TEXTBOOK ON Superconductivity and Josephson Effect: Physics ...

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The experiment by W. Meissner and R. Ochsenfeld from 1933 has revealed that superconductors do not behave as just ideal conductors. It was found that at T < Tc the field inside a superconducting specimen was always zero (B = 0) in the presence of an external field, independent of which procedure had been chosen to cool the superconductor through Tc, see Fig. 1.12. Fig. 1.12. Difference between superconductors and ideal normal conductors: for an ideal conductor, the magnetic state depends on history, for superconductor - not. This discovery was very important. Indeed, if B = 0 independent of the specimen’s history, the zero induction can be treated as an intrinsic property of the superconducting state at H0 < Hc. Furthermore, as will be explained in the next section, this implies that superconductivity appears as a result of phase transition. Facilitation very powerful thermodynamic approach for examination of superconductors. Thus, the superconducting state obeys the equations: ρ=0, B = 0. (1.3) 1.3. Magnetic flux quantization An electric current, induced in a superconducting ring, can persist for an infinitely long time. Naturally, this does not require a power supply, since there is no power dissipation in the ring. Such a persistent current can be produced as follows. Let us place the ring at T> Tc in an external magnetic field so that the magnetic field lines pass through the interior of the ring. Then the ring is cooled down to a temperature below Tc where the material is superconducting, and the external magnetic field is switched off. At the first moment after switching off the field, the magnetic flux through the ring decreases and, according to Faraday’s law of electromagnetic induction, induces a current in the ring which will be persistent from this moment on. This current prevents a further decrease of the magnetic flux through the ring, i.e., now that the external field is zero, the current itself supports the flux through the ring at the initial level. Indeed, if the ring had a finite resistance R, the flux through the ring would decay during the time of the order of L/R, where L is the inductance of the ring. In a superconducting ring, since R = 0, it takes the flux infinite time to decay. This means that the magnetic flux becomes ‘frozen’ and the ring carries a persistent 19 “supercurrent”. At first sight it may seem that the ‘frozen’ magnetic flux can take on an arbitrary value. However, in 1961-1962 an important experimental fact was established: the magnetic flux through a hollow superconducting cylinder may only assume quantized values, equal to integral multiples of the flux quantumΦ0 = 2.07 x 10 -7 G cm 2 (CGS), given by a combination of fundamental constants: Φ0 = hc/2e, [SI: Φ0 = h/2e = 2.07 x 10 -15 Wb ] (1.4) where h is Planck’s constant, c is the speed of light and e is the electron charge. Physically, the origin of the magnetic flux quantization is the same as the quantization of electron orbits in atom: the wavefunction of electrons moving along a closed orbit must contain an integral number of wavelengths over the length of the orbit. 1.4. Josephson effect Josephson effect (sometimes referred to as weak superconductivity) provides another spectacular manifestation of the quantum mechanical nature of the superconducting state. It was predicted in 1962 by a 22 year old graduate student Brian Josephson 5 and soon verified experimentally by Ivar Giaever 6 and later by many other researchers. The peculiar history of this discovery is described in ref. 7 . The term ‘weak superconductivity’ refers to a situation in which two superconductors are coupled together by a weak link. The weak link can be provided by a tunnel junction or a short constriction in the cross-section of a thin film. More generally, this can simply be a weak contact between two superconductors over a small area or other arrangements where the superconducting contact between two superconductors is somehow ‘weakened’. The requirement of “weakness” implies that the weak link should not change significantly the wavefunctions on the two sides, compared to what they had been before the link was established. There are two Josephson effects to distinguish: (i) stationary (the dc Josephson effect) and (ii) nonstationary (the ac Josephson effect). Consider first the dc effect. Let us apply a current through a weak link (or, in other words, through a Josephson junction). Then, if the current is sufficiently small, it passes through the weak link without resistance, even if the material of the weak link itself is not superconducting (for example, if it is an insulator in a tunnel junction). Here we directly come across the most important property of a superconductor: the coherent behavior of superconducting electrons. Electrons of the two superconductors, interacting through the weak link, merge into a single phase-coherent quantum state. The same can be said in a different way. Having penetrated via the weak link into the second superconductor, the wave function of electrons from the first superconductor interferes with the ‘local’ electron wave function. As a result, all superconducting electrons on both sides of the weak link are described by the same wave-function. The ac Josephson effect is even more remarkable. Let us increase the dc current through the weak link until a finite voltage appears across the junction. Then, in addition to a dc component, the voltage V will also have an ac component of angular frequency ω, so that ħω= 2eV. The ac-Josephson oscillations lead to electromagnetic wave emission from Josephson junctions, which was first detected experimentally in 1965. 1.5 Development of the theory of superconductivity In 1935 London brothers have formulated the first theory, successfully describing electrodynamic properties of superconductors. The theory was phenomenological, that is, it had 20
introduced two equations, in addition to Maxwell’s equations, governing the electromagnetic field in a superconductor. These equations provided a correct description of the two basic properties of superconductors: absolute diamagnetism and zero resistance to a dc current. The London theory did not attempt to resolve the microscopic mechanism of superconductivity on the level of electrons, that is, the question: “Why does a superconductor behave according to the London equations?” remained beyond its scope. According to the London theory, electrons in a superconductor may be considered as a mixture of two groups: superconducting electrons and normal electrons. The density of the superconducting electrons, ns decreases with increasing temperature and eventually becomes zero at T = Tc. Vice-versa, at T = 0, ns is equal to the total density of conduction electrons. These are the postulates of the so-called two-fluid model of a superconductor. A flow of superconducting electrons meets no resistance. Such a current, obviously, cannot generate a constant electric field in a superconductor because, if it did, it would cause the superconducting electrons to accelerate infinitely. Therefore, under stationary conditions, that is, without an electric field, the normal electrons are at rest. In contrast, in the presence of an ac electric field, both the normal and the superconducting components of the current are finite and the normal current obeys Ohm’s law. In this framework, a superconductor can be modeled by an equivalent circuit consisting of a normal resistor and an ideal conductor connected in parallel. The ideal conductor in the circuit must have a finite inductance to mimic the inertia of the superconducting electrons. London equations provided a simple and fairly correct description for the behavior of superconductors in both dc and ac electromagnetic fields. They also helped to understand a number of general aspects of superconductivity. However, by the end of the 1940s, it was clear that at least to one question the London theory gave a wrong answer: for an interface between normal and superconducting regions, the theory predicted negative surface energy σns < 0. This implied that a superconductor in an external magnetic field could decrease its total energy by turning into a mixture of alternating normal and superconducting regions. In order to make the total area of the interface within the superconductor as large as possible, the size of the regions must be as small as possible. This was supposed to be the case even for a long cylinder in a longitudinal magnetic field, in contradiction to experimental evidence existing at that time. Experiments showed separation of the normal and superconducting domains occurred only for specimens having a nonzero demagnetizing factor (the intermediate state, will be discussed in §2.2). In addition, domains were rather large (~1 mm, see Fig. 2.2), which could only be the case if σns > 0, in contradiction with predictions of the London theory. The above contradiction was reconciled by a theory proposed by V.L. Ginzburg and L.D. Landau (the G-L theory) which was also phenomenological but took account of quantum effects 8 . G-L assumed that a single quantum-mechanical wave function Ψ describes all superconducting electrons. Then the squared amplitude of this function (which is proportional to ns) must be zero in a normal region, increase smoothly through the normal-superconducting (NS) interface and finally reach a certain equilibrium value in a superconducting region. Therefore, a gradient of Ψ must appear at the interface. At the same time, as is well known from quantum mechanics, ⏐∇Ψ⏐ 2 is proportional to the density of the kinetic energy. Thus, by taking into account quantum effects, we also take into account an additional positive energy stored at the NS interface, which creates the opportunity to obtain σns > 0. The GL theory will be discussed in detail in Section 5. Importantly, the GL theory introduced quantum mechanics into the description of superconductors. It assigned to the entire number of superconducting electrons a wave-function depending on a single spatial coordinate [recall that, generally speaking, a wave-function of n electrons in a metal is a function of n coordinates, Ψ (r1, r2,. . . ,rn)]. By doing so, the theory established the coherent behavior of all superconducting electrons. Indeed, in quantum mechanics, a single electron in the superconducting state is described by a function Ψ(r). If we now have ns absolutely identical electrons (where ns, the superconducting electron number density, is a macroscopically large number) and all these electrons behave coherently, it is clear that the same wave-function of a single parameter is sufficient to describe all of them. This idea 21 was a breakthrough that enabled the prediction of many beautiful quantum, and at the same time macroscopic, effects in superconductivity. The GL theory was built on the basis of Landau’s theory of second-order phase transitions and is valid only in the vicinity of the critical temperature, that is, within the temperature range Tc-T 0. Those that do have σns > 0 are type-I superconductors. But the majority of superconducting alloys and chemical compounds show σns < 0; they are type-II superconductors. For type-II superconductors, there is no Meissner effect at high magnetic field and the magnetic field does penetrate inside the material, but penetrates in a very unusual way, namely, in the form of quantized vortex lines – Abrikosov vortices. Superconductivity in these materials can survive up to very high magnetic fields. Still, neither London, nor GL theory could answer the question: “What are those superconducting electrons, whose behavior they were intended to describe?”. The microscopic origin of superconductivity was finally resolved in 1957, 46 years after discovery, by the work of J. Bardeen, L. Cooper and J. Schrieffer (the BCS theory) 10 . An important contribution was also made by N.N. Bogolyubov (1958) who developed a mathematical method now widely used in studies of superconductivity. Fig. 1.13. Short summary of the microscopic BCS theory of superconductivity The decisive step in understanding the microscopic mechanism of superconductivity is due to L.Cooper (1956) 11 . The essence of his work can be outlined in Fig. 1.13. Consider a normal metal in the ground state: in k space, all states for non-interacting electrons inside the Fermi sphere are occupied, while all those outside it are empty. Then an extra pair of electrons is brought in and placed in the states (k↑) and (-k↓ ), in the vicinity of the Fermi surface (the arrows indicate the directions of electron spins). It turned out that if, for whatever reason, the two electrons become attracted to each other, they form a bound state regardless of how weak the attraction is. In real 22
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