1. Superconductivity Introduction Thermal properties Magnetic properties
London theory of the Meissner effect
Microscopic (BCS) theory
Flux quantization
Quantum tunneling
M.C. Chang
Dept of Phys

2. A brief history of low temperature (Ref: 絕對零度的探索)
1800 Charles and Gay-Lusac (from P-T relationship) proposed that the lowest temperature is -273 C (= 0 K)
1877 Cailletet and Pictet liquified Oxygen (-183 C or 90 K)
soon after, Nitrogen (77 K) is liquified
1898 Dewar liquified Hydrogen (20 K)
1908 Onnes liquified Helium (4.2 K)
G. Amontons 1700
1911 Onnes measured the resistance of metal at such a low T. To remove residual resistance, he chose mercury. Near 4 K, the resistance drops to 0. ρ T Au Hg ρR ρR
1913

3. Is the resistivity of a superconductor really zero or just very small?
A two-point probe measurement will not work because the the contact resistance and the resistance of the wires dominates everything.
A four point probe measurement can be tried but it does not work either: one is limited by the smallest voltage one can measure.

4. Is the resistivity very small or really zero?
open S2 and close S1: there is current in SC coil.
close S2 and open S1: the current in SC coil remains the same for several hours.
similar experiment years later detected no decay of current for 2 years ! (Onnes)
Such a current can be a powerful source of magnetic field (however, see later discussion on critical current).
Switch 1
Liquid He
Switch 2
SC coil
compass

5. He+ ions, Chromium as a thin film, and Platinum as a compacted powder
1.14K
3.72K
7.19K
3.40K
1.09K
2.39K
4.15K
0.39K
5.38K
0.55K
0.12K
9.50K
4.48K
0.92K
7.77K
0.01K
1.4K
0.66K
0.14K
0.88K
0.56K
0.51K
0.03K
1.37K
4.88K
0.20K
0.60K
0.0003K
Tc's given are for bulk, except for Palladium, which has been irradiated with
He+ ions, Chromium as a thin film, and Platinum as a compacted powder

6. Superconductivity in alloys and oxides
Applications of superconductor
powerful magnet
MRI, LHC...
magnetic levitation
SQUID (超導量子干涉儀)
detect tiny magnetic field
quantum bits
lossless powerline …
1910
1930
1950
1970
1990 20 40 60 80
100
120
140
160
Superconducting transition temperature (K) Hg Pb Nb
NbC
NbN
V3Si
Nb3Sn
Nb3Ge
(LaBa)CuO
YBa2Cu3O7
BiCaSrCuO
TlBaCaCuO
HgBa2Ca2Cu3O9
(under pressure)
Liquid Nitrogen
temperature (77K)
Bednorz Muller 1987
From Cywinski’s lecture note

7. Introduction
Thermal properties
Magnetic properties
London theory of the Meissner effect
Microscopic (BCS) theory
Flux quantization
Quantum tunneling

8. Thermal properties of SC: specific heat
The exponential dependence with T is called “activation” behavior and implies the existence of an energy gap above Fermi surface.
Δ ~ meV (10-4~-5 EF )

9. Connection between energy gap and Tc
Temperature dependence of Δ (obtained from Tunneling)
Universal behavior of Δ(T)
‘s scale with different Tc’s
2(0) ~ 3.5 kBTc

10. free energy Entropy Al Al FN-FS
Less entropy in SC state: more ordering
free energy Al
FN-FS
= Condensation energy ～10-8 eV per electron!
2nd order phase transition

11. More evidences of energy gap
Electron tunneling
EM wave absorption
2 suggests excitations created in “e-h” pairs

12. sc Magnetic property of the superconductor normal
Superconductivity is destroyed by a strong magnetic field. Hc for metal is of the order of 0.1 Tesla or less.
Temperature dependence of Hc(T)
All curves can be collapsed onto
a similar curve after re-scaling.
normal sc

13. Critical currents (no applied field)
Radius, a
Magnetic field Hi so
The critical current density of a long thin wire is therefore
(thinner wire has larger Jc)
From Cywinski’s lecture note
jc~108A/cm2 for Hc=500 Oe, a=500 A
Jc has a similar temperature dependence as Hc, and Tc is similarly lowered as J increases.
Cross-section through a niobium–tin cable
Phys World, Apr 2011

14. Meissner effect (Meissner and Ochsenfeld, 1933)
A SC is more than a perfect conductor
Lenz law
not only dB/dt=0
but also B=0!
Perfect diamagnetism
different
same

15. Meissner effect for a hollow cylinder
Apply a field, then lower below Tc:
There are surface currents on both inside and outside. no field inside the ring.
Remove the field:
surface current on the outside disappears;
surface current on the inside persists
Magnetic flux is trapped!
Q: what if we reduce T first, apply a field, then remove the field? (Alan Portis, Sec 8.7)

16. Superconducting alloy: type II SC
partial exclusion and remains superconducting at high B (1935)
(also called intermediate/mixed/vortex/Shubnikov state)
STM image NbSe2, 1T, 1.8K
pure In
HC2 is of the order of 10~100 Tesla (called hard, or type II, superconductor)

17. Comparison between type I and type II superconductors
Hc2
B=H+4πM
Lead + (A) 0%, (B) 2.08%, (C) 8.23%, (D) 20.4% Indium
Areas below the curves (=condensation energy) remain the same!
Condensation energy (for type I)
(Magnetic energy density)

18. Introduction
Thermal properties
Magnetic properties
London theory of the Meissner effect
Microscopic (BCS) theory
Flux quantization
Quantum tunneling

19. London theory of the Meissner effect (Fritz London and Heinz London, 1934)
Two-fluid model: nn Tc
Carrier density T ns
Superfluid density ns  = 
Normal fluid density nn
Assume
like free charges
London proposed
where
It can be shown that ▽ψ=0 for simply connected sample (See Schrieffer)

20. Temperature dependence of λL
Penetration length λL
Outside the SC, B=B(x) z
(expulsion of magnetic field)
Temperature dependence of λL
tin
Higher T, smaller nS
Predicted λL(0)=340 A, measured 510 A
also decays

21. Coherence length ξ0 (Pippard, 1939)x ns
surface
superconductor 0
In fact, ns cannot remain uniform near a surface. The length it takes for ns to drop from full value to 0 is called 0
Microscopically it’s related to the range of the Cooper pair.
The pair wave function (with range 0) is a superposition of one-electron states with energies within Δ of EF (A+M, p.742).
Energy uncertainty of a Cooper pair
Therefore, the spatial range of the variation of nS
0 ~ 1 μm >> λ for type I SC

22. For type-I SC, we need to use (see Tinkham, App.3 for details)
local in q-space
If Ay(x) is a constant within a thickness ξ0 from the surface, then it reduces to the “local” form
But in fact, it is nonzero only within λ. So

23. Penetration depth, correlation length, and surface energy
Type I superconductivity
Type II superconductivity
0 > , surface energy is positive
0 < , surface energy is negative
From Cywinski’s lecture note
smaller λ, cost more energy to expel the magnetic field.
smaller ξ0, get more “negative” condensation energy.
When ξ0 >> λ (type I), there is a net positive surface energy. Difficult to create an interface.
When ξ0 << λ (type II), the surface energy is negative. Interface may spontaneously appear.

24. Vortex state of type II superconductor (Abrikosov, 1957)
2003
Vortex state of type II superconductor (Abrikosov, 1957)
Normal core
the magnetic flux  in a vortex is always quantized (discussed later).
the vortices repel each other slightly.
the vortices prefer to form a triangular lattice (Abrikosov lattice).
isc
the vortices can move and dissipate energy (unless pinned by impurity ← Flux pinning)
Hc2
Hc1 H -M
From Cywinski’s lecture note

25. Estimation of Hc1 and Hc2 (type II)
Near Hc1, there begins with a single vortex with flux quantum 0, therefore
Near Hc2, vortex are as closely packed as the coherence length allows, therefore
Typical values, for Nb3Sn,
ξ0 ～ 34 A, λL ～ 1600 A

26. Origin of superconductivity?
Metal X can (cannot) superconduct because its atoms can (cannot) superconduct?
Neither Au nor Bi is superconductor, but alloy Au2Bi is!
White tin can, grey tin cannot! (the only difference is lattice structure)
good normal conductors (Cu, Ag, Au) are bad superconductor;
bad normal conductors are good superconductors, why?
What leads to the superconducting gap?
Failed attempts: polaron, CDW...
mercury
Isotope effect (1950):
It is found that Tc =const × M-α
α～ 1/2 for different materials
lattice vibration?

27. Brief history of the theories of superconductors
1935 London: superconductivity is a quantum phenomenon on a macroscopic scale. There is a “rigid” (due to the energy gap) superconducting wave function Ψ.
1950
Frohlich: electron-phonon interaction maybe crucial.
Reynolds et al, Maxwell: isotope effect
Ginzburg-Landau theory: ρS can be varied in space Suggested the connection
2003
and wrote down the eq. for order parameter Ψ(r) (App. I)
Ref: 1972 Nobel lectures by Bardeen, Cooper, and Schrieffer
1956 Cooper pair: attractive interaction between electrons (with the help of crystal vibrations) near the FS forms a bound state.
1957 Bardeen, Cooper, Schrieffer: BCS theory
Microscopic wave function for the condensation of Cooper pairs.
1972

28. +++ e Dynamic electron-lattice interaction → Cooper pair ～ 1 μm
Effective attractive interaction between 2 electrons (sometimes called overscreening)
(range of a Cooper pair; coherence length)

29. Cooper pair, and BCS prediction
2 electrons with opposite momenta (p↑,-p↓) can form a bound state with binding energy (the spin is opposite by Pauli principle)
2(0) ~ 3.5 kBTc
Fraction of electrons involved ～ kTc/EF ～ 10-4
Average spacing between condensate electrons ～ 10 nm
Therefore, within the volume occupied by the Cooper pair, there are approximately (1μm/10 nm)3 ～ 106 other pairs.
These pairs (similar to bosons) are highly correlated and form a macroscopic condensate state with (BCS result)
(～upper limit of Tc)

30. Energy gap and Density of states
~ O(1) meV
Electrons within kTC of the FS have their energy lowered by the order of kTC during the condensation.
On the average, energy difference (due to SC transition) per electron is

31. Families of superconductors
Cuperate
(iron-based)
T.C. Ozawa 2008
Conventional BCS
Heavy fermion
F. Steglich 1978
wiki

32. Introduction
Thermal properties
Magnetic properties
London theory of the Meissner effect
Microscopic (BCS) theory
Flux quantization
Quantum tunneling (Josephson effect, SQUID)

33. Flux quantization in a superconducting ring
(F. London 1948 with a factor of 2 error, Byers and Yang, also Brenig, 1961)
Current density operator
SC, in the presence of B
London eq. with
Inside a ring
0 ～ the flux of the Earth's magnetic field through a human red blood cell (~ 7 microns)

34. Single particle tunneling (Giaever, 1960)
dI/dV
20-30 A thick
SIS
For T>0 (Tinkham, p.77)
Ref: Giaever’s 1973 Nobel prize lecture

35. NIS junction as a refrigerator
Below 300 mk ～ boiling point of He3
NIS junction as a refrigerator
Appl. Phys. Lett. 102, (2013)

36. Josephson effect (Cooper pair tunneling) Josephson, 1962
1) DC effect:
There is a DC current through SIS in the absence of voltage.
1973
Giaever tunneling
Josephson tunneling

37. Science 290, 773 (2000)

38. 2) AC Josephson effect
Apply a DC voltage, then there is a rf current oscillation.
(see Kittel, p.290 for an alternative derivation)
An AC supercurrent of Cooper pairs with freq. ν=2eV/h, a weak microwave is generated.
ν can be measured very accurately, so tiny ΔV as small as V can be detected.
Also, since V can be measured with accuracy about 1 part in 1010, so 2e/h can be measured accurately.
JJ-based voltage standard (1990): V ≡ the voltage that produces ν=483,597.9 GHz (exact)
advantage: independent of material, lab, time (similar to the quantum Hall standard).

39. 3) DC+AC: Apply a DC+ rf voltage, then there is a DC current
Another way of providing a voltage standard
Shapiro steps (1963) given I, measure V

40. SQUID (Superconducting QUantum Interference Device)
The current of a SQUID with area 1 cm2 could change from max to min by a tiny H=10-7 gauss!
For junction with finite thickness

41. Non-destructive testing
SuperConducting Magnet
Non-destructive testing
MCG, magnetocardiography
MEG, magnetoencephlography

42. Super-sentitive photon detector
Transition edge sensor
semiconductor detector superconductor detector
科學人,2006年12月

43. 磁力梯度儀
Sci Am. Aug, 1994