1.1 Introduction

1.2 Basic properties of superconductors

1.3 Models of superconductivity

1.4 Simple superconducting systems

Chap1. A survey of superconductivity

1.1 IntroductionThe potential applications of superconductivity in engineeringFor example, the area of magnetic field production and detection.

• Superconducting Magnet : MRI (2 T), Experimental magnet (20 T)- A commercial superconducting magnetic flux density- 20 T (about 200,000 times larger than the Earth Field (0.3 ~ 0.6 G))

• SQUIDs (superconducting quantum interference devices)- Magnetometers use superconductors to detect fluxes smaller than10-15 T-m2 (10-7 Gauss-cm2 << Hearth).

• Maglev train, Electric cable, Motor, Generator, and so on…

1.2 Basic properties of superconductors

·▣Physical properties☞ Perfect conductivity (Zero-resistance)☞ Perfect diamagnetism (Meissner effect)

▣ The discovery of superconductivity☞ The science of cryogenics ☞1898, James Dewar : liquid hydrogen (20 K)☞1908, H.K. Onnes : liquid helium (4.2 K, 1atm)☞1911, (Onnes’ s assistant Gilles Holst) : superconducting state

(Figure 1.1) of Hg (mercury)☞ So far, Hg-based superconductor is the highest Tc superconductor:HgBa2Ca2Cu3O10 at 135 K (-138℃) and 150 K under high-pressure.

In this Section, we will briefly examine some of those properties of that are important to us.

Kamerlingh Onnes reported, now with complete confidence, that below a critical temperature, the mercury had “passed into a new state, which may be called the superconducting state.” in modern notation;

☜ Figure 1.1 The original resistance versus temperature curve KamerlinghOnnes reported to announce the discovery of superconductivity in mercury.

Tc: critical temperature, Jc : critical current density

Hc : critical magnetic field )1.1())(1()( 20 −−−−−≈

ccc T

THTH

where Hc0 is the critical field at T=0 K.

Perfect conductivity (Zero-resistance)

Meissner effect (perfect diamagnetism)

In 1933 - Walter Meissner and Robert Ochsenfeld

A material that, when placed in a magnetic field, tries to minimizethe magnetic flux density, B, inside itself, is known asdiamagnetic, Thus a superconductor, which seeks to maintain the condition B=0 within itself, is called a perfect diamagnetism.

The superconductor’s ability to expel flux, now known as the Meissner effect, is an indication that

Superconductivity is: - Perfect conductivity (R = 0) - Perfect diamagnetism

0)(B =+= MHB π4

From the early days of superconductivity research, it was apparent that an important goal was to find materials with high Tc.

Fig 1.2 shows an example of this observation.

☜ Figure 1.2 The critical fields and temperatures of several superconductors. Each material remains superconducting for fields less than the critical tend to have higher maximum critical fields.

))(1()( 20

ccc T

THTH −≈

· In the late 1920s and early 1930s : a whole series of superconducting materials were discovered.

· Until 1950s : A15 materials (named by their crystal structure)- V3Ga(16 K), Nb3Sn(18 K), Nb3Ge(23 K) at J = 0, H = 0.

· 1970s : Chevrel phase materials (ternary compounds)

PbMo6S8(15 K, 60 T) [fig 1.2]

· 1986 : Highest Tc superconductor Nb3Ge (23 K)

· By the end of 1986 : K. Alex Műller & J. G. Bednorz (Novel price)

- La1.85Ba0.15CuO4(35 K)

· 1987 : Paul C. W. Chu at UH (Dr. W. N. Kang’s former advisor)

YBa2Cu3O7(95 K): enough to be cooled by liquid nitrogen ( 77 K)

History of typical superconductors

1993

2001Liquid H2

1.3 Models of superconductivity

· In Chapters 2 & 3 : Classical model of superconductivity- [resistance=0] + [perfect diamagnetism] → London equations

· In Chapter 4 : the classical model is sufficient for understanding suchapplications as prototypical superconducting memories, dispersionless waveguides, and magnetically levitated trains.

· In Chapter 5 : We develop macroscopic quantum model (MQM).- type I : does not violate the bulk Meissner effect- type II : allows flux to enter the bulk of its volume - single flux quantum Φ0=2.07 x 10-15 T-m2

· In Chapters 6 & 7 : Type II superconductors are often the materials used in high field, high power applications. For this reason,we study this class extensively.

In this section, we discuss a series of models for superconductivity that are explored in this book to various degrees.

▣ BCS theory (microscopic):

- Notion : electrons that carried lossy currents in the normal (metallic) state, pair together in the superconducting state.

=> Cooper pairs (superelectrons)

- The density of superelectrons is temperature dependent=> T↓ → cooper pairs electron↑

▣ Energy Gap (2Δ):The electrons in a Cooper pair are bound with an energy 2Δ that is typically on the order of 1 meV for conventional superconductivity (Tc ≤25 K). Usually 2Δ is referred to as the energy gap of the superconductor.

· In Chapters 8 & 9 : Josephson junction and their device

· In Chapter 10 : Ginzburg – Landau theory & BCS theory

1.4 Simple superconducting systemsPhase diagram of superconductivity

Figure 1.3 The phase space for the superconducting Nb–Ti alloy. The material is superconducting inside the volume of phase space indicated.

☜ There three critical parameters; Tc, Jc, and Hc.

Hc

Tc

Jc

▣ Lumped circuit model of superconducting system

The loop is initially placed in a static magnetic field such that flux threads the hole. After the system reaches the steady state, the field is suddenly turned off.

- Induced current in the wire =>Lenz’ s law

0,0 ==−− εε dtdiLiR

To understand the time dependence of the current, it is convenient to make a lumped circuit model of the system. Notice that the model, shown in Figure 1.4b.

From elementary circuit theory, we know that the current will decay in the exponential fashion

RLteItI τ/0)( −=

RL

RL =τ

The decay of the current is experimentally observed by measuring the magnetic field produced by the loop.

where the time constant τRL is given by

ㅡ (1. 2)

ㅡ (1. 3)

dtLR

idiiR

dtdiL −=−= ,

Suppose that the phenomenon observed by Kamerlingh Onnes is one of zero dc resistance. If the loop is made out of a superconductor

∞→→ RLRτ

0lim ㅡ (1. 4)

and so

0)( 0 ≥= tforItI ㅡ (1. 5)

A persistent current will flow in the superconducting loop and the magnetic field it produces will never be observed to decay.

Kamerlingh Onnes did perform this experiment, as have many other researchers since, and there has never been an observed decay ofthe threading flux caused by a nonzero value of resistance.

=> the experimental value of the resistivity is zero.

Figure 1.5 The charging of a superconducting electromagnet that takesadvantage of the flow of persistence currents. Everything inside the box is made from superconducting material.

▣ Persistent currents with superconducting electromagnet?Consider the magnet design schematically illustrated in Figure 1. 5.

Both the magnet solenoid and the connecting thermal switch are made from superconducting material.

· The switch is opened (heating) → T > Tc

· The switch is closed (cooling) → T < Tc

If we now turn off the current source, the superconducting electromagnet will stay charged because persistent currents willflow in the closed superconducting loop.

▣ Superconducting memory element.Since the magnetic field produced by these currents never decays, we can use this loop as a memory element. The memory element can store one bit of binary information :the logical 1 and 0 states respectively represent the cases of whether a field is or is not produced by the loop.

=> Chapter 21 state (B 0state (B

:)00:)00

≠≠==

ii

Chap2. Perfect Conductivity

CONTENTS

2.2 Circuits and time constant

2.3 Field theory and time constant

2.4 Magnetoquasistatics

2.5 The first London equation

2.6 Field inside a perfect conductor

2.1 Introduction◈ In this chapter, we begin modeling the superconductor using

Maxwell equation. ◈ Superconductor will be considered to the simple R-L-C circuits.

2.2 Circuits and time constantTo understand why time constants are a useful concept, we begin by examining the R-L-C circuit, as shown in Figure 2.1.

Figure 2.1

An R-L-C Circuit

We can express this voltage-current relation for;the resistor as

the inductor as

the capacitor as (2.2) ---------

(2.2) ---------

(2.1) ---------

CC

LL

RR

vdtdCi

idtdLv

Riv

=

=

=

0 0 2

2

2

2=++⇔=++ kx

dtdxR

dtxdm

Cq

dtdqR

dtqdL

◈We now find the impedance of our R-L-C circuit as seen by the source.

The drive of the system is an oscillation current that we can represent using complex numbers :

}ˆRe{)cos( tjeitii ωφω =+= ㅡ (2. 4)

where φjeii ≡ˆ ㅡ (2. 5)

Because the system is linear and time invariant, this voltage will oscillate with the same frequency as the drive:

ㅡ (2. 6)}ˆRe{ tjevv ω=We can therefore write

ivZ ˆˆ

)( =ω ㅡ (2. 7)

where Z (impedance) can be frequency dependent.

0 0 2

2

2

2=++⇔=++ kx

dtdxR

dtxdm

Cq

dtdqR

dtqdL

LL

L LijdtdiLv ω==

CC

C vCjdt

dvCi ω==

ㅡ (2. 8)

ㅡ (2. 9)

We can solve for Z(ω) :

1)()1(

)1)(()(

)1)((

)1(

/111

)(1

2 +++

=++

+=

+

++=+

+=

RCjLCjRLj

CjLjR

CjLjR

Z

CjLjR

CjLjR

CjLjRZ

ωωω

ωω

ωω

ω

ωω

ωω

ωωω

ㅡ (2. 10)

}{ and tjtj evveiiFrom ωω ˆRe}ˆRe{ ==

We will be better able to interpret the transfer function if we write it in terms of symbols that have more physical meaning.

Let us therefore define

the inductive time constant :

the capacitive time constant :

the coupling time constant :

RL

RL ≡τ

RCRC ≡τ

RCRLLC LC τττ =≡

ㅡ (2. 11)

ㅡ (2. 12)

ㅡ (2. 13)

τLC as the inverse of the resonant frequency of an L-C network. From these definitions, we can rewrite the transfer function in the more suggestive form:

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+=

RCLC

RL

jjRZ

ωτωτωτω

))(1(1)( 2 ㅡ (2. 14)⇒

+++

=1)(

)( 2 RCjLCjRLjZωω

ωω

These regimes are illustrated in Figure 2.2.

-(2.16)-----------

:limits frequency High

-(2.15)-----------

:limits frequency Low

0

0

1)1(

1)1(

>>⇒==>>

<<⇒==<<

LCLC

LCLC

ωττ

ωω

ωττ

ωω

)14.2())(1(

1)( 2 −⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+=

RCLC

RL

jjRZ

ωτωτωτω

0 LCτω /1=

Resonance Frequency (ω0 ):

Z(ω) has maximum value at LCτω /10 =

Figure 2.3: For “low” frequencies and “small” resistances, equivalent to the series R-L circuit.

Figure 2.4: For “low” frequencies and “large” resistances, equivalent to the parallel R-C network.

Figure 2.3 A series R-L circuit

for R << ZC and R ≈ ZL

Figure 2.4 A parallel R-C circuit

for R >> ZL and R ≈ ZC

)14.2())(1(

1)( 2 −⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+=

RCLC

RL

jjRZ

ωτωτωτω

Approximation for low frequency limits

(2.24)--------------- )11()(lim

1RC

RL

jjRZ

LC ωτωτω

ωτ ++

≈<<

(2.26)--------------- )1()(lim-(2.25)--------------- 1

:resistance smallFor )1

1 RL

RC

jRZRC

ωτωωτ

ωτ+≈

<<

<<

)14.2())(1(

1)( 2 −⎟⎟⎠

⎞⎜⎜⎝

⎛+−

+=

RCLC

RL

jjRZ

ωτωτωτω

(2.28)--------------- )1

1()(lim

-(2.27)--------------- 1 :resistance largeFor )2

1 RC

RL

jRZ

RL ωτω

ωτ

ωτ +≈

<<

<<

2.3 Maxwell EquationsIn section 2.2, we saw how it was much easier to find the response of a circuit to a particular source if we knew that only certain frequencies were used to drive the system.

The goal of this section is to extend this technique to distributed or macroscopic systems.The expression describes the Maxwell’s equation.

Symbol Name Units

E electric field volts/meter

H magnetic field amps/meter

D electric displacement coulombs/(meter)2

B magnetic flux density tesla

Ρ free charge density coulombs/(meter)3

J free current density amps/(meter)20=∇

=∇

+∂∂

=×∇

∂∂

−=×∇

B

D

JDH

BE

ro

ro

rr

r

rr

ρ

t

t ㅡ (2. 29)

ㅡ (2. 31)

ㅡ (2. 30)

ㅡ (2. 32)

ㅡ Faraday’s law

ㅡ Ampère’s law

ㅡ Gauss’s electric and magnetic law

The conversion is simple if we recall two mathematical theorems.

∫ ∫ ⋅×∇=⋅C S

dd s)C(lC

∫ ∫∑∇=⋅

Vdvd )C(sC o

Stoke’s theorem =>

Gauss’s theorem =>

ㅡ (2. 35)

ㅡ (2. 36)

0sJ 0J

0sB 0B

sD D

sJ sD lH JDH

sB lE BE

=+⋅⇒=∂∂

+∇

=⋅⇒=∇

=⋅⇒=∇

⋅+⋅−=⋅⇒+∂∂

=×∇

⋅−=⋅⇒∂∂

−=×∇

∫∫

∫∫∫

∫∫∫

∫∫

∑

∑

∑

V

V

SSC

SC

dvdtdd

t

d

dvd

dddtdd

t

ddtdd

t

ρρ

ρρ

o

o

o

ㅡ (2. 37)

ㅡ (2. 39)

ㅡ (2. 38)

ㅡ (2. 40)

ㅡ (2. 41)

Integral forms of Maxwell equation:

2.5 The first London equation

dragem ffv+=

dtdm

B))(vE(fem ×+= q

The goal of this section is to develop such a model of a true perfect conductor.

We start by examining a single carrier, an electron of mass m and velocity v, and consider its motion under Newtonian mechanics;

In general, for an object of charge q moving with a velocity v in an electromagnetic field, Lorentz’s law states that the force on the object is

ㅡ (2. 115)

ㅡ (2. 116)

where the only forces assumed to be action on the particle are those resulting from an externally applied elctromagnetic field, fem, and those resulting from collision, fdrag.

vpf dragtr

mdtd

τ−=−=

* τtr : scattering (transport) time

The magnetic field will never contribute a force in the direction that the particle is moving.

Efem q≈

Drude proposed fdrag.

With our forces thus defined, we return to Equation 2.115 and write down as expression for the velocity of a carrier:

Evv qmdtdm

tr

=+τ

ㅡ (2. 117)

ㅡ (2. 118)

ㅡ (2. 119)

Ejm

qvtr

tr ˆ1

1)(ˆωτ

τ+

= ㅡ (2. 120)

where v and E have in the same direction.

Ejm

qvEm

qjv

EqjmvEqvmvmj

eEqevmevdtdm

eEev

tr

trtrtr

trtr

tjtj

tr

tj

tjtj

ˆ1

1)(ˆ,ˆ)1(ˆ

ˆ)1(ˆˆˆˆ

ˆ)ˆ()ˆ(

ˆ,ˆ

ωτττωτ

τω

τω

τωωω

ωω

+=∴=+⇒

=+⇒=+⇒

=+⇒

==

Ev

vJ nq= ㅡ (2. 121)where n, the number density of carriers in the material, is in units of particles per unit volume.Combining Equation 2.120 and 2.121 yields

E1

1

E1

1)(J

0

2

tr

tr

tr

j

jmnq

ωτσ

ωττ

+=

+=

mnq trτωσ

2

0 )0( ≡=

where σ0 defined as

Comparing this relation to the dispersive form of Ohm’s law, Equation 2.49 ( J(r,ω)=σ(ω)E(r,ω) ), we obtain an expression for the conductivity:

trjωτσωσ

+=

11)( 0 ㅡ (2. 124)

ㅡ (2. 122)

ㅡ (2. 123)

20

nqm

trστ =

Because σ0 as defined is an expression for the steady-state (dc) conductivity, a value that can be easily measured or found in tables, we can invert Equation 2.123 to get a useful expression for the scattering time:

ㅡ (2. 125)

To obtain estimates of τtr, we choose copper.

We find the scattering time of copper,

sec104.2 14−×≈trτAs a result, for frequencies as great as 1 THz, ωτtr≪1, and the conductivity of copper is independent of frequency.

• For high frequency (ωτtr≫1), σ is approximately a pure imaginary number and thus power is not dissipated in the medium.• Electromagnetic waves can propagate through the material with negligible loss.

where we define Λ as

2∗∗

∗

≡Λqn

m

We wish to find an expression similar to Ohm’s law but in the limit τtr→∞.

J)E

J)E EJ

v)J J(v Ev

Evv

Λ∂∂

=⇒

∂∂

=⇒=

=⇐==

∞→=+

(

( 2

t

nqm

tq

dtd

nqm

nqnq

qdtdm

qmdtdm tr

tr

ττ

This special form of Ohm’s law was first proposed by Heinz and Fritz London in 1935 and is referred to as the first London equation.

ㅡ (2. 131)

ㅡ (2. 132)

JΗHCC)(C

2

2

×∇=−∇=×∇×∇

∇−⋅∇∇=×∇×∇

To properly solve a problem involving a real (lossless) perfect conductor, we must use the first London instead of the usual (nondispersive) form of Ohm’s law. Let us find the equation governing the time evolution of the magnetic field in a perfectly conducting system.

ㅡ (2. 133)

The goal is to again express the current density in terms of the magnetic field, only time we use the first London equation.

0H)( 20 =∂∂

∇−Λ tμ

ㅡ (2. 134)

: a characteristic length associated with the properties of the system, .

0/ μΛ

0H

HB BH)

H)J)E

J)E BE

=∂∂

∇−Λ

=←∂∂

−=∇Λ∂∂

−

∇Λ∂∂

−=×∇Λ∂∂

=×∇

Λ∂∂

=∂∂

−=×∇

t

tt

tt

tt

)(

)((

((

(,

20

02

2

μ

μ

2.6 Field inside a perfect conductorWe look at a homogeneous, isotropic, linear, nondispersive perfectly conducting slab (ε, μ0, Λ) that is subjected on both sides to the applied magnetic field

z0app i}ˆRe{H tjeH ω=

as shown in Figure 2.24.

As in that section, our goal is to find the magnetic field and current density inside the slab.

☜Figure 2.24 An infinite slab of finite thickness. The slab is made from perfectly conducting material.

ㅡ (2. 135)

We know the time dependence of the driving field, Happ is sinusoidal and so can express Equation 2. 134 as

0

0,H

=−Λ

⇒

=∂∂

∇−Λ

(y)Hdydj

t

ˆ)(

)(

2

20

20

μω

μ

0ˆ)( 2

20 =−

Λ(y)H

dydμ

kyC(y)H coshˆ =

For nonzero frequencies, this relation simplifies to

0μλ Λ≡

where we defined the penetration depth

ㅡ (2. 136)

ㅡ (2. 137)

ㅡ (2. 138)

Λ= 02 μk ㅡ (2. 139)

λ1

=k ㅡ (2. 140)

ㅡ (2. 141)

zapp iH }ˆRe{ 0tjeH ω=

2cosh,

2sinh

xxxx eexeex−− +

=−

=

)/cosh(1ˆ

0 λaHC =

z0 i})/cosh()/cosh(ˆRe{H tje

ayH ω

λλ

=

xtje

ayH i}

)/cosh()/cosh(ˆRe{J 0

ω

λλ

=

☜ Figure 2.25 The distribution of field quantities in a perfect conductor; (a) (a/λ) « 1

and (b) (a/λ) » 1.

Boundary condition at y=(+a, -a) gives

⇒×

Such as Nb-Ti and Nb3Sn, as well as the high-temperature superconductors, λ is typically on the order of 0.1 μm.

∇= HJ

ㅡ (2. 142)

ㅡ (2. 143)

ㅡ (2. 144)

When we apply DC magnetic field:

z0app iH H= ㅡ (2. 145)

Let us carry this integration out for the specific case of our infinite slab subjected to a tangential magnetic field. From Equation 2.134, we obtain the differential equation:

0)H()(

0H)(

2

20

20

=∂∂

∂∂

−Λ

⇒

=∂∂

∇−Λ

y,tty

tμ

μ

ㅡ (2. 146)

Again we again find the spatial variation

)/cosh()(),( λytCtyHt

=∂∂

ㅡ (2. 147)

)/cosh()/cosh(),(),(

λλ

aytaH

ttyH

t ∂∂

=∂∂

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

ayaHtaHtyH +−=λλ

where C(t), though constant in space, is time dependent.

At the boundary planes, y=(+a, -a);

Notice that we now must integrate this expression in time. The general solution to our perfectly conducting slab problem is therefore

)/cosh()(),( λytCtyHt

=∂∂

ㅡ (2. 147)

ㅡ (2. 148)

ㅡ (2. 149)

☜ Figure 2.26 Thought experiments showing that a superconductor dose not behave as a perfect conductor

Perfect conductor and Superconductor

A perfect conductor is a flux conserving medium; a superconductor is a flux expelling medium.

)0,(),(),0,(),(

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

aHtaHbecauseyHtyH

yHayaHtaHtyH

==

+−=

λλ

Zero-FC FC

Since the magnetic fields are created primarily by electrical currents in an MQS system, Ampère’s law is replaced by the approximation.

JDH from 0)D( JH +∂∂

=×∇≈∂∂

≈×∇tt

Q

C and JH from J 0)()(0 =×∇∇∇=×∇∇≈∇ oooo

0D ≈=∇ ρo

Magnetoquasistatic (MQS) approximation(Good Conducting Block)

Time evolution of the magnetic field inside the ohmic conductor.

t∂∂

−=×∇×∇=∇−∇∇

∇−∇∇=×∇×∇⇐×∇=×∇×∇BE , J HH)

CC)C JH

2

2

o

o

(

(

Equation diffusionmagnetic 0H

)tH)((σH E,σJ E,σJ 0

200

⇐=∇−∂∂

∂∂

−=∇−×∇=×∇=

)( 20 t

μσ

μ

Magnetic diffusion time (τm).

If Equation 2.66 is multiplied by l 2, where l is the dimensions of our system,

20

2220 )( lll μστμσ ≡=∇−

∂∂

m 0,Ht

The expression for the inductance associated with the block is

⎟⎠⎞

⎜⎝⎛ =⇐=

dAL

dwhL μμ

mRL wdwh

hwd

RL τμσμσμστ ===== 2

02

00 l

Using this expression along with that for the equivalent lumped resistance (Equation 2.61), we find that indeed τm is equal to τRL given that we use wto characterize the size of the system.

11 20 <<⇔<< aa ωμσ

δ

Suppose we drive the system at a frequency ω such that (a/δ) « 1. in this limiting case, expanding Equation 2.88 to linear terms in (y/δ) yields the result that Ĥ≈Ĥ0. => Figure 2.14

From Equation 2.86, we find by direct substitution the equivalent statement

or ωτm « 1.

☜Figure 2.14 The decay of the field quantities inside the slab in the limit where (a/δ) « 1.

ㅡ (2. 91)

depth skin size sample <<<< ,δa

We now see a way to physically interpret the other extreme, that in which ωμσ0a2 » 1, or equivalently (a/δ) » 1. In this case, the expression cosh(ka) in both Equation 2.88 and 2.90 is extremely large.

The result is plotted in Figure 2.15.

☜ Figure 2.15 The decay of the field quantities inside the slab in the limit where (a/δ) » 1.

As we see in the next example, a perfect conduction approximation greatly simplifies the calculations needed to solve an MQS problem.

depth skin size sample >>>> ,δa