Theory of Superconductivity - Paper Example

When metal and alloys are cooled sufficiently to a low temperature, they tend to have their electrical resistivity drops suddenly to zero. Superconductivity phenomenon was first observed in 1911 in Leiden by Kemeligh Onnes; this was three years after he had liquefied helium. The specimen goes through a process of phase transition from its usual status of electrical resistivity to a superconducting state. When subjected to a temperature T.

Experiment survey

The de electrical resistivity reads zero when in superconductivity state, presenting electrical current that is observed to be flowing with no attention to superconducting rings. This was seen for more than a year until the experimentalist got tired of the experiment.

File and mills carried out a study on the decay of super current in the solenoid. They used precision nuclear magnetic resonance methods to analyze magnetic fields related to the super current. It was concluded that super current takes less than 100,000 years to decay. It is estimated that the decay time below finite decay time is observed in some Superconducting materials especially those used for superconducting elements due to an irreversible redistribution of magnetic flux in the article. Superconductivity shows dramatic electrical and magnetic properties; it is not possible to account for magnetic properties with the assumption that superconductor is had normal conductivity and zero resistivity.

Experiment have shows that when bulk conductors are subjected to the weak magnetic field, they will act as a perfects diamagnet that has no magnetic induction in its interior. Placing a specimen in a magnetic field then cooling it through transition temperature for superconductivity, the magnetic flux that was present is ejected from the sample in a process known as Meissner effect. Unique properties of superconductors play a crucial role in the characterization of the superconducting state.

Super conduction state is an ordered state of conduction electrons of metals. That order is the creation of loosely associate pairs of the particle. This electron tends to be ordered when subjected to a temperature that is below the transition temperature and disorders when subjected to a temperature above transition.

Occurrence of superconductivity

Bardee, cooper, and Schieffer were the first people to explain the nature and origin of order. Superconductivity takes place in several metallic, alloys and intermetallic elements of the periodic system and dopes semiconductors. The range of temperature changes is present from 90.0k for the compound YBa2Cu,0, to below 0.001 K for element Rb. Other material like Si which has a superconducting form at 165 bar, with T, = 8.3 K. tends to become superconscious when subjected to a high pressure. It is essential to remove trace quantities of foreign paramagnetic elements When conducting preliminary searches since they are capable of lowering the transition temperature significantly. For example, a part of Fe in 104 can destroy superconductivity of Mo, which when pure T, = 0.92 K; and one at. Percent of gadolinium lowers the transition temperature of lanthanum from 5.6 K to 0.6 K. However nonmagnetic do not have a different marked effect on the transition temperature.

Destruction of Superconductivity by Magnetic Fields

Superconductivity can be easily get destroyed by a sufficiently high magnetic field. The critical values at which the applied Magento fields can ruin superconductivity has been placed by HC(T), which is also the function of the temperature. When a critical temperature, critical temperature field is set at zero HC(TC)=0. The difference in the crucial filed with temperature for various superconducting element is as shown the fig 3

Meissner Effect

An experiment by Meissner and Ochshfied found out that when a conductor get cooled in a magnetic field below the transition temperature, this result to transition line of induction B pushed out. The effect of Meissner indicates that bulk superconductor act as b=0 inside the specimen. A useful form of the result is obtained of the specimen is limited to an extended thins specimen with long axes parallel to B2 demagnetizing field contribution to b will be negligible where;

According to the Ohm's law, E = PJ, it can be seen that of resistivity p gone to zero and J filed remain held finitely, then E should be zero. DBIdT is proportional to curl E according to Maxwell equations. Hence it implies that zero resistivity DBIdT=0, and not B=0. The discussion is not explicit; the results indicated that flux through metal cannot change on cooling through the transition. Meissner effect shows that perfect diamagnetism is a necessary property fo the superconducting state.

It is also expected that variance is existing between superconductors and perfect conductor. Describes as a conductor whose electrons have an infinite mean free path. Perfect conductors cannot produce permanent eddy current screen if it is placed in a magnetic field, the filed however goes in about 1 cm an hours. Pure specimens of various material that shows this behavior are known as types one superconductor, they were formally known as flexible super soft conductors. Values of H0 is usually shallow to the extent that Type 1 superconductors cannot have application in coils for superconducting magnets.

Materials that show magnetization curve are known as type ii superconductors. They are mostly alloys or transitions metal that contain high values of electrical resistivity in the normal state. Which is the electronic mean free path in the moral state that is short? Type ii superconductors include superconducting electrical properties that went up to a field denoted by Hrz. Between the lower critical field H, and the upper critical

field H, the flux density R # 0 and the Meissner effect is said to be incomplete.

Heat capacity

All superconductors have their entropy decrease significantly when cooled at critical temperature TC. The reduction in entropy between the normal state and superconducting indicate that superconducting state tends to be more orders that the average state which a measure of disorder system for the entropy. An electron that is thermally excited in the standard state orders in the superconducting state. The change in entropy is small, in aluminum of the order of lo-"

Kg per atom. The small entropy change must mean that only a tiny fraction (of

the law of the conductor electrons participates in the transition to

the ordered superconducting state.

B. Theoretical Survey

Several ways have been used to treat a conceptual understanding of the concept associated with superconductivity. Some results follow thermodynamics directly. Phenomenological equations come in handy to describe essential effects. Bardeen cooper and Schriefer gave a satisfactory quantum theory of superconductivity providing the perfect platform for subsequent work, on the other hand, Josephson and Anderson came up with criticism of the phase of the superconducting wave function.

Thermodynamics of superconductivity

The movement existing between the natural and superconducting state is thermodynamically reversible the same was the transition between liquid and vapor phases os substance can be reversed. Applying thermodynamics to the development give the expression for entropy difference between normal in and super conduction state in term of critical field curse HC against T, which is explained an analogous to the vapor pressure equation for liquid gas coexistence curve.

Type I superconductors are entirely treated with Meissner effect to have B-0 in the superconductors. Critical files he is, therefore, quantitative measure of free energy difference that is in between the superconducting and normal stated at a constant temperature. Hc refers to but specimen that is not thin file. Hc in the Type II superconductors is the critical thermodynamics files that are related to stabilization of free energy.

Stabilization of free energy of superconducting state concerning the normal state can be found through calorimetric measurement wherein this method heat capacity is measured as a function of temperature for superconductors and from ordinary conductors which is an indication that superconductors in magnetic field are larger than HC. It is, therefore, possible to compute free energy difference, which acts as stabilizing free energy of the superconductivity state. From the variance in the heat capacities.

London equation

As seen earlier, Meissner effect is an indication that magnetic susceptibility, X = - 1 / four ~ in the CGS in the superconducting state. Electrical conduction in the normal state of metal is explained by Ohms law j = US. Drastic modification of this helped to describe conduction and Meissner effect in the superconducting state. It is postulated that in super conduction state, current density becomes direct proportion to the vector potential A of the local Magnetic fields where B = curl A.. constant of proportionality is written as -c/4d; in CGS units.

Here c is the speed of light and A, is a constant with the dimensions of length.

In SI units we write -Up,+!. Thus C (CGS) j = --A ;

4TrA; This is the London equation. We express it another way of taking the curl of

both sides to obtain C (CGS) curl j = - -B ;

4 5 ~ ~ : - a - (SI) curl j = -- 1 B

London equation is The London equation (10) is understood to be written with the vector potential in the London gauge in which div A = 0, and A, = 0 on any external Surface through which no external current is fed. The subscript n denotes the Component normal to the surface. Thus div j = 0 and j, = 0, the actual physical Boundary conditions. The form (10) applies to a simply connected superconductor; additional terms may be present in a ring or cylinder, but (11)

holds true independent of geometry. First, we show that the London equation leads to the Meissner effect. By a

Maxwell equation we know that

45T (CGS) curl B = j ;

Under static conditions. We take the curl of both sides to obtain

(CGS) curl curl B = -V'B = curl j ;

air1 curl B = - V'B = k, curl j ;

BCS Theory of Superconductivity

The basis of the quantum theory of superconductivity was brought about by Bardeen, cooper and Schrieffer classic papers. BCS theory of superconductivity has a broad area of applicability. From Hee atoma in their condenses phase, to type 1 and type II mettallinc superconductors and to high temperature superconductors on the planes of corporate ions

When BSC wave functions consisting of particle pairs kl' and -k&, is treated by BSC theory, they tend to produce exectonic superconductivity that is sesn in the metal and shows energy gap. The pairing is knwns as s-wave pairing.

The BSC theory with wavefunction includes

Attaractive interaction between electorns results ot ground state that is separated from excited states by an energy gap. Consequences of energy gpa is presented by electromagnetic properties like critical fields, thermal properties. \

Electron lattice electron tends to produce energy gap, while indirect interaction continues to take place with one electron interaction with lattice and deforming it, the seconf electrn observed deformed lattice and sets itself to benefits formt eh deformation to lowere its energy. The second electron therefoe interacts with first electrin through the deformation of lattice

The extent of penetration comes apbout naturally as a result of the BSC theory. The lonndon equation therefore is accrued for magnetic fields that vary slowly in space. Therefore central phenomenon in superconductivity , messiner effect is obtained in a natural wasy

The The criterion for the transition temperatilre of an elcmcnt or alloy involvesthe electron density of orbitals D(eli) of one spin at the Fermi level and the electron-...