Singlet and triplet supercurrents in disordered mesoscopic systems

disordered mesoscopic systems

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op woensdag 13 juni 2007 om 10.00 uur

door

Ruurd Sibren KEIZER

Samenstelling van de promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. ir. T.M. Klapwijk Technische Universiteit Delft, promotor Prof. dr. J. Aarts Rijksuniversiteit Leiden

Prof. dr. H. Courtois Universit´e Joseph Fourier, France Prof. dr. R. A. de Groot Radboud Universiteit Nijmegen Prof. dr. H. Hilgenkamp Universiteit Twente

Prof. dr. Yu. V. Nazarov Technische Universiteit Delft

dr. F. S. Bergeret Universidad Aut´onoma de Madrid, Spain Prof. dr. H. W. M. Salemink Technische Universiteit Delft, reservelid

Published by: R. S. Keizer Printed by: Ponsen & Looijen

Cover: Measurements on the planar Hall effect in a Chromiumdioxide (CrO2)

thin film (see Chapter 4).

An electronic version of this thesis, including colour figures, is available at: http://www.library.tudelft.nl/dissertations/

Casimir Ph.D. Series, Delft-Leiden, 2007-04 ISBN: 978-90-8593-028-0

Copyright c° 2007 by R. S. Keizer

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the author.

During the final year of my M.Sc. studies in Delft, I worked for a year in the Nanophysics (NF) research group of Teun Klapwijk. During this year, I learned that real experimental physics on the frontier of human knowledge is very different from textbook physics. At the same time I was exposed to the exciting world of mesoscopic physics and superconductivity (and collective phenomena in general). All of this fuelled the appetite for more.

In mid 2002, I was asked to pursue our initial activities on controllable Joseph-son junctions, for another four years as a PhD student. Not convinced of the remaining fundamental physical questions in that area, I was having difficulties accepting the offer. After a number of discussions of what else might be inter-esting, Teun and I identified a couple of exciting new developments and made an effort to combine these with the general context of mesoscopic superconductivity. The first development was the recent experimental availability of high quality sin-gle crystal films of the 100% spin polarized ferromagnet CrO2, which had not yet

been the subject of any phasecoherent –mesoscopic– measurements, and which could have potential in a whole range of experiments. In an effort not to stray too much from the superconductivity scope, we decided to try and make interfaces between this material and superconductors, to create spin controlled Josephson junctions and study Andreev reflection in SF systems. The second new devel-opment was a theoretical one: less than a year before, a theoretical proposal for the existence of long-range superconducting (triplet) correlations through ferro-magnets had raised some eyebrows in the scientific community, and although we considered this proposal to be pretty exotic, CrO2seemed to be the ideal material

to test this theory. Taken together, these concepts provided a great foundation to start working on a PhD, which I did in December 2002. This thesis describes the results of our research during the last four years.

Many people have contributed significantly to the results in this Thesis. First of all I would like to thank my promotor, Teun Klapwijk, for his continuous support throughout my PhD. Teun, the initial reason why I started in your group is very closely linked to your enthusiastic and inspired way of thinking about

physics. Moreover, you gave me the freedom to work on the research projects of my own choice, and stimulated me along the road. Despite – or thanks to – the differences in our approach to doing physics, the results of our discussions on experiments and manuscript revisions have always, without exception, led to very satisfactory results in the end. Moreover, your broad scientific knowledge, overview, connections and advice were always a big help.

At this point, I would like to thank Sebastian Goennenwein and Machiel Flok-stra who have in different ways been driving forces for the results in this Thesis. Machiel, much of my intuitive and conceptual understanding of the microscopic theories of superconductivity is a result of our endless discussions, stemming from the refusal of two stubborn experimentalists to be stumbling in the dark when it comes to theoretical understanding. Moreover, our simple toy model simulation turned out to be the first numerical model for an inhomogeneous superconduc-tor out of thermal equilibrium, and made it into PRL in the end, causing a ‘distinguished’ theorist to speak the words: ‘and these guys are supposed to be experimentalists?’. Sebastian, in many respects we were the perfect team. I am still amazed that the majority of our CrO2 results are the result of a ‘miserable’

10 months. Your drive, dedication and patience were a large factor in this success. The ideas on further experiments are still forming a big pile in some cupboard, who knows what we could have done in another 10 months. Most important of all, the months we worked together were the best of my PhD. And finally: although you insist on using ‘Prof.’ in front of my name, I guess you will be the first of us to rightfully deserve that title.

The fruitful collaboration with the group of Arunava Gupta, the source of our high quality CrO2 films, was obviously instrumental to all our work with

CrO2. Arunava, your advice and knowledge provided a boost, especially in the

initial stages of the project. Later on, feedback was always fast, and to the point (sending a sample to New York and getting it back within the week really helps to keep things up to speed).

Ideas and collaborations followed from discussions with lots of people inside and outside NF, such as Jan Aarts, Paul Alkemade, Matthias Eschrig, Wim van Saarloos, Igor Mazin, Alexander Golubov, Yasuhiro Asano, Marco Aprili, Chris Bell, Francesco Giazotto, Alexander Brinkman, Rudolph Gross, Yaroslav Blanter, Ad Verbruggen, Jianrong Gao, Alberto Morpurgo, Jaap Caro and Sven Rogge. In particular, I would like to thank Yuli Nazarov together with Daniel Huertas Hernando and Vitaly Braude for keeping the theoretical mesoscopic superconduc-tivity scene in Delft alive, and for providing an explanation for the large signal in our triplet measurements.

professors coming and going, I have always felt very connected with my fel-low PhD students: Hon Tin, Gabri, Saverio, Frank, Ruth, Paul, Rami, Rogier, Merlijn, Diederik, Tarun, Nathan, Chris, Nick, Monica, David and Gert-Jan. I especially thank Tino for his advice on measuring, non-superconducting Nb, and being able to talk about important stuff in life (occasionally under the influence of a Leffe or two, or chinese tea during dim sum). Gabri Klemmen! Lansber-gen for bringing up the important matter of the number of quantum fluctuations in a sheesha, my roommates Saverio and Dennis for enduring my presence (or absence) and Saverio especially for introducing me to Cinderella.

I have had the pleasure to advise graduate students Maurits and Ilona. Mau-rits, you were a great student for a PhD in his first year. It’s unfortunate our experiments on controllable junctions and supercurrent enhancement didn’t work out due to bad Niobium. Ilona, partner in crime of the CrO2 dream team, I guess

your ability to manage me has led you to conclude you can manage a big corpo-ration. I think you passed the test. Apart from work, the presence of students Martin, Carlijn, Joost, Ivo and post-docs Derek and Iulian was a guarantee for keeping the C2H5OH level high on quite a number of late nights.

A largely undervalued contribution to research is made by the supporting staff. On our side of the Mekelweg, there were Matthias, Tony, Jordi, Niels, Ron, Ben, Jan, Maria, Monique and Daphne who provided the ideal safety net when something didn’t work, and created a good atmosphere for doing physics. Then there was the DIMES staff of which especially Anja, Marco, Roel, Arnold and Marc were always operational for some much needed backup.

Obviously, there is a world outside physics. Nice relaxation was presented by many weekends in Tilburg, Delft, Breda, Zeeland and Maastricht with Gait and Antonie, mostly spend on cardgames, waterskiing and the occasional night in town. I also would like to thank my roommates, de Heeren van 123–kwadraat: Luc, Mac en Niels. Luc, you provided welcome distractions from physics (al-though I’m afraid ‘Keldysh Green’s function’ was a part of your vocabulary as well at some time) in the form of sharing the same passion for music, visiting concerts (PJ!), and watching late night movies. Mac, for being one of my best friends over a large number of years, sharing countless passions inside and out of physics. And of course, Mac and Luc for wearing the monkey suit beside me.

Finally, all of this would not have been possible without the continuous and unconditional support of my family, which I thank very much for their under-standing.

1 Introduction 1

1.1 The sum and its parts . . . 2

1.2 Microscopics of collective states . . . 5

1.3 The state in the middle . . . 6

1.4 This Thesis . . . 9 References . . . 10 2 Theoretical concepts 13 2.1 Quantum interference . . . 14 2.2 Macroscopic phase . . . 17 2.3 Motion of quasiparticles . . . 20 2.4 Non-equilibrium superconductivity . . . 28 2.5 Super-ferro heterostructures . . . 30 2.6 Long-range correlations . . . 34 References . . . 39 3 Chromiumdioxide 43 3.1 CrO2: the holy grail of ferromagnetism? . . . 44

3.2 Single crystalline epitaxial films . . . 46

3.3 Device fabrication . . . 49

3.4 Measurement set-up . . . 52

References . . . 52

4 Planar Hall effect in chromium dioxide thin films 55 4.1 Introduction . . . 56

4.2 Planar Hall effect . . . 56

4.3 Biaxial magnetic anisotropy . . . 60

4.4 Conclusions . . . 62

5 Biaxial magnetic symmetry of epitaxial CrO2 films 65

5.1 Introduction . . . 66

5.2 Experimental and theoretical procedures . . . 67

5.3 Theoretical modelling . . . 69

5.4 Experimental results . . . 71

5.5 Discussion . . . 79

5.6 Conclusions . . . 80

References . . . 81

6 A spin triplet supercurrent through the half-metallic ferromag-net CrO2 83 6.1 A long range supercurrent through CrO2 . . . 84

6.2 Chromiumdioxide . . . 84

6.3 Magnetic symmetry . . . 84

6.4 Conventional singlet theory . . . 87

6.5 Long range triplet proximity effect . . . 87

6.6 Effect of an applied magnetic field . . . 88

6.7 A magneto-switchable Josephson junction . . . 88

6.8 Quantitative evaluation . . . 89

References . . . 90

7 A disorder insensitive triplet correlation 93 7.1 Introduction . . . 94

7.2 Experiment . . . 95

7.3 Diffusive transport . . . 95

7.4 Magnitude of the effect . . . 98

7.5 Conclusions . . . 99

References . . . 100

8 Critical voltage of a mesoscopic superconductor 103 8.1 Introduction . . . 104

8.2 Theoretical model . . . 104

8.3 Solution scheme . . . 107

8.4 Critical voltage of a mesoscopic superconducting wire . . . 108

8.5 The current free T-shaped system . . . 109

8.6 Stability . . . 111

8.7 Conclusions . . . 112

References . . . 113

Samenvatting 119

Curriculum Vitae 123

Introduction

Everyday life is filled with condensed states of matter. Examples include the chair you are sitting on, the coffee you are drinking, and the thesis you are reading. These states are characterized by their strong ordering; the forces between the atoms constituting a chair are so large that if you push against one atom, the rest will move along with it. It is needless to say this property of these atomic condensed states is of practical interest. What is less well known is that there is a much richer class of very similar states formed by the electrons which in metals such as copper carry the electrical current. Noteworthy states of this type include magnetism, superconductivity and the Fermi-liquid.

While a little bit more exotic than the atomic condensed states, magnetic and superconducting materials have a pronounced influence on today’s information age. Ever since the invention of computer systems the storage of information has relied on magnetic materials. Ranging from creditcards to hard-disk drives, information is stored by manipulating tiny pieces of magnetic material. Whereas superconductivity is not used yet in consumer electronics, a substantial part of satellite based global communication networks and space observatories contain superconducting electronics. Besides these commercial applications, both super-conducting as well as magnetic materials are being used today to try to establish a prototype model of a future quantum computer. Finally, the state termed Fermi-liquid is the basic underlying ordering of electrons in ’normal’ metals and semi-conductors, which makes it the dominant electronic condensed state to be found everywhere where electricity is used to power machines or manipulate in-formation. It comes as no surprise that a thorough appreciation of the mechanics of these states is essential for applications in future electronics.

1.1

The sum and its parts

relatively simple quantummechanical effects, it is usually sufficient to take into account the microscopic phase of one or at most two individual electrons.

There is a popular belief saying that all the fundamental physical processes in nature can be described under the condition that we possess a detailed under-standing of the nature of the smallest irreducible particles such as the electrons and their interactions. As it turns out, this claim only holds for a very small subset of physics in the areas of interest mentioned in the paragraph above. Over the last couple of decades an understanding has evolved that the vast majority of physical systems can be much more readily understood by studying the inter-actions between large ensembles of particles instead of the microscopic behavior of the particles themselves [4]. The bottom line is that collective processes and group interactions can not be understood by studying a single individual [5] (see Fig. 1.1).

As noted above these collective states – or phases – can develop when the peer pressure between the electrons is sizeable enough to force individual elec-trons to comply with group behavior, which can be described by a single entity known as the (macroscopic) order parameter. Transitions from states in which this parameter is zero to a more ordered state are governed by the very funda-mental principle of spontaneous symmetry breaking. The particular symmetry that is broken will to a great extent shape the form of the order parameter and the properties of the final state. Where for a chair this order parameter would be position x (breaking of translational symmetry), in the magnetic and super-conducting phases they are the magnetization M (rotational symmetry) and the superconducting gap ∆ (electromagnetic gauge symmetry) respectively. Many collective properties of these condensed states are a direct consequence of the form of the order parameter: the fact that you can push a chair around, change the direction of the magnetization of a magnetic domain on a hard-drive with a magnetic field and the shielding of external magnetic fields by a superconductor (Meissner effect) are just some examples. While the collective properties are all classical – they behave according to the laws of classical mechanics – the funda-mental microscopic processes that are at the foundation of these collective states are purely quantummechanical.

Figure 1.1: ‘Sir Isaac Newton discovers that more is different’, by Robert Laughlin

Ferromagnet (F)

Superconductor (S)

Figure 1.2: The interactions in superconductors and ferromagnets lead to completely different groundstates. In the superconducting state, pairs of electrons form with oppo-site spin (singlet) resulting in zero total magnetic moment. In a (perfect) ferromagnet, all spins align, yielding a non-zero total magnetization.

1.2

Microscopics of collective states

In the 1960’s, when it had become clear that the BCS theory could adequately describe the macroscopic properties of superconductors, the next step was to establish the validity and implications of this theory on a microscopic scale. Al-though the general understanding was significantly increased by the experiments that followed, they were hampered by two important factors, which limited them to cases in which the microscopic phase of the electrons was not important. First of all, theory predicted the microscopic phase of the electrons only becomes ap-parent on lengthscales ranging from 1 to 1000 nanometers. The necessary tools to fabricate devices this small were not available at the time, putting an experimen-tal limit on the available options. The second difficulty was the general lack of understanding of phasecoherence of even regular electrons in metals. Both issues were resolved during the 1980’s: microfabrication technology had advanced in a major way and at the same time the devices could now be measured in much more detail, due to the widespread availability of dilution refrigerators that allowed to cool samples to 10 milliKelvin or below.

The observation of electronic self-interference effects such as the Aharonov-Bohm effect, Weak-Localization, and Universal Conductance Fluctuations in metallic samples showed that the wave nature of electrons was to some extent pre-served in metallic films [8, 9]. These discoveries revived research activities in the superconductivity community, since it now seemed plausible that phasecoherent effects could be observed in superconducting materials as well.

of fermions is formed in the same way as that of a non-interacting Fermi-gas, and the elementary excitations (quasiparticles) in this liquid bear a one to one correspondence to the electronic excited states in the Fermi-gas. In most metals, these quasiparticles can be considered to be free electrons with a slightly increased effective mass. However, in superconductors the effect is much more pronounced: besides the renormalized mass, quasiparticles can no longer be considered as electronic particles, but rather as linear combinations of electrons and holes in the Fermi-gas.

The combined electron-hole character of the quasiparticles in a superconduc-tor is clearly illustrated in experiments when the superconducsuperconduc-tor is connected to a normal metal. The quasiparticles in the superconductor, when crossing the interface between the two materials will in the normal metal decompose in phasecoherent electron-hole pairs that change the transport properties and den-sity of states of the normal metal, giving rise to effects like: the reentrance of the normal state conductance, multiple Andreev reflection, the zero-bias anomaly, the ability to carry supercurrents [10], and is epitomized in the observation of a mini-gap in the density-of-states of a normal metal which is reminiscent of the order parameter in the superconductor [11].

Although the details of the interactions leading to the ferromagnetic ground-state can have several different forms, the quasiparticles in the ferromagnetic ground-state of a metallic ferromagnet are not so radically different from the elementary excita-tions of the Fermi-gas as they are in a superconductor. The main difference with the normal metallic quasiparticles is that due to an exchange interaction in the ferromagnet, spin-up and spin-down electrons differ in potential energy, which, to maintain electrochemical equilibrium, leads to a difference in kinetic energy at the Fermi level for spin-up and spin-down. This usually results in spin-dependent densities of states and spin-dependent transmission coefficients of an interface be-tween the ferromagnet and another conductor. While extraction of quasiparticles (and thus spins) from ferromagnets is the very principle behind the operation of commercial spin-valve devices, the clearest demonstrations of this principle can be found in a series of spin-injection and precession experiments [12, 13], which can be considered the ferromagnetic form of the mini-gap experiment mentioned above.

1.3

The state in the middle

HT_cCuprates Superfluidic3_{He, Sr} 2RuO4 Conventional (BCS superc.) Occurence 0 0 Disorder average d - wave (L=2) Singlet p - wave (L=1) Triplet s - wave (L=0) Singlet Orbital

Spin Orbital Occurence Disorder average

Spin + + - -+ -+ +

Figure 1.3: The types of superconductivity the Pauli principle allows to exist. An asymmetric (singlet) spin state is coupled to a symmetric (even) orbital state. The symmetric (triplet) spin state couples to asymmetric (odd) orbital state. In a dif-fusive medium, disorder (angular) averaging suppresses all states with higher orbital momentum, so that only the s-wave survives.

expect the two states to be mutually exclusive on a microscopic level. Therefore it is a valid question to wonder what happens at the border of the two collective states, a situation which can be experimentally accessed by using a complicated material in which both phenomena co-exist and compete, or by employing a het-erostructure consisting of more conventional superconducting and ferromagnetic materials.

In retrospect, the observation of the superfluid A phase of3_{He can be}

consid-ered to be the first experimental example of the coexistence of ferromagnetism and superconductivity in a single material, even tough 3_{He is technically not a}

superconductor but a superfluid [14]. Not only do the two phases exist in 3_He,

they are also indistinguishable: the material is superfluidic and magnetic at the same time, in a state of so-called triplet superconductivity – in which the total spin S of a Cooperpair is 1 instead of S = 0 for a singlet superconductor. For a long time, until after the discovery of high-temperature superconductivity in 1987,3_{He remained the only known material in which this coexistence took place.}

In the last 15 years, eventually a number of superconductors was suggested to po-tentially exist in a triplet state (Sr2RuO4, UPt3, (TMTSF)2PF6, UGe2, URhGe,

ZrZn2 and Fe). The likelihood of the presence of the triplet state in each of

bandstructure, which often obscures the details of the pairing interactions. The conundrum of the coexistence of the two states is much more easy to tackle by employing a heterostructure in which separate superconductors and ferromagnets are connected together. In this way, we know that well inside the two materials (or states), the effect of the other state is negligible, and that the interesting physics is to be found in the middle, where the two are connected. At this interface, there will be some mixed state of the two contributing orders. The spatial extent of this interface will be of the order of the phasecoherence lengths of the different states, which are typically on the nanoscale, meaning that experimentally to measure an effect the structure should be probed on this lengthscale as well. Moreover, theories describing this junction area must be mi-croscopic, since they have to calculate what happens with an elementary particle in one state when it travels into the other state in order to answer questions as: how fast does it decay, does it decay after all, are the states mutually exclusive, and so is there a zero in the order parameters somewhere along the connecting line? Experiments performed on superconductor-ferromagnet (SF) heterostruc-tures indicate that singlet Cooper pairs decay in an oscillating fashion (damped oscillation) in a ferromagnet, as is expected from theory since the two opposite spin electrons in the pair are sensitive to the strong exchange field (≈ 1000 Tesla) of the ferromagnet, which causes the lengthscale of the decay (exchange length) to be very short (≈1-10 nanometers). Still, the oscillating nature of the decay, and the possibility of having more than one zero-crossing of the superconducting pair amplitude (a measure for the amount of pairs) in the ferromagnet, brings about fascinating new effects such as the reversal of the supercurrent (π-junction behav-ior) in a SFS structure and the re-entrance of superconductivity (as a function of temperature) in SF bilayers.

would not be a forbidden state on the basis of the Pauli principle; due to the anticommutation properties of electrons, a symmetric (triplet) spin wavefunction is necessarily coupled to an asymmetric orbital part, making it sensitive to dis-order. In 2001, a theory was constructed [15, 16] wherein a new spin symmetric triplet wavefunction was proposed that does not require the orbital part of the wavefunction to be asymmetric, but instead relies on an asymmetry in time (or energyspace). The demonstration of the existence of this particular triplet wave-function – which is compatible with diffusive transport – and investigation of its properties is the main theme of this Thesis.

1.4

This Thesis

This Thesis describes a series of experiments and theoretical investigations aimed at understanding the interaction between different electronic collective states at a microscopic level. Besides the obvious fundamental focus, the results also have significant practical implications, which will be discussed in more detail below.

In this Thesis we address partly theoretically and partly experimentally -the following central questions: (1) Is -there evidence for a disorder insensitive triplet correlation at the junction between a superconductor and a ferromagnet? (2) How does a non-trivial location of the Fermi surface in a superconductor influence the formation and destruction of superconducting order?

Having discussed the scope, framework and central questions addressed in this Thesis, we proceed with the description of the detailed outline.

Chapter 2: In this Chapter we provide an introduction into the important theoretical concepts that are at the basis of this thesis.

Chapter 3: In this Chapter we will give an overview of the properties of half-metallic ferromagnetic CrO2 [17], since this material will turn out to be

instrumental to the observation of triplet correlations. Furthermore, we discuss the steps that are to be taken in order to enable us to do the measurements in later chapters. In particular, we discuss the technology and processing steps we developed to make the devices, and the measurement setup required to do the measurements.

Chapter 4: This Chapter reports on the observation of a Planar Hall effect in low-field magnetotransport experiments on epitaxial (single domain) CrO2 films,

which provide the footprint of a biaxial magnetic symmetry in the plane of the films. This is surprising, since CrO2 is believed to be uniaxial.

intrinsic uniaxial anisotropy in CrO2 and other energy terms, such as the strain

in the film or demagnetization. Furthermore, it is demonstrated that we can effectively control the qualitative magnetic symmetry in the film by changing one of these experimentally accessible energy terms. Employing a simple model, we show that the magnetization reversal in CrO2 films is taking place in two

subsequent ≈ 90◦ _{steps, with coherent rotation in between these steps.}

Chapter 6: Finally, we arrive at the Chapter in which we test for triplet corre-lations. We start with the fabrication of a weak-link in a conventional, wide gap superconductor (NbTiN), by replacing a small section by a fully spin polarized ferromagnet (CrO2). We show that we are able to force a Josephson

supercur-rent through this weak link, over distances much longer than can be understood in the conventional framework, which only takes s-wave singlet correlations into account. Therefore, this experiment provides the first clear evidence for the exis-tence of triplet pairs, as predicted in theory. Further, we show that we can switch the level of the supercurrent between high and low states, using the biaxial mag-netic anisotropy of the CrO2. This indicates we have constructed a new low-loss

superconducting transistor or memory element.

Chapter 7: In this Chapter, we quantitatively compare the results in Chapter 6 with theories that existed before our experiments, and theories inspired by it.

Chapter 8: The final Chapter is devoted to the second central question raised above. Following the discovery of superconductivity it has become clear that the superconducting state can be suppressed and/or enhanced by a number of processes, such as: temperature, magnetic fields and microwave irradiation. Here we address the influence of the non-thermal effect of an applied voltage to a short superconducting wire between normal metallic contacts, and show that the voltage causes a previously unknown hysteretic switching behavior between the superconducting and normal state, which is reminiscent of a first order phase transition. Seemingly unrelated to the discussions involving superconductivity and magnetism, we note that interestingly the shift in chemical potential between the two normal reservoirs needed to switch to the normal state, is equal to the shift in chemical potential between spin up and spin down electrons at the critical magnetic field at which superconductivity is destroyed in a type I superconductor (the paramagnetic limit).

References

1. These effects are described in almost any introductory quantum mechanics textbook, see eg. D. J. Griffiths, Introduction to quantum mechanics (Prentice-Hall, 1995).

278, 1788 (1997).

3. Tans, S. J. et al. Individual single-wall carbon nanotubes as quantum wires. Nature

(London) 386, 474 (1997).

4. Anderson, P. W. More is different. Science 177, 393 (1972). 5. Laughlin, R. A different universe (Basic Books, 2006).

6. Bardeen, J., Cooper, L. N. & Schrieffer, J. R. Microscopic theory of superconductivity.

Phys. Rev. 106, 162 (1957).

7. Leggett, A. J. What DO we know about high Tc? Nature Physics 2, 134 (2006).

8. Webb, R. A., Washburn, S., Umbach, C. P. & Laibowitz, R. B. Observation of h/e Aharonov-Bohm oscillations in normal-metal rings. Phys. Rev. Lett. 54, 2696 (1985). 9. Imry, J. Introduction to Mesoscopic Physics (Oxford University Press, 1997).

10. For a review, see e.g. T. M. Klapwijk. Proximity effect from an Andreev perspective. J.

Supercon. 17, 593 (2004).

11. Gu´eron, S., Pothier, H., Birge, N. O., Esteve, D. & Devoret, M. H. Superconducting proximity effect probed on a mesoscopic length scale. Phys. Rev. Lett. 77, 3025 (1996). 12. Jedema, F. J., Filip, A. T. & van Wees, B. J. Electrical spin injection and accumulation at

room temperature in an all-metal mesoscopic spin valve. Nature (London) 410, 345 (2001). 13. Jedema, F. J., Heersche, H. B., Filip, A. T., Baselmans, J. J. A. & van Wees, B. J. Electrical detection of spin precession in a metallic mesoscopic spin valve. Nature (London) 416, 713 (2002).

14. Leggett, A. J. A theoretical description of the new phases of liquid3_{He. Rev. Mod. Phys.} 47, 331 (1975).

15. Bergeret, F. S., Volkov, A. F. & Efetov, K. B. Long-range proximity effects in superconductor-ferromagnet structures. Phys. Rev. Lett. 86, 4096 (2001).

16. Kadigrobov, A., Shekhter, R. I. & Jonson, M. Quantum spin fluctuations as a source of long-range proximity effects in diffusive ferromagnet-superconductor structures. Europhys.

Lett. 54, 394 (2001).

Theoretical concepts

2.1

Quantum interference

In solids, the constituting atoms consist of an ionic core orbited by a number of electrons. In metallic solids, some of the outermost electrons (typically 1-4) are very weakly bound to the ionic core (delocalized) and are free to roam the complete extent of the solid. These free electrons are responsible for the characteristic metallic properties such as the ability to conduct electrical currents. The number of free electrons in good metals such as copper is astronomically high. For instance, in a piece of copper there are 1023 _{electrons per cubic}

cen-timeter. Since all these electrons are free to wander through the entire piece of metal, they all experience the same background potential. Classically speaking, this means that their orbits are identical, and they occupy a state with the lowest attainable energy, since this minimizes the total energy of the metal. However, quantummechanics tells us by means of the Pauli principle, that one particular element of phasespace can be occupied by at most one electron (and more gen-erally fermion). The lowest possible energy level can thus be populated by only one electron, or two if we count spin, and the rest of the 1023 _{initial electrons}

have to populate higher and higher energy states, until all electrons have been allocated. The energy level in which the last electron is placed, and which forms the border between filled and empty states, is called the Fermi energy, EF (see Figure 2.1a).

E

E

x

_{N (E)}

_{N (E)}

E

a

b

Figure 2.1: Every state in a metal can be occupied by at most two electrons (counting spin). Left: when filling the states, the energy of the state associated with the last added electron is the Fermi energy (or level), EF. Right: the density of states for electrons

in three dimensions depends on energy as N ∝√E. Since electrons close to EF need

little energy to be motivated to occupy higher orbitals, the electrons in a small region around EF (red shading) are the only ones taking place in the transport.

effect of the application of a voltage is represented by a corresponding shift of the chemical potential with respect to EF, when connected to another metal at zero voltage, the only net current will flow in the bias window from EF to the applied voltage. The natural conclusion is thus that by measuring effects of quan-tummechanical electronic self-interference on transport properties, we only take into account the interference of relatively monoenergetic – or monochromatic – electrons at the Fermi level (see Figure 2.1b).

In order to measure interference in experiments, the phase of the electronic wavefunction should be preserved. Unfortunately, interactions with photons and phonons are in general inelastic, leading to randomization of the electronic phase and destruction of coherence. These requirements condemn the measurements to very low temperatures, since at low temperatures the number of phonon modes decays as T3_{. Beside the electron-phonon interaction that was just mentioned,}

φ

φ

Φ

Figure 2.2: The conductance of a Aharonov-Bohm ring can be influenced by a mag-netic field. The left panel shows a SEM micrograph of a AB-ring patterned in a InAs heterostructure, including a metallic gate electrode which can be used to control the carrier density. The phases of the left and right moving paths pick up opposite phases from the vectorpotential and interfere at the ring exit leading to a first order correction to the conductance of: G ∝ 1 ± cos(2πΦ_Φ0 ) (Ref. [1]).

T2_{, causes it to become dominant over electron-phonon interactions below ≈ 1K.}

In conclusion, inelastic interactions put an upper limit on the lengthscale over which the electronic phase can be considered unperturbed, and over which the electron can be described as a free electron. This upper limit is the microscopic phasecoherence length, lφ, which serves as a demarcation between the classical and quantum world. In regimes where lφ is comparable or larger than classical lengthscales such as the mean free path, the cyclotron radius or the size of the system, the classical framework breaks down, and quantum corrections to the classical Boltzmann transport equation are necessary.

φl = _~1 R l.s.p · dl = ~1 R (mv + eA) · dl φr = 1_~ R r.s.p · dl = 1~ R (mv + eA) · dl (2.1)

in which we used the vectorpotential, A, the momentum, p, the velocity, v, and mass, m. Assuming the arms to be of the same length, the kinetic part of the phase is identical for both paths, and the phasedifference due to the magnetic field, ∆φB= φl− φr, can be shown to be equal to:

∆φB= e ~ I A · dl = e ~ I ∇ × A · dS = e ~B · S = 2π Φ Φ0 (2.2) with the flux quantum Φ0 = h_e. This expression shows that the phasedifference

is modulated by the magnetic field. This modulation directly affects the conduc-tance of the ring, since the conducconduc-tance, G, is proportional to the combined transport probability, PA→B, which is the square of the sum of the transport am-plitudes along the left and the right sides, Al, Ar, which are identical up to a phase difference; that is: G ∝ PA→B = |Al+ Ar| = |Al+ Alei∆φB| = 2|Al|[1 ± cos(2πΦ_Φ0 )].

2.2

Macroscopic phase

In a many body state the influence of the individual electronic phases is overshad-owed by the phase of the macroscopic order parameter, which provides a phase reference for the individual electrons. For conventional superconductivity and simple forms of ferromagnetism, the magnitude of these order parameters (∆ and M respectively) can be found from mean field theories, in which an interaction term – that represents the broken symmetry of the system – is added to the oth-erwise symmetric Hamiltonian in the local reference frame. This term is directly related to the number of constituents which are aligned with its phase, and can be seen as the effective field resulting from the average background of electrons acting on a single electron. This number will decrease at higher temperatures, since thermal fluctuations tend to limit the degree of alignment. At a certain temperature, the critical temperature, Tc, it is no longer possible to maintain the state with the spontaneously broken symmetry, and above this temperature the many body state can only manifest itself in the form of fluctuations.

0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 ∆(T) ∆0 T_c T

Figure 2.3: The temperature dependence of the mean field order parameter ∆ of the superconducting state (we used the BCS form ∆(T )_∆

0 = tanh[1.74

q

TC

T − 1] ). For

temperatures above TC, the ordered state is destroyed, and only superconducting

fluc-tuations remain.

average occupation of excited states and the magnitude of the resulting mean field order parameter is expressed. For conventional superconductors this equation for ∆ has the form [3]:

1 N0Vef f = Z _~ωD ∆ 1 − 2f (²) √ ²2 _{− ∆}2d² (2.3)

in which the parameter that can be easily controlled is the Fermi distribution function, f (²). The other parameters are material specific, namely the Debye frequency, ωD, the density of states at the Fermi level, N0, and the effective

attractive potential, Vef f. Figure 2.3 shows the temperature dependence of ∆. Note that temperature is not the only effect that can destroy the ordered state: it can also be destroyed by non-thermal effects, such as a critical gradient of the macroscopic phase (a critical current in a superconductor), or a critical voltage (non-thermal shape of the Fermi surface) as we will show in Chapter 8.

ab-Φ/Φ

(b)

(c)

Figure 2.4: Flux-quantization can be conveniently used to bias a Superconducting Quantum Interference Device (SQUID) [5]. A SQUID, shown in panel (a), is a su-perconducting loop intersected by two weak links (J), with a total critical current, IC = 2I0| cos(π_ΦΦ₀)|, where I0 is the critical current of the weak-links and Φ is the

mag-netic flux due to the external magmag-netic field. The inclusion of a small superconducting ring in the SQUID, as shown in panel (b), with a single fluxquantum Φ0 inside, shifts

the response of the SQUID by half a period, due to the evolution of the macroscopic phase between opposite points on this ring, see panel (c). (Ref. [6]).

k

k

-k

-k

Fermi surface

Figure 2.5: Condensation in real space and in momentum space. The picture on the left shows a 165 million times magnified model of a iron (Fe) crystal lattice (the Atomium in Bruxelles, Belgium). Fortunately, the positions of the atoms are well defined. The figure on the right shows a condensation in momentum space between electrons with momentum k and −k.

2.3

Motion of quasiparticles

One of the most intuitively easy condensates is the condensation of atoms in a crystal lattice (see Figure 2.5). Since the energy involved in the interactions is much larger than the kinetic energy of the atoms, the condensation is said to take place in real space, meaning the position of the (highly localized) atoms is a good quantum number to describe the interactions between atoms involved in the formation of the condensate.

The opposite is true for superconductivity, since superconductors (in the con-ventional BCS framework) are best described by a theory of weak coupling. This means the kinetic energy is much larger than the energy involved in the inter-actions. In this case the best basis of wavefunctions describing this condensate is not a set of localized wavepackets, but rather a set of electronic plane waves which are characterized by their wavenumber (momentum). The condensation is thus said to take place in momentum space (see Figure 2.5).

µ H0 ∆ ∆∗ _−H∗ 0 ¶ µ ψe ψh ¶ = ² µ ψe ψh ¶ (2.4)

The BdG equation is a set of coupled equations describing the influence of the superconducting pair potential ∆ on electrons, ψe, and holes, ψh, at the energy ² respectively. The coupling between the electronic and holelike wavefunctions is solely determined by ∆, as can seen by putting it to zero, after which we recover the usual Schr¨odinger equation, with Hamiltonian H0 = _2m1

¡_~

ı∇ − eA ¢₂

+V −EF. Since ∆ couples the electronic and holelike wavefunctions, it is a special kind of potential. One might remember from elementary quantum mechanics textbooks, that the main effect of simple potentials like the electrical potential V of the ion cores, is limited to the refraction of electronic waves. That is, the only effect of V on the electronic wavefunction is on the momentum k. Although this can still have very significant implications on for instance the localization of the elec-tron, the superconducting potential has a more important secondary effect. Since ∆ couples the electronic and holelike wavefunctions, electrons oscillate between electronlike and holelike character inside a superconductor. The lengthscale of the oscillation ξ is inversely proportional to the magnitude of ∆. ξ sets an upper limit to the phasecoherence of individual electrons inside a superconductor, and in most cases replaces lφ since ξ ¿ lφ. Moreover, the potential ∆ can be seen as a bandgap in the superconducting density of states, as normal electrons are excluded from the energy interval (−∆, ∆) around EF.

Dirty systems

arbitrary system

(

x

',t'

)

ψ

( )

x,t

ψ

∝

)

'

_,t'

,

(

x,t

x

G

_ψ

₍

x,t

₎

_ψ

₍

x

',t'

₎

Figure 2.6: The Green’s function as an expectation value for transport. The normal metal Green’s function, G, can be used to gain information on transport properties of single particles in an arbitrary system. It measures the wave amplitude or expectation value of finding a particle at time t and place x, when it was inserted in the system at (x0_{, t}0_{). Once G is known, all single particle properties can be derived from it.}

A solution to this problem is to go to a more natural mathematical framework for describing superconductivity in diffusive (or dirty) systems, after which major simplifications are possible which will put a lower limit on the required resolution equal to the superconducting coherence length, ξ, which is roughly speaking in the range of 10 to 1000 nanometers.

The solution is presented in the form of Green’s functions, which we will present here in a very intuitive context, without going into the actual calculations, to give the reader a feeling for the powerful possibilities offered by the Green’s function formalism [8].

(

x

',t'

)

ψ

_ψ

( )

x,t

ˆ Ψ

Ψ Fˆ

S – groundstate

_Ψ

(

x

_'

,t'

)

ψ

_ψ

( )

x,t

ˆ Ψ

Ψ Gˆ

Figure 2.7: In a superconductor, the normal Green’s function G is not sufficient and should be supplemented with an extra (anomalous) Green’s function, F , describing the Cooper pairs. The wave amplitude measured by this new Green’s function can be seen as the expectation value for destroying a Cooper pair by taking two electrons out of the system at (x0_{, t}0_{) and (x, t).}

Since G is the expectation value for transport through the system, it tells us how particles move through it. Knowing this, we also have a lot of information about related properties. For instance, by letting (x, t) approach (x0_{, t}0_{) one gets} the local density of states (since in this limit G is the expectation value of the particle number operator), and by taking the spatial derivative of this quantity one obtains the electrical current.

In normal metals, where ∆ ≡ 0, the normal Green’s function G is sufficient for calculating transport properties. However, in superconductors we know that electrical currents can flow without resistance due to the condensed Cooper pairs, so clearly we should introduce into the formalism a probe for these superconduct-ing correlations as well. A logical choice would be to try to break up a pair by taking an electron out of the system in the groundstate, which leaves an unpaired electron in the system with opposite spin, assuming singlet pairing, which we can subsequently take out at a later time (Fig. 2.7). Formally this would mean we take the expectation value, F = hΨ↓(x, t)Ψ↑(x0, t0)i, with the anomalous Green’s function F .

easier to solve. This is done mathematically by separating the Green’s functions in a rapidly oscillating part, which is a function of the difference (x − x0_{), and a} slowly varying envelope, which depends on the center of mass position (x + x0_)/2. The fast oscillation is subsequently averaged out. The next approximation entails an average of the remaining Green’s functions over a disorder potential, to model the impurity scattering involved in diffusive transport. In the end, the newly obtained G and F depend on only two coordinates: center of mass position, r, and energy, ² (through a Fourier transformation of t − t0_{). It is not surprising} that the expression for the Green’s functions in a diffusive superconductor, the Usadel equation, resembles a standard diffusion equation [11, 12]:

~D£F ∇2_{G − G∇}2_F¤_{= −2ı∆G − 2ı²F (2.5)}

On the left, we recognize the diffusion constant, D, and the gradients in the diffusing quantities, whereas on the right, we find the drain terms for the phase coherence, which are proportional to ∆ (due to the interaction with the condensate), and energy ², due to the parabolic electronic dispersion relation.

Note that because of the information loss at the scale of λF, this equation fails to describe effects related to self-interference, such as the Aharonov-Bohm effect, but also effects related to strong interactions involved in the system. Most importantly, the Usadel equation is not adequate for describing superconducting correlations in ferromagnets, except in the case of very weak exchange energies (for which the critical temperature of the ferromagnet is much smaller than the one of the superconductor and the Fermi energy). Another exception is the case of the 100% spin polarized ferromagnet or half-metal, since in that case the minority spin Fermi surface can be completely omitted.

Interface effects

Andreev reflection at a perfect interface

x V(x) ψ = eikx ψ = eiqx ψ = e-|q|x x ∆(x) ψ = eikx ψ = eiqx CP ψ = e-|q|x ε −ε k_h -k A

Regular electrostatic potential Superconducting potential

E_F E_F

Figure 2.8: Transport through a SN interface compared to scattering at a regular potential barrier. On the left of the interface, electrons are described by plane waves with momentum, ~k = p2m(² + EF). On the right, the electron is slowed down to

either ~q = p2m(² + E_F − V ), or ~q = q

2m(E_F ±√²2_{− ∆}2_{). For ² < V − E}

F or

² < ∆, the electron is decaying in the barrier. However, in the case of the SN interface a new conduction channel opens, whereby a second electron with −k can form a Cooper pair in the superconductor, leaving a back reflected hole in N (Andreev reflection), which doubles the conduction of the interface at low energies.

electronic quasiparticle state decays in the superconductor, the (electrical/mass) current does not decay, indicating the presence of a second (non-quasiparticle) current. This signals the conversion from normal current (carried by electronic and holelike quasiparticle states) to supercurrent (carried by bound states of electrons) under the influence of the superconducting pair potential. A more detailed illustration shows the mechanics of this Andreev reflection process [13] (see Figure 2.8). The electronic wavefunction describing electron A decays on a certain lengthscale – of ξ = 1 |q| = λFEF 2π∆ q ∆2_−²2

S

N

*

*

*

*

*

*

*

*

*

_*

*

h

e

Figure 2.9: Real space picture of electron and backreflected hole in an Andreev reflection event at a SN interface. The electronic dispersion in the normal metal causes (slow) dephasing between the otherwise identical electron and hole orbits. For energies ² < Eth the phase coherence is preserved in all of the normal metal.

Cooper pairs in a normal metal

Beside the doubling of the conductance of the NS interface, the existence of the retroreflected hole has another important consequence: since the hole traces back the path of the initial electron in real space, as shown in Fig. 2.9, the electron and hole are indistinguishable as long as their paths are in phase. This means that physically, there is no way to distinguish the electron-hole pair from a Cooper pair inside the superconductor unless there is some dephasing between the two paths. In a normal metal, the dephasing term depends on the energy ² of the initial electron: since the Cooper pair has to be formed at the Fermi level (zero energy), the retroreflected hole occupies the state at −². As a consequence of the time evolution in the Schr¨odinger equation, there is an increasing dynamical phase difference e2it²/~ _{between the wavefunctions of the electron and hole, where t is}

the elapsed time since the collision with the interface. When this phasedifference equals π, the two waves will be out of phase and thus distinguishable. In a diffusive system, electrons travel over a distance L = √Dt in a time t, where D is the diffusion constant. The coherence between the electron and hole will thus be preserved over a distance L² =

q

~D

² , a quantity known as the normal metal coherence length. Consequently, in a system of size L, the decoherence will not set in for energies ² < Eth where Eth = ~D

L2 is the Thouless energy of the system. For all practical purposes, electron-hole pairs with ² < Eth can hence be seen as Cooper pairs diffusing from the superconductor into the normal metal.

respon-x ∆_(x) ε

E

S

N

I

Figure 2.10: The transport through a weakly transparent (tunnel) interface between a superconductor and a normal metal depends on the energy of the incoming electron. Every electron has a chance of T to tunnel through the barrier. However, for ² < ∆, a Cooper pair should be formed, for which another electron should cross the barrier at the same time, thereby reducing the chance to T2_{. The additional effect of the}

singularity in de Density of States in S (NS = √_²2²_−∆2) for ² ≥ ∆ is not shown in the

Figure.

sible for a large number of effects related to the proximity of a superconductor to a normal metal. It is for instance possible to force a (zero resistance) su-percurrent through a superconductor – normal metal – superconductor system [14], where the maximum magnitude of the supercurrent, the critical current Ic, characteristically depends on the Thouless energy of the normal metal. Likewise, the density of states in a normal metal connected to a superconductor mimics the density of states of the superconductor for distance from the interface smaller than L² [15].

Tunnel interface

A bad interface between a normal metal and a superconductor can be adequately modelled with the well known model of quantummechanical tunneling through an insulating (vacuum) barrier. The transport probability T for tunneling through the insulator in the normal metal - insulator - superconductor system (NIS, see Figure 2.10) is exponentially suppressed with increasing barrier thickness d, as e−d/λF where λ

eV/2 -eV/2 f 0 0 1 L x N reservoir N reservoir +V/2 N wire -V/2 L x = 0 V/2 V/2

Figure 2.11: Left: Non-equilibrium in a mesoscopic (L ¿ lφ) voltage biased wire.

Right: The electronic distribution function, f , is Fermi-Dirac like in the reservoirs. Owing to the lack of inelastic scattering, f has a non-thermal two step shape in the middle of the wire.

be transferred to be able to speak about a successful tunnel event. However, for energies ² < ∆ an electrical current can only be transported when two elec-trons tunnel at the same time to create a Cooper pair in the superconductor, thereby reducing the total probability to T2_{. The Blonder-Tinkham-Klapwijk}

(BTK) model [16] describes the transport through the NIS interface for arbi-trary transparency of the insulator part, and has been one of the most successful instruments for the analysis of tunnelling barriers on the basis of experimental data, since it covers the total spectrum of transparencies. Moreover, an extended version of this model – which incorporates spin – is widely used to extract the interfacial spin polarization of ferromagnetic materials which in that case serve as the normal metallic contact.

2.4

Non-equilibrium superconductivity

x ∆_(x) ε −ε h

Energy mode

E

n

(ε) = n

(-ε)

Charge mode

E

n

(ε) = 1-n

(-ε)

Figure 2.12: The electronic distribution function, f , can be separated in symmetric (energy mode, f_L) and asymmetric (charge mode, f_T) components in particle-hole space. In a normal metal their behavior is identical, whereas in a superconductor these modes have different equations of motion.

Mathematically, the distribution function of the electrons, f (²), which we from this moment on no longer assume to be equal to the Fermi-Dirac distribution, can be separated (like any function) into a symmetric part and an asymmetric part in particle-hole space (see Fig. 2.12) [18]. Physically speaking, the symmetric part, fL, describes excitations due to an energy mode non-equilibrium which can for instance be created by thermal excitations or by excitations due to microwave sources. The asymmetric part, fT, is related to a charge mode non-equilibrium which can for example be caused by a nearby voltage source, which injects either electrons or holes into the system.

2.5

Super-ferro heterostructures

The complete disappearance of the electrical resistance below a certain tempera-ture is but one of the two defining properties of materials we nowadays call super-conductors. The second characteristic attribute is the perfect diamagnetism (or Meissner effect) which causes all magnetic fields to be expelled from the interior of a superconductor once it is cooled below the critical temperature.

A superconductor can not maintain this perfect diamagnetism up to arbitrar-ily high external magnetic fields, and above a certain field, the critical field, it returns to the normal state. There are two main effects, related to the magnetic field, that are responsible for this destruction of superconductivity. The orbital effect, caused by the interaction of the superconducting phase and the vectorpo-tential A, can destroy superconductivity when the kinetic energy of the particles becomes larger than the condensation energy of the pairs [19]. The second effect of the field is that it tries to align the spins along the field. In the normal state, this paramagnetic response leads to a gain in polarization energy. However, in the superconducting state, this energy advantage is not available since the pairs are condensed in a singlet configuration. Superconductivity is destroyed when the polarization energy exceeds the condensation energy of the pairs. In a bulk superconductor, the field at which this happens is given by µBHc = 1₂

√ 2∆0

(the Clogston limit) [20]. These effects predict bad news for the coexistence of (singlet) superconductivity and (ferro)magnetism, since the pairing interaction is heavily suppressed by the magnetic field (for a typical ∆ ≈ 0.1 − 1 meV, Hc = 1.2 − 12.2T ). What it also means is that it is impossible to study super-conducting correlations at higher magnetic fields, since the pair potential will collapse before we reach these fields.

Heterostructure experiments involving superconductors electronically coupled to ferromagnets do not suffer these drawbacks, and do allow to gain insight in the behavior of individual superconducting wavefunctions at very high magnetic fields. In ferromagnets, the exchange energy between spin up and spin down electrons can be seen as an artificial internal magnetic field. Since this is a bandstructure effect, the resulting exchange field can be gigantic. Just to give an impression: a relatively small exchange energy difference of 10 meV already gives rise to an effective exchange field in the order of 100 T. An additional consequence of the bandstructure nature of the exchange field, is that the field is only perceived by electrons in the ferromagnet, and not by those in the superconductor. This protects the source of superconducting coherence, the bulk superconductor, from being forced to revert to the normal state.

interface whose transparency has the same two limiting cases as the SN interface: fully transparent and insulating (the tunnel limit). In the latter, the transport is very similar to the same limit of the SN interface, something which is not the case for the fully transparent case.

The transparent interface

Let us start this discussion by noting that a fully transparent interface indicates the absence of any potential barrier at the SF junction. It means that every parti-cle impinging on it has a transmission probability of 1, if and only if, the materials on both sides of the interface are identical. In general this is not the case, and we find more complex behavior. The most trivial example is a transparent inter-face between two normal metals with spherical Fermi surinter-faces but different Fermi energies. Due to the mismatch in the corresponding Fermi wavevectors, even in this case the transmission will not be one. Beside a wavevector mismatch, also a difference in the density of states at the Fermi level, N, plays a role. This is especially true in ferromagnets, since N can be completely different for spin up and spin down electrons, and can be quantified by the polarization, P , for which we use the definition:

P = N↑− N↓ N↑+ N↓

(2.6)

For a normal metal, P = 0, and for a material with complete (100%) spin polarization P = ±1. Usually for ferromagnets P ≈ 30 − 50%.

Material Polarization

Iron (Fe) 42-46% [23]

Nickel (Ni) 43-47% [23]

Cobalt (Co) 42% [23]

Nickel Manganese Antimonide (NiMnSb) 58% [23]

La1−xSrxMnO3 (LSMO) 78-83% [23, 24]

Chromiumdioxide (CrO2) 90-97% [23, 25, 26]

Table 2.1: Spin polarizations of several ferromagnets as found from PCAR experi-ments.

experiments, where a superconducting (atomically sharp) tip is contacting a fer-romagnetic sample, or vice versa. The polarization, P , can be found quite easily from the ratio of the conductances at zero bias, GF S, and at high (eV À ∆) bias, GN. For several ferromagnets, the experimentally determined values of P are displayed in Table 2.1. In general, when the Fermi velocities are not equal, the value of P extracted from a PCAR experiment can not be interpreted simply in terms of N↑ and N↓ (ie. as in Eq. 2.6), since it also includes bandstructure effects [22], but it can be seen as some sort of polarization of the transport. To find the true value of P as defined in Eq. 2.6 other spectroscopic tools yield better results. We will return to this in the next Chapter.

The effect of a highly transmissive interface in combination with a finite polar-ization is thus to suppress Andreev reflection, and therefore the ability of Cooper pairs to penetrate the ferromagnet. This is clearly a very important effect, which could perhaps explain some peculiar differences in SF experiments on ferromag-nets with high and low spin polarizations [27, 28, 29], but which is strangely enough not taken into account by most modern theories when calculating the properties of SF heterostructures.

Cooper pair in a ferromagnet

S S F dF IS V_c 2 -V_c 2 S S

Figure 2.13: Left: The spatial oscillation of a Cooper pair in a SF structure can be probed with a second superconductor. The thickness of the ferromagnet in comparison with ξF determines the phase of the oscillation (Ref. [35]). Right: a similar oscillation

in a SNS junction can not be probed by varying the junction length, since the oscillation is energy dependent. Instead, by carefully blocking low energy states normally used by the supercurrent using a voltage, the oscillation can be revealed (Refs. [36, 37]).

exchange field of the ferromagnet. To remain at the Fermi energy, this energy dif-ference has to be balanced by a difdif-ference in kinetic energy, and thus momentum, which means the electrons in the pair have a different rate of acquiring kinetic phase, leading to a spatial modulation, or oscillation, of the wave amplitude of the pair [30, 31, 32, 33, 34]. Since the exchange energy is typically much larger than the magnitude of the superconducting gap, the oscillation is relatively en-ergy independent for electrons that contribute to the superconducting properties of the ferromagnet. Therefore, the coherence length in the ferromagnet in the diffusive limit, ξF, is also energy independent: ξF =

q

~D

Eex, so that contributions

at different energies are added constructively. Moreover, ξF is usually of a partic-ular short-range character. For a typical Eex ≈ 0.1 eV (which corresponds to a ferromagnet with a TC of around 1000 Kelvin), ξF is already much smaller than 10 nm.

junction, a supercurrent can be transported of which the magnitude and sign are a direct measure for the pair amplitude in the ferromagnet. By varying the thickness of the ferromagnet with respect to ξF we can determine how far we are in the oscillation (see Fig. 2.13). In this way it is even possible to measure negative pair amplitudes, which give rise to negative coupling and a negative current through the SFS junction (in the so called π-junction state) [38, 39, 40]. The spatial oscillation is also present in normal metals. However, it is much more difficult to observe. We already noted that in the normal metal the length scale of the oscillations, the normal metal coherence length L² =

q

~D

² depends on energy. In a SNS junction, all these energy dependent contributions with different phase evolution sum up to a certain value, in which positive contributions always dominate the negative ones under normal conditions. Only with clever experiments in which the positive contributions are suppressed the pair amplitude in the SNS junction can be shown to also feature negative contributions (cf. Fig. 2.13) [36].

2.6

Long-range correlations

Overall the effects described in the section above are very short ranged in compar-ison to the normal metal coherence length, the main reason is that singlet pairs, with opposite spins, can not survive for a long time in the hostile environment of the ferromagnetic exchange field, which tries to align the spins.

If somehow the singlet wavefunction could be converted into an equal spin triplet correlation (| ↑↑>), it would be much more at home in the ferromagnet, since the evolution of the phase in the field would be the same, leaving only the normal metal dephasing due to the usual electronic dispersion. As explained in the Introduction, at the same time the orbital part of the two electron wavefunc-tion should always be in a s-wave orbital configurawavefunc-tion in a diffusive material to survive disorder scattering over longer distances. This results in problems with the Pauli principle.

The Pauli principle states that no two identical fermions can occupy the same state in (phasespace) at the same time. Another way of expressing this principle, is by requiring the two electron wavefunction Ψ(r1, r2) to be antisymmetric under

the exchange of the particles in the generalized positions r1 and r2, which are

HT_c Cuprates Superfluidic3_{He, Sr}

2RuO4

Conventional (BCS superc.)

Occurence Disorder average

d - wave (even) Singlet (odd) p - wave (odd) Triplet (even) s - wave (even) Singlet (odd) Orbital Spin Orbital Spin Ψ(r1,r2) Occurence odd odd odd

SF heterostructures, this Thesis Triplet (even) s - wave (even) odd

Time even even even odd

Figure 2.14: The Pauli principle allows for two electron wavefunctions which are combinations of even (odd) orbital parts and odd (even) spin parts, so that the total wavefunction Ψ(r1, r2) is odd under exchange of particles. This defines the possible

wavefunctions for the ‘classical’ types of superconductivity. If the generalized coor-dinates of the particles include time (or energy) as well, new two electron states are allowed (the complementary set) when these states are odd in time.

since it is symmetric under exchange of particles. This seemed to be the end of the story for long-range correlations in ferromagnets.

In 2001, it was shown that the Pauli principle could be circumvented – or better: inverted – by a two particle wavefunction that is retarded in time [42, 43]. More specifically, the Pauli principle is automatically satisfied when the generalized coordinates of the particles include time, and Ψ is odd under the operation t ® t0_{, since in that case by definition Ψ(t = t}0_{) = 0 (which means the} symmetry or asymmetry of the spatial and spin part of the wavefunction plays no role of importance, since the particles do not occupy phasespace at the same time anyway). Therefore, the s-wave triplet wavefunction is no longer a forbidden wavefunction, as long as it is odd in time.