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17 Photoemission in the High-T

Superconductors

J. C. Campuzano Department of Physics, University of Illinois at Chicago, Chicago, USA

M. R. Norman Materials Science Division, Argonne National Laboratory,Argonne, USA

M. Randeria Tata Institute of Fundamental Research, Mumbai,India

17.1 Introduction .............................................................................924

17.2 Basics of Angle-Resolved Photoemission ....................................................924

17.2.1MatrixElementsandSelectionRules..................................................926

17.2.2TheOne-StepModel.................................................................927

17.2.3Single-ParticleSpectralFunction.....................................................928

17.2.4SpectralFunctionsandSumRules....................................................929

17.2.5AnalysisofARPESSpectra:EDCsandMDCs...........................................931

17.3 The Valence Band .........................................................................933

17.4 Normal State Dispersion and the Fermi Surface ..............................................935

17.4.1 Normal State Dispersion in Bi2212:A First Look . .......................................935

17.4.2ImprovedMethodsforFermiSurfaceDetermination....................................937

17.4.3SummaryofResultsontheOptimallyDopedFermiSurface:.............................944

17.4.4ExtendedSaddlePointSingularity....................................................945

17.4.5BilayerSplitting?....................................................................945

17.5 Superconducting Energy Gap ..............................................................948

17.5.1Particle–HoleMixing................................................................948

17.5.2QuantitativeGapEstimates...........................................................950

17.5.3DopingDependenceoftheSCGap....................................................952

17.6 Pseudogap ...............................................................................954

17.6.1 Pseudogap near (,0)................................................................954

17.6.2AnisotropyofthePseudogap.........................................................956

17.6.3FermiArcs.........................................................................957

17.6.4EvolutionoftheFermiSurfacewithDoping............................................960

17.6.5LowEnergyvsHighEnergyPseudogaps...............................................961

17.6.6OriginofthePseudogap? ............................................................962

17.7 Photoemission Lineshapes and the Electron Self-Energy ......................................963

17.7.1Self-EnergyExtraction...............................................................964

17.7.2TemperatureDependenceofSelf-Energy...............................................966

17.7.3ModelingtheSelf-Energy............................................................968

17.7.4Peak/Dip/Hump–Experiment........................................................971

17.7.5ModeModel........................................................................973

17.7.6DopingDependence.................................................................976

17.7.7DispersionKinkofNodalQuasiparticles ..............................................980

17.7.8CondensationEnergy................................................................986

17.8 Summary ................................................................................987

References...............................................................................988

924 J.C. Campuzano, M.R. Norman, and M. Randeria

17.1 Introduction

Angle resolved photoemission spectroscopy (ARPES)

has played a major role in the elucidation of the elec-

tronic excitations in the high temperature cuprate

superconductors. Several reasons have contributed

to this development. First, the great improvement

in experimental resolution, both in energy and mo-

mentum, aided by the large energy scales present

in the cuprates, allows one to see features on the

scale of the superconducting gap. More recently the

resolution has improved to such an extent, that now

features in traditional superconductors like Nb and

Pb, with energy scales of a meV, can be observed by

ARPES [1]. Second, most studies have focused on

CaCu

(Bi2212) and its single layer coun-

terpart, Bi

CuO

(Bi2201). These materials are

characterized by weakly coupled BiO layers, with the

longest interplanar separation in the cuprates. This

results in a natural cleavage plane, with minimal

charge transfer. This is crucial for ARPES, since it is

a surface sensitive technique; for the photon ener-

gies typically used, the escape depth of the outgoing

electrons is only of the order of ∼ 10 Å. For this

reason, we elect to concentrate on these materials in

the current article, since the data are known to be

reproducible among the various groups.

The third reason is the quasi-two-dimensionality

of the electronic structure of the cuprates,whichper-

mits one to unambiguously determine the momen-

tum of the initial state from the measured final state

momentum, since the component parallel to the sur-

face is conserved in photoemission. Moreover,in two

dimensions, ARPES directly probes the single parti-

cle spectral function, and therefore offersa complete

picture of the many body interactions inherent in

these strongly correlated systems.

There are other reasons to pointout as well.For ex-

ample, the very incoherent nature of the excitations

near theFermi energy in thenormal statedoes not al-

low the application of traditional techniques, such as

the de Haas van Alphen effect. The electrons simply

do not live long enough to complete a cyclotron or-

bit.Another important reason whyARPES has played

a major role is the highly anisotropic nature of the

electronic excitations in the cuprates, which means

that momentum resolved probes are desirable when

attempting to understand these materials.

It is satisfying to compare the earlier ARPES pa-

persto more recent ones,and observe theremarkable

progress the field has undergone.But this satisfaction

must be tempered by the knowledge that the current

literature is still full of unresolved issues. Undoubt-

edly, some of what is described here will be viewed

in a new light in the future, and we will try to point

out the contentious points, but the reader must keep

in mind that the writers, by necessity, approach this

task with their own set of biases. For a complemen-

tary view, the reader is referred to the recent review

by Damascelli, Shen and Hussain [2]. An earlier re-

view of ARPES studies of high T

superconductors

can be found in the book by Lynch and Olson [3].

17.2 Basics of Angle-Resolved

Photoemission

The basics of angle-resolved photoemission have

been described in detail in the literature [4]. We

will limit ourselves to a brief review of some salient

points, emphasizing those aspects of the technique

which will be useful in understanding ARPES stud-

ies of high T

superconductors. We start by looking

at a simple independent particle picture, and subse-

quently include the effects of strong interactions.

The simplest model of ARPES is the three step

model [5], which separates the process into photon

absorption, electron transport through the sample,

and emission through the surface. In the first step,

the incident photon with energy h is absorbed by

an electron in an occupied initial state, causing it to

be promoted to an unoccupied final state, as shown

in Fig. 17.1(a). There is conservation of energy, such

that

h = BE + ¥ + E

kin

, (17.1)

where ¥ isthe work function,andBE and E

kin

are the

binding and kinetic energies of the electron, respec-

tively.We will not discuss the subsequent two steps of

the three step model, photoelectron transport to the

surface and transmission through the surface into

the vacuum, as they only affect the number of emit-

ted electrons, and thus the absolute intensity [4].

17 Photoemission in the High-T

Superconductors 925

Fig. 17.1. (a) Independent

particle approximation

view of ARPES; to the right

is the resulting energy

distribution curve (EDC).

(b) Experimental EDCs

along a path in momentum

space in Bi2212. (c)In-

tensity map from these

data

The kinetic energy of the electrons is measured by

an electron energy analyzer.If the number of emitted

electrons is then plotted as a function of their kinetic

energy, as shown in Fig. 17.1, peaks are found when-

ever an allowed transition takes place.Eq.(17.1) then

yields the kinetic energy of the electron if the work

function is known. Measuring ¥ accurately is not a

trivial task, but fortunately, in the case of a metal-

lic sample one does not need to know ¥ .Byplacing

the sample in electrical contact with a good metal

(e.g.,polycrystalline gold) one can measure the bind-

ing energy in the sample with respect to the chemi-

cal potential (Fermi level E

) of Au. The photoemis-

sion signal from Au will simply be a Fermi function

convolved with the experimental energy resolution,

and from the mid point of its leading edge one esti-

mates E

The existence of the sample surface breaks (dis-

crete) translational invariance in the direction nor-

mal to the surface. However, two-dimensional trans-

lational invariance in the directions parallel to the

surface is still preserved, and thus k



,thecomponent

of the electron momentum parallel to the surface, is

conserved in the emission process. This allows us to

obtain the in-plane momentum of the initial state

by identifying it with the parallel momentum of the

emitted electron



2mE

kin



sin  cos  (17.2)

of the outgoing electron emitted along the (, )

direction with a kinetic energy E

kin

,asshownin

Fig. 17.2(a).

The momentum perpendicular to the sample sur-

face k

is not conserved, and thus knowledge of the

final state k

does not permit one to say anything

useful about the initial state k

,exceptinthecase

where k



= 0, in which case k

= k

.Weshouldem-

phasize that a given outgoing electron corresponds

to a fixed, but a priori unknown, k

.Inmaterials

with three-dimensional electronic dispersion it is,

of course, essential to fully characterize the initial

state, and many techniques have been developed to

estimate k

; see [4]. Aside from making k



=0as

described above, one can vary the incident photon

energy h thereby changing the kinetic energy of

the outgoing electron. For fixed k



this amounts to

changing theinitial state k

.(One also has to take into

account thechangesinintensity broughtabout bythe

photon energy dependence of the matrix elements).

We will not be much concerned about methods of

determination here,since the high T

cuprates are

quasi-two-dimensional (2D) materials with,in many

cases (e.g., Bi2212), no observable k

-dispersion. In

fact, this makes ARPES data from 2D materials much

easier to interpret, and is one of the reasons for

the great success of the ARPES technique for the

cuprates. In the remainder of this article, we will use

the symbol k to simply denote the two-dimensional

momentum parallel to the sample surface,unless ex-

plicitly stated otherwise.

In the independent particle approximation, the

ARPES intensity as a function of momentum k (in

the 2D Brillouin zone) and energy ! (measured with

respect to the chemical potential) is given by Fermi’s

Golden rule as

926 J.C. Campuzano, M.R. Norman, and M. Randeria

I(k, !) ∝



A · p



f (!)ı(! − "

) , (17.3)

where

and

are the initial and final states, p is

the momentum operator, and A the vector poten-

tial of the incident photon. (In general, the matrix

element also contains a p · A term. Above the bulk

plasma frequency,the lightpenetrates deeply into the

solid, making this term small. This “surface photoe-

mission” has not been prominent in the many stud-

ies of the cuprates, and we will henceforth ignore it.)

Here f (!)=1/[exp(!/T) + 1] is the Fermi func-

tion at a temperature T in units where  = k

=1.It

ensures the physically obvious constraint that pho-

toemission only probes occupied electronic states.

For noninteracting electrons, the emission at a

given k is at a sharp energy 

corresponding to

the initial state dispersion.As we will discuss below,

the effect of interactions (either electron–electron

or electron–phonon) is to replace the delta func-

tion by the one-particle spectral function, which is

a non-trivial function of both energy and momen-

tum. Although now the peaks will in general shift in

energy and acquire a width, the overall intensity is

still governed by the same matrix element as in the

non-interacting case. Thus the lessons learned from

studying general properties of the matrix elements

are equally applicable in the fully interacting case.

17.2.1 Matrix Elements and Selection Rules

ThedipolematrixelementM



A · p



is in

general a function of the momentum k,oftheinci-

dent photon energy h , and of the polarization A of

the incident light. Here we discuss dipole selection

rules which arise from very general symmetry con-

straints imposed on M

,and which are very useful in

interpreting ARPES data. Our approach is a simpli-

fied version of Hermanson’s analysis [6].

Let the photon beam be incident along a plane of

mirror symmetry of the sample (M), with detector

placed in the same mirror plane; see Fig. 17.2(a). The

final state

must then be even with respect to reﬂec-

tion in M, because if it were odd the wave function

would vanish at the detector. (For this discussion it

is simpler to imagine changing the photon polariza-

tion, keeping the detector fixed. In the actual exper-

iment, however, it is the polarization which is fixed

and the detector is moved when checking dipole se-

lection rules.)

The dipole transition is allowed if the entire ma-

trix element has an overall even symmetry. Thus two

possibilities arise. (1) For an initial state

which

is even with respect to M, the light polarization A

must also be even, i.e., parallel to M.(2)Foranini-

tial state odd with respect to M, A must also be odd,

i.e.,perpendicular to M. This can be summarized as:



A ·ˆp







even +| + |+⇒A even ,

odd +| − |−⇒A odd .

(17.4)

Consider hybridized Cu3d -O2p initial states, as

shown in Fig. 17.2(b), which have a d

−y

symmetry

about a Cu site. These states are even under reﬂec-

tion in (0, 0) − (, 0) (i.e., the plane defined by this

symmetry axis and the z-axis) and odd with respect

Fig. 17.2. (a)Arrangement

of the photon beam and de-

tector in order to make use

of the photoemission selec-

tion rules. (b) Symmetry of

the Cu3d

−y

2 orbitals hy-

bridized with the O2p or-

bitals.(c)EDCsshowingthe

symmetry of the orbitals in

(b) obtained at  =22 eV

17 Photoemission in the High-T

Superconductors 927

to (0, 0) − (, ). Therefore, measurement along the

(0, 0) − (, 0) direction will be dipole-allowed (for-

bidden) if the polarization vector A is parallel (per-

pendicular) to this axis. Figure 17.2(c) shows that,

consistent with an initial state which is even about

(0, 0) − (, 0), the signal is maximized when A lies

in the mirror plane and minimized when A is per-

pendicular to this plane. (The reasons for small non-

zero intensity in the dipole forbidden geometry are

the small,but finite,k-windowof the experiment and

the possibility of a small misalignment of the sam-

ple). Similarly, we have checked experimentally (for

Bi2212 in the Y-quadrant where there are no super-

lattice complications) that the initial state is consis-

tent with odd symmetry about (0, 0) − (, )(see

also [7]).

While the dipole matrix elements are strongly

photon energy dependent, the selection rules are, of

course, independent of photon energy. This has been

checked by measurements at 22 eV and 34 eV. [8]

All of these results are consistent with the fact that

we are probing Cu3d-O2p initial one-electron states

with d

−y

symmetry. We will see below that these

selection rules are extremely useful in disentangling

the main CuO

“band”from its umklapp images due

to the superlattice in Bi2212.

17.2.2 The One-Step Model

The three-step model of photoemission gives a very

useful zeroth order descriptionof the photoemission

process, but it needs to be put on a firmer footing,

both as regards the calculation of photoemission in-

tensities from ab initio electronic structure calcula-

tions, and also for developing a deeper understand-

ing of how many-body effects inﬂuenceARPES spec-

tra. We now brieﬂy review the so called “one-step”

model of photoemission with these goals in mind.

Ever since Hertz and Hallwachs discovered pho-

toemission [9], it is known that the photo–electron

current at the detector is proportional to the inci-

dent photon ﬂux, i.e., to the square of the vector

potential. Thus photoemission measures a nonlin-

ear response function, and the relevant correlation

function is a three current correlator, as first empha-

sized by Schaich and Ashcroft [10]. It is instructive

to brieﬂy review their argument. As in standard re-

sponse function calculations, let’s look at an expan-

sion of the current at the detector (the response) in

powers of the vector potential of incident photons

(the perturbation). Let R be the location of the de-

tector in vacuum and r denote points inside the sam-

ple.The zeroth order contribution0|j

(R, t)|0van-

ishesastherearenocurrentsﬂowingintheground

state |0 of the unperturbed system. The linear re-

sponse also vanishes, with 0|j

(R, t)j

(r, t



)|0 =0

and 0|j

(r, t



(R, t)|0 = 0, since there are no par-

ticles at the detector, in absence of the electromag-

netic field, and j

(R, t)|0 =0.Thustheleadingterm

which survives is

j



(R, t)∝









, t





, t



)

×0|j



, t





(R, t)j



, t



)|0 ,

(17.5)

where only current operators inside the sample act

on the unperturbed ground state on either side and

the current at the detector is sandwiched in between.

The three current correlation function can be

represented by the triangle diagram shown in

Fig. 17.3(A) [11],where the line between the two ex-

ternal photon vertices is the Greens function of the

“initial state” or “photo-hole” and the two lines con-

necting the photon vertex to the current at the detec-

tor represent the “photo-electron” which is emitted

from the solid. There is a large literature on the eval-

uation of the (bare) triangle diagram incorporating

the results of ab initio electronic structure calcula-

tions of a semi-infinite solid, including the effects

of realistic surface termination and multiple scatter-

ing effects in the photo–electron final states.Detailed

calculations using this formalism were first carried

out by Pendry [12] for a system with one atom per

unit cell,and later generalized to more complex crys-

tals [13–15].

This approach is very reliable for calculating pho-

toemission intensities,and gives important informa-

tion about “matrix element effects”, i.e., the depen-

dence of the ARPES intensity on k, and on the in-

cident photon energy and polarization. (It does not,

however, shed light on the many-body aspects of the

lineshape). Such ab initio methods have been exten-

928 J.C. Campuzano, M.R. Norman, and M. Randeria

Fig. 17.3. Diagrams contributing to the three-current cor-

relator of ARPES.The physical processes that each diagram

represents is discussed in the text

sively used by Bansil, Lindroos and coworkers [16]

for the high T

cuprates.

As an example of the usefulness of ab initio

methods, we show the comparison between the ob-

served and calculatedARPES intensitiesfor YBCOin

Fig. 17.4. Since the calculated intensity is a sensitive

function of the termination plane, such a compari-

son suggests that the crystal breaks at the chains, a

conclusion later confirmed by STM measurements

[17]. In fact, it was the complexity of the termina-

tion in YBCO, plus the fact that none of the pos-

sible cleavage planes of YBCO are charge neutral,

and thus involve significantcharge transfer,that con-

vinced us to focus primarily on the Bi-based cuprates

(Bi2212 and Bi2201).These materials have two adja-

cent BiO planes which are van der Waals bonded,

and thus cleavage leads to neutral surfaces which

are not plagued by uncontrolled surface effects. The

highly two-dimensional Bi-based cuprates are thus

ideal from the point of view of ARPES studies.

17.2.3 Single-Particle Spectral Function

Although the one step model gives a reasonable in-

terpretation of the overallintensity in the photoemis-

sion process, much of what we really want to learn

from the experiment relates to the spectral lineshape,

which, as we will show, is strongly inﬂuenced by cor-

relations. We must then ask ourselves the important

question of how the initial state lineshape enters the

ARPES intensity and to what extent this is revealed in

the observedlineshape.Althougha fully rigorous jus-

tification of a simple interpretation of ARPES spectra

is not available at the present time, a reasonable case

can be made for analyzing the data in terms of the

single particle spectral function of the initial state.

Fig. 17.4. Comparison of the photoemission intensity of states along the zone diagonal in YBCO with (a) even polarization

and (b) odd polarization, showing that this state is odd with respect to reﬂection about the diagonal to the Cu-O bond

direction. (c) Comparison of the photoemission intensity to the observed one as a function of polarization. The detailed

polarization comparison gives confidence to the interpretation of the observed peak as arising from the Cu-O plane states