## Abstract

A superconducting quantum interference device (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. The heart of a squid magnetometer is a tunnel junction between two superconductors called a Josephson junction. Understanding the work of these devices rests fundamentally on the BCS theory of superconductivity. In this chapter, we introduce the notion of local potential and confinement in superconductivity. We show how BCS ground state is formed from interaction of wave packets confined to these local potential wells. The starting point of the BCS theory of superconductivity is a phonon-mediated second-order term that describes scattering of electron pair at Fermi surface with momentum k i , − k i and energy 2 ℏ ω i to k j , − k j with energy 2 ℏ ω j . The transition amplitude is M = − d 2 ω d ω i − ω j 2 − ω d 2 , where d is the phonon scattering rate and ω d is the Debye frequency. However, in the presence of offset ω i − ω j , there is also a present transition between states k i , − k i and k j , − k i of sizable amplitude much larger than M . How are we justified in neglecting this term and only retaining M ? In this chapter, we show all this is justified if we consider phonon-mediated transition between wave packets of finite width instead of electron waves. These wave packets are in their local potentials and interact with other wave packets in the same well to form a local BCS state we also call BCS molecule. Finally, we apply the formalism of superconductivity in finite size wave packets to high Tc in cuprates. The copper electrons in narrow d-band live as packets to minimize the repulsion energy. The phonon-mediated coupling between wave packets (of width Debye energy) is proportional to the number of k-states in a packet, which becomes large in narrow d-band (10 times s-band); hence, d-wave Tc is larger (10 times s-wave). At increased doping, packet size increases beyond the Debye energy, and phonon-mediated coupling develops a repulsive part, destroying superconductivity at large doping levels.

### Keywords

- local potentials
- superconductivity
- phonons
- Josephson junctions
- squids

## 1. Introduction

There is a very interesting phenomenon that takes place in solid-state physics when certain metals are cooled below critical temperature of order of few Kelvin. The resistance of these metals completely disappears and they become superconducting. How does this happen? One may guess that maybe at low temperatures there are no phonons. That is not true, as we have low frequency phonons present. Why do we then lose all resistivity? Electrons bind together to form a molecule by phonon-mediated interaction. The essence of this interaction is that electron can pull on the lattice which pulls on another electron. This phonon-mediated bond is not very strong for only few meV, but at low temperatures, this is good enough; we cannot break it with collisions with phonons which only carry

This phenomenon whereby many materials exhibit complete loss of electrical resistance when cooled below a characteristic critical temperature [1, 2] is called superconductivity. It was discovered in mercury by Dutch physicist Onnes in 1911. For decades, a fundamental understanding of this phenomenon eluded the many scientists who were working in the field. Then, in the 1950s and 1960s, a remarkably complete and satisfactory theoretical picture of the classic superconductors emerged in terms of the Bardeen-Cooper-Schrieffer (BCS) theory [3]. Before we talk about the BCS theory, let us introduce the notion of local potentials.

Shown in Figure 1 is a bar of metal. How are electrons in this metal bar? Solid-state physics texts start by putting these electrons in a periodic potential [4, 5, 6]. But that is not the complete story.

Shown in Figure 2 is a periodic array of metal ions. Periodic arrangement divides the region into cells (region bounded by dashed lines in Figure 2 ) such that the potential in the *i*th cell has the form

where *ai *is co-ordinate of the ion in the cell *i*. Eq. 1 has part *A*, the own ion potential. This gives the periodic part of the potential shown as thick curves in Figure 2 and part

Coming back to a more realistic estimate of the kinetic energy, electron wave function is confined to length *k* state with two electrons with

With this energy bandwidth, electrons are all well confined by the confining potential. In fact we do not need a potential of depth 50 eV; to confine the electrons, we can just do it with a depth of ∼10 eV which means a length of around *L* _{0} ∼ 300 Å. It means electrons over length *L* _{0} are confined, and due to *screening* by electrons outside *L* _{0}, they simply do not see any potential from ions outside this length. Thus, we get a local confining potential, and the picture is shown in Figure 5 , many local wells. This is what we call *local potentials or local volumes*. Estimate of *L* _{0} is a 1D calculation; in 3D it comes to a well of smaller diameter. However, if we account for *electron–electron repulsion*, then our earlier 1D estimate is probably okay. In any case, these numbers should be taken with a grain of salt. They are more qualitative than quantitative. The different wave packets in a local volume do not leave the volume as they see a local potential due to positive atomic ions. They just move back and forth in a local volume. When we apply an electric field say along *x* direction, the wave packets accelerate in that direction, and the local volume moves in that direction as a whole. This is electric current. Electrons are moving at very high velocity up to 10^{5} m/s (Fermi velocity) in their local volumes, but that motion is just a back-and-forth motion and does not constitute current. The current arises when the local volume moves as a whole due to applied electric field. This is much slower at say drift velocity of ^{2}.

In this chapter, we spell out the main ideas of the BCS theory. The BCS theory tells us how to use phonon-mediated interaction to bind electrons together, so that we have big molecule and we call the BCS ground state or the BCS molecule. At low temperatures, phonons do not have energy to break the bonds in the molecule; hence, electrons in the molecule do not scatter phonons. So, let us see how BCS binds these electrons into something big.

## 2. Cooper pairs and binding

Let us take two electrons, both at the Fermi surface, one with momentum *k* _{1} and other *k* _{1} pulls/plucks on the lattice due to Coulomb attraction and in the process creates (emits) a phonon and thereby recoils to new momentum *k* _{2}. The resulting lattice vibration is sensed by electron

We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation

This gives for

The first integral averages to zero, while the second integral

Evaluating it explicitly, we get for our system that second-order integral is

which couples levels 1 and 3 and drives transition between them at rate

Observe *k* _{1} can emit to *k* _{2} or *k* _{2} or

How does all this help. Suppose

has energy

## 3. Electron-phonon collisions

Recall we are interested in studying how a BCS molecule scatters phonons. For this we first understand how a normal electron scatters of a thermal phonon. We also derive electron–phonon interaction (*Fröhlich*) Hamiltonian and show how to calculate

Consider phonons in a crystalline solid. We first develop the concept of a phonon packet. To fix ideas, we start with the case of one-dimensional lattice potential. Consider a periodic potential with period *a*:

where

The potential is shown in Figure 7 .

Now, consider how potential changes when we perturb the lattice sites from their equilibrium position, due to lattice vibrations:

For a phonon mode with wavenumber *k*

we have

where *p*(*x*) is the periodic with period *a*. Note

Using Fourier series, we can write

We can determine *a* _{0} by

where

We do not worry much about *ar *for

where

Then, we get

At temperature of *T* = 3 K and

*a* = 3 Å, we have

with

We considered one phonon mode. Now, consider a phonon wave packet (which can also be thought of as a mode, localized in space) which takes the form

where *Ak *as in Eq. (12). Then, the resulting deformation potential from Eq. (13) by summing over all phonon modes that build a packet becomes

This deformation potential due to phonon wave packet is shown in Figure 8 . The maximum value of the potential is around

### 3.1 Time dynamics and collisions

Of course phonons have a time dynamics given by their dispersion relation:

With the phonon dispersion relation

The deformation potential travels with velocity of sound and collides with an incoming electron. To understand this collision, consider a phonon packet as in Eq. (14) centred at the origin. The packet is like a potential hill. A electron comes along say at velocity

In the above, we assumed phonon packet is stationary; however, it moves with velocity

In the phonon frame the electron travels towards it with velocity

and by conservation of energy, the phonon has lost energy, lowering its temperature.

In the second case, electron and phonon are traveling in the same direction. This is shown in Figure 11 . In the frame of phonon, electron travels towards the phonon with velocity

Thus, we have shown that electron and phonon can exchange energy due to collisions. Now, everything is true as in statistical mechanics, and we can go on to derive *Fermi-Dirac* distribution for the electrons [4, 5, 6, 7].

All our analysis has been in one dimension. In two or three dimensions, the phonon packets are phonon tides (as in ocean tides). Let us fix ideas with two dimensions, and three dimensions follow directly. Consider a two-dimensional periodic potential with period *a*:

Now, consider how potential changes when we perturb the lattice sites from their equilibrium position, due to lattice vibrations:

Let us consider phonons propagating along *x* direction. Then,

We have due to

where *a*.

Note

Using Fourier series, we can write

We can determine *a* _{0} by *x* direction:

which gives us a deformation potential as before:

which is same along *y* direction and travels with velocity *x* direction except now the potential is like a tide in an ocean, as shown in Figure 12 .

Since deformation potential is a tide, electron–phonon collisions do not have to be head on; they can happen at oblique angles, as shown in Figure 13 in a top view (looking down). The velocity of electron parallel to tide remains unchanged, while velocity perpendicular to tide gets reflected. If the perpendicular velocity is large enough, the electron can jump over the tide and continue as shown by a dotted line in Figure 13 . Imagining the tide in three dimensions is straightforward. In three dimensions, the deformation potential takes the form a wind gust moving in say *x* direction.

We described how a normal electron scatters phonons. Now, let us go back to our discussion on electron-phonon interaction and recall a phonon

where *n* ^{3} is the number of lattice points.

Using *nk *quanta in the phonon), where *M* is the mass of ion,

Thus, electron-phonon coupling Hamiltonian is of form

where

Using a cosine potential with *M* ∼ 20 proton masses, we have *c* ∼ 1 V.

We just derived an expression for the electron-phonon interaction (*Fröhlich*) Hamiltonian in Eq. (24) and showed how to calculate the constant *c*.

We said there are only two states,

it has energy

The states do not have to be exactly on a Fermi surface as shown in Figure 14A ; rather, they can be in an annulus around the Fermi sphere as shown in Figure 14B . When

But before we proceed, a note of caution is in order when we use the formula

We proceed into the interaction frame of the natural Hamiltonian (system energies) by transformation:

This gives for

We evaluate the effective evolution of *H*(*t*) in period

Let us calculate

Similarly

and finally

Then, from Eq. (27), we get

where

Observe in the above the term *offset averaging*, which we develop now.

We have been talking about electron waves in this section. Earlier, we spent considerable time showing how electrons are wave packets confined to local potentials. We now look for phonon-mediated interaction between wave packets. A wave packet is built from many k-states (k-points). These states have slightly different energies (frequencies) which make the packet moves. We call these different frequencies *offsets* from the centre frequency. Denote *k* _{0}. The key idea is that due to local potential, the wave packet shuttles back and forth and comes back to its original state. This means on average that the energy difference between its k-points averages and the whole packet just evolved with frequency *stationary* in the well, evolving as *N* k-points in a packet, the original scattering rate *p* packet pairs at the Fermi surface as shown in Figure 16A , then the state formed from superposition of packet pairs

has binding energy *Np* though is really just a

The wave packet in a potential well shuttles back and forth, which averages the offsets *s* orbitals or waves, the bandwidth is ∼10 eV giving Fermi velocity of 10^{5} m/s, which for a characteristic length of the potential well as ∼300 Å, corresponds to a packet shuttling time of around

We saw how two electrons bind to form a Cooper pair. However, for a big molecule, we need to bind many electrons. How this works will be discussed now. The basic idea is with many electrons; we need space for electron wave packets to scatter to. For example, when there was only one packet pair at the Fermi surface, it could scatter into all possible other packet pairs, and we saw how we could then form a superposition of these states. Now, suppose we have 2*p* packet pairs possible on the Fermi surface. We begin with assuming *p* of these are occupied with electrons and remaining *p* empty. This way we create space for the *p* pairs to scatter into; otherwise, if all are full, how will we scatter? How do these empty spaces come about? We just form packets with twice the bandwidth as there are electrons. Then, half of these packets are empty as shown in Figure 17 .

## 4. BCS ground state

Let **m** ^{2} points in the tangential direction. Figure 18 shows such a pocket enlarged with k-points shown in black dots. We assume there are 4*p* such pockets with *s*, we form the function

*a*. From *N* points in a pocket, we can form *N* such functions by displacing *N* wave packets orthogonal as they are nonoverlapping and placed uniformly over the lattice in their local potential wells. Furthermore,

As we will see soon,

As shown in last section, the Cooper pair *N* k-points, and hence by displacement of *N* nonoverlapping lattice sites (potential wells) where we can put copies of *N* k-points, only *p* possible Cooper pairs, only *p* are filled and *p* are not present. Hence, when *p* choices for

The binding energy of this state is *p* states. Since *c* is of order 1 eV and *n* ^{3} is the total number of lattice sites in the solid. Then, *N* k-points, half of which are outside the Fermi sphere. Thus, per electron wave packet, we have a binding energy of ∼.025 eV around 20 meV. Therefore, forming a superconducting state is only favorable if *c* is too small, then forming the superconducting state is not useful, as the gain of binding energy is offsetted by the price we pay in having wave packets that have excursion outside the Fermi surface.

Next, we study how low-frequency thermal phonons try to break the BCS molecule. The electron wave packet collides with the phonon and gets deflected, which means the Cooper pair gets broken. Then, the superconducting state constitutes *q* that a Cooper pair is broken, and we say we have excited a **Bogoliubov** [8]. This probability can be calculated using Boltzmann distribution. In a superconducting state with 2*p*, wave packets on average 2*pq* will be damaged/deflected. This leaves only

which gives

When

## 5. Molecular orbitals of BCS states

We now come to interaction of neighboring BCS molecules. In our picture of *local potentials*, we have electron wave packets in each potential well that are coupled as pairs by phonon to form the BCS ground state. What is important is that there is a BCS ground state in each potential well. When we bring two such wells in proximity, the ground state wave functions overlap, and we form a molecular orbital between these BCS orbitals. This is shown in Figure 19 .

If *U* is the potential well, with 2*e* coming from the electron pair charge. Then, the linear combination of atomic orbital (LCAO)

where *d*.

We talked about two BCS states separated by a thin insulator in a Josephson junction. In an actual superconductor, we have an array (lattice) of such localized BCS states as shown in Figure 20 . Different phases

If *U* is the potential well, with 2*e* coming from the electron pair charge. Then, the LCAO

where

What we have now is a new lattice of potential wells as shown in Figure 20 with spacing of *b* ∼ 300 Å as compared to the original lattice of *a* ∼ 3 Å, which means 100 times larger. We have an electron pair at each lattice site. The state

is the ground state of new lattice (Eq. 40). A state like

has a momentum and constitutes the supercurrent. It has energy

In the presence of electric field *E*, we get on the diagonal of RHS of Eq. 40, additional potential

Now, consider the local BCS states in Figure 20 put in a loop. If we turn on a magnetic field (say in time *T*) through the centre of the loop, it will establish a transient electric field in the loop given by

where *r* is radius and *ar *the area of the loop. Then, by the above argument, the wavenumber *k* of the BCS states is shifted by

giving

This is the magnetic quantum flux. When one deals with the superconducting loop or a hole in a bulk superconductor, it turns out that the magnetic flux threading such a hole/loop is quantized [9, 10] as just shown.

Figure 21 depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors *S* _{1} and *S* _{2} are separated by thin insulators. A small flux through the SQUID creates a phase difference in the two superconductors (see discussion on

## 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 .

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 *H* _{c1} 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 resistance to the electric current as long as the current is not too large. At a second critical field strength *H* _{c2}, 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 ^{11} rad/s. Recall our packets had a width of ^{−13} 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^{−11} 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.

## 7. Giaever tunnelling

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 thin-insulating 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

Recall in our discussion of superconducting state that we had 2*p* pockets of which *p* were empty. We had a binding energy of

## 8. High *Tc *in cuprates

High-temperature superconductors (abbreviated high-*Tc *or HTS) are materials that behave as superconductors at unusually high temperatures. The first high-*Tc *superconductor was discovered in 1986 by IBM researchers Georg Bednorz and K. Alex Müller, who were awarded the 1987 Nobel Prize in Physics “for their important break-through in the discovery of superconductivity in ceramic materials”.

Whereas “ordinary” or metallic superconductors usually have transition temperatures (temperatures below which they are superconductive) below 30 K (243.2°C) and must be cooled using liquid helium in order to achieve superconductivity, HTS have been observed with transition temperatures as high as 138 K (135°C) and can be cooled to superconductivity using liquid nitrogen. Compounds of copper and oxygen (so-called cuprates) are known to have HTS properties, and the term high-temperature superconductor was used interchangeably with cuprate superconductor. Examples are compounds such as lanthanum strontium copper oxide (LASCO) and neodymium cerium copper oxide (NSCO).

Let us take lanthanum copper oxide *Cu* is in state Cu^{2+} with electrons in *d* ^{9} configuration [11]. *d* orbitals are all degenerated, but due to crystal field splitting, this degeneracy is broken, and *k* states get filled by only one electron each and form Wannier packets that localize electrons on their respective sites. This way electron repulsion is minimized, and in the limit

When we hole/electron dope, we remove/add electron to *k* states as d-bands are narrow and only 1–2 eV thick. This means *N* (k-points in a packet) is very large and we have much larger gap *Tc *. This is a way to understand high *Tc *, d-wave packets with huge bandwidths. This is as shown in Figure 26 . As we increase doping and add more electrons, the packet width further increases till it is *dome* characteristic of superconducting phase, whereby superconductivity increases and then decreases with doping. The superconducting *dome* is shown in Figure 27 .