beyond zero resistance – phenomenology of superconductivity nicholas p. breznay sass seminar –...
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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: and – 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 …
… in a periodic potential
EH
)(2
)( 22
rVm
rH
)()( KrVrV
K is a Bravais lattice vector
K
Wikipedia
Physics of Metals – Bloch’s Theorem
• Bloch’s theorem tells us that eigenstates have the form …
… where u(r) is a function with the periodicity of the lattice …
ErVm
)(2
22
Em
H 22
2
)()( ruer rki
)()( Kruru
rkiAer )(
Free particle Schro
Wikipedia
Physics of Metals – Drude Model
• Model for electrons in a metal– Noninteracting, inertial gas
– Scattering time
• Apply Fermi-Dirac statistics
)(
)(tp
Eqtpdt
d damping
term
H
E
k
E
k
EfEf
m
kE
2
22
http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg
Physics of Metals – Magnetic Response
• Magnetism in media
• Larmor/Landau diamagnetism– Weak anti-// response
• Pauli paramagnetism– Moderate // response
• Typical values –– Cu~ -1 x 10-5
– Al~ +2 x 10-5
minimal response to B fields– r ~ 1 B = 0H
)(0 MHB in SI
linear response
familiarly
H
E
k
E
k
EfEf
HM
H
H
H
r
0
0 )1(B
Physics of Metals – Drude Model Comments
• Wrong!– Lattice, e-e, e-p, defects,
– ~ 10-14 seconds MFP ~ 1 nm
• Useful!– DC, AC electrical conductivity
– Thermal transport• Lorenz number T
– Heat capacity of solids
Wikipedia
Em
neJ
2
m
nep
p
0
22
2
2
,1)(
)(
)(tp
Eqtpdt
d
3ATTCv
Electronic contribution
Lattice
1~'sfe
meas
28
2
22
1044.23 K
W
e
k
TL B
8106.21.2 measL
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: and – 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
Temperature
ResidualResistance
(impurities)
Impure metal
Electrical resistance
R0
Lattice (phonon)interactions
Pure metal
TD/3
Hallmark 1 – Zero Resistance
• Superconducting R vs T
R
Temperature
R0
Superconductor
Tc“Transition 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 high-field laboratory magnets (10-20T)
• Nonzero resistance changing current changing magnetic field
• One such measurement
SuperconductorCirculating
supercurrent
Magnetic (dipole) field
From Ustinov “Superconductivity” Lectures (WS 2008-2009)
I
1810Cu
SC
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
• Magnitude of “kinetic inductance”:
At 1 kHz, NormalRL 1210~
Vac
L
R
http://www.apph.tohoku.ac.jp/low-temp-lab/photo/FUJYO1.png
Hallmark 2 – Conductors in a Magnetic Field
Normal metal
Field off
Applyfield
t
EJB
t
BE
B
E
0
0
jE
)1(~)( /0
teBtB
RL /
Hallmark 2 – Conductors in a Magnetic Field
Applyfield
Perfect (metallic) conductor SuperconductorNormal metal
Cool Cool
Field off
Applyfield
Applyfield
Hallmark 2 – Meissner-Oschenfeld Effect
Superconductor
CoolApplyfield
• B = 0 perfect diamagnetism: M = -1
• Field expulsion unexpected; not discovered for 20 years.
HHM
MHB
0)(0
B/0
H
-M
HHc Hc
Hallmark 3 – Flux Quantization
27150 102~
2102~
2cmG
e
hcsV
e
h
Earth’s magnetic field ~ 500 mG, so in 1 cm2 of BEarth there are ~ 2 million 0’s.
first appearance of h in our description; quantum phenomenon
0nAdB
Total flux (field*area) is integer multiple of
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 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: and – Type I and II SC’s
SC Parameter Review
g(H)
HHc
gnormal state
gsc state
2
2
0cHg
• Magnetic field energy density
• Extract free energy difference between normal and SC states with Hc
• Know magnetic response important; use R = 0 + Maxwell’s equations … ?
London Theory – 1
• Newton’s law (inertial response) for applied electric field
SJdt
dE 2en
m
s
en
J
dt
dmeE
s
S svdt
dmF
sss evnJ
dt
dJ
m
Een Ss 2
dt
Jd
m
Een Ss
2
dt
Jd
dt
Bd
m
en Ss
2
02
B
m
enJ
dt
d sS
Supercurrent density is
Bm
enJ sS
2
We know B = 0 inside superconductors
Faraday’s law
Fritz & Heinz London, (1935)
London Theory – 2
SJdt
dE 2en
m
s
Bm
enJ sS
2
London Equations
t
EJB
000
JB
0
Bm
enBB s
2
02
Bm
enB s
2
02
Ampere’s law
=0; Gauss’s law for
electrostatics
Magnetic Penetration Depth -
B(z)
z
20
2
en
m
s BB
2
2 1
• Screening not immediate;
characteristic decay length
• Typical ~ 50 nm
• m,e fixed – uniquely specifies the superconducting electron density ns
Sometimes called the “superfluid
density”
/0)( zeBzB
B0
SC
Ginzburg-Landau Theory - 1
42
2 ns ff
• First consider zero magnetic field
• Order parameter
• Associate with cooper pair density:
• Expand f in powers of ||2
To make sense, > 0, (T)
Free energy ofsuperconducting state
Free energy ofnormal state
2sn
Need > -Infinity; B > 0
Free energy of SC state
~ # of cooper pairs
Ginzburg-Landau Theory - 2
42
2 ns ff
42
2 ns ff
02
2 ns ffd
d
2
• For < 0, solve for minimum in fs-fn …
http://commons.wikimedia.org/wiki/File:Pseudofunci%C3%B3n_de_onda_(teor%C3%ADa_Ginzburg-Landau).png
• Know that fn-fs is the condensation energy:
Ginzburg-Landau Theory - 3
242
2 ns ff
2
2
ns ff
2
02
1csn Bff
sn ff
2
02
1cB
2
0cB
Ginzburg-Landau Theory - 4
qAip
• Momentum term in H:
• Now – include magnetic field
• Classically, know that to include magnetic fields …
0
2242
22
2
1
2
BeAi
mff ns
ipVm
pH ,
2
2
0
2
2B
fmagnetic
42
2 ns ff
Ginzburg-Landau Theory - 5
• Free Energy Density 0
2242
22
2
1
2
BeAi
mff ns
02
22
1
2 0
2242
dVB
eAim
0F
022
1 22 eAim
eAim
eJ 2Re
2 *
Ginzburg-Landau Theory - 6
022
1 22 eAim
Take real,normalize
2 0
22
23
m
0
2)(32
2
mT
Define
mTT
2)()(
2
0)(
22
2 T
Linearize in
Superconducting coherence length -
x
(x)Vacuum SC
Superconductor
022
1 22 eAim
0)(
22
2 T
• Characteristic length scale for SC wavefunction variation
• London Theory magnetic penetration depth
• Ginzburg-Landau Theory coherence length
two kinds of superconductors!
Pause
Surface Energy: H(x)
(x)
gmagnetic(x)
2
2
0c
cond
Hg
2
2
0c
cond
Hg
energy penalty for excluding B
energy gain for being in SC state
gsc(x)
SC
Surface Energy: H(x)
(x)
gmagnetic(x)
2
2
0c
cond
Hg
2
2
0c
cond
Hg
energy penalty for excluding B
energy gain for being in SC state
net energy penalty at a surface / interface
gnet(x)
gsc(x)
SC
Surface Energy: H(x)
(x)
gmagnetic(x)
2
2
0c
cond
Hg
2
2
0c
cond
Hg
energy penalty for excluding B
energy gain for being in SC state
net energy gain at a surface / interfacegnet(x)
gsc(x)
SC
Type I Type II
H(x)
(x)
gmagnetic(x)
gnet(x)
gsc(x)
H(x)
(x)
gmagnetic(x)
gnet(x)
gsc(x)
• predicted in 1950s by Abrikosov• elemental superconductors
2
1
2
1
nm (nm) Tc (K) Hc2 (T)
Al 1600 50 1.2 .01
Pb 83 39 7.2 .08
Sn 230 51 3.7 .03
nm (nm) Tc (K) Hc2 (T)
Nb3Sn 11 200 18 25
YBCO 1.5 200 92 150
MgB2 5 185 37 14
Type II Superconductors
H
Normal state cores
Superconducting region
http://www.nd.edu/~vortex/research.html