Alexandrov A.S.,Theory of Superconductivity - From Weak to Strong Coupling

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Phenomenology

by several orders of magnitude in superconductors with a large GL parameter, as

follows from the relation

= 2

1/2

κ H

. (1.100)

At the border between type II and I superconductors, where κ = 2

−1/2

, all

critical ﬁelds should be the same, H

= H

, in agreement with

equation (1.100). The H –T phase diagram of type I and II bulk superconductors

is shown in ﬁgure 1.5. The ﬁnite size of the sample and its surface might affect

the magnetic ﬁeld penetration and the superconducting nucleation. In particular,

even type I superconductors could be in an intermediate state with normal and

superconducting domains, if the ﬁeld H < H

, while the superconducting

nucleation in type II superconductors could take place at the surface in a ﬁeld

< H < H

1.6.5 Vortex lattice

Vortex conﬁguration in the ﬁeld H

< H < H

can be found by numerical

integration of nonlinear GL equations. If H is only slightly less than H

the proﬁle of the order parameter can be established analytically [24]. In

the superconducting state close to H

(T ) line the order parameter φ(x , y) is

represented by a linear combination of degenerate solutions of the linearized GL

equation with n = k

= 0, equation (1.91),

φ(x, y) =



c(k

exp



−

(x − x

)

2ξ



. (1.101)

We can expect the minimum of the free energy to correspond to a regular

arrangement of the vortex lines. This happens because vortices repel each other.

To minimize their repulsive energy, the lines will take a regular arrangement so

that φ(x, y) is periodic in the x and y directions. To see how the repulsion occurs,

let us consider two p = 1 vortices with their cores at ρ

and ρ

in the plane

perpendicular to B in a type II superconductor with κ  1. The free energy is

given by the London expression:

≈

8π



dr (B

+ λ

|∇ × B|

)

Lλ

4π



dρ(h

+|∇×h|

) (1.102)

where the integral is taken in the plane outside the vortex cores. The magnetic

ﬁeld h(ρ) is the superposition of the ﬁelds h

(ρ) and h

(ρ) of each vortex,

h(ρ) =

(|ρ − ρ

|) + K

(|ρ − ρ

|)]. (1.103)

Ginzburg–Landau theory





Figure 1.6. Two parallel vortex lines repel each other.

Integrating equation (1.102) by parts and taking the limit κ →∞, one obtains in

the leading order

= 2

. (1.104)

The interaction energy U

is given by

Lλ

2π



· h

×∇×h

(1.105)

where the contour integral is taken over the surface of the second vortex core.

Integrating, one obtains the positive interaction energy

Lλ

(|ρ

− ρ

|). (1.106)

The free energy of the vortex state with two vortex lines is larger than twice the

single vortex energy. Hence, two vortices repel each other. The supercurrents of

the two vortices are added to each other in the outer regions but they are subtracted

from each other in the region between two vortices, ﬁgure 1.6. According to

the Bernoulli law, the pressure is higher where the velocity of an ideal liquid is

lower. Hence, the pressure is higher in the region between two vortices, creating

a repulsive force between them,

=−∇U

. (1.107)

The magnetic ﬁeld in the core of the ﬁrst vortex, created by the second vortex

is B

= 2

1/2

(|ρ

− ρ

|)/κ. Applying the Maxwell equation, ∇×B

4π j

, the repulsion force applied to the ﬁrst vortex is expressed through the

current j

of the second vortex as

= L

× n (1.108)

where n is a unit vector parallel to the ﬁeld.

Let us now assume that the order parameter in the vicinity of the H

(T ) line

is periodic with period a. The translation in the y direction by a does not change

φ(x, y) if k

= 2πl/a in the sum (1.101) with the integer l = 0, ±1, ±2,....

Hence, for H slightly less than H

,wehave

φ(x, y) =



2πily/a

exp



−

[x − πl/(ea H

)]

2ξ



. (1.109)

Phenomenology

In order that φ(x , y) also be periodic in x, it is necessary to impose a periodicity

on the coefﬁcients c

. A square lattice is obtained if c

l+1

= c

.Thenφ(x , y)

is periodic with period a = (2π)

1/2

ξ. A triangular lattice is obtained using the

condition c

l+2

= c

and c

= ic

. The latter is more stable in a perfect crystal.

With a further lowering of the magnetic ﬁeld, the density of vortices drops and the

period of the vortex lattice becomes larger than the coherence length. The vortex

lattice in type II superconductors is directly observed using ferromagnetic powder

sputtered on the sample surface and in neutron scattering experiments.

1.6.6 Critical current

If the current density is too high, the superconducting state is destroyed. The GL

phenomenology allows us to calculate the critical current density j

. The simplest

case is the uniform current distribution across the sample. This condition is found

in a thin superconducting ﬁlm, when the thickness of the ﬁlm is small compared

with the magnetic ﬁeld penetration depth and the coherence length. The magnetic

ﬁeld due to the current in the ﬁlm is proportional to the ﬁlm thickness. Hence, if

the ﬁlm is sufﬁciently thin, we can neglect the magnetic energy in the free energy

density f

≈ f

+ αn

βn

∗∗

. (1.110)

Minimizing f

as a function of the condensate density n

, we obtain

=−



α +

∗∗



. (1.111)

Hence, the supercurrent density j = e

∗

is a nonlinear function of v

j =−

∗



α +

∗∗



(1.112)

with a maximum at v

= (2m

∗∗

|α|/3)

1/2

. The maximum (i.e. critical) current

density is

4e(m

∗∗

)

1/2

|α/3|

3/2

. (1.113)

The Landau criterion of superﬂuidity allows us to understand j

at a more

microscopic level. Let us consider a superﬂuid, which ﬂows with a constant

velocity v. In a homogeneous system, the elementary excitations of the liquid

have well-deﬁned momenta k, so that their energy 

is a function of k. Because

of the friction, the kinetic energy of the condensate may dissipate and the ﬂow

would gradually stop. This process creates at least one elementary excitation.

In the coordinate frame moving with the liquid, the momentum and energy

Josephson tunnelling

conservation in a ‘collision’ with an object (such as the retaining wall) takes the

following form:

−Mv =−Mv



+ k (1.114)

and

2

+ 

. (1.115)

Here M is the mass of the object and v



is the liquid velocity after the collision.

Combining these two equations, one obtains



+ k · v +

= 0. (1.116)

The critical value of the velocity is obtained for M =∞as

= min







. (1.117)

The liquid is superﬂuid, if v

= 0 and the ﬂow is slow enough, v v

.In

BCS theory (chapter 2), the excitation spectrum has a gap near the Fermi surface

(k = k

) and min 

= >0. Hence, there is a non-zero critical velocity in

the BCS superconductor, v

= /k

. Flow with a higher velocity leads to pair

breaking and a loss of superconductivity. In the ideal Bose gas 

= k

/2m

∗∗

and v

= min[k/2m

∗∗

]=0. Hence, Bose–Einstein condensation alone is not

sufﬁcient for superﬂuidity. According to Bogoliubov [4], the repulsion between

bosons modiﬁes their excitation spectrum so that the repulsive Bose gas is a

superﬂuid (chapter 4).

The critical current in the vortex state of bulk type II superconductors does

not reach the pair-breaking limit. The current j in the direction perpendicular to

the vortex lines creates the Lorenz force applied to the vortex core, as follows

from equation (1.108) with j

= j. As a result, vortices move across the

current and the current inevitably ﬂows through their normal cores. This vortex

motion leads to the energy dissipation. Hence, an ideal type II superconductor

has zero critical current in the vortex state, when H

< H < H

.However,if

the external current is not very large, different defects of the crystal lattice ‘pin’

vortices preventing their motion. That is why the critical current of real type II

superconductors depends on the sample quality. Disordered samples can carry the

critical current density of about 10

Acm

−2

or even higher [28].

1.7 Josephson tunnelling

Let us consider a bulk superconductor separated into two parts by a thin contact

layer with different properties from those of the bulk. The layer might be

an insulator, a normal metal or any weak link with a reduced condensate

density or a small cross section. Its thickness is supposed to be small

compared with the coherence length (ﬁgure 1.7). Josephson [29] predicted

Phenomenology

that a supercurrent can ﬂow through the weak link without any voltage and

it oscillates as a function of time if there is a voltage drop on the contact.

Following Feynman [30], we can understand this effect in the framework of the

London description of superconductivity in terms of the condensate wavefunction

φ(x, t) = n

1/2

exp[i(x , t)] (x is the direction perpendicular to the contact). If

we suppose, that there is a supercurrent j = (e

∗

∗∗

d/dx along x in the

sample, the condensate phase should depend on x (ﬁgure 1.7). To keep the same

current density along the sample, the phase gradient should be larger in the weak

link, where n

is suppressed. Hence, there is a ﬁnite phase shift ϕ = 

− 

the condensate wavefunction on the right- (φ

) and left-hand (φ

) ‘banks’ of the

contact, x =±0. The carriers change their quantum state from φ

(t) to φ

(t) in

the contact with a transition amplitude K , which is a property of the weak link.

Hence, one can write

(t) = K φ

(t) (1.118)

and

(t) = K φ

(t) + e

∗

V φ

(t) (1.119)

where e

∗

V is a change in the electrostatic energy at the transition due to voltage

V . Taking time derivatives and separating the real and imaginary parts of these

equations, one obtains

˙n

= 2K (n

)

1/2

sin ϕ (1.120)

˙n

=−2K (n

)

1/2

sin ϕ

and

˙ϕ = e

∗

V + K [(n

)

1/2

− (n

)

1/2

]cos ϕ. (1.121)

The current I through the contact is proportional to the number of carriers

tunnelling per second, I ∝˙n

,sothat

I = I

sin ϕ (1.122)

where I

K (n

)

1/2

. If both banks of the Josephson contact are made from

the same superconductor, n

= n

and

˙ϕ = e

∗

As a result, we obtain

I = I

sin(ϕ

+ e

∗

Vt) (1.123)

where ϕ

is the phase difference in the absence of voltage.

We conclude that some current I < I

can ﬂow through the insulating thin

layer between two bulk superconductors with no applied voltage, if ϕ

is non-

zero. When the voltage is applied, the current oscillates with a frequency (in the

ordinary units)

2eV

. (1.124)

Josephson tunnelling

;



Φ([

[

Figure 1.7. Josephson’s weak link between two superconductors.

Figure 1.8. Magnetic ﬁeld dependence of the Josephson current.

The Josephson current also oscillates as a function of the applied magnetic ﬁeld.

Indeed, let us introduce a magnetic ﬁeld B into the dielectric layer parallel to the

contact (ﬁgure 1.7). The magnetic ﬁeld penetrates into the superconductor. The

size of the region along x with the magnetic ﬁeld and the supercurrent is about

w = t + 2λ

,wheret is the thickness of the layer.

Using the same arguments as in section 1.3 we obtain



dl · A(r) =

δ

∗

(1.125)

where the contour C crosses the contact twice as shown in ﬁgure 1.7. Here we

take into account that the contributions of the transport current ﬂowing along x are

mutually cancelled. The contribution of the screening current is negligible if the

length of the contour is much larger than w. The left-hand side of equation (1.125)

is an elementary ﬂux d

= Bw dy across the area of the contour, while δ is

the change of the phase shift along the contact area, so that

d

ϕ(y +dy) − ϕ(y)

∗

. (1.126)

We see that, in the presence of the magnetic ﬁeld, the phase shift and Josephson

current density j = j

sin ϕ(y) are not uniform within the contact area because

Phenomenology

(from equation (1.126)) ϕ(y) = e

∗

Bwy + ϕ

. The total current through the

contact is

I = j



L/2

−L/2

dy sin(e

∗

Bwy + ϕ

) = I

(sin ϕ

)



sin

(

π

/

)

π

/



(1.127)

where I

= j

S and S = L

is the contact cross section. The Josephson current

oscillates as a function of the magnetic ﬂux revealing a characteristic quantum

interference pattern (ﬁgure 1.8). The Josephson effect laid the groundwork for

designing squids (Superconducting Quantum Interference Devices). Squids can

measure magnetic ﬁelds and voltages as low as 10

−11

G and 10

−15

V, respectively.

Chapter 2

Weak coupling theory

The phenomenological Bose-gas picture rendered no quantitative account for the

critical parameters of conventional low-temperature superconductors. Its failure

stemmed from the very large diameter ξ of pairs in conventional superconductors

which can be estimated using the uncertainty principle, ξ ≈ 1/δk.The

uncertainty in the momentum δk is estimated using the uncertainty in the kinetic

energy δE  v

δk, which should be of the order of T

, the only characteristic

energy of the superconducting state. Therefore, ξ ≈ v

turns out to be about

1 µm for simple superconducting metals where T

 1 K and the Fermi velocity

 10

cm s

−1

. This means that pairs in conventional superconductors strongly

overlap and the Ogg–Shafroth model of real space pairs cannot be applied.

An ultimately convincing theory of conventional superconductors was

formulated by Bardeen, Cooper and Schrieffer [2]. In this and the next chapters

we describe the essentials of BCS theory using the Bogoliubov transformation [4]

in the weak-coupling regime and Green’s functions for the intermediate coupling

(see also the excellent books by Schrieffer [31], Abrikosov et al [32] and De

Gennes [33]).

2.1 BCS Hamiltonian

The BCS theory of superconductivity [2] relies on Fr¨ohlich’s observation [9]

that conduction electrons could attract each other due to their interaction with

vibrating ions of the crystal lattice. In a more complete analysis by Bardeen

and Pines [34] in which Coulomb effects and collective plasma excitations were

included, the interaction between electrons and the phonon ﬁeld was shown to

dominate over the matrix element of the Coulomb interaction near the Fermi

surface. The key point is the fact discovered by Cooper [35] that any attraction

between degenerate electrons leads to their pairing no matter how weak the

attraction is. One can understand the Cooper phenomenon by transforming the

corresponding Schr¨odinger equation for a pair into the momentum space. Because

of the Pauli principle, a pair of electrons can only ‘move’ along the Fermi surface

Weak coupling theory

in the momentum space. But it is well known that in two dimensions a bound

state exists for any attraction, however weak. The Cooper solution of the two-

particle problem in the presence of the Fermi sea demonstrated the instability of

an attractive Fermi liquid versus pair formation with a possible Bose–Einstein

condensation in the momentum space. Because of the Pauli principle, two

electrons with parallel spins tend to be at larger distances than two electrons with

antiparallel spins. Hence, following BCS, one can expect electrons to be paired

into singlets rather than into triplets with the zero centre-of-mass momentum of

the pairs. For these reasons, BCS introduced a model (truncated) Hamiltonian in

which the interaction term contains only the attraction between electrons with the

opposite spins and momenta,

H =



k,s=↑,↓

†



†

k↑

†

−k↓

(k) (2.1)

where

(k) =





V (k, k



−k



↓



↑

. (2.2)

The ﬁrst term describes independent Bloch electrons, where c

is the electron

annihilation operator with the momentum k and spin s, ξ

= E

− µ is the

band energy dispersion referred to the chemical potential µ. It is convenient to

consider an open system with a ﬁxed chemical potential rather than with a ﬁxed

total number of particles N

1 to avoid some artiﬁcial difference between odd

and even N

. H −µN

 has a minimum in the ground state for the open system.

That is why the electron energy in equation (2.1) is referred to as µ. We recall

(appendix C) that annihilation and creation operators anticommute for fermions:

, c



}≡c



+ c



= 0 (2.3)

and

, c

†



}≡c

†



+ c

†



= δ



. (2.4)

They lower or raise the number of fermions n

in a single-particle state |k, s

by one and multiply a many-particle wavefunction of non-interacting fermions by

±n

or ±(1 − n

) as

,...,n

,...= ±n

,...,n

− 1,... (2.5)

†

,...,n

,...= ±(1 − n

)|n

,...,n

+ 1,....

Here + or − depends on the evenness or oddness of the number of occupied

states, respectively, which precede the state |k, sin an adopted ordering of single-

particle states. Modelling the attractive potential V (k, k



), BCS took into account

the retarded character of the attraction mediated by lattice vibrations. The ions

of the lattice must have sufﬁcient time to react, otherwise there would be no

overscreening of the conventional Coulomb repulsion between electrons. In

Ground state and excitations

other words, the characteristic time τ

for the scattering of one electron with

the energy ξ

by another one should be longer than the characteristic time of

lattice relaxation which is about the inverse (Debye) frequency of vibrations

−1

. Using the energy–time uncertainty principle, we estimate the scattering

time as τ

≈|ξ

− ξ



−1

. Hence, V (k, k



) could be negative (i.e. attractive),

if |ξ

− ξ



| <ω

. To describe the essential physics of superconductors, BCS

introduced a simple approximation for the attractive interaction:

V (k, k



) =−2E

(2.6)

if the condition

|ξ

|, |ξ



| <ω

(2.7)

is satisﬁed, and zero if otherwise. Here E

is a positive energy depending on

the strength of the electron–phonon coupling. Multiplying it by the density of

electron states per unit cell at the Fermi level N(E

) (appendix B), we obtain a

dimensionless constant

λ = 2E

N(E

) (2.8)

which conveniently characterizes the strength of the coupling. The BCS theory

was originally developed for weakly coupled electrons and phonons with λ  1;

extended towards the intermediate coupling (λ

1) by Eliashberg [36]; and to

the strong coupling (λ

1) by us [10,11].

2.2 Ground state and excitations

The operators

,



†

in the BCS Hamiltonian (2.1) annihilate and create pairs

with total momentum K = 0. One can expect that the pairs condense in

the ground state with K = 0 like bosons below some critical temperature T

(appendix B). The number of condensed pairs N

should be macroscopically

large, if T < T

. The matrix elements of

 and



†

should be large as

well, like the matrix elements of the annihilation and creation boson operators,

N

− 1|b

=N

+ 1|b

†

=(N

)

1/2

1 (appendix C). That is all

commutators of

,



†

should be much smaller than the operators themselves

because [b

†

]=1 (N

)

1/2

. Hence, we can neglect the fact that

 does not

commute with



†

and c

†

. It is the same as replacing these operators by their

expectation values in the open system,

(k) ≈ 

(2.9)

where



=−2E

(ω

−|ξ





(ω

−|ξ



|)c

−k



↓



↑

 (2.10)