1. Superconductivity ertain metals and alloys exhibit almost zero resistivity (i.e. infinite conductivity), when they are cooled to sufficiently low temperatures. This effect is called superconductivity. This phenomenon was first of all discovered by H. K. Onnes in 1911 when measuring the electrical conductivity of metals at low temperatures. ρ 2 1 2 4 6 8 T (K) ritical or transition temperature Transition temperature (Or) The temperature at which the transition from normal state to superconducting state takes place on cooling in the absence of magnetic field is called critical temperature or transition temperature. 2. General properties of Superconductors:- Properties of superconductors:- 1. It is a low temperature phenomenon. 2. The transition temperature is different for different substances. 3. Materials having high normal resistivities exhibit superconductivity. 4. Materials for which superconductivity. 6 Z ρ =1 (where Z is a atomic number and ρ is resistivity) show 5. For chemically pure and structurally perfect specimen, the superconductivity is very sharp. 6. Ferro magnetic and Anti ferromagnetic materials are not superconductors. 7. Below the transition temperature the magnetic flux lines are rejected out of the superconductors. 8. Superconducting elements, in general, lie in the inner columns of the periodic table. 9. Those metallic elements having their valence electrons lies between 2 to 8 exhibit superconductivity. 1. Below the transition temperature the specific heat curve is discontinuous. Dr. P.Sreenivasula Reddy M.Sc, PhD Website: www.engineeringphysics.weebly.com Page 1
3. The Meissner effect When a weak magnetic is applied to a superconducting specimen at a temperature below transition temperature T the magnetic flux lines are expelled. This phenomenon is called Meissner effect. Under normal state the magnetic induction inside the specimen is B µ H + I = Where H is the external applied magnetic field and I is the magnetization produced inside the specimen. When the specimen is in superconducting state B = (Meissner effect) µ H + I Or = H = I I χ = = 1 H Thus the material is act as a perfectly diamagnetic (for diamagnetic material χ = 1 ). Let us consider a superconducting material is in normal state. From ohms law, the electric field E = Jρ On cooling the material to its transition temperature ρ tends to zero. If J is held finite E must be zero. From Maxwell s equations E = db dt db Under superconducting condition since E is zero = or B=constant. dt This means that the magnetic flux passing through the specimen should not change on cooling to the transition temperature. The Meissner effect contradicts the result. 4. Type I and type II superconductors. Or types of superconductors Based on the diamagnetic response superconductors can be classified into two types, they are 1. Type I superconductors 2. Type II superconductors.
Type I superconductors Superconductors which one follows a complete Meissner effect is called type I superconductors (also is known as soft superconductors). When the magnetic field strength is gradually increased from its initial value H < H, at the diamagnetism is abruptly disappear and the transition from H superconducting state to normal state is sharp as shown in figure. These superconductors are known as soft superconductors Examples: - Al, Zn, Hg and Sn M Super conducting state Normal state H Hc Type II superconductors:- Superconductors which does not follow the complete Meissner effect is called type I superconductors (also is known as hard superconductors). In type II superconductors, the specimen is in pure superconducting state up to the field (lower critical field) when the field is increased beyond (upper critical state) the magnetic flux lines start penetrating. The specimen is in mixed state between and. Above, the specimen is in normal state. This means that the Meissner effect is incomplete in the region between and. This region is known as vertex region. These superconductors are known as hard superconductors. Examples: - Zr, Nb M Super conductin g state Vortex region Normal state H 1 H 2 5. Differences between type I and Type II superconductor Type I superconductor Type II superconductor 1. It follows complete Meissner effect. 1. It does not follow the complete Meissner effect 2. It has single critical field value H 2. It has two critical field values and 3. There no mixed state. 3. There is a mixed state 4. They are soft superconductors 4. They are hard superconductors 5. Materials with pure form are type I 5. Materials with impurities or alloys are superconductors type II superconductors 6. Examples; Zn, Al, Hg and Sn 6. Examples: Zr, Nb Dr. P.Sreenivasula Reddy M.Sc, PhD Website: www.engineeringphysics.weebly.com Page 3
5. Penetration depth According to London s equations, the magnetic flux does not suddenly drop to zero at the surface of the type I superconductor, but decreases exponentially. The penetration of magnetic field through one face of the superconductor is shown in figure. According to Meissner effect the field inside the superconductor is zero, but in practice a small portion of field H o penetrates a small distance into the superconductor. The penetration of field at a distance x form the face is given by / Where λ =penetration depth When x= λ, then The penetration depth is the distance inside the superconductor at which the penetrating magnetic field is equal to 1/e times of the surface magnetic field H. Generally λ ranges from 1 to 1 nm. The variationof H w.r.t x is shown in figure. The penetration depth depends upon the temperature is given by the relation λ λ T = 1 4 2 T 1 4 T Where λ is the penetration depth at T = K 6. Josephson Effect Let us consider a thin insulation layer is sandwiched between the two superconductors in addition to normal tunneling of electrons, the super electrons tunnel through the insulation layer from one superconductor to another with dissociation, even at zero potential difference across the junction. Their wave functions on both sides are highly correlated. This is known as Josephson Effect.
D. Josephson effect According to Josephson when tunneling across through the insulator it introduces a phase difference φ between the two parts of the function on opposite sides of the junction as shown in figure The tunneling current is given by el I = I Sin φ Where I is the maximum current that flows through the junction without any potential difference across the junction. This effect is called D. Josephson effect. A. Josephson effect Let a static potential difference is applied across the junction, an additional phase is introduced by the cooper pairs during tunneling across the junction. This additional phase change ( φ ) at any time t can be calculated using quantum mechanics. Et φ = h Where E denotes the total energy of the system. In present case E = 2eV. Hence 2eV t φ = h The tunneling current can be written as
I = I Unit V Superconductivity Engineering Physics 2eVt Sin φ + h I = I Sin ( φ + ω t) 2eV Where ω = h This represents alternating current with angular frequencyω. This is A. Josephson effect. urrent voltage characteristic of a junction is shown in figure. 1. When Vo = there is a constant flow of dc current through the junction. This current is called superconducting current and the effect is called Josephson effect. 2. When V o < V c, a constant dc current I c flows. 3. When V o > V c, the junction has finite resistance, and the current oscillates with some frequency. Applications of Josephson Effect 2eV 1. Josephson effect is used to generate micro waves frequency with ω = h 2. A. Josephson effect is used to define standard volt. 3. A. Josephson effect is used to measure very low temperatures based on the variation of frequency of the emitted radiation with temperature. 4. A. Josephson effect is used for switching of signals from one circuit to another. 7. BS theory BS theory of superconductor was put forward by Bardeen, ooper and Schrieffer in 1957 and hence named as BS theory. This theory could explain the effects such as zero resistivity, Meissner effect, isotopic effect etc. Electron lattice interaction via lattice deformation. Let us consider an electron is passing through the lattice positive ions. The electron is attracted by the neighboring lattice positive ions as shown in figure 1. Due to the attraction of electron and ion core, the lattice gets deformed on scale. So electron get partially positive charge. Now if another electron passes by the side of assembly of said electron and ion core, it gets attracted towards the assembly.
The second electron interacts with the first electron due to the exchange of virtual photon q, between two electrons. The interaction process can be written in terms the wave vector k as ' ' k1 = k 1 q and k 2 = k 1 + q These two electrons together form a cooper pair and is known as cooper electron. ooper pairs To understand the mechanism of cooper pair formation, let us consider the distribution of electrons in metals as given by the Fermi-Dirac distribution function.. 1 F E = E E kt F 1+ e At T= K, all the Fermi energy states below the Fermi level are completely filled and all the states above are completely empty. Let us see what happens when two electrons are added to a metal at absolute zero. Since all the quantum states E < EF, are filled, they are forced to occupy states having E > EF. ooper showed that if there an attraction between the two electrons, they are able to form a bound state so that their total energy is less than 2 E F. These two electrons are paired to form a single system. These two electrons form a cooper pair and is known as cooper electron. 8. Flux quantization According to quantum mechanics matter, energy and charge is quantized. Similarly the magnetic flux passing the superconducting ring is also quantized. onsider a superconducting conducting ring in a magnetic field. If the temperature of the superconductor is greater than its critical temperature, the magnetic flux lines are passed through it as shown in figure (1).
When the super conducting ring temperature cooled less than of it s critical temperature, it obeys Meissner effect. As a result, persistent current comes into existence so that H= - M and all the magnetic flux lines are repelled by the superconductor as shown in figure (2). In this case we observe the flux is only inner of hollow sphere and outside of ring only. Even when the applied magnetic field is removed, some magnetic flux is inside the hollow ring as shown in figure. The flux inside the ring is given by 1,2,3,4, 2 Where h is Planck s constant and e is charge of electron. Thus the flux passing through the superconducting ring is equal to integral multiple of or quantized. ritical parameters of superconductivity Effect of magnetic field Superconductivity of a metal mainly depends on the temperature and strength of the magnetic field in which the metal is placed. Superconductivity disappears if the temperature of the specimen is raised abovet c or a strong enough magnetic field is applied. At temperatures belowt c, in the absence of magnetic field, the material is in superconducting state. When the strength of the magnetic field is applied to a critical value H c the superconductivity disappears. The dependence of critical field upon the temperature is given by 2 T H T = H 1 T
The variation of H w.r.t. T is shown in figure. Effect of current An electric current is passing through the superconducting material it self may gives rise to necessary magnetic field. For example, when the current is passing a superconducting ring, it gives rise to its own magnetic field. As the current increases to critical value I c, the associated magnetic field becomes H. And the superconductivity disappears. I = 2π rh Isotopic effect In superconducting materials the transition temperature varies with the average isotopic mass of their constituents. The variation is found to be in general form α T M α T M = constant Or Where α is the isotopic effect coefficient and is defined as lnt α = ln M The value of α is approximately.5. For example, the average mass varies from 199.5 to 23.4 atomic mass units and accordingly the transition temperature varies from 4.185K to 4.146K.