TEXTBOOK ON Superconductivity and Josephson Effect: Physics ...

TEXTBOOK ON
Superconductivity and Josephson Effect:
Physics and Applications.
Vladimir M. Krasnov
Department of Physics, Stockholm University, AlbaNova University Center,
SE-10691 Stockholm, Sweden
E-mail: Vladimir.krasnov@fysik.su.se
1
Preface:
This is a textbook for an introductory one quarter course “Superconductivity and Josephson
effect: Physics and Applications” for undergraduate and master students with specialty in physics
and microelectronics (originally given at Chalmers University of Technology in 2004, 2005 and
later at the Department of Physics at Stockholm University 2006-). It was mostly motivated by the
need to provide a single easily accessible textbook, which should satisfy the following constrains
imposed by the course:
1. This is an introductory course with a modest level of difficulty. The course is aiming to
provide a background for future diploma work and in-depth postgraduate studies. Students
are supposed to have a modest knowledge of quantum mechanics. However, they are
supposed to know well basic undergraduate mathematics and physics (thermodynamics
and, especially, electromagnetism).
2. My intention was to make a textbook (in a pedagogical sense), rather than a comprehensive
scientific literature. The main emphasis was made on basic issues. Specific problems for
in-depth analysis are formulated at the end of each chapter for individual studies and home
assignments.
3. This is a one quarter course. Consequently, the book is rather concise, but, I hope, selfconsistent.
Nevertheless, to get a deeper understanding students are advised to refer to the
complementary literature and WWW material, listed in the end of the book.
4. Material is organized in terms of two hour lectures.
5. Superconductivity, Josephson effect, physics and applications are (more or less) evenly
weighted in the book.
6. Apart from basic things, the book contains a number of selected, specific topics, which are
aiming to demonstrate how the obtained general knowledge can be used for solving some
practical problems, rather than to give a comprehensive overview of the whole area. Those
specific topics are not covered during lectures and are intended for homework. I tried to
make those parts self-consistent and self-explanatory, so that students could work them out
on there own. Otherwise, see the constrain #3.
7. Applications of superconductors and superconducting electronic devices are rapidly
developing areas of research and industry. Therefore, when possible, references are made
to interactive web-pages rather than original articles. When this was impossible I provided
references to books and recent up-to date publications, which could be easily accessed in
the library.
Upon preparing this compendium, I was most of all influenced by the book of Vadim Schmidt,
“The physics of superconductors: Introduction to fundamentals and applications”, which satisfy
most of the criteria: it is an excellent textbook from a pedagogical point of view, has a right level
of difficulty and balance between general and specific topics. Unfortunately it does not cover
sufficiently the Josephson effect and not exactly up-to date as far as applications are concerned. In
writing the compendium I was also using other books, lecture notes for similar courses, web-sites
on superconductivity (see the list below), actual research articles and conference proceedings.
CGS (Gaussian) system of units was used in the text. Expressions in SI system of units for
selected equations are given in square parentheses. The conversion table from CGS to SI units is
given in the Appendix.
Please let me know about detected bugs and misprints, as well as eventual general comments.
Your feedback is greatly appreciated!
Vladimir Krasnov Last updated autumn 2012
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Table of content
Preface: ......................................................................................................................................... 2
Definitions and abbreviations ....................................................................................................... 7
Lecture-1........................................................................................................................................... 8
1. Introduction............................................................................................................................... 8
1.1. Superconductivity: Historical overview and basic experimental facts............................ 12
1.2. The Meissner effect.......................................................................................................... 18
1.3. Magnetic flux quantization .............................................................................................. 19
1.4. Josephson effect............................................................................................................... 20
1.5 Development of the theory of superconductivity.............................................................. 20
Lecture-2: Magnetic and thermodynamic properties of superconductors ...................................... 25
Type-I and II superconductors.................................................................................................... 25
2.1 Magnetic properties of type-I superconductors ................................................................ 25
2.2 Intermediate state in type- I superconductors ................................................................... 26
2.3 Magnetic properties of type-II superconductors............................................................... 28
Thermodynamics of superconductors ......................................................................................... 29
2.4 Thermodynamic critical field............................................................................................ 30
2.5 Entropy of type-I superconductors.................................................................................... 31
2.6 Specific heat of superconductors ...................................................................................... 32
Problems to Lecture - 2............................................................................................................... 33
Lecture-3......................................................................................................................................... 34
Two-fluid model and linear electrodynamics of superconductors.............................................. 34
3.1 The London equations....................................................................................................... 34
3.2 London penetration depth ................................................................................................. 35
3.3 Quantum generalization of the 2-nd London equation. .................................................... 37
3.4. Magnetic flux quantization in superconductors............................................................... 37
3.5 Application of London equations for calculation of static magnetic field and current
distributions in superconductors ............................................................................................. 38
3.6. The image method for finding field distributions............................................................ 40
3.7. The short-circuit principle................................................................................................ 42
3.8 Free energy in the London model ..................................................................................... 43
Problems to lecture-3 .................................................................................................................. 44
Lecture 4 ......................................................................................................................................... 45
High frequency properties of superconductors........................................................................... 45
4.1 Kinetic inductance ............................................................................................................ 46
4.2. Normal skin effect............................................................................................................ 48
4.3. High frequency conductivity of superconductors............................................................ 48
4.4. Surface impedance........................................................................................................... 50
4.5. Superconducting transmission line .................................................................................. 52
4.6 Nonlocal Electrodynamics of Superconductors................................................................ 55
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Problems to lecture-4 .................................................................................................................. 56
Lecture 5 ......................................................................................................................................... 57
Ginzburg-Landau Theory............................................................................................................ 57
5.1 Landau’s theory of second order phase transition ............................................................ 57
5.2 Ginzburg-Landau formalism in zero magnetic field......................................................... 58
5.3 Ginzburg-Landau free energy density .............................................................................. 60
5.4 Derivation of Ginzburg-Landau (GL) equations .............................................................. 60
5.5 Gauge invariance of GL theory......................................................................................... 62
5.6 Two characteristic lengths ................................................................................................ 63
5.7 The proximity effect ......................................................................................................... 64
5.8 Relation between the microscopic and the Ginzburg-Landau theory............................... 66
5.9 Energy of a Normal metal-Superconductor interface ....................................................... 67
5.10 Critical current of a thin film .......................................................................................... 70
5.11 Critical field of a thin film .............................................................................................. 71
Problems to lecture 5 .................................................................................................................. 73
Lecture 6 ......................................................................................................................................... 75
Microscopic foundations of superconductivity........................................................................... 75
6.1 Electron - Phonon interaction ........................................................................................... 75
6.2 The ground state of a superconductor............................................................................... 76
6.3 Energy of the ground state. ............................................................................................... 79
6.4 Energy gap in the spectrum of elementary excitations ..................................................... 81
6.5. Density of states of elementary excitations ..................................................................... 82
6.6 The coherence length ........................................................................................................ 82
6.7 Temperature dependence of the energy gap ..................................................................... 83
6.8 Persistent currents ............................................................................................................. 84
Problems to Lecture 6 ................................................................................................................. 86
Lecture-7......................................................................................................................................... 87
Vortices in type-II superconductors............................................................................................ 87
7.1 Introduction....................................................................................................................... 87
7.2 Structure of an isolated Abrikosov vortex ........................................................................ 87
7.3 Vortex energy and the lower critical field Hc1.................................................................. 90
7.4 Vortex interaction ............................................................................................................. 91
7.5 The Upper Critical Field ................................................................................................... 92
7.6 Lorentz force..................................................................................................................... 93
7.7 Magnus force .................................................................................................................... 93
7.8 The viscous damping force ............................................................................................... 94
7.9 Flux-flow resistance.......................................................................................................... 96
7.10 The entropy force............................................................................................................ 97
7.11 Surface superconductivity and the third critical field ..................................................... 97
7.12 Reversible Magnetization of a Type-II Superconductor................................................. 98
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7.13 Low flux density ............................................................................................................. 99
7.14 Intermediate flux densities............................................................................................ 100
Problems to lecture 7 ................................................................................................................ 102
Lecture 8 ....................................................................................................................................... 104
Irreversible properties of type- II superconductors................................................................... 104
8.1. Pinning of vortices. ........................................................................................................ 104
8.2 Surface barrier................................................................................................................. 105
8.3 Magnetic pinning ............................................................................................................ 108
8.4 Surface pinning of the vortex lattice............................................................................... 110
8.5 Core pinning.................................................................................................................... 112
8.6 Critical state in type-II superconductors ......................................................................... 114
8.7 Lattice magnetization...................................................................................................... 120
8.8. Thermally activated flux creep ...................................................................................... 125
8.9 Magnetic relaxation ........................................................................................................ 127
8.10. Irreversibility line......................................................................................................... 128
8.11. Thermal instability....................................................................................................... 129
8.12. Melting of Abrikosov vortex lattice............................................................................. 129
Problems to lecture 8 ................................................................................................................ 130
Lecture 9 ....................................................................................................................................... 132
Josephson effect........................................................................................................................ 132
9.1. Phase coherence and types of weak links ...................................................................... 132
9.2 Tunneling in superconductors......................................................................................... 133
9.3Tunneling characteristics ................................................................................................. 135
9.4 DC and AC Josephson effects......................................................................................... 138
9.5 Gauge-invariant phase difference ................................................................................... 140
9.6 Barrier free energy .......................................................................................................... 140
Problems to lecture 9 ................................................................................................................ 141
Lecture 10 ..................................................................................................................................... 142
Electrodynamics of short Josephson junctions ......................................................................... 142
10.1 Resistively and Capacitively Shunted Junction (RCSJ) model .................................... 142
10.2 Tilted washboard: mechanical analog of the RCSJ model. .......................................... 143
10.3 Current-Voltage characteristics for overdamped junctions. ......................................... 144
10.4 Current-Voltage characteristics for underdamped junctions. ....................................... 145
10.5 Magnetic Field Effects.................................................................................................. 148
10.6 Sine-Gordon equation ................................................................................................... 149
10.7 Josephson penetration depth. ........................................................................................ 149
10.8 Magnetic field dependence of the critical current in a short junction........................... 150
10.9 Josepson plasma waves................................................................................................. 152
Problems to lecture 10 .............................................................................................................. 153
Lecture 11 ..................................................................................................................................... 154
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Fingerprints of the AC-Josephson effect .................................................................................. 154
11.1. Shapiro steps: interaction with external radiation........................................................ 154
11.2 Fiske steps: geometrical resonances ............................................................................. 156
11.3 Thermal and quantum fluctuations of phase................................................................. 157
Problems to lecture 11 .............................................................................................................. 161
Lecture 12 ..................................................................................................................................... 162
Fluxons in Josephson junctions ................................................................................................ 162
12.1 Fluxon structure in the static case................................................................................. 162
12.2 Fluxon energy and the lower critical field .................................................................... 163
12.3 Fluxon dynamics........................................................................................................... 166
12.4 Lorentz contraction of the fluxon ................................................................................. 166
12.5 Velocity matching step ................................................................................................. 168
12.6 Flux-flow characteristics............................................................................................... 170
12.7 Flux-flow oscillator....................................................................................................... 171
12.8 Mechanical analog of the sine-Gordon equation ......................................................... 172
12.9. Soliton interaction........................................................................................................ 173
Problems to lecture 12 .............................................................................................................. 174
Lecture 13 ..................................................................................................................................... 175
Superconducting detectors........................................................................................................ 175
13.1 SQUID-Superconducting Quantum Interference Device ............................................. 175
13.2 The two-junction SQUID (DC SQUID) ....................................................................... 175
13.3 The Single-Junction SQUID (RF SQUID) ................................................................... 178
13.4 Operation of the rf-SQUID ........................................................................................... 179
13.5. Pick-up coils and flux-transformers............................................................................. 182
13.6 Superconducting detectors............................................................................................ 183
13.7 Sensitivity and noise ..................................................................................................... 184
13.8 Incoherent radiation and particle detectors: Bolometers and Calorimeters.................. 185
13.9 Coherent Detection and Generation of Radiation: Mixers, Local Oscillators, and
Integrated Receivers.............................................................................................................. 190
Problems to lecture 13 .............................................................................................................. 196
Conversion from CGS to SI system of units............................................................................. 199
Useful web-links:...................................................................................................................... 201
Books on superconductivity...................................................................................................... 201
References .................................................................................................................................... 202
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Nomenclature
R – resistance [Ohm = Ω ( SI units) =10 9 emu (CGS units)]
ρ –resistivity [Ω cm (SI units) ]
L – inductance [Henry (SI) =10 9 emu (CGS)]
T – temperature (Kelvin =K)
V – voltage [Volt (SI)]
ω - angular frequency (rad/s or just 1/s)
f – frequency (Hz)
Tc – superconducting critical (transition) temperature
Hc – thermodynamic critical field (Tesla=T (SI) = 10 4 Oe (CGS))
Hc1 – the lower critical field (Tesla=T (SI) = 10 4 Oe (CGS))
Hc2 – the upper critical field (Tesla=T (SI) = 10 4 Oe (CGS))
Φ0 – magnetic flux quantum [2.07x10 -15 Weber (SI)]
h – Planks constant
c – the velocity of light in vacuum
e – electron charge
µ0= 4π·10 -7 H/m magnetic permeability of vacuum (used in SI system of units)
HTS – High Temperature Superconductors
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Lecture-1.
1. Introduction
Superconductivity was discovered about a century ago. This was an unexpected discovery,
which was (and still is) notoriously difficult to explain. Understanding of this exotic phenomenon
has put a lot of intellectual and technical challenges for scientists and engineers.
Superconductivity has become a playground for development of new ideas and techniques that
with time went well beyond the original scope and revolutionized many other areas of science,
including quantum physics, condensed matter physics, physics of strongly-correlated electronic
systems, and even particle and astro-physics, as well as inorganic and organic chemistry,
materials science, medical research, e.t.c. Today it is not easy to find a modern area of research
that would not have some link to superconductivity. During the past hundred years of intense
research, superconductivity indicated one of the front edges of modern science, steadily leading to
new surprises and discoveries. Among the most significant latest findings is the discovery of hightemperature
superconductivity in cuprates and magnetic iron-based pnictides. Those
unconventional types of superconductivity remain among the most acute and important unsolved
problems of modern fundamental science.
From the very beginning, physicists were thrilled by the perspectives of possible practical
application of superconductors, first for loss-free power transmission, and later for novel type of
“cryoelectronic” quantum devices, and super-sensitive detectors. If the first type of application is
straightforward: superconductors with zero resistance are ideal current leads, capable of
withstanding extreme current densities 1 ~10 6 -10 7 A/cm 2 , which should be compared to the current
carrying capability “ampacity” 2 of commercial Cu and Al cables, corresponding to the maximum
current density of ~200-300 A/cm 2 . The second type of application is less trivial and is due to
macroscopic phase coherence of superconductors, which allows observation of quantum
mechanical behavior even in large objects and thus allows fabrication of novel quantum electronic
devices using conventional microfabrication techniques.
Yet, the road from ideas to applications of superconductors was quite long and difficult, partly
due to requirement of very low temperatures, which was unacceptable for large-scale applications,
and partly due to tremendous development of semiconducting industry, which dominated (and
still is dominating) microelectronics. However, in recent years the practical importance of
superconductors is rapidly growing, as seen from the market survey in Fig. 1.1.
Fig. 1.1. Superconducting market survey by CONECTUS. Total investment in areas: RTD -
science, research and technological development; NMR- nuclear magnetic resonance; MRI-
Magnetic Resonance Imaging, LTS –low temperature superconductors; HTS – high temperature
superconductors.
Here is the description of the today’s superconducting market by CONECTUS (December
8

2007): “…Historically superconductor technology was first utilized in purely performance-driven
sectors i.e. in science, research and technological development (RTD), and in military
applications. In a next phase, medical applications where competition from non-superconducting
devices is weak, were opened up. Almost all of today’s superconducting products still use LTS
materials. So far, these markets are mostly for magnets ranging from small magnets for university
research to enormous systems for large laboratory facilities. The biggest current market is for
magnets used in medical diagnosis, Magnetic Resonance Imaging (MRI). As can be seen in the
graph, both fields, RTD and MRI, together account for most of today’s overall market…”. Fig.
1.2 shows the example of main applications of conventional, low temperature superconductors
(LTS) in MRI (left panel) and scientific project in Large Hardon Collider 2 (right panel).
Simultaneously, with traditional high-magnet applications, today new electronic applications
are emerging where superconductors are used in telecommunication, super-sensitivity devices and
detectors, high-frequency resonators, mixers, and other cryoelectronic components based on
superconducting tunnel junctions. New large scale applications based on high temperature
superconductors (HTS) are also being developed. Those new, highly cost-competitive commercial
applications are noted in Fig. 1.1 as “new electronic” and “new large scale” applications. From Fig.
1.1 it is seen that those new applications started from about 2003. Several scientific and
technological breakthroughs are staying behind those new applications: First of all, reliable HTS
cables were made, which outperform normal Cu cables by ~150 times 1 , and operate in liquid
nitrogen, which is cheap-enough in production and easy-enough in operation. Secondly, a
dramatic progress in refrigeration techniques has been achieved, which e.g., allowed construction
of cryogen-free systems, that can reach ultra-low temperatures (down to few milliKelvin) without
liquid He. Examples of those new electronic and new large scale applications are shown in Fig. 1.3
and 1.4, respectively.
by B.TenHaken, 3 SCENET school, October 2003
Fig.1.2. Application of low-Tc
superconducting cables for high field
persistent magnets. Bottom-left panel: in
Magnet Resonance Imaging (MRI) for
medical diagnostics. This is the main
industrial application today. Top-right panel
in the research and development area: Large
Hardon Collider (LHC), which in total
contains over 1600 superconducting magnets
weighing up to 27 tonnes each. Approximately
96 tonnes of liquid Helium is needed
to keep them at the operating temperature,
making the LHC the largest cryogenic
facility at liquid helium temperature.
Current research areas in superconductivity, which may lead in future to new applications of
9
superconducting materials include development of super-sensitive sensors of various kind, THz
frequency generators and detectors, metrology applications, development of superconducting
digital electronics, memory elements and super-computers, as well as development of principally
new quantum electronic devices for quantum informatics and quantum computing, or devices
operating with charge or spin of a single electron.
Such a rapid development requires proper education in the area of superconductivity, which is
today offered by many universities.
M.Hofheinz et al., Nature 454 (2008) 310
Fig. 1.3. Examples of new electronic applications of superconductors. Top row: liquid nitrogen
cooled HTS filters for telecommunication provide a dramatic enhancement in performance and
capacity of the telephone line without introduction of new stations. Middle-left: Application of
superconducting SQUID sensors for non-destructive testing of multilayered metallic
constructions. Middle-right: Superconducting Hot Electron Bolometer (HEB) mixer at the
Hershel Space Observatory. Bottom-left: a prototype of a Rapid Single Flux Quantum
microprocessor. Botom-right: Superconducting qubit- the basic element of quantum computer.
10

Fig. 1.4. Examples of new large scale applications using High Temperature Superconductors.
First two rows: Overview of today’s application of HTS cables for high power lines and Fault
Current Limiters at the power stations. Bottom-left: a prototype of 10 MW HTS transformer.
Bottom-right: Application of HTS motors in military ship engines.
11
1.1. Superconductivity: Historical overview and basic experimental facts
In 1908 Heike Kamerlingh-Onnes at Leiden University has successfully liquefied He-4, which
allowed him to reach temperatures down to 1.15K. He was aiming to clarify a controversy, which
existed at that time: how does the electric resistance of pure metals depend on temperature upon
approaching the absolute zero temperature? Kelvin argued that the resistance will increase to
infinity at T→0 because mobile electrons would bound “freeze” to their atoms. Dewar suggested
that the resistance will vanish because atomic vibrations, which is a “hindering factor” for
electron motion, would freeze-out. Matthiessen predicted that the resistance will saturate because
of impurities. Kamerlingh-Onnes observed that the resistance, R(T), saturates at low T, see
Fig.1.5. He attributed this fact to presence of impurities even in very pure gold. Hoping that liquid
mercury would be purer than gold, in 1911 he measured R(T) of mercury and observed that at a
temperature T* in the vicinity of 4 K the resistance of the sample dropped suddenly to zero and
remained unmeasurable at all attainable temperatures below T*, see Fig. 1.5. Importantly, as the
temperature decreased, the resistance disappeared instantly rather than gradually. It was obvious
that the sample had undergone a transition into a novel “superconductive” state characterized by
zero electrical resistance. Within the sensitivity of modern equipment, it can be claimed that the
resistivity of superconductors can be less than 10 -24 Ωcm, which has to be compared to 10 -9 Ωcm
for high-purity copper at 4.2 K. The temperature of the transition from the normal to the
superconducting state is called the critical temperature Tc.
Fig. 1.5. Resistance versus temperature for Pt, Au and Hg. A sudden drop in resistance at
T~4 K is observed for Hg. Data from H.Kamerlingh-Onnes Nobel lecture[3].
Soon after the discovery of superconductivity in mercury, it was found in many other
metals: tin, lead, indium, aluminum, niobium, etc. Many non-superconducting elements become
superconductive at high pressure (above atmospheric). For example, phosphorus appears to be the
Type-I superconducting element with the highest Tc of 14-22 K but at compression pressures of
2.5 Mbar. Fig. 1.6 provides the list of all known superconducting elements at the normal
atmospheric pressure. It is seen that superconductivity is not an exceptional, but a widespread
phenomenon.
12

Fig. 1.6. Superconductivity is a common, widespread phenomenon in a periodic table of elements
*Note: Carbon in usual forms of diamond and graphite is not superconducting. However, doped
diamond and carbon nano-tubes may become superconductive with a Tc of up to 15K.
Since many different materials exhibit superconductivity, the subject is inevitably complicated
by the need to classify and discriminate different factors that may affect superconductivity. Table
1.1 shows a list of properties of (pure) superconducting elements. It can be seen that there is no
simple rule to decide which elements become superconductors. However, the following common
points can be mentioned:
(i) Only metals become superconductors * .
(ii) All the critical temperatures of elements are under 10 K ** .
(iii) Some metals which are good conductors at room temperature: copper, silver and gold, do
not become superconductors at all *** .
(iv) Magnetic metals do not become superconductors **** .
(v) Lattice structure plays role in superconductivity: compare e.g. α and β-phases of Hg in
Table 1.1.
* Pristine HTS cuprates are insulators but become conducting and superconducting upon doping.
** High - Tc cuprate compounds may have critical temperatures in excess of 100 K, as indicated in table 1.2.
*** * Nobel metals have one valence electron per atom, loosely coupled to the lattice, and have a tightly
packed face centered cubic crystal lattice so that lattice vibrations essential for superconductivity are
constrained . The combination of those two factors results in very small electron-phonon interaction.
**** Some exception occur in so-called heavy-fermion compounds. Also pristine cuprate HTS have an anti-
ferromagnetic order.
13
Ele- Tc (K) Cryst. Hc(0) (Oe) Ele- Tc (K) Cryst.
ment
structure
ment
Structure
Al 1.175 FCC 104.9± 0.03 Pa 1.4 TET
Be 0.1125 FCC Pb 7.196±
0.006
Cd 0.6 HEX 28±1 Re 1.697±
0.006
Ga 1.175 ORC 59.2 ± 0.3 Ru 0.49 ±
0.015
Hf 0.2 ORC Sn 3.722±
0.001
Hg (α) 4.47 RHL(fcc) 411 ± 2 Ta 4.47 ±
0.04
Hg (β) 3.949 TET(bcc) 339 Tc 7.8
0.01
±
In 3.72 TET 281.5±2 Th 1.38±0
.02
Ir 0.1125± FCC 16±0.05 Ti 0.40±0
0.001
.04
La (α) 4.88 ± HEX 800 ± 10 Tl 2.38 ±
0.02
0.04
La (β) 6.0 ±
1096, 1600 V 5.40 ±
0.1
0.05
Lu 0.1 HEX

Table 1.2. Critical temperatures of superconducting alloys and compounds.
Fig. 1.7. Crystallographic structure of typical superconductors.
15
Fig. 1.8. History of discovery of superconducting compounds before (left) and after HTS
revolution
Interestingly, impurities may act differently on normal metals and superconductors. In
normal metals they increase the resistance. However, impurities may actually improve
superconducting characteristics. Non-magnetic impurities have only minor effect on the
superconducting critical temperature, as shown in Figure 1.9 for Sn-In alloy. The variation of Tc
with In-impurity content for Sn shows only a minor initial drop at about 1% impurity, and then
changes only slightly with further increase of impurity concentration. However, magnetic
impurities do rapidly destroy superconductivity at typical concentrations of just a few per cent. In
many cases the effect of impurities in an alloy can be described in terms of decreased electronic
mean free path l which enters into the low-temperature resistance of the normal state. Yet,
although nonmagnetic impurities and alloying do not affect the critical temperature much, they do
drastically change superconducting properties, e.g., the current-carrying and magnetic properties.
Shortly after the discovery, it was found that superconductivity can be destroyed not only by
heating the sample, but also by applying a relatively weak magnetic field, called the
thermodynamic critical field, Hc. Table 1.1 gives values of Tc and Hc for several superconducting
elements. Here Hc(0) is the critical field extrapolated to T=0 K. The temperature dependence of Hc
is well described by an empirical formula
Hc(T) = Hc(0) [1- (T/Tc) 2 ]. (1.1)
This dependence is shown in Fig. 1.10 which essentially represents the H-T phase diagram of
the superconducting state. Any point in the H-T plane below the curve corresponds to the
superconducting state.
16

Figure 1.9. Variation with In concentration of critical temperature of Sn-In alloys.
Fig. 1.10. Temperature dependence of the critical field Hc, and an overview of magnetic
properties of superconductors.
17
1.2. The Meissner effect
Are superconductors just ideal conductors, that is, metals with zero resistance? To understand
this let’s consider how an ideal conductor should behave in an external magnetic field that is weak
enough, H Tc the resistivity of the specimen is finite and, there fore, the
magnetic field penetrates into it. After cooling the specimen down through the superconducting
transition, the field remains in it, as illustrated in Fig. 1.11. Note that in the above reasoning we
always referred to the specimen characterized by ρ = 0 as an ideal conductor and not a
superconductor.
Fig. 1.11. Magnetic state of a perfect normal conductors depends on its history.
18

The experiment by W. Meissner and R. Ochsenfeld from 1933 has revealed that
superconductors do not behave as just ideal conductors. It was found that at T < Tc the field inside
a superconducting specimen was always zero (B = 0) in the presence of an external field,
independent of which procedure had been chosen to cool the superconductor through Tc, see Fig.
1.12.
Fig. 1.12. Difference between superconductors and ideal normal conductors: for an ideal
conductor, the magnetic state depends on history, for superconductor - not.
This discovery was very important. Indeed, if B = 0 independent of the specimen’s history, the
zero induction can be treated as an intrinsic property of the superconducting state at H0 < Hc.
Furthermore, as will be explained in the next section, this implies that superconductivity appears
as a result of phase transition. Facilitation very powerful thermodynamic approach for
examination of superconductors.
Thus, the superconducting state obeys the equations:
ρ=0, B = 0. (1.3)
1.3. Magnetic flux quantization
An electric current, induced in a superconducting ring, can persist for an infinitely long time.
Naturally, this does not require a power supply, since there is no power dissipation in the ring.
Such a persistent current can be produced as follows. Let us place the ring at T> Tc in an external
magnetic field so that the magnetic field lines pass through the interior of the ring. Then the ring
is cooled down to a temperature below Tc where the material is superconducting, and the external
magnetic field is switched off. At the first moment after switching off the field, the magnetic flux
through the ring decreases and, according to Faraday’s law of electromagnetic induction, induces
a current in the ring which will be persistent from this moment on. This current prevents a further
decrease of the magnetic flux through the ring, i.e., now that the external field is zero, the current
itself supports the flux through the ring at the initial level. Indeed, if the ring had a finite
resistance R, the flux through the ring would decay during the time of the order of L/R, where L is
the inductance of the ring. In a superconducting ring, since R = 0, it takes the flux infinite time to
decay. This means that the magnetic flux becomes ‘frozen’ and the ring carries a persistent
19
“supercurrent”.
At first sight it may seem that the ‘frozen’ magnetic flux can take on an arbitrary value.
However, in 1961-1962 an important experimental fact was established: the magnetic flux
through a hollow superconducting cylinder may only assume quantized values, equal to integral
multiples of the flux quantumΦ0 = 2.07 x 10 -7 G cm 2 (CGS), given by a combination of
fundamental constants:
Φ0 = hc/2e, [SI: Φ0 = h/2e = 2.07 x 10 -15 Wb ] (1.4)
where h is Planck’s constant, c is the speed of light and e is the electron charge. Physically, the
origin of the magnetic flux quantization is the same as the quantization of electron orbits in atom:
the wavefunction of electrons moving along a closed orbit must contain an integral number of
wavelengths over the length of the orbit.
1.4. Josephson effect
Josephson effect (sometimes referred to as weak superconductivity) provides another
spectacular manifestation of the quantum mechanical nature of the superconducting state. It was
predicted in 1962 by a 22 year old graduate student Brian Josephson 5 and soon verified
experimentally by Ivar Giaever 6 and later by many other researchers. The peculiar history of this
discovery is described in ref. 7 . The term ‘weak superconductivity’ refers to a situation in which
two superconductors are coupled together by a weak link. The weak link can be provided by a
tunnel junction or a short constriction in the cross-section of a thin film. More generally, this can
simply be a weak contact between two superconductors over a small area or other arrangements
where the superconducting contact between two superconductors is somehow ‘weakened’. The
requirement of “weakness” implies that the weak link should not change significantly the wavefunctions
on the two sides, compared to what they had been before the link was established.
There are two Josephson effects to distinguish: (i) stationary (the dc Josephson effect) and (ii)
nonstationary (the ac Josephson effect).
Consider first the dc effect. Let us apply a current through a weak link (or, in other words,
through a Josephson junction). Then, if the current is sufficiently small, it passes through the
weak link without resistance, even if the material of the weak link itself is not superconducting
(for example, if it is an insulator in a tunnel junction). Here we directly come across the most
important property of a superconductor: the coherent behavior of superconducting electrons.
Electrons of the two superconductors, interacting through the weak link, merge into a single
phase-coherent quantum state. The same can be said in a different way. Having penetrated via the
weak link into the second superconductor, the wave function of electrons from the first
superconductor interferes with the ‘local’ electron wave function. As a result, all superconducting
electrons on both sides of the weak link are described by the same wave-function.
The ac Josephson effect is even more remarkable. Let us increase the dc current through the
weak link until a finite voltage appears across the junction. Then, in addition to a dc component,
the voltage V will also have an ac component of angular frequency ω, so that
ħω= 2eV.
The ac-Josephson oscillations lead to electromagnetic wave emission from Josephson
junctions, which was first detected experimentally in 1965.
1.5 Development of the theory of superconductivity
In 1935 London brothers have formulated the first theory, successfully describing
electrodynamic properties of superconductors. The theory was phenomenological, that is, it had
20

introduced two equations, in addition to Maxwell’s equations, governing the electromagnetic field
in a superconductor. These equations provided a correct description of the two basic properties of
superconductors: absolute diamagnetism and zero resistance to a dc current. The London theory
did not attempt to resolve the microscopic mechanism of superconductivity on the level of
electrons, that is, the question: “Why does a superconductor behave according to the London
equations?” remained beyond its scope.
According to the London theory, electrons in a superconductor may be considered as a mixture
of two groups: superconducting electrons and normal electrons. The density of the
superconducting electrons, ns decreases with increasing temperature and eventually becomes zero
at T = Tc. Vice-versa, at T = 0, ns is equal to the total density of conduction electrons. These are
the postulates of the so-called two-fluid model of a superconductor. A flow of superconducting
electrons meets no resistance. Such a current, obviously, cannot generate a constant electric field
in a superconductor because, if it did, it would cause the superconducting electrons to accelerate
infinitely. Therefore, under stationary conditions, that is, without an electric field, the normal
electrons are at rest. In contrast, in the presence of an ac electric field, both the normal and the
superconducting components of the current are finite and the normal current obeys Ohm’s law. In
this framework, a superconductor can be modeled by an equivalent circuit consisting of a normal
resistor and an ideal conductor connected in parallel. The ideal conductor in the circuit must have
a finite inductance to mimic the inertia of the superconducting electrons.
London equations provided a simple and fairly correct description for the behavior of
superconductors in both dc and ac electromagnetic fields. They also helped to understand a
number of general aspects of superconductivity. However, by the end of the 1940s, it was clear
that at least to one question the London theory gave a wrong answer: for an interface between
normal and superconducting regions, the theory predicted negative surface energy σns < 0. This
implied that a superconductor in an external magnetic field could decrease its total energy by
turning into a mixture of alternating normal and superconducting regions. In order to make the
total area of the interface within the superconductor as large as possible, the size of the regions
must be as small as possible. This was supposed to be the case even for a long cylinder in a
longitudinal magnetic field, in contradiction to experimental evidence existing at that time.
Experiments showed separation of the normal and superconducting domains occurred only for
specimens having a nonzero demagnetizing factor (the intermediate state, will be discussed in
§2.2). In addition, domains were rather large (~1 mm, see Fig. 2.2), which could only be the case
if σns > 0, in contradiction with predictions of the London theory.
The above contradiction was reconciled by a theory proposed by V.L. Ginzburg and L.D.
Landau (the G-L theory) which was also phenomenological but took account of quantum effects 8 .
G-L assumed that a single quantum-mechanical wave function Ψ describes all superconducting
electrons. Then the squared amplitude of this function (which is proportional to ns) must be zero
in a normal region, increase smoothly through the normal-superconducting (NS) interface and
finally reach a certain equilibrium value in a superconducting region. Therefore, a gradient of Ψ
must appear at the interface. At the same time, as is well known from quantum mechanics,
⏐∇Ψ⏐ 2 is proportional to the density of the kinetic energy. Thus, by taking into account quantum
effects, we also take into account an additional positive energy stored at the NS interface, which
creates the opportunity to obtain σns > 0. The GL theory will be discussed in detail in Section 5.
Importantly, the GL theory introduced quantum mechanics into the description of
superconductors. It assigned to the entire number of superconducting electrons a wave-function
depending on a single spatial coordinate [recall that, generally speaking, a wave-function of n
electrons in a metal is a function of n coordinates, Ψ (r1, r2,. . . ,rn)]. By doing so, the theory
established the coherent behavior of all superconducting electrons. Indeed, in quantum
mechanics, a single electron in the superconducting state is described by a function Ψ(r). If we
now have ns absolutely identical electrons (where ns, the superconducting electron number
density, is a macroscopically large number) and all these electrons behave coherently, it is clear
that the same wave-function of a single parameter is sufficient to describe all of them. This idea
21
was a breakthrough that enabled the prediction of many beautiful quantum, and at the same time
macroscopic, effects in superconductivity.
The GL theory was built on the basis of Landau’s theory of second-order phase transitions and
is valid only in the vicinity of the critical temperature, that is, within the temperature range Tc-T
0. Those that do have σns > 0 are type-I superconductors. But the majority
of superconducting alloys and chemical compounds show σns < 0; they are type-II
superconductors. For type-II superconductors, there is no Meissner effect at high magnetic field
and the magnetic field does penetrate inside the material, but penetrates in a very unusual way,
namely, in the form of quantized vortex lines – Abrikosov vortices. Superconductivity in these
materials can survive up to very high magnetic fields.
Still, neither London, nor GL theory could answer the question: “What are those
superconducting electrons, whose behavior they were intended to describe?”. The microscopic
origin of superconductivity was finally resolved in 1957, 46 years after discovery, by the work of
J. Bardeen, L. Cooper and J. Schrieffer (the BCS theory) 10 . An important contribution was also
made by N.N. Bogolyubov (1958) who developed a mathematical method now widely used in
studies of superconductivity.
Fig. 1.13. Short summary of the microscopic BCS theory of superconductivity
The decisive step in understanding the microscopic mechanism of superconductivity is due to
L.Cooper (1956) 11 . The essence of his work can be outlined in Fig. 1.13. Consider a normal metal
in the ground state: in k space, all states for non-interacting electrons inside the Fermi sphere are
occupied, while all those outside it are empty. Then an extra pair of electrons is brought in and
placed in the states (k↑) and (-k↓ ), in the vicinity of the Fermi surface (the arrows indicate the
directions of electron spins). It turned out that if, for whatever reason, the two electrons become
attracted to each other, they form a bound state regardless of how weak the attraction is. In real
22

1913
Heike
Kamerlingh
Onnes
“for his
investigations on
the properties of
matter at low
temperatures
which led, inter
alia, to the
production of liquid
helium”
Discovery of
superconductivity
in
Hg
1978
Pyotr
Leonidovich
Kapitsa
"for his basic
inventions and
discoveries in the
area of lowtemperature
physics"
Discovery of
superfluidity
1962
Lev
Davidovich
Landau
“for his pioneering
theories for
condensed matter,
especially liquid
helium"
Superfluidity
Theory of
second order
phase
transitions
Nobel Prices in Physics Related to Superconductivity
1987
J. Georg
Bednorz
John
Bardeen
K.
Alexander
Müller
"for their important break-through in
discovery of the superconductivity
in ceramic Materials"
Discovery of High
Temperature Superconductors
1972
Leon
Neil
Cooper
“for their jointly
developed theory
of
superconductivity,
usually called the
BCS-theory"
Microscopic
theory of
superconductivity
Alexei A.
Abrikosov
Vortices in
type-II superconductors
John
Robert
Schrieffer
2003
Vitaly L.
Ginzburg
"for pioneering
contributions to
the theory of
superconductors
and superfluids”
1973
Brian
David
Josephson
For his theoretical
predictions of the
properties of a
supercurrent
through a tunnel
barrier, in
particular those
phenomena which
are generally
known as the
Josephson effects"
Josephson
effect
(theory)
Anthony J.
Leggett
Ginzburg- Superfluidity in
Landau theory He-3 (theory)
1973
Ivar
Giaever
"for his
experimental
discoveries
regarding
tunneling
phenomena in
superconductor
s"
Josephson
effect
(experiment)
XXXX
????
“for explanation
of High Tc
superconductivity?”
Table 1.3. Nobel price laureates, who have made important contributions in the area of
superconductivity. In addition, four other Nobel laureates should be mentioned, who contributed
to discovery of superfluidity: David M. Lee, Douglas D. Osheroff and Robert C. Richardson
(Nobel price in physics 1996 "for their discovery of superfluidity in helium-3") and Robert
B.Laughlin (Nobel price in physics 1998 “for their discovery of a new form of quantum fluid with
fractionally charged excitations”).
23
space, these electrons form a bound pair - a Cooper pair. What the BCS theory has demonstrated
is that taking into account the interaction between electrons and phonons can, under certain
circumstances, lead to electron-electron attraction. As a result, a part of the electrons form Cooper
pairs. The total spin of a Cooper pair is zero, which means that it represents a Bose particle (that
is, obeys Bose-Einstein statistics). Such particles possess a remarkable property: if the
temperature of a system falls below a certain temperature T they can all gather at the lowest
energy level (in the ground state). Furthermore, the larger the number of the particles that have
accumulated there, the more difficult it is for one of them to leave this state. This process is called
Bose condensation 12 . All the particles in the condensate have the same wave function, depending
on a single spatial coordinate. One can easily understand that the flow of such a condensate must
be superfluid, that is, dissipation-free. Indeed, it is not easy at all for one of the particles to be
scattered by, say, an impurity atom or by any other defect of the crystal lattice. In order to become
scattered, the particle would first have to overcome the ‘resistance’ of the rest of the condensate.
Thus one can briefly describe the phenomenon of superconductivity as follows. At T < Tc there
exists a condensate of the Cooper pairs. This condensate is superfluid. It means that the
dissipation-free electric current can be carried by the Cooper pairs, and the charge of an
elementary current carrier is 2e.
The microscopic theory of superconductivity was elaborated further by L.P. Gorkov
(1958) who developed a method to solve the model BCS problem using Green’s functions. He
applied this method, in particular, to find microscopic interpretations for all phenomenological
parameters of the GL theory as well as to define the theory’s range of validity (see Sect. 6). The
works by Gorkov have completed the development of the Ginzburg-Landau- Abrikosov-Gorkov
theory. Then everything seemed to be settled, and rather well understood. New superconducting
compounds were discovered, but the Tc remained below 25K, see the left panel in Fig. 1.8. In
mid-80 th many believed that electron-phonon mediated superconductivity is impossible above
~30K because this would require a so strong interaction of electrons with ions, that it would lead
to instability of the crystal lattice, which would destroy metallicity. Luckily such predictions
appeared to be incorrect and during the “high Tc revolution”, in a few years after 1986 when J.
Bednorz and K.A. Muller 4 discovered the first cuprate superconductor (LaBaCuO4, Tc~ 40 K), the
superconducting critical temperature made a dramatic jump, as shown in the right panel of Fig.
1.8. Subsequently, materials have been found that raise Tc to temperatures of up to ~ 138 K
(HgBa2Ca2Cu3O8). Many believe that the classic BCS theory is unable to account for many of the
properties of HTSC materials. The electron-phonon mechanism became questionable. New
mechanisms - such as magnetic-mediated pairing and pairing via quantum fluctuation near some
quantum phase transition were proposed. At present the question, why HTSC have such high Tc,
is still unanswered and remains one of the major challenges in modern condensed matter physics.
The overview of progress with understanding HTSC will be given in lecture-15.
24