Post on 23-Aug-2019
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Chapter Ten Superconductivity
An element, inter-metallic alloy or compound will conduct electricity without resistance below a certain temperature.
Temperature (K)
Res
ista
nce
(Ω)
Temperature (K)
Res
ista
nce
(mΩ
)
1987 U. of AlabamaOnnes H.K.,Commun. Phys. Lab. Univ. Leiden 124C (1911)
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1988 Bi- 110K
1988 Tl- 125K
1993 Hg- 135K
The world record Tc
138K
Thallium-doped mercuric –cuprate
TlHgBaCaCuO
1994 NIST-Colorado
Under extreme pressure ~300,000 atmospheres,
Tc can be increased by 25 ~ 30 K.
3
References :
Kittel, Introduction to Solid State Physics, 8th Ed., Ch.10, 2005.
Aschroft and Mermin, Solid State Physics, Harcourt, Ch.34, 1976.
Duzer and Turner, Principles of Superconductive Devices and Circuits, Elservier, NY, 1981.
Tinkham, Introduction to Superconductivity, McGraw-Hill, 1975.
Orlando, Foundations of Superconductivity, Addison-Wesley, 1991.
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The history of superconductors in brief :
1911 Superconductivity was first observed in Hg by Onnes of Leiden U.
1933 Meissner effect was discovered by Meissner and Ochsenfeld.
Superconducting materials will repel a magnetic field.
“Diamagnetism”
1935 Two fluid model was proposed to explain zero resistance behaviorand Meissner effect by Fritz and Heinz.
1950 “Phenomenological theories” (QM) were proposed to explain superconductivity by Ginzburg and Landau.
1957 The first widely-accepted theoretical understanding of superconductivity was advanced by Bardeen, Cooper, and Schrieffer. “BCS theory”
1913
1853~1926
19721908~1991 b: 1930 b: 1931
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1957 Abrikosov predicted the existence of the other kind superconductor.(Type II superconductors)
• 1960 Tunneling experiment was performed by Giaever.Experimental measurement of the energy gap.
1962 Josephson effect was proposed by Josephson.A significant theoretical advancement. “SQUID”
1973
b: 1929
1973b: 1940
2003b: 1928 b: 1916Ginzburg
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1980’s A decade of unrivaled discovery in the field of superconductivity.1964 Little suggested the possibility of organic superconductors.1980 Bechgard found the superconductor (TMTSF)2PF6. Tc=1.2K.1986 Müller and Bednorz created a brittle ceramic compound w/.
“insulator” Tc=30K.
1987
b: 1927 b: 1950
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Phenomena Occurrence, properties
Theory Why superconductivity occur ?
Why different from ordinary metals ?
Applications Why scientifically interesting ?
Why technologically important ?
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1.14K
3.72K
7.19K
3.40K
1.09K
2.39K4.15K
0.39K 5.38K
0.55K
0.12K
9.50K
4.48K
0.92K 7.77K
0.01K 1.4K 0.66K 0.14K
0.88K
0.56K0.51K
0.03K
1.37K 1.4K
New elements keep being added to the list.
In 2002, Li was shown to superconduct under pressures of 23 to 36GPa with critical temperature of 9 to 15K. V.V. Struzhkin et al., Science 298, 1213 (2002).
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Superconducting alloys
PbSn 6.5K solder
NbTi 12K magnet wire
Nb3Al 18.9K
Nb3Sn 18.3K
Nb3Ge 23K Highest “low” Tc
A-15 compounds
LaSrCuO 30K 1st high Tc oxide
YBa2Cu3O7 92K 1st Tc>77K
HgBaCaCuO 134K Highest Tc
150K (under high pressure)
High Tc materials
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Buckminsterfullerene(buckyball)
a closed sphere is formed by 60 carbon atoms and doped with Alkali metals
Organic superconductors
(TMTSF)2PF6 0.4~12K 1980
YPd2B2C 23 KLuNi2B2C 16.6 KYNi2B2C 15.5 KTmNi2B2C 11 K (resistance increases below Tc)ErNi2B2C 10.5 K (ferromagnetic)
(TMTSF)2PF6
UPt3
Ceramic superconductors
K3C60 18K 1991
Cs3C60 40K fulleride
Heavy FermionsHeavy fermions are compounds containing rare-earth elements such as Ce or Yb, or actinide elements like U.UPt3 0.48K CeRu2 6K UPd2Al3 2K
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Basic properties :
A truly zero DC resistance
Onnes ρsc < 10-8 × ρ(300K) = 10-8 × 100µΩcm = 10-14 Ωm
Modern technique : decay of supercurrent in a loopρsc < 10-18 × ρ(300K) = 10-18 × 10µΩcm = 10-25 Ωm
(NMR or SQUID)
A phase transition at Tc
Transition is reversible and very narrow (<10-4 K)
broadened by impurities, fluctuations, and inhomogeneities
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Magnetic properties more unusual than ρ=0
Superconductivity can be destroyed by magnetic field as temperature
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
T/Tc
H/Hc
Superconductingstate
Normal state
−= 2
c
2
c TT1)0(HH(T)
100Gauss/K
constantT
)0(HT )0(H
c
c
cc
≈
=
∝Also,
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Meissner effect : magnetic field is expelled. (1933)
T>TcNormal state
T<TcSuperconducting state
cooling down
In a constant H (< Hc)
When a specimen is placed in a magnetic field and is then cooled through the transition temperature for superconductivity,the magnetic flux originally present is ejected from the specimen.
A bulk superconductor in a weak magnetic field will act as a perfect diamagnet, with zero magnetic induction in the interior.
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Diamagnetism
M 4HBrrr
π+=magnetization
total field
applied field
M is paralleled with H – paramagnetismM is anti-paralleled with H – diamagnetism
Applying magnetic field H to a magnetic sample, screening currents and alignment of magnetic moments (spins) alter the internal magnetic field.
EM theoryfor B=0
H
B
normal
superconductingHc
H
-4πM
normalsu
perconductin
g
Hc
superconductor
HM4
0Brr
r
=−
=
π
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Diamagnetism can not be attributed to perfect conductivity (ρ=0).
Ohm’s law : Maxwell’s equations :
J Err
ρ=
tB E
∂∂
−=×∇r
r
0 tB hence, and 0E 0when =
∂∂
==r
rρ
constantB =r
Two different possible states for a perfect conductor.
Field kept out
Field trapped inside
Final state below Tc in the presence of H (<Hc) depends on history of the cooling process for a perfect conductor.
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Superconductor is perfectly diamagnetic.Transition to superconducting state is perfect reversible.
Meissner effect
Field is expelled.
B=0
independent of history.
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Type I and type II superconductors
In fact, perfect diamagnetism up to Hc is rare seen in less than 1% of known superconductors.
Type I Perfect diamagnets Most elements (Al, In, Sn, Pb,..)(Meissner 1933)
Type II Flux penetration Most alloys, films, dirty metals,…(Abrikosov 1957)
Meissner
Soft superconductor Hard superconductor
Type II superconductors stay superconducting at higher field than type I.
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Hc1 : lower critical field < Hc
Hc2 : upper critical field >> Hc
H<Hc1, superconducting state (Meissner effect)
Hc1<H<Hc2, vortex state “fluxoids” in the superconductor
H>Hc2, normal state current vortices
Nb3Sn, NbTi Hc2 ~ 10T
Nb3Al.7Ge.3 41T
YBa2Cu3O7 100T
Pb PbIn alloys w/. weight % of In(B) 2.08%(C) 8.23%(D) 20.4%T=4.2K
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Abrikosov vortexstate
Flux lattice in NbSe2 at 1000Oe at 0.2K
Magnetic flux enters sample and forms a triangular lattice of vortex lines.
H
Flux/Vortex
27
o
cm-Gauss1007.2 2
−×=
=Φeh
Flux Quantum
Density of vortex lines
o
BnΦ
= number/unit area
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More reality
B=0 only in bulk of a type I superconductor
Screening currents flow within depth λ of surface and B decays in that. Penetration depth
J
superconductor
H
B, J
boundary
λ
x
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Energy gap
Superconductors exhibit energy gap between the ground state and the lowest excited state.
filledEF
filled
∆
normal state superconducting state
~ 10-4EF
This gap is different from the energy gap of insulators(electron-lattice interaction)
The energy gap of superconductors is due to electron-electron interactionwhich order the electrons in k space with respect to the Fermi gas of electrons.
Electrons in the excited state above ∆ behave as normal electrons causing resistance. At DC they are shorted out by the superconducting electrons.
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Heat capacity3T T C βγ +=Normal state :
electron Phonon : Debye T3 model
normal statejump(1) A heat capacity jump occurs at Tc.
A 2nd order phase transition
(2) Below Tc,
an exponential temperature dependence
Al
TkTaT
B
c
e e C∆
−−∝∝
N.E. Phillips, Phys. Rev. 114, 676 (1959).
Superconducting state : H=0
Normal state : H = 300 Gauss
43.1C
CC
n
ns =−
(3) At Tc,
can be explained by BCS theory.
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Tc
Al
Tc
Al
Thermodynamic relations
Because of no structural change and the occurrence of Meissner effect, we can apply thermodynamics arguments to understand the superconducting transition.
2
2
T F
T S C
T F S
∂∂
=∂∂
=
∂∂
−=Entropy
Heat capacity
Fn(T) and Fs(T) merge at Tc.Fs(T) ≤ Fn(T) Ss(T) ≤ Sn(T)
Electrons are more ordered in the superconducting state than in the normal state.
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TTcba
TClog
c
−=
γTkBe C
∆−
∝
must supply enough thermal energy kBT to excite carrier across gap.
# of carrier : normal ~ kBT/EF
superconducting ~ exp[-∆/kBT]
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Electromagnetic absorption
hν ≥ ∆Photon energy is sufficient to create excitons
Ref
lect
ion
coef
ficie
nt
νh
2∆
normal
superconducting Threshold at 2∆ suggests excitations are created in pair.
Absorption is the same in normal and superconducting states for
ν >>∆/h.
( ) 480GHzHz108.4eVs106.1/1063.6
)eV10(22
5.3Tk
2
111934
3cB
=×=××
=∆
=
≈∆
−−
−
hυ
estimate 1 meV
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Iosotope Effect
Tc varies with the isotope of the same element.
This effect is also found in some compounds.
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Tc varies with isotope mass.
constantTM c =α
How does nuclear mass affect an electronic phenomena like SC ?Phonons (electron-phonon scattering)
Weak electron-phonon coupling
good normal conductor
bad superconductor
Strong electron-phonon coupling
good superconductor
0.08Nb3Sn0.50Hg
0.49Pb0.47Sn
0.33Mo0.32Cd
0.15Os0.45ZnExperimental values of α The original BCS model
gave 2/1Debyec MTT −∝∝
without the consideration of Coulomb interaction between electrons.
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Theory of superconductivity
Phenomenological models :
Two-fluid model (normal + superfluid )
London equation (EM Meissner effect)
Ginzburg-Landau (Order parameter)
Microscopic theory : BCS theory
Key evidence :
(1) Energy gap Excitations
(2) Isotope effect Phonons
(3) Phase transition Thermodynamics
Phase transition occurs when free energy of SC state lower than that of normal state.
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Thermodynamic relations
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
T/Tc
H/Hc
dHdG Μ−=change in Gibbs free energy at constant T
costing free energy to expel magnetic field
∫
∫•=
•−=−
Hd4H
HdT)(0,GT)(H,G ss
rr
rr
π
Μ
for H≤Hc
8πHT)(0,GT)(H,G
2
ss =−
G
Gn=Gn(0)=constantnormal state
superconducting stateEc
Hc
H8πHT)(0,GT)(0,GE
2c
nsc =−=
Condensation energy
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costing free energy to expel magnetic field (Meissener)
We can get Ec by measuring Hc.
o
2c
2c
c 2H
8πHE
µ==
(CGS) (SI)
Field energy density
Typical Type I superconductor
Hc ~ 100Gauss
Ec ~ 2.5×1014eV/cm3 carrier density n ~ 1022/cm3
Condensation energy per electron ~ 10-7eV/electron
Big or small ?
Fermi energy EF ~ 10eV
Superconducting energy gap ∆ ~ 10-3 eV
Condensation energy Ec ~ 10-7 eV
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London equation and two fluid model
• Fritz and Heinz London found a “modified” set of Maxwells’ equations that have the Meissner effect as a solution.
• They started by assuming “two fluid model”.
Two fluid model
Superfluid density ns – supercurrent σ = ∞Normal state density nn – dissipation
mne2τσ = flow in parallel
Tc
T
ns
nn
Car
rier d
ensi
ty
constantnnn ns ==+
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They postulated :
EJ (2)m
EendtJd )1(
nn
2ss
rr
rr
σ=
=
nnn
sss
venJ
venJrr
rr
−=
−=ns JJJrrr
+=where and
Eq.(1) suggests that the superfluid responds to electronic fluid in the same way that free charges do :
Eedtvdm
rr
−= ( ) m
Eendt
vend 2sss
rr
−=−
tB
c1E∂∂
−=×∇r
rrSubstituting Eq.(1) into
( ) tB
mcenJ
dtd 2
ss ∂
∂−=×∇
rrr
Bmc
enJ2
ss
rr−=×∇
Maxwell’s equation :
( ) Jc
4B s
×∇=×∇×∇
rrrrr π
London equation B
mcen 4B 2
2s2
rr π=∇Therefore,
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This gives the “London Penetration Depth”
B1B 2L
2 rr
λ=∇
en 4mc
2s
2
L πλ =
and is also consistent with Meissner effect.An uniform field exists in a superconductor only when B=0.
where
superconductor
In the pure superconducting state the only field allowed is exponentially damped as we go in from an external surface.
For a semi-infinite superconductor in a vacuum under a homogeneous field B)0,0,(B =
r
0
B
−=
L
xB(0)expB(x)λ
Screening currents in the region of size ~ λLx
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s2L
s2 J1J
rr
λ=∇ ( ) B
mcen B
mcenJ
2s
2s
s
rrrrrrr×∇−=
×∇−=×∇×∇
−=
Lss
x(0)expJ(x)Jλ
has the solution in the similar form
Sn 34nm
Pb 37nm
YBaCuO 150nm
λL(0)
Tc
λL Meissnereffect
Superconducting currents that screen the interior of superconductor against external fields also decay exponentially with distance into the superconductor.
Lower carrier density larger λL
en 4
mc2
s
2
L πλ =
Decay length for current and fields gives measure of superfluid fraction.
T
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BCS theory
Electron pair via attractive electron-electron interaction mediated by phonon.
“Cooper Pair”
SC is a many body problem – cannot ignore interaction of electrons
Free electron model
Band theory electrons
Pairing lowers energy to give SC phase transition.
Pairing causes electrons (Fermions) to act like “Bosons”– allowing them to condensate into a “superfluid state”.
will not give SC
Three questions :
(1)What is attractive force – don’t electrons repel by Coulomb force ?
(2)How is energy lowered ?
(3)Nature of ground state and excitons ?
36
Attractive force
Microscopic theory in 1957 by Bardeen, Cooper, and Schrieffer
Ground state of e-s near EF in a superconductor occupy pair states-- equal and opposite momenta-- spin singlet
Motions of 2e- in “Cooper pair” are highly correlated.
Source of correlation is a phonon mediated attraction between e-snear EF within hωD of EF.
k ,k ~ pair ↓−↑Ψrr
+ + + + + + +
+ + + + + + +
+ + + + + + +
e-
e-
k ↑
-k ↓
step I
step II
step III
1st electron (k↑) passes ions
lattices are polarized
Before polarization of lattices disappears, the (-k ↓) e- get attracted.
37
Net attraction between e- mediated by lattice (phonon).
vF>>vphonon e- can respond to distortion before it propagates away
k 1 ↑r
k 2 ↓r
absorbs the phonon
emits the phononvirtual
k1
k2
k1’
k2’
q
Or in the opposite way
The lattice deformation reaches its maximum at a distance,
nm100m10s10cm/s10~2v 7138F ==× −−
Dωπ
Phonon vibration period
The two electrons correlated by the lattice deformation thus have an approximate separation about 100nm.
The “size” of a Cooper pair.
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Condensation energy phase transition
Normal metal at T=0Kf(E)
Filled Fermi surface
1
T≠0K
EEF
Allow two electrons to be excited (costs Kinetic Energy)
And interact via attractive force (gains Potential Energy)
Cooper showed that this lowers system energy – normal ground state unstable to attractive force
Phase transition would occur.
Continue to allow more pairs to be excited and scatter until point of diminishing returns
KE (loss) > PE (gain)
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Two electrons [k1, E(K1)] and [k2,E(K2)] are in states just above EF.
A weak attractive interaction between these two electrons is turned on in the form of phonon exchange.
All other electrons in the Fermi sea are assumed to be non-interacting, and on account of the Pauli exclusion principle.
constantkkkk '2
'121 =+=+
rrrrKr
Momentum conservation
kF
(outer)EF+hωD
EF-hωD(inner)
EF kx
ky
1kr
2kr
−kr
∆
The number of pairs k1, k2 is proportional to the cross volume in k-space and is maximum for K=0
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Electron pairs with equal and opposite wave vectors
Two particle wave function 21 kkrr
−=)r,r( 21
rrΨ
( )( ) )r,r(2E
)r,r(E)r,r()r,r(V)r,r(m2
21F
2121212122
21
2
rr
rrrrrrrrh
Ψ+=
Ψ=Ψ+Ψ∇+∇−
ε
m2kE
2F
2
Fh
=
where ε is the energy of electron pair relative to the interaction-free state (V=0) in which two electrons at the Fermi sea would possess an energy
( ) ( ) ( )( )212211 rrkexp1rkexp1rkexp1 rrrrrrr−•
Ω=•
Ω•
ΩiiiNon-interacting
two electrons
( ) ( )( )∑ −•Ω
=−Ψk
2121 rrkexp)kg(1rrrrrrrr i
Two electrons with non-vanishing interaction V≠0
g(k) is the probability amplitudeprobability amplitude of finding the electron pair 2
)kg(r
probability
kkk 21
rrr=−=
( )k,krr
−
41
+><
=hh /)2m(Ek
kkfor 0g(k)
F
F
Dω
Substitute the wave function in Schrödinger equation
( ) )g(k2E)Vg(k'1g(k)mk
Fk'
'kk
22
ε+=∑Ω
+h
( )∫ •−−= r)'kki(drV(r)expV 'kkrrr
wheredescribes scattering of electron pair from (k,-k) to (k’,-k’)
+<<−=otherwise 0
Em2k,
m2kEfor VV F
2'222
Fok'k,
Dωhhh
In the simplest model,
∑Ω
−=
++−
k'
oF
22
)g(k'V
g(k)2Emk εh
ε++−
∑Ω−=
F
22k'
o
2Emk
)g(k'V
g(k)h
∑++−Ω
−=k
F
22o
2Emk
1V1
εh
42
∑Ω k
1Summing all states in k-space
( )∫−−
∫∇
Ω=
επ Fk
E3o 2EE2
dEE(k)
dS2
V1
( )dE
E(k)dS
2D(E)dE
constE(k) k
E3
∫
∇Ω
==π
( )F32
3
F E2
m2)D(E
hπ−=and
∫ −=
D
0Fo 'E2
dE')D(EV1ω
ε
h
( )
−−
=−=ε
εωε ω DFo0
Fo 22
)D(EV'E2
2)D(EV
1 D hll
h nn
where E’=E-EF energy respective to EF
−−=
)D(EV2exp2
FoDωε h
εωD
Fo
21)D(EV
2exp h−=
( ))D(EV/2exp12
Fo
D
−=
ωε h
For weak interaction VoN(EF)<<1
43
There thus exists a two-electron bound state, whose energy is lower than of the fully occupied Fermi sea by an amount ε = E-2EF <0.
Cooper pair ( )↓−↑ k ,krr
The two electron wavefunction is symmetric in spatial coordinate (r1,r2) under exchange of electrons 1 and 2, but the whole wavefunctionincluding spins must be antisymmetric .
consistent with the Pauli exclusion principle
−−=
)D(EV2exp2
FoDωε h
Phonon frequencye-ph coupling strength
density of state at EF
−−=+=
)D(EV2exp22E2EE
FoDFF ωε h
High phonon frequencyStrong e-ph couplingLarge density of states
Strong SC
44
The BCS ground state
dynamic states with electrons near Fermi surface scattering
k
-k
k’-k’
kx
ky
kF “annihilation” of a pair
“creation” of a pair
( )↓−↑ k ,krr
( )↓′−↑′ k ,krr
an energy reduction of Vkk’.the scattering of an electron pair with wavevectors to ( )k ,k
rr− ( k ,k −′
rr )′
okkkkiN ...),...kg(kii
φ∗↓−
∗↑
∗↓−
∗↑∑=Ψ
lll CCCC
N electrons with the Cooper pairing
vacuum stateN/2 pairs of creation operators
Grand canonical ensemble
45
( )∏=
∗↓−
∗↑+=Ψ
M1...kkkokkkkG vu φCC
BCS ground state
2kv the probability of the pair (k↑,-k↓) being occupied.
2k
2k v1u −= the probability of the pair (k↑,-k↓) being unoccupied.
∑∑ ↓−↑∗↓−
∗↑+=
llll
kkkk
kkk VnH CCCC
σσε
Pairing Hamiltonian
Determining the coefficients by Variational Method
0Vn Gk
kkkk
kkG =Ψ+Ψ ∑∑ ↓−↑∗↓−
∗↑
llll CCCC
σσεδ
-µN
l↑
-l↓k↑
-k↓
q
Set Ek’= Ek - EF single-particle energy relative to EF
∑∑ +=Ψ−Ψl
lllk
kkkk
2k
'kGG vuvuVvE2NH µ
46
the excitation energy of a quasi-particle of momentum hk( ) 2
k2
Fkk EE ∆+−=ε
∑−=∆l
lll vuVkk the minimum excitation energy, energy gap
−=∆
)D(EV2exp2
FoDωh
( )
2k
k
'k2
k
22Fk
Fk
k
'k2
k
v1E121u
EE
EE121E1
21v
−=
+=
∆+−
−−=
−=
ε
ε
47
Condensation energy
−=∆∆=−=
)D(EV2exp2 e wher)D(E
21EEE
FoD
2FSNC ωh
o
2
oFo
22
Fo
2
VPE
V)D(E211
V )D(E
21
VKE
∆−=∆
−
∆=∆−
∆=∆
1K)~ re temperatuTransition (T Tk ~ 100K)~ re temperatuDebye ( k
ccB
DDBD∆
ΘΘ=ωh EN
Es
KE PE
Ec
cost
gain
37.0~22)D(EV
DFo
∆
=ωh
ln
48
Electrons within ∆ of Fermi surface scatter # of electrons = D(EF)∆
Each pair contributes a decrease in energy by ∆
Condensation energy : 2F )D(E
21
∆
EF
∆
4
F
c
F
10~TT
E−≈
∆
Ek’=Ek-EF
Probability that e- state k is occupied.
Probability that pair state (k,-k) is occupied.
-hωD hωD
In reality, f(Tc)~vk2(0).
Electron distribution changes very little from Tc to 0K.
49
Excitation
(Ek-EF)/∆
εk/∆
1ElectronsHoles
Normal metalsEF
0
(Ek-EF)/∆
εk/∆
0
1
Superconductors
Electron-likeHole-likeEF
∆
( ) 2k
2Fkk EE ∆+−=ε
Quasi-particles
50
Ek’
0 ∆
D(Ek’)
1xDN Normal state
Density of quasi-particle state
( ) N22Fk
FkN
22'k
'k'
k DEE
EED
E
E)D(E
∆−−
−=
∆−=
States below the gap ∆ are pushed into divergence at the gap edge
Temperature dependent energy gap
T/Tc
∆(T
)/∆(0
) cBTk76.1)0( =∆
)0(TT1)T(
c
∆−=∆
51
Ginzburg-Landau Theory
phenomenological theory of phase transition
can be applied to the superconducting state
give equations for Ψ “the condensate wavefunction”
explain the difference between type I and type II superconductors
Concept of the order parameter Ψφiesn=Ψ
M=Ψ magnetizationsn=ΨΨ∗for superconductors,
for ferromagnets,
T/Tc
M(T
)/M(0
)
Ferromagnetic materials
Non-magnetic
Curie temperature
52
Expand free energy density near the transition in terms of Ψ
∫ •−Ψ
−∇−+Ψ+Ψ−=
H
0
242
ns 'dHMc
qA2m1
2(r)F(r)F hiβα
where α, β are phenomenological constants, α=αo(T-Tc), andβ >0 gives transition and stability
Fn(r) is free energy density of normal state, Ψ=0 as T>Tc
increases in energy due to spatial variation of Ψ-- like the kinetic energy density
2Ψ∇
q = -2e for an electron pair (Cooper pair), ns(r) : number density of Cooper pair
qA/c ensures that we have gauge invariance of the free energy
gives increase in F due to flux exclusion∫ •− 'dHM
53
( ) 0at 0FF ns =Ψ=Ψ∂−∂ ( )
ons at 0FF
Ψ=Ψ=Ψ∂−∂
Fs-Fn Fs-Fn
T>TcT<Tc
54
Minimize F wrt Ψ
.c.cc
qAc
qA2m1(r)F 2
s +Ψ
−∇−•Ψ
−∇−+ΨΨ+Ψ−= ∗δβαδ hh ii
Integrate by parts ( )( ) ( )∫∫ ∗∗ ΨΨ∇−=Ψ∇Ψ∇ δδ 2dVdV
If δΨ* vanishes on the boundaries, so that
.c.cc
qA2m1dV(r)dVF0
22*
s +
Ψ
−∇−+ΨΨ+Ψ−Ψ== ∫∫ hiβαδδ
= 0Ginzburg-Landau Equation
0c
qA2m1 2
2 =Ψ
−∇−+ΨΨ+Ψ− hiβα
Minimize F wrt A
( ) Amcq
2mq(r)j *
2*
s ΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗hi
BA and (r)jc
4H s
rrrr=×∇=×∇
π
55
This order parameter is closely related to the many body wavefunctionof superconducting state.
Apply these equations to (1) Hc2(T)
(2) coherence length ξ(3) penetration depth λ(4) flux quantization
(1) Set H=0 and let Ψ(r) be spatially constant
42ns 2(T)F(T)F Ψ+Ψ−=−
βαand from Ginzburg-Landau equation
so that
The earlier discussions gave
βα
=Ψ
βα2
FF2
ns −=−
π8HFF
2c
ns −=−
πβα
8H
2
2c
2
=
relates critical field to two of the phenomenological parameters
56
(2) Coherence length ξ
How does Ψ(r) vary?
02m
22
2 =Ψ∇−ΨΨ+Ψ−hβα
solution
=Ψξβ
α2
tanh)x( x
m2
2h
c
2
TT1~
)T(m2 −≡
αξ h
x
Ψ=0normal
Ψ(r)Spatiallyvarying
Ψ=(α/β)1/2
Super-conductor
x=0 x→∞
x/ξ
Ψ(x/ξ)
1 20
Ψ∞Normal
Superconducting
Ginzburg-Landau Equation :
B.C. Ψ=0 at x=0 in 1D
where the intrinsic coherence lengthαξ ≡
α(T) ~ T-Tc Hence, diverges at T=Tc
57
(3) Penetration depth λ
( ) Amcq
2mq(r)j *
2*
s
rhrΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗i
Assume that Ψ is real or φ=constant φiesn=ΨA
mcq
Amcq(r)j
22*
2
s
rrr Ψ−=ΨΨ−=
Amc
qB
4c(r)j
222
s
rr×∇
Ψ−=∇−=×∇
πB
mcq 4
B 2
222 Ψ
=∇π
απβλ 2
2
L q 4mc
= the London penetration depth
x/λ
B(x/λ)
1 20
HoSuperconducting
Normal
58
Dimensionless ratio κξλ
≡
πβ
α
απβ
ξλκ
2qmc
m2
q 4mc
2
2
2
hh==≡
21
=κ separates superconductors of type I and II.
κ>>1 Type II superconductorκ<<1 Type I superconductor
59
0.163.423Sn
1.023.93.8Nb
0.453.78.3Pb
0.011.6160Al
λo/ ξoλo (µm)ξo (µm)metal
(4) Flux quantization
( ) Amcq
2mq(r)j *
2*
s
rhrΨΨ−Ψ∇Ψ−Ψ∇Ψ−= ∗i φiesn=Ψand
ns is constant in space but the phase of the order parameter is not, φ(r).
( )A
mcnq
mnq
Amc
nqnn2mq(r)j
s2
s
s2
sss
rh
rhr
−∇=
−∇+∇−= −−
φ
φφ φφφφ iiii eeieeii
vector potential
gradient of the phase of the order parameter
60
Ac
q)r(jnq
m ss
r
h
r
h+=∇φ
For Ψ to be single valued s 2 dc
πφ =∫ •∇ lr
If path is well inside the superconductor where js=0
( ) ∫ •=∫ •×∇∫ =•=∫ •∇ σσφ rr
h
rr
hlrr
hlr
dBc
qdAc
q dAc
q dcc
where dσ is the an element area on a surface bounded by c
where s is an integer
0dB =Φ=∫ • σrr
flux through the loop is zero
If path is well inside the superconductor where js=0
cqdB
cqs 2
h
rr
h
Φ=∫ •= σπ
so that the flux is quantized in units of 2πhc/q
2,...1,0,s re whes q
c2±±==Φ
hπ
On the basis of the BCS theory q=2e, 27-o cm-Gauss102.07
ec
×==Φhπ
61
Type I superconducting ringT<TcT>Tc
flux expelled from superconductor, get “trapped” flux in the center of ring, this trapped flux is quantized costing energy to remove flux
field penetrates metal uniform flux throughout
Type II superconductors H>Hc1
Hc1 Hc2
flux penetration
In flux penetration region,we must also quantize flux
62
c regions of flux penetration
∫ •
+==∫ •∇
cs
scdA
cqj
nqms 2 d l
rr
h
r
hlr
πφ
Minimum in energy when s=1 a “fluxoid” or “fluxon”
Field penetrates (Hc1<H<Hc2) by introducing “fluxoids” into the superconductor.
-- quantum of flux
63
Fluxoids
Field in the center of fluxoid is Hc1 when applied field is Hc1.
Field extends and decays away from the center over a distance ~λ.
“Core” or center of vortex is normal.
Radius of the core ~ ξ.
Hc2 is the field where cores overlap, i.e. all normal.
2c1o H πλ=Φ
2o
c1HπλΦ
=
2o
c2HπξΦ
= κ ~82Hc1
x
B
λ
Isolated Abrikosovvortex
64
Abrikosov vortexstate
Flux lattice in NbSe2 at 1000Oe at 0.2K
Magnetic flux enters sample and forms a triangular lattice of vortex lines.
The photo was taken using scanning tunneling microscope.
Hr
vortex core
Superconducting region
sensitive to density of states
65
Single-particle tunneling
Tunneling junction : metal -insulator- metal sandwich barrier : thin enough the e-s can tunnel through
but not leak throughMetal A Metal B Tunneling measurement in SC was first
performed by Giaever.
Metal A Metal B
A V
V
I
0
When one electrode is replaced by a superconductor, it becomes a SIN junction and the IV characteristic deviates from the straight line
due to the absence of states within the energy gap.
N22'
k
'k'
kS DE
E)(ED
∆−=
66
Cooper pair tunneling (Josephson superconductor tunneling)
Super-conductor
Ψ1
Super-conductor
Ψ2
For electron get from 1 to 2 (or 2 → 1)it must tunnel through the insulating barrier.
21s22s11 n and n θθ ii ee =Ψ=Ψ
Assume tunneling rate small,
aE t
aE t
1222
2111
Ψ+Ψ=∂Ψ∂
Ψ+Ψ=∂Ψ∂
∗h
h
i
iIn the absence of tunneling
222
111
Et
E t
Ψ=∂Ψ∂
Ψ=∂Ψ∂
h
h
i
i
a describes the small coupling between S1 and S2 due to tunneling.
a=ΨΨ=ΨΨ ∗1221 VV
2111s2s111s1
s1
s11 nanEnn2
n t
θθθθ θ iiii eeieeii +=
+=
∂Ψ∂ &&
hh
67
( )12
s1
s21
s1
s11 n
naE
2nn θθθ −+=+ iei &
h&h
( )21
s2
s12
s2
s22 n
naE
2nn θθθ −+=+ iei &
h&h
Real part( )
( )12s2s1s1
12s1
s211
sinnn2an
cosnnaE
θθ
θθθ
−=
−+=
h&
&h
Imaginary part
( )
( )12s2s1s2
12s2
s122
sinnn2an
cosnnaE
θθ
θθθ
−−=
−+=
h&
&h
tn
tn s2s1
∂∂
−=∂∂
∂∂
∂∂
tnor
tn s2s1Tunneling current I is determined by
12
s2s1o nn2I
θθδ −=
=h
aδsinI
tnI o
s1 =∂∂
≈ ?where
68
condition : small enough currents (ns1and ns2 are nearly constant.)
δsinII o= is the Josephson current relation
Supercurrent through junction depends on the phase difference across junction.
For identical superconductors s1 and s2 (ns1≅ ns2)
( ) 2121 EE −=−θθ &&h
( )hh
&& eV2EE 1212 −=
−=−= θθδ
dtd
teV2)0()t(h
−= δδ
where V is the voltage across junction
For a fixed V externally,
−= teV2)0(sinII o
hδ
the current oscillates with
frequency h
eV2=ω
AC Josephson effect
69
DC Josephson effect
As V=0, 0=dtdδ
and ( ))0(sinII o δ= constant
Supercurrent flow through the junction in the absence of bias voltage.
AC Josephson effect
−= teV2)0(sinII o
hδ
h
eV2−=
dtdδ
As V≠0, oscillatingand
Supercurrent oscillate through the junction with a frequency 2eV/h by applying a constant voltage across the junction.
Volt VMHz6.483eV2
µω ×==
h
70
Superconducting Quantum Interference
Jtot
insulator
1 2
Two Josephson junctions
No voltage is applied.
Two paths from point 1 to point 2 : a and ba
b
superconductor
In the presence of magnetic field and flux is ΦΦ+=
Φ−=
cec
e
ob
oa
h
h
δδ
δδba δδ =
∫ •+∆−=∫ •
+∇−=∫ •
2
1
2
s
2
1
2
s
2
1d
mc2e
mend
mc2e
mendJ l
rrhlrrh
lrr
AA xφφ
0, in the interior of SC∫ •−=∆2
1a d
c2e
lrr
hAφ ∫ •−=∆
2
1b d
c2e
lrr
hAφ
c2e
abh
Φ=∆−∆ φφ
In the absence of magnetic field
71
Therefore,
cecossin2J
cesin
cesinJJ
oo
oootot
h
hh
Φ=
Φ
−+
Φ
+=
δ
δδ
varies with magnetic flux Φ
has maxima when w/. integer s.πs=Φc
eh
T=39.5mG
T=16mG
72
Devices based on two junction loops are called SQUIDs.magnetometer
An applied field is measured by monitoring the current as the flux increases from zero and counting the maxima.
Flux can be measured with a precision much smaller than πhc/e. If the loop area is 1cm2, induction fields less than 10-11T can be measured.
SQUIDs are also used as sensitive ammeters and voltmeters.
Current noise Voltage noiseEnergy sensitivity 10-22 Joule
HzpV/HzpA/ 0.3
73
74
Magnetic Resonance Imaging (MRI) Biomagnetism
1905-1983
1952
1912-1997
By impinging a strong superconductor-derived magnetic field into the body, hydrogen atoms that exist in the body's water and fat molecules are forced to accept energy from the magnetic field. They then release this energy at a frequency that can be detected and displayed graphically by a computer.
NMR was actually discovered in 1946 by Felix Bloch and Edward Purcell. The first MRI exam on a human being was not performed until July 3, 1977.
-- took almost five hours to produce one image! Today's faster computers process the data in much less time.
75
Small- Scale Applications of superconductors
76
Applications of superconductors
Trains can be made to "float" on strong superconducting magnets.
MLX01
Bogie
Magnetic levitation
1996-2002 Yamanashi Maglev Test Line1999 April 14 MLX01 test line, reaching 581km/hr w/. seat capacity 68
ML-500
1977 Miyazaki Maglev Test Track
1979 December reaching 517km/hr
77
Ship Propulsion :5MW HTCS Motor March 2003
8 million projectONR
70 million projectONR
78
30m long, 30KV, 2,000A HTC power line in Copenhagen, Denmarksupply 50,000 households May 2001
500m HTCS power line in Yokosuka, Japan
79