On the Zero and Low Field Vortex Dynamics: An Experimental Study of Type-II Superconductors

Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1214

On the zero and low field vortex dynamics

An experimental study of type-II superconductors

BY

O ¨

F

ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2003

Dissertation at Uppsala University to be publicly examined in H¨aggsalen, ˚Angstr¨omlaboratoriet, Wednesday, January 21, 2004 at 10:00 for the Degree of Doctor of Philosophy. The examination will be conducted in English

Abstract

Festin, ¨O. 2003. On the zero and low field vortex dynamics. Acta Universitatis Upsaliensis.

Comprehensive Summaries of Uppsala Dissertations from the Faculty of Technology and Science 1214. 51 pp. Uppsala. ISBN 91-554-5837-8

Dynamic properties of type-II superconductors have been experimentally studied in zero and low magnetic fields using SQUID magnetometry and I−V measurements.

In zero magnetic field close to the critical temperature, the physical properties of type-II superconductors are dominated by spontaneously created vortices. In three dimensions (3D) such vortices take the form of vortex loops and in two dimensions (2D) as vortex-antivortex pairs.

The 2D vortex dynamics has been probed using mutual inductance and flux noise measure- ments on YBa2Cu3O7 (YBCO) and MgB2 thin films in zero and low magnetic fields. In such measurements, information about vortex correlations is obtained through a temperature dependent characteristic frequency, below (above) which the vortex movements are uncorrelated (correlated). The results obtained in zero magnetic field indicate that sample heterogeneities influence the vortex physics and hinder the divergence of the vortex-antivortex correlation length.

In low magnetic fields the vortex dynamics is strongly dependent on the applied magnetic field and a power law dependence of the characteristic frequency with respect to the magnetic field is observed. The results indicate that there is a co-existence of thermally and field generated vortices.

The I−V characteristics of untwinned YBCO single crystals show that only a small broad- ening of the transition region influences the length scale over which the vortex movements are correlated. The dynamic and static critical exponents therefore exhibit values being larger in magnitude as compared to values predicted by relevant theoretical models. The results also suggest that the copper oxide planes in YBCO decouple slightly below the mean field critical temperature and hence, the system has a crossover from 3D to 2D behaviour as the temperature is increased.

From temperature dependent DC-magnetisation measurements performed on untwinned YBCO single crystals in weak applied fields, detailed information about the critical current density and the irreversibility line is obtained.

Keywords: high temperature superconductors, thermal fluctuations, YBCO, MgB2, anisotropic superconductors, vortex dynamics, Kosterlitz-Thouless transition, current voltage characteristics, complex conductivity, flux noise

Orjan Festin, Department of Engineering Sciences. Uppsala University. P O Box 534, SE-751¨ 21 Uppsala, Sweden

¨Orjan Festin 2003c

ISBN 91-554-5837-8 ISSN 1104-232X

urn:nbn:se:uu:diva-3907 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3344)

”Jamen du har ju hela helgen p˚a dig...”

/P. Svedlindh

”Empty spaces - what are we living for Abandoned places - I guess we know the score On and on, does anybody know what we are looking for...”

/F. Mercury

Papers included in the thesis

This thesis is based on the following papers, which are referred to in the text by their Roman numerals:

I Vortex fluctuations in High-Tc films: Flux Noise Spectrum and Complex Impedance

O. Festin, P. Svedlindh, B.M. Kim, P. Minnhagen, R. Chakalov¨ and Z. Ivanov, Phys. Rev. Lett. 83, 5567-5570 (1999)

II Zero field vortex dynamics in YBCO thin films

O. Festin, P. Svedlindh, R. Chakalov and Z. Ivanov, Physica B¨ 284-288, 963-964 (2000)

III ”Double superconducting transition” in YBCO thin films

O. Festin, P. Svedlindh and Z. Ivanov, Physica C 369, 295-299¨ (2002)

IV Magnetic flux noise in MgB2thin films

O. Festin, P. Svedlindh and S.I. Lee, Proc. SPIE 5112, 338-345¨ (2003)

V Zero field dynamic universality class of high-T_csuperconductors O. Festin, P. Svedlindh, A. Rydh and P. Minnhagen, Phys. Rev. B¨ (In progress)

VI Vortex fluctuations in high-Tcfilms close to the resistive transition O. Festin, P. Svedlindh, F. R¨onnung and D. Winkler, Phys. Rev.¨ B (submitted)

VII High temperature stability SQUID magnetometer for zero and low field dynamic measurements

O. Festin and P. Svedlindh, Rev. Sci. Instrum. (submitted)¨ VIII Low field magnetization of untwinned YBCO single crystals

P. Svedlindh, ¨O. Festin, J.R. Clem, E. Papadopoulou and A. Rydh, (in manuscript)

Comments on my participation

I Most of the experimental work, part of the analysis

II Part of the writing, all experimental work, except sample prepara- tion, part of the analysis

III All experimental work, part of the sample preparation, most of the writing, all analysis

IV,V All experimental work, except sample preparation and most of the writing, most of the analysis

VI Part of the experimental work, part of the analysis

VII All experimental work, all writing and most of the analysis VIII Part of the experimental work

Papers not included in the thesis

IX Study of Nucleation and Growth in the Organometallic Synthesis of Magnetic Alloy Nanocrystals: The Role of Nucleation Rate in size control of CoPt3Nanocrystals

E.V. Schevchenko, D.V. Talapin, H. Schnablegger, A. Kornowski, O. Festin, P. Svedlindh, M. Haase, H. Weller, Journal of American¨ Chemical Society 125, 9090 (2003)

Contents

1 Introduction . . . 1

1.1 History . . . 1

1.2 The vortex state . . . 4

1.2.1 The vortex structure . . . 5

2 The Samples . . . 7

2.1 YBa2Cu3O7−x. . . 7

2.2 MgB₂ . . . 9

3 Experimental techniques . . . 13

3.1 The experimental setups . . . 14

3.2 Flux noise and ac-susceptibility . . . 18

3.3 I-V and resistance measurements . . . 21

3.4 DC magnetisation measurements . . . 23

4 The superconducting phase transition . . . 25

4.1 2D vortex fluctuations in zero magnetic field . . . 27

4.1.1 Ordinary YBCO thin films . . . 28

4.1.2 The double transition . . . 32

4.1.3 Peak ratio . . . 34

4.1.4 MgB2 . . . 35

4.2 2D vortex fluctuations in low magnetic fields . . . 38

4.3 3D vortex fluctuations in zero magnetic field . . . 40

4.4 Low field DC magnetisation . . . 42

5 Acknowledgements . . . 45

Introduction

1.1 History

In the spring 1911, a student of professor Kamerlingh Onnes cooled a small tube filled with mercury down below 4.2 K and measured the resistance in the mercury. At 4.25 K they observed a sharp drop in the resistance, in disagree- ment with theories postulating that the resistance of a metal should remain finite to 0 K. To prove his student wrong, professor Kamerlingh Onnes forced his student to repeat the measurement for other samples. However, all mea- surements yielded the same result: There is a sharp drop in the resistance at 4.25 K (Fig. 1.1). Later the same year he reported that ”mercury had passed into a new state, which on account of its extraordinary electrical properties may be called the superconducting state” [1]. He also discovered that not only a high enough temperature, Tc, can destroy superconductivity, but also a high enough magnetic field, H_c, and a high enough current density, j_c.

Figure 1.1: Resistance versus temperature for mercury obtained by Kamer- lingh Onnes in 1911.

One of the two signs of a superconductor is zero resistivity, the second one is the ”Meissner effect”. The Meissner effect means that the applied magnetic field is expelled from the interior of a superconducting sample, implying that the susceptibility, being defined asχ = M/H, is equal to -1 for a supercon-

ductor [2]. The field inside a superconductor in the ”Meissner state” decays exponentially with the distance from the surface. This defines the London pen- etration depth [3],λL, being defined as H(λL) = (1/e)H(0), where H(0) is the field at the surface of the superconductor. Later on, a second type of su- perconductor was discovered, displaying two critical fields, Hc1 and Hc2. At Hc1 the Meissner effect is partly lost and the diamagnetic magnetisation of a sample decreases with increasing field and reaches zero at H_c2 where super- conductivity is destroyed. A type-II superconductor allows the magnetic field to enter the interior of a sample in the shape of flux lines or vortices [4]. The first vortices enter the sample at H_c1where the magnetisation starts to decrease and at Hc2 the sample is saturated with vortices and the superconductor turns into a normal conductor (Fig. 1.2).

Vortex state

Type II Meissner state H

T H_c2

Normal state

Hc1

Tc

Type I Meissner state H

T^c T Hc

Normal state

Figure 1.2: The H− T phase diagram for type I and type II superconductors.

It took about 40 years after the discovery of superconductivity before an ac- ceptable theoretical description was found that at least partly could explain su- perconductivity. It was a phenomenological theory developed by Ginzburg and Landau [5], that takes electrodynamic, quantum mechanic and thermodynamic properties into account. It expresses the ”degree of superconductivity” in a ma- terial as a complex order parameter described by the density of superelectrons n^∗_s, a phase θ at position r, as ψ(r) =

n^∗_s(r)e^iθ(r). Using this expression in an expansion of Gibbs free energy near Tc where the order parameter is small, gives the two ”GL-equations”. From these equations two fundamental length scales characterising the superconducting state can be determined. The first one is the penetration depth, which has already been mentioned, and the sec- ond one is the coherence length,ξ, which characterises the spatial variation of ψ(r). The coherence length in an ordinary superconductor can be up to a few micrometers in length. The GL theory is a very powerful tool when examining

physical properties of a superconductor, but it does not give a microscopic ex- planation to what happens inside the material as it becomes superconducting.

A microscopic theory of superconductivity was first presented in 1957 by Bardeen, Cooper and Schreiffer in the BCS theory [6]. In a simplified pic- ture of this theory, superconductivity is created by two electrons having wave vectors of opposite sign, k⁺and k⁻but being equal in magnitude|k⁺| = |k⁻|.

Their total wavevector will then be≈ 0, which corresponds to an infinite wave- length. The wavelength is much larger than the distance between the atoms in a crystal, which means that the Cooper pair will not be scattered by the lattice and thus it does not experience any ”resistance” as it flows through the ma- terial. The formation of a Cooper pair requires an attractive electron-electron interaction, being mediated through the lattice (Fig. 1.3). The first electron passes through the lattice and causes a small polarisation of the crystal struc- ture and hereby a lowering of the potential energy between the electron and the atoms. The second electron sees the track of the first electron and takes advantage of the decrease in potential energy caused by the small distortion of the lattice.

+

(1)

(2)

Figure 1.3: Schematic view of the phonon mediated electron pairing. Electron 1 modifies the vibration of the ion, which in turn interacts with electron 2. The net result is an attractive interaction between the two electrons.

Numerous possible applications for superconductors have been suggested but the low critical temperature for the early known superconductors was for a long time a major obstacle. No major breakthrough in increasing the critical temperature was made until the mid 80’s when first Bednorz and M¨uller re- ported superconductivity at 30 K in the La-Ba-Cu-O system[7] and a year later Wu et al. [8] synthesised YBa2Cu3O7−x (YBCO) having a critical tempera-

ture (T_c≈ 93K) above the boiling point of nitrogen. These findings were the starting point for a new era in the superconductivity research. Today hundreds of different copper oxide compounds, cuprates, have been found that display superconducting properties at ”high” temperatures. By putting the material under pressure the critical temperature can be raised. The highest critical tem- perature ever measured up to now (Tc= 164 K), was in the HgBa2Ca2Cu3O8

system under 30 GPa pressure [9].

The cuprates are characterised by superconducting CuO2-planes separated by non-metallic interlayers acting as charge reservoirs for the CuO₂ planes.

The cuprates have an orthorombic or tetragonal lattice structure, with an elon- gated c-axis while the a and b axes are of the same length. In YBCO, the a- and b-axes are≈ 3.85 ˚A(Fig. 2.1), and the c-axis is ≈ 11.7 ˚A. As a consequence, the superconducting properties are highly anisotropic, i.e. different in different crystallographic directions. The coherence length at 0 K of YBCO along the c-axis is around 3 ˚A and around 16 ˚A in the ab-plane.

In march 2001 a new type of superconductor was discovered, MgB2, having a critical temperature of 39 K [10]. Its structure is layered but not as anisotropic as the cuprates. Another very specific property in very pure MgB2 is that two bandgaps open up below Tc [11, 12], implying that there are two types of superelectrons in the material. It has been speculated that this can cause the creation of vortices carrying an arbitrary fraction of a magnetic flux quantum [13].

1.2 The vortex state

In 1957 A. A. Abrikosov succeeded to explain what happens to a superconduc- tor when exposed to a magnetic field [14]. He divided the materials into two groups, type-I and type-II superconductors. Type-I superconductors are found mainly among the pure elements, while type-II superconductors are mainly formed by alloys and oxides. What distinguishes the two groups from each other is the magnitude of the GL-parameter,κ = λL/ξ. If ξ is larger than λL, or more precisely,κ < 1/√

2, magnetic flux will be expelled from the interior of a sample up to a critical field Hc when superconductivity is destroyed. If κ > 1/√

2 vortices will enter the superconductor when H_c1< H < Hc2[15].

κ>>1lim H_c1= Φ0

4πµ0λ²_Lln(λL/ξ) (1.1) Hc2= Φ0

4πµ0ξ². (1.2)

At the first critical field, Hc1(T), the first vortices will enter the supercon- ductor and at Hc2(T) the sample will be saturated with vortices and becomes a

normal conductor. The vortices form a triangular lattice, having the distance a between the vortex cores. At Hc2, a= 2ξ the vortex cores start to overlap and the sample is saturated with vortices. Each vortex contains one magnetic flux quantum,Φ0= (h/2e) ≈ 2.07 × 10⁻¹⁵ Wb. Especially high-T_c superconduc- tors, which are of extreme type-II and have aκ >> 1/√

2, are very sensitive to external magnetic field. In a clean YBa2Cu3O7−x sample Hc1(0) is of the order 0.04 T, while H_c2(0) is of the order 100 T. If macroscopic defects are in- troduced into the material, the phase coherence will be destroyed and Hc1(T) will be lowered even more.

1.2.1 The vortex structure

An Abrikosov vortex has a normal core, which can be approximated by a long thin cylinder with its axis parallel to the external magnetic field. The diameter of the cylinder is of the order 2ξ and the density of Cooper pairs |Ψ|²decreases to zero at the vortex center (see Fig. 1.4).

2λ

r B

2ξ

Ψ²

Screening currents Vortex core

Figure 1.4: The vortex structure.

The direction of the circulating supercurrent around the core is such that the direction of the magnetic field generated by the current coincides with that of the external field and is parallel to the normal core. The magnetic field and the circulating currents decay radially out from the core as B(r) ∝ ln(λL/r) at short distances and as B(r) ∝√

r exp(−r/λL) at long distances from the vor- tex center. The circulating supercurrents surrounding the vortex core cause

an interaction between the vortices, which is repulsive (attractive) for parallel (antiparallel) vortices. The interaction extends to a distance of the orderλL. If a thin enough sample is used with thickness d<< λL, the magnetic screen- ing is less effective and the penetration depth is replaced by an effective two dimensional (2D) penetration depth, which is defined as [16, 17]

λ>>dlim Λ = 2λ²L/d. (1.3)

This also means that vortex interactions in 2D systems extend over larger length scales compared to bulk materials.

The Samples

2.1 YBa

Cu

O

The unit cell of the superconducting phase of YBa2Cu3O7 is orthorhombic (a= b), while YBa2Cu₃O₆ is tetragonal (a= b) and is an insulator. The oxygen content is therefore crucial for the superconducting properties in the YBa2Cu3O7system. If the oxygen content is decreased below≈ 6.4, the unit cell becomes tetragonal and superconductivity vanishes (Fig. 2.1). The c- axis length is also strongly dependent on the oxygen content and varies from c(x = 0.07) = 11.68 ˚A to c(x = 0.90) = 11.82 ˚A [18, 19].

(a)

11.6802 Å

3.8872 Å 3.8227 Å

O^2- Y³⁺

Ba²⁺

Cu ,Cu2+ 3+

a c

b

0.0 0.2 0.4 0.6 0.8 1.0

0 20 40 60 80 100

3.80 3.82 3.84 3.86 3.88

a,b [Å]3.90

X T_c [K]

YBa₂Cu₃O_7-x b

a T_c

(b)

Figure 2.1: In (a) the unit cell of YBa2Cu3O7 is shown. In (b) the critical temperature (left axis) as a function of x in YBa2Cu3O7−xis shown (filled circles). The right axis shows a- and b-axes length as a function of x (open circles and triangles).

Two kinds of YBa₂Cu₃O_7−x samples have been used in this work. Most experimental work have been performed on thin films having thicknesses in the range 50 to 1500 ˚A. These thin films can be considered to be two dimensional close to the resistive transition, since the penetration depth is larger than the thickness of the films. According to x-ray θ − 2θ and φ scans all films are highly c-axis oriented and exhibit a high degree of in-plane order. The second

kind of sample is single crystals. Being a factor 100-1000 thicker than the thin films, they can be considered as bulk materials. All thin films are square shaped with dimensions 5× 5 mm² and were fabricated with a laser ablation technique, either on SrTiO₃(a= 3.905 ˚A) or LaAlO3(a= 3.831 ˚A) substrates.

Both kinds of substrates have a cubic structure with lattice constants close to the a and b axes lengths of YBCO, which is necessary to obtain epitaxial films having the c-axis perpendicular to the sample surface. The a and b axes are randomly distributed in the film, forming twin boundaries where the a and b axes change place. Directly after the film deposition, the oxygen content is too low in the film. A higher oxygen content is obtained by annealing the films in O2atmosphere at a pressure slightly lower than the atmospheric pressure.

87 88 89 90

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

χ'/χ'SCR

T [K]

500 nm YBCO film H_DC = 0 Oe H_AC = 0.2 mOe f = 17 Hz

degraded

non-degraded

Figure 2.2: Example of the influence of oxygen degradation in a sample.

χ^/χ^_SCRversus temperature [20].

The sample is first quenched from the deposition temperature (780^◦- 800^◦C) down to the annealing temperature (400^◦C), where it is kept a few hours. The cooling rate to room temperature is crucial. During this process, the sam- ple passes through a nonequilibrium region, while temperatures are still high enough to allow for significant oxygen diffusion and/or redistribution in min- utes or even seconds. Therefore, it is difficult to obtain two thin films display- ing the same superconducting properties. It should also be pointed out that the same sample after one set of measurements when exposed to room tempera- ture air, degrades in a matter of days and displays a decrease in T_c as well as a broadening of the transition width. In Fig. 2.2 magnetic susceptibility versus temperature a thin YBCO film is shown [20]. The data were recorded at two different sessions and the time passed between the two sessions was 6 months.

The single crystals were fabricated through a flux growth technique and all crystals were taken from the same batch. The crystals were annealed at 400^◦ in O2for 5 days [21]. The untwinned crystals were obtained by choosing twin free areas in larger twinned crystal flakes and separating them from the rest of

the crystal with a small knife. The twin orientations and twin boundaries are detected by illuminating the sample with polarised light and become visible as long stripes in the sample (Fig. 2.3).

Figure 2.3: A twinned YBa₂Cu₃O_7−xsingle crystal illuminated with polarised light. The dark gray regions to the right are heavily twinned, with densely spaced parallel twin boundaries.

The typical shape of a twinned single crystal was a flake with dimensions 500×500 ×20µm³, while an untwinned single crystal has a rectangular shape of typical dimensions 400× 50 × 20µm³. For the I−V and resistance mea- surements untwinned single crystals were used. The samples were glued on a sapphire substrate. Current leads for I−V and resistance measurements were glued with silver paint at the short-end edges of the crystals and the voltage was measured on top of the crystals (R_contact < 1Ω) via 30 µm goldwires con- nected to contact pods on the sapphire substrate. The distance between voltage contacts, center to center, was∼ 100 µm for all crystals (Fig. 2.4).

2.2 MgB

MgB₂ has a layered graphite structure (hexagonal) whereas the boron atoms are arranged in layers, with layers of Mg in between two boron layers (Fig.

2.5). The lattice parameters are a= 3.086 ˚A and c = 3.524 ˚A. First principles calculations [11] show that the Fermi surface of MgB2consists of 2D cylindri- cal sheets arising fromσ antibonding states of B pxyorbitals, and 3D tubular networks arising fromπ bonding and antibonding states of B pzorbitals. In this theoretical framework two different energy gaps exist, the smaller con- nected to the 3D bands and the larger one associated with the superconduct- ing 2D bands. Point contact spectroscopy measurements [12], performed on clean enough samples, i.e. samples with enough long range microscopic or-

Figure 2.4: Two untwinned YBa₂Cu₃O_7−xsingle crystals contacted with gold wire. (Photo courtesy U. Beste.)

der and few macroscopic defects, have confirmed the theoretical prediction of two band superconductivity. Later, specific heat [22, 23] and penetration depth measurements [24] gave further evidence in favour of two band superconduc- tivity. More recently, Babaev [13] suggested, as a consequence of the two band gap nature, that several vortex configurations can exist in MgB2, not only or- dinary vortices withΦ0 magnetic flux, but also neutral vortices and vortices carrying an arbitrary fraction of a flux quantum. This theoretical prediction has not yet, however, been verified experimentally.

a = 3.086 Å c = 3.524 Å

:Mg :B

a a

c

Figure 2.5: The unit cell of MgB₂

The two samples used in this work were two square shaped (5×5 mm²) 400 nm thick MgB2 films. They were both grainy with a grain diameter of a few tenths of a micrometer and the surface roughness was≈ 3 nm. The two films were grown on (1102)Al2O3substrates [25]. First an amorphous boron film was deposited using a pulsed laser deposition technique. This film was put in a Nb-tube together with high purity Mg-metal and the tube was then sealed in an Ar atmosphere. Finally the tube was put in an oven for 30 min at 900^◦C, yielding high-quality MgB2films [26]. More detailed information

regarding the fabrication technique can be found elsewhere [25, 26].θ−2θ and φ scans show that the films are highly c-axis oriented, while the a-directions are randomly distributed in the films. Analysis of thin film samples prepared by different techniques show that secondary phases such as MgO, MgB₄and MgAl2O4easily form [27], resulting in a lowering of Tcas well as a broadening of the superconducting transition. Investigations regarding T_cas a function of Mg content show that the transition temperature decreases with decreasing Mg content [28].

Experimental techniques

Since the discovery of high-Tcmaterials, large efforts have been made to un- derstand the magnetic phase diagram of these materials [29]. In the interme- diate and high field region, numerous investigations, both theoretical and ex- perimental, regarding the different possible collective vortex states have been performed [30, 31, 32, 33, 34, 35]. A considerably less investigated area of research, at least experimentally, is the low field phase diagram of high-T_cma- terials. Below the mean field transition temperature thermal vortex fluctuations become important. In 3D such fluctuations take the form of vortex loops [36], while in 2D the relevant fluctuations correspond to thermally excited vortex- antivortex pairs [37, 38]. To be able to study the low field phase diagram, a controlled magnetic environment with a low background field together with a high temperature stability is required. This chapter describes the experimental techniques employed in this thesis and the information that they provide.

SQUID sensor

DC magnet

Temperature control

The sample space Helium bath

AC resistance bridge

PID

Current supply Keithley 2400 Relay box

SQUID control QD 5000

AC coil

Spectrum analyser HP 35670A

Lock-in ampl.

EG&G 7260/5302

Figure 3.1: Schematic view of the experimental setup

3.1 The experimental setups

The main part of the experiments reported in this thesis have been performed in non commercial SQUID magnetometers. These instruments can be used for magnetic flux noise, ac-susceptibility, DC and time dependent magnetisa- tion measurements in zero and low magnetic fields. A schematic view of the experimental setup employed in this thesis is shown in Fig. 3.1. The setups presented in Fig. 3.2 are designed to measure vortex fluctuations in high-Tc

superconducting films.

(a) (b)

Indium seal

Quarts Cu

Sapphire sample rod Sample Macor

Drive- and pick-up coils µ metal and Nb-can

Heater Thermometer SQUID sensor

Adjustable coil holder

Vacuum

Liquid Helium 20 mm

µ−metal and Nb can

20 mm Vacuum DC-coils Drive- and pick up coils Thermometer Heater SQUID sensor Nb can µ-metal can

Macor Sample Sapphire sample rod

Cu Quarts

Cu frame

Figure 3.2: Cross section of the experimental setups used for the zero (a) and low field (b) flux noise and ac-susceptibility measurements.

Setup (b) is an upgraded version of setup (a) (Fig. 3.2) and has been de- signed during the course of this thesis work [paper VII]. When designing this setup the main goal was to achieve a higher signal sensitivity and to introduce the possibility to apply a weak magnetic DC-field during the measurements.

In setup (a) an RF-SQUID is used and an output signal of 20 mV/Φ0with sen- sitivity 1.5 × 10⁻⁴Φ0/√

Hz at 100 Hz [39] is obtained. Setup (b) uses a DC- SQUID with output signal of 1 V/Φ0 [40] with sensitivity 7× 10⁻⁶Φ0/√

Hz at 100 Hz (Fig. 3.3). The DC-magnet in setup (b) is a Helmholtz coil with

diameter 18 mm and the distance between the coil centers is 10 mm. The base is made of Macor and each section of the coil has 3× 29 turns of 125µm NbTi wire. The magnet works in persistent mode.

10^-12 10^-11 10^-10 10^-9 10^-8 10^-7 10^-6

10^-2 10^-1 10⁰ 10¹ 10² 10³ S Φ [Φ 02 /Hz]

f [Hz]

Heater on

Heater off

Figure 3.3: Background noise obtained above Tcwith the heater on and below Tcwith the heater off.

To keep the coils cold and superconducting, the Macor base of the magnet is attached to a copper frame being in thermal contact with the surrounding helium bath through a copper rod. On the frame a heater made of a carbon resistor is glued, which is used as heat switch for the superconducting magnet circuit to make it possible to inject a current into the magnet. A calculation of the magnetic field inside the coil, considering the magnet as equally spaced current loops and taking the image effect of the Nb can into account [41] gives a field constant of 63 G/A. The calibration of the magnet, performed with a cryogenic Hall probe [42] placed on the sample position, gave a field constant of 64 G/A (Fig 3.4 (a)), which is in good agreement with the calculated value.

To achieve a low residual field at the sample position a µ-metal and a su- perconducting Nb can was used. The µ-metal can reduces the earth magnetic field and the Nb-can screens out temporal variations of the field. To measure the background field a YBCO-film was glued onto the sample holder and a DC-magnetic field was applied well below T_c0. The temperature of the film was then raised and the difference in SQUID voltage|∆USQU ID| between the superconducting and normal state was measured. This procedure was repeated for different low fields (±HDC) (Fig. 3.4 (b)). The residual field is found from the value of HDC when |∆USQU ID| = 0. From Fig. 3.4 (b) this field can be estimated to be lower than 0.1 mG.

The pick up coils in the setups are wound as first order gradiometers using NbTi superconducting wire. In setup (a) they are wound as 2× (8 + 7) turns and in setup (b) as 2× (6 + 5 + 5) turns.

-1.0 -0.5 0.0 0.5 1.00.0 0.5 1.0 1.5

(b)

H_DC [mOe]

∆U SQUID[ V ]

0 50 100 150

0 2 4 6 8 10

H DC [Oe]

I [mA]

(a)

Figure 3.4: Characteristics of the DC-magnet. (a) shows the magnetic field (H) versus injected current (I) into the magnet. (b) shows the test of the amplitude of the background field.

The wire diameter in setup (a) is 40 µm and in (b) the diameter is 75µm.

For both setups, the diameter of the pick-up coil is 1.2 mm with the distance 3.9 mm between the coil centers. This configuration allows the signal from the sample to the coil furthest away to be neglected. The pick-up coil is chosen to have an inductance slightly below the input inductance of the SQUIDs (1.9 µH) and it is 1.8 µH for setup (a) and 1.3 µH for setup (b).

The sample holder consists of a copper block and a sapphire rod on which the sample is glued (Fig. 3.2). This constitutes a firm thermal unit to which a thin quartz rod in thermal contact with the helium bath is glued. The con- struction allows a very high precision temperature control of the sample. To exploit this possibility, a non-commercial temperature control system is used, utilizing a tailor made copper thermometer. An insulated copper wire, 40 µm in diameter and 1 m in length (14Ω at room temperature), is wound bifilary around the lower part of the copper block. The resistance of the copper wire is measured by an ac-bridge without active electronic components, employing a seven decade voltage transformer [43]. The copper thermometer is calibrated according to the method given in ref. [44], and works down to 20 K.

The temperature control of the sample is achieved by means of a propor- tional - integrating - differentiating (PID) regulator connected to the output signal of a Lock-in amplifier used as null detector in the ac-bridge. The PID adjusts for zero output from the Lock-in amplifier by supplying an adequate amount of power to a manganine heater wound on the upper part of the sample holder [45]. In the temperature range 25-100 K a long time temperature stabil- ity of 300 µK can be obtained. The firm thermal contact between the sample and the thermometer yields negligible temperature gradients between the ther-

mometer and the sample at all cooling and heating rates achievable with the system. Looking again at Fig. 3.2, the distance between the sample and the upper pick-up coil is typically 0.3-1 mm. The ac-coil is wound outside the pick-up coil on a Macor frame, with a diameter of 2.4 mm. In both setups, 75 µm NbTi wire wound in 40 turns was used.

0 20 40 60 80 100

0.23 0.24 0.25 0.26 0.99 1.00

∆χ=0.0033 δT=0.68 mK T₂=88.0091 K χ'/χ' SCR

t [s]

T_o=88.0039 K

T₁=88.0065 K

∆χ=0.013

∆T=2.6 mK

T₃=80.3525 K T₄=80.3363 K T₅=80.2232 K

(b)

87.5 88.0 88.5

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

0 1 2 3 4 5 6 7

χ'/χ'SCR ∂χ'

∂Τ

T [K]

f =1.7 kHz H_AC=0.2 mOe

(a)

Figure 3.5: Temperature control accuracy test. In (a)χ^(T) for f = 1.7 kHz (dots, left axis) and∂χ^/∂T (line, right axis) is shown. (b) shows the time variation of χ^ at six different temperatures, where the first three data sets (T0− T2) were recorded close to the maximum of∂χ^/∂T. As a reference, three data sets were recorded at tem- peratures (T3− T5), where∂χ^/∂T is small.

To test the performance of the temperature control equipment, a supercon- ducting film with a narrow transition width (∆Tc0≈ 0.25 K) was used. The test temperatures were chosen close to where the derivative of the in-phase compo- nent of the ac-susceptibility (∂χ^/∂T) has its maximum value (Fig. 3.5 (a)). At each temperature a time sweep over 30 seconds was measured ofχ^. First the

difference in averageχ^ between two temperatures was measuered (∆T = 2.6 mK,∆χ = 0.013). By assuming that the ac-susceptibility in this narrow tem- perature region is linear, aχ^-temperature constant is obtained, ∆χ/∆T = 5 K⁻¹. The temperature stability is then obtained by measuring the fluctuation in the ac-susceptibility at ”constant” temperature and assuming that the fluc- tuation in the signal originates from temperature fluctuations. According to this method, the temperature stability at T≈ 88 K is ±0.3 mK. For a compar- ison, the fluctuations were measured where the sample is well below the tran- sition temperature, where∂χ^/∂T << ∂χ^/∂Ttransition. Here the fluctuations in χ^ were a factor≈ 1000 smaller (Fig. 3.5 (b)).

3.2 Flux noise and ac-susceptibility

Mutual inductance and flux noise experiments [paper I] [46, 47, 48] are two useful techniques to obtain information about the dynamical properties of vor- tices in a superconductor. In the mutual inductance measurement, the sample is placed above the pick-up and drive coils. A small current IDwith angular frequencyω is applied to the drive coil and the sample response is measured by the pick-up coil. The amplitude of the applied ac-field is chosen to be small enough so that the response of the sample is within the linear region, meaning that few vortices are generated by the field and movements of spontaneously created vortices dominate the measured response (Fig. 3.6).

To analyse the signal, a Lock-in amplifier is used, giving the in- and out- of-phase components (χ^ andχ^) of the frequency dependent susceptibility as output. The out-of-phase component corresponds to the dissipative part of the vortex response and the in-phase component corresponds to the shielding response in the superconductor. These data are related to the complex conduc- tance of the film via an integral equation [47]

δV = ID

_∞

0

M(x)

1+ (2/µ0h)(1/iωG)xdx (3.1) whereδV represents the in- and out-of-phase components of the ac - suscepti- bility, IDis the current in the drive coil, h is the distance between the pick-up coil and the sample plus the distance between the sample and the drive coil and ωG is the complex sheet conductance. x = qth and M(x) is the mutual induc- tance distribution between the pick-up and the drive coils and is a function of geometrical dimensions only. The measured ac-susceptibility is converted into complex conductance using an inversion algorithm described by Jeanneret et al. in Ref. [47].

The complex conductivity G(ω) is the inverse of the complex impedance Z(ω). In the presence of vortex-antivortex pairs (2D) it is written as Z(ω) =

89.4 89.5 89.6 89.7 89.8 -1.0

-0.8 -0.6 -0.4 -0.2 0.0

χ'/χ' SCR

T [K]

0.2 mG 0.5 mG 1.0 mG 2.0 mG 5.0 mG 10.0 mG f = 17 Hz H_DC= 0 G

(b) 0.00

0.05 0.10 0.15 0.20 0.25 0.30

(a) χ''/χ' SCR

0.2 mG 0.5 mG 1.0 mG 2.0 mG 5.0 mG 10.0 mG

f = 17 Hz H_DC= 0 G

Figure 3.6: ac-susceptibility versus temperature for a 500 ˚A thick film. The different curves correspond to different ac-field amplitudes.

iωLKε(ω) [46, 49, 16]. Here LKis the kinetic inductance of the sample caused by the non-dissipative motion of the superconducting charge carriers. ε(ω) describes the effect of vortex pairs and free vortices. From this expression the real and imaginary parts of the ac-conductance can be deduced as

Re[G(ω)] ∝ −1 ω Im

 1 ε(ω)



(3.2)

Im[G(ω)] ∝ 1 ω Re

 1 ε(ω)



(3.3) In a superconductor, close to the mean field transition temperature T_c0, dif- fusion of free vortices dominates, causing a Drude like form of the response function (cf. Ref. [50]).

Re

 1 ε(ω)



=1

ε ω²

ω²+ ω²_D (3.4)

Im

 1 ε(ω)



=1

ε ωDω

ω²+ ω²_D (3.5)

whereωDis the characteristic frequency scale being proportional to the vortex density and ˜ε is a static dielectric constant of the order unity caused by the vortex pairs. As the temperature is lowered toward the Kosterliz-Thouless temperature, T_KT [37, 38, 51, 52, 53], more and more vortices bind together and form vortex-antivortex pairs and eventually the response function changes into the Minnhagen phenomenology (MP), which is given by the following relations [16]

Re

 1

ε(ω)− 1 ε(0)



=1

ε ω ω + ω0

(3.6)

Im

 1 ε(ω)



= −1

ε 2 π

ω0ωln(ω/ω0)

ω²− ω²₀ (3.7)

whereω0 is a temperature-dependent characteristic frequency set by the di- vergence of the correlation lengthξ₊ for vortex- antivortex pairs above TKT

throughω0∝ ξ^−z₊, where z is a dynamic critical exponent [54]. The frequency ω corresponds to a length scale l_ω∝

D/ω [55, 56] where D is the diffusion constant for vortices. From Eq. (3.7) the flux noise can be deduced as

Im

 1

ωε(ω)



∝ S_Φ(ω) ∝ ω0ln(ω/ω0)

ω²− ω²₀ . (3.8)

Forω < ω0(l> ξ+) free vortices dominate the response and S_Φ(ω) is nearly constant and the system will appear uncorrelated. Whenω > ω0 (l < ξ₊) vortex pairs dominate the response and S_Φ(ω) ∝ ω^−x and hence, the system appears correlated. This description has been theoretically verified by calcu- lating flux noise spectra around the KT transition in simulations of the 2D resistively shunted junction model [57]. The description implies that S_Φ(ω) is frequency independent forω < ω0, has a crossover to anω^−3/2-behavior in an intermediate frequency regime aboveω0 and for high enough frequencies anω⁻²behavior is observed. Equation 3.8 is valid when the distance between the sample and the pick-up coil is larger than the microscopic length scales on which vortex movements take place in the sample. In such a case, vortices both inside and outside of the pick-up coil contribute to the measured noise, and there is a gradual change in magnetic flux detected by the pick-up coil as a vortex crosses the boundary defined by the coil area (Fig. 3.7).

The flux noise spectrum can be linked to the ac-susceptibility, or the com- plex conductance, via the fluctuation-dissipation-theorem as [57]

ωS_Φ(ω,T) ∝ TIm[χ(ω,T)]. (3.9)

For this equation to be valid, it is important that the field applied in the ac- susceptibility measurements is low enough to ensure that comparably few vor- tices are created by the field, i.e. the field is within the linear response regime.

Figure 3.7: A simulation of the signal measured by a pick-up coil as a function of vortex position. d is the distance in number of plaquettes be- tween the pick-up coil and the array [57]. The size of the array is 64×64, and the coil size is 32×32. The figure shows that when d is large, plaquettes inside and outside the pick-up loop contribute to the measured magnetic flux.

3.3 I-V and resistance measurements

While the zero field superconducting transition in 2D superconductors has been subject to some research activity, the transition in 3D materials has been considerably less investigated. The zero field phase transition is driven by ther- mal fluctuations in both 2D and 3D systems. The fluctuations in 3D take the form of vortex loops [36, 58]. In 2D the critical length scale isξ₊, the maxi- mum vortex pair correlation length, which diverges as the critical temperature (TKT) is approached. In 3D the critical length scale is denotedξGand can be considered as the length scale over which vortex movements are correlated, or as the size of a bundle of vortex loops. ξG diverges as Tc is approached according to [30]

ξG∝ |1 − T/Tc|^−ν. (3.10) Hereν is a static critical exponent, which, according to the 3D-XY model with relaxational dynamics, is expected to take the valueν ≈ 0.67 [59, 60]. Below T_c, the vortices are pinned and a true superconducting state is established for j< jc. jcis defined as the current density above which vortices become mobile (flux creep and flux flow region) and give rise to an induced electrical field E (Fig. 3.8). Above T_c the vortices are not pinned and a low applied current causes a linear resistance in the sample [51, 52, 33, 61]. For larger currents the vortices start to move (flux flow) and give rise to a power law behaviour of the

induced electrical field. At T_c the electrical field is proportional to the current through

E∝ j^Dz+1−1 (3.11)

where D is the dimensionality of the system and z is a dynamic critical expo- nent [30]. In Fig. 3.8 the different regions of the I−V isotherms are schemat- ically drawn.

10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10^-9

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3

U [V]

I [A]

Flux creep

Flux flow T>T

T<T

T

c

c

c

Figure 3.8: A schematic view of the expected I−V isotherms measured in the vicinity of Tcfor a type-II superconductor.

For the I−V and resistance measurements, the experimental setup shown in Fig. 3.2 (a) was used, but with the coils removed. Due to the sharp onset of superconductivity in YBCO single crystals a high temperature stability is nec- essary to avoid voltage fluctuations caused by a varying temperature. Another requirement is to have stable electrical contacts and as low contact resistance as possible. The voltage was measured with a standard four point contact con- figuration to be able to null out the contact resistance from the sample. As a current source, a Keithley 2400 source meter was used in the range 1 µA to 5 mA and as voltage meter a Keithley 2182 nanovoltmeter was used, having a resolution of approximately 1 nV. In the resistance measurements, I= 0.1 mA was used and the resistance at each temperaure was measured after a proper waiting time for the temperature to stabilise. The measured voltage was aver- aged 30 times with positive and negative currents. The I−V isotherms were obtained at constant temperature and going from low to high current in 20-50 current steps.

3.4 DC magnetisation measurements

Experimental studies of the low-field magnetisation of high-T_c superconduc- tors give information about the first critical field Hc1, the irreversibility field Hirrand the low field critical current density jc. However, geometrical effects due to the shape of the sample come into play and the desired information is hampered. The geometry of the sample is, in this case [paper VIII], a plate with quadratic shape and with a magnetic field applied perpendicular to the flat surface. For this configuration a barrier of geometrical origin delays the first penetration of vortices into the superconductor [62, 63]. Several experi- ments on the isothermal magnetic properties of high-Tc superconductors have been performed [64, 65, 66] and models describing the influence of geomet- rical barriers under these conditions have been developed [62, 63, 67, 68].

However, comparably few models have been developed describing the mag- netisation versus temperature at constant magnetic field, applied perpendicular (in the c-axis direction) to the sample surface.

The different measurement techniques used are low field remanent (REM), zero-field-cooled (ZFC), field-cooled-cooling (FCC) and field-cooled-warming magnetisations (FCW) [Paper VIII].

The measurements were performed using a SQUID magnetometer (Fig.

3.9) [69]. In this setup, the DC magnetic field is generated by a small super- conducting solenoid working in persistent mode during measurements. The sample space is shielded by µ-metal and Nb cans, resulting in a longitudinal residual field being smaller than 0.5 mOe at the position of the sample. The sample was glued on a sapphire rod and centered in one of the middle coils sections of a third order gradiometer. During all measurements the sample was kept stationary.

Figure 3.9: A schematic view of the experimental setup used in magnetisation versus temperature measurements [Paper VIII].

The superconducting phase transition

The focus in this thesis is on the critical phenomena in the vicinity of the su- perconducting transition temperature of extreme type-II superconductors. The aim is to describe critical vortex physics in zero and low magnetic field. This, however, is not an easy task. We now have, as described in the previous chap- ter, the proper tools to perform the measurements. The main problems are instead associated with the samples. YBCO and MgB2easily degrade and are sensitive to storage and fabrication methods. Even if the same parameters are used when fabricating different samples, the superconducting properties can still vary. In YBCO, defects such as twin boundaries, oxygen vacancies and regions with different oxygen content form easily. Defects broaden and ”smear out” the phase transition and hinder the divergence of the critical correlation length. As a consequence the vortex interactions are being cut off and instead of probing the diverging correlation length, one is probing finite size effects.

In other words, the phase transition is affected and governed to a larger extent by the sample quality than by a universal critical behaviour.

-15 -10 -5 0

-1.0 -0.8 -0.6 -0.4 -0.2

0.0 50 nm (1) 50 nm (2) 50 nm (3) 150 nm (4) 20 nm (6) 150 nm 5 nm

χ'/χ'SCR

T/T_χ-1 ^x10

-3

H_DC= 0 Oe

H_AC= 0.3 mOe (a)

-4 -3 -2 -1 0 10^-3

10^-2 10^-1 10⁰

(b)

ρ/ρ

T/Tc0-1

Q5 Q8 Q10

x10^-3 H_DC= 0 Oe

i= 0.1 mA T c0

Figure 4.1: (a) showsχ^/χ^_SCR[20] versus reduced temperature for seven dif- ferent films with thicknesses in the range 5-150 nm. T_χis the onset temperature for diamagnetic screening when f =17 Hz. (b) shows ρ/ρTc0 versus reduced temperature for three single crystals, where T_c0is the onset temperature of superconductivity.

The thickness of a sample influences the superconducting properties. This

is due to the effective penetration depth (Λ = 2λ²_L/d) being inversely propor- tional to the sheet density of superelectrons, which decreases with decreasing thickness. This leads to an increased density of thermally excited vortices in a thin film sample (d≤ λL) and as a consequence the transition is broadened.

Figure 4.1 shows the behaviour in the superconducting transition region for the YBCO samples used in the papers on which this thesis is based upon. As it is expected for the two thinnest films (Fig. 4.1 (a)), with thicknesses d= 5 nm and d= 20 nm, the phase transition is broadened compared to the transitions of the more thick films. If only the thickness would influence the supercon- ducting properties, there would also be a difference in onset temperature of the diamagnetic response, T_χ, for these two films, since they differ a factor 4 in thickness. This is however, not the case. T_χ is about the same (papers II and VI), which means that the influence of sample quality is more important than the thickness of the film. Between the 50 nm - and 150 nm films the differ- ence is small in T_χ, and the differences observed in transition width can also be related to the sample quality, as will be discussed later in more detail. More detailed information about the films used in the experimental work is found in Table 4.1.

Figure 4.1(b) shows the normalised resistance versus reduced temperature for the three YBCO single crystals investigated in paper V. Two of the samples show a kink in the resistive transition that leads to a broadening of the transition width,∆Tc≈ 6 × 10⁻⁴in units of reduced temperature, corresponding to≈ 60 mK in real temperature. Even though the kink appears in a narrow temperature range, it has huge influence on the properties around Tcand the effective critical exponents deduced from the analysis of the I−V data.

Each of these samples allows for individual analysis and for extracting im- portant parameters, such as TKT,Tc,Tc0,∆Tc, f0,ξ₊,ξG,λL,z,ν, jc,ωG and Hirr. Some of the samples investigated here show similar behaviour, that also is in accordance with existing theoretical descriptions of the phase transition. The requirements to be fulfilled for a sample are simply a sharp transition without structure in eitherχ^ or ρ, a well defined peak temperature in χ^ and as lit- tle structure as possible in the I−V isotherms. These requirements imply a phase transition governed by vortex interactions only and without influence of sample heterogeneities.

In this chapter a detailed analysis of the investigated samples is presented.

The results obtained are compared with experimental work published by others and existing theoretical descriptions of superconducting phase transition.

Table 4.1: Information on the superconducting films (1) - (6) studied in detail in this comprehensive summary.

Sample Substrate Capping layer Thickness [nm] T_χ[K]^a ∆T_χ[K]^b

(1)^c LaAlO₃ – 50 84.88^d 0.14

(2)^c LaAlO3 PBCO 50 83.95^d 0.50

(3)^c SrTiO3 Au 50 87.88 0.29

(4)^c YSZ – 150 90.13 0.15^e

(5)^f Al₂O₃ – 400 38.83 0.45

(6)^c SrTiO3 – 20 80.22 2.20

aT_χis the onset temperature for diamagnetism when f= 17Hz.

b∆T_χis the transition width (10-90%) for the onset of diamagnetism when f = 17Hz.

cYBCO film

dsee Ref. [70]

eWidth of the high temperature transition

fMgB2film

4.1 2D vortex fluctuations in zero magnetic field

In 2D superconductors, vortices have, provided that their separation distance is smaller than the effective penetration depth, a long range interaction depending logarithmically on distance [16, 17]. Because of this long range interaction, it is expected that a 2D superconductor should exhibit features reminiscent of a system undergoing a Kosterlitz-Thouless transition [37, 38, 51, 52, 53]. Be- low TKT the vortices are ”frozen” and bound in pairs consisting of vortices of opposite vorticity. At T ≥ TKT, vortices decouple provided that their separa- tion distance is larger than the correlation lengthξ₊, which in 2D increases exponentially with temperature as T→ TKT, i.e.ξ+∝ exp[b/

T/TKT− 1].

The correlation length can be linked to the temperature dependent charac- teristic frequency, f₀, via f₀∝ ξ^−z₊ , which gives

f₀∝ exp[−bz/

T/TKT− 1]. (4.1)

For temperatures close to, but above TKT, the vortex fluctuations/response will be dominated by vortex pairs, whose density decreases as the temperature is increased and more and more vortices unbind. Close to T_c0 the 2D vortex fluctuations are of Drude type, i.e. the response of a sample is dominated by free vortices.

According to the MP-description (Eq. (3.8)), the flux noise at temperatures just above T_KT follows a S_Φ( f ) ∝ f^−1.5 dependence in an intermediate fre- quency regime when f > f0, changes to a f⁻²behaviour when f >> f0and is frequency independent when f<< f0.