Superconductivity
M.C. Chang
Dept of Phys
•Introduction
•Thermal properties
•Magnetic properties
•London theory of the Meissnereffect
•Microscopic (BCS) theory
•Flux quantization
•Quantum tunneling
A brief history of low temperature (Ref: 絕對零度的探索)
•1800 Charles and GayLusac(from PTrelationship) proposed
that the lowest temperature is 273 C (= 0 K)
•1877 Cailletetand PictetliquifiedOxygen (183 C or 90 K)
•soon after, Nitrogen (77 K) is liquified
•1898 Dewar liquifiedHydrogen (20 K)
•1908 OnnesliquifiedHelium (4.2 K)
•1911 Onnesmeasured 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
G. Amontons
1700
1.14K
3.72K
7.19K
3.40K
1.09K
2.39K
4.15K
0.39K5.38K
0.55K
0.12K
9.50K
4.48K
0.92K
7.77K
0.01K1.4K
0.66K
0.14K
0.88K
0.56K0.51K
0.03K
1.37K1.4K
4.88K
0.20K0.60K
0.0003K
http://superconductors.org/Type1.htm
Tc'sgiven are for bulk, except for Palladium, which has been irradiated with
He+ ions, Chromium as a thin film, and Platinum as a compacted powder
Superconductivity in alloys and oxides
From Cywinski’slecture note
19101930195019701990
20
40
60
80
100
120
140
160
Superconducting transition temperature (K)
Hg
Pb
Nb
NbC
NbC
NbN
NbN
V3Si
V3Si
Nb3Sn
Nb3Sn
Nb3Ge
Nb3Ge
(LaBa)CuO
(LaBa)CuO
YBa2Cu3O7
YBa2Cu3O7
BiCaSrCuO
BiCaSrCuO
TlBaCaCuO
TlBaCaCuO
HgBa
2Ca2Cu3O9
HgBa2Ca2Cu3O9
HgBa
2Ca2Cu3O9
(under pressure)
HgBa2Ca2Cu3O9
(under pressure)
Liquid Nitrogen
temperature (77K)
Bednorz
Muller
1987
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
•…
•Introduction
•Thermal properties
•Magnetic properties
•London theory of the Meissnereffect
•Microscopic (BCS) theory
•Flux quantization
•Quantum tunneling
Thermal properties of SC: specific heat
For different superconductors,
at
CC
C
T
SN
N
C
−
~.143
The exponential dependence with Tis
called “activation”behavior and implies
the existence of an energy gapabove
Fermi surface.Δ
~ 0.11 meV(104~5
EF
)
Tk
el
S
B
eC
)0(Δ−
∝
•Temperature dependence of Δ
(obtained from Tunneling)
Δ‘s scale with differentTc’s
2Δ(0) ~ 3.5 kBTc
Universal behavior of Δ(T)
•Connection between
energy gap and Tc
1/2
()
1.741 for
(0)
C
C
TT
TT
T
⎛⎞
Δ
=
−≈
⎜⎟
Δ
⎝⎠
•Entropy
H
S
CT
T
∂
⎛⎞
=
⎜⎟
∂
⎝⎠
2
2
H
H
H
F
S
T
CSF
TTT
∂
⎛⎞
=−
⎜⎟
∂
⎝⎠
⎛⎞
∂∂
⎛⎞
==−
⎜⎟
⎜⎟
∂∂
⎝⎠
⎝⎠
Less entropy in SC state:
more ordering
2nd order phase
transition
FNFS
= Condensation energy
～108
eVper electron!
Al
Al
•free energy
More evidences of energy gap
•Electron tunneling
2Δsuggests excitations
created in “eh”pairs
ν
==
2
Δ
h
480 GHz (microwave)
•EM wave absorption
Magnetic property of the superconductor
All curves can be collapsed onto
a similar curve after rescaling.
()
{
}
2
()1
coc
HTHTT=−
()
{
}
2
()1
coc
HTHTT=−
sc
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)
Critical currents (no applied field)
Current
Radius, a
Magnetic field
Hi
The critical current density of a long
thin wire is therefore
jc~108A/cm2
for Hc=500 Oe, a=500 A
i
c
dH
π
4
=⋅
∫
so
cc
H
ca
i
2
=
a
cH
j
c
c
π
2
=
•Jc
has a similar temperature
dependence as Hc, and Tc
is similarly
lowered as Jincreases.
From Cywinski’slecture note
(thinner wire has larger Jc)
Crosssection through a
niobium–tin cable
Phys World, Apr 2011
Meissnereffect
(Meissnerand Ochsenfeld, 1933)
A SC is more than a perfect conductor
differentsame
not only dB/dt=0
but also B=0!
Perfect
diamagnetism
Lenz law
pure In
Superconductingalloy: type II SC
partial exclusion and remains superconducting at high B(1935)
(also called intermediate/mixed/vortex/Shubnikovstate)
•HC2
is of the order of 10~100 Tesla (called hard, or type II, superconductor)
STM image
NbSe2, 1T, 1.8K
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)
2
1
For a SC,
4
()(0)
8
S
SS
dFMd
dFHd
H
FH
H
H
F
π
π
=−⋅
=
→−=
2
()()
()(0) for nonmagnetic material
(0)
8
(0)
c
NcSc
NcN
NS
FHFH
FHF
FFF
H
π
=
=
∴
Δ=−=
(
)
is in Kittel
a
HB
(Magnetic energy density)
•Introduction
•Thermal properties
•Magnetic properties
•London theory of the Meissnereffect
•Microscopic (BCS) theory
•Flux quantization
•Quantum tunneling
()
2
2
2
s
s
s
s
s
S
ne
dB
J
dtmct
ne
J
ne
J
B
c
A
mc
m
φ
=
∂
∇×=−
∂
∇×=
−
−
+∇
It can be shown that
▽ψ=0 for simply
connected sample
(See Schrieffer)
1
Eq.(1)
B
E
ct
∂
+∇×=−
∂
2
2
22
4
s
L
ne
B
BB
mc
π
λ
∇=≡
London
proposed
()()
2
4
use and
s
BJ
c
vvv
π
∇×=
∇
×∇×=∇∇⋅−∇
London theory of the Meissnereffect
(Fritz London and Heinz London, 1934)
•Superfluiddensity ns
σ= ∞
•Normal fluid density nn
2
n
n
ne
m
τ
σ
=
nnn
sn
+
=
=
constant
Assume
2
(1)
(2)
ss
nn
dJneE
dtm
JE
σ
=
=
s
ss
nnn
J
env
J
env
=−
=−
where
like free
charges
nn
Tc
Carrier density
T
ns
Twofluid model:
•Penetration length λL
Outside the SC, B=B(x) z
2
2
2
/
0
()
L
L
x
dB
B
dx
BxBe
λ
λ
−
=
→=
2
233
2
170 if =
4
10/cm
SL
S
m
ne
An
c
λ
π
=≈
/
0
4
44
L
s
x
sy
L
BJ
c
cB
cdB
Je
dx
λ
π
ππλ
−
∇×=
∴
=−=
also
decays
•Temperature dependence of λL
()
1/2
4
(0)
()
1/
C
T
TT
λ
λ
=
⎡
⎤
−
⎣
⎦
tin
•Higher T, smaller nS
Predicted λL(0)=340 A,
measured 510A
(expulsion of
magnetic field)
Coherence length ξ0
(Pippard, 1939)
•In fact, ns
cannot remain uniform near a surface.
The length it takes for ns
to drop from full value to
0is called
ξ
0
x
ns
surface
superconductor
ξ0
0
from BCS theory
FF
vv
p
ξ
π
≈=↔
ΔΔΔ
•The pair wave function (with range ξ0) is a
superposition of oneelectron states with energies
within Δof EF
(A+M, p.742).
pp
m
Δ
≈
Δ
•Therefore, the spatial range of the variation of nS
ξ0
~ 1 μm>> λfor type I SC
•Microscopically it’s related to the range
of the Cooper pair.
Energy uncertainty
of a Cooper pair
Penetration depth, correlation length, and surface energy
•smaller ξ0, get more “negative”
condensation energy.
•smaller λ, cost more energy to
expel the magnetic field.
•
ξ0
> λ, surface energy is positive
From Cywinski’s lecture note
Type I superconductivity
•
ξ0
<
λ, surface energy isnegative
Type II superconductivity
•When ξ
0
>>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.
Vortex state of type II superconductor
(Abrikosov, 1957)
•the magnetic flux φin a vortex is
always quantized(discussed later).
•the vortices repel each other slightly.
•the vortices prefer to form a triangular
lattice (Abrikosovlattice).
Hc2
Hc1
H
0
M
From Cywinski’slecture note
2003
•the vortices can move and dissipate energy
(unless pinned by impurity ←Flux pinning)
Normal
core
isc
Estimation of Hc1
and Hc2 (type II)
2
0
101
2
cc
HH
φ
πλφ
π
λ
≈→≈
•Near Hc1, there begins with a single
vortex with flux quantum φ0, therefore
•Near Hc2, vortex are as closely packed
as the coherence length allows, therefore
2
0
0202
2
0
cc
NHNH
φ
πξφ
π
ξ
≈→≈
2
2
10
Therefore,
c
c
H
H
λ
ξ
⎛⎞
≈
⎜⎟
⎝⎠
Typical values, for Nb3Sn,
ξ0
～34 A, λL
～1600 A
mercury
Origin of superconductivity?
•Metal X can (cannot) superconductbecause its atomscan
(cannot) superconduct?
Neither Au nor Bi is superconductor, but alloy Au2Bi is!
White tin can, grey tin cannot!(the only difference is lattice structure)
•goodnormal conductors (Cu, Ag, Au) are bad superconductor;
badnormal conductors are good superconductors, why?
•What leads to the superconducting gap?
•Failed attempts: polaron, CDW...
It is found that Tc
=const ×
Mα
α～1/2 for different materials
lattice vibration?
•Isotope effect(1950):
•Frohlich: electronphonon interactionmaybe crucial.
•Reynolds et al, Maxwell: isotope effect
•GinzburgLandau theory: ρS
can be varied in space.
Suggested the connection
•1935 London: superconductivity is a quantum phenomenon
on a macroscopicscale. There is a “rigid”(due to the energy
gap) superconducting wave function Ψ.
•1950
2
()()
S
rr
ρψ
=
Brief history of the theories of superconductors
and wrote down the eq. for order parameter Ψ(r) (App. I)
2003
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
Microscopicwave function for the
condensation of Cooper pairs.
1972
Dynamic electronlattice interaction →Cooper pair
Effective attractive interaction
between 2 electrons
(sometimes called overscreening)
～1 μm
+++
e
(range of a Cooper pair;
coherence length)
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)
Δ()
()
int
02
1
=
−
ω
D
DEV
e
F
, see App. H
•Fractionof electrons involved～kTc/EF
～104
•Average spacingbetween condensate electrons ～10 nm
•Therefore, within the volume occupied by the Cooper pair, there
are approximately (1μm/10 nm)3
～106
other pairs.
int
3
2
500,()1
5
/3
500
DF
c
KDEV
TKe
ω
−
≤≤
∴
≤=
(～upper limit of T
c)
2Δ(0) ~ 3.5 kBTc
kTe
BCD
DEV
F
=
−
113
1
.
()
int
ω
•These pairs (similar to bosons) are highly correlated
and form a macroscopic condensate state with (BCS result)
Energy gap and Density of states
~ O(1) meV
D(E)
•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
8
4
1
0.110
10
C
BC
F
T
kTmeVeV
T
−
×
wiki
Families of superconductors
Conventional
BCS
Heavy fermion
Cuperate
(ironbased)
F. Steglich1978
T.C. Ozawa 2008
•Introduction
•Thermal properties
•Magnetic properties
•London theory of the Meissnereffect
•Microscopic (BCS) theory
•Flux quantization
•Quantum tunneling (Josephson effect, SQUID)
Flux quantizationin a superconducting ring
(F. London 1948 with a factor of 2 error, Byers and Yang, also Brenig, 1961)
**,
2
q
j
qe
mii
ψψψψ
⎛⎞
=
∇−∇=−
⎜⎟
⎝⎠
•Current density operator
•SC, in the presence of B
*
***
*
**
2
qqq
jAA
micic
ψψψψ
⎡
⎤
⎛⎞⎛⎞
=∇−+∇−
⎢
⎥
⎜⎟⎜⎟
⎢
⎥
⎝⎠⎝⎠
⎣
⎦
let =

and assume

var
y
slowl
y
with r
i
e
φ
ψψψ
2
2
2
then 
ee
jA
mmc
φ
ψ
⎛⎞
=−∇+
⎜⎟
⎝⎠
London eq. with
•Inside a ring
0 jd
⋅
=
∫
22
cc
Add
ee
φ
φ
⇒⋅=−∇⋅=−Δ
∫∫
2
00
7
flux ,
2
210 gausscm
2
hc
e
hc
nn
e
φφ
−
∴
Φ===×≡
•
φ
0
～the flux of the Earth's magnetic field
through a human red blood cell (~ 7 microns)
ns
=2
2

ψ
*
*
2
2
qe
mm
=
−
=
Single particletunneling
(Giaever, 1960)
•SIN
Ref: Giaever’s 1973
Nobel prize lecture
2030 A thick
dI/dV
For T>0
(Tinkham, p.77)
•SIS
Josephson effect
(Cooper pairtunneling)
Josephson, 1962
1973
1) DC effect:
There is a DC current through SIS in the absence
of voltage.
ψψ
θ
11
1
=ei
ψψ
θ
22
2
=ei
()
()
12
1221
0
01
(/2)(/2)
()()
0
20
/2
2
sin
/,
iKxdiKxd
S
ii
Kd
S
Kd
S
nee
ien
jKeee
m
j
jenKem
θθ
θθθθ
ψ
δ
δθθ
−++−
−−
−
−
=+
=−+
≡
=
≡−
2/e
Δ
Giaever
tunneling
Josephson
tunneling
2) AC Josephson effect
Apply a DC voltage, then there is a rfcurrent oscillation.
1
()/
/
ˆ
1
()/(1,2)
NN
iEEt
it
iii
NNee
tti
μ
ψ
ψ
θμθ
−
−−
−
=−∝=
→=−+=
0
1
00
2
22
n
2
si
eVeV
tjjt
eV
δδ
μ
μ
δ
⎛⎞
∴
=+⇒
−=
=+
⎜⎟
⎝⎠
−
•An AC supercurrentof Cooper pairs with freq. ν=2eV/h, a
weak microwave is generated.
•
ν
can be measuredvery accurately, so tiny
ΔV as small as
1015
Vcan be detected.
•Also, since V can be measured with accuracy about 1 part in
1010, so 2e/hcan be measured accurately.
•JJbased voltage standard (1990):
1 V ≣
the voltage that produces ν=483,597.9 GHz (exact)
•advantage: independent of material, lab, time (similar to the
quantum Hallstandard).
(see Kittel, p.290 for an
alternative derivation)
VVt=
+
0
υ
ω
捯c
〰0
0
〰
0
2
獩ss楮
2
2
†††‽(1Φ獩n
†瑨=re猠=C×rre湴=
2
==
n
n
n
e
jjVtt
eV
e
jJt
V
e
n
n
t
υ
ωδ
ω
υ
ω
ω
δ
ω
⎡⎤
⎛⎞
=++
⎜⎟
⎢⎥
⎝⎠
⎣⎦
⎛⎞
⎛⎞
−−+
⎜⎟
⎜⎟
=
⎝⎠
⎝⎠
⇒
∑
Shapiro steps
(1963)
given I, measure V
3) DC+AC:
Apply a DC+ rfvoltage, then there is a DC current
•Another way of providing a voltage standard
SQUID
(Superconducting QUantumInterference Device)
00
0
sinsin
2cossin
22
ab
abab
jjj
j
δ
δ
δ
δδδ
=
+
−+
⎛⎞⎛⎞
=
⎜⎟⎜⎟
⎝⎠⎝⎠
1
2
11
22
Similar to
2
2e
We now have
c
2e
c
ba
C
ab
C
c
A
dd
e
Ad
Ad
θ
θ
θ
θ
θ
⋅
=−∇⋅
⋅=−
⋅=−
∫∫
∫
∫
0
max0
0
2
2
2
2cos
2
ab
C
e
Ad
c
jj
φ
δ
δπ
φ
πφ
φ
⇒−=⋅=
⎛⎞
∴
=
⎜⎟
⎝⎠
∫
The current of a SQUID
with area 1 cm
2
could
change from max to min
by a tiny ΔH=107
gauss!
For junction with finite thickness
Nondestructive testing
SuperConductingMagnet
MCG, magnetocardiography
MEG, magnetoencephlography
Supersentitivephoton detector
科學人,2006年12月
semiconductor detector superconductor detector
Transition edge sensor
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