Theory of Superconductivity - MyCourses · Superconductivity Superconductivity was the ﬁrst...

$: Theory of Superconductivity - MyCourses · Superconductivity Superconductivity was the ﬁrst experimentally discovered macroscopic quantum phe-nomenon, where a sizable fraction of$
Theory of Superconductivity

PHYS-0551 Low Temperature Physics V

Aalto University, School of Science

Spring, 2017

V. B. Eltsov

Department of Applied Physics, Aalto University

email: [email protected]

https://mycourses.aalto.fi/course/view.php?id=14702

2

Contents

1 Introduction to Superconductivity 7

1.1 Discovery of superconductivity . . . . . . . . . . . . . . . . . . . . . 7

1.2 Basic experimental properties . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Thermodynamics of the superconducting transition . . . . . . . . . . 11

1.4 The London model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Meissner effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Phase coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7 Magnetic flux quantization . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 The energy gap and coherence length . . . . . . . . . . . . . . . . . . 17

1.9 London and Pippard regimes, dirty and clean limits . . . . . . . . . . 19

2 The BCS theory 23

2.1 Landau Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Landau criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Phonon-mediated electron attraction . . . . . . . . . . . . . . . . . . 28

2.4 The Cooper problem . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 The BCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6 The gap equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 Condensation energy . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.8 Bogolubov quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . 42

2.9 Heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.10 The Bogolubov – de Gennes equations . . . . . . . . . . . . . . . . . 46

2.11 Electric current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Ginzburg-Landau theory 53

3.1 Landau theory of phase transitions . . . . . . . . . . . . . . . . . . . 53

3.2 Ginzburg-Landau equations . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Coherence length and penetration depth . . . . . . . . . . . . . . . . 59

3

4 CONTENTS

3.4 Critical field of a superconducting slab . . . . . . . . . . . . . . . . . 61

3.5 Energy of the normal–superconducting boundary. Type I and type II

superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.6 Abrikosov vortices. Critical field Hc1 . . . . . . . . . . . . . . . . . 66

3.7 Interaction of an Abrikosov vortex with electric current . . . . . . . . 71

3.8 Upper critical field Hc2 . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9 Fluctuations. Applicability of the GL theory . . . . . . . . . . . . . . 76

3.9.1 Uncharged superfluid, T > Tc . . . . . . . . . . . . . . . . . 77

3.9.2 Uncharged superfluid, T < Tc . . . . . . . . . . . . . . . . . 77

3.9.3 Ginzburg number . . . . . . . . . . . . . . . . . . . . . . . . 79

3.10 The Anderson-Higgs mechanism . . . . . . . . . . . . . . . . . . . . 80

4 Andreev reflection 83

4.1 Normal-superconducting interface . . . . . . . . . . . . . . . . . . . 83

4.2 Transmission and reflection amplitudes . . . . . . . . . . . . . . . . 85

4.3 Andreev equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Andreev bound states in SNS structures . . . . . . . . . . . . . . . . 91

4.4.1 Short SNS junctions . . . . . . . . . . . . . . . . . . . . . . 95

4.4.2 Long SNS junctions . . . . . . . . . . . . . . . . . . . . . . 96

4.4.3 Negative energies . . . . . . . . . . . . . . . . . . . . . . . . 97

4.4.4 Point contact . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5 Vortex core states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6 Transmission and reflection at the NIS interface . . . . . . . . . . . . 102

4.7 Bound states in the SIS contact . . . . . . . . . . . . . . . . . . . . . 104

5 Current in superconducting junctions 107

5.1 Supercurrent through an SNS structure. Proximity effect . . . . . . . 107

5.1.1 Short junctions. Point contacts . . . . . . . . . . . . . . . . . 108

5.1.2 Long junctions . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Superconductor–Insulator–Normal-metal interface . . . . . . . . . . . 115

5.2.1 Current through the NIS junction . . . . . . . . . . . . . . . 115

5.2.2 Normal tunnel resistance . . . . . . . . . . . . . . . . . . . . 119

5.2.3 Landauer formula . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.4 Tunnel current . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.5 Excess current . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.6 NS Andreev current. Current conversion . . . . . . . . . . . . 122

5.3 Supercurrent in the SIS contact . . . . . . . . . . . . . . . . . . . . . 122

CONTENTS 5

6 Josephson effect and weak links 125

6.1 D.C. and A.C. Josephson effects . . . . . . . . . . . . . . . . . . . . 125

6.1.1 Weakly coupled quantum systems . . . . . . . . . . . . . . . 125

6.1.2 Josephson effect in the GL model . . . . . . . . . . . . . . . 127

6.2 Extended Josephson junctions . . . . . . . . . . . . . . . . . . . . . 129

6.2.1 Low field limit. Field screening . . . . . . . . . . . . . . . . 131

6.2.2 Higher fields. Josephson vortices. . . . . . . . . . . . . . . . 131

6.3 Dynamics of Josephson junctions . . . . . . . . . . . . . . . . . . . . 134

6.3.1 Resistively shunted Josephson junction . . . . . . . . . . . . 134

6.3.2 Capacitively and resistively shunted junction . . . . . . . . . 135

6.3.3 Effective inductance . . . . . . . . . . . . . . . . . . . . . . 137

6.3.4 Current–voltage relations . . . . . . . . . . . . . . . . . . . . 137

6.3.5 Voltage bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4 Thermal fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.5 Superconducting Quantum Interference Devices . . . . . . . . . . . . 143

6.6 Shapiro steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7 Quantum phenomena in Josephson junctions 149

7.1 The Hamiltonian and charge operator . . . . . . . . . . . . . . . . . 149

7.2 Conditions for quantum dynamics . . . . . . . . . . . . . . . . . . . 151

7.3 Macroscopic quantum tunnelling . . . . . . . . . . . . . . . . . . . . 152

7.3.1 Effects of dissipation on MQT . . . . . . . . . . . . . . . . . 154

7.4 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4.1 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4.2 Bloch’s theorem in Josephson devices . . . . . . . . . . . . . 156

7.4.3 Large Coulomb energy: Free-phase limit . . . . . . . . . . . 157

7.4.4 Low Coulomb energy: Tight binding limit . . . . . . . . . . . 159

7.5 Bloch oscillations in Josephson junctions . . . . . . . . . . . . . . . 160

7.6 Phase qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

8 Unconventional superconductivity 167

8.1 Classification of superconducting states . . . . . . . . . . . . . . . . 168

8.1.1 Spin structure of the paired states . . . . . . . . . . . . . . . 168

8.1.2 Superfluid phases of 3He . . . . . . . . . . . . . . . . . . . . 170

8.1.3 Superconducting states in a crystal . . . . . . . . . . . . . . . 171

8.1.4 High-Tc cuprates . . . . . . . . . . . . . . . . . . . . . . . . 173

8.2 Generalized BCS theory . . . . . . . . . . . . . . . . . . . . . . . . 173

8.2.1 Mean-field Hamiltonian . . . . . . . . . . . . . . . . . . . . 174

8.2.2 Bogolubov transformation . . . . . . . . . . . . . . . . . . . 175

6 CONTENTS

8.2.3 Energy spectra. Gap nodes . . . . . . . . . . . . . . . . . . . 176

8.3 Thermodynamic quantities at T → 0 . . . . . . . . . . . . . . . . . . 178

8.4 Paramagnetic susceptibility and Knight shift . . . . . . . . . . . . . . 179

8.5 Density of states. Volovik effect . . . . . . . . . . . . . . . . . . . . 181

8.6 Josephson effect with internal phase difference . . . . . . . . . . . . 184

8.7 Multi-component order parameter and different phases . . . . . . . . 185

8.7.1 Superfluid 3He . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.7.2 Superconducting phases in a tetragonal crystal . . . . . . . . 187

Chapter 1

Introduction to

Superconductivity

Superconductivity was the first experimentally discovered macroscopic quantum phe-

nomenon, where a sizable fraction of particles of a macroscopic object forms a coher-

ent state, described by a quantum-mechanical wave function. Developing theory of

superconductivity and of later discovered superfluidity paved road to more general un-

derstanding of such systems. Nowadays even our Universe with its quantum vacuum

can be considered as a macroscopic quantum system.

Superconductors were also the first macroscopic quantum systems to find practical

applications. At present, the main applications in everyday life include generation of

high magnetic fields, for example, in MRI systems in hospitals, and filtering of radio

signals in particular in base stations of mobile telephony networks. The superconduc-

tors are also used in ultra-sensitive magnetic field and temperature sensors, ultra-low-

noise amplifiers etc. Many more applications including, for example, basic elements

of quantum computers are being currently developed.

In this course we discuss basic properties of superconductors and their theoretical

explanation.

1.1 Discovery of superconductivity

In 1908 H. Kamerlingh Onnes in Leiden liquefied 4He and opened a new area of re-

search in physics: studying properties of materials in the temperature range close to

absolute zero. One of the properties he was interested in was the resistivity of metals.

First measurements with gold and platinum showed that the resistivity did not change

7

8 CHAPTER 1. INTRODUCTION TO SUPERCONDUCTIVITY

ρn

ρ = 0

TTc

ρ

Figure 1.1: Below the transition temperature, the resistivity drops to zero.

at all below 4 K. It was found that this residual resistivity increases with concentration

of impurities in the metal and Kamerlingh Onnes turned to mercury, the most pure of

the metals, available at that time. The first measurement in 1911 brought a surprise:

The resistivity became unmeasurably small at T = 4.15 K. Addition of impurities to

the sample did not change this picture. He called this phenomenon superconductivity.

Later Kamerlingh Onnes found superconductivity in some other metals and also

improved sensitivity of his measurements trying to detect the residual resistance in

the superconducting state. In particular, he introduced a method of measuring decay

of persistent currents in a ring, but in a led ring he could not observe any decay as

long as liquid helium was present in the vessel. Later measurements put an upper

bound for resistivity as low as ρ . 10−23� cm. Thus it was concluded that in the

superconducting state resistance is really zero.

1.2 Basic experimental properties

Transition to the superconducting state changes many properties of a conductor. Typ-

ical behavior of the resistivity versus temperature is shown in Fig. 1.1: The resistivity

drops abruptly to zero at a certain temperature, called the critical temperature Tc. The

critical temperature is a property of the material. It varies in a large range for simple

metal superconductors and even in a larger range for compound materials. It is also

notable that poor conductivity in the normal state does not prevent obtaining high Tc

values. Properties of some simple metal and compound superconductors are summa-

rized in Tables 1.1 and 1.2 respectively.

The zero resistivity is a fascinating and fundamental property, but it is not enough

to understand the nature of the superconducting state. Another important discovery

was done by W. Meissner and R. Ochsenfeld in 1933: They found that magnetic field

1.2. BASIC EXPERIMENTAL PROPERTIES 9

Table 1.1: Parameters for metallic superconductors

Tc, K Hc, Oe Hc2, Oe λL, nm ξ0, nm κ Type

Al 1.18 105 50 1600 0.01 I

Hg 4.15 400 40 I

Nb 9.25 1600 2700 47 39 1.2 II

Pb 7.2 800 39 83 0.47 I

Sn 3.7 305 51 230 0.15 I

In 3.4 300 40 300 I

V 5.3 1020 40 ∼30 ∼ 0.7 II

Li 4 · 10−4 0.009 I

Table 1.2: Parameters for some compound superconductors

Tc, K Hc2, T λL, nm ξ0, nm κ Type

Nb3Sn 18 25 ∼200 11.5 II

La0.925Sr0.072CuO4 34 150 2 75 II

YBa2Cu3O7 92.4 150 200 1.5 140 II

Bi2Sr2Ca3CuO10 111 II

Tl2Sr2Ca2Cu3O10 123 II

HgBa2Ca2Cu3O8 133 II

MgB2 36.7 14 185 5 40 II

SmFeAsO0.85F0.15 52 400(?) 200 2 100 II

is expelled from a superconductor, Fig. 1.2. Remarkably, this happens not only when

the field is increased at temeperatures T < Tc, but also when the superconductor is

cooled from the normal state with the magnetic field applied. This property is now

called the Meissner effect.

It was found already by Kamerlingh Onnes that superconductivity is destroyed by

application of electric current above some critical value Ic or magnetic field above

the critical field Hc, which turns out to be temperature-dependent. A typical phase

diagram of a superconductor is shown in Fig. 1.3. Empirically the dependence of Hc

on temperature is described reasonably well by

Hc(T ) = Hc(0)[1− (T /Tc)2] . (1.1)

If in a superconductor the zero resistivity and the Meissner effect disappear at the


T > Tc T < Tc

B B

Figure 1.2: The Meissner effect: At T < Tc the magnetic field is expelled from the

superconductor (right) even if the field was applied before reaching Tc (left).

T

H

Tc

Hc(0)

superconductor

normal

1st order

2nd order

Hc(T )

Figure 1.3: Phase diagram of a superconductor.

same critical field Hc, then it is called the superconductor of type I. With increasing

number of discovered superconducting materials it was found that there are cases when

the magnetic field starts to partially penetrate into the sample at some lower field Hc1

while the zero resistivity disappears and the magnetic field penetrates sample fully only

at higher field Hc2. Such superconductors are called type II. Comparison of magneti-

zation curves of type I and type II superconductors is provided in Fig. 1.4.

Superconducting transition affects not only electrical properties of the material, but

also its thermodynamic properties, as expected for a true phase transition. In partic-

ular, heat capacity jumps upward at the transition and at lower temperatures rapidly

decreases. When phonon heat capacity is subtracted, one finds that the electronic heat

capacity decreases at low temperatures exponentially, C ∝ exp(−1/kBT ), where the

energy 1 is a material property. Typical temperature dependence of heat capacity is

shown in Fig. 1.5.

An important property, which helped to build eventually the microscopic theory

of superconductivity, was discovered in 1950: It was found that for superconductors

which differ only by isotopic composition, the critical temperature and the critical field

1.3. THERMODYNAMICS OF THE SUPERCONDUCTING TRANSITION 11

− 4

πM

H Hc1 c2HHc

type I

type II

Figure 1.4: Full line: Magnetization of a type II superconductor. The linear part at low

fields corresponds to the full Meissner effect. Dashed line: Magnetization of a type I

superconductor. The Meissner effect persists up to the critical field Hc.

TTc

Cn = γ T

Cs ∝ exp(−1/kBT )

0

C

Figure 1.5: Heat capacity jumps upward on transition to superconducting state. At

T ≪ Tc the electronic heat capacity approaches zero exponentially.

scale with the mass of ions M in the crystalline lattice: Tc ∝ M−1/2, Hc ∝ H−1/2.

The ion mass affects the oscillations of the lattice (phonons), but does not change the

electronic properties in the normal state. Thus it was demonstrated that phonons plays

an essential role in the formation of the superconducting state.

1.3 Thermodynamics of the superconducting transition

Thermodynamics allows to draw a number of conclusions from the phase diagram in

Fig. 1.3. First let us consider the case of zero magnetic field H = 0. We denote

free energy (per unit volume) in the normal state as Fn(T ) and in the superconducting

state Fs(T ). Since the normal state is stable at T > Tc we have Fn(T ) < Fs(T ).

Correspondingly at T < Tc we have Fn(T ) > Fs(T ) and thus at the transition


Fn(Tc) = Fs(Tc).Now we consider non-zero magnetic field H > 0. The proper thermodynamic

potential in an external field H is the Gibbs free energy G = F −HB/4π , where B is

the magnetic induction (local magnetic field). (In these notes we use Gauss system of

units, as commonly used in works on superconductivity.) Thus on the phase transition

line at temperature T and field H = Hc(T ) we should have Gn(T ,Hc) = Gs(T ,Hc).We consider the normal state to be non-magnetic with susceptibility χ = 1. In this

case B = H, the energy of the magnetic field is H 2/8π and Fn(T ,H) = Fn(T , 0) +H 2/8π = Fn(T )+H 2/8π .

In the superconductor according to the Meissner effect B = 0. In fact, as we will

learn later in this chapter, in the Meissner state magnetic field penetrates into a small

layer at the surface of the superconductor, with typical depth of 10−6 − 10−5 cm. In

this layer electric currents circulate which screen the magnetic field inside the super-

conductor bulk. For sufficiently large samples we, however, can ignore the energy con-

tribution from the surface layer and write Fs(T ,H) ≈ Fs(T , 0) = Fs(T ). Combining

all contributions to the Gibbs free energy we obtain Fn −H 2c /8π = Fs or

Fn(T )− Fs(T ) =H 2c (T )

8π. (1.2)

This equation allows to find other thermodynamic quantities. The entropy S = −∂F/∂T .

Thus

Sn − Ss = −Hc

4π

dHc

dT. (1.3)

The latent heat of the transition is q = T (Sn − Ss). Since dHc/dT < 0 (see Fig. 1.3)

we have q ≥ 0, i.e. heat is absorbed on the transition from the superconducting to the

normal state.

If transition happens in zero field at T = Tc, then q = 0, i.e. the transition is of the

second order. Otherwise, it is of the first order.

For heat capacity C = T ∂S/∂T we have

Cn − Cs = −T

4π

[

Hcd2Hc

dT 2+(dHc

dT

)2]

. (1.4)

The heat capacity jump at the transition when H = 0 (cf. Fig. 1.5) is thus

Cs(Tc)− Cn(Tc) =Tc

4π

(dHc

dT

)2

. (1.5)

Experiments show that this relation is indeed obeyed with high accuracy.

1.4. THE LONDON MODEL 13

1.4 The London model

Now we turn to electromagnetic properties of superconductors. To describe those one

has to supplement Maxwell equations with an appropriate material equation which

relates currents induced in the material to applied fields. For normal metals this is just

the Ohm’s law j = σE, where j is electric current induced by electric field E and σ

is the conductivity. This equation clearly cannot be used in a superconductor where

σ →∞.

In 1935, F. London and H. London proposed a phenomenological model, which

captures some essential properties of superconductors. They postulated that in the

superconducting state a part of electrons with density ns acquires special properties:

such electrons can move through superconductor as free particles with velocity vs .

These “superconducting” electrons carry non-dissipative current

js = nsevs .

The rest of electrons with density nn = n − ns remain “normal” and are described

by the usual Ohm’s law. Here n is the total electron density which does not change

on transition to superconducting state and e is the electron’s charge. In the rest of the

section we will discuss only stationary states and will ignore normal electrons and their

current (see Problem 1.3).

Superconducting electrons as free particles with massm have kinetic energy nsmv2s/2.

If the local magnetic field h is present in the superconductor, then it provides contribu-

tion h2/8π to the energy. Thus in the London model we can write free energy of the

superconductor with currents and magnetic field as

FL =∫[

Fs +nsmv2

s

2+ h2

8π

]

dV =∫[

Fs +mj2

s

2nse2+ h2

8π

]

dV . (1.6)

Here Fs is the free energy density of the superconductor in the absence of currents and

fields introduced in the previous section. Using the Maxwell equation

js = (c/4π) curl h , (1.7)

we transform this to the following form

FL =∫[

Fs +mc2

32π2nse2(curl h)2 + h2

8π

]

dV =∫

FsdV+1

8π

∫ [

h2 + λ2L(curl h)2

]

dV .

(1.8)

Here

λL =(

mc2

4πnse2

)1/2

(1.9)


has the dimension of length and is called the London penetration depth.

The equation (1.8) expresses free energy as a functional of the field h, which in turn

depends on coordinates. This is a very common case in condensed-matter physics. To

minimize the free energy one has to take its variation with respect to h and equate it to

zero. (Wikipedia has a good article on functional derivatives, if you need to familiarize

yourself with them.)

In taking variation we will assume ns and thus λL and Fs to be constant is space and

will not differentiate them. This is a serious limitation of the London theory. In many

situations the superconducting density depends on the supercurrent or on coordinates.

In such cases the following equations are not directly applicable. With this assumption

variation of FL with respect to h gives

δFL =1

4π

∫

dV[

h · δh+ λ2L curl h · curl δh

]

.

Now we use vector identity div[A × B] = B curl A − A curl B with A = curl h and

B = δh to obtain

δFL =1

4π

∫

dV(

h+ λ2L curl curl h

)

· δh+ 1

4π

∫

dV div[δh× curl h] .

The second term here transforms to the integral over the surface which can be ignored

(we are interested in the equilibrium distribution of h in bulk and thus can set δh = 0 at

the surface). Equating δFL to zero for arbitrary δh in bulk we get the London equation:

h+ λ2L curl curl h = 0 . (1.10)

From equation (1.7) curl curl h = (4π/c) curl js and thus

curl js = −c

4πλ2L

h = −nse2

mch . (1.11)

Considering electrodynamics of ideal (zero-resistance) conductor one can derive from

the Maxwell equations that

curl∂js

∂t= −nse

2

mc

∂h

∂t. (1.12)

Note the extra time derivative compared to (1.11). By integrating (1.12) we obtain

curl js = −(nse2/mc)h + C(r), where C(r) is an arbitrary function of coordinates

(satisfying div C = 0), determined by the initial conditions. In order to be compatible

with the Meissner effect one has to set C(r) = 0 and thus arrives to equations (1.11)

and (1.10). The arguments using free energy, on the contrary, do not explicitly refer to

the Meissner effect for the derivation of equation (1.10).

1.5. MEISSNER EFFECT 15

h

h②

①0

S

λ

▲

Figure 1.6: The Meissner effect: Magnetic field penetrates into a superconductor only

over distances shorter than λL.

1.5 Meissner effect

One of successes of the London model is the description of the Meissner effect. Using

vector identity

curl curl h = ∇ div h−∇2h

and div h = 0, we transform Eq. (1.10) to

h = λ2L∇2h . (1.13)

Consider a superconductor which occupies the half-space x > 0. A magnetic field hy

is applied parallel to its surface (Fig. 1.6). We obtain from Eq. (1.13)

∂2hy

∂x2− λ−2

L hy = 0

which gives

hy = hy(0) exp(−x/λL) .

The field decays in a superconductor such that there is no field in the bulk. According

to Eq. (1.7) the supercurrent also decays and vanishes in the bulk.

Therefore,

B = H+ 4πM = 0

in a bulk superconductor, where induction B is the averaged microscopic field h and H

is the applied filed. The magnetization and susceptibility are

M = − H

4π; χ = ∂M

∂H= − 1

4π(1.14)

as for an ideal diamagnetic. We remind that the Meissner effect in type I superconduc-

tors persists up to the fieldH = Hc where superconductivity is destroyed, while in type

II superconductors ideal diamagnetizm is observed only to lower field Hc1, Fig. 1.4.


1.6 Phase coherence

The physical mechanism behind the superconductivity is that electrons (Fermi parti-

cles) form pairs (called Cooper pairs), which are bosons, and these bosons experience

Bose-Einstein condensation to a single state, which can be described by the wave func-

tion ψ = |ψ | exp(iχ). The theoretical description of this process will be presented in

the next chapter. Here we will look at the main consequences of existence of macro-

scopically coherent phase χ .

The quantum mechanical expression for the current associated with the wave func-

tion ψ is

js =e∗

2m∗

[

ψ∗pψ + ψ p†ψ∗]

. (1.15)

Here p = −ih∇−(e∗/c)A is the appropriate momentum operator for charged particles

and we took into account that charge carriers are not electrons but pairs with charge e∗

and mass m∗. Performing calculations we obtain

js = −(e∗)2

m∗c|ψ |2

(

A− hce∗

∇χ

)

= −e2ns

mc

(

A− hc2e

∇χ

)

, (1.16)

where we put for Cooper pairs e∗ = 2e,m∗ = 2m and the density of pairs |ψ |2 = ns/2(here we do not assume that it is constant in space). Note that the expression (1.16) is

compatible with the London model. Indeed, taking curl from both sides, remembering

that in the London model ns = const and noticing that curl ∇χ = 0 we obtain

curl js = −e2ns

mccurl A = −e

2ns

mch ,

which coincides with Eq. (1.11). This connection of London equations to quantum-

mechanical current expression and following from that flux quantization were noticed

already by F. London.

1.7 Magnetic flux quantization

Let us integrate the expression (1.16) along a closed contour within a superconductor

(Fig. 1.7):

−mce2

∮

n−1s js ·dl =

∮ (

A− hc2e

∇χ

)

·dl =∫

S

curl A ·dS− hc2e1χ = 8− hc

2e2πn

(1.17)

Here 8 is the magnetic flux through the contour. The phase change along the closed

contour is 1χ = 2πn where n is an integer because the wave function ψ is single

1.8. THE ENERGY GAP AND COHERENCE LENGTH 17

B

❧

Figure 1.7: Magnetic flux through the hole in a superconductor is quantized.

valued. Rearranging terms we get

8′ = 8+ 4π

c

∮

λ2L js · dl = 80n . (1.18)

F. London called the quantity 8′ fluxoid. It is seen from Eq. (1.18) that it is quantized

in superconductors: it assumes only integer multiples of the quantum of magnetic flux

80 =πhc

|e| ≈ 2.07× 10−7 Oe · cm2 . (1.19)

In SI units, 80 = πh/|e| = 2.07× 10−15T·m2.

If we select the contour in the superconductor bulk, where js = 0, then the fluxoid

is equal to flux and we obtain magnetic flux quantization 8 = 80n.

1.8 The energy gap and coherence length

Pairing of electrons to Cooper pairs occurs if it is energetically favorable. The energy

gain is 210 per pair (10 per particle). If thermal fluctuation break pairs, the super-

conductivity is destroyed. Thus we can estimate kBTc ∼ 10. We will derive exact

expression from the microscopic theory in the next chapter.

Another energy scale is seen in the heat capacity measurements, Fig. 1.5. The

exponential dependence of the electronic heat capacity C ∝ exp(−1/kBT ) shows

that in the energy spectrum of normal electrons in a superconductor the energy gap

1 appears. Generally, 1 is temperature-dependent, but at T ≪ Tc it approaches a

constant value which coincides (in the most common case) with 10. This fact is also

explained by the microscopic theory.

The energy scale 10 is connected with an important length scale. According to

the quantum-mechanical uncertainty principle, in order for electrons in the pair to be

bound with the energy 10, their interaction time should exceed τp ∼ h/10. Electrons


move with Fermi velocity vF . If an electron moves ballistically (not scattered) during

time τp, then it travels the distance τpvF . This distance determines the “size” of the

Cooper pair and is called the coherence length

ξ ∼ hvF

10. (1.20)

If the superconducting state is made from objects of size ξ , then its properties cannot

change on the scale substantially smaller than ξ . This concerns both the amplitude

of the condensate wave function |ψ | (for example, width of the transition layer be-

tween coexisting normal and superconducting regions), and also its phase χ , which

determines the supercurrent, Eq. (1.16). We can use this limitation on phase gradients

(∇χ)max ∼ 1/ξ to provide estimate for the critical field Hc.

The maximum gradient of the phase of the condensate wave function limits the

maximum supercurrent to (js)max ∼ (hnse/m)(∇χ)max according to Eq. (1.16). In

the Meissner state the maximum supercurrent is reached at the surface of the su-

perconductor and its value is js = (c/4π)| curl h| = (c/4π)H/λL, see Eq. (1.7)

and Sec. 1.5. Critical field corresponds to the maximum possible value of js . Thus

Hc = (4πλL/c)(js)max. Collecting all expressions together and using definition of the

flux quantum from Eq. (1.19) we obtain

Hc ∼80

πξλL. (1.21)

This critical field is also called the thermodynamic critical field since it is connected

to the free energy difference between normal and superconducting states in Eq. (1.2).

As a reference to the idea of Bose condensation of Cooper pairs, this energy difference

is called the condensation energy. Using Eq. (1.21) we can rewrite it in a different

form. Let us consider the case of T = 0 where it is reasonable to expect that density of

superconducting electrons is equal to the total density ns(T = 0) = n:

Fn(0)− Fs(0) =H 2c (0)

8π∼ (πhc/e)2

8π3(hvF /10)2

4πne2

mc2= n

2mv2F

120 ∼ [N(0)10]10 .

(1.22)

Here N(0) = (3/4)n/EF = 3n/(2mv2F ) is the density of electron states per unit

energy interval close to the Fermi energy EF . The physical meaning of Eq. (1.22) is

that “condensation” affects only particle states with energies within10 from the Fermi

energy EF (as typical for degenerate Fermi systems) and each such particle brings

energy gain 10.

The exact numerical coefficient in Eq. (1.20) in the microscopic theory is usually

defined so that

ξ0 =hvF

2πkBTc, (1.23)

where ξ0 is the zero-temperature coherence length in a clean material.

1.9. LONDON AND PIPPARD REGIMES, DIRTY AND CLEAN LIMITS 19

1.9 London and Pippard regimes, dirty and clean limits

We have introduced two length scales characterizing a superconductor: The London

penetration depth λL and the coherence length ξ . How do their values compare? Let

us consider a typical metallic superconductor with Tc = 1 K and the electron density n

one per ion. A typical lattice constant is a0 ∼ 4 A and thus n = a−30 ≈ 4 · 1022 cm−3

and vF = pF /m = (h/m)(3π2n)1/3 ≈ 108 cm/s. Using equations (1.23) and (1.9) we

obtain ξ ≈ 1400 nm and λL ≈ 30 nm. (Compare these values to data for aluminum

in Table 1.1.) Thus in this case ξ ≫ λL and we realize that the London model is

not applicable: It assumes local connection between supercurrent and magnetic field,

which is not possible when supercurrent is carried by Cooper pairs with extent ξ much

exceeding the characteristic scale of magnetic field variation λL.

To overcome this difficulty, Pippard suggested non-local generalization of the elec-

trodynamics of superconductors, in which supercurrent is related to some average of

the magnetic field in the region of size ξ . The Pippard’s model was an essential tool

to understand experimentally observed properties of clean-metal superconductors. We,

however, will not consider it in this course. One reason is that even in clean metals

there is a region of applicability of the London model: Since superconducting transi-

tion at T = Tc is of the second order, we should have ns → 0 and thus λL→∞ when

T → Tc.

Second, in recent decades focus of research in superconductivity shifted from clean

metals to alloys and compounds that have higher Tc and thus smaller ξ0 values and,

more importantly, are “dirty” in a sense that electrons in such materials experience a

lot of scattering. A remarkable result of the microscopic theory of superconductivity,

supported by experiments, is that static properties of a superconductor (like the critical

temperature Tc) are not affected by non-magnetic impurities (scattering centers). On

the other hand, all properties connected to the spatial variation of the superconducting

state (in particular, supercurrents and coherence length) are strongly affected.

Let us consider a superconductor where electrons have the mean free path ℓ be-

tween scattering by impurities. That is, the scattering time τs = ℓ/vF . If τs is smaller

than the characteristic time of interaction in the Cooper pair τp we should consider

an electron to move diffusively with the diffusion coefficient D ∼ v2F τs = vF ℓ. The

coherence length in the dirty case is determined by the distance the electron travels

during the time τp:

ξdirty =√

Dτp =√

vF ℓh/10 =√

ξcleanℓ , (1.24)

where ξclean = ξ0 in Eq. (1.23). Since usually ℓ is of a few interatomic distances, i.e.

ℓ≪ λL we have ξdirty < λL and the dirty materials are usually in the London regime.


Problems

Problem 1.1. Derive equation (1.2) without referring to the Gibbs free energy. Con-

sider a long cylinder in the magnetic field parallel to the axis and calculate the work

produced by an external source to increase the field from 0 to H when the cylinder is

non-magnetic normal metal and when it is a superconductor.

Problem 1.2. (a) A thermally isolated tin sample is in the superconducting state at

temperature 1 K in the magnetic field just below the critical. The field is increased

adiabatically until the whole sample becomes normal. Calculate the final temperature.

(b) A thermally isolated tin sample is in the normal state at temperature 1 K in the mag-

netic field just above the critical. The field is decreased adiabatically until the whole

sample becomes superconducting. Calculate the final temperature. Tin parameters are

in Table 1.1.

Problem 1.3. (a) Derive the equation (1.12) for an ideal conductor from the Maxwell

equations. Hint: First show that for freely moving electrons

∂

∂tjs =

nse2

mE , (1.25)

where E is the electric field.

(b) Then follow the logic presented in the notes to derive the London equation (1.10),

but take into account the current from normal electrons. Show that in this case

curl curl h+ λ−2L h = B ∂

∂th+ C ∂

2

∂t2h (1.26)

and thus in stationary states ∂h/∂t = 0 the contribution from normal electrons van-

ishes. Find constants B and C.

Problem 1.4. Consider the same settings as in Sec. 1.5: A superconductor which oc-

cupies the half-space x > 0. An alternating magnetic field H = yh0 cos�t is applied

parallel to its surface (Fig. 1.6). Using Eq. (1.26) show that with increasing � penetra-

tion depth of magnetic field into superconductor increases.

Problem 1.5. Find the distribution of the magnetic field and of the current in a super-

conducting slab of a thickness d placed in an external homogeneous magnetic field H

parallel to the slab. The longitudinal dimensions of the slab are much larger than d.

Problem 1.6. In the same settings as in the previous problem find the critical value of

H at which the slab goes normal.

Problem 1.7. Find the distribution of the magnetic field and of the current in a super-

conducting slab of a thickness d carrying a total current I along the length of the slab.

The longitudinal dimensions of the slab are much larger than d .

1.9. LONDON AND PIPPARD REGIMES, DIRTY AND CLEAN LIMITS 21

Problem 1.8. A thin superconducting film with a thickness d ≪ λL is deposited on a

dielectric filament (cylinder) of a radius R ≫ d . The filament is placed into a longi-

tudinal magnetic field at a temperature T > Tc and then cooled down below Tc. The

field is then switched off. Find out how the captured magnetic field is quantized.

Problem 1.9. The same setting as in Problem 1.8: A thin superconducting film with a

thickness d ≪ λL is deposited on a dielectric filament (cylinder) of a radius R ≫ d .

The filament is placed into a longitudinal magnetic fieldH at a temperature T > Tc and

then cooled down below Tc. Calculate the captured magnetic field inside the dielectric

cylinder and the supercurrent in the film.


Chapter 2

The BCS theory

The microscopic theory of superconductivity (BCS) was developed by Bardeen, Cooper

and Schrieffer and independently from them by Bogolubov in 1957. The theory essen-

tially uses the concept of quasiparticles, first introduced by Landau in his theory of

superfluid 4He and then in the theory of the strongly-interacting Fermi liquid (Lan-

dau Fermi liquid). Later this concept became a cornerstone of quantum description of

condensed matter.

Each quantum system has its ground state, which is occupied at T = 0 and a set of

excited levels. The idea of quasiparticles is that the lowest energy levels of a (uniform)

macroscopic system are reasonably approximated by plane waves

ψkα(r) ∝ eikr−i(ǫkα/h)t

with a suitably chosen energy ǫkα (with respect to the ground state) as a function of

the wave vector k (or momentum p = hk). Here α represents other quantum num-

bers like spin or quasiparticle type. The function ǫkα is called the dispersion law or

the energy spectrum of quasiparticles and can have rather different forms. If ψkα is

not exactly an eigenfunction of the Hamiltonian, then the plane wave will transform to

other waves with time (quasiparticle decay) or if one starts with combination of quasi-

particles then they will transform to other combinations (quasiparticle scattering). For

the quasiparticle picture to be useful such processes should be sufficiently slow, so that

the transformation time, or lifetime of a quasiparticle τlife ≫ h/ǫkα .

If the quasiparticle picture is applicable to a given system, then it provides im-

mediate benefits. First, it allows to describe a strongly-interacting many-body system

of real particles with a gas of weakly-interacting quasiparticles. Second, it provides

universality: quite often in different systems we find quasiparticles of the same type

where the energy spectrum differs only by some numeric constants, material parame-

23

24 CHAPTER 2. THE BCS THEORY

ters. One example is phonons which originate from oscillations of atoms or ions in a

condensed-matter system: Acoustic phonons have spectrum which is linear ǫphk = cs hk

up to ǫphk . hωD . Here the sound velocity cs and the Debye frequency ωD are ma-

terial parameters. Note that in Fermi systems quasiparticles can have Fermi or Bose

statistics, while in Bose systems only Bose quasiparticles exist. Quasiparticles are also

often called excitations and we will use both names interchangeably.

The BCS theory provides description of the ground state of a superconductor and

of the appropriate excitations. It explains how those are modified from the case of the

normal metal, which we briefly review next.

2.1 Landau Fermi liquid

The ground state of a system of non-interacting fermions (so-called Fermi gas) corre-

sponds to the filled states with energies E below the Fermi energy EF , determined by

the number of fermions. In a homogeneous system, one can describe particle states by

momentum p such that the spectrum becomes Ep. The condition of maximum energy

Ep = EF defines the Fermi surface in the momentum space. In an isotropic system,

this is a sphere such that its volume divided by (2πh)3

nσ =4πp3

F

3(2πh)3

gives number of particles with the spin projection σ per unit (spatial) volume of system.

For electrons with spin 12

, the total number of particles in the unit volume of the system,

i.e., the particle density is twice nσ

n =p3F

3π2h3(2.1)

Thus pF ∼ hn1/3 = h/a0, where a0 is of the interatomic scale.

The ground state of the Fermi gas corresponds to the energy E0. Excitations in the

Fermi gas that increase its energy as compared to E0 are created by moving a particle

from a state below the Fermi surface to a state above it. This process can be considered

as a superposition of two processes. First is the removal of a particle from the system

out of a state below the Fermi surface. The second is adding the particle to a state above

the Fermi surface. By taking the particle out of the state with an energy E1 < EF we

increase the energy of the system and create a hole excitation with the positive energy

ǫ1 = EF − E1. By adding the particle into a state with an energy E2 > EF we again

increase the energy and create a particle excitation with a positive energy ǫ2 = E2−EF .

The energy of the system is thus increased by ǫ1 + ǫ2 = E2 − E1.

2.1. LANDAU FERMI LIQUID 25

Fp

p−p’

p’

particlehole

Figure 2.1: Particle (shaded circle) and hole (white circle) excitations in Landau Fermi

liquid. The particle excitation is obtained by adding a particle. The hole excitation is

obtained by removing a particle (black circle) with an opposite momentum.

Shown in Fig. 2.1 are processes of creation of particle and hole excitations in a

Fermi liquid. Consider it in more detail. Removing a particle with a momentum p′ and

an energy E′ from below the Fermi surface, p′ < pF and E′ < EF changes the total

momentum of the system by −p′ and the energy by ǫ−p′ = EF − E′. These are the

momentum and energy of the hole excitation. Adding a particle with the momentum p

and the energy E above the Fermi surface, p > pF and E > EF , creates an excitation

with momentum p and energy ǫp = E − EF . For an isotropic system, the excitation

spectrum will thus have the form

ǫp ={

p2

2m− EF , p > pF

EF − p2

2m, p < pF

(2.2)

shown in Fig. 2.2. Here we put E = p2/2m as appropriate for free particles. Since

EF = p2F /2m we have ǫp = 0 at p = pF . As we can create quasiparticles with an

arbitrary small energy, in thermal equilibrium they obey Fermi distribution with zero

chemical potential

f (ǫp) =1

eǫp/kBT + 1. (2.3)

Now let us switch on the interaction between particles. Landau supposed that in

such Fermi liquid excitations have the same type of spectrum as in the gas: The limiting

momentum is connected to the density with the same expression (2.1) and the spectrum

(2.2) is applicable. Now, however,m becomes a material parameter, which is called the

effective mass m∗. Later this assumption was properly proved. The stability of Fermi

surface to the interparticle interactions is of topological origin.

It can be shown that in three and two dimensions quasiparticles have sufficiently

long lifetime to be well-defined when |p − pF | ≪ pF . (In one dimension such ideal-


(p /2m) −E2F

εp

0pp

F

particlesholes

Figure 2.2: Single-particle spectrum (p2/2m) − EF (dashed line) is transformed into

the Landau excitation spectrum ǫp in a strongly correlated Fermi liquid.

gas-like quasiparticles do not exist.) This range is sufficient for explanation of super-

conductivity since kBTc ≪ EF . In such a narrow range of momenta around pF we can

approximate the spectrum (2.2) as

ǫp =|p2 − p2

F |2m∗

= |p − pF |(p + pF )2m∗

≈ vF |p − pF | , (2.4)

where vF = pF /m∗ is a material parameter with the dimension of velocity called

Fermi velocity.

In the following we will often express some physical quantity as a sum over quasi-

particle states. The sum over momenta can be as usual converted to the integral as

A =∑

p

a(p) ≈∫

d3p

(2πh)3a(p).

If the expression under the integral is independent of the direction of p we further have

A =∫

4πp2dp

(2πh)3a(p) =

∫2m∗p d(p2/2m∗)

(2π)2h3a(p)

=∫

m∗p

2π2h3a(p)dǫp =

∫

N(ǫp)a(ǫp)dǫp. (2.5)

The quantity N(ǫ) is called the density of states: N(ǫ)dǫ is the number of states (per

single spin direction) in the energy interval from ǫ to ǫ + dǫ. So far these transfor-

mations are applicable to any system. In Landau Fermi-liquid, however, such integrals

2.2. LANDAU CRITERION 27

have physical meaning only if the integration range is limited to the vicinity of the

Fermi surface or if a(ǫp) rapidly goes to zero with increasing ǫp. In this case N(ǫp)

can be replaced with its value at ǫp = 0 (i.e. p = pF )

N(0) = m∗pF2π2h3

(2.6)

and taken out of the integral:

∑

p

a(p) = N(0)∫

a(ǫp)dǫp ,∑

pσ

a(p) = 2N(0)

∫

a(ǫp)dǫp , (2.7)

The constant density of states is one of the most important features of Landau Fermi-

liquid, which determines many of its physical properties.

Note that the care should be taken when the summation covers both particle and

hole excitations:

∑

p, ǫp<Ec

a(p) = N(0)∫

ǫp<Ec

a(ǫp)dǫp = 2N(0)

∫ Ec

0

a(ǫp)dǫp . (2.8)

The factor of two in the last expression comes from the fact that the range ǫp < Ec

maps both to the particle excitations with pF < p < pF + Ec/vF and to the hole

excitations with pF − Ec/vF < p < pF and we assumed that a(ǫp) is the same for

the particle and hole excitations.

2.2 Landau criterion

Within the quasiparticle model Landau explained superfluidity of 4He in the following

manner. Let us assume superfluid helium flows along a tube with velocity v. In order

to dissipate energy, a quasiparticle should be created with momentum p and energy ǫp,

as defined in the reference frame connected with the liquid. In the laboratory frame

the energy is E′ = E + Pv +Mv2/2, where E is the energy and P is the momentum

in the liquid reference frame and M is the mass of the liquid. Before creation of the

quasiparticle E = E0 and P = 0, and afterwards E = E0+ ǫp and P = p. The change

of the energy in the laboratory frame is thus

1E′ = ǫ′p = ǫp + pv. (2.9)

For the creation of the quasiparticle to be favorable one should have 1E′ < 0. The

minimum value of1E′ is when p is antiparallel to v, giving ǫp+pv < 0 or v > ǫp/p.

Creation of any quasiparticle is thus possible only above the critical velocity

v > vc = min (ǫp/p) . (2.10)


This is the Landau criterion for the dissipationless flow.

For example, in Bose-Einstein condensates of dilute gases the excitations are phonons

with ǫp = csp and we get vc = cs . In the strongly interacting 4He the quasiparticle

spectrum is more complex, but min (ǫp/p) is still finite giving vc ≈ 60 m/s. Now if we

apply the Landau criterion to Fermi liquid with spectrum in Fig. 2.2 we immediately

see that vc = ǫ(pF )/pF = 0. That is, electron Fermi liquid in normal metals does not

support superflow.

The idea that in a superconductor electrons form pairs which are bosons have been

suggested relatively early. However, understanding of the physical mechanism behind

pairing required about 40 years after the discovery of superconductivity.

2.3 Phonon-mediated electron attraction

The usual interaction between two electrons is the Coulomb repulsion. In a metal,

electrons, however, are moving in the lattice of positively-charged ions. The interac-

tion with the motion of ions can result in effective attraction between the conduction

electrons: An electron polarizes the lattice, attracting ions and thus creating a cloud

of the positive charge which can attract another electron. Simultaneously Coulomb re-

pulsion is reduced due to the screening of the electric field. To describe these effects

properly for the whole ensemble of electrons and ions one have to use Green functions

formalism. The basic features can however be demonstrated by considering one or two

electrons separately from the rest. (In this section we call Fermi quasiparticles in a

metal as electrons, to distinguish them from phonons.)

Let us consider a charge Q at r = 0 in the cloud of electrons. The charge produces

electrostatic potential ϕ which changes electron density by δn. Poisson equation for

the potential is

1ϕ = −4π [eδn+Qδ(r)] .

Change of the electron density is determined by Fermi distribution when electrostatic

energy is added to the electron energy ǫ → ǫ + eϕ. At low temperatures the change

of occupation occurs only in the region of energies eϕ at the Fermi energy and δn =−2N(0)eϕ. The equation for the potential becomes

1ϕ − 8πe2N(0)ϕ = −4πQδ(r) .

The easiest way to solve this equation is Fourier transform. The answer is

ϕk =4πQ

k2 + k2s

, ϕ(r) = Q

rexp(−ksr), ks =

√

8πe2N(0). (2.11)

2.3. PHONON-MEDIATED ELECTRON ATTRACTION 29

Thus Coulomb interaction becomes short-ranged in metals. We can estimate the range

of the interaction as

k−1s ∼

(

e2 n

EF

)−1/2

=(

e2n1/3

EF

)−1/2

n−1/3 ∼ n−1/3 = a0 .

Here we took into account that kinetic energy of electrons in the metal EF is of the

order of the potential energy of interaction with ions ∼ e2/a0 = e2n1/3. Thus the

interaction range is limited to the inter-electron (i.e. interatomic) distances.

The ions in the lattice can be approximately considered as doing harmonic oscil-

lations around equilibrium positions with Debye frequency ωD . The potential energy

for such an oscillator should reach atomic scale EF ∼ e2/a0 at the displacement of a0.

With the ion mass M we have Mω2Da

20 ∼ EF ∼ (h/a0)

2/m and thus

ωD ∼h

a20

1√Mm∼p2F

hm

√

m

M= EF

hb,

where b =√M/m ∼ 300. An electron moving with velocity vF spends time about

a0/vF close to each ions, which corresponds to ωDa0/vF fraction of the oscillation

period. The electron applies force e2/a20 on the ion and thus the ion shift is

x ∼ ωDa

vF

e2/a20

Mω2D

∼ EF

vFMωD∼ h

a0M

a20

√Mm

h= a0

b∼ (1/300)a0 .

The relaxation time of the shift is∼ ω−1D and the length of the tail of shifted ions behind

the electron is ∼ vFω−1D ∼ hb/pF ∼ ba0 ∼ 300a0. In this tail there is an excess of

positive charge density, which can attract another electron, while the repulsion from the

electron, creating the tail is screened at much shorter distances. Of cause, the electrical

field of the tail is also screened and thus only an electron which moves exactly along

the tail, i.e. with the momentum collinear with the momentum of the first electron, can

efficiently interact with the induced charge in the tail.

On the quantum-mechanical language one describes this process as a scattering of

two electrons via exchange of a phonon. The momenta of electrons before interaction

are p1 and p2 and after the interaction are p′1 and p′2, respectively. From the considera-

tion above we conclude that efficient interaction is only possible when p1 ≈ ±p2. All

initial and final states should be very close to the Fermi surface. Thus in the case of

p1 ≈ p2 only limited number of final states with p′1 ≈ p1 and p′2 ≈ p2 are available.

On the other hand if p1 ≈ −p2 we have p′1 ≈ −p′2 owing to momentum conservation,

but direction of p′1 with respect to p1 can be arbitrary. Thus essentially all Fermi sur-

face is available for final states, the probability of scattering is drastically larger and

only such processes are important.


p1 p′2

p1p′2

p′1 p2

p′1p2

−q q

(1) (2)

Figure 2.3: Phonon-mediated electron-electron interaction which leads to attraction

between the electrons.

Let us estimate the scattering amplitude for two electrons with initial momenta p1

and p2 to go to the states with momenta p′1 = p1+ q and p′2 = p2− q via exchange of

a phonon. Owing to the energy conservation the energy of the initial state (I) is equal

to that of the final state (II): EI = ǫp1+ ǫp2

= ǫp′1+ ǫp′2

= EII. In the lowest (second)

order of the perturbation theory there are two possible intermediate states, see Fig. 2.3:

(1) Electron with momentum p1 emits the phonon with momentum −q and energy

hω−q. The phonon is then absorbed by the electron with momentum p2. (2) Electron

with momentum p2 emits the phonon with momentum q and energy hωq. The phonon

is then absorbed by the electron with momentum p1. Since for phonons ω−q = ωq, the

energies of the intermediate states are:

E1 = ǫp′1+ ǫp2

+ hωq , E2 = ǫp1+ ǫp′2

+ hωq .

In the second-order perturbation theory the matrix element connecting initial and final

states is

〈II|He−ph−e|I〉 =1

2

∑

i=1,2

〈II|He−ph|i〉(

1

EII − Ei+ 1

EI − Ei

)

〈i|He−ph|I〉

= W ∗q

(

1

ǫp′1− ǫp1

− hωq

+ 1

ǫp′2− ǫp2

− hωq

)

Wq

= |Wq|2h

(1

ω − ωq

− 1

ω + ωq

)

= 2|Wq|2h

ωq

ω2 − ω2q

(2.12)

HereWq is the matrix element for emission of the photon with momentum q and hω =ǫp′1−ǫp1

= −(ǫp′2−ǫp2

). We find that when |ω| < ωq the matrix element in Eq. (2.12)

2.4. THE COOPER PROBLEM 31

is negative, i.e. it corresponds to the attraction between electrons. Another important

conclusion is that this attraction does not depend on the directions of p1, p2 and thus

electrons interact in a state with the orbital momentum L = 0. This means that the

orbital part of their wave function is symmetric with respect to the particle interchange.

Since the total wave function should be antisymmetric for Fermi particles, we conclude

that the spin part of the wave function is antisymmetric, i.e. the electrons interact in

the state with total spin S = 0 or, in other words, they have opposite spins. Since the

phonon density of states∝ q2 rapidly increases with q, the most important contribution

to the electron attraction comes from the largest ωq, that is ωq ∼ ωD .

The following theory is built using a simplified model of the electron interaction.

We will assume that two electrons with opposite momenta and spins attract each other

with the constant amplitude −W if ǫp1< Ec and ǫp2

< Ec (where Ec ∼ hωD) and do

not interact otherwise. The fact that this interaction is mediated by phonons will not be

important.

2.4 The Cooper problem

As we know in three dimensions arbitrarily small attraction between particles is not

sufficient to form a bound state. However in 1956 Cooper noticed that if two particles

interact in the presence of the filled Fermi sphere, then the bound state always exists.

Qualitative reason behind this feature is that in such a case 3-dimensional integrals

over momentum∫

d3p/(2πh)3 can be replaced with integrals over the energy with the

fixed density of states close to the Fermi surface N(0)∫

dǫ. Thus effectively interac-

tion problem becomes one-dimensional, where the bound state exists for any attractive

potential. Let us consider this problem in more details.

We are looking for the bound state of a pair of quasiparticles, Fig. 2.4. We denote

the energy of the state E and its wave function 9(r1, r2). The wave function satisfies

the Schrodinger equation

[

He(r1)+ He(r2)+W(r1, r2)]

9(r1, r2) = E9(r1, r2) . (2.13)

Here W(r1, r2) = W(r) is the interaction potential and r = r1 − r2 . As follows

from the previous section we look for the interaction of quasiparticles with opposite

momenta and spin, which have wave functions ψp↑(r1) ∝ eipr1/h and ψ−p↓(r2) ∝e−ipr2/h. We can thus represent the pair wave function as a linear combination of

single-particle wave functions

9(r1, r2) =∑

p

cpψp↑(r1)ψ−p↓(r2) =∑

p

apeipr/h = 9(r) .


(a) (b)

pF pF

p↑ p↑

−p↓ −p↓

Figure 2.4: In the Cooper problem two particle excitations (a) or two hole excitations

(b) with opposite momenta and spin interact in the presence of the otherwise undis-

turbed filled Fermi sphere. A bound state is found for an arbitrarily small attraction

between the interacting quasiparticles. In the self-consistent ground state of a super-

conductor particle and hole states are mixed and the binding energy is found to be twice

larger than in the Cooper problem.

The inverse transformation is

ap = V −1

∫

9(r)e−ipr/h d3r ,

where V is the volume of the system.

We take Fourier transform of both sides of Eq. (2.13), that is multiply by e−ipr/h

and integrate over the volume. Since He(r1)ψp↑(r1) = ǫpψp↑(r1) and He(r2)ψp↓(r2) =ǫ−pψp↓(r2) = ǫpψp↓(r2) [ǫp is even, see Eq. (2.4)] we obtain

[He(r1)9(r1, r2)]p = [He(r2)9(r1, r2)]p = ǫpap .

For the product W(r)9(r) we use the convolution theorem

[W(r)9(r)]p =∑

p′ap′Wp−p′ .

Here the Fourier transform of the potential is simultaneously the matrix element for the

transition of the pair of particles from states p and −p to states p′ and −p′:

Wp−p′ = V −1

∫

W(r)e−i(p−p′)r/h d3r = 〈eip′r1/he−ip′r2/h|W(r1, r2)|eipr1/he−ipr2/h〉 = Wp,p′ .

Thus equation (2.13) becomes

2ǫpap +∑

p′Wp,p′ap′ = Eap .

2.4. THE COOPER PROBLEM 33

E

8 . E/

2ǫ0

2ǫn

−W−1

Eb

Figure 2.5: The function 8(E) for a system with a discrete spectrum ǫn.

For the interaction model introduced at the end of the previous section we write

Wp,p′ ={

−W, ǫp and ǫp′ < Ec, i.e. pF − Ec/vF < p(and p′) < pF + Ec/vF0, otherwise

(2.14)

where Ec ≪ EF . We thus have

ap = −W

E − 2ǫp

∑

p′,ǫp′<Ec

ap′ (2.15)

Let us denote

C =∑

p,ǫp<Ec

ap

Eq. (2.15) yields

ap = −WC

E − 2ǫp

whence

C = −WC∑

p,ǫp<Ec

1

E − 2ǫp

This gives

− 1

W=

∑

p,ǫp<Ec

1

E − 2ǫp

≡ 8(E) (2.16)

Equation (2.16) is illustrated in Fig. 2.5. Let us put our system in a large box.

The levels ǫp will become a discrete set ǫn shown in Fig. 2.5 by vertical dashed lines.

The lowest level ǫ0 is very close to zero and will approach zero as the size of the box


increases. The function 8(E) varies from −∞ to +∞ as E increases and crosses

each ǫn > 0. However, for negative E < 0, the function 8(E) approaches zero as

E → −∞, and there is a crossing point with a negative level −1/W for negative E.

This implies that there is a bound state with E = Eb < 0 satisfying Eq. (2.16):

1

W=

∑

p,ǫp<Ec

1

2ǫp − Eb.

Here we replace the sum over momentum with the integral over energy (2.8):

1

W= 2N(0)

∫ Ec

0

dǫp

2ǫp + |Eb|= N(0) ln

( |Eb| + 2Ec

|Eb|

)

. (2.17)

From this equation we obtain

|Eb| =2Ec

e1/N(0)W − 1(2.18)

For weak coupling, N(0)W ≪ 1, we find

|Eb| = 2Ece−1/N(0)W (2.19)

For strong coupling, N(0)W ≫ 1,

|Eb| = 2N(0)WEc

We see that there exists a state of a quasiparticle pair (the Cooper pair) with an

energy |Eb| below the Fermi surface. It means that the system of normal-state particles

and holes is unstable towards formation of pairs provided there is an attraction (how-

ever small) between electrons. Note that above we considered only a single pair, while

the rest of the system remains undisturbed. In reality formation of pairs will continue

until the whole system departs sufficiently far from the normal metal so that further

pairing becomes energetically unfavorable. We will describe such self-consistent state

further in this chapter.

The Cooper pairing effect provides a basis for understanding of superconductivity.

According to this picture, the pairs, being Bose particles, form a Bose condensate in a

single state with a wave function that has a single phase for all pairs, which is the basic

requirement for existence of a spontaneous supercurrent.

2.5 The BCS model

Let us start from the description of uniform superconducting state in the absence of

electric and magnetic fields. We will mark quasiparticle states with the wave vector

2.5. THE BCS MODEL 35

k = p/h and spin σ =↑,↓. The states at k > kF = pF /h are particles and at k < kF

are holes. For non-interacting system we can write Hamiltonian as

H0 =∑

kσ,k>kF

ǫkc†kσ ckσ +

∑

kσ,k<kF

ǫkh†kσhkσ .

Here c†kσ and ckσ are creation and annihilation operators for particle excitations in

the normal metal and h†kσ and hkσ are creation and annihilation operators for hole

excitations. But creation/annihilation of a hole with the wave vector k and spin σ is

equivalent to annihilation/creation of a particle with the wave vector −k and spin −σ .

Thus h†kσ = c−k,−σ and hkσ = c†

−k,−σ and we have

H0 =∑

kσ,k>kF

ǫkc†kσ ckσ +

∑

kσ,k<kF

ǫ−kckσ c†kσ

=∑

kσ,k>kF

ǫkc†kσ ckσ −

∑

kσ,k<kF

ǫkc†kσ ckσ +

∑

kσ,k<kF

ǫk

=∑

kσ

ξkc†kσ ckσ + E0 . (2.20)

Here we used ǫk = ǫ−k and the commutation relation for fermionic operators and also

defined

ξk = sign(k − kF )ǫk =h2k2

2m− EF ≈ hvF (k − kF ) . (2.21)

The constant energy E0 of the filled Fermi sphere we will not include in the following

expressions until the very end of the section.

Now we add the pairing interaction to the Hamiltonian. Considering the model of

the interaction we are using we can write the Hamiltonian as

H =∑

kσ

ξkc†kσ ckσ +

∑

kk′Wkk′c

†k↑c

†−k↓c−k′↓ck′↑ . (2.22)

In writing this expression we took into account that the pairing interaction occurs

between quasiparticles with opposite momenta and spins and that it transforms state

(k′,−k′)→ (k,−k) without affecting the quasiparticle spin.

The Hamiltonian (2.22) is of the fourth order with respect to quasiparticle oper-

ators and thus is very difficult to analyze. The approach taken in the BCS theory is

to use mean-field approximation. For two operators A and B we approximately write

AB ≈ 〈A〉B+A 〈B〉−〈A〉〈B〉. The error in this approximation is (A−〈A〉)(B−〈B〉),i.e. quadratic in fluctuations around the average, which are relatively small in a macro-

scopic system. We use this approximation with A = c†k↑c

†−k↓ and B = c−k′↓ck′↑ to


convert (2.22) to the BCS model Hamiltonian:

HBCS =∑

kσ

ξkc†kσ ckσ +

∑

k′

(

c−k′↓ck′↑∑

k

Wkk′⟨

c†k↑c

†−k↓

⟩)

+

∑

k

(

c†k↑c

†−k↓

∑

k′Wkk′

⟨

c−k′↓ck′↑⟩

)

−∑

k

(⟨

c†k↑c

†−k↓

⟩∑

k′Wkk′

⟨


)

.

We define

1k =∑

k′Wkk′

⟨


(2.23)

so that

1∗k =∑

k′Wkk′

⟨

c†k′↑c

†−k′↓

⟩

. (2.24)

We will find that 1k is non-zero in the superconducting state. Being the expectation

value of the pairing amplitude it can be used as a wave function of the condensate of

Cooper pairs. Using this definition we write HBCS as

HBCS =∑

kσ

ξkc†kσ ckσ +

∑

k

(

1∗kc−k↓ck↑ +1kc†k↑c

†−k↓ −1k

⟨

c†k↑c

†−k↓

⟩)

. (2.25)

We want to describe the superconducting state using picture of the condensate +

new type of quasiparticles. That is, we want to transform HBCS to

HBCS =∑

kσ

Ekγ†kσγkσ + Econd , (2.26)

so that it is diagonal in new quasiparticle creation and annihilation operators γ†kσ and

γkσ . Then Ek will be the energy spectrum of the new quasiparticles and Econd will

describe the energy gain compared to the normal metal due to formation of the Cooper-

pair condensate. The new quasiparticles are called Bogolubov quasiparticles and the

transformation is called Bogolubov (or Bogolubov-Valatin) transformation. The idea is

to present new operators as linear combinations of the original quasiparticle operators

in a way which mixes particles and holes:

γk↑ = u∗kck↑ + v∗khk↑ = u∗kck↑ + v∗kc†−k↓ , (2.27)

γ†−k↓ = ukc

†−k↓ − vkh

†−k↓ = ukc

†−k↓ − vkck↑ . (2.28)

Functions uk and vk are to be found. New operators should satisfy usual Fermi com-

mutation relations

{

γkσ , γ†k′σ ′

}

= δkk′δσσ ′ , {γkσ , γk′σ ′} ={

γ†kσ , γ

†k′σ ′

}

= 0 . (2.29)

2.5. THE BCS MODEL 37

In particular,{

γk↑, γ†k↑

}

= γk↑γ†k↑ + γ

†k↑γk↑

= u∗kukck↑c†k↑ + u

∗kvkck↑c−k↓ + ukv

∗kc

†−k↓c

†k↑ + v

∗kvkc

†−k↓c−k↓

+ uku∗kc

†k↑ck↑ + ukv

∗kc

†k↑c

†−k↓ + vku

∗kc−k↓ck↑ + vkv

∗kc−k↓c

†−k↓

= |uk|2{

ck↑, c†k↑

}

+ u∗kvk

{

ck↑, c−k↓}

+ ukv∗k

{

c†−k↓, c

†k↑

}

+ |vk|2{

c†−k↓, c−k↓

}

= |uk|2 + |vk|2 = 1 . (2.30)

In the last line we use commutation relations for ckσ operators which are similar to

(2.29). It is easy to show that the obtained condition (2.30) is sufficient to satisfy all

requirements (2.29).

Using equations (2.27), (2.28) and (2.30) we express ckσ operators via γkσ opera-

tors:

ck↑ = ukγk↑ − v∗kγ†−k↓ , (2.31)

c†−k↓ = u

∗kγ

†−k↓ + vkγk↑ . (2.32)

We put these expressions to Eq. (2.25) and find (using ξ−k = ξk)

HBCS =∑

k

{

ξk

(

u∗kγ†k↑ − vkγ−k↓

) (

ukγk↑ − v∗kγ†−k↓

)

+ ξk

(

u∗kγ†−k↓ + vkγk↑

) (

ukγ−k↓ + v∗kγ†k↑

)

+1∗k(

ukγ−k↓ + v∗kγ†k↑

) (

ukγk↑ − v∗kγ†−k↓

)

+1k

(

u∗kγ†k↑ − vkγ−k↓

) (

u∗kγ†−k↓ + vkγk↑

)

−1k

⟨

c†k↑c

†−k↓

⟩ }

=∑

k

{ (

ξk|uk|2 +1∗kukv∗k +1ku

∗kvk

)

γ†k↑γk↑ + ξk|vk|2γk↑γ

†k↑

+ ξk|uk|2γ †−k↓γ−k↓ +

(

ξk|vk|2 −1∗kukv∗k −1ku

∗kvk

)

γ−k↓γ†−k↓

+(

−ξku∗kv∗k −1∗k(v∗k)2 +1k(u

∗k)

2)

γ†k↑γ

†−k↓ + ξku

∗kv∗kγ

†−k↓γ

†k↑

+(

−ξkukvk +1∗ku2k −1kv

2k

)

γ−k↓γk↑ + ξkukvkγk↑γ−k↓ −1k

⟨

c†k↑c

†−k↓

⟩ }

.

Then we use commutation relations (2.29) to switch the order of the operators

HBCS =∑

k

{ [

ξk

(

|uk|2 − |vk|2)

+1∗kukv∗k −1ku

∗kvk

] (

γ†k↑γk↑ + γ †

−k↓γ−k↓)

+[

−2ξku∗kv∗k +1k(u

∗k)

2 −1∗k(v∗k)2]

γ†k↑γ

†−k↓

+[

−2ξkukvk +1∗ku2k −1kv

2k

]

γ−k↓γk↑ (2.33)

+ 2ξk|vk|2 −1∗kukv∗k −1ku

∗kv∗k −1k

⟨

c†k↑c

†−k↓

⟩ }

. (2.34)


We see that coefficients at non-diagonal terms γ†k↑γ

†−k↓ and γ−k↓γk↑ are complex con-

jugate of each other and thus both terms will vanish if

2ξkukvk +1kv2k −1∗ku2

k = 0 . (2.35)

A solution of this equation can be found in the following form:

uk = |uk|eiχk/2, vk = |vk|e−iχk/2, 1k = |1k|eiχk . (2.36)

We obtain

2ξk|uk||vk| + |1k|eiχk |vk|2e−iχk − |1k|e−iχk |uk|2eiχk = 0

and

(2ξk|uk||vk|)2 = |1k|2(

|uk|2 − |vk|2)2= |1k|2

(

|uk|2 + |vk|2)2− 4|1k|2|uk|2|vk|2

= |1k|2 − 4|1k|2|uk|2|vk|2 .

Thus using proper arguments (2.36) we get

ukv∗k =

1k

2

√

ξ2k + |1k|2

(2.37)

and

|uk|2 − |vk|2 =2ξk|uk||vk||1k|2

= ξk√

ξ2k + |1k|2

.

Together with |uk|2 + |vk|2 = 1 this gives

|uk|2 =1

2

1+ ξk√

ξ2k + |1k|2

, (2.38)

|vk|2 =1

2

1− ξk√

ξ2k + |1k|2

. (2.39)

Inserting these definitions to Eq. (2.34) we find

HBCS =∑

k

{

ξkξk

√

ξ2k + |1k|2

+1∗k1k +1k1

∗k

2

√

ξ2k + |1k|2

(

γ†k↑γk↑ + γ †

−k↓γ−k↓)

+ ξk

1− ξk√

ξ2k + |1k|2

− |1k|2√

ξ2k + |1k|2

−1k

⟨

c†k↑c

†−k↓

⟩ }

=∑

k

Ek

(

γ†k↑γk↑ + γ †

−k↓γ−k↓)

+∑

k

[

ξk − Ek −1k

⟨

c†k↑c

†−k↓

⟩]

, (2.40)

2.6. THE GAP EQUATION 39

where

Ek =√

ξ2k + |1k|2 . (2.41)

Using plausible assumption |1−k| = |1k| (we will see this later) and thus E−k = Ek

we finally arrive to the desired diagonalized form of the BCS Hamiltonian:

HBCS =∑

kσ

Ekγ†kσγkσ + Econd ,

where (now we insert E0 back)

Econd =∑

k

[

ξk − Ek −1k

⟨

c†k↑c

†−k↓

⟩]

+∑

kσ,k<kF

ǫk

=∑

k,k<kF

(ξk + 2ǫk)+∑

k,k>kF

ξk +∑

k

[

−Ek −1k

⟨

c†k↑c

†−k↓

⟩]

.

When k < kF we have ξk+2ǫk = −ǫk+2ǫk = ǫk and when k > kF we have ξk = ǫk.

Thus

Econd =∑

k

[

ǫk − Ek −1k

⟨

c†k↑c

†−k↓

⟩]

. (2.42)

2.6 The gap equation

To complete solution of the problem we still have to determine1k. To do that we insert

Bogolubov transformation (2.31) and (2.32) into the definition (2.23). Since uk and vk

depend on 1k we will get an equation for 1k which is called the self-consistency

equation or the gap equation. We have

1k =∑

k′Wkk′

⟨


=∑

k′Wkk′

⟨(

uk′γ−k′↓ + v∗k′γ†k′↑

) (

uk′γk′↑ − v∗k′γ†−k′↓

)⟩

=∑

k′Wkk′

[

u2k′⟨

γ−k′↓γk′↑⟩

+ v∗k′uk′⟨

γ†k′↑γk′↑

⟩

− v∗k′uk′⟨

γ−k′↓γ†−k′↓

⟩

− (v∗k′)2⟨

γ†k′↑γ

†−k′↓

⟩]

.

For fermionic Bogolubov quasiparticles in thermal equilibrium we have

⟨

γ−k′↓γk′↑⟩

=⟨

γ†k′↑γ

†−k′↓

⟩

= 0 ,⟨

γ†k′↑γk′↑

⟩

= 1−⟨

γ−k′↓γ†−k′↓

⟩

= f (Ek′) (2.43)

where f (E) is the Fermi distribution function

f (E) = 1

eE/kBT + 1. (2.44)

Thus we have

1k = −∑

k′Wkk′uk′v

∗k′ [1− 2f (Ek′)] (2.45)

= −∑

k′Wkk′

1k′

2Ek′[1− 2f (Ek′)] . (2.46)


Here we used Eq. (2.37). Now we substitute Wkk′ for our model interaction (2.14)

Wkk′ ={

−W, ǫk and ǫk′ < Ec

0, otherwise(2.47)

It is clear that solution of Eq. (2.46) can then be found in the form

1k ={

1, Ek <√

E2c + |1|2 ≈ Ec

0, otherwise(2.48)

Here we assumed that |1| ≪ Ec as hinted by the solution of the Cooper problem. With

these substitutions equation (2.46) becomes

1 = W1∑

k,ǫk<Ec

1− 2f (Ek)

2Ek

= 1W2

∑

k,ǫk<Ec

1

Ek

tanhEk

2kBT. (2.49)

It has a trivial solution 1 = 0 corresponding to the normal metal. Non-trivial solution

we find replacing sum in Eq. (2.49) with the integral (2.8)

∑

k,ǫk<Ec

→∫

ǫk<Ec

d3k

(2π)3= 2N(0)

∫ Ec

0

dǫk . (2.50)

From (2.21) and (2.41) we have

dǫk = d(√

E2k − |1|2

)

= EkdEk√

E2k − |1|2

.

If ǫk changes from 0 toEc, thenEk changes between |1| and√

E2c + |1|2 ≈ Ec. Thus

we have∑

k,ǫk<Ec

→ 2N(0)

∫ Ec

|1|

EkdEk√

E2k − |1|2

, (2.51)

and the gap equation (2.49) becomes

1 = WN(0)∫ Ec

|1|

dE√

E2 − |1|2tanh

E

2kBT. (2.52)

The dimensionless parameter λ = N(0)W is called the interaction constant. The equa-

tion (2.52) in the whole temperature range can be solved numerically, see Fig. 2.6.

Here we consider some limiting cases.

At the critical temperature T = Tc the gap 1 vanishes. We have

1 = λ∫ Ec

0

tanh

(E

2kBTc

)dE

E. (2.53)

2.6. THE GAP EQUATION 41

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

T /Tc

0

|1|/

10

Figure 2.6: Dependence of the magnitude of the superconducting gap on temperature

found from numerical solution of Eq. (2.52).

This reduces to

1

λ=∫ Ec/2kBTc

0

tanh x

xdx . (2.54)

The integral∫ x0

0

tanh x

xdx = ln(x0B) .

Here B = 4γ /π ≈ 2.26 where γ = eC ≈ 1.78 and C = 0.577 . . . is the Euler

constant. Therefore,

kBTc = (2γ /π)Ece−1/λ ≈ 1.13Ece−1/λ . (2.55)

The interaction constant is usually small, being of the order of 0.1 − 0.3 in practical

superconductors. Therefore, usually Tc ≪ Ec/kB.

For zero temperature we denote |1(T = 0)| = 10. We obtain from Eq. (2.52)

[since tanh(E/2kBT )→ 1]

1

λ=∫ Ec

10

dE√

E2 −120

= arcoshEc

10= ln

(

Ec

10+√

E2c

120

− 1

)

≈ ln2Ec

10(2.56)

Therefore, at T = 0

|1| ≡ 10 = 2Ece−1/λ = (π/γ )kBTc ≈ 1.76kBTc .


2.7 Condensation energy

Let us find the condensation energy for T = 0. From equation (2.42) we have

Econd =∑

k

[

ǫk − Ek −1k

⟨

c†k↑c

†−k↓

⟩]

=∑

k,ǫk<Ec

(ǫk − Ek)−1∑

k,ǫk<Ec

⟨

c†k↑c

†−k↓

⟩

(2.57)

Here we took into account (2.48). Using transformation to the integral (2.50) we have

for the first sum in (2.57):

∑

k,ǫk<Ec

(ǫk − Ek) = 2N(0)

∫ Ec

0

(ǫk − Ek)dǫk = 2N(0)

∫ Ec

0

(ǫ −√

ǫ2 + |1|2)dǫ

= N(0)[

ǫ2 − ǫ√

ǫ2 + |1|2 − |1|2 ln

(

ǫ +√

ǫ2 + |1|2)]Ec

0

= N(0)

E2c − Ec

√

E2c +12

0 −120 ln

Ec +√

E2c +12

0

10

≈ N(0)[

E2c − E2

c

(

1+ 1

2

120

E2c

)

−120 ln

2Ec

10

]

= −1

2N(0)12

0 −12

0

W,

where the value of the logarithm is taken from (2.56).

For the second sum in (2.57) we have using (2.24) and (2.47)

∑

k,ǫk<Ec

⟨

c†k↑c

†−k↓

⟩

= −1∗

W.

Combining all expressions we get

Econd = Fs(0)− Fn(0) = −1

2N(0)12

0 −12

0

W+11

∗

W= −1

2N(0)12

0 . (2.58)

Compare this to an order-of-magnitude estimation in Eq. (1.22). Using connection to

the thermodynamic critical field (1.2) we find the critical field at T = 0

Hc(0) =√

4πN(0)10 . (2.59)

2.8 Bogolubov quasiparticles

Quasiparticles in the superconducting state are superpositions of particle and hole ex-

citations of the original normal system, see (2.27) and (2.28). One can say that |uk|2 is

2.8. BOGOLUBOV QUASIPARTICLES 43

E

p p

F

F

∆

particlesholes

E

♣❋

2∆

✭�✁

❊❋✲2∆

❊❋✰2∆

Figure 2.7: (a) The BCS spectrum of excitations in a superconductor. The solid line

shows the spectrum of quasiparticles near the Fermi surface where the Landau quasi-

particles are well defined. At higher energies closer to EF the Landau quasiparticles

are not well-defined (dotted line). The dashed line at lower energies shows the behavior

of the spectrum in the normal state ǫp = |ξp|. (b) The superconducting gap is formed

near the Fermi surface that has existed in the normal state. For zero temperatures the

states inside the inner region with energies below EF − 1 are filled while those with

energies above EF +1 are empty.

the particle “fraction” in the Bogolubov quasiparticle and |vk|2 is the hole “fraction”.

In the normal state |1k| = 0 and

|uk|2 = (1+ ξk/|ξk|)/2 = 0 for k < kF , |uk|2 = 1 for k > kF ,

|vk|2 = (1− ξk/|ξk|)/2 = 1 for k < kF , |vk|2 = 0 for k > kF .

In this case Bogolubov quasiparticles are simply holes below Fermi surface and parti-

cles above Fermi surface. In the superconducting state this, however, applies only far

from the Fermi surface. At k = kF we have γk↑ = (1/√

2)e−iχk/2ck↑+(1/√

2)eiχk/2hk↑,

i.e. equal admixture of a particle and a hole. For simplicity we will still continue to call

excitations at k < kF as holes and at k > kF as particles.

The energy spectrum (2.41) of the Bogolubov quasiparticles is shown in Fig. 2.7.

A distinct feature of the spectrum is the presence of the energy gap: Minimum energy

of |1| is reached at p = pF . This spectrum satisfies the Landau criterion for dissi-

pationless motion. It is clear that the minimum value of ǫp/p is reached close to the

minimum of the energy spectrum and thus

vc = min(Ep/p) ≈ |1|/pF . (2.60)


N(0)

N

∆ E

s

Figure 2.8: Density of states as a function of energy. There are no states for E < |1|.The DOS has a square-root singularity at E → |1|. It approaches the normal-state

value N(0) for E ≫ |1|.

In a superconductor there are no quasiparticle states with energies Ek < |1|. Thus

we can say that the density of states is zero. The density of states at energies higher

than the gap we find by comparing (2.50) and (2.51). As a result, the density of states

in the superconductor is

Ns(E) =

0, E 6 |1|,

N(0)E

√

E2 − |1|2, E > |1|.

(2.61)

The plot of this function is shown in Fig. 2.8

For a given energy E > |1| there are two possible values of ξk:

ξ±k = ±√

E2 − |1|2 = hvF (k± − kF ). (2.62)

The value ξ+k corresponds to the particle-like excitations and ξ−k to the hole-like. The

corresponding absolute values of the wave vector are

k± = kF ±1

hvF

√

E2 − |1|2 . (2.63)

The group velocity of a quasiparticle is

vg =dE

dp= k

h

dE

dk,

where k is a unit vector in the direction of momentum. Differentiating (2.62) with

respect to k we find that

± E√

E2 − |1|2dE

dk= hvF

2.9. HEAT CAPACITY 45

and thus

vparticlesg = vF

√

E2 − |1|2E

k, vholesg = −vF

√

E2 − |1|2E

k. (2.64)

We see that particle excitations propagate in the direction of their momentum, while

the hole excitations – in the opposite direction. Close to the minimum of the dispersion

curve the group velocity decreases and at k = kF reaches zero.

Finally we note that, since Ek = E−k, for a given energy E > |1| and direction

k there are four relevant Bogolubov quasiparticle states: A particle with k = k+k

and a hole with k = −k−k which propagate in the direction of k and a particle with

k = −k+k and a hole with k = k−k which propagate in the opposite direction.

2.9 Heat capacity

It is the easiest to calculate the heat capacity using entropy, since the entropy of the

Cooper-pair condensate is zero. The entropy contribution of quasiparticles is

S = −kB

∑

kσ

[(1− f (Ek)) ln(1− f (Ek))+ f (Ek) ln f (Ek)] .

The heat capacity

C = −T dSdT= −kBT

∑

kσ

d

df[(1− f ) ln(1− f )+ f ln f ] d

dTf (Ek) .

We have

d

df[(1− f ) ln(1− f )+ f ln f ] = ln

f

1− f = ln e−Ek/kBT = − Ek

kBT.

Thus

C =∑

kσ

Ekdf (Ek)

dT. (2.65)

While taking derivative in (2.65) one should remember that both f andEk =√

ξ2k + |1k(T )|2

depend on T . We also perform summation over the spin index and get

C = 2∑

k

EkeEk/kBT

(

eEk/kBT + 1)2

1

kB

[Ek

T 2− 1

T

dEk

dT

]

= 2

kB

∑

k

f (Ek)(1− f (Ek))

[

E2k

T 2− 1

2T

d|1k|2dT

]

. (2.66)

For an arbitrary temperature the heat capacity can be calculated only numerically.

Let us consider the case T ≪ Tc. In this regime f (Ek) ≈ e−Ek/kBT , 1 − f ≈ 1,


|1| ≈ 10 = const and Ek =√

ǫ2k +12

0 ≈ 10 + ǫ2k/210, since due to the exponential

factor only ǫk . kBT ≪ 10 are important in (2.66). Replacing the sum with the

integral over ǫk as in (2.50) and extending the upper limit to infinity (again due to

rapidly decreasing exponential factor) we get

C ≈ 4

kBN(0)

∫ ∞

0

exp

[

−10 + ǫ2/210

kBT

]

ǫ2 +120

T 2dǫ

≈412

0

kBT 2exp

(

− 10

kBT

)

N(0)

∫ ∞

0

exp

(

− ǫ2

210kBT

)

dǫ .

The integral here is equal√

210kBT√π/2 and thus we get for T ≪ Tc

C = 2√

2πkBN(0)10

(10

kBT

)3/2

exp

(

− 10

kBT

)

. (2.67)

Exponential suppression of the heat capacity is a characteristic feature of systems with

an energy gap.

2.10 The Bogolubov – de Gennes equations

We can extend the theoretical description to non-uniform superconductors and super-

conductors in external field in the following manner. We switch from momentum to

real-space representation of quasiparticle creation and annihilation operators:

9†(r, σ ) =∑

k

e−ikrc†kσ , 9(r, σ ) =

∑

k

eikrckσ . (2.68)

For the system we have considered so far the non-interacting Hamiltonian H0 in (2.20)

corresponds to

H0 =∑

σ

∫

d3r9†(r, σ )Hf9(r, σ )

where the free particle Hamiltonian

Hf =1

2m(−ih∇)2 − µ. (2.69)

Here µ is the chemical potential. Let us convert main results for the uniform system

to real-space representation and identify places where Hf enters. Then we declare

that these results will hold for non-uniform system in the external field provided Hf is

replaced by

He =1

2m

(

−ih∇ − ec

A)2+ U(r)− µ, (2.70)

2.10. THE BOGOLUBOV – DE GENNES EQUATIONS 47

which accounts for the magnetic field through the vector potential A and includes some

non-magnetic potential U(r). A more proper approach would be to start with Hamil-

tonian which includes He from the beginning and to build a mean-field theory around

that along the lines outlined above. This was done by Bogolubov and de Gennes. The

end result turns out to be the same.

Instead of using (2.68) we multiply uk and vk by eikr factors and thus introduce

functions of the coordinates uk(r) and vk(r):

9†(r ↓) =∑

k

[

u∗ke−ikrγ

†−k↓ + vke

ikrγk↑]

=∑

k

[

u∗k(r)γ†−k↓ + vk(r)γk↑

]

, (2.71)

9(r ↑) =∑

k

[

ukeikrγk↑ − v∗ke−ikrγ

†−k↓

]

=∑

k

[

uk(r)γk↑ − v∗k(r)γ†−k↓

]

. (2.72)

Requiring Fermi commutation relations for 9 and γ operators we obtain some condi-

tions on uk(r) and vk(r), in particular

∑

k

[

u∗k(r)uk(r′)+ v∗k(r′)vk(r)

]

= δ(r− r′), (2.73)

similar to (2.30).

We will consider only cases where the pairing interaction strength and the gap are

constant as a function of the wave vector, Wkk′ = −W and 1k = 1, which now

becomes function of the coordinates 1(r). The self-consistency equation (2.45) then

retains its simple form, since ukv∗k = uke

ikrv∗ke−ikr = uk(r)v

∗k(r):

1(r) = W∑

k,ǫk<Ec

uk(r)v∗k(r)(1− 2f (Ek)). (2.74)

The condition for the diagonal form of the effective Hamiltonian was previously

expressed by equations (2.35) and (2.37). It is easy to see that these two equations are

equivalent to the following two equations

ξkuk +1kvk = Ekuk, (2.75)

1∗kuk − ξkvk = Ekvk. (2.76)

Indeed, if we multiply (2.75) by vk and subtract (2.76) multiplied by uk we arrive to

(2.35) and if we multiply (2.75) by v∗k and add conjugated (2.76) multiplied by uk we

get (2.37).

We then multiply (2.75) and (2.76) by eikr and notice that ξkeikr = Hf e

ikr and

−ξkeikr = −H ∗f eikr. We obtain

Hf uk(r)+1(r)vk(r) = Ekuk(r), (2.77)

−H ∗f vk(r)+1∗(r)uk(r) = Ekvk(r). (2.78)


Replacing here Hf with He we arrive to the Bogolubov – de Gennes equations

− h2

2m

(

∇ − ie

hcA

)2

uk(r)+ [U(r)− µ] uk(r)+1(r)vk(r) = Ekuk(r), (2.79)

h2

2m

(

∇ + ie

hcA

)2

vk(r)− [U(r)− µ] vk(r)+1∗(r)uk(r) = Ekvk(r). (2.80)

Together with the self-consistency equation (2.74) and normalization (2.73) they form

the basis for description of non-uniform superconducting states.

Obviously, for the uniform state 1(r) = |1|eiχ and A = 0 these equations give

the solution which we already know

uk(r) = Ukeiχ/2eikr, vk(r) = Vke

−iχ/2eikr (2.81)

and

Uk =1√2

(

1+ ξk

Ek

)1/2

, Vk =1√2

(

1− ξk

Ek

)1/2

, Ek =√

ξ2k + |1|2. (2.82)

The amplitudes Uk and Vk are sometimes called coherence factors.

2.11 Electric current

With the Bogolubov – de Gennes description one can calculate response of a super-

conductor to the electromagnetic field. The full description results in complicated ex-

pressions which resemble the phenomenological non-local model of Pippard. Here we

will consider a simple case of the slowly varying vector potential and current. Our

goal is to derive Eq. (1.16) from which the London model follows. In this case the

amplitude of 1 is constant while its phase changes along the direction of the current:

1 = |1|eiχ(r) = |1|eiqr. The solution of the Bogolubov – de Gennes equations is

found in the Problem 2.4:

uk(r) = ei(k+q/2)r Uk, vk(r) = ei(k−q/2)r Vk, (2.83)

Uk =1√2

(

1+ ξk

E(0)k

)1/2

, Vk =1√2

(

1− ξk

E(0)k

)1/2

, E(0)k =

√

ξ2k + |1|2,

(2.84)

Ek = hkvs + E(0)k , (2.85)

vs =h

2m∇χ − eA

mc= hq

2m− eAmc. (2.86)

In equation (2.85) we recognize the Galilean transformation of the quasiparticle spec-

trum (2.9) with the gauge-invariant superflow velocity (2.86). The coherence factors

(2.84) turn out to be the same as in the absence of the current (2.82).

2.11. ELECTRIC CURRENT 49

The general quantum-mechanical expression for the current is

j = e

2m

∑

σ

⟨

9†(r, σ )p9(r, σ )+[

p†9†(r, σ )]

9(r, σ )⟩

, (2.87)

where

p = −ih∇ − ec

A, p† = ih∇ − ec

A. (2.88)

Note that unlike expression (1.15) we cannot switch the order of 9† and 9 operators.

Inserting into (2.87) expressions of 9 operators via γ operators from (2.71) and

(2.72) and using (2.43) to find averages of γ operators we get

j = e

m

∑

k

[

f (Ek)(

u∗k(r) puk(r)+ uk(r) p†u∗k(r))

+ (1− f (Ek))(

vk(r) pv∗k(r)+ v∗k(r) p†vk(r)) ]

.

Performing the same calculations as in Sec. 1.6 for ψ = uk(r) and for ψ = v∗k(r),

where uk(r) and vk(r) are given by (2.83), we find

j = e

m

∑

k

{

f (Ek)U2k

[

2h(

k+ q

2

)

− 2eA

c

]

+ (1− f (Ek))V2k

[

2h(

−k+ q

2

)

− 2eA

c

]}

= e

m

(

hq− 2eA

c

)∑

k

{

f (Ek)U2k + (1− f (Ek))V

2k

}

+ 2he

m

∑

k

k{

f (Ek)U2k − (1− f (Ek))V

2k

}

= e · 2vs ·n

2+ 2he

m

∑

k

k{

f (Ek)[

U2k + V 2

k

]

−[

f (E(0)k )+ V 2

k

]}

= nevs −2he

m

∑

k

k{

f (E(0)k )− f (Ek)

}

. (2.89)

Here we used (2.84) and (2.86), the expression for the particle density

n =∑

σ

⟨

9†(r, σ )9(r, σ )⟩

= 2∑

k

{

f (Ek)u∗k(r)uk(r)+ (1− f (Ek))vk(r)v

∗k(r)

}

= 2∑

k

{

f (Ek)U2k + (1− f (Ek))V

2k

}

, (2.90)

and the fact that Ak = f (E(0)k )+ V 2k is an even function of k and thus

∑

k kAk = 0.

The expression

f (E(0)k )− f (Ek) =

1

e

√

ξ2k+|1|2/kBT + 1

− 1

e

(√

ξ2k+|1|2+hkvs

)

/kBT + 1

(2.91)


depends on the angle between k and vs and is evidently the largest when k is parallel

to vs . Thus the sum over k in (2.89) is directed along vs and we can define the density

of normal electrons nn so that

2he

m

∑

k

k{

f (E(0)k )− f (Ek)

}

= nnevs . (2.92)

Then the current in expression (2.89) becomes

j = nevs − nnevs = nsevs = −e2ns

mc

(

A− hc2e

∇χ

)

, (2.93)

where ns = n − nn is the superconducting density. We obtained equation (1.16) and

hence the London model and provided the expression for calculation of ns . In general

nn and ns are vs-dependent.

For small vs such that hkvs ≪√

ξ2k + |1|2, i.e. vs ≪ |1|/(hkF ) = vc, we expand

(2.91) as

f (E(0)k )− f (Ek) = −f ′(E(0)k ) hkvs

and thus

nnevs = −2h2e

m

∑

k

k(kvs)f′(E(0)k ) = −2h2e

3mvs∑

k

k2f ′(E(0)k ), (2.94)

since for the spherically-symmetric distribution of k we have

∑

k

k(ka)g(k) = 1

3a∑

k

k2g(k).

Replacing in (2.94) the sum with the integral as in (2.51) and taking k2 out of the

integral as k2F we get

nn = −2h2

3m

∑

k

k2f ′(E(0)k ) ≈ −2h2

3m2N(0)k2

F

∫ ∞

|1|

E√

E2 − |1|2f ′(E)dE

= −2n

∫ ∞

|1|

E√

E2 − |1|2f ′(E)dE. (2.95)

Since f ′(E) < 0 we of cause have nn > 0. At the critical temperature 1 = 0 and we

have

nn = −2n

∫ ∞

0

f ′(E)dE = −2n (f (∞)− f (0)) = −2n (0− 1/2) = n.

For T ≪ Tc the normal density is exponentially small, see Problem 2.6.

2.11. ELECTRIC CURRENT 51

Problems

Problem 2.1. Calculate the mean square radius of the Cooper pair.

Problem 2.2. Find the temperature dependence of the gap for T → Tc.

Problem 2.3. Find the heat capacity jump at T = Tc and the temperature dependence

of the critical field for T → Tc.

Problem 2.4. Find the energy spectrum and the coherence factors for the case when

1 = |1|eiqr.

Problem 2.5. Derive the gap equation that determines the dependence of |1| on vs for

the case when 1 = |1|eiqr.

Problem 2.6. Find the temperature dependence of the normal density for T → 0.

Problem 2.7. Find the temperature dependence of the superconducting density for

T → Tc.


Chapter 3

Ginzburg-Landau theory

Many important properties of superconductors can be explained by the theory, devel-

oped by Ginzburg an Landau in 1950, before the creation of the microscopic theory.

The Ginzburg-Landau (GL) theory often provides much simpler and more tractable

description than the BCS theory. Originally the GL theory was developed as a phe-

nomenological model. In 1958 Lev Gor’kov demonstrated that it can be derived from

the microscopic theory within certain restrictions on the parameters. The most im-

portant restrictions are |1| ≪ kBTc, i.e. temperature should be close to the transition

temperature, and λL ≫ ξ0, i.e. electrodynamics is considered local, like in the London

model. Qualitatively, however, predictions of the GL theory are applicable in a much

wider parameter range.

The GL theory has some limitations. It deals only with the gap parameter 1 and

does not describe behavior of quasiparticles. It is also less successful in description of

non-stationary (time-dependent) states.

The starting point of the GL theory is the Landau theory of the symmetry-breaking

phase transitions, which we will briefly introduce.

3.1 Landau theory of phase transitions

Landau considered phase transitions where some symmetry is spontaneously broken:

The Hamiltonian possesses a certain symmetry on both sides of the transition, while

the ground state obeys this symmetry on one side of the transition (symmetric phase)

and looses this symmetry on another side (broken-symmetry phase). He introduced an

order parameter ψ which measures departure of the system state from the symmetric

one. That is, ψ = 0 in the symmetric phase and non-zero in the broken-symmetry

phase.

53

54 CHAPTER 3. GINZBURG-LANDAU THEORY

F

Re Ψ Im Ψχ∆ ∆

Figure 3.1: Below the transition temperature, the free energy Eq. (3.1) has a minimum

at a nonzero order parameter magnitude. The minimum energy is degenerate with

respect to the order parameter phase χ .

For example, for the transition from the paramagnetic to ferromagnetic state the

magnetic moment M can be used as an order parameter. Here the symmetry with re-

spect to space rotations is broken. For the superconducting transition the gap 1 (or a

quantity, proportional to it) is a good order parameter. Since 1 measures expectation

value of the product of two creation operators, Eq. (2.24), the superconducting state

with 1 6= 0 violates particle number conservation. The symmetry behind this con-

servation law, usually called global U(1) symmetry, is broken in the superconducting

state.

Close to the transition, where the order parameter is small, Landau suggested to

expand the free energy of the system in terms of the order parameter, keeping only

contributions which are compatible with the symmetry of the system in the symmetric

phase. For the uniform superconductor in the absence of magnetic field the proper

expansion is

Fsn = Fs − Fn = α|1|2 +β

2|1|4 . (3.1)

Here α and β are expansion parameters, generally dependent on temperature, pressure

and other relevant conditions. To find the equilibrium value of 1 we have to minimize

the free energy. In order to have the minimum at 1 = 0 when T > Tc one should have

α(T ) > 0 at T > Tc. For the minimum at 1 6= 0 one should have α(T ) < 0, β > 0 at

T < Tc. Close to the transition we can thus write

α(T ) = α′(T − Tc), β(T ) ≈ β(Tc) > 0 . (3.2)

3.2. GINZBURG-LANDAU EQUATIONS 55

Minimizing Fsn at T < Tc we obtain

|1|2 = 12GL = −

α

β= α′(Tc − T )

β, Fsn = −

α2

2β. (3.3)

Thus at T → Tc we have |1| ∝ (1 − T/Tc)1/2 as we found from the BCS theory in

Problem 2.2. Comparing (3.3) with the result of this problem we find

α

β= 8π2

7ζ(3)k2

BTc(T − Tc) . (3.4)

Comparing Fsn from (3.3) with equation (1.2) we get

α2

2β= H 2

c

8π, Hc = 2α′Tc

√

π/β

(

1− T

Tc

)

. (3.5)

Using expression for Hc from the BCS theory derived in Problem 2.3 we find

α2

β= 8π2

7ζ(3)N(0)k2

B(T − Tc)2 . (3.6)

From equations (3.4) and (3.6) we obtain expressions for α and β from the microscopic

theory

α = N(0)T − TcTc

, β = 7ζ(3)N(0)

8π2k2BT

2c

. (3.7)

Note that the condition of the minimum of the free energy fixes only the amplitude

(3.3) of the order parameter 1 = |1|eiχ . With respect to the phase χ the energy is

degenerate. This is illustrated in Fig. 3.1. The fact that the system chooses a particular

phase among a set of equivalent values is the manifestation of the broken symmetry.

(In the case of ferromagnets this is the choice of a particular direction of M among

equivalent ones.) Since the phase factor eiχ is a 1× 1 unitary matrix, this symmetry is

called U(1) symmetry.

3.2 Ginzburg-Landau equations

To describe non-uniform superconductors and superconductors in the external field,

Ginzburg and Landau added to the Landau free energy (3.1) terms depending on the

gradient of the order parameter and on the magnetic field:

FGL = α|1|2 +β

2|1|4 + γ

∣∣∣∣

(

−ih∇ − 2e

cA

)

1

∣∣∣∣

2

+ h2

8π, (3.8)

FGL[1,A] =∫

FGLdV . (3.9)


In principle, the GL model can be considered as a generalization of the London model

(1.6): Energy of the superconducting state in the absence of currents and fields is taken

from the Landau theory of phase transitions, while the gradient term resembles the

kinetic energy of superconducting electrons in view of results of Sec. 2.11 and in par-

ticular Eq. (2.86) for vs . Another way to look at the gradient term in (3.8) is to consider

it as a first term of expansion of the free energy over the gradients of the order param-

eter, written in a gauge-invariant form. Of cause, the fact that 2e enters the expression

(3.8) cannot be derived from general considerations, but reflects the microscopic origin

of superconductivity.

As in section 1.4 the free energy FGL is a functional of the fields 1 and A. To

minimize it we take the variation and equals it to zero. Since 1 is complex, 1 and

1∗ are linearly independent. One can check, though, that variations with respect to 1

and 1∗ give equations which are complex conjugate of each other. Thus we will take

variation only with respect to 1∗. We have

δ1∗(|1|2) = δ1∗(11∗) = 1δ1∗,δ1∗(|1|4) = δ1∗(121∗2) = 2121∗δ1∗ = 2|1|21δ1∗,

and

δ1∗

∣∣∣∣

(

−ih∇ − 2e

cA

)

1

∣∣∣∣

2

=[(

−ih∇ − 2e

cA

)

1

]

︸︷︷︸

X

[(

ih∇ − 2e

cA

)

δ1∗︸︷︷︸

a

]

= ihX ∇a +(

−2e

cA

)

X a = iha(−∇X)+ ih div(aX)+ a(

−2e

cA

)

X

= a(

−ih∇ − 2e

cA

)

X+ ih div(aX)

= δ1∗(

−ih∇ − 2e

cA

)2

1+ ih div

[

δ1∗(

−ih∇ − 2e

cA

)

1

]

, (3.10)

where we used vector identity X ∇a = −a div X+div(aX) and we note that notations

div X and ∇X are equivalent. Combining all expressions we have

δ1∗FGL =∫

dV δ1∗{

γ

(

−ih∇ − 2e

cA

)2

1+ α1+ β|1|21}

(3.11)

+ ihγ∫

dS δ1∗ n ·(

−ih∇ − 2e

cA

)

1 . (3.12)

Here we converted volume integral from divergence to the surface integral, n is the

normal to the surface.

3.2. GINZBURG-LANDAU EQUATIONS 57

Requiring δ1∗FGL = 0 for arbitrary δ1∗ we get from the volume term (3.11) the

Ginzburg-Landau equation

γ

(

−ih∇ − 2e

cA

)2

1+ α1+ β|1|21 = 0 (3.13)

and from the surface term (3.12) the boundary condition

n ·(

−ih∇ − 2e

cA

)

1 = 0 . (3.14)

Note that this boundary condition we obtained from FGL which does not include the

energy of interaction between the superconductor and its surroundings. Thus condition

(3.14) is applicable, say, on the boundary between the superconductor and vacuum.

More general consideration leads to the condition

n ·(

−ih∇ − 2e

cA

)

1 = i

bs1, (3.15)

where the parameter bs depends on the material which is in contact with the supercon-

ductor.

Now let us take variation with respect to A. We have

δA(h2) = δA(curl A)2 = 2

b︷︸︸︷

curl A curl

a︷︸︸︷

δA = 2 (div[δA× curl A] + δA curl curl A) .

Here we used vector identity div[a× b] = b curl a− a curl b. We also have

δA

∣∣∣∣

(

−ih∇ − 2e

cA

)

1

∣∣∣∣

2

=[

δA

(

−ih∇ − 2e

cA

)

1

] [(

ih∇ − 2e

cA

)

1∗]

+[(

−ih∇ − 2e

cA

)

1

] [

δA

(

ih∇ − 2e

cA

)

1∗]

= −2e

cδA

[

1∗(

−ih∇ − 2e

cA

)

1+1(

ih∇ − 2e

cA

)

1∗]

.

Thus we obtain

δAFGL =∫

dV δA

{j

c− 2e

cγ

[

1∗(

−ih∇ − 2e

cA

)

1+1(

ih∇ − 2e

cA

)

1∗]}

+ 1

4π

∫

dS n · [δA× curl A] . (3.16)

Here we used Maxwell equation j = (c/4π) curl h = (c/4π) curl curl A. Setting this

variation to zero we obtain the expression for the supercurrent

j = 2eγ

[

1∗(

−ih∇ − 2e

cA

)

1+1(

ih∇ − 2e

cA

)

1∗]

(3.17)

= 4eγ |1|2(

h∇χ − 2e

cA

)

, (3.18)


where 1 = |1|eiχ . By comparing this expression with results of section 2.11 and

Problem 2.7 we can find the coefficient γ in the clean limit, see Problem 3.1.

In general case the coefficient γ depends on purity of the sample. The purity is

characterized by the parameter xs = τskBTc/h ∼ τs/τp, where τs is the electronic

mean free time due to the scattering by impurities and τp is the characteristic time

of the pairing interaction, see Sec. 1.9. Superconductors are called clean when this

parameter is large, and they are dirty in the opposite case. One has

γ = N(0)πD

8hkBTcy(xs) (3.19)

where D = v2F τs/3 is the diffusion coefficient, and

y(xs) =8

π2

∞∑

n=1

1

(2n+ 1)2[(2n+ 1)2πxs + 1] (3.20)

This function is y = 1 for the dirty limit xs ≪ 1, and it is

y = 7ζ(3)

2π3xs

for the clean case xs ≫ 1. Here ζ(3) ≈ 1.202 is the Riemann zeta function. Therefore

γ ={

πN(0)D/8kBTch , dirty

7ζ(3)N(0)v2F /48π2(kBTc)

2 , clean(3.21)

so that γdirty/γclean ∼ xs ≪ 1.

In the end we note that the GL functional (3.9) can be written in a different useful

form. Using the same transformations as in (3.10) but keeping 1∗ instead of δ1∗ we

get

FGL =∫

dV

[

α|1|2 + β2|1|4 + γ1∗

(

−ih∇ − 2e

cA

)2

1+ h2

8π

]

+ ihγ∫

dS 1∗ n ·(

−ih∇ − 2e

cA

)

1

=∫

dV

[

−β2|1|4 + h2

8π+1∗

{

γ

(

−ih∇ − 2e

cA

)2

1+ α1+ β|1|21}]

=∫

dV

[

−β2|1|4 + h2

8π

]

= H 2c

8π

∫

dV

[

−( |1|1GL

)4

+(

h

Hc

)2]

(3.22)

where the surface integral is zero due to the boundary condition (3.14) and the part in

curly braces is zero due to the GL equation (3.13).

3.3. COHERENCE LENGTH AND PENETRATION DEPTH 59

3.3 Coherence length and penetration depth

Let us introduce the dimensionless order parameter ψ so that 1 = ψ1GL and the

equilibrium order-parameter magnitude

1GL =√

−αβ=√

|α|β. (3.23)

Then the equation (3.13) becomes

−ξ2

(

∇ − 2ie

hcA

)2

ψ − ψ + |ψ |2ψ = 0 . (3.24)

Here the length

ξ(T ) = h√

γ /|α| = h√

γ

α′Tc

[

1− T

Tc

]−1/2

(3.25)

defines the characteristic length scale of variation of the order parameter. It is called

the Ginzburg-Landau coherence length. Using expressions (3.7) and (3.21) and the

definition of the zero-temperature coherence length ξ0 from (1.23) we find in the clean

limit

ξ(T ) =(

7ζ(3)

12

)1/2

ξ0

[

1− T

Tc

]−1/2

(3.26)

and in the dirty limit

ξ(T ) = π

2√

3

√

ξ0ℓ

[

1− T

Tc

]−1/2

. (3.27)

Here ℓ = vF τs is the mean free path between scatterings. Since xs = ℓ/2πξ0, the

clean limit corresponds to ℓ≫ ξ0 and the dirty limit to ℓ≪ ξ0. Thus in dirty materials

ξ(T ) is generally much smaller that in clean materials.

Comparing expressions (3.18) and (1.16) for the supercurrent we see that they co-

incide if we define the Ginzburg-Landau superconducting density as

ns = 8γm|1|2 . (3.28)

We already saw that equation (1.16) [and thus equation (3.18) with definition (3.28)]

explains the Meissner effect and gives the penetration depth value

λL =(

mc2

4πnse2

)1/2

= c

4|e|√

2πγ |1|= c

4|e|

√

β

2πα′γ Tc

[

1− T

Tc

]−1/2

. (3.29)

We see that within the GL model both λL(T ) and ξ(T ) have the same temperature

dependence. Their ratio is called the Ginzburg-Landau parameter

κ = λL(T )

ξ(T )= c

4|e|hγ

√

β

2π. (3.30)


We can also express the critical field (3.5) through ξ and λL:

Hc =hc

2√

2|e|λLξ= 80

2√

2πλLξ. (3.31)

The GL model is applicable when |1(T )| ≪ |1(T = 0)| ∼ kBTc. In particular, if

the amplitude of the order parameter is at the equilibrium value 1GL the requirement

becomes Tc − T ≪ Tc. Another requirement is the local electrodynamics, so that the

penetration depth λL(T ) ≫ ξ0. As we see from (3.29), λL(T ) → ∞ when T → Tc.

So this condition is also satisfied sufficiently close to Tc.

Since ξ(T ) limits the maximum gradient of the order parameter also for its phase,

(∇χ)max ∼ 1/ξ(T ), we can define the characteristic maximum current density, the

Ginzburg-Landau critical current, from Eq. (3.18) as

jGLc = 4|e|γ12GLh(∇χ)max =

4|e|hγ12GL

ξ= hc2

8π |e|λ2Lξ∝ (1− T/Tc)3/2 . (3.32)

Sometimes it is convenient to normalize the order parameter in such a way that

|ψ |2 = ns/2, ψ = 2√γm1 . (3.33)

This is the same normalization which we used for the wave function ψ in section 1.6.

The GL free energy becomes

FGL = a|ψ |2 +b

2|ψ |4 + 1

4m

∣∣∣∣

(

−ih∇ − 2e

cA

)

ψ

∣∣∣∣

2

+ h2

8π, (3.34)

FGL =∫

FGLdV =∫[

−b2|ψ |4 + h2

8π

]

dV = H 2c

8π

∫ [

−|ψ |4 + (h/Hc)2]

dV ,

(3.35)

where

a = α

4mγ, b = β

16m2γ 2. (3.36)

The coherence length, the penetration depth and the equilibrium magnitude of the order

parameter become

ξ(T ) = h√4m|a|

, λL =(

mc2

8πe2ψ2GL

)1/2

, ψGL =√

|a|b, (3.37)

and the current

j = e

m|ψ |2

(

h∇χ − 2e

cA

)

. (3.38)

For the dimensionless order parameter ψ = ψ/ψGL equation (3.24) obviously holds.

3.4. CRITICAL FIELD OF A SUPERCONDUCTING SLAB 61

x

yz

d/2

−d/2

H

j

Figure 3.2: Superconducting slab in the parallel magnetic field.

3.4 Critical field of a superconducting slab

As an application of the GL theory let us perform calculation of the critical field for

an infinite slab with thickness d < ξ(T ) in the magnetic field H applied parallel to

the slab. This was one of the first prediction of the GL theory which was checked

experimentally and which demonstrated the superiority of the GL approach compared

to the London model. Other simple applications of the GL theory one can find in

Problems 3.3 and 3.4.

The setting is shown in Fig. 3.2: The slab occupies space between z = −d/2 and

z = d/2, the field is in the y direction and the current and the vector potential are in

the x direction. First we remind the prediction of the London model which was found

in problem 1.6:

H 2c,slab = H 2

c

[

1− 2λL

dtanh

d

2λL

]−1

. (3.39)

Here Hc is the thermodynamic critical field. In the London model the transition at

Hc,slab is always of the first order, since ns is not allowed to change. In the GL model

we may allow ns = 2|ψ |2 to deviate from the equilibrium value. Since we consider

slabs thinner than the coherence length we can assume that the order parameter is

constant in the slab: ψ = ψψGL with ψ = const but not necessarily unity. For

ψ = const we have from the equation (3.38)

−2e2

mc|ψ |2A = j = c

4πcurl curl A

or∂2A

∂z2=

8πe2ψ2GL

mc2ψ2A = ψ2

λ2L

A .

The general solution of this equation is

A = A1ezψ/λL + A2e

−zψ/λL .

The constants A1 and A2 should be determined from the boundary condition for the


magnetic field h = ∂A/∂z = H at z = ± d/2. We find

h = H cosh(ψz/λL)

cosh(ψd/2λL), A = HλL

ψ

sinh(ψz/λL)

cosh(ψd/2λL). (3.40)

In the fixed external field we should consider the Gibbs free energy G = F −HB/4π

with B = 〈h〉. We have per unit area of the slab

Gsn = Gs(H)− Gn(H) =d/2∫

−d/2

dz

[

FGL −hH

4π

]

+ d H2

8π

=d/2∫

−d/2

dz

[

a|ψ |2 + b2|ψ |4 + 1

4m

(2e

cA

)2

|ψ |2 + h2 − 2hH

8π

]

+ d H2

8π

= 1

8π

d/2∫

−d/2

dz

[

H 2c (−2ψ2 + ψ4)+ A

2ψ2

λ2L

+ h2 − 2hH

]

+ d H2

8π

= d H2c

8π(−2ψ2 + ψ4)+ H

2

8π

{

−2λL

ψtanh(ψd/2λL)+ d

}

To find ψ we minimize Gsn by setting ∂Gsn/∂ψ = 0 which gives the equation

(H

Hc

)2

= 4ψ2(1− ψ2) cosh2(ψd/2λL)

[

sinh(ψd/λL)

ψd/λL− 1

]−1

, (3.41)

which is valid for any H including H = Hc,slab. The critical field corresponds to

Gsn = 0 which gives the equation

(Hc,slab

Hc

)2

= ψ2(2− ψ2)

[

1− tanh(ψd/2λL)

ψd/2λL

]−1

. (3.42)

Equations (3.41) and (3.42) give the solution of the problem in the implicit form.

Let us consider limiting cases. If d ≫ λL then ψ ≈ 1 and from Eq. (3.42) we get

the same result as in the London model (3.39). Using condition d ≫ λL it can be

simplified as

Hc,slab = Hc (1+ λL/d) . (3.43)

Now let us consider thin films, d ≪ λL. Expanding hyperbolic functions in

Eq. (3.41) we get

H 2c,slab = H 2

c

4ψ2(1− ψ2)

(ψd/λL)2/6= 24H 2

c

λ2L

d2(1− ψ2) , (3.44)

3.4. TYPE I AND TYPE II SUPERCONDUCTORS 63

Hc,s

lab/Hc

λL(T )/d

00.2 0.4 0.6

1

2

3

2√

6λL/d

hysteresis

d

Figure 3.3: Measurements of the critical field

of the tin films of different thicknesses d versus

temperature-controlled London penetration depth

λL (symbols) compared to the GL model for

Hc,slab (solid line). Broken line shows the hys-

teresis in the transition observed for thicker films,

indicating that the transition is of the first order.

The hysteresis disappears for the thin films, which

is consistent with the transition of the second or-

der (N.V. Zavaritski, 1952).

and from Eq. (3.42) we get

H 2c,slab = H 2

c

ψ2(2− ψ2)

(ψd/2λL)2/3= 12H 2

c

λ2L

d2(2− ψ2) . (3.45)

Solving these two equations we obtain

ψ = 0 ,

i.e. the transition is of the second order and

Hc,slab = 2√

6λL

dHc ≫ Hc . (3.46)

With decreasing film thickness the transition changes from the first order to the second

order at d =√

5λL, see Problem 3.6.

These predictions of the GL theory were confirmed by the experiment, see Fig. 3.3.

3.5 Energy of the normal–superconducting boundary.

Type I and type II superconductors

If one places a large slab in the perpendicular magnetic field, then the magnetic field

starts to penetrate into the slab already at H < Hc, see Fig. 3.4, since expelling the

field in this geometry would require screening currents to exceed the critical magni-

tude. Thus normal and superconducting regions in the same material may coexist. It

is clear, that for the stable situation the field in normal regions should be equal to Hc.


S N

S

N NS S

B = H < Hc

B = 0

B = Hc

Figure 3.4: Intermediate state of type I superconductors. In the external field H < Hc

the sample is divided to normal and superconducting domains so that in the normal

phase magnetic induction B = Hc while in the superconducting domains magnetic

field is absent.

When doing calculation of the stable configuration it turns out that additionally one

has to require that normal-superconducting (NS) boundary have some extra energy,

associated with it. This energy can be determined experimentally by comparing the

experimentally observed NS patterns to the calculated ones. Where does this energy

originate?

Let us consider this problem within the GL theory. We will look at a single bound-

ary in the field H = Hc and will assume the system to be homogeneous in y and z

directions, and to be fully superconducting at x →−∞ and normal at x →+∞. The

goal is thus to solve the GL equations (3.24) and (3.38) with the boundary conditions

ψ → 1, h→ 0, at x →−∞,ψ → 0, h→ Hc, at x →+∞.

(3.47)

In the general case this can be done only numerically. But the qualitative picture of

the solution is clear: ψ changes from 1 to 0 in the layer of thickness ξ and h changes

from 0 to Hc in the layer of thickness λL, Fig. 3.5. What energy is associated with this

profile of the order parameter?

Remember that we have to consider the Gibbs free energy. For the pure super-

conducting or the pure normal state at H = Hc this energy is equal to −H 2c /8π , see

Sec. 1.3. Let us consider this value as a zero of energy. Using (3.35) we get

G =∞∫

−∞

dx

[

FGL −hH

4π+ H

2c

8π

]

= H 2c

8π

∞∫

−∞

dx

[

−|ψ |4 +(h−HcHc

)2]

= H 2c

8πδ ,

(3.48)

3.5. TYPE I AND TYPE II SUPERCONDUCTORS 65

x0x0

(a) (b)

h(x)h(x) 9(x) 9(x)

∼ ξ ∼ ξ

∼ λL

∼ λL

Figure 3.5: Change of the order parameter amplitude |ψ(x)| and of the local magnetic

field h(x) at the boundary between coexisting normal and superconducting domains.

(a) Type I superconductor with√

2λL < ξ . (b) Type II superconductor with√

2λL >

ξ.

where δ is a characteristic quantity of the length dimension. Using conditions (3.47)

we see that the expression under integral in (3.48) is zero everywhere except the NS

boundary.

In the case ξ ≫ λL, i.e. κ ≪ 1, Fig. 3.5a, we have a region of size∼ ξ −λL where

|ψ | and h are both small which makes a positive contribution to the integral (3.48)

and thus δ ∼ ξ − λL. Exact calculation in this limit (Problem 3.8) gives δ ≈ 1.89ξ .

Superconductors with the positive NS energy are called type I superconductors. In this

case the picture of reasonably thick normal and superconducting slices in Fig. 3.4 has

sense.

In the opposite case ξ ≪ λL, i.e. κ ≫ 1, Fig. 3.5b, we have a region of size∼ λL−ξ where |ψ | ∼ 1 and h ∼ Hc which makes a negative contribution to the integral (3.48)

and thus δ ∼ −(λL−ξ) = ξ−λL. Exact calculation in this limit gives δ ≈ −1.104λL.

Superconductors with negative NS energy are called type II superconductors. In this

case the picture of reasonably thick normal and superconducting slices in Fig. 3.4 can

not be stable: number of NS boundaries will tend to increase and the size of normal

regions to decrease. The limit to this process is set by the flux quantization: A normal

region with the trapped magnetic field which is surrounded by the superconductor can

at minimum carry one quantum of circulation 80 and thus it cannot disappear. Such

objects are called quantized vortices and we will consider them further in this chapter.

Special consideration shows that the transition from the positive to the negative

energy of the NS boundary occurs at

κ = 1√2.

Thus superconductors with κ < 1/√

2 are of type I, and those with κ > 1/√

2 are of

type II.

What material parameters determine value of κ? In the clean limit we have from


equations (3.30), (3.7), (3.21) and (2.6)

κ = c

4|e|hγ

√

β

2π= 3c

|e|h

√π

7ζ(3)N(0)

kBTc

v2F

= 3π2

√14ζ(3)

hc

e2

kBTc

EF

√

e2m

πhpF

= 3π2

√14ζ(3)

hc

e2

kBTc

EF

√

e2/a0

EF≈ 3π2

√14ζ(3)

hc

e2

kBTc

EF∼ 103 kBTc

EF. (3.49)

Here we used

EF =mv2

F

2=p2F

2m, a0 =

2πh

pF, pF =

p2F

pF= (2mEF )

a0

2πh

and the fact that the ratio of the potential energy of conducting electrons, e2/a0, to their

kinetic energy EF is of the order of unity for good metals, although it may become

larger for systems with strong correlations between the electrons. The last factor in

Eq. (3.49) is usually small: kBTc/EF is below 10−3 for usual superconductors, but it

is of the order of 10−1 − 10−2 for high temperature superconductors with Tc ∼ 100K

and EF ∼ 1000K .

We see that for usual clean superconductors the Ginzburg-Landau parameter is nor-

mally small, though, in some cases it may be of the order of 1. On the contrary, for

high temperature superconductors, which have a tendency to be strongly correlated

systems with a not very low ratio of kBTc/EF , the parameter κ is usually very large.

The Ginzburg-Landau parameter increases for dirty superconductors with xs ≪ 1:

κdirty ∼ κclean/xs . (3.50)

Therefore, dirty alloys normally have a large κ .

3.6 Abrikosov vortices. Critical field Hc1

Magnetic field penetrates into type II superconductors in the form of small regions

which carry one quantum of magnetic flux 80 each. As can be seen from Eq. (1.17),

this corresponds to the increase of the order-parameter phase χ by 2π on a loop around

this region. This phase winding is a topological invariant: It cannot be continuously

reduced to zero. Homotopy group theory connects this topological invariant to exis-

tence of linear defects of the order parameter: continuous lines where ψ = 0 and

around which the phase of ψ winds by 2π . In a superconductor such linear defects

carry magnetic flux and are called Abrikosov vortices, Fig. 3.6. They were theoreti-

cally predicted by A. Abrikosov in 1957. As a topological object, a vortex cannot end

in superconductor bulk: It can only end at the boundary or form a closed loop.

3.6. ABRIKOSOV VORTICES. CRITICAL FIELD HC1 67

B

Figure 3.6: Abrikosov vortices in type II superconductor are topologically-protected

linear defects of the order parameter: Order parameter is zero at the vortex axis and the

phase of the order parameter winds by 2π on a loop around the vortex. Each vortex

carries a single quantum of magnetic flux.

Let nv be density of vortex lines in a plane perpendicular to the direction of the

magnetic field (and to vortex axes). Since each vortex carries flux 80, the magnetic

induction is simply

B = 80nv (3.51)

and the magnetization of the sample

M = 80nv −H4π

= 80

4πnv +M0 ,

where M0 = −H/4π is the magnetization (1.14) with the complete Meissner effect.

From the magnetization curve of type II superconductor in Fig. 1.4 we see that in

the fields slightly above Hc1 difference M − M0 is small and it goes to zero when

H → Hc1. Thus in this limit vortex density nv is small, vortices are far apart and we

can consider each of them individually.

Let us study within the GL theory a single straight vortex. We place it at r = 0

of the cylindrical coordinate system (r, φ, z) and assume the problem to be axially-

symmetric and uniform in the z direction. Thus we can write

ψ = ψGLψ, ψ = f (r) eiφ , (3.52)

h = h(r) z . (3.53)

We can choose

A = A(r) φ (3.54)

so that

h(r) = 1

r

d

drrA(r) (3.55)


and

j = c

4πcurl h = − c

4π

dh(r)

drφ . (3.56)

Inserting (3.52) and (3.54) into the GL equation (3.24) we get the equation

ξ2

[

f ′′ + f′

r− 4e2

h2c2Q2f

]

+ f − f 3 = 0 , (3.57)

where the prime denotes derivative over r and we introduced function Q(r) so that

A(r) = Q(r)+ hc

2er. (3.58)

Note that

h(r) = 1

r

d

drrQ(r) . (3.59)

Now inserting equations (3.52) and (3.54) into the current expression (3.38) and ex-

pressing the current via Q from (3.56) and (3.59) we get the equation

Q′′ + Q′

r− Qr2− f

2Q

λ2L

= 0 . (3.60)

We obtained the system of two equations (3.57) and (3.60) for two functions f (r) and

Q(r) with boundary conditions

f → 0 and A finite, r → 0,

f → 1 and h→ 0, r →∞ .(3.61)

In general case the system can be solved only numerically. An example of the solution

is plotted in Fig. 3.7. It is clear that when λL ≫ ξ , i.e. κ ≫ 1 there are three distinct

regions in the vortex structure:

(i) r ≫ λL. Here f = 1 and h is decaying exponentially with increasing distance

from the vortex axis.

(ii) ξ ≪ r ≪ λL. Here 1− f ≪ 1 and h is changing.

(iii) r ≪ ξ . Here f ≪ 1 and increasing with r while h ≈ const.

The region r . ξ where the order parameter substantially deviates from the equilibrium

value is called the vortex core.

In the each of the regions (i)–(iii) we can solve the equations (3.57) and (3.60)

analytically. We start with equation (3.60) in the regions (i) and (ii). Here we can set

f = 1 and the Eq. (3.60) becomes the modified Bessel equation. We need the solution

which decays when r →∞. It is

Q(r) = Q0K1(r/λL) . (3.62)

3.6. ABRIKOSOV VORTICES. CRITICAL FIELD HC1 69

00

1

20

5

1

10

2

15

3

r/ξ

r/λL

9(r)/9GL

h(r)/Hc1

Figure 3.7: Dependence of the order-parameter amplitude and of the magnetic field on

the distance from the axis of the Abrikosov vortex found from numerical solution of

the GL equations for κ = 5.

HereK1 is the modified Bessel function of the second kind with the asymptotic behav-

ior1

K1(x) = 1/x, x ≪ 1,

K1(x) =√

2

πxe−x, x ≫ 1,

and the constant Q0 is determined from the boundary condition at r → 0

A = Q0λL

r+ hc

2erfinite,

which gives

Q0 = −hc

2eλL. (3.63)

Magnetic field we find from equation (3.59)

h(r) = hc

2eλ2L

K0(r/λL) = −80

2πλ2L

K0(r/λL) . (3.64)

Note that with our choice of the direction of phase winding in Eq. (3.52) the magnetic

field turns out to be directed against the z axis. The asymptotic behavior of K0 is

K0(x) = − ln x, x ≪ 1,

K0(x) =√

2

πxe−x, x ≫ 1,

1Digital Library of Mathematical Functions http://dlmf.nist.gov is highly recommended.


In the region (iii) the magnetic field is approximately constant and thus is equal to the

value given by Eq. (3.64) at r = ξ .

Now we consider equation (3.57) in the regions (ii) and (iii). Here

Q(r) = Q0λL

r= − hc

2er

and equation (3.57) becomes

ξ2

(

f ′′ + f′

r− f

r2

)

+ f − f 3 = 0. (3.65)

In the region (iii) f ≪ 1, we can omit f 3 in this equation. Then it becomes the Bessel

equation with the solution

f = CJ1(r/ξ) = f0r/ξ . (3.66)

The constant f0 ∼ 1 can be found only numerically. In the region (ii) we put g =1− f ≪ 1 into Eq. (3.65) and obtain in the first order

g = ξ2

2r2. (3.67)

We can summarize all findings in the following table

r ≪ ξ f = f0 r/ξ h = −h0 ln(λL/ξ)

ξ ≪ r ≪ λL f = 1− ξ2/(2r2) h = −h0 ln(λL/r)

λL ≪ r f = 1 h = −h0

√

2λL/πre−r/λL

(3.68)

where

h0 =80

2πλ2L

.

Now we can calculate the energy of the vortex compared to the state when there is

no vortex (f = 1, h = 0). We have from Eq. (3.35)

Fv = FvortexGL − Fno vortex

GL = H 2c

8π

∞∫

0

[

1− f 4 + (h/Hc)2]

2πrdr . (3.69)

The integral can be evaluated separately in the ranges (0, ξ), (ξ, λL) and (λL,∞) using

expressions (3.68). It is easy to see that among the six terms [(1 − f 4) and (h/Hc)2

contributions in each of the three ranges] the largest by the factor of ln κ comes from

the 1 − f 4 term in the (ξ, λL) range. The reason is that it results in a logarithmically

divergent integral, while all other terms are regular. We thus have

Fv ≈H 2c

8π

λL∫

ξ

(1− f 4) 2πr dr = H 2c

4

λL∫

ξ

4g r dr = H 2c

2ξ2 ln

λL

ξ=

820

16π2λ2L

ln κ .

(3.70)

3.7. INTERACTION OF AN ABRIKOSOV VORTEX WITH ELECTRIC CURRENT71

The first critical field Hc1 corresponds to penetration of the first vortices to the

superconductor. It is thus determined by the condition that the Gibbs free energy of the

state without vortices is equal to the energy of the state with the far separated vortices.

We have

0 = Gvortex(Hc1)−Gno vortex(Hc1) =

nvFv −1

4π(Bvortex − Bno vortex)Hc1 = nvFv −

1

4πnv80Hc1 . (3.71)

From this equation we obtain

Hc1 =80

4πλ2L

ln κ = Hcln κ√

2κ. (3.72)

Comparing to (3.68) we find that in the vortex core h(0) = 2Hc1. Note also that the

energy difference in (3.71) does not depend on the density of vortices in the approx-

imation of individual vortices. That means that at Hc1 vortices start to penetrate in

the sample until their mutual interaction breaks the condition (3.71). This corresponds

to the vertical derivative dM/dH at H = Hc1 on the magnetization curve of type II

superconductor, Fig. 1.4.

3.7 Interaction of an Abrikosov vortex with electric cur-

rent

Appearance of vortices changes the electrical and magnetic properties of a supercon-

ductor quite substantially. In particular, if vortices are free to move, then the resistivity

becomes non-zero. We can see this by considering interaction of a vortex with applied

electric current jex within the London model, Eq. (1.6). Let vortex be in the position

r0. The current around the vortex is j = j(r− r0). The current makes the contribution

to the energym

2nse2(j+ jex)2 = m

2nse2

[

j2 + (jex)2 + 2j jex]

.

The first two terms here are the energy of the vortex and of the external current. The

last term is the interaction energy

Fint =m

nse2jex j . (3.73)

For the force per unit length of the vortex we have

fL = −∇r0

∫

FintdS = −m

nse2∇r0

∫

jex j dS ,


where the integration is in the plane perpendicular to the vortex axis. Applied current

jex is independent of r0 while ∇r0j = −∇rj. We replace ∇r0

with−∇r, considering jex

to be constant, and use vector identity ∇(Ca) = (C∇)a+C× curl a valid for constant

vector C:

fL =m

nse2

∫

∇(jex j) dS = m

nse2

(∫

[jex × curl j] dS +∫

(jex∇)j dS

)

=

m

nse2

(∫

[jex × curl j] dS −∫

j div jex dS

)

= m

nse2

∫

[jex × curl j] dS .

Here we integrated by parts the second term and use div jex = 0. Using result of

Problem 3.9 we find

curl j = c80

4πλ2L

δ(r − r0)b ,

where b is the unit vector along the direction of the vortex. Finally we have

fL =80

c[jex × b] . (3.74)

This is the Lorentz force acting on the vortex. Multiplying it on the vortex density nv

we find the force FL acting per unit volume. Since magnetic induction B = nv80b we

have

FL = c−1[jex × B] . (3.75)

Under action of this force vortices may start to move with velocity vL. Let us assume

that they experience viscous damping force fv = −ηvL. In steady motion fL + fv = 0

and thus

vL = (80/ηc)[jex × b] . (3.76)

In the frame moving with vortices there is magnetic field B. Thus in the laboratory

frame electrical field appears

E = c−1[B× vL] . (3.77)

Substituting (3.76) and taking into account b = B/B and B ⊥ jex we obtain

E = (80B/ηc2)jex . (3.78)

But appearance of the electric field along the current means that the resistivity is no

longer zero:

ρ = E/j ex = 80B/ηc2 . (3.79)

Thus superconducting properties are lost.

In order to preserve the zero resistivity vortices should be immobile. This can be

achieved by pinning: In an inhomogeneous superconductor vortex energy may depend

3.8. UPPER CRITICAL FIELD HC2 73

on the vortex position. A vortex thus can settle to a local minimum of energy and then

application of the current above some critical value would be required before the vortex

will start to move.

Note that the current jex may also be produced by another vortex. This results in

the interaction between vortices. Using result of Problem 3.9 we find that two vortices

with the same direction of flux repel each other with the force fL ∝ δ−1 inversely

proportional to the distance between them δ, if δ < λL. If δ ≫ λL, then the interaction

is much smaller. In a sense this resembles the interaction between ions in a metal:

They repel each other at small distances due to Coulomb interaction while at larger

distances the Coulomb repulsion is screened. With this analogy we expect that at lower

temperatures vortices tend to form a crystalline-like lattice (or, if disorder is strong, a

glass-like state). At higher temperatures, if thermal fluctuations are strong enough and

vortices are sufficiently mobile, the lattice can melt and vortex liquid is formed. The

vortex matter including dynamics of vortices in superconductors is by itself a large area

of research, which we won’t discuss in this course.

3.8 Upper critical field Hc2

When in a type II superconductor magnetic field is increased above Hc1, distance be-

tween Abrikosov vortices decreases. At sufficiently high fields vortices may come

closer to each other then the coherence length ξ , and their cores would start to overlap.

Since in the core the amplitude of the order parameter is suppressed, the average am-

plitude of the order parameter will start to decrease in this situation, until eventually it

will become zero everywhere at some field which is called Hc2. Thus one can expect

that for the type II superconductors transition to the normal state in the field is of the

second order.

One can arrive to the same conclusion also from a different point of view. We have

seen in Sec. 3.4 that for a thin superconducting slab in the parallel field, the critical

field may significantly exceed Hc. So why the sample should go completely normal

at the field Hc, while it can split to thin normal and superconducting slabs? In type

I superconductors this scenario is prevented by positive energy of the NS interface,

but this limitation is not applicable to type II superconductors. Thus we conclude that

in the type II superconductor transition to the normal state occurs only when even

infinitesimally small superconducting region cannot exist.

We can find the corresponding critical field Hc2 using GL theory. Since in our case

|ψ | ≪ ψGL, i.e. |ψ | ≪ 1, we can linearize the GL equation (3.24):

−ξ2

(

∇ − 2ie

hcA

)2

ψ − ψ = 0 . (3.80)


Additionally, such a small superconducting region cannot affect the magnetic field.

Thus the vector potential is fully determined by the external field H = H z and we

can select A = Hxy. Then the equation (3.80) separates for variables x, y and z and

for the last two it becomes the Schrodinger equation for a free particle, which has the

plane-wave solutions. Thus we look for the overall solution in the form

ψ = f (x)eikyyeikzz . (3.81)

Inserting this into Eq. (3.80) we get

−ξ2f ′′ − ξ2

[

(ikz)2 + (iky)2 −

4ie

hcHx(iky)− 4e2

h2c2H 2x2

]

f = f

or

−ξ2f ′′ + ξ2

(

ky −2eH

hcx

)2

f = (1− k2z ξ

2)f .

Introducing

x0 =hc

2eHky

we arrive to the equation

−ξ2f ′′ +(

2eHξ

hc

)2

(x − x0)2f = (1− k2

z ξ2)f . (3.82)

We are looking for the solution of this equation which goes to zero when x → ±∞.

But Eq. (3.82) coincides with the equation of the harmonic oscillator

(

− h2

2m

d2

dx2+ 1

2mω2x2

)

ψ = Eψ

with

ω = 4|e|Hξ2

h2c,

for which the solution satisfying such boundary conditions exists only when

E = hω(

n+ 1

2

)

with integer n. Applying this to Eq. (3.82) we find

1− k2z ξ

2 = h4|e|Hξ2

h2c

(

n+ 1

2

)

or

H = hc

4|e|1/ξ2 − k2

z

n+ 1/2.

3.8. UPPER CRITICAL FIELD HC2 75

Meissner

Vortex

Normal

HT T

HH

H

c

c

c1

c2

Figure 3.8: Phase diagram of a type II superconductor.

The maximum possible value of the field obviously corresponds to kz = 0 and n = 0.

We thus obtain

Hc2 =hc

2|e|1

ξ2= 80

2πξ2∝ 1− T

Tc. (3.83)

Comparing with Eq. (3.31) we find

Hc2 = Hc√

2κ . (3.84)

For κ > 1/√

2 we have Hc2 > Hc. The phase diagram of type II superconductor is

shown in Fig. 3.8

In a type I superconductor Hc2 < Hc, but it also has a physical meaning. In this

case the superconducting transition at Hc is of the first order. As a result, normal state

can be supercooled and can exist as a metastable state at H < Hc. The reason behind

this is that owing to positive energy of NS interface, a superconducting seed of a finite

size should be formed to be stable. However, at the fieldHc2 even infinitesimally small

superconducting region becomes energetically favourable. Thus the field Hc2 put the

absolute limit for supercooling of the normal phase.

In this section we have considered formation of the superconducting seed in an

infinite system. Close to the boundary (with vacuum or dielectric), however, stable

superconducting region can exist up to the field

Hc3 = 2.4κHc = 1.7Hc2 . (3.85)

SinceHc3 > Hc2 it isHc3 which normally puts the limit on supercooling of the normal

phase.


3.9 Fluctuations. Applicability of the GL theory

The GL theory is a mean-field theory: In a homogeneous system ψ does not depend

on space coordinates or time, ψ = ψGL. Actually at non-zero temperature the system

will fluctuate around this mean value: ψ(r, t) = ψGL+δψ(r, t). Consider a particular

instantaneous realization of the fluctuation δψ(r). The fluctuation changes the energy

of the system by

δFGL[δψ] = FGL[ψGL + δψ] − FGL[ψGL] . (3.86)

The probability to encounter such a fluctuation is

P [δψ] = 1

Zexp

(

−δFGL[δψ]kBT

)

, (3.87)

where the normalization factor Z is called partition function

Z =∑

s

exp

(

−δFGL[δψs]kBT

)

. (3.88)

Here the sum is taken over all possible realizations of fluctuations labeled by the in-

dex s.

The average value of any quantity a[ψ] can be calculated as

〈a[ψ]〉 =∑

s

a[ψGL + δψs]P [δψs] . (3.89)

In particular, the free energy turns out to be

F = −kBT lnZ . (3.90)

One may also consider deviation of ψ from the equilibrium value as creation of

some (bosonic) quasiparticles or excitations, which are called collective modes of the

order parameter. In a homogeneous system we can perform Fourier transform of the

order parameter (as before, we perform calculations for a unit volume V = 1)

δψ(r) =∑

k

eikrδψk (3.91)

and present δFGL as

δFGL[δψ(r)] =∑

k

εkδψ∗kδψk .

Here εk is a contribution to the energy of the new excitations which we may call the

potential energy. To find the energy spectrum of the collective modes one would need

to consider time dependence and thus corresponding kinetic energy. We, however, will

only discuss the “rest mass” of the excitations, i.e. potential energy when k→ 0.

3.9. FLUCTUATIONS. APPLICABILITY OF THE GL THEORY 77

3.9.1 Uncharged superfluid, T > Tc

For simplicity we first look at the case of uncharged superfluid with e = 0. This might

be a hypothetical 3He superfluid with the s-wave pairing. (In actual superfluid 3He

pairing is of the p-wave type.) We will also use the simplest possible approximation,

the so-called Gaussian approximation, where in Eq. (3.86) we leave only the second-

order terms with respect to fluctuations. (Note that the first-order terms disappear since

we consider fluctuations around the equilibrium value.) We start with the case T > Tc.

Here ψGL = 0 and ψ = δψ . We obtain

δFGL[δψ(r)] =∫

d3r

[

a(δψ(r))∗δψ(r)+ 1

4m(−ih∇δψ(r))∗ (−ih∇δψ(r))

]

=∑

kk′

∫

d3r e−ikreik′r[

aδψ∗kδψk′ +1

4m(−ih ik δψk)

∗ (−ih ik′ δψk′)

]

=∑

k

(

aδψ∗kδψk +h2k2

4mδψ∗kδψk

)

=∑

k

(

a + h2k2

4m

)[

(Re δψk)2 + (Im δψk)

2]

. (3.92)

We have used relation∫

d3r e−ikreik′r = δkk′ . Since δψ(r) is complex, Re δψk and

Im δψk are independent. Thus, we have found two collective modes with potential

energy

εk = a +h2k2

4m.

Since a > 0 at T > Tc these modes have a gap. Or, in the language of field theory,

they are massive.

With the help of Eq. (3.92) one can calculate the partition function and then the

free energy and other thermodynamic quantities, like heat capacity. It turns out that the

heat capacity due to fluctuations diverges at T → Tc + 0 as

Cfluct ∝ (T − Tc)−1/2 .

3.9.2 Uncharged superfluid, T < Tc

We may expect that the situation changes in the broken-symmetry phase, since now in

the Mexican-hat potential, Fig. 3.1, the uniform change of the phase does not change

the energy of the system. We may select undisturbed order parameter to be real ψ =ψGL and with fluctuations included it becomes

ψ = (ψGL + δψ)eiφ ,


where the fluctuations δψ and φ are real. We have

FGL[ψ] =∫

d3r[

a(ψGL + δψ)2 +b

2(ψGL + δψ)4

+ 1

4m

(

− ih(∇δψ)eiφ + (−ih)i(ψGL + δψ)(∇φ)eiφ)∗

(

− ih(∇δψ)eiφ + (−ih)i(ψGL + δψ)(∇φ)eiφ)]

(3.93)

and thus up to the second order in fluctuations

δFGL =∫

d3r[

a(δψ)2 + b2

6ψ2GL(δψ)

2 + 1

4m

{

ih(∇δψ)∗e−iφ(−ih)(∇δψ)eiφ

+ hψGL(∇φ)∗e−iφ hψGL(∇φ)eiφ}]

=∫

d3r

[

2|a|(δψ)2 + h2

4m

{

|∇δψ |2 + |a|b|∇φ|2

}]

. (3.94)

Here we used ψ2GL = −a/b = |a|/b. Performing Fourier transform

δψ(r) =∑

k

eikrδψk, φ(r) =∑

k

eikrφk

we obtain

δFGL =∑

k

{(

2|a| + h2k2

4m

)

δψ∗kδψk +|a|b

h2k2

4mφ∗kφk

}

. (3.95)

Since δψ(r) and φ(r) are real, the real and imaginary parts of δψk and φk are not

independent:

δψ−k = δψ∗k , φ−k = φ∗k . (3.96)

Thus equation (3.95) describes one massive amplitude mode with the energy

εak = 2|a| + h

2k2

4m(3.97)

and one massless phase mode with the energy

εphk =

|a|b

h2k2

4m. (3.98)

The appearance of gapless (massless) Goldstone modes is a characteristic feature of

systems with broken continuous symmetries.

3.9. FLUCTUATIONS. APPLICABILITY OF THE GL THEORY 79

3.9.3 Ginzburg number

Since the GL theory neglects fluctuations, it is applicable only when the fluctuations

are small compared to the mean-field value of the order parameter. For the amplitude

we have⟨

δψ2⟩

≪ ψ2GL (3.99)

and for the phase⟨

φ2⟩

≪ 1 . (3.100)

Let us estimate the magnitude of the amplitude fluctuations. Using Eq. (3.89) we

write⟨

δψ2⟩

=∑

k

′∫

d2δψk δψ∗kδψk e

−(εak/kBT )δψ

∗kδψk

∫

d2δψk e−(εa

k/kBT )δψ∗kδψk

. (3.101)

The prime at the sum means that due to condition (3.96) we have to sum only over a

hemisphere of k values. Since the expressions under the integrals do not depend on the

argument of δψk we perform integration over the argument

∫

d2δψk →∞∫

0

2π |δψk| d|δψk|

and use integral expressions

∞∫

0

xe−ax2dx = 1

2a,

∞∫

0

x3e−ax2dx = 1

2a2

to find

⟨

δψ2⟩

=∑

k

′ 1/[2(εak/kBT )

2]1/[2εa

k/kBT ]= kBT

2

∑

k

1

εak

= kBT

2

2π/ξ∫

0

4πk2 dk

(2π)31

|a| + h2k2/4m

= kBTm

π2h2

2π/ξ∫

0

k2 dk

k2 + ξ−2= kBT

2π |a|ξ3

[

1− (2π)−1 arctan 2π]

. (3.102)

Here we converted sum over k to the integral, took into account that the shortest wave-

length of fluctuations is ∼ ξ and used expression (3.37) for ξ .

For applicability of the GL theory one requites⟨

δψ2⟩

≪ ψ2GL, i.e.

1

2π

kBTc

ξ3≪ a2

b= H 2

c

8π


orkBTc

ξ30

(

1− TTc

)−3/2≪ Hc(0)

2

(

1− T

Tc

)2

.

Thus

1− T

Tc≫ Gi , (3.103)

where the Ginzburg number

Gi =[

kBTc

Hc(0)2ξ30

]2

∼[

kBTc

N(0)120(hvF /10)3

]2

∼ (kBTc)4

E4F

. (3.104)

Here we used Eqs. (2.59) and (1.20). In typical superconductors kBTc ∼ 10−3EF and

the Ginzburg number is very small Gi ∼ 10−12. Thus the applicability range of the GL

theory starts from an almost immediate vicinity of the transition.

3.10 The Anderson-Higgs mechanism

We return to discussion of the GL model with electric charge. Fluctuations in the

charged system are evidently coupled to the electromagnetic field. We will consider the

external field to be absent, so that non-zero value of the vector potential A and magnetic

field h = curl A are due to fluctuations. We continue to use Gaussian approximation.

If T > Tc and ψ = δψ one can see from the functional (3.34) that in the second order

there is no terms that couple the field and the order parameter, so their fluctuations in

the Gaussian approximation are independent.

At T < Tc the situation is more interesting. Similar to Eqs. (3.93) and (3.94) we

have

FGL[ψ] =∫

d3r[

a(ψGL + δψ)2 +b

2(ψGL + δψ)4

+ 1

4m

∣∣− ih(∇δψ)eiφ + (−ih)i(ψGL + δψ)(∇φ)eiφ −

2e

cA(ψGL + δψ)eiφ

∣∣2

+ 1

8π| curl A|2

]

and

δFGL =∫

d3r

[

2|a|(δψ)2 + h2

4m

{

|∇δψ |2 + |a|b

∣∣∣∣∇φ − 2e

hcA

∣∣∣∣

2}

+ 1

8π| curl A|2

]

.

(3.105)

Owing to the gauge invariance we can add to the vector potential gradient of any func-

tion. We choose the transformation

A→ A′ +∇

(hc

2eφ

)

.

3.10. THE ANDERSON-HIGGS MECHANISM 81

Note that curl A = curl A′. We obtain

δFGL =∫

d3r

[

2|a|(δψ)2 + h2

4m|∇δψ |2 + |a|

b

e2

mc2|A′|2 + 1

8π| curl A′|2

]

.

The phase fluctuations have been absorbed to the vector potential. We drop prime from

A and perform Fourier transform of fluctuations. Since [curl A]k = −i(k×Ak) we get

| curl A|2 → (−ik×Ak)∗(−ik×Ak) = (k×A∗k) ·(k×Ak) = k2|Ak|2−(kA∗k)(kAk) .

Here we used vector identity (a× b) · (c× d) = (ac)(bd)− (bc)(ad). Thus we get

δFGL =∑

k

{(

2|a| + h2k2

4m

)

|δψk|2 +(

|a|b

e2

mc2+ k2

8π

)

|Ak|2 −1

8π(kA∗k)(kAk)

}

.

We split the vector potential in parts parallel to k = k/k and perpendicular to it

Ak = A‖k + A⊥k , A

‖k = k(kAk)

so that |Ak|2 = |A‖k|2 + |A⊥k |2 and (kA∗k)(kAk) = k2|A‖k|2. Finally we obtain

δFGL =∑

k

{(

2|a| + h2k2

4m

)

|δψk|2 +e2ψ2

GL

mc2|A‖k|

2 +(

e2ψ2GL

mc2+ k2

8π

)

|A⊥k |2}

.

(3.106)

We see that above the transition, when ψGL = 0, the electromagnetic field has only

transverse degrees of freedom, which are massless (photons). In the superconducting

state the longitudinal component appears and all components become massive. Com-

pare this to uncharged system where the phase mode is Goldstone, i.e. massless. This

mechanism in which the Goldstone mode acquires a mass is known as the Anderson-

Higgs mechanism. In this context the massive amplitude mode is called the Higgs

boson.

Similar mechanism is thought to be working in the high-energy physics, where the

symmetry breaking at the electroweak transition leads to appearance of the Higgs field

and the gauge W and Z bosons become massive.


Problems

Problem 3.1. Find expression for the coefficient γ in the Ginzburg-Landau free energy

(3.8) from the microscopic theory in the clean limit.

Problem 3.2. Find the Ginzburg-Landau superconducting density in the dirty limit and

compare it with ns in the clean limit.

Problem 3.3. Calculate the critical current for a thin film with thickness d ≪ ξ(T ).

Problem 3.4. A thin superconducting film with a thickness d ≪ λL and d ≪ ξ(T ) is

deposited on a dielectric filament (cylinder) of a radius R ≫ d. The filament is placed

into a longitudinal magnetic field H at a temperature T > Tc and then cooled down

below Tc. Find the dependence of Tc on H . (The Little and Parks effect.)

Problem 3.5. A thin superconducting slab with thickness d ≪ ξ(T ) is placed in the

magnetic field H oriented parallel to the slab. Find the dependence of the magnitude

of the order parameter in the slab on H for cases d ≫ λL and d ≪ λL.

Problem 3.6. A thin superconducting slab with thickness d ≪ ξ(T ) is driven form

the superconducting to normal state by the magnetic field oriented parallel to the slab.

Find the critical thickness dc so that at d > dc the transition is of the first order and at

d < dc it is of the second order.

Problem 3.7. The film with thickness d ≫ ξ has the same upper critical field as a bulk

superconductor. Find the upper critical field for a film with a thickness d ≪ ξ placed

in a magnetic field tilted by an angle 2 from the normal of the film.

Problem 3.8. Calculate the energy of the normal-superconductor boundary in an ex-

treme type I superconductor in the limit λL→ 0.

Problem 3.9. Calculate electric current around a straight Abrikosov vortex.

Problem 3.10. Abrikosov vortex is pinned on a void with diameter d so that d & ξ

and d ≪ λL. Estimate the critical current to release the vortex from the pinning site.

In the calculations ignore the unpinned part of the vortex.

Problem 3.11. Calculate the mean square of phase fluctuations 〈φ2〉 in uncharged

superfluid. Show that the conditions 〈φ2〉 ≪ 1 results in the same Ginzburg criterion

(3.103) as for the amplitude fluctuations.

Problem 3.12. Calculate fluctuation contribution Cfluct to the heat capacity. Show that

the condition Cfluct ≪ 1C, where the 1C is the heat capacity jump at the transition,

results in the Ginzburg criterion (3.103).

Chapter 4

Andreev reflection

The energy spectrum of quasiparticles in a superconductor can depend on coordinates

via spatial dependence of the energy gap 1(r). When a quasiparticle propagates in

such inhomogeneous medium, it can experience a special type of scattering process

called Andreev reflection. It is characterized by conversion from particle-like excita-

tion to hole-like and vice versa. Sequence of Andreev reflections can lead to formation

of quasiparticle standing waves, Andreev bound states, with energies smaller than the

minimum quasiparticle energy |1| in superconductor bulk. Andreev reflection is im-

portant in a number of situations, where gap substantially changes on a length scale

of the coherence length: In particular, at NS interfaces, various superconducting junc-

tions, cores of Abrikosov vortices.

4.1 Normal-superconducting interface

Let us consider propagation of quasiparticles through normal-superconducting inter-

face using Bogolubov – de Gennes equations (2.79) and (2.80):

− h2

2m

(

∇ − ie

hcA

)2

u(r)+ [Uex(r)− EF ] u(r)+1(r)v(r) = ǫu(r), (4.1)

h2

2m

(

∇ + ie

hcA

)2

v(r)− [Uex(r)− EF ] v(r)+1∗(r)u(r) = ǫv(r). (4.2)

For notation simplicity the subscript k is removed (in future we will consider states

which are not necessarily labelled with the wave vector). The external non-magnetic

potential is renamed toUex to avoid confusion with the coherence factor. Previously we

marked quasiparticle energy in normal metal as ǫ and in superconductor as E. In this

chapter we consider motion of a quasiparticle between normal and superconducting

83

84 CHAPTER 4. ANDREEV REFLECTION

x

x

0

0

∆

N S

Uex

Figure 4.1: Model of NS (Uex = 0) or NIS (Uex = Iδ(x)) interface.

sides of the interface with conservation of energy. We will use notation ǫ for this

conserved energy.

We will consider a very simple model, Fig. 4.1: The normal metal occupies half-

space x < 0, and the superconductor is in half-space x > 0. Magnetic field is absent,

A = 0. Material parameters, in particular kF and EF = h2k2F /2m, are the same at

both sides of the interface. As we know, the energy gap increases at the interface on a

length scale of the order of the coherence length ξ . We will ignore this and consider

1(r) = 0 on the normal side and 1(r) = 1 = const on the superconducting side.

Since 1 is uniform on the superconducting side and magnetic field is absent we can

select the phase of 1 to be zero: 1 = |1|eiχ with χ = 0. Finally we allow for some

insulating layer at the interface by introducing delta-like external potential at x = 0 as

Uex(r) = Iδ(x) . (4.3)

With all these simplifying assumptions, we do not really have to solve Eqs. (4.1)

and (4.2): In each of the half spaces the system is uniform and the solution is already

known to us, Eqs. (2.81) and (2.82). We have to join these solutions at x = 0.

Since the whole system is uniform in y and z directions, we can represent the

solution as (

u(r)

v(r)

)

= eikyy+ikzz(

ux(x)

vx(x)

)

. (4.4)

Inserting this into Eqs. (4.1) and (4.2) we obtain[

− h2

2m

d2

dx2− Ex + Iδ(x)

]

ux +1vx = ǫux (4.5)

4.2. TRANSMISSION AND REFLECTION AMPLITUDES 85

−[

− h2

2m

d2

dx2− Ex + Iδ(x)

]

vx +1∗ux = ǫvx (4.6)

where

Ex = EF −h2(k2

y + k2z )

2m.

At x = 0 the ux and vx functions are continuous

(

ux(0)

vx(0)

)

L

=(

ux(0)

vx(0)

)

R

. (4.7)

Here subscript L refers to left half-space with normal metal and R to right half-space

with superconductor. The derivatives, however, may be discontinuous at the interface

owing to the presence of delta function. To see this we integrate Eq. (4.5) over the

interval (−δ, δ):

− h2

2m

dux

dx

∣∣∣

δ

−δ− Ex

δ∫

−δ

ux dx + Iux(0)+1δ∫

0

vx dx =δ∫

−δ

ux dx .

When δ→ 0 all integrals here vanish and we obtain

− h2

2mu′x(0)

∣∣∣

R

L+ Iux(0) = 0 ,

and similarly from Eq. (4.6). Thus we find

(

u′x(0)

v′x(0)

)

R

−(

u′x(0)

v′x(0)

)

L

= 2mI

h2

(

ux(0)

vx(0)

)

. (4.8)

We will discuss the role of an insulating barrier further in this course. In this chapter we

consider the barrier to be absent, I = 0 and thus derivatives u′x and v′x are continuous

at the interface.

4.2 Transmission and reflection amplitudes

Consider a particle excitation with energy ǫ > |1| and wave vector

ki = (k+Nx , ky, kz), ki = k+N

approaching the NS interface from the normal side at angle θ with the normal to the

interface, k+Nx = ki cos θ , Fig. 4.2. The products of interaction of the incident particle

with the interface should have: (a) the same energy ǫ, (b) the same wave vector com-

ponents ky and kz, (c) group velocity directed from the interface. From discussion in


ki

ka

kb kc

kd

x

y

vga vgd

N S

θ

pF

pF

−pF

−pF

N

0

S

0

iab cd

ǫ ǫ

Figure 4.2: Particle (i), incident on the NS/NIS interface, is scattered at the interface

to Andreev reflected hole (a), specularly reflected particle (b), transmitted particle (c)

and transmitted hole (d). (Left) Diagram of wave vectors. Note that the group velocity

of the holes is directed against their momenta. (Middle) and (right): Energy spectra.

Sec. 2.8 we see that there are four possibilities which satisfy these conditions:

(i) Reflected particle with the wave vector

kb = (−k+Nx , ky, kz), kb = k+N

(ii) Transmitted particle with the wave vector

kc = (k+Sx , ky, kz), kc = k+S

(iii) Reflected hole with the wave vector

ka = (k−Nx , ky, kz), ka = k−N

(iv) Transmitted hole with the wave vector

kd = (−k−Sx , ky, kz), kd = k−S

Note that in the last two cases the wave vector is directed towards the interface, but the

group velocity is away from it. The magnitudes of wave vectors corresponding to the

energy ǫ are given by Eq. (2.63)

k±N = kF ±ǫ

hvF, (4.9)

k±S = kF ±√

ǫ2 − |1|2hvF

. (4.10)

Here plus sign refers to particle and minus sign to hole excitations.

Applying uniform solution of BdG equations to the left half-space we have(

ux

vx

)

L

= eik+Nx x

(

1

0

)

+ aeik−Nx x

(

0

1

)

+ be−ik+Nx x

(

1

0

)

. (4.11)


pF

pF

−pF

−pF

ǫ ǫ

N

0

S

0

ia 2 b2c2 d2

Figure 4.3: Left: The state in N region with an incoming hole (i) that is Andrev reflected

as a particle (a2) and normally reflected as a hole (b2). Right: Transmitted hole (c2)

and particle (d2) in S region.

The amplitude of the incident particle is set to 1.

In the right half-space we have(

ux

vx

)

R

= ceik+Sx x

(

U

V

)

+ de−ik−Sx x

(

V

U

)

. (4.12)

The coherence factors in uniform superconductor, Eq. (2.82), are

U = 1√2

(

1+√

ǫ2 − |1|2ǫ

)1/2

, V = 1√2

(

1−√

ǫ2 − |1|2ǫ

)1/2

. (4.13)

Note that in Eq. (2.82) the hole states have negative sign of ξk. In notation of Eq. (4.13)

the sign of√

ǫ2 − |1|2 is always positive and thus for holes we have to switchU ↔ V ,

like we have done in Eq. (4.12).

The boundary conditions (4.7) and (4.8) yield

1+ b = cU + dV (4.14)

a = cV + dU (4.15)

k+Nx − bk+Nx = ck+Sx U − dk−Sx V (4.16)

ak−Nx = ck+Sx V − dk−Sx U (4.17)

This is a system of 4 linear equations with 4 unknowns a, b, c, d and coefficients which

depend on ǫ and θ . In general, it has a unique solution. Expressing a from (4.15) and

inserting it to (4.17) and expressing b from (4.14) and inserting it to (4.16) we arrive to

equations

cV (k+Sx − k−Nx )− dU(k−Sx + k−Nx ) = 0 (4.18)

cU(k+Sx + k+Nx )− dV (k−Sx − k+Nx ) = 2k+Nx (4.19)


pF

pF

−pF

−pF

ǫ ǫ

N

0

S

0

i a bc d3 3 3 3

Figure 4.4: The states of incoming, reflected and transmitted particles and holes at a

NS/NIS interface when a incident particle approaches barrier from the superconducting

side.

In the following we will apply the semiclassical approximation. It is based on the

fact that the Fermi wave length of quasiparticles k−1F ∼ a0 is the shortest length scale

in superconductors. In particular, k−1F ≪ ξ , which is equivalent to 1≪ EF . Indeed,

1

EF∼ hvF /ξ

hkFpF /m= 1

kF ξ∼ a0

ξ.

In semiclassical approximation we omit all terms of the order of 1/EF and of higher

orders. Thus we can write

k±Sx ≈ k±Nx ≈ kF cos θ ≡ kx . (4.20)

Then from Eqs. (4.18) and (4.19) we get

d = 0, c = 1/U . (4.21)

Inserting these values to Eqs. (4.14) and (4.15) we find

a = V/U, b = 0 . (4.22)

Thus in semiclassical approximation the amplitude of specular reflection is zero.

The incident particle is reflected from the NS interface as a hole. This process is called

Andreev reflection. It has a remarkable property: Since k−Nx ≈ k+Nx we have ka ≈ ki .

The group velocity of the hole, however, is directed against its wave vector ka . There-

fore, Andreev reflected hole moves along the same trajectory as the incident particle

but in the opposite direction. In fact, since the wave vectors are not completely identi-

cal, directions of the incident and reflected trajectories are slightly different (see Prob-

lem 4.1).

If the incident excitation is a hole, Fig. 4.3, it is easy to see that the result remains

the same

a2 = a, b2 = 0, c2 = c, d2 = 0, (4.23)


pF

pF

−pF

−pF

ǫ ǫ

N

0

S

0

i abcd 4 4 4 4

Figure 4.5: The states of incoming, reflected and transmitted particles and holes at a

NS/NIS interface when a incident hole approaches barrier from the superconducting

side.

where a2 and b2 are amplitudes of the reflection as particle (Andreev) and hole (spec-

ular), respectively, and c2 and d2 are amplitudes of the transmission as hole or particle,

respectively.

For excitations, incident from the superconducting side, the solution is obtained in

Problem 4.3. The result is

a3 = a4 = −a, b3 = b4 = 0, (4.24)

c3 = c4 =√

ǫ2 − |1|2ǫ

c, d3 = d4 = 0 , (4.25)

where subscript 3 refers to the incident particle, Fig. 4.4, and subscript 4 to the incident

hole, Fig. 4.5.

Now we consider case ǫ < |1|. This is only possible for the excitations coming

from the normal side. In fact, in mathematical transformations so far we did not use

the fact the energy is above the gap, so all results should be formally valid also for the

case ǫ < |1|. In this case, however, U and V in Eq. (4.13) become complex. We can

explicitly write that as

U = 1√2

(

1+ i√

|1|2 − ǫ2

ǫ

)1/2

, V = 1√2

(

1− i√

|1|2 − ǫ2

ǫ

)1/2

. (4.26)

Andreev reflection amplitude becomes

a = V /U , (4.27)

and the probability of reflection

|a|2 = 1 , (4.28)

since |U | = |V |. The Andreev reflection is complete. Illustration of this process is

shown in Fig. 4.6.


❛��✁✂✁✄❛☎✆✝ ✂✞✄✆

♣✆�✆☎✟❛☎✁�✠ ♣❛✟☎✁✡✄✆✁�✡✁✝✆�☎ ♣❛✟☎✁✡✄✆

✟✆r✄✆✡☎✆✝ ✂✞✄✆

☛☛☛

☛

✈✠

❈✞✞♣✆✟

♣❛✁✟

◆ ❙

✲☞✌☞

✰☞✌☞

✲✍☞✌☞

✈✠

✲✍☞✌☞

Figure 4.6: Illustration of the nature of Andreev reflection for ǫ < |1| at an ideal NS

interface: An incident particle forms a Cooper pair in the superconductor together with

an annihilated hole. This hole is expelled into the normal metal and moves back as a

reflected object.

4.3 Andreev equations

In semiclassical approximation we can simplify BdG equations (4.1) and (4.2). We

consider external potential to be absent Uex(r) = 0. Let us represent functions u(r)

and v(r) as(

u(r)

v(r)

)

= eikF kr

(

u(r)

v(r)

)

. (4.29)

Here k = k/k is a unit vector in the direction of k. The idea of this transformation is

that we separate part of u(r) and v(r) which varies with the coordinates rapidly, on the

scale of Fermi wave length. The rest

(

u(r)

v(r)

)

= ei(k−kF )kr

(

U(r)

V (r)

)

(4.30)

is a slow function of coordinates, which changes on the scale of coherence length ξ .

Inserting definition (4.29) to equation (4.1) we obtain

− h2kF

2m

[

2ik∇u+ 2e

hckAu− 1

kF

2ie

hcA∇u− 1

kF

e2

h2c2A2u+ 1

kF∇

2u

]

+1v = ǫu .

(4.31)

We have

k−1F ∼ a0, ∇u ∼ u/ξ, ∇2u ∼ u/ξ2, eA/(hc) ∼ 1/ξ

4.4. ANDREEV BOUND STATES IN SNS STRUCTURES 91

since [using Eq. (1.21)]

h = curl A ⇒ A ∼ HcλL ∼80

ξ∼ hc

eξ.

Thus two first terms in square brackets in Eq. (4.31) are of the order of u/ξ . The three

last ones are of the order of (u/ξ)(a0/ξ) ∼ (u/ξ)(1/EF ) and can be neglected in

semiclassical (or WKB) approximation. Defining

vF =hkF k

m= vF k (4.32)

we obtain Andreev equations

−ihvF ·(

∇ − ie

hcA

)

u+1v = ǫu (4.33)

ihvF ·(

∇ + ie

hcA

)

v +1∗u = ǫv (4.34)

The second equation here is obtained from (4.2) in a similar way as the first one. Com-

pared to BdG equations, Andreev equations are of the first order, which may consider-

ably reduce the complexity of the problem.

4.4 Andreev bound states in SNS structures

Consider an SNS structure consisting of a normal slab in the plane (y, z) having a

thickness d in between two superconducting half-spaces x < −d/2 and x > d/2 (see

Fig. 4.7). The quasiparticles in both superconductors and in the normal metal have the

same Fermi velocity, and there are no insulating barriers between them and magnetic

field is absent. We allow for the arbitrary phase difference φ between superconductors

and write the gap in the right superconductor is 1R = |1|eiφ/2, while in the left

superconductor1L = |1|e−iφ/2. As before, we assume that electron quasiparticles do

not scatter on impurities or on other objects like phonons. We also assume that there is

no electron–electron scattering. This means that the size d is shorter than the electron

mean free path. The mean free path should also be larger than the superconducting

coherence length ξ . The latter limitation is important if d < ξ .

In the considered SNS structure Andreev equations take simple form with u(x) and

v(x) depending only on the x coordinate

−ihvFx u′(x)+1v(x) = ǫu(x) (4.35)

ihvFx v′(x)+1∗u(x) = ǫv(x) (4.36)

Here vFx = vF cos θ , where θ is the angle between k and normal to the interface.


S S

Np

h

1 = |1|eiφ/21 = |1|e−iφ/2

1 = 0

0−d/2 d/2 x

Figure 4.7: The Andreev bound states in the SNS structure.

Here we discuss the case of sub-gap energy ǫ < |1|. The energies ǫ > |1| are

considered in Problem 4.7. A particle that moves in the normal region to the right (i.e.

vFx > 0) will be Andreev reflected from the NS interface into a hole. The hole will

then move to the left and is Andreev reflected into the particle, and so on. As a result,

it is localized in the N region. It is natural to expect that such localized states have a

discrete energy spectrum, which we derive below. Similar localization happens to the

particle moving to the left.

Solving equations (4.35) and (4.36) in the normal region with 1 = 0 we get

u(x) = AeiλNx, v(x) = A2e−iλNx , (4.37)

where A and A2 are arbitrary constants and

λN =ǫ

hvFx. (4.38)

ReplacingA2 with another arbitrary constant a asA2 = Aa we write the wave function

in the N region as(

u(x)

v(x)

)

N

= A[

eiλNx

(

1

0

)

+ ae−iλNx(

0

1

)]

. (4.39)

If the momentum projection on the x axis is positive, kx > 0, the evanescent exci-

tation on the right is a particle. Solving equations (4.35) and (4.36) with 1 = 1R we

get

u(x) = A3e−λSx, v(x) = A4e

−λSx, A4 = e−iφ/2A3ǫ − i

√

|1|2 − ǫ2

|1| , (4.40)

where A3 is an arbitrary constant and

λS =√

|1|2 − ǫ2

h|vFx |. (4.41)

The sign in the exponent in solution (4.40) is chosen so that the wave function de-

cays away from the interface. Introducing another constant d1 as A3 = d1Ueiφ/4 [see


Eq. (4.26)] we find that A4 = d1V e−iφ/4. Thus the wave function in the right super-

conductor x > d/2 is

(

u(x)

v(x)

)

R

= d1e−λSx

(

Ueiφ/4

V e−iφ/4

)

. (4.42)

In the left superconductor x < −d/2 the evanescent excitation is a hole and 1 =1L. Similarly to Eq. (4.42) we obtain

(

u(x)

v(x)

)

L

= d ′1eλSx(

V e−iφ/4

Ueiφ/4

)

(4.43)

Constants A, a, d1 and d ′1 are determined from continuity of the wave function at

the interfaces and from overall normalization. Continuity at the right interface x = d/2gives

AeiλNd/2 = d1e−λSd/2Ueiφ/4 (4.44)

aAe−iλNd/2 = d1e−λSd/2V e−iφ/4 (4.45)

Dividing here the second equation by the first we find

ae−iλNd = V

Ue−iφ/2

From this we find that probability of Andreev reflection |a|2 = |V |2/|U |2 = 1 [cf.

Eq. (4.28)], as expected for sub-gap states. Continuity at the left interface gives

aeiλNd = U

Veiφ/2

Excluding a we find

e2i(λNd−φ/2) = U2

V 2= ǫ + i

√

|1|2 − ǫ2

ǫ − i√

|1|2 − ǫ2

We denote

sinα = ǫ

|1|The range of α is determined in such a way that

cosα = |1|−1√

|1|2 − ǫ2

is positive to ensure the decay of wave functions in the S regions, i.e.,

−π/2 < α < π/2


We obtain

e2i(λNd−φ/2) = sinα + i cosα

sinα − i cosα= ei(π/2−α)

e−i(π/2−α)= e−2iα+iπ .

The exponents on the left and right sides can differ by factor e2πil , where l is an arbi-

trary integer. Thus

2

(ǫ

hvFxd − φ

2

)

= −2α + π + 2πl

or

ǫ = hωx[φ

2− arcsin

ǫ

|1| + π(

l + 1

2

)]

, (4.46)

where

ωx =vFx

d= t−1

x

is the inverse time needed for a particle to fly from one end of the normal region to the

other. Integer l enumerates thus different bound states. Its possible values are limited

by our original assumption that ǫ < |1|.Consider now particles that have a negative projection kx < 0 on the x axis. In the

right superconductor x > d/2 we have evanescent hole and the wave function becomes

(

u(x)

v(x)

)

R

= d2e−λSx

(

V eiφ/4

Ue−iφ/4

)

(4.47)

The wave function in the left superconductor x < −d/2 with evanescent particle is

(

u(x)

v(x)

)

L

= d ′2eλSx(

Ue−iφ/4

V eiφ/4

)

(4.48)

Applying continuity at the right interface x = d/2 we find

ae−iλNd = U

Ve−iφ/2

Continuity at the left interface gives

aeiλNd = V

Ueiφ/2

As a result,

e2i(λNd−φ/2) = ǫ − i√

|1|2 − ǫ2

ǫ + i√

|1|2 − ǫ2

or

e2i(λNd−φ/2) = e2iα−iπ


Since λN = ǫ/hvFx = −ǫ/h|vFx | we find

ǫ = −h|ωx |[φ

2+ arcsin

ǫ

|1| + π(

l − 1

2

)]

(4.49)

Combining this with Eq. (4.46) we finally obtain

ǫ = ±h|ωx |[φ

2∓ arcsin

ǫ

|1| + π(

l ± 1

2

)]

(4.50)

The upper sign refers to kx > 0, the lower sign refers to kx < 0.

In semiclassical approximation normalization of the wave function is calculated

along the particle trajectory and not as an integral over the space, because u and v

determined from semiclassical equations are valid only in the vicinity of the trajectory

(and true wave functions decay quickly away from the trajectory). In the case of the

planar SNS junction it is more convenient to use projection of the trajectory on the

x axis, since it is this integral which we will use in calculating the electrical current

through the junction in the next chapter, where normalization is important:

∞∫

−∞

(

|u(x)|2 + |v(x)|2)

dx = 1 . (4.51)

In the normal region |u|2 + |v|2 = 2|A|2. In the right region

|u|2 + |v|2 = |d1|2e−2λSx(

|U |2 + |V |2)

= 2|d1|2e−2λSx |U |2

since |U |2 = |V |2. From Eq. (4.44) we find |A|2 = |d1|2e−λSd |U |2 and in the right

region

|u|2 + |v|2 = 2|A|2eλS (d−2x)

In the left region we similarly obtain

|u|2 + |v|2 = 2|A|2eλS (d+2x)

Calculating the integrals from −∞ to −d/2 then from −d/2 to d/2 and from d/2 to

∞ we find

|A|2 = 1

2(d + λ−1S )= 1

2

√

|1|2 − ǫ2

h|vFx | + d√

|1|2 − ǫ2(4.52)

4.4.1 Short SNS junctions

Consider the limit of a small width of normal region

d ≪ h|vFx ||1| =

hvF cos θ

|1| = ξ cos θ .


π0 2π

ε

φ

π❅ω /2

3π❅ω /2x

x

π0 2π

ε∆

φ

(a) (b)

Figure 4.8: The Andreev bound states in (a) short, hωx ≫ |1|, and (b) long, hωx ≪|1|, SNS structures. The solid lines refer to vFx > 0, while the dotted lines are for

vFx < 0.

We have h|ωx | ≫ |1|. Since ǫ < |1|, ǫ/h|ωx | is small and Eq. (4.50) gives

φ

2∓ arcsin

ǫ

|1| + π(

l ± 1

2

)

= 0

Recall that α = arcsin[ǫ/|1|] should be between −π/2 and π/2. For vFx > 0 (upper

signs) we have to choose l = −1 to have −π/2 < α < π/2 (assuming φ is in the

range from 0 to 2π ). For vFx < 0 (lower signs), the choice is l = 0. Thus we obtain

only one bound state with particle moving right (and hole left) and another state with

particle moving left (and hole right). The energy spectrum is [see Fig. 4.8 (a)]

ǫ = ∓|1| cosφ

2(4.53)

4.4.2 Long SNS junctions

For a long N region, when

d ≫ h|vFx |/|1| , hωx ≪ |1|

the bound states with small l will have energy ǫ ≪ |1|. Thus in Eq. (4.50) we can

neglect arcsin(ǫ/|1|) ≈ ǫ/|1| ≪ 1. The spectrum becomes [see Fig. 4.8 (b)]

ǫ = ±h|ωx |(φ

2− π

2

)

+ πh|ωx |l (4.54)

where l = 0, 1, . . .. (We flipped the sign of l for kx < 0.) This approximation is valid

only for small energies, i.e. when l ≪ |1|/hωx .


π0 2π

ε

φ

π❅ω /2

3π❅ω /2x

x

π0 2π

ε∆

φ

(a) (b)

−∆

−π❅ω /2

−3π❅ω /2

x

x

Figure 4.9: The Andreev bound states in (a) short, hωx ≫ |1|, and (b) long, hωx ≪|1|, SNS structures for extended energy range −|1| < ǫ < |1|. The solid lines refer

to vFx > 0, while the dotted lines are for vFx < 0.

4.4.3 Negative energies

Bogolubov – de Gennes equations (4.1) and (4.5) possess an important property, known

as particle-hole symmetry: If

(

u(r)

v(r)

)

is a solution for the energy ǫ (4.55)

then(

v(r)∗

−u(r)∗

)

is a solution for the energy − ǫ (4.56)

Thus formally we can introduce negative energies of excitations. This is similar to neg-

ative energies found in the Dirac equation, which led to the concept of Dirac sea: Un-

derstanding of quantum vacuum as including filled fermionic states with negative en-

ergy. Similar interpretation can be used in the case of superconductor: If the negative-

energy states are introduced, one have to conclude that they are filled and belong to

“vacuum” degrees of freedom.

Since complex conjugation in (4.56) changes the sign of the k vector, in our SNS

structure we notice that if a state with ǫ > 0 belongs to kx > 0 then the state with

−ǫ < 0 belongs to −kx < 0. Informally speaking, when we flip spectrum in Fig. 4.8

to negative energies we simultaneously have to flip solid and dashed lines, so that we

arrive to Fig. 4.9.


x

S S

Figure 4.10: The point contact SNS structure.

Formally, for short junctions, the spectrum becomes

ǫ = ∓|1| cos(φ/2) (4.57)

for 0 < φ < 2π with the upper sign for vx > 0 and lower sign for vx < 0. Similarly,

for long junctions,

ǫ = ±h|ωx |(φ

2− π

2

)

+ πh|ωx |l (4.58)

where l = 0,±1,±2, . . . for the entire region 0 < φ < 2π .

4.4.4 Point contact

One of the examples of SNS structures is the so-called point contact. This is a structure

where two superconductors are connected through a narrow (with a cross section of an

area S ∼ a2 where the transverse dimension is a ≪ ξ ) and short d ≪ ξ normal piece

called constriction (see Fig. 4.10). In fact, since the wave function has no possibility

to vary within the constriction, the results do not change if the constriction is also

superconducting.

The energy spectrum of sub-gap states and the wave functions are found in Prob-

lem 4.8. The spectrum is given by Eq. (4.53). The normalization of the wave function

is

|A|2 =√

|1|2 − ǫ2

2hvF(4.59)

4.5 Vortex core states

Consider the low energy Andreev states in the vortex core, Fig. 4.11. The gap function

has the form 1 = |1(r)|eiφ where r and φ are the radius and the azimuthal angle in

the cylindrical coordinate frame (r, φ, z) with the z axis along the vortex axis. The

4.5. VORTEX CORE STATES 99

b x

y

y

x

particle

hole

ac

Figure 4.11: Spatial dependence of the gap in the core of the quantized vortex (left)

leads to Andreev reflection (right) of quasiparticles and formation of bound states. In

conventional superconductors vortex core radius ac ∼ ξ . Angular momentum of a

quasiparticle hµ = pF b. Note that for the trajectories passing close to the vortex

center (b ≪ ac) the phase difference of 1 between reflection points is close to π and

one may expect almost zero-energy states, Fig. 4.8.

modulus |1(r)| goes to zero for r = 0 and saturates at |1| = 10 at distances r ≫ ξ

(see Fig. 3.7). The vector potential has only an azimuthal component, Eq. (3.54).

We put(

u

v

)

= eikzzeiµφ(

f+(r)eiφ/2

f−(r)e−iφ/2

)

where µ is the azimuthal quantum number. It should be the half-integer µ = n + 1/2

since the wave function has to be single valued. The BdG equations (4.1), (4.2) take

the form

− h2

2m

[

d2f+dr2+ 1

r

df+dr−(µ+ 1/2

r− eAφ

hc

)2

f+ + k2⊥f+

]

+ |1|f− = ǫf+

h2

2m

[

d2f−dr2+ 1

r

df−dr−(µ− 1/2

r+ eAφ

hc

)2

f− + k2⊥f−

]

+ |1|f+ = ǫf−

where k2⊥ = k2

F − k2z . We further assume that the London penetration length λL ≫ ξ .

In this limit,eAφ

hc∼ eHr

hc∼ 1

r

H

Hc2

r2

ξ2≪ 1

ξ

for r ∼ ξ . This we found also from the GL theory, Eqs. (3.58) and (3.63). Therefore,

− h2

2m

[

d2f+dr2+ 1

r

df+dr− µ

2 + 1/4+ µr2

f+ + k2⊥f+

]

+ |1|f− = ǫf+ (4.60)

h2

2m

[

d2f−dr2+ 1

r

df−dr− µ

2 + 1/4− µr2

f− + k2⊥f−

]

+ |1|f+ = ǫf− (4.61)


Consider |µ| ≪ kF ξ . Introduce rc such that µk−1F ≪ rc ≪ ξ . For r < rc we

neglect |1(r)| ≪ 10. The solutions of Eqs. (4.60), (4.61) are the Bessel functions

f± = A±Jµ±1/2[(k⊥ ± λN )r] (4.62)

where v⊥ = hk⊥/m and

λN =ǫ

hv⊥The functions f± do not have singularities at r = 0.

For r > rc we look for solution in the form

(

f+f−

)

= H (1)l (k⊥r)

(

g+g−

)

+H (2)l (k⊥r)

(

g∗+g∗−

)

(4.63)

where l =√

µ2 + 1/4 andH(1)l is the Hankel function of the first kind. The amplitudes

g± are slow function: they vary at distances of the order of ξ . For r > rc we have

dH(1)l /dr = ik⊥H (1)

l . Neglecting the second derivatives of g± we obtain

− ih2k⊥m

dg+dr+ |1|g− =

(

ǫ − µh2

2mr2

)

g+ (4.64)

ih2k⊥m

dg−dr+ |1|g+ =

(

ǫ − µh2

2mr2

)

g− (4.65)

Equations (4.64) and (4.65) are obtained in the semiclassical approximation. They are

similar to the Andreev equations in cylindrical coordinates.

Let us now put

(

g+g−

)

= C(

eiψ(r)/2−iπ/4

−ie−iψ(r)/2+iπ/4

)

e−K(r)

We obtain

hv⊥dψ

dr= 2|1| sinψ + 2

(

ǫ − µh2

2mr2

)

(4.66)

hv⊥dK

dr= |1| cosψ (4.67)

We shall see that for µk−1F ≪ ξ , the function ψ is small. Therefore,

K(r) = (hv⊥)−1

∫ r

0

|1(r ′)| dr ′ (4.68)

and

ψ(r) = −e2K(r)

∫ ∞

r

(

2λN −µ

k⊥r ′ 2

)

e−2K(r ′) dr ′

4.5. VORTEX CORE STATES 101

1

−1

1

ǫµ(kz = 0)

n = 0

nmax

µ/kF

hω0

ǫµ(kz)

kzac

Figure 4.12: Caroli – de Gennes – Matricon spectrum of the vortex-core bound states

in the extended energy range including negative energies. (Left) Spectrum at kz = 0 as

a function of discrete angular momentum µ. The main branch of the spectrum crosses

zero energy and for ǫ ≪ |1| possesses equidistant energy levels with the spacing of

minigap hω0, Eq. (4.71). In systems where the core size ac > ξ also higher-energy

branches exist with number nmax ∼ ac/ξ . (Right) Set of spectra as a function of kz at

different values of µ. These spectra are continuous.

The constant of integration here is taken to make ψ a bounded function for r → ∞.

The second term under the integral diverges for r → 0. Integrating by parts, we obtain

ψ(r) = µ

k⊥r+ 2λN r − 2e2K(r)

∫ ∞

0

(

λN +µ|1(r)|hk⊥v⊥r

)

e−2K(r) dr (4.69)

The term under the integral has now no singularities, since |1(r)| ∝ r at r ≪ ξ ,

Eq. (3.66).

Let us now match this solution Eqs. (4.63), (4.68), and (4.69) with Eq. (4.62) at

r = rc. In Eq. (4.62) we have

Jµ±1/2[(k⊥±λN )rc] = [2/πk⊥rc]1/2 cos

[

(k⊥ ± λN )rc +(µ± 1/2)2

2k⊥rc− π

2

(

µ± 1

2

)

− π4

]

We neglected λN as compared to k⊥ in the denominator. On the other hand,

H(1)l (k⊥rc) = [2/πk⊥rc]1/2 exp

[

i

(

k⊥rc +l2

2k⊥rc− πl

2− π

4

)]

Since 2J (x) = H (1)(x)+H (2)(x) the matching requires

k⊥rc+l2

2k⊥rc− πl

2− π

4+ψ(rc)

2− π

4= (k⊥+λN )rc+

(µ+ 1/2)2

2k⊥rc− π

2

(

µ+ 1

2

)

− π4


For µ≫ 1 when l = µ this gives

∫ ∞

0

(

λN +µ|1(r)|hk⊥v⊥r

)

e−2K(r) dr = 0 (4.70)

This equation determines the discrete energies

ǫµ(kz) = −µk−1⊥

∫∞0 (|1(r)|/r) e−2K(r) dr

∫∞0 e−2K(r) dr

(4.71)

of the localized states, Fig. 4.12. They form an equidistant spectrum

ǫµ = −µω0(kz) (4.72)

where the interlevel spacing is called the minigap

ω0 ≡ ω0(kz = 0) ∼ 10

pF ξ∼12

0

EF≪ 10 (4.73)

These energy states in the vortex core were first obtained by Caroli, de Gennes

and Matricon in 1964. The energy spectrum Eq. (4.71) holds for µ ≪ kF ξ . Note

that kF ξ ∼ EF /10 ≫ 1. For these µ the energy ǫ ≪ 10. For larger µ the energy

approaches ∓10. Since µ = n + 1/2 the lowest energy is nonzero: ǫ1/2(kz = 0) =ω0/2.

4.6 Transmission and reflection at the NIS interface

We extend calculations of Sec. 4.2 to the case when there is an insulating barrier at the

interface. The boundary condition (4.8) with I 6= 0 should be used.

(1) Particle with ǫ > |1| incident from the normal region, Fig. 4.2. Equations

(4.14) and (4.15) remain the same

1+ b = cU + dVa = cV + dU

while instead of Eqs. (4.16) and (4.17) we obtain

i(ck+Sx U − dk−Sx V )− i(k+Nx − bk+Nx ) = 2|kx |Z(1+ b) (4.74)

i(ck+Sx V − dk−Sx U)− aik−Nx = 2|kx |Za (4.75)

We introduce here the dimensionless barrier strength

Z = mI

h2|kx |. (4.76)

4.6. TRANSMISSION AND REFLECTION AT THE NIS INTERFACE 103

The barrier strength Z is generally a function of the incident angle θ , where kx =kF cos θ . Using semiclassical approximation (4.20) we find the solution

a = UV

U2 + (U2 − V 2)Z2(4.77)

b = − (U2 − V 2)(Z2 + iZ)

U2 + (U2 − V 2)Z2= − (U

2 − V 2)|Z|√Z2 + 1eiδ

U2 + (U2 − V 2)Z2(4.78)

c = (1− iZ)UU2 + (U2 − V 2)Z2

= − i√Z2 + 1Ueiδ

U2 + (U2 − V 2)Z2(4.79)

d = iZV

U2 + (U2 − V 2)Z2(4.80)

where the scattering phase δ is defined as

tan δ = 1/Z

In the limit of the normal state on the right, V = 0, U = 1, and a = d = 0 (that is,

there is no Andreev reflection) while

b = − iZ

1+ iZ , c =1

1+ iZ

so that

|b|2 = Z2

1+ Z2, |c|2 = 1

1+ Z2= T . (4.81)

We will use notation T for the transmission coefficient in the normal state.

Without the barrier Z = 0 and b = d = 0 while

a = V/U , c = 1/U

as we found in Sec. 4.2, Eqs. (4.21) and (4.22).

(2) The state with an incident hole in the normal region, Fig. 4.3, results in

a2 = a(−Z) , b2 = b(−Z) , c2 = c(−Z) , d2 = d(−Z) (4.82)

(3) The state with a particle incident on the barrier from the superconducting side,

Fig. 4.4, leads to

a3 = −a(−Z), b3 = b(−Z), c3 =vgS

vgNc(−Z), d3 = −

vgS

vgNd(−Z) (4.83)

Here

vgS =hkx

m

√

ǫ2 − |1|2ǫ


x

y

SS

{

{

{

{

p

h h

hh

p p

p

kx −kx

−kx kx

Figure 4.13: Structure of the bound state in the SIS contact with a particle moving

to the right (solid arrows) and to the left (dashed arrows). For the hole quasipaticles

direction of the group velocity (opposite to momentum) is shown by arrows.

is the component of the group velocity of quasiparticles in the superconductor perpen-

dicular to the interface. Similarly,

vgN =hkx

m

is the normal component of the group velocity of quasiparticles in the normal metal.

(4) The state with a hole incident on the barrier from the superconducting side,

Fig. 4.5, results in

a4 = −a(Z), b4 = b(Z), c4 =vgS

vgNc(Z), d4 = −

vgS

vgNd(Z) (4.84)

For subgap states equations (4.77)–(4.80) and (4.82) remain valid, but U and V

now becomes complex, U → U and V → V , Eq. (4.26). In particular, we find for the

probability of Andreev reflection

|a|2 = |1|2ǫ2 + (|1|2 − ǫ2)(1+ 2Z2)2

(4.85)

while the total probability of reflection as either a particle or a hole is unity

|a|2 + |b|2 = 1 . (4.86)

4.7 Bound states in the SIS contact

Now we look at one more example of a barrier structure which consists of two super-

conducting half-spaces separated by a barrier of strength Z in the plane (y, z). The gap

4.7. BOUND STATES IN THE SIS CONTACT 105

π0 2π

∆

−∆

ǫ

φ21√

1− T

Figure 4.14: The energy spectrum of the bound states in the SIS contact in the extended

energy range.

has the form 1 = |1|e±φ/2 in the right (left) superconductor, respectively. We con-

sider the states with energies ǫ < |1|. The bound state appears due to proper matching

of Andreev and specular reflection and transmission amplitudes, Fig. 4.13.

We consider case with the particle moving to the right. Similar to Eq. (4.42) we

write for the right superconductor

(

u(x)

v(x)

)

Rp

= cRe−λSx(

Ueiφ/4

V e−iφ/4

)

,

(

u(x)

v(x)

)

Rh

= dRe−λSx(

V eiφ/4

Ue−iφ/4

)

for the particle and hole states, respectively. In the left superconductor we have

(

u(x)

v(x)

)

Lp

= cLeλSx(

Ue−iφ/4

V eiφ/4

)

,

(

u(x)

v(x)

)

Lh

= dLeλSx(

V e−iφ/4

Ueiφ/4

)

.

In order to apply boundary conditions (4.7) and (4.8) we have to switch from (u, v)

to (ux, vx) representation of wave functions. Comparing Eqs. (4.4) and (4.29) we see

that this is done by multiplying by eikF kxx = eikxx or by e−ikxx , as appropriate, see

Fig. 4.13. We obtain

(

ux

vx

)

R

= cReikxx−λSx(

Ueiφ/4

V e−iφ/4

)

+ dRe−ikxx−λSx(

V eiφ/4

Ue−iφ/4

)

, (4.87)

(

ux

vx

)

L

= cLeikxx+λSx(

Ue−iφ/4

V eiφ/4

)

+ dLe−ikxx+λSx(

V e−iφ/4

Ueiφ/4

)

., (4.88)

Applying boundary conditions (4.7) and (4.8), see Problem 4.9, we find that the solu-


tion (that is, the bound state) exists only for the energy

ǫ = |1|√

1− T sin2(φ/2) (4.89)

where T = 1/(1 + Z2) is the transmission coefficient in the normal state, Eq. (4.81).

The spectrum of Eq. (4.89) is shown in Fig. 4.14.

Without a barrier Z = 0 when T = 1 we recover the spectrum of a ballistic point

contact Eq. (4.53) and Fig. 4.9a. For a final T the gap |1|√

1− T appears for φ = π .

The bound state energy is shifted towards the bulk gap |1| and merges with |1| for a

tunnel junction with very low transmission T → 0.

Problems

Problem 4.1. Find the deflection angle of the trajectory during the Andreev reflection.

Problem 4.2. Find the amplitude of the specular scattering for a particle excitation

with ǫ > |1| approaching NS interface from the normal side in the first non-vanishing

order on 1/EF .

Problem 4.3. Consider an excitation approaching the NS interface from the supercon-

ducting side. In semiclassical approximation find reflection and transmission ampli-

tudes, Eqs. (4.24) and (4.25).

Problem 4.4. Find the average probability for a quasiparticle in superconductor to

experience Andreev reflection from NS interface as a function of temperature. Consider

cases T ≪ Tc and T → Tc.

Problem 4.5. Calculate the velocity of a slow drift of an Andreev state in a SNS

structure with d ≫ ξ0 along the SN plane. Explain the origin of the drift.

Problem 4.6. Complete derivation of wave functions for an SNS structure for ǫ < |1|in Sec. 4.4. That is, starting from Eqs. (4.35), (4.36) and (4.51) derive equations (4.39),

(4.42), (4.43) and (4.52).

Problem 4.7. Find the wave functions for an SNS structure for ǫ > |1|.

Problem 4.8. Find the energy spectrum and wave functions of the superconducting

point contact for ǫ < |1|.

Problem 4.9. Find the energy spectrum in the SIS contact, Eq. (4.89).

Chapter 5

Current in superconducting

junctions

In this chapter we consider electric current in inhomogeneous structures which include

superconductors: Superconductor – normal metal – superconductor (SNS) junction and

normal metal – insulator – superconductor (NIS) interface. We will also briefly dis-

cuss superconductor – insulator – superconductor (SIS) junctions. Andreev reflection

and Andreev bound states provide microscopic explanation for phenomena observed in

such structures.

5.1 Supercurrent through an SNS structure. Proximity

effect

The current is given by equation following Eq. (2.88)

j = e

m

∑

n

[

f (ǫn)(

u∗n pun + un p†u∗n)

+ (1− f (ǫn))(

vn pv∗n + v∗n p†vn

) ]

. (5.1)

where n labels various quantum states. It is more convenient to calculate it in the N

region, where from Eqs. (4.29) and (4.39)(

u(r)

v(r)

)

= eikF krA

[

eiλNx

(

1

0

)

+ ae−iλNx(

0

1

)]

(5.2)

We insert (5.2) into (5.1) and, applying the semi-classical approximation, calculate

derivatives only of the rapidly varying functions eikF kr. As before, we consider mag-

netic field to be absent, thus

ψ∗pψ + ψ p†ψ∗ = 2h|ψ |2∇χ, ψ = |ψ |eiχ .

107

108 CHAPTER 5. CURRENT IN SUPERCONDUCTING JUNCTIONS

As a result we find for the current in the normal region

j = 2he

m

∑

n

A2[

f (ǫn)∇(kF kr)+ (1− f (ǫn))|a|2∇(−kF kr)]

.

In the following we limit consideration only to the current carried by subgap states. In

this case |a|2 = 1. The total current through the junction is

I = S jx = S2he

m

∑

n

A2 [f (ǫn)kF cos θ − (1− f (ǫn))kF cos θ]

= −2evFS∑

n

A2(1− 2f (ǫn)) cos θ (5.3)

= − ehS∑

n

(1− 2f (ǫn))hvFx

√

|1|2 − ǫ2n

h|vFx | + d√

|1|2 − ǫ2n

, (5.4)

where S is the area of the junction and we inserted A from (4.52) and used definition

(4.32). Here ǫn is the bound state energy determined by Eq. (4.50). The quantum

number n describes various states, i.e., the states that belong to various ky , kz and

kx(ǫ).

To calculate the supercurrent we assume that there is no voltage across the junction

and the distribution functions correspond to equilibrium

1− 2f (ǫ) = tanhǫ

2kBT.

For the bound states, at a given phase difference φ there is only a finite number of

states satisfying the conditions either ǫ = ǫ>(kx) or ǫ = ǫ<(kx) for different signs

of kx . We then split the sum in Eq. (5.4) in two, with vFx = |vFx | for kx > 0 and

vFx = −|vFx | for kx < 0:

I = − ehS

∑

n,kx>0

(1− 2f (ǫ>))hvFx

√

|1|2 − ǫ2>

hvFx + d√

|1|2 − ǫ2>

−∑

n,kx<0

(1− 2f (ǫ<))h|vFx |

√

|1|2 − ǫ2<

h|vFx | + d√

|1|2 − ǫ2<

(5.5)

5.1.1 Short junctions. Point contacts

One can check that for short contacts d ≪ ξ , the states in the continuum ǫ > |1| (see

Problem 4.7) do not contribute to the current: the contributions of particles flying from

the left and of those flying from the right cancel in equilibrium. Therefore, Eq. (5.5)

gives the full expression for the current.

5.1. SUPERCURRENT THROUGH AN SNS STRUCTURE. PROXIMITY EFFECT109

For the short junctions with d ≪ ξ0 one can neglect the term with d in denomina-

tors in Eq. (5.5) and in the energy spectrum. The latter is given by Eq. (4.53). Since

for a given difference φ of the superconducting phase across the junctions there exists

only one state for either vFx > 0 (for π < φ < 2π ) or vFx < 0 (for 0 < φ < π )

according to ǫ>,< = ∓ǫφ where

ǫφ = |1| cosφ

2

we find

I =

− ehN>|1| sin(φ/2) tanh

−ǫφ2kBT

, φ > π

e

hN>|1| sin(φ/2) tanh

ǫφ

2kBT, φ < π

= e

hN>|1| sin(φ/2) tanh

|1| cos(φ/2)

2kBT. (5.6)

Here N> ∝ S is the total number of states with all possible ky and kz flying through

the contact of an area S. Eq. (5.6) can be written as

I = −2eN>

h

∂ǫφ

∂φtanh

ǫφ

2kBT(5.7)

We will see that this is a very general form for a current through superconducting

junctions.

Eq. (5.6) can be applied only when the phase in the superconducting regions is

constant in space, in a sense that it does not vary at distances of the order of λ−1S ∼ ξ ,

that is, when the current is significantly smaller than the critical current in the super-

conductor (see Sec. 1.8). For a wide contact whereN> in the normal part is of the same

order as in superconductors, it is only true when the phase difference is small, φ ≪ 1.

In a general case Eq. (5.6) holds if the number of transverse modes in the normal

part N> is much smaller than that in the superconducting regions, which ensures a

small value of the current. One of the examples is the point contact, Sec. 4.4.4. The

current through the point contact is found from Eq. (5.3). The spectrum is the same

as in the SNS contact with exactly one state for either kx > 0 or kx < 0, depending

on φ and both states give the same contribution to the current. The difference is in the

normalization of the wave functions which was found in Problem 4.8. We have

I = 2evFS

√

|1|2 − ǫ2φ

2hvFtanh

ǫφ

2kBT

∑

n,kx>0

cos θ

= e

hS|1| sin(φ/2) tanh

|1| cos(φ/2)

2kBT

∑

n,kx>0

kx

kF. (5.8)


The sum here is calculated by conversion to the integral

∑

n,kx>0

c(kx) =∞∫

0

c(kx)dkx

2π

∞∫

−∞

dky

2π

∞∫

−∞

dkz

2πδ(ǫ − ǫ>)a

= a

(2π)3

∞∫

0

c(kx) dkx

∞∫

0

2πk‖ dk‖ δ

(

k2x + k2

‖2m

−k2F

2m− ǫ>

)

Here k‖ is the projection of the wave vector to (y, z) plane and a is the normalization

constant to be found. Since ǫ> ≪ EF we can calculate the integral for ǫ> = 0. The

argument of δ-function can be zero only when kx 6 kF . Additionally for δ-function

we have

δ(g(k‖)) =δ(k‖ − k0)

|g′(k0)|, g(k0) = 0 .

We apply this transformation with k0 =√

k2F − k2

x and g′ = 2k‖/(2m) and obtain

∑

n,kx>0

c(kx) =a

(2π)2

kF∫

0

c(kx) dkx

∞∫

0

k‖ dk‖δ(

k‖ −√

k2F − k2

x

)

√

k2F − k2

x/m

= am

(2π)2

kF∫

0

c(kx) dkx

The normalization constant is determined by considering c(kx) = 1. In this case the

result should be the number of states in a half of the Fermi surface. Thus

∑

n,kx>0

1 = am

(2π)2kF =

2πk2F

(2π)2, a = 2πkF

m

and

∑

n,kx>0

c(kx) =kF

2π

kF∫

0

c(kx) dkx (5.9)

Using Eq. (5.9) with c(kx) = kx/kF , we find for the current

I = N>e|1| sin(φ/2)

htanh|1| cos(φ/2)

2kBT

= π |1| sin(φ/2)

eRShtanh|1| cos(φ/2)

2kBT(5.10)

where1

RSh= N>

R0(5.11)

is the so called Sharvin conductance (inverse resistance) of the contact in the normal

state. The quantity

R0 =πh

e2≈ 12.9 k� (5.12)


0 π

2π

φ

Ι1

2

3

Figure 5.1: The supercurrent through the point contact. Curve (1) corresponds to a low

temperature T ≪ Tc, curve (3) is for a temperature close to Tc.

is the (two-spin) quantum of resistance and

N> =k2FS

4π(5.13)

is the effective number of states (for both spin projections) penetrating through the

contact. One can write the Sharvin conductance as

1

RSh= e2

πh

k2FS

4π= e2N(0)vFS

2(5.14)

The dependence Eq. (5.10) is shown in Fig. 5.1 for various temperatures. It has a

maximum which is called the critical current: A point contact cannot sustain nondissi-

pative currents larger than Ic. For low temperatures, the critical current

Ic =π |1|eRSh

(5.15)

is reached near φ = π . For temperatures close to Tc the current becomes

I = Ic sinφ (5.16)

where the critical current is

Ic =π |1|2

4kBT eRSh(5.17)

is reached at φ = π/2.

5.1.2 Long junctions

In long junctions, the states with energies larger than |1| do also contribute to the

supercurrent. However, in some cases their contribution is negligible. Here we consider


the current at low temperatures, when the main contribution comes from the bound

states.

For long junctions d ≫ ξ0 we neglect the term hvFx in the denominator in Eq.

(5.5) and find

I = −eSd

∑

n,kx>0

|vFx | (1− 2f (ǫ>))−∑

n,kx<0

|vFx | (1− 2f (ǫ<))

= −eSd

∑

kx>0

|vFx |

l0∑

l=0

tanh

[h|ωx |(φ − π)/2+ h|ωx |πl

2kBT

]

−l0∑

l=1

tanh

[−h|ωx |(φ − π)/2+ h|ωx |πl2kBT

]

= −eSd

∑

kx>0

|vFx |l0∑

l=−l0tanh

[h|ωx |(φ − π)/2+ h|ωx |πl

2kBT

]

Here l0 corresponds to ǫ = |1|, i.e., l0 = |1|/πh|ωx | ≫ 1.

Consider the limit of low temperatures T ≪ |1| and very long junction, |ωx | ≪kBT i.e., d ≫ hvF /kBT . In the sum over l one can replace the upper limit by infinity.

In this case all what happens at energies ǫ > |1| has no effect, thus the delocalized

states can be ignored. The sum over l becomes

6 =∞∑

l=−∞tanh

[h|ωx |(φ − π)/2+ h|ωx |πl

2kBT

]

If one considers the variable h|ωx |πl as continuous and replaces the summation with

integration, the sum turns to zero being a sum of an odd function. Therefore, we need

to take into account the discrete nature of summation. To do this, we use the fact the

residue of cot z is equal 1 at its poles z = πl:

∞∑

l=−∞tanh

[h|ωx |(φ − π)/2+ h|ωx |πl

2kBT

]

= 1

2πi

∫

C1

tanh

[h|ωx |(φ − π)/2+ h|ωx |z

2kBT

]

cot z dz

where the contourC1 in the complex plane is shown in Fig. 5.2. By shifting the contour


Im z

Re z

C1

C1

C2

C2

πl

zn

Figure 5.2: The contours of integration in the complex plane z.

to go around poles of tanh function we obtain

6 = − 1

2πi

∫

C2

tanh

[h|ωx |(φ − π)/2+ h|ωx |z

2kBT

]

cot z dz

= − kBT

πih|ωx |

∫

C2

tanh

[h|ωx |(φ − π)/2

2kBT+ z

]

cot

[2kBT z

h|ωx |

]

dz

= −2kBT

h|ωx |

∞∑

n=−∞cot

[2kBT zn

h|ωx |

]

= −2kBT

h|ωx |

∞∑

n=0

{

cot

[2kBT zn

h|ωx |

]

+ cot

[2kBT z−n−1

h|ωx |

]}

= −2kBT

h|ωx |

∞∑

n=0

2 sin(π − φ)cos(2iωn/|ωx |)− cos(π − φ) = −

2kBT

h|ωx |

∞∑

n=0

2 sinφ

cosφ + cosh(2ωn/|ωx |)

where

zn = iπ(

n+ 1

2

)

− h|ωx |(φ − π)/22kBT

are the poles of the tanh function, and

hωn = 2πkBT

(

n+ 1

2

)

are the so called Matsubara frequencies. We used here identities

cotα + cotβ = 2 sin(α + β)cos(α − β)− cos(α + β) , cos(iα) = coshα .

In the limit of very long junction, |ωx | ≪ kBT , i.e. d ≫ hvF /kBT , the factor

ωn/|ωx | ≫ 1 so that

cosh(2ωn/|ωx |) ≈ e2ωn/|ωx |/2


Therefore, only the term with n = 0 is important in the sum and

6 = −8kBT

h|ωx |e−2ω0/|ωx | sinφ = −8kBT d

h|vFx |e−2πkBT d/h|vFx | sinφ

The current becomes

I = 8kBT eS sinφ

h

∑

kx>0

e−2πkBT d/hvFx (5.18)

To sum over the states with kx > 0 we calculate using Eq. (5.9).

∑

kx>0

e−2πkBT dm/h2kx = kF

2π

kF∫

0

e−2πkBT dm/h2kx dkx ≈

kF

2πk2F

h2

2πkBT dme−2πkBT d/hvF

Finally,

I =2eSvF k

2F

π2de−d/ξN sinφ =

4hN(0)v2F eS

de−d/ξN sinφ = 1

2eRSh

16hvF

de−d/ξN sinφ

(5.19)

where

ξN =hvF

2πkBT

is the “normal-state” coherence length. Eq. (5.19) can be written as

I = Ic sinφ


Ic =1

2eRSh

16hvF

de−d/ξN (5.20)

where 2RSh is the resistance of two SN contacts in the normal state.

We see that for I < Ic the supercurrent can flow through the normal region that is

in contact with superconductors. This is called the proximity effect: the superconduc-

tor induces Cooper-pair-like correlations between electrons in the normal state. These

correlations decay exponentially into the normal metal over distances ξN inversely pro-

portional to temperature. The exponential decay of correlations ensures a small value

of the current which is required for the validity of our calculations.

There is not only the supercurrent in the normal metal but also an energy gap.

Indeed, the energy spectrum is given by Eq. (4.54). The lowest energy of excitation is

ǫ0 = h|ωx |(φ

2− π

2

)

It depends on the phase difference between the superconductors. The gap vanishes for

φ = π .

5.2. SUPERCONDUCTOR–INSULATOR–NORMAL-METAL INTERFACE 115

5.2 Superconductor–Insulator–Normal-metal interface

We switch to consideration of the NIS interface, where the current becomes depen-

dent on the barrier strength Z. The wave functions and transmission and reflection

amplitudes at the NIS interface have been analyzed in Sec. 4.6.

5.2.1 Current through the NIS junction

Consider the case when a voltage V is applied across the interface. We assume that the

potential of the superconductor is zero while the potential of the normal metal is V .

The current is given by Eq. (5.1). For the x component across the junction we have

jx =e

m

∑

n

[

fn

(

u∗n pxun + un p†xu∗n

)

+ (1− fn)(

vn pxv∗n + v∗n p†

xvn

) ]

(5.21)

with

px = −ih∂

∂x, p†

x = ih∂

∂x,

since magnetic field we consider absent. Thus

u∗n pxun + un p†xu∗n = 2h |uxn|2 kxn, uxn = |uxn| eikxnx,

vn pxv∗n + v∗n p†

xvn = 2h |vxn|2 (−kxn), vxn = |vxn| eikxnx .

It is easier to calculate current in the normal part. For the states (1)–(4) in Sec. 4.6 in

semiclassical approximation we have

State Wave function in the N region Contribution to jx/(2hekx/m)

(1) p→ N |S eikxx

(

1

0

)

+ aeikxx(

0

1

)

+ be−ikxx(

1

0

)

fNp (1− |b|2)+ (1− fNp )(−|a|2)

(2) h→ N |S e−ikxx(

0

1

)

+ a2e−ikxx

(

1

0

)

+ b2eikxx

(

0

1

)

fNh (−|a2|2)+ (1− fNh )(1− |b2|2)

(3) N |S← p c3e−ikxx

(

1

0

)

+ d3eikxx

(

0

1

)

f Sp (−|c3|2)+ (1− f Sp )(−|d3|2)

(4) N |S← h c4eikxx

(

0

1

)

+ d4e−ikxx

(

1

0

)

f Sh (−|d4|2)+ (1− f Sh )(−|c4|2)

Here fNp/h and f Sp/h are distribution functions of particle/hole excitations in the normal

and superconducting regions, respectively.

To get the total current through the junction we multiply the current density (5.21)

by the area of the junction S. We also convert the sum over states to the integral over

the energy [with the density of states NN in the normal metal for states (1) and (2) and


kF

kF

N S

00

ǫǫ

|1| ǫǫ − eVǫ + eV

ϕ = 0ϕ = V

Figure 5.3: Quasiparticle spectra in NIS junction with applied voltage V . Remember

that particle excitations in normal metal have charge e < 0 and hole excitations have

charge −e > 0. It is assumed that the local equilibrium in each part of the junction

is not affected by a relatively weak coupling and thus quasiparticle distributions shift

together with the spectra.

with the density of states in superconductor NS for states (3) and (4)] and momentum

direction. We obtain

INIS =2heS

m

∫

ǫ>0

dǫ

∫

kx>0

d�k

4πkx

{

NN[

fNp (1− |b|2)− (1− fNp )|a|2

− fNh |a2|2 + (1− fNh )(1− |b2|2)]

+NS[

− f Sp |c3|2 − (1− f Sp )|d3|2 − f Sh |d4|2 − (1− f Sh )|c4|2]}

(5.22)

Since in the S region electrical potential ϕ = 0 the distribution is simply

f Sp (ǫ) = f Sh (ǫ) ≡ f S(ǫ) = f0(ǫ) ≡1

eǫ/kBT + 1. (5.23)

Note that f Sh (ǫ) = 1− f Sp (−ǫ). In the N region the potential ϕ = V and the energies

are shifted by the electrostatic energy, Fig. 5.3:

fNp (ǫ) =1

eǫ−eVkBT + 1

= f0(ǫ − eV ) ≡ fN (ǫ), fNh (ǫ) =1

eǫ+eVkBT + 1

= f0(ǫ + eV ).

(5.24)

Note that again fNh (ǫ) = 1− fNp (−ǫ). Inserting distribution functions into (5.22) and

using relation between a2, b2, c3, d3, c4, d4 and a, b, c, d from Eqs. (4.82) – (4.84) we


find

INIS = 2heS

m

∫

ǫ>0

dǫ

∫

kx>0

d�k

4πkx

{

NN

[(

1− |b|2)(

fN (ǫ)+ fN (−ǫ))

−|a|2(

2− fN (ǫ)− fN (−ǫ))]

−NS[

|c|2(

f S(ǫ)+ f S(−ǫ))

+ |d|2(

2− f S(ǫ)− f S(−ǫ))] v2

gS

v2gN

}

= 2heS

m

∫

ǫ>0

NN dǫ

∫

kx>0

d�k

4πkx

[(

1− |b|2 + |a|2)(

fN (ǫ)+ fN (−ǫ)− 1)

−(

|c|2 − |d|2)(

f S(ǫ)+ f S(−ǫ)− 1) vgS

vgN

]

+2heS

m

∫

ǫ>0

NN dǫ

∫

kx>0

d�k

4πkx

[

1− |b|2 − |a|2 −(

|d|2 + |c|2) vgS

vgN

]

We have used here (see Sec. 2.8)

vgS

vgNNS = NN .

In the expression for INIS we have

f S(ǫ)+ f S(−ǫ)− 1 = f Sp (ǫ)− f Sh (ǫ) = 0

and

vgN · 1 = vgN(

|a|2 + |b|2)

+ vgS(

|c|2 + |d|2)

, (5.25)

as can be directly verified using Eqs. (4.77)–(4.80). Equation (5.25) expresses the

conservation of quasiparticle flow, where the flux towards the interface is equal to the

flux outwards from the interface.

As a result, the current becomes

INIS =2heS

m

∫

ǫ>0

NN dǫ

∫

kx>0

d�k

4πkx(

1−|b|2+|a|2)(

fN (ǫ)+fN (−ǫ)−1)

. (5.26)

Here fN (ǫ)+fN (−ǫ)−1 is an even function of ǫ. So far scattering amplitudes a and

b were defined only for ǫ > 0. We can formally extend them to negative energies as

even functions. The easiest way is to modify definitions of U and V as

U = 1√2

1+

√

ǫ2 − |1|2ǫ2

1/2

, V = 1√2

1−

√

ǫ2 − |1|2ǫ2

1/2

. (5.27)


eIR___N

∆

eV__∆

Z=0

Z=50

Z=1

2

1

01 2

Iexc

Figure 5.4: The current–voltage curves for a NIS interface at low temperatures. The

dashed line is the Ohm’s law.

With the expression under the integral being an even function of ǫ we can write

INIS =heS

m

∞∫

−∞

NN dǫ

∫

kx>0

d�k

4πkx(

1− |b|2 + |a|2)(

fN (ǫ)+ fN (−ǫ)− 1)

= heS

m

∞∫

−∞

NN dǫ

∫

kx>0

d�k

4πkx(

1− |b|2 + |a|2)(

f0(ǫ − eV )− f0(ǫ + eV ))

= 2heS

m

∞∫

−∞

NN dǫ

∫

kx>0

d�k

4πkx(

1− |b|2 + |a|2)(

f0(ǫ − eV )− f0(ǫ))

. (5.28)

The integration over angles in Eq. (5.28) requires knowledge of the dependence of the

barrier strength Z on the incident angle. We, however, can replace the strength with

some angle-averaged value Z, then the integration over angles gives

∫

kx>0

d�k

4πkx →

1

2AkF ,

where A ∼ 1 is the geometry-dependent constant. Finally we obtain

INIS = AeN(0)vFS∫ ∞

−∞dǫ(

1− |b(Z)|2 + |a(Z)|2)(

f0(ǫ − eV )− f0(ǫ))

(5.29)

Note that this expression remains valid when e is replaced with |e|.


The factor

1− |b|2 + |a|2

plays the role of the transmission coefficient for particle-hole reflection at the NIS

interface. It is by the Andreev reflection coefficient |a|2 larger that that in the normal

state. The coefficients |a|2 and |b|2 are given by Eqs. (4.77), (4.78) for |ǫ| > |1|, and

Eqs. (4.85), (4.86) for |ǫ| < |1|.The current–voltage curves for various barrier strengths at low temperatures, T ≪

Tc, are shown in Fig. 5.4.

5.2.2 Normal tunnel resistance

Consider several limiting cases. First assume that the superconductor is in the normal

state |1| = 0. We have |a|2 = 0 and

|b|2 = Z2

1+ Z2

The current becomes

ININ =AeN(0)vFS

1+ Z2

∫ ∞

−∞[f0(ǫ − eV )− f0(ǫ)] dǫ =

Ae2N(0)vFSV

1+ Z2= V

RN

where1

RN= Ae2N(0)vFS

1+ Z2(5.30)

We use here ∫ ∞

−∞[f0(ǫ − eV )− f0(ǫ)] dǫ = eV . (5.31)

To prove this we write

∫ ∞

−∞[f0(ǫ − eV )− f0(ǫ)] dǫ =

∫ ∞

−∞[f0(ǫ − eV )−2(eV − ǫ)] dǫ

−∫ ∞

−∞[f0(ǫ)−2(−ǫ)] dǫ +

∫ ∞

−∞[2(eV − ǫ)−2(−ǫ)] dǫ

Since the first integral in the r.h.s. converges, we can now make the shift ǫ − eV → ǫ

in it, after which it cancels the second integral. The third term gives

∫ eV

0

dǫ = eV

Using definition (5.30) the current through the NIS junction Eq. (5.29) can be writ-

ten as

INIS =1+ Z2

eRN

∫ ∞

−∞dǫ

[

1− |b|2 + |a|2]

[f0(ǫ − eV )− f0(ǫ)] (5.32)


5.2.3 Landauer formula

For Z = 0 we have form Eq. (5.30)

1

RN= Ae2N(0)vFS ≡

1

RSh(5.33)

It is the inverse Sharvin resistance which exists without a barrier.

To explain this result we should recall our assumption that the two conducting elec-

trodes separated by a contact have different voltages. For Z = 0 this may happen only

if the contact has an area much smaller than the cross sections of the two electrodes.

This is exactly similar to the point contacts considered earlier in this chapter.

Consider a point contact between two normal metals in more detail. We assume

that the barrier is absent so that the electrons fly freely (ballistically) through the con-

striction from one electrode to another. The current through the constriction is

I = 2e∑

px>0;py ,pz

[

|vgx |f0(ǫ − eV )− |vgx |f0(ǫ)]

= 2e∑

py ,pz

∫ ∞

0

dpx

2πh

∂ǫpy ,pz(px)

∂px[f0(ǫ − eV )− f0(ǫ)]

The factor 2 comes due to the spin. The summation runs over such py, pz whose states

penetrate through the constriction. We further have

I = 2e

h

∑

py ,pz

∫ ∞

−EFdǫ [f0(ǫ − eV )− f0(ǫ)] =

2e2N>

hV (5.34)

Here N> is the number of states with px > 0 that go through the constriction and we

used Eq. (5.31) for the energy integral.

Equation (5.34) is the well-known Landauer formula for a ballistic constriction. It

shows that the conductance of the ballistic constriction G = (2e2/h)N> is an integer

multiple of the quantum of conductance G0 = 2e2/h where

R0 =1

G0= h

2e2≈ 12.9 k� (5.35)

is the quantum of resistance. The dissipation of energy is concentrated in the electrodes

where the incoming particles relax to the local chemical potential.

The Sharvin conductance in Eq. (5.33) can be written as

1

RSh= 2e2N>

h= N>

R0

where

N> = πhAN(0)vFS = Ak2FS

2π


is the number of penetrating modes.

In general, for a contact between two normal metals separated by a constriction

with a barrier, the conductance in Eq. (5.30) can be written as

GNIN =2e2

h

N>∑

n=1

Tn (5.36)

where

Tn = 1− |bn|2 =1

1+ Z2n

(5.37)

is the transmission coefficient for the mode n. Equation (5.36) is known as the Landauer-

Buttiker formula.

5.2.4 Tunnel current

Consider a junction with a strong barrier Z2 ≫ 1. For |ǫ| > |1| we find from Eqs.

(4.77) and (4.78)

|b|2 ≈ 1− 1

Z2(U2 − V 2)= 1− ǫ

Z2√

ǫ2 − |1|2= 1− NS(ǫ)

Z2N(0)

while |a|2 ∼ Z−4. For |ǫ| < |1| Eqs. (4.85) and (4.86) yield |a|2 ∼ Z−4 thus |b|2 = 1

and 1− |b|2 + |a|2 = 0.

Therefore,

INIS =1

eRN

∫ ∞

−∞

NS(ǫ)

N(0)[f0(ǫ − eV )− f0(ǫ)] dǫ (5.38)

where we putNS(ǫ)

N(0)= ǫ√

ǫ2 − |1|22(

ǫ2 − |1|2)

with 2(x) being the Heaviside step function. This is the well known expression for

the tunnel current. Thus the contact with large barrier strength Z is equivalent to the

tunnel junction.

For low temperatures T ≪ Tc we find

INIS =√

(eV )2 − |1|2|e|RN

2(|e|V − |1|) (5.39)

5.2.5 Excess current

For large voltages, |e|V ≫ |1| the integral in Eq. (5.32) for the current through the

junction is determined by energies of the order of |e|V . Indeed, for |ǫ| ≫ |1|

|a|2 ≈ |1|24ǫ2(1+ Z2)2

, |b|2 ≈ Z2

1+ Z2− |1|2Z2

2ǫ2(1+ Z2)2


Therefore I ≈ V/RN . The curve I (V ) for large V goes parallel to the Ohm’s law, but

it is shifted by a constant current which is called the excess current (see Fig. 5.4). We

define the excess current as

Iexc(V ) = INIS(V )− IN (V )

= 1

eRN

∫ ∞

−∞dǫ[(

1+ Z2)(

1− |b|2 + |a|2)

− 1](

f0(ǫ − eV )− f0(ǫ))

The term in the brackets under the integral at ǫ ≫ |1|

(

1+ Z2)(

1− |b|2 + |a|2)

− 1 ≈ 1+ 2Z2

4(1+ Z2)

|1|2ǫ2

(5.40)

decays as ǫ−2, therefore the integral converges at |ǫ| ∼ |1|. If |e|V ≫ |1|, T , then

in the interval |ǫ| . |1| we have f0(ǫ − eV ) ≈ 0 and the excess current becomes

independent of V . The saturated value for high voltages V →∞ is

Iexc(∞) =1

eRN

∫ ∞

−∞dǫ[(

1+ Z2)(

1− |b|2 + |a|2)

− 1](

− f0(ǫ))

= 1

2|e|RN

∫ ∞

−∞dǫ[(

1+ Z2)(

1− |b|2 + |a|2)

− 1]

(5.41)

since 1 − 2f0(ǫ) is an odd function of ǫ. The current Iexc(∞) vanishes for Z → ∞,

since RN →∞, while the value of the integral is finite owing to Eq. (5.40).

5.2.6 NS Andreev current. Current conversion

One more important limit is for the Andreev-reflection mediated current at low temper-

atures for a zero barrier strength. For T ≪ Tc and low voltages |e|V ≪ |1| we need

only |ǫ| ≪ |1|. We have |a|2 = 1 while |b|2 = 0. Therefore, the current becomes

I = 2

eRSh

∫ ∞

−∞[f0(ǫ − eV )− f0(ǫ)] dǫ =

2V

RSh

The conductance is twice the normal-state conductance (see Fig. 5.4). This is due

to the fact that both particles and holes contribute to the current. The current in the

normal region is carried by the normal excitations. However, the wave function of the

normal excitations decays into the superconducting region. The normal current is then

converted into the supercurrent, Fig. 4.6.

5.3 Supercurrent in the SIS contact

Using the wave functions Eqs. (4.87), (4.88) and the spectrum Eq. (4.89) one can calcu-

late the current through the SIS junction from Eq. (5.1) for an equilibrium distribution

5.3. SUPERCURRENT IN THE SIS CONTACT 123

without applied voltage. Near the contact λSx ≪ 1 and we obtain in the right region

I = 2he

m

∑

n

kxRe[

f (ǫn)(

c∗RU∗e−ikxx + d∗RV ∗eikxx

) (

cRUeikxx − dRV e−ikxx

)

− [1− f (ǫn)](

cRV eikxx + dRUe−ikxx

) (

c∗RV∗e−ikxx − d∗RU∗eikxx

)]

= −2he

m

∑

n

kx[1− 2f (ǫn)](

|cR|2 − |dR|2)

|U |2 (5.42)

With the proper normalization of the wave function, we find similarly to Eq. (5.6), that

the supercurrent is (see Problem 5.5)

I = N>T e|1|22h

sinφ

ǫφtanh

( ǫφ

2T

)

(5.43)

Here N> is the number of channels,

N> = R0/RSh =πh

e2RSh

It is easy to see that the current can be written in the form of Eq. (5.7)

I = −2eN>

h

∂ǫφ

∂φtanh

( ǫφ

2T

)

where ǫφ is given now by Eq. (4.89).

This current is a supercurrent since it flows without voltage. The current can be

written also as

I = π |1|22eRN

sinφ

ǫφtanh

( ǫφ

2T

)

where1

RN= T

RSh

is the conductance of a contact with a transparency T .

For a ballistic contact T = 1 we recover Eq. (5.10). For a tunnel junction T ≪ 1

the current becomes

I = Ic sinφ


Ic =π |1|2eRN

tanh

( |1|2T

)

(5.44)


Problems

Problem 5.1. Compare critical current density in the superconducting point contact

with that in superconductor bulk. Consider cases T → Tc and T ≪ Tc.

Problem 5.2. Derive Eq. (5.39).

Problem 5.3. Calculate the differential conductance dINIS/dV of a tunnel NIS junc-

tion at low temperatures and show that it is proportional to the density of states in the

superconductor at the energy ǫ = |e|V .

Problem 5.4. Calculate the saturated (V →∞) excess current for Z = 0.

Problem 5.5. Using condition (4.51) find the normalization of wave functions in the

SIS contact, that is |cR|2 and |dR|2 in Eq. (4.87). Then use Eq. (5.42) to find the super-

current through the SIS junction as a function of the phase difference φ, Eq. (5.43).

Chapter 6

Josephson effect and weak links

We have seen that a supercurrent can flow through a junction of two superconductors

separated by narrow constriction, by a normal region or by a high-resistance insulating

barrier, or by combinations of these. The current is a function of the phase difference

between the two superconductors. These junctions are called weak links.

There may be various dependencies of the current on the phase difference. The

form of this dependence and the maximum supercurrent depend on the conductance of

the junction: The smaller is the conductance the closer is the dependence to a simple

sinusoidal shape. The examples considered in the previous chapter are: ballistic contact

Eq. (5.10) at temperatures close to Tc, long SNS structures, Eq. (5.20), and a tunnel

junction, problem 5.5.

The presence of a supercurrent is a manifestation of the fundamental property of

the phase coherence that exists between two superconductors separated by a weak link;

it is called the Josephson effect.

6.1 D.C. and A.C. Josephson effects

The general features of the Josephson effect can be understood using simple models of

couples superconductors.

6.1.1 Weakly coupled quantum systems

Let us use the simple description of a superconductor with the wave function ψ of

the Cooper pairs, which we used in Sec. 1.6. Assume that two superconductors are

coupled (e.g. by electron tunnelling), Fig. 6.1 but so weakly, that the state of each

125

126 CHAPTER 6. JOSEPHSON EFFECT AND WEAK LINKS

V

ϕ2= −V/2

ϕ1= V/2

Is

χ1= −φ/2

χ2= φ/2

Figure 6.1: The Josephson junction of two superconductors separated by an insulating

barrier.

superconductor is uniform

ψ1 = N1/21 eiχ1 , ψ2 = N1/2

2 eiχ2 , (6.1)

where N1 and N2 are the number of Cooper pairs in each superconductor and χ1 and

χ2 are the superconducting phases. In the absence of the coupling and of the electric

field, the wave functions are time-independent, which corresponds to the zero energy

Eα of the Cooper-pair condensate

ih∂ψα

∂t= Eαψα = 0, α = 1, 2

If V is the applied potential difference between superconductors (so that the electric

potential ϕ1 = V/2 and ϕ2 = −V/2) and there exists some coupling −K between the

superconductors, one can write Schrodinger equations as

ih∂ψ1

∂t= (E1 + e∗V/2)ψ1 −Kψ2 (6.2)

ih∂ψ2

∂t= (E2 − e∗V/2)ψ2 −Kψ1 (6.3)

Here e∗ = 2e is the charge of the Cooper pair. Inserting here Eq. (6.1) and separating

real and imaginary parts we obtain

hdN1

dt= −2K

√

N1N2 sin(χ2 − χ1)

hdN2

dt= 2K

√

N1N2 sin(χ2 − χ1)

and

hN2dχ2

dt= eVN2 +K

√

N1N2 cos(χ2 − χ1)

hN1dχ1

dt= −eVN1 +K

√

N1N2 cos(χ2 − χ1)

6.1. D.C. AND A.C. JOSEPHSON EFFECTS 127

From the first two equations we obtain the charge conservation N1 + N2 = const

together with the relation

Is = Ic sinφ (6.4)

where

Is = e∗dN2

dt= −e∗ dN1

dt

is the current flowing from the first into the second electrode,

Ic = 4eK√

N1N2/h (6.5)

is the critical Josephson current, while φ = χ2 − χ1 is the phase difference.

To interpret the second pair of equations we note that the overall phase plays no

role. Therefore we can put χ2 = φ/2 while χ1 = −φ/2. We find after subtracting the

two equations

hdφ

dt= 2eV . (6.6)

Equation (6.4) has a familiar form and describes the so called d.c. Josephson effect:

The supercurrent can flow through the insulating layer provided there is an interaction

between the superconducting regions. Equation (6.6) describes the a.c. Josephson

effect: the phase difference grows with time if there is a voltage between two super-

conductors. (Actually, dφ/dt < 0 if V > 0, since e < 0.) The d.c. and a.c. Josephson

effects are manifestations of the macroscopic quantum nature of superconductivity.

6.1.2 Josephson effect in the GL model

In the Ginzburg-Landau model the weak link can be realized as a short bridge of length

L ≪ ξ between two bulk superconductors, Fig. 6.2. The cross-section of the bridge S

is smaller than that of the bulk pieces so that the order parameter changes only within

the bridge, while in bulk it is uniform

ψ(x) = ψ(x)/ψGL ={

e−iφ/2 for x ≤ 0

eiφ/2 for x ≥ L(6.7)

We will assume magnetic field to be absent and will also ignore the magnetic field of

the small supercurrent in the bridge. To find the order parameter in the bridge one has

to solve the GL equation (3.24)

−ξ2ψ ′′ − ψ + |ψ |2ψ = 0 (6.8)

with boundary conditions (6.7). Here prime denotes derivative over x. Since L ≪ ξ ,

when φ 6= 0 the term with the derivative is the largest and we can ignore two other


0 L x

Figure 6.2: Weak link in the Ginzburg-Landau model is a narrow bridge of length

L≪ ξ between two bulk superconductors.

terms so that the solution for 0 < x < L is

ψ = L− xL

e−iφ/2 + x

Leiφ/2 = cos

φ

2+ i

(

2x

L− 1

)

sinφ

2. (6.9)

This solution is also valid for φ = 0. The current is given by Eq. (3.38)

Is = Sjx = −iheS

2mψ2GL

[

ψ∗ψ ′ − ψ(ψ∗)′]

= −ih eS2mψ2GL

(

cosφ

2

) 4i

Lsin

φ

2= Ic sinφ , (6.10)

where the critical current

Ic =hSe

mLψ2GL (6.11)

Thus we obtained the D.C. Josephson effect.

We can also find the energy of the junction (with respect to the state with no current

ψ = 1) using Eq. (3.34). Again leaving only the biggest term with the derivative we

find

FJ = S∫ L

0

1

4mψ2GL

∣∣−ihψ ′

∣∣2dx =

Sh2ψ2GL

4m

∣∣∣∣

2

Lsin

φ

2

∣∣∣∣

2

L

=Sh2ψ2

GL

2mL(1− cosφ) = EJ (1− cosφ) , (6.12)

where

EJ =hIc

2e(6.13)

If voltage V is applied to the junction, then electric field produces work IsV per

unit time. In the absence of normal resistance or capacitance of the junction (we will

consider their effects later), this work changes the energy of the junction

d

dtFJ = IsV

6.2. EXTENDED JOSEPHSON JUNCTIONS 129

y

H

x

z

W

d

Figure 6.3: A long Josephson junction in a magnetic field.

orhIc

2esinφ

dφ

dt= V Ic sinφ .

Thus we obtained the A.C. Josephson effect Eq. (6.6).

6.2 Extended Josephson junctions

Consider two large superconductors 1 and 2 separated by a thin insulating layer with a

thickness d and placed into the magnetic filed. The (x, y) plane is in the middle of the

insulating layer. The superconductor 1 is at z < −d/2, the superconductor 2 occupies

the region z > d/2. Along the y axis both superconductors occupy space 0 < y < W ,

while along the x axis the system is uniform, Fig. 6.3. The magnetic field is applied

parallel to the insulating layer along the x axis. We choose A = (0, Ay, 0). Therefore,

hx = −∂Ay

∂z

The field decays within the superconductors due to Meissner screening

hx(y, z) =

H(y)e−(z−d/2)/λ2 , z > d/2

H(y), −d/2 < z < d/2

H(y)e(z+d/2)/λ1 , z < −d/2

We denote H(y) the field in between the two superconductors and assume that it is

independent of z; λ1,2 is the London penetration length in the superconductor 1 and 2,

respectively. The vector potential deep in the superconductor 2 is

Ay,2 = −∫ ∞

0

hx dz = −∫ d/2

0

hx dz−∫ ∞

d/2

hx dz = −H(d

2+ λ2

)

(6.14)


Here we put Ay(z = 0) = 0. Similarly,

Ay,1 = −∫ −∞

0

hx dz =∫ 0

−d/2hx dz+

∫ d/2

−∞hx dz = H

(d

2+ λ1

)

(6.15)

Deep inside superconductors js = 0, thus from Eq. (1.16)

Ay =hc

2e

∂χ

∂y.

Applying this to Eqs. (6.14) and (6.15) we find

∂χ1

∂y= 2e(λ1 + d/2)H

hc,

∂χ2

∂y= −2e(λ2 + d/2)H

hc

and∂φ

∂y= −2e(λ1 + λ2 + d)H

hc(6.16)

where φ = χ2 − χ1.

Using the Maxwell equation

c

4πcurlz h = jc sinφ or − c

4π

∂H

∂y= jc sinφ

we obtainhc2

8πe(λ2 + λ1 + d)∂2φ

∂y2= jc sinφ

or

λ2J

∂2φ

∂y2= sinφ (6.17)

where

λJ =

√

hc2

8πejc(λ2 + λ1 + d)(6.18)

is called the Josephson length. Equation (6.17) is called Ferrell and Prange (1963)

equation. Note that since Ic includes e as a factor [e.g. Eqs. (6.5), (6.11)], the expres-

sion under root in Eq. (6.18) is positive independently of the sign convention for e.

We can write Eq. (6.16) in the form

H = −4πjcλ

2J

c

∂φ

∂y(6.19)

Equation (6.17) is similar to equation of motion for a pendulum. Indeed, the latter

has the formdθ2

dt2= −g

lsin θ

where the angle θ is measured from the bottom. Equation (6.17) is obtained from it by

replacing θ = π − φ and putting λ−2J = g/l. This means that the pendulum angle φ is

measured from the top, Fig. 6.4.

6.2. EXTENDED JOSEPHSON JUNCTIONS 131

φ

θ

Figure 6.4: A pendulum moves in time similar to variations of φ along the y axis.

6.2.1 Low field limit. Field screening

Consider first a low field H applied outside the junction at y = 0. In this case φ is also

small. Expanding Eq. (6.17) we find

λ2J

∂2φ

∂y2= φ

whence (see Fig. 6.5, curve 1)

φ = φ0e−y/λJ

From Eq. (6.19) we find that the magnetic field decays as

H (y) = H(0)e−y/λJ

where

H(0) = 4πjcλJφ0

c(6.20)

Magnetic field decays into the junction in a way similar to the Meissner effect. The

penetration length is λJ . This length is larger than λL because the screening current

cannot exceed the Josephson critical current jc.

6.2.2 Higher fields. Josephson vortices.

To find a solution of Eq. (6.17) for larger fields, we multiply it by ∂φ/∂y and obtain

after integration

λ2J

2

(∂φ

∂y

)2

+ cosφ = A , (6.21)

where A is a constant. Eq. (6.21) gives

λJ√2

∫ φ

φ0

dφ√A− cosφ

= y , (6.22)

where φ0 = φ|y=0.


φ

−π

0

π

2π

Y0 2Y0 y12 3Y0

Figure 6.5: The phase difference φ as a function of the distance into the junction mea-

sured from the left edge. Curve 1: small magnetic fields. Curve 2: large fields, phase

runs from φ = −π at the edge through 2πn values making Josephson vortices.

The case of the small fields, discussed above, corresponds to φ→ 0 and ∂φ/∂y →0 when y →∞, and Eq. (6.21) results in A = 1. The applied filed is then, according

to Eq. (6.19)

H (0) = −4πjcλ

2J

c

∂φ

∂y

∣∣∣∣y=0

= −4πjcλ

2J

c

√2

λJ

√

1− cosφ0 =8πjcλJ

csin

φ0

2

The sign in√

1− cosφ0 = ±√

2 sin(φ0/2) is chosen to agree with Eq. (6.20) for

small φ0. Increasing field leads to an increase in φ0 until it reaches ±π . This threshold

corresponds to the field

H1 =8π |jc|λJ

c= 80

πλJ (λ1 + λ2 + d)(6.23)

Above this field, the constant A > 1, and the phase φ can vary within unlimited

range. Consider for example the case φ0 = −π . The phase runs indefinitely through

the values 2πn producing the so called solitons (Fig. 6.5, curve 2). The phase solitons

are also called the Josephson vortices: the phase difference across the junction varies

by 2π each time as we go past one Josephson vortex. The distance between vortices is

L = 2Y0 ∼ λJ . Note that unlike Abrikosov vortices, the Josephson vortices have no

cores, where the magnitude of the order parameter is suppressed to zero. Thus there is

no analogue of the upper critical field Hc2.

Consider the case H ≫ H1. In this case A ≫ 1 and in Eq. (6.21) one can ignore

cosφ. Thus ∂φ/∂y = const and from Eq. (6.19) we see that H = const, i.e. there is no

screening. Integrating Eq. (6.19) we find

φ = φ0 −Hc

4πjcλ2J

y

6.3. DYNAMICS OF JOSEPHSON JUNCTIONS 133

−4 −2 20 4

0.2

0.4

0.6

0.8

1.0

8/80

Imax/Ic

Figure 6.6: Maximum supercurrent through the Josephson junnction Imax normalized

to the critical current Ic in zero magnetic field versus magnetic flux 8 through the

junction, Eq. (6.26).

The supercurrent becomes

js = jc sin(

φ0 + 2πy

L

)

where

L = −8π2jcλ

2J

cH= 80

(λ1 + λ2 + d)H(6.24)

is the distance between the Josephson vortices.

The total current through the junction of the width W is

I = jc∫ W

0

sin(

φ0 + 2πy

L

)

dy = jcL

2π

[

cosφ0 − cos

(

φ0 + 2πW

L

)]

= Wjcsin(πW/L)

πW/Lsin

(

φ0 + 2πW/2

L

)

= Icsin(π8/80)

π8/80sin φ0 , (6.25)

where Ic = jcW ,

8 = HW(λ1 + λ2 + d)

is the total flux through the junction and φ0 is the phase difference at the middle of the

junction y = W/2. The maximum supercurrent is

Imax =∣∣∣∣Ic

sin(π8/80)

π8/80

∣∣∣∣, (6.26)

see Fig. 6.6.


R JV

I

Figure 6.7: The resistively shunted Josephson junction.

6.3 Dynamics of Josephson junctions

6.3.1 Resistively shunted Josephson junction

Here we consider the a.c. Josephson effects in systems which carry both Josephson

and normal currents in presence of a voltage. As we know, the normal current has

a complicated dependence on the applied voltage which is determined by particular

properties of the junction. In this Section, we consider a simple model that treats the

normal current as being produced by usual Ohmic resistance subject to a voltage V .

This current should be added to the supercurrent. Therefore, the total current has the

form

I = V

R+ Ic sinφ . (6.27)

The difference of the phases at the both sides from the junction φ obeys the a.c. Joseph-

son relation (6.6). Together equations (6.27) and (6.6) describe the so called resistively

shunted Josephson junction (RSJ) model (see Fig. 6.7).

The full equation for the current is

I = h

2eR

∂φ

∂t+ Ic sinφ (6.28)

If |I | < |Ic|, the phase is stationary:

φ = arcsin(I/Ic)

and voltage is zero. The phase difference reaches ±π/2 for |I | = |Ic|.If I > |Ic|, the absolute value of the phase difference starts to grow with time, and

a voltage appears. Note that if V > 0 the phase actually decreases, since e < 0. Let

t0 be the time needed for the phase to decrease from π/2 to π/2 − 2π . The average

voltage is then

(2e/h)V = 2π/t0 ≡ ωJ (6.29)


II

V

c

Figure 6.8: The current–voltage dependence for resistively shunted Josephson junction.

Calculating t0 (see Problem 6.2) we find the current–voltage dependence

V = R√

I 2 − I 2c (6.30)

It is shown in Fig. 6.8.

6.3.2 Capacitively and resistively shunted junction

The Josephson junction has also a finite capacitance, Fig. 6.9. Let us discuss its effect

on the dynamic properties of the junction. The total current through the junction is

I = C ∂V∂t+ VR+ Ic sinφ = hC

2e

∂2φ

∂t2+ h

2eR

∂φ

∂t+ Ic sinφ (6.31)

Let us first consider the energy balance. The work of the external current source is

δA = IV δt = δ(

CV 2

2

)

+ V2

Rδt + hIc

2esinφ

∂φ

∂tδt = δFC + δQ+ δFJ . (6.32)

Here FC = CV 2/2 is the energy of the capacitor, δQ is the Joule heating in the normal

resistance and FJ is the energy of the Josephson supercurrent, Eq. (6.12)

FJ = EJ (1− cosφ) , with EJ =hIc

2e.

Eq. (6.31) can also be written as a mechanical analogue equation

M∂2φ

∂t2= −η∂φ

∂t− ∂U(φ)

∂φ(6.33)

of a particle with coordinate φ and with the “mass”

M = h2C

4e2= h2

8EC(6.34)

moving in a viscous medium with a viscosity

η = h2

4e2R= h2

8ECRC(6.35)


R J CI

Figure 6.9: The capacitively and resistively shunted Josephson junction.

under action of the potential

U(φ) = EJ (1− cosφ)− (hI/2e)φ = EJ (1− cosφ − φI/Ic) (6.36)

Here we introduce the energy

EC =e2

2C

associated with charging the capacitor C with one electron charge. The potential

Eq. (6.36) is called a tilted washboard potential, Fig. 6.10.

For small oscillations around the potential minimum 1 − cosφ = 2 sin2(φ/2) ≈φ2/2 and

U(φ) ≈ EJφ2

2= Kφ2

2

with the effective spring constant K = EJ . Thus the frequency of oscillations around

the minimum is

ωp =√

K

M=√

8EJEC

h=√

2eIc

hC(6.37)

It is called the plasma frequency.

Equation (6.31) can be written also as

ω−2p

∂2φ

∂t2+Q−1ω−1

p

∂φ

∂t+ sinφ = I

Ic(6.38)

where we introduce the quality factor

Q = ωpRC =

√

2eIcR2C

h(6.39)

that characterizes the relative dissipation in the system. This parameter is large when

resistance is large so that the normal current and dissipation are small.


U

φ

Figure 6.10: The tilted washboard potential, Eq. (6.36). The tilting angle is determined

by the ratio I/Ic and shown here for I < 0. The dot shows a particle with a coordinate

φ in a potential minimum.

6.3.3 Effective inductance

Sometimes it is convenient to introduce an effective inductance equivalent to the Joseph-

son junction if the phase variations are small. The inductance L with current I gener-

ates magnetic flux 8 = LI/c (in Gaussian units). Thus voltage across the inductance

is

V = 1

c

∂8

∂t= L

c2

∂I

∂t. (6.40)

On the other hand, for small φ we have sinφ ≈ φ and the Josephson current becomes

I = Icφ. Thus from the a.c. Josephson relation,

V = h

2e

∂φ

∂t≈ h

2eIc

∂I

∂t.

Comparing this to Eq. (6.40) we find the effective inductance of the Josephson junction

LJ =hc2

2eIc. (6.41)

In terms of the effective inductance, the plasma frequency is

ωp =√

2eIc

hC= c√

LJC,

which coincides with the resonance frequency of an LC circuit.

6.3.4 Current–voltage relations

Consider the dynamics of the Josephson junction in an increasing current. As long as

the current is below Ic, the phase φ is stationary: it is determined by I = Ic sinφ.

The junction is superconducting. In the representation of a mechanical particle with a

coordinate φ in a tilted washboard potential this means that the particle is localized in


U

1

2 φ

φ0

Figure 6.11: The tilted washboard potential for I/Ic close to unity. Dashed line: |I | <|Ic|, the potential has minima. Solid line: |I | > |Ic|, the minima disappear.

one of the minima of the potential (state φ0 in Fig. 6.11). As I increases and approaches

Ic, the tilt increases, and the minima gradually disappear as shown in Fig. 6.11. For

I > Ic the particle begins to roll down the potential relief. A nonzero velocity ∂φ/∂t

determines the voltage across the junction.

The current–voltage dependence is most simple for an overdamped junction which

corresponds to smallQ i.e., to small capacitance and small resistance (that is, large dis-

sipation). In this case we can neglect the term with the second derivative in Eqs. (6.38)

and (6.33). We thus return to the case considered in the previous section where the

current–voltage dependence is determined by Eq. (6.30).

For a finite Q the current–voltage dependence becomes hysteretic (see Fig. 6.12).

With increasing current voltage is zero and the phase φ is localized (state φ0 in Fig.

6.11) until I reaches Ic. For I > Ic the particle rolls down the potential (solid line in

Fig. 6.11), and a finite voltage appears which corresponds to a voltage jump shown by

a solid line in Fig. 6.12. However, when the current is decreased, a dissipative regime

with a finite voltage extends down to currents smaller than Ic. The current at which

the voltage disappears is called retrapping current. It corresponds to trapping of the

particle back into one of the potential minima φ0 in Fig. 6.11.

This behavior has a simple explanation. A particle with a small dissipation will roll

down the potential overcoming the potential maxima by inertia even if I < Ic provided

the loss of energy during its motion from one maximum (state 1 in Fig. 6.11) to the

next (state 2), separated by phase difference δφ = 2π , is smaller than the energy gain

(h/2e)Iδφ = πhI/e. If the dissipation is larger (i.e., Q is smaller), the energy loss

exceeds the energy gain and the particle has no energy to continue its motion, thus it

falls down into the potential minimum and remains trapped there (state φ0 in Fig. 6.11).

In a sense, this describes a transition from “insulating” to superconducting state with


II

V

cI r

Figure 6.12: The current–voltage curve for resistively and capacitively shunted Joseph-

son junction. The dotted line (coinciding with Fig. 6.8) is for resistively shunted junc-

tion, small Q. The solid lines show the hysteretic behavior of a contact with a large

Q.

increasing dissipation.

For small damping and large Q the mechanical analogue suggests that the un-

trapped solution is a very rapid slide down over potential with almost constant velocity

and thus almost constant voltage across the junction V ≈ V . We can write the phase

as

φ = ωJ t + δφ, ωJ = 2eV/h ,

where δφ ≪ 1. Inserting this form into Eq. (6.38) we find

ω−2p

∂2δφ

∂t2+Q−1ω−1

p ωJ +✘✘✘✘✘✘Q−1ω−1

p

∂δφ

∂t+ sin(ωJ t +✚✚δφ) = I/Ic , (6.42)

where the two terms are neglected because δφ ≪ 1 and Q ≫ 1. For the time-

independent component of Eq. (6.42) we have

ω−1p Q−1ωJ = I/Ic . (6.43)

Thus the average voltage across the junction

V = h

2eωJ =

h

2eωpQ

I

Ic= h

2e

√

2eIc

hC

√

2eIcR2C

h

I

Ic= IR .

We obtained linear ohmic relation. For the oscillating component of Eq. (6.42) we have

ω−2p

∂2δφ

∂t2+ sin(ωJ t) = 0


and thus

δφ =ω2p

ω2J

sin(ωJ t) .

The variation δφ is small if ωp/ωJ ≪ 1. According to Eq. (6.43) this condition reads

ωp

ωJ= Ic

IQ≪ 1 .

If the current does not satisfy this condition, δφ becomes large, and the finite voltage

regime breaks down. Therefore, the retrapping current is

Ir ∼ Ic/Q . (6.44)

It goes to zero as Q→∞.

6.3.5 Voltage bias

So far we considered Josephson junction biased by constant current. Let us consider

the case when the constant voltage V is applied to the junction. In this case

∂φ

∂t= 2e

hV = const, φ = φ0 + ωJ t,

∂2φ

∂t2= 0 .

Equation (6.31) gives

I = V

R+ Ic sin(φ0 + ωJ t)

and the average current has simple ohmic behavior for any damping

I = V/R .

Thus for observation of non-trivial dynamics of Josephson junctions the current bias is

essential.

6.4 Thermal fluctuations

Consider first overdamped junction. A particle with a coordinate φ is mostly sitting in

one of the minima of the washboard potential, see Fig. 6.13. It can go into the state in

a neighboring minimum if it receives the energy enough to overcome the barrier. This

energy can come from the heat bath, for example, from phonons. The probability of

such a process is proportional to exp(−U±/kBT ) where U± is the height of the barrier

as seen from the current state of the particle. The probability P+ to jump over the

6.4. THERMAL FLUCTUATIONS 141

U

φ

U+

P+P−

U−

2πEJI

Ic2π =

πhI

e

Figure 6.13: Thermal fluctuations in the overdamped junctions. The particle (filled

circle) is mostly sitting in a minimum of the washboard potential, but can jump over

the barrier U+ or U− to the next or previous minimum (empty circles) with probability

P+ or P−, respectively.

barrier from the state φ0 to the state φ0+2π and the probability P− to jump to the state

φ0 − 2π are

P± = ωa exp

[

−U0 ∓ (πhI/2e)kBT

]

,

where ωa is a constant attempt frequency, andU0 = (U++U−)/2. Overall, the particle

will drift with the average velocity

∂φ

dt= 2π(P+ − P−) = 4πωa exp

[

− U0

kBT

]

sinh

(πhI

2ekBT

)

.

This will produce a finite voltage

V = h

2e

∂φ

dt= 2πhωa

eexp

[

− U0

kBT

]

sinh

(πhI

2ekBT

)

.

For low currents I → 0 the barrier height is U0 ≈ 2EJ and expanding hyperbolic

sine we find

V = I π2h2ωa

e2kBTexp

(

−2EJ

kBT

)

This is a linear dependence characterized by certain resistance that depends on the

attempt frequency. One can express the attempt frequency in terms of the resistance in

the normal state R. Indeed, for T ∗ ∼ EJ /kB the Josephson barrier is ineffective thus

the exponent can be replaced by unity, and the current voltage dependence defines the

normal resistanceπ2h2ωa

e2kBT ∗= R

whence ωa = e2EJR/π2h2. Using this we find for the voltage

V = EJRI

kBTexp

(

−2EJ

kBT

)


II

V

c

Figure 6.14: The current–voltage curves of a RSJ junction in presence of thermal fluc-

tuations. The curves from top to bottom correspond to increasing EJ /kBT ; the curve

starting at I = Ic refers to EJ /kBT →∞.

This determines the effective resistance of the junction

RJ = REJ

kBTexp

(

−2EJ

kBT

)

(6.45)

It is exponentially small for low temperatures.

We see that the junction has a finite (though small) resistance even for low currents.

The current–voltage curve for an overdamped RSJ junction in presence of thermal fluc-

tuations is shown in Fig. 6.14.

In the case of underdamped junctions, the particle will roll down the potential relief

as soon as it gets above the potential barrier, Fig. 6.15. The probability of this process is

P = ωa exp(−Ub/kBT ). The attempt frequency ωa is now the frequency of undamped

oscillations in the potential minimum determined by U ′(φ0) = 0, i.e. sinφ0 = I/Ic

such that

ωa =√

U ′′(φ0)

M=√

EJ cos arcsin(I/Ic)

M= ωp

(

1− I2

I 2c

)1/4

(6.46)

The barrier height is Ub = Umax − Umin where

Umin = U(φ0), Umax = U(φ1), U ′(φ1) = 0, φ1 = πsign(I/Ic)− φ0 . (6.47)

Therefore

Ub = 2EJ

[

cos arcsinI

Ic−∣∣∣∣

I

Ic

∣∣∣∣arccos

∣∣∣∣

I

Ic

∣∣∣∣

]

= 2EJ

[√

1− I2

I 2c

−∣∣∣∣

I

Ic

∣∣∣∣arccos

∣∣∣∣

I

Ic

∣∣∣∣

]

(6.48)

The probability is more important for large currents I → Ic when the barrier is small

6.5. SUPERCONDUCTING QUANTUM INTERFERENCE DEVICES 143

U

φUmin

Umax

φ0 φ1

Figure 6.15: The barrier for particle escape in the strongly tilted washboard potential.

and Eq. (6.48) can be expanded as

Ub ≈4√

2

3EJ (1− I/Ic)3/2 (6.49)

In fact, Eq. (6.48) can be reasonably approximated in the whole range of currents as

Ub ≈ 2EJ (1− I/Ic)3/2

As the current increases from zero to Ic the probability P = ωa exp(−Ub/kBT )

of an escape from the potential minimum increases from exponentially small up to

P ∼ ωp ∼ 1010 sec−1. The voltage generated by escape processes is

V = πh

eP ≈ πh

eωa exp

[

−2EJ

kBT

(

1− I

Ic

)3/2]

, (6.50)

where ωa ∼ ωp is given by Eq. (6.46).

The threshold current on a rising-current branch for underdamped junction in Fig. 6.12

is thus

Icf = Ic(

1−[

(kBT/2EJ ) ln(πhωp/eV0)]2/3

)

,

where V0 is a characteristic voltage scale. This current is actually somewhat smaller

than Ic.

The rising part of the I–V curve in Fig. 6.14 for an overdamped junction near Ic is

also determined by an exponential dependence V = (πh/e)P+ where the probability

P+ contains the barrier from Eq. (6.49). Indeed, the probability of the reverse process

P− is now strongly suppressed by a considerably higher barrier seen from the next

potential minimum.

6.5 Superconducting Quantum Interference Devices

Equation (6.4) form a basis of SQUIDs. Consider a divice consisting of two Josephson

junctions in parallel connected by bulk superconductors, Fig. 6.16. This device is


1

2

3

4Φ

Ib

I

Ia

Figure 6.16: A SQUID of two Josephson junctions connected in parallel.

called dc SQUID. Let us integrate js defined by Eq. (1.16) along the contour that goes

clockwise all the way inside the superconductors (dashed line in Fig. 6.16). We have

χ3 − χ1 + χ2 − χ4 −2e

hc

(∫ 3

1

A · dl+∫ 2

4

A · dl

)

= 0

since js = 0 in the bulk. Neglecting the small sections of the contour between the

points 1 and 2 and between 3 and 4, we find

φa − φb =2e

hc

∮

A · dl = 2π8

80(6.51)

where φa = χ2 − χ1 and φb = χ4 − χ3.

Assuming that both Josephson junctions have the same critical current we find the

total current through the device

I = Ia + Ib = Ic sinφa + Ic sinφb = 2Ic cos

(π8

80

)

sin

(

φa −π8

80

)

. (6.52)

The maximum current thus depends on the magnetic flux through the loop

Ic,SQUID = 2

∣∣∣∣Ic cos

(π8

80

)∣∣∣∣. (6.53)

We see that when8 = (n+1/2)80, then Ic,SQUID = 0. This, however, is valid only

approximately, as long as one can neglect the inductance L of the SQUID loop. For

finite inductance the circulating current through the loop Icirc = (Ib − Ia)/2 provides

additional flux, and the total flux through the loop8 differs from the external flux8ext

8 = 8ext +L

c

Ib − Ia2= 8ext − βL

80

2πsin

(π8

80

)

cos

(

φa −π8

80

)

, (6.54)

6.6. SHAPIRO STEPS 145

0.500

1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

8ext/80I c

,SQ

UID

/2I c

βL = 0

1

2

3

Figure 6.17: Dependence of the maximum supercurrent through dc SQUID Ic,SQUID

on the applied flux through the SQUID loop 8ext for different values of the SQUID

inductance represented by parameter βL, Eq. (6.55).

where we have introduced the dimensionless parameter

βL =2eLIc

hc2= L

LJ. (6.55)

To determine the dependence of the maximum current through the SQUID Ic,SQUID on

8ext one now has to solve coupled equations (6.52) and (6.54). In general, this can be

done only numerically, Fig. 6.17. The minimum of the critical current is still observed

at8ext = (n+ 1/2)80, but with increasing L the modulation of the current is reduced.

For practical devices usually βL & 1 and this effect is important.

When the bias current I through the dc SQUID exceeds the critical value, then

the voltage appears across the loop. In this regime the current–voltage relation for a

SQUID is similar to that of a single Josephson junction, but with the critical current

which depends on the external flux, Fig. 6.18. Thus the voltage is at maximum when

8ext = (n + 1/2)80. In usual operation of a SQUID magnetometer, 8ext is kept

constant by a flux-lock loop (FLL). FLL provides the feedback flux, compensating

changes in the measured flux, through a special coil, coupled to the SQUID.

6.6 Shapiro steps

When a Josephson junction is driven by an a.c. voltage (or is subject to a microwave ir-

radiation) with a frequency ω, the d.c. component of supercurrent through the junction

exhibits the so called Shapiro steps: jumps of the current at constant voltages satisfying

V = nhω/2e.Let the voltage across the junction be

V = V0 + V1 cos(ωt)


Figure 6.18: The current–voltage relation in a SQUID, biased above the critical current.

The phase difference across the junction is then

φ = φ0 + ωJ t + a sin(ωt)

where ωJ = 2eV0/h and a = 2eV1/hω. The supercurrent becomes

I = Ic sinφ = Ic sin(φ0 + ωJ t) cos[

a sin(ωt)]

+ Ic cos(φ0 + ωJ t) sin[

a sin(ωt)]

.

We use expansions

cos[

a sin(ωt)]

= J0(a)+ 2

∞∑

k=1

J2k(a) cos(2kωt)

sin[

a sin(ωt)]

= 2

∞∑

k=1

J2k−1(a) sin(

(2k − 1)ωt)

to find

I/Ic = J0(a) sin(φ0 + ωJ t)+ 2

∞∑

k=1

J2k(a) cos(2kωt) sin(φ0 + ωJ t)

+ 2

∞∑

k=1

J2k−1(a) sin(

(2k − 1)ωt)

cos(φ0 + ωJ t) =

J0(a) sin(φ0 + ωJ t)+∞∑

k=1

J2k(a)[

sin(φ0 + ωJ t − 2kωt)+ sin(φ0 + ωJ t + 2kωt)]

+∞∑

k=1

J2k−1(a)[

− sin(

φ0 + ωJ t − (2k − 1)ωt)

+ sin(

φ0 + ωJ t + (2k − 1)ωt)]

.

6.6. SHAPIRO STEPS 147

I

V

hω/2e

2hω/2e

3hω/2e

Figure 6.19: The current–voltage curves of a RSJ junction irradiated by a microwave

with frequency ω.

Using the parity Jn(z) = (−1)nJ−n(z) of the Bessel functions we combine the sums

I = Ic∞∑

n=−∞(−1)nJn(2eV1/hω) sin(φ0 + ωJ t − nωt) .

We see that for ωJ = nω, i.e., for

V0 = nhω/2e (6.56)

the supercurrent has a dc component In = IcJn(2eV1/hω) sin(φ0 + πn). This dc

component adds to the normal dc current V0/R, Sec. 6.3.5. If the voltage is slightly

different from that, given by Eq. (6.56), the supercurrent will slowly oscillate in the

range

1In = 2IcJn(2eV1/hω) .

In the experiment Josephson junctions are usually current-biased. In this case one

observes plateaus with maximum width 1In in current-voltage dependence, Fig. 6.19,

instead of current spikes, which we found in the voltage-biased model.


Problems

Problem 6.1. Find the distance between Josephson vortices forH close toH1, Sec. 6.2.2.

Problem 6.2. Derive Eq. (6.30).

Problem 6.3. Two Josephson junctions have the critical currents Ic1 = 500µA and

Ic2 = 700µA are connected in parallel by superconductors. The total current through

both of them is I = 1 mA. Find the currents through each junction.

Problem 6.4. The junction has a critical current Ic = 1 mA and the normal resistance

R = 2 �. Find the d.c. voltage and the Josephson frequency ωJ if the current through

the junction is I = 1.2 mA.

Problem 6.5. The critical current of the junction is Ic. The current through the junction

has d.c. and a.c. components such that

I = I0 + I1 sin(ωt)

where I0 < Ic and I1 ≪ I0. Find the voltage across the junction.

Problem 6.6. Find dependence of Ic,SQUID on βL for 8ext = 80/2. Consider cases of

small and large βL.

Chapter 7

Quantum phenomena in

Josephson junctions

The mechanical analogy between dynamics of the phase in a Josephson junction and

a particle in the washboard potential can be extended to quantum-mechanical descrip-

tion. The range of quantum phenomena, observed in Josephson junctions, is rather

wide. They find usage in particular in superconducting quantum electronics, which is

one of key players in the rapidly developing field of quantum technology and quantum

informtaion processing. Here we include only a brief introduction.

7.1 The Hamiltonian and charge operator

If φ is the coordinate of the a particle, then the momentum operator is

pφ = −ih∂

∂φ, (7.1)

which has usual commutation relation with the coordinate

[pφ, φ]− = −ih .

The Shrodinger equation for the wave function 9 is

H9 =[

p2φ

2M+ U(φ)

]

9 = E9 . (7.2)

For the Josephson junction the mass M is given by Eq. (6.34) and the potential U(φ)

by Eq. (6.36). Thus the Hamiltonian is

H = −4EC∂2

∂φ2+ EJ (1− cosφ)− hI

2eφ . (7.3)

149

150 CHAPTER 7. QUANTUM PHENOMENA IN JOSEPHSON JUNCTIONS

The mass term in the mechanical analogy comes from the capacitance of the junc-

tion. Associated kinetic energy is actually the charging energy of the capacitor, Eq. (6.32).

Introducing operator of charge Q on the capacitor, we can write

p2φ

2M= Q2

2C.

Using Eqs. (7.1) and (6.34) we find

Q = −2ie∂

∂φ. (7.4)

This leads to the commutation relation

[Q, φ]− =2e

h[pφ, φ]− = −2ie . (7.5)

The eigenfunction of a state with the charge Q obeys the equation

Q9Q = Q9Q or − 2ie∂9Q

∂φ= Q9Q .

It is

9Q(φ) = 90eiQφ/2e . (7.6)

Assuming a single-valued wave function

9Q(φ + 2π) = 9Q(φ)

we obtain quantization of charge πQ/e = 2πn, i.e.,

Q = 2en ,

where n is an integer number of Cooper pairs.

Using the charge operator (7.4) the Hamiltonian (7.3) can be rewritten as

H = EC

e2Q2 + EJ (1− cosφ)− hI

2eφ .

Introducing the bias charge

q(t) =∫

I (t) dt

and using ac Josephson relation (6.6) it can be shown that the Josephson Hamiltonian

can be equivalently written as

H = EC

e2

(

Q+ q(t))2 + EJ (1− cosφ) (7.7)

7.2. CONDITIONS FOR QUANTUM DYNAMICS 151

R

R C

S Sext

extC

V

L1 L2

Figure 7.1: One possible realization of the quantum Josephson junction device: A small

Josephson junction is connected to the external leads L1 and L2 by high-resistanceRext

and low capacity Cext ≪ C contacts. Both Rext and the tunnel resistance R should be

larger than R0.

7.2 Conditions for quantum dynamics

From commutation relation (7.5) we conclude that the quantum uncertainty in phase

1φ and in charge 1Q are restricted by the charge of a Cooper pair 1φ1Q ∼ 2e.

Thus, quantum-mechanical effects in a Josephson junction become important when

charge of the order of electron charge Q ∼ e is important. This charge corresponds

to the charging energy EC = e2/2C. In order to quantum states not to be smeared

by temperature fluctuations it is necessary to have EC ≫ kBT . Quantum effects are

favored by small capacitance, like in mechanical systems quantum effects becomes

more important with decreasing particle mass.

With nanofabrication it is reasonably straightforward to make junctions with area

A ∼ (100 nm)2 and thickness d ∼ 1 nm, while typical dielectric constant of used

isolators is ǫ ∼ 10. Thus characteristic capacitance of a junction is

C = ǫA

4πd∼ 10−15 F and required T ≪ EC

kB∼ 1 K .

From this we see why cooling with dilution refrigerators to T ∼ (10 − 50)mK is a

perfect match to nanofabrication capabilities, which allowed explosive growth of the

field of nanoelectronics in recent decades.

Another condition for quantum dynamics is that the tunnel resistance in the junction

should be large enough to avoid averaging out by quantum fluctuations in the particle

number. To be observable, the charging energy e2/2C must exceed the quantum un-

certainty in energy h/1t associated with the finite lifetime 1t ∼ RC of the charge on

the capacitor. Equating e2/2C to h/RC we find that the capacitance drops out and the

condition becomes R > R0 where R0 is the resistance quantum R0 = h/2e2 ≈ 12 k�,

Eq. (5.35).

One has also to take care that leads with capacitance Cext and resistance Rext, con-

necting to the junction, do not destroy its quantum behavior. For that one should have

Cext ≪ C and Rext ≪ R while still Rext ≫ R0, Fig. 7.1.


U

E

0

φ0

φ1

φ2φ3

φ

Figure 7.2: The tilted washboard potential in the quantum case. A quantum particle

can escape from the potential minimum by tunnelling through the barrier (grey region).

Another important characteristic of the Josephons junction is the Josephson energy

EJ = hIc/2e, Eq. (6.3.2). In the quantum regime we can have EJ > EC or EJ < EC .

As we will see later, the dynamics is quite different in these two regimes.

7.3 Macroscopic quantum tunnelling

With the account of quantum effects, the behavior of the junction in presence of a high

bias current is different from that considered earlier in Sec. 6.3. Consider the Hamilto-

nian Eq. (7.3) for a representative particle in a washboard potential. The representative

particle with the coordinate φ can now escape from the potential minimum at φ0 by

tunnelling through the potential barrier, see Fig. 7.2, with maximum at φ1. If the max-

imum at φ3 in Fig. 7.2 is lower than the minimum at φ0, the particle needs only one

tunnelling through the barrier shown by a gray region in the figure.

Tunnelling of the representative particle means a tunnelling of the entire system

from one macroscopic state that contains many particles to another macroscopic state.

This process involves a macroscopic number of particles and thus its probability should

be inherently small. However, the Josephson junction provides a tool that can help us

to observe these macroscopic quantum tunnelling (MQT) events.

The easiest way to solve the Schrodinger equation (7.2) is to use the WKB approx-

imation

9 = exp

(

i

∫

λ(φ) dφ

)

(7.8)

assuming

dλ/dφ ≪ λ2 . (7.9)

Inserting definition (7.8) into (7.2) we obtain

−4EC

(

(iλ)2 +✓✓✓idλ

dφ

)

9 + U(φ)9 = E9 ,

7.3. MACROSCOPIC QUANTUM TUNNELLING 153

where the crossed term can be neglected due to condition (7.9). Thus

λ2 = E − U(φ)4EC

.

With this expression the condition (7.9) transforms to dU/dφ ≪ λ3EC . Taking into

account that dU/dφ ∼ EJ and |U − E| ∼ EJ we find that the WKB approximation

holds when EJ ≫ EC .

For the energy below the potential maximum we have

λ = iλ = i√U(φ)− E2E

1/2C

,

where the sign is chosen to ensure the decay of the wave function with increasing φ.

The transmission probability through the barrier is proportional to the square of the

transmission amplitude

exp

(

−∫ φ2

φ0

λ dφ

)

,

where φ0 and φ2 are the turning points satisfying E = U(φ0) = U(φ2). The probabil-

ity of tunnelling becomes

P ∼ ωa exp

(

−E−1/2C

∫ φ2

φ0

√

U(φ)− E dφ)

. (7.10)

The exponent is generally of the order of

(EJ /EC)1/21φ ≫ 1 ,

where 1φ = φ2 − φ0. This results in a very small probability. For zero current,

1φ ∼ π . Using definition of plasma frequency (6.37) we can present the probability

as

P ∼ ωp exp(

−√

2πEB/hωp

)

,

where EB ∼ 2EJ is the barrier height. This will transform into the Boltzmann factor

exp(−EB/kBT ) for the crossover temperature

Tcr ∼ hωp/√

2πkB .

For typical value of ωp ≈ 1011 sec−1 this corresponds to Tcr ≈ 100 mK. At T >

Tcr thermally-activated jumping over the barrier, Sec. 6.4, is more important than the

quantum tunnelling.

The tunnelling probability increases for I → Ic, when the barrier height is getting

small, see Eq. (6.49). We have

U − E ∼ EJ (1− I/Ic)3/2 ,


while [cf. Eq. (6.47)]

1φ ∼ φ1 − φ0 = 2(π

2− φ0

)

= 2 arccos(I/Ic) ∼√

1− (I/Ic)2 ,

so that the factor in the exponent for the probability becomes

∼ −(EJ /EC)1/2(1− I/Ic)5/4 .

Of cause, to calculate the probability even up to an order of magnitude, the factor in

the exponent should be determined with greater precision. Calculations in Problem 7.1

give for I → Ic

P = ωa exp

−6√

2

5

√

EJ

EC

(

1− I2

I 2c

)5/4

= ωa exp

[

−12 23/4

5

√

EJ

EC

(

1− I

Ic

)5/4]

.

(7.11)

Consideration in this section are applicable only to underdamped junctions (see below),

where ωa is given by Eq. (6.46).

7.3.1 Effects of dissipation on MQT

For low temperatures, the system occupies the low energy states in the potential mini-

mum with the oscillator frequency ωp. Consider the limit of low currents. The charac-

teristic “time” it takes for the system to tunnel through the barrier is tt ∼ 2π/ωp. The

energy dissipated during this time is

ED ∼V 2

Rtt =

h2

4e2R

(dφ

dt

)2

tt ∼2πh2ωp

4e2R.

It should be smaller than the energy itself, ED ≪ hωp/2, otherwise the system cannot

tunnel into a state in another potential minimum. This gives the condition

R ≫ R0 =h

2e2, (7.12)

R0 being the quantum of resistance. This agrees with the estimates on the barrier

resistance made in Sec. 7.2. Together with the condition EJ ≫ EC , introduced above

for validity of the WKB approximation, this requires underdamped junction. Indeed,

Q = ωpRC ≫√

8EJEC

h

πh

e2C = π

√

2EJ

EC≫ 1 .

If the condition (7.12) is fulfilled, the MQT is possible. The phase can escape

from the potential minimum, and the current driven junction will exhibit a finite volt-

age. It will not be superconducting in a strict sense. However, if the dissipation is

larger, i.e., R < R0, the phase cannot tunnel. There will be no voltage: the junction

is superconducting. Therefore, the dissipation helps the superconductivity, which is a

counterintuitive result.

7.4. BAND STRUCTURE 155

7.4 Band structure

7.4.1 Bloch’s theorem

The Band structure of the energy states in a periodic potential is a consequence of

the Bloch’s theorem known in solid state physics: Any solution of the Schrodinger

equation for a particle in a potential U(x) periodic with a period a has the form

9k(x) = uk(x)eikx (7.13)

where uk(x) is a periodic function

uk(x + a) = uk(x) (7.14)

Am equivalent formulation of the Bloch’s theorem is that for a particle in a potential

U(x) periodic with a period a there exists a quantity k such that the wave function

obeys

9k(x + a) = eika9k(x) (7.15)

The quantity k is called quasimomentum. The energy, i.e., the eigenvalue of the

Schrodinger equation

[

− h2

2m

d2

dx2+ U(x)

]

9k(x) = Ek9k(x)

depends on the quasimomentum. The energy spectrum is split into intervals continu-

ously filled by the values Ek as functions of k (energy bands) separated by intervals

where there no values of Ek (forbidden bands). These energy bands are labelled by the

band numbers n such that E = Ekn.

The quasimomentum is defined within an interval

−πa≤ k ≤ π

a

which is called the first Brillouin zone. All the quasimomenta that differ by an integer

multiple of 2π/a are equivalent, i.e., the quasimomenta

k′ = k + (2π/a)n

refer to the same quasimomentum. Indeed, Eq. (7.15) shows that 9k′(x + a) =eika9k′(x), i.e., belongs to the same quasimomentum as 9k . However, there may be

many states belonging to the same quasimomentum, so that k and k + (2π/a)n do not

necessarily belong to the same state.


7.4.2 Bloch’s theorem in Josephson devices

In the case of Josephson junctions, the coordinate is φ. If the junction is not connected

to the current source (I = 0), the period of the washboard potential is 2π . Therefore,

solutions of the Schrodinger equation with the Hamiltonian Eq. (7.3)

−4EC∂2

∂φ29k + EJ (1− cosφ)9k = Ek9k (7.16)

should obey

9k(φ + 2π) = ei2πk9k(φ) , (7.17)

where k is defined within the first Brillouin zone −1/2 < k < 1/2. Equation (7.16) is

known in mathematics as the Mathieu equation.

Without the potential we would have

9k = eikφ .

Comparing this with Eq. (7.6) we recognize that k plays the role of charge Q/2e.

Therefore, in Josephson junctions the quasimometum k is actually the quasicharge

Q = 2ek

defined within the first Brillouin zone

−e < Q < e . (7.18)

If we have an external current source, at the first glance potential in the Hamiltonian

Eq. (7.3) is not periodic. However, using equivalent form (7.7) and by making a gauge

transformation

9 = 9e−iq(t)φ/2e

we find that the function 9 satisfies now Eq. (7.16) with the periodic potential. Thus it

obeys the Bloch’s theorem, Eq. (7.17), or, written with quasicharge Q,

9Q(φ + 2π) = eiπQ/e9Q(φ) (7.19)

As a result the function 9 satisfies

9Q(φ + 2π) = eiπ [Q−q(t)]/e9Q(φ) (7.20)

Requiring it to be single valued we find

Q = q(t)+ 2en (7.21)

where n is an integer number of Cooper pairs transferred through the junction. Thus in

the presense of the bias current the quasicharge depends on time as

dQ

dt= dq

dt= I (7.22)


e 3e−3e 0 q

E

2e−2e −e 4e−4e

Figure 7.3: The energy spectrum of a free charge (in a zero Josephson potential) as a

function of the bias charge q.

7.4.3 Large Coulomb energy: Free-phase limit

This limit is realized when the Josephson energy EJ is much smaller that the charging

energy EC , i.e., EJ ≪ EC . The Schrodinger equation (7.16)

−4EC∂2

∂φ29Q + EJ (1− cosφ)9Q = EQ9Q (7.23)

It has the solutions which are close to the eigenstates for fixed charge Eq. (7.6). The

spectrum has the form of parabolas

E − EJ =(q + 2en)2

2C= EC

Q2

e2

shown in Fig. 7.3. The parabolas are shifted by integer multiple of the Cooper pair

charge 2e.

A small Josephson potential introduces small energy gaps at the boundary of the

Brillouin zone where the free-charge parabolas cross (black point in Fig. 7.3). To

calculate the first energy gap we note that the potential

−EJ cosφ = −EJ2

[

eiφ + e−iφ]

couples the states at the q = e boundary of the Brillouin zone

9q=e = eiφ/2

and the states at the q = −e boundary

9q=−e = e−iφ/2

which differ by δQ = 2e and thus belong to the same quasicharge q. The wave function

at q = e will thus be a linear combination

9q = c1eiφ/2 + c2e

−iφ/2


❡−❡ 3❡−3❡ 0 ◗

❊❊❣

Figure 7.4: The energy spectrum for a Josephson junction in the limit of nearly free

phase. The spectrum in the first Brillouin zone −e < q < e is shown by solid lines.

Inserting this into the Schrodinger equation (7.23) we find

EC

[

c1eiφ/2 + c2e

−iφ/2]

− EJ2

[

c1e3iφ/2 + c2e

iφ/2 + c1e−iφ/2 + c2e

−3iφ/2]

= (E − EJ )[

c1eiφ/2 + c2e

−iφ/2]

The harmonics with ±3iφ/2 couple to the q = 3e quasicharge. Comparing the coeffi-

cients at the ±iφ/2 harmonics we find

(EC − E + EJ )c1 −EJ

2c2 = 0

(EC − E + EJ )c2 −EJ

2c1 = 0

whence

E = (EC + EJ )±EJ

2(7.24)

This means that the energy gap has the width Eg = EJ with the middle at EC + EJ ,

see Fig. 7.4. The middle point is shifted with respect to its free-phase-limit (EJ = 0)

location at EC due to the constant component of the potential. The lowest energy is

also shifted above zero, see Problem 7.2.

Since the boundaries Q = −e and Q = e of the Brillouin zone are equivalent,

as well as they are, in general, for any Q = 2em (m is an integer), one can use the

so called extended zone scheme where the energy in each band EQ,n ≡ En(Q) is a

periodic function of Q:

En(Q+ 2em) = En(Q)

This is shown by dashed lines in Fig. 7.4.


7.4.4 Low Coulomb energy: Tight binding limit

In this limit the Josephson energy is larger than the charging energy EJ ≫ EC which

implies large capacitance. The system behavior is close to that for a particle in a series

of deep potential wells. One can expand the potential near each minimum

U(φ) = EJφ2

2

to get the oscillator potential. The Schrodinger equation (7.23) transforms into the

oscillator equation

−4ECd2ψ

dφ2+ EJφ

2

2ψ = Eψ

The energy spectrum is

En =√

8ECEJ

(

n+ 1

2

)

= hωp(

n+ 1

2

)

(7.25)

The energy spacing hωp ∼ EJ√EC/EJ ≪ EJ . The lowest energy wave function is

ψ0(φ) =(

EJ

8π2EC

)1/8

exp

(

−φ2

2

√

EJ

8EC

)

Note that this harmonic approximation is valid only for the lowest levels in the absence

of bias current. Otherwise, the anharmonicity of the true potential becomes important

and the level spacing becomes uneven. This is important, for example, for applications

of Josephson devices as qubits.

The true solution of the equation (7.23) is a Bloch wave, Eq. (7.13). In solid-state

physics there is a common approach to construct such solution in the tight-binding

limit. One can perform a Fourier transformation of the Bloch wave 9(n)k (r) belonging

to the bund number n

9(n)k (r) = 1√

N

∑

R

9(n)R (r) eikR .

Here R labels positions of all the sites in the lattice, with the total number N . The

Fourier component 9(n)R (r) is called the Wannier function. Note that 9

(n)R (r) depends

only on r − R. Indeed, performing inverse Fourier transform and using Eq. (7.14) we

obtain

9(n)R (r) = 1√

N

∑

k

9(n)k (r)e−ikR = 1√

N

∑

k

u(n)k (r)eikre−ikR

= 1√N

∑

k

u(n)k (r − R)eik(r−R) = 9(n)(r − R) .


U

0 2π−2π

9(φ + 2π) 9(φ − 2π)

E(n)b

E(n)a

φ

Figure 7.5: The energy band spectrum for a Josephson junction in the limit of large

Josephson energy (tight binding). The energy bands are widened oscillator levels for a

localized particle with the wave function ψ .

The Wannier function 9(n)R (r) is localized at the lattice site R and in the tight-binding

limit it is approximated by the solution of the single-well problem, 9(n)(r − R) ≈

ψn(r − R).

Thus, for the Josephson junction we can write the full wave function as

9(n)Q (φ) = 1√

N

∑

m

9(n)(φ − 2πm)ei(Q/2e)2πm, 9(n)(φ) ≈ ψn(φ) .

Here integer m labels all potential minima at φm = 2πm. The energy E(n)Q is found as

the average value of the Hamiltonian (7.3)

E(n)Q =

∫ ∞

−∞

(

9(n)Q (φ)

)∗H9

(n)Q (φ) dφ .

Each level is broadened into an energy band (see Problem 7.3), Fig. 7.5

E(n)Q = E

(n)a −

1

2E(n)b cos

πQ

e. (7.26)

The band width E(n)b is determined by overlaps of the Wannier functions 9(n)(φ) cen-

tered at φm and φm±1. It is exponentially small. For example, the lowest band width

is

E(0)b = 8hωp

√

EJ

πECexp

(

−√

8EJ

EC

)

.

7.5 Bloch oscillations in Josephson junctions

Consider Josephson junction biased with a low current such that the (Zener) transitions

from one band to another have low probability. The dynamics of quasichargeQ is then

described by Eq. (7.22)∂Q

∂t= I .

7.5. BLOCH OSCILLATIONS IN JOSEPHSON JUNCTIONS 161

e/C

−e/C

V

t

Figure 7.6: The voltage across the junction as a function of time for a constant current

bias in the free-phase limit EC ≫ EJ . The average voltage is zero.

The group velocity of the phase is given by the general expression

∂φ

∂t= ∂EQ

h∂k= 2e

h

∂EQ

∂Q. (7.27)

We omit here the band index (n). Therefore, for a constant current,

∂φ

∂Q= 2e

I h

∂EQ

∂Q

and thus

φ = 2e

I hEQ + φ0 .

But EQ is a periodic function of quasicharge Q, see e.g. Fig. 7.4, which is constantly

increasing due to applied bias current. Thus, the phase is performing periodic motion

known as Bloch oscillations. The amplitude of Bloch oscillations is

1φ = 2e

I hEb ,

where Eb = maxEQ − minEQ is the band width. The period of Bloch oscillations

follows from the 2e periodicity of EQ

tB =2e

I.

In the limit of large Coulomb energy (free phase), EC ≫ EJ , we have from

Eq. (7.24)

Eb ≈ ECand the amplitude of phase oscillations is large

1φ ≈ EC/EJ ∼ tB/R0C ≫ 1 .

The oscillations produce voltage

V = h

2e

∂φ

∂t= ∂EQ

∂Q≈ Q

C.


φ

U

0 2π−2π

Figure 7.7: MQT is equivalent to Landau–Zener transitions between the energy bands

up to the continuum.

Each time when the quasicharge Q approaches the boundary of the Brillouin zone

QB = +e (or QB = −e), the quasicharge changes QB → QB ± 2e and the voltage

jumps from+e/C to−e/C (or vice versa). In such Umklapp process for an electron in

the crystalline lattice, the extra momentum is transferred to the crystalline lattice. In the

Josephson junction the change in the quasicharge Q by 2e means that one Cooper pair

with the charge 2e is transferred through the junction and the integer n in Eq. (7.21)

changes by ±1.

We see that the Cooper pair transfer through the junction occurs only when the

voltage across the junction reaches a threshold value e/C. This is manifistation of the

Coulomb blockade. If the junction is shunted by the normal resistance R then at bias

currents I < Ith = e/(RC), all current will flow through the normal part, as voltage

e/C would not be reached.

On the contrary, if the capacity is high such that EJ ≫ EC , the band width is

very narrow, and the amplitude of phase oscillations is exponentially small. The phase

is essentially fixed such that the current Ic cosφ flows almost without voltage: the

junction is superconducting. A finite voltage can then appear as a result of macroscopic

quantum tunnelling considered in Sec. 7.3 within the semiclassical approach. In the

band picture, the macroscopic quantum tunnelling is equivalent to Zener transitions

from a lower band up to higher bands in Fig. 7.5 and finally to the continuum for

E > 2EJ (see Fig. 7.7).

7.6 Phase qubit

In a recent decade enabling quantum information processing has become a very active

area of research in physics. Such processing is based on qubits, which are basically

two-level quantum systems. A qubit should be well isolated from the decohering influ-

ence of the environment for storage of quantum information. Simultaneously it should

provide a good coupling for data processing and data readout. It has been suggested

that macroscopic quantum effects in superconductors allow to satsify simultaneously

7.6. PHASE QUBIT 163

R J CI

♦�✁✂❛✄☎♦✆

■➭✇

■r✝✞✟✠✡☛■✟❞

✂✁❛☞♦✌✄

❯✭✍✮

Figure 7.8: (Left) The phase qubit is a current-biased Josephson junction with EJ ≫EC and R ≫ R0. (Middle) Energy levels in the normal operation. (Right) During

readout the potential barrier is reduced to have a large probability to tunnel for state

|1〉.

these two contradictory goals. A number of possible realizations of superconduct-

ing qubits have been proposed using usual superconductors, so far considered in this

course. Even more possibilities are opened by topologically nontrivial superconductiv-

ity. In connection to the previous material, we will briefly look at the phase qubit.

A phase qubit is a Josephson junction with EJ ≫ EC , biased with the current I

approaching critical Ic, Fig. 7.8. Remember also conditions for quantum dynamics,

discussed in Sec. 7.2. In particular, resistance of the tunnel barrier R should exceed the

resistance quantum R0.

As we have found in the previous section, when EJ ≫ EC the bands are very

narrow, the band structure can be ignored and we can consider energy levels separately

in each minimum of the washboard potential. If I ≪ Ic then these levels are nearly

harmonic, Eq. (7.25) and such system is not suitable as a qubit: The nearly equal

separation between all levels prevents addressing two lowest levels individually. On

the contrary, when I → Ic the anharmonicity of the potential increases and E2 − E1

becomes progressively smaller thanE1−E0. Simultaneously, decay of the states to the

next minimum of the washboard potential increases, Eq. (7.11), which is an unwanted

side effect. The compromise is usually found when the number of states in a single

potential well Ns is about 4, Fig. 7.8 (middle).

We can estimate Ns as the height of the potential barrier Ub in the strongly tilted

washboard potential, Eq. (6.49), divided by the plasma frequency ωa , which is also

modified when I → Ic, Eq. (6.46). We find

Ns =Ub

hωa= (4

√2/3)EJ (1− I/Ic)3/2√

8EJEC(1− I/Ic)1/4(1+ I/Ic)1/4= 23/4

3

√

EJ

EC

(

1− I

Ic

)5/4

.

(7.28)

Single potential well in the strongly tilted washboard potential can be approximated by


■✴■c

� readout

Figure 7.9: Probability to switch junction in the phase qubit from superconducting

to resistive state, while current I is applied for time 1t . With proper selection of

I = Ireadout one can discriminate between |0〉 and |1〉 states with high fidelity. The

current scale is for the typical junction parameters.

a cubic potential which allows to find analytic expansions for the energy levels. The

first correction isEn

hωa= n+ 1

2− 5

72Ns

(

n2 + n+ 11

30

)

. (7.29)

The state of the qubit can be manipulated by applying microwave pulses Iµw at

the frequency (E1 − E0)/h to the current bias line. Often the Josephson junction in

the phase qubit is implemented as a SQUID loop with two junctions. In this case the

critical current Ic can be controlled with the flux applied to the loop, Eq. (6.53). This

gives an extra knob to manipulate the state of the qubit by changing energy EJ .

A phase qubit has a beneficial property of built-in readout scheme. When the bias

current is increased so that only two states are left in the potential, Fig. 7.8 (right), the

tunneling rates through the barrier differ drastically, P1/P0 ∼ 500, see Problem 7.5.

If the current is increased only for a fixed time 1t , then there is a finite probability

to observe a phase tunneling event. Let’s divide the total time 1t to m intervals. The

probability to observe tunnelling by the end of time1t is one minus the probability not

to observe tunneling in all those small intervals

Pn,1t = 1− limm→∞

(

1− Pn1t

m

)m

= 1− exp(−Pn1t) . (7.30)

Owing to big difference between P1 and P0 and their strong dependence on the bias

current I it is possible to find such a current Ireadout and the interval 1t , so that the

tunnelling will be practically always observed if the qubit was in the state |1〉 and

almost never observed if the qubit was in the state |0〉, Fig. 7.9. As we discussed in

Sec. 6.3.4, in an underdamped junction, which is the case of the phase qubit, releasing

the phase from the minimum of the washboard potential causes it to continuously roll

7.6. PHASE QUBIT 165

down and switches the junction state from superconducting to resistive, Fig. 6.12. This

switch will thus be an experimental signature of the tunneling event.

Problems

Problem 7.1. Find the quasiclassical probability of MQT from the minimum of the

washboard potential for I close to Ic, Eq. (7.11).

Problem 7.2. Find the lowest energy in the Josephson junction in the nearly free-phase

limit, EJ ≪ EC . This is found in the lowest band at Q = 0, see Fig. 7.4.

Problem 7.3. Derive the general form of a band in the tight-binding limit, Eq. (7.26).

There is no need to calculate values of E(n)a and E

(n)b .

Problem 7.4. In a phase qubit, made from aluminum tunnel junction and operated at

dilution refrigerator temperature T = 20 mK, what are the constraints for the transition

frequency (E1 − E0)/h? Al parameters are in Table 1.1.

Problem 7.5. In a phase qubit, roughly estimate the ratio of tunneling probabilities

for levels n + 1 and n, using simple approach of Sec. 7.3. Adjust expression (7.11) to

upper levels assuming distance the phase tunnel under the barrier and the barrier height

change linearly with the level number n.


Chapter 8

Unconventional

superconductivity

The microscopic theory of superconductivity, based on the BCS model, is able to de-

scribe properties of many superconducting materials. The weak-coupling regime, con-

sidered in this course, is directly applicable only in a limited amount of cases, but

strong-coupling generalizations of the BCS theory have been developed reasonably

early. Also the effects of a possibly anisotropic crystalline lattice, realistic compli-

cated Fermi surfaces and band structure, and of the defects and disorder have been

considered. The fundamental origin of superconductivity, formation of Cooper pairs of

electrons, mediated by interaction with oscillations of the crystalline lattice, phonons,

remained, however, the same. We remind that such interaction occurs between elec-

trons with opposite momenta and spins (i.e. in a singlet or S = 0 state), it does not

depend on the electrons spin and momenta direction, and the interaction happens al-

most at the same coordinate in space (i.e. approximately flat in momentum space),

so-called s-wave pairing, Sec. 2.3. Superconductors with this kind of electron pairing

are called conventional.

The growing number of superconducting materials show properties which are in-

compatible with behavior of conventional superconductors and thus are called uncon-

ventional. Two big classes of such materials are heavy-fermion superconductors and

high-Tc cuprates, but more are known already and definitely even more will be dis-

covered or engineered in future. In this chapter we will make a brief look at some

properties of unconventional superconductors.

Historically the first experimentally discovered system with unconventional Cooper

pairing was not a superconductor, but superfluid 3He. Interaction of 3He atoms at short

167

168 CHAPTER 8. UNCONVENTIONAL SUPERCONDUCTIVITY

distances is hard repulsion, so it is easy to see that conventional s-wave pairing is unfa-

vorable. At larger distances there is van der Waals attraction. If a Cooper pair forms in

a state with orbital momentum l = 2 then the maximum of the wave function of such d-

wave state will be in the range of attraction and one might expect the Cooper instability

to occur. It turns out that interaction of 3He atoms with magnetic fluctuations provides

stronger pairing mechanism which favors pairing in l = 1, S = 1 state. Note that with

conventional pairing in superconductors electrons interact via ions, which are relatively

independent from the electrons. In 3He fluctuations are provided by the same atoms

which pair, so the situation is microscopically more involved. In the end, different

kinds of fluctuations contribute to the pairing, including those which are responsible

for the van der Waals force.

Superfluid 3He is a conceptually simpler system than a solid-state superconductor

owing to the absence of crystalline lattice and impurities, and simple spherical Fermi

surface in the normal state. Moreover, at low pressures 3He is close to the weak-

coupling regime of the BCS theory. Thus developing theory of superfluid 3He was

instrumental for understanding of unconventional superconductors as well.

8.1 Classification of superconducting states

In all known materials Cooper pairing occurs between particles with spin 1/2. Thus

the wave function of a pair can correspond to spin S = 0 (singlet state) or spin S = 1

(triplet state). In the materials, which possess inversion symmetry, mixture of such

states is prohibited. We limit our consideration only to such cases. Note, though,

that superconductors without inversion symmetry are known, like CePt3Si. Also we

assume that Cooper pairing occurs between particles with opposite momenta k and

−k, i.e. with zero total momentum. Again there is a possibility that in some known

materials this is not the case.

8.1.1 Spin structure of the paired states

Let us start with an isotropic system, where states can be classified by definite angular

momentum l. The wave function of the pair is

9 lpair = ψl(r1 − r2)χ12 = χ12

∑

k

al(k)eik(r1−r2) . (8.1)

Here al(k) is the Fourier transform of the coordinate part ψl(r1 − r2) of the wave

function (cf. Sec. 2.4) and χ12 is the spin part. Functions al(k) can be expanded in

8.1. CLASSIFICATION OF SUPERCONDUCTING STATES 169

spherical harmonics Ylm(k)

al(k) =l∑

m=−lηlmYlm(k) . (8.2)

They are odd function of k for odd l and even for even l

al(−k) = (−1)lal(k) (8.3)

The total wave function 9 lpair should be antisymmetric for particle exchange r1 ↔ r2,

which corresponds, Eq. (8.1), to k↔ −k. Using (8.3) we conclude that χ12 should be

antisymmetric, i.e. S = 0, for even l and symmetric, S = 1, for odd l.

For S = 0 we have

χ12 =|↑1↓2〉− |↓1↑2〉 =(

1

0

)(

0 1)

−(

0

1

)(

1 0)

=(

0 1

−1 0

)

= iσy .

We remind definition of Pauli matrices

σx =(

0 1

1 0

)

, σy =(

0 −ii 0

)

, σz =(

1 0

0 −1

)

. (8.4)

Thus the wave function for even l = 0, 2, 4, . . . (called s, d , g, . . . states as traditional

from atomic physics) is

9 lpair(k) = al(k)iσy =l∑

m=−lηlmYlm(k)iσy . (8.5)

The set of the coefficients {ηl−l, . . . , ηll} forms superconducting order parameter. In a

spatially inhomogeneous case it is r-dependent. For the s-wave pairing, l = 0, the

order parameter is a single complex number η00, as we know.

For the case S = 1, the state is a combination of the states with three different spin

projections Sz on the quantization axis

Sz = 1 χ112 =|↑1↑2〉 =

(

1 0

0 0

)

,

Sz = 0 χ012 =|↑1↓2〉+ |↓1↑2〉 =

(

0 1

1 0

)

,

Sz = −1 χ−112 =|↓1↓2〉 =

(

0 0

0 1

)

,

and the wave function is

9 lpair(k) =1∑

α=−1

aαl (k)χα12 =

(

a1l (k) a0

l (k)

a0l (k) a−1

l (k)

)

(8.6)


It is convenient to define vector d(k) so that

a1l = −dx + idy, a0

l = dz, a−1l = dx + idy . (8.7)

Using vector d, the wave function Eq. (8.6) can be represented as

9 lpair(k) =(

−dx(k)+ idy(k) dz(k)

dz(k) dx(k)+ idy(k)

)

= [dx(k)σx + dy(k)σy + dz(k)σz]iσy = i[d(k)σ ]σy . (8.8)

Vector d can be expanded in spherical harmonics

dα(k) =l∑

m=−lηlαmYlm(k) . (8.9)

Set of coefficients ηlαm plays the role of the order parameter for the case with triplet

pairing S = 1 and odd l = 1, 3, . . .. In the case of p-wave pairing, l = 1, the order

parameter contains nine complex functions η1αm, α = x, y, z, m = −1, 0, 1.

8.1.2 Superfluid phases of 3He

As an example, let us consider the case of superfluid 3He with pairing in l = 1, S = 1

state, Fig. 8.1. We have

Y11(k) = −√

3

2√

2π(kx+iky), Y10(k) =

√3

2√πkz, Y1,−1(k) =

√3

2√

2π(kx−iky).

Thus according to Eq. (8.9) each component of d(k) is a linear combination of compo-

nents of k and we can write (assuming summation over repeated indices)

dα(k) = Aαi ki . (8.10)

The 3× 3 matrix Aαi is the order parameter in p-wave Fermi superfluids.

Experimentally observed phases of superfluid 3He are

B phase:

dB(k) ∝ k, ABαi ∝ δαi (8.11)

9Bpair ∝ (−kx + iky) |↑↑〉 + kz(|↑↓〉+ |↓↑〉)+ (kx + iky) |↓↓〉 (8.12)

The B phase contains pairs with |Sz = +1, m = −1〉, |Sz = 0, m = 0〉 and |Sz =−1, m = +1〉.

A phase:

dA(k) ∝ (kx + iky, 0, 0), AAαi ∝ dα(mi + ini), (8.13)

8.1. CLASSIFICATION OF SUPERCONDUCTING STATES 171

1

0

1

2

3

0.5

30

20

10

A

A

B

B

B NORMAL

NORMAL

A1A1

P ✭�✁✂✄

❚ ✭☎✆✄

❍ ✭✝✄

Figure 8.1: Phase diagram of bulk superfluid 3He.

where d = (1, 0, 0) is the spin anisotropy vector and m = (1, 0, 0) and n = (0, 1, 0)

are mutually orthogonal orbital anisotropy vectors. The pair wave function has the

form

9Apair ∝ (kx + iky)(

− |↑↑〉+ |↓↓〉)

(8.14)

The A phase contains pairs with |Sz = +1, m = 1〉 and |Sz = −1, m = 1〉. The pairing

amplitude is zero at kx = ky = 0, i.e. at the poles kz = ±1.

A1 phase: This phase is observed in the magnetic field close to Tc.

dA1(k) ∝ (kx + iky, i(kx + iky), 0), 9A1

pair ∝ (kx + iky) |↑↑〉 (8.15)

The A1 phase has only |Sz = +1, m = 1〉 pairs.

Polar phase: This phase is observed when 3He is placed in a nanostructured mate-

rial, consisting of long solid strands with diameter ≈ 9 nm and spacing 30− 50 nm, all

oriented in the same direction z.

dpolar(k) ∝ (0, 0, kz), 9polarpair ∝ kz

(

|↑↓〉+ |↓↑〉)

(8.16)

The polar phase has only |Sz = 0, m = 0〉 pairs. The pairing amplitude is zero at

kz = 0 circle.

8.1.3 Superconducting states in a crystal

In general, superconducting materials are anisotropic and the pairing states cannot be

classified with the orbital momentum l. We can generalize classification of the pairing

states in the following way. Isotropic system is invariant under arbitrary space rota-

tions which form SO(3) group. Spherical harmonics Ylm for a given l form a basis of


d Ŵ ψŴg (k)

1 A1g c1k2z + c2(k

2x + k2

y)

1 A2g kx ky(k2x − k2

y)

1 B1g k2x − k2

y

1 B2g kx ky

2 Eg kzkx , kzky

Figure 8.2: Point symmetry group D4h. (Left) The group includes one 4th order ro-

tation axis, four second-order axes, and inversion. (Right) Table of even irreducible

representations of the group D4h. These representations are compatible with singlet

(S = 0) pairing.

an irreducible representation of this group. This means that under action of any trans-

formation g, belonging to this group, these functions transform as linear combinations

of themselves:

Ylm(gk) =l∑

m′=−lŴmm′Ylm′(k) .

We can treat now Eqs. (8.2) and (8.9) as expansions in a basis of an irreducible repre-

sentation of the relevant symmetry group.

We can thus generalize classification of pairing states in a crystal with point sym-

metry group G using different irreducible representations Ŵ of the group G. For the

singlet pairing we can use only representations Ŵg where basis functions ψŴgi are even

9spair(k) = a(k)iσy, a(k) =

dŴ∑

i=1

ηiψŴgi (k), ψ

Ŵgi (−k) = ψŴgi (k) . (8.17)

Here dŴ is the dimensionality of the particular representation Ŵ and set of coefficients

ηi forms the order parameter.

For the triplet pairing we can use only representations Ŵu where basis functions

ψŴui are odd

9 tpair(k) = i[d(k)σ ]σy, d(k) =

dŴ∑

i=1

ηiψŴui (k), ψ

Ŵui (−k) = −ψŴui (k) . (8.18)

The full symmetry group G of the normal state includes also globalU(1) symmetry,

time reversal symmetry R and, possibly spin rotation group SO(3)S , if one can neglect

spin-orbit interaction. On the transition to superconducting state at least U(1) symme-

try is broken and thus the symmetry group in the superconducting state H is only a

8.2. GENERALIZED BCS THEORY 173

subgroup G. We can provide an alternative definition of unconventional superconduc-

tivity: If the superconducting group H contains all elements of G except U(1) we call

the superconductivity conventional. If some additional symmetries are broken on the

transition to superconducting state, then the superconductivity is unconventional.

8.1.4 High-Tc cuprates

As an example let us consider high-Tc metal-oxide superconducors. Many of those, in-

cluding the first discovered La2−xSrxCuO4, have crystal lattices with tetragonal sym-

metry and G = D4h = D4 × I , where I is the inversion.

The groupD4 has one rotation axis of the fourth order and four of the second order.

It has five irreducible representations, four of them are one-dimensional A1, A2, B1,

B2, and one two-dimensional E. Inversion in the group D4h doubles the number of

representations, so that for every representation of the group D4 there is one even and

one odd representation of the group D4h, Fig. 8.2.

There is a lot of experimental evidence, including phase-sensitive SQUID mea-

surements, which we will discuss later in Sec. 8.6, that the superconducting state of

high-Tc cuprates belongs to even non-trivial one-dimensional representation B1g with

ψB1g = k2x − k2

y and thus corresponds to singlet pairing, Eq. (8.17) with

a(k) = (k2x − k2

y)η . (8.19)

We see that the order parameter in this case is given by a single complex function η,

like in the conventional case. The pairing amplitude, however, is zero at diagonals

kx = ky . This has important consequences for spectra of Bogolubov quasiparticles and

for all quasiparticle-determined properties of the superconducting state.

Owing to similarity of Eq. (8.19) to a spherical harmonic belonging to l = 2, it is

traditional to say that high-Tc superconductors have d-wave pairing. As we see, this is

not strictly correct, since superconducting states in a crystal are not classified with the

orbital momentum.

8.2 Generalized BCS theory

Here we briefly outline how the BCS theory for pairing interaction which might depend

on direction of quasiparticle momenta and their spins is constructed. In general, it

follows procedure of Chapter 2.


8.2.1 Mean-field Hamiltonian

Similar to Eq. (2.22) we can write generalized Hamiltonian as

H =∑

k

ξkc†kσ ckσ +

1

2

∑

kk′Wαβ,λµ(k,k′)c†

kαc†−kβc−k′λck′µ . (8.20)

We assume summation over repeated spin indices (Greek letters). This fourth order

Hamiltonian is transformed using mean-field approximation to [cf. Eq. (2.25)]

HBCS =∑

k

ξkc†kσ ckσ+

1

2

∑

k

(

1†k,αβc−kαckβ +1k,αβc

†kαc

†−kβ −1k,αβ

⟨

c†kαc

†−kβ

⟩)

,

(8.21)

with gap functions [cf. Eqs. (2.23), (2.24)]

1k,αβ =∑

k′Wαβ,λµ(k,k′)

⟨

c−k′λck′µ⟩

, (8.22)

1†k,λµ =

∑

k′Wαβ,µλ(k

′,k)⟨

c†k′αc

†−k′β

⟩

. (8.23)

Now the gap is 2× 2 matrix

1k =(

1k↑↑ 1k↑↓1k↓↑ 1k↓↓

)

. (8.24)

It has the same structure as the Cooper pair wave function, discussed above. For singlet

pairing, Eq. (8.17)

1k = 1(

0 a(k)

−a(k) 0

)

= 1a(k)iσy , (8.25)

and for triplet pairing, Eq. (8.18)

1k = 1(

−dx(k)+ idy(k) dz(k)

dz(k) dx(k)+ idy(k)

)

= i1[d(k)σ ]σy . (8.26)

Here we separated the amplitude 1, so that the rest is normalized to unity

∫d�k

4π|a(k)|2 = 1 ,

∫d�k

4π|d(k)|2 = 1 . (8.27)

In the following we will discuss only so-called unitary phases, where 1†k1k is

proportional to the unity matrix (12k is the scalar proportionality coefficient)

1†k1k = 12

kσ0, σ0 =(

1 0

0 1

)

. (8.28)


Among examples, which we have discussed so far, only A1 phase of 3He is non-unitary.

Note that it appears in the magnetic field and the discussion here ignores magnetic field

anyway.

Introducing vector

Ck =

ck↑ck↓c

†−k↑c

†−k↓

(8.29)

we can rewrite the Hamiltonian (8.21) as

HBCS =∑

k

C†k4kCk +K , (8.30)

where 4× 4 matrix

4k =1

2

(

ξkσ0 1k

1†k −ξkσ0

)

(8.31)

andK is the part which does not depend on quasiparticle operators (and will contribute

to the condensate energy).

8.2.2 Bogolubov transformation

The Hamiltonian (8.30) is diagonalized using Bogolubov transformation

Ck = UGk , (8.32)

whereGk is a vector of γkσ operators, similar to (8.29) and Uk is a unitary 4×4 matrix

Uk =(

uk vk

v∗−k u∗−k

)

. (8.33)

Here uk and vk are 2× 2 matrices.

From the condition that the Hamiltonian is diagonal in γkσ operators one finds

uk =(Ek + ξk)σ0

√

(Ek + ξk)2 +12k

, (8.34)

vk = −1k

√

(Ek + ξk)2 +12k

, (8.35)

where

Ek =√

ξ2k +12

k . (8.36)


++

–

–

kx

ky |∆k|

+

+

–

–

kx

ky

Figure 8.3: Energy gap corresponding to B1g representation of the D4h group. This is

also called dx2−y2 pairing. The gap changes sign as a function of the azimuthal angle

of k. There exist line nodes in the energy spectra, where diagonals kx = ±ky cross

the Fermi surface. (Left) Energy gap with the isotropic Fermi surface. (Right) More

realistic Fermi surface of high-Tc copper-oxide superconductors.

The transformed Hamiltonian is

HBCS =∑

k

G†kEkGk +K , (8.37)

where

Ek =1

2

(

Ekσ0 0

0 −Ekσ0

)

. (8.38)

8.2.3 Energy spectra. Gap nodes

From Eqs. (8.37) and (8.38) we see thatEk is the energy spectrum of Bogolubov quasi-

partices.

For singlet pairing using Eqs. (8.28) and (8.25) we find

Ek =√

ξ2k +12|a(k)|2 . (8.39)

In the case of conventional fully isotropic s-wave pairing a(k) belongs to l = 0 rep-

resentation of SO(3) group, Y00 = const, so that 1a(k) = |1|eiφ and we recover

isotropic energy gap

Ek =√

ξ2k + |1|2 .

In the case of high-Tc cuprates, where a(k) =√

2(k2x − k2

y) is given by Eqs. (8.19),

(8.27), we find

Ehtck =

√

ξ2k + 2|1|2(k2

x − k2y)

2 =√

ξ2k + 2|1|2 cos2 2ϕ , (8.40)


n

m

^^

∆Asinθ

θ∆AEF

∆B

EF

A phase B phase

Figure 8.4: Energy gaps in the A and B phases of superfluid 3He. In 3He-A point

gap nodes exist at the north and south poles of the Fermi sphere, while in 3He-B the

gap is isotropic. In the equatorial plane the gap in 3He-A is larger than in 3He-B:

1A =√

3/21B .

where ϕ is the azimuthal angle of k. So we obtain four line nodes, Ek = 0, where

planes kx = ±ky cross the Fermi surface, i.e. at ϕ = π(n/2 + 1/4), n = 0, 1, 2, 3,

Fig. 8.3.

For triplet pairing using Eqs. (8.28) and (8.26) we find

Ek =√

ξ2k +12|d(k)|2 . (8.41)

For the 3He-B we have d(k) = k from (8.11), (8.27). Thus

EBk =√

ξ2k + |1|2 ,

like in the isotropic s-wave case, Fig. 8.4.

In 3He-A from (8.13), (8.27) we obtain d =√

3/2(kx + iky, 0, 0) and

EAk =√

ξ2k +

3

2|1|2(k2

x + k2y) =

√

ξ2k +

3

2|1|2 sin2 θ , (8.42)

where θ is the polar angle of k. Thus we obtain two point nodes at the poles of the

Fermi sphere k = (0, 0,±1), Fig. 8.4.

Existence of line or point nodes in the energy spectra of Bogolubov quasiparticles

is a characteristic feature of unconventional superconductors and superfluids. In prin-

ciple, nodes may be accidental. For example identity representation A1g of the group

D4h has a basis function

ψA1g = c1k2z + c2(k

2x + k2

y) .

Conventional superconductivity, corresponding to this representation, would have ani-

sotropic energy gap which might vanish if one of the parameters c1 or c2 is zero. On

the contrary, if we consider unconventional superconductivity of high-Tc cuprates, the


zero in the basis function of B1g representation (8.19) can not be removed by choice

of free parameters. It is protected by a particular symmetry breaking, leading to this

representation. Such gap nodes are called symmetry nodes.

Among symmetry nodes there are those which are topologically protected. Such

nodes would not disappear even if the symmetry is reduced further. It can be shown

that point symmetry nodes, like in 3He-A, are topological, while line nodes, like in

high-Tc superconductors, are not.

8.3 Thermodynamic quantities at T → 0

In conventional s-wave superconductors the gap in the energy spectrum of Bogolubov

quasiparticles leads to exponential temperature dependence of many parameters at tem-

peratures T ≪ Tc. In this course we have seen this on the examples of the heat capacity,

Sec. 2.9, and the normal density, Problem 2.6. But this applies also to the London pen-

etration depth, connected with the superconducting density, Eq. (1.9), paramagnetic

susceptibility, electronic thermal conductivity, ultrasonic attenuation coefficient, etc.

The gap nodes, which may exsist in unconventional superconductors and superfluids,

allow excitation of quasiparticles even at very low temperatures and the exponential

temperature dependence of thermodynamic and kinetic parameters is replaced with the

power-law dependence.

As an example, let us calculate the heat capacity of 3He-A which has two gap

nodes at north and south poles of the Fermi sphere, Eq. (8.42), at θ = 0, π . For

simplicity of notation we introduce scaled gap 12A = (3/2)|1|2. At low temperatures

only quasiparticles close to the nodes are important. Thus in general expression (2.66)

we can neglect the term with derivative of |1k|, since |1k| = 1A sin θ is small. We

also note that

f (Ek)(

1− f (Ek))

= 1

4 cosh2(Ek/2kBT )

and thus

CA =1

2kBT 2

∑

k

E2k

cosh2(Ek/2kBT ).

We change sum over k to the integral, but now the expression under the sum depends

on the direction of k. Thus instead of Eq. (2.50) we have

∑

k,ǫk<Ec

→ 2N(0)

∫ Ec

0

dǫk

∫d�k

4π= N(0)

2π

∫ Ec

0

dǫk

∫ 2π

0

dϕ

∫ π

0

sin θdθ . (8.43)

In our case we can extend integration over energies to infinity, since kBT ≪ 1A ≪ Ec,

perform integration over azimuthal angle ϕ, and use symmetry of northern and southern

8.4. PARAMAGNETIC SUSCEPTIBILITY AND KNIGHT SHIFT 179

hemispheres. We obtain

CA =N(0)

kBT 2

∫ ∞

0

dǫk

∫ π/2

0

sin θdθE2

k

cosh2(Ek/2kBT ).

Close to nodes dǫk ≈ dEk. One should remember, though, that for Ek < 1A the

range of possible θ values is limited to θ < arcsin(Ek/1A). We obtain (omitting index

k)

CA =N(0)

kBT 2

∫ ∞

0

E2

cosh2(E/2kBT )dE

∫ arcsin(min(1,E/1A))

0

sin θdθ .

Since important energies are E ∼ kBT ≪ 1A, thus θ ≪ 1 and

∫ arcsin(min(1,E/1A))

0

sin θdθ ≈∫ E/1A

0

θdθ = 1

2

(E

1A

)2

.

Finally we obtain

CA =N(0)

2kBT 212A

∫ ∞

0

E4

cosh2(E/2kBT )dE

= N(0)

2kBT 212A

(2kBT )5

∫ ∞

0

z4

cosh2 zdz = 7π4k4

BN(0)

1512A

T 3 . (8.44)

The value of the last integral is 7π4/240.

We conclude that heat capacity of a system with (first-order) point nodes is pro-

portional to T 3 at T → 0. Similarly one can show that in a system with line nodes

C ∝ T 2 when T → 0. Such power-law dependencies at low temperatures are ob-

served not only in the heat capacity, but in other thermodynamic and kinetic quantities.

One should have in mind that impurities in an unconventional superconductor with gap

nodes may modify the value of the exponent, but dependencies still remain power-law.

8.4 Paramagnetic susceptibility and Knight shift

Although magnetic field is screened in the superconductor bulk, the effects of para-

magnetic (Pauli) susceptibility χ in the superconducting state can be observed. One

popular way is to measure nuclear magnetic resonance on some element in the su-

perconductor lattice. Since the wave function of electrons is generally non-zero at the

position of a magnetic ion, the magnetic momenta of nuclei and electrons interact. This

interaction results in the shift of the NMR precession frequency from the Larmor value

γH , so-called Knight shift. The shift is proportional to the electron susceptibility χ .


Figure 8.5: Knight shift, measured on O17 in Sr2RuO4 superconductor. The broken line

shows behavior expected for singlet dx2−y2 pairing, Eq. (8.45). The experimental result

(symbols) does not depend on temperature, which provides evidence for the triplet

pairing. (K. Ishida et al, Nature 396, 658 (1998))

In the case of a superconductor with singlet pairing, Cooper pairs do not contribute

to the magnetic response. Thus all magnetic moment is due to excitations. In Prob-

lem 8.4 we find

χ(T ) = χNY (T ) , (8.45)

where susceptibility of the normal state is χN = 2µ2BN(0) (we ignore Fermi-liquid

effects) and Y (T ) is Yosida function

Y (T ) =∫d�k

4πY(k, T ), Y (k, T ) = 1

2kBT

∫ ∞

0

dξk

cosh2(Ek/2kBT ). (8.46)

If the pairing is triplet, then from Eq. (8.74) in Problem 8.1 we find that projection

of the Cooper pair spin on the direction of d is zero. This can also be seen from

Eqs. (8.6) and (8.7). If the magnetic field is oriented in the direction of d, then χ in the

superconducting state will be reduced as in the case of the singlet pairing. On the other

hand, if the field is in the plane perpendicular to d, then both excitations and Cooper

pairs contribute into the magnetic moment and χ = χN . In this case the Knight shift

does not change on the transition to superconducting state. If such behavior is observed,

like in Sr2RuO4, then this is an evidence for the triplet pairing, Fig. 8.5.

Note that in general d is a function of k, so there is no guaranty, that there exists

such a magnetic field direction which is perpendicular to d(k) for all Cooper pairs. For

example, in 3He-B d(k) = k covers all possibles directions and thus susceptibility is

suppressed in the superfluid state. On the other hand, in 3He-A, Eq. (8.13), d for all

8.5. DENSITY OF STATES. VOLOVIK EFFECT 181

Cooper pairs is in the x direction, and there is a whole plane of the field orientations

which are perpendicular to d. If the spin anisotropy axis is free to rotate, as is the case

in liquid 3He-A, then it will orient to minimize the energy and thus to maximize χ . As

a result, in 3He-A apparent χ = χN .

In a crystal, in principle, one can have free rotation of the spin anisotropy axis

(weak spin-orbit coupling) and thus χ = χN . Alternatively, the anisotropy axis may

be fixed by the crystalline lattice. Then χ would be anisotropic function of the field

direction.

8.5 Density of states. Volovik effect

Caroli – de Gennes – Matricon bound fermion states in the core of Abrikosov vortex,

Sec. 4.5, contribute to the total density of states of a superconductor

Ns(E) =∑

k

δ(E − Ek) . (8.47)

In an ideal conventional superconductor without vortices Ns(0) = 0, Eq. (2.61). Since

minigap in the spectrum of the core-bound states is small, Eq. (4.73), contribution to the

density of states from these fermions can be estimated as if the cores of the coherence

length ξ radius would be filled with the normal phase

Ns,bound ∼ N(0)πξ2nv (8.48)

If H ≫ Hc1 then magnetic induction in the superconductor B ≈ H , and density of

vortices nv we find from Eq. (3.51)

nv =B

80≈ 1

2πξ2

H

Hc2. (8.49)

Here we used expression (3.83) for the upper critical field. Inserting this into (8.48)

yields

Ns,bound ∼ N(0)H

Hc2. (8.50)

Thus at low temperatures and large magnetic fields the heat capacity of a superconduc-

tor is largely determined by the core-bound states

Cs ∼H

Hc2Cn . (8.51)

Now let us consider unconventional superconductor with nodes in the energy spec-

trum, like high-Tc cuprates. There are two consequences of the existence of nodes.


Figure 8.6: Volovik effect in high-Tc superconductor La2−xSrxCuO4. (Left) Tempera-

ture dependence of the heat capacity shows contribution Cs = γ (H)T ∝ Cn (intercept

at the y axis). (Right) Proportionality coefficient γ depends on the magnetic field H as

predicted by Eq. (8.56). (S.J. Chen et al, PRB 58, 14753 (1998))

First, in the direction of nodes the bound states “leak” from the vortex core, i.e. ex-

tent of their wave function becomes much larger than the core size. This increases

contribution to the density of states compared to (8.50). The second effect, connected

to delocalized states outside the vortex core, is even bigger. Remember that in the

presence of the supercurrent the energy spectrum of quasiparticles is shifted by hkvs ,

Eq. (2.85). Let us calculate the average density of states in the unit cell of the vortex

lattice

Ns(0) = nv∫ R

ξ

r dr

∫ 2π

0

dϕv∑

k

δ(Ek + hkvs) , (8.52)

where Ek is given by Eq. (8.40). Here R = (πnv)−1/2 is the radius of the unit cell. We

consider case H ≫ Hc1, thus R < λL. At these distances vs is given simply by (see

problem 3.9)

vs =h

2mrϕv = vs(r)(− sinϕv, cosϕv, 0), vs =

h

2mr. (8.53)

The sum in Eq. (8.52) we convert to the integral as in Eq. (2.51), taking into account

the angular k dependence

∑

k

δ(Ek + hkvs) = 2N(0)

∫d�k

4π

∫ ∞

|1k|

Ek dEk√

E2k −12

k

δ(Ek + hkvs)

= 2N(0)

∫

−hkvs>|1k|

d�k

4π

h|kvs |√

(hkvs)2 −12k

,

where we took into account that for δ-function to give non-zero contribution hkvs

shoud be sufficiently negative and thus −hkvs = h|kvs |.

8.5. DENSITY OF STATES. VOLOVIK EFFECT 183

Since

k = kF (sin θ cosϕ, sin θ sinϕ, cos θ)

we have

hkvs = pF vs sin θ sin(ϕ − ϕv)and |1k| =

√21| cos 2ϕ|. Thus the integration limit is

−vs sin θ sin(ϕ − ϕv) > (√

21/pF ) | cos 2ϕ| .

Far outside vortex core vs ≪ vc ∼ 1/pF . Thus this condition can be satisfied only in

the nearest vicinity of the gap nodes ϕ ≈ ϕn = π(n/2 + 1/4), n = 0, 1, 2, 3. In this

vicinity cos 2ϕ ≈ 2(ϕn − ϕ), so that

−|δϕn| < ϕ − ϕn < |δϕn|, δϕn =vspF

2√

21sin θ sin(ϕn − ϕv) .

The integral over ϕ close to each node is∫ ϕn+|δϕn|

ϕn−|δϕn|dϕ

h|kvs |√

(hkvs)2 −12k

=∫ |δϕn|

−|δϕn|dϕ′

|δϕn|√

δϕ2n − ϕ′2

= π |δϕn| .

Note that only those nodes make contribution where sin(ϕn − ϕv) < 0. That is, contri-

bution of each node is π(|δϕn| − δϕn)/2. We thus have

∑

k

δ(Ek + hkvs) =N(0)

2

∫ π

0

sin θdθ

3∑

n=0

|δϕn| − δϕn2

= N(0)

2

vspF

2√

21

(

| sin(π/4− ϕv)| + | sin(π/4+ ϕv)|)∫ π

0

sin2 θdθ

= πN(0)vspF

8√

21

(

| sin(π/4− ϕv)| + | sin(π/4+ ϕv)|)

Inserting this into Eq. (8.52) we find that the integral over ϕv is equal 8 and thus

Ns(0) = nvπN(0)pF√

21

∫ R

ξ

rh

2mrdr = N(0)

2

√

πnv

2

hvF

1= N(0)

2

√

πnv

2ξ . (8.54)

Inserting here nv from Eq. (8.49) we obtain

Ns(0) =N(0)

4

√

H

Hc2. (8.55)

Thus if the field H ≪ Hc2 the main contribution to the density of states comes from

the gap nodes in the regions outside of the vortex core. The heat capacity have similar

behavior

Cs ∝ Cn√

H/Hc2 . (8.56)

This behavior, predicted by G.E. Volovik, was indeed observed in high-Tc and heavy

fermion superconductors, Fig. 8.6.


8.6 Josephson effect with internal phase difference

In section 6.1.1 we considered a simple model of the Josephson effect in conventional

superconductors. We started with the wave function of the Cooper-pair condensate in

superconductors 1 and 2

ψ1 = |ψ1|eiχ1 , ψ2 = |ψ2|eiχ2

and obtained dc Josephson current

I = Ic sin(χ2 − χ1), Ic = 4eK|ψ1||ψ2|/h ,

where K is the coupling. Let us now consider superconductors with general singlet

pairing and single-component order parameter. We have, Eq. (8.17)

ψ1 = ψŴ1(k)|η1|eiχ1 , ψ2 = ψŴ2(k)|η2|eiχ2 , (8.57)

where ψŴ1 and ψŴ2 are the basis functions of appropriate one-dimensional representa-

tions. It is natural to assume that the relevant k direction is the direction of the interface

normal, which we denote n1 and n2 in superconductors 1 and 2. Comparing with the

previous expression for I we can write

I = Ic sin(χ2 − χ1), Ic = 4eKψŴ1(n1)ψŴ2(n2)|η1||η2|/h . (8.58)

One consequence of this expression is that if n1 or n2 is oriented along the gap node

direction, then we expect the critical current to be zero.

Let superconductor 1 be conventional superconductor belonging to identity repre-

sentation A1g and ψŴ1 = 1. The superconductor 2 is a tetragonal crystal with uncon-

ventional superconducting state, belonging to B1g representation of group D4h with

ψŴ2 = k2x − k2

y , like a high-Tc cuprate superconductor. If the Josephson junction a

is made at the surface of the tetragonal crystal where n2a = x and another junction b

is at the surface where n2b = y, then we find from Eq. (8.58) that at the same phase

difference the current in these junctions flows in the different direction

Ica = −Icb (8.59)

For a SQUID, made from these two junctions, Fig. 8.7, we find similar to Eq. (6.53)

I = Ia + Ib = Ica sinφa + Icb sinφb = Ica sinφa + Icb sin(φa − 2π8/80) , (8.60)

but now instead of Ica = Icb we use Eq. (8.59), which results in

I = Ica [sinφa − sin(φa − 2π8/80)] = 2Ica sinπ8

280cos

(

φa −π8

280

)

. (8.61)

8.7. MULTI-COMPONENT ORDER PARAMETER AND DIFFERENT PHASES185

++

–

–

kx

ky

+

+8

I

I

Ia

Ib

Figure 8.7: SQUID made from two junctions between conventional s-wave supercon-

ductor and dx2−y2 superconductor shows internal π phase shift.

The maximum current is

Ic,SQUID = 2

∣∣∣∣Ica sin

(π8

80

)∣∣∣∣. (8.62)

Compared to Eq. (6.53) we see that now at 8 = n80 we have minimum of the critical

current instead of maximum in the conventional SQUID. This behavior was indeed ob-

served in high-Tc superconductors like YBa2Cu3O7−x , and was regarded as the definite

experimental proof of unconventional pairing.

8.7 Multi-component order parameter and different phases

If an unconventional superconducting state belongs to a multidimensional representa-

tion of the symmetry group G, then the order parameter has several components ηi ,

Eqs. (8.17), (8.18). In this case the free energy of the system at T < Tc may have a

number of local minima at non-zero values of ηi . These different minima correspond

to different superconducting phases. Which phase is stable, i.e. corresponds to the

global minimum, depends on parameters of the system, which in turn may depend on

temperature, pressure, magnetic field and other relevant conditions.

At temperatures close to Tc analysis can be performed using Landau expansion

for the free energy, Sec. 3.1. The expression for the free energy should be invariant

under all symmetry transformations of the normal state, i.e. under the action of the

group G = G×U(1)×R[×SO(3)S]. Here R represents time-reversal symmetry. The


symmetry with respect to spin rotations SO(3)S is included for systems with spin and

negligible spin-orbit coupling.

For isotropic system (G = SO(3) of space rotations) and s-wave pairing (one-

dimensional representation ψŴ = Y00) the free energy is given by Eq. (3.1)

F = α|η|2 + β2|η|4 .

As we know, the Landau energy functional can be derived from the microscopic

theory when 1(T )≪ 1(0). From the weak-coupling BCS theory one finds

F = N(0)T − TcTc

12 + 7ζ(3)

16π2N(0)

∫d�k

4π(a(k)a∗(k))2

14

T 2c

(8.63)

for the singlet pairing and

F = N(0)T − TcTc

12 + 7ζ(3)

16π2N(0)

∫d�k

4π(d(k)d∗(k))2

14

T 2c

(8.64)

for the triplet pairing with the unitary order parameter.

8.7.1 Superfluid 3He

As one example of multi-component order parameter we use superfluid 3He, which

is isotropic (G = SO(3)) and p-wave (3-dimensional representation ψŴm = Y1m). It

is conventional instead of expansion ηαm of d over Y1m, Eq. (8.9), to use expansion

Aαi over ki as the order parameter, Eq. (8.10). The symmetry of the normal state G

transforms the order parameter as

Aαi → Aαi = RSαβROij eiφRTAβj ,

where RS and RO are matrices of arbitrary rotations of spin and orbital spaces, respec-

tively, and time reversal is represented by the operator RTAβj = A∗βj .

In order for the free energy to be invariant under such transformation, it should

include A and A∗ in pairs and spin indices should be contracted with spin indices, and

the orbital indices with orbital. With all possible terms up to the fourth order the free

energy has the form

F = αA∗αiAαi + β1|AαiAαi |2 + β2(A∗αiAαi)

2 + β3A∗αiAαjA

∗βiAβj

+ β4A∗αiAβiA

∗βjAαj + β5A

∗αiAβiA

∗αjAβj . (8.65)

The coefficients βi depend on temperature and pressure.


0 10 20 30

Pressure, bar

1

1.2

1.4

1.6

1.8

2

Co

effi

cien

t in

th

e fr

ee e

ner

gy

A phase: (β2+β

4+β

5)/β

0

B phase:

(β1+β

2+[β

3+β

4+β

5]/3)/β

0

Figure 8.8: Coefficients in the free energy of the A phase of superfluid 3He (8.67) and

of the B phase (8.68) as a function of pressure (H. Choi et al, JLTP 148, 507 (2007)).

One can verify that the A and B phases of superfluid 3He with the order parameters

(see Sec. 8.1.2 and for normalization Sec. 8.2.3)

AA =√

3

21(T )

1 i 0

0 0 0

0 0 0

, AB = 1(T )

1 0 0

0 1 0

0 0 1

(8.66)

correspond to the local minima of the free energy (8.65). The values of the free energy

are

FA = 3α12 + 914(β2 + β4 + β5) , (8.67)

FB = 3α12 + 914

(

β1 + β2 +β3 + β4 + β5

3

)

(8.68)

From the weak-coupling BCS theory, Eq. (8.64), one obtains that

β2 = β3 = β4 = −β5 = −2β1 = 2β0, β0 =7ζ(3)

240

N(0)

(πkBTc)2.

In this case it is easy to check that FA > FB . In reality strong coupling effects

are important in 3He at elevated pressures. This changes the values of βi parameters,

Fig. 8.8, so that the A phase becomes stable close to Tc at pressures above 21 bar.

8.7.2 Superconducting phases in a tetragonal crystal

As another example of the multi-component order parameter, we examine the case of

singlet pairing in a tetragonal crystal with the point symmetry group G = D4h. The

only even two-dimensional representation of this group, Eg , has basis functions kzkx


-4 -2 2 4

-4

-2

2

4

0

A

B

C(1, ±i )

(1, 0) or (0, 1)

(1, ±1)

β3/β

1

β2/β

1

Figure 8.9: Phase diagram of unconventional superconducting states corresponding

to the two-dimensional representation Eg of the symmetry group D4h of a tetragonal

crystal. In the white area on the left the 4th order term in the free energy (8.70) is

negative and thus higher-order terms are required to find the energy minimum.

and kzky . The wave function

a(k) = η1 kzkx + η2 kzky (8.69)

corresponds to the two-component order parameter η = (η1, η2). From the shape of

basis functions we see that under action of the groupG the order parameter transforms

as a vector lying in the basal plane x–y. Under action of U(1) the order parameter

transforms as η→ ηeiφ and under time reversal as η→ η∗.

Free energy, invariant under all these transformations, can be written as

F = α|η|2 + β1|η|4 +β2

2

(

η∗21 η22 + η2

1η∗22

)

+ β3|η1|2|η2|2 , (8.70)

where |η|2 = ηη∗ = |η1|2 + |η2|2.

Minimization of the free energy, Problem 8.8, leads to three possible superconduct-

ing phases

ηA = 1(1,±i), FA = 2α|1|2 + (4β1 − β2 + β3)|1|4, (8.71)

ηB = 1(1,±1), FB = 2α|1|2 + (4β1 + β2 + β3)|1|4, (8.72)

ηC = 1(1, 0) or 1(0, 1) FC = α|1|2 + β1|1|4 . (8.73)

The phase diagram is shown in Fig. 8.9.

We see that these three phases are indeed unconventional, since in each of them

additional symmetry besides U(1) is broken. The most interesting is the A phase,

where the time reversal symmetry is broken, i.e. (ηA)∗ 6= ηAeiφ . The same situation


is observed in 3He-A. Spontaneous violation of time-reversal symmetry can lead to

appearance of spontaneous magnetization.

In the B and C phases, the 4th order rotational symmetry around the z axis is re-

duced to the second-order, i.e. the symmetry breaking is D4h→ D2h.

Finally, we note that all the phases are doubly degenerate which may lead to the

formation of domains.

Problems

Problem 8.1. Using expressions (8.5) and (8.8) for the wave function of the singlet

and triplet state, derive the formulas for

(a) the expectation value of the spin of a pair in the triplet state

S = −ih∫d�k

4πd∗(k)× d(k) ; (8.74)

(b) the expectation value of the orbital momentum of a pair

Lsinglet = −ih∫d�k

4πa∗(k)

(

k× ∂

∂k

)

a(k) , (8.75)

Ltriplet = −ih∫d�k

4π

∑

α

d∗α(k)

(

k× ∂

∂k

)

dα(k) . (8.76)

Problem 8.2. Using expressions from the previous problem, find the average spin and

the orbital momentum of a Cooper pair in the A, B and A1 phases of superfluid 3He

and in the dx2−y2 state of high-Tc superconductors.

Problem 8.3. A nanowire (a one-dimensional conductor) with spin-orbit coupling of

the so-called Rashba type is placed on the top of a conventional superconductor. The

Hamiltonian of such wire is given by

H =∑

k

c†kα(ξkσ0 − αRσyk + Bσz)αβckβ +1

∑

k

[

ck↑c−k↓ + c†−k↓c

†k↑

]

. (8.77)

Here αR is the strength of the Rashba coupling, B is the applied magnetic field, which

provides the Zeeman splitting, and 1 is the proximity-induced superconducting pair-

ing.

(a) Using unitary transformation ckα = Uαβgkβ diagonalize the Hamiltonian (8.77) in

the case of 1 = 0

H |1=0 =∑

k

[

ε+k g†k↑gk↑ + ε

−k g

†k↓gk↓

]

and find the upper and lower energy bands ε±k .

(b) Apply the same transformation to the full Hamiltonian with 1 6= 0 and derive the


effective Hamiltonian in the case where only the lower band is filled (that is, when gk↑operators are ignored). Explain why the result describes unconventional superconduc-

tivity.

(c) In the effective Hamiltonian express the gap matrix 1k,αβ via a(k) [Eq. (8.25)] or

d(k) [Eq. (8.26)], as appropriate. Use expressions from Problem 8.1 to find effective

spin and orbital momentum.

Problem 8.4. Derive equations (8.45) and (8.46) for paramagnetic susceptibility in the

case of singlet pairing.

Problem 8.5. Calculate the Yosida function (8.46) for dx2−y2 pairing. Consider cases

T → Tc and T ≪ Tc.

Problem 8.6. Show that in a superconductor with the gap nodes at isolated points at

the Fermi surface, the heat capacity at low temperatures as a function of magnetic field

is described by the formula

Cs ∝ CnH

Hc2lnHc2

H.

Problem 8.7. Suggest how to use crystalline boundaries in a crystal with dx2−y2 pair-

ing, Sec. 8.6, to realize a half-quantum vortex.

Problem 8.8. Minimize the free energy (8.70) to find superconducting phases (8.71)–

(8.73), and build the phase diagram, Fig. 8.9.

Problem 8.9. Find the energy spectrum of quasiparticles in the superconducting phases

from the previous problem.

Problem 8.10. Using expression (8.63) find which phase in the phase diagram Fig. 8.9

is realized close to Tc in the weak-coupling limit.

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