between the dominant features (open symbols in
Fig. 3(b). Clearly, since the kink energy is a sum of the
superconducting gap and a spectrum of bosonic modes,
it should not be taken as a precise measurement of the
energy of any particular bosonic mode. This casts doubts
to the analysis of the doping dependent properties of the
kink in the nodal band dispersion based on the single
bosonic mode coupling scenario7,8. More importantly,
this illustrates the complex nature of lattice effects in
these oxides.
C. Interband Scattering
Borisenko et al.8,9 observed that the scattering rate
of the bonding and antibonding bands along the nodal
direction cross each other near the energy of the Van
Hove singularity, suggesting a strong inter-band scatter-
ing between the bonding and antibonding bands. They
argued that only a mode with ”odd” symmetry, such as
spin resonance mode, can mediate such inter-band scat-
tering. The question whether phonons can induce such
inter-band scattering has also been raised by these au-
thors.
First, we note that recent high energy and momentum
resolution ARPES experiments on Bi2212 using low en-
ergy photons( <10 eV) have better resolved the bilayer
splitting at the nodal point25. However, as shown in Fig.
2 of Ref. 25, the scattering rate of the bonding and anti-
bonding band does not exhibit a crossover behavior as
reported by Borisenko at al.. The inconsistency of the
data between the two groups implies that more experi-
ments and better analysis are needed to verify whether
this inter-band scattering effect is genuine.
Second, empirically, it has been known for over 15 years
that interband electron-phonon coupling in the cuprates
is very large. The evidence comes from the strong reso-
nance profiles of many Raman active phonons, which dis-
play large intensity variations26. This is generally under-
stood as a result of strong interband coupling, whereby
phonons can be brought in and out of resonances via tun-
ing of the incident photon energy27. Since, in general,
phonons can also provide momentum to scatter electrons
along the c-axis, direct inter bi-layer scattering can occur
which involves mixing of different symmetries of phonons.
This can be viewed in a simplified way even if we first
neglect direct interband scattering and consider a bilayer
system coupled to c−axis phonons. For qz = 0, a simple
classification of c-axis modes is possible:
k,σ,α=1,2
ǫα(k)c
k,α,σ
ck,α,σ +
t⊥(k)
k,1,σck,2,σ + h.c.
k,q,σ,α=1,2,ν
gν,α(k, q)
k+q,α,σck,α,σ
aν(−q) + a
+ h.c.
, (1)
where α is the index for the electronic states of different
layers, ǫ1(k) = ǫ2(k), t⊥(k) describes the hopping of elec-
trons between two layers, and the index ν can be either
gerade or ungerade active c-axis modes, with symmetry
classification with respect to the displacement eigenvec-
tors to the inversion center of the cell, depicted in Fig.
4. After diagonalizing the first two terms by canonical
transformation, the electron-phonon coupling can be re-
cast as
(g,u)
k,q,σ
g(g,u)(k, q)
a(g,u)(q) + a
(g,u)
k+q,+,σck,(+,−),σ + c
k+q,−,σck,(−,+),σ
+ h.c.
..(2)
We have used the c+ and c− for the even and odd
linear combination of c1 and c2, and subscript g and u
for the gerade and ungerade mode, respectively. Thus
for qz = 0, where this classification is possible, ger-
ade phonons induce intra-band scattering (even chan-
nel), while the ungerade phonons mediate the inter-
band scattering process (odd channel) even without di-
rect electron-phonon coupling across the layers. Yet for
qz = π/c, the classification inverts, where gerade modes
become ungerade and vice-versa, as illustrated in Fig 4.
Thus, even in this simple case, modes at different qz con-
tribute both to intra and interband scattering, and the
net weight of the coupling appearing in the self energy is
then largely determined by the specific momentum struc-
ture g(k, q). Since the self-energy generally involves sums
over qz, and coupling directly of electrons in adjacent lay-
ers via phonons are non-negligible, clearly the inter-band
scattering phenomena can not be used to argue against
FIG. 4: The illustration of the gerade and ungerade c−axis
phonons. The eigenmode of the gerade (ungerade) phonons
is even (odd) with respect to the mirror plane between two
CuO2 layers at qz = 0, while their definition swapped at qz =
π/c. The black, grey, and white circles represent the Cu, Ca,
and O atoms, respectively.
the phonons being important to the electronic states.
We also add a remark concerning the electron-
phonon coupling derived from Raman measurements28
in YBa2Cu3O7 and Bi-2212 compared to that obtained
from ARPES. While one might naively expect the cou-
plings to be comparable from Raman and ARPES, we re-
mark that this situation is remarkably different if the cou-
pling is strongly moment dependent and whenever corre-
lations are appreciable. Since Raman measures phonons
with net zero momentum transfer and ARPES involves a
sum over all transfers, a sizeable coupling difference may
be discernable. This is specifically the case for the B1g
phonon, where scattering involving momentum transfers
across the necks of the Fermi surface near (π, 0)14, further
enhanced via correlations29, yields a strong contribution
to the electron self-energy that is absent in phonon self-
energies. Moreover, a sum rule analysis presented in Ref.
30 highlights in general how electron and phonon self-
energies may be qualitatively different in strongly corre-
lated systems.
D. ARPES Experiments on Zn and Ni substituted
Bi2212
In this section, we comment on recent experiments
about the renormalization effects in Zn and Ni substi-
tuted Bi2212 crystal31,32. The strength of sharp renor-
malizations in these substituted crystals is found to be
weakened compared to the pristine crystals. Since the
magnetic properties are expectedly modified due to the
Cu substitution by these impurities, the authors con-
cluded that the sharp renormalization effects are induced
by magnetic-related modes, not phonons.
In fact, a close examination of the data published by V.
B. Zabolotnyy et al.31 and K. Terashima et al.32 implies
that the magnetic property is not the only modification
due to the substitution by Zn and Ni. First, although
both sets of data are consistent in the antinodal region
where the strength of the band renormalization is re-
duced, they are inconsistent with each other on the kink
strength along the nodal direction. In the data set of V.
B. Zabolotnyy et al., the kink strength is weaker in the
Zn or Ni doped samples, whereas there is no detectable
change in the data set reported by K. Terashima et al..
Second, the data from K. Terashima et al. (Fig. 1(d)-
(f) in Ref. 32) suggest that the bilayer-splitting struc-
ture is much clearer in the pristine crystals than in the
Zn and Ni doped crystals. Since the authors have ruled
out the possibility of a significant doping level difference
between pristine and impurity-doped crystals, the dis-
tinct visibility variation of the bilayer structure implies
a impurity-related change in the electronic structure.
From these two observations on their data, it implies
that not only the magnetic properties could change, the
band structure and scattering behaviors could also be
affected due to these impurities. It is possible that
these changes of the electronic structures could “weaken”
the renormalizaton features observed in the ARPES
spectrum. Furthermore, we note that the strength of
electron-phonon coupling could also be modified by the
substituted impurities: this can be inferred from the
change of the Fano spectra lineshape of the B1g 340
cm−1 phonon in Raman spectral for Zn-doped YBCO33
and Th-doped YBCO34 resulting from an increase in the
phonon linewidth due to impurity scattering. Therefore,
the experiments on Ni and Zn substituted Bi2212 crys-
tals are inconclusive experiments to distinguish phonon
and magnetic modes as the origin of the renormalization
effects.
III. CONCLUSION
We have shown that the temperature and doping de-
pendence of the renormalization effects, inter-band band
scattering, and the results of Zn and Ni doped materials
can be understood in the framework of electron-phonon
coupling. On the other hand, the issues that make it
not plausible for the sharp kink being of spin origin, es-
pecially the spin resonance mode, remain: i) the nearly
constant energy scale as a function of doping in small
gap system12; ii) the multiple modes12; iii) the presence
of clear kink in the normal state4,13,16 iv) the detailed
agreement between B1g phonon based explanation of the
mode coupling as a function of momentum11,14, while
the spin resonance with tiny spectral weight (2%) is un-
likely to give an explanation for both nodal and antin-
odal renormalization; v) the accumulated evidence for
lattice polaron effect in underdoped and deeply under-
doped systems35,36. With these weaknesses of the spin
resonance interpretation, lattice effect is a more plausible
explanation of the renormalization effects. It remains a
possibility that the spin-fluctuation and other strong cor-
relation effects are also very important to determine the
electronic structure of cuprates; they likely contribute to
a smooth renormalization of the band and may be more
relevant to the higher binding energy. However, opti-
cal phonons are the most probable origin for the renor-
malization effects due to sharp modes near 40-70 meV,
which is also supported by the recent finding of STM
experiments37,38.
Acknowledgments
W.S. Lee acknowledge the support from SSRL which
is operated by the DOE Office of Basic Energy Science,
Division of Chemical Science and Material Science under
contract DE-AC02-76SF00515. T. P. Devereaux would
like to acknowledge support from NSERC, ONR grant
N00014-05-1-0127 and the A. von Humboldt Foundation.
APPENDIX A: MIGDAL-ELIASHBERG BASED
APPROACH
In the calculations presented herein, we evaluate elec-
tronic self energies and spectral functions via Migdal-
Eliashberg treatment, as discussed in Ref. 39. The
dressed Green’s function in the superconducting state is
given in Nambu notation by
Ĝ(k, ω) =
ωZ(k, ω)τ̂0 + [ǫ(k) + χ(k, ω)]τ̂3 + φ(k, ω)τ̂1
[ωZ(k, ω)]2 − [ǫ(k) + χ(k, ω)]2 − φ2(k, ω)
, (A1)
from which the spectral function follows A(k, ω) = − 1
G′′1,1(k, ω) as shown in Figs. 1c,1d, and 2. The momentum-
dependent components of the Nambu self energy are given as generalizations of those found in Ref. 39:
ωZ2(k, ω) =
|gν(k,p− k)|
[nb(Ων) + nf(Ep)][δ(ω +Ων − Ep) + δ(ω − Ων + Ep)]
+[nb(Ων) + nf (−Ep)][δ(ω − Ων − Ep) + δ(ω +Ων + Ep)]
χ2(k, ω) = −
|gν(k,p− k)|
[nb(Ων) + nf (Ep)][δ(ω +Ων − Ep)− δ(ω − Ων + Ep)]
+[nb(Ων) + nf (−Ep)][δ(ω − Ων − Ep)− δ(ω +Ων + Ep)]
φ2(k, ω) =
|gν(k,p− k)|
[nb(Ων) + nf (Ep)][δ(ω +Ων − Ep)− δ(ω − Ων + Ep)]
+[nb(Ων) + nf(−Ep)][δ(ω − Ων − Ep)− δ(ω +Ων + Ep)]
where ν denotes the phonon mode index, and nf and nb
are the Fermi and Bose occupation factors. gν(k,q) are
the corresponding electron-phonon couplings for mode
ν, given in reference14 for the B1g and breathing modes.
We choose to model the A1g coupling via a momentum
independent coupling. Further details can be found in
Ref. 17.
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