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.

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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].

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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')

55

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')