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

$$
\hbar \frac{d}{dt} \begin{bmatrix} a \\ b \end{bmatrix} = -i \begin{bmatrix} \epsilon\_1 - 2\Delta & d \\ d & \epsilon\_2 - 2\Delta \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} \tag{39}
$$

has a momentum and constitutes the supercurrent. It has energy

ð*T* 0

Δ*k*ð Þ¼ 2*πr*

In the presence of electric field *E*, we get on the diagonal of RHS of Eq. 40, additional potential *eExi*. How does *ϕ<sup>k</sup>* evolve under this field? Verify with

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

Ð*t* <sup>0</sup> *<sup>E</sup>*ð Þ*<sup>τ</sup> <sup>d</sup><sup>τ</sup>* <sup>ℏ</sup> *:*. 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

> *<sup>E</sup>*ð Þ¼ *<sup>τ</sup> Bar* 2*πr ,*

where *r* is radius and *ar* the area of the loop. Then, by the above argument, the

2*e*Φ<sup>0</sup>

2*e :*

<sup>Φ</sup><sup>0</sup> <sup>¼</sup> *Bar* <sup>¼</sup> *<sup>h</sup>*

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

*Depiction of the schematic of a SQUID where two superconductors* S*<sup>1</sup> and* S*<sup>2</sup> are separated by thin insulators.*

Ð *T* <sup>0</sup> *<sup>E</sup>*ð Þ*<sup>τ</sup>* <sup>ℏ</sup> <sup>¼</sup> ð Þ <sup>2</sup>*<sup>e</sup> Bar*

<sup>ℏ</sup> <sup>¼</sup> <sup>2</sup>*π,* (42)

<sup>2</sup>*πr*<sup>ℏ</sup> . Since we have

<sup>ℏ</sup> ; *ϕk t*ð Þ is a solution with eigenvalue ϵ*k t*ð Þ. In general, in the presence of

ϵ*<sup>k</sup>* ¼ ϵ<sup>0</sup> � 2Δ þ 2*d* cos *kb*.

closed loop with Φ<sup>0</sup> ¼ *Bar*

giving

**Figure 21.**

**87**

time varying *E t*ð Þ, we have *k t*ðÞ¼ *<sup>k</sup>* � <sup>2</sup>*<sup>e</sup>*

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

transient electric field in the loop given by

wavenumber *<sup>k</sup>* of the BCS states is shifted by <sup>Δ</sup>*<sup>k</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*<sup>e</sup>*

such a hole/loop is quantized [9, 10] as just shown.

*k t*ðÞ¼ *<sup>k</sup>* � *eEt*

where ϵ1*,* ϵ<sup>2</sup> is the Fermi energy of the electron pair in two orbitals. ϵ<sup>1</sup> ¼ ϵ<sup>2</sup> unless we apply a voltage difference *v* between them; then, ϵ<sup>1</sup> � ϵ<sup>2</sup> ¼ *v*. Let us start with *v* ¼ 0 and *a b* � � <sup>¼</sup> <sup>1</sup> 1 � �, and then nothing happens, but when there is phase difference *a b* � � <sup>¼</sup> <sup>1</sup> exp ð Þ *iϕ* � �, then *a b* � � evolves and we say we have supercurrent *I*∝sin *ϕ* between two superconductors. This constitutes the Josephson junction (in a Josephson junction, there is a thin insulator separating two BCS states or superconductors). Applying *v* generates a phase difference *<sup>d</sup><sup>ϕ</sup> dt* ∝*v* between the two orbitals which then evolves under *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 *ϕ<sup>i</sup>* induce a supercurrent as in Josephson junction.

If Φ*<sup>i</sup>* denotes the local Cooper pairs, then their overlap creates a transition *d* ¼ Φ*<sup>i</sup>* h i j2*eU*jΦ*<sup>i</sup>*þ<sup>1</sup> , where *U* is the potential well, with 2*e* coming from the electron pair charge. Then, the LCAO ∑*ai*Φ*<sup>i</sup>* evolves as

$$
\hbar \frac{d}{dt} \begin{bmatrix} a\_1 \\ \vdots \\ a\_n \end{bmatrix} = -i \begin{bmatrix} \epsilon\_1 - 2\Delta & d & 0 \\ d & \ddots & \vdots \\ & d & \epsilon\_n - 2\Delta \end{bmatrix} \begin{bmatrix} a\_1 \\ \vdots \\ a\_n \end{bmatrix} \tag{40}
$$

where ϵ*<sup>i</sup>* ¼ ϵ<sup>0</sup> is the Fermi energy of the electron pair in BCS state Φ*i*. ϵ*<sup>i</sup>* is same unless we apply a voltage difference *v* to the superconductor. Eq. 40 is the tightbinding approximation model for superconductor.

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

$$
\begin{bmatrix} a\_1 \\ \vdots \\ a\_n \end{bmatrix} = \frac{1}{\sqrt{n}} \begin{bmatrix} \mathbf{1} \\ \vdots \\ \mathbf{1} \end{bmatrix} \tag{41}
$$

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

$$
\begin{bmatrix} a\_1 \\ \vdots \\ a\_n \end{bmatrix} = \frac{1}{\sqrt{n}} \begin{bmatrix} \exp\left(ik\mathbf{x}\_1\right) \\ \vdots \\ \exp\left(ik\mathbf{x}\_n\right) \end{bmatrix}
$$

**Figure 20.** *Depiction of array of local BCS states in local potential wells with different phases.*

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

has a momentum and constitutes the supercurrent. It has energy ϵ*<sup>k</sup>* ¼ ϵ<sup>0</sup> � 2Δ þ 2*d* cos *kb*.

In the presence of electric field *E*, we get on the diagonal of RHS of Eq. 40, additional potential *eExi*. How does *ϕ<sup>k</sup>* evolve under this field? Verify with *k t*ðÞ¼ *<sup>k</sup>* � *eEt* <sup>ℏ</sup> ; *ϕk t*ð Þ is a solution with eigenvalue ϵ*k t*ð Þ. In general, in the presence of time varying *E t*ð Þ, we have *k t*ðÞ¼ *<sup>k</sup>* � <sup>2</sup>*<sup>e</sup>* Ð*t* <sup>0</sup> *<sup>E</sup>*ð Þ*<sup>τ</sup> <sup>d</sup><sup>τ</sup>* <sup>ℏ</sup> *:*.

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

$$\int\_{0}^{T} E(\pi) = \frac{Ba\_r}{2\pi r},$$

where *r* is radius and *ar* the area of the loop. Then, by the above argument, the wavenumber *<sup>k</sup>* of the BCS states is shifted by <sup>Δ</sup>*<sup>k</sup>* <sup>¼</sup> ð Þ <sup>2</sup>*<sup>e</sup>* Ð *T* <sup>0</sup> *<sup>E</sup>*ð Þ*<sup>τ</sup>* <sup>ℏ</sup> <sup>¼</sup> ð Þ <sup>2</sup>*<sup>e</sup> Bar* <sup>2</sup>*πr*<sup>ℏ</sup> . Since we have closed loop with Φ<sup>0</sup> ¼ *Bar*

$$
\Delta k(2\pi r) = \frac{2\epsilon \Phi\_0}{\hbar} = 2\pi,\tag{42}
$$

giving

ℏ *d dt*

*v* ¼ 0 and

*a b* � �

ence

*a b* � �

<sup>¼</sup> <sup>1</sup>

which then evolves under *d*.

Josephson junction.

**Figure 20.**

**86**

<sup>¼</sup> <sup>1</sup> 1 � �

exp ð Þ *iϕ* � �

*a b* � �

*Magnetometers - Fundamentals and Applications of Magnetism*

, then

ductors). Applying *v* generates a phase difference *<sup>d</sup><sup>ϕ</sup>*

pair charge. Then, the LCAO ∑*ai*Φ*<sup>i</sup>* evolves as

2 6 4

*a*1 ⋮ *an*

binding approximation model for superconductor.

3 7 <sup>5</sup> ¼ �*<sup>i</sup>*

2 6 4

100 times larger. We have an electron pair at each lattice site. The state

*a*1 ⋮ *an* 3 7 <sup>5</sup> <sup>¼</sup> <sup>1</sup> ffiffiffi *n* p

2 6 4

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

2 6 4

*a*1 ⋮ *an*

*Depiction of array of local BCS states in local potential wells with different phases.*

3 7 <sup>5</sup> <sup>¼</sup> <sup>1</sup> ffiffiffi *n* p

ℏ *d dt* ¼ �*i*

*a b* � � ϵ<sup>1</sup> � 2Δ *d*

where ϵ1*,* ϵ<sup>2</sup> is the Fermi energy of the electron pair in two orbitals. ϵ<sup>1</sup> ¼ ϵ<sup>2</sup> unless we apply a voltage difference *v* between them; then, ϵ<sup>1</sup> � ϵ<sup>2</sup> ¼ *v*. Let us start with

*I*∝sin *ϕ* between two superconductors. This constitutes the Josephson junction (in a Josephson junction, there is a thin insulator separating two BCS states or supercon-

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 *ϕ<sup>i</sup>* induce a supercurrent as in

If Φ*<sup>i</sup>* denotes the local Cooper pairs, then their overlap creates a transition *d* ¼ Φ*<sup>i</sup>* h i j2*eU*jΦ*<sup>i</sup>*þ<sup>1</sup> , where *U* is the potential well, with 2*e* coming from the electron

> ϵ<sup>1</sup> � 2Δ *d* 0 *d* ⋱ ⋮

where ϵ*<sup>i</sup>* ¼ ϵ<sup>0</sup> is the Fermi energy of the electron pair in BCS state Φ*i*. ϵ*<sup>i</sup>* is same unless we apply a voltage difference *v* to the superconductor. Eq. 40 is the tight-

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

> 1 ⋮ 1

exp ð Þ *ikx*<sup>1</sup> ⋮ exp ð Þ *ikxn* 3 7 5

2 6 4 3 7

2 6 4

*d* ϵ*<sup>n</sup>* � 2Δ

3 7 5

2 6 4

*a*1 ⋮ *an* 3 7

<sup>5</sup> (41)

<sup>5</sup> (40)

*d* ϵ<sup>2</sup> � 2Δ � � *a*

, and then nothing happens, but when there is phase differ-

evolves and we say we have supercurrent

*dt* ∝*v* between the two orbitals

*b* � �

(39)

$$
\Phi\_0 = Ba\_r = \frac{h}{2e}.
$$

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.** *Depiction of the schematic of a SQUID where two superconductors* S*<sup>1</sup> and* S*<sup>2</sup> are separated by thin insulators.*

**Figure 21** depicts the schematic of a superconducting quantum interference device (SQUID) where two superconductors *S*<sup>1</sup> and *S*<sup>2</sup> are separated by thin insulators. A small flux through the SQUID creates a phase difference in the two superconductors (see discussion on Δ*k* above) leading to the flow of supercurrent. If an initial phase, *δ*<sup>0</sup> exists between the superconductors. Then, this phase difference after application of flux is from Eq. (42), *<sup>δ</sup><sup>a</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>e</sup>*Φ<sup>0</sup> <sup>ℏ</sup> across top insulator and *<sup>δ</sup><sup>a</sup>* <sup>¼</sup> *<sup>δ</sup>*<sup>0</sup> � *<sup>e</sup>*Φ<sup>0</sup> <sup>ℏ</sup> across bottom insulator (see **Figure 21**). This leads to currents *Ja* ¼ *I*<sup>0</sup> sin *δ<sup>a</sup>* and *Jb* ¼ *I*<sup>0</sup> sin *δ<sup>b</sup>* through top and down insulators. The total current *<sup>J</sup>* <sup>¼</sup> *Ja* <sup>þ</sup> *Jb* <sup>¼</sup> <sup>2</sup>*I*<sup>0</sup> sin *<sup>δ</sup>*<sup>0</sup> cos *<sup>e</sup>*Φ<sup>0</sup> <sup>ℏ</sup> . This accumulates charge on one side of SQUID and leads to a potential difference between the two superconductors. Therefore, the flux is converted to a voltage difference. The voltage oscillates as the phase difference *<sup>e</sup>*Φ<sup>0</sup> ℏ goes in integral multiples of *π* for every flux quanta Φ0. SQUID is the most sensitive magnetic flux sensor currently known. The SQUID can be seen as a flux to voltage converter, and it can generally be used to sense any quantity that can be transduced into a magnetic flux, such as electrical current, voltage, position, etc. The extreme sensitivity of the SQUID is utilized in many different fields of applications, including biomagnetism, materials science, metrology, astronomy and geophysics.
