18 Superconducting thin films

Introduction

H. Kamerlingh Onnes in 1908, measures the electrical resistance of metals at very low temperatures by successfully liquefy helium. In 1911, scientists discovered that as the temperature of Hg was reduced at around 4.2 K, the resistance of Hg did not fall continuously as expected, but in its place dropped suddenly to zero over a range of a few hundredths of a degree. This phenomenon of non-zero resistance of a metal was termed as superconductivity. Later on the same phenomena was also discovered for other metal elements such as Pb, Sn and Al at critical temperatures lying between 4 to10K. However, the microscopic mechanism of such an unexpected behaviour of the metals found at some critical temperatures known as superconductivity was not discovered until 1957. In 1957, three of the scientists Bardeen, Cooper and Schrieffer try to investigate this microscopic behaviour. In 1933, Meissner & Ochsenfeld was also discovered one another important property of superconductors called Meissner effect. Superconductors exhibits the property of perfect conductivity, such that the magnetic flux should be excluded from entering a superconductor, whereas it is also found that when the superconducting material was cooled through its transition temperature, the magnetic flux was expelled from the interior of the material. This phenomenon is known as ‘Meissner effect’.

As compare to the bulk materials the superconducting properties of thin films are found to be much more different and superior. The superconducting transition temperature (Tc) of thin films changes remarkably with the film thickness thus it can certainly affect the superconducting behaviour of the deposited material. In the nanometer scale range of the film thickness, the motions of the electrons of the film surface and boundary is confined and leading to the formation of isolated electronic states called quantum well states. The over all electronic structure of the film changes due to this quantum size effect. Also, the physical properties of the films vary considerably at small thicknesses. The latest investigations have also demonstrated the variations of the properties of deposited films with film thickness such as the electronic density of states, electron-phono-coupling, surface energy, and thermal stability.

Distinct behaviours exhibited by Superconducting thin films

1.  Zero resistance: Below a specific temperature called critical temperature (Tc), the electric resistance suddenly falls to zero in a superconductor. The perfect conduction of the current will be achieved at zero resistance. Although, it does not happen in reality due to the defects and vibrations of the atoms which causes the resistance in the material when the electrons flow through it. However, after all such defects and vibrations existence in a superconductor, the electric resistance is equal to zero.

2. Persistent currents: Persistent current is the current which is set up in a superconductor forever i.e without any driving voltage a current will flow forever in the superconductors. As a consequence, magnetic flux that passes through a continuous loop of superconducting materials remains constant. (fig. 3).

3.       Perfect diamagnetism: In atomic orbitals, the current is induced by the application of an applied magnetic field which causes diamagnetism. The applied field is opposed by the diamagnetic material and the induced currents produce a magnetisation within the material, also in the absence of the applied field the induced magnetisation disappears. The relation of the magnetic filed in diamagnetic materials is given as B = μμ0H, here the relative permeability μ is slightly less than unity. In a superconductor the field B is zero in the interior i.e. the field is completely screened from the interior of the material, this is the extreme consequence of the diamagnetic effect in the superconductors. Hence, in superconductors the relative permeability is zero.

4. Energy gap: The energy spectrum of the superconducting materials is found to have the gap in it i.e. there is an existence energy interval with no allowed eigen energies in this region. The evidence of this phenomenon is lying in the fact that the most thermodynamic properties of a superconducting material are found to vary as e−∆/(kBT). Thus, the unoccupied states above the gap can be filled by the exponentially smaller number of particles which have enough thermal energy equal to the energy of the gap. When the energy of the external magnetic field is equal to the energy gap of the superconductor, electrons in the superconducting state are able to absorb this energy and bridge the energy gap to a normal conducting state resulting in a dissipative resistance initiated quench for the superconductor.

Various Theories of Superconductivity

Phenomenological Theories

Ø   London Theory

A phenomenological theory by assuming the essentiality of the diamagnetism and the describing the electromagnetic properties is proposed by London and London. The theory assumed that there is somehow a consistency or stiffness in the superconducting state such that the in the presence of the magnetic filed the wave functions are not modified very much, this kind of approach is suggested by F. London and is known as a quantum theoretic approach. The presented theory gives a two fluid picture namely, the electrons from a normal fluid of concentration and a super fluid of concentration.

London suggests some equations based the defined theory which permits the absence of magnetic field in side the material,

Ø   Ginzburg-Landau Theory

Landau theory describes the second order phase transitions with symmetry reduction of the materials. It says that there is a thermodynamic parameter, which takes zero value in the symmetric phase occurring at high temperatures and in the less symmetric phase it becomes non-zero continuously. Superconductors are categorized by the Free Energy density parameter (r) which vanishes above Tc. Non-Uniform field superconductors, effects of surface, quantization of flux and Type II superconductors are described successfully by Landau theory.

Some unanswered questions after Phenomenological Theories

The Phenomenological Theories successfully describe the various phenomenon related to the superconducting state of the material and are well established but the major mechanism of microscopic features of the superconductor is still unquestioned. A large category of the physicists working in 1900’s attempt to answer this question some of those are- Bohr, Einstein, Feynman, Born and Heisenberg. However, in 1957, Bardeen, Cooper, and Schrieffer give a significant contribution to this with the famous theory called BCS theory. The theory came around after 50 years of the experimental discovery by Kamerlingh-Onnes. The discontinuity in the energy spectrum of the superconductors is appeared as a consequence of the exponential decay of the specific heat at low temperatures. Since an electron-hole pair excitation in case of metal near the Fermi surface requires very little energy, which contributes to the continuous spectrum of the metal and found to be in contrast to the superconducting materials. The isotope effect is found to be another major experiment. If M is the mass of the isotope then from the fundamental study of the superconducting materials containing different isotope it is observed that Tc follows the relation M−1/2, where Tc is the transition temperature of the superconductors. Since, mass M is connected to the ions forming the lattice, therefore this study suggests that the lattice and hence the phonons play a crucial role in the structure of the superconducting state. This attractive interation between the electron-electron or electron- phonons generate Cooper pairs i.e. bound states formed by two electrons of opposite spins and momenta is considered as the key parameter of the BCS theory. The discontinuous energy spectrum and perfect diamagnetism of the superconductors is presented by the macroscopic ground state formed by Cooper pairs. The existence of a well-defined Fermi surface is considered as the main feature of the formation of Cooper pairs.

Before describing the microscopic BCS theory understanding of some terms is important. The terms are as follows:

Electron-phonon interaction

An effective attraction between conduction electrons gives rise to the superconductivity. When the two electrons are placed in a metallic environment there exist additional attractive forces between the electrons with the repulsive Coulomb force. This attractive force in case of standard superconductors arises due to the interaction with the ionic system. In case of a normal metal, homogeneous positive background of ions is considered generates charge neutrality in the system. This is the polarised medium. An electron lying close to the Fermi surface moves with a much larger velocity than the velocity of the ions. Hence, during the process of polarisation of the ions (10−13 sec), first electron covers a distance of ∼100◦A, while before the ionic fluctuation relaxes the second electron can occur by to lower its energy with the concentration of positive charge. The effective attraction generated by this process is sufficient enough to overcome the force of repulsion between two electrons. Frolich in 1950, discovers the mechanism of electron-phonon “pairing”, which was later confirmed by the discovery of “isotope effect”.

Cooper pair

A simple quantum mechanics problem can most efficiently describe the physic involved in the BCS theory. Let V(r1-r2) is an attractive potential for the two electrons system interacting with each other. The Schrodinger equation for the two body problem is given as

By reducing the two body problem to one body problem with the help of the concept of reduced mass, the above equation can be solved to get the wave-function Ψ (r1, r2) and the energy E.

Regardless of the amount of attractive interaction, a bound state will be generated from the respective solutions of the wave function and energy, such a bound state is called a Cooper pair. In case of the free electron the attractive interaction has to overcome a threshold to create a bound state, thus the formation of cooper pair is fundamentally different from the formation of free electron. The existence of a well-defined Fermi surface is found to be the basic feature responsible for this different behaviour.

Josephson Effect

Josephson junction describes the contact between two superconductors, in which there is found to be a thin (< 2 nm) dielectric tunnel barrier between the two superconductors. The effect explains the tunneling of Cooper pairs through a barrier. The dc Josephson effects, relates the tunnel current to critical current  ?? through a Josephson junction as,

? = ?? sin ∅

The maximum Josephson tunnel current flowing through the barrier is called the critical current. The value of the tunnel current is measured by the density of Cooper pairs, tunnel barrier thickness, area of the tunnel junction and by the phase difference ∅.

The ac Josephson effect relates the voltage across a Junction to the temporal change of the phase difference thus a voltage across a Josephson junction leads to a current.

Microscopic Theory

BCS Theory of Superconductivity

BCS theory is a microscopic theory that should describe the phenomenology of superconducting materials based on first principles namely electron and crystal structure of the material and the Hamiltonian of the system. According to the investigations of this theory:

• The physical nature of the command parameters should be identified.
• Special non-classical features of the superconducting phase should be outlined.
• The fundamental understanding of superconductivity should not depend on the band-structure of the material since the superconductivity is observed in many metals. Also the theory demands the detailed form of electron-electron and electron-lattice-interaction.
• A temperature dependent description should make use of thermal Green’s functions.

BCS many-particle state

• A reduced Hamiltonian is required to find a many-particle state. The Hamiltonian is reduced under the considerations of forming the cooper-pairs and respects the Fermi character of the electron. During this process of forming the Hamiltonian it is taken into account that the structure of the solution is not affected much due to the missing parts of the Hamiltonian and the contributions in the normal and in the superconducting state is maintained to be same.
• The variational principle id used to get the correlations in the many-particle state. For the application of the principle we need the matrix elements of the reduced Hamiltonian and after some tedious algebra the desired wavefunction is achieved.

Applications of superconducting thin films

• Maglev (magnetic levitation) trains: Since the superconductors repel a magnetic field therefore practically get rid of the frictional effect between the train and the track. However, due to the strong magnetic fields the safety concerns are there which leads to a risk to human health.
• Particle accelerators/Large hadron colliders: Rutherford Appleton Laboratory in Oxfordshire, UK developed this use of superconductors in 1960s. By a coalition of scientific organisations from several countries, the biggest large hadron collider is built in Switzerland. In case of accelerators, to accelerate charged particles very fast (around the speed of light) extremely powerful electromagnets were developed using superconductors.
• Superconducting Quantum Interference Devices (SQUIDs): These devices are used to detect very weakest magnetic fields. In mine detection equipments for the removal of land mines, these devices are found to be helpful.
• E-bombs: USA is trying to develop these devices by making use of the strong magnetic fields derived by superconducting materials which creates fast and high-intensity electromagnetic pulse to disable electronic equipments of the enemies. In March 2003 wartime E-bombs were first used by USA forces to attack Iraqi broadcast facility. E-bombs can release a huge amount of energy around two billion watts at once.

Following uses of superconductors are under development:

• Efficient generation of the electricity.
• Prompt computing processes.

Features

• Bose condensation
• Off-diagonal long-range order
• Meissner effect and Flux Quantization
• Josephson effects
• Temperature Dependence
• Crystal structure,
• Fermi surface
• Electron-Phonon interaction

Nobel Prizes in the field of superconductivity

• H. Kamerlingh-Onnes in 1913: Investigations on “The properties of matter at low temperatures (production of liquid helium)”.
• Lev Davidovich Landau in 1962: “Theories for condensed matter (especially liquid helium)”.
• John Bardeen, Leon N. Cooper and J. Robert Schrieer in 1972: “Theory of superconductivity (BCS-theory)”.
• Ivar Giaever in 1973: Experimental discoveries concerning “Tunnelling phenomena in superconductors”.
• Brian D. Josephson in 1973: Theoretical calculations of “The properties of a supercurrent through a tunnel barrier (Josephson effects)”.
• Pyotr Leonidovich Kapitza in 1978: Inventions and discoveries in the area of “low-temperature physics”.
•  J Georg Bednorz, K. Alexander Muller in 1987: Breakthrough in the discovery of “superconductivity in ceramic materials”.
• David M. Lee, Douglas D. Oshero , Robert C. Richardson in 1996: Discovery of “superfluidity in helium-3”.

 REFERENCES

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