**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 phononmediated 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 *kBT* amount of energy which is small at low temperatures. Then, electrons do not travel alone; they travel in a bunch, as a big molecule; and you cannot scatter them with phonon collisions.

*V x*ð Þ¼ �*Ze* 4*π*ϵ<sup>0</sup>

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

potential has magnitude ln *n V*<sup>0</sup> at its ends and 2ln *<sup>n</sup>*

ture, the kinetic energy of the electrons in <sup>1</sup>

expand its eigenfunctions exp *i* <sup>2</sup>*π<sup>m</sup>*

state with two electrons with �*<sup>π</sup>*

bandwidth of 5–10 eV.

subtract). For *<sup>Z</sup>* <sup>¼</sup> 1 and *<sup>a</sup>* <sup>¼</sup> <sup>3</sup>*A*<sup>∘</sup>

deeper by � ln *<sup>n</sup>*

0 to <sup>ℏ</sup>2*π*<sup>2</sup>

**Figure 3.**

**69**

1 ∣*x* � *ai*∣ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} *A*

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

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 *B*, the other ion potential. This is shown as dashed curve (trough or basin) in **Figure 2**. We do not talk much about this potential in solid-state physics texts though it is prominent (for infinite lattice, part B is constant that we can

, we have *<sup>V</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*

*<sup>n</sup>* <sup>¼</sup> <sup>10</sup>9; then, we are talking about a trough that is �50 V deep. At room tempera-

Coming back to a more realistic estimate of the kinetic energy, electron wave function is confined to length *L* ¼ *na* (due to confining potential); then, we can

<sup>2</sup>*a*2*<sup>m</sup>* � 5 eV. This spread of kinetic energy is modified in the presence of periodic potential. We then have an energy band as shown in **Figure 4** with a

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*<sup>0</sup> � 300 Å. It means electrons over length *L*<sup>0</sup> 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*<sup>0</sup> 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

<sup>2</sup>*<sup>a</sup>* <sup>≤</sup> *<sup>k</sup>*<sup>≤</sup> *<sup>π</sup>*

*Depiction of Fermi gas of electrons in a metal moving in a confining potential.*

*na <sup>x</sup>* � � <sup>¼</sup> exp ð Þ *ikx* with energies <sup>ℏ</sup>2*k*<sup>2</sup>

quently, Fermi-Dirac statistics give much higher kinetic energies), the trough is deep enough to confine these electrons. It is this trough, basin or confining potential that we talk about in this chapter. Electrons move around in this potential as wave packets as shown in **Figure 3A**. Electrons can then be treated as a Fermi gas as shown in **Figure 3B**. While gas molecules in a container rebound of wall, the confining potential ensures electrons roll back before reaching the ends.

<sup>2</sup> *V*0. Let us say metal block is of length 30 cm with total sites

<sup>þ</sup> �*Ze* 4*π*ϵ<sup>0</sup> ∑ *j*6¼*i*

1 ∣*x* � *aj*∣

*,* (1)

<sup>4</sup>*π*ϵ0*<sup>a</sup>* � 3*V*. Then, the other ion

<sup>2</sup> *kBT* ¼ *:*02*eV* (as we will see subse-

<sup>2</sup>*<sup>a</sup>*, and kinetic energy of electrons goes from

<sup>2</sup> *V*<sup>0</sup> in the centre. The centre is

<sup>2</sup>*<sup>m</sup>* . We fill each *k*


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? Solidstate physics texts start by putting these electrons in a periodic potential [4–6]. 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

**Figure 2.** *Depiction of the potential due to metal ions. A rapidly varying periodic part* A *and slowly varying part* B*.*

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

$$V(\mathbf{x}) = \underbrace{\frac{-Ze}{4\pi\epsilon\_0}\frac{1}{|\mathbf{x} - a\_i|}}\_{A} + \underbrace{\frac{-Ze}{4\pi\epsilon\_0}\sum\_{j\neq i} \frac{1}{|\mathbf{x} - a\_j|}}\_{B},\tag{1}$$

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 *B*, the other ion potential. This is shown as dashed curve (trough or basin) in **Figure 2**. We do not talk much about this potential in solid-state physics texts though it is prominent (for infinite lattice, part B is constant that we can subtract). For *<sup>Z</sup>* <sup>¼</sup> 1 and *<sup>a</sup>* <sup>¼</sup> <sup>3</sup>*A*<sup>∘</sup> , we have *<sup>V</sup>*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>* <sup>4</sup>*π*ϵ0*<sup>a</sup>* � 3*V*. Then, the other ion potential has magnitude ln *n V*<sup>0</sup> at its ends and 2ln *<sup>n</sup>* <sup>2</sup> *V*<sup>0</sup> in the centre. The centre is deeper by � ln *<sup>n</sup>* <sup>2</sup> *V*0. Let us say metal block is of length 30 cm with total sites *<sup>n</sup>* <sup>¼</sup> <sup>10</sup>9; then, we are talking about a trough that is �50 V deep. At room temperature, the kinetic energy of the electrons in <sup>1</sup> <sup>2</sup> *kBT* ¼ *:*02*eV* (as we will see subsequently, Fermi-Dirac statistics give much higher kinetic energies), the trough is deep enough to confine these electrons. It is this trough, basin or confining potential that we talk about in this chapter. Electrons move around in this potential as wave packets as shown in **Figure 3A**. Electrons can then be treated as a Fermi gas as shown in **Figure 3B**. While gas molecules in a container rebound of wall, the confining potential ensures electrons roll back before reaching the ends.

Coming back to a more realistic estimate of the kinetic energy, electron wave function is confined to length *L* ¼ *na* (due to confining potential); then, we can expand its eigenfunctions exp *i* <sup>2</sup>*π<sup>m</sup> na <sup>x</sup>* � � <sup>¼</sup> exp ð Þ *ikx* with energies <sup>ℏ</sup>2*k*<sup>2</sup> <sup>2</sup>*<sup>m</sup>* . We fill each *k* state with two electrons with �*<sup>π</sup>* <sup>2</sup>*<sup>a</sup>* <sup>≤</sup> *<sup>k</sup>*<sup>≤</sup> *<sup>π</sup>* <sup>2</sup>*<sup>a</sup>*, and kinetic energy of electrons goes from 0 to <sup>ℏ</sup>2*π*<sup>2</sup> <sup>2</sup>*a*2*<sup>m</sup>* � 5 eV. This spread of kinetic energy is modified in the presence of periodic potential. We then have an energy band as shown in **Figure 4** with a bandwidth of 5–10 eV.

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*<sup>0</sup> � 300 Å. It means electrons over length *L*<sup>0</sup> 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*<sup>0</sup> 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

**Figure 3.** *Depiction of Fermi gas of electrons in a metal moving in a confining potential.*

**1. Introduction**

phonon collisions.

that is not the complete story.

**Figure 1.**

**Figure 2.**

**68**

*Depiction of a metallic bar.*

the potential in the *i*th cell has the form

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.

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 phononmediated 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 *kBT* amount of energy which is small at low temperatures. Then, electrons do not travel alone; they travel in a bunch, as a big molecule; and you cannot scatter them with

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? Solidstate physics texts start by putting these electrons in a periodic potential [4–6]. But

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

*Depiction of the potential due to metal ions. A rapidly varying periodic part* A *and slowly varying part* B*.*

The resistance of these metals completely disappears and they become

*Magnetometers - Fundamentals and Applications of Magnetism*

*Magnetometers - Fundamentals and Applications of Magnetism*

**Figure 4.** *The dispersion curve and energy band for electrons in a periodic potential.*

by electron �*k*<sup>1</sup> which absorbs this oscillation and is thrown back to momentum �*k*2. The total momentum is conserved in the process. This is depicted in **Figure 6A**. The corresponding Feynman diagram for this process is shown in **Figure 6B**. The above process where two electrons interact with exchange of phonon can be represented as a three-level atomic system. Level 1 is the initial state of the electrons *k*1*,* � *k*1, level 3 is the final state of the electrons *k*2*,* � *k*<sup>2</sup> and the level 2 is the intermediate state *k*2*,* � *k*1. There is a transition with strength Ω ¼ ℏ*d* between levels 1 and 2 involving emission of a phonon and a transition with

*(A) Depiction of the Fermi sphere and how electron pair k*1*,* � *k*<sup>1</sup> *at Fermi sphere scatters to k*2*,* � *k*<sup>2</sup> *at the Fermi sphere. (B) How this is mediated by exchange of a phonon in a Feynman diagram. (C) A three-level*

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

strength Ω between levels 2 and 3 involving absorption of a phonon. Let *E*1*, E*2*, E*<sup>3</sup> be the energy of the three levels. Since pairs are at Fermi surface, *E*<sup>1</sup> ¼ *E*3. The state of

We proceed into the interaction frame of the natural Hamiltonian (system

*E*<sup>1</sup> Ω 0 Ω<sup>∗</sup> *E*<sup>2</sup> Ω<sup>∗</sup> 0 Ω *E*<sup>1</sup>

> *E*<sup>1</sup> 0 0 0 *E*<sup>2</sup> 0 0 0 *E*<sup>1</sup>

3 7

> 3 7 5

<sup>Δ</sup>*E t* � �<sup>Ω</sup> <sup>0</sup>

<sup>Δ</sup>*E t* � �<sup>Ω</sup> <sup>0</sup>

1

ℏ

<sup>Δ</sup>*E t* � �Ω<sup>∗</sup>

<sup>5</sup>*ψ:* (2)

CA*<sup>ψ</sup>:* (3)

*ϕ:* (4)

the three-level system evolves according to the Schrödinger equation:

2 6 4

> *it* ℏ

0

B@

2 6 4

ℏ

ℏ


<sup>Δ</sup>*E t* � �Ω<sup>∗</sup> 0 exp *<sup>i</sup>*

*<sup>ψ</sup>*\_ <sup>¼</sup> �*<sup>i</sup>* ℏ

*system that captures the various transitions involved in this process.*

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

*ϕ* ¼ exp

0 exp � *<sup>i</sup>*

0 exp � *<sup>i</sup>*

energies) by transformation

This gives for Δ*E* ¼ *E*<sup>2</sup> � *E*<sup>1</sup>

*i* ℏ

exp

*<sup>ϕ</sup>*\_ <sup>¼</sup> �*<sup>i</sup>* ℏ

**71**

**Figure 6.**

**Figure 5.**

*Depiction of local potentials due to metal ions that locally confine electrons.*

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 105 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 10<sup>3</sup> m/s for an ampere current through a wire of cross section 1 mm2 .

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*<sup>1</sup> and other *k*1. Let us see how they interact with phonons. Electron *k*<sup>1</sup> 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

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

#### **Figure 6.**

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 105 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 10<sup>3</sup> m/s

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

Let us take two electrons, both at the Fermi surface, one with momentum *k*<sup>1</sup> and other *k*1. Let us see how they interact with phonons. Electron *k*<sup>1</sup> 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

.

for an ampere current through a wire of cross section 1 mm2

*The dispersion curve and energy band for electrons in a periodic potential.*

*Magnetometers - Fundamentals and Applications of Magnetism*

*Depiction of local potentials due to metal ions that locally confine electrons.*

binds these electrons into something big.

**2. Cooper pairs and binding**

**Figure 4.**

**Figure 5.**

**70**

*(A) Depiction of the Fermi sphere and how electron pair k*1*,* � *k*<sup>1</sup> *at Fermi sphere scatters to k*2*,* � *k*<sup>2</sup> *at the Fermi sphere. (B) How this is mediated by exchange of a phonon in a Feynman diagram. (C) A three-level system that captures the various transitions involved in this process.*

by electron �*k*<sup>1</sup> which absorbs this oscillation and is thrown back to momentum �*k*2. The total momentum is conserved in the process. This is depicted in **Figure 6A**. The corresponding Feynman diagram for this process is shown in **Figure 6B**. The above process where two electrons interact with exchange of phonon can be represented as a three-level atomic system. Level 1 is the initial state of the electrons *k*1*,* � *k*1, level 3 is the final state of the electrons *k*2*,* � *k*<sup>2</sup> and the level 2 is the intermediate state *k*2*,* � *k*1. There is a transition with strength Ω ¼ ℏ*d* between levels 1 and 2 involving emission of a phonon and a transition with strength Ω between levels 2 and 3 involving absorption of a phonon. Let *E*1*, E*2*, E*<sup>3</sup> be the energy of the three levels. Since pairs are at Fermi surface, *E*<sup>1</sup> ¼ *E*3. The state of the three-level system evolves according to the Schrödinger equation:

$$
\psi = \frac{-i}{\hbar} \begin{bmatrix} E\_1 & \Omega & 0 \\ \Omega^\* & E\_2 & \Omega^\* \\ 0 & \Omega & E\_1 \end{bmatrix} \psi. \tag{2}
$$

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

$$\phi = \exp\left(\frac{it}{\hbar} \begin{bmatrix} E\_1 & 0 & 0\\ 0 & E\_2 & 0\\ 0 & 0 & E\_1 \end{bmatrix}\right) \nu. \tag{3}$$

This gives for Δ*E* ¼ *E*<sup>2</sup> � *E*<sup>1</sup>

$$\dot{\boldsymbol{\phi}} = \frac{-i}{\hbar} \begin{bmatrix} 0 & \exp\left(-\frac{i}{\hbar} \Delta E \ t\right) \Omega & 0 \\\\ \exp\left(\frac{i}{\hbar} \Delta E \ t\right) \Omega^\* & 0 & \exp\left(\frac{i}{\hbar} \Delta E \ t\right) \Omega^\* \\\\ 0 & \exp\left(-\frac{i}{\hbar} \Delta E \ t\right) \Omega & 0 \end{bmatrix} \boldsymbol{\phi}. \tag{4}$$

*H t*ð Þ is periodic with period <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>* <sup>Δ</sup>*E*. After Δ*t*, the system evolution is

$$\phi(\Delta t) = \left( I + \int\_0^{\Delta t} H(\sigma) d\sigma + \int\_0^{\Delta t} \int\_0^{\sigma\_1} H(\sigma\_1) H(\sigma\_2) d\sigma\_2 d\sigma\_1 + \dots \right) \phi(0). \tag{5}$$

The first integral averages to zero, while the second integral

$$\int\_{0}^{\Delta t} \int\_{0}^{\sigma\_{1}} H(\sigma\_{1})H(\sigma\_{2})d\sigma\_{2}d\sigma\_{1} = \frac{1}{2} \int\_{0}^{\Delta t} \int\_{0}^{\sigma\_{1}} [H(\sigma\_{1}), H(\sigma\_{2})]d\sigma\_{2}d\sigma\_{1}.\tag{6}$$

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

$$\frac{-i\Delta t}{\hbar} \begin{bmatrix} \mathbf{0} & \frac{|\Omega|^2}{E\_1 - E\_2} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \frac{|\Omega|^2}{E\_1 - E\_2} & \mathbf{0} & \mathbf{0} \end{bmatrix},\tag{7}$$

*U x*ð Þ¼ ∑

The potential is shown in **Figure 7**.

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

their equilibrium position, due to lattice vibrations:

<sup>Δ</sup>*U x*ð Þ¼ *Ak* exp ð Þ *ikx* <sup>1</sup>

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

ð Þ ¼� *x V*<sup>0</sup>

*V*0

Using Fourier series, we can write

*Depiction of the periodic potential in Eq. (1).*

<sup>¼</sup> *Ak* exp ð Þ *ikx* <sup>1</sup>

For a phonon mode with wavenumber *k*

where

we have

**Figure 7.**

**73**

*n l*¼1

*V x*ð Þ¼ *<sup>V</sup>*<sup>0</sup> cos <sup>2</sup> *<sup>π</sup><sup>x</sup>*

Δ*U x*ð Þ¼ ∑*V*<sup>0</sup>

Δ*al* ¼ *Ak*

*V x*ð Þ¼ � *al* ∑*V x*ð Þ � *la ,*

2

ð Þ *x* � *al* Δ*al:*

<sup>2</sup> <sup>≤</sup>*x*<sup>≤</sup>

*a*

*<sup>n</sup>* <sup>p</sup> exp ð Þ *ikal ,* (11)

ð Þ *x* � *al* exp ð Þ �*ik x*ð Þ � *al* |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} *p x*ð Þ

> �*a* <sup>2</sup> <sup>≤</sup> *<sup>x</sup>* <sup>≤</sup>

2*πrx a* � �*:*

ð Þ *x* � *la* exp ð Þ �*ik x*ð Þ � *la* |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} *p x*ð Þ

> *a* 2

<sup>2</sup> (9)

*,*

*,*

*:* (10)

*a* � �*,* �*<sup>a</sup>*

<sup>¼</sup> <sup>0</sup> <sup>∣</sup>*x*∣ ≥ *<sup>a</sup>*

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

Now, consider how potential changes when we perturb the lattice sites from

1 ffiffiffi

ffiffiffi *<sup>n</sup>* <sup>p</sup> <sup>∑</sup>*V*<sup>0</sup>

> ffiffiffi *<sup>n</sup>* <sup>p</sup> <sup>∑</sup>*V*<sup>0</sup>

2*π a*

> 2 *:*

> > *r*

<sup>¼</sup> <sup>0</sup>∣*x*∣ ≥ *<sup>a</sup>*

*p x*ð Þ¼ *a*<sup>0</sup> þ ∑

sin <sup>2</sup>*π<sup>x</sup> a* � �*,*

*ar* exp *i*

which couples levels 1 and 3 and drives transition between them at rate <sup>M</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> *E*1�*E*<sup>2</sup> .

Observe *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> and *E*<sup>3</sup> ¼ 2ϵ<sup>2</sup> and the electron energies *E*<sup>2</sup> ¼ ϵ<sup>1</sup> þ ϵ<sup>2</sup> þ ϵ*d*, where ϵ*<sup>d</sup>* ¼ ℏ*ω<sup>d</sup>* is the energy of emitted phonon. We have ϵ<sup>1</sup> ¼ ϵ<sup>2</sup> ¼ ϵ. Then, the transition rate between levels 1 and 3 is <sup>M</sup> ¼ � <sup>Ω</sup><sup>2</sup> ϵ*d* . Therefore, due to interaction mediated through lattice by exchange of phonons, the electron pair *k*1*,* � *k*<sup>1</sup> scatters to *<sup>k</sup>*2*,* � *<sup>k</sup>*<sup>2</sup> at rate � <sup>Ω</sup><sup>2</sup> ϵ*d* . The scattering rate is in fact <sup>Δ</sup>*<sup>b</sup>* ¼ � <sup>4</sup>Ω<sup>2</sup> <sup>ϵ</sup>*<sup>d</sup>* as *k*<sup>1</sup> can emit to *k*<sup>2</sup> or �*k*2. Similarly, �*k*<sup>1</sup> can emit to *k*<sup>2</sup> or �*k*2, making it a total of four processes that can scatter *k*1*,* � *k*<sup>1</sup> to *k*2*,* � *k*2.

How does all this help. Suppose ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i are only two states around. Then, a state like

$$\phi = \frac{|k\_{1\circ} - k\_1\rangle + |k\_{2\circ} - k\_2\rangle}{\sqrt{2}}\tag{8}$$

has energy 2ϵ þ Δ*b*. That has lower energy than the individual states in the superposition. Δ*<sup>b</sup>* is the binding energy. Now, let us remember what is Ω. It comes from electron–phonon interaction. Let us pause, develop this a bit in the next section and come back to our discussion.

## **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 Ω in the above. The following section is a bit lengthy as it develops ways to visualize how a thermal phonon scatters an electron.

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*:

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

$$U(\mathbf{x}) = \sum\_{l=1}^{n} V(\mathbf{x} - a\_l) = \sum V(\mathbf{x} - l a),$$

where

*H t*ð Þ is periodic with period <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

ð<sup>Δ</sup>*<sup>t</sup>* 0

*H*ð Þ *σ dσ* þ

*Magnetometers - Fundamentals and Applications of Magnetism*

ð<sup>Δ</sup>*<sup>t</sup>* 0

The first integral averages to zero, while the second integral

*<sup>H</sup>*ð Þ *<sup>σ</sup>*<sup>1</sup> *<sup>H</sup>*ð Þ *<sup>σ</sup>*<sup>2</sup> *<sup>d</sup>σ*2*dσ*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

�*i*Δ*t* ℏ

transition rate between levels 1 and 3 is <sup>M</sup> ¼ � <sup>Ω</sup><sup>2</sup>

ϵ*d*

section and come back to our discussion.

visualize how a thermal phonon scatters an electron.

Consider a periodic potential with period *a*:

**3. Electron-phonon collisions**

ð*<sup>σ</sup>*<sup>1</sup> 0

� �

2 ð<sup>Δ</sup>*<sup>t</sup>* 0

<sup>0</sup> j j <sup>Ω</sup> <sup>2</sup>

000

0 0

ϵ*d*

*E*<sup>1</sup> � *E*<sup>2</sup>

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

j j <sup>Ω</sup> <sup>2</sup> *E*<sup>1</sup> � *E*<sup>2</sup> M

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

Observe *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> and *E*<sup>3</sup> ¼ 2ϵ<sup>2</sup> and the electron energies *E*<sup>2</sup> ¼ ϵ<sup>1</sup> þ ϵ<sup>2</sup> þ ϵ*d*, where ϵ*<sup>d</sup>* ¼ ℏ*ω<sup>d</sup>* is the energy of emitted phonon. We have ϵ<sup>1</sup> ¼ ϵ<sup>2</sup> ¼ ϵ. Then, the

mediated through lattice by exchange of phonons, the electron pair *k*1*,* � *k*<sup>1</sup> scatters

or �*k*2. Similarly, �*k*<sup>1</sup> can emit to *k*<sup>2</sup> or �*k*2, making it a total of four processes that

How does all this help. Suppose ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i are only two states

*<sup>ϕ</sup>* <sup>¼</sup> <sup>∣</sup>*k*1*,* � *<sup>k</sup>*1i þ <sup>∣</sup>*k*2*,* � *<sup>k</sup>*2<sup>i</sup> ffiffi 2

has energy 2ϵ þ Δ*b*. That has lower energy than the individual states in the superposition. Δ*<sup>b</sup>* is the binding energy. Now, let us remember what is Ω. It comes from electron–phonon interaction. Let us pause, develop this a bit in the next

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 Ω in the above. The following section is a bit lengthy as it develops ways to

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.

. The scattering rate is in fact <sup>Δ</sup>*<sup>b</sup>* ¼ � <sup>4</sup>Ω<sup>2</sup>

ð*<sup>σ</sup>*<sup>1</sup> 0

*ϕ*ð Þ¼ Δ*t I* þ

ð<sup>Δ</sup>*<sup>t</sup>* 0

<sup>M</sup> <sup>¼</sup> <sup>Ω</sup><sup>2</sup> *E*1�*E*<sup>2</sup> .

**72**

to *<sup>k</sup>*2*,* � *<sup>k</sup>*<sup>2</sup> at rate � <sup>Ω</sup><sup>2</sup>

can scatter *k*1*,* � *k*<sup>1</sup> to *k*2*,* � *k*2.

around. Then, a state like

ð*<sup>σ</sup>*<sup>1</sup> 0

<sup>Δ</sup>*E*. After Δ*t*, the system evolution is

*ϕ*ð Þ 0 *:* (5)

½ � *H*ð Þ *σ*<sup>1</sup> *; H*ð Þ *σ*<sup>2</sup> *dσ*2*dσ*1*:* (6)

*,* (7)

. Therefore, due to interaction

p (8)

<sup>ϵ</sup>*<sup>d</sup>* as *k*<sup>1</sup> can emit to *k*<sup>2</sup>

*H*ð Þ *σ*<sup>1</sup> *H*ð Þ *σ*<sup>2</sup> *dσ*2*dσ*<sup>1</sup> þ …

$$V(x) = V\_0 \cos^2\left(\frac{\pi x}{a}\right), \quad \frac{-a}{2} \le x \le \frac{a}{2} \tag{9}$$

$$= \mathbf{0} \quad |\mathbf{x}| \ge \frac{a}{2}. \tag{10}$$

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:

$$
\Delta U(\mathbf{x}) = \sum V'(\mathbf{x} - a\_l) \Delta a\_l \dots
$$

For a phonon mode with wavenumber *k*

$$
\Delta a\_l = A\_k \frac{1}{\sqrt{n}} \exp\left(ik a\_l\right),
\tag{11}
$$

we have

$$\begin{aligned} \Delta U(\mathbf{x}) &= A\_k \exp\left(ik\mathbf{x}\right) \frac{1}{\sqrt{n}} \underbrace{\sum V'(\mathbf{x} - a\_l) \exp\left(-ik(\mathbf{x} - a\_l)\right)}\_{p(\mathbf{x})}, \\ &= A\_k \exp\left(ik\mathbf{x}\right) \frac{1}{\sqrt{n}} \underbrace{\sum V'(\mathbf{x} - la) \exp\left(-ik(\mathbf{x} - la)\right)}\_{p(\mathbf{x})}, \end{aligned}$$

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

$$\begin{aligned} V'(\mathbf{x}) &= -V\_0 \frac{2\pi}{a} \sin\left(\frac{2\pi\mathbf{x}}{a}\right), \frac{-a}{2} \le \mathbf{x} \le \frac{a}{2}, \\ &= \mathbf{0}|\mathbf{x}| \ge \frac{a}{2}. \end{aligned}$$

Using Fourier series, we can write

$$p(\mathbf{x}) = a\_0 + \sum\_r a\_r \exp\left(i\frac{2\pi r \mathbf{x}}{a}\right).$$

**Figure 7.** *Depiction of the periodic potential in Eq. (1).*

We can determine *<sup>a</sup>*<sup>0</sup> by *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *a* Ð *a* 2 �*a* 2 *p x*ð Þ*dx*, giving

$$a\_0 = i \frac{V\_0}{a} \int\_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{2\pi}{a} \sin\left(\frac{2\pi\chi}{a}\right) \sin\left(k\chi\right),$$

where *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>m</sup> na* . This gives

$$a\_0 = i \frac{2V\_0}{a} \frac{1}{1 - \left(\frac{m}{n}\right)^2} \sin \frac{m\pi}{n}.$$

We do not worry much about *ar* for *r* 6¼ 0 as these excite an electron to a different band and are truncated by the band-gap energy. Now, note that, using equipartition of energy, there is *kBT* energy per phonon mode, giving

$$A\_k = \sqrt{\frac{k\_B T}{m}} \frac{\mathbf{1}}{\alpha\_k} = \sqrt{\frac{k\_B T}{m}} \frac{\mathbf{1}}{\alpha\_d \sin\left(\frac{\pi m}{n}\right)},\tag{12}$$

This deformation potential due to phonon wave packet is shown in **Figure 8**.

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

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

With the phonon dispersion relation *ω<sup>k</sup>* � *υk*, where *υ* is the velocity of sound,

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

climb the hill, it will go past the phonon as in (b) in **Figure 9**, or else it will slide back, rebound of the hill and go back at the same velocity *vg* as in (a) in **Figure 9**.

*Depiction on how an incoming electron goes past the deformation potential (b); if its velocity is sufficient, else it*

sin <sup>2</sup> *<sup>π</sup>*ð Þ *<sup>x</sup>*�*υ<sup>t</sup>* 2*a* � �

> *π*ð Þ *x*�*υt* 2*a*

2 <sup>p</sup> .

ð Þ *Ak* exp ð Þ *ikal* exp ð Þþ �*iωkt h:c :* (15)

� � *:* (16)

<sup>2</sup> *mv*<sup>2</sup> *<sup>g</sup>*<sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2 <sup>p</sup> ) to

The maximum value of the potential is around *<sup>V</sup>*<sup>0</sup>ffiffi

Δ*al*ðÞ¼ *t*

1 *<sup>n</sup>* <sup>∑</sup> *k*>0

Δ*U* � *V*<sup>0</sup>

along say at velocity *vg* . If the velocity is high enough (kinetic energy <sup>1</sup>

**3.1 Time dynamics and collisions**

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

we get

**Figure 8.**

**Figure 9.**

**75**

*slides back and rebounds as in (a).*

*Depiction of the deformation potential in Eq. (14).*

where *ω<sup>d</sup>* is the Debye frequency. Then, we get

$$
\Delta U(\mathbf{x}) = \frac{i}{\sqrt{n}} \underbrace{\left(\frac{2V\_0}{a} \sqrt{\frac{k\_B T}{m}} \frac{\mathbf{1}}{\omega\_d}\right)}\_{\nabla\_0} \exp\left(ikx\right). \tag{13}
$$

At temperature of *<sup>T</sup>* = 3 K and *<sup>ω</sup><sup>d</sup>* <sup>¼</sup> <sup>10</sup><sup>13</sup> rad/s, we have. ffiffiffiffiffiffi *kBT m* q <sup>1</sup> *<sup>ω</sup><sup>d</sup>* � *:*03 ̊A; with *a* = 3 Å, we have

$$
\Delta U \sim \frac{i}{\sqrt{n}} .01 V\_0 \exp\left(ik\alpha\right);
$$

and with *V*<sup>0</sup> ¼ 10*V*, we have

$$
\Delta U \sim \frac{\cdot \mathbb{1} i}{\sqrt{n}} \exp\left(ikx\right) V.
$$

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

$$
\Delta a\_l = \frac{1}{n} \sum\_k A\_k \exp\left(ika\_l\right),
$$

where *<sup>k</sup>* <sup>¼</sup> *<sup>m</sup>*<sup>Δ</sup> and <sup>Δ</sup> <sup>¼</sup> <sup>2</sup>*<sup>π</sup> na* and *Ak* as in Eq. (12). Then, the resulting deformation potential from Eq. (13) by summing over all phonon modes that build a packet becomes

$$
\Delta U \sim \overline{V}\_0 \frac{\sin^2 \left(\frac{\pi x}{2a}\right)}{\left(\frac{\pi x}{2a}\right)}.\tag{14}
$$

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

This deformation potential due to phonon wave packet is shown in **Figure 8**. The maximum value of the potential is around *<sup>V</sup>*<sup>0</sup>ffiffi 2 <sup>p</sup> .

### **3.1 Time dynamics and collisions**

We can determine *<sup>a</sup>*<sup>0</sup> by *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>

*na* . This gives

where *<sup>k</sup>* <sup>¼</sup> <sup>2</sup>*π<sup>m</sup>*

Then, we get

*kBT m* q <sup>1</sup>

becomes

**74**

*a*<sup>0</sup> ¼ *i*

*Magnetometers - Fundamentals and Applications of Magnetism*

*Ak* ¼

<sup>Δ</sup>*U x*ð Þ¼ *<sup>i</sup>*

*<sup>ω</sup><sup>d</sup>* � *:*03 ̊A; with *a* = 3 Å, we have

where *ω<sup>d</sup>* is the Debye frequency.

and with *V*<sup>0</sup> ¼ 10*V*, we have

where *<sup>k</sup>* <sup>¼</sup> *<sup>m</sup>*<sup>Δ</sup> and <sup>Δ</sup> <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

*a* Ð *a* 2 �*a* 2

> 2*π a*

2*V*<sup>0</sup> *a*

equipartition of energy, there is *kBT* energy per phonon mode, giving

*ωk* ¼

> 2*V*<sup>0</sup> *a*

ffiffiffiffiffiffiffiffi *kBT m* r 1

> ffiffiffi *n* p

At temperature of *<sup>T</sup>* = 3 K and *<sup>ω</sup><sup>d</sup>* <sup>¼</sup> <sup>10</sup><sup>13</sup> rad/s, we have. ffiffiffiffiffiffi

<sup>Δ</sup>*<sup>U</sup>* � *<sup>i</sup>*

ffiffiffi

<sup>Δ</sup>*<sup>U</sup>* � *:*1*<sup>i</sup>*

can also be thought of as a mode, localized in space) which takes the form

*n* ∑ *k*

potential from Eq. (13) by summing over all phonon modes that build a packet

Δ*U* � *V*<sup>0</sup>

<sup>Δ</sup>*al* <sup>¼</sup> <sup>1</sup>

ffiffiffi

*V*<sup>0</sup> *a* ð*a* 2 �*a* 2

*a*<sup>0</sup> ¼ *i*

*p x*ð Þ*dx*, giving

sin <sup>2</sup>*π<sup>x</sup> a* � �

1 <sup>1</sup> � *<sup>m</sup> n*

We do not worry much about *ar* for *r* 6¼ 0 as these excite an electron to a different band and are truncated by the band-gap energy. Now, note that, using

� �<sup>2</sup> sin *<sup>m</sup><sup>π</sup>*

ffiffiffiffiffiffiffiffi *kBT m* r 1

ffiffiffiffiffiffiffiffi *kBT m* r 1

!


*<sup>n</sup>* <sup>p</sup> *:*01*V*<sup>0</sup> exp ð Þ *ikx ;*

*<sup>n</sup>* <sup>p</sup> exp ð Þ *ikx <sup>V</sup>:*

*Ak* exp ð Þ *ikal ,*

sin <sup>2</sup> *<sup>π</sup><sup>x</sup>* 2*a* � � *πx* 2*a*

*na* and *Ak* as in Eq. (12). Then, the resulting deformation

� � *:* (14)

We considered one phonon mode. Now, consider a phonon wave packet (which

*ωd*

sin ð Þ *kx ,*

*n :*

*ω<sup>d</sup>* sin *<sup>π</sup><sup>m</sup> n*

� � *,* (12)

exp ð Þ *ikx :* (13)

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

$$
\Delta a\_l(t) = \frac{1}{n} \sum\_{k>0} \left( A\_k \exp\left( i k a\_l \right) \exp\left( -i a\_k t \right) + h.c \right). \tag{15}
$$

With the phonon dispersion relation *ω<sup>k</sup>* � *υk*, where *υ* is the velocity of sound, we get

$$
\Delta U \sim \overline{V}\_0 \frac{\sin^2 \left( \frac{\pi (\mathbf{x} - \imath t)}{2a} \right)}{\left( \frac{\pi (\mathbf{x} - \imath t)}{2a} \right)}.\tag{16}
$$

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 *vg* . If the velocity is high enough (kinetic energy <sup>1</sup> <sup>2</sup> *mv*<sup>2</sup> *<sup>g</sup>*<sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2 <sup>p</sup> ) to climb the hill, it will go past the phonon as in (b) in **Figure 9**, or else it will slide back, rebound of the hill and go back at the same velocity *vg* as in (a) in **Figure 9**.

**Figure 8.** *Depiction of the deformation potential in Eq. (14).*

#### **Figure 9.**

*Depiction on how an incoming electron goes past the deformation potential (b); if its velocity is sufficient, else it slides back and rebounds as in (a).*

In the above, we assumed phonon packet is stationary; however, it moves with velocity *υ*. Now, consider two scenarios. In the first one, the electron and phonon are moving in opposite direction and collide. This is shown in **Figure 10**.

In the phonon frame the electron travels towards it with velocity *vg* þ *υ*. If the velocity is high enough (kinetic energy <sup>1</sup> <sup>2</sup> *m vg* <sup>þ</sup> *<sup>υ</sup>* � �<sup>2</sup> <sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2 <sup>p</sup> ) to climb the hill, it will go past the phonon with velocity *vg* þ *υ* (the resulting electron velocity in lab frame is just *vg*). Otherwise, it slides back, rebounds and goes back with velocity *vg* þ *υ* (the velocity in lab frame is *vg* þ 2*υ*). Therefore, electron has gained energy:

$$
\Delta E = m\upsilon\_{\mathfrak{g}}\upsilon\tag{17}
$$

Now, consider how potential changes when we perturb the lattice sites from

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Let us consider phonons propagating along *x* direction. Then, Δ*al* constitutes longitudinal phonons, while Δ*am* constitutes transverse phonons. Transverse phonons do not contribute to deformation potential as can be seen in the following. Let

> 1 ffiffiffi

Δ*am* ¼ *Ak*

1 ffiffiffi

2*π a*

<sup>¼</sup> <sup>0</sup> <sup>∣</sup>*x*∣*,* <sup>∣</sup>*y*∣ ≥ *<sup>a</sup>*

*p x*ð Þ¼ *; y a*<sup>0</sup> þ ∑

which gives us a deformation potential as before:

sin <sup>2</sup>*π<sup>y</sup> a*

> 2 *:*

*r, s*

*a*2 Ð *a* 2 �*a* 2 Ð *a* 2 �*a* 2

<sup>Δ</sup>*al* <sup>¼</sup> <sup>1</sup>

Δ*U x*ð Þ� *; y V*<sup>0</sup>

except now the potential is like a tide in an ocean, as shown in **Figure 12**.

*n* ∑ *k*

� � cos *<sup>π</sup><sup>x</sup>*

*ars* exp *<sup>i</sup>* <sup>2</sup>*πrx*

*Ak* exp ð Þ *ikal ,*

sin <sup>2</sup> *<sup>π</sup>*ð Þ *<sup>x</sup>*�*υ<sup>t</sup>* 2*a* � �

> *π*ð Þ *x*�*υt* 2*a*

which is same along *y* direction and travels with velocity *υ* along the *x* direction

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

verse phonons do not contribute. The contribution of longitudinal phonons is same as in 1D case. As before consider a wave packet of longitudinal phonons propagating

*Vx x* � *al* ð Þ *; y* � *am* Δ*al* þ *Vy x* � *al* ð Þ *; y* � *am* Δ*am:*

*<sup>n</sup>* <sup>p</sup> exp ð Þ *ikxal :* (20)

*,*

*<sup>n</sup>* <sup>p</sup> <sup>∑</sup>*Vy <sup>x</sup>* � *al* ð Þ *; <sup>y</sup>* � *am* exp ð Þ �*ik x*ð Þ � *al* |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} *p x*ð Þ *;y*

> *a* � �<sup>2</sup>

*a* þ 2*πsy a* � � � � *:*

*,* �*<sup>a</sup>*

<sup>2</sup> <sup>≤</sup>*x, y*<sup>≤</sup>

*p x*ð Þ *; y dxdy*, giving *a*<sup>0</sup> ¼ 0. Hence, trans-

� � *,* (21)

*a* 2

their equilibrium position, due to lattice vibrations:

*lm*

us focus on the transverse phonons. Then,

Δ*U x*ð Þ¼ *; y Ak* exp ð Þ *ikxx*

where *p x*ð Þ *; y* is the periodic with period *a*.

*Vy*ð Þ¼� *x; y V*<sup>0</sup>

Using Fourier series, we can write

We can determine *<sup>a</sup>*<sup>0</sup> by *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>

Δ*U x*ð Þ¼ *; y* ∑

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

We have due to Δ*am*

Note

along *x* direction:

**77**

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 *vg* � *<sup>υ</sup>*. If the velocity is high enough (kinetic energy <sup>1</sup> <sup>2</sup> *m vg* � *<sup>υ</sup>* � �<sup>2</sup> <sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2 <sup>p</sup> ) to climb the hill, it will go past the phonon with velocity *vg* � *υ*. The velocity in lab frame is *vg* . Otherwise, it slides back, rebounds and goes back with velocity *vg* � *υ*. Then, the velocity in lab frame is *vg* � 2*υ*. Therefore, electron has lost energy, and by conservation of energy, the phonon has gained energy, raising its temperature.

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 [4–7].

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*:

$$U(\mathbf{x}, \mathbf{y}) = \sum\_{lm} V(\mathbf{x} - a\_l, \mathbf{y} - a\_m) = \sum\_{lm} V(\mathbf{x} - la, \mathbf{y} - ma). \tag{18}$$

$$\begin{split} V(\mathbf{x}, \mathbf{y}) &= V\_0 \cos^2 \left( \frac{\pi \mathbf{x}}{a} \right) \cos^2 \left( \frac{\pi \mathbf{y}}{a} \right), \quad \frac{-a}{2} \le \mathbf{x}, \mathbf{y} \le \frac{a}{2} \\ &= \mathbf{0} \quad |\mathbf{x}|, |\mathbf{y}| \ge \frac{a}{2}. \end{split} \tag{19}$$

#### **Figure 10.**

*Depiction on how an electron and phonon traveling towards each other collide.*

**Figure 11.** *Depiction on how an electron and phonon traveling in same direction collide.*

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

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

$$
\Delta U(\mathbf{x}, \mathbf{y}) = \sum\_{lm} V\_x(\mathbf{x} - a\_l, \mathbf{y} - a\_m) \Delta a\_l + V\_\mathbf{y}(\mathbf{x} - a\_l, \mathbf{y} - a\_m) \Delta a\_m.
$$

Let us consider phonons propagating along *x* direction. Then, Δ*al* constitutes longitudinal phonons, while Δ*am* constitutes transverse phonons. Transverse phonons do not contribute to deformation potential as can be seen in the following. Let us focus on the transverse phonons. Then,

$$
\Delta a\_m = A\_k \frac{1}{\sqrt{n}} \exp\left(ik\_x a\_l\right). \tag{20}
$$

We have due to Δ*am*

In the above, we assumed phonon packet is stationary; however, it moves with velocity *υ*. Now, consider two scenarios. In the first one, the electron and phonon

In the phonon frame the electron travels towards it with velocity *vg* þ *υ*. If the

past the phonon with velocity *vg* þ *υ* (the resulting electron velocity in lab frame is just *vg*). Otherwise, it slides back, rebounds and goes back with velocity *vg* þ *υ* (the

and by conservation of energy, the phonon has lost energy, lowering its tem-

to climb the hill, it will go past the phonon with velocity *vg* � *υ*. The velocity in lab frame is *vg* . Otherwise, it slides back, rebounds and goes back with velocity *vg* � *υ*. Then, the velocity in lab frame is *vg* � 2*υ*. Therefore, electron has lost energy, and by conservation of energy, the phonon has gained energy, raising its temperature. 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

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

*V x* � *al* ð *; y* � *am*Þ ¼ ∑

*a* � �

> 2 *:*

<sup>¼</sup> <sup>0</sup> <sup>∣</sup>*x*∣*,* <sup>∣</sup>*y*∣ ≥ *<sup>a</sup>*

*lm*

*,* �*<sup>a</sup>*

<sup>2</sup> <sup>≤</sup>*x, y*<sup>≤</sup>

cos <sup>2</sup> *<sup>π</sup><sup>y</sup> a* � �

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

<sup>2</sup> *m vg* <sup>þ</sup> *<sup>υ</sup>* � �<sup>2</sup>

<sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2

Δ*E* ¼ *mvgυ* (17)

<sup>p</sup> ) to climb the hill, it will go

<sup>2</sup> *m vg* � *<sup>υ</sup>* � �<sup>2</sup>

*V x*ð Þ � *la; y* � *ma :* (18)

*a* 2

<sup>&</sup>gt; *eV*<sup>0</sup>ffiffi 2 <sup>p</sup> )

(19)

are moving in opposite direction and collide. This is shown in **Figure 10**.

velocity in lab frame is *vg* þ 2*υ*). Therefore, electron has gained energy:

with velocity *vg* � *<sup>υ</sup>*. If the velocity is high enough (kinetic energy <sup>1</sup>

derive *Fermi-Dirac* distribution for the electrons [4–7].

*lm*

*V x*ð Þ¼ *; <sup>y</sup> <sup>V</sup>*<sup>0</sup> cos <sup>2</sup> *<sup>π</sup><sup>x</sup>*

*Depiction on how an electron and phonon traveling towards each other collide.*

*Depiction on how an electron and phonon traveling in same direction collide.*

periodic potential with period *a*:

*U x*ð Þ¼ *; y* ∑

velocity is high enough (kinetic energy <sup>1</sup>

*Magnetometers - Fundamentals and Applications of Magnetism*

perature.

**Figure 10.**

**Figure 11.**

**76**

$$
\Delta U(\mathbf{x}, \mathbf{y}) = A\_k \exp\left(ik\_\mathbf{x}\mathbf{x}\right) \frac{\mathbf{1}}{\sqrt{n}} \underbrace{\sum\_{\mathbf{y}} V\_\mathbf{y} \left(\mathbf{x} - a\_l, \mathbf{y} - a\_m\right) \exp\left(-ik(\mathbf{x} - a\_l)\right)}\_{p(\mathbf{x}, \mathbf{y})},
$$

where *p x*ð Þ *; y* is the periodic with period *a*. Note

$$V\_{\mathcal{V}}(\mathbf{x}, \mathbf{y}) = -V\_0 \frac{2\pi}{a} \quad \sin\left(\frac{2\pi y}{a}\right) \cos\left(\frac{\pi \mathbf{x}}{a}\right)^2, \ \frac{-a}{2} \le \mathbf{x}, \mathbf{y} \le \frac{a}{2}$$

$$= \mathbf{0} \quad |\mathbf{x}| , |\mathbf{y}| \ge \frac{a}{2}.$$

Using Fourier series, we can write

$$p(\mathbf{x}, \mathbf{y}) = a\_0 + \sum\_{r\_2s} a\_{rs} \exp\left(i\left(\frac{2\pi r \mathbf{x}}{a} + \frac{2\pi s \mathbf{y}}{a}\right)\right).$$

We can determine *<sup>a</sup>*<sup>0</sup> by *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *a*2 Ð *a* 2 �*a* 2 Ð *a* 2 �*a* 2 *p x*ð Þ *; y dxdy*, giving *a*<sup>0</sup> ¼ 0. Hence, transverse phonons do not contribute. The contribution of longitudinal phonons is same as in 1D case. As before consider a wave packet of longitudinal phonons propagating along *x* direction:

$$
\Delta a\_l = \frac{1}{n} \sum\_k A\_k \exp\left(ika\_l\right),
$$

which gives us a deformation potential as before:

$$
\Delta U(\mathbf{x}, \boldsymbol{y}) \sim \overline{V}\_0 \frac{\sin^2 \left(\frac{\pi(\mathbf{x} - \boldsymbol{w})}{2a}\right)}{\left(\frac{\pi(\mathbf{x} - \boldsymbol{w}t)}{2a}\right)},\tag{21}
$$

which is same along *y* direction and travels with velocity *υ* along the *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

*c* ffiffiffiffiffi *<sup>n</sup>*<sup>3</sup> <sup>p</sup> |ffl{zffl} Ω

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

we form the state

annulus N to be <sup>N</sup>

**Figure 14.**

**79**

it has energy 2ϵ þ ð Þ N � 1 Δ*b*.

shown in detail below) is modified to <sup>Δ</sup>*<sup>b</sup>* <sup>¼</sup> <sup>4</sup>ϵ*d*Ω<sup>2</sup>

*<sup>n</sup>*<sup>3</sup> � *<sup>ω</sup><sup>d</sup> ωF*

long as <sup>Δ</sup>*<sup>ω</sup>* <sup>&</sup>lt;*ωd*, in BCS theory, we approximate <sup>Δ</sup>*<sup>b</sup>* � � <sup>4</sup>Ω<sup>2</sup>

where *b, b*† are the annihilation and creation operators for phonon.

tial. Then, with *<sup>a</sup>* � <sup>3</sup>*A*<sup>∘</sup> and *<sup>M</sup>* � 20 proton masses, we have *<sup>c</sup>* � 1 V.

Hamiltonian in Eq. (24) and showed how to calculate the constant *c*.

*<sup>ϕ</sup>* <sup>¼</sup> <sup>1</sup> ffiffiffiffi *N* p ∑ *i*

Using a cosine potential with *V*<sup>0</sup> � 10 V, we can approximate Coulomb poten-

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

We just derived an expression for the electron-phonon interaction (*Fröhlich*)

We said there are only two states, ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i. In general we have for *i* ¼ 1*,* …*,* N , ∣*ki,* � *ki*i states on the Fermi sphere as shown in **Figure 14A**, and if

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 ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i are not both on the Fermi surface (rather in an annulus) such that the energy of ∣*k*1*,* � *k*1i is *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> and the energy of

> Δϵ2�ϵ<sup>2</sup> *d*

, where *n*<sup>3</sup> is the total number of *k* points in the Fermi sphere

*E*<sup>3</sup> ¼ ∣*k*2*,* � *k*2i is 2ϵ2, with *ε*<sup>1</sup> 6¼ *ε*2, then the formula of scattering amplitude (as

annulus in **Figure 14B** to be of width *ωd*, we get a total number of states in the

Fermi energy ϵ*<sup>F</sup>* � 10 eV, the binding energy is �meV. Thus, we have shown how phonon-mediated interaction helps us bind an electron pair with energy �meV.

*(A) Electron pairs on the fermi surface and (B) electron pairs in an annulus around the fermi surface.*

and <sup>ϵ</sup>*<sup>F</sup>* <sup>¼</sup> <sup>ℏ</sup>*ω<sup>F</sup>* is the Fermi energy. This gives a binding energy <sup>Δ</sup>*<sup>b</sup>* ¼� *<sup>c</sup>*<sup>2</sup>

*i b* exp ð Þ� *ikx <sup>b</sup>*† exp ð Þ �*ikx* � �*,* (24)

∣*ki,* � *ki*i*,* (25)

, where ϵ<sup>1</sup> � ϵ<sup>2</sup> ¼ *Δϵ* ¼ ℏ*Δω*. As

<sup>ϵ</sup>*<sup>d</sup>* . Therefore, if we take an

ϵ*F*

. With the

**Figure 12.**

*Depiction of the deformation potential tide as shown in Eq. (21).*

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 exp ð Þ *ikx* which produces a deformation potential Δ*U x*ð Þ as in Eq. 13. Now in three dimensions, it is

$$
\Delta U(\infty) = \frac{i}{\sqrt{n^3}} \frac{V\_0}{a} A\_k(\exp\left(ik\infty\right) \exp\left(-i\alpha\_k t\right) - h.c), \tag{22}
$$

where *n*<sup>3</sup> is the number of lattice points.

Using <sup>1</sup> <sup>2</sup> *Mω*<sup>2</sup> *kA*<sup>2</sup> *<sup>k</sup>* ¼ *nk*ℏ*ω<sup>k</sup>* (there are *nk* quanta in the phonon), where *M* is the mass of ion, *ω<sup>k</sup>* phonon frequency and replacing *Ak*, we get

$$
\Delta U(\mathbf{x}) = \frac{i}{\sqrt{n^3}} \underbrace{\frac{V\_0}{a}}\_{c} \sqrt{\frac{2\hbar}{\text{Ma}\nu\_k}} (\sqrt{n\_k} \exp\left(ikx\right) \exp\left(-i\nu\_k t\right) - h.c.) \tag{23}
$$

Thus, electron-phonon coupling Hamiltonian is of form

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

$$\underbrace{\frac{c}{\sqrt{n^3}}}\_{\Omega} i \big( b \exp\left(ik\alpha\right) - b^\dagger \exp\left(-ik\alpha\right) \big), \tag{24}$$

where *b, b*† are the annihilation and creation operators for phonon.

Using a cosine potential with *V*<sup>0</sup> � 10 V, we can approximate Coulomb potential. Then, with *<sup>a</sup>* � <sup>3</sup>*A*<sup>∘</sup> and *<sup>M</sup>* � 20 proton masses, we have *<sup>c</sup>* � 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, ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i. In general we have for *i* ¼ 1*,* …*,* N , ∣*ki,* � *ki*i states on the Fermi sphere as shown in **Figure 14A**, and if we form the state

$$
\phi = \frac{1}{\sqrt{N}} \sum\_{i} |k\_i \ -k\_i\rangle,\tag{25}
$$

it has energy 2ϵ þ ð Þ N � 1 Δ*b*.

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 ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i are not both on the Fermi surface (rather in an annulus) such that the energy of ∣*k*1*,* � *k*1i is *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> and the energy of *E*<sup>3</sup> ¼ ∣*k*2*,* � *k*2i is 2ϵ2, with *ε*<sup>1</sup> 6¼ *ε*2, then the formula of scattering amplitude (as shown in detail below) is modified to <sup>Δ</sup>*<sup>b</sup>* <sup>¼</sup> <sup>4</sup>ϵ*d*Ω<sup>2</sup> Δϵ2�ϵ<sup>2</sup> *d* , where ϵ<sup>1</sup> � ϵ<sup>2</sup> ¼ *Δϵ* ¼ ℏ*Δω*. As long as <sup>Δ</sup>*<sup>ω</sup>* <sup>&</sup>lt;*ωd*, in BCS theory, we approximate <sup>Δ</sup>*<sup>b</sup>* � � <sup>4</sup>Ω<sup>2</sup> <sup>ϵ</sup>*<sup>d</sup>* . Therefore, if we take an annulus in **Figure 14B** to be of width *ωd*, we get a total number of states in the annulus N to be <sup>N</sup> *<sup>n</sup>*<sup>3</sup> � *<sup>ω</sup><sup>d</sup> ωF* , where *n*<sup>3</sup> is the total number of *k* points in the Fermi sphere and <sup>ϵ</sup>*<sup>F</sup>* <sup>¼</sup> <sup>ℏ</sup>*ω<sup>F</sup>* is the Fermi energy. This gives a binding energy <sup>Δ</sup>*<sup>b</sup>* ¼� *<sup>c</sup>*<sup>2</sup> ϵ*F* . With the Fermi energy ϵ*<sup>F</sup>* � 10 eV, the binding energy is �meV. Thus, we have shown how phonon-mediated interaction helps us bind an electron pair with energy �meV.

**Figure 14.** *(A) Electron pairs on the fermi surface and (B) electron pairs in an annulus around the fermi surface.*

dimensions, the deformation potential takes the form a wind gust moving in say *x*

We described how a normal electron scatters phonons. Now, let us go back to our discussion on electron-phonon interaction and recall a phonon exp ð Þ *ikx* which produces a deformation potential Δ*U x*ð Þ as in Eq. 13. Now in three dimensions, it is

*<sup>k</sup>* ¼ *nk*ℏ*ω<sup>k</sup>* (there are *nk* quanta in the phonon), where *M* is the

*Ak*ð Þ exp ð Þ *ikx* exp ð Þ� �*iωkt h:c ,* (22)

ð Þ exp ð Þ *ikx* exp ð Þ� �*iωkt h:c :* (23)

direction.

**Figure 13.**

**Figure 12.**

Using <sup>1</sup>

**78**

<sup>2</sup> *Mω*<sup>2</sup> *kA*<sup>2</sup>

<sup>Δ</sup>*U x*ð Þ¼ *<sup>i</sup>*

<sup>Δ</sup>*U x*ð Þ¼ *<sup>i</sup>*

*Depiction of the deformation potential tide as shown in Eq. (21).*

*Magnetometers - Fundamentals and Applications of Magnetism*

where *n*<sup>3</sup> is the number of lattice points.

ffiffiffiffiffi *<sup>n</sup>*<sup>3</sup> <sup>p</sup> *V*<sup>0</sup> *a*

ffiffiffiffiffi *<sup>n</sup>*<sup>3</sup> <sup>p</sup>

mass of ion, *ω<sup>k</sup>* phonon frequency and replacing *Ak*, we get

s


ffiffiffiffiffiffiffiffiffiffi 2ℏ *Mω<sup>k</sup>*

Thus, electron-phonon coupling Hamiltonian is of form

ffiffiffiffiffi *nk* p

*V*<sup>0</sup> *a*

*The top view collision of an electron with a deformation potential tide at an angle θ.*

This paired electron state is called Cooper pair. Now, the plan is we bind many electrons and make a big molecule called BCS ground state.

But before we proceed, a note of caution is in order when we use the formula <sup>Δ</sup>*<sup>b</sup>* <sup>¼</sup> <sup>4</sup>ϵ*d*Ω<sup>2</sup> Δϵ2�ϵ<sup>2</sup> *d* . For this we return to phonon scattering of ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i. Consider when *E*<sup>1</sup> 6¼ *E*3. Observe *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> ¼ 2ℏ*ω*1, *E*<sup>3</sup> ¼ 2ϵ<sup>2</sup> ¼ 2ℏ*ω*2, *E*<sup>2</sup> ¼ ϵ<sup>1</sup> þ ϵ<sup>2</sup> and ℏ*ω<sup>d</sup>* is the energy of the emitted phonon. All energies are with respect to Fermi surface energy ϵ*F*. The state of the three-level system evolves according to the Schrödinger equation:

$$\psi = -i \begin{bmatrix} 2a\_1 & d \exp\left(-i\omega\_d t\right) & 0\\ d \exp\left(i a\_d t\right) & a\_1 + a\_2 & d \exp\left(i\omega\_d t\right) \\ 0 & d \exp\left(-i\omega\_d\right) & 2a\_2 \end{bmatrix} \psi. \tag{26}$$

*<sup>H</sup>*<sup>13</sup> <sup>¼</sup> �*d*<sup>2</sup> 2Δ*t*

> <sup>¼</sup> *id*<sup>2</sup> 2Δ*t*

� �*i*

*ψ*ð Þ¼ Δ*T* exp �*i*Δ

where

*<sup>H</sup>eff* ¼ �*<sup>i</sup>*

� *d*2 *ωd*

**81**

ð<sup>Δ</sup>*<sup>t</sup>* 0

1 *<sup>ω</sup><sup>d</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>ω</sup>* <sup>þ</sup>

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

�*d*<sup>2</sup> exp ð Þ *<sup>i</sup>*Δ*ω*Δ*<sup>t</sup> ωd*

> 2 6 4

*<sup>d</sup>* exp � *<sup>i</sup>*

�*d*<sup>2</sup> *ωd*

. But observe *d*≪Δ*ω*, that is, term *Heff*

Observe in the above the term *Heff*

2 Δ*ω*Δ*t*

Then, from Eq. (27), we get

0

B@

� <sup>Δ</sup>*<sup>ω</sup> ωd*

exp ð Þ �*i*Δ*E*<sup>1</sup> *t*

� � ð<sup>Δ</sup>*<sup>t</sup>*

ð*τ* 0

0

1 *ω<sup>d</sup>* � Δ*ω*

*:*

2*ω*<sup>1</sup> 0 0 0 *ω*<sup>1</sup> þ *ω*<sup>2</sup> 0 0 02*ω*<sup>2</sup>

<sup>2</sup>*ω*<sup>1</sup> � <sup>Δ</sup>*<sup>ω</sup>*

exp ð Þ� *i*Δ*E*<sup>2</sup> *t*

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

3 7 5

*ωd*

Δ*ω ωd*

superconductivity [3]. We say the electron pair *k*1*,* � *k*<sup>1</sup> scatters to *k*2*,* � *k*<sup>2</sup> at rate

then, how are we justified in neglecting these terms. This suggests our calculation of scattering into an annulus around the Fermi surface requires caution and our expression for the binding energy may be high as binding deteriorates in the presence of offset. However, we show everything works as expected if we move to a wave-packet

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 exp ð Þ *ik*0*x p x*ð Þ as a wave packet centred at momentum *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 *ω*ð Þ *k*<sup>0</sup> . We may say the packet is *stationary* in the well, evolving as exp ð Þ *ik*0*x p x*ð Þ! exp ð Þ �*iω*ð Þ *k*<sup>0</sup> *t* exp ð Þ *ik*0*x p x*ð Þ. **Figure 15** picturizes the offsets getting averaged by showing wave packets standing in the local potential. All we are saying is that now the whole packet has energy ϵ0. So now, we can study how packet pair (at fermi surface) centred at *k*1*,* � *k*<sup>1</sup> scatters to *k*2*,* � *k*2. This is shown in **Figure 16A**. The scattering is through a phonon packet with width localized to the local well. Let us say our electron packet width is Debye frequency *ωd*. If there are *N* k-points in a packet, the original scattering rate Δ*<sup>b</sup>* gets modified

picture. The key idea is what we call *offset averaging*, which we develop now.

� � *<sup>ω</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*d* exp

*d* exp

*i* 2 Δ*ω*Δ*t*

*i* 2 Δ*ω*Δ*t*

1

ð<sup>Δ</sup>*<sup>t</sup>* 0

� �

exp ð Þ¼� *i*2Δ*ω t i*

exp ð Þ *i*Δ*E*<sup>2</sup> *t*

�*d*<sup>2</sup>

*ω*2 *<sup>d</sup>* � Δ*ω*<sup>2</sup>

CA exp *<sup>H</sup>*Δ*<sup>t</sup>* � � *<sup>ψ</sup>*ð Þ¼ <sup>0</sup> exp *<sup>H</sup>eff* <sup>Δ</sup>*<sup>t</sup>* � �*ψ*ð Þ <sup>0</sup> *,*

� � �*d*<sup>2</sup>

� � <sup>2</sup>*ω*<sup>2</sup>

<sup>13</sup> gives us the attractive potential responsible for

<sup>13</sup> is much smaller than terms *<sup>H</sup>eff*

Δ*ω ωd*

ð*τ* 0

*ω<sup>d</sup>* exp ð Þ *i*Δ*ω*Δ*t*

*ωd*

2 Δ*ω*Δ*t* � �

*<sup>d</sup>* exp � *<sup>i</sup>*

exp ð Þ �*i*Δ*E*<sup>1</sup> *t*

(32)

(33)

23 :

<sup>12</sup> and *<sup>H</sup>eff*

*:*

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

$$\phi = \exp\left(\begin{bmatrix} 2\alpha\_1 & 0 & 0\\ 0 & \alpha\_1 + \alpha\_2 & 0\\ 0 & 0 & 2\alpha\_2 \end{bmatrix}\right) \varphi. \tag{27}$$

This gives for Δ*E*<sup>1</sup> ¼ *ω<sup>d</sup>* � ð Þ *ω*<sup>1</sup> � *ω*<sup>2</sup> <sup>Δ</sup>*<sup>ω</sup>* and Δ*E*<sup>2</sup> ¼ *ω<sup>d</sup>* þ Δ*ω*

$$\dot{\phi} = -i \underbrace{\begin{bmatrix} 0 & \exp\left(-i\Delta E\_1 \ t\right)d & 0\\ \exp\left(i\Delta E\_1 \ t\right)d & 0 & \exp\left(i\Delta E\_2 \ t\right)d\\ 0 & \exp\left(-i\Delta E\_2 \ t\right)d & 0 \end{bmatrix}}\_{H(t)} \phi. \tag{28}$$

We evaluate the effective evolution of *<sup>H</sup>*(*t*) in period <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> ωd* . After Δ*t*, the system evolution is

$$\phi(\Delta t) = \exp\left(\underbrace{\int\_0^{\Delta t} H(\sigma)d\sigma + \frac{1}{2}\int\_0^{\Delta t} \left[H(\sigma\_1), \int\_0^{\sigma\_1} H(\sigma\_2)d\sigma\_2\right] d\sigma\_1}\_{\overline{H}\_{\Delta t}}\right) \phi(0). \tag{29}$$

Let us calculate *H*12. Assuming ∣Δ*ω*∣≪*ωd*, then

$$\overline{H}\_{12} = \frac{-i}{\Delta t} \int\_0^{\Delta t} \exp\left(-i\Delta E\_1 \cdot t\right) d\!\!/ = i \frac{\Delta \phi}{\phi\_d} d\!\!/ \tag{30}$$

Similarly

$$\overline{H}\_{23} = \frac{-i}{\Delta t} \int\_0^{\Delta t} \exp\left(-i\Delta E\_2 \cdot t\right) d\!\!/ = -i\frac{\Delta \alpha}{\alpha\_d} d,\tag{31}$$

and finally

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

$$\begin{split} \overline{H}\_{13} &= \frac{-d^2}{2\Delta t} \left\{ \int\_0^{\Delta t} \exp\left(-i\Delta E\_1 \ t\right) \int\_0^r \exp\left(i\Delta E\_2 \ t\right) - \int\_0^{\Delta t} \exp\left(i\Delta E\_2 \ t\right) \int\_0^r \exp\left(-i\Delta E\_1 \ t\right) \right\} \\ &= \frac{id^2}{2\Delta t} \left(\frac{1}{\alpha\_d + \Delta \alpha} + \frac{1}{\alpha\_d - \Delta \alpha}\right) \int\_0^{\Delta t} \exp\left(i2\Delta \alpha \ t\right) = -i \frac{-d^2 \alpha\_d \cdot \exp\left(i\Delta \alpha \Delta t\right)}{\alpha\_d^2 - \Delta \alpha^2} \\ &\sim -i \frac{-d^2 \exp\left(i\Delta \alpha \Delta t\right)}{\alpha\_d} .\end{split}$$

Then, from Eq. (27), we get

$$\begin{aligned} \boldsymbol{\psi}(\Delta T) &= \exp\left(-i\Delta \begin{bmatrix} 2\alpha\_1 & 0 & 0\\ 0 & \alpha\_1 + \alpha\_2 & 0\\ 0 & 0 & 2\alpha\_2 \end{bmatrix}\right) \exp\left(\overline{\boldsymbol{H}}\Delta t\right) \boldsymbol{\psi}(0) = \exp\left(\boldsymbol{H}^{\text{eff}}\Delta t\right) \boldsymbol{\psi}(0), \end{aligned} \tag{32}$$

where

This paired electron state is called Cooper pair. Now, the plan is we bind many

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

Consider when *E*<sup>1</sup> 6¼ *E*3. Observe *E*<sup>1</sup> ¼ 2ϵ<sup>1</sup> ¼ 2ℏ*ω*1, *E*<sup>3</sup> ¼ 2ϵ<sup>2</sup> ¼ 2ℏ*ω*2, *E*<sup>2</sup> ¼ ϵ<sup>1</sup> þ ϵ<sup>2</sup> and ℏ*ω<sup>d</sup>* is the energy of the emitted phonon. All energies are with respect to Fermi surface energy ϵ*F*. The state of the three-level system evolves according to the

. For this we return to phonon scattering of ∣*k*1*,* � *k*1i and ∣*k*2*,* � *k*2i.

2*ω*<sup>1</sup> *d* exp ð Þ �*iωdt* 0 *d* exp ð Þ *iωdt ω*<sup>1</sup> þ *ω*<sup>2</sup> *d* exp ð Þ *iωdt* 0 *d* exp ð Þ �*iω<sup>d</sup>* 2*ω*<sup>2</sup>

> 2*ω*<sup>1</sup> 0 0 0 *ω*<sup>1</sup> þ *ω*<sup>2</sup> 0 0 02*ω*<sup>2</sup>

3 7 5

1

We proceed into the interaction frame of the natural Hamiltonian (system

0 exp ð Þ �*i*Δ*E*<sup>1</sup> *t d* 0 exp ð Þ *i*Δ*E*<sup>1</sup> *t d* 0 exp ð Þ *i*Δ*E*<sup>2</sup> *t d* 0 exp ð Þ �*i*Δ*E*<sup>2</sup> *t d* 0


*H*ð Þ *σ*<sup>1</sup> *;*

exp ð Þ �*i*Δ*E*<sup>1</sup> *t d* ¼ *i*

exp ð Þ �*i*Δ*E*<sup>2</sup> *t d* ¼ �*i*


ð*<sup>σ</sup>*<sup>1</sup> 0

� �

*H*ð Þ *σ*<sup>2</sup> *dσ*<sup>2</sup>

Δ*ω ωd*

> Δ*ω ωd*

2 6 4

0

B@

This gives for Δ*E*<sup>1</sup> ¼ *ω<sup>d</sup>* � ð Þ *ω*<sup>1</sup> � *ω*<sup>2</sup> <sup>Δ</sup>*<sup>ω</sup>* and Δ*E*<sup>2</sup> ¼ *ω<sup>d</sup>* þ Δ*ω*

We evaluate the effective evolution of *<sup>H</sup>*(*t*) in period <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup>*

ð<sup>Δ</sup>*<sup>t</sup>* 0

ð<sup>Δ</sup>*<sup>t</sup>* 0

3 7

<sup>5</sup>*ψ:* (26)

CA*<sup>ψ</sup>:* (27)

*ϕ:* (28)

. After Δ*t*, the

*ϕ*ð Þ 0 *:* (29)

*d:* (30)

*d,* (31)

3 7 5

*ωd*

1

CCCA

*dσ*<sup>1</sup>

electrons and make a big molecule called BCS ground state.

*Magnetometers - Fundamentals and Applications of Magnetism*

<sup>Δ</sup>*<sup>b</sup>* <sup>¼</sup> <sup>4</sup>ϵ*d*Ω<sup>2</sup> Δϵ2�ϵ<sup>2</sup> *d*

Schrödinger equation:

*ψ*\_ ¼ �*i*

energies) by transformation:

*<sup>ϕ</sup>*\_ ¼ �*<sup>i</sup>*

system evolution is

*ϕ*ð Þ¼ Δ*t* exp

Similarly

and finally

**80**

ð<sup>Δ</sup>*<sup>t</sup>* 0

0

BBB@

*H*ð Þ *σ dσ* þ

Let us calculate *H*12. Assuming ∣Δ*ω*∣≪*ωd*, then

*<sup>H</sup>*<sup>12</sup> <sup>¼</sup> �*<sup>i</sup>* Δ*t*

*<sup>H</sup>*<sup>23</sup> <sup>¼</sup> �*<sup>i</sup>* Δ*t*

2 6 4 2 6 4

*ϕ* ¼ exp *it*

$$H^{\mathcal{J}\dagger} = -i \begin{bmatrix} 2\alpha\_1 & -\frac{\Delta\alpha}{\alpha\_d} d \exp\left(\frac{i}{2} \Delta\alpha \Delta t\right) & \frac{-d^2}{\alpha\_d} \\\\ -\frac{\Delta\alpha}{\alpha\_d} d \exp\left(-\frac{i}{2} \Delta\alpha \Delta t\right) & \alpha\_1 + \alpha\_2 & \frac{\Delta\alpha}{\alpha\_d} d \exp\left(-\frac{i}{2} \Delta\alpha \Delta t\right) \\\\ \frac{-d^2}{\alpha\_d} & \frac{\Delta\alpha}{\alpha\_d} d \exp\left(\frac{i}{2} \Delta\alpha \Delta t\right) & 2\alpha\_2 \end{bmatrix}. \tag{33}$$

Observe in the above the term *Heff* <sup>13</sup> gives us the attractive potential responsible for superconductivity [3]. We say the electron pair *k*1*,* � *k*<sup>1</sup> scatters to *k*2*,* � *k*<sup>2</sup> at rate � *d*2 *ωd* . But observe *d*≪Δ*ω*, that is, term *Heff* <sup>13</sup> is much smaller than terms *<sup>H</sup>eff* <sup>12</sup> and *<sup>H</sup>eff* 23 : then, how are we justified in neglecting these terms. This suggests our calculation of scattering into an annulus around the Fermi surface requires caution and our expression for the binding energy may be high as binding deteriorates in the presence of offset. However, we show everything works as expected if we move to a wave-packet picture. The key idea is what we call *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 exp ð Þ *ik*0*x p x*ð Þ as a wave packet centred at momentum *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 *ω*ð Þ *k*<sup>0</sup> . We may say the packet is *stationary* in the well, evolving as exp ð Þ *ik*0*x p x*ð Þ! exp ð Þ �*iω*ð Þ *k*<sup>0</sup> *t* exp ð Þ *ik*0*x p x*ð Þ. **Figure 15** picturizes the offsets getting averaged by showing wave packets standing in the local potential. All we are saying is that now the whole packet has energy ϵ0. So now, we can study how packet pair (at fermi surface) centred at *k*1*,* � *k*<sup>1</sup> scatters to *k*2*,* � *k*2. This is shown in **Figure 16A**. The scattering is through a phonon packet with width localized to the local well. Let us say our electron packet width is Debye frequency *ωd*. If there are *N* k-points in a packet, the original scattering rate Δ*<sup>b</sup>* gets modified

#### **Figure 15.**

*The offsets getting averaged by showing wave packets standing in the local potential, bound by phonon-mediated interaction. Offset average so effectively packets are standstill; they do not move.*

to *N*Δ*b*. If there are *p* packet pairs at the Fermi surface as shown in **Figure 16A**, then the state formed from superposition of packet pairs

$$\phi = \frac{1}{\sqrt{p}} \sum\_{i} |k\_i, -k\_i\rangle \tag{34}$$

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,

*Packets at the Fermi surface that has twice the number of k-states as electrons. This means half the packet sites*

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

Let *kF* denote wavevector radius of the Fermi sphere. We describe electron wave

<sup>2</sup> inside and same outside the Fermi sphere) and **<sup>m</sup>**<sup>2</sup> points in the tan-

packets formed from k-points (wavevectors) near the Fermi sphere surface. **Figure 18** shows the Fermi sphere with surface as thick circle and an annulus of thickness *ω<sup>d</sup>* (energy units) shown in dotted lines. Pockets have **n**, k-points in radial

gential direction. **Figure 18** shows such a pocket enlarged with k-points shown in

*Depiction of the Fermi sphere with surface as a thick circle and an annulus of thickness ω<sup>d</sup> (energy units) shown in dotted lines. Pockets with k-points are shown in black dots. Superposition of these k-points in a pocket forms a*

half of these packets are empty as shown in **Figure 17**.

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

**4. BCS ground state**

direction (**<sup>n</sup>**

**Figure 18.**

*wave packet.*

**83**

**Figure 17.**

*will be empty.*

has binding energy *pN*Δ*b*. What is *Np* though is really just a N number of points in the annulus around the Fermi surface which we counted earlier. Hence, we recover the binding energy we derived between electrons earlier, but now it is binding of packets. Really, this way we do not have to worry about offsets when we bind; in packet land they average in local potential.

The wave packet in a potential well shuttles back and forth, which averages the offsets *ω<sup>i</sup>* � *ω<sup>j</sup>* to zero. Then, the question of interest is how fast do we average these offsets compared to packet width which we take as Debye frequency *ωd*. For *s* orbitals or waves, the bandwidth is �10 eV giving Fermi velocity of 105 m/s, which for a characteristic length of the potential well as �300 Å, corresponds to a packet shuttling time of around � <sup>10</sup>�13s, which is same as the Debye frequency *<sup>ω</sup>d*, so we may say we average the offsets in a packet. By the time offsets have evolved significantly, the packet already returns to its original position, and we may say there are no offsets.

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

**Figure 16.** *(A) Packet pairs k*1*,* � *k*<sup>1</sup> *and k*2*,* � *k*<sup>2</sup> *on the Fermi surface and (B) many such pairs.*

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

#### **Figure 17.**

to *N*Δ*b*. If there are *p* packet pairs at the Fermi surface as shown in **Figure 16A**,

*The offsets getting averaged by showing wave packets standing in the local potential, bound by phonon-mediated*

in the annulus around the Fermi surface which we counted earlier. Hence, we recover the binding energy we derived between electrons earlier, but now it is binding of packets. Really, this way we do not have to worry about offsets when we

has binding energy *pN*Δ*b*. What is *Np* though is really just a N number of points

The wave packet in a potential well shuttles back and forth, which averages the offsets *ω<sup>i</sup>* � *ω<sup>j</sup>* to zero. Then, the question of interest is how fast do we average these offsets compared to packet width which we take as Debye frequency *ωd*. For *s* orbitals or waves, the bandwidth is �10 eV giving Fermi velocity of 105 m/s, which for a characteristic length of the potential well as �300 Å, corresponds to a packet shuttling time of around � <sup>10</sup>�13s, which is same as the Debye frequency *<sup>ω</sup>d*, so we may say we average the offsets in a packet. By the time offsets have evolved significantly, the packet already returns to its original position, and we may say

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

*(A) Packet pairs k*1*,* � *k*<sup>1</sup> *and k*2*,* � *k*<sup>2</sup> *on the Fermi surface and (B) many such pairs.*

∣*ki,* � *ki*i (34)

*<sup>ϕ</sup>* <sup>¼</sup> <sup>1</sup> ffiffiffi *<sup>p</sup>* <sup>p</sup> <sup>∑</sup> *i*

then the state formed from superposition of packet pairs

*Magnetometers - Fundamentals and Applications of Magnetism*

*interaction. Offset average so effectively packets are standstill; they do not move.*

bind; in packet land they average in local potential.

there are no offsets.

**Figure 15.**

**Figure 16.**

**82**

*Packets at the Fermi surface that has twice the number of k-states as electrons. This means half the packet sites will be empty.*

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 *kF* denote wavevector radius of the Fermi sphere. We describe electron wave packets formed from k-points (wavevectors) near the Fermi sphere surface. **Figure 18** shows the Fermi sphere with surface as thick circle and an annulus of thickness *ω<sup>d</sup>* (energy units) shown in dotted lines. Pockets have **n**, k-points in radial direction (**<sup>n</sup>** <sup>2</sup> inside and same outside the Fermi sphere) and **<sup>m</sup>**<sup>2</sup> points in the tangential direction. **Figure 18** shows such a pocket enlarged with k-points shown in

#### **Figure 18.**

*Depiction of the Fermi sphere with surface as a thick circle and an annulus of thickness ω<sup>d</sup> (energy units) shown in dotted lines. Pockets with k-points are shown in black dots. Superposition of these k-points in a pocket forms a wave packet.*

black dots. We assume there are 4*<sup>p</sup>* such pockets with *<sup>N</sup>* <sup>¼</sup> **nm**<sup>2</sup> points in each pocket. In pocket *s*, we form the function

$$\phi\_s(r) = \frac{1}{\sqrt{N}} \sum\_j \exp\left(ik\_j \cdot r\right) \tag{35}$$

which means the Cooper pair gets broken. Then, the superconducting state constitutes *p* � 1 pairs and damaged pair. Each term in the state in Eq. (37) will scatter to

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

superconducting electron collides carries an energy *kBT*. When this energy is less than Δ, the electron cannot be deflected and will not scatter as we cannot pay for the increase of energy. Therefore, superconducting electrons do not scatter phonons. However, this is not the whole story as the phonon can deflect the electron slightly from its course and over many such collisions break the Cooper pair; then, the energy budget Δ is paid in many increments of *kBT*. These are small angle scattering events. Thus, there will be a finite probability *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 *p*<sup>0</sup> ¼ *p*ð Þ 1 � 2*q* good pairs which give an average energy per pair to be 1ð Þ � 2*q* Δ. The probability

> *<sup>q</sup>* <sup>¼</sup> exp ð Þ �ð Þ <sup>1</sup> � <sup>2</sup>*<sup>q</sup>* <sup>Δ</sup>*=kBT* <sup>1</sup> <sup>þ</sup> exp ð Þ �ð Þ <sup>1</sup> � <sup>2</sup>*<sup>q</sup>* <sup>Δ</sup>*=kBT ,*

> > <sup>¼</sup> ð Þ <sup>1</sup> � <sup>2</sup>*<sup>q</sup>* <sup>Δ</sup>

that the superconducting electron will be deflected of phonon is

ln <sup>1</sup> *<sup>q</sup>* � <sup>1</sup> 

orbital between these BCS orbitals. This is shown in **Figure 19**.

*Depiction of two BCS ground states in local potential wells separated in a weak link.*

*Tc* is the critical temperature where superconducting transition sets in.

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

If Φ<sup>1</sup> and Φ<sup>2</sup> are the orbitals of Cooper pair in individual potential wells, then the overlap creates a transition *d* ¼ h i Φ1j2*eU*jΦ<sup>2</sup> , where *U* is the potential well, with 2*e* coming from the electron pair charge. Then, the linear combination of atomic

When *kBTc* <sup>¼</sup> <sup>Δ</sup>, the above equation gives *<sup>q</sup>* <sup>¼</sup> <sup>1</sup>

**5. Molecular orbitals of BCS states**

orbital (LCAO) *a*Φ<sup>1</sup> þ *b*Φ<sup>2</sup> evolves as

**Figure 19.**

**85**

*Np p*ð Þ �1

*<sup>ω</sup><sup>d</sup>* , the superconducting gap. The phonon with which the

*<sup>ω</sup><sup>d</sup>* . The total binding energy has

*kBT :* (38)

<sup>2</sup> and gap 1ð Þ � 2*q* Δ goes to zero.

*p p*ð Þ � <sup>1</sup> states, and the binding energy is � <sup>4</sup>ℏ*d*<sup>2</sup>

*Np*

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

reduced by <sup>Δ</sup> <sup>¼</sup> <sup>4</sup>ℏ*d*<sup>2</sup>

which gives

*ϕ<sup>s</sup>* has characteristic width *a*. From *N* points in a pocket, we can form *N* such functions by displacing *ϕ<sup>s</sup>* to *ϕs*ð Þ *r* � *ma* , and putting these functions uniformly spaced over the whole lattice in their local potential wells. Thus, each pocket gives *N* wave packets orthogonal as they are nonoverlapping and placed uniformly over the lattice in their local potential wells. Furthermore, *ϕ<sup>s</sup>* and *ϕs*<sup>0</sup> are orthogonal as they are formed from mutually exclusive k-points. The wave packet *ϕ<sup>s</sup>* moves with a group velocity *υ*, which is the Fermi velocity in a radial direction to the pocket from which it is formed. From wave packet *ϕ<sup>s</sup>* and its antipodal packet *ϕ*�*s*, we form the joint wave packet:

$$
\Phi\_{\mathfrak{s}}(r\_1, r\_2) = \phi\_{\mathfrak{s}}(r\_1)\phi\_{-\mathfrak{s}}(r\_2). \tag{36}
$$

As we will see soon, Φ*<sup>s</sup>* will be our Cooper pair.

As shown in last section, the Cooper pair <sup>Φ</sup>*<sup>s</sup>* scatters to <sup>Φ</sup>*<sup>s</sup>*<sup>0</sup> with rate <sup>M</sup> ¼ � <sup>4</sup>*d*<sup>2</sup> *N <sup>ω</sup><sup>d</sup>* . Now, we study how to form BCS state with many wave packets. We present a counting argument. Observe *ϕ<sup>s</sup>* is made of *N* k-points, and hence by displacement of *ϕs*, we have *N* nonoverlapping lattice sites (potential wells) where we can put copies of *ϕ<sup>s</sup>* as described in Section 4. However of the *N* k-points, only *<sup>N</sup>* <sup>2</sup> are inside the Fermi sphere so we only have only *<sup>N</sup>* <sup>2</sup> wave packets. Therefore, <sup>1</sup> <sup>2</sup> of the wavepacket sites are empty. Hence, of the 2*p* possible Cooper pairs, only *p* are filled and *p* are not present. Hence, when Φ*<sup>s</sup>* scatters to Φ*<sup>s</sup>*0, we have *p* choices for *s* 0 . Then, we can form a joint state of Cooper pairs present and write it as Φ*<sup>i</sup>*1Φ*<sup>i</sup>*2…Φ*ip* . Doing a superposition of such states, we get the superconducting state:

$$
\Psi = \sum \Phi\_{i\_1} \Phi\_{i\_2} \dots \Phi\_{i\_p} \,. \tag{37}
$$

The binding energy of this state is � <sup>4</sup>ℏ*d*<sup>2</sup> *Np*<sup>2</sup> *<sup>ω</sup><sup>d</sup>* as each index in Ψ scatters to *p* states. Since 4*pN* are the total k-points in the annulus surrounding the Fermi surface. We have <sup>Ω</sup> <sup>¼</sup> *<sup>c</sup>* ffiffiffi *<sup>n</sup>*<sup>3</sup> <sup>p</sup> where *<sup>c</sup>* is of order 1 eV and *<sup>n</sup>*<sup>3</sup> is the total number of lattice sites in the solid. Then, <sup>4</sup>*Np <sup>n</sup>*<sup>3</sup> � *<sup>ω</sup><sup>d</sup> ωF* . Thus, the binding energy is �*pc <sup>c</sup>* <sup>ℏ</sup>*ω<sup>F</sup>* (*<sup>c</sup> <sup>c</sup>* <sup>2</sup>ℏ*ω<sup>F</sup>* per wave packet/ electron). The Fermi energy ℏ*ω<sup>F</sup>* is of order 10 eV, while ℏ*ω<sup>d</sup>* � *:*1 eV. Thus, binding energy per wave packet is of order .05 eV. The average kinetic energy of the wave packet is just the Fermi energy ϵ*<sup>F</sup>* as it is made of superposition of *<sup>N</sup>* <sup>2</sup> points inside and *<sup>N</sup>* <sup>2</sup> points outside the Fermi sphere. If the wave packet was just formed from kpoint inside the Fermi surface, its average energy would be <sup>ϵ</sup>*<sup>F</sup>* � <sup>ℏ</sup>*ω<sup>d</sup>* <sup>4</sup> . Thus, we pay a price of <sup>ℏ</sup>*ω<sup>d</sup>* <sup>4</sup> � *:*025 eV per wave packet; when we form our wave packet out of *N* kpoints, 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 <sup>c</sup>* <sup>ℏ</sup>*ω<sup>F</sup>* <sup>&</sup>gt; <sup>ℏ</sup>*ω<sup>d</sup>* <sup>4</sup> . Observe 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 *p* � 1 pairs and damaged pair. Each term in the state in Eq. (37) will scatter to *p p*ð Þ � <sup>1</sup> states, and the binding energy is � <sup>4</sup>ℏ*d*<sup>2</sup> *Np p*ð Þ �1 *<sup>ω</sup><sup>d</sup>* . The total binding energy has reduced by <sup>Δ</sup> <sup>¼</sup> <sup>4</sup>ℏ*d*<sup>2</sup> *Np <sup>ω</sup><sup>d</sup>* , the superconducting gap. The phonon with which the superconducting electron collides carries an energy *kBT*. When this energy is less than Δ, the electron cannot be deflected and will not scatter as we cannot pay for the increase of energy. Therefore, superconducting electrons do not scatter phonons. However, this is not the whole story as the phonon can deflect the electron slightly from its course and over many such collisions break the Cooper pair; then, the energy budget Δ is paid in many increments of *kBT*. These are small angle scattering events. Thus, there will be a finite probability *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 *p*<sup>0</sup> ¼ *p*ð Þ 1 � 2*q* good pairs which give an average energy per pair to be 1ð Þ � 2*q* Δ. The probability that the superconducting electron will be deflected of phonon is

$$q = \frac{\exp\left(-(1-2q)\Delta/k\_B T\right)}{1+\exp\left(-(1-2q)\Delta/k\_B T\right)},$$

which gives

black dots. We assume there are 4*<sup>p</sup>* such pockets with *<sup>N</sup>* <sup>¼</sup> **nm**<sup>2</sup> points in each

1 ffiffiffiffi *N* p ∑ *j*

*ϕ<sup>s</sup>* has characteristic width *a*. From *N* points in a pocket, we can form *N* such functions by displacing *ϕ<sup>s</sup>* to *ϕs*ð Þ *r* � *ma* , and putting these functions uniformly spaced over the whole lattice in their local potential wells. Thus, each pocket gives *N* wave packets orthogonal as they are nonoverlapping and placed uniformly over the lattice in their local potential wells. Furthermore, *ϕ<sup>s</sup>* and *ϕs*<sup>0</sup> are orthogonal as they are formed from mutually exclusive k-points. The wave packet *ϕ<sup>s</sup>* moves with a group velocity *υ*, which is the Fermi velocity in a radial direction to the pocket from which it is formed. From wave packet *ϕ<sup>s</sup>* and its antipodal packet *ϕ*�*s*, we form the

As shown in last section, the Cooper pair <sup>Φ</sup>*<sup>s</sup>* scatters to <sup>Φ</sup>*<sup>s</sup>*<sup>0</sup> with rate <sup>M</sup> ¼ � <sup>4</sup>*d*<sup>2</sup>

packet sites are empty. Hence, of the 2*p* possible Cooper pairs, only *p* are filled and

*Np*<sup>2</sup>

*<sup>n</sup>*<sup>3</sup> <sup>p</sup> where *<sup>c</sup>* is of order 1 eV and *<sup>n</sup>*<sup>3</sup> is the total number of lattice sites in the

Since 4*pN* are the total k-points in the annulus surrounding the Fermi surface. We

electron). The Fermi energy ℏ*ω<sup>F</sup>* is of order 10 eV, while ℏ*ω<sup>d</sup>* � *:*1 eV. Thus, binding energy per wave packet is of order .05 eV. The average kinetic energy of the wave

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

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

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,

<sup>2</sup> points outside the Fermi sphere. If the wave packet was just formed from k-

<sup>4</sup> � *:*025 eV per wave packet; when we form our wave packet out of *N* k-

<sup>ℏ</sup>*ω<sup>F</sup>* <sup>&</sup>gt; <sup>ℏ</sup>*ω<sup>d</sup>*

. Thus, the binding energy is �*pc <sup>c</sup>*

packet is just the Fermi energy ϵ*<sup>F</sup>* as it is made of superposition of *<sup>N</sup>*

point inside the Fermi surface, its average energy would be <sup>ϵ</sup>*<sup>F</sup>* � <sup>ℏ</sup>*ω<sup>d</sup>*

can form a joint state of Cooper pairs present and write it as Φ*<sup>i</sup>*1Φ*<sup>i</sup>*2…Φ*ip* . Doing a

Now, we study how to form BCS state with many wave packets. We present a counting argument. Observe *ϕ<sup>s</sup>* is made of *N* k-points, and hence by displacement of *ϕs*, we have *N* nonoverlapping lattice sites (potential wells) where we can put

copies of *ϕ<sup>s</sup>* as described in Section 4. However of the *N* k-points, only *<sup>N</sup>*

*p* are not present. Hence, when Φ*<sup>s</sup>* scatters to Φ*<sup>s</sup>*0, we have *p* choices for *s*

superposition of such states, we get the superconducting state:

exp *ikj* � *<sup>r</sup>* � � (35)

*N <sup>ω</sup><sup>d</sup>* .

<sup>2</sup> are inside

. Then, we

<sup>2</sup> of the wave-

0

<sup>2</sup>ℏ*ω<sup>F</sup>* per wave packet/

<sup>2</sup> points inside

<sup>4</sup> . Thus, we pay a

Φ*s*ð Þ¼ *r*1*;r*<sup>2</sup> *ϕs*ð Þ *r*<sup>1</sup> *ϕ*�*<sup>s</sup>*ð Þ *r*<sup>2</sup> *:* (36)

<sup>2</sup> wave packets. Therefore, <sup>1</sup>

Ψ ¼ ∑Φ*<sup>i</sup>*1Φ*<sup>i</sup>*2…Φ*ip :* (37)

<sup>ℏ</sup>*ω<sup>F</sup>* (*<sup>c</sup> <sup>c</sup>*

*<sup>ω</sup><sup>d</sup>* as each index in Ψ scatters to *p* states.

<sup>4</sup> . Observe if *c* is too small, then

*ϕs*ð Þ¼ *r*

As we will see soon, Φ*<sup>s</sup>* will be our Cooper pair.

the Fermi sphere so we only have only *<sup>N</sup>*

The binding energy of this state is � <sup>4</sup>ℏ*d*<sup>2</sup>

superconducting state is only favorable if *c <sup>c</sup>*

pocket. In pocket *s*, we form the function

*Magnetometers - Fundamentals and Applications of Magnetism*

joint wave packet:

have <sup>Ω</sup> <sup>¼</sup> *<sup>c</sup>*

and *<sup>N</sup>*

**84**

price of <sup>ℏ</sup>*ω<sup>d</sup>*

the Fermi surface.

solid. Then, <sup>4</sup>*Np*

ffiffiffi

*<sup>n</sup>*<sup>3</sup> � *<sup>ω</sup><sup>d</sup> ωF*

$$\ln\left(\frac{\mathbf{1}}{q} - \mathbf{1}\right) = \frac{(\mathbf{1} - \mathbf{2}q)\Delta}{k\_B T}.\tag{38}$$

When *kBTc* <sup>¼</sup> <sup>Δ</sup>, the above equation gives *<sup>q</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> and gap 1ð Þ � 2*q* Δ goes to zero. *Tc* is the critical temperature where superconducting transition sets in.
