3.1 Introduction

3.2 The Second London Equation

3.3 Superconducting Lumped Circuits

3.4 The Two-Fluid Model

Chap 3. The Classical Model

field DCfor )0,(),( ),0,(),(

)0,()/cosh()/cosh()]0,(),([),(

aHtaHbecauseyHtyH

yHayaHtaHtyH

3.1 Introduction

◈

Purpose

: This chapter is included

the Meissner

effect with classical

electrodynamics, and thus called a classical model of superconductivity.

Eq.London Second B)J(

Eq.London First JE

0B J,H

E ,BE

t

t

◈

Fundamental Equations for Classical Model

3.2 The Second London Equation☜ An infinite slab of finite thickness. (made from a superconducting material)

The goal is to examine how the additional complication of

Meissner effect changes the results found in Section 2.6.

For ω → 0 (ħω << 2Δ), λ

is independent of frequency.

The flux distribution (B) inside the material is not frequency dependent.

Superconductor in B field derived by first London Equation

22 **

*

0**

*

0

22

ΛΛdepth n penetratio where

1)-(3 ------ 0B1

qnm

μqnm

μλ

t

If B field satisfies following relation (3-2) then B will satisfy Eq. (3-1) as well.

2)-(3 ------ 0B1 22

If we postulate that this expression holds for all driving frequencies, we will account for the perfect diamagnetism of a bulk superconductor!!

i)/sinh()/sinh(

HReJ

HJ

i)/cosh()/cosh(

HReH

x0

z0

eay

eay

jwt

jwt

Review: Solution in a perfect conductor for (a) (a/λ) ≪

1, (b) (a/λ) ≫

1

1 / a

1 / a

HTS and NbTifor nm 100

Second London Equation

EquationLondon Second BJ)(

BJ J,HB

HB J,H B,B

BB1B)B(B

nPostulatio with LE 1 B1B 0B1

0000

00

02

2

st2

222

▶

2nd

LE is time independent and explains Meissner

effect.

▶

The first and the second London equation are the

fundamental equations that describe superconductivity.

(3.7) --------- B)J(

)J(E

BE and JE

:law sFaraday' and LE 1 combiningBy

EquationLondon Second BJEquationLondon Secondofformderivative Time

st

tt

t

tt

i

i

iH If

}sin)211(Re{

}cos)1(Re{)(H (ii)

ctorsupercondu inside ,0)(H (i)field Magnetic (1)

}Re{

3

3

0

r3

3

0

z0app

erRH

erRH

eH

jwt

jwt

jwt

Rr

Rr

A Spherical Bulk Superconductor in a Uniform Magnetic Field

Example 3.2.1Consider the spherical bulk superconductor of radius R. (λ≪R)

0B1 22

i

i

}sin23Re{K

0H ),(HH

(2.77) ----- ,n )H(HnKcurrent Surface (2)

0

12

r12

eH jwt

Rr

i}sin)211(Re{ i}cos)1(Re{)(H

3

3

0r3

3

0 erRHe

rRHRr jwtjwt

Fields produced by superconducting sphere in a magnetic field.

iK }sin23Re{

current Surface

0 eH jwt

Example 3.2.2

A Superconducting Cylinder in a Uniform Magnetic Field

Consider the long superconducting cylinder of radius R and length h. (R≪h)

iK

0H ,iH Where

)H(HnK

iH

0

102

12

0

H

H

H

z

zapp

The field distribution of a long superconducting cylinder in an applied field.

iK ,iH ,iH 0020 HHH zzapp

☜ A Hollow Superconducting Cylinder in a Uniform Magnetic Field

Example 3.2.3Consider solid body contains one or more holes in it.

☜ Zero-field cooled state: The field is applied after the cylinder is in the superconducting state.

A superposition argument showing the flux though the hole is zero.

☜ Field-cooled state: The field is applied before the cylinder is in the superconducting state.

A superposition argument showing how the flux through the hole is maintained. The field is applied before the cylinder is in the superconducting state.

☜ After turn off field,

Sequence Showing How Flux may Be Permanently Stored

3.3 Superconducting

Lumped Circuits

In this section, three such examples are presented:

(1) A flux transformer

(2) A memory element

(3) A magnetic monopole detector

☜ A Lumped Circuit Illustration of Flux Conservation

0)( , 0 Rctor supercondu aFor

where)(

flux Magnetic :inductance self& Mutual :L & M

,

221

112221

222

2212

iLdtd

MiRiiLdtd

Ridtdv

iLMi

coil second in theinitially isflux no if over time. conserved isctor supercondu aby enclosedflux The

221 iL

Memory Element

A Superconducting Memory Element

☜ Storing information (0 ->1) in a single memory element

Cryotron switch

(3) Magnetic Monopole Detector

A new set of electromagnetic equation

m

e

t

t

B, D

, JD B

, JBE

e

m

“e” and “m” refer to quantities associated with the electric and magnetic field respectively.

A Magnetic Monopole Detector

The conservation law of magnetic charge.

0J 0Jm

tt m

Consider a single loop of superconducting wire. (there are no field inside it)

charge Magnetic : where

,0)(

0 lE because 0

sJsB lE m

m

m

cm

s sc

Q

Qdtd

dIdtd

dddtdd

If magnetic charge(Qm

) passes through the loop, trapped flux (Ф) should be changed,

3.4 The Two-Fluid Model

※ The purpose of this section is to complete the classical model so that

it includes the notion superconductivity involves a thermodynamic

change of state.

Two fluid = normal electrons + superelectrons

★ normal electrons: responsible for the currents in a normal material.

★ superelectrons: responsible for the currents in the superconductor.

※ The issues of thermodynamic transition

To answer by studying an experiment.

λ

= λ(T) : penetration depth changes as a function of temperature.

)(1~

)()(

*0

2**

*

0 TnqnmT

☜ Data showing the temperature dependence of the London penetration depth, λ(T), in Mercury.

c

40

TTfor )/(1

)(

ally,Experiment

cTTT

λ0

: the value of the penetration depth at T = 0 K.

※ Superconductivity is thermodynamic transition

Λ

= λ(T) : penetration depth as a function of temperature.

Temperature dependence of Magnetization

022

0

*0

4*0

*

240

02**

*

0

*)(* , ]1[)(

)(Λ ,

)(1

)(

qmnTTfor

TTnTn

q*n*m*

T/Tc-qnm

cc

For temperatures above the Tc , the number density of superelectrons

remain zero.

Temperature dependence of superelectron density (n*)

totnn21*

0 (ntot : the density of normal electron at normal state)

Superelectron (Cooper pair) is in reality a correlation of two actual electrons and its density is given by,

Assume : T = 0 K → there are no unpaired electrons left in the material. ntot is not a function of temperature.

cc

tot

ctottottot

ctottot

cc

tot

TTforTTnTn

TTnnTnn

TallforTTnTnTnTnn

TTforTTnTn

)(

electrons unpaired ofdensity theisn Where

/)(

]/1[212)( )(2)(

]1[21)(

4

4

4*

4*

,T Tfor JJ (J)density current Total csn

]1[)( From4

*0

*

cTTnTn

The thermodynamic relations of current density in the superconducting state.

)J)((E

E1

J

JJJ

s

0n

sn

Tt

j tr

The normal current is given by Ohmic law:

)J(E

t

The supercurrent is given by the first London equation:

(λ(T) : temperature dependent penetration depth)

E)11

(JJJ 20

0sn

jj tr

tjeJ 0ReJ

trj

1)( ,EJ 0

B)J)(( s T

Let us rewrite the 2nd London Eq. in terms of the supercurrent.

=>

The normal and superelectron densities change as a function of temperature.

42

0 )(

ctot

tr

TTnTnand

mnq Equations; yields the result

42

0 )(

c

trtot

TT

menT

Using equations , we express Λ(T) as))(1(21)( and

)(4*

2**

*

ctot T

TnTnqn

m

cctot

TTforTTen

mT

)/(1

1)( 42

The superelectrons

are really pairs of electrons so that

m* = 2m and

q* = 2q = -2e

① ---- 0B]2)1(1[

0B))(1(

222

22200

22

trtr

trtr

jj

t

sn JJJH

The magnetic diffusion equation

(See next page detailed solution)

This equation correctly describes both the electrodynamics and thermodynamics of the flux inside a superconductor, assuming ω≪ωpair

.

Solving this expression for Jn , we then substitute it into Equation n0

J)1()(

1EtT tr

By taking the curl of the resulting relation

We rewrite (under sinusoidal steady-state conditions)

86)-(2 ---- depth)(skin 2 where0

tjeB 0ReB

0B)(1

)BB)(1(BΛ ofon substitutiAfter

)BBΛ)(1(BΛ

)ΛBB/)(1(1Blaw Faraday' and BΛJ ,0B using

curl a taking )JH)(1(1E

)JH)(1(1J)1(1E

J)1(1E ,JJJH

22200

22

2200

2

20

02

00

02

0

s

s0

s0

n0

n0

sn

t

tt

tt

/tt

t

tt

t

trtr

tr

tr

tr

tr

trtr

tr

222

2222

1

001

20

0sn

21 and 0B)1(0B))(21(lim

limits freq. lowfor E)))((

1)((Jlim

49)-(3 ---- E)))((

11

)((JJJ

2

jkk

j

TjT

TjjT

tr

tr

tr

①

An infinite slab of finite thickness.

icoshcoshB 00 zka

kyHμ

If we used the two-fluid model when solving for the field distribution in the SC slab

where the complex wave number is given by

This depends on temperature and is consistent with two-fluid model.

22

2

)(2

)(1)(

Tj

TTk

Pairing energy εpair : There must be some energy that is characteristic of the attraction between two paired electrons.

cBpair Tk

(There are no paired electrons above the Tc .)

(kB

:Boltzmann’s constant)

This same pairing energy is called the “energy gap” in the BCS theory and is given by the symbol 2△.

2h

the energy of an object to its frequency ( in quantum mechanics)

h : Plank’s constant

=> We can determine the energy of an individual electron by its frequency.

ccB

pair TThk 111039.12

Ex) In the high critical temperature superconductors, Tc

~ 100K

Maximum frequency: f (=ω/2π) ≈ 2.2 THz