Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 15 Proof# 1 - 29/07/09

456 15 Superconductivity

t1.1 Table 15.1. Transition temperatures of some selected superconducting elements

t1.2 Elements Al Sn Hg In FCC La HCP La Nb Pb

t1.3 T

(K) 1.2 3.7 4.2 3.4 6.6 4.9 9.3 7.2

Fig. 15.1. Temperature dependence of the resistivity of typical superconducting

metals

NORMAL STATE

SUERCONDUCTING

STATE

Fig. 15.2. Temperature dependence of the critical magnetic ﬁeld of a typical

superconducting material

Magnetic Properties 29

There is a critical magnetic ﬁeld H

(T ), which depends on temperature. When 30

H is above H

(T ), the material is in the normal state; when H<H

(T )itis 31

superconducting. A plot of H

(T )versusT is sketched in Fig. 15.2. 32

In a type I superconductor, the magnetic induction B must vanish in the 33

bulk of the superconductor for H<H

(T ). But we have 34

B = H +4πM =0 for H<H

(T ), (15.1)

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15.1 Some Phenomenological Observations of Superconductors 457

Fig. 15.3. Magnetic ﬁeld dependence of the magnetization M and magnetic induc-

tion B of a type I superconducting material

Fig. 15.4. Magnetic ﬁeld dependence of the magnetization M and magnetic induc-

tion B of a type II superconducting material

which implies that 35

M = −

4π

for H<H

(T ). (15.2)

This behavior is illustrated in Fig. 15.3.

In a type II superconductor, the magnetic ﬁeld starts to penetrate the 37

sample at an applied ﬁeld H

lower than the H

. The Meissner eﬀect is 38

incomplete yet until at H

.TheB approaches H only at an upper critical 39

ﬁeld H

. Figure 15.4 shows the magnetic ﬁeld dependence of the magneti- 40

zation, −4πM, and the magnetic induction B in a type II superconducting 41

material. Between H

and H

ﬂux penetrates the superconductor giving a 42

mixed state consisting of superconductor penetrated by threads of the mate- 43

rial in its normal state or ﬂux lines. Abrikosov showed that the mixed state 44

consists of vortices each carrying a single ﬂux Φ =

. These vortices are 45

arranged in a regular two-dimensional array. 46

Speciﬁc Heat 47

The speciﬁc heat shows a jump at T

and decays exponentially with an energy 48

Δ of the order of k

as e

−Δ/k

below T

, as is shown in Fig. 15.5. There is 49

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458 15 Superconductivity

Fig. 15.5. Temperature dependence of the speciﬁc heat of a typical superconducting

material

Fig. 15.6. Tunneling current behavior for (a) a normal metal–oxide–normal metal

structure and (b) a superconductor–oxide–normal metal structure

a second order phase transition (constant entropy, constant volume, no latent 50

heat) with discontinuity in the speciﬁc heat. 51

Tunneling Behavior 52

If one investigates tunneling through a thin oxide, in the case of two nor- 53

mal metals, one obtains a linear current–potential diﬀerence curve, as is 54

sketched in Fig. 15.6a. For a superconductor–oxide–normal metal structure, 55

a very diﬀerent behavior of the tunneling current versus potential diﬀerence 56

is obtained. Fig. 15.6b shows the tunneling current–potential diﬀerence curve 57

of a superconductor–oxide–normal metal structure. 58

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15.2 London Theory 459

Fig. 15.7. Temperature dependence of the damping constant of low frequency sound

waves in a superconducting material

Acoustic Attenuation 59

For T<T

and ω<2Δ, there is no attenuation of sound due to electron 60

excitation. In Fig. 15.7, the damping constant α of low frequency sound waves 61

in a superconductor is sketched as a function of temperature. 62

15.2 London Theory 63

Knowing the experimental properties of superconductors, London introduced 64

a phenomenological theory that can be described as follows: 65

1. The superconducting material contains two ﬂuids below T

(T )

is the 66

fraction of the electron ﬂuid that is in the super ﬂuid state.

(T )

= 67



1 −

(T )



is the fraction in the normal state. The total density of electrons

in the superconducting material is n = n

+ n

. 69

2. Both the normal ﬂuid and super ﬂuid respond to external ﬁelds, but the 70

superﬂuid is dissipationless while the normal ﬂuid is not. We can write the 71

electrical conductivities for the normal and super ﬂuids as follows: 72

(15.3)

but τ

→∞giving σ

→∞. 73

(T ) → n as T → 0,

(T ) → 0asT → T

4. To explain the Meissner eﬀect, London proposed the London equation

∇×j +

B =0. (15.4)

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460 15 Superconductivity

How does the London equation arise? Let us consider the equation of motion 76

of a super ﬂuid electron, which is dissipationless, in an electric ﬁeld E that is 77

momentarily present in the superconductor: 78

= −eE

where v

is the mean velocity of the super ﬂuid electron caused by the ﬁeld 79

E. But the current density j is simply 80

j = −n

. (15.5)

Notice that this gives the relation

= −n

E. (15.6)

Equation (15.6) describes the dynamics of collisionless electrons in a perfect

conductor, which cannot sustain an electric ﬁeld in stationary conditions. 83

Now, from Faraday’s induction law, we have 84

∇×E = −

B. (15.7)

Combining this with (15.6) gives us



∇×j +



=0. (15.8)

The solution of (15.8) is that

∇×j +

B = constant.

Because in the bulk of a superconductor the magnetic induction B must be

zero, London proposed that for superconductors, the “constant” had to be 88

zero and j = −

A [called the London gauge] giving (15.4). The London 89

equation implies that, in stationary conditions, a superconductor cannot sus- 90

tain a magnetic ﬁeld in its interior, but only within a narrow surface layer. If 91

we use the relation 92

∇×B =

4π

j, (15.9)

(This is the Maxwell equation for ∇×B in stationary conditions without the

displacement current

E.), we can obtain

∇×(∇×B)=

4π

∇×j = −

4π

B. (15.10)

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BookID 160928 ChapID 15 Proof# 1 - 29/07/09

15.2 London Theory 461

But, ∇×(∇×B)=∇(∇·B) −∇

B giving 95

∇

B =

4πn

B ≡

B. (15.11)

The solutions of (15.11) show a magnetic ﬁeld decaying exponentially with a 97

characteristic length Λ. One can also obtain the relation ∇

j =

4πn

j.The 98

quantity Λ

4πn

is called the London penetration depth.Fortypical 99

semiconducting materials, Λ

∼ 10 − 10

nm.Ifwehaveathinsupercon- 100

ducting ﬁlm ﬁlling the space −a<z<0 as shown in Fig. 15.8a, then the 101

magnetic ﬁeld B parallel to the superconductor surface has to fall oﬀ inside 102

the superconductor from B

, the value outside, as 103

B(z)=B

−|z|/Λ

near the surface z =0

and

104

B(z)=B

−|z+a|/Λ

near the surface z = −a.

Figure 15.8b shows the schematic of the ﬂux penetration in the supercon-

105

ducting ﬁlm. The ﬂux penetrates only a distance Λ

≤ 10

nm. One can show 106

that it is impossible to have a magnetic ﬁeld B normal to the superconductor 107

surface but homogeneous in the x − y plane. 108

Fig. 15.8. A superconducting thin ﬁlm (a) and the magnetic ﬁeld penetration (b)

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462 15 Superconductivity

15.3 Microscopic Theory–An Introduction 109

In the early 1950s Fr¨olich suggested that the attractive part of the electron– 110

phonon interaction was responsible for superconductivity predicting the iso- 111

topic eﬀect. The isotope eﬀect, the dependence of T

on the mass of the 112

elements making up the lattice was discovered experimentally independent 113

of Fr¨olich’s work, but it was in complete agreement with it. Both Fr¨olich, and 114

later Bardeen, attempted to describe superconductivity in terms of an elec- 115

tron self-energy associated with virtual exchange of phonons. Both attempts 116

failed. In 1957, Bardeen, Cooper, and Schrieﬀer (BCS) produced the ﬁrst cor- 117

rect microscopic theory of superconductivity.

The critical idea turned out to 118

be the pair correlations that became manifest in a simple little paper by L.N. 119

Cooper.

120

Let us consider electrons in a simple metal described by the Hamiltonian 121

H = H

+ H

,whereH

and H

are, respectively the unperturbed 122

Hamiltonian for a Bravais lattice and the interaction Hamiltonian of the elec- 123

trons with the screened ions. Here we neglect the eﬀect of the periodic part 124

of the stationary lattice to write H

by 125



k,σ

†

kσ



q,s

¯hω

q,s

†

q,s

where σ and s denote, respectively, the spin of the electrons and the three

126

dimensional polarization vector of the phonons, and a

q,s

annihilates a phonon 127

of wave vector q and polarization s,andc

kσ

annihilates an electron of wave 128

vector k and spin σ. We will show the basic ideas leading to the microscopic 129

theory of superconductivity. 130

15.3.1 Electron–Phonon Interaction 131

The electron–phonon interaction can be expressed as 132



k,q,σ



†

−q

+ a



†

k+qσ

kσ

, (15.12)

133

where M

is the electron–phonon matrix element deﬁned, in a simple model 134

discussed earlier, by 135

N¯h

2Mω

| q | V

Here V

is the Fourier transform of the potential due to a single ion at the 136

origin, and the phonon spectrum is assumed isotropic for simplicity. (In this 137

case, only the longitudinal modes of s parallel to q give ﬁnite contribution 138

J. Bardeen, L.N. Cooper, J.R. Schrieﬀer, Phys. Rev. 108, 1175 (1957)

L.N. Cooper, Phys. Rev. 104, 1189 (1956).

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15.3 Microscopic Theory–An Introduction 463

Fig. 15.9. Electron–phonon interaction (a) Eﬀective electron–electron interaction

through virtual exchange of phonons (b)and(c): Two possible intermediate states

in the eﬀective electron–electron interaction

to H

.) This H

can give rise to an eﬀective electron–electron interaction 139

associated with virtual exchange of phonons as denoted in Fig. 15.9a. The 140

ﬁgure shows that an electron polarizes the lattice and another electron inter- 141

acts with the polarized lattice. There are two possible intermediate states in 142

this process as shown in Fig. 15.9b and c. In Fig. 15.9b the initial energy is 143

= ε

+ ε



and the intermediate state energy is E

= ε

+ ε



−q

+¯hω

.In 144

Fig. 15.9c the initial energy is the same, but the intermediate state energy is 145

= ε

k+q

+ ε



+¯hω

. We can write this interaction in the second order as 146



f | H

| mm | H

| i

− E

. (15.13)

This gives us the interaction part of the Hamiltonian as follows:

147







σσ



| M

f|c

†

k+qσ

kσ

|mm|c

†



−qσ



†

|i

ε(k



)−[ε(k



−q)+¯hω

]

f|c

†



−qσ



−q

|mm|c

†

k+qσ

kσ

†

−q

|i

ε(k)−[ε(k+q)+¯hω

]

(15.14)

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464 15 Superconductivity

One can take the Hamiltonian 148

H = H

+ H





kσ

†

kσ



¯hω

†



k,q,σ



†

−q

+ a



†

k+qσ

kσ

(15.15)

and make a canonical transformation

149

−S

(15.16)

where the operator S is deﬁned by

150

S =



kqσ



αa

†

−q

+ βa



†

k+qσ

kσ

(15.17)

to eliminate the a

and a

†

−q

operators to lowest order. To do so, α and β in 151

(15.17) must be chosen, respectively, as 152

α =[ε(k) − ε(k + q) − ¯hω

]

−1

β =[ε(k) − ε(k + q)+¯hω

]

−1

(15.18)

Then, the transformed Hamiltonian is

153



kσ

†

kσ



kσ,k



W (k, q)c

†

k+qσ

†



−qσ



kσ

, (15.19)

154

where W

is deﬁned by 155

| M

¯hω

[ε(k + q) − ε(k)]

− (¯hω

)

. (15.20)

156

Note that when ΔE = ε(k + q) − ε(k) is smaller than ¯hω

, W

is negative. 157

This results in an eﬀective electron–electron attraction. 158

15.3.2 Cooper Pair 159

Leon Cooper investigated the simple problem of a pair of electrons interact- 160

ing in the presence of a Fermi sea of “spectator electrons”. He took the pair 161

to have total momentum P =0andspinS = 0. The Hamiltonian is written as 162

H =



,σ



†

σ

−







†





†

−



¯σ

−¯σ

σ

, (15.21)

163

where ε



¯h



, {c

σ

†





} = δ





σσ



, and the strength of the interaction, V , 164

is taken as a constant for a small region of k-space close to the Fermi surface. 165

The interaction term allows for pairscattering from (σ, −¯σ)to(



σ, −



¯σ). 166

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15.3 Microscopic Theory–An Introduction 465

Cooper took a variational trial function 167

Ψ=



†

kσ

†

−k¯σ

| G>, (15.22)

where | G is the Fermi sea of spectator electrons, | G = Π

k<k

|k|

†

kσ

†

−k¯σ

| 0 >. 168

If we evaluate 169

Ψ | H | Ψ = E, (15.23)

we get

170

E =2





∗



− V







∗







. (15.24)

The coeﬃcient a



is determined by requiring E{a



} to be minimum subject 171

to the constraint





∗



= 1. This can be carried out using a Lagrange 172

multiplier λ as follows: 173

∂

∂a

∗

(





∗



− V







∗







− λ





∗



=0. (15.25)

This gives

174

2ε

− V





− λa

=0. (15.26)

This can be written

175





2ε

− λ

. (15.27)

Deﬁne a constant C =





. Then, we have 176

2ε

− λ

. (15.28)

Summing over k and using the fact C =





we have 177

C = VC



2ε

− λ

, (15.29)

178

f(λ)=



2ε

− λ

. (15.30)

The values of ε

form a closely spaced quasi continuum extending from the 179

energy E

to roughly E

+¯hω

where ω

is the Debye frequency. In Fig. 15.10 180