PhD Thesis Trap

Katholieke Universiteit Leuven 1425

FACULTEIT WETENSCHAPPEN DEPARTEMENT NATUURKUNDE

Transport Properties of Underdoped Cuprates

in High Magnetic Fields Promotoren: Prof. Dr. V.V. Moshchalkov Dr. J. Vanacken

Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen door Lieven Trappeniers

2000

Dit werk is tot stand gekomen dankzij de intensieve samenwerking met een

aantal collega's die ik dan ook expliciet wens te bedanken. In de eerste plaats

mijn promotoren Victor Moshchalkov en Johan Vanacken. Victor, bedankt

voor de frisse ideeën, de hulp wanneer de modellen me boven het hoofd

dreigden te groeien en het nakijken van dit manuscript.

De weg die Johan en ik aflegden om supergeleiders te bestuderen in pulsvelden

was lang en voerde ons van het opbouwen van de apparatuur via

resistiviteitsmetingen op supergeleiders naar magnetisatiemetingen. Na een

aantal jaar zetten we terug de stap naar transportmetingen, wat uiteindelijk

resulteerde in deze thesis. Onderweg werden we meer vrienden dan collega's.

Bedankt Johan voor de motiverende samenwerking en het luisterend oor.

De vier cryostaten, vijf magneten en een 8 tal meetprobes die we onderweg

nodig hadden werden voor een groot stuk mogelijk dankzij de hulp van Luc

Grammet, Freddy Gentens, Johan Morren, Philippe Mispelter en Jef Haesaerts.

Jean-Pierre Locquet (IBM), bedankt voor de samples en de interessante

discussies.

De leden en ex-leden van het supergeleiders-team: Johan, Patrick, Gerd, Kris en

Liesbet zijn bedankt voor de goede sfeer en de hulp. Met natuurlijk een

speciale vermelding voor Johan en Gerd die me in de laatste rush om

experimentele data bijstonden om "in ploegen" te meten. Bedankt Patrick, voor

de samples, het nalezen van de thesis en de gesprekjes tussendoor. Bedankt ook

de goede collega's die ik onderweg mocht tegenkomen: Igne, Paul, "Mr. Li",

Alexei, Alexander, Willy, Fritz, Tony en Manus.

Ik ben ook Professor Yvan Bruynseraede erkentelijk voor de financiële steun

tijdens het moeizame begin (en einde) van mijn IWT mandaat en voor de

mogelijkheid om elk jaar een prima conferentie te kunnen bijwonen. Uiteraard

ben ik ook het IWT erkentelijk voor de vier jaar financiële steun.

Mijn dank gaat ook uit naar mijn vrienden en familieleden die gedurende de

voorbije jaren interesse toonden in mijn werk. Bedankt moeke, vake en Koen,

voor de steun.

Ik apprecieer uiteraard ook de uitzonderlijke steun die ik krijg van Els en ik

besef ten volle dat het niet evident is om in het labo mee pulsvelden-

experimenten te komen uitvoeren de avond voor je bevalling. Ik denk niet dat

veel vrouwen "zo zot" zouden zijn.

Bedankt ook Hanne, om mij te doen beseffen dat de moeilijkheden bij het

krijgen van nieuwe tandjes en het zetten van je eerste stapjes minstens even

belangrijk zijn als de thesis-perikelen van je vake.

Introduction

The origin of high-temperature superconductivity in cuprate

materials is one of the biggest puzzles in physics,

but the behaviour of these materials when they are not

superconducting is an even bigger mystery. [Batlogg2000]

The discovery of superconductivity in 1986 [Bednorz86] in a layered copper-

oxide compound with a critical temperature Tc above 30 K was the beginning of

new area of research: high-Tc superconductivity. After this initial discovery, a

whole family of high-Tc's was discovered, all having a layered structure with

CuO2 planes and intermediate building blocks. Generally, it is accepted that the

CuO2 planes are important for the superconducting properties while the

intermediate blocks act as charge reservoirs.

The temperature versus doping T-p phase diagram of the high-Tc cuprates

contains a rich variety of phase lines and crossover lines (figure 1 below).

Apart from the superconducting phase, showing up at low temperatures in a

certain range of doping, these copper oxides are antiferromagnetic (AF) Mott

insulators at low doping. At a certain level of hole doping, the long-range AF

correlations of the Cu2+ spins in the CuO2 planes are destroyed, leaving only

short range AF correlations in the material. Experimental evidence on

underdoped cuprates indicates (i) the existence of short range

antiferromagnetic fluctuations [Shirane87, Birgenau88, Rossat91, 93 & 94],

(ii) the opening of a pseudo-gap in the density of states [Ong96, Hanaguri99,

Timusk99] at temperatures far above the superconducting critical temperature

Tc and (iii) the formation of stripes in the CuO2 planes (charge stripes

intercalated by a hole free antiferromagnetic Mott insulator) [Thurston89,

Cheong91, Mason92 & 94, Yamada97, Tranquada97 & 97b, Hunt99, Aeppli97,

Dai98, Kao99, Arai99].

The occurrence of superconductivity in materials showing such exotic magnetic

and electronic properties is still the subject of intense scientific debate.

INTRODUCTION

Moreover, any model explaining the normal-state properties or the possibility of

superconductivity must account for these recent observations. The

understanding of mechanisms responsible for the appearance of these normal-

state properties may provide key information about the nature of the

superconducting mechanism itself.

This conclusion has motivated scientists to study the normal-state properties

extensively and the recent observation of the pseudo-gap and the possibility of

the presence of charge stripes can be regarded as a consequence of this renewed

interest.

Figure 1: The properties of the high-Tc cuprates vary with temperature (vertical axis) and doping of the CuO2 planes [Batlogg2000].

One physical property that has attracted much attention is the normal-state charge transport, both the resistivity and the Hall-effect. The temperature

dependence of the zero-field resistivity ρ(T) for the optimally doped cuprates

was shown to exhibit a robust linearity down to the critical temperature. When

these compounds are underdoped, a remarkable super-linear ρ(T) curve

INTRODUCTION

develops at intermediate temperatures while at higher temperatures (T > T*) the

linear ρ(T) dependence persists. It was demonstrated that, even in rather

strongly underdoped samples, showing an insulator-like ρ(T) behaviour at low

temperatures, the occurrence of superconductivity is possible. The Hall-

coefficient was found to be dependent upon doping and temperature. A

systematic study of the zero-field normal-state transport properties of

underdoped YBa2Cu3Ox thin films at T > Tc, revealed that for all samples both

ρ(T) and the Hall-data can be scaled using the same scaling temperature

[Wuyts94 & 96]. Since these two properties are closely related to scattering of

charge carriers, this suggest that scattering in these materials has a common

origin and that only the energy scale differs.

Although widely studied, the transport properties of the underdoped cuprates

still retain some mysterious features that remain to be solved. What are the

microscopic scattering mechanisms responsible for the robust linear ρ(T)

behaviour ? What is the origin of the S-shaped super-linear ρ(T) curve in

underdoped samples ? Can it be related to the opening of a pseudo (spin) gap or

the occurrence of charge stripes intercalating hole-free AF regions ? What is

the influence of hole doping on these features ? It is thus clear that the

completion of the temperature versus doping T-p phase diagram needs more

experimental investigations to be carried out.

Oddly enough, the study of the normal ground-state properties is hindered by the superconducting phase itself, since the normal-state at T < Tc is hidden

behind the zero sample resistance in the superconducting state !

Therefore, inspired by earlier work on La2-xSrxCuO4 and Bi2Sr2CuOy [Ando95,

96, 96b & 96c], in this work a dual-track approach is chosen. In a first stage,

underdoped cuprates with varying levels of doping were prepared:

YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox epitaxial thin films [Wagner99] and

strained La1.9Sr0.1CuO4 epitaxial thin films [Locquet98]. Doing so, the whole

underdoped region of the phase diagram can be covered while the reduced

critical temperatures allow easier access to the normal-state transport properties.

In a second stage, in order to gain access to the normal-state properties below

Tc, the transport measurements were carried out in pulsed high magnetic fields up to 50 T to suppress superconductivity in a reversible way.

INTRODUCTION

Chapter 1 reviews the most important physical properties of the cuprate

superconductors, which are essential to understand the discussion made in this

work. Furthermore, it gives a review of the state-of-the-art knowledge of the

high-temperature superconductors.

In chapter 2, the employed experimental techniques and their modifications

carried out during this work are reviewed. The experimental setup and

procedure for measuring the magnetoresistivity and Hall-effect in pulsed

magnetic fields are discussed.

Chapter 3 reports our measurements of the zero-field and high-field resistivity and magnetoresistance of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox

epitaxial thin films [Wagner99] and strained La1.9Sr0.1CuO4 epitaxial thin films

[Locquet98]. It will be checked whether the scaling, reported for the normal

state properties above Tc [Wuyts94 & 96], also holds for the high field transport

data. By measuring the resistivity of these cuprates at very high magnetic

fields, the normal state ρ(T) curve will be constructed, even at T < Tc. Doing so,

a statement about whether the ground state (i.e. in the absence of superconductivity) is metallic or insulating will be made. These findings will

be confronted with available models for charge transport and superconductivity

in the CuO2 planes and an experimental T(x) phase diagram for YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox will be constructed.

In chapter 4, the findings of chapter 3 will be further substantiated with the

results of high-field Hall-effect measurements on the same thin films. Also

for the Hall-effect, the use of very high magnetic fields is essential in order to

suppress superconductivity. We will report Hall-effect measurements on

YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films at temperatures extending to below the critical temperature Tc. From the measurements of the high-field Hall-

resistivity ρyx(H), the Hall coefficient RH(T) at fixed field will be calculated.

The combination of these RH(T) and ρab(T) curves then allows the derivation of

the Hall-angle. Finally, the carrier density nH that can be extracted from our

Hall-data will enable us to construct a generic T(p) phase diagram for the

YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds.

INTRODUCTION

The data discussed in chapters 3 and 4 will be used in chapter 5 to investigate

the effect of the short-range antiferromagnetic fluctuations and possible stripe formation in the CuO2 plane on the normal state transport properties. In this

charge-stripe picture [Emery97b & 99b], dynamic metallic [Ichikawa99,

Noda99, Tajima99] stripes are thought to dominate the transport properties. To

check this idea, an existing model [Moshchalkov93, 98b], describing transport

both in the 2D Heisenberg regime (above T*) as in the 1D striped regime (below

T*) where the pseudo gap develops, will be used as a framework for the

interpretation of our high-field normal-state transport data. Finally, a generic temperature versus hole doping T(p) phase diagram will be constructed.

Contents Chapter 1 Cuprate Superconductors ....................................................1

1.1 Introduction ......................................................................................................1 1.2 Structural properties .........................................................................................1 1.3 Doping of high-temperature superconductors ..................................................4 1.4 Evolution of physical properties with doping ................................................12 1.5 Generic T(p) phase diagram ...........................................................................26

Chapter 2 Experimenting in pulsed high magnetic fields..................29 2.1 Generation of pulsed magnetic fields .............................................................29 2.2 Cryogenics .....................................................................................................31 2.3 Transport measurements ................................................................................33

Chapter 3 Normal-state resistivity of YBa2Cu3Ox, (Y1-yPry)Ba2Cu3Ox and (La1.9Sr0.1)CuO4 ...........................................................43

3.1 Introduction ....................................................................................................43 3.2 Zero-field resistivity at T > Tc ........................................................................46 3.3 Suppression of superconductivity by high magnetic fields ............................59 3.4 Normal-state resistivity at T < Tc in high magnetic fields ..............................68 3.5 Comparison with the La2-xSrxCuO4 system ....................................................81 3.6 Localisation effects at T → 0 in the YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and

La1.9Sr0.1CuO4 samples...................................................................................85 3.7 Conclusions ....................................................................................................94

Chapter 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films ...................................................................................97

4.1 Introduction ....................................................................................................97 4.2 Hall-effect in the normal state below Tc .........................................................99 4.3 Phase diagram ..............................................................................................113 4.4 Conclusions ..................................................................................................115

Chapter 5 Effect of stripe formation on the transport properties of underdoped cuprates........................................................119

5.1 Charge ordering revisited - Stripes...............................................................120 5.2 Spin ladders, a magnetic structure between 1D & 2D..................................123 5.3 Quantum transport in doped 1D and 2D Heisenberg systems......................125 5.4 Stripe ordering at low temperatures .............................................................139 5.5 Construction of the T(p) phase diagram .......................................................143 5.6 Conclusions ..................................................................................................146

Summary .................................................................................151 Appendices .................................................................................161 Bibliography .................................................................................173 Nederlandstalige samenvatting ..........................................................185 Publicatielijst .................................................................................195

Symbols

Crystallographic a, b, c Crystallographic axes / lattice parameters

d Separation of the CuO2 planes

P, Px Hydrostatic and uniaxial pressure (x = a, b or c)

εx Strain (x = a, b or c)

Temperatures and energy scales Tc, Tc,onset, Tc,mid, Tc,offset critical temperature, at the onset, middle and offset of

the transition

TN Neèl temperature for an antiferromagnet

TRVB, TB Spinon and holon pairing temperatures in the RVB model

To Temperature below which local AF correlations appear

T* Temperature below which the spin-gap opens

TMI Boundary between the metallic and insulating ρ(T)

θD Debye temperature for the phonon spectrum

∆, ∆s, ∆p, ∆SC Gap, spin-gap, pseudo-gap, superconducting gap

Ec1, Ec2 Mobility edges

EF Fermi energy

Electronic properties g(E), g(EF) Density of states (DOS), at the Fermi level

Cel, γ = Cel/T Electronic specific heat, -coefficient

n, nH Total density of charge carriers, determined from the Hall effect

p Hole concentration in the CuO2 plane

q Elementary charge, either positive or negative

µ, µH Mobility of the charge carriers, determined by the Hall-effect

Magnetic properties

J Exchange coupling

t Hopping amplitude in the t-J model

1/(T1T) Normalised spin relaxation rate (NMR)

KS Knight shift

SYMBOLS

nc Number of chains constituting a spin-ladder

σ, σ1D, σ2D Conductivity, in 1 and 2 dimensions

H, µoH Applied magnetic field expressed in respectively Gauss & Tesla

B Magnetic induction

Electric properties J (or I), Ji Current, along the i-direction

j Current density

Ei Electric field along the i-direction

ρij Element i,j of the resistivity tensor

σij Element i,j of the conductivity tensor

ρo Residual resistivity

RH, θH Hall coefficient, Hall angle

R Sheet resistance, also called resistance per square

Length scales

eBm

h=λ Magnetic length

λ Penetration depth in a superconductor

ξ Coherence length in a superconductor

ξm, ξm1D, ξm2D Magnetic correlation length, in 1D and 2D

l Mean free path for scattering of charge carriers

Lφ Inelastic length for scattering of charge carriers

CHAPTER 1 Cuprate Superconductors

1

Chapter 1

Cuprate Superconductors

1.1 Introduction

The discovery of superconductivity in 1986 [Bednorz86] in a layered copper-oxide compound came as a great surprise, not only because of the record transition temperatures Tc, but also because these materials are relatively poor conductors in the "normal" (i.e. non-superconducting) state. Indeed, these superconductors are obtained by doping parent compounds that are antiferromagnetic (AF) Mott insulators, materials in which both the antiferromagnetism and the insulating behaviour are the result of strong electron-electron interactions. At a certain level of hole doping, long-range AF correlations of the Cu2+ spins are destroyed, leaving only short range AF correlations in the material. Even now, almost fifteen years after the start of cuprate-superconductivity research, the influence of the AF correlations and the gradual doping with holes on the transport properties and the occurrence of superconductivity remains a hot topic of research.

1.2 Structural properties

All cuprate superconductors known up to now are layered perovskites

containing CuO2 layers alternated by intermediate building blocks.

Superconductivity takes place in the doped CuO2 layers while the other layers

act as charge reservoirs. One possible way of classifying these cuprates is by

the number of CuO2 planes and the nature of the intermediate building blocks.

The La2-xAxCuO4 type of cuprate superconductors (with A being Ba, Sr,

Ca, …), also known as the "2-1-4" group, contains 2 crystallographic CuO2


2

planes per unit cell, sandwiched with La-O layers. In this system, doping is

done by partly substituting La3+ by A2+ in the La-O layers, leading to a

maximum bulk Tc of 40 K. The RBa2Cu3Ox group (R being a rare-earth Y, Gd,

Eu, Tm, …), also known as the "1-2-3" group, also contains two CuO2 planes

per crystallographic unit cell, alternated with building blocks of Ba-O, R and

CuO-chain layers. In this system, doping is mostly done by changing the

oxygen content in the Cu-O layers, leading to a maximum Tc of 92 K at the

optimal doping of x = 6.95. The (Bi, Tl, Hg)mM2Can-1CunOm+2n+2 materials (M

being Ba or Sr), with m = 1 or 2 and n = 1, 2 or 3 can reach Tc values of about

133 K. The most common members of this family are "2-2-0-1" (containing 1

CuO2 layer), "2-2-1-2" (with 2 CuO2 layers) and "2-2-2-3" (having 3 CuO2

layers), but the highest Tc of 133 K is reached in HgBa2Ca2Cu3O8+δ (1-2-2-3),

without any external pressure. For obvious reasons they are sometimes called

telephone-book compounds. Here, the block layers responsible for doping are

the (Bi, Tl or Hg)-O layers. This work will focus on the YBa2Cu3Ox and

La2-xSrxCuO4 cuprate superconductors.

YBa2Cu3O7

CuO

BaO

CuO2

Y

CuO2

BaO

CuO

O(1)

Cu(1)

O(5)

O(4)

O(3)

Cu(2)

O(2)

BaCuO2

YCuO2

BaCuO3

La2-xSrxCuO4

CuO2

LaO

LaO

CuO2

LaO

LaO

CuO2

Y

Ba/La/Sr

O

Cu

O vacancy

ab

c

YBa2Cu3O7

CuO

BaO

CuO2

Y

CuO2

BaO

CuO

O(1)

Cu(1)

O(5)

O(4)

O(3)

Cu(2)

O(2)

BaCuO2

YCuO2

BaCuO3

La2-xSrxCuO4

CuO2

LaO

LaO

CuO2

LaO

LaO

CuO2

La2-xSrxCuO4

CuO2

LaO

LaO

CuO2

LaO

LaO

CuO2

Y

Ba/La/Sr

O

Cu

O vacancy

ab

c

Figure 1.1: Unit cell of YBa2Cu3O7 (left) and La2-xSrxCuO4 (right) with the identification of the non-equivalent crystallographic sites for oxygen and copper atoms in the case of YBa2Cu3O7.

Neutron [Jorgensen87, 88 & 90, Hinks87] and X-ray [Schuller87, Grant87]

diffraction measurements revealed the unit cell of the YBa2Cu3Ox system to be


3

built of three stacked perovskite blocks stacked on top of each other (BaCuO3,

YCuO2 and BaCuO2) as shown in Figure 1.1. The central Y atom is

sandwiched between two CuO2 planes. These planes are separated from the top

and bottom (basal) CuOy layers by a BaO block. In the unit cell, the Cu-atoms

in the basal CuOy layers are named Cu(1) while the Cu in the CuO2 layers is

denoted as Cu(2). Five non equivalent positions could be identified

[Jorgensen87] for the oxygen atoms. The oxygen atoms in the CuO2 planes are

referred to as O(2) and O(3), whereas the O atoms in the BaO layer are labelled

O(4). In the basal CuOy plane, two positions were found: O(1) along the b-axis

and O(5) along the a-axis. The equivalence of these two positions in the

tetragonal phase is destroyed in the orthorhombic case where oxygen atoms are

preferably located in the O(1) positions. The basal CuOy layers have therefore a

partial filling with oxygen and the total number of oxygen atoms in a unit cell

varies from 6 to 7. For x = 6, the basal planes contain no oxygen and the

structure is tetragonal. For higher oxygen contents, the basal plane is gradually

filled with oxygen, resulting in a structural phase transition from tetragonal to

orthorhombic at x ~ 6.4. Fully oxygenated compounds with x = 7 have an

orthorhombic unit cell and the material is metallic. The onset of metallic

behaviour coincides approximately with the concentration corresponding to the

structural phase transition. The evolution of the lattice parameters of bulk

YBa2Cu3Ox with varying oxygen content is summarised in figure 1.2. Because

of the presence of ab-twinning in this compound, only high resolution neutron

or X-ray techniques or the use of untwinned single crystals yield accurate

estimates for the lattice parameters.

The La2-xSrxCuO4 system was also shown [Radaelli94, Tarascon87, Takagi87

& 89, Fleming87] to have a layered structure composed of two CuO2

crystallographic layers per unit cell (figure 1.1), in this case alternated by La-O

layers. However, in this system the layers are not aligned in the c-direction as

observed in the "123" system. Here, the O atoms of successive CuO2 layers are

stacked above each other whereas the Cu atoms alternate between (0,0,0) and

(1/2,1/2,1/2) sites in adjacent layers. Moreover, although from a

crystallographers point of view there are two CuO2 layers in a unit cell, they are

separated by two LaO layers and electronically, the La214 system is considered

being a 1-layer compound. No Cu-O chains are present in this system. Doping


4

is done by partly substituting the trivalent La by the bivalent Sr (x ranging from

0 to 0.4), resulting in a structural phase transition from orthorhombic (at x = 0)

to tetragonal (at x = 0.4) symmetry. The evolution of the lattice parameters of

bulk La2-xSrxCuO4 with changing Sr content is summarised in figure 1.2.

3.75

3.80

3.85

13.013.113.213.313.413.513.6

3.75

3.80

3.85

11.5

11.6

11.7

11.811.9

12.0

0.0 0.1 0.2 0.3 0.46.0 6.2 6.4 6.6 6.8 7.0x x

a

b

c

a

b

c

YBa2Cu3Ox La2-xSrxCuO4

a,b,

c (Å

)

RT

10 Kb

a

Figure 1.2: Lattice parameters for bulk YBa2Cu3Ox (left) [Jorgensen90] and bulk La2-xSrxCuO4 (right) [Takagi89, Radaelli94], at different doping levels, determined by neutron diffraction and X-ray diffraction. All data were taken at room-temperature, except the 10 K data for La2-xSrxCuO4.

1.3 Doping of high-temperature superconductors

The undoped parent compounds of the cuprate superconductors are

antiferromagnetic insulators. The introduction of holes in the CuO2 planes

transforms the 3d9 Cu2+ state (that is fourfold co-ordinated with the O(3) oxygen

atoms) into effective S = 0 Cu3+ sites (bound state of a S = 1/2 Cu2+ with a hole

residing mainly on the 4 surrounding O 2p orbitals) thus destroying long range

AF order and providing hole-type charge carriers which, for large hole

concentrations, are mobile in the plane (by transfer of electrons). The doping of

the CuO2 plane is generally realised in one of the following ways: by changing

the oxygen content, by chemical substitution of the cations or by the application

of strain.


5

1.3.1 Doping by varying the oxygen content

This method of doping is mainly used in the YBa2Cu3Ox system where the

oxygen content can be systematically changed from x = 6 to x = 7, resulting in a

change from an AF insulator to an underdoped superconductor at x ~ 6.3 and

gradually approaching the optimally doped Tc = 92 K case at x = 6.95. The

oxygen content of the YBa2Cu3Ox samples is set to the desired value starting

from a sample with optimal oxygen content. This sample is then kept at fixed

temperature and oxygen pressure as to yield the desired oxygen content. After

that, a slow cooldown at controlled temperature and oxygen pressure is

initiated, carefully guided by a "constant oxygen-content line" in the p-T phase

diagram [Tetenbaum89]. Details of the technical realisation of the different

oxygen contents are described in appendix B.

As introduced above, the variable part in the oxygen content in the YBCO unit

cell is entirely located in the basal CuOy plane (CuO chains). At x = 6, no

oxygen is present in the basal planes, the Cu(1) atoms have a twofold co-

ordination (by O(4)) and they are in the Cu1+ state [Rushan90]. When oxygen is

added to this insulating YBa2Cu3O6 compound, the O-atoms have an equal

probability to locate themselves in the O(1) or the O(5) positions. At x ~ 6.4

however, a structural phase transition to the orthorhombic unit cell is induced

by the addition of extra oxygen and the added oxygen is located preferably in

the O(1) sites, thus forming Cu-O-Cu chain fragments along the

crystallographic b-axis.

A simplified picture of the creation of the Cu-O fragments can be described as

follows: when a neutral O-atom is introduced into the basal plane, it needs to

extract two electrons from the surrounding structure to convert into O2-. These

electrons are provided by the 2 adjacent Cu1+ ions by transforming into 3-fold

coordinated Cu2+ [Rushan90] (Cu1+-.-Cu1+ + O → Cu2+-O2--Cu2+). No holes are

introduced in the structure, so far, by creating such Cu-O-Cu fragments. At

higher doping levels, the Cu-O-Cu fragments are lengthened by adding extra O-

atoms. When binding to the added O-atom, the 3-fold coordinated Cu2+

transforms in a 4-fold coordinated Cu2+ and one hole is created in order to

provide the second electron for the O2- and thus keep the charge balance

constant (Cu2+-O2--Cu2+-.-Cu1+ + O → Cu2+-O2--Cu2+-O2--Cu2+ + hole). If the


6

added O-atom connects two Cu-O-Cu-… fragments, 2 holes are created because

the Cu-atoms already have a +2 valence (…-Cu2+-O2--Cu2+-.-Cu2+-O2--Cu2+-… +

O → …-Cu2+-O2--Cu2+-O2--Cu2+-O2--Cu2+-… + 2 holes) [Rushan90, Veal91].

The precise amount of holes created during the filling of the basal plane with

oxygen is thus strongly dependent upon the distribution of the O-atom over the

three different processes described above. At low doping levels (6.0 < x

< 6.25), the introduced O-atoms mostly create Cu-O-Cu monomers in the CuOy

basal planes (both along the a and the b axis) and almost no holes are produced.

At intermediate oxygen contents (6.25 < x < 6.8), the oxygen atoms start to

enlarge the length of the …-Cu-O-Cu-… fragments (preferably along the b-

axis) thus reducing the number of monovalent Cu(1) atoms and creating holes.

6.0 6.2 6.4 6.6 6.8 7.00.0

0.2

0.4

0.6

0.8

1.0

hole

s / u

nit c

ell

x

YBa2Cu3Ox

Figure 1.3: The number of holes per unit cell in the YBa2Cu3Ox system [Veal91].

At near optimum doping (6.5 < x ≤ 6.95), isolated oxygen vacancies are further

filled, releasing additional holes. As oxygen is added, the decreasing

probability of O(5) occupation leads to the formation of distinct equilibrium

phases for the oxygen ordering in the basal CuO plane. At low oxygen

concentrations and moderate temperatures, an imperfect tetragonal oxygen

ordering is present, while at higher concentrations orthorhombic oxygen

ordering is established by the formation of Cu-O chains along the b-axis with a

superstructure period of one (ortho I phase), two (ortho II phase) or more a-axis


7

lattice spacings, as observed by electron microscopy [Cava90, Reyes89,

Andersen99]. Based on structural calculations [deFontaine90] and assuming a

double-cell superstructure (ortho II) of the Cu-O chains at x = 6.5, the (non-

monotonic) increase of the number of holes per unit cell by the addition of

oxygen was calculated, as shown in figure 1.3 [Veal91].

The holes that are thus created in the basal CuOy planes ( by adding oxygen and

changing the valence of Cu(1)) are moved to the CuO2 planes by the transfer of

electrons (charge transfer model) to the Cu-O chains [Cava90]. The mean

valence of the Cu(2) atoms is thus increased above 2, by creating a finite

fraction of Cu ions with a formal valence of 3+ [Cava90, Brown90].

1.3.2 Doping by chemical substitution of cations

Changing the hole concentration in the CuO2 planes can also be realised by

performing a chemical substitution of the cations Y3+ and La3+ in the

YBa2Cu3Ox ("123") or La2CuO4 ("214") unit cell. If we take a closer look at

these unit cells (figure 1.1) , we can choose between substitutions in the CuO2

planes, the CuO chains (in YBa2Cu3Ox) or at the La, Ba or the Y sites.

Depending on the nature of the substituting atom and its position in the unit cell

the influence of the chemical substitution on the normal-state and

superconducting properties will differ substantially.

In the YBa2Cu3Ox system, chemical doping is generally done by substituting

Y3+ by Pr4+,3+,.. or Ca2+ or Cu by a metal like Co, Fe, Al or Zn. Co, Fe and Al

will be mainly located at the basal Cu(1) locations (the chains) while (for small

concentrations) Zn is known to locate itself on the Cu(2) sites in the CuO2 plane

[Walker95, Tarascon88, Xiao88] giving rise to a rapid reduction of Tc.

Substitution of the trivalent Y by the bivalent Ca is known to introduce extra

holes, resulting in overdoping [Neumeier89]. An interesting case, however, is

the Y1-yPry substitution, being an exception to the general observation that

almost all members of the RBa2Cu3O7 family exhibit about the same critical

temperature Tc ~ 90 K; regardless of the type of rare-earth ion R that is used.

Y1-yPryBa2Cu3Ox is widely reported to show a gradual deterioration of its

superconducting and normal state properties as the Pr content y is increased,

becoming an insulator at y > 0.55 [Dalichaouch88, Neumeier89, Xu92,


8

François96, Jiang97, Tang99]. To complicate things even further,

superconductivity has been predicted [Blackstead95] and reported

[Blackstead96] in the PrBa2Cu3O7 compound; with Pr at the rare-earth position

between the CuO2 planes !

In order to elucidate the origin of these conflicting observations, we will

elaborate on the role of the location of the Pr ions in the YBa2Cu3O7 unit cell

for the mechanisms of destroying or enabling superconductivity.

Several reasons for the destruction of superconductivity by the presence of Pr-

atoms in the YBa2Cu3O7 unit cell are reported in literature. Often, hole filling

by the (supposed tetravalent) Pr-ion is used to explain the destruction of

superconductivity while others claim magnetic pair-breaking through an

exchange interaction between the spin of the charge carriers and the spin of the

paramagnetic Pr-ion to be important. Recent investigations [Tang99,

Dalichaouch88, Xu92, Neumeier89] on Y1-yPryBa2Cu3Ox, Y1-2yPryCayBa2Cu3Ox

and YBa2-yPryCu3O7 showed that Pr at Y sites suppresses superconductivity by

both hole filling and magnetic pair-breaking. The Y-Pr sheets then act as

"internal" charge reservoirs, inside the CuO2 bilayer, with the magnetic moment

of the paramagnetic Pr atom interacting with the carriers in the CuO2 planes.

When Pr is located at the Ba sites, only hole filling occurs, which is plausible

since the distance between the CuO2 planes and the Ba site is longer than to the

R-sites in between the planes and thus the magnetic interaction should be

weaker.

In general, it is observed that PrBa2Cu3O7 has a substantial solubility for Pr at

the Ba sites and the Pr/Ba occupancy strongly depends on the substrate

temperature during thin film deposition [Tang99, Blackstead95]. Thus, in many

cases, a finite Ba/Pr mixture in the Y1-yPryBa2Cu3Ox system can therefore not be

excluded. Blackstead and Dow attribute the often-claimed absence of

superconductivity in the PrBa2Cu3O7 compound to this finite Pr/Ba substitution

[Blackstead95 & 96]; a material with Pr only located at the Y sites, in between

the planes, should then be superconducting.

Apart from the possible hole filling and magnetic pair breaking effects, the

introduction of Pr in the unit cell, substituting Y and possibly also Ba, will


9

induce appreciable disorder in the crystal (the atomic radius of Pr lies in

between those of Y and Ba).

In the La2CuO4 system, chemical doping is generally achieved by partly

substituting the trivalent La3+ by the divalent Sr2+ (or Ba2+) thus obtaining

La2-xSrxCuO4; with x = 0 to 0.4. In the undoped case, the LaO plane has one

electron to donate to the surrounding structure since only two electrons of the

three, supplied in the formation of La3+, are required in the La3+O2- plane. The

two electrons, donated by two adjacent LaO layers, are used for the charge

balance of the intermediate Cu2+(O2-)2 plane. When a fraction of the La3+ is

substituted by Sr2+, a shortage of electrons is created. The resulting reduced

transfer of electrons to the CuO2 planes leads to the addition of holes into the

CuO2 plane by increasing the mean valence of copper from +2 (x = 0) to

Cu Cu12 3−+ +

x x (x > 0).

1.3.3 Influence of epitaxial strain

The copper oxide superconductors exhibit a remarkable range of pressure

dependencies of the superconducting transition temperature (dTc/dP). The

observed values for dTc/dP can be very high (up to +7 K/GPa), zero or negative,

depending on the doping level and the precise member of the cuprate family.

The pressure and strain dependencies of Tc contain both in-plane (ab) and out-

of-plane (c) contributions and can (for a tetragonal unit cell) be written as

equation 1.1 with ε = (dbulk - dstrained)/dbulk the strain.

cc

cab

ab

cc

c

c

ab

ccc

TTT

PP

TP

P

TTT

εε∂

∂ε

ε∂∂

∂∂

∂∂

++=

++=

2)0(

2)0(

(1.1)

Thus, the dTc/dP, as observed under hydrostatic conditions, is the result of a

subtle balance between the in-plane and out-of-plane derivatives, which often

have opposite signs [Locquet98 & 98b, Fietz 96]. Since the effect of applied

pressure on the superconducting properties depends on the crystal structure and

the pressure induced deformations, it does not come as a surprise that

YBa2Cu3O7 and La2-xSrxCuO4 show a very different dTc/dP behaviour. The


10

influence of applied pressure on the critical temperature Tc of the YBa2Cu3Ox

system, which by itself has a tetragonal to orthorhombic transition at x ~ 6.4, is

shown in figure 1.4. The interpretation of this plot is still the subject of

discussion and must account for oxygen ordering and pressure effects in the

CuO chains, charge redistribution in the CuO2 plane and for the fact that in the

orthorhombic phase there is an anisotropic dTc/dP for the a and b-axis [Fietz96,

Benischke92, Kraut93, Welp92, Welp94, Meingast91, Jorgensen90b, Pickett97

and 97b]. In any case, it is clear from figure 1.4 that the non-monotonic, and

qualitatively different behaviour of dTc/dP in the a and b directions makes

epitaxial ab strain not suitable to dope this material. Even a substrate like

SrLaAlO4, with a lattice parameter of approximately 3.76 Å at room

temperature (see also figure 1.5), will not necessarily yield a higher critical

temperature because of the compressive strain in both a and b directions.

6.5 6.6 6.7 6.8 6.9 7.0-4

-2

0

2

4

6

8

10

a

b

c

hydrostatic

YBa2Cu3Ox

x

dTc/

dP (

K/G

Pa)

Figure 1.4: The variation of dTc/dP versus doping level x for hydrostatic pressure [Beniscke92] and a, b and c-axis uniaxial pressure [Kraut93, Welp94]. All data were taken below 105 K to reduce the influence of oxygen ordering.

The bulk La2-xSrxCuO4 system however, has an orthorhombic structure over

almost the whole range of doping (x < 0.2); as long as the temperature is of the

order of Tc (see figures 1.2 and 1.10). As such, no structural phase transitions

will complicate dTc/dP, although the transition is close. Moreover, no CuO


11

chains are present in this compound, taking away the problem of oxygen

reordering and pressure effects in these chains. The problem of the different

dTc/dP response in a and b directions for the orthorhombic phase - for doping

by the application of epitaxial ab strain - was in the La2-xSrxCuO4 compound

successfully solved by the use of strained ultra-thin films [Locquet96, 98 & 98b,

Sato97]. It was shown [Locquet98b & 96b] that when the thickness of a thin-

film does not exceed 500 Å, the square symmetry of the unit cell of the

substrate can be imposed onto the film.

tensile

compressive

SrTiO3

SrLaAlO4

La1.9Sr0.1CuO4

3.76

3.78

3.80

3.82

3.92

3.94

0 200 400 600 800 1000

Temperature (ºC)

Lat

tice

para

met

er (

Å)

growthtemperature

Figure 1.5: Temperature dependence of the ab-plane lattice parameters of La1.9Sr0.1CuO4, SrTiO3 and SrLaAlO4.

Locquet and co-workers have prepared strained La1.9Sr0.1CuO4 ultra-thin films

of typical thickness of 100 Å by molecular beam epitaxy with block-by-block

deposition [Locquet94]. The choice of the substrate - SrLaAlO4 (SLAO) or

SrTiO3 (STO) - enabled them to induce compressive (SLAO) or tensile (STO)

strain in the ab-plane (see figure 1.5). This deformation - which essentially

keeps the volume of the unit cell constant - increases (compressive) or decreases

(tensile) the critical temperature significantly. A doubling of the critical

temperature has been achieved [Locquet98]. Although the precise mechanism

responsible for this strong enhancement of Tc is still under discussion, it was

suggested [Locquet98b] that a reduced charge transfer to the CuO2 planes


12

(which are more separated by the compressive ab-plane stress and volume

conservation) leads to a self-doping of the plane by an increased Cu valence.

However, tentative Hall measurements [Locquet00] seem to indicate that not

only the charge carrier density but also scattering effects might play an

important role for the critical temperature Tc. Moreover, the increased (or

decreased) orbital overlap between the CuO2-plane Cu and O atoms upon the

application of compressive (or tensile) strain might also play a relevant role that

goes beyond a simple doping picture.

1.4 Evolution of physical properties with doping

1.4.1 Electronic properties

Changing the carrier concentration in the copper oxide materials will

profoundly influence the normal state electronic properties and the occurrence

of the superconducting state.

The electronic properties of a superconductor - and more specifically the

electron density of states versus energy g(E) - can be looked upon in two ways:

the semiconductor view and the bosonic approach. In the semiconductor representation (figure 1.6, left), a superconductor below its critical temperature

Tc is considered to have for simple quasiparticles an energy gap of 2∆ between a

filled lower band and an upper band which is empty at T = 0. Excitations at

finite temperature can create quasi-particles by increasing the energy of an

electron with 2∆, thus creating a hole in the valence band. This single-electron

picture does not take into account electron pairing leading to the formation of

bosons (Cooper pairs). The bosonic representation (figure 1.6, right) includes

the bose condensation in such that at T = 0 a single energy level of paired

electrons (bosons) is separated from an empty band by a gap of energy ∆. At

finite temperatures, electron pairs can break up and go to the upper band, both

gaining an energy ∆ (since the total binding energy 2∆ is divided over the two

electrons).


13

E E

EF

EF+∆

EF+∆

T = 0 T = 0

Figure 1.6: Schematic view of the superconducting gap in the semiconductor representation (left) and the boson-condensation approach with Cooper pairs (right)

In order to gain knowledge about the normal-state electronic properties of the

copper-oxide superconductors and their evolution with doping, band structure

calculations [Pickett87 & 90] and extensive studies of the optical properties of

these materials have been performed [Yu93, Kircher91]. One can represent the

normal-state density of states (DOS) versus energy schematically for different

levels of doping (figure 1.7). The empty upper band (upper "Hubbard band") is formed by non-occupied Cu- 22 yx

3d−

states while the filled lower band is

created by a hybridisation of the O- yx,p2 band and the Cu- 22 yx3d

− lower

Hubbard band [Yu93]. The undoped system is a Mott insulator, a system with

all except one (which is half-filled) orbitals in the (Cu) 3d-shell filled, which is

insulating rather than metallic by virtue of the strong electron-electron

repulsion. Doping the material (adding holes to the CuO2 planes) can be

thought of as introducing holes into the valence band and thus lowering the

Fermi energy to within that band.

However, an alternative view [Moshchalkov88 & 90, Quitmann92], inspired by

the physics of lightly doped semiconductors, considers the dopants as impurities

that create an electron-acceptor level close to the valence band (figure 1.7a).

This impurity band grows as more (randomly distributed) dopant atoms are

added - the width being determined by the mean distance between the impurities


14

- and the Fermi-level enters the band (figure 1.7a). The electrons in this band

remain localised since the impurities are still at a large distance.

At intermediate doping levels, the wave functions of the impurities have an

increasing overlap and a narrow band with metallic conductivity appears

(figure 1.7b). At the same time, due to the disorder introduced by the

impurities, the tails of the band (below and above the two so-called mobility-

edges Ec1 and Ec2) correspond to the localised states [Anderson58] and only

hopping can take place. At higher doping levels, the Fermi-level EF crosses the

upper mobility edge Ec2 and an insulator to metal transition (IMT) sets in

(figure 1.7c). From that point on, the localised (hopping) tails coexist with -and

are shunted by- the central band in which the delocalised states show metallic

conduction.

occupied

localised

extendedE

g(E)

Ec1 Ec2

EF

Ec1 Ec2

EF

(c)

(b)

(a)

E

g(E)

E

g(E)

EF

dopi

ng

Figure 1.7: Schematic representation of the density of states g(E) of a high-temperature superconductor for light doping, in the insulating regime (a), intermediate doping (b) and heavy doping where metallic transport sets in .

From a certain level of hole doping on, a superconducting gap 2∆ between

single-electron bands opens around the Fermi energy when the material is

cooled below the critical temperature Tc (figure 1.6). As a result, the density of

states g(E) will increase at EF-∆ and EF+∆. This was microscopically described

by the famous theory of Bardeen, Cooper and Schrieffer (BCS [Bardeen57])

who considered a net attractive potential V appearing between the electrons due


15

to the electron phonon interaction. The critical temperature Tc and the gap ∆

both depend exponentially on the coupling constant V and the density of states

g(EF) at the Fermi-level (with ωD the Debye frequency for phonons), leading to

a constant ratio between ∆ and Tc (equation 1.2); when assuming a not too

strong coupling (the weak coupling limit).

53.3~)0(2

13.1

2

)(

1

)(

1

cBVEgDcB

VEgD

TkeTk

e

F

F ∆

=

=∆−

−

ω

ω

h

h (1.2)

This relation between the critical temperature Tc, the potential V and the DOS

g(EF), provides a framework for a possible explanation for the very high critical

temperatures observed in the novel cuprate superconductors. One way is to turn

to a very strong coupling V between the electrons while others focus on the

enhancement of g(EF) by postulating either novel coupling interactions (e.g.

bipolarons [Alexandrov88, Mott90]) or an inhomogeneous distribution of

charge carriers within the CuO2 planes [Bianconi96, 97 & 98, Valetta97]. Only

the latter scenario seems to escape the problem of the increasing ratio 2∆/kBTc

(beyond 3.53) and the onset of lattice instabilities as the coupling gets stronger.

The opening of the superconducting gap ∆ exactly at Tc was not observed in the

novel cuprate superconductors. In these compounds a partial (pseudo) gap was

found to open at temperatures far above the onset of the macroscopic

superconducting state as some kind of precursor. This pseudo-gap will be

discussed in more detail later on in this chapter.

For various high-Tc compounds, the evolution of the superconducting critical

temperature Tc with doping was shown to obey a universal curve for the

dependence on the hole concentration p in the CuO2 plane [Tallon90, 93 & 95].

This concentration (expressed as a fraction of holes per Cu atom in the plane) is

in La2-xSrxCuO4 directly given by the strontium content x whereas in the

YBa2Cu3Ox system it is the result of bond valence calculations. The universal

Tc(p) curve can be described by the parabolic-like empirical relation

Tc(p)/Tc,max=1 - 82.6 (p - 0.16)2 (figure 1.8).


16

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.0

0.2

0.4

0.6

0.8

1.0

1.2

Tc/

Tc,

max

p, hole concentration

YBa2Cu3Ox

Y1-yCayBa2Cu3O6.96

Y1-yCayBa2Cu3O6

Y0.9Ca0.1Ba2Cu3Ox

La2-xSrxCuO4

Figure 1.8: Reduced critical temperature versus hole concentration (fraction of holes per Cu-atom in the CuO2 plane) for different high-Tc compounds, after [Tallon95, Cava90].

1.4.2 Magnetic structure

During the past few years, the evolution of the magnetic structure of the high Tc

cuprates upon hole doping has been studied extensively using inelastic neutron scattering (INS) (YBa2Cu3Ox [Rossat91, 93 & 94, Tranquada97, Dai98,

Arai99, Kao99], La2-xSrxCuO4 [Shirane87, Birgenau88, Endoh88, Thurston89,

Cheong91, Mason92 & 94, Yamada97, Tranquada97]) and nuclear magnetic resonance (NMR) [Alloul89, Kitaoka91, Yasuoka94 & 97, Berthier97,

Carretta99]. It was found that, at low levels of doping, these oxides are 3D

antiferromagnets (AF) with a Neèl temperature of about 410 K for the undoped

YBa2Cu3O6 compound and TN ~ 250-300 K for La2CuO4. In the undoped

compounds, the CuO2 planes are built up from S = ½ 3d9 Cu2+ and S = 0 2p6 O2-

states (figure 1.9). This results in an AF ordering of the Cu spins, mediated by

the electrons forming the S = 0 2p O-orbital (super-exchange). At temperatures

TN < T < J ~ 1700 K, the 3D AF order is destroyed but 2D AF correlations

persist. The magnetic properties of the CuO2 planes can thus be described as a

2D Heisenberg system on a square lattice by the Hamiltonian

∑ ⋅=ji

ji SSJH,

rr (1.3)


17

with i and j labelling lattice sites, iSr

are spin ½ operators and <i,j> denotes

nearest neighbours sites. J (>0), the AF exchange coupling determining the

energy scale of the problem, was estimated to be JCu-Cu ~ 125-170 meV

(previous references and [Mizuno98]).

S=1/2 Cu2+

S=1/2 Cu2+

S=0 Cu3+

(formal)

AF

disturbed AF order

Slightlydoped

Undoped

Figure 1.9: Schematic view of the CuO2 plane and the influence of hole doping on the antiferromagnetic (AF) order in the plane.

When holes are introduced into de CuO2 plane, they go to the oxygen orbitals (the O yx,p2 band lies slightly above the Cu 22 yx

3d−

lower Hubbard band) and

a formal S = 0 Cu3+ state is formed (figure 1.9). This so-called Zhang-Rice

singlet [Zhang88, Tjeng97] is a bound state of a S=1/2 Cu2+ with a hole residing

mainly on the 4 surrounding O 2p orbitals. The introduction of these effective

S = 0 sites represents a significant disturbance of the AF ordering and as a

result, the Neèl temperature is drastically decreased upon doping (figure 1.10).

At a concentration of p ~ 0.02 holes/Cu-atom, the long-range 3D AF state is

completely suppressed; although short range AF correlations still persist for

higher doping levels. In the La2-xSrxCuO4 system, these AF correlations are

purely 2D and have a correlation length of ~ 50 Å for the undoped plane and

8 Å for the CuO2 plane near optimum doping. In YBa2Cu3Ox, the neighbouring

CuO2 planes remain coupled for these AF fluctuations, even up to optimal

doping [Rossat93].


18

La2-xSrxCuO4

0 0.10 0.20 0.30

Sr content x

0

20

40

0

50

150

200

250

100

TN (K) Tc (K)

Antiferrom

agnetic

Orthorhombic Tetragonal

Superconducting

I M

Ortho Tetra

0

50

100

oxygen content x6 6.2 6.4 6.6 6.8 7

0

100

200

300

400

TN (K) Tc (K)

OrthorhombicMetallic

TetragonalInsulating

Antiferro-magnetic Superconducting

YBa2Cu3Ox

Figure 1.10: Schematic phase diagram for the magnetic, superconducting and structural properties of bulk La2-xSrxCuO4 and YBa2Cu3Ox as a function of Sr or oxygen content x [Rossat93, Fleming87, Takagi89].

Once holes are present in the plane, transfer of electrons allows a S = 1/2 Cu2+

and S = 0 Cu3+ to exchange positions and the Mott-insulator state is destroyed at

a certain critical level of doping (figure 1.10). The model describing the motion

of the S = 0 Cu3+ ions in a background of Heisenberg coupled S = 1/2 Cu2+ ions

is known as the t-J model, represented by its Hamiltonian:

( )∑∑ ++ +−⋅=σ

σσσσ,,

,,,,, ji

ijjiji

ji cccctSSJHrr

(1.4)


19

with t > 0 a "hopping amplitude" and +σ,ic and σ,ic the electron creation and

annihilation operators respectively; no double occupancy is allowed. The

exchange interaction was estimated JCu-Cu ~ 125-170 meV (see above) whereas

the hopping amplitude was calculated to be tCu-O ~ 1 - 1.24 eV [Mizuno98].

This picture is complicated by two fairly recent observations: the existence of a

pseudo spin-gap at temperatures T > Tc (for a review see [Timusk99]) and the

observation of incommensurate spin fluctuations in both La2-xSrxCuO4

[Thurston89, Cheong91, Mason92 & 94, Yamada97, Tranquada97, Hunt99] and

YBa2Cu3Ox [Aeppli97, Tranquada97, Dai98, Kao99, Arai99] pointing towards

the formation of dynamic stripes [Bianconi96, 97 &98, Emery97b & 99b,

Zaanen99, Valetta97].

1.4.2.1 The pseudo gap

The pseudo-gap is observed as a gradual and partial decrease in the density of

states near the Fermi level, at temperatures by far exceeding the

superconducting critical temperature [Ong96, Hanaguri99, Timusk99]. This

pseudo-gap has been found in underdoped La2-xSrxCuO4 as well as in

YBa2Cu3Ox as reflected in (i) a reduction of the spin-lattice relaxation rate 1/T1T

and Knight-shift KS in NMR [Alloul89, Yasuoka94 & 97, Berthier97], (ii) the

development of a gap around EF in the dynamic spin susceptibility in inelastic

neutron scattering (INS) [Rossat93, Thurston89], (iii) a decrease in the

electronic specific heat coefficient γ = Cel/T ~ g(EF) [Loram93 & 98,

Momono99], (iv) a loss of spectral weight in Raman spectroscopy

measurements [Chen97, Naeini99 & 99b], (v) the development of a gap in angle

resolved photo emission spectroscopy (ARPES) [Loeser96, Ino98 &99], (vi)

tunnelling spectroscopy [Oda98] and (vii) resistivity (see paragraph 1.4.3).

These measurements all show the opening of a partial energy gap ∆ in the DOS

below a temperature T*. ARPES measurements [Loeser96, Ino98 &99,

Timusk99] showed this gap to be consistent with a d-wave symmetry, showing

the development of a gap in the (π,0) directions in reciprocal space with arcs of

a gapless Fermi surface in the (π,π) directions in underdoped samples.

In the scientific literature, several crossover temperatures T* (To, Tco, Ts, Tsg) and

various gaps ∆ (∆s, ∆p, ∆o, …) have been used. In La2-xSrxCuO4 two crossover


20

temperatures were found: To marks the onset of AF correlations whereas below

T* the pseudo-gap opens (see figure 1.11). In YBa2Cu3Ox, the observed pseudo-

gap in the DOS was shown to be very sensitive to specific physical property

used to define the pseudo gap. Probing charge excitations (like ARPES) is

reported to yield a gap ∆p, approximately twice as large as the spin-excitation

gap ∆s as observed in NMR and INS experiments [Nakano98, Mihailovic99].

This points in the direction of two possible (complementary) mechanisms, both

leading to the opening of a depleted region in the DOS near EF. Numerous

models were proposed to address the origin of such gaps; the precise relation

between these two gaps and the superconducting gap is, however, not yet

cleared out.

p

T

3DAF

TN

Tc

SC

To

T*

spingap

AF correlations

Figure 1.11: Schematic and simplified diagram showing the superconducting and 3D antiferromagnetic (AF) phases, together with the To and T* crossover lines for the onset of AF fluctuations and pseudo gap behaviour, respectively.

One possible paradigm for explaining the opening of the spin-gap is the

resonating valence bond (RVB) model [Anderson73 & 88, Suzumura88,

Nagaosa90] in which the attributes of the electrons -spin and charge- are

separated into two types of quasiparticles carrying spin (spinons with zero

charge) and charge (holons without spin). The formation of the

superconducting state then requires both the spinons (fermions) and holons

(bosons) to be paired into Cooper pairs. In the RVB model, below a

temperature TRVB (= T*), due to the strong AF correlations, the spinons are


21

paired into so-called Zhang-Rice singlets [Zhang88, Tjeng97] and the holons

are responsible for the (non-superconducting) transport of electric current. At a

temperature TB < TRVB, also the holons undergo Bose-condensation and the

superconducting state is established (Tc = TB). The phase line for the pairing of

the spinons shows a decreasing TRVB as the level of hole doping is increased,

whereas the temperature TB for the bose-condensation of the holons increases.

The prediction by the RVB model of the existence of two separate mechanisms

for the opening of a gap is in a nice agreement with the different pseudo gaps

reported when probing charge- or spin-excitations [Nakano98, Mihailovic99].

The spin-bag mechanism [Schrieffer88] is, like the RVB model, also based on

the notion of spin-charge separation. In this model, the introduction of a hole in

an AF correlated region disturbs the AF order and a bag (a sort of a spin-

polarised cloud or an AF spin-polaron) is created in the AF in which the hole is

trapped. When another hole is added to the structure, another hole-bag is

created and the two holes are attracted to share a common bag. The resulting

pairing interaction then leads to the opening of a superconducting gap.

A model that uses similar concepts as the RVB model and spin-bag model is the

precursor-pairing model [Emery97, 97b & 99, Randeira97] combined with the

concept of charge stripes [Cheong91, Tranquada97, Arai99, Bianconi96, 97

&98, Zaanen99,]. Below a temperature To, AF correlations set in and holes are

expelled from the AF regions [Schrieffer88] when cooling down further. In that

way hole-free AF Mott-insulating regions are created, intercalated by hole-rich

metallic charge-stripes (topological doping [Kivelson96]). At this point, no

superconductivity is established yet (in contrast to the spin-bag model described

above). When lowering the temperature even further, the reduced

dimensionality resulting from the confinement of the AF correlations, allows a

spin-gap to be established [Dagotto96] at temperatures far above the

superconducting critical temperature Tc. This energy gap is then imposed upon

the metallic stripes by pair hopping of holes between the stripe and the AF

surroundings (magnetic proximity effect). The superconducting state is

recovered when the spin-gaps, that are formed locally, acquire macroscopic

coherence by Josephson coupling between the stripes. However, the precise


22

link between the J (125-170 meV) scale local physics and the Tc (10 to 20 meV)

scale long-range physics of superconductivity still remains unclear.

1.4.2.2 Formation of stripes

The above mentioned incommensurate fluctuations in the microscopic spin

ordering have been observed both in La2-xSrxCuO4 [Thurston89, Cheong91,

Mason92 & 94, Yamada97, Tranquada97, Hunt99] and YBa2Cu3Ox [Aeppli97,

Tranquada97, Dai98, Kao99, Arai99], at temperatures above the

superconducting critical temperature Tc. They show up in the structure factor

derived from inelastic neutron scattering experiments as intensity peaks that are

incommensurate with the crystal lattice spacing. These observations can be

interpreted as an argument in favour of a separation of the CuO2 planes into AF

hole-free regions (a few lattice spacings wide), alternated with metallic stripes

of holes (a superlattice of quantum stripes [Bianconi96, 97 &98]). This

interpretation is supported by 63Cu nuclear quadrupole resonance (NQR)

measurements which are sensitive to the local gradient of the electric field and

are wiped out by the strong gradients in the alternating hole free and hole rich

stripes. The observation of incommensurate fluctuations in the microscopic

magnetic ordering forms a strong experimental support for the models based on

stripe formation [Bianconi96, 97 &98, Emery97b & 99b, Zaanen99].

1.4.3 Transport properties

In a free electron system, the electrical resistivity can be written as [Ashcroft76]

τ

ρ2

*

ne

m= (1.5)

with m* the effective mass, n the density and τ the scattering time of the

electrons. Although being a naïve single electron picture, it already provides

the intimate connection between the electrical resistivity and the density and the

microscopic scattering of the electrons. This indicates that the study of the

transport properties of a material can give key information about the nature of

the charge carrier scattering.

Scattering of electrons can be due to lattice imperfections, magnetic and non-

magnetic impurities or the interactions with thermal lattice vibrations


23

(phonons). In conventional metals, the transport properties are governed by this

electron-phonon scattering. There are two relevant temperature regions relative

to the Debye temperature θD. For high temperatures T >> θD, all phonon modes

are excited and the resistivity ρ varies linearly with temperature. However, as

the temperature decreases the resistivity drops faster with temperature and

obeys a T 5 law. This is due to the fact that below θD an increasing number of

phonon modes with high excitation energy starts to freeze out, leading to a

reduced electron-phonon scattering [Ashcroft76].

>∝<<∝

+=−

−− )(

)(5

Dphel

Dphelphelo TT

TT

θρθρ

ρρρ (1.6)

The residual resistivity ρo remaining at low temperatures is then a measure for

the amount of impurities in the crystal lattice. The Debye temperature θD for

the high-temperature superconductors lies between 300 K and 450 K and the

deviation from linear resistivity should certainly be expected to show up at

T ≤ 0.2 θD [Poole95].

In the cuprate high-temperature superconducting compounds, an enormous

effort has been put in the exploration of the resistivity ρ(T) at various levels of

hole doping and different geometrical configurations between the transport

current and the crystallographic axes. It was found that these materials exhibit a

very unusual behaviour of the resistivity when doping is changed from

underdoped to optimally doped and overdoped samples [Takagi92, Kimura92,

Ito93, Batlogg94, Ando96b, Wuyts94 & 96].

At optimal doping, the in-plane resistivity ρab(T) shows a robust linear

behaviour that extends from the critical temperature Tc up to very high

temperatures; although Tc is significantly lower than the Debye temperature θD.

The absence of an excess conductivity indicates that another scattering

mechanism, besides electron-phonon scattering, may play a prominent role in

these materials. When the concentration of holes is lowered, the resistance

increases and the linear ρab(T) behaviour transforms into a super-linear

behaviour at low temperatures T < T* (figure 1.12); the so-called S-shape

behaviour. This excess conductivity was studied extensively in YBa2Cu3Ox

[Ito93, Wuyts94 & 96] and La2-xSrxCuO4 [Suzuki91, Kimura92, Takagi92,


24

Batlogg94] and was (by comparison with magnetic measurements like for

instance NMR) attributed to the opening of the spin pseudo-gap below a

temperature T*. At even lower levels of hole doping, the conductivity decreases

even further and at low temperatures, the slope dρ/dT becomes negative, a sign

for the onset of insulating (or semiconducting) behaviour (see also the phase

diagram in figure 1.13). The temperature at which dρab/dT changes sign to an

insulating behaviour decreases as the doping level is increased. The

temperature range, where the in-plane resistivity shows an insulating behaviour,

thus shrinks as doping is elevated and it is therefore camouflaged by the onset

of the superconducting phase, as can be seen in the phase diagram in

figure 1.13.

underdopednear optim

aloverdoped

ρab

Tρab

Tρab

T

ρc

T

ρc

T

ρc

T

II

Figure 1.12: Schematic overview of the transport properties of the high-Tc cuprates in the underdoped (top), optimally doped (middle) and overdoped (bottom) regime with the transport current in the ab plane (left) and parallel to the c axis (right).

So one can ask then whether the traditional description of the metal to insulator

(MI) transition by a simple vertical line in the T(p) diagram (figure 1.10) is

physically correct. This question becomes of particular relevance when

considering the fact that the out-of-plane c-axis resistance ρc(T) exhibits an

insulating alike behaviour (dρ/dT <0) in the whole underdoped to optimally

doped region of the phase diagram (figure 1.13). Only in the overdoped regime


25

a Fermi-liquid alike regime is entered, characterised by an approach of the ρ ~

T 2 behaviour, a sign of electron-electron scattering becoming dominant in

charge-carrier scattering.

To address this issue, the experiments, initiated by Ando to measure the normal-

state transport properties of Bi2Sr2CuOy (Bi2201) [Ando96c, 97, 97b] and

La2-xSrxCu4 (La214) [Ando95, 96, 96b, 97, 97b, Boebinger96] below Tc, are of

great importance. Ando and co-workers were able to suppress the masking

superconducting phase by applying very high magnetic fields and thus to reveal

the normal-state properties below Tc. In that way, it was shown that in La214

and Bi2201 both the ab-plane and c-axis transport show a diverging behaviour

of ρ(T) at low temperatures. The traditional MI phase line of figure 1.10 at a

certain critical concentration of doped holes, was therefore nuanced to yield the

phase diagram shown in figure 1.13, accounting for the results of Ando and co-

workers and the observed anisotropy.

The consequences of this puzzling behaviour for the microscopic mechanism

for conduction and scattering of the charge-carriers are still the subject of

intense debate and might be connected with the opening of the (spin) pseudo-

gap at T* and the presence of charge stripes in the CuO2 planes.

p

T

3DAF

Fermi Liquid

TN

Tc

Insulating

Metallic ρab

Insulating ρc

SC

Figure 1.13: Schematic phase diagram for the transport properties of the high-Tc cuprates. The notions metallic and insulating are defined by dρ/dT > 0 and dρ/dT <0 respectively.


26

1.5 Generic T(p) phase diagram

The copper-oxide high-temperature superconductors form a class of materials

showing unusual features in their electronic and magnetic properties which are

as fascinating as uncomprehended. Following the discussion in the previous

paragraphs we will construct a generic T(p) phase diagram for this class of

layered CuO2 materials and focus on specific points where knowledge is still

lacking. In figure 1.14 the major regimes for the electronic and magnetic

properties are sketched.

At low levels of hole doping the 3D AF phase on the T(p) diagram is rapidly

shrinking as the doping is increased and only short range AF correlations

persist, when cooling below a temperature To. Below a temperature T* a partial

gap in the DOS is formed around EF. Although being still under discussion,

experimental evidence exists for this pseudo gap to be a spin gap

(paragraph 1.4.2.1, and for a review see [Timusk99]). As revealed by

measurements in high magnetic fields on La2-xSrxCuO4 and Bi2Sr2CuOy, at low

temperatures (mostly below Tc), both the in plane and the out of plane c-axis

resistivity in the underdoped compounds show a diverging resistivity

(dρ/dT < 0), pointing to an insulating or semiconducting ground-state. Thus, in

these compounds the metal to insulator (MI) transition line was shown to

penetrate into the superconducting phase rather than being a vertical line in the

T(p) phase diagram (figure 1.14). The superconducting phase -showing up

below Tc in a very limited range of doping levels- is superimposed with this

normal-state behaviour and masks some of its anomalous features. The critical

temperature Tc has a parabolic dependence on the concentration of holes in the

CuO2 plane. At near optimum doping and above Tc, the high-Tc's are anomalous

metals showing a linear ρ(T) behaviour that is not in agreement with the

transition from ρ(T) ~ T 5 to ρ(T) ~ T at T ~ θD in conventional metals.

This phase diagram is however not complete and evokes some questions.

What happens in the two layer compound YBa2Cu3Ox ? Does it also exhibit a

MI phase line entering into the superconducting phase rather than being a

vertical line at some critical concentration of holes ?


27

p

T

3DAF

SC

spingap

anomalous metal

Fermi liquid

I M

TN

Tc

T*

AF correlations

To

MI

p

T

3DAF

SC

spingap

anomalous metal

Fermi liquid

I M

TN

Tc

T*

AF correlations

To

MI

Figure 1.14: Schematic overview of the T(p) phase diagram for the high-Tc cuprates, showing the 3D antiferromagnetic (AF) and superconducting (SC) region. The additional phase lines are explained in the text.

What can we extract from the S-shape behaviour of ρ(T) below T* in the pseudo

spin-gap regime and can we make the experimental and theoretical correlation

between these two phenomena more specific ? Can it be compared to what

happens in other techniques in the pseudo gap regime ? Are these phenomena

purely dependent upon the charge carrier density or is the situation more

complicated ? To answer these important question, high field transport

experiments (both magneto-resistivity and Hall-effect) were initiated on thin

films of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox with varying oxygen content and

on ultra thin films of La1.9Sr0.1CuO4 under epitaxial strain.

CHAPTER 2 Experimenting in pulsed high magnetic fields

29

Chapter 2

Experimenting in pulsed high

magnetic fields

Experimenting in high pulsed magnetic fields is not an easy task because of the

necessary miniaturisation and the transient signals in the experiment. The large

sweep rates of the magnetic field produce unwanted voltages in the wiring and

induce eddy currents in the metallic parts of the set-up. The main experimental

techniques and their modifications carried out during this work are reviewed in

this chapter. The experimental set-up and procedure for measuring the

magnetoresistivity and Hall-effect in pulsed magnetic fields are discussed.

2.1 Generation of pulsed magnetic fields

The generation of a magnetic field can be achieved by simply putting a current

through some conducting windings. Evidently, the more windings and the

higher the current, the higher the magnetic field will be. Constraints of heating,

mechanical stability and the need for enormous electric power block this

roadmap for DC fields at around 36 Tesla, achieved at the hybrid magnet of

NRIM in Tsukuba [Herlach95].

Some of these constraints can be relieved by turning to pulsed magnetic fields

using non-destructive, liquid nitrogen cooled, coils. The transient nature of

such field pulses reduces the thermal load and takes away the need for a

constant feed of a high electric current. The mechanical stability of the coil

however is in this case more complicated and the forces are far above what can

be sustained by wires with a reasonable conductivity (like copper). Therefore,


30

at the K.U.Leuven, the choice was made to turn to solenoids with a non-uniform

density of windings - in order to distribute the stress over all layers -, alternated

by glass- or carbon-fibre reinforcement thus providing the necessary strength

[Li98, Herlach89 & 96]. Figure 2.1 shows a schematic drawing of a Leuven-

type coil.

0

10

20

30

40

50

0 5 10 15 20-10

0

10

20

30

40

50

µ oH

(T

)

t (ms)

µo dH

/dt (kT/s)

MeL83Ucap = 4750 VC = 24 mF

epoxycontacts

Stainless steelcylinder

carbon-fibrecomposite

glass -fibrecomposite

conductor

studding

8 - 17 mm

Figure 2.1: Schematic drawing of a coil for the generation of pulsed magnetic fields. (plot) Magnetic field and sweep rate versus time for MeL83.

The generation of the necessary high pulsed currents is achieved by the

discharge of a high-voltage capacitor bank. The switching gear that is

necessary to permit such a discharge and the diagnosis of the coil before and

after the shot is presented in figure 2.2. Four coils (g) can be powered

sequentially, with a charging time of at maximum 2 minutes in between. This

discharge is done either by a mechanical switch (f) or a stack of thyristors (f).

The switching set-up also contains safety discharge switches (d) allowing a

quick and controlled disposal of the energy in case of an emergency. The

maximum energy is 475 kJ, provided by a 38 mF capacitor bank charged to

5000 V, although the default value for daily experiments is set to 24 mF. The


31

oscillations in the LRC circuit that is formed by the coil (g), the wiring and the

capacitor bank (c) are strongly damped by the 0.08 Ω crowbar circuit (e). A

typical field profile (using coil MeL83) is shown in figure 2.1. This 60 T-class

coil, used to produce the experimental results in this work, is capable of

generating 57 T with a pulse duration of ~ 17 ms and was the first user-coil in

Leuven to have carbon-fibre reinforcement between the layers of CuAg wire. It

has a bore diameter of 17.4 mm and the design constitutes of 10 layers, each

with 18 windings of CuAg-wire, resulting in L = 512 µH and R = 46 mΩ at

liquid nitrogen temperature. These values are checked before and after the field

pulse to ensure that no deterioration of the mechanical or geometrical properties

of the coil has occurred.

(a) (c) (d) (f)(e)

(g)

(h) L/R measurement

(b)

Figure 2.2: Simplified electrical scheme of the high-voltage switching-gear that enables the discharge of the capacitor bank in a chosen coil, permitting a pre- and post-shot diagnosis of the coil.

The whole pulsed field facility is controlled by a central measuring computer

that operates the high-voltage bank, performs the discharge, diagnoses the coils,

makes stringent safety checks, collects experimental data, controls the

temperature of the experiments and allows to perform a first analysis of the

data. It is evident that for safety reasons all connections of this computer to the

high-voltage switching gear or the experimental equipment are made by optical

fibre. Safety measures also include strict grounding prescriptions and the use of

shielded Faraday cages containing the experimental equipment.

2.2 Cryogenics

The miniaturisation imposed by the constraints during the design of the pulsed

field coils and the pulsed nature of the magnetic field have severe implications


32

on the cryogenic technology. The tail of the cryostat (figure 2.3) can have a

maximum outer diameter of about 17 mm while leaving enough free space

inside to allow physical experiments to be carried out. Moreover, this tail is

preferably constructed out of a non-conducting (plastic, ceramic) or poorly

conducting (stainless steel) material as to reduce eddy-currents. At the

K.U.Leuven pulsed field facility it is now possible to perform measurements

from room temperature down to 360 mK [Li98, Herlach89 & 96].

vacuumspace

vacuumpump

He flowout

Cu-block+ heater

Cu-block

He contact-gas inletsample room

vacuum

sam

ple

room

15.38mm

He flow

in

flow cryostat

vibrationinsulation

N2 vessel

60 teslamagnet

magnetmountingrack

cryostatmountingplate

Figure 2.3: Schematic view of the 4He cryostat (top panel) and the magnet support with vibration insulation (bottom panel).


33

The home-made 4He flow cryostat that was used to establish the data in this

work has a tail of outer diameter 16 mm and consists of 4 coaxial stainless-steel

tubes (figure 2.3, top panel). The gas flow is first conducted through a heated

copper block, then guided through the tail after which it passes another copper

block that thermally isolates the inner tube from the surroundings. This cryostat

allows experiments from room-temperature down to 4.2 Kelvin.

Another requirement for fruitful experiments in pulsed fields is an effective

insulation for mechanical vibrations between the cryostat and the magnet. This

is achieved by rigidly attaching the coil to a fixed support system. The cryostat

itself is then mounted on top of this rack using cylinders of elastic silicone

rubber (figure 2.3, bottom panel) for damping. This construction enables

experimenting in an environment with a vibrational noise level that is an order

of magnitude lower than in the absence of vibration insulation.

2.3 Transport measurements

The presence of parasitic effects like mechanical vibrations, transient voltages

induced by the pulsed magnetic fields and a high-voltage discharge in the

neighbourhood of the experimental equipment also asks for specific counter

measures to be taken.

As a first precaution, the home-made measuring probe (based on a 1 meter glass

tube of diameter 6 mm) is attached to the cryostat using vibration insulation.

Damping of the mechanical vibrations is necessary since any movement of the

measuring probe in the, in reality non-uniform, magnetic field will induce

unwanted voltages. Simulations show the inhomogeneity of the magnetic field

to be less than 0.01 % at 1 mm from the centre of the magnet, yielding parasitic

voltages in the circuit that are an order of magnitude smaller than the "direct"

µodH/dt voltages induced in the wires. However, since they are essentially

random, they cannot be compensated completely and efficient damping of

vibrations remains important.

Secondly, the metallic cryostat, together with the shielding of the connecting

measuring-cable and the shielded measuring box forms one single Faraday cage


34

(figure 2.4). This provides efficient screening of spurious transient

electromagnetic effects. The various signals of the physical measurement are

led into the box containing the instrumentation devices by separate coaxial

cables.

The actual set-up for the measurement of the magneto-resistance and the Hall

effect is depicted in figure 2.4. The thin film samples are patterned in a 1 mm

long 50 µm strip as to generate a high signal even at low currents. Electric

contacts are realised by means of gold wire, attached with silver paint to thin

(< 1000 Å) gold pads deposited and annealed on the film. The constant current

through the sample is generated using a battery-operated current source,

independent of the rest of the set-up, with a series resistor of about 600 kΩ to

limit the variation of the current (caused by a varying resistance of the sample)

to less than 0.5 %. The signal, generated by the sample then is amplified by a

home-made instrumentation amplifier (based on the Burr-Brown INA103 and

INA110 low noise operational amplifiers) with a bandwidth ranging from DC

up to 150 kHz. No electronic filtering is performed. In this amplifier, the

physical signal is mixed with the µodH/dt voltage that is induced in a small

pick-up coil in the vicinity of the sample. This permits to subtract spurious

signals that are induced in the wiring and to obtain the useful physical signal.

The magnetic field is measured by monitoring the induced voltage in a second

pick-up coil with a calibrated area (S = 66.83 mm2 for the measurements in this

work). The different voltages, generated during the 17 ms field pulse, are

measured with a 4 channel, 12 bit, transient recorder (BE256 Bakker

Electronics) operating at 1 MHz.

The temperature is monitored by means of a silicon miniature diode temperature sensor (SMDT v3.0, Institute of Cryogenics, Southampton), which,

with a constant current of 10 µA applied, gives signals from 0.56 V at room

temperature up to 1.71 V at 4.2 Kelvin. This voltage is then measured using a

digital voltmeter (DVM, Keithley 2000).

All data taken by this voltmeter and the transient recorder are transferred to the

central measuring computer by means of an optical-fibre IEEE-488 data bus,

using special bus extension devices (Hewlett-Packard 37204 HP-IB extender).


35

Cryostat Shielded measuring box

x1

I

SMDT

dB/dt

comp

Shieldedcable

x1

230 V

filter

IEEE

TransientRecorder

optical

Hall

MR

I

DVM

Cryostat Shielded measuring box

x1

I

SMDT

dB/dt

comp

Shieldedcable

x1x1

230 V

filter

IEEE

TransientRecorder

optical

Hall

MR

I

DVM

Figure 2.4: Schematic overview of the set-up used for transport measurements in pulsed magnetic fields. The cryostat (left) is connected to the shielded measuring box (right) by means of a shielded cable. The parts with grey background depict the wiring for measurements of respectively the magnetoresistance (MR) and the Hall-effect.

All measurements, reported in this work, were taken in the in-plane transverse

configuration (figure 2.5), with the applied current in the ab-plane and the

magnetic field applied perpendicular to this current and the ab-plane (H // c).

H

H

H

HH

I

I

I

I

I

a

b

c

in-planeI // ab

out-of-planeI // c

H // c H // ab

transverse

transverse

transverse

longitudinal

longitudinal

Figure 2.5: Geometrical configurations for transport measurements on layered materials, with I the transport current, H the external magnetic field and a, b and c the crystallographic axes.


36

Although the voltage versus time traces that are transferred to the computer are

almost completely compensated for induced voltages µodH/dt, the combination

of multiple field pulses is required in order to obtain the real physical signal.

2.3.1 Magnetoresistance

For measurements of the magnetoresistance, two shots of different field

polarity and positive current are averaged (equation 2.1). Since µodH/dt is an

odd function of the magnetic field (see figure 2.11), this removes induced

µodH/dt influences that have survived the hardware compensation in the

amplifier.

)0(2

1 00 >+

== <> IVV

II

VR HH (2.1)

This experimental procedure can be illustrated with a real experiment on a

1770 Å thick YBa2Cu3O6.7 epitaxial thin film, data that will be presented in

detail in chapter 3. Figure 2.6 presents the voltage, generated in the 1 mm x

50 µm strip at 4.2 K (top) and 70 K (bottom), during a 50 T field pulse of

positive (VH>0) and negative (VH<0) polarity. The applied current was 146 µA,

resulting in a current density of approximately 1.6· 107 A/m2 (much smaller than

the critical current density of the order 109 A/m2 and the depairing current

~ 1013 A/m2) and the signal was amplified 200 times. The relevant physical

signal that is plotted in figure 2.6 after amplification corresponds to a voltage

over the sample that lies in the range 1 to 100 mV (depending on the sample and

the current). From these plots the equivalent level of noise at the sample (both

of vibrational and electronic origin), can be estimated at 70 µV at low fields and

1 mV at 50 T (the width of the noise band). The two shots with opposite

polarity of the magnetic field show a significant hysteresis at low magnetic

fields where the asymmetry in the sweep rate µodH/dt is high (figure 2.6). This

imperfect compensation is removed by combining the VH>0 and VH<0 traces

using equation 2.1, resulting in the third trace on the two figures. Both below

and above Tc the resulting traces are of a remarkable quality, showing no

significant sign of voltages induced due to mechanical vibrations. Moreover,

although the sweep rate of the magnetic field varies from 15 kT/s at rising field

to -5 kT/s at lowering magnetic field, no hysteresis was observed in these traces.


37

This proves that the heating of the metallic parts (sample, contacts and cryostat) is of minimal influence on the physical measurement. Furthermore, the perfect reproducibility of all measurements reported in this

work by smaller field pulses (typically between 15 and 20 T) shows that the

transport properties, derived from pulsed field measurements are not

significantly influenced by the high sweep rate.

0 10 20 30 40 507

8

9

10

11

µoH (T)

VH > 0

VH < 0

YBa2Cu3O6.7T = 70 K

V (

x200

, Vol

t)

0 10 20 30 40 50

0

2

4

6

8

µoH (T)

V (

x200

, Vol

t)

VH > 0

VH < 0

YBa2Cu3O6.7T = 4.2 K

Figure 2.6: Transport measurements in an YBa2Cu3O6.7 epitaxial thin film at T = 4.2 K << Tc (top) and T = 70 K > Tc (bottom). The two shots VH>0 and VH<0 are shifted up by one unit with respect to the final trace.

The final traces, as presented in figure 2.6, can then be recalculated to the true

resistivity ρ by dividing by the applied current and the amplification factor and

taking into account the thickness and width of the strip and the distance between

the voltage probes. The resulting ρ(H) plots at various temperatures are

presented as one summarising plot (figure 2.7) in which the data were smoothed

by adjacent averaging over 20 points. Taking into account the 3.3 µs sampling


38

rate this results in an averaging time window of 66 µs thus reducing the

influence of frequency components above 16 kHz. This frequency is still much

higher than the approximate 50 Hz at which the physical properties of interest

vary during the field pulse. From figure 2.7 it can be noted that no additional

hysteresis is induced by the low-pass filtering procedure. During the past few

years, it was shown that such an excellent quality of transport measurements

can only be achieved by a careful implementation of the precautions discussed

above.

The collection of one ρ(H) trace, including the reproducing low-field shots,

takes about 2.5 hours (depending on the temperature stability) resulting in 3 to 4

ρ(H) traces per day. Thus the construction of the summarising plot in figure 2.7

takes as a minimum one week of intensive experimenting.

0 10 20 30 40 500

100

200

300

400

500

µoH (T)

ρ (µ

Ωcm

)

4.2 K

127 KT

YBa2Cu3O6.7

Figure 2.7: Resistivity ρ versus applied field up to 50 T (up and down field-sweeps) for an epitaxial thin film of YBa2Cu3O6.7 at temperatures T = 4.2 K, 11.4 K, 19.4 K, 30.1 K, 38.8 K, 46.8 K, 52.4 K, 62.6 K, 70 K, 81 K, 100.8 K and 127 K. Note the absence of hysteresis effects and induced voltages due to mechanical vibrations.

2.3.2 Hall effect

The transverse electric field arising from the Hall-effect is measured using a

standard 5-terminal configuration where a virtual contact pad can be moved by

varying a 100 kΩ resistor, placed between the two contacts at the same side of


39

the strip (figures 2.4 and 2.8). This allows the virtual contact to be placed at

exactly the opposite side of the 3rd contact pad (figure 2.8). The variable

resistor has to be of high resistance such that no significant current is flowing

through it. Doubling the value of this variable resistor was shown not to

improve the quality of the measurements but rather to complicate offset

adjustments of the amplifiers. A proper adjustment of the compensation of the

system involves three successive steps, which have to be iterated until the

desired level of compensation is obtained. As a first step, the zero current

offsets of the four operational amplifiers have to be set to zero. Step two

involves the alignment of the virtual Hall contact pad, with the current set to the

value used in the later experiments. In the third step the overall compensation

of the system for the induced voltages µodH/dt is optimised using a number of

small (6 T) field pulses. Since all three steps are interdependent, at least one

additional iteration is necessary.

VHI I

Figure 2.8: Electric scheme for a 5 terminal measurement of the Hall effect.

A proper measurement of the Hall-voltage requires four field pulses to be

combined and averaged. From figure 2.11 it can be seen that the Hall-voltage

can be obtained in two independent ways (VHall,a and VHall,b in equation 2.2), both

removing the parasitic µodH/dt and magnetoresistive signals.

+−=

+=+

=><<>

<<>>

2

22 0,00,0

,

0,00,0,,,

IHIHbHall

IHIHaHall

bHallaHallHall VV

V

VVVVV

V (2.2)

Also here, the experimental procedure can very well be illustrated with a real

experiment on a 850 Å thick Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film, data that

will be presented in detail below in chapter 4. Figure 2.9 shows four voltage


40

traces (VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0), generated in the 1 mm x 50 µm

Hall pattern at 84.6 K during a 17 T field pulse with the polarity of both the

field H and the current I changing in between the pulses. These traces were

taken after amplification by a factor 1250 and with an applied current 600 µA,

resulting in a current density of approximately 7· 107 A/m2 (much smaller than

the critical current density (of the order 109 A/m2) and the depairing current

~ 1013 A/m2).

-3

-2

-1

0

1

2

3

V (

x125

0, V

olt)

µoH (T)

VH I> >0 0,

VH I< <0 0,

VV V

Hall aH I H I

,, ,=

+> > < <0 0 0 0

2

Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K

0 5 10 15 20

-3

-2

-1

0

1

2

3V

V VHall b

H I H I,

, ,= −+> < < >0 0 0 0

2

µoH (T)

VH I> <0 0,

VH I< >0 0,Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K

0 5 10 15 20

V (

x125

0, V

olt)

Figure 2.9: Measurements of the voltage over a Hall pattern in a Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film at 84.6 K in pulsed fields up to 17 T: (top) VHall,a as an average of VH>0, I>0 and VH<0, I<0 and (bottom) VHall,b as an average of VH<0, I>0 and VH>0, I<0. The four shots VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0 are shifted down by one unit with respect to the final trace. No filtering or numerical smoothing was applied.

The relevant physical signal that is plotted in figure 2.9 after amplification

corresponds to a Hall voltage across sample that reaches at maximum a few

100 µV (depending on the sample and the current). Also here, the raw traces


41

VH>0, I>0, VH<0, I<0, VH<0, I>0 and VH>0, I<0 show an appreciable hysteresis over the

whole field region. Just as in the case of the magnetoresistance experiments this

imperfect compensation is removed by combining these traces in order to give 2/)( 0,00,0, <<>> += IHIHaHall VVV and 2/)( 0,00,0, ><<> +−= IHIHbHall VVV using

equation 2.2 (figure 2.9). These two traces almost perfectly overlap and show a

linear behaviour. However, when comparing these two traces in detail, small

symmetric deviations from the 4-pulse-averaged VHall were observed. This

might be due to an effect that is an even function of both field- and current

polarity; the precise origin of this effect is still under investigation. In any case

this artefact is completely removed by performing the average over four pulses

(figure 2.10).

0 5 10 15 200.0

0.5

1.0

1.5

µoH (T)

Y0.6Pr0.4Ba2Cu3O6.7T = 84.6 K

VH (

Vol

t)

VV V

Hall

Hall a Hall b=+, ,

2

0.00

0.02

0.04

0.06

0.08

0.10

0 5 10 15 20

µoH (T)

Y0.6Pr0.4Ba2Cu3O6.7

T = 84.6 K

VH/µ

oH (

V/T

)

Figure 2.10: The Hall voltage VHall in a Y0.6Pr0.4Ba2Cu3O6.7 epitaxial thin film at 84.6 K in fields up to 17 T as an average of VHall,a and VHall,b (top). The ratio VH/µoH which is proportional to the Hall coefficient RH (bottom).


42

The final VH curve in the upper panel of figure 2.10 is of a remarkable quality,

showing no significant sign of mechanical vibrations or hysteresis; this curve

was smoothed with a 20-point adjacent averaging (16 kHz). Also here the high

field shots (40 to 50 T) were reproduced by smaller (10 to 20 T) pulses and the

underlying physics seems not to be significantly influenced by the sweep rate of

the magnetic field.

From this Hall voltage versus applied field, the ratio VH/µoH (proportional to the

Hall coefficient RH) can be calculated (lower panel of figure 2.10). This gives

(above Tc) an approximately constant value over the applied field range. From this ratio, we can calculate the Hall coefficient zxyH HjER ≈ by taking into

account the value of the applied current and the width of the strip.

The collection of one VH(H) trace, including the reproducing low-field shots,

takes about 4 to 5 hours (depending on the temperature stability) yielding in 2 to

3 traces/day. A reasonable characterisation of the Hall coefficient of a sample

thus takes at least 2 weeks of intensive experimenting.

µoH (T)

Induced µodH/dt

µoH (T)

Magnetoresistance

I > 0

I < 0

µoH (T)

Hall-voltage

I > 0

I < 0

µoH (T)

Induced µodH/dt

µoH (T)

Induced µodH/dt

µoH (T)

Magnetoresistance

I > 0

I < 0

µoH (T)

Magnetoresistance

I > 0

I < 0

µoH (T)

Hall-voltage

I > 0

I < 0

µoH (T)

Hall-voltage

I > 0

I < 0

Figure 2.11: Schematic view of the magnetoresistance, the Hall voltage and the parasitic µodH/dt voltage versus applied magnetic field. The full line is for positive currents while the dashed line represents the situation for negative measuring currents.

CHAPTER 3 Normal-state resistivity of Y123, YPr123 & La214

43

Chapter 3

Normal-state resistivity of

YBa2Cu3Ox, (Y1-yPry)Ba2Cu3Ox

and (La1.9Sr0.1)CuO4

3.1 Introduction

Although widely studied, the transport properties of the underdoped cuprates

still retain some mysterious features that remain to be solved. What are the

microscopic scattering mechanisms responsible for the robust linear ρ(T)

behaviour ? What is the origin of the S-shaped ρ(T) curve below T* ? Can it be

related to the opening of a pseudo (spin) gap or the occurrence of charge stripes

intercalating hole-free AF regions ? What is the influence of hole doping on

these features ? Where in the temperature (T) versus hole doping (p) phase

diagram should the boundary between the metallic and insulator-like behaviour

be drawn ? It is clear that, as discussed also in chapter 1, the completion of the

T(p) phase diagram needs more experimental investigations to be carried out.

Recently, it was shown that the zero-field normal-state ρ(T) curves for

YBa2Cu3Ox at various doping levels exhibit an almost perfect scaling onto a

universal curve by linearly scaling both temperature and resistivity [Wuyts94 &

96]. The linear part of this universal curve was interpreted in terms of

scattering of charge carriers in a 2D antiferromagnet (AF) [Moshchalkov93,

Wuyts96]. The super-linear behaviour at lower temperatures was not

understood yet.


44

This work [Moshchalkov93, Wuyts94 & 96] on the normal-state resistivity

above Tc can in a natural way be extended to the region below the critical

temperature Tc by suppressing the "unwanted" superconducting phase by the use

of strong magnetic fields (cf. the work of Ando and co-workers on La214 and

Bi2201 cited earlier). In our work we will do this on the "123" system in which

the doping level is changed by varying the oxygen content x (YBa2Cu3Ox) and

chemically substituting Pr for Y ((Y0.6Pr0.4)Ba2Cu3Ox). This dual-track

approach - lowering Tc by changing the hole content p on one side and using

very high magnetic fields on the other side - allows us to cover the whole

underdoped to optimally doped region of the T-p phase diagram and thus

address the questions raised above.

x t

(Å)

Tc,mid

(K)

∆∆T

(K)

ρρo

(µΩµΩcm)

R

(kΩΩ)

ρρ290 K

(µΩµΩcm)

ρρ290K/ρρo

#1Y 6.95 1770 92.2 1.1 48 0.8 579 ~12

#2Y 6.8 1300 73.7 3.5 35 0.6 426 ~12

#1Y 6.7 1770 58.2 2.7 209 3.6 1171 ~5.6

#2Y 6.5 1300 52.9 2.2 95 1.6 835 ~8.7

#1Y 6.45 1770 41.7 7.7 607 10.4 2237 ~3.7

#3Y 6.4 2300 17.9 13.2 1900 32 4910 ~2.6

#4YPr 6.95 850 41.4 5.3 278 4.8 658 ~2.4

#4YPr 6.85 850 31.8 6.7 459 7.8 1205 ~2.6

#4YPr 6.7 850 22.3 12.7 805 13.8 1904 ~2.4

Table 3.1: Overview of the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films selected for this work with their thickness t, critical temperature Tc, width of the transition ∆Tc, residual resistivity ρo, sheet resistance per square R (calculated from the residual resistivity and to be compared to the quantum of resistance h/4e2 ≈ 6.45 kΩ), resistivity at 290 K ρ290 K and the ratio ρ290 K/ρo. The method to determine ρo will be discussed in paragraph 3.2.2.


45

For this purpose, a total of approximately 70 epitaxial thin films was deposited

by sputtering on SrTiO3 substrates [Wagner99]. A selection of these films was

patterned by wet chemical etching after which the oxygen content was adjusted

to the desired nominal value. The appendices give more details about the

sample preparation (A, B & C) and characterisation (D). For the YBa2Cu3Ox

compound, two films were selected for experiments in pulsed magnetic fields

(labelled #1 and #2). They were subsequently made oxygen deficient (see

table 3.1 and appendix B). Film #3 was not used for experiments in pulsed

fields, and thus only the zero-field data for this x = 6.4 sample will be presented.

For the (Y0.6Pr0.4)Ba2Cu3Ox compound, one film was selected and the oxygen

content was consecutively set to nominal values x = 6.95, x = 6.85 and x = 6.7.

The main physical properties such as thickness, critical temperature Tc,

resistivity at 290 K and residual resistivity ρo of the samples are summarised in

table 3.1.

Experimenting in high DC magnetic fields is possible up to 36 tesla (at the

NRIM in Tsukuba); when higher fields are necessary one has to turn to the

technique of pulsed magnetic fields [Herlach95]. The physical measurements

reported here, were carried out at the pulsed magnetic fields facility of the

K.U.Leuven [Herlach89 & 96] that allows measurements in magnetic fields up

to 60 T at temperatures from room temperature down to 360 mK. The transport

measurements (both magnetoresistivity and Hall-effect) were performed in a

home-made cryostat and using a home-made high-field coil. All measurements

reported in this work were performed in the transverse geometry (H ⊥ I) with

the magnetic field perpendicular to the film (H // c) and the current sent along

the ab-plane (I // ab). A detailed discussion of the experimental procedure and

the special issues arising when experimenting in transient magnetic fields was

given in chapter 2.

In this chapter we will report the normal-state resistivity of YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox thin films at various oxygen contents and La1.9Sr0.1CuO4

ultra thin films under epitaxial strain, above and below their critical temperature

Tc. For the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox compounds, we will make a

scaling analysis of both the zero-field and the high magnetic field data. The

low-temperature divergence of the resistivity will be examined.


46

3.2 Zero-field resistivity at T > Tc

The zero-field resistivity of the samples was measured using a standard 4 point

measuring technique on a 1000 x 50 µm strip patterned onto the thin film. This

well defined geometry allows to obtain a high signal and to recalculate it to the

resistivity. The precise experimental procedure is described in chapter 2.

3.2.1 Temperature dependence of the zero-field resistivity

As explained on the previous pages, the temperature dependence of the in-plane

resistivity ρab(T) was measured on two YBa2Cu3Ox and one (Y0.6Pr0.4)Ba2Cu3Ox

epitaxial thin films at various levels of oxygen content. The results for the x =

6.45, 6.5, 6.7, 6.8 and x = 6.95 YBa2Cu3Ox films are summarised in figure 3.2

while the data for the (Y0.6Pr0.4)Ba2Cu3Ox x = 6.7, 6.85 and x = 6.95 films are

presented in figure 3.3.

From these plots, the resistivity at 290 K can be extracted, yielding values

between 420 and 4900 µΩcm that increase while lowering the oxygen content,

for each sample individually. The ρ(290 K) values are summarised in table 3.1

and they are in good agreement with the values reported in literature, ranging

from 300 µΩcm up to 104 µΩcm. Thus, the high-Tc superconductors are

located somewhere between the very good metallic conductors (e.g. like Cu)

with ρ(290 K) ~ 1 - 10 µΩcm and the poor conductors ρ290 K ~ 10 - 100 µΩcm

on one side and the semiconductors with ρ290 K ~ 104 to 1015 µΩcm on the other

side. The resistivity of film #1 is slightly higher than film #2, possibly because

of the uncertainty in the measurement of the thickness (see appendix C) or

possible over-etching (same appendix). However, the smaller value of the ratio

ρ290 K/ρo (see below) for film #1, compared to film #2, indicates that film #1

apparently contains more disorder than film #2.

These resistivities can be interpreted in term of the sheet resistance (or

resistance per square) R = (ρa)/(ad) = ρ/d, with a the side of the square and d

the thickness of the sheet. This resistance per square can, for the Y123

compounds, be estimated by considering them as a stacking of thin conducting

layers (e.g. the CuO2 planes) and setting d = c/2 ~ 5.85 Å. However, since the

two CuO2 planes are lying very close together in the Y123 unit cell, setting


47

d = c would also be an acceptable choice. This apparent uncertainty in R is no

problem, since the resistance per square can be regarded as an estimate for the

maximum sheet resistance for a metallic system. The estimated sheet resistance

can be compared to the quantum of resistance h/4e2 ≈ 6.45 kΩ. (equation 3.1).

⇒<<⇒>>

==insulator1

metal14

quantum econductanc

e/squareconductanc

21

2

c

ehlk

abF ρ

(3.1)

This comparison is preferably made on the basis of the low-temperature

resistivity (residual resistivity) where temperature dependent scattering (like the

phonon contribution) is small. Although this is just a rough estimate, the values

for the sheet resistance in table 3.1 already show that the samples with the

lowest doping approach the 6.45 kΩ limit, a strong hint towards an insulating

behaviour at low temperatures.

A remarkable observation concerning the Pr doped YBa2Cu3Ox is that, although

the ρ290 K values are equivalent with the undoped compound, the residual

resistivities ρo are significantly higher. This is probably related with the fact

that, as discussed in paragraph 1.3.2, the introduction of Pr in the YBa2Cu3Ox

unit cell causes, apart from hole filling and magnetic pair breaking effects,

appreciable disorder in the crystal. The relatively low values of ρ290 K/ρo for

these films (~ 2.5 for (Y0.6Pr0.4)Ba2Cu3Ox and 3 to 12 for YBa2Cu3Ox) indicate

their relatively high impurity concentration compared to normal metals where

ρ290 K/ρo ~ 1000. The method used for obtaining a reliable estimate for ρo will

be discussed in paragraph 1.2.2.

The plots in figure 3.2 and 3.3 reveal a temperature dependency of the

resistivity that contains a rich variety of features that, moreover, are

pronouncedly sensitive to the doping level. At optimal doping (YBa2Cu3O6.95)

the ρ(T) exhibits a remarkable linearity that extends down to just above the

critical temperature Tc. At lower levels of hole doping, the regime of linear ρ(T)

shifts to higher temperatures and an S-shape, super-linear, behaviour emerges

between Tc and T* (T* is commonly used to denote the temperature at which the

linear ρ(T) transforms into the super-linear behaviour). This development of a

region T < T* with a reduced resistivity is accompanied by an increase of the

slope in the region of linear ρ(T) and an overall increase of the resistivity (both


48

ρ290 K and ρo, see also table 3.1). This is because the in-plane transport is

determined by the number of charge carriers in the plane (and hence the doping

level). The stronger increase of ρo is an indication that reducing the oxygen

content or substituting Pr for Y, apart from lowering the charge carrier density,

also induces appreciable disorder (in Pr-doped YBa2Cu3Ox, there are at least

three effects playing: hole-filling changes the density of charge carriers,

magnetic scattering by the paramagnetic Pr ions and disorder).

0

200

400

600

800

1000

ρ ab

(µΩ

cm)

0 50 100 150 200 250 3000

500

1000

1500

2000

T (K)

ρab (µΩ

cm)

YBa2Cu3Ox

#1

#2

#2

#1

#1

6.456.56.76.86.95

x

Figure 3.2: The zero-field resistivity ρ(T) for the YBa2Cu3Ox films selected for this work.

These qualitative changes in the behaviour of ρ(T) for YBa2Cu3Ox,

(Y1-yPry)Ba2Cu3Ox and also La2-xSrxCuO4 are by now widely documented in the

literature for both thin films and single crystals [Almasan97, Batlogg94,

Boebinger96, Jiang97, Levin97, Takagi92, Wuyts94 & 96]. It was also shown

that in strongly underdoped samples, at low temperatures, an increase of the

resistivity occurs upon decreasing temperature. This insulating ρ(T) behaviour

is found to be compatible with the onset of superconductivity.

The critical temperature Tc at which superconductivity sets in is also closely

correlated with the hole doping (e.g. the oxygen content or the level of Y/Pr

substitution). When reducing the oxygen content of the thin films, the critical

temperature decreases, both in YBa2Cu3Ox (figure 3.2) and (Y0.6Pr0.4)Ba2Cu3Ox

(figure 3.3).


49

0 50 100 150 200 250 3000

500

1000

1500

2000

T (K)

ρ ab

(µΩ

cm)

Y0.6Pr0.4Ba2Cu3Ox

6.76.856.95

x

Figure 3.3: The zero-field resistivity ρ(T) for the (Y0.6Pr0.4)Ba2Cu3Ox thin films selected for this work.

In figure 3.4, the Tc(x) values for the YBa2Cu3Ox thin films studied in this work

are plotted together with the values obtained for bulk samples [Beyers89,

Cava90] for which the oxygen content was accurately determined by chemical

and thermogravimetric analyses. As already reported extensively in the

literature, the Tc(x) data exhibit a systematic decrease with decreasing oxygen

content x with a plateau near Tc ~ 60 K. From this plot, a good agreement

between our thin film samples and the bulk Tc(x) data is demonstrated.

A simple experimental phase diagram is obtained by plotting the Tc(x) data for

the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films in one graph (figure 3.5). As

noted above, both systems show a decreasing Tc(x) as x is reduced. In the

(Y1-yPry)Ba2Cu3Ox system however the substitution of 40 % of the Y atom by Pr

already suppresses Tc to about 41 K and a further reduction of the charge carrier

density by oxygen desorption induces an additional reduction of Tc. From this

phase-diagram, it is clear that the oxygen content x is not an appropriate

parameter for constructing a phase diagram for both YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox, since (Y0.6Pr0.4)Ba2Cu3Ox is already underdoped by the

40 % Y/Pr substitution. The precise mechanism of doping the 123 system by an

Y/Pr substitution is not yet clear (see paragraph 1.3.2). However, the solubility

for Pr at the Ba sites suggests a finite Ba/Pr substitution in the Y1-yPryBa2Cu3Ox


50

system to be possible. The suppression of superconductivity by adding Pr is

most probably a mixture of both hole filling and magnetic pair-breaking

[Tang99, Dalichaouch88, Xu92, Neumeier89] (see paragraph 1.3.2).

6.2 6.4 6.6 6.8 7.00

20

40

60

80

100

Bulk, Beyers et al. Bulk, Cava et al.

Thin Films, this work

x

Tc (

K)

YBa2Cu3Ox

Figure 3.4: Critical temperature Tc for YBa2Cu3Ox in the form of bulk material [Beyers89, Cava90] and thin films [this work].

6.0 6.2 6.4 6.6 6.8 7.00

100

200

300

400

500

T (

K)

x

YBa2Cu3Ox

AF

Y0.6Pr0.4Ba2Cu3Ox

SC

Tc

Tc

Tc

Figure 3.5: Experimental phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system with the critical temperature Tc, mid. The antiferromagnetic region is only indicative.


51

3.2.2 Scaling of the zero-field resistivity for different doping levels

The distinct features in the temperature dependence of the zero-field resistivity

of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox discussed above, i.e. a linear ρ(T) at

temperatures above, a super-linear ρ(T) behaviour below T* and a saturating or

increasing ρ(T) at low temperatures are universal for all the curves reported in

figure 3.2 and 3.3 (except for some samples the low temperature saturation).

The only difference is the temperature at which these features set in. This

observation has motivated scientists to make a successful scaling analysis of the

transport properties of YBa2Cu3Ox [Wuyts94 & 96].

A similar scaling analysis can be applied to our ρ(T) data of figures 3.2 and 3.3.

In figure 3.6 the scaled ρ(T) curve is plotted for the YBa2Cu3Ox thin films for

which the temperature is rescaled with a parameter ∆ and the resistivity is given

by (ρ-ρo)/(ρ∆-ρo). The parameter ∆ then defines the energy scale controlling the

linear and super-linear behaviour. The residual resistivity ρo is subtracted from

ρ and the resulting resistivity is then divided by ρ∆-ρo, with ρ∆ the resistivity at

T = ∆. This procedure and the nomenclature differ from the approach used by

Wuyts et al. who did not subtract the residual resistivity and used a crossover

temperature To rather than T*.

Similar to the previous reports [Wuyts94 & 96], a very nice scaling is obtained

for all YBa2Cu3Ox samples with oxygen contents from x = 6.4 up to the optimal

value of 6.95 (figure 3.6). The scaling reported in figure 3.6 is of an even better

quality than the previous attempts, due to the subtraction the residual resistivity.

This near-perfect quality of the scaling of the ρ(T) curves is most convincingly

demonstrated by the good overlap of the derivatives as illustrated in figure 3.7.

Within the noise level, the overlap is perfect; the upward deviations at low

temperatures mark the onset of superconductivity.


52

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T/∆

ρ ρρ ρ

−−

o

o∆

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

YBa2Cu3Ox

Figure 3.6: Scaling of the zero field ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆.

0 1 2 30

1

2

3

4

5

∆

T/∆

dρ/d

T

YBa2Cu3Ox

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

Figure 3.7: Derivative dρ/dT of the scaled resistivity of the YBa2Cu3Ox thin films shown in figure 3.6.

At this point, the question arises whether a similar scaling analysis works

equally well for the temperature dependence of the resistivity for the

(Y0.6Pr0.4)Ba2Cu3Ox thin films. This seems to be probable since qualitatively the


53

same features were observed in these films (a linear regime shifting to higher

temperatures and an S-shape region developing as the oxygen content is

decreased). The result of this scaling analysis is shown in figure 3.8. Also here,

a very good scaling of the ρ(T) data is obtained by subtracting the residual

resistivity and then linearly scaling the temperature with ∆ and the resistivity

with ρ∆. Moreover, the (Y0.6Pr0.4)Ba2Cu3Ox data collapse perfectly with the

YBa2Cu3Ox data shown in figure 3.6. Also here, the temperature derivatives of

the resistivity are in good agreement (figure 3.9).

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ ρρ ρ

−−

o

o∆

T/∆

Y0.6Pr0.4Ba2Cu3Ox

x = 6.7 / 6.85 / 6.95

Figure 3.8: Scaling of the zero field ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.7, 6.85 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆.

A remarkable observation from these rescaled ρ(T) data is however that,

although the (Y0.6Pr0.4)Ba2Cu3Ox and the YBa2Cu3Ox data fall on the same

universal curve, the energy scale ∆ in the two systems differs substantially. For

the same oxygen content, the scaling parameter ∆ that is needed to obtain the

universal curve is significantly higher for the (Y0.6Pr0.4)Ba2Cu3Ox films,

although the critical temperature is about a factor of 2 lower (see table 3.10).

This apparent higher energy scale ∆ in the (Y0.6Pr0.4)Ba2Cu3Ox system is further

illustrated by the rescaled ρ(T) data in figure 3.8 for which the linear regime is


54

only reached at room temperature (whereas for the YBa2Cu3Ox system the linear

regime is entered easily for the samples with the highest oxygen content).

0 1 2 3-1

0

1

2

3

4

T/∆

dρ/d

T

Y0.6Pr0.4Ba2Cu3Ox

x = 6.7 / 6.85 / 6.95

Figure 3.9: Derivative dρ/dT of the scaled resistivity of the (Y0.6Pr0.4)Ba2Cu3Ox thin films shown in figure 3.8.

composition x Tc, mid

(K)

∆∆

(K)

T*

(K)

ρρo

(µΩµΩcm)

YBa2Cu3Ox 6.95 92.2 86.5 173 48

6.8 73.7 126.9 254 34.9

6.7 58.2 174.3 349 209

6.5 52.9 198.6 397 95.3

6.45 41.7 215.1 430 607

6.4 17.9 248.5 497 1900

(Y0.6Pr0.4)Ba2Cu3Ox 6.95 41.4 158.7 317 277.7

6.85 31.8 185.9 372 458.5

6.7 22.3 184.5 369 804.8

Table 3.10: Oxygen content x, critical temperature Tc, scaling parameter ∆, crossover temperature T* ≈ 2∆ and residual resistivity ρo for the studied thin films. The energy scale ∆ and the residual resistivity ρo are scaling parameters used in figure 3.6 and 3.8.


55

Having established the perfect scaling of the zero field ρ(T) data of both the

YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system, a more detailed discussion is

appropriate. In figures 3.11 and 3.12, the rescaled data are plotted again, but

now with some extra indications to facilitate the discussion.

At high temperatures T > T*, a linear regime is observed. This regime is

universal for all samples, at least for those were T* is within our experimental

temperature interval (< 300 K). For the samples with the lowest oxygen content

this is not the case. For YBa2Cu3Ox, the temperature T* is approximately equal

to 2∆, the energy scale of the conduction process (see table 3.10). The

crossover temperature T* increases as the doping level is decreased. These

general observations are valid for both the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox system and the region of linear ρ(T) is labelled I in

figure 3.11. Since in (Y0.6Pr0.4)Ba2Cu3Ox we were not able to enter this region

unambiguously, no label was introduced there, and the temperature T* is defined as T* ≈≈ 2∆∆ (as observed experimentally for YBa2Cu3Ox).

At lower temperatures T < T*, the ρ(T) curves deviate from their linearity and

the curved, super-linear, regime sets in. Although already observed in the

individual ρ(T) curves, our scaling analysis implies this enhanced conductivity

to be a universal feature for the conduction (and hence the scattering process) in

the high-Tc samples. This region is labelled II in figures 3.11 and 3.12. The

increased conductivity is often explained by the opening of a pseudo spin-gap

below T* (for a review see [Timusk99]) and for that reason we have adopted this

nomenclature. The implications of our scaling analysis however are far more

general and are not restricted to any specific theoretical model.

At very low temperatures T < 0.25∆ for YBa2Cu3Ox and T < 0.35∆ for the

(Y0.6Pr0.4)Ba2Cu3Ox system, a saturation of the ρ(T) curves sets in, with a small

increase of ρ for the samples with the lowest oxygen content. This is the onset

of a region (labelled III) were the ρ(T) curves have a negative slope (see also

the derivatives in figure 3.7 and 3.9) and that might tentatively be identified

with the onset of localisation.


56

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T*

∆

T/∆

ρ ρρ ρ

−−

o

o∆

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

YBa2Cu3Ox

III II I

Figure 3.11: Scaling of the zero-field ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. Three regions of different ρ(T) behaviour are indicated together with the energy scale ∆ and the crossover temperature T*.

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ ρρ ρ

−−

o

o∆

T/∆

Y0.6Pr0.4Ba2Cu3Ox

III II

∆

x = 6.7 / 6.85 / 6.95

Figure 3.12: Scaling of the zero field ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.7, 6.85 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. The regions II and II of different ρ(T) behaviour are indicated, together with the energy scale ∆.


57

From the scaling, presented in figures 3.11 and 3.12, another parameter, the

residual resistivity ρρo, can be extracted. This property, that is otherwise

hardly accessible (unless by turning to some risky extrapolation), can be easily

obtained here since it's just a scaling parameter used in the construction of

figures 3.11 and 3.12. The residual resistivities ρo are summarised in table 3.10.

The perfect scaling of the metallic resistivity in at least region I and region II of

the ρ(T) plot in figures 3.11 and 3.12 is a strong indication that in all high-Tc

samples, the transport properties are dominated by the same underlying

scattering mechanism; from the strongly underdoped to the near optimally

doped samples. The only thing that changes when lowering the doping level, is

the energy scale on which this process is applicable. Since this scaling also

exhibits a good agreement with the pseudo spin-gap crossover temperature T*,

one is tended to pinpoint the origin of this dominant scattering mechanism to be

magnetic. Although this idea will be further elaborated in chapter 5, a first

phase diagram can already be constructed (figure 3.13).

6.0 6.2 6.4 6.6 6.8 7.00

100

200

300

400

500

T (

K)

x

II

I

AF

SCTc

T*Y0.6Pr0.4Ba2Cu3Ox

Tc T*

YBa2Cu3Ox

Tc T*

Figure 3.13: Experimental phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system with the critical temperature Tc, mid and the crossover temperature T*. The antiferromagnetic region is indicative.

In figure 3.13, the experimental phase diagram for the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox systems is now complemented with the crossover

temperature T* that marks the transition from the linear (region I) to the super-


58

linear (region II) ρ(T) behaviour on cooling down. Since the tendency of

localisation at low temperatures is still weak, region III is omitted in this phase

diagram.

From this experimental T(x) diagram it is clear that a substantial difference

exists between the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox systems. This is

probably related with the fact that the introduction of Pr in the YBa2Cu3Ox unit

cell possibly causes hole filling as well as magnetic pair breaking (and disorder

in the crystal). In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the

Y atom by Pr suppresses Tc to about 41 K but elevates ∆ and hence the

crossover temperature T*. This augmentation of T* is of such an order that it

exceeds the T* observed in YBa2Cu3Ox samples with the same oxygen content,

although the critical temperature of (Y0.6Pr0.4)Ba2Cu3Ox is about a factor of two

lower.

Therefore, when checking the phase diagram of figure 3.13 more carefully, it

seems that the oxygen content x is not a suitable parameter for constructing a

generic phase diagram but one should better turn to the "real" density of charge

carriers (accounting for possible hole-filling effects). From these data, it is not

clear whether the shift of the (Y0.6Pr0.4)Ba2Cu3Ox data to lower temperature in

the T(p) diagram, due to the reduction of the charge carrier density by the Y/Pr

substitution, suffices to obtain a reasonable overlap of the boundary lines in this

T(p) diagram. At least the Tc(p) boundary for (Y0.6Pr0.4)Ba2Cu3Ox is influenced

not only by the density of charge-carriers but also by additional magnetic pair-

breaking. This question -the existence of a single T(p) phase diagram for

YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox- will therefore be addressed later in this

work in the context of the Hall-effect data (chapter 4).

Another important question is what happens below the critical temperature Tc,

when superconductivity is suppressed by a high magnetic field ? Do all

samples show a saturating or insulating ρ(T) behaviour at low temperatures ? Is

it possible to construct a metal to insulator boundary ? Will the scaling of the

ρ(T) data still hold for the normal state below Tc ? These questions will be

addressed within the next paragraphs.


59

3.3 Suppression of superconductivity by high magnetic fields

In order to make more precise statements about the normal-state transport

properties below Tc it is necessary to suppress the superconducting state. The

application of a very high magnetic field is well-suited for this task since

experiments can be carried out during the process of the suppression of

superconductivity and, moreover, it has the advantage of being a fully reversible

process.

H

T

Hc2

Hc3

Hc1

Hirr

Tc

Hc2(0)

Figure 3.14: Schematic field versus temperature (H-T) phase diagram for type II superconductors.

The relation between type II superconductivity and externally applied magnetic

fields is quite complicated (see figure 3.14). At very small fields, all magnetic

flux is expelled and a perfect diamagnetic response is established (Meissner-

Ochsenfeld effect). Above the first critical field Hc1 however, flux lines enter a

type II superconductor individually and distribute themselves by minimising

their energy with respect to their mutual repulsion and their attraction to defects

in the superconductor. This results in a magnetic response that deviates

substantially from the ideal diamagnetic case but is of technological importance

in that a finite critical current can be accommodated due to the magnetic

gradient sustained by the flux lines that are pinned on the defects. Above the

so-called irreversibility field Hirr the flux lines become unpinned. This liquid of

individual flux lines penetrating the superconductor persists up to the second

critical field Hc2 that is significantly higher than Hc1 (see figure 3.14). Above


60

Hc2, bulk superconductivity is destroyed and only surface superconductivity

survives up to Hc3, at surfaces that are parallel to the applied field and that have

a roughness smaller than the coherence length ξ.

The suppression of bulk superconductivity at Hc2 can be attributed to two main

mechanisms. The first mechanism, known as orbital pair-breaking, is

important not too far from the critical temperature Tc (and hence low magnetic

fields) and the description using the linearised Ginzburg-Landau equations

yields a linear Hc2(T). The second mechanism, known as spin pair-breaking, is

active at lower temperatures and breaks up pairs of electrons or holes by simple

Zeeman splitting. This mechanism of breaking up the Cooper pairs is of

particular relevance in the type II superconductors in which above Hc1 the

magnetic field penetrates, couples with the spins and modifies the shape of the

Hc2(T) curve [Werthamer66] to bend down (figure 3.14). The upper limit of the

second critical field Hc2(0), the so-called Clogston-Chandrasekhar or Pauli

paramagnetic limit, in a spin-singlet superconductor can then be estimated by

comparing the energy of the magnetic Zeeman splitting µBgB with the energy

gap of the superconductor 3.53 kBTc (the BCS value), with µB the Bohr

magneton. When taking into account an extra factor 2 introduced by more

detailed calculations [Clogston62, Chandrasekhar62], the Pauli paramagnetic

limit equals to µoHP/Tc ~ 1.84 T/K. The Pauli pair-breaking will obviously

become an important issue at very high magnetic fields.

Experimentally, the Hc2(T) curve was found to have a negative curvature at high

temperatures. The Hc2(T) line transforms to an upward curvature at lower

temperatures and does not exhibit any saturation at low temperatures

[Osofsky93, Affronte94, Mackenzie94, Ando99]. These findings are in

disagreement with the conventional Werthamer-Helfand-Hohenberg (WHH)

theory [Werthamer66] (a refinement of the BCS electron-phonon theories by

including impurity scattering, electron spin and spin-orbit effects) that predicts a

negative curvature over the whole temperature range with a saturation at low

temperatures. This WHH Hc2(T) curve is plotted in figure 3.14. A lot of

theoretical effort has been put in obtaining a description of the Hc2(T) curve that

does agree with the experimental data for high-Tc's. Among the proposed

models are strong [Schossmann86] and very strong [Marsiglio87] coupling


61

models, (bi-) polaron models [Alexandrov87, 92 & 94] and theoretical work

[Tešanovic91, Rasolt92] predicting re-entrant superconductivity at high

magnetic fields H > Hc∞ (at which all charge carriers are in the lowest Landau

level) and low temperatures in very pure systems with a low density of charge

carriers.

The effect of the suppression of superconductivity, by applying a high magnetic

field, on the transport properties can not be described accurately by one single

mechanism or physical regime. At low magnetic fields (H < Hirr) all vortices in

the superconductor are pinned and a zero voltage drop occurs over the sample

(figure 3.15). When this irreversibility field is exceeded, vortices are de-pinned

and become mobile, thus giving rise to a finite dissipation. Hence, when a

small measuring current is applied, a finite voltage drop will appear due this so-

called flux creep (FC). At higher magnetic fields, the flux lines will move

freely and the Bardeen-Stephen flux flow model [Bardeen65] predicts the

resistivity to increase linearly with field (figure 3.15).

ρ

Flow

TAFF

H

FC

Hc2Hirr

ρ

Flow

TAFF

H

FC

Hc2Hirr

Figure 3.15: Schematic view of the ρ(H) transition below Tc from a irreversible superconductor to the normal state, traversing the regimes of thermally assisted flux flow (TAFF), flux creep (FC) and flux flow. These regimes are explained in the text.

At even higher fields H > Hc2 bulk superconductivity is destroyed and the

normal-state resistivity is recovered. At finite temperatures, the changes at Hirr

and Hc2 are smoothed over a finite range of magnetic fields, resulting in a

rounding of the corners of the superconducting to normal-state transition. At

Hirr, thermally assisted flux flow (TAFF) will allow flux lines to jump to


62

neighbouring potential wells in the landscape of pinning sites and thus give rise

to a finite dissipation, even below Hirr. At Hc2 the transition from the

superconducting to the normal state is smoothed by the existence of

superconducting fluctuations, even above Hc2 (figure 3.15).

The experimental resistivity versus field curves for the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox thin films are presented in figures 3.16 to 3.23. All curves

were taken in the transverse H // c and I // ab configuration at the pulsed

magnetic fields facility at the K.U.Leuven. The applied measuring current was

in the range 100-200 µA, resulting in a current density of approximately 1 to

2· 107 A/m2, much smaller than the critical current density (of the order

109 A/m2) and the depairing current 1013 A/m2. The signal was amplified 200

times and two traces with opposite field polarity were combined in order to

remove spurious inductive effects due to the high sweep rate of the magnetic

field. A description of the experimental procedure was given in chapter 2.

Figures 3.16 to 3.20 present the ρab(H) curves measured at temperatures varying

from T >> Tc down to 4.2 K for the YBa2Cu3Ox films with x = 6.95, 6.8, 6.7, 6.5

and 6.45.

0

100

200

300

400

ρ ab

(µΩ

cm)

µοH (T)

41.4 K

121.4 K

51 K

102.6 K

90.9 K

186 K

60.3 K55 K

YBa2Cu3O6.95

0 10 20 30 40 50

Figure 3.16: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.95 thin film at temperatures 41.4, 51, 55, 60.3, 65.8, 70, 75.9, 79, 81.5, 84, 86.6, 88.8, 90.9, 94.9, 102.6, 111, 121.4, 139.3, 157.9 and 186 K. Only the data taken at rising magnetic field are shown, the data were smoothed over 20 points.


63

In the optimally doped case (figure 3.16), above Tc a small positive

magnetoresistivity is present when sweeping the magnetic field up to 50 T. For

clarity, the data presented in figure 3.16 were smoothed over 20 points and only

show the sample response during rising magnetic field. When lowering the

temperature, even just above Tc a small low-field excess-conductivity develops

that is suppressed by high magnetic fields. The metallic tendency of decreasing

resistivity with lowering temperature seems to continue itself below Tc, when

considering the ρ(µoH = 50 T) data. This is in agreement with the resistance per

square, estimated in table 3.1 and paragraph 3.2.1 to be 0.8 kΩ, far below the

limit of h/4e2 ≈ 6.45 kΩ. When reducing the temperature to below Tc, a

superconducting transition develops in the ρ(H) curves. While close to Tc the

ρ(H) transition is narrow, it broadens significantly when lowering the

temperature. This indicates that the irreversibility field Hirr(T) and the second

critical field Hc2(T) separate from each other when going to lower temperature

in the H-T phase diagram (schematically drawn in figure 3.14). Note that below

40 K the 50 tesla pulsed magnetic field does not suffices to recover the normal,

non-superconducting, state.

The YBa2Cu3O6.8 data are presented in figure 3.17, no smoothing was applied

and the data taken during both rising and lowering magnetic field are shown.

Although exhibiting more noise, no important distortion of the data can be

observed and the two branches of the ρab(H) curves coincide almost perfectly.

Only at certain temperatures a small hysteresis can be observed, in most cases

due to a small drift of the electronic equipment between the field pulses. To

illustrate the fine quality of the measurements in pulsed magnetic fields, all

ρab(H) data presented below in this work will display the data-sets taken during

both rising and lowering magnetic field. From the YBa2Cu3O6.8 data, presented

in figure 3.17, similar observations can be made as for the optimally doped case.

Also here a small magnetoresistivity effect is present above Tc and the metallic

tendency for the ρ(µoH = 50 T ) data also seems to continue below Tc. When

reducing the temperature to below Tc, a superconducting behaviour develops in

the ρ(H) curves. However, while close to Tc the ρ(H) transition is narrow and it

significantly broadens when lowering the temperature, at the lowest

temperatures the transition steepens again ! This indicates the irreversibility

line Hirr(T) and the second critical field Hc2(T) to come closer again after their


64

initial separation at intermediate temperatures (see the schematic H-T phase

diagram in figure 3.14).

0

50

100

150

200

250

ρ ab

(µΩ

cm)

µοH (T)

0 10 20 30 40 50

4.2 K

153.7 K

141.5 K

119.5 K

100.3 K91.4 K78.4 K

YBa2Cu3O6.8

Figure 3.17: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.8 thin film at temperatures 4.2, 15, 20, 25, 30.2, 35.2, 40.5, 45.2, 51.9, 60.2, 65, 70, 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K. The data taken during both rising and falling magnetic field are shown, no smoothing was performed.

0 10 20 30 40 500

100

200

300

400

500

ρ ab

(µΩ

cm)

µοH (T)

YBa2Cu3O6.7

4.2 K

126.9 K

100.8 K

81 K

11.4 K

Figure 3.18: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.7 thin film at temperatures 4.2, 11.4, 19.4, 30, 38.8, 46.8, 52.4, 62.6, 69.9, 81, 100.8 and 126.9 K. The data taken during both rising and falling magnetic field are shown, a 20 point smoothing was performed.


65

Although at first sight the YBa2Cu3O6.7 data, presented in figure 3.18, are

similar to the x = 6.95 and x = 6.8 data (small magneto-resistive effect above Tc,

the transition width increasing and decreasing again as temperature is lowered)

an important difference becomes clear when taking a closer look. The obvious

metallic tendency (decreasing resistivity upon lowering temperature) above Tc is

for the YBa2Cu3O6.7 sample not unambiguously extended into the normal state

below Tc. On the contrary, according to the ρ(µoH = 50 T ) data, a saturation

(or even a mild increase) is present at around 50 K. From this one might

anticipate the YBa2Cu3O6.7 sample to be on the limit of having a localised

ground state at low temperatures.

0 10 20 30 40 500

50

100

150

ρ ab

(µΩ

cm)

µοH (T)

YBa2Cu3O6.5

73 K

4.2 K

8.9 K

14.2 K

66.9 K

Figure 3.19: Field dependence of the resistivity ρab(H) for an epitaxial YBa2Cu3O6.5 thin film at T = 4.2, 8.9, 14.2, 17.6, 22.6, 26.3, 30.5, 33.1, 36.8, 41.3, 45.9, 50.3, 54, 58.7, 62, 66.9 and 73 K. The data taken during both rising and falling magnetic field are also shown for certain temperatures, a 20 point smoothing was performed.

The ρab(H) data for the YBa2Cu3O6.5 sample at various temperatures are shown

in figure 3.19. Also in this x = 6.5 case, a ρab(H) transition from the

superconducting to the normal state develops below Tc, exhibiting an initial

broadening on cooling down, with a sharpening at the lowest temperatures.

However, this ρab(H) picture is very different from the data at higher oxygen

content in that at low temperatures the high field ρab(H) data show a strong

increase causing a crossing of the ρab(H) curves. This is a true sign of the


66

normal state resistivity at 50 T increasing strongly with decreasing temperature.

The initial tendency of metallic ρab(T) behaviour at high temperatures thus

transforms into a more insulating behaviour at low temperatures. For the

YBa2Cu3Ox sample with the lowest oxygen content, x = 6.45, this tendency is

even more pronounced (figure 3.20).

0 10 20 30 40 500

200

400

600

800

1000

98.7 K

4.2 K

ρ ab

(µΩ

cm)

µοH (T)

YBa2Cu3O6.45

8.7 K

124.7 K

73 K

13.7 K

Figure 3.20: Field dependence of the in-plane resistivity ρab(H) for an epitaxial YBa2Cu3O6.45 thin film at temperatures 4.2, 8.7, 13.7, 15.8, 20.7, 24.7, 28.7, 32.1, 40.5, 43.8, 55.3, 73, 98.7 and 124.7 K. The data taken during both rising and falling magnetic field are shown without smoothing.

In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the Y atom by Pr

already suppresses Tc to about 41 K and a further reduction of the charge carrier

density by oxygen desorption induces an additional decrease of Tc. The

disorder introduced by the Y/Pr substitution reflects in the ρab(T) curves shown

earlier in figure 3.3: a higher residual resistivity is obtained, accompanied by a

tendency for localisation at low temperatures (except for the x = 6.95 sample).

The question is what will happen below Tc when suppressing

superconductivity ? Figures 3.21, 3.22 and 3.23 present the ρab(H) curves at

temperatures varying from T >> Tc down to 4.2 K for the (Y0.6Pr0.4)Ba2Cu3Ox

films with x = 6.95, 6.85 and 6.7. In all three cases an increasing ρab(50 T) with

lowering temperature is observed. For the x = 6.95 sample this is a striking

observation since the zero-field ρab(T) curve shows no such tendency.


67

Y0.6Pr0.4Ba2Cu3O6.95

0 10 20 30 40 500

100

200

300

400

500

4.2 K8.1 K

12.7 K19.8 K

90.5 K

ρ ab

(µΩ

cm)

µοH (T)

Figure 3.21: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.95 thin film at temperatures 4.2, 8.1, 12.7, 19.8, 23.4, 27, 32.8, 35.5, 40.5, 45.1, 49.9, 55.5, 59.8, 70, 80.9 and 90.5 K. The data taken during both rising and falling magnetic field are shown without smoothing.

0 10 20 30 40 500

200

400

600

800

10004.2 K4.75 K

9.8 K

80 K

12.5 K16.1 K18.6 K


ρ ab

(µΩ

cm)

µοH (T)

Figure 3.22: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.85 thin film at temperatures 4.2, 4.75, 9.8, 12.5,16.1, 18.6, 21.2, 24.8, 27.9, 32.1, 37.6, 42.8, 51.6, 63.5, 68.7 and 80 K. The data taken during both rising and falling magnetic field are shown without smoothing.


68

0 10 20 30 40 500

500

1000

1500 Y0.6Pr0.4Ba2Cu3O6.7

ρ ab

(µΩ

cm)

µοH (T)

4.2 K

7.7 K

12.0 K

18.3 K23 K28 K36.7 K

Figure 3.23: Field dependence of the in-plane resistivity ρab(H) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.7 thin film at temperatures 4.2, 7.7, 12, 18.3, 23, 28, 36.7, 42, 53.9 and 69.6 K. The data taken during both rising and falling magnetic field are shown, no smoothing was performed.

Thus, even at optimal oxygen content, the 40 % Y/Pr substitution introduces

enough disorder to induce a low temperature localisation. This observation of

an insulating ground state is in agreement with the sheet resistances calculated

from the residual resistivities, reported in table 3.1, that are of the order of the

limiting value of h/4e2 ≈ 6.45 kΩ for a metallic system.

3.4 Normal-state resistivity at T < Tc in high magnetic fields

On the basis of the high field ρab(H) curves for the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox thin films, presented in figures 3.16 to 3.23, it is possible to

gain access to the normal-state resistivity below Tc. However it is not

immediately clear what criterion to take in order to obtain a ρab(T) that is a true

reflection of the underlying normal-state.

Above the critical temperature Tc only a small (but nevertheless finite)

magneto-resistive effect is observed, as illustrated in figure 3.24 for the Tc,mid =

73.7 K YBa2Cu3O6.8 sample. When approaching the critical temperature,

superconducting fluctuations create an excess conductivity that can only be


69

removed by applying a high magnetic field. In this T > Tc case one can thus

choose for the normal state resistivity between the ρab(0 T) and the ρab(50 T)

data. The ρab(0 T) data have the advantage of coinciding with the zero field

ρab(T) curve but contain an excess conductivity due to superconducting

fluctuations. The ρab(50 T) data are less susceptible to the influence of the

excess conductivity but contain a small magneto resistive contribution and are

therefore slightly overestimated. This can be taken into account by truly referring to these data as the "50 T normal-state transport properties" and not simply the normal state properties.

0 10 20 30 40 50

80

100

120

140

160

180

200

220

240

78.4 K

91.4 K

100.3 K

119.5 K

141.5 K

153.7 K

YBa2Cu3O6.8

ρ ab

(µΩ

cm)

µοH (T)

Figure 3.24: Illustration of the various possible methods for obtaining the normal-state resistivity above Tc from the ρab(H) curves. The data are for an epitaxial YBa2Cu3O6.8 thin film at temperatures 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K.

Below the critical temperature Tc, the picture is blurred by the existence of a

superconducting transition at fields H < Hc2 and superconducting fluctuations

even above this second critical field (figure 3.25). Here one does not have

simple access to the normal-state ρab(0 T) data and one is tempted into turning

to an extrapolation to zero magnetic field to obtain the "true" normal-state

resistivity. A simple linear extrapolation is easy to perform reproducibly by

using the quasi-linear regime at high magnetic fields. However, as can be seen

in both figure 3.24 and 3.25, such an extrapolation is bound to fail since it

depends heavily upon the precise nature of the real magnetoresistivity in this


70

region. From figure 3.24 it is clear that the normal state magnetoresistivity has

rather a quadratic-like behaviour than a simple linear one and a linear

extrapolation would yield values that would be significantly underestimated.

0

50

100

150

200

250

ρ ab

(µΩ

cm)

µοH (T)

0 10 20 30 40 50

4.2 K

153.7 K

141.5 K

119.5 K

100.3 K91.4 K78.4 K

YBa2Cu3O6.8

Figure 3.25: Illustration of the various possible methods for obtaining the normal-state resistivity below Tc from the ρab(H) curves. The data are for an epitaxial YBa2Cu3O6.8 thin film at temperatures 4.2, 15, 20, 25, 30.2, 35.2, 40.5, 45.2, 51.9, 60.2, 65, 70, 78.4, 91.4, 100.3, 119.5, 141.5 and 153.7 K.

This becomes particularly clear from figure 3.26 were the linear extrapolation is

shown to suffer from the excess conductivity at T ~ Tc and the systematic

underestimation at T < Tc. A quadratic extrapolation, however, is not a good

alternative due to the inherent uncertainty during its construction.

Therefore, in this work, the choice was made to take the 50 T ρab data as a true

reflection of the normal-state transport properties. This has the additional

advantage that by similarly gathering the ρab data at 0, 10, 20, 30, 35, 40, 45 and

50 T, a smooth transition from a superconductor to a metal or insulator can be

depicted. Moreover, by comparing the 45 T and the 50 T data it is possible to

see whether the ρab(T) curves at 45 and 50 T separate or not. A near-

coincidence (because of a finite magnetoresistivity) of these curves indicates

that the normal-state is fully reached and that the high field data reflect the "true" normal-state.


71

0

100

200

300

400YBa2Cu3O6.8

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

@ 50 T

0 T linear extrapolation

H

I a

bc

Figure 3.26: Influence of the criterion on the "normal state" resistivity below Tc for the YBa2Cu3O6.8 thin film.

The experimental ρab(T) resistivity versus temperature curves for the

YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox thin films are shown in figures 3.27 to

3.34. These plots contain the zero field ρab(T) combined with the high field 10,

20, 30, 35, 40, 45 and 50 tesla ρab(T) curves, constructed as described above,

and for the (Y0.6Pr0.4)Ba2Cu3Ox thin films, the value of the sheet resistance

corresponding to R = h/4e2 ≈ 6.45 kΩ is indicated on the plots. All high-field

curves were taken in the transverse H // c and I // ab configuration.

Figures 3.27 to 3.31 present the ρab(T) curves for the YBa2Cu3Ox films with x =

6.95, 6.8, 6.7, 6.5 and 6.45. In the optimally doped x = 6.95 case (figure 3.27),

the metallic behaviour (dρ/dT > 0) is extended to below Tc and no saturation is

present. Moreover, below 50 K the high magnetic fields do not manage to

destroy superconductivity and the superconducting state is recovered. The

magnetic field here simply broadens the ρab(T) transition.

When the oxygen content in the YBa2Cu3Ox samples is lowered (figures 3.28 to

3.31), the ρab(T) transition is broadened even further and the reduced energy

scale (lower Tc and hence lower critical fields) causes the high field ρab(T) data

to reflect the normal-state more closely. Indeed, the 45 T ρab(T) curve follows

the 50 T trace down to low temperatures for the samples with x ≤ 6.8.


72

T (K)

0 50 100 150 200 250 3000

100

200

300

400

500

600ρ a

b (µ

Ωcm

)

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

YBa2Cu3O6.95

Figure 3.27: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3O6.95 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T.

0

100

200

300

400

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

YBa2Cu3O6.8


As anticipated in the ρab(H) curves in figure 3.18, at an oxygen content x = 6.7 a

plateau in the ρab(T) curve starts to develop (figure 3.29). At even lower oxygen

contents x = 6.5 (figure 3.30) and x = 6.45 (figure 3.31), the plateau transforms

into an increasingly insulator-like behaviour (dρ/dT < 0) that diverges at

x = 6.45, corresponding to the strong tendency of crossing ρab(H) curves in


73

figures 3.19 and 3.20. This is a strong indication that, even in the region of the

YBa2Cu3Ox T(x) phase diagram were superconductivity shows up, the ground

state for x ≤ 6.7 has an insulating or semiconducting nature. This observation

agrees with our estimates above (table 3.1) for the sheet resistance of these

materials that become of the order of the limiting value of h/4e2 ≈ 6.45 kΩ.

0

200

400

600

800

1000

1200

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

YBa2Cu3O6.7


0

200

400

600

800

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

YBa2Cu3O6.5



74

0

500

1000

1500

2000

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

YBa2Cu3O6.45


In the (Y1-yPry)Ba2Cu3Ox system, the substitution of 40 % of the Y atoms by Pr

suppresses Tc to about 41 K for an optimal content of oxygen. The appreciable

disorder introduced by the Y/Pr substitution is reflected in the low ρ290 K/ρ0

values: 2.4 for the (Y0.6Pr0.4)Ba2Cu3O6.95 (optimally oxygenated) compared to

3.4 and 12 for the x = 6.45 and x = 6.95 YBa2Cu3Ox samples respectively. This

is especially remarkable since the (Y0.6Pr0.4)Ba2Cu3O6.95 (ρ290 K/ρ0 ~ 2.4) and the

YBa2Cu3O6.45 (ρ290 K/ρ0 ~ 3.4) samples exhibit almost identical critical

temperatures Tc,mid of respectively 41.4 K and 41.7 K.

Figures 3.32 to 3.34 present the ρab(T) curves for the (Y0.6Pr0.4)Ba2Cu3Ox films

with x = 6.95, 6.85 and 6.7. Already in the fully oxygenated sample

(figure 3.32), the metallic behaviour (dρ/dT > 0) above Tc transforms into an

insulating behaviour (dρ/dT < 0) when suppressing the superconducting state

and its fluctuations. This tendency is stronger in the oxygen deficient x = 6.85

and x = 6.7 samples (figures 3.33 and 3.34). In these last two cases the normal

state is attained easily (even at 40 tesla) and the low temperature state of

insulating (diverging !) ρab(T) is clearly demonstrated in all these

(Y0.6Pr0.4)Ba2Cu3Ox samples, although no sign of a saturating ρab(T) is present

above Tc in the x = 6.95 sample.


75

Also for the (Y0.6Pr0.4)Ba2Cu3Ox films, the insulating behaviour agrees with the

rough estimates made in table 3.1 for the sheet resistance of these materials. On

the plots (3.32 to 3.34), the limiting value of R = h/4e2 ≈ 6.45 kΩ is indicated.

0

100

200

300

400

500

600

700

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)


Rÿ = 6.45 kΩ

Figure 3.32: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.95 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T. The value of the sheet resistance corresponding to R = h/4e2 ≈ 6.45 kΩ is indicated on the plot.

0

200

400

600

800

1000

1200Y0.6Pr0.4Ba2Cu3O6.85

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

Rÿ = 6.45 kΩ

Figure 3.33: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.85 thin film at 0, 10, 20, 30, 35, 40, 45 and 50 T. The value of the sheet resistance corresponding to R = 6.45 kΩ is indicated on the plot.


76

45 / 40 / 35 / 30 / 20 / 10 / 0 T

T (K)

0

500

1000

1500

2000

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

Y0.6Pr0.4Ba2Cu3O6.7

Rÿ = 6.45 kΩ

Figure 3.34: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3O6.7 thin film at 0, 10, 20, 30, 35, 40 and 45 T. The value of the sheet resistance corresponding to R = 6.45 kΩ is indicated on the plot.

3.4.1 Scaling of the metallic normal-state resistivity

Now that the normal-state transport properties below Tc for the YBa2Cu3Ox and

the (Y0.6Pr0.4)Ba2Cu3Ox samples have been determined experimentally, it is

good to take a second look at the zero-field scaling and to check whether it is

also valid for the high-field normal-state below Tc.

In figure 3.35, the scaled ρab(T) is given for the YBa2Cu3Ox; x = 6.4, 6.45, 6.5,

6.7, 6.8 and the x = 6.95 samples. For all these samples (except the x = 6.4

sample) the ρab(50 T) data were added, after being scaled by the same

parameters ∆, ρo and ρ∆ (see paragraph 3.2.2). For the optimally doped sample,

the ρab(50 T) curve follows the zero field ρab(T) accurately, until below T/∆ ~ 1

the fields necessary to suppress the superconducting state go beyond the 50 T

we can access experimentally. Therefore, below this temperature the ρab(50 T)

curve starts to deviate from the zero field ρab(T) by showing the onset of the

superconducting state. The YBa2Cu3O6.8 and YBa2Cu3O6.7 samples demonstrate

a ρab(50 T) behaviour that follows the universal curve, until at a certain

temperature there is a clear tendency towards a saturating ρab(T). Also the

YBa2Cu3O6.5 and YBa2Cu3O6.45 samples show this saturating behaviour and


77

even exhibit a diverging ρab(T) at the lowest temperatures. The data for these

last two samples (x = 6.45 and 6.5) however seem to follow the universal curve

to lower temperatures and a resistance minimum is observed at a temperature

TMI with TMI/∆ ~ 0.25. For the samples with x = 6.7 and x = 6.8, this simple

ratio is a lower limit, since the tendency for ρab(T) to saturate is already present

slightly above T ~ 0.25 ∆.

x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T/∆

ρ ρρ ρ

−−

o

o∆

T*

∆

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

YBa2Cu3Ox

III II I

x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T/∆

ρ ρρ ρ

−−

o

o∆

T*

∆

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

YBa2Cu3Ox

III II I

Figure 3.35: Scaling of the zero field and 50 tesla ρ(T) data for the YBa2Cu3Ox thin films with x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. The three regions of different ρ(T) behaviour are indicated as well as the energy scale ∆ and the crossover temperature T*.

We can conclude that the resistivity versus temperature data for the YBa2Cu3Ox

system scale - in regions I and II - perfectly onto a universal curve. For lower

temperatures, i.e. region III, the samples with the lowest oxygen content deviate

from the universal behaviour.

In figure 3.36, the scaled ρab(T) is shown for the (Y0.6Pr0.4)Ba2Cu3Ox sample

with x = 6.7, 6.85 and x = 6.95. The high field data were added after scaling

them with the same ∆, ρo and ρ∆ parameters employed for the zero-field scaling

(see paragraph 3.2.2).


78

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ ρρ ρ

−−

o

o∆

T/∆

Y0.6Pr0.4Ba2Cu3Ox

III II

∆

x = 6.7 / 6.85 / 6.95

x = 6.7x = 6.85x = 6.95

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ ρρ ρ

−−

o

o∆

T/∆

Y0.6Pr0.4Ba2Cu3Ox

III II

∆

x = 6.7 / 6.85 / 6.95

x = 6.7x = 6.85x = 6.95

Figure 3.36: Scaling of the zero field and 50 tesla ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films with x = 6.85 and x = 6.95. For the x = 6.7 sample, only the 45 T data were experimentally accessible. The temperature is rescaled with a parameter ∆ (an energy scale) and the resistivity is given by (ρ-ρo)/(ρ∆-ρo) in which the residual resistivity ρo is subtracted and ρ∆ is the resistivity at T = ∆. Two of the three regions of different ρ(T) behaviour are indicated as well as the energy scale ∆.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-4

-2

0

2

T/∆

dρ/d

T

Y0.6Pr0.4Ba2Cu3Ox

x = 6.7 / 6.85 / 6.95

Figure 3.37: Derivative dρ/dT of the scaled resistivity of the (Y0.6Pr0.4)Ba2Cu3Ox thin films shown in figure 3.36, also including the high field data.


79

From figure 3.36, it is clear that in the (Y0.6Pr0.4)Ba2Cu3Ox case a reasonable

scaling was obtained for the low temperature diverging resistivity in region III

(apart from the already discussed scaling in regimes I and II). This is illustrated

by the good overlap of the derivatives dρ/dT of the scaled resistivity

(figure 3.37). In the (Y0.6Pr0.4)Ba2Cu3Ox case, the resistance minimum was

observed at T/∆ ~ 0.35 (see also the derivatives in figure 3.37), somewhat

higher than the ratio 0.25 found for YBa2Cu3Ox.

It is not a priori clear whether such a universal scaling should be fulfilled for the

low temperature regime in region III. The behaviour of the resistivity in this

region is dominated by the localisation of charge carriers due to the disorder. It

would rather come as a surprise if the energy scale ∆ describing the scattering

process in the metallic regions I and II would also be able to give a good

description of these insulator-like features in regime III.

The reasonable scaling of the high-field data of the (Y0.6Pr0.4)Ba2Cu3Ox samples

can be understood in view of the high energy-scale ∆ for these samples at fixed

oxygen content (table 3.10 and figure 3.36) which, combined with the higher

value T/∆ ~ 0.35 for the resistance minimum, yields a temperature TMI for the

onset of localisation that lies significantly above the TMI values for YBa2Cu3Ox

(with a lower ∆ and T/∆ ~ 0.25). This implies that the Y/Pr substitution seems

to enhance disorder considerably, resulting in a rather high TMI. The variations

in ρ(T) at low temperatures, due to the additional disorder arising from the

oxygen desorption, are made less clear by the high value of ∆, squeezing the

low-temperature data together. In the YBa2Cu3Ox system these deviations are

neither obscured by other disorder-contributions, nor compressed by the T/∆

rescaling and so significant deviations clearly show up at low temperatures.

The substantial discrepancy between the YBa2Cu3O6.7 and YBa2Cu3O6.8 samples

has an additional explanation in that the normal state magnetoresistivity ρ(H)

exhibits a strong field dependence at around 40 to 50 Kelvin (figures 3.17 and

3.18). This can also be seen from the ρ(T) plots in figures 3.28 and 3.29 where

the 40 T and 50 T separate slightly. This additional high-field effect,

superimposed onto the normal-state transport properties hinders a proper

analysis.


80

In the remaining part of this paragraph the focus will be on the metallic part of

the universal ρ(T) curve, where the scaling works fine i.e. region I and II. The

low-temperature regime of insulating behaviour (region III) will be discussed

later in this chapter.

The linear temperature dependence of the resistivity ρ(T), as observed in region

I (at high temperatures) has puzzled the high-Tc community for the last decade.

Numerous models were proposed to explain this robust linear resistivity. The

models based on the electron-phonon coupling, however, are bound to fail since

for the optimally doped compounds the absence of any deviation from linearity

down to Tc implies a coupling factor that is unable to explain the high values for

Tc. Other models involve real-space paring of polarons [Alexandrov87 & 88,

Mott90], the existence of a narrow, metallic, impurity band [Moshchalkov88 &

90, Quitmann92] or a spin-charge separation into spinons and holons in the

Resonating Valence Bond (RVB) model [Nagaosa90, Suzumura88,

Anderson88] (see also Chapter 1). However, any model trying to accurately

describe the normal-state transport properties of the cuprate superconductors

should also account for the S-shaped super-linear behaviour observed in

region II. Recent experimental evidence [Ito93, Wuyts94 & 96, Batlogg94]

shows the super-linear resistivity in this regime to be closely connected to the

antiferromagnetic (AF) fluctuations in the underdoped cuprates.

Recent models therefore include these AF fluctuations explicitly [Pines90 & 97,

Hlubina95, Chubukov96, Li99, Yanase99] and also take into account the

presence of stripes [Emery97, 97b & 99, Randeira97, Moshchalkov93, 98, 99,

99b & 99c] that are observed both in La2-xSrxCuO4 and YBa2Cu3Ox

([Thurston89, Cheong91, Mason92 & 94, Yamada97, Tranquada97, Hunt99]

and [Aeppli97, Tranquada97, Dai98, Kao99, Arai99] respectively).

The nearly antiferromagnetic Fermi-liquid model (NAFLM) for cuprates

[Hlubina95, Chubukov96, Pines97, Li99] assumes no spin-charge separation

but considers the interactions between Fermionic charge carriers and the

localised spins. Calculations in the framework of this NAFLM yield a ρ(T) ~ T

at high temperatures and an approximate ρ(T) ~ T 2 behaviour in the low

temperature region. The precise position of the crossover between these two

regimes is not yet established and the present degree of refinement of the NAFL


81

model does not yet provide a sufficient agreement with experimental data to be

conclusive [Li99]. Along the same tracks, Moshchalkov proposed a model for

quantum transport in 2 dimensional (2D) Heisenberg [Moshchalkov93] and

1 dimensional (1D) systems [Moshchalkov98b], which was successfully applied

for YBa2Cu3Ox and YBa2Cu4O8 [Moshchalkov99, 99b & 99c, Trappeniers99].

This model implicitly accounts for the recently established striped structure of

the CuO2 planes and contains the spin pseudo-gap as a simple fitting parameter.

The model yields an excellent fit of the available experimental data. In

chapter 5, the model and its applicability will be discussed in more detail.

3.5 Comparison with the La2-xSrxCuO4 system

It is interesting to compare these findings on YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox thin films with the results from resistivity experiments on

ultra thin La2-xSrxCuO4 layers in which Tc was additionally changed by applying

in-plane strain instead of chemical doping.

For these experiments, at the IBM research labs in Zürich, three ultra-thin

La1.9Sr0.1CuO4 films of fixed stoichiometry were prepared by molecular beam

epitaxy with block-by-block deposition [Locquet94 & 98]. The choice of the

substrate - SrLaAlO4 or SrTiO3 - determines the presence of compressive

(SrLaAlO4) or tensile (SrTiO3) epitaxial strain in the ab-plane (see figure 1.5 in

paragraph 1.3.3).

Figure 3.38 presents the ρab(T) curves for these three strained thin films. The

zero-field ρab(T) data show similar features upon the application of epitaxial

strain as were observed in the YBa2Cu3Ox system upon doping by changing the

oxygen content (or in La2-xSrxCuO4 itself upon changing Sr-content x

[Takagi92]). All three samples have a xSr = 0.1 stoichiometry which is -

ignoring strain effects- on the underdoped side of the phase diagram. Bulk

material of this composition is free of strain effects and has a Tc of about 25 K.

Samples A and B were deposited on SrLaAlO4 and have an effective thickness t

of 125 Å and 150 Å respectively. The compressive strain in these films results

in Tc values of respectively 47.1 K and 35.4 K that are significantly higher than


82

the bulk value and they even exceed the value of 38 K of the optimally doped

samples (table 3.39). The application of compressive epitaxial strain induces an

almost linear ρab(T), contrasting with the more S-shaped ρab(T) in unstrained

material [Takagi92]. Further inspection also reveals a trace of a cross-over T*

around 250 K for the sample with the largest compressive strain (sample A),

contrasting with T* ~ 400 K for the unstrained material [Takagi92]. These

observations (higher Tc, linear ρab(T) and reduced T*) suggest that the

compressive strain possibly leads to an additional doping of the CuO2 planes.

0 50 100 150 200 250 3000

200

400

600

800

T (K)

La1.9Sr0.1CuO4

ρ (µ

Ωcm

)

sample C, 150 Å on SrTiO3

sample B, 150 Å on SrLaAlO4

sample A, 125 Å on SrLaAlO4

Figure 3.38: Zero-field and 50 T resistivity versus temperature ρab(T) for the ultra-thin La1.9Sr0.1CuO4 films with tensile strain (sample C, up triangles) and compressive strain (sample B, circles and A, down triangles).

Name Substr. t

(Å)

Tc, mid

(K)

∆∆T

(K)

LC416 SrTiO3 150 14.1 7.2

LC391 SrLaAlO4 150 35.4 1.7

LC438 SrLaAlO4 125 47.1 1.7

Bulk ~ 25

Table 3.39: Overview of the ultra-thin La1.9Sr0.1CuO4 films with the substrate, thickness t and critical temperature Tc.


83

Sample C consists of 15 unit cells (~ 150 Å), deposited on SrTiO3 and the

induced tensile strain results in a critical temperature Tc ~ 14.1 K, significantly

lower than the bulk value (see table 3.39). The ρab(T) curve for this sample is

slightly curved and already around Tc a tendency towards insulating behaviour

is present. A comparison with the results on polycrystalline and thin film

La2-xSrxCuO4 [Takagi92] without strain-effects suggests that tensile strain

possibly causes an effective underdoping of the samples.

No scaling behaviour of the temperature dependence of the resistivity was

found for these three strained samples since the features in region I and II

(linear, super-linear) are not sufficiently pronounced.

Tentative results from Hall measurements on ultra-thin films under epitaxial

strain from the same research group [Locquet98] indicate a possible trend of a

decreasing carrier concentration in the CuO2 plane (lower doping) as

compressive strain is applied [Locquet2000]. If these findings are confirmed,

the origin of the marked changes in the superconducting and normal-state

properties upon compressive or tensile strain might be due to in-plane processes

like a changing orbital overlap, magnetic coupling or altered scattering

processes.

0 10 20 30 40 500

50

100

150

ρ (µ

Ωcm

)

125 Å La1.9Sr0.1CuO4 deposited on SrLaAlO4

µoH (T)

72.7 K

61.6 K

46.5 K

14.7 K

Figure 3.40: Field dependence of the in-plane resistivity ρab(H) for the 125 Å La1.9Sr0.1CuO4 film on SrLaAlO4, with compressive strain (sample A) at temperatures 14.7, 17, 20, 25, 30.2, 34.4, 39, 42, 46.5, 61.6, 72.7 K. For some traces the branch at lowering magnetic field was omitted, the data were smoothed over 20 points.


84

In order to gain knowledge upon the normal-state transport properties of these

strained ultra-thin films at temperatures below their critical temperature Tc,

transport measurements in pulsed magnetic fields were performed. The ρ(H)

traces at various temperatures above and below Tc are summarised in

figures 3.40 (sample A), 3.41 (sample B) and 3.42 (sample C). Whereas samples A and B exhibit metallic features (decreasing resistivity upon lowering temperature), sample C (tensile strained) shows a strong tendency towards an insulating ρρ(T), with a crossing of the ρ(H) traces as temperature is

lowered (figure 3.42).

0 10 20 30 40 500

50

100

150

200 150 Å La1.9Sr0.1CuO4 deposited on SrLaAlO4

4.2 K

62.6 K

43.7 K

ρ (µ

Ωcm

)

µoH (T)

Figure 3.41: Field dependence of the in-plane resistivity ρab(H) for the 150 Å La1.9Sr0.1CuO4 film with compressive strain, deposited on SrLaAlO4 (sample B) at temperatures 4.2, 13, 20.2, 30, 35.8, 43.7 and 62.6 K. The traces during both rising and lowering field are shown, no smoothing was performed.

From these ρ(H) plots, the temperature dependence of the resistivity in the

normal state (the normal state at 50 T) can be reconstructed at temperatures

below Tc. This normal state resistivity is, for all three samples, given in

figure 3.38. For the samples that show a metallic ρ(T) above Tc (samples A and

B) this behaviour is simply continued below Tc by the high field data. For

sample C, that already demonstrated a slight tendency towards a saturating

resistivity, a strong divergence of the low-temperature resistivity is seen

(figure 3.38). These observations further support that, apart from possibly


85

modifying the effective doping of the CuO2 plane, adding epitaxial strain also

causes disorder-induced scattering becoming important in materials with tensile

in-plane strain.

0

100

200

300

400

500

0 10 20 30 40 50

150 Å La1.9Sr0.1CuO4 deposited on SrTiO3

4.2 K

ρ (µ

Ωcm

)

µoH (T)

53.4 K62.5 K

Figure 3.42: Field dependence of the in-plane resistivity ρab(H) at temperatures 4.2, 7.6, 10.1, 13.7, 20.3, 26, 32, 40, 46.6, 53.4 and 62.5 K for the 150 Å La1.9Sr0.1CuO4 film with tensile strain, deposited on SrTiO3 (sample C). Only the traces during rising field are shown, the data were smoothed over 20 points.

3.6 Localisation effects at T →→ 0 in the YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and La1.9Sr0.1CuO4 samples

The low-temperature in-plane resistivity of the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox samples strongly increases as temperature is lowered. This

diverging ρ(T) behaviour (region III) was in the previous paragraph shown to

deviate from the universal scaling in the metallic regime (region I and II).

The ρ(T) curves for the strongly underdoped samples exhibiting such an

insulating tendency are plotted in figure 3.43 (for the x = 6.45 and 6.5

YBa2Cu3Ox films), figure 3.44 (for the x = 6.7, 6.85 and 6.95

(Y0.6Pr0.4)Ba2Cu3Ox thin film) and figure 3.38 (for sample C, the tensile strained

La1.9Sr0.1CuO4 ultra-thin film). For the two YBa2Cu3Ox samples with the lowest

oxygen content a diverging ρ(T) is seen; although the onset of the

superconducting state becomes visible at the lowest temperatures. Due to their


86

significantly lower critical temperatures, the tensile strained La1.9Sr0.1CuO4

ultra-thin film and the (Y0.6Pr0.4)Ba2Cu3Ox film (at all levels of oxygen content)

show a strongly diverging ρ(T) down to the lowest experimentally accessible

temperatures (4.2 K).

0

500

1000

1500

0 50 100 150 2000

200

400

600

T (K)

ρ ab

(µΩ

cm) YBa2Cu3Ox

x = 6.45

x = 6.5

ρab (µΩ

cm)

Figure 3.43: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial YBa2Cu3Ox thin film without an external magnetic field and at 50 tesla with x = 6.45 and 6.5.

0

500

1000

1500

2000

x = 6.95

x = 6.85

x = 6.7

T (K)

0 50 100 150 200 250 300

ρ ab

(µΩ

cm)

Y0.6Pr0.4Ba2Cu3Ox

Figure 3.44: Temperature dependence of the in-plane resistivity ρab(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3Ox thin film without an external magnetic field and at 45 tesla (for x = 6.7) and 50 tesla (for x = 6.85 and 6.95).


87

The origin of such a low temperature increase of the resistivity can be attributed

to several mechanisms. At low temperatures, a simple thermally activated

process yields a diverging ρ(T) ~ exp(1/T) behaviour. In many cases, however,

a weaker dependency is observed. The general equation 3.2 describes the so-

called hopping of charge carriers between two localised states, with To a

characteristic temperature and α a constant that is determined by the precise

conditions of this hopping process. The simple case of Mott variable range

hopping [Mott79] yields a power α = 1/3 or 1/4 in two- and three dimensions,

respectively, by assuming an energy-independent density of states near the

Fermi level (n = 0 in equation 3.3). The 3D α = 1/4 case transforms into

α = 1/3 in very high magnetic fields when the magnetic length λm becomes of

the order of the localisation radius. At low carrier concentrations, due to poor

screening, a soft Coulomb-gap can develop around the Fermi-level (0 < n ≤ 2,

n = 2 for electron-electron repulsion) thus yielding α = 1/2 in 3 dimensions

[Shklovskii84]. For arbitrary n (nature of the gap) and D (dimension of the

problem), the power α is given by equation 3.4.

=α

ρρT

TT o

o exp)( (3.2)

n

FEEEg −~)( (3.3)

1

1

+++

=Dn

nα (3.4)

To check the validity of such activated or hopping type of charge transport, all

low-temperature data with an insulating tendency (figures 3.43, 3.44 and 3.38)

were analysed. Figure 3.45 shows such an analysis for the low-temperature part

of the 50 T ρab(T) curve of the YBa2Cu3O6.45 sample; this picture is

representative for all other data. In this plot, the resistivity is plotted

logarithmically versus T -α with α ranging from 0.1 up to 0.5. Any behaviour

obeying equation 3.2 should then give a linear plot at some value of α. From

this figure, it is clear that no good correspondence is obtained for any value of

α; only in the α < 0.1 case some agreement is found. This tendency is observed

in all the low-temperature resistivity data.


88

YBa2Cu3O6.45

0.0 0.2 0.4 0.6 0.8

T-α

103

ρ ab

(µΩ

cm)

α = 0.5 0.4 0.33 0.25 0.2 0.1

50 T

high T low T

Figure 3.45: Low-temperature part of the ρab(T) curve, plotted logarithmically versus T -α for an epitaxial YBa2Cu3O6.45 thin film at 50 tesla.

The fact itself, that no correspondence is found with the known α = 1/4, 1/3 or

1/2 coefficients, is not sufficient to rule out the possibility of a hopping process

being relevant; as the precise value of this parameter is determined by assuming

a rather soft g(E) around the Fermi-level (equation 3.3). However, the linearity

observed only at α < 0.1 (by squeezing the data together) implies the physics to

obey a severely modified form of equation 3.2. This limit of α < 0.1 yields a

much smoother ρ(T) than the generic hopping-transport and our data can thus

not be accurately described by equation 3.2.

Recently, high field experiments by Ando and co-workers on the normal-state

transport properties of Bi2Sr2CuOy (Bi2201) [Ando96c, 97, 97b] and

La2-xSrxCuO4 (La214) [Ando95, 96, 96b, 97, 97b, Boebinger96] showed both

the in-plane ab as the out-of-plane c-axis high-field resistivity to diverge as

~ ln(1/T) at low-temperatures. This is indeed a much smoother divergence than

proposed by the hopping models. To check the validity of this observation for

our data on the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox samples, in figures 3.46

and 3.47 the in-plane resistivities were re-plotted versus ln(T). From these two

plots, a nice linear behaviour ρ(ln(T)) can be observed thus confirming the

ln(1/T) divergence for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox samples.

Also in the tensile strained La1.9Sr0.1CuO4 ultra-thin film, a good agreement of


89

the high-field data with the ln(1/T) divergence was found. This agreement of

the high-field ρ(T) data with the ln(1/T) divergence is better than the

correspondence with a simple power law T-α [Hao00].

0

500

1000

1500

ln(T)

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

200

400

600

x = 6.45

x = 6.5

ρ ab

(µΩ

cm)

YBa2Cu3Ox

ρab (µΩ

cm)

50 T

Figure 3.46: Temperature dependence of the in-plane resistivity ρab versus ln(T) for an epitaxial YBa2Cu3Ox thin film without an external magnetic field and at 50 tesla with x = 6.45 and 6.5.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.50

500

1000

1500

2000

ln(T)

ρ ab

(µΩ

cm)

Y0.6Pr0.4Ba2Cu3Ox

x = 6.95

x = 6.85

x = 6.750 T

Figure 3.47: Temperature dependence of the in-plane resistivity ρab versus ln(T) for an epitaxial (Y0.6Pr0.4)Ba2Cu3Ox thin film without external field, at 45 tesla (for x = 6.7) and at 50 tesla (for x = 6.85 and 6.95).


90

Although the low-temperature (high-field) ln(1/T) divergence is by now an

experimental fact for both the ab-plane and the c-axis resistivities in Bi2201

[Ando96c, 97, 97b] and La214 [Ando95, 96, 96b, 97, 97b, Boebinger96] and at

least the ab-plane resistivity in Y123 and YPr123 [this work and

Trappeniers99], no consensus exists on the nature of this "localisation" of the

charge carriers. A whole variety of models can provide us with a divergence of

the resistivity at low temperatures.

However, some predictions for the resistivity of the high-Tc cuprates violate the

available experimental data [Ando, Boebinger96, Trappeniers99, this work] in

the low-temperature limit and their relevance for these low-temperature normal-

state transport properties of the high-Tc cuprates can thus be questioned.

The ln(1/T) dependence of the resistivity as predicted from spin-flip scattering

in the Kondo framework fails at low temperatures where µBgB > kBT and the

"impurity" spins are aligned by the external magnetic field. For a 50 T

magnetic field, the spin-flip process starts to diminish at around T < 50 K and

the resistivity should saturate. This contradicts the experimental findings.

The model described earlier in this chapter, explaining the linear ρ(T) at high

temperatures by assuming the development of a narrow metallic impurity band

with localised edges upon doping [Moshchalkov88 & 90, Quitmann92] also

predicts a diverging resistivity at low temperatures. The disagreement of the

proposed ρ(T) ~ exp(1/T) with the experimental high-field data might be due to

the simplification by approximating the impurity band by a square density of

states g(E).

Some models cannot be ruled out immediately on the basis of experimental data

at this point.

Without being complete, it is worthwhile to mention the phenomenological c-

axis resistivity model [Zha96] predicting a divergent ρc and another model

accounting for electron interactions in 2D disordered systems [Altshuler80]

predicting a ln(1/T) dependence in both ab and c directions. Both models rely

on the suppression of the 2D in-plane density of states as measured in Knight

shift experiments. Also the bipolaron model gives a logarithmically diverging

resistivity by assuming a temperature dependent scattering time for the


91

bipolarons [Alexandrov97]. The 2D Luttinger liquid model [Anderson91,

Clarke95] predicts a resistivity that diverges as a power law, both in ρab(T) as in

ρc(T), by assuming the existence of a confined spin-charge separated liquid with

incoherent hopping between the planes. The agreement of our high-field data

with such a power law is less convincing than the correspondence with the

ln(1/T) behaviour.

In the 2D Anderson weak localisation theory, a charge carrier is localised due to

the interference with its coherently backscattered wave function. This model

was shown to yield an ln(T) correction to the conductivity [Lee85]. This

correction then transforms into an ln(1/T) correction for the resistivity (inverse

conductivity). However, coherent backscattering can be frustrated by an

external magnetic field and at 50 T might possibly result in a negative

magnetoresistivity.

A model that is of particular interest is the charge-stripe picture [Cheong91,

Tranquada97, Arai99]. The dynamic incommensurate stripes, observed also in

YBa2Cu3Ox [Aeppli97, Tranquada97, Dai98, Kao99, Arai99], are shown to

yield metallic transport properties [Noda99, Ichikawa99, Tajima99,

Moshchalkov99 & 99c]. When the 1D charge stripes are pinned and the stripe

order increases, the movement of the charge carriers is restricted to these fixed

1D paths and the resistivity is reported to increase [Noda99, Lavrov99]. These

findings for the insulating regime (region III) complement the model for

quantum transport in 1D and 2D Heisenberg systems [Moshchalkov93, 98, 99,

99b & 99c, Trappeniers99] that already gives a proper description for the

metallic (striped) regime (region II) and 2D Heisenberg regime (region I). The

possible relevance of the stripe pinning to the low temperature divergence of the

resistivity will be discussed in detail in chapter 5.

In any case, whatever the origin of the insulating-alike behaviour, we can

complement the phase diagram (figure 3.13) with the temperature TMI at which

the metallic dρ/dT > 0 transforms into an insulating dρ/dT < 0 behaviour

(figure 3.48). This boundary TMI was (arbitrarily) taken at the resistance

minimum and it separates region II and III, introduced earlier. For the lowest

oxygen contents in YBa2Cu3Ox this yields TMI ~ 0.25 ∆. As discussed above,

the insulating resistivity part of the samples with x = 6.7, 6.8 and 6.95 does not


92

scale very well onto the universal curve and therefore has no such simple

expression for TMI. Moreover, in these samples, the resistance minimum was

not observed unambiguously and a tendency to a saturating ρ(T) starts already

above TMI. Therefore, the TMI ~ 0.25 ∆ data for the x = 6.7, 6.8 and 6.95

samples, plotted in figure 3.48, represent a lower limit and must be considered with some reservations. In contrast to this, the resistivity data of the x = 6.7,

x = 6.85 and x = 6.95 (Y0.6Pr0.4)Ba2Cu3Ox samples do fall onto each other and

the resistance minimum can be approximated by TMI ~ 0.35 ∆.

6.0 6.2 6.4 6.6 6.8 7.0

0

100

200

300

400

500

T (

K)

x

AF

SCTc

T*

TMI

TN Y0.6Pr0.4Ba2Cu3Ox

YBa2Cu3Ox

Tc T* TMI

Tc T* TMI

0

100TMI

I

II

IIIII

III

SC

6.0 6.2 6.4 6.6 6.8 7.06.0 6.2 6.4 6.6 6.8 7.0

0

100

200

300

400

500

0

100

200

300

400

500

T (

K)

x

AF

SCTc

T*

TMI

TN Y0.6Pr0.4Ba2Cu3Ox

YBa2Cu3Ox

Tc T* TMITc T* TMI

Tc T* TMITc T* TMI

0

100

0

100TMI

II

IIII

IIIIIIIIII

IIIIII

SC

Figure 3.48: Experimental T(x) phase diagram for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox system, with the superconducting critical temperature Tc, mid and the crossover temperature T* between the linear and super-linear ρ(T). At this point, the temperature of the resistance minimum, indicating a transition from a metal to an insulator-like behaviour TMI is added. The antiferromagnetic region is only indicative. The Tc, mid(x) and TMI(x) lines for (Y0.6Pr0.4)Ba2Cu3Ox were shifted down by 100 K.

When checking this phase diagram more closely, it is clear that the MI

boundary between the metallic region II and the insulating region III for both

compounds is a decreasing function of the doping. Both TMI(x) lines penetrate

the underlying superconducting region below Tc(x) that can thus be thought of

as masking the underlying insulating ground state.


93

The TMI(x) line for the YBa2Cu3Ox compound seems to vanish at optimal

doping. This observation however strongly relies on the assumption that the

TMI ~ 0.25 ∆ relation holds also for the higher levels of oxygen content (x = 6.7,

6.8 and 6.95) in YBa2Cu3Ox. In any case, for the YBa2Cu3Ox compound the

TMI(x) line is below Tc(x) for all oxygen concentrations above x = 6.5, meaning

that the zero field ρ(T) will not show any tendency of an insulating behaviour.

It is only at high magnetic fields, by suppressing the superconductivity below

Tc(x), that the underlying insulating ground-state can be accessed.

The TMI(x) line for the (Y0.6Pr0.4)Ba2Cu3Ox compound does not vanish at optimal

oxygen content since these cuprates are already strongly underdoped by the

40 % Y/Pr substitution. What is truly remarkable for this compound is that the

TMI(x) line is above Tc(x) for all oxygen contents. Thus, the tendency towards

an insulating behaviour (saturation or divergence) of the resistivity is already

visible above Tc and is only enhanced by the application of a high external

magnetic field.

A final remark about this phase diagram was already made in the previous

paragraphs. From this phase diagram, it is clear that the oxygen content is not a

good parameter to construct a generic phase diagram for the YBa2Cu3Ox and

the (Y0.6Pr0.4)Ba2Cu3Ox compounds. A better parameter would be the true

charge carrier density, accessible by measurements of the Hall-effect. This

would result in a shift to the left of the Tc, T* and TMI boundaries for

(Y0.6Pr0.4)Ba2Cu3Ox with respect to the line for YBa2Cu3Ox, since at least the

supposed hole-filling effect of the Y/Pr substitution would then be taken into

account. It is not clear beforehand what will be the role of the claimed magnetic

pair-breaking in the (Y/Pr)Ba2Cu3Ox compound.

In the next chapter an attempt will be made to construct the generic T(p) phase

diagram for the YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox compounds based on Hall-

effect measurements in very high pulsed magnetic fields.


94

3.7 Conclusions

The temperature dependent resistivity of a set of YBa2Cu3Ox, (Y0.6Pr0.4)Ba2Cu3Ox and strained La1.9Sr0.1CuO4 epitaxial thin films was

measured in zero field and at very high pulsed magnetic fields. The zero-

field data show that these samples are rather poor conductors having relatively

low ratios ρ290 K/ρo (ρ290 K/ρo ~ 2.5 for (Y0.6Pr0.4)Ba2Cu3Ox and 2.6 to 12.2 for

YBa2Cu3Ox) compared to pure normal metals where ρ290 K/ρo ~ 1000. This

agrees with previous findings on all superconducting high-Tc cuprates and

indicates their relative impurity.

Furthermore, it was shown that the zero-field normal-state resistivity above Tc

for various levels of hole doping -both for the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox members of the cuprate superconductors- can be scaled

onto one single universal curve. An energy scale ∆, the resistivity ρ∆ and the

residual resistivity ρo are used as scaling parameters. The universal curve

exhibits a region (labelled I) of linear ρ(T) at high temperatures T > T*, a

super-linear ρ(T) at intermediate temperatures T <T* (region II) and a low

temperature insulating ρ(T) regime (labelled III) at T < TMI. This low-

temperature regime is masked by the onset of superconductivity at T = Tc. The

distinct features in the temperature dependence of the metallic zero-field

resistivity of YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox in regions I and II are

universal for all the reported zero-field curves, the only difference is the

temperature at which these features set in. The existence of a universal metallic

ρ(T) curve was interpreted as a strong indication of one single mechanism

dominating the scattering of the charge carriers in these materials. This

mechanism then determines the energy scale ∆.

In order to gain access to the low-temperature T < Tc part of the normal-state

transport properties (regions II & III), experiments in very high magnetic fields

were performed. These experiments allowed to suppress the superconducting

state thus retaining the normal-conducting high field normal-state properties.

These experiments revealed the ground state of YBa2Cu3Ox (for x ≤ 6.8),

(Y0.6Pr0.4)Ba2Cu3Ox (for all levels of oxygen content) and the tensile strained


95

La1.9Sr0.1CuO4 ultra-thin film to be of an insulating nature with the resistivity

increasing as temperature is lowered.

Performing the same scaling analysis as in the zero-field case, it was shown that

for the YBa2Cu3Ox system the high-field ρρ(T) data for region III -the insulating regime- do not scale, whereas they more or less do scale in the

(Y0.6Pr0.4)Ba2Cu3Ox system (by squeezing the data together). It was argued that

such a scaling is not very likely since it would imply that the same energy scale

∆ controls both the metallic (region I and II) and the insulating (region III) part

of the ρ(T) curves; the insulating regime most likely being dominated by

disorder.

The insulating tendency in region III was shown not to obey a simple activated

or hopping mechanism. It was argued that, most probably, spin-flip scattering

in the framework of the Kondo model or the existence of a narrow metallic

impurity band with localised edges do not play an important role in producing a

divergent resistivity upon lowering temperatures. Our low-temperature ρ(T)

data were shown to obey the ln(1/T) divergence in high magnetic fields, as

was first observed in the cuprates by Ando.

The comparison of the transport data on strained La1.9Sr0.1CuO4 ultra-thin films

with data on samples without epitaxial strain and our YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox samples demonstrated that epitaxial strain probably not only

influences the doping of the CuO2 planes but also modifies the disorder

scattering.

In the final section of this chapter, an experimental T(x) phase diagram was

constructed including the superconducting critical temperature Tc(x), the

temperature T*(x) marking the opening of the pseudo spin-gap and also TMI(x),

the boundary between the metallic regime (regions I and II) and the insulating

regime (region III). It was argued that a generic T(p) phase diagram (with p

the number of holes per CuO2 plane) can only be obtained when combining

these data with Hall-effect measurements. This will be realised in the next

chapter.

CHAPTER 4 Hall-effect in YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films

97

Chapter 4

Hall-effect in YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox thin films

4.1 Introduction

For a 3 dimensional system, the current density jr

and the electric field Er

are

related by Ejrr

⋅= σ (and by jErr

⋅= ρ ) with σ and ρ the conductivity and

resistivity tensors, respectively. In a magnetic field along the z-axis, this can be

written in more detail (equation 4.1).

=

z

y

x

zz

yyyx

xyxx

z

y

x

E

E

E

j

j

j

σσσσσ

00

0

0

(4.1)

The Onsager relations for the symmetry of the kinetic coefficients ( )()( HH jiij −= σσ ) yield yyxx σσ = and xyyx σσ −= for the diagonal and off-

diagonal components of σ . The resistivity tensor ρ is the inverse of the

conductivity tensor σ and is given by equation 4.2:

−

+=

xxxy

xyxx

xyxxyyyx

xyxx

σσσσ

σσρρρρ

22

1 and

zzzz σ

ρ1

= (4.2)

The diagonal resistivity tensor components ρxx and ρyy represent the

magnetoresistance in the transverse (H ⊥ j) configuration whereas the third

diagonal component ρzz describes the longiudinal (H // J) configuration.

The off-diagonal elements ρxy and ρyx describe the Hall effect [Hall1879] which

occurs when a current flowing through a conductor is subject to a magnetic field


98

with a component perpendicular to it (H ⊥ J) (figure 4.1). This perpendicular

component of the magnetic field tends to deflect the charge carriers by means of

the Lorentz force. In the steady state, the accumulating charges at the border of

the wire give rise to a finite electric field opposing their motion so that the lines

of current flow are straightened. This transverse electric field EH is known as

the Hall field. The relation between the applied current density jx and the

transverse electric field Ey = EH is defined by the off-diagonal element of the

resistivity tensor ρyx = Ey/jx. Since the transverse electric field EH exactly

opposes the Lorentz force it is expected (and in most cases also observed) to

vary linearly with the magnetic induction B. This leads in a natural way to the

definition of the quantity z

yx

zx

yH BBj

ER

ρ== as the Hall coefficient.

In a free electron system, this coefficient is related to the charge carrier density n by the simple nqRH 1= relation, were q is the elementary charge. In this

simple picture, the Hall coefficient is directly determined by the density of

charge carriers, whereas its sign is set by the their nature (positive or negative).

This is a surprising result since in general the Hall coefficient is dependent on

temperature as well as on magnetic field and one would expect a more

complicated expression. This more elaborate description is provided by the

semi-classical model for electron dynamics [Ashcroft76]. The description reproduces the single electron nqRH 1= result in the more general high-field

limit of clean samples in which the entire current is carried by charge carriers

from a single band.

H

jx+ + + + + + + + + + + + + + + + +

- - - - - - - - - - - - - - - - - -EyEx

xy

z

vx

-|e|v x B

t w

Figure 4.1: Schematic view of the configuration of the current density j and the applied magnetic field H in which the Lorentz force induces a transverse electric field EH known as the Hall field. The thickness t and width w of the specimen are indicated.


99

The Hall-effect in high-temperature superconductors has been reported

extensively in the literature for the regime T >> Tc [Harris92, Xiong93,

Almasan94, Wuyts94 & 96] where there it is not affected by superconducting

fluctuations close to Tc or magnetic vortices in the mixed state [Hagen93,

Kopnin99].

In this chapter, we will report Hall-effect measurements on YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox thin films at temperatures extending to below the critical

temperature Tc. From the measurements of the high-field Hall-resistivity

ρyx(H), the Hall coefficient RH(T) at fixed field will be calculated. The

combination of these RH(T) and ρab(T) curves then allows the derivation of the

Hall-angle. Finally, the carrier density nH, that can be extracted from our Hall

data, enables us to construct a generic T(p) phase diagram for the YBa2Cu3Ox

and Y0.6Pr0.4Ba2Cu3Ox compounds.

4.2 Hall-effect in the normal state below Tc

Commonly, the Hall resistivity ρyx, arising in the presence of a magnetic field,

is measured in magnetic fields of the order of 1 to 10 tesla. Such an approach

suffers from evident complications in the vicinity of the superconducting

transition. Even above the critical temperature Tc, superconducting fluctuations

change the electronic properties of the high-Tc materials. Below Tc, in the

presence of magnetic fields H < Hc2, a type II superconductor is in the so-called

mixed state with magnetic vortices penetrating the material. The presence of

these vortices strongly influences the response of both ρxx(H) and ρxy(H)

[Bardeen65, Vinokur93, Kopnin99, Hagen93]. Thus, it is clear that, in order to

gain access to the normal-state Hall effect, it is of crucial importance to use

very high magnetic fields H > Hc2(T).

In this work, the Hall-effect was measured on the same thin films whose in-

plane transverse magnetoresistance ρxx(T,H) data were presented in Chapter 3.

In the next paragraphs we will present the Hall-effect data for the YBa2Cu3Ox

thin films with x = 6.45, 6.5 and 6.95 and the Y0.6Pr0.4Ba2Cu3Ox films with

x = 6.7, 6.85 and 6.95.


100

The measurements of the Hall-effect were performed in magnetic fields up to

50 T at the pulsed fields facility of the K.U. Leuven. All measurements were

performed in the transverse (H ⊥ I) in-plane (I // ab) configuration (H // c), with

a standard 5-terminal configuration over a Hall pattern of length 1 mm, and

width w = 50 µm; the thickness t of the thin films was of the order of 1000 to

2000 Å (table 3.1 in chapter 3). The Hall signal was amplified 1250 times and

the applied current was set such that the current density did not exceed

~ 7 107 A/m2. This yields Hall-voltages across the sample that are at maximum

a few 100 µV in amplitude. This is three orders of magnitude smaller than the

typical signal in the magnetoresistivity experiments (see chapter 2).

The meticulous implementation of vibration insulation and shielding for electric

interference, combined with a low contact resistance (below 1 Ω) allowed these

measurements to be carried out with large-bandwidth amplifiers and without

any electronic filtering. The Hall-voltage reported here is the result of the

combination of four field pulses with a changing polarity of field and current

used to eliminate spurious induced µodH/dt voltages and unwanted

contributions of the magnetoresistance. A more detailed discussion of the

experimental issues coming into play when experimenting in pulsed magnetic

fields is given in chapter 2.

In figures 4.2 and 4.3, the off-diagonal Hall resistivity ρyx(H) is given for an

YBa2Cu3O6.45 film of thickness t ≅ 1770 Å. This resistivity was calculated with

equation 4.3 from the Hall voltage VH across the sample (a four-pulse average)

by taking into account the applied current Jx and the thickness t of the film. The

width of the sample does not enter the equation directly but, however, strongly

determines the acceptable current that can be applied in order to obtain a

reasonable value for the current density jx. The ρyx(H) curves presented in

figures 4.2 and 4.3 were obtained without any electronic filtering, only a 20

point adjacent averaging was performed, reducing the influence of frequency

components above 16 kHz.

tJ

V

twJ

wV

j

E

x

H

x

H

x

yyx =

==ρ (4.3)


101

0 5 10 15 20

0

5

10

15

µoH (T)

YBa2Cu3O6.45ρ y

x (µ

Ωcm

)

4.2 K

20 K

25 K

29 K

35 K

79.4 K53.3 K

t ≈ 1770 Å

Figure 4.2: The off-diagonal resistivity ρyx versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures. The magnetic field was swept to about 17 T.

0 5 10 15 20 25 30 35 40 45 50

0

10

20

30

40

µoH (T)

YBa2Cu3O6.45

ρ yx

(µΩ

cm) t ≈ 1770 Å

4.2 K

12.9 K

20 K

64.9 K

35 K47.6 K

Figure 4.3: The off-diagonal resistivity ρyx versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures. The magnetic field was swept to about 42 T.

The low- and high-field ρyx(H) curves for the YBa2Cu3O6.45 film at temperatures

ranging from T << Tc up to T >> Tc as presented in figures 4.2 and 4.3 show a

field-induced transition from ρyx ~ 0 at low temperatures and fields to a regime


102

with a finite ρyx ~ H. It is only at high magnetic fields H > Hc2 or high temperatures T > Tc that a Hyx ∝ρ (expected from the balance of the Lorentz

force with the force arising from the transverse electric field) is reached. This

further illustrates the need for high magnetic fields when probing the Hall-effect

around or below Tc.

The quality of these experiments is illustrated by taking a closer look at the

ρyx(H) curve at 4.2 K. It is only at around 35 T that a finite ρyx is induced in the

sample. Below this field, within the high-frequency noise band, the ρyx(H)

curve exhibits an almost perfect ρyx = 0.

Although in this work the focus is on the normal state (e.g. T > Tc or H > Hc2)

Hall effect, it is worthwhile to mention that the fact that the ρyx(H) curves

exhibit a transition, resembling the ρxx(H) magnetoresistive transition, has

inspired scientists to look for an eventual scaling between these two transport

properties [Hagen93, Vinokur93, Wang94, Casaca99, Kopnin99] in the mixed

state. The scaling can be written in the general form of equation 4.4

[Vinokur93, Hagen93].

Bo

xxxy Φ

=βρ

αρ (4.4)

In this equation, α(T,B) is a microscopic parameter that is predicted both to be

independent [Vinokur93] or dependent [Wang94] on flux pinning. In both

models, the exponent β is set to 2. However, recently, Kopnin and Vinokur

[Kopnin99] revisited the previous analysis of Vinokur [Vinokur93] and showed

the scaling exponent β not to be universal and to vary from β ≈ 2 to β ≈ 1

depending on the magnetic field and the concentration of defects. The β ≈ 2

case is then recovered in the limit of weak pinning. To complicate things even

further, when pinning becomes important (e.g. at low temperatures) the

parameter α can become proportional to the applied field [Vinokur93],

neutralising the 1/B in equation 4.4 and thus yielding 2xxxy ρρ ∝ . However,

since experimental evidence on this subject is conflicting, this issue is still

under debate.


103

We have used equation 4.4 in an attempt to scale our ρyx(H) and ρxx(H) curves.

The main problem in such an analysis is that it can only be performed in the

narrow field range where both ρyx and ρxx are finite and that one should not

leave the mixed state (e.g. H < Hc2). This narrow field range (~ 10 T) makes it

difficult to distinguish between β ≈ 1 or 2, or to comment upon the eventual 1/B

dependency in equation 4.4. Our data, suffering also from minor temperature

differences between ρyx(H) and ρxx(H) experiments, allow no conclusive

statement about this scaling.

Experimentally, a sign reversal of the Hall resistivity the ρyx can be observed in

moderate magnetic fields and not too far from Tc. It is not yet established

whether this is an intrinsic effect or it is just a consequence of flux pinning. In

our experiments, this sign reversal was not observed.

4.2.1 Hall coefficient

From the ρyx(H) traces introduced in the previous paragraph, the Hall-

coefficient RH can be calculated by applying equation 4.5. This coefficient is

obtained by simply dividing the off-diagonal resistivity by the magnetic

induction. In analysing our data, we have taken the external magnetic µoH to

approximate the induction B, a common assumption that is however only valid

above the Hc2(T) line where the diamagnetic response of the material vanishes.

Since, in this work, emphasis is put on the normal-state properties, this is an

acceptable procedure.

tHJ

V

Bt

J

V

BBj

ER

ox

H

zx

H

z

yx

zx

yH µ

ρ≈

===

1 (4.5)

Figure 4.4 shows the RH versus field curves at various temperatures for the same

thin YBa2Cu3O6.45 film for which the ρyx(H) traces were shown in the previous

paragraph. Also here, a field-induced transition from the superconducting to the

normal-state response is observed. The 29.1 K curve for example starts at

RH ~ 0, rises with magnetic field and reaches an approximately constant value at

high fields (where the ρyx(H) curves become linear (figure 4.3)). It is this

regime of constant RH(H) at H > Hc2 that is of importance for our normal-state

analysis.


104

0 5 10 15 20 25 30 35 40 45 50

0

5

10

µoH (T)

YBa2Cu3O6.45

t ≈ 1770 Å

RH (

10-9

m3 /

C)

4.2 K12.9 K

20 K29.1 K

35 K

53.3 K41.9 K

Figure 4.4: The Hall coefficient RH versus applied magnetic field for a thin YBa2Cu3O6.45 film of thickness 1770 Å at various temperatures.

From the RH(H) curves, the temperature dependence of the Hall-coefficient at

fixed magnetic field can be extracted by applying the same procedure as used

for the normal-state resistivity in the previous chapter.

The RH(T) curves at 10, 20, 30, 40 and 50 T were constructed for the

YBa2Cu3Ox thin films with x = 6.45, 6.5 and 6.95 and the Y0.6Pr0.4Ba2Cu3Ox

films with x = 6.7, 6.85 and 6.95. These RH(T) curves are plotted together with

their ρxx(T) equivalent (= ρab(T) in our transverse, in-plane, configuration) in

figures 4.5 to 4.10. In this work, our goal was to access the Hall-effect in the

normal state at T < Tc and the experimental efforts were therefore concentrated

on the low-temperature part of the RH(T) curve.

From these combined RH(T) and ρab(T) plots, the gradual transition from a

superconductor to the normal-state, induced by applying a high magnetic field,

can be seen. The changes upon varying hole doping of the temperature

dependence of ρab(T) at low temperatures, were discussed in the previous

chapter. The direct comparison between the ρab(T) and RH(T) curves allows us

to describe the region where the magnetic field is able to suppress the

superconducting state and where the normal-state RH(T) is fully attained.


105

0 50 100 150 200 250 3000

5

10

T (K)

0

500

1000

1500

2000

2500

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)

YBa2Cu3O6.45

Figure 4.5: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin YBa2Cu3O6.45 film. The data were taken at 10, 20, 30, 40 and 50 T.

A general observation that can be made from these RH(T) plots is that, when

plotted on the same scale, strongly underdoped samples seem to show a weaker

temperature dependence of RH(T) than compounds with a higher doping level.

This observation agrees with earlier work on YBa2Cu3Ox and Y1-yPryBa2Cu3Ox

thin films [Wuyts94 & 96, Xiong93] where, however, it was shown that this is

just a consequence of "squeezing" the data together. When plotted in separate

plots, at every level of hole doping a significant temperature dependence of RH

remains, in the temperature range from Tc up to room temperature.

The RH(T) curves for the YBa2Cu3O6.45 sample show a transition to the normal-

state value of RH as temperature is increased (figure 4.5). This transition shifts

to lower temperatures as the field is increased to 40 tesla. At these high fields,

the low-temperature normal-state RH(T) is recovered, exhibiting little structure

on changing temperature. From the comparison with the in-field ρab(T) curves,

it is clear that at 40 T the normal state is entered unambiguously down to 25 K.

Below this temperature, the ρab(T) curves at 40 T and 50 T separate slightly,

indicating the onset of superconductivity. Thus, by using high pulsed magnetic

fields, we were able to extend the knowledge of the Hall-effect of this

Tc,mid = 41.7 K sample to a temperature T ~ 25 K << Tc.


106

0

200

400

600

800

0 50 100 150 200 250 3000

5

10

T (K)

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)

YBa2Cu3O6.5


0

200

400

600

0 50 100 150 200 250 3000

5

10

T (K)

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)

YBa2Cu3O6.95


Similar observations can be made for all the presented RH(T) plots. However, it

is worthwhile to focus for a while on the RH(T) curves for the two samples with

the lowest critical temperature Tc, Y0.6Pr0.4Ba2Cu3O6.7 and Y0.6Pr0.4Ba2Cu3O6.85,


107

presented in figures 4.8 and 4.9. For these two compounds, the 50 T magnetic

field suffices largely to suppress the superconducting state over the entire range

of temperatures that was explored experimentally. For these plots then, the

entire 50 T RH(T) curve represents the normal-state response.

0

500

1000

1500

2000

0 50 100 150 200 250 3000

5

10

T (K)

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)

Y0.6Pr0.4Ba2Cu3O6.7

Figure 4.8: The in-plane resistivity ρab (upper panel) and the Hall coefficient RH (lower panel) versus temperature for a thin Y0.6Pr0.4Ba2Cu3O6.7 film. The data were taken at 10, 20, 30, 40 and 50 T.

0

500

1000

0 50 100 150 200 250 3000

5

10

T (K)

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)




108

0

200

400

600

0 50 100 150 200 250 3000

5

10

T (K)

10 T 20 T 30 T 40 T 50 T

RH (

10-9

m3 /

C)

ρ ab

(µΩ

cm)



Although not in our experimental window, it can be remarked here that the high

temperature Hall coefficient was shown to exhibit a change in slope at about the

temperature T*, marking the crossover in the resistivity [Ilonca93, Wuyts94].

For the samples with the lowest level of hole doping, the relative insensitivity to temperature variations of the low-temperature Hall-coefficient RH is in sharp contrast with the strongly divergent resistivity ρρab(T) found in these compounds. This nearly temperature independent Hall

coefficient, also reported for Bi2201 crystals and La214 thin films [Ando97], is

an important challenge to the existing models already explaining a low-

temperature divergence of ρab(T).

Among these models, some indeed predict a constant RH at low temperatures.

The 2D Luttinger model predicts a constant RH [Anderson96b] but its predicted

power-law divergence does not agree as good with the high-field ρab(T) data as

does the ln(1/T). The conventional 2D weak localisation also yields a

temperature independent RH [Lee85] and was already shown to reproduce the

ln(1T) divergence of ρab(T) (chapter 3, paragraph 3.6).


109

4.2.2 Hall angle

A property that is closely related to the Hall-coefficient, is the Hall-angle. This

quantity can be computed by combining the Hall-coefficient with the results

from magnetoresistivity experiments by applying equation 4.6.

HRBRE

E

oH

ab

zH

ab

yx

xx

y

xH µ

ρρρρ

θ ≈===cot (4.6)

The Hall-angle thus basically is the ratio of the longitudinal and transverse

electric fields and hence also the ration between the diagonal and off-diagonal

elements of the resistivity tensor (equation 4.2); also in this equation, the

Bz ≈ µoH approximation was made. The Hall-angle is related to the Hall-

mobility by:

Hzxx

HH B

R

θρµ

cot

1== (4.7)

The importance of this Hall-angle for the high-Tc cuprates was pointed out by

Anderson [Anderson91] as a means of accounting for the deviations from the

linear temperature dependencies for ρ and RH upon changing doping level. He

argued that the Hall-angle should depend quadratically upon temperature

CTH += 2cot αθ .

This prediction comes about in the so-called 2D Luttinger liquid model, in

which spin and charge are separated and are described by spinon and holon

quasiparticles. The quasiparticle excitations are then assumed to obey two

different relaxation times. The longitudinal charge transport (the resistivity) is

determined by the relaxation time τtr, whereas the Hall-resistivity is determined

by both τtr and a transverse scattering time τH. The transverse relaxation rate

(τH)-1 is governed by spinon-spinon scattering and is proportional to T 2

[Anderson91]. Computing the Hall-angle then yields a T 2 dependence

(equation 4.8) where the constant C is introduced by a temperature independent

scattering process at magnetic impurities by the simple application of

Matthiessen's rule.

CTimpuritiesHtr

tr

yx

xxH +∝+∝= 21

cot ατττ

τρρ

θ (4.8)


110

The quadratic temperature dependence of the Hall-angle was confirmed

experimentally in a wide variety of compounds (among these are YBa2Cu3Ox

[Chien91, Wuyts94 & 96, Xiong93], (YPr)Ba2Cu3Ox [Almasan94, Xiong93]

and La2-xSrxCuO4 [Ando97]) and the correlation of the constant C in

equation 4.8 with the presence of magnetic impurities was indeed confirmed

(Zn [Chien91], Pr [Xiong93]). Wuyts et al. demonstrated a nice scaling

behaviour of the Hall-angle when using the same scaling temperature as for the

longitudinal transport data [Wuyts94 & 96] and found indications for deviations

from the T2 dependence. They also showed the slope α to be closely related to

the density of charge carriers in the plane.

The Hall-angle versus temperature curves were constructed using the high-field

RH(T) and ρab(T) data for the YBa2Cu3Ox thin films with x = 6.45, 6.5 and 6.95

and the Y0.6Pr0.4Ba2Cu3Ox films with x = 6.7, 6.85 and 6.95 and are plotted

versus T 2 in figure 4.11. From this plot, it is clear that we are able to extend the

existing studies at T > Tc to temperatures T << Tc. The region of a diverging

Hall-angle (mostly seen around T ~ Tc) was significantly shifted to lower

temperatures in the high field limit. Since our experimental effort was however

mainly concentrated on the transport properties at these low temperatures

(T < Tc), the temperature window is rather limited in comparison to most other

studies, where it extends up to T 2 ~ 10⋅104 K2. Nevertheless, a tentative

agreement with the often cited quadratic temperature dependence can be

observed, although our data do not allow a reliable determination of the slope α or to distinguish between a possible scaling or a mere shift of Hθcot .

A statement can however be made about the zero-temperature intercept C of

Hθcot . The C values estimated from figure 4.11, and the estimates for the

error, are summarised in table 4.12. The low value for the YBa2Cu3O6.5 thin

film is representative for YBa2Cu3Ox films that are not too strongly underdoped

[Wuyts94, Xiong93]. For the lowest oxygen contents (x = 6.45 in our case), the

parameter C was shown to increase, supposedly due to scattering by the

antiferromagnetic fluctuations showing up in these samples (see the discussion

in chapter 1). This is confirmed by our high-field data on the x = 6.45 and 6.5

samples. This temperature independent contribution C, arising from magnetic

scattering, was also shown to increase markedly upon a gradual substitution of


111

Y by the magnetic Pr [Xiong93]. The high-field Hθcot data for the

Y0.6Pr0.4Ba2Cu3Ox films studied in this work (table 4.12) indeed show a

significantly higher value for the intercept C when comparing to YBa2Cu3Ox

samples with a similar oxygen content.

This argument suggests that the presence of the magnetic Pr atoms, apart from

the reduction of the critical temperature Tc, causes a significant temperature-

independent contribution to the scattering of the charge carriers. This is in

agreement with the high residual resistivity ρo and the small ρ290 K/ρo resistivity

ratio, reported in chapter 3.

0

20

40

60

0.0 0.5 1.0 1.5 2.0 2.50

20

40

cot θ

H

T2 (104 K2)


Y0.6Pr0.4Ba2Cu3O6.7


YBa2Cu3O6.5

YBa2Cu3O6.45

Figure 4.11: The Hall angle plotted versus T2 for thin films of YBa2Cu3O6.45, YBa2Cu3O6.5, Y0.6Pr0.4Ba2Cu3O6.7, Y0.6Pr0.4Ba2Cu3O6.85 and Y0.6Pr0.4Ba2Cu3O6.95. The Y123 data were taken at 40 T whereas the YPr123 data are at 50 T. The arrows indicate the critical temperature Tc; the dashed lines were constructed by linearly extrapolating (the weakly temperature dependent) RH.

4.2.3 Carrier density

From the Hall-coefficient RH, the simple qnR HH 1= relation gives an estimate

for the charge carrier density. However, the marked temperature dependence of

this property over the temperature range Tc → 300 K [Xiong93, Wuyts94 & 96]

in the high-Tc's, introduces some doubt at which temperature to define nH and

about the validity of this simple approach.


112

However, in the case of our high-field Hall data, we were able to construct the

RH(T) curve down to temperatures T << Tc were the temperature dependence of

the Hall-coefficient is weak. Moreover, the use of very high magnetic fields

(50 T) at temperatures T << Tc fulfils at least two out of three conditions for the simple qnR HH 1= limit of the semi-classical model for electron dynamics

[Ashcroft76] to be valid. The demand for clean samples remains of course a

problematic issue in the high-Tc compounds (see the discussion of the residual

resistivity ρo and the resistance ratio ρ290 K/ρo in chapter 3).

In view of this, for these high-field data, the qnR HH 1= relation can be taken

as a reasonable approximation. Indeed, in this limit, the low temperature RH(T)

shows only little temperature dependence (figures 4.5 to 4.10), as also observed

in Bi2201 crystals and La214 thin films [Ando97]. Therefore it seems

appropriate to take nH(T = Tc) as an approximation for the charge carrier

density. These values are summarised in table 4.12; the error bars account for

the small temperature dependence of RH.

From these nH estimates, p, the fraction of holes per Cu-atom in the CuO2 plane,

can be calculated by simply multiplying nH by the volume of the unit cell

a· b· c ≈ 173.24 Å3 and divide it by 2, the number of Cu atoms per unit cell in the

CuO2 plane. These values are also reported in table 4.12.

Tc,mid

(K)

nH

(1027 m-3)

p

(#/Cu)

C

YBa2Cu3O6.95 92.2 2.36 ± 0.15 0.210 ± 0.01 N/A

YBa2Cu3O6.5 52.9 0.83 ± 0.03 0.073 ± 0.003 2.1 ± 2

YBa2Cu3O6.45 45.5 0.71 ± 0.02 0.062 ± 0.002 15.4 ± 2

Y0.6Pr0.4Ba2Cu3O6.95 41.4 1.24 ± 0.01 0.108 ± 0.001 7 ± 2

Y0.6Pr0.4Ba2Cu3O6.85 31.8 0.98 ± 0.01 0.086 ± 0.001 13 ± 6

Y0.6Pr0.4Ba2Cu3O6.7 22.3 0.93 ± 0.01 0.081 ± 0.001 24 ± 2

Table 4.12: Superconducting critical temperature Tc,mid, charge carrier density nH from the Hall data at T = Tc, the fraction p of holes per Cu-atom in the CuO2 plane and the zero-temperature intercept C of the quadratic part of the Hall-angle.


113

4.3 Phase diagram

It was discussed in chapter 3 that the oxygen content x is not a good parameter

to construct a phase diagram that is valid for both YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox (a so-called generic phase-diagram). It was shown that none

of the three characteristic boundary lines Tc(x), T*(x) or TMI(x) coincided for the

two systems. Also, it was argued that turning to a charge carrier density, as

obtained from the Hall data, would at least account for the supposed hole-filling

effect of the Y/Pr substitution and shift the boundary lines of the

Y0.6Pr0.4Ba2Cu3Ox system to the left in the T(x) plane with respect to the

YBa2Cu3Ox lines. The influence of the claimed magnetic pair-breaking in the

(Y/Pr)Ba2Cu3Ox compound would then become clear by a non-coincidence of

the Tc(x) lines.

The values for the hole fraction in the CuO2 planes p, as calculated from the

high-field Hall data, enable us then to construct such a generic T(p) phase

diagram for both the YBa2Cu3Ox and the Y0.6Pr0.4Ba2Cu3Ox systems. In

figure 4.13, the critical temperatures Tc,mid and the boundary TMI between the

metallic and insulating ρ(T) regimes are plotted versus p. It is clear from this

plot that the Tc(p) lines for the two compounds do not coincide but that they

show a qualitatively similar behaviour, shifted to a lower Tc for the

Y0.6Pr0.4Ba2Cu3Ox system. The fact that Tc(p)YPr < Tc(p)Y is a strong indication

that the magnetic pair-breaking in the (Y/Pr)Ba2Cu3Ox compound might play an

important role in the reduction of the superconducting critical temperature Tc.

As far as the boundary TMI(p) between the metallic and insulating ρ(T) regimes

is concerned, the discussion in paragraph 3.6 concerning the T(x) phase diagram

(figure 3.48) is still valid for this T(p) plot. For the YBa2Cu3Ox system, this

TMI(p) line is below the Tc(p) line for almost all levels of hole doping. In this

system, the insulating state is thus effectively masked by the onset of

superconductivity and is only made visible by high-field transport

measurements. For the Y0.6Pr0.4Ba2Cu3Ox compound (dotted lines in

figure 4.13), the TMI(p) line is above the Tc(p) line for all levels of oxygen

content and the insulating tendency can therefore already be seen in the zero-

field ρ(T) measurements above Tc. This strong tendency towards an insulating


114

low-temperature resistivity behaviour is in agreement with the additional

disorder introduced by the Y/Pr substitution and large residual resistivity ρo and

small resistance ratio ρ290 K/ρo for this Y0.6Pr0.4Ba2Cu3Ox compound.

0

50

100

0 0.05 0.10 0.15 0.20 0.25

T (

K)

p (holes/Cu-atom in the plane)

TMI

TMI

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox Tc TMI

Tc TMI

Tc

Tc

Figure 4.13: Superconducting critical temperature Tc (open symbols) and the boundary TMI (filled symbols) between the metallic and the insulating regimes for ρ(T), plotted versus the fraction of holes per Cu-atom in the CuO2 plane, for the YBa2Cu3Ox (diamonds) and the Y0.6Pr0.4Ba2Cu3Ox (circles) thin films. The arrow indicates the TMI = Tc point for YBa2Cu3Ox.

When we complement this T(p) plot with the crossover line T*(p) between the

linear ρ(T) at T > T* and the super-linear ρ(T) at T < T*, we obtain the phase

diagram as shown in figure 4.14. A striking observation from this plot is that

the crossover temperature T*(p)YPr ~ T*(p)Y, in contrast to the Tc(p)YPr < Tc(p)Y

discussed above. It seems that the crossover at T* is fully determined by the

density of charge carriers in the CuO2 plane, whereas for the critical

temperature Tc, additional effects come into play (for example magnetic pair

breaking).


115

0 0.5 1.0 1.5 2.0 2.5

0 0.05 0.10 0.15 0.20 0.25

0

100

200

300

400

T (

K)


nH (1027 m-3)

T*

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

Tc TMIT*

TcTMI

TMI

0

100

Figure 4.14: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane (bottom axis). The equivalent density of charge carriers, obtained from the Hall data is indicated on the upper axis. The Tc(x) and TMI(x) boundaries for Y0.6Pr0.4Ba2Cu3Ox were shifted down by 100 K.

4.4 Conclusions

In this chapter, Hall-effect measurements on YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox

thin films at temperatures extending to below the critical temperature Tc are

reported. These measurements were performed at very high pulsed magnetic

fields in order to fully access the normal state (H > Hc2).

These Hall measurements yield a signal that is three orders of magnitude

smaller than in standard magnetoresistivity experiments. The successful

accomplishment of these measurements was possible by a careful


116

implementation of vibration insulation and shielding for electric interference,

combined with a low contact resistance.

From the measurements of the high-field Hall-resistivity ρyx(H), the field

dependence of the Hall coefficient RH(H) at various temperatures was

calculated. By concentrating on the normal-state part of the data (H > Hc2), the

Hall coefficient RH(T) curves at fixed (high) field were constructed. These RH(T)

curves show a transition to the normal-state value of RH as temperature is

increased. This transition shifts to lower temperatures in high fields. At these

high fields, the low-temperature normal-state RH(T) is recovered, exhibiting

only a weak temperature dependence. By combining these RH(T) curves with

the high-field ρab(T) curves for the same samples, it is clear that at these high

fields the normal state is entered unambiguously down to temperatures T << Tc.

Thus, by using pulsed high magnetic fields, we were able to extend the

temperature range for the observation of the Hall-effect of the YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox thin films to temperatures T << Tc.

The insensitivity of the low-temperature Hall-coefficient RH to temperature variations is in sharp contrast with the strongly divergent ρρab(T) observed in these compounds. The approximately temperature-independent Hall

coefficient, is an important test for the existing models already explaining a low-

temperature divergence of ρab(T). Among the models still applicable for the

description of both ρab(T) and RH(T) is the weak localisation-model, which

explains both the constant RH at low temperatures and the ln(1/T) divergence of the high-field ρρab(T) data.

A further combination of the Hall data with the ρab(T) resistivity curves then

allows the derivation of the Hall-angle. The Hall-angle is predicted, and widely

observed, to obey a quadratic temperature dependence up to room temperature,

with a constant offset that is introduced by temperature independent scattering

at magnetic impurities. Also here, the existing studies at T > Tc are extended to

temperatures T << Tc by adding our high-field data. The region of diverging

Hall-angle (mostly observed around T ~ Tc) was significantly shifted to lower

temperatures in this high field limit. Although our experimental temperature

window is limited, a tentative agreement with the often cited quadratic

temperature dependence was observed. However, our data do not allow a


117

reliable determination of the slope or to distinguish between a possible scaling or a simple shift of Hθcot . The zero-temperature intercept of Hθcot , giving

information about the presence of magnetic scatterers, was, in the samples with

the lowest oxygen content, shown to be elevated, supposedly due to

antiferromagnetic fluctuations.

Also in the Y/Pr substituted samples, a higher offset of the Hall-angle was

observed. This suggests that the presence of the magnetic Pr atoms, apart from

the reduction of the critical temperature Tc, causes a significant temperature-

independent contribution to the scattering of the charge carriers. This is in

agreement with the high residual resistivity ρo and the small ρ290 K/ρo resistivity

ratio, reported in chapter 3.

Finally, the carrier density nH (and thus also p, the fraction of holes per Cu-atom

in the CuO2 plane) enables us to construct a generic T(p) phase diagram for the

YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds. Turning to this hole fraction p

accounts for the expected hole-filling effect of the Y/Pr substitution and shifts

the boundary lines of the Y0.6Pr0.4Ba2Cu3Ox system on the T-p plane to the left

with respect to the YBa2Cu3Ox lines.

The critical temperature Tc,mid(p) and the boundary TMI between the metallic and

insulating-alike ρ(T) regimes do not coincide for the YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox compounds; although demonstrating a qualitatively similar

behaviour, shifted to a lower Tc for the Y0.6Pr0.4Ba2Cu3Ox system. The fact that

Tc(p)YPr < Tc(p)Y is an indication of the magnetic pair-breaking in the

(Y/Pr)Ba2Cu3Ox compound, which might play an important role in the reduction

of the superconducting critical temperature Tc.

For the YBa2Cu3Ox system, the TMI(p) line is below the Tc(p) line for almost all

levels of hole doping and thus the insulating state at zero-field is effectively

masked by the onset of superconductivity. For the Y0.6Pr0.4Ba2Cu3Ox

compound, the TMI(p) line is above the Tc(p) line for all levels of oxygen content

and the insulating tendency can therefore already be seen in the zero-field ρ(T)

measurements above Tc. This tendency towards an insulating low-temperature

resistivity behaviour is in agreement with the additional disorder introduced by


118

the Y/Pr substitution and the large residual resistivity ρo and small resistance

ratio ρ290 K/ρo for this Y0.6Pr0.4Ba2Cu3Ox compound.

The T*(p) lines, for the crossover between the linear ρ(T) at T > T* and the

super-linear ρ(T) at T < T*, coincide for the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox

thin films. It seems that the crossover at T* is fully determined by the density of charge carriers in the CuO2 plane, whereas for the critical temperature Tc, additional effects come into play (for example magnetic pair breaking).

CHAPTER 5 Stripe formation and transport properties of underdoped cuprates

119

Chapter 5

Effect of stripe formation on

the transport properties of

underdoped cuprates

"The problem with the spin-gap is

that there are too many right ways to understand it …, not too few.

When one realises what is going on, it seems all too obvious

in several ways that one should have known all along.

(P.W. Anderson [Anderson96])

The universal ρ(T) behaviour in the underdoped YBa2Cu3Ox and

Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, strongly points into the

direction of a single scattering mechanism being dominant over the whole

underdoped regime in the Y123 system. Bearing in mind the complex magnetic

phase diagram discussed in chapter 1 with short-range AF correlations and a

pseudo gap showing up at temperatures far above the superconducting critical

temperature Tc and reminding the strong indications of stripe formation in the

CuO2 planes, it is tempting to assign the origin of this dominant scattering

mechanism to the microscopic magnetic and charge ordering.


120

5.1 Charge ordering revisited - Stripes

The undoped HTS are Mott insulators (with a half-filled upper orbital in the Cu

3d-shell) that are insulating rather than metallic by virtue of the strong electron-

electron repulsion. Doping the material (adding holes to the CuO2 planes)

results in the local destruction of the AF order. In the absence of long-range

Coulomb repulsion, a moderately doped AF tends to expel holes and the

material exhibits phase separation into hole-rich (metallic) regions and hole free

AF areas [Emery90]. This tendency towards local phase separation into hole-

free and hole-rich phases is frustrated when the Coulomb repulsion between the

holes is taken into account [Kivelson96]; a matter of particular relevance in the

underdoped HTS that are poor metals and in which screening is thus not

guaranteed.

The competition between the long-range Coulomb repulsion and the short-range

magnetic dipole interaction and their influence on the ordering of the holes in

the AF were studied by molecular dynamics simulations [Stojkovic99]. The

main result of these simulations is shown in figure 5.1 where the various phases

are drawn as a function of the magnetic dipole interaction (~ J) and the doping

level of the CuO2 plane. At weak dipole interactions, the Coulomb interaction

pushes the system to form a Wigner crystal. At moderate dipole interactions,

the Wigner phase is frustrated and stripes of holes (diagonal and vertical-

horizontal) show up at higher levels of hole doping. At very strong magnetic

interactions the holes forms a glassy clumped phase. The introduction of

thermal or crystallographic disorder changes this phase diagram and gives the

Wigner-phase a more glassy nature. The striped phases however retain their

main features although the stripes become of finite length (due to

crystallographic disorder) and are dynamic (at high enough temperatures).

Other groups predict exotic striped phases, including liquids crystals and glass

phases [Kivelson98].

These simulations confirm earlier experimental work showing the formation of

magnetic domains, incommensurate with the crystal lattice, both in

La2-xSrxCuO4 [Thurston89, Cheong91, Mason92 & 94, Yamada97,

Tranquada97, Hunt99] and YBa2Cu3Ox [Aeppli97, Tranquada97 & 97b, Dai98,


121

Kao99, Arai99]. This shows that, in reality, charge might be neither

homogeneously spread over the CuO2 planes, nor phase-separated into 2 phases

but that some complicated 1D-like structure is more probable.

diagonalstripes

clumpsWignercrystal

Dop

ing

Magnetic dipole interaction ~ J

geometricphase

Figure 5.1: Schematic phase diagram of the arrangement of holes, doped into a planar AF, in the presence of a long-range Coulomb interaction. As the magnetic interaction is becoming more important, the Wigner crystal becomes unstable and other phases show up (adapted from [Stojkovic99]).

In this charge-stripe picture [Emery97b & 99b, Moshchalkov93, 98b & 99 and

refs. above], dynamic metallic [Ichikawa99, Noda99, Tajima99] stripes are

thought to dominate the transport properties. Below a certain temperature the

tendency towards local phase separation results in the formation of dynamic

charge stripes, acting as domain walls for the antiferromagnetic (AF)

surroundings. The confinement of the AF regions leads to the development of a

spin-gap that is transferred to the holes in the stripe by hopping of electron pairs

perpendicular to the stripe, or, from an alternative point of view, by the coherent

transverse fluctuation of the charge stripe thus interacting with the underlying

AF [Emery97b & 99b]. This might result in a local pairing (with a gap equal to

the spin-gap) of charges, the so-called pre-formed pairs. Thus, the "magnetic

proximity effect" [Emery97b & 99b] then imposes the spin-gap onto the holes

in the charge-stripes. At lower temperatures, T = Tc, it is predicted that the local

pairs could acquire macroscopic coherence, an imperative condition for the

onset of bulk superconductivity [Emery97b & 99b].


122

Figure 5.2 presents a schematic view on the formation of dynamic charge stripes

in the underdoped CuO2 plane. On a short length scale (top) the straight, 1D,

metallic charge stripes form a domain wall for the surrounding antiferromagnet.

At intermediate length scales (middle), the holes recover a 2D motion (stripe

meandering), while at the macroscopic level (i.e. the sample size) the effective

1D conduction is recovered. Indeed, in this macroscopic view, the metallic

wires retain their 1D character as the meandering takes place on a much smaller

length scale.

Figure 5.2: A schematic view on the formation of dynamic charge stripes in the underdoped CuO2 plane (after [Zaanen99]), on a short length scale (top), an intermediate length scale (middle) and the macroscopic level (sample size).


123

Neutron diffraction experiments on La2-xSrxCuO4 revealed the inverse stripe

distance 1/ds to increase approximately linearly with Sr doping [Tranquada97b,

Yamada98] and start to saturate at around xSr = 0.12 where the inter-stripe

distance ds equals 4a, a being the crystallographic lattice parameter (figure 5.3 a

and b). This suggests the increasing doping to cause the stripes to be packed

denser, essentially keeping the same doping level within the stripe. In the

underdoped region, a linear correspondence between the inverse stripe distance

1/ds and the critical temperature Tc was found [Yamada98]. In the overdoped

regime (c and d in figure 5.3), the distance between the stripes was shown to be

almost constant upon doping, suggesting the holes to enter the regions between

the stripes and the doping contrast of the stripes with respect to their

surroundings to decrease.

p

1

ds

(a)

(c)

(b)

(d)

Figure 5.3: (a)-(d) A schematic view on the formation of charge stripes in the CuO2 plane upon doping (on a short length scale) (after [Moshchalkov99d]). The curve represents the inverse distance between the stripes versus the doping level of the plane.

5.2 Spin ladders, a magnetic structure between 1D & 2D

Low-dimensional quantum Heisenberg antiferromagnets are reported to exhibit

fascinating properties [Dagotto96 & 99]. The simple 1D spin ½ nearest

neighbour Heisenberg chain does not show an ordered ground state, due to

quantum fluctuations [Bethe31]. Instead, this system shows power-law


124

correlations with gapless excitations. The 2D Heisenberg spin ½ AF does

exhibits long range order at low temperature and they too accommodate a

gapless creation of an S = 1 excitation.

Spin-ladders (SL) are arrays of coupled chains and thus present structures that

interpolate between the 1D and the 2D case. By assembling a varying number

of these 1D chains into a SL, the influence of the dimensionality of the AF

region on the correlation length and spin-gap can be studied.

Numerical calculations revealed this crossover to be not as smooth as expected.

Ladders made of an even number of chains have so-called spin-liquid ground

states with purely short-range spin correlations that have an exponential decay.

In this even-chain SL, a finite gap exist for the creation of spin S = 1 excitations

[Dagotto92]. A ladder with an odd number of legs exhibits quite opposite

features and seems to retain some of the properties of the purely 1D single

chain, namely gapless spin excitations and a power-law decay of the AF

correlations.

Figure 5.4: Schematic drawing of the Cu2O3 sheet of SrCu2O3 (left) and the Cu3O5 sheet of Sr2Cu3O5 (right) containing a 2-leg (left) and three leg ladder (right) (adapted from [Azuma94]).

One class of materials, known to show intrinsically a SL behaviour, is formed

by the cuprates with a modified (and mixed) coordination of the CuO4 squares

in the CuO2 planes. The alternating edge- and corner-sharing between the CuO4

squares in these materials such as Srx-1Cux+1O2x, SrCu2O3, Sr2Cu3O5 and


125

(SrCa)14Cu24O41 [Azuma94, Dagotto96, Uehara96, Nagata97, Takano97] results

in the formation of n-leg spin ladders with nc = 1,2 or 3 (figure 5.4). The spin-

gap, predicted by the numerical work to exist in even-chain SL, was observed

experimentally whereas the 3-leg SL were shown to be gapless. Moreover,

doping 2-leg ladders by chemical substitution and external pressure induces a

crossover to more metallic transport properties, creates a super linear ρ(T)

behaviour similar to that observed in the HTS and even produces

superconductivity [Uehara96, Nagata97].

5.3 Quantum transport in doped 1D and 2D Heisenberg systems

The discussion in the previous paragraphs makes clear that the transport

properties of the high-Tc cuprates should be extremely sensitive to the

underlying microscopic magnetic structure. In order to account for the possible

inhomogeneous intercalation of AF insulating regions and metallic hole-rich

stripes, a physical model was developed [Moshchalkov93, 98, 98b, 99, 99b &

99c]. The model describes the transport both in the 2D Heisenberg regime

(above T*) as in the 1D striped regime (below T*) where the pseudo gap

develops.

A rapidly growing experimental evidence ([Dagotto96, Emery97, Tranquada97]

and previous paragraphs) indicates that the 1D scenario might be also relevant

for the description of the underdoped high-Tc cuprates where 1D stripes can be

eventually formed. Since mobile carriers in this case are expelled from the

surrounding Mott-insulator phase into the stripes, the latter then provide the

lowest resistance paths. This makes the transport properties very sensitive to

the formation of the stripes, both static and dynamic.

5.3.1 The model

The importance of the CuO2 planes for the transport properties is a widely

documented feature of the high-Tc cuprates. The confinement of the charge

carriers in these planes reduces the dimensionality for charge transport to 2

dimensions (or less) and makes the conductivity σ in such 2D metallic system


126

to be dominated by quantum transport. At moderate temperatures T* < T < To,

AF correlations will develop in the CuO2 planes, yielding a 2D Heisenberg

regime. At lower temperatures T < T*, local charge inhomogeneities (1D charge

stripes) will confine the AF regions, resulting in the formation of a pseudo spin-

gap. Any model trying to explain the unusual transport properties should

account for these features (i.e. the magnetic structure and the dimensionality).

The proposed approach [Moshchalkov93, 98b & 99] is based on three basic

assumptions:

1. the dominant scattering mechanism in HTS in the whole temperature

range is of magnetic origin;

2. the specific temperature dependence of the resistivity ρ(T) can be

described by the inverse quantum conductivity σ-1 with the inelastic

length Lφ being fully controlled, (via a strong interaction of holes

with Cu2+ spins, due to the magnetic proximity effect [Emery97b &

99b]) by the magnetic correlation length ξm, and, finally,

3. the proper 1D or 2D expressions should be used for calculating the

quantum conductivity with Lφ ~ ξm.

The quantum conductivity in 2D is proportional to ln(Lφ) whereas the quantum

conductivity of a single 1D wire is a linear function of the inelastic length Lφ

[Abrikosov88]. They are represented by

=−

lLe

bTT DD

φσρ ln1

~)()(2

21

2h

(5.1)

φσρ Le

bTT DD

h

2

211

11

~)()( =− (5.2)

with l the elastic length and b the thickness of the 2D layer or the diameter of

the 1D wire. These expressions for the resistivity of the 1D wires and 2D layers

can be detailed by assuming that the dominant scattering mechanism is of

magnetic origin and thus enforce the Lφ ~ ξm condition (ξm being the magnetic

correlation length). Equations 5.1 and 5.2 for the 2D and 1D resistivities can

then be modified as to give:


127

=

=−

lTe

bTT Dm

LDDm

)(ln1

~)()( 22

21

2ξσρ

ξφ h (5.3)

)(1

~)()( 1

2

211

1 Te

bTT DmLDD

mξσρ

ξφ h=− = (5.4)

The determination of the precise behaviour of the resistivity in the 2D

Heisenberg (T > T*) and the 1D striped (T < T*) regimes thus requires the

calculation of the magnetic correlation lengths in a 1D (ξm1D) and a 2D (ξm2D)

antiferromagnetic structure.

In the 2D Heisenberg case, the temperature dependence of the correlation length

ξm2D was calculated for the isotropic case [Hasenfratz91] and is expressed as

( )

⋅−

⋅=

T

F

F

T

F

ceTDm

2

222

2exp

221

28

πππ

ξh

(5.5)

with c being the spin velocity and F 2 a parameter that can be directly related to

the exchange interaction J. For simplicity we adopt the JF =22π relation that

is valid over a wide temperature range. For the 1D striped phase, the similarity

with the 1D even-chain Heisenberg AF spin-ladder compounds can be

employed and the spin-correlation length found by Monte Carlo simulations

[Greven96] can be taken for ξm1D :

∆−

∆+=∆ −

T

TADm exp

2)( 1

1 πξ (5.6)

where A ≈ 1.7 and ∆ is the spin-gap.

The combination of these expressions for the 1D and 2D spin correlation

lengths with the proper expression for the quantum resistance then gives the

temperature dependence of the resistivity. For temperatures T > T*, in the 2D

Heisenberg regime, surprisingly, the resistivity is a linear function of

temperature [Moshchalkov93] due to the mutual cancellation in the limit T <<

2J between the logarithmic ρ(ξm) dependency and the exponential temperature

dependence of ξm. For T < T* the striped 1D phase yields equation 5.8, with J//

the intra-chain coupling and a the spacing between the 1D wires (J// comes in to

recalculate the theorist units) [Moshchalkov98b & 99].


128

[ ] ( )[ ]J

T

e

b

T

JTT mDD 2

1

1122 ~expln~ln~)()(

h−

−−

= ξσρ (5.7)

[ ]

∆

−+∆

== −

TJ

TA

Jae

bTT DD exp

2)()(

////2

21

11 πσρ

h (5.8)

To verify the validity of the proposed 1D spin ladder model, a crucial test is its

application to the resistivity data obtained on the even-chain spin-ladder

compound Sr2.5Ca11.5Cu24O41 described above [Nagata97]. This compound, due

to its specific crystalline structure, definitely contains a two-leg (nc = 2) Cu2O3

ladder and therefore its resistivity along the ladder direction should indeed obey

the 1D expression 5.8. The results of the ρ(T) fit with equation 5.8 are shown in

figure 5.5. This fit demonstrates a remarkable quality over the whole

temperature range T ~ 25-300 K, except for the lowest temperatures where the

onset of the localisation effects, not considered here, is clearly visible in the

experiment. Moreover, the used fitting parameters ρo, C and ∆ all show very

reasonable values.

ρ ρ(T) CTT

= + −

0

exp∆

0 50 100 150 200 250 3000

1

2

3

4

ρ (1

0-4 Ω

cm)

T (K)

0 2 4 6 8200

220

240

∆ (K

)

pressure (GPa)

4.5 GPa

8 GPa

Sr2.5Ca11.5Cu24O41+δ

Figure 5.5: Temperature dependence of the resistivity for a Sr2.5Ca11.5Cu24O41 even-chain spin-ladder single crystal at 4.5 GPa and 8 GPa (experimental data points after [Nagata97]). The solid line represents a fit using equation 5.8 describing transport in 1D SL's.


129

The expected residual resistance )()2( 2//

20 aeJb πρ h∆= for b ~ 2a ~7.6 Å,

∆ ~ 200 K and J// ~ 1400 K (the normal value for the CuO2 planes) is

ρo ~ 0.5· 10-4 Ωcm which is in good agreement with ρo ~ 0.83· 10-4 Ωcm found

from the fit. The fitted gap ∆ ~ 216 K (at 8 GPa) (figure 5.5) is close to

∆ ~ 320 K determined for the undoped SL SrCu2O3 from inelastic neutron

scattering experiments [Takano97]. In doped systems it is natural to expect a

reduction of the spin gap upon doping. Therefore the difference between the

fitted value (216 K) and the one measured in an undoped system (320 K) seems

to be quite fair. Finally the calculated fitting parameter 0103.0)2()( =∆= oAC ρπ (in units of 10-4 Ωcm/K) is to be compared with

C = 0.013 (from the 8 GPa fit on figure 5.5). Using the fitting procedure for the

two pressures 4.5 GPa (∆ ~ 219 K) and 8 GPa (∆ ~ 216 K), we have obtained a

weak suppression of the spin-gap under pressure d∆/dp ~ -1 K/GPa.

The super linear ρ(T) behaviour observed in this doped even-chain SL under

external pressure indicates, by its similarity with the S-shaped ρ(T) in

underdoped HTS, that the picture of 1D transport might be relevant to the HTS.

To investigate the possibility of using the 1D scenario for describing transport

properties of the 2D CuO2 planes of the high-Tc superconductors, it is

appropriate to compare the temperature dependency of the resistivity of a

typical underdoped high-Tc material YBa2Cu4O8 with that of the even-chain SL

compound Sr2.5Ca11.5Cu24O41. The crystal structure of the YBa2Cu4O8

compound ('124') differs substantially from that of the more common

YBa2Cu3O7 ('123'), since 124 contains double CuO chains stacked along the c-

axis and shifted by b/2 along the b axis [Karpinski88]. These chains are

believed to act as charge reservoirs, therefore they may have a strong influence

on the transport in the CuO2 planes themselves. In the 124 case, the 1D features

of this double CuO chain can be expected to induce an intrinsic doping

inhomogeneity in the neighbouring CuO2 planes thus enhancing in a natural

way the formation of the 1D stripes. A weak coupling of 1D chains to 2D

planes might be sufficient to reduce the effective dimensionality by

preferentially orienting the stripes in the CuO2 planes along the chains. But

even in pure 2D planes, without their coupling to the 1D structural elements, the

formation of the 1D stripes is possible. Using a simple scaling parameter To, a


130

perfect overlap of the two sets of data was found: (ρ-ρo)/ρ(To) versus T/To (with

ρo being the residual resistance) for YBa2Cu4O8 and Sr2.5Ca11.5Cu24O41

(figure 5.6). Note that ρo should be subtracted from ρ(T) since ρo may contain

contributions from several scattering mechanisms depending on the sample

quality.

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

T/To

YBa2Cu4O8

∆ = (224 ± 5) K

Sr2.5Ca11.5Cu24O41+δ at 8 GPa

∆ = (216 ± 5) K

(ρ-ρ

o)/ρ

(To)

Figure 5.6: Scaling analysis on the temperature dependence of the resistivity of the underdoped high-Tc superconductor YBa2Cu4O8 and the even-leg spin-ladder Sr2.5Ca11.5Cu24O41.

This perfect scaling of the ρ(T) data of an underdoped HTS on one side and an

even-leg spin-ladder on the other side has severe implications for the nature of

the charge transport and the scattering in the high-Tc cuprates' CuO2 layers. It

convincingly demonstrates that resistivity vs. temperature dependencies of

underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL

with a spin-gap ∆ are governed by the same underlying 1D (magnetic)

mechanism.

Early experiments on twinned high-Tc samples however, created an illusion that

all planar Cu sites in the CuO2 planes are equivalent. Recent experiments on

perfect untwinned single crystals have strongly nuanced this belief. A very

large anisotropy in the ab-plane of twin-free samples has been reported for

resistivity (ρa/ρb(YBa2Cu2O7) = 2.2 [Gagnon94, Friedman90] and

ρa/ρb(YBa2Cu4O8) = 3.0 [Bucher95]), for thermal conductivity


131

(κa/κb(YBa2Cu4O8) = 3-4 [Cohn98]), for superfluid density [Yu92, Wang98]

and for optical conductivity [Wang98, Tajima96]. In all these experiments,

much better metallic properties have been clearly seen along the direction of the

chains (b-axis). And what is truly remarkable, that this in-plane anisotropy can

be partly suppressed by a small (only 0.4 %) amount of Zn [Wang98], which is

known to replace copper, at least for Zn concentrations up to 4 %, only in the

CuO2 planes [Tarascon88, Xiao88] ! The latter suggests that the ab-anisotropy

can not only be explained just by assuming the existence of highly conducting

CuO-chains. Instead, the observation of the anisotropy in the transport

properties in the ab-plane for YBa2Cu4O8 [Bucher95] and YBa2Cu3O7

[Gagnon94], interpreted as a large contribution of strongly metallic Cu-O chains

ρchain(T), might be re-interpreted taking into account the fact that the in-plane

anisotropy is caused by certain processes in the CuO2 planes themselves, where

the substitution of Cu by Zn takes place. In this situation we may expect that

the chains are actually imposing certain directions in the CuO2 planes for the

formation of 1D stripes.

However, inelastic neutron scattering experiments on YBa2Cu3O7 [Dai98,

Kao99, Arai99] show evidence for the existence of rather dynamic stripes and

the observation of 1D features in the transport properties should therefore not be

limited to the Cu-O chain-direction only. Moreover, although the 1D stripes are

dynamic, no averaging of the transport properties will occur, since, even for

dynamic stripes, the charge will automatically follow the most conducing paths,

i.e. stripes, even if they are moving fast. Fitting the 1D quantum transport

model [Moshchalkov93, 98b & 99] to the in-plane ρ(T) curve for YBa2Cu4O8

(equation 5.8) results in a very nice fit [Moshchalkov98b, 99, 99b & 99c], yielding a spin-gap ( )K5224 ±=∆ (figure 5.7) were the slope of ln[(ρ-ρo)/T]

versus 1/T (see inset in figure 5.7) defines the spin gap value. Therefore, we

can conclude that the resistivity of underdoped cuprates below T* (see inset in

figure 5.7) simply reflects the temperature dependence of the magnetic correlation length DDm 11 /1 ρξ = in the even-chain SL's and the pseudo-gap is

the spin-gap formed in the 1D stripes.


132

4 6 8 10 12 146

7

8

9

10

11

1/T (10-3 K-1)

−−

lnρ ρ

o

T

slope ∆ = 224 K

0.0

0.1

0.2

0.3

0.4ρ

(10-4

Ωcm

)

T (K)

0 50 100 150 200 250 300

ρ ρ(T) C TT

= + −

0 exp

∆

0 200 400 600 8000246810

T (K)

ρ (1

00 µ

Ωcm

)

T *

Figure 5.7: Temperature dependence of the resistivity of a YBa2Cu4O8 single crystal (open circles); the solid line represents the fit using equation 5.8. The fit parameters were ρo = 0.024 10-4 Ωcm, C = 0.00242 10-4 Ωcm/K and ∆ = 224 K. The high-temperature data taken on another crystal [Bucher94], shown in the inset, illustrate the crossover at T* to 2D (linear behaviour). Insert (upper left): fit of experimental data using equation 5.8.

0 100 200 300 400 5000

5

10

15

20

25

30

T (K)

Ksp

in (

10-2

%)

K T K K TTD( ) ( ) exp= + −

−0 1

12

∆

YBa2Cu4O8

Figure 5.8: Knight shift data KS for the YBa2Cu4O8 system [Bucher94] fitted with equation 5.9 for 2-leg spin-ladders [Troyer94]. The resulting fitting parameters are K(0) = (0.6 ± 2) 10-2 %, K1D = (870 ± 40) 10-2 % and ∆ = (222 ± 20) K [Moshchalkov99 & 99c].


133

In order to substantiate these observations, we can use similar ideas in the

analysis of other physical properties. Since in underdoped cuprates the scaling

temperature To, used in the scaling of ρ(T), works equally well for resistivity as

for Knight-shift data KS [Wuyts96], these KS data can also be used for fitting

with the expressions derived from the 1D SL models. For a 2-leg SL, the

temperature dependence of the Knight shift KS should obey the following

expression [Troyer94]:

( )TTTKS∆−− exp~)( 2

1

(5.9)

Fitting the KS(T) data [Bucher94] for YBa2Cu4O8 with this expression gives an excellent result (figure 5.8) with a spin-gap ( )K20222 ±=∆ which is very

close to the value ( )K5224 ±=∆ derived from the resistivity data (see above).

Therefore, for the underdoped HTS, we have related the linear ρ(T) behaviour

above T* with quantum transport in a 2D AF Heisenberg system and the S-

shaped super-linear behaviour below T* with a 1D quantum transport model for

even-chain spin ladders (the striped phase). In the next paragraphs we will

identify the universal ρ(T) behaviour reported in chapter 3 with this 1D/2D

model, extract the spin gap ∆ and construct an experimental generic T(p) phase

diagram.

5.3.2 Application to the data

In chapter 3, the in plane resistivity ρab(T) of underdoped YBa2Cu3Ox,

(Y0.6Pr0.4)Ba2Cu3Ox and La2-xSrxCuO4 thin films was shown to exhibit a linear

ρ(T) dependence at high temperatures T > T*, a super-linear behaviour at T < T*

and an insulating resistivity at the lowest temperatures for strongly underdoped

samples. This insulating behaviour was made evident by the application of very

high magnetic fields in order to suppress superconductivity. Doping the high-Tc

materials reduces the tendency towards insulating behaviour and lowers the

crossover temperature T* such that the super linear ρab(T) makes way for the

linear region, extending to lower temperatures. These general observations are,

for the YBa2Cu3Ox compound, nicely illustrated by the composite ρab(T) plots

given in figure 5.9.


134

0

100

200

300

400

500

600

0

200

400 0

100

200

300

0

200

400

0 50 100 150 200 250 3000

500

1000

T (K)

ρ ( µ

Ωcm

)

x = 6.5

50 / 45 / 40 / 35 / 30 / 20 / 10 / 0 T

x = 6.7

x = 6.8

x = 6.95

x = 6.45

YBa2Cu3Ox

Figure 5.9: Resistivity versus temperature for the YBa2Cu3Ox thin films studied in this work at zero-field and at 10, 20, 30, 35, 40, 45 and 50 T.

These in-plane resistivities were, both for the YBa2Cu3Ox and the

(Y0.6Pr0.4)Ba2Cu3Ox compound, shown to scale onto one universal curve

(figures 5.10 and 5.11). From these plots, a perfect scaling in regimes I (linear

part) and II (curved, super-linear ρ(T)) was observed for the zero-field curves.

In the insulating regime (III), the scaling is of less good quality.

The perfect scaling of the metallic in-plane resistivities for these compounds is a

strong indication that one scattering mechanism is dominant for the strongly

underdoped up to the near-optimally doped samples. Only the energy scale (the

scaling parameter ∆ and the crossover temperature T* ≈ 2∆) varies upon

doping.


135

x = 6.45x = 6.5x = 6.7x = 6.8x = 6.95

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

T/∆

ρ ρρ ρ

−−

o

o∆

T*

∆

x = 6.4 / 6.45 / 6.5 / 6.7 / 6.8 / 6.95

YBa2Cu3Ox

III II I

Figure 5.10: Scaled zero field and 50 T ρ(T) for the YBa2Cu3Ox films (x = 6.4, 6.45, 6.5, 6.7, 6.8 and x = 6.95). The regions of different ρ(T) behaviour are indicated as well as the energy scale ∆ and the crossover temperature T* ≈ 2∆; ρo is the residual resistivity and ρ∆ is the resistivity at T = ∆. Two additional lines represent the predictions from the 1D/2D quantum transport model for T < T* (eq. 5.8) and T > T* (eq. 5.7).

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ ρρ ρ

−−

o

o∆

T/∆

Y0.6Pr0.4Ba2Cu3Ox

III II

∆

x = 6.7 / 6.85 / 6.95

x = 6.7x = 6.85x = 6.95

Figure 5.11: Scaled zero field and 50 T ρ(T) data for the (Y0.6Pr0.4)Ba2Cu3Ox thin films (x = 6.85 and 6.95). For the x = 6.7 sample, only the 45 T data were experimentally accessible. The regions of different ρ(T) behaviour are indicated as well as the energy scale ∆; ρo is the residual resistivity and ρ∆ is the resistivity at T = ∆. The additional line is the prediction from the 1D/2D quantum transport model for T < T* (eq 5.8).


136

In the framework of the previous paragraph, it is almost evident to try to

correlate this "dominant process" with the magnetic scattering mechanisms in

1D and 2D, introduced there.

For T > T*, where short range AF fluctuations are seen in inelastic neutron

scattering experiments, the resistivity is observed to have a linear temperature

dependency (region I). This regime is thus perfectly described by equation 5.7

for quantum transport in a 2D Heisenberg system with the inelastic length

determined by the magnetic (2D) correlation length.

For T < T* ≈≈ 2∆∆, were charge stripes can develop, the resistivity is observed to

have a super-linear temperature dependence (region II). This regime should

then be accurately described by equation 5.8, describing quantum transport in a

1D striped material with again the inelastic length determined by the magnetic

(1D) correlation length. To check this, the ρ(T) curve described by this

expression for 1D conduction (equation 5.8) was plotted together with the data

in figures 5.10 and 5.11. A perfect overlap with the data is established up to

slightly above T/∆ = 1. The scaling of the data was performed such that the data

fall onto the universal ρ(T) = ρo+ CTexp(-∆/T) curve with C = exp(1) =

2.7183… . In that way, the scaling parameters necessary to obtain the collapsing ρρab(T) traces directly yield estimates for the spin pseudo-gap ∆∆ within this model for transport in a 1D striped case.

Also for the tensile strained La1.9Sr0.1CuO4 ultra-thin film (150 Å deposited on

SrTiO3, sample C), showing an S-shaped in-plane resistivity ρab(T), this ρab(T)

can be fitted with expression 5.8 for 1D quantum transport (figure 5.12). The

agreement is quite reasonable for temperatures TMI < T < ∆ (as was observed

also on the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox compounds) and a spin gap

of ∆ = (125 ± 20) K was obtained.

In figure 5.13, the estimates for the spin pseudo-gap ∆ and the crossover

temperature T* ≈ 2∆ are, for the YBa2Cu3Ox system, plotted versus the oxygen

content x. Like T*, the spin-gap decreases upon doping, approaching the critical

temperature Tc near the optimally doped case. This is a well documented trend

for the pseudo-gap and is not restricted to the YBa2Cu3Ox or the

(Y0.6Pr0.4)Ba2Cu3Ox compounds (for a review, see [Timusk99]).


137

0 50 100 150 200 250 3000

200

400

600

800

T (K)

La1.9Sr0.1CuO4

ρ (µ

Ωcm

)

tensile strained150 Å on SrTiO3

∆ = (125 ± 20) K

Figure 5.12: Temperature dependence of the in-plane resistivity ρab(T) for a tensile-strained 150 Å La1.9Sr0.1CuO4 ultra-thin film fitted with the expression for 1D quantum transport. The arrow indicated the gap ∆.

6.2 6.4 6.6 6.8 7.00

100

200

300

400

500

T (

K)

x

YBa2Cu3Ox

∆ from ρab(T)

T*≈2∆ from ρab(T)T*

∆

Figure 5.13: Spin gap ∆ and crossover temperature T* ≈ 2∆ for the for the YBa2Cu3Ox thin films, as derived from the scaling of their in-plane resistivities ρab(T) with the curve for 1D quantum transport.


138

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.00

100

200

300

∆(K

)

x

YBa2Cu3Ox

ρab(T) on thin filmsthis work Wuyts et. al

ρab(T) on crystalstwinned detwinned

KS on 17O on 63Cu

6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.00

100

200

300

∆(K

)

x

YBa2Cu3Ox

ρab(T) on thin filmsthis work Wuyts et. al

ρab(T) on crystalstwinned detwinned

KS on 17O on 63Cu

Figure 5.14: The spin gap ∆ of YBa2Cu3Ox versus oxygen content x, from the scaling of the ρab(T) data with the curve for 1D quantum transport for the thin films in this work (open diamonds) and a direct fit on the films from [Wuyts94 & 96] (down triangles), twinned crystals [Ito93] (up triangles) and de-twinned crystals [Gagnon94] (squares). The spin gap obtained from a fit of the Knight-shift on 17O [Martindale96] (filled diamonds) and on 63Cu and 17O [Takigawa91] (circles) is also added.

A last, crucial, check for the 1D conductivity model [Moshchalkov98b & 99] is

the direct comparison of our values for the pseudo-gap with estimates from the

literature. In figure 5.14, we have re-plotted our ∆(x) data on thin films (open

diamonds) together with estimates from resistive measurements on other

YBa2Cu3Ox thin films [Wuyts94 &96], twinned [Ito93] and de-twinned

[Gagnon94] single crystals. Within the error bars, these data agree well.

Additionally, we have plotted estimates of the pseudo-gap as derived from

CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders

[Takigawa91, Martindale96]. Also these data, although obtained with a totally

different technique, yield estimates for the spin-gap that are in good agreement

with our ∆(x) data. This proves that the 1D quantum transport model

[Moshchalkov98b & 99], used to describe the transport in underdoped cuprates

at T < T* in a 1D striped manner, not only agrees qualitatively, but also yields

values for the pseudo spin-gap ∆ that agree well with independent estimates.

Although this correspondence is quite convincing, it should be mentioned that

experimental techniques probing charge excitations (like ARPES, quasi-particle


139

relaxation measurements and tunnelling experiments) yield values of the

pseudo-gap ∆p, that are significantly higher (about a factor 2) than the spin-

excitation gap ∆s as observed in NMR and INS experiments [Timusk99,

Mihailovic99]. In the 1D quantum transport model, where the inelastic length

is presumed to be dominated by the magnetic correlation length, the agreement

of our data with the gap-value determined from NMR experiments then seems to

be natural.

The only dissonance in this discussion comes from the often-cited 89Y NMR

data on underdoped YBa2Cu3Ox reported by Alloul and co-workers [Alloul88 &

89]. These Knight-shift data were shown earlier to scale very well, using the

same scaling temperature To that was derived from the scaling of ρab(T)

[Wuyts96]. This was interpreted as a strong indication that the magnetic

structure and the opening of the spin-gap are relevant also for transport

measurements, motivating the development of the 1D/2D quantum transport

model [Moshchalkov93, 98b, 99]. This argument still stands. However, when

fitting the expression for the Knight shift KS(T) (as in figure 5.8) to these data,

the resulting values for the pseudo-gap are about a factor 2 higher than the gap

values determined from resistivity measurements or data on in-plane 17O and 63Cu Knight-shift measurements on aligned powders [Takigawa91,

Martindale96]. The origin of this deviation is not clear but could be due to the

use of non-aligned powders [Alloul88 & 89] or possible differences between

NMR measurements probing inter-plane 89Y on one side and in-plane 17O and 63Cu on the other side.

5.4 Stripe ordering at low temperatures

At low temperatures, T < TMI, the metallic behaviour of the resistivity in regions

I and II transforms into an insulating, diverging, ρ(T) (region III). The

diverging high-field ρ(T) data were shown to agree better with the ln(1/T)

divergence than with a simple power law T-α. Although the origin of such a

logarithmic divergence is still strongly debated (see chapter 3), it is interesting

to analyse our data for the normal-state resistivity and Hall-effect within the

framework of the model considering stripe formation in the CuO2 plane.


140

In this charge-stripe picture [Cheong91, Tranquada97, Arai99, Emery97b,

Moshchalkov98b & 99], dynamic metallic [Noda99, Ichikawa99, Tajima99]

stripes are thought to dominate the transport properties. So, within this model,

one expects a strong influence on the transport properties when, for some

reason, the 1D charge stripes are fragmented or pinned. Then the movement of

the charge carriers is restricted to these fixed 1D paths (in contrast to the

dynamic stripes) and the resistivity is reported to increase [Noda99, Ichikawa99,

Lavrov99]. Moreover, in the presence of stripe fragmentation, charge carriers

have to hop to another metallic stripe, passing the intercalating Mott-insulator,

also resulting in an increased resistivity. The occurrence of fragmentation or

pinning of the otherwise dynamic stripes can be understood when one realises

that any scattering process (or local scattering centre) yielding an inelastic

length smaller than Lφ~ξm1D will destroy the fragile regime of a striped CuO2

plane. By inserting the temperature dependency of this Lφ into the conductivity

expression with the proper dimensionality, one is then, in principle, able to

describe the low-temperature ln(1/T) divergence of the high-field resistivity (see

below).

One possible type of pinning centre is the crystallographic disorder in the CuO2

plane, in the form of dislocations. These dislocations will also alter the local

electronic and magnetic structure in the plane and at low temperatures, when the

stripes are less mobile, they can be expected to pin the magnetic domain walls

formed by the charge stripes (figure 5.2). Moreover, in the case of strong

pinning, stripe fragmentation is predicted to occur [Kivelson98].

Experimentally, the pinning of charge stripes has been seen by neutron

diffraction experiments on Nd-doped and pure La2-xSrxCuO4 [Tranquada97 &

97b]. The striking observation from these data is that, although the

incommensurate features (i.e. the stripes) are almost identical, the scattering in

the pure, near optimally doped, (La2-xSrx)CuO4 system is inelastic (dynamic

stripes) whereas in the (La1.6-xNd0.4Srx)CuO4 system elastic scattering is

observed, corresponding to static stripes. In general, pinning of these stripes is

correlated with the onset of an increasing resistivity [Noda99], although stripe

pinning has been found in underdoped samples that are metallic (but close to the


141

metal-insulator transition) [Ichikawa99], suggesting stripe fragmentation to be

as important as pinning for the creation of an insulating state.

So, for dynamic stripes, the resistivity will be quasi 1D metallic and the Hall

response in a magnetic field will be finite since dynamic charge stripes are able

to respond to the transverse electric field acting on the charge carriers. For

pinned stripes that are not fragmented, the resistivity can be expected to

remain essentially metallic since the 1D metallic wires remain. However, such

a reduced mobility of the stripes can be expected to have a noticeable influence

on the Hall effect. When the stripes are pinned, they cannot properly react to

the Lorentz force on the charge carriers and only a reduced Hall field (and thus

Hall resistivity ρyx) is built up. However, in the presence of stripe fragmentation or inter-stripe hopping, also an effect will be present due to the

charge inter-stripe hopping across the Mott-insulator phase. This will result in

an insulating longitudinal resistivity and a small but finite Hall effect.

Recently, based on Hall effect and X-ray measurements on Nd-doped

La2-xSrxCuO4 crystals [Noda99], it was argued that the Hall conductivity σxy

(equation 5.10) is related to the inverse stripe order.

( )

( )H

HRBRH

ab

oH

ab

zH

xx

yx

xy

yxxy

xx

22222 ρ

µ

ρρ

ρ

ρρ

ρσ ≈=≈

+=

(5.10)

In order to check this idea, we have combined our high-field ρab(T) and RH(T)

data above and below Tc to calculate the Hall conductivity σxy using

equation 5.10. The results are summarised in figure 5.15, for the samples

showing a pronounced divergence of the low-temperature resistivity.

From the plots in figure 5.15, it is clear that, once the resistivity starts increasing

on lowering temperature (at T < TMI), also the Hall conductivity goes down

rapidly and hence, according to the analysis made in [Noda99], stripe order in

these underdoped YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox samples increases.

However, a significant difference with the data published on the Nd-doped

La2-xSrxCuO4 crystals [Noda99] becomes clear when comparing them with our

high-field ρab(T) and RH(T) data (figures 4.5 to 4.10 in chapter 4). In our data,

the decreasing Hall conductivity σxy is almost completely due to the strongly


142

diverging longitudinal resistivity ρab(T) whereas the Hall response RH(T)

remains finite (and approximately constant) down to the lowest temperatures in

our experiments.

T (K)

σ xy

(arb

. uni

ts)

Y0.6Pr0.4Ba2Cu3O6.7

Y0.6Pr0.4Ba2Cu3O6.85 Y0.6Pr0.4Ba2Cu3O6.95

YBa2Cu3O6.45 YBa2Cu3O6.5

0 50 100 150

( )( )

σµ

ρxy

o H

ab H

HH R

≈2

0 50 100 150

TMI

TMITMI

TMI TMI

Figure 5.15: The off-diagonal conductivity σxy, calculated by combining the Hall coefficient RH and the in-plane resistivity ρab at 40 tesla (equation 5.10). The σxy,(T) data are presented for the YBa2Cu3O6.45, YBa2Cu3O6.5, Y0.6Pr0.4Ba2Cu3O6.7, Y0.6Pr0.4Ba2Cu3O6.85 and Y0.6Pr0.4Ba2Cu3O6.95 thin films. The arrows indicate TMI where the resistivity starts increasing on lowering temperature and the x-axis is drawn at σxy = 0.

When combining this result with the discussion about dynamic versus pinned

stripes, it becomes clear that, at low temperatures, the charge stripe picture can

only be brought in agreement with our normal-state transport data by assuming

stripe fragmentation or inter-stripe hopping effects. This causes an effective

recovery of the two-dimensional (2D) regime. By inserting the temperature

dependence of the inelastic length Lφ, of the scattering mechanisms working in

the intercalating insulating phase, into the conductivity expression for 2D

quantum transport (equation 5.7), one can calculate the low-temperature ln(1/T)


143

divergence of the high-field resistivity. For example, the inelastic length for

electron-electron or electron-phonon scattering, Lφ ~ 1/T€α [Abrikosov88],

combined with equation 5.7 for 2D quantum transport gives an ln(1/T)

correction to the low-temperature resistivity. Also electron interference effects

in the 2D weak localisation theory predict an ln(1/T) behaviour. Moreover, this

2D weak localisation model also agrees with our finding of a constant Hall

coefficient RH(T) at low temperatures.

5.5 Construction of the T(p) phase diagram

The construction of a so-called generic T(p) phase diagram, describing the

superconducting and normal-state transport properties of both the YBa2Cu3Ox

and the (Y0.6Pr0.4)Ba2Cu3Ox compounds, requires the combination of our high-

field transport data (chapter 3) and the estimates for the carrier concentration

from the Hall effect (chapter 4). This experimental phase diagram, already

constructed in chapter 4, can now be re-investigated in the framework of the

1D/2D quantum transport model [Moshchalkov93, 98b & 99]. Of course,

regardless of this interpretation, the experimental T(p) phase diagram, including its crossover lines remains valid.

In figure 5.16, the spin-gap ∆ as derived by applying the equation for 1D

quantum transport is plotted versus p, the fraction of holes per Cu atom in the

CuO2 plane. From this plot, it is clear that the ∆(p) data for YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox coincide very well, contrasting with the previous

disagreement of the T*(x) ≈ 2∆(x) crossover line for the two compounds. Thus,

the energy scale ∆(p) (for the 1D quantum transport) is well described by the

carrier density in the CuO2 plane. Moreover, since within the quantum transport

model [Moshchalkov93, 98b & 99], applied here for YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox, this energy scale equals the pseudo spin-gap, also this

pseudo spin-gap ∆ is well described by the carrier density in the CuO2 plane.


144

0

5

10

15

20

0

50

100

150

200

250

∆(K

) ∆(m

eV)

0 0.05 0.10 0.15 0.20 0.25


YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

0

5

10

15

20

0

5

10

15

20

0

50

100

150

200

250

0

50

100

150

200

250

∆(K

) ∆(m

eV)

0 0.05 0.10 0.15 0.20 0.250 0.05 0.10 0.15 0.20 0.25


YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

Figure 5.16: The pseudo-gap ∆ for the YBa2Cu3Ox and the Y0.6Pr0.4Ba2Cu3Ox thin films, derived from scaling the ρab(T) data to the expression for 1D quantum transport, plotted versus the fraction of holes per Cu atom in the CuO2 plane.

Having established the relevance of the carrier density as a suitable parameter

for creating a phase diagram for the normal-state properties within the 1D/2D

quantum transport model, the experimental T(p) phase diagram can now be re-

interpreted in the framework of this model. In figure 5.17, the T(p) phase

diagram is re-plotted with some extra indications to facilitate the discussion.

Region I, where a metallic linear temperature dependence of the resistivity is

observed (T > T*), is accurately described by the expression for a 2D Heisenberg system where short range AF fluctuations are observed in inelastic

neutron scattering experiments. When an underdoped high-Tc cuprate is cooled

below T*, an S-shaped ρ(T) develops, that can be scaled onto a single universal

curve for both compounds. This curve is accurately described by the model for

transport in a 1D striped regime (region II) and yields values for the spin-gap

that agree well with estimates in literature. This gap is well described as a

function of the carrier density.


145

0 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)

metallic2D Heisenberg

metallic1D striped

T*

200

300

400

0

100 TcTMI

0

100

I

II

IIISC

SC

≈ ≈TMI

Tc

II

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

III

0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)


metallic1D striped

T*

200

300

400

0

100

200

300

400

0

100 TcTMI

0

100

0

100

II

IIII

IIIIIISC

SC

≈ ≈TMI

Tc

IIII

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

IIIIII

Figure 5.17: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the 2D/1D crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane. The data for Y0.6Pr0.4Ba2Cu3Ox were shifted down by 100 K

The 1D striped regime is defined by 4 boundaries in the T(p) diagram. At low doping levels, the bulk antiferromagnetic order is recovered and the stripes

disappear. At high doping levels, the distance between stripes is expected to

decrease, charges leak into the Mott insulator phase between the stripes and as a

result, the charge stripes collapse completely (see also figure 5.3). At high temperatures, stripe meandering is expected to destroy the 1D regime,

recovering the 2D regime with antiferromagnetic fluctuations. At low


146

temperatures T < TMI, stripe pinning, fragmentation and inter-stripe hopping

effects establish a 2D insulating regime (region III). In the T(p) diagram, the

onset of this insulating-alike regime is indicated by TMI, below which the

resistivity increases with lowering temperature. The TMI(p) line for the

Y0.6Pr0.4Ba2Cu3Ox compound lies significantly higher than its equivalent for the

YBa2Cu3Ox samples (figure 5.17). This might, within the stripe scenario, be

due to the presence of additional disorder caused by the Y/Pr substitution,

resulting in stronger stripe pinning and fragmentation effects. At low temperatures T < Tc, the onset of macroscopic coherence between the so-

called pre-formed pairs [Emery97b & 99b] is predicted to result in the recovery

of bulk superconductivity (in the absence of high magnetic fields).

5.6 Conclusions

The universal ρ(T) behaviour in the underdoped YBa2Cu3Ox and

Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, is a strong indication of one

single scattering mechanism being dominant over the whole underdoped regime

in the Y123 system. Only the energy scale (the scaling parameter ∆ and the

crossover temperature T* ≈ 2∆) varies upon doping.

Any model trying to explain the extraordinary features of the normal-state

transport properties of the high-Tc's (linear ρ(T) at high temperatures, S-shaped

ρ(T) at intermediate temperatures and logarithmically diverging ρ(T) and

constant Hall coefficient RH at low temperatures) should also account for the

complex magnetic phase diagram for these high-Tc cuprates. In the underdoped

region of this diagram, at moderate temperatures T* < T < To, short-range

antiferromagnetic correlations develop in the CuO2 planes. Moreover, an

increasing amount of experimental and theoretical indications are in favour of

the existence of dynamic one-dimensional charge stripes in the CuO2 planes at

T < T*, acting as domain walls for the antiferromagnetic fluctuations. These

local charge inhomogeneities (1D charge stripes) will confine the AF regions,

resulting in the formation of a pseudo spin-gap at temperatures far above the

superconducting critical temperature Tc.


147

It is then tempting to assign the origin of the dominant scattering mechanism for

charge transport to the microscopic magnetic ordering in the planes of the high-

Tc cuprates. The importance of the CuO2 planes for the transport properties is a

widely documented feature of the high-Tc cuprates. The confinement of the

charge carriers in these planes reduces the dimensionality for charge transport to

2 dimensions (or less) and makes the conductivity σ in such 2D metallic system

to be dominated by quantum transport. In this case the approach based on the

following three basic assumptions [Moshchalkov93, 98b & 99] can be used: (i)

the dominant scattering mechanism in HTS in the whole temperature range is of

magnetic origin; (ii) the specific temperature dependence of the resistivity ρ(T)

can be described by the inverse quantum conductivity σ-1 with the inelastic

length Lφ being fully controlled by the magnetic correlation length ξm, and

finally, (iii) the proper 1D or 2D expressions should be used for calculating the

quantum conductivity with Lφ ~ ξm.

At high temperatures T* < T < To, in the 2D Heisenberg regime, the

combination of the expressions for the 2D spin correlation length with the

proper expression for the quantum resistance gives a linear temperature

dependence of the resistivity, due to the mutual cancellation between the

logarithmic ρ(ξm) dependence and the exponential temperature dependence of

ξm. This result is in perfect agreement with our finding of a linear ρ(T) at high

temperatures for all measured underdoped YBa2Cu3Ox and Y0.4Pr0.4Ba2Cu3Ox

thin films.

At intermediate temperatures TMI < T < T*, in the 1D striped regime, inelastic

neutron scattering experiments show evidence for the existence of dynamic

stripes and the observation of 1D features in the transport properties should

therefore not be limited to the Cu-O chain-direction only. Moreover, although

the 1D stripes are dynamic, no averaging of the transport properties will occur,

since, even for dynamic stripes, the charge will automatically follow the most

conducing paths, i.e. stripes, even if they are moving fast. So, in transport

experiments the magnetic correlation length ξm1D of a dynamic insulating stripe

permanently imposes the constraint Lφ~ξm1D, thus providing a persistent 1D

character of the charge transport in underdoped cuprates. Inserting this inelastic

length in the expression for 1D quantum conductivity yields an S-shaped ρ(T)


148

that perfectly describes the resistivity data obtained on the even-chain spin-

ladder compound Sr2.5Ca11.5Cu24O41. This compound, due to its specific

crystalline structure, definitely contains a 1D spin-ladder and therefore its

resistivity along the ladder direction should indeed obey the expression for 1D

quantum transport.

As a next step, a convincing scaling was pointed out between the resistivity of

this 1D spin-ladder compound and a typical underdoped high-Tc material,

YBa2Cu4O8, demonstrating that the resistivity versus temperature dependencies

of underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL

with a spin-gap ∆ are governed by the same underlying 1D (magnetic)

mechanism. This magnetic origin of the scattering of the charge carriers is

further confirmed by the fact that the scaling temperature To, used in the scaling

of ρ(T), works equally well for resistivity as for Knight-shift data KS and by the

nice fit of these KS data with the expressions derived from the 1D SL models.

The ρ(T) data of a La1.9Sr0.1CuO4 thin film under tensile epitaxial strain and

YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films with varying oxygen content,

scaled onto one universal curve, are all perfectly described by the expression for

1D quantum transport. The values of the spin-gap ∆, estimated from this fitting,

are in agreement with an independent determination of ∆ from resistive

measurements on other YBa2Cu3Ox thin films, twinned and de-twinned single

crystals. Moreover, they agree with estimates of the pseudo-gap as derived

from CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders.

In the 1D quantum transport model, where the inelastic length is presumed to be

dominated by the magnetic correlation length, the agreement of our data with

the gap-value determined from NMR experiments seems only natural. This

proves that our analysis, describing the transport in underdoped cuprates at

T < T* by taking into account the presence of 1D stripes, not only agrees

qualitatively, but also yields values for the pseudo spin-gap ∆ that agree well

with independent estimates.

At low temperatures T < TMI, the metallic behaviour of the resistivity at high

temperatures transforms into an insulating-alike, diverging, ρ(T) that was shown

to agree with a ln(1/T) divergence. In this chapter, our data for the normal-state

resistivity and Hall-effect was confronted with the possibility of stripe


149

formation in the CuO2 plane. In this charge-stripe picture, dynamic, metallic

stripes are thought to dominate the transport properties. So, within this model,


reason, the 1D charge stripes are fragmented or pinned. Such a fragmentation

or pinning of the else dynamic stripes will be caused by any scattering process

(or local scattering centre) yielding an inelastic length smaller than Lφ ~ ξm1D

(disorder). It was found that, at low temperatures, the charge stripe picture can

only be brought in agreement with our normal-state transport data by assuming stripe fragmentation or inter-stripe hopping effects.

These processes invoke a strong influence of the intercalating Mott insulator

phase on the charge transport, yielding a 2D insulating resistivity and a finite Hall response. By inserting the temperature dependence of the inelastic length

Lφ, of the scattering mechanisms working in the intercalating insulating phase,

into the conductivity expression for 2D quantum transport (equation 5.7), one

can calculate the low-temperature ln(1/T) divergence of the high-field

resistivity. For example, the inelastic length for electron-electron or electron-

phonon scattering, Lφ ~ 1/T€α [Abrikosov88], combined with the expression for

2D quantum transport gives an ln(1/T) correction to the low-temperature

resistivity. Also electron interference effects in the 2D weak localisation theory

predict an ln(1/T) behaviour. Moreover, this 2D weak localisation model also

agrees with our finding of a constant Hall coefficient RH(T) at low temperatures.

The construction of a so-called generic T(p) phase diagram, describing the

superconducting and normal-state transport properties of both the YBa2Cu3Ox

and the (Y0.6Pr0.4)Ba2Cu3Ox compounds was possible by combining our high-

field transport data with the estimates for the carrier concentration from the Hall

effect. This experimental phase diagram, was in this chapter re-investigated in

the framework of the 1D/2D quantum transport model. It was shown that the

energy scale ∆(p) (for the 1D quantum transport) is well described by the carrier

density in the CuO2 plane. Moreover, since within the quantum transport model

this energy scale equals the pseudo spin-gap, also this pseudo spin-gap ∆ is well

described by the carrier density in the CuO2 plane.

The 1D striped regime is defined by four boundaries in the T(p) diagram. At

low doping levels, the bulk antiferromagnetic order is recovered and the stripes


150

disappear. At high doping levels, the distance between stripes is expected to

decrease and the Mott insulator phase between stripes collapses. At high temperatures, stripe meandering is expected to destroy the 1D regime,

recovering the 2D regime with antiferromagnetic fluctuations. At low temperatures T < TMI, stripe pinning, fragmentation and inter-stripe hopping

effects establish a 2D insulating regime. In the T(p) diagram, the onset of this

insulating regime is indicated by TMI, below which the resistivity increases with

lowering temperature. The TMI(p) line for the Y0.6Pr0.4Ba2Cu3Ox compound lies

significantly higher than its equivalent for the YBa2Cu3Ox samples , which

might, within the stripe picture, be due to the presence of additional disorder

due to the Y/Pr substitution, resulting in stronger stripe pinning and

fragmentation effects. At low temperatures T < Tc, the onset of macroscopic

coherence between the so-called pre-formed pairs [Emery97b & 99b] is

predicted to result in the recovery of bulk superconductivity.

SUMMARY

151

Summary

The normal-state transport properties of a set of YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox thin films with various oxygen concentrations and strained

La1.9Sr0.1CuO4 epitaxial thin films were measured in zero field and at very high pulsed magnetic fields. This dual-track approach - lowering the critical

temperature Tc by changing the hole content p on one side and using very high

magnetic fields on the other side - allows us to cover the whole underdoped to

optimally doped region of the T-p (temperature versus hole content in the CuO2

plane) phase diagram (figure 1) and thus explore the normal-state transport

properties for these compounds -usually studied at temperatures above the

critical temperature Tc- for the first time at temperatures T < Tc.

Figure 1: The properties of the high-Tc cuprates vary with temperature (vertical axis) and doping of the CuO2 planes [Batlogg2000].

SUMMARY

152

As a first step, inspired by earlier work [Wuyts94 & 96], it was shown that the

metallic zero-field normal-state resistivity ρρ(T) above Tc for various levels of

hole doping -both for the YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox members of

the cuprate superconductors- can be scaled onto one single universal curve.

An energy scale ∆, the resistivity ρ∆ at T = ∆ and the residual resistivity ρo are

used as scaling parameters. The inclusion of the residual resistivity ρo in this

scaling allowed a scaling that is of a better quality than the earlier analysis

[Wuyts94 & 96]. The universal curve exhibits three regions with a qualitatively

different ρ(T) behaviour:

(I) a region of linear ρ(T) at high temperatures T > T*,

(II) a super-linear ρ(T) at intermediate temperatures T < T*

(III) a low temperature insulating-alike ρ(T) regime at T < TMI.

The low-temperature regime (III) is to a large extent masked by the onset of

superconductivity at T = Tc. The distinct features in the temperature

dependence of the metallic zero-field resistivity ρ(T) of YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox in regions I and II are universal for all the reported curves,

the only difference is the temperature scale ∆∆ at which these features occur.

As a second step, the magnetoresistivity of these YBa2Cu3Ox and

(Y0.6Pr0.4)Ba2Cu3Ox epitaxial thin films was measured in pulsed high magnetic fields up to 50 T. This allowed us to investigate the normal-state ρρ(T) behaviour at temperatures below the critical temperature Tc. It was shown

that these high field data in the metallic regime scale reasonably good with the

universal ρ(T/∆) curve. However, the ρ(T) data in the low-temperature

insulating regime (of diverging resistivity) do not scale satisfactorily when

using the same scaling parameters as in the zero-field scaling.

The existence of a universal ρ(T/∆) curve for the metallic normal-state

resistivity was interpreted as a strong indication that one single mechanism

dominates the scattering of the charge carriers in these materials. This

mechanism then determines the energy scale ∆, which is dependent upon

doping.

SUMMARY

153

These experiments revealed the ground state at T < Tc of YBa2Cu3Ox (for

x ≤ 6.8), (Y0.6Pr0.4)Ba2Cu3Ox (for all levels of oxygen content) and the tensile

strained La1.9Sr0.1CuO4 ultra-thin film to be insulating, which leads to an increase of the low-temperature resistivity upon decreasing temperature.

The insulating-alike ρ(T) behaviour at low temperatures was shown not to obey

a simple activated or hopping mechanism. It was argued that, most probably,

spin-flip scattering in the framework of the Kondo model or the existence of a

narrow metallic impurity band with localised edges do not play an important

role in producing a divergent resistivity upon lowering temperatures. Our low-

temperature data were shown to have a ln(1/T) divergence in high magnetic fields, as was first observed in the cuprate La2-xSrxCuO4 by Ando [Ando95].

Models predicting a divergent resistivity, which are not conflicting with our

experimental data, include the phenomenological c-axis resistivity model

[Zha96], a model accounting for electron interactions in 2D disordered systems

[Altshuler80], the bipolaron model [Alexandrov97], the 2D Luttinger model

[Anderson91, Clarke95], the 2D Anderson weak localisation theory [Lee85] or

the pinning of the dynamic charge stripes [Kivelson98, Noda99, Ichikawa99,

Tranquada97 & 97b].

The comparison of the transport data on epitaxially strained La1.9Sr0.1CuO4

ultra-thin films with data on samples without epitaxial strain and our

YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox samples demonstrated that epitaxial strain

probably influences not only the doping of the CuO2 planes but also the disorder

scattering.

In order to facilitate further interpretation of the normal-state resistivity data,

Hall-effect measurements on the same YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin

films at temperatures extending to below the critical temperature Tc were

reported. These measurements were performed at very high pulsed magnetic

fields in order to fully access the normal state (H > Hc2).

The Hall measurements yield a signal that is three orders of magnitude smaller

than in standard magnetoresistivity experiments. The successful

accomplishment of these measurements was possible by a careful

SUMMARY

154

implementation of vibration insulation and shielding for electric interference,

combined with a low contact resistance to the sample.

From the measurements of the high-field Hall-resistivity ρyx(H), the field

dependence of the Hall coefficient RH(H) at various temperatures was

calculated. By concentrating on the normal-state part of the data (H > Hc2), the

Hall coefficient RH(T) curves at fixed (high) field were constructed. These RH(T)

curves show a transition to the normal-state value of RH as temperature is

increased. This transition shifts to lower temperatures in high fields. At these

high fields, the low-temperature normal-state RH(T) is recovered, exhibiting only a very weak temperature dependence. By combining the RH(T) curves

with the high-field ρab(T) curves for the same samples, it is clear that at these

high fields the normal state is entered unambiguously down to temperatures

T << Tc. Thus, by using very high magnetic fields, we were able to access the

normal-state Hall-effect in the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films in

the temperature range down to temperatures T << Tc.

The quasi temperature independence of the low-temperature Hall-coefficient RH is in sharp contrast with the strongly divergent ρρab(T) observed in these compounds. The approximately temperature-independent

Hall coefficient, is an important test for the existing models already explaining a

low-temperature divergence of ρab(T). Among the models, still applicable for

the description of both ρab(T) and RH(T), the weak localisation-model explains

both the constant RH at low temperatures and the ln(1/T) divergence of the high-

field ρab(T) data.

A further combination of the Hall data with the ρab(T) resistivity curves then

allows the derivation of the Hall-angle. The Hall-angle is predicted, and widely

observed, to obey a quadratic dependence upon temperature, in the temperature

range from Tc up to room temperature, with a constant offset that is related to

the temperature independent scattering by magnetic impurities. Also here, the

existing studies at T > Tc are extended to temperatures T << Tc by our high-field

data. The region of diverging Hall-angle (mostly observed around T ~ Tc) was

significantly shifted to lower temperatures in this high field limit. Although our

experimental temperature window is limited, a tentative agreement with the

often cited quadratic temperature dependence was seen. The zero-temperature

SUMMARY

155

intercept of the Hall-angle cotθH , giving information about the presence of

magnetic scatterers, was, in the samples with the lowest oxygen content, shown

to be elevated, supposedly due to antiferromagnetic fluctuations. Also in the

Y/Pr substituted samples, a higher offset of the Hall-angle was observed. This

suggests that the presence of the magnetic Pr atoms, apart from the reduction of

the critical temperature Tc, causes a significant temperature-independent

contribution to the scattering of the charge carriers. This is in agreement with

the high residual resistivity ρo and the small ρ290 K/ρo resistivity ratio.

The carrier density nH (and thus also p, the fraction of holes per Cu-atom in

the CuO2 plane) calculated from the Hall-data enabled us to construct a generic T(p) phase diagram for the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox compounds.

The universal ρρ(T) behaviour in the underdoped YBa2Cu3Ox and

Y0.4Pr0.4Ba2Cu3Ox thin films, reported in chapter 3, strongly supports the idea

that a single scattering mechanism is dominant over the whole underdoped

regime in the Y123 system. Bearing in mind the complex magnetic phase

diagram, with short-range AF correlation and a pseudo gap showing up at

temperatures far above the superconducting critical temperature Tc and

reminding the strong indications of stripe formation in the CuO2 planes (an

inhomogeneous intercalation of AF insulating regions and metallic hole-rich

stripes), it is tempting to assign the origin of this dominant scattering mechanism to the microscopic magnetic and charge ordering.

Therefore, our high-field transport data (both the resistivity and the Hall-effect)

were used to investigate the effect of the short-range antiferromagnetic

fluctuations and possible stripe formation in the CuO2 plane on the normal

state transport properties. In this charge-stripe scenario [Emery97b & 99b],

dynamic metallic [Ichikawa99, Noda99, Tajima99] stripes are thought to

dominate the transport properties. Since mobile carriers in this case are

expelled from the surrounding Mott-insulator phase into the stripes, the latter

then provide the lowest resistance paths. This makes the transport properties

very sensitive to the formation of the stripes, both static and dynamic. To check

this idea, an existing model [Moshchalkov93, 98b], describing transport both in the 2D Heisenberg regime (above T*) as in the 1D striped regime

SUMMARY

156

(below T*) where the pseudo gap develops, was used as a framework for the

interpretation of our high-field normal-state transport data.

The proposed approach [Moshchalkov93, 98b] is based on three basic

assumptions:

1. the dominant scattering mechanism in HTS in the whole

temperature range is of magnetic origin;

2. the specific temperature dependence of the resistivity ρ(T)

can be described by the inverse quantum conductivity σ-1

with the inelastic length Lφ being fully controlled, (via a

strong interaction of holes with Cu2+ spins) by the magnetic

correlation length ξm, and, finally,

3. the proper 1D or 2D expressions should be used for

calculating the quantum conductivity with Lφ ~ ξm.

At high temperatures T* < T < To, in the 2D Heisenberg regime, the

combination of the expressions for the 2D spin correlation length with the

proper expression for the quantum resistance gives a linear temperature dependence of the resistivity, due to the mutual cancellation between the

logarithmic ρ(ξm) dependence and the exponential temperature dependence of

ξm. This result is in perfect agreement with our finding of a linear ρ(T) at high

temperatures for all measured underdoped YBa2Cu3Ox and Y0.4Pr0.4Ba2Cu3Ox

thin films.

At intermediate temperatures Tc < T < T*, in the 1D striped regime, although

the 1D stripes are dynamic, no averaging of the transport properties will occur,

since, even for dynamic stripes, the charge will automatically follow the most

conducing paths, i.e. stripes, even if they are moving fast. So, in transport

experiments the magnetic correlation length ξm1D of a dynamic insulating stripe

permanently imposes the constraint Lφ~ξm1D, thus providing a persistent 1D

character of the charge transport in underdoped cuprates. Inserting this inelastic

length in the expression for 1D quantum conductivity yields an S-shaped ρρ(T) that perfectly describes the resistivity data obtained on the even-chain spin-

ladder compound Sr2.5Ca11.5Cu24O41. This compound, due to its specific

crystalline structure, definitely contains a 1D spin-ladder and therefore its

SUMMARY

157

resistivity along the ladder direction should indeed obey the expression for 1D

quantum transport.

As a next step, a convincing scaling was found between the resistivity of this

1D spin-ladder compound and a typical underdoped high-Tc material,

YBa2Cu4O8, demonstrating that the resistivity versus temperature dependencies of underdoped cuprates in the pseudo-gap regime at T < T* and even-chain SL with a spin-gap ∆∆ are governed by the same underlying 1D (magnetic) mechanism. This magnetic origin of the scattering of the

charge carriers is further confirmed by the fact that the scaling temperature To,

used in the scaling of ρ(T), works equally well both for resistivity and for

Knight-shift data KS. A good fit of these KS data is achieved by using the

expressions derived from the 1D spin-ladder models.

The ρ(T) data of a La1.9Sr0.1CuO4 thin film under tensile epitaxial strain and

YBa2Cu3Ox and (Y0.6Pr0.4)Ba2Cu3Ox thin films with varying oxygen content,

scaled onto one universal curve, are all perfectly described by the expression for 1D quantum transport. The values of the spin-gap, found from this

fitting, are in agreement with independent estimates from resistive

measurements on other YBa2Cu3Ox thin films, twinned and de-twinned single

crystals. Moreover, they agree with estimates of the pseudo-gap as derived

from CuO2-plane 17O and 63Cu Knight-shift measurements on aligned powders.

In the 1D quantum transport model, where the inelastic length is presumed to be

dominated by the magnetic correlation length, the agreement of our data with

the gap-value determined from NMR experiments seems to be quite natural.

This proves that our approach, describing the transport in underdoped cuprates at T < T* in a 1D striped manner, not only agrees qualitatively, but also yields values for the pseudo spin-gap ∆∆ that agree well with independent estimates.

In the final chapter of this work, our data for the normal-state resistivity and

Hall-effect were analysed by considering the possibility of stripe pinning by

disorder. The metallic behaviour of the resistivity at high temperatures,

transforms at low temperatures T < TMI into an insulating-alike, diverging, ρ(T)

that was shown to agree with a ln(1/T) law, whereas the Hall-coefficient is

temperature independent at low temperatures. Within the charge-stripe picture,

SUMMARY

158


reason, the 1D charge stripes are fragmented or pinned. Such a fragmentation

or pinning of the else dynamic stripes will be caused by any scattering process

(or local scattering centre) yielding an inelastic length smaller than Lφ ~ ξm1D. It

was found that, at low temperatures, the charge stripe picture can only be

brought into agreement with our normal-state transport data by assuming stripe fragmentation, promoting inter-stripe hopping effects. These processes

invoke a strong influence of the intercalating Mott insulator phase on the charge

transport, yielding a (2D) insulating resistivity and a finite Hall response.

By inserting the temperature dependence of the inelastic length Lφ, of the

scattering mechanisms working in the intercalating insulating phase, into the

conductivity expression for 2D quantum transport, one can calculate the low-

temperature ln(1/T) divergence of the high-field resistivity. For example, the

inelastic length for electron-electron or electron-phonon scattering, Lφ ~ 1/T€α

[Abrikosov88], combined with the expression for 2D quantum transport gives

an ln(1/T) correction to the low-temperature resistivity. Also electron

interference effects in the 2D weak localisation theory predict an ln(1/T)

behaviour. Moreover, this 2D weak localisation model also agrees with our

finding of a constant Hall coefficient RH(T) at low temperatures.

The construction of a so-called experimental generic T(p) phase diagram,

describing the superconducting and normal-state transport properties of both the

YBa2Cu3Ox and the (Y0.6Pr0.4)Ba2Cu3Ox compounds was possible by combining

our high-field transport data with the estimates for the carrier concentration

from the Hall effect. This experimental phase diagram (see figure 2 below, to

be compared with the schematic phase diagram on figure 1 above) was

investigated in the framework of the 1D/2D quantum transport model

[Moshchalkov93, 98b]. It was shown that the energy scale ∆(p) (for the 1D

quantum transport) is well described by the carrier density in the CuO2 plane.

Moreover, since within the quantum transport model this energy scale equals

the pseudo spin-gap, also this pseudo spin-gap ∆ is well described by the

carrier density in the CuO2 plane.

The 1D striped regime is defined by four boundaries in the T(p) diagram.

At low doping levels, the bulk antiferromagnetic order is recovered and the

SUMMARY

159

stripes disappear. At high doping levels, the distance between stripes is

expected to decrease and the Mott insulator phase between stripes collapses. At

high temperatures, stripe meandering results in the destruction of the 1D

regime, thus recovering the 2D regime with antiferromagnetic fluctuations. At

low temperatures T < TMI, stripe pinning, fragmentation and inter-stripe

hopping effects establish a 2D insulating regime. In the T(p) diagram, the onset

of this insulating-alike regime is indicated by TMI, below which the resistivity

increases with lowering temperature. The TMI(p) line for the Y0.6Pr0.4Ba2Cu3Ox

compound lies significantly higher than its equivalent for the YBa2Cu3Ox

samples , which might, within the stripe picture, be due to the presence of

additional disorder caused by the Y/Pr substitution, which enhances stripe

pinning and fragmentation effects. For the YBa2Cu3Ox system, the TMI(p) line

is below the Tc(p) line for almost all levels of hole doping and thus the

insulating state at zero-field is effectively masked by the onset of

superconductivity. For the Y0.6Pr0.4Ba2Cu3Ox compound, the TMI(p) line is

above the Tc(p) line for all levels of oxygen content and the insulating tendency

can therefore already be seen in the zero-field ρ(T) measurements above Tc. At temperatures T < Tc, the onset of macroscopic coherence between the so-

called pre-formed pairs [Emery97b & 99b] is predicted to result in the recovery

of bulk superconductivity.

The critical temperature Tc,mid(p) and the boundary TMI between the metallic and

insulating-alike ρ(T) regimes do not coincide for the YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox compounds; although showing a qualitatively similar

behaviour, shifted to a lower Tc for the Y0.6Pr0.4Ba2Cu3Ox system. The fact that,

for the same hole concentration p, the critical temperature Tc is lower in the

Y0.6Pr0.4Ba2Cu3Ox system in comparison with YBa2Cu3Ox, is an indication that

the magnetic pair-breaking in the (Y/Pr)Ba2Cu3Ox compound might play an

important role in the reduction of the superconducting critical temperature Tc.

So, by performing magnetoresistivity and Hall effect measurements in very high

magnetic fields on a selection of high-Tc superconducting thin films with

varying level of hole doping, we were able to complement the schematic T-p phase diagram in figure 1 with three experimental phase boundaries: T*(p)

describing a 2D to 1D crossover, the superconducting critical temperature Tc(p)

SUMMARY

160

and TMI(p) marking the onset of an insulating-alike behaviour. The normal-state generic phase diagram was discussed in terms of a 2D metallic

Heisenberg regime, a metallic 1D stripe region where the pseudo gap develops

and a low-temperature insulating regime.

0 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)


metallic1D striped

T*

200

300

400

0

100 TcTMI

0

100

I

II

IIISC

SC

≈ ≈TMI

Tc

II

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

III

0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)


metallic1D striped

T*

200

300

400

0

100

200

300

400

0

100 TcTMI

0

100

0

100

II

IIII

IIIIIISC

SC

≈ ≈TMI

Tc

IIII

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

IIIIII

Figure 2: Generic T(p) phase diagram for the YBa2Cu3Ox (diamonds, solid line) and the Y0.6Pr0.4Ba2Cu3Ox (circles, dotted line) thin films. Indicated are the 2D/1D crossover temperature T* (filled symbols), the superconducting critical temperature Tc (open symbols) and the boundary TMI between the metallic and the insulating-alike regimes for ρ(T). All are plotted versus the fraction of holes per Cu-atom in the CuO2 plane. The data for the Y0.6Pr0.4Ba2Cu3Ox system are shifted down by 100 K.

APPENDICES

161

Appendices

Sample preparation and characterisation

Any proper characterisation of the normal state transport properties of high-Tc

superconductors -as is the aim of this work- requires the use of high-quality

single crystalline samples. This leaves the choice to either single-crystals or

epitaxial thin films. From the viewpoint of magnetoresistance and Hall effect

measurements, the use of epitaxial thin films is desirable for reasons of a higher

resistance and Hall-signal and the possibility of patterning the films in a well-

defined geometry.

In this appendix, the preparation of these thin films by high-pressure DC-sputtering will be discussed in appendix A. The procedure for changing the oxygen content of these thin films to a certain nominal value is discussed in

appendix B, while the patterning and contacting of the films is treated in

appendix C. Finally, the techniques employed for the characterisation of the

structural and superconducting properties are introduced in appendix D.

Appendix A: Thin film deposition by sputtering

The preparation of high quality thin films of YBa2Cu3Ox requires two

constraints to be fulfilled. The first constraint involves the stability of the

YBa2Cu3Ox compound during deposition. From the thermodynamic phase diagram (oxygen pressure

2OP versus temperature T), presented in figure A.1, it

is clear that the YBa2Cu3Ox compound is stable only in a narrow T-2OP

window. Combining this with the need for relatively high deposition

temperatures, in order for the deposited atoms to have enough mobility to eventually form a crystalline film, oxygen partial pressures of a few hPa are

required.

APPENDICES

162

This short discussion shows that ultra high vacuum (UHV) techniques like

molecular beam epitaxy (MBE) are only suited to deposit thin films of these

copper oxides when using differential pumping or a local flux of oxygen and

accurate control of the different sources [Locquet94]. Locquet and co-workers

have prepared the strained La1.9Sr0.1CuO4 ultra-thin films (of typical

thickness of 100 Å) used in this work by molecular beam epitaxy with block-by-block deposition [Locquet94]. The choice of the substrate - SrLaAlO4

(SLAO) or SrTiO3 (STO) - enabled them to induce compressive (SLAO) or

tensile (STO) strain in the ab-plane [Locquet96, 98 & 98b, Sato97], see also

paragraph 1.3.3.

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

900 800 700 600 500 400103

102

101

100

10-1

10-2

P O2 (

Pa)

1000/T (K-1)

T (oC)

YBa2Cu3OxY2BaCuO5

+BaCuO2

+CuO2

x = 6.0

x = 6.5x = 6.9deposition

Figure A.1: Thermodynamic phase diagram (oxygen partial pressure 2OP versus

temperature T), indicating the stability region for the YBa2Cu3Ox compound [Hammond89]. The shaded ellipse indicates the region of oxygen pressure and substrate temperature where the actual deposition takes place.

For the preparation of the YBa2Cu3Ox and (YPr)Ba2Cu3Ox thin films studied

in this work, it was chosen to use the high-pressure DC-sputtering technique.

For this purpose, a dedicated setup was developed, implementing the necessary

precautions to perform a high temperature deposition in an oxygen atmosphere

APPENDICES

163

[Wagner99, Teniers99]. Sputtering involves the creation of a plasma in the

sputtering gas (in our case 2.5 hPa of oxygen) by a glow-discharge maintained

by an electric field (280 V over a distance of 2 cm) between the substrate and

the target material, a water-cooled ceramic disk of stoichiometric YBa2Cu3Ox

or Y0.6Pr0.4Ba2Cu3Ox (99.99 %, 50 mm diameter, 3 mm thickness,

Superconductive Components Inc.). In this plasma, ionised +2O molecules are

created and accelerated to the target material -kept at a negative potential- and

their collision with the target releases neutral atoms from the stoichiometric

disk. These atoms condense on the substrate (heated to a temperature of

approximately 840 oC on a grounded holder) and form a thin film at a

deposition rate of 20 Å/min.

To avoid the deposition of off-stoichiometric films due to the different

sputtering yield of the four constituents of the YBa2Cu3Ox compound, all new

targets were pre-sputtered 24 hours before the first use and 45 minutes before

each deposition. This procedure creates an "altered" layer that is depleted of the

atoms with the highest sputtering yield. As such, the sputtered atoms will

regain the desired stoichiometry. The possible back-sputtering of the

deposited film by negatively charged ions is prevented by using a very high

oxygen partial pressure (2.5 hPa) that neutralises and thermalises these ions.

An important parameter in the deposition of thin films is the choice of the

substrate. In table A.2 the lattice parameters and thermal expansion

coefficients are summarised for YBa2Cu3Ox and commonly used substrates. In

this work SrTiO3, cut perpendicular to the c-axis, was chosen as a substrate

because of the good matching of both the lattice parameters and the thermal

expansion coefficient. Moreover, this substrate material does result in

interdiffusion like MgO. For reasons of thermal homogeneity (and hence the

homogeneity of the thin film) rather thick substrates (1.2 mm), surrounded by

platelets of Al2O3, were glued to the heater with silver epoxy.

APPENDICES

164

Material ab-plane symmetry

Lattice parameters

(Å)

Thermal expansion coefficient

(10-6/oC)

YBa2Cu3Ox orthorhombic a = 3.823 b = 3.886 c = 11.65

12

SrTiO3 (100) square a = 3.905 11

MgO (100) square a = 4.2 13

LaAlO3 trigonal a = 5.375

SrLaAlO4 square a = 3.76

Table A.2: Lattice parameters and thermal expansion coefficients for YBa2Cu3Ox and various substrate materials. Also the symmetry in the ab-plane of the materials is indicated.

The deposition procedure starts with 45 minutes of pre-sputtering at 2.5 hPa

oxygen pressure with the sample shielded. The actual deposition at ~ 20 Å/min

is also carried out at 2.5 hPa with a substrate temperature of 840 oC. After this,

an oxygen annealing is performed at 700 hPa and 700 oC for 15 minutes, then at

600 oC for another 15 minutes after which a slow cooldown (30 minutes) to

room temperature is performed, still keeping an oxygen pressure of 700 hPa. In

this way, an epitaxial thin film, optimally loaded with oxygen, is obtained.

Appendix B: Procedure to vary the oxygen content

As can already be seen on the thermodynamic phase diagram of figure A.1, the

YBa2Cu3Ox compound exists with variable oxygen contents (6 < x < 7),

depending on temperature and the partial pressure of oxygen in the

environment. In this diagram of oxygen partial pressure versus temperature,

phase lines of constant oxygen content for the YBa2Cu3Ox compound can be

drawn (figure B.1). In principle, at room temperature, this compound has no

stable oxygen content since these ambient conditions are far above the x = 6.9

line and samples with a reduced oxygen content will take up oxygen. The

dynamics is however so slow that at a time scale of weeks one has an effective

constant oxygen concentration.

APPENDICES

165

A simple method for changing the oxygen content in the YBa2Cu3Ox compound

was already introduced above (App. A): a slow cooldown at a constant oxygen

pressure, crossing the P(T) lines of constant oxygen content. When the desired

P(T) line is reached, the temperature is kept constant for a certain time after

which a fast quenching of the sample to room temperature is performed. Since,

during this quenching procedure, multiple P(T) lines have to be crossed, the

homogeneity of the oxygen distribution in the sample is strongly dependent

upon this quenching step.

In this work a refined method was used [Maenhoudt95], in which the P(T)

diagram is traversed as indicated in figure B.1.

108

106

102

100

10-2

10-4

104

0 200 400 600 800 1000

6.1

6.5

6.86.6

6.9

6.2

6.7

6.36.4

P O2 (

Pa)

T (oC)

Figure B.1: Phase diagram (oxygen partial pressure 2OP versus temperature T),

adapted from [Gallagher87], indicating the P(T) phase lines of constant oxygen content for the YBa2Cu3Ox compound.

As a first step, the optimally oxygenated thin film sample -covered in an

optimally oxygenated bulk YBa2Cu3Ox container- is put in a quartz tube that is

evacuated to a pressure of 10-6 mbar and consecutively put at approximately

13 hPa of oxygen pressure and heated up to the P(T) line corresponding to the

desired nominal oxygen content (x = 6.6 in figure B.1). After keeping the

system at these conditions for a few hours, a slow, computer controlled,

cooldown (1 oC/min) is initiated, carefully following the P(T) phase line of

APPENDICES

166

constant oxygen concentration (figure B.1). At low temperatures and pressure,

when the dynamics of oxygen desorption is slow, the system is quenched to

room temperature (figure B.1).

Several P(T) phase diagrams were proposed in literature, all qualitatively in

agreement with the one shown in figure B.1 [Gallagher87]. In this work it was chosen to use the

2OP -T diagram of Tetenbaum and co-workers [Tetenbaum89]

which, after comparison with other phase diagrams [Maenhoudt95], gives a

good correspondence between the critical temperatures of thin film samples

prepared according to this scheme and bulk samples [Maenhoudt95] (see

figure 3.4 for the comparison of our thin films with published data on bulk

samples). Therefore, the values for the oxygen content indicated for our thin

film samples is the nominal value from the oxygen desorption procedure.

Appendix C: Patterning and contacting of thin films

In order to obtain a well defined geometry for the magnetoresistivity and Hall-

effect measurements on the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films, they

were etched into the pattern shown in figure C.1. The dimensions of the bridges

were chosen in a way to ensure sufficiently large signals.

Before the actual etching, gold contacts were evaporated (thickness ~ 500 Å)

and annealed at 500oC in an oxygen flow. This resulted in contacts with a

resistance below 1 Ω; no influence of this procedure on the superconducting

transition was observed.

As a first step in the etching procedure, a thin layer of photo resist (OLIN

HPR504) is spinned onto the film, which is then hardened by baking at 90oC for

about 12 minutes. While covered with the proper chromium-on-glass mask, the

film is then illuminated by UV light during 10 seconds. After that, the resist

layer is developed (OLIN HPRD407) and the film is rinsed with distilled water

and dried with nitrogen gas. The etching itself is performed with HNO3 during

10 to 30 seconds, depending on the thickness of the film. After checking

whether the etching was successful, the resist layer is removed with acetone.

APPENDICES

167

1 mm

11

22

33 44 55

Figure C.1: The pattern etched onto the YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films. The dimensions of the bridges are (1) 3000 x 300 µm, (2) 1000 x 50 µm, (3) 100 x 10 µm, (4) 50 x 5 µm and (5) 30 x 5 µm.

For the measurements in pulsed magnetic fields, bridge 2 was selected

(figure C.1) and cut out of the film using a diamond wire-saw. Due to its length

to width ratio (l/w = 20), this narrow 1000 x 50 µm strip has the advantage of

giving a high signal in transport experiments (important for measurements in

pulsed fields), even when a small current is applied.

The disadvantage of these dimensions is that even a small over-etching can

result in deviations of the transport properties. However, for all films studied in

this work, no broadening of the resistive superconducting transition, a sign of

local degradation, was observed after etching.

Finally, the small 3 x 3 mm piece of the film, containing strip 2, was mounted

on a plexi-glass holder and contacted with gold wire (0.1 mm diameter) using

silver paint.

The thickness of the films was determined by atomic force microscopy (AFM)

on the patterned strips (see table C.2). The edge of the strip was traversed

several times at different positions along the strip. The indicated thickness t is

the average of these measurements and the error was estimated to be

approximately 100 Å.

APPENDICES

168

Sample composition t (Å)

#1 YBa2Cu3Ox 1770

#2 YBa2Cu3Ox 1300

#3 YBa2Cu3Ox 2300

#4 Y0.6Pr0.4Ba2Cu3Ox 850

Figure C.2: The thickness of the films studied in this work as determined by AFM measurements. More information on these samples is given in paragraph 3.1

Appendix D: Characterisation

D.1 Structural characterisation by X-ray diffraction (XRD)

The structural quality of the thin films was checked by x-ray diffraction (XRD)

measurements. This allows the characterisation of the texturisation of the thin

films. Since the wavelength of the x-rays is of the same order as the lattice

parameters, they are diffracted by the lattice and the condition for constructive interference is given by Bragg's law λθ nd =sin2 with d being the interlayer

spacing, θ the angle of incidence, λ the wavelength of the x-rays and n an

integer number.

θ

2θX-raysource

Detector

Thin film

Figure D.1: The Bragg-Brentano XRD geometry with fixed and equal distance between the sample and the detector and the sample and the source. The sample has an angular degree of freedom (θ) while the detector moves in a circle (2θ.)

The x-ray diffraction measurements in this work were carried out in the Bragg-

Brentano geometry (figure D.1) using a Rigaku 12 kW rotating anode x-ray

diffractometer using Cu-Kα radiation (wavelength 1.5408 Å, mostly used at

7.5 kW). Figures D.2 and D.3 show a θθ-2θθ scan for a YBa2Cu3Ox and a

APPENDICES

169

Y0.6Pr0.4Ba2Cu3Ox thin film, in which the sample rotates with an angle θ while

the detector performs a 2θ rotation. In that way, all directions are scanned in

which constructive interference can give rise to intensity peaks (Bragg's law).

0 10 20 30 40 50 60 700

5

10

15

20

25

(007

)

(200

) SrT

iO3 (

006)

(005

)

(004

)

(100

) SrT

iO3 (

003)

(002

)

(001

)

(008

)

2θ (degree)

Inte

nsity

(a.

u.)

0 20 40 60

log(

inte

nsity

)

2θ (degree)

YBa2Cu3O7 on SrTiO3

(007

)

(200

) SrT

iO3 (

006)

(005

)

(004

)

(100

) SrT

iO3 (

003)

(002

)

(001

)

(008

)

Figure D.2: XRD spectrum of a YBa2Cu3Ox thin film on SrTiO3, measured during a θ€-2θ scan. The inset shows the same data, with a logarithmic intensity scale.

0 10 20 30 40 50 60 700

5

10

15

20

25

30

(007

)

(200

) SrT

iO3 (

006)

(005

)

(004

)

(100

) SrT

iO3 (

003)

(002

)

(001

)

Inte

nsity

(a.

u.)

2θ (degree)

Y0.6Pr0.4Ba2Cu3O7 on SrTiO3

0 20 40 60

2θ (degree)

log(

inte

nsity

)

(007

)

(200

) SrT

iO3 (

006)

(0

05)

(004

)

(100

) SrT

iO3 (

003)

(0

02)

(001

)

(008

)

Figure D.3: XRD spectrum of a Y0.6Pr0.4Ba2Cu3Ox thin film on SrTiO3, measured during a θ -2θ scan. The inset shows the same data, with a logarithmic intensity scale.

APPENDICES

170

From these figures (D.2 and D.3) it is clear that mainly (00l) peaks are visible,

indicating a strong c-axis orientation of the material. The absence of any other

significant peaks proves the single phase character of the films. The small

peaks in between the (00l) peaks (three orders of magnitude smaller than the

(00l) peaks) were shown to originate either from the substrate or the sample

holder (aluminium & plasticine), since they are observed also in measuring the

XRD pattern of a virgin substrate.

The distribution of the crystallographic directions around the c-axis is

characterised by a rocking-scan in which the detector is kept fixed at an

intensity peak while the sample is moved around the original angle θ. The

rocking-curves for the same YBa2Cu3Ox and Y0.6Pr0.4Ba2Cu3Ox thin films are

shown in figures D.4 and D.5, respectively. The crystallographic orientations in

the material have a narrow distribution around the c-axis with a full width at

half maximum (FWHM) of 0.22o for the YBa2Cu3Ox and 0.24o for the

Y0.6Pr0.4Ba2Cu3Ox thin films. This indicates that our YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox thin films are epitaxial, apart from possible ab-twinning

effects due to the small a-b difference.

18.5 19.0 19.5 20.00

5

10

15

20(005)-peak rocking curve2θ = 38.56o

FWHM = 0.22o

θ (degree)

Inte

nsity

(a.

u.)

YBa2Cu3O7 on SrTiO3

Figure D.4: Rocking curve around the (005) line for the YBa2Cu3Ox thin film of figure D.2.

APPENDICES

171

18.5 19.0 19.5 20.00

2

4

6

8

10(005)-peak rocking curve2θ = 38.61o

FWHM = 0.24o

Y0.6Pr0.4Ba2Cu3O7 on SrTiO3

θ (degree)

Inte

nsity

(a.

u.)

Figure D.5: Rocking curve around the (005) line for the Y0.6Pr0.4Ba2Cu3Ox thin film of figure D.3.

Also the La1.9Sr0.1CuO4 ultra thin films, in which epitaxial strain was induced by

depositing them on SrLaAlO4 (compressive strain) or SrTiO4 (tensile strain),

were studied by XRD [Locquet94, 96, 96b, 98 & 98b]. The θ -2θ scans for

these films only show (00l) peaks -indicating their good epitaxy and c-axis

orientation- and finite size peaks -suggesting a film roughness of ± 1 unit cell.

Structural refinements allow to estimate the lattice parameters and to show that

the tensile and compressive strain basically keep the volume of the unit cell

constant.

D.2 Superconducting properties

The quality of the as-deposited, fully oxygenated, YBa2Cu3Ox and

Y0.6Pr0.4Ba2Cu3Ox thin films was checked by measuring the superconducting

transition by AC susceptibility measurements. This was done with a 77.7 Hz

AC field of amplitude 3.6 G generated in an excitation coil coupled to a

compensation coil separated from the detection coil by the (unpatterned) thin

film. Any diamagnetic response to this small magnetic field changes the

coupling between these coils. The difference between the (now unbalanced)

compensation and detection coils is proportional to the AC susceptibility. This

APPENDICES

172

method involves a much stricter quality check than resistive R(T) measurements

since it involves the whole film whereas R(T) measurements only probe the first

superconducting path (percolation). For our optimally oxygenated

YBa2Cu3O6.95 films a χAC transition width of less than 1.5 K is typical (see

figure D.6).

90 91 92 93 94 95

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

T (K)

χ ac / |

χ ac(

90 K

)|

YBa2Cu3Ox thin film3.6 G @ 77.7 Hz

χ′′

χ′

Figure D.6: The AC susceptibility as a function of temperature for a YBa2Cu3O6.95 thin film. The measurement was carried out with an excitation of 3.6 G at a frequency of 77.7 Hz; χac' and χac'' denote the in-phase and out-of-phase signal.

After patterning and depositing gold contacts, the superconducting transition

was checked by measuring the resistive R(T) transition. For the samples

discussed in this work, these R(T) data and the resulting values for Tc are

summarised in paragraphs 3.1 and 3.2.

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NEDERLANDSTALIGE SAMENVATTING

185

Nederlandstalige samenvatting

De eigenschappen van het elektrisch transport in een set van YBa2Cu3Ox en

(Y0.6Pr0.4)Ba2Cu3Ox dunne filmen met verschillende zuurstofinhoud en

La1.9Sr0.1CuO4 ultra dunne filmen onder epitaxiale spanning werden bestudeerd

in de normaal-geleidende toestand. Hiertoe werden de transport

eigenschappen (weerstand en Hall-effect) opgemeten in nulveld en in gepulste

hoge magneetvelden. De keuze voor een tweevoudige aanpak - enerzijds het

verlagen van de kritische temperatuur Tc door de ladingsdragers-concentratie p

te veranderen en anderzijds het gebruik van hoge magneetvelden - laat toe het

gehele ondergedopeerde gebied van het T-p (temperatuur versus gaten-

concentratie in het CuO2 vlak) fasediagram (figuur 1) te bestrijken. Het wordt

dan mogelijk om de transporteigenschappen van deze materialen in de normale

toestand (d.w.z. niet supergeleidend) te bestuderen beneden de kritische temperatuur Tc.

Figuur 1: De eigenschappen van de hoge-Tc koperoxides veranderen sterk met veranderende temperatuur (vertikale as) en dopering van het CuO2 vlak [Batlogg2000].


186

In een eerste stap (geïnspireerd door vroeger werk [Wuyts94 & 96]) werd

aangetoond dat de nulveld normaal-geleidende resistiviteit ρρ(T) boven Tc

schaalt op een universele curve voor verschillende zuurstofinhouden x, zowel

voor YBa2Cu3Ox als voor (Y0.6Pr0.4)Ba2Cu3Ox. Deze schaling werd uitgevoerd

met drie schalingsparameters: een energieschaal ∆, de resistiviteit ρ∆ bij T = ∆

en de residuele resistiviteit ρo. Het expliciet inbrengen van de residuele

resistiviteit ρo in deze schaling zorgde voor een betere schaling dan de vroegere

analyse [Wuyts94 & 96]. De universele curve vertoont drie gebieden met een

kwalitatief verschillend ρ(T) gedrag:

(I) een gebied van lineaire ρ(T) bij hoge temperaturen T > T*,

(II) een super-lineaire ρ(T) bij temperaturen TMI < T < T*

(III) een isolerend ρ(T) regime bij T < TMI.

Het lage-temperatuur regime (III) zit voor een groot gedeelte verstopt achter de

supergeleidende fase die optreedt bij T < Tc. De karakteristieken in de

temperatuursafhankelijkheid van de metallische nulveld resistiviteit ρ(T) van

YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox in gebieden I en II zijn universeel voor alle

gemeten curves. Het enige verschil is de temperatuursschaal ∆∆ bij dewelke de

regimes optreden.

In een tweede stap werd de magnetoresistiviteit van deze YBa2Cu3Ox en

(Y0.6Pr0.4)Ba2Cu3Ox epitaxiale dunne filmen gemeten in gepulste hoge magneetvelden tot 50 T. Dit liet ons toe het ρρ(T) gedrag in de normale toestand bij temperaturen onder de kritische temperatuur Tc te meten.

Deze hoge veld gegevens in het metallische regime schalen goed met de

universele ρ(T/∆) curve. Alleen de ρ(T) data in het isolerende regime bij lage

temperaturen schalen niet bevredigend wanneer dezelfde parameters gebruikt

worden als in de schaling van de nulveld gegevens.

Het bestaan van een universele ρ(T/∆) curve voor de metallische resistiviteit in

de normale toestand werd geïnterpreteerd als een sterke indicatie dat in deze

materialen één enkel mechanisme de verstrooiing van de ladingsdragers

domineert. Het is dit mechanisme dat de energieschaal ∆ (die afhankelijk is van

de dopering) bepaalt.


187

Deze experimenten toonden aan dat de grondtoestand bij T < Tc van

YBa2Cu3Ox (voor x ≤ 6.8), (Y0.6Pr0.4)Ba2Cu3Ox (voor alle niveaus van

zuurstofinhoud) en epitaxiaal uitgerekt La1.9Sr0.1CuO4 eerder isolerend is. De

divergentie van de resistiviteit bij lage temperaturen volgt een ln(1/T) wet in hoge magneetvelden en de implicaties voor de theoretische modellen werden

besproken.

De vergelijking van de transportgegevens voor epitaxiaal gespannen

La1.9Sr0.1CuO4 ultra-dunne filmen met gegevens op monsters zonder deze

spanning en met onze meetgegevens op YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox

monsters wijst er op dat epitaxiale spanning waarschijnlijk niet alleen de

dopering van de CuO2 vlakken maar ook de verstrooiing aan wanorde

beïnvloedt.

Ten einde een verdere interpretatie van de resistiviteitsgegevens in de normale

toestand mogelijk te maken, werden Hall-effect metingen onder de kritische

temperatuur Tc uitgevoerd op dezelfde YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox dunne

filmen. Deze metingen werden in zeer hoge gepulste magneetvelden uitgevoerd

om zo de normale-toestand (d.w.z. niet-supergeleidend) te bestuderen (H > Hc2).

De Hall metingen geven een signaal dat drie grootte-ordes kleiner is dan in

gewone magnetoresistiviteitsexperimenten. De succesvolle uitvoering van

dergelijke experimenten was slechts mogelijk door een zorgvuldige

implementatie van vibratie-isolatie, afscherming voor elektromagnetische

interferentie en het realiseren van een lage contactweerstand met het

meetmonster.

Uit de metingen van de hoge-veld Hall-resistiviteit ρyx(H) werd de

veldafhankelijkheid van de Hall-coëfficiënt RH(H) bij verschillende

temperaturen berekend. Door ons te concentreren op het gedeelte in de normale

toestand (H > Hc2), werd de temperatuursafhankelijkheid van de Hall coëfficiënt

RH(T) bij een vast (hoog) magneetveld geconstrueerd. Bij deze hoge velden

wordt de Hall-coëffciënt RH(T) in de normale toestand gemeten, zelfs bij

T < Tc. Deze RH(T) vertoont slechts een zeer zwakke temperatuursafhankelijkheid bij lage temperaturen. Door de RH(T) curves te

combineren met de hoge-veld ρab(T) curves voor de zelfde samples wordt


188

duidelijk dat bij deze hoge velden de normale toestand volledig wordt bereikt,

zelfs bij temperaturen T << Tc.

De temperatuurs-onafhankelijkheid van de lage-temperatuur Hall-coëfficiënt RH staat in scherp contrast met de sterk divergerende ρρab(T) in deze materialen. Dit is een belangrijke test voor de bestaande modellen die

reeds een lage-temperatuur divergentie van ρab(T) voorspellen. Van deze

modellen verklaart het zwakke lokalisatie-model zowel de constante RH bij lage

temperaturen als de ln(1/T) divergentie van de hoge-veld ρab(T) gegevens.

De ladingsdragersdichtheid nH (en aldus ook p, de fractie van gaten per Cu-

atoom in de CuO2 vlakken), berekend uit de Hall-data, liet toe een generisch T(p) fasediagram op te stellen voor de YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox materialen.

Het universele ρρ(T) gedrag in de ondergedopeerde YBa2Cu3Ox en

Y0.4Pr0.4Ba2Cu3Ox dunne filmen is een sterk argument voor het idee dat één

enkel verstrooiingsmechanisme het hele ondergedopeerde regime in het Y123

systeem domineert. Met het complexe magnetische fasediagram in het

achterhoofd (korte-afstands AF correlaties en een pseudo energiekloof bij

temperaturen ver boven de supergeleidende kritische temperatuur Tc) en

geconfronteerd met de sterke aanwijzingen voor de vorming van ladingsstrepen

in de CuO2 vlakken is het redelijk om de oorsprong van dit dominant verstrooiingsmechanisme toe te wijzen aan de microscopische ordening van magnetisme en ladingen.

Daarom gebruikten we onze hoge-veld transport gegevens (zowel de resistiviteit

als het Hall-effect) om het effect op de transport-eigenschappen in de normale

toestand van de korte-afstands antiferromagnetische fluctuaties en de mogelijke

vorming van ladingsstrepen in het CuO2 vlak na te gaan. In dit ladingsstreep

scenario [Emery97b & 99b], worden dynamische, metallische [Ichikawa99,

Noda99, Tajima99] strepen geacht de transporteigenschappen te bepalen.

Vermits mobiele ladingsdragers in dit geval uit de Mott isolator naar de strepen

gestoten worden, vormen deze strepen de paden met de laagste weerstand.

Daardoor zijn de transporteigenschappen zeer gevoelig voor de vorming van

zowel statische als dynamische strepen. Om deze ideeën te verifiëren werd een


189

bestaand model [Moshchalkov93, 98b & 99] gebruikt. Dit model beschrijft het

elektrisch transport zowel in het tweedimensionale (2D) Heisenberg regime (boven T*) als in het ééndimensionale (1D) regime (onder T*) met ladingsstrepen, waar zich de pseudo energiekloof ontwikkelt.

Deze benadering [Moshchalkov93, 98b & 99] is gebaseerd op drie aannames:

1. het dominante verstrooiingsmechanisme in hoge-Tc materialen in

het hele temperatuursgebied is van magnetische oorsprong;

2. de specifieke temperatuursafhankelijkheid van de resistiviteit ρ(T)

kan worden beschreven door de inverse kwantumgeleidbaarheid

σ -1 met de inelastische lengte Lφ bepaald door de magnetische

correlatielengte ξm (via een sterke interactie van gaten met Cu2+

spins);

3. de gepaste 1D of 2D uitdrukkingen moeten worden gebruikt om de

kwantum geleidbaarheid uit te rekenen met Lφ ~ ξm.

Bij hoge temperaturen T* < T < To, in het 2D Heisenberg regime, geeft de

combinatie van de uitdrukkingen voor de 2D spin correlatielengte met de

gepaste uitdrukking voor de kwantumweerstand een lineaire temperatuursafhankelijkheid van de resistiviteit. Dit resultaat is in perfecte

overeenstemming met de lineaire ρ(T) bij hoge temperaturen voor alle gemeten

ondergedopeerde YBa2Cu3Ox en Y0.4Pr0.4Ba2Cu3Ox dunne filmen.

Bij matige temperaturen Tc < T < T*, in het 1D gestreept regime, geeft het

invullen van de 1D magnetische correlatielengte in de uitdrukking voor 1D

kwantumgeleidbaarheid een gebogen, super-lineaire ρρ(T) die de resistiviteit

van Sr2.5Ca11.5Cu24O41, een spin-ladder met een even aantal benen, perfect

beschrijft. Dit materiaal bevat door zijn specifieke kristalstructuur zeker een 1D

spin-ladder en daarom zou de resistiviteit volgens de richting van de ladder

zeker de uitdrukking voor 1D kwantumtransport moeten volgen.

In een volgende stap werd een overtuigende schaling aangetoond tussen de

resistiviteit van deze 1D spin-ladder en een typisch ondergedopeerd hoge-Tc

materiaal, YBa2Cu4O8. Dit wijst erop dat de temperatuursafhankelijkheden van de resistiviteit van ondergedopeerde koperoxides (in het regime met


190

een pseudo-energiekloof bij T < T*) enerzijds en spin ladders met een even aantal benen met een spin-energiekloof ∆∆ anderzijds, bepaald worden door hetzelfde onderliggende 1D (magnetisch) mechanisme. De magnetische

oorsprong van de verstrooiing van de ladingsdragers wordt verder bevestigd

door het feit dat de schalingstemperatuur, gebruikt in de schaling van ρ(T), even

goed werkt voor resistiviteits- als voor Knight-shift gegevens KS. Deze KS

gegevens kunnen goed gefit worden met de uitdrukkingen die afgeleid werden

in de 1D spin-ladder modellen.

De ρ(T) gegevens van een La1.9Sr0.1CuO4 dunne film onder epitaxiale

trekspanning en YBa2Cu3Ox en (Y0.6Pr0.4)Ba2Cu3Ox dunne filmen met

verschillende zuurstof inhoud (geschaald op een universele curve) zijn alle

perfect beschreven door de uitdrukking voor 1D kwantum transport. De

waardes van de spin-energiekloof, die gevonden werden uit deze fitting, zijn in

overeenstemming met onafhankelijke afschattingen op basis van resistieve

metingen op andere YBa2Cu3Ox dunne filmen, "getwinde" en ge-"detwinde"

éénkristallen. Bovendien komen ze overeen met afschattingen van de pseudo-

energiekloof uit 17O en 63Cu Knight-shift metingen op opgelijnde poeders. In

het 1D kwantumtransport model, waar de inelastische lengte bepaald wordt

door de magnetische correlatielengte, lijkt de overeenstemming van onze data

met de energiekloof bepaald uit NMR experimenten eerder evident. Dit bewijst

dat onze aanpak (het beschrijven van het transport in ondergedopeerde koperoxides bij T < T* in een 1D gestreepte manier) niet alleen kwalitatief overeenkomt maar ook waardes geeft voor de pseudo spin-energiekloof ∆∆ die goed overeenkomen met onafhankelijke afschattingen.

In het laatste hoofdstuk van dit werk werden onze data voor de resistiviteit en

het Hall-effect in de normale toestand geconfronteerd met de mogelijkheid van

de hechting van de ladingsstrepen ("stripe pinning") door wanorde. Het

metallische gedrag van de resistiviteit bij hoge temperaturen gaat bij lage

temperaturen T < TMI over in een meer isolerend, divergerend, ρ(T) gedrag dat

overeenkomt met een ln(1/T) wet. De Hall-coëfficiënt daarentegen is

temperatuursonafhankelijk bij lage temperaturen. Binnen het ladingsstreep

beeld kan men een sterke invloed verwachten op de transport eigenschappen

wanneer de 1D ladingsstrepen gefragmenteerd zijn of vastgehecht worden.


191

Zulke fragmentatie of hechting van de ladingsstrepen kan worden veroorzaakt

door elk verstrooiingsproces dat een inelastische lengte oplegt die kleiner is dan

Lφ ~ ξm1D.

Bij lage temperaturen, kan het ladingsstreep beeld alleen in overeenstemming worden gebracht met onze transportgegevens in de normale toestand door aan te nemen dat er streep-fragmentatie optreedt, zodat inter-streep hopping effecten belangrijk worden. Deze processen veroorzaken een sterke invloed

van de tussenliggende Mott isolator fase op het ladingstransport wat resulteert

in een (2D) isolerende resistiviteit en een klein Hall signaal. Door de

inelastische lengte van het proces dat werkzaam is in de tussenliggende AF in te

vullen in de uitdrukking voor 2D kwantumtransport zou dan het geobserveerde

ln(1/T) gedrag bekomen moeten worden. Ook elektron interferentie effecten in

het kader van 2D zwakke lokalisatie geven een ln(1/T) divergentie voor de

resistiviteit en een constante Hall-coëfficiënt RH(T) bij lage temperaturen, in

overeenstemming met onze resultaten.

De constructie van een zogenaamd experimenteel generisch T(p) fase diagram,

dat de transport eigenschappen in de normale toestand en de supergeleidende

eigenschappen van zowel de YBa2Cu3Ox en de (Y0.6Pr0.4)Ba2Cu3Ox materialen

beschrijft, was mogelijk door onze hoge-veld transportgegevens te combineren

met de afschattingen voor de ladingsdragersdichtheid uit het Hall effect. Dit

experimentele fasediagram (zie figuur 2 hieronder, te vergelijken met het

schematische fasediagram op figuur 1 hierboven) werd onderzocht in het kader

van het 1D/2D kwantumtransport model [Moshchalkov93, 98b & 99]. De

energie schaal ∆(p) (voor 1D kwantum transport) is goed beschreven door de

ladingsdragersdichtheid in de CuO2 vlak. Vermits deze energieschaal binnen

het kwantumtransport model precies de pseudo spin-energiekloof is, is ook deze

pseudo spin-energiekloof ∆ goed beschreven door de ladingsdragersdichtheid

in het CuO2 vlak.

Het 1D gestreept regime is gedefinieerd door vier grenzen in het T(p) diagram. Bij lage dopering wordt de bulk antiferromagnetische orde

herwonnen en verdwijnen de strepen. Bij sterke dopering zal de afstand tussen

de strepen verminderen en zal de tussenliggende Mott isolator fase verdwijnen.

Bij hoge temperaturen, kunnen de ladingsstrepen meanderen zodat het 1D


192

regime vernietigd wordt en een 2D regime met antiferromagnetische fluctuaties

optreedt. Bij lage temperaturen T < TMI, kunnen de ladingsstrepen

vastgehecht ("pinned") of gefragmenteerd worden en inter-streep hopping

effecten veroorzaken dan een 2D isolerend regime. In het T(p) diagram wordt

dit isolerend gebied aangeduid met TMI, onder dewelke de resistiviteit toeneemt

met afnemende temperatuur. De TMI(p) lijn voor het Y0.6Pr0.4Ba2Cu3Ox

materiaal ligt beduidend hoger dan de TMI(p) lijn voor de YBa2Cu3Ox samples.

Dit kan binnen het ladingsstrepen-model verklaard worden door de bijkomende

wanorde veroorzaakt door de Y/Pr substitutie, waardoor de hechting van de

strepen en fragmentatie effecten versterkt worden. Voor het YBa2Cu3Ox

systeem ligt de TMI(p) lijn onder de Tc(p) lijn voor bijna alle niveaus van

gatendopering en aldus wordt de isolerende toestand bij nulveld effectief

gemaskeerd door het optreden van supergeleiding. Voor het Y0.6Pr0.4Ba2Cu3Ox

materiaal ligt TMI(p) lijn boven de Tc(p) lijn voor alle niveaus van

zuurstofinhoud en de isolerende tendens kan daarom reeds worden gezien in de

nulveld ρ(T) metingen boven Tc. Bij temperaturen T < Tc, kan het optreden

van macroscopische coherentie tussen de zogenaamde voorgevormde paren

[Emery97b & 99b] resulteren in bulk supergeleiding.

Alhoewel ze een kwalitatief gelijkaardig gedrag vertonen, vallen de kritische

temperatuur Tc,mid(p) en de grens TMI tussen de metallische en meer isolerende

ρ(T) regimes niet samen voor de YBa2Cu3Ox en Y0.6Pr0.4Ba2Cu3Ox materialen.

Het feit dat voor dezelfde gatenconcentratie p, de kritische temperatuur Tc lager

is in het Y0.6Pr0.4Ba2Cu3Ox systeem in vergelijking met YBa2Cu3Ox, is een

indicatie dat naast wanorde ook magnetische paar-breking in het

(Y/Pr)Ba2Cu3Ox materiaal een belangrijke rol zou kunnen spelen in de reductie

van de supergeleidende kritische temperatuur Tc.

Door magnetoresistiviteitsmetingen en Hall-effect metingen uit te voeren in

zeer hoge magnetische velden op een selectie van hoge-Tc supergeleidende

dunne filmen met verscheidene niveaus van gatendopering, konden we het

schematisch T-p fase diagram uit figuur 1 complementeren met drie

experimentele faselijnen: de T*(p) lijn die een 2D naar 1D overgang beschrijft,

de supergeleidende kritische temperatuur Tc(p) en TMI(p) voor het begin van een

isolerend gedrag. Het generisch fasediagram voor de normale toestand werd


193

besproken in het kader van een 2D metallisch Heisenberg regime, een

metallisch gebied met 1D ladingsstrepen waar de pseudo energiekloof zich

ontwikkelt en een lage-temperatuur isolerend regime.

0 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)


metallic1D striped

T*

200

300

400

0

100 TcTMI

0

100

I

II

IIISC

SC

≈ ≈TMI

Tc

II

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

III

0 0.10 0.15 0.20 0.250.050 0.10 0.15 0.20 0.250.05

T (

K)

p (#/in-plane Cu)


metallic1D striped

T*

200

300

400

0

100

200

300

400

0

100 TcTMI

0

100

0

100

II

IIII

IIIIIISC

SC

≈ ≈TMI

Tc

IIII

AF

YBa2Cu3Ox

Y0.6Pr0.4Ba2Cu3Ox

IIIIII

Figuur 2: Generisch T(p) fasediagram voor de YBa2Cu3Ox (ruiten, volle lijn) en de Y0.6Pr0.4Ba2Cu3Ox (cirkels, puntlijn) dunne filmen. De 2D/1D overgangstemperatuur T* (volle symbolen), de supergeleidende kritische temperatuur Tc (open symbolen) en de grens TMI tussen de metallische en de meer isolerende regimes voor ρ(T) zijn aangeduid op de figuur. Alle gegevens zijn uitgezet ten opzichte van de fractie van gaten per Cu-atoom in het CuO2 vlak. De gegevens voor het Y0.6Pr0.4Ba2Cu3Ox systeem zijn 100 K verschoven.

PUBLICATIELIJST

195

Publicatielijst

Lieven Trappeniers (1994-1999)

Publicaties in internationale tijdschriften Study of High-Temperature Superconducting Thin Films in Magnetic Fields up to 50 Tesla, J. Vanacken, L. Trappeniers, A.S. Lagutin, P. Wagner, U. Frey, H. Adrian, F. Herlach, V.V. Moshchalkov en Y. Bruynseraede, Institute of Physics Conference Series 148 (1995) 971-974.

Strong Pinning in Melt-textured YBa2Cu3O7 with non superconducting Y2BaCuO5 inclusions, K. Rosseel, D. Dierickx, J. Lapin, V.V. Metlushko, W. Boon, L. Trappeniers, J. Vanacken, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede and O. Vanderbiest, Institute of Physics Conference Series 148 (1995) 279-282.

Hopping conductivity of magnetic polarons in epitaxial Pr0.5Sr0.5MnO3 films, P. Wagner, V. Metlushko, J. Vanacken, L. Trappeniers, V.V. Moshchalkov, Y. Bruynseraede, Czechoslovak Journal of Physics 46 (1996), 2004-2006.

Magnetic Transitions and Magneto-Transport in Epitaxial Pr0.5Sr0.5MnO3 Thin Films, P. Wagner, V. Metlushko, M. Van Bael, R.J.M. Vullers, L. Trappeniers, A. Vantomme, J. Vanacken, G. Kido, V.V. Moshchalkov and Y. Bruynseraede, Journal de Physique IV 6 (1996) C3-309-343.

Observation of Magneto-Thermal Instabilities in YxTm1-xBa2Cu3O7 Single Crystal in Pulsed Magnetic Field Magnetization Measurements, L. Trappeniers, J. Vanacken, I.N. Goncharov, K. Rosseel, A.S. Lagutin, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Journal of Low Temperature Physics 105 (1996) 1029-1034.

Experimental Techniques for Pulsed Magnetic Fields, F. Herlach, C.C. Agosta, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, A. Lagutin, L. Li, L. Trappeniers, J. Vanacken, L. Van Bockstal and A. Van Esch, Physica B 216 (1996) 161-165.

Magnets, semiconductors and organic conductors at the Leuven pulsed field laboratory, F. Herlach, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, L. Li, P. Reinders, K. Rosseel, L. Trappeniers, J. Vanacken, L. Van Bockstal, A. Van Esch, A. House, J. Singleton, M. Kartsovnik, J. De Boeck, G. Borghs, "High Magnetic Fields on the Physics of Semiconductors II", G. Landwehr, W. Ossau (eds.) II, 905-914 (1997) ISBN 981-02-3076-1

PUBLICATIELIJST

196

Magneto-Transport in Epitaxial Thin Films of the Magnetic Perovskite Pr0.5Sr0.5MnO3, P.H. Wagner, V. Metlushko, L. Trappeniers, A. Vantomme, J. Vanacken, G. Kido, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev.B 55 (1997) 3699.

Anomalous Hall effect in thin films of Pr0.5Sr0.5MnO3, P. Wagner, D. Mazilu, L. Trappeniers, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev. B 55 (1997) 14721-14724.

Influence of Processing Parameters on Critical Currents and Irreversibility Fields of Fast Melt Processed YBa2Cu3O7 with Y2BaCuO5 inclusions, K. Rosseel, D. Dierickx, J. Vanacken, L. Trappeniers, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 1587-1590.

Construction of the Current-Voltage Characteristic in a 12 Decade Voltage Window using Magnetisation Measurements, J. Vanacken, K. Rosseel, L. Trappeniers, M. Van Bael, A.S. Lagutin, D. Dierickx, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 985-988.

Influence of Columnar Defects on Critical Currents and Irreversibility Fields in (YxTm1-x)Ba2Cu3O7 Single Crystals, L. Trappeniers, J. Vanacken, K. Rosseel, A.Yu. Didyk, I.N. Goncharov, L.I. Leonyuk, W. Boon, F. Herlach, V.V. Moshchalkov, and Y. Bruynseraede, Inst. Phys. Conf. Ser. 158 (1997) 1591-1594.

Irreversible Magnetic Properties of Ceramic HgBa2Ca2Cu3O8 at Very High Pulsed Magnetic Fields, L. Trappeniers, J. Vanacken, K. Rosseel, S. Reich, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Journal of Superconductivity 11 (1998) 35.

Scaling Behavior of the Normal State Properties of the Underdoped High Tc Cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, “Superconducting and related oxides: Physics and nano-engineering III”, Edited by D. Pavuna and I. Bozovic, SPIE proc. 3481 (SPIE, Bellingham, 1998) 10-16.

Spin Dependent Hopping and Colossal Negative Magnetoresistance in Epitaxial Nd0.52Sr0.48MnO3 Films in Fields up to 50 T, P. Wagner, I. Gordon, L. Trappeniers, J. Vanacken, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Phys. Rev. Lett. 81 (1998) 3980.

Magnetic Phase Diagram of the Spin-Peierls Compound CuGeO3 doped with Al and Sn, S.V. Demishev, L. Weckhuysen, J. Vanacken, L. Trappeniers, F. Herlach, Y. Bruynseraede, V.V. Moshchalkov, A.A. Pronin, N.E. Sluchanko, N.A. Samarin, J. Meersschaut en L.I. Leonyuk, Phys. Rev. B 58 (1998) 6321-6329.

Critical Currents, Pinning Forces and Irreversibility Fields in (Tm1-xYx)Ba2Cu3O7 Single Crystals with Columnar Defects in Fields up to 50 T, L. Trappeniers, J. Vanacken, L. Weckhuysen, K. Rosseel, A.Yu. Didyk, I.N. Goncharov, L.I. Leonyuk, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Physica C 313 (1999) 1-10.

PUBLICATIELIJST

197

1D Quantum Transport in the Even-Chain Spin-Ladder Compound Sr2.5Ca11.5Cu24O21 and YBa2Cu4O8 , V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Europhys. Lett. 46 (1999) 75-81.

Influence of Columnar Defects on Critical Currents, Pinning Forces and Irreversibility Fields in (Tm1-xYx)Ba2Cu3O7 Single Crystals up to 50 T, L. Trappeniers, J. Vanacken, L. Weckhuysen, K. Rosseel, W. Boon, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede, A. Didyk, I. Goncharov and L. Leonyuk, “Physics and Material Science of Vortex States, Flux Pinning and Dynamics” NATO ASI series 345-355, Ed. R. Kossowsky.

Simulation and Calibration of an Open Inductive Sensor for Pulsed Field Magnetization Measurements, L. Weckhuysen, J. Vanacken, L. Trappeniers, M.J. Van Bael, W. Boon, K. Rosseel, F. Herlach, V.V. Moshchalkov, Y. Bruynseraede, Rev. Sci. Instr. 70 (1999) 2708-2710.

Normal State Resistivity of Underdoped YBa2Cu3Ox thin films and La2-xSrxCuO4 ultra-thin films under epitaxial strain, L. Trappeniers, J. Vanacken, P. Wagner, G. Teniers, S. Curras, J. Perret, P. Martinoli, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede, J. Low Temp. Phys. 117 (1999) 681-685.

Doped CuO2 planes in High-Tc Cuprates: 2D or not 2D ?, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, J. Low Temp. Phys. 117 (1999) 1283-1287.

Ter perse 1D Quantum Transport in the Even-Chain Spin-Ladder Compound Sr2.5Ca11.5Cu24O21 and YBa2Cu4O8 , V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Proc. of the second int. conf. on “Stripes and High Tc Superconductors”, Rome, June 2-6, 1998 to be published in Journal of Superconductivity

Pseudo-Gap and Crossover from the 2D Heisenberg to the even - leg Spin Ladder regime in underdoped cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, Proc. of the NATO ARW on “Symmetry and Pairing in Superconductors”, Yalta , April 28-May 2, 1998.

Magnetic Field Induced Localisation and Current driven Delocalisation in Y0.5Pr0.5Ba2Cu3Ox Films, Z. Hao, B.R. Zhao, B.Y. Zhu, L. Trappeniers, J. Vanacken and V.V. Moshchalkov, submitted to Phys. Rev. Lett.

Stripe formation and disorder induced stripe fragmentation, V.V. Moshchalkov, L. Trappeniers, G. Teniers, J. Vanacken, P. Wagner and Y. Bruynseraede, submitted to Physica C.

Stripes and dimensional crossovers in high-Tc cuprates, V.V. Moshchalkov, L. Trappeniers and J. Vanacken, submitted to Physica C.

Normal State Resistivity of Underdoped YBa2Cu3O7-d Films and La2-XSrXCuO4 Ultra -Thin Films in Fields up to 60 T, L. Trappeniers, J. Vanacken, P. Wagner,

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198

S. Curras, G. Teniers, J. Perret, J.-P. Locquet, V.V. Moshchalkov and Y. Bruynseraede, submitted to Physica C.

Zonder leescomité Critical Parameters of High Tc Cuprates at 60 T, V.V. Moshchalkov, J. Vanacken, L. Trappeniers, K. Rosseel, D. Dierickx, P. Wagner, W. Boon, I.N. Goncharov, A. Yu Didyk, A.S. Lagutin, L.I. Leonyuk, N. Harrison, F. Herlach and Y. Bruynseraede, Physicalia 19 (1997) 205-218.

The K.U.Leuven pulsed magnet facility, L. Van Bockstal, R. Bogaerts, W. Boon, A. De Keyser, M. Hayne, L. Liang, L. Trappeniers, P. Reinders, K. Rosseel, J. Vanacken, M. Van Cleemput and A. Van Esch, Physicalia 19 (1997) 165-175

Publicaties op internationale conferenties Influence of columnar pinning defects on the critical currents in (YxTm1-x)Ba2Cu3O7 single crystals, L. Trappeniers, J. Vanacken, I.N. Goncharov, W. Boon, F. Herlach, V.V. Moshchalkov and Y. Bruynseraede, Paper presented at NATO Advanced Study Institute "Materials Aspects of High Tc superconductivity: 10 years after the discovery", Delphi, Griekenland (1996).

Experimental techniques for pulsed magnetic fields in the 60 T/20 ms range, L. Van Bockstal, R. Bogaerts, W. Boon, I. Deckers, A. De Keyser, N. Harrison, A. Lagutin, L. Li, L. Trappeniers, J. Vanacken, A. Van Esch, F. Herlach, Proceedings of Physical Phenomena at High Magnetic Fields - II, ed. Z. Fisk, L. Gor'kov, D. Meltzer and R. Schrieffer,(Conference Paper) World Scientific, Singapore (1996) 755-760. 505

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