Specialized Scientic Council on Condensed Matter at the High Attestation Commis sion
ON THE THEORY OF HIGHTEMPERATURE
SUPERCONDUCTIVITY:
FROM ELECTRONIC STRUCTURE TO FLUCTUATIONAL
PROPERTIES AND ELECTRODYNAMIC BEHAVIOR
TODOR MIHAYLOV MISHONOV
Department of Theoretical Physics,St.Clement of Ohrid University o f Soa
DISSERTATION
for awarding of scientic degree
DOCTOR of SCIENCE (DSc) in Physics
Bulgarian PACS:01.03.26 Electrical,Magnetic and Optical Properties of Condensed Matter
Referees:
Prof.Jordan Brankov,DSc
Prof.Vesselin Kovachev,DSc
Prof.Nikolay Tonchev,DSc
Soa
2007
CONTENTS
Contents
i
Introduction,purpose and methods
v
Description and organization of the dissertation work
vii
General conclusions for the hightemperature superconductors
xi
1 Tightbinding modelling of the electronic band structure of layered superconducting per
ovskites
1
1.1 Introduction
.........................................1
1.1.1 Appology to the band theory
............................2
1.2 Layered cuprates
......................................3
1.2.1 Model
........................................3
1.2.2 Effective Hamiltonians
...............................5
1.3 Conduction bands of the RuO
2
plane
............................10
1.4 Discussion
..........................................11
1.5 Determination of density of states of thin highT
c
lms by FET type microstructures
..13
1.5.1 Determination of logarithmic derivative of density of states by electronic mea
surements
......................................13
2 Superconductivity of overdoped cuprates:the modern face of the ancestral twoelectron
exchange
17
2.1 Introduction
.........................................17
2.2 Lattice Hamiltonian
.....................................18
2.2.1 The fourband model in a nutshell
.........................18
2.2.2 The HeitlerLondon &SchubinWonsowskyZener interaction
..........21
2.3 Reduced Hamiltonian and the BCS gap equation
......................23
2.4 Separable sd model
....................................24
2.5 Antiferromagnetic character of J
sd
............................28
2.5.1 Intraatomic correlations
..............................29
2.5.2 Indirect sd exchange
................................30
2.5.3 Effect of mixing wave functions
..........................30
2.5.4 Cooper and Kondo singlet formation
........................31
2.6 Dogmatics,Discussion,Conclusions and Perspectives
...................31
2.6.1 Aesthetics and frustrations of the central dogmas
.................32
2.6.2 Discussion
.....................................34
2.6.3 The reason for the success of the CuO
2
plane
...................36
2.6.4 Perspectives:if Tomorrow comes...
.......................37
i
ii Contents
3 Specic heat and penetration depth of anisotropicgap BCS supe rconductors for a fac
torizable pairing potential
41
3.1 Specic heat
.........................................41
3.2 Electrodynamic behavior
..................................49
3.3 The case for Sr
2
RuO
4
...................................53
3.4 Discussion and conclusions
.................................54
4 Plasmons and the Cooper pair mass
57
4.1 Plasmons:prediction
....................................57
4.2 In search for the vortex charge and the Cooper pair mass
.................58
4.2.1 Introduction
.....................................58
4.2.2 Model
........................................59
4.2.3 TypeII superconductors
..............................59
4.2.4 Experimental setup for measuring the vortex charge
...............62
4.2.5 How to measure the Cooper pair mass
.......................63
4.2.6 Discussion and conclusions
............................69
5 Thermodynamics of Gaussian uctuations and paraconductivity in layered superconduc
tors
71
5.1 Introduction
.........................................71
5.2 Weak magnetic elds
....................................73
5.2.1 Formalism
.....................................73
5.2.2 EulerMacLaurin summation for the free energy
.................76
5.2.3 Layering operator
ˆ
L illustrated on the example of paraconductivity
.......78
5.2.4 Power series for the magnetic moment within the LD model
...........83
5.2.5 The epsilon algorithm
...............................84
5.2.6 Power series for differential susceptibility
.....................87
5.3 Strong magnetic elds
...................................88
5.3.1 General formula for the free energy
........................88
5.3.2 Fluctuation part of thermodynamic variables
...................90
5.3.3 Selfconsistent approximation for the LD model
..................94
5.3.4 3D test example
...................................96
5.4 Some remarks on the tting of the GL parameters
.....................98
5.4.1 Determination of the cutoff energy ε
.......................98
5.4.2 Determination of the coherence length ξ
ab
(0)
...................100
5.4.3 Determination of the Cooper pair lifetime constant τ
0
..............101
5.4.4 Determination of the Ginzburg number and penetration depth λ
ab
(0)
.......102
5.5 Discussion and conclusions
.................................102
6 Kinetics and Boltzmann kinetic equation for uctuation Cooper pairs
105
6.1 Introduction
.........................................105
6.2 FromTDGL equation via Boltzmann equation to Newton equation
............105
6.3 Fluctuation conductivity in different physical condition
..................108
6.3.1 High frequency conductivity
............................108
6.3.2 Hall effect
......................................109
6.3.3 Magnetoconductivity
................................109
6.3.4 Strong electric elds
................................110
6.4 Current functional:selfconsistent approximation and energy cutoff
...........111
6.5 Fluctuation conductivity in nanowires
...........................113
Contents iii
6.6 Discussion and Conclusion
.................................115
7 Fluctuation conductivity in superconductors in strong electric elds
117
7.1 Introduction
.........................................117
7.2 Boltzmann equation and formula for the current
......................118
7.3 Dimensionless variables
..................................120
7.4 Paraconductivity in a layered metal
.............................122
7.5 AslamazovLarkin conductivity for Ddimensional superconductors
...........124
7.5.1 Strong electric eld expansion
...........................125
7.5.2 Weak electric elds below T
c
...........................126
7.6 Striped superconductors and thick lms
..........................128
7.7 Determination of the lifetime constant τ
0
..........................129
7.8 Conductivity correction by detection of 3rd harmonics
..................131
7.9 Discussion and conclusions
.................................132
8 A model for the linear temperature dependence of the electrical resistivity of layered
cuprates
135
8.1 Introduction
.........................................135
8.2 Model
............................................136
8.3 Numerical example
.....................................138
8.4 Discussion and conclusions
.................................140
9 Theory of terahertz electric oscillations by supercooled superconductors
143
9.1 Introduction
.........................................143
9.2 Physical model
.......................................143
9.2.1 Qualitative consideration and analogies
......................143
9.2.2 Formulas for the differential conductivity
.....................145
9.3 Description of the oscillations
...............................147
9.4 Performance of the generator
................................148
9.5 Possible applications
....................................149
Appendices
151
A Calculation of OOhopping amplitude by the surface integral method
151
B T
c
ǫ
s
correlations:a hint toward the mechanismof superconductivity in cuprates
153
C Order parameter equation for anisotropicgap superconductors
155
D The epsilon algorithmin FORTRAN90
163
E Solution to the Boltzmann equation
167
F Basic contributions
171
G Publications of Dr.Mishonov used as a basis for this DSc thesis
173
Bibliography
175
Acknowledgments,retrospect
199
Index
200
INTRODUCTION,PURPOSE AND METHODS
In the last 20 years the investigation of hightemperature superconductivity took central place in con
densed matter physics.About 100,000 scientic papers were published,Fig.
1
,and it was one of the
largestscale investigations in the history of science in general.In such a way hightemperature super
conductors became some of the best investigated materials.Their technology was developed for the
purposes of producing cables,transformers,generators and powerful motors.Their possible applications
in future superconducting electronics and generators of terahertz oscillations are intensively investigated.
On the other hand,the mechanism of hightemperature superconductivity remained unclear and this in
tellectual puzzle has attracted the attention of many theoreticians fromall areas of physics.
The purpose of the present dissertation is to give a unied picture of the th eory of hightemperature
superconductivity,with the attention being focused on those wellinvestigated properties which can be
understood within the framework of established models;it is assumed that if some phenomenon can be
understood on the basis of textbook methods there is signicant chance this phenomenon to enter the
next generation textbooks.Of course,this approach reveals the professional deformation of a teacher
for whom working with students is a technology for obtaining new scientic re sults.Some theoretical
predictions,e.g.,the infrared transparency of bismuth cuprates,plasmons in thin superconducting lms,
and the rst determination of the effective mass of Cooper pairs,have be en experimentally conrmed
during the longtime scientic research whose results are described in the p resent dissertation.Physics
is an experimental science,and one of the challenges of theoretical physics,in particular,is to predict
the outcome of experiments that nobody has done yet.In this sense,the experimental support for the
theoretical results of this dissertation is an important hint that we are on the right track.
A broad spectrum of methods has been used to achieve the goals of this thesis.For the theoretical
description of hightemperature superconductivity the standard methods of statistical physics,quantum
eld theory,and theoretical chemistry are used,while as a mathematical method,a routine for numer
ical summation of divergent series is developed.Since the world is made of atoms,we will start with
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Year
1
10
100
1000
10000
Number of articles
1913
Figure 1:Number of articles published in the period 19452006 containing`superconduct*'in the title
(◦),or any of the title,abstract or the list of keywords (✷) according to Science Citation Index Expanded
(www.isiknowledge.com).
v
vi Introduction,purpose and methods
the atomic wave functions,and within the framework of this basis set we will consider the exchange
interaction and the electron spectrum.Further,the solution of the statistical problem gives analytical
formulas for the heat capacity and electromagnetic response of hightemperature cuprates,as well as the
coefcients of the GinzburgLandau (GL) theory.Special attention is pa id to the dynamics and statistics
of the effective GL function as for hightemperature superconductors (as opposed to conventional) the
uctuation phenomena are strongly expressed and contribute signicantly to the kinetic coefcients and
thermodynamic behavior.
The nal aimof the dissertation is to described novel physical effects,s uch as polariton propagation
in the bulk superconducting phase,determination of the effective mass of charge carriers by standard
methods of electronics,and to investigate the perspective of using hightemperature superconductors for
creation of novel generators for electromagnetic waves in the still not used for technical applications
terahertz range.
DESCRIPTION AND ORGANIZATION OF THE
DISSERTATION WORK
The dissertation contains 201 pages,27 gures,547 references,9 c hapter and 7 appendices.It is based
upon 25 publications of the dissertator,published in accessible journals,having about 70 citations (this
number does not include works from the PhD theses of the author).As a whole the dissertator has
about 70 works in different areas of physics,if taking into account also those uploaded to the
arXiv
eprint server.The total number of citations of the dissertator is around 150.The pdf le of the current
version of the dissertation is accessible from
http://www.phys.unisoa.bg/∼mishonov/
(please,send
critical comments and remarks to mishonov@phys.unisofia.bg).
The dissertation is organized as follows:
1.In the 1st chapter is investigated the electronic band structure of the layered superconducting per
ovskites.It is demonstrated that tight binding model,known in the quantum chemistry also as
method of linear combination of atomic orbitals (LCAO) depicted for CuO
2
plane at Fig.1.1 is an
adequate tool for investigation of single particle spectrum.In section 1.2 are considered layered
cuprates,while in section 1.3 layered ruthenates.It is used the standard basis of valence orbitals for
metal and oxygen ions.There are derived analytical formulas for electron dispersion which give
the relation between the energy and electron momentum (1.5) and (1.26).It is shown that these
formulas describe well the ab initio calculations Fig.1.2 and Fig.1.3 and also the experiments
for angular resolved photoemission spectroscopy (ARPES) Figs.1.4,1.5,1.6.In Appendix A it
is shown how the amplitudes of single electron hopping can be calculated by the known from the
atomic physics method of asymptotic wave functions.In the conclusion section of this chapter is
concluded that in nearest years the two times difference for the band width between the ab initio
calculations and ARPES data is insurmountable.That is why the hopping amplitudes and single
site energies have to be treated as parameters of LCAO theory which has tte d by the experiment.
It is concluded that tight binding parameters are the meeting point between th e computer physics
and the experiment.The theory gives the adequate basis orbitals but the numerical values are de
termined by the comparison with the experiment.It is analyzed that the ab initio theory describe
correctly the shape of the Fermi surface (the contour of Fermi for strongly anisotropic perovskites)
and actually it is necessary only one parameter to be tted the density of states for the charge car
riers.In section 1.5 is suggested newmethod for determination of the logarithmic derivative of the
density of states which requires only standard electronics but not unique accelerators and electron
spectrometers of the class of Scienta 200.The suggested method requires the measurement of the
second harmonics of the work function when a harmonic current is applied trough a thin lm;see
Fig.1.7.
2.In the 2nd chapter exchange interaction is considered as possible pairing mechanism in high
temperature cuprates.In section 2.2 are analyzed as HeitlerLondon inter atomic two electron
exchange as well as the intraatomic sd exchange by SchubinWonsovskyZenerKondo.Spe
cial attention is devoted on the history of creation of the ideas of electron pairing and exchange.
In section 2.3 is derived the reduced Hamiltonian of BardeenCooperSc hrieffer (BCS) and it is
vii
viii Description and organization of the dissertation work
derived BCS equation for superconducting gap (2.25).The exchange interaction is considered
in the LCAO basis and it is shown that necessary gap symmetry,which agrees with the experi
ment,can be derived only for intraatomic exchange with antiferromagnetic sign.In section 2.4
is sequentially derived a separable model corresponding to sd pairing interaction schematically
presented at Figs.2.1 and 2.2.This separable model derived by rst p rinciples coincides by form
with model Hamiltonians phenomenologically postulated for the description of the available ex
perimental data.The results of the model calculations with the help of the derived analytical
formulas for electron dispersion (1.5),the velocity of charge carriers (2.40),the superconducting
gap (2.30) as function of the electron quasimomentumare shown in Fig.2.3.The correspondence
of the theoretical calculation with the photoemission measurements is depicted in Fig.2.3 (d),and
it is concluded that a fundamental understanding of the pairing mechanism in hightemperature
cuprates is already achieved.The twoelectron 3d4s exchange is one the biggest amplitudes in
condensed matter physics and can be found to be not only the mechanism of the ferromagnetism
of the iron,but also the mechanism of cuprate superconductivity.In the section 2.5 is analyzed
that necessary antiferromagnetic sign of the exchange interaction is rather rule than exception not
only in the atomic physics but also in the condensed matter.In section 2.6 is juxtaposed the de
veloped microscopic theory for hightemperature superconductivity with the dogmatic dominated
this domain over 15 years.Esthetics and frustrations of the central dogmas are considered.The
Ginzburg periodization of the physics of superconductivity is analyzed and it is considered the
possibility that epoch tomorrow is already came.As important criterion for th e validity of the
different theoretical models for hightemperature superconductivity in the appendix B is consid
ered the explanation of the subtle correlation between copper 4s level in the cuprate crystal and
the critical temperature of the superconducting transition.This correlation without any theoretical
interpretation was investigated for almost all hightemperature cuprates in the group on electron
structure in MPI in Stuttgart.
3.In 3rd chapter is considered thermodynamic and electrodynamic behavior of anisotropic gap su
perconductors.The solution of equation for superconducting order parameter for the separable sd
model turned out to be important for understanding thermodynamics and electrodynamics for all
anisotropic gap superconductors.Mathematical details are described in the appendix C,and in the
appendix D is given a program realizing epsilon algorithm for summation of divergent series.In
this chapter 3 is derived general formula for the heat capacity (3.29) for the case of separable re
duced BCS Hamiltonian.It is shown that this newmicroscopic result embraces as special cases all
known formulas by GorterCasimir,GinzburgLandau,Gor'kovMelikB arkhudarov,Pokrovsky
and Moskalenko.In parallel this thermodynamic consideration gives a new method for the deriva
tion of BCS formulas for the coefcients of GinzburgLandau theory.It is shown how these
formulas can be generalized when a Van Hove singularity s near to the Fermi level (3.33).Us
ing this formula are tted the data for MgB
2
one hightemperature superconductor with phonon
mechanism of the superconductivity,see Fig.3.3.For completeness is given also the formula for
the penetration depth.The temperature dependence of the skin depth as function of the reduced
temperature are compared for:yttriumcuprate,magnesiumdiboride,dwave superconductor,and
isotropic gap superconductor;Fig.3.4.The derived formulas are applicable also to t the data
for strontium ruthenate Fig.3.7 and practically for all superconductors.In such a way for high
temperature superconductors is repeated the analysis for conventional BCS superconductors and
it is demonstrated that main thermodynamic and electrodynamic properties can be derived from
rst principles.Coauthors of some basic works from the dissertation ar e former students of the
dissertator which have listen his lectures on statistical mechanics,superconductivity and solid state
theory.This demonstrates that hightemperature superconductivity is not a mystery and its theory
can be trivialized to this level of simplicity fromwhich starts the nal understan ding in the science.
Description and organization of the dissertation work ix
The whole richness of the condensed matter physics can be found in the cuprate physics.Espe
cially for the underdoped case when the translation invariance is broken one can observe charge
and spin density waves.The whole cuprate physics is far from nalization,but we consider that
the pairing mechanismis revealed in the analysis of overdoped cuprates given in chapter 3.
4.Some new electrodynamic properties of superconductors,and especially hightemperature super
conductors are considered in chapter 4.A new prediction is made in section 4.1 the infrared
transparency of bismuth cuprates at low temperatures.Due to strong anisotropy of the material
the plasma waves related to Cooper pair motion perpendicularly to the copper oxygen plane have
frequency smaller than superconducting gap (4.1).At such conditions the damping is small and
polaritons can propagate in the bulk of the superconductor.This prediction was conrmed in many
labs in Japan and USA.In the same section 4.1 are described the predicted two dimensional plas
mons in thin superconducting lms;this prediction was experimentally conrmed in Grenoble.
Section 4.2 is devoted on the problem how to measure the effective mass of Cooper pairs.The
development of this direction of the physics of superconductivity was retained for decades by the
opinion of famous theorists that the effective mass of superuid particles is experimentally inac
cessible parameter.Several different methods for creation of Cooper pair massspectroscopy.The
formula (4.43) derived by the dissertator was used in the Bell Laboratories for the rst determi
nation of the effective mass of Cooper pairs.It is shown that Bernoulli effect gives even better
experimental method for determination of Cooper pair mass,and it is concluded that without
Cooper pair mass the physics of superconductivity is Hamlet without the Prince.It is theoretically
concluded that all methods for measurements of effective mass of Cooper pairs are related to elec
tric eld effects in superconductors.It is derived that Bernoulli effe ct creates electric charge of
superconducting vortices (4.12) following their motion.
5.For the conventional superconductors uctuations have often only a cademic interest.Due to small
coherence length,however,uctuations plays signicant role for high temperature superconduc
tors and they contribution have to be taken into account at the interpretation of many experimental
data.Chapter 4 is devoted on thermodynamics of Gaussian uctuations and uctuation conductiv
ity.In selfconsistent approximation is calculated the free energy of Gaussian uctuations (5.129),
and also following from it formulas for the uctuational magnetization (5.133) and heat capacity
(5.150).It is shown how at analytically solved problem for two dimensional superconductor one
can obtain analytical solution for layered superconductor and bulk one.The general procedure is
illustrated by the examples of magnetic susceptibility and heat capacity (5.5860).Special atten
tion is devoted on three dimensional case,where results for the free energy (5.178) and uctuation
magnetization (5.180) are presented by Hurwitz zetafunctions.At the derivation of these results
was used the known from the quantum eld theory zetafunction technique for ultraviolet regu
larization.It is shown that divergent series have initially to be summated by EulerMacLaurin
method.Then the described in appendix D epsilonalgorithm gives fast convergent Pad´e approx
imants.In sections 5.1 and 5.2 are considered the cases of week and strong magnetic eld,and
in section 5.4 are suggested some new methods for determination of the parameters of time de
pendent GinzburgLandau (TDGL) theory.It is derived that time cons tant of this theory can be
determined by ratio of uctuation conductivity and uctuation susceptibility (5.6 1).With the help
of this formula for rst time was checked that for cuprates relaxation times ag ree with BCS theory.
6.In chapter 6 are considered the kinetics and Boltzmann kinetic equation for uctuation Cooper
pairs.In section 6.2 Boltzmann kinetic equation is derived in the framework of TDGL theory.In
section 6.3,having review character,the general formula for the uctua tion conductivity (6.15) is
applied for different physical conditions:high frequencies (6.22),perpendicular to the layers mag
netic eld and strong electric eld.There are obtained formulas for magneto conductivity (6.36),
x Description and organization of the dissertation work
uctuational Hall conductivity (6.30),and uctuation conductivity in arbitr ary dimensions (6.29);
those formulas agree with obtained yet results derived by other methods.The new derivations
obtained by solution of Boltzmann equation are,however,signicantly simpler,clearer and in this
sense esthetically more attractive.In section 6.4 the inuence of electric eld end energy cutoff
are taken into account in the considered selfconsistent equation for the reduced temperature;see
also (5.167).In section 6.5 is considered uctuational conductivity of na nowires which can be
investigated in nearest future when the technology for sample preparation becomes routine.
7.The development of technology give the possibility to make superconductor nanostructures for
which heating effect is not so important when huge current densities are applied.In chapter 7 effect
of strong electric eld are investigated using Boltzmann equation.The formula for the uctuation
current (7.15) is a correction to the result by Gor'kov obtained directly by the microscopic theory.
In section 7.3 are introduced appropriate dimensionless variables and whit their help in section 7.4
is considered uctuation conductivity for layered metal (7.29).There ar e analyzed the important
special cases for very strong electric elds (7.62),and the opposite ca se of weak electric elds
below the critical temperature (7.72).In section 7.8 is considered the possible generation of 3rd
harmonics created by nonlinear uctuation conductivity at constant temper ature.
8.The chapter 8 is devoted on one of the most important properties of the normal phase of supercon
ductors the linear temperature dependence of the resistivity above the cr itical temperature.In the
early stages of the development of hightemperature superconductivity this property was specied
as a proof of exotic and unconventional behavior and for inapplicability of standard theoretical
method for these materials.The model evaluation made in chapter 8 demonstrated that this lin
ear behavior can be explained as consequence of the strong anisotropy and indispensable in this
case thermodynamic uctuations of the electric eld perpendicular to the layer s.The electric eld
oscillations are connected with the uctuations of the electron density.Analog ous density uctua
tions of the air make the sky blue.Pictorially one can say that resistivity is linear because the sky
is blue;the electrons from perovskite planes are scattered by uctuations of their own density.In
the bulk metal such an effect is impossible because the electric eld is screen ed.
9.The dissertation is completely theoretical,but has completely experimentally oriented methodol
ogy;it is considered that physics is an experimental science.The purpose is to analyze experi
mental data and as criterion of reached understanding new experiments to be suggested for which
the result is qualitatively predicted.For all other chapters we have good agreement between the
theory and experiment and even some successful predictions.Not of such type is however the case
of chapter 9 when an extrapolation is made.The theory of uctuation condu ctivity predicts that
for superconductors supercooled under constant electric voltage the differential conductivity can
become negative and this effective negative friction can be used for creation of principally new
generators of electromagnetic waves in the terahertz range.Up to nowthe use of hightemperature
superconductors to ll the terahertz gap is only a theoretical idea.In this h ypothesis expressed
in 9th chapter the dissertator optimistically believes as after 8 sunny days he stops carrying his
umbrella.
GENERAL CONCLUSIONS FOR THE
HIGHTEMPERATURE SUPERCONDUCTORS
After 20 year comprehensive investigations it was reliably veried,that th ere are no other perovskites
with comparable critical temperature.No matter what chemical substitutions have been tried,other high
temperature perovskite was not synthesized;meanwhile,the discovered phonon superconductors have
reached critical temperature comparable to the early samples by Bednorz and Mueller.Another problem
gradually emerged:what is so unique about the cuprate CuO
2
plane in giving rise to superconductivity,
so that for 20 years nothing comparable came to existense?As a rule,all models for hightemperature
superconductivity use quite general properties and system of notions which can be realized in many
materials.
Within the framework of the developed in the dissertation theory exceptionality of CuO
2
plane nds
a natural explanation:the pairing is mediated by a 4s3d interaction localized in copper ions,but in order
to have maximal sd hybridization of the conduction band those orbitals have to be hybridized with the
oxygen 2p orbitals.In this sense the exceptionality of the CuO
2
plane is created by a triple coincidence:
if we look at the periodic table of the elements we will observe that for copper and oxygen those three
levels 3d,4s,2p are maximally close to each other.Even for the cuprates alone,the location of the
4s level clearly correlates with the critical temperature.This correlation can be easily calculated in the
suggested sd model,but remains unexplained in the framework of all other theoretical models.That
is why one of our early publications was entitled the 3dto4sby2p highway to superconductivity in
cuprates.
For two chemical elements three levels can be near,but outside this optimization all the other prop
erties of hightemperature superconductors can be understood within the framework of the standard
theoretical approaches.The whole analysis performed in the dissertation revealed that we are close to
a victory of the traditionalism,as happened half a century ago in quantum electrodynamics.Of course,
physics of superconductivity is a living science,and many of the considerations are not yet for a museum,
but we consider that most of the conclusions made here will not be qualitatively revised.
If one makes a Google search for`mystery of high temperature superconductivity',there will be
thousands of webpage hits.It was a real intellectual challenge for contemporary physics.What has been
achieved in the dissertation can be summarized as completing'the Bardeen pr ogram' the mystery is
revealed;we have the microscopic theory for the hightemperature superconductivity,we understand the
investigated properties and can predict technical applications.
xi
CHAPTER
1
TIGHTBINDING MODELLING OF THE
ELECTRONIC BAND STRUCTURE OF LAYERED
SUPERCONDUCTING PEROVSKITES
1.1 Introduction
After the discovery of the highT
c
superconductors the layered cuprates became one of the most studied
materials in the solid state physics.A vast range of compounds were synthesised and their properties
comprehensively investigated.The electron band structure is of particular importance for understanding
the nature of superconductivity in this type of perovskites [
1
].Along this line one can single out the
signicant success achieved in the attempts to reconcile the photoelectron sp ectroscopy data [
2
,
3
] and
the band structure calculations of the Fermi surface (FS) especially for compounds with simple structure
such as Nd
2−x
Ce
x
CuO
4−δ
[
4
,
5
].A qualitative understanding,at least for the selfconsistent electron
picture,has been achieved and for the most electron processes in the layered perovskites one can employ
adequate lattice models.
There is not much analysis of the electronic band structures of the highT
c
materials in the terms of
single analytical expressions available.This is something for which there is a clear need,in particular
to help in the construction of more realistic manybody Hamiltonians.The aim of this Chapter is to
analyse the common features in the electron band structure of the layered perovskites within the tight
binding (TB) method (for a nice review see references [
6
,
7
,
8
]).In the following we shall focus on the
metallic (being eventually superconducting) phase only,with the reservation that the antiferromagnetic
correlations,especially in the dielectric phase,could substantially change the electron dispersions.It
is shown that the linear combination of atomic orbitals (LCAO) approximation can be considered as
an adequate tool for analysing energy bands.Within the latter exact analytic results are obtained for
the constant energy contours (CEC).These expressions are used to t the FS of Nd
2−x
Ce
x
CuO
4−δ
[
4
],
Pb
0.42
Bi
1.73
Sr
1.94
Ca
1.3
Cu
1.92
O
8+x
[
9
,
10
,
11
,
12
,
13
],and Sr
2
RuO
4
[
14
] measured in angleresolved
photoemission/angleresolved ultraviolet spectroscopy (ARPES/ARUPS) experiments.
In particular,by applying the L¨owdin perturbative technique for the CuO
2
plane we give the LCAO
wave function of the states near the Fermi energy ǫ
F
.These states could be useful in constructing the
pairing theory for CuO
2
plane.For the layered cuprates we nd an alternative concerning the Fe rmi
level locationCu 3d
x
2
−y
2 vs.O2pσ character of the conduction band.It is shown that analysis of
1
2 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
extra spectroscopic data is needed in order for this dilemma to be resolved.As regards the RuO
2
plane,
the existence of three pockets of the FS unambiguously reveals the Ru4dε character of the conduction
bands [
15
,
16
].
To address the conduction bands in the layered perovskites we start from a common Hamiltonian
including the basis of valence states O2p and Ru4dε,or Cu3d
x
2
−y
2
,Cu4s respectively for cuprates.
Despite the equivalent crystal structure of Sr
2
RuO
4
[
17
] and La
2−x
Ba
x
CuO
4
[
18
,
19
],the states in
their conduction band(s) are,in some sense,complementary.In other words,for the CuO
2
plane the
conduction band is of σcharacter while for the RuO
2
plane the conduction bands are determined by π
valence bonds.This is due to the separation into σ and πpart of the Hamiltonian H = H
(σ)
+H
(π)
in
rst approximation.The latter two Hamiltonians are studied separately.
Accordingly,this Chapter is structured as follows.In Sec.
1.2
we consider the generic H
(4σ)
Hamil
tonian of the CuO
2
plane [
20
,
21
,
22
] and H
(π)
= H
(xy)
+H
(z)
is then studied in Sec.
1.3
.The results of
the comparison with the experimental data are summarised in Sec.
1.4
.Before embarking on a detailed
analysis,however,we give an account of some clarifying issues concerning the applicability of the TB
model and the band theory in general.
1.1.1 Appology to the band theory
It is wellknown that the electron band theory is a selfconsistent treatment of the electron motion in the
crystal lattice.Even the classical 3body problem demonstrates strongly correlated solutions,so it is a
priori unknown whether the selfconsistent approximation is applicable when describing the electronic
structure of every new crystal.However,the oneparticle band picture is an indispensable stage in the
complex study of materials.It is the analysis of experimental data using a conceptually clear band theory
that reveals nontrivial effects:how strong the strongly correlated electronic effects are,whether it is
possible to take into account the inuence of some interactioninduced orde r parameter back into the
electronic structure etc.Therefore the comparison of the experiment with the band calculations is not an
attempt,as sometimes thought,to hide the relevant issuesit is a tool to reveal inte resting and nontrivial
properties of the electronic structure.
Many electron band calculations have been performed for the layered perovskites and results were
compared to data due to ARPES experiments.The shape of the Fermi surface is probably the simplest
test to check whether we are on the right track or some conceptually newtheory should be used fromthe
very beginning.
The tightbinding interpolation of the electronic structure is often used for ttin g the experimental
data.This is because the accuracy of that approximation is often higher than the uncertainties in the
experiment.Moreover,the tightbinding method gives simple formulae which could be of use for experi
mentalists to see howfar they can get with such a simple minded approach.The tightbinding parameters,
however,have in a sense their own life independent of the ab initio calculations.These parameters can
be tted directly to the experiment even when,by some reasons,the electron band calculations could give
wrong predictions.In this sense the tightbinding parameters are the appropriate intermediary between
the theory and experiment.As for the theory,establishing of reliable oneparticle tightbinding parame
ters is the preliminary step in constructing more realistic manybody Hamiltonians.The role of the band
theory is,thus,quite ambivalent:on one hand,it is the nal language us ed in efforts towards under
standing a broad variety of phenomena;on the other hand,it is the starting point in developing realistic
interaction Hamiltonians for sophisticated phenomena such as magnetismand superconductivity.
The tightbinding method is the simplest one employed in the electron band calculations and it is
described in every textbook in solid state physics;the layered perovskites are now probably the best
investigated materials and the Fermi surface is a fundamental notion in the physics of metals.There is a
consensus that the superconductivity of layered perovskites is related to electron processes in the CuO
2
and RuO
2
planes of these materials.It is not,however,fair to criticise a given study,employing the
1.2.Layered cuprates 3
tightbinding method as an interpolation scheme to the rst principles calculations,for not thoroughly
discussing the manybody effects.The criticism should rather be readdressed to the ab initio band cal
culations.An interpolation scheme cannot contain more information than the underlying theory.It is not
erroneous if such a scheme works with an accuracy high enough to adequately describe both the theory
and experiment.
In view of the above,we nd it very strange that there are no simple interpo lation formulae for
the Fermi surfaces available in the literature and the experimental data are being published without an
attempt towards simple interpretation.One of the aims of the present Chapter is to help interpret the
experimental data by the tightbinding method as well as setting up notions in the analysis of the ab
initio calculations.
1.2 Layered cuprates
1.2.1 Model
The CuO
2
plane appears as a common structural detail for all layered cuprates.Therefore,in order to
retain the generality of the considerations,the electronic properties of the bare CuO
2
plane will be ad
dressed without taking into account structural details such as dimpling,orthorhombic distortion,double
planes,surrounding chains etc.For the square unit cell with lattice constant a
0
threeatomic basis is as
sumed {R
Cu
,R
O
a
,R
O
b
} = {0,(a
0
/2,0),(0,a
0
/2)}.The unit cell is indexed by vector n = (n
x
,n
y
),
where n
x
,n
y
= integer.Within such an idealized model the LCAO wave function spanned over the
Cu3d
x
2
−y
2i,Cu4si,O
a
2p
x
i,O
b
2p
y
i states reads
ψ
LCAO
(r) =
X
n
h
X
n
ψ
O
a
2p
x
(r −R
O
a
−a
0
n) +Y
n
ψ
O
b
2p
y
(r −R
O
b
−a
0
n) (1.1)
+S
n
ψ
Cu4s
(r −R
Cu
−a
0
n) +D
n
ψ
Cu3d
x
2
−y
2
(r −R
Cu
−a
0
n)
i
,
where Ψ
n
= (D
n
,S
n
,X
n
,Y
n
) is the tightbinding wave function in lattice representation.
The neglect of the differential overlap leads to an LCAO Hamiltonian of the CuO
2
plane
H =
X
n
n
D
†
n
[−t
pd
(−X
n
+X
x−1,y
+Y
n
−Y
x,y−1
) +ǫ
d
D
n
]
+S
†
n
[−t
sp
(−X
n
+X
x−1,y
−Y
n
+Y
x,y−1
) +ǫ
s
S
n
]
+X
†
n
[−t
pp
(Y
n
−Y
x+1,y
−Y
x,y−1
+Y
x+1,y−1
) (1.2)
−t
sp
(−S
n
+S
x+1,y
) −t
pd
(−D
n
+D
x+1,y
) +ǫ
p
X
n
]
+Y
†
n
[−t
pp
(X
n
−X
x−1,y
−X
x,y+1
+X
x−1,y+1
)
−t
sp
(−S
n
+S
x,y+1
) −t
pd
(D
n
+D
x,y+1
) +ǫ
p
Y
n
]
o
,
where the components of Ψ
n
should be considered as being Fermi operators.The notations ǫ
d
,ǫ
s
,and ǫ
p
stand respectively for the Cu3d
x
2
−y
2,Cu4s and O2pσ singlesite energies.The direct O
a
2p
x
→O
b
2p
y
exchange is denoted by t
pp
and similarly t
sp
and t
pd
denote the Cu4s → O2p and O2p → Cu3d
x
2
−y
2
hoppings respectively.The sign rules for the hopping amplitudes are sketched in Figure
1.1
the bonding
orbitals enter the Hamiltonian with a negative sign.The latter follows directly from the surface integral
approximation for the transfer amplitudes,given in Appendix
A
.
For the Bloch states diagonalizing the Hamiltonian (
1.2
)
Ψ
n
≡
D
n
S
n
X
n
Y
n
=
1
√
N
X
p
D
p
S
p
e
iϕ
a
X
p
e
iϕ
b
Y
p
e
ipn
,(1.3)
4 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
+
+


+
+


+
+


+
O2p
x
+

O2p
y
++


Cu3d
x y
22
Cu4s
+
+

+

+

+

+

+

+

+

(x,y)
(x+1,y)
(x1,y)
(x1,y+1)
(x,y+1)
(x,y1) (x+1,y1)
t
pp
t
pp
t
sp
t
sp
t
pd
t
pd
Figure 1.1:Schematic of a CuO
2
plane (only orbitals relevant to the discussion are depicted).The solid
square represents the unit cell with respect to which the positions of the other cells are determined.The
indices of the wave function amplitudes involved in the LCAO Hamiltonian (
1.2
) are given in brackets.
The rules for determining the signs of hopping integrals t
pd
,t
sp
,and t
pp
are shown as well.
where N is the number of the unit cells,we use the same phases as in references [
20
,
21
,
22
]:ϕ
a
=
1
2
(p
x
− π),ϕ
b
=
1
2
(p
y
− π).This equation describes the Fourier transformation between the coor
dinate representation Ψ
n
= (D
n
,S
n
,X
n
,Y
n
),with n being the cell index,and the momentum rep
resentation ψ
p
= (D
p
,S
p
,X
p
,Y
p
) of the TB wave function (when used as an index,the electron
quasimomentum vector is denoted by p).Hence,the Schr¨odinger equation i~d
t
ˆ
ψ
p,α
= [
ˆ
ψ
p,α
,
ˆ
H] for
ψ
p,α
(t) = e
−iǫt/
ψ
p,α
,with α being the spin index (↑,↓) (suppressed hereafter),takes the form
H
(4σ)
p
−ǫ11
ψ
p
=
−ε
d
0 t
pd
s
X
−t
pd
s
Y
0 −ε
s
t
sp
s
X
t
sp
s
Y
t
pd
s
X
t
sp
s
X
−ε
p
−t
pp
s
X
s
Y
−t
pd
s
Y
t
sp
s
Y
−t
pp
s
X
s
Y
−ε
p
D
p
S
p
X
p
Y
p
= 0,(1.4)
where
ε
d
= ǫ −ǫ
d
,ε
s
= ǫ −ǫ
s
,ε
p
= ǫ −ǫ
p
,
and
s
X
= 2 sin(
1
2
p
x
),s
Y
= 2 sin(
1
2
p
y
),x = sin
2
(
1
2
p
x
),y = sin
2
(
1
2
p
y
)
0 p
x
,p
y
2π.
This 4σband Hamiltonian is generic for the layered cuprates,cf.Ref.[
22
].We have also included the
direct oxygenoxygen exchange t
pp
dominated by the σ amplitude.The secular equation
det
H
(4σ)
p
−ǫ11
= Axy +B(x +y) +C = 0 (1.5)
gives the spectrumand the canonical formof the CEC with energydependent coefcients
A(ǫ) = 16(4t
2
pd
t
2
sp
+2t
2
sp
t
pp
ε
d
−2t
2
pd
t
pp
ε
s
−t
2
pp
ε
d
ε
s
)
B(ǫ) = −4ε
p
(t
2
sp
ε
d
+t
2
pd
ε
s
) (1.6)
C(ǫ) = ε
d
ε
s
ε
2
p
.
1.2.Layered cuprates 5
,Z
,Z
Z,
Z,
X
C
D
Figure 1.2:LDA Fermi contour of Nd
2−x
Ce
x
CuO
4−δ
(dotted line) calculated by Yu and Freeman [
5
]
(with the kind permission of the authors),and the LCAOt (solid line) accord ing to Eq.(
1.5
).The tting
procedure uses C and D as reference points.
Hence,the explicit CEC equation reads as
p
y
= ±2 arcsin
√
y,if 0 ≤ y = −
Bx +C
Ax +B
1.(1.7)
This equation reproduces the rounded squareshaped FS,centered at the (π,π) point,inherent for all
layered cuprates.The best t is achieved when A,B and C are considered as tting parameters.Thus,for
a CEC passing through the D = (p
d
,p
d
) and C = (p
c
,π) reference points,as indicated in Figure
1.2
,the
tting coefcients (distinguished by the subscript f) in the canonical equation A
f
xy+B
f
(x+y)+C
f
= 0
have the form
A
f
= 2x
d
−x
c
−1,x
d
= sin
2
(p
d
/2)
B
f
= x
c
−x
2
d
,x
c
= sin
2
(p
c
/2)
C
f
= x
2
d
(x
c
+1) −2x
c
x
d
,
(1.8)
and the resulting LCAOFermi contour is quite compatible with the LDAcalculations for Nd
2−x
Ce
x
CuO
4−δ
[
23
,
5
].Due to the simple shape of the FS the curves just coincide.We note also that the canonical equa
tion (
1.5
) would formally correspond to 1band TB Hamiltonian of a 2D square lattice of the form
ǫ(p) = −2t(cos p
x
+cos p
y
) +4t
′
cos p
x
cos p
y
,
with strong energy dependence of the hopping parameters,where t
′
is the antibonding hopping between
the sites along the diagonal,cf.references [
24
,
25
,
26
].
1.2.2 Effective Hamiltonians
Studies of the electronic structure of the layered cuprates have unambiguously proved the existence of
a large hole pocketa rounded square centred at the (π,π) point.This observation is indicative for a
Fermi level located in a single band of dominant Cu3d
x
2
−y
2 character.To address this band and the
6 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
related wave functions it is therefore convenient an effective CuHamiltonian to be derived by L¨owdin
downfolding of the oxygen orbitals.This is equivalent to expressing the oxygen amplitudes from the
third and fourth rows of Eq.(
1.4
)
X =
1
η
p
h
t
pd
s
X
1 +
t
pp
ε
p
s
2
Y
D+t
sp
s
X
1 −
t
pp
ε
p
s
2
Y
S
i
Y =
1
η
p
h
−t
pd
s
Y
1 +
t
pp
ε
p
s
2
X
D+t
sp
s
Y
1 −
t
pp
ε
p
s
2
X
S
i
,
(1.9)
where η
p
= ε
p
−
t
2
pp
ε
p
s
2
X
s
2
Y
,and substituting back into the rst and the second rows of the same equation.
Such a downfolding procedure results in the following energydependent copper Hamiltonian
H
Cu
(ǫ) =
ǫ
d
+
(2t
pd
)
2
η
p
x +y +
8t
pp
ε
p
xy
(2t
pd
)(2t
sp
)
η
p
(x −y)
(2t
pd
)(2t
sp
)
η
p
(x −y) ǫ
s
+
(2t
pd
)
2
η
p
x +y −
8t
pp
ε
p
xy
,(1.10)
which enters the effective Schr¨odinger equation H
Cu
D
S
= ǫ
D
S
.Thus,from Eq.(
1.9
) and Eq.(
1.10
)
one can easily obtain an approximate expression for the eigenvector corresponding to a dominant Cu3d
x
2
−y
2
character.Taking D ≈ 1,in the lowest order with respect to the hopping amplitudes t
ll
′ one has
Cu3d
x
2
−y
2i =
D
S
X
Y
≈
1
(t
sp
t
pd
/ε
s
ε
p
)(s
2
X
−s
2
Y
)
(t
pd
/η
p
)s
X
−(t
pd
/η
p
)s
Y
,(1.11)
i.e.,X
2
+Y 
2
+S
2
≪D
2
≈ 1.We note that within this Cu scenario the Fermi level location and the
CEC shape are not sensitive to the t
pp
parameter.Therefore one can neglect the oxygenoxygen hopping
as was done,for example,by Andersen et al.[
20
,
21
,
22
] (the importance of the t
pp
parameter has been
considered by Markiewicz [
27
]) and the band structure of the Hamiltonian (
1.10
) for the same set of
energy parameters as used in Ref.[
22
] is shown in Figure
1.3
(a).In this case the FS can be tted by its
diagonal alone,i.e.using only D as a reference point.Hence an equation for the Fermi energy follows,
A(ǫ
F
)x
2
d
+2B(ǫ
F
)x
d
+C(ǫ
F
) = 0,which yields ǫ
F
= 2.5 eV.As seen in Figure
1.3
(b),the deviation
from the twoparametr t,discussed in Sec.
1.2.1
is almost vanishing thus justifying the neglect of t
pp
and using oneparametr t.
However,despite the excellent agreement between the LDA calculations,the LCAO t and the
ARPES data regarding the FS shape,the theoretically calculated conduction band width w
c
in the lay
ered cuprates is overestimated by a factor of 2 or even 3 [
4
].Such a discrepancy may well point to some
alternative interpretations of the available experimental data.In the following section we shall consider
the possibility for a Fermi level lying in an oxygen band.
1.2.2.1 Oxygen scenario:the AbrikosovFalkovsky model
Various hints currently exist in favor of O2p character of the states near the Fermi level [
28
,
29
,
30
,
31
,
32
,
33
].We consider that these arguments cannot be a priori ignored.This is best see if,following Abrikosov
and Falkovsky [
34
],the experimental data are interpreted within an alternative oxygen scenario.
Accordingly,the oxygen 2p level is assumed to lie above the Cu3d
x
2
−y
2 level,and the Fermi level
to fall into the upper oxygen band,ǫ
d
< ǫ
p
< ǫ
F
< ǫ
s
.The Cu3d
x
2
−y
2 band is completely lled in the
metallic phase and the holes are found to be in the approximately halflled O 2pσ bands.To inspect
such a possibility in detail we use again the L¨owdin downfolding procedure now applied to Cu orbitals.
1.2.Layered cuprates 7
X
M
(a)
5
2.5
0
2.5
5
7.5
10
Energy(eV)
X
(b)
Y
D
C
Figure 1.3:(a) Electron band structure of the generic for the CuO
2
plane 4σband Hamiltonian using
the parameters from Ref.[
22
] and the Fermi level ǫ
F
= 2.5 eV tted from the LDA calculation by Yu
and Freeman [
5
];(b) The LCAOFermi contour (solid line) tted to the LDAFermi surface (d ashed line)
for Nd
2−x
Ce
x
CuO
4−δ
[
5
] using only D as a reference point.The deviation of the t at the C point is
negligible.
Fromthe rst and second rows of Eq.(
1.4
) we express the copper amplitudes
D =
t
pd
ε
d
(s
X
X −s
Y
Y )
S =
t
sp
ε
s
(s
X
X +s
Y
Y )
(1.12)
and substitute them in the third and the fourth rows.This leads to an effective oxygen Hamiltonian of
the form
H
O
(ǫ) = B
s
X
s
X
s
X
s
Y
s
Y
s
X
s
Y
s
Y
−t
eﬀ
0 s
X
s
Y
s
Y
s
X
0
(1.13)
with spectrum
ǫ(p) = 2B(ǫ)(x +y)
"
−1 ±
s
1 +(2τ +τ
2
)
4xy
(x +y)
2
#
,(1.14)
where
B(ǫ) = −
t
2
pd
ε
d
+
t
2
sp
(−ε
s
)
,t
eﬀ
(ǫ) = t
pp
+2
t
2
pd
ε
d
,τ(ǫ) = t
eﬀ
/B (1.15)
−ε
s
,ε
d
> 0 (1.16)
and the conduction band dispersion rate ǫ
c
(p) corresponds to the + sign for τ < 1.It should be
noted that Eq.(
1.14
) is an exact result within the adopted 4σband model.As a consequence,it is
easily realised that along the (0,0)(π,0) direction the conduction band is dispersionless,ǫ
c
(p
x
,0) = 0.
This corresponds to the extended Van Hove singularity observed in the ARPES experiment [
35
] and we
consider it being a hint in favour of the oxygen scenario (the copper model would give instead the usual
Van Hove scenario).
Depending on the τ value two different limit cases occur.For τ ≪ 1 one gets a simple Pad´e
approximant
ǫ
c
(p) = 4t
eﬀ
(ǫ
c
)
2xy
x +y
(1.17)
8 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
0
0
2
2
4
4
6
6
8
8
10
10
12
12
14
14
16
16
18
18
20
20
22
22
x = 0.15
ARPES
Crossing
M
Y
Y
X
X
(a) (b)
Y
X
M
Figure 1.4:(a) Energy dispersion of the nonbonding oxygen band ǫ
c
(p),equation Eq.(
1.17
).A few
cuts through the energy surface,i.e.,CEC,are presented together with the dispersion along the high
symmetry lines in the Brilloiun zone;(b) The Fermi surface of Nd
2−x
Ce
x
CuO
4−δ
(solid line) determined
by equation Eq.(
1.17
) for x = 0.15 (shaded slice in panel (a) and compared with experimental data
(points with error bars) for the same value of x after King et al.[
4
].θ and ϕ denote the polar and
azimuthal emission angles,respectively,measured in degrees.The empty dashed circles show kspace
locations where ARPES experiments have been performed (cf.Fig.2 in Ref.[
4
]) and their diameter
corresponds to 2
◦
experimental resolution.
and eigenvector of H
(4σ)
ci =
D
S
X
Y
≈
1
q
s
2
X
+s
2
Y
2t
pd
ε
d
s
X
s
Y
0
−s
Y
s
X
,(1.18)
normalized according to the inequality D
2
+ S
2
≪ X
2
+ Y 
2
≈ 1.This limit case acceptably
describes the experimental ARPES data,e.g.,for Nd
2−x
Ce
x
CuO
4−δ
material with single CuO
2
planes
and no other complicating structural details.Schematic representation of the energy surface dened
by Eq.(
1.17
) is shown in Figure
1.4
(a).In Figure
1.4
(b) we have presented a comparison between the
ARPES data from Ref.[
4
] and the Fermi contour calculated according to Eq.(
1.17
) for x = 0.15.Note
that no tting parameters are used and this contour should be referred to as an ab initio calculation of the
FS.
The opposite limit case t
eﬀ
≫ B,i.e.τ ≫ 1,has been analysed in detail by Abrikosov and
Falkovsky [
34
].The conduction band dispersion rate ǫ
c
and the corresponding eigenvector of the Hamil
tonian H
O
(
1.13
) now take the form
ǫ
c
(p) = 4t
eﬀ
(ǫ
c
)
√
xy (1.19)
ci ≈
1
√
2
(t
pd
/ε
d
)(s
X
+s
Y
)
(t
sp
/ε
s
)(s
X
−s
Y
)
1
−1
,(1.20)
provided that D
2
+S
2
≪X
2
+Y 
2
≈ 1.In other words,the last approximation,τ ≫1,corresponds
to a pure oxygen model where only hoppings between oxygen ions are taken into account.Clearly,this
model is the complementary to the copper scenario and is based on an effect completely neglected in
1.2.Layered cuprates 9
(a)
X
(b)
Z
Y
Figure 1.5:(a) ARUPS Fermi surface of Pb
0.42
Bi
1.73
Sr
1.94
Ca
1.3
Cu
1.92
O
8+x
by Schwaller et al.[
13
];
(b) LCAO t to (a) according to the AbrikosovFalkovsky model [
34
] using the D reference point with
p
d
= 0.171 ×2π.
its copper counterpart,where t
pp
≡ 0.This limit case of the oxygen scenario suitably describes the
ARUPS experimental data for Pb
0.42
Bi
1.73
Sr
1.94
Ca
1.3
Cu
1.92
O
8+x
[
9
,
10
,
11
,
12
,
13
].The FS of the latter
is tted by its diagonal (the D point) according to the AbrikosovFalkovsky relation (
1.19
) and the result
is shown in Figure
1.4
.
There exist a tremendous number of ARPES/ARUPS data for layered cuprates which makes the
reviewing of all those spectra impossible.To illustrate our TB model we have chosen data for the Pb
substitution for Bi in Bi
2
Sr
2
CaCu
2
O
8
,see Figure
1.5
.In this case the CuO
2
planes are quite at and
the ARPES data are not distorted by structural details.When present,distortions were misinterpreted as
a manifestation of strong antiferromagnetic correlations.We believe,however,that the experiment by
Schwaller et al.[
13
] reveals the main feature of the CuO
2
plane band structurethe large hole pocket
found to be in agreement with the oneparticle band calculations.Strong support to this viewcomes from
a recent paper by Campuzano et al.[
36
] where the ARPES Fermi surface of pure Bi
2
Sr
2
CaCu
2
O
8+δ
has
been mapped [cf.the inset of Fig.1 (a) therein].This experimental ndin g is in excellent agreement with
our tightbinding t to the Fermi surface of Pb
0.42
Bi
1.73
Sr
1.94
Ca
1.3
Cu
1.92
O
8+x
,studied by P.Schwaller
and coworkers in Ref.[
13
],Figure
1.5
.The remarkable coincidence of the Fermi surfaces of these two
compounds is a nice conrmation that Pb substitution for Bi is irrelevant for th e band structure of the
CuO
2
plane and the Fermi surface of the latter is therefore revealed to be a common feature.
Besides the good agreement between the theory and the experiment,regarding the FS shape,we
should also point out the compatibility between the calculated and the experimental conduction band
width.Indeed,within the AbrikosovFalkovsky model [
34
],according to Eq.(
1.19
),one gets for the
conduction bandwidth 0 ε
c
(p) w
c
≈ 4t
pp
,which coincides with the value obtained fromEq.(
1.17
)
provided that t
2
pd
≪ t
pp
(ǫ
F
− ǫ
d
).The ab initio calculation of t
pp
as a surface integral (see Ap
pendix
A
),making use of atomic wave functions standard for the quantum mechanical calculations,
gives t
pp
≈ 200350 meV in different estimations.This range is in acceptable agreement with the ex
perimental w
c
≃ 1 eV [
4
];within the LCAO model an exact analytic result for w
c
can be obtained from
the equation w
c
= 4t
pp
+8t
2
pd
/(w
c
−ǫ
d
).
We note also that the TB analysis allows the bands to be unambiguously classie d with respect to
the atomic levels fromwhich they arise.Within such terms,for the oxygen scenario one can describe the
metal→insulator transition as being the charge transfer Cu
1+
O
1
1
2
−
2
→ Cu
2+
O
2−
2
.The possibility for
monovalent copper Cu
1+
in the superconducting state is discussed,for example,by Romberg et al.[
37
,
38
].
10 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
1.3 Conduction bands of the RuO
2
plane
Sr
2
RuO
4
is the rst coperfree perovskite superconductor isostructural to the highT
c
cuprates [
17
].The
layered ruthenates,just like the layered cuprates,are strongly anisotropic and in a rst approximation the
nature of the conduction band(s) can be understood by analysing the bare RuO
2
plane.One should repeat
the same steps as in the previous section but now having Ru instead of Cu and the Fermi level located in
the metallic bands of Ru4dπ character.To be specic,the conduction bands arise formthe hybridis ation
between the Ru4d
xy
,Ru4d
yz
,Ru4d
zx
and O
a
2p
y
,O
b
2p
x
,O
a,b
2p
z
πorbitals.The LCAOwave function
spanned over the four perpendicular to the RuO
2
plane orbitals reads
Ψ
(z)
LCAO
(r) =
1
√
N
X
p
X
n
h
D
zx,n
ψ
Ru4d
zx
(r −a
0
n) +D
zy,n
ψ
Ru4d
zy
(r −a
0
n) (1.21)
+ e
iϕ
a
Z
a,n
ψ
O
a
2p
z
(r −R
O
a
−a
0
n) +e
iϕ
b
Z
b,n
ψ
O
b
2p
z
(r −R
O
b
−a
0
n)
i
e
ipn
Hence,the πanalog of Eq.(
1.4
) takes the form
H
(z)
p
−ǫ11
ψ
(z)
p
=
−ε
zx
0 t
z,zx
s
X
0
0 −ε
zy
0 t
z,zy
s
Y
t
z,zx
s
X
0 −ε
za
−t
zz
c
X
c
Y
0 t
z,zy
s
Y
−t
zz
c
X
c
Y
−ε
zb
D
zx
D
zy
Z
a
Z
b
= 0,(1.22)
where
ε
zx
= ǫ −ǫ
zx
,ε
za
= ǫ −ǫ
za
,c
X
= 2 cos(p
x
/2),
ε
zy
= ǫ −ǫ
zy
,ε
zb
= ǫ −ǫ
zb
,c
Y
= 2 cos(p
y
/2),
(1.23)
and ǫ
zx
,ǫ
zy
,ǫ
za
,and ǫ
zb
are the single site energies respectively for Ru4d
zx
,Ru4d
zy
and O
a
2p
z
,O
b
2p
z
orbitals.t
zz
stands for the hopping between the latter two orbitals and,if a negligible orthorhom
bic distortion is assumed,the metaloxygen πhopping parameters are equal,t
z,zy
= t
z,zx
and also
ǫ
z
= ǫ
za
= ǫ
zb
.The phase factors e
iϕ
a,b
in Eq.(
1.21
) are chosen in compliance with Ref.[
22
],see
equation Eq.(
1.3
).
Identically,writing the LCAO wave function spanned over the three inplane πorbitals Ru4d
xy
,
O
a
2p
y
,and O
b
2p
x
in the way in which Eq.(
1.21
) is designed one has for the inplane Schr ¨odinger
equation
H
(xy)
p
−ǫ11
ψ
(xy)
p
=
−ε
xy
t
pdπ
s
X
t
pdπ
s
Y
t
pdπ
s
X
−ε
ya
t
′
pp
s
X
s
Y
t
pdπ
s
Y
t
′
pp
s
X
s
Y
−ε
xb
D
xy
Y
a
X
b
= 0,(1.24)
where t
pdπ
denotes the hopping Ru4d
xy
→O
a,b
2pπ and t
′
pp
,respectively,O
a
2p
y
→O
b
2p
x
.The deni
tions for the other energy parameters are in analogy to Eq.(
1.23
) (for negligible orthorhombic distortion
ǫ
ya
= ǫ
xb
6= ǫ
z
).Thus,the πHamiltonian of the RuO
2
plane takes the form
H
(π)
=
X
p,α=↑,↓
ψ
(z)†
p,α
H
(z)
p
ψ
(z)
p,α
+ψ
(xy)†
p,α
H
(xy)
p
ψ
(xy)
p,α
.(1.25)
In an earlier paper [
39
,
40
] we have derived the corresponding secular equations and now we shall only
provide the nal expressions in terms of the notations used here
det(H
(z,xy)
p
−ǫ11) = A
(z,xy)
xy +B
(z,xy)
(x +y) +C
(z,xy)
= 0,
A
(z)
= 16(t
4
z,zx
−t
2
zz
ε
2
zx
) A
(xy)
= 32t
′
pp
t
2
pdπ
−16ε
xy
t
′2
pp
B
(z)
= −16t
2
zz
ε
2
zx
−4t
2
z,zx
ε
zx
ε
z
,B
(xy)
= −t
2
pdπ
ε
ya
C
(z)
= ε
2
zx
(ε
2
z
−16t
2
zz
) C
(xy)
= ε
xy
ε
2
ya
.(1.26)
1.4.Discussion 11
Z X
(a)
5
4
3
2
1
0
Energy(eV)
Z
(b)
X
Figure 1.6:(a) LCAO band structure of Sr
2
RuO
4
according to Eq.(
1.25
).The Fermi level (dashed line)
crosses the three Ru4dε bands of the RuO
2
plane;(b) LCAO t (solid lines) to the ARPES data (circles)
by Lu et al.[
14
],cf.also Ref.[
39
,
40
].
The three sheets of the Fermi surface in Sr
2
RuO
4
tted to the ARPES data given by Lu et al.[
14
] are
shown in Figure
1.6
(b).To determine the Hamiltonian parameters we have made use of the dispersion
rate values at the highsymmetry points of the Brillouin zone.To the best of our knowledge,the TB
analysis of the Sr
2
RuO
4
band structure was rst performed in Ref.[
39
,
40
] (subsequently,the latter
results were reproduced in Ref.[
41
] without referring to Ref.[
39
,
40
]).The RuO
2
plane band structure
resulting fromthe set of parameters
t
zz
= t
′
pp
= 0.3 eV,ε
z
= −2.3 eV,ε
xy
= −1.62 eV,
t
pdπ
= t
z,zx
= 1 eV,ε
zx
= −1.3 eV,ε
ya,xb
= −2.62 eV.
(1.27)
is shown in Figure
1.6
(a).This t is subjected to the requirement of providing as good as possib le a de
scription of the narrowenergy interval around ǫ
F
whereas the lled bands far belowthe Fermi level match
only qualitatively to the LDA calculations by Oguchi [
15
] and Singh [
16
].In addition we note that the
de Haasvan Alphen (dHvA) measurements [
42
] of the Sr
2
RuO
4
FS differ fromthe ARPES results [
14
].
Thus,tting the dHvA data by using modied TB parameters is a natural rene ment of the proposed
model.We note that the diamondshaped hole pocket,centred at the X point [see Figure
1.6
(b)],is very
sensitive to the`game of parameters'.For that band the van Hove energy is fairly close to the Fermi
energy.As a result,a minor change in the parameters could drive a van Hove transition transforming
this hole pocket to an electron one,centred at the Γ point.Indeed,such a band conguration has been
recently observed also in the ARPES revision of the Sr
2
RuO
4
Fermi surface [
43
].This can be easily
traced already from the energy surfaces ǫ(p) calculated earlier in Ref.[
39
,
40
].The comparison of the
ARPES data with TB energy surfaces could be a subject of a separate study.
1.4 Discussion
The LCAOanalysis of the layered perovskites band structure,performed in the preceding sections,mani
fests a good compatibility with the experimental data and the band calculations as well.Due to the strong
anisotropy of these materials,their FS within a reasonable approximation is determined by the properties
of the bare CuO
2
or RuO
2
planes.
12 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
Despite these planes having identical crystal structure,their electronic structures are quite differ
ent.While for the RuO
2
plane the Fermi level crosses metallic πbands,the conduction band of the
CuO
2
plane is described by a σHamiltonian (
1.4
).The latter gives for the CuO
2
plane a large hole
pocket centered at the (π,π) point.Its shape,if no additional sheets exist,is well described by the
exact analytic results within the LCAO model,Eq.(
1.5
),as found for Nd
2−x
Ce
x
CuO
4−δ
[
5
,
4
] and
Pb
0.42
Bi
1.73
Sr
1.94
Ca
1.3
Cu
1.92
O
8+x
[
9
,
10
,
11
,
12
,
13
].For a number of other cuprates,YBa
2
Cu
3
O
7−δ
[
44
],
YBa
2
Cu
4
O
8
[
35
],Bi
2
Sr
2
CaCu
2
O
8
[
45
,
46
,
47
],Bi
2
Sr
2
CuO
6
[
48
],the innitelayered superconductor
Sr
1−x
Ca
x
CuO
2
[
49
],HgBa
2
Ca
2
Cu
3
O
8+δ
[
50
],HgBa
2
CuO
4+δ
[
51
],HgBa
2
Ca
n−1
Cu
n
O
2n+2+δ
[
52
],
Tl
2
Ba
2
Ca
n−1
Cu
n
O
4+2n
,[
1
],Sr
2
CuO
2
F
2
,Sr
2
CuO
2
Cl
2
,Ca
2
CuO
2
Cl
2
[
53
],this large hole pocket is eas
ily identied.For all of the above compounds,however,its shape is usually deformed due to appearance
of additional sheets of the Fermi surface originating fromaccessoires of the crystal structure.
As the most important implication for the CuO
2
plane we should point out the intrinsic alternative
about the Fermi level location (see Sec.
1.2
).It is commonly believed that the states at the FS are of
dominant Cu3d
x
2
−y
2 character (see e.g.Ref.[
22
]).Nevertheless,the spectroscopic data for the FS can
be equally well interpreted within the oxygen scenario,according to which the FS states are of dominant
O2pσ character.A number of indications exist in favour of the oxygen model and the importance of the
t
pp
hopping amplitude [
27
,
28
,
29
,
30
]:
(i) O1s → O2p transitions observed in EELS experiments for the metallic phase of the layered
cuprates,which reveal an unlled O 2p atomic shell;
(ii) the oxygen scenario reproduces in a natural way the extended Van Hove singularity observed in
the ARPES experiments while the Cu scenario fails to describe it;
(iii) the metalinsulator transition can be easily described;
(iv) the width of the conduction band is directly related to the atomic wave functions.
Some authors even wager that the oxygen model will win [
33
].If the oxygen scenario is corroborated,
due to the cancellation of the largest amplitude t
sp
the small hoppings t
pd
and t
pp
should be properly
evaluated eventually as surface integrals (see Appendix
A
) and some band calculations may well need a
revision.It would be quite valuable if a mufntin calculation of H
+
2
ion was performed and compared
with the exact results when the hopping integral is comparatively small,of the order of the one that ts the
ARPES data t
pp
∼ 200 meV.We also note that even the copper model gives an estimation for t
pp
closer to
the experiment than the LDAcalculations.The smallness of t
pp
within the oxygen scenario,on the other
hand,is guaranteed by the nonbonding character of the conduction band.This scenario,therefore,can
easily display heavy fermion behaviour,i.e.,effective mass m
eﬀ
t
pp
→0
−→ huge,and density of states (DOS)
∝ m
eﬀ
∝ 1/t
pp
(we note that no realistic band calculations for heavy fermion systems can be performed
without employing the asymptotic methods fromthe atomic physics).It is also instructive to compare the
TB analyses of heavy fermion systems and layered cuprates.The alternative for the Fermi level location
(metallic vs.oxygen band) exist for the cubic bismuthates as well [
54
,
55
].When the Fermi level falls
into heavy fermion oxygen bands,one of the isoenergy surfaces is a rounded cube [
54
].Indeed,such a
isoenergy surface has been recently conrmed by the LMTO method applie d to Ba
0.6
K
0.4
BiO
3
[
56
].
Due to the equally good t of the results for the FS of the layered cuprates w ithin the two models
we can infer that at present any nal judgement about this alternative w ould be premature.Thus far we
consider that the oxygen model should be taken into account in the interpretation of the experimental
data.Moreover,the angular dependence of the superconducting order parameter Δ(p) ∝ cos(p
x
) −
cos(p
y
) is readily derived within the standard BCS treatment of the oxygenoxygen superexchange [
57
,
58
,
59
].Analysis of some extra spectroscopic data by means of different models would nally solve this
dilemma.This cannot be done within the framework of the TB method.A coherent picture requires a
thorough study,where the TB model is just an useful tool to test the properties of a given solution.
1.5.Determination of density of states of thin highT
c
lms by FET type microstructures 13
Up to now,the applicability of the LCAO approximation to the electron structure of the layered
cuprates can be considered as being proved.The basis function of the LCAO Hamiltonian can be in
cluded in a realistic oneelectron part of the lattice Hamiltonians for the layered perovskites.This is
an indispensable step preceding the inclusion of the electronelectron superexchange,electronphonon
interaction or any other kind of interaction between conducting electrons.
1.5 Determination of density of states of thin highT
c
lms by FET type
microstructures
The importance of density of states (DOS) for the physics of highT
c
cuprates was discussed in many
papers [
60
,
61
,
62
,
63
,
64
,
65
,
27
,
66
,
67
,
68
].The purpose of this section is to suggest a simple electronic
method for determination of DOS.The proposed experiment requires the preparation of eldeffect tran
sistor (FET) type microstructure and require standard electronic measurements.The FET controls the
current between two points but does so differently than the bipolar transistor.The FET relies on an elec
tric eld to control the shape and hence the conductivity of a channel in a semiconductor material.The
shape of the conducting channel in a FET is altered when a potential difference is applied to the gate
terminal (potential relative to either source or drain).It causes the electrons owto change it's width and
thus controls the voltage between the source and the drain.If the negative voltage applied to the gate is
high enough,it can remove all the electrons fromthe gate and thus close the conductive channel in which
the electrons ow.Thus the FET is blocked.
The system,considered in this section is in hydrodynamic regime,which means low frequency
regime where the temperature of the superconducting lm adiabatically follows the dissipated Ohmic
power.All working frequencies of the lockins say up to 100 kHz are actually low enough.The investi
gations of superconducting bolometers show that only in MHz range it is necessary to take into account
the heat capacity of the superconducting lm.As an example there is a publica tion,corresponding to this
topic [
69
] as well as the references therein.In this work we propose an experiment with a FET,for which
we need to measure the second harmonic of the sourcegate voltage and the third harmonic of the source
drain voltage.Other higher harmonics will be present in the measurements (e.g.from the leads),but in
principle they can be also used for determination of the density of states.An analogous experimental
research has been already performed for investigation of thermal interface resistance [
70
].The suggested
experiment can be done using practically the same experimental setup,only the gate electrodes should
be added to the protected by insulator layer superconducting lms.
1.5.1 Determination of logarithmic derivative of density of states by electronic
measurements
Here we suggest a simple electronic experiment,determining the logarithmic derivative of the density of
states by electronic measurements using a thin lmof the material Tl:2201.The thic kness of the samples
should be typical for the investigation of highT
c
lms,say 50200 nm.Such lms demonstrate already
the properties of the bulk phase.The numerical value of this parameter
ν
′
(E
F
) =
dν(ǫ)
dǫ
,(1.28)
will ensure the absolute determination of hopping integrals.
We propose a eld effect transistor (FET) from Tl:2201 Fig.
1.7
to be investigated electronically
with lockin at second and third harmonics.Imagine a strip of Tl:2201 and between the ends of the strip,
between the source (S) and the drain (D) is applied an AC current
I
SD
(t) = I
0
cos(ωt).(1.29)
14 Tightbinding modelling of the electronic band structure of layered superconducting perovskites
S D
U
(3f)
SD
V
insulator
V
Gate
thinfilm
U
(2f)
SG
I0 tcos( )ω
Figure 1.7:A eld effect transistor (FET) is schematically illustrated.The cu rrent I(t),applied between
the source (S) and the drain (D) has frequency ω.Running through the transistor the electrons create
voltage U
SG
with double frequency 2ω between the source (S) and the gate (G).The sourcedrain voltage
U
SD
is measured on the triple frequency 3ω.
For low enough frequencies the ohmic power P increases the temperature of the lm T above the
ambient temperature T
0
P = RI
2
SD
= α(T −T
0
),(1.30)
where the constant α determines the boundary thermoresistance between the Tl:2201 lm and the s ub
strate,and R(T) is the temperature dependent sourcedrain (SD) resistance.We suppose that for thin
lm the temperature is almost homogeneous across the thickness of the lm.In s uch a way we obtain
for the temperature oscillations
(T −T
0
) =
RI
2
SD
α
=
RI
2
0
α
cos
2
(ωt).(1.31)
As the resistance is weakly temperature dependent
R(T) = R
0
+(T −T
0
)R
0
′
,R
0
′
(T
0
) =
dR(T)
dT
T
0
.(1.32)
A substitution here of the temperature oscillations from Eq.(
1.31
) gives a small time variations of the
resistance
R(t) = R
0
1 +
R
′
0
α
I
2
0
cos
2
(ωt)
.(1.33)
Now we can calculate the sourcedrain voltage as
U
SD
(t) = R(t)I
SD
(t).(1.34)
Substituting here the SD current fromEq.(
1.29
) and the SD resistance fromEq.(
1.33
) gives for the SD
voltage
U
SD
(t) = U
(1f)
SD
cos(ωt) +U
(3f)
SD
cos(3ωt).(1.35)
The coefcient in front of the rst harmonic U
(1f)
SD
≈ R
0
I
0
is determined by the SDresistance R
0
at low
currents I
0
,while for the third harmonic signal using the elementary formula cos
3
(ωt) = (3 cos (ωt) +
cos (3ωt))/4 we obtain
U
(3f)
SD
=
U
(1f)
SD
4α
I
2
0
R
′
0
.(1.36)
1.5.Determination of density of states of thin highT
c
lms by FET type microstructures 15
Fromthis formula we can express the boundary thermoresistance by electronic measurements
α =
U
(1f)
SD
4U
(3f)
SD
I
2
0
R
′
0
.(1.37)
The realization of the method requires tting of R(T) and numerical differentiation at working tem
perature T
0
;the linear regression is probably the simplest method if we need to know only one point.
Given α,we can express the time oscillations of the temperature substituting in Eq.(
1.31
)
T =T
0
+
RI
2
0
2α
(1 +cos(2ωt))≈T
0
1+
R
SD
I
2
0
2αT
0
cos(2ωt)
.(1.38)
In this approximation terms containing I
4
0
are neglected and also we consider that shift of the average
temperature of the lmis small.
The variations of the temperature lead to variation of the work function of the lm according to the
wellknown formula fromthe physics of metals
W(T) = −
π
2
6e
ν
′
ν
k
2
B
T
2
,ν
′
(E
F
) =
dν
dǫ
E
F
,(1.39)
where the logarithmic derivative of the density of states ν(ǫ) taken for the Fermi energy E
F
has dimen
sion of inverse energy,the work function W has dimension of voltage,T is the temperature in Kelvins
and k
B
is the Boltzmann constant.For an introduction see the standard text books on statistical physics
and physics of metals.[
71
,
72
] Substituting here the temperature variations fromEq.(
1.38
) gives
W = −
π
2
k
2
B
6e
ν
′
ν
T
2
0
1 +
R
0
I
2
0
αT
0
cos(2ωt)
+O(I
4
0
),(1.40)
where Ofunction again marks that the terms having I
4
0
are negligible.
The oscillations of the temperature creates AC oscillations of the sourcegate (SG) voltage.We
suppose that a lockin with a preamplier,having high enough internal res istance is switched between
the source and the gate.In these conditions the second harmonics of the workfunction and of the SG
voltage are equal
U
(2f)
SG
= −
π
2
k
2
B
6e
ν
′
ν
T
2
0
R
0
I
2
0
αT
0
,(1.41)
U
SG
(t) = U
(2f)
SG
cos(2ωt) +U
(4f)
SG
cos(4ωt) +...(1.42)
Substituting α fromEq.(
1.37
) we have
U
(2f)
SG
= −
4π
2
k
2
B
6e
ν
′
ν
U
(3f)
SD
I
0
T
0
R
′
0
.(1.43)
Fromthis equation we can nally express the pursued logarithmic derivativ e of the density of states
dlnν(ǫ)
dǫ
E
F
=
ν
′
ν
= −
3e
2π
2
k
2
B
I
0
T
0
U
(2f)
SG
U
(3f)
SD
dR
dT
.(1.44)
In such way the logarithmic derivative of the density of states can be determined by fully electronic
measurements with a FET.This important energy parameter can be used for absolute determination of
the hopping integrals in the generic LCAO model.The realization of the experiment can be considered
as continuation of already published detailed theoretical and experimental investigations and having a set
of complementary researches we can reliably determine the LCAO parameters.
CHAPTER
2
SUPERCONDUCTIVITY OF OVERDOPED
CUPRATES:THE MODERN FACE OF THE
ANCESTRAL TWOELECTRON EXCHANGE
2.1 Introduction
The discovery of hightemperature superconductivity [
18
,
19
] in cuprates and the subsequent research
rush have led to the appearance of about 100 000 papers to date [
73
] (cf.Fig.
1
on page
v
).Virtually
every fundamental process known in condensed matter physics was probed as a possible mechanism
of this phenomenon.Nevertheless,none of the theoretical efforts resulted in a coherent picture [
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