Superconductivity
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Transcript Superconductivity
Beyond Zero Resistance –
Phenomenology of Superconductivity
Nicholas P. Breznay
SASS Seminar – Happy 50th!
SLAC
April 29, 2009
Preview
• Motivation / Paradigm Shift
• Normal State behavior
• Hallmarks of Superconductivity
– Zero resistance
– Perfect diamagnetism
– Magnetic flux quantization
• Phenomenology of SC
– London Theory, Ginzburg-Landau Theory
– Length scales: l and x
– Type I and II SC’s
Physics of Metals - Introduction
• Atoms form a periodic lattice
• Know (!) electronic states key for
the behavior we are interested in
• Solve the Schro …
H E
2 2
H (r )
V (r )
2m
… in a periodic potential
V (r ) V (r K )
K
K is a Bravais lattice vector
Wikipedia
Physics of Metals – Bloch’s Theorem
2 2
V ( r ) E
2m
• Bloch’s theorem tells us that
eigenstates have the form …
ik r
(r ) e u(r )
… where u(r) is a function with the
periodicity of the lattice …
u(r ) u(r K )
Free particle Schro
2 2
H
E
2m
ik r
(r ) Ae
Wikipedia
Physics of Metals – Drude Model
• Model for electrons in a metal
– Noninteracting, inertial gas
– Scattering time t
p (t )
d
damping term
p(t ) qE
dt
t
• Apply Fermi-Dirac statistics
E
Ef
2k 2
E
2m
k
http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg
Physics of Metals – Magnetic Response
in SI
B m0 ( H M )
• Magnetism in media
• Larmor/Landau diamagnetism
M cH
linear response
B m0 (1 c ) H
m0 m r H
mH
– Weak anti-// response
• Pauli paramagnetism
– Moderate // response
familiarly
• Typical c values –
– cCu~ -1 x 10-5
– cAl~ +2 x 10-5
minimal response to B fields
– mr ~ 1 B = m0H
E
E
Ef
Ef
H
k
k
Physics of Metals – Drude Model Comments
• Wrong!
– Lattice, e-e, e-p, defects,
– t ~ 10-14 seconds MFP ~ 1 nm
p (t )
d
p(t ) qE
dt
t
ne 2t
J
m
E
p2
ne2
2
( ) 1 2 , p
m0m
• Useful!
– DC, AC electrical conductivity
– Thermal transport
k 2k B 2
8 W
L
2
.
44
10
sT
3e2
K2
Lmeas 2.1 2.6 108
• Lorenz number k/sT
– Heat capacity of solids
Cv T AT 3
Electronic
contribution
Lattice
meas
~1
fe's
Wikipedia
Preview
• Motivation / Paradigm Shift
• Normal State behavior
• Hallmarks of Superconductivity
– Zero resistance
– Perfect diamagnetism
– Magnetic flux quantization
• Phenomenology of SC
– London Theory, Ginzburg-Landau Theory
– Length scales: l and x
– Type I and II SC’s
Hallmark 1 – Zero Resistance
• Metallic R vs T
– e-p scattering (lattice interactions) at high temperature
– Impurities at low temperatures
R
Lattice (phonon)
interactions
Electrical resistance
Residual
Resistance
(impurities)
R0
TD/3
Temperature
Hallmark 1 – Zero Resistance
• Superconducting R vs T
R
R0
“Transition temperature”
Tc
Temperature
Hallmark 1 – Zero Resistance
• Hard to measure “zero” directly
• Can try to look at an effect of the
zero resistance
• Current flowing in a SC ring
– Not thought experiment –
standard configuration for highfield laboratory magnets (1020T)
• Nonzero resistance changing
current changing magnetic
field
• One such measurement
Magnetic (dipole) field
I
Circulating
supercurrent
Superconductor
SC
1018
Cu
From Ustinov “Superconductivity” Lectures (WS 2008-2009)
Hallmark 1 – Zero Resistance Notes
• R = 0 only for DC
• AC response arises from kinetic
inductance of superconducting
electrons
– Changing current electric field
• Model: perfect resistor (normal
electrons), inductor (SC electrons) in
parallel
Vac
L
R
• Magnitude of “kinetic inductance”:
At 1 kHz,
L ~ 1012 RNormal
http://www.apph.tohoku.ac.jp/low-temp-lab/photo/FUJYO1.png
Hallmark 2 – Conductors in a Magnetic Field
Normal metal
E
B 0
B
E
t
E
B m0 J
t
E j
Apply
field
Field off
B(t ) ~ B0 (1 et /t )
t L/R
Hallmark 2 – Conductors in a Magnetic Field
Normal metal
Apply
field
Perfect (metallic) conductor
Apply
field
Field off
Cool
Superconductor
Apply
field
Cool
Hallmark 2 – Meissner-Oschenfeld Effect
Superconductor
• B = 0 perfect diamagnetism: cM = -1
B m0 ( H M ) 0
M cH H
Apply
field
• Field expulsion unexpected; not discovered for
20 years.
B/m0
-M
Hc
H
Hc
H
Cool
Hallmark 3 – Flux Quantization
B dA n0
Earth’s magnetic field ~ 500 mG, so in
1 cm2 of BEarth there are ~ 2 million 0’s.
h
hc
0
~ 2 10 15V s
~ 2 10 7 G cm 2
2e
2e
first appearance of h in
our description; quantum
phenomenon
Total flux (field*area)
multiple of 0
is integer
Hallmark 3 – Flux Quantization
Apply uniform field
Measure flux
Aside – Cooper Pairing
• In the presence of a weak
attractive interaction, the filled
Fermi sphere is unstable to the
formation of bound pairs electrons
• Can excite two electrons d above
Ef, obtain bound-state energy <
2Ef due to attraction
• New minimum-energy state
allows attractive interaction (e-p
scattering) by smearing the FS
The physics of superconductors Shmidt, Müller, Ustinov
Preview
• Motivation / Paradigm Shift
• Normal State behavior
• Hallmarks of Superconductivity
– Zero resistance
– Perfect diamagnetism
– Magnetic flux quantization
• Phenomenology of SC
– London Theory, Ginzburg-Landau Theory
– Length scales: l and x
– Type I and II SC’s
SC Parameter Review
• Magnetic field energy density
• Extract free energy difference
between normal and SC states
with Hc
H
g m0 c
2
2
• Know magnetic response
important; use R = 0 + Maxwell’s
equations … ?
g(H)
gnormal state
gsc state
Hc
H
London Theory – 1
• Newton’s law (inertial response) for applied electric field
d
F m vs
dt
d JS
eE m
dt ns e
d
E J S
dt
m
ns e2
Supercurrent density is
J s ns evs
2
ns e E dJS
m
dt
d
dt
Faraday’s law
ns e2 E dJ S
m
dt
ns e 2
J S m B 0
n e dB
dJ
s
S
m dt
dt
2
ns e 2
JS
B
m
We know B = 0 inside superconductors
Fritz & Heinz London, (1935)
London Theory – 2
London Equations
d
E J S
dt
m
ns e2
ns e 2
JS
B
m
=0; Gauss’s law
for electrostatics
Ampere’s
law
E
B m0 J m0 0
t
B m0 J
2
ns e 2
B B m0
B
m
2
ns e2
B m0
B
m
Magnetic Penetration Depth - l
• Screening not immediate;
characteristic decay length
2 1
B 2B
m
l
m0ns e2
2
l
• Typical l ~ 50 nm
• m,e fixed – l uniquely specifies
the superconducting electron
density ns
Sometimes called
the “superfluid
density”
B( z) B0e z / l
SC
B(z)
B0
l
z
Ginzburg-Landau Theory - 1
• First consider zero magnetic field
• Order parameter
• Associate with cooper pair
2
density:
ns
Free energy of
superconducting state
Free energy of
normal state
fs fn
2
2
4
• Expand f in powers of ||2
To make sense, > 0, (T)
Free energy of
SC state ~ #
of cooper pairs
Need > -Infinity; B > 0
Ginzburg-Landau Theory - 2
fs fn
2
2
4
• For < 0, solve for minimum
in fs-fn …
fs fn
2
d
d
2
4
2
f
f
0
s
n
2
2
http://commons.wikimedia.org/wiki/File:Pseudofunci%C3%B3n_de_onda_(teor%C3%ADa_Ginzburg-Landau).png
Ginzburg-Landau Theory - 3
fs fn
2
2
4
2
• Know that fn-fs is the condensation energy:
2
f s fn
2
fn f s
1
2
Bc
2 m0
fn fs
Bc
1
2
Bc
2 m0
m0 2
Ginzburg-Landau Theory - 4
p2
H
V ,
2m
• Momentum term in H:
p i
2
• Now – include magnetic field
f magnetic
p i qA
• Classically, know that to include
magnetic fields …
fs fn
2
2
4
B
2 m0
2
B
1
2
i 2eA
fs fn
2
2m
2m0
2
4
Ginzburg-Landau Theory - 5
2
B
1
• Free Energy Density f s f n
i 2eA 2
2
2m
2m0
2
dF 0
4
2
B
4 1
2
2
i 2eA dV 0
d
2
2m
2 m0
1
i 2eA 2 0
2m
2e
J
Re * i 2eA
m
2
Ginzburg-Landau Theory - 6
2
Take real,
normalize
1
i 2eA 2 0
2m
2
Define
Linearize in
x (T )
3
2 2
0
2m
2
2 3 0
(T ) 2m
2
(T ) 2m
2
2
0
x (T )
2
Superconducting coherence length - x
2
1
i 2eA 2 0
2m
2
2
0
x (T )
2
• Characteristic length scale for SC
wavefunction variation
Vacuum
SC
(x)
x
Superconductor
x
Pause
• London Theory
magnetic penetration depth l
• Ginzburg-Landau Theory
coherence length x
lx
two kinds of superconductors!
Surface Energy and “Type II”
H(x)
H(x)
(x)
l
(x)
x
l
x
x
x
x l
l x
Surface Energy: x l
H(x)
(x)
x
l
SC
energy penalty for excluding B
gmagnetic(x)
gsc(x)
2
gcond
H
m0 c
2
2
gcond
H
m0 c
2
energy gain for being in SC state
Surface Energy: x l
H(x)
(x)
x
l
SC
energy penalty for excluding B
gmagnetic(x)
gsc(x)
gnet(x)
2
gcond
H
m0 c
2
2
gcond
H
m0 c
2
energy gain for being in SC state
net energy penalty at a surface / interface
Surface Energy: x l
H(x)
(x)
x
SC
l
energy penalty for excluding B
gmagnetic(x)
gsc(x)
2
gcond
H
m0 c
2
2
gcond
H
m0 c
2
energy gain for being in SC state
gnet(x)
net energy gain at a surface / interface
Type I
Type II
k
H(x)
(x)
l x
l
x
l
x
H(x)
(x)
l x
x
l
gmagnetic(x)
gmagnetic(x)
1
k
2
gsc(x)
k
gsc(x)
gnet(x)
1
2
gnet(x)
• elemental superconductors
x nm
l (nm)
Tc (K)
Hc2 (T)
Al
1600
50
1.2
.01
Pb
83
39
7.2
Sn
230
51
3.7
• predicted in 1950s by Abrikosov
x nm
l (nm)
Tc (K)
Hc2 (T)
Nb3Sn
11
200
18
25
.08
YBCO
1.5
200
92
150
.03
MgB2
5
185
37
14
Type II Superconductors x l
Normal state cores
Superconducting region
H
http://www.nd.edu/~vortex/research.html
The End
• London Theory
magnetic penetration depth l
• Ginzburg-Landau Theory
coherence length x
lx
two kinds of superconductors