2 Experimentally Determined Properties of Superconductors²

2.1 Zero Resistance, Narrow Transition

Onnes' original data showed that the resistance of mercury dropped by at least ten orders of magnitude when cooled from room temperature to 4K. Later, Onnes sent a current pulse through a superconducting ring and observed that the current flowed with no energy loss for over 24 hours. In 1912, Onnes concluded that the resistance of mercury was at least a million times smaller than the best room-temperature conductors. More modern measurements have been unable to measure any resistance in a superconductor, and have been able to demonstrate that the resistance drops by at least fourteen orders of magnitude through a temperature change as little as $3\times 10^{-4}$K.³ A current pulse in a superconducting ring can circulate for years with no observable deterioration.

2.2 Critical Magnetic Field, Critical Current

Despite Onnes' hope for the practical applications of superconductivity, he soon discovered that superconductivity is very fragile and can be destroyed by magnetic fields or electric currents. With no external magnetic field, superconductivity starts at a critical temperature $(T_{c})$, but as the external magnetic field increases, the temperature at which superconductivity begins decreases. Magnetic fields greater than the critical field $(H_{c})$ destroy superconductivity, and fields greater than $H_{0}$ destroy superconductivity even at absolute zero. Experimentally it was determined that $H_{c}$ varies parabolicly with temperature:

$$H_{c}(T)\simeq H_{0}\left(1-\left(\frac{T}{T_{c}}\right)^{2}\right),$$ (1)

where $T$ is the temperature and $T_{c}$ the critical temperature (see figure 4).

Figure 4: Critical temperature vs. magnetic field. $H_{c}(T)$ defines a curve that separates the superconducting phase and the normal phase. The curve is approximately parabolic.
Source: Vidali (1993), page 53.

Similarly, electrical currents greater than the critical current ($I_{c}$) also destroys superconductivity.

2.3 The Meissner Effect

It was not until 1933 that Walther Meissner and Robert Ochsenfeld discovered what is now considered a fundamental property of superconductors: perfect diamagnetism, or the expulsion of external magnetic fields (see figure 5).

Figure 5: A diamagnet expels magnetic fields: magnetic field lines are bent around a perfect diamagnet.
Source: Vidali (1993), page 4.

The importance of this discovery for theories of superconductivity can be demonstrated with a basic thought experiment: place a perfect conductor - or what was originally thought to be a superconductor - with a temperature above $T_{c}$ in a magnetic field, and the magnetic field will penetrate the metal (see figure 6).

Figure 6: Magnetic fields inside a perfect conductor in response to lowering the temperature and magnetic field. The first path corresponds to turning off the magnetic field and then lowering the temperature (A $\rightarrow$D $\rightarrow$C); the second path corresponds to lowering the temperature and then turning off the magnetic field (A $\rightarrow$B $\rightarrow$C).

Turning off the magnetic field (A $\rightarrow$D) will, by Faraday's law of induction, induce a current that will try to maintain the original magnetic field. This induced current will decay away because of the finite resistance of the metal, and no magnetic field will penetrate the material. Lowering the temperature below $T_{c}$ will make the metal a perfect conductor (D $\rightarrow$C) with no magnetic field penetrating the metal. If, however, we lower the temperature first, then the metal becomes a perfect conductor in the presence of a magnetic field (A $\rightarrow$B), but the magnetic properties remain unchanged and the magnetic field continues to penetrate the metal. Lowering the magnetic field (B $\rightarrow$C) will again induce a current to maintain the magnetic field, but this current will not decay because now the metal has no resistance. Therefore, the magnetic field will be maintained inside the metal if we take the path A $\rightarrow$B $\rightarrow$C, but there will be no magnetic field inside the metal if we take the path A $\rightarrow$D $\rightarrow$C. The process of going from A to C is path dependent; a perfect conductor cannot be described solely by its state (temperature and magnetic field), nor is the process of becoming a perfect conductor governed by equilibrium thermodynamics, a major obstacle for theories of superconductivity.

Performing the same thought experiment, but taking into account Meissner and Ochsenfeld's discovery that a superconductor is also a perfect diamagnet, eliminates this problem (see figure 7).

Figure 7: Magnetic fields inside a superconductor in response to lowering the temperature and magnetic field. The first path corresponds to turning off the magnetic field and then lowering the temperature (A $\rightarrow$D $\rightarrow$C); the second path corresponds to lowering the temperature and then turning off the magnetic field (A $\rightarrow$B $\rightarrow$C).

The path A $\rightarrow$D $\rightarrow$C remains the same: lowering the magnetic field will induce a current that will decay away and no magnetic field will penetrate the metal. If, however, we lower the temperature below $T_{c}$ in the presence of a magnetic field (A $\rightarrow$B), then the magnetic field will be expelled by the Meissner effect. Lowering the external magnetic field (B $\rightarrow$C) will not affect the metal because the magnetic field does not penetrate the metal. The superconductor ends in a state with no internal magnetic field at point C regardless of the path taken. The Meissner effect was a breakthrough for theories of superconductivity because it allowed superconductivity to be treated thermodynamically and, as we shall see later, it helped the development of the London equations.⁴

2.4 Thermodynamic Properties

The transition to superconductivity by lowering the temperature with no external magnetic field was found to be a second-order phase transition, characterized by a discontinuous change in the specific heat at $T_{c}$ and no latent heat. If the transition occurs in the presence of a magnetic field, the phase transition is first-order, characterized by latent heat and a singularity in the specific heat. Experimental data indicated that the specific heat of a superconductor was proportional to $T^{3}$. Later it was determined that an exponential curve fit the data better.

2.5 The Isotope Effect

By studying the critical temperature of different isotopes of mercury, in 1950 Maxwell, and independently Reynolds and Seitz, discovered that $T_{c}\propto M^{-1/2}$ where M is the atomic mass of the material. Since the Debye frequency⁵ $(\omega _{D})$ is proportional to $M^{-1/2}$, the isotope effect implies that, holding other factors constant, $\omega _{D}\propto T_{c}$. The fact that the atomic mass of the metal affects its critical temperature shows that superconductivity involves some interaction between the electrons and the metal lattice.

2.6 Penetration Depth

Experimental evidence showed that the magnetic field near the surface of a superconductor decayed exponentially as it penetrated into the superconductor. This decay length, called the penetration depth, is usually on the order of 1000Å.

2.7 Energy Gap⁶

As early as the 1930's, an energy gap between the superconducting ground state and the first excited state had been hypothesized by Fritz and Heinz London as means of trapping a superconductor in its ground state at low temperatures. Early experimental evidence bounded the the size of the energy gap, however, direct experimental evidence of an energy gap ($\sim$10$^{-4}$eV) was not presented until 1956 using microwave techniques developed during World War II.

Footnotes

... Superconductors²

The main sources used for this section are chapters 2-5 of Vidali (1993, p.13-65), and chapter 3 of Dahl (1992, p.50-65). Thermodynamic properties come primarily from chapter 1 of Parks (1969, p.19-26). Additional information comes came from Buckel (1991); Kresin and Wolf (1990); Ibach and Lüth (1996).

...K.³

The width of the transition can be much broader in some superconductors. In 1965, Neighbor measured the width of the superconducting transition to be less than $3\times 10^{-4}$K in lead (Parks: 1969, p.2).

...⁴

Note that the Meissner effect was discovered twenty years after superconductivity because the effect is unobservable in the hollow spheres that Onnes and others used in their experiments and because of the difficulties in making sensitive magnetic field measurements on superconductors.

... frequency⁵

The Debye frequency is the high-end cutoff frequency for the vibrational modes of atoms, based on treating the atoms in a metal as trapped in a harmonic potential.

...sub:Energy-Gap⁶

Source: Dahl (1992, p.242-255)