Transcript 投影片 1 - National Cheng Kung University
10. Superconductivity
• • • Experimental Survey – – – – – – – Occurrence of Superconductivity Destruction of Superconductivity by Magnetic Fields Meisner Effect Heat Capacity Energy Gap Microwave and Infrared Properties Isotope Effect Theoretical Survey – – – – – – – – – – – – – – – Thermodynamics of the Superconducting Transition London Equation Coherence Length BCS Theory of Superconductivity BCS Ground State Flux Quantization in a Superconducting Ring Duration of Persistent Currents Type II Superconductors Vortex State Elimination of
H c
1 and
H c
2 Single Particle Tunneling Josephson Superconductor Tunneling DC Josephson Effect AC Josephon Effect Macroscopic Quantum Interference High-Temperature Superconductors
K.Onnes (1911) :
ρ
→ 0 as
T
→
T C
Experimental Survey
1. ρ → 0 for
T < T C
. Persistent current in ring lasts > 1 yr.
2. NMR: supercurrent decay time > 10 5 yrs.
3. Meissner effect: superconductor = perfect diamagnet.
Normal state SuperC state 4. BCS theory : Cooper pairs (
k
,
–k
). See App. H & I.
Occurrence of Superconductivity
Occurrence : Metallic elements, alloys, intermetallic compounds, doped semiconductors, organic metals, … Range of
T C
: 90K for YBa 2 Cu 3 O 7 .
.001K for Rh.
Si:
T C
= 8.3K at
P
= 165 Kbar
Destruction of Superconductivity by Magnetic Fields
Magnetic field destroys superconductivity.
H C
B aC
in CGS units
H C
C
0
H C
H C
1
T C
2 Magnetic impurities lower
T C
10 –4 : Fe destroys superC of Mo (
T C
1% Gd lowers
T C
= 0.92K ).
of La from 5.6K to 0.6K.
Non-magnetic impurities do not affect
T C .
Meissner Effect
B
= 0 inside superC Normal state SuperC state For a long thin specimen with long axis //
H
a
,
H
is the same inside & outside the specimen (depolarizing field ~ 0)
B
H
a
4
M
0 →
M H a
1 4 Caution : A perfect conductor (ρ = 0) may not exhibit Meissner effect. Ohm’s law
E
j
→
E
0
if
0 1
c
B
t
0 →
B
t
0 (
B
is frozen, not expelled.) Also, a perfect conductor cannot maintain a permanent eddy current screen →
B
penetrates ~1 cm/hr.
Most elements Alloys / Transition metals with high ρ.
ρ = 0 but
B
0 in vortex state .
H C
2
H C
2 ~ 41T for Nb 3 (Al 0.7
~ 54T for PbMo 6 S 8 .
Ge 0.3
).
Commercial superconducting magnets of ~1T are readily available.
C N
T
T
S N
T
→
S N
T
Heat Capacity
S S
S N
→ superC state is more ordered Δ
S
~ 10 –4
k B
→ only 10 –4 per atom e’s participate in transition.
Al
C e S
e
→ energy gap
Energy Gap
Comparison with optical & tunneling measurements →
C e S
e
E g
/ 2 not
C e S
e
E g
/
E g
4
F
For
H a
= 0, n-s transition is 2 nd order ( no latent heat, discontinuity in
C e
,
E g
→0 at
T C
).
Microwave and Infrared Properties
EM waves are mostly reflected due to impedance mismatch at metal-vacuum boundary.
They can penetrate about ~20A into the metal.
Photons with ω <
E g
are not absorbed → surface penetration is greater in superC than in normal state. For
T
For
T
<<
T C ρ s ρ s
T C
,
ρ s
, = 0 for ω <
E g
ρ n
for ω >
E g
.
. (sharp threshold at
E g
) 0 for all ω 0 ( screening of
E
incomplete due to finite inertia of
e
)
Isotope Effect
Isotope effect:
M T C
const
→ e-phonon interaction involved in superC. Original BCS:
k T B C
2
D e
1 /
N V
→
T C
Debye
M
1/2 Deviation from α = ½ can be caused by coulomb interaction between e’s. 1 2 Absence of isotope effect due to band structure.
Theoretical Survey
Thermodynamics of the Superconducting Transition London Equation Coherence Length BCS Theory of Superconductivity BCS Ground State Flux Quantization in a Superconducting Ring Duration of Persistent Currents Type II Superconductors Vortex State Estimation of H c1 and H c2 Single Particle Tunneling Josephson Superconductor Tunneling DC Josephson Effect AC Josephon Effect Macroscopic Quantum Interference • • • Thermodynamics Considerations Phenomenological Models Quantum Theory
Type I superC:
Thermodynamics of the Superconducting
B
4
M
W
0
B a
M
d
B
a dF d
B
a F N F N
S
F S
F N
2
B aC
8
F N F S d F S
1 4
B d a B a
a
F S
B a
2 8
F S
2
B aC
8
dF N dT T C
dF S dT T C
→ no latent heat ( 2 nd order transition) (continuous transition)
London model: → → 4
j
c
London Equation
j
c
4
L
2
A
→ in London gauge:
c
4 2
L
A
4
j
c c
4
L
2
B
1 2
L
A
1 2
L
B
2
B
0 2
B
1
L
2
B
London equation
A
n
surface with
j
0 0
λ L
= London penetration length
B
/ /
B
/ /
e
x
/
L
L
mc
2 4
nq
2 see flux quantization • • Meissner effect not complete in thin enough films.
H C
of thin films in parallel fields can be very high.
Coherence Length
Coherence length ξ ~ distance over which
n S
remains relatively uniform.
See Landau-Ginzburg theory, App I., for exact definition.
Local properties = Average of non-local properties over regions ~ ξ .
Minimum thickness of normal-superC interface ~ ξ .
Let → Spatial variation of
ψ
increases K.E.
→ High spatial variation of
ψ S
can destroy superC.
e i k x
1 2
e
e i k x
1
L L
/ 2 / 2
p
2 2
m
2
k
2 2
m
p
2 2
m
2 1 2 1 2 2
m
dx
1 2 2
e i q x
e
i q x
2 sin
q Lq
2
L L
/2 /2 1
e
i q x
dx
e
i k x
2
m
2
k
q
2
e
k
q
2 1
e i q x
k
1 cos
qx
2
k e i k x
1
e
i q x
2 2
m k q
2 2 2
m
k
q
2
k
2 2 2
m
k
2
k q
for
q
<<
k
2 2
m k q
→ Critical modulation for destroying superC is Intrinsic coherence length : 0 2
k F
2
mE g
v F
2
E g
see Table 5 ξ in impure material is smaller than ξ 0 . (built-in modulation) 2 2
m k q
0
E g
ξ & λ depend on normal state mfp . 1 1 0
l
1 Pure sample: 0
L
see Tinkham, p.7 & 113.
→
L
0 Dirty sample: 0
l L
0
l
→
L l ξ
0 = 10
λ L
1 2 1 2 Type I Type II
BCS Theory of Superconductivity
BCS = Bardeen, Cooper, Schrieffer BCS wavefunction = Cooper pairs of electrons
k
and –
k
(
s
-wave pairing) • • • • • Features & accomplishments of BCS theory : Attractive e-e interaction –
U
→
E g
between ground & excited states.
E g
–
U
dictates
H C
, thermal & EM properties.
is due to effective e-ph-e interaction.
λ , ξ , London eq. (for slowly varying
B
), Meissner effect, … Quantization of magnetic flux involves unit of charge 2
e
.
U D
(
ε F
) << 1 :
T C
1.14
exp
U D
1
F
E g
3.528
k T B C θ
= Debye temperature Higher
ρ
→ Higher
T C
(worse conductor → better superC)
BCS Ground State
T
= 0 Normal state Cooper pair: 1-e occupancy with
T
eff =
T C
Super state: Cooper pair mixes e’s from below & above ε F normal super but
E
normal
E
super due to –
U
.
Cooper pair : (
k
, –
k
) → spin = 0 (boson) see App. H
Energy intensity for large number of photons: →
Flux Quantization in a Superconducting Ring
E
4
n e i
I
1 4
E
* 2 1 4 4
n e
i
2
n
Let
ψ
(
r
) be the super state wave function.
Particle density
n
=
ψ
*
ψ n
= constant → Velocity operator:
n e i
v
1
m
p
q
A
c
* 1
n e
i
q
A
c
Particle flux: *
v
n m e
i
i
q
A
c e i
n m q c
A
Electric current density:
j
n q m c
j
A
q
*
v
n q
2
m c
B
n q m q
A
c
London eq. with
L
mc
2 4
nq
2
j
n q m q
A
c
Meissner effect:
B
=
j
= 0 inside superC →
q
A
c
C c d
l
q c
C
A
d
l
q
S d
σ
c q
S
B
d
σ
c ψ
measurable →
ψ
single-valued → Δ = 2
π s
h c s q
Flux quantization
q
= –2e → Flux through ring :
h c
2
e
7
gauss cm
2
ext sc s
0 = fluxoid or fluxon Φ ext not quantized → Φ sc must adjust
s
Z
see Tinkham, p.121, for a derivation via Sommerfeld quantization rule
Duration of Persistent Currents
Thermal fluctuation : superC → normal : fluxoid escapes from ring Transition rate
W
= ( attempt freq ) ( Boltzmann factor for activation barrier ) Boltzmann factor for activation barrier = exp( −
β
Δ
F
) Free energy of barrier = Δ
F
= (minimum volume) (excess free energy density of normal state) minimum volume
R
ξ 2 .
R =
wire thickness excess free energy density of normal state =
H C
2 / 8
π.
R
= 10 −4 cm,
ξ
= 10 −4 cm,
H C
= 10 3 G, gives Δ
F
10 −7 erg.
R
8 2
H C
2 Note: estimate is good for
T
= 0 to 0.8
T C
while Δ
F
→ 0 as
T
→
T C
− β 10 −15 erg at
T
= 10K →
e
F
e
10 8 10 7 Attempt freq
E g
/ 10 −15 / 10 −27 10 12 s −1
W
10 12 10 7 10 7
s
1 Age of universe ~ 10 18 s Exceptions: Near
T C
or in Type II materials.
Type II Superconductors
H a < H C1 H C1 < H a < H C2 H C2 < H a B = 0 B 0 fluxoid penetration M = 0
Type I
ξ > λ 2
Type II
ξ < λ 2 0
l
Electronic structure not much affected
Normal-Super Conductor Interface
Ref: W.Buckel, “Superconductivity” Lowering of energy due to field penetration
B
A H
8 2 Increase of energy due to destruction of Cooper pairs: Normal:
B
C
0 Bulk superC:
B
C
C
A H
8
C
2
H
8
C
2
V
Interface energy at
H
=
H C
C
B
A H
8
C
2 0 0
Type I for Type II
H C2 for Nb 3 Sn ~ 100kG.
Type I
surface energy > 0
Type II
surface energy < 0 Thin films with H normal to surface Type I: Intermediate state Type II: Vortex state Fluxoid penetration reduces increase of energy due to flux repulsion.
Vortex State
Meissner effect starts breaking down when a normal core can be substained. Normal core radius is always Fluxoids well separated: fluxoid radius λ
H C
1 ξ ; otherwise it’ll be bridged by surrounding ψ S 0 2 = Field for nucleation of single fluxoid .
Closed-packed fluxoids: fluxoid radius ξ
H C
2 0 2 Type II: κ > 1 → λ > ξ Type I: κ < 1 → λ < ξ →
H C
1 <
H C
2 →
H C
1 >
H C
2 Vortex state allowed.
SuperC destroyed before fluxoid allowable → no vortex state.
Flux lattice in NbSe 2 at 1000 G & 0.2K.
STM showing DOS at ε F .
E N – E S
= Stabilization energy →
f
core 1 8
H C
2 2 Decrease of
E
for allowing
H
penetration:
f
mag 1 8
B a
2 2 Total core energy wrt super-state
f
Threshold for stable fluxoid:
f
= 0 at
B a
=
H C
1 .
H C
1
H C
2 0 0 2 2 →
H C
2 2
H C
1
H C
H C
1
H C
2
f
core
f mag
→ 1 8
H C
2 2
B a
2 2
H C
H C
1
H C
1
H C
H C
2 1
H C
2
Single Particle Tunneling
2 metals I and II separated by insulator
C
.
T C
Al Sn 1.14K 3.72K
Glass + In contact + Al strip 1mm wide 2000A thick + Al 2 O 3 20-30 A: S.P.T.
10A: J.T.
+ Sn strip Direct measurement of J.T. requires double junctions. See W.Buckel, “Superconductivity”, p.85.
I, II both normal: line 1
N S
N n
2 2 T 0 I normal, II super: T = 0, line 2 T 0, line 3
I , II both super: T 0
Josephson Superconductor Tunneling
• • • DC Josephson effect: DC current when
E = B = 0
AC Josephson effect: rf oscillation for DC V.
Macroscopic long-range quantum interference:
B
across 2 junctions → interference effects on I S
j
→
DC Josephson Effect
i
t
1
T
2
i
t
2
n e j i
j
1
t
1 2
n
1
n
1
t
i
t
2 1 2
n
2
n
2
t
t
i
t
1 2
i T
2
i T
1
T
1
T
= transfer frequency 1 2
n
1
n
1
t
i
1
t
i T
2 1 1 2
n
2
n
2
t
i
2
t
i T
1 2
i T
i T
2 1
n
2
n
1
e i
n
1
n
2
e
i
Real parts: Imaginary parts: 1 2
n
1
n
1
t
T n
2
n
1 sin 1 2
n
2
n
2
t
T
1
t
T n
1
n
2 sin
n
2
n
1 cos 2
t
T n
1
n
2 cos →
n
1
t
2
T
n
2
t
2
T n n
1 2 sin
n n
1 2 sin
t
1
n
1 1
n
2
n
1
n
2
n n
1 2 cos 0
n
1
t
n
2
t J
n
1
t
→
J
J
0 sin
n
1
n
2 → DC current up to
i C
while
V
= 0.
AC Josephson Effect
V
across junction:
i
t
1
T
2
eV
1 1
t
1 2
n
1
t
2 1 2
n
2
n
1
t
i
t
n
2
t
i
t
1 2
i T
2
i e V
1
i T
1
i e V
2 Real parts: 1 2
n
1
n
1
t
T
1 2
n
2
n
2
t
T n
2
n
1 sin
n
1
n
2 sin →
i
t
2
T
1
eV
2 1 2
n
1
n
1
t
i
1
t
i T
1 2
n
2
n
2
t
i
2
t
i T n
2
n
1
e i
i e V q
2
e n
1
e
i
n
2
i e V
n
1
t
2
T
n
2
t
2
T n n
1 2 sin
n n
1 2 sin
n
1
t
n
2
t
Imaginary parts: 1
t
T
2
t
T n
2
n
1 cos
e V n
1
n
2 cos
e V
t
1
n
1 1
n
2
n
1
n
2
n n
1 2 cos 2
e V
0 2
e V t J
J
0 sin
J
0 sin 0 2
e V t
AC current with 2
e V
483.6
Mhz
for
V
= 1 μV Precision measure of
e
/
Macroscopic Quantum Interference
Around closed loop enclosing flux Φ: For
B
= 0, 2
s
2
q h c
2
e c
a
2
a
1
a
b
2
b
1
b
For
B
0, or
b
a
2
e c
b
0
e c
a
0
e c
J tot
J b
J a
J
0 sin 0
e c
sin 0
e c
2
J
0 sin 0 cos
e
c
periodicity = 39.5 mG I max = 1 mA zero offset due to background B Junction area = 3 mm 0.5 mm periodicity = 16 mG I max = 0.5 mA Prob 6
High-Temperature Superconductors
T C ceiling for intermetallic compounds = 23K.