Patterson, Bailey - Solid State Physics Introduction to theory

Problems 457

(with A, B, C, and u0 being positive constants). Generalization to higher dimension have been made. The solitary wave owes its stability to the competition of dispersion and nonlinear effects (such as a tendency to steepen waves). The solitary wave propagates with a velocity that depends on amplitude.

Problems

7.1Calculate the demagnetization factor of a sphere.

7.2In the mean-field approximation in dimensionless units for spin 1/2 ferromagnets the magnetization (m) is given by

m = tanh m

t ,

where t = T/Tc and Tc is the Curie temperature. Show that just below the Curie temperature t < 1,

m = 3 1−t .

7.3Evaluate the angular momentum L and magnetic moment μ for a sphere of mass M (mass uniformly distributed through the volume) and charge Q (uniformly distributed over the surface), assuming a radius r and an angular velocity ω. Thereby, obtain the ratio of magnetic moment to angular momentum.

7.4Derive Curie’s law directly from a high-temperature expansion of the partition function. For paramagnets, Curie’s law is

χ= CT (The magnetic susceptibility),

where Curie’s constant is

C = μ0 Ng 2μB2 j( j +1) .

3k

N is the number of moments per unit volume, g is Lande’s g factor, μB is the Bohr magneton, and j is the angular momentum quantum number.

7.5Prove (7.155).

7.6Prove (7.156).

8 Superconductivity

8.1 Introduction and Some Experiments (B)

In 1911 H. Kamerlingh Onnes measured the electrical resistivity of mercury and found that it dropped to zero below 4.15 K. He could do this experiment because he was the first to liquefy helium and thus he could work with the low temperatures required for superconductivity. It took 46 years before Bardeen, Cooper, and Schrieffer (BCS) presented a theory that correctly accounted for a large number of experiments on superconductors. Even today, the theory of superconductivity is rather intricate and so perhaps it is best to start with a qualitative discussion of the experimental properties of superconductors.

Superconductors can be either of type I or type II, whose different properties we will discuss later, but simply put the two types respond differently to external magnetic fields. Type II materials are more resistant, in a sense, to a magnetic field that can cause destruction of the superconducting state. Type II superconductors are more important for applications in permanent magnets. We will introduce the Ginzburg–Landau theory to discuss the differences between type I and type II.

The superconductive state is a macroscopic state. This has led to the development of superconductive quantum interference devices that can be used to measure very weak magnetic fields. We will briefly discuss this after we have laid the foundation by a discussion of tunneling involving superconductors.

We will then discuss the BCS theory and show how the electron–phonon interaction can give rise to an energy gap and a coherent motion of electrons without resistance at sufficiently low temperatures.

Until 1986 the highest temperature that any material stayed superconducting was about 23 K. In 1986, the so-called high-temperature ceramic superconductors were found and by now, materials have been discovered with a transition temperature of about 140 K (and even higher under pressure). Even though these materials are not fully understood, they merit serious discussion. In 2001 MgB2, an intermetallic material was discovered to superconduct at about 40 K and it was found to have several unusual properties. We will also discuss briefly so-called heavyelectron superconductors.

Besides the existence of superconductivity, Onnes further discovered that a superconducting state could be destroyed by placing the superconductor in a large enough magnetic field. He also noted that sending a large enough current through the superconductor would destroy the superconducting state. Silsbee later suggested that these two phenomena were related. The disruption of the superconductive state is caused by the magnetic field produced by the current at the surface of

8.1 Introduction and Some Experiments (B) 461

flux rather than excluded flux. A plot of critical field versus temperature typically (for type I as we will discuss) looks like Fig. 8.2. The equation describing the critical fields dependence on temperature is often empirically found to obey

In 1950, H. Fröhlich discussed the electron–phonon interaction and considered the possibility that this interaction might be responsible for the formation of the superconducting state. At about the same time, Maxwell and Reynolds, Serin, Wright, and Nesbitt found that the superconducting transition temperature depended on the isotopic mass of the atoms of the superconductor. They found MαTcconstant. This experimental result gave strong support to the idea that the elec- tron–phonon interaction was involved in the superconducting transition. In the simplest model, α = 1/2.

In 1957, Bardeen, Cooper, and Schrieffer (BCS) finally developed a formalism that contained the correct explanation of the superconducting state in common superconductors. Their ideas had some similarity to Fröhlich’s. A key idea of the BCS theory was developed by Cooper in 1956. Cooper analyzed the electron– phonon interaction in a different way from Fröhlich. Fröhlich had discussed the effect of the lattice vibrations on the self-energy of the electrons. Cooper analyzed the effect of lattice vibrations on the effective interaction between electrons and showed that an attractive interaction between electrons (even a very weak attractive interaction at low enough temperature) would cause pairs of the electrons (the Cooper pairs) to form bound states near the Fermi energy (see Sect. 8.5.3). Later, we will discuss the BCS theory and show the pairing interaction causes a gap in the density of single-electron states.

Fig. 8.3. (a) Type I and (b) Type II superconductors

As we have mentioned a distinction is made between type I and type II superconductors. Type I have only one critical field while type II have two critical fields. The idea is shown in Fig. 8.3a and b. 4πM is the magnetic field produced by the surface superconducting currents induced when the external field is applied. Type I superconductors either have flux penetration (normal state) or flux exclusion

8.1 Introduction and Some Experiments (B) 463

8.1.1 Ultrasonic Attenuation (B)

The BCS theory of the ratio of the normal to the superconducting absorption coefficients (αn to αs) as a function of temperature variation of the energy gap (discussed in detail later) can be interpreted in such a way as to give information on the temperature variation of the energy gap. Some experimental results on (αn/αs) versus temperature are shown in Fig. 8.5. Note the close agreement of experiment and theory, and that the absorption of superconductors is much lower than for the normal case when well below the transition temperature.

8.1.2 Electron Tunneling (B)

There are at least two types of tunneling experiments of interest. One involves tunneling from a superconductor to a superconductor with a thin insulator separating the two superconductors. Here, as will be discussed later, the Josephson effects are caused by the tunneling of pairs of electrons. The other type of tunneling (Giaever) involves tunneling of single quasielectrons from an ordinary metal to a superconducting metal. As will be discussed later, these measurements provide information on the temperature dependence of the energy gap (which is caused by the formation of Cooper pairs in the superconductor), as well as other features.

8.1.3 Infrared Absorption (B)

The measurement of transmission or reflection of infrared radiation through thin films of a superconductor provides direct results for the magnitude of the energy gap in superconductors. The superconductor absorbs a photon when the photon’s energy is large enough to raise an electron across the gap.

8.1.4 Flux Quantization (B)

We will discuss this phenomenon in a little more detail later. Flux quantization through superconducting rings of current provides evidence for the existence of paired electrons as predicted by Cooper. It is found that flux is quantized in units of h/2e, not h/e.

8.1.5 Nuclear Spin Relaxation (B)

In these experiments, the nuclear spin relaxation time T1 is measured as a function of temperatures. The time T1 depends on the exchange of energy between the nuclear spins and the conduction electrons via the hyperfine interaction. The data for T1 for aluminum looks somewhat as sketched in Fig. 8.6. The rapid change of T1 near T = Tc can be explained, at least quantitatively, by BCS theory.

8.2 The London and Ginzburg–Landau Equations (B) 465

For many years, superconducting transition temperatures (well above 20 K) had never been observed. With the discovery of the new classes of high-temperature superconductors, transition temperatures (well above 100 K) have now been observed. We will discuss this later, also. The exact nature of the interaction mechanism is not known for these high-temperature superconductors, either.

8.2 The London and Ginzburg–Landau Equations (B)

We start with a derivation of the Ginzburg–Landau (GL) equations, from which several results will follow, including the London equations. Originally, these equations were proposed on intuitive, phenomenological lines. Later, it was realized they could be derived from the BCS theory. Gor’kov showed the GL theory was a valid description of the BCS theory near Tc. He also showed that the wave function ψ of the GL theory was proportional to the energy gap. Also, the density of superconducting electrons is |ψ|2. Due to spatial inhomogeneities ψ = ψ(r), where ψ(r) is also called the order parameter. This whole theory was developed further by Abrikosov and is often known as the Ginzburg, Landau, Abrikosov, and Gor’kov theory (for further details, see, e.g., Kuper [8.20]

Near the transition temperature, the free energy density in the phenomenological GL theory is assumed to be (gaussian units)

where N and S refer to normal and superconducting phases. The coefficients α and β are phenomenological coefficients to be discussed. h2/8π is the magnetic energy density (h = h(r) is local and the magnetic induction B is determined by the spatial average of h(r), so A is the vector potential for h, h = × A). m* = 2m (for pairs of electrons), q = 2e is the charge and is negative for electrons, and ψ is the complex superconductivity wave function. Requiring (in the usual calculus of variations procedure) δFS/δψ* to be zero (δFS/δψ = 0 would yield the complex conjugate of (8.3)), we obtain the first Ginzburg–Landau equation

FS can be regarded as a functional of ψ and A, so requiring δFS/δA = 0 we obtain the second GL equation for the current density: