Superconductivity and Low Temperature Physics

Transcription

1 Superconductivity and Low Temperature Physics Part I: Superconductivity Lecture Notes of the Academic Year 2013/14 Rudi Hackl and Dietrich Einzel Walther-Meissner-Institut Bayerische Akademie der Wissenschaften Walther-Meissner-Strasse 8 D Garching hackl@wmi.badw.de c preliminary Garching, November 21, 2013

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3 Contents 1 Introduction A brief history of low-temperature physics Condensates Present Status of Superconductivity Materials Areas of Application Solved and Open Problems Basic Experiments and Understanding Key Experiments Zero Resistance Perfect diamagnetism Shubnikov phase (mixed state) Quantization of the flux Josephson effects Summary Thermodynamics Condensation Energy Entropy Specific heat capacity Electrodynamics The Drude model in the limit τ Generalized London theory The London equations Some conclusions iii

4 iv R. HACKL AND D. EINZEL CONTENTS 3 Microscopic Theory The Cooper Instability Origin of the interaction The BCS wave function Coherent states in a boson field Properties of fermions A coherent state of fermions Determination of the ground state The BCS Hamilonian in second quantization Some expectation values Determination of the energy minimum at T = The general solution Approximation of the four-operator term The Bogoliubov-Valatin transformation Solution of the gap equation for T Connection to experiments Thermodynamic properties Single particle response Ginzburg-Landau Theory Phase transitions Application to superconductivity Density of the free energy Functional of the free energy The Ginzburg-Landau equations Two new length scales Screening Ginzburg-Landau coherence length Energy of the normal-superconductor interface States with internal flux The upper critical field B c The nucleation field B c3 on the surface The thermodynamic critical field B c The lower critical field B c The Abrikosov lattice (1957) c Walther-Meißner-Institut

5 CONTENTS SUPERCONDUCTIVITY v 5 The Josephson Effect Weakly coupled superconductors The Josephson equations The RCSJ model Josephson contact in a microwave field Josephson effect in a magnetic field Ring with a single weak link Ring with two weak links: Quantum interference Quantum interference in a long junction Unconventional Materials Classification The iron-age of superconductivity Copper-oxygen compounds History Materials Physical properties Superconductivity Summary and perspectives An overview of applications Potential areas Economic considerations Areas of application Passive applications Physical and technical challenges Examples Active devices

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7 Preface This is a first version of lecture notes on superconductivity on an introductory level. It is intended to be used along with the lectures I deliver at the Technical University Munich. The manuscript is the result of several courses on the subject during the last years and should not be considered final. In fact, it will need a few more iterations. In particular, the referencing is not yet complete. The notes are inspired by several textbooks and lecture notes on superconductivity or Condensed Matter Physics including the books by J. F. Annett, N. Ashcroft and D. Mermin, W. Buckel and R. Kleiner, P.-G. de Gennes, R. Groß and A. Marx, H. Kinder (notes of A. Heinrich), C. Kittel, A. Sudbø and C. Fossheim, V. V. Schmidt (edited by P. Müller and A. Ustinov), and M. Tinkham. We profitted from numerous discussions with our colleagues working in the field, in particular at the Technical University Munich and at the Walther-Meissner-Institut. The notes are not for distribution. Garching, October 16, 2013 Rudi Hackl vii

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9 Chapter 1 Introduction Imagine a macroscopic object which obeys the rules of quantum mechanics and can be described by a single wave function. What would be the consequences? This question is at the heart of the phenomena to be discussed here. An intuitive answer can be found by considering the usual microscopic quantum world. The wave function describing the electron in a hydrogen atom, for instance, has an amplitude and a phase. The amplitude corresponds to the probability of finding the electron in a volume element around the pointr. The phase determines the type of the stationary state dictated by the uniqueness and the number nodes of the wave function. In the stationary state there is no dissipation. The superposition of stationary states leads to interference phenomena and transitions with typically very sharp energies. Superconductivity and superfluidity along with Bose condensation or density wave order (periodic modulations of the magnetization or charge density) are macroscopic quantum phenomena. In all cases the establishment to a coherent quantum state facilitates charge, mass or spin transport without energy dissipation. For appropriately prepared experimental settings interference phenomena can be observed. In the case of superconductivity the characteristic energy is solely determined by constants of nature. Electromagnetic radiation can be created and extremely sensitive detectors can be constructed which exploit the rigid phase of the wave function. The emergence of quantum phenomena on a macroscopic scale, i.e. in a large ensemble which can be described by statistical methods, can be expected when temperature is not the dominant energy scale any further. Therefore, the discovery of macroscopic quantum phenomena in the early 20 th century is closely related to the development of low temperature techniques although meanwhile a variety of manifestations have been found to exist also at room temperature and above. In this chapter we first sketch briefly the hallmarks of physics at low temperature (with the focus placed on superconductivity), then qualitatively describe examples of condensates and finally provide an overview over the current status of superconductivity. 1.1 A brief history of low-temperature physics The liquefaction of air in 1895 [?] can be considered the starting point of low-temperature physics. It took only 13 years until Heike Kamerlingh 1 Onnes in Leiden succeeded to liquefy Helium, the gas with the lowest boiling point [?]. Not only does He have the lowest boiling point of all elements but it is also the only material which does not solidify upon further cooling. Only at an applied pressure of approximately 30 bar the solidification can be induced. This phenomenon is among the first manifestations of quantum 1 Privy Councillor ( Hofrat ) 1

10 2 R. HACKL AND D. EINZEL Introduction (a) (b) Figure 1.1: Resistances of metals at low temperature. Note the different temperature scales in (a) and (b). (a) Proposals for the low-temperature resistivity of metals as of Temperatures below 15 K were unaccessible. Curves 1 through 3 were suggested by Dewar, Matthiessen, and Kelvin, respectively. From W. Buckel, Supraleitung VCH Weinheim 5th Edition (1994), Fig. 1. (b) Resistance of Hg close to the superconducting transition. From Ref. [1]. effects directly determining the macroscopic properties. If one estimates the amplitude of the zero-point oscillations on the basis of the uncertainty principle values on the order of 10 to 20% of the atomic diameter are found which are big enough to prevent the establishment of a crystal lattice. Only three years later Onnes observed vanishing resistance upon cooling mercury (Hg) below 4.2 K. [1] At that time the low temperature properties of metals were discussed intensively since the experimental data ended at some 15 K being close to the solidification point of liquid hydrogen. There were three main concepts Fig. 1.1 (a): (1) Probably inspired by the Drude model 2 Dewar argued in 1904 that ρ should asymptotically approach 0 in the limit T 0 since all scattering mechanisms might freeze out. (2) Matthiessen 3 observed that the increase of the resistivity of a metal as a consequence of a small concentration of another metal in a solid solution was temperature independent. This implies that the resistivity of a metal can stem from more than one source, for example a constant one from impurities, ρ 0, and a temperature dependent one of different origin, ρ(t ), which add up. 4. As a consequence the resistivity saturates at low temperature at a finite ρ 0, the residual resistivity. (3) Alternatively, the resistivity was proposed to diverge due to electron localization (Lord Kelvin 1902). 5 In the experiment on very clean Hg, Onnes and his collaborators unexpectedly found the resistance to become abruptly unmeasurably small at a finite critical temperature T c as shown in Fig. 1.1 (b) rather than asymptotically in the limit T 0. He called this qualitatively new phenomenon superconduction. It was clear from the beginning that none of the three scenarios discussed above was capable of describing the observation. It took until 1957 until Bardeen, Cooper, and Schrieffer succeeded in presenting a microscopic theory of the superconducting 2 P. Drude A. Matthiessen, Ann. Phys. Chem. (Pogg. Folge) 110, 190 (1860) and A. Matthiessen and G. Vogt, Ann. Phys. Chem. (Pogg. Folge) 122, 19 (1864) 4 Matthiessen s rule is purely empirical and valid only if the different scattering mechanisms are independent and isotropic. For a discussion see Ashcroft and Mermin Solid State Physics (Holt-Saunders International Edition 1981) 5 Anderson showed in 1958 (Phys. Rev. 109, 1492) that even a small amount of impurities can localize electrons due to interference of Bloch waves. Even in a completely pure system (with low carrier density) the resistivity can diverge due to the crystallization of the electrons (Wigner). c Walther-Meißner-Institut

11 A brief history of low-temperature physics SUPERCONDUCTIVITY 3 state [2]. A discovery of enormous importance and influence was the demonstration of thermodynamic nature of the superconducting phase by Ochsenfeld and Meissner in Soon thereafter in 1935, Fritz and Heinz London suggested the first phenomenology [?] and proposed a first quantum mechanical approach in In the same year Ginsburg and Landau applied Landau s theory for phase transitions to superconductors using a complex order parameter rather than a real one such as for magnetism or lattice distortions. In the year of publication of the microscopic theory Abrikosov used the Ginsgurg Landau phenomenology to derived the existence of a mixed state accommodating superconductivity and a magnetic field on a mesoscopic scale [?] as observed by Shubnikov and coworkers already in 1936 [?]. This discovery paved the way towards a huge variety of applications since electric currents and very large magnetic fields can coexist here with superconductivity facilitating the quasi-dissipationless maintenance of magnetic fields in excess of 20 T. Finally in 1960, Eliashberg made connection to real materials in his celebrated strong-coupling approach [3] which is used to date for analyzing interaction potentials in superconductors. In 1938 a phenomenon analogous to superconductivity was found in an uncharged system. Here, mass flows without viscosity. This property is called superfluidity and was independently discovered by Kapitsa 6 and by Allen and Misener 7 in Helium below 2.17 K. Rotation corresponds to the magnetic field and vortices of electric current around lines of the field (see section??) to vortices of the liquid. If two lumps of a superconductor or two reservoirs of a superfluid are weakly connected in a way that only the supracomponent can pass, the eventually occurring voltage and pressure drops, respectively, correspond. The spectacular fountain effect (Fig. 1.2) is a result of the pressure difference across a socalled weak link (realized by fine powder in the lower part of the nozzle which is transparent only for the superfluid) induced by a temperature difference between the reservoirs (by heating the upper part close to the narrow nozzle). The corresponding phenomenon in a superconductor is the Josephson effect predicted 1962 [?] and observed one year later [?]. It is one of the most spectacular manifestation of the quantum nature of the superconducting condensate and became the basis of a highly sensitive detector of magnetic fields with a broad range of applications [?]. In 1972, Lee, Osheroff, and Richardson discovered superfluidity also in the lighter isotope 3 He. The transition temperature T c = 2.2 mk is three orders of magnitude smaller than that of the of the heavier isotope. 8 At first glance, superfluidity in the two isotopes appears to be only quantitatively different. However, quantum effects dominate the behavior at these temperatures and the nuclear spin of 1/2 prevents 3 He from Bose condensation occurring in S = 0 4 He. As opposed to a condensate of strongly interacting 4 He atoms 3 He forms a pair condensate similar to that of a superconductor. In either case the condensate is neutral. Only one year later A. Leggett proposed a full theoretical description of the various phases of 3 He [4]. One of the most important technical innovations, proposed in 1962 and first demonstrated in 1965, was the realization of the dilution refrigerator using mixtures of 3 He and 4 He. Here, the cooling is achieved by dissolving 3 He in 4 He. The minimal temperature of this continuously working cryo-system is below 10 mk. The dilution refrigerator is the basis of many ultra-low-temperature experiments and replaced the cooling by paramagnetic salts almost completely. If a nuclear demagnetization stage is attached a record base temperature of 8-10 µk was realized. Lower temperatures can be reached only in cold gases (see below). In the range between 10 µk and 1 K quite a few new superconductors and unexpected phases of matter were found as described in the textbooks and to be discussed partially in the second part of the lecture series. In these days dilution refrigerators do not need pre-cooling with liquid 4 He any further as in the early days. With the continuous improvement of closed-cycle refrigerators µk can now be generated by 6 P. Kapitsa, Nature 141, 74 (1938) 7 J.F. Allen and A.D. Misener, Nature 141, 75 (1938) 8 Lee, Osheroff, and Richardson, Phys. Rev. Lett. vv, ppp (1972) 2013

12 4 R. HACKL AND D. EINZEL Introduction Figure 1.2: Bose-Einstein condensation. (left) Demonstration of the fountain effect in superfluid He. The Photograph was taken by Jack Allen in 1970 who co-discovered superfluidity. From R.J. Donnelly, Physics Today, July 1995, p. 30. (right) Bose-Einstein condensation of 87 Rb at 500 nk. The series of pictures shows the momentum distribution of Rubidium atoms in a trap at different temperatures. If the majority of the atoms is in the lowest state the total momentum approaches zero. From cms/projects/bec/. combining a pulse-tube unit [?,?] with an optimized dilution system. The prototype of this innovation was developed at the Walther-Meissner-Institut [?] and is now commercially available. Another important discovery at low temperature is the integer and fractional quantum Hall effect which, in connection with the Josephson effect, completes, for instance, the metrological triangle. Spin and charge density waves SDW/CDW are ordering phenomena very similar to the superconducting state. Here, the ordering produces a periodic modulation of the carrier or spin density [5]. The modulation amplitude can vary between a fraction of a percent and several ten percent of the average charge density. While in a superconductor two electrons form a Cooper pair having vanishing net momentum density waves are characterized by a finite ordering vector connecting particle and hole states. The most spectacular discoveries of the late 20 th and the early 21 st century are the Bose-Einstein condensation of trapped gases [?], the observation of superconductivity above 100 K and 40 K in the cuprates [?, 6] and, respectively, in iron pnictides [7, 8]. These most recent developments demonstrate that low-temperature physics in general and superconductivity in specific remain vibrant fields of condensed matter physics. A timetable displaying the history in consecutive order is planned to be provided soon in the appendix. 1.2 Condensates Schrödinger was the first to elaborate on a coherent state of Bosons [?] and constructed a coherent wave function from a superposition of an infinite number of harmonic oscillator wave functions ψ n = 1/ n!(â ) n 0 with â the creation operator and 0 denoting the ground state, α = e α 2 2 n=0 α n ψ n = e α 2 2 n! n=0 (αâ ) n 0. (1.2.1) n! c Walther-Meißner-Institut

13 Present Status of Superconductivity SUPERCONDUCTIVITY 5 Here, α = α e iϕ is a complex number in polar representation having the amplitude α and the phase ϕ. As shown in problem 1 α obeys a Poisson distribution which is sharply peaked at the average occupation number n = N. In addition, one finds an uncertainty relation between the average occupation number N and the phase ϕ to hold implying that for a large number of particle ϕ/ϕ = 1/ N. Coherent states have this structure in general. Here the wave function is constructed for Bosons and can directly applied for the description of the laser field or for Bose-condensed cold atoms. In principle, it is also applicable for 4 He but strong coupling effects must be considered in this case and make affairs much more complicated. Is the wave function also useful for Fermionic systems? It is the achievement of Schrieffer to construct a coherent state similar to that in Eq. (1.2.1) from the Fermionic wave function of electrons. The nature of Schrieffer s state will be derived in detail in chapter 3. The origins of a superconducting and a Bose condensate are quite different. Superconductivity originates from pair correlations between the electrons 9 In the case of Helium or, similarly, for vapors of alkali metals 10 the individual atoms undergo a Bose-Einstein condensation at sufficiently low temperatures, which means that a macroscopic number of particles assumes the lowest possible state. The difference between the alkalis and Helium is just the interaction strength V e f f. In the weak-coupling limit the first excited state has essentially the same energy as the ground state. With increasing coupling a gap develops between the ground state and the first excited state. Then, following the argumentation of Landau, condensed particles can be accelerated to a finite critical velocity v c before they leave the ground state and dissipate energy. In spite of the fundamental differences between superconductors and Bose condensed systems all form a coherent ground state which can be described by a single quantum mechanical wave function. 1.3 Present Status of Superconductivity Materials The development was continuous but quite slow in the beginning. While many elements become superconducting as shown in Fig. 1.3 the transition tempertures are moderate, and Nb having T c = 9.26 K holds the record. If pressure is applied even semiconductors may become superconducting. At pressure values in the range of GPa ( Mbar) even insulators or alkali metals occasionally reach T c values between 15 and 25 K (see Table??). In Fig. 1.3, the elements with pressure-induced superconducting transitions are marked with red points. On the other hand, compounds soon exceeded the transition temperatures of the elements as shown in Fig. 1.4 and Table??. This was the time period when materials sciences flourished and many compounds and new properties and effects were discovered which demonstrated the potential of applications while the transition temperatures T c increased at a very low rate of 0.4 K/year (Fig. 1.4). The discovery of superconductivity in the copper oxygen system La 2 x Ba x CuO 4 was a qualitative change in many respects. 11 The slow increase of T c which in a sense was the linear analogue of Moore s law was interrupted in the non-murphyian direction. Most importantly, the superconducting state turned out to be unconventional meaning that in addition to the gauge symmetry also the rotational symmetry is broken at the transition. In other words, the order parameter and the energy gap have a lower symmetry than the crystal. Unconventional superconductivity was not completely new 12 but was considered an exception occurring only in a few exotic systems such as 9 We note here, that Bogoliubov had results equivalent to those of BCS at approximately the same time but the publication was delayed. 10 Cornell, Ketterle, and Wieman 11 J.G. Bednorz and K.A. Müller, Z. Phys. B (1986) 12 F. Groß et al., Z. Phys. B (1986). It is a coincidence with some charm that the article of Bednorz and Müller on La 2 x Ba x CuO 4 directly follows the first clear evidence of an unconventional gap with non-zero orbital momentum in UPt 3 and UBe

14 6 R. HACKL AND D. EINZEL Introduction Figure 1.3: Transition temperature T c of the elements. The latest record is T c 25 K in Ca at 210 MPa [?]. Nb having T c = 9.26 K is the elements with the highest transition under normal conditions. Higher values were obtained only with applied pressure. From perat ture T c (K) tran nsition n tem 150 HgBa 2Ca 2Cu 3O 8+δ Tl 2 Ba 2 Ca 2 Cu 3 O 10+δ Bi 2 Sr 2 Ca 2 Cu 3 O 10+δ NdFeAsO 1-x F x YBa 2 Cu 3 O 6+x 2 Ba 1-x K x Fe 2 As LaFePO La 2-x Ba x CuO 4 MgB 2 3 NbTi V 3 Si Nb 3 Ge 2 Hg Cs 2 RbC 60 Ba 0.6 K 0.4 BiO year of fdiscovery Figure 1.4: Maximal transition temperature T c vs. year of discovery. The discovery of superconductivity in the cuprates is a historical hallmark. Superconductivity in Fe-based compounds was similarly unexpected. 13:56:26] the heavy fermion compounds (e.g., UPt 3 ) or, more recently, in the ruthenates (e.g., Sr 2 RuO 4 ) 13 or in superfluid 3 He. The cuprates, on the other hand, cover a very broad class of materials which have maximal transition temperatures Tc max in a range from about 20 K in Bi 2 Sr 2 CuO 6 to 160 K in HgBa 2 Ca 2 Cu 3 O 8. Surprisingly enough, apart from Tc max the temperature versus doping phase diagrams are similar: all are antiferro- 13 G.M. Luke et al., Nature 394, 558 (1998) c Walther-Meißner-Institut

15 Present Status of Superconductivity SUPERCONDUCTIVITY 7 magnetic insulators at half filling and superconductivity exists for 0.05 < p < 0.27 with p the number of holes per CuO 2 formula unit (away from the Cu3d 9 configuration). 14 While the first applications are already on the market, the orgin of superconductivity is still an open question at least for the majority of the people working in the field. High-T c research is indeed among the most vibrant fields in solid state physics in these days, and we devote a complete chapter to it. Since the discovery of the cuprates several other quite interesting materials were found to superconduct. The biggest surprise was certainly MgB 2 with T c = 39.5 K. 15 Very recently, also elements were found to become superconducting at temperatures close to 20 K under sufficiently high pressures. 16 In January 2008 superconductivity was reported in Nd(O 1 x Fe x )FeAs compounds. 17 By exchanging the rare earth a maximal T c of 55 K was obtained Areas of Application It is immediately clear that the state of zero resistance offers enormous opportunities for applications. The first realization was a solenoid for fields in the range of 5 T in a volume of a few cubic-centimeters. In these days, the maximal permanent field maintained by a superconductor is slightly above 21 T, and the largest volume is several m 3 in coils for fusion experiments. Applications include now power transmission, generators, motors, energy storage, fault-current limiters, and filters. The most popular application which many people have already encountered are superconducting coils for magnetic resonance imaging. Beyond the state of R = 0 there are various other properties of superconductors which can be exploited. For instance, the behavior of weakly coupled superconductors opens a wide field of active devices which rest on the Josephson effects. For sufficiently low currents there is no voltage drop across the weak link (dc Josephson effect), and the phase difference between the two superconductors remains constant. For higher currents there is also a voltage drop making the phase difference time dependent. It follows from the Josephson equations that the current has now oscillating components (ac Josephson effect). Exploiting the dc effect extremely sensitive detectors for magnetic fields can be made (Superconducting QUantum Interference Device) which allow one to observe and to locate the currents in the heart and in the brain. The ac effect can be used to produce and detect electromagnetic waves in the hundred GHz range. Other very important applications are the voltage standard and, along with the quantized Hall effect, the determination of Planck s constant h and the elementary charge e Solved and Open Problems The London brothers could clarify the phenomenology and electrodynamics of superconductors relatively soon after the discovery. In 1935 they proposed a model which was capable of describing the exclusion of magnetic fields from bulk superconductors and discussed the exponential decay of the field in a surface sheeth of thickness λ L which is now called the London penetration depth. They understood that the effect was quantum mechanical in origin and describable in terms of a single wave function being closely related to coherent states as proposed by Schrödinger in However, the microscopic foundation of this wave function was completely unclear in particular since Schrödinger s proposal was for Bosons and not for Fermions. It took another 22 years until Schrieffer could derive a coherent state wave function also for Fermions which finally completed the BCS theory of superconductivity [2]. 14 We note that the AF ordering (Néel) temperature T N and Tc max do not scale. 15 Akimitsu et al., Nature vvv, ppp (2001). 16 J.J. Schilling, Schrieffer s book, ppp (2006). 17 abc et al., J. Am. Chem. Soc. vvv, ppp (2008). 18 abc et al., Nature vvv, ppp (2008). 19 E. Schrödinger, Naturwissenschaften 28, (1926). 2013

16 8 R. HACKL AND D. EINZEL Introduction The BCS theory considered only the limit of weak coupling. Eliashberg succeed to establish a connection between the interaction potential V k,k and spectra density of phonons. This became possible after Migdal s observation that the electron-phonon vertex is of order m/m with the electron and ion masses m and M. Physically speaking this means that the electron-phonon interaction is retarded or that the respective energy scales differ by orders of magnitude and can be considered independent. Eliashberg s approach can be generalized to other type of interaction as long as the interaction between two electrons is bosonic. Therefore, the scheme is used today to analyze the superconducting properties even of cuprates and pnictides. However, while qualitative conclusions are enlightening a quantitative description including the theoretical derivation of the transition temperature of an existing superconductor or the prediction of a new superconducting material are still at their infancy. Some progress was made for conventional systems recently [9]. In spite of enormous progress the understanding of the cuprates is still limited. One of the major problems is the similarity of all relevant energy scales such as the Fermi energy E F, the exchange coupling J, the energy gap k, or the phonon energy hω q. The solution of these type of problems is among the most tantalizing questions in condensed matter physics. c Walther-Meißner-Institut

17 Chapter 2 Basic Experiments and Understanding Many of the basic observations in superconductors to be described in the first section of this chapter could be understood phenomenologically long before the microscopic theory was finally presented in In particular the London theory highlights the different aspects of the zero-resistance state and the perfect diamagnetism. If derived from the Schrödinger equation time-dependent phenomena such as the Josephson effects follow directly highlighting the quantum mechanical nature of the phenomenon in a simple though instructive way. 2.1 Key Experiments Zero Resistance After Onnes succeeded to liquefy Helium (1908) he started immediately to study the resistivities of metals. Hg was the purest material available at that time. Instead of finding support for one of the proposals being discussed at his time (see chapter I) he discovered a state of vanishingly small resistance (R < 10 5 Ω) below T = 4.2 K, the boiling point of liquid He (Fig.??). 1 The determination of the actual resistivity of a superconductor requires new techniques since the precision of a direct four-probe measurement is orders of magnitude too low. Already Onnes designed an experiment which uses the magnetic moment of a persistent current I in a ring to determine the decay rate of I rather than the resistance itself. In the experiment there are two concentric superconducting rings. The outer one is fixed, the inner one is suspended on a quartz filament. As long as there is no current in the rings the inner ring turns with the filament. Now we get supercurrents I i and I o to flow in the rings. To this end a (for simplicity) homogeneous field B 0 B c (T) is turned on with the axis parallel to the axes of the concentric rings. If the rings are above their transition temperature T c the field penetrates the cross sections S i and S o as well as the material homogeneously. Now the rings are being cooled below T c and the field is expelled from the material due to the Ochsenfeld-Meissner effect (see next paragraph), i.e. is slightly distorted around the solid superconductors but more or less unchanged in most of the free space. What happens if the field is switched off? This can be derived easily from Faraday s law for either of the rings, E = B t. (2.1.1) 1 H. Kammerlingh Onnes, Leiden Comm. 120b, 122b, 124c (1911) 9

18 10 R. HACKL AND D. EINZEL Basic Experiments and Understanding We integrate over the cross section S of the respective ring and apply Stokes theorem, B S t S da = E ds (2.1.2) = E dl. Γ Here Γ is the edge of S and runs approximately along the center of the ring. Since the resistance of the superconductor vanishes for sufficiently small fields (see below) there is no voltage drop along the ring and, consequently, t Φ = 0 (2.1.3) In the last equation Φ BS is the flux penetrating the ring, and the voltage drop for one circulation around the ring is zero since R = 0. Hence the flux through a closed superconducting loop is constant and cannot escape after the external field is switched off. The related field is maintained by a current I in the ring. In turn, we would be unable to get the current running when applying the field at T < T c. For this reason it is misleading to think of inducing a current by changing an external field. The current is rather a result of maintaining the flux fixed. If the filament is now twisted there will be a restoring force proportional to µ i µ o with the moments given by µ = SI. Once the equilibrium position is reached any turn of the inner ring as a consequence of a variation of I i,o can be measured with great precision. In order to determine a limit for the resistivity ρ from I we consider the ring as a loop with a resistance R and an inductance L for which RI + L di dt = 0. This differential equation can e integrated by separating the variables, and we find (2.1.4) R = L t ln I(t) I(0), (2.1.5) where I(t) is directly proportional to the torque. Onnes and his collaborators kept the experiment running for a year - quite heroic at that time. With modern NMR techniques the field produced by a current can be measured with much higher precision than with a torque meter, and a lower limit of 10 5 years was found for I to become significant Perfect diamagnetism Given the perfect conductivity of a superconductor it is obvious that a magnetic field is completely shielded when it is applied below the transition temperature. The induced screening currents do not decay as in conventional metals with finite resistivity. However, what happens if the superconducting state is entered by lowering the temperature in a finite external field? The question was answered by Meissner and Ochsenfeld in 1933 when they studied the force between two parallel tin wires. If the currents were applied above and below T c the forces were different. In the latter case the mutual exclusion of the fields due to screening explained the result. However, the same effect was observed when the wires were cooled through T c with the currents on implying that the field was also expelled without induction to start the screening currents 2. 2 W. Meißner and R. Ochsenfeld, Naturwissenschaften 81, 787 (1933) c Walther-Meißner-Institut

19 Key Experiments SUPERCONDUCTIVITY 11 B (a) B i (b) Type I SC T 0 B c (0) B c (T) field cooled (f.c.) B i = i = B a B 0 B i =0 i = 0 zero field cooled (z.f.c.) T c T Meissner state B c normal ( ) state B 0 Figure 2.1: Field-temperature phase diagram of a superconductor. A given point in the superconducting can be reached independent of the path. In this way the superconducting state is established as a thermodynamic phase. Therefore, the second actually defining property of a superconductor is its perfect diamagnetism: a magnetic field which is sufficiently small to not destroy the new state is expelled from a singly connected specimen irrespective of the route the superconducting state is entered (Fig. 2.1). Hence, superconductivity is a thermodynamic phase which, in turn, implies the existence of a critical field B c (T ) defining the phase boundary between the normal and superconducting states in the presence of a field and determining the condensation energy (see section 2.2). The field free state is usually called Meissner state. The magnitude of critical field can be approximated by a parabola, [ ( ) ] T 2 B c (T ) = B c (0) 1. (2.1.6) T c The microscopic theory arrives at a similar phenomenology but there are deviation which depend on the electron-phonon coupling strength. The path-independent complete exclusion of magnetic flux from the interior of a bulk superconductor in the Meissner state raises several questions: (i) How does the field (induction B = µ 0 H) change at the surface? (ii) How are the screening currents set off if not according to Faraday s law? (iii) Is it possible that the flux penetrates partially to reduce the magnetic energy? (iv) What happens in a material which is not simply connected, i.e. has voids? The level of complication in finding answers to these questions differs remarkably. (ii)-(iv) can be answered only qualitatively here and will be discussed in detail in the following chapters. (i) follows from the phenomenological London theory (see section??). Electrodynamics shows directly that the applied field decays exponentially away from the surface into the bulk. The same holds true for the screening currents. The characteristic length scale is called the London penetration depth λ L and is of order nm. The range can be traced back to the carrier concentration. Pippard observed a dependence of the penetration depth on the purity of a material and concluded that non-local electrodynamics must be used for properly explaining this effect. This means that the response at a point r depends on the perturbation in a material dependent volume of order r ξ 0 3. The new characteristic length scale ξ 0 was explained microscopically 20 years later by Bardeen, Cooper, and Schrieffer (BCS theory) but also anticipated in the framework of the Ginzburg-Landau phenomenology. (ii) is also a consequence of the London equation but the microscopic foundation was found only in 1957 by BCS. The reason for the Meissner effect is the existence of a uniform phase θ of the superconducting electrons being established at the transition. The current is proportional to this phase modified by the vector potential A of the applied field B 0 = A, j ( θ ea). From a classical point of view the force on the electrons is transverse as for the Lorentz force but exits only in quantum-mechanical systems with a coherent phase for all carriers, where a finite vector potential sets off a current irrespective of 2013

20 12 R. HACKL AND D. EINZEL Basic Experiments and Understanding the history. The uniform phase was anticipated by Fritz and Heinz London already in 1935 without a theoretical justification. Also Ginzburg and Landau used a complex order parameter with a single-valued phase to construct the Free-Energy functional but did not discuss its origin. It needed the Geistesblitz (flash of genius) of Bob Schrieffer to realize the necessity of a coherent state of fermions and to construct it in a fashion similar to that proposed by Schrödinger for bosons in So we have to understand the full microscopic theory (see chapter 3) to properly explain the Meissner effect. A different approach is possible via a thermodynamic argumentation which we shall discuss in chapter 4. The full exclusion of magnetic flux is associated with a rapidly increasing energy E = V /2µ 0 B 2 with V the sample volume. If we recall that the critical fields are moderate in elemental superconductors the new phase will not survive sufficiently long to make the materials particularly useful for applications. Is there a way around as insinuated in (iii)? There are two qualitatively different answers. The first one is an effect of the demagnetization as described in Appendix 1. Around superconducting samples having shapes different from infinite slabs or cylinders oriented along H 0 the field changes close to the surface once the Meissner state is established. This phenomenon originates in the usual electrodynamic relation B = 0 and the condition n (H 0 H i ) = 0 describing the continuity of normal and, respectively, the tangential components of the fields around a superconductor. Whenever the demagnetization factor n is different from 0 the surface enhancement of the field around a generally shaped superconductor implies that the critical field is reached earlier in locations with enhanced field than in those without. In the case of a sphere the enhancement is 1/3 at the equator with the equatorial plane perpendicular to the homogeneous field with magnitude B 0. Consequently the field starts penetrating for B 0 = 2/3B c. In the range 2/3B c < B 0 < B c, the sample is in the intermediate state with normal and superconducting regions coexisting on a macroscopic scale. If a slab (for instance a thin-film sample) with area S a 2 and thickness t much smaller than a is aligned perpendicular to the field n is close to unity and the field becomes very large at the edges rapidly exceeding the critical field, and flux penetrates even for very small applied fields B 0 B c. For stability reasons the field in the normal regions must be B c and vanishes in the superconducting parts. For satisfying the conservation of flux ( B = 0) everywhere the density of the superconducting regions scales with the ratio of the applied to the critical field as ρ sc = 1 (B 0 /B c ). If the intermediate state is studied with a decoration experiment, with magneto-optics or neutrons one finds a regular pattern of the normal and superconducting regions even in a perfect crystal. The regions are aligned along high-symmetry lines of the crystal lattice. It is in fact the electronic structure which defines preferential orientations as will become plausible in chapter 3. The second part of the answer refers to a new state which is sufficiently important to be discussed in a new paragraph Shubnikov phase (mixed state) The energy needed to keep the magnetic field completely outside the superconducting volume increases quadratically with the field. In turn, a complete penetration quenches superconductivity. Shubnikov showed 1936 that a small amount of Tl in Pb leads to a new mixed state with some flux starting to penetrate the material long before superconductivity vanishes [?] The experimental magnetization curves for a Pb single crystal with 5%Pb replaced by Tl are shown in Fig For the cylindrical shape of the sample a demagnetization effect can be excluded. Later it turned out that practically all non-elemental superconductors and Nb share the property of normal regions encircled by superconducting screening currents around the field lines [panel (b)]. Each vortex carries only the smallest amount of flux possible (see next paragraph). There is a lower critical field B c1 at which the field starts to penetrate and an upper critical field B c2 at which the material becomes a normal metal. The creation of the first flux line at B c1 does not cost energy. Therefore the magnetization decays at an infinite rate as shown in panel (c) of Fig At B c2 the magnetization vanishes at a finite rate. c Walther-Meißner-Institut

21 Key Experiments SUPERCONDUCTIVITY 13 (a) (b) (c) - 0 M Type II SC mixed state B c1 Meissner state B c2 B 0 Figure 2.2: Type II superconductor in a magnetic field. (a) Induction B vs. applied field of a cylindrical sample of Pb 95 Tl 5 [?]. There is no jump of the induction at a critical field but rather a sharp onset at B c1 and a continuous approach to the normal state induction. At B c2 the superconducting induction branches of the normal one with a different slope. (b) Above B c1 the partially penetrates the material. Instead of running around the perimeter of the samples the screening currents encircle individual flux lines, each carrying Φ 0, and form vortices. (c) Schematic plot of the magnetization µ 0 M = B i B 0 vs. B 0. Below B c1 B i = 0. The rate of reduction of µ 0 M is logarithmically divergent indicating that the energy need for the first flux line is vanishingly small. At B c2 the slope is finite. The existence of the mixed state separates the superconductors in two classes. Those with complete Meissner effect are usually called type I while type II materials are characterized by a Meissner state at low fields and a mixed state between B c1 and B c2. The infinite slope of µ 0 M at B c1 and the complete reversibility of the magnetization indicate that the flux lines can move freely in the mixed state. Consequently, even though the field at which superconductivity vanishes is occasionally very high, also type II superconductors still appear to be of little practical use since any current leads to a movement of the flux lines due to the Lorentz force and to dissipation. Only when the pinning of flux lines in disordered alloys such as NbTi or by the introduction of artificial pinning center was achieved applications emerged rapidly. Materials with strong pinning are called magnetically hard or sometimes type III superconductors. The applications aspects were among the reasons why Abrikosov earned the 2003 Nobel prize for his theoretical description of the mixed state on the basis of the Ginzburg-Landau theory (chapter 4). The study of the flux line lattice remains an important field of research into superconductors. As briefly mentioned above the vortices carry a flux which, for energetic considerations, is small. From a topological point of view a superconductor perforated by normal conducting regions is not simply connected any further. Hence, if a persistent current having a quantum mechanical nature encircles a vortex quantization effects can be expected. Similar effects can be expected when a macroscopic sample is not simply connected. This brings us finally to a discussion of question (iv) of paragraph Quantization of the flux When electrons move freely they do not change their quantum state. Therefore, if they form a nondissipative current in a ring one can expect the angular momentum to be quantized according to the Sommerfeld condition. This was the reasoning following Fritz London s prediction of 1950 when the 2013

22 14 R. HACKL AND D. EINZEL Basic Experiments and Understanding B.S. Deaver and W.M. Fairbank, PRL 7, 43 (1961) Figure 2.3: Experimental setup for (left) and results of flux quantization measurements [?,?]. The magnetic flux trapped in the superconducting lead cylinder is measured with a resonance method. Once the R. Doll and M. Näbauer, PRL 7, 51 (1961) cylinder is below T c and the static field in y-direction B 0 is switched off the field B M starts oscillating at the resonance frequency of the cylinder. The oscillation amplitude as detected with a mirror is proportional to the magnetic moment of the cylinder, hence the included flux. experiments on the flux quantization in multiply connected superconductors were discussed in the late 1950ies and early 1960ies. The experiments were performed simultaneously by two groups which did not have information about each other. Deaver and Fairbanks [?] in Stanford used a vibrating sample (Fonertype) magnetometer for determining the trapped flux in a small cylinder. In the experiment performed by Doll and Näbauer at the Bavarian Academy of Sciences and Humanities (shown in Fig. 2.3 l.h.s.) the magnitude of the magnetic field B 0 stored in a hollow cylinder with cross section S was studied using a resonance method. For freezing in the flux a homogeneous field B 0 is applied along the axis of a lead cylinder. Then the cylinder is cooled below T c. The trapped flux and, consequently, the oscillation amplitude are found to increase in steps. The results demonstrate that the flux in the cylinder Φ = B 0 S can assume only multiples of h/2e = (46) Wb. Similarly important as the observation of the quantization in itself is the magnitude of the flux quantum. In contrast to the expectation of London it is only h/2e rather than h/e. Onsager was probably the first to consider this possibility soon after the publication of the BCS theory in The meaning can hardly be overestimated: the wave function which imposes the quantization condition is that of electron pairs, exactly as derived by BCS, rather than single electrons Josephson effects Another manifestation of the quantum nature of the superconducting state is the observation of coherence phenomena between two weakly connected superconductors. The setting is similar to that of a tunneling contact and the basic condition is that the wave functions of the superconducting condensates rather than those of the single particles overlap. Then one can observe all phenomena characteristic of two coupled quantum mechanical oscillators as will be derived in detail in section??. The main results are c Walther-Meißner-Institut

23 Thermodynamics SUPERCONDUCTIVITY 15 two equations which describe the statics (dc) and dynamics of carriers across a weak link [?], I(γ) = I c sin(ϕ 1 ϕ 2 ) (2.1.7) γ = 2π U (2.1.8) t Φ 0 where ϕ 1,2, γ, and Φ 0 are, respectively, the phases in the weakly coupled superconductors 1 and 2, the phase difference and the flux quantum. U is the voltage across the junction. In the case of the dc Josephson effect there is a dissipationless supercurrent without a voltage drop across, e.g., an insulator. The second equation describes the dynamics if the current exceeds the critical current I c which is the only material dependent quantity. The Josephson frequency 1/Φ 0 = 2e/h contains only constants and has the value (11) MHz/µV Summary Superconductivity was discovered in 1911 in Hg three years after the first liquefaction of He. Until now the resistivity in the superconducting state is zero to within the experimental precision of approximately Ωm. The flux quantization and the Josephson effects demonstrate the quantum nature of superconductivity. In addition, the magnitude of the flux quantum shows directly that the wave function of the condensate must correspond to pairs of electrons. The perfect diamagnetism as discovered by Meissner and Ochsenfeld in 1933 demonstrates that superconductivity is a thermodynamic phase allowing us to derive relations for the condensation energy, the entropy, and the specific heat capacity in the superconducting state. 2.2 Thermodynamics The thermodynamic properties provide us with a great deal of important information on a material in particular on the energetics. Clearly, we learn something about the bulk while many spectroscopies (except for neutrons) suffer from surface sensitivities. However, it is not always trivial to isolate the desired quantity out of a large variety of contributions. In addition, in the presence of magnetism the thermodynamic potentials are not uniquely defined (see Appendix 3) and the energy of the field B itself needs to be included in a way appropriate for the experimental circumstances. For a superconductor in the Meissner state, for instance, the field energy stored in the sample volume equals the condensation energy. For simplicity we use the magnitudes of B, H, and M rather than the vectorial quantities. This corresponds to a specialization to cylindrical symmetry and homogeneous media Condensation Energy Since superconductivity is a thermodynamic phase the energy difference between the normal (n) and the superconducting (s) state must be finite. Using the densities of the free energies f n,0 and f s,0, respectively, the energy difference for zero field (index 0) can be expressed as f n,0 f s,0 = B2 s 2µ 0. (2.2.9) Here, the condensation energy is expressed in terms of the so far unknown field B s. As long as the applied field is smaller than the critical field B c and completely excluded from the sample volume the condensation energy does not change, hence f s,0 = f s,b. Now we assume that a cylindrical sample of 2013

24 16 R. HACKL AND D. EINZEL Basic Experiments and Understanding a superconducting material is put in the homogeneous field B 0 of a long solenoid. In this idealized configuration there is no distortion of the field around the specimen since the demagnetization vanishes (n = 0). The total volume-integrated free energies F for the interior of the solenoid with the sample in the normal and the superconducting states for a field B 0 < B c read, respectively, F n,b0 = V f n,0 +V B2 0 2µ 0 +V ext B 2 0 2µ 0 (2.2.10) F s,b0 = V f s,0 +V ext B 2 0 2µ 0, (2.2.11) where V = A L is the volume of the sample having cross section A and length L and V ext is the volume inside the coil surrounding the sample, V ext = V coil V. For T < T c (B 0 ) (and B 0 < B c ) the difference of the free energies becomes F n F s = V ( f n,0 f s,0 ) +V B2 0 2µ 0. (2.2.12) If we increase B 0 towards B c from below and use Eq. (2.2.9) we obtain F n F s = V B2 s 2µ 0 +V B2 c 2µ 0. (2.2.13) Upon crossing the critical field superconductivity collapses and the field B 0 = B c enters V. Both the condensation and the field energy induce an electromotive force E in the solenoid while the generator maintains the constant current I c required for the field B c. If there are N windings of the solenoid over the length L the field and the current are related as µ 0 NI c = B c L and E = N Φ with Φ the time derivative of the flux through one winding. The total electrical energy W generated at the transition is given by the time integral over the power, W = = b a a b = B cl A µ 0 N ΦI c dt (NI c )dφ Bc 0 db = V B2 c µ 0. (2.2.14) For the flux only the sample cross section is relevant since the field around the sample does not change. Since there is no dissipation ( S = 0) W is equal to the free energy difference at B c given in Eq. (2.2.13) and consequently B s B c. Hence the condensation energy is given by the critical field, and the free energy changes abruptly by twice the condensation energy upon crossing the B c line. While the (Helmholtz) free energy is useful for studying the electromagnetic energy released at the transition it is less useful for practical purposes for depending on the volume V and, practically always, on the magnetization M. In an experiment it is next to impossible to control these variables. Therefore, a potential which depends on the pressure p and the applied magnetic field B 0 = µ 0 H 0 is desirable. The Gibbs potential (Gibbs free energy) G =U T S pv BH 0 having the differential dg = SdT V d p V MdB 0 (upon using the proper U) depends only on controllable parameters and, in addition, is the most appropriate function for studying phase transition. Since G is the macroscopic version of the chemical c Walther-Meißner-Institut

25 Thermodynamics SUPERCONDUCTIVITY 17 potential µ, G = Nµ, G 1 (T,B c ) and G 2 (T,B c ) must be equal at the transition between phase 1 and 2. Therefore, we replace now F by G. For the densities we can write g = f BH 0 and get G n,b0 = V f n,0 V B2 0 2µ 0 V ext B 2 0 2µ 0 (2.2.15) G s,b0 = V f s,0 V ext B 2 0 2µ 0. (2.2.16) The second term on the r.h.s. of Eq. (2.2.15) is given only by the external field since M is usually vanishingly small in a superconducting material, and B = B 0. The corresponding term in Eq. (2.2.16) is zero since M = H 0 hence B = 0 in the Meissner state. The difference G n G s = V ( f n f s ) V B2 0 2µ 0 (2.2.17) obviously vanishes at B 0 = B c. For zero field the difference of both potentials yields the condensation energy, G n G s = V B2 c 2µ 0 = F n F s. (2.2.18) Obviously, the field energy is irrelevant in this case and F can be replaced by F, which does not include the vacuum field. F is preferable for theoretical considerations for containing only the magnetization as a microscopic quantity. Experimentally, F corresponds to the situation when a superconductor is first cooled below T c,b and then moved into the field. Then, if we recall the setting from above the generator has to supply only the energy of the excluded field equalling the condensation energy. What happens if the field is now cranked up with the sample inside until it reaches B c (T )? One may have guessed, it is exactly the case described above and twice the condensation energy is released. This is no violation of energy conservation. Rather, the second half of the energy corresponds to the mechanical energy needed to move the sample into the filed from the field-free region. Using m = V M, the differentials of F and G read, df(t,v, M) = SdT pdv + Bdm (2.2.19) dg(t, p,b 0 ) = SdT +V d p mdb 0, (2.2.20) where the index 0 at B 0 is added to unambiguously denote that the field is controlled from outside. This is only possible in the case of G. In F, B depends on all other variables and on the location. However, for a homogeneous material in the Meissner state B = B 0 is a good approximation. For constant T and p dg s can be integrated, Bc 0 yielding Bc dg s = mdb 0 (2.2.21) 0 G s (T,B c ) G s (T,0) = V B2 c 2µ 0. (2.2.22) In the normal state the magnetization is small and G n (T,B c ) G n (T,0) 0. (2.2.23) 2013

26 18 R. HACKL AND D. EINZEL Basic Experiments and Understanding At the the phase boundary G s (T,B c ) = G n (T,B c ), and the difference of Eqs. (2.2.23) and (2.2.22) yields again the l.h.s. of Eq. (2.2.18). Now, without including of the field energy, F G +V B 0 M yields F = G at B 0 = 0, and also the r.h.s. of Eq. (2.2.18) is recovered. At the critical field one finds F s (T,B c ) = G s (T,B c ) +V B c M c = G s (T,B c ) V B c H c (2.2.24) F n (T,B c ) = G n (T,B c ). (2.2.25) Since the Gibbs potentials at B c are equal the discontinuity of F is recovered. Finally, we calculate the variation of G s (T,B < B c ), G s (T,B) G s (T,0) = V 2µ 0 B 2, (2.2.26) G s (T,B) G n (T,0) + V 2µ 0 B 2 c = V 2µ 0 B 2, (2.2.27) G s (T,B) = G n (T,0) V 2µ 0 ( B 2 c B 2) (2.2.28) which is sometimes called the Meissner parabola and shown in Fig. 2.4 (a). Before deriving other quantities we note that B2 c 2µ 0 is an energy density having the unit of a pressure. This allows us to get a feeling for the order of magnitude of the condensation energy. For Nb with T c = 9.2 K, B c = 0.2 T, and a lattice constant a = 3.3 Å we get B 2 c 2µ 0 = 16.5 kj m 3 (2.2.29) = 16.5 kpa (2.2.30) corresponding to the pressure at 1.6 m under a water surface and a condensation energy of 2 µev/atom Entropy The entropy measures the degree of order in a system and can be derived from both thermodynamic potentials (see Eqs. (2.2.19) and (2.2.20)). From an experimental point of view G is more convenient. For the consideration below, we can ignore the difference and write down the entropy S for either constant magnetization and volume or constant applied field and pressure, S(T ) = F T = G M,V T B0,p. (2.2.31) In many cases one is interested in the temperature dependence of the entropy and other thermodynamic quantities. For the entropy change upon entering the superconducting state we obtain for F S(T ) = S s S n = T (F s F n ) = ( ) V B 2 T 2µ c(t ) 0 = V B c (T ) B c(t ). (2.2.32) µ 0 T c Walther-Meißner-Institut

27 Thermodynamics SUPERCONDUCTIVITY 19 Strictly speaking we can use only F here since the volume appears in the condensation energy. However, in a solid the differences between quantities measured for constant volume or constant pressure are on the order of a few percent and will not be considered below. Apart from this subtlety Eq. (2.2.32) is extremely useful as soon as the temperature dependence of the critical field (or of F) is known. In fact the microscopic theory provides predictions which could be use. However, for getting qualitative insight we use the phenomenological temperature dependence of B c given in Eq. (2.1.6). From Eq. (2.1.6) we see and from experiments we know that B c(t ) T has no divergence at any temperature and the following qualitative conclusions can be derived: B c (T T c ) = 0 implies that the entropy is continuous at T c. In other words, there is no latent heat in zero field. In the limit T 0 the entropy difference vanishes according to the Nernst theorem. Since B c (0) 0 the derivative B c(t 0) T approaches zero. Obviously, Eq. (2.1.6) has the proper limiting behavior for T 0. However, systematic studies show that the curvature is not correct. In the range 0 < T < T c B c (T ) decreases implying that B c(t ) T < 0. As a consequence the entropy in the superconducting state is smaller than in the normal state. This means that the superconducting state has a higher degree of order and can transport heat less efficiently. For further considerations the derivatives of Eq. (2.1.6) are useful. We get B c (T ) T = B c (0) 2T T 2 c = B [= B T ] (2.2.33) 2 B c (T ) T 2 = B c (0) 2 Tc 2 = B = const. (2.2.34) allowing us the calculate the latent heat as a function of temperature. The latent heat is defined as the entropy change times the temperature, L = Q = T S. Using Eq. (2.2.33) the density of the latent heat reads ( ) L V = 4B c(0)b c (T ) T 2 (2.2.35) 2µ 0 T c Except for T = 0 and T = T c there is a latent heat at the phase transition, and the transition becomes first order Specific heat capacity The specific heat capacity c V,p at either constant volume or pressure is an extensively used quantity which reflects the bulk properties. In superconductors the low-temperature part is particularly important for characterizing the energy gap. Close to T c the heat capacity has a discontinuity which reflects details of the transition and the coupling strength. There are various experimental complications such as defects or the superposition of strong other contributions from, e.g., the lattice. At the moment we wish to focus only on the concept. 2013

28 20 R. HACKL AND D. EINZEL Basic Experiments and Understanding Usually the heat capacity is measured at constant pressure. However, c V is more desirable since all energies depend on the Volume. Fortunately the difference c p c V is small in a solid and can be ignored here. The specific heat capacity is defined as c V,p = 1 V Q T [ ] J V,p m 3 K (2.2.36) where Q is the thermal energy supplied in the interval between T and T + T. The molar heat capacity which is frequently used in earlier publications is then given by c M mol ρ with M mol the molar mass and ρ the mass density. Using Q = T S and taking the limit we get c x = T V ( ) S. T x,... (2.2.37) It depends on the thermodynamic potential which of the variables x,... is taken constant (see above). For M and V being constant we obtain c V = T V ( 2 ) F T 2. (2.2.38) M,V In the following the reference to constant variables is dropped for simplicity. The difference of the superconducting and the normal state heat capacities can be calculated right away from the entropy difference, c s c n = T V (S s S n ) T = T V 2 T 2 (F s F n ). (2.2.39) From the condensation energy the famous Rutgers equation is obtained, [ c s c n = T ( Bc ) (T ) 2 ( 2 ) ] B c (T ) + B c (T ) µ 0 T T 2. (2.2.40) This is a general result that can be used whenever B c (T ) is known from a microscopic treatment. If the BCS predictions are used the data of Al, a prototypical weak-coupling superconductor, can be described well. Similarly well works the Eliashberg theory for strong coupling Pb. For a qualitative visualization of the thermodynamic functions (Fig. 2.4) and for demonstrating a few important limiting cases we use the parabolic approximation for B c (T ) [Eq. (2.1.6)]. In the limit T T c B c (T ) has a finite slope and the heat capacity in the superconducting state is bigger than in the normal state, c s = c n + 8 T c B 2 c(0) 2µ 0. (2.2.41) The discontinuity can be calculated on the basis of the low-temperature limes as will be done below. If we disregard the parabolic variation of B c (T ) for a moment we realize that the discontinuity depends on the rate of variation of the entropy below T c with respect to that above or, sloppily speaking, on the sharpness of the kink in the entropy at T c. c Walther-Meißner-Institut

29 Thermodynamics SUPERCONDUCTIVITY 21 Thermodynamic functions 2 0 [B c (0)] -2 [g s (T,B)-g n (T,B)] B F(T,0.3B c ) T c,b (a) B/B c (0) T/T c,0 [s s -s n ]/ T c (b) T c,b T/T c 2.0 (c) B ( T ) c Bc (0) 1 T T c 2 (c s -c n )/c n T c,b T/T c Figure 2.4: Thermodynamic function calculated for the parabolic approximation of the critical field [Eq. (2.1.6)]. Shown are the respective differences of the superconducting and the normal state functions for the electronic part of (a) the Gibbs free energy g s (T,B) g n (T,B), (b) the entropy s s (T,B) s n (T,B), and (c) heat capacity c s (T,B) c n (T,B) normalized to the volume and the condensation energy and related quantities. In (a) the Gibbs potential is plotted for various fields as indicated. For zero field the Gibbs potential and the free energy F coincide (red line) and the slopes close to T c,0 vanish indicating a second order transition. For finite field the transition at T c,b is first order. While g s (0,B) g n (0,B) becomes smaller with increasing field for all temperatures and vanishes at B = B c F starts to increase with field as shown for B = 0.3B c (dash-dotted green line). The free energy according to Eq. (2.2.19) stays constant at T = 0 (not shown). The entropy and the heat capacity are shown for zero field (full line) and for B/B c = 0.5 (blue dashed line). For finite field the entropy has a discontinuity corresponding to a latent heat L [Eq. (2.2.35)] and a δ-like contribution to the heat capacity (thick blue vertical line). Since the entropy difference vanishes at T = 0 and T = T c and since the superconducting state has a higher order than the normal state S is negative for 0 < T < T c and has a minimum. Consequently, c = c s c n crosses zero or, in other words, c s and c n have an intersection point. For directly comparing c s and c n we need an approximation for c n and take the Sommerfeld model which describes quasi-free electrons in a solid (see, e.g., Ashcroft and Mermin). Here, c n = γt, and the constant of proportionality γ is given by the electronic density of states for both spin projections at the Fermi energy N(E F ), c n = γt = π2 3 k2 BN(E F )T. (2.2.42) In a conventional metal with only the lattice and the electrons contributing to the specific heat, c(t ) = γt + bt 3, γ can be determined experimentally. If the total heat capacity c/t is plotted as a function of T 2 the intersection point at T = 0 yields γ. The discontinuity at T c can now be 2013

30 22 R. HACKL AND D. EINZEL Basic Experiments and Understanding expressed as c(t c ) c n (T c ) = c s c n c n = 1 γt c 8 T c B 2 c(0) 2µ 0. (2.2.43) One could arrive at a complete phenomenology including a prediction for c(t c ) if γ could be derived in some way. There is in fact a possibility opening up by further analyzing c s is in the low-temperature limit [10]. To this end we rewrite Eq. (2.2.40) using the results of Eqs. (2.2.33) and (2.2.34), [ c n = γt = c s T (B T ) [ ( ) ] 2 T 2 + Bc (0) 1 ]B. (2.2.44) µ 0 Since B is a constant the last term approaches zero linearly for T 0 while T (B T ) 2 vanishes rapidly. Also c s varies also faster than T as will be shown immediately and is found experimentally for basically all known superconductors 3 hence the limit T 0 yields T c γ = 4 B 2 c(0) Tc 2 [ N(E F )]. (2.2.45) 2µ 0 With γ plugged back in Eq. (2.2.44) the superconducting heat capacity at low temperature can be expressed without a free parameter, c s (T 0) = 12 T 2 c B 2 ( ) c(0) T 3. (2.2.46) 2µ 0 T c Using quite simple operations we derived several predictions demonstrating the importance of both the thermodynamic arguments and experiments. Eq. (2.2.45) demonstrates the relation between the density of electronic states N(E F ), T c, and B c (0) hence the condensation energy which is also found in the microscopic theory. Even though the constant 4 is model dependent, i.e. depends here on the slope of B c (T ) near T c, Eq. (2.2.45) conveys the important message of an interrelation between the normal and the superconducting state allowing one to check the plausibility and consistency of experimental results. According to Eq. (2.2.46) c s (T 0) approaches zero as T 3. Trivially, this justifies our assumption c s (T 0) T 1+α with α > 0. Less trivially Eq. (2.2.46) predicts a power law for c s (T ) close to T = 0. For this reason all early data for the low-temperature specific of superconductors were compared with T 3. However the results for tin or vanadium [?] clearly show an exponential variation (see Fig.) which was immediately recognized as a manifestation of an energy gap in the electronic excitation spectrum. The discrepancy between the prediction of Eq. (2.2.46) and the observed variation or the BCS result originates, as mentioned, in the small differences between the parabolic and the proper temperature dependence of B c (T 0). From an experimental point of view affairs are even more complicated. Exponential variations are in fact rare and are found only in a few weak coupling superconductors such as Al, Sn or V and some other elements. More recently, exponential temperature dependences were also used for explaining c s (T ) in MgB 2 or some iron-based systems. In either case multi-band effects make the analysis rather complicated and less stringent. In strong coupling conventional materials like Nb or Pb one finds in fact T 3 while in the copper-oxygen superconductors c s T 2 is established (see chapter 6). Different reasons are at the origin of these deviations: For Pb and Nb strong coupling effects lead to electronic states inside the gap even at T = 0 making the exponential temperature dependence to vanish. The T 3 power law can be derived in the framework of the Eliashberg theory [?, 3]. In contrast 3 For a fully gapped superconductor like Al the T dependence is exponential, turning into T 3 in the strong coupling limit. In the case of a gap with line nodes c s T 2. Only in the case of osculating (kissing) nodes c s may vary linearly with T. These important new developments will be discussed in chapter 6. c Walther-Meißner-Institut

31 Electrodynamics SUPERCONDUCTIVITY 23 to the conventional metals, the cuprates do not have a constant gap. Rather, the gap vanishes on lines on the Fermi surface (line nodes). Close to these nodes there are always thermal excitations except for T = 0. The density of those excitations and, hence, c s depends on the k-dependence of the gap close to the nodes. For the cuprates the variation is linear, i.e. proportional to (ϕ ϕ 0 ), where ϕ is the angle on the Fermi surface away from the position of the node at ϕ 0. Finally, we plug γ of Eq. (2.2.45) in Eq. (2.2.43) and obtain a universal number, c(t c ) c n (T c ) = 2 (2.2.47) which is larger in comparison to the weak coupling BCS value of Obviously, the approximation of B c (T ) used for all our considerations here mimics strong coupling effects in general. This was shown directly in detailed studies of the critical field [?] the main result of which is reproduced in Fig. In Al one finds 1.42(2) for the discontinuity in excellent agreement with the weak coupling result while 2.6 and 2.1 are observed for Pb (T c = 7.2 K) and Nb (T c = 9.26 K), respectively. The BCS value is also universal and is derived from the entropy of the quasiparticle excitations that reflect the gap. Larger values for the discontinuity are strong coupling effects. The coupling strength in this context refers to the interaction between the electrons and bosonic excitations such as phonons (considered by BCS) or charge and spin fluctuations (believed to play a role in unconventional superconductors) for instance mediating the electron pairing below T c. In the Landau-Fermi liquid picture the coupling is characterized by a dimensionless parameter λ which measures the interaction related reduction of the Fermi velocity v F and the effective electron mass m at the Fermi level (for an elementary discussion see, e.g., Ashcroft and Mermin). With increasing interaction v F is reduced as (1 + λ) 1 while m is enhanced as (1 + λ) with λ varying between 0 (weak coupling) and values of order unity (strong coupling). For Al one finds λ 0.1. In the strong coupling elements Pb and Nb λ is close to 2. We see that the discussion of thermodynamics provides deep insight into the microscopic origin of superconductivity without, however, explaining the origin of the ordered or condensed state. Nevertheless, the experimental observation of the entropy reduction below T c (via the heat capacity around T c ) and of the existence of an energy gap in the electronic excitation spectrum (via the heat capacity close to zero temperature) are hallmarks of research paving the way towards the microscopic explanation. In addition, the power laws found for the heat capacity of unconventional superconductors highlight the relevance of thermodynamic studies in contemporary work. Here, we have to pour some water on the vine. As already mentioned the heat capacity does not reveal the desired isolated properties of the subsystem of interest which are the electrons here. Except for very low temperatures the contribution of the phonons dominates the heat capacity, and sophisticated methods are necessary to isolate the electronic contribution (see Loram). In magnetic systems spin excitations may contribute enormously and in materials with disorder, even though very little, the contribution of two-level systems (Schottky anomaly) is very strong. For these reasons, various other thermodynamic quantities such as thermal conductivity or expansion are derived in order to bypass the problems inherent to the heat capacity. Concerning the copper and iron-base systems the study of the latter quantities revealed qualitatively new insights (Zaini, Ando, Taillefer, Meingast, Hardy). 2.3 Electrodynamics Naïvely, perfect conductivity occurs if the time between two collisions of an electron τ or the mean free path l diverge. It is a different story how this can be realized since even in an ideally clean material perfect conductivity is a quantum mechanical effect and can be expected only at zero temperature. It is instructive nevertheless to consider the transition to perfect conductivity before discussing the possible 2013

32 24 R. HACKL AND D. EINZEL Basic Experiments and Understanding origin in a quantum mechanical context. In 1935, after the discovery of the Meissner-Ochsenfeld effect Fritz and Heinz London considered electrodynamic consequences of a two fluid model consisting of a mixture of two currents moving with and without dissipation. They did not further specify the origin of the superfluid current and used the limit τ but were aware that quantum mechanics must be at work. The Maxwell equations as well as some useful identities and the implications of gauge transformations required here and in the following will be summarized in Appendix The Drude model in the limit τ The first description of metallic conduction is due to Paul Drude [11] proposed shortly after the discovery of the electron by Thomson in In this model the conduction electron having charge q move freely between the atoms of a metal with an average drift velocity v D. Between two collisions at times t and t + τ the electron is accelerated by the electric field E driving the current j which can be expressed as j = nqv D = nq m p, (2.3.48) where n, m, and p are, respectively the electronic density, mass, and momentum. In an incremental time interval dt the electron s momentum changes from p(t) to p(t + dt), p(t + dt) = [ p(t) + f (t)dt + O(dt 2 ) ]( 1 dt ), (2.3.49) τ driven by the force f (t) = qe. Contributions of higher than linear order in dt will be neglected as expressed by O(dt 2 ). Since the scattering time τ is finite the probability of an electron to not contribute to the average momentum change is dt/τ reducing the total acceleration by (1 dt/τ). Neglecting terms of order dt 2 and higher Eq can be rewritten as d p dt = p(t) + f (t) τ (2.3.50) which, after multiplication by nq/m and substitution of f yields ( d dt + 1 ) τ j = nq2 E. (2.3.51) m This result of Drude shows directly that the current increases linearly with time in the presence of a field E if τ becomes large constituting a slightly embarrassing statement. Therefore, before studying the limit τ, we solve the differential equation (2.3.51). First we define the conductivity σ(ω) as j = σe and then assume E to vary harmonically as E(t) = E 0 e iωt, ( iω + 1 )σe 0 e iωt = nq2 τ m E 0e iωt, yielding directly the final result for the conductivity, σ(ω) = nq2 m 1 τ 1 iω, (2.3.52) or, upon separating real and imaginary part, σ(ω) = nq2 m τ 1 + iω τ 2 + ω 2 = nq2 m τ 1 + iωτ 1 + (ωτ) 2. (2.3.53) c Walther-Meißner-Institut

33 Electrodynamics SUPERCONDUCTIVITY 25 The real part with q replaced by the electronic charge e (e = Cb) assumes the usual form Rσ(ω) = ne2 m τ 1 + (ωτ) 2 (2.3.54) which is proportional to the lifetime τ. In the static limit the usual dc conductivity σ 0 = ne2 m τ is recovered. The frequency dependence is given by a Lorentzian having a width τ 1. The integral over all frequencies obeys the famous f -sum rule, dωrσ(ω) = π ne2 m (2.3.55) which states here nothing else but charge conservation. Now we are prepared to derive the limit τ 0. Eq. (2.3.52) shows directly that σ becomes purely imaginary, ne2 lim σ(ω) = i τ ωm, (2.3.56) and Eq. (2.3.55) can be interpreted as the Lorentz representation of the Dirac δ with spectral weight π ne2 m and the width vanishing as τ 1. Since the conductivity is an analytic causal function, meaning that the system responds to a perturbation at time t 0 only at t t 0, real and imaginary part of σ are related by Kramers-Krönig transformations, e.g. for the real part, Rσ(ω) = 1 π dξ Iσ(ξ ) ξ ω, (2.3.57) where means taking the principal value. Using the imaginary part of σ as derived in Eq. (2.3.56) the δ function centered at ω = 0 for the real part is recovered after some calculation and the complex conductivity in the collisionless limit becomes σ s (ω) = ne2 m [ ( )] 1 πδ(ω) + i. (2.3.58) ω The preceding qualitative study of the Drude conductivity in the limit of diverging scattering time in the context of superconductivity is meaningful only for ω / h where is the energy gap of the individual material. is, as we remember, the energy reduction of the electronic system that drives the phase transition and will be derived microscopically in the next chapter. We will see then that the infinite conductivity at ω = 0 is a central result for superconductors. However, at finite frequency Rσ vanishes only for hω / h and approaches the normal state conductivity asymptotically for ω > / h. The functional form of the imaginary part turns out to be very useful (once again for hω < / h) if the complex conductivity can be determined experimentally. This is either possible by reflectivity measurements over wide energy ranges and exploitation of the analyticity of the response, here the reflectivity (Tanner, Basov), or by measuring reflection and transmission in thin samples or by ellipsometry experiments which directly return the real and imaginary part of the dielectric function ε = ε 1 + iε 2 and, after some simple algebra, the conductivity. Once σ is derived and Iσ is divided by ω a constant is obtained which directly reflects the plasma frequency ω 2 pl = ne2 ε 0 m (2.3.59) and, as we will derive below, the London penetration depth λ L for a magnetic field. 2013

34 26 R. HACKL AND D. EINZEL Basic Experiments and Understanding The reasoning can be carried on a bit further by taking the curl of Ampère s law [see Maxwell s equations (problem 3 set 2 and problem 2 set 3)], and one obtains ( 2 1c 2 2 t 2 ) B = µ 0 j (2.3.60) = µ 0 σ(ω)e = µ 0 σ(ω) B. (2.3.61) t Note that we must use vectors here since derivatives with respect to r will be essential. If we solve this differential equation by using the harmonic ansatz B(r,t) = B 0 (r)e iωt as above for E we obtain 2 B 0 (r) = ω2 pl ω2 c 2 iωτ 1 iωτ B 0(r). (2.3.62) Here the focus is placed on small frequencies ω < ω pl since superconductivity is destroyed otherwise and is of order mev while ω pl is in the ev range. Therefore ω can be neglected, and the screening equation is directly obtained 2 B 0 (r) = B 0(r) [δ(ω)] 2, (2.3.63) where δ(ω) is the frequency dependent skin depth of a normal metal, δ(ω) = c ω pl 1 + i ωτ. (2.3.64) Formally, the London penetration depth can be obtained in the limit τ. But wait a minute! This is only possible as long as the frequency is finite. If we take the hydrodynamic or static limit ω 0 which we are interested in for a superconductor in a constant magnetic field the penetration depth diverges. This is exactly what happens in a normal metal where τ determines how fast the field penetrates. One could argue that the limit τ should be taken first. Then, however, the result is valid for all frequencies including those well above. Bottom line: A superconductor is not a perfect metal! While the results for the conductivity are instructive and useful if the limitations are kept in mind they are not capable to provide a genuine understanding of the superconducting state. To this end a quantum mechanical treatment is necessary which yields the results also in the proper limits Generalized London theory The quantum-mechanical nature of the condensate was anticipated by the London brothers already in 1935 and was discussed in more detail in 1950 when, for instance, the possibility of flux quantization was mentioned. This implies that the condensate is described by a single wave function ψ(r,t) = a(r,t)e iθ(r,t) (2.3.65) having a real amplitude a(r,t) and a single rigid phase θ(r,t) 4 rather than O(10 21 ) electronic wave functions in a macroscopic solid. ψ(r, t) will be inserted in the Schrödinger equation allowing us to derive an expression for a quantum mechanical current. 4 Rigid does not mean constant. In the presence of fields and currents θ depends on position and time as does the amplitude. c Walther-Meißner-Institut

35 Electrodynamics SUPERCONDUCTIVITY 27 Lorentz force and canonical momentum With a magnetic field B present the kinematic momentum mv has to be replaced by the canonical momentum p = mv + qa where q is the charge of a point-like particle (q = e, with e = Cb) and A is the vector potential. The Lorentz force F q is directly related to this substitution as can be seen by replacing E and B by A, F q = m v t = q(e + v B) = q( Φ A t + (v A) (v )A). (2.3.66) where all quantities depend on r and t. Using the total derivative of A da dt = A t + (v )A and reordering terms one finds a relation between the time derivative of the canonical momentum and a new scalar potential Φ = Φ v A at the position of the particle, d (mv + qa) = q (Φ v A). (2.3.67) dt Thus the Lorentz force is the force on a particle in the co-moving coordinate system and can be derived from a generalization of Newton s law [Eq. (2.3.67)] with the force q Φ. Quantum mechanical derivation of the supercurrent In quantum mechanics we use the correspondence principle and replace the momentum by the momentum operator, p h canonical momentum i mv h qa. kinematic momentum i Note that the kinematic momentum is not gauge invariant any further. For writing down the Schrödinger equation we substitute q and m for the charge and the mass to indicate that the formulation (here and later in the Ginzburg-Landau theory, chapter 4) is more general and applicable also for electron pairs, for instance, i hψ = 1 2m ( ) 2 h i qa ψ + qφψ + Ξψ. (2.3.68) The Schrödinger equation is, of course, gauge invariant since a gauge transformation has to be applied all quantities including Φ and ψ. Eq. (2.3.68) describes a particle having charge q. In a neutral system (q = 0) only the potential Ξ = Ξ(r,t) survives which can be the chemical potential or a general potential energy. For q = 0 and Ξ = 0 Eq. (2.3.68) describes a free uncharged particle. However, it is useful to keep track of possible interactions. We insert now the single-particle wave function Eq. (2.3.65) into Eq. (2.3.68) (Madelung transformation) and define the action S(r, t) = hθ(r, t). Using S and taking the real part of Eq. (2.3.68) yields (see Problem 3 in set 4) S ( S qa)2 + + qφ + Ξ = h2 2 a t 2m 2ma. (2.3.69) 2013

36 28 R. HACKL AND D. EINZEL Basic Experiments and Understanding The second term on the l.h.s. is the kinetic energy 1 2 mv2 and the sum of the second, third and forth terms is the Hamilton function. The term on the r.h.s. of Eq.(2.3.69) indicates the quantum mechanical origin of the equation and is proportional to the spatial variation of the amplitude of ψ. The quasi-classical limit can be obtained by making the transition h 2 0 directly leading to the classical Hamilton-Jacobi equation, S t = H. The imaginary part of Eq. (2.3.68) yields a 2 t { } a 2 = ( S qa) m (2.3.70) (2.3.71) which is a continuity equation for the probability-current density j p = a2 ( h θ qa) m (2.3.72) since a 2 = ψ ψ is a probability density the multiplication of Eq. (2.3.72) with q or m yields the respective charge and mass densities a 2 q and a 2 m and current densities j p q and j p m. The result is, of course, equivalent to the usual quantum mechanical current density, which was already derived by Schrödinger and follows also from the Ginzburg-Landau equations, ( ) j p = {ψ a2 hi m θ qa ψ + ψ ( hi } )ψ θ qa (2.3.73) and can be obtained by using Eq. (2.3.65) for ψ. The supercurrent in a charged system can be written as j s = a2 q m ( h θ qa) n sq 2 m ( h θ A) (2.3.74) q where n s = a 2 is the density of superconducting carriers having mass m and charge q. This equation can be considered the central result of the London theory. All London equations can be derived directly from this quantum mechanical expression The London equations As shown experimentally (see paragraph 2.1.4) and derived theoretically (see chapter 3) the charge in a superconducting condensate is 2e (for traditional reasons the positive sign is used) implying m 2m and n s n/2 for the mass and the density, respectively. These values will be used from now on. For practical purposes it is more convenient to have a direct relationship between the current and the magnetic field. By applying the curl to Eq. (2.3.74) and remembering that θ = 0 the second London equation j(r) = ne2 m B 0(r). (2.3.75) follows immediately. It describes the existence of a current in the presence of a static field B 0 and explains the Meissner-Ochsenfeld effect phenomenologically. In the static limit and for an isolated piece of a superconductor the charge density does not change and the continuity equation (2.3.71) yields j = c Walther-Meißner-Institut

37 Electrodynamics SUPERCONDUCTIVITY 29 ρ = 0. Then the screening equations for the field B and the current density j can be derived directly from Eq. (2.3.75) by either substituting j = µ 0 1 B or taking the curl, respectively, 2 B 0 (r) = B 0(r) λl 2, (2.3.76) where we substituted m/µ 0 ne 2 by λl 2, the square of a length, and 2 j(r) = 1 λl 2 j(r). (2.3.77) The expressions demonstrate that the field and the supracurrent exist only in a surface sheath of thickness λ L as will be discussed in detail in the following paragraph. Upon substituting B 0 = A in Eq. (2.3.75) the resulting equation j(r) = ne2 m A(r) (2.3.78) is the not gauge invariant version of the second London equation. A closer look reveals that Eqs. (2.3.74) and (2.3.78) are equivalent modulo a gauge transformation. Using Eq. (2.3.78) one can derive a screening equation also for A. Finally, by taking the time derivative of Eq. (2.3.78), one gets the acceleration equation ne2 j(r) = t m E(r), (2.3.79) which was discussed earlier and is sometimes called the first London equation. Even if we finally arrive at equations, which are formally equivalent to those derived from the Drude model in the limit τ the argumentation here is qualitatively different. Most importantly, the quantum mechanical nature is properly taken into account allowing us to avoid artificial and unphysical arguments such as an infinite lifetime at finite temperature. 5 In addition, the result is valid in the right limit of 0 ω which particularly includes the static limit. The upper limit will be derived in chapter Some conclusions We discuss first consequences of the screening and then demonstrate the the main conclusions from the quantum mechanical nature of the Eq. (2.3.74) before returning to a special case of screening. The penetration of field and current into a semi-infinite superconductor Eq. (2.3.76) describes the screening of a magnetic field by a superconductor. Here we present a solution for the simplest possible geometry being a semi-infinite solid with the y z plane at x = 0 separating the superconductor (x 0) from the vacuum. The homogeneous field B 0 = B z,0 is aligned along the z-axis making the problem one-dimensional with B z (x) only varying along x, d 2 B z (x) dx 2 = B z(x) λl 2. (2.3.80) 5 This is a point of view which became clear only after the formulation of the microscopic theory and is not intended to criticize earlier proposals. However, it is the only sensible approach in these days. 2013

38 30 R. HACKL AND D. EINZEL Basic Experiments and Understanding 1 B z,0 B z,0 B z (x) A y (x)/ L B z (0) 0-1 A y (x<0) A y (x>0) x/ L Figure 2.5: Theoretical prediction for the penetration of the induction B = B z (x), the current density j s = j y (s) (x), and the vector potential A = A y (x). The usual ansatz for the differential equation (2.3.80) is an exponential, B z (x) = be αx, yielding the characteristic equation α = ±λl 1. The boundary conditions B z(0) = B z,0 and B z (x ) = 0 determine the solution completely, B z (x) = B z,0 e x λ L. (2.3.81) The current density follows from Ampère s law, µ 0 j s = B ê z B z (x), and is oriented along the y-axis represented by the unit vector ê y. The vectorial relation reads j s (x) = ê y B z,0 µ 0 λ L e x λ L. (2.3.82) Eq. (2.3.78) gives one possible choice for the vector potential being anti-parallel to the current density, A(x) = m ne 2 j m B z,0 s = ê y µ 0 ne 2 e x λ L. (2.3.83) λ L B = A leads back to Eq. (2.3.81). The relationship between the various quantities is summarized in Fig The characteristic length λ L is called the London penetration depth and is given by m λ L (0) = µ 0 ne 2 = c ε0 m ne 2 = c ω pl. (2.3.84) The first equality in Eq. (2.3.76) is found frequently, probably because it follows directly from the screening equation (2.3.76), but obscures the meaning a little. In fact, Eq. (2.3.84) is a remarkable result since it relates a genuinely superconducting length scale with the velocity of light and a plasma frequency which is orders of magnitude above the energies relevant for the condensate. In addition, ω pl is composed only of quantities characterizing the normal state. Note, however, that one can substitute the values of q, m, and n s without changing ω pl. Typical plasma frequencies for metals are in the visible range and above (1-5 ev) corresponding to approximately 2π( )10 15 s 1 and yield a range from 40 to 200 nm for λ L in overall agreement with experiment (see Table 2.1). Table 2.1: Experimental magnetic penetration depths in nm of selected superconductors. Al Pb Nb 3 Sn YBa 2 Cu 3 O 7 30 ± ± 10 c Walther-Meißner-Institut

39 Electrodynamics SUPERCONDUCTIVITY 31 ω pl results from the broken gauge symmetry or the unique phase and phase stiffness of the condensate and is closely related to the Higgs mechanism which makes the photons to acquire a finite mass in the presence of a gauge field being equivalent to the vector potential Ahere. [?, 12] Hence, the Meissner effect is the Higgs mechanism in a superconductor and ω pl reflects the rigidity of the phase rather than the relevant energies. At T c λ L (T ) diverges, the field penetrates, and the photons lose the mass they acquired due to fixed phase of the condensate. is proportional to n s which depends on temperature and vanishes at T c. If there is a gap in the electronic excitation spectrum n s saturates exponentially at low temperature while if the gap becomes zero for specific k-points n s can be described by characteristic power laws. For this reason precise measurements of λ L (T ) play an important role for characterizing superconductors. λl 1 There are various methods which are particularly sensitive to changes rather than absolute magnitudes. Prominent examples are microwave, µ spin rotation, optical (see above) but also magnetometry techniques. For example if a sample is put in a maximum of the B field of a superconducting resonator the quality factor Q 2 changes when the penetration depth of the sample changes as a function of temperature. However, Q 2 can be determined with very high precision using state of the art equipment. The results of this technique on very pure YBa 2 Cu 3 O 7 samples (Hardy) finally convinced the majority of the people that superconductivity in the cuprates is unconventional. In a similar fashion by measuring the changes of the penetration depth as a function of temperature in UPt 3 and UBe 13 with a magnetometer the proposal of unconventional pairing was put forward for the first time [13]. Note that all previous equations here are strictly local in that the current in rdepends only on the field in r. Upon studying the penetration depth of PbIn alloys having various concentrations of defects and mean free paths l Pippard found the penetration depth to depend systematically on l. He concluded that there must be another length scale characterizing the superconducting state and called it coherence length ξ 0. Whenever l and ξ 0 are of the same order of magnitude the modification of λ L turned out to be particularly strong. in fact, ξ 0 measures the volume around rover which one has to integrate to properly calculate the response to a perturbation in a point r. In other words, the response in a superconductor is not local but depends on the neighborhood of a point. This purely experimental observation, to which we return when discussing the London vortex, was reproduced in an excellent fashion by the BCS theory [2]. Vanishing canonical momentum If the velocity v in the canonical momentum Eq. (2.3.67) is expressed in terms of the supracurrent j s = nev s one obtains ( m ) p = ne j s + ea (2.3.85) and, upon using Eq. (2.3.78), p = 0. (2.3.86) This is another remarkable result that predicts the existence of superconducting charge carriers with vanishing total momentum and consequently anticipates Cooper pairing of electrons at momenta k and k. A simply connected superconductor For an isolated bulk sample of a superconductor the charge density ρ s is constant. This corresponds to the absence of currents j s flowing in or out, 0 = ρ ( ) s = j s = e2 n s h t m 2e θ A. (2.3.87) 2013

40 32 R. HACKL AND D. EINZEL Basic Experiments and Understanding Fluxoid Quantisierung S S B 0 S QN s s J Q QA M 2 s A LJQ Q 0 dr ( r, t) 2 n ; n 0, 1, 2,... ' S S n h Q 2 dr A J 2 n Q 2 ds B J 0 s 0 L Q 0 Flux(oid) Quant Figure 2.6: Determination of the flux through a hollow cylinder. The homogeneous applied field B 0 is parallel to the cylinder axis and perpendicular to the surface S with boundary S being inside the superconductor. 0 L s Q Except for the term θ Eqs. (2.3.78) and (2.3.87) look similar. The question is therefore how they can be reconciled. The problem was already realized by BCS, and they argue that θ vanishes in the right gauge. It is elucidating to argue the other way around and use the London gauge Ȧ = 0. Then 2 θ follows immediately and θ = const is a direct consequence. Since the starting point was an isolated superconductor with no currents crossing the surface θ has to vanish on the surface since the current is proportional to θ and θ surface = 0. With θ = const θ 0 follows immediately demonstrating the rigidity of the phase independent of statistical arguments. It is interesting to study Eq. (2.3.87) without using the London gauge. To keep things simple we assume a harmonic spatial variation of all fields having the form f (r) = f 0 e ikr yielding the replacement ik and arriving at j s = q2 n s m ( θ A) = q2 n s m (ikθ A). Taking the divergence in the static limit as above one finds ik A + k 2 θ = 0 allowing one to eliminate θ. The resulting current j s = q2 n s m ( k(k A) k 2 ) A (2.3.88) is now gauge invariant and equivalent to the second London equation [Eq. (2.3.75)]. k is the wave vector characterizing position dependent fields on length scales of λ L. k A is the longitudinal projection of the field A on the direction of the fastest change of θ, i.e. θ, and vanishes in the London gauge. Quantization of the fluxoid The quantum mechanical form of the supercurrent [Eq. (2.3.74)] provides us with the proper tool to explain the flux quantization experiment described in paragraph and to get additional insight into the necessary experimental conditions. The setting, this time from a theoretical point of view, is sketched in Fig We rearrange [Eq. (2.3.74)] and integrate over the entire boundary S of the surface S (see Fig. 2.6) on both sides, { h θ dl = A + m } q S S n s q 2 j s dl. (2.3.89) c Walther-Meißner-Institut

41 Electrodynamics SUPERCONDUCTIVITY 33 The line integral over a gradient of a field θ is given by the difference of the fields in the starting and end points. Since θ is the phase of the wave function Eq. (2.3.65), which has to be defined uniquely in a position r, θ is defined only up to integer multiples of 2π, ψ(r) = a(r)e iθ(r) = a(r)e i(θ(r)+2πn) = ψ(r), and the integral over the phase gradient yields S θ dl = 2πn. n Z (2.3.90) Using this result and Stokes theorem Eq. (2.3.89) can be recast, h q S n = ( A) ˆndS + µ 0 λl 2 j s dl, (2.3.91) S where ˆn is a unit vector perpendicular to S. The expression on the r.h.s. is the fluxoid since it depends not only on the flux through the cross section of the hollow cylinder but also on the screening currents in the cylinder walls. The quantization of the fluxoid was demonstrated by Little and Parks [?] in Only if the walls are much thicker than λ L one can find an integration path which is sufficiently buried inside the wall so as to j s 0 along S and to make the second term on the r.h.s. of Eq. (2.3.91) vanish. Only in the case j s = 0 the flux is quantized, Φ = B ˆndS = h S q n. (2.3.92) The London vortex Finally we return to the screening equation in a mathematically more complicated geometry and calculate the energy in a single vortex line for getting a first idea of how the mixed state (see section 2.1.3) develops. First we determine the asymptotic behavior of the field around the line as a function of the distance r. The second step is a lengthy calculation (see Fossheim and Sudbø), and we write down only the result. A single vortex can penetrate a simply connected superconducting material for instance in a type II superconductor very close to B c1. Then there is a normal filament with radius ξ λ L (vortex core) oriented along the field, e.g. ê z, where the field is maximal and supercurrents circulating around the filament which screen the field over a length scale of λ L. By replacing 2 B we can write the screening equation as B + λ 2 L ( B) = ê z Φ L δ(r) (2.3.93) where δ(r) is the two dimensional Dirac δ function. The meaning of Eq. (2.3.93) becomes immediately clear if we integrate over circular surface S perpendicular to ê z having a radius r λ L, S BdS + λ 2 L S ( B)dl = ê z Φ L. (2.3.94) For the second term we used again Stokes theorem. With B = µ 0 j s and j s = 0 for r λ L Φ L is the total flux along the line. If, on the other hand, ξ < r < λ L only a fraction r 2 /λl 2 of the total flux is screened, and we set S B ˆndS zero. The remaining part can be further simplified by exploiting the cylindrical symmetry, B = B r = Φ L 2πrλL 2, (2.3.95) 2013

42 34 R. HACKL AND D. EINZEL Basic Experiments and Understanding where 2πr results from the line integral, yielding B(r) = Φ [ ( )] L r 2πλL 2 c ln. (2.3.96) λ L Since this approximate expression diverges for r 0 ξ has the role of a cut-off below which B(r) becomes constant. The full functional form is the solution of a Bessel differential equation resulting from ( B) in cylindrical spherical coordinates and B = ê z B z, 1 r r ( r B z r ) B z λ 2 L = 0 (2.3.97) that leads to a 0 th -order Bessel or Hankel function K 0, B(r) = Φ ( ) L r 2πλL 2 K 0 λ L (2.3.98) which, for r < λ L, has the asymptotic behavior described above and decays exponentially for r > λ L. If one determines the energy of a vortex line by integrating over the field and the current density L = 1 2µ 0 B 2 + λ 2 L ( B) 2 dr (2.3.99) one finds for the energy per unit length L [ ( )] L = Φ2 L ξ 4πλL 2µ ε ln ; ε 0.1 ( ) 0 λ L where the term ε originates in the condensation energy. For a qualitative argument (see Annett) one can neglect the condensation energy and confine the integration to the region ξ r λ L and substitute B) by j = Φ L /(2πrλ 2 L )ê φ )in Eq. (2.3.99). Since we are interested in the energy per unit length the integration dr = d 2 r = 2πrdr yields Eq. (2.3.99) with ε = 0. The significance of Eq. ( ) is in the proportionality of the energy L to the square of the number of flux quanta, n 2. This means that the costs of getting more than one flux quantum into a vortex increase more rapidly than those for creating n vortices, nl 1 < L n. Eq. ( ) but also the derivation of the asymptotic behavior highlight the existence of a second length scale ξ over which superconductivity is almost completely suppressed while the supracurrents and the field can coexist between ξ and several λ L. ξ is essentially the same length scale as ξ 0 derived by Pippard upon studying the penetration depth in disordered superconductors (see above). ξ ξ 0 will turn out to be related to the length scale over which coherence in the condensed state can be maintained and which reflects the diameter of a Cooper pair. Further details will be discussed in the frameworks of the BCS and the Ginzburg-Landau theory. c Walther-Meißner-Institut

43 Chapter 3 Microscopic Theory The microscopic theory was finally presented in The last break-through was the derivation of a coherent wave function using a superposition of Fermions by Schrieffer. Cooper, on the other hand, showed that an infinitesimally small attractive interaction between Fermions having E > E F leads to a reduction in their energy by an amount proportional to the cut-off energy of the interaction. The possibility of an attraction between electrons mediated by phonons was already proposed by Fröhlich and studied in detail by Pines and Bardeen. Under the supervision of John Bardeen the three ingredients were combined. The plan of the chapter is as follows: We first prove the existence of the Cooper instability. In the second paragraph we show that the electron-phonon interaction leads indeed to an attractive interaction. The third paragraph is devoted to the derivation of the coherent wave function. Finally, we show how the Hamiltonian of the interacting system can be minimized at zero and at finite temperature. In the last part we calculate a few response functions in order to show how a superconductor reacts to external perturbations. 3.1 The Cooper Instability We assume that the Fermi sea is completely filled at zero temperature. We now add two particles with momenta k 1,k 2 > k F as shown in Fig If we construct a state of two Fermions the total wave function has to obey anti-commutation relations so as to satisfy the Pauli principle, ψ(r 1,σ 1,r 2,σ 2 ) = ψ(r 2,σ 2,r 1,σ 1 ). (3.1.1) We can separate the wavefunction ψ into three factors if we introduce relative coordinates, r 1 = R + r/2 and r 2 = R r/2, with R and r describing the center of mass and the distance of the two electrons. With spin part represented by χ(σ 1,σ 2 ) and the propagating part by plane waves ψ reads ψ(r 1,σ 1,r 2,σ 2 ) = 1 V e ik R e ik r χ(σ 1,σ 2 ), (3.1.2) which can be reformulated as ψ = 1 V e ik R Φ(r 1 r 2 )χ(σ 1,σ 2 ). (3.1.3) 35

44 36 R. HACKL AND D. EINZEL Microscopic Theory Cooper-Paare T = 0 ( r,, r, ) e ik R ( r r ) - ( r,, r, ) spin 1, 2 k, E F spin 1, Singulett E F + D spin 1 k, 1, 2 2 Figure 3.1: Cooper instability of two particles having opposite momenta and spins in a shell with thickness hω 0 above the Fermi energy E F. At T = 0 the Fermi sea is completely filled hence the additional particles have an energy above E F. Triplett It was shown before that the London theory predicts vanishing canonical momentum of the superconducting carriers [Eqs. (2.3.78) and (2.3.85)] on purely electrodynamic reasons. This momentum can now be identified as the center of mass of two electrons implying a state k 2 = k 1. Consequently, the first term in Eq is essentially unity in equilibrium. The second term shows that the carriers can have internal relative momentum. The third term describes the spins. In a system with inversion symmetry where parity is a good good quantum number the two possible relative orientations of the spins are parallel and anti-parallel, and one arrives at triplet and singlet wave functions χ t (σ 1,σ 2 ) = χ s (σ 1,σ 2 ) = 1 2 ( + ) (3.1.4) 1 ( ). (3.1.5) 2 If the two spins are exchanged χ t (σ 1,σ 2 ) and χ s (σ 1,σ 2 ) conserve or change sign, respectively. Since the center-of-mass part is constant the product of the functions χ and Φ must be anti-symmetric to fulfill Eq. (3.1.1), Φ has to be symmetric and anti-symmetric for singlet and triplet spin wave functions, respectively. Possible representations are Φ s = cos[k (r 1 r 2 )] and Φ t = sin[k (r 1 r 2 )]. Recently, superconductivity was discovered in CePt 3 Si [?] and other systems without inversion symmetry. As a consequence the distinction between singlet and triplet states is not possible any further and there are always mixtures. In addition the Fermi surface splits up into two sheets. However, we won t dwell on this exotic though exciting and well studied case [?] but use it just as an appetizer for the part on novel superconductors. Now, we derive the energy gain for the case of a small attractive interaction between the two extra electrons outside the Fermi sea. If we leave out all complications this is a simple exercise in perturbation theory. We assume that the unperturbed system has only kinetic energy ˆT and count the energy from the chemical potential µ. From a statistical point of view we take the grand canonical potential with the particle number being variable. Subtracting the energy of the Fermi sea means using the Landau-Fermi quasiparticle concept and excitation energies ξ k = ε k µ rather than band energies ε k. With ˆV being an attractive interaction between two electron states k and k having magnitudes larger than the Fermi momentum, k > k F and energy ξ k > 0, the eigen energies E of the perturbed Hamiltonian Ĥ = Ĥ 0 + ˆV can be found from the eigen states of the unperturbed system, Ĥ 0 k = 2ξ k k (3.1.6) c Walther-Meißner-Institut

45 The Cooper Instability SUPERCONDUCTIVITY 37 where the functions k are a complete set of eigen functions obeying k k = δ kk. Then Ĥ ψ = E ψ ψ = ϕ k k. k (3.1.7) (3.1.8) Multiplication with k from the left and collecting terms yields k (Ĥ 0 + ˆV ) k ϕ k k = k E ϕ k k k ϕ k (2ξ k E) = k ϕ k k ˆV k. (3.1.9) The calculation of the matrix element on the right hand side is usually complicated since the potential between two electrons is unknown and since there are only approximations to the wave functions. Eliashberg was the first to present a realistic model [3] in terms of a material-specific electron-phonon interaction. Only recently, the wave functions could be approximated for a few conventional systems [?]. Bardeen, Cooper and Schrieffer, although having a phenomenology for the attractive interaction, assumed simply that V kk k ˆV k is attractive for energies smaller than a typical phonon energy hω 0 and zero otherwise, V kk = { g 2 eff ξ k hω 0 otherwise Now Eq. (3.1.9) can be solved right away by noting that, since k is a complete set, k ϕ k = k ϕ k, ϕ k = g 2 k ϕ k eff 2ξ k E ϕ k = g 2 eff k ϕ k k k 2ξ k E 1 = g 2 eff k 1 2ξ k E. (3.1.10) The k-summation in Eq. (3.1.10) can be transformed into an energy integral as k = N F hω0 0 dξ where N F is the density of states for both spin projections yielding { E = 2 hω 0 exp 2 } N F g 2 eff (3.1.11) for the case N F g 2 eff 1. N Fg 2 eff is a dimensionless parameter which is often abbreviated by the coupling constant λ. In the BCS weak coupling approximation λ is much smaller than 1. In the strong coupling case discussed by Eliashberg λ can be of order 1 or even larger. Here we focus on the weak-coupling limit and note that even in this case the energy gain E cannot be expanded in powers of λ. Via the cutoff energy hω 0, which will be discussed in more detail in the next section, the energy gain depends on a material property. While there are various other possibilities, in the treatment of BCS hω 0 was a typical phonon energy directly explaining the experimentally observed isotope effect T c M β where β is expected to be 0.5 for a harmonic oscillator (Fig. 3.2). 2013

46 38 R. HACKL AND D. EINZEL Microscopic Theory BCS-Theorie Figure 3.2: Isotope effect in tin [10]. The different symbols refer to the results of different authors. The table shows the exponent β for various elements. 3.2 Origin of the interaction Although suggested by the isotope effect (Fig. 3.2) an interaction depending on the lattice is not the most obvious way towards an attraction between electrons. Rather, the electro-static force seems to prevail by far. In order to understand the reasoning at the time of discovery and to get an idea of the relevant physics we first have a closer look at the Coulomb interaction in a metal which is distinctly different from that of point charges given by V C = e2 4πε 0 r (3.2.12) where r = r 2 r 1 is the distance between two electrons at r 1 and r 2. The Fourier transform is given by V C (q) = drv C e iq r = e2 ε 0 q 2. (3.2.13) In a metal, in contrast, the Coulomb interaction is screened. For instance, a charged impurity is hardly visible just a few lattice constants away. The exact distance depends on the charge density and is described ba the Thomas-Fermi theory. The screened potential was proposed by Yukawa in the context of nuclear matter, reads V C,TF (r) = and has the Fourier transform e2 4πε 0 r e r r T F, (3.2.14) e 2 V C,TF (q) = ε 0 (q 2 + ktf 2 (3.2.15) ). If there is one electron per lattice site r T F a and k T F π a close to the Fermi momentum. The potential does not decay strictly exponentially but oscillates around the zero (Friedel oscillations). In any case c Walther-Meißner-Institut

47 Origin of the interaction SUPERCONDUCTIVITY 39 k+q (a) -k-q k+q (b) -k-q E i -q q t E i k -k k -k Figure 3.3: Feynman diagrams for the interaction between two electrons via the emission and reabsorption of a bosonic particle, here a phonon. the perturbation of the charge density is screened over distances of a few lattice constants, and other interactions can gain influence. In conventional metals the most important additional interaction is the electron lattice interaction. Forces from charge and spin modulations are important in systems like cuprates or iron-based superconductors but will not be studied here at the moment. In most of the cases the energy scales of the electronic degrees of freedom are much higher than those of the secondary interactions implying the existence of distinctly different time scales. In the case of the phonons this difference is quite obvious since the masses of the involved particles are roughly 4 orders of magnitude different. While the electrons follow a perturbation practically immediately the ions react slowly but remember the perturbation for longer. This is called the retardation effect and can be understood as follows: An electron with momentum k moves in the lattice and attracts the ions close to its trajectory. This distortion of the lattice survives on time scales much longer than the lifetime of an electron between collisions and provides an attractive potential for the other electrons especially (for phase space reasons) for those having opposite momenta k. Pines and Bardeen have studied the resulting potentials between the electrons in the framework of the jellium model (electrons on a homogeneous background of positive charge) and found the result used for the BCS theory [?,?]. The derivation can be found in the textbooks [14] and will not be repeated here. The result for the full potential is expressed in terms of the dielectric constant one obtains V kk = e 2 ε 0 q 2 ε(q,ω), (3.2.16) and if the Thomas-Fermi screening and the electron-ion interaction are included in the expression for ε(q,ω) the full result can be written down as V kk = ( ) e 2 ε 0 (q 2 + kt 2 F ) 1 + ω2 q ω 2 ωq 2. (3.2.17) One can get an idea (not a rigorous derivation) of the quantum mechanical processes by considering the energy change due to emission and reabsorption of phonons by electrons as suggested by de Gennes and shown in Fig To this end we have to calculate in second order perturbation theory the matrix element k ˆV e p e k from two transitions of the form k Ĥ e p i = W q (i) and sum over all intermediate 2013

48 40 R. HACKL AND D. EINZEL Microscopic Theory states i. Since there are two time-reversed processes we get k ˆV e p e k = 1 2 i { 1 k Ĥ e p i + 1 } i Ĥ e p k. (3.2.18) E k E i E k E i If we ignore the influence of the small differences between the states k and k on the electron phonon matrix elements the second element on the r.h.s. is just W q (i). The initial and final energies are E k = 2ξ k and E k = 2ξ k, respectively, since ξ k is a symmetric function of k as is the phonon energy hω q w.r.t. q. In the intermediate state the crystal momentum and the energy are conserved, yielding equal energies for diagram (a) diagram (b) momentum energy momentum energy electron 1 k = k + q ξ k k ξ k electron 2 k ξ k k = (k + q) ξ k phonon q hω q +q hω q both processes in the intermediate states, E (a) i = E (b) i = ξ k +ξ k + hω q. After replacing ±(ξ k ξ k ) by hω the energy denominators in Eq. (3.2.18) read ± hω hω q. Since there is no reference to the intermediate state any further the summation is only over the matrix elements and the total expression for the energy change due to electron-phonon coupling becomes k ˆV e p e k = W q 2 h { }. (3.2.19) 2 ω ω q ω + ω q For the full interaction including Coulomb part we use the Thomas-Fermi result [Eq. (3.2.15)] in addition to Eq. (3.2.19) and get k ˆV k = V C,T F (q) + W q 2 ω q h ω 2 ωq 2. (3.2.20) Although this result is anything else but quantitative it provides a feeling for the underlying physics: the energy dependence of the effective electron-electron interaction results from a phonon-mediated perturbational correction to the electronic energies (Fig. 3.4). It is limited to a range of the order of the Debye energy hω D. At energies well above hω D the usual screened but repulsive Coulomb interaction prevails. For small energies the effect of the Coulomb interaction is overcompensated by the lattice, sometimes called overscreening effect, and can lead to a appreciable attractive potential that provides the interaction needed for Cooper pairing. 3.3 The BCS wave function As outlined in chapter 1 [Eq. (1.2.1)] the construction of a coherent state of bosons is relatively straightforward. With bosons the macroscopic occupation of the ground state can be realized directly since all particles can be in the same state of vanishing momentum at sufficiently low temperatures. With Fermions the Pauli principle precludes a direct solution. Before we present Schrieffer s solution to the problem we have a brief look at the coherent state proposed by Schrödinger [?] since it has already all relevant properties needed later and is easier tractable. We only summarize the results and leave the detailed calculations as an exercise (problem 1 of set 1). c Walther-Meißner-Institut

49 The BCS wave function SUPERCONDUCTIVITY 41 6 (V C,T F ) -1 < k V k > e -p in te ra c tio n B C S a p p ro x. V ω/ω q Figure 3.4: Effective electron-phonon interaction for a mode hω q. For the plot Eq. (3.2.17) is used with a small damping γ/ω q = Both the energy and the potential are normalized. The red curve visualizes the BCS approximation for the attractive potential using an arbitrary magnitude of V 0 = Coherent states in a boson field Bosonic and fermionic properties can be formulated in second quantization using creation (a k ) and annihilation (a k ) operators which usually depend on momentum k. The operators obey commutation or anti-commutation relations [a,b] = ab ba reflecting the symmetry properties of the wave functions. In the case of bosons they read, ] [a k,a k = a k a k a k a k (3.3.21) = δ k,k (3.3.22) [a k,a k ] = 0 (3.3.23) [a k,a k ] = 0 (3.3.24) where δ k,k is the Kronecker δ. The simplest case of a coherent state is a superposition of an infinite number of harmonic oscillator wave functions without momentum dependence, ψ n = 1/ n!(â ) n 0 with 0 denoting the ground state, α = e α 2 2 n=0 (αâ ) n n! 0 = e α 2 2 e (αâ ) 0. (3.3.25) After discussing the Madelung transformation of the Schrödinger equation and deriving the supracurrent [Eq. (2.3.74)] it becomes clear that writing the complex number α as α = α e iϕ means establishing a new wave function. Since α is an eigenvalue of the annihilator â, â α = α α the expectation values of the number operator N ˆn = α â â α and all its moments can be calculated right away without referring to Eq. (3.3.25) yielding the following main results: N N = ˆn 2 ˆn 2 N N = α 2, (3.3.26) = 1 N, and (3.3.27) N ϕ 1 2. (3.3.28) 2013

50 42 R. HACKL AND D. EINZEL Microscopic Theory Kohärente Zustände <n> 1/2 p(<n>,n) 0.4 0,4 <n> = 10 <n> = ,3 <n> = , , ,0 n <n> n/<n> Figure 3.5: Poisson distribution P n (N) for different expectation values N = ˆn as a function of the occupation number n. The n-axis is normalized to N. For small N the asymmetry is still visible. The ordinate axis is normalized to N 1 since the maximum of P n (N) scales as 2πN 1 in the limit N. In the sum O( N) distributions P n (N) contribute for large N. In other words, in absolute units the width increases while the relative width decreases. Obviously, α 2 corresponds to the average occupation number of the state α. The distribution around N becomes increasingly sharp with increasing N and the amplitude and the phase are conjugate variables. The phase becomes infinitely rigid in the large N limit. Although all properties are describe in this way it is instructive to write down the expectation value for the number operator explicitly using Eq. (3.3.25), N = ˆn = n=0 n α 2n e α 2 (3.3.29) n! = α 2 e α 2 n=1 = α 2. n α 2(n 1) n(n 1)! The r.h.s. of Eq. (3.3.29) is a Poisson distribution for n, ˆn = n=0 np n(n) [see also Eq. (3.3.26)]. If P n (N) is plotted for different values of N as a function of n Eq. (3.3.27) can be visualized directly (Fig. 3.5). If the k dependence is restored wave functions of atoms in a trap or laser fields can be described, ( α = exp α k a k 1 k 2 a k ). 2 (3.3.30) for instance, being, however, beyond our interest here. Rather we focus now on fermions Properties of fermions Fermions are governed by the Pauli principle stating that each quantum state can only be occupied once. For this reason there is only the ground state 0 and a singly occupied state 1, and the wave functions are anti-symmetric as formulated in Eq. (3.1.1). Using creators (c k ) and annihilators (c k), the Pauli c Walther-Meißner-Institut

51 The BCS wave function SUPERCONDUCTIVITY 43 principle corresponds to the following properties, c k 0 = 0 (3.3.31) c k 0 = 1 c k 1 = 0 c k 1 = 0. (3.3.32) (3.3.33) (3.3.34) For the anti-commutation relations we use braces {} + in order to make affairs as clear as possible. In addition, we have to now take care of the spin σ, {c kσ,c k σ } + = c kσ c k σ + c k σ c kσ = δ kk δ σσ (3.3.35) {c kσ,c k σ } + = 0 (3.3.36) {c kσ,c k σ } + = 0. (3.3.37) We define now ˆn kσ = c kσ c kσ and find 0 ˆn kσ 0 = 0, 1 ˆn kσ 1 = 1, etc. These properties are sufficient to derive a coherent state of fermions A coherent state of fermions The target is to derive a fermion state similar to that of Eq: (3.3.30). To this end we first define pair annihilation and creation operators P k = c k c k and (3.3.38) P k = c k c k, (3.3.39) respectively, and calculate their properties. P k 0 corresponds to the creation of a state with an electron at k having up-spin and simultaneously one at k having down-spin. Since the two involved fermions have opposite spin it is sensible to look at commutators rather than anti-commutators, yielding (as shown in problem 3 of set 5) [P k,p k ] = 0, (3.3.40) ] [P k,p k = 0, and (3.3.41) ] [P k,p k = δ kk (1 ˆn k ˆn k ). (3.3.42) The latter relation is neither a commutator nor an anti-commutator in the usual sense for being different from 1 or 0 in the general case. We finally need the powers of P k for calculating the exponential series, P k P k = ( ) 2 P k = c k c k c k c k = c k c k c k c k = 0. (3.3.43) 2013

52 44 R. HACKL AND D. EINZEL Microscopic Theory Note that we exchanged c k c k in the second line producing a minus sign and a series of two equal creators acting on the ground state thus making the whole operator to vanish. Using this and the anticommutator in Eq. (3.3.41) two properties, Schrieffer s ground state can now be written down as ( ) BCS = a exp = c exp k = c k k α k P k ( α k P k 0 ) 0 ( ) 1 + α k P k 0. (3.3.44) For the last transformation we used that all powers of P k beyond linear vanish. We set BCS BCS = 1 for determining the constant c and find 1 BCS BCS = c 0 (1 + αkp k )(1 + α k P k ) 0. k (3.3.45) The normalization condition can be satisfied if all factors are unity, 1 = c 0 (1 + α kp k )(1 + α k P k ) 0 = c 2 (1 + α k 2 ), yielding the constant c. Using this normalization the BCS ground state can be written down as BCS = (u k + v kp k ) 0 with (3.3.46) k u k = αk 2 and v k = α k 1 + αk 2 where u k 2 + v k 2 = 1. (3.3.47) BCS is a wave function of a coherent state with the complex number α k having an amplitude and a phase. The same holds true for the coherence factors u k and v k which will be derived in detail later. Using the conjugate complex follows the convention in Annett s book and is the most popular but not the only one in the literature. The BCS wave function describes the coherent superposition of the vacuum and 2, 4, 6... electrons. Below we derive the average number of particles in the condensate and its fluctuations and find full agreement with the coherent state Eq. (3.3.25). 3.4 Determination of the ground state Using BCS we have to find the ground state of the Hamiltonian Ĥ BCS of the system by minimizing E = BCS Ĥ BCS BCS. There are various ways. BCS used real numbers, u k = sinθ k and v k = cosθ k, which obviously satisfy the normalization [Eq. (3.3.47)]. The most direct approach is the Bogoliubov transformation. However, using Lagrange multiplicators helps visualizing several relevant details (Annett). Therefore, we make this detour before deriving the Bogoliubov transformations. The problem we wish to solve is minimizing the ground state energy, E = BCS Ĥ BCS BCS. To this end we switch to momentum space which is equivalent to Fourier transform the Hamiltonian. The technique is named second quantization. c Walther-Meißner-Institut

53 Determination of the ground state SUPERCONDUCTIVITY The BCS Hamilonian in second quantization As already demonstrated in Problem 2 of set 5 the position dependent wave functions ˆψ σ (r) can be expanded in plan waves. The transformations and the corresponding inverse transformations read ˆψ σ (r) = 1 V ĉ kσ e ik r ĉ kσ = 1 k V ˆψ σ(r) = 1 V ĉ kσ e ik r ĉ kσ = 1 V k ˆψ σ (r)e ik r dr (3.4.48) ˆψ σ(r)e ik r dr. (3.4.49) The Hamiltonian in configuration space is a sum over all N electrons and has a kinetic energy term, one from the external fields such as the potentials φ(r) or A(r), and the interaction between the electrons which is supposed to lead to superconductivity, Ĥ = σ N i [ h 2 ] 2m 2 i V ext (r) σ N i, j V (r i r j ) (3.4.50) The factor 1/2 in front of the interaction tern takes care of double counting. If the expansions Eqs. (3.4.48) and (3.4.49) are inserted the sum over the particles transforms into a k sum, and integrating over all space completes the Fourier transformation, Ĥ = [ h ˆψ σ(r) 2 ] 2m 2 V ext (r) ˆψ σ (r)dr+ 1 2 V (r r ) ˆψ σ(r) ˆψ σ (r) ˆψ σ(r ) ˆψ σ (r )drdr. (3.4.51) While the evaluation of the kinetic energy term is straightforward that of the other contributions is more time consuming and will be left as an exercise. The kinetic energy ˆT works as follows: ˆT = = 1 V σ [ h ˆψ σ(r) 2 k,k 2m 2 ] ˆψ σ (r)dr [ĉ k σ e ik r h 2 k 2 2m ] ik r ĉkσe dr h = 2 k 2 σ k 2m ĉ kσĉkσ. (3.4.52) After Fourier transforming the interaction term we can finally write Ĥ = k,σ ε k ĉ kσĉkσ + V q ĉ k 1 +q,σ 1 ĉ k 2 q,σ 2 ĉ k2 σ 2 ĉ k1 σ 1. (3.4.53) k 1,k 2,q,σ 1,σ 2 The interaction term can be simplified by using the assumptions for a Cooper pair, k 2 = k 1 and, for a singlet state, σ 2 = σ 1 and the resulting BCS Hamiltonian reads Ĥ BCS = ε k ĉ kσĉkσ + V k,k ĉ k,σ k,k,σ k ĉ k ĉ k ĉ k. (3.4.54) The BCS Hamiltonian is a rather general and allows, for instance, for anisotropies in the interaction potential. It does neither cover triplet pairing nor is it sufficient to deal with exotic cases such as noncentrosymmetric systems with mixtures of singlet and triplet pairing and spin-orbit splitting of the Fermi surface. However, for all our purposes below, including the discussion of the cuprates, the iron-based, and heavy fermion systems with nodes and sign changes of the energy gap k. Before deriving the ground state energy we determine a few expectation values for the BCS wave function. 2013

54 46 R. HACKL AND D. EINZEL Microscopic Theory Some expectation values We now calculate the expectation value for the momentum distribution function for one spin projection, ˆn k = BCS ĉ k ĉk BCS = 0 (u k + v k ĉ k ĉ k )ĉ k ĉk (u k + v kĉ k ĉ k ) 0 = u k 2 0 ĉ k ĉk 0 + u k v k 0 ĉ k ĉk ĉ k ĉ k 0 + +v k u k 0 ĉ k ĉ k ĉ k ĉk 0 + v k 2 0 ĉ k ĉ k ĉ k ĉk ĉ k ĉ k 0 = u k u k v k 0 + v k u k 0 + v k 2 (1 0 ĉ k ĉ k 0 ) = v k 2. (3.4.55) The term with the prefactor v k 2 can be broken down as 0 ĉ k ĉ k ĉ k ĉk ĉ k ĉ k 0 = 0 ĉ k (1 ĉ k ĉk )(1 ĉ k ĉk )ĉ k 0 = 0 ĉ k ĉ k 0 0 ĉ k ĉ k ĉk ĉ k 0 0 ĉ k ĉ k ĉk ĉ k ĉ k ĉ k ĉk ĉ k ĉk ĉ k 0. The first term is 1 either since the operators in this order yield 1 1 or can be rearranged as above. In all other cases the last two operators anti-commute, and...c 0 always vanishes. The total number of particles is just the sum over all k-points, N = k,σ v k 2 = 2 k v k 2 = (1 u k 2 + v k 2 ). (3.4.56) k With the expectation value of the four-particle interaction operator given by ĉ k c k ĉ k ĉ k = v k v k u k u k the energy expectation value of the BCS Hamilton operator can be written as E E = 2 ε k v k 2 + V k,k v k v k u k u k. (3.4.57) k k,k Before starting to determine the minimum of E we write down two other expectation values which will be useful below and are christened Gorkov amplitudes after one of the pioneers of the theory of superconductivity, ĉ k ĉ k = u k v k g k (3.4.58) ĉ k ĉ k = u kv k g k, (3.4.59) which can easily be recognized as the expectation values of the pair annihilator and creator, respectively. Finally, we look at the fluctuations N of N. To this end we need quantities such as N = N 2 N 2 = 2 k ( ˆn 2 k ˆn k 2 ) = 2 k v k 2 2 k v k 4 = 2 k v k 2 2 k (1 u k 2 ) v k 2 = 2 k u k 2 v k 2. This expression is proportional to N. So we find immediately that N/N = N 1 and realize that the relative fluctuations of the particle number vanish in the limit N (whereas the fluctuations N diverge as N). In other words the particle number has the statistical behavior of a coherent state [Eqs. (3.3.25) (3.3.27)] meaning that order N particles fluctuate in an out of the unpaired part of the electrons. Where are they going to? To find that out we determine the fluctuations of the condensate which have to be determined from ĉ k ĉ k ± ĉ k ĉ k. The amplitude or particle number fluctuations are described by the + sign, and N is proportional to N as above and the relative fluctuations vanish as N 1. Correspondingly, the absolute fluctuations of the phase to be derived from ĉ k ĉ k ĉ k ĉ k vanish in limit N as expected for a coherent state. (I have to prove that mathematically but I believe it is true.) c Walther-Meißner-Institut

55 Determination of the ground state SUPERCONDUCTIVITY Determination of the energy minimum at T = 0 We show here how the energy minimum for the BCS ground state at T = 0 can be found. The solution will be included in that of the following section but the derivation here is almost free of formalities and instructive. The minimum will be calculated with the constraints of (1) the conservation of the total number of particles and (2) of u k 2 + v k 2 = 1. The method used for this procedure is that of the Lagrangian multipliers. Since there are two constraints we have two factors λ 1 and λ 2 the meaning of which will become transparent during the derivation. Although returning a result identical to that Bardeen, Cooper, and Schrieffer obtained by using sine and cosine trigonometric functions for u k and v k, fixing the condition Eq. (3.3.47), the method used in the textbook by Annett has the advantage of clarifying the physical meaning of the quantities µ and E k and highlighting the quasiparticle concept. The two constraints are ϕ 1 = N (1 u k 2 + v k 2 ) = 0 k ϕ 2 = u k u k + v k v k + 1 = 0, and upon setting zero the partial derivatives of E + λ 1 φ 1 + λ 2 φ 2 w.r.t. u k and v k one obtains the eigenvalue equations (ε k + λ 1 )u k + k v k = λ 2 u k (3.4.60) ku k (ε k + λ 1 )v k = λ 2 v k, where k = V k,k u k v k k V k,k g k k (3.4.61) was used along with the definition in Eq. (3.4.58). k is the gap parameter which has the full momentum dependence here and corresponds to the binding energy of the Cooper pairs and the energy gain due to the reduction of the electronic energy. The first multiplier shifts the energy, and we identify λ 1 = µ where µ is the chemical potential. The second multiplier λ 2 = ±E k corresponds to the energy of the quasiparticles excited out of the condensate across the gap (Bogolubov quasiparticles, see next paragraph), and upon solving the eigenvalue equation we find E k = (ε k µ) 2 + k 2 = u k 2 = 1 [ 1 + ξ ] k 2 E k v k 2 = 1 [ 1 ξ ] k 2 E k u k v k = k 2E k, ξ 2 k + k 2 (3.4.62) (3.4.63) (3.4.64) (3.4.65) where we have also defined the quasiparticle energy ξ k = ε k µ in the normal state. Eqs. (3.4.63) and (3.4.64) are the coherence factors and describe the occupation probability of unpaired holes and electrons, 2013

56 48 R. HACKL AND D. EINZEL Microscopic Theory respectively. Eq. (3.4.65) describes the occupation of the condensate. Upon inserting Eq. (3.4.65) into Eq. (3.4.61) the fully momentum dependent gap equation, k = k V k,k k, (3.4.66) 2 ξ 2k + k 2 is obtained. This equation cannot be solved analytically in the general case. If the gap and the interaction potential are assumed to be k-independent one obtains an equation similar to (3.1.10). After transforming the sum into an integral the solution for λ = N F g eff 2 (attractive interaction V = g eff 2 ) and the cut-off of the attractive range hω 0 reads 1 = λ ln hω 0 + ( hω 0 ) [ ] ] 2 hω 0 [ hω 0 ln ln 2 hω 0 = 2 hω 0 exp, yielding ( 1 λ ). (3.4.67) 3.5 The general solution In the preceding section it was shown how the ground state of a superconductor can be constructed from the coherent wave functions of the condensate [Eq. (3.3.46)]. For deriving the properties at finite temperature we have to deal with the 4-operator term ĉ k ĉ k ĉ k ĉ k in the interaction energy Approximation of the four-operator term Wick s theorem states that four-operator terms can be approximated by a sum of pairs of operators times the expectation value of the corresponding pair, ĉ k ĉ k ĉ k ĉ k ĉ k ĉ k ĉ k ĉ k +... This can be made plausible by using the identities ĉ k ĉ k = g k + ĉ k ĉ k g k = g k + δg k ĉ k ĉ k = g k + ĉ k ĉ k g k = g k + δg k with the Gorkov amplitudes defined in Eqs. (3.4.58) and (3.4.59). The interaction energy can now be expressed as E int = k,k V k,k (g k + δg k )(g k + δg k ) V k,k (g k δg k + g k δg k + g k g k ) k,k ) = (g kĉ k ĉ k + g k ĉ k ĉ k g k g k = k,k V k,k [ k ĉ k ĉ k + k k ( k ĉ k ĉ k g k k) ]. (3.5.68) c Walther-Meißner-Institut

57 The general solution SUPERCONDUCTIVITY 49 Here all terms higher than linear in δg ( ) k were neglected, and Eq. (3.4.61) and its conjugate complex were used. On noting that the quasiparticle energies are symmetric in momentum, ξ k = ξ k, the kinetic energy ˆT can be rewritten as ˆT = ξ k ĉ k,σĉk,σ k,σ = ξ k (ĉ k ĉk + ĉ k ĉ k k = k ) ( ) ξ k ĉ k ĉk ξ k ĉ k ĉ k + ξ k, (3.5.69) where Eq. (3.3.35) was applied in the last step. If the preceding two equations are added the mean field Hamiltonian Ĥ MF µn can be written in matrix form as proposed first by Nambu, Ĥ MF µn = k ( )( ) ξ k + g k k + ĉ k,ĉ ξk k k }{{} k ξ k }{{} Ĉ k Ξ k ( ĉk ĉ k } {{ } Ĉ k ). (3.5.70) The operators Ĉ k and Ĉ k are vectors of two creators/annihilators and are called spinors. Ξ k is the energy matrix with the quasiparticle energies in the diagonal and the gap as off-diagonal elements. For finding the minimum of Ĥ MF µn the energy matrix has to be diagonalized by a unitary transformation. The problem is similar to that of a two-level system with finite coupling between quantum mechanical oscillators The Bogoliubov-Valatin transformation For diagonalization we first insert the transformation matrices B k and B k into Eq. (3.5.70) and then determine their coefficients, [ ] Ĥ MF µn = ξ k + g k k +Ĉ k (B k k }{{} B k )Ξ k(b k }{{} B k )Ĉ k. (3.5.71) }{{} For a unitary transformation the product of B k and B k has to yield the unity matrix, B kb k = 1. Consequently one can write the transformation in terms of u k and v k etc. with u k 2 + v k 2 = 1 as above, B k = B k (B k) = ( uk v k v k u k ( u k v k v k u k ) (3.5.72) ). (3.5.73) This is the Bogoliubov-Valatin transformation which yields new operators ˆB k = Ĉ k B k and ˆB k = B kĉk reading ˆβ k = u k ĉ k v kĉ k (3.5.74) ˆβ k = v kĉ k + u kĉ k (3.5.75) 2013

58 50 R. HACKL AND D. EINZEL Microscopic Theory and, respectively, ˆβ k = u kĉ k v kĉ k (3.5.76) ˆβ k = v k ĉ k + u k ĉ k, (3.5.77) and the new energy matrix ( B k Ξ Ek D kb k = k D k E k ). (3.5.78) The coefficients u k and v k are found by the condition D k = 0 and the normalization and lead to the same result as the minimization of the energy as displayed in Eqs. (3.4.62) (3.4.65). The result is equivalent to the solution of the eigenvalue equation Eq. (3.4.60) and corresponds to finding the ground state. Here we made an important additional step by deriving the new ˆβ operators. The two equations (3.5.74) and (3.5.76) define these new operators as a coherent superposition of creators and annihilators, hence electrons and holes, at k and k. It can be shown directly by insertion that the ˆβ operators obey fermion anti-commutation relations, { ˆβ kσ, ˆβ k σ } + = δ kk δ σσ (3.5.79) { ˆβ kσ, ˆβ k σ } + = 0 (3.5.80) { ˆβ kσ, ˆβ k σ } + = 0. (3.5.81) Consequently, in the superconductors there are no electrons or hole any further but, rather, composite fermions carrying the name Boguliubov quasiparticles and the condensate of Cooper pairs after their inventors. The probability of finding one of those objects in energy and momentum space is given by the coherence factors u k and v k which can be visualized particularly easily by assuming a linear electronic dispersion around the Fermi energy and momentum, ξ k hv F (k k F ) with v F and k F the Fermi velocity and the Fermi momentum. In the state k, k with particle-hole mixing every band crossing the chemical potential µ is reflected about µ. Since the particles in the reflected band are those of the original one the two bands mix in the usual quantum mechanical fashion and the levels repel each other by virtue of the interaction energy 2 k. At the intersection point k F u k 2 and v k 2 are both 1/2 and the character of the Boguliubov quasiparticles is exactly 50% electronic and 50% hole-like. Away from k F the character changes between 0 and 100%. Fig. 3.6 shows a theoretical result for particlehole mixing in a strong coupling material, here the cuprate Bi 2 Sr 2 CaCu 2 O 8. (From [15].) The band is reflected about µ and the color-coded occupation probability is given by the coherence factors shown in Fig If the new ˆβ operators [Eqs. (3.5.74) and (3.5.76)] and the coherence factors [Eqs. (3.4.63) and (3.4.64)] are plugged back into Eq. (3.5.71) Ĥ MF µn can be separated into a ground state part and a part of thermally excited quasiparticles, Ĥ MF µn = k = k = k = k Ĥ MF µn = k [ ξ k + g k k + ( ) ξ k + g k k ( ) ξ k + g k k ( ) ξ k + g k k ( ˆβ k, ˆβ ) ( E k 0 k 0 E k + k + k + k,σ ) ( ξ k E k + g k k ( E k ˆβ k ˆβ k E k ˆβ k ˆβ k ( E k ˆβ k ˆβ k + E k ˆβ ( E k ˆβ k,σ ˆβ k,σ E k ) ) ( ˆβk ) ˆβ k k ˆβ k E k ) )] + E k ˆβ ˆβ k,σ k,σ. (3.5.82) k,σ c Walther-Meißner-Institut

59 The general solution SUPERCONDUCTIVITY = 0 mev = 20 mev Energy k (ev) a a Figure 3.6: Dispersion ξ k along the (0,0) (π,0) (Γ X) direction, typical for a cuprate superconductor, above (left) and below T c. The chemical potential (yellow horizontal line) is at zero energy in the quasiparticle picture, the Fermi momentum is indicated in red. (The blue vertical line is an artifact.) Shown here is the result of a strong coupling calculation in false color representation with warm colors indicating the maxima of the spectral functions. Above T c the band crosses µ = 0 at k = k F. Below T c the band is reflected about µ. The two bands intersecting at k F interact similarly as states in a two-level system. The spectral weight changes according to the coherence factors plotted in Fig For every fixed k the integral over all energies is unity corresponding to one quasiparticle. From [15]. u k 2, v k 2, u*v ( 0 ) -1 /k F = /t = 0.2 u k 2, v k 2, u*v, f( k ) u*u v*v u*v f k ( /a) Quasiparticle energy k ( ) Figure 3.7: Coherence factors u k 2, v k 2, and u k v k for electron, hole, and pair occupation, respectively. (a) shows the coherence factors as a function of momentum. The underlying band structure corresponds to the 2D tight-binding representation ξ k = t[cos(k x a) + cos(k y a)] µ (a is the lattice constant) along k y = 0 at half filling, µ t. For better visualization the gap is as large as 10% of µ. For a conventional metallic superconductor /µ = O(0.001) in the cuprates the maximal gap may reach 5 10% of µ. The half width (FWHM) k of u k v k defines the BCS coherence length, ξ BCS ξ 0 = 2π/ k. (b) shows the coherence factors as a function of energy. The Cooper pairs live around the Fermi energy but extend substantially into the band. As a result all electrons including the unpaired ones are enslaved and have the same phase. The Fermi function at T = 1.76T c, f (ξ k,1.76t c ), has an energy dependence similar to that of v k 2. The ˆβ operators obey fermionic anti-commutation relations [Eqs. (3.5.79) (3.5.81)]. Hence the expectation values of the corresponding number operators are described by Fermi statistics, ˆβ k ˆβ k = ν(e k ) 1 exp( Ek k B T ) + 1 (3.5.83) ˆβ k ˆβ k = 1 ν(e k). (3.5.84) Here we have written the Fermi distribution function as ν(e k ) rather than f (E k ) as usual in order to 2013

60 52 R. HACKL AND D. EINZEL Microscopic Theory highlight that the energy of a Bogoliubov quasiparticle is in the argument. These equations show that the second term on the r.h.s. of Eq. (3.5.82) describes the kinetic energy of free Fermions having dispersion E k. It vanishes in the limit T 0 and in the limit 0. (Note that for correctly determining the transition into the normal state the limit is lim 0 E k = ξ k at any temperature.) Since the second term vanishes in the limit T 0 the first term is the ground state energy U BCS. Using these limits appropriately Eq. (3.5.82) is the basis for determining the energy gain in the superconducting state (see Problem 2 in set 7) yielding E T =0 = Ĥ MF µn Ĥ n = 1 4 N F 2 (3.5.85) for a constant gap and N F the density of states at the Fermi energy E F for both spin projections (which is twice the number of k points). From Eq. (3.5.83) the expectation value of the Gorkov amplitude at finite temperature can be derived (see problem 1, set 7), ĉ k ĉ k = {1 2ν(E k )}u k v k ( ) Ek = tanh u k v k (3.5.86) 2k B T with u k v k given by Eq. (3.4.65). This result enables us to write down the gap equation at finite temperature Solution of the gap equation for T 0 Together with Eq. (3.5.86) the gap equation (3.4.61) with the definition Eq. (3.4.58) can be written down as a major result of this section, k = V k,k ĉ k ĉ k k,k = k,k V k,k u k v k tanh k = k,k V k,k k 2E k ( Ek ) 2k B T ( Ek tanh 2k B T ). (3.5.87) This equation reproduces the result of Eq. (3.4.67) of section if the limit T = 0 is taken. At finite temperature one can calculate the limit 0 and derive the transition temperature T c (where the gap vanishes). To simplify we use the BCS approximation, as above, and assume that k = is constant, the dispersion is parabolic and isotropic, and that the interaction is attractive and also isotropic yielding 1 N F g eff 2 1 λ = hω0 0 dξ E tanh ( E 2k B T The limit E ξ and the substitution x = ξ /2k B T c yield ). (3.5.88) 1 λ hω0 /2k B T c = 0 ( 2e γ = ln π dx tanhx x hω 0 2k B T c ), (3.5.89) c Walther-Meißner-Institut

61 The general solution SUPERCONDUCTIVITY 53 where γ = is the Euler constant. Combining this result and that of Eq. (3.4.67), k B T c = eγ π 2 hω 0e 1 λ (3.5.90) = 2 hω 0 e 1 λ returns the famous universal BCS relationship between the gap and the transition temperature, k B T c = π e γ = (3.5.91) In the majority of cases the ratio of twice the gap to T c is given, 2 /k B T c = Note that any dependence on materials properties, in particular the cutoff energy hω 0, corresponding to the Debye frequency hω D in the case of electron-phonon coupling, and the electron-phonon coupling parameter λ drop out entirely. While this is typical for a mean field theory as here it is not realistic for most of the materials. Only a few of the elements such as Al, Sn or V are close to the BCS prediction. Elements like Pb or Nb and most of the alloys have gap ratios well above 3.53 and reach 5.6 in Nb 3 Sn. This means that the influence of the cutoff or, more physically, of details of the interaction cannot be ignored any further. Eliashberg solved that problem in 1960 [3]. In essence the solution takes into account the full energy dependence of the interaction potential leading to energy dependent gap functions as well. The treatment requires the knowledge of Green s function techniques, and the discussion is far beyond the scope of these lecture notes. If the interaction potential V k,k is either repulsive or strongly momentum dependent the gap function k may become momentum dependent as well and may even change sign on the Fermi surface [16, 17]. What means a sign change of the gap? Formally, this is a simple question since we can write down the gap in terms of k-dependent functions such as k = 0 2 [cos(k xa) cos(k y a)], (3.5.92) where a is the lattice constant of a square lattice. Eq. (3.5.92) reproduces the d x 2 y2 gap in the copperoxygen compounds, for instance. Fossheim and Sodbø describe in some detail how this gap follows from a specific choice of V k,k. The problem with visualizing such a gap lies in the way it is measured experimentally. In the typical spectroscopic experiments such as tunneling, photoemission, optical or Raman spectroscopy or thermodynamic experiments such as heat capacity, heat transport or penetration depth only the magnitude of the gap k enters. In addition, it is hard to imagine how an energy reduction below the chemical potential can be both positive and negative. The solution is that the gap has a phase deriving from the wave function of the coherent state. This phase enters whenever the relative phases of two coupled condensates are probed. Also remember that the expression for the current contains a phase gradient. Consequently, whenever experiments with coupled superconductors are performed the phase and, hence, the sign of k becomes intuitively clearer. The seminal phase-sensitive experiments in the cuprates [18, 19] will be discussed in more detail in chapter 6. A momentum-dependent gap raises additional questions as to the possible outcome of thermodynamic experiments or the condensation energy. In the latter case Eq. (3.5.87) has to be solved for the respective momentum-dependent gap. In the case of d x 2 y 2 gap in Eq. (3.5.92) the mean field condensation energy leads to a maximal gap larger than the canonical BCS value, and 2 0 = In the cuprates one finds 2 8 indicating very strong coupling. If there is a multiband system the smallest gap can be well below 3.53 while the largest one always exceeds 3.53 since otherwise the condensation energy is not reached. Concerning thermodynamics the answer is similarly complicated. For a system with a finite gap everywhere on one of the Fermi surfaces one typically finds activated behavior, i.e. a response being characterized by a gap. Then the smallest gap dominates (but does not exclusively determines) the 2013

62 54 R. HACKL AND D. EINZEL Microscopic Theory activation energy since, with increasing temperature, the branches with the smaller Bogolyubov energy E k are populated first. If the gap has nodes, (k 0 ) = 0, there is no activated behavior any further because unpaired particles exist whenever T 0, hence always. Then the specific heat, the penetration depth or the thermal conductivity are described by temperature power laws rather than exponentials. Prominent examples are UPt 3, UBe 13 [13] and YBa 2 Cu 3 O 7 [20]. For predictions the explicit variation with temperature of the gap and of the condensate density is required. Neither the temperature dependence of the gap nor that of other thermodynamic quantities can be written down explicitly. Rather the implicit Eq. (3.5.88) needs to be solved. Mühlschlegel derived numerical values which are tabulated [21]. In addition, interpolation schemes can be derived [22] which provide an analytic expression for 0 < T T c which agrees with the exact numerical solution to better than a fraction of a percent and is sufficient for most of the practical purposes, 0 (T ) 0 (0) = tanh ( πk B T c 2 0 (0) 3 c s c n c n [ 0 (0)] 2 Tc [ k (0)] 2 FS [ Tc T T T c ] ). (3.5.93) Note that the T c /T term is not defined at T = 0 while the limit T 0 exits. The result of Eq. (3.5.93), as plotted in Fig. 3.8, has the right asymptotic behavior and can be used for estimates of other properties. It is derived for the weak-coupling BCS limit but it is also valid approximately for the strong coupling case. In particular, it is also valid for superconductors with anisotropic gaps. Then the gap ratio (T )/ (0 ) Figure 3.8: Temperature dependence of the superconducting energy gap according to Eq. (3.5.93). The deviations from the numerical solution [21] is of the order of a percent. 0 (0)/k B T c (see above), the discontinuity of the heat capacity c s c n c n at T c, and the Fermi surface average of [ k / 0 (0)] 2 have to be used accordingly. In the case of an isotropic gap one finds 1.764, 1.426, and 1, respectively, for the so-called BCS-Mühlschlegel parameters. For the case of d x 2 y2 gap [Eq. (3.5.92)] in a tetragonal system, which, cum grano salis, applies for the cuprates, the weak-coupling values are, respectively, 2.14, and 0.5 [22]. The influence of replacing the BCS-Mühlschlegel parameters on 0 (T ) is mild. Other quantities may change substantially. T /T c 3.6 Connection to experiments In this section we derive a few quantities which follow from the BCS theory and can be observed experimentally including thermodynamic properties, the tunneling density of states, and two-particle properties for which the coherence factors play an important role. c Walther-Meißner-Institut

63 Connection to experiments SUPERCONDUCTIVITY Thermodynamic properties The solution of the gap equation for all temperatures enables us to determine the energy o the Bogoliubov quasiparticles E k (T ) = [ξ 2 k + k(t ) 2 ] 1/2 and their thermal occupation ν(e k (T )) leading directly to the thermodynamic functions. For transforming the k sums into integrals the momentum dependence will be dropped occasionally. Since the Bogoliubov quasiparticles are a system of free fermions and since the condensate does not contribute for being a fully ordered state the density of electronic entropy in the superconducting state comes only from the quasiparticle and can be written in terms of ν(e k (T )), s s = 2k B [ν k lnν k + (1 ν k )ln(1 ν k )]. (3.6.94) k From Eq. (3.6.94) the heat capacity c s follows directly as c s = T ds s dt. (3.6.95) Some writing can be saved if one takes the derivative w.r.t. β = (k B T ) 1, c s = β ds s dβ = 2βk B ν k k E k [ Ek β d ] dβ k(t ) 2. (3.6.96) Here the derivative of the occupation number w.r.t. the quasiparticle energy leads directly to the Yosida function Y (T ) which enters many thermodynamic properties, N F Y (T ) = ν k = k E k N F 4k B T c µ dξ [ )] 2 (3.6.97) cosh( E k B T which is exponentially small at low temperature for an isotropic gap and shows directly that the heat capacity show activated behavior then. At T c the heat capacity of the normal state is also needed and can be derived from Eq. (3.6.96) by setting the gap zero reproducing the result (2.2.42), c n = γt = π2 3 k2 BN(E F )T. From that and Eq. (3.6.96) one finds for an isotropic gap, ( ) d c(t c ) = N(E F ) (T ) 2 dt and for the ratio to the normal-state value another universal number, c c n = (3.6.98) Using the result in Eq. (3.6.96) the internal energy and finally the free energy density in the normal and the superconducting state and, thereof, the critical field as a function of temperature can be found. Now, by using the BCS expressions, the variation with T is slightly different from the parabola as shown in Fig As mentioned earlier weak coupling superconductors are described well by the BCS prediction while those with strong coupling first get closer to the parabola and then even go beyond. Fig shows the temperature dependence of the thermodynamic functions as derived above. 2013

64 56 R. HACKL AND D. EINZEL Microscopic Theory Figure 3.9: Temperature dependence of the critical field. The lower curve follows from the BCS prediction the upper one represents the parabola from the two fluid model proposed first by Gorter and Casimir. From Ref. [2]. Figure 3.10: Temperature dependence of the thermodynamic function according to the BCS model. From Tinkham Single particle response Since the condensate is a coterie only pairs of electrons can be added or removed (Andreev). Therefore, if single particle properties are being studied, only the quasiparticle contribute and the response from the condensate can be disregarded. This holds for (single particle as opposed to Josephson) tunneling and photoemission spectroscopy for instance whenever the final state is outside the superconductor. One of the most important quantities in this is the single particle density of states N(E) which is a momentum integrated function. Since the number of particles above and below T c is conserved on can simply equate N(ξ )dξ = N(E)dE and, using the approximation N(ξ ) N(0) in the normal state, finds for the c Walther-Meißner-Institut

65 Connection to experiments SUPERCONDUCTIVITY 57 superconducting density of states, N(E) N(0) { = dξ de = E E E > 0 E <, (3.6.99) exhibiting the typical square root singularity at the gap edge. This result is well reproduced in weakcoupling isotropic superconductors. 2013

66

67 Chapter 4 Ginzburg-Landau Theory The model of Ginzburg and Landau (GL) proposed 1950 is an extension of the London theory and allows the treatment of spatially inhomogeneous mixtures of normal and superconducting regions in a material. In this way all configurations of currents and fields in a superconductor can be treated. Therefore the GL theory has a wide range of applications in practice. Gorkov showed in the beginning of the 1960ies that the BCS theory and the GL model are equivalent in the temperature range around T c where the GL theory is valid. 4.1 Phase transitions Before discussing the results for superconductors we briefly summarize the theory for second order phase transition originally proposed for magnets by L. D. Landau in Landau expanded the density of the free energy in powers of the magnetization M being a natural order parameter different from zero in the ordered state (T < T c ) and identical zero above. Hence the description of magnetism is a clear-cut problem. The purpose of the theory, formulated a long time before numerical methods were successfully introduced to the problem of magnetism, was essentially to derive thermodynamical properties in the ordered state and later also above in the fluctuation regime from the simplest possible assumptions. Then the density of the free energy can be written as f M = f n + α(p,t )M(r) 2 + β(p,t ) M(r) 4. (4.1.1) 2 Without external fields only the magnitude of M(r) = M(r) enters. In the simplest case (no position dependence) the free energy becomes minimal if the first derivative of f vanishes and the second derivative is positive, 0 = f M M = 2α(p,T )M + 2β(p,T )M3 (4.1.2) 0 < 2 f M M 2 = 2α(p,T ) + 6β(p,T )M 2. (4.1.3) A non-trivial solution of Eq. (4.1.2) is M 2 = α β (4.1.4) which has a real solution only if α and β have opposite sign and β 0. This is fulfilled at T < T c for α(t ) = α 0 (T T c ) (4.1.5) 59

68 60 R. HACKL AND D. EINZEL Ginzburg-Landau Theory with α 0 and β positive constants. If M 2 from Eq. (4.1.4) is inserted into Eq. (4.1.3) 4α is obtained being positive in the ordered phase, and the extremum of f M is a minimum if and only if α < 0 and β > 0. The evolution of the free energy landscape for various temperatures is shown in Fig The f(p,t,m ) α = 0.1 α = α = 0 α = α = M /M 0 Figure 4.1: Free energy density in a magnetic system as a function of the temperature. The temperature is expressed in terms of α. For negative α the minimum of f M is found at finite magnetization M. difference f M f n = α2 2β (4.1.6) corresponds to a reduction of the free energy in the ordered state and opens the way towards the derivation of all thermodynamic properties as in section 2.2. With α given in Eq. (4.1.5) the transition is of second order, implying that there is no volume and entropy change and, consequently, no latent heat at T c. Then the Ehrenfest relations [?] between the discontinuities in the heat capacity c p, the thermal expansion, ᾱ T, and the compressibility, κ T can be derived (Landau-Lifshitz, Statistical Physics), ᾱ Tc = d p dt κ Tc (4.1.7) Tc c p = d p T c dt ᾱ Tc. (4.1.8) Tc The pressure dependence of the transition temperature T c, dt c /d p is closely (but certainly not trivially) related to microscopic mechanisms at the origin of a phase transition, as can be seen for instance from the Eq. (3.5.90) in the case of superconductivity, and is therefore a highly desirable quantity. Since c p and ᾱ T are experimentally accessible Eq. (4.1.8) can be used to indirectly determine dt c /d p. This is sometimes simpler than a direct measurement of T c in a pressure cell. In any case, one obtains an internal thermodynamic consistency check and can get the compressibility from Eq. (4.1.7). Obviously, this type of approach proves very useful already in its simplest version. Ginzburg and Landau showed that it can be applied to superconductivity and even well beyond [12]. 4.2 Application to superconductivity A major problem is the proper definition of an order parameter. One could argue that the temperature dependent condensate density of the London theory would be the right choice. However, it becomes immediately clear that one would not even get a current where the phase gradient plays a crucial role. This c Walther-Meißner-Institut

69 Application to superconductivity SUPERCONDUCTIVITY 61 was the motivation for Ginzburg to suggest an order parameter in the spirit of the Madelung substitution for the wave function, Eq. (2.3.65), ψ(r,t) = a(r,t)e iθ(r,t), where a(r,t) and θ(r,t) are real functions of position and time. In the GL theory the function ψ depends on also temperature T is normalized as 2 ψ(r,t,t ) 2 = 2a(r,t,T ) 2 = n s (T ) = n f s (T ). (4.2.9) where n is the constant density of electrons in a material, and n s (T ) is the number of condensed electrons that is twice the number of Cooper pairs. f s (T ) is a dimensionless function that vanishes at T c and reaches unity if all electrons are condensed in the limit T 0. Note that the GL model is valid only for T T c Density of the free energy The density of the free energy of a superconductor can be derived in the spirit of Eq. (4.2.16) as long as there are no external fields or currents, f s = f n + α(p,t ) ψ(r,t) 2 + β(p,t ) ψ(r,t) 4 2 (4.2.10) f n + α(p,t ) a(r,t) 2 + β(p,t ) a(r,t) 4, 2 (4.2.11) which has solutions equivalent to those of Eq. (4.2.16). Here, only the amplitude is important for finding the minimum, and the phase can still be chosen arbitrarily as shown in Fig In the figure this f s ( ) Re Figure 4.2: Free energy density in a superconductor with out field and currents. The energy minimum depends only on the amplitude of ψ but is independent of the phase θ. corresponds to moving the order parameter around the bottom of the valley. Moving out of the valley is an amplitude fluctuation for which energy must be invested. As soon as there are currents and fields the kinetic energy of the electrons and the field energy becomes important. In addition there is an interrelation. The kinetic and field energy are given by 1 2 M v2 1 2M h i ψ(r,t) QAψ(r,t) 2 (4.2.12) and, respectively, by W = 1 B 2 2µ MdV = 1 0 2µ 0 [B(r) B 0 ] 2 dv (4.2.13) 2013 Im

70 62 R. HACKL AND D. EINZEL Ginzburg-Landau Theory neglecting demagnetization effects. The magnitude is taken for arriving always at real numbers for the energies. Nowadays one would call that an educated guess since the result is correct but there is no justification otherwise. Since all functions depend now on position only appropriate integrals over the sample and over all space lead to the proper energy minimum corresponding to deriving and minimizing the functional of the free energy Functional of the free energy To this end the energies related to the order parameter are integrated over the sample volume while the magnetic energies have to be taken in the entire space yielding F s = F s + sample αψ ψ + β 2 (ψ ψ) 2 + h2 2M }{{} γ ψ + Q 2 i h Aψ dv + 1 [B(r) B 0 ] 2 dv. (4.2.14) 2µ 0 Originally the parameter γ was used in order to indicate that the third term in the first integral is another expansion coefficient. It will turn out, as expected, that M and Q are the mass and the charge of a Cooper pair having being unknown at the time the GL theory was proposed. For finally deriving the GL equations the variation of Eq. (4.2.14) has to be determined. Since the absolute value is inconvenient in this context we rewrite the γ term, h 2 2M sample ψ + Q 2 ( i h Aψ dv = h2 ψ + Q ) 2M sample i h A ψdv + i h ψ (i h + QA)ψ ˆndS, 2M S (4.2.15) where S is the sample surface and ˆn the surface normal in point r. In addition we will need the variation of the magnetization reading δ 1 2µ 0 B 2 MdV = 1 µ 0 = 1 µ 0 = 1 µ 0 = 1 µ 0 B M δb M dv B M ( δa M )dv ( B M ) δa M (B M δa M )dv ( B M ) δa M dv (4.2.16) The Ginzburg-Landau equations For the deriving the GL differential equations the variational derivative of F has to be evaluated. This means finding the change δf if ψ or A are replaced by ψ + δψ and A + δa. If F is minimal F ψ δψ = 0; F A δa = 0 where for traditional reasons the variation of ψ is considered, and the subscript M referring to the magnetization has been dropped for simplicity. The variation of ψ is obviously calculated only inside c Walther-Meißner-Institut

71 Two new length scales SUPERCONDUCTIVITY 63 the first integral which vanishes only if the integrand vanishes for all possible δψ yielding, ( ) 1 2 h 2M i QA ψ + (α + βψ ψ)ψ = 0 (4.2.17) ( ) h i ˆn QAˆn ψ = 0 (4.2.18) ( ) h i ˆn QAˆn ψ = bψ. (4.2.19) Eq. (4.2.17) is the first Ginzburg-Landau equation (GL1) from which most of the properties can be derived. Eq. (4.2.18) results from the second integral of Eq. (4.2.15) over the sample surface and says that the gradient of the wave function perpendicular to the surface vanishes in zero field meaning that no current flows in or out and that there are no variations of the pair density close to the surface as we have seen already before. If one attaches a metal to the surface of a superconductor one arrives at Eq. (4.2.19). Note that deriving such an equation implies that Eq. (4.2.15) is valid also outside the superconductor, at least in a small distance. This observation is called proximity effect and is consistent with the microscopic theory in that the coherent wave function of the pairs leaks out of the material over a length scale given by the coherence length. Along with the proximity effect, which means that superconductivity can be induced in a normal metal attached to a superconductor, the Josephson effect is another important consequence. The variation of the vector potential δ A leads to the second Ginzburg-Landau equation (GL2) which describes the current density in the presence of superconductivity, j s = h Q i 2M (ψ ψ ψ ψ ) Q2 M Aψ ψ (4.2.20) Q ( ) ] hi [ψ 2M QA ψ + c.c. (4.2.21) where c.c. means complex conjugate and indicates that the current density is a real quantity. If we insert again the Madelung wave function we obtain j s = a2 Q 2 M ( ) h Q θ A and with the usual replacement Q 2e (traditionally no minus sign here), M 2m, and 2a 2 n, j s = ne2 m ( ) h 2e θ A reproducing the London result Eq. (2.3.74). (4.2.22) (4.2.23) 4.3 Two new length scales The equations for the current (4.3.25) and (4.2.22) and the first GL equation (4.2.17) without field lead to two new length scales tha turn out to be similar to the scales λ L and ξ 0. However, though being related some care has to be taken to keep the definitions clear and to understand the meaning properly. We shall outline the relation without derivation later. 2013

72 64 R. HACKL AND D. EINZEL Ginzburg-Landau Theory Screening For a moment we go back to Eq. (4.2.22) and take the curl, j s = a2 Q 2 and find upon using Ampère s law M ( A) = a2 Q 2 M B (4.3.24) ( B) = µ 0 j s 2 B = a 2 Q 2 µ 0 B. M (4.3.25) Eq. (4.3.25) is formally equal to Eq. (2.3.76), and if a 2 = α β, Q = 2e, M = 2m are substituted the GL penetration depth is obtained, λ 2 GL = mβ 2µ 0 α e 2. (4.3.26) It is tempting to use 2a 2 = n as above to recover the London penetration depth but then we get the zero temperature limit while the GL theory is valid only close to T c. However, we can insert the temperature dependence of α and find λ GL to diverge as T c T 1 at T c and, in the clean limit, λ GL (T,l ) = λ L(0) 2 T c T T c. (4.3.27) Ginzburg-Landau coherence length If A = 0 in the first GL equation (4.2.17) and a 2 0 = α/β is substituted for a real equilibrium order parameter GL1 can be made dimensionless by dividing the entire equation by a 0, h 2 2 f 4m α x 2 + f + f 3 = 0. (4.3.28) Here the problem is confined to one dimension and f = a/a 0. For dimensional reasons the prefactor of the second derivative w.r.t. to x must by a length squared and defines already the GL coherence length ξ GL, ξ 2 GL(T ) = h 2 4m α 0 (T T c ) (4.3.29) the meaning of which becomes transparent by looking at small deviations from the equilibrium, δ f = f 1. By neglecting all terms of order (δ f ) 2 and higher one finds a differential equation for δ f, 2 x 2 (δ f ) = 2 ξgl 2 (δ f ), (4.3.30) which identifies ξ GL as the healing length of a perturbation to the equilibrium order parameter. With this length scale we have a phenomenological argument in favor of Eq. (4.2.19) and observe that the order parameter ψ does not change abruptly but rather varies continuously over a characteristic distance ξ GL. However, similarly as in the case of the penetration depth, ξ GL is different from the BCS ξ 0 in both magnitude and meaning. While the magnitudes of ξ GL and ξ 0 differ only by a factor of order 1 c Walther-Meißner-Institut

73 Two new length scales SUPERCONDUCTIVITY 65 the interpretation of ξ GL is purely phenomenological. It has neither a connection to Cooper pairs nor to non-locality: the GL theory is strictly local. However, in contrast to the London limit with the coherence length exceeding the penetration depth ξ GL is the smallest length scale in those cases where the GL theory develops its full power. Hence, effects of non-locality can be ignored safely. The ratio of the penetration depth and the coherence length is an important quantity and we define the dimension less GL parameter, κ = λ GL ξ GL = β 2µ 0 2m he = β 2µ 0 1 µ B. (4.3.31) where µ B is the Bohr magneton. κ depends only weakly on temperature through β, so in a first approximation it can be considered constant close to T c. However, Eq. (4.3.31) and, similarly, the equations (4.3.29) and (4.3.27) conceal that ξ GL and λ GL are not solely given by constants of nature but, rather, depend in subtle ways on the BCS coherence length ξ 0 and the mean free path l of the electrons. The functional dependence is obtained by comparing the results of BCS and GL close to T c. While the dependences of ξ GL (ξ 0,l) and λ GL (ξ 0,l) are very important whenever real materials are to be analyzed quantitatively they will not particularly further the basic understanding here. We study now the consequences of varying κ. In practice this means looking at different (clean) materials covering a range of approximately 0.1 < κ Energy of the normal-superconductor interface We consider now an infinite superconductor. At x < 0 the magnetic field is homogeneous, points along the z-axis, and just reaches the critical field B c. For x 0 the field decays, hence the surface x = 0 separates the normal from the superconducting part. Then superconductivity is almost completely suppressed at x = 0 implying that the order parameter, i.e. the density of pairs a 2, at the surface is vanishingly small and negligible in comparison to the equilibrium value a 2 0 deep inside the superconductor for x. On the other hand, the magnetic field is screened over a typical length scale of a few λ GL. What is energetically more favorable, small or large κ? Small κ means that the field is pushed out over a short length scale while condensation energy is lost in the range ξ GL λ GL needed for a/a 0 to recover to unity. In the opposite case the field can penetrate over a much larger range than that needed for a to approach a 0. Obviously, condensation energy is lost here in a smaller volume than in the previous case, and less energy has to be invested to keep the field out. This situation is energetically more favorable and can be written down by considering the free energy density for x 0, ( ) f s dx = f n B2 c dx + (ξ GL λ GL ) B2 c. (4.3.32) 2µ 0 2µ 0 The last term is an estimate for the trade-off between condensation and field energy. The energy is readily written down since the condensation energy has exactly the same magnitude as the the energy stored in the field. The signs of ξ GL and λ GL are determined in a way that the free energy increases with ξ GL and decreases with λ GL. In this way a surface energy σ can be defined. The sign of σ determines as to whether or not the field penetrates, σ = f δ B2 c 2µ 0 (ξ GL λ GL ) (4.3.33) where a negative sign corresponds to an energy gain. For calculating numbers the functional form of the field penetration and the recovery of a/a 0 cannot be neglected but will not be considered here since it 2013

74 66 R. HACKL AND D. EINZEL Ginzburg-Landau Theory will be derived in a very simple way below. The result yields a limiting value for κ for which σ changes sign, κ < 1 2 typei (4.3.34) κ > 1 2 typeii (4.3.35) dividing the world into type I superconductors which exclude the field always completely except for the thin surface layer carrying the screening current and type II superconductors into which the field can penetrate. We have seen earlier that vortices are formed which carry one flux quantum each. Since κ reaches values in the range of 100 for copper-oxygen superconductors the upper critical field up to which superconductivity survives can be orders of magnitude higher than the field B c1 at which the first flux line penetrates and the thermodynamical critical field B th c (to be defined later). 4.4 States with internal flux For κ > 1 it becomes increasingly favorable for the superconductor to let the field penetrate as observed first by Shubnikov and coworkers (see Fig. 2.2). The consequences are far-reaching and the basis of many applications such as coils for generating high magnetic fields. The way the field penetrates first was discussed in section (2.3.4), and the field penetration depth λ L was found to be the characteristic dimension of the vortex while a mysterious cutoff ξ λ L was postulated to keep the problem tractable by avoiding a divergence in the center of the vortex. It is intuitively clear now that the scale ξ ξ 0 introduced there is related to (but not equal to) the GL coherence length ξ GL. We show now that ξ GL is intimately related to the upper critical field B c2 at which superconductivity collapses The upper critical field B c2 To this end the first GL equation (4.2.17) has to be analyzed. Assuming that the applied field B 0 is only slightly below B c2 Eq. (4.2.17) can be linearized since the order parameter ψ ψ is suppressed substantially below its equilibrium value ψ 0 2 = α/β and βψ ψ α. The linearized equation reads h 2 2M ( i + QA)2 ψ = α ψ. (4.4.36) Eq. (4.4.36) is formally equal to a Schrödinger equation for a particle with mass M and charge Q in a field B = A, and the problem can be mapped on a known one. For doing so, the field is assumed to point along the z-axis, B = B 0 ê z, and a gauge is chosen for which A = ê y B 0 x. With the abbreviations ω c = QB 0 M eb 0 m x 0 = hk y M ω c [ = k yφ 0 2πB 0 (e > 0); (4.4.37) ] and the ansatz (4.4.38) ψ = exp(i[k y y + k z z]) f (x) (4.4.39) where ω c is the cyclotron frequency, the equation of a one-dimensional harmonic oscillator, ( ) h2 2 f (x) 2M x 2 + M ω c (x x 0 ) 2 f (x) = α h2 kz 2 f (x) E f (x), (4.4.40) 2 2M c Walther-Meißner-Institut

75 States with internal flux SUPERCONDUCTIVITY 67 is found which is entirely equivalent to finding the Landau levels of an electron in a normal metal. x 0 is the equilibrium position, and ω c is the eigenfrequency. The harmonic oscillator has the spectrum ( E = n + 1 ) hω c 2 (4.4.41) where n is an integer. The final step is to find the largest field B 0 for which Eq. (4.4.41) has a solution. With the usual substitutions for M and Q, and E given in Eq. (4.4.40) one obtains ( n + 1 ) h eb 0 2 m + h2 kz 2 4m = α 0(T c T ). (4.4.42) The field becomes maximal for n = 0 and k z = 0, hence B c2 = 4mα 0(T c T ) h h 2 2e = Φ 0 2πξ 2 GL (T ). (4.4.43) (4.4.44) The second equation shows immediately that the vortex lines are essentially at a distance of ξ GL (T ) before superconductivity collapses. As a consequence, for κ 1, the field and the Cooper pair density a 2 oscillate weakly on similar length scales. For large κ the field penetrates the superconductor almost homogeneously and a 2 oscillates between zero in the vortex core and a value much smaller than a 2 0. The temperature dependence of B c2 is linear close to T c, as can be seen directly from Eq. (4.4.43), hence is consistent with Eq. (2.1.6). To make it consistent with the BCS prediction the prefactors become important. This problem has been addressed by Gorkov and will be discussed in a later version of the lecture notes. In a finite field B < B c2 there is no phase transition at T c but only at a lower temperature T c (B) which can be obtained by inverting Eq. (4.4.43), [ T c (B) = T c (0) 1 2πξ (0) GL B Φ 0 ], (4.4.45) where ξ (0) GL = h2 /4mα 0 T c (0) was used. Not unexpectedly, the transition temperature into the superconducting state decreases linearly with increasing field. It should be noted, however, that some care should be taken again upon extrapolating the result to low temperatures and identify Φ 0 /2πξ (0) GL with the critical field at zero temperature. In other words, mind the range of validity! The nucleation field B c3 on the surface So far only an infinite superconductor was considered. If there is a surface, for instance at x = 0, the boundary conditions (4.2.18) or (4.2.19) have to be taken into account depending on whether the superconductor is either in vacuum or covered by an insulator or, respectively, by a metal. In the case of vacuum, which is usually the case for a typical experimental situation, (or in the case of an insulator) the gauge invariant current across the surface must vanish, ( ) h i x 2eA(x) ψ(x) = 0. (4.4.46) x=0 2013

76 68 R. HACKL AND D. EINZEL Ginzburg-Landau Theory In the above paragraph we found that the maximal field corresponds to vanishing momentum along the z-axis, k z = 0, and the lowest level of the harmonic oscillator, n = 0. The corresponding eigenfunction reads f (x) = C exp ( [ x x0 2ξGL ] 2 ), (4.4.47) which, however, satisfies the boundary condition only for x 0 = 0 and x 0 = giving the eigenvalue above. The question is as to whether or not eigenfunctions and -values can be found in the intervall 0 < x 0 < for which the field is different from B c2. Saint-James and de Gennes [?] found a solution in 1963 and showed that for x 0 = 0.59ξ GL 0.59 heb 0 2m = α 0(T c T ) (4.4.48) replaces Eq. (4.4.42) (in the case of k z = 0 and n = 0) leading to the surface critical field B c3 = 1.695B c2. The exact solution is complicated and requires numerics. In particular, the eigenfunction must have a vanishing derivative at x = 0 which is not the case for Eq. (4.4.47) at x 0 0. Kittel and de Gennes suggested variational approaches using analytical functions which are described briefly in the textbooks by Tinkham and de Gennes. Fossheim and Sudbø give a quite detailed derivation. Physically speaking the enhanced surface field corresponds to an enhanced Cooper pair density at the surface being enforced by the boundary condition Eq.(4.4.46). Without the boundary condition the order parameter would have a negative slope at x = 0 whenever x 0 > 0 while it is pushed up otherwise. The boundary condition can be satisfied with any function which is symmetric about x = 0. The simplest ground-state wave function in agreement with this requirement is a Gaussian centered at x = 0. For x 0 > 0 symmetry can be imposed by a superposition of two Gaussians at ±x 0 or, simpler, by a trial wave function f (x) = (1 + ax 2 )exp( bx 2 ). If the potential energy in Eq. (4.4.40) is also symmetrized about x = 0 one arrives at a double well potential. For appropriate values of x 0 the maximum at x = 0 is low and the potential is wide enough to facilitate a lower eigenvalue. The modulus of corresponding eigenfunction has then a small depression at x = 0 but maxima at x x 0 just as the trial wave function for a < 0. This symmetry considerations make the reasoning behind the surface effects more plausible and show how boundary conditions can be imposed. Needless to say that all parts for x < 0 have no physical meaning and are constructed in a similar fashion as mirror charges for instance. The existence of the surface critical field B c3 shows directly that the transition at B c2 cannot be sharp under realistic conditions but, rather, must be a crossover. In particular, if the surface is rough on the order of the coherence length the maximal value of B c3 is not reached and, in addition, vortices are pinned to the surface making affairs even more complicated. In other words, for determining the range of superconductivity in the presence of magnetic fields and, similarly important, currents, which we have disregarded here for the sake of simplicity, become very important. Hence, for characterizing superconductors and determining their intrinsic properties the conditions of the experiments have to be controlled. What we discuss here are the intrinsic properties of clean materials in the limit of full reversibility in order to understand the thermodynamic and microscopic properties at the origin of the condensed state. Deviations from the described behavior are equally important since they are the basis of applications and will be discussed there The thermodynamic critical field B c While the thermodynamic critical field is intuitively clear in a type I superconductor the definition is less obvious when the flux can penetrate. However, the GL approach provides a direct interpretation since c Walther-Meißner-Institut

77 States with internal flux SUPERCONDUCTIVITY 69 the difference between the free energy densities in the superconducting and the normal state is given by f s f n = α2 2β = B2 c 2µ 0. (4.4.49) Recalling the expressions for ξ GL and λ GL in Eqs. (4.3.29) and (4.3.27) we find that the ratio α /β can be derived from λ GL while ξ GL depends on α alone and obtain α = α β = B 2 c = = B c = h 2 4mξ 2 GL (4.4.50) m 2µ 0 λ 2 GL e2 (4.4.51) h 2 2(2e) 2 ξ 2 GL λ 2 GL Φ 2 0 2(2π) 2 ξgl 2 λ GL 2 Φ 0. (4.4.52) 22πξGL λ GL Along with the definition of κ in Eq. (4.3.31) and the expression for the upper critical field in Eq. (4.4.44) we find, B c2 = 2κB c, (4.4.53) immediately proving the dichotomy spelled out in Eqs. (4.3.34) and (4.3.35): Whenever 2κ < 1 the nucleation field B c2 is smaller than the condensation field making it energetically unfavorable for the flux to penetrate into the material. Rather the field is completely expelled at B c. We may ask as to whether B c2 has still a physical meaning. In fact, neglecting surface effects, B c2 is now the supercooling field, and (in very clean samples) an applied field decreasing from B 0 > B c cannot be expelled before reaching B c2 < B c. Once the field is expelled B c can be approached from below. In optimal conditions, including nearly atomically flat surfaces and the suppression of edge effects, the field can be cranked up further and penetrates only at B c1 [?]. Consequently, as expected for a first order phase transition, there is a hysteresis. It is also clear from the discussion of thermodynamics that the integral over the magnetization corresponds to the condensation energy The lower critical field B c1 In the opposite case the nucleation takes place at B c2 > B c but what happens at B c? Unfortunately nothing, making the determination of the condensation energy particularly complicated in all type II superconductors with high B c2. For this reason there is still a debate in the case of the cuprates after more than two decades. However, since the thermodynamical critical field is still determined by the integral over the magnetization there must be a compensation of the contributions to the integral at high fields in the low-field range in the spirit of a Maxwell construction. In other words, the field is completely excluded up to a critical field B c1 and then starts to penetrate. The first flux line penetrates without energy since the Gibbs potentials with and without flux should be the same in this point of the phase diagram. Then, as a first guess, one would expect B 2 c B c1 B c2 and B c 2κB c1. In reality and for κ 1, the core enhances the energy stored in a single vortex line by a factor of order lnκ + ε [see Eq. ( )], and the better estimate for B c1 is given by (see textbooks) B c1 = B c 4πλ 2 GL (lnκ ) κ 1 (4.4.54) yielding B c B c1 B c2 O(lnκ). The relation between the various bulk fields is sketched in Fig

78 70 R. HACKL AND D. EINZEL Ginzburg-Landau Theory - 0 M B c1 B c B c2 B Figure 4.3: Magnetization of type I and type II superconductors as a function of the applied field B for various GL parameters κ (sketch). For κ < 1/ 2 the field is excluded completely (black). The red and the blue curves correspond to κ 0.8 and 2, respectively. As indicated for κ 2 the shaded areas above and below the thermodynamical field B c are equal (Maxwell construction). The infinite slope at B c1 is only observed in ideally clean samples and indicates that the first flux line can penetrate free of energy The Abrikosov lattice (1957) For κ 1 and parallel flux lines along, e.g. ê z, it can be shown relatively easily (Tinkham) that the force density f on one flux line from all other flux lines and a potential transport current is given by f = j s ê z Φ 0 (4.4.55) where j s is the sum of all contributions to the supercurrent density at the location of the flux line, and Φ 0 is the flux quantum. f is repulsive for two neighboring flux lines in equilibrium. If there is a transport current j tr perpendicular to the flux lines the screening currents around the vortices have components parallel and anti-parallel to j tr leading to a Lorentz force on the vortices and making them move perpendicular to j tr and ê z. This movement results in a voltage drop parallel to j tr hence a finite resistance and is potentially detrimental if large currents have to transported in a magnetic field such as in a solenoid. Only if the movement can be suppressed efficiently type II superconductors will be useful for this kind of applications. As will be discussed in chapter 6 the flux lines can be pinned in various ways, and the type II materials with pinning (sometimes called type III superconductors) can in fact be used for high current and high field applications. Here we focus on the problem of having forces on a flux line whenever the line is not in a fully symmetric environment where all screening currents from neighboring lines cancel out by symmetry. Abrikosov found a solution to this problem by deriving a general solution ψ L for the linearized first GL equation which is strictly valid only in the limit B 0 B c2. With the field in ê z direction we found B c2 by evaluating the lowest eigenvalue of Eq. (4.4.40) at n = 0 and k z = 0. Hence, only k y has to be considered in Eq. (4.4.39) and will be abbreviated by k in the following. Eq. (4.4.38) shows that for each k there exists a slab in the yz-plane centered at x 0 carrying one flux quantum. Since we need a symmetric, i.e. a periodic solution, we set k = nq (4.4.56) with a fixed q yielding a real space periodicity y = 2π/q and x n = nqφ 0 2πB 0 (4.4.57) showing directly that the spacing in x-direction is x = qφ 0 2πB 0 (4.4.58) c Walther-Meißner-Institut

79 States with internal flux SUPERCONDUCTIVITY 71 and that B 0 x y = Φ 0 as to be expected. Since the functions ( [ ] ) x 2 xn ψ n = exp(inqy)exp 2ξGL (4.4.59) are orthogonal for different n because of the factor exp(inqy) the general solution is a superposition of all functions ψ n, ( [ ] ) x 2 xn ψ L = C n exp(inqy)exp (4.4.60) n 2ξGL with coefficients C n to be determined. While periodicity in y-direction is built in via Eq. (4.4.56) the periodicity in x-direction is recovered only if the coefficients C n are periodic, C n+m = C n. From the discussion above a square lattice follows immediately for index-independent coefficients C n or, equivalently, m = 1. For C 1 = ic 0 and m = 2 a triangular lattice is found. For deciding as to which of the possible lattices is being realized Abrikosov found that the linearized GL equation is insufficient and observed that the normalization independent parameter β A = ψ4 L ψ 2 L 2 (4.4.61) should be as close to 1 as possible. β A = 1 obviously holds for constant ψ L corresponding to the field free case. All other variations of ψ L (dictated by flux penetrating the material) lead to larger values of β A. For the square lattice β A = 1.18 for the triangular lattice β A = Hence the triangular lattice wins by a very small margin. Since numerical calculations are necessary it is not that surprising that Abrikosov, in his original paper in 1957, found the square lattice to prevail. Kleiner, Roth, and Autler showed in 1964 that the triangular lattice out of all periodic solutions has in fact the optimal value of β A. Tinkham notices that, for the same density of flux lines, the distance a i between the flux lines is slightly larger in the triangular lattice (closest packing) than in the square lattice, a triangle = ( ) asquare (4.4.62) 3 favoring the first one since it reduces the positive repulsive energy between the lines. The flux line lattice has been visualized first by Essmann and Träuble in 1967 by decoration and, more recently, by scanning techniques as shown in Fig The difference between the possible periodic solutions is in fact sufficiently small to allow solutions other than triangular in many cases were either the underlying lattice or the symmetry of the order parameter introduce additional anisotropies. For instance in Nb one finds various transitions between different flux line configurations as a function of temperature and field [?]. In YBa 2 Cu 3 O 7 the d x 2 y2 symmetry of the order parameter is sufficient to make the square lattice more favorable [?]. 2013

80 72 R. HACKL AND D. EINZEL Ginzburg-Landau Theory Abricosov vortex lattice Cne n inqy 1 n exp x 2 2 ( T ) Bc q (a) (b) (c) Figure 4.4: Flux line lattice. (a) and (b) show the different filling for the square (a) and the triangular (b) flux line lattice. From Tinkham [?]. (c) Scanning tunneling map of NbSe 2 in the Shubnikov phase [23]. The dark areas correspond to the vortex cores where the superconducting gap and, hence, the order parameter vanishes. c Walther-Meißner-Institut

81 Chapter 5 The Josephson Effect Elementary optics and quantum mechanics show that waves can tunnel through regions of space even in the case of a potential barrier. For instance, light can travel across a sufficiently narrow gap between two dielectrica even if the incoming wave is totally reflected from the inner surface of one of the materials. Another example are electrons tunneling through an insulator sandwiched between two metals. In all cases the wavelength and the height of the barrier govern the transmitted intensity. Typically, for distances larger than a few wavelengths the transmitted intensity vanishes. What happens in these circumstances with the Cooper pairs of a condensate, and is it worthwhile to bother? If we could couple two condensates weakly it appears very intriguing to utilize the rigid phases on either side of the junction so as to generate beat frequencies in the MHz or GHz range having unprecedented stability and watch quantum mechanics at work on a macroscopic scale. When Josephson considered this possibility in 1962 [?] the experts were skeptical. John Bardeen for instance argued that the tunneling probability would be orders of magnitude too small for being the square of the tunneling efficiency of a single electron. It turned out that the rules for condensates are different and that the single particle probability still determines the order of magnitude of the matrix element. 5.1 Weakly coupled superconductors The Ginzburg-Landau model gives us an idea of how we should think of pair tunneling at least in the case of a metallic interface between two superconductors. Eq. (4.2.19) explicitly describes how far the wave function extends into the normal metal via the proximity effect, and Eq. (4.3.29) shows ξ GL (T ) to be the approximate length scale. A schematic view of the experimental setup for observing the Josephson effects is shown in Fig The two superconductors which may be made of different materials are separated by a thin insulating or metallic layer (grey). Other realizations of weak links are either narrow superconducting bridges with lateral dimensions close to ξ GL ξ 0 or boundaries between differently oriented single crystals or intrinsic contacts in natural very anisotropic materials [?] or hetero structures. Most importantly, the critical supercurrent density across the weak link must be orders of magnitude smaller than in the bulk and coherence between the two condensates must be maintained. The junctions can be either voltage or current biased. In most of the cases the current I is determined from out side and the voltage U is the dependent quantity. Nevertheless, the current-voltage characteristics will be plotted the other way around partially for traditional reasons. However, since the energy is given eu this is also more instructive as we will see below. 73

82 74 R. HACKL AND D. EINZEL The Josephson Effect superconductor insulator/ metal superconductor ~ 0 U I Figure 5.1: Schematics of a weak link. The grey slab separating the superconductors on the left and the right can be either an insulator or a semi conductor or a metal. If the cross section is reduced to dimensions of the coherence length ξ 0 even a superconductor can be used a weak link. The thickness of the layer should be on the order of ξ 0 or less. Usually the contact is current biased and the voltage drop U is the dependent quantity. 5.2 The Josephson equations The dynamics of the Cooper pairs can be described by time-dependent Schrödinger equations for the condensate wave functions ψ 1,2 (r,t) for superconductors 1 and 2 having stationary energy eigenvalues E 1,2. Then the most physical approach to a weak link is to introduce an interaction W 12 E 1,2 between the the two superconductors leading to an equation system of two coupled harmonic oscillators [?,?], i h ψ 1 (r,t) = E 1 ψ 1 (r,t) +W 12 ψ 2 (r,t) i h ψ 2 (r,t) = E 2 ψ 1 (r,t) +W 21 ψ 1 (r,t), where ψ is the partial derivative of ψ w.r.t. time. For solving these equations we use again the Madelung representation of the wave function [Eq. (2.3.65)]. In contrast to the chapter on GL theory we substitute here a 2 1,2 (r,t) = Qn pair(r,t) ρ Q (r,t) for the amplitude. After separating imaginary and real part we get, respectively, ȧ 2 1 = 2W 12a 1 a 2 sin(θ 2 θ 1 ) = ȧ 2 2 and (5.2.1) h ha 2 θ 1 1 = E 1 a 2 1 +W 12 a 1 a 2 cos(θ 2 θ 1 ) (5.2.2) ha 2 2 θ 2 = E 2 a 2 2 W 21 a 1 a 2 cos(θ 1 θ 2 ). (5.2.3) (5.2.4) From the first equation we obtain that ȧ 2 1 = ȧ2 2 meaning that the rate of change of the charge density on the left side is equal but opposite to that on the right side of the junction. In addition, the time derivative of the density is related to a current density by virtue of the continuity equation. However, it is clear that an imbalance of charges is not possible. Hence, so long as no current is supplied from outside there won t be a current across the junction, and the phase difference vanishes. In turn, if an external current from a source is supplied Cooper pairs may move from one side to the other side, and θ 2 θ 1 will be different from zero. The maximal current is proportional to the matrix element W 12 W 21 and is expected to be small. These considerations allow us to write down the first Josephson equation, j = j c sin(θ 2 θ 1 ) j c sin( θ), (5.2.5) c Walther-Meißner-Institut

83 The Josephson equations SUPERCONDUCTIVITY 75 where j c is the critical Josephson current density, which states that there is a finite supercurrent across, e.g., an insulator without a voltage drop. The current is proportional to the sine of the relative phase of the condensates in superconductor 1 and 2 or directly to θ for small currents. An estimate of the critical current can be obtained from GL theory or from microscopic considerations. The latter was achieved by Ambegaokar and Baratoff [?] who found the full temperature dependence of the product of the critical current I c and the normal resistance R n of the junction which can be approximated as R n (T c ), I c R n = π 2e tanh ( ). (5.2.6) 2k B T This important result is universal, i.e. independent of the type of junction. In the limit T 0 it reduces to I c R n = π (0)/2e while close to T c the temperature dependence becomes linear and I c R n = 2.34πk B /e(t c T ) with the constant being 635 µv/k. The phase difference θ is not gauge invariant implying that the current would depend on the selected gauge of the vector potential A in an applied magnetic field. Therefore, before proceeding, we introduce the gauge-invariant phase difference γ, γ = θ 2π Φ A ds, (5.2.7) where the integration limits correspond to the two sides of the weak link. Applying the gauge transformations A A + χ and θ θ + 2e χ immediately proves Eq. (5.2.7). γ is generally valid with and h without field and has to used later when we derive properties of junctions in a field and study quantum interference effects. The gauge invariant first Josephson equation the reads j = j c sinγ. (5.2.8) For the analysis of Eqs. (5.2.2) and (5.2.3) we assume the same material on each side and get a 1 = a 2 and W 12 = W 21. Since the cosine is an even function, the sum of the two expressions yields the second Josephson equation, d dt γ = 2e h U, (5.2.9) where U = E 1 E 2 and Q = 2e. Obviously, upon exceeding j c there is a finite voltage across the junction, and the phase difference becomes time dependent. The differential equation can be integrated right away yielding γ(t) = γ(0) + t 0 2π Φ 0 Udt = γ(0) + 2π U Φ 0 t. (5.2.10) If we insert γ(t) into the first Josephson equation we obtain a current ( j(t) = j c sin γ(0) + 2π U ) t. (5.2.11) Φ 0 that oscillates at the frequency U/Φ 0 where Φ 0 is the flux quantum. The inverse of the flux quantum is the celebrated Josephson frequency 1 Φ 0 = (11) MHz µv, (5.2.12) meaning that for a voltage drop of 1 µv a frequency of (11) MHz can be observed that depends only on constants of nature. Along with the resistance normal given by the von-klitzing constant h/e 2 = R K = (84)Ω the Josephson frequency plays an important role in metrology since the voltage can be linked to elementary constant. 2013

84 76 R. HACKL AND D. EINZEL The Josephson Effect 5.3 The RCSJ model While the fundamental properties are easily derived the details important for the design and understanding of sensors and other devices require the study of individual junctions. For that we note that a weak link is not only a highly non-linear resistance but has an Ohmic and a capacitive shunt which are of crucial importance for the dynamics. Therefore we analyze the properties of the device sketched in Fig The bias current I b is distributed between the three branches across the resistor, the weak link, and the ca- Figure 5.2: Equivalent circuit of a resistively and capacitively shunted junction (RCSJ). (a) The weak link is symbolized by a cross. The total bias current I b is distributed between the three branches as I R = U/R, I J = I c sinγ, and I C = C U. (b) and (c) The dynamics of the RSCJ model can be visualized by a washboard potential where U(δ) = E J cosδ I b /I c δ where δ is the coordinate. (b) If the I b < T c the phase is time independent and a supercurrent flows across the junction. (c) Above the critical current δ becomes time dependent but does not vary linearly as expected for a harmonic oscillator. From Clarke. pacity as I R = U/R, I J = I c sinγ, and I C = C U. We use the second Josephson equation (5.2.9) to eliminate U and U and get I b I c = sinγ + Φ 0 2πI c R γ + Φ 0C 2πI c γ. (5.3.13) Eq. (5.3.13) is equivalent to the second order differential equation of a driven pendulum. The term I c sinγ replaces the linear term of the harmonic oscillator equation with small amplitude and allows for full rotations. The mechanical analogue reads D = mgl sinϕ + Γ ϕ + Θ ϕ (5.3.14) where D is the driving torque, mgl sinϕ is the gravitational energy if the pendulum is rotated out of equilibrium, Γ is the damping, and Θ is the moment of inertia. In the case of the RCSJ model it is customary (and convenient) to introduce the parameters τ c = Φ 0 2πI c R, (5.3.15) β c = 2πI cr 2 C Φ 0 Stewart McCumber parameter (5.3.16) = RC τ c τ RC τ c Q 2 quality factor, τ = t τ c, and dt = τ c dτ, (5.3.17) c Walther-Meißner-Institut

85 The RCSJ model SUPERCONDUCTIVITY 77 and obtain the dimensionless sine-gordon equation 1 i = sinγ + γ τ + β 2 γ c τ 2. (5.3.18) The Stewart-McCumber parameter β c corresponds to a quality factor and indicates how freely the system can oscillate. However, independent of β c the time dependences of i(τ) and γ(τ) will never be harmonic. Rather, as we will see below, β c indicates as to whether or not the I V characteristics is hysteretic. For β c = 0 we find i = sinγ + γ τ (5.3.19) which can be integrated for i > 0 over one period allowing us to derive an average voltage from (5.2.9). Separation of the variables and integration between 0 and 2π and, respectively, 0 and the period duration τ 0 yields 2π i 2 1 = τ 0 (5.3.20) and after restoring units and inserting the average oscillation frequency 2π/T = 2π(τ c τ 0 ) 1 in Eq. (5.2.9) we find U = R Ib 2 I2 c. (5.3.21) This result shows that the voltage is zero for I b = I c and then asymptotically approaches Ohmic behavior for I b I c. For I b < I c integral (5.3.20) diverges implying that the phase γ does not depend on time and U vanishes. It is not possible to solve Eq. (5.3.18) analytically for β c > 0, and the I V characteristics can only be obtained numerically. For β c > 0.5 the I V curves become increasingly hysteretic as shown in Fig This means that the voltage assumes a finite value immediately at I c jumping directly to the Ohmic line for β c 1. On reducing I b the voltage follows Ohm s law down to I < where U finally jumps back to zero and the current flows free of losses. If we integrate over one period we find that Figure 5.3: I V curves for differently damped Josephson contacts. The numerical simulation is according to the model of a resistively and capacitively shunted junction (RCSJ) for β c as indicated. From R. Kleiner (PhD thesis, TUM 1092). R. Kleiner Dissertation TUM 1 The equation is the non-linear analogue of the one-dimensional Klein-Gordon equation, being the relativistic form of the Schrödinger equation. Goldstein, Poole, and Safko (Addison Wesley 2002) speculate that the name is a frivolous pun. 2013

86 78 R. HACKL AND D. EINZEL The Josephson Effect T 0 Udt = Φ 0 (5.3.22) meaning that the phase changes exactly by one flux quantum per cycle. The energy is radiated off the junction with a frequency on the order of the Josephson frequeny times the voltage. 5.4 Josephson contact in a microwave field We go now the other way around and irradiate the junction with a microwave field U HF = U 0 + U 1 cos(ω 1 t) with U 1 U 0. Then the phase varies as γ(t) = γ 0 + 2e h U 0t + 2e U 1 sin(ω 1 t) (5.4.23) h ω 1 which has to be inserted into Eq. (5.2.8). The general solution is given by Bessel functions of order n, J n as ( ) ( ) 2eU1 2eU0 j(t) = j c J n sin n=0 hω 1 h t ± nω 1t + γ 0. (5.4.24) We search now for the dc contributions to j(t) and observe that the time integral over the sum is different from zero only if ω 1 = 2e h U 0 (5.4.25) and integer multiples thereof. Hence the distance on the voltage axis U n between different steps is an integer multiple of U 0, U n = n hω 2e = nu 0 (5.4.26) The way the calculation works can be visualized relatively easily for the first step (see Problem 2 in set 10). 5.5 Josephson effect in a magnetic field In a magnetic field the gauge invariant phase difference (5.2.7) across the junction becomes relevant as one may anticipate already from the supercurrent (2.3.74) Ring with a single weak link We consider now the setup sketched in Fig. (5.4). There is a ring having a central hole. The total diameter of the ring is at least an order of magnitude larger than the penetration depth λ eff of the magnetic field B = A implying that a contour Γ can be found along which the screening currents vanish. The field can penetrate the weak link between points 1 and 2 and the superconductor above and below up to λ eff. The distance of 1 and 2 from the weak link is larger than λ eff. Then Eq. (2.3.74) yields 1 2 θ ds = 2e 1 A ds θ 1 θ 2 = 2e 1 A ds (5.5.27) h 2 h 2 c Walther-Meißner-Institut

87 Josephson effect in a magnetic field SUPERCONDUCTIVITY 79 (ds) 1 2 J s = 0 Figure 5.4: Ring with a weak link in a magnetic field. The field B = A points out of the plane. The ring walls are much thicker than the penetration depth λ eff of the field. Therefore the current J s along the contour Γ (integration variable ds) vanishes. The field penetrates the weak link and the adjacent superconductor as long as the distance from the weak link is smaller than λ eff. Points 1 and 2 are in the field free part. and θ 2 θ 1 = γ + 2e 2 A ds. (5.5.28) h 1 Upon adding Eq. (5.5.27) and (5.5.28) one obtains 2πn = γ + 2e h = γ + 2e h Γ A ds B ds (5.5.29) where S is the total are inclosed by Γ. Since the phase is unique only modulo 2πn the factor 2πn, where n is an integer, has to be added. The last integral is the magnetic flux Φ through the ring. If we insert the flux quantum Φ 0 we find that the gauge invariant phase ) γ = 2π (n ΦΦ0 (5.5.30) changes proportional to the flux through the ring. The rate of change is given by Φ 0 which is, as we know, a very small number. Hence, a ring as shown in Fig. (5.4) can in principle be a very sensitive sensor if the γ can be measured Ring with two weak links: Quantum interference To this end we consider a ring having two weak links as shown in Fig. 5.5 (a). With this setup the total flux penetrating the ring is proportional to the phase differences γ 1 and γ 2 of links 1 and 2, γ 1 γ 2 = 2π Φ Φ 0. (5.5.31) 2013

88 80 R. HACKL AND D. EINZEL The Josephson Effect Figure 5.5: Superconducting quantum interference device (SQUID). (a) Schematic view. (b) I V curve as a function of the applied field. (c) Modulation of the maximal critical current as a function of the field. Note that the Josephson junctions have to have a relatively small β c so as to avoid hysteresis. If the two contacts have exactly the same properties the total current across the two links is simply the sum of the individual currents, I tot = I c (sinγ 1 + sinγ 2 ). With the substitution γ 1 = γ 0 + πφ Φ 0 γ 2 = γ 0 πφ Φ 0 where γ 0 takes care of all problems with screening currents (which are a complicated issue in practice) we find immediately I max (B) = 2I c sinγ 0 cos πφ Φ 0. (5.5.32) Eq. (5.5.32) states that the current is maximal for vanishing field. This is not surprising since the two currents add symmetrically. In all other cases there is a superposition of the bias and the screening current which lead to a pattern of I max (B) which is equivalent to that of an optical interference experiment with a double slit. Therefore the ring with two weak links is named superconducting quantum interference device (SQUID) 2. If β c of the two contacts is in the right range I max (B) oscillates around a mean value but does not reach zero. In addition all hysteresis effects vanish. Then the voltage in the resistive state is a unique function of B as long as 2Φ < Φ 0. If the field through the SQUID is compensated in the most sensitive part of the I V curve [Fig. 5.5 (b)] one has a very sensitive field detector (10 6 Φ 0 typically). If not the voltage oscillates [Fig. 5.5 (c)] Quantum interference in a long junction So far we assumed that the current is constant across the entire junction. However, the width of the superconductor perpendicular to the field B and the current I is usually as large as several penetration depths λ as shown in Fig Hence, the gauge invariant phase may vary substantially in the direction perpendicular to the current and the field. The width of the contact a is much larger than the penetration depth λ, and the field points into the positive z-direction. Note that d ξ can be larger than λ in type I superconductors. Then the flux through the area enclosed by the curve Γ (blue) is given by Φ(x) = Bx(d + 2λ) (5.5.33) 2 John Clarke who invented the SQUID in 1966 is a gourmet. c Walther-Meißner-Institut

89 Josephson effect in a magnetic field SUPERCONDUCTIVITY 81 yielding the gauge invariant phase, γ(x) = γ(0) + 2π Φ(x) Φ 0. (5.5.34) For getting the maximal supercurrent I max in y direction Eq. (5.5.34) will be inserted in the first Josephson x B d a/2 0 x y -a/2 Figure 5.6: Long (wide) weak link. The width a in x-direction is much larger than the penetration depth λ. The current I is along y, and the field B points along the z-direction. For getting the flux Φ(x) one has to integrate in the x y plane inside the curve Γ (blue). The screening currents j screen (λ) are indicated schematically (red). Here they are larger than the critical current across the weak link. in y direction the sample is larger than shown. equation (5.2.8) that is then integrated in x and z direction in the limits [±a/2] and [0,c] where c is the thickness of the sample: ( ) sin πφ(x) Φ I max = I c sin(γ(0)) 0. (5.5.35) πφ(x) Φ 0 Eq. (5.5.35) is equivalent to the Fraunhofer pattern in optics describing the intensity distribution of the diffracted light after a single slit while the SQUID corresponds to the double slit experiments. In practice the flux dependence of the maximal current across a SQUID is a superposition of a sin and cos modulation since the legs of the SQUID have also a finite width. If Eq. (5.5.34) is inserted in the Eq. (5.2.8) an x- dependent current density as shown in Fig. 5.7 is found which explains the zeros in the maximal current physically. If the field is reduced it can be expelled from the junction. However, since the critical current across the junction is very small in comparison to that in the bulk the Josephson penetration depth λ J, ( ) 1 Φ0 2 λ J =, 2πµ 0 j c d (5.5.36) is usually much larger than the London penetration depth λ L. Eq. (5.5.36) is derived from the Ferrell- Prange equation [?]. Typical values for λ J exceed those for λ L by at least one order of magnitude. 2013

90 82 R. HACKL AND D. EINZEL The Josephson Effect I max /I c e Magnetic field B = 0 Magnetic field B = B 0 / / 0 f Magnetic field B = B 0 Magnetic field B = 3B 0 /2 Figure 5.7: Current in a long junction. (a) (d) Current distribution for different field/flux values where B 0 corresponds to one flux quantum. (e) Maximal current (in units of the critical current) as a function of the flux penetrating the junction according to Eq. (5.5.35). Φ 0 is the flux quantum. (f) Experimental results from a Pb/YBa 2 Cu 3 O 7 junction. From [?]. c Walther-Meißner-Institut

91 Chapter 6 Unconventional Materials The entire field of superconductivity is driven by the discovery and development of new compounds bringing the materials scientists into the center of the game. As opposed to semiconductor physics it is extremely hard to predict or at least guess the tendency towards superconductivity. In addition, any prediction of transition temperatures from first principles, even for known materials, is difficult and was successful only for a few elements and simple compounds with conventional electron-phonon coupling [9]. To our knowledge BKBO is the only material having a T c close to 30 K which was predicted on the basis of the isostructural low-t c sister compound BaPbBiO3 [?]. In all other cases empirical rules such as those of Bernd Matthias [?] or highly complex quantum chemistry considerations have little or no predictive power yet meaning that our understanding is still insufficient and requires new and bright ideas. Hence, the purpose of this chapter is twofold in that the zest for understanding on the theoretical side is highlighted and the inventions of the materials scientist are celebrated by describing some of the fascinating properties of unconventional superconductors. 6.1 Classification Electron-phonon coupling was considered the only mechanism leading to superconductivity for a long time although magnetic polarizability was recognized as possible alternative [24]. However, only the discovery of systems with heavy electrons [25] and superconductivity therein [26] opened new vistas. 1 It was pointed out from the beginning that the heavy mass (large heat capacity) may originate in the Kondo-like interaction of the conduction electrons with the spins of the 4 f -electrons of cerium. The discovery of superconductivity at approximately 0.5 K in CeCu 2 Si 2 in 1979 [26] was a new paradigm and opened a completely new field of research that remained active and vibrant until now [27]. The heavy fermion systems were certainly the first examples of materials belonging to the class of unconventional superconductors which include now also the copper-oxygen [6] and the iron-based compounds [7] as compiled in Table 6.1. What means unconventional? The distinction between conventional and unconventional superconductors is not clear-cut. Sometimes all electron-phonon superconductors carry the name conventional while all others are unconventional. In the terminology of Anderson [?] dividing rule is given by the Fermi surface average of the gap k : Whenever k k = 0 superconductivity is unconventional. Considering the modern developments it makes also sense to ask as to whether or not there 1 The name heavy electrons (or fermions) derives from measurements of the electronic heat capacity c p = γt with the γ the Sommerfeld constant. As shown first for CeAl 3 c p is approximately three orders of magnitude larger than that of a usual metal. Since γ is proportional to the density of states at the Fermi energy N F and, hence, to the effective mass m of the electrons a large γ was associated with a large m. Maurice Rice pointed out early that, by definition, N F v F 1 implying a large γ to be more indicative of slow rather than heavy electrons. He added that slow electrons might sound less exciting and important. 83

92 84 R. HACKL AND D. EINZEL Unconventional Materials Table 6.1: The main catagories of superconductors. Conventional includes all metallic compounds with electron-phonon coupling where a lattice instability driven by strong coupling is the only additional phase. MgB 2 is listed separately for holding the record of the conventional systems due to a mixture of intra- and interband pairing. CuO 2 represents all superconductors on copper-oxygen basis. Fe-based systems include also those with Fe replaced by Rh. f -electron systems are typically those having a high electronic specific heat at low temperature and are also known as heavy fermion compounds. The following abbreviations are used: electron (el.), instability (inst.), antiferromagnetism (AF), density wave (DW), spin density wave (SDW), ferromagnetism (FM), and cylinder (cyl.). The missing entry in the last column reflects the existence of various types of pairing states in f -electron systems. The years in brackets indicate the first chance for observing superconductivity (see text). References can be found in the text. conventional MgB 2 CuO 2 Fe-based f -electron structural anisotropy isotropic isotropic normal state metal metal strange metal metal heavy el. competing phases (lattice inst.) AF, DW SDW AF/FM Fermi surface multi-band cyl. + sphere cyl. 1 band cyl. 5 bands multi-band T c,max (K) H c2,max (T) ξ 0 (nm) κ GL (nm) superconductivity isotropic s anisotropic s d x 2 y 2 s ± year of discovery (1952) 1986(1978) 2006(1995) 1979 is one or more other phases competing with superconductivity. The heavy fermions, the cuprates, the Fe-based materials, and organic metals share this property. In the vast majority of the cases magnetism competes with superconductivity. Less frequently, charge order plays a role. Very often the competing phase is suppressed continuously as a function of a parameter r other than temperature such as applied pressure or magnetic field or doping. If the transition temperature to the ordered state T o (r) approaches zero for a specific r c the material belongs to the large class of quantum critical systems [?, 28]. Here, the physics is dominated by the fluctuations of the spin or charge density in a wide range of temperature above r c which are considered to contribute substantially to superconductivity in both the cuprates and the Fe-based compounds [29, 30]. In fact, since the electron-phonon coupling is too small to explain transition temperatures in the range of 100 K other types of exchange bosons have to take over at least so long as a the interaction is retarded (meaning that the timescale is much longer than that of the electronic screening). Although the definitions are not generally accepted and/or used one has at least an intuitive idea along which lines the divide runs. Nevertheless, when reading the literature one should be aware of the slightly fuzzy definitions. For the present purposes it is enough to know that there are systems having properties fundamentally different from those of the usual metallic superconductors. In most of the cases they are characterized by rich phase diagrams and the proximity of magnetism and superconductivity. In the following we provide an overview of the Fe-based and copper-oxygen systems and a brief summary of the heavy fermions and the organic metals. c Walther-Meißner-Institut

93 The iron-age of superconductivity SUPERCONDUCTIVITY The iron-age of superconductivity High-temperature superconductivity in compounds with iron 2 was not considered seriously before Yoichi Kamihara and coauthors in Hideo Hosono s group discovered a transition at T c = 26 K in La(O 1 x F x )FeAs. [31] It is interesting to note that LaFePO having T c = 6 K was synthesized already in 1995 by Barbara Zimmer in the group of Wolfgang Jeitschko 3. The observation of superconductivity was only mentioned in Zimmer s thesis. Since the publication of Kamihara s results in 2008 several ten thousand papers have appeared, and the highest T c so far exceeds 50 K. Even though the materials contain arsenic or other rather toxic elements many laboratories started with the preparation of polyand single-crystalline samples. Generally the Fe atoms form two-dimensional layers and are coordinated with atoms from the fifth column in most of the cases (with Se being an exception) as shown in Fig The family name pnictides (Pn) comes from Greek πνiχτıκóς (asphyxiant, suffocative) for the column of nitrogen. In particular Chinese scientists contributed a lot of important results and found Nd(O 1 x F x )FeAs with the so far highest transition at 55 K [?]. While in the beginning many people believed the iron pnictides to be another class of oxides Dirk Johrendt and his group [8] demonstrated that high transition temperatures can also be obtained in purely intermetallic compounds. At optimal doping with x 0.4 Ba 1 x K x Fe 2 As 2 reaches a T c of 38 K. In contrast to the oxifluorides (see Fig. 6.1) large single crystals can be grown although the homogeneity and quality is not in all cases satisfactory. However, crystals of BaFe 2 (As 1 x P x ) 2 and LaFePO are already sufficiently clean for the observation of quantum oscillations [32, 33]. Figure 6.1: Structures of iron pnictide compounds (by courtesy of D. Johrendt). LaFeAsO 1 x F x (1111; Tc max = 28 K) was the first pnictide superconductor with high T c [31]. With Pr, Nd or Sm replacing La T c exceeds 50 K [?]. LaFeAsO 1 x F x is isostructural to LaFePO having the transition at 6 K at the stoichiometric composition. BaFe 2 As 2 (122) develops a spin-density wave (SDW). When doped with K for Ba [8], Co for Fe [34] or P for As the SDW is suppressed and superconductivity appears. LiFeAs (111) has a maximal T c of 18 K. The simplest of the materials, FeSe (11), with the same structural elements but Se for As has a maximal T c of 8 K with 9% Se deficiency at ambient pressure and reaches 27 K at 1.5 GPa [35, 36]. The question as to the origin of superconductivity arose immediately. Are the iron pnictides similar to the cuprates or to MgB 2 with T c = 39 K due to electron-phonon coupling or are they a material class on their own? The spin-density-wave (SDW) order of the parent compound indeed suggests a proximity to the cuprates, where the superconducting phase emerges from a Mott insulator. With doping p away from 2 This section is partially copied from a contribution of one of us (R.H.) to the Annual Report 2009 of the WMI. The title is borrowed from a Viewpoint by Michelle Johannes in Physics 1, 28 (2008). 3 Note that the undoped parent compounds of both the cuprates and the pnictides were well known quite some time ahead of the first observation of superconductivity in doped variants. 2013

94 86 R. HACKL AND D. EINZEL Unconventional Materials half filling spin and charge fluctuations as well as superconductivity follow antiferromagnetic long range order. However, in contrast to the cuprates there is no universal phase diagram in the pnictides (Fig. 6.2). In the essentially hole doped oxifluorides there is an abrupt transition from a magnetically ordered phase to BaFea superconducting one with T c only weakly depending on doping. The electron-doped intermetallic 2 As 2 compounds have a smooth transition, and SDW order and superconductivity may even coexist [37]. The phase boundary of superconductivity is dome shaped. Ba(Fe 1-x Co x ) 2 As 2 Figure 6.2: Phase diagrams of LaFeAsO 1 x F x (left) [38] and Ba(Fe 1 x Co x ) 2 As 2 (right) [37]. While SDW und superconductivity (SC) overlap in 122 there is a strict separation in La The structural transition at T α always precedes spin density wave order at T Chu et al. β. PRB 79, (2009) The differences in the phase diagrams of the pnictides are surprising since the electronic structures are Mandrus, Canfield, Büchner, Klauss, Dai,. remarkably similar. There are 5 bands derived from the Fe 3d orbitals. Two (α 1,2 ) form concentric holelike Fermi cylinders around the center of the Brillouin zone (BZ), two (β 1,2 ) have FSs which encircle the corner of the small BZ derived from the 2Fe crystallographic unit cell (Fig. 6.3). Since the cross sections of the resulting Fermi surfaces are nearly equal the α and β sheets are nested with the vector Q (π,π). Consequently, the electronic susceptibility as described by the Lindhard function is strongly momentum dependent [30] and is believed to be at the origin of the SDW. The pronounced peaks in the susceptibility make the strong variations of the properties upon small changes of the electronic and lattice structures at least plausible. June 16, 2009; 4 The real part of the susceptibility is also considered a possible origin of superconductivity [30, 39 41] while the electron-phonon coupling is probably weak [42]. From this point of view, the pnictides and the cuprates appear to be cousins in the same family even if the strong metallicity of the parent phases of the FePn compounds may argue otherwise. But how can the coordinates of the pnictides be determined? In a recent optical transport study the authors conclude from the reduced band width that the pnictides are half way between normal metals and the cuprates [43]. However, the related spectral redistribution to be expected upon doping is not observed by angle-resolved photoemission (ARPES) and x-ray absorption (XAS) [44, 45]. Perhaps one of the most telling similarities would be if the pnictides had the signature property of all cuprates an energy gap k having nodes and a sign change along the Fermi surface [19]. In the 6 years after the discovery the understanding and the experimental knowledge has advanced substantially. The results from the usual spectroscopic methods such as tunneling and photoemission start to converge. In most of the compounds the modulation of the energy on the Fermi surface is not supportive of gap nodes on individual Fermi surfaces [47]. However, it becomes more likely that the sign change between the hole and the electron surfaces leaves imprints in the tunneling spectra in an applied magnetic field [?] supporting the early suggestion of Mazin and coworkers [30] of an s ± gap. Here the phases of the gaps on the hole- and electron-like Fermi surfaces differ by π. This would imply that, for c Walther-Meißner-Institut

95 The iron-age of superconductivity SUPERCONDUCTIVITY 87 Generic band structure and Fermi surfaces 1Fe k y 2Fe Q X M k x Figure 6.3: Iron plane (left), Brillouin zone (BZ), and real part of the non-interacting susceptibility χ 0 (q,ω) [30] (right) of FePn materials. The cell relevant for the electronic structure contains 1 Fe atom (dashes) and is smaller November 30, by2009 a factor of 2 SPP and1458- rotated Iron Pnictides by 45 with page respect 14 to the crystal cell (full line and axes a and b). The BZ of the unit cell (full line) and the first quadrant of the Fe plane (dashed line) are shown along with the FS cross sections at π/c = 0 (adopted from Ref. [46]). The dotted FSs are obtained by downfolding the 1 Fe BZ. Even with the ellipsoidal elongation of the M barrels the α and β bands are approximately nested. Reχ 0 (q,ω) controls the pairing strength [30]. ideal conditions of equal cross sections of the respective Fermi surfaces and magnitudes of the gaps, the Josephson current in c-direction perpendicular to the Fe planes would vanish for a conventional counter electrode. While this Gedankenexperiment is good for visualizing the consequences of the s ± gap (and also other gaps) the technical realization is hampered by the real shape of the Fermi surfaces and the remaining variation of the gaps. Here the coherence factors come into play. As was shown in problem 2 of set 5 external perturbations having the potentials φ or A are either independent of the particle momentum or proportional to the momentum or its square. This and the spin introduce a sensitivity to the sign of k in the response since the Bogoliubov particles in a superconductor are coherent superpositions of electrons and holes and the amplitudes of the perturbation matrix elements have to be added before being squared. (The problem is outlined in the books of Tinkham and de Gennes and will be discussed in detail in a later version of chapter 3 of this manuscript.) The bottom line is that two particle response functions given by a superposition of occupied and unoccupied states such as the spin susceptibility, the ultrasound absorption, the NMR properties, the optical conductivity or the light scattering response assume characteristic energy and temperature dependences depending on the related perturbation operator. In the case of the spin susceptibility and of the longitudinal ultrasound absorption there is no momentum involved. For NMR and optical absorption (one dipole transition) k enters linearly via k A and for light scattering the perturbation is proportional to k 2 (two dipole transitions) via A 2. Since the full momentum dependence of the gap enters the coherence factors the different types of responses assume energy and temperature dependences characteristic for individual k. In the Fe-based superconductors a resonance in the spin correlation has been observed by neutron scattering [48] which may in fact be indicative of gaps having opposite sign on the electron and hole bands. The NMR and optical experiments are not conclusive yet. Also the angle-resolved photoemission (ARPES) is more complicated than in the cuprates due to the relatively strong electronic k z dispersion following from the weak anisotropy (see Table 6.1). In fact ARPES and Andreev tunneling favor large, essentially constant gaps on all Fermi surfaces [49, 50]. In some compounds such as Ba 1 x K x Fe 2 As 2 the modulation of the gap is stronger along k z than in the k x k y -plane [?]. Nevertheless, some of the materials have very small gaps on parts of the Fermi surface in addition to the possible sign change [47, 51, 52]. Some experiments indicate a strong modulation of k and even true nodal behavior [53, 54]. The strong material dependence inferred from the experiments is also expected from theoretical consideration that indicate the close proximity of various ground states [?]. 2013

96 88 R. HACKL AND D. EINZEL Unconventional Materials The possible anisotropies in the superconducting state are accompanied by band dependent carrier dynamics in the normal state as observed by quantum oscillatory phenomena [32, 33] and the analysis of Hall data [55]. For the clarification of these fundamental questions at the heart of the physics of the pnictides bulk sensitive spectroscopies with band and momentum resolution will be instrumental. Recently, the critical current density was determined in magnetic fields up to 45 T and found to be in the range of 10 4 A cm 2 up to at least 30 T (Fig. 6.4). These properties open the possibility to use Ba(Fe 1 x Co x ) 2 As 2 for static fields in the range above 21 T the limiting value of Nb 3 Sn. It became Figure 6.4: Critical currents j c and pinning forces of Ba(Fe 1 x Co x ) 2 As 2 in a magnetic field. The transport current is in the a b plane. (a) The field B that points in z-direction. (b) The field B is in the a b-plane. From Ref. [?]. apparent relatively early that the pnictides are much harder a problem to solve than, e.g., MgB 2. In fact, electron-spin or direct electron-electron interactions moved into the main focus of research. Therefore, the people working on the CuO 2 compounds were naturally attracted. The hope is that the results in the pnictides pave the way also towards a better understanding of the cuprates and of high-temperature superconductivity in general. 6.3 Copper-oxygen compounds When superconductivity was discovered in LaBaCuO [6] in 1986 and the proper phase La 2 x Ba x CuO 4 isolated soon thereafter [?] an unprecedented goldrush started. Now almost 3 decades and some 300,000 publications later the puzzle is still among the most important issues of solid state research but the knowledge of superconductivity, competing phases, strange metals, Mott and charge transfer insulators and the theoretical description thereof has grown tremendously. In addition, several experimental methods were pushed close to perfection and opened completely new insights. Here, we just give a brief summary of what has been revealed in this still exciting field. 4 4 The section is partially copied from the article in Zeitschrift für Kristallographie published by one of us (R.H.) [56]. c Walther-Meißner-Institut

97 Copper-oxygen compounds SUPERCONDUCTIVITY History It is enlightening to look at the activities preceding the final successful synthesis of Ba x La 5 x Cu 5 O 5(3 y) [6]. As in many other cases there was a program where to search for higher transition temperatures. The famous empirical rules of Bernd Matthias, who succeeded to discover more than 1000 new superconducting materials, are probably the best documented example. They include, among other statements, Stay away from oxides which had its origin in the unsuccessful attempt to find an oxide with a T c substantially above 10 K. Therefore, the search for new superconductors was focused on cubic intermetallic compounds with the hope to possibly reach a T c close to 40 K, the theoretically expected upper limit for isotropic metals [57]. In fact, such a compound would have been very advantageous for applications with liquid hydrogen sufficing as a cryogen. Beyond the very successful mainstream research on intermetallic superconductors there were also various innovative ideas on the market. Little, picking up a comment of F. London, theoretically studied the possibility of superconductivity in organic materials with the conducting and polarizable structural elements being separated in space [58]. Allender, Bray, and Bardeen continued partially in this direction [59] but considered also metal-semiconductor hetero-structures [60]. The basic idea is that electron-phonon coupling is expected to be stronger in highly polarizable poorly conducting materials. Then, the electrons in an adjacent thin metallic layer are being coupled to Cooper pairs via the polarizability of the insulator or semiconductor over a length scale on the order of the superconducting coherence length ξ 0 as defined in the microscopic theory of Bardeen, Cooper, and Schrieffer (BCS) [2]. Alternatively, the role of polarons was explored for essentially homogeneous materials with low carrier density [61]. On this substrate and with a sound background in ferroelectricity, Bednorz and Müller (Fig. 6.5) started their search for strong coupling superconductors in the early 1980ies which lead to the discovery of the cuprates. Figure 6.5: J. Georg Bednorz (left) and K. Alex Müller. By courtesy of the Nobel Foundation Materials Within a few years almost ten families of cuprates have been discovered. They have rather different crystal structures with occasionally huge unit cells accommodating between 8 and over 100 atoms. The most popular families are compiled in Table 6.2 along with their maximal superconducting transition temperatures T max c. 2013

98 90 R. HACKL AND D. EINZEL Unconventional Materials Table 6.2: Important families of copper oxygen superconductors. R is for Nd, Sm or even La in the case of thin films. n is the number of adjacent CuO 2 planes; three seems to be best for T c [62]. Hg-1223 has the highest T c so far. With applied pressure, 150 K can be reached [63]. number of CuO 2 planes n chemical formula nickname T max c (n) [K] Ref. La 2 x Ba x CuO 4 LBCO 30 [?, 6] La 2 x Sr x CuO 4 LSCO 39 [64, 65] R 2 x Ce x CuO 4 RCCO 29 [66, 67] YBa 2 Cu 3 O 6+x Y [68] Bi 2 Sr 2 Ca n 1 Cu n O 2n+4+δ Bi-22(n-1)(n) [?] Tl 2 Ba 2 Ca n 1 Cu n O 2n+4+δ Tl-22(n-1)(n) [69] TlBa 2 Ca n 1 Cu n O 2n+3+δ Tl-12(n-1)(n) [69] HgBa 2 Ca n 1 Cu n O 2n+2+δ Hg-12(n-1)(n) [62, 69] Structure and chemistry The copper-oxygen compounds have nearly tetragonal crystal structures with b/a ratios between 1.00 and The parent compounds La 2 CuO 4 and Nd 2 CuO 4 (see insets of Fig. 6.8) have been known for a long time [70 72] and have K 2 NiF 4 structure [73] with 3 approximately cubic perovskite-like blocks per unit cell stacked along the crystallographic c-axis. The resulting c/a ratio is close to 3. According to the valence count (and also to band structure calculations) the compounds should be metals with a half filled conduction band (1 electron per unit cell) but the strong electronic correlations block the transport (for details see section 6.3.3). Therefore, the number of free carriers is zero at half filling (n = p = 0). If Nd is partially replaced by Ce or La by Sr(Ba) the materials are doped away from half filling, 1 + n or 1 p, respectively, and become conductors with a small number n or p of mobile carriers directly given by x. The solubility limits for Ce and Sr(Ba) are x 0.18 and 0.33, respectively. Both compounds have CuO 2 planes at distances c/2 which are offset by (1/2,1/2) in tetragonal lattice units a. In La 2 CuO 4 the Cu atom is in the center of an oxygen octahedron, in Nd 2 CuO 4 there is no oxygen in apex position above and below the CuO 2 plane. In contrast to K 2 NiF 4, La 2 CuO 4 is tetragonal only at temperatures above 530 K [74]. Below, the structure is slightly orthorhombic with the octahedra tilted about their basal axis which corresponds to the orthorhombic a -axis. The transition temperature to the orthorhombic structure decreases with Sr doping and vanishes at x Nd 2 CuO 4 remains tetragonal for all temperatures. In doped Nd 2 x Ce x CuO 4 (NCCO) and in La 2 CuO 4 there is excess oxygen in the structures which has to be removed for optimal physical properties. La 2 x Sr x CuO 4 (LSCO) for x > 0.05 has an oxygen deficit after preparation. It is not clear yet whether fully stoichiometry La 2 CuO 4 and Nd 2 CuO 4 exist at all. Nevertheless, La 2 x Sr x CuO 4 is one of the most intensively studied cuprates, since the entire doping range is accessible with a single relatively well ordered compound. YBa 2 Cu 3 O 6+x (Y-123) is the only material which exists at the stoichiometric composition. Crystals with the exact cation ratio 1:2:3 can be grown from the flux. When prepared in BaZrO 3 crucibles they are virtually free of defects [75 77]. The structures for the limiting doping levels YBa 2 Cu 3 O 6 and YBa 2 Cu 3 O 7 are shown in Fig YBa 2 Cu 3 O 6 has a half filled conduction band but is an antiferromagnetic (AF) insulator for the same reasons as La 2 CuO 4 and Nd 2 CuO 4. The copper atoms in the chains have valence 1+, and holes on the CuO 2 planes are generated only when one chain Cu has two oxygen neighbors. For certain intermediate compositions in the range 0 < x < 1 highly ordered phases can be prepared with domain sizes of several 100 Å in all 3 crystallographic directions [78 80]. Within small limits doping can also be achieved by replacing Y 3+ by Ca 2+ [81]. Y-123 has two neighboring CuO 2 planes at a distance c Walther-Meißner-Institut

99 Copper-oxygen compounds SUPERCONDUCTIVITY 91 a YBa 2 Cu 3 O 6 YBa 2 Cu 3 O 7 Figure 6.6: Structure of YBa 2 Cu 3 O 6+x (Y-123). (left) YBa 2 Cu 3 O 6 and (right) fully oxygenated YBa 2 Cu 3 O 7. The small (blue) sphere in the center is Y, all other small (red) spheres are Cu. Ba and O are represented as large (yellow) spheres and open circles, respectively. In spite of a half filled conduction band (1 electron per unit cell) YBa 2 Cu 3 O 6 is insulating due to electronic correlations and has no mobile carriers (p = 0). Doping in Y-123 is achieved by adding O to the CuO chains along the crystallographic b-axis. In this way electrons are removed from the CuO 2 planes. YBa 2 Cu 3 O 7 has 0.82 electrons or 0.18 mobile holes per CuO 2 formula unit (p = 0.18). This is already slightly above the doping level optimal for superconductivity (see Fig. 6.8). The CuO 2 planes (shaded, see Fig. 6.9) are common to all cuprates with Y-123 having two of them at a distance c/3. The maximal T c is reached in triple-layer compounds (see Table 6.2). Details about most of the crystal structures of the cuprates can be found in Shaked et al. [69]. c b c/3 a which share one octahedron and are in symmetric positions with respect to the central Y atom of the conventional unit cell. Bi-based cuprates, in particular Bi 2 Sr 2 CaCu 2 O 8+δ (Bi-2212) [?], are very popular although they grow only far off stoichiometry and have complex modulated crystal structure according to Ref. [82]. However, since the two adjacent BiO 2 layers are bound by van der Waals interaction, they can be cleaved easily. The surfaces obtained in this way are stable, electrically neutral, and typically atomically flat over mm ranges. This makes Bi-2212 the workhorse of surface sensitive experimental methods such as angleresolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). The doping level can be changed via the oxygen content in the Bi O layer and by replacing Ca 2+ by Y 3+. The highest Tc max at p 0.16 is obtained with 8 % Y doping and reaches typically 96 K [83]. By annealing the crystals at oxygen partial pressures in excess of 1000 bar [84] p = 0.23 can be obtained. In order to reach the underdoped range below p = 0.16 annealing protocols with low oxygen partial pressure and low temperatures are required in contrast to what is expected from the phase diagram [85]. In addition to these three extremely well studied families, the thallium and mercury-based cuprates are also important since the highest transition temperatures and the most extreme doping levels can be reached here in materials with Tc max above 90 K. In either case, compounds with up to five neighboring CuO 2 planes exist [69] all of them becoming superconducting above 90 K. The record breaking HgBa 2 Ca 2 Cu 3 O 8+δ has three adjacent CuO 2 planes and reaches a T c close to 150 K at 25 GPa applied pressure [63]. Three CuO 2 planes are apparently optimal for T c, for four and more layers T c decreases again. Very recently, HgBa 2 CuO 4+δ (Hg-1201) was demonstrated to be another model system in that the doping can be tuned in the range 0.07 p 0.24 while a very high degree of order can be maintained manifesting itself in a Meissner effect of almost 100 % [86]. Single layer Tl 2 Ba 2 CuO 6+δ (Tl-2201) can be driven non-superconducting metallic on the overdoped side by high-pressure oxygen annealing [87]. Since optimal doping is slightly below the accessible 2013

100 92 R. HACKL AND D. EINZEL Unconventional Materials range the maximal transition temperature at p = 0.16 can only be extrapolated to be close to 95 K in the latest generation of crystals [88]. Hence, Tl-2201 is expected to display the properties of a true high-t c compound in contrast to LSCO. Growth of crystals All materials were first prepared in crucibles. The main complications are that the liquid phases and fluxes corroded all standard materials and that the desired compositions do not normally correspond to stable points in the high-temperature phase diagrams. Y-123 alone can be grown stoichiometrically from the flux. Instead of Y other rare earth metals can be used but only crystals with Y are free of site defects. In all other cases a finite number of rare earth atoms change position with Ba thus reducing the degree of order. The only stable crucible material is BaZrO 3 which had to be developed first [75 77]. All other materials get partially dissolved and contaminate the crystals. Crystals grown in BaZrO 3 are shown in Fig Figure 6.7: Y-123 single crystals in BaZrO 3 crucibles. BaZrO 3 is not corroded by the flux, which is aggressive to all conventional crucible materials, and had to be developed first [75]. After decanting the flux freestanding crystals, many of them with atomically flat surfaces, are left behind on the walls. From [76] with permission. Single crystals of NCCO, LSCO and of the Bi family were relatively soon grown in optical furnaces using the [traveling solvent] floating zone technique ([TS]FZ) with a small volume of appropriate flux between the feed and the seed rod. When the flux is liquefied the material in the feed rod is dissolved and diffuses to the colder seed where it crystallizes. Normally, several tricks are necessary to obtain a single crystal or at least sufficiently large grains. The temperature of the melt, the external pressure of the atmosphere and the oxygen partial pressure are all very critical for a successful growth and have to be determined systematically [83,89,90]. In this way large high-quality single crystals can be prepared in several places of the world having very similar properties. However, in contrast to Y-123 full reproducibility cannot be achieved. For the high vapor pressures of Hg and Tl at elevated temperatures the TSFZ technique cannot be applied for the preparation of single crystal of the respective families. Using sealed crucibles and a second containment the volatility and also the toxicity can be controlled reasonably well at the price of a reduced parameter space for the growth conditions. In spite of these challenges, single crystals of Tl-2201 and Hg-1201 have now a reproducibly high quality and reach sizes in the mm range [86 88,91]. The potential of these materials was already demonstrated and can hardly be overestimated. c Walther-Meißner-Institut

101 Copper-oxygen compounds SUPERCONDUCTIVITY 93 Thin films Along with these bulk methods thin-film techniques were developed and applied successfully. With a few exceptions all applications rely on thin films. There are various methods, (i) chemical vapor deposition (CVD), (ii) sputtering, (iii) pulsed laser deposition (PLD), (iv) molecular beam epitaxy (MBE), (v) thermal co-evaporation, and (vi) jet printing. All have been known before the advent of the cuprates, (ii)-(vi) are still used for the cuprates. In a PLD machine a high-energy UV laser evaporates material from a target with the desired composition. The substrate on which the film is supposed to be grown is in the center of the plume of the evaporating atoms. PLD is very efficient and fast and is probably the most widely used technique. In contrast to PLD, the MBE technique allows full access to the ratio of the cations. The regulation of the respective ion currents is practically instantaneous facilitating layerby-layer growth and an extremely high film perfection [92]. Artificial multi-layers with digital interfaces can be produced. Thermal co-evaporation is a comparably simple and cheap approach which turned out to be very successful for the production of large-scale films of Y-123. Since the crucibles with the starting materials are heated resistively the regulation is only important for the long-term control of the cation ratio. In all cases the substrates have to be heated to the appropriate temperature. Usually one uses mono-crystalline material with lattice parameters close to those of the deposited films. Within certain limits the atmosphere in the preparation chamber can be adjusted. In most of the cases the deposited cuprate films have oxygen deficits which are fixed by in-situ post-annealing protocols [93]. The quality of MBE films can come very close to those of single crystals although the structures are not normally fully relaxed. On the other hand, materials can be prepared which do not exist in equilibrium. This holds particularly true for electron-doped cuprates [66] Physical properties As mentioned above the purity of the samples has crucial influence on the properties. In particular, the integrity of the CuO 2 planes is essential to reach the maximal transition temperature of a material class. In turn, the effect of impurities in the CuO 2 plane can be studied systematically [94] and demonstrates that the majority of the properties derives from the planes in accordance with band structure calculations. Here, I try to focus on the properties of virtually clean materials and widely ignore defect-related and preparation problems. Phase diagram The most remarkable property of the cuprates, in my opinion right after the high T c, is the universality of the phase diagrams as shown in Fig Here, the phases are only described separately; possible interrelations will be postponed to section At least on the hole-doped side, the shape of the superconducting dome is almost independent of the material class. In the doping range 0.05 p 0.27 the transition temperature is well reproduced by [98] T c Tc max = (p 0.16) 2. (6.3.1) The maximum is always close to 16 % doping but T max c can vary between 38 K for La 1.84 Sr 0.16 CuO 4 [64] and approximately 150 K in HgBa 2 Ca 2 Cu 3 O 8+δ at 25 GPa [63]. A variety of materials following the universal curve are compiled in Ref. [95]. The relation is valid only in clean materials. In disordered samples the dome shrinks in that superconductivity occurs in a narrower doping range and at lower 2013

102 94 R. HACKL AND D. EINZEL Unconventional Materials 300 metallic insulating metallic Te empera ature (K) Nd 2-xCe xcuo 4 Cu O Ln AF La 2-xSr xcuo 4 T* T O n SC 0 Doping SC p 0.30 Figure 6.8: Generic phase diagrams of electron (left) and hole-doped (right) cuprates. The insets show the tetragonal unit cells of the prototypical compounds Nd 2 CuO 4 and La 2 CuO 4, respectively. The shaded range in the center of the diagram indicates long-ranged antiferromagnetism (AF). The superconducting dome on the hole-doped side reaches from 0.05 to 0.27 in all clean cuprates [95] independent of the maximal transition temperature Tc max close to Tc max varies between 30 K in La 2 x Ba x CuO 4 [?] and 150 K in HgBa 2 Ca 2 Cu 3 O 8+δ at an applied pressure of 25 GPa [63]. T and T 0 schematically indicate the onset temperatures of the pseudogap range [96] and of charge and spin ordering [97]. temperatures. This effect can be studied systematically by substituting Zn for Cu, for instance, or by irradiation [99 102]. Also single-layer Bi 2 Sr 1.61 La 0.39 CuO 6+δ has a narrower dome and is not superconducting for p < 0.10 [103]. In the following we discuss only the clean limit, where all correlation lengths of ordering phenomena are much smaller than the electronic mean free path l. As has been shown for Y-123 l depends more on the sample quality than on the doping and can be in the micrometer range in YBa 2 Cu 3 O 6.53 with alternating completely filled and empty chains [104]. Given that zero doping (half filling) can be reached, the antiferromagnetism is similarly universal as superconductivity. The maximal Néel temperatures T N range between 280 and 420 K. As-grown La 2 CuO 4 has T N 280 K and only post-annealing in Ar yields T N = 325 K [74, 105]. In Y-123 T N reaches 420 K [97]. There is no direct scaling between T N and T c. Long-ranged AF disappears rapidly on the hole-doped side and somewhat slower for electron-doping, in any case much faster than one would expect from the percolation limit [106]. However, spin correlations without long-ranged order can be observed well above T N and up to very high doping levels. In LSCO, Wakimoto and coworkers find them to disappear along with superconductivity above p sc2 = 0.27 [107]. Presently, there is not enough experimental material available to decide whether or not spin fluctuations and superconductivity generally coexist up to p sc2. In the range up to p 0.20 there is another transition or crossover at a doping dependent temperature T (p) which is usually referred to as the pseudogap line [96, 108]. The nomenclature pseudogap is related to the observation of a reduced spectral weight in the ARPES spectra measured below T (p) on c Walther-Meißner-Institut

103 Copper-oxygen compounds SUPERCONDUCTIVITY 95 fractions of the Fermi surface [109, 110]. This gap in the single-particle properties leaves imprints on practically all types of responses. In some material classes, and here individual behavior sets in, other crossover phenomena can be observed below T (p). Particularly in the La-based compounds, static spin and charge textures develop [97, ]. More recently, indications of fluctuating ordering phenomena were also observed in Bi-2212 and Y-123 [ ]. Finally, in some compounds a spin glass phase at low doping can be observed below some 10 K. For the rich variety of materials available, the hole-doped side is more exhaustively studied than electrondoped cuprates. Nevertheless, it is well established that the AF range is broader and the superconducting dome is narrower for n-doping [66, 90]. A pseudogap was observed by various methods [66, ] and may be a signature of the back-folding of the conduction band at the AF Brillouin zone [124]. The temperature range of the results from different methods is not yet consistent. While ARPES indicates the opening of the pseudogap above T c [125], tunneling reveals the pseudogap only below T c when superconductivity is suppressed by a high magnetic field [121]. Systematic transport studies on thin films uncover a quantum critical point at n [123] which could be the end point of the T (n) line. Similarly, a crossover from a small to a large Fermi surface close to n = 0.17 may be interpreted as a closing of the SDW-like gap [126]. Electronic structure In conventional superconductors, the band width, the Fermi, phonon, and gap energies are well separated, and superconductivity can be treated as a small perturbation of the normal state. In the cuprates, all energy scales are in close proximity including the magnetic exchange coupling J. The correlation effects originate in the large Coulomb repulsion U, lead to a substantial incoherent part of the electronic spectral functions, and k is not a good quantum number any further. Consequently, interaction effects are interrelated and cannot be observed independently, a fact which still creates confusion. Instead of dealing with these complications it appears more fruitful to search for the origin of the variation of Tc max in an otherwise rather universal phase diagram. A natural starting point seems the electronic structure of the CuO 2 plane as the basic building unit in the individual environment of a given material class. High energies The most transparent access to the electronic structure of the CuO 2 plane is through La 2 CuO 4 since with the valences of La and O, given as 3+ and 2, respectively, Cu, residing only in the plane, is in a 2+ state. The 4s orbital is not relevant for the plane and, on an atomic level, we are dealing with oxygen 2p and copper 3d states. In the tetragonal environment of the cuprates the degeneracy of the nine 3d electrons is lifted and the d x 2 y2 orbital happens to be the highest occupied one hybridizing with the oxygen p x,y orbitals as shown in Fig. 6.9 (a). The resulting conduction band is half filled and therefore, on this level of sophistication, La 2 CuO 4 should be a metal. This is also predicted by band structure calculation in local density approximation (LDA). However, already a Hartree-Fock calculation shows that the exchange energy is higher than the kinetic energy and blocks the metallic transport. This effect is usually referred to as a Mott metal-insulator transition. The appropriate description is the Hubbard model which sets the kinetic and the Coulomb energy in relation. If next-nearest neighbor hopping t is included in addition to the nearest neighbor integral t for a more realistic description of the cuprates [see Eq. (6.3.3)] the one-band Hubbard Hamilonian reads [127], H = t(c iσ c jσ) +t (c iσ c jσ) +U n i n i, (6.3.2) i, j σ i where c iσ and c iσ creates and, respectively, annihilates an electron with spin σ on site i and n iσ = c iσ c iσ is the density operator. i, j indicates that the sum is restricted to nearest- and next nearest neighbor hopping in the case of t and t, respectively. 2013

104 96 R. HACKL AND D. EINZEL Unconventional Materials (a) O Cu π (b) ϕ M k y Γ X a π π k x π Figure 6.9: Electronic structure at E F = µ(t 0) of an idealized quadratic CuO 2 plane. Panel (a) shows the orbital character of Cu and O (without phases) at low quasiparticle energy ξ k = ε k µ 0. (b) Brillouin zone and Fermi surface of a single CuO 2 plane. The Fermi surface encircles the empty states around the M points at (±π,±π) for unit lattice spacing a. The diagonal (Γ M) is usually called the nodal direction since the superconducting gap k crosses 0 here (see section 6.3.4). Correspondingly, the neighborhood of X is called antinode. The number of carriers, electrons or holes, correspond to the imbalance of the areas around M and Γ. At half filling (1 electron per CuO 2 ) the areas equal. The plane should be metallic but the correlations make it insulating. If one considers a situation with one hole per copper site the hole tries to hop from site i via the bridging oxygen to one of the four nearest-neighbor sites j. The kinetic energy which can be expressed in terms of the transfer integral t is much smaller than the Coulomb repulsion U for double occupancy. In this way the transport is blocked. In addition, since the Pauli principle is also at work, hopping is only possible when the spins on sites i and j are anti-parallel hence antiferromagnetically ordered. For t U the magnetic coupling between the neighboring spins is given as J = t 2 /U in second order perturbation theory. If one constructs a tight-binding Fermi surface with only nearest-neighbor hopping (t = 0) and half filling in the idealized quadratic Brillouin zone (BZ) of the CuO 2 plane a square results covering half of the BZ area and being rotated by 45 [see Fig. 6.9 (b)] which coincides with the magnetic BZ. Due to the extended parallel parts, the configuration is unstable and a gap would open at the Fermi energy such as in charge or spin density wave (CDW/SDW) systems [5]. It is instructive to have a closer look at the effect of the Hubbard U on the electronic structure as shown in Fig On an atomic level, the Cu 3d and the O 2p orbitals at ε d and ε p, respectively, are split by some 2 ev. Crystal field splitting and hybridization broaden the atomic levels considerably, and the Cu 3d x 2 y 2 and O 2p x,y orbitals are mixed covalently. The Fermi energy E F is in the middle of the anti-bonding (AB) band indicating metallicity. The dispersionless non-bonding (NB) and the bonding (B) bands are approximately 3 ev below E F. The correlation energy U opens a gap at E F, which was first introduced by Mott, and the system becomes an insulator. In the case of the cuprates U is approximately 8 ev hence much larger than ε p ε d, and all three bands are needed for a proper description. Now the system will be doped by replacing La 3+ by Sr 2+. In a one-band picture, part of the copper is nominally transformed into Cu 3+ for compensation, and hopping becomes possible into empty d x 2 y 2 orbitals thus opening a channel for transport. This picture is quite useful but according to the preceding paragraph not quite true. In fact the first hole is created on oxygen as demonstrated experimentally very early [128,130] since the charge-transfer energy ε p ε d is smaller than U [?]. More precisely, the extra positive charge forms a cloud around the central Cu 2+ which compensates also the copper spin and is therefore called a Zhang-Rice singlet (ZRS) [129]. With minor modification, it still obeys the dynamics of the one-band Hubbard model as if the charge would reside on the copper. The fact that ε p ε d < U can only be taken into account properly - rather than approximately as in the one-band Hubbard model sketched in the preceding paragraph - if the Cu 3d x 2 y 2, O 2p x, and c Walther-Meißner-Institut

105 Copper-oxygen compounds SUPERCONDUCTIVITY 97 ε UHB E F AB ε d ε p ε d U NB ZRS ε p E B E F crystal field covalency LHB t U Figure 6.10: Schematic band structure of the CuO 2 plane at ev energies in electron representation. The levels ε p and ε d correspond, respectively, to the O 2p and Cu 3d atomic levels. The bonding band (B) at approximately 6 ev below the Fermi level E F (long dashes), the dispersionless non-bonding (NB) and the anti-bonding (AB) band originate from the covalently overlapping Cu 3d x 2 y 2 and O 2p x,y orbitals. All other orbitals are neglected. Since the hopping integral t is much smaller than the Coulomb energy U the configuration is insulating at half filling, and the AB conduction band is split by U into the upper (UHB) and lower (LHB) Hubbard band separated by the Mott gap U. For small U the first hole goes on copper. If U exceeds ε p ε d the material becomes a charge transfer insulator [?] and the first doped hole is created on the oxygen [128]. The new state is called a Zhang-Rice singlet (ZRS) [129]. (The figure is inspired by lecture notes of D. Einzel.) O 2p y orbitals are included in a three-band Hubbard model [131, 132]. It has been shown via numerical solutions of the two models that both give rather similar results for the phase diagrams at T = 0 including superconductivity [133], since the physics apparently does not change qualitatively if the role of the Hubbard U is taken over by ε p ε d. Beyond the phase diagrams on either side of half filling also the electron-hole asymmetry [see Fig. 6.8] can be captured. The holes go on the oxygen atoms and quench the superexchange coupling between the antiferromagnetically ordered Cu spins while the electrons appear first on the Cu site and just dilute the spins. Consequently, the antiferromagnetism survives much longer on the electron-doped side. To which extent the substantial differences between the material classes in Tc max and other ordering phenomena can be explained through the material dependent fine tuning of ε p ε d is a matter of present research [133]. Energies in the range k B T In order to arrive at energies in the range of a few k B T around E F, where the relevant physics such as superconductivity occurs, the high-energy degrees of freedom have to be integrated out [133, 134]. This procedure is a further idealization and is only qualitative in that individual properties cannot be captured. Nevertheless, a thorough understanding of the coherent part of the electrons spectral properties is at the origin of the explanation of the relevant interactions and of superconductivity. In this spirit, a downfolded LDA band structure [135] can be derived which reproduces the experimental results from ARPES [136,137]. This holds particularly true for the shape of the Fermi surface which can be obtained to within a few percent from the tight binding band structure (now in momentum space), ξ k = 2t(cos(k x ) + cos(k y )) + 4t cos(k x )cos(k y ) µ. (6.3.3) Data from slightly overdoped Bi-2212 can be fitted satisfactorily using t = 250 mev, t /t = 0.35 and neglecting band-splitting effects. With µ/t = 1.1 one arrives at a filling close to p = With minor changes in the parameters the CuO 2 Fermi surfaces of all other cuprates can be reproduced. The Brillouin zone with the Fermi surface for a single CuO 2 plane is shown in Fig The parametrization of the 2013

106 98 R. HACKL AND D. EINZEL Unconventional Materials band structure in terms of hopping integrals is frequently used as a basis for phenomenological modeling [ ] and microscopic calculations [133, 142, 143]. The experimental dispersion yields a Fermi velocity v F cm/s [144] substantially smaller than the LDA prediction indicating strong interaction effects. A hallmark is the kink-like change of the slope v k = h 1 ε k / k at approximately 70 mev observed in the nodal ARPES spectra [145,146] which turns out to be quite universal [144]. In the presence of conventional electron-phonon interaction a kink is expected in the strong coupling limit [14]. In the cuprates, the origin of the interactions is still controversial. The usual electron-phonon coupling [ ], the coupling to spin excitations [151,152], fluctuations of the charge density [142, 153, 154] and orbital currents [155, 156] have been proposed. The strong renormalisation is in accordance with the short lifetime τ k (ω) of the electrons which decreases rapidly away from the Fermi surface [145, 157]. For a while it appeared that the imaginary part Σ k (ω) = 2[τ k(ω)] 1 of the electronic self energy Σ = Σ + iσ is scale free and varies linearly with energy ω and temperature T [145] as expected when the electrons scatter on critical fluctuations of any origin (such as spin or charge density or orbital currents). As a consequence, the quasiparticle weight Z k = [1 Σ k (ω)] 1 at the Fermi energy E F vanishes and the electron s spectral function is distributed over very large energy scales in contrast to the properties of a Landau Fermi liquid with 0 < Z k < 1. This phenomenology is called marginal Fermi liquid (mfl) [158] and has a big share in the discussion of the cuprates. With the continuously improved resolution of ARPES experiments, various substructures were found at low energies and analyzed in terms of phonons [147, 148] and spin fluctuations [151, 157]. In all cases, Σ k (ω) varies faster than linear at low energies and crosses over to the more linear behavior in the 50 mev range. This phenomenology is expected if the electrons couple to a dispersionless (Einstein) mode which must not necessarily be a lattice vibration. These recent observations let it appear very likely that several interactions contribute to the coupling spectrum. At a given energy ω, τ k (ω) depends strongly on the position on the Fermi surface. Along the diagonal Γ M line of the BZ (nodal direction) Σ k (ω) is relatively small and does not depend substantially on doping [136, 137]. In the vicinity of the X point (antinode) things are more complicated. At very high doping as in Tl-2201, the lifetime of antinodal quasiparticles may even be longer than that of nodal ones [159]. With decreasing doping the lifetime and the weight of the antinodal quasiparticles decreases continuously. Slightly below optimal doping the interactions along the principle directions become strong enough to completely suppress coherence even in the superconducting state [160, 161]. At elevated temperatures no quasiparticle develops any more for p < 0.18 indicating a loss of coherence in the pseudogap regime on parts of the Fermi surface [137]. Bridging the gap At first glance one would expect that the renormalized conduction band with a width in the range of an ev and the UHB and LHB split by U 8 ev do not overlap. However, the strong correlations leave only a small fraction Z k 1 of the quasiparticles spectral weight close to E F and distribute 1 Z k over large energy scales. This incoherent part can be observed experimentally between the LHB and the conduction band as a faint structure with almost vertical dispersion [ ] and is reproduced theoretically using Monte-Carlo techniques for solving the Hubbard model [165,166]. Thus, there is one more indication that the Hubbard model is a reasonable starting point for the description of the CuO 2 planes in the cuprates. Two-particle dynamics The CuO 2 planes determine the majority of the physical properties and, in particular, carry the currents in the normal and in the superconducting states. Owing to the layered structure, the cuprates are electrically c Walther-Meißner-Institut

107 Copper-oxygen compounds SUPERCONDUCTIVITY 99 highly anisotropic [167]. The anisotropies of the resistivities ρ c /ρ ab, with the subscripts indicating the crystallographic directions, range from 30 to 10 5 [168, 169]. Transport At optimal doping all clean cuprates have an essentially linear resistivity in the a b plane down to T c, ρ ab (T ) T, which saturates only at very low temperature, as demonstrated for LSCO in magnetic fields [170]. Below a doping dependent temperature Tρ (p) T (p) there is a reduction of ρ ab (T ) below the linear variation which is associated with a pseudogap [108, 168, 169, 171, 172]. If the high-temperature part is extrapolated to T = 0 the residual resistivity is very small and approaches 0 for the cleanest optimally doped crystals. If superconductivity is suppressed by high magnetic fields ρ ab (T ) saturates at a finite value [170]. For p > 0.16, also the out-of-plane resistivity becomes purely metallic and ρ c (T ) ρ ab (T ). The ratio ρ c /ρ ab (T 1.5T c ) at optimal doping is approximately 30 in Y-123 [168] and 5000 in Bi-2212 [169] having the same Tc max and close to 500 in LSCO with Tc max = 38 K [170]. There is a dichotomy between the under- and the over-doped ranges on the hole-doped side which becomes particularly clear at low temperatures when superconductivity is suppressed by magnetic fields. Various systematic studies have been carried out recently by Ando and coworkers [64]. The results for LSCO and Y-123 are shown in Fig In both Y-123 and LSCO the resistivity generally turns in- (a) (b) (c) (d) Figure 6.11: Resistivity vs T for different doping levels p for LSCO (a,b) and Y-123 (c,d). The comparison indicates the similarity of the different classes as long as the crystals are sufficiently clean. Note the deviation from linearity below a doping dependent temperature T (p) which is particularly clearly seen in Y-123 at 0.11 p 0.15 corresponding to 6.60 y From Ref. [64] with permission. sulating, corresponding to dρ ab (T )/dt < 0, at low temperatures and doping before superconductivity appears at p sc For p > p sc1 dρ ab (T )/dt becomes essentially positive for T > T c. If a magnetic field is applied which is high enough to suppress superconductivity completely an upturn is observed with a logarithmic divergence towards zero temperature. With increasing p the minimum shifts to lower temperature and approaches T = 0 close to p = 0.17 [170]. For p > 0.17 the resistivity remains metallic and exhibits a T α variation over extended temperature ranges with α 1.5. Apparently, full metallicity develops above optimal doping. In strongly over-doped Tl-2201 with T c 15 K, 1.5 < α < 2 is found [173]. If the resistivity is fitted to ρ(t ) = ρ 0 + AT + BT 2 the coefficient A of the linear term approaches zero as p sc2 p for p p sc2 [174, 175]. The details become more transparent when the dynamics is studied as a function of the electron momentum k as discussed already briefly in the context of the electronic structure and single-particle lifetimes in paragraph Concerning transport, two-particle properties have to be considered where an electron is scattered from an occupied into an empty state leading to the usual restrictions and corrections. In an early nuclear magnetic resonance (NMR) experiment, deviations from the Korringa law, (T 1 T ) 1 = const, with T 1 the spin lattice relaxation time and T the temperature was found well above T c for p 0.17 [176], indicating the loss of a relaxation channel for the electrons. The NMR 2013

108 100 R. HACKL AND D. EINZEL Unconventional Materials Γ μ 0 (K) B 1g B 2000 T = 200K opt. 2g Bi-2212 doping T = 200 K 1000 X X Γ M X Γ M Γ μ 0 / T (K K/K) (a) (b) Γ-M Γ-M X doping p Figure 6.12: In-plane anisotropy of the electronic relaxation Γ µ = [τ µ ] 1 at 200 K as seen by electronic Raman scattering [183]. (11.6K = 8cm 1 = 1meV) Γ µ is closely related to a resistivity ρ, and τ 1 = ne 2 m ρ in a Drude model. As an additional information from Raman scattering, the regions around the X points (diamonds) and along the Γ M line (squares) of the Brillouin zone (see Fig. 6.9) can be projected independently with different light polarizations µ [181]. (a) There is little doping dependence along the nodal directions (Γ M). At X there is an abrupt change at p = 0.21 ± The crossover seems to be universal since it is also seen in LSCO and in Tl-2201 [184] and by NMR [185]. It is predicted by the Hubbard-Holstein model [143]. (b) The temperature dependence of Γ µ, Γ µ / T 200K is isotropic above p = 0.21 ± Below the crossover, the antinodal derivative decreases continuously and changes sign close to p = form factors suggest that particles close to (π, 0) may experience a gap which was directly observed by ARPES [109, 110]. Similarly, optical transport (IR) results in Y-123 show that the electrons with momenta along the diagonal relax differently from those at the X points of the BZ. The distinction is possible for the specific crystal and band structure of Y-123 which facilitates to project diagonal and (π, 0) momenta for in-plane and out-of-plane polarizations, respectively [135]. For this reason, the pseudogap as an anti-nodal property was discovered first by c-axis polarized IR spectroscopy [177]. However, the projection in optical spectroscopy with in-plane polarizations is incomplete with finite sensitivity everywhere in the Brillouin zone, and the pseudogap is clearly visible below optimal doping also for E a,b [178, 179]. The electronic Raman response measures a quantity similar to the conductivity [180, 181] but has inplane selection rules which facilitate independent access to nodal and antinodal electrons by appropriately selecting the light polarizations [181, 182]. Results for Raman relaxation rates Γ µ (µ is for the polarizations corresponding to symmetry projections [181]) of differently doped Bi-2212 at 200 K are shown in Fig In the nodal configuration the doping dependence of the spectra and of the corresponding carrier relaxation is weak for p sc1 < p < p sc2 [183, 186, 187]. Above p 0.21 no polarization dependence corresponding to a relaxation anisotropy can be observed. The relaxation rate for antinodal electrons increases abruptly below p 0.21, and approximately 30 % of the spectral weight is lost in the energy range up to 250 mev. This was traced back to a doping and momentum dependent correlation gap c Walther-Meißner-Institut

109 Copper-oxygen compounds SUPERCONDUCTIVITY 101 extrapolating to 2 C 200 mev at p = 0 [183]. A similar phenomenology emerges with a progressive loss of quasiparticle coherence starting at X and proceeding to the node upon reducing p as observed by ARPES [109, 110, 137, 188] and studied also in the context of light scattering [186, 189, 190]. The correlation gap, the pseudogap and, finally, the superconducting gap (see section 6.3.4), have different energy scales and their interrelation has to be determined yet. The onset of anomalies in the doping range around p = 0.16 are also seen in the Hall effect [191] and, particularly clearly in the Nernst signal [175, ] constituting, respectively, a transverse voltage in response to a charge and heat current in a perpendicular magnetic field. The Hall effect exhibits a maximum close to T [175, 194] but only the quantum oscillations of the Hall resistivity at very low temperature indicate that the anomaly may be related to a reconstruction of the Fermi surface [117]. Very recently, indications of a Fermi surface reconstruction were also discovered on the electron-doped side in NCCO [126]. The Nernst effect is sensitive to superconducting vortices [195] and density-wave order [196]. For a long time the Nernst signal observed between T and T c in LSCO was considered a signature of vortex motion above the coherence temperature T c in the spirit of a 2D Kosterlitz-Thouless transition [192]. Only recent results in Eu-doped LSCO showed that the onset of the Nernst voltage coincides with the charge-ordering temperature [194,197] found in various other experiments [?,113]. In Y-123 the onset of the anisotropic Nernst signal [175, 192, 193] coincides with various other indications of broken rotational symmetry such as Kerr rotation [198] or incommensurable peaks in the dynamic spin susceptibility [118, 119]. New frequencies in the quantum oscillations indicate that a partial reconstruction of the Fermi surface goes along with the ordering phenomena. Hence, the superstructures found first in the the spin channel in LSCO [199, 200] and Nd-doped LSCO [111] seem to be a generic phenomenon of all cuprates that is accompanied by charge order. Since the superstructures are static only in exceptional cases signatures of them in the transport escaped observation for a long time, in particular in the compounds with high T c. Their importance is being unveiled only slowly. Further details will be discussed at the end of the following subsection and in sections and Spin dynamics Homogeneous magnetism exists in wide doping ranges. At 0 p 0.03 the antiferromagnetism is long-ranged in LSCO. In NCCO three-dimensional (3D) antiferromagnetism exists below n = The exchange coupling J 130 mev is among the largest ones existing. The order is truly 3D but the coupling along the c-axis is orders of magnitude smaller than along a. The dynamics at high energy was studied early by Raman scattering. The photon flips essentially two neighboring spins breaking six bonds with energy J [201]. More accurately, a two-magnon density of states is measured and projects the flat parts of the dispersion in the vicinity (π,0). The maximal energy observed is therefore at E 2M 6Js. In a spin 1/2 system the peak is close to 3J (2.7J for quantum corrections) [?, 201, 202]. In LSCO, Y-123 and Bi-2212 spin correlations can be observed by Raman scattering up to approximately p = 0.20 [?, 187, 203]. In LSCO and NCCO magnetic short-range order was observed by inelastic neutron scattering up to p 0.27 [107] and n 0.17 [204], respectively. For the large magnitude of J the full dispersion of the spin excitations was studied with neutrons only recently. The spectrum extends beyond J well above the energy of thermal neutrons. Results for various compounds up to approximately 200 mev are shown in Fig [97]. If the energy axis is normalized to the exchange coupling J the dispersions collapse on top of each other lending evidence to the universality of the spin excitations. At intermediate and low energies spin excitations were studied in detail by neutron scattering and NMR. The decrease of the Knight shift below T c indicated spin singlet pairing [207] (see section 6.3.4). For T > T c, the spin-lattice relaxation rate T1 1 is proportional to T compatible with Fermi liquid-like carriers [176,207] only close to optimal doping and above. At low doping a spin gap is found below T c by neutron scattering [208] and magnetic resonance [176,209] putting magnetism and superconductivity in relation. It was conjectured early that most of the spin susceptibility results from itinerant electrons rather than 2013

110 102 R. HACKL AND D. EINZEL Unconventional Materials E / J La 1.90 Sr 0.10 CuO 4 La Ba CuO 4 La 1.84 Sr 0.16 CuO 4 YBa 2 Cu 3 O 6.5 YBa 2 Cu 3 O (0.5+h, 0.5) (rlu) Figure 6.13: Dispersion of the spin excitations of La-based cuprates and in Y-123. If the energies are normalized to the exchange coupling J the spectra are universal. Low and high-energy parts are compatible with fluctuating charge and spin order [205, 206]. From [97] with permission. localized Cu moments. This inspired the model of a nearly antiferromagnetic Fermi liquid [210]. The interaction of carriers and spin fluctuations can be studied systematically in the fluctuation exchange (FLEX) scheme [?,?]. However, in which way spin fluctuations participate in the Cooper pairing is still an open question (see section 6.3.4). With polarized neutrons an intriguing narrow mode with wavevector Q = (π,π) was found at low temperatures [208, ] which is usually referred to as the π-resonance. For p 0.16 the energy of the resonance E R is proportional to T c. In the underdoped range the mode appears already between T and T c when the pseudogap opens up. The spectral weight of the mode is between 1 and 6 % of the integrated spectral weight of the spin susceptibility [216] and its origin is controversial. Its role as a mediator of superconductivity has been explored in various studies [152, ]. The results, however, did not generate general agreement yet. In spite of that the π-mode is characteristic of the cuprates and possibly other superconductors in close proximity to a magnetic phase [?]. In the La-based compounds incommensurate peaks shifted by δ(0,π) and δ(π,0) (in the square unit cell of the CuO 2 planes) away from the AF reflex at (π/2,π/2) were discovered early in the inelastic channel indicating a dynamic superstructure on top of the antiferromagnetic order for p = and 0.14 [199]. If part of the La is replaced by Nd the superstructure becomes static [111]. Charge order accompanied by a lattice distortion with a periodicity of four unit cells appear before the eight unit cell superstructure of the spins is established. Below the onset point of superconductivity, p p sc1 0.05, static diagonal stripe order is observed in LSCO. At p sc1 the stripes rotate by 45 and start to fluctuate meaning they can only be observed at finite energy [199, 220]. Generally, charge order precedes spin order upon cooling [111, 113, 221]. Fluctuating modulations of the charge density cannot normally be observed by neutron scattering but can be visualized by tunneling spectroscopy due to interference effects [114]. Fig shows that charge and spin order in Bi-2212 and LSCO have the same orientation above p sc1. Recently, equally oriented nematic order was also observed in underdoped superconducting Y-123 in the dynamic spin susceptibility [118, 119]. Assuming a stripe-like superstructure of the spins the dispersion (Fig. 6.13) can be predicted quantitatively [205, 206, 222]. Hence, evidence mounts that dynamic phase c Walther-Meißner-Institut

111 Copper-oxygen compounds SUPERCONDUCTIVITY 103 La 2-x Sr x CuO Bi 2 Sr 2 CaCu 2 O 8+ (c) Figure 6.14: Spin and charge ordering in LSCO (a,b) and Bi-2212 (c), respectively. Parts (a,b) display incommensurate neutron reflexes at (h,k) = (0.5 ± ε,0.5 ± ε) below p sc and at (0.5 ± δ,0.5) and (0.5,0.5±δ) above p sc1. They originate from a static and dynamic spin superstructure, respectively, corresponding to static and fluctuating stripes [111,199,220]. Above p sc1 the spin and charge superstructures are observed to have the same orientation. Superstructures with similar orientations including the rotation [see panel (b)] are also seen in Y-123 [116, 118, 119, 175]. From [220] with permission. The result on Bi-2212 (c) is a filtered STS image and shows a modulation of the charge density which becomes visible due to interferences. The arrows indicate the orientation of the CuO 2 planes. From [114] with permission. separation and the related ordering phenomena are generic properties of the cuprates contributing to anomalies such as critical fluctuations, Fermi surface reconstruction and the pseudogap in the electronic excitation spectrum. Competing phases The pseudogap range is one of the most intensively studied areas of the phase diagram being observed below the T line (see Fig. 6.8) and for doping levels below approximately p = 0.21 [96, 108, 172, 223]. Various properties discussed above indicate a gap in the quasiparticle excitation spectrum and, hence, an instability above the transition to superconductivity. It is clear from ARPES [109, 110] that the Fermi surface is not fully gapped above T c. Concomitantly, the materials remain metallic, and the resistivity even decreases slightly below T since the strongly interacting quasiparticles are gapped out (see Fig. 6.11). To some extent the resistivity of the cuprates (Fig. 6.11) indicates similarities to CDW or SDW systems such as the recently discovered FeAs superconductors [7,8,31], where the resistivity also drops upon entering the ordered phase in the undoped parent compounds [37]. Similarly, in several f -electron systems a magnetically ordered phase is suppressed as a function of either doping or pressure giving room for superconductivity [27]. In either case, superconductivity is in close proximity to other ordered phases, in complete contrast to conventional materials [?]. Clearly, there is another instability above the superconducting phase, and it is of pivotal importance to understand the relationship of the phases as to whether they compete or cooperate [96]. In most of the cases, in particular in the compounds with high transition temperatures, T is a crossover rather than a phase transition. There are various indications of a broken symmetry [114, 142, ] 2013

112 104 R. HACKL AND D. EINZEL Unconventional Materials with the fluctuations of incipient order widely considered important and responsible for many of the anomalous properties of the cuprates including superconductivity. Very early Varma and coworkers introduced the concept of a quantum critical point above which temperature is the only energy scale rather than collective excitations such as spin waves or phonons. The related fluctuations lead to an almost complete collapse of Landau s Fermi liquid model and to the marginal Fermi liquid phenomenology [158], where an electron has vanishingly small coherent weight even at E F. The quantum critical point (QCP), at which the fluctuations suppress any phase transition above absolute zero, is somewhere buried below the superconducting dome in the range 0.15 < p < 0.22 on the hole-doped side [175] and close to n = for electron-doped systems [123]. Order or incipient order (for reviews see Ref. [28] or the book by Sachdev [229]) are expected below T (p) and between n = p = 0 and the critical doping. Similarly as in many other systems the QCP cannot be accessed directly since it is protected by superconductivity. The appearance of superconductivity above a QCP is one of the reasons why the fluctuations are considered a possibility to mediate Cooper pairing. It is controversial which types of fluctuations dominate. Inspired by the NMR results Anderson proposed the resonating valence bond (RVB) model where fluctuating spin singlets are formed at high temperature and condense below T c [230]. A similar phenomenology follows if polarons condense into bi-polarons [61]. Ong and coworkers interpreted the onset of the Nernst signal between T c and up to maximally 3 T c T at p 0.10 in terms of superconducting fluctuations which survive even below p sc1 [192]. Recent studies in La 2 x y Eu y Sr x CuO 4 (LEuSCO) show that the Nernst signal sets in along with the formation of a CDW-like superstructure and may originate from the related charge ordering [194, 197] rather than from vortex motion [196]. While the superstructure is static in LEuSCO, LNdSCO [111], LBCO and La 2 x (Ba 1 y Sr y ) x CuO 4 [221, 231] below T 0 (Fig. 6.8) fluctuating order is observed in LSCO [220] and also in Y-123, at least at specific doping levels [116, 117, 119, 191]. Fermi surface reconstruction has also been observed recently in NCCO. It can be described in terms of band folding resulting from the AF order [89, 126]. Yet, the details and the origin behind the reconstruction remain important problems to solve on either side of zero doping. As already mentioned, to some extent there is a similarity to CDW and SDW materials with dimension d greater than one, where only part of the Fermi surface is gapped while the rest sustains metallicity [232, 233] or even superconductivity such as in 2H-NbSe 2 [234]. Beyond these similarities the type of order in the cuprates has many new features. In particular in the high-t c compounds, an ordered phase is not established, and it is probably sensible to speak of nematic order [114, 228], including spontaneous deformations of the Fermi surface [235], with only the rotational symmetry broken. The lattice appears to play a crucial role in stabilizing the order. It has been shown for La-based compounds that the tilting angle θ t of the copper-oxygen octahedra may be a parameter to quantify the proximity to static order [112]. If θ t exceeds a critical value static order is established by kind of a lock-in transition and superconductivity disappears. Hence, in realistic models electron-phonon interaction should be included in the Hubbard model [142] to bring the derived phase diagrams closer to the experiments [143, 185]. There is no evidence whether and which fluctuations contribute to superconductivity (see also section 6.3.4). However, it was shown for LBCO at 1/8 doping and for LEuSCO that static order quenches the 3D phase transition [112, 236]. In the case of LBCO the CuO 2 planes decouple and the phase transition to 2D superconductivity is suppressed by fluctuations [236]. Upon applied pressure the static order becomes nematic and superconductivity is restored [237]. There are two conclusions. (i) The pseudogap phase has many signatures of a broken symmetry other than the gauge symmetry of superconductivity. There are many experimental indications that the rotational symmetry is broken and that the electronic states partially reconstruct due to incipient charge order driven by the strong correlations. The contribution of superconducting fluctuations to the pseudogap is c Walther-Meißner-Institut

113 Copper-oxygen compounds SUPERCONDUCTIVITY 105 small as can be seen independently from the spectral weight redistribution in the optical conductivity [238]. (ii) As soon as the order becomes static superconductivity is quenched. In this sense there is a competition between the two phases, and the possible coexistence is clearly different from that in conventional CDW and SDW systems [234, 239]. To which extent the critical fluctuations of incipient order contribute to or drive superconductivity needs to be clarified [96] Superconductivity Superconductivity in the cuprates mesmerized nearly all solid state scientists for its unprecedented robustness. The essential parameters of Y-123 are summarized in Table 6.3 and compared with those of Al and Nb 3 Sn. Applications on a large scale were expected to be realized within a few years. The transition temperatures were high enough to make cooling with liquid nitrogen an option. Upper critical fields in the 100 T range and critical current densities of j c 10 7 A cm 2 close to those of the best metallic alloys and three to four orders of magnitude above the maximal capacity of Cu triggered expectations of completely replacing power transmission lines, storing energy or constructing magnets with fields in excess of 30 or even 40 T virtually free of energy consumption. However, a brief look at Table 6.3 shows were the problems are buried. Nevertheless, substantial progress could be made since 1986 and several of the ideas have become commercial products (see chapter 7). Table 6.3: Superconducting parameters of Al, Nb 3 Sn, and Y-123. The data for Al and Nb 3 Sn are taken from Ref. [10]. The references for Y-123 are indicated in the last column. Some entries are estimated using the relations ξ BCS = hv F (π 0 ) 1 and B c2 = Φ 0 ( 2πξ GL ) 2 with v F the Fermi velocity and Φ 0 = Wb the flux quantum. The critical current for Al is determined via the Silsbee criterion [10] for a wire with 1 mm diameter. All derived quantities should be considered order of magnitude estimates. Note that ξ BCS and ξ GL are related but different quantities. λ has no index since it can be measured with some accuracy (see, e.g., Ref. [104]) Also in the case of λ, the London and the Ginzburg-Landau (GL) definitions should be distinguished. quantity unit Al Nb 3 Sn Y-123 comment Ref. Tc max K [81] 0 mev [240] 0 k B T c ξ ab BCS Å B c c1 T B c B c c2 T B c [241] B ab c2 T 240 B ab [10] ξgl ab ξgl c Å 15 Å λ ab Å ± 100 B ab [104] κ GL j c (5K,10T) A/cm [242] 2013

114 106 R. HACKL AND D. EINZEL Unconventional Materials Experiments One can spot the short coherence length ξ 0 (used if specialization to ξ BCS or ξ GL is not necessary) to be among the major problems to deal with, since it prevents effectively the pinning of flux-lines. The resulting flux flow goes along with energy dissipation and kills all applications in high fields and with large currents. Typically, one pinning center per coherence volume ξ 2 0,ab ξ 0,c is needed. Hence, in conventional alloys flux flow can be suppressed by a moderate density of defects (see Table 6.3). In addition, (non-magnetic) impurities have little impact on superconductivity in an s-wave superconductor [243]. In contrast, T c is rapidly reduced in the cuprates, and a high density of pinning centers is required. Fortunately, the structure helps, but one had to learn to keep the pinning centers away from the CuO 2 planes. For instance, oxygen clusters on the CuO chain sites of Y-123 pin effectively [?] and have only mild influence on T c [78]. For practical purposes the proper distribution of pinning centers is among the major challenges, and the dynamics of flux lines remains an important field of research [?,?]. What is the origin of the short coherence length and of the sensitivity to disorder? One could phrase it this way: you have the choice between Skylla and Charybdis. The high transition temperature goes along with a large energy gap which results in a short coherence length, ξ BCS = hv F (π ) 1. The high transition temperatures in turn, come from an exotic coupling mechanism which makes superconductivity in the cuprates unconventional. Following the definition proposed by Pitaevski [17] and Brueckner et al. [16] unconventional means that k k = 0. 2 ξ 2k + k 2 (6.3.4) Hence the gap k is strongly anisotropic, changes sign and has nodes on the Fermi surface making T c highly susceptible to defects. The sign change is topologically different from a strongly anisotropic but generally positive gap with vanishingly small minima and goes along with a discontinuous transition from a four-fold to a two-fold rotational symmetry which implies a change in the phase of the gap similar to the structure of atomic orbitals with l 1. Since spin singlet pairing was identified early by NMR [207], odd internal angular momentum of the Cooper pairs going along with spin triplet states can be excluded. Hence, l = 2 is the lowest possible angular momentum and d x 2 y2 is realized. Experimentally, the sign change of the gap can be demonstrated only in a phase-sensitive experiment [18,19] and not by spectroscopy probing the magnitude of the gap k. The d x 2 y2 character was pinned down by Wollman and coworkers [18] and consecutively corroborated in various ways for both electron and hole-doped cuprates [19]. On the Fermi surface k = k F the d-wave gap is simply given by ϕ = 0 cos(2ϕ) with 0 the gap maximum and ϕ the azimuthal angle which is zero on the M-X line [see Fig. 6.9] with the origin in M. On the tight-binding band structure, as given in Eq. (6.3.3), the gap is parameterized as k = 0 2 [cos(k xa) cos(k y a)] (6.3.5) for a quadratic unit cell with lattice parameter a. It is an enchanting coincidence that the paper on the unconventional gap in UBe 13 [13] directly precedes the article on superconductivity in La-Ba-Cu-O [6]. There, the magnetic penetration depth λ(t ) was used as a diagnostic tool which also brought the break-through for the cuprates [20]. Although there were very early indications that the gap is strongly anisotropic and may even have nodes [ ], only the experiment of Hardy and coworkers on high-quality Y-123 single crystals [20] triggered an avalanche of activities including the first phase sensitive experiment by Wollman et al. [18]. c Walther-Meißner-Institut

115 Copper-oxygen compounds SUPERCONDUCTIVITY 107 To map out the magnitude of the gap in the cuprates spectroscopically, resolution in k-space is needed. For this reason ARPES became particularly important since one can map electronic single-particle energies with a resolution E 2 mev as a function of the in-plane momentum k with a resolution of the Fermi surface angle ϕ of better than one degree. The cuprates and ARPES profited mutually from each other, in kind of a symbiosis, since Bi-2212, due to its extremely two-dimensional structure and the exceptional cleaving plane between the Bi-O layers, facilitated deep insights into the physics of the cuprates by photoemission and thus enormously fueled the method itself [137]. The most important results are the observation of the Fermi surface and of the momentum dependence of k [248, 249] as shown in Fig. 6.15, of the pseudogap [109, 110] having the same momentum dependence as k, the doping dependence of the dispersion [136, 188], and various renormalization effects on the band structure which are believed to be in close but not yet understood relationship with the Cooper pairing [146, 151, 250]. Fermi-surface angle (deg) Figure 6.15: The magnitude of the gap k as a function of the Fermi surface angle as defined in Fig Note that the labels on the Brillouin zone in the inset correspond to the reciprocal lattice of Bi The inset shows experimental points for the Fermi surface (circles) and the tight-binding fit (full line). The hairlines indicate the replica originating from the superstructure of the Bi-O layers. From [249] with permission. Electronic Raman scattering [251] is among the few other possibilities to see the gap anisotropy directly since different parts of the Fermi surface are projected independently by appropriately adjusting the polarizations of the incoming and outgoing photons [181, 182]. Since light scattering is a two-particle method both the gap in the excitation spectrum and the condensate are seen. At optimal doping the results agree with those from ARPES. Additional information is obtained predominantly at more extreme doping levels closer to the onset points of superconductivity. It turns out that the gap close to the nodal direction scales with T c in very wide doping ranges [?,190,240, ] in qualitative agreement with low-energy tunneling [257] and recent ARPES results [258]. In the latter experiment the particle-hole mixing typical for (k,-k) pairing can be seen below but close to T c at E > E F. This identifies the observed gap as the superconducting one. On the electron-doped side ARPES [125] and Raman scattering [259, 260] reveal gap magnitudes 0 /k B T c in the range 2 to 2.5, much smaller than for hole-doping. Phase sensitive experiments show that the gap changes sign similarly as on the hole-doped side [19, 261]. The gap appears to vary nonmonotonically [125, 260]. However, the interference with the pseudogap may influence the magnitude of 2013

116 108 R. HACKL AND D. EINZEL Unconventional Materials the superconducting gap and needs to be clarified further. The relatively small gap ratios signal intermediate to weak coupling, and it is not overly surprising that 0 approximately follows T c in the relatively small doping range where superconductivity exists [262]. For p-doping, the approximate scaling of the gap extracted from extended portions of the Fermi surface around the nodes as 0 4.5k B T c in the under- and over-doped ranges [181, 190, 254, 257] is rather surprising as one would expect the gap to reflect the supposedly doping dependent coupling strength. In fact, most of the electron-electron interactions such as those from spin fluctuations or Coulomb repulsion increase towards p = 0. However, all these coupling potentials are not isotropic but dominate along the principle axes so that the nodal part could be less influenced. It has been argued that the increase of the gap may be compensated by the loss of major parts of the Fermi surface due to the interaction itself [137, 189, 190, ]. The reduction of the superfluid density towards low doping may be a a fingerprint of this phenomenon [172, 266, 267] but there is no quantitative understanding yet. One would expect that this problem could be clarified by looking at the environment of the X points (antinode) where the strong interactions prevail. However, the loss of coherent quasi-particles and the opening of the pseudogap for p 0.19 progressively shroud the pairing dynamics. This is further complicated by the emergence of inhomogeneities which can be observed by scanning tunneling spectroscopy (STS) as shown in Fig In the spectra of Bi-2212 large and small gaps are spatially separated, and the large gaps line up with oxygen defects where the doping level is expected to be reduced [268]. The pseudogap and the superconducting gap may even mix in some doping ranges [223, 269, 270]. di/dv (arb) (a) (b) Sample Bias (mv) Figure 6.16: Inhomogeneity of the tunneling spectra of slightly underdoped (p = 0.15 ± 0.01) Bi-2212 as seen by STS. The spectra in panel (a) are measured at the spots in panel (b) having the same color. Note that the slope close to zero bias depends only little on the position. The asymmetry for positive and negative bias results from the charge order. From [268] with permission. Affairs do not simplify on the overdoped side. While the condensation energy has a maximum close to p = 0.19 and the superfluid density tends to saturate [172, 267] the nodal and anti-nodal gaps continue to develop independently. In addition, the interpretation of the coherence peaks remains controversial. There are particularly enlightening experiments. (i) Electronic Raman scattering with applied pressure demonstrates that the superconductivity-induced anti-nodal structures decouple from T c already at optimal doping [271]. (ii) In STS the energy of the coherence peaks in the range of 0 is lower than one would expect from the slope close to zero bias [257, 268, 270] and depend on the location on the sample [268]. The π-mode at energy E R (see section 6.3.3) follows T c (p) and E R (p) (p) on the overdoped side [?,211, ,272]. On the underdoped side, data on Y-123 [213,272] show that the scaling with T c is not valid any more. As to whether or not the proportionality to 0 takes over depends on the definition of the gap which, in my opinion, remains problematic, in particular in the presence of the pseudogap. Hence, the π-resonance proves to be in close relationship to unconventional superconductivity [?, 216, 219], while the more stringent question as to its relationship with the magnitude of the gap and with the origin of the Cooper pairing is not settled. c Walther-Meißner-Institut

117 Copper-oxygen compounds SUPERCONDUCTIVITY 109 In summary, not all microscopic properties in the superconducting state are clarified. While the symmetry of the energy gap is found to be universally d x 2 y2 [19] the magnitude, which could be instrumental to identify the origin of Cooper pairing, is hard to pin down. Close to the node scaling with T c seems likely but around X discrepancies as large as ±50 % between different probes are typical. The condensation energy seems to peak at p = 0.19 [172] making superconductivity most robust slightly above optimal doping. Origin of superconductivity The basic notions of the superconducting state are a d-wave gap and a universal dome-shaped dependence of the transition temperature T c on doping p. Whenever p = 0 is accessible the cuprates are AF insulators highlighting the importance of strong electronic correlations. In the range 0 < p < 0.20 a competition of ordering phenomena is observed. These facts should be captured at least qualitatively by a theoretical approach towards superconductivity. In conventional superconductors the condensation of electrons into Cooper pairs is an instability of the normal metallic state. Cooper showed that an infinitesimally weak attractive potential, V 0, which is non-zero only for energies ξ k hω 0 with hω 0 the energy of the coupling boson (phonon in conventional metals), makes two electrons with opposite momenta and spins living above a filled Fermis sphere, ξ k = ε k E F > 0, to pair and to reduce their energy by [273]. Bardeen, Cooper and Schrieffer (BCS) derived how N = O(10 23 ) electrons can exploit this energy gain by forming a condensate which is characterized by a single wave function similar to that of an electron in an isolated atom or an infinite plane wave [2]. The energy gain and the transition temperature T c depend linearly on the cutoff hω 0 and exponentially on the coupling strength λ = N F V 0 with N F the density of electronic states at E F. In the BCS approximation, λ 1, there is no direct relationship to real materials. This open problem was solved by Eliashberg who showed how λ can be derived from the phonon spectrum and reach values in excess of 1 [3, 274]. Coupling spectra α 2 (ω)f(ω) with α 2 (ω) the energy dependent electronphonon interaction and F(ω) the phonon density of states have been derived for elements and alloys from electron tunneling spectra [275]. Since F(ω) can be measured directly by neutron scattering α and F can be derived independently and serve as a basis for a quantitative comparison with theoretical predictions [276] and, therefore, provide key information for a microscopic understanding of the pairing. Generalizations including momentum dependent coupling and bosonic excitations other than phonons have been put forward but require various approximations (similary as Eliashberg s original approach) [151, 218, 250, 277, 278]. In all cases the characteristic bosonic energy must be much smaller than the electronic energies, hω 0 E F, [279] implying that the interaction is retarded. This means that one deals with two fairly different time scales. The electronic one reacts instantaneously to a perturbation. The other one maintains the polarization field created by one electron sufficiently long so as to allow a second electron to experience it. This condition holds excellently in conventional metals having hω 0 /E F = O(10 2 ). In the beginning (see section 6.3.1) strong electron phonon coupling with λ > 1 was considered to lead to sufficiently stable Cooper pairing in the cuprates. The limiting case is the formation of polarons which, at low temperature, condense into bi-polarons [61]. Polaronic behavior was indeed observed at low doping [280, 281] but it is hard to pin down at optimal doping and beyond. This does not imply that electron-phonon coupling can be disregarded. Actually, there are experimental indications such as doping dependent shifts in the phonon spectra [282, 283], isotope effects at low doping [284], kinks in the electronic dispersion in the entire doping range [146, 147, 149] or strong coupling effects of specific phonons [285, 286]. However, the derived overall coupling constants are considered to be too small to support superconductivity in the 100 K range [62,250], and the way the electron-lattice interaction enters is probably different from the situation in conventional superconductors [62, 143]. 2013

118 110 R. HACKL AND D. EINZEL Unconventional Materials Coupling mechanisms other than phononic can arise from low-energy spin [?,?, 210, ] or charge fluctuations [153] or from high-energy instantaneous interactions such as the Coulomb repulsion U or the exchange coupling J [230, 291, 292]. In all cases the Hubbard model is a useful starting point that predicts antiferromagnetism, phase separation and superconductivity [29, 133] as shown in Fig or rder pa aramet ter antiferromagnetic superconducting AF+SC pseudogap SC doping p Figure 6.17: Phases predicted by the 3-band Hubbard model on the p-doped side. Similar to the 1-band version (oxygen orbitals not explicitly taken into account) antiferromagnetic (AF) and superconducting (SC) correlations corresponding to finite order parameters are found. In the overlap region a pseudogap appears in the derived spectral functions. SC vanishes only at p = 0. Additional crossover lines are found at higher doping if electron-phonon interaction is included [142, 143]. From [133] with permission. For t < U superconductivity can be obtained even though the interaction is repulsive, since the d- wave gap changes sign and facilitates a solution of the gap equation. In a real-space argument one would say that the two electrons avoid the repulsive part of U by arranging in a d-wave pair function which vanishes when the potential is repulsive [29, 133] as already pointed out by Pitaevskii and Brueckner et al. [16, 17]. This type of interaction is instantaneous since it is purely electronic and on a very high energy or short time scale [292]. In the limit U the Coulomb repulsion is integrated out and J 130meV becomes the highest energy scale right after the band width. For p 0.03 the nearest-neighbor coupling J leads to the usual Heisenberg-type long-ranged AF order (given that there is finite coupling in c-direction). At higher doping only short range order and paramagnetism survive. There are indications that the coupling between 2 spins survives beyond optimal doping [107]. On this basis Anderson formulated the RVB approach [230], where local singlet pairs start to couple well above T c via the exchange energy J and condense into Cooper pairs below T c. Then, the pseudogap is the energy reduction in the RVB state and phase fluctuations prevent the singlets from condensing above T c. Upon proceeding to lower energies the pairing becomes more conventional in the sense that the interaction is retarded. Then, the Eliashberg theory can be applied [3, 274, 277], and a coupling spectrum should be derivable from any type of electronic response or, turning the argument around, observable by an appropriate independent experimental method such as neutron scattering in the case of spin fluctuations. There are quantitative studies of how the spin spectrum could provide enough coupling for the cuprates [152] but there is no consensus yet since the interaction between a spin fluctuation and an electron can be treated only phenomenologically. Alternatively, the electrons can also interact via fluctuations of orbital currents [155, 156] or of the charge density [153, 154]. Small magnetic moments as possible indications of orbital currents were discovered recently below T [?] but it is as complicated as in the case of spins to determine the coupling. Traces of charge fluctuations are even harder to pin down since there is no independent probe for the related excitations [114]. Caprara et al. propose to study the c Walther-Meißner-Institut

119 Copper-oxygen compounds SUPERCONDUCTIVITY 111 related Aslamazov-Larkin fluctuations by light scattering [293]. Since charge fluctuations couple to the lattice, the phonons may be back in the game in an indirect fashion [142]. Presently, spectra measured with different methods are analyzed in order to find fingerprints of the relevant bosons. These include ARPES [146, 147, 151, 152], neutron scattering [152], STS [138], IR [218, ], and Raman spectroscopy [?, 139, 184]. There are indications for both electronphonon [146, 147] and electron-spin [151] interaction in the nodal ARPES spectra. With adjusted coupling constants the nodal electron dispersion can be reproduced by and large on the basis of the spin susceptibility [152]. Away from the node high-resolution ARPES data do not exist yet. The optical and Raman spectra always show two well separated energy scales at approximately 50 and 200 mev [184, 296] which cannot a priori be identified with specific excitation. Possibly the strong polarization dependence in the Raman results [184] may help to assign the modes via the selection rules. Then charge and spin fluctuations would dominate at low and high energies, respectively [?] making the exchange coupling J an important player, as suggested by Anderson [292], along with coupled charge-phonon excitations. However, as in the other cases the experiments have been done above T c and the coupling constants can at best be guessed. Is there any independent criterion to foster a decision? Poilblanc and Scalapino derived a partial sum rule for the complex Eliashberg gap function Φ(k, ω), I(k,Ω) = f k (Φ) (ω Ω), varying between 0 for Ω = 0 and approximately 1 for Ω which measures the contributions to the pairing interaction at a given momentum k as a function of the cut-off energy Ω [297]. When all the coupling (attractive or repulsive) is exhausted at high energies I(k,Ω) approaches 1. In Pb for instance, I(Ω) increases most rapidly at the transversal and longitudinal phonon frequencies, exceeds unity above the energy range of the phonons due to the unretarded repulsion in normal metals and approaches 1 asymptotically in the high energy limit [298]. This type of analysis requires high resolution data for the gap function Φ(k, ω) which do not exist for the cuprates yet. However, theoretical models can be studied, and in the Hubbard model 80 % of the coupling occurs in the energy range of the spin fluctuations. In contrast to Pb, I(Ω) does not exceed unity in the Hubbard model indicating instantaneous pairing interactions at higher energies such as J and U. Their relative weight has still to be clarified [133, ]. It will be an important step forward if the relevant interactions or energy scales in the cuprates can be identified. In spite of enormous progress both experimental and theoretical, the main question as to the relative contribution of the various possible pairing mechanisms is not yet answered. Probably, it is the right mixture which allows one to explain not only the phase diagram but also the material dependence. It has been noticed that the maximal T c and the ratio t /t depend in a systematic way on the distance of the apical oxygen from the CuO 2 plane reflecting the individual electronic structure of the compounds [301]. Johnston and coworkers derived the corresponding electron-phonon coupling λ ph and find that T c can be tuned substantially by varying the ratio λ s /λ ph with λ s the coupling via spin fluctuations [62]. Similarly, since the three-band Hubbard model includes p d charge fluctuations which depend sensitively on details of the materials it may supplement the plain vanilla one-band model [302] by adding a channel for tuning T max c [133]. In any case, we have to further sharpen our diagnostic tools to finally tackle the proper origin(s) of superconductivity in the cuprates and, maybe, get ideas towards novel materials. Although new superconductors have usually been found through the intuition of the materials scientists, the search was always guided by concepts Summary and perspectives The highest transition temperatures to superconductivity so far are observed in copper-oxygen compounds with CuO 2 planes as the common building elements. The planes are separated by perovskite-like 2013

120 112 R. HACKL AND D. EINZEL Unconventional Materials blocks. T c can reach 135 K under normal conditions and exceed 150 K with 25 GPa applied pressure. The upper critical field B c2 (T 0) reaches values well above 100 T in compounds with T c 100 K. The cuprates share the proximity of superconductivity and other ordered phases or states with incipient order with various materials such as f -electron systems, organic metals and the recently discovered ironbased superconductors. It is a crucial question as to which extent superconductivity gets fueled by the neighboring instabilities and their fluctuations. Apparently, the dimensionality plays a role since low dimensions favor fluctuations and reduce the screening [303]. At present the Hubbard model seems a viable way towards a microscopic description since it captures antiferromagnetism, competing phases and superconductivity. However, it is an open question which of the interactions emerging from the model, i.e. spin fluctuations, exchange coupling and Coulomb repulsion, dominate in driving superconductivity. Even phonons may enter in a couple of ways yet different from those in conventional systems. So far neither theory nor experiment are in a state to suggest search strategies for new superconducting materials. However, from what we know from the comparison of the cuprates, iron-pnictides and f - electron systems, nearly planar systems at the brink of stability of a magnetic or charge-ordered phase seem to be favorable. After almost three decades of research into cuprates various applications emerge (see next chapter). The most popular passive devices are frequency filters, fault current limiters, power transmission lines, dynamical capacitors, high field magnets, efficient eddy current heaters, and low-noise pick-up coils. Presently, only SQUID magnetometers are among the cuprate-based applications using the dynamic properties (Josephson effect). Although cycle frequencies in the THz range would be possible computing with Josephson junctions appears very unlikely an application of the cuprates at the moment for the enormous and continuous progress of semiconductor devices relying on established technology. c Walther-Meißner-Institut

121 Chapter 7 An overview of applications Materials and applications have a strong impact on the development of a field since they generate funding at a high rate and from many sources. The potential for applications of superconductivity was recognized already by Kamerlingh-Onnes but it was a long way to go before the first solenoid was presented in 1955 by Yntema [304], providing a field of 0.71 T. The conductor was Nb, a weak type II superconductor having κ 0.9. The development of the Ginzburg-Landau theory was one major step towards the understanding of the kind of problems that had to be solved for this application: high currents in high fields. In addition, the metallurgy of superconducting materials, connections between superconductors and between superconductors and normal metals, and efficient cooling at the temperature of liquid helium were on the agenda. NbTi was a big leap since it has an upper critical field of approximately 16 T while maintaining a high ductility. It is still the material that dominates high field applications. Only for fields in excess of 12 T Nb 3 Sn [305] and YBa 2 Cu 3 O 7 conductors 1 are used and developed, respectively. Apart from big magnetic coils high-power applications on a large scale are still at their infancy. On the other hand active devices for detecting small voltages and magnetic fields are widely used and commercially available. These include SQUID magnetometers for research, exploration and military, filters for mobile telecommunication, and Josephson elements for metrology. To get superconducting technology started the advantages must be striking, and other solutions have to be outperformed technically and economically. 7.1 Potential areas Economic considerations By and large 2 new technologies prevail only if the production and operation costs are substantially smaller and if the reliability is comparable or better in comparison to an established solution. In addition, the inertia of running systems can hardly be overestimated. While the operation costs can be estimated and controlled reasonably well the production costs may include a completely new infrastructure. A well known example is hydrogen technology. Similarly, for power transmission over larger distances with superconductors requiring cooling with liquid He the costs He recovery and liquefaction systems are prohibitively high. Although operative cables were and are available there was no progress in installation before the cuprate superconductors permitted cooling with liquid nitrogen. Here, the liquefaction and distribution is orders of magnitude simpler and cheaper than in the case of He. Consequently, also the 1 Boebinger ASC1 coil final.pdf 2 For details search for the Peter principle. 113

122 114 R. HACKL AND D. EINZEL An overview of applications reliability is much higher. For the grid, the requirements are approximately 1/365 days for maintenance as can be realized with nitrogen technology but not with He. If, on the other hand, the operating costs exceed the those for installation new approaches are desirable. As an example we compare the costs for a conventional and a superconducting magnet for a laboratory. Similar considerations apply for a magnet used for magnetic resonance imaging (MRI) in medical applications or chemical analysis. Table 7.1 shows a comparison. This example shows that experiments in Table 7.1: Comparison of the approximated overall cost of a laboratory magnet providing a field of 10 T in a bore of 50 mm. k e is 1,000 e, LHe is for liquid helium. In modern MRI machines there is no LHe refill any more since the evaporating gas is reliquefied with a closed-cycle refrigerator consuming approximately 5 kw corresponding to 40 e/d for electricity. NbTi solenoid Cu solenoid installation 100 k e 30 k e infrastructure 0 10 k e cooling 3 l LHe/d 1 m 3 water/min 30 e/d 3 k e energy 0 5,000 kw 0 40,000 k e/d maintenance LHe refill electricity/water reliability 1/365 d? conductor 1 e(ka m) e (ka m) 1 high fields were out of reach for standard laboratories before the advent of superconducting coils similarly as medical examination using MRI. There is just one exception for which resistive magnets are still needed: the generation of continuous fields above 21 T. The power consumption of the 45 T magnet in the National High Field Laboratory in Tallahassee (FL, USA) or of the 33 T magnet in the CNRS High Field Laboratory in Grenoble (France) is accordingly high and reaches values in the 30 MW range requiring cooling facilities with capacities on the order of 1,000 m 3 per hour. We may conclude that superconductivity will preferably be used for local applications since cooling of extended installations is complicated and costly. However, with the advent of the cuprates the restrictions become more relaxed since LHe can be replaced by liquid nitrogen (LN2) entailing a cost reduction by approximately a factor of 100. For instance, cooling with LN2 costs approximately 0.5 e (m d) 1 for a high power cable. Nevertheless, the production costs for the conductors remain high Areas of application It is clear that the so far available superconductors will not completely replace conventional materials. Even if a material with a T c above room temperature will successfully be synthesized in the future problems such as flux pinning (see below), AC losses, connection with other materials, and, last but not least, production costs will be an issue. On the basis of current technology the following applications are realized or feasible. Some will be discussed in more detail below. High current applications This includes high-field magnets for research, medicine, fusion, and accelerators, the improvement of local grids, the interconnection of independent grids, fault current limiters, dynamic synchronous condensers, motors, and generators. c Walther-Meißner-Institut

123 Passive applications SUPERCONDUCTIVITY 115 Filter technology Filter characteristics can be improved substantially with an enhanced conductivity. Here the dynamic conductivity in the MHz and predominantly GHz range is relevant. Using cuprates at K at least an order of magnitude improvement in performance and size can be reached in comparison to copper. Here mobile telecommunication and defense prevail. Sensors SQUID magnetometers are widely used in research and development but also in Geology for exploration. The Josephson effect is used for metrology for instance. Superconducting computers In the beginning one was thinking of superconducting logic circuits switching between zero and finite resistance. This concept was outperformed by Si technology before any realization. The rapid single flux quantum logics (RSFQ) is still competitive concerning frequency but is not pursued any further. Currently SQUIDS are being studied as to whether they can be used as Qbits in a quantum computer. Prior to all applications many physical concepts had to be developed and technical challenges had to be solved. We shall discuss the important aspects before providing an overview of realized applications in a field. A broad discussion will follow in the lecture Applied Superconductivity in the Summer Term. 7.2 Passive applications In most passive applications one wants to exploit the dissipation free transport of high currents sometimes in a magnetic field. In type I superconductors the critical currents and fields are very small limiting the range of applications to the screening of small static fields and electromagnetic noise in the sub GHz range ( hω < where is the gap). In type II materials the fields destroying superconductivity are much higher but in the presence of a transport current the vortices may move giving rise to dissipation. We discuss no the origin of the dissipation and possible remedies Physical and technical challenges If a transport current j tr flows in a superconductor in the presence of a magnetic field as shown in Fig. 7.1 there is a force on the vortices originating from the Lorentz force. We saw in the discussion of the Abrikosov lattice that the vortices assume a symmetric arrangement without a transport current since the net forces vanish. For a finite transport current perpendicular to the field B the symmetry is broken since the transport and the screening currents add vectorially, j(r) total = j(r) tr + j(r) screen resulting in a net projection of the currents around the flux lines on the y-direction and a force F L in positive x-direction. The symmetry breaking can also be seen in a different way: the applied current j tr creates a field around the superconductor which enhances the field on the l.h.s. and leads to a reduction of B on the opposite side. As a consequence the vortex density starts to become location dependent thus establishing a force to restore equilibrium. The density of the Lorentz force reads f = F V = nev tr B j tr B. (7.2.1) If we integrate over the volume of one flux line, cπλ 2, we get F 1 c j tr B πλ 2 (7.2.2) 2013

124 116 R. HACKL AND D. EINZEL An overview of applications F L c B b z j tr a y x Figure 7.1: Superconductor in a magnetic field. The field B that points in z-direction is in the range B c1 < B < B c2. The transport current j tr and the screening currents add resulting in an imbalance of the screening currents around the flux lines. The net force on the flux lines originates in this asymmetry. since most of the flux is in a cylinder having a radius of order λ, and since the current is perpendicular to the field. Inside the cylinder the force is approximately constant, and we arrive at the result derived already earlier, F 1 = c j tr B d 2 r. (7.2.3) πλ 2 or, more generally, F 1 /c = j tr ê z Φ 0 since a flux line carries one flux quantum. The power dissipated by one flux line is then dq 1 dt = d dt x 0 F 1 dx F 1 dx dt = F 1 v FL (7.2.4) where v FL is the (unknown) drift velocity of the flux lines in x-direction. The last part of the derivation is counting the number of flux lines N, and the total dissipation is P = N Q 1 P = UI = B ab Φ 0 F 1 v FL (7.2.5) with U the voltage drop along y and I = ac j tr. Using Eq. (7.2.3) the voltage drop over the length b of the superconductor can be expressed as U = B v FL b, (7.2.6) meaning that, for homogeneous conditions, the electric field is E = B v FL. The drift velocity v FL can be measured by virtue of Eq. (7.2.6). We can estimate v FL by defining the maximally acceptable voltage drop. Usually one defines the onset of the resistive state by occurrence of one µv along the direction of the transport current implying that in a field of 1 T and and b = 10 2 m the drift velocity reaches only v FL = 10 4 m/s. Whenever v FL 0 energy is dissipated. For understanding how the energy is dissipated we can follow two ways. Qualitatively we observe that Cooper pair are broken up in the center of the vortex and turn into normal particles. As long as the vortex is at rest the normal electrons do so as well making the mixed state dissipationless after the relaxation time of the normal electrons. However, if the vortices move the unpaired electrons are dragged on with velocity v FL and dissipate energy via collisions. Finally, vortices appear on one side and get annihilated on the opposite side giving rise to an ac component in U. If the spacing of the vortices is d the average voltage drop over the distance d is Ū d = B v FL d, and a frequency ω = 2π v FL d (7.2.7) c Walther-Meißner-Institut

125 Passive applications SUPERCONDUCTIVITY 117 can be found. If we solve this for v FL and substitute it into Ū d we find Ū d = ωbd2 2π. (7.2.8) Bd 2 is the flux through one vortex that is just the flux quantum Φ 0, and we find Ū d = hω 2e (7.2.9) and recover the second Josephson equation. Very sloppily one can say that the current trajectory gets interrupted by a vortex when it moves through and causes a phase slip of 2π that gives rise to a voltage pulse. The important question is as to whether or not the vortices can be prevented from moving since otherwise the mixed state (Shubnikov phase) is next to useless for applications. We now discuss how the pinning of the vortices can in fact be accomplished. The most direct way of describing pinning (see V. V. Schmidt) is by realizing that a vortex has a normal core inside which the condensation energy vanishes implying that the energy density inside the core is enhanced by approximately B 2 c/2µ 0 where B c is the thermodynamic critical field or by B 2 c 2µ 0 πξ 2 (7.2.10) per unit length over that of the superconducting material at distances larger than ξ from the center of the core. The same loss of condensation energy occurs if a hole with diameter 2ξ is drilled into the superconductor. This can be realized experimentally by shooting accelerated heavy atoms on the superconductor. Now, if the vortex core is centered around this columnar defect, no additional condensation energy is lost. If, however, the vortex is moved away from the hole the free energy of the material is pushed up again by the same amount. Therefore a restoring force f p (per unit length) exists trying to pull the vortex back into the cylindrical cavity. At the edge of the cavity f p ξ should be approximately equal to the condensation energy, f p B2 c 2µ 0 πξ, (7.2.11) yielding f p,c B2 c 2µ 0 πξ c, (7.2.12) for the slab of thickness c (see Fig. 7.1). Whenever f p,c < F 1 (see above) the restoring force is big enough to the vortex pinned. If we equate f p,c and F 1 we find for the current density j tr = B2 c 2µ 0 Φ 0 πξ, or, by using the GL expression (4.4.52) for the thermodynamic critical field, (7.2.13) j tr = B cξ 4 2µ 0 λ, (7.2.14) which is comparable to the pair-breaking critical current (not derived yet from the GL theory). In other words, given efficient pinning a superconductor in in field can carry a high current. However, the total field is substantially enhanced by the current. Therefore, the critical current is reduced substantially upon approaching B c

126 118 R. HACKL AND D. EINZEL An overview of applications Examples Conventional materials The most popular application with the biggest market are solenoids for MRI, and many of us had an opportunity to experience one of the machines. A field of up to 3 T with well-defined small gradients in a volume of approximately 1 m 3 is provided by a system of superconducting coils. The multifilamentary wires are made of NbTi embedded in a matrix of alloyed Cu. They can be fabricated by established extrusion techniques and wound up [10]. Hydrogen is the only nucleus being probed here and the spatial resolution is approximately 0.5 mm. Coils for 4 8 T are in the experimental phase since the resolution can be improved and other nuclei such as phosphorus can be used as a probe. The other big market is laboratory magnets. As mentioned above relatively inexpensive NbTi standard laboratory magnets are in wide use. For chemical analysis high-field NMR has an increasing share. After a very long experimental phase Nb 3 Sn coils providing fields in excess of 21 T became available in 1980ies and are standard for high-field NMR magnets only since the nineties [306]. Here, hybrid coils are used. The outer coil is a NbTi solenoid providing some 10 T the inner part is a Nb 3 Sn solenoid adding another 10 or 11 T, and a field 21 T can be reached in a bore of 50 mm having a homogeneity of better than 10 6 in a volume of 1 cm 3. Similarly, the 45 T magnet in Tallahassee is a hybrid having a NbTi, a Nb 3 Sn, and a resistive coil with the latter one providing an additional field of some 25 T at the price of 30 MW electrical power and 1440 m 3 per hour cooling water. On of the biggest installations of superconducting magnets is the accelerator of the CERN. In Karlsruhe coils for fusion reactors were developed and built. The coils have complicated shapes and are as big as a two-story building. Since a while prototypes for toroidal coils for energy storage are being developed. Beyond the conventional metallic systems there is an increasing number of applications of high-t c cuprates but none of the established techniques can be used for the cuprates. 3 Cuprates with high transition temperature It is not only the higher transition temperatures which relax the requirements for cooling but also the enhanced robustness of superconductivity as quantified by the condensation energy F B 2 c Tc 2 with B c the thermodynamical critical field. In Y-123 for instance, the upper critical field B c2 (T ), at which superconductivity collapses, is in the range 140 T in the low-t limit and still some 40 T at 77 K [241]. The critical current densities exceed 100 A mm 2 and 10,000 A mm 2 at 77 and 4.2 K, respectively. For active devices a switching frequency in the THz range can be reached for the large energy gap, τ 1 / h. Given these advantages, why took it more than 20 years until the first application was commercialized? It is rather complicated to get sufficiently homogeneous large-scale products at competitive costs since the materials have to be synthesized at C and are brittle. Y-123, the workhorse in applications, is a quaternary compound and small deviations from stoichiometry reduce T c substantially. On the other hand, defects at average distances close to the coherence length ξ 0 20 Å are necessary for high critical fields and currents. The enormous a c-anisotropy and the critical current of in-plane grain boundaries which decreases exponentially with the misalignment angle [307] require essentially mono-crystalline specimens. For coils or for power transmission lines the quality must be maintained over hundreds of meters. In spite of the difficulties various products are in the test phase now (for recent overviews see, e.g., Ref. [ ] or the web-links in the references). Wires with Bi-2223 can be bought from the shelf with lengths up to 1.5 km [311]. For this product Bi-2223 powder is filled in metal tubes, mostly Ag, and 3 The following section is an almost literal quote from a publication of one of us (R.H.) [56] c Walther-Meißner-Institut

127 Active devices SUPERCONDUCTIVITY 119 then rolled and reheated several times to get the platelet-like micro-crystals aligned. Very good results are obtained with oriented films of Y-123, which are deposited on strained metallic ribbons on top of various buffer layers [312, 313]. There are various techniques to grow thin films.for industrial production costs play a central role. Thermal co-evaporation of the metallic elements followed by in-situ post-annealing in oxygen [93] is among the promising methods. It was developed in a spin-off company of the Technical University Munich which now sells superconductors as well as production and test equipment [312]. For some applications jet-printing is used to deposit liquid precursors which are processed afterwards [314]. In this way complicated geometries can be realized, possibly at the price of a slightly reduced critical current. On the other hand, the technique is extremely simple and cost effective. In wider collaborations fault current limiters have been developed. In contrast to conventional technology the device can reversible interrupt the connection between the generator and the grid in the case of an overload. Inductive fault current limiters can cut spikes without fully interrupting the transmission. The first device produced by Zenergy Power was delivered in 2010 to CE Electric in the United Kingdom. In the United States dynamic synchronous condensers are used to compensate variable inductive and capacitive loads resulting from and enhancing rapid fluctuations of the power consumption in the grid. Filters for base stations of mobile communication have a much better selectivity and are substantially smaller. The superconducting filter element has an area of only a few quare centimeters. Thousands of these filters which fit into a 19 rack have been installed already [315]. Motors and generators having rotors with superconducting coils have a slightly better efficiency and substantially reduced size and weight. This helps to reduce the material consumption and makes the technology favorable for, e.g., ships or upcoming applications such as wind turbines. It has been demonstrated that thin-film Y-123 receiver coils improve the signal-to-noise ratio of MRI by a factors between 2 and 9 compared to that achievable with copper [316] as shown in Fig. 7.2 At the National High Field Laboratory in Tallahasse solenoids for the 30 to 40 T range are under development. For many experiments these magnets can replace the 45 T Hybrid magnet with a superconducting outer and a conventional inner coil at a price of approximately 20 MW power consumption. All these applications are local, and cryogen-free cooling is possible and is actually used widely. The development and optimization of pulse-tube cryo-coolers in the last decade simplified the refrigeration substantially and improved the reliability. The maintenance intervals are years and the the base temperature of a single stage engine is close to 30 K. For the filters miniature Stirling coolers have been developed with a power consumption in the 100 W range [315]. For non-local application such as power transmission cooling is still an issue. It is very much relaxed by the use of liquid nitrogen but still complicated and subject to failure [310]. Therefore, superconductors are mainly considered for specialized applications, where conventional techniques cannot carry the increased load, although the development of cables is very advanced. Nevertheless, using a cable manufactured by AmSC, the Long Island Power Authority started to transmit electricity for 300,000 households in April 2008 as shown in Fig. 7.3 [313]. Demonstration projects have been started in various other places of the world. 7.3 Active devices The sensing technique with SQUIDS on the basis of conventional superconductors was already mature at the end of the 1980ies while the use of cuprates operated at 77 K has still to overcome some problems [308]. Due to low pinning potentials the noise is the main problem to fix. While the noise of approximately 50 ft Hz 1/2 at 1 Hz is still too high for magneto-encephalography the study of the heart is feasible and has enough resolution [317]. There exist small start-up companies producing integrated solutions [318]. SQUIDs are now used mainly in the laboratory but also in military applications, in geology, and in quality control. Bits for quantum computing are so far only made of conventional metals. 2013

128 120 R. HACKL AND D. EINZEL An overview of applications tissue B m He compressor c-shaped open magnet plastic tube Figure 7.2: Schematic view of a setup for magnetic resonance imaging (MRI). The c-shaped open magnet is fabricated with Bi-2223 tape wire. The sensor has an Y-123 receiver. Magnet and receiver are cooled cryogen free with closed-cycle He refrigerators. The lower image on the r.h.s. shows the MRI image of the little finger with a conventional Cu receiver. The image on top is the result obtained with the Y-123 receiver. The improvement of the signal to noise ratio for the same scanning time is approximately a factor of 5. Reproduced from [316]. Figure 7.3: Superconducting power transmission line (front) operated by the Long Island Power Authority (LIPA). The cables have a core of Y-123 conductor including Cu for protection, a channel for liquid nitrogen and thermal insulation. The superconductors carry the same energy as the over-head lines in the background. The system is operative since April Courtesy of American Superconductors [313]. c Walther-Meißner-Institut