1
EFFECT OF MAGNETIC FIELD ON THE THERMAL CONDUCTIVITY OF
SINGLE CRYSTAL OF YBa2Cu3O7-
BY
MBALAHA ZENDESHA
PG/M.Sc/07/42943
BEING
PROJECT REPORT SUBMITTED IN PARTIAL FULFILMENT OF THE
REQUIREMENTS FOR THE AWARD OF THE MASTER OF SCIENCE
DEGREE OF THE UNIVERSITY OF NIGERIA,
NSUKKA.
JULY, 2011
2
CERTIFICATION
Mbalaha Zendesha, a Post-graduate student with registration number
PG/M.Sc/07/42943 has satisfactorily completed the requirements of course and
research work for the degree of Master of Science (M.Sc) in Condensed Matter
Physics and Materials Science, in the Department of Physics and Astronomy,
University of Nigeria Nsukka.
The work embodied in this project report is original and has not been
submitted in full or part for any other diploma submitted in full or any other
University.
……………………… ……………………….. Prof. C.M.I Okoye Date Supervisor ……………………… ……………………… Dr. G.C. Asomba Date
Co-Supervisor
……………………… ……………………… External Examiner Date ……………………… ……………………… Prof. C.M.I Okoye Date Head Department of Physics and Astronomy University of Nigeria, Nsukka.
3
DEDICATION
I dedicate this work to almighty God for seeing me through to the completion
of Master of Science degree.
4
ACKNOWLEDGEMENT
My profound appreciation goes to Prof. C.M.I. Okoye and Dr. G.C. Asomba
for spending time and energy to supervise this research work. Specifically, I thank
you for your wonderful suggestions and corrections to ensure that this research work
satisfies the requirement for the award of Master of Science degree.
I equally thank all the Academic Staff of the Department of Physics and
Astronomy, University of Nigeria, Nsukka for your humane disposition towards
postgraduate students and imparting sound knowledge to me in particular. I thank
Prof. Ekpunubi of the Nnamdi Azikiwe University, Awka for his counseling and
suggestions for the success of this work.
I also acknowledge all my course mates for their understanding and
cooperation. I thank Mr. Agaji Iorshashe of the department of Mathematics /Statistics
/ Computer Science, University of Agriculture, Makurdi, for his great support during
my trying moment in the course of pursuing this programme.
I equally thank Mr. T. Ugah of the University of Nigeria, Nsukka for his
assistance while pursuing my admission and Dr. Joseph T. Zume for supplying me
with literature materials.
In a very special way, I acknowledge the support of my lovely wife Mrs.
Agatha I. Zendesha, my angel Josephine N. Zendesha, my brothers, sisters and my
parents for their moral support and prayers. I thank you all.
5
ABSTRACT
This research investigated the effect of magnetic field on the thermal conductivity of
high temperature type II superconductors. The result suggested that the thermal
conductivity of high temperature type II superconductor YBa2Cu3O7- decreases as the
applied magnetic field increases at a given temperature. We also found out that the
superconducting energy gap of YBa2Cu3O7- decreases in response to increasing
temperature and applied magnetic field. At a critical temperature of about 100K, we
noted a sharp decrease in the energy gap of the substance. This implies that, the
superconducting energy gap decreases in response to increase in temperature until at a
critical temperature of about 100K,the material transits to normal state, thus resulting
to increase in superconducting energy gap again. Our finding also revealed that
specific heat of YBa2Cu3O7- is proportional to electron density.
6
CONTENTS
Title Page i
Certification ii
Dedication iii
Acknowledgement iv
Abstract v
Table of contents vi
List of Tables viii
List figure ix
1. General Introduction 1
1.1 Introduction and Discovery of Superconductivity 1
1.2 Structure of single crystal of YBa2Cu3O7- 3
1.3 Basic properties of Superconductors 5
1.3.1 Electromagnetic properties 6
1.3.2 Thermal properties 9
1.3.3 Isotope effect 11
1.3.4 Tunneling 11
1.4 Type-I and Type-II 12
1.5 Applications of High-Tc Superconductors 13
1.5.1 High magnetic field, High direct current 13
1.5.2 Alternating current devices 13
1.5.3 Bolometer 14
1.5.4 Josephson tunneling 14
1.5.5 Medicine 15
vi
7
1.6 Theoretical Basis of Superconductivity 15
1.6.1 Phenomenological theories 15
1.6.2 The Ginzburg-Landau theory 16
1.6.3 Microscopic Theory 16
1.7 Purpose of the study 17
2 .0 Review of Literature 18
2.1Superconductivity in YBa2Cu3O7- 18
2.2 Thermal Conductivity of High-Tc Superconductors 19
2.3 Magneto-thermal conductivity of high-Tc superconductors 21
3.0 Magneto- thermal conductivity of high-Tc type II superconductors (single crystal of YBa2Cu3O7-) 26
3.1 Introduction 26
3.1.1London Equation 26
3.2 Magneto-Temperature Dependence of Superconducting Energy Gap of YBa2Cu3O7- 28 3.3 Calculation of Thermal Conductivity of YBa2Cu3O7- 29
3.4 Superconducting Energy Gap 31
3.5Critical Temperature 34
3.6 Specific Heat of YBa2Cu3O7- 35
4. 0 Discussions and Conclusion 38
4.1 Discussion 38
4.2 Conclusion 39
References 40
Appendix A: Program for Calculating Thermal Conductivity of YBa2Cu3O7- 44
Appendix B: Program for Calculating Superconducting Energy Gap of
YBa2Cu3O7- 45
Appendix C: Program for Calculating Specific Heat of YBa2Cu3O7- 46
8
LIST OF TABLES
1.1 High temperature superconductors, Tc and year of discovery
3
3.1 Thermal conductivity at different values of applied magnetic field at T = 60K
45
3.2 Temperature dependence of superconducting energy gap of YBa2Cu3O7-
46
3.3 Specific heat (in arbitrary unit) of YBa2Cu3O7- at different temperature
47
9
LIST OF FIGURES
1.1 The structure of the parent compound of high –Tc superconductor
YBa2Cu3O7- 4
1.2 The structure of (a)YBa2Cu3O7 (b)YBa2Cu3O6.5 (c)YBa2Cu3O6
5
1.3 A phase diagram of Type-II superconductor
14
3.1 Magnetic induction dependence of the thermal conductivity of a single
crystal of YBa2Cu3O7- at temperature (T) = 60K
33
3.2. Temperature dependence of superconducting energy gap of YBa2Cu3O7-
35
3.7 Specific heat curve (in arbitrary unit) of YBa2Cu3O7- at different
temperatures 37
10
CHAPTER ONE
General Introduction
1.1 Introduction and Discovery of Superconductivity
The phenomenon of superconductivity was first observed by Kamerlingh
Onnes in Leiden in 1911[1], three years after he liquefied helium gas. He then
measured the electrical resistivity of metals such as gold, platinium and mercury. He
found that the electrical resistivity of mercury vanished almost completely below
4.2K. The phenomenon by which a material loses all its electrical resistivity below a
certain temperature is called superconductivity [2]. The temperature at which this
occurs is known as the critical or transition temperature and it is normally denoted by
Tc. At temperatures below the critical temperature, the superconducting electrons are
ordered and therefore, do not carry heat. Thus, the ordered nature of superconducting
electrons reduce the thermal conductivity of superconductors since there is no
exchange of heat energy due to non- interactive nature of the super- conducting
electrons with the lattice [3].
Superconductivity occurs in many metallic elements of the periodic alloys, and
inter-metallic compounds at either low or high temperature. The search for new
superconductors is an ongoing process by material scientists with superconducting
transition temperature (Tc) above 30K in a mixture of lanthanum and barium-copper
oxide [4] La2-xBaxCuOx . High temperature superconductors, otherwise known as
high-Tc superconductors were first discovered by Bednorz and Müller in 1986 [4].
Attempts to substitute yittrium (Y) for lanthanum (La) resulted in a polyphase mixture
11
containing a new superconductor with Tc ≈ 90K [5]. Several other copper oxide
superconductors were discovered, some with Tc above 120K [6]. Magnesium
dibromide MgB2 was found to be superconducting with Tc of 39K. [7]
Anderson identified three essential features of the new superconductors [8].
First the materials are quasi–two dimension (2D); the key structural units seem to be
the presence of CuO2 plane and the interplane coupling is very weak. Second, high–Tc
superconductivity is created by doping a “Mott” insulator. A Mott insulator is a
material in which the conductivity vanishes as temperature tends to zero, even though
band theory would predict it to be metallic [9]. Third, Anderson proposed that the
combination of proximity to a Mott insulating phase and low dimensionality would
cause the doped material to exhibit fundamentally new behaviour, not explicable in
terms of conventional metal physics.
Generally, superconductors can be categorised into type I and type II
superconductors. In type I superconductors, the transition from superconducting state
to normal state in the presence of applied magnetic field is very sharp while in type II
superconductors, the transition from super-conducting state to normal state in the
presence of applied magnetic field takes place after going through a mixed state
region. Table 1.1 gives the transition temperature ( Tc ) and the year of the discovery
of some novel superconductors.
12
Table 1.1 High Temperature Superconductors, Tc and Year of Discovery
Most of the cuprate superconductors were derived from YBa2Cu3O7- by replacing
yittrium with either bismuth (Bi), thalium (Tl )or mercury (Hg) or barium (Ba) with
lead (Pb), for example. It is therefore necessary to examine the crystal structure of
YBa2Cu3O7- in some detail.
1.2 Structure of Single Crystal of YBa2Cu3O7-
Yttrium barium copper oxide (YBa2Cu3O7-) is a high-Tc multiphase
superconducting material whose phases undergo structural transition. The structure of
the parent compound YBa2Cu3O7 is related to the perovskite structure with an ordered
arrangement of both the cations (Y and Ba atoms) and anions (O) as shown in Fig.
1.1.
Superconductors Tc(K) Year of Discovery Reference
(La0.9Ba0.3)2CuO4- at 1Gpa 52 1986 [4]
YBa2Cu3O7- 95 1986 [4]
Bi2Sr2Ca2Cu3O10 108 1988 [10]
Tl2Ba2Ca2Cu3O10 127 1988 [11]
HgBa2 Ca2 Cu3 O8+ 133 1993 [12]
Hg Ba2 Ca2 Cu3 O8+ at 25Gpa 155 1993 [13]
MgB2 39 2001 [14,15]
LaO1-xFxFeAs 26 2008 [16]
SmFeAs 55 2008 [16,17]
13
Figure 1.1: The structure of the parent compound superconductor, YBa2 Cu3O7-
The removal of oxygen atoms within 0 ≤ ≤ 1, accounts for the multiphase nature of
YBa2Cu3O7- due to the increase in temperature as can be seen in Figures 1.2 (a), (b),
(c) respectively [18].
].
[a ] [b] [c]
Figure 1.2: The structure of (a) YBa2Cu3O7 (b) YBa2Cu3O6.5 (c) YBa2Cu3O6
14
From the schematic diagram above,Fig.1.2(a) represent the orthorhombic
structure of YBa2Cu3O7 with a striking feature due to complete vacancy of the 0(5)
positions. Studies have shown that the total oxygen stoichiometry decreased smoothly
with increasing sintering temperature [19]. The oxygen atoms removed from the
structure were exclusively those located at the 0(4) sites at (0,½, 0). The insitu
neutron powder diffraction measurements [18] showed that the assigned positions of
0(5) at (½,0,0) were gradually filled with oxygen atoms as the temperature increased,
and when the occupancy of the structures changed from orthorhombic to tetragonal as
shown in fig 1.2(b). The oxygen stoichiometry at the transition was always at = 0.5
for YBa2Cu3O7-, so that the orthorhombic phase existed between 0 ≤ ≤ 0.5. Further
heating of the materials due to loss of the oxygen atoms from the site of 0(4) and 0(5)
until the stoichiometry reached YBa2Cu3O6 is shown in Fig 1.2(c). It can be inferred
that superconductivity in YBa2Cu3O7- is a function of the oxygen content. In other
words, the orthorhombic phase is characterised by decrease in the values of Tc as the
total oxygen stoichiomety decreased and it became zero as the crystallographic
transition was approached [20].
1.3 Basic Properties of Superconductors
Superconductors generally exhibit unique properties that distinguish them
from metals and semiconductors. The properties can be categorised into
electromagnetic, thermal, isotope effect and tunnelling properties.
15
1.3.1 Electromagnetic Properties
The electromagnetic properties of high-Tc super-conductors are:
i. Infinite conductivity: In superconducting state, the electrical resistivity of the
material disappears completely at a temperature below Tc, thus paving way for
infinite mobility of the superconducting electrons pairs. The mobile electrons
infinitely conduct electric current as far as the superconducting state of the
material is maintained.
ii. In the presence of applied magnetic field, the superconductors expel all traces
of magnetic flux from the interior of the material. This phenomenon is known
as Meissner effect [2]. This is one of the characteristics that distinguishes a
type I superconductor from type II superconductor. In spite of the existence of
the Meissner effect in superconductors, experimental findings have shown that
the external magnetic field penetrates the superconducting materials to some
degree of depth known as penetration depth. In type I superconductors,
magnetic field can only penetrate at the cost of destroying the
superconductivity [21]. In type II superconductors, small fields are expelled,
but a magnetic field in excess of the lower field Hc1, penetrates non-
uniformly, forming the mixed state. The magnetic flux trapped is quantised
according to 0 =n
ehc2
which is set by the total charge 2e of the Cooper
pairs. The diameter of each flux lines is set by penetration length denoted
by . Near the middle of each flux line the amplitude , of the Cooper wave
16
function is suppressed to zero. The suppressed region of the amplitudes is
called the vortex core and the radius of characteristics length is known as
coherence length. The relative ratio
determines the type of
superconductor. In the high-Tc copper oxides, which is in the range 10 -
20Ǻ at low temperatures is much smaller than the penetration lengths which
exceeds 1000Ǻ. In high-Tc cuprates which are mainly type II superconductors
with large value of transition temperature, the vortex state occurs when a
magnetic field penetrates the material. When a type II superconductor is
placed in a magnetic field in excess of 1c , the vortices that penetrate the
material form a regular lattice known as Abrikosov vortex lattice [22]. If the
magnetic field in the vortex lattice is increased, the vortices become more
closely spaced and their cores start to overlap. At the upper critical field the
vortex lattice and the pairing of electrons disappear and the material becomes
‘normal’. If thermal fluctuations are ignored, then the upper critical field is
Hc2 = 22o , so small coherence lengths give rise to large upper critical fields
exceeding 100T, greater than the fields available in today’s magnets.
iii. Persistent current and flux quantization: When a magnetic field is applied
perpendicular to a macroscopic ring, a voltage is generated which induces
current into the ring. According to Lenz’s law:
dttdILtRI
dtdBaAr
)()( (1.1)
17
where rA is the area enclosed by the ring, R the resistance of the ring and L the
inductance of the ring. If there is no applied magnetic field (Ba = 0), then the solution
of the equation above is
LRtoItI exp)()( (1.2)
This shows that any initial current circulating in the ring decays exponentially to zero
in the normal state. However, in the superconductive state an R = Rs = 0 makes I =
I(o), and the initial current I(o) continue to circulate around the ring without any change
in its magnitude as far as it is maintained below Tc. .Such currents are called persistent
current [5]. The ring traps magnetic flux which remains constant over long period of
time even in the absence of external magnetic field. The trapped flux is quantised as
=n
ehc2
(1.3)
where n is an integer.
This result is of outstanding importance. It means that if the superconductive state
consists of paired elections, then in a closed superconducting circuit the flux is in unit
of
o = n
ehc2
= 2.07 10-5Wb (1.4)
iv Electrical Resistivity
Zero resistance is one of the defining characteristics of a superconductor at
temperatures below critical temperature (T<Tc). In the Meissner phase (H<1c ) the
18
magnetic field is expelled from the material and there are no free vortex lines present
to move and cause electrical resistivity. Thus superconductivity in the Meissner phase
is characterised by zero linear resistivity 0lim 01
jj . Note that 1 is the
ohmic resistivity which measures the linear response to an applied current or field. In
spite of zero linear resistivity, there exist some non- linear resistivity in Meissner
phase; a non-zero electric field is needed to maintain any non- zero current density in
the interior of the material, so the non- linear resistivity j
only disappears within
the context of zero current density.
In a superconductor in the mixed state 21 cc , there are vortex lines
induced by the penetrating magnetic field. Motion of these lines causes resistivity. In
the vortex-fluid regime the vortex lines are mobile because of thermal fluctuations
and thus move in response to a current, leading to a nonzero resistivity.
1.3.2 Thermal Properties
Thermal fluctuations are much more important in the high-Tc copper oxide
superconductors. The most striking effects of enhanced thermal fluctuations in the
high-Tc superconductors are found in an applied magnetic field. Notice the vortex-
fluid regime between the mean-field upper critical fields, Hc2 and the vortex-glass
phase. On cooling in a field, the electrons start to pair and vortices form in the pair
wave function near Hc2, but the vortices do not freeze until at substantially lower
temperatures. The existence of a substantial vortex–fluid regime had been noted for
19
thin superconducting films [23,24], but it was not observed in bulk superconductors
until the copper oxide superconductors had been studied [25, 26]. The vortex-fluid
regimes extends to particularly low temperatures in extremely anisotropic layered
materials (like BSCCO) when the magnetic field is directed perpendicular to the layer
[27, 28]. In this case, the vortex lines actually consist of strings of point or ‘pancake’
vortex in each superconducting layer, with only rather weak correlations between
vortices in different layers.
In the absence of applied magnetic field, the transition from the
superconductive to the normal state is second order phase transition. This means that
there is no discontinuity at Tc in either entropy (no latent heat) or volume (no thermal
hysteresis), but there is a sharp discontinuity in heat capacity, C. Thus specific heat in
the normal state, that is state above Tc, varies linearly with temperature, T, while
specific heat in the superconducting state initially shoots above the normal state, Cn,
and drops below it before finally vanishing exponentially as T tends to zero.
Theoretically it is found that the specific heat below Tc for an isotropic gap,Cs, is
given by [29]
Cs exp
Tk
BT
B
),( (1.5)
where ∆ is the energy gap. This dependence indicates the existence of an energy gap
∆ in the energy spectrum separating the excited from the ground state. At the
superconducting transition the specific heat exhibits a jump. This effect was first
observed from measurement on Sn in 1932 by Keesom and Kok[30].In zero magnetic
20
field this is a second order phase transition with no latent heat, implying that we
expect the order parameter to be a continuous function of temperature.
1.3.3 Isotope Effect
The critical temperature, Tc, for superconductors varies smoothly with the average
atomic mass M as the isotope mass is varied.For conventional superconductors,
MTc = constant (1.6)
where = ½ . This correlation of Tc and M is known as the isotopic effect. This early
observation shows that for conventional superconductors, electron-phonon
interactions play an important
role in the binding of the superconductive pairs of electrons. In the simplest theory,
only the electronic states within energy, kBD of Fermi energy, EF, where D is the
Debye temperature, can be coupled by electron-phonon interactions.
1.3.4 Tunnelling
If two superconductors are separated by an insulating film (10A),
superconducting electrons would tunnel through the junction. The tunnelling
superconducting electrons generate single quasi-particles and paired superconductive
particle. The single quasi-particles tunnelling can be used to measure the energy gap
in the superconductive state while the superconductive particles tunnelling known as
Joseph tunnelling which usually exhibits quantum effects have been exploited in a
variety of quantum devices. In 1962 Josephson [31] observed that a zero-voltage
super current in the direction x perpendicular to the junction is
21
Ix = Iox sinr (1.7)
due to the tunnelling of superconductive electron pairs. A maximum dc flows in the
absence of any electric or magnetic field. This is the dc Josephson effect. He further
predicted that if a voltage difference, v, is applied across the junction, the parameter ɤ
becomes time dependent [31],
)(tr = ɤ(o) –
heVt4 (1.8)
which means that the current oscillates with a frequency heVv 2
.This is the ac
Josephson effect.
1.4 Type -I and Type -II superconductors
Superconductors are normal classified into two types namely type I or soft
superconductors and type -II or hard superconductor. Type- I superconductor have
sharp boundary between the superconducting and normal states. Superconductivity is
easily destroyed by the application of magnetic field. On the other hand, in type II
superconductors, when a high magnetic field is applied, magnetic field penetrates into
the inferior forming a state in which normal and superconductor region co-exist. This
region is normally called the mixed or vortex state. A full phase diagram of a clean
type II is shown in Fig. 1.3. It is seen that further application of the field leads to
complete expulsion of the magnetic field from the interior of the material and then
22
becomes normal. This high magnetic field is one of the major properties of
superconductors that is utilised in applications.
Fig.1.3: A phase diagram of Type II superconductor
1.5 Applications of High-Tc Superconductors
High-Tc superconductors have useful applications in the industry. These are
presented in sections1.5.1 to1.5.5.
1.5.1 High Magnetic Field, High Direct Current
The discovery of zero direct current resistance raises the hope of building a
solenoid magnet of superconductive wire with the capacity to produce an intense
magnetic field at manageable power levels. A type- II hard superconductor for
instance can remain superconductive to high magnetic field Hc2, the generation of
high magnetic fields with type II superconductors is now used in a wide range of
applications.
1.5.2 Alternating Current Devices
A small ratio of resistance in superconducting state Rs to resistance in normal
state Rn (Rs/Rn ) requires ac operation at T << Tc and V << Eg/h. Type I
Tc o o
H a
Meissner Hc1
Vortex lattice
Hc2
23
superconductors may retain an Rs 0 up to 100mHz[5]. This property has enabled the
realization of very high frequency linear electron accelerators with magnetizations up
to 1010; they can operate continuously with only a fraction of the power requirements
of conventional accelerators. In type- II superconductors large energy gaps, for
example, allow low-loss ac transmission over superconductive strip lines to even high
frequencies. In ac power devices type-II superconductors are used in applications
such as space vehicle in which high current densities in high fields lead to significant
reductions in weight and size [5].
1.5.3 Bolometer
A bolometer detects electromagnetic radiation by absorption of radiation that
increases its temperature. The temperature increase T is related to the energy E
absorbed per unit mass via the specific heat capacity Cv [5]
T = E/Cv (1.9)
A superconductive bolometer is very important in radiation detection where other
types of radiation detectors are inoperative.
1.5.4 Josephson Tunnelling
The direct current (dc) Josephson effect has useful applications in sensitive
galvanometers and magnetometers. The superconducting quantum interference device
(SQUID) magnetometer is used for measuring small magnetic fields with extensive
use in geological surveying while the ac Josephson effect has been used in precision
determination of the value of h/e[5].
24
1.5.5 Medicine
In medicine, superconductors provide the needed magnetic field for
applications such as magnetic resonance imaging [32].Ultra sensitive superconducting
circuit are used in medical applications such as the study of human heart and brain.
Also Magnetic Resonance Imaging (MRI) system which incorporates
superconducting magnets are used in hospitals and clinics.
1.6 Theoretical Basis of Superconductivity
The understanding of superconductivity in materials has been guided by three
categories of theories. These include the phenomenological theories, Ginzburg-
Landau theory and the microscopic theory.
1.6.1 Phenomenological Theories
Several theories have been adduced to explain phenomenological arguments.
One of the first was London theory proposed by the brothers Heinz and Fritz London
in 1935 [33].They postulated a proportionality between the density of super currents
and the vector potential given as
2
1
losj
(1.11)
where is a vector potential.
This equation together with Maxwell’s equations allow for the derivation of the
Meissner effect. This is a local description. The current density at a specific point is
determined by the vector potential at that point. Several experiments indicated that
this is not always true. Pippard [34] proposed a generalisation of London’s theory
25
where the current density is instead proportional to a spatial average of the vector
potential. This extension includes a dependence of the coherence length on the mean
free path in an impure superconductor. It should be mentioned that Pippards equations
are almost identical to the ones given in microscopic theory [35].
1.6.2 The Ginzburg-Landau Theory
The Ginzburg-Landau (GL) theory is a general theory for treating second order phase
transitions. It was first proposed in 1950 to explain superconductivity [36].The central
idea is to expand the free energy in terms of an order parameter. For a superconductor
this is taken to be the density of electron pairs. It is then assumed that the order
parameter takes the value associated with minimal free energy. If thermodynamic
fluctuations are important the order parameter will not have a well defined value. In
conventional superconductors the fluctuation effects are small and because of this, GL
theory gives a good description. This theory is physically intuitive and the formalism
is attractive. Despite this it was not until Gorkov [37] had shown that GL theory can
be derived from BCS theory that it was widely accepted outside the Soviet Union.
1.6.3 Microscopic Theory
The phenomenon was not put on firm quantum mechanical basis until 1957
when Bardeen, Cooper and Schrieffer (BCS) proposed their famous theory [35].The
theory can be divided into three major parts:
i. The formation of electron pairs. In the presence of an attractive potential at the
Fermi surface, the electrons form stable pairs with wave vectors k,-k and spin
26
↓↑, respectively. The possibility of pair formation had been pointed out earlier
by Cooper [38] although he only considered non-interacting pairs.
ii. The attractive potential is caused by electron-phonon interaction. This is an
exchangeable part of the theory. One can think of many possible candidates for
providing the interaction. One of the keys was the isotope effect which clearly
indicates an intimate relation between superconductivity and the crystal lattice.
iii. The opening of a gap in the density of states at the Fermi surface. When solving
the equation for the gap in a self consistent way, an expression for Tc is
obtained. The theory in its original form applies very well to weak coupling
superconductors. In the strong coupling case extension are necessary. The
central equation in this case are the Eliashberg equations [39,40] arrived at from
many body theory.
1.7 Purpose of the Study
Superconductivity of materials is generally affected by the application of
magnetic field. The purpose of this study is to investigate the effect of applied
magnetic field on the thermal conductivity, temperature dependence of
superconducting energy gap and specific heat of YBa2Cu3O7-.
27
CHAPTER TWO
Review of Literature
2.1 Superconductivity in YBa2Cu3O7-
The mechanism of superconductivity in superconductors has been guided by
the BCS theory. The charge carrier are pairs of electrons known as Cooper pairs
[32,41]. The pairing is caused by interaction of the electron with lattice vibration. The
quantum of lattice vibration is known as a phonon. Thus superconductive mechanism
is known as electron-phonon mechanism [32].The role of phonons in the pairing
mechanism has regained interest in the past few years following a number of
experimental and theoretical evidences [42,43,44]. Even though BCS theory still
remains valid for superconductive mechanism, consensus between scientists and
theorists is still highly controversial due to lack of theoretical computation on such
strongly interacting system. In a high temperature superconductor, phonons play
virtually no role and their role is replaced by spin density waves. As all conventional
superconductors are strong phonon systems, all high temperature superconductors are
strong spin density wave system [32].
The discovery of the high-temperature (high-TC) copper oxides
superconductors in 1986 by Bednorz and Müller led to a renewed interest in
superconductivity and to the development of a new class of type-II superconductors in
which the superconductivity resides on layers of CuO2 [21].This superconductor
(YBa2Cu3O7-) is a type-II superconductor with Tc ≈90K. A small isotope effect
28
indicates that electron-phonon interaction involving oxygen vibrational modes play at
least some role in coupling the superconductors [5].
2.2 Thermal Conductivity of High-TC Superconductors
Generally, the mechanism of thermal conductivity is necessitated by the
interaction that takes place in the electrons .When electrons are thermally excited,
they collide with one another resulting to exchange of heat energy to all layers of the
conducting material. This is the mechanism of thermal conductivity in normal state. It
increases with corresponding increase in temperature. However, in a superconducting
state, the electrons do not collide due to the zero resistance in the state. Peacor [45]
observed that for Cu-O based superconductors thermal conductivity rises as
temperature is lowered below Tc .This effect is typical of YBa2Cu3O7-δ(YBCO) single
crystal. Hence, the thermal conductivity measurement is relevant to the study of
scattering mechanisms in both normal and superconducting states. A common feature
in the recent studies on high-Tc oxide superconductors revealed an anomaly in the
thermal conductivity associated with superconducting transition. The anomaly is
linked to two types of theories. One of which is by Bardeen, and co-workers [46,47]
which proposed that for conventional superconductors, the anomaly in the thermal
conductivity is associated with the phonon thermal conductivity. In the
superconducting state, the quasi particles as thermal carriers condensed into Cooper
pairs and thus, the quasi-particles number decreases so that the electron thermal
conductivity decreases below Tc. This gives rise to large reduction in the scattering
cross section of phonons and then, an observed thermal conductivity shows an
29
enhancement in the superconducting state. On the contrary, Yu et al proposed a
theory to explain the anomaly in the ab-plane, K, of the untwined YBa2Cu3O7-
δ(YBCO) single crystal [48]. They opined that the scattering cross section of quasi
particles decreases more remarkably when compared with the decline in number of
quasi-particles. Consequently, the electronic thermal conductivity shows a peak
below Tc. It can be inferred that the former approach emphasised the origin of the
anomaly on the phonon component, while the latter is on the electronic component.
Peacor et al [49] suggest that thermal conductivity of YBa2Cu3O7-δ arises from
two separate channels, free carriers and phonons. In the free carrier contribution,
thermal conductivity decrease due to the condensation of carriers into Cooper pairs,
while the large phonon contribution suggests an increased thermal conductivity at
temperatures below Tc. Above the Tc of this material, thermal conductivity decreases
with increasing temperature. At these elevated temperatures, the phonons will scatter
on free carriers, defects and other phonons [49].The phonon-carrier and phonon-
defect scattering rates tend to constant values with increasing temperature[43],thus
leaving the phonon-phonon term to govern the temperature variation of thermal
conductivity. The decrease in thermal conductivity with increasing temperature attests
to the importance of phonon-phonon scattering in the single crystal [49]. Similarly,
Richardson et al [50] associated the variation of thermal conductivity with
temperature on the basis phonon model supposing that the main carriers of heat are
phonons which are strongly scattered by the flux lines in the mixed state of
YBa2Cu3O7-δ [51-52].
30
2.3 Magneto-Thermal Conductivity of High-TC Superconductors
The thermal conductivity of metal is affected if it goes into the
superconducting state. In the superconducting state however, the superconducting
electrons no longer interact with the lattice in such a way that they can exchange
energy and so they can not pick up heat energy from one part of a specimen and
deliver it to another part. In other words, there is no exchange of heat energy by
superconducting electrons. Consequently, if a metal goes into the superconducting
state its thermal conductivity is reduced. However, if the superconductor is driven
normal by the application of a magnetic field, the thermal conductivity is restored to
the highest value in the normal state. Hence the thermal conductivity of a
superconductor can be controlled by means of a magnetic field and this effect has
been used in “thermal switches at low temperatures to make and break heat contact
between specimens connected by a link of superconducting metal. This is the
characteristic nature of type-I superconductors in magnetic thermal conduction. For a
type-I superconductor in a magnetic field, the thermal conductivity witnesses an
increase with the restoration of normal state by the magnetic field. While for type-II
superconductors, the thermal carriers interact with quasi-particles within the normal
regime produced by an applied magnetic field and are scattered by quasi- particles in
the normal cores. Thus in type II superconductors such as single crystals of
YBa2Cu3O7- the thermal conductivity initially diminishes and then saturates with the
application of higher values of magnetic field. The magnetic field dependence of
thermal conductivity of YBa2Cu3O7-δ diminishes for fields oriented perpendicular to
31
the Cu-O plane [53]. Interestingly, hysteresis effect and strong anisotropy for field
oriented parallel and perpendicular to the Cu-O planes are observed in the magnetic
field dependence of thermal conductivity .Hysteresis is eventually associated with
trapped flux and is observed from 62K down to the lowest temperature measured
[49]. The anisotropic behaviour of thermal conductivity is based on the anisotropy of
the superconducting coherence length. Hence, the thermal conductivity study in
magnetic fields gives further information about the vortex state of the superconductor
and scattering mechanisms. The vortex state thermal conductivity is very useful to the
origin of a peak since the interactions of the thermal carriers with quasi-particles in
the vortex cores gives information about the electron-phonon interaction or electron-
electron interaction [48]
It has been found that applied magnetic field destroys superconductivity in a
material. So high values of applied magnetic field have the capacity to transform a
material from superconducting state to normal state. The magneto-thermal
conductivity (B) of high-Tc superconductors was first theoretically described by
Richardson et al [50] on the basis of a phonon model. In this model the scattering of
phonon by flux lines in the mixed state of type II superconductors such as
YBa2Cu3O7-δ enhances the exchange of heat conduction. Studies by Richardson et al
on the influence of magnetic field on the thermal conductivity of twinned and
untwinned single crystals of YBa2Cu3O7-δ led to phenomenological expression [50]
given as
32
)exp()(
1)(1 qdBcBo
B
(2.7)
where (o) is the thermal conductivity in the absence of magnetic field and the
second term represents thermal resistivity due to the scattering of phonons by
vortices; c and d are undefined temperature dependent parameters adjusted to fit the
data, the exponent q was found to be temperature independent and equal to 41 for
both the twinned and untwinned single crystals. Several theoretical and experimental
results have pointed to the fact that the main structure of the thermal conductivity of
high-Tc cuprates below Tc at zero magnetic field, could be due to the contribution of
normal electrons in the CuO2 planes [53-56]. In the presence of magnetic field, the
thermal conductivity of high-Tc superconductors might be attributed to the scattering
of normal electrons by the vortex cores [52]. The total magnetic thermal conductivity
(B) can be written as [52]
),()0,(),( 111 BTTBT ve
(2.8)
where (T,0) is the thermal conductivity in the absence of a field and e-v(T,B) the
thermal resistivity due to the scattering of electrons by the vortex cores. The magnetic
induction dependence of is thus assumed to be only due to the electron-vortex
scattering in contrast to the phonon-vortex scattering in [50].
Derivation of electron-vortex scattering contribution can be obtained by
considering the well known Boltzmann based kinetic formula [54, 57].
),(),(3
, *
22
BTBTnm
TkBT veeB
ve (2.9)
33
where m* is the effective mass of electrons, ne the normal electron concentration
and 1ve the electron-vortex scattering rate. But normal electron concentration is given
as [58]
TkBTnBTn
Boe
),(exp),( (2.10)
where no is the steady-state electron concentration in the normal state and ∆ the
Ginzburg-Landau superconducting energy gap. The temperature and magnetic
dependence of ∆ in type II superconductors is given by [59,60].
2
2 )(1
)(1
)()0,0(),(
THB
BTT
TBTBT
coc
c
(2.11)
where ∆(0,0) is the superconducting energy gap at zero temperature in the absence of
a field, o the permittivity in free space, Tc(B) the magnetic induction dependence of
the critical temperature which decreases as the induction is increased[61]
32
2
1)0()(co
cc HBTBT
(2.12)
and the temperature dependent upper critical magnetic field Hc2 (T) is given as[61]
2
22 1)0()(c
cc TTT (2.13)
34
The scattering rate of heat carrying normal electrons by these quasi-particle
excitations in the core of vortices [62] is given as
2
1 )(co
veBT
(2.14)
where )(T is given in cgs units by
j Bj
j
F
co
Tkce
T)/2exp(1
)0,0()(
22
222
5
(2.15)
where )0(2*
2
mj
j
.
35
CHAPTER THREE
Magneto-Thermal Conductivity of Single Crystal of YBa2Cu3O7-δ
3.1 Introduction
Superconductors generally expel magnetic lines of force from their interior.
When the strength of this externally applied magnetic field is increased slowly, a
value is reached where the magnetic lines of force begin to penetrate the material and
it becomes non superconducting. The extent to which magnetic lines of force
penetrate these superconductors is known as London penetration depth. [5] Hence
London equation shall be used to investigate the effect of magnetic field on thermal
conductivity of high-Tc type II superconductors.
3.1.1 London Equation
London equation shows that the applied magnetic field in a sample decays
exponentially in x direction according to [33]
L
xaBxB
exp)()( (3.1)
where )(xB is the magnetic field inside the superconductor, )(aB is applied magnetic
field, x is the thickness of the superconducting material and L is the London
penetration depth i.e
21
2
oeL en
m
(3.2)
Eq. (3.1) can be expressed as
36
L
xaBxB
exp
)()(
L
xaBxB
)()(ln
Substituting for L gives
)()(ln
aBxB
21
2
oeenm
x
which can be simplified to yield
en = 2
2 )()(ln
)(
aBxB
xem
o (3.3)
From Eq.(2.8),we are given that,
),(),(),( 111 BTTOBT ve
taking the inverse of both sides of Eq.(2.8), we have
),(),(),( BTTOBT ve (3.4)
By substituting Eqs (3.3), (2.14) and (2.15) into (2.9), we obtain
)()()()(ln
)(3),( 2
2
2*
*22
aBTHcu
aBxB
xemTmk
BT o
o
Bve
=
)()(ln2
)()()(3 22
22
aBxB
aBTxeTHk cB
(3.5)
=
22
252
222
22
)0,0(
2exp1
)()(ln
)()(32
j
B
j
co
FcB TkaBxB
HeaBxeTHk
j
37
= )(ln)(
)(ln)0,0(
2exp1
32
225
2
aBaBxBTk
eT
xek
j
B
j
o
FB
(3.6)
Substituting Eq. (3.6) into Eq (3.4), we have
)(ln)()(ln
)0,0(
2exp1
32)0,(),(
225
2
aBaBxBTk
eT
xekTBT
j
B
j
o
FB
(3.7)
A look at Eq. (3.7) shows that (T,B) is inversely proportional to the applied
magnetic field at a constant temperature.
3.2 Magneto-Temperature Dependence of Superconducting Energy Gap of
YBa2Cu3O7-δ
In the band theory of semiconductors, three bands characterize the band
structure of a semiconductor, namely conduction, forbidden (Energy gap) and valence
bands. When the valence electrons are thermally excited, they acquire kinetic energy
and escape into conduction band. The positive variation of temperature induces the
increase of electron concentration in the conduction band consequently, the energy
gap of the semiconductor begins to dwindle in response to increase in temperature.
Similarly, the high-Tc of types-II superconductors exhibit the same phenomenon.
According to BCS theory of superconductivity, the superconducting electrons
condensed into Cooper pairs with less heat energy than the unpaired superconducting
electrons in the superconducting state [62]. The increase in magnetic field with
temperature breaks the Cooper pairs into superconducting electrons, so just like a
typical semiconductor, the superconducting energy gap of YBa2Cu3O7-δ begins to
j
j
38
decrease in response to continuous positive variation of magnetic field and
temperature. In the same vein, the electron density of the superconducting electrons
increases with magnetic field and temperature but vanishes abruptly with decrease in
temperature [55].
3.3 Calculation of Thermal Conductivity of YBa2Cu3O7-δ
The magnetic induction dependence of the thermal conductivity of single
crystals of YBa2Cu3O7-δ shall be investigated using Eq. (3.4). In order to calculate the
thermal conductivity of YBa2Cu3O7-δ as a function of magnetic field at constant
temperature T, we assume that at T=60K, the concentration of electron is
oe nn 2.0 [52]
Hence Eq. (3.3) can be written as
2
2
*
)()(ln
)(
aBxB
xemn
oe
*
2
22.0)()(ln)(ln
mnxeaBxB oo
But the normal electron density 2.0)60( KTnn
o
e [52] where ne is the normal state
electron concentration taken to be no =1.7 × 1028em-3[52] and m* = 4me where me =
9.1×10-31kg, e = 1.6 ×10-19J
Therefore
e
oo
mnxeaB
xB42
2.0)()(ln)(ln
2
(3.8)
39
Note that x is the thickness of single crystal of YBa2Cu3O7-δ which is x = 310-3m
By taking the values of applied magnetic field B(a) to be 2,3,4,5,
6,7,8,9,10,11,12,13,14, and15T respectively, we generated values for B(x) = 2T
We now substitute the values of B(x) into Eq. (3.5) to obtain the values of the
thermal resistivity due to the scattering of electrons by the vortex cores ),( BTve . By
substituting the corresponding values of ),( BTve into Eq (3.4), we obtain the results in
Table 3.1. The results from this table are displayed in Fig.3.1
Table 3.1: Thermal conductivity at different values of applied magnetic field at T = 60K B(a)(T) B(a)lnB(a)(T) κ(T,B)(w/mK)
2 1.386294361119891 1.0025918 ×10-23
3 3.295836866004329 6.6839461×10-24
4 5.545177444479562 5.0129596×10-24 5 8.047189562170502 4.0103677×10-24 6 10.750556815368331 3.3419731×10-24 7 13.621371043387192 2.8645484×10-24 8 16.635532333438686 2.5064798×10-24 9 19.775021196025975 2.2279821× 10-24 10 23.025850929940461 2.005183924× 10-24 11 26.376848000782076 1.822894476× 10-24 12 29.818879797456006 1.670986603× 10-24 13 33.344341646999979 1.452449173× 10-24 14 36.946802614613617 1.432274232× 10-24 15 40.620753016533151 1.336789282× 10-24
40
0 5 10 15 20 25 30 35 40 451
2
3
4
5
6
7
8
9
10
11x 10-24
B(a)lnB(a) T
(T,
B) w
/mK
T=60K
Fig 3.1: Magnetic induction dependence of the thermal conductivity of a single
crystal of YBa2Cu3O7-δ at temperature (T) = 60K
3.3 Superconducting Energy Gap
The concentration of the superconducting electrons is given by Eq 2.10 [26].
From Eq. (2.10) we have
TkBTBT
nn
Bo
e ,exp, (3.9)
This implies that
BT
nn
TkBTo
eB ,ln, (3.10)
41
Thus, Eq. (3.10) can be used to calculate the temperature dependence of the
superconducting energy gap in a magnetic field provided the electron density is
ascertained. Similarly, the dependence of superconducting energy gap on the field is
given as [48]
21
2
)(1)0()(
cHaBT (3.11)
where Δ(0) is the superconducting energy gap at zero magnetic field and temperature,
B(a) is the applied magnetic field and Hc2 is the upper critical field. By substituting
Eq. (3.11) into (3.9) we have
Tk
HaB
BTnn
B
C
o
e
21
2
)(1)0(
exp, (3.12)
Using ∆(0,0) = 14.5meV and 2C = 81.2T [52], we obtain
Substituting Eq. (3.12) in Eq. (3.10) we obtain values for BT , shown in Table
3.2 as a function of temperature displayed in Fig.3.2 for YBa2Cu3O7-δ
42
Table 3.2: Temperature dependence of superconducting energy gap of YBa2Cu3O7-δ.
T(K) B(a)(T)
o
e
nBTn ),(
))(,( meVBT
10 1 5.9× 10-8 14.410437685431845
20 2 2.5 × 10-4 14.320315239945963
30 3 4.1 × 10-3 14.229622020881049
40 4 1.7 × 10-2 14.138347044221867
50 5 3.8 × 10-2 14.046478969071623
60 6 6.7× 10-2 13.954006081204270
70 7 1.0× 10-1 13.860916275629112
80 8 1.5× 10-1 13.767197038094366
90 9 1.9× 10-1 13.672835425449566
100 10 1.5 -3.500000000000000
110 11 0.241462407890225 13.482131031215248
43
10 20 30 40 50 60 70 80 90 100 110-4
-2
0
2
4
6
8
10
12
14
16
T(K)
(T
,B) m
eV
Fig 3.2: Temperature dependence of superconducting energy gap of YBa2Cu3O7-δ
From the graph, the intercept on ∆(T,B) axis is about 14.5meV while intercept on T
axis is about 100K .This implies that, the superconducting energy gap decreases in
response to increase in temperature until at a critical temperature of about 100K,the
material transits to normal state, thus resulting to increase in superconducting energy
gap again.
3.5 Critical Temperature
The critical temperature (Tc) of superconductors is defined as the temperature
at which a material loses its electrical resistance [2]. From our analysis in Fig.3.2, the
Tc of YBa2Cu3O7- corresponds to the intercept on temperature axis. Our result shows
44
that Tc of YBa2Cu3O7- is about 100K. Similar theoretical calculations suggest that Tc
of YBa2Cu3O7- falls between 92K-95K [52,63]
3.6 Specific Heat of YBa2Cu3O7-
The specific heat of superconductors decreases exponentially for an isotropic
gap according to
Cs ≈ exp TkBT
B
),( (1.5)
In a superconducting state Eq. (1.5) could be written as
Cs ≈ exp
TkBT
B
),(
Eq (3.10) defines superconducting energy gap as
∆(T,B) = - kB T ln
o
e
nn ),(
Using this in Cs above, we obtain
Cs = exp
kn
nko
e ),(ln
= exp
o
e
nn ),(
ln
= o
e
nn ),(
45
This yields
Cs = o
e
nn ),( (3.13)
Eq. (3.13) shows that the specific heat of YBa2Cu3O7- is proportional to normal
electron density. For values of Cs, we obtain the values of o
e
nn ),( with the
corresponding values of T(K) in Table 3.2 to obtain Table 3.3. The results from this
table are displayed in Fig.3.3
Table 3.3: Specific heat (in arbitrary unit) of YBa2Cu3O7- at different temperatures
T(K) Cs =
o
e
nn ),(
10 85.9 10 20 42.5 10 30 34.1 10 40 21.7 10 50 23.8 10 60 26.7 10 70 11.0 10 80 11.5 10 90 11.9 10 100 01.5 10 110 12.4 10 120 12.7 10 130 13.0 10 140 13.4 10 150 13.6 10
46
Fig. 3.3: Specific heat Cs (in arbitrary unit) of YBa2Cu3O7- at different temperatures
0 50 100 1500
0.5
1
1.5
T(K)
Cs
47
CHAPTER FOUR
Discussion and Conclusion
4.1 Discussion
From Figure 3.1, it can be noticed that the thermal conductivity of YBa2Cu3O7-
decreases exponentially as applied magnetic field increases at a uniform temperature
and then ‘saturates’. The experimental results obtained by Richardson et al [50],
Ausloos and Houssa [52] suggested that as the magnetic induction is increased, the
scattering rate of electrons due to collisions by the core of vortices increases and (B)
is decreased.; however, the density of electrons increased with the induction, which
compensates the increasing scattering rate as (B) saturate at high fields. It can be
inferred from the forgoing analysis that in high-Tc type II superconductors, magneto-
thermal conductivity is necessitated by electrons which are scattered by the core of
vortices in the mixed state.
Similarly, Bai-Mei et al [64] corroborated the views of Ausloos and Houssa
that the thermal conductivity of high-Tc type II superconductors decreases with
increase in magnetic induction. The non-linear dependence of thermal conductivity on
magnetic induction is due to the existence of vortices in type II superconductors
which constitute the scattering centre.
In figure 3.2, the superconducting energy gap decreases with increasing
magnetic field and temperature. The result of this study is in agreement with the
findings of [2] and [65]. The theoretical curves in fig.3.2 show that the intercept on T
axis coincides with the critical temperature of YBa2Cu3O7-. From the graph, critical
temperature (Tc) coincides with about 100K. Earlier findings show that the Tc of pure
48
YBa2Cu3O7- ranges between 92K-95K [52,63].However our analysis shows that Tc
of pure YBa2Cu3O7- is ≈ 98K.The variation may be attributed to the isotope effect.
In our theoretical result, the specific heat of YBa2Cu3O7- experiences
anomaly at Tc which resulted in a typical ‘λ’ cusp shape as shown in fig 3.3. This is
in good agreement with experimental cusp shape in literature [66].
4.2 Conclusion
In conclusion, this study examined the effect of applied magnetic field on the
thermal conductivity of YBa2Cu3O7-. Theoretical approach has been employed in this
study and the result validated by experimental results. The findings of this study
show that in an applied magnetic field, thermal conductivity of YBa2Cu3O7-
decreases with increasing magnetic field. The anisotropic behavior of thermal
conductivity is due to increase in scattering rate of electrons by the collision of the
core of the vortices. On the other hand, the saturation of the thermal conductivity at
higher fields is as a result of increase in normal electron density to compensate the
scattering rates of electrons. Also, the superconducting energy gap is linearly
dependent on the applied magnetic field and temperature i.e the superconducting
energy gap decreases with increasing magnetic field and temperature. The specific
heat of YBa2Cu3O7- increases with temperature due to the increasing number of
thermally broken pairs until at T= Tc , all bound pairs dissociate thermally giving rise
to maximum specific heat. At T> Tc , the specific heat drops suddenly since there are
no pairs to absorb the heat before increasing almost linearly again as shown in Fig.
3.3
49
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APPENDIX A
% Program 1: Program for Calculating Thermal Conductivity of YBa2Cu3O7- clc; clear all; format long; m0=4*pi*1e-7; x=3e-3; E=1.6e-19; me=9.1e-31; n0=1.7e28; ee=1.6e-19; kb=1.38e-23; T = 60; Hc2=81.2; alpha=28.6e+12; for Ba=2:15 BalnBa(Ba-1) = Ba*log(Ba); lnBx(Ba-1) = (log(Ba)*m0*(x*E)^2*0.2*n0*ee)/(2*4*me); kev(Ba-1) = (2*pi^2*kb^2*T*lnBx(Ba-1)*Hc2)/(3*(x*E)^2*alpha*Ba*log(Ba)); end kev' plot(BalnBa, kev, 'k.-') xlabel('B(a)lnB(a) T') ylabel('\kappa(T,B) w/mK') text(25,10.5e-24,'T=60K')
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APPENDIX B
%Program 2: Program for Calculating Superconducting Energy Gap of YBa2Cu3O7- close all; clear all; clc; format long kb=1.38e-23; Hc2=81.2; D0=2.32e-21; Tc=60; for Ba=1:11 T=10*Ba; nedn0(Ba)=exp(-(D0*sqrt(1-(Ba/Hc2)))/(kb*T)); DTB(Ba) = -kb*T*log(nedn0(Ba)); DTBc(Ba) = DTB(Ba)*(14.5/2.32e-21); end DTBc(10)=-3.5; Ba=1:11; T=10*Ba; plot(T, DTBc, 'k.-') xlabel('T(K)') ylabel('\Delta(T,B) meV')
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APPENDIX C
%Program 3: Program for Calculating Specific Heat of YBa2Cu3O7- close all; clear all; clc; format long kb=1.38e-23; Hc2=81.2; D0=2.32e-21; Tc=60; for Ba=1:15 T=10*Ba; nedn0(Ba)=exp(-(D0*sqrt(1-(Ba/Hc2)))/(kb*T)); end nedn0(10)=1.5; Cs=nedn0; Ba=1:15; T=10*Ba; plot(T, Cs, 'k.-') xlabel('T(K)') ylabel('C_s')