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

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