In this article we will discuss about:- 1. Introduction to Superconductors 2. Persistent Currents in Superconductors 3. Effect of Magnetic Fields 4. The Meissner Effect 5. Types 6. Josephson Effects 7. The London Equation 8. Squid.
Introduction to Superconductors:
The electrical resistivity of many metals and alloys drops suddenly to zero when their specimens are cooled to a sufficiently low temperature. This phenomenon was first observed by K. Ones in 1911 in Hg: the resistivity of Hg vanished completely below 4.2 k (Fig. 7.29), the transition from normal conductivity occurring over a very narrow range of temperature of the order of 0.05 K.
Similar results were obtained by using various other metals such as Pb, Sn, Nb, In etc. The phenomenon of disappearance of electrical resistance below a certain temperature was called superconductivity by Ones and superconductors are the specimens under this condition.
The discovery of superconductivity aroused considerable interest in this field since the materials with no electrical resistance and hence negligible heat losses could be exploited to fabricate powerful and economical devices which consume very little amount of electrical energy e.g. electromagnet made up of a superconducting material can function for years together even after removal of supply voltage.
ADVERTISEMENTS:
However, due to the requirement of very low temperature, it was not feasible to manufacture such devices. Thus, soon after the discovery of superconductivity, a lot of research work was undertaken to develop a superconducting material having as high critical (or transition) temperature as possible. A number of materials including various metals, alloys, intermetallic and ceramics were employed for this purpose.
The superconductivity is not a classical phenomenon like ferromagnetism, it is a state in which metallic electrons display macroscopic properties qualitatively different from those of an ideal Fermi gas, and it is an essential quantum mechanical consequence of interactions between electrons that are ignored in the independent particle model.
In fact, the superconducting state is a distinct phase of matter having characteristic electrical, magnetic, thermodynamic and other physical properties. The most easily observed characteristics of bulk superconductors are the zero electrical resistance and the perfect diamagnetism.
ADVERTISEMENTS:
The theoretical explanation of the phenomenon was given by Bardeen, Cooper and Schrieffer in 1957 and is known as the BCS theory. According to this theory an electron moving through an elastic crystal lattice creates a slight distortion of the lattice as a result of coulomb forces between the positively charged lattice and the negatively charged electrons. If this distortion persists for a finite time it can affect a second passing electron.
In 1956 Cooper showed that the effect of this phenomenon is for the current to be carried in superconductors not by individual electrons but by bound Paris of electrons called Cooper pairs. The BCS theory is based on a wave function in which all the electrons are paired. Because the total momentum of a Cooper pair is unchanged by the interaction between one of its electrons and the lattice, the flow of electrons continues indefinitely.
Persistent Currents in Superconductors:
If a superconductor has the form of ring, a current can be induced in it by electromagnetic induction; we have to simply cool the ring in a magnetic field from a temperature tc to below tc and then to remove the field. Now, it has been observed that this current continue to persist with undiminished strength for days. Such currents are called the persistent currents and are quite expected of superconductors.
The complete disappearance of resistivity may be demonstrated by making use of the properties of rings. Consider a ring exposed to an external magnetic field or flux density b which is changing as a function of time. The current I(t) flowing in the ring at time t is given by Lenz law which states that
where A is area of ring R is its resistance and L is inductance of the ring. If there is no external applied magnetic field then above equation becomes;
and which has the general solution;
I(t) = I(0) e-Rt/L
ADVERTISEMENTS:
Thus, the current in the ring decays exponentially in the absence of an external changing magnetic field and it eventually vanishes.
However, if the ring becomes superconductor, then R = 0 and I(t) = I(0). In other words, the current does not decay with time. This current is called persistent current and circulation of the current gives rise to a magnetic field. This property is very useful for high field magnets.
Effect of Magnetic Fields on Superconductors:
Application of a sufficiently strong magnetic field to superconductors causes the destruction of their superconductivity, i.e., the restoration of their normal conducting state. The critical value of the magnetic field for the destruction of superconductivity is denoted by Hc and is functionally related to temperature as
where Hc (0) is the critical field at 0 K, has a specific value for each material. Note that the lower the temperature, the higher the value of Hc and the highest critical temperature occurs when there is no magnetic field, i.e., T = Tc when Hc = 0. A typical plot of critical magnetic field versus temperature for lead is shown in Fig. 7.30. Thus we find that the superconducting state is stable only in some definite ranges of magnetic fields and temperature. For higher fields and temperatures, the normal state is more stable.
Some of the important properties of superconductors are as follows:
(a) The current in the superconductor’s persists for a very long time. This is demonstrated by placing a loop of the superconductor in a magnetic field, lowering its temperature below transition temperature Tc and then removing the field. The current which is setup is found to persist over a period longer than year without any attenuation.
(b) The magnetic field does not penetrate into the body of the superconductor. The property known as the Meissner effect is the fundamental characterization of superconductivity. However, when the magnetic field H is greater than a critical field Hc, the superconductor becomes a normal conductor.
(c) When a current through the superconductor is increased beyond critical value lc (T), the superconductor again becomes a normal conductor, i.e., the magnetic field which causes a superconductor become normal from a superconducting state is not necessarily an external magnetic field, it may arise as a result of electric current flow in the conductor, the superconductivity may be destroyed when the current exceeds the critical value Ic, which at the surface of the wire will produce a critical field Hc given by-
Ic = 2πrHc
This is known as Silsbee’s rule.
(d) The specific heat of the material shows an abrupt change at T = Tc jumping to a large value for T < Tc.
(e) In all cases involving transition metals, the variation of Tc with number of valence electrons shows sharp maxima for Z = 3, 5 and 7.
(f) A rather striking correlation exists between Tc and Z2 for elements Hg, La, Pb, Nb, Zn, Tl, In, Sn, V, Tc, Cd, Ga and Al.
(g) For a given value of Z, certain crystal structures seem more favourable than others, e.g.,β-tungsten and α-Mn structures are conductive to the phenomenon of superconductivity.
(h) Tc increases with a high power of the atomic volume and inversely as the atomic mass and is known is isotope effect.
(i) Superconductivity occurs in materials having high normal resistivities. The condition np > 106 is a good criterion for the existence of superconductivity, where n is number of valence electrons per c.c. and p is the resistivity in use at 20°C.
If one observes the total list of superconducting materials, the general features to be noted are:
(a) Monovalent metals are generally not superconductors.
(b) Ferromagnetic and antiferromagnetic metals are not superconductors.
(c) Good conductors at room temperature are not superconductors and superconducting metals are not good conductors at room temperature as the normal metals.
(d) Amorphous thin films of Be, Bi and Fe show superconductivity.
(e) Bi, Te and Sb become superconductor under high pressure.
The Meissner Effect on Superconductors:
Meissner and Ochsenfeld in 1933 found that if a long superconductor is cooled in a longitudinal magnetic field to below the value of critical temperature corresponding to that field, then the lines of induction B are pushed out of the body of the superconductor at the transition (Fig. 7.31).
This phenomenon is called the Meissner effect. Such flex exclusion is also observed if the superconductor is first cooled below Tc and then placed in the magnetic field. The effect is of fundamental importance as it shows that a bulk superconductor behaves in an external magnetic field H as if inside the specimen B = μ 0 (H + M) = 0 or X = -1; that is, a superconductor exhibits perfect diamagnetism.
The perfect diamagnetism of superconductors is an independent property, not at all related to zero resistivity. Let us try to relate the two properties together and see what happens. From Ohm’s Law E = rj, we see that if the resistivity r goes to zero while j is held infinite, then E must be zero. Using the Maxwell’s equation
∇ × E = -∂B/∂t
We obtain B = constant. This concludes that the flux through the specimen cannot changes on cooling through the transition. This means that when a perfect conductor (p = 0) is cooled in the magnetic field until its resistance becomes zero, the magnetic field in the material gets frozen in and cannot change subsequently irrespective of the applied field. This is obviously in contradiction with the Meissner effect. Thus, perfect diamagnetism and zero resistivity are the two independent essential properties of the superconducting state (Fig. 7.32).
Type I and Type II Superconductors:
Let us consider the magnetization of the superconductors as shown in Fig. 7.33. For one group of superconductors the magnetization curve is of the type I; that is, they are completely diamagnetic and hence the flux is completely excluded. The superconductors of this group are called type I superconductors, or soft superconductors because these superconductors give away their superconductivity at lower field strengths. They are usually pure specimens of some elements and the value of Hc for them is always too low to have any useful technical application in coils for superconducting magnets.
For the other group of superconductors the magnetization curve is of the type II. For this group, for applied fields below Hc1, the specimen is diamagnetic and hence the flux is completely excluded in this range of field; Hc1 is called the lower critical field. At Hc1 the flux begins to penetrate the specimen, and the penetration increases until Hc2 is reached.
At Hc2 the magnetization vanishes and the specimen becomes normal conductor; Hc2 is called the upper critical field. Moreover, the magnetization of this group of superconductors vanishes gradually as the field is increased, rather than suddenly as for the type I superconductors.
However, they are completely superconducting for all fields below Hc2. The superconductors of this group are called type II superconductors. They tend to be alloys or transition metals with high values of the electrical resistivity in the normal state. Type II superconductors are technically very useful materials, in contrast to type I superconductors.
Let us finally add that if the magnetization curves in Fig. 7.33 are reversible, the superconductor is an ideal one. Both the type I and type II superconductors can be ideal in this respect. Non-ideal superconductors are those which exhibit irreversible magnetization behaviour, i.e., hysteresis.
Type II superconductors with a large amount of magnetic hysteresis induced by mechanical treatment are called hard superconductors, i.e., these superconductors require large fields to bring them back to the normal state. Hence these materials can be used to manufacture superconducting wires which can be used to produce high magnetic field of the order of 10 T.
Intermediate State:
Let us consider the Meissner effect in a sphere of superconducting material. When the specimen becomes superconducting, the flux is concentrated at the sides of the specimen. This means that the field at the sides of the specimen is higher than at the top or bottom. Thus, if the field is increased, it will reach at value Hc, at the sides before the field at the top and bottom has this value.
Accordingly, at the sides of the specimen the initiation of the transition from superconducting state to normal state may set in before than that at the top and bottom. As a result, the specimen must exist as a complex mixture of normal and superconducting regions, called the intermediate state.
The intermediate state is considered completely equivalent to a mixture of the two states. If the field is increased further, the proportion of normal regions grows at the expense of the superconducting region until, when external field equals Hc all over the surface, the specimen is wholly normal and the transition is complete.
Energy Gap in Superconductors:
Experiments have shown that in superconductors, for temperatures in the neighborhood of absolute zero, a forbidden energy gap just above the Fermi level is observed. Fig. 7.34 (a) shows the conduction band in the normal state, while Fig. 7.34 (b) depicts an energy gap equal to 2Δ at the Fermi level in the superconducting state.
Thus, the Fermi level in a superconductor is midway between the ground state and the first excited state so that each lies on energy distance A away from the Fermi level. Electrons in excited state above the gap behave as normal electrons. At absolute zero, there are no electrons above the gap.
The energy gap in superconductors signified that the concentration of electrons is more in superconducting state than in normal metal, i.e., electrons are condensed below critical temperature Tc. It is observed that the energy gap decreases continuously to zero as the temperature is increased to Tc. This means A is found to be a function of temperature T. Thus, Δ (T) represents energy gap at temperature T and is given by
Experimentally, it is found that 2Δ(0)/KBTC = 3.5 for most of the metals (see Table 7.6). The variation of energy gap as a function of temperature is shown in Fig. 7.35.
Critical Currents:
The minimum current that can be passed in a sample without destroying its superconductivity is called critical current lc. If a wire of radius r and of a type-I superconductor carries a current I, there is a surface magnetic field H1 = I/2r associated with the current. If H1 exceeds Hc, the material will go normal. If in addition, a transverse magnetic field H is applied to a wire, the condition for the transition to the normal state at the surface is that the sum of the applied field and the field due to the current should equal the critical field.
The critical current Ic will decrease linearly with increase of the applied field until it reaches zero at H = Hc/2. If the applied field is zero, Ic = 2πrHc. Similar procedure may be applied to pure type II superconductors for H < Hc1 that is when the superconductor is not in the mixed state.
Josephson Effects:
Josephson observed some remarkable effects associated with the tunneling of superconducting electron pair through a very thin insulator (1~5 mm) sandwiched between two superconductors. Such an insulating layer forms a weak link between the superconductors which is referred to as the Josephson junction.
The effects of pair tunneling include:
(i) The DC Josephson Effect:
According to this effect, a dc current flows across the junction when no voltage is applied across it.
(ii) The AC Josephson Effect:
If a DC voltage is applied across the junction, r.f. current oscillations of frequency f = 2eV/h are set up across it, e.g. a DC voltage of 1μV produces a frequency of ~484 MHz. By measuring the frequency and the voltage, the value of e/h can be determined. Hence this effect has been utilized to measure e/h very precisely and may be used as a means of establishing a voltage standard Also an application of rf voltage along with the DC voltage can result in the flow of direct current through the junction.
(iii) Macroscopic Quantum Interference:
This effect describes the influence of the applied magnetic field on the super current flowing through the junction. According to this effect, if a DC magnetic field is applied through a superconducting circuit containing two junctions, the maximum super current shows interference effects which depend on the intensity of the magnetic field.
The London Equation:
In Meissner effect that there is complete expulsion of the magnetic flux out of the body of the superconductor at the transition. In fact, it is not exactly so, the magnetic flux decays from a constant value at the surface to zero value in the interior somewhat gradually rather than abruptly.
Flux penetration refers to the persistence afflux through a small volume along the surface of the superconductor. It may be noted that the distance of penetration below the surface is typically of the order of 500 Å. This indicates that the phenomenon is of importance only for thin films having thickness less than this value.
The above theory is based upon the model given by F. London and H. London. The two new equations have come out of their work which, besides explaining Meisnner effect, also explains the observations on thin films. These equations are known as London equations.
The London theory is based on rather old ideas of the two-fluid model, according to which a superconductor can be thought to be composed of both normal and superconducting electrons. Let nn, vn and ns, vs be respectively the density and velocity of normal and superconducting electrons. If n0 is the number of electrons per unit volume then on the average
n0 = nn + ns
In the following discussion we will consider only superconducting electrons and may be replaced by n.
The assumption of zero resistivity in superconductivity leads to the acceleration equation;
m (dv/dt) = -eE
Also the current density j, number of electrons per unit volume, is;
j = -nev
Therefore, we have;
This is known as first london equation.
It must be mentioned that here only the superconducting electrons ae under consideration and not all the electrons: a superconductor can be supposed as composed of both normal and superconducting electrons. The normal electrons behave like electrons in a non-conductor, and thus are of no interest to us. Further, the superconducting electrons are being assumed to respond to electric field just as free electrons do. Now taking the curl of both sides of Eqn. (1) we have, with;
This is the London equation. Sometimes this equation together with eqn. (I) are called the London equations.
London Penetration Depth:
Using Maxwell’s equation
The penetration depth is also found to depend strongly on temperature and to become much larger as T-approaches Tc. The observation can be fitted extremely well by a simple expression of the form
This equation implies that the number of superconducting electrons n varies in the following manner-
i.e., the density of superconducting electrons increases from zero at Tc to n0 at absolute zero which also depicts the temperature variation of penetration depth Δ.
Eqn. (XI) is a simple differential equation which has the solution;
B = B0 e-x/Λ … (XII)
where x is the distance from the surface measured into the specimens and B0 is the field at the surface. This means that the field does not drop to zero abruptly at the surface but runs into a certain region below the surface over which it gradually decays from the constant value B0 at the surface, i.e., with the finite penetration of fields.
The thickness of the penetrated region is measured by Δ. The Meissner effect is also accounted for as the specimens having dimensions much greater than Δ can be considered with good approximation as if they have Δ = 0, i.e., the area in which the filed has not penetrated at all.
Squid:
Squid is an acronym for superconducting quantum interference device and is an arrangement as shown in Fig. 7.36. All squids make use of the fact that the maximum current in a superconducting ring that contains a Josephson junction varies periodically as the magnetic flux through the ring changes. The periodicity is interpreted as an interference effect involving the wave functions of the cooper pairs.
It consists of a ring of superconducting material having two side arms A and B which act as an entrance and exit for the super current respectively. The insulating layers P and Q may have different thickness and let the currents through these layers be I1 and I2 respectively. The variation I1 and I2 versus the magnetic field is shown in Fig. 7.37.
Both I1 and I2 vary periodically with the magnetic field, the periodicity of I1, being greater than that of l2. The variation of I2 is an interference effect of the two junctions while that of I1 is a diffraction effect that arises from the finite dimension of each junction. Since the current is sensitive to very small changes in the magnetic field, the squid can be used as a very sensitive galvanometer.
The Josephson effects are the consequence of the fact that the superconductor is characterized by a single wave function. The flow of a super current takes place between any two points where the wave function has different phases. The change in phase can be brought about by the applied electric and magnetic fields. The states having different phases can be superimposed by using an arrangement in Fig. 7.36. Thus the Josephson effects exhibit the quantum interference phenomenon on a macroscopic scale.
The word quantum signifies that the entire superconductor is in a single quantum state and interference signifies that the measured properties depend on the phase of the state which can be changed by applying electric and magnetic fields and the states with different phases can produce interferences effects. The Squids find their application in medical diagnostics, under sea communications, submarine detection and geophysical prospecting. Magnetic fields changes as small as 10-21 T can be detected by Squids.