(I) Microscopic BCS theory of superconductivity - Ioan Kosztin

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Nov 15, 2013 (3 years and 9 months ago)

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Superconductivity
•Brief history of superconductivity
•Physical properties of conventional superconductors
•Microscopic BCS theory of superconductivity
•Superconducting energy gap spectroscopy
•Josephsoneffect (tunneling)
Physics 215
Winter 2002
Prof. Ioan Kosztin
Introduction to Modern Physics
Lecture #26-27
Brief History of Superconductivity
•1911 KamerlinghOnnesdiscovers superconductivity in Hg at T
c=4 K
•1913 KamerlinghOnnesthe Nobel Prize in Physics
•1933 Meissnerand Ochsenfelddiscover the
MeissnerEffect
•1941 Superconductivity is reported in Nbnitride at T
c=16 K
•1953 Superconductivity is reported in V
3Si at T
c=17.5K
•1957 Microscopic BCS theory of superconductivity is developed
•1962 The Josephsoneffect is predicted based on the BCS theory
•1962 Development of first superconducting wire (Westinghouse)
•1972 Bardee, Cooper & Schriefferwin the Nobel Prize in Physics
•1973 Josephsonwins the Nobel Prize in Physics
•1986 Müllerand Bednorz(IBM-Zurich) discover High Temperature
Superconductivity in La-Ba-Cu-O at T
c=35K !
•1987 Müllerand Bednorzwin the Nobel Prize in Physics
•1987 Superconductivity found in YBCO copper oxide at T
c=92K !!!
•1988 T
c
is pushed to 120K in a ceramic containing Ca and Tl
•1993 HgBa
2Ca2Cu3O8
is found to superconductat T
c=133K
•…………
What is superconductivity ?
= superfluidityof the conduction electrons in a metal/conductor
superfluidity= frictionless/dissipationlessflow of a
quantum
fluid,
e.g., He
4
at low temperatures (T < 2K)
•He
4
atoms are
bosons
which undergo Bose-Einstein (BE)
condensation as T -> 0K
•BE condensate exhibits superfluidity
BUT
electrons are fermions which do not BE condense !!??
⇒Then, how can conduction electrons superconduct??
•At sufficiently low T, conduction electrons can pair up,
forming
Cooper pairs
, which are
charged
(q=2e) bosons !
•Cooper pairs undergo BE condensation
•The condensate of Cooper pairs (with 2e electric charge)
exhibits superfluidityand, therefore, superconductivity
What is superconductivity ?
Attractive interaction
between fermions
Form bosons
(Cooper pairs)
[e=0]
/
Superfluidity
[e=0]
Bose condensation
Superconductivity
Formation of Cooper pairs
k
-k
Fermi Sea
ion
electron
There is a time-retarded,
effective attraction
between two electrons in a
crystal lattice (virtual
electron-phonon interaction)
The Cooper problem (1956)
If two electrons attract each other
(V<0) in the presence of a Fermi sea
then they always form a bound state,
regardless how weak the attraction is!
The bound state energy:
2/||(0)
22,
VN
c
F
EEe
ω

=−∆∆=

energy gap
DOS
at FS
Microscopic theory of superconductivity
ardeen
BC
ooperchrieffer
S
BCS theory
(1957) –Nobel Prize in Physics 1972
•Mean field approximation + pairing approximation
•Ground state = condensate of Cooper pairs
•Elementary excitations = gapped fermions
(quasi electrons and holes)
Physical Properties of Superconductors
•Critical temperature T
c
•Isotope effect
•Meissnereffect
•Persistent currents (more than just zero resistivity)
•Electronic specific heat
•Energy gap
•Coherence length
•Critical magnetic fields (B
c, Bc1, and Bc2)
•Magnetic penetration length
•Flux quantization
Superconducting Critical Temperature T
c
= the temperature at which the system (sample) undergoes a
phase transition
from a normal conducting state into a super-
conducting state, characterized by zero dc electrical
resistivity
Hg
Pt
superconductornormal
Isotope effect
Tc
varies with the isotopic mass of the sample, according to the
empirical law:
~
c
T
M
α

Exp
:
3
Substance
0.45
0.50
0.49
0.08
1/2
Zn
Hg
Pb
NbSn
BCStheory
α
Meissnereffect
B

c
T
T
>
c
T
T
<
When a superconducting sample is cooled below T
c
in the
presence of an external magnetic field, the magnetic field
(i.e., lines of the induction B) are pushed out
⇒a superconductor is a perfect diamagnet!
Persistent currents
Due to vanishing electrical resistivity, once a current is
established in a circular superconducting ring, the current will
flow indefinitely (at least for a very long time ~ years …), without
dissipation through the ring.
The persistent current in the ring can be established by cooling
the sample in an external magnetic field (cfMeissnereffect)
Why does the permanent
magnet levitate above the
(high temperature) super-
conductor immersed in liquid
nitrogen (T=77 K) ?
superconducting loop at
T>T
c
in the presence of
an external magnetic
field
the magnetic flux is
trapped inside the SC
loop, through which an
induced supercurrentis
established
Electronic specific heat CV
CV
has a characteristic jump (discontinuity) at T
c
V
C
T
c
T
s
C
n
C
()
n
CTT
γ
=
0
/
3/2
()
B
kT
s
CTATe
−∆

=
c
snc
V
TT
CCCconstT
=
∆=−=⋅
Excitation Gap and Order Parameter
h
e
E
k
kF
E
k
k
F

h-likee-like
2
2
F
k
k
E
m
ε
=−
22
kk
E
ε
=+∆
Normal
state
Superconducting
state
Excitation
gap ∆
Superconducting
order parameter ∆sc
()()
sc
TT
∆=∆
BCS gap equation:
12()
10
2
k
k
k
fE
V
E

+=→

BCS
Tin
Tantalum
Niobium
0.20.40.60.8
0.2
0.4
0.6
0.8
1.0
1.0
0.0
0.0
∆(Τ)/∆(0)
Τ/Τ
c
Coherence Length ~ size of the Cooper Pairs
0
~
(0)
F
v
ξ


Why does the Mean Field Approximation work so well ?
A:
Because the size of Cooper pairs >> the inter particle
distance
~
F
F
v
a
E

k-k
ξ
ο
a
0
~1
(0)(0)
FFF
F
vvE
E
a
ξ
=→
∆∆


Numerical example:
4
3
0
(0)~10K,~10K,~1Å
~10Å
F
Ea
a
ξ



Critical magnetic field
A sufficiently strong external magnetic field can destroy the
superconducting state
superconducting
phase
normal
phase
B
T
()
c
BT
phase diagram of a
type-I superconductor
thermodynamic
critical magnetic field
experimental
data
Critical magnetic field: Type-I and II SC
Type-I SC
Type-II SC
lower critical field
upper critical field
the magnetic field can penetrate the SC through vortices (normal
filament regions)
superconducting
phase
normal
phase
B
T
()
c
BT
superconducting
normal
T
1
c
B
B
2
c
B
vortex
Magnetic penetration depth
Magnetic fields are expelled from the bulk of a type-I
superconductor by the formation of surface currents
Both surface currents and magnetic field penetrate the
superconductor to a small, but finite (~100nm), extent
The magnetic field B
inside a superconductor
decays exponentially
with the distance x
from the surface
/
0
()
x
BxBe
λ

=
magneticpenetrationdepth
λ
=
2
0
()1
c
T
T
T
λλ

=−


Temperature dependence of the
penetration depth:
Flux quantization
B

Φ
After removing the external magnetic field surrounding a
multiply connected superconducting sample (e.g., ring), the
trapped magnetic flux (due to persistent currents) will be
quantized
15-2
00
,2.0710Tm
22
hh
nn
ee
n

Φ==ΦΦ==×


magnetic flux quantum
Energy gap spectroscopy
experimental techniques to measure the superconducting energy gap 2∆
Single particle tunneling
Absorption of EM radiation
/2/
tg
V
Eee
==∆
(Giaever1960)
electrons can tunnel through the thin insulating
layer (I) in both N-I-N and S-I-N junctions
N-I-N
S-I-N
Superconductors can absorb
photons only if their energy
is larger than the energy
gap 2∆
The photon energy E > 2∆is
absorbed and used to break
up a Cooper pair