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The quantum mechanics of
superconductors
Subir Sachdev
Talk online:
http://pantheon.yale.edu/~subir
The quantum mechanics of
•Insulators
•Metals
•Semiconductors
•Superconductors
Technology in
the 19th century
Technology of
the 20th century
Technology of
the 21st century ?
Electrical resistivity of Cu at 273 K: 1.56 10
Electrical resistivity of C (diamond) at 273 K:
10
Ohm meter
9 1021 Ohm meter
size of atom ~ 1010 meters
Ratio is comparable to the ratio
size of galaxy ~ 1021 meters
Resistance of Hg measured by Kammerlingh Onnes in 1911
6 108 Ohm meter
K
Quantum theory of metals
Valence electrons occupy plane wave states which extend
across the entire material
k2
k 
2 m*
i2
H  
*
2
m
i
2
e
e
2
k
e
e
e

kF
k
1
4 3
n  2
  kF
3
 2  3
kF
Fermi velocity vF  * ~ 108 cm/sec
m
Fermi surface of copper
Electrical conduction occurs by acceleration of
electrons near the Fermi surface
Quantum theory of metals
i2
H  
 VQ cos  Qxi 
*
2m
i
i
2
e
e
e
e
Quantum theory of metals
i2
H  
 VQ cos  Qxi 
*
2m
i
i
2
Electrons can scatter off VQ and change their momentum by  Q
k
Q
Q
k
Fermi surface of copper
Deviation in shape from sphere is caused by
non-resonant scattering off periodic potential
Quantum theory of insulators
i2
H  
 VQ cos  Qxi 
*
2m
i
i
2
Electrons can scatter off VQ and change their momentum by  Q
Resonant scattering of electrons between momenta k  Q / 2
k
Q
Q

2
Q
2
k
Quantum theory of insulators
i2
H  
 VQ cos  Qxi 
*
2m
i
i
2
Electrons can scatter off VQ and change their momentum by  Q
Resonant scattering of electrons between momenta k  Q / 2
Energy gap:
k
No electron
states in a
certain range
of energies !
Q

2
Q
2
k
Quantum theory of insulators
Insulators have their Fermi
level exactly at the edge of the
band gap. Electrical
conduction requires excitation
of electrons across the band
gap.
k

Q
2
Q
2
k
Quantum theory of semiconductors
Semiconductors are insulators with a small
energy gap
Si
k

Q
2
Q
2
k
Quantum theory of semiconductors
Doping with a small concentration of P
introduces mobile charge carriers
Si
k
P

Q
2
Q
2
k
We have so far considered the quantum theory
of a large number of fermions moving in a
periodic potential and found ground states
which are
metals, insulators and semiconductors
Now consider the quantum theory of a large
number of bosons moving in a periodic potential
87Rb
bosonic atoms in a magnetic trap and an optical lattice potential
The strength of the period potential can be varied in the experiment
M. Greiner, O. Mandel, T.
Esslinger, T. W. Hänsch, and I.
Bloch, Nature 415, 39 (2002).
Strong periodic potential
“Eggs in an egg carton”
Tunneling between neighboring minima is negligible and
atoms remain localized in a well. However, the total
wavefunction must be symmetric between exchange
Strong periodic potential
“Eggs in an egg carton”
Tunneling between neighboring minima is negligible and
atoms remain localized in a well. However, the total
wavefunction must be symmetric between exchange
Strong periodic potential
“Eggs in an egg carton”
Tunneling between neighboring minima is negligible and
atoms remain localized in a well. However, the total
wavefunction must be symmetric between exchange
Insulator =
+
+
+
+
+
Weak periodic potential
A single atom can tunnel easily between
neighboring minima
Weak periodic potential
A single atom can tunnel easily between
neighboring minima
Weak periodic potential
A single atom can tunnel easily between
neighboring minima
Weak periodic potential
A single atom can tunnel easily between
neighboring minima
Weak periodic potential
G
= e
i

+
+
The ground state of a single particle is a zero
momentum state, which is a quantum superposition of
states with different particle locations.

The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
The Bose-Einstein condensate in a weak periodic potential
Lowest energy state for many atoms
BEC = G G G
= e3i
+
(
+
+
+
+
+
+ ....27 terms
Large fluctuations in number of atoms in each potential well
– superfluidity (atoms can “flow” without dissipation)
)
87Rb
bosonic atoms in a magnetic trap and an optical lattice potential
The strength of the period potential can be varied in the experiment
M. Greiner, O. Mandel, T.
Esslinger, T. W. Hänsch, and I.
Bloch, Nature 415, 39 (2002).
Superfluid-insulator transition
V0=0Er
V0=13Er
V0=3Er
V0=7Er
V0=10Er
V0=14Er
V0=16Er
V0=20Er
In any subvolume of the superfluid, there are large fluctuations in
the number of atoms but a definite phase 
Physical meaning of 
Consider two superfluids connected by a weak link
Phase 1 ;
iN11
e
w
N1
Phase 2 ;
H   w  N1  1, N 2  1 N1 , N 2  N1  1, N 2  1 N1 , N 2
(Super)current in weak link =
=
=
iw
 N ,N
1
2w
2
eiN22 N2

d N1
i
   N 1, H 


dt
N1  1, N 2  1  N1 , N 2 N1  1, N 2  1 
sin 1  2 
A superfluid state is characterized by the Ginzburg-Landau
complex “order parameter” Y(x)
Y  x ~ e
i ( x )
Supercurrent ~ 
Persistent currents

C
C
 dx  2 n
No local change of the
wavefunction can change
the value of n
Supercurrent flows
“forever”
How do metals form superconductors ?
Electrons form (Cooper) pairs at low temperatures,
and these pairs act like bosons
Pair wavefunction
ky
kx

Y    k    

S 0
(Bose-Einstein) condensation of Cooper pairs
The quantum mechanics of
•Insulators
•Metals
•Semiconductors
•Superconductors