Download 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
– Weak anti-// response
 m0 m r H
• Pauli paramagnetism
mH
– 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
 ne2t 
 E
J  
m


p 2
ne2
2
 ( )  1  2 ,  p 

m0m
• Useful!
– DC, AC electrical conductivity
– Thermal transport
k  2k B 2
8 W
L


2
.
44

10
sT
3e 2
K2
Lmeas  2.1  2.6  108
• 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
 1018
 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 ~ 10 12 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  e  t /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  n0
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)  is integer
multiple of 0
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
Hc
g  m0
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 e 2
Supercurrent density is
J s  ns evs
2
ns e E dJ S

m
dt
Faraday’s law


 ns e2 E  dJ S

 
m
dt
d    ns e2
  JS 

dt 
m

B  0




n e dB
dJ
 s
  S
m dt
dt
2
 
ns e2 
  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 e 2
 
ns e2 
  JS  
B
m
=0; Gauss’s law
for electrostatics
Ampere’s
law

 
E
  B  m0 J  m0 0
t
  

    B  m0   J
   2 
ns e2 
  B    B   m0
B
m
2 
ns e 2 
 B  m0
B
m
Magnetic Penetration Depth - l
• Screening not immediate;
characteristic decay length
2 1 
 B 2B
m
l 
m0 n s e 2
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  fs 
1
2
Bc
2 m0
fn  fs
1
2
Bc
2 m0
m0 2
Bc 

Ginzburg-Landau Theory - 4
p2
H
V ,
2m
• Momentum term in H:
p  i
2
B
f magnetic  
2 m0
• Now – include magnetic field
p   i  qA
• Classically, know that to include
magnetic fields …
fs  fn    
2

2

4

2
B
1
2
 i  2eA 
f s  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
lx
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
g cond
H
 m0 c
2
2
g cond
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
g cond
H
 m0 c
2
2
g cond
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
g cond
H
 m0 c
2
2
g cond
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)
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
1
2
• 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
lx
two kinds of superconductors