Journal ArticleDOI

# Intraband optical spectral weight in the presence of a van Hove singularity: Application to Bi 2 Sr 2 CaCu 2 O 8+δ

22 Nov 2006-Physical Review B (American Physical Society (APS))-Vol. 74, Iss: 17, pp 174516

Abstract: The Kubo single-band sum rule is used to determine the optical spectral weight of a tight-binding band with farther than nearest-neighbor hopping. We find for a wide range of parameters and doping concentrations that the change due to superconductivity at low temperature can be either negative or positive. In contrast, the kinetic energy change is always negative. We use an angle-resolved-photoemission-spectroscopy-determined tight-binding parametrization of Bi2Sr2CaCu2O8 δ to investigate whether this can account for recent observations of a positive change in the spectral weight due to the onset of superconductivity. With this band structure we find that in the relevant doping regime a straightforward BCS calculation of the optical spectral weight cannot account for the experimental observations.
Topics: Van Hove singularity (57%)

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Reference
Intraband optical spectral weight in the presence of a van Hove
singularity: Application to Bi
2
Sr
2
CaCu
2
O
8+δ
MARSIGLIO, Frank, et al.
Abstract
The Kubo single-band sum rule is used to determine the optical spectral weight of a
tight-binding band with farther than nearest-neighbor hopping. We find for a wide range of
parameters and doping concentrations that the change due to superconductivity at low
temperature can be either negative or positive. In contrast, the kinetic energy change is
always negative. We use an angle-resolved-photoemission-spectroscopy-determined
tight-binding parametrization of Bi2Sr2CaCu2O8 δ to investigate whether this can account for
recent observations of a positive change in the spectral weight due to the onset of
superconductivity. With this band structure we find that in the relevant doping regime a
straightforward BCS calculation of the optical spectral weight cannot account for the
experimental observations.
MARSIGLIO, Frank, et al. Intraband optical spectral weight in the presence of a van Hove
singularity: Application to Bi
2
Sr
2
CaCu
2
O
8+δ
. Physical review. B, Condensed matter and
materials physics, 2006, vol. 74, no. 17, p. 174516
DOI : 10.1103/PhysRevB.74.174516
Available at:
http://archive-ouverte.unige.ch/unige:24317
Disclaimer: layout of this document may differ from the published version.
1 / 1

Intraband optical spectral weight in the presence of a van Hove singularity:
Application to Bi
2
Sr
2
CaCu
2
O
8+
F. Marsiglio,
1,2
F. Carbone,
1
A. B. Kuzmenko,
1
and D. van der Marel
1
1
DPMC, Université de Genève, 24 Quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
2
Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1
Received 26 June 2006; revised manuscript received 7 September 2006; published 22 November 2006
The Kubo single-band sum rule is used to determine the optical spectral weight of a tight-binding band with
farther than nearest-neighbor hopping. We ﬁnd for a wide range of parameters and doping concentrations that
the change due to superconductivity at low temperature can be either negative or positive. In contrast, the
kinetic energy change is always negative. We use an angle-resolved-photoemission-spectroscopy-determined
tight-binding parametrization of Bi
2
Sr
2
CaCu
2
O
8+
to investigate whether this can account for recent observa-
tions of a positive change in the spectral weight due to the onset of superconductivity. With this band structure
we ﬁnd that in the relevant doping regime a straightforward BCS calculation of the optical spectral weight
cannot account for the experimental observations.
DOI: 10.1103/PhysRevB.74.174516 PACS numbers: 74.25.Gz, 74.20.Fg, 74.72.h, 79.60.i
I. INTRODUCTION
Recent optical experiments on several high-T
c
cuprates at
optimal and low doping levels
14
have shown an increase in
the low-frequency spectral weight when the system goes su-
perconducting. These observations are at odds with the sim-
plest expectation based on BCS theory,
57
where the kinetic
energy is expected to increase in the superconducting state;
however, they conform with the general notion of “kinetic-
energy-driven” superconductivity.
8
Since the optical spectral
weight is just the negative of the kinetic energy for a single
band with nearest-neighbor hopping only, a decrease in spec-
tral weight is expected to occur below the superconducting
transition temperature. Several alterations to the standard
BCS picture have been proposed, the most minor of which
involve an alteration to the boson spectrum when the system
goes superconducting.
912
More recent measurements
13,14
have shown a continuous
evolution with doping from “non-BCS-like” low doping to
“BCS-like” high doping behavior; an understanding of this
doping dependence has been suggested in Refs. 1417 based
on strong electron-electron correlations and in Refs. 1820
based on the doping dependence of a transition driven by
pairing versus phase coherence.
While all these proposals remain interesting possibilities
to explain the observations, the purpose of this paper is to
revisit the question of what “BCS-like” behavior is, by tak-
ing into account band structure details. It is important to do
this, since our intuition is based on the behavior of the ki-
netic energy which as we illustrate below always increases
in the superconducting state. However, the optical spectral
weight for a single band is given by
21,22
WT =
2
2
e
2
0
+
d
Re
xx

2
N
k
2
k
k
x
2
n
k
, 1
whereas the negative of the band kinetic energy is given by a
somewhat different expression; in the simplest case, it is
given by
K =−
2
N
k
k
n
k
, 2
where
k
is the tight-binding dispersion which takes into
account already Hartree-Fock-type corrections and n
k
is the
single-spin momentum distribution function we take the lat-
tice spacing to be unity. The sum over k is over the ﬁrst
Brillouin zone and, in the case with bilayer splitting see
below, includes a summation over the two split bands. Note
that this is not the total kinetic energy of all the electrons, but
just the kinetic energy of the electrons in the given tight-
binding bands; furthermore, only in the case of nearest-
neighbor hopping is W proportional to K. In the presence
of more complicated interactions, the expectation value of
the kinetic energy has more complicated terms.
We ﬁrst review the expectation for the kinetic energy,
based on Eq. 2, since this correspondence has been used to
build intuition concerning the optical spectral weight. First,
what happens when the system goes superconducting? The
momentum distribution function changes as discussed
previously
7
—it goes from a Fermi-like distribution function
in the absence of strong correlations to a distribution
smeared by the presence of a superconducting order param-
eter. For an order parameter with d-wave symmetry, the mo-
mentum distribution is no longer a function of the band
structure energy
k
alone. For example, for a BCS order pa-
rameter with simple nearest-neighbor pairing form,
k
=cos k
x
cos k
y
/ 2; then, as k varies from 0,0 to
,0,
the magnitude of the order parameter changes from zero to
. On the other hand, as k varies along the diagonal from
the bottom of the band to the top, the order parameter is
zero and constant. In any event, even at zero temperature,
BCS-like superconductivity raises the kinetic energy of the
electrons see Fig. 1b of Ref. 7. This is as expected, since
for noninteracting electrons the normal state at zero tempera-
ture corresponds to a state with the lowest possible kinetic
energy. Therefore, any modiﬁcation to this state for ex-
ample, because of a superconducting instability can only
increase the kinetic energy expectation value.
PHYSICAL REVIEW B 74, 174516 2006
1098-0121/2006/7417/17451610 ©2006 The American Physical Society174516-1

The question, partially answered in Refs. 7 and 14 is, does
this behavior remain at all electron densities? Furthermore,
with farther than nearest-neighbor hopping, does the spectral
weight given by Eq. 1兲兴 also follow the same trend as the
negative of the kinetic energy? Perhaps not surprisingly, we
ﬁnd that the spectral weight does not qualitatively follow the
kinetic energy near a van Hove singularity. However, as will
be discussed further below, we ﬁnd that for the band struc-
ture and doping regime thought to be applicable in
Bi
2
Sr
2
CaCu
2
O
8+
BSCCO,
24
the spectral weight should de-
crease in the superconducting state relative to the normal
state. That is, correlations, phase ﬂuctuations, scattering rate
collapse, or some other scenario is required to understand the
“anomalous” behavior. We will also address the temperature
dependence in the normal state; in some ways, this is a more
easily measured quantity than the change below T
c
.
In the next section we examine the optical spectral weight
for a model with nearest-neighbor hopping only. This simple
band structure yields an optical spectral weight that is di-
rectly proportional to the expectation value of the negative of
the kinetic energy. We examine the behavior of the optical
spectral weight as a function of electron density. Note that
we will use the symbol n to denote electron density; for a
single band, this quantity will span values from 0 to 2. It will
be used when systematic investigations of the spectral
weight for a given band structure are carried out. When com-
paring with experiments in the cuprates, we will use the sym-
bol
to denote doping away from half-ﬁlling—i.e., n =1.
Thus,
=1−n and the regime of experimental interest is
roughly 0
0.25. We use the phrase “hole doping” to
refer speciﬁcally to the value of
.
Following this section we introduce next-nearest-neighbor
hopping into the band structure t-t
model. This moves the
van Hove singularity away from half ﬁlling and also causes
the spectral weight to deviate from the kinetic energy; hence,
both will be plotted in the ensuing plots. We ﬁnd already in
this simple extension signiﬁcant departures from the “stan-
dard BCS” description based solely on the kinetic energy.
Finally, following Ref. 24, we also introduce a next-next-
nearest-neighbor hopping and a bilayer splitting term; these
are required for a quantitatively accurate description of the
angle-resolved photoemission spectroscopy ARPES results.
We ﬁnd that these terms have signiﬁcant effects on the opti-
cal sum rule. First, the van Hove singularity is split into two
singularities; second, the ﬁrst of these occurs at a much
lower hole doping level than in the t-t
model.
As discussed in the summary, the end result is that i the
change in the optical spectral weight due to superconductiv-
ity can be either positive or negative, depending on the band
structure and electron density, and ii if a parametrization of
the band structure is adopted from ARPES studies, then the
optical spectral weight decreases in the superconducting
state. The observed increase for optimal and underdoped
samples then requires additional ingredients. Some possibili-
ties are brieﬂy mentioned.
II. NEAREST-NEIGHBOR HOPPING ONLY
For nearest-neighbor hopping only, the band structure is
given by
k
nn
=−2tcos k
x
+ cos k
y

and we have that 2W=−K in two dimensions. In Fig. 1 we
show plots of the spectral weight versus T
2
for two represen-
tative electron densities n=1 and n =0.5. The ﬁrst places the
Fermi level right on the van Hove singularity, while the sec-
ond is well removed from all van Hove singularities. These
are computed through the usual procedure: ﬁrst, even in the
normal state, the chemical potential must be determined at
FIG. 1. Color online Spectral weight vs T
2
for a half-ﬁlling
and b n = 0.5. The normal state is given by the solid red curve, and
the superconducting state with d-wave s-wave symmetry by the
short-dashed blue dashed green curve. In both cases the normal-
state result is almost linear in T
2
and the superconducting state
shows a decrease in the spectral weight increase in the kinetic
energy as expected. We used t =0.4 eV and BCS values for order
parameters, etc., with T
c
=69 K.
MARSIGLIO et al. PHYSICAL REVIEW B 74, 174516 2006
174516-2

each temperature to ensure that the electron density remains
constant as a function of temperature. This is the common
procedure, though it is true that in complicated systems for
which one is using some “low-energy” tight-binding Hamil-
tonian to describe the excitations that it is not immediately
obvious that the electron number density should remain ﬁxed
as a function of temperature; nonetheless, we adopt this pro-
cedure here. In Eq. 1 the chemical potential enters the mo-
mentum distribution function, which, in the normal state, is
replaced by the Fermi-Dirac distribution function, n
k
f
k
. In the superconducting state, we simply adopt a model
temperature dependence for the order parameter, following
Ref. 7. This has been tested for both s-wave and d-wave
symmetries by comparing to fully self-consistent solutions to
BCS equations with separable potentials.
7
One still has to
determine the chemical potential self-consistently for each
temperature, which is done by solving the number equation
in the superconducting state for a ﬁxed chemical potential
and order parameter, and iterating until the desired number
density is achieved. We use the standard BCS expression
n =1−
1
N
k
k
E
k
1−2fE
k

where E
k
k
2
+
k
2
and
k
can take on both s-wave
and d-wave forms. In the normal state this expression re-
duces to the simple Fermi function; even above T
c
, however,
iteration for the correct value of the chemical potential is
required. Interestingly, if one incorrectly adopts the same
chemical potential as a function of temperature, then the ef-
fects discussed here become more pronounced for example,
in Fig. 6c below, the variation above T
c
and the reversal
below T
c
is stronger.
The value of the zero-temperature order parameter is ﬁxed
by the weak-coupling BCS values, 2
0
=
k
B
T
c
where
=4.23.5 for the d-wave s-wave case. Further details are
provided in Ref. 7. For the electron densities studied in the
ﬁrst part of this paper, we simply take T
c
=69 K for all elec-
tron densities. In Sec. IV we will adopt T
c
values as observed
from experiment.
Both plots in Fig. 1 show somewhat linear behavior with
T
2
, though in Fig. 1a there is some noticeable upward cur-
vature due to the van Hove singularity which is present at the
Fermi level for this electron density. The decrease in spectral
weight at the transition is more pronounced for s-wave sym-
metry dashed green curves than for d-wave symmetry dot-
ted blue curves. The normal-state results show a decreasing
value with increasing temperature, indicative of an increas-
ing kinetic energy. This is a “textbook” example of the tem-
perature dependence of the spectral weight through a super-
conducting transition.
57
In Fig. 2 we examine both the spectral weight difference
W
s
W
N
and W
d
W
N
for s- and d-wave symmetry,
respectively—N here stands for “normal” at zero tempera-
ture, and the slope of WT with respect to T
2
at T
c
versus
electron density n. These plots make evident several impor-
tant points. First, the van Hove singularity clearly plays a
role; it enhances the overall magnitude of the effect, whether
we examine the difference between the superconducting and
normal state at zero temperature or the slope at T
c
. In fact the
latter tracks the former, indicating that both are related to one
another. One can understand this qualitatively by the obser-
vation that in both cases warming up or going superconduct-
ing the momentum distribution function broadens, though
for different reasons.
7
The most important point to learn from
this plot is that the difference is always negative, indicating
that, for nearest-neighbor hopping only, the opening of a gap
does indeed increase the kinetic energy and decrease the
spectral weight in a superconductor.
III. NEXT-NEAREST-NEIGHBOR HOPPING
When next-nearest-neighbor hopping is included in the
band structure, one obtains the so-called t-t
model. This
model has band structure
k
nnn
=−2tcos k
x
+ cos k
y
+4t
cos k
x
cos k
y
5
and goes a long way towards understanding the Fermi
surface of Bi
2
Sr
2
CaCu
2
O
8+
BSCCO, as determined by
ARPES,
23,24
at least for the doping levels studied. On the
theoretical side, the presence of t
shifts the van Hove sin-
gularity to an energy given by
=−4t
. For the sake of this
study one can study all electron densities; however, one must
bear in mind that most experiments on BSCCO are at doping
levels such that the van Hove singularity is not crossed; i.e.,
FIG. 2. Color online The difference W
d
W
N
in the spectral
weight between the superconducting state with d-wave symmetry
and the normal state at zero temperature vs doping dotted blue
curve. The dashed green curve shows the same quantity for s-wave
symmetry, and the pink points indicate the slope with respect to T
2
of the spectral weight near T
c
. All three quantities are always nega-
tive and show an enhancement near half-ﬁlling due to the van Hove
singularity. In fact, the pink points are almost a perfect inverted
image of the density of states see the minus sign in Eq. 8兲兴, except
for the small density regime near half-ﬁlling, where the van Hove
singularity makes the Sommerfeld expansion invalid.
INTRABAND OPTICAL SPECTRAL WEIGHT IN PHYSICAL REVIEW B 74, 174516 2006
174516-3

the Fermi surfaces are always hole like. We will also study
see next section a band structure more pertinent to
BSCCO,
23,24
which uses a next-next-nearest-neighbor hop-
k±
=
k
nnn
−2t
cos 2k
x
+ cos 2k
y
± t
cos k
x
cos k
y
2
/4,
6
which we will refer to as the t-t
-t
model. Note that we
allow for a bilayer splitting term as well, following Kordyuk
et al.
24
However, they actually adjust hopping parameters for
each doping, while we simply adopt the ones used for their
overdoped sample: t =0.40 eV, t
=0.090 eV, t
=0.045 eV,
and t
=0.082 eV. Illustrative plots of the band structures are
shown in Fig. 3.
Returning now to the t-t
model, the van Hove singularity
occurs at an electron density n=0.60—i.e., a hole doping
away from half-ﬁlling of
=1−n= 0.4. As mentioned
above, this high level of doping is never realized in samples
of BSCCO.
25
In any event, we are interested in the more
generic behavior of the spectral weight, given a reasonably
representative band structure for the cuprates.
In Fig. 4 we show a summary of the doping dependence
of the various quantities with the t-t
band structure. In both
Figs. 4a and 4b we have plotted the density of states at
the Fermi level as a function of doping this is possible for a
doping-independent band to illustrate where the van Hove
singularity is. The remarkable feature in Fig. 4a, for elec-
tron densities below the van Hove singularity, is that the
spectral weight change in the superconducting state is posi-
tive. Similarly, in Fig. 4b the actual slope of the spectral
weight above T
c
is positive. Note that our intuition about the
kinetic energy change remains correct; it is indeed negative,
for all electron densities, for both s-wave and d-wave sym-
metries. Moreover, the slope is also everywhere negative,
which establishes a deﬁnite correlation between the slope
above T
c
and the change at T= 0. Note that in Ref. 7 see Fig.
4 of that reference the doping parameters were such that the
optical sum rule and the negative of the kinetic energy were
qualitatively and even quantitatively similar.
26
Here, in the
FIG. 3. Illustrative plots of the band structure for a nearest-
neighbor hopping only, b the t-t
model, and c parametrization
of Kordyuk et al. Ref. 24 of the band structure with bilayer split-
ting. The van Hove singularities occur where the band dispersion
ﬂattens.
FIG. 4. Color online兲共a The difference W
d
W
N
in the spec-
tral weight between the superconducting state with d-wave symme-
try and the normal state at zero temperature vs doping solid red
curve, for the t-t
band structure, with t =0.4 eV and t
=0.09 eV.
The dashed green curve shows the same quantity for s-wave sym-
metry. Both exhibit positive values to the left of the van Hove
singularity the density of states at the Fermi level is indicated, as a
function of doping, by the dot-dashed cyan curve. The negative of
the kinetic energy for d-wave blue short-dashed curve and for
s-wave dotted pink curve symmetry behaves as expected, always
negative, and peaks in absolute value at the van Hove singularity.
b The normal-state slope taken at T
c
=69 K of the spectral
weight vs doping solid red curve. The dashed green curve shows
the same quantity for the negative of the kinetic energy. These
behave in very similar fashion to the differences taken at zero
temperature shown in a.
MARSIGLIO et al. PHYSICAL REVIEW B 74, 174516 2006
174516-4

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Journal ArticleDOI
M. R. Norman1, Mohit Randeria1, Hong Ding2, Hong Ding1  +2 moreInstitutions (2)
TL;DR: Tight-binding fits to the normal-state dispersion and superlattice modulation effects are described and the anisotropic s-wave gap cos is found, which within a one-band BCS framework suggests the importance of next-near-neighbor Cu-Cu interactions.
Abstract: Recently, high-resolution angle-resolved photoemission spectroscopy has been used to determine the detailed momentum dependence of the superconducting gap in the high-temperature superconductor ${\mathrm{Bi}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8}. In this paper, we first describe tight-binding fits to the normal-state dispersion and superlattice modulation effects. We then discuss various theoretical models in light of the gap measurements. We find that the simplest model which fits the data is the anisotropic s-wave gap cos({\mathit{k}}_{\mathit{x}})cos({\mathit{k}}_{\mathit{y}}), which within a one-band BCS framework suggests the importance of next-near-neighbor Cu-Cu interactions. Various alternative interpretations of the observed gap are also discussed, along with the implications for microscopic theories of high-temperature superconductors. 181 citations Journal ArticleDOI TL;DR: A peak in the real part of the sub-band-gap conductivity {sigma}{sub 1} similar to the coherence peak expected in an {ital s}-wave BCS superconductor is observed. Abstract: We measure directly both the real and imaginary parts of the sub-band-gap conductivity \ensuremath{\sigma} in {\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7}$ in the 15--80 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ frequency range using coherent time-domain terahertz spectroscopy. We observe a peak in the real part ${\mathrm{\ensuremath{\sigma}}}_{1}$ similar to the coherence peak expected in an s-wave BCS superconductor. The absence of such a peak in NMR relaxation-rate experiments suggests that its origin lies in a strongly temperature-dependent inelastic-scattering rate rather in coherence factors.

179 citations

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