Chapter 10: Superconductivity

References:

1. C. Kittel ; Introduction to solid state physics

2. M. Tinkham: Introduction to superconductivity
3. Paul Hansma, Tunneling Spectroscopy
 Table of Contents
(1)

(2)

(3)
Periodic Table of Superconductivity

M. Debessai et al., J. Phys.: Conf. Series 215, 012034 (2010).

 Experimental Survey of
Superconductivity Phenomenon
 Helium Liquefaction in 1908

July 10, 1908

Heike Kamerlingh Onnes

Nobel Prize, 1913

“Door meten tot weten”

(Knowledge through measurement)

© Leiden Institute of Physics

 WHAT IS A SUPERCONDUCTOR?
1. Zero resistance

2. Complete expulsion of magnetic flux

MH7699A.18
 Discovery of Superconductivity in 1911

April 8, 1911

Tc
Resistance()

Heike Kamerlingh Onnes

Nobel Prize, 1913

“Door meten tot weten”

(Knowledge through measurement)

Temperature (K)
H. Kamerlingh Onnes, Commun. Phys. Lab. Univ. Leiden. Suppl. 29 (Nov. 1911).
 SUPERCONDUCTIVITY

Type of material What happens in a wire? Result

Conductor Collisions cause

dissipation (heat)
Electrons flow easily
(like water through a
garden hose)
No current flow
Insulator at all

Electrons are tightly bound no flow

(like a hose plugged with cement)

Superconductor No collisions
No dissipation
Electrons bind into pairs and No heat
cannot collide No resistance
(a frictionless hose)
HOW SMALL IS THE RESISTANCE?

Copper Cylinder
1) Induce current
2) Current decays in about 1/1000 second

Superconducting Cylinder
1) Induce current
2) Current does not decay
(less than 0.1% in a year)
so, resistance is smaller than copper
1000 years
by ────────────
1/1000 second
i.e., at least 1 trillion times!
 The Meissner Effect in 1933

Perfect diamagnetism Walther Meißner Robert Ochsenfeld

B=0

© PTB Berlin Institute

 Meissner Effect
M 1
B = Ba + 4𝜋M = 0 ; =－ Eq.(1)
Ba 4𝜋
Perfect Diamagnetism
The magnetic properties cannot be accounted for by the
assumption that a superconductor is a normal conductor with zero
electrical resistivity.
The result B = 0 cannot be derived from the characterization of
a super-conductor as a medium of zero resistivity.
From Ohm’s law, E = ρj, we see that if the resistivity ρ goes to
zero, while j is held finite, then E must be zero. By a Maxwell
equation dB/dt is proportional to curl E, so that zero resistivity
implies dB/dt = 0. This argument is not entirely transparent, but the
result predicts that the flux through the metal cannot change on
cooling through the transition. The Meissner effect contradicts
this result, and suggests that perfect diamagnetism is an
essential property of the superconducting state.
Type I superconductor Type II superconductor

Perfect Diamagnetism
 Basic Properties of Superconductors

Zero electrical resistance + Meissner effect

Resistance()

B=0

Temperature (K)
 Type I & II Superconductors

Type I : Al, Pb… Type II : Nb, NbTi, Nb3Sn and HTSC Lev V. Shubnikov

Vortex

found type-II SC in
Pb-Bi alloy in 1935.

J. N. Rjabinin, L.W. Schubnikow, Physikalische Zeitschrift der Sowjetunion 7, 122 (1935)

 Superconducting Vortices in type II SC

Decoration image of vortex lattice Alexei A. Abrikosov

Nobel Prize 2003



U. Essmann and H. Trauble, Physics Letters 24A, 526 (1967)

 radius

A. A. Abrikosov, Doklady Akademii Nauk SSSR 86, 489 (1952)

A. A. Abrikosov, Sov. Phys. JETP 5,1174 (1957)
Why Superconductivity is so fascinating ?

 Fundamental SC mechanism

 Novel collective phenomenon at low temp

 Applications

Bulk: - Persistent current, power storage

- Magnetic levitation
- High field magnet, MRI

Electronics:
- SQUID magnetometer
- Josephson junction electronics
 POSSIBLE IMPACT OF
SUPERCONDUCTIVITY
● Energy
- Superconductivity generators & motors
- Power transmission & distribution
- Energy storage systems
- Magnets for fusion power
- Magnets for magneto-hydrodynamic power

● Transportation
- Magnets for levitated trains
- Electro-magnetic powered ships
- Magnets for automobiles

● Health care
- Magnetic resonance imaging
MH7699A.11
 Normal Metallic State
Electrons in wave-like states in momentum-space (k-space)

Free electron gas Fermi liquid 𝑘

ky ky E

kx kx

kz kz
Fermi surface k

𝑝 = ℏ𝑘 = ℎ/𝜆

ℏ2 𝑘 2
𝐸=
2𝑚
 BCS Theory in 1957
for Low Tc Superconductivity

Microscopic theory for SC

Cooper Pairs

John Bardeen Leon Cooper Robert Schrieffer

Exchange boson:
Lattice Vibration Mode
Nobel Prize 1972

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

 Superconducting Ground State

Normal state Superconducting

Cooper Pairs
ground state

ky ky

Δ
𝑘

kx kx

kz Exchange boson: kz
−𝑘 Lattice Vibration Mode

• Spin singlet
• L=0; S=0
• Binding energy: Δ
 Fundamental Mechanism
The superconducting state is an ordered state of the conduction electrons of
the metal.

Electron-Phonon Coupling

Cooper Pair formed by two electrons k, and –k

with opposite spins near the Fermi level, as
coupled through phonons of the lattice
The nature and origin of the ordering was explained by Bardeen, Cooper, and
Schrieffer.3
BCS Theory, 1957
J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162 (1957); 108, 1175 (1957).
The Discovery of Superconductivity
• Early 90’s -- elemental SP metals like
Hg, Pb, Al, Sn, Ga, etc.
• Middle 90’s -- transitional metals, alloys, and
compounds like Nb, NbN, Nb3Sn, etc.
• Late 90’s -- perovskite oxides

A-15
B1
HTSC
 A-15 compound A3B, with Tc = 15-23 K
In the so called β–W structure
With three perpendicular linear chains of A atoms on the cubic face,
and B atoms are at body centered cubic site,
With the presence of a sharp peak of N(E) at EF

1973 discovery of Nb3Ge, 23K !

how about Nb3Si ??
 Low temperature Superconductors

-- Mediated by Electron phonon coupling

-- McMillian formula for Tc

 : electron phonon coupling constant

* : Coulomb repulsion of electrons

  N(0) < I2 >/ 2

Are electrons or phonons more important?

 The Phonon Spectrum of the low Tc
A-15 compound Nb3Al

Soft Phonons
 History of Conventional SC

150

125

100

BCS in 1957
TC(K)

75

50

Nb3Al1-xGex
Nb3Sn
25 NbN
Pb Nb
Hg
4.2K
0
1900 1920 1940 1960 1980 2000 2020
Year
 History of Conventional SC

150

125

100

BCS in 1957
TC(K)

75

50 ~Year of 2915
To reach 300K
Nb3Al1-xGex
Nb3Sn
25 NbN
Pb Nb
Hg
4.2K
0
1900 1920 1940 1960 1980 2000 2020
Year
 Can we raise the Tc higher
than 30K?

Are we reaching the limitation

of the BCS Theory ?
 Matthias’s Rules for Searching High TC SC

Bernd Matthias
1. Stay away from insulators; transition metals are

better.

2. There are favorable electron/atom ratios.

3. High symmetry is good; cubic symmetry is best.

4. Stay away from Oxygen

5. Stay away from magnetism

6. Stay away from theorists.

W. E. Pickett , Physica B 296, 112 (2001)

I. I. Mazin, Nature 464, 183 (2010)
A legacy of Superconductivity
Ted H. Geballe

Stanford, April, 2015

The Beginning of Unconventional SC: Heavy Fermion SC

Enormous effective mass of their

charge carriers. This is achieved by a
sharp spike in the DOS at the Fermi
surface, to as much as 1000 times the
density of states in Cu.
Frank Steglich
© Max Planck Institute

F. Steglich et al., Phys. Rev. Lett. 43, 1892 (1979)

Breakthrough in late 1986
By Bednorz and Muller

Start the HTSC Era !

 Discovery of High Tc Cuprates

Z. Phys. B – Condensed Matter 64,189 (1986)

La2-xBaxCuO4 , Tc=30K

J. Georg Bednorz K. Alex Müller

Nobel Prize 1987

 Discovery of High Tc Cuprates

Tc > 77K ! YBa2Cu3O7-, Tc~93K

朱經武 吳茂昆 CuO chain

BaO

CuO plane

M. K. Wu et al., PRL 58, 908 (1987)

High Temperature Superconductor YBa2Cu3O7
(90K)

CuO2 plane 2-D

Cu-O chain 1-D

Invention of Oxide Molecular Beam Epitaxy

For HTSC Single Crystal Films.
 Woodstock of Physics - March Meeting 1987

“The stores and the bars were all ‘Physicists welcome,’ ” said Paul M. Grant, who
headed the superconductivity research at I.B.M.’s Almaden Research Center in
San Jose. He recalled a discotheque in Chelsea with a long line of people waiting
to get in. “The bouncers took anybody that had a physical society badge on to
the front,” Dr. Grant recalled, “and we got in gratis. Can you imagine what a
culture shift? We had a hell of a good time.” – NY Times

© American Institute of Physics

 La2-xBaxCuO4
Perovskite oxide structure
 At small x cation doping,
Antiferromagnetic Mott
Insulator
 For SC state, the Tc is
maximum at x = 0.15
Cu La/Ba
O

Z. Phys. Rev. B 64 189 (1986)

 La2-xBaxCuO4

Each Ba atom substituted

for the captures, and
electron from CuO2 plane
leaving p holes per unit cell

Ba
Cu
O

Z. Phys. Rev. B 64 189 (1986)

 High Tc Cuprate Superconductors (CuSC)

CuO2
CuO2 Ca
CuO2
Y Sr
Bi
Ba O
La(Sr)
CuO

La2-xSrxCuO4 YBa2Cu3Oy Bi2Sr2CaCu2Oy

(LSCO) (YBCO) (Bi2212 or BSCCO)
(Tcmax ~ 40 K) (Tcmax ~ 93 K) (Tcmax ~ 95 K)
 History of Superconductors

150
HgBaCaCuO
125 TlSrBaCuO

Bi2Sr2CaCu2O9
100
YBa2Cu3O7

77K
TC(K)

75

50
La2-xBaxCuO4
Nb3Al1-xGex
Nb3Sn
25 NbN
Pb Nb
Hg
4.2K
0
1900 1920 1940 1960 1980 2000 2020
Year
 Honorable Mention : MgB2 in 2001
Tc=39K
Two superconducting gaps
Strong sp2 bonding and hybridization
Jun Akimitsu E2g phonon and  bond coupling leads to high Tc
秋光純
-bond 𝑝𝑥
B

Mg -bond 𝑝𝑦

© 青山学院大学

-bond 𝑝𝑧

J. Nagamatsu et al., Nature 410, 63 (2001)

Amy Liu et al., PRL 87, 087005 (2001)
H.J. Choi et al., Nature 418, 758 (2002)
The Discovery of Fe-based Superconductors (FeSC) in 2006

2006 : LaFeP(O1-xFx): TC~5K

2007 : LaNiPO: TC~3K
2008 : LaFeAs(O1-xFx) , TC~26K

Hideo Hosono

Y. Kamihara et al., JACS. 128, 10012 (2006)

T. Watanabe et al., JACS. 46, 7719 (2007)
Y. Kamihara et al., JACS. 130, 3296 (2008)
History of Conventional and High TC Superconductors

150
HgBaCaCuO
125 TlSrBaCuO

Bi2Sr2CaCu2O9
100
YBa2Cu3O7
FeSe/STO 77K
TC(K)

75

SmFeAsO
50
La2-xBaxCuO4 FeSe at 8.9GPa
Nb3Al1-xGex
Nb3Sn LaOFeAs
25 NbN
Pb Nb
Hg FeSe
4.2K
0
1900 1920 1940 1960 1980 2000 2020
Year
 Honorable Mention : H3S in 2015

TC=203K under High Pressure

Likely H-rich H3S
Conventional BCS superconductor ?
Mikhail Eremets

© Max-Planck-Institut für Chemie A. P. Drozdov et al., Nature 525, 73 (2015)

 The results are the work of Mikhail Eremets, Alexander Drozdov and
their colleagues at the Max Planck Institute for Chemistry in Mainz,
Germany in 2015. They find that when they subject samples of
hydrogen sulfide to extremely high pressures — around 1.5 million
atmospheres (150 Gigapascals) — and cool them below 203 K, the
samples display the classic hallmarks of superconductivity: zero
electrical resistance and the Meissner effect.

 Other hydrogen compounds may be good candidates for high Tc too.

For instance, compounds that pair hydrogen with Pt, K, Se, Te,
instead of sulfur.

 Zhang in Dallas and Yugui Yao of the Beijing Institute of Technology in

China predict that substituting 7.5% of the sulfur atoms in hydrogen
sulfide with phosphorus, and upping the pressure to 2.5 million
atmospheres (250 GPa) could raise the superconducting transition
temperature all the way to 280 K, above water's freezing point.
Will all non magnetic metal become SC at low T?
(I) Destruction of Superconductivity by Magnetic Impurities
It is important to eliminate from the specimen even trace
quantities of foreign paramagnetic elements
(II) Destruction of Superconductivity by Magnetic fields
At the critical temperature the critical field is zero: Hc(Tc)=0
Type I superconductor Type II superconductor

Perfect Diamagnetism
Type II Superconductors
1. A good type I superconductor excludes a magnetic field until
superconductivity is destroyed suddenly, and then the field penetrates
completely.
2. (a) A good type II superconductor excludes the field completely up to a
field Hc1.
(b) Above Hc1 the field is partially excluded, but the specimen remains
electrically superconducting.
(c) At a much higher field, Hc2, the flux penetrates completely and
superconductivity vanishes.
(d) An outer surface layer of the specimen may remain superconducting
up to a still higher field Hc3.
3. An important difference in a type I and a type II superconductor is in the
mean free path of the conduction electrons in the normal state. are type I,
with κ < 1, will be type II. is the situation when κ = λ / ξ > 1.
1. A superconductor is type I if the surface energy is always positive as the
magnetic field is increased, For H < Hc
2. And type II if the surface energy becomes negative as the magnetic field
is increased. For Hc1 < H < Hc2
The free energy of a bulk superconductor is increased when the magnetic
field is expelled. However, a parallel field can penetrate a very thin film
nearly uniformly (Fig. 17), only a part of the flux is expelled, and the energy
of the superconducting film will increase only slowly as the external magnetic
field is increased.

S N S N S
 Ginsburg Landau
Parameter

k << 1 k>> 1
Type I Type II
Vortex State
In such mixed state, called the vortex state, the external magnetic field
will penetrate the thin normal regions uniformly, and the field will also
penetrate somewhat into the surrounding superconducting materials
Normal Core
of Vortex


k=/>1
 The term vortex state describes the circulation of superconducting currents in
vortices throughout the bulk specimen,

Flux lattice Abrikosov triangular

at 0.2K of NbSe2 lattice as imaged by
LT-STM, H. Hess et al

The vortex state is stable when the penetration of the applied field into the
superconducting material causes the surface energy become negative. A type II
superconductor is characterized bv a vortex state stable over a certain range of
magnetic field strength; namely, between Hc1 and Hc2.
 Vortex Imaging of NbSe2 by LT-STM

Harald F. Hess 2H-NbSe2 : Tc = 7.1 K, TCDW = 29 K

© www.janelia.org

Abrikosov triangular lattice

H. F. Hess et al., PRL 62, 214 (1989).

H. F. Hess et al., PRL 64, 2711 (1990).
Doping Pb with some In

Type I SC becomes type II SC

 Measured by pulsed magnetic field
Chevrel Phase (Ternary Sulfides)

Hc2

Hc2 vs T in
A-15 compound

T
 Entropy S vs T for Aluminum

The small entropy change must mean that only a small fraction (of the order
of 10-4) of the conduction electrons participate in the transition to the ordered
superconducting state.
 Free energy vs T for Aluminum

dFN/ dT = dFS/ dT at TC
 FN =  FS at TC
Zero latent heat,
2nd order phase transition

So that the phase transition is second order (there is no latent heat of transition at T c ).
heat capacity of an electron gas is
1
Cel = π2 D(ϵF) kB2 T (34)
3
D(ϵF) = 3N/2ϵF = 3N/2 kBTF (35)
1
Cel = π2 NkBT/TF. (36) Compare with CV = 2NkBT/TF
2

TF is called the Fermi temperature, where EF = kBTF

K metal

1
γ = π2 NkBT/TF Since ϵF  TF  1/m ∴γm (See Eq. 17)
2

At temperatures much below both the Debye temperature and the Fermi
temperature, the heat capacity of metals may be written as the sum of electron
and phonon contributions: C = γT + AT 3
C/T = γ + AT 2 (37)
γ , called the Sommerfeld parameter At low T, the electronic term dominates.
 Heat Capacity of Ga at low T

Discontinuous change of
C at Tc, C/ Tc =1.43

Electronic part of heat capacity in SC state: Ces/ γTc  a exp (-bTc /T)
Proportional to -1/T, suggestive of excitation of electrons across an energy gap.
 Evidence for Energy Gap in 1953

Another motivation for the BCS theory of superconductivity.

Linear

A. Brown, M. W. Zemansky, and H. A. Boorse, Phys. Rev. 92, 52 (1953)

B. B. Goodman, Proc. Phys. Soc. (London) A66, 217 (1953)
 Energy Gap

In a superconductor the important interaction is the electron-electron

interaction via phonons, which orders the electron in the k space with
respect to the Fermi gas of electrons.

The exponential factor in the electron heat capacity of a superconductor

Is found to be –Eg/2kBT
Ces = γTc exp(-1.76 Tc/T)

The transition in zero magnetic field from the superconducting state to

the normal state is observed to be a second-order phase transition.

Energy Gap of superconductors in Table 3

Eg(0)/kBTc = 3.52 Weak electron-phonon coupling

Eg(0)/kBTc > 3.52 Strong electron-phonon coupling

= 2
 Eg ~ 10-4 F
1-5 meV 3-10 eV

(T)/(0)= (1- T/Tc)1/2

Mean field theory

Sn
Ta
Nb

Eg(T) as the order parameter,

goes smoothly to zero at Tc
-- second order phase transition
 Isotope Effect in 1950
• Lattice vibration is a part of the SC process.
• A crucial step to a microscopic theory.

Emanuel Maxwell Bernard Serin & Charles Reynolds

𝑇𝑐 ∝ 1 𝑀
Tc

© MIT

© Rutgers University
Hg

Average Mass Number

Emanuel Maxwell, Phys. Rev. 78, 477 (1950)

C.A. Reynolds et al., Phys. Rev. 78, 487 (1950)
 Isotope Effect

α~ 0.5

    M-1/2
3. The penetration depth and the coherence length emerge as natural
consequence of the BCS theory. The London equation is obtained for
magnetic fields that vary slowly in space. Thus, the central phenomenon in
superconductivity, the Meissner effect, is obtained in a natural way.

penetration depth (λ) ; coherence length (ξ)

4. The electron density of orbitals D(EF) of one spin at the Fermi level, and
the electron –lattice interaction U. For UD(EF) << 1, the BCS theory predicts:

2∆ /kBTc = 3.52
Where  is the Debye temperature, and U is an attractive interaction
(electron-phonon interaction).

For dirty metal (a poor conductor) → ρ(300)↑, U↑, Tc↑ (but a good SC)

5. Magnetic flux through a superconducting ring is quantized and the effective

unit of charge is 2e rather than e.
Evidence of pairing of electrons
 Perfect Conductor vs Superconductor
Perfect Conductor SC 完美導體
Perfect Conductor 超導體
SC

Apply B
Cooled

T<Tc
Cooled T<Tc
Apply B

B0 B0
 Vortex-Current Interaction

• Lorentz force on JS due to the interaction

between JS and B.

• Vortex motion implies that the vortex is subject

to a power input per unit volume of vortex of
characteristic radius rB

Magnus force
Magnus force

• Vortex motion leads to dissipation! R0 !

• Vortex pinning is crucial for applications.

drag force
 Quantum Levitation

Magnetic flux pinning is key.

Unstable for type I superconductors.

Superconductor

S
N
Magnet S

S N S

© Quantum Experience ltd. © NHMFL

A legacy of Superconductivity

Bob Hammond
 30K 40K 90K

120K 80K 40K