### Freezing of the Flux Liquid in High Temperature

### Superconductors

### A thesis submitted for the degree

### of Doctor of Philosophy of

### the Australian National University

### Damian Justin Charles Jackson

### The work contained in this thesis is my own original research, produced in col

### laboration with my supervisor — Dr M. P. Das. Any material taken from other

### references is explicitly acknowledged as such.

*( *

*j*

### Abstract

The work presented in this thesis is concerned w ith exam ining th e freezing tra n sition of th e m agnetic flux lines, and the effect of this tran sitio n on the critical current. It is believed th a t in the region of low m agnetic field th e freezing line corresponds well w ith the so called irreversible line th a t appears in th e

*H -T*

phase
diagram of the HTcS m aterials. This therefore provides a useful reference to com
pare th e results of this approach.
C hapter 1 presents a general introduction to th e phenom enon of superconduc tiv ity of b oth the conventional and high tem p eratu re superconductors. It is argued th a t m any of th e m ore im portant properties of th e HTcS m aterials, such as the critical current, are heavily dependent on the interactions of th e flux lines, and therefore th e real world usefulness of these m aterials relies on a good understanding of th eir properties.

In C hapter 2 a brief review of the ideas th a t have be pu t forward to explain th e n atu re of th e irreversible line is presented. Predictions m ade by these theories are th en briefly com pared with the results of recent experim ents. This leads to the proposal of th e irreversible line representing a freezing of th e flux liquid into a flux solid.

To exam ine the freezing transition, a relatively straightforw ard theory has been developed, known as density functional theory. The basis for this theory is presented in C hapter 3, and the formalism developed. The freezing of th e flux lattice can then be calculated using a phenomenological form for th e interaction potential of the flux lines. The results obtained for the freezing line enable a com parison between experim ent and theory.

In C hapter 4 an a tte m p t to b e tte r understand the interaction po ten tial of the flux lines is considered via a simple two-layer model. Such a m odel enables a b e tte r understanding of the n atu re of the full 3D interaction. Having calculated th e inter action potential, it is then possible to repeat the calculation of C hapter 3, replacing th e original phenomenological potential. The results are again com pared w ith both experim ent and earlier results.

T h e ap p ro x im a te lo catio n of th is s ta te in th e phase diag ram is calcu lated , and its consequences discussed in te rm s of ex p erim en tally m easu rab le effects.

F inally, C h a p te r 6 exam ines th e critical cu rren t by using a m odel of a g ran u lar su p erco n d u cto r. T h e Josephson cu rren t is calcu lated by finding th e G reen ’s fu n c tions for th e sy stem in th e absence of m ag n etic field, and th e n using th ese in a linear response theory, valid for sm all values of applied field. W hile th is work has n o t yet been com pleted, p relim in ary resu lts show th a t th e calcu lated form of th e critical cu rren t is consistent w ith previously pu b lished results.

T he w ork in C h a p te r 3 has been p u b lish ed as,

Jackson, D. J. C. an d D as, M. P. 1994. *Solid State Comm.* 9 0 , 479.
G ross, E. K. U. and D reizier, R. M. (eds). 1995. *D ensity Functional*

### Acknowledgements

### I would first like to thank my supervisor, Dr M. P. Das, for the assistance that

### he has provided throughout the period of my research. Thanks should also go to

### Dr A. J. Davies, who during his time in the department provided invaluable help

### and encouragement.

### The Department of Theoretical Physics has proven to be a pleasant and interest

### ing place to work. I would like to thank my fellow students for making my time here

### so memorable. I would especially like to thank Derek Daniel, for many late night

### discussions, and Alex Vagov, who has shown considerable restraint to my continual

### questioning of him, not only on physics, but also Russian language and culture.

### In addition I would like to express my gratitude to Prof. N. M. Plakida for his

### kind invitation to visit the Joint Institute for Nuclear Research in Dubna, Russia,

### and his associate Viktor Udovenko, who was of immeasurable help during my stay.

### Finally I would like to dedicate this thesis to my parents, to whom I have been

### more trouble than I’m worth throughout the years.

### While in Australia my work has been supported by both an ANU scholarship

### and a Special Overseas Postgraduate Research Award provided by the DEET.

### Honi soit qui mal y pense.

### Contents

**1 In tr o d u c tio n ** **1**

### 1.1 Properties of the Superconducting S t a t e ...

### 2

### 1.1.1

### Measurable Quantities for Conventional Superconductors . .

### 3

### 1.1.2

### Measurable Quantities for H T cS ...

### 4

### 1.2 Phenomenological Theories...

### 6

### 1.2.1

### Two-Fluid M o d e l...

### 6

### 1.2.2

### London T h e o r y ...

### 7

### 1.2.3

### Pippard T h eo ry ...

### 8

### 1.2.4

### Ginzburg-Landau Theory ( G L ) ...

### 9

### 1.3 Microscopic T h eo ry ...

### 11

### 1.3.1

### Conventional Superconductors...

### 11

### 1.3.2

### High Temperature Superconductors...

### 14

### 1.4 Critical Values and Flux D y n a m ic s...

### 15

### 1.5 Motivation of this T h e sis...

### 17

**2 A R e v ie w o f th e P r e se n t S ta tu s o f F lu x L a ttic e M e ltin g ** **19**

### 2.1 Anderson-Kim Flux C r e e p ...

### 19

### 2.2 Giant Flux C r e e p ...

### 23

### 2.3 Collective Pinning...

### 25

### 2.4 Thermal Fluctuations and Flux Cutting ...

### 28

### 2.5 Vortex E ntanglem ent...

### 31

### 2.6 Vortex Glass P h a s e ...

### 34

### 2.7 What the Experiments S a y ...

### 36

### 2.7.1

### Decoration E x p erim e n ts...

### 36

### 2.7.2

### Mechanical M easurem ents...

### 37

### 2.7.3

### Resistivity vs. T em p e ra tu re ...

### 37

### 2.7.4

### Small Angle Neutron Scattering (S A N S )...

### 38

### 2.7.5

### Lorentz Microscopy...

### 38

### 2.8 S u m m a r y ...

### 39

### x

*Contents*

**3 D e n sity F u n ctio n a l A p p ro a ch to F reezin g ** **41**

### 3.1

### Density Functional T h e o ry ...

### 42

### 3.1.1

### The F u n c tio n a l...

### 42

### 3.1.2

### Basis of D F T ...

### 43

### 3.2 Equilibrium theory of Freezing...

### 47

### 3.3 Calculation of the Freezing L in e ...

### 48

### 3.3.1

### Choosing a Solid D ensity...

### 48

### 3.3.2

### The Two Particle Correlation F u n c tio n ...

### 51

### 3.4 Determination of the Freezing lin e ...

### 53

### 3.5 S u m m a r y ...

### 56

**4 P r o p e r tie s o f th e P a n ca k e V o r tex L a ttic e ** **59**

### 4.1

### Ginzburg-Landau Energy F u n c tio n a l...

### 59

### 4.2 The Two-Layer P ro b le m ...

### 62

### 4.2.1

### Magnetic Field due to Pancake Vortices ...

### 63

### 4.3 Single Layer Potential ...

### 65

### 4.3.1

### Rotationally Symmetric Solution ...

### 66

### 4.3.2

### Non-Aligned Vortex Position...

### 69

### 4.4 Multi-Potential Density Functional Theory (M P D F T )...

### 70

### 4.5 S u m m a r y ...

### 73

**5 E x iste n c e o f th e V o r tex G as ** **75**

### 5.1

### The Vortex Gas P h a s e ...

### 75

### 5.2 Experimental C onsequences...

### 77

### 5.3 S u m m a r y ...

### 78

**6 C ritica l C u rren t in a G ranular S u p e rc o n d u c to r ** **79**

### 6.1

### Second Q u a n tis a tio n ...

### 79

### 6.2 Green’s Function Approach to Superconductivity ...

### 84

### 6.3 Current in a Small External F i e ld ...

### 86

### 6.4 Zero Field Green’s Functions...

### 88

### 6.5 Simplification of the Current Expressions...

### 90

### 6.6 Calculation of the C u r r e n t...

### 93

### 6.7 S u m m a r y ...

### 94

**7 C o n clu sio n ** **97**
**A ** **101**

### A.l Properties of the Energy Functional...

### 101

CHAPTER 1

### Introduction

*Successful research impedes further successful research.*

— *K eith J. Pendred, The Bulletin o f the A tom ic Scientists, 1963*

Prior to 1911, it was believed th a t there was nothing special to be found in the behaviour of th e conductivity of m etals. Their resistance to th e flow of electrical current was to be expected, even if it was not particularly appreciated. This all changed when a D utch physicist called Heike K am erlingh Onnes decided to use his knowledge of liquefying Helium to study m aterials at very low tem peratures. He found (Onnes, 1911) th a t when m ercury was cooled to around 4K it lost all electrical resistance. This loss of resistance was quite sudden, and happened over a very small range of tem p eratu re.

Since the discovery of this effect in mercury, it has been found th a t m any el
em ents and com pounds also possess sim ilar behaviour when th eir tem p eratu re is
lowered sufficiently. Superconductivity is m ainly associated w ith this ra th e r dra
m atic loss of electrical resistance, although there are m any other effects th a t these
m aterials m anifest, some of which are m entioned below. W hile this phenom enon
was well known and categorised by experim entalists, it took nearly 45 years for
a satisfactory theory to emerge — the famous Bardeen-Cooper-Schrieffer (BCS)
theory of superconductivity (Bardeen *et al.*, 1957).

Ju st when theorists thought they had an understanding of superconductivity, another discovery in 1986 by Bednorz and Müller (Bednorz and M üller, 1986) tu rn ed everything upside down. This was the discovery of th e so called high tem p eratu re superconductors (HTcS). Before these compounds were found, it was believed by m any th a t th e highest critical tem perature had already been reached, and th a t new com pounds would only change this by a fraction of a degree. This is not to say th a t th e search stopped. The goal of a higher critical te m p eratu re led to th e study of m any different, exotic m aterials such as m etallic alloys, interm etallics, nitrides and carbides, in addition to the standard metallics. The breakthrough was finally m ade w ith the synthesis of the new oxide m aterials. These new m aterials, therefore, produced quite a shock when they were found to have critical tem p eratu res of around

**2** *1. Introduction*

35K, com pared w ith th e conventional m aterials of around 23K.

Once again, these m aterials are one step ahead of th e theorists. W hile m any more m aterials are being found w ith higher and higher critical tem peratures, and m any HTcS m aterials are finding their way into real world applications, there is still no com plete theory for this phenom ena. This chapter will provide a brief introduction to bo th experim entally m easurable effects of superconductivity and provide a to u r of th e theories th a t have been used to m odel them . The chapter ends w ith a discussion of th e m otivation for this work. This is in no way intended to be a rigorous introduction to the subject, which can be found in alm ost any book on the subject (Rickayzen, 1965; de Gennes, 1966; Schrieffer, 1964; T inkham , 1975; Cyrot and Pavuna, 1992).

### 1.1 Properties of the Superconducting State

Conventional superconductors appear in two forms, known as type-I and type-II.
A m ajor difference is th a t type-I m aterials allow no m agnetic flux to enter the sample
interior as long as the tem p eratu re is m aintained below the critical tem p eratu re, *Tc.*
All the m agnetic flux is expelled until the applied m agnetic field reaches the critical
field, *H*c, when superconductivity is destroyed and the m aterial reverts to its norm al
state. This to tal exclusion of m agnetic flux is called the M eissner-Ochsenfeld effect,
and in this regime th e m aterial exhibits perfect conductivity. These m aterials tend
to have quite low values of b o th critical tem p eratu re and critical field.

Type-II m aterials have two distinct critical fields. The lower critical field, 77cl,
and th e upper critical field, *Hc2j* which plays a role analogous to the critical field in
type-I m aterials in th a t it denotes the field at which superconductivity is destroyed.
For m agnetic fields between these values, th e field penetrates the sample as lines
of m agnetic flux parallel to th e applied m agnetic field. For m agnetic fields between
*H*ci and *H c2* perfect conductivity is lost, for reasons th a t are discussed later. Each
m agnetic flux line contains a m agnetic field of size equal to the m agnetic flux quan
tu m , *<f*)o = *h e /*2e. These flux lines have a m utual repulsion due to their m agnetic
fields, and have been shown to combine to form a lattice of flux lines inside the
m aterial, known as an Abrikosov lattice (Abrikosov, 1957). A schem atic is shown in
Figure 1.1. The phase diagram for a typical conventional superconductor is shown
in Figure 1.2. All th e HTcS m aterials are of type-II, and only this type will be
considered throughout this thesis.

*1.1. Properties o f the Superconducting State* 3

Figure 1.1: A schematic drawing of the Abrikosov lattice. The flux lines
(tubes) can be seen to form an hexagonal lattice in the *x -y* plane, where the
magnetic field is applied parallel to the 2-axis.

### 1.1.1 Measurable Quantities for Conventional Superconductors

T h ere are q u ite a few interesting properties ex h ib ited by m a te ria ls w hen th e y becom e sup erco n d u ctin g . A b rief description of a few of th ese p ro p erties is given below, an d m an y m ore can be found in th e references:• As m e n tio n ed above, th e M eissner-Ochsenfeld effect enables th e m a te ria l to ex h ib it b o th zero resistance and perfect diam ag n etism .

• W hen th e specific h eat is m easured, it is found to ex h ib it a ju m p a t th e
tra n s itio n te m p e ra tu re . As the te m p e ra tu re is raised, th e specific h eat s ta rts
to increase q u ite sharply. As it approaches *Tc* it s ta rts to curve over, and
undergoes a ra p id drop down to th e value of th e n o rm al m a te ria l, whose
b eh av io u r it th e n continues to follow.

• M icrow ave an d infrared experim ents can be used to show th e existen ce of an energy gap in th e ex citatio n spectrum . It can be shown th a t th is energy gap decreases w ith increasing te m p eratu re, u n til it vanishes a t th e critica l te m p e ra tu re . T he physical reason for th e energy gap is discussed below in rela tio n to BCS theory.

[image:11.546.125.434.91.295.2]4 _{1. Introduction}

**Normal Phase**

**Abrikosov Lattice Phase**

**Meissner Phase**

**Temperature**

Figure 1.2: Typical magnetic phase diagram for a conventional low tem pera ture superconductor.

rela tio n sh ip betw een isotopic m ass an d tra n s itio n te m p e ra tu re helped to p oint th e way to phonon m e d iate d electro n -electro n a ttra c tio n .

• T h e w avefunction of th e electro n p air ap pears to be a spin singlet.

• T h e m a te ria ls are found to possess a London m om en t of value —2m w /e , w here
*u)* is th e an g u lar frequency of th e ro ta tin g su p erco n d u ctin g disc.

T hese are ju s t a few of th e stran g e p ro p erties possessed by th ese m a teria ls. As it can be seen, th e re is a rem arkable diversity of effects th a t w ould need to be explained by a th e o ry of sup erco ndu ctiv ity .

### 1.1.2 Measurable Quantities for HTcS

[image:12.546.113.422.90.301.2]**1.1. Properties of the Superconducting State**

5
**1.1. Properties of the Superconducting State**

**Normal Phase**
**Flux Fluid Phase**

**Irreversible Line**

**Abrikosov Lattice Phase**

**Temperature**

Figure 1.3: Typical magnetic phase diagram for a HTcS.

So while there is some sim ilarity between conventional and HTcS m aterials, m any of the properties of the HTcS are anomalous. In addition, th ere are several properties th a t are very hard to explain in term s of the theories th a t were developed for th e conventional superconductors, some of which are listed below:

• The observed values of th e critical tem p eratu re are found to be very high — of th e order of 150K com pared with around 20K. This is hard to explain w ithin th e fram ework of the electron-phonon interaction of the BCS model.

• The coherence length is very small and anisotropic, w ith th e com ponent paral lel to the copper oxide planes being larger th a n th e perpendicular com ponent. In addition to this anisotropy, the core of th e flux lines are also m uch th in n er in HTcS m aterials, so they are less likely to get caught at pinning centres present inside the m aterial. These small values also increase th e effect of fluc tuations on the physical properties. In addition, th e m agnetic pen etratio n depth is found to be m uch greater th an the coherence length while possessing the sam e anisotropy.

[image:13.546.125.440.97.306.2]**6** *1. Introduction*

*•* W hen plotting out the phase diagram in th e *H - T* plane, it is found th a t a
new line appears betw een the upper and lower critical tem peratures. This
line is called the m elting line, and m arks the division between the flux liquid
and flux solid phases. Experim entally, this line m arks a distinction between
phases of reversible and irreversible m agnetic behaviour.

These are ju st a few of th e features th a t are hard to explain with the theories for conventional superconductors. It would seem th a t both types of superconductors still have several properties in common, and so any theory put forward to describe HTcS m ight be expected to possess sim ilarities to those for conventional m aterials. However, it is expected th a t in the HTcS m aterials th e physics of the flux line lattice becomes much more im p o rtan t. M any of th e m easurable features of these m aterials derive from th e behaviour of the flux lines. Therefore it would appear th a t a knowledge of the physics of the flux lines is ju st as im p o rtan t as a microscopic description.

As a first approxim ation it would seem to make sense to first try to apply con ventional theories to these new m aterials, and to “tw eak” them where required. If this is the case, w hat then are th e theories used to explain conventional supercon ductivity?

### 1.2 Phenomenological Theories

There were m any theories originally pu t forward in an a tte m p t to explain su perconductivity. This is in no way intended to be a thorough list, bu t only those th a t can be considered to have had a m ajor im pact on the evolution of the field.

### 1.2.1 Two-Fluid Model

One of th e first atte m p ts to exam ine superconductivity as a two-fluid m odel
was m ade by G orter and Casim ir (G orter and Casim ir, 1934). Their idea was
th a t as the superconducting tran sitio n was a second order phase change, an order
param eter could be introduced whose value would decrease gradually w ith increasing
tem p eratu re until it vanished for *T > Tc.* This order param eter could be thought
of as m easuring th e fraction of m aterial th a t is in the superconducting phase. To
find th e behaviour of th e order param eter the following form for the free energy was
postulated,

*F ( T )* = *x F , ( T )* + (1 — x )1/ 2 *Fn(T),*

where *x* plays th e role of the order param eter. These term s are chosen so th a t *x ( T )*
can be determ ined in term s of *Fn(T)* and *Fa(T)* from the equilibrium condition

*1.2. Phenomenological Theories* *7*

This choice is equivalent to having the two phases m utually dependent. A fter m in imising, one finds th a t

*x(t)* = 1 — t 4,

where *t* is the reduced tem perature, *t = T / T c.* Even though th eir choice for the
free energy had little physical justification, th e expression for *x ( T )* gave a very
good description of th e tem perature dependence of the p en etratio n depth when
su b stitu ted as th e electron density in the London model.

### 1.2.2 London Theory

One of th e m ost comprehensive of the early theories was th a t proposed by F. Lon
don and H. London (London, 1950). Their idea was th a t the local m agnetic field
was th e factor th a t controlled the supercurrent. T he to tal current inside a supercon
ductor is the sum of th e supercurrent, and th e norm al current, *j n .* The equations
used to describe this system are,

V x (Aj.) = —— c

*I t* = e

**J ** **Jn T Js**

**jn **= 0-e,

where *p* is th e electric charge density, *o* the electrical conductivity and A is a
p aram eter to be determ ined. In addition to these expressions, M axwell’s equations
are also required. By combining all the above equations, several of the variables
can be elim inated. If, in addition, it is assumed th a t there is charge neutrality, the
static case reduces to

V 2h = / *An*

### lx?

**h,**

w ith sim ilar expressions for e and **j. **It can also be shown th a t th e London equations
im ply perfect conductivity. The last equation is very im p o rtan t, in th a t it shows
th e existence of a m agnetic penetration depth,

A *— c*

This implies th a t the m agnetic field is not discontinuous across th e surface of a superconductor, bu t instead decays exponentially w ith distance. These equations can be solved analytically for many different situations to give th e behaviour of th e m agnetic field w ithin the superconductor.

### 8

*1. Introduction*

achieve th is, shielding cu rren ts are prod u ced w ithin a skin of d e p th A aro u nd th e edge of th e su p erco n d u cto r th a t produ ce a field equal and o p posite to th e applied m ag n etic field. H owever, if th e su p erco n d u cto r contained a hole of som e sort, th e m ag n etic flux w ould be able to pass th ro u g h th is hole, b u t still n o t p e n e tra te th e su p erco n d u cto r itself.

T he final p a rt of th e London th e o ry of in terest is th e London kernel. In London th e o ry th e re is a connection betw een th e cu rren t density an d th e vecto r p o te n tia l of th e ap plied m ag n etic field,

**A(r) = —cAjs(r),**

w hich can be w ritte n in Fourier space as

### j‘(q) = " ( X )

A(q)-U sing b o th th is rela tio n , and th e M axw ell relatio n betw een m ag n etic field and cu r ren t density, th e following expression can be derived,

**j»(q) = - ( £ : ) ^ ( q ) A ( q ) ,**

w here *Kl(*q ) is th e London kernel, an d is given by

**K**

**K**

**l****M**

**M**

**4-7T**

X ? '

It can be seen th a t th is kernel is in fact q-independent. To in tro d u ce a depen d en cy
on *q* in to th is expression leads to an ex am in atio n of th e P ip p a rd kernel, *K p {*q ).

### 1.2.3

### P ip p a rd Theory

W hile th e London th e o ry gave a good general descriptio n of th e su p erco n d u cto rs, th e re was ex p e rim e n ta l evidence of b ehav io ur not explain ab le w ith in th e London fram ew ork, such as th e v ariatio n of A w ith o rien tatio n , an d th e effect of im p u rities. P ip p a rd in tro d u ced th e concept of th e “range of coherence” (P ip p a rd , 1953) to describe th e long range in teractio n s of th e electrons in th ese m a teria ls, an d from th is p ro d u ced a set of nonlocal relation s.

P ip p a rd m easu red th e effect of im p u rities on th e p e n e tra tio n d e p th and found th a t it decreased ex p o n en tially w ith increasing im p u rity co n cen tratio n . F rom th is, P ip p a rd w rote down, in analogy w ith th e anom alous skin effect, th e following no n local relatio n ,

**3 ** **/• j _ r (r - A ) e x p ( - r / g )**

**s**

**s**

**47tc£0A**

**J**

**J**

**4**

In spherical co o rd in ates it is possible to in te g ra te this expression, an d so derive an expression for th e P ip p a rd kernel in F ourier space,

* K P (q)* 3£

*1.2. Phenomenological Theories* 9

If this form is com pared w ith the London kernel, the following relations can be shown,

A = *\l*

*X = X *

*J U T ( k*

*\ 1/ 3*

A i(2x)i/3 ( aJ

### £3

### «

**toll**

**toll**

*e*

**» €oA|.**

In addition, th e following form for £ was found to agree well w ith experim ent,

1

### _

1 1*W) ~To + ^i’*

where *t* is the m ean free p ath and a is a fitting param eter. This m odel was then
able to account for most of the behaviour not described by the London theory. Thus
P ip p ard introduced th e idea th a t any pertu rb atio n would not be a local effect, but
would instead have its influence extended over a region of thickness £. This coher
ence length, along w ith the nonlocal n atu re of the theory, were th e m ost im p o rtan t
concepts introduced by Pippard. There were still some effects th a t had not been ex
plained satisfactorily, such as the tem perature dependence of th e p en etratio n depth.
One such theory th a t sought to do so was th a t developed by G inzburg and Landau.

### 1.2.4 Ginzburg-Landau Theory (GL)

This theory derives from work done by Landau on exam ining second order phase transitions. The central idea being th a t when a substance undergoes a phase change, th ere is some physical property of the system th a t differs between th e two phases. If it is possible to identify this property, it can th en be used to study the tran si tion. Landau theory studies the phase transition through th e change in sym m etry th a t occurs between the two phases. This loss of sym m etry can be m easured by introducing th e idea of an order param eter, which is a m easure of th e departure of th e new configuration from the original, higher sym m etry phase. In addition to th e order param eter, the Landau free energy is also defined, from which physical properties can th en be derived. The Landau theory of phase transitions has since been applied to m any problem s, and its foundations and applications are studied in detail in the book by Toledano and Toledano (Toledano and Toledano, 1987) as well as th e original work of Landau (Ginzburg and Landau, 1950; Landau, 1965). It is im p o rtan t to note th a t the theory is a phenomenological theory, in th a t it assumes the existence of a phase transition and a change in sym m etry, and no a tte m p t is m ade to explain the microscopic origin.

A good startin g point is the definition of th e Landau free energy functional, which is given by

**10** *1. Introduction*

w here *ip* is an order p a ra m e te r and *a* an d *ß* are unknow n coefficients, w ith a cho
sen to be te m p e ra tu re d ep en d en t. T h e te rm linear in *ip* can be shown to be zero
th ro u g h sy m m e try arg u m en ts. B o th *e** and *m** represen t effective electron charge
an d effective m ass respectively. T his is done to tak e into account possible electro n
pairing or som e o th e r m echanism . T his fu n ctio nal will be discussed in m uch m ore
d etail in a la te r ch ap ter.

By m inim ising th e to ta l free energy w ith resp ect to b o th *ip** and A th e following
set of eq uations are gen erated ,

**— I**

**2771* '**

**— V x h =**

*An*

*e** \ 2

*-ih\7* -|---A J *ip* + *a ip* + *ßip\ip\2 =* 0

*e*h * *(e*)2*

*(ip*Vip — ipV ip*)*--- *\ip* J2 A.,

2m*i *m e*

( l . i )

along w ith th e following b o u n d ary conditions*,

**n **

• **f - i h v + j A ) 4> =**

**f - i h v + j A ) 4> =**

### 0

**n x (h - H) = 0,**

w here

**H **

is a uniform e x tern al field. It is possible to solve th ese eq u atio n s in certain
geom etries, an d doing so yields two im p o rta n t length scales — th e p e n e tra tio n d e p th
and th e coherence leng th. Expressions for th ese q u an tities are respectively
*m =*

A(T) =

**1 / 2**

*2m*ot!(Tc* — *T ) J*
*m*c2ß*
*An* (e*)2 *a' (Tc* — T )

**1 / 2**

w here th e te m p e ra tu re dependence of *a* has been chosen to be of th e form *ot!(T — Tc).*
From th ese expressions th e te m p e ra tu re dependence of th ese q u an tities is clearly
visible. T hese lengths can be com bined to form a te m p e ra tu re in d ep en d en t scaleless
q u an tity , th e G inzb u rg -L an d au p a ra m e te r,

*k* = *m*

*m* *'*

T his p a ra m e te r can be shown to differen tiate betw een th e two types of sup erco nd u c
tors. For *k < l / y / 2* th e eq uation s describe a ty p e-I m a te ria l, w hereas for *k > l/y / 2*

th e y describe a ty p e -II m ateria l.

*1.3. Microscopic Theory-* **11**

Finally, it should be noted th a t this theory also nicely describes th e quantisation of m agnetic flux. If the order param eter is w ritten in th e form

th en when this is su b stitu ted into equation (1.1) and integrated around a closed curve, Stokes’ theorem can be used to show

### /

**d S - h +**

*m c*

(e*):

### /

*di***• j ** *he*

### Ivf = ±7V ’

where th e fundam ental flux quantum is defined as
*he*

**2**

### ?

**(**

**1**

**.**

**2**

**)**

where it has been assum ed th a t *e* =* 2e.

The beauty of this approach lies in its simplicity, which should not be confused
w ith its usefulness. Some of the power of GL theory has been m entioned above,
and m ore can be found in the references. The application of GL theory to type-
II superconductors is covered in considerably m ore detail in th e book by Saint-
Jam es (Saint-Jam es *et al*., 1969). The theory gives th e ability to step back from
th e microscopic n atu re of the problem , and atte m p t to make predictions based on a
m ore macroscopic viewpoint. This m eans th a t even though no microscopic theory
m ay be available, an understanding of some of the underlying m echanism s can
still be a ttain ed through this approach. It can be particularly useful in exam ining
the behaviour of the flux lines, which would be a daunting task startin g from a
microscopic theory (Das, 1989).

A microscopic theory is still needed, however, to explain, for exam ple, th e origin of electron pairing. Such a theory is available for conventional superconductors, and is presented next.

### 1.3 Microscopic Theory

### 1.3.1 Conventional Superconductors

**12** *1. Introduction*

One of th e precursors to the form ulation of this theory was the discovery by Cooper (Cooper, 1956) th a t it was possible for the norm al state to be unstable to a certain kind of electron pairing. It was shown th a t the binding energy for a single pair of electrons could, under certain conditions, include negative values. This effect can be shown to be due to an over-screening of the Coulomb repulsion by th e ions. The m odel th a t was considered consisted of pairs of electrons possessing equal and opposite m om enta and so lying on opposite sides of the Fermi surface. The H am iltonian for this model is given by,

*H = Y , 2 e k b+k bk* + £ W t i f c .

*k * *kk'*

where

*Vek * *= ( k ' , - k ' \ V \ k , - k )*

is the interaction potential describing the scattering of particles from one state ( k | , k ^ ) to another w ith different m om entum ( k '^ j k 'l ) , and

*K = c k ^ c - k i*

*W* = C - 4 Cf c|.

To find a solution to this H am iltonian the following trial wavefunction was used,

W hen this is su b stitu ted into the H am iltonian, the following expression is m inimised,
*5(tP\H - * *=* 0,

where *N* is th e to tal num ber of particles. A fter perform ing the m inim isation, one
finds th a t th e energy required to create a quasiparticle of m om entum *k* is given by

*E 2k* = (e* - M)2 + A 2,

w ith *Ak* being the “energy gap” which satisfies th e relation

**A k**

**y Vkk'**

**y Vkk'**

*k'*

**Ak>**

2 *E k>*

Two approxim ations are generally m ade to solve these equations. One is th a t *fi* = 0,
and the other is

*Vkk'* *- V*

0

for *\ek \* and |efc/| < *ljc*

otherwise.

*1.3. Microscopic Theory-* *13*

of the Fermi distribution function. The energy gap can be found as a function of tem p eratu re from the tem perature dependent version of the earlier definition

1 **r u* ****d e**

0) *V * * Jo* tanh £ (e2 + A 2)1/2

*N ( 0 ) V * *Jo* (e2 + A 2)1/ 2

In th e weak coupling lim it, several predictions are m ade by BCS theory, 2 A 0

*kß Tc* *=* 27re-7 « 3.52
*Tc C J T c)* e27 A i „ o

### W

1 ^ " 0 -168*H C{T)* « *Hc{*0)

*H C(T)* « 1.74 # c(0) (
*C , - C n*

* r p* \

**2**"

1 - 1 . 0 6 1 — T

**Te ** 7C(3) ^ L43
*Tc =* 0.85 0d*e~1/N(°) v .*

*T* -> 0

*T ^ T r*

These predictions were found to be in reasonable agreem ent w ith m any of th e m a terials th a t were known to be superconducting. However certain m aterials such as lead and m ercury were found to be in disagreement. For m ore details on the agreem ent between theory and experim ent the reader is referred to th e book by Parks (Meservey and Schwartz, 1969).

BCS theory has since been rew ritten in the language of field theory (Abrikosov
*et al.,* 1963). W hile this obviously changes none of th e results, it does allow for a
m ore concise form ulation of the theory. It should also be noted th a t it has been
shown (Gorkov, 1959) th a t close to *Tc* the BCS and GL theories are equivalent.
This work helps relate th e GL functions to the param eters th a t appear in th e BCS
equation, for exam ple the order param eter can be rew ritten as

= *( .* 7C(3)r N 1/2

**V>(r) =**

V8 *n 2( k BT)-*

**A(r),**

and the GL phenomenological param eters *a*and

*ß*as

67r2(fcs T )2 / *T \* 6tt

“ 7C(3)e» l *T j '* P 7 C (3 )e J n ‘

This new form ulation has also been a stepping stone for fu rth er extensions to th e theory. A set of integral equations were derived (Eliashberg, 1960) in an a tte m p t to introduce a frequency dependence for the energy gap — to help produce a retard ed interaction th a t was dependent on the phonon velocity. The theory of the electro m agnetic properties was also given a sounder footing (Rickayzen, 1959; Bogoliubov,

14 *1. Introduction*

### 1.3.2

### High T em perature S uperconductors

U n fo rtu n ately , as of yet this section rem ains q u ite sh ort. T h ere still is no com p le te th e o ry for H TcS in th e sense th a t BCS th e o ry was a com plete th eo ry for th e conventional su p erco n d u cto rs. As m en tio n ed above, due to th e sim ilarities of sev eral of th e effects ex h ib ited by th e tw o types of m a teria ls, it was hoped th a t som e form of ex tension of th e BCS th e o ry could be used. Some of th e questions th a t m u st be answ ered by any such th e o ry are:

• A re phonons th e m ed iato rs of th e in te ra c tio n betw een electrons, or do o th e r effects need to be considered, such as spin or charge flu ctu atio n s, or a com bi n a tio n of th ese effects?

• How im p o rta n t is th e crossover from 2D to 3D th a t is presen t in th ese m a te ri
als? T his s ta te m e n t requires a little elab o ratio n on th e s tru c tu ra l p ecu liarity
of th e oxide su p erco n d u cto rs. In these m ateria ls it is believed m ost of th e
cu rren t flow occurs w ith in th e C u -0 layers. T hese layers are sep arated from
th e ir n eighbours by a d istan ce th a t is d ep en d en t on th e chem ical com position
of th e m a te ria l. T hese system s can therefo re be considered as a stack of *N*
coupled layers, w hich in th e lim itin g case of zero in terlay er coupling reduces
to a tw o d im ensional problem .

• Is th is a singlet spin sta te , or a trip le t? T his is a very im p o rta n t q uestion, and has sparked m uch in terest, b o th th e o re tic ally and exp erim entally. As of yet th e re is still no consensus of opinion one way or th e o ther.

• A re all th e novel n o rm al s ta te p ro p erties p red icted ? It is im p o rta n t th a t n o t only th e su p erco n d u ctin g phase of th ese m a teria ls be explained, b u t any such th e o ry p u t forw ard m u st also provide an ad eq u a te descrip tion of th e n o rm al sta te .

*1.4. Critical Values and Flux Dynamics* 15

100 T _{Conventional Type-II}

HTcS

Figure 1.4: The relationship between the critical values of the magnetic held, current density and temperature for both the conventional and high temper ature superconductors.

### 1.4 Critical Values and Flux Dynamics

[image:23.546.103.447.97.333.2]16 *1. Introduction*

*al.* (B latter *et al.,* 1994) and B randt (B randt, 1995). Neglecting questions about
m echanism , the behaviour of the critical current can be studied from the viewpoint
of the flux lattice, if it is assumed th a t it is this phenom enon which has the greatest
effect on th e current.

It is known th a t each m agnetic flux carries one flux quantum of m agnetic field.
This flux line can be thought of as a tu b e of norm al m aterial w ith a radius approx
im ately equal to the coherence length. Around this core th e supercurrent circulates
forming a m agnetic sheath extending out to a radius given by the m agnetic pene
tratio n depth (Caroli *et al.,* 1964). This model for the flux line is quite simple to
picture, and useful for m aking predictions from theory. The exact n atu re of the
core is very difficult to describe due to the nonlocality of th e theory. Throughout
this work, this qualitative description will be used for th e flux lines, and the exact
n atu re of the core and other details will be neglected.

The reason th e flux lattice tends to suppress the critical current is simple to
understand. If th e flux lines are in a m aterial which is relatively free of any form of
pinning centres, then they are free to move around under th e influence of an external
force. This movement of the flux lines, along w ith th e dissipation caused by th eir
movem ent, has been exam ined by several authors (Anderson and Kim, 1964) and
more details will be given in the next chapter. W hen the current starts to flow, there
is an interaction between the current and the m agnetic field of the flux line. This
*J* X *B* force tends to move the flux lines perpendicular to the direction of current

flow, and therefore scatter the electrons, hence reducing th e m axim um current th a t can flow. If, however, th e m aterial contains a lot of defects, then the entire lattice tends to become pinned as the flux lines become trap p ed inside pinning centres. This enables a much higher current to flow and helps to explain why th e flux lattice is so im p o rtan t as regards the critical current. It would seem th a t to have as large a critical current as possible requires th a t th e m aterial used be able to effectively pin the flux lattice. One problem is th a t due to the small core and large m agnetic “sh eath ” , th e flux lines in HTcS are much m ore flexible th an their conventional counterparts. This m eans th a t th e concept of the rigid Abrikosov lattice is not really applicable, instead it is a more softer lattice, as shown schem atically in Figure 1.5. U nfortunately this added flexibility introduces m any complications, due to a higher probability of processes such as vortex cutting or entanglem ent, which are discussed in a little m ore detail in the next chapter.

**1.5. Motivation of this Thesis**

**1.5. Motivation of this Thesis**

**17**

**17**

**Q **

**Ö**

### Figure 1.5: In HTcS the flux lines are much more flexible, giving rise to a

### softer form of the Abrikosov lattice.

### explain the behaviour of the current, an understanding of the irreversible line is

### required.

### 1.5 Motivation of this Thesis

### This thesis is therefore involved with the study of the flux liquid within a HTcS

### material. Most specifically it is concerned with examining the irreversible line. It

### is important to note that opinion is divided as to what the irreversible line actually

### represents. In the present author’s opinion, this does represent some form of melting

### of the flux lattice, as it is hoped the following work will show. So as not to bias

### anyone’s point of view, Chapter 2 is a brief review of the most recent theoretical

### ideas put forward to explain the origin of the irreversible line.

### In Chapter 3 a density functional calculation is presented. This calculation uses

### a first principle approach to the examination of the melting/freezing transition.

### Using this method, the freezing point can be found as a function of both magnetic

### field and temperature and plotted out on the phase diagram to be compared with

### experiment. This is a first principle examination, requiring no free parameters to

### “fit the data.”

[image:25.546.109.447.91.302.2]18 *1. Introduction*

to the irreversible line.

In C h ap ter 5 the low density phase of the vortex liquid is studied. The exis
tence of th e vortex gas is postulated, and its position in the *H - T* phase diagram
calculated.

In C h ap ter 6 the critical current is exam ined in term s of the Josephson effect.
A sim ple m odel of a granular superconductor is used, w ith each grain being con
sidered as an island of superconductor in a sea of norm al m aterial. The grains are
Josephson coupled, and this intergrain current is w hat determ ines *J c.* Using the
Gorkov form alism both th e G reen’s function and anomalous G reen’s function are
calculated, and th en used to study the linear response of th e system.

CHAPTER 2

### A Review of the Present Status of Flux

### Lattice Melting

The m agnetic phase diagram for a conventional type-II superconductor has sev
eral im p o rtan t features, m any of which were discussed in th e last chapter. It was
shown th a t a new feature appeared in the m agnetic phase diagram of th e HTcS
m aterials. This new line is known as the irreversible line. Its nam e derives from
th e fact th a t on th e lower side of this line, the m agnetic behaviour of th e m aterial
is reversible, b u t above the line it becomes irreversible. The effect is thought to
be due to the m elting of the Abrikosov lattice into a flux liquidL This m elting
tran sitio n can take place more easily in the HTcS m aterials due to th e higher tem
peratures th a t are available for m easurem ents. T he tran sitio n is of great interest, as
th e critical current for a m aterial is highly dependent on th e state of th e flux lines.
If th e Abrikosov lattice does undergo a m elting transition, th en th e flux liquid will
correspond to a lower J c, due to stronger scattering. The following is a b rief review
of th e current theories used to describe th e n atu re of the irreversible line, startin g
w ith th e early theory of flux creep of Anderson and Kim, and progressing to the
newer theories pu t forward by B randt, Nelson, MalozemofF and Fisher *et al.*

### 2.1 Anderson-Kim Flux Creep

For th e flux lattice to be able to m elt, it m ust be able to break away from any
pinning centres, due to dislocations, vacancies, etc. To break free from th e pinning
centre, th e flux line needs to overcome th e potential barrier, which in th e HTcS
m aterials is possible due to the relatively high tem p eratu res involved. T herm al
activation also occurs in conventional superconductors, bu t due to th e relatively
low values of *Tc* the effect is less pronounced. One of the first theories to

exam-tin the case of low magnetic field the melexam-ting line and the irreversible line are in very close proximity to each other. Therefore melting is sometimes considered as the cause of the irreversible line.

20 *2. A R eview o f the Present Status o f Flux Lattice M elting*

ine th is p h en o m en o n for th e conventional sup erco ndu cto rs is due to A nderson an d K im (A n d erso n and K im , 1964). T h e m ain idea b eh in d th e ir p ap er is to ex am in e th e th e rm a l a c tiv atio n of flux lines p ast pinning centres. T h e pinning centres are used as a source for th e p o te n tia l b arrie r, and th e ir exact n a tu re was not im p o rta n t.

T h e ir m o d el consisted of a su p erco n d u cto r u n d er a m ag n etic field, *H*, ap p lied
p e rp e n d ic u la r to th e m a te ria l, and carrying a bulk cu rren t J = (c/47r)V **x** H . It
can be seen th a t due to th e presence of th e cu rren t in th e m a teria l, th e d en sity of
th e flux lines will be nonuniform . B ecause of th e m u tu a l repulsion of th e flux lines,
one can th in k of th e re being a m ag n etic pressure ex erted by th e flux lines on one
a n o th e r. T h is p ressu re is re la te d to th e m ag n etic energy p er u n it volum e, *H 2/8tt.* If
th e re w ere no p in ning sites, th e n th is pressure w ould act so as to m ake th e d en sity
of th e flux lines uniform , w hich w ould th e n lead to *J* = 0. An assu m p tio n m a d e
in th e ir d eriv a tio n is th a t th e in te rn a l and e x tern al m ag n etic fields are equal, i.e.
*B = H*, w hich is valid for all b u t low fields.

To begin to calcu late th e ra te of th is th e rm a lly activ ated m otion, know ledge of th e d riv in g force of th e flux lines, due to th e m ag n etic pressure, and of th e n a tu re of th e p o te n tia l b arriers is req uired . T h e in te ra c tio n energy betw een flux lines was c a lc u la te d by A brikosov (A brikosov, 1957), an d was w ritte n as

*Fint*

### ( i ) H ( H - ' 2V2H)

(**2**.**1**)

w here *k* is th e G inzb u rg -L an d au p a ra m e te r defined in th e last c h ap ter and *Kq* is a
B essel fu n ctio n . In m ost of th e conventional sup erco ndu cto rs th e d istan ce **|r» — i*j|**

is u su ally less th a n A, *V 2H* is sm all, so th e expression for free energy is sim p ly
given by th e m a g n etic energy. T his m eans th a t th e force on th e flux lines is ju s t th e
L orentz force, i.e. V F = J x H /c . T he force per flux line per u n it len g th is th e n
ju s t J x * <j>o/c,* w here

*is given by eq u atio n (1.2).*

**(f)0***2.1. Anderson-Kim Flux Creep* *21*

To find th e free energy of one of these bundles, a startin g point is th e Lorentz
force equation shown above, which for a bundle containing *rib* flux lines is given by
*J(f>0 ribl/c,* where *l* is an effective interaction length, and can be taken as th e distance
betw een pinning centres. As a function of the bundle position th e free energy can
be w ritten as

*J H X 2lx*

* * f o r c e* — •

**C**

Taking the size of the potential barrier to be approxim ately £0, th en its energy scales
as *( H 2/ 8tt)£q.* Assuming only a fraction *p* of this pinning force is acting, th en the
to ta l barrier energy can be w ritten as

**pHZ$\ _ ( J H ^ o**

**pHZ$\ _ ( J H ^ o**

8tt j \ *c*

T he ra te of barrier penetration per second can then be w ritten as

*R* = *u 0 e~Fb/kBT.* (2.3)

To find the ra te of diffusion of flux density [B|, th e rate at which flux enters and leaves a small volume is required. This is given by

**f f l . - v . ( M ) ,**

where V is a two dimensional gradient and R is th e vector, whose m agnitude is given by (2.3), in the direction of the gradient of m agnetic pressure,

### a = Vp = v ( £ ) = J x 7-

### (2-5>

However this simple idea will not be valid near the critical fields *H c\* and *H c2.*
N ear the lower critical field the flux lines will have a separation greater th a n A,
and as nt, 1 it would be expected th a t the force would become equal to th a t
on a single flux line. Near the upper critical field, the flux lines are forced very
close together, and under these conditions it is probable th a t the bundle idea m ay
no longer rem ain valid, and th a t core interactions would be expected to becom e
im p o rtan t. It is also suggested th a t the flux lattice m ay become rigid, and so th e
bundles would be unable to slip past one another.

E quation (2.4) can be used to derive both the critical current curve and the creep rate equation. The critical current is derived by assum ing th a t th e critical param eters are those for which the creep rate becomes im m easurably slow. If this ra te is denoted by i?c, then,

**22**

**2. A Review of the Present Status of Flux Lattice Melting**

**2. A Review of the Present Status of Flux Lattice Melting**

*{ J H U t * *pH? * *k BT * *{ R c*

*crlt * *c * *Sir \ 2l* A2/£o Va>o
To keep th e derivation clear, redefine *F0 = p H 2£***o/87r so** th a t

***o(0)**

(**2**.**6**)

^cnt(O) —

**foA****2l**** '**

which is the critical m agnetic force at *T* = 0. This means th a t the value of *a* giving
th e m inim um detectable creep can be defined as

**<*crit{T) _ F0{T)**

**<*crit{T) _ F0{T)**

**_**

**kBT**

**kBT**

**/**

**v0 \**

**v0 \**

a(0)

**F0(**

0) **F0(**

**F0(**

0) n \ v minJ ’ (2.7)
**F0(**

where *v* is th e flux creep velocity. This is simply equation (2.6) w ritten in a slightly
different form, where it can be seen th a t the creep rate depends exponentially on a .

So from above, we have

*l/oLl*

w ith

*v = v 0 e*

*J H * *1 d H * *d ( H 2\*
*a =* ---- = ---— ~ — ---- ,

**c****47T ** **dr ****dr \ S i****t**** J**

where use has been m ade of

and

**Je =**

**Je =**

*c d H*

*OL* 1 =

47r *dr*

**kBT**

**kBT**

Fo(0) a c(0).

If th e derivation is restricted to one dim ension, as in the flux line wall, one can w rite
*d 2*

*d a _ H 2*

47t ° *d x 2*

«/ati _{(}_{2}_{.}_{8}_{)}

It should be noted th a t th e spatial derivatives of *H* have been neglected com pared to
those of *v.* This is because the exponential dependence of *v* on the derivatives of *H*
are of the order of o:c( 0 ) /a i which is ~ 300 tim es larger th an the direct dependence.
It is possible to solve equation (2.8) for a , giving

*a* = *f [ x*) — q:i In *t*,

where *f ( x )* is an unspecified function of distance. Using th a t fact th a t the creep
ra te is unobservably slow unless *a* ~ *a c,* then *f ( x )* ~ a c, so th a t

*a = a c — a*i In t, (2.9)

*2.2. Giant Flux Creep*

### 23

### a hollow cylinder by Kim

*et al.*

### (Kim

*et*

### a/., 1964) This theory also helped to give

### a microscopic view of the critical state model proposed by Bean (Bean, 1962), by

### showing that this model is based on a balance between the flux density gradient, due

### to the applied field, and pinning forces inside the material. This work was further

### extended by Beasley

*et al.*

### (Beasley

*et al*

### ., 1969), where the creep rate equation was

### solved for different geometries. For a cylinder of radius

*r*

### it was found that

*dM*

### _ /

*r j c\ ( kBT*

### \

*l i t ~*

### V^k /

*[ I T )*

### ’

### where

*U*

### represents the pinning potential of the material. It is this equation that

### Malozemoff and Yeshurun (Yeshurun and Malozemoff, 1988) made use of in their

### giant flux creep model described below.

### One important question to ask is how applicable this idea is to HTcS? The

### Anderson-Kim model of flux creep considers the flux line to be rigid along the c-

### axis, and so the system becomes basically 2D in nature. An obvious question is what

### would happen if the flux lines are allowed to be more flexible, giving the system a

### more 3D nature? By having the ability to flex, it is conceivable that the effect of

### the pinning centres could be altered. These types of effect are considered in more

### detail by the work of Brandt and Nelson discussed later.

### 2.2 Giant Flux Creep

### Malozemoff used an idea based on the Anderson-Kim model, to try to describe

### the form of the irreversible line in HTcS. From experiments it is seen that the

### irreversible line can be characterised in the

*H -T*

### plane close to

*Tc*

### by

*H*

### oc (1 —

*t*

### )3/2,

### where

*t = T /T c.*

### The work was motivated by the experiments of Worthington

*et*

*al.*

### (Worthington

*et al*

### ., 1987), who were able to measure a logarithmic dependence

### on time for the magnetisation of a YBCO sample. This dependence then suggested

### a flux creep picture, based on random pinning sites in the crystal. The model was

### based on equation (2.10), where

*r*

### was taken as half the sample dimension, with

*H*

### || c. They also derived a similar expression for a slab geometry,

*dM _*

### /

*aJc\ I k BT*

### \

*I t ~*

### U

f### J

*[ I T )*

### ’

### where

*a*

### is the slab thickness, with

*H*

### _L

*c.*

### Both equations (2.10) and (2.11) are

### valid only in the region where the critical state model holds, which requires that

### the applied field is larger than

*Hc\.*

### In the calculation a demagnetising factor was

### assumed which modifies the applied field by a factor of 1/(1 —

*N*

### ), where

*N*

### = 0.7

### was chosen to give an effective field larger that

*Hc\.*

### The expressions, as they stand, are able to explain the anisotropy of the relax

### ation measured experimentally. This arises because with

*H\\c, Jc*

### is larger (Dinger

**(**

**2**

**.**

**11**

**)**

24 *2. A Review o f the Present Status o f Flux Lattice Melting*

*et al.*, 1987), and in th e geom etry chosen, *r > a* . This means th a t *d M / d l n t* is
larger for *H* || c, in agreem ent w ith experim ent. Since a higher *J c* corresponds to
greater pinning, it is expected th a t this will lead to a weaker relaxation. But in the
critical state model, a larger *J c* corresponds to a larger flux gradient, and hence an
increase in the rate of flux jum ping.

To carry the calculation further, the following phenomenological forms were
assumed. F irst, *J c =* J c(0) (1 — *t ) n*, where th e typical range in *n* is 1 to 5/2, and
secondly t/0 oc (1 *—t ) 1/ 2.* This means th a t the im p o rtan t quantity *J c/Uo* oc (1 — £)m,
where *m = n —* | . These were then su b stitu ted into equations (2.10) and (2.11),
and fitted to the experim ental data. It is found th a t th e best fit corresponds to
*m — 2* w ith *Uq =* 0.6 eV for *H* || c, and *Uq =* 0.1 eV for *H* _L c. It was m entioned
th a t due to the fact th a t *Uo,\\* > £/**o**,_**l**, tw in boundaries will play an im p o rtan t role as
pinning centres for *H* || c. The values found for *Uo* are to be treated w ith “factor-of-
tw o” accuracy, although their values seem reasonable if you consider the following
argum ent. In th e original Anderson-Kim model, th e pinning potential was said to
scale as *H 2£3/ Stt.* If this is transferred over to the new superconductors, then one
would expect £3 = *£2b* £c, and using published values for these coherence lengths one
gets *Uo —* 0.15 eV, which is consistent w ith th e previously quoted values.

The calculation to param eterise th e irreversible line starts w ith th e following expression for th e critical current density (Cam pbell and Everts, 1972),

where *J*co is the critical current density in the absence of therm al activation, *d* is the
distance between pinning centres, *B* is the m agnetic induction, *Q,* is some oscillation
frequency of a pinned flux line and *E c* is a m inim um m easurem ent voltage per m eter.
For a conventional type-II superconductor, the logarithm term is small, and so the
therm al activation contribution is negligible. B ut for the new HTcS, th e term is a
much larger correction, and should be taken into account. For this equation to be
useful, a knowledge of th e tem p eratu re dependence of *Uo* is required. Malozemoff
adopted a general scaling approach to obtain an order of m agnitude estim ate. To
this end, it was considered th a t *T* ~ *Tc* and th a t the applied field was small, enabling
th e Anderson-K im form for the pinning potential to be used. This is combined w ith
th e following G inzburg-Landau expressions,

**(2.12)**

*H c* = 1.73 *H c0* (1 — *t)*
f = 0.74

### 6,(1

*- t ) ~ 1/2.*

*2.3. Collective Pinning*

### 25

### When the flux lattice spacing becomes less than the penetration depth, then pinning

### due to collective effects becomes important. Assume for simplicity that this happens

### for

*a0 =*

### / f , where

*a0*

### is the lattice spacing and is given by

*a0 =*

### 1.075(</>o

*/ B ) 1^2.*

### This means that for a field above that given by a0, the potential is limited in the

### plane by

*do*

### and along the c-axis by £, so that

*Uo*

### is expected to scale as

*H**

*clq*

*£*

### / 87T

*f 2.*

### One can then substitute for both

*Hc*

### and a0 to finally arrive at

**Uo**

### 2.56

**H?**

**H?**

**q****M**

o
**M**

*&irf2B*

### (1 -

*t f /2.*

### Using this expression for

*Uo*

### in equation (2.12), the condition for the critical current

### to vanish is

*(Sir}2BkBTc \n(BdQ/Ec) \ 2/3*

### 1

### 2.56

*H2M o *

*)*

### (2.13)

### which reduces to

*B*

### oc (1 -

*i f ' 2,*

### as required. Thus it can be seen that by using as simple a model as flux creep, the

### main characteristics of the irreversible line can be derived.

### One must suppose that similar arguments to before also remain valid in this

### case. This result draws heavily on the work of Anderson and Kim, along with other

### standard results for the conventional type-II superconductors. At some point in the

### phase diagram for the HTcS, a changeover from 2D to 3D behaviour is expected.

### This means that the 3D nature of the system will dominate part of the phase

### diagram. How accurately an inherently 2D theory can explain this region is open

### to question.

### 2.3 Collective Pinning

### An idea similar to that of Anderson-Kim is one put forward by Larkin and

### Ovchinnikov (Larkin and Ovchinnikov, 1979). This is based on the phenomenon of

### collective flux creep. This occurs when the pinning in the system is weak, which

### leads to collective pinning. The early work on this idea was directed towards conven

### tional superconductors, but has been extended to include the new HTcS (Feigel’man

*et al*

### ., 1989). The basic idea is that the flux lines do not jump between pinning sites

### individually, rather they jump in groups of several flux lines, and sometimes as a

### collection of such groups. The experimentally accessible predictions produced by

### this theory are similar to those produced from the vortex glass theory, which is

### discussed in more detail below.

### It has been shown (Larkin and Ovchinnikov, 1979) that if a flux lattice is sub

### jected to a random array of pinning sites, and the pinning force is short range, i.e.

### of the form

26 *2. A Review o f the Present Status o f Flux Lattice M elting*

where th e function *K ( x , y )* decreases rapidly for *x , y > r*p, w ith *rp* being some
characteristic length of th e system , then th e long range order of the lattice breaks
down beyond a certain distance, often denoted as *Lp.* It was also shown th a t the
critical current was dependent upon both *R c* and Lc, pinning lengths along the
ab and c direction respectively. A basis for this idea is in exam ining the current
dependence of the pinning potential, th a t is, *U{ J).* The sim plest case is for *B* ~ *H c*i,
w ith an isotropic m edium where the elastic constants have approxim ately th e same
value, * C n* ~ C 4 4 ~

*~ C, where C u ,*

**C****qg***and*

**C 44***6 are th e bulk, shear and tilt elasticity m odulus respectively. Assuming also th a t*

**C****g***J*~

*J c,*th e distance th a t a flux bundle hops is

*Uhop*~ £, giving

*Uc*~

*C ( £ / R C)2VC,*where

*Vc = L CR 2C.*

A more interesting case is when *H * *H ci,* still for *J* ~ *J c.* In this situation C n *=*
*C44* (766, so shear and tilt deform ations dom inate *Jc.* It is shown (Feigel’man

*et* a/., 1989) th a t in this case

It is also shown th a t using *L c* ~ *R c*(C4 4/C6 6) a n d *R c* ~
reduces to

*Uc* **r 3 / 2 r 3 / 2**

° 66

*w*

’
*cii2c3*

*6'2e/w,*

this
where

*d K { u , r )*
*d u2*

and *K [ u ,r )* is th e function defined earlier.

If this is com pared w ith th e earlier expression, it can be seen, upon substituting
for *Rc*and Lc, th a t the two expressions differ only by a factor of ( C n /*Cgg*)1^2• The
explanation for this difference is th a t in the second case, the flux jum ps in the
form of a bundle of approxim ately

### ( C n /

*Cgg) 1^2*subbundles of volume

*Vc.*Due to th e relatively large value of

### Cn,

th e flux lines prefer to ju m p in flux groups, and individual subbundle jum ping is energetically unfavourable.An interesting question arises when one considers *J* «C *Jc*• Under this condition,
the hopping distance of a bundle will be much larger th an for *J* ~ J c, and can be
estim ated as

*J B u hop(J)* ~ C66 (2.14)
where *R \* is th e bundle size in the plane perpendicular to both the applied field and
th e hopping vector. This arises from the definition of *Jc* (Larkin and Ovchinnikov,
1979).

It can also be shown th a t the fluctuations in a system of random pinning sites increase like