Chapter 4: Superconductivity Flashcards
Superconductivity
(Overview: 3 points)
- Below critical temperature TC:
- Perfect conductor: resistivity vanishes
- Perfect diamagnet: repels magnet field inside with χ = −1
Superconductivity
(Types: 6 points)
-
Type-I (Conventional):
- Can be described with BCS Theory
- Can completely expel external field below Bcth
-
Type-II (Unconventional):
- Cannot be described with BCS Theory
- Have second critical field
Basic Properties
(Perfect Conductor: 2 points)
- resistivity vanishes below TC
- Drops within ∆T = T(R = 0.9Rn)−T(R = 0.1Rn) for normal resistivity Rn
Basic Properties
(Perfect Conductor [Measurement]: 3 points + diagram)
- Measure current inductively by measuring induced magnetic field
- B ∝ I
- Lifetimes upto 1014 s have beeb measured
Basic Properties
(Perfect Diamagnet [Meissner Effect: Overview]: 2 point + diagram)
Can distinguish between perfect and superconductor by looking at response in magnetic field below TC
- Meissner Effect: ability of superonductor to expel magnetic fields below critical field Bcth when supercooled
Basic Properties
(Perfect Diamagnet [Meissner Effect: Supercooled with B = 0]: 4 points + diagram)
- Same response when B-field switched on
- Lenz Law → induced surface currents shield magnetic field
- Same response when B-field switched off
- No magnetic moment
Basic Properties
(Perfect Diamagnet [Meissner Effect: Supercooled with B > 0]: 6 points)
- Different response when supercooled
- Perfect conductor: magnetic field penetrates
- Superconductor: magnetic field expelled
- Different response when B-field switched off
- Perfect conductor: persistent magnetization becuase of Lenz currents
- Superconductor: no magnetization
Basic Properties
(Critical Field [Overview]: 2 point + diagram)
- Because Meissner effect is reversible → superconductivity can be destroyed by critical magnetic field Bcth
- Otherwise, superconductor could do infinite work to push out magnetic field
Basic Properties
(Critical Field [Temperature Dependence])
Basic Properties
(Critical Field [Magnetization Inside]: 2 points + graph)
- Bcth < 0 → can increase with extenral field
- Bcth > 0 → magnetization breaks down
Basic Properties
(Critical Field [Field Inside]: 2 points + graph)
- Bcth < 0 → external field shielded
- Bcth > 0 → external field penetrates
Basic Properties
(Flux Quantization [Overview]: 2 points)
- Experiments show that magnetic flux through a superconducting ring is an interger of a flux quantum (see below)
- Experimental evidence of Cooper pairs
Basic Properties
(Flux Quantization [Experiment]: 1 point + graph)
- Trap magnetic flux in supercooled lead tube and measure torque it causes on mirror
Thermodynamic Properties
(Overview: 3 points)
- For type-I and -II superconductors in magnetic field, look at
- Enthalpy
- Entropy
- Specific Heat
Thermodynamic Properties
(Type-I [Enthalpy]: 3 points + graph)
- Phase transition at B = Bcth
- For B < Bcth → change in enthalpy ∆G ∝ B2 (Meissner parabola)
- B >= Bcth → enthalpy is that or normal state (≈ contst.)
Thermodynamic Properties
(Type-I [Enthalpy: Condensation Energy]: 3 points)
- Difference between normal and superconducting enthalpy density at B = 0
- ∆g = gn(0,T)−gs(0,T) = Bcth2/(2µo)
- field repulsion energy needed to push external field out
Thermodynamic Properties
(Type-I [Entropy: Overview]: 2 points + graph)
-
Recall:
- entropy of normal state Sn ∝ T
- enthalpy of normal state gn ∝ T2
Thermodynamic Properties
(Type-I [Entropy: Take-Away]: 2points)
- In B-field: first-order transition
- No B-field: second-order transition
Thermodynamic Properties
(Type-I [Specific Heat: Overview]: 3 points)
- Specific heat is measurable
-
Rutger’s formula gives different between normal and superconducting state ∆C = CN - CS
- For T = TC → ∆C < 0
Thermodynamic Properties
(Type-I [Specific Heat: Take-Away]: 1 point + graph)
- Heat capacity is greater in superconducting state
Thermodynamic Properties
(Type-I [Summary]: 3 points)
- Enthalpy → condensation energy
- Entropy → first-(second-)order phase transition with(out) B-field
- Specific heat → superconducting state has higher heat capacity
Thermodynamic Properties
(Type-II [Overview]: 3 points)
- There exist two critical field Bc1, Bc2
- <em></em>B < Bc1 → same as type-I (Meissner effect)
- B > Bc1 → external field is not fully repelled
Thermodynamic Properties
(Type-II [Enthalpy]: 2 points + graph)
-
Bc1 < B < Bc2 → Shupnikov effect
- enthalpy density increases slower than B2
Thermodynamic Properties
(Type-II [Entropy])
Thermodynamic Properties
(Type-II [Specific Heat])
London Equations
(Overview: 3 points)
- Describe perfect conductor and perfect diamagneti with classical electrodynamics
- Can account for Meissner effect
- Can use to derive supercurrent carrier density ns
London Equations
(Assumptions: 4 points)
- EOM from Drude model with no scattering τ, σ → ∞
- charge carrier density n(T) = nn(T ) + ns(T ) consists of normal and supercarriers
- T < TC → nn = 0 and ns = n
- T > TC → nn = n and ns = 0
London Equations
(First Equation: 2 points)
Relates supercurrent density Js = nsqsvs to external field E
London Equations
(Second Equation: 3 points)
- From Faraday’s law of induction ∂<em>t</em> [∇ × (ΛJs) + B] = 0
- From Meissner effect, change in flux and static field have to induce screening
London Equations
(London Penetration Depth: 3 points)
- from field screening equation ∇ × B = µoJ
- _London penetration depth_ λL2 = Λ/µo
London Equations
(London Penetration Depth [Temperature Dependence])
- Empirical relation
London Equations
(Take-Away [Overview]: 2 points)
- Consider superconducting material for x >= 0
- There exist B-field penetration and supercurrent in superconducting material
London Equations
(Take-Away [B-field]: 1 point + 1 graph)
Permits exponentially decaying B-fields (e.g. type-II)
London Equations
(Take-Away [Supercurrent]: 1 point + 1 graph)
Supercurrent along surface
London Equations
(Take-Away [Notes]: 4 points)
- Thin films (∼ λL) are not field free
- Superconducitivity is DC phenomenon
- For AC, normal carrier start to scatter
- Can generalize London equations by assuming macroscopic wave function ψ(r,t) such that |ψ(r,t)|2 ≡ ns(r,t)
Ginzburg-Landau Theory
(Overview: 5 points)
- Describes spartial vartiation of ns by extending Landau’s theory for second-order phase transition
- Order parameter gs is complex and spatially varying
- Distinguish between:
- Spatially homogeneous with B = 0
- Spartially varying with B > 0
Ginzburg-Landau Theory
(Spatially Homogeneous, B = 0 [Assumptions]: 4 points)
- |ψ(r,t)|2 = |ψ(r,0)|2 = |ψo(r)|2 = constant
- Expand gs in terms of |ψo(r)|2
- β > 0 otherwise large |ψo(r)| → gs < g
- α must change sign at TC → α(T) = α*(1 − T /TC)
Ginzburg-Landau Theory
(Spatially Homogeneous, B = 0 [Take-Away]: 5 points)
- Can relate the following to α, β
- supercurrent carrier density ns
- condensation energy ∆g
- Meaning of α
- condesation energy for forming one Cooper pair
Ginzburg-Landau Theory
(Spatially Homogeneous, B = 0 [∆g vs |ψo(r)|]: 3 points + graph)
- Left: above TC
- MIddle: below TC
- Right: complex |ψo(r)|
Ginzburg-Landau Theory
(Spatially Varying, B > 0 [Assumptions])
- Again, expand gs around |ψo(r)|2, but include terms for field repulsion and spartial variance of order
Ginzburg-Landau Theory
(Spatially Varying, B > 0 [Take-Away]: 2 points)
- Last term is measure of stiffness → large variation over short scale require lots of energy
- Finding ground state ψo(r) by minimizing ∆g leads to First and Second GL Equations
Ginzburg-Landau Theory
(Penetration Depth: 2points)
- λGL = λL/2
Ginzburg-Landau Theory
(Coherence Length: 2 points)
Local perturbations in density decay expoentially with this characterstic length
Ginzburg-Landau Theory
(Ginzburg-Landau Parameter: 3 points)
- κ < 1 → type-I
- κ > 1 → type-II
Ginzburg-Landau Theory
(Spatial Ordering at Interface: 2 points + graph)
- No external field
- No diffusion of carriers
Microscopic Description
(Goal)
Microscopic description of superconducting electron
Microscopic Theory
(BCS Theory [Overview]: 5 points + equation)
- There exists attractive interaction that forms Cooper pair
- bosonic interation mediated by exchange boson
- exchange of “virtual phonon”
- Look for when Coulomb potential in medium with Thomas-Fermi screening is negative
- screened plasma frequency Ωp
Microscopic Theory
(BCS Theory [Take-Away]: 5 points + diagram)
- Consider energy difference between two electrons ∆E = Ek,1 − Ek,2
- for ∆E/(hbar) = ω < Ωp → VC(q,ω) < 0
- Pairing happens in k-space → retarded dynamics interaction
- describes how attractive potential can arise
- allows electrons to be far apart, otherwise Coulomb repulsion would dominate
Micoscopic Theory
(BCS Theory [Interaction Diagram])
BCS Theory
(Cooper Pairs [Assumptions]: 3 points)
- Free electron gas
- T = 0
- all states up to EF are filled
BCS Theory
(Cooper Pairs [Overview]: 5 points + diagram)
- Add two electrons that interact via lattice by exchange of virtual phonon qD
- Attraction maximized for ∆k = 0 → k1 = −k2
- Cooper pair
- Describe with sum over all two-particle wavefunctions
- |Ak|2 ≡ probably of finding pair in state (k, −k)
- Cooper pair
BCS Theory
(Cooper Pairs [Take-Away]: 2 points + equation)
- Difference in energy of paired electrons ∆E makes Fermi sea unstable
- Drives system to new phase → BCS groundstate
BCS Theory
(BCS Groundstate: 3 points + graph)
- Cooper pairs are bosons → can condense into macroscopic groundstate
- Gap opens up at EF ± ∆
- Even at T=0, there exist free states below EF → allows for continuous breaking of Cooper pairs
BCS Theory
(BCS Groundstate [Condensation Energy])
BCS Theory
(BCS Groundstate [Density of States]: 3 points + graph)
- Divergent for Ek = ∆
- Ek >> ∆ → DOS of free electron gas
- Cooper pairs are δ-function at Ek = 0
BCS Theory
(BCS State for T > 0: 4 points + graph)
- As T → TC then ∆ → 0
- Thermal breaking of Cooper pairs and exitation of quasi-particles
- Hinders exchange of virtual phonons
- Thermal breaking of Cooper pairs and exitation of quasi-particles
- Can show ∆(0) = 1.764kBTC
Measuring ∆
(Indirectly: 4 points + 2 graphs)
- Can measure indirectly by looking at specific heat
- Heating superconductor → breaking of Cooper pairs and excitation of quasi-particles
- Use Boltzmann statistics for population of excited quasi-particles
- Specific heat C ∝ exp[−∆/(kBT)] only, since Cooper pairs do not contribute (zero-entropic state)
Measuring ∆
(Directly: 2 points)
- Tunneling spectroscopy
- Absorption spectroscopy
Measuring ∆
(Optical Properties [Reflectivity]: 2 points + graph)
- Reflectivity jumps from metal curve to R = 1 for ω ∝ 2∆
- No absorption for ω < 2∆ → total reflectivity
Measuring ∆
(Optical Properties [Spectral Weight]: 2 points + 2 graphs)
- Spectral weight of σ1 jumps to δ-peak at ω = 0 after opening of gap
- Induvtive response in σ2 → Meissner effect
Measuring ∆
(Tunneling Spectroscopy [Setup]: 1 point + 1 diagram)
- Two thin conducting regions separated by small insulating region
Measuring ∆
(Tunneling Microscopy [Main Idea]: 3 points + graph)
- Look at SC-Ins-NC junction
- With correct bias |eU| ≥ ∆ electrons from SC can tunnel to quasi-particle states in NC
- Allows for measurement of energy gap via current measurement
Josephson Junction
(Overview: 4 points + diagram)
- Sc-Ins-Sc with very thin insulating gap
- Macroscopic wavefunctions in each side reach over into the other side
- Phase difference between each wavefunction (hbar)(φ˙2−φ˙1) = −(E2− E1) = 2eU
- Look at cases U = 0 and |U | > 0
Josephson Junction
(Case U = 0: 3 points + graph)
- Constant phase difference drives current Is = IJ sin(φ2 − φ1)
- DC-Josephson effect: DC-current Is goes through insulator without voltage drop
- Source delivers charges
Josephson Junction
(Case |U | > 0: 2 points + graph)
- Quasi-particles begin tunneling
- AC-Josephson effect: Current jumps from Josephson current IJ to finite value
Josephson Junction
(B > 0: 2 points + diagram)
- Use pair of Josephson junctions → SQUID
- Allows precise magnetic flux measurements
Unconventional Superconductors
(Overview: 6 points)
- Virtual phonon is not only possible exchange boson
- Could be slowing decaying spin fluctuation or paramagnon
- Different ways to define “unconventonal”
- Electron-phonon interaction (yes/no)
- Mean pair potential is (not) zero [conventional/unconventional
- order parameter is same (lower) than underlying lattice symmetry [conventional/unconventional]
Unconventional Superconductors
(Completing Order Classification: 3 points)
- Effective interaction requires strong polarizability (structurally, electrically, etc.) → close to (structurally, electrically, etc.) instability → close to phase transition
- Can tune mediating interaction by external control parameter (doping, pressure, etc.)
- If tuning can result make order parameter zero → system is quantum critical and fluctuations near quantum critical point are important
Cuprates
(Structure: 2 points + 2 diagrams)
- Think of as stacks of Josephson junctions
- Apex oxygen strongly controls TC
Cuprates
(Electronic Structure: 5 points + 2 graphs)
- Expectation: chemical potential µ in middle of anti-bonding band → should be metal
- Reality: experimentally shown to be good insulator
-
Explanation:
- Strong crystal field splits anti-bonding level into upper-/lower Hubbard bands
- µ now in gap
- Similar to Mott insulator
Cuprates
(Electronic Structure [Doping]: 3 points)
- Doping by replacing L3+ → C4+
-
Adds extra electrion to CuO layer, even though that’s not where dopant goes
- Intrinsic modulation doped system
-
Adds extra electrion to CuO layer, even though that’s not where dopant goes
Cuprates
(Fermi Surface: graph)
Cuprates
(Spin Structure and Phase Diagram: 3 points + diagram)
- t << U → no hopping
- Uncertainty principle → virtual hopping
- Spin fluctuations are possible “glue”
Cuprates
(Superconducting Properties)
Highly anisotropic due to layers
