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

TL;DR: In this article, the authors used angle-resolved-photo-emission-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.

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.

Summary

I. INTRODUCTION

• Recent optical experiments on several high-T c cuprates at optimal and low doping levels [1] [2] [3] [4] have shown an increase in the low-frequency spectral weight when the system goes superconducting.
• These observations are at odds with the simplest expectation based on BCS theory, [5] [6] [7] where the kinetic energy is expected to increase in the superconducting state; however, they conform with the general notion of "kineticenergy-driven" superconductivity.
• The authors first review the expectation for the kinetic energy, based on Eq. ͑2͒, since this correspondence has been used to build intuition concerning the optical spectral weight.
• Note that the authors will use the symbol n to denote electron density; for a single band, this quantity will span values from 0 to 2.
• The observed increase for optimal and underdoped samples then requires additional ingredients.

II. NEAREST-NEIGHBOR HOPPING ONLY

• The first places the Fermi level right on the van Hove singularity, while the second is well removed from all van Hove singularities.
• The normal state is given by the solid red curve, and the superconducting state with d-wave ͑s-wave͒ symmetry by the short-dashed blue ͑dashed green͒ curve.
• 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.
• These plots make evident several important points.
• First, the van Hove singularity clearly plays a role; it enhances the overall magnitude of the effect, whether the authors examine the difference between the superconducting and normal state at zero temperature or the slope at T c .

III. NEXT-NEAREST-NEIGHBOR HOPPING

• When next-nearest-neighbor hopping is included in the band structure, one obtains the so-called t-tЈ model.
• 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., the Fermi surfaces are always hole like.

IV. BILAYER SPLITTING

• It is evident that the characteristics of the optical spectral weight will be very dependent on the band structure and the doping level.
• As an ex- ample of what the authors consider a remote possibility, Fig. 8 indicates that if the doping levels for BSCCO are not as indicated, but rather lie in the regime between the two van Hove singularities ͑i.e., approximately between n = 0.5 and n = 0.7͒, then the results will be very different.
• Thus, the anomaly below T c would agree with experiment, 13, 14 including a crossover from a positive change for underdoped samples to a negative change for overdoped samples.
• This was also found in C-DMFT calculations, 14 and further theoretical work and experiments would be required, however, to disentangle band structure effects from strong correlation effects.
• Finally, one can ask about the doping dependence of the magnitude of the optical spectral weight.

V. SUMMARY

• The primary result of this paper is the revelation that the single-band optical spectral weight may behave very differently from the kinetic energy, both in the normal state and in the superconducting state.
• This occurs when one uses a band structure more complicated than one involving nearestneighbour ͑NN͒ hopping only, since, with NN hopping only, the two are identical.
• On the other hand, if one accepts the band structure for, say, BSCCO, as determined by ARPES, then the spectral weight observations [1] [2] [3] [4] 13, 14 remain anomalous-i.e., cannot be explained by BCS theory alone.
• The authors have advanced a couple of possibilities, and many others have been proposed in the literature: doping levels may be shifted slightly compared to what they think they are, in which case strong correlations well beyond BCS theory are required to explain the observed trend with doping. [14] [15] [16] [17] [18] [19] [20].
• Finally, including a scattering rate collapse below T c also qualitatively accounts 7 for the data.

Figures

FIG. 5. Color online The finite-temperature correction to the optical spectral weight thick curves , for various values of t , as indicated. Also plotted is the analogous quantity for the kinetic energy which, by Eq. 8 , is just −8tg F . Note that the latter is always negative, while the spectral weight becomes positive for a finite electron concentration for electron hole densities below above the van Hove singularity. The magnitude of the anomalous region increases with increasing t .

FIG. 6. Color online Spectral weight for four representative densities for the t-t model see Fig. 5 . The progression from top to bottom is from high electron concentration to low electron concentration. Note the qualitative change from negative slope lowersuperconducting vs normal-state difference to positive slope higher-superconducting vs normal-state difference as the electron concentration changes such that the chemical potential sweeps through the van Hove singularity.

FIG. 1. Color online Spectral weight vs T2 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 normalstate result is almost linear in T2 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 Tc=69 K.

FIG. 8. Color online Spectral weight difference solid red curve as a function of electron concentration for the band structure determined by ARPES Ref. 24 . For simplicity we put Tc=70 K for all electron densities. The normal-state slope shows very similar behavior. Also plotted is the negative of the change in the kinetic energy dashed green curve and the density of states at the Fermi level dot-dashed cyan curve , for reference. Note that “anomalies” in the spectral weight occur at the two van Hove singularities, while the change in kinetic energy is simply gently modulated by the density of states. Also shown with solid blue symbols is the hole doping regime of relevance to the BSCCO experiments in Ref. 14 and considered in the previous figure. For clarity only d-wave results are shown. The three open squares show the doping concentrations considered in the next figure, of possible relevance for overdoped samples with chemical potential that crosses the van Hove singularity Ref. 30 .

FIG. 7. Color online Spectral weight for the four hole doping concentrations along with their measured Tc’s considered in the experiments Ref. 14 . These calculations use the ARPESdetermined band structure parameters from Ref. 24, including the bilayer splitting. Slopes are always negative, and the change at Tc is also always negative. Note that in this case the s-wave order parameter gives roughly the same value as the d-wave parameter.

FIG. 2. Color online The difference Wd−WN 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 T2 of the spectral weight near Tc. All three quantities are always negative 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 , except for the small density regime near half-filling, where the van Hove singularity makes the Sommerfeld expansion invalid.

FIG. 10. Color online Spectral weight at zero temperature in the normal state vs electron concentration for four different band structures: that applicable to BSCCO solid red curve , the t-t model with t=0.40 eV and t =0.09 eV long-dashed green curve and with t =−0.09 eV short-dashed blue curve , and the nearestneighbor hopping only model, with t=0.40 eV. The three models that are physically relevant to the hole-doped cuprates LSCO and BSCCO have non-negative next nearest-neighbor hopping t and show a spectral weight that decreases with increasing hole doping over the relevant hole doping regime see thick solid red line along the bottom axis . See text for further discussion.

FIG. 9. Color online Spectral weight for three hole doping concentrations that move the chemical potential across the first van Hove singularity see Fig. 8 . Note the dramatic change that occurs as one dopes across the van Hove singularity see text .

FIG. 4. Color online a The difference Wd−WN in the spectral weight between the superconducting state with d-wave symmetry 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 symmetry. 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 Tc=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 .

FIG. 3. Illustrative plots of the band structure for a nearestneighbor hopping only, b the t-t model, and c parametrization of Kordyuk et al. Ref. 24 of the band structure with bilayer splitting. The van Hove singularities occur where the band dispersion flattens.

Full text

Article
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
共Received 26 June 2006; revised manuscript received 7 September 2006; published 22 November 2006
兲
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 number共s兲: 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
1–4
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,
5–7
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.
9–12
More recent measurements
13,14
have shown a continuous
evolution with doping from “non-BCS-like” 共low doping兲 to
“BCS-like” 共high doping兲 behavior; an understanding of this
doping dependence has been suggested in Refs. 14–17 based
on strong electron-electron correlations and in Refs. 18–20
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 共as we illustrate below兲 always increases
in the superconducting state. However, the optical spectral
weight for a single band is given by
21,22
W共T兲 =
2ប
2
␲
e
2
冕
0
+⬁
d
␯
Re关
␴
xx
共
␯
兲兴 =
2
N
兺
k
⳵
2
⑀
k
⳵
k
x
2
n
k
, 共1兲
whereas the negative of the band kinetic energy is given by a
somewhat different expression; in the simplest case, it is
given by
− 具K典 =−
2
N
兺
k
⑀
k
n
k
, 共2兲
where
⑀
k
is the tight-binding dispersion 共which takes into
account already Hartree-Fock-type corrections兲 and n
k
is the
single-spin momentum distribution function 共we take the lat-
tice spacing to be unity兲. The sum over k is over the first
Brillouin zone and, in the case with bilayer splitting 共see
below兲, 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 band共s兲; furthermore, only in the case of nearest-
neighbor hopping is W proportional to −具K典. 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. 共2兲, 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
共in the absence of strong correlations兲 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
=⌬共cos k
x
−cos k
y
兲/ 2; then, as k varies from 共0,0兲 to 共
␲
,0兲,
the magnitude of the order parameter changes from zero to
⌬. On the other hand, as k varies along the diagonal 共from
the bottom of the band to the top兲, the order parameter is
zero 共and constant兲. In any event, even at zero temperature,
BCS-like superconductivity raises the kinetic energy of the
electrons 关see Fig. 1共b兲 of Ref. 7兴. 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 共for ex-
ample, because of a superconducting instability兲 can only
increase the kinetic energy expectation value.
PHYSICAL REVIEW B 74, 174516 共2006兲
1098-0121/2006/74共17兲/174516共10兲 ©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 关given by Eq. 共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+
␦
共BSCCO兲,
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 共t-t
⬘
model兲. 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 共ARPES兲 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 共i兲 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 共ii兲 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
=−2t共cos k
x
+ cos k
y
兲共3兲
and we have that 2W=−具K典 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. 共Color online兲 Spectral weight vs T
2
for 共a兲 half-filling
and 共b兲 n = 0.5. The normal state is given by the solid red curve, and
the superconducting state with d-wave 共s-wave兲 symmetry by the
short-dashed blue 共dashed green兲 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 共increase in the kinetic
energy兲 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 共2006兲
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. 共1兲 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共
⑀
k
−
␮
兲. 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
兺
k
⑀
k
−
␮
E
k
关1−2f共E
k
兲兴, 共4兲
where E
k
⬅
冑
共
⑀
k
−
␮
兲
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 共incorrectly兲 adopts the same
chemical potential as a function of temperature, then the ef-
fects discussed here become more pronounced 关for example,
in Fig. 6共c兲 below, the variation above T
c
and the reversal
below T
c
is stronger兴.
The value of the zero-temperature order parameter is fixed
by the weak-coupling BCS values, 2⌬
0
=
␩
k
B
T
c
where
␩
=4.2共3.5兲 for the d-wave 共s-wave兲 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. 1共a兲 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 共dashed green curves兲 than for d-wave symmetry 共dot-
ted blue curves兲. 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.
5–7
In Fig. 2 we examine both the spectral weight difference
共W
s
−W
N
and W
d
−W
N
for s- and d-wave symmetry,
respectively—N here stands for “normal”兲 at zero tempera-
ture, and the slope of W共T兲 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 共warming up or going superconduct-
ing兲 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
=−2t共cos k
x
+ cos k
y
兲 +4t
⬘
cos k
x
cos k
y
共5兲
and goes a long way towards understanding the Fermi
surface of Bi
2
Sr
2
CaCu
2
O
8+
␦
共BSCCO兲, 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. 共Color online兲 The difference 共W
d
−W
N
兲 in the spectral
weight between the superconducting state with d-wave symmetry
and the normal state at zero temperature vs doping 共dotted blue
curve兲. The dashed green curve shows the same quantity for s-wave
symmetry, and the pink points indicate the slope 共with respect to T
2
兲
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 关see the minus sign in Eq. 共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 共2006兲
174516-3

the Fermi surfaces are always hole like. We will also study
共see next section兲 a band structure more pertinent to
BSCCO,
23,24
which uses a next-next-nearest-neighbor hop-
ping amplitude in addition:
⑀
k±
=
⑀
k
nnn
−2t
⬙
共cos 2k
x
+ cos 2k
y
兲 ± t
⬜
共cos k
x
− cos k
y
兲
2
/4,
共6兲
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
共away from half-filling兲 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. 4共a兲 and 4共b兲 we have plotted the density of states at
the Fermi level as a function of doping 共this is possible for a
doping-independent band兲 to illustrate where the van Hove
singularity is. The remarkable feature in Fig. 4共a兲, 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. 4共b兲 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 共see Fig.
4 of that reference兲 the doping parameters were such that the
optical sum rule and the negative of the kinetic energy were
qualitatively 共and even quantitatively兲 similar.
26
Here, in the
FIG. 3. Illustrative plots of the band structure for 共a兲 nearest-
neighbor hopping only, 共b兲 the t-t
⬘
model, and 共c兲 parametrization
of Kordyuk et al. 共Ref. 24兲 of the band structure with bilayer split-
ting. The van Hove singularities occur where the band dispersion
flattens.
FIG. 4. 共Color online兲共a兲 The difference 共W
d
−W
N
兲 in the spec-
tral weight between the superconducting state with d-wave symme-
try and the normal state at zero temperature vs doping 共solid red
curve兲, 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 共the density of states at the Fermi level is indicated, as a
function of doping, by the dot-dashed cyan curve兲. The negative of
the kinetic energy for d-wave 共blue short-dashed curve兲 and for
s-wave 共dotted pink curve兲 symmetry behaves as expected, always
negative, and peaks 共in absolute value兲 at the van Hove singularity.
共b兲 The normal-state slope 共taken at T
c
=69 K兲 of the spectral
weight vs doping 共solid red curve兲. The dashed green curve shows
the same quantity for the negative of the kinetic energy. These
behave in very similar fashion to the differences 共taken at zero
temperature兲 shown in 共a兲.
MARSIGLIO et al. PHYSICAL REVIEW B 74, 174516 共2006兲
174516-4

Frequently asked questions

Q1. What have the authors contributed in "Intraband optical spectral weight in the presence of a van hove singularity: application to bi2sr2cacu2o8+" ?

The authors 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. The authors 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 the authors find that in the relevant doping regime a straightforward BCS calculation of the optical spectral weight can not account for the experimental observations. 

Q2. What is the spectral weight of a tight-binding band?

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 spectral weight is expected to occur below the superconducting transition temperature. 

Q3. What is the kinetic energy of the d-wave symmetry?

For an order parameter with d-wave symmetry, the momentum distribution is no longer a function of the band structure energy k alone. 

Q4. What is the simplest way to determine the chemical potential?

In the normal state this expression reduces to the simple Fermi function; even above Tc, however, iteration for the correct value of the chemical potential is required. 

Q5. What is the band structure for nearest-neighbor hopping?

For nearest-neighbor hopping only, the band structure is given byk nn = − 2t cos kx + cos ky 3and the authors have that 2W=− K in two dimensions. 

Q6. What is the spectral weight of the n-body?

Using t =0.10 eV, for example, would result in a very narrow range of electron densities for which the optical spectral weight has behavior opposite to that of the negative of the kinetic energy see Fig. 5, blue dashed curves . 

Q7. How can one apply the Sommerfeld expansion to W T?

If one defines the quantitygxx 1Nk 2 k kx 2 − k , 10then the Sommerfeld expansion can be applied to W T as was done for the kinetic energy. 

Q8. What is the spectral weight of the van Hove singularity?

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. 

Q9. What is the spectral weight in the superconducting state?

The remarkable feature in Fig. 4 a , for electron densities below the van Hove singularity, is that the spectral weight change in the superconducting state is positive. 

Q10. What is the doping dependence of the optical spectral weight?

In Fig. 8 the doping dependence of the optical spectral weight slope is shown as a function of electron concentration n for the hole-doped region with respect to half-filling . 

Q11. What is the impact of the correlation effects on the spectral weight?

This has a significant impact on the interpretation of experimental results, as doping dependence due to correlation effects, for instance, would have to be separated out either experimentally or theoretically. 

Q12. What is the slope of the spectral weight anomaly?

The corresponding slope above Tc would, however, be inconsistent with experi-ment not shown , but the slope is a purely normal-state property and, like all other normal state properties, undoubtedly requires electron correlations for a proper understanding. 

Q13. What is the kinetic energy of the electrons in the given tightbinding band?

Note that this is not the total kinetic energy of all the electrons, but just the kinetic energy of the electrons in the given tightbinding band s ; furthermore, only in the case of nearestneighbor hopping is W proportional to − K .