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LIBRARY
Mlchlgan State
University

 

 

 

This is to certify that the

dissertation entitled

The Effect of Columnar Defects on the
Vortex Melting Transition in YB32Cu3O7,5

presented by

Robert James Olsson

has been accepted towards fulfillment
of the requirements for

PhD. Physics

degree in

flaw/wear

 

 

Major professor

Date SQ/Sfl/“Z 000

 

MSU is an Affirmative Action "Equal Opportunity Institution 0- 12771

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE 'DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

moo mm.“

 

 

The Efiec

The Effect of Columnar Defects on the Vortex Melting Transition in YBa2Cu307.z5

By

Robert James Olsson

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2000

The Effect

The
crystals isi
crystals. A
Confirmed i
critical polr
melting tra,
irradiation.
results in 3
samples. |
a Continuo.

fluctUati0n

ABSTRACT

The Effect of Columnar Defects on the Vortex Melting Transition in YBaZCU3O7.5

BY

Robert James Olsson

The vortex melting transition in high quality, untwinned YBa2Cu307.5single
crystals is investigated in the presence of columnar defect tracks imbedded in the
crystals. A first order melting transition from a vortex lattice to liquid has been
confirmed in clean samples, with the melting line ending at an upper and lower
critical point, beyond which a continuous transition is presumed. In this work the
melting transition and associated critical points are altered by the controlled
irradiation of the samples by high energy uranium and lead ions. The irradiation
results in a random distribution of straight, continuous defect tracks in the
samples. For high defect densities the first order melting transition is replaced by
a continuous Bose glass phase transition. The transition is described by a critical
fluctuation regime. Current-voltage measurements were obtained within this
non-ohmic regime, and successfully scaled according to the Bose glass model.
The evolution of the melting from a first order to continuous transition is
investigated by the introduction of low densities of columnar defects and point-
like defects created by proton irradiation. At low magnetic fields below the critical
point a Bose glass transition is confirmed from the data from the columnar defect

crystals. With increasing field the lower critical point is crossed and the vortex

mwsm
solid-to—ord
from the pr:
increasing l
columnar d.
suggests th
resulting in

induce wan<

lattice is reestablished. Thus, this implies the possible existence of a disordered
solid-to-ordered vortex lattice phase transition within the solid state. The data
from the proton irradiation shows a decrease in the upper critical field with
increasing point disorder, whereas the opposite is found in a crystal with
columnar defects, irradiated at a dose matching field of 1000 Gauss. This
suggests that columnar defects tend to restrain the meandering of vortices
resulting in a higher upper critical point, whereas the point defect pinning sites

induce wandering and possible vortex entanglement.

This thesis

This thesis is lovingly dedicated to Kristy Olsson. Thanks! Without your help and
strength, this work would never have been completed.

The
profession
National 8
Superconc
through the
This work i
taught me
colleagues
Mazilu, Go
Reginald R
MIChIgan S
Slump! anc
DOSItion 0ft
0°”Slant su

ACKNOWLEDGMENTS

There are many who have been helpful, directly and indirectly, in my
professional development. The funding for my research was provided by the
National Science Foundation Science and Technology Center for
Superconductivity, under contract #DMR91-20000, and also from Harold Myron
through the Division of Educational Programs at Argonne National Laboratory.
This work is a continuation of the overall work of Wai-Kwong Kwok, who has
taught me a level of thoroughness I had not previously known. The other major
colleagues in this effort are Lisa Paulius and Andra Petrean, with help from Ana
Mazilu, Goran Karapetrov, Valentina Tobos, David Hofman, Bruce Glagola,
Reginald Ronningen, and George Crabtree. Thanks to my committee at
Michigan State University: Simon Billinge, Mark Dykman, Wayne Repko, Daniel
Stump, and especially to Phillip Duxbury, who was kind enough to take on the
position of major advisor. Thanks also to the Argonne Soccer Club, for their
constant support. Finally, a special thanks to Jerry Cowen (posthumously) and

Alan Meltzer, for noticing.

LlSi of T
List of F r
CHAPTE
CHAPTE
2.1
2.2
2 3
CHAPTER
3.1
3.2
3.3
3.4
3.5
CHAPTER .
4.1

4.2

 

TABLE OF CONTENTS

List of Tables .................................................... viii
List of Figures .................................................... ix
CHAPTER 1. Introduction .......................................... 1
CHAPTER 2. Vortex states ........................................ 4
2.1 Introduction to superconductivity ........................... 4
2.2 Ginzburg-Landau theory ................................ 10
2.3 High temperature superconductors ........................ 14
CHAPTER 3. Vortex motion and vortex pinning ....................... 18
3.1 Lorentz force and dissipation ............................ 18
3.2 Dynamics of vortex motion .............................. 20
3.3 Thermally-activated flux flow ............................. 21
3.4 Anderson-Kim vortex creep model ........................ 23
3.5 Vortex glass transition .................................. 24
CHAPTER 4. The vortex phase diagram of clean YBazcuaom crystals ..... 25
4.1 First order vortex solid to liquid melting transition ............. 25
4.2 Critical points ......................................... 35
CHAPTER 5. Vortex pinning by defects in YBazcu307.,s single crystals ..... 44
5.1 Pinning in the vortex liquid state by twin boundaries .......... 44
5.2 Point defects and the Vortex Glass theory .................. 47
5.3 Highly viscous Vortex Molasses model ..................... 50
5.4 Bose Glass theory .................................... 50
CHAPTER 6. Crystal growth and preparation ......................... 58

vi

CHAPTE

71

CHAPTER
inc
81
82
83

8 4
CHAPTER

91

92
93
CHAPTER

Blinograph

CHAPTER 7. Experimental setup and heavy ion irradiation .............. 68

7.1 System configuration ................................... 68
(7.1.a) Cryogenic and Superconducting Magnet System ........... 68
(7.1.b) Sample Probe ...................................... 71
(7.1.0) Electronics ........................................ 73

7.2 Heavy ion irradiation ................................... 74

CHAPTER 8. The effect of high densities of columnar defects on vortex motion

in clean, untwinned YBazcu;»,07.,s single crystals ................... 90
8.1 Introduction .......................................... 90
8.2 Uranium ion irradiation: U1, U2, and U4 .................... 94
8.3 Lead ion irradiation: Pb1 ............................... 107
8.4 Angular dependence for PM ............................ 1 13

CHAPTER 9. The effect of low doses of columnar defects on vortices in

untwinned YBA20U307.8 crystals ......................... 121

9.1 Introduction ......................................... 121

9.2 Low densities of columnar defects ....................... 123

9.3 Comparison with low densities of point defects .............. 137
CHAPTER 10. Conclusion ....................................... 148
Bibliography ................................................... 152

vii

Table 5.1

Table 7.1

 

 

LIST OF TABLES

Table 5.1 Charge bosons-vortex lines analogy .......................... 53

Table 7.1 Columnar defect tracks in YBCO ............................ 87

viii

Figure 2 1
Figure 2.2
me23

me24

 

 

 

Figure 3.1
me32
FlQure41
me42
HWW43
me44
“Wm45
Figme 4.6
Figme 5'1
Figme 52
Flgure 5.3
“Wm54
mee1
FIQLire 52
FIQUre 53 I
me64-
FiQUre 65 I

LIST OF FIGURES

Figure 2.1 Type I superconductors ................................... 5
Figure 2.2 Type II superconductors ................................... 8
Figure 2.3 Vortex lines ............................................. 9
Figure 2.4 Resistivity measurements, low and high temperatures ........... 16
Figure 3.1 Lorentz force ........................................... 19
Figure 3.2 E-J curves for YBCO ..................................... 22
Figure 4.1 Low, high temperature phase diagrams ...................... 27
Figure 4.2 Resistivity versus temperature, untwinned YBCO ............... 28
Figure 4.3 E-J curves for untwinned YBCO ............................ 31
Figure 4.4 R vs T, experimental phase diagram for YBCO ................. 34
Figure 4.5 Vortex liquid, glass, and lattice phases for YBCO ............... 36
Figure 4.6 Elastic versus pinning energies ............................ 38
Figure 5.1 Viscous damping due to twin boundaries ..................... 45
Figure 5.2 Angular dependence of the R v. T data, twinned YBCO ......... 47
Figure 5.3 Vortex lines in the presence of columnar defects ............... 54
Figure 5.4 Angular dependence of the Bose glass transition .............. 56
Figure 6.1 YBa2Cu307.5 unit cell ..................................... 59
Figure 6.2 High temperature phase diagram ........................... 60

Figure 6.3 Phase diagram for YBCO as a function of oxygen stoichiometry. . .60
Figure 6.4 Twin boundaries in YBCO ................................. 64

Figure 6.5 Detwinning device ....................................... 66

 

Figure 7.5

Figure 7.6
Figure 7.7

meTB

 

Figure 7.9
Figure 7.1C
“Me 7.11
Figure 8.1
Figure 8.2
Figure 8 3
FlQure 34
Figure 8.5
Flgure 3.6
FigUre 87
Figufees
F'We 8.9
FIQUre 810
Filere 8.1 1

Figtlre 8.12

Figure 7.1 Helium cryostat system ................................... 69

Figure 7.2 Gas handling system .................................... 71
Figure 7.3 Sample probe and holder schematics ........................ 72
Figure 7.4 The creation of columnar defects via heavy ion irradiation ........ 76
Figure 7.5 Stopping power for U, Pb, and Au in a YBCO target ............. 79
Figure 7.6 Defect formation schematic ............................... 79

Figure 7.7 Heavy ion irradiation chamber, orientation, and sample holder ..... 81

Figure 7.8 Radiographic film image .................................. 84
Figure 7.9 Alpha particle counting electronics .......................... 86
Figure 7.10 Resistivity versus temperature data in zero field ............... 88
Figure 7.11 TEM image of columnar defects in YBCO .................... 89
Figure 8.1 Resistivity versus temperature for U0, U1, U2, and U4 .......... 95
Figure 8.2 Pre- and postirradiation melting transition .................... 97
Figure 8.3 Angular dependence of the resistivity ........................ 98
Figure 8.4 Onset of colomnar pinning in the liquid ....................... 98
Figure 8.5 R vs. T, E vs. J data for U4 ............................... 100
Figure 8.6 Irreversibility lines vs. temperature, for U1, U2, and U4 ......... 102
Figure 8.7 Critical current versus temperature for U1, U2, and U4 .......... 104
Figure 8.8 S analysis of R vs. T data, crystal U1 ....................... 106
Figure 8.9 R vs. T data for PhD and PM ............................. 108

Figure 8.10 Onset of non-ohmic behavior in the liquid state, for crystal Pb1. .109
Figure 8.11 E-J curves for PM and scaling of the curves ................ 110

Figure 8.12 E-J curves and scaling for H=0.2,0.5, and 2T ................ 112

Figure 8.1
me8‘
Figure 8.1
Figure 8.1-
me91
me92
me93
me94
me95
me96
“Wee?
“mesa
Figure 9.9 r
“Wmem
“WES“

FlSlure 9.12

 

Figure 8.13 Irreversibility, critical currents for PM and U1 ................ 114

Figure 8.14 Angular dependence of the resistivity ...................... 116
Figure 8.15 E-J curves and scaling for H=1T at one degree .............. 117
Figure 8.16 E-J curves for H=1T, applied at large angles ................ 119
Figure 9.1 Phase diagram for YBCO ................................ 122
Figure 9.2 Data for preirradation and for 50 Gauss irradiation ............. 124
Figure 9.3 E-J curves for pre- and post-irradiation, 50 Gauss irradiation ..... 127
Figure 9.4 R vs. T, Phase diagrams for 100 Gauss irradiation ............ 129

Figure 9.5 E-J curves and scaling for 500, 1000 Gauss irradiation ......... 131

Figure 9.6 Angular dependence of the non-ohmic onset ................. 132

Figure 9.7 Normalized R vs. T for 100, 500, and 1000 Gauss irradiation. . . . 134

Figure 9.8 H-T diagram for 1000 Gauss irradiation ..................... 136
Figure 9.9 R vs. T, dRIdT for H=4T, proton irradiation data ............... 139
Figure 9.10 dR/dT, H-T diagram for proton irradiation data ............... 140
Figure 9.11 H vs. T, comparison between columnar and point defects ...... 143
Figure 9.12 E-J curves, various point defect densities, Jc vs. T ........... 145

xi

In tr
there is a ti
and solids.
field, knowr
associated r

of a crystal 1

 

Present. Fu

icelike melt;

 

Chapter 1

INTRODUCTION

In the magnetic phase diagram of high temperature superconductors,
there is a wide field and temperature regime in which there exists novel liquids
and solids, consisting entirely of magnetic field lines. These lines of magnetic
field, known as flux lines or vortices, possess many of the characteristics
associated with normal matter. Like solids, these flux lines can acquire the form
of a crystal with a lattice structure, or form a disordered glass when defects are
present. Furthermore at high temperatures the vortices are even capable of an
ice-like melting transition to a vortex liquid state.

Vortices interact with the environment in a number of controllable ways.
The density of vortices can be varied by an external applied magnetic field.
They can be localized or 'pinned' in place by defects in the underlying crystal.
They may be thermally set in motion about their equilibrium position by heat, and
even be driven to move in a preferential direction by an applied current.
Although vortices freeze via a first order transition from a liquid to a lattice state
in the absence of disorder, several novel vortex glassy phases can be obtained
by freezing the vortex liquid in a variety of defect environments. The
environment may be a random set of weak/strong point defects, line defects, or
may even be a periodic defect array. Thus vortex matter can be studied in a
number of experimentally controllable ways which may not all be possible in

atomic solids. One such problem in atomic solids is the transformation of a first

order trans
disorder. '
systems, h
in introduci
study of vo
provide a p
transitions.
The ‘.
0f the first 0
crystals in ii
Observed tol
order melting
critical point,
EVOIUIIOn CHI

 

'1an I00 I";

FIrSt tl

 

We" meltin

Phase is foun
and the CIIllca
glass and a la
from a first 0rd

IIIle defects W”

order transition into a second order or continuous transition in the presence of
disorder. This problem has theoretically been studied extensively in magnetic
systems, however, related experiments have been hampered due to the difficulty
in introducing quenched disorder into the system in a controlled way. Thus, the
study of vortex matter can be a powerful tool for the study of real matter, and also
provide a platform to investigate fundamental problems in the physics of phase
transitions.

The focus of this thesis is on the fundamental issue of the transformation
of the first order vortex melting transition in superconducting YBazCuaoy.ls
crystals in the presence of disorder. In the absence of disorder, this transition is
observed to be first order, as in the case of very high quality samples. The first
order melting transition line terminates at both ends at an upper and a lower
critical point, whereby the transition becomes continuous in nature. The
evolution of the first order solid to liquid vortex transition will be investigated with
respect to the introduction of line defects into the superconductor via high energy
heavy ion irradiation.

First, the case for a high density of defects will be studied, where the
vortex melting transition is continuous at all investigated fields, and the solid
phase is found to be a disordered glassy state. Second, the melting transition
and the critical points are studied in the dilute line defect limit. Here both a vortex
glass and a lattice state are observed, and a defect density-dependent evolution
from a first order to a continuous melting transition is found. Finally, results from

line defects will be compared with results from point defects. Hopefully this work

will shed ll
of phase tr
The
framework
transition 0.
growth and
samples, T
of both high
Phase trans
With that of p

IIne In the V0

 

will shed light not only on vortex physics, but also aid in a broader understanding
of phase transitions.

The first part of this thesis provides a theoretical and experimental
framework of the physics of vortex matter, focusing especially on the melting
transition observed in YBa2Cu307.5. The second part describes the crystal
growth and preparation processes and the details of line defect creation in these
samples. The main part of the thesis details the melting transition as a function
of both high and low densities of line defects and analyzes each of the novel
phase transition lines and their evolution. The final section compares the results
with that of point defects and proposes the existence of a new phase transition

line in the vortex solid.

2.1 lntr:

 

 

 

The phenc

Onnes in 1

observed a‘
PTOPerties

From meas
constant WE
essentially l
SupercondL
from the sa
Melssner 8‘
ObseWed ir
supermndL
EXIErnai fie
Zero by Set

applied fiel.

Chapter 2

VORTEX STATES

2.1 Introduction to superconductivity

The phenomenon of superconductivity was discovered by H. Kammerlingh
Onnes in 1911[1] in mercury, in which a sudden drop to zero resistivity was
observed at a critical transition temperature T6 = 4.1 K. In order to gauge the
properties of this state, a persistent current was set up in a superconducting ring.
From measurements of the field induced by this current, the current decay time
constant was extrapolated to be greater than 100,000 years, thus establishing
essentially Iossless current flow in the ring. It was later recognized that the
superconducting state is also characterized by the expulsion of magnetic field
from the sample, a phenomena akin to perfect diamagnetism, called the
Meissner effect[2]. This phenomena of diamagnetism in superconductors is
observed in magnetization measurements at temperatures below the
superconducting transition temperature, as shown in Figure 2.1(a). As an
external field H is applied, the superconducting material keeps the internal field at
zero by setting up a diamagetic field with a circulating current to offset the
applied field. Thus there is negative bulk magnetization. Upon the application of
a magnetic field above a critical field Hc, the energy necessary to expel the field
becomes greater than the condensation energy, and the sample returns to the
normal state. The shaded region in Figure 2.1(a) is equal to the condensation

energy Hc2181r, which represents the lower energy of the superconducting state

(a) Magnetization curve for type | superconductor

M

 

 

 

 

(b) Phase diagram for type I superconductor

HcIO)

   

normal state

superconducting Hc(T)
Meissner state

 

 

Figure 2.1 Type I superconductors.

 

  
 

field at zer
the normal
at T= 0 an

The

 

established
thin layer fn
5UP€rC0ndu
maximum C

results are .

as compared to the normal state. The critical field is temperature dependent,
and represented empirically by Hc=Hc(0)-(1-(T/Tc)2), where Hc(0) is the critical
field at zero temperature, as shown in Figure 2.1(b). In type I superconductors,
the normal to superconducting state transition is of first order everywhere except
at T= 0 and T= Tc(H=0), where the transition is continuous.

The zero magnetic field within the bulk of the superconductor is
established by a counterflowing circulating supercurrentjs, which flows within a
thin layer from the surface, producing a magnetic field which shields the
superconductor from the applied field. Thus a critical field is equivalent to a
maximum critical current jc, beyond which the sample is driven normal. These
results are described within the two London equations for superconductivity[3]:

= 471712 a

2 3,0.)

E

 

(2.1)

C

2
h =--‘”"1 Vx (1,) (2.2)

C

 

where E is the electric field induced by the superconducting current jg, h is the
local magnetic field, A is the field penetration depth and c is the speed of light. A
nonzero magnetic field exists only where there is a gradient in the supercurrent
density (eqn. (2.2)), and an electric field (and thus loss) exists only where there is
a change in the supercurrent density in time (eqn. (2.1)).

Materials that display a single transition from the diamagnetic to the
normal state are known as type I superconductors. However, most commercial
superconductors used today are type II superconductors, which have an

intermediate mixed state distinguished by the absence of total flux expulsion, as

 

shown by
usually on
critical fieh
penetrates
containing
described ;

SUpercondL

 

maximum 3
core the ma
from the Col
Vorti
s“PerCUl’rer
Abrikosov v
mEOreticall}
for CIOSe to
a: ((1)0 / B I
Via ferrOrria
sea"Who tu
vollices the
the ”What
Dotti/antic”;
For;
“Wally beW

shown by the magnetization curve in Figure 2.2(a). Type II superconductors are
usually characterized by a low field Meissner state which terminates at a lower
critical field HM, and by an intermediate or mixed state where the magnetic field
penetrates the superconductor in tubes of quantized flux known as vortices, each
containing one flux quantum (Do. The vortices (Figure 2.3(a)) can be qualitatively
described as containing a core region of radius 5, where the density of
superconducting quasiparticles is zero at the center and increases sharply to a
maximum across the distance 5 described as a coherence length. Within the
core the magnetic field is at its maximum value and decays over a distance A
from the core center.

Vortices repel one another via the Lorentz interaction between the
supercurrents and the extended field, forming a lattice structure known as the
Abrikosov vortex lattice (Figure 2.3(b)), named after Alexei Abrikosov, who
theoretically predicted its existence in Type II superconductors in 1957. Except

for close to Hc1 the intervortex spacing a can be approximately given by

1/2

a = (tbol B) , with B e H. This triangular lattice configuration has been observed
via ferromagnetic filament surface decoration[4], neutron diffraction[5], and
scanning tunneling microscopy[6]. As the applied field is increased the density of
vortices increases, until overlap of the normal core drives the superconductor into
the normal state at an upper critical field Hg. The phase diagram for
conventional type II superconductors is shown in Figure 2.2(b).

For pure type I superconductors, Hc(0) and T0 are quite low, with Hc(0)

usually between 200 - 800 Gauss, and T6 < 10 K, resulting in a rather small

(a) Magnetization curve for a type II superconductor

M

Hcl Hc2

 

 

(b) Phase diagram for a type II superconductor

Hc2(0)

no rrnal state

 
   

‘mixed state

 

Hc1(0) 1‘ .1 1» ~. ~- .. , . ,. _ .
~ , ;. r: I, . n. , ,. .;1\ .
3’: ‘2": € 7 w MM~ ”-1 ”2‘ ~
‘5 .' ‘ ‘ 1" .’ ’ ' "'1 ~ 3‘ ‘WI
‘ .1 '.A / . s, . _. .'_ ‘ 1 'F“ i" '
*-~ ‘... I c. 'l . .. .. I; .. l- r .- ‘-

 

Figure 2.2 Type II superconductors.

 

 

 

 

 

 

 

 

Vortex line, with nonsuperconducting core, radius é

(a) Profiles of the field and order parameter of a vortex line.

triangular vortex lattice, spacing a = (4:9)

(b) Vortices forming a vortex lattice solid state. View
is along the vortex lines.

Figure 2.3 Vortex lines.

 

superconr
much higt
material u
approximal

superconc

USCFUI for p

2.2 Ginz

The superc

 

hi can be or
theory 0f se
(”the Syste
transition te
Order Pararr
parameter is

electrons. n

free energy

Where F, is
maQIIEtic fie
llhenomeW)I
matron Win

the aDOVe eq

superconducting phase space. On the other hand, type II superconductors have
much higher upper critical fields ch(0). For Nin wire, a low temperature type II
material used in windings of superconducting magnets, Tc is ~9.7 K, but ch(0) is
approximately 100,000 Gauss (10 Tesla). Correspondingly, type II
superconductors exhibit significantly higher critical currents, thus making them

useful for practical applications.

2.2 Ginzburg-Landau theory

The superconducting transition, in the presence of a local magnetic field density
h, can be described by the phenomenological Ginzburg-Landau theory[7]. This
theory of second-order phase transitions describes a gradual change in the state
of the system followed by a discontinuity in the symmetry of the system at the
transition temperature. It describes the process by using an expansion of an
order parameter near the transition temperature. For superconductors, the order

parameter is a wavefunction w defined by the local density of superconducting
electrons, n, = |ip|2 which exists below the transition temperature at T < Tc. The

free energy density in the presence of zero field is given by
p; = P; + 04le sugm‘ (2.3)

Where F. is the free energy density of the superconductor in the absence of a
magnetic field, Fn is its free energy in the normal state and a and B are
phenomenological material dependent expansion coefficients. Minimizing the
equation with respect to Ile yields lull2 = «Jr/fl. Substituting this expression into

the above equation yields If, - If, = az/Zfi 5 Hi /87r, establishing a lower free

10

 

energy in
given by t
the transit
(I I? that r

positive ar

 

temperatur
2 I
(ma) ) nee
For a

field. the GI

 

Where the fc
9' moving in
of the C00pe
is the applier

.F-l...

The first differ

S
UDerCOndUCt‘

evaluating (2 r

WarjeS.

energy in the superconducting state than in the normal state, with the difference
given by the condensation energy. Since the order parameter must be zero at
the transition temperature T = Tc and nonzero below Tc, it follows from IVIZ = -
all? that a(T= Tc) = 0 and or(T< 7,) < 0. Thus to first order, or ~ (T-Tc) and B is
positive and temperature independent. Furthermore, since H} = 41ra2/B, the
temperature dependence of or correlates with the empirical formula H¢2=Hc(0)(1-
(ma?) near Tc.

For an inhomogeneous superconductor in a uniform external magnetic

field, the Gibbs free energy near Tc can be written as:

 

 

2
l h e. h2 h-H
G=G +al I2+El l4+ , -V——A +—- 0 2.4
’ " W 2 W 2m |(i c )‘l’l 87: 47: ( )

where the fourth term describes the kinetic energy density for a particle of charge
9* moving in the field with a vector potential A, m * is the superconducting mass
of the Cooper pair (charge 2e), Gn is the normal state Gibbs free energy, and H0
is the applied external magnetic field. Evaluation of eqn. (2.4) with respect to

variations of w, iy*, and A, yields the Ginzburg-Landau differential equations:

 

 

2 1 h e* 2
ary+fi|w| w+ (7V-—A) ry=0 (2.5)
2m* 1 c
c e‘h e"‘2
—V h=—— *V -W * - * A: 2.6

The first differential equation provides the first characteristic length scale for
superconductors: the coherence length 5. The coherence length is obtained by
evaluating (2.5) in zero field, and provides a measure of the length over which

ytvaries:

11

 

The CODS‘r

made cle
imaginary

equation b

This IS the 1

little variatic

 

 

along the v
a Simple bor

Choosing the

Where 3» des

penetration I

Ean. (28) a

as git/eh by (:

Note that SITlc

the Ginszrg‘

h2

2=———-— 2.7
5 2m*la(T)| ‘ ’

The consequences of the second Ginzburg-Landau differential equation can be
made clear by the substitution of I]!(r) = |iy(r)|e"""’, thus separating the real and

imaginary parts of the order parameter. With this substitution in (2.6), the

equation becomes
a: II!
J = fjlmfhva - e—A) (2.8)
m C

This is the equation for the supercurrent density. Within a region where (phas
little variation, the supercurrent density J is then proportional to, and directed
along the vector potential A, Le. perpendicular to the field direction. By applying
a simple boundary condition for a superconducting boundary ( J=0 across), and

choosing the London gauge as V -A = 0, then eqn. (2.8) can be rewritten as

V2A+% = o (2.9)

where 2. describes the penetration depth of the vector potential, and thus the field
penetration length:

12 _ m*02

- 474w|2e *2

Eqns. (2.8) and (2.9) can be used to describe the Meissner effect: Eqn. (2.9)

(2.10)

shows that over a characteristic length A, the field is screened by supercurrents
as given by (2.8), beyond which the material superconducts in a zero field state.
Note that since both 1.2 and 52 are proportional to 1/|or|, they both diverge as

T —) Tc. Finally, one other important parameter in evaluating superconductors is

the Ginzburg-Landau parameter x:

12

"swam

 

type i or 1',
Th

considenn.

l Superconc

 

minimum St
for tVile l su
35 (H: / 87: 3.:
material Sup
depth (Meis:
efiends bey
surface is “E
to “Win;
the magneuc
Possible iota
The G

and prOVldes
an applied fie

x:- (2.11)

It is the magnitude of this parameter which defines a material as being either a
type I or type II superconductor.

The criterion for a type I versus a type II superconductor is established by
considering an interface between normal and superconducting regions. For type
I superconductors, the total energy of the boundary is positive, so that a
minimum surface area is the energetically favorable condition. This is because,

for type I superconductors, A < 6. Since the surface energy can be approximated
as (Hf; /81r)(§- it), this results in a positive surface energy. Thus the bulk of the

material superconducts, with no internal magnetic field beyond the penetration
depth (Meissner state). For type II superconductors the penetration length
extends beyond the coherence length. In this case the energy of the boundary
surface is negative, specifically for K > 1/1/2 [8]. Thus it is energetically favorable
to maximize the total surface energy, which is accomplished by the localization of
the magnetic flux into single, quantized flux lines, each equaling the lowest
possible total flux, co = hc/2e = 2.1x10'"G-m2.

The Ginzburg-Landau theory describes the region close to the transition
and provides a framework and valid predictions for the superconducting state in
an applied field, including the Meissner and mixed states, from simple energy

considerations.

13

 

2.3 Hi:

 

High lem
MalledQ].

high we

characteris
SUpercond
have result

convention;

 

One

'W T; and I
35 the SQua
high Tc Squ

Fer example
”Q ~ 120 1,
compared 1C
large dlfiere,

The h
short Cohere

2.3 High temperature superconductors

High temperature superconductivity was discovered in 1987 by Bednorz and
Milller[9]. High temperature superconducting materials are characterized by their
high superconducting temperatures, very short coherence length, and a large
superconducting anisotropy owing to their layered structure. These
characteristic features have led to a new understanding of type II
superconductors and furthermore, with the addition of disorder into the system,
have resulted in a wealth of new vortex phases which were unobservable in
conventional lower temperature superconductors.

One of the characteristic parameters used in discerning the difference in
low To and high Tc superconductors is the Ginzburg number Gi, which is defined
as the squared ratio of the thermal energy kaT and the condensation energy. In
high Tc superconductors, the high transition temperature, small coherence length

and large anisotropy leads to a large Ginzburg number Gi

2
-_ l 772
G' ' 8 [H32<0)x’€3<0>] (2'12)

 

For example, for YBa20u307.,5 (YBCO) with H H c, Tc ~ 90 K, §~ 16 A,
He; ~ 120 T, y~ 7, and rc~ 60, yielding a Ginzburg number of Gi =5 10'2
compared to Gi ~ 10‘8 for conventional low temperature superconductors. This
large difference leads to many novel phenomena.

The higher temperatures result in large thermal fluctuations, while the very
short coherence length dramatically reduces the effective pinning strength of

naturally-occurring defects. The large anisotropy weakens the correlation of the

14

vortex alc

effective r

This paran
superconc
i; =2 A a'
line is stror
most ciroun
anisotropy -
behave mor

In Va
energy. the
compete in y
”flex Sysler

 

Energy Can b
calefill intr0d

transpon CU"
Figure

19’ -. '
SlSley In ti)

vortex along the length of the line. The anisotropy parameter 7, is defined by the

effective mass ratio mo/mab

 

 

 

7%“) = ’1‘ =5“ (2.13)

This parameter is a reflection of the planar structure of the high temperature
superconductors. For YBCO (xz 60), 1.40) z 1000 A, 55), z 16 A, 7 ~ 7, and
6c ==I 2 A and M0) 3 7000 A. For this material, the correlation along the vortex
line is strong enough to still be considered a three-dimensional elastic line in
most circumstances; however for BiZSr2CaCuz08 (BSCCO), due to its extreme
anisotropy r150, the c-axis vortex correlation is very weak, making the vortices
behave more like 2 dimensional objects, called pancake vortices.

In various parts of the phase diagram, the vortex-vortex interaction
energy, the thermal energy, and the pinning energy can be comparable and
compete in ways to produce new transitions and phases. One advantage of the
vortex system is that all the relevant parameters can be carefully controlled in the
laboratory. The vortex density can be varied by the magnetic field, the thermal
energy can be varied with temperature, the pinning energy can be varied by
careful introduction of pinning sites via irradiation, and furthermore, the driving
force to depin the vortices from the pinning sites can be controlled with the
transport current.

Figure 2.4 presents a comparison of the temperature dependence of the

resistivity in the superconducting state for a Nb3Ti wire and an untwinned YBCO

15

 

(a) Resista‘

22

 

(b) Re
crystal

6 .

p (UH-cm)

(a) Resistance vs. temperature for low temperature type I I superconductor.
NbaTl wire
200 r...l....'....,rrr.l....

 

 

150 "

‘
in d
I- «1
I ‘
50 - 1 Jr -
D d
b
D
b ) J
o .3-:‘:-- 1 1 L4 1 n l n n n 1

7.0 8.0 9.0 10.0 11.0 12.0
T (K)
(b) Resistivity vs. temperature for high temperature type II YBCO
crystal. The kink in the resistivity is marked by the arrow.
60 - - - - - - - -

I ’v ‘ ~— 1'— T I

 

 

 

 

 

 

 

 

 

p (HQ-cm)

 

76 80 84 88 92

T (K)

Figure 2.4 Resistivity measurements of the normal to
superconducting state transition of low and high Tc

superconductors in various magnetic fields.

16

crystal at various applied magnetic fields. While both are type II
superconductors, the behavior is dramatically different. In Nb3Ti, a sharp
transition is seen at Tcz, which corresponds to the upper critical field ch, marked
by a sharp drop in resistivity to zero. For YBCO, the transition at Tcz is
characterized by a broadening of the resistive transition with increasing applied
field, with a large temperature span between the onset of superconductivity
where the resistivity initially drops and the zero resistivity temperature. For a
high quality, relatively defect free YBCO crystal as the one represented here, a
sharp drop in the resistivity is observed near the zero resistivity temperature.
This sudden drop or 'kink' in resistivity is a new feature associated with some
high temperature superconductors and reflects a vortex phase transition from a
vortex liquid state to a vortex solid state with lowering temperatures. It is
observed in high temperature superconductors due to their large characteristic
superconducting critical temperature and large Ginzburg number which promotes
the 'melting' of the vortex lattice, leading to the existence of a vortex liquid state
over a large portion of the phase diagram. The vortex liquid state is sandwiched
between the upper critical field line Ham and the vortex lattice melting line
Hm(T). In conventional low temperature type II superconductors, Hm(T) is
believed to be very close to ch and thus experimentally very difficult to

discern[10].

17

3.1 Lorentz

Although supe

 

is not the case
presence of a '

force is induce

where n is the
WSW. the von
Electric field le
the motiOn of \
vortices. FOr E
the Sample wh
Case_ the VORe
its condensatic
unseat 0' 'depi

referred to as .
¢

Chapter 3

VORTEX MOTION AND VORTEX PINNING

3.1 Lorentz force and dissipation
Although superconductivity is usually associated with the loss of resistance, this
is not the case in the vortex state of defect free type II superconductors. In the
presence of a transport current and a perpendicular magnetic field, a Lorentz
force is induced on the vortices given by:

fL=J x n <I>olc (3.1)
where n is the unit vector in the direction of the vortices, see Figure 3.1. As a
result, the vortices are set in motion by the Lorentz force, inducing a transverse
electric field leading to the observation of a finite resistance. In order to prevent
the motion of vortices, one needs to introduce effective defect sites to 'pin' the
vortices. For example, pinning can be initiated in a small microscopic region of
the sample which is non-superconducting due to a crystalline defect. In this
case, the vortex core would naturally sit on the normal state defect site to lower
its condensation energy, and a finite level of Lorentz force will be required to
unseat or 'depin' the vortex. The applied current required to depin the vortices is
referred to as a critical current jc. Thus for applications such as superconducting
wires and magnets where large amount of currents are required to flow without
resistance, effective pinning sites must be fabricated into the material to prevent
vortex motion. Such defects sites can be made through substitution, irradiation,

or even mechanical damage inflicted upon the material.

18

 

Figure 3.1 Lorentz force due to a current J applied
perpendicular to the field B.

The pinning strength of a superconductor is dependent upon defects
within the underlying crystal structure, and hence a perfect crystal would not be
able to pin vortices. However, even the highest quality YBCO crystals yet
produced contain enough ‘point’ defects to be able to pin the vortices. Point
defects in YBazcuaoy.as are typically oxygen vacancies in the copper-oxygen
chains or interstitials, usually less than 10 A in effective radius, which act as
weak pinning sites. For low defect density crystals, the vortex solid crosses from
a weakly-pinned vortex lattice to an unpinned, moving vortex solid as the current
density is increased beyond the critical current jc.

Resistivity measurements represent a dynamic non-equilibrium vortex
environment, and therefore cannot extract any thermodynamic quantities.
However, it will be shown in later chapters that the location of different vortex

thermodynamic transitions can be determined from such measurements.

19

 

3.2 Dynarr
For an appliea

on an isolatec‘

where y(t) des

the underlying

 

 

 

the applied cur
E = v x B/c, is
the applied cur

motion. The d

3.2 Dynamics of vortex motion
For an applied current in the presence of a perpendicular applied field, the forces

on an isolated vortex in a type II superconductor can be given by:

"f° - 70w. = 0 (3.2)

 

where 7(t) describes the frictional drag force coefficient between the vortex and
the underlying sample. In this simple model, the vortex velocity is proportional to
the applied current density. The induced electric field, given by Faraday’s law

E = v x B/c, is perpendicular to the Lorentz force but parallel to the direction of
the applied current, thus generating a finite voltage drop transverse to the vortex
motion. The dissipative flux flow resistivity is given by:

_ voB _ B<I>0
3 JC ‘yc2

 

(3.3)

If the drag force is independent of the Lorentz force, the resultant resistivity is
then current independent, i.e. ohmic. This resistivity can be related to the normal
state resistivity pn via the Bardeen-Stephen model[11]. In this approach, the
motion of a vortex results in an electric field existing within the vortex core with
radius 5, and directed along the direction of the applied current. The resultant
drag force is due to current moving within the ‘normal’ core. This current is
induced by the electric field, with 7 given by y = (D3, /(21r§2c2p,, ). Since

ch = (Do/21:52, the Barden-Stephen model approximates the resistivity by

p” z puB/ch. Ignoring any intervortex interactions, this model predicts an ohmic

response to the applied current.

20

 

3.3 Therrr
Vortices are 2

I
for the vortex

currents. the

 

no longer valir
regime is desr
can be repres
vortices[13]. |
energy barrier

The vortex mc

Where the ffac
densities Such
”WEI assume

#187), Whare A

3.3 Thermally-activated flux flow

Vortices are also affected by defects within the crystal, which act as pinning sites
for the vortex lines, tending to arrest the vortex motion. For low transport
currents, the Lorentz force may approach the pinning strength, and eqn. (3.2) is
no longer valid since the pinning force must now be included. The low current
regime is described by the thermally-activated flux flow (T AF F) model[12], which
can be represented in terms of a perturbation to the flow velocity of the
vortices[13]. In this model, the retardation of vortex motion is described by an
energy barrier to motion Up, which is large compared to the thermal energy kBT.

The vortex motion is retarded by an amount 8v<<v, reducing the resistivity,

p=-i’£— (3.4)

”5‘1
v

where the fraction 8v/v depends on the applied current density. At low current
densities such that the Lorentz force is less than the pinning force, the TAF F
model assumes a thermally-activated vortex motion described by 5vlv z A exp(Up

/k3T), where A>>1, and eqn. (3.4) becomes

p s fife'UWM (3.5)

At low current densities the TAF F model predicts ohmic behavior, with a small
resistivity p << pg. For large current densities the Lorentz force overwhelms the
pinning force such that 8vlv —> O and flux flow resistivity is recovered.

The schematic shown in Figure 3.2 describes this behavior. At high
temperatures where the thermal energy is always greater than the vortex pinning

energy. the vortex is in the flux flow state, and the Bardeen-Stephen model

21

 

predicts ohrr
the tempera?

pinning energh
TAFF flow at
Previous won
measurement
transition. It r

a temperature

vortices are We

 

Figu re
Shows
repreSe
to a liq i

predicts ohmic behavior, represented by a linear E-J curve with a slope of 1. As
the temperature is lowered and the thermal energy becomes comparable with the
pinning energy Up, the E-J curves develops an S shape, as the flow crosses from
TAFF flow at low currents to Bardeen-Stephen type flow at high currents.
Previous work[14-16] has observed an S shape in current-voltage
measurements, in a very narrow temperature window close to the melting
transition. It has been suggested[15] that the S shape appears below the kink, in
a temperature window just below the vortex melting temperature where the

vortices are weakly pinned and can be easily thermally depinned, i.e. Up = kBT.

 

flux flow

   
  
 

vortex liquid

logE

I'd
T AFF vortex so i

 

 

 

logj

Figure 3.2 Transport measurments in YBCO. The schematic
shows current-voltage behavior on a log-log plot. Tm

represents a melting or crossover transition from a vortex solid
to a liquid state.

22

 

3.4 Ander
Vortex motioi
investigated
vortex hoppin
the Lorentz f
is the Lorentz
of the vortex
vortioes, h0p p

force. the resr

In order to rel;
EStimated as

current densil

Thus the thec
DOHZero tern;
barrier, energ
Vonex mOtion
temperatllre.
te'“l’erature t

3.4 Anderson-Kim vortex creep model

Vortex motion for the case of strong pinning (pinning energy Up >> k7) has been
investigated via the flux creep model[17, 18], which describes thermally-activated
vortex hopping over the pinning barriers. Again, for low enough current densities
the Lorentz force is less than the pinning force, and so Up > U), where U) = jBV.,rp
is the Lorentz force energy due to a displacement rp, which is the pinning radius
of the vortex volume. In the presence of an applied current, a net hopping of
vortices, hopping rate v0, would produce a flow in the direction of the Lorentz

force, the resultant resistivity given by

p = (212,,3/ j )6” "Tsinm ijcrp NJ) (3.6)
In order to relate the pinning energy with the critical current, the pinning energy is

estimated as U, = U, -(l- j/jc ). Thus at low current densities Up is constant,

and the argument of the hyperbolic function replaces the function, and the

current density cancels out, leading to an ohmic resistivity:
pm z (ZvoBchrp/kT)e'”°m (3.7)

Thus the theory predicts a finite linear resistivity p ~ e’“°”‘T for low currents at any
nonzero temperature T, due to the vortex hopping over the constant pinning
barrier, energy U0. This prediction that any Lorentz force would produce some
vortex motion would imply that there is no true critical current except at zero
temperature. Evidence of this vortex creep has been obtained for low
temperature type II superconductors[18]. However, measurements of YBCO
polycrystalline[19]and thin films[20] find no evidence of vortex creep. Instead,

the E-J curves drop off dramatically below the transition temperature, providing

23

evidence the
force. This I

measuremer

3.5 Vorte)
An explanatir
solid, where I

applied curre

mep~1n
decreases the
a vanishingly
existence of a

reSisfiViTY at lc

evidence that the effective pinning energy diverges in the limit of small Lorentz
force. This has also been confirmed in high resolution contactless

measurements of a twinned sample[21], at temperatures further below Tm.

3.5 Vortex glass transition
An explanation for this behavior was proposed by Fisher for a disordered vortex
solid, where he predicted the dependence of the effective pinning energy with the

applied current to be[22]

U z erU‘H)” (3.8)
where ,1: ~ 1 is called the glassy exponent. Thus as the current density j
decreases the pinning energy diverges. Assuming p ~ e'u’", eqn. (3.8) leads to
a vanishingly small resistivity at low currents and finite temperatures, and the

existence of a true superconducting state, defined by the absence of linear

resistivity at low currents.

24

THE V0

4.1 First 01
One of the mi
superconduct
Abrikosov latt
transition to a
recently Brézi
condensate d.
of first order it
the high temp.
bngih 5. and I
Much theoretir
of a Undeman
melted When t

alraction CL of

The brackets <

directions. T

Within the vorte

Chapter 4

THE VORTEX PHASE DIAGRAM OF CLEAN YBa2C0307-5 CRYSTALS

4.1 First order vortex solid to liquid melting transition
One of the most salient characteristics of the vortex state in high temperature
superconductors is the existence of a first order melting transition of the
Abrikosov lattice in clean single crystals. Although a continuous freezing
transition to a vortex lattice state at He; was predicted by Abrikosov, more
recently Brézin, Nelson, and Thiaville have shown that when fluctuations in the
condensate density are taken into account, the freezing transition at He; can be
of first order just below ch in low temperature type II superconductors[23]. For
the high temperature superconductors, due to their high Tc, small coherence
length 5, and large anisotropic mass ratio, the melting line can be far from Hez.
Much theoretical work has been conducted to investigate this transition in terms
of a Lindemann criterion for melting[24], where the vortex lattice is considered
melted when the mean-square vortex line thermal displacement <u2> is equal to
a fraction or, of the lattice spacing a, similar to the melting of solids[25-27]:

(112) = cja2 (4.1)
The brackets <....> represent integration over the wavevectors k, and in all
directions. The vortex is expressed as an elastic object with elastic deformations

within the vortex lattice, described by a Hamiltonian which includes the vortex

25

 

bulk modulus

applied along

I

By investigat:
vortex di5plac

criterion, the n

 

where Si is th
parameter 59 :

CWSiHIIOQraphi.
predicted at ve
Vortex interacti
Shear and bulk
strength weake
fraCtiOn fTOm th

are exPGCted tc

bulk modulus cu, tilt modulus C44, and shear modulus 055 [26, 28] (here for a field

applied along the c axis):

H = %2ua(—k)[c,,(k)kak, + 6“,,{c,,,(k)[ka2 + k: ] + cukczjlub(k) (4.2)
I

By investigating positional fluctuations of this Hamiltonian with temperature, the
vortex displacement is obtained. From these calculations, and the Lindemann

criterion, the melting line has been estimated to be

B (T,0)=-5-£iH (0)[1—T/T]’3 (4.3)
m 8 Ci c2 c

0

where Gi is the Ginzburg number, 8,, is the superconducting anisotropy

 

parameter s, = Hy? cos2 0 + sin2 6, 6 is the angle between the field and the

crystallographic c-axis, and )6 z 1-2[27, 29]. A melting transition has also been
predicted at very low fields near Hc1(T). In this dilute vortex limit the vortex-
vortex interaction becomes exponentially small, substantially decreasing the
shear and bulk moduli[25, 30]. As the field is decreased the lattice interaction
strength weakens exponentially, allowing the vortex lines to wander a substantial
fraction from their equilibrium positions and finally melt. The two melting lines
are expected to meet at a temperature below Tc, as shown in Figure 4.1(b).

The first experimental indication of a vortex melting transition in high
temperature superconductors was observed in resistivity measurements on
untwinned, single crystals of YBCO[31, 32]. An extremely sharp drop in the
resistivity was observed in the tail of the temperature dependent superconducting
resistive transition in the presence of a magnetic field applied along the

crystallographic c-axis. A typical example is shown in Figure 4.2(a). A kink, or

26

C
H

I 20.“. 0:0:an

we

H02

: .22... 0.326%:

HCI'

1 986

       
        

I
O
M

I .
g 06 v

z 006

.§ 00 Mixed
c Abrikosov Vortex State
8, Lattice

E

 

Lower Critical Field

Temperature
(a)

e

Magnetic Fleld H

9}

 

Temperature TO
(b)

Figure 4.1 Phase diagram (a) for low temperature type II
superconductors, and (b) for high temperature type II superconductors.

27

-cm)
( H“
p

Fierr
YBCC

(a)
cry)
ass

-cm)
tun
p

 

 

 

 

(a) Resistivity vs. temperature for high quality, untwinned YBCO
crystal.The arrow shows the kink in the resistivity at H=4T,
associated with first order melting transition.

 

v v v fir I v v v v 1 v f r f r v v v

100’

p (rm-cm)
O)
O

k
C

 

 

 

0 .
75 80 85 90 95
TlK)

(b) Enhanced view of the melting transition region, showing
non-ohmic behavior below the kink.

  
  
  
  
  
  
 

 

1o fI I f‘ififirfI I I I I I I I I I I
p
t
, Hllc
8 c
, -----2r 6.7 Alan”
’ ---2T 0.67 Alan”
A l
E 6 '--'--1r 6.7 Alcm’
9 r
C: i ---1T 0.67 A/cm’
1 l
v 4
°- T
t
2 .
0

 

 

86 e7 88 89 90
'T(K)

Figure 4.2 Resistivity versus temperature of an untwinned
YBCO single crystal.

28

sharp drop i

arrow). Althr.
unusual aspe
superconducl
behavior abo
the splitting 0

different mea:

linked to 3 lat«

 

field sweeps c
transition from

One as
3 Peak effect
narrow tempe
a SOflening 01
ismperamre 1
lattice to dist:
Cllticaj Currer
lattice are de
zero as the v
observed in 1
order melting
measuring (2

in
”9386 in U

sharp drop in resistivity, is observed at Tm ~ 84 K for H = 4T (indicated by an
arrow). Although a drop in resistivity can also be attributed to pinning, the
unusual aspect of the sharp drop is that it is typically narrower than the zero field
superconducting transition. Furthermore, the resistivity is characterized by ohmic
behavior above the kink and non-ohmic behavior below the kink as indicated by
the splitting of the two resistivity curves shown in Figure 4.2(b), obtained with two
different measuring currents for H =1 and 2T. In addition, hysteretic behavior
linked to a latent heat has also been observed in both temperature and magnetic
field sweeps of the resistivity near the melting temperature, suggesting that the
transition from vortex solid to liquid is of first order.

One aspect of pinning near a vortex melting transition is the occurrence of
a peak effect in the critical current, where the critical current is enhanced within a
narrow temperature window. The enhancement in critical current is attributed to
a softening of the vortex shear modulus as one approaches the melting
temperature from below. This relaxation of the shear modulus allows for the
lattice to distort and accommodate nearby pinning sites, thus increasing the
critical current[33]. Above the peak effect temperature T,,, the shear bonds of the
lattice are destroyed due to vortex melting and the critical current plummets to
zero as the vortex solid is transformed to a liquid. This peak effect has been
observed in untwinned and weakly twinned single crystals of YBCO near the first
order melting transition. In Figure 4.2(b), the resistivity curve obtained with a
measuring current density of 6.7 Alcm2 for H = 1T demonstrates a sharp

increase in the resistivity with increasing temperature as one approaches the

29

 

‘kink‘ associa
increasing th
However, jus
decreases to
temperature
lattice are de
softening of
Gifted is seen
the E-J curves
Curves. Belov
sudden increa
Shear mOdUlu:
OftemPeraturi
1045 V / Cm.
The VOI
studied USing
usual to”, my
a Crl’Stal, fer a
and the iOp ar

Since the Cum

‘kink’ associated with the first order melting transition. This is expected since
increasing thermal energy would tend to decrease the effectiveness of pinning.
However, just below the transition temperature Tm, the resistivity rapidly
decreases to nearly zero indicating that pinning is enhanced within this narrow
temperature region. At Tm, the resistivity rises sharply as the shear bonds of the
lattice are destroyed. Thus the peak effect in YBCO is associated with a
softening of the vortex lattice shear modulus prior to melting[34-36]. The peak
effect is seen as a crossing in the E-J curves as shown in Figure 4.3. For T> Tm,
the E-J curves show ohmic behavior as indicated by the linear behavior of the
curves. Below Tm, a sharp downturn in the E-J curves is obtained, indicating a
sudden increase in vortex pinning associated with the appearance of a finite
shear modulus. The inset of Figure 4.3 shows the critical current jc as a function
of temperature, obtained from the E-J curves using a voltage criterion of

10" V I cm.

The vortex melting transition for an untwinned YBCO sample was also
studied using a flux transformer geometry. Four contacts are attached in the
usual four probe geometry, but symmetrically to the top and bottom a-b plane of
a crystal, for a total of eight contacts. The current is passed on the top surface,
and the top and bottom voltages Viop and Vbonom are measured simultaneously.
Since the current density is larger at the top surface where the current is injected
than at the bottom surface, a gradient in the Lorentz force along the vortex lines
is established. lnforrnation about the correlation of the vortex structure along its

length can be obtained by monitoring the temperature dependence of the

30

E (V/nm)

dissipative
axis. It We
defined by
esiablishjr
Measu,em
n0 C‘axis o
transjtl'onv l
he field d”

The,

 

 

vvvvvvvvvvvvvvvv

H-2Tllc

 

 

 

A 6 8% 6'7
E . —I—88.02 K T (K)
o 10 :'—'—87.82 K
s I +8162 K

V ; +8743 K

, —o—87.22 K

LIJ l +8102 K
+8652 K .
’ +8571 K '

1 +1 1111:]

 

I

; «l
1‘21]. A... ii ti"

0.01 0.1 1
J (A/cmz)

Figure 4.3 Current-voltage curves showing ohmic behavior
above, end sharply non-ohmic behavior below the melting transition.
The data is from the same sample as measured in Figure 4.2.

 

 

 

 

dissipative voltage near the melting temperature for an applied field along the c-
axis. It was reported[37] that the correlation of the vortex line along the c-axis,
defined by Viop = Vbotm, appears just below the melting temperature,
establishing a three-dimensional vortex configuration within the solid.
Measurements of the c-axis resistivity for samples from 15 to 100 pm thick found
no c-axis coherence in the liquid state, and coherence just below the melting
transition, providing evidence of a simultaneous loss of vortex correlation along
the field direction at the melting transition for clean, untwinned samples.

These earlier transport measurements established the location of the first

order vortex melting transition on the H-T phase diagram. However, since these

31

 

were nonequ
thermodynam

Accord:
magnetic syste

derivative of it

The res

 

S correspondlr

Tilelrnodynarr
in magnetizati
magnetizatior
VS- T measure
lie, Mam >1
Similar 10 me‘
tiErnsition frOr

VorteX Der do

Hm, de/dT a

mating tranSi

were non-equilibrium measurements no further information of equilibrium
thermodynamic state could be obtained.

According to thermodynamics, for a first order phase transition of a
magnetic system to occur, there must be a discontinuity of the first partial
derivative of the Gibbs free energy G

G(T,H) = U — TS - MH (4.4)

The resultant Clausius—Clapeyron equation predicts a jump in the entropy

S corresponding to a discontinuity in the magnetization at the transition:

AMdH'" = —AS (4.5)
dT

 

Thermodynamic evidence of a first order phase transition in YBCO was obtained

in magnetization measurements[38, 39]. For H = 4T, a jump was found in the

magnetization 47rAM = 47r(M,.qu,d - M,o,,d)=0.3 Gauss[39] in both M vs. H and M

I

vs. T measurements. The increase in magnetization in the liquid state

(i.e., MW > Mm“) indicates that the vortex density is lower in the solid state,

similar to melting of ice. A convenient way to compare the entropy of the
transition from one sample to another is to calculate the entropy increase per

vortex per double Cu-O plane using the Clausius-Clapeyron equation

AS =-——e——°— (4.6)

where d=11.7 A, the c axis parameter for YBCO, k3 is Boltzmann’s constant, and
Hm, dHanT are obtained from Hm = 99.7(1-T/Tc)"36, which is the best fit of the

melting transition data[39], obtaining a value of AS, = 0.65-0.7 kg for YBCO.

32

 

A dire
thennodynar
simultaneous
untwinned Yl
coincided wit
were also pe
measured fro
wide range 01
measuremen

More r
vortex melting
measuremen-
one crystal. |
onset of the k
lwc Crystals c
in the resistivl
and torque d2
Gun/es fir Si 8)-

NIElting trans)

A direct correlation between the kink in the resistivity and the
thermodynamic first order vortex melting transition was obtained through a
simultaneous measurement of the resistivity and the magnetization on an
untwinned YBCO crystal[40], confirming that the jump in the magnetization
coincided with the onset of the kink in the resistivity. Calorimetric measurements
were also performed on the same untwinned sample[41]. The entropy was

measured from the latent heat of melting L = TAS, yielding AS, ~ 0.45ke for a

wide range of fields, from 1 to 8 Tesla in good agreement with the magnetization
measurements.

More recently, in an experimental collaboration with ET H, the first order
vortex melting transition was observed in high-resolution torque-magnetometry
measurements[42, 43], which extended the melting line down to 0.09 Tesla in
one crystal. In this collaboration, the melting transition was determined from the
onset of the kink in the resistivity and torque magnetometry measurements on
two crystals cleaved from the same parent piece. Figure 4.4 shows (a) the peak
in the resistivity data and (b) the resultant melting transition from the resistivity
and torque data. Also shown is the temperature where DC current-voltage
curves first exhibit non-ohmic behavior. All three methods of establishing the
melting transition are in good agreement with each other, and exhibit the same
Power law dependence with temperature, as given by eqn. (4.3). For this data
6 = 1.34 i .04, in agreement with data from other high quality untwinned YBCO

crystals[32, 44].

33

 

10

(a) Resistivity and the derivative of the resistivity data.

 

     
   
  
  
 

 

 

 

 

 

120 I I I I I I I I I I I I I I I I I I I U I I I I I I I I I ' I I I I‘ 7o
:H'1T H C 89.80 K I
100 . peak In the derivative «— 5°
.- 50
60
A
g i 40 Q
r. ‘ '0
d; 60 , \
3. ' . O.
V ‘ 30 -I
Q
40
20
20 . : 10
0 0
66 89 90 91 92 93 94 95

T (K)

(b) Melting line from resistivity and torque measurements.

 

1O I I I II I I ' I I r 1 I I 1 I I I I
I -IA' T I I
w'vr- ' "

 

 

 

 

 

 

: n-1,--- rr‘..1fi.
‘ l
6
8 - .
l .1
l- a
l-
A 6 i" "
1- .
v
E I
I 4 ‘
101*(1-T/Tc)"3° T d
l' I
~ h
\
2 r—fi—H II c: klnk In reslstlvlty 1i
_ O H II c: Torque-magnetometry “ -
)--l--H ll ab: klnk In reslstlvlty .3 .-
1: H II c: onset of nonohmlc I-V curves I ‘
o ...L...1..a1a..1.4. a..1...1

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
T/T
c

Figure 4.4

34

The order 0‘

simulations.

I
entropy jum;

 

 

the latent he

the jump[47]

4.2 Critica
Recent trans
first order tra
magnetic fiel
point Hum 56
Order melting
the 'kink’ in t:
ma‘eretizatic

the peak in t.
ath Fun
point Can Val
‘Optimany dz
CrYStalsfs‘I' 5
Can be drann.
is GEDEndem

The te

can be (mamE

The order of the melting transition has been investigated using Monte Carlo
simulations. Although earlier work predicted somewhat smaller values in the
entropy jump[45, 46], more recent work has found a somewhat higher value of
the latent heat a 0.4 kaT, as well as the temperature dependence for the size of

the jump[47], and is seen as confirmation of the experimental results.

4.2 Critical points
Recent transport, magnetization, and calorimetry measurements indicate that the
first order transition in some clean untwinned YBCO crystals terminates at high
magnetic fields, giving way to a continuous transition above an upper critical
point Hm, see Figure 4.5. In transport measurements, the termination of the first
order melting transition at high fields is demonstrated by the disappearance of
the ‘kink’ in the resistive transition at an upper critical point, Hucp. Similarly, in
magnetization and calorimetry measurements, the jump in the magnetization and
the peak in the specific heat denoting the first order transition is also suppressed
at Hum. Furthermore, it has been shown that the magnitude of this upper critical
point can vary widely from sample to sample ranging from ~5—12 Tesla in
~optimally doped YBCO[48-50] to beyond 30 Tesla in overdoped YBCO
crystals[51, 52], whereas in underdoped YBCO crystals the upper critical point
can be dramatically reduced[53]. These results imply that the upper critical point
is dependent upon the density of point defects within a sample.

The termination of the first order phase transition at the upper critical point

can be qualitatively explained by considering the competition between the vortex

35

 

b-----

 

      
      
  
   

ntinuous or crossover
transition

normal state

vortex liquid

entangled
vortex glass

enhanced pinning ling, ""

.'
-‘.'.--

vortex lattice

H01
Meissner state

 

 

T 7°

Figure 4.5 Possible three phases in the mixed state: Vortex liquid,

lattice, and glass, in clean YBCO crystals.

elastic energy, the pinning energy, and the thermal energy[54, 55]. Along the

first order vortex lattice melting line, the elastic energy of the vortex structure E.,

can be described by a cage potential:

5,, = CwuzL + £,(u2 /L)

where 055 is the shear modulus, u is the displacement of a vortex line from its
equilibrium position, L is the length of the vortex and s. = fl so is the vortex line
tension with y= mIM and so = (tho/41:12)? Along the melting line, we can use the

Lindemann criterion u: = 01,282 to obtain E., = yea cLza, where a is the vortex

36

spacing. Th

 

(Lo/Lcil’sis4
to ~ 23/8 is

by minimizin
upper critica
energy, lead

-mo ,
Em ~ B c

 

energy of the
neighboan \
energy Earn ‘

The e
the vortex sc
vortex solid 1
lattice to an
ofweak poir
Simplified (:2
which inves-
lattjce 0f lint
configuratio
sum of the !
keep the V0
Dointdefec1

spacing. The pinning energy can be estimated by Epm = (ye. 0L2 52 )1’3

(Lol Le)"5[54, 56] where the first term represents the depinning temperature,

Lo ~ 27a is the characteristic size of the longitudinal vortex fluctuation obtained
by minimizing eqn. (4.7) with respect to L, and LC is the pinning length. At the
upper critical point, E, = EM, and the pinning energy overcomes the vortex lattice
energy, leading to the loss of the first order transition. Basically, E, ~ 3'"2 and
Epm ~ B“"° and thus Em, dominates at high fields. In this scenario, the elastic
energy of the vortex Ed, tries to keep the vortex approximately aligned with its
neighboring vortices while the isotropic point defects, with effective pinning
energy Epm , promote line wandering.

The existence of an upper critical point suggests a transformation within
the vortex solid phase from an ordered lattice state below Hum to a disordered
vortex solid state above Hoop. Indeed, a field-driven transition from a vortex
lattice to an entangled disordered solid has been predicted, due to the presence
of weak point disorder[54, 56-60]. The transition can be described using a
simplified cage potential description of the vortex lattice[54, 55] (see Figure 4.6),
which investigates the energetics for a fluctuation u(z) of a vortex line within a
lattice of lines, and the influence of point pinning centers on the line
configuration. For this model the elastic energy of the vortex E.,, which is the
sum of the single vortex tilt energy and the vortex-vortex interactions, tries to
keep the vortex approximately aligned within the cage. However, the isotropic
point defects, with effective pinning energy Epm, promote line wandering. The

vortex solid is expected to transform from a lattice to an entangled vortex

37

 

 

 

Vortex line
L

disc

d

Elastic energy of the vortex:

Eel = intervortex energy + tilt energy
=066u2 L + 044 u2/L

 

 

{ Vortex lattice:
Eel > Epin

J u<<a

a
vortex line caged by 6 vortices

 

 

 

Entangled vortex glass:

£9! < Epin
u ~ a

 

 

disorder-induced
dislocations

Figure 4.6 Vortex line fluctuation u in a vortex lattice.

38

solid[57] when E.) = Epin. At this point the transverse wandering length is of the
order of the lattice spacing, and so dislocations can proliferate, leading to
entanglement (see Figure 4.6). This entanglement transition line is expected to
be marked by a sharp increase in the measured critical current[59]. Above this
line the vortices are no longer tightly bound as a lattice, and so adjust to the local
pinning potential, becoming more effectively pinned by the defects. The fact that
entangled vortex lines would have to be cut in order to move independently,
results in the increase in pinning efficiency and critical current. For this new
state, an increase in temperature may produce a second-order transition[29] from
a vortex glass to a liquid.

Evidence of a phase transition from a vortex lattice to a disordered solid
has been obtained from neutron diffraction data on BSCCO[61]. Measurements
at low fields and temperatures show the expected diffraction peaks for a
hexagonal lattice. The peaks disappeared with increasing temperature (vortex
lattice melting) as well as with increasing field (vortex lattice to disordered solid).
Magnetization measurements in BSCCO[62, 63] and YBCO[53, 64-67] samples
also show a sharp increase in the critical currents represented as the onset of a
second magnetization peak as the magnetic field is raised within the solid-solid
transition is crossed, as expected for a transition from a weakly pinned lattice to
strongly pinned glass. This transition is shown as an enhanced pinning line in
Figure 4.5. It is an open question whether this ordered-disordered transition line
meets the melting line at Hm, which would then suggest HM, to be a multicritical

point. At higher fields the magnetization reaches a maximum, known as the 2"d

39

magnetization peak, and then decreases monotonically with increasing field.
Recent measurements[50, 67, 68] suggest that the 2"" magnetization peak line
may be related to the critical current peak effect seen in YBCO crystals. In this
scenario the upper critical point would not be a multicritical point, and the
disordered solid state would also exist in the region where the peak effect is
seen, which is close to but just below the first-order melting line. As the
temperature or field is lowered, the vortex lattice would then be established.

In the transformer configuration measurements[49], the data suggests
strong evidence of flux line cutting in the solid phase above Hm, where
mem = O Volts, while V.” at 0. This suggests that the part of the vortex lines
near the top are in motion, while the part near the bottom are in a strongly pinned
solid phase, with vortex cutting occurring within the bulk of the sample. This
work also finds evidence of an entangled state above H...» from c-axis resistivity
measurements. The data show dissipation in the c-axis resistivity at
temperatures where the transverse resistivity has dropped to zero, implying the
existence of vortex loops and entanglement above Hm.

With increasing temperature, in the region above Hm thermal fluctuations
induce a transition from the entangled solid state to the liquid state. The vortex
glass theory[22] predicts a continuous transition in the case of isotropic disorder,
with the current-voltage curves obeying critical scaling rules within the critical
fluctuation regime[29]. Measurements on clean untwinned YBCO have yet to
produce this scaling, however the vortex-glass theory also predicts a power law

behavior for relatively strong driving currents near the transition:

40

I) ~ (T - Tm)" (43)
By evaluating the ohmic tail of the resistivity above How, a power 3 = 5-7 has
been found[48, 64], with good agreement between Tye, obtained by a fit of the
measured resistivity[48] to eqn. (4.8), and T)”, which is defined as the onset of
irreversibility in the magnetization measurements[64]. Nevertheless, since no
critical fluctuation regime is observed, then the above data cannot clearly be
associated with a phase transition, and thus no definitive observation of the
vortex glass transition has been found.

More recently, analogous to the upper critical point, a lower critical point
has been identified in YBCO single crystals. For example, in both the resistivity
and magnetometry data in Figure 4.4(b) inset, the melting line appears to
terminate at a lower critical point H“, = 0.1 Tesla. Similar to the upper critical
point, this point is defined by the field below which the kink in the resistivity, jump
in magnetization, or peak in the specific heat is no longer seen in the data.
There is a wide variation of the location of this point[14, 16, 43, 69-72], but for
high quality untwinned samples the lower critical field is consistently below 0.5T,
with the lowest measured at 0.09T. The lack of a measurable kink at lower fields
may be due to its close proximity to the critical fluctuation regime of the
superconducting order parameter within 1 K of Tc, or simply due to a lack of
resolution of the experimental techniques at very low fields. Also, if the sample is
not perfectly homogeneous, the position of the melting transition may be
smeared due to a distributed melting temperature. Thus a true lower critical point

may not exist in very high purity, low point defect crystals.

41

A lower critical point would possibly mark the onset of a higher order
quuid-to-glass transition below Hm. if a glass state does exist below Hm
thermodynamics demands that there be yet another solid-solid phase transition,
from a disordered glass to an ordered lattice. (For an alternative view, which
predicts a liquid-liquid transition below a critical end point and no first-order
melting line, see[73].) The existence of the new lower critical point and the
vortex phase transition below Hm,p will be discussed in Chapter 9.

In summary, the first order vortex lattice melting transition in clean
untwinned crystals of YBCO has been firmly established via dynamic and
thermodynamic measurements. This first order line is shown to terminate at both
high and low magnetic fields. Above and below the upper and lower critical
points, the first order transition transforms into a continuous higher order
transition. A consequence of the critical points is that novel vortex phases may
exist within the vortex solid state at high and low magnetic fields. Some of the
novel phases, such as an entangled vortex solid phase at high fields, has been
the focus of extensive experimental and theoretical research. On the other hand,
not much is known about the vortex solid state below the lower critical point at
low fields.

in order to shed light on this interesting regime of the vortex phase
diagram, we examine the effect of induced columnar defects on the first order
vortex melting transition. The advantage over point defects is that the density of
columnar defects can be carefully controlled and monitored experimentally.

Furthermore, being line defects, these defects are dimensionally well matched

42

with vortex
temperalur
aiundame
transition c2
defects. Th

on induced

 

 

 

with vortex lines and have been proven to be strong pinning sites at high
temperatures near the melting transition. in addition, this study will shed light on
a fundamental issue in the physics of phase transitions, namely how a first order
transition can transform into a second or higher transition with the introduction of
defects. These results will be discussed and compared with collaborative work

on induced point defects in YBCO via electron and proton irradiation.

43

Chapter 5

VORTEX PINNING BY DEFECTS IN Y332CU307-5 SINGLE CRYSTALS

5.1 Pinning in the vortex liquid state by twin boundaries

As discussed in Chapter 4, in transport measurements on untwinned, clean
YBCO crystals, strong pinning and non-ohmic behavior was only seen below the
melting transition. However, interesting behavior in the liquid state has been
found in as-grown YBCO crystals with twin boundaries. Twin boundaries are
strain fields along the <100> and <110> which occur during the growth process
of YBCO when the crystal undergoes a tetragonal to orthorhombic transition. To
the left and right of a twin plane, the basal structure of the crystal is rotated by
ninety degrees. These planar objects, are of the order of 10 A wide, and contain
atomic scale defects such as oxygen vacancies and atomic scale ion
displacements. These twin boundaries can be spaced from anywhere between
1000 A and several microns apart and, in homogenous crystals, extend
throughout the thickness of the sample, hence serving as intrinsic pinning sites in
as-grown YBCO crystals prior to detwinning[74].

The viscosity of the vortex liquid state in the presence of twin boundaries
has been studied by Marchetti and Nelson using a hydrodynamic approach[75].
The model describes the flow of vortices confined in channels made of parallel
twin boundaries as shown in Figure 5.1. The viscosity of the vortex liquid within
the channels is described in terms of a healing length extending from the twin

boundaries where the vortices are localized, to the center of the channel where

 

Fig

the vortices
velocity is '
walls, loca
Contained
described
Lorentl tr

interactic

 

 

 

 

 

A

2yMI- 4 L
T o —>
w e D v(y)

0- e b
0 $
' —> 8
-zy/w- - u .- --.~ u».

 

 

 

Figure 5.1 \fiscous damping due to twin boundaries. The curve in
the figure shows the velocity distribution, given by eqn. (5.1).

the vortices are less restricted. For a twin boundary spacing W, the vortex
velocity is predicted to depend on the position y with respect to the twin boundary
walls, located at y = :l:Wl 2, see Figure 5.1. The viscous damping coefficient is

contained in a characteristic healing length 6 = 1/11/y, where y is the previously

described single vortex frictional drag, n is the intervortex viscosity, and v0 is the
Lorentz force dependent single vortex velocity in the absence of intervortex

interactions (recall eqn. (3.3)). The resultant velocity profile is predicted to be:

 

cosh(y/6) ] (5'1)

v(y) = ”[1 _ cosh(W/25)
Here the vortices in the twin boundaries are assumed to be stationary. in this
model, the viscosity 1] is large enough to effect the vortex flow only when the
possibility of vortex entanglement (i.e. crossing along the lines) is taken into

account. This is seen as an increase in the viscosity at Tm, becoming very large

as the temperature is decreased.

45

The interplay between the first order vortex melting transition discussed in
Chapter 4 and twin boundary pinning in the liquid state mentioned above is
shown in a YBCO crystal with only two twin boundaries space 140pm apart[34],
as shown in Figure 5.2. When the field was applied parallel to the twin boundary
planes, a ‘shoulder’ appeared at Tm, where the resistivity starts to drop rapidly.
Below Tm the current-voltage measurements were found to be weakly non-
ohmic. By rotating the field with respect to the twin boundaries and thereby
weakening the effect of twin boundary pinning on the vortices[74], the sharp
melting transition Tm was reestablished at a temperature below Tm. In this case,
the resistivity is non-ohmic only below Tm. This is strong evidence that there is
significant intervortex viscosity in the liquid state: When individual vortex lines
become pinned by the twin boundary planes, the motion of vortices between the
planes is reduced by viscous damping.

The twin boundary shoulder has also been investigated in a heavily
twinned crystal, using the flux transformer configuration[31, 76]. In this work,
evidence of c-axis vortex correlation is first seen in the liquid state at Tn, ~ Tm.
Below Ty, and at low currents Vrop = Vbouom, implying either a disentangled vortex
liquid state or a topologically entangled liquid where the vortex cutting energy is
large. The Tn, transition also marks the disappearance of linear c—axis resistivity,
providing evidence of vortex correlation along its entire length in the liquid state
below Tn,[77]. For high applied transport currents however, V.» > Vmom,
indicating evidence of vortex line cutting[30, 78-80]. In this model the force

necessary to cut the vortex lines[80] becomes less than the Lorentz force, and so

46

 

 

resistivity

 

 

 

 

Temperature

Figure 5.2 Angular dependence in resistivity vs temperature data, H=4T.
After W. Kwok, et aI., Peak Efi’ect as a Precursor to Vortex Lattice Melting in
Single Crystal YBaz Cu307.a, Physical Review Letters 73, 2614 (1994).

vortex cutting occurs, with the vortex motion faster at the top surface than the
bottom, resulting in a higher voltage at the top surface of the crystal. The
temperature dependence of Tn, for the establishment of c-axis correlation is also
found to be thickness dependent in these twinned samples, as expected since
the energy to drag the vortices should increase with line length[77]. Both the two
twin and multiple twin studies above provide evidence that the extended twin
planes enhance the vortex coherence along the c-axis, and thus vortex line

straightening.

5.2 Point defects and the Vortex Glass theory
The effect of isotropic point defects on the melting transition has been studied
theoretically[22, 29]. The transition is predicted to be a continuous phase

transition to a glass state known as the Vortex Glass (VG) phase, with a critical

47

fluctuation regime close to the Vortex Glass transition temperature Tye. Within
this regime fluctuations in the vortex line correlation length Ive are expected to
scale as

1V6 ~l T — Tm I‘V’ (5.2)
where V is the Vortex Glass static critical exponent. (For the Vortex Glass
model, the critical exponents are primed in this thesis, in order to distinguish
between this model and the Bose Glass model described below.) In this isotropic
model the vortex correlation lengths both parallel and perpendicular to the
applied field are expected to diverge at the glass transition temperature, following
eqn. (5.2). The relaxation time of a fluctuation is expected to scale with the

dynamic scaling exponent 2’, as 1 ~ IVGZ'. The fluctuation regime is characterized
by a non-ohmic current-voltage (E-j) behavior close to the transition, which
scales both above and below Tye according to:
Ezra" = 6910336) (5.3)

Using eqn. (5.3), the E-j curves collapse onto the smooth functions ei for
temperatures above and below TVG. In addition, for T > TVG, and within the
critical temperature regime but at low current densities, ohmic behavior is
expected. Here the resistivity scales as

p(T-eTJG.j—>0)~<T—Tm>""""’ (5.4)
At temperatures above the critical regime, a relatively free vortex liquid is
predicted, characterized by ohmic behavior for all currents. Below Tye for low
current densities, the theory predicts diverging barriers to vortex motion, resulting

in a true superconducting state, defined here as p( j -—> O) —> 0. In terms of a

48

measured electric field, the theory predicts a sharp decrease with decreasing

current: E cc WWW, where u is the glassy exponent, predicted to lie between
0.25 and 1[22, 36].

Although a number of earlier papers claimed to experimentally observe a
Vortex Glass transition, all of them involved YBCO samples with correlated
disorder in the form of twin boundaries. Transport measurements on untwinned
YBCO single crystals with point defects created by electron[16] and proton[81,
82] irradiation have so far failed to observe the nonohmic E-j curves predicted by
the Vortex Glass theory. For all these measurements on irradiation induced
disordered crystals, the kink in the resistivity is replaced by a monotonically
decreasing ohmic resistivity. However, the tail of the resistivite transition has
successfully been scaled to eqn. (5.4) in the case of a high density of defects
induced by proton irradiation[82], with the resultant resistive scaling exponent
z’(v’-1) being independent of the field and field direction, in agreement with the
isotropic Vortex Glass model. The determination of the non-linear E-J behavior
above and below the critical regime has so far eluded measurements, probably
due to the high voltage sensitivity needed to probe the vortex glass state at the
transition. Apparently, point defects shift the melting transition to lower
temperatures in a monotonic fashion, whereby the resistivity gradually decays to
zero value but at temperatures where the measured voltage signal becomes

immeasurably small.

49

 

5.3 Hi9
Transport
produced l
the expechL
Glass theo
glass trans ,,

overdampe ,

equation fo

divergence

from above.

 

This equatic

solid transiti

5.3 Highly viscous Vortex Molasses model

Transport measurements on untwinned YBCO single crystals with point defects
produced by proton irradiation have shown purely linear E-J curves, in contrast to
the expected nonohmic behavior for diverging barriers as predicted by the Vortex
Glass theory. An alternative theory[83], based on the general theories of liquid-
glass transitions, has recently been introduced. It describes a highly viscous,
overdamped “Vortex Molasses" model, characterized by a VogeI-Fulcher

equation for the liquid relaxation time 1' ~ exp[l/(T - 7;)]. This exponential

divergence leads to a rapidly decreasing resistivity as the glass is approached
from above:

p<0=p.e“”‘”"‘ (5.5)
This equation describes ohmic behavior of the viscous liquid, near the liquid to
solid transition. The general theory is independent of whether there is any
diverging coherence length scale, because measurements are overshadowed by
the effects of the rapidly increasing viscosity, resulting in purely ohmic resistivity.
However, it is difficult to discern between eqns. (5.4) and (5.5) from resistivity
measurements[83]. Thus at this time it is an open question as to whether a
simple viscous vortex liquid akin to a window glass or a true Vortex Glass

transition is occurring in the case of point defect pinning.
5.4 Bose Glass theory
Columnar defects provide one of the most efficient vortex pin sites in high

temperature superconductors[84, 85]. These straight cylindrical defect tracks,

50

consisting of a normal core approximately 40 to 100 A in diameter and up to

50 pm long, can be created by the impingement of high energy, heavy ions upon
a YBCO sample. Since the defect diameter is comparable to that of the core
radius of the vortex lines, these defect tracks are highly suitable as anisotropic
pinning sites. The density of columnar defect sites is usually defined by the
equivalent dose matching field B,, necessary to produce the same vortex line
density as columnar defects: rim = B,D I (Do, resulting in a mean defect spacing
d = 1/J’ = «EV—B; .

Columnar defects have significant affect on the vortex motion, which are
strongly correlated with the ratio of the vortex to columnar defect density. For
applied fields H on the order of the defect matching field B, and parallel to the
defects, the vortex lines are strongly localized to the columnar defects for
temperatures well below the melting transition. The random defect sites act to
destroy the long-range order of the vortex lattice, producing a glassy state known
as the Bose glass phase[86, 87]. This solid state is characterized by localization
of vortex lines on the individual columnar defects, resulting in a disentangled
vortex phase with the vortices randomly positioned transverse to the applied
field. Wrth increasing temperature the lines become progressively delocalized
from the defects, and above the Bose glass melting temperature an entangled
vortex liquid is predicted. The transition from glass to liquid is continuous, and
from the theory of critical phenomena[88, 89] it is predicted that there exists a

critical fluctuation region located close to, and both above and below T36. Within

51

this critical region it is possible to collapse E-j curves by scaling the data with
appropriate diverging coherence lengths.

The vortex free energy of a system of vortices in a random configuration of
straight columnar defects is presented for a system of N vortex lines, length L,
aligned along the z = c axis = defect direction, in terms of transverse (ab plane)

fluctuations in position R,(z), subject to a Lorentz force fL[87]:

L N dR 2 .
Adm 45”] camera-mar

 

paw

N N
+2 UrlRu(Z)] — fI. ° 2 Ru(Z)} (5.6)
" u

The first term includes the vortex tilt energy, where a, = (1/ y)2(<l>0 /47r,1,,,,)2 ln(x) is

the line energy. The intervortex pair interactions are given by V“, which decays
logarithmically at short distances r < 3.1,, and exponentially for r > M. The
columnar defect interaction is given by the z-independent potential U,, which is
approximated by a cylindrical potential constant well depth U0, radius b. The
term Bose glass arises because eqn. (5.6) is analogous to a system of N
charged bosons in a 2 dimensional space, with randomly distributed defects
acting as local pinning sites[86, 87]. A vortex line ‘wandering’ along the length L
of the vortices, given by the thickness of the crystal corresponds to a boson
wandering in time. A strongly pinned vortex line can then be described as a
boson localized (in space and time) to a defect. The boson/vortex mapping is

given in Table 5.1.

52

 

Bosons mass it h/kT Pair Charge Electric Current
potential field

 

Vortices 81 T L 280K°(r//1) 4,0 sz/c E

 

 

 

 

 

 

 

 

 

 

Table 5.1 Charged bosons-vortex lines analogy. After D. R. Nelson, V. M.
Vinokur, Boson localization and correlated pinning of superconducting
vortex arrays, Physical Revew B 48, 13060 (1993).

The melting transition may be evaluated within the Bose glass theory. As
the temperature increases, thermal fluctuations tend to promote vortex
wandering via the creation of vortex kinks, see Figure 5.3. The theory predicts a
wandering length from the original pin center, called the transverse wandering
length li(T). The ‘time’ scale for a boson to wander a distance I](T) is given by
the longitudinal wandering length along 2, l"(T). Close to the transition these two
lengths diverge as T36 is approached from both above and below the transition:

t] ~1/lTBG-Tl" (5.7)
and the characteristic length of a fluctuation, along the vortex line,

12,, ~ I: (5.8)
where Tm is the Bose glass transition temperature, defined as the temperature
where the transverse fluctuation ll(T) is approximately equal to the intervortex
distance d, and v is the critical scaling exponent. By this definition, at the Bose
glass melting transition the vortices become entangled, leading to extended

wandering. The relationship in eqn. (5.8) is due to the anisotropic length scales

53

of a fiuctuation[90], and has been confirmed using Monte Carlo simulations[91].

columnar
defects

at
g
g

 

Figure 5.3 Vortex lines in the presence of columnar defects.

The relaxation time of a fluctuation is expected to diverge at T”, with a dynamic
scaling exponent z:

1' ~ 81 (5-9)
From the diverging relations given in eqns. (5.7-5.9), it is possible to scale both E
and j. The appropriate scaling relations are obtained via the anisotropic
Ginzburg-Landau free energy, eqn. (2.4): j oc af/aA], E cc 3A] /a:, with H H c,
A i is the vector potential in the ab plane, and A i ~ 1 / 11(1). The current density
is then found to scale as j ~ 1/ 2,4,, while the resultant electric field produced by
vortex motion scales as E ~1/llr, leading to the scaling hypothesis describing
the critical regime:

Eli” = allege, lcT) (5.10)

where F; is the universal scaling function. Mile F+ and F; are unknown, the
scaling hypothesis allows current-voltage data for different temperatures to be
collapsed onto F+ above, and F_ below Tm. Since the response is finite at the
transition, the two divergent lengths must cancel there, giving a power law
dependence E ~ J““"’3 when T= 11,6. For temperatures near butjust aboveTB ,
in the limit of very low current density, the resistivity is expected to be ohmic, in
agreement with the TAFF model. Thus in this limit the function 110:) ~ x. From
eqn. (5.10) the resistivity should then vanish as

p(T —> Tga, J —> 0) ~ (T - germ” (5.10)
The Bose glass theory differs dramatically from the Vortex Glass theory for fields
applied at an angle 0 from the c-axis. Specifically, the Bose glass theory predicts
anisotropic pinning, as opposed to the isotropic pinning characteristics of the
Vortex Glass theory. This difference is inherent in the two diverging length
scales (eqns. (5.7) and (5.8)) in the Bose glass model, which predicts a
maximum in pinning strength for a field applied along the columnar defect
direction, and thus a cusp in the transition temperature, see Figure 5.4. Because
of this, for T=Tas(0=0°) the constant current angular resistivity (i.e. along the line
marked A in Figure 5.4) is expected to decrease with decreasing tilt as
p~Hi‘z‘2’[87, 92]. Vlfithin the Bose glass phase an infinite tilt modulus is
expected[87]. In this ‘transverse Meissner phase', the application of a small
perpendicular field H i will not affect the vortex system, as the vortices remain
localized along the defect tracks. At some finite, temperature dependent H ]°(T),

a lock-in transition occurs, where the strongly pinned Bose glass state is

55

 

 

  

 

liquid
Tm \ /
\\\ / ’//

vortex crystal,

not localized

to the defects Bose glass

< b
H].

Figure 5.4 The Bose glass transition in a constant applied field H aligned
with the columnar defects, as a function of a perpendicular applied field HL.

After D.R. Nelson and V.M. Vinokur, Boson localization and correlated pinning
of superconducting vortex arrays, Physical Review B 48, 13060 (1993).

transformed continuously into a kinked vortex configuration[93]. The transition is
characterized by a sharp increase in the measured resistivity, due to the
relatively free motion of the kinks as they move along the columnar defects,
producing a net vortex flow in the Lorentz force direction. The lock-in transition is
expected to scale from above as[92]

TBG(0) — 730(9) ~l H j I”" (5.12)
The transition cannot be scaled from below, since I" is essentially infinite (equal
to the sample thickness) below the transition. For large angles, the vortices are
not expected to remain localized to the defects at low temperatures, leading to a
vortex crystal state with different characteristics than the Bose glass, but this

state is not investigated. In Chapter 8, the Bose glass theory is studied via

56

transport measurements of YBCO crystals containing columnar defects. Both

the non-ohmic critical regime as well as the lock-in transition are investigated.

57

Chapter 6

CRYSTAL GROWTH AND PREPARATION

The crystal structure of YBa2Cu307.5 with 8:0 is shown in Figure 6.1. YBazCuao-I
belongs to the AB03 (A=Yttrium; B=Barium; C=Copper; O=Oxygen) structure of
the perovskite family. The unit cell is constructed of a simple stacking of the
AB03 perovskite structure with some missing oxygen atoms. The oxygen atom
vacancies are located around the Yttrium site and in the basal plane along the a
direction. A structure without oxygen deficiency will have the formula
YBazcuaog, while the structure with the missing oxygen gives YBa2Cu307. The
missing oxygen atoms surrounding the Yttrium atom are responsible for the
layered structure of the buckled two dimensional Cu(2)-O(2)/O(3) planes
sandwiching the Y site. An interesting unique feature is the square planar two
dimensional chains along the basal b direction and the missing oxygen atoms
along the a direction which are responsible for the orthorhombic structure in this
compound.

The superconducting state is dependent on the hole concentration[94, 95]
in the Cqu planes (see Figure 6.2). As the density of holes increases, the
system transforms from an insulating anti-ferromagnetic state to the
superconducting paired state, both states characterized by long-range
interactions. For YBazcu307 .5, the hole concentration is controlled by the
oxygen stoichiometry. as shown in the phase diagram for YBCO[96], see Figure

6.3. According to the charge transfer model[97], the CuO chains act as charge

58

@ Yttrium . Barium Copper 0 Oxygen

 

 

 

 

Cu(2)

 

0(1) q

 

 

 

 

 

 

>b

 

Figure 6.1 YBa20307-5 unit cell shown here for 8=0.

59

 

 

 

 

 

 

400
25 300
9
a metallic
E
8. 200
E
a)
F.
100
superconducting
o 0.1 0.2 0.3 0.4
hole density nh

Figure 6.2 High temperature phase diagram.

 

 

Temperature (K)

 

 

 

 

60 32 54 63 &8 10

oxygen stoichiometry 6

Figure 6.3 Phase diagram for YBCO as a function of oxygen
stoichiometry.

60

reservoirs for the CuO4 planes. The initial development of the chains, which
occurs at the tetragonal-orthorhombic transition at ~680° C[98] (8~6.4 in the
figure), produces a sharp increase in the hole concentration and the
superconducting state at low temperatures. The hole density is strongly
dependent upon the Cu-O bond lengths of the system, both in the CuO..
plane[95] and perpendicular to them[99]. As the 0(4) atom sites become
progressively filled up with increasing 8, the Cu(1)-0(1) bond length increases.
The net effect is a change in the Cu(1) coordination[97]. This lowers the Cu(1)
charge state, with the charge transferred from the CuO4 planes. The increase in
oxygen also leads to a shortening in the length between the planes and chains,
allowing for more charge transfer between them. The hole concentration, or hole
doping, reaches an optimum value given by the maximum transition temperature
Tc, at 5=6.94. For 5 > 6.94, the hole concentration becomes overdoped,
characterized by a small drop in Tc.

The occupancy of the 0(4) and 0(5) sites, as well as the superconducting
transition Tc, depend critically on the oxygen content of the crystal[100-102].
The oxygen concentration varies widely in YBCO samples, and can be adjusted
by annealing the sample in an oxygen environment. As noted in Figure 6.3,
there is a tetragonal to orthorhombic transition as a function of oxygen
concentration. For 5<0.1, the 0(5) site is vacant and 0(4) is fully occupied. For
this stoichiometry, the unit cell dimensions are a=3.8227(1) A, b=3.8872(2) A,
and c=11.6802(2) A[102]. For a lower oxygen content, the 0(4) occupancy

decreases and 0(5) increases, until both reach an equivalent 20% occupancy at

61

=0.65, wherein the crystal becomes tetragonal[102]. As the oxygen content is
decreased further, the occupancy of the now equivalent 0(4) and 0(5) sites drop,
and for 5=1.0, both positions are unoccupied.

Single crystals of YBa2Cu307.5 were grown using the self-flux
method[103]. The process begins with the mixing of 99.99% Y203, 99.99%
BaCOa, and 99.99% CuO to a ratio of 1.00:9.59:9.45, which is equivalent to a
Y:Ba:Cu ratio of 5:27:68. The mixture is pulverized for approximately an hour,
using an agate mortar and pestle, achieving a fine gray mixture. Small pellets of
diameter ~1I2” and mass ~7 grams are formed by pressing the material under a
pressure of 3000 lbs. A single pellet is placed in the center of a gold boat, which
has a raised center to enhance the flow of the mixture, which exists in a partial
liquid form between 935 and 1000° C. The pellet and boat are then heated, in
air and at atmospheric pressure, in temperature steps of 120° C/hr to a maximum
temperature of 985° C. The furnace is maintained at this temperature for 1 1/2
hours, and then the temperature is lowered at a rate of 1° C/hour, down to 880°
C. The cooling rate is a critical feature in the growth of the crystals[104]. The
crystals are formed during this slow-cooling stage, with some of the crystals
forming at the surface of the pellet, and others forming in the liquid which has
puddled in the bottom edges of the boat.

After cooling to room temperature, the crystals are harvested from the flux.
The best crystals, identified by their clean rectangular platelet shapes are usually
found in the comers and edges of the boat, and also in proximity to the melted

pellet. The resultant batch usually contains 5-10 rectangular single crystals on

62

the order of 1(w) x 10) mm in the ab plane, and a thickness ranging from 20-
200pm, as well as 20-30 smaller crystals. In order to achieve optimally high
superconducting transition, the crystals undergo a post anneal in flowing oxygen
(at 1 atm) at 410° C for ten days. This elevated temperature enables oxygen
diffusion into the crystal.

After annealing the crystals typically display a To ~93 K, but contain planar
defects in the form of twin boundaries (see Figure 6.4). These naturally
occurring, quasi-two dimensional structures are the result of the tetragonal-to-
orthorhombic phase transition which it underwent near 680° C in the growth
process. The tetragonal phase consists of an equal distribution of the 0(4) and
0(5) sites. At the transition to the orthorhombic phase, oxygen atoms in the 0(5)
position move to the 0(4) sites, resulting in the Cu-O chain structure along the b
axis shown in Figure 6.1. This produces an increase in the length of the lattice
parameter b, and a subsequent distortion strain in the basal plane[103]. The
strain is taken up by the formation of twin boundaries which are strain fields
along the <110> and <110> direction and act as sinks for atomic displacements
and oxygen vacancies in the crystal. The twin boundaries in the as-grown
crystals can be viewed with a polarized light microscope, where they separate
two different colored regions where the basal plane is rotated by 90 degrees, as
shown in Figure 6.4.

Twin boundaries act as highly anisotropic defect planes. For our studies,
it is necessary to isolate them from other defects which we artificially induce by

irradiation. This is done via the application of uniaxial pressure in the ab

63

Twin boundary
\

\ b| E
0 Oxygen 4‘. e . a

O . .

\ Yo I o O o o

. copper . °\.,.\ 0 e e e
I \ O O o O

. o . o ‘ . e e

a \ o o o

\ ° . .. °

. o . o . o .. O. -.“-:‘;=.

o .

.dlsordered

e o e o e o e ,, ‘='°'>'region

(a)

Cu(1) and 0(4) (CuO chains) shown.

it» <°)

   

Figure 6.4 Twin boundaries in YBCO. (a) Schematic of twin
plane. (b) A twinned crystal. (c) A twin-free crystal.

plane[105, 106]. The crystal is placed within a detwinning device where uniaxial
pressure is applied with a spring and adjusting micrometer, see Figure 6.5. A
section of the device containing the crystal, sandwiched between two quartz
plates with a soft gold leaf buffer is placed within a furnace and heated to 420° C
in flowing oxygen to ensure that the oxygen content in the crystal does not vary
during the detwinning process; the spring and micrometer are kept outside the
furnace and cooled with flowing air. The applied uniaxial pressure is typically of
the order of ~107 Nlm. The pressure causes the oxygen atoms in the direction of
the uniaxial pressure to move to the transverse direction and consequently
leaves the uniaxial pressure direction to form the shorter crystallographic a-axis
and the transverse direction becomes the longer b-axis. This alignment of the a
and b axes throughout the entire crystal eradicates the twin boundaries. The
removal of twin boundaries takes minutes to achieve under the appropriate
pressure and heat. However, it is important to continue the process ~24 hours
after the crystal appears fully detwinned, in order to allow the crystal to come to
equilibrium. The superconducting transition temperature is usually unaffected by
this detwinning process. These detvvinned crystals invariably exhibit a very sharp
first order vortex lattice to liquid melting transition, usually observed as a sharp
'kink' in the resistivity in the presence of a magnetic field.

For heavy ion irradiation experiments, these crystals must be thinned to
less than 30 pm along the c-axis, in order to ensure that the heavy ions produce
continuous and straight defect tracks throughout the sample. A crystal is thinned

by mounting it with crystal bond onto a metal holder along with small glass plates

65

 

 

*—

10' l

 

 

 

 

 

crystal
micrometer metal late

‘I‘

5522:3222: as

glass plates --

' nclosed)
spnng (8 steel retention
quartz rod bands

Figure 6.5 Detwinning device.

to support the crystal edges and then grinding it, using 1-30 pm grit polishing
discs and diamond-in-oil suspensions. Since the glass is harder than the crystal,
it provides a support along the crystal edges, as well as providing a horizontal
polishing plane. Final polishing is conducted using 0.1 pm diamond suspension.
With this method, flat millimeter sized crystals less than 20 pm thick with no
discernible scratches down to a scale of 0.1 pm can be obtained.

Transport measurements are conducted using the standard four-probe
technique. One of the challenging aspects of this type of measurement on single
crystals is the fabrication of low resistance contacts. Using an evaporation
chamber, the sample surface is first cleaned with argon gas plasma etching.
Four gold contacts, approximately 2000 A thick, are then deposited on the crystal
by gold evaporation through a metal mask. As deposited, the gold contacts have
high resistances. Hence the gold contacts are subsequently sintered at 410° C
in flowing oxygen for approximately six hours to ensure good bonding of the gold

to the ceramic crystal surface. Half mil annealed gold wires are attached to the

66

gold pads using Epo-TeK H20E silver epoxy, then cured for 5-10 minutes at
relatively low temperatures, usually around 150° C, resulting in final contact

resistances of about 1 n or less.

67

Chapter 7
EXPERIMENTAL SETUP AND HEAW ION IRRADIATION

7.1 System configuration

(7.1.a) Cryogenic and Superconducting Magnet System

The cryostat (Figure 7.1) is a home built 3He system capable of achieving
temperatures as low as 0.47 K, but adapted for use in our experiments with ‘He.
The dewar used in conjunction with the cryostat is a superinsulated design built
by Precision Cryogenics, and fabricated mainly of aluminum with a 12 inch G-10
neck. The main sections of the 3He cryostat are shown in Figure 7.1. It is atop
loading system, with a sample chamber diameter of 0.75“. The midsection of the
sample chamber is surrounded and in contact with a 1 K pot which can hold 1
liter of liquid helium, used mainly for low temperature experiments below 1 K.
The tail of the sample chamber is constructed of non-magnetic stainless steel
and extends approximately 12 inches below the 1 K pot, enabling the tail of the
cryostat to be thermally isolated from the 1 K pot when used in the 3He mode.
Both the sample chamber and the 1 K pot are isolated from the liquid helium bath
by an inner vacuum can. Two superconducting magnets surround the tail of the
cryostat: A 1.5 Tesla transverse split coil NbTi superconducting magnet which
resides in the 2.75" diameter bore of an 8.0 Tesla longitudinal superconducting
magnet. The two superconducting magnets supply the two orthogonal fields,
which can be programmed to give a desired resultant vector field. Each

superconducting magnet is controlled with a Lakeshore Cryotronic Model 622

68

14. 8"

 

  
   
 
  
 
  
 

1 K Pot pump vent port

Menard?”

Sample chamber
7 7/8“

 

Magnet Support (4)

 

Vapor Cooled
Magnet Current
Leads

Supports (3)

 

 

1.5T Transverse 3
Superconducting '
Magnet ~

 

.5'

I‘ .7.
I ‘ ' 1
" h '9'}: ,‘

 

 

/1 K Pot fill line

1 K Pot pump out port

<— Inner Vacuum Can
pump out port

3 — 77K Plate

 

He Level

Sensor
Magnet
Support

 

1 K Pot Plate

 

Inner Vacuum Can

 

I
I
e
. nl
I
I
I
I
I
I
I
I
V .
I
I
e
' -
I
I
i' I g .l .‘ '1» c
. -. _ I
U

\ Supports (3)

; —— Liquid

. helium bath
Superinsulated
Dewar

Outer Vacuum
Can

 

 

V I' I '7 VVVVVVVV

 

 

7:7“ 8T Longitunidal
- Superconductin

 

g Magnet

Figure 7.1 3He, 4He cryostat system.

69

power supply capable of delivering 120 A at up to 5 V. The current through each
magnet is independently monitored by a Keithley 196 digital voltmeter connected
across a copper shunt (0.0010), which is in series with the magnet high current
cables. The following steps define the cooling operation down to about 0.47K
using 3He as an exchange gas in the sample chamber. The sample chamber,
inner, and outer vacuum cans are evacuated with a rotary pump/diffusion pump
system to about ~10‘6 torr (see Figure 7.2). The bath of the dewar is then filled
to capacity with liquid nitrogen to pre-cool the superconducting magnets.
Subsequently, the liquid nitrogen is removed by supplying a back pressure of
nitrogen gas. When all the liquid is displaced, the bath is flushed several times
with ‘He gas to ensure that no liquid nitrogen remains at the bottom of the dewar.
Next, the bath is filled with liquid ‘He to capacity. The 1 K pot is then filled with
liquid 4He through a supplied transfer line and the sample chamber is filled with
3He gas to about 400 mtorr. By pumping on the 1 K pot through a large rotary
pump, and thereby reducing the vapor pressure of liquid 4He, the temperature in
the 1 K pot falls to about 1.19 K, allowing the 3He gas to first condense at the
inner wall of the sample chamber where it contacts the 1 K pot. The condensed
liquid then drips down and collects at the bottom of the sample chamber.
Sufficient 3He gas is introduced into the sample chamber to completely immerse
the sample with liquid 3He when the gas is condensed. By pumping on the liquid
3He, we can reduce the vapor pressure of the liquid and the temperature
decreases to about 0.47 K. For the data presented in this thesis, the system is

operated using liquid 4He: The sample chamber is filled with 4He gas to a

70

 

 

 

 

 

 

 

 

 

Figure 7.2 Gas handling system.

pressure of ~100 mm Hg. The 1 K pot is filled with 4He gas, and cooling occurs
via conduction and radiation from the liquid 4He in the bath. Temperature control
is maintained locally by a non-inductively wound phosphor bronze heater wire

wrapped around a copper cap which surrounds the sample.

(7.1.b) Sample Probe

The sample probe consists of a long G-10 (fiberglass reinforced plastic) tube
interrupted with two OFHC copper heat sinks located in positions such that they
are in contact with the top and bottom of the 1 K pot (see Figure 73(3)). The tail
of the resistivity probe (Figure 7.3(b)) consists of a Lakeshore Cemox

thermometer calibrated from 300K to 0.33 K, a standard T08 eight pin IC socket

71

    

rotation axis

G10

(a)

(b) (C)

Crystal mounted on 8 pin G10 holder.

 

removable copper /
cap with heater wire

 

 

 

 

themlorneter

(cornered within

the T08 IC
socket)

8 pin receptacle

SpinGIO ., ,

II hOIdOI’ 6.: ; (.1

a sample ~

 

rotation angle.
Figure 7.3 (a) Sample probe used to insert the samples in the cryostat. The
probe can be rotated about its long axis. (b) Expanded view of the probe end.
The sample is located at the bottom of the figure. A thermometer is located
close to the underside of the sample holder. Also shown is the copper cap used

to control the local temperature. (c) View of sample and sample holder. Here
four contacts are shown, with gold wires connecting the contacts to the pins.

for the sample, and a cap heater consisting of an OFHC copper cap wrapped
non-inductively with 36 gauge Phosphor Bronze heater wire.

To facilitate quick sample exchanges, we developed a removable sample
holder to fit in the IC socket (Figure 7.3(c)) for four-probe resistivity
measurements. The sample holder consists of a G—10 disk with either four or
eight Phosphor Bronze posts which fits the IC socket. For resistivity

measurements the YBCO single crystals are typically mounted on a sapphire

72

substrate with silver epoxy on both ends of the sample which also serves as the
current contacts. The sapphire substrate with the sample is then mounted onto
the G-10 disk with GE varnish. Contacts from the sample to the posts are made
with 0.5 mil annealed gold wire which are attached with silver epoxy to the
sample, and attached to the posts with indium solder. The G-10 disk with the
sample plugs into to the 8 pin lC socket at the bottom of the probe. Connection
from the sample to the top of the probe is made with 24 twisted pairs of 38 gauge
copper wire, terminating at a hermetically sealed Amphenol 26 pin connector at
the top. An O-ring slip connection, which mates the probe to the cryostat,
enables the height of the probe to be adjusted with the respect to the center of
the superconducting magnets and also allows the probe to be rotated 360°. This
latter capability allows an extra degree of sample orientation with respect to the
transverse magnetic field provided by the 1.5T superconducting split coil magnet.
Temperature is controlled by a Lakeshore Cryotronics DRC-93A
temperature controller connected to the Cernox thermometer and with the
Phosphor Bronze heater wire wrapped around the OFHC copper cap as shown

in Figure 7.3(b).

(7.1.c) Electronics

A schematic diagram of the standard four probe geometry for measuring the ac
resistivity is shown in Figure 7.3(c). AC current is generated with a Wavetek
Model 90 function generator, in series with a 1 kn resistor which enables the

function generator to provide a constant current source to the sample in a

73

transport measurement as long as any changes in the resistivity of the sample
upon cooling remain much smaller than 1 kn. An additional 0.1 Q resistor in
series with the sample is used to monitor the sample current. The sample
voltage and current (measured across the 0.1 o resistor) leads are each
connected to a Stanford Research model 554 transformer which acts as a step
up 1:100 isolation pre-amplifier. These transformers are then connected to
Stanford Research 830 lock-in amplifiers.

For DC resistivity measurements, the function generator and the
associated resistors are replaced with a single Keithley model 220 current source
and the DC voltage is measured using a Keithley 182 nanovoltmeter, making
sure to reverse the current direction several times to avert thermal drift voltages.

The data acquisition system consists of a UMAX (Apple clone) desktop
computer connected with the laboratory instruments via an IEEE 488.2 GPIB

interface, running either C++ or LabVlew data acquisition programs.

7.2 Heavy ion irradiation

The superconducting coherence length, for YBa2Cu30n crystals for fields
parallel to the crystallographic c-axis at zero temperature, is approximately

§o= 16 A[107]. Since the coherence length varies with temperature as §~ 50

I (1-T/Tc)"2, the coherence length for YBCO varies from ~ 16 to 100 A
throughout most of its superconducting temperature range, up to ~0.98Tc. This

length is a measure of the radius of the normal vortex core. In the creation of the

74

nonsuperconducting core, the free energy per unit length increases by the vortex

condensation energy:

' hsz-H: 2 71
a]. -§;fl'§ (-)

Since this is the energy to suppress the superconducting state within the core,
any nonsuperconducting defect would act to pin the vortex line at the defect,
since some of the condensation energy would then be conserved. A favorable
pinning site will be a nonsuperconducting column of defect material, aligned with
the vortex line, with a diameter comparable with the vortex core size. Such
columnar defects can be produced by high energy heavy ion irradiation of the
crystals. Magnetization[108, 109], and transport[44, 110, 111] data have shown
sharp increases in the critical current, as well as an upward temperature shift in
the irreversibility line in YBCO, as a result of heavy ion irradiation. So far, these
defects produce the most effective pinning sites in YBCO[84].

Columnar defects are created in YBCO by the impingement of high energy
ions, which traverse the thickness of the crystal, producing defect tracks (see
Figure 7.4). In order to produce these defects in a linear column geometry, the
ions must have enough energy to pass completely through the sample with only
negligible stray from the bombardment direction, as well as transfer enough
energy to the material to create the track; this energy is defined as the stopping
power, dE/dx. The energy transfer from the ion, moving at velocity v, to the

target material occurs via nuclear scattering and electronic interactions, which

75

 

   

columnar defect tracks

crystal

 

 

heavy Ion beam

Figure 7.4 The creation of columnar defect tracks via
heavy ion irradiation.

can be calculated using Monte Carlo calculations[112, 113]. The ion energy loss
per unit length is given as the combination of the two types of processes:
(7.2)

dE/dx=dE/dx) +dE/dx)

nuclear electronic

where dE/dx),,u¢,.,, describes ion-atomic elastic scattering collisions,
characterized by large deflections and energy losses per collision, and
dE/dx).m,~c describes the interaction of the ion with the electronic structure of
the target, characterized by small ion deflections and energy losses per collision.
These are commonly referred to as the nuclear and electronic stopping power Sn
and 8.. They depend not only on the ion type and energy, but also on the target
material. For high ion energies (> 200 keV/amu), Sn can be neglected relative to

the size of S... For the high energy ions, the electronic stopping power so is

76

calculated by treating the solid as a charged electron plasma, with a charge
density p that varies with position. The interaction between the electron plasma
and moving ion is calculated using the Lindhard particle—plasma interaction
function l(v, p), which relates the interaction of the ion, velocity v, with the
electron distribution function, integrated over all wavelengths. The density is
convoluted with l(v, p), and the heavy ion’s effective charge Z*(v, Z.)=Z y(v,Z.),
where Z is the ion atomic number, Z. is a target atom atomic number, and y is
the fractional effective charge of the ion:

8. = i l(v.Z> (2 nv.z.»"’ p dv (7.3)
The ion is said to be effectively stripped of a (target material dependent) fraction
7 of its total electrons due to the interaction of the outer shell electrons with the
material.

For heavy ions, the stopping power is calculated in the following way: The
predictions of equation (7.3) are compared to experimental data for hydrogen
and helium ions. From this comparison, eqn. (7.3) is corrected to give more
accurate predictions. Thus a resultant empirically corrected electronic stopping
function is obtained. Finally, for heavy ions, a scaling procedure is applied,
where the heavy ion stopping power is related to the hydrogen stopping power in
the same material:

Sheavy ion = Shydrogen Zzheavy ion 72 (7.4)
The resulting predictions have been shown to be accurate to within 10% of actual

results.

77

The effect of high energy heavy ion irradiation in YBCO is calculated using
the Transport of Ions in Matter (TRIM) Monte Carlo calculation program[114].
This program takes as inputs the ion type, its initial energy, and the target
material, producing a statistical distribution of the final ion penetration depth and
energy transferred to the target material (ion energy loss per unit length) along
the ion path. This energy is the stopping power S=Sn+S., in units of eV l A. A
typical distribution includes over 5000 ion simulated trajectories. The program
assumes a random distribution of target atoms, with a density of 6.54 glcm3 for
YBCO in the fully oxygenated state. Figure 7.5 shows TRIM calculations for
three ions, Au, U, and Pb, at varying initial energies, for a YBCO target. The
figure plots the stopping power (ion energy loss per Angstrom) of the impinging
ion, as a function of target depth. The ion energies in the figure are for the ions
before entering the material. As the ion transfers energy to the target, its energy
is lowered, thus changing the prediction of the stopping power.

The process of columnar defect track formation is described by the ion
explosion spike model[115-117]. As the high-energy ion traverses the
superconductor, it has been found to interact almost exclusively with the
electrons, causing local ionization along the path. The atoms remaining within
the now electron-depleted region are thus affected by a net repulsive force and, if
this force is greater than the atomic bond strength, the atoms are displaced along
the path, as shown schematically in Figure 7.6. In this manner a cylindrical
region of defects is established. In order to produce an effectively continuous

path of defects, certain criteria must be satisfied. The number of ionizations must

78

 

eV / ion / Angstrom

 

Target depth (pm)

Figure 7.5 Stopping power for U, Pb, and Au in a YBCO target.

hoax ion

   

Figure 7.6 Defect Formation: Atoms (white circles) are first
ionized by the ion, then mutually repulsed from the electron
depleted region.

79

be at least one per ab-plane, with an energy transfer high enough so that the net
ion repulsion will be greater than the atomic bond strength. The mobility of the
created defects must be less than an interatomic distance, otherwise the defects
created would disperse over time. It must also be possible to deplete the region
of electrons for enough time as to allow the atoms to repulse one another. This
last criterion is not satisfied for most metals, excepting for those with a density of

conduction electrons less than 102°lcm2[115] .

A concern in the production of ‘straight’ columnar defects is the stopping
power. From electron microscopy measurements, a lower bound S... in the
stopping power has been established for the production of continuous defect
tracks. Szenes[118] examined thirteen data on YBCO, obtaining Sm a 1900 eV/
A for this minimum stopping power, in agreement with other work[119, 120].
Data on 0.58 GeV Sn[121] and 2.29 GeV Xe[122] finds somewhat higher values
for this threshold, Sm >28 eV I A and 23 eV I A, respectively. For stopping
energies below the threshold, aligned spherical defects have been produced
instead of continuous columnar defects[121].

Our crystals were irradiated with high energy heavy ions at two facilities:
The 36” diameter ATSCAT chamber at the Argonne Tandem Linear Accelerator
System (ATLAS), located at Argonne National Laboratory, and the N3 chamber
at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State
University. The basic chamber configuration is shown in Figure 7.7. The
samples are placed on a ladder mount which is located at the center of the

chamber on a rotatable table which is also capable of raising and lowering the

80

 

 

battery

 

‘°" 993'“ beamline
direction

gold foil

 
 
    
  

vaCUU t

chambe electron

suppressor
with hole,
radius R

    
 
 
    
     
      

sample
mount
ladder

* downstream

 

e|ectrometer ladder, front view

 

 

 

 

 

hole, radius r
removable

Faraday cup
crystals

 

Figure 7.7 Heavy ion irradiation vacuum chamber, orientation,
and sample ladder.

81

sample. The ladder mount consists of several rectangular aluminum plates
which are stacked vertically on top of one other. Samples are mounted onto the
aluminum plates, except for the topmost plate, which contains a fixed diameter
hole drilled into the plate. This hole is used to align the beam on the ladder and
is also used as the reference diameter of the beam (i.e. the beam size is
adjusted by the beam controller such that the entire beam barely passes through
this hole). A telescope located outside the chamber and downstream from the
beam tube is used to align the sample with the beam direction and record the
position of each sample on the ladder. After the alignment is completed, a
Faraday cup is placed at the back end of the chamber and connected to an
electrometer to measure the beam current. This 'downstream‘ Faraday cup is
used to record the beam current through the hole on the ladder mount. During
irradiation, the beam current impinging on the samples on the ladder is also
monitored with an electrometer. Heavy ions striking the samples with over

1 GeV of energy produce ejection of surface electrons from the target. To
counter this effect and to obtain an accurate record of the beam current, a -300V
suppressor plate with a collimator hole is placed in close proximity in front of the
target samples. For the high energy ions used in the irradiation, the mean
ejected electron energy from the target is on the order of 250 eV[123]. Thus the
negative charge of -300V at the suppressor plate is enough to repel the electrons
back onto the target. Monitoring the beam current on the conducting suppressor
plate can assist the beam tuner to steer the beam onto the target. The

downstream Faraday cup is geometrically suppressed, is. any ejected electrons

82

will be collected by the long cylindrical wall of the cup, thereby yielding an
accurate reading.

One of the main issues in using heavy ion irradiation to create columnar
tracks is the control of the irradiation dose which depends heavily on the stability
and uniformity of the beam intensity. In most high energy heavy ion accelerators,
the beam energy can be controlled to a high degree of accuracy. However, the
beam intensity and the homogeneity of the beam over a few millimeter square
region can vary dramatically during the time it takes to irradiate a single sample.
One technique to homogenize the beam pattern is to place a thin gold foil in front
of the beamline. The foil acts to diffuse the beam via Rutherford scattering and
eliminate possible hot spots within the spatial profile of the beam. The beam
profile can be checked by exposing a piece of Gafchromic'”I radiographic film for
a few seconds (see Figure 7.8). These films are sensitive to ion irradiation, and
change from a light blue hue to a dark blue color after exposure to ion irradiation.
If the beam is spatially inhomogeneous, a hot spot indicating a very high dose
section of the beam will appear as a reddish ‘burn‘ mark on the film.

After sample alignment has been completed, the chamber is closed and
pumped down to ~1045 torr. The heavy ions enter the chamber in a stripped
state, with a net positive charge. Electrons are further stripped as the ions
traverse through the gold foil. The charge state can be calculated for a given ion
energy[124].

Given the final ion charge Q, the beam flux can be calculated in the

following manner: The beam is adjusted such that the entire beam barely passes

83

ATLAS April 18-20 1997

  

(143+ -, 057+
e E with gold foil
f, E 1.4 GeV, 70 epA
In
'0 v‘ 10 sec

Figure 7.8 Radiographic film image.

through the hole with radius r on the topmost plate of the ladder, and the current
In; is measured by the downstream Faraday Cup. The ion flux F, defined as the
numbers of ions per cm2 per second, is obtained from this current:
F = rec . (1charge/1.6x10'19 C) . (1 ion/charge Q) I (in?) (7.5)
The columnar defect density n is chosen to be equivalent to a matching vortex
density Bo,
n= Boltbo (75)
where d>o=2.07 x 10"5Tlm2 is the flux quantum and B], is in units of Tesla. The
time of irradiation, t, to obtain a specific matching field B... is given by:
t (sec) = (6.1%) . (m2) . o . 1.6 x10:19 c / rec (7.7)
The beam is monitored using an alpha particle detector, pointed towards

the gold foil at a fixed angle from the beam direction. Thus the detector counts

the elastically (Rutherford) scattered particles allowing for the calculation of the
beam current intensity at off-angles from the beam direction. The detector output
is a voltage pulse proportional to the alpha particle incident energy. Figure 7.9
shows the detector and the electronics. An incident particle produces electron-
ion pairs within the particle detector. The bias voltage acts to collect the free
electrons before they recombine. The signal is amplified, and the single channel
analyzer (SCA) converts all of the voltage pulses above a threshold into constant
amplitude voltage pulses, which can then be measured by the counter. The
measured counts per unit time is proportional to the beam current. Besides
calculating the predicted particle counts for a given beam intensity from
Rutherford scattering off the gold foil we can also obtain the corresponding
conversion from actual beam current to off-angle detector counts by measuring
the alpha detector counts and the downstream Faraday cup current
simultaneously for one minute. The average current on the Faraday cup is first
converted to equivalent irradiation dose B.(1min), using eqn. (7.7). The number
of counts measured is then defined as being equal to B,,. To obtain a necessary
dose Bo, the crystal is irradiated until the counter measures the corresponding
number of counts needed to attain B.., La, #counts(B,,) = #counts(1min) - Bol
B,,(1 min).

For the heavy ion irradiation experiment, three large untwinned
YBa2Cu30-H5 single crystals, each < 20 pm thick, were grown using the self-flux
method, described earlier. Two of the crystals were each cleaved down the c-

axis into four smaller pieces using a clean razor blade and the third crystal was

85

SI Detector:
EG&G ORTEC

 

model
. 050 Tennelec model T0178
BU-01 10° quad preamplifier
ll—l
HV IN

 

 

 

 

 

 

Bias voltage
source +80V

ORTEC model 551
ORTEC model 572 sin to channel

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IN i: ' .‘ % _- .
PREAMP 4.3. . i ‘ r 1:
" 0c . i INPUT r "- "mm
3 ; INPUT .. ——e .g , I:
um Bl J Sou? '1'; (.1 0mm”: f. INPUT
* r k f :1: °—T”T‘_°
ORTEC model ORTEC model
706 prescaler 771 timer-counter
r/L
oscilloscope

Figure 7.9 Alpha particle counting electronics

cleaved into five pieces. This approach ensures that each of the pieces for
irradiation will have the same underlying starting quality. One piece from each
large cleaved crystal was kept as an unirradiated reference. The crystals were
irradiated with 1.4 GeV 23°05“ and 1.4 GeV 2°"13656+ ions at ATLAS, to a dose
matching field B,,, = 1, 2, and 4 Tesla for the uranium ions, and B0 = 50 Gauss,
100 Gauss, 500 Gauss, 1000 Gauss, and 1 Tesla for the lead ions. The stopping
powers for these ions and incident energies are presented in Fig. 7.5. Resistivity

vs. temperature data for the crystals irradiated with high doses is shown in

86

Figure 7.10, for zero applied field. The data shows an increase in the normal
state resistivity, a decrease in T30, and broadening of the transition ATCO with
increasing irradiation dose. For this work, Tea is defined as the peak in the
derivative dp/dT (Fig. 4.4), and ATCa is defined as 90%-10% of pmmai, where
pm"... is the resistivity at the onset of the peak in the derivative. These results

are summarized in Table 7.1.

Table 7.1 Columnar defect tracks in YBCO.

 

 

 

 

 

 

 

Crystal Tc (K) Tch(unirrad) 8Tc (K) p(95K)/p(unirr,95K) Calc. defect Measured
(approximate) separation (A) debct
core (A)
PhD (0T) 93.83 1 0.28 1 - -
PM (11') 92.57 0.987 0.46 1.13 458 ~70-96
U0 (01') 92.59 1 0.38 1 - -
U1 (11) 90.79 0.98 1.20 1.14 458 90
U2 (2T) 89.67 0.968 1.95 1.90 324 90
U4 (4T) 88.0 0.95 3.40 2.86 229 90

 

 

 

 

 

 

 

 

 

The defect track information was obtained from electron microscopy
measurements (Figure 7.11). For the 1.4 GeV Pb ion defects the 70-96 A value
in Table 7.1 was estimated from High Resolution Electron Microscopy (HREM)
measurements on YBCO samples irradiated with 0.9 GeV Pb ions[111] and 1.1
GeV Au ions[125]. This diameter agrees with other TEM measurements on
YBCO crystals irradiated with 1.08 GeV Au ions[84, 126, 127], which found
continuous homogeneous defects, with a diameter of 60-70 A. The stopping

powers shown in Figure 7.5 are comparable for the Au and Pb ions at equivalent

87

mum '0'“ 1.4 GeV 2“wa Ions
200 ~ L ...................

      

:° U0 (unirradiated) 120 I Pbo (unirradiated)
:I U1 (B.=1T) E. Pb1 (B.=1T)
150 :. U2 (3.321) 100
E :' U4 (80:41.) 2‘ 80
9 . 9
g 100 - g 60
Z a.

50’

 

 

 

 

Figure 7.10 Resistivity versus temperature data in zero field.

ion energies; since the two ions are nearly the same mass, comparable track
formation is expected.

TEM measurements on tracks in YBCO produced by 1.3 GeV U ions have
been performed[122]. From this and other work[128, 129], a more complex
picture of the damaged tracks has emerged, in which three types of damaged
regions have been observed: At the center of the defect is the amorphous core
region, d a 10 nm. Beyond the core is an oxygen-reordered region extending
approximately 10 nm around the core, and an extended stressed region, which
extends up to 30 nm further beyond the core. Within the extended stressed
region, ‘nanotwin’ structures may form between two defects when they are within
50 nm of one another. However, angle dependent measurements of the
resistivity of the uranium irradiated crystals failed to record any anisotropic

pinning due to the nanotvvins. The net conclusion is that the effective damage

88

radius extends beyond the core, although it is unclear whether this affects the
pinning behavior of the defects.

Also, recent data on 2.25 GeV Au[122] and 1.1 GeV Au ions[125, 130]
show significant defect core diameter modulation (4—11 nm for the 2.25 GeV
data) in measurements at varying depths. TEM measurements on 1.3 GeV U
irradiated crystals (S>50 eV IA) did not show this modulation, thus it is as yet

unclear if this occurs in the 1.4 GeV Pb (S='47 eV IA) irradiated crystals.

 

Figure 7.11 TEM image of columnar defects in YBCO, looking
along the defect direction. Defects were created by 1.3 GeV
uranium ions.

89

Chapter 8

THE EFFECT OF HIGH DENSITIES OF COLUMNAR DEFECTS ON VORTEX
MOTION IN CLEAN, UNTWINNED YBazCU307.5 SINGLE CRYSTALS

8.1 Introduction

The motion of vortices in YBazcuaOm can be dramatically affected by the
bombardment of high energy, heavy ions, which produce amorphous defect
tracks through the sample. These columnar defect tracks produce anisotropic
pinning centers which are energetically favorable pinning sites for vortex lines.
The result is significant vortex pinning in the vortex solid state and the slowing
down of vortex dynamics in the vortex liquid state. The nature of the vortex
melting transition is also altered by the defects. In the presence of columnar
defects the melting transition is predicted to be a continuous, Bose glass
transition[86, 87]. Unlike isotropic defects which are responsible for a purported
Vortex Glass transition[22, 29], the Bose glass transition predicts a sharp cusp in
the transition temperature as the applied field is rotated about the columnar
defects, clearly discerning it from the isotropic pinning case.

For very clean YBa2Cu307.5 (Y 800) single crystals, in the absence of
correlated disorder, a first order melting transition has been found in transport
measurements[32, 131] and confirmed by magnetization[38-40] and specific
heat[41] measurements. In transport measurements, a sharp drop, or 'kink' near
the tail of the resistivity curve measured in a magnetic field has been associated
with the first order melting transition. For crystals with a significant number of

either point or correlated defects, the first order transition transforms into a

90

continuous transition, with the transition predicted to be either to a Bose glass,
vortex glass, or polymer glass (vortex molasses), depending on the type of
defects induced into the sample.

Several experiments purporting to demonstrate the existence of these
phases have been reported. However, earlier experiments have been plagued
by inherent defects such as twin boundaries in the as grown sample leading to
some ambiguity as to the nature of the glassy phase. In many cases, what was
reported as a vortex glass phase[20, 132, 133] in twinned YBCO thin films and
crystals may in fact be related to a Bose glass phase due to the existence of
correlated defects in the samples[134]. Twin boundaries can behave as
correlated defects, and introduce added complications in transport
measurements due to such phenomena as guided motion of the vortices parallel
to a twin plane[135, 136]. A number of earlier studies on YBCO crystals with
columnar defects induced by heavy ion irradiation were also performed on
twinned crystals, compounding the difficulty in separating the behavior of vortex
localization by columnar defects from twin boundary pinning. Thus a definitive
investigation of the Bose glass transition in YBCO is still lacking.

In this chapter, we investigate the melting transition in pristine untwinned
YBCO crystals in the presence of columnar defects created by Pb and U ions,
and compare the results with the predictions of the Bose glass theory. We
determine the static and dynamic critical exponents v and 2, respectively,

associated with the Bose glass scaling theory and show direct evidence of the

91

anisotropic pinning nature of the columnar defects. In addition, we find a
systematic kink in the irreversibility line which tracks the defect matching field.

Single crystals of YBCO were prepared using the flux growth method
described elsewhere[103]. The as grown and annealed crystals were detwinned
by applying uniaxial pressure along one side of the ab-plane at 420° C in flowing
oxygen and then polished down along the crystallographic c-axis to less than
30pm to ensure that the heavy ions traverse through the entire cross-section of
the sample according to TRIM calculations[112]. No vestiges of twins were
observed by polarized microscopy after detwinning.

Three large detwinned crystals were cleaved into three sets of several
pieces. The first crystal (thickness 19 um) was cleaved into four crystals, three of
which were irradiated with 1.4 GeV 20“Pb'r’fi’” ions along the crystallographic c-
axis to dose matching fields of B0 = 100 G(Pb0.01), 1000G (Pb0.1), and 1T (Pb1)
and the fourth piece from the same crystal was kept as a reference (PbO). The
second crystal (thickness 17.5 pm) was cleaved into five pieces, three of which
were irradiated in like manner with 1.4 GeV 23"U ions to dose matching fields of
8., = 1T (U1), 2T (U2), and 4T (U4) and the other two were irradiated with 4 GeV
297Au ions to dose matching fields of Bo=1 (Au1) and 4T (Au4) at the National
Superconducting Cyclotron Laboratory at Michigan State University. For this set
no reference crystal was kept. Instead U1 was precharacterized before
irradiation (U0). The third crystal (thickness 10 um) was cleaved into four pieces,
three of which were irradiated with 1.4 GeV 2°°Pb5°"* ions to dose matching

fields of B,=50 (Pb50G), 100 (Pb100G), 500 Gauss (Pb500G). Cleaving several

92

crystals from a larger piece for this experiment ensures that the starting
underlying quality of the crystals prior to irradiation are equivalent. In this
Chapter I will report on the crystals irradiated at high defect doses, namely Pb1.
U1, U2, and U4.

Transport measurements were carried out using the standard four-probe
technique. Gold contacts were first evaporated onto the surface of the crystal
and sintered at 420° C. Gold wires were subsequently attached to the contacts
with silver epoxy, resulting in contact resistances of about 1 n. AC resistivity
measurements were performed with transport current densities typically in the
range of 2 to 20 Mom2 at 23 Hz directed in the crystallographic ab-plane of the
crystal. DC resistivity and l-V measurements were carried out with a nano-
voltmeter with current reversal to minimize thermal effects. The crystal was
placed in the bore of a 1.5 T superconducting split coil 'transverse' magnet which
resides in the bore of an 8 T 'longitudinal' superconducting solenoid magnet. The
magnetic field could be rotated with respect to the sample by energizing the
magnets independently. For the irradiated crystals, 8 is defined as the angle
between the applied magnetic field and the columnar defect direction. The
measuring current was always applied in the ab-plane and perpendicular to the
applied field/columnar defect plane, ensuring maximum Lorentz force on the

vortices for all orientations.

93

8.2 Uranium ion irradiation: U1, U2, and U4

The zero field resistivity data for these crystals were plotted in the last chapter,
Figure 7.10. A broadening in the superconducting transition temperature width, a
lowering of the transition temperature, and an increase in the normal state
resistivity were observed in the irradiated crystals. Figure 8.1 shows resistivity
versus temperature measurements for the U0, U1, U2, and U4 samples in
applied fields of 8,7,6,5,4,3,2,1,0.5, and 0 Tesla aligned with the crystallographic
c-axis. The data in Figure 8.1 has been normalized with respect to the resistivity
at 95 K and To in order to compare the relative effects of the irradiation. Before
irradiation, a ‘kink' in the tail of the superconducting resistive transition associated
with the first order vortex solid to liquid melting transition can be clearly observed
in U0 for H = 1-8T. After irradiation, the kink has been replaced by a smooth,
monotonic decrease in the resistivity, indicating that the first order transition has
been completely suppressed by the irradiation and transformed perhaps to a
higher order transition.

Figure 8.2(a) shows that the kink in the resistivity for the unirradiated
crystal also marks the onset of non-ohmic behavior, with a sharp minimum in
resistivity at ~86 K (peak effect). In contrast, the uranium-ion irradiated crystals
show ohmic behavior throughout the entire superconductive resistive transition
for all fields. This is clearly shown in Figure 8.2(b) for the U1 crystal where the
resistivity curves measured with two different current densities lie on top of each

other for all fields investigated. For these current densities, the data is ohmic

94

0.8 Crystal H II c
B =OT #1.";I’.
(P A

 

   

 

 

 

. . = 4 T ,
(p
' 81
0.2 . l
0.0

‘ 0.84 0.88 0.92 0.6 1.00
T/ TC

Flgure 8.1 Resistvity vs. temperature, for crystals irradiated with
237067+ ions, to a dose matching field 3(1) = o, 1, 2, and 4T.
Shown are for applied fields of 8,7,6,5,4,3,2,1,0.5, and 0 Tesla.

95

down to the experimental resolution. The result is also found in data for crystals
U2 and U4. The absence of a kink and the presence of ohmic behavior suggest
that the vortices remain in the liquid state down to the zero resistivity limit.

The dramatic anisotropic pinning behavior of columnar defects is
demonstrated in Figure 8.3 where we show the angular dependence of the
resistivity measured at a fixed field value and at several temperatures. At high
temperatures, we observe a broad maximum at 8 = 0° when the magnetic field is
aligned with the crystallographic c-axis and a minimum at 8 = 90° when the field
is parallel with the ab-plane of the crystal. This behavior at high temperatures
reflects the anisotropic superconducting behavior of YBCO due to its effective
mass anisotropy. With lowering temperatures, the broad maximum observed at
8 = 0° slowly transforms into a shallow minimum as the columnar defects induced
by heavy ion irradiation along the c-axis begin to trap the vortices in the liquid
state. Wrth further lowering of the temperature, the shallow dip turns into a sharp
minimum. We define the onset of the anisotropic pinning due to columnar
defects as the temperature where the dip centered around 8=0° first appears.
This onset temperature can also be determined by the temperature where the
two resistivity curves in Figure 8.3 inset, taken at two different angles of the
applied field, first deviate from each other.

The magnetic field dependence of the onset temperature of anisotropic
pinning by columnar defects Tom... for U1, U2, and U4 is shown in Figure 8.4,
where here the data is normalized to Tc. At low applied fields (H<1 Tesla), the

To"... curves for all three crystals lie almost lie on top of each other, regardless of

96

100

 

1O

 

:3 (HQ-cm)

0.1

0.01

 

 

 

Hllc
881T

10

p (HQ-cm)

0.1

 

 

76 78 so s2 s4 86 88 so 9‘2
700

Figure 8.2 (a) Preirradiation melting transition, marking the

onset of nonohmic behavior. (b) After irradiation only ohmic
behavior is observed.

97

 

50 II'UI'V‘IIII'rIUIITUIUUIIU'I'IIU'II

Ut B°=1T A defect

H
llllLllJ

  
 

 

 

lLlllllll

 

 

 

 
 

 

 

 

3:
10 °: “

‘. 89.21K

2‘.

8
07.L1.. . .1..l..i..1..|..
-90 -60 -3o 30 so 90

0
9 (deg)
Figure 8.3 Angular dependence of the resistivity, demonstrating
the anisotropic pinning strength of the columnar defects.

 

 

 

7 b""1'fffl""l""I'"'I"TTTT"'1"*"

. -l

i d

6 - 1

i I

5 .’ S

i :

E 4 t :

:1: t :

3 L :

P a

t +8-1T '

p Q 1

2 b .

: +B.-2T :

+3 INT ‘

1 L . .

0 ’ . .IrLI
0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02
/T
onset C

Figure 8.4 Onset of columnar pinning in the vortex liquid as a
function of applied field, for defect matching fields of 1, 2, and 4T.

98

the dose matching field. In this field regime, the defect density is greater than the
vortex density for all three crystals since the lowest dose matching field is B°=1T.
As the field is increased, Tonw decreases, as the ratio of vortices to pinning sites
increases. For H = 1T and above, Tomi becomes strongly dependent on the
irradiation dose, with higher onset temperatures with higher doses. These
results demonstrate the remarkably controlled change in the vortex pinning
behavior as the ratio of columnar defect to vortex density is varied.

We verified the ohmic behavior observed in the liquid state in Figure 8.2
with DC current-voltage curves. Figure 8.5(b) show the E-J curves in a log plot
for the B4, = 4T crystal, taken at H=2T || 0 at various temperatures. The
corresponding temperatures are marked with arrows in the resistivity vs.
temperature data, in Figure 8.5(a). The measurement resolution is
E = 3x10”7 V/cm. For current densities below ~20 Alcmz, the electric field shows
ohmic behavior. Non-ohmic behavior is observed above 20 Alcmz. However, this
appears to be caused by sample heating at the current contacts, appearing as
the applied current is increased. This was further confirmed by ac resistivity
measurements at current densities up to 195 Alcm2 as shown in Figure 8.5(c)
where we observed a systematic shift to higher resistivity with increasing
measuring current density due to heating of the sample. Thus within a DC
measuring current density of 20 A/cmz, we observe ohmic behavior in all the
crystals irradiated with U ions. The results suggest that after irradiation, the
vortex liquid state persists down to temperatures below which we are unable to

measure the resistivity. We define an irreversibility temperature as the

99

E (V/cm)

 

    
    
 

 

 

 

    
  
  
       
   

   
 
 
     
   

 

 

 

P H 8 2T II defects
‘60 " U4 3.“ Tesla
A120 I
E .
9 .
4
Ci so .
v 4
O. 4
«I
40 .,
d
I
o .
82 84 86 88 90 92 94
T (K)
(b) (c)
10" 7' "w' r' "" ' ' "'"‘ ‘7 """ ' '" 160
’ l-I-2Tllc : ,
m a .4 Tesla | . I U4 (3:47)
"’4 -o.—ee 59K / i
; -.-..Zm l I 150 y H = 6T ll defects
4» —o—88.08K I . .
10 r—e—essex I A I
; -—-e5.oek : g 140 b
~ -0-u.52k I. . , ,
w‘ -o—esrie K G I
. +8313 K e 130 ,
I #0293 k ..
10" r " O' t I 3 3.9 Alcm’
I I 120 I ° ] = 19.5 A/cm2
’ I = 195 A/cm’
1 110 '
w .
0,01 0.1 1 ,0 100 10’ 87 88 89 90 91 92 93 94
2
J (A/cm ) T (K)

Figure 8.5 (a) RvT for U4, H=4T. Arrows indicate temperatures where
E-J curves were obtained as shown in (b). In (b) the dashed line
indicates the crossover current for ohmic behavior below and non-ohmic
above the line. (c) Contact heating at high current densities.

100

temperature where p = 0.01 pQ-cm, the resolution of the experiment. This
represents an upper bound of the irreversibility behavior. Figure 8.6(a) shows
the irreversibility line on an H-T phase diagram for the irradiated (B,,,=1, 2, and
41') crystals using this criterion. For comparison, the temperature axis is
normalized to the zero resistance temperature at zero field for each crystal. The
plot shows an upward shift of the irreversibility line with irradiation dose. Also
plotted for comparison is the melting transition for U0, defined by the onset of the
peak in the temperature derivative of the resistivity, dp / dT. All the irradiated
samples show a change in slope of the irreversibility line, with linear behavior
above, and positive curvature below their respective dose matching field 3...
indicated by the arrows in the figure. Best fits of the low field data below the
matching field to |1-t|°' are presented in Figure 8.6(b) and linear fits to the high
field data above the dose matching field for each irradiated crystal are shown in
Figure 8.6(c). Above the matching field, the linear slope increases with dose
(Fig. 8.6(b)). If we extrapolate the first order vortex melting line with the fitted
irreversibility lines after irradiation, we find that they intersect at 47.5T, 29.5T,
and 15.5T for crystals U4, U2, and U1 respectively (see Figure 8.6(d)). This
suggests a possible recovery of the first order melting transition at high fields.
Indeed, we will show in Chapter 9 for crystals irradiated with a low dose matching
field, the first order transition is recovered at a measurable field.

The abrupt change in slope of the irreversibility line at an applied magnetic
field related to the matching field has been previously reported, by Smith et al., at

H: 8,,[137] as well as by Krusin-Elbaum et al., at H=1/ZB,,[127], and by

101

 

 

 

    

 

 

 

 

  
    
 
  

 

 

8 l- (a)
6 .
4 C
P o
2 ’ ' ’
: +U2:B.-2T
' -D-U4:B.I4T
0 - ‘ - l - A l
0.84 088 0.92 096
T/ T
p0
(C)
8 . '. U1. 56 (‘l TIT”).
'. ‘I'UZ: 65'(1-TIT'0)‘
.3. '.‘-JI|--U4: 74'(1-T/Tw).
a .. q
p ". \ .g 1
. i. l
4 _ ‘ .
I .\. 4
2 . Preirradiation ‘3, .
_ _ _ 1.33 ‘
_Hm 87.8(1 TIT”) ‘1‘ ‘
8.33”“ ‘ ‘ 6.6 ““““ 6.95 “““ 1
T T
I 90

      

Ht'u1: 3100'l1-tl"°
0.5 ,""U2: 410'l1-tl1'39
I""U4: 342'l1-tl ‘3“

o -1---1---l-,;|-

 

 

 

 

 

 

 

 

Figure 8.6 Irreversibility lines versus normalized temperature,

for U1, U2, and U4.

102

Paulius et al. at H=ZB°[44]. The linear behavior above 8,, can be interpreted as
a weakening of the overall pinning efficiency when the vortices outnumber the
columnar defects as the vortex-vortex interaction begins to dominate[138, 139].

It has been predicted[139] that the rate of increase of the critical current
with decreasing temperature will be lower for fields above B,b compared with
fields below 3., as a result of the weakening of the pinning strength above B...
Since our data only show ohmic behavior, we are not able to determine the
critical current. However, an upper bound can be determined from our data if we
use a criterion of E = 106 Vlcm, the limit of our resolution for DC measurements,
to define the critical current density. In this manner we identify Jc as the
measurable vortex motion in the E vs. J curves. Plotted in Figure 8.7(a) is Jc
vs. T for crystal U2, for fields above and below B,,=2T. The data shows no
abrupt change in the slopes A Jcl ATfor all applied fields near 2T. Samples U1
and U4 also display similar behavior.

However, a comparison among the irradiated crystals with the unirradiated
crystal demonstrates an increase in the pinning efficiency with increasing dose
matching field. Plotted in Figures 8.7(b) through 8.7(t) are Jc vs. T/Tpo data for
U1, U2, and U4, in applied fields of 0.5, 1, 2, 4, and GT, respectively. The critical
current for the irradiated crystals lie above the critical current of the unirradiated
(pristine crystal) crystal, at all magnetic fields. In Figure 8.7(b) a comparison of
the three irradiated crystals taken at a magnetic field value of H = 0.5T< 8,, show
that all the curves lie almost on top of each other and seem to be independent of

the defect dose. Likewise, we observe similar behavior for H = 1T in all the

103

 

 

   
    

 

 

 

 

         
    
  

 

  
    
   
 

 

 

  

 

 

 

   
 
 
  

   
  

 

 

 

 

 

(a) , 30,, - - - - (b)
I ' U2 B =21- : D U1: B.=1T .
200 . o I — . 1
t that Hllc 250.4u2:a.=zr ”—0511”.
: 47 I 0m: 3.41 :
«7‘ 150 . " A 200'. J
E : 27 “E I Unirradated :
31‘ : 1T 0.5T g 150 E I:
'3 100 1' 1| V0 l .
. '1 ; :
I ‘00 r 1
so 1 50E 3
o b - , . . . a l . . . l A n ‘
80 82 84 86 88 90 0.92 0.94 0.96 0.98 1 1.02
T(K) T/Tpo
300 (c) ,0, (d)
’---,---,---,fi--,---: Bunsen
250: H=1Tllc ; LUZ: agar
EUnlnadlated 3 300 ° 0* Bf“
A 200} a U1: 3,-11 .1 A
NE 1 A uz 30.21, : NE 250 Unlnadlated
g ‘50:, om: 8.-4T ; ‘2’ 200
:° E 1 3°150
100: 1
I j 100
5°? '1 so '
z a , . l
892‘ ‘ 594‘ ‘ 3,93 0,93 . 1,02 0.86 0.88 0.9 0.92 0.94 0.96 0.93 1 1.02
TIT T/T
90 PO
(e) (f)
m ' 'rr'""‘1[""""'"t'jtit'i‘ m I ' ' V U ri ' U V—f '7‘ fijfij"
250:- H=4TII c i I H=6T II c ;
i l 300 r ‘
‘rzoo- u m: 3,-11 «A Unlrradlated III u1- e .11 .
E ‘Uz 30m 8 . ’ auz eI-zr 1
2150; Omega: 2200» bums,“ -
V0 p I V” ’ 4
'7 100 : 1 .3 E 1
L' '. . .
I 1 100 ~ -
50:- -j I
F 1‘ o ’
8.86 0.88 0.9 0.92 0.94 . . 1.02 0,34 033 0.92 0,93 1
T / T pa T/ T N

Figure 8.7 Critical current versus temperature for U1, U2, and U4.

104

irradiated crystals. However, for H = 2T, the Jc curve for sample U1 (B. = 11')
begins to fall below the Jc curves of samples U2 (8,, = 21') and U4 (B, = 41'),
which were irradiated with higher dose matching fields, as shown in figs. 8.7(c).
At even higher fields, the Jc curve for U2 begins to fall below that of U4, as
shown in figs. 8.7(e) and (f). Thus the pinning efficiency decreases as the
number of vortices outnumbers the columnar defects, although the functional
dependence of Jc with temperature remains the same.

As previously discussed in Chapter 5, a Bose glass transition is predicted
for crystals irradiated with columnar defects. This theory predicts a power law
dependence on the tail of the resistivity data: p ~ (T Jae)". where s = v(z-2), and
v and z are the static and dynamic scaling exponents respectively. Recall this
ansatz is for the case of an ohmic resistivity at low currents, which is applicable
for these crystals where we find only ohmic behavior. We determine T56 and s
by fitting the tail of the resistivity transition, by evaluating the linear region of the
p / (dp/dT) vs. T plot, as shown in Figure 8.8(a). The inverse of the slope yields
3 = 3.6, while the temperature intercept gives T35 =85.78 K. In order to check
the quality of the fit, we compared the result with the measured experimental
data, as shown in Figure 8.8(b). To within an error of As=iO.3, the fit is in
excellent agreement with the measured data. Figure 8.8(c) shows the results of
this analysis for U1, U2, and U4, in fields up to 8 Tesla. Our values for the
scaling exponents vary widely in field from 2.5 — 6. Previous measurements[44]
of an untwinned YBCO single crystal irradiated by 1 GeV U ions also find ohmic

behavior in the resistivity data at all temperatures. From these measurements a

105

2.5 vvv"vvvv'vvvv'vvvv'vvvvi'vvvv

 

        

 

 

 

 

; 3.17 (a) :
2 '- ° -‘
l H=3T ll detects .
_ t ls=0.1mA(23Hz) ‘
l- 1.5 '-
‘O l .
:‘3‘ I 1
t 1 '. .'
c=- : l
D 1
0.5 ' .1
slope of llne .1/3.3 ;
O - - - l - - - - l - - - - l - - _ J 1_._. - A‘
85.5 86 86.5 87 87.5 88 88.5
T K
. f
I Bo=1T I, I
5 S H =37 ll defects . j
_ l=0.1mA (23Hz) ' 1
’E‘ 4 : 1
o - --- p=1.2'(T-85.78)3-° ;
I
c: 3 E J
3 : :
a. 2 i .‘
1 : J
l- d

 

 

 

0 A . . . n - . .
85.8 86 86.2 86.4 86.6 86.8 87 87.2 87.4

 

 

 

 

T (K) (C)
8,...,--.,..-.......
., E H u detects "'Bf”
E '°O°B.=2T 1‘,
6 I } "‘"B=4T ";
(‘1‘ 5 ’ ."}.-.§ . 2. o. '5‘: I i
a .. r. +2525! 2
i : .‘!"‘*-ooi.. 3. E
w 3%- ‘r" "1.. :
2 E. .I...i i
1 E-

00 1 1 .21 g 1 L 1 LA 1 61 n n 1 8| .

4
H (T )
Figure 8.8 (a), (b) Evaluation the tail of the resistivity data in

terms of the Bose glass transition (T-TBG)S. (c) Results of s for
U1, U2, and U4.

106

value of s = 5.2 :l: 0.4 was obtained for applied fields up to 7 Tesla, in general

agreement with our results.

8.3 Lead ion irradiation: PM
The sample Pb1. irradiated with 1.4 GeV Pb ions to 3., = 1 Tesla, is compared
with an unirradiated piece of the same crystal, sample PbO, in Figure 8.9. Similar
to the uranium-irradiated samples, the kink in the resistivity associated with the
first order vortex melting transition in the unirradiated crystal is replaced by a
smooth continuous transition after irradiation. However, unlike the uranium
irradiated samples, Pb1 show non-ohmic behavior at measurable resistances, as
shown in Figure 8.10. The onset of non-ohmic behavior is represented by the
temperature where the resistivity curves, measured at two different currents 1
mA and 0.1mA in a magnetic field, begin to deviate from each other. This point
marks the onset of the Bose glass critical regime expected for a continuous
liquid-to-glass transition. At high magnetic field, the onset of non-ohmic behavior
shifts to lower temperatures and the corresponding resistivity values also
decreases. For fields above 4 Tesla the non-ohmic behavior is below the
sensitivity of our apparatus.

As described in Chapter 5, in the presence of correlated defects, the Bose
glass theory predicts a continuous transition which is described by a transverse
and a longitudinal correlation length given by: e, ~1/ITBG - T I” and 1?, ~ If:

respectively, where Tm is the Bose glass transition temperature, and v is a critical

scaling exponent. The relaxation time of a fluctuation is expected to diverge with

107

  
   
   

 

120.r....,....,..4.
Preirradiation (thin lines):
H=8,7,6,5,4,3,2,1, and 0.5T
f Prostirradiation (thick lines):
H=6,5,4,3,2,1, and 0.5T

r I I

100

 

 

 

- 1
E80' 1
‘3 2
0:60 _
3 I
CL40 I
20' 1
0p 1 ...l. .l
80 85 90 95
T(K)

Figure 8.9 Resistivity versus temperature for PhD (unirradiated)
and PM (1T defect dose).

a dynamic scaling exponent 2, given by: 1 ~ I: . From the anisotropic Ginzburg-
Landau free energy, the current density scales as J ~ II! 113,, while the resultant
electric field produced by vortex motion scales as E ~1/e,r, leading to the
scaling ansatz for the critical current, describing the critical regime:

E131” = F,(€,£,J¢olcT) where P; is the universal scaling function. While F; and
F_ are unknown, the scaling hypothesis allows current-voltage data for different

temperatures to be collapsed onto F; above, and F_ below T36. Since the

108

    

100 r H II defects
EH=5. 4. 3. 2.6. 2. 1, and 0.5T

1or

p (HQ-cm)

o.1_r

 

 

 

1
84.0 85.0 86.0 87.0 88.0 89.0 90.0 91.0 92.0

T (K)

Figure 8.10 Onset (arrows) of non-ohmic behavior in the liquid
state, for crystal Pb1.

response is finite at the transition, the two divergent lengths must cancel there,
giving a power law dependence E ~ WW3 when T= T”.
Figure 8.11(a) shows DC current-voltage curves for Pb1. taken above and

within the non-ohmic critical regime, for H = 1 Tesla. The curves are linear for

T _>_91.70 K, where the resistive response is ohmic, and become progressively
non-ohmic with decreasing temperature. In Figure 8.11(b) the nonlinear curves
have been scaled according to the scaling ansatz, with an excellent collapse of
the data both above and below Tag. The critical exponents v and 2, as well as

the Bose glass temperature were varied to obtain the best overall fit. From this

109

 

10'2 10'1 10° 10‘ 102
J (A/cmz)
(b)

 

-2
10
10°10‘ 10210310‘10510‘10710"

3v
J/(T-TBG)

Figure 8.11 (a) E-J curves for PM (H=1T || defects), showing a high
temperature linear regime, and a non-ohmic critical fluctuation regime
surrounding the Bose glass transition temperature TBG. (b) Scaling of the

non-ohmic curves.

110

analysis we obtain v=1.67i0.10, z=3.44¢0.10, and s=v(z-2)=2.4:l:0.2. Similar
analysis for H = 0.2T, 0.5T, and 2T, yielded the same critical exponents, as
shown in Figure 8.12.

Also plotted in Figures 8.11 and 8.12 is the expected power law

dependence E~J"“7 at Tag. where we have used 2 = 3.44. This power law

behavior is consistent with the experimental data, and serves as a consistency
check for the scaling analysis. Figure 8.13(a) shows the vortex phase diagram
for Pb1. The Bose glass transition temperature, obtained from the above E-J
scaling (circles) shifts to lower temperatures as the applied field is increased.
Beyond H = 4T, the non-ohmic behavior characterizing the onset of the Bose
glass critical regime is shifted to lower temperatures and lies below the
measurable voltage resolution of the experiment. For these fields, we resort to
fitting the tail of the ohmic resistivity data with the power law p ~ (T— Tac)’. as was
done with the U ion data to obtain the Bose glass transition temperature. The tail
of the p vs. T measurements for H = 4-6T was fit to this power law, in a
temperature range of 0.5-0.7 K. From this analysis we obtained
3 = v(z-2) = 4410.5, a value substantially higher than that obtained from the E-J
scaling procedure. A possible reason for this discrepancy may be that our
resistivity data are not within the true critical fluctuation regime for these fields,
which perhaps lie at even lower temperatures. The obtained values for 8
however are in agreement with the values obtained from the U data above.

The values of Tea obtained from scaling the E-J curves are in very good

agreement with the Two curve where the resistivity goes to zero, using a criterion

111

H=0.2T ll detects

H=0.2T

TBG = 91.85 K i

1
i

 

 

100 10’ 1o‘ 19' 10' 10' 19'
J/(T-Tg)5

 

   
 
 

H .Tll tects 4
’05 d° H=0.5T

T33 = 91.60 K

      

. E ~J1.47

11:21 11 defects ;

1.47
10" ' E ~ J

- --...—l n. _.-l A--_‘ .‘AA-‘ AAA“.

 

10'7 . ‘ . .
10 10 10 10 10
J (A/cmz)

 

Flgure 8.12 E-J curves and scaling for H=0.2, 0.5, and 2T.

112

of p = 0.01 uQ-cm, as shown in Figure 8.13(a). A plot of the irreversibility line
using the T2m criterion shows an abrupt change in slope of the Bose glass
transition line at the dose matching field for the Pb1. similar to the one observed
in the uranium irradiated crystals, as shown in Figure 8.13(b). Also plotted in the
same phase diagram is the first order vortex melting line before irradiation. An
extrapolation of the linear behavior above H = B,,, = 1T shows that the first order
melting line intersects the irreversibility line after irradiation near H = 26T, much
bigger than the value of 15.5T reported for the uranium irradiated sample, U1. A
comparison of the irreversibility line of PM and U1 in Figure 8.13(a) shows that
Hm for PM lies above that of U1, indicating that the solid phase is enhanced
more with the Pb ion irradiation than with U ions for a 1T irradiation dose.
Furthermore, the critical current density for PM is also enhanced over that of U1
as shown in Figure 8.13(d), where the temperature dependent critical current

curves for PM always lies above that of U1 for all fields.

8.4 Angular dependence for PM

The anisotropic pinning behavior of the defects predicted from the Bose glass
theory is exemplified by the detailed plot of the resistivity when the magnetic field
is tilted away from the columnar defect direction, as shown in Figure 8.14(a).
The minimum dissipation at 9 = 0° indicates that columnar defects are most
effective as pinning sites when the vortices are aligned along the defect direction.

Dissipation increases with increasing tilt angle, up to 0 = 9,, beyond which

113

(a)

 

 

   
  
 

 

 

 

  
     

 
 
 

  

 

 

 

 

 

 

 

 

 

6
Hm defined
5 - by 0.01un-cm criterion
4 .-
A
t:
3 .
I
2 ~+HI PM
ff
0 H99 PM: From Vvl data scaling
1 --o-H Pb1
non-ohrnlconut
+H U1 .
IL! 2 * 2 2
8.88 0.92 T/T 0.96 1
6 _ ....... 9° so _
PM 1
5 '. .1 25
4 J 20 '
E 3 '. u: b 15
I E 3 I
. l
2 .H '278'11-11“‘° 1 1° .
14-68-11-7/7 l 1
1 L 9° J 5 '
. : 93'l1-T/T 1‘-35 J .
1 m“ w ‘ 0
8.86 0.88 0.9 0.92 0.94 0.96 0.98 0,5
TIT”o
300‘
; (d) B.-1TUlons: opensymbols
250 '_ B.-1TPblons: closedsymbols
IdrcleszH-4T
i ares: 2T
fi‘ 200 Td‘lgumonds: 1T
E {trianglecz 0.5T T
0 1
2 150E
4100; ‘\ n
l
so ’- “
I \ ‘
o ’ - ' , , ' , ‘
84 85 6 8 88.9 :1 91
T(K)

Figure 8.13 (a) Transition lines for Pb1. U1. (b) and (c) Transition
line for Pb1. as compared to the preirradiation melting line. (d) Jc v. T

for PM and U1.

114

pinning by columnar defects disappears and dissipation is dictated by the
intrinsic superconducting anisotropy.

As the temperature is lowered below T8009 = 0), the Bose glass phase is
entered, characterized by strong vortex localization along the defect tracks.
Within this so called ‘transverse Meissner’ phase[86, 87] an infinite tilt modulus is
expected, where the vortices remain aligned to the tracks even as the applied
field is tilted. However, at some finite H10), a lock-in transition[87] occurs,
where the strongly pinned Bose glass state is transformed continuously into a
kinked vortex configuration[93]. The transition is characterized by a sharp
increase in the measured resistivity, due to the relatively free motion of the kinks
as they move along the columnar defects. As the temperature is lowered, the
angle where the transition occurs increases. The lock-in transition is expected to
scale from above as [86, 87, 92] TBG(O) — 786(0) ~I H 1‘ l"”. The transition cannot
be scaled from below, since I" is essentially infinite (equal to the sample
thickness) below the transition.

At H=1.0T and at very small angles -1°<9 <1°, we measured several E-J
curves at various temperatures near T39. We can collapse the E-J data with the
same scaling procedure as the 9=0° data, obtaining s= v(z-2)=2.4:l:0.10,
v=1 5710.4, and z=3.55:l:0.5, in excellent agreement with our previous results.
This is demonstrated in Figure 8.15, which shows scaled data for 9=1°. Thus
within the 1° window the vortices behave as if they are perfectly aligned with the
columnar pins, illustrating the effective localization of the vortices to the defect

sites. Beyond 1° the data no longer can be scaled in this manner.

115

 

P (HQ-cm)

 

 

 

0 30 60 90
9 (deg)

 

   
    
 

E

‘9

G

3

o. ' P " 9(1-2)
0-1 g 91.49 K: Ilnear

5 (b)

+9140 K: Ilnear

+9130 K: 2:13.16
——91.20 K: z=3.36
+9110 K: 2:3.65

 

A 90.8 -

90.6 '
90 6 '

90.5:

 

(C
90.9;

A
or 111*[vvvlr1va—rr

 

 

phase near TBG. as

-12 -8 -4 0 4 8 12

9 (deg)

Figure 8.14 Angular dependence of the resistivity, for H=0.5T, crystal
Pb1. (a) The arrow denotes the accomodation angle, beyond which the
vortex lines are no longer confined to the defects. (b) Fits of the liquid
a function of angle. (c) Fits to the lock-in transition.

116

 

p /(T-T )Vu'z)

 

9'
10°10‘10210310‘1051o°1o’1o°
J/(T-T )av
as

Figure 8.15 E-J curves and resultant scaling for H=1T, applied
1' from the defect direction.

117

As the angle between the applied field and the columnar defect tracks is
increased, the current-voltage measurements no longer display the negative
curvature that characterized temperatures below T33. This is shown in

Figure 8.16, where E vs.J data is plotted for 9 = 5, 25, 70, and 90°. While there
appears to be a slight downward curvature in the 6 = 5° data, none can be seen
at the larger angles, although the low temperature curves still are non-ohmic.
Since the negative curvature, clearly seen in the 9 = 0-1° E vs. J data, is a
criterion of the strongly pinned Bose glass transition, the lack of this negative
curvature can be described as a weakening of the defects to effectively pin the
vortex lines, and thus we cannot qualify the nature of the solid-liquid transition at
large angles.

Shown in Figure 8.14(b) is p vs 0 data for H = 0.5T. For the temperatures
shown the vortex lines are in the liquid state, but at temperatures close to the
Bose glass transition temperature T33 = 91.10 K, here defined by the
p = 0.01 un-cm criterion. According to the Bose glass theory[87, 92], just at
T3G(6=0°) the resistivity is expected to increase with increasing tilt as p~9("2’. For
T > T3G(9=0°), we find that the resistivity can be fit with a linear function and
close to T33(9=0°) the resistivity can be fit with p ~ 9(2'2’ with z = 3.4 i 0.1, in good
agreement with the 2 value obtained from E-J scaling above.

In Figure 8.14(a) a sharp increase in resistivity is seen as the field is tilted
away from the defect track direction. We equate this sharp jump with the lock-in
transition, here determined by using a p = 0.01 uD-cm criterion to establish the

onset of a measurable resistance. This same criterion was used earlier to obtain

118

(b)

 

H=1T 9:5 ° i _ H=1T 0:25 °

___A
d
o

--..-.l --__L-

   

    
   

 

 

1:4 mm 111.1 (1 ll 891K:
1 10 100 10’

J (A/cmz)

  
  
  

H=1T 0:70 °

 

 

‘9' I. [.3811
0.1 1 10 100
J (A/cm2 )

 

Figure 8.16 E-J curves for H=1T, applied at large angles.

Tm," and since the Tm, line showed consistent agreement with the Bose Glass
transition line obtained from scaling the E—J curves below H=3T, it may be a
reasonable estimate of the glass-liquid transition at finite angle. The transition

line is shown in the Figure 8.14(c), along with best fits to the lock-in transition

function 56(0) - T,G(9) ~l H f l"", where H 1‘ / H z 0 for small angles. As a check,

other resistive criteria were also investigated, with consistent results for v. From
this analysis, using resistivity criteria ranging from 0.05 to 0.1 uQ-cm, we

obtained 0 = 1.7 i 0.4, in excellent agreement with the earlier analysis. Recent

119

magnetization measurements carried out on untwinned YBCO samples irradiated
with 4 GeV Au ions[140] found 1) z 1.7 - 1.9 for 9=1-10°, in agreement with our
measurements. Data from twinned crystals[134, 141] find a sharper peak in the
transition when compared to these results, demonstrating that the glass phase is
more effectively stabilized with columnar defects than with twin boundaries.

In summary, a non-ohmic regime was observed in transport data on an
untwinned YBCO crystal with columnar defects produced by Pb ion irradiation.
The non-ohmic E-J curves were scaled following the Bose glass theory, both
above and below a critical transition temperature T33, from which the critical
exponents v and 2 were obtained. The values of these exponents are in
agreement with results obtained from the angular dependence of the resistivity.
Finally, a sharp change in the slope of the irreversibility line was found at the
matching field for crystals U1, U2, U4, and Pb1. with a weakening in the pinning
efficiency for applied fields above B3. Whereas a Bose glass transition is firmly
established in the PM crystal, the data from the uranium irradiated crystal show
at best an upper bound of the transition. Nevertheless, a change in slope at the
matching field is observed in data as well as enhanced pinning for both U and Pb
irradiated samples. This consistency implies the existence of a Bose glass
transition for the uranium samples, at temperatures where the resistivity has

fallen below the experimental resolution.

120

Chapter 9

THE EFFECT OF LOW DOSES OF COLUMNAR DEFECTS ON VORTICES IN
UNTWINNED YBa200307.3 CRYSTALS

9.1 Introduction

With new experiments on the first order melting transition line in YBa20u307.5
single crystals, both an upper (H333) and a lower (H133) critical point of this melting
line has been found. These critical points mark a pronounced change in the
vortex behavior with disorder. The position of these critical points vary widely
from sample to sample, even among so called 'clean' crystals. Furthermore, in
some overdoped YBCO crystals, no evidence of an upper critical point has been
found, although these crystals exhibit a fairly large lower critical point. On the
other hand, in optimally doped YBCO crystals, the lower critical point can extend
down to a few hundred Gauss and very close to the zero field superconducting
transition temperature, although the upper critical point in this case is usually
below 15 Tesla.

Within the vortex solid state, a second magnetization peak has been
observed in magnetization loop data, shown schematically in Figure 9.1 inset.
The onset of an increase in the magnetization at H' as a function of increasing
field indicates an enhancement of pinning within the solid state, as shown in
Figure 9.1. The locus of points on the phase diagram in the vortex solid state
which indicates this change in the vortex behavior is usually defined by the onset
of increased magnetization[142], the peak in the derivative dMldH[143], or, for

very clean crystals, by the onset of hysteretic behavior[66, 144]. Above I-f lies

121

 

 

 

 

 

 

  
  
    

  
   
 

\
\Hinline l 3p H,”
\ M l
\
disordered \
entangled \
phase \ Hucp H
H l'1p2nd"lag“efizammpeak vortex
’ llqurd
I
enhanced first order
P'Pn'm \ melting
rne
vortex /
H‘ lattice H
lop

 

 

 

 

T

Figure 9.1 Phase diagram for YBCO. Inset shows a typical

magnetization loop, for temperatures below Tucp.

the peak in the magnetization data Hp, known as the 2"d magnetization peak (see
Figure 9.1 inset). This line has been associated with a dislocation-mediated
phase transition[59] from a vortex lattice to a disordered glass phase, with the
line possibly extending to the upper critical point[145]. A recent study[146] has
raised the possibility that the peak in measured magnetization Hp marks an
additional transition from a disordered glass to a highly disordered entangled
phase. This peak line also appears to approach Hoop. Thus the upper critical

point may in fact be a multicritical point where the liquid, lattice and glass phase

merge.

122

The positions of both the upper and lower critical points can be altered by
the controlled introduction of point-like defects, created by oxygen vacancies,
proton irradiation, and electron irradiation, as well as by columnar defects
produced by heavy-ion irradiation. In the last chapter, I presented data on a
single crystal where the first order melting transition was suppressed completely

and replaced by a continuous Bose glass transition due to columnar defect

,'I
h

pinning sites within the sample. In this chapter, the evolution of the melting line

x‘h.._.

from first to second order is investigated via the introduction of low densities of
columnar defects. Data is presented on crystals irradiated with 1.4 GeV Pb ions,
to dose matching fields of 50 Gauss (PbSOG), 100 Gauss (Pb1OOG), 500 Gauss
(Pb500G), and 0.1 Tesla (Pb0.1T)[147]. These results are compared with YBCO
crystals irradiated with protons and electrons where the major defects are point-

like[148].

9.2 Low densities of columnar defects
The samples studied were prepared in the same manner as previously described
in Chapter 8. Each sample, (except Pb0.1) was precharacterized before
irradiation to allow for comparative studies of the same crystal before and after
irradiation. An unirradiated crystal was also kept as a reference sample for
general comparisons.

Figure 9.2(a) shows the tail of the resistivity vs. temperature data for
PbSOG at H=2 Tesla. Here the squared ratio of the defect spacing to the vortex

spacing is H/Bq, = 400. At high fields the kink in the resistivity associated with the

123

 

 

H =2 T kink ln reslstlvlty

+Prenadatlon
+8350 Gauss

p (HO-CM)

 

 

 

 

0
78 80 82 84 86 88 80 92 94

 

T (K)

 

 

 

 

91.5

   
   

    
    
  

 

 

     
    
   

T (K)

(0

I-o—P v "" 1:01 r'réc'r.‘ 1mm H =:01 T """ L nonohmic on'set ' 1
P I

C -rB°=50 Gauss I

)- q

j ] +Unirradiated ;

i 11“ (Postlrradlatlon) 3

:l-l'cp (Prelrrad) I

 

r"""
I.

 

 

 

 

. : .4 2.892.8 93 9 .2 o -----
T (K)9 3 90 90.5 91 91.5 92 92.5 93 93.5 94
T (K)

Flgure 9.2 Data for preirradiation and for B¢=50Gauss.

124

first order melting transition is observed in both the unirradiated and the
irradiated sample. Note that while the position of the kink has remained
unchanged, the resistive drop has sharpened for the irradiated sample.

Figure 9.2(b) shows that the resultant first order melting transition lines for the
unirradiated and irradiated sample, determined from the temperature of the onset
of non-ohmic behavior, which coincides with the temperature of the onset of the
resistive ‘kink’, lie virtually on top of one another.

The kink in the temperature dependence of the resistivity can be tracked
down to H=0.8-1T (Figure 9.2(0) and (d)) in the unirradiated sample and defines
the pre-irradiation lower critical point H133. After irradiation the kink can be
tracked down to lower fields, with H.¢(Pb50G) = 0.3T, thus establishing a lower
Hrq, after irradiation. This is quite surprising, and at first sight would suggest that
irradiation actually improves the quality of the crystal by maintaining the vortex
lattice state to lower fields and higher temperatures. However, transport
measurements are non—equilibrium measurements. The 'kink' in the resistivity
which is associated with the first order melting transition will only be observable if
the current density used to measure the resistivity is lower than the critical
current of the sample. If the measuring current is greater than the critical current
at the melting temperature, a finite resistivity will be recorded at the position of
the melting temperature due to flux flow. This could explain the decrease in the
lower critical point from 0.8T before irradiation to 0.3T after irradiation in sample
Pb5OG. If the measuring current was above the critical current before irradiation

and below the 'enhanced' critical current after irradiation, the new position of the

125

lower critical point is a better representation of the true lower critical point of the
crystal. In other words, the introduction of a minute number of columnar disorder
is able to increase the critical current of the crystal by 'carpet tacking' a few
vortices.

Although the first order transition is recovered above the lower critical
point, vortex pinning and hence the critical current is dramatically altered by the
columnar defects. Current-voltage (E-J) curves before (a) and after (b)
irradiation for PbSOG are plotted in Figure 9.3 for an applied field of 1T. The E-J
curves display ohmic behavior above the melting temperature Tm.11=90.0K.
Before irradiation, non-ohmic behavior is measured within an 11 K-wide
temperature range, but after irradiation this range has decreased to a mere
1.2 K, establishing strong pinning just below the melting transition for the
irradiated crystal. This sharp increase in pinning after irradiation can be
illustrated by comparing the critical current density vs. temperature, shown in
Figure 9.3(0) where the critical current density is defined by the onset of
measurable voltage, E = 10'6 Vlcm. The fact that the critical currents are
significantly shifted to higher values for fields as high as 2T for a B3 of only
50 Gauss is a measure of the remarkable pinning ability of the defects once the
vortex solid state is established. Combining this with the fact that the first order
vortex solid to liquid transition exists above H133 after irradiation demonstrates
that the vortices are pinned elastically, without compromising the periodicity of

the lattice.

126

~‘\ -‘t—‘l R‘~.‘mb 4’1:
. I. -

 

 

(I. 7 31m;
b _.

100

E 1 0 --o-- Preirradiation
° —B :50 G
2 0
1
do +2T

+1T
+0.3T

+0.1T
+0.05T

0.1 ,

 

0.01
78 80 82 84 86 88 90 92 94

T (K)
Figure 9.3 E-J curves for (a) pre- and (b) post-irradiation, H=1T>H|cp.

(c) Dramatic increase in jc after low dose irradiation, here shown as a
semi-log plot.

127

Below the lower critical point of 0.3T, no 'kink' in the resistivity is observed as
shown in Figure 9.2(e). A comparison of the resistivity curves before and after
irradiation measured at two different currents show an upward shift in the zero
resistivity temperature after irradiation. Although the resistivity curves are
monotonic and no vestige of a 'kink' associated with the first order melting
transition is observed, the curves display non-ohmic behavior at a resistivity
value approximately Pnon-ohmlc/Pn = 90%, near the onset of the superconducting
transition. Moreover, the resistive drop is significantly sharper for the irradiated
sample. The irreversibility line near the lower critical point defined by the onset of
non-ohmic behavior near the lower critical point is shown in Figure 9.2(f). Below
0.3T the irreversibility line for Pb5OG shifts to higher temperatures, thus showing
the enhancement of the solid phase due to the columnar defects.

Similar behavior is observed for the Pb1OOG crystal as presented in
Figure 9.4. For this crystal the lower critical point is observed at the same field,
H.¢(preirrad) = H.33(Pb1OOG) = 0.3T. Figures 9.4(a) and (b) show data above
and below H13, for both the irradiated and unirradiated case. In (a) a sharp peak
in the derivative in the resistivity is clearly seen, while in (b) at H = 0.1T, the kink
disappears in both data sets. Plotted in Figure 9.4(c) is H... at fields less than
1 Tesla, again obtained from the onset of non—ohmic behavior. Similar to Pb5OG,
the irreversibility line after irradiation lies above the pre-irradiation first order
melting line below the lower critical point. At still higher matching fields, the lower
critical point shifts up significantly. Plotted in Figure 9.4(d) is the phase diagram

obtained for PbSOOG. Here, the lower critical point shifts up to H1¢=1.5T, and the

128

 

 

 

  

 

 
 
    
   

 

 

 

 

1
+Prelrrsdatlon ,
, +8 2100 Gauss .
0.8 D . I
1 H for both unlrrad.

0.6 l- W 1
E 1 and lrrad. data :
I l 1

0.4 b 1

: 4

0.2 '- .

l 4
l 1
l 4
90 90.5 91 91.5 92 92.5 93 93.5

T(K)

 

 

'vv'vvvrvvv

H=0.1T

      
 

50
400.
'O
O.
30_|
20

10

 

" ' ................ o
91.8 '2.2 92.4 92.8 92.8 93 93.2
T (K)

 

 

Flgure 9.4 (a) Kink associated with the first order melting transition.
(b) Low field, continuous transition, for H=0.1T. (c) Resultant phase
diagram, for both pre- and post-irradiation, B¢=100 Gauss. (d) Phase

diagram for crystal irradiated to B¢=500 Gauss.

129

. 3353»

K. .i- a!)

irreversibility line is enhanced over the pre-irradiation first order melting line
below the lower critical point. Finally, for Pb0.1, irradiated to a dose matching
field of 3,, = 0.1T, the lower critical point increases to H109 = 4T.

In order to investigate the transition below pr in the presence of columnar
defects, current-voltage measurements were taken at constant temperature for
crystals Pbsooe and Pb0.1 in an applied field of 1T, see Figure 9.5. This data
can be scaled using the Bose glass theory, with values for v and 2 close to those 3
found from the PM analysis in Chapter 8 as shown in the inset of the two figures.
This provides evidence that below the lower critical point the vortices freeze via a
continuous Bose glass transition. Further confirmation is found by evaluating the
Bose glass irreversibility line as a function of angle at an applied field below Hm
as shown in Figure 9.6. Plotted is the normalized melting temperature before
irradiation, and the irreversibility temperature determined from the onset of non-
ohmic behavior after irradiation, as a function of the tilt angle between the applied
magnetic field and along the axis of the columnar defects. The irradiated crystal
displays a cusp in the irreversibility temperature centered about the columnar
defect, in agreement with the prediction of the Bose glass theory. Beyond 9310°
the kink in the resistivity is recovered in the irradiated crystal and the
irreversibility line transforms into a first order melting line which joins smoothly
with the melting line of the pre-irradiated sample. The solid line is a fit of the
angular dependence of the first order melting line to the anisotropic London

formula, given by kBTm = [(DocL]/[47r2/1§bB"2£(6)”‘], where

8(9) = 72 cosz(9) + sin2(0), and yis the superconducting mass anisotropy. The fit

130

 

fifififi'fifi“ ‘fififi

Teas-89.5 K /1 H=1T

1‘ 8 =500 Gauss
1 o

v=1.531
z=3.w1

 

 

 

   
    

 

  

 

 

. (P)
r. I "I
10-2{21 H=1T 1
5A,, B =1ooo G a
P}:- (b I
,u -’t '
E1°3FZ 1.
§ ; 3
V10"? 1° 41
UJ _r JI(T-Tg)' 1
10"“,- 1
I I
391.27K i
-e' o9.9K'
10 ~ .2 2 ~
10 10

Flgure 9.5 (a) E—J curves for (a) PDSOOG and (b) Pb0.1T. The
insets show the scaling of these curves to the Bose glass theory.

131

 

     
     
 

 

 

 

 

1.02 P 1 ' ' ' I ‘ ' ' ' I I If f r 1 v v v 1
_ 0 before irradiation .
B after irradiation 84) =500 G ' d
A 1.015 1 5,2 1’
2R — kBT ¢° c L
m = 2 .
8 1.01 - 4152ow"2 6(9) 114 -:
E .
l: I
E 1.005 f 1
'- 'i
l- 1 00 r (:1 '1 D
H=0 5T ° 09°“
1:7.9 1
0.995 L 1 1 1 l n 1 l 1 1 l 1 n n 1
-100 ~50 0 50 100

9 (degrees from c-axis)

Figure 9.6 Angular dependence of the melting transition
before irradiation, and of the irreversibility line after irradiation.

yields a Lindemann criterion number of cL = 0.18 and a mass anisotropy of
y= 7.9, in good agreement with previous measurements on untwinned YBCO
single crystals[32]. Thus, by increasing the tilt angle of the magnetic field with
the columnar defects, the effective pinning strength of the defects is sufficiently
weakened as to reestablish the first order melting transition. Our results clearly
demonstrate that the vortex solid below the irreversibility line is a Bose glass.
The data from Pb50, Pb100, Pb500 and Pb0.1T demonstrate that the first
order melting transition is recovered when the applied magnetic field is above
30—608,,,, corresponding to ~6-8 vortices between two given columnar defects.
This is in qualitative agreement with the data from PM (8,, = 11'), where by
extrapolating the pre-irradiation first order melting curve and the Bose glass

transition curve to higher magnetic fields showed an intersection at a field of 26T.

132

Our results suggest that the lower critical point is a vortex pinning saturation
point. At low fields, the vortex interaction energy is less than the vortex pinning
energy, resulting in the randomizing of the vortex structure due to columnar
defect pinning (loss of long range translational and orientational order), leading to
a Bose glass state. Vlfith increasing field, the density of vortices increases. At
the lower critical point, the vortex interaction energy becomes greater than the

pinning energy and the vortex lattice is recovered, along with the first order solid

-. XENIA-v.11

to liquid transition.

The phase diagram for the four irradiated crystals also shows the
evolution of the first order melting line to a Bose glass line. This occurs through
a shift in the lower critical point to higher temperatures, and with a corresponding
shift of the irreversibility line below the lower critical point to higher temperatures.
The increasing shift in the irreversibility line with dose is in agreement with the
measurements on crystals U1, U2, U4, and PM and confirms that the defects not
only change the nature of the transition, but also act to stabilize the vortex solid
phase. The enlargement of the vortex solid regime is shown by the cross-
hatched areas in Figure. 9.4(c) and Figure 9.4(d).

The vortex motion in the liquid state is also retarded by the introduction of
columnar defects. This is seen in Figure 9.7, where the normalized resistivity is
plotted for Pb1OOG, PbSOOG, and Pb0.1. For Pb0.1 the temperature is also
normalized since the transition temperature shifts down slightly after irradiation.
While there is virtually no difference between the unirradiated and irradiated data

sets for Pb1OOG, there is significant lowering of the resistivity above the melting

133

    
    

 

1 ' ' I r j ' I ' ' ' I " ' ' I "

  
      
   

 

 

 

  
       
  
 

 

 

 

  

 

 

_ H=8,6,5,4,3,2,1,0.5 T
0.8 . .
3? 0 6 : Preirradiation ‘
m , . _ .
a: - 8 =100 Gauss -
v D o
i; 0.4 _
0.2 '
0 ' , . , , 1
T (K) '
1F"t"'r**'r"fr' ‘1
: H = 8,7,6,5,4,3,2,1.5,1T 3‘
0.8 I. Preirradiation -
I — B = 500 Gauss '
A 0
fi
02. 0.6
l- .
o .
E 0.4
Q l-
0.2'
80 84 88 92
T K
1"
:H = 8,7,6,5,4,3,2,1,0.5T
0.8 _-
Q 0.6 '_ ----Preirradiation '_ '1 1‘
a : —B°=1000G (0.1T) x .
Q 0 4 ' /
E. ' . _. . ' .
0.2 '
o b
0.85 0.9 0.95 1
T/‘l'c

Figure 9.7 Normalized resistivity versus temperature for (a) 100 G,
(b) 500 G, and (c) 1000 G. For (a) and (b) the data compares pre-
and post-irradiation on the same sample. In (0) the 1000 G data is
compared to a reference sample.

134

temperature for the other two crystals irradiated at a higher dose. This can be
attributed to an increase in the viscosity of the vortex liquid due to a retardation of
the vortex velocity from pinning in the liquid state[75, 149]. If the healing length
for vortex flow originating from one pinning site as described in Chapter 5 is cut
off by another nearby columnar defect, the vortex velocity in the liquid can be
greatly retarded.

The crystals were also investigated at high fields at the National High

 

Magnetic Field Laboratory at Tallahassee, Florida. Shown in Figure 9.8 is the
temperature derivative of the resistivity for Pb0.1 (upper inset) and a comparison
with the reference crystal at 1OT (lower inset). The vortex melting lines for these
two crystals are shown in the main figure, obtained from the peak in the
temperature derivative of the resistivity. As noted previously, the lower critical
point for Pb0.1 shifts up to H1.» = 4T, and for fields below this field the
irreversibility line of the irradiated crystal is shifted to higher temperatures. For
the reference crystal, the kink associated with first order vortex melting is
detected up to H...» = 9T, but after irradiation the kink is observed up to

Hm:p = 11T. These are the first measurements which show an increase in the
upper critical point as a result of imbedding columnar defects within the sample.
The upper critical point marks the transformation of the first order vortex melting
line into a higher order transition. This transformation is speculated to occur due
to increased fluctuation of the vortex lines at high magnetic fields brought upon
by increased pinning of weak random point defects. Previous measurements on

twinned and untwinned crystals[49] have provided evidence that for fields above

135

 

 

 

 

 

 

 

20 fi--fi...,..
:Hllc
[
15 —
C10 - Hucp
I - (84:01

 

H=10T 11 c f

 

 

 

 

dP/dT (HQ-CWK)

 

 

0 .
73747576777879
T(K) T(K)

Figure 9.8 H-T diagram for crystal PBO.1T, where both the unirradiated
(circles) and irradiated (triangles) crystal melting transition lines are shown.
The upper inset shows the derivative of the resistivity dp/dT versus T after

irradiation. The lower inset compares dp/dT unirradiated (triangles) and
irradiated data (squares) for H=10T, with a peak in the data after irradiation.

Hm the vortices become entangled, due to meandering of the vortex lines
promoted by the presence of point defects. Figure 9.8 demonstrates that the
columnar defects can inhibit line wandering of the vortex line by pinning the
vortex along its entire length. Consequently, the vortices remain unentangled,
and thus the vortex lattice structure is preserved to even higher fields, resulting in
an upward shift of the critical point. Also plotted in fig. 9.8 above Hum are the

Tm, lines, defined where the resistivity goes to zero, using a p = 0.01un-cm

136

criterion. As a result of the increase in Hum, this solid-liquid transition line above

Hucp is also shifted to higher temperatures and fields after irradiation.

9.3 Comparison with low densities of point defects
The effect of low density of point defects induced by proton irradiation on
untwinned YBach307-a has been investigated[148]. A clean, untwinned YBCO

T1

crystal, with dimensions 880(l)x440(w)x40(t) pma, was irradiated with 9 MeV

protons at the tandem accelerator at Western Michigan University (WMU).
Transport measurements were performed using the standard four-probe
technique, with the current directed along the ab-plane of the crystal. The crystal
was first pre-characterized and then sent to VVMU for proton irradiation.
Successive irradiations and characterization with transport measurements were
performed on the same sample, with total doses amounting to 0.25, 0.5, 1.0, 1.5,
and 2.0x1015 protons/cmz. The irradiation produced random point-like defect
sites, with diameters <20 A in addition to cluster defects ~30 A in diameter. The
created defects are isotropic in nature without any apparent correlation along the
irradiation direction. It is estimated[148] that the resultant defect densities are
approximately one defect per ~29000 unit cells for 0.25x1015 plcm2 dose,
increasing to one defect per ~3600 for 2x1015 p/cmz, corresponding to defect
separations from 175 A for the lowest dose, to 85 A for the highest dose. Room
temperature annealing of ~30% of the oxygen defects has been taken into
account in this calculation. As was the case for low dose columnar defects,

successive proton irradiations at these low doses have virtually no effect on the

137

zero field transition temperature Tc, with only about ~6% increase in the normal
state resistivity for the highest dose.

The unirradiated crystal for this study displayed a zero-field transition
temperature of Te = 92.8 K, with a transition width ATc< 300 mK. Similar to our
previous YBCO reference crystals, the unirradiated sample showed a sharp kink
in the resistivity measurements associated with the first order melting transition,

in fields of 0.05-8T, the maximum field of our measurement. Figure 9.9 shows

 

resistivity measurements for H = 4T applied parallel to the c-axis, for an applied
current density of 6 A/cm2 in the ab-plane of the crystal. Fig. 9.9(a) shows an
expanded view of the melting transition after each subsequent proton irradiation;
the inset shows the full data up to 95 K. Above the transition the resistivity
clearly decreases with increasing dose. The figure also shows a lowering of the
melting transition temperature with increasing dose, the position of which is
tracked by the onset of the peak in the temperature derivative of the resistivity,
see Fig. 9.9(b). The onset of the peak clearly shifts down with increasing dose.
The height of the peak also decreases with dose, and for doses of 1.5 and
2.0x1015 plcmz, a peak is no longer discemable. However no broadening of the
temperature width of the transition is observed[148].

In Figure 9.10(a) the derivative of the resistivity data is shown for
H = 3-8T, for a dose of 1.5x1015 plcmz. While a peak in the derivative is
observed for H = 4—7T (shown as hatched in the figure), the peak has
disappeared above and below these fields, establishing an upper critical point of

Hm = 7T and a lower critical point of H”, = 4T. For a dose of 1.0x1015 plcmz, the

138

 

0.2'rfr'l

 

 

 

 

t-D- Preirradiation
' _’--°--0.25x10'5p/cm2
.-"-0.5x10‘5p/cm2
“+1.0x1015p/cm2
0.05 '_"'—1.5x10‘~"p/cm2
r+2.0x10‘5p/cm2

p/ p(95K)

  
   
  

 

 
 
   

 

 

 

. (a) .
0 . ‘1 r a ', .
4o. . ' (b) l
52 1 l
\ ' .
(E) 30 1- .-
2. ' .
v _ 1
. ' 1
t; 20_ 1
Q.
'0 \' ‘

 

 

h ’
' l
10- , . 'L
p i ‘ ‘ ‘ .1 _ u
h"" v " - . . .
h I 1.; .‘o“°:.o ‘3'
‘ ‘-s" “ I
I. 0 O O . 1 A A ‘ l t A + A

0.89 0.896 0.902 0.907 0.913
T/T
c

 

Figure 9.9 (a) Resistivity versus temperature for H=4T, here
shown after successive proton irradiations. (b) The derivative of
the resistivity data, showing a peak at all but the highest dose.

139

F: 3.3.].- ‘JMI. ufi

 

N 0 # 0'1 0: \l on
' "

dp/dT (pQ-cm/K)

 

 

8T

   
  

1
4
1
1

 

 

 

 

-o— Preirradiation

 

 

 

(b) +1.0 x 1015 p/cm2
6
i +1.5x10'5p/cm2
A ----- 2.0 x 1015 p/cm2
b 4 _
I
2 l-
o l l l ‘ .
0.80 0.85 0.90 0.95 1.0
Tm [TC

Figure 9.10 (a) Plot showing the derivative of the resisitivity for a dose of

1.5x1015p/cm2, for H=3,4,5,6,7, and 8T. A peak in the derivative is seen
for H=4-7T, shown as hatched in the figure. (b) Phase transition lines for

the proton irradiation. For the preirradiation, 1.0, and 1.5x1015plcm2
data, the lines show the first order melting, defined by the onset of the kink
in the resistivity data. The dashed line shows a resistivity criterion for the

2x1015p/cm2 dose.

140

I1

upper critical point is above H = 8T, in agreement with previous measurements
on an untwinned crystal[150], which found that for the same dose the upper
critical point was observed at 9T, lowered from an unirradiated value of 12.5T.
Figure 9.10(b) shows the resultant vortex phase diagram. The first order
melting transition is shifted downward in temperature, and there is an upward
shift in Hg, and a downward shift in H“up with increasing dose. At a dose of
2x1015 plcmz the critical points have converged and no first order transition is

seen; for this dose the Tum (p = 0.01pQ-cm criterion) line is shown. The .1

fl

behavior of Hm,p with defect density agrees with other measurements of YBCO
samples in which point defects were produced by oxygen vacancies, created by
lowering the oxygen concentration[52, 53, 71]. The Hum point in an electron
irradiated crystal was also investigated by coworkers[148]. In that work Hucp was
observed to increase via the lowering of the defect density through successive
heat treatments of the sample, which is consistent with our proton irradiation
results. For the electron irradiation, the strong defect sites were presumed to be
point-like vacancies in the Cu-O planes[151], with the heat treatments acting to
gradually anneal out the defects.

The lowering of the melting transition at high defect densities has been
observed previously in proton[81, 82] and electron[16] irradiated YBCO crystals.
A very recent study[145] of an electron irradiated crystal has confirmed the
lowering of Huq, and the first order melting line, as well as a shift to lower fields of
the ‘enhanced pinning line’ H' with increasing defect density, sketched in

Figure 9.11, upper plot. The downward shift in H' as well as in the magnetization

141

peak Hp as a function of increased oxygen vacancies has also been
confirmed[53, 144].
Whereas the proton-created defects induced shifts in both upper and
lower critical points, this is not the case for electrons, leading to a possible
explanation based on the difference in defect production for the two cases[148]:
While protons produce both large cluster defects as well as smaller vacancies, 1

TEM studies[151] show that electrons produce random defects less than 20 A in

I Imam—fa...”

diameter, without creating the larger clustered defects. Since both ions act to
lower Hum, we speculate that the smaller point-like defects are responsible for
vortex disorder at these higher fields, possibly by promoting line meandering
leading to entanglement. The fact that Hucp can also be lowered by increasing
the density of oxygen vacancies supports this view. The behavior of qu,
however is controlled by the larger cluster defects, which may act as stronger
pinning sites as compared to electrons. Previous magnetization
measurements[151] show stronger pinning strength for proton versus electron
irradiated YBCO samples. The cluster defects are able to pin the vortex lines at
low vortex densities and high temperatures, possibly resulting in an unentangled
vortex glass below the lower critical point: At these low fields the intervortex
distance is much larger than (>3x) the defect spacing, which may lead to
individual vortex pinning. With increasing dose more cluster defects are created
which can pin more vortex lines, and thus the lower critical point shifts upward

with dose.

142

 

   
 

    

 

\ \ point defects created
\ by proton irradiation
\
\
\
\ vortex
disordered \ Hucp liquid
entangled
phase
K first order
melting
vortex
lattice

an"... m

" vortex glass? \\\

 

 

 

  
   
 
 
 

\
\ columnar defects created
\ by heavy-ion irradiation
\
\
\
\ vortex
disordered \K H liquid
entangled

    

phase

first order

vortex
lattice
R BE

fl.—
"" \

Bose glass \

 

 

Figure 9.11 The effect of (a) point defects and (b) columnar
defects on the vortex phase diagram of YBCO.

143

As discussed earlier, it is theorized that the peak effect measured in unirradiated
YBCO crystals just below the melting transition was possibly due to a disordered
phase, which lies just between the liquid and solid states[50, 67, 68], and
extends up into the disordered solid phase above the enhanced pinning line. In
this scenario, the magnetization peak line H, meets the transport critical current
peak in Jc below the upper critical point, and the upper critical point would not be

a multicritical point. Recent magnetization data[143] on YBCO crystals irradiated

gnu-nut

with electrons appears to show a disassociation between H...» and the peak line,
lending some support to this idea. While the resistivity data on crystals irradiated L“
with low doses of columnar defects show a complete suppression of the peak

effect, this is not the case for the low dose proton irradiation data. Shown in

Figure 9.12 is E-J measurements for H = 2T. A clear transition from linear to

negative curvature of the E-J temperature curves can be seen in (a) the pre-

irradiation curves and (b) the 0.5x1015 plcm2 dose curves. At the highest dose

no transition is seen and the curves all show ohmic behavior (fig. 9.12(c)).

Plotted in fig. 9.12(d) is jc vs. TITc, where jc is defined by an E = 10‘6 V/cm

criterion; in fig. 9.12(e) it is replotted in semilog. The sharp jump in jc (clearly

seen in fig. 9.12(e)) for the unirradiated crystal and for the first two proton doses

reflects the first order vortex lattice freezing transition where a non-zero shear

modulus first appears in the vortex solid state. The jc line increases above the

jump, indicating enhanced pinning, which would be expected for point pins in a

disordered vortex phase. Once the first order melting transition is suppressed (at

a dose of 1x1015 p/cm2 for H = 21), then the curves shift downward with

144

 

‘0 ' """‘I ' """‘I ' v""'l ' """I ' ""

(a) (b)

 
   

- --....l - ---
A
O

10" 0.5x101"’p/cm2
H=2T

g 10" Preirradiation
E H=2T
Lu

  

E (V/cm)

07.94K
/

 

 

   

2.0x101 5p/cm2

=2T

 

 

 

  
   
  
 
 

E-D-Preinadiation
;"°"o.25x10"‘p/cm2
aoo E.'°'0.5x10"’p/cm2
5""1.0x10“p/cm2
20° '+1.5x10"plcm2
*2.0x10“p/cm’

h

r 1 iii

100

 

 

 

   

AAALA

AAAAJ‘AAA

o ‘ 0.1 -- - - - . -
0.87 0.88 0.89 0.9 0.91092 0.93 . .95 0.9 0.91 0.92 0.93 0.94 0.95

T/T T/T
c c

 

Figure 9.12 (a)-(c) E-J curves, as a function of proton
irradiation dose. (d), (e) Critical current density versus
temperature as a function of proton irradiation dose.

145

increasing dose. This reflects the downward shift in temperature of the solid
state with increasing point defects, as seen in the vortex phase diagram

(fig. 9.10(b)). The ability of the point defects to destabilize the solid state, due to
the enhanced vortex wandering induced by the random defect sites, is in direct
contrast with columnar pinning sites, which tend to stabilize the solid state and
shift the transition line to higher temperatures and fields.

For the case of point defects, a vortex glass has been predicted[22, 29], in
which a non-ohmic critical fluctuation regime is expected near the transition.
However, looking at fields below the lower critical point, no feature of a liquid to
solid transition was seen in the current-voltage curves (see Figure 9.12(c)). and
thus the data presumably reflects only the liquid state. If there is a vortex glass
transition, it lies below our experimental resolution. Recent data on the angular
dependence of the ohmic tail of the resistivity for a crystal irradiated to a higher
proton dose (3x10‘° p/cmz), does show evidence of a possible vortex glass
transition in YBCO[82].

In summary, the vortex phase diagram of high temperature
superconductors in the presence of disorder contain a rich variety of novel vortex
phases, which evolve as the density and dimensionality of the defects are
increased. The general results of low densities of point and columnar defects are
summarized in Figure 9.11. For the case of point defects created by proton
irradiation the upper critical point is shifted downward with increasing dose, while
for a crystal with a columnar density of 8,, = 1000G the measured Hm shifted

upwards. Hence we conclude that columnar defects tend to prevent vortex line

146

 

meandering which is normally promoted by weak random point defects, resulting
in an upward shift in the upper critical point. However, as discussed previously,
the upper critical point in YBCO will be established at higher fields, at the point in
the phase diagram where the pinning energy of the underlying point defects
overcome the vortex tilt energy and the vortex line begins to meander, leading to
a possible entangled vortex state.

This type of novel behavior was also seen near the lower critical point.

. . _vamyufit‘

For both types of defects, H109 is observed to shift upwards with increasing W
disorder, with the first order melting transition transformed into a continuous I
transition below Hep. This transformation occurs due to the energy balance

between the pinning energy and the vortex elastic energy, specifically the vortex

shear modulus energy. The pinning energy of the point-like defect clusters

induced by proton irradiation, and the columnar defects created by heavy ion

irradiation, both act to randomize the vortex structure. For the case of the

columnar defects, a Bose glass transition was established below Hum. It is

unclear as to the nature of the transition below H1“, in the case of proton

irradiated samples. Above Hiq, the first order melting transition was

reestablished. This occurs when the vortex-vortex interaction energy overcomes

the pinning energy, restoring the lattice state at fields above Hicp. Our work

implies a possible existence of a low field disorder-order transition, shown in

Figure 9.11 as dashed lines just below the vortex lattice. Whether this new line is

related to the H' or HD line determined from recent magnetization measurements

remains to be seen.

147

Chapter 10

CONCLUSION

We have investigated the effects of defects on the vortex melting transition in
YBCO single crystals. Columnar defect densities at matching fields 8,, = 1, 2,
and 4 Tesla were investigated, in applied fields up to 8 Tesla. For these high
defect densities, the kink in the resistivity data associated with a first order
melting transition is replaced by a monotonically decreasing function of
temperature. In the presence of a density of columnar defects in which the
applied field H ~ 8., a continuous transition to a Bose glass has been predicted.
This glassy phase is characterized by vortex localization within the defects, for an
applied field applied along the defect direction. The glass-to liquid transition thus
represents a localization-delocalization transition of the vortex lines. This
transition has been investigated within the framework of the Bose glass theory, in
which a scaling ansatz for E-J curves is expected for temperatures within the
critical fluctuation regime. The results of the irradiation confirm a continuous
transition, with a scalable non-ohmic critical regime. From scaling of resistivity
data the static and dynamic critical exponents v and 2 were obtained, with
consistent values of these exponents at fields above and below 8,.

Due to the anisotropic nature of the defects, the effective pinning strength
of the defects can be weakened by tilting the field with respect to the defect
direction, resulting in a sharp cusp in the melting transition as a function of field

angle. From the Bose glass theory a lock-in transition is predicted, which

148

describes vortex kink proliferation resulting in a cross-over from a Bose glass to
liquid, induced by rotating the vortices at large angles away from the defects.
This is studied experimentally by locating the onset of a measurable resistance
as a function of angle and temperature. Results from angular measurements
agree with the scaling results obtained for a field applied along the defects. Thus
the existence of a transition to a Bose glass state is firmly established. Finally,
the defects act to shift the melting transition to higher temperatures, thereby
stabilizing the vortex solid state at higher thermal fluctuations. However, for
vortex line densities above the columnar defect density the Bose glass transition
line begins to approach the melting transition line of the pristine sample, although
the merging of these lines is expected at significantly higher fields.

The evolution of the melting line from a first order transition to a
continuous transition was evaluated by introducing low densities of columnar
defects into the crystals. The lower critical point is shown to shift upwards with
increasing columnar defect density, with the critical point occurring at Hiq, z 30-
608... For fields below H1.» a Bose glass transition is experimentally confirmed.
Angular measurements identify the expected cusp predicted by the Bose glass
theory, as well as a crossover from a Bose glass to a vortex lattice with
increasing tilt angle. For 8,, = 1000 Gauss, an upward shift in the upper critical
field H...» is observed. Thus the vortex lattice can itself be stabilized by columnar
defects. Above Hm an entangled vortex configuration has been predicted, and
much experimental evidence points to this scenario. An upward shift in Hue,D

then implies that the columnar defects act to maintain straight vortices, thereby

149

inhibiting vortex wandering and the production of topological defects expected for
an entangled phase.

In comparison to anisotropic columnar defects, low doses of isotropic
defects created by proton irradiation were investigated. The defects take the
form of both point-like defects and somewhat larger cluster defects. The
resultant behavior of the melting transition is quite different in this case. Both the
upper and lower critical points are shifted towards one another, while the first
order melting line is shifted to lower temperatures. Thus for proton irradiated
samples the vortex lattice is destabilized by the defects at both low and high
fields, as well as at lower temperatures. At high fields the defects are expected
to produce an enhancement of vortex line wandering, and thus a lowering of the
upper critical point. This is due to the random isotropic nature of the defects,
which tend to tilt and deform the lattice in directions both parallel and
perpendicular to the field direction. The upward shift in the lower critical point is
surprising, and has not been observed in electron irradiated crystals or in crystals
with varying oxygen content. In the presence of point-like defects, a possible

phase transition below H1“, has not yet been established.

150

BIBLIOGRAPHY

151

 

_‘r. .
1

.1

10

11

12

13

Bibliography

H. K. Onnes, Leiden Comm. 120b, 122b, 124c (1911).

W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933).

F. London and H. London, Proc. Roy. Soc. (London) A149, 71 (1935).

U. Essmann and H. Trauble, Phys. Lett. 24A, 526 (1967).

M. Yethiraj, H. A. Mook, G. D. Wignall, et al., Phys. Rev. Lett. 70, 857
(1993).

H. F. Hess, R. B. Robinson, and J. V. Waszczak, Phys. Rev. Lett. 64,
271 1 (1990).

V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i. Teor 20, 1064 (1950).

A. A. Abrikosov, Sov. Phys.aJETP 5, 1 174 (1957).

J. G. Bednorz and K. A. Miller, Zeitschrift fur Physik B 64, 189 (1986).

M. F. Schmidt, V. E. Israeloff, and A. M. Goldman, Phys. Rev. Lett. 70,
2162(1993)

J. Bardeen and M. J. Stephen, Physical Review 140, A1197 (1965).

P. H. Kes, J. Aarts, J. v. d. Berg, et al., Superconductivity, Science and
Technology 1, 242 (1989).

V. M. Vinokur, M. V. Feigel'man, V. B. Geshkenbein, etal., Phys. Rev.
Lett. 65, 259 (1990).

152

14

15

16

17

18

19

20

21

22

23

24

25

26

27

W. K. Kwok, J. Fendrich, S. Fleshler, et al., Phys. Rev. Lett. 72, 1092
(1994).

W. K. Kwok, J. Fendrich, U. Welp, et al., Phys. Rev. Lett. 72, 1088 (1994).

J. A. Fendrich, W. K. Kwok, J. Giapintzakis, et al., Phys. Rev. Lett. 74,
1210(1995)

P. W. Anderson, Phys. Rev. Lett. 9, 309 (1962).

Y. B. Kim, C. F. Hempstead, and A. R. Stmad, Phys. Rev. Lett. 9, 306
(1 962).

T. K. Worthington, E. Olsson, C. S. Nichols, et al., Phys. Rev. B 43, 10538
(1991)

R. H. Koch, V. Foglietti, W. J. Gallagher, at al., Phys. Rev. Lett. 63, 1511
(1989).

M. Charalambous, R. H. Koch, T. Masselink, et al., Phys. Rev. Lett. 75,
2578 (1995).

M. P. A. Fisher, Phys. Rev. Lett. 62, 1415 (1989).

E. Brézin, D. R. Nelson, and A. Thiaville, Phys. Rev. B 31, 7124 (1985).

F. A. Lindemann, Zeitschrift fur Physik 11, 609 (1910).

D. R. Nelson and H. S. Seung, Phys. Rev. B 39, 9153 (1989).

A. Houghton, R. A. Pelcovits, and A. Sudbo, Phys. Rev. B 40, 6763
(1989)

L. I. Glazman and A. E. Koshelev, Phys. Rev. B 43, 2835 (1991).

153

28

29

30

31

32

33

35

36

37

38

39

40

41

E. H. Brandt, Journal of Low Temperature Physics 26, 709 (1977).

D. S. Fisher, M. P. A. Fisher, and D. A. Huse, Phys. Rev. B 43, 130
(1991).

D. R. Nelson, Phys. Rev. Lett. 60, 1973 (1988).

H. Safar, P. L. Gammel, D. A. Huse, et al., Phys. Rev. Lett. 72, 1272
(1994).

W. K. Kwok, S. Fleshler, U. Welp, et al., Phys. Rev. Lett. 69, 3370 (1992).

A. B. Pippard, Philosophical Magazine 19, 217 (1969).

W. K. Kwok, J. A. Fendrich, C. J. v. d. Beek, et al., Phys. Rev. Lett. 73,
2614 (1994).

W. K. Kwok, J. A. Fendrich, V. M. Vinokur, et al., Phys. Rev. Lett. 76, 4596
(1996).

G. Blatter, M. V. Feigel'man, V. B. Geshkenbein, etal., Rev. Mod. Phys.
66, 1125 (1994).

D. Lepez, E. F. Righi, G. Nieva, et al., Phys. Rev. Lett. 76, 4034 (1996).

R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. Lett. 76, 835 (1996).

U. Welp, J. A. Fendrich, W. K. Kwok, et al., Phys. Rev. Lett. 76,4809
(1996).

J. A. Fendrich, U. Welp, W. K. Kwok, et al., Phys. Rev. Lett. 77, 2073
(1996).

A. Schilling, R. A. Fisher, N. E. Phillips, etal., Nature 382, 791 (1996).

154

 

42

43

45

46

47

48

49

50

51

52

53

54

M. Willemin, A. Schilling, H. Keller, et al., Phys. Rev. Lett. 81, 4236
(1998)

A. Schilling, M. Willemin, C. Rossel, etal., Phys. Rev. B 61, 3592 (2000).

L. M. Paulius, J. A. Fendrich, W.-K. Kwok, et al., Phys. Rev. B 56, 913
(1997)

R. E. Hetzel, A. Sudbo, and D. A. Huse, Phys. Rev. Lett. 69, 518 (1992).

u Hi
”.2...”

R. Sasik and D. Stroud, Phys. Rev. Lett. 75, 2582 (1995).

t .n'n- 'L‘Lf -1- - ‘ 4‘-

M. J. W. Dodgson, V. B. Geshkenbein, H. Nordborg, et al., Phy. Rev. B
57,14498(1998)

H. Safar, P. L. Gammel, D. A. Huse, etal., Phys. Rev. Lett. 70, 3800
(1993).

D. Lepez, L. Krusin-Elbaum, H. Safar, et al., Phys. Rev. Lett. 80, 1070
(1998).

G. W. Crabtree. W. K. Kwok, U. Welp, et al., in Physics and Materials
Science of Vortex States, Flux Pinning and Dynamics, edited by R.
Kossowsky and S. Bose (Kluwer Academic Publishers, Amsterdam,

1 999).

C. Marcenat, R. Calemczuk, A. Erb, et al., Physica C 282-287, 2059
(1997).

K. Shibata, T. Nishizaki, T. Naito, et al., Physica C 317-318, 540 (1999).

K. Deligiannis, P. A. J. d. Groot, M. Oussena, et al., Phys. Rev. lett. 79,
2121(1997)

V. \finokur, B. Khaykovich, E. Zeldov, et al., Physica C 295, 209 (1998).

155

55

56

57

58

59

60

61

62

63

65

66

67

68

69

G. W. Crabtree and D. R. Nelson, Physics Today April, 38-45 (1997).

D. Ertas and D. R. Nelson, Physica C 272, 79 (1996).

T. Giamarchi and P. L. Doussal, Phys. Rev. Lett. 72, 1530 (1994).

T. Giamarchi and P. L. Doussal, Phys. Rev. B 52, 1242 (1995).

T. Giamarchi and P. L. Doussal, Phys. Rev. B 55, 6577 (1997).

J. Kierfeld and V. Vinokur, cond-mat/9909190 .

R. Cubitt, E. M. Forgan, G. Yang, et al., Nature 365, 407 (1993).

B. Khaykovich, E. Zeldov, D. Majer, et al., Phys. Rev. Lett. 76,2555
(1 996).

B. Khaykovich, M. Konczykowski, E. Zeldov, et al., Phys. Rev. B 56, 517
(1997)

T. Nishizaki, R. Naito, and N. Kobayashi, Phys. Rev. B 58, 11169 (1998).

D. Giller, A. Shaulov, Y. Yeshurun, et al., Phys. Rev. B 60, 1076 (1999).

S. Kokkaliaris, P. A. J. d. Groot, S. N. Gordeev, et al., Phys. Rev. Lett. 82,
5116(1999)

H. KLlpfer, T. Wokf, C. Lessing, et al., Phys. Rev. B 58, 2886 (1998).

D. L0pez and et al., unpublished .

D. E. Farrell, J. P. Rice, and D. M. Ginsberg, Phys. Rev. Lett. 67, 1165
(1991).

156

 

70

71

72

73

74

75

76

77

78

79

80

81

82

83

W. K. Kwok, U. Welp, G. W. Crabtree, et al., Phys. Rev. Lett. 64, 966
(1990)

M. Roulin, A. Junod, A. Erb, et al., Phys. Rev. Lett. 80, 1722 (1998).

A. Schilling, R. A. Fisher, N. E. Phillips, etal., Phys. Rev. Lett. 78, 4833
(1997)

A. K. Kienappel and M. A. Moore, cond-mat/9804314 (1998).

S. Fleshler, W. K. Kwok, U. Welp, et al., Phys. Rev. B 47, 14448 (1993).

M. C. Marchetti and D. R. Nelson, Phys. Rev. B 42, 9938 (1990).

D. L0pez, G. Nieva, and F. d. L. Cruz, Phys. Rev. B 50, 7219 (1994).

D. Lepez, E. J. Righi, G. Nieva, etal., Phys. Rev. B 53, 8895 (1996).

A. Schdnenberger, V. Geskenbein, and G. Blatter, Phys. Rev. Lett. 75,
1 380 (1995).

C. Carraro and D. S. Fisher, Phys. Rev. B 51, 534 (1995).

M. A. Moore and N. K. VVIlkin, Phys. Rev. B 50, 10294 (1994).

W. Jiang, N.-C. Yeh, T. A. Tomgrello, et al., J. Phys. Condens. Mat. 9,
8085 (1997).

A. M. Petrean, L. M. Paulius, W. K. Kwok, et al., accepted by Phys. Rev.
Lett., in press.

C. Reichhardt, A. van Otterlo, and G. T. Zimanyi, Phys. Rev. Lett. 84,
1994(2000)

157

85

86

87

88

89

90

91

92

93

95

96

97

L. Civale, Supercond. Sci. Technol. 1 0, A1 1 (1997).

M. Konczykowski, F. Rullier-Albenque, E. R. Yacoby, et al., Phys. Rev. B
44, 7167 (1991).

D. R. Nelson and V. M. Wnokur, Phys. Rev. Lett. 68, 2398 (1992).

D. R. Nelson and V. M. Vlnokur, Phys. Rev. B 48, 13060 (1993).

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena
(Oxford Unversity Press, New York, 1971).

H. E. Stanley, Rev. Mod. Phys. 71, 358 (1999).

M. P. A. Fisher, P. B. Weichman, B. Grinstein, et al., Phys. Rev. B 40, 546
(1989)

M. Wallin, E. S. Serensen, S. M. Girvin, et al., Phys. Rev. B 49, 12115
(1994).

J. Lidmar and M. Wallin, Europhys. Lett. 47, 494-500 (1999).

T. Hwa, D. R. Nelson, and V. M. VII‘IOKUI’, Phys. Rev. B 48, 1167 (1993).

Z. Z. Wang, J. Clayhold, and N. P. Ong, Phys. Rev. B 36, 7222 (1987).

M.-H. Whangbo and C. C. Torardi, Science 249, 1143 (1990).

J. Rossat-Mig nod et al., in Frontiers in Solid State Sciences, edited by L.
C. Gupta and M. S. Mutani (World Scientific, Singapore, 1993), Vol. 1.

G. Ceder, R. McCormack, and D. Fontaine, Phys. Rev. B 44, 2377 (1991).

158

98

99

100

101

102

103

104

105

106

107

108

109

110

111

T. B. Lindemer, J. F. Hurley, J. E. Gates, at al., J. Am. Ceram. Soc. 72,
1175(1989)

L. H. Greene and B. G. Bagley, in Physical Properties of High
Temperature Superconductors, edited by D. M. Ginsberg (World Scientific,
Singapore, 1990), Vol. 2.

T. Siegrist, S. Sunshine et al., Phys. Rev. B 35, 7137 (1987).

W

F. Beech, S. Miraglia, A. Santoro, et al., Phys. Rev. B 35, 8778 (1987).

\—’~."'A

J. Jorgensen, B. W. Veal et al., Phys. Rev. B 41, 1863 (1990). 9" .

D. L. Kaiser, F. Holtzberg, M. F. Chisholm, et al., Journal of Crystal
Growth 85, 593 (1987).

H. Zheng, M. Jiang, B. W. Veal, et al., Physica C 301, 147 (1998).
U. Welp, M. Grimsditch, H. You, at al., Physica C 161, 1 (1989).
H. Schmid, E. Burkhardt, B. N. Sun, et al., Physica C 157, 555 (1989).

U. Welp, W. K. Kwok, G. W. Crabtree, et al., Phys. Rev. Lett. 62, 1908
(1989)

L. Civale, A. Manrvick et al., Phys. Rev. Lett. 67, 648 (1991).

M. Konczykowski, V. M. Vinokur, F. Rullier—Albenque, et al., Physical
Review B 47, 5531 (1993).

T. K. Worthington, M. P. A. Fisher, D. A. Huse, et al., Phys. Rev. B 46,
11854(1992)

W. Jiang, N.-C. Yeh, D. S. Reed, etal., Phys. Rev. Lett. 72, 550 (1994).

159

 

112

113

114

115

116

117

118

119

120

121

122

123

124

125

J. F. Ziegler, J. P. Biersack, and U. Littlemark, The Stopping and Range of
Ions in Solids (Pergamon, New York, 1985).

J. F. Ziegler, Ion Implantation Technology (North-Holland, Amsterdam,
1 992).

J. P. Biersack and J. F. Ziegler, http://www.research.ibm.comlionbeams .

R. L. Fleischer, P. B. Price, and R. M. Walker, Journal of Applied Physics
36, 3645 (1965).

J. H. O. Varley, Journal of Nuclear Energy 1, 130-143 (1954).

S. A. Durrani and R. K. Bull, Solid State Track Detection (Pergamon, New
York, 1987).

G. Szenes, Phys. Rev. B 54, 12458 (1996).

B. Hensel, B. Roas, S. Henke, et al., Phys. Rev. B 42, 4135 (1990).

V. Hardy, C. Groult, M. Hervieu, et al., Nuclear Instruments and Methods
in Physics Research 854, 472 (1991).

R. Wheeler, M. A. Kirk, R. Brown, et al., in Materials Research Society
Symposium Proceedings, 1992), p. 683.

Y. Yan and M. A. Kirk, Phys. Rev. B 57, 6152 (1998).

G. Schiwietz, in NA TO ASI Series B, edited by R. A. Baragiola (Plenum
Press, New York, 1993), Vol. 306, p. 197-214.

Shima and et al., Nucl. Inst. Math. 200, 605 (1982).

R. Wheeler and M. A. Kirk, unpublished.

160

 

126 A. D. Marwick, L. Civale, L. Krusin-Elbaum, et al., Nucl. Inst. Meth. Phys.
Res. B 80l81, 1143 (1993).

127 L. Krusin-Elbaum, L. Civale, G. Blatter, et al., Physical Review Letters 72,
1914(1994)

128 Y. Yan, R. A. Doyle, A. M. Campbell, et al., Phil. Mag. Lett. 73, 299
(1996).

129 Y. Yan and M. A. Kirk, Phil. Mag. Lett. 79, 841 (1999).
130 M. A. Kirk and Y. Yan, Micron 30, 507 (1999).

131 H. Safar, P. L. Gammel, D. A. Huse, et al., Phys. Rev. Lett. 69, 824
(1992)

132 P. L. Gammel, L. F. Schneemeyer, and D. J. Bishop, Physical Review
Letters 66, 953 (1991).

133 w. Jiang, N.-C. Yeh, D. 5. Reed, et al., Physical Review B 47, 8308
(1993)

134 S. A. Grigera, E. Morré, E. Osquiguil, et al., Phys. Rev. Lett. 81, 2348
(1998).

135 U. Welp, T. Gardiner, D. Gunter, et al., Physica C 235-240, 241 (1994).
136 M. Oussena, P. A. J. d. Groot etal., Phys. Rev. B 51, 1389 (1995).

137 A. W. Smith, H. M. Jaeger, T. F. Rosenbaum, et al., Phys. Rev. B 59,
11665 (1999).

138 A. l. Larkin and V. M. Vinokur, Phys. Rev. Lett. 75, 4666 (1995).

161

139

140

141

142

143

144

145

146

147

148

149

150

151

L. Radzihovsky, Phys. Rev. Lett. 74, 4923 (1995).

A. W. Smith, J. M. Jaeger, T. F. Rosenbaum, et al., to be published
(2000).

W. K. Kwok, J. A. Fendrich, S. Fleshler, et al., in Proceedings of the 7th
lntemational Workshop on Critical Currents in Superconductors, Alpach,
Austria, 1994).

H. Klipfer, T. Wolf, C. Lessing, et al., Phys. Rev. B 58, 2886 (1998).
T. Nishizaki, T. Naito, S. Okayasu, et al., Phys. Rev. B 61, 3649 (2000).

S. Kokkaliaris, A. A. Zhukov, P. A. J. d. Groot, et al., Phys. Rev. B 61,
3655 (2000).

T. Nishizaki and N. Kobayashi, Supercon. Sci. Technol. 13, 1 (2000).

A. A. Zhukov, S. Kokkaliaris, P. A. J. d. Groot, et al., Phys. Rev. B 61, 886
(2000).

W. K. Kwok, R. J. Olsson, G. Karapetrov, et al., Phys. Rev. Lett. 84, 3706
(2000).

L. M. Paulius, W. K. Kwok, R. J. Olsson, et al., Phys. Rev. B 61, in press.
M. C. Marchetti and D. R. Nelson, Physica 174, 40 (1991).

W. K. Kwok, L. Paulius, D. Lopez, et al., Materials Science Forum 315-
317, 314 (1999).

J. Giapintzakis, W. C. Lee, J. P. Rice, et al., Phys. Rev. B 45, 10677
(1992)

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