Superconductivity

Superconductivity was discovered by the Dutch physicist Heike Kamerlingh Onnes in 1911 while investigating the electrical resistance of mercury at extremely low temperatures.

He observed that the electrical resistance of mercury decreased abruptly to an immeasurably small value when its temperature was lowered below approximately 4.2 K.

This sudden disappearance of measurable electrical resistance is called superconductivity.

The materials that exhibit superconductivity below a characteristic temperature are known as superconductors.

The temperature below which a material enters the superconducting state is called its critical temperature or transition temperature, denoted by \(T_c\).

Definition: Superconductivity is the phenomenon in which a material exhibits zero DC electrical resistance and expels magnetic flux from its interior under appropriate conditions when cooled below its critical temperature.

The critical temperature depends on the nature, composition, crystal structure, and other physical properties of the superconducting material.

Critical Temperatures of Some Superconductors

Superconductor	Critical Temperature \(T_c\) (K)
Mercury (Hg)	4.15
Aluminium (Al)	1.18
Tin (Sn)	3.72
Lead (Pb)	7.19
Niobium (Nb)	9.25
Nb₃Ge	Approximately 23

Variation of Electrical Resistivity with Temperature

In an ordinary metal, electrical resistivity generally decreases gradually as the temperature decreases because the scattering of conduction electrons by lattice vibrations is reduced.

However, in a superconductor, the resistivity decreases abruptly to zero when the temperature falls below the critical temperature \(T_c\).

The electrical resistivity of a superconductor can be represented schematically as

\[ \rho(T)= \begin{cases} 0, & TT_c, \end{cases} \]

where \(\rho_n(T)\) represents the electrical resistivity of the normal state.

The transition occurs over a narrow temperature interval in a sufficiently homogeneous sample.

Fig. Schematic variation of electrical resistivity with temperature for a superconductor.

Below \(T_c\), the DC electrical resistivity becomes zero, allowing an electrical current to persist without ordinary resistive energy loss.

However, superconductivity can be destroyed if the temperature, applied magnetic field, or current exceeds its corresponding critical value.

Isotope Effect

Experimental investigations have shown that the critical temperature of several conventional superconductors depends on the isotopic mass of the constituent atoms.

For isotopes of the same element, the critical temperature approximately follows the relation

\[ \boxed{ T_c \propto M^{-\alpha} } \]

where

\[ \begin{aligned} T_c &= \text{Critical temperature},\\ M &= \text{Isotopic mass},\\ \alpha &= \text{Isotope-effect exponent}. \end{aligned} \]

For an ideal conventional BCS superconductor under the simplest harmonic approximation,

\[ \boxed{\alpha \approx \frac{1}{2}} \]

Therefore,

\[ T_c \propto \frac{1}{\sqrt{M}} \]

or

\[ \boxed{ T_c M^{1/2}=\text{constant} } \]

This dependence of critical temperature on isotopic mass is called the isotope effect.

The isotope effect provides important evidence for the involvement of lattice vibrations (phonons) in the formation of Cooper pairs in conventional superconductors.

However, the isotope-effect exponent is not universally equal to 0.5. Its value may differ because of electronic structure, strong coupling, anharmonicity, and other physical effects.

Meissner Effect

In 1933, Walther Meissner and Robert Ochsenfeld discovered that a superconducting material expels magnetic flux from its interior when cooled below its critical temperature in a sufficiently weak applied magnetic field.

This phenomenon is called the Meissner effect.

Consider a superconducting specimen placed in an external magnetic field.

When the specimen is in its normal state, magnetic flux can penetrate its interior. However, when the temperature is lowered below \(T_c\), the specimen enters the superconducting state and expels magnetic flux from its bulk.

For an ideal superconductor in the Meissner state, the magnetic flux density inside the bulk is

\[ \boxed{B=0} \]

This property indicates perfect diamagnetism in the ideal Meissner state.

In SI units,

\[ B=\mu_0(H+M) \]

For \(B=0\),

\[ \boxed{M=-H} \]

where \(H\) is the magnetic field intensity and \(M\) is the magnetization.

Thus, the magnetic susceptibility is

\[ \boxed{ \chi=\frac{M}{H}=-1 } \]

This expression applies to an ideal specimen with negligible demagnetizing effects in the complete Meissner state.

Important: The Meissner effect distinguishes a superconductor from an ordinary perfect conductor. Zero electrical resistance alone does not imply that magnetic flux already present inside a specimen will be expelled. Magnetic flux penetrates a real superconductor over a small distance called the London penetration depth.

Critical Magnetic Field

Superconductivity can be destroyed by applying a sufficiently strong magnetic field, even when the temperature remains below the critical temperature.

For a type-I superconductor, the magnetic field at which the superconducting state is destroyed is called the critical magnetic field, denoted by \(H_c\).

The critical magnetic field decreases as the temperature increases and becomes zero at the critical temperature \(T_c\).

For many type-I superconductors, the temperature dependence of the critical magnetic field can be approximated by

\[ \boxed{ H_c(T)=H_c(0) \left[ 1-\left(\frac{T}{T_c}\right)^2 \right] } \]

where

\[ \begin{aligned} H_c(T) &= \text{Critical field at temperature }T,\\ H_c(0) &= \text{Critical field at }0\,\text{K},\\ T_c &= \text{Critical temperature}. \end{aligned} \]

At absolute zero,

\[ H_c(0)=H_c(0) \left[1-0\right]. \]

Therefore, the critical magnetic field has its maximum value at \(T=0\).

At the critical temperature,

\[ H_c(T_c)=H_c(0)(1-1)=0. \]

Thus,

\[ \boxed{H_c(T_c)=0} \]

Fig. Approximate variation of critical magnetic field with temperature for a type-I superconductor.

The region below the critical-field curve corresponds to the superconducting state, whereas the region above the curve corresponds to the normal state.

Type-II superconductors exhibit two critical fields, \(H_{c1}\) and \(H_{c2}\). Between these fields, magnetic flux penetrates the material in the form of quantized vortices, producing a mixed state.

BCS Theory of Superconductivity

The microscopic theory of conventional superconductivity was developed by John Bardeen, Leon Cooper, and John Schrieffer in 1957.

This theory is known as the BCS theory.

According to BCS theory, an effective attractive interaction mediated by lattice vibrations can cause electrons near the Fermi surface to form correlated pairs called Cooper pairs.

In conventional superconductors, these pairs typically consist of electrons with opposite momenta and opposite spins.

A macroscopic quantum-coherent state is formed, and an energy gap develops in the electronic excitation spectrum.

The energy gap at absolute zero for an ideal weak-coupling BCS superconductor is

\[ \boxed{ 2\Delta(0)\approx3.52\,k_B T_c } \]

where

\[ \begin{aligned} \Delta(0)&=\text{Superconducting energy gap at }0\,\text{K},\\ k_B&=\text{Boltzmann constant},\\ T_c&=\text{Critical temperature}. \end{aligned} \]

The coherent superconducting state supports electrical current without ordinary DC resistive dissipation under appropriate conditions.