Quantum Critical Behavior in the Superfluid Density of High-Temperature Superconducting Thin Films

QUANTUM CRITICAL BEHAVIOR IN THE SUPERFLUID DENSITY OF HIGH-TEMPERATURE SUPERCONDUCTING THIN FILMS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Iulian N. Hetel, Maitrise de Physique et Applications, M.S. Physics

*****

The Ohio State University

2008

Dissertation Committee: Approved by

Professor Thomas R. Lemberger, Adviser Professor Klaus Honscheid Adviser Professor R. Sooryakumar Physics Graduate Program Professor Nandini Trivedi

ABSTRACT

A central question in the physics of high-temperature superconductors is how superconductivity is lost at the extreme ends of the superconducting phase diagram, underdoping and overdoping. When mobile holes are removed from optimally doped cuprates, the transition temperature TC and superﬂuid density nS(0) decrease in a surprisingly correlated fashion. I succeeded in producing and measuring homogeneous underdoped high-temperature superconducting ﬁlms by partially substituting

+2 +3 Ca for Y in Y Ba2Cu3O7−δ ﬁlms with reduced oxygen concentrations in the CuO chains.

I test the idea that the physics of underdoped cuprates is dominated by phase ﬂuc- tuations by measuring the temperature dependence of superﬂuid density nS(T ) and by changing the dimensionality of the system from 3D thick samples to 2D ultrathin

films. Thick Y1−xCaxBa2Cu3O7−δ films are in agreement with previous measurements of pure Y Ba2Cu3O7−δ samples and do not show any 2D or 3D-XY critical regimes in the temperature dependence of superfluid density. Moreover, the transition temperature has a square-root dependence on absolute superfluid density at zero temperature, rather than showing the predicted linear dependence in the case of strong thermal phase fluctuations. When superfluid density is measured in underdoped 2D Y1−xCaxBa2Cu3O7−δ films as thin as only 2 unit cells for all doping levels, nS(T ) has a dramatic downturn consistent with a 2D vortex-antivortex pair unbinding transition and, at severe underdoping, TC is linearly proportional to nS(0). This

ii dimensionality-dependent scaling relation is the result of quantum phase ﬂuctuations that suppress superconductivity near a Quantum Critical Point at zero temperature.

Further measurements in an additional family of high-temperature superconductors,

La2−xSrxCuO4, are consistent with my results in underdoped Y1−xCaxBa2Cu3O7−δ.

La2−xSrxCuO4 also provided the opportunity to study the other extreme end of the superconducting phase diagram. In the overdoped region, as carrier density increases, supefluid density and the transition temperature are both suppressed. While it still remains uncertain what produces this suppression, a plausible interpretation is that only a small fraction of the hole carriers contribute to the superfluid density and the pair breaking effects are so important that they destroy superconductivity.

iii To my wife Laura

iv ACKNOWLEDGMENTS

Along the years I had the privilege to work with a number of people who taught me a lot about what it means to be a researcher. I want to thank my PhD adviser Tom

Lemberger for welcoming me in his laboratory and providing me with an interesting and challenging research project. I have learned a lot from our interaction and I appreciated his guidance and patience along the diﬃcult parts of this study. Nandini

Trivedi, R. Sooryakumar, and Klaus Honscheid were kind enough to serve as reviewers of my project and gave insightful comments that I have incorporated in the ﬁnal version of this dissertation. My discussions with Mohit Randeria and Zlatko Tesanovic helped me to get a better understanding of the direction and implications of my experimental results. Also, Rita Roknlin was very generous in oﬀering technical support whenever I needed it.

Before my coming to Ohio State University, several people introduced me to the fascinating ﬁeld of superconductivity: Mircea Crisan and Utu Deac from Babes-Bolyai

University, Cluj Napoca, Romania, Manuel Nunez-Regueiro from CRTBT-CNRS,

Grenoble, France, and my advisers during my master’s degree in Sherbrooke, Patrick

Fournier and Mario Poirier. I am grateful to all of them for the knowledge and passion for physics that they shared with me.

v I was fortunate to share the Lemberger lab with two great colleagues, Mun Seog

Kim and Yuri Zuev, who guided and welcomed me during my ﬁrst months in the lab and supported me throughout the years. I am especially thankful to my (high- energy) colleagues from the high-energy group - Don Burdette, James Morris, Joe

Regensburger and Evan Frodermann - who made lunchtime fun and included me in their clandestine Friday pizza meetings. My parents Felicia and Nicolae were always supportive and encouraged me to pursue my education even when it meant being far away from home. My younger brother Laurentiu shares my passion for science and managed to get to the PhD ﬁnish line before me: I couldn’t be prouder of him.

My wife Laura was a constant support and inspiration. Not only did she proofread my papers, but she also contributed to my experiments by changing the voltage in the drive coil (.002V). With one of us working on a PhD in physics and the other writing a PhD in the humanities at the same time, we never had a dull moment and we learned a lot from and about each other.

This research project beneﬁted from the support of several institutions. The Ohio

State University Presidential Fellowship allowed to focus solely on my research during my last year of studies. The OSU Alumni Research Grant and the Sigma Xi Grant for Aid-in-Research provided funding for purchasing materials that were crucial for my project. Also, with the support of I2Cam and the Council of Graduate Students,

I was able to share and test my research at scientiﬁc conferences.

vi VITA

July 18, 1976 ...... Born - Sibiu, Romania

1995-1999 ...... Engineering Physics, Babes-Bolyai University, Cluj-Napoca, Romania 1999-2000 ...... Agence Universitaire de la Franco- phonie Fellow, Universite Grenoble I, France 2001-2002 ...... Graduate Research Associate, Univer- site de Sherbrooke, Quebec, Canada 2002-2006 ...... Graduate Research Associate, The Ohio State University, USA 2007-2008 ...... Presidential Fellow, The Ohio State University, USA

PUBLICATIONS

Research Publications

I. Hetel, T. R. Lemberger and M. Randeria. Quantum critical behaviour in the superﬂuid density of strongly underdoped ultrathin cuprate ﬁlms Nature Physics, 3, 700 (2007.

I. Hetel, T. R. Lemberger, A. Tsukada and M. Naito. Doping dependent superﬂuid

density in La2−xSrxCuO4 thin ﬁlms (in preparation).

T. R. Lemberger, I. Hetel, A.J. Hauser, and F. Y. Yang. Superﬂuid density of superconductor-ferromagnet bilayers J. Appl. Phys. 103, 07C701 (2008).

T. R. Lemberger, I. Hetel, J. W. Knepper, and F. Y. Yang. Penetration depth study of very thin superconducting Nb ﬁlms Phys. Rev. B 76, 094515 (2007).

vii Y. L. Zuev, I. Hetel, T. R. Lemberger. Search for Cooper-pair ﬂuctuations in severely underdoped YBCO ﬁlms Phys. Rev. B 74, 012502 (2006).

R. Prozorov, D. D. Lawrie, I. Hetel, P. Fournier, and R. W. Giannetta. Field-

dependent diamagnetic transition in magnetic superconductor Sm1.85Ce0.15CuO4−δ Phys. Rev. Lett. 93, 147001 (2004).

P. Richard, G. Riou, I. Hetel, S. Jandl, M. Poirier and P. Fournier. Role of oxygen

nonstoichiometry and the reduction process on the local structure of Nd2−xCexCuO4±δ Phys. Rev. B 70, 064513 (2004).

P. Fournier, M.-E. Gosselin, S. Savard, I. Hetel, P. Richard and G. Riou. Fourfold oscillations and irreversibility of the magnetoresistance in the nonmetallic regime of electron-doped cuprates Phys. Rev. B 69, 220501(R) (2004).

FIELDS OF STUDY

Major Field: Physics

Studies in: Experimental Condensed Matter Physics Low Temperature Physics Superconductivity

viii TABLE OF CONTENTS

Page

Abstract ...... i

Dedication ...... iii

Vita...... vi

List of Tables ...... xi

List of Figures ...... xii

Chapters:

1. INTRODUCTION ...... 1

2. BASIC CONCEPTS ...... 6

2.1 Superconductivity and superﬂuid density ...... 6

2.2 General properties of high-temperature superconductors ...... 9

2.3 Phase ﬂuctuations in high-temperature superconductors ...... 11

3. EXPERIMENTAL METHODS ...... 15

ix 3.1 Preparation of atomically ﬂat SrT iO3 substrates ...... 15

3.2 Thin ﬁlm deposition ...... 23

3.3 Superﬂuid density measurements ...... 27

4. UNDERDOPING Y Ba2Cu3O7−δ ...... 31

4.1 Chain disorder in underdoped Y Ba2Cu3O7−δ ...... 31

4.2 Y1−xCaxBa2Cu3O7−δ underdoping while adding holes ...... 33

4.2.1 Oxygen reduction in Y0.80Ca0.2Ba2Cu3O7−δ ...... 34

4.2.2 Doping dependence, homogeneity, and superﬂuid density . . 36

5. UTRATHIN Y1−xCaxBa2Cu3O7−δ FILMS ...... 41

5.1 Challenges in the fabrication of ﬁlms with reduced thickness . . . . 41

5.2 Temperature dependence of superﬂuid density ...... 47

5.2.1 KTB transition in two-dimensional samples ...... 49

5.2.2 Frequency dependence of KTB transition ...... 54

5.2.3 TC vs nS(0). Scaling next to a Quantum Critical Point . . . 57

5.2.4 Temperature dependence of 1/λ2: 2D vs. 3D ...... 63

6. SUPERFLUID DENSITY MEASUREMENTS IN La2−xSrxCuO4 FILMS 67

6.1 Evolution of transition temperature and superﬂuid density with doping 69

6.2 2D ﬂuctuations in ultrathin La1.85Sr0.15CuO4 ...... 78

6.3 Frequency and current dependence of the 2D transition in ultrathin

La1.85Sr0.15CuO4 ...... 82

x 7. CONCLUSIONS ...... 84

Appendices:

A. Appendix: Y1−xCaxBa2Cu3O7−δ ﬁlms summary ...... 86

B. Appendix: Atomic Force Microscopy ...... 89

Bibliography ...... 91

xi LIST OF TABLES

Table Page

5.1 The upper limit on the transition temperatures for several 2 unit cell

Y1−xCaxBa2Cu3O7−δ films when thermal phase fluctuations effects are suppressed by quantum fluctuations...... 65

6.1 Properties of La2−xSrxCuO4−δ ﬁlms as a function of doping. x is the nominal Sr doping, TC (ρ = 0) sets the upper limit on the transition temperature, ρab(50K) is the ab-plane resistivity just above the onset −2 of transition, and ρab(50K)×λ (0)/µ0 is related to the scattering rate τ −1...... 77

A.1 Y1−xCaxBa2Cu3O7−δ ﬁlms: deposition parameters and basic properties. 86

A.2 Deposition parameters and transition temperatures for Y1−xCaxBa2Cu3O7−δ ﬁlms 2 unit cell thick, grown between protective P rBa2Cu3O7−δ layers and with an optional 3uc amorphous P rBa2Cu3O7−δ cap layer. . . . 87

A.3 Deposition parameters and transition temperatures for Y1−xCaxBa2Cu3O7−δ ﬁlms 2 unit cell thick, grown between protective P rBa2Cu3O7−δ layers and with an additional 3uc amorphous P rBa2Cu3O7−δ cap layer deposited at 300oC...... 88

xii LIST OF FIGURES

Figure Page

2.1 The crystallographic structure of one-unit cell of Y Ba2Cu3O7−δ, when the oxygen chains are full (δ = 0) or totally unoccupied (δ = 1), and

the structure of two-unit cell La2−xSrxCuO4 [16]...... 10

3.1 Perovskite structure of SrT iO3 with the two possible termination layers, SrO and T iO2 [24]...... 16

3.2 Sketch of the Y BCO/ST O interface [26] ...... 18

3.3 (a) AFM image of a mechanically repolished SrT iO3 substrate (b) after etching in BHF, pH ≈ 5 for 30 seconds (c) 5 minutes (d) 12 minutes. The brightest spots in all four images are 10 nm from the surface. . . 19

3.4 Zoomed image of the repolished and 12-minute BHF, pH ≈ 5, etched

SrT iO3 substrate from ﬁgure 3.3. Even if the surface is not perfect, a cross section perpendicular to the steps, along the black line, allows us to distinguish the one-unit cell steps. The vertical distance of approximately 4A˚ was measured between the two red triangles on adjacent terraces...... 20

3.5 AFM image of a SrT iO3 substrate etched in buﬀered HF, pH ≈ 5, for 12 minutes and annealed in oxygen at 900oC for one hour. The one-unit cell steps between the two red triangles are clearly visible in the section analysis along the black line...... 21

3.6 SrT iO3 substrates after H2O soaking and BHF etching (a) old substrate (b) zoom (c) new substrate (d) old substrate with hole pits. The surface roughness is less than 2nm...... 22

3.7 AFM image of the surface of a 20 unit cell (23nm) Y1−xCaxBa2Cu3O7−δ ﬁlm right after deposition. Islands with a height of one-unit cell(1uc ≈ 11.7A˚) can be observed. The RMS roughness of the ﬁlms is approximately 19A˚...... 26

xiii 3.8 Typical temperature dependence of: (a) real and imaginary components of mutual inductance M (black and red curves, respectively) (b) −2 corresponding superfluid density nS ∝ λ = µ0ωσ2(T ) and dissipation peak in real conductivity µ0ωσ1(T ) for two superconducting films of the same thickness and different transition temperatures...... 29

4.1 Temperature dependence of superﬂuid density (black curves) and real

conductivity σ1(T ) (red curves) for two oxygen depleted Y Ba2Cu3O7−δ ﬁlms, 20 unit cells (top) and 10 unit cells (bottom) thick. A lower transition temperature is obtained in the 10 uc ﬁlm after oxygen is disordered in the chains. The intersection with the dotted lines indi-

cates the temperature where a 2D transition should occur if the CuO2 bilayers were coupled...... 32

4.2 Temperature dependence of superﬂuid density λ−2 and the real con-

ductivity σ1 (red curves and inset in (c)) for a Y0.80Ca0.2Ba2Cu3O7−δ ﬁlm, 140nm thick when: (a) oxygen chains are occupied and (d) oxygen chains are empty. Intermediate steps in the reduction process as result of repeated annealing in argon atmosphere for (b) 10 minutes at 200 − 250oC and for (c) 30 minutes 250oC (black), 300oC (blue), 350oC (green) are shown...... 35

4.3 Temperature dependence of superﬂuid density λ−2 (black curves) and

the real conductivity peak σ1 (red curves) for Y1−xCaxBa2Cu3O7−δ films in-situ annealed after growth in reduced O2 content. Oxygen pressure and anneal length is marked by the curves. (a)-(b) different films with x = 0.30 and 80 uc, 40uc and 20 uc thick. The second curve in (a) is the same film after an oxygen reduction annealing 300oC/0.5 hours in Argon atmosphere. (c)-(d) two x = 0.168 films 40 uc thick and with different oxygen content...... 37

4.4 Temperature dependence of superﬂuid density λ−2 (black curves) and

the real conductivity peak σ1 (red curves) for a Y0.85Ca0.15Ba2Cu3O7−δ film as grown (top curve) and remeasured after 1 week, 1 month and 6 months. Changes likely due to oxygen diffusion outside the film at 300K...... 38

4.5 Temperature dependence of superﬂuid density λ−2 (black curves) and

the real conductivity peak σ1 (red curves) for a Y0.80Ca0.20Ba2Cu3O7−δ film as grown (top curve) and remeasured after 1 month and 6 months. Changes likely due to oxygen diffusion outside the film at 300K.... 39

xiv −2 5.1 Evolution of superfluid density nS ∝ λ (T ) and the real conductivity σ1(T ) for a P rBa2Cu3O7−δ/Y0.80Ca0.20Ba2Cu3O7−δ/P rBa2Cu3O7−δ film (PCaYP) 10/2/10 uc as grown (top curve) and after the film self-annealed at room temperature and oxygen diffused outside the samples. Each measurement is made at intervals of approximately 15 hours...... 43

5.2 Oxygen reordering eﬀects in a P CaY P/aP ﬁlm 10/2/10/3 uc measured in a 120-hour interval...... 45

5.3 Superﬂuid density (black line - left axis) and electrical resistance (blue line - right axis) in a P Y P/aP ﬁlm 10/2/10/3 uc at high oxygen concentration ...... 46

−2 5.4 Temperature dependence of the superfluid density nS ∝ λ (left axis - black) and the real conductivity µ0ωσ1 (right axis - red) in 2 uc Y1−xCaxBa2Cu3O7−δ films with various oxygen concentrations. The films are grown between protective P rBa2Cu3O7−δ layers and with an additional 3uc amorphous P rBa2Cu3O7−δ cap layer. Dissipation peaks only few degrees wide indicate the high degree of homogeneity in reduced dimensionality and unprecedented reduced doping levels. . 48

−2 5.5 Superfluid density nS ∝ λ (T ) (left axis) and real conductivity µ0ωσ1(T ) (right axis) for the two most underdoped 2 uc Y1−xCaxBa2Cu3O7−δ films. The temperature dependence of µ0ωσ1(T ) is an indicator of good film homogeneity. Intersection of dashed lines with 1/λ−2(T ) is approx- 2 2 imately where a 2D transition is predicted: 1/λ (T ) = 8πµ0kBT/dΦ0 where d = film thickness...... 50

5.6 The Kosterlitz-Thouless-Berezinski (KTB) transition in 1/λ2(T ) for

a wide range of doping in 2 uc Y1−xCaxBa2Cu3O7−δ films. Intersec- tion with the dashed line indicates the theoretical prediction for the occurrence of this transition when the entire film behaves like a 2D system. Top three (brown) curves are the same film in three conditions: as-grown and with two other doping levels, after some oxygen diffused out of the film or chain oxygen annealed at room temperature. Similarly, the next 8 (blue) curves represent the same film at different

doping levels (as-grown has the highest TC ). Additional curves represent diﬀerent as-grown ﬁlms...... 51

0 5.7 Helicity modulus ∝ nS in a J − J − J junction (blue- white - blue areas in the inset) in a XY model when the size of the J 0 barrier goes from 0 to the full system size. Here, J 0 = 0.4J [48] ...... 53

xv −2 5.8 The superﬂuid density λ (left axis) and the real conductivity µ0ωσ1 (right axis) in a 2 uc Y1−xCaxBa2Cu3O7−δ ﬁlm measured at four different frequencies...... 55

5.9 The real R and imaginary ωL components of the impedance at 2kHz

in the same 2 uc Y1−xCaxBa2Cu3O7−δ film from p figure 5.8. The intersection between dR and dωL indicates the position of the σ1 peak measured at the same frequency in the previous figure...... 56

2 5.10 Scaling of TC with absolute superﬂuid density ns ∝ 1/λ (0) on a log- log scale for Y1−xCaxBa2Cu3O7−δ 2 unit cells thick (red dots) and 40 uc thick (green dots). Error bars deﬁned in text. For reference we include Uemura’s µSR results on YBCO powders (open black circles),

Hc1 measurements on clean YBCO crystals (open orange squares), microwave measurements on ultra-clean YBCO crystals (open blue squares), and data on 20-40 uc YBCO films (black dots). Solid lines 2 illustrate the linear relationship, TC ∝ 1/λ (0), that describes our strongly underdoped ultrathin films and is expected near a 2D quantum critical point. The dashed line illustrates a square root relation- q 2 ship, TC ∝ 1/λ (0), that describes strongly underdoped 3D samples (crystals and thick films) and is consistent with 3D quantum criticality. 58

5.11 Normalized temperature dependencies of superﬂuid density in four

severely underdoped 2uc Y1−xCaxBa2Cu3O7−δ films (Tc = 26 K blue, 15 K green, 12 K pink, 5.3 K red) that fit on the linear dependence 2 TC ∝ 1/λ (0). The universal temperature dependence of superfluid −2 −2 density nS(T )/nS(0) ∝ λ (T )/λ (0) = f(T/TC ) strongly supports the existence of a Quantum Critical Point when superconductivity is lost with underdoping...... 62

5.12 On normalized scales we compare five ultrathin 2 uc Y0.7Ca0.3Ba2Cu3O7−δ films (TC = 54K orange , 36 K red, 26 K blue, 15 K green, 12 K pink), an underdoped 20 uc Y0.7Ca0.3Ba2Cu3O7−δ film (black circles) and the 2 quadratic dependence [1 − (T/TC0) ] in red. The normalising factor for the temperature is the quadratic extrapolation to zero of the low

temperature of superﬂuid density, equivalent to a mean-ﬁeld type TC0 transition temperature...... 63

xvi 6.1 Temperature dependence of ab plane resistivity ρ(T ) in La2−xSrxCuO4 films (all samples in the left figure, overdoped samples in the right figure). With continuous and dotted lines we identify films on underdoped and overdoped sides of the phase diagram. The red continuous line represents the optimal doping x = 0.15. Sr concentrations are labeled on the figures...... 68

6.2 Underdoped to optimal doped La2−xSrxCuO4 films. Temperature dependence of superfluid density λ−2(T ) (dark curves) and the fluctuation

peaks in real conductivity σ1(T ) (red curves). The Sr concentrations are: (a) x = 0.06, (b) x = 0.09, (c) x = 0.12, (d) x = 0.15. Dashed lines represent quadratic ﬁts to the low-temperature dependence and the intersection with the dotted lines indicate where a 2D transition

should exist in individual decoupled CuO2 layers...... 70

6.3 Overdoped La2−xSrxCuO4 films. Temperature dependence of superfluid density (dark curves) and the fluctuation peaks in real conduc-

tivity σ1(T ) (red curves) for films with nominal Sr concentrations: (a) x = 0.18, (b) x = 0.21, (c) x = 0.24, 0.27 and 0.30, (d) x = 0.27. Dashed lines are quadratic fits to the low-temperature dependence, and in (d) the intersection with the dotted line indicates where the 2D transition takes place in the entire thickness of the film...... 71

6.4 The dependence of superﬂuid density λ−2(0) (red) and of transition

temperature TC (black) on Sr concentration. Error bars in TC described in text...... 74

−2 6.5 TC versus λ (0) for La1−xSrxCuO4 (blue circles) is compared with the same dependence for Y1−xCaxBa2Cu3O7−δ thin films in figure 5.10. Er- ror bars represent the full width of the transition, from the resistive TC to the lower end of the fluctuations peak, and the blue arrow indicates the increase of doping from underdoping to overdoping...... 76

−2 6.6 (a) Dependence of superfluid density λ and real conductivity σ1 versus temperature for an La1.85Sr0.15CuO4 film only 45A˚ thick. The dashed lines are corresponding KTB lines for one decoupled CuO2 layer, 6.6A˚, (black) and for the entire thickness of the film (red). (b) Corrected superfluid density based on the assumption that only a fraction of the film is superconducting and that the absolute value should be the same as in thicker films. The downturn in λ−2(T ) at approximate 40 K is consistent with our previous results in 2 uc

Y1−xCaxBa2Cu3O7−δ ﬁlms and with a 2D transition that takes place in the entire thickness...... 80

xvii 6.7 Frequency and current variation effects in the real and imaginary components of the impedance R and ωL (top figure) and the corresponding −2 λ (T ) = µ0ωσ2 and µ0ωσ1(T ) (bottom figure). d is the thickness of the film. 5 mV at 50kHz for the dotted line, 100 mV at 50kHz for the continuous line, 5 mV at 10kHz for the dashed line...... 81

−2 −2 −1 6.8 λ (T ) = µ0ωσ2 (black curve) and λ (T ) = L µ0/d based on the assumption that the inductive response is equivalent to the bare inductance of the superﬂuid density. d is the thickness of the ﬁlm. . . . 83

B.1 Thickness calibration. AFM images of Ca-YBCO (a) and PBCO (b) thin ﬁlms (bright side of the picture) deposited as a result of 1,000 laser pulses on STO substrates (dark side of the picture). The deposition rates are approximately 0.71A˚/pulse for YBCO and Ca-doped YBCO and approximately 0.77A˚/pulse for PBCO...... 89

B.2 AFM images of the surface of Y1−xCaxBa2Cu3O7−δ films grown directly on SrT iO3 without protective layers (a) 40 unit cell thick (b) 8 unit cell thick. Both films have RMS roughness around 17A˚. In the thinner film it is possible to observe how the 4A˚ (one unit cell) steps

on the surface of the SrT iO3 substrate are preserved on the surface of the ﬁlm...... 90

B.3 3D AFM image of a Y1−xCaxBa2Cu3O7−δ ﬁlm grown between protective P rBa2Cu3O7−δ layers, 10/2/10 unit cells thick. The steps on the surface are approximately 1uc ≈ 11.7A˚...... 90

xviii CHAPTER 1

INTRODUCTION

Twenty years after the discovery of superconductivity in high-temperature copper oxide superconductors, there is still no viable explanation as to what holds the electrons paired together in a Cooper pair. The main characteristics of these high- temperature superconductors (HTSc) are very similar to the ones observed in classical superconductors, i.e they show zero electrical resistivity and magnetic shielding

(Meissner eﬀect). However, from the very beginning of their physical investigation, it became obvious that such high transition temperatures cannot be explained by electron-phonon coupling and the classical Bardeen-Schreiﬀer-Cooper (BCS) theory of superconductivity, so it was thought that pairing might have an electronic or magnetic nature.

It is not only the higher transition temperature that diﬀerentiates cuprate superconductors from the classical superconductors; HTSc show a very complex nature across all their physical properties and exhibit new fundamental physics. Supercon- ductivity is produced by introducing doping in ceramic Mott insulators, so it is not surprising that these materials are dominated by strong electron correlations (in other words, the on-site Coulomb repulsion between electrons is stronger than the kinetic energy). Phase sensitive experiments pointed out that the gap symmetry observed in cuprates is the anisotropic d-wave as opposed to the isotropic s-wave pairing that

1 characterizes classical superconductors [1, 2]. This gap does not vanish at the transition temperature in low-doped supercondutors and persists up to higher temperatures in the pseodo-gap state [3].

Although an impressive number of theories have been proposed, there is still no satisfactory all-encompassing explanation concerning various aspects of cuprates and their changing behavior over the entire phase diagram.

Another very important difference between classical superconductors and HTSc is the extent to which their transition temperatures are affected by phase fluctuations of the order parameter. In the case of the classical superconductors, phase fluctuations are not important and the transition temperature is determined by the energy gap

∆. In the case of HTSc, which have quasi-two-dimensional layered structure and short coherence length between the layers, small superfluid density/reduced superfluid stiffness and high transition temperatures, phase fluctuations become important and set an upper limit for the transition temperature [4]. In addition, the effects of

ﬂuctuations are expected to be more pronounced in the underdoped region and it is generally believed that phase ﬂuctuations are largely responsible for the suppression of superconductivity when holes are removed from optimally doped CuO2 planes.

In one of its most popular description the pseudo-gap is considered a precursor for superconductivity. In the pseodo-gap state there are preformed superconducting pairs but strong phase ﬂuctuations prevent the occurence of long-range phase coherence until TC [5].

The main purpose of this study is to explore the physics of underdoped cuprate superconductors near the super-to-insulator transition, with TC < 10K and a super-

ﬂuid density less than 1% of the values at optimal doping. Moreover, by studying real two-dimensional superconducting samples, I investigate the properties within a

2 single copper oxide layer or bilayer, the fundamental structural unit in cuprates. The

underdoped region is one of the less explored parts of the phase diagram of cuprates.

While the study of superconductivity at underdoping is crucial for understanding su-

perconductivity, its exploration was previously limited due to the lack of homogeneous

superconducting samples in this region of the phase diagram. Because no such sam-

ples were available, there is an ongoing controversy between theoretical predictions

and experimental results.

The belief in the existence of a universal linear correlation between transition

temperature and superﬂuid density, TC ∝ nS, with underdoping as a result of strong

thermal phase ﬂuctuations became very popular in HTSc physics once superﬂuid

density was measured in moderately underdoped cuprates [6]. The ﬂuctuations hy-

pothesis also accounted for the extremely wide critical region near TC (around 10

K) in Y Ba2Cu3O6.95 crystals [7]. Therefore, scientists were surprised when recent

measurements showed that the linear dependence is not followed anymore in strongly

underdoped Y Ba2Cu3O7−δ crystals [8, 9] and ﬁlms [10]. In fact, the scaling rela- q tion is close to: TC ∝ nS(0). Moreover, the critical region near TC was much

smaller than in moderately underdoped samples [11]. The accurate determination of

penetration depth in my films clarifies the superfluid density puzzle in underdoped

superconductors.

In my project, the challenge of developing a consistent method for obtaining homo-

geneous underdoped samples is matched by the diﬃculty of performing measurements

and of understanding the physical properties of superconductors in the controversial

underdoped regime. In this region of the phase diagram, even small alterations in

the chemical composition can have a dramatic impact on physical properties. For a

while, it was believed that disorder and magnetic impurities caused by the proximity

3 to the AFM phase will produce strong pair breaking eﬀects and will totally suppress superconductivity when TC < 20K in cuprates [12, 13]. I found a stable more consistent way to produce low-doped superconductors by substituting Ca for Y in

Y Ba2Cu3O7−δ. By extending this research to the unprecedented study of real two- dimensional underdoped superconductors, the complexity of the problem increases even more. I succeeded in growing ﬁlms that are as thin as only two molecular layers

(23.5A˚ thick) that have transition temperatures as low as 3 Kelvin, which is at the very vicinity of the superconductor to non-superconductor transition at zero Kelvin.

In order to reveal the importance of phase ﬂuctuations, these samples were characterized using a two-coil mutual inductance measurement of magnetic penetration depth.

In a follow-up project, I tested the consistency of the results obtained in Ca −

YBCO by performing similar measurements on an additional family of superconductors, La1−xSrxCuO4. The La1−xSrxCuO4 thin films produced by our collaborators at NTT Tokyo are superconducting over an extended doping interval from x = 0.06 to x = 0.30. Even if these samples where not as homogeneous in the underdoped region as the Y1−xCaxBa2Cu3O7−δ films I grew, they gave me the opportunity to measure superfluid density at the other extreme of the phase diagram, i.e. the overdoped region.

This thesis is organized as follows. In Chapter 2, I review the basic properties of superconductors, the link between these properties and penetration depth, as well as the main characteristics of cuprates with a special focus on phase fluctuations. Chap- ter 3 describes the experimental techniques used for substrate preparation, thin film deposition, and superfluid density measurements. In Chapter 4, I present the results of calcium doping and oxygen reduction in YBCO films. This study is extended in

4 Chapter 5 to the unprecedented measurement of superfluid density in two-dimensional underdoped HTSc films. Finally, Chapter 6 focuses on penetration depth measurements in LSCO films.

5 CHAPTER 2

BASIC CONCEPTS

2.1 Superconductivity and superﬂuid density

The phenomenon of superconductivity was discovered in 1911 by K.Onnes while

observing the disappearance of electrical resistance in mercury cooled below 4.2K. The

fundamental properties of superconductors are the lack of resistivity and the expulsion

of any exterior magnetic ﬂux (or perfect diamagnetism). These two properties can be

directly used in order to determine the physical characteristics of superconductors.

Before a microscopical description of superconductors became available, Ginzburg

and Landau [14] used a phenomenological approach to describe the normal-to- super-

conductor transition in the general frame of thermodynamical second-order phase

transitions. Thus we can deﬁne a complex order parameter Ψ that is zero in the

non-superconducting state (the disordered phase) and has a ﬁnite value below the TC

(in the ordered phase). Ψ characterizes the degree of superconductivity.

In the Ginzburg-Landau theory, Ψ is analogous with an eﬀective wavefunction of superconducting electrons, and the magnitude |Ψ(x)|2 gives the local density of the center of mass of superconducting electrons or the superﬂuid density, nS, at the

2 nS given location x as |Ψ(x)| = 2 . In their theory, the free energy is an expansion of powers of |Ψ(x)|2 and |∇Ψ(x)|2 and, when this expansion is minimized with respect

6 to the order parameter, we can obtain an equation for Ψ(~r) similar to the Schrodinger

equation to describe the superconducting state.

The order parameter Ψ will direct us to one of the most striking and unexpected

properties of superconductors that places them into the quantum phenomenon: the

quantum coherence/the phase stiﬀness. The superconducting carriers are all in a

single quantum ground state similar to a Bose-Einstein condensate, a state that is

described by the same macroscopic wave function/order parameter. The common

ground state and the phase coherence explain the zero DC resistivity in a supercon-

ductor. The system will stay coherent unless external perturbation will have enough

energy to excite quasiparticles out of the ground state and produce a gradient in

the phase. These decoherence eﬀects should be directly observed in the superﬂuid

density. We can therefore consider the order parameter and implicitly superﬂuid den-

sity to be a very important physical quantity when describing the strength of the

superconducting state and its stiﬀness to phase ﬂuctuations.

The linking of the order parameter to measurable quantities is done by using the

London equations [15]. These equations were the ﬁrst to describe the electrodynamic

properties of superconductors.

We are most interested in the second London equation, which describes the low-

frequency electrodynamics: n e2 ∇ × j~ = − S B~ (2.1) S m∗c

~ ∗ where j~S is the supercurrent density, B is an external magnetic field and m is the effective mass of superconducting electrons. When combined with the Maxwell equa- ~ ~ tion, ∇ × B = µ0j~S in the steady state, equation 2.1 leads to a very important consequence: an external magnetic field vanishes exponentially inside a superconductor over a finite length called penetration depth, λ.

7 2 1 ~ (∇ − λ2 )B = 0 (2.2) ~ ~ −x/λ B = B0e with the penetration depth λ related to the superﬂuid density by:

m∗ nS = 2 2 (2.3) µ0e λ

The dynamic of superconducting electrons can be treated similarly to free electrons

in a non-dissipative conductor. The resulting current density

e2n j~ = S E~ sinωt (2.4) S m∗ω 0

~ ~ is always out of phase with the electric ﬁeld E = E0cosωt and the conductivity is

purely imaginary: e2n σ = S (2.5) 2 m∗ω

Therefore, from conductivity measurements at low enough frequencies ω, it is possible

to deduce the temperature dependence of penetration depth and superﬂuid density:

2 e nS(T ) 1 −2 σ2(T ) = ∗ = λ (T ) (2.6) m ω µ0ω

−2 From this dependence we can obtain the absolute value of nS ∝ λ at zero Kelvin and essential information about microscopic properties of superconductors. This absolute value is particularly hard to measure and most experiments present only relative changes in penetration depth. In the temperature dependence, it is possible to compare experimental data with constraints imposed on this quantity by diﬀerent theoretical models. At low-T, λ−2(T ) is dominated by the density of state, therefore

we can obtain information about the pairing state and the symmetry of the order

parameter. When temperatures are near TC , vortex dynamics and thermal phase

8 ﬂuctuations are signiﬁcant and control this temperature dependence. In our exper-

iment we are able to measure both the relative changes and the absolute value of

superﬂuid density in superconducting ﬁlms.

2.2 General properties of high-temperature superconductors

The parent compounds (without any doping) for high-temperature superconduc-

tors are Mott insulators. Their valence band is only half ﬁlled, so we would expect

them to present conducting properties. However, due to strong on-site Coulomb re-

pulsion, a second electron cannot be added on the sites resulting in an insulator. In

addition to these insulating properties, all HTSc are characterized by the existence of

the anti-ferromagnetic order at very low doping levels. When carriers are added by

ion substitution or by oxygen insertion, conductivity is restored and these materials

become superconducting at low temperatures above a critical doping. When doping

levels are increased, the critical temperature where these materials become supercon-

ductors rises from absolute zero up to a maximum value obtained at optimal doping.

In bulk samples optimal doping occurs at x = 0.16 and TC = 37K for La2−xSrxCuO4, and x = 0.19 and TC = 94K for Y Ba2Cu3O7−δ.

The most important common feature of high-temperature superconductors is the

existence of CuO2 planes and other atoms in-between them. This structure is similar

to the perovskite structure ABX3. If we look at the Y Ba2Cu3O7−δ elementary cell

in ﬁgure 2.1 we observe the superposition of three ABX3 unit cells, with the A atom

alternating between barium and yttrium, B being copper and X being oxygen. There

are however significant differences: there is oxygen deficiency in the middle plane that

contains the Y atom and two missing oxygen atoms on the top and bottom of CuO2

layers that will form the CuO chains. Since these chains are not a shared characteristic

9 Figure 2.1: The crystallographic structure of one-unit cell of Y Ba2Cu3O7−δ, when the oxygen chains are full (δ = 0) or totally unoccupied (δ = 1), and the structure of two-unit cell La2−xSrxCuO4 [16].

of all cuprates, they are not expected to play an important part in the superconducting mechanism, but they do play an important part in doping. Because oxygen atoms in the chains are more mobile than oxygen atoms in the CuO2 planes, their conﬁguration

is changed when doping is added or removed or when the chains are disordered. While

in Y Ba2Cu3O7 all the chains are occupied, in the case of Y Ba2Cu3O6 the oxygen

sites are empty and these chains don’t exist (see ﬁgure 2.1 middle). The distance

between the middle copper oxide layers CuO2 that form a bilayer is only 3.842A˚.

La2−xSrxCuO4 has a much simpler structure similar to K2NiF e4 one-unit cell (uc)

containing one CuO2 monolayer. In ﬁgure 2.1 (picture on the right) we observe the

superposition of two uc with the CuO2 planes 6.6A˚ apart.

10 This highly anisotropic structure will produce very anisotropic physical proper-

ties. There is a signiﬁcant diﬀerence between the values of fundamental properties

(electrical resistivity, penetration depth, coherence lenght) in the ab plane and along

the c−axis perpendicular to the plane. In YBCO the anisotropy ratio in penetration depth is λab vs. λc ≈ 10000A˚ [17] and ρc/ρab ≈ 100. The layered structure and the

high anisotropy lead to the general belief that cuprates are quasi two-dimensional

materials and that they can be very well described by two-dimensional models and

theories. Therefore, it is natural to want to study superconductivity in the most basic

building block for a superconductor, i.e. a one-unit cell, and to see if the properties

of real 2D superconductors coincide with previous observations in quasi-2D thicker

systems.

2.3 Phase ﬂuctuations in high-temperature superconductors

Thermal phase ﬂuctuations are a controversial subject in HTSc physics. The

limits where ﬂuctuations are becoming important is set by the Ginzburg criterion,

which evaluates when the thermal energy kBT starts to be the order of ﬂuctuation

energy and when the size of the ﬂuctuations of the order parameter are comparable

with the magnitude of the order parameter:

2 2 2 2 2π µ0kB λ (0) 3 |T − TC0| < 4 2 TC0 Φ0 ξ (2.7) 4 3 −13 λ Tc0 −2 −2 |T − TC0| < 4.7 × 10 ξ2 nm K 2 where Φ0 = h/4e is the superconducting ﬂux quantum, ξ the Ginzburg-Landau

coherence length and TC0 is a mean-ﬁeld transition temperature. In classical super-

conductors, ﬂuctuations are important only in a small µKelvin interval below TC .

Because cuprates have a shorter coherence length, a smaller superﬂuid density (bigger

penetration depth) and higher transition temperatures, ﬂuctuations are expected to

11 have an increased impact on these materials and dominate the physical properties of

HTSc over a much larger temperature interval - from at least 0.1K to a few degrees

Kelvin [18]. This eﬀect should be enhanced even more in the underdoped region

where superﬂuid density decreases with reduced doping levels.

The upper limit for superﬂuid density in the vicinity of transition is set by the

−2 mean-ﬁeld behavior with λ ∝ (TC − T ). Because of quasi two-dimensional (2D)

structure and short coherence length in the c-axis direction, ﬂuctuations in HTSc

are expected to be 2D and to produce deviations from the mean-ﬁeld behavior. In a

pure 2D system, a Kosterlitz-Thouless-Berezinski transition should occur as a spon-

taneous phase transition due to thermally activated vortex-antivortex unbinding, and

a discontinuous drop is observed in superﬂuid density. In other words, this transi-

tion is a balance between the thermal energy kBT and the condensation free energy

2 FN (T ) − FS(T ) in the coherent volume πξ d. The transition temperature in a two-

dimensional system with thickness d is set by:

1 8πµ0 kBT2D 2 = 2 × (2.8) λ (T2D) Φ0 d

where Φ0 = 2πh/¯ 2e is the ﬂux quantum. Measurements on samples with ﬁnite

thickness and with weak coupling between the CuO2 planes, disorder, as well as measurements in a ﬁnite frequency regime will change the appearance of this discontinuous drop into a sudden downturn with increased slope. When the CuO layers inside a superconductor are decoupled, the eﬀective thickness for this transition should be of one-unit cell.

A substantial eﬀort was dedicated to trace the signatures of 2D ﬂuctuations in high-temperature superconductors but, up to now, the only report of a KTB transition in a cuprate was seen in high-frequency conductivity measurements in

12 Bi2Sr2CaCu2O8+δ [5]. In these measurements, the onset of a frequency dependence in the conductivity of the films was interpreted as evidence for two-dimensional behavior. At such high frequency values, no downturn in superfluid density was observed since the discontinuity is a characteristic of the static regime. The experimental evidence of 2D fluctuations was also missing in very thin films. Rufenacht et al [19] measured the superfluid density in a La2−xSrxCuO4 film as thin as 26A˚ without seen any trace of fluctuations effects although, in this sample, fluctuation should appear naturally as a result of reduced dimensionality. The authors argued that the superfluid jump is suppressed by disorder.

An explanation for the absence of the KTB transition in cuprates is an interlayer coupling stronger than expected. In this case we should observe a crossover from

−2 2/3 a 2D regime to a 3D regime. A 3D − XY behavior with λ ∝ (TC − T ) that replaces the discontinuity in superﬂuid density in the critical region. There is ex- tensive experimental evidence for 3D-XY behavior near the transition. Kamal et. al

[7] were the first to observ a very wide critical region in the microwave penetration depth measurements in ultra-pure YBCO crystals. This conclusion was supported by thermal expansion measurements performed by Meingast et al. [20], yet recent measurements of superfluid density in underdoped thin films [10] and crystals [9] are not consistent with the 3D − XY fluctuation hypothesis. In these last measurements superfluid density has a linear temperature dependence over a wide range interval below TC ( e.g. for TC ≈ 16K this interval extends down to 4K). This type of behavior corresponds to a mean-field like dependence in the absence of fluctuations.

Another point where the hypothesis of ﬂuctuations can be tested is the relation between the absolute value nS(0) and TC . As doping is removed in HTSc, both the

13 transition temperature TC and superﬂuid density nS decrease in a surprisingly correlated manner. This idea originated in the early µSR measurements of the penetration depth in moderately underdoped cuprates, measurements that suggested a propor-

−2 tionality relation, TC ∝ λ(0) ∝ nS(0), the ”Uemura relation” [6, 21]. The most popular explanation for this universal relation was given by the hypothesis of thermal phase fluctuations destroying the superconductivity when both the superfluid density and the transition temperature become small as a result of underdoping [4]. This relation was extended to a large variety of cuprate families and to other exotic superconductors, but it was never tested in strongly underdoped samples (TC ≤ 10K) and extremely low values of superfluid density.

The same measurements that showed a limited fluctuation region next to the critical temperature [10, 9] suggest a significant deviation from Uemura’s linear dependence. In fact, YBCO films and crystals agree with each other and both show the

2 1/2 TC ∝ [1/λ (0)] , despite the fact that the absolute values of superfluid densities in crystals are several times larger than those of the films. These observations motivated the new hypothesis that underdoping leads to the disappearance of superconductivity at a 3D quantum critical point (QCP), as opposed to a first-order quantum phase transition [22, 23]. Since this relation is an important link to theoretical models, it needs to be tested further in the most underdoped samples in the very vicinity of critical doping and in samples with reduced dimensionality.

14 CHAPTER 3

EXPERIMENTAL METHODS

Recent developments in the deposition methods for superconducting thin ﬁlms

have signiﬁcantly improved their homogeneity. As a result, the properties of the

ﬁlms have become comparable with the properties of the crystals. Since crystals are

three-dimensional and can never be reduced to an actual two-dimensional system,

thin ﬁlms are the only suitable choice for the investigation of superconductivity in

reduced dimensionality

The main goals of my project were to grow underdoped superconducting thin ﬁlms

as close as possible to the two-dimensional limit and to measure their superﬂuid den-

sity. To this purpose, I developed a method for obtaining atomically ﬂat substrates,

improved the deposition method, and measured the superﬂuid density of my ﬁlms

using a two-coil mutual inductance technique. In addition, I analyzed the surface

morphology of substrates and of thin ﬁlms immediately after deposition with the use

of a commercial Atomic Force Microscope (AFM) Nanoscope III in tapping mode.

3.1 Preparation of atomically ﬂat SrT iO3 substrates

The substrates I used in my project are SrT iO3 (100) single crystals. This material is one of the best available choices for growing YBCO ﬁlms. When growing epitaxial

ﬁlms with particular crystallographic orientations, it is essential to have a substrate

15 with crystallographic structure and lattice parameters very similar to those of the

deposited material: the closer the match between the substrate and the ﬁlm, the

better the obtained epitaxy. SrT iO3 has the typical cubic perosvkite structure (a

= 3.906 A˚) with Ti in the corners and Sr in the center of the cube (YBCO lattice

parameters a = 3.82A˚, b = 3.89A˚, c/3 = 3.89A˚), as seen in ﬁgure 3.1 [24].

Figure 3.1: Perovskite structure of SrT iO3 with the two possible termination layers, SrO and T iO2 [24].

With a melting point at approximately 2080oC, STO is very stable at temper-

atures above 700oC. This property makes it very suitable for epitaxial deposition

and eliminates the risk of a chemical reaction between the substrate and the ﬁlm.

In addition, the similar thermal expansion coeﬃcients in STO and YBCO prevent

the formation of cracks in the ﬁlms during thermal cycles (STO 11 × 10−6/K and

13.4 × 10−6/K for YBCO [25]). A template buﬀer layer between the ﬁlm and the

16 substrate will relax any stress induced to the ﬁlm that might result from small lattice

mismatches between them. In the case of YBCO, I used P rBa2Cu3O7−δ insulating

protective layers between the ﬁlm and the substrates as well as on the top of the ﬁlm.

When growing superconducting ﬁlms that are continuous layers only several nm

thick, one important concern should be to start ﬁlm growing on atomically ﬂat sur-

faces free of any contamination. The ideal substrate roughness must be much smaller

that 1uc YBCO, 11.7A˚. A (100) STO substrate has two possible termination layers:

SrO (basic oxide) and T iO2 (acidic oxide). Commercially polished substrates have a random mixture of both termination layers and they are not smooth enough for growing atomic scale films. However, since T iO2 is not reacting with most acids and SrO is very soluble in acids and even water, this difference in solubility between the two possible terminations makes it possible to obtain the ”ideal” substrate, a substrate which is terminated uniquely with T iO2 and is atomically flat. The only noticeable features when imaging the surface are one-unit cell steps of around 3.9A˚ that are due to the

miscut angle on the substrate. On a perfectly terminated substrate such as this one,

it is most probable that the ﬁlm will continue the perovskite stacking sequence from

the substrate: - SrO −T iO2 −BaO −CuO −BaO −CuO2 −Y −CuO2 −BaO −CuO

[24], as seen in ﬁgure 3.2 [26].

The standard recipe consists of using a sequence of etching substrates in diﬀerent acid solutions or/and annealing them above the threshold temperature for surface recrystallization. I have optimized this method for SrT iO3, but variations of this preparation method can be used for other substrates such as NdGaO3, SrLaAlO3, or LaAlO3. The optimal etching/annealing process is highly dependent on the initial condition of the substrate (e.g., polishing stress, oxygen deﬁciency, crystallinity), its miscut angle, and the acid used.

17 Figure 3.2: Sketch of the Y BCO/ST O interface [26]

The surface preparation method used here is based on etching the substrate in commercial ammonia-buffered HF, NH4F + HF , with a pH around 5 (BHF) [27, 28] and annealing it in flowing oxygen at temperatures slightly above the recrystallization threshold of 800OC [29] (the annealing takes place in the deposition chamber right before growing the film). It was previously observed that it is unlikely to obtain only one single type of surface termination by heat treatment on smooth substrates

(roughness below 0.2nm) with both types of surface terminations [30], therefore etching is mandatory in this case. However, annealing is very eﬀective at removing any surface contamination with CO2, H2O, or organic traces.

18 Figure 3.3: (a) AFM image of a mechanically repolished SrT iO3 substrate (b) after etching in BHF, pH ≈ 5 for 30 seconds (c) 5 minutes (d) 12 minutes. The brightest spots in all four images are 10 nm from the surface.

This method requires strict control of the pH of the acid (pH from 4.3 to 5). If

the pH is too low (acid too strong), then etch pits will form. If the pH is too high

(acid too weak), then SrO residues will precipitate on the surface or the etching

will mostly take place where surface defects exist. In the ’step-ﬂow’ regime SrO is

removed and T iO2 is lifted oﬀ from uneven parts of the surface by side etching of

SrO until terraces (≈ 100nm wide) and one-unit cell steps (3.905A˚) are formed (see

19 ﬁgure 3.4). The measured etching rates along the (100) directions are 0.1nm/s for

pH 4.3 and 0.9nm/s for pH 3.7 [29].

Figure 3.4: Zoomed image of the repolished and 12-minute BHF, pH ≈ 5, etched SrT iO3 substrate from ﬁgure 3.3. Even if the surface is not perfect, a cross section perpendicular to the steps, along the black line, allows us to distinguish the one-unit cell steps. The vertical distance of approximately 4A˚ was measured between the two red triangles on adjacent terraces.

In ﬁgure 3.3 the AFM pictures were taken after repeated etching over diﬀerent lengths of time on a repolished substrate. The bright yellow spots on the surface are

10nm high points originating from less perfect polishing and from SrO precipitation

on the surface. The density of these spots is reduced with increasing etching time.

Nearly perfectly terminated surfaces start to form after 12 minutes of etching (ﬁgure

3.3d) and we observe the one-unit cell steps on the zoom of the same image in ﬁgure

3.4. Then, the perfect termination was observed after annealing a similar substrate

at 900oC for 1 hour in ﬂowing oxygen (ﬁgure 3.5). The etching time may vary with

20 the quality of the substrates, but it is recommended to keep it as short as possible in

order to prevent the formation of etch pits. It should start with 30 seconds then be

increased until atomically ﬂat terraces become noticeable.

Figure 3.5: AFM image of a SrT iO3 substrate etched in buﬀered HF, pH ≈ 5, for 12 minutes and annealed in oxygen at 900oC for one hour. The one-unit cell steps between the two red triangles are clearly visible in the section analysis along the black line.

The method was further improved by soaking the substrates in demineralized water for 10 to 30 minutes in an ultrasonic bath, before a shorter 1-2 minutes dip in BHF acid [31, 32]. When the substrate is removed from the soak in water, the

SrO layers on its surface will have formed Sr(OH)2 and SrCO3. The formation of

SrCO3 is unavoidable because of the exposure to the carbon dioxide in the air. Both

compounds can be easily removed using the BHF acid.

This intermediate water-soaking step brings consistency to the etching process

and produces surfaces that are less than 0.2 nm rough over an extended area (ﬁgure

21 Figure 3.6: SrT iO3 substrates after H2O soaking and BHF etching (a) old substrate (b) zoom (c) new substrate (d) old substrate with hole pits. The surface roughness is less than 2nm.

3.6). As I previously mentioned, the etching process is highly dependent on the initial state of the substrate. Substrates with diﬀerent origins and history show somehow diﬀerent results after an identical etching (10 minutes water soaking and 1.5 minutes

BHF etching).

The first substrate is an old repolished substrate and, as seen in figure 3.6a and in the zoom in figure 3.6b, the one-unit cell steps between 100nm wide terraces are very well defined. The second substrate is a new substrate and figure 3.6c shows

22 that the surface is smooth and the steps are visible but not completely etched. In

the third substrate, presented in ﬁgure 3.6d, square etch pits start to form due to

repeated cycles of etching and ﬁlm deposition. The ﬁnal step that helps the surface

to recrystallize is the annealing of the substrate at 800oC for 10 minutes in an oxygen

atmosphere. This last step takes place inside the deposition chamber right before

the deposition and guarantees a smooth fresh surface free of contamination. A less

perfect surface (with a mixture of both possible terminations, SrO and T iO2) such as the one in ﬁgure 3.6c can be improved by repetitive etching but, as far a I can tell, superconducting ﬁlms can be grown homogeneously in molecular size layers as long as the surface roughness is less than 0.2nm. When etch pits appear on the surface, the substrate needs to be mechanically repolished.

3.2 Thin ﬁlm deposition

The cuprate superconductors are very complex materials consisting of at least four chemical elements. Growing these superconductors is a complex process that requires growth methods and conditions that are signiﬁcantly diﬀerent from the ones that were previously used for classical superconductors like Pb or Nb. The success of this process depends on obtaining the right stoichiometry with reduced impurity inclusions, the right crystallographic alignment, and the right level of doping that sets the value of the transition temperature.

One of the most successful methods is Pulse Laser Deposition (PLD). In this method, an EXICMER laser that operates at 248nm generates pulses at 1-50Hz

(length of the pulse in the order of 25 nanoseconds). These pulses are used to ablate a target that has the same chemical composition as the desired ﬁlm inside a chamber

23 with controlled atmosphere. Because of the high-energy laser pulses, a thermally- emitted ion and electron plasma will be generated from the target. The thermal energy is converted into kinetic energy and the ablated material will adiabatically expand in space forming an elliptic plume that will stick to a substrate positioned on a heater at 760 − 780oC.

Optical alignments and several adjustments are necessary before growing the ﬁlms.

The most important parameters for a successful growth are the following: the laser spot on the target, the energy density, the temperature of the substrate, and the in-situ gas composition and pressure during both film deposition and post-deposition annealing. These parameters have to be individually optimized for any particular deposition system and, by varying them, we can adjust and control the physical properties of the films. The most homogeneous films are the as-grown films and additional treatments (ex-situ annealing, oxygenation, or oxygen reduction) are likely to decrease homogeneity.

The laser spot size on the target controls the thickness uniformity of the deposition and the density of the energy that produces the ablation. Most of the material will be deposited perpendicular to this spot. There are two distinct regimes: spherical expansion for small spots (in the order of 100µm) and linear expansion for spots bigger than 1cm. The thickness of the films is uniform over large substrate areas when only one of these regimes dominates the deposition [33]. As far as energy density is concerned, levels less than 1.5J/cm2 are too low and will not produce ablation while energy levels above a critical value (around 3J/cm2) will produce heating in the target and removal of micron sized pieces of the target that will contaminate the surface of the film. If particulates are observed on the surface, their density and size can be reduced by reducing the laser’s energy or by doing off-axis deposition (the lateral

24 expansion is smaller for the heavier components of the plasma). High density targets, with high heat conductivity, will minimize the heating on the surface of the target and will result in growing of ﬁlms with smooth surfaces.

To obtain a uniform energy density, only the center part of the laser beam is used.

The laser spot on the target (approximately 3×3mm2) is the image of a 9mm×13mm aperture placed in the front of the beam through a 0.5m focal lens and will produce an energy density of 1.8 − 2J/cm2.

The reduced thickness of the samples makes them extremely sensitive to their chemical environment and their properties can be quickly altered. All superconducting ﬁlms are deposited on BHF etched substrates with thin insulating layers of

P rBa2Cu3O7−δ protecting the films above and below. At the deposition parameters mentioned above, we deduced the thin film deposition rates for Ca doped YBCO and PBCO by measuring the thickness of deposited films after 1,000 laser pulses. As seen in figure B.1 in Appendix B, we deposit approximately 0.71A˚/pulse for YBCO and Ca-doped YBCO and approximately 0.77A˚/pulse for PBCO. At these deposition rates, 17 pulses are necessary to complete one Ca-YBCO molecular layer (and respectively 14 pulses for a PBCO molecular layer), and we observed a variation less than

5% for ﬁlms that are 20 and 80 unit cell thick. The temperature of the substrates during the deposition of YBCO ﬁlms with the CuO2 planes parallel to the substrate is between 760 − 780oC.

There are two ways of broadly adjusting doping levels inside our ﬁlms. The ﬁlms can be grown out of a mixture of two targets of pure YBCO and 30% Ca doped YBCO

(30% Ca is the maximum solubility for YBCO ﬁlms). In addition we can change the oxygen gas pressure inside the deposition chamber during growth and post-growth in situ annealing. Since our purpose is to obtain high-quality underdoped ﬁlms,

25 the necessary doping will be ideally provided by the calcium atoms, and the oxygen pressure will be kept only high enough to maintain the stability of the sample. In the case of ultrathin films I discovered that oxygen will diffuse outside the samples at room temperature - see section 5.1. To minimize this loss of doping, I deposited an amorphous PBCO diffusion barrier at a temperature of 760 − 780oC, which is too low for the crystallographic structure to form.

Figure 3.7: AFM image of the surface of a 20 unit cell (23nm) Y1−xCaxBa2Cu3O7−δ ﬁlm right after deposition. Islands with a height of one-unit cell(1uc ≈ 11.7A˚) can be observed. The RMS roughness of the ﬁlms is approximately 19A˚.

The surface image of a Ca-YBCO film observed with AFM indicates that our films have a smooth surface morphology (figure 3.7). The roughness of 1-2nm is produced by one-unit cell steps islands of materials that start to nucleate before the previous

26 layer was completely ﬁlled and are intrinsic to the growth of YCBO by PLD [34].

This island type of growth was observed in YBCO superconducting ﬁlms with the

best superconducting properties and is believed to be intrinsic to the growth of YBCO

thin ﬁlms by PLD [34].

Some authors suggest that these terraces are evidence of screw dislocations and

might be at the origin of the ﬂux pinning and of high critical currents observed in

YBCO [35, 36]. However I never observed any spiral structures on the ﬁlms that I

grew.

3.3 Superﬂuid density measurements

Superﬂuid density in superconducting ﬁlms is measured using a two-coil mutual

inductance measurement [37, 38]. With this method, it is possible to measure not

only the relative changes in superﬂuid density due to changes in temperature, but also

the absolute value at zero Kelvin. The experiment consists in measuring the mutual

inductance between two small coils (drive and pick-up coils) placed on opposite sides

of the ﬁlm. When an AC ﬁeld produces the current Idrive in the drive coil, an induced

current Vpick−up is measured in the pick-up coil and the complex mutual inductance

is:

M = M1 + iM2 = Vpick−up/iωIdrive (3.1)

The typical temperature dependences of both the real and the imaginary compo-

nents of mutual inductance for two ﬁlms with identical lateral dimensions and thick-

ness but different transition temperatures (TC1 = 48K dashed line and TC2 = 36K continuous line) are displayed in figure 3.8a. When a film is in the normal state, it has no effect on the applied field, therefore the value of real part of the mutual inductance (black curve) is set by the distance between the coils. At temperatures below

27 TC inductive currents in the surface of the superconductor will produce a magnetic

field that opposes the applied field. The applied field will be screened by the film and,

as a consequence, the real component of the mutual inductance is reduced by a factor

that depends on the total thickness of the ﬁlm and on the complex conductivity of

the ﬁlm. The imaginary component (red curve) is zero at every temperature except

near transition where we can observe a negative peak as result of energy dissipation.

Since we intend to measure the absolute value of penetration depth in ﬁnite size

ﬁlms, it is very important to distinguish the contribution to the mutual inductance

due to the coupling between the coils around the ﬁlm (zero position) from the total

mutual inductance. This contribution can be measured by replacing the ﬁlm with

a similarly shaped thick lead foil. By subtracting this background value from the

mutual inductance we can very well approximate the size of our ﬁlms as inﬁnite [37].

Zero position mutual inductance turned out not to be very important for ultrathin

ﬁlms since the value of coupling around the ﬁlm is only around 1% of the smallest

value of mutual inductance measured in the superconducting state.

In the inﬁnite-area approximation and when the precise geometry of the coils is

known, we can obtain a table that links mutual inductance to the ﬁlms’ sheet complex

conductivity σ(ω)d = σ1(ω)d − iσ2(ω)d [38]. The penetration depth is deduced from

the imaginary part: 1 = µ ωσ (3.2) λ2 0 2

As observed in the relation 3.2, 1/λ−2 is directly related to the superﬂuid density. Since their dependences are equivalent, we can alternate between these two quantities to illustrate the same physics and we can call λ−2 superﬂuid density. The

−2 corresponding superﬂuid density nS ∝ λ = µ0ωσ2(T ) and the real part of complex

conductivity σ1 curves are displayed in ﬁgure 3.8b for the same two ﬁlms for which

28 Figure 3.8: Typical temperature dependence of: (a) real and imaginary components of mutual inductance M (black and red curves, respectively) (b) corresponding su- −2 perfluid density nS ∝ λ = µ0ωσ2(T ) and dissipation peak in real conductivity µ0ωσ1(T ) for two superconducting films of the same thickness and different transition temperatures.

we previously analyzed the mutual inductance dependencies (ﬁgure 3.8a). Since both

ﬁlms have the same thickness, it is possible to observe a reduced value of mutual

inductance in the superconducting state for the film with the higher TC , due to an increased screening factor. As a result this film has an increased superfluid density.

29 The main parameter used to assess samples’ level of homogeneity is the width of the superconducting transition that is the same as the width of the dissipation peak in σ1. When a sample is homogeneous, all parts of the film become superconducting at the same temperature, resulting in a narrow transition interval and a sharp σ1 peak. If σ 6= 0 in the superconducting state below TC , the samples are not entirely superconducting and it is also possible that the temperature dependence of superfluid density will be dominated by inhomogeneity effects.

Most of the measurements in this project were done using a dipole drive coil (1 mm in diameter with 2 layers of 4 turns) and a quadrupole pick-up coil (2 mm in diameter with 3-4 layers of 24 turns). Both coils were made out of 0.05mm diameter copper wire on a nylon form. The typical frequency in our measurements is 50kHz and the frequency-dependent measurements are in the range of 500Hz-2MHz.

The drive circuit has an AC signal generator of 1-100kHz, a high resistor to maintain constant current in the circuit when the resistance of the coils is changing with temperature, and a reference mutual inductor 28µH used to measure the current in the drive circuit. Both the voltage drop across the reference mutual inductor and the voltage drop on the pick-up coil are measured using lock-in amplifiers. Precautions were taken to ensure that all data were measured in the linear-response regime, i.e. mutual inductance is independent of the magnitude of current in the drive coil. Near the transition temperature, the AC field needs to be kept low enough so that we have only thermally exited vortices. Typical induced currents into the films are two order of magnitudes smaller than the critical current in YBCO of approximately 106A/cm2.

30 CHAPTER 4

UNDERDOPING Y Ba2Cu3O7−δ

4.1 Chain disorder in underdoped Y Ba2Cu3O7−δ

It is already very well known that, as doping is reduced, both the transition temperatures and the superﬂuid density decrease. The most obvious method for underdoping Y Ba2Cu3O7−δ is reducing the oxygen concentration. Y Ba2Cu3O7−δ has two oxygen sites: plane and chain. Oxygen reduction in the underdoped Y Ba2Cu3O7−δ only aﬀects the chains, resulting in empty oxygen sites and disorder.

The result of my attempts to improve the homogeneity of underdoped Y Ba2Cu3O7−δ

ﬁlms grown in reduced oxygen atmosphere are very similar to the previous ﬁndings

−2 of Zuev et al. [10]. In figure 4.1 I plot λ (T ) and σ1(T ) for two of my most homogeneous films 20 and 10 unit cells (uc) thick. The second curve for the 10 uc film is the effect of induced disorder in CuO chains by heating the film at 100oC. The

−2 8πµ0 dotted lines are defined as λ (T2D) = 2 × kBT2D/d, where d is the full thickness Φ0 of the film, and the intersection between this line and λ−2(T ) marks the point where a fluctuation-driven transition should occur in the films when the interlayer coupling is significant. There are no significant changes in the temperature dependence of λ−2 before this intersection, which indicates that the CuO2 bilayers are coupled.

31 Figure 4.1: Temperature dependence of superfluid density (black curves) and real conductivity σ1(T ) (red curves) for two oxygen depleted Y Ba2Cu3O7−δ films, 20 unit cells (top) and 10 unit cells (bottom) thick. A lower transition temperature is obtained in the 10 uc film after oxygen is disordered in the chains. The intersection with the dotted lines indicates the temperature where a 2D transition should occur if the CuO2 bilayers were coupled.

The temperature dependence of real conductivity σ1 displays wide dissipation peaks, ∆TC ≈ 5K. Obviously, the transition widths are broadened by chain doping inhomogeneity. Zuev et al. [10] studied a large number of Y Ba2Cu3O7−δ films with transition temperatures between 5K and 30K. Similar to my findings, the transition temperatures in their films are reduced even more, without changing the doping, when

32 the oxygen chains are deliberately disordered. However, this process was not totally

reversible and the transition temperatures were broadened by disordering the chains.

Better homogeneity at these reduced doping levels was observed in a high-purity

YBCO crystal. In this sample it was possible to reversibly reduce the transition from

TC = 17K to TC ≈ 3K by controlled induced disorder [9]. The higher transition

temperatures are obtained by self-annealing that will allow chain oxygen to order

in the Ortho-II phase and increase the plane doping at reduced oxygen levels. This

sample provided the opportunity to study many diﬀerent transition temperatures

in the same sample but, since chain oxygen can be ordered and disordered easily,

the doping homogeneity can be altered resulting in transition temperatures that are

not very stable. Therefore, no chain oxygen content and a diﬀerent approach to

dope YBCO should lead to improved homogeneity. One option for replacing the

common oxygen doping method is the introduction of doping in-between the planes

by substitution of Y atoms.

4.2 Y1−xCaxBa2Cu3O7−δ underdoping while adding holes

Doping Y Ba2Cu3O7−δ with calcium might seem to be the wrong approach when

obtaining underdoped superconductors because, when Ca2+ replaces Y +3, additional

doping holes are introduced in the CuO2 planes. When the oxygen stoichiometry is

7 − δ ≈ 7 samples are overdoped as a result of the extra Ca, but when 7 − δ ≈ 6 the calcium will compensate for oxygen deﬁciency and will provide the necessary carriers to attain superconductivity even when no oxygen remains in the CuO chains. When most of the chain oxygens are missing, the ﬁlms are tetragonal and therefore free of twin planes. Inhomogeneity from chain oxygens is minimized since chains don’t contribute much to doping.

33 Two important questions need to be asked. Are the samples still stable at reduced

oxygen content in the chains? And will any additional disorder result from calcium

doping?

4.2.1 Oxygen reduction in Y0.80Ca0.2Ba2Cu3O7−δ

To answer these questions, I ﬁrst tested the eﬀects of oxygen reduction in a

Y0.80Ca0.2Ba2Cu3O7−δ film 140nm thick produced by Pulsed Laser Deposition in Pro- fessor Mannhart’s group at University of Augsburg, Germany. This group is known to have significantly optimized the deposition of Ca doped YBCO films with the purpose of increasing the critical current across grain boundaries [39].

2 In figure 4.2a we observe the 1/λ and the dissipation peak in σ1 for the film as- grown. The film is close to optimal doping TC ≈ 81.5K and shows a sharp transition width ∆TC ≈ 0.5K (see inset). Ca doping produce a decrease in the maximum transition temperature compared with TCmax. ≈ 91K in optimally doped Y Ba2Cu3O7.

Since the film is very thick and the value of mutual inductance in the superconducting state at optimal doping is very close to the zero position (coupling between the coils around the film), the determination of the absolute value of superfluid density is not very accurate. I will therefore not comment on the low-temperature dependence. However, once doping is reduced, the film will screen less and will have better resolution on the superfluid density.

In figure 4.2b the top curve is identical with 4.2a. The other three curves present the same film after its oxygen concentration was reduced by repeated annealing in argon atmosphere at 200 − 250oC for 10 minutes. The oxygen reduction continues in figure 4.2c, but I accelerated the oxygen diffusion process by increasing both the annealing temperature and the annealing time.

34 Figure 4.2: Temperature dependence of superﬂuid density λ−2 and the real conductivity σ1 (red curves and inset in (c)) for a Y0.80Ca0.2Ba2Cu3O7−δ ﬁlm, 140nm thick when: (a) oxygen chains are occupied and (d) oxygen chains are empty. Intermediate steps in the reduction process as result of repeated annealing in argon atmosphere for (b) 10 minutes at 200 − 250oC and for (c) 30 minutes 250oC (black), 300oC (blue), 350oC (green) are shown.

−2 As expected, both superﬂuid density λ and TC evolve with oxygen reduction.

This process is not uniform. The σ1 dependencies (inset ﬁgure 4.2c) indicate that, while some parts of the ﬁlm are underdoped and start to develop a peak around 28K, there is a very broad transition interval in the rest of the sample that extends up to

80K. It is surprising to see how little change there is in the absolute value λ−2(0) with

35 oxygen reduction between the black curve and the green curve. This might indicate that the CuO chains have only a small contribution to superﬂuid density.

After the ﬁnal annealing, when almost all the chain oxygen atoms are gone or completely disordered and only calcium doping contributes to superconductivity the

film regains its homogeneity and has only a single transition as indicated by the green curve in figure 4.2c and figure 4.2d. An additional annealing at the same temperature will not change the film and increasing the temperature will destroy the structural stability. Both the transition width and the superfluid density are comparable to previous results from pure Y Ba2Cu3O7−δ films with TC ≈ 25K [10].

This result proves that it is possible to obtain superconducting Y1−xCaxBa2Cu3O7−δ

films with low transition temperatures and reduced oxygen doping. This type of samples should be more stable since there are no chains left to disorder, δ ≈ 1. However, reducing doping in films with high oxygen content can produce other unknown disorder effects (such as decomposition) during repeated annealing. We therefore speculate that the best samples with the most homogeneous properties will be as-grown with no chain oxygen and needing no additional heat treatments.

4.2.2 Doping dependence, homogeneity, and superﬂuid density

Using Pulsed Laser Deposition I grew Y1−xCaxBa2Cu3O7−δ ﬁlms (140mtorr O2 @

o 760 C) over the entire range of Ca doping from a mixture of two targets Y Ba2Cu3O7−δ and Y0.7Ca0.3Ba2Cu3O7−δ (x = 0.30 is the maximum solubility of Ca). Ca diﬀuses easily at the growth temperature so I expected to obtain uniform doping in the entire

ﬁlm. Since I wanted most of the doping to originate from Ca and because I considered that long annealings would be a source of disorder, I gradually decreased the anneal

36 time from the 24 hours previously used in YBCO to 1-2 hour intervals and then to 1-

10 minute intervals. A summary of all the ﬁlms grown at diﬀerent Ca concentrations

together with their growing conditions and their transition temperatures is presented

in Appendix A.

Figure 4.3: Temperature dependence of superfluid density λ−2 (black curves) and the real conductivity peak σ1 (red curves) for Y1−xCaxBa2Cu3O7−δ films in-situ annealed after growth in reduced O2 content. Oxygen pressure and anneal length is marked by the curves. (a)-(b) different films with x = 0.30 and 80 uc, 40uc and 20 uc thick. The second curve in (a) is the same film after an oxygen reduction annealing 300oC/0.5 hours in Argon atmosphere. (c)-(d) two x = 0.168 films 40 uc thick and with different oxygen content.

The first samples in figure 4.3(a)-(b) prove that Y0.70Ca0.3Ba2Cu3O7−δ films have

properties comparable to pure YBCO and that it is possible to study a wide range

37 of transition temperatures at underdoping by controlling the oxygen concentration

during in-situ annealing (at 600oC).

A Y0.85Ca0.15Ba2Cu3O7−δ ﬁlm annealed in-situ in 1torr O2 was not superconduct-

ing while for a Y0.832Ca0.168Ba2Cu3O7−δ ﬁlm annealed in only 10mtorr the transition

temperature was TC = 8K (ﬁgure 4.3d). The lack of superconductivity was due

undoubtedly to insuﬃcient doping since these sample became superconducting af-

ter a short annealing in O2/O3. Therefore, the minimum calcium concentration for a

superconducting sample when the chains are oxygen-free is between 0.15 < x < 0.168.

Figure 4.4: Temperature dependence of superfluid density λ−2 (black curves) and the real conductivity peak σ1 (red curves) for a Y0.85Ca0.15Ba2Cu3O7−δ film as grown (top curve) and remeasured after 1 week, 1 month and 6 months. Changes likely due to oxygen diffusion outside the film at 300K.

It is worth noticing the improved homogeneity at very reduced doping in the

TC = 8K sample as compared to YBCO ﬁlms without Ca [10] and the lack of any

38 sign of fluctuations effects in figure 4.3d. In fact, in this sample with extremely low

−2 −2 superﬂuid density λ (0) ≈ 0.22µm and TC ≈ 8K, superconductivity should be

totally suppressed above about 2.5 K in the hypothesis of strong ﬂuctuations in de-

−2 coupled CuO2 bilayers. The temperature dependence of superfluid density λ (T ) in all thick Y1−xCaxBa2Cu3O7−δ films I grew confirms that there are no 2D transitions in individual decoupled CuO2 bilayers. This result is in agreement with the observation in thick Y Ba2Cu3O7−δ films [10] and high-purity Y Ba2Cu3O7−δ crystals [9] and suggests that the interlayer coupling is strong enough to make the films three dimensional, even when TC < 10K and at extremely small values of superfluid density.

Figure 4.5: Temperature dependence of superfluid density λ−2 (black curves) and the real conductivity peak σ1 (red curves) for a Y0.80Ca0.20Ba2Cu3O7−δ film as grown (top curve) and remeasured after 1 month and 6 months. Changes likely due to oxygen diffusion outside the film at 300K.

39 The homogeneity and stability of the Ca doped samples is still related to their

oxygen content and the transition width is sharpest for as-grown ﬁlms. These ﬁlms

have good homogeneity with ∆TC less than 0.5-1 Kelvin as observed in the real

conductivity peak. Additional annealing broadens the transition, so I stopped pur-

suing this path. Moreover I discovered that, if the concentration of oxygen is above

7 − δ ≈ 6, the transition temperatures decrease after long time intervals. These changes are most likely due to oxygen diffusing outside the films at room temperature. In figure 4.4 and figure 4.5 we observe the effect of oxygen loss in two different

ﬁlms Y0.85Ca0.15Ba2Cu3O7−δ and Y0.80Ca0.20Ba2Cu3O7−δ 40 uc thick in a six-month

time interval. This oxygen reduction process does not introduce any additional chain

disorder as in the case of annealing, and the sharp transition widths are almost pre-

served when TC decreases. If the Ca doping is low enough, the diﬀusion will likely produce the entire range of transition temperatures down to zero Kelvin. However, since this process takes such a long time, samples risk to deteriorate before diﬀusion is complete.

−2 The measured values of λ (0) in all Y1−xCaxBa2Cu3O7−δ ﬁlms presented above

agree quantitatively with results from Y Ba2Cu3O7−δ ﬁlms, which indicates that scat-

tering from Ca is not particularly important. Also, their characteristics ﬁt very well

2 in the TC vs 1/λ (0) dependence from Zuev et al. [10] and indicate that both samples,

with or without Ca, share the same phenomenology. The temperature dependence

−2 λ (T ) and the relation between TC and absolute superﬂuid density nS(0) are com-

pared to results from Y Ba2Cu3O7−δ and two-dimensional Y1−xCaxBa2Cu3O7−δ ﬁlms

in the next chapter.

40 CHAPTER 5

UTRATHIN Y1−xCaxBa2Cu3O7−δ FILMS

5.1 Challenges in the fabrication of ﬁlms with reduced thickness

Although the mechanism of superconductivity is not yet fully understood, there is a general agreement regarding the important role played in these materials by the

CuO2 planes and the fact that superconductivity is intrinsic to any individual CuO2 layer-bilayer.

Having a layered structure, weakly coupled layers with coherence length in the c direction smaller than the interlayer separation, and high anisotropy of physical properties, HTSc are considered quasi-two dimensional systems. Since theoretical work on HTSc relies on this last property, most of the microscopical models proposed to explain HTSc are two-dimensional. A two-dimensional treatment simpliﬁes calcula- tions by excluding from the analysis the strength of interlayer coupling and additional complications associated with it. This is the main motivation for my studying superconductivity in the most basic building block for a superconductor, 1 molecular layer

(1 unit cell = 1 uc), and my measuring superﬂuid density in real 2D samples. These samples are an important experimental test of the validity of theoretical models and the results should have implications in the future understanding of cuprates.

41 Fluctuations are unavoidable when reducing the dimensionality from 3D to 2D.

Previous measurements using the same method successfully detected fluctuations effects in very thin MoGe and In/InO3 films [40], therefore ultrathin Y1−xCaxBa2Cu3O7−δ superconducting films allow us to test if 2D criticality can be detected or not in cuprates. In addition, by comparing 2D with 3D samples, we can address the role of interlayer coupling in high-temperature superconductors.

Previous results showed that it is possible to grow optimally-doped Y Ba2Cu3O7−δ

ﬁlms 1-2 uc thick between insulating P rBa2Cu3O7−δ layers using pulsed laser deposition [41, 42, 43]. However, our attempts always failed when the thickness of the

films was reduced below 10 uc (117.6A˚) even for the films annealed in high oxygen pressure. Also, additional post-annealing in ozone did not make them superconducting. At that time I was not unsure what the cause is but, later in the chapter, I argue that oxygen diffuses outside ultrathin films and, most probably, these samples did not have enough doping to superconduct. No reports of ultrathin underdoped samples were found.

+3 +2 As seen in Chapter 4, by partially replacing Y with Ca in Y1−xCaxBa2Cu3O7−δ

(maximum x = 0.30), we obtained doping levels high enough to attain superconductivity even if the oxygen concentration in the CuO chains is nearly zero (overall oxygen stoichiometry is 7 − δ ≈ 6). The cation doping is also stable when the thickness of the ﬁlms is reduced. Preliminary attempts resulted in superconducting 8uc and 4uc

Y0.8Ca0.2Ba2Cu3O7−δ films, but these films were inhomogeneous and had broad transitions. I then optimized the deposition conditions and I grew Y0.8Ca0.2Ba2Cu3O7−δ and Y0.7Ca0.3Ba2Cu3O7−δ (Ca-YBCO) superconducting films as thin as only 1 and 2 unit cells (1 uc = 1.17 nm) by pulsed laser deposition onto atomically flat SrT iO3 substrates as described in section 3.2 (laser spot size 3 × 3mm2, energy density 2J/cm2,

42 substrate temperature 760oC and in situ annealing at 500oC/5-10 minutes). In these

ﬁlms, only a small concentration of oxygen is suﬃcient to change the transition temperatures in the strongly underdoped region and I achieved a wide range of hole doping with greatly reduced inhomogeneity in the CuO chains.

−2 Figure 5.1: Evolution of superfluid density nS ∝ λ (T ) and the real conductivity σ1(T ) for a P rBa2Cu3O7−δ/Y0.80Ca0.20Ba2Cu3O7−δ/P rBa2Cu3O7−δ film (PCaYP) 10/2/10 uc as grown (top curve) and after the film self-annealed at room temperature and oxygen diffused outside the samples. Each measurement is made at intervals of approximately 15 hours.

The ﬁlms are protected above and below by thin insulating layers of P rBa2Cu3O7−δ.

Some Ca might diﬀuse into adjacent P rBa2Cu3O7−δ during the growing process and

the actual Ca concentration may be lower than its nominal value. In the case of

ultra-thin ﬁlms, the monotonic relation between oxygen pressure during in-situ an-

nealing and transition temperature is not valid anymore. New parameters become

43 very important: the thickness of the protective layer, the cooling rate of the ﬁlm be-

low 400o (thermoelectric measurements showed that oxygen diﬀusion outside YBCO has a maximum value around 400oC), and the time taken between producing and measuring the sample. To avoid alteration of the properties with time, I cooled the samples and performed the measurements as soon as the ﬁlms were grown. A one-unit cell is obtained with only 17 laser pulses. The AFM pictures in section 3.2 show that

Ca-YBCO deposition by PLD has an island-type growth mode and that substrates have 4A˚ steps. Since my experimental goal was to obtain a continuous superconducting film (with no pinholes) at least 1-uc thick, I performed all measurements on 2-uc thick films. Their quantitative analysis will confirm that, in fact, both unit cells are superconducting and contribute to the superfluid density.

The superﬂuid density λ−2(T ) and real conductivity curves in ﬁgure 5.1 show the evolution with time for a P rBa2Cu3O7−δ/Y0.80Ca0.20Ba2Cu3O7−δ/P rBa2Cu3O7−δ

ﬁlm (PCaYP) that is 10/2/10 uc thick. The top black plot is the as-grown ﬁlm.

Each data set was taken at intervals of approximately 15 hours while the ﬁlm was

at room temperature, so oxygen diﬀusion away from the ﬁlm resulted in changes in

their doping concentration. This type of sample provided the opportunity to study

many diﬀerent transition temperatures in the same sample but, since oxygen diﬀusion

is highly dependent on external parameters such as temperature, this dependency

introduces unpredictable outcomes especially at reduced TC .

In order to overcome this diﬃculty and to obtain more stable samples, I im-

proved the growing procedures by adding an amorphous P rBa2Cu3O7−δ cap layer

deposited below 300oC, a temperature that is too low for crystallographic structures

to form. This layer acts like a diﬀusion barrier, so most of the changes are due to

oxygen reordering effects as seen in figure 5.2. As a consequence, no significant TC

44 reduction was observed in P rBa2Cu3O7−δ/Y0.80Ca0.20Ba2Cu3O7−δ/P rBa2Cu3O7−δ

/amorphous P rBa2Cu3O7−δ films (PCaYP/aP) after a 120-hour interval. It is not clear why TC increased after 22 hours, but these variations can represent the lateral doping differences in the film on a 2-3 millimeter scale when the drive coil, 1 millimeter in diameter, is not perfectly centered on the film.

Figure 5.2: Oxygen reordering eﬀects in a P CaY P/aP ﬁlm 10/2/10/3 uc measured in a 120-hour interval.

The superﬂuid density 1/λ2 and the electrical resistance versus T for a P Y P/aP

10/2/10/3 film are plotted in figure 5.3. This was the sample where I achieved the highest doping level in an ultrathin Y1−xCaxBa2Cu3O7−δ film. The superfluid density turns on right when resistance is close to zero. In my experience, this indicates a homogeneous sample. Otherwise, if only parts of the film were superconducting, the resistance would drop to zero through percolation paths but λ−2(T ) would start to

45 grow only at a lower temperature when most of the ﬁlm is superconducting. The transition temperature TC ≈ 52K is comparable to the maximum TC observed in

2 uc YBCO films without Ca [41, 42, 43]. The integrity and high quality of this sample is confirmed by the values of normal state resistivity measured right above the transition temperature. The values of ρ(TC+) ≈ 75−100µΩcm in this film correspond to the values measured in 1000A˚ thick Y Ba2Cu3O7−δ films when 7 − δ ≈ 6.60 and

TC ≈ 50K [44] (the imprecision originates from the irregular shape of the pattern used for the resistivity measurement). There are some technical diﬃculties associated with making electrical contacts through P rBa2Cu3O7−δ without changing the properties of the ﬁlm, therefore electrical resistivity measurements succeeded only in the samples with the highest doping levels.

Figure 5.3: Superﬂuid density (black line - left axis) and electrical resistance (blue line - right axis) in a P Y P/aP ﬁlm 10/2/10/3 uc at high oxygen concentration

46 5.2 Temperature dependence of superﬂuid density

A large number of underdoped 2uc Y1−xCaxBa2Cu3O7−δ ﬁlms with transition temperatures between 52K and 3K were produced and measured (see Appendix A

2 tables A.2 and A.3). The 1/λ (T ) and σ1(T ) dependencies are displayed in figure 5.4 for six films that are representative of the entire set of samples. All the films show one narrow dissipation peak and σ1(T ) only a few degrees wide. σ1 is zero below the transition, which indicates good sample homogeneity without any secondary superconducting transitions. A non-zero σ1 in the superconducting state could produce dramatic changes in the 1/λ2(T ). Therefore, this feature is one important criteria in selecting samples so that we can distinguish intrinsic electronic structure effects from extrinsic effects originating in locally varying properties due to inhomogeneities.

−2 For the highly-doped sample PCaYP120106, TC = 52K, we observe λ (0) ≈

−2 14µm , which is comparable to values obtained in thick Y Ba2Cu3O7−δ ﬁlms and

40 uc thick Y1−xCaxBa2Cu3O7−δ films with comparable critical temperatures. On this basis, we conclude that thin and thick films have similar structural and stoichio- metrical qualities and that our samples have continuous superconducting layers. The comparison of ultrathin films, thick films, and ultraclean crystals will further support this assertion.

The temperature dependence of λ−2 for all samples shows the same basic features: it is approximately quadratic at low T and has a downturn as the temperature increases close to the critical region. Precautions were taken to ensure that data was taken in the linear-response regime, i.e. the value of mutual inductance is independent of the size of the 50 kHz magnetic ﬁeld produced by the drive coil. The induced currents in the ﬁlms are in the order of 104A/cm2. Therefore, the drop

47 −2 Figure 5.4: Temperature dependence of the superfluid density nS ∝ λ (left axis - black) and the real conductivity µ0ωσ1 (right axis - red) in 2 uc Y1−xCaxBa2Cu3O7−δ films with various oxygen concentrations. The films are grown between protective P rBa2Cu3O7−δ layers and with an additional 3uc amorphous P rBa2Cu3O7−δ cap layer. Dissipation peaks only few degrees wide indicate the high degree of homogeneity in reduced dimensionality and unprecedented reduced doping levels.

48 in 1/λ2 is not produced by supercurrents exceeding the critical current in our ﬁlms

6 2 (JYBCO ≈ 10 A/cm ).

The following subsections are organized as follows. In the first subsection I discuss the temperature dependence of superfluid density in the critical region and the nature of the finite temperature phase transition. The second subsection focuses on the frequency dependence of KTB transition. In the third subsection I analyze the scaling relation between TC and superfluid density and, finally in the fourth subsection, I compare the changes in the 1/λ2(T ) dependence with doping and discuss the differences between 2uc and thicker Y1−xCaxBa2Cu3O7−δ films.

5.2.1 KTB transition in two-dimensional samples

Frequency-dependent conductivity in thick Bi2Sr2CaCu2O8+δ ﬁlms has been in-

terpreted as a signature of 2D ﬂuctuations in HTSc [5]. However, the existence of 2D

ﬂuctuations was never directly observed in superﬂuid density measurements.

The key quantitative feature in the temperature dependence of 1/λ2 that I follow

is the downturn as the temperature approaches the critical region. The Kosterlitz-

Thouless-Berezinski (KTB) theory applied to 2D superconducting ﬁlms predicts a dis-

continuous drop in superﬂuid density at the superconductor-to-normal phase transi-

tion as a result of thermally excited vortex-antivortex (V-aV) pairs [45]. The theoreti-

2 2 cal transition temperature T2D is the temperature where 1/λ (T2D) = 8πµ0kBT2D/dΦ0

(d =ﬁlm thickness and Φ0 = 2πh/¯ 2e is the ﬂux quantum). The discontinuous drop

strictly applies only if 1/λ2 is measured at zero frequency and it should change into

rapid downturn at ﬁnite frequencies.

2 1/λ and the real conductivity σ1 vs. T for the most underdoped 2-uc ﬁlms are

plotted in ﬁgure 5.5. From these two dependencies we can see the unprecedented low

49 −2 Figure 5.5: Superfluid density nS ∝ λ (T ) (left axis) and real conductivity µ0ωσ1(T ) (right axis) for the two most underdoped 2 uc Y1−xCaxBa2Cu3O7−δ films. The temperature dependence of µ0ωσ1(T ) is an indicator of good film homogeneity. Intersec- tion of dashed lines with 1/λ−2(T ) is approximately where a 2D transition is predicted: 2 2 1/λ (T ) = 8πµ0kBT/dΦ0 where d = film thickness.

critical temperatures, TC ≈ 2.8K and TC ≈ 5K, and the unprecedented homogeneity at such low doping levels. Figure 5.6 extends the 1/λ2(T ) display over a wide range of doping in underdoped 2-uc Y1−xCaxBa2Cu3O7−δ films. The top 3 curves (brown) represent measurements on the same film and are identical with figure 5.1. Similarly, the next 8 curves (blue) represent the same film at different oxygen concentrations

(same as figure 5.2). All other curves represent different as-grown samples. Among all of the films shown in figure 5.6 some have narrower transitions than others; some seem to have two closely-spaced transitions; and some show an extended ”foot” above

50 TC . Details vary from sample to sample due in part to small doping inhomogeneity on the CuO chains, but their main features (an almost ﬂat low-temperature behavior ending in an abrupt downturn in λ−2(T )) are quite reproducible. The downturn is also

2 present in ﬁlms that show upward curvature in 1/λ ’s next to TC , therefore limited inhomogeneity does not mask the 2D transition.

Figure 5.6: The Kosterlitz-Thouless-Berezinski (KTB) transition in 1/λ2(T ) for a wide range of doping in 2 uc Y1−xCaxBa2Cu3O7−δ films. Intersection with the dashed line indicates the theoretical prediction for the occurrence of this transition when the entire film behaves like a 2D system. Top three (brown) curves are the same film in three conditions: as-grown and with two other doping levels, after some oxygen diffused out of the film or chain oxygen annealed at room temperature. Similarly, the next 8 (blue) curves represent the same film at different doping levels (as-grown has the highest TC ). Additional curves represent different as-grown films.

51 −2 The right-hand side of the λ (T2D) relation is plotted as a dashed line in ﬁgures

5.5 and 5.6. Its intersection with λ−2(T ) measured at ﬁnite frequency approximates

the predicted TC . In the simplest scenario, we would expect the intersection to occur

at the onset of the downturn in 1/λ2, as it does in 2D ﬁlms of superﬂuid He4 [46, 47]

measured at 5 kHz. Instead, it consistently occurs closer to the middle of the drop

in 1/λ2. Given the complexities of cuprate ﬁlms such as grain boundaries, vortex

pinning, and residual inhomogeneity, these eﬀects should be taken into account in

addition to the classical KTB theory when quantitatively ﬁtting our results.

A similar result is observed in numerical simulations in the XY model for an

inhomogeneous superconductor. In this model, strong superconducting regions char-

acterized by the Josephson coupling J (blue areas, inset ﬁgure 5.7) are separated by

a weaker region with J 0 = 0.4J (white area, inset ﬁgure 5.7) and legth d [48]. The

temperature dependence of helicity modulus in the direction perpendicular to the

junction displayed in the ﬁgure 5.7 is equivalent to the temperature dependence of

superﬂuid density. The homogeneous systems endpoints, only J (d = 0) or only J 0

π (d = 128), undergo classic KTB phase transitions with a discontinuity at TC = 2 J. As soon as d > 0, the downturn in helicity modulus starts at lower temperatures than

π in the theoretical prediction TC = 2 J and the intersection with KTB line takes place in the middle of the downturn as in my samples in figure 5.6. In my case this will be equivalent to small doping variations and areas of fluctuating superconductivity inside the films.

An alternate explanation for the occurrence of a two-dimensional transition in cuprates comes from Benfatto et al. [49, 50]. In their model, due to the ﬁnite interlayer coupling, the temperature dependence of superﬂuid density does not follow anymore the universal KTB behavior with a transition in one CuO2 bilayer (one unit

52 0 Figure 5.7: Helicity modulus ∝ nS in a J − J − J junction (blue- white - blue areas in the inset) in a XY model when the size of the J 0 barrier goes from 0 to the full system size. Here, J 0 = 0.4J [48]

cell). In this case, the expected discontinuity is replaced by a downturn curvature at

a temperature highly dependent on the vortex core energy µ [49]. In 3D samples, µ is

bigger than typical vortex-core energy in the anisotropic 2DXY model µxy and the

temperature dependence of superﬂuid density has a small drop close to TC , similar

to previous observations in pure YBCO ﬁlms [11]. When µ/µxy ≈ 6 there is no trace

of the downturn in supeﬂuid density.

Based on this theory, Benfatto et al. consider two distinctive regimes in my un-

derdoped 2uc Y1−xCaxBa2Cu3O7−δ ﬁlms. The doping dependent vortex core energy

53 µ was estimated from the temperature dependencies of superﬂuid density in ﬁgure 5.6

[50]. In the ﬁrst regime, at low doping levels, the value of µ/µxy is large and the down-

turn in superﬂuid density is consistent with a transition in entire 2 uc thickness. In

the second regime at higher doping (25K < TC < 50K) µ/µxy is close to 1, therefore

we will observe a change in superﬂuid density at a temperature corresponding to a

transition that takes place in each individual bilayer followed by the abrupt downturn

at TKTB for the entire coupled ﬁlm.

The two points that I wish to draw from these results are that, regardless of

details, 2 uc thick ﬁlms with a wide range of doping levels are consistent among

themselves and their temperature dependence of superﬂuid density points to a 2D

super-to-normal transition mediated by unbinding of V-aV pairs.

5.2.2 Frequency dependence of KTB transition

Another legitimate question to ask is how changing frequency will aﬀect the ap-

parent KTB transition.

−2 Figure 5.8 shows λ (T ) and the σ1 peak for a 2uc Y1−xCaxBa2Cu3O7−δ ﬁlm measured at several frequencies from 0.5kHz to 50kHz, as labeled on the picture

(between data sets the ﬁlm was hold at temperatures below 100K). The intersection

−2 between the dashed line and λ = µ0ωσ2(T ) indicates the theoretical prediction for

the KTB transition in the entire ﬁlm thickness. For the data set measured at 50kHz

the predicted transition temperature is approximately 15 K. As frequency decreases,

both the dissipation peak and the downturn in λ−2 shift to lower temperatures.

This frequency dependence can be explained if we analyze the sheet complex con-

ductivity dσ = d(σ1 − iσ2) (d is the ﬁlm thickness) in relation to the sheet impedance

of the ﬁlm Z = R + iωL ≡ 1/dσ. R represents the dissipative contribution near

54 TC from vortex-antivortex pairs that start to unbind and from doping inhomogene-

ity. L combines contributions from superﬂuid density (superﬂuid density is purely

inductive) and from the inductance produced by vortex-antivortex pairs.

−2 Figure 5.8: The superfluid density λ (left axis) and the real conductivity µ0ωσ1 (right axis) in a 2 uc Y1−xCaxBa2Cu3O7−δ film measured at four different frequencies.

For the same ﬁlm in ﬁgure 5.8 we display the resistance and the impedance at

2 2kHz in the vicinity of the transition in ﬁgure 5.9. Superfuid density 1/λ = µ0ωσ2 and σ1 are deduced from:

1 R − iωL d(σ − iσ ) = = (5.1) 1 2 Z R2 + ω2L2

For T < TC , the vortex contribution is small and R << ωL. All the impedance

2 −1 2 is from the superﬂuid and 1/λ ≈ µ0/d × L . The value 1/λ is independent of

55 Figure 5.9: The real R and imaginary ωL components of the impedance at 2kHz in the same 2 uc Y1−xCaxBa2Cu3O7−δ film from p figure 5.8. The intersection between dR and dωL indicates the position of the σ1 peak measured at the same frequency in the previous figure.

frequency and the low-temperature dependencies from ﬁgure 5.8 overlap. The vortex

-antivortex pairs unbind near TC (starting from around 13 K) and R and ωL start to

grow. Therefore, the changes in λ−2(T ) at diﬀerent frequencies are resulting from the

frequency-dependent contribution of vortices to the impedance. If the frequency is

increased enough, the vortex contribution would become unimportant and the entire

inductive response would be equivalent with the inductance of bare superﬂuid density.

2 The peak in σ1 in ﬁgure 5.8 occurs when R = ωL (the maximum of R/(R +

ω2L2). As the frequency increases, the intersection between R and ωL shifts at higher

temperatures and so does the σ1 peak. The downturn in superﬂuid density measured

56 at 2kHz starts at a temperature around 13K (ﬁgure 5.8) where R is approximately

10% of ωL (ﬁgure 5.9). We can conclude that the temperature where the downturn

is observed is set by the dissipation that starts to become important near TC and

that the frequency-dependend inductance is only a signature of frequency-dependent

vortex dynamics.

5.2.3 TC vs nS(0). Scaling next to a Quantum Critical Point

Researchers have been long trying to ﬁnd a simple, quantitative, and general re-

lation that can relate the main characteristics of superconductors and can indicate

a universal behavior in cuprates. The most known scaling is Uemura’s linear depen-

−2 dence between the transition temperature TC and the superﬂuid density: TC ∝ λ (0).

Even if this phenomenological dependence was not really tested in severely underdoped samples (TC < 50K), Uemura’s scaling had a high impact and motivated the strong phase ﬂuctuations hypothesis in underdoped high-temperature superconduc-

+ tors [4, 6]. An additional phenomenological scaling nS(0) ∝ TC /ρ(TC ) was proposed by Homes et al. [51, 52]. More rigorous scaling relations are deduced in quantum crit-

ical theories that imply the existence of deﬁned dependencies and critical exponents

[23].

2 In figure 5.10 I show my results for TC vs. 1/λ (0) for thick and thin films. TC is defined from the midpoint of the drop in 1/λ2(T ) and error bars extend from the top of the drop to where 1/λ2 is 5% of its value at T = 0 Kelvin. We first note that, at moderate underdoping (transition temperatures from 20 to 50 K), data on 2 unit cell Y1−xCaxBa2Cu3O7−δ films (red circles) overlap with data on thick

Y1−xCaxBa2Cu3O7−δ ﬁlms presented in chapter 4 (green circles) and with data on thick Y Ba2Cu3O7−δ ﬁlms from Zuev et al. [10]. This is an additional proof of the

57 2 Figure 5.10: Scaling of TC with absolute superfluid density ns ∝ 1/λ (0) on a log-log scale for Y1−xCaxBa2Cu3O7−δ 2 unit cells thick (red dots) and 40 uc thick (green dots). Error bars defined in text. For reference we include Uemura’s µSR results on YBCO powders (open black circles), Hc1 measurements on clean YBCO crystals (open orange squares), microwave measurements on ultra-clean YBCO crystals (open blue squares), and data on 20-40 uc YBCO films (black dots). Solid lines illustrate the 2 linear relationship, TC ∝ 1/λ (0), that describes our strongly underdoped ultrathin films and is expected near a 2D quantum critical point. The dashed line illustrates q 2 a square root relationship, TC ∝ 1/λ (0), that describes strongly underdoped 3D samples (crystals and thick films) and is consistent with 3D quantum criticality.

58 high ﬁlm quality at reduced thickness, where grain boundaries or steps in the surface

of the substrate might have had severe eﬀects.

Thick (20 - 40 uc) YBCO ﬁlms and our thick (40 uc) Ca-YBCO ﬁlms agree quan-

2 α titatively with each other at all dopings and both show the scaling TC ∝ [1/λ (0)]

where α ≈ 0.5 (see dashed line in ﬁgure 5.10). We emphasize that this scaling is

apparently insensitive to disorder in our ﬁlms. This is because high-purity YBCO

single crystals (orange and blue squares) exhibit the same scaling [8, 9] despite the

fact that their superﬂuid densities at T = 0 are several times larger than those of

the ﬁlms. We included for reference Uemura’s original µSR results on moderately

underdoped YBCO powders [6], that were at the origin of the ﬂuctuation hypothesis

in underdoped superconductors, for reference (open black circles). Because of the

limited doping interval it seemed reasonable to ﬁt the data with a linear dependence

but, as seen above, this dependence is not conﬁrmed at more severe underdoping.

The most important part of ﬁgure 5.10 is the part related to strong underdoping

where a striking diﬀerence between the 2D and 3D samples emerges. For ultrathin

ﬁlms, TC drops more rapidly with underdoping and the relationship between TC and

2 2 α 1/λ (0) is close to linear: TC ∝ [1/λ (0)] , where α ≈ 1 (see solid lines in ﬁgure

5.10). As a consequence, the superfluid densities of 2uc films exceed not only the values measured in thick films, but also those of clean YBCO crystals with similar transition temperatures.

2 When analyzing the origin of this dependence between TC and 1/λ in two-

dimensional samples, one is tempted to say that it is just the long time expected

Uemura relation where strong 2D thermal phase ﬂuctuations set an upper limit on

the transition temperature below the mean field TC . However, thermal phase fluctuations alone can explain neither the difference in scaling for 2D and 3D samples nor the

59 insensitivity of scaling to disorder between the ﬁlms and crystals, so the mechanism

must be more complicated than Uemura’s prediction. Also, an agreement between

my results and the scaling proposed by Homes et al. [51, 52] is unlikely to be possible.

+ + For nS(0) ∝ TC /ρ(TC ) scaling to ﬁt my data, ρ(TC ) would have to be independent

−1/2 of TC in 2D ﬁlms and proportional with TC in 3D samples, but there is no reason to expect such dependencies.

In order to understand scaling differences and their relation to dimensionality, I will look at theoretical predictions on superfluid density assuming that underdoping destroys superconductivity at a quantum critical point (QCP) [53]. There is a clear distinction between the quantum phase transition (zero temperature phase transitions) and the finite-temperature classical phase transitions. The quantum phase transitions are not produced by changing the temperature (or other external perturbation such as pressure or magnetic field), but they are the result of some fundamental changes in the ground state Hamiltonian of the system. In cuprates we assume a quantum phase transition between the underdoped superconducting phase to the non-superconducting state produced by varying the carrier concentration x.

The quantum fluctuations effects are important at the microscopical scale next to a zero temperature quantum critical point, but they don’t have any influence on the critical behavior next to the transition temperature [54]. These effects can be observed at low but finite temperatures in the vicinity of QCP, and result in anomalous physical properties and scaling relations. The finite temperature behavior and the scaling relation next to a QCP in D dimensions can be treated as a classical system by introducing an imaginary time at zero temperature. This imaginary time is equivalent to an extra spatial dimension resulting in a classical system with D + 1 dimensions [55].

60 As a superconductor approaches the QCP at the critical doping x = xC , the

zν transition temperature vanishes as TC ∝ δ where δ = |x − xc| and z and ν represent the quantum dynamical and the correlation length exponents respectively. Josephson scaling [56] near a QCP implies that the T = 0 superﬂuid density vanishes as nS(0) ∝

δ(z+D−2)ν where D is the spatial dimensionality and δ is the hole doping. If I eliminate

z/(z+D−2) δ from these two relations, I obtain the scaling relationship TC ∝ nS(0) between the two quantities measured in my experiment [23, 22]. This relation is independent of ν and the value of quantum z is diﬀerent from the critical dynamic exponent measured next to TC in a ﬁnite temperature phase transition.

z/(z+1) In D = 3 dimensions, theory ﬁnds TC ∝ nS(0) . If z lies between 1 and 2,

α then we expect TC ∝ nS(0) where α is between 1/2 and 2/3. This is consistent with all of the data on 3D samples, films and crystals. Turning now to the 2 unit cell Y1−xCaxBa2Cu3O7−δ films, in D = 2 dimensions, theory predicts linearity, TC ∝ nS(0), independent of the value of z. These scaling dependencies that we observe in Y1−xCaxBa2Cu3O7−δ are a strong argument in favor of the existence of quantum phase fluctuations near a quantum critical point, fluctuations that are responsible for the suppression of superconductivity at underdoping.

The TC ∝ nS relation in 2D samples is not the only evidence supporting the existence of a quantum critical point. The universality applies to the entire temperature dependence of superﬂuid density and thus we expect:

−2 nS(T ) λ (T ) T ∝ −2 = f( ) (5.2) nS(0) λ (0) TC where f is a universal function [53]. It is therefore not surprising that underdoped 2

unit cell thick Y1−xCaxBa2Cu3O7−δ ﬁlms that show TC ∝ nS also satisfy the equation

5.2. Figure 5.11 shows that, on normalized scales, the temperature dependence of su-

−2 −2 perﬂuid density λ (T )/λ (0) vs. T/TC of four severely underdoped ﬁlms (transition

61 Figure 5.11: Normalized temperature dependencies of superfluid density in four severely underdoped 2uc Y1−xCaxBa2Cu3O7−δ films (Tc = 26 K blue, 15 K green, 12 2 K pink, 5.3 K red) that fit on the linear dependence TC ∝ 1/λ (0). The universal tem- −2 −2 perature dependence of superfluid density nS(T )/nS(0) ∝ λ (T )/λ (0) = f(T/TC ) strongly supports the existence of a Quantum Critical Point when superconductivity is lost with underdoping.

temperatures between 26K and 5.3K) overlap as a consequence of sharing the same universal behavior.

Given the overwhelming agreement between theoretical predictions and the experimental results presented above, I conclude that the superconductor-to-normal transition in severely underdoped high-temperature superconductors is the result of

62 quantum phase ﬂuctuations that exist in the proximity of a Quantum Critical Point.

The normal state above TC is most likely a phase disordered superconductor [57].

Figure 5.12: On normalized scales we compare five ultrathin 2 uc Y0.7Ca0.3Ba2Cu3O7−δ films (TC = 54K orange , 36 K red, 26 K blue, 15 K green, 12 K pink), an underdoped 20 uc Y0.7Ca0.3Ba2Cu3O7−δ film (black circles) 2 and the quadratic dependence [1 − (T/TC0) ] in red. The normalising factor for the temperature is the quadratic extrapolation to zero of the low temperature of superfluid density, equivalent to a mean-field type TC0 transition temperature.

5.2.4 Temperature dependence of 1/λ2: 2D vs. 3D

Diﬀerences between thick and thin ﬁlms are naturally present in the T-dependence

−2 −2 −2 of λ (T ). Figure 5.2.4 shows normalized data, λ (T )/λ (0) vs. T/TC0, for one

63 −2 underdoped 20 uc Y0.7Ca0.3Ba2Cu3O7−δ ﬁlm (black circles, TC = 34K, λ (0) =

−2 −2 3.8µm ) and several 2uc ﬁlms at various doping levels. λ (0) and TC0 are ob-

tained from quadratic ﬁts to low-T data. TCO provides a good estimate of mean-ﬁeld

behavior when ﬂuctuations are ignored.

The symbol on each curve indicates where the KTB theory predicts a transition,

assuming that the film fluctuates as a single 2D entity. For the thickest film, the

onset of the transition occurs at a small superﬂuid density, λ−2(T )/λ−2(0) ≈ 0.05.

As seen in section 4.2.2, interlayer coupling in thick ﬁlms is enough to make them

3D even when strongly underdoped, hence TC ≈ TC0. For the most doped ultrathin

film in the figure (TC = 50K), the onset of the transition marked by the change in the slope of superfluid density occurs at λ−2(T )/λ−2(0) ≈ 0.36. As doping decreases,

the transition moves to higher superﬂuid density so that at severe underdoping TC is

−2 established (controlled) by the λ (0) rather than the ”mean-ﬁeld” TC0. For the most

underdoped ultrathin ﬁlms shown in the ﬁgure, the onset of the transition occurs at

−2 −2 a high superﬂuid density, λ (T )/λ (0) ≈ 0.70, so TC lies signiﬁcantly below TC0.

It is not surprising that data sets overlap for these ﬁlms since quantum ﬂuctuations

−2 −2 are dominant and λ (T )/λ (0) = f(T/TC ), as seen in ﬁgure 5.11. I emphasize

that there is no critical thickness that separates thick ﬁlms from thin ﬁlms. As far as

ﬂuctuations are concerned, all ﬁlms behave as 2D superconductors with a thickness

equal to the ﬁlm thickness.

There is a remarkable agreement between all the ﬁlms and the quadratic ﬁt in

the low-temperature dependence of superﬂuid density. The simplest interpretation

for this behavior is that it originates from strong impurity scattering in a d-wave

superconductor [58]. However, since quantum ﬂuctuations are important, they should

also have a contribution in the low temperature dependence.

64 Classical phase ﬂuctuations are known to produce a linear suppression in the su-

perﬂuid density by a factor proportional to kBTL(0)/4¯hRQ [59, 60], where RQ =

2 2 h/¯ 4e ≈ 1000Ω is the quantum resistance and L(0) = µ0λ (0)/d. The quantum

suppression of thermal phase ﬂuctuations was deduced by Lemberger et al. in anal-

ogy with resistively shunted Josephson junctions arrays [61]. In their model quan-

tum eﬀects become important when kBT/h¯ is smaller than the frequency of the ﬁlm

RN /L(0). The combined eﬀect should suppress the superﬂuid density as follows:

n (T ) 1 k T k T S ≈ 1 − B × B (5.3) nS(0) 4 hR¯ Q/L(0) hR¯ N /L(0)

where RN is obtained from the normal state resistivity RN = ρN /d. The energy

scale in the denominator in equation 5.3 sets an upper limit on the transition tem-

perature (Tqf ) when thermal phase ﬂuctuations are suppressed by quantum eﬀects.

√ −2 −2 TC (K) λ (µm ) Tqf (K) Tqf (K)/ 2π

26 2.8 115 45.89

15 1.4 57 22.74

12 1.15 47 18.75

5.3 0.34 14 5.58

Table 5.1: The upper limit on the transition temperatures for several 2 unit cell Y1−xCaxBa2Cu3O7−δ films when thermal phase fluctuations effects are suppressed by quantum fluctuations.

In table 5.1 I estimate this temperature for several severely underdoped 2 uc

Y1−xCaxBa2Cu3O7−δ using the superﬂuid densities and transition temperatures from

65 ﬁgure 5.11 and approximating the value of normal state resistivity ρN ≈ 500µΩcm.

The values of Tqf are roughly in the order of magnitude and are projecting a similar phase diagram with experiments that proved the persistence of vorticies and ﬂuc- tuating superconductivity above TC [62, 63] but, at this point I don’t quantitatively understand these results. If a prefactor of 2π is added to the numerator in relation 5.3, in this case, I obtain an upper limit on the transition temperature in agreement with

TC0 from figure . Yet, a more detailed investigation of possible quantum effects in the temperature dependence of superfluid density in high temperature superconductors has to be done.

66 CHAPTER 6

SUPERFLUID DENSITY MEASUREMENTS IN La2−xSrxCuO4 FILMS

La2−xSrxCuO4−δ (LSCO) is one of the few cuprate superconductors where we

should be able to map the entire phase diagram from the superconductor to non-

superconductor transition in the underdoped region until the total suppression of

superconductivity by disorder in the overdoped region. Since LSCO has a CuO

monolayer crystallographic structure, ﬂuctuations can be studied in this compound

in a more anisotropic superconductor than Y Ba2Cu3O7−δ. However, in practice, it

is extremely diﬃcult to obtain homogeneous underdoped samples. The LSCO ﬁlms

are less homogeneous than Y1−xCaxBa2Cu3O7−δ ﬁlms when TC < 20K, but it was

possible to obtain homogeneous ﬁlms at doping levels on the overdoped side. When

trying to grow them two-dimensionally, La2−xSrxCuO4−δ ﬁlms with reduced thickness

are superconducting only when their doping is close to optimal.

Starting from La2CuO4, superconducting ﬁlms are obtained by tuning the con-

centration of Sr from x = 0.06 to around x = 0.30. When Sr+2 replaces La+3 in

+2 La2−xSrxCuO4, an electron is extracted from the CuO2 planes for any Sr ion and, as a result, the planes are doped with holes. In addition, doping can be varied by the oxygen vacancies that are forming in the CuO2 planes and are compensating for the

Sr doping.

67 I measured superfluid density in LSCO films to study the doping dependence of this quantity and to confirm if, similar to my findings in Y1−xCaxBa2Cu3O7−δ, a

Quantum Critical Point (QCP) and quantum phase ﬂuctuations dominate the physical properties next to the superconductor to non-superconductor transition at low doping in an additional family of high-temperature superconductors. Another goal was to investigate the overdoped end of the superconducting phase diagram and to see if the two-dimensional transition observed in two unit-cell Y1−xCaxBa2Cu3O7−δ

ﬁlms exists in ultrathin LSCO ﬁlms.

Figure 6.1: Temperature dependence of ab plane resistivity ρ(T ) in La2−xSrxCuO4 films (all samples in the left figure, overdoped samples in the right figure). With continuous and dotted lines we identify films on underdoped and overdoped sides of the phase diagram. The red continuous line represents the optimal doping x = 0.15. Sr concentrations are labeled on the figures.

In the past, superﬂuid density measurements were used to study the carrier concentration dependence in bulk LSCO [64] and to identify the symmetry of the order

68 parameter in LSCO ﬁlms [65]. In the low-temperature superﬂuid density, it was

possible to distinguish between the quadratic λ−2(T ) seen in LSCO ﬁlms, consistent

with d-wave symmetry when disorder is important and the exponential dependence

characterizing an isotropic s-wave superconductor.

The LSCO ﬁlms were produced by molecular-beam epitaxy (MBE) on 10mm ×

10mm × 0.35mm LaSrAlO4 (001) substrates by our collaborators at Nippon Tele-

graph and Telephone Corporation, Basic Research Laboratories (NTT) in Tokyo.

Details of their ﬁlm growth method can be found in Naito et al. [66, 67]. Induced

compressive strains into the ﬁlms due to lattice mismatch between ﬁlms and sub-

strates (a = 3.754A˚ for LaSrAlO4 and a = 3.777A˚ for bulk LSCO) make it possible to increase the maximum transition temperature to 44K, which is about 7K above the maximum transition temperature in La2−xSrxCuO4 crystals.

6.1 Evolution of transition temperature and superﬂuid density with doping

Two samples were grown simultaneously at each Sr concentration, one to measure electrical resistivity and one to measure superﬂuid density. The values of doping are nominal and they are set by deposition conditions. The electrical resistivity ρab was measured with a standard four-probe method at NTT. The results are displayed in

ﬁgure 6.1. The underdoped (continuous lines) and the overdoped (dotted lines) are separated by the continuous red line representing the optimal doped ﬁlm.

The resistive transitions are reasonably sharp and we observe increasing transition temperatures up to optimal doping at x = 0.15. The normal state resistivity ρ(50K) decreases monotonically when x increases, even when TC starts to decrease. This is a conﬁrmation that samples above x ≥ 0.18 (ﬁgure 6.1 right) are in fact overdoped.

69 The normal state resistivity values in table 6.1 are comparable with values measured

in high-quality untwinned LSCO crystals [68].

Figure 6.2: Underdoped to optimal doped La2−xSrxCuO4 films. Temperature dependence of superfluid density λ−2(T ) (dark curves) and the fluctuation peaks in real conductivity σ1(T ) (red curves). The Sr concentrations are: (a) x = 0.06, (b) x = 0.09, (c) x = 0.12, (d) x = 0.15. Dashed lines represent quadratic fits to the low-temperature dependence and the intersection with the dotted lines indicate where a 2D transition should exist in individual decoupled CuO2 layers.

In figure 6.2 and figure 6.3 the superfluid density λ−2 and the real conductivity

σ1 are shown for all the ﬁlms with nominal doping ranging between x = 0.06 −

0.30. The thickness of each film is labeled on the figure. Blue dashed lines are quadratic fits over extended low-temperature intervals consistent with a disordered d-wave superconductor. The width of the transition is equivalent with the width of

70 Figure 6.3: Overdoped La2−xSrxCuO4 films. Temperature dependence of superfluid density (dark curves) and the fluctuation peaks in real conductivity σ1(T ) (red curves) for films with nominal Sr concentrations: (a) x = 0.18, (b) x = 0.21, (c) x = 0.24, 0.27 and 0.30, (d) x = 0.27. Dashed lines are quadratic fits to the low-temperature dependence, and in (d) the intersection with the dotted line indicates where the 2D transition takes place in the entire thickness of the film.

σ1(T ) dissipation peak. The underdoped La2−xSrxCuO4 films (TC < 25K) are not as homogeneous as the underdoped Y1−xCaxBa2Cu3O7−δ films studied in the previous chapters. In figure 6.2a we observe that, at x = 0.06 doping, the 45nm thick film is inhomogeneous and has two transitions: the first transition takes place between

14−17K and the second transition happens between 5−8K. A corresponding feature is observed in λ−2 at the second transition temperature as if the ﬁlm had two layers with diﬀerent transition temperatures. At higher doping levels, all the other samples

71 have only one peak, with some variations in the width of the transition due to small Sr

variations in doping. It is interesting to note the behavior of the overdoped samples

with x > 0.21 where transitions are suddenly extremely sharp, ∆TC ≤ 1K (ﬁgure

6.3c and d).

The onset of superfluid density coincides fairly well with the temperature where resistivity becomes zero inside the films, except for the film with x = 0.09 in figure

−2 6.2b. In this ﬁlm, the diﬀerence between the resistive TC and the TC from λ is the

result of a percolative transition and λ−2 drops at 30K, but some superconductivity still survives up to 40K. This behavior is due to the somehow granular nature of the ﬁlm, similar to the observations of Leemann et al. in Y Ba2Cu3O7 [69]. When

the weak link among the grains becomes normal, the ability of the ﬁlm to screen

the probing magnetic ﬁeld decreases dramatically, resulting in a drop in superﬂuid

density. I exclude the interpretation of this drop as a 2D transition in individual

decoupled CuO2 layers since the corresponding KTB line (the dashed line in ﬁgure

6.2b) intersects the superﬂuid density before the transition and some small diamagnetic screening persists up to 40K.

I tested further the hypotheses of strong ﬂuctuations and of the possible existence of a KTB transition in the rest of the La2−xSrxCuO4 ﬁlms 450A˚ thick. Assuming that

−2 the CuO2 layers are decoupled, the only ﬁlm that showed any change in λ in the

critical region is the x = 0.15 sample in ﬁgure 6.2d. The intersection between λ−2 and

2 2 ˚ the KTB line 1/λ (TKTB) = 8πµ0kBT2D/dΦ0 (d = 6.6A the distance between CuO2

monolayers and Φ0 = 2πh¯2e is the ﬂux quantum) is at a slightly lower temperature

than the change in temperature dependence. However, since this behavior is not

reproducible in other underdoped to optimal doped samples, I consider it to be a mere

inhomogeneity eﬀect. The transition is also missing in overdoped samples, therefore

72 La1−xSrxCuO4 thick ﬁlms behave similarly to Y1−xCaxBa2Cu3O7 thick samples as

far as 2D thermal ﬂuctuations are concerned.

Moreover, when it comes to the most overdoped sample with the lowest TC and

−2 −2 the lowest value of absolute superﬂuid density (TC = 4K and λ (0) = 0.15µm ) in

ﬁgure 6.3 d, it is possible to observe a sharp drop in λ−2(T ) that is consistent with a KTB transition occuring in the entire thickness of the ﬁlm. This suggests that, at least when it comes to overdoping, the CuO2 layers are coupled. The same conclusion was reached when performing microwave-conductivity measurements in underdoped

La2−xSrxCuO4, x = 0.07 − 0.14, where the effective thickness for the KTB transition is approximately half of the entire film thickness [70]. I present more arguments in favor of the 2D transition in coupled CuO2 layers in the next section where I discuss superfluid density measurements in a 4.5nm thick La1.85Sr0.15CuO4 film.

The low-temperature dependence of all ﬁlms regardless of their doping level can be

ﬁtted with a quadratic over an extended interval. At higher temperatures, the most homogeneous samples show a crossover to a linear mean-ﬁeld type dependence up to

TC . This behavior is consistent with the strong scattering in a d-wave superconductor

[58] previously observed in LSCO ﬁlms [65]. Above x = 0.21, all samples develop an upward curvature. This type of temperature dependence was already seen in overdoped n-type superconductors [71, 72] and, since it occurs in the samples with the sharpest ∆TC intervals, it cannot be a result of inhomogeneity. This change in curvature takes place in a temperature range where superﬂuid density is related to the gap ∆(T ) and might indicate a change in the temperature dependence of the gap.

Others have speculated that this behavior results from a change in the symmetry of the order parameter by the opening of a second superconducting gap at overdoping

[73].

73 Figure 6.4: The dependence of superﬂuid density λ−2(0) (red) and of transition temperature TC (black) on Sr concentration. Error bars in TC described in text.

−2 The dependence of superfluid density λ (0) and transition temperature TC on the nominal doping is illustrated in figure 6.4. The error bars for the transition temperature are limited by the resistive TC and the lower end of the fluctuation peak in σ1. The transition temperature increases very fast up to 10% of its maximum value of 43.8K(x = 0.15) in a very short doping range from 0.06 to 0.12. Absolute values of

λ−2(0) are obtained from quadratic extrapolation of the low-temperature dependence to zero Kelvin. The maximum superfluid densities around λ−2(0) ≈ 20µm−2 are measured in x = 0.18 and x = 0.21 samples and these values are only 10% lower than in previous measurements in aligned LSCO powders [64]. This confirms the high-quality of the samples and the reduced effect of grain boundaries within them.

74 These ﬁlms with the higher values of superﬂuid density are also the samples that

display linear normal state resistivities ρab(T ) ∝ T .

In the overdoped region, the last two samples with the highest Sr concentrations

have a slightly higher TC since the actual doping might be lower as a result of oxygen

vacancies. For doping above x = 0.20 up to x = 0.25 the supeﬂuid density and the

transition temperature are both suppressed, unlike the case of powders where TC was

suppressed but λ−2 saturates at approximately 25µm−2 [64]. It is uncertain what suppresses the superﬂuid density while the carrier density increases in the overdoped side of the phase diagram. A similar result was observed in overdoped T l2Ba2CuO6+δ

[74, 75]. I will return to this point below.

−2 In ﬁgure 6.5, I evaluate the correlation between superﬂuid density λ (0) and TC in

La1−xSrxCuO4 (blue plots) against my previous results in Y1−xCaxBa2Cu3O7−δ from

ﬁgure 5.10. The blue arrow indicates the direction of the doping increase. We can

−2 notice that La1−xSrxCuO4 ﬁlms have higher λ (0) values than Y1−xCaxBa2Cu3O7−δ

ﬁlms at the same transition temperatures. In the underdoped region, the dependence

−2 of TC on λ (0) is not entirely certain since data with TC < 15K is missing, but

the present samples agree with the underdoped Y1−xCaxBa2Cu3O7−δ ﬁlms and the q −2 sublinear dependence TC ∝ λ (0) (dashed line in ﬁgure 6.5).

−2 After a maximum in superﬂuid density, both λ (0) and TC decrease in the over-

doped region. The origin of these decreases is an important issue to be resolved. TC vs λ−2 follows a path approximately parallel with the underdoped sublinear dependence. Due to the similarity with thick Y1−xCaxBa2Cu3O7−δ ﬁlms, we can speculate that LSCO has two quantum critical points in the phase diagram at both extreme doping levels, x = 0.05 and x = 0.30. More 3D and 2D samples need to be measured in order to remove any uncertainty on this particular scaling dependence.

75 −2 Figure 6.5: TC versus λ (0) for La1−xSrxCuO4 (blue circles) is compared with the same dependence for Y1−xCaxBa2Cu3O7−δ thin films in figure 5.10. Error bars represent the full width of the transition, from the resistive TC to the lower end of the fluctuations peak, and the blue arrow indicates the increase of doping from underdoping to overdoping.

76 −2 2 x TC (ρ = 0) TC (ns = 0) λ (0) ρab(50K) ρab(50K)/λ (0)µ0

(K) (K) (µm−2) (µΩcm) (K)

0.06 17.5 16.00 1.30 590 46.97

0.06 24 23.00 4.50 377 103.9

0.09 39.7 33 6.0 170 62.46

0.09 40 38 10.50 160 102.89

0.12 40.1 38.75 12.54 140 107.52

0.15 43.8 42 17.40 90 95.90

0.18 41 38 21.46 54 70.97

0.21 33 32 20.30 48 59.6

0.24 19 18.5 11.10 37 25.15

0.27 21 20 6.8 35 14.57

0.30 9 8.5 1.6 28 2.74

0.27 4 4 0.15 31 0.28

Table 6.1: Properties of La2−xSrxCuO4−δ ﬁlms as a function of doping. x is the nominal Sr doping, TC (ρ = 0) sets the upper limit on the transition temperature, ρab(50K) is the ab-plane resistivity just above the onset of transition, and ρab(50K)× −2 −1 λ (0)/µ0 is related to the scattering rate τ .

It is possible that disorder plays a role in the simultaneous decrease of TC and

λ−2(0). In an attempt to relate λ−2(0) to normal state resistivity ρ(50K) and to obtain a quasiparticle scattering rate τ at TC , in a simple Drude model:

−2 2 ∗ λ (0) nS(0)e m 1/τ nS(0) ρ(50K) × ≈ ∗ × 2 ≈ × 1/τ (6.1) µ0 m nne nn 77 where nS is the superﬂuid density and nn is the hole density. The hole scattering rates in the approximation nS/nn ≈ 1 can be seen in the last column in Table 6.1, together with a summary of all measured parameters for all the ﬁlms. For most of the samples, the scattering rate in units of temperature is roughly 2 × TC , which is surprisingly large. In the overdoped region scattering rates become unphysically

−1 2 ∗ small. A larger scattering rate is obtained directly from τ ≈ ρ(TC +)nne /m (for

−1 2 x = 30, we obtain τ ≈ 40K versus ρab(50K)/λ (0)µ0 ≈ 2K) and scattering rates at

least three order of magnitude larger are generally observed in infrared measurements

[76]. The only plausible interpretation is that only a small fraction of the hole carriers

contribute to the superﬂuid density nS/nn << 1 and the pair breaking eﬀects are

important.

6.2 2D ﬂuctuations in ultrathin La1.85Sr0.15CuO4

The presence of ﬂuctuations is unavoidable in two-dimensional systems and, as

−2 seen in 2uc thick Y1−xCaxBa2Cu3O7−δ ﬁlms, a drop in λ (0) will be observed in

highly homogeneous samples. In a previous attempt by Rufenacht et al.[19] to mea-

sure superﬂuid density in a 2D La1−xSrxCuO4 ﬁlm only 26.6A˚ thick, TC = 8K and

x = 0.10, the superﬂuid density was substantially reduced by disorder and the au-

thors observed no drop in the superﬂuid density contrary to general expectations for

a 2D system. They explained this paradox by arguing that disorder smoothed the

transition. The mechanism for smoothing the transition is unclear.

−2 In ﬁgure 6.6a λ (T ) and σ1(T ) are measured in an optimal doped La1.85Sr0.15CuO4

ﬁlm that is 45A˚ thick. The resistive TC ≈ 42K coincides with the onset of super-

ﬂuid density and the ﬁlm shows better doping homogeneity than thicker samples.

In fact, some of the width of σ1 peak can be due to the onset of two-dimensional

78 fluctuations. The low-temperature dependence of λ−2(T ) fits with the quadratic dependence previously observed in thicker films but, if we extrapolate this dependence at zero Kelvin, the absolute value λ−2(0) ≈ 9.25µm−2 is approximately half of the measured superfluid density in the optimal doped 450A˚ film with the same transition doping/transition temperature (λ−2(0) ≈ 17.4µm−2).

I believe that the explanation for the reduced superfluid density is that the effective superconducting film is only a fraction of the entire thickness, which is a plausible explanation since La2−xSrxCuO4 films are grown directly on the substrate and their surface is not protected. This conclusion is also supported by the resistivity measurements that show a value of normal state resistivity ρ(50K) = 120µΩcm twice as big when compared with the 450A˚ film. Corrected superfluid density, assuming that both 45A˚ and 450A˚ have the same absolute value λ−2(0), is plotted in figure 6.6b.

If we analyze λ−2 in the critical region, we discover an abrupt change in its slope around 40K similar with my ﬁndings in 2uc Y1−xCaxBa2Cu3O7−δ, and this change is consistent with a two-dimensional vortex-antivortex unbinding transition. Dashed

2 2 lines in figure 6.6a and b represent the KTB line 1/λ (T2D) = 8πµ0kBT/dΦ0 when the CuO2 layers are coupled in the entire thickness of the film (red dashed line) and when the CuO2 layers are decoupled (d = 6.6A˚ - black dashed line). Independent of my previous supposition of the effective superconducting thickness of the film, in both figures the intersection of the red line with 1/λ2 occurs at a lower temperature than the drop and the intersection with the black dashed line is close to the middle of the drop, just like in 2D Y1−xCaxBa2Cu3O7−δ films.

When rescaling λ−2 to a reduced eﬀective thickness, the intersection between

−2 −2 −2 λ (T ) and λ (T2D) for decoupled CuO2 layers is closer to the downturn in λ (T ).

79 −2 Figure 6.6: (a) Dependence of superfluid density λ and real conductivity σ1 versus temperature for an La1.85Sr0.15CuO4 film only 45A˚ thick. The dashed lines are corresponding KTB lines for one decoupled CuO2 layer, 6.6A˚, (black) and for the entire thickness of the film (red). (b) Corrected superfluid density based on the assumption that only a fraction of the film is superconducting and that the absolute value should be the same as in thicker films. The downturn in λ−2(T ) at approximate 40 K is consistent with our previous results in 2 uc Y1−xCaxBa2Cu3O7−δ films and with a 2D transition that takes place in the entire thickness.

This takes place because the measured superﬂuid density depends on the thickness

of the ﬁlm and the KTB transition describes a universal behavior in 2D system.

It is possible to bring the intersection of λ−2 with the KTB line for decoupled

CuO2 layers at the onset of the downturn in superﬂuid density if we consider that

80 even a smaller fraction of the 45A˚ is superconducting. In this case, the superfluid density in the ultrathin film would be 50% higher than in thick films, which would be highly suprising. It is most likely that the two-dimensional transition takes place in the entire film and therefore the coupling between CuO2 should be taken into account.

Figure 6.7: Frequency and current variation effects in the real and imaginary compo- −2 nents of the impedance R and ωL (top figure) and the corresponding λ (T ) = µ0ωσ2 and µ0ωσ1(T ) (bottom figure). d is the thickness of the film. 5 mV at 50kHz for the dotted line, 100 mV at 50kHz for the continuous line, 5 mV at 10kHz for the dashed line.

81 6.3 Frequency and current dependence of the 2D transition in ultrathin La1.85Sr0.15CuO4

We can analyze the critical behavior next to T2D by looking at the sheet impedance of the ﬁlm Z = R + iωL. In ﬁgure 6.7 top we show the temperature dependence of

2 2 2 sheet resistance dR = σ1/(σ1 + σ2) and sheet impedance impedance dωL = σ2/(σ1 +

2 σ2) in three diﬀerent measurements: amplitude of drive voltage 5 mV at 50kHz, 100mV at 50 kHz and, 5mV at 10kHz (d =thickness of the ﬁlm). The corresponding

−2 λ (T ) = µ0ωσ2 and µ0ωσ1(T ) are displayed in ﬁgure 6.7 bottom. The resistance is

negligible until the temperature is close to the transition, then it starts to grow rapidly

due to the increasing number of vortex-antivortex pairs. The diﬀerence between ωL

measured at diﬀerent frequencies is exactly the frequency ratio.

Similar to 2uc Y1−xCaxBa2Cu3O7−δ ﬁlms, when frequency decreases from 50 kHz

to 10 kHz (dotted lines vs. dashed lines), the intersection between R and ωL shifts

from 42K to approximately 38 K and so does the σ1 peak.

If we analyze the current dependence of the impedance, we observe that the vortex

contribution is signiﬁcantly increased when the current in the drive coil is increased

by a factor of 20 (continuous lines vs. dotted lines). As a consequence the resistivity

increases almost one order of magnitude. The vortex contribution to impedance is

considered to be of the same order of magnitude with the vortex contribution to

the resistance [77]. However, we do not observe any change in ωL between the two

currents, so I consider this contribution not to be signiﬁcant until the sudden change

in the slope at approximately 42K. By neglecting the dissipative eﬀects in resistance

before the transition we can obtain a better estimate of λ−2 directly from L−1.

−2 −1 In ﬁgure 6.8 I compare the superﬂuid density λ = µ0ωσ2 with L µ0/d. The

diﬀerence between them in the critical region is striking and indicates that the drop

82 −2 −2 −1 Figure 6.8: λ (T ) = µ0ωσ2 (black curve) and λ (T ) = L µ0/d based on the assumption that the inductive response is equivalent to the bare inductance of the superﬂuid density. d is the thickness of the ﬁlm.

−2 we observe in λ = µ0ωσ2 (black line) is partially due to dissipative effects when the vortex contribution is important. If these effects are subtracted (green line), the downturn in superfluid density occurs at a higher temperature slightly below the intersection with the KTB line for the entire film thickness. This result is consistent with previous observation of KTB transition in superfluid helium films [47, 46] and

−2 suggests that the inconsistency between the TKTB line and the downturn in λ (T ) that starts at a lower temperature is related to the vortex-antivortex contribution to conductivity.

83 CHAPTER 7

CONCLUSIONS

Producing homogenous samples is the biggest experimental challenge in study-

ing strongly underdoped cuprate superconductors. I have produced underdoped

Y Ba2Cu3O7−δ ﬁlms by reducing the oxygen concentration in the CuO chain layers of

this compound close to zero and by replacing 20% to 30% of Y 3+ cations with Ca2+.

These films have transition temperatures near the non-superconducting to superconducting transition at critical doping and they are more stable since no chain oxygen is left to disorder them when δ ≈ 6. I succeded in growing Y1−xCaxBa2Cu3O7−δ films as thin as only 2 unit cells, films that allowed the unprecedented study of the effects of phase fluctuations in real two-dimensional underdoped superconductors (TC as low as 3K). I measured the temperature dependence of superfluid density in these films by using a two-coil mutual inductance method.

The temperature dependence of superfluid density and the absolute value of superfluid density in thick Y1−xCaxBa2Cu3O7−δ films agree quantitatively with results from pure Y Ba2Cu3O7−δ films, which indicates that scattering from Ca is not particularly important. No signature of 2D or 3D-XY criticality was observed in thick

films, therefore I conclude that fluctuation effects are much weaker than expected and that interlayer coupling should be taken into account. In 2 uc thick films with a wide range of doping levels, the temperature dependence of superfluid density shows an

84 abrupt downturn at TC that points to a 2D super-to-normal transition mediated by the unbinding of vortex - antivortex pairs in the entire thickness of the ﬁlms and con-

ﬁrms that there is nothing about cuprates that invalidates the notion that vortex-pair unbinding mediates the transition in 2D.

If we analyze the dependence of TC on the absolute value of superﬂuid density, the most important result is obtained at severe underdoping where we observe a diﬀerence

2 0.3 between 3D thicker samples showing TC ∝ [1/λ (0)] and 2D ultrathin ﬁlms showing

2 a linear dependence TC ∝ 1/λ (0). This scaling relation dependent on dimensionality and independent of disorder is explained by the existence of a quantum critical point and quantum phase ﬂuctuations that destroy superconductivity with underdoping.

This hypothesis is further supported by the universal temperature dependence between normalized superfluid densities and temperatures. Additional measurements in underdoped-to-optimally doped La2−xSrxCuO4 films point out that these superconductors behave similarly as far as phase fluctuations are concerned. Further measurements of superfluid density in the most anisotropic cuprate (Bi2Sr2CaCu2O8+δ)

will clarify if this behavior is universal for all high-temperature superconductors.

The overdoped side of the phase diagram was studied in La2−xSrxCuO4 ﬁlms with

increased homogeneity. Above x = 0.21, all samples developed an upward curvature in

the temperature dependence of superﬂuid density that points toward a change in the

temperature dependence of the gap. The density of carriers increases with overdoping

while the absolute values of superﬂuid density are decreasing, so I conclude that only

a small fraction of the carriers contribute to the superﬂuid density and the overdoped

region is dominated by microscopic phase separations and pair breaking eﬀects.

85 APPENDIX A

Appendix: Y1−xCaxBa2Cu3O7−δ ﬁlms summary

−2 Doping Sample Thickness Annealing TC λ o −2 x u.c. in O2 at 600 C K µm 0.30 CYBCO110405 80 700 torr/24h 72-80 24

CYBCO110905 80 30 torr/2h 67 17

CYBCO111605 40 1 torr/1.5h 64.5 14

CYBCO110906 20 20 torr/1 min 33 3.8 0.15 CYBCO111805 40 1 torr/1.5h - -

CYBCO112105 40 8 torr/1.5h 30 4.2 0.168 CYBCO112805 40 140mtorr/1.5 71 17

CYBCO112905 40 30mtorr/5min 56 11.4

CYBCO113005 40 10mtorr/1min 8 0.24 0.20 CYBCO120505 40 10mtorr/1min - -

CYBCO120805 40 140mtorr/5min at 500C 48 11.8

CYBCO120905 8 300mtorr/5min at 500C 10-20 1.1

PYPBCO013106 4 1torr/5min at 500C 5-25 1

Table A.1: Y1−xCaxBa2Cu3O7−δ ﬁlms: deposition parameters and basic properties.

86 Sample Thickness Annealing TC u.c. in O2 K PCaYP020106 10/2/10 1 atm/5min @600oC 15

PCaYP021406 10/2/15 500 torr/5min @600oC 3

PCaYP021506 10/2/15 700 torr/5min @600oC -

PCaYP021606 10/2/15 700 torr/5min @600oC 10-20

PCaYP022406 10/2/15 1 atm/5min @500oC -

PCaYP022606 10/2/15 1 atm/5min @500oC 5

PCaYP031806 10/2/15 3 torr /5min @500oC -

PCaYP031906 10/2/15 3 torr /5min @500oC 6-14

PCaYP032606 10/2/10 1 torr /5min @500oC -

PCaYP032706 10/2/10 1 torr /5min @500oC -

PCaYP040406a 10/2/10 200 torr /5min @500oC 40

PCaYP040406b 10/2/10 200 torr /5min @500oC 47-50

PCaYP042606 10/1/10 200 torr /5min @500oC -

PCaYP042706 10/2/10/3 25 torr /5min @500oC 36

PCaYP050106 0/1/10/3 10 torr /5min @500oC -

PCaYP050206 2/1/10/3 25 torr /5min @500oC 4-12

PCaYP050806 2/1/10/3 30 torr /5min @500oC -

PCaYP080906 2/2/6/3 700 torr/ no dwell -

PCaYP081506 2/2/6/3 1 atm/5min @500oC -

Table A.2: Deposition parameters and transition temperatures for Y1−xCaxBa2Cu3O7−δ ﬁlms 2 unit cell thick, grown between protective P rBa2Cu3O7−δ layers and with an optional 3uc amorphous P rBa2Cu3O7−δ cap layer. 87 Sample Thickness Annealing TC u.c. in O2 K PCaYP081806 3/2/10/3 6 torr/10min @600oC 28

PCaYP082206 3/2/10/3 1 torr/10min @600oC -

PCaYP082306 3/2/10/3 3 torr/10min @600oC 12

PCaYP091106 3/2/10/3 2.5 torr/10min @600oC 11-15

PCaYP091506 3/2/10/3 2 torr/10min @600oC 3

PCaYP092006 3/2/10/3 2.4 torr/10min @600oC 28

PCaYP092106 3/2/10/3 2.2 torr/10min @600oC 15

PCaYP102606 3/2/10/3 2 torr/10min @600oC 3

PCaYP112706 3/2/10/3 2.2 torr/10min @600oC -

PCaYP120106 3/2/10/3 1 atm/10min @600oC 54

PCaYP091707 3/2/10/3 50 torr/10min @600oC 54

PCaYP092907 3/2/10/3 8 torr/10min @600oC 34

Table A.3: Deposition parameters and transition temperatures for Y1−xCaxBa2Cu3O7−δ ﬁlms 2 unit cell thick, grown between protective P rBa2Cu3O7−δ layers and with an additional 3uc amorphous P rBa2Cu3O7−δ cap layer deposited at 300oC.

88 APPENDIX B

Appendix: Atomic Force Microscopy

Figure B.1: Thickness calibration. AFM images of Ca-YBCO (a) and PBCO (b) thin ﬁlms (bright side of the picture) deposited as a result of 1,000 laser pulses on STO substrates (dark side of the picture). The deposition rates are approximately 0.71A˚/pulse for YBCO and Ca-doped YBCO and approximately 0.77A˚/pulse for PBCO.

89 Figure B.2: AFM images of the surface of Y1−xCaxBa2Cu3O7−δ films grown directly on SrT iO3 without protective layers (a) 40 unit cell thick (b) 8 unit cell thick. Both films have RMS roughness around 17A˚. In the thinner film it is possible to observe how the 4A˚ (one unit cell) steps on the surface of the SrT iO3 substrate are preserved on the surface of the film.

Figure B.3: 3D AFM image of a Y1−xCaxBa2Cu3O7−δ ﬁlm grown between protective P rBa2Cu3O7−δ layers, 10/2/10 unit cells thick. The steps on the surface are approximately 1uc ≈ 11.7A˚.

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