**6. Meissner effect**

When a superconductor placed in an magnetic field is cooled below its critical *Tc*, we find it expel all magnetic field from its inside. It does not like magnetic field in its interior. This is shown in **Figure 22**.

resistance to the electric current as long as the current is not too large. At a second critical field strength *Hc*2, no magnetic field expulsion takes place. How can we

*(A) Depiction on how packets shuttle back and forth in local potential and (B) how they execute a cyclotron*

*SQUID Magnetometers, Josephson Junctions, Confinement and BCS Theory of Superconductivity*

they do not shuttle. Instead, they do cyclotron motion with frequency *<sup>ω</sup>* <sup>¼</sup> *eB*

expulse the magnetic field. This explains the critical field.

which *<sup>p</sup>* were empty. We had a binding energy of � <sup>4</sup>ℏ*d*<sup>2</sup>

shown in **Figure 23**. At a field of about 1 Tesla, this is about 10<sup>11</sup> rad/s. Recall our packets had a width of *<sup>ω</sup><sup>D</sup>* � <sup>10</sup><sup>13</sup> Hz and the shuttling time of packets was 10�<sup>13</sup> s, so that the offsets in a packet do not evolve much in the time the packet is back. But, when we are doing cyclotron motion, it takes 10�<sup>11</sup> s (at 1 T field) to come back, and by that time, the offsets evolve a lot, which means poor binding. It means in the presence of magnetic field we cannot bind well. Therefore, physics wants to get rid of magnetic field, bind and lower the energy. Magnetic field hurts binding and therefore it is expelled. But if the magnetic field is increased, then the cyclotron frequency increases, and at a critical value, our packet returns home much faster, allowing for little offset evolution; therefore, we can bind and there is no need to

We talked about how wave packets shuttle back and forth in local potentials and get bound by phonons to form a BCS molecule. In the presence of a magnetic field,

When we bring two metals in proximity, separated by a thin-insulating barrier, apply a tiny voltage nd then the current will flow in the circuit. There is a thininsulating barrier, but electrons will tunnel through the barrier. Now, what will happen if these metals are replaced by a superconductor? These are a set of experiments carried out by Norwegian-American physicist Ivar Giaever who shared the Nobel Prize in Physics in 1973 with Leo Esaki and Brian Josephson "for their discoveries regarding tunnelling phenomena in solids". What he found was that if one of the metals is a superconductor, the electron cannot just come in, as there is an energy barrier of Δ, the superconducting gap. Your applied voltage has to be at least as big as Δ for tunneling to happen. This is depicted in **Figure 24**. Let us see

Recall in our discussion of superconducting state that we had 2*p* pockets of

*Np*<sup>2</sup>

*<sup>ω</sup><sup>d</sup>* . What will it cost to

*<sup>m</sup>*. This is

explain Meissner effect?

*motion in the presence of a magnetic field.*

*DOI: http://dx.doi.org/10.5772/intechopen.83714*

**Figure 23.**

**7. Giaever tunnelling**

why this is the case.

**89**

German physicists Walther Meissner and Robert Ochsenfeld discovered this phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples canceled nearly all interior magnetic fields. A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In type-I superconductor if the magnetic field is above certain threshold *Hc*, no expulsion takes place. In type-II superconductors, raising the applied field past a critical value *Hc*<sup>1</sup> leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no

**Figure 22.**

*Depiction of the Meissner effect whereby the magnetic field inside a superconductor is expulsed when we cool it below its superconducting temperature* Tc*.*

*SQUID Magnetometers, Josephson Junctions, Confinement and BCS Theory of Superconductivity DOI: http://dx.doi.org/10.5772/intechopen.83714*

**Figure 23.**

**Figure 21** depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors *S*<sup>1</sup> and *S*<sup>2</sup> are separated by thin insulators. A small flux through the SQUID creates a phase difference in the two superconductors (see discussion on Δ*k* above) leading to the flow of supercurrent. If an initial phase, *δ*<sup>0</sup> exists between the superconductors. Then, this phase difference

<sup>ℏ</sup> across bottom insulator (see **Figure 21**). This leads to currents *Ja* ¼ *I*<sup>0</sup> sin *δ<sup>a</sup>* and *Jb* ¼ *I*<sup>0</sup> sin *δ<sup>b</sup>* through top and down insulators. The total current

leads to a potential difference between the two superconductors. Therefore, the flux is converted to a voltage difference. The voltage oscillates as the phase difference *<sup>e</sup>*Φ<sup>0</sup>

goes in integral multiples of *π* for every flux quanta Φ0. SQUID is the most sensitive magnetic flux sensor currently known. The SQUID can be seen as a flux to voltage converter, and it can generally be used to sense any quantity that can be transduced into a magnetic flux, such as electrical current, voltage, position, etc. The extreme sensitivity of the SQUID is utilized in many different fields of applications, including biomagnetism, materials science, metrology, astronomy and geophysics.

When a superconductor placed in an magnetic field is cooled below its critical *Tc*, we find it expel all magnetic field from its inside. It does not like magnetic field

German physicists Walther Meissner and Robert Ochsenfeld discovered this

*Depiction of the Meissner effect whereby the magnetic field inside a superconductor is expulsed when we cool it*

phenomenon in 1933 by measuring the magnetic field distribution outside superconducting tin and lead samples. The samples, in the presence of an applied magnetic field, were cooled below their superconducting transition temperature, whereupon the samples canceled nearly all interior magnetic fields. A superconductor with little or no magnetic field within it is said to be in the Meissner state. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In type-I superconductor if the magnetic field is above certain threshold *Hc*, no expulsion takes place. In type-II superconductors, raising the applied field past a critical value *Hc*<sup>1</sup> leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no

<sup>ℏ</sup> . This accumulates charge on one side of SQUID and

<sup>ℏ</sup> across top insulator and

ℏ

after application of flux is from Eq. (42), *<sup>δ</sup><sup>a</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>e</sup>*Φ<sup>0</sup>

*Magnetometers - Fundamentals and Applications of Magnetism*

*<sup>δ</sup><sup>a</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>0</sup> � *<sup>e</sup>*Φ<sup>0</sup>

*<sup>J</sup>* <sup>¼</sup> *Ja* <sup>þ</sup> *Jb* <sup>¼</sup> <sup>2</sup>*I*<sup>0</sup> sin *<sup>δ</sup>*<sup>0</sup> cos *<sup>e</sup>*Φ<sup>0</sup>

**6. Meissner effect**

**Figure 22.**

**88**

*below its superconducting temperature* Tc*.*

in its interior. This is shown in **Figure 22**.

*(A) Depiction on how packets shuttle back and forth in local potential and (B) how they execute a cyclotron motion in the presence of a magnetic field.*

resistance to the electric current as long as the current is not too large. At a second critical field strength *Hc*2, no magnetic field expulsion takes place. How can we explain Meissner effect?

We talked about how wave packets shuttle back and forth in local potentials and get bound by phonons to form a BCS molecule. In the presence of a magnetic field, they do not shuttle. Instead, they do cyclotron motion with frequency *<sup>ω</sup>* <sup>¼</sup> *eB <sup>m</sup>*. This is shown in **Figure 23**. At a field of about 1 Tesla, this is about 10<sup>11</sup> rad/s. Recall our packets had a width of *<sup>ω</sup><sup>D</sup>* � <sup>10</sup><sup>13</sup> Hz and the shuttling time of packets was 10�<sup>13</sup> s, so that the offsets in a packet do not evolve much in the time the packet is back. But, when we are doing cyclotron motion, it takes 10�<sup>11</sup> s (at 1 T field) to come back, and by that time, the offsets evolve a lot, which means poor binding. It means in the presence of magnetic field we cannot bind well. Therefore, physics wants to get rid of magnetic field, bind and lower the energy. Magnetic field hurts binding and therefore it is expelled. But if the magnetic field is increased, then the cyclotron frequency increases, and at a critical value, our packet returns home much faster, allowing for little offset evolution; therefore, we can bind and there is no need to expulse the magnetic field. This explains the critical field.
