Diploma thesis

FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS
COMENIUS UNIVERSITY BRATISLAVA
Density of states in the vortex core
of cuprate superconductors
DIPLOMA THESIS
Bratislava 2009
Lucia Komendová
Department of Experimental Physics
Faculty of Mathematics, Physics and Informatics
Comenius University Bratislava
Density of states in the vortex core
of cuprate superconductors
Diploma thesis
Lucia Komendová
4.1.1 Physics
Diploma thesis advisor
Dr. Richard Hlubina
BRATISLAVA 2009
By this I declare that I wrote this diploma thesis by myself, only with the
help of the referenced literature under the careful supervision of my thesis
advisor.
Bratislava 24th April 2009
Lucia Komendová
I would like to express my deep thanks to all the people who have supported
me during the writing of this thesis, especially to my thesis advisor, my
family and friends.
3
Abstrakt (SK)
Komendová Lucia: Hustota stavov v jadre vı́ru v kuprátových supravodičoch.
[Diplomová práca]. Univerzita Komenského v Bratislave. Fakulta matematiky, fyziky a informatiky; Katedra experimentálnej fyziky. Školiteľ: Doc.
RNDr. Richard Hlubina, DrSc. Stupeň odbornej kvalifikácie: Magister v
odbore 4.1.1 Fyzika. Bratislava: FMFI UK, 2009.
Práca sa zaoberá numerickým výpočtom hustoty stavov v rôznych homogénnych aj nehomogénnych supravodivých systémoch.
Začı́na teoret-
ickým úvodom do problematiky vysokoteplotnej supravodivosti. Samotné
riešenie vychádza z výpočtu pomocou Greenových funkciı́. Úloha sa rieši
numericky použitı́m štandardných maticových operáciı́. Výstupom sú mapy
hustoty stavov v okolı́ vı́ru zobrazené v grafoch v poslednej kapitole.
Kľúčové slová: supravodiče, vı́ry, hustota stavov
4
Abstract (EN)
Komendová Lucia: Density of states in the vortex core of cuprate superconductors. [Diploma thesis]. Comenius University Bratislava. Faculty of
mathematics, physics and informatics; Department of experimental physics.
Thesis advisor: Dr. Richard Hlubina; Thesis is a part of requirement for
obtaining a Master’s degree in Physics. Bratislava: FMFI UK, 2009.
The work is devoted to the numerical studies of the density of states in
various homogeneous and inhomogeneous superconducting systems. It begins with a theoretical introduction to the topic of the high-temperature
superconductivity. The solution itself is based on a calculation using the
Green’s functions. The problem is solved numerically using the standard
matrix operations. The main output are the maps of the density of states in
the vicinity of the vortex shown in the plots in the last chapter.
Keywords: superconductors, vortices, density of states
Contents
1 Introduction
11
1.1
Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2
The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3
High temperature superconductors . . . . . . . . . . . . . . . 20
1.4
Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Calculation of DOS
29
2.1
Green’s function, spectral function and density of states . . . . 29
2.2
Non-interacting case . . . . . . . . . . . . . . . . . . . . . . . 32
2.3
Noninteracting Green’s functions . . . . . . . . . . . . . . . . 35
2.4
Equations for the Green’s functions . . . . . . . . . . . . . . . 39
2.5
Solving the Gor’kov equations . . . . . . . . . . . . . . . . . . 43
2.6
Translationally invariant case . . . . . . . . . . . . . . . . . . 44
2.7
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Results and discussion
55
3.1
Usual vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2
Split vortex I . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3
The cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4
Split vortex II . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Conclusion
64
5
List of Figures
1.1
The BCS energy gap in the electronic density of states . . . . 13
1.2
The effective electron-electron interaction potential . . . . . . 15
1.3
The doping phase diagram . . . . . . . . . . . . . . . . . . . . 20
1.4
The crystal structures of high temperature superconductors . . 21
1.5
The phase diagram of the type 2 superconductors . . . . . . . 23
1.6
The possible gaps for tetragonal superconductor . . . . . . . . 26
1.7
The vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8
The STM images of vortices in BSCCO - Hoogenboom . . . . 27
2.1
The tight binding hopping . . . . . . . . . . . . . . . . . . . . 34
2.2
The electronic dispersion relation . . . . . . . . . . . . . . . . 36
2.3
The DOS in the non-interacting case . . . . . . . . . . . . . . 37
2.4
The DOS for the extended s-wave and d-wave superconductor
2.5
The non-interacting Green’s function - convergence . . . . . . 50
2.6
The d-wave DOS for different lattice sizes and values of Γ . . . 51
2.7
How is the G0 computed . . . . . . . . . . . . . . . . . . . . . 52
3.1
The vortex core DOS for singly and doubly quantized vortices
3.2
The vortex core DOS - Berthod . . . . . . . . . . . . . . . . . 57
3.3
The DOS for a simple vortex in a d-wave superconductor . . . 58
3.4
The zero bias DOS for a split vortex - standard ∆ij . . . . . . 59
3.5
The zero bias DOS for a split vortex - standard ∆ij . . . . . . 60
3.6
The zero-bias DOS for a cut in the d-wave superconductor . . 61
3.7
The LDOS for a cut in the d-wave superconductor . . . . . . . 62
6
48
56
LIST OF FIGURES
7
3.8
The zero bias DOS for a split vortex - mixed ∆ij . . . . . . . . 63
3.9
The zero bias DOS for a split vortex - mixed ∆ij . . . . . . . . 63
List of abbreviations
HTS
High temperature superconductors
BCS
Bardeen, Cooper, Schrieffer or their theory of superconductivity
DOS
Density of states
LDOS
Local density of states
STM
Scanning tunneling microscopy
ARPES
Angle resolved photoemission spectroscopy
YBCO
YBa2 Cu3 O7−δ , yttrium barium copper oxide
BSCCO
Bi2 Sr2 CaCu2 O8+δ , bismuth strontium calcium copper oxide
LSCO
ATLAS
LAPACK
FFTW
La2−x Srx CuO4 , lantanum strontium copper oxide
Automatically Tuned Linear Algebra Subprograms
Linear Algebra Package
Fastest Fourier Transform in the West
8
Preface
One thing was certain, that the WHITE kitten had had nothing to do
with it . . .
Lewis Carrol, Through the Looking Glass
Though the high temperature cuprate superconductors are known already
for more than 20 years, the precise mechanism behind their remarkable properties is still the matter of discussions and research. During the years various
theories aimed at finding the microscopic origin of this new phase of matter
have been proposed like Anderson’s resonating valence bond theory and the
quantum critical point scenario, to name a few. For the time being none
of them has been really successful in explaining the variety of phenomena
involved, so the subject stays a rich source of open problems.
In this thesis we compute the densities of states for superconducting systems. The density of states is an important physical quantity overwhelmingly
encountered in solid state physics. It is measured by means of modern experimental methods like the scanning tunneling microscopy (STM) and the
angle resolved photoemission spectroscopy (ARPES). In this work, we first
introduce the most important concepts in the field of conventional and high
temperature superconductivity. After that we derive the equations valid for
the Green’s functions of the superconducting system using a model BCS-like
Hamiltonian. These Gor’kov equations are then numerically solved for the
case of an inhomogeneous d-wave superconductor. The generality of this
approach enables us to allow for the presence of the normal or split vortex,
which is basically a topological singularity in the phase of the superconduct9
LIST OF FIGURES
10
ing order parameter. The vortices are usually formed when the magnetic
field penetrates the type 2 superconductors. As the final step, the information about the density of states is easily obtained from the spectral function
A(i, ω) once the Green’s function is known completely.
Due to the computational complexity of this calculation which needs to
perform several Fourier transforms, matrix inversions and matrix multiplications on matrices with a few thousands of rows and columns, we needed to
use some of the specialized numerical libraries: FFTW and Atlas + Lapack
bundle. Despite these optimizations we stayed limited to lattice sizes of up
to approximately 100 lattice constants and had to struggle a bit for sufficient
resolution.
Chapter 1
Introduction
This introductory chapter contains some background information on superconductivity, a short sketch of the BCS theory, and a description of the
structure and electrical properties of the cuprate superconductors.
1.1
Superconductivity
The electrical resistance of many metals and compounds goes abruptly to
zero when we cool them to sufficiently low temperatures. The existence of
this zero resistance superconducting state was first observed by H. K. Onnes
in Leiden in 1911, three years after he was the first to liquefy helium. At
some critical temperature Tc (which was 4.2 K for the Hg sample Onnes had)
the material undergoes a phase transition from a state with finite resistance
to the superconducting state with zero DC resistance. This means that when
we connect the superconducting sample to the current source, the measured
voltage difference between the contacts will be zero. On the other hand, if
we inductively create an electrical current in a ring-shaped superconductor,
it does not decay and it persists essentially forever. The magnetic properties
of the superconductors are at least equally amazing. Experiments on bulk
superconducting samples in small magnetic fields have found them to be ideally diamagnetic (B = 0 inside the sample). If we put the sample inside the
11
CHAPTER 1. INTRODUCTION
12
magnetic field and cool it under Tc , we observe that the originally present
magnetic field is expelled out. The magnetic field lines would rather be deformed than forced to enter the sample. This is called the Meissner effect.
The magnetic field thus enters the superconductor just in a thin surface layer
of width of the order of the magnetic penetration depth λ. We understand
now that microscopically the superconducting state is an ordered state of
the conduction electrons in a metal. This ordering (proven by the decrease
in entropy) is connected with the creation of bound electron pairs. These
bound electrons are separated in space by average distance ξ, called the coherence length, which can be surprisingly big. The pairs of electrons obey
Bose-Einstein statistics, which means that under some temperature Tc they
can condense into a single quantum state coherent throughout the whole
volume of the superconductor. Very similarly the bosonic 4 He atoms condense to form a superfluid. The origin and nature of this phenomenon for
the conventional superconductors were explained by Bardeen, Cooper and
Schrieffer.
1.2
The BCS theory
In 1957 the first truly microscopic theory of superconductivity was developed
by Bardeen, Cooper and Schrieffer (BCS). In this section we try to present
it shortly, following [1] and [2].
One of the main points included in the BCS theory is the existence of
an energy gap 2∆ in the density of states at the Fermi level, sketched in
Fig. 1.1. The gap is always formed around the Fermi energy, but it does not
lower the conductivity like the gap in insulators or semiconductors. The 2∆
is the energy needed to break up a Cooper pair into two free electrons.
Another contribution of the BCS theory was the correct explanation of the
isotope effect, the dependence of the transition temperature on the mass
CHAPTER 1. INTRODUCTION
13
Figure 1.1: The BCS energy gap in the electronic density of states N (ε). There are
no single electron states available around the Fermi energy. The missing spectral weight
forms two coherence peaks above and under the gap, which are typical for superconducting
spectra (STM, ARPES).
of ions forming the crystal lattice Tc ∝ M −α with the exponent α = 1/2.1
The isotope effect gives us a hint about the role which phonons play in the
superconductivity at least in some substances.
The BCS theory is built upon the three major insights:
1. The idea by Fröhlich that the electron - electron interaction may be
attractive,
2. Cooper’s finding that two electrons added to a filled Fermi sea would
form a stable pair bound state, no matter how weak is the attractive
force,
3. Schrieffer’s proposal for a many particle wave function in a coherent
state form, with all the electrons near the Fermi surface paired up.
1
Most superconductors agree well with this prediction, though there are exceptions
showing a reduced isotope effect (α < 1/2, e.g., in molybdenum or osmium) or even no
isotope effect at all (α = 0, i.e., the mass independent critical temperature in ruthenium,
zirconium, YBCO and others)[1].
CHAPTER 1. INTRODUCTION
14
In the following text we want to focus on these three points.
Electron-electron attraction
Two electrons normally strongly repel each other, because of the Coulomb
force. However, if we take into account also other quasiparticles like the
phonons, the electrons may effectively attract each other. Let us discuss
more deeply the mechanism of the electron-phonon interaction.
First it is important to realize that the time scales of the movement of the
electrons and those of the phonons are very different. Phonons are quantized
vibrations of the ionic crystal lattice. Since the mass of ions M is much
greater than the mass of electrons m, the electrons move much more rapidly.
Imagine an electron travelling through the lattice. In its position, the overall electrical charge neutrality is disturbed and the place becomes slightly
negatively charged. This negative charge forms an attractive potential for
the positively charged ions, so they start to move toward it. But they are
so slow that when the ionic charge accumulates, the electron is long gone.
As a result, the electrical neutrality is disturbed, this time making a slightly
positively charged place. This in turn attracts the other electrons. Thus
effectively the electrons attract each other. We see immediately that this
interaction must be retarded in time.
The whole idea can be rather easily formalized using the notion of the
dielectric function ε(q, ω) dependent on the wave vector q and frequency ω
which is a straightforward generalization of the static dielectric function ε.
This approach leads to the following formula for the effective potential [3, 2]:
ωq2
4πe2
4πe2
Veff (q, ω) = 2
= 2
1+ 2
(1.1)
q ε(q, ω)
kT F + q 2
ω − ωq2
where kT F is the Thomas-Fermi wave vector and ωq is the frequency of the
phonon. The latter is of course limited from above by the Debye frequency
ωD and thus the effect of phonons is negligible for the electrons with large
CHAPTER 1. INTRODUCTION
15
Figure 1.2: The effective electron-electron interaction potential Veff (ω) as defined in
(1.1) for fixed q. The potential tends to zero in the static limit, it is attractive for ω < ωq
and repulsive for ω > ωq .
energies (the ions would not move so quickly). Figure 1.2 shows the graph
of Veff (ω) where we see how the phonon mediated interaction between
electrons would be attractive for ω < ωq .
The BCS modeled this attraction with the simplified step potential
Veff (q, ω) = −|geff |2 ,
| ω| < ωD
(1.2)
that is with the constant value for | ω| < ωD and zero above the cut-off at
Debye frequency ωD . This comes together with the corresponding effective
Hamiltonian
Ĥ = −|geff |2
X
c†k1 +q,σ1 c†k2 −q,σ2 ck1 σ1 ck2 σ2 ,
(1.3)
k1 ,σ1 ,k2 ,σ2 ,q
where the sum is restricted to the electron states in the range ±~ωD around
the Fermi energy, and c†kσ and ckσ are the usual electron creation and anni-
CHAPTER 1. INTRODUCTION
16
hilation operators. They obey the fermionic anticommutation rules
{ckσ , c†k0 σ0 } = δkk0 δσσ0
(1.4)
{ckσ , ck0 σ0 } = {c†kσ , c†k0 σ0 } = 0
(1.5)
where the anticommutator of two operators X and Y is defined as
{X, Y } ≡ XY + Y X.
(1.6)
Cooper has considered what would happen if we add two new electrons to
the filled Fermi sea, if such attractive interaction is present. He has shown
that the interaction would make the energy of this Cooper pair lower compared to the normal metallic case. Hence it would be energetically favourable
to take some electrons from the top of the Fermi sea and raise their kinetic
energy, because the system will profit from the binding energy of the Cooper
pairs
E = −2~ωD e−2/λ
(1.7)
where the electron-phonon coupling parameter, λ is defined as
λ = |geff |2 N (εF ),
(1.8)
i.e., the strength of the electron-phonon interaction times the density of states
at the Fermi energy. The result was obtained in the weak coupling limit,
λ 1.
BCS wavefunction
After the BCS trio knew this, it was obvious that all the electrons within the
attractive part of k-space around the Fermi energy would like to form such
pairs. The hard part was to write the wave function of such state and it was
Schrieffer who was successful at this task. Let us define the pair creation
operator
Pk† = c†k↑ c†−k↓
(1.9)
CHAPTER 1. INTRODUCTION
17
whose role is to create a pair of electrons in the conduction band with a
zero net momentum and opposite spins. Schrieffer proposed the following
many-body wavefunction,
!
|ΨBCS i = C. exp
X
αk Pk†
|0i
(1.10)
k
The complex numbers αk allow for variational optimization of the energy, the
vacuum state |0i is the state with an unoccupied conduction band and C is
just a normalization constant. We note that the commutator [Pk† , Pk†0 ] = 0 for
k 6= k0 and this is why we may utilize the relation exp(A+B) = exp A·exp B,
which holds for the commuting operators A and B. Then we write
Y
|ΨBCS i = C.
exp αk Pk† |0i
(1.11)
k
Thanks to the Pauli principle (Pk† )2 = 0, so only the first two terms in the
Taylor expansion of the exponential function are nonzero:
Y
|ΨBCS i = C.
1 + αk Pk† |0i
(1.12)
k
In order for this wave function to be properly normalized, i.e., hΨBCS |ΨBCS i =
1, the constant C must be
C=
Y
k
1
1 + |αk |2
Finally, the BCS wave function is usually written in the form
Y
|ΨBCS i =
u∗k + vk∗ Pk† |0i,
(1.13)
(1.14)
k
where u∗k , vk∗ ∈ C are
1
,
1 + |αk |2
αk
=
,
1 + |αk |2
u∗k =
(1.15)
vk∗
(1.16)
since αk were so far arbitrary complex numbers, it suffices to remember that
|uk |2 + |vk |2 = 1,
∀k.
(1.17)
CHAPTER 1. INTRODUCTION
18
It is important to note that |ΨBCS i does not have a definite number of electrons, since we add pairs neither with zero probability, nor with the unit
probability, but has a definite complex phase originating from the complex
numbers αk .
The mean-field Hamiltonian
The interaction that we have studied so far was the one defined in (1.2)
with the Hamilton operator (1.3). Now we shall consider just the terms
describing the scattering of Cooper pairs formed by two electrons with the
opposite spin. This is sufficient, since here we are only interested in the
singlet superconductivity. Furthermore we assume zero net momentum of
the Cooper pairs. On the other hand, we allow for the momentum-dependent
interaction potential Vkk0 . With all these changes incorporated, Ĥ becomes
X
1X
εk c†kσ ckσ −
Ĥ =
Vkk0 c†k0 ↑ c†−k0 ↓ c−k↓ ck↑ ,
(1.18)
Ω k,k0
k,σ
where Ω denotes the system volume. The BCS theory then proceeds to
minimize the total energy E,
E = hΨBCS |Ĥ|ΨBCS i,
(1.19)
with respect to the parameters uk and vk . In this process of minimization we
have to ensure that the mean total number of particles hN i ≡ hΨBCS |N̂ |ΨBCS i
stays constant as well as the condition (1.17) fulfilled, which we do by introducing the Lagrange multipliers µ and Ek . Finally this gives us a pair of
linear homogeneous equations for the parameters uk and vk , which we shall
write in a single matrix eigenvalue equation
!
!
εk − µ
∆k
uk
∆∗k
−(εk − µ)
vk
= Ek
uk
!
vk
where we have defined the BCS gap parameter ∆k as
1X
∆k =
Vkk0 bk0 ,
Ω k0
(1.20)
CHAPTER 1. INTRODUCTION
19
while the superconducting order parameter bk is
bk ≡ hΨBCS |c−k↓ ck↑ |ΨBCS i.
(1.21)
It is non-zero only in the superconducting state, because it is measuring the
probability of taking away one Cooper pair from |Ψi and the state |Ψi staying
the same. This is zero in the normal state, because there the state vectors
with different particle numbers are mutually orthogonal. The eigenvalues of
the matrix equation above are ±Ek , where
Ek =
p
(εk − µ)2 + |∆k |2
(1.22)
These are the excitations energies for an extra electron or hole added to the
BCS ground state.
Summary
The phenomenon of superconductivity originates from the effective attractive interaction between the electrons possible under some specific conditions.
This attraction leads to the creation of bound pairs consisting of two electrons with opposite momentum. The pairs are called Cooper pairs after the
theoretician who proposed the idea. The two spin-1/2 electrons forming the
pair can produce either the singlet state with total spin S = 0 or a triplet
state with total spin S = 1. In the following we consider the first case, which
is appropriate for the cuprates. The wavefunction of the superconducting
state is thus formed only from the single particle states with the property
that either both states |+k ↑i and |−k ↓i are occupied or are both empty.
The elementary excitations of the system are called bogoliubons and have an
p
energy dispersion law of the form Ek = (εk − µ)2 + |∆k |2 as opposed to
the ordinary metal’s Ek = εk − µ. Hence there exists a minimal ‘gap’ energy
needed to excite the system, which is equal to the minimum of |∆k |.
CHAPTER 1. INTRODUCTION
20
Figure 1.3: Schematic temperature - hole doping phase diagram of high-Tc superconductors. AF is the antiferromagnetic Mott insulator and SC labels the superconducting region.
Remaining portions of the diagram are the pseudogap phase and the strange metal phase.
The line separating them is dashed because it is probably not a real thermodynamical
phase transition line. Adapted from [15].
1.3
High temperature superconductors
Ever since the original discovery of the phenomenon of superconductivity
by Heike Kammerlingh Onnes in 1911 the researchers around the globe were
looking for superconducting materials with higher and higher transition temperature Tc . Despite the efforts they could not go beyond Tc of 23 K of the
Nb3 Ge for quite a long time. Some theoretical arguments existed based on
the BCS theory which predicted that significant improvement was not even
possible. Then the discovery of high temperature superconductors came as
a big surprise in 1986, when Bednorz and Muller found superconductivity
in the LaBaCuO system with Tc = 35 K. Soon after that (early 1987) Paul
Chu et al. overcame the liquid nitrogen barrier when they found 90 K superconductivity in YBCO. Further variations of crystal structure by total or
partial substitution of another atomic species (doping) and applied pressure
led to the observation of transition temperatures up to 160 K. For a review
of these developments see e.g. [4, 5].
CHAPTER 1. INTRODUCTION
21
a
b
c
Figure 1.4: The crystal structures of some of the important high temperature superconductors (a) LSCO (here with zero Sr doping), (b) YBCO, (c) BSCCO; source: [17]. We
see the common trait: the presence of the CuO2 planes in each of these compounds.
CHAPTER 1. INTRODUCTION
22
The original undoped compound is usually an antiferromagnetic Mott insulator with one hole in every unit cell. The doping introduces free charge
carriers into the system and in some range of dopant concentrations the resulting compound is superconducting - the so called ‘superconducting dome’
region of the phase diagram in Fig. 1.3. At temperatures above the superconducting critical temperature in the hole-doped case there is a pseudogap
phase and a strange metal phase. The strange metal refers to a normal
(non-superconducting) state of the compound which shows characteristics of
a non-Fermi liquid behavior. The nature of the pseudogap phase is not really
understood yet, though it is believed that it may be an incoherent state of
Cooper pairs.
All high temperature superconductors known until February 2008 were
structurally similar, their basic constituents being the CuO2 planes separated by the insulating spacer layers; this is where the name cuprates comes
from. The crystal structures of few of the important cuprate superconductors are depicted in Fig. 1.4. In the copper oxide plane each copper ion
has four neighbouring oxygens. The transition temperature is determined by
the exact chemical composition and stoichiometry, but after years of study,
the prevailing opinion is that the main effect of everything apart from CuO2
planes themselves is simply to tune the electronic properties of these planes.
In particular the CuO chains in YBCO and the La, Bi or Sr atoms provide
some electrons and thus modify the in-plane electronic density. In February
2008 a new class of iron based superconductors with a relatively high Tc was
discovered by Takahashi et al. [6]. The initial hope was that the study of
this new class of superconducting materials may help to finally solve also the
mysteries of the cuprates. At second glance it seems that here we see yet
another different mechanism of how materials may become superconducting,
despite some similarities seen in the crystal structure. From now on we will
concentrate only on the cuprates.
CHAPTER 1. INTRODUCTION
23
Figure 1.5: The phase diagram of the type 2 superconductors.
The specific layered crystal structure of high temperature superconductors
leads to an extreme anisotropy in their physical properties. For instance
the normal state electrical resistance along c axis is higher by a factor of
ρc (T )/ρab (T ) ∼ 102 - 105 (Y resp. Bi-Tl based cuprates) compared to the
electrical resistance in ab plane [7],[8]. The experiments on cuprates also
show that the superconducting ‘particles’ have charge 2e, so the pairs of
electrons are formed - the Cooper pairs. These then undergo a Bose-Einstein
condensation to a superconducting state. The size of the Cooper pair - the
coherence length ξ - is small (approximately 20 Å in the plane and just
around 2 Å in c direction, which is less than the lattice spacing, see Fig. 1.4)
as opposed to the hundreds of Å in the low temperature superconductors.
One reason to expect this is that ξ ∝ ~vF /Tc and the critical temperature
Tc of the cuprates is very large. Since the parent compound is insulator, the
carrier density and thus the Fermi velocity vF in the cuprates is small. From
this we can expect the normal state resistance and the magnetic penetration
depth λ to be large. The experiments have confirmed λ ≈ 1400 Å. Thus the
high temperature superconductors are strongly type 2 materials characterized
CHAPTER 1. INTRODUCTION
24
by a large value of the Ginzburg-Landau parameter κ = λ/ξ 1.
Apart from its anisotropy, the superconducting state of the cuprates is
in many ways very similar to that of the low temperature type 2 superconductors (see the type 2 mean-field phase diagram in Fig. 1.5). There is a
region of phase space with zero resistance and a complete Meissner effect under the lower critical field Hc1 , though Hc1 in the cuprates is typically very
small (around 70 mT, according to [9]). Above it there is not only the simple
Abrikosov vortex lattice phase but also some more complicated phases with
a nontrivial structure formed by vortices. Depending on the interplay among
the thermal fluctuations, quantum fluctuations and the pinning the resulting
state may be the regular lattice or Bragg glass where the structure is regular,
or the vortex glass state where the vortices are pinned to random positions.
Another possibility is that the vortices remain mobile and are in a vortex
liquid state.
The Fermi surface in cuprates is highly two-dimensional. Therefore to
determine the gap function, it is sufficient to consider a two-dimensional
square Brillouin zone. There are only a few possible types of gap functions,
because they have to be consistent with the symmetry group of the square.
Furthermore they have to be scalar, because the pairing in the cuprates
has been shown by early nuclear magnetic resonance experiments to be spin
singlet. These gap functions shall be considered:
∆k = ∆
(s)
(1.23)
∆k = ∆(cos(kx a) + cos(ky a))
(s− )
(1.24)
∆k = ∆(cos(kx a) − cos(ky a))
(dx2 −y2 )
(1.25)
(dxy )
(1.26)
∆k = ∆ sin(kx a) sin(ky a)
∆k = ∆ sin(kx a) sin(ky a)(cos(kx a) − cos(ky a))
(gxy(x2 −y2 ) ) (1.27)
The first gap function is just a constant, which is exactly what is considered
in the standard BCS theory. It is obviously invariant under all possible
symmetry operations of the square Brillouin zone (rotation by 90◦ , mirror
reflections and parity), hence it is conventional or s-wave pairing. Let us now
CHAPTER 1. INTRODUCTION
25
have a look at Fig. 1.6, where the next three gap functions are depicted. The
second gap function is constant under all possible symmetries as well. Thus
it is a conventional s-wave pairing too, even though the magnitude of this gap
function becomes zero at eight points on the Fermi surface. This gap function
is called ’extended-s’. The other two gap functions are unconventional, they
both change sign under rotation of the square by 90◦ . They have the same
symmetry as the atomic d spherical harmonic functions dx2 −y2 and dxy , so
they are termed d-wave. In this case the quasiparticle gap 2|∆k | vanishes
at four points at the Fermi surface.
The different gap functions are possible to distinguish experimentally.
For instance, if the gap is everywhere finite, there will be a true gap, i.e.,
no states around the Fermi energy in the measured quasiparticle density of
states. The presence of the gap nodes is also visible in the measurements of
the penetration depth λ and specific heat capacity CV . This way the s-wave
possibilities have been rejected. The d-wave states only differ in the location
of gap nodes with respect to the Brillouin zone axes. Two independent types
of experiments have convinced us that the pairing in the cuprates is dx2 −y2 .
First was the angle resolved photoemission (ARPES), which can tell the
k-space location of nodes and the other was based on a clever use of the
Josephson effect, which also helped to confirm the sign changes of ∆k .
In the calculation presented in next chapters we chose ∆, in a
sense as it is used in (1.25), as our unit of energy.
1.4
Vortex
The magnetic field passes through a type 2 superconductor through the socalled vortices, see Fig. 1.7. These are essentially line singularities in the
superconducting order parameter around which the superconductor’s phase
winds [3]. By winding we mean that it changes by 2nπ, n ∈ Z, when going
around. The value of n is determined by the number of magnetic flux quanta
Φ0 passing through the vortex. The vortices in the cuprate HTS have a
CHAPTER 1. INTRODUCTION
26
Figure 1.6: The possible gaps for tetragonal superconductor: the extended-s (left) and
two d-wave gap functions: dx2 −y2 (middle) and dxy (right). The gap in the cuprates has
been experimentally determined to be dx2 −y2 , analytically described by Eq. (1.25).
highly two-dimensional ‘pancake’ character, because they are to a high degree
confined just to the CuO2 planes.
In the vortex core there can exist some localized low-energy vortex-core
states, which are marked by a presence of the zero-bias peak in the LDOS in
the vortex core. In [11] Berthod argues that the suppression of the amplitude
of the superconducting order parameter, i.e., of density of superconducting
electrons is not necessary for these vortex core states to form. However the
existence of the topological defect in the phase of the order parameter is
needed.
The scanning tunneling spectroscopy measurements done by Hoogenboom
et al. [12] on the BSCCO samples show the vortex cores with very irregular
distribution and shape, ‘patches of various sizes and shapes scattered over
the surface’, see Fig. 1.8, interpreted as caused by a strong pinning of vortices to the defects and inhomogeneities. Some of them were observed to
consist of two or more randomly distributed smaller elements separated by
the small zones with the LDOS having the coherence peaks typical for the
superconducting state and absent in the vortex core DOS. The authors attribute this splitting to the quantum tunneling of the vortices between the
different pinning sites.
Different explanation has been proposed by Hlubina in [13]. His hypoth-
CHAPTER 1. INTRODUCTION
27
Figure 1.7: The vortices in the low-temperature type 2 superconductor Nb and in
cuprates. The profile of magnetic field b(~r) and order parameter Ψ(~r). From [10].
Figure 1.8: The STM image of vortices in BSCCO - Hoogenboom [12].
CHAPTER 1. INTRODUCTION
28
esis is that the vortices in BSCCO may really exist in the form of pairs of
two half vortices. He had shown that this should be possible, if there is not
only the normal Josephson current present, but also its second harmonics.
The computation was done in the framework of the Ginzburg Landau theory
and predicted the magnetic field profile for the split vortex. It was not clear
whether this result holds true also in the microscopic view, thus the need
arose to do the calculation of the density of states using the Green’s function
formalism - the inhomogeneous BCS theory.
Chapter 2
Calculation of DOS
2.1
Green’s function, spectral function and
density of states
We define the retarded Green’s function Gσ (i, j, t) as
oE
−i Dn
Gσ (i, j, t) ≡
ciσ (t), c†jσ
θ(t)
(2.1)
~
This Green’s function is closely connected to the measurable physical properties of the material and is used to make the theoretical predictions about
them. In the definition the braces {} denote the anticommutator,
{X, Y } ≡ XY + Y X,
(2.2)
and the angle brackets hi denote either the average over the ground state or
the thermal average. The ciσ (t) is the electron annihilation operator at the
time t and as such can be expressed using the Heisenberg picture:
ciσ (t) = e
iHt
~
ciσ e
−iHt
~
(2.3)
In this section our aim is to show the connection between this Green’s function and the spectral function A(i, ω). To this end we use the definition
of the thermal average of the quantity X,
1 X −βEn
hXi =
e
hn|X|ni,
Z n
29
(2.4)
CHAPTER 2. CALCULATION OF DOS
where β =
1
.
kT
30
We will actually work with the time Fourier transform of the
Green’s function Gσ (i, j, t), that is with Gσ (i, j, ω)
Z
−i ∞
Gσ (i, j, ω) =
dtei(ω+iΓ)t h{ciσ (t), c†jσ }i
~ 0
(2.5)
where we have moved ω a bit away from the real axis by introducing a small
positive imaginary part iΓ. This is needed to avoid the divergence in further
calculations1 . Let us continue by deciphering the operator part
i
1 X −βEn h
†
†
†
h{ciσ (t), cjσ }i =
hn|ciσ (t)cjσ |ni + hn|cjσ ciσ (t)|ni
e
Z n
We insert the resolution of identity in the form 1 =
P
m
(2.6)
|mihm|, which is the
completeness relation for the spectrum of states |ni, in between the creation
and annihilation operators. We have obtained two terms like this
hn|ciσ (t)|mi = hn|e
iHt
~
ciσ e
−iHt
~
i
|mi = e ~ (En −Em )t hn|ciσ |mi
(2.7)
We take a glimpse of (2.5) now
Z
i
−i ∞
1 X −βEn
†
Gσ (i, j, ω) =
dtei(ω+iΓ)t e ~ (En −Em )t +
e
hn|ciσ |mihm|cjσ |ni
Z n,m
~ 0
Z
i
1 X −βEn
−i ∞
dtei(ω+iΓ)t e ~ (Em −En )t
+
e
hn|c†jσ |mihm|ciσ |ni
(2.8)
Z n,m
~ 0
The time integral is easy to do:
Z
i
−i ∞
1
dtei(ω+iΓ)t e ~ (En −Em )t =
~ 0
~(ω + iΓ) + En − Em
(2.9)
After having used this we arrive to the expression for the Green’s function
in the Lehmann representation
"
#
hn|ciσ |mihm|c†jσ |ni
hn|c†jσ |mihm|ciσ |ni
1 X −βEn
e
+
Gσ (i, j, ω) =
Z n,m
~(ω + iΓ) − (Em − En ) ~(ω + iΓ) + (Em − En )
(2.10)
1
This has actually a deeper meaning connected with the causality, but we shall postpone
the discussion of this until later in this chapter.
CHAPTER 2. CALCULATION OF DOS
31
Let us now consider the diagonal part Gσ (i, i, ω)
|hn|ciσ |mi|2
1 X −βEn
|hm|ciσ |ni|2
Gσ (i, i, ω) =
e
+
Z n,m
~(ω + iΓ) − (Em − En ) ~(ω + iΓ) + (Em − En )
(2.11)
+
We consider now the limit Γ → 0 , so that we can make use of the SokhatskyWeierstrass theorem,2 which tells us that
1
x+i0+
= P( x1 ) − iπδ(x). We use
this to evaluate the quantity Aσ (i, ω)
1
Aσ (i, ω) ≡ − Im Gσ (i, i, ω)
π
(2.12)
and get
Aσ (i, ω) =
1 X −βEn
e
[|hn|c†iσ |mi|2 δ(~ω − (Em − En )) +
Z n,m
+|hm|ciσ |ni|2 δ(~ω + (Em − En ))]
(2.13)
Let us consider for simplicity just the zero temperature version of the previous
line:
ATσ =0 (i, ω) =
X
N +1
[|hmN +1 |c†iσ |0N i|2 δ(~ω − (Em
− E0N )) +
m
N −1
+|hmN −1 |ciσ |0N i|2 δ(~ω + (Em
− E0N ))]
(2.14)
where we have added the upper indices denoting the number of particles in
each state so that the quantum mechanical expectation values in front of
delta functions are non-zero. Because T = 0 the initial state of the system is
the ground state, which we denoted by |0i and its number of particles we call
N . The state hm| in first part shall be any state of the system with N + 1
particles. In the second part the same symbol may be any state with N − 1
particles. The expectation value hmN +1 |c†iσ |0N i gives us the overlap between
the ground state to which we have added one extra electron and some excited
2
Sokhatsky-Weierstrass theorem: Let f be a complex-valued function which is defined
and continuous on the real line, and let a and b be real constants with a < 0 < b. Then
Rb
R b f (x)
the theorem states that limε→0+ a x±iε
dx = ∓iπf (0) + P a f (x)
x dx, where P denotes the
Cauchy principal value. After [16].
CHAPTER 2. CALCULATION OF DOS
32
state |mi. The amplitude of this squared, |hmN +1 |c†iσ |0N i|2 , is the probability
of adding electron to the N -particle ground state and finishing in this excited
state. Since in Eq. (2.14) we sum over all possible states |mi we measure
the total probability of having unoccupied states at energy higher by ~ω at
some position in the first part. The second part counts the occupied states
from which we can steal one electron at energy cost ~ω. We see that Aσ (i, ω)
is the spectral function, because it measures the spectral density at point
Ri . In this work we prefer to call it the local density of states (LDOS).
This is because another similar type of the spectral function exist, A(k, ω)
Aσ (k, ω) =
1 X −βEn
e
[|hn|c†kσ |mi|2 δ(~ω − (Em − En )) +
Z n,m
+|hm|ckσ |ni|2 δ(~ω + (Em − En ))]
(2.15)
which counts the states with the equal momentum k, rather than position.
The local density of states A(i, ω) is the physical quantity measured in the
STM experiments, whereas the ARPES experiments measure the momentum
spectral function A(k, ω). The ARPES is actually capable of capturing just
the second part of the last relation, because in the experiment the electrons
are usually coming out of the material after they have been hit by the X-ray
photon. In the STM experiments the direction of the tunneling current can
be controlled by applying a positive or negative bias voltage, so both parts
of (2.13) are experimentally accesible.
2.2
Non-interacting case
Let us study the non-interacting grand-canonical Hamiltonian Ĥ0 :
Ĥ0 = −
X
tij c†iσ cjσ − µN̂
(2.16)
ijσ
The first term is the kinetic energy of the electrons. The annihilation operator
cjσ destroys one electron with the spin σ from the site Rj and c†iσ creates one
electron with the same spin σ at position Ri . Hence by acting with c†iσ cjσ
CHAPTER 2. CALCULATION OF DOS
33
on some state we move one electron from Rj to Ri , provided there was an
electron at Rj at the beginning and the place Ri was not already occupied
by the electron with the same spin, because such movement is not allowed
by the Pauli principle. The sole purpose of the term −µN̂ is to set the zero
of the energy scale to the value of chemical potential µ. Since this term is
much simpler, we forget it for a while. We will resurrect it by the end of
this calculation. Let us now discuss the values of the hopping energies tij .
It seems that to model the band structure of the high-Tc superconductors
we need to take non-zero only the nearest neighbor interactions with the
hopping energy t and the next-nearest neighbors with the hopping energy
t1 < 0. The bonds between sites to which t or t1 applies are painted in Fig.
2.1. Using the knowledge about the hopping energies we may rewrite Ĥ0 to
the following form:
i
Xh †
†
†
†
Ĥ0 = −
t(ciσ ci+x̂,σ + ciσ ci+ŷ,σ ) + t1 (ciσ ci+x̂+ŷ,σ + ciσ ci+x̂−ŷ,σ ) + h.c.
i
(2.17)
where x̂ = (a, 0) is the vector to the nearest neighbor in the positive x
direction and ŷ = (0, a) is the vector to the nearest neighbor in the positive
y direction. The a denotes the lattice constant. We deal here with the noninteracting and thus obviously also translationally invariant case. We may
take advantage of this and use the Fourier transform to find the spectrum of
Ĥ0 . Let us define
1 X −ik·Rj
e
cjσ
N j
1 X ik·Rj
=
e
ckσ
N k
ckσ =
(2.18)
cjσ
(2.19)
√
where 1/N = 1/ N 2 and N 2 is the total number of the points Rj in the two
dimensional lattice. We use the periodic boundary conditions, thus values of
k are discrete: kx =
πnx
Na
where nx ∈ N, |nx | ≤ N , analogously for ky . We
use the above definition of the creation and annihilation operators in k-space
CHAPTER 2. CALCULATION OF DOS
34
t
t1
Figure 2.1: The tight binding hopping. The nearest neighbor hopping energy t applies
to the horizontal and vertical bonds, the next nearest neighbor hopping energies t1 belong
to the diagonal bonds. All the other hoppings are neglected and the hopping integrals tij
for them are assumed to be zero.
and obtain
−
i
t X h −ik·Ri ik0 ·(Ri +x̂)
†
ik0 ·(Ri +ŷ)
0
e
e
+
e
c
c
kσ k σ −
N 2 i,k,k0
0
i
Xh
0
e−ik·Ri eik ·(Ri +x̂+ŷ) + eik ·(Ri +x̂−ŷ) c†kσ ck0 σ + h.c.
Ĥ0 = −
(2.20)
t1
N2
(2.21)
i,k,k0
Now we use the identity (valid if both k a k0 are from the 1st Brillouin zone):
X
0
ei(k−k )·Rj = N 2 δk,k0
(2.22)
j
We are left with
Ĥ0 = −
X
t · (eikx a + e−ikx a + eiky a + e−iky a )c†k,σ ck,σ +
(2.23)
+ e−i(kx +ky )a + ei(kx −ky )a + e−i(kx −ky )a ]c†k,σ ck,σ
(2.24)
k
i(kx +ky )a
+ t1 · [e
This is equivalent to:
X
Ĥ0 =
[−2t(cos kx a + cos ky a) − 4t1 cos kx a cos ky a] c†k,σ ck,σ
(2.25)
k
Now we want to identify the zero of energy with the chemical potential µ.
To achieve this we have to subtract the term µN̂ , where N̂ is the number
CHAPTER 2. CALCULATION OF DOS
operator N̂ =
P
†
k,σ ck,σ ck,σ ,
35
which counts the total number of electrons.
Finally we shall write simply:
Ĥ0 =
X
ε̃k c†k,σ ck,σ
(2.26)
k
with the dispersion relation
ε̃k ≡ −2t(cos kx a + cos ky a) − 4t1 cos kx a cos ky a − µ
(2.27)
We see that the states created by c†kσ and destroyed by ckσ are the eigenstates
of H0 with energy eigenvalues ε̃k . We can imagine this as adding or removing
the plane wave electrons from the energy band described by the dispersion
relation ε̃k and depicted in Fig. 2.2. We have set the parameters of our
model band to t = 5 and t1 = −1.5 in the units of parameter ∆, which was
introduced in (1.25), which we chose as our unit of energy. The proper value
of µ can be determined numerically. We know the in-plane electronic density
n has the value between 0.8 and 0.9 in the cuprates. Because of the two
possible values of spin, it turns out that 40−45% of the points in the Brillouin
zone are occupied. We start occupying k-space from the lowest energy up and
find at what energy we have to stop to have the right number of electrons.
By such procedure we have obtained the value µ ≈ −5. Had we used this
value, we would have to separate the effects of superconductivity around
the Fermi surface (ε̃k = 0) from the effects of the van Hove singularities at
ε̃k = −1 for µ = −5. Since the non-interacting density of states forms just a
background to the superconducting effects, it makes sense to move away to
µ = 5 (as did Berthod in [11]) or even to µ = 10, which we used for most of
our calculations. Here the van Hove singularities lie safely below the Fermi
surface. The density of states for our model band is shown in Fig. 2.3.
2.3
Noninteracting Green’s functions
We will now find the explicit formula for the non-interacting Green’s functions G0 :
G0 (i, j, t) ≡
−i
h{ciσ (t), c†jσ }iΘ(t)
~
(2.28)
CHAPTER 2. CALCULATION OF DOS
Figure 2.2:
36
The electronic dispersion relation ε̃k = −2t(cos(kx a) + cos(ky a)) −
4t1 cos(kx a) cos(ky a) − µ with t = 5, t1 = −1.5, and µ = 10. Plotted contours are
placed at ε̃k equal to the multiples of 2 in units of ∆. The Fermi surface corresponds
to ε̃k = 0. The van Hove singularities in the density of states occur when the dispersion
relation has the saddle points, that is in our case at ε̃k = −16. The minimal ε̃k is -24 at
the centre of the Brillouin zone (point [0, 0]). The maximum occurs at the corners of the
Brillouin zone [kx , ky ] = [± πa , ± πa ] and its value is 16.
CHAPTER 2. CALCULATION OF DOS
37
Figure 2.3: The density of states for the non-interacting case, computed for the dispersion relation (2.27) with parameters set to t = 5, t1 = −1.5, and µ = 10. Size of lattice
was N = 1024 and Γ = 0.02. The zero in the plot corresponds to the value of chemical
potential µ, the van Hove singularity cusp is at ω = −16. The density of states is non-zero
in the range from ω = −24 to ω = 16.
where the time evolution of the system is given by the Hamiltonian Ĥ0 :
X
Ĥ0 =
ε̃k c†kσ ckσ
(2.29)
k
We would like to find an expression for G0 , which is expressed in terms of
real space creation and annihilation operators, but Ĥ0 is simple in k-space.
Let us work for a while in momentum space, since it is an easier option. The
ckσ (t) is connected with ckσ by the Heisenberg relation:
ckσ (t) = e
iH0 t
~
ckσ e
−iH0 t
~
(2.30)
By differentiating the previous relation with respect to the time we get:
i~
iH0 t
−iH0 t
∂
ckσ (t) = e ~ [ckσ , H0 ]e ~
∂t
(2.31)
The calculation of the commutator [ckσ , Ĥ0 ] is straightforward and leads to
the result [ckσ , Ĥ0 ] = ε̃k ckσ . Hence we have
i~
∂
ckσ (t) = ε̃k ckσ (t)
∂t
(2.32)
CHAPTER 2. CALCULATION OF DOS
38
and
ckσ (t) = e−iε̃k t/~ ckσ
(2.33)
We put this back in (2.28):
oE
−i Dn
Θ(t)
ciσ (t), c†jσ
~
oE
i X ik·Ri −ik0 ·Rj Dn
†
G0 (i, j, t) = −
e
c
(t),
c
Θ(t)
kσ
k0 σ
~N 2 k,k0
G0 (i, j, t) =
(2.34)
(2.35)
n
o
We now use (2.33) and the basic anticommutator ckσ , c†k0 σ = δk,k0 to arrive
to
G0 (i, j, t) =
−i X ik·(Ri −Rj ) −iε̃k t/~
e
e
Θ(t)
~N 2 k
(2.36)
We wish to compute G0 (i, j, ω), the time Fourier transform of the above
expression:
0
Z
+∞
dtei(ω+iΓ)t G0 (i, j, t) =
−∞
Z
−i X ik·(Ri −Rj ) ∞ i((ω+iΓ)−ε̃k /~)t
=
e
e
dt
~N 2 k
0
G (i, j, ω) =
(2.37)
(2.38)
To make this integral converge we have replaced again ω by ω + iΓ, Γ is a
small positive real number:
∞
−1 X ik·(Ri −Rj ) ei(ω+iΓ−ε̃k /~)t
1 X eik·(Ri −Rj )
G (i, j, ω) = 2
e
= 2
N k
~(ω + iΓ) − ε̃k 0
N k ~(ω + iΓ) − ε̃k
(2.39)
0
The last expression shall be compared with the definition of Fourier transformed G0 (k, ω)
G0 (i, j, ω) ≡
1 X ik·(Ri −Rj ) 0
e
G (k, ω)
N2 k
(2.40)
By comparing two last relations we see that the expression for G0 (k, ω) is
extraordinarily simple:
G0 (k, ω) =
1
~(ω + iΓ) − ε̃k
(2.41)
CHAPTER 2. CALCULATION OF DOS
2.4
39
Equations for the Green’s functions
The local density of states is calculated using the Green’s function formalism
A(i, ω) = −
1
Im G(i, i, ω + iΓ)
π
(2.42)
That means that we compute the density of states in the point Ri (thus the
adjective local) from the imaginary part of the diagonal of Green’s matrix
G(i, i, ω) at the the same point i at the frequency ω (equivalently energy
E) to which we add a small imaginary part iΓ. This small imaginary part
ensures the analytical convergence and smooths the DOS which would be a
set of δ-functions at possible energies instead of continuous function.
We consider the following lattice Hamiltonian:
Ĥ = −
X
ij
tij c†iσ cjσ −
1X
1X ∗
∆ij (c†i↑ c†j↓ − c†i↓ c†j↑ ) −
∆ (cj↓ ci↑ − cj↑ ci↓ )
2 ij
2 ij ij
(2.43)
where the indices i and j number the lattice sites. The first part of (2.43) is
the non-interacting Hamiltonian
Ĥ0 = −
X
tij c†iσ cjσ
(2.44)
ij
which describes the hopping of electrons between the sites of the lattice
(the atoms) and corresponds to the kinetic energy in the language of second
quantization, that we are using here. We have discussed it thoroughly in the
previous section. The second and third part of Ĥ correspond to the superconducting material, where it is possible to create and destroy the singlet
electron pairs (the Cooper pairs). Since the last two terms are hermitian
conjugated, the Hamiltonian is hermitian as it should be. Next we recall
the definition of the retarded Green’s function (the ordinary propagator)
Gσ (i, j, t)
oE
−i Dn
ciσ (t), c†jσ
θ(t)
(2.45)
~
We first attempt to calculate the time evolution of Gσ (i, j, t). Starting
Gσ (i, j, t) ≡
from its definition, we can apply the partial derivative with respect to time
CHAPTER 2. CALCULATION OF DOS
40
and we obtain:
oE
oE
∂Gσ (i, j, t)
−i Dn
i Dn
=
ciσ , c†jσ
δ(t) −
ċiσ (t), c†jσ
θ(t)
∂t
~
~
(2.46)
Now the first term is easy to evaluate, n
because owe deal with the equal time
anticommutator which is known to be ciσ , c†jσ = δij , for the second term
we need to exploit further the equation (2.3), namely to partially differentiate
it with respect to time. The result is
i~
−iHt
iHt
∂ciσ (t)
= e ~ [ciσ , H]e ~
∂t
(2.47)
In order to proceed we need to know the commutator of the electron annihilation operator with the Hamiltonian of the system (2.43). In preparation
to this it is comfortable first to compute only the commutator with the typical terms of the Hamiltonian. Furthermore we fix σ =↑. This is possible
without loss of generality, because the spins occur in the Hamiltonian in a
symmetric way and thus our choice would not affect the results. The only
nonzero commutators are these:
[c†i↑ cj↑ , ck↑ ] = −δik cj↓
(2.48)
[c†i↑ c†j↓ , ck↑ ] = −δik c†j↓
(2.49)
[c†i↓ c†j↑ , ck↑ ] = δjk c†i↓
(2.50)
Putting the results together:
[Ĥ, ck↑ ] =
X
ij
tij δik cj↑ +
1X
∆ij (δik c†j↓ + δjk c†i↓ )
2 ij
(2.51)
which we use to get ċk↑ needed to express the time evolution of G↑ (k, l, t)
(compare (2.46)):
Dn
oE
∂G↑ (k, l, t)
iX
†
i~
= δkl δ(t) +
tij δik
cj↑ (t), cl↑ θ(t)+
∂t
~ ij
Dn
oE
Dn
oE
i X
†
†
†
†
+
∆ij δik
cj↓ (t), cl↑ θ(t) + δjk
ci↓ (t), cl↑ θ(t)
2~ ij
(2.52)
CHAPTER 2. CALCULATION OF DOS
41
We observe that another type of Green’s function occurs here, this time
with both operators being creation ones, leading us to define the so-called
anomalous propagator:
F↑ (i, j, t) ≡
oE
−i Dn †
ci↓ (t), c†j↑ θ(t)
~
(2.53)
Using this new definition and the assumption that ∆ik = ∆ki we get:
i~
X
X
∂G↑ (k, l, t)
= δkl δ(t) −
tkj G↑ (j, l, t) −
∆kj F↑ (j, l, t)
∂t
j
j
(2.54)
It seems at first that we haven’t proceeded much further since we now know
the time evolution of G but we have expressed it as a function of another
unknown function F . The good news is that using a very similar procedure
we can obtain an equation for the time evolution of F which in turn will be
coupled to G. Then we will have to solve them simultaneously for G and
F . But let us first derive the equation for F . Partially differentiating (2.53)
with respect to time we get:
∂F↑ (k, l, t)
−i Dn † † oE
i Dn † † oE
=
ck↓ , cl↑ δ(t) −
ċk↓ , cl↑ θ(t)
∂t
~
~
(2.55)
The first term is zero because it’s the equal time anticommutator of two
creation operators. To evaluate the second term, we express the operator c†k↓
in the Heisenberg picture:
c†k↓ (t) = e
iHt
~
c†k↓ e
−iHt
~
(2.56)
For its time derivative it follows
∂c†k↓ (t)
−iHt
iHt
−i~
= e ~ [Ĥ, c†k↓ ]e ~
∂t
(2.57)
We see that we need the commutator [Ĥ, c†k↓ ] for which we use in complete
analogy to the previous case:
[c†i↓ cj↑ , c†k↓ ] = δjk c†i↓
(2.58)
[cj↓ ci↑ , c†k↓ ] = −δjk ci↑
(2.59)
[cj↑ ci↓ , c†k↓ ] = δik cj↑
(2.60)
CHAPTER 2. CALCULATION OF DOS
42
Thus the commutator of electronic creation operator with our Hamiltonian
is
1X ∗
∆ [δjk ci↑ + δik cj↑ ] =
2 ij ij
ij
X
1X ∗
1X ∗
=−
tik c†i↓ +
∆ik ci↑ +
∆kj cj↑
2 i
2 j
i
[Ĥ, c†k↓ ] = −
X
tij δjk c†i↓ +
(2.61)
(2.62)
Now we can express the time derivative of the Green’s function F↑ (k, l, t)
using the equation (2.55):
i~
X
∂F↑ (k, l, t) X
=
tki F↑ (i, l, t) −
∆∗ik G↑ (i, l, t)
∂t
i
i
(2.63)
We thus arrived to the second of the set of two coupled equations: (2.54)
and (2.63). Just for convenience let us rewrite the equation (2.54) here so
that we see them together:
i~
X
X
∂G↑ (k, l, t)
= δkl δ(t) −
tkj G↑ (j, l, t) −
∆kj F↑ (j, l, t)
∂t
j
j
(2.64)
We can transform these two differential equations into the algebraic ones if
we define the Fourier transform from the time domain into the frequency
domain:
Z
G(i, j, ω) = dtei(ω+iΓ)t G(i, j, t)
Z
dω −i(ω+iΓ)t
G(i, j, t) =
e
G(i, j, ω)
2π
(2.65)
(2.66)
We have used again a slightly imaginary frequency ω + iΓ to avoid problems
with convergence and at the same time to preserve causality. This allows us
to replace the time derivative i~ ∂G(t)
by ~(ω + iΓ)G(ω) and we get
∂t
[~(ω + iΓ)1 + t]G(ω) + ∆F (ω) = 1
(2.67)
[~(ω + iΓ)1 − t]F (ω) + ∆∗ G(ω) = 0
(2.68)
This system of equations is commonly called Gor’kov equations (compare
e.g. Appendix B in [14]).
CHAPTER 2. CALCULATION OF DOS
2.5
43
Solving the Gor’kov equations
Let us define two auxiliary matrices A and B
A−1 ≡ [~(ω + iΓ)1 + t]
(2.69)
B −1 ≡ [~(ω + iΓ)1 − t]
(2.70)
This allows us to rewrite the Gor’kov equations to the form
A−1 G(ω) + ∆F (ω) = 1
(2.71)
B −1 F (ω) + ∆∗ G(ω) = 0
(2.72)
From the second Gor’kov equation we extract the expression for F (ω):
F (ω) = −B · ∆∗ · G(ω)
(2.73)
We insert this into the first equation:
A−1 · G(ω) − ∆ · B · ∆∗ · G(ω) = 1
(2.74)
We multiply by A and this leads us to:
[1 − A · ∆ · B · ∆∗ ]G(ω) = A
(2.75)
Finally we have found out how to compute G(ω):
G(ω) = [1 − A · ∆ · B · ∆∗ ]−1 · A
(2.76)
We see that we will need to perform 4 matrix multiplications and 1 matrix
inversion for every frequency ω. The computational complexity of each of
these operations is N 3 summations and multiplications. The matrices A and
B will be obtained by means of Fourier transforms which scale as N log N .
We can thus expect the matrix operations to be the bottleneck of the computation. From the point of view of the storage space needed in the memory
it is again the matrix operations which are worse. We have to store all the
matrices on which we do the operations. This poses the most important
constraint on the system sizes which we are able to explore.
CHAPTER 2. CALCULATION OF DOS
44
There is one thing which we still owe to the reader and that is the calculation of matrices A and B. To this end let us again look at first Gor’kov
equation:
A−1 G(ω) + ∆F (ω) = 1
(2.77)
The equation will simplify considerably, if we take ∆ ≡ 0, which physically corresponds to the normal, non-superconducting state. To make visible
whether we are talking about the general Green’s function G(ω), or about
the one in the normal state, we add a superscript 0 to the latter. Then we
have A−1 G0 (ω) = 1 and thus
A = G0 (i, j, ω)
(2.78)
But we have already found an easy way to compute the noninteracting
Green’s function G0 (i, j, ω) as a single Fourier transform (FT) of G0 (k, ω),
see Eq. (2.41). Thus
A = FT
1
~(ω + iΓ) − ε̃k
Similar relation is used to compute B
1
B = FT
~(ω + iΓ) + ε̃k
(2.79)
(2.80)
This holds true since the only difference between A and B was in the sign
right in front of the kinetic energy matrix element t and the only effect of this
different sign is again just a different sign right in front of the corresponding
k-space kinetic energy term ε̃k .
2.6
Translationally invariant case
We now turn our attention to the translationally invariant case when the
Hamilton operator density is independent of position in space, i.e. same for
every lattice point. That means that all tij and ∆ij can be only functions of
vector connecting the points i and j and are independent of the individual
CHAPTER 2. CALCULATION OF DOS
45
positions of the points i and j in the lattice.
When considering this case we have to use the periodic boundary conditions, because otherwise we would have some points on the border and some
in the middle of the sample and since then they can be distinguished, the
translational invariance would be lost.
Why do we devote this whole section to the translationally invariant superconducting system? First of all, the simple d-wave and s-wave superconductors are translationally invariant in the zero applied magnetic field, so this
describes real physical systems. Next, because of translational invariance the
full solution is easily obtainable by means of a single Fourier transform. This
allows comparison of our numerical results to these special cases and thus
provides us with some tests to check correctness and accuracy of our program.
Apart from this, we also use the results from the non-interacting case, which
is translationally invariant by definition, to obtain the interacting Green’s
function by some matrix manipulations even in the most complicated state
we have considered: a superconductor with the vortex.
The general equations coupling the Green’s functions G(ω) and F (ω) were
[~(ω + iΓ)1 + t]G(ω) + ∆F (ω) = 1
(2.81)
[~(ω + iΓ)1 − t]F (ω) + ∆∗ G(ω) = 0
(2.82)
or when we write sums and indices explicitly:
~(ω + iΓ)G(k, l, ω) = δkl −
X
tkj G(j, l, ω) −
j
~(ω + iΓ)F (k, l, ω) =
X
j
tkj F (j, l, ω) −
X
∆kj F (j, l, ω)
(2.83)
j
X
∆∗jk G(j, l, ω)
(2.84)
j
We now define the Fourier expansions from the real space to k-space for
the functions G(ω) and F (ω) as well as for the Dirac δ-function, hopping
amplitudes tij and the superconducting order parameter ∆ij . The N 2 is the
total number of the lattice points. The forward transformation for Green’s
CHAPTER 2. CALCULATION OF DOS
46
functions is
1 X
Gp eip·(Ri −Rj )
N2 p
1 X
F (i, j, ω) = 2
Fp eip·(Ri −Rj )
N p
G(i, j, ω) =
(2.85)
(2.86)
These functions explicitly depend only on the relative position vector R =
Ri − Rj . The corresponding backward transformation or the equation for
the Fourier components Gp and Fp is then:
X
Gp =
G(i, j, ω)e−ip·R
(2.87)
R
Fp =
X
F (i, j, ω)e−ip·R
(2.88)
R
The Kronecker δ transforms as usually:
1 X ip·(Rk −Rl )
e
δkl = 2
N p
(2.89)
The Fourier transform for ∆ij is:
1 X
∆k eik·(Ri −Rj )
N2 k
X
∆k =
∆ij e−ik·R
∆ij =
(2.90)
(2.91)
R
The transform of tij we define with the additional minus sign3 :
1 X
tij = − 2
εk eik·(Ri −Rj )
N k
X
εk = −
tij e−ik·R
(2.92)
(2.93)
R
Now we introduce these expressions to the equation (2.83) and we get:
X
X
1 X
~(ω + iΓ)
Gp eip·(Rk −Rl ) =
eip·(Rk −Rl ) + 2
εk Gp eik·Rk e−ip·Rl ei(p−k)·Rj
N
p
p
jkp
(2.94)
3
This minus sign comes from the fact that the non-interacting Hamiltonian is conP
ventionally written as Ĥ0 = − ij tij c†iσ cjσ in the real space representation but as
P
Ĥ0 = k εk c†kσ ckσ in k-space representation.
CHAPTER 2. CALCULATION OF DOS
47
In the last exponential summed over j we recognize the Kronecker δ. Using
it, the sum over k is performed easily and we arrive to:
~(ω + iΓ)Gp = 1 + εp Gp − ∆p Fp
(2.95)
Analogously we proceed with equation (2.84) to obtain:
~(ω + iΓ)Fp = −εp Fp − ∆∗p Gp
(2.96)
The equations (2.95) and (2.96) form a simple system of 2 linear equations.
To see it more clearly we shall write it in the matrix form:
!
!
~(ω + iΓ) − εp
∆p
Gp
=
~(ω + iΓ) + εp
∆∗p
Fp
1
!
0
We can immediately write the solution:
~(ω + iΓ) + εp
~2 (ω + iΓ)2 − ε2p − |∆p |2
−∆∗p
Fp = 2
~ (ω + iΓ)2 − ε2p − |∆p |2
Gp =
(2.97)
(2.98)
Now we are fully able to compute the Green’s function G(i, j, ω) and thus
the density of states for any translationally invariant superconducting system.
We do it just by using the above solution for our given dispersion relation
εk and superconducting order parameter ∆ and returning by the Fourier
transform (Eq. 2.85) to the real space. The most important examples of the
cases which we are able to compute are the extended s-wave case and the
d-wave case shown in Fig. 2.4.
2.7
Implementation
The calculation was implemented in the C programming language. Its
main advantages are its speed and the good availability of numerical libraries.
CHAPTER 2. CALCULATION OF DOS
48
Figure 2.4: The density of states for the extended s-wave (top) and d-wave superconductor (bottom). Computed for the square lattice of size 1024 × 1024 and the electronic
energy dispersion relation (2.27) with t = 5, t1 = −1.5, µ = 10 and Γ = 0.05.
CHAPTER 2. CALCULATION OF DOS
49
We used the FFTW library [18] for the Fourier transform. Specifically
it was version 3.1.2, which was current at the time of writing the code. The
origin of the name FFTW is ’the Fastest Fourier Transform in the West’. It is
a C subroutine library for computing the discrete Fourier transform of real or
complex data in one or more dimensions. It also provides its own implementation of data type complex. We have used the function fftw plan dft 2d(),
which creates an optimized plan for doing a Fourier transform and the actual
call for Fourier transform, the method fftw execute(), which executes this
plan. For details on use, see the library documentation available at [18].
The matrix operations are done by the CLapack [19], which is a C clone
of the popular Fortran LAPACK library. The name LAPACK is a shorthand
for Linear Algebra PACKage. As the name suggests it provides basically all
the linear algebra operations on real and complex vectors and matrices. The
user may even choose between single and double precision. The LAPACK
relies on using the optimized basic subroutines, such as the multiplication of
a vector by scalar etc. These are usually provided by the BLAS, which is a de
facto standard interface for such collections of common algebraic functions.
BLAS means Basic Linear Algebra Subprograms. Several implementations
of BLAS are available. We used the ATLAS [20], the Automatically Tuned
Linear Algebra Software. It is available precompiled for some of the common
computer architectures, however it is recommended to compile it from the
source, so that it can reach its optimum performance.
The matrix operations needed were the matrix-matrix multiplication, that
is the function cblas zgemm() and the matrix inversion, which is provided
by the successive use of clapack zgetrf() and clapack zgetri().
Together these libraries proved to be very fast, memory efficient and reliable. Still the needed matrix calculations scale very badly with the lattice
size, so they limited the system sizes which we were able to compute. We
needed bigger lattices, because then the calculation is more precise, see Fig.
CHAPTER 2. CALCULATION OF DOS
50
Figure 2.5: The imaginary part of the non-interacting Green’s function G0 (i, i, ω). We
see the convergence towards the exact values as a function of lattice size. Apparent is the
need to use the lattice size at least of the order of 100. Computed for the electronic energy
dispersion relation (2.27) with t = 5, t1 = −1.5, µ = 10 and Γ = 0.5, ω = −1.
2.5. The reason for this is that with bigger lattice we have more points in kspace4 , so their energies are more closely spaced. The spectrum is more and
more dense and thus smaller values of the parameter Γ are needed to make
the density of states - originally a sum of Dirac δ-functions - continuous.
However, at some lattice size we necessarily run out of computer memory.
This is typically not more than roughly a 100 × 100 lattice, if we have 8 GB
memory at our disposal. So we need to use a finite value of Γ, but too big Γ
is bad, because we unnecessarily blur the details of the density of states. To
gain some insight into this issue we included Fig. 2.6, where we studied the
effects of changing Γ or N on a simple homogeneous d-wave case.
Fortunately there is a way how to optimize our efforts - described in the
article [11] by Berthod. Since we can easily compute the Fourier transform
4
The number of possible states does not depend on the representation and thus is equal
to N 2 as in the real space lattice.
CHAPTER 2. CALCULATION OF DOS
51
Figure 2.6: The d-wave DOS for different lattice sizes and values of Γ. a) N = 1024,
optimal Γ ≈ 0.02, b) N = 92, optimal Γ ≈ 0.25. Computed for the electronic energy
dispersion relation (2.27) with t = 5, t1 = −1.5, µ = 10.
CHAPTER 2. CALCULATION OF DOS
52
Figure 2.7: The illustration shows how is the G0 computed for a larger lattice adapted
to the smaller one by cutting the corner squares and glueing them together.
and thus also the non-interacting Green’s function G0 (i, j, ω), we do this on
a much bigger lattice. Due to the specifics of the Fast Fourier Transform
algorithm it works most effectively with arrays of size 2N , where N ∈ N.
Hence we compute the non-interacting Green’s function, e.g., on a lattice
1024 × 1024 where the finite size effects are already negligible. We may use
a relatively small Γ too, because the energy spacing is small. Now we have to
take the corresponding parts of this Green’s function matrix and build a new
matrix appropriate for a smaller lattice on which we finish the interacting
calculation, see Fig. 2.7.
During the writing and debugging of the code of the program we used
various methods to ensure that the produced results are correct. First it
was necessary to test the libraries if they work correctly in a way that we
expected them to work. We tested the matrix inversion and multiplication
first on a simple small matrices, where it was possible to do an independent
calculation on a paper. Then we tested if the functions work also on the
huge matrices, e.g., how much the result of multiplying a given matrix by its
inverse differs from the unit matrix. Interesting information was also how
much time does such calculation take.
CHAPTER 2. CALCULATION OF DOS
53
The whole program was tested against the independent calculation for
the translationally invariant case of the homogeneous extended-s wave and
d-wave superconductor. Here it is important to implement correctly the
periodic boundary conditions. These have not been used later for the vortex
calculations. There we have used the open boundaries instead and we have
put away the densities of states near the ends of the lattice.
We further tested the program by introducing a global constant phase
to the ∆, which should not change the results. At this point we run first
inhomogeneous calculations with a nontrivial phase field corresponding to a
single vortex, which were compared with the results published in [11]. The
densities of states away from the vortex core were similar to the usual d-wave
DOS. The calculations were then done on a largest possible lattice 96 × 96
without changing its size. All parameters except for the lattice size kept, we
repeated the calculation using G0 from the 1024 × 1024 lattice and finishing
on a lattice 64 × 64. The results turn out to be almost the same (Γ has to
be big enough for this, we used Γ = 0.5).
Finally we include here the Table 2.1 showing the program structure.
CHAPTER 2. CALCULATION OF DOS
54
1. Include libraries, initialize global variables
2. Functions
epsk - the electronic energy dispersion
delta - the amplitude of the order parameter
theta - the phase of the order parameter
3. Method cutCorners - cuts out the appropriate parts of G0 and glues
them together
4. The main: Open the phase input file, read out the phases, allocate
space for the matrices, construct ∆ matrix
5. for ω ∈ hmin, maxi do {
G0k = 1/(ω − εk + I ∗ Γ)
Fourier transform, cutCorners → G0plus
G0k = 1/(ω + εk + I ∗ Γ)
Fourier transform, cutCorners → G0minus
Go from G0plus and G0minus of the form (x, y)
(one value for every lattice point)
to the Gplus, Gminus of the form ((x1 , y1 ), (x2 , y2 ))
(one value for every possible pair of lattice points)
Evaluate the Eq. (2.76) - 4 multiplications and 1 matrix inversion
Extract the DOS from the Green’s matrix
Save computed data
}
6. Free memory
Table 2.1: Program structure
Chapter 3
Results and discussion
In this place we wish to summarize the applications of the method described
in the previous chapter. These are all some special cases of the general gap
function ∆ij , introduced in (2.90) as a Fourier transform of ∆k . Since ∆ij is
a complex function, we shall write it as
∆ij = |∆ij | · exp iθij exp iχij ,
(3.1)
where |∆ij | is the amplitude and the rest is the phase factor, which we have
divided into two parts. In θij the slow changes of phase connected with the
presence of vortex are incorporated. In contrast, the underlying d-wave gap
symmetry is obtained by setting the phase χij equal to zero on horizontal and
to π on vertical bonds. This fast d-wave modulation was used in all the following calculations. Furthermore we kept the gap amplitude |∆ij | constant,
|∆ij | ≡ ∆, thus all the modifications were done only to the phase θij . There
are two supporting arguments for neglecting the changes in |∆ij |. First, the
creation of Cooper pairs probably occurs at larger energy scale (in the pseudogap phase) and thus the changes in amplitude of ∆ij should be negligible
here, because what we do is essentially a zero-temperature calculation. Second, Berthod’s results [11] have convinced us that it is almost completely the
phase singularity that is responsible for the LDOS in the vortex core and the
vanishing of |∆ij | is not necessary.
55
CHAPTER 3. RESULTS AND DISCUSSION
56
Figure 3.1: The density of states in the vortex core for singly and doubly quantized vortices. Computed for the square lattice of size 51 × 51 and the electronic energy dispersion
relation (2.27) with t = 5, t1 = −1.5, µ = 5, Γ = 0.02 and G0 from lattice 1024 × 1024.
3.1
Usual vortex
The first non-trivial system we chose to study was a simple vortex with one or
two flux quanta. We were able to reproduce the results of Berthod from [11].
Our results are shown in Fig. 3.1. For comparison, the results published in
[11] are reprinted in Fig. 3.2. The behavior of LDOS as a function of distance
from the vortex core is presented in Fig. 3.3. One intention behind redoing
the work of Berthod was to test the correctness of our program. Since the
results do not differ significantly, we hope that it is correct in all important
aspects.
There are however some differences between our method and that of
Berthod: we use the phases θij computed from London’s theory as described
in [13]1 and we are able also to run the full inhomogeneous computation for
1
They are approximately the same as the angle measured from the positive x axis
counterclockwise, but have have been adjusted iteratively so that they do not violate the
continuity equation ∇ · j = 0 in the discrete lattice case.
CHAPTER 3. RESULTS AND DISCUSSION
57
Figure 3.2: The vortex core DOS - from Berthod in [11]. Computed for the square lattice
of size 51 × 51 and the electronic energy dispersion relation (2.27) with t = 5, t1 = 0,
µ = 5 and Γ = 0.02. The ν = −1 corresponds in our notation to the singly quantized
vortex and ν = −2 to the doubly quantized vortex.
somewhat bigger lattices (92 × 92 vs. 51 × 51).
3.2
Split vortex I
We next moved on to compute the density of states for a d-wave superconductor with a system of two half-vortices described in [13]. This split solution
minimizes the Josephson energy for a particular modification of the normal
Josephson current relation.2 We used the original phase field with two singularities - the half-vortices - around each the phase field grows by π. The
total phase was given by a sum of these two contributions. In the case of the
usual vortex the only discontinuity in phase was the jump by 2π located at
the positive x-axis. This was actually equivalent to no jump at all, because of
the periodicity of the goniometric functions. In contrast for two half-vortices
there is a real discontinuity, a jump in phase by π on the line connecting the
2
This is explained more later in the Section Split vortex II and in the original article.
CHAPTER 3. RESULTS AND DISCUSSION
58
Figure 3.3: The DOS for a simple vortex in a d-wave superconductor. The density is
depicted in growing distances from the centre of the vortex. We see the zero-bias-peak
right at the vortex core, which changes to the bulk DOS of a simple d-wave superconductor
with increased distance. Computed for the square lattice of size 64 × 64 and the electronic
energy dispersion relation (2.27) with t = 5, t1 = −1.5, µ = 10 and Γ = 0.5.
CHAPTER 3. RESULTS AND DISCUSSION
59
Figure 3.4: The zero bias DOS for a split vortex, the result was obtained using the
standard ∆ij (3.1). Computed for the square lattice of size 96 × 96 and the electronic
energy dispersion relation (2.27) with t = 5, t1 = −1.5, µ = 10 and Γ = 0.5.
two half-vortices. We were interested in the behaviour of the local density
of states in the cores of half-vortices, as well as on the line joining them.
Computation was done with the usual relation for ∆ij (3.1). The results are
shown in the Figures 3.4 and 3.5. The density of states shows a zero-bias
peak on the phase-jump line and also in the cores.
3.3
The cut
One new question arose from the results of Section Split vortex I. Is it the
phase jump by π that causes a zero-bias peak? To find the answer we have
studied the lattice with a ’cut’ on a diagonal where we have introduced
the phase shift by π: under the diagonal, the horizontal links have phase 0
and the vertical ones have phase π whereas above the diagonal the phases
are exchanged. The resulting spatial distribution of the zero-bias density of
states is in Fig. 3.6. For the standard choice (3.1) the densities of states
right at the cut and away from it look like in Fig. 3.7. The local densities
CHAPTER 3. RESULTS AND DISCUSSION
60
-30
min
-40
-50
max
-60
-60
-50
-40
-30
Figure 3.5: The zero bias DOS for a split vortex, the result was obtained using the
standard ∆ij (3.1). The plot shows color-coded map of the same data set as in Fig. 3.4.
of states on the cut are very similar to those at the vortex core. The bulk
densities of states are almost identical to each other and to the normal dwave DOS. From this fact we conclude that the zero bias peak observed in
the LDOS on the line connecting the half-vortices is a result of the presence
of the phase cut line. Let us consider what would happen if we moved the
half-vortices nearer and nearer. The likely answer is that finally they would
be indistinguishable from the usual vortex. Thus we concluded that the zero
bias peak in a normal core results probably from the same mechanism as
described above, but the phase cut by π is present only in the single point,
right at the vortex core.
3.4
Split vortex II
Seeing the results presented in the previous paragraphs, we were led to the
speculation of using a different prescription for ∆ij , which would be consistent
with the idea of two Josephson harmonics present in the current equation in
CHAPTER 3. RESULTS AND DISCUSSION
61
-20
min
-30
-40
max
-50
-50
-40
-30
-20
Figure 3.6: The map of the zero-bias density of states for a cut in the d-wave superconductor. The insets show the phases connecting the lattice points. Computed for the
square lattice of size 64 × 64 and the electronic energy dispersion relation (2.27) with
t = 5, t1 = −1.5, µ = 10 and Γ = 0.5 at E = 0.
[13]. Namely we considered
J1
J2
∆ij = ∆
exp(iθij ) +
exp(2iθij ) exp iχij ,
J1 + J2
J1 + J2
(3.2)
because the split vortex solution was originaly found for the Hamiltonian
P
H = hiji e(θi − θj ) defined as the sum of the Josephson energies,
e(θ) = −J1 cos θ − J2 cos 2θ
(3.3)
Doing the calculation with the parameters equivalent to those used in [13],
we have shown that in a superconductor with such mixed behaviour it should
be possible to observe a split vortex. The zero bias local density of states
together with its map is in Figures 3.8 and 3.9. The half-vortices appear
split.
CHAPTER 3. RESULTS AND DISCUSSION
62
Figure 3.7: The LDOS for a cut in the d-wave superconductor. The first line is the
LDOS right on the cut, the second and third one show the density of states far from the
cut, on the sites symmetrical with respect to the diagonal on which is the cut located.
We see that the two choices for phases are absolutely equivalent for the latter two in that
the both graphs lie almost exactly on top of each other. Nevertheless right at the cut the
situation is different and we observe the zero bias peak. Computed for the square lattice
of size 64 × 64 and the electronic energy dispersion relation (2.27) with t = 5, t1 = −1.5,
µ = 10 and Γ = 0.5.
CHAPTER 3. RESULTS AND DISCUSSION
63
Figure 3.8: The zero bias DOS for a split vortex, the result was obtained using the
mixed ∆ij with 17% normal content. Computed for the square lattice of size 64 × 64 with
G0 from 1024 × 1024 calculation and the electronic energy dispersion relation (2.27) with
t = 5, t1 = −1.5, µ = 10 and Γ = 0.5.
-20
min
-30
-40
max
-50
-50
-40
-30
-20
Figure 3.9: The zero bias DOS for a split vortex, the result was obtained using the
mixed ∆ij with 17% normal content. The plot shows color-coded map of the same data
set as in Fig. 3.8.
Chapter 4
Conclusion
We numerically solved the Gor’kov equations for the inhomogeneous superconductor. We studied various possible configurations of the superconducting
phase. First we reproduced the results of Berthod from [11] on singly and
doubly quantized vortices. Then we have studied the lattice with a cut on a
diagonal where we introduced the phase shift by π. From this we have gained
some understanding of the origin of the zero-bias-peak at the centre of the
usual vortex core. Finally we have shown that in a superconductor with a
mixed behaviour, that is when the Josephson energy has a non-negligible
second harmonics, it should be possible to observe a split vortex in the STM
measurements.
64
Zhrnutie (SK)
V práci sme počı́tali hustotu stavov A(i, ω) pre modelový vysokoteplotný
supravodič pomocou formalizmu Greenových funkciı́ G(ω). Tieto dve veličiny
spája vzťah A(i, ω) ≡ − π1 Im G(i, i, ω). Na výpočet Greenovej funkcie G(ω)
sme použili Gorkovove rovnice, z ktorých pre ňu vyplýva vzťah (2.76). Na
výpočet podľa neho sme použili numerické knižnice: FFTW na Fourierovu
transformáciu a knižnice LAPACK a ATLAS na maticovú algebru - násobenie
matı́c a hľadanie inverznej matice.
Vykonali sme kontrolné výpočty pre dva známe translačne invariantné
systémy, rozšı́rený s-vlnový a d-vlnový supravodič. Jeden raz pomocou nášho
programu a druhý raz pomocou kontrolného programu, ktorý využı́va pri
výpočte translačnú invariantnosť, aby bolo možné overiť funkčnosť Fourierovej transformácie a maticových operáciı́. Overili sme tiež, že výsledok je
rovnaký bez ohľadu na zmenenú globálnu fázu.
Po dôkladnom testovanı́ sme pristúpili k samotným výpočtom hustoty
stavov v jadre vı́ru vo vysokoteplotných supravodičoch. Zreprodukovali sme
výsledky z článku [11], ktoré sa týkajú vı́rov s jedným a dvoma kvantami
magnetického toku. Čo však nebolo dosiaľ známe, bolo ako by vyzerali
výsledky STM experimentu merajúceho hustotu stavov v prı́pade, že by pole
fáz zodpovedalo vı́ru rozštiepenému na dva polvı́ry. V zodpovedajúcom rozloženı́ fáz je totiž prı́tomná netriviálna nespojitosť, skok o π pozdĺž úsečky
spájajúcej stredy polvı́rov. Výpočet nakoniec ukázal, že pozdĺž celej tejto
úsečky by mala byť pozorovateľná relatı́vne vysoká hustota stavov pri nulovej
energii (tzv. zero-bias peak). Pravdepodobnou prı́činou tejto anomálie hus-
65
CHAPTER 4. CONCLUSION
66
toty stavov je práve už spomı́naný skok fázy o π.
Či je práve toto dôvodom sme sa rozhodli otestovať v jednoduchšom modeli, kde sme mriežku rozdelili diagonálou na dve polovice, medzi ktorými je
jediný rozdiel a to otočené fázové pole o π. Takýto model naozaj vykazuje
takmer rovnaké hustoty stavov ako rozštiepený polvı́r a predstavuje vlastne
limitu nekonečne vzdialených polvı́rov. Samozrejme keby Josephsonov člen v
energii obsahoval nie člen úmerný cos θ, ale len členy typu cos 2θ, skok fázy o
π by nič neovplyvnil. V tejto súvislosti nás napadlo, že parameter usporiadania ∆ by teoreticky mohol v sebe obsahovať nielen klasickú Josephsonovskú
zložku, ale aj túto jej druhú harmonickú t.j. zložku s dvojnásobnou fázou,
ktorá rotáciu fázy o π nevnı́ma. S takýmto ansatzom sa vı́r naozaj ukazuje
vo výpočte rozštiepený.
Bibliography
[1] Annett, J. F.: Superconductivity, Superfluids and Condensates, Oxford
University Press, New York (2004)
[2] Hlubina, R.: Lectures on Superconductivity, accessed 16th April 2009
at http://www.dep.fmph.uniba.sk/~hlubina/STUD_MATER/lecture.
pdf
[3] Sigrist, M.:
Festkörperphysik II, lecture notes ETH Zürich (Win-
tersemester 2005/06)
[4] Tinkham, M.: Introduction to Superconductivity, 2d ed., McGraw-Hill,
Inc. (1996)
[5] Norman, M. R., Pépin, C.: The electronic nature of high temperature
cuprate superconductors, Rep. Prog. Phys. 66, 1547-1610 (2003)
[6] Takahashi, H., et al.: Superconductivity at 43 K in an iron-based layered
compound LaO1−x Fx FeAs, Nature 453, 376-378 (2008)
[7] Mohammadizadeh, M. R., Akhavan, M.: Two dimensionality aspects of
HTSC, IOP Supercond. Sci. Technol. 16, 12161223 (2003)
[8] Kumar, N., et al.: Normal state c-axis resistivity of the high-Tc cuprate
superconductors, arXiv:cond-mat/9704242v1 [cond-mat.supr-con]
[9] Blatter, G., et al.: Vortices in high-temperature superconductors, Rev.
Mod. Phys. 66, 1125 - 1388 (1994)
67
BIBLIOGRAPHY
68
[10] Ong, N. P.: Vorticity and phase diagram of cuprates (2006), powerpoint talk, accessed 27th April 2009 at http://www.princeton.edu/
~npo/index.html
[11] Berthod, C.: Vorticity and vortex-core states in type-II superconductors,
Phys. Rev. B 71, 134513 (2005)
[12] Hoogenboom, B. W., et al.: Shape and motion of vortex cores in
Bi2 Sr2 CaCu2 O8+δ , Phys. Rev. B 62, 9179 - 9185 (2000)
[13] Hlubina, R.: Possible vortex splitting in high temperature cuprate superconductors, Phys. Rev. B 77, 094503 (2008)
[14] Sigrist, M., Ueda, K.: Phenomenological theory of unconventional superconductivity, Rev. Mod. Phys. 63, 239 - 311 (1991)
[15] Damascelli, A., Hussain, Z., Shen, Z.: Angle-resolved photoemission
studies of the cuprate superconductors, Rev. Mod. Phys. 75, 473 - 541
(2003)
[16] Sokhatsky-Weierstrass theorem accessed 19th April 2009 at http://en.
wikipedia.org/wiki/Sokhatsky-Weierstrass_theorem
[17] Superconducting cuprates accessed 7th March 2009 at http://hoffman.
physics.harvard.edu/research/SCmaterials.php
[18] FFTW Home Page accessed 15th April 2009 at http://www.fftw.org/
[19] CLAPACK (f2c’ed version of LAPACK) accessed 15th April 2009 at
http://www.netlib.org/clapack/
[20] Automatically Tuned Linear Algebra Sofware (ATLAS) accessed 15th
April 2009 at http://math-atlas.sourceforge.net/

Download Report

Diploma thesis

Paperzz.com

Your Paperzz